Nuprl Lemma : hd-copathAgree
∀[A:𝕌']. ∀[B:A ⟶ Type]. ∀[w:coW(A;a.B[a])]. ∀[x,y:copath(a.B[a];w)].
(copath-hd(x) = copath-hd(y) ∈ coW-dom(a.B[a];w)) supposing
(0 < copath-length(y) and
0 < copath-length(x) and
copathAgree(a.B[a];w;x;y))
Proof
Definitions occuring in Statement :
copathAgree: copathAgree(a.B[a];w;x;y),
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,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
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,
uimplies: b supposing a,
copath: copath(a.B[a];w),
copath-length: copath-length(p),
pi1: fst(t),
copathAgree: copathAgree(a.B[a];w;x;y),
all: ∀x:A. B[x],
nat: ℕ,
decidable: Dec(P),
or: P ∨ Q,
less_than: a < b,
and: P ∧ Q,
less_than': less_than'(a;b),
top: Top,
true: True,
squash: ↓T,
not: ¬A,
implies: P ⇒ Q,
false: False,
prop: ℙ,
coPath: coPath(a.B[a];w;n),
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
guard: {T},
copath-hd: copath-hd(p),
pi2: snd(t),
coPathAgree: coPathAgree(a.B[a];n;w;p;q),
sq_type: SQType(T),
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
btrue: tt,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
bfalse: ff
Lemmas referenced :
decidable__lt,
top_wf,
less_than_wf,
copath-length_wf,
copathAgree_wf,
eq_int_wf,
less_than_transitivity2,
le_weakening2,
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,
sqequalHypSubstitution,
productElimination,
thin,
sqequalRule,
extract_by_obid,
dependent_functionElimination,
setElimination,
rename,
hypothesisEquality,
hypothesis,
unionElimination,
because_Cache,
lessCases,
isectElimination,
sqequalAxiom,
isect_memberEquality,
independent_pairFormation,
voidElimination,
voidEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
lambdaFormation,
imageElimination,
independent_functionElimination,
lambdaEquality,
applyEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
intEquality,
instantiate,
cumulativity,
impliesFunctionality
Latex:
\mforall{}[A:\mBbbU{}']. \mforall{}[B:A {}\mrightarrow{} Type]. \mforall{}[w:coW(A;a.B[a])]. \mforall{}[x,y:copath(a.B[a];w)].
(copath-hd(x) = copath-hd(y)) supposing
(0 < copath-length(y) and
0 < copath-length(x) and
copathAgree(a.B[a];w;x;y))
Date html generated:
2018_07_25-PM-01_41_00
Last ObjectModification:
2018_06_01-AM-11_33_40
Theory : co-recursion
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