### Nuprl Lemma : combine-list-as-reduce

`∀[A:Type]. ∀[f:A ⟶ A ⟶ A].`
`  (∀[L:A List]`
`     combine-list(x,y.f[x;y];L) = outl(reduce(λx,y. case y of inl(z) => inl f[x;z] | inr(z) => inl x;inr ⋅ ;L)) ∈ A `
`     supposing 0 < ||L||) supposing `
`     (Comm(A;λx,y. f[x;y]) and `
`     Assoc(A;λx,y. f[x;y]))`

Proof

Definitions occuring in Statement :  combine-list: `combine-list(x,y.f[x; y];L)` length: `||as||` reduce: `reduce(f;k;as)` list: `T List` comm: `Comm(T;op)` assoc: `Assoc(T;op)` outl: `outl(x)` less_than: `a < b` it: `⋅` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` so_apply: `x[s1;s2]` lambda: `λx.A[x]` function: `x:A ⟶ B[x]` decide: `case b of inl(x) => s[x] | inr(y) => t[y]` inr: `inr x ` inl: `inl x` natural_number: `\$n` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` all: `∀x:A. B[x]` or: `P ∨ Q` assert: `↑b` ifthenelse: `if b then t else f fi ` btrue: `tt` less_than: `a < b` squash: `↓T` less_than': `less_than'(a;b)` false: `False` and: `P ∧ Q` cons: `[a / b]` top: `Top` bfalse: `ff` not: `¬A` implies: `P `` Q` prop: `ℙ` exists: `∃x:A. B[x]` nat: `ℕ` ge: `i ≥ j ` satisfiable_int_formula: `satisfiable_int_formula(fmla)` subtype_rel: `A ⊆r B` isl: `isl(x)` colength: `colength(L)` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` guard: `{T}` decidable: `Dec(P)` nil: `[]` it: `⋅` so_lambda: `λ2x.t[x]` so_apply: `x[s]` sq_type: `SQType(T)` nat_plus: `ℕ+` true: `True` uiff: `uiff(P;Q)` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` append: `as @ bs` so_lambda: `so_lambda(x,y,z.t[x; y; z])` so_apply: `x[s1;s2;s3]` outl: `outl(x)` le: `A ≤ B` combine-list: `combine-list(x,y.f[x; y];L)` comm: `Comm(T;op)` infix_ap: `x f y`
Lemmas referenced :  last_lemma list-cases null_nil_lemma length_of_nil_lemma product_subtype_list null_cons_lemma length_of_cons_lemma false_wf nat_properties full-omega-unsat 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 length_wf equal-wf-T-base nat_wf colength_wf_list int_subtype_base reduce_nil_lemma 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 decidable__equal_int reduce_cons_lemma bool_wf bool_subtype_base reduce_wf unit_wf2 it_wf btrue_wf list_wf comm_wf assoc_wf last_wf append_wf cons_wf nil_wf length-append add_nat_plus length_wf_nat nat_plus_wf nat_plus_properties decidable__lt add-is-int-iff outl_wf assert_of_tt squash_wf true_wf combine-list-append subtype_rel_self iff_weakening_equal list_ind_cons_lemma list_ind_nil_lemma combine-list_wf non_neg_length assert_wf isl_wf list_induction reduce_hd_cons_lemma reduce_tl_cons_lemma list_accum_nil_lemma list_accum_cons_lemma list_accum_wf combine-list-cons and_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality dependent_functionElimination independent_isectElimination hypothesis unionElimination sqequalRule imageElimination productElimination voidElimination promote_hyp hypothesis_subsumption isect_memberEquality voidEquality lambdaFormation setElimination rename intWeakElimination natural_numberEquality approximateComputation independent_functionElimination dependent_pairFormation lambdaEquality int_eqEquality intEquality independent_pairFormation axiomSqEquality applyEquality because_Cache equalityTransitivity equalitySymmetry applyLambdaEquality dependent_set_memberEquality addEquality baseClosed instantiate cumulativity unionEquality inlEquality inrEquality axiomEquality functionEquality universeEquality hyp_replacement imageMemberEquality pointwiseFunctionality baseApply closedConclusion functionExtensionality

Latex:
\mforall{}[A:Type].  \mforall{}[f:A  {}\mrightarrow{}  A  {}\mrightarrow{}  A].
(\mforall{}[L:A  List]
combine-list(x,y.f[x;y];L)
=  outl(reduce(\mlambda{}x,y.  case  y  of  inl(z)  =>  inl  f[x;z]  |  inr(z)  =>  inl  x;inr  \mcdot{}  ;L))
supposing  0  <  ||L||)  supposing
(Comm(A;\mlambda{}x,y.  f[x;y])  and
Assoc(A;\mlambda{}x,y.  f[x;y]))

Date html generated: 2019_06_20-PM-01_30_30
Last ObjectModification: 2018_08_21-PM-01_55_59

Theory : list_1

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