### Nuprl Lemma : swapped_select

`∀[T:Type]. ∀[L1,L2:T List]. ∀[i,j:ℕ||L1||].`
`  {(((L2[i] = L1[j] ∈ T) ∧ (L2[j] = L1[i] ∈ T)) ∧ (||L2|| = ||L1|| ∈ ℤ) ∧ (L1 = swap(L2;i;j) ∈ (T List)))`
`  ∧ (∀[x:ℕ||L2||]. (L2[x] = L1[x] ∈ T) supposing ((¬(x = j ∈ ℤ)) and (¬(x = i ∈ ℤ))))} `
`  supposing L2 = swap(L1;i;j) ∈ (T List)`

Proof

Definitions occuring in Statement :  swap: `swap(L;i;j)` select: `L[n]` length: `||as||` list: `T List` int_seg: `{i..j-}` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` guard: `{T}` not: `¬A` and: `P ∧ Q` natural_number: `\$n` int: `ℤ` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` guard: `{T}` and: `P ∧ Q` prop: `ℙ` int_seg: `{i..j-}` squash: `↓T` true: `True` subtype_rel: `A ⊆r B` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` implies: `P `` Q` cand: `A c∧ B` lelt: `i ≤ j < k` all: `∀x:A. B[x]` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` not: `¬A` top: `Top` le: `A ≤ B` less_than: `a < b` flip: `(i, j)` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` uiff: `uiff(P;Q)` ifthenelse: `if b then t else f fi ` bfalse: `ff` sq_type: `SQType(T)` bnot: `¬bb` assert: `↑b` nequal: `a ≠ b ∈ T `
Lemmas referenced :  not_wf equal_wf int_seg_wf length_wf list_wf swap_wf squash_wf true_wf swap_length iff_weakening_equal select_wf int_seg_properties decidable__le satisfiable-full-omega-tt intformand_wf intformnot_wf intformle_wf itermConstant_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_var_lemma int_formula_prop_wf decidable__lt intformless_wf intformeq_wf int_formula_prop_less_lemma int_formula_prop_eq_lemma swap_swap and_wf less_than_wf lelt_wf swap_select eq_int_wf bool_wf equal-wf-T-base assert_wf bnot_wf eqtt_to_assert assert_of_eq_int eqff_to_assert bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot neg_assert_of_eq_int uiff_transitivity iff_transitivity iff_weakening_uiff assert_of_bnot
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut sqequalRule sqequalHypSubstitution productElimination thin independent_pairEquality axiomEquality hypothesis isect_memberEquality isectElimination hypothesisEquality extract_by_obid intEquality setElimination rename equalityTransitivity equalitySymmetry natural_numberEquality cumulativity because_Cache universeEquality applyEquality lambdaEquality imageElimination imageMemberEquality baseClosed independent_isectElimination independent_functionElimination hyp_replacement applyLambdaEquality dependent_set_memberEquality dependent_functionElimination unionElimination dependent_pairFormation int_eqEquality voidElimination voidEquality independent_pairFormation computeAll lambdaFormation equalityElimination promote_hyp instantiate impliesFunctionality

Latex:
\mforall{}[T:Type].  \mforall{}[L1,L2:T  List].  \mforall{}[i,j:\mBbbN{}||L1||].
\{(((L2[i]  =  L1[j])  \mwedge{}  (L2[j]  =  L1[i]))  \mwedge{}  (||L2||  =  ||L1||)  \mwedge{}  (L1  =  swap(L2;i;j)))
\mwedge{}  (\mforall{}[x:\mBbbN{}||L2||].  (L2[x]  =  L1[x])  supposing  ((\mneg{}(x  =  j))  and  (\mneg{}(x  =  i))))\}
supposing  L2  =  swap(L1;i;j)

Date html generated: 2017_10_01-AM-08_38_08
Last ObjectModification: 2017_07_26-PM-04_26_53

Theory : list!

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