### Nuprl Lemma : reduce2_shift

`∀[A,T:Type]. ∀[L:T List]. ∀[k:A]. ∀[i:ℕ]. ∀[f:T ⟶ {i..i + ||L||-} ⟶ A ⟶ A].`
`  (reduce2(f;k;i;L) = reduce2(λx,i,l. (f x (i - 1) l);k;i + 1;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]` apply: `f a` lambda: `λx.A[x]` function: `x:A ⟶ B[x]` subtract: `n - m` add: `n + m` natural_number: `\$n` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` so_lambda: `λ2x.t[x]` nat: `ℕ` ge: `i ≥ j ` all: `∀x:A. B[x]` decidable: `Dec(P)` or: `P ∨ Q` uimplies: `b supposing a` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` implies: `P `` Q` not: `¬A` top: `Top` and: `P ∧ Q` prop: `ℙ` int_seg: `{i..j-}` lelt: `i ≤ j < k` guard: `{T}` less_than: `a < b` squash: `↓T` uiff: `uiff(P;Q)` so_apply: `x[s]` le: `A ≤ B` subtype_rel: `A ⊆r B` true: `True` iff: `P `⇐⇒` Q` rev_implies: `P `` Q`
Lemmas referenced :  list_induction uall_wf nat_wf int_seg_wf length_wf equal_wf reduce2_wf nat_properties decidable__le satisfiable-full-omega-tt intformand_wf intformnot_wf intformle_wf itermConstant_wf itermAdd_wf itermVar_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_add_lemma int_term_value_var_lemma int_formula_prop_wf le_wf subtract_wf int_seg_properties itermSubtract_wf int_term_value_subtract_lemma decidable__lt add-is-int-iff intformless_wf int_formula_prop_less_lemma false_wf lelt_wf list_wf length_of_nil_lemma reduce2_nil_lemma length_of_cons_lemma reduce2_cons_lemma squash_wf true_wf non_neg_length subtype_rel_dep_function int_seg_subtype subtype_rel_self add-subtract-cancel decidable__equal_int intformeq_wf int_formula_prop_eq_lemma iff_weakening_equal
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut thin extract_by_obid sqequalHypSubstitution isectElimination hypothesisEquality sqequalRule lambdaEquality cumulativity hypothesis functionEquality setElimination rename because_Cache addEquality functionExtensionality applyEquality dependent_set_memberEquality natural_numberEquality dependent_functionElimination unionElimination independent_isectElimination dependent_pairFormation int_eqEquality intEquality isect_memberEquality voidElimination voidEquality independent_pairFormation computeAll productElimination pointwiseFunctionality equalityTransitivity equalitySymmetry promote_hyp imageElimination baseApply closedConclusion baseClosed independent_functionElimination axiomEquality lambdaFormation imageMemberEquality 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)  =  reduce2(\mlambda{}x,i,l.  (f  x  (i  -  1)  l);k;i  +  1;L))

Date html generated: 2017_10_01-AM-08_35_03
Last ObjectModification: 2017_07_26-PM-04_25_39

Theory : list!

Home Index