Nuprl Lemma : sv-bag-only-combine

[A,B:Type]. ∀[b:bag(A)]. ∀[f:A ⟶ bag(B)].
(sv-bag-only(⋃x∈b.f[x]) sv-bag-only(f[sv-bag-only(b)]) ∈ B) supposing
((∀a:A. 0 < #(f[a])) and
(∀a:A. single-valued-bag(f[a];B)) and
0 < #(b) and
single-valued-bag(b;A))

Proof

Definitions occuring in Statement :  sv-bag-only: sv-bag-only(b) single-valued-bag: single-valued-bag(b;T) bag-combine: x∈bs.f[x] bag-size: #(bs) bag: bag(T) less_than: a < b uimplies: supposing a uall: [x:A]. B[x] so_apply: x[s] all: x:A. B[x] function: x:A ⟶ B[x] natural_number: \$n universe: Type equal: t ∈ T
Definitions unfolded in proof :  uall: [x:A]. B[x] member: t ∈ T uimplies: supposing a squash: T exists: x:A. B[x] so_apply: x[s] true: True and: P ∧ Q cand: c∧ B so_lambda: λ2x.t[x] all: x:A. B[x] subtype_rel: A ⊆B nat: decidable: Dec(P) or: P ∨ Q less_than: a < b le: A ≤ B satisfiable_int_formula: satisfiable_int_formula(fmla) false: False implies:  Q not: ¬A top: Top prop: guard: {T} iff: ⇐⇒ Q rev_implies:  Q single-valued-bag: single-valued-bag(b;T) sq_or: a ↓∨ b similar-bags: similar-bags(A;as;bs) sq_stable: SqStable(P) uiff: uiff(P;Q) rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :  bag-member-sv-bag-only bag-member-implies-hd-append sv-bag-only_wf single-valued-bag-combine bag-combine-size-bound2 decidable__lt bag-size_wf bag-combine_wf nat_wf satisfiable-full-omega-tt intformand_wf intformnot_wf intformless_wf itermConstant_wf itermVar_wf intformle_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_less_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_le_lemma int_formula_prop_wf single-bag_wf single-valued-bag_wf squash_wf true_wf iff_weakening_equal bag-member_wf bag-combine-single-left all_wf less_than_wf bag_wf equal_wf bag-combine-append-left sv-bag-only-append bag-member-append bag-size-bag-member sq_stable__bag-member bag-member-size bag-member-combine
Rules used in proof :  cut introduction extract_by_obid sqequalHypSubstitution sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isectElimination thin hypothesisEquality independent_isectElimination hypothesis because_Cache cumulativity imageElimination productElimination applyEquality functionExtensionality natural_numberEquality sqequalRule lambdaEquality independent_pairFormation dependent_functionElimination equalityTransitivity equalitySymmetry setElimination rename unionElimination dependent_pairFormation int_eqEquality intEquality isect_memberEquality voidElimination voidEquality computeAll imageMemberEquality baseClosed independent_functionElimination functionEquality universeEquality isect_memberFormation axiomEquality productEquality lambdaFormation inrFormation hyp_replacement applyLambdaEquality

Latex:
\mforall{}[A,B:Type].  \mforall{}[b:bag(A)].  \mforall{}[f:A  {}\mrightarrow{}  bag(B)].
(sv-bag-only(\mcup{}x\mmember{}b.f[x])  =  sv-bag-only(f[sv-bag-only(b)]))  supposing
((\mforall{}a:A.  0  <  \#(f[a]))  and
(\mforall{}a:A.  single-valued-bag(f[a];B))  and
0  <  \#(b)  and
single-valued-bag(b;A))

Date html generated: 2017_10_01-AM-08_55_59
Last ObjectModification: 2017_07_26-PM-04_38_01

Theory : bags

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