Nuprl Lemma : div-search-lemma-ext
∀a:ℤ. ∀b:{a + 1...}. ∀f:ℤ ⟶ 𝔹.
∃x:{a..b-} [((∀y:{a..x + 1-}. (¬↑(f y))) ∧ (∀z:{x + 1..b + 1-}. (↑(f z))))]
supposing ∃x:{a..b-} [((∀y:{a..x + 1-}. (¬↑(f y))) ∧ (∀z:{x + 1..b + 1-}. (↑(f z))))]
Proof
Definitions occuring in Statement :
int_upper: {i...},
int_seg: {i..j-},
assert: ↑b,
bool: 𝔹,
uimplies: b supposing a,
all: ∀x:A. B[x],
sq_exists: ∃x:A [B[x]],
not: ¬A,
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
member: t ∈ T,
subtract: n - m,
genrec-ap: genrec-ap,
ifthenelse: if b then t else f fi ,
div-search-lemma,
divide-and-conquer,
decidable__assert,
uniform-comp-nat-induction,
decidable__lt,
decidable__squash,
decidable__and,
decidable__less_than',
decidable_functionality,
squash_elim,
sq_stable_from_decidable,
any: any x,
iff_preserves_decidability,
sq_stable__from_stable,
stable__from_decidable,
uall: ∀[x:A]. B[x],
so_lambda: so_lambda(x,y,z,w.t[x; y; z; w]),
so_apply: x[s1;s2;s3;s4],
so_lambda: λ2x.t[x],
top: Top,
so_apply: x[s],
uimplies: b supposing a
Lemmas referenced :
div-search-lemma,
lifting-strict-decide,
istype-void,
strict4-decide,
lifting-strict-less,
divide-and-conquer,
decidable__assert,
uniform-comp-nat-induction,
decidable__lt,
decidable__squash,
decidable__and,
decidable__less_than',
decidable_functionality,
squash_elim,
sq_stable_from_decidable,
iff_preserves_decidability,
sq_stable__from_stable,
stable__from_decidable
Rules used in proof :
introduction,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
cut,
instantiate,
extract_by_obid,
hypothesis,
sqequalRule,
thin,
sqequalHypSubstitution,
equalityTransitivity,
equalitySymmetry,
isectElimination,
baseClosed,
Error :isect_memberEquality_alt,
voidElimination,
independent_isectElimination
Latex:
\mforall{}a:\mBbbZ{}. \mforall{}b:\{a + 1...\}. \mforall{}f:\mBbbZ{} {}\mrightarrow{} \mBbbB{}.
\mexists{}x:\{a..b\msupminus{}\} [((\mforall{}y:\{a..x + 1\msupminus{}\}. (\mneg{}\muparrow{}(f y))) \mwedge{} (\mforall{}z:\{x + 1..b + 1\msupminus{}\}. (\muparrow{}(f z))))]
supposing \mexists{}x:\{a..b\msupminus{}\} [((\mforall{}y:\{a..x + 1\msupminus{}\}. (\mneg{}\muparrow{}(f y))) \mwedge{} (\mforall{}z:\{x + 1..b + 1\msupminus{}\}. (\muparrow{}(f z))))]
Date html generated:
2019_06_20-PM-01_15_56
Last ObjectModification:
2019_03_12-PM-09_04_30
Theory : int_2
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