Nuprl Lemma : fset_for_when_unique
∀s:DSet. ∀g:IAbMonoid. ∀f:|s| ⟶ |g|. ∀b:|s| ⟶ 𝔹. ∀u:|s|.
((↑b[u])
⇒ (∀as:FiniteSet{s}
((↑(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,
finite_set: FiniteSet{s},
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,
iabmonoid: IAbMonoid,
grp_car: |g|,
dset: DSet,
set_car: |p|
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
dset: DSet,
so_lambda: λ2x.t[x],
so_apply: x[s],
finite_set: FiniteSet{s},
iabmonoid: IAbMonoid,
imon: IMonoid,
guard: {T}
Lemmas referenced :
all_wf,
set_car_wf,
assert_wf,
mset_mem_wf,
equal_wf,
finite_set_wf,
bool_wf,
grp_car_wf,
iabmonoid_wf,
dset_wf,
fset_properties,
mset_for_when_unique
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
hypothesis,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
sqequalRule,
lambdaEquality,
functionEquality,
applyEquality,
dependent_functionElimination,
independent_functionElimination
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:FiniteSet\{s\}
((\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:
2016_05_16-AM-07_51_44
Last ObjectModification:
2015_12_28-PM-05_59_52
Theory : mset
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