Nuprl Lemma : pcw-consistent-steps_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])].
(pcw-consistent-steps(P;p.A[p];p,a.B[p;a];p,a,b.C[p;a;b];s1;s2) ∈ ℙ)
Proof
Definitions occuring in Statement :
pcw-consistent-steps: pcw-consistent-steps(P;p.A[p];p,a.B[p; a];p,a,b.C[p; a; b];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: 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-step: pcw-step(P;p.A[p];p,a.B[p; a];p,a,b.C[p; a; b]),
pcw-consistent-steps: pcw-consistent-steps(P;p.A[p];p,a.B[p; a];p,a,b.C[p; a; b];s1;s2),
spreadn: spread3,
prop: ℙ,
and: P ∧ Q,
subtype_rel: A ⊆r B,
so_apply: x[s1;s2;s3],
so_apply: x[s1;s2],
so_apply: x[s],
so_lambda: λ2x.t[x],
so_lambda: λ2x y.t[x; y],
so_lambda: so_lambda(x,y,z.t[x; y; z]),
ext-family: F ≡ G,
all: ∀x:A. B[x],
guard: {T},
uimplies: b supposing a,
implies: P ⇒ Q,
ext-eq: A ≡ B,
pi2: snd(t),
pi1: fst(t)
Lemmas referenced :
param-co-W_wf,
equal_wf,
param-co-W-ext,
subtype_rel_weakening,
equal_functionality_wrt_subtype_rel2,
assert_wf,
isl_wf,
unit_wf2,
subtype_rel_union,
subtype_rel-equal,
pcw-step_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
productElimination,
sqequalRule,
productEquality,
cumulativity,
applyEquality,
because_Cache,
lambdaEquality,
functionExtensionality,
dependent_functionElimination,
functionEquality,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
hypothesis_subsumption,
applyLambdaEquality,
unionEquality,
axiomEquality,
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{}[s1,s2:pcw-step(P;p.A[p];p,a.B[p;a];p,a,b.C[p;a;b])].
(pcw-consistent-steps(P;p.A[p];p,a.B[p;a];p,a,b.C[p;a;b];s1;s2) \mmember{} \mBbbP{})
Date html generated:
2017_04_14-AM-07_42_00
Last ObjectModification:
2017_02_27-PM-03_13_56
Theory : co-recursion
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