Nuprl Lemma : mon_for_when_none
∀s:DSet. ∀g:IMonoid. ∀f:|s| ⟶ |g|. ∀b:|s| ⟶ 𝔹. ∀as:|s| List.
((∀x:|s|. ((↑(x ∈b as)) ⇒ (¬↑b[x]))) ⇒ ((For{g} x ∈ as. (when b[x]. f[x])) = e ∈ |g|))
Proof
Definitions occuring in Statement :
mem: a ∈b as,
mon_for: For{g} x ∈ as. f[x],
list: T List,
assert: ↑b,
bool: 𝔹,
so_apply: x[s],
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
function: x:A ⟶ B[x],
equal: s = t ∈ T,
mon_when: when b. p,
imon: IMonoid,
grp_id: e,
grp_car: |g|,
dset: DSet,
set_car: |p|
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
implies: P ⇒ Q,
prop: ℙ,
dset: DSet,
so_apply: x[s],
imon: IMonoid,
top: Top,
assert: ↑b,
ifthenelse: if b then t else f fi ,
bfalse: ff,
false: False,
infix_ap: x f y,
mon_when: when b. p,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
or: P ∨ Q,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
not: ¬A,
squash: ↓T,
exists: ∃x:A. B[x],
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
true: True,
subtype_rel: A ⊆r B
Lemmas referenced :
list_induction,
all_wf,
set_car_wf,
assert_wf,
mem_wf,
not_wf,
equal_wf,
grp_car_wf,
mon_for_wf,
mon_when_wf,
grp_id_wf,
mem_nil_lemma,
istype-void,
mon_for_nil_lemma,
mem_cons_lemma,
mon_for_cons_lemma,
bor_wf,
set_eq_wf,
list_wf,
bool_wf,
imon_wf,
dset_wf,
equal-wf-T-base,
bnot_wf,
eqtt_to_assert,
uiff_transitivity,
eqff_to_assert,
assert_of_bnot,
or_wf,
member_wf,
iff_transitivity,
iff_weakening_uiff,
assert_of_bor,
assert_of_dset_eq,
squash_wf,
true_wf,
istype-universe,
mon_ident,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
subtype_rel_self,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
because_Cache,
sqequalRule,
lambdaEquality_alt,
functionEquality,
setElimination,
rename,
hypothesis,
dependent_functionElimination,
hypothesisEquality,
applyEquality,
universeIsType,
independent_functionElimination,
isect_memberEquality_alt,
voidElimination,
functionIsType,
inhabitedIsType,
equalityIsType1,
equalityTransitivity,
equalitySymmetry,
baseClosed,
unionElimination,
equalityElimination,
productElimination,
independent_isectElimination,
unionIsType,
independent_pairFormation,
inlFormation_alt,
inrFormation_alt,
imageElimination,
universeEquality,
dependent_pairFormation_alt,
promote_hyp,
instantiate,
cumulativity,
natural_numberEquality,
imageMemberEquality
Latex:
\mforall{}s:DSet. \mforall{}g:IMonoid. \mforall{}f:|s| {}\mrightarrow{} |g|. \mforall{}b:|s| {}\mrightarrow{} \mBbbB{}. \mforall{}as:|s| List.
((\mforall{}x:|s|. ((\muparrow{}(x \mmember{}\msubb{} as)) {}\mRightarrow{} (\mneg{}\muparrow{}b[x]))) {}\mRightarrow{} ((For\{g\} x \mmember{} as. (when b[x]. f[x])) = e))
Date html generated:
2019_10_16-PM-01_02_49
Last ObjectModification:
2018_10_08-PM-00_29_15
Theory : list_2
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