### Nuprl Lemma : tuple-sum-wf-partial

`∀[P:Type]. ∀[G:P ⟶ Type]. ∀[f:i:P ⟶ (G i) ⟶ partial(ℕ)]. ∀[as:P List]. ∀[x:tuple-type(map(G;as))].`
`  (tuple-sum(f;as;x) ∈ partial(ℕ))`

Proof

Definitions occuring in Statement :  tuple-sum: `tuple-sum(f;L;x)` tuple-type: `tuple-type(L)` map: `map(f;as)` list: `T List` partial: `partial(T)` nat: `ℕ` uall: `∀[x:A]. B[x]` member: `t ∈ T` apply: `f a` 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` or: `P ∨ Q` tuple-sum: `tuple-sum(f;L;x)` ifthenelse: `if b then t else f fi ` btrue: `tt` le: `A ≤ B` less_than': `less_than'(a;b)` 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` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` subtype_rel: `A ⊆r B` bfalse: `ff` bool: `𝔹` unit: `Unit`
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 null_nil_lemma tupletype_nil_lemma nat-partial-nat istype-false istype-le unit_wf2 product_subtype_list colength-cons-not-zero colength_wf_list decidable__le intformnot_wf int_formula_prop_not_lemma 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 null_cons_lemma tupletype_cons_lemma null-map null_wf istype-nat list_wf partial_wf nat_wf istype-universe add-wf-partial-nat tuple-type_wf map_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 Error :isectIsTypeImplies,  Error :inhabitedIsType,  Error :functionIsTypeImplies,  unionElimination Error :dependent_set_memberEquality_alt,  promote_hyp hypothesis_subsumption productElimination Error :equalityIstype,  because_Cache instantiate applyLambdaEquality imageElimination baseApply closedConclusion baseClosed applyEquality intEquality sqequalBase equalityElimination Error :functionIsType,  universeEquality Error :productIsType,  cumulativity

Latex:
\mforall{}[P:Type].  \mforall{}[G:P  {}\mrightarrow{}  Type].  \mforall{}[f:i:P  {}\mrightarrow{}  (G  i)  {}\mrightarrow{}  partial(\mBbbN{})].  \mforall{}[as:P  List].
\mforall{}[x:tuple-type(map(G;as))].
(tuple-sum(f;as;x)  \mmember{}  partial(\mBbbN{}))

Date html generated: 2019_06_20-PM-02_03_24
Last ObjectModification: 2019_02_22-AM-11_13_33

Theory : tuples

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