Nuprl Lemma : copath-eta
∀[A:𝕌']. ∀[B:A ⟶ Type]. ∀[w:coW(A;a.B[a])]. ∀[p:copath(a.B[a];w)].
(0 < copath-length(p) ⇒ (copath-cons(copath-hd(p);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: 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,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
nat: ℕ,
copath: copath(a.B[a];w),
coPath: coPath(a.B[a];w;n),
copath-length: copath-length(p),
pi1: fst(t),
false: False,
guard: {T},
uimplies: b supposing a,
all: ∀x:A. B[x],
top: Top,
or: P ∨ Q,
sq_type: SQType(T),
uiff: uiff(P;Q),
and: P ∧ Q,
ifthenelse: if b then t else f fi ,
btrue: tt,
iff: P ⇐⇒ Q,
not: ¬A,
rev_implies: P ⇐ Q,
bfalse: ff
Lemmas referenced :
less_than_wf,
copath-length_wf,
nat_wf,
copath_wf,
coW_wf,
copath-cons-hd-tl,
eq_int_wf,
less_than_transitivity1,
le_weakening,
less_than_irreflexivity,
assert_wf,
bnot_wf,
not_wf,
equal-wf-T-base,
bool_cases,
subtype_base_sq,
bool_wf,
bool_subtype_base,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
hypothesis,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
hypothesisEquality,
sqequalRule,
lambdaEquality,
applyEquality,
setElimination,
rename,
dependent_functionElimination,
axiomEquality,
because_Cache,
isect_memberEquality,
instantiate,
cumulativity,
functionEquality,
universeEquality,
productElimination,
independent_pairEquality,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
independent_functionElimination,
voidElimination,
intEquality,
baseClosed,
voidEquality,
unionElimination,
independent_pairFormation,
impliesFunctionality
Latex:
\mforall{}[A:\mBbbU{}']. \mforall{}[B:A {}\mrightarrow{} Type]. \mforall{}[w:coW(A;a.B[a])]. \mforall{}[p:copath(a.B[a];w)].
(0 < copath-length(p) {}\mRightarrow{} (copath-cons(copath-hd(p);copath-tl(p)) = p))
Date html generated:
2018_07_25-PM-01_40_16
Last ObjectModification:
2018_06_14-AM-10_48_50
Theory : co-recursion
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