Nuprl Lemma : copath-eta2
∀[A:𝕌']. ∀[B:A ⟶ Type]. ∀[w:coW(A;a.B[a])]. ∀[p:copath(a.B[a];w)]. ∀[q:coW-dom(a.B[a];w)].
(0 < copath-length(p)
⇒ (q = copath-hd(p) ∈ coW-dom(a.B[a];w))
⇒ (copath-cons(q;copath-tl(p)) = p ∈ copath(a.B[a];w)))
Proof
Definitions occuring in Statement :
copath-cons: copath-cons(b;x),
copath-tl: copath-tl(x),
copath-hd: copath-hd(p),
copath-length: copath-length(p),
copath: copath(a.B[a];w),
coW-dom: coW-dom(a.B[a];w),
coW: coW(A;a.B[a]),
less_than: a < b,
uall: ∀[x:A]. B[x],
so_apply: x[s],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
prop: ℙ,
subtype_rel: A ⊆r B,
nat: ℕ,
squash: ↓T,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
copath-eta,
equal_wf,
coW-dom_wf,
copath-hd_wf,
less_than_wf,
copath-length_wf,
nat_wf,
copath_wf,
coW_wf,
copath-tl_wf,
subtype_rel-equal,
coW-item_wf,
subtype_rel_self,
iff_weakening_equal,
squash_wf,
true_wf,
copath-cons_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
lambdaFormation,
sqequalRule,
lambdaEquality,
applyEquality,
independent_isectElimination,
natural_numberEquality,
setElimination,
rename,
instantiate,
cumulativity,
functionEquality,
universeEquality,
functionExtensionality,
independent_functionElimination,
imageElimination,
because_Cache,
equalityTransitivity,
equalitySymmetry,
imageMemberEquality,
baseClosed,
productElimination
Latex:
\mforall{}[A:\mBbbU{}']. \mforall{}[B:A {}\mrightarrow{} Type]. \mforall{}[w:coW(A;a.B[a])]. \mforall{}[p:copath(a.B[a];w)]. \mforall{}[q:coW-dom(a.B[a];w)].
(0 < copath-length(p) {}\mRightarrow{} (q = copath-hd(p)) {}\mRightarrow{} (copath-cons(q;copath-tl(p)) = p))
Date html generated:
2018_07_25-PM-01_40_20
Last ObjectModification:
2018_06_14-AM-10_58_10
Theory : co-recursion
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