Nuprl Lemma : bag-in-subtype
∀[A,B:Type]. ∀[b:bag(B)]. b ∈ bag(A) supposing ∀x:B. (x ↓∈ b ⇒ (x ∈ A)) supposing strong-subtype(A;B)
Proof
Definitions occuring in Statement :
bag-member: x ↓∈ bs,
bag: bag(T),
strong-subtype: strong-subtype(A;B),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
member: t ∈ T,
sq_stable: SqStable(P),
implies: P ⇒ Q,
uiff: uiff(P;Q),
and: P ∧ Q,
squash: ↓T,
prop: ℙ,
all: ∀x:A. B[x],
respects-equality: respects-equality(S;T),
exists: ∃x:A. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
guard: {T},
nat: ℕ,
false: False,
ge: i ≥ j ,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
or: P ∨ Q,
subtype_rel: A ⊆r B,
cons: [a / b],
le: A ≤ B,
less_than': less_than'(a;b),
colength: colength(L),
nil: [],
it: ⋅,
sq_type: SQType(T),
less_than: a < b,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
decidable: Dec(P),
bag-append: as + bs,
append: as @ bs,
list_ind: list_ind,
single-bag: {x},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
sq_or: a ↓∨ b
Lemmas referenced :
sq_stable__respects-equality,
strong-subtype-iff-respects-equality,
respects-equality-bag,
bag_wf,
strong-subtype_wf,
istype-universe,
bag-member_wf,
equal-wf,
bag_to_squash_list,
all_wf,
member_wf,
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
istype-less_than,
list-cases,
list-subtype-bag,
nil_wf,
product_subtype_list,
colength-cons-not-zero,
colength_wf_list,
istype-le,
list_wf,
subtract-1-ge-0,
subtype_base_sq,
intformeq_wf,
int_formula_prop_eq_lemma,
set_subtype_base,
int_subtype_base,
spread_cons_lemma,
decidable__equal_int,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
itermAdd_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
int_term_value_add_lemma,
decidable__le,
le_wf,
cons_wf,
istype-nat,
bag-append_wf,
single-bag_wf,
bag-member-append,
bag-member-single
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
independent_functionElimination,
because_Cache,
productElimination,
independent_isectElimination,
sqequalRule,
imageMemberEquality,
baseClosed,
imageElimination,
universeIsType,
inhabitedIsType,
instantiate,
universeEquality,
functionIsType,
equalityIstype,
dependent_functionElimination,
promote_hyp,
equalitySymmetry,
hyp_replacement,
applyLambdaEquality,
lambdaEquality_alt,
functionEquality,
isectEquality,
cumulativity,
rename,
lambdaFormation_alt,
setElimination,
intWeakElimination,
natural_numberEquality,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
axiomEquality,
equalityTransitivity,
functionIsTypeImplies,
unionElimination,
voidEquality,
closedConclusion,
applyEquality,
hypothesis_subsumption,
dependent_set_memberEquality_alt,
baseApply,
intEquality,
sqequalBase,
inlFormation_alt,
inrFormation_alt
Latex:
\mforall{}[A,B:Type].
\mforall{}[b:bag(B)]. b \mmember{} bag(A) supposing \mforall{}x:B. (x \mdownarrow{}\mmember{} b {}\mRightarrow{} (x \mmember{} A)) supposing strong-subtype(A;B)
Date html generated:
2019_10_15-AM-11_01_23
Last ObjectModification:
2019_08_15-PM-03_41_06
Theory : bags
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