Nuprl Lemma : copathAgree_transitivity
∀[A:𝕌']. ∀[B:A ⟶ Type]. ∀[w:coW(A;a.B[a])].
∀x,y,z:copath(a.B[a];w).
((copath-length(x) ≤ copath-length(y))
⇒ (copath-length(y) ≤ copath-length(z))
⇒ copathAgree(a.B[a];w;x;y)
⇒ copathAgree(a.B[a];w;y;z)
⇒ copathAgree(a.B[a];w;x;z))
Proof
Definitions occuring in Statement :
copathAgree: copathAgree(a.B[a];w;x;y),
copath-length: copath-length(p),
copath: copath(a.B[a];w),
coW: coW(A;a.B[a]),
uall: ∀[x:A]. B[x],
so_apply: x[s],
le: A ≤ B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
sq_stable: SqStable(P),
gt: i > j,
subtype_rel: A ⊆r B,
so_apply: x[s],
so_lambda: λ2x.t[x],
guard: {T},
le: A ≤ B,
uimplies: b supposing a,
uiff: uiff(P;Q),
prop: ℙ,
false: False,
not: ¬A,
squash: ↓T,
true: True,
top: Top,
less_than': less_than'(a;b),
and: P ∧ Q,
less_than: a < b,
or: P ∨ Q,
decidable: Dec(P),
nat: ℕ,
member: t ∈ T,
pi1: fst(t),
implies: P ⇒ Q,
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
copath: copath(a.B[a];w),
copath-length: copath-length(p),
copathAgree: copathAgree(a.B[a];w;x;y)
Lemmas referenced :
sq_stable__le,
coPathAgree_le,
coPathAgree_transitivity,
coW_wf,
coPath_wf,
nat_wf,
le_wf,
not-gt-2,
le_weakening2,
coPath_subtype,
coPathAgree_wf,
less_than_irreflexivity,
less_than_transitivity1,
not-lt,
less_than_wf,
top_wf,
decidable__lt
Rules used in proof :
universeEquality,
functionEquality,
cumulativity,
instantiate,
productEquality,
applyEquality,
lambdaEquality,
lessEquality,
spreadEquality,
independent_isectElimination,
independent_functionElimination,
imageElimination,
baseClosed,
imageMemberEquality,
natural_numberEquality,
voidEquality,
voidElimination,
independent_pairFormation,
isect_memberEquality,
sqequalAxiom,
isectElimination,
lessCases,
because_Cache,
unionElimination,
hypothesis,
hypothesisEquality,
rename,
setElimination,
dependent_functionElimination,
extract_by_obid,
introduction,
cut,
sqequalHypSubstitution,
thin,
productElimination,
lambdaFormation,
isect_memberFormation,
computationStep,
sqequalTransitivity,
sqequalReflexivity,
sqequalRule,
sqequalSubstitution
Latex:
\mforall{}[A:\mBbbU{}']. \mforall{}[B:A {}\mrightarrow{} Type]. \mforall{}[w:coW(A;a.B[a])].
\mforall{}x,y,z:copath(a.B[a];w).
((copath-length(x) \mleq{} copath-length(y))
{}\mRightarrow{} (copath-length(y) \mleq{} copath-length(z))
{}\mRightarrow{} copathAgree(a.B[a];w;x;y)
{}\mRightarrow{} copathAgree(a.B[a];w;y;z)
{}\mRightarrow{} copathAgree(a.B[a];w;x;z))
Date html generated:
2018_07_25-PM-01_40_52
Last ObjectModification:
2018_06_16-AM-11_57_20
Theory : co-recursion
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