Nuprl Lemma : decidable-predicate-and
∀[T,S:Type]. ∀[A:T ⟶ ℙ]. ∀[B:S ⟶ ℙ].
((∃t:T. (A t)) ⇒ (∃s:S. (B s)) ⇒ (∀p:T × S. Dec(predicate-and(A;B) p) ⇐⇒ (∀x:T. Dec(A x)) ∧ (∀y:S. Dec(B y))))
Proof
Definitions occuring in Statement :
predicate-and: predicate-and(A;B),
decidable: Dec(P),
uall: ∀[x:A]. B[x],
prop: ℙ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
product: x:A × B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
rev_implies: P ⇐ Q,
predicate-and: predicate-and(A;B),
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
cand: A c∧ B,
false: False
Lemmas referenced :
all_wf,
decidable_wf,
predicate-and_wf,
and_wf,
exists_wf,
not_wf,
decidable__and2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
independent_pairFormation,
sqequalHypSubstitution,
productElimination,
thin,
hypothesisEquality,
cut,
lemma_by_obid,
isectElimination,
productEquality,
sqequalRule,
lambdaEquality,
applyEquality,
hypothesis,
functionEquality,
cumulativity,
universeEquality,
dependent_functionElimination,
independent_pairEquality,
unionElimination,
inlFormation,
inrFormation,
introduction,
independent_functionElimination,
voidElimination,
because_Cache,
isect_memberEquality
Latex:
\mforall{}[T,S:Type]. \mforall{}[A:T {}\mrightarrow{} \mBbbP{}]. \mforall{}[B:S {}\mrightarrow{} \mBbbP{}].
((\mexists{}t:T. (A t))
{}\mRightarrow{} (\mexists{}s:S. (B s))
{}\mRightarrow{} (\mforall{}p:T \mtimes{} S. Dec(predicate-and(A;B) p) \mLeftarrow{}{}\mRightarrow{} (\mforall{}x:T. Dec(A x)) \mwedge{} (\mforall{}y:S. Dec(B y))))
Date html generated:
2016_05_14-PM-04_09_02
Last ObjectModification:
2015_12_26-PM-07_54_50
Theory : fan-theorem
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