Nuprl Lemma : mon_for_when_unique
∀s:DSet. ∀g:IMonoid. ∀f:|s| ⟶ |g|. ∀b:|s| ⟶ 𝔹. ∀u:|s|.
((↑b[u])
⇒ (∀as:|s| List
((↑distinct{s}(as))
⇒ (↑(u ∈b as))
⇒ (∀v:|s|. ((↑b[v]) ⇒ (↑(v ∈b as)) ⇒ (v = u ∈ |s|)))
⇒ ((For{g} x ∈ as. (when b[x]. f[x])) = f[u] ∈ |g|))))
Proof
Definitions occuring in Statement :
distinct: distinct{s}(ps),
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],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
equal: s = t ∈ T,
mon_when: when b. p,
imon: IMonoid,
grp_car: |g|,
dset: DSet,
set_car: |p|
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
prop: ℙ,
dset: DSet,
so_apply: x[s],
imon: IMonoid,
top: Top,
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
bfalse: ff,
false: False,
infix_ap: x f y,
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
band: p ∧b q,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
ball: ball,
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
mon_when: when b. p,
not: ¬A
Lemmas referenced :
list_induction,
assert_wf,
distinct_wf,
mem_wf,
all_wf,
set_car_wf,
equal_wf,
grp_car_wf,
mon_for_wf,
mon_when_wf,
distinct_nil_lemma,
istype-void,
mem_nil_lemma,
mon_for_nil_lemma,
true_wf,
distinct_cons_lemma,
mem_cons_lemma,
mon_for_cons_lemma,
bor_wf,
set_eq_wf,
ball_wf,
bnot_wf,
infix_ap_wf,
bool_wf,
eqtt_to_assert,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
bfalse_wf,
list_wf,
imon_wf,
dset_wf,
or_wf,
equal-wf-T-base,
not_wf,
squash_wf,
istype-universe,
mon_ident,
subtype_rel_self,
iff_weakening_equal,
uiff_transitivity,
assert_of_bnot,
assert_of_band,
iff_transitivity,
iff_weakening_uiff,
assert_of_bor,
assert_of_dset_eq,
grp_op_wf,
mon_for_when_none,
ball_char,
grp_id_wf,
member_wf,
assert_functionality_wrt_uiff
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
because_Cache,
sqequalRule,
lambdaEquality_alt,
functionEquality,
dependent_functionElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
applyEquality,
inhabitedIsType,
universeIsType,
independent_functionElimination,
isect_memberEquality_alt,
voidElimination,
functionIsType,
equalityIsType1,
unionElimination,
equalityElimination,
productElimination,
independent_isectElimination,
dependent_pairFormation_alt,
equalityTransitivity,
equalitySymmetry,
promote_hyp,
instantiate,
unionIsType,
baseClosed,
imageElimination,
universeEquality,
natural_numberEquality,
imageMemberEquality,
independent_pairFormation,
inlFormation_alt,
inrFormation_alt
Latex:
\mforall{}s:DSet. \mforall{}g:IMonoid. \mforall{}f:|s| {}\mrightarrow{} |g|. \mforall{}b:|s| {}\mrightarrow{} \mBbbB{}. \mforall{}u:|s|.
((\muparrow{}b[u])
{}\mRightarrow{} (\mforall{}as:|s| List
((\muparrow{}distinct\{s\}(as))
{}\mRightarrow{} (\muparrow{}(u \mmember{}\msubb{} as))
{}\mRightarrow{} (\mforall{}v:|s|. ((\muparrow{}b[v]) {}\mRightarrow{} (\muparrow{}(v \mmember{}\msubb{} as)) {}\mRightarrow{} (v = u)))
{}\mRightarrow{} ((For\{g\} x \mmember{} as. (when b[x]. f[x])) = f[u]))))
Date html generated:
2019_10_16-PM-01_02_59
Last ObjectModification:
2018_10_08-AM-11_32_18
Theory : list_2
Home
Index