Nuprl Lemma : copathAgree-extend
∀[A:𝕌']. ∀[B:A ⟶ Type]. ∀[w:coW(A;a.B[a])].
∀p:copath(a.B[a];w). ∀b:coW-dom(a.B[a];copath-at(w;p)). copathAgree(a.B[a];w;p;copath-extend(p;b))
Proof
Definitions occuring in Statement :
copathAgree: copathAgree(a.B[a];w;x;y),
copath-extend: copath-extend(q;t),
copath-at: copath-at(w;p),
copath: copath(a.B[a];w),
coW-dom: coW-dom(a.B[a];w),
coW: coW(A;a.B[a]),
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
cand: A c∧ B,
assert: ↑b,
bnot: ¬bb,
exists: ∃x:A. B[x],
it: ⋅,
unit: Unit,
bool: 𝔹,
bfalse: ff,
sq_type: SQType(T),
coPath-at: coPath-at(n;w;p),
coPath: coPath(a.B[a];w;n),
btrue: tt,
ifthenelse: if b then t else f fi ,
coPath-extend: coPath-extend(n;p;t),
eq_int: (i =z j),
coPathAgree: coPathAgree(a.B[a];n;w;p;q),
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
or: P ∨ Q,
decidable: Dec(P),
guard: {T},
true: True,
le: A ≤ B,
subtype_rel: A ⊆r B,
subtract: n - m,
uimplies: b supposing a,
uiff: uiff(P;Q),
prop: ℙ,
and: P ∧ Q,
false: False,
squash: ↓T,
not: ¬A,
less_than: a < b,
less_than': less_than'(a;b),
copath-at: copath-at(w;p),
top: Top,
nat: ℕ,
copathAgree: copathAgree(a.B[a];w;x;y),
copath-extend: copath-extend(q;t),
copath: copath(a.B[a];w),
implies: P ⇒ Q,
sq_stable: SqStable(P),
so_apply: x[s],
so_lambda: λ2x.t[x],
member: t ∈ T,
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x]
Lemmas referenced :
coW-item_wf,
neg_assert_of_eq_int,
assert-bnot,
bool_cases_sqequal,
equal_wf,
assert_of_bnot,
iff_weakening_uiff,
iff_transitivity,
eqff_to_assert,
assert_of_eq_int,
eqtt_to_assert,
bool_subtype_base,
bool_wf,
subtype_base_sq,
bool_cases,
int_subtype_base,
not_wf,
bnot_wf,
assert_wf,
less_than_irreflexivity,
le_weakening,
less_than_transitivity1,
eq_int_wf,
sq_stable__le,
primrec-wf2,
less_than_wf,
set_wf,
le-add-cancel2,
coPath_subtype,
coPath-extend_wf,
coPathAgree_wf,
add_functionality_wrt_le,
minus-minus,
zero-add,
less-iff-le,
not-le-2,
decidable__le,
subtract_wf,
all_wf,
le_weakening2,
coPath_wf,
le_wf,
coPath-at_wf,
coW_wf,
copath_wf,
copath-at_wf,
coW-dom_wf,
le-add-cancel,
add-commutes,
add-associates,
add-zero,
zero-mul,
add-mul-special,
minus-one-mul-top,
add-swap,
minus-one-mul,
nat_wf,
minus-add,
condition-implies-le,
not-lt-2,
equal-wf-base,
member_wf,
false_wf,
squash_wf,
top_wf,
copath-extend_wf,
sq_stable__copathAgree
Rules used in proof :
promote_hyp,
dependent_pairFormation,
equalityElimination,
impliesFunctionality,
equalitySymmetry,
equalityTransitivity,
unionElimination,
independent_pairFormation,
dependent_set_memberEquality,
functionExtensionality,
universeEquality,
functionEquality,
cumulativity,
instantiate,
baseClosed,
imageMemberEquality,
multiplyEquality,
minusEquality,
independent_isectElimination,
intEquality,
productEquality,
imageElimination,
voidEquality,
voidElimination,
isect_memberEquality,
sqequalAxiom,
natural_numberEquality,
because_Cache,
addEquality,
rename,
setElimination,
lessCases,
productElimination,
independent_functionElimination,
hypothesis,
dependent_functionElimination,
applyEquality,
lambdaEquality,
sqequalRule,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[A:\mBbbU{}']. \mforall{}[B:A {}\mrightarrow{} Type]. \mforall{}[w:coW(A;a.B[a])].
\mforall{}p:copath(a.B[a];w). \mforall{}b:coW-dom(a.B[a];copath-at(w;p)).
copathAgree(a.B[a];w;p;copath-extend(p;b))
Date html generated:
2018_07_25-PM-01_41_29
Last ObjectModification:
2018_07_24-PM-05_49_18
Theory : co-recursion
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