Nuprl Lemma : pcw-steprel_wf

[P:Type]. ∀[A:P ⟶ Type]. ∀[B:p:P ⟶ A[p] ⟶ Type]. ∀[C:p:P ⟶ a:A[p] ⟶ B[p;a] ⟶ P].
[s1,s2:pcw-step(P;p.A[p];p,a.B[p;a];p,a,b.C[p;a;b])].
  (StepRel(s1;s2) ∈ ℙ)


Proof




Definitions occuring in Statement :  pcw-steprel: StepRel(s1;s2) 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: 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-steprel: StepRel(s1;s2) pcw-step: pcw-step(P;p.A[p];p,a.B[p; a];p,a,b.C[p; a; b]) spreadn: spread3 so_lambda: λ2x.t[x] so_apply: x[s] so_lambda: λ2y.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] ext-family: F ≡ G all: x:A. B[x] ext-eq: A ≡ B and: P ∧ Q subtype_rel: A ⊆B pi1: fst(t) implies:  Q prop:
Lemmas referenced :  param-co-W-ext unit_wf2 pcw-step-agree_wf true_wf equal_wf pcw-step_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut sqequalRule sqequalHypSubstitution productElimination thin hypothesis_subsumption extract_by_obid isectElimination hypothesisEquality lambdaEquality applyEquality dependent_functionElimination hypothesis because_Cache equalityTransitivity equalitySymmetry unionEquality lambdaFormation unionElimination independent_functionElimination axiomEquality isect_memberEquality functionEquality cumulativity 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{}[s1,s2:pcw-step(P;p.A[p];p,a.B[p;a];p,a,b.C[p;a;b])].
    (StepRel(s1;s2)  \mmember{}  \mBbbP{})



Date html generated: 2019_06_20-PM-00_35_35
Last ObjectModification: 2018_08_21-PM-01_53_32

Theory : co-recursion


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