Nuprl Lemma : copathAgree-last
∀[A:𝕌']. ∀[B:A ⟶ Type]. ∀[w:coW(A;a.B[a])]. ∀[p,q:copath(a.B[a];w)].
(((fst(copath-last(w;q))) = copath-at(w;p) ∈ coW(A;a.B[a]))
∧ (copath-at(w;q) = coW-item(copath-at(w;p);snd(copath-last(w;q))) ∈ coW(A;a.B[a]))) supposing
((copath-length(q) = (copath-length(p) + 1) ∈ ℤ) and
copathAgree(a.B[a];w;p;q))
Proof
Definitions occuring in Statement :
copathAgree: copathAgree(a.B[a];w;x;y),
copath-last: copath-last(w;p),
copath-length: copath-length(p),
copath-at: copath-at(w;p),
copath: copath(a.B[a];w),
coW-item: coW-item(w;b),
coW: coW(A;a.B[a]),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
pi1: fst(t),
pi2: snd(t),
and: P ∧ Q,
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
coPathAgree: coPathAgree(a.B[a];n;w;p;q),
rev_uimplies: rev_uimplies(P;Q),
copath-tl: copath-tl(x),
pi2: snd(t),
copath-hd: copath-hd(p),
eq_int: (i =z j),
it: ⋅,
unit: Unit,
bool: 𝔹,
bfalse: ff,
btrue: tt,
ifthenelse: if b then t else f fi ,
copath-last: copath-last(w;p),
coPath-at: coPath-at(n;w;p),
copath-at: copath-at(w;p),
coPath: coPath(a.B[a];w;n),
less_than: a < b,
pi1: fst(t),
copath-length: copath-length(p),
copathAgree: copathAgree(a.B[a];w;x;y),
copath: copath(a.B[a];w),
sq_type: SQType(T),
squash: ↓T,
sq_stable: SqStable(P),
exists: ∃x:A. B[x],
true: True,
less_than': less_than'(a;b),
le: A ≤ B,
top: Top,
subtract: n - m,
uiff: uiff(P;Q),
rev_implies: P ⇐ Q,
not: ¬A,
iff: P ⇐⇒ Q,
or: P ∨ Q,
decidable: Dec(P),
guard: {T},
ge: i ≥ j ,
false: False,
implies: P ⇒ Q,
nat: ℕ,
prop: ℙ,
subtype_rel: A ⊆r B,
so_apply: x[s],
so_lambda: λ2x.t[x],
all: ∀x:A. B[x],
cand: A c∧ B,
and: P ∧ Q,
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
coPathAgree_wf,
and_wf,
coPath_wf,
or_wf,
not-le-2,
coPath_subtype,
coPath-at_wf,
minus-zero,
le-add-cancel2,
not-equal-2,
decidable__int_equal,
uiff_transitivity,
assert_of_bnot,
iff_weakening_uiff,
iff_transitivity,
eqff_to_assert,
assert_of_eq_int,
eqtt_to_assert,
bool_subtype_base,
bool_cases,
add-subtract-cancel,
bool_wf,
equal-wf-T-base,
not_wf,
bnot_wf,
assert_wf,
le_antisymmetry_iff,
eq_int_wf,
sq_stable__equal,
sq_stable__and,
add-is-int-iff,
equal-wf-base,
member_wf,
top_wf,
iff_weakening_equal,
subtype_rel_self,
true_wf,
squash_wf,
subtype_rel-equal,
pi2_wf,
coW-item_wf,
le-add-cancel-alt,
not-lt-2,
decidable__lt,
subtype_base_sq,
one-mul,
zero-mul,
mul-distributes-right,
two-mul,
add-mul-special,
copath-at_wf,
pi1_wf,
coW-dom_wf,
int_subtype_base,
set_subtype_base,
copath-last_wf,
sq_stable__le,
le_weakening,
le_transitivity,
le-add-cancel,
add-zero,
add_functionality_wrt_le,
add-commutes,
add-swap,
add-associates,
minus-minus,
minus-add,
minus-one-mul-top,
zero-add,
minus-one-mul,
condition-implies-le,
less-iff-le,
not-ge-2,
false_wf,
subtract_wf,
decidable__le,
le_wf,
less_than_wf,
ge_wf,
less_than_irreflexivity,
less_than_transitivity1,
nat_properties,
coW_wf,
copath_wf,
copathAgree_wf,
nat_wf,
equal_wf,
le_reflexive,
copath-length_wf
Rules used in proof :
hyp_replacement,
applyLambdaEquality,
dependent_pairEquality,
orFunctionality,
addLevel,
inrFormation,
inlFormation,
dependent_set_memberEquality,
equalityElimination,
impliesFunctionality,
closedConclusion,
baseApply,
sqequalAxiom,
lessCases,
levelHypothesis,
equalityUniverse,
multiplyEquality,
productEquality,
promote_hyp,
sqequalIntensionalEquality,
functionExtensionality,
imageElimination,
baseClosed,
imageMemberEquality,
dependent_pairFormation,
minusEquality,
voidEquality,
unionElimination,
voidElimination,
independent_functionElimination,
intWeakElimination,
lambdaFormation,
universeEquality,
functionEquality,
cumulativity,
instantiate,
equalitySymmetry,
equalityTransitivity,
isect_memberEquality,
natural_numberEquality,
addEquality,
rename,
setElimination,
intEquality,
axiomEquality,
independent_pairEquality,
independent_pairFormation,
productElimination,
because_Cache,
independent_isectElimination,
hypothesis,
applyEquality,
lambdaEquality,
sqequalRule,
hypothesisEquality,
isectElimination,
extract_by_obid,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
introduction,
cut,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[A:\mBbbU{}']. \mforall{}[B:A {}\mrightarrow{} Type]. \mforall{}[w:coW(A;a.B[a])]. \mforall{}[p,q:copath(a.B[a];w)].
(((fst(copath-last(w;q))) = copath-at(w;p))
\mwedge{} (copath-at(w;q) = coW-item(copath-at(w;p);snd(copath-last(w;q))))) supposing
((copath-length(q) = (copath-length(p) + 1)) and
copathAgree(a.B[a];w;p;q))
Date html generated:
2018_07_25-PM-01_41_25
Last ObjectModification:
2018_07_24-AM-11_49_57
Theory : co-recursion
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