Nuprl Lemma : coW-equiv-iff3
∀[A:𝕌']
∀B:A ⟶ Type. ∀w,w':coW(A;a.B[a]).
(coW-equiv(a.B[a];w;w')
⇐⇒ ∀p:maximal-copath(a.B[a];w')
∃q:maximal-copath(a.B[a];w)
∀n:ℕ
((∀i:ℕn. (copath-length(p i) = i ∈ ℤ))
⇒ (∀i:ℕn. ((copath-length(q i) = i ∈ ℤ) ∧ coW-equiv(a.B[a];copath-at(w;q i);copath-at(w';p i))))))
Proof
Definitions occuring in Statement :
maximal-copath: maximal-copath(a.B[a];w),
coW-equiv: coW-equiv(a.B[a];w;w'),
copath-length: copath-length(p),
copath-at: copath-at(w;p),
coW: coW(A;a.B[a]),
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
copath-length: copath-length(p),
copath: copath(a.B[a];w),
btrue: tt,
eq_int: (i =z j),
ifthenelse: if b then t else f fi ,
coPath-at: coPath-at(n;w;p),
copath-nil: (),
copath-at: copath-at(w;p),
less_than: a < b,
cand: A c∧ B,
coWmem: coWmem(a.B[a];z;w),
exists: ∃x:A. B[x],
guard: {T},
sq_type: SQType(T),
so_apply: x[s1;s2;s3],
true: True,
top: Top,
subtract: n - m,
squash: ↓T,
sq_stable: SqStable(P),
uiff: uiff(P;Q),
or: P ∨ Q,
decidable: Dec(P),
not: ¬A,
false: False,
less_than': less_than'(a;b),
le: A ≤ B,
uimplies: b supposing a,
pi1: fst(t),
so_lambda: so_lambda(x,y,z.t[x; y; z]),
subtype_rel: A ⊆r B,
lelt: i ≤ j < k,
int_seg: {i..j-},
maximal-copath: maximal-copath(a.B[a];w),
nat: ℕ,
rev_implies: P ⇐ Q,
so_apply: x[s],
so_lambda: λ2x.t[x],
prop: ℙ,
member: t ∈ T,
implies: P ⇒ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x]
Lemmas referenced :
minus-zero,
not-equal-2,
equal-wf-base-T,
copathAgree_refl,
pi1_wf,
copath-nil-Agree,
top_wf,
it_wf,
coW-equiv_transitivity,
coW-equiv_inversion,
copath-at-extend,
length-copath-extend,
copathAgree-extend,
copath-extend_wf,
coW-equiv_weakening,
less_than_wf,
coW-item_wf,
copath-last_wf,
coW-dom_wf,
pi2_wf,
add_functionality_wrt_eq,
copathAgree-last,
coW-equiv-iff,
sq_stable__copathAgree,
sq_stable__all,
minus-minus,
zero-mul,
add-mul-special,
subtract_wf,
iff_weakening_equal,
subtype_rel_self,
lelt_wf,
le-add-cancel2,
less-iff-le,
not-lt-2,
decidable__lt,
true_wf,
squash_wf,
decidable__int_equal,
decidable__all_int_seg,
int_subtype_base,
subtype_base_sq,
equal-wf-T-base,
copath_length_nil_lemma,
copath-nil_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,
decidable__le,
false_wf,
int_seg_subtype_nat,
copathAgree_wf,
copath_wf,
dependent-choice,
coW_wf,
copath-at_wf,
le_wf,
copath-length_wf,
equal_wf,
int_seg_wf,
nat_wf,
exists_wf,
maximal-copath_wf,
all_wf,
coW-equiv_wf
Rules used in proof :
hyp_replacement,
levelHypothesis,
equalityUniverse,
multiplyEquality,
dependent_pairFormation,
axiomEquality,
independent_pairEquality,
dependent_pairEquality,
minusEquality,
voidEquality,
isect_memberEquality,
imageElimination,
baseClosed,
imageMemberEquality,
independent_functionElimination,
voidElimination,
unionElimination,
equalitySymmetry,
equalityTransitivity,
independent_isectElimination,
functionExtensionality,
addEquality,
universeEquality,
cumulativity,
instantiate,
productEquality,
because_Cache,
productElimination,
dependent_set_memberEquality,
intEquality,
rename,
setElimination,
natural_numberEquality,
functionEquality,
dependent_functionElimination,
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:maximal-copath(a.B[a];w')
\mexists{}q:maximal-copath(a.B[a];w)
\mforall{}n:\mBbbN{}
((\mforall{}i:\mBbbN{}n. (copath-length(p i) = i))
{}\mRightarrow{} (\mforall{}i:\mBbbN{}n
((copath-length(q i) = i)
\mwedge{} coW-equiv(a.B[a];copath-at(w;q i);copath-at(w';p i))))))
Date html generated:
2018_07_29-AM-09_21_45
Last ObjectModification:
2018_07_25-PM-03_17_39
Theory : co-recursion
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