Nuprl Lemma : fix_wf_coW_parameter
∀[P:Type]. ∀[A:𝕌']. ∀[B:A ⟶ Type]. ∀[G:⋂W:𝕌'. ((P ⟶ W) ⟶ P ⟶ (a:A × (B[a] ⟶ W)))]. ∀[p:P].
(fix(G) p ∈ 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,
apply: f a,
fix: fix(F),
isect: ⋂x:A. B[x],
function: x:A ⟶ B[x],
product: x:A × B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
so_apply: x[s],
so_lambda: λ2x.t[x],
isect2: T1 ⋂ T2,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
top: Top,
bfalse: ff,
ext-eq: A ≡ B,
and: P ∧ Q
Lemmas referenced :
fix_wf_corec_parameter,
top_wf,
bool_wf,
coW-corec
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
applyEquality,
hypothesis,
sqequalHypSubstitution,
sqequalRule,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
hypothesisEquality,
isect_memberEquality,
isectElimination,
thin,
because_Cache,
isectEquality,
universeEquality,
cumulativity,
functionEquality,
productEquality,
instantiate,
extract_by_obid,
lambdaEquality,
unionElimination,
equalityElimination,
functionExtensionality,
voidElimination,
voidEquality,
productElimination
Latex:
\mforall{}[P:Type]. \mforall{}[A:\mBbbU{}']. \mforall{}[B:A {}\mrightarrow{} Type]. \mforall{}[G:\mcap{}W:\mBbbU{}'. ((P {}\mrightarrow{} W) {}\mrightarrow{} P {}\mrightarrow{} (a:A \mtimes{} (B[a] {}\mrightarrow{} W)))]. \mforall{}[p:P].
(fix(G) p \mmember{} coW(A;a.B[a]))
Date html generated:
2019_06_20-PM-00_56_05
Last ObjectModification:
2019_01_02-PM-01_32_43
Theory : co-recursion-2
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