Nuprl Lemma : bag-rep-size-restrict
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[x:T]. ∀[b:bag(T)]. (bag-rep(#((b|x));x) = (b|x) ∈ bag(T))
Proof
Definitions occuring in Statement :
bag-restrict: (b|x),
bag-rep: bag-rep(n;x),
bag-size: #(bs),
bag: bag(T),
deq: EqDecider(T),
uall: ∀[x:A]. B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
squash: ↓T,
exists: ∃x:A. B[x],
bag-restrict: (b|x),
bag-size: #(bs),
bag-rep: bag-rep(n;x),
bag-filter: [x∈b|p[x]],
so_lambda: λ2x.t[x],
all: ∀x:A. B[x],
prop: ℙ,
deq: EqDecider(T),
subtype_rel: A ⊆r B,
uimplies: b supposing a,
so_apply: x[s],
implies: P ⇒ Q,
top: Top,
empty-bag: {},
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
eqof: eqof(d),
ifthenelse: if b then t else f fi ,
bfalse: ff,
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
not: ¬A,
ge: i ≥ j ,
le: A ≤ B,
satisfiable_int_formula: satisfiable_int_formula(fmla),
cons-bag: x.b,
true: True
Lemmas referenced :
bag_to_squash_list,
list_induction,
equal_wf,
bag_wf,
primrec_wf,
length_wf_nat,
filter_wf5,
l_member_wf,
empty-bag_wf,
cons-bag_wf,
int_seg_wf,
length_wf,
list-subtype-bag,
list_wf,
filter_nil_lemma,
length_of_nil_lemma,
primrec0_lemma,
nil_wf,
filter_cons_lemma,
bag-rep_wf,
bag-size_wf,
bag-restrict_wf,
deq_wf,
bool_wf,
eqtt_to_assert,
safe-assert-deq,
length_of_cons_lemma,
primrec-unroll,
eq_int_wf,
assert_of_eq_int,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
non_neg_length,
eqof_wf,
satisfiable-full-omega-tt,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformeq_wf,
itermAdd_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
int_formula_prop_wf,
squash_wf,
true_wf,
add-subtract-cancel
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
hypothesisEquality,
imageElimination,
productElimination,
promote_hyp,
hypothesis,
rename,
sqequalRule,
lambdaEquality,
cumulativity,
lambdaFormation,
setElimination,
applyEquality,
setEquality,
natural_numberEquality,
independent_isectElimination,
independent_functionElimination,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
hyp_replacement,
equalitySymmetry,
applyLambdaEquality,
axiomEquality,
universeEquality,
unionElimination,
equalityElimination,
equalityTransitivity,
addEquality,
dependent_pairFormation,
instantiate,
int_eqEquality,
intEquality,
independent_pairFormation,
computeAll,
imageMemberEquality,
baseClosed
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[x:T]. \mforall{}[b:bag(T)]. (bag-rep(\#((b|x));x) = (b|x))
Date html generated:
2018_05_21-PM-09_52_42
Last ObjectModification:
2017_07_26-PM-06_32_01
Theory : bags_2
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