`∀[f1,f2,g1,g2:ℕ+ ⟶ ℤ].  (bdd-diff(f1;f2) `` bdd-diff(g1;g2) `` bdd-diff(λn.(f1[n] + g1[n]);λn.(f2[n] + g2[n])))`

Proof

Definitions occuring in Statement :  bdd-diff: `bdd-diff(f;g)` nat_plus: `ℕ+` uall: `∀[x:A]. B[x]` so_apply: `x[s]` implies: `P `` Q` lambda: `λx.A[x]` function: `x:A ⟶ B[x]` add: `n + m` int: `ℤ`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` implies: `P `` Q` bdd-diff: `bdd-diff(f;g)` exists: `∃x:A. B[x]` member: `t ∈ T` nat: `ℕ` ge: `i ≥ j ` all: `∀x:A. B[x]` decidable: `Dec(P)` or: `P ∨ Q` uimplies: `b supposing a` satisfiable_int_formula: `satisfiable_int_formula(fmla)` false: `False` not: `¬A` top: `Top` and: `P ∧ Q` prop: `ℙ` le: `A ≤ B` so_apply: `x[s]` so_lambda: `λ2x.t[x]` subtype_rel: `A ⊆r B` nat_plus: `ℕ+` uiff: `uiff(P;Q)` rev_uimplies: `rev_uimplies(P;Q)` guard: `{T}`
Lemmas referenced :  nat_properties decidable__le satisfiable-full-omega-tt intformand_wf intformnot_wf intformle_wf itermConstant_wf itermAdd_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_add_lemma int_term_value_var_lemma int_formula_prop_wf le_wf less_than'_wf nat_plus_wf all_wf absval_wf subtract_wf bdd-diff_wf nat_wf nat_plus_properties less_than_wf decidable__equal_int intformeq_wf itermSubtract_wf itermMinus_wf int_formula_prop_eq_lemma int_term_value_subtract_lemma int_term_value_minus_lemma add-is-int-iff subtract-is-int-iff false_wf and_wf equal_wf le_functionality le_weakening int-triangle-inequality add_functionality_wrt_le
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation lambdaFormation sqequalHypSubstitution productElimination thin dependent_pairFormation dependent_set_memberEquality addEquality setElimination rename cut hypothesisEquality hypothesis introduction extract_by_obid isectElimination dependent_functionElimination natural_numberEquality unionElimination independent_isectElimination lambdaEquality int_eqEquality intEquality isect_memberEquality voidElimination voidEquality sqequalRule independent_pairFormation computeAll independent_pairEquality because_Cache axiomEquality equalityTransitivity equalitySymmetry applyEquality functionExtensionality functionEquality pointwiseFunctionality promote_hyp baseApply closedConclusion baseClosed setEquality hyp_replacement Error :applyLambdaEquality

Latex:
\mforall{}[f1,f2,g1,g2:\mBbbN{}\msupplus{}  {}\mrightarrow{}  \mBbbZ{}].
(bdd-diff(f1;f2)  {}\mRightarrow{}  bdd-diff(g1;g2)  {}\mRightarrow{}  bdd-diff(\mlambda{}n.(f1[n]  +  g1[n]);\mlambda{}n.(f2[n]  +  g2[n])))

Date html generated: 2016_10_26-AM-09_02_44
Last ObjectModification: 2016_07_12-AM-08_13_01

Theory : reals

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