Nuprl Lemma : int-dot-mul-right

[x:ℤ]. ∀[as,bs:ℤ List].  (as ⋅ bs as ⋅ bs)


Proof




Definitions occuring in Statement :  int-vec-mul: as integer-dot-product: as ⋅ bs list: List uall: [x:A]. B[x] multiply: m int: sqequal: t
Definitions unfolded in proof :  uall: [x:A]. B[x] member: t ∈ T all: x:A. B[x] nat: implies:  Q false: False ge: i ≥  guard: {T} uimplies: supposing a prop: subtype_rel: A ⊆B or: P ∨ Q int-vec-mul: as top: Top cons: [a b] colength: colength(L) so_lambda: λ2y.t[x; y] so_apply: x[s1;s2] nil: [] it: so_lambda: λ2x.t[x] so_apply: x[s] sq_type: SQType(T) squash: T sq_stable: SqStable(P) uiff: uiff(P;Q) and: P ∧ Q le: A ≤ B not: ¬A less_than': less_than'(a;b) true: True decidable: Dec(P) iff: ⇐⇒ Q rev_implies:  Q subtract: m less_than: a < b
Lemmas referenced :  nat_properties less_than_transitivity1 less_than_irreflexivity ge_wf less_than_wf equal-wf-base nat_wf list_subtype_base int_subtype_base list-cases int_dot_nil_left_lemma mul-zero product_subtype_list spread_cons_lemma equal_wf subtype_base_sq set_subtype_base le_wf colength_wf_list sq_stable__le le_antisymmetry_iff add_functionality_wrt_le add-associates add-zero zero-add le-add-cancel equal-wf-T-base decidable__le false_wf not-le-2 condition-implies-le minus-add minus-one-mul minus-one-mul-top add-commutes subtract_wf not-ge-2 less-iff-le minus-minus add-swap map_nil_lemma int_dot_cons_nil_lemma map_cons_lemma int_dot_cons_lemma mul-swap mul-distributes integer-dot-product_wf list_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut thin lambdaFormation extract_by_obid sqequalHypSubstitution isectElimination hypothesisEquality hypothesis setElimination rename intWeakElimination natural_numberEquality independent_isectElimination independent_functionElimination voidElimination sqequalRule lambdaEquality dependent_functionElimination isect_memberEquality sqequalAxiom because_Cache baseApply closedConclusion baseClosed applyEquality intEquality unionElimination voidEquality promote_hyp hypothesis_subsumption productElimination equalityTransitivity equalitySymmetry instantiate cumulativity applyLambdaEquality imageMemberEquality imageElimination addEquality dependent_set_memberEquality independent_pairFormation minusEquality multiplyEquality

Latex:
\mforall{}[x:\mBbbZ{}].  \mforall{}[as,bs:\mBbbZ{}  List].    (as  \mcdot{}  x  *  bs  \msim{}  x  *  as  \mcdot{}  bs)



Date html generated: 2017_04_14-AM-08_55_49
Last ObjectModification: 2017_02_27-PM-03_39_51

Theory : omega


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