Nuprl Lemma : mset_for_when_unique
∀s:DSet. ∀g:IAbMonoid. ∀f:|s| ⟶ |g|. ∀b:|s| ⟶ 𝔹. ∀u:|s|.
((↑b[u])
⇒ (∀as:MSet{s}
((∀x:|s|. ((x #∈ as) ≤ 1))
⇒ (↑(u
∈b as))
⇒ (∀v:|s|. ((↑b[v]) ⇒ (↑(v ∈b as)) ⇒ (v = u ∈ |s|)))
⇒ ((msFor{g} x ∈ as. when b[x]. f[x]) = f[u] ∈ |g|))))
Proof
Definitions occuring in Statement :
mset_for: mset_for,
mset_mem: mset_mem,
mset_count: x #∈ a,
mset: MSet{s},
assert: ↑b,
bool: 𝔹,
so_apply: x[s],
le: A ≤ B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
natural_number: $n,
equal: s = t ∈ T,
mon_when: when b. p,
iabmonoid: IAbMonoid,
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,
dset: DSet,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
nat: ℕ,
so_apply: x[s],
prop: ℙ,
iabmonoid: IAbMonoid,
imon: IMonoid,
sq_stable: SqStable(P),
mset: MSet{s},
quotient: x,y:A//B[x; y],
and: P ∧ Q,
squash: ↓T,
mset_for: mset_for,
mset_mem: mset_mem,
mset_count: x #∈ a,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
sq_stable__all,
all_wf,
set_car_wf,
le_wf,
mset_count_wf,
nat_wf,
assert_wf,
mset_mem_wf,
equal_wf,
grp_car_wf,
mset_for_wf,
mon_when_wf,
sq_stable__equal,
mset_wf,
bool_wf,
iabmonoid_wf,
dset_wf,
squash_wf,
equal-wf-base,
list_wf,
permr_wf,
mem_wf,
count_wf,
mon_for_when_unique,
distinct_iff_counts_le_one
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
sqequalRule,
lambdaEquality,
dependent_functionElimination,
applyEquality,
natural_numberEquality,
because_Cache,
functionEquality,
functionExtensionality,
independent_functionElimination,
pointwiseFunctionalityForEquality,
pertypeElimination,
productElimination,
imageMemberEquality,
baseClosed,
productEquality,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}s:DSet. \mforall{}g:IAbMonoid. \mforall{}f:|s| {}\mrightarrow{} |g|. \mforall{}b:|s| {}\mrightarrow{} \mBbbB{}. \mforall{}u:|s|.
((\muparrow{}b[u])
{}\mRightarrow{} (\mforall{}as:MSet\{s\}
((\mforall{}x:|s|. ((x \#\mmember{} as) \mleq{} 1))
{}\mRightarrow{} (\muparrow{}(u
\mmember{}\msubb{} as))
{}\mRightarrow{} (\mforall{}v:|s|. ((\muparrow{}b[v]) {}\mRightarrow{} (\muparrow{}(v \mmember{}\msubb{} as)) {}\mRightarrow{} (v = u)))
{}\mRightarrow{} ((msFor\{g\} x \mmember{} as. when b[x]. f[x]) = f[u]))))
Date html generated:
2017_10_01-AM-10_00_39
Last ObjectModification:
2017_03_03-PM-01_02_20
Theory : mset
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