### Nuprl Lemma : bag-map-equal

`∀[T,A:Type].`
`  ∀f,g:T ⟶ A. ∀P:T ⟶ 𝔹.`
`    ((∀x:T. ((¬↑(P x)) `` ((f x) = (g x) ∈ A)))`
`    `` (∀as:bag(T). ((↑null([x∈as|P x])) `` (bag-map(f;as) = bag-map(g;as) ∈ bag(A)))))`

Proof

Definitions occuring in Statement :  bag-filter: `[x∈b|p[x]]` bag-map: `bag-map(f;bs)` bag: `bag(T)` null: `null(as)` assert: `↑b` bool: `𝔹` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` not: `¬A` implies: `P `` Q` apply: `f a` function: `x:A ⟶ B[x]` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  bag-null: `bag-null(bs)` uall: `∀[x:A]. B[x]` member: `t ∈ T` all: `∀x:A. B[x]` implies: `P `` Q` prop: `ℙ` so_lambda: `λ2x.t[x]` so_apply: `x[s]` bag: `bag(T)` quotient: `x,y:A//B[x; y]` and: `P ∧ Q` subtype_rel: `A ⊆r B` uimplies: `b supposing a` bag-filter: `[x∈b|p[x]]` bag-map: `bag-map(f;bs)` nat: `ℕ` ge: `i ≥ j ` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` not: `¬A` top: `Top` uiff: `uiff(P;Q)` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` cand: `A c∧ B` int_seg: `{i..j-}` lelt: `i ≤ j < k` le: `A ≤ B` guard: `{T}` rev_uimplies: `rev_uimplies(P;Q)`
Lemmas referenced :  assert_wf bag-null_wf bag-filter_wf bag_wf all_wf not_wf equal_wf bool_wf bag-map_wf list_wf permutation_wf equal-wf-base list-subtype-bag map_equal select_wf nat_properties 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 less_than_wf length_wf nat_wf assert_of_null filter_wf5 l_member_wf member_filter select_member lelt_wf assert_functionality_wrt_uiff eta_conv null_nil_lemma btrue_wf member-implies-null-eq-bfalse and_wf null_wf btrue_neq_bfalse
Rules used in proof :  sqequalSubstitution sqequalRule sqequalReflexivity sqequalTransitivity computationStep isect_memberFormation introduction cut lambdaFormation hypothesis extract_by_obid sqequalHypSubstitution isectElimination thin dependent_functionElimination setEquality cumulativity hypothesisEquality applyEquality functionExtensionality lambdaEquality functionEquality axiomEquality because_Cache isect_memberEquality independent_functionElimination pointwiseFunctionalityForEquality pertypeElimination productElimination equalityTransitivity equalitySymmetry rename productEquality independent_isectElimination setElimination unionElimination natural_numberEquality dependent_pairFormation int_eqEquality intEquality voidElimination voidEquality independent_pairFormation computeAll dependent_set_memberEquality applyLambdaEquality

Latex:
\mforall{}[T,A:Type].
\mforall{}f,g:T  {}\mrightarrow{}  A.  \mforall{}P:T  {}\mrightarrow{}  \mBbbB{}.
((\mforall{}x:T.  ((\mneg{}\muparrow{}(P  x))  {}\mRightarrow{}  ((f  x)  =  (g  x))))
{}\mRightarrow{}  (\mforall{}as:bag(T).  ((\muparrow{}null([x\mmember{}as|P  x]))  {}\mRightarrow{}  (bag-map(f;as)  =  bag-map(g;as)))))

Date html generated: 2017_10_01-AM-08_45_41
Last ObjectModification: 2017_07_26-PM-04_30_51

Theory : bags

Home Index