### Nuprl Lemma : bag-map-filter

`∀[T,A:Type]. ∀[f:T ⟶ A]. ∀[P:T ⟶ 𝔹]. ∀[Q:A ⟶ 𝔹].`
`  ∀[L:bag(T)]. (bag-map(f;[x∈L|P[x]]) = [x∈bag-map(f;L)|Q[x]] ∈ bag(A)) supposing ∀x:T. Q[f x] = P[x]`

Proof

Definitions occuring in Statement :  bag-filter: `[x∈b|p[x]]` bag-map: `bag-map(f;bs)` bag: `bag(T)` bool: `𝔹` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` so_apply: `x[s]` all: `∀x:A. B[x]` apply: `f a` function: `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` bag: `bag(T)` quotient: `x,y:A//B[x; y]` and: `P ∧ Q` all: `∀x:A. B[x]` implies: `P `` Q` prop: `ℙ` so_lambda: `λ2x.t[x]` so_apply: `x[s]` bag-map: `bag-map(f;bs)` bag-filter: `[x∈b|p[x]]` nat: `ℕ` false: `False` ge: `i ≥ j ` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` not: `¬A` top: `Top` subtype_rel: `A ⊆r B` guard: `{T}` or: `P ∨ Q` cons: `[a / b]` colength: `colength(L)` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` decidable: `Dec(P)` nil: `[]` it: `⋅` sq_type: `SQType(T)` less_than: `a < b` squash: `↓T` less_than': `less_than'(a;b)` bool: `𝔹` unit: `Unit` btrue: `tt` uiff: `uiff(P;Q)` ifthenelse: `if b then t else f fi ` bfalse: `ff` bnot: `¬bb` assert: `↑b`
Lemmas referenced :  bag_wf list_wf permutation_wf equal_wf equal-wf-base all_wf bool_wf map-filter map_wf nat_properties satisfiable-full-omega-tt intformand_wf intformle_wf itermConstant_wf itermVar_wf intformless_wf int_formula_prop_and_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_less_lemma int_formula_prop_wf ge_wf less_than_wf equal-wf-T-base nat_wf colength_wf_list less_than_transitivity1 less_than_irreflexivity list-cases filter_nil_lemma product_subtype_list spread_cons_lemma intformeq_wf itermAdd_wf int_formula_prop_eq_lemma int_term_value_add_lemma decidable__le intformnot_wf int_formula_prop_not_lemma le_wf subtract_wf itermSubtract_wf int_term_value_subtract_lemma subtype_base_sq set_subtype_base int_subtype_base decidable__equal_int filter_cons_lemma eqtt_to_assert eqff_to_assert bool_cases_sqequal bool_subtype_base assert-bnot quotient-member-eq permutation-equiv bag-filter_wf bag-map_wf list-subtype-bag subtype_rel_bag assert_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut sqequalHypSubstitution pointwiseFunctionalityForEquality extract_by_obid isectElimination thin cumulativity hypothesisEquality hypothesis sqequalRule pertypeElimination productElimination equalityTransitivity equalitySymmetry lambdaFormation because_Cache rename dependent_functionElimination independent_functionElimination productEquality isect_memberEquality axiomEquality lambdaEquality applyEquality functionExtensionality functionEquality independent_isectElimination setElimination intWeakElimination natural_numberEquality dependent_pairFormation int_eqEquality intEquality voidElimination voidEquality independent_pairFormation computeAll sqequalAxiom unionElimination promote_hyp hypothesis_subsumption applyLambdaEquality dependent_set_memberEquality addEquality baseClosed instantiate imageElimination equalityElimination setEquality hyp_replacement

Latex:
\mforall{}[T,A:Type].  \mforall{}[f:T  {}\mrightarrow{}  A].  \mforall{}[P:T  {}\mrightarrow{}  \mBbbB{}].  \mforall{}[Q:A  {}\mrightarrow{}  \mBbbB{}].
\mforall{}[L:bag(T)].  (bag-map(f;[x\mmember{}L|P[x]])  =  [x\mmember{}bag-map(f;L)|Q[x]])  supposing  \mforall{}x:T.  Q[f  x]  =  P[x]

Date html generated: 2017_10_01-AM-08_46_01
Last ObjectModification: 2017_07_26-PM-04_31_04

Theory : bags

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