Nuprl Lemma : bag-count-remove1
∀[T:Type]
∀eq:EqDecider(T). ∀x:T. ∀bs:bag(T).
((x ↓∈ bs ∧ ((#x in outl(bag-remove1(eq;bs;x))) = ((#x in bs) - 1) ∈ ℕ)) ∨ (¬x ↓∈ bs))
Proof
Definitions occuring in Statement :
bag-remove1: bag-remove1(eq;bs;a),
bag-count: (#x in bs),
bag-member: x ↓∈ bs,
bag: bag(T),
deq: EqDecider(T),
nat: ℕ,
outl: outl(x),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
not: ¬A,
or: P ∨ Q,
and: P ∧ Q,
subtract: n - m,
natural_number: $n,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
implies: P ⇒ Q,
decidable: Dec(P),
or: P ∨ Q,
and: P ∧ Q,
cand: A c∧ B,
exists: ∃x:A. B[x],
not: ¬A,
false: False,
prop: ℙ,
guard: {T},
uimplies: b supposing a,
subtype_rel: A ⊆r B,
sq_type: SQType(T),
true: True,
nat: ℕ,
single-bag: {x},
deq: EqDecider(T),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
eqof: eqof(d),
ifthenelse: if b then t else f fi ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
bfalse: ff,
bnot: ¬bb,
assert: ↑b,
squash: ↓T,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
isl: isl(x),
outl: outl(x)
Lemmas referenced :
decidable__bag-member,
decidable-equal-deq,
bag-remove1-property,
not_wf,
bag-member_wf,
equal_wf,
nat_wf,
bag-count_wf,
bag_wf,
deq_wf,
subtype_base_sq,
int_subtype_base,
subtract_wf,
single-bag_wf,
cons_wf,
nil_wf,
list-subtype-bag,
ifthenelse_wf,
bool_wf,
eqtt_to_assert,
safe-assert-deq,
decidable__equal_int,
satisfiable-full-omega-tt,
intformnot_wf,
intformeq_wf,
itermSubtract_wf,
itermAdd_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_subtract_lemma,
int_term_value_add_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
eqff_to_assert,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
squash_wf,
true_wf,
bag-count-append,
iff_weakening_equal,
add_functionality_wrt_eq,
bag-count-single,
outl_wf,
assert_wf,
isl_wf,
unit_wf2,
and_wf,
btrue_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
introduction,
extract_by_obid,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_functionElimination,
dependent_functionElimination,
unionElimination,
inlFormation,
independent_pairFormation,
because_Cache,
productElimination,
voidElimination,
cumulativity,
sqequalRule,
inrFormation,
productEquality,
equalityTransitivity,
equalitySymmetry,
universeEquality,
instantiate,
intEquality,
independent_isectElimination,
hyp_replacement,
applyLambdaEquality,
applyEquality,
natural_numberEquality,
lambdaEquality,
setElimination,
rename,
equalityElimination,
dependent_pairFormation,
int_eqEquality,
isect_memberEquality,
voidEquality,
computeAll,
promote_hyp,
imageElimination,
imageMemberEquality,
baseClosed,
unionEquality,
dependent_set_memberEquality
Latex:
\mforall{}[T:Type]
\mforall{}eq:EqDecider(T). \mforall{}x:T. \mforall{}bs:bag(T).
((x \mdownarrow{}\mmember{} bs \mwedge{} ((\#x in outl(bag-remove1(eq;bs;x))) = ((\#x in bs) - 1))) \mvee{} (\mneg{}x \mdownarrow{}\mmember{} bs))
Date html generated:
2018_05_21-PM-09_48_21
Last ObjectModification:
2017_07_26-PM-06_30_37
Theory : bags_2
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