Nuprl Lemma : detach_msubset
∀s:DSet. ∀a,b:MSet{s}. ((↑(a ⊆b b)) ⇒ (b = ((b - a) + a) ∈ MSet{s}))
Proof
Definitions occuring in Statement :
bsubmset: a ⊆b b,
mset_diff: a - b,
mset_sum: a + b,
mset: MSet{s},
assert: ↑b,
all: ∀x:A. B[x],
implies: P ⇒ Q,
equal: s = t ∈ T,
dset: DSet
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
squash: ↓T,
subtype_rel: A ⊆r B,
nat: ℕ,
true: True,
uimplies: b supposing a,
guard: {T},
dset: DSet,
le: A ≤ B,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
bfalse: ff
Lemmas referenced :
assert_wf,
bsubmset_wf,
mset_wf,
dset_wf,
eq_mset_iff_eq_counts,
mset_sum_wf,
mset_diff_wf,
equal_wf,
squash_wf,
true_wf,
mset_count_wf,
nat_wf,
mset_count_sum,
add_functionality_wrt_eq,
ndiff_wf,
mset_count_diff,
iff_weakening_equal,
set_car_wf,
ndiff_add_eq_imax,
count_bsubmset,
imax_unfold,
le_int_wf,
bool_wf,
uiff_transitivity,
equal-wf-T-base,
le_wf,
eqtt_to_assert,
assert_of_le_int,
decidable__equal_int,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformeq_wf,
itermVar_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
lt_int_wf,
less_than_wf,
bnot_wf,
eqff_to_assert,
assert_functionality_wrt_uiff,
bnot_of_le_int,
assert_of_lt_int
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
hypothesis,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
dependent_functionElimination,
hypothesisEquality,
productElimination,
independent_functionElimination,
applyEquality,
lambdaEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
intEquality,
setElimination,
rename,
sqequalRule,
because_Cache,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination,
unionElimination,
equalityElimination,
dependent_pairFormation,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll
Latex:
\mforall{}s:DSet. \mforall{}a,b:MSet\{s\}. ((\muparrow{}(a \msubseteq{}\msubb{} b)) {}\mRightarrow{} (b = ((b - a) + a)))
Date html generated:
2017_10_01-AM-10_00_24
Last ObjectModification:
2017_03_03-PM-01_01_49
Theory : mset
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