Nuprl Lemma : coPathAgree_wf
∀[A:𝕌']. ∀[B:A ⟶ Type]. ∀[n:ℕ]. ∀[w:coW(A;a.B[a])]. ∀[p,q:coPath(a.B[a];w;n)]. (coPathAgree(a.B[a];n;w;p;q) ∈ ℙ)
Proof
Definitions occuring in Statement :
coPathAgree: coPathAgree(a.B[a];n;w;p;q)
,
coPath: coPath(a.B[a];w;n)
,
coW: coW(A;a.B[a])
,
nat: ℕ
,
uall: ∀[x:A]. B[x]
,
prop: ℙ
,
so_apply: x[s]
,
member: t ∈ T
,
function: x:A ⟶ B[x]
,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x]
,
member: t ∈ T
,
nat: ℕ
,
implies: P
⇒ Q
,
false: False
,
ge: i ≥ j
,
guard: {T}
,
uimplies: b supposing a
,
prop: ℙ
,
so_lambda: λ2x.t[x]
,
so_apply: x[s]
,
coPathAgree: coPathAgree(a.B[a];n;w;p;q)
,
eq_int: (i =z j)
,
all: ∀x:A. B[x]
,
bool: 𝔹
,
unit: Unit
,
it: ⋅
,
btrue: tt
,
uiff: uiff(P;Q)
,
and: P ∧ Q
,
ifthenelse: if b then t else f fi
,
bfalse: ff
,
exists: ∃x:A. B[x]
,
or: P ∨ Q
,
sq_type: SQType(T)
,
bnot: ¬bb
,
assert: ↑b
,
subtract: n - m
,
nequal: a ≠ b ∈ T
,
not: ¬A
,
le: A ≤ B
,
less_than': less_than'(a;b)
,
decidable: Dec(P)
,
iff: P
⇐⇒ Q
,
rev_implies: P
⇐ Q
,
subtype_rel: A ⊆r B
,
top: Top
,
true: True
,
coPath: coPath(a.B[a];w;n)
,
squash: ↓T
Lemmas referenced :
nat_properties,
less_than_transitivity1,
less_than_irreflexivity,
ge_wf,
less_than_wf,
coPath_wf,
coW_wf,
btrue_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
true_wf,
eqff_to_assert,
eq_int_wf,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
false_wf,
le_wf,
decidable__le,
subtract_wf,
not-ge-2,
less-iff-le,
condition-implies-le,
minus-one-mul,
zero-add,
minus-one-mul-top,
minus-add,
minus-minus,
add-associates,
add-swap,
add-commutes,
add_functionality_wrt_le,
add-zero,
le-add-cancel,
le_weakening2,
nat_wf,
int_subtype_base,
assert_wf,
bnot_wf,
not_wf,
equal-wf-base,
coW-dom_wf,
coW-item_wf,
subtype_rel-equal,
not-le-2,
not-equal-2,
and_wf,
bool_cases,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
sqequalRule,
intWeakElimination,
lambdaFormation,
natural_numberEquality,
independent_isectElimination,
independent_functionElimination,
voidElimination,
lambdaEquality,
dependent_functionElimination,
isect_memberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
cumulativity,
applyEquality,
functionExtensionality,
because_Cache,
instantiate,
unionElimination,
equalityElimination,
productElimination,
dependent_pairFormation,
promote_hyp,
dependent_set_memberEquality,
independent_pairFormation,
addEquality,
voidEquality,
intEquality,
minusEquality,
functionEquality,
universeEquality,
baseClosed,
productEquality,
imageElimination,
applyLambdaEquality,
imageMemberEquality,
impliesFunctionality
Latex:
\mforall{}[A:\mBbbU{}']. \mforall{}[B:A {}\mrightarrow{} Type]. \mforall{}[n:\mBbbN{}]. \mforall{}[w:coW(A;a.B[a])]. \mforall{}[p,q:coPath(a.B[a];w;n)].
(coPathAgree(a.B[a];n;w;p;q) \mmember{} \mBbbP{})
Date html generated:
2018_07_25-PM-01_38_01
Last ObjectModification:
2018_06_01-AM-09_58_31
Theory : co-recursion
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