Nuprl Lemma : pcw-consistent-paths_wf
∀[P:Type]. ∀[A:P ⟶ Type]. ∀[B:p:P ⟶ A[p] ⟶ Type]. ∀[C:p:P ⟶ a:A[p] ⟶ B[p;a] ⟶ P]. ∀[f,g:Path].
(pcw-consistent-paths(P;p.A[p];p,a.B[p;a];p,a,b.C[p;a;b];f;g) ∈ ℙ)
Proof
Definitions occuring in Statement :
pcw-consistent-paths: pcw-consistent-paths(P;p.A[p];p,a.B[p; a];p,a,b.C[p; a; b];f;g),
pcw-path: Path,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2;s3],
so_apply: x[s1;s2],
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,
pcw-consistent-paths: pcw-consistent-paths(P;p.A[p];p,a.B[p; a];p,a,b.C[p; a; b];f;g),
so_lambda: λ2x.t[x],
implies: P ⇒ Q,
prop: ℙ,
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],
pcw-path: Path
Lemmas referenced :
all_wf,
nat_wf,
not_wf,
pcw-final-step_wf,
equal_wf,
pcw-step_wf,
pcw-path_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
lambdaEquality,
functionEquality,
cumulativity,
hypothesisEquality,
applyEquality,
functionExtensionality,
because_Cache,
setElimination,
rename,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality,
universeEquality
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]. \mforall{}[f,g:Path].
(pcw-consistent-paths(P;p.A[p];p,a.B[p;a];p,a,b.C[p;a;b];f;g) \mmember{} \mBbbP{})
Date html generated:
2017_04_14-AM-07_42_02
Last ObjectModification:
2017_02_27-PM-03_13_47
Theory : co-recursion
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