Nuprl Lemma : decidable__equal_product
∀[A:Type]. ∀[B:A ⟶ Type].
((∀a,b:A. Dec(a = b ∈ A)) ⇒ (∀a:A. ∀u,v:B[a]. Dec(u = v ∈ B[a])) ⇒ (∀x,y:a:A × B[a]. Dec(x = y ∈ (a:A × B[a]))))
Proof
Definitions occuring in Statement :
decidable: Dec(P),
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
product: x:A × B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x],
member: t ∈ T,
decidable: Dec(P),
or: P ∨ Q,
so_apply: x[s],
prop: ℙ,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
guard: {T},
not: ¬A,
false: False,
pi2: snd(t),
pi1: fst(t),
uimplies: b supposing a
Lemmas referenced :
all_wf,
decidable_wf,
equal_wf,
subtype_rel_self,
subtype_rel_wf,
not_wf,
equal_functionality_wrt_subtype_rel2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
productElimination,
thin,
cut,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
hypothesisEquality,
unionElimination,
productEquality,
cumulativity,
applyEquality,
functionExtensionality,
introduction,
extract_by_obid,
isectElimination,
sqequalRule,
lambdaEquality,
functionEquality,
universeEquality,
equalitySymmetry,
hyp_replacement,
Error :applyLambdaEquality,
inlFormation,
dependent_pairEquality,
inrFormation,
independent_functionElimination,
voidElimination,
because_Cache,
equalityUniverse,
levelHypothesis,
equalityTransitivity,
independent_isectElimination
Latex:
\mforall{}[A:Type]. \mforall{}[B:A {}\mrightarrow{} Type].
((\mforall{}a,b:A. Dec(a = b)) {}\mRightarrow{} (\mforall{}a:A. \mforall{}u,v:B[a]. Dec(u = v)) {}\mRightarrow{} (\mforall{}x,y:a:A \mtimes{} B[a]. Dec(x = y)))
Date html generated:
2016_10_21-AM-09_43_38
Last ObjectModification:
2016_07_12-AM-05_04_12
Theory : equality!deciders
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