Nuprl Lemma : coW-equiv-iff2
∀[A:𝕌']
∀B:A ⟶ Type. ∀w,w':coW(A;a.B[a]).
(coW-equiv(a.B[a];w;w')
⇐⇒ ∀p:copath(a.B[a];w')
∃q:copath(a.B[a];w)
((copath-length(q) = copath-length(p) ∈ ℤ) ∧ coW-equiv(a.B[a];copath-at(w';p);copath-at(w;q))))
Proof
Definitions occuring in Statement :
coW-equiv: coW-equiv(a.B[a];w;w'),
copath-length: copath-length(p),
copath-at: copath-at(w;p),
copath: copath(a.B[a];w),
coW: coW(A;a.B[a]),
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
function: x:A ⟶ B[x],
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
nequal: a ≠ b ∈ T ,
assert: ↑b,
bnot: ¬bb,
it: ⋅,
unit: Unit,
bool: 𝔹,
copath-cons: copath-cons(b;x),
coWmem: coWmem(a.B[a];z;w),
bfalse: ff,
sq_type: SQType(T),
coPath: coPath(a.B[a];w;n),
btrue: tt,
ifthenelse: if b then t else f fi ,
copath-nil: (),
eq_int: (i =z j),
coPath-at: coPath-at(n;w;p),
cand: A c∧ B,
exists: ∃x:A. B[x],
true: True,
top: Top,
subtract: n - m,
uiff: uiff(P;Q),
or: P ∨ Q,
decidable: Dec(P),
uimplies: b supposing a,
guard: {T},
not: ¬A,
false: False,
less_than': less_than'(a;b),
le: A ≤ B,
copath-at: copath-at(w;p),
pi1: fst(t),
copath-length: copath-length(p),
copath: copath(a.B[a];w),
nat: ℕ,
subtype_rel: A ⊆r B,
rev_implies: P ⇐ Q,
prop: ℙ,
so_apply: x[s],
so_lambda: λ2x.t[x],
member: t ∈ T,
implies: P ⇒ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x]
Lemmas referenced :
not-equal-2,
neg_assert_of_eq_int,
assert-bnot,
bool_cases_sqequal,
length-copath-cons,
copath-cons_wf,
coW-item-coWmem,
coW-item_wf,
coW-equiv-iff,
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,
equal-wf-base,
not_wf,
bnot_wf,
assert_wf,
less_than_irreflexivity,
le_weakening,
less_than_transitivity1,
eq_int_wf,
coW-equiv_inversion,
copath_length_nil_lemma,
copath-nil_wf,
primrec-wf2,
less_than_wf,
set_wf,
coPath-at_wf,
int_subtype_base,
equal-wf-T-base,
le-add-cancel,
add-zero,
add_functionality_wrt_le,
add-commutes,
add-swap,
add-associates,
minus-minus,
minus-add,
minus-one-mul-top,
zero-add,
minus-one-mul,
condition-implies-le,
less-iff-le,
not-le-2,
decidable__le,
subtract_wf,
le_weakening2,
le_wf,
false_wf,
coPath_wf,
coW_wf,
copath-at_wf,
nat_wf,
copath-length_wf,
equal_wf,
exists_wf,
all_wf,
coW-equiv_wf,
copath_wf
Rules used in proof :
dependent_pairEquality,
equalityElimination,
promote_hyp,
independent_pairEquality,
impliesFunctionality,
equalitySymmetry,
equalityTransitivity,
dependent_pairFormation,
baseClosed,
closedConclusion,
baseApply,
minusEquality,
voidEquality,
isect_memberEquality,
addEquality,
independent_functionElimination,
voidElimination,
unionElimination,
independent_isectElimination,
dependent_functionElimination,
natural_numberEquality,
dependent_set_memberEquality,
functionExtensionality,
productElimination,
universeEquality,
functionEquality,
cumulativity,
instantiate,
because_Cache,
rename,
setElimination,
intEquality,
productEquality,
hypothesis,
applyEquality,
lambdaEquality,
sqequalRule,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
independent_pairFormation,
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')
\mLeftarrow{}{}\mRightarrow{} \mforall{}p:copath(a.B[a];w')
\mexists{}q:copath(a.B[a];w)
((copath-length(q) = copath-length(p))
\mwedge{} coW-equiv(a.B[a];copath-at(w';p);copath-at(w;q))))
Date html generated:
2018_07_25-PM-01_49_03
Last ObjectModification:
2018_07_24-PM-03_46_18
Theory : co-recursion
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