Nuprl Lemma : bag-count-drop-trivial
∀[T:Type]. ∀eq:EqDecider(T). ∀[x,y:T]. ∀[bs:bag(T)]. (#y in bag-drop(eq;bs;x)) = (#y in bs) ∈ ℕ supposing ¬(x = y ∈ T)
Proof
Definitions occuring in Statement :
bag-drop: bag-drop(eq;bs;a),
bag-count: (#x in bs),
bag: bag(T),
deq: EqDecider(T),
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
not: ¬A,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
uimplies: b supposing a,
bag-drop: bag-drop(eq;bs;a),
or: P ∨ Q,
exists: ∃x:A. B[x],
and: P ∧ Q,
squash: ↓T,
prop: ℙ,
label: ...$L... t,
decidable: Dec(P),
false: False,
uiff: uiff(P;Q),
satisfiable_int_formula: satisfiable_int_formula(fmla),
implies: P ⇒ Q,
not: ¬A,
top: Top,
nat: ℕ,
guard: {T},
ge: i ≥ j ,
true: True,
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
bag-count: (#x in bs),
count: count(P;L),
reduce: reduce(f;k;as),
list_ind: list_ind,
single-bag: {x},
cons: [a / b],
ifthenelse: if b then t else f fi ,
nil: [],
it: ⋅,
deq: EqDecider(T),
bool: 𝔹,
unit: Unit,
btrue: tt,
eqof: eqof(d),
le: A ≤ B,
less_than': less_than'(a;b),
bfalse: ff,
bnot: ¬bb,
assert: ↑b
Lemmas referenced :
bag-remove1-property,
equal_wf,
squash_wf,
true_wf,
nat_wf,
bag-count_wf,
bag-count-append,
single-bag_wf,
decidable__equal_int,
add-is-int-iff,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformeq_wf,
itermVar_wf,
itermAdd_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_formula_prop_wf,
false_wf,
decidable__le,
nat_properties,
intformle_wf,
itermConstant_wf,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
le_wf,
iff_weakening_equal,
subtype_base_sq,
set_subtype_base,
int_subtype_base,
not_wf,
bool_wf,
eqtt_to_assert,
safe-assert-deq,
eqff_to_assert,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
decide_bfalse_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
dependent_functionElimination,
hypothesisEquality,
unionElimination,
productElimination,
hypothesis,
sqequalRule,
applyEquality,
lambdaEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
cumulativity,
pointwiseFunctionality,
rename,
promote_hyp,
baseApply,
closedConclusion,
baseClosed,
independent_isectElimination,
natural_numberEquality,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
dependent_set_memberEquality,
applyLambdaEquality,
setElimination,
imageMemberEquality,
universeEquality,
independent_functionElimination,
instantiate,
addEquality,
hyp_replacement,
axiomEquality,
equalityElimination
Latex:
\mforall{}[T:Type]
\mforall{}eq:EqDecider(T)
\mforall{}[x,y:T]. \mforall{}[bs:bag(T)]. (\#y in bag-drop(eq;bs;x)) = (\#y in bs) supposing \mneg{}(x = y)
Date html generated:
2018_05_21-PM-09_48_38
Last ObjectModification:
2017_07_26-PM-06_30_43
Theory : bags_2
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