### Nuprl Lemma : fst-recode-tuple

`∀[T:Type]. ∀[f:T ⟶ (T List × Top × Top)]. ∀[L:T List].`
`  ((fst((recode-tuple(f) L))) = reduce(λT,X. ((fst((f T))) @ X);[];L) ∈ (T List))`

Proof

Definitions occuring in Statement :  recode-tuple: `recode-tuple(f)` append: `as @ bs` reduce: `reduce(f;k;as)` nil: `[]` list: `T List` uall: `∀[x:A]. B[x]` top: `Top` pi1: `fst(t)` apply: `f a` lambda: `λx.A[x]` function: `x:A ⟶ B[x]` product: `x:A × B[x]` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  recode-tuple: `recode-tuple(f)` 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` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` not: `¬A` top: `Top` and: `P ∧ Q` prop: `ℙ` subtype_rel: `A ⊆r B` guard: `{T}` or: `P ∨ Q` so_lambda: `so_lambda(x,y,z.t[x; y; z])` so_apply: `x[s1;s2;s3]` pi1: `fst(t)` cons: `[a / b]` colength: `colength(L)` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` decidable: `Dec(P)` nil: `[]` it: `⋅` 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)` spreadn: spread3 ifthenelse: `if b then t else f fi ` btrue: `tt` bfalse: `ff` bool: `𝔹` unit: `Unit` uiff: `uiff(P;Q)` bnot: `¬bb` assert: `↑b`
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-T-base nat_wf colength_wf_list less_than_transitivity1 less_than_irreflexivity list-cases reduce_nil_lemma list_ind_nil_lemma nil_wf product_subtype_list spread_cons_lemma intformeq_wf itermAdd_wf int_formula_prop_eq_lemma int_term_value_add_lemma decidable__le intformnot_wf int_formula_prop_not_lemma le_wf equal_wf subtract_wf itermSubtract_wf int_term_value_subtract_lemma subtype_base_sq set_subtype_base int_subtype_base decidable__equal_int reduce_cons_lemma list_ind_cons_lemma list_wf top_wf list_ind_wf null_nil_lemma null_cons_lemma append_wf null_wf3 subtype_rel_list bool_wf eqtt_to_assert assert_of_null append-nil btrue_wf bfalse_wf and_wf btrue_neq_bfalse eqff_to_assert bool_cases_sqequal bool_subtype_base assert-bnot pi1_wf_top subtype_rel_product
Rules used in proof :  sqequalSubstitution sqequalRule sqequalReflexivity sqequalTransitivity computationStep 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 independent_pairFormation computeAll independent_functionElimination axiomEquality cumulativity applyEquality because_Cache unionElimination promote_hyp hypothesis_subsumption productElimination equalityTransitivity equalitySymmetry applyLambdaEquality dependent_set_memberEquality addEquality baseClosed instantiate imageElimination functionExtensionality productEquality independent_pairEquality equalityElimination functionEquality universeEquality

Latex:
\mforall{}[T:Type].  \mforall{}[f:T  {}\mrightarrow{}  (T  List  \mtimes{}  Top  \mtimes{}  Top)].  \mforall{}[L:T  List].
((fst((recode-tuple(f)  L)))  =  reduce(\mlambda{}T,X.  ((fst((f  T)))  @  X);[];L))

Date html generated: 2018_05_21-PM-08_03_45
Last ObjectModification: 2017_07_26-PM-05_39_50

Theory : general

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