Nuprl Lemma : sub-bag_antisymmetry
∀[T:Type]. ∀[as,bs:bag(T)]. (as = bs ∈ bag(T)) supposing (sub-bag(T;as;bs) and sub-bag(T;bs;as))
Proof
Definitions occuring in Statement :
sub-bag: sub-bag(T;as;bs),
bag: bag(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
sub-bag: sub-bag(T;as;bs),
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
exists: ∃x:A. B[x],
subtype_rel: A ⊆r B,
nat: ℕ,
top: Top,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
empty-bag: {},
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
guard: {T},
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
implies: P ⇒ Q,
not: ¬A,
and: P ∧ Q
Lemmas referenced :
bag-subtype-list,
bag-append-empty,
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,
decidable__le,
bag-size-zero,
bag-append_wf,
equal_wf,
bag_wf,
exists_wf,
bag-size-append,
nat_wf,
bag-size_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
productElimination,
thin,
hypothesis,
applyEquality,
lambdaEquality,
lemma_by_obid,
isectElimination,
hypothesisEquality,
setElimination,
rename,
because_Cache,
isect_memberEquality,
voidElimination,
voidEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
universeEquality,
independent_isectElimination,
dependent_functionElimination,
unionElimination,
setEquality,
intEquality,
natural_numberEquality,
dependent_pairFormation,
int_eqEquality,
independent_pairFormation,
computeAll
Latex:
\mforall{}[T:Type]. \mforall{}[as,bs:bag(T)]. (as = bs) supposing (sub-bag(T;as;bs) and sub-bag(T;bs;as))
Date html generated:
2016_05_15-PM-02_35_51
Last ObjectModification:
2016_01_16-AM-08_52_13
Theory : bags
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