### Nuprl Lemma : reduce2_wf

`∀[A,T:Type]. ∀[L:T List]. ∀[k:A]. ∀[i:ℕ]. ∀[f:T ⟶ {i..i + ||L||-} ⟶ A ⟶ A].  (reduce2(f;k;i;L) ∈ A)`

Proof

Definitions occuring in Statement :  reduce2: `reduce2(f;k;i;as)` length: `||as||` list: `T List` int_seg: `{i..j-}` nat: `ℕ` uall: `∀[x:A]. B[x]` member: `t ∈ T` function: `x:A ⟶ B[x]` add: `n + m` 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` 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` reduce2: `reduce2(f;k;i;as)` so_lambda: `so_lambda(x,y,z.t[x; y; z])` so_apply: `x[s1;s2;s3]` 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)` int_seg: `{i..j-}` lelt: `i ≤ j < k` le: `A ≤ 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 int_seg_wf length_wf equal-wf-T-base nat_wf colength_wf_list less_than_transitivity1 less_than_irreflexivity list-cases length_of_nil_lemma list_ind_nil_lemma 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 length_of_cons_lemma list_ind_cons_lemma non_neg_length decidable__lt lelt_wf subtype_rel_dep_function int_seg_subtype subtype_rel_self list_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut thin lambdaFormation extract_by_obid sqequalHypSubstitution isectElimination hypothesisEquality hypothesis setElimination rename sqequalRule intWeakElimination natural_numberEquality independent_isectElimination dependent_pairFormation lambdaEquality int_eqEquality intEquality dependent_functionElimination isect_memberEquality voidElimination voidEquality independent_pairFormation computeAll independent_functionElimination axiomEquality equalityTransitivity equalitySymmetry functionEquality cumulativity addEquality because_Cache applyEquality unionElimination promote_hyp hypothesis_subsumption productElimination applyLambdaEquality dependent_set_memberEquality baseClosed instantiate imageElimination functionExtensionality universeEquality

Latex:
\mforall{}[A,T:Type].  \mforall{}[L:T  List].  \mforall{}[k:A].  \mforall{}[i:\mBbbN{}].  \mforall{}[f:T  {}\mrightarrow{}  \{i..i  +  ||L||\msupminus{}\}  {}\mrightarrow{}  A  {}\mrightarrow{}  A].
(reduce2(f;k;i;L)  \mmember{}  A)

Date html generated: 2017_10_01-AM-08_35_01
Last ObjectModification: 2017_07_26-PM-04_25_38

Theory : list!

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