### Nuprl Lemma : zip_wf

`∀[T1,T2:Type]. ∀[as:T1 List]. ∀[bs:T2 List].  (zip(as;bs) ∈ (T1 × T2) List)`

Proof

Definitions occuring in Statement :  zip: `zip(as;bs)` list: `T List` uall: `∀[x:A]. B[x]` member: `t ∈ T` product: `x:A × B[x]` universe: `Type`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` all: `∀x:A. B[x]` nat: `ℕ` implies: `P `` Q` false: `False` ge: `i ≥ j ` uimplies: `b supposing a` not: `¬A` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` top: `Top` and: `P ∧ Q` prop: `ℙ` guard: `{T}` or: `P ∨ Q` zip: `zip(as;bs)` list_ind: list_ind nil: `[]` it: `⋅` cons: `[a / b]` le: `A ≤ B` less_than': `less_than'(a;b)` colength: `colength(L)` so_lambda: `λ2x.t[x]` so_apply: `x[s]` sq_type: `SQType(T)` less_than: `a < b` squash: `↓T` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` decidable: `Dec(P)` subtype_rel: `A ⊆r B` so_lambda: `so_lambda(x,y,z.t[x; y; z])` so_apply: `x[s1;s2;s3]`
Lemmas referenced :  nat_properties full-omega-unsat intformand_wf intformle_wf itermConstant_wf itermVar_wf intformless_wf istype-int int_formula_prop_and_lemma istype-void 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 intformeq_wf int_formula_prop_eq_lemma list-cases nil_wf list_wf product_subtype_list colength-cons-not-zero colength_wf_list istype-false le_wf subtract-1-ge-0 subtype_base_sq set_subtype_base int_subtype_base spread_cons_lemma decidable__equal_int subtract_wf intformnot_wf itermSubtract_wf itermAdd_wf int_formula_prop_not_lemma int_term_value_subtract_lemma int_term_value_add_lemma decidable__le list_ind_cons_lemma list_ind_nil_lemma cons_wf nat_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity Error :isect_memberFormation_alt,  introduction cut thin Error :lambdaFormation_alt,  extract_by_obid sqequalHypSubstitution isectElimination hypothesisEquality hypothesis setElimination rename sqequalRule intWeakElimination natural_numberEquality independent_isectElimination approximateComputation independent_functionElimination Error :dependent_pairFormation_alt,  Error :lambdaEquality_alt,  int_eqEquality dependent_functionElimination Error :isect_memberEquality_alt,  voidElimination independent_pairFormation Error :universeIsType,  axiomEquality equalityTransitivity equalitySymmetry applyLambdaEquality Error :functionIsTypeImplies,  Error :inhabitedIsType,  unionElimination productEquality promote_hyp hypothesis_subsumption productElimination Error :equalityIsType1,  because_Cache Error :dependent_set_memberEquality_alt,  instantiate imageElimination Error :equalityIsType4,  baseApply closedConclusion baseClosed applyEquality intEquality independent_pairEquality universeEquality

Latex:
\mforall{}[T1,T2:Type].  \mforall{}[as:T1  List].  \mforall{}[bs:T2  List].    (zip(as;bs)  \mmember{}  (T1  \mtimes{}  T2)  List)

Date html generated: 2019_06_20-PM-01_47_12
Last ObjectModification: 2018_10_06-PM-06_09_35

Theory : list_1

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