Definition: regular-int-seq
The factor ⌜2⌝ on the right of the inequality is essential to prove accelerate_wf⋅
k-regular-seq(f) == ∀n,m:ℕ+. (|(m * (f n)) - n * (f m)| ≤ ((2 * k) * (n + m)))
Lemma: regular-int-seq_wf
∀[k:ℤ]. ∀[f:ℕ+ ⟶ ℤ]. (k-regular-seq(f) ∈ ℙ)
Definition: strong-regular-int-seq
strong-regular-int-seq(a;b;f) == ∀n,m:ℕ+. ((a * |(m * (f n)) - n * (f m)|) ≤ (b * (n + m)))
Lemma: strong-regular-int-seq_wf
∀[a,b:ℤ]. ∀[f:ℕ+ ⟶ ℤ]. (strong-regular-int-seq(a;b;f) ∈ ℙ)
Lemma: regular-int-seq-weakening
∀[n,k:ℤ]. ∀[f:ℕ+ ⟶ ℤ]. (k-regular-seq(f)) supposing (n-regular-seq(f) and (n ≤ k))
Lemma: sq_stable__regular-int-seq
∀[k:ℕ+]. ∀[f:ℕ+ ⟶ ℤ]. SqStable(k-regular-seq(f))
Definition: bdd-diff
bdd-diff(f;g) == ∃B:ℕ. ∀n:ℕ+. (|(f n) - g n| ≤ B)
Lemma: bdd-diff_wf
∀[f,g:ℕ+ ⟶ ℤ]. (bdd-diff(f;g) ∈ ℙ)
Lemma: trivial-bdd-diff
∀[f,g:ℕ+ ⟶ ℤ]. bdd-diff(f;g) supposing ∀n:ℕ+. ((f n) = (g n) ∈ ℤ)
Lemma: bdd-diff-equiv
EquivRel(ℕ+ ⟶ ℤ;f,g.bdd-diff(f;g))
Lemma: bdd-diff_weakening
∀a,b:ℕ+ ⟶ ℤ. ((a = b ∈ (ℕ+ ⟶ ℤ)) ⇒ bdd-diff(a;b))
Lemma: bdd-diff_inversion
∀a,b:ℕ+ ⟶ ℤ. (bdd-diff(a;b) ⇒ bdd-diff(b;a))
Lemma: bdd-diff_transitivity
∀a,b,c:ℕ+ ⟶ ℤ. (bdd-diff(a;b) ⇒ bdd-diff(b;c) ⇒ bdd-diff(a;c))
Lemma: bdd-diff_functionality
∀x1,x2,y1,y2:ℕ+ ⟶ ℤ. (bdd-diff(x1;x2) ⇒ bdd-diff(y1;y2) ⇒ (bdd-diff(x1;y1) ⇐⇒ bdd-diff(x2;y2)))
Lemma: bdd-diff-add
∀[f1,f2,g1,g2:ℕ+ ⟶ ℤ]. (bdd-diff(f1;f2) ⇒ bdd-diff(g1;g2) ⇒ bdd-diff(λn.(f1[n] + g1[n]);λn.(f2[n] + g2[n])))
Lemma: bdd-diff-regular-int-seq
∀[k,b:ℕ]. ∀[f:{f:ℕ+ ⟶ ℤ| k-regular-seq(f)} ]. ∀[g:ℕ+ ⟶ ℤ].
k + b-regular-seq(g) supposing ∀n:ℕ+. (|(f n) - g n| ≤ (2 * b))
Lemma: bdd-diff-iff-eventual
∀f,g:ℕ+ ⟶ ℤ. (∃m:ℕ+. ∃B:ℕ. ∀n:{m...}. (|(f n) - g n| ≤ B) ⇐⇒ bdd-diff(f;g))
Lemma: eventually-equal-implies-bdd-diff
∀f,g:ℕ+ ⟶ ℤ. ((∃m:ℕ+. ∀n:{m...}. ((f n) = (g n) ∈ ℤ)) ⇒ bdd-diff(f;g))
Definition: real
ℝ == {f:ℕ+ ⟶ ℤ| regular-seq(f)}
Lemma: real_wf
ℝ ∈ Type
Lemma: real-regular
∀[x:ℝ]. ∀[k:ℕ+]. k-regular-seq(x)
Lemma: implies-equal-real
∀[x,y:ℝ]. x = y ∈ ℝ supposing ∀n:ℕ+. ((x n) = (y n) ∈ ℤ)
Lemma: real-has-valueall
∀[x:ℝ]. has-valueall(x)
Lemma: real-has-value
∀[x:ℝ]. (x)↓
Definition: accelerate
accelerate(k;f) == eval k2 = 2 * k in λn.eval m = k2 * n in (f m) ÷ k2
Lemma: accelerate_wf
∀[k:ℕ+]. ∀[f:{f:ℕ+ ⟶ ℤ| k-regular-seq(f)} ]. (accelerate(k;f) ∈ ℝ)
Lemma: accelerate-bdd-diff
∀k:ℕ+. ∀[f:{f:ℕ+ ⟶ ℤ| k-regular-seq(f)} ]. bdd-diff(accelerate(k;f);f)
Lemma: accelerate-real-strong-regular
∀k:ℕ+. ∀x:ℝ. strong-regular-int-seq(2 * k;(2 * k) + 1;accelerate(k;x))
Lemma: accelerate1-real-strong-regular
∀x:ℝ. strong-regular-int-seq(2;3;accelerate(1;x))
Lemma: regular-consistency
∀[x,y:ℝ]. ∀[n,m:ℕ+]. ((m * |(x n) - y n|) ≤ ((n * |(x m) - y m|) + (4 * n) + (4 * m)))
Lemma: bdd-diff-regular
∀[x,y:ℕ+ ⟶ ℤ]. ∀[k,l:ℕ+].
(∀n:ℕ+. (|(x n) - y n| ≤ ((2 * k) + (2 * l)))) supposing (bdd-diff(x;y) and k-regular-seq(x) and l-regular-seq(y))
Lemma: regular-less
∀[x,y:ℝ]. ∀n:ℕ+. ((x n) + 4 < y n ⇒ (∀m:ℕ+. ((x m) ≤ ((y m) + 4))))
Lemma: regular-less-iff
∀[x,y:ℝ]. (∃n:ℕ+ [(x n) + 4 < y n] ⇐⇒ ∀b:{4...}. ∃n:ℕ+. ∀m:{n...}. (x m) + b < y m)
Definition: reg-less
reg-less(x;y) == eval x' = x in eval y' = y in find-ge(λn.(x n) + 4 <z y n;1)
Lemma: reg-less_wf
∀[x:ℝ]. ∀[y:{y:ℝ| ∃n:ℕ+ [(x n) + 4 < y n]} ]. (reg-less(x;y) ∈ {n:ℕ+| (x n) + 4 < y n} )
Lemma: fun-not-int
∀[f:ℕ+ ⟶ ℤ]. (isint(f) ~ ff)
Definition: int-to-real
r(n) == λk.(2 * k * n)
Lemma: int-to-real_wf
∀[n:ℤ]. (r(n) ∈ ℝ)
Lemma: real-list-has-valueall
∀[x:ℝ List]. has-valueall(x)
Lemma: real-valueall-type
valueall-type(ℝ)
Lemma: valueall-type-real-list
valueall-type(ℝ List)
Definition: reg-seq-list-add
reg-seq-list-add(L) == λn.cbv_list_accum(s,a.s + (a n);0;L)
Lemma: reg-seq-list-add-as-l_sum
∀[L:(ℕ+ ⟶ ℤ) List]. (reg-seq-list-add(L) = (λn.l_sum(map(λx.(x n);L))) ∈ (ℕ+ ⟶ ℤ))
Lemma: reg-seq-list-add_wf
∀[L:ℝ List]. (reg-seq-list-add(L) ∈ {f:ℕ+ ⟶ ℤ| ||L||-regular-seq(f)} )
Lemma: reg-seq-list-add_functionality_wrt_permutation
∀[L,L':ℝ List].
reg-seq-list-add(L) = reg-seq-list-add(L') ∈ {f:ℕ+ ⟶ ℤ| ||L||-regular-seq(f)} supposing permutation(ℝ;L;L')
Lemma: reg-seq-list-add_functionality_wrt_bdd-diff
∀L,L':(ℕ+ ⟶ ℤ) List.
(((||L|| = ||L'|| ∈ ℤ) ∧ (∀i:ℕ||L||. bdd-diff(L[i];L'[i]))) ⇒ bdd-diff(reg-seq-list-add(L);reg-seq-list-add(L')))
Definition: reg-seq-add
reg-seq-add(x;y) == λn.((x n) + (y n))
Lemma: reg-seq-add_wf
∀[x,y:ℕ+ ⟶ ℤ]. (reg-seq-add(x;y) ∈ ℕ+ ⟶ ℤ)
Lemma: reg-seq-add_functionality_wrt_bdd-diff
∀x,x',y,y':ℕ+ ⟶ ℤ. (bdd-diff(x;x') ⇒ bdd-diff(y;y') ⇒ bdd-diff(reg-seq-add(x;y);reg-seq-add(x';y')))
Definition: radd-list
radd-list(L) == let L' ⟵ L in eval k = ||L'|| in if (k =z 0) then r0 else accelerate(k;reg-seq-list-add(L')) fi
Lemma: radd-list_wf
∀[L:ℝ List]. (radd-list(L) ∈ ℝ)
Lemma: radd-list_functionality_wrt_permutation
∀[L1,L2:ℝ List]. radd-list(L1) = radd-list(L2) ∈ ℝ supposing permutation(ℝ;L1;L2)
Lemma: radd-list_wf-bag
∀[L:bag(ℝ)]. (radd-list(L) ∈ ℝ)
Definition: radd
a + b == accelerate(2;reg-seq-list-add([a; b]))
Lemma: radd_wf
∀[a,b:ℝ]. (a + b ∈ ℝ)
Lemma: radd-bdd-diff
∀a,b:ℝ. bdd-diff(a + b;λn.((a n) + (b n)))
Lemma: radd-approx
∀[a,b:ℝ]. ∀[n:ℕ+]. ((a + b) n ~ ((a (4 * n)) + (b (4 * n))) ÷ 4)
Definition: rminus
-(x) == λn.(-(x n))
Lemma: rminus_wf
∀[x:ℝ]. (-(x) ∈ ℝ)
Lemma: rminus-int
∀[n:ℤ]. (-(r(n)) = r(-n) ∈ ℝ)
Definition: reg_seq_mul
reg_seq_mul(x;y) == λn.[(x n) * (y n) ÷ 2 * n]
Lemma: reg_seq_mul_wf
∀[x,y:ℕ+ ⟶ ℤ]. (reg_seq_mul(x;y) ∈ ℕ+ ⟶ ℤ)
Lemma: reg_seq_mul-regular-eventually
∀[x,y:ℝ].
∀B,b:ℕ+.
∀n,m:{b...}. (|(m * (reg_seq_mul(x;y) n)) - n * (reg_seq_mul(x;y) m)| ≤ ((2 * B) * (n + m)))
supposing ∀n,m:{b...}. ((2 * ((m * |x n|) + (n * |y m|))) ≤ ((n * m) * ((4 * B) - 1)))
Lemma: reg_seq_mul-regular
∀[x,y:ℝ].
∀B:ℕ+
B-regular-seq(reg_seq_mul(x;y)) supposing ∀n,m:ℕ+. ((2 * ((m * |x n|) + (n * |y m|))) ≤ ((n * m) * ((4 * B) - 1)))
Definition: reg-seq-mul
reg-seq-mul(x;y) == λn.(((x n) * (y n)) ÷ 2 * n)
Lemma: reg-seq-mul_wf
∀[x,y:ℕ+ ⟶ ℤ]. (reg-seq-mul(x;y) ∈ ℕ+ ⟶ ℤ)
Lemma: reg-seq-mul-regular-eventually
∀[x,y:ℝ].
∀B,b:ℕ+.
∀n,m:{b...}. (|(m * (reg-seq-mul(x;y) n)) - n * (reg-seq-mul(x;y) m)| ≤ ((2 * B) * (n + m)))
supposing ∀n,m:{b...}. ((2 * ((m * |x n|) + (n * |y m|))) ≤ ((n * m) * ((4 * B) - 2)))
Lemma: reg-seq-mul-regular
∀[x,y:ℝ]. ∀k:ℕ+. k + 1-regular-seq(reg-seq-mul(x;y)) supposing ∀n:ℕ+. ((|x n| ≤ (n * k)) ∧ (|y n| ≤ (n * k)))
Definition: canon-bnd
canon-bnd(x) == |x 1| + 3
Lemma: canon-bnd_wf
∀[x:ℝ]. (canon-bnd(x) ∈ {k:{3...}| ∀n:ℕ+. (|x n| ≤ (n * k))} )
Definition: canonical-bound
canonical-bound(r) == (|r 1| + 4) ÷ 2
Lemma: canonical-bound_wf
∀[r:ℝ]. (canonical-bound(r) ∈ {k:{2...}| ∀n:ℕ+. (|r n| ≤ ((2 * n) * k))} )
Lemma: reg-seq-mul_wf2
∀[x,y:ℝ]. (reg-seq-mul(x;y) ∈ {f:ℕ+ ⟶ ℤ| imax(|x 1|;|y 1|) + 4-regular-seq(f)} )
Lemma: reg-seq-mul-comm
∀[x,y:ℕ+ ⟶ ℤ]. (reg-seq-mul(x;y) = reg-seq-mul(y;x) ∈ (ℕ+ ⟶ ℤ))
Lemma: reg-seq-mul-assoc
∀x,y,z:ℝ. bdd-diff(reg-seq-mul(reg-seq-mul(x;y);z);reg-seq-mul(x;reg-seq-mul(y;z)))
Lemma: reg-seq-mul_functionality_wrt_bdd-diff
∀x1:ℝ. ∀[x2,y1:ℕ+ ⟶ ℤ]. ∀y2:ℝ. (bdd-diff(y1;y2) ⇒ bdd-diff(x1;x2) ⇒ bdd-diff(reg-seq-mul(x1;y1);reg-seq-mul(x2;y2)))
Definition: rmul
a * b == eval a1 = a in eval b1 = b in accelerate(imax(|a1 1|;|b1 1|) + 4;reg-seq-mul(a1;b1))
Lemma: rmul_wf
∀[a,b:ℝ]. (a * b ∈ ℝ)
Lemma: rmul-bdd-diff-reg-seq-mul
∀a,b:ℝ. bdd-diff(a * b;reg-seq-mul(a;b))
Definition: req
x = y == ∀n:ℕ+. (|(x n) - y n| ≤ 4)
Lemma: req_wf
∀[x,y:ℝ]. (x = y ∈ ℙ)
Lemma: stable_req
∀[x,y:ℝ]. Stable{x = y}
Lemma: req-iff-bdd-diff
∀[x,y:ℝ]. uiff(x = y;bdd-diff(x;y))
Lemma: req_witness
∀[x,y:ℝ]. ((x = y) ⇒ (λn.<λ_.Ax, Ax, Ax> ∈ x = y))
Lemma: sq_stable__req
∀[x,y:ℝ]. SqStable(x = y)
Lemma: accelerate-req
∀[k:ℕ+]. ∀[x:ℝ]. ((accelerate(k;x) ∈ ℝ) ∧ (accelerate(k;x) = x))
Lemma: req-equiv
EquivRel(ℝ;x,y.x = y)
Lemma: req_weakening
∀[a,b:ℝ]. a = b supposing a = b ∈ ℝ
Lemma: req-same
∀[a:ℝ]. (a = a)
Lemma: req_inversion
∀[a,b:ℝ]. b = a supposing a = b
Lemma: req_transitivity
∀[a,b,c:ℝ]. (a = c) supposing ((b = c) and (a = b))
Lemma: req_functionality
∀[x1,x2,y1,y2:ℝ]. (uiff(x1 = y1;x2 = y2)) supposing ((y1 = y2) and (x1 = x2))
Lemma: req-int
∀[a,b:ℤ]. uiff(r(a) = r(b);a = b ∈ ℤ)
Lemma: radd-list_functionality
∀[L1,L2:ℝ List]. radd-list(L1) = radd-list(L2) supposing (||L1|| = ||L2|| ∈ ℤ) ∧ (∀i:ℕ||L1||. (L1[i] = L2[i]))
Lemma: radd_list_nil_lemma
radd-list([]) ~ r0
Lemma: radd-list-cons
∀[L:ℝ List]. ∀[x:ℝ]. (radd-list([x / L]) = (x + radd-list(L)))
Lemma: radd-as-radd-list
∀[a,b:ℝ]. ((a + b) = radd-list([a; b]) ∈ ℝ)
Lemma: radd_functionality
∀[a1,a2,b1,b2:ℝ]. ((a1 + b1) = (a2 + b2)) supposing ((a1 = a2) and (b1 = b2))
Lemma: rminus_functionality
∀[x,y:ℝ]. -(x) = -(y) supposing x = y
Lemma: rminus_functionality_wrt_bdd-diff
∀x,y:ℕ+ ⟶ ℤ. (bdd-diff(x;y) ⇒ bdd-diff(-(x);-(y)))
Lemma: rminus-rminus
∀[x:ℝ]. (-(-(x)) = x)
Lemma: rminus-rminus-eq
∀[x:ℝ]. (-(-(x)) = x ∈ ℝ)
Lemma: radd_comm_eq
∀[a,b:ℝ]. ((a + b) = (b + a) ∈ ℝ)
Lemma: radd_comm
∀[a,b:ℝ]. ((a + b) = (b + a))
Lemma: radd_assoc
∀[a,b,c:ℝ]. (((a + b) + c) = (a + b + c))
Lemma: radd-assoc
∀[x,y,z:ℝ]. ((x + y + z) = ((x + y) + z))
Lemma: radd-ac
∀[a,b,c:ℝ]. ((a + b + c) = (b + a + c))
Lemma: rmul_assoc
∀[a,b,c:ℝ]. (((a * b) * c) = (a * b * c))
Lemma: rmul-assoc
∀[a,b,c:ℝ]. ((a * b * c) = ((a * b) * c))
Lemma: rmul_comm
∀[a,b:ℝ]. ((a * b) = (b * a))
Lemma: rmul_functionality
∀[r1,r2,s1,s2:ℝ]. ((r1 * s1) = (r2 * s2)) supposing ((s1 = s2) and (r1 = r2))
Lemma: rmul-ac
∀[a,b,c:ℝ]. ((a * b * c) = (b * a * c))
Lemma: radd-int
∀[a,b:ℤ]. ((r(a) + r(b)) = r(a + b))
Lemma: radd-zero
∀[x:ℝ]. ((x + r0) = x)
Lemma: radd-zero-both
∀[x:ℝ]. (((x + r0) = x) ∧ ((r0 + x) = x))
Lemma: radd-rminus
∀[x:ℝ]. ((x + -(x)) = r0)
Lemma: radd-rminus-both
∀[x:ℝ]. (((x + -(x)) = r0) ∧ ((-(x) + x) = r0))
Lemma: radd-rminus-assoc
∀[x,y:ℝ]. (((x + -(x) + y) = y) ∧ ((-(x) + x + y) = y))
Lemma: radd-list-append
∀[L1,L2:ℝ List]. (radd-list(L1 @ L2) = (radd-list(L1) + radd-list(L2)))
Lemma: rmul-int
∀[a,b:ℤ]. ((r(a) * r(b)) = r(a * b))
Lemma: rmul-zero
∀[x:ℝ]. ((x * r0) = r0)
Lemma: rmul-zero-both
∀[x:ℝ]. (((x * r0) = r0) ∧ ((r0 * x) = r0))
Lemma: rmul-one
∀[x:ℝ]. ((x * r1) = x)
Lemma: rmul-identity1
∀[x:ℝ]. ((r1 * x) = x)
Lemma: rmul-one-both
∀[x:ℝ]. (((x * r1) = x) ∧ ((r1 * x) = x))
Lemma: rmul-distrib1
∀[x,y,z:ℝ]. ((x * (y + z)) = ((x * y) + (x * z)))
Lemma: rmul-distrib2
∀[x,y,z:ℝ]. (((y + z) * x) = ((y * x) + (z * x)))
Lemma: rmul-distrib
∀[a,b,c:ℝ]. (((a * (b + c)) = ((a * b) + (a * c))) ∧ (((b + c) * a) = ((b * a) + (c * a))))
Lemma: rminus-as-rmul
∀[r:ℝ]. (-(r) = (r(-1) * r))
Lemma: rmul-minus
∀[r:ℝ]. ∀[n:ℤ]. ((r * r(-n)) = (-(r) * r(n)))
Lemma: rminus-zero
-(r0) = r0
Lemma: rmul_over_rminus
∀[a,b:ℝ]. (((-(a) * b) = -(a * b)) ∧ ((a * -(b)) = -(a * b)))
Lemma: rminus-radd
∀[r,s:ℝ]. (-(r + s) = ((r(-1) * s) + (r(-1) * r)))
Lemma: radd-list-linearity
∀[T:Type]. ∀[x,y:T ⟶ ℝ]. ∀[a,b:ℝ]. ∀[L:T List].
(radd-list(map(λk.((a * x[k]) + (b * y[k]));L)) = ((a * radd-list(map(λk.x[k];L))) + (b * radd-list(map(λk.y[k];L)))))
Lemma: radd-list-linearity1
∀[T:Type]. ∀[x,y:T ⟶ ℝ]. ∀[L:T List].
(radd-list(map(λk.(x[k] + y[k]);L)) = (radd-list(map(λk.x[k];L)) + radd-list(map(λk.y[k];L))))
Lemma: radd-list-linearity2
∀[T:Type]. ∀[x:T ⟶ ℝ]. ∀[a:ℝ]. ∀[L:T List]. (radd-list(map(λk.(a * x[k]);L)) = (a * radd-list(map(λk.x[k];L))))
Lemma: radd-list-linearity3
∀[T:Type]. ∀[x:T ⟶ ℝ]. ∀[a:ℝ]. ∀[L:T List]. (radd-list(map(λk.(x[k] * a);L)) = (radd-list(map(λk.x[k];L)) * a))
Lemma: radd-list-one
∀[L:Top List]. (radd-list(map(λk.r1;L)) = r(||L||))
Lemma: rnonzero-lemma1
∀[x:ℝ]. ∀[n:ℕ+]. ∀m:ℕ+. ((n ≤ m) ⇒ (m ≤ (n * |x m|))) supposing 5 ≤ |x n|
Definition: reg-seq-inv
reg-seq-inv(x) == λm.((4 * m * m) ÷ x m)
Lemma: reg-seq-inv_wf
∀[b:ℕ+]. ∀[x:{f:ℕ+ ⟶ ℤ| b-regular-seq(f)} ]. ∀[k:ℕ+].
reg-seq-inv(x) ∈ {f:ℕ+ ⟶ ℤ| b * ((k * k) + 1)-regular-seq(f)} supposing ∀m:ℕ+. ((2 * m) ≤ (k * |x m|))
Definition: reg-seq-adjust
reg-seq-adjust(n;x) == λi.if (i) < (n) then 2 else (x i)
Lemma: reg-seq-adjust_wf
∀[n:ℕ+]. ∀[x:ℝ]. reg-seq-adjust(n;x) ∈ {f:ℕ+ ⟶ ℤ| if (n =z 1) then 1 else 4 fi -regular-seq(f)} supposing ∀i:ℕ+. (i <\000C n ⇒ (|x i| ≤ 4))
Lemma: reg-seq-adjust-property
∀[n:ℕ+]. ∀[x:ℝ]. ∀m:ℕ+. (m ≤ (n * |reg-seq-adjust(n;x) m|)) supposing 5 ≤ |x n|
Definition: rnonzero
rnonzero(x) == ∃n:ℕ+. 4 < |x n|
Lemma: rnonzero_wf
∀[x:ℕ+ ⟶ ℤ]. (rnonzero(x) ∈ ℙ)
Lemma: rnonzero_functionality
∀x,y:ℝ. ((x = y) ⇒ (rnonzero(x) ⇐⇒ rnonzero(y)))
Definition: rinv
rinv(x) ==
eval n = mu-ge(λn.eval k = x n in
4 <z |k|;1) in
if (n =z 1)
then accelerate(5;reg-seq-inv(x))
else accelerate(4 * ((4 * n * n) + 1);reg-seq-inv(reg-seq-adjust(n;x)))
fi
Lemma: rinv_wf
∀[x:ℝ]. (rnonzero(x) ⇒ (rinv(x) ∈ ℝ))
Lemma: rinv-functionality-lemma
∀x,y:ℤ. ∀a,b,n:ℕ+.
((n ≤ (a * |x|)) ⇒ (n ≤ (b * |y|)) ⇒ (|x - y| ≤ 4) ⇒ (|((4 * n * n) ÷ x) - (4 * n * n) ÷ y| ≤ (2 + (16 * a * b))))
Lemma: rinv_functionality
∀[x,y:ℝ]. (rnonzero(x) ⇒ (x = y) ⇒ (rinv(x) = rinv(y)))
Lemma: rinv_functionality-tst
∀[x,y:ℝ]. (rnonzero(x) ⇒ (x = y) ⇒ (rinv(x) = rinv(y)))
Lemma: rmul-rinv1
∀[x:ℝ]. (rnonzero(x) ⇒ ((x * rinv(x)) = r1))
Lemma: rinv1
rinv(r1) = r1
Definition: rsub
x - y == x + -(y)
Lemma: rsub_wf
∀[x,y:ℝ]. (x - y ∈ ℝ)
Lemma: rsub-int
∀[n:ℤ]. ∀m:ℤ. ((r(n) - r(m)) = r(n - m))
Lemma: rsub_functionality
∀[x1,x2,y1,y2:ℝ]. ((x1 - y1) = (x2 - y2)) supposing ((y1 = y2) and (x1 = x2))
Lemma: rmul-rsub-distrib
∀[a,b,c:ℝ]. (((a * (b - c)) = ((a * b) - a * c)) ∧ (((b - c) * a) = ((b * a) - c * a)))
Lemma: rsub-approx
∀[a,b:ℝ]. ∀[n:ℕ+]. ((a - b) n ~ ((a (4 * n)) - b (4 * n)) ÷ 4)
Lemma: radd-preserves-req
∀[x,y,z:ℝ]. uiff(x = y;(z + x) = (z + y))
Definition: real_term_value
real_term_value(f;t) ==
int_term_ind(t;
itermConstant(n)⇒ r(n);
itermVar(v)⇒ f v;
itermAdd(l,r)⇒ rl,rr.rl + rr;
itermSubtract(l,r)⇒ rl,rr.rl - rr;
itermMultiply(l,r)⇒ rl,rr.rl * rr;
itermMinus(x)⇒ r.-(r))
Lemma: real_term_value_wf
∀[f:ℤ ⟶ ℝ]. ∀[t:int_term()]. (real_term_value(f;t) ∈ ℝ)
Lemma: real_term_value_add_lemma
∀b,a,f:Top. (real_term_value(f;a (+) b) ~ real_term_value(f;a) + real_term_value(f;b))
Lemma: real_term_value_sub_lemma
∀b,a,f:Top. (real_term_value(f;a (-) b) ~ real_term_value(f;a) - real_term_value(f;b))
Lemma: real_term_value_minus_lemma
∀a,f:Top. (real_term_value(f;"-"a) ~ -(real_term_value(f;a)))
Lemma: real_term_value_mul_lemma
∀b,a,f:Top. (real_term_value(f;a (*) b) ~ real_term_value(f;a) * real_term_value(f;b))
Lemma: real_term_value_const_lemma
∀c,f:Top. (real_term_value(f;"c") ~ r(c))
Lemma: real_term_value_var_lemma
∀c,f:Top. (real_term_value(f;vc) ~ f c)
Definition: req_int_terms
t1 ≡ t2 == ∀f:ℤ ⟶ ℝ. (real_term_value(f;t1) = real_term_value(f;t2))
Lemma: req_int_terms_wf
∀[t1,t2:int_term()]. (t1 ≡ t2 ∈ ℙ)
Lemma: req_int_terms_functionality
∀[x1,x2,y1,y2:int_term()]. (uiff(x1 ≡ y1;x2 ≡ y2)) supposing (y1 ≡ y2 and x1 ≡ x2)
Lemma: req_int_terms_weakening
∀[t1,t2:int_term()]. t1 ≡ t2 supposing t1 = t2 ∈ int_term()
Lemma: req_int_terms_transitivity
∀[t1,t2,t3:int_term()]. (t1 ≡ t3) supposing (t2 ≡ t3 and t1 ≡ t2)
Lemma: req_int_terms_inversion
∀[t1,t2:int_term()]. t1 ≡ t2 supposing t2 ≡ t1
Lemma: itermAdd_functionality_wrt_req
∀[a,b,c,d:int_term()]. (a (+) c ≡ b (+) d) supposing (a ≡ b and c ≡ d)
Lemma: itermSubtract_functionality_wrt_req
∀[a,b,c,d:int_term()]. (a (-) c ≡ b (-) d) supposing (a ≡ b and c ≡ d)
Lemma: itermMultiply_functionality_wrt_req
∀[a,b,c,d:int_term()]. (a (*) c ≡ b (*) d) supposing (a ≡ b and c ≡ d)
Lemma: itermMinus_functionality_wrt_req
∀[a,b:int_term()]. "-"a ≡ "-"b supposing a ≡ b
Lemma: imonomial-req-lemma
∀f:ℤ ⟶ ℝ. ∀ws:ℤ List. ∀t:int_term().
(real_term_value(f;accumulate (with value t and list item v):
t (*) vv
over list:
ws
with starting value:
t))
= (real_term_value(f;t)
* real_term_value(f;accumulate (with value t and list item v):
t (*) vv
over list:
ws
with starting value:
"1"))))
Lemma: imonomial-term-linear-req
∀f:ℤ ⟶ ℝ. ∀ws:ℤ List. ∀c:ℤ. (real_term_value(f;imonomial-term(<c, ws>)) = (r(c) * real_term_value(f;imonomial-term(<1,\000C ws>))))
Lemma: imonomial-term-add-req
∀vs:ℤ List. ∀a,b:ℤ-o. ∀f:ℤ ⟶ ℝ. ((real_term_value(f;imonomial-term(<a, vs>)) + real_term_value(f;imonomial-term(<b, vs\000C>))) = real_term_value(f;imonomial-term(<a + b, vs>)))
Lemma: ipolynomial-term-cons-req
∀[m:iMonomial()]. ∀[p:iMonomial() List]. ipolynomial-term([m / p]) ≡ imonomial-term(m) (+) ipolynomial-term(p)
Lemma: add-ipoly-req
∀p,q:iMonomial() List. ipolynomial-term(add-ipoly(p;q)) ≡ ipolynomial-term(p) (+) ipolynomial-term(q)
Lemma: minus-poly-req
∀p:iPolynomial(). ipolynomial-term(minus-poly(p)) ≡ "-"ipolynomial-term(p)
Lemma: imonomial-cons-req
∀v:ℤ List. ∀u,a:ℤ. ∀f:ℤ ⟶ ℝ. (real_term_value(f;imonomial-term(<a, [u / v]>)) = ((f u) * real_term_value(f;imonomial-t\000Cerm(<a, v>))))
Lemma: mul-monomials-req
∀[m1,m2:iMonomial()]. imonomial-term(mul-monomials(m1;m2)) ≡ imonomial-term(m1) (*) imonomial-term(m2)
Lemma: mul-mono-poly-req
∀[m:iMonomial()]. ∀[p:iMonomial() List].
ipolynomial-term(mul-mono-poly(m;p)) ≡ imonomial-term(m) (*) ipolynomial-term(p)
Lemma: mul-ipoly-req
∀p,q:iMonomial() List. ipolynomial-term(mul-ipoly(p;q)) ≡ ipolynomial-term(p) (*) ipolynomial-term(q)
Lemma: real_term_polynomial
∀t:int_term(). ipolynomial-term(int_term_to_ipoly(t)) ≡ t
Lemma: real_polynomial_null
∀t:int_term(). t ≡ "0" supposing inl Ax ≤ null(int_term_to_ipoly(t))
Lemma: real_polynomial_null_orig
∀t:int_term(). t ≡ "0" supposing null(int_term_to_ipoly(t)) = tt
Lemma: req-iff-rsub-is-0
∀[a,b:ℝ]. uiff(a = b;(a - b) = r0)
Lemma: test-ring-req
∀v,v1,v2,v3,v4,v5,v6,v7,v8,v9:ℝ.
((((((v6 * v3) + (v2 * v5) + (v4 * v7)) - (v7 * v2) + (v3 * v4) + (v5 * v6))
* (((v6 * v9) + (v8 * v1) + (v * v7)) - (v7 * v8) + (v9 * v) + (v1 * v6)))
+ ((((v * v7) + (v6 * v5) + (v4 * v1)) - (v1 * v6) + (v7 * v4) + (v5 * v))
* (((v6 * v9) + (v8 * v3) + (v2 * v7)) - (v7 * v8) + (v9 * v2) + (v3 * v6)))
+ ((((v * v3) + (v2 * v7) + (v6 * v1)) - (v1 * v2) + (v3 * v6) + (v7 * v))
* (((v6 * v9) + (v8 * v5) + (v4 * v7)) - (v7 * v8) + (v9 * v4) + (v5 * v6))))
= r0)
Definition: rat_termco
rat_termco() ==
corec(X.lbl:Atom × if lbl =a "Constant" then ℤ
if lbl =a "Var" then ℤ
if lbl =a "Add" then left:X × X
if lbl =a "Subtract" then left:X × X
if lbl =a "Multiply" then left:X × X
if lbl =a "Divide" then num:X × X
if lbl =a "Minus" then X
else Void
fi )
Lemma: rat_termco_wf
rat_termco() ∈ Type
Lemma: rat_termco-ext
rat_termco() ≡ lbl:Atom × if lbl =a "Constant" then ℤ
if lbl =a "Var" then ℤ
if lbl =a "Add" then left:rat_termco() × rat_termco()
if lbl =a "Subtract" then left:rat_termco() × rat_termco()
if lbl =a "Multiply" then left:rat_termco() × rat_termco()
if lbl =a "Divide" then num:rat_termco() × rat_termco()
if lbl =a "Minus" then rat_termco()
else Void
fi
Definition: rat_termco_size
rat_termco_size(p) ==
fix((λsize,p. let lbl,x = p
in if lbl =a "Constant" then 0
if lbl =a "Var" then 0
if lbl =a "Add" then 1 + (size (fst(x))) + (size (snd(x)))
if lbl =a "Subtract" then 1 + (size (fst(x))) + (size (snd(x)))
if lbl =a "Multiply" then 1 + (size (fst(x))) + (size (snd(x)))
if lbl =a "Divide" then 1 + (size (fst(x))) + (size (snd(x)))
if lbl =a "Minus" then 1 + (size x)
else 0
fi ))
p
Lemma: rat_termco_size_wf
∀[p:rat_termco()]. (rat_termco_size(p) ∈ partial(ℕ))
Definition: rat_term
rat_term() == {p:rat_termco()| (rat_termco_size(p))↓}
Lemma: rat_term_wf
rat_term() ∈ Type
Definition: rat_term_size
rat_term_size(p) ==
fix((λsize,p. let lbl,x = p
in if lbl =a "Constant" then 0
if lbl =a "Var" then 0
if lbl =a "Add" then 1 + (size (fst(x))) + (size (snd(x)))
if lbl =a "Subtract" then 1 + (size (fst(x))) + (size (snd(x)))
if lbl =a "Multiply" then 1 + (size (fst(x))) + (size (snd(x)))
if lbl =a "Divide" then 1 + (size (fst(x))) + (size (snd(x)))
if lbl =a "Minus" then 1 + (size x)
else 0
fi ))
p
Lemma: rat_term_size_wf
∀[p:rat_term()]. (rat_term_size(p) ∈ ℕ)
Lemma: rat_term-ext
rat_term() ≡ lbl:Atom × if lbl =a "Constant" then ℤ
if lbl =a "Var" then ℤ
if lbl =a "Add" then left:rat_term() × rat_term()
if lbl =a "Subtract" then left:rat_term() × rat_term()
if lbl =a "Multiply" then left:rat_term() × rat_term()
if lbl =a "Divide" then num:rat_term() × rat_term()
if lbl =a "Minus" then rat_term()
else Void
fi
Definition: rtermConstant
"const" == <"Constant", const>
Lemma: rtermConstant_wf
∀[const:ℤ]. ("const" ∈ rat_term())
Definition: rtermVar
rtermVar(var) == <"Var", var>
Lemma: rtermVar_wf
∀[var:ℤ]. (rtermVar(var) ∈ rat_term())
Definition: rtermAdd
left "+" right == <"Add", left, right>
Lemma: rtermAdd_wf
∀[left,right:rat_term()]. (left "+" right ∈ rat_term())
Definition: rtermSubtract
left "-" right == <"Subtract", left, right>
Lemma: rtermSubtract_wf
∀[left,right:rat_term()]. (left "-" right ∈ rat_term())
Definition: rtermMultiply
left "*" right == <"Multiply", left, right>
Lemma: rtermMultiply_wf
∀[left,right:rat_term()]. (left "*" right ∈ rat_term())
Definition: rtermDivide
num "/" denom == <"Divide", num, denom>
Lemma: rtermDivide_wf
∀[num,denom:rat_term()]. (num "/" denom ∈ rat_term())
Definition: rtermMinus
rtermMinus(num) == <"Minus", num>
Lemma: rtermMinus_wf
∀[num:rat_term()]. (rtermMinus(num) ∈ rat_term())
Definition: rtermConstant?
rtermConstant?(v) == fst(v) =a "Constant"
Lemma: rtermConstant?_wf
∀[v:rat_term()]. (rtermConstant?(v) ∈ 𝔹)
Definition: rtermConstant-const
rtermConstant-const(v) == snd(v)
Lemma: rtermConstant-const_wf
∀[v:rat_term()]. rtermConstant-const(v) ∈ ℤ supposing ↑rtermConstant?(v)
Definition: rtermVar?
rtermVar?(v) == fst(v) =a "Var"
Lemma: rtermVar?_wf
∀[v:rat_term()]. (rtermVar?(v) ∈ 𝔹)
Definition: rtermVar-var
rtermVar-var(v) == snd(v)
Lemma: rtermVar-var_wf
∀[v:rat_term()]. rtermVar-var(v) ∈ ℤ supposing ↑rtermVar?(v)
Definition: rtermAdd?
rtermAdd?(v) == fst(v) =a "Add"
Lemma: rtermAdd?_wf
∀[v:rat_term()]. (rtermAdd?(v) ∈ 𝔹)
Definition: rtermAdd-left
rtermAdd-left(v) == fst(snd(v))
Lemma: rtermAdd-left_wf
∀[v:rat_term()]. rtermAdd-left(v) ∈ rat_term() supposing ↑rtermAdd?(v)
Definition: rtermAdd-right
rtermAdd-right(v) == snd(snd(v))
Lemma: rtermAdd-right_wf
∀[v:rat_term()]. rtermAdd-right(v) ∈ rat_term() supposing ↑rtermAdd?(v)
Definition: rtermSubtract?
rtermSubtract?(v) == fst(v) =a "Subtract"
Lemma: rtermSubtract?_wf
∀[v:rat_term()]. (rtermSubtract?(v) ∈ 𝔹)
Definition: rtermSubtract-left
rtermSubtract-left(v) == fst(snd(v))
Lemma: rtermSubtract-left_wf
∀[v:rat_term()]. rtermSubtract-left(v) ∈ rat_term() supposing ↑rtermSubtract?(v)
Definition: rtermSubtract-right
rtermSubtract-right(v) == snd(snd(v))
Lemma: rtermSubtract-right_wf
∀[v:rat_term()]. rtermSubtract-right(v) ∈ rat_term() supposing ↑rtermSubtract?(v)
Definition: rtermMultiply?
rtermMultiply?(v) == fst(v) =a "Multiply"
Lemma: rtermMultiply?_wf
∀[v:rat_term()]. (rtermMultiply?(v) ∈ 𝔹)
Definition: rtermMultiply-left
rtermMultiply-left(v) == fst(snd(v))
Lemma: rtermMultiply-left_wf
∀[v:rat_term()]. rtermMultiply-left(v) ∈ rat_term() supposing ↑rtermMultiply?(v)
Definition: rtermMultiply-right
rtermMultiply-right(v) == snd(snd(v))
Lemma: rtermMultiply-right_wf
∀[v:rat_term()]. rtermMultiply-right(v) ∈ rat_term() supposing ↑rtermMultiply?(v)
Definition: rtermDivide?
rtermDivide?(v) == fst(v) =a "Divide"
Lemma: rtermDivide?_wf
∀[v:rat_term()]. (rtermDivide?(v) ∈ 𝔹)
Definition: rtermDivide-num
rtermDivide-num(v) == fst(snd(v))
Lemma: rtermDivide-num_wf
∀[v:rat_term()]. rtermDivide-num(v) ∈ rat_term() supposing ↑rtermDivide?(v)
Definition: rtermDivide-denom
rtermDivide-denom(v) == snd(snd(v))
Lemma: rtermDivide-denom_wf
∀[v:rat_term()]. rtermDivide-denom(v) ∈ rat_term() supposing ↑rtermDivide?(v)
Definition: rtermMinus?
rtermMinus?(v) == fst(v) =a "Minus"
Lemma: rtermMinus?_wf
∀[v:rat_term()]. (rtermMinus?(v) ∈ 𝔹)
Definition: rtermMinus-num
rtermMinus-num(v) == snd(v)
Lemma: rtermMinus-num_wf
∀[v:rat_term()]. rtermMinus-num(v) ∈ rat_term() supposing ↑rtermMinus?(v)
Lemma: rat_term-induction
∀[P:rat_term() ⟶ ℙ]
((∀const:ℤ. P["const"])
⇒ (∀var:ℤ. P[rtermVar(var)])
⇒ (∀left,right:rat_term(). (P[left] ⇒ P[right] ⇒ P[left "+" right]))
⇒ (∀left,right:rat_term(). (P[left] ⇒ P[right] ⇒ P[left "-" right]))
⇒ (∀left,right:rat_term(). (P[left] ⇒ P[right] ⇒ P[left "*" right]))
⇒ (∀num,denom:rat_term(). (P[num] ⇒ P[denom] ⇒ P[num "/" denom]))
⇒ (∀num:rat_term(). (P[num] ⇒ P[rtermMinus(num)]))
⇒ {∀v:rat_term(). P[v]})
Lemma: rat_term-definition
∀[A:Type]. ∀[R:A ⟶ rat_term() ⟶ ℙ].
((∀const:ℤ. {x:A| R[x;"const"]} )
⇒ (∀var:ℤ. {x:A| R[x;rtermVar(var)]} )
⇒ (∀left,right:rat_term(). ({x:A| R[x;left]} ⇒ {x:A| R[x;right]} ⇒ {x:A| R[x;left "+" right]} ))
⇒ (∀left,right:rat_term(). ({x:A| R[x;left]} ⇒ {x:A| R[x;right]} ⇒ {x:A| R[x;left "-" right]} ))
⇒ (∀left,right:rat_term(). ({x:A| R[x;left]} ⇒ {x:A| R[x;right]} ⇒ {x:A| R[x;left "*" right]} ))
⇒ (∀num,denom:rat_term(). ({x:A| R[x;num]} ⇒ {x:A| R[x;denom]} ⇒ {x:A| R[x;num "/" denom]} ))
⇒ (∀num:rat_term(). ({x:A| R[x;num]} ⇒ {x:A| R[x;rtermMinus(num)]} ))
⇒ {∀v:rat_term(). {x:A| R[x;v]} })
Definition: rat_term_ind
rat_term_ind(v;
rtermConstant(const)⇒ Constant[const];
rtermVar(var)⇒ Var[var];
rtermAdd(left,right)⇒ rec1,rec2.Add[left;
right;
rec1;
rec2];
rtermSubtract(left,right)⇒ rec3,rec4.Subtract[left;
right;
rec3;
rec4];
rtermMultiply(left,right)⇒ rec5,rec6.Multiply[left;
right;
rec5;
rec6];
rtermDivide(num,denom)⇒ rec7,rec8.Divide[num;
denom;
rec7;
rec8];
rtermMinus(num)⇒ rec9.Minus[num; rec9]) ==
fix((λrat_term_ind,v. let lbl,v1 = v
in if lbl="Constant" then Constant[v1]
if lbl="Var" then Var[v1]
if lbl="Add"
then let left,v2 = v1
in Add[left;
v2;
rat_term_ind left;
rat_term_ind v2]
if lbl="Subtract"
then let left,v2 = v1
in Subtract[left;
v2;
rat_term_ind left;
rat_term_ind v2]
if lbl="Multiply"
then let left,v2 = v1
in Multiply[left;
v2;
rat_term_ind left;
rat_term_ind v2]
if lbl="Divide"
then let num,v2 = v1
in Divide[num;
v2;
rat_term_ind num;
rat_term_ind v2]
if lbl="Minus" then Minus[v1; rat_term_ind v1]
else Ax
fi ))
v
Lemma: rat_term_ind_wf
∀[A:Type]. ∀[R:A ⟶ rat_term() ⟶ ℙ]. ∀[v:rat_term()]. ∀[Constant:const:ℤ ⟶ {x:A| R[x;"const"]} ].
∀[Var:var:ℤ ⟶ {x:A| R[x;rtermVar(var)]} ]. ∀[Add:left:rat_term()
⟶ right:rat_term()
⟶ {x:A| R[x;left]}
⟶ {x:A| R[x;right]}
⟶ {x:A| R[x;left "+" right]} ]. ∀[Subtract:left:rat_term()
⟶ right:rat_term()
⟶ {x:A| R[x;left]}
⟶ {x:A| R[x;right]}
⟶ {x:A|
R[x;left "-" right]} ].
∀[Multiply:left:rat_term() ⟶ right:rat_term() ⟶ {x:A| R[x;left]} ⟶ {x:A| R[x;right]} ⟶ {x:A| R[x;left "*" right]} \000C].
∀[Divide:num:rat_term() ⟶ denom:rat_term() ⟶ {x:A| R[x;num]} ⟶ {x:A| R[x;denom]} ⟶ {x:A| R[x;num "/" denom]} ].
∀[Minus:num:rat_term() ⟶ {x:A| R[x;num]} ⟶ {x:A| R[x;rtermMinus(num)]} ].
(rat_term_ind(v;
rtermConstant(const)⇒ Constant[const];
rtermVar(var)⇒ Var[var];
rtermAdd(left,right)⇒ rec1,rec2.Add[left;right;rec1;rec2];
rtermSubtract(left,right)⇒ rec3,rec4.Subtract[left;right;rec3;rec4];
rtermMultiply(left,right)⇒ rec5,rec6.Multiply[left;right;rec5;rec6];
rtermDivide(num,denom)⇒ rec7,rec8.Divide[num;denom;rec7;rec8];
rtermMinus(num)⇒ rec9.Minus[num;rec9]) ∈ {x:A| R[x;v]} )
Lemma: rat_term_ind_wf_simple
∀[A:Type]. ∀[v:rat_term()]. ∀[Constant,Var:var:ℤ ⟶ A]. ∀[Add,Subtract,Multiply,Divide:num:rat_term()
⟶ denom:rat_term()
⟶ A
⟶ A
⟶ A]. ∀[Minus:num:rat_term()
⟶ A
⟶ A].
(rat_term_ind(v;
rtermConstant(const)⇒ Constant[const];
rtermVar(var)⇒ Var[var];
rtermAdd(left,right)⇒ rec1,rec2.Add[left;right;rec1;rec2];
rtermSubtract(left,right)⇒ rec3,rec4.Subtract[left;right;rec3;rec4];
rtermMultiply(left,right)⇒ rec5,rec6.Multiply[left;right;rec5;rec6];
rtermDivide(num,denom)⇒ rec7,rec8.Divide[num;denom;rec7;rec8];
rtermMinus(num)⇒ rec9.Minus[num;rec9]) ∈ A)
Definition: rat_term_to_ipolys
rat_term_to_ipolys(t) ==
rat_term_ind(t;
rtermConstant(c)⇒ <if c=0 then [] else [<c, []>], [<1, []>]>;
rtermVar(v)⇒ <[<1, [v]>], [<1, []>]>;
rtermAdd(left,right)⇒ rl,rr.let p1,q1 = rl
in let p2,q2 = rr
in <add_ipoly(mul_ipoly(p1;q2);mul_ipoly(p2;q1)), mul_ipoly(q1;q2)>;
rtermSubtract(left,right)⇒ rl,rr.let p1,q1 = rl
in let p2,q2 = rr
in <add_ipoly(mul_ipoly(p1;q2);mul_ipoly(minus-poly(p2);q1))
, mul_ipoly(q1;q2)
>;
rtermMultiply(left,right)⇒ rl,rr.let p1,q1 = rl
in let p2,q2 = rr
in <mul_ipoly(p1;p2), mul_ipoly(q1;q2)>;
rtermDivide(num,denom)⇒ rl,rr.let p1,q1 = rl
in let p2,q2 = rr
in <mul_ipoly(p1;q2), mul_ipoly(q1;p2)>;
rtermMinus(num)⇒ rx.let p1,q1 = rx
in <minus-poly(p1), q1>)
Lemma: rat_term_to_ipolys_wf
∀[t:rat_term()]. (rat_term_to_ipolys(t) ∈ iPolynomial() × iPolynomial())
Definition: int-radd
k + x == λn.(((2 * k) * n) + (x n))
Lemma: int-radd_wf
∀[k:ℤ]. ∀[x:ℝ]. (k + x ∈ ℝ)
Lemma: int-radd-req
∀[k:ℤ]. ∀[x:ℝ]. (k + x = (r(k) + x))
Definition: int-rsub
k - x == λn.(((2 * k) * n) - x n)
Lemma: int-rsub_wf
∀[k:ℤ]. ∀[x:ℝ]. (k - x ∈ ℝ)
Lemma: int-rsub-req
∀[k:ℤ]. ∀[x:ℝ]. (k - x = (r(k) - x))
Definition: int-rmul
k1 * a == eval k = k1 in λn.if (k) < (0) then -(a ((-k) * n)) else if (0) < (k) then a (k * n) else 0
Lemma: int-rmul_wf
∀[k:ℤ]. ∀[a:ℝ]. (k * a ∈ ℝ)
Lemma: int-rmul-req
∀[k:ℤ]. ∀[a:ℝ]. (k * a = (r(k) * a))
Lemma: int-rmul_functionality
∀[k1,k2:ℤ]. ∀[a,b:ℝ]. (k1 * a = k2 * b) supposing ((k1 = k2 ∈ ℤ) and (a = b))
Lemma: int-rmul-one
∀[a:ℝ]. (1 * a = a)
Definition: int-rdiv
(a)/k1 == eval k = k1 in λn.((a n) ÷ k)
Lemma: int-rdiv_wf
∀[k:ℤ-o]. ∀[a:ℝ]. ((a)/k ∈ ℝ)
Lemma: int-rdiv-int-rmul
∀[k:ℤ-o]. ∀[a:ℝ]. (k * (a)/k = a)
Lemma: int-rdiv-one
∀[a:ℝ]. ((a)/1 = a)
Definition: alt-int-rdiv
alt-int-rdiv(x;k) == if k=1 then x else (λn.[x n ÷ k])
Lemma: alt-int-rdiv_wf
∀[x:ℝ]. ∀[k:ℕ+]. (alt-int-rdiv(x;k) ∈ {y:ℝ| k * y = x} )
Definition: rat-to-real
r(a/b) == (r(a))/b
Lemma: rat-to-real_wf
∀[a:ℤ]. ∀[b:ℤ-o]. (r(a/b) ∈ ℝ)
Definition: reg-seq-nexp
reg-seq-nexp(x;k) == λn.(x n^k ÷ 2 * n^k - 1)
Lemma: reg-seq-nexp_wf
∀[x:ℝ]. ∀[k:ℕ+]. (reg-seq-nexp(x;k) ∈ {f:ℕ+ ⟶ ℤ| (k * ((canon-bnd(x)^k - 1 ÷ 2^k - 1) + 1)) + 1-regular-seq(f)} )
Definition: rnexp
x^k1 ==
eval k = k1 in
if (k =z 0)
then r1
else eval B = canon-bnd(x) in
accelerate((k * ((B^k - 1 ÷ 2^k - 1) + 1)) + 1;λn.(x n^k ÷ 2 * n^k - 1))
fi
Lemma: rnexp_wf
∀[k:ℕ]. ∀[x:ℝ]. (x^k ∈ ℝ)
Lemma: rnexp-req
∀[k:ℕ]. ∀[x:ℝ]. (x^k = if (k =z 0) then r1 else x * x^k - 1 fi )
Lemma: rnexp_zero_lemma
∀x:Top. (x^0 ~ r1)
Lemma: rnexp-int
∀[k:ℕ]. ∀[z:ℤ]. (r(z)^k = r(z^k))
Lemma: rnexp0
∀[k:ℕ+]. (r0^k = r0)
Definition: rmax
rmax(x;y) == λn.imax(x n;y n)
Lemma: rmax_wf
∀[x,y:ℝ]. (rmax(x;y) ∈ ℝ)
Lemma: rmax_functionality_wrt_bdd-diff
∀[x1,x2,y1,y2:ℕ+ ⟶ ℤ]. (bdd-diff(y1;y2) ⇒ bdd-diff(x1;x2) ⇒ bdd-diff(rmax(x1;y1);rmax(x2;y2)))
Lemma: rmax_functionality
∀[x1,x2,y1,y2:ℝ]. (rmax(x1;y1) = rmax(x2;y2)) supposing ((x1 = x2) and (y1 = y2))
Lemma: rmax-com
∀[x,y:ℝ]. (rmax(x;y) = rmax(y;x))
Lemma: rmax-assoc
∀[x,y,z:ℝ]. (rmax(rmax(x;y);z) = rmax(x;rmax(y;z)))
Definition: rmin
rmin(x;y) == λn.imin(x n;y n)
Lemma: rmin_wf
∀[x,y:ℝ]. (rmin(x;y) ∈ ℝ)
Lemma: rmin_functionality_wrt_bdd-diff
∀[x1,x2,y1,y2:ℕ+ ⟶ ℤ]. (bdd-diff(y1;y2) ⇒ bdd-diff(x1;x2) ⇒ bdd-diff(rmin(x1;y1);rmin(x2;y2)))
Lemma: rmin_functionality
∀[x1,x2,y1,y2:ℝ]. (rmin(x1;y1) = rmin(x2;y2)) supposing ((x1 = x2) and (y1 = y2))
Lemma: rmin-com
∀[x,y:ℝ]. (rmin(x;y) = rmin(y;x))
Lemma: rmin-assoc
∀[x,y,z:ℝ]. (rmin(rmin(x;y);z) = rmin(x;rmin(y;z)))
Lemma: rminus-rmax
∀[x,y:ℝ]. (-(rmax(x;y)) = rmin(-(x);-(y)))
Lemma: rminus-rmin
∀[x,y:ℝ]. (-(rmin(x;y)) = rmax(-(x);-(y)))
Lemma: rmin-req-rminus-rmax
∀[x,y:ℝ]. (rmin(x;y) = -(rmax(-(x);-(y))))
Lemma: rmin-rmax-distrib-strong
∀a,b,c:ℝ. (rmin(a;rmax(b;c)) = rmax(rmin(a;b);rmin(a;c)) ∈ ℝ)
Lemma: rmin-rmax-distrib
∀a,b,c:ℝ. (rmin(a;rmax(b;c)) = rmax(rmin(a;b);rmin(a;c)))
Lemma: rmin-rmax-absorption-strong
∀b,a:ℝ. (rmin(b;rmax(b;a)) = b ∈ ℝ)
Lemma: rmax-rmin-absorption-strong
∀b,a:ℝ. (rmax(b;rmin(b;a)) = b ∈ ℝ)
Lemma: rmin-rmax-absorption
∀b,a:ℝ. (rmin(b;rmax(b;a)) = b)
Lemma: rmax-rmin-absorption
∀b,a:ℝ. (rmax(b;rmin(b;a)) = b)
Definition: rabs
|x| == λn.|x n|
Lemma: rabs-as-rmax
∀[x:Top]. (|x| ~ rmax(x;-(x)))
Lemma: rabs_wf
∀[x:ℝ]. (|x| ∈ ℝ)
Lemma: rabs-approx
∀[x,n:Top]. (|x| n ~ |x n|)
Lemma: rabs_functionality
∀[x,y:ℝ]. |x| = |y| supposing x = y
Lemma: rabs_functionality_wrt_bdd-diff
∀x,y:ℕ+ ⟶ ℤ. (bdd-diff(x;y) ⇒ bdd-diff(|x|;|y|))
Lemma: rabs-rminus
∀[x:ℝ]. (|-(x)| = |x| ∈ ℝ)
Lemma: rabs-difference-symmetry
∀[x,y:ℝ]. (|x - y| = |y - x|)
Lemma: rabs-rmul
∀[x,y:ℝ]. (|x * y| = (|x| * |y|))
Lemma: rmin-rmax-real-decomp
∀[r:ℝ]. ((rmin(|r|;rmax(r0;r)) + rmax(-(|r|);rmin(r0;r))) = r)
Definition: rpositive
rpositive(x) == ∃n:ℕ+ [4 < x n]
Lemma: rpositive_wf
∀[x:ℕ+ ⟶ ℤ]. (rpositive(x) ∈ ℙ)
Definition: rpositive2
rpositive2(x) == ∃n:ℕ+. ∀m:ℕ+. ((n ≤ m) ⇒ (m ≤ (n * (x m))))
Lemma: rpositive2_wf
∀[x:ℕ+ ⟶ ℤ]. (rpositive2(x) ∈ ℙ)
Lemma: rpositive2_functionality
∀x,y:ℕ+ ⟶ ℤ. (bdd-diff(x;y) ⇒ (rpositive2(x) ⇐⇒ rpositive2(y)))
Lemma: rpositive-iff
∀[x:ℝ]. (rpositive(x) ⇐⇒ rpositive2(x))
Lemma: rpositive_functionality
∀x,y:ℝ. rpositive(x) ⇐⇒ rpositive(y) supposing x = y
Lemma: rpositive-radd
∀x,y:ℝ. (rpositive(x) ⇒ rpositive(y) ⇒ rpositive(x + y))
Lemma: rpositive-rmax
∀x,y:ℝ. (rpositive(rmax(x;y)) ⇐⇒ rpositive(x) ∨ rpositive(y))
Lemma: rpositive-rmul
∀x,y:ℝ. (rpositive(x) ⇒ rpositive(y) ⇒ rpositive(x * y))
Definition: rnonneg
rnonneg(x) == ∀n:ℕ+. ((-2) ≤ (x n))
Lemma: rnonneg_wf
∀[x:ℕ+ ⟶ ℤ]. (rnonneg(x) ∈ ℙ)
Definition: rnonneg2
rnonneg2(x) == ∀n:ℕ+. ∃N:ℕ+. ∀m:{N...}. (((-2) * m) ≤ (n * (x m)))
Lemma: rnonneg2_wf
∀[x:ℕ+ ⟶ ℤ]. (rnonneg2(x) ∈ ℙ)
Lemma: rnonneg2_functionality
∀x,y:ℕ+ ⟶ ℤ. (bdd-diff(x;y) ⇒ (rnonneg2(x) ⇐⇒ rnonneg2(y)))
Lemma: rnonneg-iff
∀[x:ℝ]. (rnonneg(x) ⇐⇒ rnonneg2(x))
Lemma: rnonneg_functionality
∀x,y:ℝ. rnonneg(x) ⇐⇒ rnonneg(y) supposing x = y
Lemma: rpositive-implies-rnonneg
∀[x:ℝ]. (rpositive(x) ⇒ rnonneg(x))
Lemma: rnonneg-radd
∀x,y:ℝ. (rnonneg(x) ⇒ rnonneg(y) ⇒ rnonneg(x + y))
Lemma: rnonneg-int
∀[n:ℤ]. uiff(rnonneg(r(n));0 ≤ n)
Lemma: rpositive-int
∀[n:ℤ]. uiff(rpositive(r(n));0 < n)
Lemma: rpositive-radd2
∀x,y:ℝ. (rpositive(x) ⇒ rnonneg(y) ⇒ rpositive(x + y))
Lemma: rnonneg-rmul
∀x,y:ℝ. (rnonneg(x) ⇒ rnonneg(y) ⇒ rnonneg(x * y))
Lemma: sq_stable__rnonneg
∀[r:ℝ]. SqStable(rnonneg(r))
Lemma: rminus-nonneg
∀[x:ℝ]. (x = r0) supposing (rnonneg(-(x)) and rnonneg(x))
Lemma: rabs-nonneg
∀[x:ℝ]. rnonneg(|x|)
Lemma: rmax-positive
∀x,y:ℝ. ((rpositive(x) ∨ rpositive(y)) ⇒ rpositive(rmax(x;y)))
Lemma: rmax-nonneg
∀[x,y:ℝ]. rnonneg(rmax(x;y)) supposing rnonneg(x) ∨ rnonneg(y)
Lemma: rmin-positive
∀x,y:ℝ. ((rpositive(x) ∧ rpositive(y)) ⇒ rpositive(rmin(x;y)))
Lemma: rmin-nonneg
∀[x,y:ℝ]. rnonneg(rmin(x;y)) supposing rnonneg(x) ∧ rnonneg(y)
Definition: rless
x < y == ∃n:ℕ+ [(x n) + 4 < y n]
Lemma: rless_wf
∀[x,y:ℝ]. (x < y ∈ ℙ)
Lemma: rless-by-computation1
∀[x,y:ℝ]. x < y supposing (x 1000) + 4 < y 1000
Lemma: rpositive-rless
∀[x:ℝ]. (r0 < x ⇐⇒ rpositive(x))
Lemma: rless-iff
∀x,y:ℝ. (x < y ⇐⇒ ∃n:ℕ+. ∀m:ℕ+. ((n ≤ m) ⇒ (m ≤ (n * ((y m) - x m)))))
Lemma: rless-iff-large-diff
∀x,y:ℝ. (x < y ⇐⇒ ∀b:ℕ+. ∃n:ℕ+. ∀m:ℕ+. ((n ≤ m) ⇒ (b ≤ ((y m) - x m))))
Lemma: rless-iff-rpositive
∀x,y:ℝ. (x < y ⇐⇒ rpositive(y - x))
Lemma: rless-iff4
∀x,y:ℝ. (x < y ⇐⇒ ∃n:ℕ+. ∀m:{n...}. (x m) + 4 < y m)
Lemma: rless-iff8
∀x,y:ℝ. (x < y ⇐⇒ ∃m:ℕ+ [(x m) + 8 < y m])
Lemma: rless-iff2
∀x,y:ℝ. (x < y ⇐⇒ ∃n:ℕ+. (x n) + 4 < y n)
Lemma: rless-iff2-ext
∀x,y:ℝ. (x < y ⇐⇒ ∃n:ℕ+. (x n) + 4 < y n)
Definition: rlessw
rlessw(x;y) == quick-find(λn.(x n) + 4 <z y n;1)
Lemma: rlessw_wf
∀x:ℝ. ∀y:{y:ℝ| x < y} . (rlessw(x;y) ∈ x < y)
Lemma: int-rdiv-rless-witness
∀k:ℕ+. (k ∈ (r1)/k < (r(4))/k)
Lemma: int-rdiv-rless-witness2
∀k:ℕ+. (3 * k ∈ (r1)/k < (r(2))/k)
Lemma: rless-content
∀[x:ℝ]. ∀[y:{y:ℝ| x < y} ]. (rlessw(x;y) ∈ x < y)
Lemma: sq_stable__rless
∀x,y:ℝ. SqStable(x < y)
Lemma: sq_stable__rless-or2
∀x,y,a,b:ℝ. SqStable((x < y) ∨ (a < b))
Lemma: sq_stable__rless-or3
∀x,y,a,b,c,d:ℝ. SqStable(((x < y) ∨ (a < b)) ∨ (c < d))
Definition: rleq
x ≤ y == rnonneg(y - x)
Lemma: rleq_wf
∀[x,y:ℝ]. (x ≤ y ∈ ℙ)
Lemma: rleq-implies
∀[x,y:ℝ]. ∀n:ℕ+. ((x (4 * n)) ≤ ((y (4 * n)) + 11)) supposing x ≤ y
Lemma: rleq-iff4
∀[x,y:ℝ]. (x ≤ y ⇐⇒ ∀n:ℕ+. ((x n) ≤ ((y n) + 4)))
Lemma: rleq-iff-not-rless
∀[x,y:ℝ]. uiff(y ≤ x;¬(x < y))
Lemma: rless_complement
∀x,y:ℝ. (¬(x < y) ⇐⇒ y ≤ x)
Lemma: canonical-bound-property
∀x:ℝ. ((r(-canonical-bound(x)) ≤ x) ∧ (x ≤ r(canonical-bound(x))))
Lemma: rless_functionality
∀x1,x2,y1,y2:ℝ. (x1 < y1 ⇐⇒ x2 < y2) supposing ((y1 = y2) and (x1 = x2))
Lemma: rless-int
∀n,m:ℤ. (r(n) < r(m) ⇐⇒ n < m)
Lemma: rleq-int
∀n,m:ℤ. (r(n) ≤ r(m) ⇐⇒ n ≤ m)
Lemma: stable__rleq
∀[x,y:ℝ]. Stable{x ≤ y}
Lemma: sq_stable__rleq
∀[x,y:ℝ]. SqStable(x ≤ y)
Lemma: rleq_functionality
∀[x1,x2,y1,y2:ℝ]. (uiff(x1 ≤ y1;x2 ≤ y2)) supposing ((y1 = y2) and (x1 = x2))
Definition: rgt
x > y == y < x
Lemma: rgt_wf
∀[x,y:ℝ]. (x > y ∈ ℙ)
Definition: rge
x ≥ y == y ≤ x
Lemma: rge_wf
∀[x,y:ℝ]. (x ≥ y ∈ ℙ)
Lemma: rless_transitivity
∀x,y,z:ℝ. ((x < y) ⇒ (y < z) ⇒ (x < z))
Lemma: rless_transitivity1
∀x,y,z:ℝ. ((x < y) ⇒ x < z supposing y ≤ z)
Lemma: rless_transitivity2
∀x,y,z:ℝ. (y < z) ⇒ (x < z) supposing x ≤ y
Lemma: rleq_transitivity
∀[x,y,z:ℝ]. (x ≤ z) supposing ((y ≤ z) and (x ≤ y))
Lemma: rleq_antisymmetry
∀[x,y:ℝ]. (x = y) supposing ((y ≤ x) and (x ≤ y))
Lemma: req_fake_le_antisymmetry
∀[x,y:ℝ]. (x = y) supposing ((y ≤ x) and (x ≤ y))
Lemma: rleq_weakening
∀[x,y:ℝ]. x ≤ y supposing x = y
Lemma: rleq_weakening_equal
∀[x,y:ℝ]. x ≤ y supposing x = y ∈ ℝ
Lemma: rleq_functionality_wrt_implies
∀[a,b,c,d:ℝ]. ({a ≤ d supposing b ≤ c}) supposing ((c ≤ d) and (b ≥ a))
Lemma: rless_functionality_wrt_implies
∀a,b,c,d:ℝ. ({(b < c) ⇒ (a < d)}) supposing ((c ≤ d) and (b ≥ a))
Lemma: rleq_weakening_rless
∀[x,y:ℝ]. x ≤ y supposing x < y
Definition: rbetween
x≤y≤z == (x ≤ y) ∧ (y ≤ z)
Lemma: rbetween_wf
∀[x,y,z:ℝ]. (x≤y≤z ∈ ℙ)
Lemma: radd_functionality_wrt_rleq
∀[x,y,z,t:ℝ]. ((x + y) ≤ (z + t)) supposing ((y ≤ t) and (x ≤ z))
Lemma: radd_functionality_wrt_rless1
∀x,y,z,t:ℝ. (y < t) ⇒ ((x + y) < (z + t)) supposing x ≤ z
Lemma: radd_functionality_wrt_rless2
∀x,y,z,t:ℝ. ((x < z) ⇒ (x + y) < (z + t) supposing y ≤ t)
Lemma: radd-preserves-rless
∀x,y,z:ℝ. (x < y ⇐⇒ (z + x) < (z + y))
Lemma: radd-preserves-rleq
∀[x,y,z:ℝ]. uiff(x ≤ y;(z + x) ≤ (z + y))
Lemma: rless-implies-rless
∀a,b:ℝ. ∀[c,d:ℝ]. (a < b) supposing ((c < d) and ((d - c) = (b - a)))
Lemma: rleq-implies-rleq
∀[a,b,c,d:ℝ]. (a ≤ b) supposing ((c ≤ d) and ((d - c) = (b - a)))
Lemma: req-implies-req
∀[a,b,c,d:ℝ]. (a = b) supposing ((c = d) and ((d - c) = (b - a)))
Lemma: radd-of-nonneg-is-zero
∀[a,b:{x:ℝ| r0 ≤ x} ]. uiff((a + b) = r0;(a = r0) ∧ (b = r0))
Lemma: trivial-rless-radd
∀a,d:ℝ. (uiff(a < (a + d);r0 < d) ∧ uiff(a < (d + a);r0 < d))
Lemma: trivial-rsub-rless
∀a,d:ℝ. uiff((a - d) < a;r0 < d)
Lemma: trivial-rleq-radd
∀[a,d:ℝ]. (uiff(a ≤ (a + d);r0 ≤ d) ∧ uiff(a ≤ (d + a);r0 ≤ d))
Lemma: trivial-rsub-rleq
∀[a,d:ℝ]. uiff((a - d) ≤ a;r0 ≤ d)
Lemma: radd-list_functionality_wrt_rleq
∀[L1,L2:ℝ List]. radd-list(L1) ≤ radd-list(L2) supposing (||L1|| = ||L2|| ∈ ℤ) ∧ (∀i:ℕ||L1||. (L1[i] ≤ L2[i]))
Lemma: rmul_functionality_wrt_rleq
∀[x,y,z:ℝ]. ((x * y) ≤ (z * y)) supposing ((r0 ≤ y) and (x ≤ z))
Lemma: rmul_functionality_wrt_rleq2
∀[x,y,z,w:ℝ]. ((x * w) ≤ (z * y)) supposing ((w ≤ y) and (x ≤ z) and (((r0 ≤ x) ∧ (r0 ≤ y)) ∨ ((r0 ≤ w) ∧ (r0 ≤ z))))
Lemma: rminus-reverses-rless
∀x,y:ℝ. ((x < y) ⇒ (-(y) < -(x)))
Lemma: rminus-reverses-rleq
∀[x,y:ℝ]. -(y) ≤ -(x) supposing x ≤ y
Lemma: rsub_functionality_wrt_rleq
∀[x,y,z,t:ℝ]. ((x - y) ≤ (z - t)) supposing ((y ≥ t) and (x ≤ z))
Lemma: rsub_functionality_wrt_rless
∀x,y,z,t:ℝ. ((x < z) ⇒ (x - y) < (z - t) supposing t ≤ y)
Lemma: rmul_functionality_wrt_rless
∀x,y,z:ℝ. ((x < z) ⇒ (r0 < y) ⇒ ((x * y) < (z * y)))
Lemma: rmul_functionality_wrt_rless2
∀x,y,z,w:ℝ. ((r0 < w) ⇒ (x < z) ⇒ (x * w) < (z * y) supposing w ≤ y supposing r0 ≤ z)
Lemma: rless_irreflexivity
∀[e:ℝ]. False supposing e < e
Lemma: radd-non-neg
∀x,y:ℝ. ((r0 ≤ x) ⇒ (r0 ≤ y) ⇒ (r0 ≤ (x + y)))
Lemma: rsub-rmin-rleq-rabs
∀[a,b:ℝ]. ((b - rmin(a;b)) ≤ |a - b|)
Definition: rneq
x ≠ y == (x < y) ∨ (y < x)
Lemma: rneq_wf
∀[x,y:ℝ]. (x ≠ y ∈ ℙ)
Lemma: rneq-iff
∀x,y:ℝ. (x ≠ y ⇐⇒ ∃n:ℕ+. 4 < |(x n) - y n|)
Lemma: radd-preserves-rneq
∀x,y,z:ℝ. (x ≠ y ⇐⇒ z + x ≠ z + y)
Lemma: sq_stable__rneq
∀x,y:ℝ. SqStable(x ≠ y)
Lemma: sqs-rneq-or
∀x,y:ℝ. SqStable(x ≠ r0 ∨ y ≠ r0)
Definition: rneq-zero-or
rneq-zero-or(x;y) ==
eval m = find-ge-val(λp.let a,b = p
in 4 <z |a| ∨b4 <z |b|;λn.eval a = x n in
eval b = y n in
<a, b>;1) in
let v,m1 = m
in let v1,v2 = v
in if (4) < (v1)
then inl (inr m1 )
else if (v1) < (-4)
then inl inl m1
else if (4) < (v2) then inr inr m1 else if (v2) < (-4) then inr (inl m1) else Ax
Lemma: sq_stable__rneq-or
∀x,y:ℝ. SqStable(x ≠ r0 ∨ y ≠ r0)
Lemma: rneq-zero-or_wf
∀[x,y:ℝ]. rneq-zero-or(x;y) ∈ x ≠ r0 ∨ y ≠ r0 supposing x ≠ r0 ∨ y ≠ r0
Lemma: rneq-int
∀n,m:ℤ. (r(n) ≠ r(m) ⇐⇒ ¬(n = m ∈ ℤ))
Lemma: rneq_functionality
∀x1,x2,y1,y2:ℝ. (x1 ≠ y1 ⇐⇒ x2 ≠ y2) supposing ((y1 = y2) and (x1 = x2))
Lemma: rneq-zero
∀x:ℝ. (x ≠ r0 ⇐⇒ rpositive(x) ∨ rpositive(-(x)))
Lemma: rnonzero-iff
∀x:ℝ. (rnonzero(x) ⇐⇒ x ≠ r0)
Lemma: rneq_irreflexivity
∀[e:ℝ]. False supposing e ≠ e
Lemma: rneq_irrefl
∀[e:ℝ]. (¬e ≠ e)
Lemma: absval-implies-rneq
∀x,y:ℝ. ((∃n:ℕ+. 4 < |(x n) - y n|) ⇒ x ≠ y)
Definition: case-real
case-real(a;b;f) == accelerate(3;λn.if 4 <z |(a n) - b n| ∧b (f n) then a n else b n fi )
Lemma: case-real_wf
∀[P:ℙ]. ∀[a,b:ℝ]. ∀[f:{n:ℕ+| 4 < |(a n) - b n|} ⟶ ((↓P) ∨ (↓¬P))].
(case-real(a;b;f) ∈ {z:ℝ| (P ⇒ (z = a)) ∧ ((¬P) ⇒ (z = b))} )
Definition: case-real2
case-real2(a;b;f) == case-real(a;b;λn.if (a n) + 4 <z b n then f (inl n) else f (inr n ) fi )
Lemma: case-real2_wf
∀[P:ℙ]. ∀[a,b:ℝ]. ∀[f:a ≠ b ⟶ ((↓P) ∨ (↓¬P))]. (case-real2(a;b;f) ∈ {z:ℝ| (P ⇒ (z = a)) ∧ ((¬P) ⇒ (z = b))} )
Lemma: case-real2_wf2
∀[d,a,b:ℝ]. ∀[f:a ≠ b ⟶ ((↓r0 < d) ∨ (↓¬(r0 < d)))].
(case-real2(a;b;f) ∈ {z:ℝ| ((r0 < d) ⇒ (z = a)) ∧ ((d ≤ r0) ⇒ (z = b))} )
Definition: case-real3-seq
case-real3-seq(a;b;f) == λn.if f n then a n else b n fi
Lemma: case-real3-seq_wf
∀[f:ℕ+ ⟶ 𝔹]. ∀[b:ℝ]. ∀[a:ℝ supposing ∃n:ℕ+. (↑(f n))].
case-real3-seq(a;b;f) ∈ {s:ℕ+ ⟶ ℤ| 3-regular-seq(s)} supposing ∀n,m:ℕ+. ((↑(f n)) ⇒ (¬↑(f m)) ⇒ (|(a m) - b m| ≤ \000C4))
Definition: case-real3
case-real3(a;b;f) == accelerate(3;case-real3-seq(a;b;f))
Lemma: case-real3_wf
∀[f:ℕ+ ⟶ 𝔹]. ∀[b:ℝ]. ∀[a:ℝ supposing ∃n:ℕ+. (↑(f n))].
case-real3(a;b;f) ∈ ℝ supposing ∀n,m:ℕ+. ((↑(f n)) ⇒ (¬↑(f m)) ⇒ (|(a m) - b m| ≤ 4))
Lemma: case-real3-req1
∀[f:ℕ+ ⟶ 𝔹]
∀[b,a:ℝ]. case-real3(a;b;f) = a supposing ∀n,m:ℕ+. ((↑(f n)) ⇒ (¬↑(f m)) ⇒ (|(a m) - b m| ≤ 4))
supposing ∃n:ℕ+. (↑(f n))
Lemma: case-real3-req2
∀[f:ℕ+ ⟶ 𝔹]. ∀[b:ℝ]. ∀[a:Top]. (case-real3(a;b;f) = b) supposing ∀n:ℕ+. (¬↑(f n))
Lemma: rinv_wf2
∀[x:ℝ]. (x ≠ r0 ⇒ (rinv(x) ∈ ℝ))
Lemma: rinv_functionality2
∀[x,y:ℝ]. (x ≠ r0 ⇒ (x = y) ⇒ (rinv(x) = rinv(y)))
Lemma: rmul-rinv
∀[x:ℝ]. (x * rinv(x)) = r1 supposing x ≠ r0
Lemma: rmul-rinv2
∀[x:ℝ]. (rinv(x) * x) = r1 supposing x ≠ r0
Lemma: rmul-rinv3
∀[x,a:ℝ]. (x * rinv(x) * a) = a supposing x ≠ r0
Lemma: rmul_reverses_rless
∀x,y,z:ℝ. ((x < z) ⇒ (y < r0) ⇒ ((z * y) < (x * y)))
Lemma: rinv-positive
∀x:ℝ. ((r0 < x) ⇒ (r0 < rinv(x)))
Lemma: rmul-inverse-is-rinv
∀[x:ℝ]. ∀[t:ℝ]. t = rinv(x) supposing (x * t) = r1 supposing x ≠ r0
Lemma: rmul-neq-zero
∀x,y:ℝ. (x ≠ r0 ⇒ y ≠ r0 ⇒ x * y ≠ r0)
Lemma: rminus-neq-zero
∀x:ℝ. (x ≠ r0 ⇒ -(x) ≠ r0)
Lemma: rinv-rminus
∀[x:ℝ]. -(rinv(x)) = rinv(-(x)) supposing x ≠ r0
Lemma: rinv-negative
∀x:ℝ. ((x < r0) ⇒ (rinv(x) < r0))
Lemma: rinv-neq-zero
∀x:ℝ. (x ≠ r0 ⇒ rinv(x) ≠ r0)
Lemma: rinv-of-rmul
∀[x,y:ℝ]. (rinv(x * y) = (rinv(x) * rinv(y))) supposing (y ≠ r0 and x ≠ r0)
Lemma: rinv-of-rinv
∀[x:ℝ]. rinv(rinv(x)) = x supposing x ≠ r0
Lemma: rmul_preserves_rless
∀x,y,z:ℝ. ((r0 < y) ⇒ (x < z ⇐⇒ (x * y) < (z * y)))
Lemma: rmul_preserves_rleq
∀[x,y,z:ℝ]. uiff(x ≤ z;(x * y) ≤ (z * y)) supposing r0 < y
Lemma: rmul_preserves_rleq2
∀[x,y,z:ℝ]. ((x * y) ≤ (z * y)) supposing ((x ≤ z) and (r0 ≤ y))
Lemma: rmul_preserves_rleq3
∀[x,y,a,b:ℝ]. ((x * y) ≤ (a * b)) supposing ((x ≤ a) and (y ≤ b) and ((r0 ≤ x) ∧ (r0 ≤ y)))
Lemma: rmul_reverses_rleq
∀[x,y,z:ℝ]. ((z * y) ≤ (x * y)) supposing ((y ≤ r0) and (x ≤ z))
Lemma: rminus_functionality_wrt_rleq
∀[x,y:ℝ]. -(x) ≤ -(y) supposing x ≥ y
Lemma: rabs-positive
∀x:ℝ. (x ≠ r0 ⇒ rpositive(|x|))
Lemma: rabs-neq-zero
∀x:ℝ. (x ≠ r0 ⇒ (r0 < |x|))
Lemma: not-rpositive
∀[x:ℝ]. rnonneg(-(x)) supposing ¬rpositive(x)
Lemma: not-rless
∀[x,y:ℝ]. y ≤ x supposing ¬(x < y)
Lemma: rless_complement-RelRST-test
∀[x,y,z:ℝ]. ((x ≤ y) ⇒ (¬(z < y)) ⇒ (x ≤ z))
Lemma: rleq-iff-bdd
∀[x,y:ℝ]. (x ≤ y ⇐⇒ ∃B:ℕ. ∀n:ℕ+. ((x n) ≤ ((y n) + B)))
Lemma: not-rneq
∀[x,y:ℝ]. x = y supposing ¬x ≠ y
Lemma: rneq-symmetry
∀x,y:ℝ. (x ≠ y ⇒ y ≠ x)
Lemma: req-iff-not-rneq
∀[x,y:ℝ]. uiff(x = y;¬x ≠ y)
Lemma: rmul_preserves_rneq
∀a,b,x:ℝ. (x ≠ r0 ⇒ a ≠ b ⇒ x * a ≠ x * b)
Lemma: radd-rmax
∀[x,y,z:ℝ]. ((x + rmax(y;z)) = rmax(x + y;x + z))
Lemma: rmax-int
∀[a,b:ℤ]. (rmax(r(a);r(b)) = r(imax(a;b)))
Lemma: rmin-int
∀[a,b:ℤ]. (rmin(r(a);r(b)) = r(imin(a;b)))
Lemma: rabs-int
∀[a:ℤ]. (|r(a)| = r(|a|) ∈ ℝ)
Lemma: rabs-int-rmul
∀[k:ℤ]. ∀[x:ℝ]. (|k * x| = |k| * |x|)
Lemma: rabs-int-rmul-unit
∀[k:ℕ]. ∀[x:ℝ]. (|-1^k * x| = |x|)
Lemma: radd-rmin
∀[x,y,z:ℝ]. ((x + rmin(y;z)) = rmin(x + y;x + z))
Lemma: rabs-abs
∀[a:ℤ]. (|r(a)| = r(|a|))
Lemma: zero-rleq-rabs
∀[x:ℝ]. (r0 ≤ |x|)
Lemma: square-nonneg
∀[x:ℝ]. (r0 ≤ (x * x))
Lemma: r-triangle-inequality
∀[x,y:ℝ]. (|x + y| ≤ (|x| + |y|))
Lemma: r-triangle-inequality2
∀[x,y,z:ℝ]. (|x - z| ≤ (|x - y| + |y - z|))
Lemma: r-triangle-inequality-rsub
∀[x,y:ℝ]. (|x - y| ≤ (|x| + |y|))
Lemma: radd-list-rabs
∀L:ℝ List. (|radd-list(L)| ≤ radd-list(map(λx.|x|;L)))
Lemma: rabs-rmul-rleq
∀[x,y,a,b:ℝ]. (|x * y| ≤ (a * b)) supposing ((|y| ≤ b) and (|x| ≤ a))
Lemma: rabs-rmul-rleq-rabs
∀[x,y,a,b:ℝ]. (|x * y| ≤ |a * b|) supposing ((|y| ≤ |b|) and (|x| ≤ |a|))
Lemma: rleq-rmax
∀[x,y:ℝ]. ((x ≤ rmax(x;y)) ∧ (y ≤ rmax(x;y)))
Lemma: rleq-iff
∀x,y:ℝ. (x ≤ y ⇐⇒ ∀n:ℕ+. ∃N:ℕ+. ∀m:{N...}. (((-2) * m) ≤ (n * ((y m) - x m))))
Definition: rleq2
rleq2(x;y) == ∀n:ℕ+. ∃N:ℕ+. ∀m:{N...}. (((-2) * m) ≤ (n * ((y m) - x m)))
Lemma: rleq2_wf
∀[x,y:ℕ+ ⟶ ℤ]. (rleq2(x;y) ∈ ℙ)
Lemma: rleq2-iff-rnonneg2
∀[x,y:ℕ+ ⟶ ℤ]. (rleq2(x;y) ⇐⇒ rnonneg2(reg-seq-add(y;-(x))))
Lemma: rleq2_functionality
∀x1,y1,x2,y2:ℕ+ ⟶ ℤ. (bdd-diff(x1;x2) ⇒ bdd-diff(y1;y2) ⇒ (rleq2(x1;y1) ⇐⇒ rleq2(x2;y2)))
Lemma: rleq-iff-rleq2
∀x,y:ℝ. (x ≤ y ⇐⇒ rleq2(x;y))
Lemma: rless-implies
∀x,y:ℝ. ((x < y) ⇒ (∃n:ℕ+. ∀m:ℕ+. ((n ≤ m) ⇒ (5 ≤ ((y m) - x m)))))
Lemma: rmax-req
∀[x,y:ℝ]. rmax(x;y) = y supposing x ≤ y
Lemma: rmin-req
∀[x,y:ℝ]. rmin(x;y) = y supposing y ≤ x
Lemma: rmax-req2
∀[x,y:ℝ]. rmax(x;y) = x supposing y ≤ x
Lemma: rmin-req2
∀[x,y:ℝ]. rmin(x;y) = x supposing x ≤ y
Lemma: rmin-max-cases
∀a,b:ℝ. (a ≠ b ⇒ (((rmin(a;b) = a) ∧ (rmax(a;b) = b)) ∨ ((rmin(a;b) = b) ∧ (rmax(a;b) = a))))
Lemma: rmin-classical-cases
∀a,b:ℝ. (¬¬((rmin(a;b) = a) ∨ (rmin(a;b) = b)))
Lemma: rmin-stable-cases
∀a,b:ℝ. ∀P:Type. (Stable{P} ⇒ (((rmin(a;b) = a) ⇒ P) ∧ ((rmin(a;b) = b) ⇒ P)) ⇒ P)
Lemma: rmax_ub
∀[x,y,z:ℝ]. z ≤ rmax(x;y) supposing (z ≤ x) ∨ (z ≤ y)
Lemma: rmax_strict_lb
∀x,y,z:ℝ. ((x < z) ∧ (y < z) ⇐⇒ rmax(x;y) < z)
Lemma: rmax_strict_ub
∀x,y,z:ℝ. ((z < x) ∨ (z < y) ⇐⇒ z < rmax(x;y))
Lemma: rmax-rneq
∀x,y,z,w:ℝ. (rmax(x;y) ≠ rmax(z;w) ⇒ (x ≠ z ∨ y ≠ w))
Lemma: rmax_lb
∀[x,y,z:ℝ]. uiff((x ≤ z) ∧ (y ≤ z);rmax(x;y) ≤ z)
Lemma: rmax_functionality_wrt_rleq
∀[x1,x2,y1,y2:ℝ]. (rmax(x1;y1) ≤ rmax(x2;y2)) supposing ((y1 ≤ y2) and (x1 ≤ x2))
Lemma: rmin-rleq
∀[x,y:ℝ]. ((rmin(x;y) ≤ x) ∧ (rmin(x;y) ≤ y))
Lemma: rmin_strict_ub
∀x,y,z:ℝ. ((z < x) ∧ (z < y) ⇐⇒ z < rmin(x;y))
Lemma: rmin_ub
∀x,y,z:ℝ. ((z ≤ x) ∧ (z ≤ y) ⇐⇒ z ≤ rmin(x;y))
Lemma: rmin_lb
∀[x,y,z:ℝ]. rmin(x;y) ≤ z supposing (x ≤ z) ∨ (y ≤ z)
Lemma: rmin_functionality_wrt_rleq
∀[x1,x2,y1,y2:ℝ]. (rmin(x1;y1) ≤ rmin(x2;y2)) supposing ((y1 ≤ y2) and (x1 ≤ x2))
Lemma: rmin_strict_lb
∀x,y,z:ℝ. ((x < z) ∨ (y < z) ⇐⇒ rmin(x;y) < z)
Lemma: rmax-minus-rmin
∀a,b:ℝ. (|a - b| = (rmax(a;b) - rmin(a;b)))
Lemma: rmin-rleq-rmax
∀a,b:ℝ. (rmin(a;b) ≤ rmax(a;b))
Lemma: rmin-lb-convex
∀a,b,t:ℝ. ((r0 ≤ t) ⇒ (t ≤ r1) ⇒ (rmin(a;b) ≤ ((t * a) + ((r1 - t) * b))))
Lemma: rmax-ub-convex
∀a,b,t:ℝ. ((r0 ≤ t) ⇒ (t ≤ r1) ⇒ (((t * a) + ((r1 - t) * b)) ≤ rmax(a;b)))
Lemma: rabs-difference-rmax
∀[a,b,x,y:ℝ]. (|rmax(x;y) - rmax(a;b)| ≤ rmax(|x - a|;|y - b|))
Lemma: rabs-of-nonneg
∀[x:ℝ]. |x| = x supposing r0 ≤ x
Lemma: rmul-nonneg-rabs
∀[x,y:ℝ]. (x * |y|) = |x * y| supposing r0 ≤ x
Lemma: rabs-rabs
∀[x:ℝ]. (||x|| = |x|)
Lemma: rabs-of-nonpos
∀[x:ℝ]. |x| = -(x) supposing x ≤ r0
Lemma: rneq-if-rabs
∀x,y:ℝ. x ≠ y supposing r0 < |x - y|
Lemma: rneq-if-rabs2
∀x,y:ℝ. ((r0 < |x - y|) ⇒ x ≠ y)
Lemma: rneq-iff-rabs
∀x,y:ℝ. (x ≠ y ⇐⇒ r0 < |x - y|)
Lemma: sq_stable_rneq
∀x,y:ℝ. SqStable(x ≠ y)
Lemma: rabs-positive-iff
∀x:ℝ. (x ≠ r0 ⇐⇒ r0 < |x|)
Lemma: rnexp_unroll
∀[x:ℝ]. ∀[n:ℕ]. (x^n = if (n =z 0) then r1 if (n =z 1) then x else x^n - 1 * x fi )
Lemma: rnexp_step
∀[x:ℝ]. ∀[n:ℕ+]. (x^n = (x^n - 1 * x))
Lemma: rnexp_functionality
∀[n:ℕ]. ∀[x,y:ℝ]. x^n = y^n supposing x = y
Lemma: rnexp-add
∀[n,m:ℕ]. ∀[x:ℝ]. ((x^n * x^m) = x^n + m)
Lemma: rnexp1
∀[x:ℝ]. (x^1 = x)
Lemma: rnexp2
∀[x:ℝ]. (x^2 = (x * x))
Lemma: rnexp3
∀[x:ℝ]. (x^3 = (x * x * x))
Lemma: rnexp-one
∀n:ℕ. (r1^n = r1)
Lemma: rnexp-add1
∀[n:ℕ]. ∀[x:ℝ]. (x^n + 1 = (x * x^n))
Lemma: rnexp-minus-one
∀n:ℕ. (r(-1)^n = if (n rem 2 =z 0) then r1 else r(-1) fi )
Lemma: rnexp-mul
∀[n,m:ℕ]. ∀[x:ℝ]. (x^m^n = x^m * n)
Lemma: rnexp-nonneg
∀x:ℝ. ((r0 ≤ x) ⇒ (∀n:ℕ. (r0 ≤ x^n)))
Lemma: rnexp-even-nonneg
∀n:ℕ. (((n rem 2) = 0 ∈ ℤ) ⇒ (∀x:ℝ. (r0 ≤ x^n)))
Lemma: rnexp2-nonneg
∀x:ℝ. (r0 ≤ x^2)
Lemma: rnexp-positive
∀x:ℝ. ((r0 < x) ⇒ (∀n:ℕ. (r0 < x^n)))
Lemma: rnexp2-positive
∀x:ℝ. (x ≠ r0 ⇒ (r0 < x^2))
Lemma: rnexp-rless
∀x,y:ℝ. ((r0 ≤ x) ⇒ (x < y) ⇒ (∀n:ℕ+. (x^n < y^n)))
Lemma: rnexp-rleq
∀x,y:ℝ. ((r0 ≤ x) ⇒ (x ≤ y) ⇒ (∀n:ℕ. (x^n ≤ y^n)))
Lemma: rnexp_functionality_wrt_rleq
∀x,y:ℝ. ((r0 ≤ x) ⇒ (x ≤ y) ⇒ (∀n:ℕ. (x^n ≤ y^n)))
Lemma: rnexp-rleq-iff
∀x,y:ℝ. ((r0 ≤ x) ⇒ (r0 ≤ y) ⇒ (∀n:ℕ+. (x ≤ y ⇐⇒ x^n ≤ y^n)))
Lemma: rnexp-req-iff
∀n:ℕ+. ∀x,y:ℝ. ((r0 ≤ x) ⇒ (r0 ≤ y) ⇒ (x = y ⇐⇒ x^n = y^n))
Lemma: rabs-rnexp
∀[n:ℕ]. ∀[x:ℝ]. (|x^n| = |x|^n)
Lemma: rabs-rnexp2
∀[x:ℝ]. (|x|^2 = x^2)
Lemma: square-rleq-implies
∀x,y:ℝ. ((r0 ≤ y) ⇒ (x^2 ≤ y^2) ⇒ (x ≤ y))
Lemma: rmax-idempotent
∀[x:ℝ]. (rmax(x;x) = x)
Lemma: rmin-idempotent
∀[x:ℝ]. (rmin(x;x) = x)
Lemma: rmin-idempotent-eq
∀[x:ℝ]. (rmin(x;x) = x ∈ ℝ)
Lemma: rpower-one
∀[x:ℝ]. (x^1 = x)
Lemma: rpower-two
∀[x:ℝ]. (x^2 = (x * x))
Lemma: rpower-nonzero
∀x:ℝ. (x ≠ r0 ⇒ (∀n:ℕ. x^n ≠ r0))
Lemma: rpower-greater-one
∀x,q:ℝ. ((r0 < x) ⇒ ((r1 + x) < q) ⇒ (∀n:ℕ. (r1 + (r(n) * x)) < q^n supposing 1 < n))
Definition: rdiv
(x/y) == x * rinv(y)
Lemma: rdiv_wf
∀[x,y:ℝ]. (x/y) ∈ ℝ supposing y ≠ r0
Lemma: rinv-as-rdiv
∀[y:ℝ]. rinv(y) = (r1/y) supposing y ≠ r0
Lemma: rinv-mul-as-rdiv
∀[y,a:ℝ]. (rinv(y) * a) = (a/y) supposing y ≠ r0
Lemma: mul-rinv-as-rdiv
∀[y,a:ℝ]. (a * rinv(y)) = (a/y) supposing y ≠ r0
Lemma: int-rinv-cancel
∀[a:ℤ]. ∀[b:ℤ-o]. ∀[x:ℝ]. ((r(a * b) * rinv(r(b)) * x) = (r(a) * x))
Lemma: int-rinv-cancel2
∀[a:ℤ]. ∀[b:ℤ-o]. ((r(a * b) * rinv(r(b))) = r(a))
Lemma: rdiv_functionality
∀[x1,x2,y1,y2:ℝ]. ((x1/y1) = (x2/y2)) supposing ((y1 = y2) and (x1 = x2) and y1 ≠ r0)
Lemma: rdiv_functionality_wrt_rleq2
∀[x,y,z,w:ℝ]. ((x/w) ≤ (z/y)) supposing ((x ≤ z) and (y ≤ w) and ((r0 < y) ∧ ((r0 ≤ x) ∨ (r0 ≤ z))))
Lemma: rdiv-self
∀[x:ℝ]. (x/x) = r1 supposing x ≠ r0
Lemma: rdiv-zero
∀[x:ℝ]. (r0/x) = r0 supposing x ≠ r0
Lemma: rmul-ident-div
∀[r,s:ℝ]. ((r/r) * s) = s supposing r ≠ r0
Lemma: rmul-zero-div
∀[x,y:ℝ]. ((r0/y) * x) = r0 supposing y ≠ r0
Lemma: rmul_preserves_req
∀[x,y,z:ℝ]. uiff(x = z;(x * y) = (z * y)) supposing y ≠ r0
Definition: rat_term_to_real
rat_term_to_real(f;t) ==
rat_term_ind(t;
rtermConstant(n)⇒ <True, r(n)>;
rtermVar(v)⇒ <True, f v>;
rtermAdd(left,right)⇒ rl,rr.let P,x = rl
in let Q,y = rr
in <P ∧ Q, x + y>;
rtermSubtract(left,right)⇒ rl,rr.let P,x = rl
in let Q,y = rr
in <P ∧ Q, x - y>;
rtermMultiply(left,right)⇒ rl,rr.let P,x = rl
in let Q,y = rr
in <P ∧ Q, x * y>;
rtermDivide(num,denom)⇒ rl,rr.let P,x = rl
in let Q,y = rr
in <(P ∧ Q) ∧ y ≠ r0, (x/y)>;
rtermMinus(num)⇒ rx.let P,x = rx
in <P, -(x)>)
Lemma: rat_term_to_real_wf
∀[t:rat_term()]. ∀[f:ℤ ⟶ ℝ]. (rat_term_to_real(f;t) ∈ P:ℙ × ℝ supposing P)
Definition: req_rat_term
r ≡ p/q ==
∀f:ℤ ⟶ ℝ
let P,x = rat_term_to_real(f;r)
in P ⇒ (real_term_value(f;q) ≠ r0 ∧ (x = (real_term_value(f;p)/real_term_value(f;q))))
Lemma: req_rat_term_wf
∀[r:rat_term()]. ∀[p,q:int_term()]. (r ≡ p/q ∈ ℙ)
Lemma: rat_term_polynomial
∀r:rat_term(). let p,q = rat_term_to_ipolys(r) in r ≡ ipolynomial-term(p)/ipolynomial-term(q)
Definition: rat-term-eq
rat-term-eq(r1;r2) ==
let p1,q1 = rat_term_to_ipolys(r1)
in let p2,q2 = rat_term_to_ipolys(r2)
in null(add-ipoly(mul-ipoly(p1;q2);mul-ipoly(minus-poly(p2);q1)))
Lemma: rat-term-eq_wf
∀[r1,r2:rat_term()]. (rat-term-eq(r1;r2) ∈ 𝔹)
Lemma: assert-rat-term-eq
∀r1,r2:rat_term().
((↑rat-term-eq(r1;r2))
⇒ (∀f:ℤ ⟶ ℝ. let p,x = rat_term_to_real(f;r1) in let q,y = rat_term_to_real(f;r2) in p ⇒ q ⇒ (x = y)))
Lemma: assert-rat-term-eq2
∀[r1,r2:rat_term()]. ∀[f:ℤ ⟶ ℝ].
((snd(rat_term_to_real(f;r1))) = (snd(rat_term_to_real(f;r2)))) supposing
((fst(rat_term_to_real(f;r1))) and
(fst(rat_term_to_real(f;r2))) and
(inl Ax ≤ rat-term-eq(r1;r2)))
Lemma: rmul-rdiv-cancel
∀[a,b:ℝ]. (a * (b/a)) = b supposing a ≠ r0
Lemma: rmul-rdiv-cancel2
∀[a,b:ℝ]. ((b/a) * a) = b supposing a ≠ r0
Lemma: rmul-rdiv-cancel3
∀[a,b,c:ℝ]. (a * (b/a) * c) = (b * c) supposing a ≠ r0
Lemma: rmul-rdiv-cancel4
∀[a,b,c:ℝ]. ((b/a) * a * c) = (b * c) supposing a ≠ r0
Lemma: another-test-ring-req
∀a,b,c,d,e,x:ℝ. (b ≠ r0 ⇒ d ≠ r0 ⇒ x ≠ r0 ⇒ (((a/b) * (c/d) * (b * e/x)) = ((a * c/d) * e/x)))
Lemma: rmul_reverses_rleq_iff
∀[x,y,z:ℝ]. uiff(x ≤ z;(z * y) ≤ (x * y)) supposing y < r0
Lemma: rmul_reverses_rless_iff
∀x,y,z:ℝ. ((y < r0) ⇒ (x < z ⇐⇒ (z * y) < (x * y)))
Lemma: rmul_preserves_rneq_iff
∀a,b,x:ℝ. (x ≠ r0 ⇒ (a ≠ b ⇐⇒ x * a ≠ x * b))
Lemma: rmul_preserves_rneq_iff2
∀a,b,x:ℝ. (x ≠ r0 ⇒ (a ≠ b ⇐⇒ a * x ≠ b * x))
Lemma: one-rdiv-rmul
∀[x,y:ℝ]. ((r1/y) * x) = (x/y) supposing y ≠ r0
Lemma: rmul-rdiv-cancel5
∀[a,b,c:ℝ]. ((b/a) * (a/c)) = (b/c) supposing a ≠ r0 ∧ c ≠ r0
Lemma: rmul-rdiv-cancel6
∀[a,b,c:ℝ]. (a * (b/a * c)) = (b/c) supposing a ≠ r0 ∧ c ≠ r0
Lemma: rmul-rdiv-cancel7
∀[a,b:ℝ]. (a * b/b) = a supposing b ≠ r0
Lemma: rmul-rdiv-cancel8
∀[a,b,c:ℝ]. (a * b/b * c) = (a/c) supposing b ≠ r0 ∧ c ≠ r0
Lemma: rmul-rdiv-cancel9
∀[a,b,c:ℝ]. (a * b/c * b) = (a/c) supposing b ≠ r0 ∧ c ≠ r0
Lemma: rmul-rdiv-cancel10
∀[a,b,c:ℝ]. (b * a/b * c) = (a/c) supposing b ≠ r0 ∧ c ≠ r0
Lemma: req-rdiv
∀x,y,z:ℝ. (z ≠ r0 ⇒ (x = (y/z) ⇐⇒ (x * z) = y))
Lemma: req-rdiv2
∀x,y,z,u:ℝ. (z ≠ r0 ⇒ u ≠ r0 ⇒ ((x/u) = (y/z) ⇐⇒ (x * z) = (y * u)))
Lemma: radd-rdiv
∀[a,b,c:ℝ]. ((b/a) + (c/a)) = (b + c/a) supposing a ≠ r0
Lemma: rsub-rdiv
∀[a,b,c:ℝ]. ((b/a) - (c/a)) = (b - c/a) supposing a ≠ r0
Lemma: rmul-rdiv
∀[x,y,a,b:ℝ]. (((x/a) * (y/b)) = (x * y/a * b)) supposing (b ≠ r0 and a ≠ r0)
Lemma: rmul-rdiv-one
∀[a,b:ℝ]. (((r1/a) * (r1/b)) = (r1/a * b)) supposing (b ≠ r0 and a ≠ r0)
Lemma: rmul-rdiv2
∀[x,a,b:ℝ]. ((x/a * b) = ((x/a) * (r1/b))) supposing (b ≠ r0 and a ≠ r0)
Lemma: rdiv-rdiv
∀[x,a,b:ℝ]. (((x/a)/b) = (x/a * b)) supposing (b ≠ r0 and a ≠ r0)
Lemma: rminus-rdiv
∀[x,a:ℝ]. -((x/a)) = (-(x)/a) supposing a ≠ r0
Lemma: rminus-rdiv2
∀[x,a:ℝ]. -((x/a)) = (x/-(a)) supposing a ≠ r0
Lemma: radd-rneq0
∀x,y:ℝ. (x + y ≠ r0 ⇐⇒ x ≠ -(y))
Lemma: int-rdiv-req
∀[k:ℤ-o]. ∀[a:ℝ]. ((a)/k = (a/r(k)))
Lemma: int-rdiv_functionality
∀[k1,k2:ℤ-o]. ∀[a,b:ℝ]. ((a)/k1 = (b)/k2) supposing ((k1 = k2 ∈ ℤ) and (a = b))
Lemma: rsub-rmin
∀[x,y,z:ℝ]. ((x - rmin(y;z)) = rmax(x - y;x - z))
Lemma: rleq-iff-all-rless
∀[x,y:ℝ]. uiff(x ≤ y;∀e:{e:ℝ| r0 < e} . (x ≤ (y + e)))
Lemma: rat-to-real-req
∀[a:ℤ]. ∀[b:ℤ-o]. (r(a/b) = (r(a)/r(b)))
Lemma: rinv_preserves_rless
∀a,b:ℝ. ((r0 < a) ⇒ (a < b) ⇒ ((r1/b) < (r1/a)))
Lemma: rinv_preserves_rneq
∀a,b:ℝ. (a ≠ r0 ⇒ b ≠ r0 ⇒ a ≠ b ⇒ (r1/a) ≠ (r1/b))
Lemma: rinverse-nonzero
∀x:ℝ. (x ≠ r0 ⇒ (r1/x) ≠ r0)
Lemma: inverse-rpower
∀[x:ℝ]. ∀[n:ℕ]. ((r1/x^n) = (r1/x)^n) supposing x ≠ r0
Lemma: rnexp-rmul
∀[n:ℕ]. ∀[x,y:ℝ]. (x * y^n = (x^n * y^n))
Lemma: rnexp-rdiv
∀[y,x:ℝ]. ∀[n:ℕ]. ((y^n/x^n) = (y/x)^n) supposing x ≠ r0
Lemma: rabs-bounds
∀[x:ℝ]. ((-(|x|) ≤ x) ∧ (x ≤ |x|))
Lemma: rabs-rinv
∀y:ℝ. (y ≠ r0 ⇒ (|rinv(y)| = rinv(|y|)))
Lemma: rabs-rdiv
∀x,y:ℝ. (y ≠ r0 ⇒ (|(x/y)| = (|x|/|y|)))
Lemma: rleq-int-fractions
∀[a,b:ℤ]. ∀[c,d:ℕ+]. uiff((r(a)/r(c)) ≤ (r(b)/r(d));(a * d) ≤ (b * c))
Lemma: req-int-fractions
∀[a,b:ℤ]. ∀[c,d:ℤ-o]. uiff((r(a)/r(c)) = (r(b)/r(d));(a * d) = (b * c) ∈ ℤ)
Lemma: req-int-fractions2
∀[a,b:ℤ]. ∀[c:ℤ-o]. uiff((r(a)/r(c)) = r(b);a = (b * c) ∈ ℤ)
Lemma: rleq-int-fractions2
∀[a,b:ℤ]. ∀[d:ℕ+]. uiff(r(a) ≤ (r(b)/r(d));(a * d) ≤ b)
Lemma: rleq-int-fractions3
∀[a,b:ℤ]. ∀[d:ℕ+]. uiff((r(b)/r(d)) ≤ r(a);b ≤ (a * d))
Lemma: rless-int-fractions
∀a,b:ℤ. ∀c,d:ℕ+. ((r(a)/r(c)) < (r(b)/r(d)) ⇐⇒ a * d < b * c)
Lemma: decidable__rless-int-fractions
∀a,b:ℤ. ∀c,d:ℕ+. Dec((r(a)/r(c)) < (r(b)/r(d)))
Lemma: rless-int-fractions2
∀a,b:ℤ. ∀d:ℕ+. (r(a) < (r(b)/r(d)) ⇐⇒ a * d < b)
Lemma: rless-int-fractions3
∀a,b:ℤ. ∀d:ℕ+. ((r(b)/r(d)) < r(a) ⇐⇒ b < a * d)
Lemma: rmul-int-rdiv
∀[x:ℝ]. ∀[a,b:ℤ]. ((r(a) * (r(b)/x)) = (r(a * b)/x)) supposing x ≠ r0
Lemma: rmul-int-rdiv2
∀[x:ℝ]. ∀[a,b:ℤ]. (((r(b)/x) * r(a)) = (r(a * b)/x)) supposing x ≠ r0
Lemma: radd-int-fractions
∀[a,b:ℤ]. ∀[c,d:ℕ+]. (((r(a)/r(c)) + (r(b)/r(d))) = (r((a * d) + (b * c))/r(c * d)))
Lemma: rsub-int-fractions
∀[a,b:ℤ]. ∀[c,d:ℕ+]. (((r(a)/r(c)) - (r(b)/r(d))) = (r((a * d) - b * c)/r(c * d)))
Lemma: rmul-int-fractions
∀[a,b:ℤ]. ∀[c,d:ℕ+]. (((r(a)/r(c)) * (r(b)/r(d))) = (r(a * b)/r(c * d)))
Lemma: rdiv-int-fractions
∀a,b:ℤ. ∀c,d:ℕ+. ((r(a)/r(c))/(r(b)/r(d))) = (r(a * d)/r(c * b)) supposing ¬(b = 0 ∈ ℤ)
Lemma: int-rdiv-int-rdiv
∀[k,j:ℤ-o]. ∀[x:ℝ]. (((x)/k)/j = (x)/j * k)
Lemma: clear-denominator1
∀[a,b,c,d:ℝ]. uiff(((a/b) * c) = d;(c * a) = (d * b)) supposing b ≠ r0
Lemma: clear-denominator2
∀[a,b,c,d,e:ℝ]. uiff((((a/b) * c) * e) = d;(c * e * a) = (d * b)) supposing b ≠ r0
Lemma: fractions-req
∀[a,b,c,d:ℝ]. (c ≠ r0 ⇒ d ≠ r0 ⇒ uiff((a/c) = (b/d);(a * d) = (b * c)))
Lemma: fractions-rless
∀a,b,c,d:ℝ. ((r0 < c) ⇒ (r0 < d) ⇒ ((a/c) < (b/d) ⇐⇒ (a * d) < (b * c)))
Lemma: fractions-rleq
∀a,b,c,d:ℝ. ((r0 < c) ⇒ (r0 < d) ⇒ ((a/c) ≤ (b/d) ⇐⇒ (a * d) ≤ (b * c)))
Definition: rat2real
rat2real(q) == if isint(q) then r(q) else let a,b = q in (r(a))/b fi
Lemma: rat2real_wf
∀[q:ℚ]. (rat2real(q) ∈ ℝ)
Lemma: rleq-rat2real
∀[q1,q2:ℚ]. uiff(rat2real(q1) ≤ rat2real(q2);q1 ≤ q2)
Lemma: rless-rat2real
∀q1,q2:ℚ. uiff(rat2real(q1) < rat2real(q2);q1 < q2)
Lemma: rneq-rat2real
∀q1,q2:ℚ. uiff(rat2real(q1) ≠ rat2real(q2);¬(q1 = q2 ∈ ℚ))
Lemma: req-rat2real
∀q1,q2:ℚ. uiff(rat2real(q1) = rat2real(q2);q1 = q2 ∈ ℚ)
Lemma: rat2real-qdiv
∀a:ℚ. ∀b:ℤ-o. (rat2real((a/b)) = (rat2real(a)/r(b)))
Lemma: rat2real-qdiv2
∀a:ℤ. ∀b:ℤ-o. (rat2real((a/b)) = (r(a)/r(b)))
Lemma: rat2real-qadd
∀[a,b:ℚ]. (rat2real(a + b) = (rat2real(a) + rat2real(b)))
Lemma: rat2real-qmul
∀[a,b:ℚ]. (rat2real(a * b) = (rat2real(a) * rat2real(b)))
Lemma: rat2real-qsub
∀[a,b:ℚ]. (rat2real(a - b) = (rat2real(a) - rat2real(b)))
Lemma: rat2real-qavg
∀[a,b:ℚ]. (rat2real(qavg(a;b)) = (rat2real(a) + rat2real(b)/r(2)))
Lemma: rat2real-qavg-2
∀[a,b:ℚ]. ((r(2) * rat2real(qavg(a;b))) = (rat2real(a) + rat2real(b)))
Lemma: rat2real-qmax
∀[a,b:ℚ]. (rat2real(qmax(a;b)) = rmax(rat2real(a);rat2real(b)))
Lemma: rat2real-qmin
∀[a,b:ℚ]. (rat2real(qmin(a;b)) = rmin(rat2real(a);rat2real(b)))
Lemma: radd-positive-implies
∀x,y:ℝ. ((r0 < (x + y)) ⇒ ((r0 < x) ∨ (r0 < y)))
Lemma: rabs-difference-bound-iff
∀x,y,z:ℝ. (|x - y| < z ⇐⇒ ((y - z) < x) ∧ (x < (y + z)))
Lemma: rabs-difference-bound-rleq
∀x,y,z:ℝ. (|x - y| ≤ z ⇐⇒ ((y - z) ≤ x) ∧ (x ≤ (y + z)))
Lemma: rabs-rleq
∀x,z:ℝ. (|x| ≤ z ⇐⇒ (-(z) ≤ x) ∧ (x ≤ z))
Lemma: rabs-difference-is-zero
∀x,y:ℝ. (|x - y| = r0 ⇐⇒ x = y)
Lemma: rabs-rleq-iff
∀x,z:ℝ. (|x| ≤ z ⇐⇒ (-(z) ≤ x) ∧ (x ≤ z))
Lemma: rabs-is-zero
∀x:ℝ. (|x| = r0 ⇐⇒ x = r0)
Lemma: rabs-rless-iff
∀x,z:ℝ. (|x| < z ⇐⇒ (-(z) < x) ∧ (x < z))
Lemma: rabs-difference-lower-bound
∀x,y,z:ℝ. (z < |x - y| ⇐⇒ ((z + y) < x) ∨ ((z + x) < y))
Lemma: rneq-rabs
∀a,v:ℝ. (v ≠ a ⇒ -(v) ≠ a ⇒ v ≠ |a|)
Lemma: rless-cases1
∀x,y:ℝ. ((x < y) ⇒ (∀z:ℝ. ((x < z) ∨ (z < y))))
Lemma: rnexp-rminus
∀x:ℝ. ∀n:ℕ. (-(x)^n = if isOdd(n) then -(x^n) else x^n fi )
Lemma: rnexp-rless2
∀x,y:ℝ. ((x < y) ⇒ (r0 < y) ⇒ (∀n:ℕ+. ((↑isOdd(n)) ⇒ (x^n < y^n))))
Lemma: rnexp-rless-odd
∀n:ℕ+. ((↑isOdd(n)) ⇒ (∀x,y:ℝ. ((x < y) ⇒ (x^n < y^n))))
Lemma: rnexp-req-iff-odd
∀n:ℕ+. ∀x,y:ℝ. ((↑isOdd(n)) ⇒ (x = y ⇐⇒ x^n = y^n))
Lemma: rnexp-req-iff-even
∀n:ℕ+. ∀x,y:ℝ. ((↑isEven(n)) ⇒ (|x| = |y| ⇐⇒ x^n = y^n))
Lemma: square-req-iff
∀x,y:ℝ. ((r0 ≤ x) ⇒ (r0 ≤ y) ⇒ (x = y ⇐⇒ x^2 = y^2))
Lemma: squares-req
∀x,y:ℝ. (y ≠ r0 ⇒ (x^2 = y^2 ⇐⇒ (x = y) ∨ (x = -(y))))
Lemma: rabs-ub
∀a:ℝ. ((r0 < a) ⇒ (∀x:ℝ. (a ≤ |x| ⇐⇒ (a ≤ x) ∨ (a ≤ -(x)))))
Lemma: rabs-strict-ub
∀a:ℝ. ((r0 ≤ a) ⇒ (∀x:ℝ. (a < |x| ⇐⇒ (a < x) ∨ (a < -(x)))))
Lemma: rabs-rneq
∀a,b:ℝ. (|a| ≠ |b| ⇒ a ≠ b)
Lemma: real-approx
∀[x:ℝ]. ∀[n:ℕ+]. (|(r(2 * n) * x) - r(x n)| ≤ r(2))
Definition: rational-approx
(x within 1/n) == (r(x n))/2 * n
Lemma: rational-approx_wf
∀[x:ℕ+ ⟶ ℤ]. ∀[n:ℕ+]. ((x within 1/n) ∈ ℝ)
Lemma: rational-approx-property
∀x:ℝ. ∀n:ℕ+. (|x - (x within 1/n)| ≤ (r1/r(n)))
Lemma: rational-approx-property-alt
∀x:ℝ. ∀n:ℕ+. (|(r(2 * n) * x) - r(x n)| ≤ r(2))
Lemma: rational-approx-property-alt2
∀x:ℝ. ∀n:ℕ+. (|r(x n)| ≤ ((r(2 * n) * |x|) + r(2)))
Lemma: rational-approx-property-ext
∀x:ℝ. ∀n:ℕ+. (|x - (x within 1/n)| ≤ (r1/r(n)))
Lemma: rational-approx-property1
∀x:ℝ. ∀n:ℕ+. (x ≤ ((x within 1/n) + (r1/r(n))))
Lemma: rational-approx-property2
∀x:ℝ. ∀n:ℕ+. (((x within 1/n) - (r1/r(n))) ≤ x)
Definition: rational-lower-approx
(below x within 1/n) == eval m = 2 * n in eval a = (x m) - 2 in (r(a))/2 * m
Lemma: rational-lower-approx_wf
∀[x:ℕ+ ⟶ ℤ]. ∀[n:ℕ+]. ((below x within 1/n) ∈ ℝ)
Lemma: rational-lower-approx-property
∀x:ℝ. ∀n:ℕ+. (((below x within 1/n) ≤ x) ∧ (x ≤ ((below x within 1/n) + (r1/r(n)))))
Definition: rational-upper-approx
above x within 1/n == eval m = 2 * n in eval a = (x m) + 2 in (r(a))/2 * m
Lemma: rational-upper-approx_wf
∀[x:ℕ+ ⟶ ℤ]. ∀[n:ℕ+]. (above x within 1/n ∈ ℝ)
Lemma: rational-upper-approx-as-rat2real
∀x:ℝ. ∀n:ℕ+. ∃q:ℚ. (above x within 1/n = rat2real(q))
Lemma: rational-lower-approx-as-rat2real
∀x:ℝ. ∀n:ℕ+. ∃q:ℚ. ((below x within 1/n) = rat2real(q))
Lemma: rational-upper-approx-property
∀x:ℝ. ∀n:ℕ+. (((above x within 1/n - (r1/r(n))) ≤ x) ∧ (x ≤ above x within 1/n))
Definition: rational-inner-approx
rational-inner-approx(x;n) ==
eval m = 2 * n in
eval z = x m in
eval a = if 4 <z z then z - 2
if z <z -4 then z + 2
else 0
fi in
(r(a))/2 * m
Lemma: rational-inner-approx_wf
∀[x:ℕ+ ⟶ ℤ]. ∀[n:ℕ+]. (rational-inner-approx(x;n) ∈ ℝ)
Lemma: rational-inner-approx-property
∀x:ℝ. ∀n:ℕ+. ((|rational-inner-approx(x;n)| ≤ |x|) ∧ (|x - rational-inner-approx(x;n)| ≤ (r(2)/r(n))))
Lemma: rational-inner-approx-int
∀x:ℝ. ∀n:ℕ+. ∃z:ℤ. ((|(r(z)/r(4 * n))| ≤ |x|) ∧ (|x - (r(z)/r(4 * n))| ≤ (r(2)/r(n))))
Lemma: integer-between-reals
∀a,b:ℝ. ((r(2) ≤ (b - a)) ⇒ (∃k:ℤ. ((a < r(k)) ∧ (r(k) < b))))
Lemma: close-reals-iff
∀[x,y:ℝ]. ∀[k:ℕ+]. uiff(|x - y| ≤ (r1/r(k));∀m:ℕ+. ((|(x m) - y m| * k) ≤ ((4 * k) + (2 * m))))
Lemma: implies-close-reals
∀[x,y:ℝ]. ∀[m:ℕ+]. ∀[k:ℕ]. ((|(x m) - y m| ≤ (2 * k)) ⇒ (|x - y| ≤ (r(2 + k)/r(m))))
Lemma: close-reals-implies
∀[x,y:ℝ]. ∀[m:ℕ+]. |(x m) - y m| ≤ 4 supposing |x - y| ≤ (r1/r(3 * m))
Definition: ravg
ravg(x;y) == (x + y/r(2))
Lemma: ravg_wf
∀[x,y:ℝ]. (ravg(x;y) ∈ ℝ)
Lemma: ravg_comm
∀[x,y:ℝ]. (ravg(x;y) = ravg(y;x))
Lemma: ravg-between
∀x,y:ℝ. ((x < y) ⇒ ((x < ravg(x;y)) ∧ (ravg(x;y) < y)))
Lemma: ravg-weak-between
∀x,y:ℝ. ((x ≤ y) ⇒ ((x ≤ ravg(x;y)) ∧ (ravg(x;y) ≤ y)))
Lemma: ravg-dist
∀x,y:ℝ. ((|ravg(x;y) - x| = ((r1/r(2)) * |y - x|)) ∧ (|ravg(x;y) - y| = ((r1/r(2)) * |y - x|)))
Lemma: ravg-dist-when-rleq
∀[x,y:ℝ]. ((ravg(x;y) - x) = ((r1/r(2)) * (y - x))) ∧ ((y - ravg(x;y)) = ((r1/r(2)) * (y - x))) supposing x ≤ y
Definition: regular-upto
regular-upto(k;n;f) ==
bdd-all(n;i.bdd-all(n;j.|((i + 1) * (f (j + 1))) - (j + 1) * (f (i + 1))| ≤z (2 * k) * ((i + 1) + j + 1)))
Lemma: regular-upto_wf
∀[k,n:ℕ]. ∀[f:ℕ+ ⟶ ℤ]. (regular-upto(k;n;f) ∈ 𝔹)
Lemma: assert-regular-upto
∀[k,n:ℕ]. ∀[f:ℕ+ ⟶ ℤ]. (↑regular-upto(k;n;f) ⇐⇒ ∀i,j:ℕ+n + 1. (|(i * (f j)) - j * (f i)| ≤ ((2 * k) * (i + j))))
Definition: strong-regular-upto
strong-regular-upto(a;b;n;f) ==
bdd-all(n;i.bdd-all(n;j.a * |((i + 1) * (f (j + 1))) - (j + 1) * (f (i + 1))| ≤z b * ((i + 1) + j + 1)))
Lemma: strong-regular-upto_wf
∀[a,b,n:ℕ]. ∀[f:ℕ+ ⟶ ℤ]. (strong-regular-upto(a;b;n;f) ∈ 𝔹)
Lemma: assert-strong-regular-upto
∀[a,b,n:ℕ]. ∀[f:ℕ+ ⟶ ℤ].
(↑strong-regular-upto(a;b;n;f) ⇐⇒ ∀i,j:ℕ+n + 1. ((a * |(i * (f j)) - j * (f i)|) ≤ (b * (i + j))))
Definition: seq-min-upper
seq-min-upper(k;n;f) ==
primrec(n;1;λi,r. if ((i + 1) * (f r)) + ((2 * k) * (i + 1)) ≤z (r * (f (i + 1))) + ((2 * k) * r)
then r
else i + 1
fi )
Lemma: seq-min-upper_wf
∀[k,n:ℕ]. ∀[f:ℕ+ ⟶ ℤ]. (seq-min-upper(k;n;f) ∈ ℕ+)
Lemma: seq-min-upper-le
∀[k:ℕ]. ∀[n:ℕ+]. ∀[f:ℕ+ ⟶ ℤ]. (seq-min-upper(k;n;f) ≤ n)
Lemma: seq-min-upper-property
∀[k,n:ℕ]. ∀[f:ℕ+ ⟶ ℤ].
∀i:ℕ+n + 1. (((i * (f seq-min-upper(k;n;f))) - seq-min-upper(k;n;f) * (f i)) ≤ ((2 * k) * (seq-min-upper(k;n;f) - i)))
Definition: seq-max-lower
seq-max-lower(k;n;f) ==
primrec(n;1;λi,r. if ((i + 1) * (f r)) - (2 * k) * (i + 1) ≤z (r * (f (i + 1))) - (2 * k) * r then i + 1 else r fi )
Lemma: seq-max-lower_wf
∀[k,n:ℕ]. ∀[f:ℕ+ ⟶ ℤ]. (seq-max-lower(k;n;f) ∈ ℕ+)
Lemma: seq-max-lower-le
∀[k:ℕ]. ∀[n:ℕ+]. ∀[f:ℕ+ ⟶ ℤ]. (seq-max-lower(k;n;f) ≤ n)
Lemma: seq-max-lower-property
∀[k,n:ℕ]. ∀[f:ℕ+ ⟶ ℤ].
∀i:ℕ+n + 1. (((seq-max-lower(k;n;f) * (f i)) - i * (f seq-max-lower(k;n;f))) ≤ ((2 * k) * (seq-max-lower(k;n;f) - i)))
Lemma: regular-upto-iff
∀k,b:ℕ+. ∀x:ℕ+ ⟶ ℤ.
(↑regular-upto(k;b;x)
⇐⇒ ∀n,m:ℕ+b + 1.
let j = seq-min-upper(k;b;x) in
let z = (r((x j) + (2 * k))/r((2 * k) * j)) in
(((r((x n) - 2 * k)/r((2 * k) * n)) ≤ z) ∧ (z ≤ (r((x n) + (2 * k))/r((2 * k) * n))))
∧ ((r((x m) - 2 * k)/r((2 * k) * m)) ≤ z)
∧ (z ≤ (r((x m) + (2 * k))/r((2 * k) * m))))
Lemma: regular-upto-iff2
∀k,b:ℕ+. ∀x:ℕ+ ⟶ ℤ.
(↑regular-upto(k;b;x)
⇐⇒ ∀n,m:ℕ+b + 1.
let j = seq-max-lower(k;b;x) in
let z = (r((x j) - 2 * k)/r((2 * k) * j)) in
(((r((x n) - 2 * k)/r((2 * k) * n)) ≤ z) ∧ (z ≤ (r((x n) + (2 * k))/r((2 * k) * n))))
∧ ((r((x m) - 2 * k)/r((2 * k) * m)) ≤ z)
∧ (z ≤ (r((x m) + (2 * k))/r((2 * k) * m))))
Lemma: regular-iff-all-regular-upto
∀k:ℕ+. ∀x:ℕ+ ⟶ ℤ. (k-regular-seq(x) ⇐⇒ ∀b:ℕ+. (↑regular-upto(k;b;x)))
Lemma: regular-int-seq-iff
∀k:ℕ+. ∀x:ℕ+ ⟶ ℤ.
(k-regular-seq(x)
⇐⇒ ∀n,m:ℕ+.
∃z:ℝ
((((r((x n) - 2 * k)/r((2 * k) * n)) ≤ z) ∧ (z ≤ (r((x n) + (2 * k))/r((2 * k) * n))))
∧ ((r((x m) - 2 * k)/r((2 * k) * m)) ≤ z)
∧ (z ≤ (r((x m) + (2 * k))/r((2 * k) * m)))))
Lemma: regular-iff
∀x:ℕ+ ⟶ ℤ
(regular-seq(x)
⇐⇒ ∀n,m:ℕ+.
∃z:ℝ
((((r((x n) - 2)/r(2 * n)) ≤ z) ∧ (z ≤ (r((x n) + 2)/r(2 * n))))
∧ ((r((x m) - 2)/r(2 * m)) ≤ z)
∧ (z ≤ (r((x m) + 2)/r(2 * m)))))
Definition: ddr
ddr(x;n) == eval N = 10^n in eval N2 = N ÷ 2 in eval a = x N2 in r(a/N)
Lemma: ddr_wf
∀x:ℝ. ∀n:ℕ+. (ddr(x;n) ∈ {y:ℝ| |x - y| ≤ (r1/r(5 * 10^(n - 1)))} )
Lemma: neg-approx-of-nonneg-real
∀x:ℝ. ((r0 ≤ x) ⇒ (∀n:ℕ+. (((x n) ≤ 0) ⇒ (|x n| ≤ 2))))
Lemma: rmul-rmax
∀[x,y,z:ℝ]. ((r0 ≤ z) ⇒ ((z * rmax(x;y)) = rmax(z * x;z * y)))
Lemma: rmul-rmin
∀[x,y,z:ℝ]. ((r0 ≤ z) ⇒ ((z * rmin(x;y)) = rmin(z * x;z * y)))
Lemma: rmax-rnexp
∀[n:ℕ]. ∀[x,y:ℝ]. ((r0 ≤ x) ⇒ (r0 ≤ y) ⇒ (rmax(x^n;y^n) = rmax(x;y)^n))
Lemma: rmin-rnexp
∀[n:ℕ]. ∀[x,y:ℝ]. ((r0 ≤ x) ⇒ (r0 ≤ y) ⇒ (rmin(x^n;y^n) = rmin(x;y)^n))
Lemma: rless-property
∀x,y:ℝ. ∀n:x < y. (x n) + 4 < y n
Lemma: rless-cases-proof
∀x,y:ℝ. ((x < y) ⇒ (∀z:ℝ. ((x < z) ∨ (z < y))))
Definition: rless-case
rless-case(x;y;n;z) == eval m = (12 * n) + 1 in if ((x m) + 4) < (z m) then inl m else (inr m )
Lemma: rless-cases
∀x,y:ℝ. ((x < y) ⇒ (∀z:ℝ. ((x < z) ∨ (z < y))))
Lemma: rless-case_wf
∀[x,y:ℝ]. ∀[n:x < y]. ∀[z:ℝ]. (rless-case(x;y;n;z) ∈ (x < z) ∨ (z < y))
Lemma: rless-cases-sq
∀x:ℝ. ∀y:{y:ℝ| x < y} . ∀z:ℝ. ((x < z) ∨ (z < y))
Lemma: reals-close-or-rneq
∀K:ℕ+
(∃g:ℝ ⟶ ℝ ⟶ ℕ3 [(∀x,y:ℝ.
((((g x y) = 1 ∈ ℤ) ⇒ ((r1/r(K)) < (y - x)))
∧ (((g x y) = 2 ∈ ℤ) ⇒ ((r1/r(K)) < (x - y)))
∧ (((g x y) = 0 ∈ ℤ) ⇒ (|x - y| < (r(2)/r(K))))))])
Definition: rclose-or-sep
rclose-or-sep(K;x;y) ==
eval x1 = (36 * K) + 1 in
if (((r1)/K x1) + 4) < (|(x - y) x1|)
then if (-((x - y) ((x1 * 20) + 1))) < ((x - y) ((x1 * 20) + 1)) then 2 else 1
else 0
Lemma: reals-close-or-rneq-ext
∀K:ℕ+
(∃g:ℝ ⟶ ℝ ⟶ ℕ3 [(∀x,y:ℝ.
((((g x y) = 1 ∈ ℤ) ⇒ ((r1/r(K)) < (y - x)))
∧ (((g x y) = 2 ∈ ℤ) ⇒ ((r1/r(K)) < (x - y)))
∧ (((g x y) = 0 ∈ ℤ) ⇒ (|x - y| < (r(2)/r(K))))))])
Lemma: rclose-or-sep_wf
∀[K:ℕ+]. ∀[x,y:ℝ].
(rclose-or-sep(K;x;y) ∈ {i:ℕ3|
((i = 1 ∈ ℤ) ⇒ ((r1/r(K)) < (y - x)))
∧ ((i = 2 ∈ ℤ) ⇒ ((r1/r(K)) < (x - y)))
∧ ((i = 0 ∈ ℤ) ⇒ (|x - y| < (r(2)/r(K))))} )
Definition: small-real-test
small-real-test(k;z) == rless-case((r1)/k;(r(4))/k;k;z)
Lemma: small-real-test_wf
∀[k:ℕ+]. ∀[z:ℝ]. (small-real-test(k;z) ∈ ((r1)/k < z) ∨ (z < (r(4))/k))
Lemma: rneq-cases
∀x,y:ℝ. (x ≠ y ⇒ (∀z:ℝ. (x ≠ z ∨ y ≠ z)))
Lemma: rneq-cotrans
∀x,y,z:ℝ. (x ≠ y ⇒ (x ≠ z ∨ y ≠ z))
Lemma: rneq-radd
∀x,y,a,b:ℝ. (x + y ≠ a + b ⇒ (x ≠ a ∨ y ≠ b))
Lemma: rmul-is-negative1
∀x,y:ℝ. (((x * y) < r0) ⇒ (x ≠ r0 ∨ y ≠ r0))
Lemma: rmul-is-negative
∀x,y:ℝ. (((x * y) < r0) ⇒ ((x < r0) ∨ (y < r0)))
Lemma: rmul-is-positive
∀x,y:ℝ. (r0 < (x * y) ⇐⇒ ((r0 < x) ∧ (r0 < y)) ∨ ((x < r0) ∧ (y < r0)))
Lemma: combine-rless
∀a,b,c,d:ℝ. ((a < b) ⇒ (c < d) ⇒ (((b * c) + (a * d)) < ((b * d) + (a * c))))
Lemma: rmul-negative-iff
∀x,y:ℝ. ((x * y) < r0 ⇐⇒ ((r0 < x) ∧ (y < r0)) ∨ ((x < r0) ∧ (r0 < y)))
Lemma: rneq-rmul
∀x,y,a,b:ℝ. (x * y ≠ a * b ⇒ (x ≠ a ∨ y ≠ b))
Lemma: square-req-self-iff
∀x:ℝ. ((x * x) = x ⇐⇒ (x = r1) ∨ (x = r0))
Lemma: int-rmul-is-positive
∀k:ℤ. ∀y:ℝ. (r0 < k * y ⇐⇒ (0 < k ∧ (r0 < y)) ∨ (k < 0 ∧ (y < r0)))
Lemma: rdiv-is-positive
∀x,y:ℝ. (y ≠ r0 ⇒ (r0 < (x/y) ⇐⇒ ((r0 < x) ∧ (r0 < y)) ∨ ((x < r0) ∧ (y < r0))))
Lemma: int-rdiv-is-positive
∀x:ℝ. ∀k:ℤ-o. (r0 < (x)/k ⇐⇒ ((r0 < x) ∧ 0 < k) ∨ ((x < r0) ∧ k < 0))
Lemma: epsilon/2-lemma
∀[k:ℕ+]. (((r1/r(2 * k)) + (r1/r(2 * k))) ≤ (r1/r(k)))
Lemma: epsilon/n-lemma
∀[k,n:ℕ+]. ((r(n) * (r1/r(n * k))) ≤ (r1/r(k)))
Definition: real-ratio-bound
real-ratio-bound(M;x;y;a;b) ==
eval c = rclose-or-sep(M;x;y) in
if (c =z 1) then (a/y - x)
if (c =z 2) then (b/x - y)
else (r(M)/r(2)) * rmin(a;b)
fi
Lemma: real-ratio-bound_wf
∀[M:ℕ+]. ∀[x,y:ℝ]. ∀[a,b:{r:ℝ| r0 < r} ].
(real-ratio-bound(M;x;y;a;b) ∈ {r:ℝ| ((x < y) ⇒ (r ≤ (a/y - x))) ∧ ((y < x) ⇒ (r ≤ (b/x - y))) ∧ (r0 < r)} )
Lemma: real-ratio-bound-cases
∀M:ℕ+. ∀x,y:ℝ.
∀[a,b:{r:ℝ| r0 < r} ].
((x < y) ∧ (real-ratio-bound(M;x;y;a;b) = (a/y - x))) ∨ ((y < x) ∧ (real-ratio-bound(M;x;y;a;b) = (b/x - y)))
supposing (r(2)/r(M)) ≤ |x - y|
Lemma: nearby-cases
∀n:ℕ+. ∀x,y:ℝ. ((x < y) ∨ (y < x) ∨ (|x - y| ≤ (r1/r(n))))
Lemma: nearby-cases-ext
∀n:ℕ+. ∀x,y:ℝ. ((x < y) ∨ (y < x) ∨ (|x - y| ≤ (r1/r(n))))
Lemma: by-nearby-cases
∀[P:ℝ ⟶ ℝ ⟶ ℙ]
∀n:ℕ+. ∀x:ℝ.
((∀y:{y:ℝ| x < y} . P[x;y])
⇒ (∀y:{y:ℝ| y < x} . P[x;y])
⇒ (∀y:{y:ℝ| |x - y| ≤ (r1/r(n))} . P[x;y])
⇒ (∀y:ℝ. P[x;y]))
Lemma: by-nearby-cases-ext
∀[P:ℝ ⟶ ℝ ⟶ ℙ]
∀n:ℕ+. ∀x:ℝ.
((∀y:{y:ℝ| x < y} . P[x;y])
⇒ (∀y:{y:ℝ| y < x} . P[x;y])
⇒ (∀y:{y:ℝ| |x - y| ≤ (r1/r(n))} . P[x;y])
⇒ (∀y:ℝ. P[x;y]))
Lemma: int-rdiv-cancel
∀a:ℤ. ∀n,m:ℕ+. ((r(m * a))/m * n = (r(a))/n ∈ ℝ)
Lemma: near-inverse-of-increasing-function
∀f:ℝ ⟶ ℝ. ∀n,M:ℕ+. ∀z:ℝ. ∀a,b:ℤ.
∀k:ℕ+
(∃c:ℤ. (∃j:ℕ+ [((|f[(r(c))/j] - z| ≤ (r1/r(n))) ∧ ((r(a))/k ≤ (r(c))/j) ∧ ((r(c))/j ≤ (r(b))/k))])) supposing
((z ≤ f[(r(b))/k]) and
(f[(r(a))/k] ≤ z) and
(∀x,y:ℝ.
(((r(a))/k ≤ x)
⇒ (x < y)
⇒ (y ≤ (r(b))/k)
⇒ ((f[x] ≤ f[y]) ∧ (((y - x) ≤ (r1/r(M))) ⇒ ((f[y] - f[x]) ≤ (r1/r(n))))))))
supposing a < b
Lemma: near-inverse-of-increasing-function-ext
∀f:ℝ ⟶ ℝ. ∀n,M:ℕ+. ∀z:ℝ. ∀a,b:ℤ.
∀k:ℕ+
(∃c:ℤ. (∃j:ℕ+ [((|f[(r(c))/j] - z| ≤ (r1/r(n))) ∧ ((r(a))/k ≤ (r(c))/j) ∧ ((r(c))/j ≤ (r(b))/k))])) supposing
((z ≤ f[(r(b))/k]) and
(f[(r(a))/k] ≤ z) and
(∀x,y:ℝ.
(((r(a))/k ≤ x)
⇒ (x < y)
⇒ (y ≤ (r(b))/k)
⇒ ((f[x] ≤ f[y]) ∧ (((y - x) ≤ (r1/r(M))) ⇒ ((f[y] - f[x]) ≤ (r1/r(n))))))))
supposing a < b
Lemma: rbetween-convex
∀x,a,b:ℝ. ((a < b) ⇒ a≤x≤b ⇒ (∃t:ℝ. ((r0 ≤ t) ∧ (t ≤ r1) ∧ (x = ((t * a) + ((r1 - t) * b))))))
Lemma: rnexp-is-positive
∀i:ℕ+. ∀x:ℝ. ((r0 < |x^i|) ⇒ (r0 < |x|))
Lemma: square-rless-implies
∀x,y:ℝ. ((r0 ≤ y) ⇒ (x^2 < y^2) ⇒ (x < y))
Lemma: rnexp2-positive-iff
∀x:ℝ. (r0 < x^2 ⇐⇒ x ≠ r0)
Lemma: square-positive-iff
∀x:ℝ. (r0 < (x * x) ⇐⇒ x ≠ r0)
Lemma: rmul-nonneg
∀[x,y:ℝ]. r0 ≤ (x * y) supposing ((r0 ≤ x) ∧ (r0 ≤ y)) ∨ ((x ≤ r0) ∧ (y ≤ r0))
Lemma: rmul-nonneg-case1
∀[x,y:ℝ]. r0 ≤ (x * y) supposing (r0 ≤ x) ∧ (r0 ≤ y)
Lemma: rmul-nonzero
∀x,y:ℝ. (x * y ≠ r0 ⇐⇒ x ≠ r0 ∧ y ≠ r0)
Lemma: rdiv-nonzero
∀x,y:ℝ. (y ≠ r0 ⇒ ((x/y) ≠ r0 ⇐⇒ x ≠ r0))
Lemma: square-nonzero
∀x:ℝ. (x * x ≠ r0 ⇐⇒ x ≠ r0)
Lemma: square-is-zero
∀x:ℝ. ((x * x) = r0 ⇐⇒ x = r0)
Lemma: square-rless-1-iff
∀x:ℝ. (x^2 < r1 ⇐⇒ |x| < r1)
Lemma: square-rleq-1-iff
∀x:ℝ. (x^2 ≤ r1 ⇐⇒ |x| ≤ r1)
Lemma: square-rge-1-iff
∀x:ℝ. (r1 ≤ x^2 ⇐⇒ r1 ≤ |x|)
Lemma: square-is-one
∀x:ℝ. (x^2 = r1 ⇐⇒ (x = r1) ∨ (x = -(r1)))
Lemma: square-req-1-iff
∀x:ℝ. (x ≠ -(r1) ⇒ (x^2 = r1 ⇐⇒ x = r1))
Lemma: imonomial-nonneg-lemma
∀[k:ℕ+]. ∀[m,m':iMonomial()].
∀f:ℤ ⟶ ℝ. (r0 ≤ real_term_value(f;imonomial-term(m))) supposing mul-monomials(m';m') = mul-monomials(m;<k, []>) ∈ iMo\000Cnomial()
Lemma: imonomial-nonneg
∀[m:iMonomial()]. ∀f:ℤ ⟶ ℝ. (r0 ≤ real_term_value(f;imonomial-term(m))) supposing ↑nonneg-monomial(m)
Lemma: ipolynomial-nonneg
∀[p:iPolynomial()]. ∀f:ℤ ⟶ ℝ. (r0 ≤ real_term_value(f;ipolynomial-term(p))) supposing ↑nonneg-poly(p)
Lemma: real-term-nonneg
∀[t:int_term()]. ∀f:ℤ ⟶ ℝ. (r0 ≤ real_term_value(f;t)) supposing ↑nonneg-poly(int_term_to_ipoly(t))
Lemma: rationals-dense
∀x:ℝ. ∀y:{y:ℝ| x < y} . ∃n:ℕ+. ∃m:ℤ. ((x < (r(m)/r(n))) ∧ ((r(m)/r(n)) < y))
Lemma: rationals-dense-ext
∀x:ℝ. ∀y:{y:ℝ| x < y} . ∃n:ℕ+. ∃m:ℤ. ((x < (r(m)/r(n))) ∧ ((r(m)/r(n)) < y))
Lemma: small-reciprocal-real
∀x:{x:ℝ| r0 < x} . ∃k:ℕ+. ((r1/r(k)) < x)
Lemma: small-reciprocal-real-ext
∀x:{x:ℝ| r0 < x} . ∃k:ℕ+. ((r1/r(k)) < x)
Lemma: small-reciprocal-rneq-zero
∀x:ℝ. (x ≠ r0 ⇒ (∃k:ℕ+. ((r1/r(k)) < |x|)))
Definition: positive-lower-bound
positive-lower-bound(x) == rlessw(r0;x) + 1
Lemma: positive-lower-bound_wf
∀[x:{x:ℝ| r0 < x} ]. (positive-lower-bound(x) ∈ {k:ℕ+| (r1/r(k)) < x} )
Definition: integer-approx
integer-approx(x;k) == (x k) ÷ 2 * k
Lemma: integer-approx_wf
∀[x:ℝ]. ∀[k:ℕ+]. (integer-approx(x;k) ∈ {n:ℤ| |x - r(n)| ≤ (r1 + (r1/r(k)))} )
Definition: reduce-real
reduce-real(x;b;k) == integer-approx((x/b);k)
Lemma: reduce-real_wf
∀[k:ℕ+]. ∀[x:ℝ]. ∀[b:{b:ℝ| r0 < b} ]. (reduce-real(x;b;k) ∈ {n:ℤ| |x - r(n) * b| ≤ (b + (b/r(k)))} )
Lemma: implies-regular
∀[k:ℕ+]. ∀[x:ℕ+ ⟶ ℤ].
k-regular-seq(x) supposing ∀n,m:ℕ+. (|(x within 1/n) - (x within 1/m)| ≤ ((r(k)/r(n)) + (r(k)/r(m))))
Lemma: implies-real
∀[x:ℕ+ ⟶ ℤ]. x ∈ ℝ supposing ∀n,m:ℕ+. (|(x within 1/n) - (x within 1/m)| ≤ ((r1/r(n)) + (r1/r(m))))
Definition: converges-to
lim n→∞.x[n] = y == ∀k:ℕ+. (∃N:ℕ [(∀n:ℕ. ((N ≤ n) ⇒ (|x[n] - y| ≤ (r1/r(k)))))])
Lemma: converges-to_wf
∀[x:ℕ ⟶ ℝ]. ∀[y:ℝ]. (lim n→∞.x[n] = y ∈ ℙ)
Lemma: rational-approx-converges-to
∀[x:ℝ]. lim n→∞.(x within 1/n + 1) = x
Lemma: rinv-converges-to-0
lim n→∞.(r1/r(n + 1)) = r0
Lemma: converges-to_functionality
∀x1,x2:ℕ ⟶ ℝ. ∀y1,y2:ℝ. ({lim n→∞.x1[n] = y1 ⇒ lim n→∞.x2[n] = y2}) supposing ((y1 = y2) and (∀n:ℕ. (x1[n] = x2[n])))
Lemma: converges-to_functionality2
∀x1,x2:ℕ ⟶ ℝ. ∀y1,y2:ℝ. (lim n→∞.x1[n] = y1 ⇒ lim n→∞.x2[n] = y2) supposing ((y1 = y2) and (∀n:ℕ. (x1[n] = x2[n])))
Definition: converges
x[n]↓ as n→∞ == ∃y:ℝ. lim n→∞.x[n] = y
Lemma: converges_wf
∀[x:ℕ ⟶ ℝ]. (x[n]↓ as n→∞ ∈ ℙ)
Definition: diverges
n.x[n]↑ == ∃e:ℝ. ((r0 < e) ∧ (∀k:ℕ. ∃m,n:ℕ. ((k ≤ m) ∧ (k ≤ n) ∧ (e ≤ |x[m] - x[n]|))))
Lemma: diverges_wf
∀[x:ℕ ⟶ ℝ]. (n.x[n]↑ ∈ ℙ)
Lemma: infinitesmal-zero
∀[x:ℝ]. uiff(x = r0;∀[k:ℕ+]. (|x| ≤ (r1/r(k))))
Lemma: infinitesmal-difference
∀[x,y:ℝ]. uiff(x = y;∀[k:ℕ+]. (|x - y| ≤ (r1/r(k))))
Lemma: unique-limit
∀[x:ℕ ⟶ ℝ]. ∀[y1,y2:ℝ]. (y1 = y2) supposing (lim n→∞.x[n] = y2 and lim n→∞.x[n] = y1)
Definition: bounded-sequence
bounded-sequence(n.x[n]) == ∃b:ℝ. ∀n:ℕ. (|x[n]| ≤ b)
Lemma: bounded-sequence_wf
∀[x:ℕ ⟶ ℝ]. (bounded-sequence(n.x[n]) ∈ ℙ)
Lemma: converges-implies-bounded
∀x:ℕ ⟶ ℝ. (x[n]↓ as n→∞ ⇒ bounded-sequence(n.x[n]))
Lemma: integer-bound
∀x:ℝ. ∃n:ℕ+. (|x| ≤ r(n))
Definition: r-bound
r-bound(x) == fst((TERMOF{integer-bound:o, 1:l} x))
Lemma: r-bound_wf
∀[x:ℝ]. (r-bound(x) ∈ ℕ+)
Lemma: r-bound-property
∀[x:ℝ]. ((r(-r-bound(x)) ≤ x) ∧ (x ≤ r(r-bound(x))))
Lemma: r-strict-bound-property
∀x:ℝ. ((r(-(r-bound(x) + 1)) < x) ∧ (x < r(r-bound(x) + 1)))
Definition: cauchy
cauchy(n.x[n]) == ∀k:ℕ+. (∃N:ℕ [(∀n,m:ℕ. ((N ≤ n) ⇒ (N ≤ m) ⇒ (|x[n] - x[m]| ≤ (r1/r(k)))))])
Lemma: cauchy_wf
∀[x:ℕ ⟶ ℝ]. (cauchy(n.x[n]) ∈ ℙ)
Lemma: converges-iff-cauchy
∀x:ℕ ⟶ ℝ. (x[n]↓ as n→∞ ⇐⇒ cauchy(n.x[n]))
Lemma: converges-iff-cauchy-ext
∀x:ℕ ⟶ ℝ. (x[n]↓ as n→∞ ⇐⇒ cauchy(n.x[n]))
Definition: cauchy-limit
cauchy-limit(n.x[n];c) == accelerate(2;λn.(x[c n] n))
Lemma: cauchy-limit_wf
∀[x:ℕ ⟶ ℝ]. ∀[c:cauchy(n.x[n])]. (cauchy-limit(n.x[n];c) ∈ ℝ)
Lemma: converges-to-cauchy-limit
∀x:ℕ ⟶ ℝ. ∀c:cauchy(n.x[n]). lim n→∞.x[n] = cauchy-limit(n.x[n];c)
Lemma: converges-cauchy-witness
∀[x:ℕ ⟶ ℝ]. ∀[y:ℝ]. ∀[cvg:lim n→∞.x[n] = y]. (λk.(cvg (2 * k)) ∈ cauchy(n.x[n]))
Lemma: req-from-converges
∀[x:ℕ ⟶ ℝ]. ∀[y:ℝ]. ∀[cvg:lim n→∞.x[n] = y]. (y = cauchy-limit(n.x[n];λk.(cvg (2 * k))))
Definition: real-from-approx
real-from-approx(n.x[n]) == cauchy-limit(n.x[n + 1];λk.(2 * k))
Lemma: real-from-approx_wf
∀[a:ℝ]. ∀[x:k:ℕ+ ⟶ {v:ℝ| |v - a| ≤ (r1/r(k))} ]. (real-from-approx(n.x[n]) ∈ {b:ℝ| b = a} )
Definition: almost-positive
almost-positive(x) == ¬¬(r0 < x)
Lemma: almost-positive_wf
∀[x:ℝ]. (almost-positive(x) ∈ ℙ)
Definition: pseudo-positive
pseudo-positive(x) == ∀y:ℝ. ((¬¬(r0 < y)) ∨ (¬¬(y < x)))
Lemma: pseudo-positive_wf
∀[x:ℝ]. (pseudo-positive(x) ∈ ℙ)
Lemma: pseudo-positive-almost-positive
∀[x:ℝ]. almost-positive(x) supposing pseudo-positive(x)
Lemma: pseudo-positive-iff
∀x:ℝ. ((r0 ≤ x) ⇒ (pseudo-positive(x) ⇐⇒ ∀y:ℝ. ((¬(x = y)) ∨ (¬(y = r0)))))
Lemma: limit-shift
∀m:ℕ. ∀X:ℕ ⟶ ℝ. ∀a:ℝ. (lim n→∞.X[n] = a ⇒ lim n→∞.X[n + m] = a)
Lemma: limit-shift-iff
∀m:ℕ. ∀X:ℕ ⟶ ℝ. ∀a:ℝ. (lim n→∞.X[n] = a ⇐⇒ lim n→∞.X[n + m] = a)
Lemma: not-diverges-converges
∀[x:ℕ ⟶ ℝ]. (¬(x[n]↓ as n→∞ ∧ n.x[n]↑))
Lemma: subsequence-converges
∀a:ℝ. ∀x,y:ℕ ⟶ ℝ.
((∃N:ℕ. ∀n:ℕ. ∃m:ℕ. ((n ≤ m) ∧ (y[n] = x[m])) supposing N ≤ n) ⇒ lim n→∞.x[n] = a ⇒ lim n→∞.y[n] = a)
Definition: converges-to-infinity
lim n →∞.x[n] = ∞ == ∀k:ℕ+. ∀large(n).r(k) ≤ x[n]
Lemma: converges-to-infinity_wf
∀[x:ℕ ⟶ ℝ]. (lim n →∞.x[n] = ∞ ∈ ℙ)
Lemma: rnexp-converges
∀x:ℝ. ((|x| < r1) ⇒ lim n→∞.x^n = r0)
Lemma: rinv-exp-converges
∀M:ℕ+. ∀N:{2...}. lim n→∞.(r1/r(M * N^n)) = r0
Lemma: rnexp-converges-ext
∀x:ℝ. ((|x| < r1) ⇒ lim n→∞.x^n = r0)
Lemma: rinv-exp-converges-ext
∀M:ℕ+. ∀N:{2...}. lim n→∞.(r1/r(M * N^n)) = r0
Lemma: rpowers-converge
∀x:ℝ. (((|x| < r1) ⇒ lim n→∞.x^n = r0) ∧ ((r1 < x) ⇒ lim n →∞.x^n = ∞))
Lemma: rpowers-converge-ext
∀x:ℝ. (((|x| < r1) ⇒ lim n→∞.x^n = r0) ∧ ((r1 < x) ⇒ lim n →∞.x^n = ∞))
Lemma: r-archimedean
∀x:ℝ. ∃n:ℕ. ((r(-n) ≤ x) ∧ (x ≤ r(n)))
Lemma: simple-converges-to
∀x:ℕ ⟶ ℝ. ∀a,c:ℝ. ((∀n:ℕ. (|(x n) - a| ≤ ((r1/r(2^n)) * c))) ⇒ lim n→∞.x n = a)
Lemma: common-limit-squeeze
∀a,b,c:ℕ ⟶ ℝ.
((∀n:ℕ. ((a[n] ≤ a[n + 1]) ∧ (a[n + 1] ≤ b[n + 1]) ∧ (b[n + 1] ≤ b[n])))
⇒ lim n→∞.c[n] = r0
⇒ (∀n:ℕ. r0≤b[n] - a[n]≤c[n])
⇒ (∃y:ℝ. (lim n→∞.a[n] = y ∧ lim n→∞.b[n] = y)))
Lemma: common-limit-squeeze-ext
∀a,b,c:ℕ ⟶ ℝ.
((∀n:ℕ. ((a[n] ≤ a[n + 1]) ∧ (a[n + 1] ≤ b[n + 1]) ∧ (b[n + 1] ≤ b[n])))
⇒ lim n→∞.c[n] = r0
⇒ (∀n:ℕ. r0≤b[n] - a[n]≤c[n])
⇒ (∃y:ℝ. (lim n→∞.a[n] = y ∧ lim n→∞.b[n] = y)))
Lemma: common-limit-midpoints
∀a,b:ℕ ⟶ ℝ.
((∀n:ℕ
(((a[n + 1] = a[n]) ∧ (b[n + 1] = (a[n] + b[n]/r(2)))) ∨ ((a[n + 1] = (a[n] + b[n]/r(2))) ∧ (b[n + 1] = b[n]))))
⇒ (∃y:ℝ. (lim n→∞.a[n] = y ∧ lim n→∞.b[n] = y)))
Definition: rsum'
rsum'(n;m;k.x[k]) ==
eval n' = n in
eval m' = m in
eval k = (m' - n') + 1 in
if (k) < (1) then r0 else eval k2 = 2 * k in λj.eval m = k2 * j in Σ(x[n + i] m | i < k) ÷ k2
Definition: rsum
Σ{x[k] | n≤k≤m} == eval n' = n in eval m' = m in let xs ⟵ map(λk.x[k];[n', m' + 1)) in radd-list(xs)
Lemma: rsum_wf
∀[n,m:ℤ]. ∀[x:{n..m + 1-} ⟶ ℝ]. (Σ{x[k] | n≤k≤m} ∈ ℝ)
Lemma: rsum'-eq-rsum
∀[n,m:ℤ]. ∀[x:{n..m + 1-} ⟶ ℝ]. (rsum'(n;m;k.x[k]) = Σ{x[k] | n≤k≤m} ∈ (ℕ+ ⟶ ℤ))
Lemma: rsum'_wf
∀[n,m:ℤ]. ∀[x:{n..m + 1-} ⟶ ℝ]. (rsum'(n;m;k.x[k]) ∈ ℝ)
Lemma: rsum'-rsum
∀[n,m:ℤ]. ∀[x:{n..m + 1-} ⟶ ℝ]. (Σ{x[k] | n≤k≤m} = rsum'(n;m;k.x[k]))
Lemma: rsum_unroll
∀[n,m:ℤ]. ∀[x:{n..m + 1-} ⟶ ℝ].
(Σ{x[k] | n≤k≤m} = if m <z n then r0 if (m =z n) then x[n] else Σ{x[k] | n≤k≤m - 1} + x[m] fi )
Lemma: rsum_single
∀[n:ℤ]. ∀[x:{n..n + 1-} ⟶ ℝ]. (Σ{x[k] | n≤k≤n} = x[n])
Lemma: rsum-telescopes
∀[n:ℤ]. ∀[m:{n...}]. ∀[x,y:{n..m + 1-} ⟶ ℝ].
Σ{x[k] - y[k] | n≤k≤m} = (x[m] - y[n]) supposing ∀i:{n..m-}. (y[i + 1] = x[i])
Definition: rpolynomial
(Σi≤n. a_i * x^i) == Σ{(a i) * x^i | 0≤i≤n}
Lemma: rpolynomial_wf
∀[n:ℕ]. ∀[a:ℕn + 1 ⟶ ℝ]. ∀[x:ℝ]. ((Σi≤n. a_i * x^i) ∈ ℝ)
Lemma: rpolynomial_unroll
∀[n:ℕ]. ∀[a:ℕn + 1 ⟶ ℝ]. ∀[x:ℝ].
((Σi≤n. a_i * x^i) = if (n =z 0) then a 0 else ((a n) * x^n) + (Σi≤n - 1. a_i * x^i) fi )
Definition: pointwise-req
x[k] = y[k] for k ∈ [n,m] == ∀k:ℤ. ((n ≤ k) ⇒ (k ≤ m) ⇒ (x[k] = y[k]))
Lemma: pointwise-req_wf
∀[n,m:ℤ]. ∀[x,y:{n..m + 1-} ⟶ ℝ]. (x[k] = y[k] for k ∈ [n,m] ∈ ℙ)
Definition: pointwise-rleq
x[k] ≤ y[k] for k ∈ [n,m] == ∀k:ℤ. ((n ≤ k) ⇒ (k ≤ m) ⇒ (x[k] ≤ y[k]))
Lemma: pointwise-rleq_wf
∀[n,m:ℤ]. ∀[x,y:{n..m + 1-} ⟶ ℝ]. (x[k] ≤ y[k] for k ∈ [n,m] ∈ ℙ)
Lemma: rsum_functionality
∀[n,m:ℤ]. ∀[x,y:{n..m + 1-} ⟶ ℝ]. Σ{x[k] | n≤k≤m} = Σ{y[k] | n≤k≤m} supposing x[k] = y[k] for k ∈ [n,m]
Lemma: rsum_functionality2
∀[n,m:ℤ]. ∀[x,y:{n..m + 1-} ⟶ ℝ].
Σ{x[k] | n≤k≤m} = Σ{y[k] | n≤k≤m} supposing ∀k:ℤ. ((n ≤ k) ⇒ (k ≤ m) ⇒ (x[k] = y[k]))
Lemma: rsum_functionality_wrt_rleq
∀[n,m:ℤ]. ∀[x,y:{n..m + 1-} ⟶ ℝ]. Σ{x[k] | n≤k≤m} ≤ Σ{y[k] | n≤k≤m} supposing x[k] ≤ y[k] for k ∈ [n,m]
Lemma: rsum_functionality_wrt_rleq2
∀[n,m:ℤ]. ∀[x,y:{n..m + 1-} ⟶ ℝ].
Σ{x[k] | n≤k≤m} ≤ Σ{y[k] | n≤k≤m} supposing ∀k:ℤ. ((n ≤ k) ⇒ (k ≤ m) ⇒ (x[k] ≤ y[k]))
Lemma: rsum_functionality_wrt_rleq3
∀[n,m:ℤ]. ∀[x,y:{n..m + 1-} ⟶ ℝ].
Σ{x[k] | n≤k≤m} ≤ Σ{y[k] | n≤k≤m} supposing ∀k:ℤ. ((n ≤ k) ⇒ (k ≤ m) ⇒ (y[k] ≥ x[k]))
Lemma: rpolynomial_functionality
∀[n:ℕ]. ∀[a,b:ℕn + 1 ⟶ ℝ]. ∀[x,y:ℝ].
((Σi≤n. a_i * x^i) = (Σi≤n. b_i * y^i)) supposing ((x = y) and a[k] = b[k] for k ∈ [0,n])
Lemma: rsum-telescopes2
∀[n:ℤ]. ∀[m:{n...}]. ∀[x,y:{n..m + 1-} ⟶ ℝ].
Σ{x[k] - y[k] | n≤k≤m} = (x[n] - y[m]) supposing ∀i:{n..m-}. (x[i + 1] = y[i])
Lemma: rsum-rewrite-test
Σ{r(i) | 1≤i≤10} ≤ Σ{r(1 + i) | 1≤i≤10}
Lemma: rsum-split
∀[n,m:ℤ]. ∀[x:{n..m + 1-} ⟶ ℝ]. ∀[k:ℤ].
(Σ{x[i] | n≤i≤m} = (Σ{x[i] | n≤i≤k} + Σ{x[i] | k + 1≤i≤m})) supposing ((k ≤ m) and (n ≤ k))
Lemma: rsum-split2
∀[n,m:ℤ]. ∀[k:{n..m + 1-}]. ∀[x:{n..m + 1-} ⟶ ℝ]. (Σ{x[i] | n≤i≤m} = (Σ{x[i] | n≤i≤k} + Σ{x[i] | k + 1≤i≤m}))
Lemma: rsum-shift
∀[k,n,m:ℤ]. ∀[x:Top]. (Σ{x[i] | n≤i≤m} ~ Σ{x[i + k] | n - k≤i≤m - k})
Lemma: rsum-split-shift
∀[k,n,m:ℤ]. ∀[x:{n..m + 1-} ⟶ ℝ].
(Σ{x[i] | n≤i≤m} = (Σ{x[i] | n≤i≤k} + Σ{x[k + i + 1] | 0≤i≤m - k + 1})) supposing ((k ≤ m) and (n ≤ k))
Lemma: rsum-single
∀[n:ℤ]. ∀[x:{i:ℤ| i = n ∈ ℤ} ⟶ ℝ]. (Σ{x[i] | n≤i≤n} = x[n])
Lemma: rsum-empty
∀[n,m:ℤ]. ∀[x:Top]. Σ{x[i] | n≤i≤m} ~ r0 supposing m < n
Lemma: rsum-split-first
∀[n,m:ℤ]. ∀[x:{n..m + 1-} ⟶ ℝ]. Σ{x[i] | n≤i≤m} = (x[n] + Σ{x[i] | n + 1≤i≤m}) supposing n ≤ m
Lemma: rsum-split-first-shift
∀[n,m:ℤ]. ∀[x:{n..m + 1-} ⟶ ℝ]. Σ{x[i] | n≤i≤m} = (x[n] + Σ{x[i + 1] | n≤i≤m - 1}) supposing n ≤ m
Lemma: rsum-split-last
∀[n,m:ℤ]. ∀[x:{n..m + 1-} ⟶ ℝ]. Σ{x[i] | n≤i≤m} = (Σ{x[i] | n≤i≤m - 1} + x[m]) supposing n ≤ m
Lemma: rsum-difference
∀[k,n,m:ℤ]. ∀[x:{k..m + 1-} ⟶ ℝ].
((Σ{x[i] | k≤i≤m} - Σ{x[i] | k≤i≤n}) = Σ{x[i] | n + 1≤i≤m}) supposing ((n ≤ m) and (k ≤ n))
Lemma: rsum-difference2
∀[k,n,m:ℤ]. ∀[x,y:{k..m + 1-} ⟶ ℝ].
((Σ{x[i] | k≤i≤m} - Σ{y[i] | k≤i≤n}) = Σ{x[i] | n + 1≤i≤m}) supposing
((∀i:{k..n + 1-}. (x[i] = y[i])) and
(n ≤ m) and
(k ≤ n))
Lemma: rsum_linearity1
∀[n,m:ℤ]. ∀[x,y:{n..m + 1-} ⟶ ℝ]. (Σ{x[k] + y[k] | n≤k≤m} = (Σ{x[k] | n≤k≤m} + Σ{y[k] | n≤k≤m}))
Lemma: rsum_linearity2
∀[n,m:ℤ]. ∀[x:{n..m + 1-} ⟶ ℝ]. ∀[y:ℝ]. (Σ{y * x[k] | n≤k≤m} = (y * Σ{x[k] | n≤k≤m}))
Lemma: rsum_linearity3
∀[n,m:ℤ]. ∀[x:{n..m + 1-} ⟶ ℝ]. ∀[y:ℝ]. (Σ{x[k] * y | n≤k≤m} = (Σ{x[k] | n≤k≤m} * y))
Lemma: rsum_linearity-rsub
∀[n,m:ℤ]. ∀[x,y:{n..m + 1-} ⟶ ℝ]. (Σ{x[k] - y[k] | n≤k≤m} = (Σ{x[k] | n≤k≤m} - Σ{y[k] | n≤k≤m}))
Lemma: rsum-rminus
∀[n,m:ℤ]. ∀[x:{n..m + 1-} ⟶ ℝ]. (Σ{-(x[k]) | n≤k≤m} = -(Σ{x[k] | n≤k≤m}))
Lemma: rsum-constant
∀[n,m:ℤ]. ∀[a:ℝ]. (Σ{a | n≤k≤m} = (a * Σ{r1 | n≤k≤m}))
Lemma: rsum-constant2
∀[n,m:ℤ]. ∀[a:ℝ]. (Σ{a | n≤k≤m} = (a * if m <z n then r0 else r((m - n) + 1) fi ))
Lemma: rsum-zero
∀[n,m:ℤ]. (Σ{r0 | n≤k≤m} = r0)
Lemma: rsum-zero-req
∀[n,m:ℤ]. ∀[f:{n..m + 1-} ⟶ ℝ]. Σ{f[k] | n≤k≤m} = r0 supposing ∀k:{n..m + 1-}. (f[k] = r0)
Lemma: rsum-one
∀[n,m:ℤ]. (Σ{r1 | n≤k≤m} = if m <z n then r0 else r((m - n) + 1) fi )
Lemma: rsum_product
∀[a,b,c,d:ℤ]. ∀[x:{a..b + 1-} ⟶ ℝ]. ∀[y:{c..d + 1-} ⟶ ℝ].
((Σ{x[i] | a≤i≤b} * Σ{y[j] | c≤j≤d}) = Σ{Σ{x[i] * y[j] | c≤j≤d} | a≤i≤b})
Lemma: rsum_nonneg
∀[n,m:ℤ]. ∀[y:{n..m + 1-} ⟶ ℝ]. r0 ≤ Σ{y[k] | n≤k≤m} supposing r0 ≤ y[k] for k ∈ [n,m]
Lemma: rsum_int
∀[n:ℕ]. ∀[y:ℕn ⟶ ℤ]. (Σ{r(y[k]) | 0≤k≤n - 1} = r(Σ(y[k] | k < n)))
Lemma: rabs-rsum
∀[n,m:ℤ]. ∀[x:{n..m + 1-} ⟶ ℝ]. (|Σ{x[i] | n≤i≤m}| ≤ Σ{|x[i]| | n≤i≤m})
Lemma: rsum-positive-implies
∀n,m:ℤ. ∀x:{n..m + 1-} ⟶ ℝ. ((r0 < Σ{x[i] | n≤i≤m}) ⇒ (∃i:{n..m + 1-}. (r0 < |x[i]|)))
Lemma: rsum-of-nonneg-positive-iff
∀n,m:ℤ. ∀x:{n..m + 1-} ⟶ ℝ. ((∀i:{n..m + 1-}. (r0 ≤ x[i])) ⇒ (r0 < Σ{x[i] | n≤i≤m} ⇐⇒ ∃i:{n..m + 1-}. (r0 < x[i])))
Lemma: rsum-of-nonneg-zero-iff
∀[n,m:ℤ]. ∀[x:{n..m + 1-} ⟶ ℝ].
uiff(Σ{x[i] | n≤i≤m} = r0;∀i:{n..m + 1-}. (x[i] = r0)) supposing ∀i:{n..m + 1-}. (r0 ≤ x[i])
Lemma: item-rleq-rsum-of-nonneg
∀n,m:ℤ. ∀x:{n..m + 1-} ⟶ ℝ. ((∀i:{n..m + 1-}. (r0 ≤ x[i])) ⇒ (∀i:{n..m + 1-}. (x[i] ≤ Σ{x[i] | n≤i≤m})))
Lemma: rmul-rpolynomial
∀[n:ℕ]. ∀[a:ℕn + 1 ⟶ ℝ]. ∀[x,b:ℝ]. ((b * (Σi≤n. a_i * x^i)) = (Σi≤n. λi.(b * (a i))_i * x^i))
Lemma: rpolynomial-rmul
∀[n:ℕ]. ∀[a:ℕn + 1 ⟶ ℝ]. ∀[x,b:ℝ]. (((Σi≤n. a_i * x^i) * b) = (Σi≤n. λi.((a i) * b)_i * x^i))
Lemma: rdiv-rpolynomial
∀[n:ℕ]. ∀[a:ℕn + 1 ⟶ ℝ]. ∀[x,b:ℝ]. ((Σi≤n. a_i * x^i)/b) = (Σi≤n. λi.(a i/b)_i * x^i) supposing b ≠ r0
Lemma: shift-rpolynomial
∀[n:ℕ]. ∀[a:ℕn + 1 ⟶ ℝ]. ∀[x:ℝ].
((x * (Σi≤n. a_i * x^i)) = (Σi≤n + 1. λi.if (i =z 0) then r0 else a (i - 1) fi _i * x^i))
Lemma: add-rpolynomials
∀[n,m:ℕ]. ∀[a:ℕn + 1 ⟶ ℝ]. ∀[b:ℕm + 1 ⟶ ℝ]. ∀[x:ℝ].
((Σi≤n. a_i * x^i) + (Σi≤m. b_i * x^i)) = (Σi≤n. λi.if i ≤z m then (a i) + (b i) else a i fi _i * x^i) supposing m ≤ n
Lemma: subtract-rpolynomials
∀[n,m:ℕ]. ∀[a:ℕn + 1 ⟶ ℝ]. ∀[b:ℕm + 1 ⟶ ℝ]. ∀[x:ℝ].
((Σi≤n. a_i * x^i) - (Σi≤m. b_i * x^i)) = (Σi≤n. λi.if i ≤z m then (a i) - b i else a i fi _i * x^i) supposing m ≤ n
Lemma: add-rpolynomials-same-degree
∀[n:ℕ]. ∀[a,b:ℕn + 1 ⟶ ℝ]. ∀[x:ℝ]. (((Σi≤n. a_i * x^i) + (Σi≤n. b_i * x^i)) = (Σi≤n. λi.((a i) + (b i))_i * x^i))
Lemma: subtract-rpolynomials-same-degree
∀[n:ℕ]. ∀[a,b:ℕn + 1 ⟶ ℝ]. ∀[x:ℝ]. (((Σi≤n. a_i * x^i) - (Σi≤n. b_i * x^i)) = (Σi≤n. λi.((a i) - b i)_i * x^i))
Definition: rpolydiv
rpolydiv(n;a;z) == λi.primrec(n - 1 - i;a n;λj,r. ((a (n - j + 1)) + (z * r)))
Lemma: rpolydiv_wf
∀[n:ℤ]. ∀[a:ℕn + 1 ⟶ ℝ]. ∀[z:ℝ]. (rpolydiv(n;a;z) ∈ ℕn ⟶ ℝ)
Lemma: rpolydiv-rec
∀[n:ℤ]. ∀[a,z:Top].
((rpolydiv(n;a;z) (n - 1) ~ a n) ∧ (∀i:ℕn - 1. (rpolydiv(n;a;z) i ~ (a (i + 1)) + (z * (rpolydiv(n;a;z) (i + 1))))))
Lemma: rpolydiv-property
∀[n:ℕ+]. ∀[a:ℕn + 1 ⟶ ℝ]. ∀[z,x:ℝ].
((Σi≤n. a_i * x^i) = (((x - z) * (Σi≤n - 1. rpolydiv(n;a;z)_i * x^i)) + (Σi≤n. a_i * z^i)))
Lemma: rpolynomial-linear-factor
∀n:ℕ+. ∀a:ℕn + 1 ⟶ ℝ. ∀z:ℝ.
∃b:ℕn ⟶ ℝ. ((∀[x:ℝ]. ((Σi≤n. a_i * x^i) = ((x - z) * (Σi≤n - 1. b_i * x^i)))) ∧ ((b (n - 1)) = (a n)))
supposing (Σi≤n. a_i * z^i) = r0
Definition: rprod
rprod(n;m;k.x[k]) == if m <z n then r1 else rprod(n;m - 1;k.x[k]) * x[m] fi
Lemma: rprod_wf
∀[n,m:ℤ]. ∀[x:{n..m + 1-} ⟶ ℝ]. (rprod(n;m;k.x[k]) ∈ ℝ)
Lemma: rprod-empty
∀[n,m:ℤ]. ∀[x:Top]. rprod(n;m;k.x[k]) ~ r1 supposing m < n
Lemma: rprod_functionality
∀[n,m:ℤ]. ∀[x,y:{n..m + 1-} ⟶ ℝ]. rprod(n;m;k.x[k]) = rprod(n;m;k.y[k]) supposing x[k] = y[k] for k ∈ [n,m]
Lemma: rprod-single
∀[n:ℤ]. ∀[x:{n..n + 1-} ⟶ ℝ]. (rprod(n;n;k.x[k]) = x[n])
Lemma: rprod-of-positive
∀n,m:ℤ. ∀x:{n..m + 1-} ⟶ ℝ. r0 < rprod(n;m;k.x[k]) supposing ∀k:{n..m + 1-}. (r0 < x[k])
Lemma: rprod-rminus
∀n,m:ℤ. ∀x:{n..m + 1-} ⟶ ℝ. rprod(n;m;k.-(x[k])) = (r(-1)^(m - n) + 1 * rprod(n;m;k.x[k])) supposing n ≤ m
Lemma: rprod-of-negative
∀n,m:ℤ. ∀x:{n..m + 1-} ⟶ ℝ.
(((m - n rem 2) = 1 ∈ ℤ) ⇒ (r0 < rprod(n;m;k.x[k]))) ∧ (((m - n rem 2) = 0 ∈ ℤ) ⇒ (rprod(n;m;k.x[k]) < r0))
supposing (∀k:{n..m + 1-}. (x[k] < r0)) ∧ (n ≤ m)
Lemma: rprod-rsub-symmetry
∀n,m:ℤ. ∀x,y:{n..m + 1-} ⟶ ℝ.
rprod(n;m;k.x[k] - y[k]) = (r(-1)^(m - n) + 1 * rprod(n;m;k.y[k] - x[k])) supposing n ≤ m
Lemma: rprod-split
∀[n,m:ℤ]. ∀[x:{n..m + 1-} ⟶ ℝ]. ∀[i:ℤ].
rprod(n;m;k.x[k]) = (rprod(n;i;k.x[k]) * rprod(i + 1;m;k.x[k])) supposing (i ≤ m) ∧ (n ≤ (i + 1))
Lemma: rprod-split-last
∀[n,m:ℤ]. ∀[x:{n..m + 1-} ⟶ ℝ]. rprod(n;m;k.x[k]) = (rprod(n;m - 1;k.x[k]) * x[m]) supposing n ≤ m
Lemma: rprod-split-first
∀[n,m:ℤ]. ∀[x:{n..m + 1-} ⟶ ℝ]. rprod(n;m;k.x[k]) = (x[n] * rprod(n + 1;m;k.x[k])) supposing n ≤ m
Lemma: rprod-is-zero
∀[n,m:ℤ]. ∀[x:{n..m + 1-} ⟶ ℝ]. rprod(n;m;k.x[k]) = r0 supposing ∃k:{n..m + 1-}. (x[k] = r0)
Lemma: rprod-as-itop
∀[n,m:ℤ]. ∀[x:Top]. (Π(λx,y. (x * y),r1) n ≤ k < m. x[k] ~ rprod(n;m - 1;k.x[k]))
Lemma: rsum-as-itop
∀[n,m:ℤ]. ∀[x:{n..m-} ⟶ ℝ]. (Π(λx,y. (x + y),r0) n ≤ k < m. x[k] = Σ{x[k] | n≤k≤m - 1})
Definition: r-list-sum
r-list-sum(L) == reduce(λx,y. (x + y);r0;L)
Lemma: r-list-sum_wf
∀[L:ℝ List]. (r-list-sum(L) ∈ ℝ)
Lemma: r-list-sum_functionality
∀[L1,L2:ℝ List]. r-list-sum(L1) = r-list-sum(L2) supposing (||L1|| = ||L2|| ∈ ℤ) ∧ (∀i:ℕ||L1||. (L1[i] = L2[i]))
Lemma: rpolynomial-complete-factors
∀n:ℕ+. ∀a:ℕn + 1 ⟶ ℝ. ∀z:ℕn ⟶ ℝ.
((∀i,j:ℕn. ((¬(i = j ∈ ℤ)) ⇒ z i ≠ z j))
⇒ ∀[x:ℝ]. ((Σi≤n. a_i * x^i) = ((a n) * rprod(0;n - 1;j.x - z j))) supposing ∀j:ℕn. ((Σi≤n. a_i * z j^i) = r0))
Lemma: rpolynomial-complete-factors-ordered
∀n:ℕ+. ∀a:ℕn + 1 ⟶ ℝ. ∀z:ℕn ⟶ ℝ.
((∀j:ℕn - 1. ((z j) < (z (j + 1))))
⇒ ∀[x:ℝ]. ((Σi≤n. a_i * x^i) = ((a n) * rprod(0;n - 1;j.x - z j))) supposing ∀j:ℕn. ((Σi≤n. a_i * z j^i) = r0))
Lemma: rpolynomial-complete-roots-unique
∀[n:ℕ+]. ∀[a:ℕn + 1 ⟶ ℝ].
∀[z,y:ℕn ⟶ ℝ].
(∀[j:ℕn]. ((z j) = (y j))) supposing
((∀j:ℕn. ((Σi≤n. a_i * z j^i) = r0)) and
(∀j:ℕn. ((Σi≤n. a_i * y j^i) = r0)) and
(∀j:ℕn - 1. ((y j) < (y (j + 1)))) and
(∀j:ℕn - 1. ((z j) < (z (j + 1)))))
supposing a n ≠ r0
Lemma: exists-rneq-iff
∀n:ℕ. ∀a,b:ℕn ⟶ ℝ. (∃i:ℕn. a[i] ≠ b[i] ⇐⇒ r0 < Σ{|a[i] - b[i]| | 0≤i≤n - 1})
Lemma: sq_stable_ex_rneq
∀n:ℕ. ∀a,b:ℕn ⟶ ℝ. SqStable(∃i:ℕn. a[i] ≠ b[i])
Lemma: sq_stable_double_ex_rneq
∀m,n:ℕ. ∀a,b:ℕm ⟶ ℕn ⟶ ℝ. SqStable(∃i:ℕm. ∃j:ℕn. a[i;j] ≠ b[i;j])
Lemma: sq_stable_ex_nonzero
∀n:ℕ. ∀a:ℕn ⟶ ℝ. SqStable(∃i:ℕn. a[i] ≠ r0)
Lemma: rsum-triangle-inequality1
∀[n,m:ℤ]. ∀[x,y:{n..m + 1-} ⟶ ℝ]. ((Σ{|x[i]| | n≤i≤m} - Σ{|y[i]| | n≤i≤m}) ≤ Σ{|x[i] + y[i]| | n≤i≤m})
Lemma: rsum-triangle-inequality2
∀[n,m:ℤ]. ∀[x,y:{n..m + 1-} ⟶ ℝ]. ((Σ{|y[i]| | n≤i≤m} - Σ{|x[i]| | n≤i≤m}) ≤ Σ{|x[i] + y[i]| | n≤i≤m})
Lemma: real-binomial
∀[n:ℕ]. ∀[a,b:ℝ]. (a + b^n = Σ{r(choose(n;i)) * a^n - i * b^i | 0≤i≤n})
Lemma: rpolynomial-composition1
∀n:ℕ. ∀a:ℕn + 1 ⟶ ℝ. ∀b,c,d:ℝ. ∃a':ℕn + 1 ⟶ ℝ. ∀x:ℝ. ((Σi≤n. a'_i * x^i) = ((Σi≤n. a_i * ((c - b) * x) + b^i) - d))
Definition: rmaximum
rmaximum(n;m;k.x[k]) == primrec(m - n;x[n];λi,s. rmax(s;x[n + i + 1]))
Lemma: rmaximum_wf
∀[n,m:ℤ]. ∀[x:{n..m + 1-} ⟶ ℝ]. (rmaximum(n;m;k.x[k]) ∈ ℝ) supposing n ≤ m
Lemma: rmaximum_functionality
∀[n,m:ℤ].
∀[x,y:{n..m + 1-} ⟶ ℝ].
rmaximum(n;m;k.x[k]) = rmaximum(n;m;k.y[k]) supposing ∀k:ℤ. ((n ≤ k) ⇒ (k ≤ m) ⇒ (x[k] = y[k]))
supposing n ≤ m
Lemma: rmaximum_functionality_wrt_rleq
∀[n,m:ℤ].
∀[x,y:{n..m + 1-} ⟶ ℝ].
rmaximum(n;m;k.x[k]) ≤ rmaximum(n;m;k.y[k]) supposing ∀k:ℤ. ((n ≤ k) ⇒ (k ≤ m) ⇒ (x[k] ≤ y[k]))
supposing n ≤ m
Lemma: rmaximum-split
∀[n,m:ℤ]. ∀[x:{n..m + 1-} ⟶ ℝ]. ∀[k:ℤ].
(rmaximum(n;m;i.x[i]) = rmax(rmaximum(n;k;i.x[i]);rmaximum(k + 1;m;i.x[i]))) supposing (k < m and (n ≤ k))
Lemma: rmaximum-shift
∀[k,n,m:ℤ]. ∀[x:Top]. rmaximum(n;m;i.x[i]) ~ rmaximum(n - k;m - k;i.x[i + k]) supposing n ≤ m
Lemma: rmaximum_ub
∀[k,n,m:ℤ]. ∀[x:{n..m + 1-} ⟶ ℝ]. (x[k] ≤ rmaximum(n;m;i.x[i])) supposing ((k ≤ m) and (n ≤ k))
Lemma: rmaximum-lub
∀n,m:ℤ. ∀x:{n..m + 1-} ⟶ ℝ. ∀r:ℝ. (∀k:{n..m + 1-}. (x[k] ≤ r)) ⇒ (rmaximum(n;m;i.x[i]) ≤ r) supposing n ≤ m
Lemma: rmaximum-select
∀n,m:ℤ. ∀x:{n..m + 1-} ⟶ ℝ. ∀e:ℝ. ((r0 < e) ⇒ (∃i:{n..m + 1-}. ((rmaximum(n;m;i.x[i]) - e) < x[i]))) supposing n ≤ m
Lemma: rmaximum-constant
∀[n,m:ℤ]. ∀[x:{n..m + 1-} ⟶ ℝ]. ∀[r:ℝ]. rmaximum(n;m;i.x[i]) = r supposing ∀i:{n..m + 1-}. (x[i] = r) supposing n ≤ m
Definition: rminimum
rminimum(n;m;k.x[k]) == primrec(m - n;x[n];λi,s. rmin(s;x[n + i + 1]))
Lemma: rminimum_wf
∀[n,m:ℤ]. ∀[x:{n..m + 1-} ⟶ ℝ]. (rminimum(n;m;k.x[k]) ∈ ℝ) supposing n ≤ m
Lemma: rminimum_functionality
∀[n,m:ℤ].
∀[x,y:{n..m + 1-} ⟶ ℝ].
rminimum(n;m;k.x[k]) = rminimum(n;m;k.y[k]) supposing ∀k:ℤ. ((n ≤ k) ⇒ (k ≤ m) ⇒ (x[k] = y[k]))
supposing n ≤ m
Lemma: rminimum_functionality_wrt_rleq
∀[n,m:ℤ].
∀[x,y:{n..m + 1-} ⟶ ℝ].
rminimum(n;m;k.x[k]) ≤ rminimum(n;m;k.y[k]) supposing ∀k:ℤ. ((n ≤ k) ⇒ (k ≤ m) ⇒ (x[k] ≤ y[k]))
supposing n ≤ m
Lemma: rminimum-split
∀[n,m:ℤ]. ∀[x:{n..m + 1-} ⟶ ℝ]. ∀[k:ℤ].
(rminimum(n;m;i.x[i]) = rmin(rminimum(n;k;i.x[i]);rminimum(k + 1;m;i.x[i]))) supposing (k < m and (n ≤ k))
Lemma: rminimum-shift
∀[k,n,m:ℤ]. ∀[x:Top]. rminimum(n;m;i.x[i]) ~ rminimum(n - k;m - k;i.x[i + k]) supposing n ≤ m
Lemma: rminimum_lb
∀[k,n,m:ℤ]. (∀[x:{n..m + 1-} ⟶ ℝ]. (rminimum(n;m;i.x[i]) ≤ x[k])) supposing ((k ≤ m) and (n ≤ k))
Lemma: rminimum-glb
∀n,m:ℤ. ∀x:{n..m + 1-} ⟶ ℝ. ∀r:ℝ. (∀k:{n..m + 1-}. (r ≤ x[k])) ⇒ (r ≤ rminimum(n;m;i.x[i])) supposing n ≤ m
Lemma: rminimum-cases
∀n,m:ℤ.
∀x:{n..m + 1-} ⟶ ℝ. (¬¬(∃a:{n..m + 1-}. ((rminimum(n;m;i.x[i]) = x[a]) ∧ (∀j:{n..m + 1-}. (x[a] ≤ x[j])))))
supposing n ≤ m
Lemma: rminimum-select
∀n,m:ℤ. ∀x:{n..m + 1-} ⟶ ℝ. ∀e:ℝ. ((r0 < e) ⇒ (∃i:{n..m + 1-}. (x[i] < (rminimum(n;m;i.x[i]) + e)))) supposing n ≤ m
Lemma: rminimum-positive
∀n,m:ℤ. ∀x:{n..m + 1-} ⟶ ℝ. (r0 < rminimum(n;m;i.x[i]) ⇐⇒ ∀i:{n..m + 1-}. (r0 < x[i])) supposing n ≤ m
Lemma: rminimum-constant
∀[n,m:ℤ]. ∀[x:{n..m + 1-} ⟶ ℝ]. ∀[r:ℝ]. rminimum(n;m;i.x[i]) = r supposing ∀i:{n..m + 1-}. (x[i] = r) supposing n ≤ m
Lemma: Cauchy-Schwarz1
∀[n:ℕ]. ∀[x,y:ℕn + 1 ⟶ ℝ].
((Σ{x[i] * y[i] | 0≤i≤n} * Σ{x[i] * y[i] | 0≤i≤n}) ≤ (Σ{x[i] * x[i] | 0≤i≤n} * Σ{y[i] * y[i] | 0≤i≤n}))
Lemma: Cauchy-Schwarz1-strict
∀n:ℕ. ∀x,y:ℕn + 1 ⟶ ℝ.
((∃i,j:ℕn + 1. x[j] * y[i] ≠ x[i] * y[j])
⇒ ((Σ{x[i] * y[i] | 0≤i≤n} * Σ{x[i] * y[i] | 0≤i≤n}) < (Σ{x[i] * x[i] | 0≤i≤n} * Σ{y[i] * y[i] | 0≤i≤n})))
Lemma: Cauchy-Schwarz1-strict-iff
∀n:ℕ. ∀x,y:ℕn + 1 ⟶ ℝ.
(∃i,j:ℕn + 1. x[j] * y[i] ≠ x[i] * y[j]
⇐⇒ (Σ{x[i] * y[i] | 0≤i≤n} * Σ{x[i] * y[i] | 0≤i≤n}) < (Σ{x[i] * x[i] | 0≤i≤n} * Σ{y[i] * y[i] | 0≤i≤n}))
Lemma: Cauchy-Schwarz2
∀[n:ℕ]. ∀[x,y:ℕn ⟶ ℝ].
((Σ{x[i] * y[i] | 0≤i≤n - 1} * Σ{x[i] * y[i] | 0≤i≤n - 1}) ≤ (Σ{x[i] * x[i] | 0≤i≤n - 1}
* Σ{y[i] * y[i] | 0≤i≤n - 1}))
Lemma: Cauchy-Schwarz2-strict
∀n:ℕ. ∀x,y:ℕn ⟶ ℝ.
(∃i,j:ℕn. x[j] * y[i] ≠ x[i] * y[j]
⇐⇒ (Σ{x[i] * y[i] | 0≤i≤n - 1} * Σ{x[i] * y[i] | 0≤i≤n - 1}) < (Σ{x[i] * x[i] | 0≤i≤n - 1}
* Σ{y[i] * y[i] | 0≤i≤n - 1}))
Lemma: radd-limit
∀x,y:ℕ ⟶ ℝ. ∀a,b:ℝ. (lim n→∞.x[n] = a ⇒ lim n→∞.y[n] = b ⇒ lim n→∞.x[n] + y[n] = a + b)
Lemma: constant-limit
∀a,b:ℝ. (lim n→∞.a = b ⇐⇒ a = b)
Lemma: rminus-limit
∀x:ℕ ⟶ ℝ. ∀a:ℝ. (lim n→∞.x[n] = a ⇒ lim n→∞.-(x[n]) = -(a))
Lemma: rsub-limit
∀x,y:ℕ ⟶ ℝ. ∀a,b:ℝ. (lim n→∞.x[n] = a ⇒ lim n→∞.y[n] = b ⇒ lim n→∞.x[n] - y[n] = a - b)
Lemma: rmax-limit
∀x,y:ℕ ⟶ ℝ. ∀a,b:ℝ. (lim n→∞.x[n] = a ⇒ lim n→∞.y[n] = b ⇒ lim n→∞.rmax(x[n];y[n]) = rmax(a;b))
Lemma: const-rmul-limit-with-bound
∀x:ℕ ⟶ ℝ. ∀a,c:ℝ. ∀m:ℕ+. ((|c| ≤ r(m)) ⇒ lim n→∞.x[n] = a ⇒ lim n→∞.c * x[n] = c * a)
Lemma: rmul-limit
∀x,y:ℕ ⟶ ℝ. ∀a,b:ℝ. (lim n→∞.x[n] = a ⇒ lim n→∞.y[n] = b ⇒ lim n→∞.x[n] * y[n] = a * b)
Lemma: rleq-limit
∀[x,y:ℕ ⟶ ℝ]. ∀[a,b:ℝ]. (a ≤ b) supposing ((∀n:ℕ. (x[n] ≤ y[n])) and lim n→∞.y[n] = b and lim n→∞.x[n] = a)
Lemma: constant-rleq-limit
∀[x:ℝ]. ∀[y:ℕ ⟶ ℝ]. ∀[a:ℝ]. (x ≤ a) supposing ((∀n:ℕ. (x ≤ y[n])) and lim n→∞.y[n] = a)
Lemma: rleq-limit-constant
∀[x:ℝ]. ∀[y:ℕ ⟶ ℝ]. ∀[a:ℝ]. (a ≤ x) supposing ((∀n:ℕ. (y[n] ≤ x)) and lim n→∞.y[n] = a)
Lemma: rless-iff-rleq
∀x,y:ℝ. (x < y ⇐⇒ ∃m:ℕ+. (x ≤ (y - (r1/r(m)))))
Lemma: rless-implies-rleq
∀x,y:ℝ. ((x < y) ⇒ (∃m:ℕ+. (x ≤ (y - (r1/r(m))))))
Lemma: req-iff-rabs-rleq
∀x,y:ℝ. (x = y ⇐⇒ ∀m:ℕ+. (|x - y| ≤ (r1/r(m))))
Lemma: req-iff-rabs-rleq-bound
∀x,y:ℝ. (x = y ⇐⇒ ∃B:ℕ+. ∀m:ℕ+. (|x - y| ≤ (r(B)/r(m))))
Lemma: near-real-implies-real
∀[x:ℕ+ ⟶ ℤ]. ∀[y:ℝ]. x ∈ {x:ℝ| x = y} supposing ∀n:ℕ+. (|(x within 1/n) - y| ≤ (r1/r(n)))
Lemma: rational-approx-implies-req
∀[k:ℕ+]. ∀[x:ℝ]. ∀[a:ℕ+ ⟶ ℤ].
((∀n:ℕ+. (|x - (r(a n)/r(2 * n))| ≤ (r(k)/r(n)))) ⇒ {k-regular-seq(a) ∧ (accelerate(k;a) = x)})
Lemma: accelerate-rational-approx
∀[k:ℕ+]. ∀[x:ℝ]. ∀[a:ℕ+ ⟶ ℤ]. ((∀n:ℕ+. (|x - (r(a n)/r(2 * n))| ≤ (r(k)/r(n)))) ⇒ (accelerate(k;a) ∈ {y:ℝ| y = x} ))
Lemma: req-iff-rational-approx
∀[x:ℝ]. ∀[a:ℕ+ ⟶ ℤ]. (regular-seq(a) ∧ (a = x) ⇐⇒ ∀n:ℕ+. (|x - (r(a n)/r(2 * n))| ≤ (r1/r(n))))
Lemma: rabs-diff-rmul
∀[a,b,c,d,x,y:ℝ]. ((|a - b| ≤ x) ⇒ (|c - d| ≤ y) ⇒ (|(a * c) - b * d| ≤ ((|a| * y) + (|d| * x))))
Lemma: rabs-diff-rdiv
∀[a,b,c,d,x,y:ℝ].
(c ≠ r0 ⇒ d ≠ r0 ⇒ (|a - b| ≤ x) ⇒ (|(r1/c) - (r1/d)| ≤ y) ⇒ (|(a/c) - (b/d)| ≤ ((|a| * y) + (|(r1/d)| * x))))
Definition: ireal-approx
j-approx(x;M;z) == |x - (r(z)/r(2 * M))| ≤ (r(j)/r(M))
Lemma: ireal-approx_wf
∀[j:ℕ]. ∀[x:ℝ]. ∀[M:ℕ+]. ∀[z:ℤ]. (j-approx(x;M;z) ∈ ℙ)
Lemma: ireal-approx_functionality
∀[j:ℕ]. ∀[M:ℕ+]. ∀[z:ℤ]. ∀[x,y:ℝ]. j-approx(x;M;z) ⇐⇒ j-approx(y;M;z) supposing x = y
Lemma: ireal-approx-1
∀[x:ℝ]. ∀[M:ℕ+]. 1-approx(x;M;x M)
Lemma: ireal-approx-radd
∀[x,y:ℝ]. ∀[j,i:ℕ]. ∀[M:ℕ+]. ∀[a,b:ℤ]. (j-approx(x;M;a) ⇒ i-approx(y;M;b) ⇒ j + i-approx(x + y;M;a + b))
Lemma: ireal-approx-radd-int
∀[x,y:ℝ]. ∀[j:ℕ]. ∀[M:ℕ+]. ∀[a,n:ℤ]. (j-approx(x;M;a) ⇒ j-approx(x + r(n);M;a + (2 * n * M)))
Lemma: ireal-approx-rmul
∀[x,y:ℝ]. ∀[j,M:ℕ+]. ∀[a,b:ℤ].
∀k:ℕ+
((|x| ≤ (r1/r(4)))
⇒ ((2 * |b|) ≤ (k * M))
⇒ j-approx(x;k * M;a)
⇒ j-approx(y;M;b)
⇒ j-approx(x * y;M;(a * b) ÷ 2 * k * M))
Lemma: ireal-approx-rmul2
∀[x,y:ℝ]. ∀[j,M:ℕ+]. ∀[a,b:ℤ].
∀k:ℕ+. ((|x| ≤ (r1/r(2))) ⇒ (|b| ≤ k) ⇒ j-approx(x;k;a) ⇒ j-approx(y;2 * M;b) ⇒ j-approx(x * y;M;(a * b) ÷ 4 * k))
Definition: poly-approx-aux
poly-approx-aux(a;x;xM;M;n;k)
==r if (k =z 0)
then a n M
else eval u = poly-approx-aux(a;x;xM;M;n + 1;k - 1) in
eval b = ((2 * |u|) ÷ M) + 1 in
eval z = if (b =z 1) then xM else x (b * M) fi in
eval v = (u * z) ÷ 2 * b * M in
eval t = a n M in
v + t
fi
Lemma: poly-approx-aux_wf
∀[k:ℕ]. ∀[a:ℕ ⟶ ℝ]. ∀[x:ℝ]. ∀[xM:ℤ]. ∀[M:ℕ+]. ∀[n:ℕ]. (poly-approx-aux(a;x;xM;M;n;k) ∈ ℤ)
Lemma: poly-approx-aux-property
∀[k:ℕ]. ∀[a:ℕ ⟶ ℝ]. ∀[x:ℝ]. ∀[xM:ℤ]. ∀[M:ℕ+]. ∀[n:ℕ].
((|x| ≤ (r1/r(4))) ⇒ 1-approx(x;M;xM) ⇒ k + 1-approx(Σ{(a (n + i)) * x^i | 0≤i≤k};M;poly-approx-aux(a;x;xM;M;n;k)))
Definition: poly-approx
poly-approx(a;x;k;N) ==
eval B = 2 * (k + 1) in
eval M = B * N in
eval xM = x M in
eval z = poly-approx-aux(a;x;xM;M;0;k) in
z ÷ B
Lemma: poly-approx_wf
∀[k:ℕ]. ∀[a:ℕ ⟶ ℝ]. ∀[x:ℝ]. ∀[N:ℕ+]. (poly-approx(a;x;k;N) ∈ ℤ)
Lemma: poly-approx-property
∀[k:ℕ]. ∀[a:ℕ ⟶ ℝ]. ∀[x:ℝ]. ∀[N:ℕ+]. ((|x| ≤ (r1/r(4))) ⇒ 1-approx(Σ{(a i) * x^i | 0≤i≤k};N;poly-approx(a;x;k;N)))
Definition: rless-witness
rless-witness(x;y;p) == fst((TERMOF{rless-implies-rleq:o, 1:l} x y p))
Lemma: rless-witness_wf
∀[x,y:ℝ]. ∀[p:x < y]. (rless-witness(x;y;p) ∈ ℕ+)
Lemma: rless-witness-property
∀[x,y:ℝ]. ∀[p:x < y]. ((x ≤ (y - (r1/r(rless-witness(x;y;p))))) ∧ ((x + (r1/r(rless-witness(x;y;p)))) ≤ y))
Lemma: rinv-limit
∀x:ℕ ⟶ ℝ. ∀a:ℝ. (lim n→∞.x[n] = a ⇒ (∀n:ℕ. x[n] ≠ r0) ⇒ a ≠ r0 ⇒ lim n→∞.(r1/x[n]) = (r1/a))
Definition: regularize
regularize(k;f) ==
λn.if regular-upto(k;n;f)
then f n
else eval m = mu(λn.(¬bregular-upto(k;n;f))) - 1 in
eval j = seq-min-upper(k;m;f) in
k * ((n * ((f j) + (2 * k))) ÷ j * k)
fi
Lemma: regularize_wf
∀[k:ℕ+]. ∀[f:ℕ+ ⟶ ℤ]. (regularize(k;f) ∈ ℕ+ ⟶ ℤ)
Lemma: regular-upto-regularize
∀f:ℕ+ ⟶ ℤ. ∀k,m:ℕ. ((↑regular-upto(k;m + 1;f)) ⇒ {∀n:ℕ. ((n ≤ m) ⇒ (regularize(k;f) n ~ f n))})
Lemma: regularize-k-regular
∀k:ℕ+. ∀f:ℕ+ ⟶ ℤ. k-regular-seq(regularize(k;f))
Lemma: regularize-regular
∀k:ℕ+. ∀x:{f:ℕ+ ⟶ ℤ| k-regular-seq(f)} . (regularize(k;x) = x ∈ (ℕ+ ⟶ ℤ))
Lemma: regularize-real
∀k:ℕ+. ∀x:ℝ. (regularize(k;x) = x ∈ ℝ)
Lemma: weak-continuity-principle-real
∀x:ℝ. ∀F:ℝ ⟶ 𝔹. ∀G:n:ℕ+ ⟶ {y:ℝ| x = y ∈ (ℕ+n ⟶ ℤ)} . (∃n:ℕ+ [F x = F (G n)])
Lemma: weak-continuity-principle-real-double
∀x:ℝ. ∀F,H:ℝ ⟶ 𝔹. ∀G:n:ℕ+ ⟶ {y:ℝ| x = y ∈ (ℕ+n ⟶ ℤ)} . (∃n:ℕ+ [(F x = F (G n) ∧ H x = H (G n))])
Lemma: weak-continuity-principle-real-nat
∀x:ℝ. ∀F:ℝ ⟶ ℕ. ∀G:n:ℕ+ ⟶ {y:ℝ| x = y ∈ (ℕ+n ⟶ ℤ)} . (∃n:ℕ+ [((F x) = (F (G n)) ∈ ℕ)])
Lemma: pseudo-positive-is-positive
∀x:ℝ. r0 < x supposing pseudo-positive(x)
Lemma: real-weak-Markov
∀x,y:ℝ. x ≠ y supposing ∀z:ℝ. ((¬(z = x)) ∨ (¬(z = y)))
Lemma: rneq-by-function
∀x,y,a,b:ℝ. ∀f:ℝ ⟶ ℝ. (a ≠ b ⇒ (f[x] = a) ⇒ (f[y] = b) ⇒ (∀x,y:ℝ. ((x = y) ⇒ (f[x] = f[y]))) ⇒ x ≠ y)
Lemma: test-rat-term-poly
∀[x,y:ℝ]. (r1/r1 + (y/x * x * x)) = (x * x * x/y + (x * x * x)) supposing (y < -(x * x * x)) ∧ x ≠ r0
Lemma: real-weak-Markov-ext
∀x,y:ℝ. x ≠ y supposing ∀z:ℝ. ((¬(z = x)) ∨ (¬(z = y)))
Lemma: pseudo-positive-is-positive-proof2
∀x:ℝ. r0 < x supposing pseudo-positive(x)
Lemma: rneq-function
∀f:ℝ ⟶ ℝ. ((∀x,y:ℝ. ((x = y) ⇒ (f[x] = f[y]))) ⇒ (∀x,y:ℝ. (f[x] ≠ f[y] ⇒ x ≠ y)))
Lemma: cantor-lemma
∀x,y,e,z:ℝ.
(∃x',y':ℝ. ((x ≤ x') ∧ (x' < y') ∧ (y' ≤ y) ∧ ((z < x') ∨ (y' < z)) ∧ ((y' - x') < e))) supposing
((x < y) and
(r0 < e))
Lemma: cantor-lemma2
∀z:ℕ ⟶ ℝ
∃f:ℕ ⟶ {p:ℝ × ℝ| (fst(p)) < (snd(p))} ⟶ {p:ℝ × ℝ| (fst(p)) < (snd(p))}
∀n:ℕ. ∀p:{p:ℝ × ℝ| (fst(p)) < (snd(p))} .
let x,y = p
in let x',y' = f n p
in (x ≤ x') ∧ (x' < y') ∧ (y' ≤ y) ∧ (((z n) < x') ∨ (y' < (z n))) ∧ ((y' - x') < (r1/r(n + 1)))
Lemma: reals-uncountable
∀z:ℕ ⟶ ℝ. ∀x,y:ℝ. ((x < y) ⇒ (∃u:ℝ. ((x ≤ u) ∧ (u ≤ y) ∧ (∀n:ℕ. u ≠ z n))))
Lemma: reals-uncountable-simple
∀f:ℕ ⟶ ℝ. (¬Surj(ℕ;ℝ;f))
Definition: series-sum
Note that in Σn.x[n] = a, the index n is in ℕ (i.e, 0,1,2,...) rather
than ℕ+ (1,2,3,...) as in Bishop & Bridges.⋅
Σn.x[n] = a == lim n→∞.Σ{x[i] | 0≤i≤n} = a
Lemma: series-sum_wf
∀[x:ℕ ⟶ ℝ]. ∀[a:ℝ]. (Σn.x[n] = a ∈ ℙ)
Lemma: series-sum_functionality
∀x,y:ℕ ⟶ ℝ. ∀a,b:ℝ. ({Σn.x[n] = a ⇒ Σn.y[n] = b}) supposing ((a = b) and (∀n:ℕ. (x[n] = y[n])))
Lemma: series-sum-unique
∀[x:ℕ ⟶ ℝ]. ∀[a,b:ℝ]. (a = b) supposing (Σn.x[n] = b and Σn.x[n] = a)
Lemma: series-sum-0
Σi.r0 = r0
Lemma: series-sum-constant
∀x:ℝ. Σi.if (i =z 0) then x else r0 fi = x
Lemma: series-sum-linear1
∀x,y:ℕ ⟶ ℝ. ∀a,b:ℝ. (Σn.x[n] = a ⇒ Σn.y[n] = b ⇒ Σn.x[n] + y[n] = a + b)
Lemma: series-sum-linear2
∀x:ℕ ⟶ ℝ. ∀a,c:ℝ. (Σn.x[n] = a ⇒ Σn.c * x[n] = c * a)
Lemma: series-sum-linear3
∀x:ℕ ⟶ ℝ. ∀a,c:ℝ. (Σn.x[n] = a ⇒ Σn.x[n] * c = a * c)
Lemma: triangular-reciprocal-series-sum
Σn.(r1/r(t(n + 1))) = r(2)
Definition: series-converges
Σn.x[n]↓ == ∃a:ℝ. Σn.x[n] = a
Lemma: series-converges_wf
∀[x:ℕ ⟶ ℝ]. (Σn.x[n]↓ ∈ ℙ)
Lemma: series-converges-tail
∀x:ℕ ⟶ ℝ. (Σn.x[n]↓ ⇒ (∀y:ℕ ⟶ ℝ. ((∃N:ℕ. ∀n:{N...}. (y[n] = x[n])) ⇒ Σn.y[n]↓)))
Lemma: series-converges-tail2
∀N:ℕ. ∀x:ℕ ⟶ ℝ. (Σn.x[n + N]↓ ⇒ Σn.x[n]↓)
Lemma: series-converges-tail2-ext
∀N:ℕ. ∀x:ℕ ⟶ ℝ. (Σn.x[n + N]↓ ⇒ Σn.x[n]↓)
Lemma: series-converges-rmul
∀c:ℝ. ∀x:ℕ ⟶ ℝ. (Σn.x[n]↓ ⇒ Σn.x[n] * c↓)
Lemma: series-converges-rmul-ext
∀c:ℝ. ∀x:ℕ ⟶ ℝ. (Σn.x[n]↓ ⇒ Σn.x[n] * c↓)
Lemma: series-converges-limit-zero
∀x:ℕ ⟶ ℝ. (Σn.x[n]↓ ⇒ lim n→∞.x[n] = r0)
Definition: converges-absolutely
converges-absolutely(n.x[n]) == Σn.|x[n]|↓
Lemma: converges-absolutely_wf
∀[x:ℕ ⟶ ℝ]. (converges-absolutely(n.x[n]) ∈ ℙ)
Lemma: comparison-test
∀y:ℕ ⟶ ℝ. (Σn.y[n]↓ ⇒ (∀x:ℕ ⟶ ℝ. Σn.x[n]↓ supposing ∀n:ℕ. (|x[n]| ≤ y[n])))
Lemma: comparison-test-ext
∀y:ℕ ⟶ ℝ. (Σn.y[n]↓ ⇒ (∀x:ℕ ⟶ ℝ. Σn.x[n]↓ supposing ∀n:ℕ. (|x[n]| ≤ y[n])))
Lemma: converges-absolutely-converges
∀x:ℕ ⟶ ℝ. (converges-absolutely(n.x[n]) ⇒ Σn.x[n]↓)
Definition: series-diverges
Σn.x[n]↑ == n.Σ{x[i] | 0≤i≤n}↑
Lemma: series-diverges_wf
∀[x:ℕ ⟶ ℝ]. (Σn.x[n]↑ ∈ ℙ)
Lemma: series-diverges_functionality
∀[x,y:ℕ ⟶ ℝ]. {Σn.x[n]↑ ⇒ Σn.y[n]↑} supposing ∀n:ℕ. (x[n] = y[n])
Lemma: series-diverges-rmul
∀[x:ℕ ⟶ ℝ]. (Σn.x[n]↑ ⇒ (∀c:ℝ. (c ≠ r0 ⇒ Σn.c * x[n]↑)))
Lemma: series-diverges-trivially
∀z:ℝ. ((r0 < z) ⇒ (∀x:ℕ ⟶ ℝ. ((∀k:ℕ. ∃n:ℕ. ((k ≤ n) ∧ (z ≤ |x[n]|))) ⇒ Σn.x[n]↑)))
Lemma: series-diverges-tail
∀x:ℕ ⟶ ℝ. ∀N:ℕ. (Σn.x[N + n]↑ ⇒ Σn.x[n]↑)
Lemma: series-diverges-tail-iff
∀x:ℕ ⟶ ℝ. ∀N:ℕ. (Σn.x[N + n]↑ ⇐⇒ Σn.x[n]↑)
Lemma: harmonic-series-diverges
Σn.(r1/r(n + 1))↑
Lemma: harmonic-series-diverges-to-infinity
∀n:ℕ. ((r1 + (r(n)/r(2))) ≤ Σ{(r1/r(i)) | 1≤i≤2^n})
Lemma: comparison-test-for-divergence
∀x,y:ℕ ⟶ ℝ. (Σn.y[n]↑ ⇒ (∃N:ℕ. ∀n:{N...}. ((r0 ≤ y[n]) ∧ (y[n] ≤ x[n]))) ⇒ Σn.x[n]↑)
Lemma: partial-geometric-series
∀n:ℕ. ∀c:ℝ. (c ≠ r1 ⇒ (Σ{c^i | 0≤i≤n} = (r1 - c^n + 1/r1 - c)))
Lemma: geometric-series-converges
∀c:{c:ℝ| (r0 ≤ c) ∧ (c < r1)} . Σn.c^n = (r1/r1 - c)
Lemma: geometric-series-one-half
We have Σn.(r1/r(2)^n) = r(2) because n = 0 ∈ ℤ is included in the series.⋅
Σn.(r1/r(2)^n) = r(2)
Lemma: geometric-series-converges-ext
∀c:{c:ℝ| (r0 ≤ c) ∧ (c < r1)} . Σn.c^n = (r1/r1 - c)
Lemma: ratio-test
∀x:ℕ ⟶ ℝ. ∀N:ℕ.
((∀c:{c:ℝ| (r0 ≤ c) ∧ (c < r1)} . ((∀n:{N...}. (|x[n + 1]| ≤ (c * |x[n]|))) ⇒ Σn.x[n]↓))
∧ (∀c:{c:ℝ| r1 < c} . ((∀n:{N...}. ((c * |x[n]|) < |x[n + 1]|)) ⇒ Σn.x[n]↑)))
Lemma: ratio-test-ext
∀x:ℕ ⟶ ℝ. ∀N:ℕ.
((∀c:{c:ℝ| (r0 ≤ c) ∧ (c < r1)} . ((∀n:{N...}. (|x[n + 1]| ≤ (c * |x[n]|))) ⇒ Σn.x[n]↓))
∧ (∀c:{c:ℝ| r1 < c} . ((∀n:{N...}. ((c * |x[n]|) < |x[n + 1]|)) ⇒ Σn.x[n]↑)))
Lemma: ratio-test-corollary
∀x:ℕ ⟶ ℝ
((∀n:ℕ. x[n] ≠ r0) ⇒ (∀L:ℝ. (lim n→∞.|(x[n + 1]/x[n])| = L ⇒ (((L < r1) ⇒ Σn.x[n]↓) ∧ ((r1 < L) ⇒ Σn.x[n]↑)))))
Lemma: Kummer-criterion
∀a,x:ℕ ⟶ ℝ.
((lim n→∞.a[n] * x[n] = r0
⇒ (∃c:{c:ℝ| r0 < c}
∃N:ℕ
((∀n:{N...}. ((r0 < a[n]) ∧ (r0 < x[n])))
∧ (∀n:{N...}. ((r0 < a[n]) ∧ (c ≤ ((a[n] * x[n]/x[n + 1]) - a[n + 1]))))))
⇒ Σn.x[n]↓)
∧ ((∃N:ℕ
((∀n:{N...}. ((r0 < a[n]) ∧ (r0 < x[n])))
∧ (∀n:{N...}. (((a[n] * x[n]/x[n + 1]) - a[n + 1]) ≤ r0))
∧ Σn.(r1/a[N + n])↑))
⇒ Σn.x[n]↑))
Lemma: r-archimedean-implies
∀x:ℝ. ∀m:ℕ+. ∃M:ℕ+. (((r1/r(M)) * x) ≤ (r1/r(m)))
Lemma: r-archimedean-implies2
∀x:ℝ. ∀d:{d:ℝ| r0 < d} . ∃M:ℕ+. ((x/r(M)) ≤ d)
Lemma: r-archimedean-rabs
∀x:ℝ. ∃n:ℕ. (|x| ≤ r(n))
Lemma: r-archimedean-rabs-ext
∀x:ℝ. ∃n:ℕ. (|x| ≤ r(n))
Lemma: r-archimedean2
∀x:ℝ. ∃N:ℕ. ∀n:{N...}. (|(x/r(n + 1))| ≤ (r1/r(2)))
Lemma: Raabe-lemma
∀y:ℕ ⟶ ℝ. ∀c:ℝ.
((r0 < c)
⇒ (∀N:ℕ+. ((∀n:{N...}. (r0 < y[n])) ⇒ (∀n:{N...}. (c ≤ (r(n) * ((y[n]/y[n + 1]) - r1)))) ⇒ lim n→∞.y[n] = r0)))
Lemma: Raabe-test
∀x:ℕ ⟶ ℝ. ∀L:ℝ.
((∀n:ℕ. (r0 < x[n]))
⇒ lim n→∞.r(n) * ((x[n]/x[n + 1]) - r1) = L
⇒ (((r1 < L) ⇒ Σn.x[n]↓) ∧ ((L < r1) ⇒ Σn.x[n]↑)))
Lemma: alternating-series-tail-bound
∀x:ℕ ⟶ ℝ. ∀M:ℕ.
((∀n:ℕ. ((M ≤ n) ⇒ ((r0 ≤ x[n]) ∧ (x[n + 1] ≤ x[n]))))
⇒ lim n→∞.x[n] = r0
⇒ (∀a:{M...}. ∀b:ℕ. (|Σ{-1^i * x[i] | a≤i≤b}| ≤ x[a])))
Lemma: alternating-series-converges
∀x:ℕ ⟶ ℝ. ((∃M:ℕ. ∀n:ℕ. (M < n ⇒ ((r0 ≤ x[n]) ∧ (x[n + 1] ≤ x[n])))) ⇒ lim n→∞.x[n] = r0 ⇒ Σn.-1^n * x[n]↓)
Lemma: alternating-series-converges-ext
∀x:ℕ ⟶ ℝ. ((∃M:ℕ. ∀n:ℕ. (M < n ⇒ ((r0 ≤ x[n]) ∧ (x[n + 1] ≤ x[n])))) ⇒ lim n→∞.x[n] = r0 ⇒ Σn.-1^n * x[n]↓)
Lemma: rdiv-factorial-lemma1
∀n:ℕ. ((r((2 * (n + 1))!)/r((2 * n)!)) = r(((n * 2) + 2) * ((n * 2) + 1)))
Lemma: rdiv-factorial-lemma2
∀x:ℝ. ∀b:ℕ. (((x * x) ≤ r(b * b)) ⇒ (∀n:ℕ. (((b ÷ 2) ≤ n) ⇒ ((x * x) ≤ (r((2 * (n + 1))!)/r((2 * n)!))))))
Lemma: rdiv-factorial-lemma3
∀x:ℝ. ∀b:ℕ.
(((x * x) ≤ r(b * b)) ⇒ (∀n:ℕ. (((b ÷ 2) ≤ n) ⇒ ((x * x) ≤ (r(((2 * (n + 1)) + 1)!)/r(((2 * n) + 1)!))))))
Lemma: rdiv-factorial-limit-zero-from-bound
∀x:ℝ. ∀n:ℕ. ((|x| ≤ r(n)) ⇒ lim n→∞.(|x|^n/r((n)!)) = r0)
Lemma: rdiv-factorial-limit-zero-from-bound2
∀x:ℝ. ∀n:ℕ. ((|x| ≤ r(n)) ⇒ lim n→∞.(|x|^n/r((n + 1)!)) = r0)
Lemma: rdiv-factorial-limit-zero
∀x:ℝ. lim n→∞.(|x|^n/r((n)!)) = r0
Lemma: sine-exists
∀x:ℝ. ∃a:ℝ. Σi.-1^i * (x^(2 * i) + 1)/((2 * i) + 1)! = a
Lemma: sine-exists-ext
∀x:ℝ. ∃a:ℝ. Σi.-1^i * (x^(2 * i) + 1)/((2 * i) + 1)! = a
Definition: sine
sine(x) == fst((TERMOF{sine-exists-ext:o, 1:l} x))
Lemma: sine_wf
∀[x:ℝ]. (sine(x) ∈ ℝ)
Lemma: sine-is-limit
∀x:ℝ. Σi.-1^i * (x^(2 * i) + 1)/((2 * i) + 1)! = sine(x)
Lemma: sine_functionality
∀[x,y:ℝ]. sine(x) = sine(y) supposing x = y
Lemma: sine0
sine(r0) = r0
Lemma: sine-rminus
∀x:ℝ. (sine(-(x)) = -(sine(x)))
Definition: sine-approx
sine-approx(x;k;N) ==
eval u = poly-approx(λi.(r(if (i rem 2 =z 0) then 1 else -1 fi ))/((2 * i) + 1)!;x^2;k;2 * N) in
eval b = |u| + 1 in
eval z = x b in
(u * z) ÷ 4 * b
Lemma: sine-approx_wf
∀[x:ℝ]. ∀[k:ℕ]. ∀[N:ℕ+]. (sine-approx(x;k;N) ∈ ℤ)
Lemma: sine-approx-property
∀[x:ℝ]. ∀[k:ℕ]. ∀[N:ℕ+].
((|x| ≤ (r1/r(2))) ⇒ 1-approx(Σ{-1^i * (x^(2 * i) + 1)/((2 * i) + 1)! | 0≤i≤k};N;sine-approx(x;k;N)))
Lemma: sine-poly-approx-1
∀[x:{x:ℝ| (r0 ≤ x) ∧ (x ≤ r1)} ]. ∀[k:ℕ].
(|sine(x) - Σ{-1^i * (x^(2 * i) + 1)/((2 * i) + 1)! | 0≤i≤k}| ≤ (x^(2 * k) + 3/r(((2 * k) + 3)!)))
Lemma: sine-poly-approx
∀[x:{x:ℝ| |x| ≤ (r1/r(2))} ]. ∀[k:ℕ]. ∀[N:ℕ+].
(|sine(x) - (r(sine-approx(x;k;N))/r(2 * N))| ≤ ((|x|^(2 * k) + 3/r(((2 * k) + 3)!)) + (r1/r(N))))
Lemma: sine-approx-lemma
∀a:{2...}. ∀N:ℕ+. (∃k:ℕ [(N ≤ (a^((2 * k) + 3) * ((2 * k) + 3)!))])
Lemma: sine-approx-lemma-bad
∀a:{2...}. ∀N:ℕ. (∃k:ℕ [(N ≤ (a^((2 * k) + 3) * ((2 * k) + 3)!))])
Lemma: sine-approx-lemma-bad-ext
∀a:{2...}. ∀N:ℕ. (∃k:ℕ [(N ≤ (a^((2 * k) + 3) * ((2 * k) + 3)!))])
Lemma: cosine-approx-lemma
∀a:{2...}. ∀N:ℕ+. (∃k:ℕ [(N ≤ (a^((2 * k) + 2) * ((2 * k) + 2)!))])
Comment: extract-sine-lemma-non-uniform
Using CompNatInd' we got this extract for sine-approx-lemma :
⌜λa,N. eval aa = a * a in
eval b = 6 * a^3 in
if (b) < (N)
then letrec rec(d)=λk,b. eval k' = k + 1 in
eval b' = b * aa * ((2 * k) + 4) * ((2 * k) + 5) in
if (b') < (N) then rec (d - 1) k' b' else k' in
rec
N
0
b
else 0⌝⋅
Lemma: sine-approx-lemma-ext
∀a:{2...}. ∀N:ℕ+. (∃k:ℕ [(N ≤ (a^((2 * k) + 3) * ((2 * k) + 3)!))])
Lemma: cosine-approx-lemma-ext
∀a:{2...}. ∀N:ℕ+. (∃k:ℕ [(N ≤ (a^((2 * k) + 2) * ((2 * k) + 2)!))])
Lemma: sine-approx-for-small
∀a:{2...}. ∀N:ℕ+. ∀x:{x:ℝ| |x| ≤ (r1/r(a))} . (∃z:ℤ [(|sine(x) - (r(z)/r(2 * N))| ≤ (r(2)/r(N)))])
Lemma: sine-approx-for-small-ext
∀a:{2...}. ∀N:ℕ+. ∀x:{x:ℝ| |x| ≤ (r1/r(a))} . (∃z:ℤ [(|sine(x) - (r(z)/r(2 * N))| ≤ (r(2)/r(N)))])
Definition: sine-small
sine-small(x) == accelerate(2;λN.sine-approx(x;genfact-inv(N;48;k.4 * ((2 * k) + 3) * ((2 * k) + 2));N))
Lemma: sine-small_wf
∀[x:{x:ℝ| |x| ≤ (r1/r(2))} ]. (sine-small(x) ∈ {y:ℝ| y = sine(x)} )
Lemma: cosine-exists
∀x:ℝ. ∃a:ℝ. Σi.-1^i * (x^2 * i)/(2 * i)! = a
Lemma: cosine-exists-ext
∀x:ℝ. ∃a:ℝ. Σi.-1^i * (x^2 * i)/(2 * i)! = a
Definition: cosine
cosine(x) == fst((TERMOF{cosine-exists-ext:o, 1:l} x))
Lemma: cosine_wf
∀[x:ℝ]. (cosine(x) ∈ ℝ)
Lemma: cosine-is-limit
∀x:ℝ. Σi.-1^i * (x^2 * i)/(2 * i)! = cosine(x)
Lemma: cosine_functionality
∀[x,y:ℝ]. cosine(x) = cosine(y) supposing x = y
Lemma: cosine0
cosine(r0) = r1
Lemma: cosine-rminus
∀x:ℝ. (cosine(-(x)) = cosine(x))
Definition: cosine-approx
cosine-approx(x;k;N) == poly-approx(λi.(r(if (i rem 2 =z 0) then 1 else -1 fi ))/(2 * i)!;x^2;k;N)
Lemma: cosine-approx_wf
∀[x:ℝ]. ∀[k:ℕ]. ∀[N:ℕ+]. (cosine-approx(x;k;N) ∈ ℤ)
Lemma: cosine-approx-property
∀[x:ℝ]. ∀[k:ℕ]. ∀[N:ℕ+]. ((|x| ≤ (r1/r(2))) ⇒ 1-approx(Σ{-1^i * (x^2 * i)/(2 * i)! | 0≤i≤k};N;cosine-approx(x;k;N)))
Lemma: cosine-poly-approx-1
∀[x:{x:ℝ| (r0 ≤ x) ∧ (x ≤ r1)} ]. ∀[k:ℕ].
(|cosine(x) - Σ{-1^i * (x^2 * i)/(2 * i)! | 0≤i≤k}| ≤ (x^(2 * k) + 2/r(((2 * k) + 2)!)))
Lemma: cosine-poly-approx
∀[x:{x:ℝ| |x| ≤ (r1/r(2))} ]. ∀[k:ℕ]. ∀[N:ℕ+].
(|cosine(x) - (r(cosine-approx(x;k;N))/r(2 * N))| ≤ ((|x|^(2 * k) + 2/r(((2 * k) + 2)!)) + (r1/r(N))))
Lemma: cosine-approx-for-small
∀a:{2...}. ∀N:ℕ+. ∀x:{x:ℝ| |x| ≤ (r1/r(a))} . (∃z:ℤ [(|cosine(x) - (r(z)/r(2 * N))| ≤ (r(2)/r(N)))])
Lemma: cosine-approx-for-small-ext
∀a:{2...}. ∀N:ℕ+. ∀x:{x:ℝ| |x| ≤ (r1/r(a))} . (∃z:ℤ [(|cosine(x) - (r(z)/r(2 * N))| ≤ (r(2)/r(N)))])
Definition: cosine-small
cosine-small(x) == accelerate(2;λN.cosine-approx(x;genfact-inv(N;8;k.4 * ((2 * k) + 2) * ((2 * k) + 1));N))
Lemma: cosine-small_wf
∀[x:{x:ℝ| |x| ≤ (r1/r(2))} ]. (cosine-small(x) ∈ {y:ℝ| y = cosine(x)} )
Lemma: exp-series-converges
∀x:ℝ. Σn.(x^n)/(n)!↓
Lemma: exp-exists
∀x:ℝ. ∃a:ℝ. Σn.(x^n)/(n)! = a
Lemma: exp-exists-ext
∀x:ℝ. ∃a:ℝ. Σn.(x^n)/(n)! = a
Definition: rexp
e^x == fst((TERMOF{exp-exists-ext:o, 1:l} x))
Lemma: rexp_wf
∀[x:ℝ]. (e^x ∈ ℝ)
Lemma: rexp-is-limit
∀x:ℝ. Σn.(x^n)/(n)! = e^x
Lemma: rexp_functionality
∀[x,y:ℝ]. e^x = e^y supposing x = y
Lemma: rexp0
e^r0 = r1
Lemma: rexp-of-nonneg-stronger
∀x:ℝ. ((r0 ≤ x) ⇒ ((r1 + x) ≤ e^x))
Lemma: rexp-of-nonneg
∀x:ℝ. ((r0 ≤ x) ⇒ (r1 ≤ e^x))
Lemma: rexp-of-positive
∀x:ℝ. ((r0 < x) ⇒ (r1 < e^x))
Definition: rexp-approx
rexp-approx(x;k;N) == poly-approx(λi.(r1)/(i)!;x;k;N)
Lemma: rexp-approx_wf
∀[x:ℝ]. ∀[k:ℕ]. ∀[N:ℕ+]. (rexp-approx(x;k;N) ∈ ℤ)
Lemma: rexp-approx-property
∀[x:ℝ]. ∀[k:ℕ]. ∀[N:ℕ+]. ((|x| ≤ (r1/r(4))) ⇒ 1-approx(Σ{(x^i)/(i)! | 0≤i≤k};N;rexp-approx(x;k;N)))
Definition: rset
Set(ℝ) == {A:ℝ ⟶ ℙ| ∀x,y:ℝ. ((x = y) ⇒ (A x) ⇒ (A y))}
Lemma: rset_wf
Set(ℝ) ∈ 𝕌'
Definition: rset-member
x ∈ A == A x
Lemma: rset-member_wf
∀[A:Set(ℝ)]. ∀[x:ℝ]. (x ∈ A ∈ ℙ)
Definition: rseteq
rseteq(A;B) == ∀x:ℝ. (x ∈ A ⇐⇒ x ∈ B)
Lemma: rseteq_wf
∀[A,B:Set(ℝ)]. (rseteq(A;B) ∈ ℙ)
Lemma: rseteq_weakening
∀[A,B:Set(ℝ)]. rseteq(A;B) supposing A = B ∈ Set(ℝ)
Lemma: rset-member_functionality
∀A,B:Set(ℝ). ∀x,y:ℝ.
((∀y:ℝ. SqStable(y ∈ A)) ⇒ (∀y:ℝ. SqStable(y ∈ B)) ⇒ rseteq(A;B) ⇒ {(x ∈ A) ⇒ (y ∈ B)} supposing x = y)
Definition: mk-rset
{x:ℝ | P[x]} == λx.P[x]
Lemma: mk-rset_wf
∀[P:ℝ ⟶ ℙ]. {x:ℝ | P[x]} ∈ Set(ℝ) supposing ∀x,y:ℝ. ((x = y) ⇒ P[x] ⇒ P[y])
Lemma: member_mk_rset_lemma
∀z,P:Top. (z ∈ {x:ℝ | P[x]} ~ P[z])
Definition: upper-bound
A ≤ b == ∀x:ℝ. ((x ∈ A) ⇒ (x ≤ b))
Lemma: upper-bound_wf
∀[A:Set(ℝ)]. ∀[b:ℝ]. (A ≤ b ∈ ℙ)
Lemma: upper-bound_functionality
∀[A:Set(ℝ)]. ∀[b,c:ℝ]. {A ≤ c supposing A ≤ b} supposing b ≤ c
Definition: strict-upper-bound
A < b == ∀x:ℝ. ((x ∈ A) ⇒ (x < b))
Lemma: strict-upper-bound_wf
∀[A:Set(ℝ)]. ∀[b:ℝ]. (A < b ∈ ℙ)
Lemma: strict-upper-bound_functionality
∀A:Set(ℝ). ∀b,c:ℝ. {A < b ⇒ A < c} supposing b ≤ c
Definition: lower-bound
lower-bound(A;b) == ∀x:ℝ. ((x ∈ A) ⇒ (b ≤ x))
Lemma: lower-bound_wf
∀[A:Set(ℝ)]. ∀[b:ℝ]. (lower-bound(A;b) ∈ ℙ)
Definition: bounded-above
bounded-above(A) == ∃b:ℝ. A ≤ b
Lemma: bounded-above_wf
∀[A:Set(ℝ)]. (bounded-above(A) ∈ ℙ)
Lemma: bounded-above-strict
∀[A:Set(ℝ)]. (bounded-above(A) ⇒ (∃b:ℝ. A < b))
Definition: bounded-below
bounded-below(A) == ∃b:ℝ. lower-bound(A;b)
Lemma: bounded-below_wf
∀[A:Set(ℝ)]. (bounded-below(A) ∈ ℙ)
Definition: sup
sup(A) = b == A ≤ b ∧ (∀e:ℝ. ((r0 < e) ⇒ (∃x:ℝ. ((x ∈ A) ∧ ((b - e) < x)))))
Lemma: sup_wf
∀[A:Set(ℝ)]. ∀[b:ℝ]. (sup(A) = b ∈ ℙ)
Lemma: sup-unique
∀[A:Set(ℝ)]. ∀[b,c:ℝ]. (b = c) supposing (sup(A) = c and sup(A) = b)
Definition: inf
inf(A) = b == lower-bound(A;b) ∧ (∀e:ℝ. ((r0 < e) ⇒ (∃x:ℝ. ((x ∈ A) ∧ (x < (b + e))))))
Lemma: inf_wf
∀[A:Set(ℝ)]. ∀[b:ℝ]. (inf(A) = b ∈ ℙ)
Lemma: inf_functionality
∀[A,A':Set(ℝ)]. ∀b,b':ℝ. ((b = b') ⇒ rseteq(A;A') ⇒ (inf(A) = b ⇐⇒ inf(A') = b'))
Lemma: sup_functionality
∀[A,A':Set(ℝ)]. ∀b,b':ℝ. ((b = b') ⇒ rseteq(A;A') ⇒ (sup(A) = b ⇐⇒ sup(A') = b'))
Lemma: rleq_inf
∀[A:Set(ℝ)]. ∀[b,c:ℝ]. (inf(A) = b ⇒ (c ≤ b ⇐⇒ ∀x:ℝ. ((x ∈ A) ⇒ (c ≤ x))))
Lemma: inf-rleq
∀[A:Set(ℝ)]. ∀b,c:ℝ. (inf(A) = b ⇒ (b ≤ c ⇐⇒ ∀e:ℝ. ((r0 < e) ⇒ (∃x:ℝ. ((x ∈ A) ∧ ((x - e) ≤ c))))))
Lemma: inf-rless
∀[A:Set(ℝ)]. ∀b,c:ℝ. (inf(A) = b ⇒ (b < c ⇐⇒ ∃x:ℝ. ((x ∈ A) ∧ (x < c))))
Lemma: inf-unique
∀[A:Set(ℝ)]. ∀[b,c:ℝ]. (inf(A) = b ⇒ inf(A) = c ⇒ (b = c))
Definition: member-closure
y ∈ closure(A) == ∃x:ℕ ⟶ ℝ. (lim n→∞.x[n] = y ∧ (∀n:ℕ. (A x[n])))
Lemma: member-closure_wf
∀[A:ℝ ⟶ ℙ]. ∀[y:ℝ]. (y ∈ closure(A) ∈ ℙ)
Definition: closed-rset
closed-rset(A) == ∀y:ℝ. (y ∈ closure(A) ⇒ (A y))
Lemma: closed-rset_wf
∀[A:ℝ ⟶ ℙ]. (closed-rset(A) ∈ ℙ)
Definition: upper-bounds
upper-bounds(A) == λb.A ≤ b
Lemma: upper-bounds_wf
∀[A:Set(ℝ)]. (upper-bounds(A) ∈ Set(ℝ))
Definition: strict-upper-bounds
strict-upper-bounds(A) == λb.A < b
Lemma: strict-upper-bounds_wf
∀[A:Set(ℝ)]. (strict-upper-bounds(A) ∈ Set(ℝ))
Lemma: upper-bounds-closed
∀[A:Set(ℝ)]. closed-rset(upper-bounds(A))
Lemma: sup-iff-closure
∀[A:Set(ℝ)]. ∀x:ℝ. (sup(A) = x ⇐⇒ A ≤ x ∧ x ∈ closure(A))
Lemma: closures-meet
∀[P,Q:ℝ ⟶ ℙ].
((∃a,b:ℝ. ((P a) ∧ (Q b) ∧ (a ≤ b)))
⇒ (∃c:ℝ
(((r0 ≤ c) ∧ (c < r1))
∧ (∀a,b:ℝ.
(((P a) ∧ (Q b) ∧ (a ≤ b))
⇒ (∃a',b':ℝ. ((P a') ∧ (Q b') ∧ (a ≤ a') ∧ (a' ≤ b') ∧ (b' ≤ b) ∧ ((b' - a') ≤ ((b - a) * c))))))))
⇒ (∃y:ℝ. (y ∈ closure(P) ∧ y ∈ closure(Q))))
Lemma: closures-meet-sq
∀[P,Q:ℝ ⟶ ℙ].
((∃a:{a:ℝ| P a} . (∃b:ℝ [((Q b) ∧ (a ≤ b))]))
⇒ (∃c:{c:ℝ| (r0 ≤ c) ∧ (c < r1)}
∀a:{a:ℝ| P a} . ∀b:{b:ℝ| (Q b) ∧ (a ≤ b)} .
∃a':{a':ℝ| P a'} . (∃b':{b':ℝ| (Q b') ∧ (a' ≤ b')} [((a ≤ a') ∧ (b' ≤ b) ∧ ((b' - a') ≤ ((b - a) * c)))]))
⇒ (∃y:ℝ. (y ∈ closure(λz.(↓P z)) ∧ y ∈ closure(λz.(↓Q z)))))
Lemma: closures-meet-sq-ext
∀[P,Q:ℝ ⟶ ℙ].
((∃a:{a:ℝ| P a} . (∃b:ℝ [((Q b) ∧ (a ≤ b))]))
⇒ (∃c:{c:ℝ| (r0 ≤ c) ∧ (c < r1)}
∀a:{a:ℝ| P a} . ∀b:{b:ℝ| (Q b) ∧ (a ≤ b)} .
∃a':{a':ℝ| P a'} . (∃b':{b':ℝ| (Q b') ∧ (a' ≤ b')} [((a ≤ a') ∧ (b' ≤ b) ∧ ((b' - a') ≤ ((b - a) * c)))]))
⇒ (∃y:ℝ. (y ∈ closure(λz.(↓P z)) ∧ y ∈ closure(λz.(↓Q z)))))
Lemma: closures-meet'
∀P,Q:Set(ℝ).
((∃a,b:ℝ. ((a ∈ P) ∧ (b ∈ Q) ∧ (a < b)))
⇒ (∃c:ℝ
(((r0 ≤ c) ∧ (c < r1))
∧ (∀a,b:ℝ.
(((a ∈ P) ∧ (b ∈ Q) ∧ (a < b))
⇒ (∃a',b':ℝ. ((a' ∈ P) ∧ (b' ∈ Q) ∧ (a ≤ a') ∧ (a' < b') ∧ (b' ≤ b) ∧ ((b' - a') ≤ ((b - a) * c))))))))
⇒ (∃y:ℝ. (y ∈ closure(P) ∧ y ∈ closure(Q))))
Lemma: closures-meet-sq'
∀[P,Q:ℝ ⟶ ℙ].
((∃a:{a:ℝ| P a} . (∃b:ℝ [((Q b) ∧ (a < b))]))
⇒ (∃c:{c:ℝ| (r0 ≤ c) ∧ (c < r1)}
∀a:{a:ℝ| P a} . ∀b:{b:ℝ| (Q b) ∧ (a < b)} .
∃a':{a':ℝ| P a'} . (∃b':{b':ℝ| (Q b') ∧ (a' < b')} [((a ≤ a') ∧ (b' ≤ b) ∧ ((b' - a') ≤ ((b - a) * c)))]))
⇒ (∃y:ℝ. (y ∈ closure(λz.(↓P z)) ∧ y ∈ closure(λz.(↓Q z)))))
Lemma: closures-meet-sq'-ext
∀[P,Q:ℝ ⟶ ℙ].
((∃a:{a:ℝ| P a} . (∃b:ℝ [((Q b) ∧ (a < b))]))
⇒ (∃c:{c:ℝ| (r0 ≤ c) ∧ (c < r1)}
∀a:{a:ℝ| P a} . ∀b:{b:ℝ| (Q b) ∧ (a < b)} .
∃a':{a':ℝ| P a'} . (∃b':{b':ℝ| (Q b') ∧ (a' < b')} [((a ≤ a') ∧ (b' ≤ b) ∧ ((b' - a') ≤ ((b - a) * c)))]))
⇒ (∃y:ℝ. (y ∈ closure(λz.(↓P z)) ∧ y ∈ closure(λz.(↓Q z)))))
Lemma: least-upper-bound
∀[A:Set(ℝ)]
((∃x:ℝ. (x ∈ A))
⇒ bounded-above(A)
⇒ (∃b:ℝ. sup(A) = b ⇐⇒ ∀x,y:ℝ. ((x < y) ⇒ ((∃a:ℝ. ((a ∈ A) ∧ (x < a))) ∨ A ≤ y))))
Definition: totally-bounded
totally-bounded(A) ==
∀e:ℝ. ((r0 < e) ⇒ (∃n:ℕ+. ∃a:ℕn ⟶ ℝ. ((∀i:ℕn. (a i ∈ A)) ∧ (∀x:ℝ. ((x ∈ A) ⇒ (∃i:ℕn. (|x - a i| < e)))))))
Lemma: totally-bounded_wf
∀[A:Set(ℝ)]. (totally-bounded(A) ∈ ℙ)
Lemma: totally-bounded-implies-nonvoid
∀[A:Set(ℝ)]. (totally-bounded(A) ⇒ (∃x:ℝ. (x ∈ A)))
Lemma: totally-bounded-bounded-above
∀[A:Set(ℝ)]. (totally-bounded(A) ⇒ bounded-above(A))
Lemma: totally-bounded-sup
∀[A:Set(ℝ)]. (totally-bounded(A) ⇒ (∃b:ℝ. sup(A) = b))
Definition: rset-neg
-(A) == λx.(-(x) ∈ A)
Lemma: rset-neg_wf
∀[A:Set(ℝ)]. (-(A) ∈ Set(ℝ))
Lemma: member_rset_neg_lemma
∀x,A:Top. (x ∈ -(A) ~ -(x) ∈ A)
Lemma: inf-as-sup
∀[A:Set(ℝ)]. ∀b:ℝ. (inf(A) = b ⇐⇒ sup(-(A)) = -(b))
Lemma: totally-bounded-neg
∀[A:Set(ℝ)]. (totally-bounded(A) ⇐⇒ totally-bounded(-(A)))
Lemma: totally-bounded-inf
∀[A:Set(ℝ)]. (totally-bounded(A) ⇒ (∃b:ℝ. inf(A) = b))
Definition: interval
An interval has two endpoint. There are three kinds of endpoints
that we represent with the type ⌜ℝ + ℝ + Top⌝.
An endpoint ⌜inr anything ⌝ represents "infinity"
(positive or negative, depending on whether it is the right or left endpoint).
An endpoint of the form ⌜inl x⌝ where ⌜x ∈ ℝ + ℝ⌝ is a "finite" endpoint.
If ⌜x = (inl a) ∈ (ℝ + ℝ)⌝ then the interval is "closed" at that endpoint (written [a,..
or ...,a] ) and if ⌜x = (inl a) ∈ (ℝ + ℝ)⌝ then the interval is "open" at
that endpoint (written (a,... or ...,a) ).
Therefore, the type ℝ + ℝ + Top × (ℝ + ℝ + Top)
represents nine kinds of intervals:
[a,b] (a,b], (-infinity, b]
[a,b), (a,b), (-infintiy, b)
[a, infinity), (a,infinity), (-infinity, infinity).⋅
Interval == ℝ + ℝ + Top × (ℝ + ℝ + Top)
Lemma: interval_wf
Interval ∈ Type
Definition: i-finite
i-finite(I) == let l,u = I in (↑isl(l)) ∧ (↑isl(u))
Lemma: i-finite_wf
∀[J:Interval]. (i-finite(J) ∈ ℙ)
Lemma: decidable__i-finite
∀J:Interval. Dec(i-finite(J))
Definition: endpoints
endpoints(I) == let l,u = I in <case outl(l) of inl(a) => a | inr(a) => a, case outl(u) of inl(b) => b | inr(b) => b>
Lemma: endpoints_wf
∀[I:Interval]. endpoints(I) ∈ ℝ × ℝ supposing i-finite(I)
Definition: left-endpoint
left-endpoint(I) == fst(endpoints(I))
Lemma: left-endpoint_wf
∀[I:Interval]. left-endpoint(I) ∈ ℝ supposing i-finite(I)
Definition: right-endpoint
right-endpoint(I) == snd(endpoints(I))
Lemma: right-endpoint_wf
∀[I:Interval]. right-endpoint(I) ∈ ℝ supposing i-finite(I)
Definition: iproper
iproper(I) == i-finite(I) ⇒ (left-endpoint(I) < right-endpoint(I))
Lemma: iproper_wf
∀[I:Interval]. (iproper(I) ∈ ℙ)
Lemma: sq_stable__iproper
∀I:Interval. SqStable(iproper(I))
Definition: i-length
|I| == right-endpoint(I) - left-endpoint(I)
Lemma: i-length_wf
∀[I:Interval]. |I| ∈ ℝ supposing i-finite(I)
Lemma: iproper-length
∀I:Interval. ((i-finite(I) ∧ iproper(I)) ⇒ (r0 < |I|))
Lemma: iproper-length-iff
∀I:Interval. (i-finite(I) ⇒ (r0 < |I| ⇐⇒ iproper(I)))
Definition: i-closed
i-closed(I) == let l,u = I in (↑((¬bisl(l)) ∨bisl(outl(l)))) ∧ (↑((¬bisl(u)) ∨bisl(outl(u))))
Lemma: i-closed_wf
∀[I:Interval]. (i-closed(I) ∈ ℙ)
Lemma: decidable__i-closed
∀I:Interval. Dec(i-closed(I))
Definition: i-member
r ∈ I ==
let l,u = I
in case l
of inl(x) =>
case x
of inl(a) =>
case u of inl(y) => case y of inl(b) => (a ≤ r) ∧ (r ≤ b) | inr(b) => (a ≤ r) ∧ (r < b) | inr(y) => a ≤ r
| inr(a) =>
case u of inl(y) => case y of inl(b) => (a < r) ∧ (r ≤ b) | inr(b) => (a < r) ∧ (r < b) | inr(y) => a < r
| inr(x) =>
case u of inl(y) => case y of inl(b) => r ≤ b | inr(b) => r < b | inr(y) => True
Lemma: i-member_wf
∀[I:Interval]. ∀[x:ℝ]. (x ∈ I ∈ ℙ)
Lemma: i-member_functionality
∀I:Interval. ∀a,b:ℝ. a ∈ I ⇐⇒ b ∈ I supposing a = b
Lemma: sq_stable__i-member
∀I:Interval. ∀r:ℝ. SqStable(r ∈ I)
Lemma: i-member-finite
∀[I:Interval]. ∀[r:ℝ]. left-endpoint(I)≤r≤right-endpoint(I) supposing r ∈ I supposing i-finite(I)
Lemma: i-finite-iff-bounded
∀I:Interval. (i-finite(I) ⇐⇒ ∃a,b:ℝ. ∀[r:ℝ]. a≤r≤b supposing r ∈ I)
Definition: rccint
[l, u] == <inl inl l, inl inl u>
Lemma: rccint_wf
∀[a,b:ℝ]. ([a, b] ∈ Interval)
Lemma: member_rccint_lemma
∀r,u,l:Top. (r ∈ [l, u] ~ (l ≤ r) ∧ (r ≤ u))
Definition: rcoint
[l, u) == <inl inl l, inl (inr u )>
Lemma: rcoint_wf
∀[l,u:ℝ]. ([l, u) ∈ Interval)
Lemma: member_rcoint_lemma
∀r,u,l:Top. (r ∈ [l, u) ~ (l ≤ r) ∧ (r < u))
Definition: rciint
[l, ∞) == <inl inl l, inr ⋅ >
Lemma: rciint_wf
∀[l:ℝ]. ([l, ∞) ∈ Interval)
Lemma: member_rciint_lemma
∀r,l:Top. (r ∈ [l, ∞) ~ l ≤ r)
Definition: rocint
(l, u] == <inl (inr l ), inl inl u>
Lemma: rocint_wf
∀[l,u:ℝ]. ((l, u] ∈ Interval)
Lemma: member_rocint_lemma
∀r,u,l:Top. (r ∈ (l, u] ~ (l < r) ∧ (r ≤ u))
Definition: rooint
(l, u) == <inl (inr l ), inl (inr u )>
Lemma: rooint_wf
∀[l,u:ℝ]. ((l, u) ∈ Interval)
Lemma: member_rooint_lemma
∀r,u,l:Top. (r ∈ (l, u) ~ (l < r) ∧ (r < u))
Definition: roiint
(l, ∞) == <inl (inr l ), inr ⋅ >
Lemma: roiint_wf
∀[l:ℝ]. ((l, ∞) ∈ Interval)
Lemma: member_roiint_lemma
∀r,l:Top. (r ∈ (l, ∞) ~ l < r)
Definition: ricint
(-∞, u] == <inr ⋅ , inl inl u>
Lemma: ricint_wf
∀[u:ℝ]. ((-∞, u] ∈ Interval)
Lemma: member_ricint_lemma
∀r,u:Top. (r ∈ (-∞, u] ~ r ≤ u)
Definition: rioint
(-∞, u) == <inr ⋅ , inl (inr u )>
Lemma: rioint_wf
∀[u:ℝ]. ((-∞, u) ∈ Interval)
Lemma: member_rioint_lemma
∀r,u:Top. (r ∈ (-∞, u) ~ r < u)
Definition: riiint
(-∞, ∞) == <inr ⋅ , inr ⋅ >
Lemma: riiint_wf
(-∞, ∞) ∈ Interval
Lemma: member_riiint_lemma
∀r:Top. (r ∈ (-∞, ∞) ~ True)
Lemma: left_endpoint_rccint_lemma
∀b,a:Top. (left-endpoint([a, b]) ~ a)
Lemma: right_endpoint_rccint_lemma
∀b,a:Top. (right-endpoint([a, b]) ~ b)
Lemma: iproper-roiint
∀x:ℝ. iproper((x, ∞))
Lemma: iproper-riiint
iproper((-∞, ∞))
Definition: closed-rational-interval
closed-rational-interval(a;b;c;d) == [r(a/b), r(c/d)]
Lemma: closed-rational-interval_wf
∀[a:ℤ]. ∀[b:ℤ-o]. ∀[c:ℤ]. ∀[d:ℤ-o]. (closed-rational-interval(a;b;c;d) ∈ Interval)
Lemma: interval-totally-bounded
∀a:ℝ. ∀b:{b:ℝ| a ≤ b} . totally-bounded(λx.(x ∈ [a, b]))
Definition: lower-right-endpoint
lower-right-endpoint(a;b;n) == (a + n + 1 * b)/n + 2
Lemma: lower-right-endpoint_wf
∀[a,b:ℝ]. ∀[n:ℕ+]. (lower-right-endpoint(a;b;n) ∈ ℝ)
Definition: raise-left-endpoint
raise-left-endpoint(a;b;n) == (n + 1 * a + b)/n + 2
Lemma: raise-left-endpoint_wf
∀[a,b:ℝ]. ∀[n:ℕ+]. (raise-left-endpoint(a;b;n) ∈ ℝ)
Lemma: lower-right-endpoint-rless
∀a,b:ℝ. ∀n:ℕ+. ((a < b) ⇒ ((a < lower-right-endpoint(a;b;n)) ∧ (lower-right-endpoint(a;b;n) < b)))
Lemma: raise-left-endpoint-rless
∀a,b:ℝ. ∀n:ℕ+. ((a < b) ⇒ ((a < raise-left-endpoint(a;b;n)) ∧ (raise-left-endpoint(a;b;n) < b)))
Lemma: raise-lower-endpoints-rless
∀a,b:ℝ. ∀n:ℕ+. ((a < b) ⇒ (raise-left-endpoint(a;b;n) < lower-right-endpoint(a;b;n)))
Definition: old-i-approx
old-i-approx(I;n) ==
let l,u = I
in case l
of inl(x) =>
case x
of inl(a) =>
case u of inl(y) => case y of inl(b) => [a, b] | inr(b) => [a, b - (r1/r(n))] | inr(y) => [a, r(n)]
| inr(a) =>
case u
of inl(y) =>
case y of inl(b) => [a + (r1/r(n)), b] | inr(b) => [a + (r1/r(n)), b - (r1/r(n))]
| inr(y) =>
[a + (r1/r(n)), r(n)]
| inr(x) =>
case u of inl(y) => case y of inl(b) => [r(-n), b] | inr(b) => [r(-n), b - (r1/r(n))] | inr(y) => [r(-n), r(n)]
Definition: i-approx
i-approx(I;n) ==
let l,u = I
in case l
of inl(x) =>
case x
of inl(a) =>
case u of inl(y) => case y of inl(b) => [a, b] | inr(b) => [a, b - (r1/r(n))] | inr(y) => [a, r(n)]
| inr(a) =>
case u
of inl(y) =>
case y of inl(b) => [a + (r1/r(n)), b] | inr(b) => [a + (r1/r(n)), b - (r1/r(n))]
| inr(y) =>
[a + (r1/r(n)), r(n)]
| inr(x) =>
case u of inl(y) => case y of inl(b) => [r(-n), b] | inr(b) => [r(-n), b - (r1/r(n))] | inr(y) => [r(-n), r(n)]
Lemma: i-approx_wf
∀[n:ℕ+]. ∀[I:Interval]. (i-approx(I;n) ∈ Interval)
Lemma: i-finite-approx
∀n,m:ℕ+. ∀I:Interval. (i-finite(i-approx(I;n)) ⇐⇒ i-finite(i-approx(I;m)))
Lemma: i-approx-monotonic
∀I:Interval. ∀n,m:ℕ+. ∀x:ℝ. ((x ∈ i-approx(I;n)) ⇒ (x ∈ i-approx(I;m))) supposing n ≤ m
Lemma: i-member-implies
∀I:Interval. ∀r:ℝ.
((r ∈ I)
⇒ (∃n,M:ℕ+
((r ∈ i-approx(I;n))
∧ (∀y:{y:ℝ| y ∈ I} . ((|y - r| ≤ (r1/r(M))) ⇒ (y ∈ i-approx(I;n))))
∧ (iproper(I) ⇒ iproper(i-approx(I;n))))))
Lemma: i-member-iff
∀I:Interval. ∀r:ℝ. (r ∈ I ⇐⇒ ∃n:ℕ+. (r ∈ i-approx(I;n)))
Lemma: i-member-proper-iff
∀I:Interval. (iproper(I) ⇒ (∀r:ℝ. (r ∈ I ⇐⇒ ∃n:ℕ+. (iproper(i-approx(I;n)) ∧ (r ∈ i-approx(I;n))))))
Lemma: i-member-approx
∀I:Interval. ∀x:ℝ. ∀n:ℕ+. ((x ∈ i-approx(I;n)) ⇒ (x ∈ I))
Lemma: i-member-witness
∀I:Interval. ∀r:ℝ. ((r ∈ I) ⇒ (∃n:ℕ+. (r ∈ i-approx(I;n))))
Lemma: i-member-finite-closed
∀I:Interval. (i-closed(I) ⇒ i-finite(I) ⇒ (∀r:ℝ. (r ∈ I ⇐⇒ left-endpoint(I)≤r≤right-endpoint(I))))
Lemma: i-member-between
∀I:Interval. ∀a,b:ℝ. ((a ∈ I) ⇒ (b ∈ I) ⇒ (∀r:ℝ. ((a ≤ r) ⇒ (r ≤ b) ⇒ (r ∈ I))))
Definition: rfun
I ⟶ℝ == {x:ℝ| x ∈ I} ⟶ ℝ
Lemma: rfun_wf
∀[I:Interval]. (I ⟶ℝ ∈ Type)
Lemma: rfun_subtype_1
∀[a,b,c:ℝ]. ((a ≤ c) ⇒ (c ≤ b) ⇒ ([a, b] ⟶ℝ ⊆r [a, c] ⟶ℝ))
Lemma: rfun_subtype_2
∀[a,b,c:ℝ]. ((a ≤ c) ⇒ (c ≤ b) ⇒ ([a, b] ⟶ℝ ⊆r [c, b] ⟶ℝ))
Lemma: rfun_subtype_3
∀[a,b,c,d:ℝ]. ((a ≤ c) ⇒ (c ≤ d) ⇒ (d ≤ b) ⇒ ([a, b] ⟶ℝ ⊆r [c, d] ⟶ℝ))
Definition: r-ap
f(x) == f x
Lemma: r-ap_wf
∀[I:Interval]. ∀[f:I ⟶ℝ]. ∀[x:ℝ]. f(x) ∈ ℝ supposing x ∈ I
Definition: rfun-eq
rfun-eq(I;f;g) == ∀x:{x:ℝ| x ∈ I} . (f(x) = g(x))
Lemma: rfun-eq_wf
∀[I:Interval]. ∀[f,g:I ⟶ℝ]. (rfun-eq(I;f;g) ∈ ℙ)
Lemma: rfun-eq_weakening
∀[I:Interval]. ∀[f,g:I ⟶ℝ]. ((f = g ∈ I ⟶ℝ) ⇒ rfun-eq(I;f;g))
Lemma: rfun-eq_inversion
∀[I:Interval]. ∀[f,g:I ⟶ℝ]. rfun-eq(I;g;f) supposing rfun-eq(I;f;g)
Lemma: rfun-eq_transitivity
∀[I:Interval]. ∀[f,g,h:I ⟶ℝ]. (rfun-eq(I;f;h)) supposing (rfun-eq(I;g;h) and rfun-eq(I;f;g))
Lemma: r-ap_functionality
∀[I:Interval]. ∀[f,g:I ⟶ℝ]. ∀[x:{x:ℝ| x ∈ I} ]. f(x) = g(x) supposing rfun-eq(I;f;g)
Definition: i-witness
i-witness(I;r;p) == fst((TERMOF{i-member-witness:o, 1:l} I r p))
Lemma: i-witness_wf
∀[I:Interval]. ∀[r:ℝ]. ∀[p:r ∈ I]. (i-witness(I;r;p) ∈ ℕ+)
Lemma: i-witness-property
∀I:Interval. ∀r:ℝ. ∀p:r ∈ I. (r ∈ i-approx(I;i-witness(I;r;p)))
Definition: i-type
i-type(I) == n:ℕ+ × {r:ℝ| r ∈ i-approx(I;n)}
Lemma: i-type_wf
∀[I:Interval]. (i-type(I) ∈ Type)
Definition: i-real
real(p) == snd(p)
Lemma: i-real_wf
∀[I:Interval]. ∀[p:i-type(I)]. (real(p) ∈ ℝ)
Definition: i-nonvoid
i-nonvoid(I) == ∃r:ℝ. (r ∈ I)
Lemma: i-nonvoid_wf
∀[I:Interval]. (i-nonvoid(I) ∈ ℙ)
Lemma: iproper-nonvoid
∀I:Interval. (iproper(I) ⇒ i-nonvoid(I))
Lemma: i-finite-closed-is-rccint
∀[I:Interval]. I ~ [left-endpoint(I), right-endpoint(I)] supposing i-finite(I) ∧ i-closed(I)
Definition: icompact
icompact(I) == i-nonvoid(I) ∧ i-closed(I) ∧ i-finite(I)
Lemma: icompact_wf
∀[I:Interval]. (icompact(I) ∈ ℙ)
Lemma: icompact-is-rccint
∀[I:Interval]. I ~ [left-endpoint(I), right-endpoint(I)] supposing icompact(I)
Lemma: rccint-icompact
∀a,b:ℝ. (a ≤ b ⇐⇒ icompact([a, b]))
Lemma: i-approx-closed
∀I:Interval. ∀n:ℕ+. i-closed(i-approx(I;n))
Lemma: i-approx-finite
∀I:Interval. ∀n:ℕ+. i-finite(i-approx(I;n))
Lemma: i-approx-compact
∀I:Interval. ∀n:ℕ+. ∀r:ℝ. ((r ∈ i-approx(I;n)) ⇒ icompact(i-approx(I;n)))
Lemma: implies-approx-compact
∀I:Interval. ∀n:ℕ+. ((∃r:ℝ. (r ∈ i-approx(I;n))) ⇒ icompact(i-approx(I;n)))
Lemma: i-approx-approx
∀I:Interval. ∀n,m:Top. (i-approx(i-approx(I;n);m) ~ i-approx(I;n))
Lemma: i-approx-of-compact
∀I:Interval. (icompact(I) ⇒ (∀n:ℕ+. (i-approx(I;n) = I ∈ Interval)))
Lemma: icompact-length-nonneg
∀[I:Interval]. r0 ≤ |I| supposing icompact(I)
Lemma: icompact-endpoints-rleq
∀[I:Interval]. left-endpoint(I) ≤ right-endpoint(I) supposing icompact(I)
Lemma: icompact-endpoints
∀I:Interval. (icompact(I) ⇒ ((left-endpoint(I) ∈ I) ∧ (right-endpoint(I) ∈ I)))
Lemma: i-member-compact
∀I:Interval. (icompact(I) ⇒ (∀r:ℝ. (r ∈ I ⇐⇒ left-endpoint(I)≤r≤right-endpoint(I))))
Lemma: iproper-approx
∀I:Interval
(iproper(I) ⇒ (∀n:ℕ+. (icompact(i-approx(I;n)) ⇒ (icompact(i-approx(I;n + 1)) ∧ iproper(i-approx(I;n + 1))))))
Lemma: i-approx-elim
∀I:Interval. ∀n:ℕ+. (i-nonvoid(i-approx(I;n)) ⇒ (∃a:ℝ. ∃b:{b:ℝ| a ≤ b} . (i-approx(I;n) = [a, b] ∈ Interval)))
Lemma: i-member-diff-bound
∀I:Interval. (icompact(I) ⇒ (∀x,y:{x:ℝ| x ∈ I} . (|x - y| ≤ |I|)))
Lemma: rmin-i-member
∀I:Interval. ∀a,b:ℝ. ((a ∈ I) ⇒ (b ∈ I) ⇒ (rmin(a;b) ∈ I))
Lemma: rmax-i-member
∀I:Interval. ∀a,b:ℝ. ((a ∈ I) ⇒ (b ∈ I) ⇒ (rmax(a;b) ∈ I))
Definition: real-closed-interval-lattice
For real numbers ⌜a ≤ b⌝, the reals in the closed interval ⌜[a, b]⌝
form a distributive lattice with meet ⌜rmin(x;y)⌝ and join ⌜rmax(x;y)⌝.
But they do not form a bounded distributive lattice with 0 = a and 1 = b
because we can't prove that ⌜rmin(a;x)⌝ is the same real as ⌜a⌝.
We can only prove that it is an "equivalent" real using the `req` relation
(that we display ⌜x = y⌝).
Thus the closed interval forms a generalized-bounded-distributive-lattice.⋅
real-closed-interval-lattice(a;b) == mk-general-bounded-dist-lattice({r:ℝ| r ∈ [a, b]} ;λ2x y.rmin(x;y);λ2x y.rmax(x;y)\000C;a;b;λ2x y.x = y)
Lemma: real-closed-interval-lattice_wf
∀[a,b:ℝ]. real-closed-interval-lattice(a;b) ∈ GeneralBoundedDistributiveLattice supposing a ≤ b
Definition: real-interval-lattice
real-interval-lattice(I) == mk-distributive-lattice({r:ℝ| r ∈ I} ; λ2x y.rmin(x;y); λ2x y.rmax(x;y))
Lemma: real-interval-lattice_wf
∀[I:Interval]. (real-interval-lattice(I) ∈ DistributiveLattice)
Lemma: i-member-convex
∀I:Interval. ∀a,b:ℝ. ((a ∈ I) ⇒ (b ∈ I) ⇒ (∀t:ℝ. ((r0 ≤ t) ⇒ (t ≤ r1) ⇒ ((t * a) + ((r1 - t) * b) ∈ I))))
Lemma: i-member-convex'
∀I:Interval. ∀a,b:ℝ.
((a ∈ I) ⇒ (b ∈ I) ⇒ (∀z:{z:ℝ| r0 < z} . ∀x,y:{z:ℝ| r0 ≤ z} . (((x + y) = z) ⇒ (((x * a) + (y * b)/z) ∈ I))))
Lemma: i-member-convex2
∀I:Interval. ∀a,b:ℝ. ((a ∈ I) ⇒ (b ∈ I) ⇒ (∀n:ℕ+. ∀i,j:ℕ. (((i + j) = n ∈ ℤ) ⇒ ((i * a + j * b)/n ∈ I))))
Lemma: i-member-convex3
∀I:Interval. ∀a,b:ℝ. ((a ∈ I) ⇒ (b ∈ I) ⇒ (∀n:ℕ+. ∀i:ℕ. (((i + 1) = n ∈ ℤ) ⇒ ((i * a + b)/n ∈ I))))
Lemma: i-member-convex4
∀I:Interval. ∀a,b:ℝ. ((a ∈ I) ⇒ (b ∈ I) ⇒ (∀n:ℕ+. ∀i:ℕ. (((1 + i) = n ∈ ℤ) ⇒ ((a + i * b)/n ∈ I))))
Lemma: sq_stable__icompact
∀I:Interval. SqStable(icompact(I))
Lemma: i-closed-finite-rep
∀I:Interval. (i-closed(I) ⇒ i-finite(I) ⇒ (∃a,b:ℝ. (I = [a, b] ∈ Interval)))
Lemma: i-approx-rep
∀I:Interval. ∀n:ℕ+. ∀r:ℝ. ((r ∈ i-approx(I;n)) ⇒ (∃a,b:ℝ. ((a ≤ b) ∧ (i-approx(I;n) = [a, b] ∈ Interval))))
Lemma: i-approx-rep2
∀I:Interval. ∀n:ℕ+. ∃a,b:ℝ. (i-approx(I;n) = [a, b] ∈ Interval)
Lemma: i-type-member
∀I:Interval. ∀p:i-type(I). (real(p) ∈ I)
Lemma: member-i-type
∀[I:Interval]. ∀[r:ℝ]. ∀[p:r ∈ I]. (<i-witness(I;r;p), r> ∈ i-type(I))
Definition: subinterval
I ⊆ J == ∀r:ℝ. ((r ∈ I) ⇒ (r ∈ J))
Lemma: subinterval_wf
∀[I,J:Interval]. (I ⊆ J ∈ ℙ)
Lemma: subinterval-riiint
∀[I:Interval]. I ⊆ (-∞, ∞)
Lemma: i-finite-subinterval
∀I,J:Interval. (I ⊆ J ⇒ i-finite(J) ⇒ i-finite(I))
Lemma: subinterval_transitivity
∀[I,J,K:Interval]. (I ⊆ J ⇒ J ⊆ K ⇒ I ⊆ K )
Lemma: subinterval-subtype
∀[I,J:Interval]. {x:ℝ| x ∈ I} ⊆r {x:ℝ| x ∈ J} supposing I ⊆ J
Lemma: i-approx-is-subinterval
∀I:Interval. ∀n:ℕ+. i-approx(I;n) ⊆ I
Lemma: compact-subinterval
∀I:Interval. ∀J:{J:Interval| icompact(J)} . (J ⊆ I ⇒ (∃n:{n:ℕ+| icompact(i-approx(I;n))} . J ⊆ i-approx(I;n) ))
Lemma: iproper-subinterval
∀I,J:Interval. (I ⊆ J ⇒ iproper(I) ⇒ iproper(J))
Lemma: rcc-subinterval
∀I:Interval. ∀a,b:ℝ. ([a, b] ⊆ I ⇐⇒ (a ≤ b) ⇒ ((a ∈ I) ∧ (b ∈ I)))
Lemma: trivial-subinterval
∀I:Interval. [left-endpoint(I), right-endpoint(I)] ⊆ I supposing icompact(I)
Lemma: subinterval-trivial
∀I:Interval. I ⊆ [left-endpoint(I), right-endpoint(I)] supposing icompact(I)
Lemma: between-rmin-rmax
∀[x,y,t:ℝ]. uiff((rmin(x;y) ≤ t) ∧ (t ≤ rmax(x;y));(|t - x| ≤ |y - x|) ∧ (|t - y| ≤ |y - x|))
Lemma: rmin-rmax-subinterval
∀I:Interval. ∀a,b:ℝ. ((a ∈ I) ⇒ (b ∈ I) ⇒ [rmin(a;b), rmax(a;b)] ⊆ I )
Lemma: rmin3-rmax3-subinterval
∀I:Interval. ∀a,b,c:ℝ. ((a ∈ I) ⇒ (b ∈ I) ⇒ (c ∈ I) ⇒ [rmin(a;rmin(b;c)), rmax(a;rmax(b;c))] ⊆ I )
Lemma: member-rcc-min-max
∀[x,y,t:ℝ]. uiff(t ∈ [rmin(x;y), rmax(x;y)];(|t - x| ≤ |y - x|) ∧ (|t - y| ≤ |y - x|))
Lemma: i-approx-containing2
∀I:Interval. ∀a,b:ℝ. (((a ∈ I) ∧ (b ∈ I)) ⇒ (∃n:ℕ+. ((a ∈ i-approx(I;n)) ∧ (b ∈ i-approx(I;n)))))
Lemma: i-approx-containing
∀I:Interval. ∀x:ℝ. ((x ∈ I) ⇒ (∃m:ℕ+. (icompact(i-approx(I;m)) ∧ (x ∈ i-approx(I;m)))))
Lemma: compact-proper-interval-near-member
∀J:Interval
(icompact(J)
⇒ iproper(J)
⇒ (∀x:ℝ. ((x ∈ J) ⇒ (∀r:ℝ. ((r0 < r) ⇒ (∃y:ℝ. ((y ∈ J) ∧ (|y - x| ≤ r) ∧ (r0 < |y - x|))))))))
Lemma: icompact-is-subinterval
∀I:Interval. (icompact(I) ⇒ (∃n:ℕ+. I ⊆ i-approx((-∞, ∞);n) ))
Definition: sublevelset
sublevelset(I;f;c) == λx.((x ∈ I) ∧ (f(x) ≤ c))
Lemma: sublevelset_wf
∀[I:Interval]. ∀[f:I ⟶ℝ]. ∀[c:ℝ]. (sublevelset(I;f;c) ∈ ℝ ⟶ ℙ)
Definition: superlevelset
superlevelset(I;f;c) == λx.((x ∈ I) ∧ (c ≤ f(x)))
Lemma: superlevelset_wf
∀[I:Interval]. ∀[f:I ⟶ℝ]. ∀[c:ℝ]. (superlevelset(I;f;c) ∈ ℝ ⟶ ℙ)
Lemma: rfun_subtype
∀[I,J:Interval]. I ⟶ℝ ⊆r J ⟶ℝ supposing J ⊆ I
Definition: frs-non-dec
frs-non-dec(L) == ∀i,j:ℕ||L||. ((i ≤ j) ⇒ (L[i] ≤ L[j]))
Lemma: frs-non-dec_wf
∀[p:ℝ List]. (frs-non-dec(p) ∈ ℙ)
Lemma: sq_stable__frs-non-dec
∀[p:ℝ List]. SqStable(frs-non-dec(p))
Definition: frs-refines
frs-refines(p;q) == (∀x∈q.(∃y∈p. x = y))
Lemma: frs-refines_wf
∀[p,q:ℝ List]. (frs-refines(p;q) ∈ ℙ)
Lemma: frs-refines_weakening
∀[p,q:ℝ List]. ((p = q ∈ (ℝ List)) ⇒ frs-refines(p;q))
Lemma: frs-refines_transitivity
∀[p,q,r:ℝ List]. (frs-refines(p;q) ⇒ frs-refines(q;r) ⇒ frs-refines(p;r))
Definition: frs-increasing
frs-increasing(p) == ∀i,j:ℕ||p||. (i < j ⇒ (p[i] < p[j]))
Lemma: frs-increasing_wf
∀[p:ℝ List]. (frs-increasing(p) ∈ ℙ)
Lemma: frs-increasing-non-dec
∀[p:ℝ List]. (frs-increasing(p) ⇒ frs-non-dec(p))
Lemma: frs-increasing-sorted-by
∀[p:ℝ List]. (frs-increasing(p) ⇐⇒ sorted-by(λx,y. (x < y);p))
Lemma: frs-non-dec-sum
∀p:ℝ List. (1 < ||p|| ⇒ frs-non-dec(p) ⇒ (Σ{p[i + 1] - p[i] | 0≤i≤||p|| - 2} = (last(p) - p[0])))
Lemma: frs-increasing-cons
∀p:ℝ List. ∀c:ℝ. (frs-increasing([c / p]) ⇐⇒ (0 < ||p|| ⇒ (c < p[0])) ∧ frs-increasing(p))
Lemma: frs-non-dec-sorted-by
∀[p:ℝ List]. (frs-non-dec(p) ⇐⇒ sorted-by(λx,y. (x ≤ y);p))
Definition: frs-separated
frs-separated(p;q) == (∀x∈p.(∀y∈q.x ≠ y))
Lemma: frs-separated_wf
∀[p,q:ℝ List]. (frs-separated(p;q) ∈ ℙ)
Lemma: frs-increasing-separated-common-refinement
∀p,q:ℝ List.
(frs-increasing(p)
⇒ frs-increasing(q)
⇒ frs-separated(p;q)
⇒ (∃r:ℝ List. (frs-increasing(r) ∧ frs-refines(r;p) ∧ frs-refines(r;q) ∧ frs-refines(p @ q;r))))
Definition: frs-mesh
frs-mesh(p) == if ||p|| <z 2 then r0 else rmaximum(0;||p|| - 2;i.p[i + 1] - p[i]) fi
Lemma: frs-mesh_wf
∀[p:ℝ List]. (frs-mesh(p) ∈ ℝ)
Lemma: frs-mesh-nonneg
∀[p:ℝ List]. (frs-non-dec(p) ⇒ (r0 ≤ frs-mesh(p)))
Lemma: adjacent-frs-points
∀[p:ℝ List]. ∀[i:ℕ||p|| - 1]. (frs-non-dec(p) ⇒ r0≤p[i + 1] - p[i]≤frs-mesh(p))
Lemma: frs-mesh-bound
∀[p:ℝ List]. ∀[x:ℝ]. ((r0 ≤ x) ⇒ (∀[i:ℕ||p|| - 1]. ((p[i + 1] - p[i]) ≤ x)) ⇒ (frs-mesh(p) ≤ x))
Lemma: avoid-real
∀a,b,x:ℝ. ((a < b) ⇒ (∃a',b':ℝ. ((a ≤ a') ∧ (b' ≤ b) ∧ (a' < b') ∧ ((x < a') ∨ (b' < x)))))
Lemma: avoid-reals
∀L:ℝ List. ∀a,b:ℝ. ((a < b) ⇒ (∃a',b':ℝ. ((a ≤ a') ∧ (b' ≤ b) ∧ (a' < b') ∧ (∀x∈L.(x < a') ∨ (b' < x)))))
Lemma: avoid-reals-simple
∀L:ℝ List. ∀a,b:ℝ. ((a < b) ⇒ (∃c:ℝ. ((a ≤ c) ∧ (c < b) ∧ (∀x∈L.c ≠ x))))
Definition: partitions
partitions(I;p) == frs-non-dec(p) ∧ (0 < ||p|| ⇒ ((left-endpoint(I) ≤ p[0]) ∧ (last(p) ≤ right-endpoint(I))))
Lemma: partitions_wf
∀[I:Interval]. ∀[p:ℝ List]. partitions(I;p) ∈ ℙ supposing icompact(I)
Lemma: sq_stable__partitions
∀I:Interval. ∀p:ℝ List. (icompact(I) ⇒ SqStable(partitions(I;p)))
Definition: partition
partition(I) == {p:ℝ List| partitions(I;p)}
Lemma: partition_wf
∀[I:Interval]. partition(I) ∈ Type supposing icompact(I)
Lemma: partition-point-member
∀I:Interval. (icompact(I) ⇒ (∀p:partition(I). (∀x∈p.x ∈ I)))
Definition: nearby-partitions
nearby-partitions(e;p;q) == (||p|| = ||q|| ∈ ℤ) ∧ (∀i:ℕ||p||. (|p[i] - q[i]| ≤ e))
Lemma: nearby-partitions_wf
∀[e:ℝ]. ∀[p,q:ℝ List]. (nearby-partitions(e;p;q) ∈ ℙ)
Lemma: nearby-partitions_functionality
∀p,q:ℝ List. ∀e1,e2:ℝ. ((e1 ≤ e2) ⇒ {nearby-partitions(e1;p;q) ⇒ nearby-partitions(e2;p;q)})
Lemma: nearby-frs-mesh
∀e:{e:ℝ| r0 < e} . ∀p,q:ℝ List. (nearby-partitions(e;q;p) ⇒ (frs-mesh(p) ≤ (frs-mesh(q) + (r(2) * e))))
Lemma: nearby-increasing-partition
∀I:Interval
((icompact(I) ∧ iproper(I))
⇒ (∀p:partition(I). ∀e:{e:ℝ| r0 < e} . ∃q:partition(I). (frs-increasing(q) ∧ nearby-partitions(e;p;q))))
Lemma: nearby-increasing-partition-avoids
∀I:Interval
((icompact(I) ∧ iproper(I))
⇒ (∀p:partition(I)
(frs-increasing(p)
⇒ (∀L:ℝ List. ∀e:{e:ℝ| r0 < e} .
∃q:partition(I). (frs-increasing(q) ∧ nearby-partitions(e;p;q) ∧ frs-separated(q;L))))))
Definition: full-partition
full-partition(I;p) == [left-endpoint(I) / (p @ [right-endpoint(I)])]
Lemma: full-partition_wf
∀[I:Interval]. ∀[p:partition(I)]. (full-partition(I;p) ∈ ℝ List) supposing icompact(I)
Lemma: last-full-partition
∀[I:Interval]. ∀[p:partition(I)]. (last(full-partition(I;p)) = right-endpoint(I) ∈ ℝ) supposing icompact(I)
Lemma: full-partition-non-dec
∀[I:Interval]. ∀[p:partition(I)]. frs-non-dec(full-partition(I;p)) supposing icompact(I)
Lemma: full-partition-point-member
∀I:Interval. (icompact(I) ⇒ (∀p:partition(I). (∀x∈full-partition(I;p).x ∈ I)))
Definition: partition-mesh
partition-mesh(I;p) == frs-mesh(full-partition(I;p))
Lemma: partition-mesh_wf
∀[I:Interval]. ∀[p:partition(I)]. (partition-mesh(I;p) ∈ ℝ) supposing icompact(I)
Lemma: partition-mesh-nil
∀[I:Top]. (partition-mesh(I;[]) ~ |I|)
Lemma: partition-mesh-nonneg
∀[I:Interval]. ∀[p:partition(I)]. (r0 ≤ partition-mesh(I;p)) supposing icompact(I)
Lemma: adjacent-partition-points
∀[I:Interval]
∀[p:partition(I)]
(((¬0 < ||p||) ⇒ r0≤right-endpoint(I) - left-endpoint(I)≤partition-mesh(I;p))
∧ (0 < ||p||
⇒ (r0≤p[0] - left-endpoint(I)≤partition-mesh(I;p)
∧ (∀i:ℕ||p|| - 1. r0≤p[i + 1] - p[i]≤partition-mesh(I;p))
∧ r0≤right-endpoint(I) - last(p)≤partition-mesh(I;p))))
supposing icompact(I)
Lemma: adjacent-full-partition-points
∀[I:Interval]
∀[p:partition(I)]
∀i:ℕ||full-partition(I;p)|| - 1. r0≤full-partition(I;p)[i + 1] - full-partition(I;p)[i]≤partition-mesh(I;p)
supposing icompact(I)
Lemma: partition-endpoints
∀[I:Interval]
∀[p:partition(I)]. ∀[x:ℝ]. full-partition(I;p)[0]≤x≤full-partition(I;p)[||full-partition(I;p)|| - 1] supposing x ∈ I
supposing icompact(I)
Lemma: partition-split-cons
∀[I:Interval]
∀[a:ℝ]. ∀[bs:ℝ List].
(partitions(I;[a / bs]) ⇒ (partitions([left-endpoint(I), a];[]) ∧ partitions([a, right-endpoint(I)];bs)))
supposing icompact(I)
Lemma: partition-split-cons-mesh
∀[I:Interval]
∀[a:ℝ]. ∀[bs:ℝ List].
(partitions(I;[a / bs])
⇒ (partitions([left-endpoint(I), a];[])
∧ partitions([a, right-endpoint(I)];bs)
∧ (left-endpoint(I) ≤ a)
∧ (a ≤ right-endpoint(I))
∧ (partition-mesh([left-endpoint(I), a];[]) ≤ partition-mesh(I;[a / bs]))
∧ (partition-mesh([a, right-endpoint(I)];bs) ≤ partition-mesh(I;[a / bs]))))
supposing icompact(I)
Lemma: partition-lemma
∀e:ℝ
((r0 < e)
⇒ (∀n:ℕ+. ∀f:ℕn ⟶ ℝ.
∀x:ℝ. ∃i:ℕn. (|x - f i| ≤ e) supposing f 0≤x≤f (n - 1) supposing ∀i:ℕn - 1. r0≤(f (i + 1)) - f i≤e))
Lemma: firstn-partition
∀I:Interval
(icompact(I) ⇒ (∀a:ℝ. ∀p:partition(I). ∀i:ℕ||p||. ((a = p[i]) ⇒ (firstn(i;p) ∈ partition([left-endpoint(I), a])))))
Lemma: nth_tl-partition
∀I:Interval
(icompact(I)
⇒ (∀a:ℝ. ∀p:partition(I). ∀i:ℕ||p||. ((a = p[i]) ⇒ (nth_tl(i + 1;p) ∈ partition([a, right-endpoint(I)])))))
Lemma: mesh-property
∀I:Interval
(icompact(I)
⇒ (∀p:partition(I). ∀e:ℝ.
((r0 < e)
⇒ ∀x:ℝ. ((x ∈ I) ⇒ (∃i:ℕ||full-partition(I;p)||. (|x - full-partition(I;p)[i]| ≤ e)))
supposing partition-mesh(I;p) ≤ e)))
Definition: trivial-partition
trivial-partition(I) == []
Lemma: trivial-partition_wf
∀[I:Interval]. trivial-partition(I) ∈ partition(I) supposing i-finite(I)
Lemma: mesh-trivial-partition
∀[I:Top]. (partition-mesh(I;[]) ~ |I|)
Definition: uniform-partition
uniform-partition(I;k) ==
mklist(k - 1;λi.(((r(k) - r(i + 1)) * left-endpoint(I)) + (r(i + 1) * right-endpoint(I))/r(k)))
Lemma: uniform-partition_wf
∀[I:Interval]. ∀[k:ℕ+]. (uniform-partition(I;k) ∈ partition(I)) supposing icompact(I)
Lemma: uniform-partition-point
∀[I:Interval]
∀[k:ℕ+]
∀i:ℕk + 1
((full-partition(I;uniform-partition(I;k))[i] * r(k))
= (((r(k) - r(i)) * left-endpoint(I)) + (r(i) * right-endpoint(I))))
supposing icompact(I)
Lemma: mesh-uniform-partition
∀[I:Interval]. ∀[k:ℕ+]. (partition-mesh(I;uniform-partition(I;k)) = (|I|/r(k))) supposing icompact(I)
Lemma: partition-exists
∀I:Interval. (icompact(I) ⇒ (∀e:ℝ. ((r0 < e) ⇒ (∃p:partition(I). (partition-mesh(I;p) ≤ e)))))
Lemma: simple-partition-exists
∀a,b:ℝ.
((a ≤ b)
⇒ (∀e:ℝ
((r0 < e)
⇒ (∃M:ℕ+
∃g:ℕM + 1 ⟶ {x:ℝ| x ∈ [a, b]}
(((g 0) = a ∈ ℝ) ∧ ((g M) = b ∈ ℝ) ∧ (∀i:ℕM. (((g i) ≤ (g (i + 1))) ∧ (((g (i + 1)) - g i) ≤ e))))))))
Definition: is-partition-choice
is-partition-choice(p;x) == ∀i:ℕ||p|| - 1. (x i ∈ [p[i], p[i + 1]])
Lemma: is-partition-choice_wf
∀[p:ℝ List]. ∀[x:ℕ||p|| - 1 ⟶ ℝ]. (is-partition-choice(p;x) ∈ ℙ)
Lemma: sq_stable__is-partition-choice
∀[p:ℝ List]. ∀[x:ℕ||p|| - 1 ⟶ ℝ]. SqStable(is-partition-choice(p;x))
Definition: partition-choice
partition-choice(p) == i:ℕ||p|| - 1 ⟶ {x:ℝ| x ∈ [p[i], p[i + 1]]}
Lemma: partition-choice_wf
∀[p:ℝ List]. (partition-choice(p) ∈ Type)
Lemma: partition-choice-subtype
∀[p:ℝ List]. ({f:ℕ||p|| - 1 ⟶ ℝ| is-partition-choice(p;f)} ⊆r partition-choice(p))
Lemma: partition-choice-indep-funtype
∀[I:Interval]
∀[p:partition(I)]. (partition-choice(full-partition(I;p)) ⊆r (ℕ||p|| + 1 ⟶ {x:ℝ| x ∈ I} )) supposing icompact(I)
Lemma: implies-is-partition-choice
∀I:Interval
(icompact(I)
⇒ (∀p:partition(I). ∀x:partition-choice(full-partition(I;p)). is-partition-choice(full-partition(I;p);x)))
Lemma: partition-choice-member
∀I:Interval
(icompact(I)
⇒ (∀p:partition(I). ∀x:partition-choice(full-partition(I;p)). ∀i:ℕ||full-partition(I;p)|| - 1. (x i ∈ {x:ℝ| x ∈ I} )\000C))
Definition: partition-choice-ap
x[i] == x i
Lemma: partition-choice-ap_wf
∀I:Interval
(icompact(I) ⇒ (∀p:partition(I). ∀x:partition-choice(full-partition(I;p)). ∀i:ℕ||p|| + 1. (x[i] ∈ {x:ℝ| x ∈ I} )))
Definition: partition-sum
S(f;p) == Σ{(f (x i)) * (p[i + 1] - p[i]) | 0≤i≤||p|| - 2}
Lemma: partition-sum_wf
∀I:Interval
(icompact(I)
⇒ (∀p:partition(I). ∀f:I ⟶ℝ. ∀x:partition-choice(full-partition(I;p)). (S(f;full-partition(I;p)) ∈ ℝ)))
Lemma: partition-sum_functionality
∀I:Interval
(icompact(I)
⇒ (∀p:partition(I). ∀q:ℝ List.
((||q|| = ||p|| ∈ ℤ)
⇒ (∀i:ℕ||q||. (q[i] = p[i]))
⇒ (∀f:I ⟶ℝ. ∀x:partition-choice(full-partition(I;p)).
(S(f;full-partition(I;p)) = S(f;full-partition(I;q)))))))
Lemma: constant-partition-sum
∀I:Interval
(icompact(I) ⇒ (∀p:partition(I). ∀f:I ⟶ℝ. ∀z:ℝ. ((z ∈ I) ⇒ (S(f;full-partition(I;p)) = ((f z) * |I|)))))
Definition: partition-refines
P refines Q == frs-refines(P;Q)
Lemma: partition-refines_wf
∀I:Interval. ∀P,Q:partition(I). (P refines Q ∈ ℙ) supposing icompact(I)
Lemma: partition-refines_weakening
∀I:Interval. ∀P,Q:partition(I). ((P = Q ∈ partition(I)) ⇒ P refines Q) supposing icompact(I)
Lemma: partition-refines_transitivity
∀I:Interval. ∀P,Q,R:partition(I). (P refines Q ⇒ Q refines R ⇒ P refines R) supposing icompact(I)
Lemma: partition-refines-cons
∀I:Interval
(icompact(I)
⇒ (∀a:ℝ. ∀bs:ℝ List.
(partitions(I;[a / bs])
⇒ (0 < ||bs|| ⇒ (a < hd(bs)))
⇒ (∀p:partition(I)
(p refines [a / bs]
⇒ (∃q:partition([left-endpoint(I), a])
∃r:partition([a, right-endpoint(I)])
(q refines []
∧ r refines bs
∧ (∃x:ℝ. ((x = a) ∧ (p = (q @ [x / r]) ∈ (ℝ List))))
∧ ||r|| + ||q|| < ||p||
∧ (∀x:partition-choice(full-partition(I;p))
(is-partition-choice(full-partition([left-endpoint(I), a];q);x)
∧ is-partition-choice(full-partition([a, right-endpoint(I)];r);λi.(x (i + ||q|| + 1))))))))))))
Lemma: uniform-partition-refines
∀a:ℝ. ∀b:{b:ℝ| a ≤ b} . ∀k,m:ℕ+. uniform-partition([a, b];m * k) refines uniform-partition([a, b];k)
Lemma: uniform-partition-increasing
∀a,b:ℝ. ((a < b) ⇒ (∀k:ℕ+. frs-increasing(uniform-partition([a, b];k))))
Definition: rrange
f[x](x∈I) == λy.∃x:ℝ. ((x ∈ I) ∧ (f[x] = y))
Lemma: rrange_wf
∀[I:Interval]. ∀[f:I ⟶ℝ]. (f[x](x∈I) ∈ Set(ℝ))
Lemma: rset-member-rrange
∀I:Interval. ∀[f:I ⟶ℝ]. ∀r:ℝ. ((r ∈ I) ⇒ (f[r] ∈ f[x](x∈I)))
Definition: continuous
Bishop defines f[x] to be continuous on compact interval J if
for every ∈ > 0 there is a delta > 0 such that
∀x,y:ℝ. ((x ∈ J) ⇒ (y ∈ J) ⇒ (|x - y| ≤ delta) ⇒ (|f[x] - f[y]| ≤ ∈))
Bishop then defines f[x] to be continuous on an arbitrary interval I if
if it is continuous on every compact subinterval of I.
The constructive content of this definition is called the modulus of continuity
of f.
In order to keep the modulus of continuity simpler (but equivalent),
we modify this definition in two ways.
First, rather than quantify over all ∈ > 0 we can quantify over n > 0
and use (r1/r(n)) in place of ∈.
Second, rather than quantify over all compact subintervals of I, we
define a uniform family of compact subintervals i-approx(I;m), indexed by
m > 0, that "fill up" the interval I.
This modification give the modulus of continuity the type like
ℕ+ ⟶ ℕ+ ⟶ ℝ⋅
f[x] continuous for x ∈ I ==
∀m:{m:ℕ+| icompact(i-approx(I;m))} . ∀n:ℕ+.
(∃d:ℝ [((r0 < d)
∧ (∀x,y:ℝ. ((x ∈ i-approx(I;m)) ⇒ (y ∈ i-approx(I;m)) ⇒ (|x - y| ≤ d) ⇒ (|f[x] - f[y]| ≤ (r1/r(n))))))])
Lemma: continuous_wf
∀[I:Interval]. ∀[f:I ⟶ℝ]. (f[x] continuous for x ∈ I ∈ ℙ)
Definition: proper-continuous
f[x] (proper)continuous for x ∈ I ==
∀m:{m:ℕ+| icompact(i-approx(I;m)) ∧ iproper(i-approx(I;m))} . ∀n:ℕ+.
(∃d:ℝ [((r0 < d)
∧ (∀x,y:ℝ. ((x ∈ i-approx(I;m)) ⇒ (y ∈ i-approx(I;m)) ⇒ (|x - y| ≤ d) ⇒ (|f[x] - f[y]| ≤ (r1/r(n))))))])
Lemma: proper-continuous_wf
∀[I:Interval]. ∀[f:I ⟶ℝ]. (f[x] (proper)continuous for x ∈ I ∈ ℙ)
Lemma: proper-continuous-implies
∀[I:Interval]. ∀[f:I ⟶ℝ].
(f[x] (proper)continuous for x ∈ I
⇒ (∀m:ℕ+. (icompact(i-approx(I;m)) ⇒ iproper(i-approx(I;m)) ⇒ f[x] continuous for x ∈ i-approx(I;m))))
Lemma: continuous-rneq
∀I:Interval. ∀f:I ⟶ℝ. (f[x] continuous for x ∈ I ⇒ (∀a,b:{x:ℝ| x ∈ I} . (f[a] ≠ f[b] ⇒ a ≠ b)))
Definition: dense-in-interval
dense-in-interval(I;X) == ∀a,b:{r:ℝ| r ∈ I} . ((a < b) ⇒ (∃x:ℝ. (((a < x) ∧ (x < b)) ∧ (X x))))
Lemma: dense-in-interval_wf
∀[I:Interval]. ∀[X:{a:ℝ| a ∈ I} ⟶ ℙ]. (dense-in-interval(I;X) ∈ ℙ)
Definition: metric
metric(X) == {d:X ⟶ X ⟶ ℝ| (∀x,y,z:X. ((d x z) ≤ ((d x y) + (d z y)))) ∧ (∀x:X. ((d x x) = r0))}
Lemma: metric_wf
∀[X:Type]. (metric(X) ∈ Type)
Lemma: metric-on-void
∀[X:Type]. Top ⊆r metric(X) supposing ¬X
Lemma: same-metric
∀[X:Type]. ∀[d:metric(X)]. ∀[d':X ⟶ X ⟶ ℝ]. d' ∈ metric(X) supposing ∀x,y:X. ((d' x y) = (d x y))
Definition: metric-eq
metric-eq(X;d;d') == ∀x,y:X. ((d' x y) = (d x y))
Lemma: metric-eq_wf
∀[X:Type]. ∀[d,d':X ⟶ X ⟶ ℝ]. (metric-eq(X;d;d') ∈ ℙ)
Lemma: metric-symmetry
∀[X:Type]. ∀[d:metric(X)]. ∀x,y:X. ((d x y) = (d y x))
Lemma: metric-nonneg
∀[X:Type]. ∀[d:metric(X)]. ∀[x,y:X]. (r0 ≤ (d x y))
Lemma: metric-on-subtype
∀[X,Y:Type]. metric(X) ⊆r metric(Y) supposing Y ⊆r X
Definition: meq
x ≡ y == (d x y) = r0
Lemma: meq_wf
∀[X:Type]. ∀[d:metric(X)]. ∀[x,y:X]. (x ≡ y ∈ ℙ)
Lemma: stable__meq
∀[X:Type]. ∀[d:metric(X)]. ∀[x,y:X]. Stable{x ≡ y}
Lemma: sq_stable__meq
∀[X:Type]. ∀[d:metric(X)]. ∀[x,y:X]. SqStable(x ≡ y)
Lemma: meq-equiv
∀[X:Type]. ∀[d:metric(X)]. EquivRel(X;x,y.x ≡ y)
Lemma: meq-same
∀[X:Type]. ∀[d:metric(X)]. ∀[x:X]. x ≡ x
Lemma: meq_inversion
∀[X:Type]. ∀[d:metric(X)]. ∀[x,y:X]. y ≡ x supposing x ≡ y
Lemma: meq_weakening
∀[X:Type]. ∀[d:metric(X)]. ∀[x,y:X]. x ≡ y supposing x = y ∈ X
Lemma: meq_transitivity
∀[X:Type]. ∀[d:metric(X)]. ∀[x,y,z:X]. (x ≡ z) supposing (x ≡ y and y ≡ z)
Lemma: meq_functionality
∀[X:Type]. ∀[d:metric(X)]. ∀[x1,x2,y1,y2:X]. (uiff(x1 ≡ y1;x2 ≡ y2)) supposing (y1 ≡ y2 and x1 ≡ x2)
Definition: mdist
mdist(d;x;y) == d x y
Lemma: mdist_wf
∀[X:Type]. ∀[d:metric(X)]. ∀[x,y:X]. (mdist(d;x;y) ∈ ℝ)
Lemma: mdist-nonneg
∀[X:Type]. ∀[d:metric(X)]. ∀[x,y:X]. (r0 ≤ mdist(d;x;y))
Lemma: mdist-symm
∀[X:Type]. ∀[d:metric(X)]. ∀[x,y:X]. (mdist(d;x;y) = mdist(d;y;x))
Lemma: mdist-triangle-inequality1
∀[X:Type]. ∀[d:metric(X)]. ∀[x,y,z:X]. (mdist(d;x;z) ≤ (mdist(d;x;y) + mdist(d;z;y)))
Lemma: mdist-triangle-inequality
∀[X:Type]. ∀[d:metric(X)]. ∀[x,y,z:X]. (mdist(d;x;z) ≤ (mdist(d;x;y) + mdist(d;y;z)))
Lemma: mdist-same
∀[X:Type]. ∀[d:metric(X)]. ∀[x:X]. (mdist(d;x;x) = r0)
Lemma: mdist_functionality
∀[X:Type]. ∀[d:metric(X)]. ∀[x,y,x',y':X]. (mdist(d;x;y) = mdist(d;x';y')) supposing (x ≡ x' and y ≡ y')
Lemma: meq-iff-mdist-rleq
∀[X:Type]. ∀[d:metric(X)]. ∀[x,y:X]. (x ≡ y ⇐⇒ ∀k:ℕ+. (mdist(d;x;y) ≤ (r1/r(k))))
Definition: msep
x # y == r0 < mdist(d;x;y)
Lemma: msep_wf
∀[X:Type]. ∀[d:metric(X)]. ∀[x,y:X]. (x # y ∈ ℙ)
Lemma: msep_functionality
∀[X:Type]. ∀d:metric(X). ∀x,y,x',y':X. (x ≡ x' ⇒ y ≡ y' ⇒ {x # y ⇐⇒ x' # y'})
Lemma: sq_stable__msep
∀[X:Type]. ∀d:metric(X). ∀x,y:X. SqStable(x # y)
Lemma: msep-symm
∀[X:Type]. ∀d:metric(X). ∀x,y:X. uiff(x # y;y # x)
Lemma: msep-or
∀[X:Type]. ∀d:metric(X). ∀x,y:X. (x # y ⇒ (∀z:X. (x # z ∨ z # y)))
Lemma: not-msep
∀[X:Type]. ∀[d:metric(X)]. ∀[x,y:X]. x ≡ y supposing ¬x # y
Lemma: msep-not-meq
∀[X:Type]. ∀d:metric(X). ∀x,y:X. (x # y ⇒ (¬x ≡ y))
Definition: rmetric
rmetric() == λx,y. |x - y|
Lemma: rmetric_wf
rmetric() ∈ metric(ℝ)
Lemma: rmetric-meq
∀[x,y:ℝ]. uiff(x ≡ y;x = y)
Definition: induced-metric
induced-metric(d;f) == λx,y. (d (f x) (f y))
Lemma: induced-metric_wf
∀[X:Type]. ∀[d:metric(X)]. ∀[Y:Type]. ∀[f:Y ⟶ X]. (induced-metric(d;f) ∈ metric(Y))
Definition: induced-rmetric
induced-rmetric(f) == λx,y. |(f x) - f y|
Lemma: induced-rmetric_wf
∀[Y:Type]. ∀[f:Y ⟶ ℝ]. (induced-rmetric(f) ∈ metric(Y))
Definition: real-subset-metric
real-subset-metric() == induced-rmetric(λp.(fst(p)))
Lemma: real-subset-metric_wf
∀[P:ℝ ⟶ ℙ]. (real-subset-metric() ∈ metric(x:ℝ × P[x]))
Definition: scale-metric
c*d == λx,y. (c * (d x y))
Lemma: scale-metric_wf
∀[X:Type]. ∀[c:{c:ℝ| r0 ≤ c} ]. ∀[d:metric(X)]. (c*d ∈ metric(X))
Definition: metric-leq
d1 ≤ d2 == ∀x,y:X. (mdist(d1;x;y) ≤ mdist(d2;x;y))
Lemma: metric-leq_wf
∀[X:Type]. ∀[d1,d2:metric(X)]. (d1 ≤ d2 ∈ ℙ)
Lemma: scale-metric-leq-iff
∀[X:Type]. ∀[d1,d2:metric(X)]. ∀[c:{c:ℝ| r0 < c} ]. (c*d1 ≤ d2 ⇐⇒ d1 ≤ (r1/c)*d2)
Lemma: metric-leq-meq
∀[X:Type]. ∀[d1,d2:metric(X)]. (d1 ≤ d2 ⇒ (∀x,y:X. (x ≡ y ⇒ x ≡ y)))
Definition: metric-space
MetricSpace == X:Type × metric(X)
Lemma: metric-space_wf
MetricSpace ∈ 𝕌'
Definition: mk-metric-space
X with d == <X, d>
Lemma: mk-metric-space_wf
∀[X:Type]. ∀[d:metric(X)]. (X with d ∈ MetricSpace)
Definition: prod-metric
prod-metric(k;d) == λv,w. Σ{mdist(d i;v i;w i) | 0≤i≤k - 1}
Lemma: prod-metric_wf
∀[k:ℕ]. ∀[X:ℕk ⟶ Type]. ∀[d:i:ℕk ⟶ metric(X[i])]. (prod-metric(k;d) ∈ metric(i:ℕk ⟶ X[i]))
Lemma: prod-metric-meq
∀[k:ℕ]. ∀[X:ℕk ⟶ Type]. ∀[d:i:ℕk ⟶ metric(X[i])]. ∀[p,q:i:ℕk ⟶ X[i]]. uiff(p ≡ q;∀i:ℕk. p i ≡ q i)
Definition: prod2-metric
prod2-metric(dX;dY) == λv,w. let x1,y1 = v in let x2,y2 = w in mdist(dX;x1;x2) + mdist(dY;y1;y2)
Lemma: prod2-metric_wf
∀[X,Y:Type]. ∀[dX:metric(X)]. ∀[dY:metric(Y)]. (prod2-metric(dX;dY) ∈ metric(X × Y))
Lemma: prod2-metric-meq
∀[X,Y:Type]. ∀[dX:metric(X)]. ∀[dY:metric(Y)]. ∀[p,q:X × Y]. uiff(p ≡ q;fst(p) ≡ fst(q) ∧ snd(p) ≡ snd(q))
Definition: prod-metric-space
prod-metric-space(k;X) == <i:ℕk ⟶ (fst((X i))), prod-metric(k;λi.(snd((X i))))>
Lemma: prod-metric-space_wf
∀[k:ℕ]. ∀[X:ℕk ⟶ MetricSpace]. (prod-metric-space(k;X) ∈ MetricSpace)
Definition: union-metric-space
union-metric-space(T;eq;X) ==
i:T × (fst((X i))) with λp,q. let i,v = p in let j,w = q in if eq i j then rmin(mdist(snd((X i));v;w);r1) else r1 fi
Lemma: union-metric-space_wf
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[X:T ⟶ MetricSpace]. (union-metric-space(T;eq;X) ∈ MetricSpace)
Definition: interval-metric-space
interval-metric-space(I) == <{x:ℝ| x ∈ I} , rmetric()>
Lemma: interval-metric-space_wf
∀[I:Interval]. (interval-metric-space(I) ∈ MetricSpace)
Definition: unit-interval-ms
I == interval-metric-space([r0, r1])
Lemma: unit-interval-ms_wf
I ∈ MetricSpace
Definition: unit-prod
I x (X;d) == prod-metric-space(2;λi.if (i =z 0) then I else <X, d> fi )
Lemma: unit-prod_wf
∀[X:Type]. ∀[d:metric(X)]. (I x (X;d) ∈ MetricSpace)
Lemma: mdist-difference
∀[X:Type]. ∀[d:metric(X)]. ∀[x,a,b:X]. (|mdist(d;x;a) - mdist(d;x;b)| ≤ mdist(d;a;b))
Lemma: mdist-difference2
∀[X:Type]. ∀[d:metric(X)]. ∀[x,a,b,y:X]. (|mdist(d;x;y) - mdist(d;a;b)| ≤ (mdist(d;x;a) + mdist(d;y;b)))
Definition: is-mfun
f:FUN(X;Y) == ∀x1,x2:X. (x1 ≡ x2 ⇒ f[x1] ≡ f[x2])
Lemma: is-mfun_wf
∀[X,Y:Type]. ∀[d:metric(X)]. ∀[d':metric(Y)]. ∀[f:X ⟶ Y]. (f:FUN(X;Y) ∈ ℙ)
Lemma: is-mfun-compose
∀[X,Y,Z:Type]. ∀[d:metric(X)]. ∀[d':metric(Y)]. ∀[d'':metric(Z)]. ∀[f:X ⟶ Y]. ∀[g:Y ⟶ Z].
(g o f:FUN(X;Z)) supposing (g:FUN(Y;Z) and f:FUN(X;Y))
Lemma: stable__is-mfun
∀[X,Y:Type]. ∀[d:metric(X)]. ∀[d':metric(Y)]. ∀[f:X ⟶ Y]. Stable{f:FUN(X;Y)}
Lemma: sq_stable__is-mfun
∀[X,Y:Type]. ∀[d:metric(X)]. ∀[d':metric(Y)]. ∀[f:X ⟶ Y]. SqStable(f:FUN(X;Y))
Definition: mfun
FUN(X ⟶ Y) == {f:X ⟶ Y| f:FUN(X;Y)}
Lemma: mfun_wf
∀[X,Y:Type]. ∀[d:metric(X)]. ∀[d':metric(Y)]. (FUN(X ⟶ Y) ∈ Type)
Lemma: mfun-subtype
∀[X,Y:Type]. ∀[d:metric(X)]. ∀[d':metric(Y)]. ∀[A:Type]. FUN(X ⟶ A) ⊆r FUN(X ⟶ Y) supposing A ⊆r Y
Lemma: mfun-subtype2
∀[X,Y:Type]. ∀[d:metric(X)]. ∀[d':metric(Y)]. ∀[A:Type]. FUN(X ⟶ Y) ⊆r FUN(A ⟶ Y) supposing A ⊆r X
Lemma: id-mfun
∀[X:Type]. ∀[d:metric(X)]. (λx.x ∈ FUN(X ⟶ X))
Lemma: compose-mfun
∀[X,Y,Z:Type]. ∀[dX:metric(X)]. ∀[dY:metric(Y)]. ∀[dZ:metric(Z)]. ∀[f:FUN(X ⟶ Y)]. ∀[g:FUN(Y ⟶ Z)].
(g o f ∈ FUN(X ⟶ Z))
Definition: mcompose
mcompose(f;g) == g o f
Lemma: mcompose_wf
∀[X,Y,Z:Type].
∀dx:metric(X). ∀dy:metric(Y). ∀dz:metric(Z). ∀[f:FUN(X ⟶ Y)]. ∀[g:FUN(Y ⟶ Z)]. (mcompose(f;g) ∈ FUN(X ⟶ Z))
Lemma: simple-glueing
∀[X:Type]. ∀[dX:metric(X)]. ∀[A,B:X ⟶ ℙ].
((∀x1,x2:X. (x1 ≡ x2 ⇒ A[x1] ⇒ A[x2]))
⇒ (∀x1,x2:X. (x1 ≡ x2 ⇒ B[x1] ⇒ B[x2]))
⇒ (∀x:X. (A[x] ∨ B[x]))
⇒ (∀[Y:Type]. ∀[dY:metric(Y)].
∀f:FUN({x:X| A[x]} ⟶ Y). ∀g:FUN({x:X| B[x]} ⟶ Y).
∃h:FUN(X ⟶ Y). ∀x:X. ((A[x] ⇒ h x ≡ f x) ∧ (B[x] ⇒ h x ≡ g x))
supposing ∀x:X. ((A[x] ∧ B[x]) ⇒ f x ≡ g x)))
Definition: homeomorphic
homeomorphic(X;dX;Y;dY) == ∃f:FUN(X ⟶ Y). (∃g:FUN(Y ⟶ X) [((∀x:X. g (f x) ≡ x) ∧ (∀y:Y. f (g y) ≡ y))])
Lemma: homeomorphic_wf
∀[X,Y:Type]. ∀[dX:metric(X)]. ∀[dY:metric(Y)]. (homeomorphic(X;dX;Y;dY) ∈ ℙ)
Lemma: homeomorphic_functionality
∀[X,Y,X',Y':Type].
(∀[dX:metric(X)]. ∀[dY:metric(Y)]. homeomorphic(X;dX;Y;dY) ≡ homeomorphic(X';dX;Y';dY)) supposing (X ≡ X' and Y ≡ Y')
Lemma: homeomorphic_weakening
∀[X,Y:Type]. ∀[dX:metric(X)]. ∀[dY:metric(Y)]. (X ≡ Y ⇒ (dX = dY ∈ metric(X)) ⇒ homeomorphic(X;dX;Y;dY))
Definition: homeo-inv
homeo-inv(h) == let f,g = h in <g, f>
Lemma: homeo-inv_wf
∀[X,Y:Type]. ∀[dX:metric(X)]. ∀[dY:metric(Y)]. ∀[h:homeomorphic(X;dX;Y;dY)]. (homeo-inv(h) ∈ homeomorphic(Y;dY;X;dX))
Lemma: homeomorphic_inversion
∀[X,Y:Type]. ∀[dX:metric(X)]. ∀[dY:metric(Y)]. (homeomorphic(X;dX;Y;dY) ⇒ homeomorphic(Y;dY;X;dX))
Lemma: homeomorphic_transitivity
∀[X,Y,Z:Type]. ∀[dX:metric(X)]. ∀[dY:metric(Y)]. ∀[dZ:metric(Z)].
(homeomorphic(X;dX;Y;dY) ⇒ homeomorphic(Y;dY;Z;dZ) ⇒ homeomorphic(X;dX;Z;dZ))
Definition: homeomorphic+
homeomorphic+(X;dX;Y;dY) ==
∃f:FUN(X ⟶ Y)
((∃g:FUN(Y ⟶ X) [((∀x:X. g (f x) ≡ x) ∧ (∀y:Y. f (g y) ≡ y))])
∧ (∃B:ℕ+ [(∀x1,x2:X. (mdist(dY;f x1;f x2) ≤ (r(B) * mdist(dX;x1;x2))))]))
Lemma: homeomorphic+_wf
∀[X,Y:Type]. ∀[dX:metric(X)]. ∀[dY:metric(Y)]. (homeomorphic+(X;dX;Y;dY) ∈ ℙ)
Definition: is-msfun
is-msfun(X;d;Y;d';f) == ∀x1,x2:X. (f[x1] # f[x2] ⇒ x1 # x2)
Lemma: is-msfun_wf
∀[X,Y:Type]. ∀[d:metric(X)]. ∀[d':metric(Y)]. ∀[f:X ⟶ Y]. (is-msfun(X;d;Y;d';f) ∈ ℙ)
Lemma: sq_stable__is-msfun
∀[X,Y:Type]. ∀d:metric(X). ∀[d':metric(Y)]. ∀[f:X ⟶ Y]. SqStable(is-msfun(X;d;Y;d';f))
Definition: msfun
msfun(X;d;Y;d') == {f:X ⟶ Y| is-msfun(X;d;Y;d';f)}
Lemma: msfun_wf
∀[X,Y:Type]. ∀[d:metric(X)]. ∀[d':metric(Y)]. (msfun(X;d;Y;d') ∈ Type)
Definition: m-cont-real-fun
m-cont-real-fun(X;d;x.f[x]) ==
∀e:{e:ℝ| r0 < e} . ∃delta:{e:ℝ| r0 < e} . ∀x,x':X. ((mdist(d;x;x') < delta) ⇒ (|f[x] - f[x']| < e))
Lemma: m-cont-real-fun_wf
∀[X:Type]. ∀[d:metric(X)]. ∀[f:X ⟶ ℝ]. (m-cont-real-fun(X;d;x.f[x]) ∈ ℙ)
Lemma: m-cont-real-fun-is-mfun
∀[X:Type]. ∀[d:metric(X)]. ∀[f:X ⟶ ℝ]. (m-cont-real-fun(X;d;x.f[x]) ⇒ λx.f[x]:FUN(X;ℝ))
Lemma: mdist-m-cont
∀[X:Type]. ∀d:metric(X). ∀a:X. m-cont-real-fun(X;d;x.mdist(d;a;x))
Definition: m-cont-metric-fun
m-cont-metric-fun(X;dX;Y;dY;x.f[x]) ==
∀e:{e:ℝ| r0 < e} . ∃delta:{e:ℝ| r0 < e} . ∀x,x':X. ((mdist(dX;x;x') < delta) ⇒ (mdist(dY;f[x];f[x']) < e))
Lemma: m-cont-metric-fun_wf
∀[X:Type]. ∀[dX:metric(X)]. ∀[Y:Type]. ∀[dY:metric(Y)]. ∀[f:X ⟶ Y]. (m-cont-metric-fun(X;dX;Y;dY;x.f[x]) ∈ ℙ)
Definition: meqfun
meqfun(d;A;f;g) == ∀a:A. f a ≡ g a
Lemma: meqfun_wf
∀[A,X:Type]. ∀[d:metric(X)]. ∀[f,g:A ⟶ X]. (meqfun(d;A;f;g) ∈ ℙ)
Lemma: meqfun-equiv-rel
∀[A,X:Type]. ∀[d:metric(X)]. EquivRel(A ⟶ X;f,g.meqfun(d;A;f;g))
Lemma: meqfun-equiv-rel-mfun
∀[A,X:Type]. ∀[dA:metric(A)]. ∀[d:metric(X)]. EquivRel(FUN(A ⟶ X);f,g.meqfun(d;A;f;g))
Definition: metric-subspace
metric-subspace(X;d;A) == strong-subtype(A;X) ∧ (∀a:A. ∀x:X. (x ≡ a ⇒ (x ∈ A)))
Lemma: metric-subspace_wf
∀[X:Type]. ∀[d:metric(X)]. ∀[A:Type]. (metric-subspace(X;d;A) ∈ ℙ)
Lemma: sq_stable__metric-subspace
∀[X:Type]. ∀[d:metric(X)]. ∀[A:Type]. SqStable(metric-subspace(X;d;A))
Lemma: set-metric-subspace
∀[X:Type]. ∀[d:metric(X)]. ∀[P:X ⟶ ℙ]. metric-subspace(X;d;{x:X| P[x]} ) supposing ∀x,y:X. (P[x] ⇒ y ≡ x ⇒ P[y])
Lemma: set-metric-subspace2
∀[X:Type]. ∀[d:metric(X)]. ∀[P,Q:X ⟶ ℙ].
(metric-subspace({x:X| Q[x]} ;d;{x:X| P[x]} )) supposing ((∀x,y:X. (P[x] ⇒ y ≡ x ⇒ P[y])) and (∀x:X. (P[x] ⇒ Q[x])\000C))
Lemma: stable-union-metric-subspace
∀[X:Type]. ∀[d:metric(X)]. ∀[T:Type]. ∀[P:T ⟶ X ⟶ ℙ].
metric-subspace(X;d;stable-union(X;T;i,x.P[i;x])) supposing ∀i:T. ∀x,y:X. (P[i;x] ⇒ y ≡ x ⇒ P[i;y])
Definition: m-interior-point
m-interior-point(X;d;A;p) == ∃M:ℕ+. ∀x:X. ((mdist(d;x;p) ≤ (r1/r(M))) ⇒ (x ∈ A))
Lemma: m-interior-point_wf
∀[X,A:Type]. ∀[d:metric(X)]. ∀[p:A]. (m-interior-point(X;d;A;p) ∈ ℙ) supposing strong-subtype(A;X)
Definition: m-interior
m-interior(X;d;A) == {p:A| m-interior-point(X;d;A;p)}
Lemma: m-interior_wf
∀[X,A:Type]. ∀[d:metric(X)]. (m-interior(X;d;A) ∈ Type) supposing strong-subtype(A;X)
Definition: m-boundary
m-boundary(X;d;A) == {p:A| ¬m-interior-point(X;d;A;p)}
Lemma: m-boundary_wf
∀[X,A:Type]. ∀[d:metric(X)]. (m-boundary(X;d;A) ∈ Type) supposing strong-subtype(A;X)
Definition: m-ball
m-ball(X;d;c;r) == {x:X| mdist(d;c;x) ≤ r}
Lemma: m-ball_wf
∀[X:Type]. ∀[d:metric(X)]. ∀[c:X]. ∀[r:ℝ]. (m-ball(X;d;c;r) ∈ Type)
Definition: m-open-ball
m-open-ball(X;d;c;r) == {x:X| mdist(d;c;x) < r}
Lemma: m-open-ball_wf
∀[X:Type]. ∀[d:metric(X)]. ∀[c:X]. ∀[r:ℝ]. (m-open-ball(X;d;c;r) ∈ Type)
Definition: m-sphere
m-sphere(X;d;c;r) == {x:X| mdist(d;c;x) = r}
Lemma: m-sphere_wf
∀[X:Type]. ∀[d:metric(X)]. ∀[c:X]. ∀[r:ℝ]. (m-sphere(X;d;c;r) ∈ Type)
Lemma: m-ball-boundary-subtype-m-sphere
∀[X:Type]. ∀[d:metric(X)]. ∀[c:X]. ∀[r:ℝ]. (m-boundary(X;d;m-ball(X;d;c;r)) ⊆r m-sphere(X;d;c;r))
Lemma: m-sphere-subtype-m-ball-boundary
∀[X:Type]. ∀[d:metric(X)]. ∀[c:X]. ∀[r:ℝ].
m-sphere(X;d;c;r) ⊆r m-boundary(X;d;m-ball(X;d;c;r))
supposing ∀c,x:X. ∀M:ℕ+.
∃y:X. ((mdist(d;c;y) = (mdist(d;c;x) + mdist(d;x;y))) ∧ (r0 < mdist(d;y;x)) ∧ (mdist(d;y;x) ≤ (r1/r(M))))
Lemma: m-ball-boundary
∀[X:Type]. ∀[d:metric(X)]. ∀[c:X]. ∀[r:ℝ].
m-boundary(X;d;m-ball(X;d;c;r)) ≡ m-sphere(X;d;c;r)
supposing ∀c,x:X. ∀M:ℕ+.
∃y:X. ((mdist(d;c;y) = (mdist(d;c;x) + mdist(d;x;y))) ∧ (r0 < mdist(d;y;x)) ∧ (mdist(d;y;x) ≤ (r1/r(M))))
Definition: m-closed-subspace
m-closed-subspace(X;d;A) == ∀x:X. ((∀k:ℕ+. ∃a:A. (mdist(d;x;a) ≤ (r1/r(k)))) ⇒ (x ∈ A))
Lemma: m-closed-subspace_wf
∀[X:Type]. ∀[d:metric(X)]. ∀[A:Type]. m-closed-subspace(X;d;A) ∈ ℙ supposing metric-subspace(X;d;A)
Lemma: mfun-strong-subtype
∀[X,Y:Type]. ∀[d:metric(X)]. ∀[d':metric(Y)]. ∀[A:Type].
strong-subtype(FUN(X ⟶ A);FUN(X ⟶ Y)) supposing metric-subspace(Y;d';A)
Definition: mfun-class
mfun-class(X;dx;Y;dy) == f,g:FUN(X ⟶ Y)//meqfun(dy;X;f;g)
Lemma: mfun-class_wf
∀[X:Type]. ∀[dx:metric(X)]. ∀[Y:Type]. ∀[dy:metric(Y)]. (mfun-class(X;dx;Y;dy) ∈ Type)
Lemma: mfun-class-strong-subtype
∀[X,Y:Type]. ∀[d:metric(X)]. ∀[d':metric(Y)]. ∀[A:Type].
strong-subtype(mfun-class(X;d;A;d');mfun-class(X;d;Y;d')) supposing metric-subspace(Y;d';A)
Definition: image-space
f[X] == y:Y × {x:X| y ≡ f x}
Lemma: image-space_wf
∀[X,Y:Type]. ∀[dY:metric(Y)]. ∀[f:X ⟶ Y]. (f[X] ∈ Type)
Definition: image-ap
f[x] == <f x, x>
Lemma: image-ap_wf
∀[X,Y:Type]. ∀[d:metric(Y)]. ∀[f:X ⟶ Y]. ∀[x:X]. (f[x] ∈ f[X])
Definition: image-metric
image-metric(d) == λp,q. (d (fst(p)) (fst(q)))
Lemma: image-metric_wf
∀[X,Y:Type]. ∀[f:X ⟶ Y]. ∀[d:metric(Y)]. (image-metric(d) ∈ metric(f[X]))
Definition: m-unif-cont
UC(f:X ⟶ Y) == ∀k:ℕ+. ∃delta:{d:ℝ| r0 < d} . ∀x,y:X. ((mdist(dx;x;y) ≤ delta) ⇒ (mdist(dy;f x;f y) ≤ (r1/r(k))))
Lemma: m-unif-cont_wf
∀[X:Type]. ∀[dx:metric(X)]. ∀[Y:Type]. ∀[dy:metric(Y)]. ∀[f:X ⟶ Y]. (UC(f:X ⟶ Y) ∈ ℙ)
Definition: m-ptwise-cont
PtwiseCONT(f:X ⟶ Y) ==
∀x:X. ∀k:ℕ+. ∃delta:{d:ℝ| r0 < d} . ∀y:X. ((mdist(dx;x;y) ≤ delta) ⇒ (mdist(dy;f x;f y) ≤ (r1/r(k))))
Lemma: m-ptwise-cont_wf
∀[X:Type]. ∀[dx:metric(X)]. ∀[Y:Type]. ∀[dy:metric(Y)]. ∀[f:X ⟶ Y]. (PtwiseCONT(f:X ⟶ Y) ∈ ℙ)
Definition: m-retraction
Retract(X ⟶ A) == {f:X ⟶ A| f:FUN(X;A) ∧ (∀a:A. f a ≡ a)}
Lemma: m-retraction_wf
∀[X,A:Type]. ∀[d:metric(X)]. Retract(X ⟶ A) ∈ Type supposing A ⊆r X
Lemma: m-retraction_functionality
∀[X,Y:Type].
∀[A,B:Type]. ∀[d:metric(X)]. Retract(X ⟶ A) ≡ Retract(Y ⟶ B) supposing A ⊆r X supposing A ≡ B supposing X ≡ Y
Lemma: m-retraction-subtype1
∀[X,Y,A:Type]. ∀[d:metric(X)]. Retract(X ⟶ A) ⊆r Retract(Y ⟶ A) supposing A ⊆r Y supposing Y ⊆r X
Lemma: m-retraction-subtype2
∀[X,A,B:Type]. ∀[d:metric(X)]. Retract(X ⟶ A) ⊆r Retract(X ⟶ B) supposing B ⊆r X supposing A ≡ B
Definition: homeo-image
homeo-image(A;Y;dY;h) == {y:Y| ∃a:A. y ≡ (fst(h)) a}
Lemma: homeo-image_wf
∀[X,Y:Type]. ∀[dX:metric(X)]. ∀[dY:metric(Y)]. ∀[h:homeomorphic(X;dX;Y;dY)]. ∀[A:Type].
homeo-image(A;Y;dY;h) ∈ Type supposing A ⊆r X
Lemma: homeo-image-inverse
∀[X,Y:Type]. ∀[dX:metric(X)]. ∀[dY:metric(Y)]. ∀[h:homeomorphic(X;dX;Y;dY)]. ∀[A:Type].
homeo-image(homeo-image(A;Y;dY;h);X;dX;homeo-inv(h)) ≡ A supposing metric-subspace(X;dX;A)
Lemma: trivial-homeo-image
∀[X,Y:Type]. ∀[dX:metric(X)]. ∀[dY:metric(Y)]. ∀[h:homeomorphic(X;dX;Y;dY)]. homeo-image(X;Y;dY;h) ≡ Y
Lemma: homeo-image-homeomorphic-subtype
∀[X,Y:Type]. ∀[dX:metric(X)]. ∀[dY:metric(Y)]. ∀[h:homeomorphic(X;dX;Y;dY)]. ∀[A:Type].
h ∈ homeomorphic(A;dX;homeo-image(A;Y;dY;h);dY) supposing metric-subspace(X;dX;A)
Lemma: homeo-image-homeomorphic
∀[X,Y:Type]. ∀[dX:metric(X)]. ∀[dY:metric(Y)].
∀h:homeomorphic(X;dX;Y;dY). ∀[A:Type]. homeomorphic(A;dX;homeo-image(A;Y;dY;h);dY) supposing metric-subspace(X;dX;A)
Lemma: homeo-image-boundary
∀[X,Y:Type]. ∀[dX:metric(X)]. ∀[dY:metric(Y)].
∀h:homeomorphic(X;dX;Y;dY)
∀[A:Type]
homeo-image(m-boundary(X;dX;A);Y;dY;h) ≡ m-boundary(Y;dY;homeo-image(A;Y;dY;h)) supposing metric-subspace(X;dX;A)
supposing PtwiseCONT(fst(h):X ⟶ Y) ∧ PtwiseCONT(snd(h):Y ⟶ X)
Lemma: homeomorphic-retraction
∀[X,A:Type]. ∀[d:metric(X)].
Retract(X ⟶ A) ⇒ (∀Y:Type. ∀d':metric(Y). ∀h:homeomorphic(X;d;Y;d'). Retract(Y ⟶ homeo-image(A;Y;d';h)))
supposing metric-subspace(X;d;A)
Definition: fixedpoint-property
FP(X) == ∀f:FUN(X ⟶ X). ∀n:ℕ+. ∃x:X. (mdist(d;f x;x) ≤ (r1/r(n)))
Lemma: fixedpoint-property_wf
∀[X:Type]. ∀[d:metric(X)]. (FP(X) ∈ ℙ)
Definition: m-open
m-open(X;d;x.A[x]) == ∀x:X. (A[x] ⇒ (∃k:ℕ+. ∀y:X. ((mdist(d;x;y) ≤ (r1/r(k))) ⇒ A[y])))
Lemma: m-open_wf
∀[X:Type]. ∀[d:metric(X)]. ∀[A:X ⟶ ℙ]. (m-open(X;d;x.A[x]) ∈ ℙ)
Definition: m-set
m-set(X;d;x.A[x]) == ∀x,y:X. (x ≡ y ⇒ (A[x] ⇐⇒ A[y]))
Lemma: m-set_wf
∀[X:Type]. ∀[d:metric(X)]. ∀[A:X ⟶ ℙ]. (m-set(X;d;x.A[x]) ∈ ℙ)
Lemma: m-open-set
∀[X:Type]. ∀[d:metric(X)]. ∀[A:X ⟶ ℙ]. (m-open(X;d;x.A[x]) ⇒ m-set(X;d;x.A[x]))
Definition: m-open-cover
m-open-cover(X;d;I;i,x.A[i; x]) == (∀i:I. m-open(X;d;x.A[i; x])) ∧ (∀x:X. ∃i:I. A[i; x])
Lemma: m-open-cover_wf
∀[X:Type]. ∀[d:metric(X)]. ∀[I:Type]. ∀[A:I ⟶ X ⟶ ℙ]. (m-open-cover(X;d;I;i,x.A[i;x]) ∈ ℙ)
Lemma: m-open-cover-iff
∀[X:Type]. ∀[d:metric(X)]. ∀[I:Type]. ∀[A:I ⟶ X ⟶ ℙ].
(m-open-cover(X;d;I;i,x.A[i;x])
⇐⇒ (∀i:I. m-open(X;d;x.A[i;x])) ∧ (∃b:X ⟶ ℕ+. ∃c:X ⟶ I. ∀x,y:X. ((mdist(d;x;y) ≤ (r1/r(b x))) ⇒ A[c x;y])))
Definition: mcauchy
mcauchy(d;n.x[n]) == ∀k:ℕ+. (∃N:ℕ [(∀n,m:ℕ. ((N ≤ n) ⇒ (N ≤ m) ⇒ (mdist(d;x[n];x[m]) ≤ (r1/r(k)))))])
Lemma: mcauchy_wf
∀[X:Type]. ∀[d:X ⟶ X ⟶ ℝ]. ∀[x:ℕ ⟶ X]. (mcauchy(d;n.x[n]) ∈ ℙ)
Definition: mconverges-to
lim n→∞.x[n] = y == ∀k:ℕ+. (∃N:ℕ [(∀n:ℕ. ((N ≤ n) ⇒ (mdist(d;x[n];y) ≤ (r1/r(k)))))])
Lemma: mconverges-to_wf
∀[X:Type]. ∀[d:X ⟶ X ⟶ ℝ]. ∀[x:ℕ ⟶ X]. ∀[y:X]. (lim n→∞.x[n] = y ∈ ℙ)
Lemma: mconverges-to_functionality
∀[X:Type]. ∀d:metric(X). ∀x1,x2:ℕ ⟶ X. ∀[y:X]. {lim n→∞.x1[n] = y ⇒ lim n→∞.x2[n] = y} supposing ∀n:ℕ. x1[n] ≡ x2[n]
Lemma: constant-mconverges-to
∀[X:Type]. ∀[d:metric(X)]. ∀[y:X]. lim n→∞.y = y
Lemma: subsequence-mconverges-to
∀[X:Type]. ∀[d:metric(X)]. ∀[a:X].
∀x,y:ℕ ⟶ X. (subsequence(a,b.a ≡ b;n.x[n];n.y[n]) ⇒ lim n→∞.x[n] = a ⇒ lim n→∞.y[n] = a)
Lemma: m-unique-limit
∀[X:Type]. ∀d:metric(X). ∀x:ℕ ⟶ X. ∀[y1,y2:X]. (y1 ≡ y2) supposing (lim n→∞.x[n] = y1 and lim n→∞.x[n] = y2)
Definition: mconverges
x[n]↓ as n→∞ == ∃y:X. lim n→∞.x[n] = y
Lemma: mconverges_wf
∀[X:Type]. ∀[d:X ⟶ X ⟶ ℝ]. ∀[x:ℕ ⟶ X]. (x[n]↓ as n→∞ ∈ ℙ)
Lemma: mconverges-implies-mcauchy
∀[X:Type]. ∀[d:metric(X)]. ∀[x:ℕ ⟶ X]. (x[n]↓ as n→∞ ⇒ mcauchy(d;n.x[n]))
Lemma: subsequence-mconverges
∀[X:Type]. ∀[d:metric(X)]. ∀[a:X]. ∀x,y:ℕ ⟶ X. (subsequence(a,b.a ≡ b;n.x[n];n.y[n]) ⇒ x[n]↓ as n→∞ ⇒ y[n]↓ as n→∞)
Definition: mdiverges
n.x[n]↑ == ∃e:ℝ. ((r0 < e) ∧ (∀k:ℕ. ∃m,n:ℕ. ((k ≤ m) ∧ (k ≤ n) ∧ (e ≤ mdist(d;x[m];x[n])))))
Lemma: mdiverges_wf
∀[X:Type]. ∀[d:X ⟶ X ⟶ ℝ]. ∀[x:ℕ ⟶ X]. (n.x[n]↑ ∈ ℙ)
Definition: mcomplete
mcomplete(M) == let X,d = M in ∀x:ℕ ⟶ X. (mcauchy(d;n.x n) ⇒ x n↓ as n→∞)
Lemma: mcomplete_wf
∀[M:MetricSpace]. (mcomplete(M) ∈ ℙ)
Definition: cauchy-mlimit
cauchy-mlimit(cmplt;x;c) == fst((cmplt x c))
Lemma: cauchy-mlimit_wf
∀[X:Type]. ∀[d:metric(X)]. ∀[cmplt:mcomplete(X with d)]. ∀[x:ℕ ⟶ X]. ∀[c:mcauchy(d;n.x n)].
(cauchy-mlimit(cmplt;x;c) ∈ X)
Lemma: converges-to-cauchy-mlimit
∀[X:Type]
∀d:metric(X). ∀cmplt:mcomplete(X with d). ∀x:ℕ ⟶ X. ∀c:mcauchy(d;n.x n). lim n→∞.x n = cauchy-mlimit(cmplt;x;c)
Lemma: cauchy-mlimit-unique
∀[X:Type]. ∀[d:metric(X)]. ∀[cmplt:mcomplete(X with d)]. ∀[x:ℕ ⟶ X]. ∀[c:mcauchy(d;n.x n)]. ∀[z:X].
cauchy-mlimit(cmplt;x;c) ≡ z supposing lim n→∞.x n = z
Lemma: reals-complete
mcomplete(ℝ with rmetric())
Lemma: reals-complete-ext
mcomplete(ℝ with rmetric())
Lemma: real-interval-complete
∀a,b:ℝ. mcomplete({x:ℝ| x ∈ [a, b]} with rmetric())
Lemma: metric-leq-cauchy
∀[X:Type]. ∀[d1,d2:metric(X)]. ∀c:{c:ℝ| r0 < c} . (d1 ≤ c*d2 ⇒ (∀x:ℕ ⟶ X. (mcauchy(d2;n.x n) ⇒ mcauchy(d1;n.x n))))
Lemma: scale-metric-cauchy
∀[X:Type]. ∀[d:metric(X)]. ∀c:{c:ℝ| r0 < c} . ∀x:ℕ ⟶ X. (mcauchy(d;n.x n) ⇐⇒ mcauchy(c*d;n.x n))
Lemma: scale-metric-converges
∀[X:Type]. ∀[d:metric(X)]. ∀c:{c:ℝ| r0 < c} . ∀x:ℕ ⟶ X. ∀y:X. (lim n→∞.x n = y ⇐⇒ lim n→∞.x n = y)
Lemma: scale-metric-complete
∀[X:Type]. ∀[d:metric(X)]. ∀c:{c:ℝ| r0 < c} . (mcomplete(X with d) ⇐⇒ mcomplete(X with c*d))
Lemma: prod-metric-space-complete
∀k:ℕ. ∀X:ℕk ⟶ MetricSpace. ((∀i:ℕk. mcomplete(X i)) ⇒ mcomplete(prod-metric-space(k;X)))
Lemma: union-metric-space-complete
∀T:Type. ∀eq:EqDecider(T). ∀X:T ⟶ MetricSpace. ((∀i:T. mcomplete(X i)) ⇒ mcomplete(union-metric-space(T;eq;X)))
Lemma: metric-weak-Markov
∀[X:Type]. ∀d:metric(X). (mcomplete(X with d) ⇒ (∀x,y:X. ((∀z:X. ((¬z ≡ x) ∨ (¬z ≡ y))) ⇒ x # y)))
Lemma: metric-strong-extensionality
∀[X:Type]
∀d:metric(X)
(mcomplete(X with d)
⇒ (∀Y:Type. ∀d':metric(Y). ∀f:X ⟶ Y.
((∀x1,x2:X. (x1 ≡ x2 ⇒ f[x1] ≡ f[x2])) ⇒ (∀x1,x2:X. (f[x1] # f[x2] ⇒ x1 # x2)))))
Lemma: m-strong-extensionality
∀[X:Type]. ∀d:metric(X). (mcomplete(X with d) ⇒ (∀Y:Type. ∀d':metric(Y). ∀f:FUN(X ⟶ Y). is-msfun(X;d;Y;d';f)))
Lemma: msfun-ext-mfun
∀[X,Y:Type]. ∀[d:metric(X)]. ∀[d':metric(Y)]. (mcomplete(X with d) ⇒ msfun(X;d;Y;d') ≡ FUN(X ⟶ Y))
Lemma: punctured-homeomorphism
∀[X,Y:Type]. ∀[d:metric(X)]. ∀[d':metric(Y)].
(mcomplete(X with d)
⇒ mcomplete(Y with d')
⇒ (∀h:homeomorphic(X;d;Y;d'). ∀p:Y. (h ∈ homeomorphic({x:X| x # (snd(h)) p} ;d;{y:Y| y # p} ;d'))))
Lemma: metric-leq-converges-to
∀[X:Type]. ∀[d1,d2:metric(X)]. (d2 ≤ d1 ⇒ (∀x:ℕ ⟶ X. ∀y:X. (lim n→∞.x[n] = y ⇒ lim n→∞.x[n] = y)))
Lemma: metric-leq-complete
∀[X:Type]. ∀[d1,d2:metric(X)].
(d2 ≤ d1
⇒ (∀x:ℕ ⟶ X. (mcauchy(d2;n.x n) ⇒ (∃y:ℕ ⟶ X. (subsequence(a,b.a ≡ b;n.x n;n.y n) ∧ mcauchy(d1;n.y n)))))
⇒ mcomplete(X with d1)
⇒ mcomplete(X with d2))
Lemma: equiv-metrics-preserve-complete
∀[X:Type]. ∀[d1,d2:metric(X)].
((∃c1,c2:{s:ℝ| r0 < s} . (c1*d1 ≤ d2 ∧ c2*d2 ≤ d1)) ⇒ (mcomplete(X with d1) ⇐⇒ mcomplete(X with d2)))
Lemma: m-closed-iff-complete
∀[X:Type]
∀d:metric(X)
(mcomplete(X with d) ⇒ (∀[A:Type]. (metric-subspace(X;d;A) ⇒ (m-closed-subspace(X;d;A) ⇐⇒ mcomplete(A with d)))))
Definition: m-TB
This is a useful formulation of "metric total boundedness".
The lemma m-TB-iff proves that it is equivalent to
∀k:ℕ. ∃n:ℕ+. ∃xs:ℕn ⟶ X. ∀x:X. ∃i:ℕn. (mdist(d;x;xs i) ≤ (r1/r(k + 1)))
This says that for every epsilon (1/(k+1)) there is a finite collection, xs,
of points in X (some might call it a "subfinite" collection) such that
every point in X is within distance epsilon from some point in xs.
The constructive content of this is
1) the function B from k in ℕ to n in ℕ+
2) for given k, the xs:ℕB k ⟶ X
3) the "decider" or "chooser" c that for a given x in X finds
the index in ℕB k of the point in the xs for which the distance to x is
less than epsilon.⋅
m-TB(X;d) ==
{tb:B:ℕ ⟶ ℕ+ × k:ℕ ⟶ ℕB k ⟶ X × (X ⟶ k:ℕ ⟶ ℕB k)|
let B,xs,c = tb in
∀p:X. ∀k:ℕ. (mdist(d;p;xs k (c p k)) ≤ (r1/r(k + 1)))}
Lemma: m-TB_wf
∀[X:Type]. ∀[d:metric(X)]. (m-TB(X;d) ∈ Type)
Lemma: m-TB-iff
∀[X:Type]. ∀[d:metric(X)]. (m-TB(X;d) ⇐⇒ ∀k:ℕ. ∃n:ℕ+. ∃xs:ℕn ⟶ X. ∀x:X. ∃i:ℕn. (mdist(d;x;xs i) ≤ (r1/r(k + 1))))
Lemma: real-interval-m-TB
∀a:ℝ. ∀b:{b:ℝ| a ≤ b} . m-TB({x:ℝ| x ∈ [a, b]} ;rmetric())
Lemma: m-TB-product
∀m:ℕ. ∀[X:ℕm ⟶ Type]. ∀[d:i:ℕm ⟶ metric(X[i])]. ((∀i:ℕm. m-TB(X[i];d[i])) ⇒ m-TB(i:ℕm ⟶ X[i];prod-metric(m;d)))
Lemma: continuous-image-m-TB
∀[X:Type]
∀dX:metric(X). ∀[Y:Type]. ∀dY:metric(Y). ∀f:X ⟶ Y. (UC(f:X ⟶ Y) ⇒ m-TB(X;dX) ⇒ m-TB(f[X];image-metric(dY)))
Lemma: m-TB-sup-and-inf
∀[X:Type]
∀dX:metric(X)
(m-TB(X;dX)
⇒ (∀f:X ⟶ ℝ. (UC(f:X ⟶ ℝ) ⇒ ((∃a:ℝ. inf(λr.∃x:X. (r = (f x))) = a) ∧ (∃b:ℝ. sup(λr.∃x:X. (r = (f x))) = b)))))
Definition: m-inf
m-inf{i:l}(d;mtb;f;mc) == fst(fst((TERMOF{m-TB-sup-and-inf:o, 1:l, i:l} d mtb f mc)))
Lemma: m-inf_wf
∀[X:Type]. ∀[d:metric(X)]. ∀[mtb:m-TB(X;d)]. ∀[f:X ⟶ ℝ]. ∀[mc:UC(f:X ⟶ ℝ)]. (m-inf{i:l}(d;mtb;f;mc) ∈ ℝ)
Lemma: m-inf-property1
∀[X:Type]
∀d:metric(X). ∀mtb:m-TB(X;d). ∀f:X ⟶ ℝ. ∀mc:UC(f:X ⟶ ℝ). inf(λr.∃x:X. (r = (f x))) = m-inf{i:l}(d;mtb;f;mc)
Lemma: m-inf-property
∀[X:Type]
∀d:metric(X). ∀mtb:m-TB(X;d). ∀f:X ⟶ ℝ. ∀mc:UC(f:X ⟶ ℝ).
((∀x:X. (m-inf{i:l}(d;mtb;f;mc) ≤ (f x))) ∧ (∀e:ℝ. ((r0 < e) ⇒ (∃x:X. ((f x) < (m-inf{i:l}(d;mtb;f;mc) + e))))))
Definition: m-sup
m-sup{i:l}(d;mtb;f;mc) == fst(snd((TERMOF{m-TB-sup-and-inf:o, 1:l, i:l} d mtb f mc)))
Lemma: m-sup_wf
∀[X:Type]. ∀[d:metric(X)]. ∀[mtb:m-TB(X;d)]. ∀[f:X ⟶ ℝ]. ∀[mc:UC(f:X ⟶ ℝ)]. (m-sup{i:l}(d;mtb;f;mc) ∈ ℝ)
Lemma: m-sup-property1
∀[X:Type]
∀d:metric(X). ∀mtb:m-TB(X;d). ∀f:X ⟶ ℝ. ∀mc:UC(f:X ⟶ ℝ). sup(λr.∃x:X. (r = (f x))) = m-sup{i:l}(d;mtb;f;mc)
Lemma: m-sup-property
∀[X:Type]
∀d:metric(X). ∀mtb:m-TB(X;d). ∀f:X ⟶ ℝ. ∀mc:UC(f:X ⟶ ℝ).
((∀x:X. ((f x) ≤ m-sup{i:l}(d;mtb;f;mc))) ∧ (∀e:ℝ. ((r0 < e) ⇒ (∃x:X. ((m-sup{i:l}(d;mtb;f;mc) - e) < (f x))))))
Definition: mtb-cantor
When mtb is witness to metric space X being totally bounded,
the first component of mtb is a function B of type ⌜ℕ ⟶ ℕ+⌝.
This gives us a "general Cantor space" k:ℕ ⟶ ℕB[k]⋅
mtb-cantor(mtb) == k:ℕ ⟶ ℕ(fst(mtb)) k
Lemma: mtb-cantor_wf
∀[X:Type]. ∀[d:metric(X)]. ∀[mtb:m-TB(X;d)]. (mtb-cantor(mtb) ∈ Type)
Definition: mtb-seq
When mtb is a witness that metric space X is totally bounded
and s is a point in the general Cantor space, mtb-cantor(mtb),
then at each natural number k there are finitely many points xs(k)
(that every other point in X is within 1/(k+1) of one of them),
and s(k) is the index of one of these points.
Thus we get a sequence of points in X.
Not every such sequence will converge in X, but for every point p
in X there is always one, coming from mtb-point-cantor(mtb;p),
that does converge to p (see mtb-seq-mtb-point-cantor-mconverges-to).
⋅
mtb-seq(mtb;s) == let B,xs,c = mtb in λk.(xs k (s k))
Lemma: mtb-seq_wf
∀[X:Type]. ∀[d:metric(X)]. ∀[mtb:m-TB(X;d)]. ∀[s:mtb-cantor(mtb)]. (mtb-seq(mtb;s) ∈ ℕ ⟶ X)
Definition: mtb-point-cantor
For a point p in X, this gives the sequence of indexes to the points
that are better and better "approximations" to p.
So, it is a point s in the general Cantor space mtb-cantor(mtb)
and mtb-seq(mtb;s) is a Cauchy sequence (in fact, a regular sequence) that
converges to p.⋅
mtb-point-cantor(mtb;p) == let B,xs,c = mtb in c p
Lemma: mtb-point-cantor_wf
∀[X:Type]. ∀[d:metric(X)]. ∀[mtb:m-TB(X;d)]. ∀[p:X]. (mtb-point-cantor(mtb;p) ∈ mtb-cantor(mtb))
Lemma: mtb-seq-mtb-point-cantor-mconverges-to
∀[X:Type]. ∀d:metric(X). ∀mtb:m-TB(X;d). ∀x:X. lim n→∞.mtb-seq(mtb;mtb-point-cantor(mtb;x)) n = x
Definition: m-k-regular
This is the "metric" version of the notion of a (k)regular sequence
(that was used to define the real numbers).
Every (k)regular sequence is a Cauchy sequence (m-k-regular-mcauchy)
so it will converge if the metric space X is complete.⋅
m-k-regular(d;k;s) == ∀n,m:ℕ. (mdist(d;s n;s m) ≤ ((r(k)/r(n + 1)) + (r(k)/r(m + 1))))
Lemma: m-k-regular_wf
∀[X:Type]. ∀[d:metric(X)]. ∀[k:ℕ]. ∀[s:ℕ ⟶ X]. (m-k-regular(d;k;s) ∈ ℙ)
Lemma: m-k-regular-monotone
∀[n,k:ℕ]. ∀[X:Type]. ∀[d:metric(X)]. ∀[s:ℕ ⟶ X]. m-k-regular(d;k;s) supposing m-k-regular(d;n;s) supposing n ≤ k
Lemma: m-k-regular-mcauchy
∀[X:Type]. ∀[d:metric(X)]. ∀[s:ℕ ⟶ X]. ∀b:ℕ+. (m-k-regular(d;b;s) ⇒ mcauchy(d;n.s n))
Definition: m-reg-test
For a sequence of integers, s, we can decide is s is regular upto some bound b
because n ≤ m is decidable on integers.
For the metric version of regularity, we won't be able to decide
whether the distance z = mdist(d;x;y) is or is not less than or equal to a given
(rational) real number.
But if a < b then we can decide (a < z) ∨ (z < b).
So we set a to be the bound for 2-regularity and b the bound for 3-regularity.
Then to test whether we can add point x onto sequence s,
we will either find that the extended sequence is not 2-regular,
m-not-reg(d;s;n),or that it is (still) 3-regular.
A 2-regular sequence, s, has ∀n:ℕ. m-not-reg(d;s;n) = ff,
(see m-regular-not-m-not-reg)
and ∀n:ℕb. m-not-reg(d;s;n) = ff implies that s is at least 3-regular
(see not-m-not-reg-3regular).⋅
m-reg-test(d;b;s;x) ==
int-seg-case(0;b;λn.rless-case((r(2)/r(n + 1)) + (r(2)/r(b + 1));(r(3)/r(n + 1)) + (r(3)/r(b + 1));rlessw((r(2)/r(n
+ 1))
+ (r(2)/r(b + 1));(r(3)/r(n + 1)) + (r(3)/r(b + 1)));mdist(d;s n;x)))
Lemma: m-reg-test_wf
∀[X:Type]
∀d:metric(X). ∀b:ℕ. ∀s:ℕb ⟶ X. ∀x:X.
(m-reg-test(d;b;s;x) ∈ (∃n:ℕb. (((r(2)/r(n + 1)) + (r(2)/r(b + 1))) < mdist(d;s n;x)))
∨ (∀n:ℕb. (mdist(d;s n;x) < ((r(3)/r(n + 1)) + (r(3)/r(b + 1))))))
Definition: m-not-reg
m-not-reg(d;s;n) == isl(m-reg-test(d;n;s;s n))
Lemma: m-not-reg_wf
∀[X:Type]. ∀[d:metric(X)]. ∀[n:ℕ]. ∀[s:ℕn + 1 ⟶ X]. (m-not-reg(d;s;n) ∈ 𝔹)
Lemma: m-regular-not-m-not-reg
∀[X:Type]. ∀[d:metric(X)]. ∀[s:ℕ ⟶ X]. (m-k-regular(d;2;s) ⇒ (∀n:ℕ. m-not-reg(d;s;n) = ff))
Lemma: not-m-not-reg-3regular
∀[X:Type]. ∀[d:metric(X)]. ∀[s:ℕ ⟶ X]. ∀[b:ℕ].
((∀n:ℕb. m-not-reg(d;s;n) = ff) ⇒ (∀n,m:ℕb. (mdist(d;s n;s m) ≤ ((r(3)/r(n + 1)) + (r(3)/r(m + 1))))))
Definition: first-m-not-reg
We can search for the first place, up to bound k,
where sequence s is not 2-regular.
We either locate the first failure of 2-regularity
or we find that the sequence is at least 3-regular up to k.
(see first-m-not-reg-property). ⋅
first-m-not-reg(d;s;k) == search(k;λn.m-not-reg(d;s;n))
Lemma: first-m-not-reg_wf
∀[X:Type]. ∀[d:metric(X)]. ∀[k:ℕ]. ∀[s:ℕk ⟶ X]. (first-m-not-reg(d;s;k) ∈ ℕk + 1)
Lemma: first-m-not-reg-property
∀[X:Type]
∀d:metric(X). ∀k:ℕ. ∀s:ℕk ⟶ X.
((first-m-not-reg(d;s;k) = 0 ∈ ℤ ⇐⇒ ∀n:ℕk. m-not-reg(d;s;n) = ff)
∧ let i = first-m-not-reg(d;s;k) - 1 in
(∀n:ℕi. m-not-reg(d;s;n) = ff) ∧ m-not-reg(d;s;i) = tt
supposing 0 < first-m-not-reg(d;s;k))
Definition: m-regularize
For any sequence of points s, we "regularize" s by testing if there
is a failure of 2-regularity. If so, we make the regularized sequence
become constant from that index on. Otherwise, the sequence will be
unchanged, and will be at least 3-regular.
We know that any 2-regular sequence s has m-regularize(d;s) = s
(see m-regularize-of-regular)
and the regularization of any sequence s is a Cauchy sequence
(with modulus of Cauchyness λk.(6 * k) )
(see m-regularize-mcauchy)⋅
m-regularize(d;s) == λn.eval m = first-m-not-reg(d;s;n + 1) in if 0 <z m then s (m - 2) else s n fi
Lemma: m-regularize_wf_finite
∀[X:Type]. ∀[d:metric(X)]. ∀[b:ℕ]. ∀[s:ℕb ⟶ X]. (m-regularize(d;s) ∈ ℕb ⟶ X)
Lemma: m-regularize_wf
∀[X:Type]. ∀[d:metric(X)]. ∀[s:ℕ ⟶ X]. (m-regularize(d;s) ∈ ℕ ⟶ X)
Lemma: m-regularize-of-regular
∀[X:Type]. ∀[d:metric(X)]. ∀[s:ℕ ⟶ X]. m-regularize(d;s) = s ∈ (ℕ ⟶ X) supposing m-k-regular(d;2;s)
Lemma: m-regularize-mcauchy
∀[X:Type]. ∀[d:metric(X)]. ∀[s:ℕ ⟶ X]. (λk.(6 * k) ∈ mcauchy(d;n.m-regularize(d;s) n))
Lemma: mtb-point-cantor-seq-regular
∀[X:Type]. ∀[d:metric(X)]. ∀mtb:m-TB(X;d). ∀p:X. m-k-regular(d;1;mtb-seq(mtb;mtb-point-cantor(mtb;p)))
Definition: mtb-cantor-map
Given a point p in mtb-cantor(mtb), the sequence mtb-seq(mtb;p) may not
be Cauchy, but its regularization m-regularize(d;mtb-seq(mtb;p)) is
(and λk.(6 * k) is a witness -- modulus of Cauchyness -- for that).
Thus (given a witness cmplt for the completenss of X) it converges
to a point in X.
So we get a map from the general Cantor space, mtb-cantor(mtb), to X.
It is onto X (mtb-cantor-map-onto) and is uniformly
continuous (mtb-cantor-map-continuous).
Hence, every compact metric space is the continuous image of a Cantor
space.
We get an even stronger version mtb-cantor-map-onto-common
of the onto property of this map. It says that for points x and y in X
that are withing 1/(n+1) of each other (mdist(d;x;y) ≤ (r1/r(n + 1)))
then we can find points p and q in mtb-cantor(mtb) such that
p maps to x (mtb-cantor-map(d;cmplt;mtb;p) ≡ x) and q maps to y
(mtb-cantor-map(d;cmplt;mtb;q) ≡ y) and p and q agree on their first
n values (they are equal in the type i:ℕn ⟶ ℕ(fst(mtb)) i).
Because of the theorem general-cantor-to-int-uniform-continuity,
we can then prove that every FUNCTION
(f in X ⟶ ℝ for which ∀x,y:X. (x ≡ y ⇒ ((f x) = (f y))) )
from a compact metric space to the reals is uniformly continuous.
(see compact-metric-to-real-continuity).
Then, using the fact that ⌜X × X⌝ is also a compact metric space
(see m-TB-product and prod-metric-space-complete)
we can easily prove that every FUNCTION from X to a metric space Y
is uniformly continuous. compact-metric-to-metric-continuity⋅
mtb-cantor-map(d;cmplt;mtb;p) == cauchy-mlimit(cmplt;m-regularize(d;mtb-seq(mtb;p));λk.(6 * k))
Lemma: mtb-cantor-map_wf
∀[X:Type]. ∀[d:metric(X)]. ∀[cmplt:mcomplete(X with d)]. ∀[mtb:m-TB(X;d)]. ∀[p:mtb-cantor(mtb)].
(mtb-cantor-map(d;cmplt;mtb;p) ∈ X)
Lemma: mtb-cantor-map-onto-common
∀[X:Type]
∀d:metric(X). ∀cmplt:mcomplete(X with d). ∀mtb:m-TB(X;d). ∀n:ℕ. ∀x,y:X.
((mdist(d;x;y) ≤ (r1/r(n + 1)))
⇒ (∃p,q:mtb-cantor(mtb)
((p = q ∈ (i:ℕn ⟶ ℕ(fst(mtb)) i)) ∧ mtb-cantor-map(d;cmplt;mtb;p) ≡ x ∧ mtb-cantor-map(d;cmplt;mtb;q) ≡ y)))
Definition: mcompact
mcompact(X;d) == mcomplete(X with d) × m-TB(X;d)
Lemma: mcompact_wf
∀[X:Type]. ∀[d:metric(X)]. (mcompact(X;d) ∈ Type)
Lemma: compact-metric-to-real-continuity
∀[X:Type]. ∀d:metric(X). ∀c:mcompact(X;d). ∀f:FUN(X ⟶ ℝ). UC(f:X ⟶ ℝ)
Definition: compact-mc
compact-mc{i:l}(d;c;f) == TERMOF{compact-metric-to-real-continuity:o, 1:l, i:l} d c f
Lemma: compact-mc_wf
∀[X:Type]. ∀d:metric(X). ∀c:mcompact(X;d). ∀f:FUN(X ⟶ ℝ). (compact-mc{i:l}(d;c;f) ∈ UC(f:X ⟶ ℝ))
Definition: compact-sup
compact-sup{i:l}(d;c;f) == m-sup{i:l}(d;snd(c);f;compact-mc{i:l}(d;c;f))
Lemma: compact-sup_wf
∀[X:Type]. ∀d:metric(X). ∀c:mcompact(X;d). ∀f:FUN(X ⟶ ℝ). (compact-sup{i:l}(d;c;f) ∈ ℝ)
Lemma: compact-sup-property
∀[X:Type]
∀d:metric(X). ∀c:mcompact(X;d). ∀f:FUN(X ⟶ ℝ).
((∀x:X. ((f x) ≤ compact-sup{i:l}(d;c;f))) ∧ (∀e:ℝ. ((r0 < e) ⇒ (∃x:X. ((compact-sup{i:l}(d;c;f) - e) < (f x))))))
Definition: compact-inf
compact-inf{i:l}(d;c;f) == m-inf{i:l}(d;snd(c);f;compact-mc{i:l}(d;c;f))
Lemma: compact-inf_wf
∀[X:Type]. ∀d:metric(X). ∀c:mcompact(X;d). ∀f:FUN(X ⟶ ℝ). (compact-inf{i:l}(d;c;f) ∈ ℝ)
Lemma: compact-inf-property
∀[X:Type]
∀d:metric(X). ∀c:mcompact(X;d). ∀f:FUN(X ⟶ ℝ).
((∀x:X. (compact-inf{i:l}(d;c;f) ≤ (f x))) ∧ (∀e:ℝ. ((r0 < e) ⇒ (∃x:X. ((f x) < (compact-inf{i:l}(d;c;f) + e))))))
Lemma: compact-metric-to-metric-continuity
∀X:Type. ∀d:metric(X). (mcompact(X;d) ⇒ (∀Y:Type. ∀dY:metric(Y). ∀f:FUN(X ⟶ Y). UC(f:X ⟶ Y)))
Lemma: fixedpoint-property_functionality
∀X:Type. ∀d:metric(X). ∀Y:Type. ∀d':metric(Y). (homeomorphic(X;d;Y;d') ⇒ mcompact(X;d) ⇒ FP(X) ⇒ FP(Y))
Lemma: mtb-cantor-map-onto
∀[X:Type]. ∀[d:metric(X)]. ∀[cmplt:mcomplete(X with d)]. ∀[mtb:m-TB(X;d)]. ∀[x:X].
mtb-cantor-map(d;cmplt;mtb;mtb-point-cantor(mtb;x)) ≡ x
Lemma: mtb-cantor-map-continuous
∀[X:Type]. ∀[d:metric(X)]. ∀[cmplt:mcomplete(X with d)]. ∀[mtb:m-TB(X;d)]. ∀[k:ℕ+]. ∀[p,q:mtb-cantor(mtb)].
mdist(d;mtb-cantor-map(d;cmplt;mtb;p);mtb-cantor-map(d;cmplt;mtb;q)) ≤ (r1/r(k))
supposing ∀i:ℕ. ((i ≤ (18 * k)) ⇒ ((p i) = (q i) ∈ ℤ))
Lemma: compact-metric-to-int-bounded
∀[X:Type]. ∀d:metric(X). ∀cmplt:mcomplete(X with d). ∀mtb:m-TB(X;d). ∀f:X ⟶ ℤ. ∃B:ℕ. ∀x:X. ∃y:X. (x ≡ y ∧ (|f y| ≤ B))
Definition: dist-fun
dist-fun(d;x) == λa.mdist(d;x;a)
Lemma: dist-fun_wf
∀[X:Type]. ∀[d:metric(X)]. ∀[x:X]. (dist-fun(d;x) ∈ FUN(X ⟶ ℝ))
Definition: compact-dist
dist(x;A) == compact-inf{i:l}(d;c;dist-fun(d;x))
Lemma: compact-dist_wf
∀[X:Type]. ∀[d:metric(X)]. ∀[A:Type]. ∀[c:mcompact(A;d)]. ∀[x:X]. (dist(x;A) ∈ ℝ) supposing A ⊆r X
Lemma: compact-dist-nonneg
∀[X:Type]. ∀[d:metric(X)]. ∀[A:Type]. ∀[c:mcompact(A;d)]. ∀[x:X]. (r0 ≤ dist(x;A)) supposing A ⊆r X
Lemma: compact-dist-positive
∀[X:Type]
∀d:metric(X). ∀A:Type.
∀c:mcompact(A;d). ∀x:X. (r0 < dist(x;A) ⇐⇒ ∃n:ℕ+. ∀a:A. ((r1/r(n)) ≤ mdist(d;x;a))) supposing A ⊆r X
Lemma: compact-dist-zero
∀[X:Type]
∀d:metric(X). ∀A:Type.
∀c:mcompact(A;d). ∀x:X. (dist(x;A) = r0 ⇐⇒ ∀n:ℕ+. ∃a:A. (mdist(d;x;a) < (r1/r(n)))) supposing A ⊆r X
Lemma: compact-dist-zero-in-complete
∀[X:Type]
∀d:metric(X)
(mcomplete(X with d)
⇒ (∀[A:Type]. (metric-subspace(X;d;A) ⇒ (∀c:mcompact(A;d). ∀x:X. (dist(x;A) = r0 ⇐⇒ x ∈ A)))))
Lemma: not-in-compact-separated
∀[X:Type]
∀d:metric(X)
∀[A:Type]
∀c:mcompact(A;d). ∀x:X. (¬(x ∈ A) ⇐⇒ ¬¬(∃n:ℕ+. ∀a:A. ((r1/r(n)) ≤ mdist(d;x;a))))
supposing mcomplete(X with d) ∧ metric-subspace(X;d;A)
Lemma: mcomplete-stable-union
∀[X:Type]. ∀[d:metric(X)]. ∀[T:Type]. ∀[P:T ⟶ X ⟶ ℙ].
finite(T) ⇒ mcomplete(X with d) ⇒ (∀i:T. mcompact({x:X| P[i;x]} ;d)) ⇒ mcomplete(stable-union(X;T;i,x.P[i;x]) with \000Cd)
supposing ∀i:T. ∀x,y:X. (P[i;x] ⇒ y ≡ x ⇒ P[i;y])
Lemma: mcompact-stable-union
∀[X:Type]
∀d:metric(X)
∀[T:Type]. ∀[P:T ⟶ X ⟶ ℙ].
finite(T) ⇒ mcomplete(X with d) ⇒ (∀i:T. mcompact({x:X| P[i;x]} ;d)) ⇒ T ⇒ mcompact(stable-union(X;T;i,x.P[i;x\000C]);d)
supposing ∀i:T. ∀x,y:X. (P[i;x] ⇒ y ≡ x ⇒ P[i;y])
Lemma: mcompact_functionality_wrt_homeomorphic+
∀X,Y:Type. ∀d:metric(X). ∀d':metric(Y). (homeomorphic+(Y;d';X;d) ⇒ mcompact(X;d) ⇒ mcompact(Y;d'))
Lemma: mcompact_functionality
∀[X,Y:Type]. ∀d:metric(X). (mcompact(X;d) ⇐⇒ mcompact(Y;d)) supposing X ≡ Y
Lemma: mcompact-product
∀k:ℕ. ∀X:ℕk ⟶ Type. ∀d:i:ℕk ⟶ metric(X i). ((∀i:ℕk. mcompact(X i;d i)) ⇒ mcompact(i:ℕk ⟶ (X i);prod-metric(k;d)))
Lemma: mcompact-interval
∀a,b:ℝ. mcompact({x:ℝ| x ∈ [a, b]} ;rmetric()) supposing a ≤ b
Lemma: interval-to-int-bounded
∀a:ℝ. ∀b:{b:ℝ| a ≤ b} . ∀f:{x:ℝ| x ∈ [a, b]} ⟶ ℤ.
∃B:ℕ. ∀x:{x:ℝ| x ∈ [a, b]} . ∃y:{x:ℝ| x ∈ [a, b]} . ((x = y) ∧ (|f y| ≤ B))
Lemma: fixedpoint-property-iff
∀X:Type. ∀d:metric(X). (mcompact(X;d) ⇒ (FP(X) ⇐⇒ ∀f:FUN(X ⟶ X). (¬(∀x:X. f x # x))))
Lemma: mcompact-finite-subcover
∀[X:Type]
∀d:metric(X)
(mcompact(X;d)
⇒ (∀[I:Type]. ∀[A:I ⟶ X ⟶ ℙ]. (m-open-cover(X;d;I;i,x.A[i;x]) ⇒ (∃n:ℕ+. ∃L:ℕn ⟶ I. ∀x:X. ∃j:ℕn. A[L j;x]))))
Lemma: finite-subcover-implies-m-TB
∀[X:Type]
∀d:metric(X)
((∀[I:Type]. ∀[A:I ⟶ X ⟶ ℙ]. (m-open-cover(X;d;I;i,x.A[i;x]) ⇒ (∃n:ℕ+. ∃L:ℕn ⟶ I. ∀x:X. ∃j:ℕn. A[L j;x])))
⇒ m-TB(X;d))
Definition: incr-binary-seq
IBS == {s:ℕ ⟶ ℕ2| ∀i:ℕ. ((s i) ≤ (s (1 + i)))}
Lemma: incr-binary-seq_wf
IBS ∈ Type
Lemma: ibs-property
∀[s:IBS]. ∀[m:ℕ]. ∀[n:ℕ]. (s n) = 1 ∈ ℤ supposing m ≤ n supposing (s m) = 1 ∈ ℤ
Definition: mkibs
mkibs(n.p[n]) == λn.if p[n] then 1 else 0 fi
Lemma: mkibs_wf
∀[p:ℕ ⟶ 𝔹]. mkibs(n.p[n]) ∈ IBS supposing ∀n:ℕ. ((↑p[n]) ⇒ (↑p[n + 1]))
Definition: rless_ibs
rless_ibs(x;y) == λn.if (∃m∈upto(n + 1).(x (m + 1)) + 4 <z y (m + 1))_b then 1 else 0 fi
Lemma: rless_ibs_wf
∀[x,y:ℝ]. (rless_ibs(x;y) ∈ IBS)
Lemma: rless_ibs_property
∀x,y:ℝ.
((x < y ⇐⇒ ∃n:ℕ. ((rless_ibs(x;y) n) = 1 ∈ ℤ))
∧ (∀n:ℕ
(((rless_ibs(x;y) n) = 0 ∈ ℤ)
⇒ (((y < x) ∧ (∀m:ℕ. ((rless_ibs(x;y) m) = 0 ∈ ℤ))) ∨ (|x - y| ≤ (r(4)/r(n + 1)))))))
Definition: real-fun
real-fun(f;a;b) == ∀x,y:{x:ℝ| x ∈ [a, b]} . ((x = y) ⇒ ((f x) = (f y)))
Lemma: real-fun_wf
∀[a,b:ℝ]. ∀[f:[a, b] ⟶ℝ]. (real-fun(f;a;b) ∈ ℙ)
Definition: real-cont
real-cont(f;a;b) == ∀k:ℕ+. ∃d:{d:ℝ| r0 < d} . ∀x,y:{x:ℝ| x ∈ [a, b]} . ((|x - y| ≤ d) ⇒ (|(f x) - f y| ≤ (r1/r(k))))
Lemma: real-cont_wf
∀[a,b:ℝ]. ∀[f:[a, b] ⟶ℝ]. (real-cont(f;a;b) ∈ ℙ)
Lemma: real-continuity
∀a,b:ℝ. ∀f:[a, b] ⟶ℝ. real-cont(f;a;b) supposing real-fun(f;a;b) supposing a ≤ b
Lemma: stable__real-fun
∀[a,b:ℝ]. ∀[f:[a, b] ⟶ℝ]. Stable{real-fun(f;a;b)}
Lemma: sq_stable__real-fun
∀[a,b:ℝ]. ∀[f:[a, b] ⟶ℝ]. SqStable(real-fun(f;a;b))
Lemma: real-fun-iff-continuous
∀a,b:ℝ. ∀f:[a, b] ⟶ℝ. (real-fun(f;a;b) ⇐⇒ real-cont(f;a;b)) supposing a ≤ b
Lemma: real-cont-iff-continuous
∀a,b:ℝ. ((a ≤ b) ⇒ (∀f:[a, b] ⟶ℝ. (real-cont(f;a;b) ⇐⇒ f[x] continuous for x ∈ [a, b])))
Definition: ifun
ifun(f;I) == real-fun(f;left-endpoint(I);right-endpoint(I))
Lemma: ifun_wf
∀[I:Interval]. ∀[f:I ⟶ℝ]. ifun(f;I) ∈ ℙ supposing icompact(I)
Lemma: ifun-alt
∀I:Interval. ∀[f:I ⟶ℝ]. (ifun(f;I)) supposing ((∀x,y:{x:ℝ| x ∈ I} . ((x = y) ⇒ ((f x) = (f y)))) and icompact(I))
Lemma: sq_stable__ifun
∀[I:Interval]. ∀[f:I ⟶ℝ]. SqStable(ifun(f;I)) supposing icompact(I)
Lemma: ifun_subtype_1
∀[a,b,c:ℝ]. ((a ≤ c) ⇒ (c ≤ b) ⇒ ({f:[a, b] ⟶ℝ| ifun(f;[a, b])} ⊆r {f:[a, c] ⟶ℝ| ifun(f;[a, c])} ))
Lemma: ifun_subtype_2
∀[a,b,c:ℝ]. ((a ≤ c) ⇒ (c ≤ b) ⇒ ({f:[a, b] ⟶ℝ| ifun(f;[a, b])} ⊆r {f:[c, b] ⟶ℝ| ifun(f;[c, b])} ))
Lemma: ifun_subtype_3
∀[a,b,c,d:ℝ]. ((a ≤ c) ⇒ (c ≤ d) ⇒ (d ≤ b) ⇒ ({f:[a, b] ⟶ℝ| ifun(f;[a, b])} ⊆r {f:[c, d] ⟶ℝ| ifun(f;[c, d])} ))
Lemma: ifun_subtype_subinterval
∀[I,J:{J:Interval| icompact(J)} ]. {f:I ⟶ℝ| ifun(f;I)} ⊆r {f:J ⟶ℝ| ifun(f;J)} supposing J ⊆ I
Lemma: ifun-continuous
∀I:Interval. (icompact(I) ⇒ (∀f:{f:I ⟶ℝ| ifun(f;I)} . f[x] continuous for x ∈ I))
Lemma: function-is-continuous
∀I:Interval. ∀f:I ⟶ℝ. ((∀x,y:{t:ℝ| t ∈ I} . ((x = y) ⇒ (f[x] = f[y]))) ⇒ f[x] continuous for x ∈ I)
Lemma: ifun-iff-continuous
∀I:Interval. (icompact(I) ⇒ (∀f:I ⟶ℝ. (ifun(λx.f[x];I) ⇐⇒ f[x] continuous for x ∈ I)))
Definition: real-sfun
real-sfun(f;a;b) == ∀x,y:{x:ℝ| x ∈ [a, b]} . (f x ≠ f y ⇒ x ≠ y)
Lemma: real-sfun_wf
∀[a,b:ℝ]. ∀[f:[a, b] ⟶ℝ]. (real-sfun(f;a;b) ∈ ℙ)
Lemma: real-fun-implies-sfun-general
∀[I:Interval]. ∀[f:I ⟶ℝ].
∀x,y:{x:ℝ| x ∈ I} . (f x ≠ f y ⇒ x ≠ y) supposing ∀x,y:{x:ℝ| x ∈ I} . ((x = y) ⇒ ((f x) = (f y)))
Lemma: strict-increasing-implies-inv-strict-increasing
∀[I:Interval]. ∀[f:I ⟶ℝ].
(∀x,y:{x:ℝ| x ∈ I} . (((f x) < (f y)) ⇒ (x < y))) supposing
((∀x,y:{x:ℝ| x ∈ I} . ((x < y) ⇒ ((f x) < (f y)))) and
(∀x,y:{x:ℝ| x ∈ I} . ((x = y) ⇒ ((f x) = (f y)))))
Lemma: real-fun-implies-sfun
∀[a:ℝ]. ∀[b:{b:ℝ| a ≤ b} ]. ∀[f:[a, b] ⟶ℝ]. real-sfun(f;a;b) supposing real-fun(f;a;b)
Lemma: real-fun-implies-sfun-ext
∀[a:ℝ]. ∀[b:{b:ℝ| a ≤ b} ]. ∀[f:[a, b] ⟶ℝ]. real-sfun(f;a;b) supposing real-fun(f;a;b)
Lemma: function-on-compact
∀a:ℝ. ∀b:{b:ℝ| a ≤ b} . ∀f:[a, b] ⟶ℝ.
((∀x,y:{t:ℝ| t ∈ [a, b]} . ((x = y) ⇒ (f[x] = f[y])))
⇒ (∀n:ℕ+
(∃d:ℝ [((r0 < d)
∧ (∀x,y:ℝ. ((x ∈ [a, b]) ⇒ (y ∈ [a, b]) ⇒ (|x - y| ≤ d) ⇒ (|f[x] - f[y]| ≤ (r1/r(n))))))])))
Definition: discontinuous
discontinuous(f;x) ==
∃epsilon:{e:ℝ| r0 < e} . ∀delta:{e:ℝ| r0 < e} . ∃y:ℝ. ((|x - y| < delta) ∧ (epsilon < |(f x) - f y|))
Lemma: discontinuous_wf
∀[f:ℝ ⟶ ℝ]. ∀[x:ℝ]. (discontinuous(f;x) ∈ ℙ)
Lemma: not-discontinuous
∀[f:ℝ ⟶ ℝ]. ∀[x:ℝ]. (¬discontinuous(f;x)) supposing ∀x,y:ℝ. ((x = y) ⇒ ((f x) = (f y)))
Lemma: extensional-discrete-real-fun-is-constant
∀a,b:ℝ. ∀f:{x:ℝ| x ∈ [a, b]} ⟶ ℤ.
∀x,y:{x:ℝ| x ∈ [a, b]} . ((f x) = (f y) ∈ ℤ) supposing ∀x,y:{x:ℝ| x ∈ [a, b]} . ((x = y) ⇒ ((f x) = (f y) ∈ ℤ))
Lemma: extensional-real-to-bool-constant
∀f:ℝ ⟶ 𝔹. ∀x,y:ℝ. f x = f y supposing ∀x,y:ℝ. ((x = y) ⇒ f x = f y)
Lemma: extensional-interval-to-bool-constant
∀a:ℝ. ∀b:{b:ℝ| a ≤ b} . ∀f:{x:ℝ| x ∈ [a, b]} ⟶ 𝔹.
∀x,y:{x:ℝ| x ∈ [a, b]} . f x = f y supposing ∀x,y:{x:ℝ| x ∈ [a, b]} . ((x = y) ⇒ f x = f y)
Definition: discrete-type
discrete-type(T) == ∀f:ℝ ⟶ T. ((∀x,y:ℝ. ((x = y) ⇒ ((f x) = (f y) ∈ T))) ⇒ (∀x,y:ℝ. ((f x) = (f y) ∈ T)))
Lemma: discrete-type_wf
∀[T:Type]. (discrete-type(T) ∈ ℙ)
Lemma: strong-subtype-discrete-type
∀[A,B:Type]. (discrete-type(A)) supposing (discrete-type(B) and strong-subtype(A;B))
Lemma: int-discrete
discrete-type(ℤ)
Lemma: decidable-equality-implies-discrete
∀[T:Type]. ((∀x,y:T. Dec(x = y ∈ T)) ⇒ discrete-type(T))
Lemma: product-discrete
∀A:Type. ∀B:A ⟶ Type. (discrete-type(A) ⇒ (∀a:A. discrete-type(B[a])) ⇒ discrete-type(a:A × B[a]))
Lemma: prod-discrete
∀A,B:Type. (discrete-type(A) ⇒ discrete-type(B) ⇒ discrete-type(A × B))
Lemma: function-discrete
∀A:Type. ∀B:A ⟶ Type. ((∀a:A. discrete-type(B[a])) ⇒ discrete-type(a:A ⟶ B[a]))
Lemma: fun-discrete
∀A,B:Type. (discrete-type(B) ⇒ discrete-type(A ⟶ B))
Lemma: union-discrete
∀A,B:Type. (discrete-type(A) ⇒ discrete-type(B) ⇒ discrete-type(A + B))
Lemma: partial-int-not-discrete
¬discrete-type(partial(ℤ))
Lemma: base-not-discrete
¬discrete-type(Base)
Definition: real-disjoint
real-disjoint(x.A[x];y.B[y]) == ∀x,y:ℝ. ((x = y) ⇒ (¬(A[x] ∧ B[y])))
Lemma: real-disjoint_wf
∀[A,B:ℝ ⟶ ℙ]. (real-disjoint(x.A[x];y.B[y]) ∈ ℙ)
Definition: real-separation
real-separation(x.A[x];y.B[y]) == real-disjoint(x.A[x];y.B[y]) ∧ (∃x:ℝ. A[x]) ∧ (∃y:ℝ. B[y]) ∧ (∀r:ℝ. (A[r] ∨ B[r]))
Lemma: real-separation_wf
∀[A,B:ℝ ⟶ ℙ]. (real-separation(x.A[x];y.B[y]) ∈ ℙ)
Lemma: no-real-separation
∀[A,B:ℝ ⟶ ℙ]. (¬real-separation(x.A[x];y.B[y]))
Lemma: no-real-separation-corollary
∀[A,B:ℝ ⟶ ℙ]. ((∃x:ℝ. A[x]) ⇒ (∃y:ℝ. B[y]) ⇒ (∀r:ℝ. (A[r] ∨ B[r])) ⇒ (¬¬(∃x,y:ℝ. ((x = y) ∧ A[x] ∧ B[y]))))
Lemma: no-nontrivial-decidable-real-prop
∀[A:ℝ ⟶ ℙ]. ((∀x,y:ℝ. ((x = y) ⇒ (A[x] ⇐⇒ A[y]))) ⇒ (∀r:ℝ. (A[r] ∨ (¬A[r]))) ⇒ ((∀x:ℝ. A[x]) ∨ (∀x:ℝ. (¬A[x]))))
Lemma: separated-decider-not-extensional
∀a,b:ℝ. ((a < b) ⇒ (∀d:∀u:ℝ. ((a < u) ∨ (u < b)). (¬¬(∃x,y:ℝ. ((x = y) ∧ (↑isl(d x)) ∧ (↑isr(d y)))))))
Lemma: separated-decider-not-extensional2
∀a,b:ℝ. ((a < b) ⇒ (∀d:∀u:ℝ. ((a < u) ∨ (u < b)). (¬(∀x,y:ℝ. ((x = y) ⇒ isl(d x) = isl(d y))))))
Definition: cover-seq
cover-seq(d;a;b;n) ==
primrec(n;<a, b>;λi,p. let a,b = p in case d (a + b/r(2)) of inl(z) => <(a + b/r(2)), b> | inr(z) => <a, (a + b/r(2))>\000C)
Lemma: cover-seq_wf
∀[A,B:ℝ ⟶ ℙ]. ∀[d:r:ℝ ⟶ (A[r] + B[r])]. ∀[a,b:ℝ]. ∀[n:ℕ]. (cover-seq(d;a;b;n) ∈ ℝ × ℝ)
Lemma: cover-seq-0
∀[d,a,b:Top]. (cover-seq(d;a;b;0) ~ <a, b>)
Lemma: cover-seq-property
∀[A,B:ℝ ⟶ ℙ].
∀d:r:ℝ ⟶ (A[r] + B[r]). ∀a,b:ℝ.
(A[a]
⇒ B[b]
⇒ (∀n:ℕ
(A[fst(cover-seq(d;a;b;n))]
∧ B[snd(cover-seq(d;a;b;n))]
∧ ((cover-seq(d;a;b;n + 1) = let a,b = cover-seq(d;a;b;n) in <a, (a + b/r(2))> ∈ (ℝ × ℝ))
∨ (cover-seq(d;a;b;n + 1) = let a,b = cover-seq(d;a;b;n) in <(a + b/r(2)), b> ∈ (ℝ × ℝ))))))
Lemma: cover-seq-property-ext
∀[A,B:ℝ ⟶ ℙ].
∀d:r:ℝ ⟶ (A[r] + B[r]). ∀a,b:ℝ.
(A[a]
⇒ B[b]
⇒ (∀n:ℕ
(A[fst(cover-seq(d;a;b;n))]
∧ B[snd(cover-seq(d;a;b;n))]
∧ ((cover-seq(d;a;b;n + 1) = let a,b = cover-seq(d;a;b;n) in <a, (a + b/r(2))> ∈ (ℝ × ℝ))
∨ (cover-seq(d;a;b;n + 1) = let a,b = cover-seq(d;a;b;n) in <(a + b/r(2)), b> ∈ (ℝ × ℝ))))))
Lemma: inhabited-covers-reals-implies
∀[A,B:ℝ ⟶ ℙ].
((∃a:ℝ. A[a])
⇒ (∃b:ℝ. B[b])
⇒ (∀r:ℝ. (A[r] ∨ B[r]))
⇒ (∃f,g:ℕ ⟶ ℝ. ∃x:ℝ. ((∀n:ℕ. A[f n]) ∧ (∀n:ℕ. B[g n]) ∧ lim n→∞.f n = x ∧ lim n→∞.g n = x)))
Lemma: inhabited-covers-real-implies
∀[A,B:ℝ ⟶ ℙ].
((∃a:ℝ. A[a])
⇒ (∃b:ℝ. B[b])
⇒ (∀r:ℝ. (A[r] ∨ B[r]))
⇒ (∃f,g:ℕ ⟶ ℝ. ∃x:ℝ. ((∀n:ℕ. A[f n]) ∧ (∀n:ℕ. B[g n]) ∧ lim n→∞.f n = x ∧ lim n→∞.g n = x)))
Definition: cover-real
cover-real(d; a; b; cb) == accelerate(2;λn.((fst(cover-seq(d;a;b;log(2;n * (cb + 1))))) n))
Lemma: inhabited-covers-real-implies-ext
∀[A,B:ℝ ⟶ ℙ].
((∃a:ℝ. A[a])
⇒ (∃b:ℝ. B[b])
⇒ (∀r:ℝ. (A[r] ∨ B[r]))
⇒ (∃f,g:ℕ ⟶ ℝ. ∃x:ℝ. ((∀n:ℕ. A[f n]) ∧ (∀n:ℕ. B[g n]) ∧ lim n→∞.f n = x ∧ lim n→∞.g n = x)))
Definition: interval-retraction
interval-retraction(u;v;r) == rmin(v;rmax(u;r))
Lemma: interval-retraction_wf
∀[u,v,x:ℝ]. interval-retraction(u;v;x) ∈ {x:ℝ| x ∈ [u, v]} supposing u ≤ v
Lemma: interval-retraction_functionality
∀[u,v,x,u',v',x':ℝ].
(interval-retraction(u;v;x) = interval-retraction(u';v';x')) supposing ((x = x') and (v = v') and (u = u'))
Lemma: interval-retraction-req
∀[u,v:ℝ]. ∀[x:{x:ℝ| x ∈ [u, v]} ]. (interval-retraction(u;v;x) = x)
Lemma: weak-continuity-principle-interval
∀u,v:ℝ. ∀x:{x:ℝ| x ∈ [u, v]} .
∃x':{x':ℝ| x' = x}
∀F:{x:ℝ| x ∈ [u, v]} ⟶ 𝔹. ∀G:n:ℕ+ ⟶ {y:ℝ| (x = y ∈ (ℕ+n ⟶ ℤ)) ∧ (y ∈ [u, v])} .
∃z:{x:ℝ| x ∈ [u, v]} . (∃n:ℕ+ [(F x' = F z ∧ (z = (G n)))])
Definition: WCPR
WCPR(F;x;G) == WCP(λf.(F regularize(1;f));x;G)
Lemma: WCPR_wf
∀[F:ℝ ⟶ 𝔹]. ∀[x:ℝ]. ∀[G:n:ℕ+ ⟶ {y:ℝ| x = y ∈ (ℕ+n ⟶ ℤ)} ]. (WCPR(F;x;G) ∈ ℕ+)
Lemma: weak-continuity-principle-real-ext
∀x:ℝ. ∀F:ℝ ⟶ 𝔹. ∀G:n:ℕ+ ⟶ {y:ℝ| x = y ∈ (ℕ+n ⟶ ℤ)} . (∃n:ℕ+ [F x = F (G n)])
Lemma: int-int-retraction-reals-1
∀k:{2...}. ∃r:(ℤ ⟶ ℤ) ⟶ ℝ. ∀x:ℝ. (accelerate(k;x) = (r (λi.if i <z 1 then x 1 else x i fi )) ∈ ℝ)
Lemma: nat-int-retraction-reals-1
∀k:{2...}. ∃r:(ℕ ⟶ ℤ) ⟶ ℝ. ∀x:ℝ. (accelerate(k;x) = (r (λn.(x (n + 1)))) ∈ ℝ)
Lemma: int-int-retraction-reals
∃r:(ℤ ⟶ ℤ) ⟶ ℝ. ∀x:ℝ. (x = (r (λi.if i <z 1 then x 1 else x i fi )))
Lemma: nat-int-retraction-reals
∃r:(ℕ ⟶ ℤ) ⟶ ℝ. ∀x:ℝ. (x = (r (λn.(x (n + 1)))))
Definition: blend-seq
blend-seq(k;x;y) == λi.if i <z k then x i else y i fi
Lemma: blend-seq_wf
∀[k:ℕ+]. ∀[x,y:ℕ+ ⟶ ℤ]. (blend-seq(k;x;y) ∈ ℕ+ ⟶ ℤ)
Lemma: blend-close-reals
∀[k:ℕ+]. ∀[x,y:ℝ]. ((|x - y| ≤ (r1/r(k))) ⇒ 3-regular-seq(blend-seq(k;x;y)))
Definition: blended-real
blended-real(k;x;y) == accelerate(3;blend-seq(k;x;y))
Lemma: blended-real_wf
∀[k:ℕ+]. ∀[x,y:ℝ]. blended-real(k;x;y) ∈ ℝ supposing |x - y| ≤ (r1/r(k))
Lemma: blended-real-req
∀[k:ℕ+]. ∀[x,y:ℝ]. blended-real(k;x;y) = y supposing |x - y| ≤ (r1/r(k))
Lemma: blended-real-agrees
∀[k:ℕ+]. ∀[x,y:ℝ]. ∀n:ℕ+k ÷ 6. ((blended-real(k;x;y) n) = (accelerate(3;x) n) ∈ ℤ)
Lemma: connectedness-main-lemma
∀x:ℝ. ∀g:ℕ ⟶ ℝ. (lim n→∞.g n = x ⇒ (∀P:ℝ ⟶ 𝔹. ∃z:{z:ℝ| P z = P accelerate(3;x)} . (∃n:ℕ [(z = (g n))])))
Lemma: better-continuity-for-reals
∀x:ℝ. ∃x':{x':ℝ| x' = x} . ∀g:ℕ ⟶ ℝ. (lim n→∞.g n = x ⇒ (∀P:ℝ ⟶ 𝔹. ∃z:{z:ℝ| P z = P x'} . (∃n:ℕ [(z = (g n))])))
Lemma: not-all-nonneg-or-nonpos
¬(∀x:ℝ. ((r0 ≤ x) ∨ (x ≤ r0)))
Lemma: union-of-intervals-not-interval
¬(∀t:{t:ℝ| t ∈ [r(-1), r1]} . ((↓t ∈ [r(-1), r0]) ∨ (↓t ∈ [r0, r1])))
Lemma: dense-in-reals-iff
∀X:ℝ ⟶ ℙ. (dense-in-interval((-∞, ∞);X) ⇐⇒ ∀x:ℝ. ∀n:ℕ+. ∃y:ℝ. ((X y) ∧ (|x - y| < (r1/r(n)))))
Lemma: dense-in-reals-implies-accel
∀X:ℝ ⟶ ℙ
(dense-in-interval((-∞, ∞);X)
⇒ (∀x:ℝ. ∀y:{y:ℝ| y = x} . ((X x) ⇒ (X y)))
⇒ (∀x:ℝ. ∀k:ℕ+. ∃y:ℝ. ((y = accelerate(3;x) ∈ (ℕ+k ⟶ ℤ)) ∧ (X y))))
Lemma: dense-in-reals-implies
∀X:ℝ ⟶ ℙ
(dense-in-interval((-∞, ∞);X)
⇒ (∀x:ℝ. ∀y:{y:ℝ| y = x} . ((X x) ⇒ (X y)))
⇒ (∀x:ℝ. ∃x':ℝ. ((x' = x) ∧ (∀k:ℕ+. ∃y:ℝ. ((y = x' ∈ (ℕ+k ⟶ ℤ)) ∧ (X y))))))
Lemma: connectedness-main-lemma-ext
∀x:ℝ. ∀g:ℕ ⟶ ℝ. (lim n→∞.g n = x ⇒ (∀P:ℝ ⟶ 𝔹. ∃z:{z:ℝ| P z = P accelerate(3;x)} . (∃n:ℕ [(z = (g n))])))
Definition: connected
Connected(X) ==
∀[A,B:X ⟶ ℙ].
((∀x:X. ∀y:{y:X| x = y} . (A[y] ⇒ A[x]))
⇒ (∀x:X. ∀y:{y:X| x = y} . (B[y] ⇒ B[x]))
⇒ (∃a:X. A[a])
⇒ (∃b:X. B[b])
⇒ (∀r:X. (A[r] ∨ B[r]))
⇒ (∃r:X. (A[r] ∧ B[r])))
Lemma: connected_wf
∀[X:Type]. Connected(X) ∈ ℙ' supposing X ⊆r ℝ
Lemma: reals-connected
Connected(ℝ)
Lemma: closed-interval-connected
∀u,v:ℝ. Connected({x:ℝ| x ∈ [u, v]} )
Lemma: IVT-from-connected
∀u,v:ℝ. ∀f:[u, v] ⟶ℝ.
((u ≤ v)
⇒ (∀x,y:{x:ℝ| x ∈ [u, v]} . ((x = y) ⇒ (f(x) = f(y))))
⇒ (f(u) < r0)
⇒ (r0 < f(v))
⇒ (∀e:{e:ℝ| r0 < e} . ∃c:{c:ℝ| c ∈ [u, v]} . (|f(c)| < e)))
Lemma: interval-connected
∀I:Interval. Connected({x:ℝ| x ∈ I} )
Lemma: real-subset-connected-lemma
∀X:ℝ ⟶ ℙ
(dense-in-interval((-∞, ∞);X)
⇒ (∀a,b:{x:ℝ| X x} ⟶ 𝔹.
((∃x:{x:ℝ| X x} . (↑(a x)))
⇒ (∃x:{x:ℝ| X x} . (↑(b x)))
⇒ (∀x:{x:ℝ| X x} . ((↑(a x)) ∨ (↑(b x))))
⇒ (∃f,g:ℕ ⟶ {x:ℝ| X x}
∃x:ℝ. ((∀n:ℕ. (↑(a (f n)))) ∧ (∀n:ℕ. (↑(b (g n)))) ∧ lim n→∞.f n = x ∧ lim n→∞.g n = x)))))
Lemma: real-subset-connected
∀X:ℝ ⟶ ℙ
((∀x:ℝ. SqStable(X x))
⇒ (∀x:ℝ. ∀y:{y:ℝ| x = y} . ((X y) ⇒ (X x)))
⇒ dense-in-interval((-∞, ∞);X)
⇒ (∀Q:{x:ℝ| X x} ⟶ 𝔹. ∃Q':ℝ ⟶ 𝔹. ∀x:{x:ℝ| X x} . Q' x = Q x)
⇒ Connected({x:ℝ| X x} ))
Definition: VesleySchema
VesleySchema ==
∀P:(ℕ+ ⟶ ℤ) ⟶ ℙ
((∀f:ℕ+ ⟶ ℤ. ∀k:ℕ+. ∃g:{x:ℕ+ ⟶ ℤ| ¬(P x)} . (g = f ∈ (ℕ+k ⟶ ℤ)))
⇒ (∀Q:{x:ℕ+ ⟶ ℤ| ¬(P x)} ⟶ 𝔹. ∃Q':(ℕ+ ⟶ ℤ) ⟶ 𝔹. ∀x:{x:ℕ+ ⟶ ℤ| ¬(P x)} . Q' x = Q x))
Lemma: VesleySchema_wf
VesleySchema ∈ ℙ'
Definition: VesleySchema2
VesleySchema2 ==
∀P:(ℕ+ ⟶ ℤ) ⟶ ℙ
((∀f:ℕ+ ⟶ ℤ. ∀k:ℕ+. ∃g:{x:ℕ+ ⟶ ℤ| ¬(P x)} . (g = regularize(1;f) ∈ (ℕ+k ⟶ ℤ)))
⇒ (∀Q:{x:ℕ+ ⟶ ℤ| ¬(P x)} ⟶ 𝔹. ∃Q':(ℕ+ ⟶ ℤ) ⟶ 𝔹. ∀x:{x:ℕ+ ⟶ ℤ| ¬(P x)} . Q' x = Q x))
Lemma: VesleySchema2_wf
VesleySchema2 ∈ ℙ'
Definition: VesleyAxiom
VesleyAxiom ==
∀P:ℝ ⟶ ℙ
(dense-in-interval((-∞, ∞);λx.(¬(P x)))
⇒ (∀x:ℝ. ∀y:{y:ℝ| x = y} . ((P y) ⇒ (P x)))
⇒ (∀Q:{x:ℝ| ¬(P x)} ⟶ 𝔹. ∃Q':ℝ ⟶ 𝔹. ∀x:{x:ℝ| ¬(P x)} . Q' x = Q x))
Lemma: VesleyAxiom_wf
VesleyAxiom ∈ ℙ'
Lemma: Vesley-subset-connected
VesleyAxiom
⇒ (∀P:ℝ ⟶ ℙ
((∀x:ℝ. ∀y:{y:ℝ| x = y} . (P[y] ⇒ P[x])) ⇒ dense-in-interval((-∞, ∞);λx.(¬P[x])) ⇒ Connected({x:ℝ| ¬P[x]} )))
Lemma: Vesley-connected-rationals
VesleyAxiom ⇒ Connected({x:ℝ| ¬¬(∃a:ℤ. ∃b:ℤ-o. (x = (r(a)/r(b))))} )
Definition: KripkeSchema
KripkeSchema == ∀P:ℙ. ∃f:ℕ ⟶ ℕ. (((∀n:ℕ. ((f n) = 0 ∈ ℕ)) ⇒ (¬P)) ∧ ((∃n:ℕ. 0 < f n) ⇒ P))
Lemma: KripkeSchema_wf
KripkeSchema ∈ ℙ'
Definition: decdr-to-bool
bool(d) == λx.if d x then inl Ax else inr Ax fi
Lemma: decdr-to-bool_wf
∀[T:Type]. ∀[A,B:T ⟶ ℙ]. ∀[d:x:T ⟶ (A[x] + B[x])]. (bool(d) ∈ T ⟶ 𝔹)
Definition: cover-search-left
cover-search-left(d;a;b;x) ==
WCPR(λx.isl(d x);accelerate(3;cover-real(bool(d); a; b; x
));λk.blended-real(6
* k;cover-real(bool(d); a; b; x
);snd(cover-seq(bool(d);a;b;imax(log(2;(4 * 2 * 6 * k)
* (x + 1))
+ 1;log(2;(2 * 6 * k)
* (x + 1)))))))
Definition: cover-search-right
cover-search-right(d;a;b;x) ==
6
* WCPR(λx.isr(d x);accelerate(3;cover-real(bool(d); a; b; x
));λk.blended-real(6 * k;cover-real(bool(d); a; b; x
);fst(cover-seq(bool(d);a;b;log(2;(4
* 6
* k)
* (x + 1))
+ 1))))
Definition: altered-seq1
altered-seq1(d; a; b; x; n) == blended-real(6 * cover-search-left(d;a;b;x);cover-real(bool(d); a; b; x);n)
Definition: altered-seq2
altered-seq2(d; a; b; x; n) == blended-real(cover-search-right(d;a;b;x);cover-real(bool(d); a; b; x);n)
Lemma: reals-connected-ext
Connected(ℝ)
Definition: non-rational
non-rational() == {x:ℝ| ∀a:ℤ. ∀b:ℕ+. (¬(x = (r(a)/r(b))))}
Lemma: non-rational_wf
non-rational() ∈ Type
Definition: real-continuity-principle
real-continuity-principle() == ∀I:Interval. ∀f:I ⟶ℝ. f[x] continuous for x ∈ I
Lemma: real-continuity-principle_wf
real-continuity-principle() ∈ ℙ
Definition: proper-real-continuity
proper-real-continuity() ==
∀a,b:ℝ.
((a < b)
⇒ (∀f:[a, b] ⟶ℝ. ∀n:ℕ+.
(∃d:ℝ [((r0 < d)
∧ (∀x,y:ℝ. ((x ∈ [a, b]) ⇒ (y ∈ [a, b]) ⇒ (|x - y| ≤ d) ⇒ (|(f x) - f y| ≤ (r1/r(n))))))])))
Lemma: proper-real-continuity_wf
proper-real-continuity() ∈ ℙ
Lemma: cantor-middle-third-lemma
∀x,a,b:ℝ. ((x ∈ [(2 * a + b)/3, b]) ∨ (x ∈ [a, (a + 2 * b)/3])) supposing ((x ∈ [a, b]) and (a < b))
Lemma: cantor-common-middle-third-lemma
∀x,y,a,b:ℝ.
(((x ∈ [(2 * a + b)/3, b]) ∧ (y ∈ [(2 * a + b)/3, b]))
∨ ((x ∈ [a, (a + 2 * b)/3]) ∧ (y ∈ [a, (a + 2 * b)/3]))) supposing
(((x ∈ [a, b]) ∧ (y ∈ [a, b]) ∧ (|x - y| ≤ (b - a/r(6)))) and
(a < b))
Definition: unit-interval-fan
unit-interval-fan(f;n) ==
primrec(n;<0, 1>;λi,p. let a,b = p
in if f i
then eval a' = (2 * a) + b in
eval b' = 3 * b in
<a', b'>
else eval a' = 3 * a in
eval b' = a + (2 * b) in
<a', b'>
fi )
Lemma: unit-interval-fan_wf
∀[f:ℕ ⟶ 𝔹]. ∀[n:ℕ]. (unit-interval-fan(f;n) ∈ ℤ × ℤ)
Definition: cantor_ivl
cantor_ivl(a;b;f;n) ==
let k,j = unit-interval-fan(f;n)
in eval d = 3^n in
<eval k' = d - k in (k' * a + k * b)/d, eval j' = d - j in (j' * a + j * b)/d>
Lemma: cantor_ivl_wf
∀[a,b:ℝ]. ∀[f:ℕ ⟶ 𝔹]. ∀[n:ℕ]. (cantor_ivl(a;b;f;n) ∈ ℝ × ℝ)
Definition: cantor-interval
cantor-interval(a;b;f;n) ==
primrec(n;<a, b>;λi,p. let a',b' = p in if f i then <(2 * a' + b')/3, b'> else <a', (a' + 2 * b')/3> fi )
Lemma: cantor-interval_wf
∀[a,b:ℝ]. ∀[n:ℕ]. ∀[f:ℕn ⟶ 𝔹]. (cantor-interval(a;b;f;n) ∈ ℝ × ℝ)
Lemma: cantor-interval-length
∀[a,b:ℝ]. ∀[n:ℕ]. ∀[f:ℕn ⟶ 𝔹]. (((snd(cantor-interval(a;b;f;n))) - fst(cantor-interval(a;b;f;n))) = (2^n * b - a)/3^n)
Lemma: cantor-interval-rleq
∀[a,b:ℝ]. ∀[n:ℕ]. ∀[f:ℕn ⟶ 𝔹]. ((fst(cantor-interval(a;b;f;n))) ≤ (snd(cantor-interval(a;b;f;n)))) supposing a ≤ b
Lemma: cantor-interval-rless
∀a,b:ℝ. ((a < b) ⇒ (∀n:ℕ. ∀f:ℕn ⟶ 𝔹. ((fst(cantor-interval(a;b;f;n))) < (snd(cantor-interval(a;b;f;n))))))
Lemma: cantor-interval-inclusion
∀[a,b:ℝ].
∀[f:ℕ ⟶ 𝔹]. ∀[n:ℕ]. ∀[m:{n...}].
(((fst(cantor-interval(a;b;f;n))) ≤ (fst(cantor-interval(a;b;f;m))))
∧ ((fst(cantor-interval(a;b;f;m))) ≤ (snd(cantor-interval(a;b;f;m))))
∧ ((snd(cantor-interval(a;b;f;m))) ≤ (snd(cantor-interval(a;b;f;n)))))
supposing a ≤ b
Lemma: cantor-interval-inclusion2
∀[a,b:ℝ].
∀[m:ℕ]. ∀[n:ℕm]. ∀[f:ℕm ⟶ 𝔹].
(((fst(cantor-interval(a;b;f;n))) ≤ (fst(cantor-interval(a;b;f;m))))
∧ ((fst(cantor-interval(a;b;f;m))) ≤ (snd(cantor-interval(a;b;f;m))))
∧ ((snd(cantor-interval(a;b;f;m))) ≤ (snd(cantor-interval(a;b;f;n)))))
supposing a ≤ b
Lemma: cantor-interval-cauchy
∀a,b:ℝ. ∀[f:ℕ ⟶ 𝔹]. cauchy(n.fst(cantor-interval(a;b;f;n))) supposing a ≤ b
Definition: cantor_cauchy
cantor_cauchy(a;b;k) ==
eval x = eval y' = 0 + (b 4) in
eval y' = y' + (-(a 4)) in
y' ÷ 4 in
if (0) < (x)
then if (x + 4) ÷ 2=0 then 1 else (2 * log(2;((x + 4) ÷ 2) * k))
else if ((-x) + 4) ÷ 2=0 then 1 else (2 * log(2;(((-x) + 4) ÷ 2) * k))
Lemma: cantor-interval-cauchy-ext
∀a,b:ℝ. ∀[f:ℕ ⟶ 𝔹]. cauchy(n.fst(cantor-interval(a;b;f;n))) supposing a ≤ b
Lemma: cantor-interval-converges
∀a,b:ℝ. ∀f:ℕ ⟶ 𝔹. fst(cantor-interval(a;b;f;n))↓ as n→∞ supposing a ≤ b
Lemma: cantor-interval-req
∀a,b:ℝ. ∀f:ℕ ⟶ 𝔹. ∀n:ℕ.
(((fst(cantor-interval(a;b;f;n))) = (fst(cantor_ivl(a;b;f;n))))
∧ ((snd(cantor-interval(a;b;f;n))) = (snd(cantor_ivl(a;b;f;n)))))
Lemma: cantor_ivl-converges
∀a,b:ℝ. ∀f:ℕ ⟶ 𝔹. fst(cantor_ivl(a;b;f;n))↓ as n→∞ supposing a ≤ b
Lemma: cantor-interval-converges-ext
∀a,b:ℝ. ∀f:ℕ ⟶ 𝔹. fst(cantor-interval(a;b;f;n))↓ as n→∞ supposing a ≤ b
Lemma: cantor_ivl-converges-ext
∀a,b:ℝ. ∀f:ℕ ⟶ 𝔹. fst(cantor_ivl(a;b;f;n))↓ as n→∞ supposing a ≤ b
Definition: cantor-to-interval
cantor-to-interval(a;b;f) == λn.eval m = 4 * n in let x,y = cantor-interval(a;b;f;cantor_cauchy(a;b;m)) in (x m) ÷ 4
Lemma: cantor-to-interval_wf1
∀a,b:ℝ. ∀f:ℕ ⟶ 𝔹. (cantor-to-interval(a;b;f) ∈ {x:ℝ| lim n→∞.fst(cantor-interval(a;b;f;n)) = x} ) supposing a ≤ b
Lemma: cantor-to-interval_wf
∀a,b:ℝ. ∀f:ℕ ⟶ 𝔹. (cantor-to-interval(a;b;f) ∈ {x:ℝ| (a ≤ x) ∧ (x ≤ b)} ) supposing a ≤ b
Lemma: cantor-to-interval-req
∀a,b,x:ℝ. ∀f:ℕ ⟶ 𝔹.
((∀n:ℕ. (x ∈ [fst(cantor-interval(a;b;f;n)), snd(cantor-interval(a;b;f;n))])) ⇒ (cantor-to-interval(a;b;f) = x))
Lemma: cantor-to-interval-onto-lemma
∀a,b:ℝ.
∀x:ℝ. ∀n:ℕ. ∀f:{f:ℕn ⟶ 𝔹| x ∈ [fst(cantor-interval(a;b;f;n)), snd(cantor-interval(a;b;f;n))]} .
∃g:{g:ℕn + 1 ⟶ 𝔹| x ∈ [fst(cantor-interval(a;b;g;n + 1)), snd(cantor-interval(a;b;g;n + 1))]} . (g = f ∈ (ℕn ⟶ 𝔹))\000C
supposing a < b
Lemma: cantor-to-interval-onto-proper
∀a,b:ℝ. ∀x:ℝ. ((a ≤ x) ⇒ (x ≤ b) ⇒ (∃f:ℕ ⟶ 𝔹. (cantor-to-interval(a;b;f) = x))) supposing a < b
Lemma: cantor-to-interval-onto-common
∀a,b:ℝ.
∀x,y:ℝ.
((x ∈ [a, b])
⇒ (y ∈ [a, b])
⇒ (∀n:ℕ
((|x - y| ≤ (2^n * b - a)/6 * 3^n)
⇒ (∃f,g:ℕ ⟶ 𝔹
(((cantor-to-interval(a;b;f) = x) ∧ (cantor-to-interval(a;b;g) = y)) ∧ (f = g ∈ (ℕn ⟶ 𝔹)))))))
supposing a < b
Lemma: proper-interval-to-int-bounded
∀a,b:ℝ.
∀f:{x:ℝ| x ∈ [a, b]} ⟶ ℤ. ∃B:ℕ. ∀x:{x:ℝ| x ∈ [a, b]} . ∃y:{x:ℝ| x ∈ [a, b]} . ((x = y) ∧ (|f y| ≤ B)) supposing a < \000Cb
Definition: cantor_to_interval
cantor_to_interval(a;b;f) == a + ((b - a/r1 + (b - a)) * (cantor-to-interval(a;b + r1;f) - a))
Lemma: cantor_to_interval_wf
∀[a:ℝ]. ∀[b:{b:ℝ| a ≤ b} ]. ∀[f:ℕ ⟶ 𝔹]. (cantor_to_interval(a;b;f) ∈ {x:ℝ| x ∈ [a, b]} )
Lemma: old-proof-of-real-continuity
∀a,b:ℝ. ∀f:[a, b] ⟶ℝ. real-cont(f;a;b) supposing real-fun(f;a;b) supposing a ≤ b
Definition: real-cont-ps
real-cont-ps(k;a;b;f;x;N) ==
uniform-continuity-pi-search(
λm@0.(primrec(x;λ%4,m,%5. ⋅;λi,x,%7,m,%8. case FiniteCantorDecide(λs.if f
cantor-to-interval(a;b;λx.if (x) < ((i + 1) - 1)
then s x
else tt)
(N * k)=f
cantor-to-interval(a;b;λx.if (x) < ((i + 1)
- 1)
then s x
else ff)
(N * k)
then inr (λ_.⋅)
else (inl (λ_.⋅));(i + 1) - 1;λx.x)
of inl(x1) =>
let x1,f = x1
in inr (λ%14.⋅)
| inr(x1) =>
if m=(i + 1) - 1 then inl (λf,g,%3. ⋅) else (x (λf,g,%3. ⋅) m ⋅))
(λf,g,eq. ⋅)
m@0
⋅);
x;0)
Definition: real-cont-br
real-cont-br(a; b; f; k; N) ==
FAN(λn,s. if primrec(n;eval v = Kleene-M(λf@0.int2nat(f cantor-to-interval(a;b;λx.if f@0 x=0 then ff else tt)
(N * k)))
n
(λx.if s x then 1 else 0 fi ) in
if v is an integer then inl v
else inr v ;λi,r. eval v = Kleene-M(λf@0.int2nat(f
cantor-to-interval(a;b;λx.if f@0 x=0
then ff
else tt)
(N * k)))
i
(λx.if s x then 1 else 0 fi ) in
if v is an integer then ff
else r)
then tt
else inr (λx.⋅)
fi )
Lemma: function-proper-continuous
∀I:Interval. ∀f:I ⟶ℝ.
((iproper(I) ⇒ (∀x,y:{t:ℝ| t ∈ I} . ((x = y) ⇒ (f[x] = f[y])))) ⇒ f[x] (proper)continuous for x ∈ I)
Lemma: proper-continuous-is-continuous
∀I:Interval. (iproper(I) ⇒ (∀f:I ⟶ℝ. (f[x] (proper)continuous for x ∈ I ⇒ f[x] continuous for x ∈ I)))
Lemma: countable-Heine-Borel-proper
∀a:ℝ. ∀b:{b:ℝ| a < b} .
∀[C:ℕ ⟶ {x:ℝ| x ∈ [a, b]} ⟶ ℙ]
((∀n:ℕ. ∀x:{x:ℝ| x ∈ [a, b]} . ∀y:{y:{x:ℝ| x ∈ [a, b]} | x = y} . (C[n;x] ⇒ C[n;y]))
⇒ (∀x:{x:ℝ| x ∈ [a, b]} . ∃n:ℕ. C[n;x])
⇒ (∃k:ℕ. ∀x:{x:ℝ| x ∈ [a, b]} . ∃n:ℕk. C[n;x]))
Lemma: countable-Heine-Borel-weak
∀a:ℝ. ∀b:{b:ℝ| a ≤ b} .
∀[C:ℕ ⟶ {x:ℝ| x ∈ [a, b]} ⟶ ℙ]
((∀n:ℕ. ∀x:{x:ℝ| x ∈ [a, b]} . ∀y:{y:{x:ℝ| x ∈ [a, b]} | x = y} . (C[n;x] ⇒ C[n;y]))
⇒ (∀x:{x:ℝ| x ∈ [a, b]} . ∃n:ℕ. C[n;x])
⇒ (∃k:ℕ. ∀x:{x:ℝ| x ∈ [a, b]} . (¬¬(∃n:ℕk. C[n;x]))))
Lemma: continuous-implies-functional
∀I:Interval. ∀f:I ⟶ℝ. (f[x] continuous for x ∈ I ⇒ (∀a,b:{x:ℝ| x ∈ I} . ((a = b) ⇒ (f[a] = f[b]))))
Lemma: continuous_functionality_wrt_subinterval
∀I:Interval. ∀[f:I ⟶ℝ]. ∀J:Interval. (J ⊆ I ⇒ f[x] continuous for x ∈ I ⇒ f[x] continuous for x ∈ J)
Lemma: continuous_functionality_wrt_rfun-eq
∀I:Interval. ∀[f1,f2:I ⟶ℝ]. (rfun-eq(I;λx.f1[x];λx.f2[x]) ⇒ f1[x] continuous for x ∈ I ⇒ f2[x] continuous for x ∈ I)
Lemma: proper-continuous-implies-functional
∀I:Interval. ∀f:I ⟶ℝ.
(f[x] (proper)continuous for x ∈ I ⇒ iproper(I) ⇒ (∀a,b:{x:ℝ| x ∈ I} . ((a = b) ⇒ (f[a] = f[b]))))
Lemma: continuous-limit
∀I:Interval. ∀f:I ⟶ℝ. ∀y:ℝ. ∀x:ℕ ⟶ ℝ.
(f(x) continuous for x ∈ I ⇒ lim n→∞.x[n] = y ⇒ (y ∈ I) ⇒ (∀n:ℕ. (x[n] ∈ I)) ⇒ lim n→∞.f(x[n]) = f(y))
Lemma: function-limit
∀I:Interval. ∀f:I ⟶ℝ. ∀y:ℝ. ∀x:ℕ ⟶ ℝ.
((∀x,y:{x:ℝ| x ∈ I} . ((x = y) ⇒ (f(x) = f(y))))
⇒ lim n→∞.x[n] = y
⇒ (y ∈ I)
⇒ (∀n:ℕ. (x[n] ∈ I))
⇒ lim n→∞.f(x[n]) = f(y))
Lemma: dense-in-interval-implies
∀I:Interval
∀[X:{a:ℝ| a ∈ I} ⟶ ℙ]
(dense-in-interval(I;X)
⇒ (∃u,v:{a:ℝ| a ∈ I} . u ≠ v)
⇒ (∀a:{a:ℝ| a ∈ I} . ∃x:ℕ ⟶ {a:ℝ| a ∈ I} . ((∀n:ℕ. (X (x n))) ∧ lim n→∞.x n = a)))
Lemma: functions-equal-on-dense
∀I:Interval
∀[X:{a:ℝ| a ∈ I} ⟶ ℙ]
(dense-in-interval(I;X)
⇒ (∃u,v:{a:ℝ| a ∈ I} . u ≠ v)
⇒ (∀f,g:I ⟶ℝ.
((∀x,y:{x:ℝ| x ∈ I} . ((x = y) ⇒ (f(x) = f(y))))
⇒ (∀x,y:{x:ℝ| x ∈ I} . ((x = y) ⇒ (g(x) = g(y))))
⇒ (∀x:{x:ℝ| x ∈ I} . ((X x) ⇒ (f(x) = g(x))))
⇒ (∀a:{x:ℝ| x ∈ I} . (f(a) = g(a))))))
Lemma: rationals-dense-in-interval
∀I:Interval. dense-in-interval(I;λr.∃z:ℤ. ∃d:ℕ+. (r = (r(z)/r(d))))
Lemma: interior-dense-in-interval
∀a,b:ℝ. dense-in-interval([a, b];λr.((a < r) ∧ (r < b)))
Lemma: dyadic-rationals-dense
dense-in-interval((-∞, ∞);λr.∃n:ℤ. ∃m:ℕ+. (r = (r(n)/r(2^m))))
Lemma: dyadic-scaled-rationals-dense
∀a:ℝ. ((r0 < a) ⇒ dense-in-interval((-∞, ∞);λr.∃n:ℤ. ∃m:ℕ+. (r = (r(n) * (a/r(2^m))))))
Lemma: functions-equal-on-rationals
∀I:Interval
((∃u,v:{a:ℝ| a ∈ I} . u ≠ v)
⇒ (∀f,g:I ⟶ℝ.
((∀x,y:{x:ℝ| x ∈ I} . ((x = y) ⇒ (f(x) = f(y))))
⇒ (∀x,y:{x:ℝ| x ∈ I} . ((x = y) ⇒ (g(x) = g(y))))
⇒ (∀z:ℤ. ∀d:ℕ+. (((r(z)/r(d)) ∈ I) ⇒ (f((r(z)/r(d))) = g((r(z)/r(d))))))
⇒ (∀a:{x:ℝ| x ∈ I} . (f(a) = g(a))))))
Lemma: functions-equal-on-interior
∀a:ℝ. ∀b:{b:ℝ| a < b} . ∀f,g:[a, b] ⟶ℝ.
((∀x,y:{x:ℝ| x ∈ [a, b]} . ((x = y) ⇒ (f(x) = f(y))))
⇒ (∀x,y:{x:ℝ| x ∈ [a, b]} . ((x = y) ⇒ (g(x) = g(y))))
⇒ (∀x:{x:ℝ| x ∈ (a, b)} . (f(x) = g(x)))
⇒ (∀x:{x:ℝ| x ∈ [a, b]} . (f(x) = g(x))))
Lemma: total-function-limit
∀f:ℝ ⟶ ℝ. ∀y:ℝ. ∀x:ℕ ⟶ ℝ. ((∀x,y:ℝ. ((x = y) ⇒ (f[x] = f[y]))) ⇒ lim n→∞.x[n] = y ⇒ lim n→∞.f[x[n]] = f[y])
Lemma: total-function-rational-approx
∀f:ℝ ⟶ ℝ. ∀y:ℝ. ((∀x,y:ℝ. ((x = y) ⇒ (f[x] = f[y]))) ⇒ lim n→∞.f[(y within 1/n + 1)] = f[y])
Lemma: real-path-comp-exists
∀f,g:[r0, r1] ⟶ℝ.
((∀x,y:{x:ℝ| x ∈ [r0, r1]} . ((x = y) ⇒ (f(x) = f(y))))
⇒ (∀x,y:{x:ℝ| x ∈ [r0, r1]} . ((x = y) ⇒ (g(x) = g(y))))
⇒ (f(r1) = g(r0))
⇒ (∃h:[r0, r1] ⟶ℝ
((∀t:{x:ℝ| x ∈ [r0, (r1/r(2))]} . (h(t) = f(r(2) * t)))
∧ (∀t:{x:ℝ| x ∈ [(r1/r(2)), r1]} . (h(t) = g((r(2) * t) - r1)))
∧ (∀x,y:{x:ℝ| x ∈ [r0, r1]} . ((x = y) ⇒ (h(x) = h(y)))))))
Lemma: continuous-const
∀[I:Interval]. ∀[a:ℝ]. a continuous for x ∈ I
Lemma: continuous-id
∀[I:Interval]. x continuous for x ∈ I
Lemma: continuous-add
∀[I:Interval]. ∀[f,g:I ⟶ℝ].
(f[x] continuous for x ∈ I ⇒ g[x] continuous for x ∈ I ⇒ f[x] + g[x] continuous for x ∈ I)
Lemma: continuous-add-ext
∀[I:Interval]. ∀[f,g:I ⟶ℝ].
(f[x] continuous for x ∈ I ⇒ g[x] continuous for x ∈ I ⇒ f[x] + g[x] continuous for x ∈ I)
Lemma: continuous-sum
∀I:Interval. ∀n,m:ℤ. ∀f:{n..m + 1-} ⟶ I ⟶ℝ.
((∀i:{n..m + 1-}. f[i;x] continuous for x ∈ I) ⇒ Σ{f[i;x] | n≤i≤m} continuous for x ∈ I)
Lemma: continuous-max
∀[I:Interval]. ∀[f,g:I ⟶ℝ].
(f[x] continuous for x ∈ I ⇒ g[x] continuous for x ∈ I ⇒ rmax(f[x];g[x]) continuous for x ∈ I)
Lemma: continuous-minus
∀[I:Interval]. ∀[f:I ⟶ℝ]. (f[x] continuous for x ∈ I ⇒ -(f[x]) continuous for x ∈ I)
Lemma: continuous-sub
∀[I:Interval]. ∀[f,g:I ⟶ℝ].
(f[x] continuous for x ∈ I ⇒ g[x] continuous for x ∈ I ⇒ f[x] - g[x] continuous for x ∈ I)
Lemma: continuous-abs
∀[I:Interval]. ∀[f:I ⟶ℝ]. (f[x] continuous for x ∈ I ⇒ |f[x]| continuous for x ∈ I)
Lemma: continuous-abs-ext
∀[I:Interval]. ∀[f:I ⟶ℝ]. (f[x] continuous for x ∈ I ⇒ |f[x]| continuous for x ∈ I)
Lemma: continuous-abs-subtype
∀[I:Interval]. ∀[f:I ⟶ℝ]. (f[x] continuous for x ∈ I ⊆r |f[x]| continuous for x ∈ I)
Lemma: continuous-range-totally-bounded
∀I:Interval. ∀f:I ⟶ℝ.
(f[x] continuous for x ∈ I ⇒ (∀m:ℕ+. (i-nonvoid(i-approx(I;m)) ⇒ totally-bounded(f[x](x∈i-approx(I;m))))))
Lemma: continuous-compact-range-totally-bounded
∀I:Interval. ∀f:I ⟶ℝ. (icompact(I) ⇒ f[x] continuous for x ∈ I ⇒ totally-bounded(f[x](x∈I)))
Lemma: sup-range
∀I:{I:Interval| icompact(I)} . ∀f:I ⟶ℝ. (f[x] continuous for x ∈ I ⇒ (∃y:ℝ. sup(f(x)(x∈I)) = y))
Lemma: sup-range-no-mc
sup-range is Bishop's theorem that the range of a continuous function
has a supremum. To apply that theorem we need a modulus of continuity for f.
Since we can prove Brouwer's theorem Error :real-continuity1
that all functions on a proper interval are continuous,
we can prove this version of the theorem that does not require a
modulus of continuity.
We have to take some care to handle the general case where the interval
may not be proper.
The constructive content of this theorem will be terribly inefficient
since it will first construct a modulus of continuity (using bar recursion).
But, the constructive content of the original theorem of Bishop was already
very inefficient. We should think of the supremum as the "specification" of
a (well defined) real number, that one may be able to compute efficiently
by other means.
⋅
∀I:{I:Interval| icompact(I)} . ∀f:{f:I ⟶ℝ| ifun(f;I)} . ∃y:ℝ. sup(f(x)(x∈I)) = y
Lemma: inf-range
∀I:{I:Interval| icompact(I)} . ∀f:I ⟶ℝ. (f[x] continuous for x ∈ I ⇒ (∃y:ℝ. inf(f(x)(x∈I)) = y))
Lemma: inf-range-no-mc
∀I:{I:Interval| icompact(I)} . ∀f:{f:I ⟶ℝ| ifun(f;I)} . ∃y:ℝ. inf(f(x)(x∈I)) = y
Definition: range-sup
sup{f[x]|x ∈ I} == fst((TERMOF{sup-range:o, 1:l} I (λx.f[x]) mc))
Lemma: range-sup_wf
∀[I:{I:Interval| icompact(I)} ]. ∀[f:I ⟶ℝ]. ∀[mc:f[x] continuous for x ∈ I]. (sup{f[x]|x ∈ I} ∈ ℝ)
Lemma: range-sup-property
∀I:{I:Interval| icompact(I)} . ∀f:I ⟶ℝ. ∀mc:f[x] continuous for x ∈ I. sup(f[x](x∈I)) = sup{f[x]|x ∈ I}
Definition: range_sup
sup{f[x] | x ∈ I} == fst((TERMOF{sup-range-no-mc:o, 1:l} I (λx.f[x])))
Lemma: range_sup_wf
∀[I:{I:Interval| icompact(I)} ]. ∀[f:{x:ℝ| x ∈ I} ⟶ ℝ].
sup{f[x] | x ∈ I} ∈ ℝ supposing ∀x,y:{x:ℝ| x ∈ I} . ((x = y) ⇒ (f[x] = f[y]))
Definition: range_inf
inf{f[x] | x ∈ I} == fst((TERMOF{inf-range-no-mc:o, 1:l} I (λx.f[x])))
Lemma: range_inf_wf
∀[I:{I:Interval| icompact(I)} ]. ∀[f:{x:ℝ| x ∈ I} ⟶ ℝ].
inf{f[x] | x ∈ I} ∈ ℝ supposing ∀x,y:{x:ℝ| x ∈ I} . ((x = y) ⇒ (f[x] = f[y]))
Lemma: range_sup-property
∀I:{I:Interval| icompact(I)} . ∀f:{x:ℝ| x ∈ I} ⟶ ℝ.
sup(f[x](x∈I)) = sup{f[x] | x ∈ I} supposing ∀x,y:{x:ℝ| x ∈ I} . ((x = y) ⇒ (f[x] = f[y]))
Lemma: range_sup-bound
∀I:{I:Interval| icompact(I)} . ∀f:{x:ℝ| x ∈ I} ⟶ ℝ.
∀[c:ℝ]. sup{f[x] | x ∈ I} ≤ c supposing ∀x:ℝ. ((x ∈ I) ⇒ (f[x] ≤ c))
supposing ∀x,y:{x:ℝ| x ∈ I} . ((x = y) ⇒ (f[x] = f[y]))
Lemma: range_sup-const
∀I:{I:Interval| icompact(I)} . ∀[c:ℝ]. (sup{c | x ∈ I} = c)
Lemma: rleq-range_sup
∀I:{I:Interval| icompact(I)} . ∀f:{x:ℝ| x ∈ I} ⟶ ℝ.
∀[c:{c:ℝ| c ∈ I} ]. (f[c] ≤ sup{f[x] | x ∈ I}) supposing ∀x,y:{x:ℝ| x ∈ I} . ((x = y) ⇒ (f[x] = f[y]))
Lemma: rleq-range_sup-2
∀I:{I:Interval| icompact(I)} . ∀f:{x:ℝ| x ∈ I} ⟶ ℝ.
∀[c:ℝ]. c ≤ sup{f[x] | x ∈ I} supposing ∃x:ℝ. ((x ∈ I) ∧ (c ≤ f[x]))
supposing ∀x,y:{x:ℝ| x ∈ I} . ((x = y) ⇒ (f[x] = f[y]))
Lemma: range_sup_functionality_wrt_subinterval
∀I:{I:Interval| icompact(I)} . ∀f:{x:ℝ| x ∈ I} ⟶ ℝ.
∀J:{J:Interval| icompact(J)} . (J ⊆ I ⇒ (sup{f[x] | x ∈ J} ≤ sup{f[x] | x ∈ I}))
supposing ∀x,y:{x:ℝ| x ∈ I} . ((x = y) ⇒ (f[x] = f[y]))
Definition: range-inf
inf{f[x]|x ∈ I} == fst((TERMOF{inf-range:o, 1:l} I (λx.f[x]) mc))
Lemma: range-inf_wf
∀[I:{I:Interval| icompact(I)} ]. ∀[f:I ⟶ℝ]. ∀[mc:f[x] continuous for x ∈ I]. (inf{f[x]|x ∈ I} ∈ ℝ)
Lemma: range-inf-property
∀I:{I:Interval| icompact(I)} . ∀f:I ⟶ℝ. ∀mc:f[x] continuous for x ∈ I. inf(f[x](x∈I)) = inf{f[x]|x ∈ I}
Lemma: range_inf-property
∀I:{I:Interval| icompact(I)} . ∀f:{x:ℝ| x ∈ I} ⟶ ℝ.
inf(f[x](x∈I)) = inf{f[x] | x ∈ I} supposing ∀x,y:{x:ℝ| x ∈ I} . ((x = y) ⇒ (f[x] = f[y]))
Lemma: range_inf-bound
∀I:{I:Interval| icompact(I)} . ∀f:{x:ℝ| x ∈ I} ⟶ ℝ.
∀[c:ℝ]. c ≤ inf{f[x] | x ∈ I} supposing ∀x:ℝ. ((x ∈ I) ⇒ (c ≤ f[x]))
supposing ∀x,y:{x:ℝ| x ∈ I} . ((x = y) ⇒ (f[x] = f[y]))
Lemma: range_inf_functionality
∀I:{I:Interval| icompact(I)} . ∀f:{x:ℝ| x ∈ I} ⟶ ℝ.
∀g:{x:ℝ| x ∈ I} ⟶ ℝ. inf{f[x] | x ∈ I} = inf{g[x] | x ∈ I} supposing ∀x:{x:ℝ| x ∈ I} . (f[x] = g[x])
supposing ∀x,y:{x:ℝ| x ∈ I} . ((x = y) ⇒ (f[x] = f[y]))
Lemma: range_sup_functionality2
∀I,J:{I:Interval| icompact(I)} . ∀f:{x:ℝ| x ∈ I} ⟶ ℝ.
∀g:{x:ℝ| x ∈ J} ⟶ ℝ
(sup{f[x] | x ∈ I} = sup{g[x] | x ∈ J}) supposing
(((∀x:{x:ℝ| x ∈ I} . ∃y:{x:ℝ| x ∈ J} . (f[x] = g[y])) ∧ (∀x:{x:ℝ| x ∈ J} . ∃y:{x:ℝ| x ∈ I} . (f[y] = g[x]))) and
(∀x,y:{x:ℝ| x ∈ J} . ((x = y) ⇒ (g[x] = g[y]))))
supposing ∀x,y:{x:ℝ| x ∈ I} . ((x = y) ⇒ (f[x] = f[y]))
Lemma: range_sup_functionality
∀I:{I:Interval| icompact(I)} . ∀f:{x:ℝ| x ∈ I} ⟶ ℝ.
∀g:{x:ℝ| x ∈ I} ⟶ ℝ. sup{f[x] | x ∈ I} = sup{g[x] | x ∈ I} supposing ∀x:{x:ℝ| x ∈ I} . (f[x] = g[x])
supposing ∀x,y:{x:ℝ| x ∈ I} . ((x = y) ⇒ (f[x] = f[y]))
Definition: Inorm
||f[x]||_I == sup{|f[x]||x ∈ I}
Lemma: Inorm_wf
∀[I:{I:Interval| icompact(I)} ]. ∀[f:I ⟶ℝ]. ∀[mc:f[x] continuous for x ∈ I]. (||f[x]||_I ∈ ℝ)
Lemma: Inorm-bound
∀[I:{I:Interval| icompact(I)} ]. ∀[f:I ⟶ℝ]. ∀[mc:f[x] continuous for x ∈ I]. ∀[x:{r:ℝ| r ∈ I} ]. (|f[x]| ≤ ||f[x]||_I)
Lemma: Inorm-non-neg
∀[I:{I:Interval| icompact(I)} ]. ∀[f:I ⟶ℝ]. ∀[mc:f[x] continuous for x ∈ I]. (r0 ≤ ||f[x]||_I)
Lemma: function-diff-small-or-interval-proper
∀I:Interval. ∀f:I ⟶ℝ. ∀x,y:{x:ℝ| x ∈ I} . ∀e:{e:ℝ| r0 < e} .
(f[x] continuous for x ∈ I ⇒ ((|f[x] - f[y]| ≤ e) ∨ (r0 < |x - y|)))
Definition: I-norm
||f[x]||_x:I == sup{|f[x]| | x ∈ I}
Lemma: I-norm_wf
∀[I:{I:Interval| icompact(I)} ]. ∀[f:{x:ℝ| x ∈ I} ⟶ ℝ].
||f[x]||_x:I ∈ ℝ supposing ∀x,y:{x:ℝ| x ∈ I} . ((x = y) ⇒ (f[x] = f[y]))
Lemma: I-norm-bound
∀[I:{I:Interval| icompact(I)} ]. ∀[f:{x:ℝ| x ∈ I} ⟶ ℝ].
∀[x:{r:ℝ| r ∈ I} ]. (|f[x]| ≤ ||f[x]||_x:I) supposing ∀x,y:{x:ℝ| x ∈ I} . ((x = y) ⇒ (f[x] = f[y]))
Lemma: I-norm-rleq
∀[I:{I:Interval| icompact(I)} ]. ∀[f:{x:ℝ| x ∈ I} ⟶ ℝ].
∀[c:ℝ]. ||f[x]||_x:I ≤ c supposing ∀[x:{r:ℝ| r ∈ I} ]. (|f[x]| ≤ c)
supposing ∀x,y:{x:ℝ| x ∈ I} . ((x = y) ⇒ (f[x] = f[y]))
Lemma: I-norm_functionality_wrt_subinterval
∀[I:{I:Interval| icompact(I)} ]. ∀[f:{x:ℝ| x ∈ I} ⟶ ℝ].
∀[J:{J:Interval| icompact(J)} ]. ||f[x]||_x:J ≤ ||f[x]||_x:I supposing J ⊆ I
supposing ∀x,y:{x:ℝ| x ∈ I} . ((x = y) ⇒ (f[x] = f[y]))
Lemma: I-norm_functionality
∀[I:{I:Interval| icompact(I)} ]. ∀[f:{x:ℝ| x ∈ I} ⟶ ℝ].
∀[g:{x:ℝ| x ∈ I} ⟶ ℝ]. ||f[x]||_x:I = ||g[x]||_x:I supposing ∀x:{x:ℝ| x ∈ I} . (f[x] = g[x])
supposing ∀x,y:{x:ℝ| x ∈ I} . ((x = y) ⇒ (f[x] = f[y]))
Lemma: I-norm-non-neg
∀[I:{I:Interval| icompact(I)} ]. ∀[f:{x:ℝ| x ∈ I} ⟶ ℝ].
r0 ≤ ||f[x]||_x:I supposing ∀x,y:{x:ℝ| x ∈ I} . ((x = y) ⇒ (f[x] = f[y]))
Lemma: I-norm-positive-implies
∀I:{I:Interval| icompact(I)} . ∀f:{x:ℝ| x ∈ I} ⟶ ℝ.
(r0 < ||f[x]||_x:I) ⇒ (∃c:{c:ℝ| c ∈ I} . (r0 < |f[c]|)) supposing ∀x,y:{x:ℝ| x ∈ I} . ((x = y) ⇒ (f[x] = f[y]))
Lemma: real-fun-bounded
∀a:ℝ. ∀b:{b:ℝ| a ≤ b} . ∀f:[a, b] ⟶ℝ.
∃bnd:ℝ. ((r0 ≤ bnd) ∧ (∀x:ℝ. ((x ∈ [a, b]) ⇒ (|f x| ≤ bnd))))
supposing ∀x,y:{x:ℝ| x ∈ [a, b]} . ((x = y) ⇒ (f[x] = f[y]))
Lemma: real-fun-uniformly-positive
∀a:ℝ. ∀b:{b:ℝ| a ≤ b} . ∀f:[a, b] ⟶ℝ.
(real-fun(f;a;b) ⇒ (∀x:{x:ℝ| x ∈ [a, b]} . (r0 < (f x))) ⇒ (∃c:{c:ℝ| r0 < c} . ∀x:{x:ℝ| x ∈ [a, b]} . (c < (f x))))
Lemma: real-fun-uniformly-greater
∀a:ℝ. ∀b:{b:ℝ| a ≤ b} . ∀f:[a, b] ⟶ℝ.
(real-fun(f;a;b)
⇒ (∀c:ℝ. ((∀x:{x:ℝ| x ∈ [a, b]} . (c < (f x))) ⇒ (∃c':{c':ℝ| c < c'} . ∀x:{x:ℝ| x ∈ [a, b]} . (c' ≤ (f x))))))
Lemma: real-fun-uniformly-less
∀a:ℝ. ∀b:{b:ℝ| a ≤ b} . ∀f:[a, b] ⟶ℝ.
(real-fun(f;a;b)
⇒ (∀c:ℝ. ((∀x:{x:ℝ| x ∈ [a, b]} . ((f x) < c)) ⇒ (∃c':{c':ℝ| c' < c} . ∀x:{x:ℝ| x ∈ [a, b]} . ((f x) ≤ c')))))
Lemma: continuous-mul
∀I:Interval. ∀f,g:I ⟶ℝ. (f[x] continuous for x ∈ I ⇒ g[x] continuous for x ∈ I ⇒ f[x] * g[x] continuous for x ∈ I)
Lemma: continuous-rnexp
∀I:Interval. ∀n:ℕ. x^n continuous for x ∈ I
Lemma: continuous-rnexp2
∀I:Interval. ∀f:I ⟶ℝ. (f[x] continuous for x ∈ I ⇒ (∀n:ℕ. f[x]^n continuous for x ∈ I))
Lemma: continuous-rpolynomial
∀n:ℕ. ∀a:ℕn + 1 ⟶ ℝ. ∀I:Interval. (Σi≤n. a_i * x^i) continuous for x ∈ I
Definition: nonzero-on
f[x]≠r0 for x ∈ I ==
∀m:{m:ℕ+| icompact(i-approx(I;m))} . (∃c:ℝ [((r0 < c) ∧ (∀x:ℝ. ((x ∈ i-approx(I;m)) ⇒ (c ≤ |f[x]|))))])
Lemma: nonzero-on_wf
∀[I:Interval]. ∀[f:I ⟶ℝ]. (f[x]≠r0 for x ∈ I ∈ ℙ)
Lemma: const-nonzero-on
∀[I:Interval]. ∀a:ℝ. (a ≠ r0 ⇒ a≠r0 for x ∈ I)
Lemma: nonzero-on-implies
∀I:Interval. ∀f:I ⟶ℝ. (f[x]≠r0 for x ∈ I ⇒ (∀x:ℝ. ((x ∈ I) ⇒ f[x] ≠ r0)))
Definition: convex-on
convex-on(I;x.f[x]) ==
∀x,y,t:ℝ. ((x ∈ I) ⇒ (y ∈ I) ⇒ (t ∈ [r0, r1]) ⇒ (f[(t * x) + ((r1 - t) * y)] ≤ ((t * f[x]) + ((r1 - t) * f[y]))))
Lemma: convex-on_wf
∀[I:Interval]. ∀[f:I ⟶ℝ]. (convex-on(I;x.f[x]) ∈ ℙ)
Lemma: implies-convex-on
∀[I:Interval]. ∀[f:I ⟶ℝ].
((∀x,y:ℝ. ((x ∈ I) ⇒ (y ∈ I) ⇒ (x = y) ⇒ (f[x] = f[y])))
⇒ (∀x,y:ℝ.
((x < y)
⇒ (∀t:ℝ
((x ∈ I)
⇒ (y ∈ I)
⇒ (t ∈ [r0, r1])
⇒ (f[(t * x) + ((r1 - t) * y)] ≤ ((t * f[x]) + ((r1 - t) * f[y])))))))
⇒ convex-on(I;x.f[x]))
Definition: concave-on
concave-on(I;x.f[x]) ==
∀x,y,t:ℝ. ((x ∈ I) ⇒ (y ∈ I) ⇒ (t ∈ [r0, r1]) ⇒ (((t * f[x]) + ((r1 - t) * f[y])) ≤ f[(t * x) + ((r1 - t) * y)]))
Lemma: concave-on_wf
∀[I:Interval]. ∀[f:I ⟶ℝ]. (concave-on(I;x.f[x]) ∈ ℙ)
Lemma: concave-positive-nonzero-on
∀I:Interval. ∀f:I ⟶ℝ.
((∀x,y:ℝ. ((x ∈ I) ⇒ (y ∈ I) ⇒ (x = y) ⇒ (f[x] = f[y])))
⇒ (∀x:ℝ. ((x ∈ I) ⇒ (r0 < f[x])))
⇒ concave-on(I;x.f[x])
⇒ f[x]≠r0 for x ∈ I)
Lemma: convex-negative-nonzero-on
∀I:Interval. ∀f:I ⟶ℝ.
((∀x,y:ℝ. ((x ∈ I) ⇒ (y ∈ I) ⇒ (x = y) ⇒ (f[x] = f[y])))
⇒ (∀x:ℝ. ((x ∈ I) ⇒ (f[x] < r0)))
⇒ convex-on(I;x.f[x])
⇒ f[x]≠r0 for x ∈ I)
Lemma: rmul-nonzero-on
∀I:Interval. ∀f,g:I ⟶ℝ. (f[x]≠r0 for x ∈ I ⇒ g[x]≠r0 for x ∈ I ⇒ f[x] * g[x]≠r0 for x ∈ I)
Lemma: continuous-rinv
∀I:Interval. ∀f:I ⟶ℝ. (f[x] continuous for x ∈ I ⇒ f[x]≠r0 for x ∈ I ⇒ (r1/f[x]) continuous for x ∈ I)
Lemma: continuous-rdiv
∀I:Interval. ∀f,g:I ⟶ℝ.
(f[x] continuous for x ∈ I ⇒ g[x] continuous for x ∈ I ⇒ g[x]≠r0 for x ∈ I ⇒ (f[x]/g[x]) continuous for x ∈ I)
Definition: maps-compact
maps-compact(I;J;x.f[x]) ==
∀n:{n:ℕ+| icompact(i-approx(I;n))}
∃m:{m:ℕ+| icompact(i-approx(J;m))} . ∀x:{x:ℝ| x ∈ i-approx(I;n)} . (f[x] ∈ i-approx(J;m))
Lemma: maps-compact_wf
∀[I,J:Interval]. ∀[f:I ⟶ℝ]. (maps-compact(I;J;x.f[x]) ∈ ℙ)
Definition: interval-fun
interval-fun(I;J;x.f[x]) == (∀x:{x:ℝ| x ∈ I} . (f[x] ∈ J)) ∧ (∀x,y:{x:ℝ| x ∈ I} . f[x] = f[y] supposing x = y)
Lemma: interval-fun_wf
∀[I,J:Interval]. ∀[f:I ⟶ℝ]. (interval-fun(I;J;x.f[x]) ∈ ℙ)
Lemma: interval-fun-real-fun
∀[a,b:ℝ]. ∀[f:[a, b] ⟶ℝ]. real-fun(f;a;b) supposing ∃J:Interval. interval-fun([a, b];J;x.f[x])
Lemma: interval-fun-maps-compact
∀I,J:Interval. ∀f:I ⟶ℝ. (interval-fun(I;J;x.f[x]) ⇒ maps-compact(I;J;x.f[x]))
Definition: maps-compact-proper
maps-compact-proper(I;J;x.f[x]) ==
∀n:{n:ℕ+| icompact(i-approx(I;n)) ∧ iproper(i-approx(I;n))}
∃m:{m:ℕ+| icompact(i-approx(J;m)) ∧ iproper(i-approx(J;m))} . ∀x:{x:ℝ| x ∈ i-approx(I;n)} . (f[x] ∈ i-approx(J;m))
Lemma: maps-compact-proper_wf
∀[I,J:Interval]. ∀[f:I ⟶ℝ]. (maps-compact-proper(I;J;x.f[x]) ∈ ℙ)
Lemma: proper-maps-compact
∀I,J:Interval. ∀f:I ⟶ℝ. (iproper(J) ⇒ maps-compact(I;J;x.f[x]) ⇒ maps-compact-proper(I;J;x.f[x]))
Lemma: proper-continuous-maps-compact
∀I:Interval. ∀f:I ⟶ℝ. (f[x] (proper)continuous for x ∈ I ⇒ maps-compact-proper(I;(-∞, ∞);x.f[x]))
Lemma: continuous-maps-compact
∀I:Interval. ∀f:I ⟶ℝ. (iproper(I) ⇒ f[x] (proper)continuous for x ∈ I ⇒ maps-compact(I;(-∞, ∞);x.f[x]))
Lemma: function-maps-compact
∀I:Interval. ∀f:I ⟶ℝ.
(iproper(I) ⇒ (∀x,y:{t:ℝ| t ∈ I} . ((x = y) ⇒ (f[x] = f[y]))) ⇒ maps-compact(I;(-∞, ∞);x.f[x]))
Lemma: monotone-maps-compact
∀I,J:Interval. ∀f:I ⟶ℝ.
(((∀x,y:{t:ℝ| t ∈ I} . ((x ≤ y) ⇒ (f[x] ≤ f[y]))) ∨ (∀x,y:{t:ℝ| t ∈ I} . ((x ≤ y) ⇒ (f[y] ≤ f[x]))))
⇒ (∀x:{t:ℝ| t ∈ I} . (f[x] ∈ J))
⇒ maps-compact(I;J;x.f[x]))
Lemma: has-minimum-maps-compact
∀I:Interval. ∀l:ℝ. ∀f:I ⟶ℝ.
((∀x,y:{t:ℝ| t ∈ I} . ((x = y) ⇒ (f[x] = f[y])))
⇒ (∀x:{t:ℝ| t ∈ I} . (l < f[x]))
⇒ (∀a:{a:ℝ| a ∈ I} . ∀b:{b:ℝ| (b ∈ I) ∧ (a ≤ b)} . ∃c:{t:ℝ| t ∈ [a, b]} . ∀x:{t:ℝ| t ∈ [a, b]} . (f[c] ≤ f[x]))
⇒ maps-compact(I;(l, ∞);x.f[x]))
Lemma: continuous-composition-from-maps-compact
∀I,J:Interval. ∀f:I ⟶ℝ. ∀g:J ⟶ℝ.
(maps-compact(I;J;x.f[x]) ⇒ f[x] continuous for x ∈ I ⇒ g[x] continuous for x ∈ J ⇒ g[f[x]] continuous for x ∈ I)
Lemma: continuous-composition
Bishop's proof continuous-composition-from-maps-compact
that a composition of continuous functions is continuous requires
an extra hypothesis, maps-compact(I;J;x.f[x]).
But because we can prove Brouwer's theorem on continuity,
Error :real-continuity1
we can prove this theorem that does not require the extra hypothesis.⋅
∀I:Interval
(iproper(I)
⇒ (∀J:Interval. ∀f:{x:ℝ| x ∈ I} ⟶ {y:ℝ| y ∈ J} . ∀g:J ⟶ℝ.
(f[x] continuous for x ∈ I ⇒ g[x] continuous for x ∈ J ⇒ g[f[x]] continuous for x ∈ I)))
Lemma: function-values-near-same-sign
∀I:Interval. ∀f:{x:ℝ| x ∈ I} ⟶ ℝ.
(icompact(I)
⇒ (∀x,y:{x:ℝ| x ∈ I} . ((x = y) ⇒ (f[x] = f[y])))
⇒ (∀x:{x:ℝ| x ∈ I}
((r0 < |f[x]|)
⇒ (∃d:{d:ℝ| r0 < d}
∀y:{x:ℝ| x ∈ I} . ((|x - y| ≤ d) ⇒ ((r0 < f[x] ⇐⇒ r0 < f[y]) ∧ (f[x] < r0 ⇐⇒ f[y] < r0)))))))
Lemma: i-closed-closed
∀I:Interval. (i-closed(I) ⇒ closed-rset(λx.(x ∈ I)))
Lemma: superlevelset-closed
∀I:Interval. ∀[f:I ⟶ℝ]. ∀[c:ℝ]. (i-closed(I) ⇒ f(x) continuous for x ∈ I ⇒ closed-rset(superlevelset(I;f;c)))
Lemma: sublevelset-closed
∀I:Interval. ∀[f:I ⟶ℝ]. ∀[c:ℝ]. (i-closed(I) ⇒ f(x) continuous for x ∈ I ⇒ closed-rset(sublevelset(I;f;c)))
Lemma: intermediate-value-theorem
∀I:Interval. ∀f:I ⟶ℝ.
(f[x] continuous for x ∈ I
⇒ (∀a,b:{x:ℝ| x ∈ I} .
((f(a) < f(b))
⇒ (∀y:{y:ℝ| y ∈ [f(a), f(b)]} . ∀e:{e:ℝ| r0 < e} . ∃x:{x:ℝ| x ∈ [rmin(a;b), rmax(a;b)]} . (|f(x) - y| < e)))))
Lemma: intermediate-value-lemma
∀a:ℝ. ∀b:{b:ℝ| a < b} . ∀f:[a, b] ⟶ℝ.
∀c:{c:ℝ| (f(a) ≤ c) ∧ (c ≤ f(b))}
((∃d:{d:ℝ| (r0 ≤ d) ∧ (d < r1)}
∀a1:{a1:ℝ| (a1 ∈ [a, b]) ∧ (f(a1) ≤ c)} . ∀b1:{b1:ℝ| (b1 ∈ [a, b]) ∧ (c ≤ f(b1)) ∧ (a1 ≤ b1)} .
∃a2:{a2:ℝ| (a2 ∈ [a, b]) ∧ (f(a2) ≤ c)} . (∃b2:{b2:ℝ| (b2 ∈ [a, b]) ∧ (c ≤ f(b2))} [((a1 ≤ a2) ∧ (a2 ≤ b2) ∧ (\000Cb2 ≤ b1) ∧ ((b2 - a2) ≤ ((b1 - a1) * d)))]))
⇒ (∃x:ℝ [((x ∈ [a, b]) ∧ (f(x) = c))]))
supposing ∀x,y:{x:ℝ| x ∈ [a, b]} . ((x = y) ⇒ (f[x] = f[y]))
Lemma: intermediate-value-lemma-ext
∀a:ℝ. ∀b:{b:ℝ| a < b} . ∀f:[a, b] ⟶ℝ.
∀c:{c:ℝ| (f(a) ≤ c) ∧ (c ≤ f(b))}
((∃d:{d:ℝ| (r0 ≤ d) ∧ (d < r1)}
∀a1:{a1:ℝ| (a1 ∈ [a, b]) ∧ (f(a1) ≤ c)} . ∀b1:{b1:ℝ| (b1 ∈ [a, b]) ∧ (c ≤ f(b1)) ∧ (a1 ≤ b1)} .
∃a2:{a2:ℝ| (a2 ∈ [a, b]) ∧ (f(a2) ≤ c)} . (∃b2:{b2:ℝ| (b2 ∈ [a, b]) ∧ (c ≤ f(b2))} [((a1 ≤ a2) ∧ (a2 ≤ b2) ∧ (\000Cb2 ≤ b1) ∧ ((b2 - a2) ≤ ((b1 - a1) * d)))]))
⇒ (∃x:ℝ [((x ∈ [a, b]) ∧ (f(x) = c))]))
supposing ∀x,y:{x:ℝ| x ∈ [a, b]} . ((x = y) ⇒ (f[x] = f[y]))
Lemma: intermediate-value-lemma'
∀a:ℝ. ∀b:{b:ℝ| a < b} . ∀f:[a, b] ⟶ℝ.
∀c:{c:ℝ| (f(a) ≤ c) ∧ (c ≤ f(b))}
((∃d:{d:ℝ| (r0 ≤ d) ∧ (d < r1)}
∀a1:{a1:ℝ| (a1 ∈ [a, b]) ∧ (f(a1) ≤ c)} . ∀b1:{b1:ℝ| (b1 ∈ [a, b]) ∧ (c ≤ f(b1)) ∧ (a1 < b1)} .
∃a2:{a2:ℝ| (a2 ∈ [a, b]) ∧ (f(a2) ≤ c)} . (∃b2:{b2:ℝ| (b2 ∈ [a, b]) ∧ (c ≤ f(b2))} [((a1 ≤ a2) ∧ (a2 < b2) ∧ (\000Cb2 ≤ b1) ∧ ((b2 - a2) ≤ ((b1 - a1) * d)))]))
⇒ (∃x:ℝ [((x ∈ [a, b]) ∧ (f(x) = c))]))
supposing ∀x,y:{x:ℝ| x ∈ [a, b]} . ((x = y) ⇒ (f[x] = f[y]))
Lemma: intermediate-value-lemma'-ext
∀a:ℝ. ∀b:{b:ℝ| a < b} . ∀f:[a, b] ⟶ℝ.
∀c:{c:ℝ| (f(a) ≤ c) ∧ (c ≤ f(b))}
((∃d:{d:ℝ| (r0 ≤ d) ∧ (d < r1)}
∀a1:{a1:ℝ| (a1 ∈ [a, b]) ∧ (f(a1) ≤ c)} . ∀b1:{b1:ℝ| (b1 ∈ [a, b]) ∧ (c ≤ f(b1)) ∧ (a1 < b1)} .
∃a2:{a2:ℝ| (a2 ∈ [a, b]) ∧ (f(a2) ≤ c)} . (∃b2:{b2:ℝ| (b2 ∈ [a, b]) ∧ (c ≤ f(b2))} [((a1 ≤ a2) ∧ (a2 < b2) ∧ (\000Cb2 ≤ b1) ∧ ((b2 - a2) ≤ ((b1 - a1) * d)))]))
⇒ (∃x:ℝ [((x ∈ [a, b]) ∧ (f(x) = c))]))
supposing ∀x,y:{x:ℝ| x ∈ [a, b]} . ((x = y) ⇒ (f[x] = f[y]))
Definition: locally-non-constant
locally-non-constant(f;a;b;c) == ∀u,v:ℝ. ((a ≤ u) ⇒ (u < v) ⇒ (v ≤ b) ⇒ (∃z:ℝ. ((u ≤ z) ∧ (z ≤ v) ∧ f(z) ≠ c)))
Lemma: locally-non-constant_wf
∀[a,b,c:ℝ]. ∀[f:[a, b] ⟶ℝ]. (locally-non-constant(f;a;b;c) ∈ ℙ)
Definition: locally-non-constant-rational
locally-non-constant-rational(f;a;b;c) ==
∀u,v:ℝ. ((a ≤ u) ⇒ (u < v) ⇒ (v ≤ b) ⇒ (∃n:ℕ+. ∃z:ℤ. ((u ≤ (r(z))/n) ∧ ((r(z))/n ≤ v) ∧ f((r(z))/n) ≠ c)))
Lemma: locally-non-constant-rational_wf
∀[a,b,c:ℝ]. ∀[f:[a, b] ⟶ℝ]. (locally-non-constant-rational(f;a;b;c) ∈ ℙ)
Lemma: locally-non-constant-via-rational
∀a,b,c:ℝ. ∀f:[a, b] ⟶ℝ.
(f[x] continuous for x ∈ [a, b] ⇒ locally-non-constant(f;a;b;c) ⇒ locally-non-constant-rational(f;a;b;c))
Lemma: IVT-locally-non-constant
∀a:ℝ. ∀b:{b:ℝ| a < b} . ∀f:[a, b] ⟶ℝ.
∀c:ℝ. ((f(a) ≤ c) ⇒ (c ≤ f(b)) ⇒ locally-non-constant(f;a;b;c) ⇒ (∃x:ℝ [((x ∈ [a, b]) ∧ (f(x) = c))]))
supposing ∀x,y:{x:ℝ| x ∈ [a, b]} . ((x = y) ⇒ (f[x] = f[y]))
Lemma: strict-monotonic-is-locally-non-constant
∀I:Interval. ∀f:I ⟶ℝ.
(((∀x,y:{x:ℝ| x ∈ I} . ((x < y) ⇒ ((f x) < (f y)))) ∨ (∀x,y:{x:ℝ| x ∈ I} . ((x < y) ⇒ ((f y) < (f x)))))
⇒ (∀a,b:{x:ℝ| x ∈ I} . ∀x:ℝ. locally-non-constant(f;a;b;x)))
Lemma: IVT-strict-increasing
∀I:Interval. ∀f:I ⟶ℝ.
((∀x,y:{x:ℝ| x ∈ I} . ((x < y) ⇒ ((f x) < (f y))))
⇒ (∀x,y:{t:ℝ| t ∈ I} . ((x = y) ⇒ ((f x) = (f y))))
⇒ (∀a,b:{x:ℝ| x ∈ I} .
((a < b) ⇒ (∀x:ℝ. ((((f a) ≤ x) ∧ (x ≤ (f b))) ⇒ (∃c:ℝ [(((a ≤ c) ∧ (c ≤ b)) ∧ ((f c) = x))]))))))
Lemma: IVT-strict-decreasing
∀I:Interval. ∀f:I ⟶ℝ.
((∀x,y:{x:ℝ| x ∈ I} . ((x < y) ⇒ ((f y) < (f x))))
⇒ (∀x,y:{t:ℝ| t ∈ I} . ((x = y) ⇒ ((f x) = (f y))))
⇒ (∀a,b:{x:ℝ| x ∈ I} .
((a < b) ⇒ (∀x:ℝ. ((((f b) ≤ x) ∧ (x ≤ (f a))) ⇒ (∃c:ℝ. (((a ≤ c) ∧ (c ≤ b)) ∧ ((f c) = x))))))))
Lemma: inverse-of-strict-increasing-function
∀I:Interval. ∀f:I ⟶ℝ. ∀J:Interval. ∀g:x:{x:ℝ| x ∈ J} ⟶ {x:ℝ| x ∈ I} .
((∀t:{t:ℝ| t ∈ I} . (f t ∈ J))
⇒ (∀x,y:{x:ℝ| x ∈ I} . ((x < y) ⇒ ((f x) < (f y))))
⇒ (∀x,y:{t:ℝ| t ∈ I} . ((x = y) ⇒ ((f x) = (f y))))
⇒ (∀x:{x:ℝ| x ∈ J} . ((f (g x)) = x))
⇒ ((∀x:{x:ℝ| x ∈ I} . ((g (f x)) = x))
∧ (∀x,y:{x:ℝ| x ∈ J} . ((x < y) ⇒ ((g x) < (g y))))
∧ (∀x,y:{t:ℝ| t ∈ J} . ((x = y) ⇒ ((g x) = (g y))))))
Lemma: inverse-of-strict-increasing-function-exists
∀I:Interval. ∀f:I ⟶ℝ.
((∀x,y:{x:ℝ| x ∈ I} . ((x < y) ⇒ ((f x) < (f y))))
⇒ (∀x,y:{t:ℝ| t ∈ I} . ((x = y) ⇒ ((f x) = (f y))))
⇒ (∀J:Interval
((∀t:{t:ℝ| t ∈ I} . (f t ∈ J))
⇒ (∀x:{x:ℝ| x ∈ J} . ∃a,b:{t:ℝ| t ∈ I} . ((a < b) ∧ ((f a) ≤ x) ∧ (x ≤ (f b))))
⇒ (∃g:{x:ℝ| x ∈ J} ⟶ {x:ℝ| x ∈ I}
((∀x:{x:ℝ| x ∈ J} . ((f (g x)) = x))
∧ (∀x:{x:ℝ| x ∈ I} . ((g (f x)) = x))
∧ (∀x,y:{x:ℝ| x ∈ J} . ((x < y) ⇒ ((g x) < (g y))))
∧ (∀x,y:{t:ℝ| t ∈ J} . ((x = y) ⇒ ((g x) = (g y)))))))))
Lemma: inverse-of-strict-decreasing-function
∀I:Interval. ∀f:I ⟶ℝ. ∀J:Interval. ∀g:x:{x:ℝ| x ∈ J} ⟶ {x:ℝ| x ∈ I} .
((∀t:{t:ℝ| t ∈ I} . (f t ∈ J))
⇒ (∀x,y:{x:ℝ| x ∈ I} . ((x < y) ⇒ ((f y) < (f x))))
⇒ (∀x,y:{t:ℝ| t ∈ I} . ((x = y) ⇒ ((f x) = (f y))))
⇒ (∀x:{x:ℝ| x ∈ J} . ((f (g x)) = x))
⇒ ((∀x:{x:ℝ| x ∈ I} . ((g (f x)) = x))
∧ (∀x,y:{x:ℝ| x ∈ J} . ((x < y) ⇒ ((g y) < (g x))))
∧ (∀x,y:{t:ℝ| t ∈ J} . ((x = y) ⇒ ((g x) = (g y))))))
Lemma: inverse-of-strict-decreasing-function-exists
∀I:Interval. ∀f:I ⟶ℝ.
((∀x,y:{x:ℝ| x ∈ I} . ((x < y) ⇒ ((f y) < (f x))))
⇒ (∀x,y:{t:ℝ| t ∈ I} . ((x = y) ⇒ ((f x) = (f y))))
⇒ (∀J:Interval
((∀t:{t:ℝ| t ∈ I} . (f t ∈ J))
⇒ (∀x:{x:ℝ| x ∈ J} . ∃a,b:{t:ℝ| t ∈ I} . ((a < b) ∧ ((f b) ≤ x) ∧ (x ≤ (f a))))
⇒ (∃g:{x:ℝ| x ∈ J} ⟶ {x:ℝ| x ∈ I}
((∀x:{x:ℝ| x ∈ J} . ((f (g x)) = x))
∧ (∀x:{x:ℝ| x ∈ I} . ((g (f x)) = x))
∧ (∀x,y:{x:ℝ| x ∈ J} . ((x < y) ⇒ ((g y) < (g x))))
∧ (∀x,y:{t:ℝ| t ∈ J} . ((x = y) ⇒ ((g x) = (g y)))))))))
Lemma: IVT-strict-monotonic
∀I:Interval. ∀f:I ⟶ℝ.
(((∀x,y:{x:ℝ| x ∈ I} . ((x < y) ⇒ ((f x) < (f y)))) ∨ (∀x,y:{x:ℝ| x ∈ I} . ((x < y) ⇒ ((f y) < (f x)))))
⇒ (∀x,y:{t:ℝ| t ∈ I} . ((x = y) ⇒ ((f x) = (f y))))
⇒ (∀a,b:{x:ℝ| x ∈ I} .
(a ≠ b
⇒ (∀x:ℝ
(((((f a) ≤ x) ∧ (x ≤ (f b))) ∨ (((f b) ≤ x) ∧ (x ≤ (f a))))
⇒ (∃c:ℝ. ((((a ≤ c) ∧ (c ≤ b)) ∨ ((b ≤ c) ∧ (c ≤ a))) ∧ ((f c) = x))))))))
Lemma: inverse-of-strict-monotonic-function
∀I:Interval. ∀f:I ⟶ℝ.
(((∀x,y:{x:ℝ| x ∈ I} . ((x < y) ⇒ ((f x) < (f y)))) ∨ (∀x,y:{x:ℝ| x ∈ I} . ((x < y) ⇒ ((f y) < (f x)))))
⇒ (∀x,y:{t:ℝ| t ∈ I} . ((x = y) ⇒ ((f x) = (f y))))
⇒ (∀J:Interval
((∀t:{t:ℝ| t ∈ I} . (f t ∈ J))
⇒ (∀x:{x:ℝ| x ∈ J} . ∃a,b:{t:ℝ| t ∈ I} . (a ≠ b ∧ ((((f a) ≤ x) ∧ (x ≤ (f b))) ∨ (((f b) ≤ x) ∧ (x ≤ (f a))))))
⇒ (∃g:{x:ℝ| x ∈ J} ⟶ {x:ℝ| x ∈ I}
((∀x:{x:ℝ| x ∈ J} . ((f (g x)) = x))
∧ (∀x:{x:ℝ| x ∈ I} . ((g (f x)) = x))
∧ ((∀x,y:{x:ℝ| x ∈ J} . ((x < y) ⇒ ((g x) < (g y)))) ∨ (∀x,y:{x:ℝ| x ∈ J} . ((x < y) ⇒ ((g y) < (g x)))\000C))
∧ (∀x,y:{t:ℝ| t ∈ J} . ((x = y) ⇒ ((g x) = (g y)))))))))
Lemma: near-fixpoint-on-0-1
∀f:[r0, r1] ⟶ℝ
((∀x:ℝ. ((x ∈ [r0, r1]) ⇒ (f[x] ∈ [r0, r1])))
⇒ f[x] continuous for x ∈ [r0, r1]
⇒ (∀e:ℝ. ((r0 < e) ⇒ (∃x:ℝ. ((x ∈ [r0, r1]) ∧ (|f[x] - x| < e))))))
Definition: separated-partitions
separated-partitions(P;Q) == frs-increasing(P) ∧ frs-increasing(Q) ∧ frs-separated(P;Q)
Lemma: separated-partitions_wf
∀I:Interval. ∀P,Q:partition(I). (separated-partitions(P;Q) ∈ ℙ) supposing icompact(I)
Lemma: separated-partitions-have-common-refinement
∀I:Interval
∀P,Q:partition(I).
(separated-partitions(P;Q) ⇒ (∃R:partition(I). (frs-increasing(R) ∧ frs-refines(R;P) ∧ frs-refines(R;Q))))
supposing icompact(I)
Lemma: nearby-separated-partitions
∀I:Interval
((icompact(I) ∧ iproper(I))
⇒ (∀p,q:partition(I). ∀e:{e:ℝ| r0 < e} .
∃p',q':partition(I). (separated-partitions(p';q') ∧ nearby-partitions(e;p;p') ∧ nearby-partitions(e;q;q'))))
Lemma: nearby-partition-choice
∀I:Interval
(icompact(I)
⇒ (∀p,q:partition(I). ∀e:{e:ℝ| r0 < e} .
(nearby-partitions(e;p;q)
⇒ (∀x:partition-choice(full-partition(I;p))
∃y:partition-choice(full-partition(I;q)). ∀i:ℕ||p|| + 1. (|x[i] - y[i]| ≤ e)))))
Definition: decimal-digits
(d digits of x) == eval N = 10^d in eval N2 = N ÷ 2 in eval D = x N2 in eval R = D rem N in eval Q = D ÷ N in <Q, R>
Lemma: decimal-digits_wf
∀d:ℕ+. ∀x:ℝ. ((d digits of x) ∈ {p:ℤ × ℤ| let n,m = p in |x - r(n) + (r(m)/r(10^d))| ≤ (r(2)/r(10^d))} )
Definition: dd
For a = (x N), a/2N is within 1/N of x. So, if N = (10^n)/2, a will be the
integer that has n decimal digits of x.
We actually compute <"display-as", "decimal-rational", x, n, a>
because the "display hook" will display this result as =mmm.kkkkkkk
by using the n and the a to place the decimal point appropriately.
(One can explode the term so displayed to see the computed x, n, and a)⋅
n decimal digits of x ==
let answer ⟵ <"display-as", "decimal-rational", x, n, eval N = 10^n in eval N2 = N ÷ 2 in x N2>
in answer
Lemma: dd_wf
∀d:ℕ+. ∀x:ℝ.
(d decimal digits of x ∈ {a:Atom| a = "display-as" ∈ Atom}
× {a:Atom| a = "decimal-rational" ∈ Atom}
× {z:ℝ| z = x}
× {n:ℕ+| n = d ∈ ℤ}
× {n:ℤ| |x - (r(n)/r(10^d))| ≤ (r(2)/r(10^d))} )
Lemma: rnexp-convex
∀a,b:ℝ. ((r0 ≤ b) ⇒ (b ≤ a) ⇒ (∀n:ℕ+. (a - b^n ≤ (a^n - b^n))))
Lemma: rnexp-convex2
∀a,b:ℝ. ((r0 ≤ a) ⇒ (r0 ≤ b) ⇒ (∀n:ℕ+. (|a - b|^n ≤ |a^n - b^n|)))
Lemma: rnexp-convex3
∀a,b:ℝ. ((((r0 ≤ a) ∧ (r0 ≤ b)) ∨ ((a ≤ r0) ∧ (b ≤ r0))) ⇒ (∀n:ℕ+. (|a - b|^n ≤ |a^n - b^n|)))
Comment: constructing rroot
We didn't know whether the simple program for the i-th root of real x
was correct. Attempts to prove it correct had failed, so we made the
construction in this subdirectory and extracted a correct-by-construction
program extracted-rroot.
This program is fairly efficient. Using it, we compute
50 decimal digits of the 12-root of 2, ⌜50 decimal digits of extracted-rroot(12;r(2)) ⌝
in 531,134 computation steps, taking about 30 seconds.
However, the program rroot is much simpler, and computes
⌜50 decimal digits of rroot(12;r(2)) ⌝
in only 7,690 steps, almost instantly.
We used the computations find-real-neq and find-real-neq2
to experimentally check that the known correct ⌜extracted-rroot([i];[x])⌝
and the simpler ⌜rroot([i];[x])⌝ always
(on many reals of various sizes and signs) computed the same integers
(or integers that differed by at most one).
This gave us confidence that the simpler program was also correct
and we were then able to prove that is was correct!.
⋅
Lemma: near-root-rational
∀k:{2...}. ∀p:{p:ℤ| (0 ≤ p) ∨ (↑isOdd(k))} . ∀q,n:ℕ+.
(∃r:ℤ × ℕ+ [let a,b = r
in (0 ≤ p ⇐⇒ 0 ≤ a) ∧ (|(r(a))/b^k - (r(p)/r(q))| < (r1/r(n)))])
Definition: near-root
near-root(k;p;q;n) ==
if q=1
then if n=1
then eval c = 2^k - 1 in
eval a = p * 2 * c in
eval d = (2 * c) - 1 in
eval M = iroot(k;|a| + d) + 1 in
eval y = ((k * 2 * M^k - 1) ÷ d) + 1 in
eval x = iroot(k;(|a| + d) * y^k) ÷ 2 in
<if (p) < (0) then -x else x, y>
else eval b = q * n in
eval c = b^k - 1 in
eval a = p * n * c in
eval d = c - 1 in
eval M = iroot(k;|a| + d) + 1 in
eval y = ((k * b * M^k - 1) ÷ d) + 1 in
eval x = iroot(k;(|a| + d) * y^k) ÷ b in
<if (p) < (0) then -x else x, y>
else eval b = q * n in
eval c = b^k - 1 in
eval a = p * n * c in
eval d = c - 1 in
eval M = iroot(k;|a| + d) + 1 in
eval y = ((k * b * M^k - 1) ÷ d) + 1 in
eval x = iroot(k;(|a| + d) * y^k) ÷ b in
<if (p) < (0) then -x else x, y>
Lemma: near-root-rational-ext
∀k:{2...}. ∀p:{p:ℤ| (0 ≤ p) ∨ (↑isOdd(k))} . ∀q,n:ℕ+.
(∃r:ℤ × ℕ+ [let a,b = r
in (0 ≤ p ⇐⇒ 0 ≤ a) ∧ (|(r(a))/b^k - (r(p)/r(q))| < (r1/r(n)))])
Lemma: near-root_wf
∀k:{2...}. ∀p:{p:ℤ| (0 ≤ p) ∨ (↑isOdd(k))} . ∀q,n:ℕ+.
(near-root(k;p;q;n) ∈ {r:ℤ × ℕ+| let a,b = r in (0 ≤ p ⇐⇒ 0 ≤ a) ∧ (|(r(a))/b^k - (r(p)/r(q))| < (r1/r(n)))} )
Lemma: rroot-exists-part1
∀i:{2...}. ∀x:{x:ℝ| (↑isEven(i)) ⇒ (r0 ≤ x)} .
∃q:ℕ ⟶ ℝ
(lim n→∞.q n^i = x
∧ (∀n,m:ℕ. (((r0 ≤ (q n)) ∧ (r0 ≤ (q m))) ∨ (((q n) ≤ r0) ∧ ((q m) ≤ r0))))
∧ ((↑isEven(i)) ⇒ (∀m:ℕ. (r0 ≤ (q m)))))
Lemma: rroot-exists1
∀i:{2...}. ∀x:{x:ℝ| (↑isEven(i)) ⇒ (r0 ≤ x)} .
∃q:{q:ℕ ⟶ ℝ|
(∀n,m:ℕ. (((r0 ≤ (q n)) ∧ (r0 ≤ (q m))) ∨ (((q n) ≤ r0) ∧ ((q m) ≤ r0))))
∧ ((↑isEven(i)) ⇒ (∀m:ℕ. (r0 ≤ (q m))))}
lim n→∞.q n^i = x
Lemma: rroot-exists1-ext
∀i:{2...}. ∀x:{x:ℝ| (↑isEven(i)) ⇒ (r0 ≤ x)} .
∃q:{q:ℕ ⟶ ℝ|
(∀n,m:ℕ. (((r0 ≤ (q n)) ∧ (r0 ≤ (q m))) ∨ (((q n) ≤ r0) ∧ ((q m) ≤ r0))))
∧ ((↑isEven(i)) ⇒ (∀m:ℕ. (r0 ≤ (q m))))}
lim n→∞.q n^i = x
Lemma: rroot-exists-part2
∀i:{2...}. ∀x:{x:ℝ| (↑isEven(i)) ⇒ (r0 ≤ x)} . ∀q:{q:ℕ ⟶ ℝ|
(∀n,m:ℕ.
(((r0 ≤ (q n)) ∧ (r0 ≤ (q m))) ∨ (((q n) ≤ r0) ∧ ((q m) ≤ r0))))
∧ ((↑isEven(i)) ⇒ (∀m:ℕ. (r0 ≤ (q m))))} .
(lim n→∞.q n^i = x ⇒ cauchy(n.q n))
Lemma: rroot-exists
∀i:{2...}. ∀x:{x:ℝ| (↑isEven(i)) ⇒ (r0 ≤ x)} . (∃y:ℝ [(((↑isEven(i)) ⇒ (r0 ≤ y)) ∧ (y^i = x))])
Lemma: rroot-exists-ext
∀i:{2...}. ∀x:{x:ℝ| (↑isEven(i)) ⇒ (r0 ≤ x)} . (∃y:ℝ [(((↑isEven(i)) ⇒ (r0 ≤ y)) ∧ (y^i = x))])
Definition: extracted-rroot
extracted-rroot(i;x) == TERMOF{rroot-exists-ext:o, 1:l} i x
Lemma: extracted-rroot_wf
∀[i:{2...}]. ∀[x:{x:ℝ| (↑isEven(i)) ⇒ (r0 ≤ x)} ].
(extracted-rroot(i;x) ∈ {y:ℝ| ((↑isEven(i)) ⇒ (r0 ≤ y)) ∧ (y^i = x)} )
Definition: find-real-neq
find-real-neq(a;b) ==
eval x = a in
eval y = b in
mu-ge-print(λn.eval xn = x n in
eval yn = y n in
eval d = xn - yn in
1 <z |d|;0)
Definition: find-real-neq2
find-real-neq2(a;b) ==
eval x = a in
eval y = b in
find-xover-print(λn.eval xn = x n in
eval yn = y n in
eval d = xn - yn in
1 <z |d|;0;0;1)
Lemma: exp-2-3-fact
∀n:ℕ. ((2 ≤ n) ⇒ 2 * 2^n < 3^n)
Lemma: rroot-regularity-lemma
∀[k:{2...}]. ∀[n,m:ℕ+]. ∀[a,b,c,d:ℤ].
(((m ≤ a) ∨ ((a = 0 ∈ ℤ) ∧ (c = 0 ∈ ℤ)))
⇒ ((n ≤ b) ∨ ((b = 0 ∈ ℤ) ∧ (d = 0 ∈ ℤ)))
⇒ (a^k ≤ c)
⇒ c < (a + m)^k
⇒ (b^k ≤ d)
⇒ d < (b + n)^k
⇒ (|c - d| ≤ (2^k * (n^k + m^k)))
⇒ (|a - b| ≤ (2 * (n + m))))
Definition: rroot-abs
rroot-abs(i;x) == eval b = 2^i - 1 in λn.eval k = n^i in eval z = |x| k in iroot(i;b * z)
Lemma: rroot-abs_wf
∀i:{2...}. ∀x:ℝ. (rroot-abs(i;x) ∈ ℝ)
Lemma: rroot-abs-property
∀i:{2...}. ∀x:ℝ. (rroot-abs(i;x)^i = |x|)
Lemma: rroot-abs-non-neg
∀i:{2...}. ∀x:ℝ. ∀n:ℕ+. (0 ≤ (rroot-abs(i;x) n))
Definition: rroot-odd
rroot-odd(i;x) ==
eval b = 2^i - 1 in
λn.eval k = n^i in
eval z = x k in
if z <z 0 then -iroot(i;b * (-z)) else iroot(i;b * z) fi
Lemma: rroot-odd_wf
∀i:{2...}. ∀x:ℝ. (rroot-odd(i;x) ∈ ℕ+ ⟶ ℤ)
Lemma: rroot-odd-2-regular
∀i:{2...}. ∀x:ℝ. 2-regular-seq(rroot-odd(i;x))
Definition: rroot
rroot(i;x) == if isEven(i) then rroot-abs(i;x) else accelerate(2;rroot-odd(i;x)) fi
Lemma: rroot_wf
∀[i:{2...}]. ∀[x:{x:ℝ| (↑isEven(i)) ⇒ (r0 ≤ x)} ]. (rroot(i;x) ∈ {y:ℝ| ((↑isEven(i)) ⇒ (r0 ≤ y)) ∧ (y^i = x)} )
Lemma: rroot_functionality
∀[i:{2...}]. ∀[x:{x:ℝ| (↑isEven(i)) ⇒ (r0 ≤ x)} ]. ∀[y:ℝ]. rroot(i;x) = rroot(i;y) supposing x = y
Definition: rsqrt
rsqrt(x) == rroot(2;x)
Lemma: rsqrt_wf
∀[x:{x:ℝ| r0 ≤ x} ]. (rsqrt(x) ∈ {r:ℝ| (r0 ≤ r) ∧ ((r * r) = x)} )
Lemma: rsqrt_functionality
∀[x:{x:ℝ| r0 ≤ x} ]. ∀[y:ℝ]. rsqrt(x) = rsqrt(y) supposing x = y
Lemma: rsqrt_nonneg
∀[x:{x:ℝ| r0 ≤ x} ]. (r0 ≤ rsqrt(x))
Lemma: rsqrt_squared
∀[x:{x:ℝ| r0 ≤ x} ]. ((rsqrt(x) * rsqrt(x)) = x)
Lemma: rsqrt-rleq-iff
∀[x:{x:ℝ| r0 ≤ x} ]. ∀[c:ℝ]. uiff(rsqrt(x) ≤ c;(r0 ≤ c) ∧ (x ≤ c^2))
Lemma: rsqrt-is-zero
∀[x:{x:ℝ| r0 ≤ x} ]. (rsqrt(x) = r0 ⇐⇒ x = r0)
Lemma: rsqrt-rnexp-2
∀[x:{x:ℝ| r0 ≤ x} ]. (rsqrt(x)^2 = x)
Lemma: rsqrt-unique
∀[x,s:{x:ℝ| r0 ≤ x} ]. s = rsqrt(x) supposing (s * s) = x
Lemma: rsqrt-unique2
∀[x:{x:ℝ| r0 ≤ x} ]. ∀[s:ℝ]. uiff((s * s) = x;¬¬((s = rsqrt(x)) ∨ (s = -(rsqrt(x)))))
Lemma: rsqrt1
rsqrt(r1) = r1
Lemma: rsqrt-is-one
∀[x:ℝ]. uiff((r0 ≤ x) ∧ (rsqrt(x) = r1);x = r1)
Lemma: implies-rsqrt-is-one
∀[x:ℝ]. rsqrt(x) = r1 supposing x = r1
Lemma: rsqrt0
rsqrt(r0) = r0
Lemma: rsqrt-of-square
∀[x:{x:ℝ| r0 ≤ x} ]. (rsqrt(x * x) = x)
Lemma: rsqrt_square
∀[x:ℝ]. (rsqrt(x * x) = |x|)
Lemma: rsqrt-positive
∀x:{x:ℝ| r0 < x} . (r0 < rsqrt(x))
Lemma: rsqrt-positive-iff
∀x:{x:ℝ| r0 ≤ x} . (r0 < rsqrt(x) ⇐⇒ r0 < x)
Lemma: rsqrt-rmul
∀x:{x:ℝ| r0 ≤ x} . ∀[y:{x:ℝ| r0 ≤ x} ]. ((rsqrt(x) * rsqrt(y)) = rsqrt(x * y))
Lemma: rsqrt-rdiv
∀x:{x:ℝ| r0 ≤ x} . ∀y:{x:ℝ| r0 < x} . ((rsqrt(x)/rsqrt(y)) = rsqrt((x/y)))
Lemma: rsqrt_functionality_wrt_rleq
∀[x:{x:ℝ| r0 ≤ x} ]. ∀[y:ℝ]. rsqrt(x) ≤ rsqrt(y) supposing x ≤ y
Lemma: rsqrt_functionality_wrt_rless
∀x:{x:ℝ| r0 ≤ x} . ∀y:ℝ. ((x < y) ⇒ (rsqrt(x) < rsqrt(y)))
Lemma: rsqrt-rless-iff
∀x,y:{x:ℝ| r0 ≤ x} . (x < y ⇐⇒ rsqrt(x) < rsqrt(y))
Lemma: rsqrt-rneq
∀x,y:{x:ℝ| r0 ≤ x} . (rsqrt(x) ≠ rsqrt(y) ⇒ x ≠ y)
Definition: quadratic1
quadratic1(a;b;c) == (-(b) + rsqrt((b * b) - r(4) * a * c)/r(2) * a)
Lemma: quadratic1_wf
∀[a,b,c:ℝ]. (quadratic1(a;b;c) ∈ ℝ) supposing ((r0 ≤ ((b * b) - r(4) * a * c)) and a ≠ r0)
Lemma: quadratic1_functionality
∀[a,b,c,a',b',c':ℝ].
(quadratic1(a;b;c) = quadratic1(a';b';c')) supposing
((c = c') and
(b = b') and
(a = a') and
(r0 ≤ ((b * b) - r(4) * a * c)) and
a ≠ r0)
Lemma: quadratic1-zero
∀[a,b,c:ℝ]. (quadratic1(a;b;c) = r0) supposing ((r0 ≤ b) and (c = r0) and a ≠ r0)
Lemma: quadratic1-one
∀[a,b,c:ℝ]. (quadratic1(a;b;c) = r1) supposing ((c ≤ a) and (b = -(a + c)) and (r0 < a))
Definition: quadratic2
quadratic2(a;b;c) == (-(b) - rsqrt((b * b) - r(4) * a * c)/r(2) * a)
Lemma: quadratic2_wf
∀[a,b,c:ℝ]. (quadratic2(a;b;c) ∈ ℝ) supposing ((r0 ≤ ((b * b) - r(4) * a * c)) and a ≠ r0)
Lemma: quadratic2_functionality
∀[a,b,c,a',b',c':ℝ].
(quadratic2(a;b;c) = quadratic2(a';b';c')) supposing
((c = c') and
(b = b') and
(a = a') and
(r0 ≤ ((b * b) - r(4) * a * c)) and
a ≠ r0)
Lemma: quadratic2-zero
∀[a,b,c:ℝ]. (quadratic2(a;b;c) = r0) supposing ((b ≤ r0) and (c = r0) and a ≠ r0)
Lemma: quadratic-1-2-equal
∀[a,b,c:ℝ].
({(quadratic1(a;b;c) = quadratic2(a;b;c)) ∧ (quadratic1(a;b;c) = (-(b)/r(2) * a))}) supposing
(((b * b) = (r(4) * a * c)) and
a ≠ r0)
Lemma: quadratic-1-2-equal-implies
∀[a,b,c:ℝ].
((b * b) = (r(4) * a * c)) supposing
((quadratic1(a;b;c) = quadratic2(a;b;c)) and
(r0 ≤ ((b * b) - r(4) * a * c)) and
a ≠ r0)
Lemma: quadratic-1-2-equal-implies2
∀[a,b,c:ℝ].
({((b * b) = (r(4) * a * c)) ∧ (quadratic1(a;b;c) = (-(b)/r(2) * a))}) supposing
((quadratic1(a;b;c) = quadratic2(a;b;c)) and
(r0 ≤ ((b * b) - r(4) * a * c)) and
a ≠ r0)
Lemma: quadratic-formula1
∀a,b,c:ℝ.
(a ≠ r0
⇒ (r0 ≤ ((b * b) - r(4) * a * c))
⇒ (∀x:ℝ. (((x = quadratic1(a;b;c)) ∨ (x = quadratic2(a;b;c))) ⇒ (((a * x^2) + (b * x) + c) = r0))))
Lemma: quadratic-formula-simple
∀a,b,c:ℝ.
((r0 < a)
⇒ (c < r0)
⇒ (∀x:ℝ. (((x = quadratic1(a;b;c)) ∨ (x = quadratic2(a;b;c))) ⇒ (((a * x^2) + (b * x) + c) = r0))))
Lemma: quadratic-formula2
∀a,b,c:ℝ.
(a ≠ r0
⇒ (r0 ≤ ((b * b) - r(4) * a * c))
⇒ (∀x:ℝ
((((a * x^2) + (b * x) + c) = r0)
⇒ ((¬¬((x = quadratic1(a;b;c)) ∨ (x = quadratic2(a;b;c))))
∧ ((((r(2) * a) * x) + b ≠ r0 ∨ (r0 < ((b * b) - r(4) * a * c)))
⇒ ((x = quadratic1(a;b;c)) ∨ (x = quadratic2(a;b;c))))))))
Lemma: arith-geom-mean-inequality-simple
∀x:{t:ℝ| r0 ≤ t} . ∀y:ℝ. ((r(2) * x * y) ≤ (x^2 + y^2))
Lemma: arith-geom-mean-inequality
∀x,y:{t:ℝ| r0 ≤ t} . ((rsqrt(x * y) ≤ (x + y/r(2))) ∧ (x ≠ y ⇒ (rsqrt(x * y) < (x + y/r(2)))))
Lemma: Cauchy-Schwarz3
∀[n:ℕ]. ∀[x,y:ℕn ⟶ ℝ].
(|Σ{x[i] * y[i] | 0≤i≤n - 1}| ≤ (rsqrt(Σ{x[i] * x[i] | 0≤i≤n - 1}) * rsqrt(Σ{y[i] * y[i] | 0≤i≤n - 1})))
Lemma: Cauchy-Schwarz3-strict
∀n:ℕ. ∀x,y:ℕn ⟶ ℝ.
(∃i,j:ℕn. x[j] * y[i] ≠ x[i] * y[j]
⇐⇒ |Σ{x[i] * y[i] | 0≤i≤n - 1}| < (rsqrt(Σ{x[i] * x[i] | 0≤i≤n - 1}) * rsqrt(Σ{y[i] * y[i] | 0≤i≤n - 1})))
Definition: real-vec
ℝ^n == ℕn ⟶ ℝ
Lemma: real-vec_wf
∀[n:ℕ]. (ℝ^n ∈ Type)
Lemma: real-vec-subtype
∀[n,m:ℕ]. ℝ^m ⊆r ℝ^n supposing n ≤ m
Definition: req-vec
req-vec(n;x;y) == ∀i:ℕn. ((x i) = (y i))
Lemma: req-vec_wf
∀[n:ℕ]. ∀[x,y:ℝ^n]. (req-vec(n;x;y) ∈ ℙ)
Lemma: stable__req-vec
∀n:ℕ. ∀x,y:ℝ^n. Stable{req-vec(n;x;y)}
Lemma: req-vec_functionality
∀[n:ℕ]. ∀[x1,x2,y1,y2:ℝ^n]. (uiff(req-vec(n;x1;y1);req-vec(n;x2;y2))) supposing (req-vec(n;y1;y2) and req-vec(n;x1;x2))
Lemma: req-vec_weakening
∀[n:ℕ]. ∀[x,y:ℝ^n]. req-vec(n;x;y) supposing x = y ∈ ℝ^n
Lemma: req-vec_inversion
∀[n:ℕ]. ∀[x,y:ℝ^n]. req-vec(n;y;x) supposing req-vec(n;x;y)
Lemma: req-vec_transitivity
∀[n:ℕ]. ∀[x,y,z:ℝ^n]. (req-vec(n;x;z)) supposing (req-vec(n;y;z) and req-vec(n;x;y))
Definition: real-vec-add
X + Y == λs.((X s) + (Y s))
Lemma: real-vec-add_wf
∀[n:ℕ]. ∀[X,Y:ℝ^n]. (X + Y ∈ ℝ^n)
Lemma: real-vec-add-com
∀[n:ℕ]. ∀[X,Y:ℝ^n]. req-vec(n;X + Y;Y + X)
Lemma: real-vec-add-assoc
∀[n:ℕ]. ∀[X,Y,Z:ℝ^n]. req-vec(n;X + Y + Z;X + Y + Z)
Lemma: real-vec-add_functionality
∀[n:ℕ]. ∀[X1,Y1,X2,Y2:ℝ^n]. (req-vec(n;X1 + Y1;X2 + Y2)) supposing (req-vec(n;X1;X2) and req-vec(n;Y1;Y2))
Definition: real-vec-sub
X - Y == λs.((X s) - Y s)
Lemma: real-vec-sub_wf
∀[n:ℕ]. ∀[X,Y:ℝ^n]. (X - Y ∈ ℝ^n)
Lemma: real-vec-sub_functionality
∀[n:ℕ]. ∀[X1,Y1,X2,Y2:ℝ^n]. (req-vec(n;X1 - Y1;X2 - Y2)) supposing (req-vec(n;X1;X2) and req-vec(n;Y1;Y2))
Lemma: real-vec-add-cancel
∀[n:ℕ]. ∀[p,a,b:ℝ^n]. req-vec(n;a;b) supposing req-vec(n;p + a;p + b)
Definition: real-vec-mul
a*X == λi.(a * (X i))
Lemma: real-vec-mul_wf
∀[n:ℕ]. ∀[X:ℝ^n]. ∀[a:ℝ]. (a*X ∈ ℝ^n)
Lemma: real-vec-mul-linear
∀[n:ℕ]. ∀[X,Y:ℝ^n]. ∀[a:ℝ]. req-vec(n;a*X + Y;a*X + a*Y)
Lemma: real-vec-mul-linear-sub
∀[n:ℕ]. ∀[X,Y:ℝ^n]. ∀[a:ℝ]. req-vec(n;a*X - Y;a*X - a*Y)
Lemma: real-vec-mul-mul
∀[n:ℕ]. ∀[X:ℝ^n]. ∀[a,b:ℝ]. req-vec(n;a*b*X;a * b*X)
Lemma: real-vec-mul_functionality
∀[n:ℕ]. ∀[X,Y:ℝ^n]. ∀[a,b:ℝ]. (req-vec(n;a*X;b*Y)) supposing ((a = b) and req-vec(n;X;Y))
Definition: real-vec-sum
Σ{x[k] | n≤k≤m} == λi.Σ{x[k] i | n≤k≤m}
Lemma: real-vec-sum_wf
∀[n,m:ℤ]. ∀[k:ℕ]. ∀[x:{n..m + 1-} ⟶ ℝ^k]. (Σ{x[k] | n≤k≤m} ∈ ℝ^k)
Lemma: real-vec-sum_functionality
∀[n,m:ℤ]. ∀[k:ℕ]. ∀[x,y:{n..m + 1-} ⟶ ℝ^k].
req-vec(k;Σ{x[k] | n≤k≤m};Σ{y[k] | n≤k≤m}) supposing ∀i:ℤ. ((n ≤ i) ⇒ (i ≤ m) ⇒ req-vec(k;x[i];y[i]))
Lemma: real-vec-sum-single
∀[n,m:ℤ]. ∀[k:ℕ]. ∀[x:{n..m + 1-} ⟶ ℝ^k]. req-vec(k;Σ{x[k] | n≤k≤m};x[n]) supposing m = n ∈ ℤ
Lemma: real-vec-sum-empty
∀[n,m:ℤ]. ∀[x:Top]. Σ{x[k] | n≤k≤m} ~ λi.r0 supposing m < n
Lemma: real-vec-sum-split
∀[j,n,m:ℤ]. ∀[k:ℕ]. ∀[x:{n..m + 1-} ⟶ ℝ^k].
(req-vec(k;Σ{x[i] | n≤i≤m};Σ{x[i] | n≤i≤j} + Σ{x[i] | j + 1≤i≤m})) supposing ((j ≤ m) and (n ≤ j))
Lemma: real-vec-sum-shift
∀[k,n,m:ℤ]. ∀[x:Top]. (Σ{x[i] | n≤i≤m} ~ Σ{x[i + k] | n - k≤i≤m - k})
Lemma: real-vec-sum-split-first
∀[n,m:ℤ]. ∀[k:ℕ]. ∀[x:{n..m + 1-} ⟶ ℝ^k]. req-vec(k;Σ{x[i] | n≤i≤m};x[n] + Σ{x[i + 1] | n≤i≤m - 1}) supposing n ≤ m
Definition: real-vec-indep
real-vec-indep(k;L) == ∀a:ℕ||L|| ⟶ ℝ. (req-vec(k;Σ{a i*L[i] | 0≤i≤||L|| - 1};λi.r0) ⇒ (∀i:ℕ||L||. ((a i) = r0)))
Lemma: real-vec-indep_wf
∀[k:ℕ]. ∀[L:ℝ^k List]. (real-vec-indep(k;L) ∈ ℙ)
Lemma: stable__real-vec-indep
∀[k:ℕ]. ∀[L:ℝ^k List]. Stable{real-vec-indep(k;L)}
Lemma: sq_stable__real-vec-indep
∀[k:ℕ]. ∀[L:ℝ^k List]. SqStable(real-vec-indep(k;L))
Definition: real-vec-between
a-b-c == ∃t:ℝ. ((t ∈ (r0, r1)) ∧ req-vec(n;b;t*a + r1 - t*c))
Lemma: real-vec-between_wf
∀[n:ℕ]. ∀[a,b,c:ℝ^n]. (a-b-c ∈ ℙ)
Lemma: real-vec-between_functionality
∀n:ℕ. ∀a1,a2,b1,b2,c1,c2:ℝ^n. (req-vec(n;a1;a2) ⇒ req-vec(n;b1;b2) ⇒ req-vec(n;c1;c2) ⇒ (a1-b1-c1 ⇐⇒ a2-b2-c2))
Lemma: real-vec-between-symmetry
∀n:ℕ. ∀a,b,c:ℝ^n. (a-b-c ⇒ c-b-a)
Lemma: real-vec-between-inner-trans
∀n:ℕ. ∀a,b,c,d:ℝ^n. (a-b-d ⇒ b-c-d ⇒ a-b-c)
Definition: real-vec-be
real-vec-be(n;a;b;c) == ∃t:ℝ. ((t ∈ [r0, r1]) ∧ req-vec(n;b;t*a + r1 - t*c))
Lemma: real-vec-be_wf
∀[n:ℕ]. ∀[a,b,c:ℝ^n]. (real-vec-be(n;a;b;c) ∈ ℙ)
Definition: dot-product
x⋅y == Σ{(x i) * (y i) | 0≤i≤n - 1}
Lemma: dot-product_wf
∀[n:ℕ]. ∀[x,y:ℝ^n]. (x⋅y ∈ ℝ)
Lemma: dot-product_functionality
∀[n:ℕ]. ∀[x1,y1,x2,y2:ℝ^n]. (x1⋅y1 = x2⋅y2) supposing (req-vec(n;x1;x2) and req-vec(n;y1;y2))
Lemma: dot-product-comm
∀[n:ℕ]. ∀[x,y:ℝ^n]. (x⋅y = y⋅x)
Lemma: dot-product-linearity1
∀[n:ℕ]. ∀[x,y,z:ℝ^n]. ((x + y⋅z = (x⋅z + y⋅z)) ∧ (z⋅x + y = (z⋅x + z⋅y)))
Lemma: dot-product-linearity1-sub
∀[n:ℕ]. ∀[x,y,z:ℝ^n]. ((x - y⋅z = (x⋅z - y⋅z)) ∧ (z⋅x - y = (z⋅x - z⋅y)))
Lemma: dot-product-linearity2
∀[n:ℕ]. ∀[x,y:ℝ^n]. ∀[a:ℝ]. ((x⋅a*y = (a * x⋅y)) ∧ (a*x⋅y = (a * x⋅y)))
Lemma: dot-product-nonneg
∀[n:ℕ]. ∀[x:ℝ^n]. (r0 ≤ x⋅x)
Lemma: dot-product-zero
∀[n:ℕ]. ∀[x:ℝ^n]. (x⋅λi.r0 = r0)
Lemma: r2-dot-product
∀[a,b:ℝ^2]. (a⋅b = (((a 0) * (b 0)) + ((a 1) * (b 1))))
Lemma: r3-dot-product
∀[a,b:ℝ^3]. (a⋅b = (((a 0) * (b 0)) + ((a 1) * (b 1)) + ((a 2) * (b 2))))
Lemma: dot-product-split
∀[n:ℕ]. ∀[k:ℕn]. ∀[x,y:ℝ^n]. (x⋅y = (x⋅y + λi.(x (k + i))⋅λi.(y (k + i))))
Lemma: dot-product-split-last
∀[n:ℕ+]. ∀[x,y:ℝ^n]. (x⋅y = (x⋅y + ((x (n - 1)) * (y (n - 1)))))
Lemma: dot-product-split-first
∀[n:ℕ+]. ∀[x,y:ℝ^n]. (x⋅y = (((x 0) * (y 0)) + λi.(x (i + 1))⋅λi.(y (i + 1))))
Definition: real-vec-norm
||x|| == rsqrt(x⋅x)
Lemma: real-vec-norm_wf
∀[n:ℕ]. ∀[x:ℝ^n]. (||x|| ∈ ℝ)
Lemma: real-vec-norm_functionality
∀[n:ℕ]. ∀[x,y:ℝ^n]. ||x|| = ||y|| supposing req-vec(n;x;y)
Lemma: real-vec-norm-nonneg
∀[n:ℕ]. ∀[x:ℝ^n]. (r0 ≤ ||x||)
Lemma: real-vec-norm-positive-iff
∀n:ℕ. ∀x:ℝ^n. (r0 < ||x|| ⇐⇒ ∃i:ℕn. r0 ≠ x i)
Lemma: real-vec-norm-positive-iff2
∀n:ℕ. ∀x:ℝ^n. (∃i:ℕn. x i ≠ r0 ⇐⇒ r0 < ||x||)
Lemma: real-vec-norm-squared
∀[n:ℕ]. ∀[x:ℝ^n]. (||x||^2 = x⋅x)
Lemma: real-vec-norm-diff-squared
∀[n:ℕ]. ∀[x,y:ℝ^n]. (||x - y||^2 = ((||x||^2 + ||y||^2) + (r(-2) * x⋅y)))
Lemma: real-vec-norm-eq-iff
∀[n:ℕ]. ∀[x:ℝ^n]. ∀[r:ℝ]. uiff(||x|| = r;(x⋅x = r^2) ∧ (r0 ≤ r))
Lemma: real-vec-norm-equal-iff
∀[n:ℕ]. ∀[x,y:ℝ^n]. uiff(||x|| = ||y||;x⋅x = y⋅y)
Lemma: real-vec-norm-mul
∀[n:ℕ]. ∀[x:ℝ^n]. ∀[a:ℝ]. (||a*x|| = (|a| * ||x||))
Lemma: component-rleq-real-vec-norm
∀n:ℕ. ∀i:ℕn. ∀x:ℝ^n. (|x i| ≤ ||x||)
Lemma: real-vec-norm-is-0
∀[n:ℕ]. ∀[x:ℝ^n]. uiff(||x|| = r0;req-vec(n;x;λi.r0))
Lemma: real-vec-norm-0
∀[n:ℕ]. (||λi.r0|| = r0)
Lemma: real-vec-norm-1-exists
∀[n:ℕ+]. ∃b:ℝ^n. (||b|| = r1)
Lemma: real-vec-norm-diff
∀[n:ℕ]. ∀[x,y:ℝ^n]. (||x - y|| = ||y - x||)
Lemma: real-vec-norm-dim1
∀[x:ℝ^1]. (||x|| = |x 0|)
Lemma: implies-real-vec-norm-rleq
∀[n:ℕ]. ∀[x:ℝ^n]. ∀[c:ℝ]. ((∀i:ℕn. (|x i| ≤ c)) ⇒ (||x|| ≤ (rsqrt(r(n)) * c)))
Comment: real_matrix_theory
We have some matrix theory developed for matrices over a commutative ring.
The reals do not form that kind of commutative ring
So we form ⌜real-ring()⌝ that uses the quotient type ⌜x,y:ℝ//(x = y)⌝.
Now we can instantiate theorems in the matrix theory
and use them to prove things about the unquotiented reals,
by introducing the equivalence relation ⌜X ≡ Y⌝ on the type ⌜ℝ(n × m)⌝
of n by m real matrices.
For example, from the general Cramer's rule proved for commutative rings
we get a version for the reals:
real-Cramers-rule⋅
Definition: real-ring
real-ring() == <x,y:ℝ//(x = y), λx,y. ff, λx,y. ff, λx,y. (x + y), r0, λx.-(x), λx,y. (x * y), r1, λx,y. (inr ⋅ )>
Lemma: real-ring_wf
real-ring() ∈ CRng
Lemma: real-rng_lsum-sq
∀[L,A:Top]. (Σ{real-ring()} x ∈ L. A[x] ~ reduce(λx,y. (x + y);r0;map(λx.A[x];L)))
Definition: rmatrix
ℝ(a × b) == ℕa ⟶ ℕb ⟶ ℝ
Lemma: rmatrix_wf
∀[a,b:ℕ]. (ℝ(a × b) ∈ Type)
Definition: rcolumn
col(b) == λi,j. (b i)
Lemma: rcolumn_wf
∀[n:ℕ]. ∀[b:ℝ^n]. (col(b) ∈ ℝ(n × 1))
Definition: reqmatrix
X ≡ Y == ∀i:ℕa. ∀j:ℕb. ((X i j) = (Y i j))
Lemma: reqmatrix_wf
∀[a,b:ℕ]. ∀[X,Y:ℝ(a × b)]. (X ≡ Y ∈ ℙ)
Lemma: stable__reqmatrix
∀[a,b:ℕ]. ∀[X,Y:ℝ(a × b)]. Stable{X ≡ Y}
Lemma: reqmatrix_weakening
∀[a,b:ℕ]. ∀[X,Y:ℝ(a × b)]. X ≡ Y supposing X = Y ∈ ℝ(a × b)
Lemma: reqmatrix_inversion
∀[a,b:ℕ]. ∀[X,Y:ℝ(a × b)]. Y ≡ X supposing X ≡ Y
Lemma: reqmatrix_transitivity
∀[a,b:ℕ]. ∀[X,Y,Z:ℝ(a × b)]. (X ≡ Z) supposing (X ≡ Y and Y ≡ Z)
Lemma: reqmatrix_functionality
∀[a,b:ℕ]. ∀[X1,X2,Y1,Y2:ℝ(a × b)]. (uiff(X1 ≡ Y1;X2 ≡ Y2)) supposing (Y1 ≡ Y2 and X1 ≡ X2)
Definition: real-matrix-times
(A*B) == λx,y. Σ{(A x i) * (B i y) | 0≤i≤n - 1}
Lemma: real-matrix-times_wf
∀[n,a,b:ℕ]. ∀[A:ℝ(a × n)]. ∀[B:ℝ(n × b)]. ((A*B) ∈ ℝ(a × b))
Lemma: real-matrix-times_functionality
∀[n,a,b:ℕ]. ∀[A1,A2:ℝ(a × n)]. ∀[B1,B2:ℝ(n × b)]. ((A1*B1) ≡ (A2*B2)) supposing (A1 ≡ A2 and B1 ≡ B2)
Lemma: matrix-times-req-real-matrix-times
∀[n,a,b:ℕ]. ∀[A:ℝ(a × n)]. ∀[B:ℝ(n × b)]. (A*B) ≡ (A*B)
Definition: real-matrix-add
A + B == λx,y. ((A x y) + (B x y))
Lemma: real-matrix-add_wf
∀[a,b:ℕ]. ∀[A,B:ℝ(a × b)]. (A + B ∈ ℝ(a × b))
Lemma: real-matrix-add_functionality
∀[a,b:ℕ]. ∀[A1,A2,B1,B2:ℝ(a × b)]. (A1 + B1 ≡ A2 + B2) supposing (A1 ≡ A2 and B1 ≡ B2)
Lemma: matrix-plus-sq-real-matrix-add
∀[A,B:Top]. (A + B ~ A + B)
Definition: real-matrix-scalar-mul
c*A == λx,y. (c * (A x y))
Lemma: real-matrix-scalar-mul_wf
∀[a,b:ℕ]. ∀[c:ℝ]. ∀[A:ℝ(a × b)]. (c*A ∈ ℝ(a × b))
Lemma: real-matrix-scalar-mul_functionality
∀[a,b:ℕ]. ∀[c1,c2:ℝ]. ∀[A,B:ℝ(a × b)]. (c1*A ≡ c2*B) supposing ((c1 = c2) and A ≡ B)
Lemma: matrix-scalar-mul-sq-real-matrix-scalar-mul
∀[c,A:Top]. (c*A ~ c*A)
Definition: real-det
|M| ==
r-list-sum(map(λf.let k = rprod(0;n - 1;i.M i (f i)) in
if permutation-sign(n;f)=1 then k else -(k);permutations-list(n)))
Lemma: real-det_wf
∀[n:ℕ]. ∀[M:ℕn ⟶ ℕn ⟶ ℝ]. (|M| ∈ ℝ)
Lemma: real-det-sq-matrix-det
∀n:ℕ. ∀M:Top. (|M| ~ |M|)
Lemma: real-det-req-matrix-det
∀n:ℕ. ∀M:ℕn ⟶ ℕn ⟶ ℝ. (|M| = |M|)
Lemma: real-Cramers-rule1
∀[n:ℕ]. ∀[A:ℕn ⟶ ℕn ⟶ ℝ]. ∀[c:{c:ℝ| (c * |A|) = r1} ]. ∀[b:ℝ^n].
(A*c*col(λj.|λx,y. if y=j then b x else (A x y)|)) ≡ col(b)
Lemma: real-Cramers-rule
∀[n:ℕ]. ∀[A:ℝ(n × n)]. ∀[b:ℝ^n]. (A*col(λj.(|λx,y. if y=j then b x else (A x y)|/|A|))) ≡ col(b) supposing |A| ≠ r0
Definition: r2-unit-circle
r2-unit-circle(p) == (p 0^2 + p 1^2) = r1
Lemma: r2-unit-circle_wf
∀[p:ℝ^2]. (r2-unit-circle(p) ∈ ℙ)
Definition: circle-param
circle-param(t) == λi.if (i =z 0) then (r1 - t * t/r1 + (t * t)) else (r(2) * t/r1 + (t * t)) fi
Lemma: circle-param_wf
∀[t:ℝ]. (circle-param(t) ∈ {p:ℝ^2| r2-unit-circle(p)} )
Lemma: circle-param-one-one
∀[t1,t2:ℝ]. t1 = t2 supposing req-vec(2;circle-param(t1);circle-param(t2))
Lemma: circle-param-onto
∀p:{p:ℝ^2| r2-unit-circle(p)} . ((r(-1) < (p 0)) ⇒ (∃t:ℝ. req-vec(2;circle-param(t);p)))
Definition: real-vec-extend
a++z == λi.if i <z k then a i else z fi
Lemma: real-vec-extend_wf
∀[k:ℕ+]. ∀[a:ℝ^k - 1]. ∀[z:ℝ]. (a++z ∈ ℝ^k)
Lemma: req-vec-extend
∀[k:ℕ+]. ∀[a,b:ℝ^k - 1]. ∀[z,u:ℝ]. uiff(req-vec(k;a++z;b++u);(z = u) ∧ req-vec(k - 1;a;b))
Definition: interval-vec
I^n == {v:ℝ^n| ∀i:ℕn. (v i ∈ I)}
Lemma: interval-vec_wf
∀[I:Interval]. ∀[n:ℕ]. (I^n ∈ Type)
Lemma: interval-vec-subtype
∀[I:Interval]. ∀[n,m:ℕ]. I^m ⊆r I^n supposing n ≤ m
Lemma: real-vec-extend_wf_interval
∀[I:Interval]. ∀[k:ℕ]. ∀[a:I^k]. ∀[z:{z:ℝ| z ∈ I} ]. (a++z ∈ I^k + 1)
Definition: infn
infn(n;I) == primrec(n;λf.(f ⋅);λi,r,f. inf{r (λa.(f a++z)) | z ∈ I})
Lemma: infn_wf
∀[I:{I:Interval| icompact(I)} ]
∀n:ℕ
(infn(n;I) ∈ {F:{f:I^n ⟶ ℝ| ∀a,b:I^n. (req-vec(n;a;b) ⇒ ((f a) = (f b)))} ⟶ ℝ|
∀f,g:{f:I^n ⟶ ℝ| ∀a,b:I^n. (req-vec(n;a;b) ⇒ ((f a) = (f b)))} .
((∀x:I^n. ((f x) = (g x))) ⇒ ((F f) = (F g)))} )
Lemma: infn_functionality
∀n:ℕ. ∀I:{I:Interval| icompact(I)} . ∀f,g:I^n ⟶ ℝ.
((∀x,y:I^n. (req-vec(n;x;y) ⇒ ((f x) = (f y)))) ⇒ (∀x:I^n. ((f x) = (g x))) ⇒ ((infn(n;I) f) = (infn(n;I) g)))
Lemma: rleq-infn
∀[I:{I:Interval| icompact(I)} ]
∀n:ℕ. ∀f:{f:I^n ⟶ ℝ| ∀a,b:I^n. (req-vec(n;a;b) ⇒ ((f a) = (f b)))} . ∀c:ℝ.
c ≤ (infn(n;I) f) supposing ∀a:I^n. (c ≤ (f a))
Lemma: infn-rleq
∀[I:{I:Interval| icompact(I)} ]
∀n:ℕ. ∀f:{f:I^n ⟶ ℝ| ∀a,b:I^n. (req-vec(n;a;b) ⇒ ((f a) = (f b)))} . ∀x:I^n. ((infn(n;I) f) ≤ (f x))
Lemma: infn-nonneg
∀[I:{I:Interval| icompact(I)} ]
∀n:ℕ. ∀f:{f:I^n ⟶ ℝ| ∀a,b:I^n. (req-vec(n;a;b) ⇒ ((f a) = (f b)))} .
r0 ≤ (infn(n;I) f) supposing ∀a:I^n. (r0 ≤ (f a))
Lemma: infn-property
∀I:{I:Interval| icompact(I)} . ∀n:ℕ. ∀f:{f:I^n ⟶ ℝ| ∀a,b:I^n. (req-vec(n;a;b) ⇒ ((f a) = (f b)))} . ∀e:{e:ℝ| r0 < e} \000C.
∃x:I^n. ((f x) ≤ ((infn(n;I) f) + e))
Lemma: approx-zero
∀I:{I:Interval| icompact(I)} . ∀n:ℕ. ∀f:{f:I^n ⟶ ℝ| ∀a,b:I^n. (req-vec(n;a;b) ⇒ ((f a) = (f b)))} .
((¬(∀x:I^n. f x ≠ r0)) ⇒ (∀e:{e:ℝ| r0 < e} . ∃x:I^n. (|f x| < e)))
Lemma: classical-exists-n-implies-approx
∀I:{I:Interval| icompact(I)} . ∀n:ℕ. ∀f:{f:I^n ⟶ ℝ| ∀a,b:I^n. (req-vec(n;a;b) ⇒ ((f a) = (f b)))} .
((¬¬(∃x:I^n. ((f x) = r0))) ⇒ (∀e:{e:ℝ| r0 < e} . ∃x:I^n. (|f x| < e)))
Lemma: classical-exists-implies-approx
∀I:{I:Interval| icompact(I)} . ∀f:{f:I ⟶ℝ| ifun(f;I)} .
((¬¬(∃x:{x:ℝ| x ∈ I} . (f[x] = r0))) ⇒ (∀e:{e:ℝ| r0 < e} . ∃x:{x:ℝ| x ∈ I} . (|f[x]| < e)))
Lemma: Cauchy-Schwarz
∀[n:ℕ]. ∀[x,y:ℝ^n]. (|x⋅y| ≤ (||x|| * ||y||))
Lemma: Cauchy-Schwarz-strict
∀n:ℕ. ∀x,y:ℝ^n. (∃i,j:ℕn. (x j) * (y i) ≠ (x i) * (y j) ⇐⇒ |x⋅y| < (||x|| * ||y||))
Lemma: Cauchy-Schwarz-not-strict
∀[n:ℕ]. ∀[x,y:ℝ^n]. (¬(|x⋅y| < (||x|| * ||y||)) ⇐⇒ ∀i,j:ℕn. (((x j) * (y i)) = ((x i) * (y j))))
Lemma: Cauchy-Schwarz-equality
∀[n:ℕ]. ∀[x,y:ℝ^n]. ((r0 < ||y||) ⇒ (|x⋅y| = (||x|| * ||y||)) ⇒ req-vec(n;x;(x⋅y/||y||^2)*y))
Lemma: Cauchy-Schwarz-proof2
∀[n:ℕ]. ∀[x,y:ℝ^n]. (|x⋅y| ≤ (||x|| * ||y||))
Lemma: Cauchy-Schwarz-equality2
∀n:ℕ. ∀x,y:ℝ^n. (¬(|x⋅y| < (||x|| * ||y||)) ⇐⇒ (r0 < ||y||) ⇒ (∃t:ℝ. req-vec(n;x;t*y)))
Definition: rv-pos-angle
rv-pos-angle(n;a;b;c) == |a - b⋅c - b| < (||a - b|| * ||c - b||)
Lemma: rv-pos-angle_wf
∀[n:ℕ]. ∀[a,b,c:ℝ^n]. (rv-pos-angle(n;a;b;c) ∈ ℙ)
Lemma: rv-pos-angle_functionality
∀n:ℕ. ∀a1,b1,c1,a2,b2,c2:ℝ^n.
(req-vec(n;a1;a2) ⇒ req-vec(n;b1;b2) ⇒ req-vec(n;c1;c2) ⇒ {rv-pos-angle(n;a1;b1;c1) ⇐⇒ rv-pos-angle(n;a2;b2;c2)})
Lemma: Minkowski-inequality1
∀[n:ℕ]. ∀[x,y:ℝ^n]. (||x + y|| ≤ (||x|| + ||y||))
Lemma: Minkowski-inequality2
∀[n:ℕ]. ∀[x,y:ℝ^n]. (||x - y|| ≤ (||x|| + ||y||))
Lemma: Minkowski-equality
∀n:ℕ. ∀x,y:ℝ^n.
((r0 < ||y||) ⇒ (||x + y|| = (||x|| + ||y||)) ⇒ (∃t:ℝ. ((r0 ≤ t) ∧ req-vec(n;x;t*y) ∧ ((r0 < ||x||) ⇒ (r0 < t)))))
Definition: real-vec-dist
d(x;y) == ||x - y||
Lemma: real-vec-dist_wf
∀[n:ℕ]. ∀[x,y:ℝ^n]. (d(x;y) ∈ {d:ℝ| r0 ≤ d} )
Lemma: real-vec-dist-dim0
∀[x,y:Top]. (d(x;y) = r0)
Lemma: real-vec-dist-dim1
∀[x,y:ℝ^1]. (d(x;y) = |(x 0) - y 0|)
Lemma: real-vec-dist-nonneg
∀[n:ℕ]. ∀[x,y:ℝ^n]. (r0 ≤ d(x;y))
Lemma: real-vec-dist_functionality
∀[n:ℕ]. ∀[x1,x2,y1,y2:ℝ^n]. (d(x1;y1) = d(x2;y2)) supposing (req-vec(n;x1;x2) and req-vec(n;y1;y2))
Lemma: real-vec-dist-symmetry
∀[n:ℕ]. ∀[x,y:ℝ^n]. (d(x;y) = d(y;x))
Lemma: real-vec-dist-identity
∀[n:ℕ]. ∀[x,y:ℝ^n]. uiff(d(x;y) = r0;req-vec(n;x;y))
Lemma: real-vec-dist-same-zero
∀[n:ℕ]. ∀[x:ℝ^n]. (d(x;x) = r0)
Lemma: real-vec-dist-equal-iff
∀[n:ℕ]. ∀[x,y,a,b:ℝ^n]. uiff(d(x;y) = d(a;b);x - y⋅x - y = a - b⋅a - b)
Lemma: real-vec-dist-translation
∀[n:ℕ]. ∀[x,y,z:ℝ^n]. (d(x - z;y - z) = d(x;y))
Lemma: real-vec-dist-translation2
∀[n:ℕ]. ∀[x,y,z:ℝ^n]. (d(z + x;z + y) = d(x;y))
Lemma: real-vec-dist-sub-zero
∀[n:ℕ]. ∀[p,q:ℝ^n]. (d(p - q;λi.r0) = d(p;q))
Lemma: real-vec-dist-from-zero
∀[n:ℕ]. ∀[p:ℝ^n]. (d(p;λi.r0) = ||p||)
Lemma: real-vec-dist-from-zero2
∀[n:ℕ]. ∀[p:ℝ^n]. (d(λi.r0;p) = ||p||)
Lemma: real-vec-dist-dilation
∀[n:ℕ]. ∀[x,y:ℝ^n]. ∀[a:ℝ]. (d(a*x;a*y) = (|a| * d(x;y)))
Lemma: real-vec-dist-minus
∀[n:ℕ]. ∀[x,y:ℝ^n]. (d(r(-1)*x;r(-1)*y) = d(x;y))
Lemma: real-vec-dist-vec-mul
∀[n:ℕ]. ∀[x:ℝ^n]. ∀[a,b:ℝ]. (d(a*x;b*x) = (|a - b| * ||x||))
Lemma: implies-real-vec-dist-rleq
∀[n:ℕ]. ∀[x,y:ℝ^n]. ∀[c:ℝ]. ((∀i:ℕn. (|(x i) - y i| ≤ c)) ⇒ (d(x;y) ≤ (rsqrt(r(n)) * c)))
Lemma: rleq-real-vec-dist
∀[n:ℕ]. ∀[x,y:ℝ^n]. ∀[i:ℕn]. (|(x i) - y i| ≤ d(x;y))
Lemma: real-vec-dist-monotone-in-dim
∀[m:ℕ+]. ∀[p,q:ℝ^m]. (d(p;q) ≤ d(p;q))
Lemma: real-vec-triangle-inequality
∀[n:ℕ]. ∀[x,y,z:ℝ^n]. (d(x;z) ≤ (d(x;y) + d(y;z)))
Definition: rn-metric
rn-metric(n) == λx,y. d(x;y)
Lemma: rn-metric_wf
∀[n:ℕ]. (rn-metric(n) ∈ metric(ℝ^n))
Lemma: rn-metric-meq
∀[n:ℕ]. ∀[x,y:ℝ^n]. uiff(x ≡ y;req-vec(n;x;y))
Lemma: real-vec-dist-lower-bound
∀[n:ℕ]. ∀[x,y:ℝ^n]. (|||y|| - ||x||| ≤ d(x;y))
Definition: rn-prod-metric
rn-prod-metric(n) == prod-metric(n;λi.rmetric())
Lemma: rn-prod-metric_wf
∀[n:ℕ]. (rn-prod-metric(n) ∈ metric(ℝ^n))
Lemma: mcomplete-rn-prod-metric
∀n:ℕ. mcomplete(ℝ^n with rn-prod-metric(n))
Lemma: mdist-rn-prod-metric
∀[k,x,y:Top]. (mdist(rn-prod-metric(k);x;y) ~ Σ{|(x i) - y i| | 0≤i≤k - 1})
Lemma: meq-rn-prod-metric
∀[k:ℕ]. ∀[x,y:ℝ^k]. uiff(x ≡ y;req-vec(k;x;y))
Definition: max-metric
max-metric(n) == λp,q. primrec(n;r0;λi,r. rmax(r;|(p i) - q i|))
Lemma: max-metric_wf
∀[n:ℕ]. (max-metric(n) ∈ metric(ℝ^n))
Lemma: mdist-max-metric-ub
∀[n:ℕ]. ∀[x,y:ℝ^n]. ∀i:ℕn. (|(x i) - y i| ≤ mdist(max-metric(n);x;y))
Lemma: mdist-max-metric-strict-lb
∀c:{c:ℝ| r0 < c} . ∀n:ℕ. ∀x,y:ℝ^n. ((∀i:ℕn. (|(x i) - y i| < c)) ⇒ (mdist(max-metric(n);x;y) < c))
Lemma: max-metric-mdist-from-zero-rless
∀c:{c:ℝ| r0 < c} . ∀n:ℕ. ∀x:ℝ^n. ((∀i:ℕn. (|x i| < c)) ⇒ (mdist(max-metric(n);x;λi.r0) < c))
Lemma: max-metric-mdist-from-zero
∀[c:{c:ℝ| r0 ≤ c} ]. ∀[n:ℕ]. ∀[x:ℝ^n]. uiff(mdist(max-metric(n);x;λi.r0) ≤ c;∀i:ℕn. (x i ∈ [-(c), c]))
Lemma: max-metric-mdist-from-zero-strict
∀c:{c:ℝ| r0 < c} . ∀n:ℕ. ∀x:ℝ^n. (mdist(max-metric(n);x;λi.r0) < c ⇐⇒ ∀i:ℕn. (x i ∈ (-(c), c)))
Lemma: max-metric-mdist-from-zero-2
∀[c:{c:ℝ| r0 ≤ c} ]. ∀[n:ℕ]. ∀[x:ℝ^n]. uiff(mdist(max-metric(n);λi.r0;x) ≤ c;∀i:ℕn. (x i ∈ [-(c), c]))
Lemma: max-metric-leq-rn-metric
∀[n:ℕ]. max-metric(n) ≤ rn-metric(n)
Lemma: rn-metric-leq-rn-prod-metric
∀[n:ℕ]. rn-metric(n) ≤ rn-prod-metric(n)
Lemma: rn-prod-metric-le-max-metric
∀[n:ℕ]. rn-prod-metric(n) ≤ r(n)*max-metric(n)
Lemma: rn-metric-leq-max-metric
∀[n:ℕ]. rn-metric(n) ≤ r(n)*max-metric(n)
Lemma: max-metric-complete
∀n:ℕ. mcomplete(ℝ^n with max-metric(n))
Lemma: rn-metric-complete
∀n:ℕ. mcomplete(ℝ^n with rn-metric(n))
Lemma: mdist-max-metric-mul2
∀[n:ℕ]. ∀[p,q:ℝ^n]. ∀[c:ℝ]. (mdist(max-metric(n);c*p;c*q) = (|c| * mdist(max-metric(n);p;q)))
Lemma: mdist-max-metric-mul-rleq
∀[n:ℕ]. ∀[x:ℝ^n]. ∀[c,c':ℝ]. (mdist(max-metric(n);c*x;c'*x) ≤ (|c - c'| * mdist(max-metric(n);x;λi.r0)))
Lemma: mdist-rn-metric-mul
∀[n:ℕ]. ∀[p:ℝ^n]. ∀[c:ℝ]. (mdist(rn-metric(n);c*p;λi.r0) = (|c| * mdist(rn-metric(n);p;λi.r0)))
Lemma: meq-max-metric-iff-meq-rn-metric
∀[n:ℕ]. ∀[x,y:ℝ^n]. uiff(x ≡ y;x ≡ y)
Lemma: meq-max-metric
∀[n:ℕ]. ∀[x,y:ℝ^n]. uiff(x ≡ y;req-vec(n;x;y))
Lemma: mdist-max-metric-mul
∀[n:ℕ]. ∀[p:ℝ^n]. ∀[c:ℝ]. (mdist(max-metric(n);c*p;λi.r0) = (|c| * mdist(max-metric(n);p;λi.r0)))
Lemma: req-vec-meq-max-metric
∀[n:ℕ]. ∀[v,w:ℝ^n]. v ≡ w supposing req-vec(n;v;w)
Lemma: real-vec-norm-diff-bound
∀[n:ℕ]. ∀[x,y:ℝ^n]. (|||x|| - ||y||| ≤ d(x;y))
Lemma: real-vec-angle-lemma
∀n:ℕ. ∀x,z:ℝ^n. (d(z;x) < d(z;r(-1)*x) ⇐⇒ r0 < x⋅z)
Lemma: real-vec-angle-lemma2
∀n:ℕ. ∀x,z:ℝ^n. (d(z;r(-1)*x) < d(z;x) ⇐⇒ x⋅z < r0)
Lemma: real-vec-right-angle-lemma
∀[n:ℕ]. ∀[x,z:ℝ^n]. uiff(x⋅z = r0;d(z;x) = d(z;r(-1)*x))
Lemma: rv-pos-angle-symmetry
∀n:ℕ. ∀a,b,c:ℝ^n. (rv-pos-angle(n;a;b;c) ⇒ rv-pos-angle(n;c;b;a))
Lemma: rv-pos-angle-permute-lemma
∀n:ℕ. ∀x,y:ℝ^n. ((|x⋅y| < (||x|| * ||y||)) ⇒ (|x⋅y - x| < (||x|| * ||y - x||)))
Lemma: rv-pos-angle-permute
∀n:ℕ. ∀a,b,c:ℝ^n. (rv-pos-angle(n;a;b;c) ⇒ rv-pos-angle(n;c;a;b))
Lemma: rv-pos-angle-linearity
∀n:ℕ. ∀a,b,c:ℝ^n. (rv-pos-angle(n;a;b;c) ⇒ (∀t:ℝ. ((r0 < |t|) ⇒ rv-pos-angle(n;b + t*a - b;b;c))))
Lemma: not-rv-pos-angle
∀n:ℕ. ∀a,b,c:ℝ^n.
((r0 < d(a;b)) ⇒ (r0 < d(c;b)) ⇒ (¬rv-pos-angle(n;a;b;c)) ⇒ (∃t:ℝ. ((r0 < |t|) ∧ req-vec(n;c;b + t*a - b))))
Definition: rv-congruent
ab=cd == d(a;b) = d(c;d)
Lemma: rv-congruent_wf
∀[n:ℕ]. ∀[a,b,c,d:ℝ^n]. (ab=cd ∈ ℙ)
Lemma: rv-congruent-sym
∀[n:ℕ]. ∀[a,b:ℝ^n]. ab=ba
Lemma: rv-congruent-trans
∀[n:ℕ]. ∀[a,b,p,q,r,s:ℝ^n]. (pq=rs) supposing (ab=rs and ab=pq)
Lemma: stable_rv-congruent
∀[n:ℕ]. ∀[a,b,c,d:ℝ^n]. Stable{ab=cd}
Lemma: rv-congruent_functionality
∀n:ℕ. ∀a1,a2,b1,b2,c1,c2,d1,d2:ℝ^n.
(req-vec(n;a1;a2) ⇒ req-vec(n;b1;b2) ⇒ req-vec(n;c1;c2) ⇒ req-vec(n;d1;d2) ⇒ (a1b1=c1d1 ⇐⇒ a2b2=c2d2))
Definition: real-vec-sep
a ≠ b == r0 < d(a;b)
Lemma: real-vec-sep_wf
∀[n:ℕ]. ∀[a,b:ℝ^n]. (a ≠ b ∈ ℙ)
Lemma: sq_stable__real-vec-sep
∀n:ℕ. ∀a,b:ℝ^n. SqStable(a ≠ b)
Lemma: sq_stable-dist-rless
∀n:ℕ. ∀a,b,c,d:ℝ^n. SqStable(d(c;d) < d(a;b))
Lemma: real-vec-sep_functionality
∀n:ℕ. ∀a1,a2,b1,b2:ℝ^n. (req-vec(n;a1;a2) ⇒ req-vec(n;b1;b2) ⇒ (a1 ≠ b1 ⇐⇒ a2 ≠ b2))
Lemma: real-vec-sep-symmetry
∀n:ℕ. ∀a,b:ℝ^n. (a ≠ b ⇐⇒ b ≠ a)
Lemma: real-vec-sep_inversion
∀n:ℕ. ∀x,y:ℝ^n. (x ≠ y ⇒ y ≠ x)
Lemma: real-vec-sep-cases
∀n:ℕ. ∀a,b:ℝ^n. (a ≠ b ⇒ (∀c:ℝ^n. (a ≠ c ∨ b ≠ c)))
Lemma: real-vec-sep-cases-alt
∀n:ℕ. ∀x,y,z:ℝ^n. (x ≠ y ⇒ (x ≠ z ∨ y ≠ z))
Lemma: r2-sep-or
∀a:ℝ^2. ∀b:{b:ℝ^2| d(a;a) < d(a;b)} . ∀c:ℝ^2. ((d(a;a) < d(a;c)) ∨ (d(b;b) < d(b;c)))
Lemma: not-real-vec-sep-iff-eq
∀[n:ℕ]. ∀[a,b:ℝ^n]. uiff(¬a ≠ b;req-vec(n;a;b))
Lemma: not-real-vec-sep-refl
∀[n:ℕ]. ∀[a:ℝ^n]. (¬a ≠ a)
Lemma: real-vec-sep-implies
∀n:ℕ. ∀a,c:ℝ^n. (a ≠ c ⇒ (∃i:ℕn. (r0 < |(a i) - c i|)))
Lemma: real-vec-sep-iff
∀n:ℕ. ∀a,c:ℝ^n. (a ≠ c ⇐⇒ ∃i:ℕn. (r0 < |(a i) - c i|))
Lemma: real-vec-sep-msep-prod-metric
∀n:ℕ. ∀a,c:ℝ^n. (a ≠ c ⇐⇒ a # c)
Lemma: real-vec-sep-iff-rneq
∀n:ℕ. ∀a,c:ℝ^n. (a ≠ c ⇐⇒ ∃i:ℕn. a i ≠ c i)
Lemma: rn-metric-sep
∀n:ℕ. ∀x,y:ℝ^n. (r0 < mdist(rn-metric(n);x;y) ⇐⇒ ∃i:ℕn. x i ≠ y i)
Lemma: rn-metric-msep
∀n:ℕ. ∀x,y:ℝ^n. (x # y ⇐⇒ x ≠ y)
Lemma: max-metric-sep
∀n:ℕ. ∀x,y:ℝ^n. (r0 < mdist(max-metric(n);x;y) ⇐⇒ ∃i:ℕn. x i ≠ y i)
Lemma: real-vec-sep-add
∀n:ℕ. ∀x,x',y,y':ℝ^n. (x + y ≠ x' + y' ⇒ (x ≠ x' ∨ y ≠ y'))
Lemma: real-vec-sep-mul
∀n:ℕ. ∀a,b:ℝ. ∀y,y':ℝ^n. (a*y ≠ b*y' ⇒ (a ≠ b ∨ y ≠ y'))
Lemma: real-vec-sep-0-iff
∀n:ℕ. ∀x:ℝ^n. (x ≠ λi.r0 ⇐⇒ r0 < x⋅x)
Lemma: real-vec-sep-iff-dot-product
∀n:ℕ. ∀x,y:ℝ^n. (x ≠ y ⇐⇒ r0 < y - x⋅y - x)
Definition: real-vec-free
real-vec-free(k;L) == ∀a:ℕ||L|| ⟶ ℝ. (a ≠ λi.r0 ⇒ Σ{a i*L[i] | 0≤i≤||L|| - 1} ≠ λi.r0)
Lemma: real-vec-free_wf
∀[k:ℕ]. ∀[L:ℝ^k List]. (real-vec-free(k;L) ∈ ℙ)
Lemma: sq_stable__real-vec-free
∀k:ℕ. ∀L:ℝ^k List. SqStable(real-vec-free(k;L))
Definition: is-simplex
is-simplex(k;L) == real-vec-free(k;map(λv.v - hd(L);tl(L)))
Lemma: is-simplex_wf
∀[k:ℕ]. ∀[L:ℝ^k List]. is-simplex(k;L) ∈ ℙ supposing 0 < ||L||
Lemma: sq_stable__is-simplex
∀k:ℕ. ∀L:ℝ^k List. SqStable(is-simplex(k;L)) supposing 0 < ||L||
Definition: geometric-simplex
geometric-simplex(k;n) == {L:ℝ^k List| (||L|| = (n + 1) ∈ ℤ) ∧ is-simplex(k;L)}
Lemma: geometric-simplex_wf
∀[k,n:ℕ]. (geometric-simplex(k;n) ∈ Type)
Lemma: real-vec-mul-cancel
∀[n:ℕ]. ∀[v:ℝ^n]. ∀[x,y:ℝ]. (v ≠ λi.r0 ⇒ x = y supposing req-vec(n;x*v;y*v))
Lemma: length-rneq-real-vec-sep
∀n:ℕ. ∀x,v:ℝ^n. (||v|| ≠ ||x|| ⇒ x ≠ v)
Definition: real-vec-infinity-norm
||v||∞ == mdist(max-metric(n);v;λi.r0)
Lemma: real-vec-infinity-norm_wf
∀[n:ℕ+]. ∀[v:ℝ^n]. (||v||∞ ∈ ℝ)
Lemma: real-vec-infinity-norm-req
∀[n:ℕ+]. ∀[v:ℝ^n]. (||v||∞ = if (n =z 1) then |v 0| else rmax(||v||∞;|v (n - 1)|) fi )
Lemma: infinity-norm-rneq-real-vec-sep
∀n:ℕ+. ∀x,v:ℝ^n. (||v||∞ ≠ ||x||∞ ⇒ x ≠ v)
Lemma: Cauchy-Schwarz-non-equality1
∀n:ℕ. ∀x,y:ℝ^n. ((r0 < ||y||) ⇒ (∀a:ℝ. x ≠ a*y) ⇒ (|x⋅y| < (||x|| * ||y||)))
Lemma: Cauchy-Schwarz-non-equality
∀n:ℕ. ∀x,y:ℝ^n. ((r0 < ||y||) ⇒ x ≠ (||x||/||y||)*y ⇒ x ≠ (-(||x||)/||y||)*y ⇒ (|x⋅y| < (||x|| * ||y||)))
Lemma: real-vec-perp-exists
∀n:{2...}. ∀x:ℝ^n. (x ≠ λi.r0 ⇒ (∃y:ℝ^n. (y ≠ λi.r0 ∧ (x⋅y = r0))))
Lemma: rv-congruent_functionality2
∀n:ℕ. ∀a1,a2,b1,b2,c1,c2,d1,d2:ℝ^n. ((¬a1 ≠ a2) ⇒ (¬b1 ≠ b2) ⇒ (¬c1 ≠ c2) ⇒ (¬d1 ≠ d2) ⇒ (a1b1=c1d1 ⇐⇒ a2b2=c2d2))
Lemma: rv-congruent-preserves-sep
∀n:ℕ. ∀a,b,c,d:ℝ^n. (ab=cd ⇒ a ≠ b ⇒ c ≠ d)
Lemma: rv-congruent-implies-eq
∀n:ℕ. ∀a,b,c:ℝ^n. (aa=bc ⇒ req-vec(n;b;c))
Definition: rv-between
a-b-c == a-b-c ∧ a ≠ c
Lemma: rv-between_wf
∀[n:ℕ]. ∀[a,b,c:ℝ^n]. (a-b-c ∈ ℙ)
Lemma: rv-pos-angle-implies-separated
∀n:ℕ. ∀a,b,c:ℝ^n. (rv-pos-angle(n;a;b;c) ⇒ a ≠ c)
Lemma: rv-pos-angle-implies-separated2
∀n:ℕ. ∀a,b,c:ℝ^n. (rv-pos-angle(n;a;b;c) ⇒ a ≠ b)
Lemma: rv-pos-angle-shift
∀n:ℕ. ∀a,b,c:ℝ^n. (rv-pos-angle(n;a;b;c) ⇒ (∀z:ℝ^n. (z ≠ b ⇒ (¬rv-pos-angle(n;a;b;z)) ⇒ rv-pos-angle(n;z;b;c))))
Lemma: rv-pos-angle-not-between
∀[n:ℕ]. ∀[a,b,c:ℝ^n]. ¬a-b-c supposing rv-pos-angle(n;a;b;c)
Lemma: rv-pos-angle-lemma
∀n:ℕ. ∀x,y:ℝ^n. ((||x|| = ||y||) ⇒ (r0 < d(x;y)) ⇒ (r0 < d(r(-1)*x;y)) ⇒ (|x⋅y| < (||x|| * ||y||)))
Lemma: implies-rv-pos-angle
∀n:ℕ. ∀a,b,c,a':ℝ^n. (a-b-a' ⇒ ab=a'b ⇒ ab=cb ⇒ c ≠ a ⇒ c ≠ a' ⇒ rv-pos-angle(n;a;b;c))
Lemma: rv-pos-angle-implies
∀n:ℕ. ∀a,b,c:ℝ^n.
(rv-pos-angle(n;a;b;c)
⇒ (rv-pos-angle(n;c;b;a)
∧ rv-pos-angle(n;c;a;b)
∧ a ≠ c
∧ (¬a-b-c)
∧ (∀z:ℝ^n. (z ≠ b ⇒ (¬rv-pos-angle(n;a;b;z)) ⇒ rv-pos-angle(n;z;b;c)))))
Lemma: not-rv-pos-angle-implies
∀[n:ℕ]. ∀a,b,c:ℝ^n. ((¬rv-pos-angle(n;a;b;c)) ⇒ (¬(a ≠ b ∧ b ≠ c ∧ c ≠ a ∧ (¬a-b-c) ∧ (¬b-c-a) ∧ (¬c-a-b))))
Lemma: not-rv-pos-angle-iff
∀[n:ℕ]. ∀[a,b,c:ℝ^n]. uiff(¬rv-pos-angle(n;a;b;c);¬(a ≠ b ∧ b ≠ c ∧ c ≠ a ∧ (¬a-b-c) ∧ (¬b-c-a) ∧ (¬c-a-b)))
Lemma: rv-between_functionality
∀n:ℕ. ∀a1,a2,b1,b2,c1,c2:ℝ^n. (req-vec(n;a1;a2) ⇒ req-vec(n;b1;b2) ⇒ req-vec(n;c1;c2) ⇒ (a1-b1-c1 ⇐⇒ a2-b2-c2))
Lemma: rv-between_functionality2
∀n:ℕ. ∀a1,a2,b1,b2,c1,c2:ℝ^n. ((¬a1 ≠ a2) ⇒ (¬b1 ≠ b2) ⇒ (¬c1 ≠ c2) ⇒ (a1-b1-c1 ⇐⇒ a2-b2-c2))
Lemma: rv-between_functionality3
∀n:ℕ. ∀a1,a2,b1,b2,c1,c2:ℝ^n. ((¬a1 ≠ a2) ⇒ (¬b1 ≠ b2) ⇒ (¬c1 ≠ c2) ⇒ a1-b1-c1 ⇒ a2-b2-c2)
Lemma: rv-between-symmetry
∀n:ℕ. ∀a,b,c:ℝ^n. (a-b-c ⇒ c-b-a)
Lemma: rv-between-translation
∀n:ℕ. ∀a,b,c,z:ℝ^n. (z + a-z + b-z + c ⇐⇒ a-b-c)
Lemma: rv-between-vec-mul
∀n:ℕ. ∀a,b,c:ℝ. ∀z:ℝ^n. ((r0 < ||z||) ⇒ (a*z-b*z-c*z ⇐⇒ ((a < b) ∧ (b < c)) ∨ ((c < b) ∧ (b < a))))
Lemma: real-vec-dist-between-1
∀n:ℕ. ∀a,c:ℝ^n. ∀t:ℝ. (d(a;t*a + r1 - t*c) = (|r1 - t| * d(a;c)))
Lemma: real-vec-dist-between-2
∀n:ℕ. ∀a,c:ℝ^n. ∀t:ℝ. (d(t*a + r1 - t*c;c) = (|t| * d(a;c)))
Lemma: real-vec-dist-between
∀n:ℕ. ∀a,b,c:ℝ^n. (a-b-c ⇒ (d(a;c) = (d(a;b) + d(b;c))))
Lemma: real-vec-dist-be
∀n:ℕ. ∀a,b,c:ℝ^n. (real-vec-be(n;a;b;c) ⇒ (d(a;c) = (d(a;b) + d(b;c))))
Lemma: real-vec-triangle-equality
∀n:ℕ. ∀x,y,z:ℝ^n. ((r0 < d(y;z)) ⇒ (d(x;z) = (d(x;y) + d(y;z))) ⇒ (real-vec-be(n;x;y;z) ∧ ((r0 < d(x;y)) ⇒ x-y-z)))
Lemma: real-vec-right-angle
∀[n:ℕ]. ∀[x,y,z:ℝ^n].
uiff(x - y⋅z - y = r0;∀x':ℝ^n. ((d(x';y) = d(x;y)) ⇒ real-vec-be(n;x;y;x') ⇒ (d(z;x) = d(z;x'))))
Lemma: real-vec-acute-angle
∀n:ℕ. ∀x,y,z:ℝ^n. (r0 < x - y⋅z - y ⇐⇒ ∀x':ℝ^n. ((d(x';y) = d(x;y)) ⇒ real-vec-be(n;x;y;x') ⇒ (d(z;x) < d(z;x'))))
Lemma: real-vec-obtuse-angle
∀n:ℕ. ∀x,y,z:ℝ^n. (x - y⋅z - y < r0 ⇐⇒ ∀x':ℝ^n. ((d(x';y) = d(x;y)) ⇒ real-vec-be(n;x;y;x') ⇒ (d(z;x') < d(z;x))))
Lemma: rv-between-simple
∀n:ℕ. ∀c,d:ℝ^n. ((r0 < ||d||) ⇒ c - d-c-c + d)
Lemma: rv-between-small-expand
∀n:ℕ+. ∀a,c:ℝ^n. ∀k:ℕ+. (a ≠ c ⇒ (r(k + 1)/r(k))*a + (r(-1)/r(k))*c-a-(r(k + 1)/r(k))*c + (r(-1)/r(k))*a)
Lemma: rv-between-sep
∀n:ℕ. ∀a,b,c:ℝ^n. (a-b-c ⇒ a ≠ b)
Lemma: rv-sep-between
∀n:ℕ. ∀a,b:ℝ^n. (a ≠ b ⇒ (∃m:ℝ^n. a-m-b))
Lemma: rv-Tsep
∀n:ℕ. ∀a,b:ℝ^n. ∀c:{c:ℝ^n| ¬(b ≠ c ∧ (¬a-b-c))} . (a ≠ b ⇒ a ≠ c)
Lemma: rv-between-inner-trans
∀n:ℕ. ∀a,b,c,d:ℝ^n. (a-b-d ⇒ b-c-d ⇒ a-b-c)
Lemma: rv-nontrivial
∀n:{2...}. ∃a,b,c:ℝ^n. (a ≠ b ∧ b ≠ c ∧ c ≠ a ∧ (¬a-b-c) ∧ (¬b-c-a) ∧ (¬c-a-b))
Lemma: rv-sep-exists
∀n:{1...}. ∃a,b:ℝ^n. a ≠ b
Lemma: r2-nontrivial
∃a:ℝ^2. (∃b:ℝ^2 [(d(a;a) < d(a;b))])
Lemma: rv-extend-1
∀n:ℕ. ∀a,b:ℝ^n. (a ≠ b ⇒ (∀s:{s:ℝ| r0 ≤ s} . ∃x:ℝ^n. ((d(b;x) = s) ∧ ((r0 < s) ⇒ a-b-x))))
Lemma: rv-extend
∀n:ℕ. ∀a,b,c,d:ℝ^n. (a ≠ b ⇒ (∃x:{x:ℝ^n| bx=cd} . (c ≠ d ⇒ a-b-x)))
Lemma: rv-extend-2
∀n:ℕ. ∀a,b,c,d:ℝ^n. (a ≠ b ⇒ (∃x:{x:ℝ^n| bx=cd} . (c ≠ d ⇒ a-b-x)))
Lemma: rv-inner-Pasch
∀n:ℕ. ∀a,b,c,p,q:ℝ^n. (a-p-c ⇒ b-q-c ⇒ (∃x:ℝ^n. ((a ≠ q ⇒ a-x-q) ∧ (b ≠ p ⇒ b-x-p))))
Lemma: rv-tarski-parallel
∀n:ℕ. ∀a,b,c:ℝ^n. (a ≠ b ⇒ a ≠ c ⇒ (∀d,p:ℝ^n. (b-d-c ⇒ a-d-p ⇒ (∃x,y:ℝ^n. (a-b-x ∧ x-p-y ∧ a-c-y)))))
Lemma: rv-five-segment-lemma
∀n:ℕ. ∀A,B,C,D:ℝ^n. ∀t:ℝ.
((t ∈ (r0, r1))
⇒ req-vec(n;B;t*A + r1 - t*C)
⇒ let a = d(D;B) in
let b = d(A;D) in
let c = d(B;C) in
let d = d(A;B) in
let x = d(D;C) in
let s = (t/t - r1) in
x^2 = (c^2 + a^2 + (s * (b^2 - d^2 + a^2))))
Lemma: rv-five-segment
∀[n:ℕ]. ∀[a,b,c,d,A,B,C,D:ℝ^n]. (cd=CD) supposing (bd=BD and ad=AD and bc=BC and ab=AB and A-B-C and a-b-c)
Lemma: stable_real-vec-be
∀n:ℕ. ∀a,c:ℝ^n. (a ≠ c ⇒ (∀b:ℝ^n. Stable{real-vec-be(n;a;b;c)}))
Lemma: rv-non-strict-between-iff
∀n:ℕ. ∀a,b,c:ℝ^n. (a ≠ c ⇒ (¬(a ≠ b ∧ b ≠ c ∧ (¬a-b-c)) ⇐⇒ real-vec-be(n;a;b;c)))
Definition: rv-T
rv-T(n;a;b;c) == (a ≠ c ⇒ real-vec-be(n;a;b;c)) ∧ ((¬a ≠ c) ⇒ (¬b ≠ c))
Lemma: rv-T_wf
∀[n:ℕ]. ∀[a,b,c:ℝ^n]. (rv-T(n;a;b;c) ∈ ℙ)
Lemma: rv-T-iff
∀n:ℕ. ∀a,b,c:ℝ^n. (rv-T(n;a;b;c) ⇐⇒ ¬(a ≠ b ∧ b ≠ c ∧ (¬a-b-c)))
Lemma: rv-T-dist
∀n:ℕ. ∀a,b,c:ℝ^n. (rv-T(n;a;b;c) ⇒ (d(a;c) = (d(a;b) + d(b;c))))
Lemma: rv-between-iff
∀n:ℕ. ∀a,b,c:ℝ^n. (a-b-c ⇐⇒ a ≠ b ∧ b ≠ c ∧ a ≠ c ∧ rv-T(n;a;b;c))
Lemma: sq_stable__rv-between
∀n:ℕ. ∀a,b,c:ℝ^n. SqStable(a-b-c)
Lemma: rv-Tsep-alt
∀n:ℕ. ∀a,b,c,d:ℝ^n. ((¬¬(∃u,v,w:ℝ^n. (rv-T(n;u;v;w) ∧ ab=uv ∧ cd=uw))) ⇒ a ≠ b ⇒ c ≠ d)
Lemma: rv-Tsep-alt'
∀n:ℕ. ∀a,b,c,d:ℝ^n. ((¬¬(∃u,v,w:ℝ^n. ((¬(u ≠ v ∧ v ≠ w ∧ (¬u-v-w))) ∧ ab=uv ∧ cd=uw))) ⇒ a ≠ b ⇒ c ≠ d)
Lemma: not-rv-pos-angle-implies2
∀n:ℕ. ∀a,b,c:ℝ^n. ((¬rv-pos-angle(n;a;b;c)) ⇒ (¬((¬rv-T(n;a;b;c)) ∧ (¬rv-T(n;b;c;a)) ∧ (¬rv-T(n;c;a;b)))))
Lemma: rv-Gsep
∀n:ℕ. ∀a,b,c:ℝ^n. ∀d:{d:ℝ^n| ¬¬(∃u,v,w:ℝ^n. ((¬(u ≠ v ∧ v ≠ w ∧ (¬u-v-w))) ∧ ab=uv ∧ cd=uw))} . (a ≠ b ⇒ c ≠ d)
Lemma: rv-inner-Pasch3
∀n:ℕ. ∀a,b,c,p,q:ℝ^n.
(a ≠ p
⇒ b ≠ c
⇒ rv-T(n;a;p;c)
⇒ rv-T(n;b;q;c)
⇒ (∃x:ℝ^n
(rv-T(n;a;x;q)
∧ rv-T(n;b;x;p)
∧ (a ≠ q ⇒ x ≠ a)
∧ ((a ≠ q ∧ p ≠ c ∧ b ≠ q) ⇒ x ≠ q)
∧ ((b ≠ p ∧ b ≠ q) ⇒ x ≠ b)
∧ ((b ≠ p ∧ q ≠ c) ⇒ x ≠ p))))
Lemma: rv-inner-Pasch2
∀n:ℕ. ∀a,b,c,p,q:ℝ^n.
(a ≠ p
⇒ b ≠ c
⇒ rv-T(n;a;p;c)
⇒ rv-T(n;b;q;c)
⇒ (∃x:ℝ^n
((((a ≠ q ∧ p ≠ c ∧ b ≠ q) ⇒ a-x-q) ∧ ((b ≠ p ∧ q ≠ c ∧ b ≠ q) ⇒ b-x-p)) ∧ rv-T(n;a;x;q) ∧ rv-T(n;b;x;p))))
Lemma: rv-inner-Pasch'
∀n:ℕ. ∀a,b,c,p,q:ℝ^n.
(a-p-c ⇒ b-q-c ⇒ (∃x:ℝ^n. (((a ≠ q ⇒ a-x-q) ∧ (b ≠ p ⇒ b-x-p)) ∧ rv-T(n;a;x;q) ∧ rv-T(n;b;x;p))))
Lemma: rv-inner-Pasch''
∀n:ℕ. ∀a,b,c,p,q:ℝ^n.
(a-p-c
⇒ b-q-c
⇒ (∃x:ℝ^n. ((¬(a ≠ x ∧ x ≠ q ∧ (¬a-x-q))) ∧ (¬(b ≠ x ∧ x ≠ p ∧ (¬b-x-p))) ∧ (a ≠ q ⇒ a-x-q) ∧ (b ≠ p ⇒ b-x-p))))
Definition: rv-T'
rv-T'(n;a;b;c) == ∀x,y:ℝ^n. (x-a-y ⇒ x-c-y ⇒ x-b-y)
Lemma: rv-T'_wf
∀[n:ℕ]. ∀[a,b,c:ℝ^n]. (rv-T'(n;a;b;c) ∈ ℙ)
Lemma: rv-T'-implies-rv-T
∀n:ℕ+. ∀a,b,c:ℝ^n. (rv-T'(n;a;b;c) ⇒ rv-T(n;a;b;c))
Lemma: rv-T-partially-implies-rv-T'
∀n:ℕ+. ∀a,b,c:ℝ^n. ((a ≠ c ∨ (¬a ≠ c)) ⇒ rv-T(n;a;b;c) ⇒ rv-T'(n;a;b;c))
Lemma: rv-line-circle-lemma0
∀n:ℕ. ∀r:ℝ. ∀p,q:ℝ^n. ((||p|| ≤ r) ⇒ (r0 ≤ (p⋅q - p^2 - ||q - p||^2 * (||p||^2 - r^2))))
Lemma: rv-line-circle-lemma
∀n:ℕ. ∀r:ℝ. ∀p,q:ℝ^n.
(p ≠ q
⇒ (||p|| ≤ r)
⇒ let v = q - p in
(r0 ≤ (((r(2) * p⋅v) * r(2) * p⋅v) - r(4) * ||v||^2 * (||p||^2 - r^2)))
∧ (||p + quadratic1(||v||^2;r(2) * p⋅v;||p||^2 - r^2)*v|| = r)
∧ (||p + quadratic2(||v||^2;r(2) * p⋅v;||p||^2 - r^2)*v|| = r))
Lemma: punctured-ball-boundary-retraction
∀n:ℕ. ∀p:{p:ℝ^n| ||p|| < r1} . Retract({x:ℝ^n| x ≠ p} ⟶ {x:ℝ^n| ||x|| = r1} )
Lemma: rv-line-circle-0
∀n:ℕ. ∀a,b,p,q:ℝ^n.
(p ≠ q
⇒ (d(a;p) ≤ d(a;b))
⇒ (d(a;b) ≤ d(a;q))
⇒ (∃u:{u:ℝ^n| ab=au ∧ (¬(q ≠ u ∧ u ≠ p ∧ (¬q-u-p)))}
(∃v:ℝ^n [(ab=av
∧ (¬(q ≠ p ∧ p ≠ v ∧ (¬q-p-v)))
∧ ((d(a;p) < d(a;b)) ⇒ (q-p-v ∧ ((d(a;b) < d(a;q)) ⇒ q-u-p)))
∧ ((d(a;p) = d(a;b))
⇒ ((u ≠ v ⇒ ((req-vec(n;u;p) ∧ (r0 < p - a⋅q - p)) ∨ (req-vec(n;v;p) ∧ (p - a⋅q - p < r0))))
∧ (req-vec(n;u;v) ⇒ ((p - a⋅q - p = r0) ∧ req-vec(n;u;p))))))])))
Definition: rvlinecircle0
rvlinecircle0(n;a;b;p;q) ==
<p + quadratic1(d(q - a;p - a)^2;r(2) * p - a⋅q - a - p - a;||p - a||^2 - d(a;b)^2)*q - a - p - a
, p + quadratic2(d(q - a;p - a)^2;r(2) * p - a⋅q - a - p - a;||p - a||^2 - d(a;b)^2)*q - a - p - a
>
Lemma: rvlinecircle0_wf
∀[n:ℕ]. ∀[a,b,p,q:ℝ^n].
(rvlinecircle0(n;a;b;p;q) ∈ u:{u:ℝ^n| ab=au ∧ (¬(q ≠ u ∧ u ≠ p ∧ (¬q-u-p)))} × {v:ℝ^n|
ab=av
∧ (¬(q ≠ p ∧ p ≠ v ∧ (¬q-p-v)))
∧ ((d(a;p) < d(a;b)) ⇒ (q-p-v ∧ ((d(a;b) < d(a;q)) ⇒ q-u-p)))
∧ ((d(a;p) = d(a;b))
⇒ ((u ≠ v
⇒ ((req-vec(n;u;p) ∧ (r0 < p - a⋅q - p))
∨ (req-vec(n;v;p) ∧ (p - a⋅q - p < r0))))
∧ (req-vec(n;u;v)
⇒ ((p - a⋅q - p = r0) ∧ req-vec(n;u;p)))))} ) supposing
((d(a;b) ≤ d(a;q)) and
(d(a;p) ≤ d(a;b)) and
p ≠ q)
Lemma: rv-line-circle-0-ext
∀n:ℕ. ∀a,b,p,q:ℝ^n.
(p ≠ q
⇒ (d(a;p) ≤ d(a;b))
⇒ (d(a;b) ≤ d(a;q))
⇒ (∃u:∃u:ℝ^n [(ab=au ∧ (¬(q ≠ u ∧ u ≠ p ∧ (¬q-u-p))))]
(∃v:ℝ^n [(ab=av
∧ (¬(q ≠ p ∧ p ≠ v ∧ (¬q-p-v)))
∧ ((d(a;p) < d(a;b)) ⇒ (q-p-v ∧ ((d(a;b) < d(a;q)) ⇒ q-u-p)))
∧ ((d(a;p) = d(a;b))
⇒ ((u ≠ v ⇒ ((req-vec(n;u;p) ∧ (r0 < p - a⋅q - p)) ∨ (req-vec(n;v;p) ∧ (p - a⋅q - p < r0))))
∧ (req-vec(n;u;v) ⇒ ((p - a⋅q - p = r0) ∧ req-vec(n;u;p))))))])))
Lemma: rv-line-circle-1
∀n:ℕ. ∀a,b,p,q:ℝ^n.
(a ≠ b
⇒ p ≠ q
⇒ (d(a;p) ≤ d(a;b))
⇒ (d(a;b) ≤ d(a;q))
⇒ (∃u:{u:ℝ^n| ab=au ∧ (¬(q ≠ u ∧ u ≠ p ∧ (¬q-u-p)))}
∃v:{v:ℝ^n| ab=av ∧ (¬(q ≠ p ∧ p ≠ v ∧ (¬q-p-v)))}
((d(a;p) < d(a;b)) ⇒ (q-p-v ∧ ((d(a;b) < d(a;q)) ⇒ q-u-p)))))
Lemma: rv-line-circle
∀n:ℕ. ∀a,b,p,q:ℝ^n.
(p ≠ q
⇒ (∀x:{x:ℝ^n| ap=ax ∧ (¬(a ≠ x ∧ x ≠ b ∧ (¬a-x-b)))} . ∀y:{y:ℝ^n| aq=ay ∧ (¬(a ≠ b ∧ b ≠ y ∧ (¬a-b-y)))} .
∃u:{u:ℝ^n| ab=au ∧ (¬(q ≠ u ∧ u ≠ p ∧ (¬q-u-p)))}
∃v:{v:ℝ^n| ab=av ∧ (¬(q ≠ p ∧ p ≠ v ∧ (¬q-p-v)))} . (a-x-b ⇒ (q-p-v ∧ (a-b-y ⇒ q-u-p)))))
Definition: rv-ge
cd ≥ ab == ¬¬(∃u,v,w:ℝ^n. ((¬(u ≠ v ∧ v ≠ w ∧ (¬u-v-w))) ∧ ab=uv ∧ cd=uw))
Lemma: rv-ge_wf
∀[n:ℕ]. ∀[c,d,a,b:ℝ^n]. (cd ≥ ab ∈ ℙ)
Definition: rv-be
a_b_c == ¬(a ≠ b ∧ b ≠ c ∧ (¬a-b-c))
Lemma: rv-be_wf
∀[n:ℕ]. ∀[a,b,c:ℝ^n]. (a_b_c ∈ ℙ)
Lemma: stable__rv-be
∀[n:ℕ]. ∀[a,b,c:ℝ^n]. Stable{a_b_c}
Lemma: sq_stable__rv-be
∀[n:ℕ]. ∀[a,b,c:ℝ^n]. SqStable(a_b_c)
Lemma: rv-be-inner-trans
∀a,b,c,d:ℝ^2. (a_b_d ⇒ b_c_d ⇒ a_b_c)
Lemma: rv-be-symmetry
∀n:ℕ. ∀a,b,c:ℝ^n. (a_b_c ⇒ c_b_a)
Lemma: rv-be-five-segment
∀a,b,c,d,A,B,C,D:ℝ^2. (a ≠ b ⇒ a_b_c ⇒ A_B_C ⇒ ab=AB ⇒ bc=BC ⇒ ad=AD ⇒ bd=BD ⇒ cd=CD)
Lemma: rv-be-dist
∀[n:ℕ]. ∀[a,b,c:ℝ^n]. (a_b_c ⇒ (d(a;c) = (d(a;b) + d(b;c))))
Lemma: rv-pos-angle-not-be
∀[n:ℕ]. ∀[a,b,c:ℝ^n]. ¬a_b_c supposing rv-pos-angle(n;a;b;c)
Lemma: rv-ge-dist
∀n:ℕ. ∀a,b,c,p:ℝ^n. (d(a;b) ≤ d(c;p) ⇐⇒ cp ≥ ab)
Lemma: rv-gt-dist
∀n:ℕ. ∀a,b,q:ℝ^n. ((d(a;q) < d(a;b)) ⇒ (∃w:ℝ^n. (a_w_b ∧ aw=aq ∧ w ≠ b)))
Lemma: rv-line-circle-2
∀n:ℕ. ∀a,b:ℝ^n. ∀p:{p:ℝ^n| ab ≥ ap} . ∀q:{q:ℝ^n| aq ≥ ab} .
(a ≠ b ⇒ p ≠ q ⇒ (∃u:{u:ℝ^n| ab=au ∧ q_u_p} . (∃v:ℝ^n [(ab=av ∧ q_p_v)])))
Lemma: rv-line-circle-3
∀n:ℕ. ∀c,b,d,q:ℝ^n.
(c_b_d
⇒ (d(c;d) < d(c;q))
⇒ (∃u:{u:ℝ^n| cd=cu ∧ q_u_b}
(∃v:ℝ^n [(cd=cv
∧ q_b_v
∧ (b ≠ d ⇒ (u ≠ v ∧ u ≠ b ∧ v ≠ b))
∧ (req-vec(n;b;d)
⇒ ((u ≠ v ⇒ ((req-vec(n;u;b) ∧ (r0 < b - c⋅q - b)) ∨ (req-vec(n;v;b) ∧ (b - c⋅q - b < r0))))
∧ (req-vec(n;u;v) ⇒ ((b - c⋅q - b = r0) ∧ req-vec(n;u;b))))))])))
Lemma: rv-line-circle-3-ext
∀n:ℕ. ∀c,b,d,q:ℝ^n.
(c_b_d
⇒ (d(c;d) < d(c;q))
⇒ (∃u:{u:ℝ^n| cd=cu ∧ q_u_b}
(∃v:ℝ^n [(cd=cv
∧ q_b_v
∧ (b ≠ d ⇒ (u ≠ v ∧ u ≠ b ∧ v ≠ b))
∧ (req-vec(n;b;d)
⇒ ((u ≠ v ⇒ ((req-vec(n;u;b) ∧ (r0 < b - c⋅q - b)) ∨ (req-vec(n;v;b) ∧ (b - c⋅q - b < r0))))
∧ (req-vec(n;u;v) ⇒ ((b - c⋅q - b = r0) ∧ req-vec(n;u;b))))))])))
Lemma: better-r2-straightedge-compass
∀c,d,a:ℝ^2. ∀b:{b:ℝ^2| b ≠ a ∧ c_b_d} .
∃u:{u:ℝ^2| cu=cd ∧ a_b_u}
(∃v:ℝ^2 [(cv=cd
∧ v_b_u
∧ (¬((¬a_b_v) ∧ (¬b_v_a) ∧ (¬v_a_b)))
∧ (b ≠ d ⇒ v ≠ u)
∧ ((¬d ≠ b)
⇒ ((v ≠ u ⇒ ((¬u ≠ b) ∨ (¬v ≠ b)))
∧ ((¬v ≠ u) ⇒ ((∀a':ℝ^2. ((d(a';b) = d(a;b)) ⇒ a'_b_a ⇒ (d(c;a) = d(c;a')))) ∧ (¬u ≠ b))))))])
Lemma: r2-straightedge-compass
∀c,d,a:ℝ^2. ∀b:{b:ℝ^2| b ≠ a ∧ c_b_d} .
∃u:{u:ℝ^2| cu=cd ∧ a_b_u}
(∃v:ℝ^2 [(cv=cd ∧ v_b_u ∧ (¬((¬a_b_v) ∧ (¬b_v_a) ∧ (¬v_a_b))) ∧ (b ≠ d ⇒ (v ≠ u ∧ u ≠ b ∧ v ≠ b)))])
Lemma: r2-straightedge-compass-ext
∀c,d,a:ℝ^2. ∀b:{b:ℝ^2| b ≠ a ∧ c_b_d} .
∃u:{u:ℝ^2| cu=cd ∧ a_b_u}
(∃v:ℝ^2 [(cv=cd ∧ v_b_u ∧ (¬((¬a_b_v) ∧ (¬b_v_a) ∧ (¬v_a_b))) ∧ (b ≠ d ⇒ (v ≠ u ∧ u ≠ b ∧ v ≠ b)))])
Lemma: r2-straightedge-compass-simple1
∀c,d,a:ℝ^2. ∀b:{b:ℝ^2| b ≠ a ∧ c_b_d} . (∃u:ℝ^2 [(cu=cd ∧ a_b_u ∧ (b ≠ d ⇒ b ≠ u))])
Lemma: rv-weak-triangle-inequality
∀n:ℕ. ∀a,b,x,p:ℝ^n. (ax=ab ⇒ a-x-p ⇒ p ≠ b)
Lemma: rv-weak-triangle-inequality2
∀n:ℕ. ∀a,b,x,p:ℝ^n. (ax=ab ⇒ a_x_p ⇒ bp ≥ xp)
Lemma: rv-circle-circle-lemma
∀n:ℕ. ∀r1,r2:{r:ℝ| r0 ≤ r} . ∀b:ℝ^n.
((r0 < ||b||)
⇒ (∀b':ℝ^n
((b⋅b' = r0)
⇒ (||b'|| = ||b||)
⇒ ((r1^2 - r2^2) + ||b||^2^2 ≤ (r(4) * ||b||^2 * r1^2))
⇒ let c = ((r1^2 - r2^2) + ||b||^2/r(2)) in
let d = (||b||^2 * r1^2) - c^2 in
∀x:ℝ^n
((req-vec(n;x;(r1/||b||^2)*c*b + rsqrt(d)*b') ∨ req-vec(n;x;(r1/||b||^2)*c*b - rsqrt(d)*b'))
⇒ ((||x|| = r1) ∧ (||x - b|| = r2))))))
Lemma: rv-circle-circle-lemma2
∀n:{2...}. ∀r1,r2:{r:ℝ| r0 ≤ r} . ∀b:ℝ^n.
((r0 < ||b||)
⇒ ((r1^2 - r2^2) + ||b||^2^2 ≤ (r(4) * ||b||^2 * r1^2))
⇒ (∃u,v:ℝ^n
(((||u|| = r1) ∧ (||u - b|| = r2))
∧ ((||v|| = r1) ∧ (||v - b|| = r2))
∧ (((r1^2 - r2^2) + ||b||^2^2 < (r(4) * ||b||^2 * r1^2)) ⇒ u ≠ v))))
Lemma: rv-circle-circle-lemma3
∀n:{2...}. ∀a,b,c,d:ℝ^n. ∀p:{p:ℝ^n| ab=ap} . ∀q:{q:ℝ^n| cd=cq} . ∀x:{x:ℝ^n| cp=cx ∧ (¬(c ≠ x ∧ x ≠ d ∧ (¬c-x-d)))} .
∀y:{y:ℝ^n| aq=ay ∧ (¬(a ≠ y ∧ y ≠ b ∧ (¬a-y-b)))} .
(a ≠ c ⇒ (∃u,v:{p:ℝ^n| ab=ap ∧ cd=cp} . (((d(a;y) < d(a;b)) ∧ (d(c;x) < d(c;d))) ⇒ u ≠ v)))
Lemma: rv-circle-circle
∀n:{2...}. ∀a,b,c,d:ℝ^n. ∀p:{p:ℝ^n| ab=ap} . ∀q:{q:ℝ^n| cd=cq} . ∀x:{x:ℝ^n| cp=cx ∧ (¬(c ≠ x ∧ x ≠ d ∧ (¬c-x-d)))} .
∀y:{y:ℝ^n| aq=ay ∧ (¬(a ≠ y ∧ y ≠ b ∧ (¬a-y-b)))} .
(a ≠ c ⇒ (∃u,v:{p:ℝ^n| ab=ap ∧ cd=cp} . ((x ≠ d ∧ y ≠ b) ⇒ u ≠ v)))
Definition: r2-det
|pqr| == (((p 0) * (q 1)) + ((q 0) * (r 1)) + ((r 0) * (p 1))) - ((p 1) * (q 0)) + ((q 1) * (r 0)) + ((r 1) * (p 0))
Lemma: r2-det_wf
∀[p,q,r:ℝ^2]. (|pqr| ∈ ℝ)
Lemma: r2-det-is-dot-product
∀[a,b,c:ℝ^2]. (|abc| = a - b⋅λi.if (i =z 0) then -(c - b 1) else c - b 0 fi )
Lemma: r2-det_functionality
∀[p,q,r,p',q',r':ℝ^2]. (|pqr| = |p'q'r'|) supposing (req-vec(2;r;r') and req-vec(2;q;q') and req-vec(2;p;p'))
Lemma: r2-det-symmetry
∀[p,q,r:ℝ^2]. (|pqr| = |qrp|)
Lemma: r2-det-antisymmetry
∀[p,q,r:ℝ^2]. (|pqr| = -(|prq|))
Lemma: r2-det-identity
∀[p,q,r,t:ℝ^2]. (|pqr| = (|tqr| + |ptr| + |pqt|))
Lemma: another-r2-det-identity
∀[p,q,r,t,s:ℝ^2]. (((|tqr| * |tsp|) + (|ptr| * |tsq|) + (|pqt| * |tsr|)) = r0)
Lemma: r2-det-syzygy
∀[p,q,r,u,v,w,x,y,z:ℝ^2].
(((|pqw| * |rpv| * |qry| * |prx| * |quz|)
+ (|rpv| * |qru| * |rpz| * |qpy| * |qwx|)
+ (|qru| * |pqw| * |rpz| * |prx| * |qvy|)
+ (|qru| * |pqw| * |rpz| * |qpy| * |rvx|)
+ (|pqw| * |rpv| * |pqx| * |rqz| * |ruy|)
+ (|rpv| * |qru| * |pqx| * |qpy| * |rwz|)
+ (|rpv| * |qru| * |pqx| * |rqz| * |pwy|)
+ (|qru| * |pqw| * |qry| * |prx| * |pvz|)
+ (|pqw| * |rpv| * |qry| * |rqz| * |pux|))
= r0)
Lemma: r2-det-syzygy-proof2
∀[p,q,r,u,v,w,x,y,z:ℝ^2].
(((|pqw| * |rpv| * |qry| * |prx| * |quz|)
+ (|rpv| * |qru| * |rpz| * |qpy| * |qwx|)
+ (|qru| * |pqw| * |rpz| * |prx| * |qvy|)
+ (|qru| * |pqw| * |rpz| * |qpy| * |rvx|)
+ (|pqw| * |rpv| * |pqx| * |rqz| * |ruy|)
+ (|rpv| * |qru| * |pqx| * |qpy| * |rwz|)
+ (|rpv| * |qru| * |pqx| * |rqz| * |pwy|)
+ (|qru| * |pqw| * |qry| * |prx| * |pvz|)
+ (|pqw| * |rpv| * |qry| * |rqz| * |pux|))
= r0)
Lemma: r2-det-add
∀[p,q,r,t:ℝ^2]. (|p + tqr| = (|pqr| + |tqr| + (((q 1) * (r 0)) - (q 0) * (r 1))))
Lemma: r2-det-mul
∀[p,q,r:ℝ^2]. ∀[a:ℝ]. (|a*pqr| = ((a * |pqr|) + ((r1 - a) * (((q 0) * (r 1)) - (q 1) * (r 0)))))
Lemma: r2-det-convex1
∀[p,q,r,t:ℝ^2]. ∀[a:ℝ]. (|a*p + r1 - a*tqr| = ((a * |pqr|) + ((r1 - a) * |tqr|)))
Lemma: r2-det-convex2
∀[p,q,r,t,s:ℝ^2]. ∀[a,b,c:ℝ].
|a*p + b*q + c*rts| = ((a * |pts|) + (b * |qts|) + (c * |rts|)) supposing ((a + b + c) = r1) ∧ r1 - a ≠ r0
Lemma: r2-det-convex3
∀[p,q,r,t,s:ℝ^2]. ∀[a,b,c:ℝ].
|a*p + b*q + c*rts| = ((a * |pts|) + (b * |qts|) + (c * |rts|)) supposing (a + b + c) = r1
Lemma: r2-det-nonzero
∀p,q,r:ℝ^2. (rv-pos-angle(2;p;q;r) ⇒ |pqr| ≠ r0)
Definition: r2-left
r2-left(p;q;r) == r0 < |pqr|
Lemma: r2-left_wf
∀[p,q,r:ℝ^2]. (r2-left(p;q;r) ∈ ℙ)
Lemma: sq_stable__r2-left
∀a,b,c:ℝ^2. SqStable(r2-left(a;b;c))
Lemma: sq_stable__r2-left-or
∀a,b,c:ℝ^2. SqStable(r2-left(a;b;c) ∨ r2-left(a;c;b))
Lemma: r2-left-cases
∀a,b:ℝ^2. ∀c:{c:ℝ^2| |a + r(-1)*b⋅c + r(-1)*b| < (||a + r(-1)*b|| * ||c + r(-1)*b||)} .
(r2-left(a;b;c) ∨ r2-left(a;c;b))
Lemma: r2-left-pos-angle
∀a,b,c:ℝ^2. (r2-left(a;b;c) ⇒ rv-pos-angle(2;a;b;c))
Lemma: r2-left-or-right--pos-angle
∀a,b,c:ℝ^2. (r2-left(a;b;c) ∨ r2-left(a;c;b) ⇐⇒ rv-pos-angle(2;a;b;c))
Lemma: rv-circle-circle-lemma2'
∀r1,r2:{r:ℝ| r0 ≤ r} . ∀b:ℝ^2.
((r0 < ||b||)
⇒ ((r1^2 - r2^2) + ||b||^2^2 ≤ (r(4) * ||b||^2 * r1^2))
⇒ (∃u,v:ℝ^2
(((||u|| = r1) ∧ (||u - b|| = r2))
∧ ((||v|| = r1) ∧ (||v - b|| = r2))
∧ (((r1^2 - r2^2) + ||b||^2^2 < (r(4) * ||b||^2 * r1^2)) ⇒ (r2-left(u;b;λi.r0) ∧ r2-left(v;λi.r0;b))))))
Lemma: rv-circle-circle-lemma3'
∀a,b,c,d:ℝ^2. ∀p:{p:ℝ^2| ab=ap} . ∀q:{q:ℝ^2| cd=cq} . ∀x:{x:ℝ^2| cp=cx ∧ (¬(c ≠ x ∧ x ≠ d ∧ (¬c-x-d)))} .
∀y:{y:ℝ^2| aq=ay ∧ (¬(a ≠ y ∧ y ≠ b ∧ (¬a-y-b)))} .
(a ≠ c
⇒ (∃u,v:{p:ℝ^2| ab=ap ∧ cd=cp} . (((d(a;y) < d(a;b)) ∧ (d(c;x) < d(c;d))) ⇒ (r2-left(u;c;a) ∧ r2-left(v;a;c)))))
Lemma: r2-circle-circle
∀a,b,c,d:ℝ^2. ∀p:{p:ℝ^2| ab=ap} . ∀q:{q:ℝ^2| cd=cq} . ∀x:{x:ℝ^2| cp=cx ∧ (¬(c ≠ x ∧ x ≠ d ∧ (¬c-x-d)))} .
∀y:{y:ℝ^2| aq=ay ∧ (¬(a ≠ y ∧ y ≠ b ∧ (¬a-y-b)))} .
(a ≠ c ⇒ (∃u,v:{p:ℝ^2| ab=ap ∧ cd=cp} . ((x ≠ d ∧ y ≠ b) ⇒ (r2-left(u;c;a) ∧ r2-left(v;a;c)))))
Lemma: rv-compass-compass-lemma
∀a,b,c,d:ℝ^2.
(a ≠ c
⇒ (↓∃p,q:ℝ^2. (((d(a;b) = d(a;p)) ∧ (d(c;d) = d(c;q))) ∧ (d(c;p) ≤ d(c;d)) ∧ (d(a;q) ≤ d(a;b))))
⇒ (∃u,v:{p:ℝ^2| ab=ap ∧ cd=cp}
((↓∃p,q:ℝ^2. (((d(a;b) = d(a;p)) ∧ (d(c;d) = d(c;q))) ∧ (d(c;p) < d(c;d)) ∧ (d(a;q) < d(a;b))))
⇒ (r2-left(u;c;a) ∧ r2-left(v;a;c)))))
Lemma: r2-compass-compass
∀a,b:ℝ^2. ∀c:{c:ℝ^2| a ≠ c} . ∀d:{d:ℝ^2|
↓∃p,q:ℝ^2
((ab=ap ∧ (¬¬(∃w:ℝ^2. (c_w_d ∧ cw=cp)))) ∧ cd=cq ∧ (¬¬(∃w:ℝ^2. (a_w_b ∧ aw=aq))))} .
∃u:{u:ℝ^2| ab=au ∧ cd=cu}
(∃v:ℝ^2 [((ab=av ∧ cd=cv)
∧ ((↓∃p,q:ℝ^2. ((ab=ap ∧ (↓∃w:ℝ^2. (c_w_d ∧ cw=cp ∧ w ≠ d))) ∧ cd=cq ∧ (↓∃w:ℝ^2. (a_w_b ∧ aw=aq ∧ w ≠ b))))
⇒ (r2-left(u;a;c) ∧ r2-left(v;c;a))))])
Lemma: r2-compass-compass-simple1
∀a,b:ℝ^2. ∀c:{c:ℝ^2| a ≠ c} . ∀d:{d:ℝ^2|
↓∃p,q:ℝ^2
((ab=ap ∧ (↓∃w:ℝ^2. (c_w_d ∧ cw=cp ∧ w ≠ d)))
∧ cd=cq
∧ (↓∃w:ℝ^2. (a_w_b ∧ aw=aq ∧ w ≠ b)))} .
(∃u:ℝ^2 [(ab=au ∧ cd=cu ∧ r2-left(u;a;c))])
Lemma: r2-left-sep
∀a,b,c:ℝ^2. (r2-left(a;b;c) ⇒ b ≠ c)
Lemma: r2-left-separated
∀a,b,c,d:ℝ^2. (r2-left(a;c;d) ⇒ r2-left(b;d;c) ⇒ a ≠ b)
Lemma: r2-left-convex
∀a,b,x,y,z:ℝ^2. (r2-left(x;a;b) ⇒ r2-left(z;a;b) ⇒ rv-T(2;x;y;z) ⇒ r2-left(y;a;b))
Lemma: r2-left-extend
∀a,b,x,y:ℝ^2. (r2-left(x;a;b) ⇒ rv-T(2;b;x;y) ⇒ r2-left(y;a;b))
Lemma: r2-left-between
∀a,b,x,y:ℝ^2. (r2-left(x;a;b) ⇒ rv-T(2;b;y;x) ⇒ b ≠ y ⇒ r2-left(y;a;b))
Lemma: r2-left-right-lemma
∀a,b,x,y:ℝ^2. (r2-left(x;a;b) ⇒ r2-left(y;b;a) ⇒ (∃t:ℝ. ((t ∈ [r0, r1]) ∧ (|t*x + r1 - t*yab| = r0))))
Lemma: r2-left-right
∀a,b,x,y:ℝ^2. (r2-left(x;a;b) ⇒ r2-left(y;b;a) ⇒ (∃z:ℝ^2. (rv-T(2;x;z;y) ∧ (¬rv-pos-angle(2;z;a;b)))))
Lemma: r2-plane-separation1
∀a,b:ℝ^2. ∀u:{u:ℝ^2| r2-left(u;a;b)} . ∀v:{v:ℝ^2| r2-left(v;b;a)} .
(∃x:ℝ^2 [((¬((¬a_b_x) ∧ (¬b_x_a) ∧ (¬x_a_b))) ∧ u_x_v)])
Lemma: r2-not-left-right
∀a,b,c:ℝ^2. ((¬r2-left(a;b;c)) ⇒ (¬r2-left(a;c;b)) ⇒ (¬((¬rv-T(2;a;b;c)) ∧ (¬rv-T(2;b;c;a)) ∧ (¬rv-T(2;c;a;b)))))
Lemma: r2-not-left-right-iff
∀a,b,c:ℝ^2. (¬(r2-left(a;b;c) ∨ r2-left(a;c;b)) ⇐⇒ ¬((¬a_b_c) ∧ (¬b_c_a) ∧ (¬c_a_b)))
Lemma: r2-be-iff
∀u,v,x:ℝ^2. (u_x_v ⇐⇒ ((¬(d(u;v) < d(u;x))) ∧ (¬(d(u;v) < d(x;v)))) ∧ (¬(r2-left(u;x;v) ∨ r2-left(u;v;x))))
Lemma: real-vec-sep-iff2
∀n:ℕ. ∀a,b:ℝ^n. (b ≠ a ⇐⇒ d(b;b) < d(b;a))
Lemma: rv-congruent-iff2
∀n:ℕ. ∀a,b,c,d:ℝ^n. (ab=cd ⇐⇒ ¬((d(a;b) < d(c;d)) ∨ (d(c;d) < d(a;b))))
Lemma: r2-plane-separation
∀a,b:ℝ^2. ∀u:{u:ℝ^2| r2-left(u;a;b)} . ∀v:{v:ℝ^2| r2-left(v;b;a)} .
(∃x:ℝ^2 [((¬(r2-left(a;b;x) ∨ r2-left(a;x;b)))
∧ ((¬(d(u;v) < d(u;x))) ∧ (¬(d(u;v) < d(x;v))))
∧ (¬(r2-left(u;x;v) ∨ r2-left(u;v;x))))])
Lemma: r2-straightedge-compass-simple
∀c,d,a:ℝ^2. ∀b:{b:ℝ^2|
(d(b;b) < d(b;a))
∧ ((¬(d(c;d) < d(c;b))) ∧ (¬(d(c;d) < d(b;d))))
∧ (¬(r2-left(c;b;d) ∨ r2-left(c;d;b)))} .
(∃u:ℝ^2 [((¬((d(c;u) < d(c;d)) ∨ (d(c;d) < d(c;u))))
∧ (((¬(d(a;u) < d(a;b))) ∧ (¬(d(a;u) < d(b;u)))) ∧ (¬(r2-left(a;b;u) ∨ r2-left(a;u;b))))
∧ ((d(b;b) < d(b;d)) ⇒ (d(b;b) < d(b;u))))])
Lemma: r2-compass-compass-simple
∀a,b:ℝ^2. ∀c:{c:ℝ^2| d(a;a) < d(a;c)} . ∀d:{d:ℝ^2|
↓∃p,q:ℝ^2
(((¬((d(a;b) < d(a;p)) ∨ (d(a;p) < d(a;b)))) ∧ (d(c;p) < d(c;d)))
∧ (¬((d(c;d) < d(c;q)) ∨ (d(c;q) < d(c;d))))
∧ (d(a;q) < d(a;b)))} .
(∃u:ℝ^2 [((¬((d(a;b) < d(a;u)) ∨ (d(a;u) < d(a;b)))) ∧ (¬((d(c;d) < d(c;u)) ∨ (d(c;u) < d(c;d)))) ∧ r2-left(u;a;c))])
Definition: rcp
(a x b) == λi.[((a 1) * (b 2)) - (a 2) * (b 1); ((a 2) * (b 0)) - (a 0) * (b 2); ((a 0) * (b 1)) - (a 1) * (b 0)][i]
Lemma: rcp_wf
∀[a,b:ℝ^3]. ((a x b) ∈ ℝ^3)
Lemma: rcp_functionality
∀[a1,b1,a2,b2:ℝ^3]. (req-vec(3;(a1 x b1);(a2 x b2))) supposing (req-vec(3;b1;b2) and req-vec(3;a1;a2))
Lemma: rcp-same
∀[a:ℝ^3]. req-vec(3;(a x a);λi.r0)
Lemma: rcp-0
∀[a:ℝ^3]. req-vec(3;(a x λi.r0);λi.r0)
Lemma: rcp-anti-comm
∀[a,b:ℝ^3]. req-vec(3;(b x a);r(-1)*(a x b))
Lemma: rcp-Jacobi
∀[a,b,c:ℝ^3]. req-vec(3;(a x (b x c)) + (b x (c x a)) + (c x (a x b));λi.r0)
Lemma: rcp-distrib1
∀[a,b,c:ℝ^3]. req-vec(3;(a x b + c);(a x b) + (a x c))
Lemma: rcp-distrib2
∀[a,b,c:ℝ^3]. req-vec(3;(b + c x a);(b x a) + (c x a))
Lemma: rcp-mul1
∀[a,b:ℝ^3]. ∀[c:ℝ]. req-vec(3;(c*a x b);c*(a x b))
Lemma: rcp-mul2
∀[a,b:ℝ^3]. ∀[c:ℝ]. req-vec(3;(a x c*b);c*(a x b))
Lemma: rcp-Binet-Cauchy
∀[a,b,c,d:ℝ^3]. ((a x b)⋅(c x d) = ((a⋅c * b⋅d) - a⋅d * b⋅c))
Lemma: rcp-Binet-Cauchy-corollary
∀[a,b:ℝ^3]. ((a x b)⋅(a x b) = ((a⋅a * b⋅b) - a⋅b * b⋅a))
Lemma: rcp-perp1
∀[a,b:ℝ^3]. (a⋅(a x b) = r0)
Lemma: rcp-perp2
∀[a,b:ℝ^3]. (b⋅(a x b) = r0)
Definition: r3-dp-prim
r3-dp-prim() == (vec={v:ℝ^3| v ≠ λi.r0} , sep=λa,b. (a x b) ≠ λi.r0, perp=λa,b. (a⋅b = r0))
Lemma: r3-dp-prim_wf
r3-dp-prim() ∈ DualPlanePrimitives
Definition: r-rational
r-rational(x) == ∃a:ℤ. ∃b:ℕ+. (x = (r(a)/r(b)))
Lemma: r-rational_wf
∀[x:ℝ]. (r-rational(x) ∈ ℙ)
Definition: irrational
irrational(x) == ∀a:ℤ. ∀b:ℕ+. x ≠ (r(a)/r(b))
Lemma: irrational_wf
∀[x:ℝ]. (irrational(x) ∈ ℙ)
Lemma: irrational-sqrt-number-lemma
∀a:ℤ. ∀b:ℕ+. ∀n:ℕ. (((a * a) = (n * b * b) ∈ ℤ) ⇒ (∃m:ℕn + 1. ((m * m) = n ∈ ℤ)))
Lemma: rsqrt-irrational
∀n:ℕ. (irrational(rsqrt(r(n))) ∨ (∃m:ℕn + 1. ((m * m) = n ∈ ℤ)))
Lemma: rsqrt_2-irrational
irrational(rsqrt(r(2)))
Lemma: rsqrt2-repels-rationals
∀n:ℕ+. ∀m:ℤ. ((r1/r(3 * n * n)) ≤ |rsqrt(r(2)) - (r(m)/r(n))|)
Definition: fun-converges-to
Like our definition of continuous, we quantify only over ∈ of the form (r1/r(k))
and compact subintervals of the form i-approx(I;m).⋅
lim n→∞.f[n; x] = λy.g[y] for x ∈ I ==
∀m:{m:ℕ+| icompact(i-approx(I;m))} . ∀k:ℕ+.
∃N:ℕ+. ∀x:{x:ℝ| x ∈ i-approx(I;m)} . ∀n:{N...}. (|f[n; x] - g[x]| ≤ (r1/r(k)))
Lemma: fun-converges-to_wf
∀[I:Interval]. ∀[f:ℕ ⟶ I ⟶ℝ]. ∀[g:I ⟶ℝ]. (lim n→∞.f[n;x] = λy.g[y] for x ∈ I ∈ ℙ)
Lemma: fun-converges-to_functionality
∀I:Interval. ∀f1,f2:ℕ ⟶ I ⟶ℝ.
∀[g:I ⟶ℝ]
((∀n:ℕ. ∀x:{x:ℝ| x ∈ I} . (f1[n;x] = f2[n;x]))
⇒ {lim n→∞.f1[n;x] = λy.g[y] for x ∈ I ⇒ lim n→∞.f2[n;x] = λy.g[y] for x ∈ I})
Lemma: fun-converges-to_functionality2
∀I:Interval. ∀f:ℕ ⟶ I ⟶ℝ. ∀g1:I ⟶ℝ.
∀[g2:I ⟶ℝ]
((∀x:{x:ℝ| x ∈ I} . (g1[x] = g2[x])) ⇒ {lim n→∞.f[n;x] = λy.g1[y] for x ∈ I ⇒ lim n→∞.f[n;x] = λy.g2[y] for x ∈ I}\000C)
Lemma: fun-converges-to-rsub
∀I:Interval. ∀f1,f2:ℕ ⟶ I ⟶ℝ. ∀g1,g2:I ⟶ℝ.
(lim n→∞.f1[n;x] = λy.g1[y] for x ∈ I
⇒ lim n→∞.f2[n;x] = λy.g2[y] for x ∈ I
⇒ lim n→∞.f1[n;x] - f2[n;x] = λy.g1[y] - g2[y] for x ∈ I)
Lemma: fun-converges-to-rminus
∀I:Interval. ∀f:ℕ ⟶ I ⟶ℝ. ∀g:I ⟶ℝ. (lim n→∞.f[n;x] = λy.g[y] for x ∈ I ⇒ lim n→∞.-(f[n;x]) = λy.-(g[y]) for x ∈ I)
Lemma: fun-converges-to-pointwise
∀I:Interval. ∀f:ℕ ⟶ I ⟶ℝ. ∀g:I ⟶ℝ.
(lim n→∞.f[n;x] = λy.g[y] for x ∈ I ⇒ {∀x:ℝ. ((x ∈ I) ⇒ lim n→∞.f[n;x] = g[x])})
Lemma: fun-converges-to-continuous
∀[I:Interval]. ∀[f:ℕ ⟶ I ⟶ℝ]. ∀[g:I ⟶ℝ].
(lim n→∞.f[n;x] = λy.g[y] for x ∈ I ⇒ (∀n:ℕ. f[n;x] continuous for x ∈ I) ⇒ g[y] continuous for y ∈ I)
Definition: fun-cauchy
λn.f[n; x] is cauchy for x ∈ I ==
∀a:{a:ℕ+| icompact(i-approx(I;a))} . ∀k:ℕ+.
∃N:ℕ+. ∀x:{x:ℝ| x ∈ i-approx(I;a)} . ∀n,m:{N...}. (|f[n; x] - f[m; x]| ≤ (r1/r(k)))
Lemma: fun-cauchy_wf
∀[I:Interval]. ∀[f:ℕ ⟶ I ⟶ℝ]. (λn.f[n;x] is cauchy for x ∈ I ∈ ℙ)
Definition: fun-converges
λn.f[n; x]↓ for x ∈ I) == ∃g:I ⟶ℝ. lim n→∞.f[n; x] = λy.g y for x ∈ I
Lemma: fun-converges_wf
∀[I:Interval]. ∀[f:ℕ ⟶ I ⟶ℝ]. (λn.f[n;x]↓ for x ∈ I) ∈ ℙ)
Lemma: fun-converges-converges-to
∀I:Interval. ∀f:ℕ ⟶ I ⟶ℝ. ∀g:I ⟶ℝ.
((∀x:{x:ℝ| x ∈ I} . lim n→∞.f[n;x] = g[x]) ⇒ λn.f[n;x]↓ for x ∈ I) ⇒ lim n→∞.f[n;x] = λx.g[x] for x ∈ I)
Lemma: fun-converges_functionality
∀[I:Interval]. ∀f,g:ℕ ⟶ I ⟶ℝ. ((∀n:ℕ. rfun-eq(I;f n;g n)) ⇒ λn.f[n;x]↓ for x ∈ I) ⇒ λn.g[n;x]↓ for x ∈ I))
Lemma: fun-converges-iff-cauchy
∀I:Interval. ∀f:ℕ ⟶ I ⟶ℝ. (λn.f[n;x]↓ for x ∈ I) ⇐⇒ λn.f[n;x] is cauchy for x ∈ I)
Lemma: fun-converges-on-compact
∀I:Interval. ∀f:ℕ ⟶ I ⟶ℝ.
((∀m:{m:ℕ+| icompact(i-approx(I;m))} . λn.f[n;x]↓ for x ∈ i-approx(I;m))) ⇒ λn.f[n;x]↓ for x ∈ I))
Lemma: const-fun-converges
∀I:Interval. ∀f:ℕ ⟶ ℝ. (f[n]↓ as n→∞ ⇒ λn.f[n]↓ for x ∈ I))
Lemma: fun-converges-rmul
∀I:Interval. ∀f:ℕ ⟶ I ⟶ℝ.
(λn.f[n;x]↓ for x ∈ I) ⇒ (∀g:I ⟶ℝ. (g[x] continuous for x ∈ I ⇒ λn.f[n;x] * g[x]↓ for x ∈ I))))
Definition: fun-series-converges
Σn.f[n; x]↓ for x ∈ I == λn.Σ{f[i; x] | 0≤i≤n}↓ for x ∈ I)
Lemma: fun-series-converges_wf
∀[I:Interval]. ∀[f:ℕ ⟶ I ⟶ℝ]. (Σn.f[n;x]↓ for x ∈ I ∈ ℙ)
Lemma: fun-series-converges-on-compact
∀I:Interval. ∀f:ℕ ⟶ I ⟶ℝ.
((∀m:{m:ℕ+| icompact(i-approx(I;m))} . Σn.f[n;x]↓ for x ∈ i-approx(I;m)) ⇒ Σn.f[n;x]↓ for x ∈ I)
Definition: fun-series-sum
Σn.f[n](z) == (fst(cnv)) z
Lemma: fun-series-sum_wf
∀[I:Interval]. ∀[f:ℕ ⟶ I ⟶ℝ]. ∀cnv:Σn.f[n;x]↓ for x ∈ I. ∀z:{z:ℝ| z ∈ I} . (Σn.f[n](z) ∈ ℝ)
Lemma: continuous-series-sum
∀I:Interval. ∀f:ℕ ⟶ I ⟶ℝ. ∀cnv:Σn.f[n;x]↓ for x ∈ I.
((∀n:ℕ. f[n;x] continuous for x ∈ I) ⇒ Σn.f[n](y) continuous for y ∈ I)
Definition: fun-series-converges-absolutely
Σn.f[n; x]↓ absolutely for x ∈ I == Σn.|f[n; x]|↓ for x ∈ I
Lemma: fun-series-converges-absolutely_wf
∀[I:Interval]. ∀[f:ℕ ⟶ I ⟶ℝ]. (Σn.f[n;x]↓ absolutely for x ∈ I ∈ ℙ)
Lemma: fun-comparison-test
∀I:Interval. ∀f,g:ℕ ⟶ I ⟶ℝ.
(Σn.g[n;x]↓ for x ∈ I ⇒ (∀n:ℕ. ∀x:{x:ℝ| x ∈ I} . (|f[n;x]| ≤ g[n;x])) ⇒ Σn.f[n;x]↓ for x ∈ I)
Lemma: fun-series-converges-absolutely-converges
∀I:Interval. ∀f:ℕ ⟶ I ⟶ℝ. (Σn.f[n;x]↓ absolutely for x ∈ I ⇒ Σn.f[n;x]↓ for x ∈ I)
Lemma: fun-series-converges-tail
∀M:ℕ. ∀I:Interval. ∀f:ℕ ⟶ I ⟶ℝ. (Σn.f[n + M;x]↓ for x ∈ I ⇒ Σn.f[n;x]↓ for x ∈ I)
Lemma: fun-ratio-test
∀I:Interval. ∀f:ℕ ⟶ I ⟶ℝ.
((∀n:ℕ. f[n;x] continuous for x ∈ I)
⇒ (∀m:{m:ℕ+| icompact(i-approx(I;m))}
∃c:ℝ. ((r0 ≤ c) ∧ (c < r1) ∧ (∃N:ℕ. ∀n:{N...}. ∀x:{x:ℝ| x ∈ i-approx(I;m)} . (|f[n + 1;x]| ≤ (c * |f[n;x]|)))))
⇒ Σn.f[n;x]↓ absolutely for x ∈ I)
Lemma: fun-ratio-test-everywhere
∀f:ℕ ⟶ ℝ ⟶ ℝ
((∀n:ℕ. ∀x,y:ℝ. ((x = y) ⇒ (f[n;x] = f[n;y])))
⇒ (∀m:ℕ+. ∃c:ℝ. ((r0 ≤ c) ∧ (c < r1) ∧ (∃N:ℕ. ∀n:{N...}. ∀x:{x:ℝ| |x| ≤ r(m)} . (|f[n + 1;x]| ≤ (c * |f[n;x]|)))))
⇒ Σn.f[n;x]↓ absolutely for x ∈ (-∞, ∞))
Lemma: fun-series-converges-to-everywhere
∀f:ℕ ⟶ ℝ ⟶ ℝ
((∀n:ℕ. ∀x,y:ℝ. ((x = y) ⇒ (f[n;x] = f[n;y])))
⇒ (∀m:ℕ+. ∃c:ℝ. ((r0 ≤ c) ∧ (c < r1) ∧ (∃N:ℕ. ∀n:{N...}. ∀x:{x:ℝ| |x| ≤ r(m)} . (|f[n + 1;x]| ≤ (c * |f[n;x]|)))))
⇒ (∀g:ℝ ⟶ ℝ. ((∀x:ℝ. lim n→∞.Σ{f[i;x] | 0≤i≤n} = g[x]) ⇒ lim n→∞.Σ{f[i;x] | 0≤i≤n} = λx.g[x] for x ∈ (-∞, ∞))))
Lemma: power-series-converges
∀a:ℕ ⟶ ℝ. ∀b:ℝ. ∀r:{r:ℝ| r0 < r} . ∀N:ℕ.
((∀n:{N...}. (|a[n + 1]| ≤ (|a[n]|/r))) ⇒ Σn.a[n] * x - b^n↓ absolutely for x ∈ (b - r, b + r))
Lemma: power-series-converges-to
∀a:ℕ ⟶ ℝ. ∀b:ℝ. ∀r:{r:ℝ| r0 < r} . ∀N:ℕ.
((∀n:{N...}. (|a[n + 1]| ≤ (|a[n]|/r)))
⇒ (∀g:(b - r, b + r) ⟶ℝ
((∀x:{x:ℝ| x ∈ (b - r, b + r)} . Σi.a[i] * x - b^i = g[x])
⇒ lim n→∞.Σ{a[i] * x - b^i | 0≤i≤n} = λx.g[x] for x ∈ (b - r, b + r))))
Lemma: power-series-converges-everywhere-to
∀a:ℕ ⟶ ℝ. ∀b:ℝ. ∀g:ℝ ⟶ ℝ.
((∀x:ℝ. Σi.a[i] * x - b^i = g[x])
⇒ (∀r:{r:ℝ| r0 < r} . ∃N:ℕ. ∀n:{N...}. (|a[n + 1]| ≤ (|a[n]|/r)))
⇒ lim n→∞.Σ{a[i] * x - b^i | 0≤i≤n} = λx.g[x] for x ∈ (-∞, ∞))
Lemma: power-series-converges-everywhere
∀a:ℕ ⟶ ℝ. ∀g:ℝ ⟶ ℝ.
((∀x:ℝ. Σi.a[i] * x^i = g[x])
⇒ (∀r:{r:ℝ| r0 < r} . ∃N:ℕ. ∀n:{N...}. (|a[n + 1]| ≤ (|a[n]|/r)))
⇒ lim n→∞.Σ{a[i] * x^i | 0≤i≤n} = λx.g[x] for x ∈ (-∞, ∞))
Lemma: fun-converges-to-rexp
lim n→∞.Σ{(x^i)/(i)! | 0≤i≤n} = λx.e^x for x ∈ (-∞, ∞)
Lemma: fun-converges-to-cosine
lim n→∞.Σ{-1^i * (x^2 * i)/(2 * i)! | 0≤i≤n} = λx.cosine(x) for x ∈ (-∞, ∞)
Lemma: fun-converges-to-sine
lim n→∞.Σ{-1^i * (x^(2 * i) + 1)/((2 * i) + 1)! | 0≤i≤n} = λx.sine(x) for x ∈ (-∞, ∞)
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