Nuprl Lemma : W-ext
∀[A:Type]. ∀[B:A ⟶ Type]. W(A;a.B[a]) ≡ a:A × (B[a] ⟶ W(A;a.B[a]))
Proof
Definitions occuring in Statement :
W: W(A;a.B[a]),
ext-eq: A ≡ B,
uall: ∀[x:A]. B[x],
so_apply: x[s],
function: x:A ⟶ B[x],
product: x:A × B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
so_apply: x[s],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
ext-family: F ≡ G,
all: ∀x:A. B[x],
W: W(A;a.B[a]),
ext-eq: A ≡ B,
and: P ∧ Q,
subtype_rel: A ⊆r B
Lemmas referenced :
param-W-ext,
unit_wf2,
it_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
sqequalRule,
lambdaEquality,
hypothesisEquality,
applyEquality,
dependent_functionElimination,
productElimination,
independent_pairEquality,
axiomEquality,
functionEquality,
cumulativity,
universeEquality,
isect_memberEquality,
because_Cache
Latex:
\mforall{}[A:Type]. \mforall{}[B:A {}\mrightarrow{} Type]. W(A;a.B[a]) \mequiv{} a:A \mtimes{} (B[a] {}\mrightarrow{} W(A;a.B[a]))
Date html generated:
2018_05_21-PM-00_05_34
Last ObjectModification:
2018_05_14-AM-10_38_01
Theory : co-recursion
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