Nuprl Lemma : fix_wf_coW
∀[A:𝕌']. ∀[B:A ⟶ Type]. ∀[G:⋂W:𝕌'. (W ⟶ (a:A × (B[a] ⟶ W)))]. (fix(G) ∈ coW(A;a.B[a]))
Proof
Definitions occuring in Statement :
coW: coW(A;a.B[a]),
uall: ∀[x:A]. B[x],
so_apply: x[s],
member: t ∈ T,
fix: fix(F),
isect: ⋂x:A. B[x],
function: x:A ⟶ B[x],
product: x:A × B[x],
universe: Type
Definitions unfolded in proof :
and: P ∧ Q,
ext-eq: A ≡ B,
subtype_rel: A ⊆r B,
so_apply: x[s],
so_lambda: λ2x.t[x],
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
coW-corec,
fix_wf_corec-alt-proof
Rules used in proof :
productElimination,
because_Cache,
isect_memberEquality,
isectEquality,
equalitySymmetry,
equalityTransitivity,
axiomEquality,
hypothesis,
universeEquality,
applyEquality,
cumulativity,
functionEquality,
hypothesisEquality,
productEquality,
lambdaEquality,
sqequalRule,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
instantiate,
thin,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[A:\mBbbU{}']. \mforall{}[B:A {}\mrightarrow{} Type]. \mforall{}[G:\mcap{}W:\mBbbU{}'. (W {}\mrightarrow{} (a:A \mtimes{} (B[a] {}\mrightarrow{} W)))]. (fix(G) \mmember{} coW(A;a.B[a]))
Date html generated:
2018_07_29-AM-09_21_30
Last ObjectModification:
2018_07_27-PM-03_53_01
Theory : co-recursion
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