Nuprl Lemma : param-co-W-ext
∀[P:Type]. ∀[A:P ⟶ Type]. ∀[B:p:P ⟶ A[p] ⟶ Type]. ∀[C:p:P ⟶ a:A[p] ⟶ B[p;a] ⟶ P].
pco-W ≡ λp.(a:A[p] × (b:B[p;a] ⟶ (pco-W C[p;a;b])))
Proof
Definitions occuring in Statement :
param-co-W: pco-W,
ext-family: F ≡ G,
uall: ∀[x:A]. B[x],
so_apply: x[s1;s2;s3],
so_apply: x[s1;s2],
so_apply: x[s],
apply: f a,
lambda: λx.A[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,
param-co-W: pco-W,
so_apply: x[s],
so_apply: x[s1;s2],
so_apply: x[s1;s2;s3],
uimplies: b supposing a,
and: P ∧ Q,
cand: A c∧ B,
ext-family: F ≡ G,
all: ∀x:A. B[x],
ext-eq: A ≡ B,
subtype_rel: A ⊆r B,
type-family-continuous: type-family-continuous{i:l}(P;H),
sub-family: F ⊆ G,
isect-family: ⋂a:A. F[a],
nat: ℕ,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
implies: P ⇒ Q,
prop: ℙ,
top: Top,
pi1: fst(t),
pi2: snd(t),
family-monotone: family-monotone{i:l}(P;H),
so_lambda: λ2x.t[x],
guard: {T}
Lemmas referenced :
corec-family-ext,
nat_wf,
false_wf,
le_wf,
pair-eta,
equal_wf,
subtype_rel_self,
subtype_rel_wf,
subtype_rel_product,
subtype_rel_dep_function,
sub-family_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
lambdaEquality,
productEquality,
applyEquality,
functionExtensionality,
cumulativity,
functionEquality,
universeEquality,
independent_isectElimination,
independent_pairFormation,
hypothesis,
sqequalRule,
dependent_functionElimination,
productElimination,
independent_pairEquality,
axiomEquality,
isect_memberEquality,
because_Cache,
lambdaFormation,
isectEquality,
dependent_set_memberEquality,
natural_numberEquality,
equalityTransitivity,
equalitySymmetry,
voidElimination,
voidEquality,
independent_functionElimination,
dependent_pairEquality,
hyp_replacement,
applyLambdaEquality
Latex:
\mforall{}[P:Type]. \mforall{}[A:P {}\mrightarrow{} Type]. \mforall{}[B:p:P {}\mrightarrow{} A[p] {}\mrightarrow{} Type]. \mforall{}[C:p:P {}\mrightarrow{} a:A[p] {}\mrightarrow{} B[p;a] {}\mrightarrow{} P].
pco-W \mequiv{} \mlambda{}p.(a:A[p] \mtimes{} (b:B[p;a] {}\mrightarrow{} (pco-W C[p;a;b])))
Date html generated:
2017_04_14-AM-07_41_57
Last ObjectModification:
2017_02_27-PM-03_13_44
Theory : co-recursion
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