`∀x:Top. ∀n,m:ℕ.  (bag-rep(n + m;x) ~ bag-rep(n;x) + bag-rep(m;x))`

Proof

Definitions occuring in Statement :  bag-rep: `bag-rep(n;x)` bag-append: `as + bs` nat: `ℕ` top: `Top` all: `∀x:A. B[x]` add: `n + m` sqequal: `s ~ t`
Definitions unfolded in proof :  member: `t ∈ T` top: `Top` nat: `ℕ` all: `∀x:A. B[x]` uall: `∀[x:A]. B[x]` implies: `P `` Q` false: `False` ge: `i ≥ j ` uimplies: `b supposing a` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` not: `¬A` and: `P ∧ Q` prop: `ℙ` bag-rep: `bag-rep(n;x)` eq_int: `(i =z j)` subtract: `n - m` ifthenelse: `if b then t else f fi ` btrue: `tt` decidable: `Dec(P)` or: `P ∨ Q` bool: `𝔹` unit: `Unit` it: `⋅` uiff: `uiff(P;Q)` bfalse: `ff` sq_type: `SQType(T)` guard: `{T}` bnot: `¬bb` assert: `↑b` nequal: `a ≠ b ∈ T ` bag-append: `as + bs` empty-bag: `{}` cons-bag: `x.b` primrec: `primrec(n;b;c)` append: `as @ bs` nil: `[]` cons: `[a / b]` list_ind: list_ind bottom: `⊥`
Lemmas referenced :  nat_wf top_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 primrec-unroll empty_bag_append_lemma zero-add decidable__le subtract_wf intformnot_wf itermSubtract_wf int_formula_prop_not_lemma int_term_value_subtract_lemma eq_int_wf bool_wf eqtt_to_assert assert_of_eq_int intformeq_wf itermAdd_wf int_formula_prop_eq_lemma int_term_value_add_lemma eqff_to_assert equal_wf bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot neg_assert_of_eq_int cons_bag_append_lemma general_arith_equation1
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity natural_numberEquality isect_memberEquality voidElimination voidEquality cut introduction extract_by_obid hypothesis addEquality hypothesisEquality sqequalHypSubstitution setElimination thin rename because_Cache lambdaFormation isectElimination intWeakElimination independent_isectElimination dependent_pairFormation lambdaEquality int_eqEquality intEquality dependent_functionElimination sqequalRule independent_pairFormation computeAll independent_functionElimination sqequalAxiom unionElimination equalityElimination equalityTransitivity equalitySymmetry productElimination promote_hyp instantiate cumulativity

Latex:
\mforall{}x:Top.  \mforall{}n,m:\mBbbN{}.    (bag-rep(n  +  m;x)  \msim{}  bag-rep(n;x)  +  bag-rep(m;x))

Date html generated: 2017_10_01-AM-08_52_03
Last ObjectModification: 2017_07_26-PM-04_33_46

Theory : bags

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