### Nuprl Lemma : ap2-tuple_wf

`∀[n:ℕ]. ∀[A,B:Type List].`
`  ∀[C:Type]. ∀[x:C]. ∀[f:tuple-type(map(λp.(C ⟶ (fst(p)) ⟶ (snd(p)));zip(A;B)))]. ∀[t:tuple-type(A)].`
`    (ap2-tuple(n;f;x;t) ∈ tuple-type(B)) `
`  supposing (||A|| = n ∈ ℤ) ∧ (||B|| = n ∈ ℤ)`

Proof

Definitions occuring in Statement :  ap2-tuple: `ap2-tuple(len;f;x;t)` tuple-type: `tuple-type(L)` zip: `zip(as;bs)` length: `||as||` map: `map(f;as)` list: `T List` nat: `ℕ` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` pi1: `fst(t)` pi2: `snd(t)` and: `P ∧ Q` member: `t ∈ T` lambda: `λx.A[x]` function: `x:A ⟶ B[x]` int: `ℤ` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` and: `P ∧ Q` nat: `ℕ` implies: `P `` Q` false: `False` ge: `i ≥ j ` not: `¬A` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` all: `∀x:A. B[x]` top: `Top` prop: `ℙ` or: `P ∨ Q` ap2-tuple: `ap2-tuple(len;f;x;t)` eq_int: `(i =z j)` subtract: `n - m` ifthenelse: `if b then t else f fi ` btrue: `tt` cons: `[a / b]` le: `A ≤ B` bool: `𝔹` unit: `Unit` it: `⋅` uiff: `uiff(P;Q)` bfalse: `ff` sq_type: `SQType(T)` guard: `{T}` bnot: `¬bb` assert: `↑b` rev_implies: `P `` Q` iff: `P `⇐⇒` Q` pi1: `fst(t)` pi2: `snd(t)` subtype_rel: `A ⊆r B` so_lambda: `λ2x.t[x]` so_apply: `x[s]` nequal: `a ≠ b ∈ T ` decidable: `Dec(P)` tuple-type: `tuple-type(L)` list_ind: list_ind
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 list-cases tupletype_nil_lemma zip_nil_lemma map_nil_lemma length_of_nil_lemma product_subtype_list length_of_cons_lemma non_neg_length intformeq_wf itermAdd_wf int_formula_prop_eq_lemma int_term_value_add_lemma tupletype_cons_lemma null_wf eqtt_to_assert assert_of_null eqff_to_assert bool_cases_sqequal subtype_base_sq bool_wf bool_subtype_base assert-bnot iff_weakening_uiff assert_wf equal-wf-T-base list_wf tuple-type_wf map_wf zip_wf istype-universe length_wf_nat set_subtype_base le_wf int_subtype_base subtract-1-ge-0 zip_cons_cons_lemma map_cons_lemma eq_int_wf assert_of_eq_int less_than_transitivity1 le_weakening less_than_irreflexivity neg_assert_of_eq_int null_nil_lemma null_cons_lemma zip_cons_nil_lemma btrue_wf bfalse_wf btrue_neq_bfalse nil_wf intformnot_wf int_formula_prop_not_lemma nat_wf bnot_wf not_wf cons_wf decidable__equal_int add-is-int-iff itermSubtract_wf int_term_value_subtract_lemma false_wf length_wf subtract_wf subtype_rel_self ifthenelse_wf bool_cases iff_transitivity assert_of_bnot
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity Error :isect_memberFormation_alt,  introduction cut sqequalHypSubstitution productElimination thin extract_by_obid isectElimination hypothesisEquality hypothesis setElimination rename sqequalRule intWeakElimination Error :lambdaFormation_alt,  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 universeEquality instantiate unionElimination promote_hyp hypothesis_subsumption Error :inhabitedIsType,  equalityElimination because_Cache Error :equalityIsType1,  cumulativity baseClosed Error :equalityIsType3,  productEquality functionEquality Error :productIsType,  Error :equalityIsType4,  applyEquality intEquality Error :equalityIsType2,  baseApply closedConclusion Error :dependent_set_memberEquality_alt,  applyLambdaEquality independent_pairEquality pointwiseFunctionality addEquality

Latex:
\mforall{}[n:\mBbbN{}].  \mforall{}[A,B:Type  List].
\mforall{}[C:Type].  \mforall{}[x:C].  \mforall{}[f:tuple-type(map(\mlambda{}p.(C  {}\mrightarrow{}  (fst(p))  {}\mrightarrow{}  (snd(p)));zip(A;B)))].
\mforall{}[t:tuple-type(A)].
(ap2-tuple(n;f;x;t)  \mmember{}  tuple-type(B))
supposing  (||A||  =  n)  \mwedge{}  (||B||  =  n)

Date html generated: 2019_06_20-PM-02_03_15
Last ObjectModification: 2018_10_06-AM-11_41_45

Theory : tuples

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