Nuprl 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|
Proof
Definitions occuring in Statement :
real-ratio-bound: real-ratio-bound(M;x;y;a;b),
rdiv: (x/y),
rleq: x ≤ y,
rless: x < y,
rabs: |x|,
rsub: x - y,
req: x = y,
int-to-real: r(n),
real: ℝ,
nat_plus: ℕ+,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
or: P ∨ Q,
and: P ∧ Q,
set: {x:A| B[x]} ,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
member: t ∈ T,
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B,
and: P ∧ Q,
real-ratio-bound: real-ratio-bound(M;x;y;a;b),
implies: P ⇒ Q,
prop: ℙ,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
rless: x < y,
sq_exists: ∃x:A [B[x]],
nat_plus: ℕ+,
int_seg: {i..j-},
lelt: i ≤ j < k,
less_than: a < b,
squash: ↓T,
decidable: Dec(P),
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_stable: SqStable(P),
sq_type: SQType(T),
eq_int: (i =z j),
ifthenelse: if b then t else f fi ,
bfalse: ff,
btrue: tt,
cand: A c∧ B,
uiff: uiff(P;Q),
rge: x ≥ y,
rgt: x > y,
req_int_terms: t1 ≡ t2
Latex:
\mforall{}M:\mBbbN{}\msupplus{}. \mforall{}x,y:\mBbbR{}.
\mforall{}[a,b:\{r:\mBbbR{}| r0 < r\} ].
((x < y) \mwedge{} (real-ratio-bound(M;x;y;a;b) = (a/y - x)))
\mvee{} ((y < x) \mwedge{} (real-ratio-bound(M;x;y;a;b) = (b/x - y)))
supposing (r(2)/r(M)) \mleq{} |x - y|
Date html generated:
2020_05_20-AM-11_07_19
Last ObjectModification:
2019_11_06-PM-06_28_08
Theory : reals
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