Nuprl Lemma : bag-parts_wf
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[bs:bag(T)]. (bag-parts(eq;bs) ∈ bag(bag(T) List+)) supposing valueall-type(T)
Proof
Definitions occuring in Statement :
bag-parts: bag-parts(eq;bs),
bag: bag(T),
listp: A List+,
deq: EqDecider(T),
valueall-type: valueall-type(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
member: t ∈ T,
universe: Type
Definitions unfolded in proof :
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x],
less_than: a < b,
has-valueall: has-valueall(a),
has-value: (a)↓,
so_apply: x[s],
so_lambda: λ2x.t[x],
callbyvalueall: callbyvalueall,
bag-parts: bag-parts(eq;bs),
less_than': less_than'(a;b),
le: A ≤ B,
or: P ∨ Q,
decidable: Dec(P),
lelt: i ≤ j < k,
int_seg: {i..j-},
subtype_rel: A ⊆r B,
guard: {T},
prop: ℙ,
and: P ∧ Q,
top: Top,
not: ¬A,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
ge: i ≥ j ,
false: False,
implies: P ⇒ Q,
nat: ℕ,
true: True,
squash: ↓T,
pi2: snd(t),
pi1: fst(t),
uiff: uiff(P;Q),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
listp: A List+
Lemmas referenced :
valueall-type_wf,
deq_wf,
bag_wf,
int_term_value_add_lemma,
itermAdd_wf,
lelt_wf,
decidable__lt,
product-valueall-type,
bag-valueall-type,
bag-partitions_wf,
evalall-reduce,
int_formula_prop_eq_lemma,
intformeq_wf,
false_wf,
int_seg_subtype,
decidable__equal_int,
int_term_value_subtract_lemma,
int_formula_prop_not_lemma,
itermSubtract_wf,
intformnot_wf,
subtract_wf,
decidable__le,
int_seg_properties,
int_seg_wf,
nat_wf,
bag-size_wf,
le_wf,
less_than_wf,
ge_wf,
int_formula_prop_wf,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_and_lemma,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformand_wf,
satisfiable-full-omega-tt,
nat_properties,
valueall-type-has-valueall,
pi2_wf,
pi1_wf_top,
bag-append_wf,
equal_wf,
bag-member_wf,
true_wf,
squash_wf,
bag-member-partitions,
bag-settype,
bag-combine_wf,
listp_wf,
istype-universe,
istype-nat,
subtype_rel_self,
iff_weakening_equal,
bag-size-append,
decidable__assert,
bag-null_wf,
assert-bag-null,
iff_weakening_uiff,
assert_wf,
equal-wf-T-base,
empty-bag_wf,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
easy-member-int_seg,
bag-size-zero,
full-omega-unsat,
istype-int,
add-is-int-iff,
set-valueall-type,
list_wf,
length_wf,
list-valueall-type,
ifthenelse_wf,
single-bag_wf,
cons-listp,
nil_wf,
bag-map_wf
Rules used in proof :
universeEquality,
because_Cache,
isect_memberEquality,
cumulativity,
isectElimination,
extract_by_obid,
axiomEquality,
sqequalRule,
equalitySymmetry,
hypothesis,
equalityTransitivity,
hypothesisEquality,
dependent_functionElimination,
sqequalHypSubstitution,
thin,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
addEquality,
productEquality,
callbyvalueReduce,
dependent_set_memberEquality,
hypothesis_subsumption,
applyLambdaEquality,
unionElimination,
productElimination,
applyEquality,
independent_functionElimination,
computeAll,
independent_pairFormation,
voidEquality,
voidElimination,
intEquality,
int_eqEquality,
lambdaEquality,
dependent_pairFormation,
independent_isectElimination,
natural_numberEquality,
intWeakElimination,
rename,
setElimination,
lambdaFormation,
independent_pairEquality,
setEquality,
baseClosed,
imageMemberEquality,
imageElimination,
Error :memTop,
lambdaEquality_alt,
inhabitedIsType,
setIsType,
productIsType,
equalityIstype,
universeIsType,
instantiate,
lambdaFormation_alt,
equalityElimination,
dependent_pairFormation_alt,
promote_hyp,
closedConclusion,
approximateComputation,
pointwiseFunctionality,
baseApply
Latex:
\mforall{}[T:Type]
\mforall{}[eq:EqDecider(T)]. \mforall{}[bs:bag(T)]. (bag-parts(eq;bs) \mmember{} bag(bag(T) List\msupplus{}))
supposing valueall-type(T)
Date html generated:
2020_05_20-AM-09_04_29
Last ObjectModification:
2020_01_28-PM-03_23_08
Theory : bags_2
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