Nuprl 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)} )
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,
rsub: x - y,
int-to-real: r(n),
real: ℝ,
nat_plus: ℕ+,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
member: t ∈ T,
set: {x:A| B[x]} ,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
nat_plus: ℕ+,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
prop: ℙ,
real-ratio-bound: real-ratio-bound(M;x;y;a;b),
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
sq_type: SQType(T),
guard: {T},
eq_int: (i =z j),
ifthenelse: if b then t else f fi ,
bfalse: ff,
btrue: tt,
rneq: x ≠ y,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
cand: A c∧ B,
uiff: uiff(P;Q),
req_int_terms: t1 ≡ t2,
rless: x < y,
sq_exists: ∃x:A [B[x]],
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
sq_stable: SqStable(P),
rdiv: (x/y),
top: Top
Latex:
\mforall{}[M:\mBbbN{}\msupplus{}]. \mforall{}[x,y:\mBbbR{}]. \mforall{}[a,b:\{r:\mBbbR{}| r0 < r\} ].
(real-ratio-bound(M;x;y;a;b) \mmember{} \{r:\mBbbR{}|
((x < y) {}\mRightarrow{} (r \mleq{} (a/y - x)))
\mwedge{} ((y < x) {}\mRightarrow{} (r \mleq{} (b/x - y)))
\mwedge{} (r0 < r)\} )
Date html generated:
2020_05_20-AM-11_07_06
Last ObjectModification:
2020_01_06-PM-00_39_12
Theory : reals
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