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)


Definitions occuring in Statement :  swap: swap(L;i;j) select: L[n] length: ||as|| list: List int_seg: {i..j-} uimplies: supposing a uall: [x:A]. B[x] guard: {T} not: ¬A and: P ∧ Q natural_number: $n int: universe: Type equal: t ∈ T
Definitions unfolded in proof :  uall: [x:A]. B[x] member: t ∈ T uimplies: supposing a guard: {T} and: P ∧ Q prop: int_seg: {i..j-} squash: T true: True subtype_rel: A ⊆B iff: ⇐⇒ Q rev_implies:  Q implies:  Q cand: 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 then else fi  bfalse: ff sq_type: SQType(T) bnot: ¬bb assert: b nequal: a ≠ b ∈ 
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

\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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