### Nuprl Lemma : select-as-reduce

`∀[n:ℕ]. ∀[L:Top List].`
`  L[||L|| - n + 1] `
`  ~ outr(reduce(λu,x. case x of inl(i) => if (i =z n) then inr u  else inl (i + 1) fi  | inr(u) => x;inl 0;L)) `
`  supposing n < ||L||`

Proof

Definitions occuring in Statement :  select: `L[n]` length: `||as||` reduce: `reduce(f;k;as)` list: `T List` nat: `ℕ` outr: `outr(x)` ifthenelse: `if b then t else f fi ` eq_int: `(i =z j)` less_than: `a < b` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` top: `Top` lambda: `λx.A[x]` decide: `case b of inl(x) => s[x] | inr(y) => t[y]` inr: `inr x ` inl: `inl x` subtract: `n - m` add: `n + m` natural_number: `\$n` sqequal: `s ~ t`
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` select: `L[n]` nil: `[]` it: `⋅` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` cons: `[a / b]` colength: `colength(L)` decidable: `Dec(P)` 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)` bool: `𝔹` unit: `Unit` btrue: `tt` uiff: `uiff(P;Q)` ifthenelse: `if b then t else f fi ` bfalse: `ff` bnot: `¬bb` assert: `↑b` le: `A ≤ B` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` int_seg: `{i..j-}` lelt: `i ≤ j < k` outr: `outr(x)`
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 top_wf less_than_transitivity1 less_than_irreflexivity list-cases reduce_nil_lemma length_of_nil_lemma stuck-spread base_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 length_of_cons_lemma list_wf length_wf le_int_wf bool_wf assert_wf lt_int_wf bnot_wf eqtt_to_assert assert_of_le_int eq_int_wf assert_of_eq_int eqff_to_assert bool_cases_sqequal bool_subtype_base assert-bnot neg_assert_of_eq_int add-is-int-iff false_wf not_wf uiff_transitivity assert_functionality_wrt_uiff bnot_of_le_int assert_of_lt_int iff_transitivity iff_weakening_uiff assert_of_bnot general_arith_equation1 select_cons_tl_sq decidable__lt lelt_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut lambdaFormation thin 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 sqequalRule independent_pairFormation computeAll independent_functionElimination sqequalAxiom applyEquality because_Cache unionElimination baseClosed promote_hyp hypothesis_subsumption productElimination equalityTransitivity equalitySymmetry applyLambdaEquality dependent_set_memberEquality addEquality instantiate cumulativity imageElimination equalityElimination pointwiseFunctionality baseApply closedConclusion impliesFunctionality

Latex:
\mforall{}[n:\mBbbN{}].  \mforall{}[L:Top  List].
L[||L||  -  n  +  1]  \msim{}  outr(reduce(\mlambda{}u,x.  case  x
of  inl(i)  =>
if  (i  =\msubz{}  n)  then  inr  u    else  inl  (i  +  1)  fi
|  inr(u)  =>
x;inl  0;L))
supposing  n  <  ||L||

Date html generated: 2018_05_21-PM-06_51_51
Last ObjectModification: 2017_07_26-PM-04_58_00

Theory : general

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