Nuprl Lemma : intuitionistic-pigeonhole1
∀[A,B:ℕ ⟶ ℙ].
((∀s:StrictInc. ∃n:ℕ. A[s n]) ⇒ (∀s:StrictInc. ∃n:ℕ. B[s n]) ⇒ (∀s:StrictInc. ∃n:ℕ. (A[s n] ∧ B[s n])))
Proof
Definitions occuring in Statement :
strict-inc: StrictInc,
nat: ℕ,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x],
so_lambda: λ2x.t[x],
member: t ∈ T,
so_apply: x[s],
strict-inc: StrictInc,
prop: ℙ,
compose: f o g,
exists: ∃x:A. B[x],
and: P ∧ Q,
cand: A c∧ B,
guard: {T}
Lemmas referenced :
unary-strong-almost-full-has-strict-inc,
nat_wf,
strict-inc_wf,
all_wf,
exists_wf,
compose-strict-inc,
and_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
sqequalRule,
lambdaEquality,
applyEquality,
hypothesisEquality,
setElimination,
rename,
hypothesis,
independent_functionElimination,
because_Cache,
functionEquality,
cumulativity,
universeEquality,
dependent_functionElimination,
productElimination,
dependent_pairFormation,
independent_pairFormation
Latex:
\mforall{}[A,B:\mBbbN{} {}\mrightarrow{} \mBbbP{}].
((\mforall{}s:StrictInc. \mexists{}n:\mBbbN{}. A[s n])
{}\mRightarrow{} (\mforall{}s:StrictInc. \mexists{}n:\mBbbN{}. B[s n])
{}\mRightarrow{} (\mforall{}s:StrictInc. \mexists{}n:\mBbbN{}. (A[s n] \mwedge{} B[s n])))
Date html generated:
2016_05_14-PM-09_48_39
Last ObjectModification:
2015_12_26-PM-09_47_08
Theory : continuity
Home
Index