Nuprl Lemma : bag-member-strong-subtype
∀[A,B:Type]. ∀b:bag(A). ∀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],
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
implies: P ⇒ Q,
squash: ↓T,
exists: ∃x:A. B[x],
prop: ℙ,
subtype_rel: A ⊆r B,
strong-subtype: strong-subtype(A;B),
cand: A c∧ B,
bag-member: x ↓∈ bs,
and: P ∧ Q,
bag: bag(T),
quotient: x,y:A//B[x; y],
guard: {T},
iff: P ⇐⇒ Q,
l_member: (x ∈ l),
nat: ℕ,
rev_implies: P ⇐ Q,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
top: Top
Lemmas referenced :
bag_to_squash_list,
bag-member_wf,
subtype_rel_bag,
bag_wf,
strong-subtype_wf,
member-permutation,
member_wf,
list_wf,
subtype_rel_list,
permutation_wf,
strong-subtype-implies,
select_wf,
nat_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
imageElimination,
productElimination,
promote_hyp,
hypothesis,
equalitySymmetry,
hyp_replacement,
Error :applyLambdaEquality,
cumulativity,
applyEquality,
independent_isectElimination,
sqequalRule,
rename,
lambdaEquality,
dependent_functionElimination,
axiomEquality,
equalityTransitivity,
because_Cache,
isect_memberEquality,
universeEquality,
pertypeElimination,
independent_functionElimination,
productEquality,
setElimination,
natural_numberEquality,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll
Latex:
\mforall{}[A,B:Type]. \mforall{}b:bag(A). \mforall{}x:B. (x \mdownarrow{}\mmember{} b {}\mRightarrow{} (x \mmember{} A)) supposing strong-subtype(A;B)
Date html generated:
2016_10_25-AM-10_26_55
Last ObjectModification:
2016_07_12-AM-06_43_11
Theory : bags
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