Nuprl Lemma : pcw-final-step_wf
∀[P:Type]. ∀[A:P ⟶ Type]. ∀[B:p:P ⟶ A[p] ⟶ Type]. ∀[C:p:P ⟶ a:A[p] ⟶ B[p;a] ⟶ P].
∀[s:pcw-step(P;p.A[p];p,a.B[p;a];p,a,b.C[p;a;b])].
(pcw-final-step(s) ∈ ℙ)
Proof
Definitions occuring in Statement :
pcw-final-step: pcw-final-step(s),
pcw-step: pcw-step(P;p.A[p];p,a.B[p; a];p,a,b.C[p; a; b]),
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-final-step: pcw-final-step(s),
pcw-step: pcw-step(P;p.A[p];p,a.B[p; a];p,a,b.C[p; a; b]),
spreadn: spread3,
isr: isr(x),
so_lambda: λ2x.t[x],
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]
Lemmas referenced :
assert_wf,
bfalse_wf,
btrue_wf,
pcw-step_wf,
istype-universe
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
sqequalRule,
sqequalHypSubstitution,
productElimination,
thin,
extract_by_obid,
isectElimination,
unionElimination,
hypothesis,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
Error :universeIsType,
hypothesisEquality,
Error :lambdaEquality_alt,
applyEquality,
because_Cache,
Error :isect_memberEquality_alt,
Error :functionIsType,
Error :inhabitedIsType,
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{}[s:pcw-step(P;p.A[p];p,a.B[p;a];p,a,b.C[p;a;b])].
(pcw-final-step(s) \mmember{} \mBbbP{})
Date html generated:
2019_06_20-PM-00_35_33
Last ObjectModification:
2018_10_06-AM-11_20_36
Theory : co-recursion
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