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

`∀[A:Type]. ∀[f:A ⟶ A ⟶ A].`
`  (∀[L:A List]. ∀[z:A].  (reduce(f;z;L) = combine-list(x,y.f[x;y];[z / L]) ∈ A)) 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)` reduce: `reduce(f;k;as)` cons: `[a / b]` list: `T List` comm: `Comm(T;op)` assoc: `Assoc(T;op)` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` so_apply: `x[s1;s2]` lambda: `λx.A[x]` function: `x:A ⟶ B[x]` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  so_apply: `x[s1;s2]` combine-list: `combine-list(x,y.f[x; y];L)` comm: `Comm(T;op)` assoc: `Assoc(T;op)` infix_ap: `x f y` all: `∀x:A. B[x]` member: `t ∈ T` uall: `∀[x:A]. B[x]` uimplies: `b supposing a` nat: `ℕ` implies: `P `` Q` false: `False` ge: `i ≥ j ` not: `¬A` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` and: `P ∧ Q` prop: `ℙ` or: `P ∨ Q` so_lambda: `λ2x y.t[x; y]` cons: `[a / b]` le: `A ≤ B` less_than': `less_than'(a;b)` 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` decidable: `Dec(P)` subtype_rel: `A ⊆r B` true: `True` iff: `P `⇐⇒` Q` rev_implies: `P `` Q`
Lemmas referenced :  reduce_hd_cons_lemma reduce_tl_cons_lemma nat_properties full-omega-unsat intformand_wf intformle_wf itermConstant_wf itermVar_wf intformless_wf istype-int 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 istype-less_than list-cases reduce_nil_lemma list_accum_nil_lemma product_subtype_list colength-cons-not-zero colength_wf_list istype-void 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 intformnot_wf itermSubtract_wf itermAdd_wf int_formula_prop_not_lemma int_term_value_subtract_lemma int_term_value_add_lemma decidable__le le_wf reduce_cons_lemma list_accum_cons_lemma equal_wf squash_wf true_wf reduce_wf subtype_rel_self iff_weakening_equal istype-nat list_wf istype-universe
Rules used in proof :  sqequalSubstitution sqequalRule sqequalReflexivity sqequalTransitivity computationStep cut introduction extract_by_obid sqequalHypSubstitution dependent_functionElimination thin Error :memTop,  hypothesis isect_memberFormation_alt lambdaFormation_alt isectElimination hypothesisEquality setElimination rename intWeakElimination natural_numberEquality independent_isectElimination approximateComputation independent_functionElimination dependent_pairFormation_alt lambdaEquality_alt int_eqEquality independent_pairFormation universeIsType voidElimination isect_memberEquality_alt axiomEquality isectIsTypeImplies inhabitedIsType functionIsTypeImplies unionElimination because_Cache promote_hyp hypothesis_subsumption productElimination equalityIstype dependent_set_memberEquality_alt instantiate equalityTransitivity equalitySymmetry applyLambdaEquality imageElimination baseApply closedConclusion baseClosed applyEquality intEquality sqequalBase imageMemberEquality isectIsType functionIsType universeEquality hyp_replacement

Latex:
\mforall{}[A:Type].  \mforall{}[f:A  {}\mrightarrow{}  A  {}\mrightarrow{}  A].
(\mforall{}[L:A  List].  \mforall{}[z:A].    (reduce(f;z;L)  =  combine-list(x,y.f[x;y];[z  /  L])))  supposing
(Comm(A;\mlambda{}x,y.  f[x;y])  and
Assoc(A;\mlambda{}x,y.  f[x;y]))

Date html generated: 2020_05_19-PM-09_45_07
Last ObjectModification: 2019_12_31-PM-00_13_15

Theory : list_1

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