### Nuprl Lemma : tuple-type-continuous

`∀[P:Type]. ∀[X:ℕ ⟶ P ⟶ Type]. ∀[L:P List].  ((⋂n:ℕ. tuple-type(map(X n;L))) ⊆r tuple-type(map(λp.⋂n:ℕ. (X n p);L)))`

Proof

Definitions occuring in Statement :  tuple-type: `tuple-type(L)` map: `map(f;as)` list: `T List` nat: `ℕ` subtype_rel: `A ⊆r B` uall: `∀[x:A]. B[x]` apply: `f a` lambda: `λx.A[x]` isect: `⋂x:A. B[x]` function: `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` subtype_rel: `A ⊆r B` or: `P ∨ Q` cons: `[a / b]` decidable: `Dec(P)` colength: `colength(L)` nil: `[]` it: `⋅` guard: `{T}` so_lambda: `λ2x.t[x]` so_apply: `x[s]` sq_type: `SQType(T)` less_than: `a < b` squash: `↓T` less_than': `less_than'(a;b)` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` bool: `𝔹` unit: `Unit` btrue: `tt` uiff: `uiff(P;Q)` ifthenelse: `if b then t else f fi ` bfalse: `ff` bnot: `¬bb` assert: `↑b` rev_implies: `P `` Q` iff: `P `⇐⇒` Q` pi1: `fst(t)` pi2: `snd(t)`
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 istype-less_than list-cases map_nil_lemma tupletype_nil_lemma product_subtype_list colength-cons-not-zero colength_wf_list decidable__le intformnot_wf int_formula_prop_not_lemma istype-le subtract-1-ge-0 subtype_base_sq intformeq_wf int_formula_prop_eq_lemma set_subtype_base int_subtype_base spread_cons_lemma decidable__equal_int subtract_wf itermSubtract_wf itermAdd_wf int_term_value_subtract_lemma int_term_value_add_lemma le_wf map_cons_lemma tupletype_cons_lemma null-map null_wf eqtt_to_assert assert_of_null length_wf length_of_nil_lemma subtype_rel_self ifthenelse_wf btrue_wf nat_wf tuple-type_wf map_wf eqff_to_assert bool_cases_sqequal bool_wf bool_subtype_base assert-bnot iff_weakening_uiff assert_wf equal-wf-T-base list_wf nil_wf istype-nat istype-universe unit_wf2 subtype_rel_transitivity subtype_rel_product pi1_wf pi2_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 intWeakElimination natural_numberEquality independent_isectElimination approximateComputation independent_functionElimination Error :dependent_pairFormation_alt,  Error :lambdaEquality_alt,  int_eqEquality dependent_functionElimination Error :isect_memberEquality_alt,  voidElimination sqequalRule independent_pairFormation Error :universeIsType,  axiomEquality Error :functionIsTypeImplies,  Error :inhabitedIsType,  unionElimination promote_hyp hypothesis_subsumption productElimination Error :equalityIstype,  because_Cache Error :dependent_set_memberEquality_alt,  instantiate equalityTransitivity equalitySymmetry applyLambdaEquality imageElimination baseApply closedConclusion baseClosed applyEquality intEquality sqequalBase equalityElimination universeEquality isectEquality productEquality cumulativity Error :isectIsTypeImplies,  Error :functionIsType,  Error :isectIsType,  Error :productIsType,  independent_pairEquality

Latex:
\mforall{}[P:Type].  \mforall{}[X:\mBbbN{}  {}\mrightarrow{}  P  {}\mrightarrow{}  Type].  \mforall{}[L:P  List].
((\mcap{}n:\mBbbN{}.  tuple-type(map(X  n;L)))  \msubseteq{}r  tuple-type(map(\mlambda{}p.\mcap{}n:\mBbbN{}.  (X  n  p);L)))

Date html generated: 2019_06_20-PM-02_03_53
Last ObjectModification: 2019_02_20-PM-00_52_55

Theory : tuples

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