Nuprl Lemma : coW-wfdd_functionality
∀[A:𝕌']. ∀B:A ⟶ Type. ∀w,w':coW(A;a.B[a]). (coW-equiv(a.B[a];w;w') ⇒ (coW-wfdd(a.B[a];w) ⇐⇒ coW-wfdd(a.B[a];w')))
Proof
Definitions occuring in Statement :
coW-equiv: coW-equiv(a.B[a];w;w'),
coW-wfdd: coW-wfdd(a.B[a];w),
coW: coW(A;a.B[a]),
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
nequal: a ≠ b ∈ T ,
assert: ↑b,
bnot: ¬bb,
sq_type: SQType(T),
bfalse: ff,
ifthenelse: if b then t else f fi ,
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
exists: ∃x:A. B[x],
maximal-copath: maximal-copath(a.B[a];w),
guard: {T},
lelt: i ≤ j < k,
int_seg: {i..j-},
top: Top,
true: True,
less_than': less_than'(a;b),
le: A ≤ B,
subtract: n - m,
sq_stable: SqStable(P),
uimplies: b supposing a,
uiff: uiff(P;Q),
false: False,
rev_implies: P ⇐ Q,
not: ¬A,
and: P ∧ Q,
iff: P ⇐⇒ Q,
or: P ∨ Q,
decidable: Dec(P),
nat: ℕ,
subtype_rel: A ⊆r B,
so_apply: x[s],
so_lambda: λ2x.t[x],
prop: ℙ,
squash: ↓T,
coW-wfdd: coW-wfdd(a.B[a];w),
implies: P ⇒ Q,
member: t ∈ T,
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x]
Lemmas referenced :
coW-equiv_inversion,
decidable__int_equal,
decidable__all_int_seg,
exists_wf,
nequal_wf,
neg_assert_of_eq_int,
decidable__assert,
not-all-int_seg,
assert_wf,
assert_of_bnot,
not_wf,
bool_cases,
minus-minus,
zero-mul,
add-mul-special,
assert_of_eq_int,
assert-bdd-all,
copathAgree_refl,
copathAgree-nil,
copath-nil-Agree,
copath-nil_wf,
assert-bnot,
bool_subtype_base,
subtype_base_sq,
bool_cases_sqequal,
eqff_to_assert,
eqtt_to_assert,
bool_wf,
eq_int_wf,
bdd-all_wf,
int_seg_wf,
add-member-int_seg2,
iff_weakening_equal,
subtype_rel_self,
lelt_wf,
le-add-cancel2,
less-iff-le,
not-lt-2,
decidable__lt,
true_wf,
squash_wf,
subtract_wf,
int_seg_subtype_nat,
coW-equiv-iff3,
coW_wf,
coW-equiv_wf,
coW-wfdd_wf,
copathAgree_wf,
le_wf,
le-add-cancel,
add-zero,
add_functionality_wrt_le,
add-commutes,
add-swap,
add-associates,
minus-one-mul-top,
zero-add,
minus-one-mul,
minus-add,
condition-implies-le,
sq_stable__le,
not-le-2,
false_wf,
decidable__le,
copath-length_wf,
equal_wf,
all_wf,
copath_wf,
nat_wf,
set_wf
Rules used in proof :
independent_pairEquality,
existsFunctionality,
addLevel,
multiplyEquality,
levelHypothesis,
equalityUniverse,
allFunctionality,
promote_hyp,
dependent_pairFormation,
equalityElimination,
equalitySymmetry,
equalityTransitivity,
instantiate,
voidEquality,
isect_memberEquality,
setEquality,
minusEquality,
independent_isectElimination,
independent_functionElimination,
productElimination,
voidElimination,
independent_pairFormation,
unionElimination,
dependent_functionElimination,
natural_numberEquality,
addEquality,
dependent_set_memberEquality,
rename,
setElimination,
intEquality,
because_Cache,
functionExtensionality,
applyEquality,
lambdaEquality,
cumulativity,
isectElimination,
extract_by_obid,
baseClosed,
thin,
imageMemberEquality,
sqequalRule,
imageElimination,
hypothesis,
sqequalHypSubstitution,
introduction,
universeEquality,
hypothesisEquality,
functionEquality,
cut,
lambdaFormation,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[A:\mBbbU{}']
\mforall{}B:A {}\mrightarrow{} Type. \mforall{}w,w':coW(A;a.B[a]).
(coW-equiv(a.B[a];w;w') {}\mRightarrow{} (coW-wfdd(a.B[a];w) \mLeftarrow{}{}\mRightarrow{} coW-wfdd(a.B[a];w')))
Date html generated:
2018_07_29-AM-09_21_53
Last ObjectModification:
2018_07_25-PM-03_34_21
Theory : co-recursion
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