Nuprl Lemma : coW-item_wf
∀[A:𝕌']. ∀[B:A ⟶ Type]. ∀[w:coW(A;a.B[a])]. ∀[b:coW-dom(a.B[a];w)]. (coW-item(w;b) ∈ coW(A;a.B[a]))
Proof
Definitions occuring in Statement :
coW-item: coW-item(w;b),
coW-dom: coW-dom(a.B[a];w),
coW: coW(A;a.B[a]),
uall: ∀[x:A]. B[x],
so_apply: x[s],
member: t ∈ T,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
coW-item: coW-item(w;b),
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
all: ∀x:A. B[x],
implies: P ⇒ Q,
prop: ℙ,
coW-dom: coW-dom(a.B[a];w),
pi1: fst(t),
ext-eq: A ≡ B,
and: P ∧ Q
Lemmas referenced :
coW-ext,
subtype_rel_weakening,
coW_wf,
pi2_wf,
equal_wf,
coW-dom_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
hypothesis_subsumption,
thin,
instantiate,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
lambdaEquality,
cumulativity,
applyEquality,
because_Cache,
hypothesis,
productEquality,
functionEquality,
independent_isectElimination,
productElimination,
independent_pairEquality,
lambdaFormation,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
independent_functionElimination,
axiomEquality,
isect_memberEquality,
universeEquality
Latex:
\mforall{}[A:\mBbbU{}']. \mforall{}[B:A {}\mrightarrow{} Type]. \mforall{}[w:coW(A;a.B[a])]. \mforall{}[b:coW-dom(a.B[a];w)].
(coW-item(w;b) \mmember{} coW(A;a.B[a]))
Date html generated:
2018_07_25-PM-01_37_34
Last ObjectModification:
2018_06_01-AM-09_50_00
Theory : co-recursion
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