Nuprl Lemma : divide-and-conquer-ext
∀[Q:a:ℤ ⟶ {a...} ⟶ ℙ]
∀s:{2...}
((∀a:ℤ. ∀b:{a..a + s-}. Q[a;b])
⇒ (∀a,b,c:ℤ. (Q[a;c] ⇒ Q[a;b]) ∨ (Q[c;b] ⇒ Q[a;b]) supposing a < c ∧ c < b)
⇒ (∀a:ℤ. ∀b:{a...}. Q[a;b]))
Proof
Definitions occuring in Statement :
int_upper: {i...},
int_seg: {i..j-},
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
implies: P ⇒ Q,
or: P ∨ Q,
and: P ∧ Q,
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,
divide-and-conquer,
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 :
divide-and-conquer,
lifting-strict-decide,
istype-void,
strict4-decide,
lifting-strict-less,
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{}[Q:a:\mBbbZ{} {}\mrightarrow{} \{a...\} {}\mrightarrow{} \mBbbP{}]
\mforall{}s:\{2...\}
((\mforall{}a:\mBbbZ{}. \mforall{}b:\{a..a + s\msupminus{}\}. Q[a;b])
{}\mRightarrow{} (\mforall{}a,b,c:\mBbbZ{}. (Q[a;c] {}\mRightarrow{} Q[a;b]) \mvee{} (Q[c;b] {}\mRightarrow{} Q[a;b]) supposing a < c \mwedge{} c < b)
{}\mRightarrow{} (\mforall{}a:\mBbbZ{}. \mforall{}b:\{a...\}. Q[a;b]))
Date html generated:
2019_06_20-PM-01_15_43
Last ObjectModification:
2019_03_12-PM-09_29_39
Theory : int_2
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