Nuprl Lemma : mset_for_when_dom_shift
∀s:DSet. ∀g:IAbMonoid. ∀f:|s| ⟶ |g|. ∀c:|s| ⟶ 𝔹. ∀p,q:MSet{s}.
((↑(p ⊆b q))
⇒ (∀x:|s|. ((↑(x ∈b q - p)) ⇒ (¬↑c[x])))
⇒ ((msFor{g} x ∈ p. when c[x]. f[x]) = (msFor{g} x ∈ q. when c[x]. f[x]) ∈ |g|))
Proof
Definitions occuring in Statement :
bsubmset: a ⊆b b,
mset_diff: a - b,
mset_for: mset_for,
mset_mem: mset_mem,
mset: MSet{s},
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,
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],
iabmonoid: IAbMonoid,
imon: IMonoid,
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
infix_ap: x f y
Lemmas referenced :
all_wf,
set_car_wf,
assert_wf,
mset_mem_wf,
mset_diff_wf,
not_wf,
bsubmset_wf,
mset_wf,
bool_wf,
grp_car_wf,
iabmonoid_wf,
dset_wf,
equal_wf,
squash_wf,
true_wf,
mset_for_wf,
mon_when_wf,
mset_for_functionality,
mset_sum_wf,
detach_msubset,
iff_weakening_equal,
mset_for_mset_sum,
grp_op_wf,
mset_for_when_none,
mon_ident
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
hypothesis,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
sqequalRule,
lambdaEquality,
functionEquality,
dependent_functionElimination,
applyEquality,
functionExtensionality,
because_Cache,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
independent_functionElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination,
productElimination
Latex:
\mforall{}s:DSet. \mforall{}g:IAbMonoid. \mforall{}f:|s| {}\mrightarrow{} |g|. \mforall{}c:|s| {}\mrightarrow{} \mBbbB{}. \mforall{}p,q:MSet\{s\}.
((\muparrow{}(p \msubseteq{}\msubb{} q))
{}\mRightarrow{} (\mforall{}x:|s|. ((\muparrow{}(x \mmember{}\msubb{} q - p)) {}\mRightarrow{} (\mneg{}\muparrow{}c[x])))
{}\mRightarrow{} ((msFor\{g\} x \mmember{} p. when c[x]. f[x]) = (msFor\{g\} x \mmember{} q. when c[x]. f[x])))
Date html generated:
2017_10_01-AM-10_00_46
Last ObjectModification:
2017_03_03-PM-01_02_05
Theory : mset
Home
Index