Nuprl Lemma : copathAgree-tl
∀[A:𝕌']. ∀[B:A ⟶ Type]. ∀[w:coW(A;a.B[a])].
∀p,q:copath(a.B[a];w).
(copathAgree(a.B[a];w;p;q)
⇒ 0 < copath-length(p)
⇒ 0 < copath-length(q)
⇒ copathAgree(a.B[a];coW-item(w;copath-hd(p));copath-tl(p);copath-tl(q)))
Proof
Definitions occuring in Statement :
copathAgree: copathAgree(a.B[a];w;x;y),
copath-tl: copath-tl(x),
copath-hd: copath-hd(p),
copath-length: copath-length(p),
copath: copath(a.B[a];w),
coW-item: coW-item(w;b),
coW: coW(A;a.B[a]),
less_than: a < b,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
copath: copath(a.B[a];w),
copathAgree: copathAgree(a.B[a];w;x;y),
copath-tl: copath-tl(x),
copath-length: copath-length(p),
pi1: fst(t),
member: t ∈ T,
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,
false: False,
prop: ℙ,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
uimplies: b supposing a,
coPathAgree: coPathAgree(a.B[a];n;w;p;q),
bfalse: ff,
exists: ∃x:A. B[x],
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
subtract: n - m,
subtype_rel: A ⊆r B,
le: A ≤ B,
so_lambda: λ2x.t[x],
so_apply: x[s],
coPath: coPath(a.B[a];w;n),
pi2: snd(t),
copath-hd: copath-hd(p),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
decidable__lt,
top_wf,
less_than_wf,
lt_int_wf,
subtract_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
not-lt-2,
condition-implies-le,
add-associates,
nat_wf,
minus-add,
minus-minus,
minus-one-mul,
add-swap,
minus-one-mul-top,
add-commutes,
zero-add,
less-iff-le,
add_functionality_wrt_le,
le-add-cancel-alt,
copath-length_wf,
copathAgree_wf,
copath_wf,
coW_wf,
eq_int_wf,
less_than_transitivity1,
le_weakening,
less_than_irreflexivity,
assert_wf,
bnot_wf,
not_wf,
equal-wf-T-base,
less_than_transitivity2,
le_weakening2,
bool_cases,
assert_of_eq_int,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
sqequalRule,
cut,
introduction,
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,
imageElimination,
independent_functionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
dependent_pairFormation,
promote_hyp,
instantiate,
addEquality,
minusEquality,
applyEquality,
lambdaEquality,
intEquality,
cumulativity,
functionEquality,
universeEquality,
impliesFunctionality
Latex:
\mforall{}[A:\mBbbU{}']. \mforall{}[B:A {}\mrightarrow{} Type]. \mforall{}[w:coW(A;a.B[a])].
\mforall{}p,q:copath(a.B[a];w).
(copathAgree(a.B[a];w;p;q)
{}\mRightarrow{} 0 < copath-length(p)
{}\mRightarrow{} 0 < copath-length(q)
{}\mRightarrow{} copathAgree(a.B[a];coW-item(w;copath-hd(p));copath-tl(p);copath-tl(q)))
Date html generated:
2018_07_25-PM-01_41_10
Last ObjectModification:
2018_06_01-AM-11_59_21
Theory : co-recursion
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