Nuprl Lemma : bag-no-repeats-settype
∀[T:Type]. ∀[bs:bag(T)]. uiff(bag-no-repeats({x:T| x ↓∈ bs} ;bs);bag-no-repeats(T;bs)) supposing ∀x,y:T. Dec(x = y ∈ T\000C)
Proof
Definitions occuring in Statement :
bag-member: x ↓∈ bs,
bag-no-repeats: bag-no-repeats(T;bs),
bag: bag(T),
decidable: Dec(P),
uiff: uiff(P;Q),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
set: {x:A| B[x]} ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
uiff: uiff(P;Q),
and: P ∧ Q,
bag-no-repeats: bag-no-repeats(T;bs),
squash: ↓T,
prop: ℙ,
all: ∀x:A. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
exists: ∃x:A. B[x],
subtype_rel: A ⊆r B,
cand: A c∧ B,
guard: {T},
implies: P ⇒ Q,
l_all: (∀x∈L.P[x]),
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
top: Top,
less_than: a < b
Lemmas referenced :
bag-no-repeats_wf,
bag-member_wf,
bag-subtype,
all_wf,
decidable_wf,
equal_wf,
bag_wf,
subtype_rel_list,
subtype_rel_bag,
equal_functionality_wrt_subtype_rel2,
no_repeats-settype,
list-subtype-bag,
no_repeats_wf,
subtype_rel_self,
list-set-type2,
bag-member-select,
select_wf,
int_seg_properties,
length_wf,
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,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
int_seg_wf,
no_repeats-subtype,
bag-settype,
and_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
sqequalRule,
sqequalHypSubstitution,
imageElimination,
hypothesis,
imageMemberEquality,
hypothesisEquality,
thin,
baseClosed,
extract_by_obid,
isectElimination,
setEquality,
cumulativity,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
independent_pairEquality,
isect_memberEquality,
because_Cache,
lambdaEquality,
universeEquality,
dependent_pairFormation,
applyEquality,
independent_isectElimination,
setElimination,
rename,
independent_functionElimination,
productEquality,
lambdaFormation,
hyp_replacement,
applyLambdaEquality,
natural_numberEquality,
unionElimination,
int_eqEquality,
intEquality,
voidElimination,
voidEquality,
computeAll,
dependent_set_memberEquality,
addLevel,
levelHypothesis
Latex:
\mforall{}[T:Type]. \mforall{}[bs:bag(T)]. uiff(bag-no-repeats(\{x:T| x \mdownarrow{}\mmember{} bs\} ;bs);bag-no-repeats(T;bs)) supposing \mforall{}x\000C,y:T. Dec(x = y)
Date html generated:
2017_10_01-AM-09_01_12
Last ObjectModification:
2017_07_26-PM-04_42_21
Theory : bags
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