Nuprl Lemma : coW-equiv-iff
∀[A:𝕌']
∀B:A ⟶ Type. ∀w,w':coW(A;a.B[a]).
(coW-equiv(a.B[a];w;w') ⇐⇒ ∀z:coW(A;a.B[a]). (coWmem(a.B[a];z;w) ⇐⇒ coWmem(a.B[a];z;w')))
Proof
Definitions occuring in Statement :
coWmem: coWmem(a.B[a];z;w),
coW-equiv: coW-equiv(a.B[a];w;w'),
coW: coW(A;a.B[a]),
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
label: ...$L... t,
copath-cons: copath-cons(b;x),
coPathAgree: coPathAgree(a.B[a];n;w;p;q),
copathAgree: copathAgree(a.B[a];w;x;y),
sg-legal2: Legal2(x;y),
top: Top,
less_than': less_than'(a;b),
le: A ≤ B,
ext-eq: A ≡ B,
exists: ∃x:A. B[x],
coW-dom: coW-dom(a.B[a];w),
coW-item: coW-item(w;b),
coWmem: coWmem(a.B[a];z;w),
bfalse: ff,
btrue: tt,
ifthenelse: if b then t else f fi ,
subtract: n - m,
eq_int: (i =z j),
coPath: coPath(a.B[a];w;n),
copath: copath(a.B[a];w),
squash: ↓T,
sq_type: SQType(T),
uimplies: b supposing a,
true: True,
false: False,
not: ¬A,
guard: {T},
cand: A c∧ B,
or: P ∨ Q,
decidable: Dec(P),
nat: ℕ,
subtype_rel: A ⊆r B,
copath-nil: (),
copath-length: copath-length(p),
sg-init: InitialPos(g),
pi2: snd(t),
sg-legal1: Legal1(x;y),
pi1: fst(t),
sg-pos: Pos(g),
coW-game: coW-game(a.B[a];w;w'),
coW-equiv: coW-equiv(a.B[a];w;w'),
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 :
coW-game-step-isom,
isom-preserves-win2,
sg-normalize_wf,
sg-normalize-win2,
simple-game_wf,
copath-nil_wf,
copath-cons_wf,
top_wf,
sg-legal2_wf,
coPathAgree_wf,
equal-wf-base,
coPathAgree0_lemma,
sg-change-init_wf,
win2_wf,
false_wf,
coW-equiv_weakening,
subtype_rel_weakening,
coW-ext,
coW-item_wf,
le_wf,
set_subtype_base,
pi1_wf,
coPath_wf,
pi2_wf,
iff_weakening_equal,
subtype_rel_self,
true_wf,
squash_wf,
equal_wf,
int_subtype_base,
subtype_base_sq,
copath_wf,
equal-wf-T-base,
nat_wf,
copath-length_wf,
decidable__int_equal,
sg-init_wf,
sg-legal1_wf,
sg-pos_wf,
set_wf,
coW-game_wf,
win2-iff,
coW-equiv_inversion,
coW-equiv-implies,
coWmem_wf,
iff_wf,
coW_wf,
all_wf,
coW-equiv_wf
Rules used in proof :
voidEquality,
isect_memberEquality,
independent_pairEquality,
dependent_set_memberEquality,
dependent_pairEquality,
dependent_pairFormation,
hypothesis_subsumption,
applyLambdaEquality,
imageMemberEquality,
imageElimination,
promote_hyp,
independent_isectElimination,
equalityTransitivity,
functionExtensionality,
levelHypothesis,
equalityUniverse,
voidElimination,
inrFormation,
baseClosed,
intEquality,
productEquality,
equalitySymmetry,
inlFormation,
unionElimination,
natural_numberEquality,
rename,
setElimination,
productElimination,
independent_functionElimination,
dependent_functionElimination,
universeEquality,
functionEquality,
because_Cache,
cumulativity,
instantiate,
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{}z:coW(A;a.B[a]). (coWmem(a.B[a];z;w) \mLeftarrow{}{}\mRightarrow{} coWmem(a.B[a];z;w')))
Date html generated:
2018_07_25-PM-01_48_57
Last ObjectModification:
2018_07_11-PM-00_21_34
Theory : co-recursion
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