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 :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
rev_implies: P ⇐ Q,
coW-equiv: coW-equiv(a.B[a];w;w'),
coW-game: coW-game(a.B[a];w;w'),
sg-pos: Pos(g),
pi1: fst(t),
sg-legal1: Legal1(x;y),
pi2: snd(t),
sg-init: InitialPos(g),
copath-length: copath-length(p),
copath-nil: (),
subtype_rel: A ⊆r B,
nat: ℕ,
decidable: Dec(P),
or: P ∨ Q,
cand: A c∧ B,
uimplies: b supposing a,
not: ¬A,
false: False,
true: True,
sq_type: SQType(T),
guard: {T},
squash: ↓T,
copath: copath(a.B[a];w),
coPath: coPath(a.B[a];w;n),
eq_int: (i =z j),
subtract: n - m,
ifthenelse: if b then t else f fi ,
btrue: tt,
bfalse: ff,
coWmem: coWmem(a.B[a];z;w),
coW-item: coW-item(w;b),
coW-dom: coW-dom(a.B[a];w),
exists: ∃x:A. B[x],
ext-eq: A ≡ B,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
sg-legal2: Legal2(x;y),
copathAgree: copathAgree(a.B[a];w;x;y),
copath-cons: copath-cons(b;x),
label: ...$L... t
Lemmas referenced :
coW-equiv_wf,
coWmem_wf,
coW_wf,
istype-universe,
coW-equiv_inversion,
coW-equiv-implies,
win2-iff,
coW-game_wf,
sg-pos_wf,
sg-legal1_wf,
sg-init_wf,
copath_length_nil_lemma,
decidable__equal_int,
copath-length_wf,
set_subtype_base,
le_wf,
int_subtype_base,
istype-int,
subtype_base_sq,
equal_wf,
squash_wf,
true_wf,
copath_wf,
istype-nat,
subtype_rel_self,
iff_weakening_equal,
pi2_wf,
nat_wf,
coPath_wf,
pi1_wf,
coW-item_wf,
coW-ext,
subtype_rel_weakening,
coW-equiv_weakening,
decidable__le,
full-omega-unsat,
intformnot_wf,
intformle_wf,
itermConstant_wf,
int_formula_prop_not_lemma,
istype-void,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
istype-le,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
win2_wf,
sg-change-init_wf,
coPathAgree0_lemma,
sg-legal2_wf,
simple-game_wf,
sg-normalize-win2,
sg-normalize_wf,
isom-preserves-win2,
copath-cons_wf,
copath-nil_wf,
coW-game-step-isom
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
Error :lambdaFormation_alt,
independent_pairFormation,
Error :universeIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
Error :lambdaEquality_alt,
applyEquality,
hypothesis,
Error :functionIsType,
because_Cache,
Error :productIsType,
instantiate,
cumulativity,
universeEquality,
Error :inhabitedIsType,
independent_functionElimination,
dependent_functionElimination,
productElimination,
Error :setIsType,
setElimination,
rename,
equalityTransitivity,
equalitySymmetry,
natural_numberEquality,
unionElimination,
Error :inlFormation_alt,
Error :equalityIstype,
intEquality,
independent_isectElimination,
baseClosed,
sqequalBase,
Error :inrFormation_alt,
voidElimination,
imageElimination,
imageMemberEquality,
applyLambdaEquality,
hypothesis_subsumption,
productEquality,
functionEquality,
Error :dependent_pairFormation_alt,
Error :dependent_pairEquality_alt,
Error :dependent_set_memberEquality_alt,
approximateComputation,
Error :isect_memberEquality_alt,
independent_pairEquality,
promote_hyp
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:
2019_06_20-PM-01_11_42
Last ObjectModification:
2019_01_20-PM-01_19_45
Theory : co-recursion-2
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