Nuprl Lemma : zip-append

[A,B:Type]. ∀[xs1:A List]. ∀[ys1:B List]. ∀[xs2:A List]. ∀[ys2:B List].
  zip(xs1 xs2;ys1 ys2) zip(xs1;ys1) zip(xs2;ys2) supposing ||xs1|| ||ys1|| ∈ ℤ


Proof




Definitions occuring in Statement :  zip: zip(as;bs) length: ||as|| append: as bs list: List uimplies: supposing a uall: [x:A]. B[x] int: universe: Type sqequal: t equal: t ∈ 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 ≥  uimplies: supposing a satisfiable_int_formula: satisfiable_int_formula(fmla) exists: x:A. B[x] not: ¬A top: Top and: P ∧ Q prop: subtype_rel: A ⊆B guard: {T} or: P ∨ Q append: as bs so_lambda: so_lambda(x,y,z.t[x; y; z]) so_apply: x[s1;s2;s3] 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) le: A ≤ B decidable: Dec(P) less_than: a < b squash: T less_than': less_than'(a;b) uiff: uiff(P;Q)
Lemmas referenced :  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 length_wf list_wf equal-wf-T-base nat_wf colength_wf_list less_than_transitivity1 less_than_irreflexivity list-cases equal-wf-base-T length_of_nil_lemma list_ind_nil_lemma zip_nil_lemma equal-wf-base product_subtype_list spread_cons_lemma subtype_base_sq set_subtype_base le_wf int_subtype_base length_of_cons_lemma non_neg_length intformeq_wf itermAdd_wf int_formula_prop_eq_lemma int_term_value_add_lemma cons_wf decidable__le intformnot_wf int_formula_prop_not_lemma subtract_wf itermSubtract_wf int_term_value_subtract_lemma decidable__equal_int list_ind_cons_lemma zip_cons_cons_lemma add-is-int-iff false_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 dependent_pairFormation lambdaEquality int_eqEquality intEquality dependent_functionElimination isect_memberEquality voidElimination voidEquality sqequalRule independent_pairFormation computeAll independent_functionElimination sqequalAxiom cumulativity equalityTransitivity equalitySymmetry because_Cache applyEquality unionElimination baseClosed promote_hyp hypothesis_subsumption productElimination instantiate applyLambdaEquality dependent_set_memberEquality addEquality imageElimination pointwiseFunctionality baseApply closedConclusion universeEquality

Latex:
\mforall{}[A,B:Type].  \mforall{}[xs1:A  List].  \mforall{}[ys1:B  List].  \mforall{}[xs2:A  List].  \mforall{}[ys2:B  List].
    zip(xs1  @  xs2;ys1  @  ys2)  \msim{}  zip(xs1;ys1)  @  zip(xs2;ys2)  supposing  ||xs1||  =  ||ys1||



Date html generated: 2018_05_21-PM-06_37_56
Last ObjectModification: 2017_07_26-PM-04_53_05

Theory : general


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