Nuprl Lemma : bag-restrict-size-bound
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[x:T]. ∀[b:bag(T)].
(((#((b|x)) + #((b|¬x))) = #(b) ∈ ℤ) ∧ (#((b|¬x)) ≤ #(b)) ∧ (#((b|x)) ≤ #(b)))
Proof
Definitions occuring in Statement :
bag-co-restrict: (b|¬x),
bag-restrict: (b|x),
bag-size: #(bs),
bag: bag(T),
deq: EqDecider(T),
uall: ∀[x:A]. B[x],
le: A ≤ B,
and: P ∧ Q,
add: n + m,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
and: P ∧ Q,
cand: A c∧ B,
top: Top,
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
nat: ℕ,
guard: {T},
prop: ℙ,
ge: i ≥ j ,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
implies: P ⇒ Q,
not: ¬A,
le: A ≤ B
Lemmas referenced :
bag-restrict-split,
deq_wf,
bag_wf,
nat_wf,
less_than'_wf,
int_formula_prop_wf,
int_term_value_add_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_not_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_and_lemma,
itermAdd_wf,
intformeq_wf,
intformnot_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformand_wf,
satisfiable-full-omega-tt,
le_wf,
nat_properties,
bag-restrict_wf,
bag-co-restrict_wf,
decidable__le,
bag-size_wf,
bag-size-append
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
isect_memberEquality,
voidElimination,
voidEquality,
hypothesis,
applyEquality,
lambdaEquality,
imageElimination,
because_Cache,
natural_numberEquality,
imageMemberEquality,
hypothesisEquality,
baseClosed,
independent_pairFormation,
dependent_functionElimination,
unionElimination,
equalityTransitivity,
equalitySymmetry,
setElimination,
rename,
setEquality,
intEquality,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
computeAll,
productElimination,
independent_pairEquality,
axiomEquality,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[x:T]. \mforall{}[b:bag(T)].
(((\#((b|x)) + \#((b|\mneg{}x))) = \#(b)) \mwedge{} (\#((b|\mneg{}x)) \mleq{} \#(b)) \mwedge{} (\#((b|x)) \mleq{} \#(b)))
Date html generated:
2016_05_15-PM-08_10_57
Last ObjectModification:
2016_01_16-PM-01_27_59
Theory : bags_2
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