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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