Nuprl Lemma : decidable__sub-bag
∀[T:Type]. ((∀x,y:T. Dec(x = y ∈ T)) ⇒ (∀as,bs:bag(T). Dec(sub-bag(T;as;bs))))
Proof
Definitions occuring in Statement :
sub-bag: sub-bag(T;as;bs),
bag: bag(T),
decidable: Dec(P),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x],
member: t ∈ T,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
nat: ℕ,
so_apply: x[s],
prop: ℙ,
decidable: Dec(P),
or: P ∨ Q,
guard: {T},
le: A ≤ B,
not: ¬A,
false: False,
uiff: uiff(P;Q),
uimplies: b supposing a,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
bag-member_wf,
int_formula_prop_wf,
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,
intformnot_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformand_wf,
satisfiable-full-omega-tt,
nat_properties,
decidable__le,
bag-member-count,
assert_of_le_int,
decidable-equal-deq,
decidable__bag-member,
less_than'_wf,
assert-bag-all,
not_wf,
equal_wf,
decidable_wf,
bag_wf,
le_int_wf,
bag-all_wf,
decidable__assert,
nat_wf,
bag-count_wf,
le_wf,
all_wf,
sub-bag_wf,
decidable_functionality,
sub-bag-iff,
deq-exists
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
productElimination,
independent_functionElimination,
hypothesis,
rename,
promote_hyp,
dependent_functionElimination,
sqequalRule,
lambdaEquality,
applyEquality,
setElimination,
because_Cache,
universeEquality,
unionElimination,
inlFormation,
inrFormation,
independent_pairFormation,
introduction,
independent_pairEquality,
voidElimination,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
natural_numberEquality,
setEquality,
intEquality,
dependent_pairFormation,
int_eqEquality,
isect_memberEquality,
voidEquality,
computeAll
Latex:
\mforall{}[T:Type]. ((\mforall{}x,y:T. Dec(x = y)) {}\mRightarrow{} (\mforall{}as,bs:bag(T). Dec(sub-bag(T;as;bs))))
Date html generated:
2016_05_15-PM-08_09_06
Last ObjectModification:
2016_01_16-PM-01_27_53
Theory : bags_2
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