### Nuprl Lemma : quicksort-int-prop1

`∀[L:ℤ List]. ∀[i:ℕ||L||]. ∀[j:ℕi].  (quicksort-int(L)[j] ≤ quicksort-int(L)[i])`

Proof

Definitions occuring in Statement :  quicksort-int: `quicksort-int(L)` select: `L[n]` length: `||as||` list: `T List` int_seg: `{i..j-}` uall: `∀[x:A]. B[x]` le: `A ≤ B` natural_number: `\$n` int: `ℤ`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` all: `∀x:A. B[x]` squash: `↓T` prop: `ℙ` and: `P ∧ Q` sorted-by: `sorted-by(R;L)` le: `A ≤ B` int_seg: `{i..j-}` lelt: `i ≤ j < k` guard: `{T}` 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` less_than: `a < b` sq_stable: `SqStable(P)`
Lemmas referenced :  length_wf int_seg_wf less_than'_wf int_term_value_constant_lemma int_formula_prop_le_lemma itermConstant_wf intformle_wf decidable__le select_wf lelt_wf int_formula_prop_wf int_formula_prop_eq_lemma int_term_value_var_lemma int_formula_prop_less_lemma int_formula_prop_not_lemma int_formula_prop_and_lemma intformeq_wf itermVar_wf intformless_wf intformnot_wf intformand_wf satisfiable-full-omega-tt decidable__lt int_seg_properties sq_stable__le permutation_wf l_member_wf le_wf sorted-by_wf and_wf list_wf quicksort-int_wf quicksort-int-length
Rules used in proof :  cut lemma_by_obid sqequalHypSubstitution sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isectElimination thin hypothesisEquality dependent_functionElimination because_Cache applyEquality lambdaEquality setElimination rename hypothesis sqequalRule imageMemberEquality baseClosed setEquality intEquality introduction imageElimination productElimination dependent_set_memberEquality independent_pairFormation natural_numberEquality unionElimination equalityTransitivity equalitySymmetry independent_isectElimination dependent_pairFormation int_eqEquality isect_memberEquality voidElimination voidEquality computeAll independent_functionElimination isect_memberFormation independent_pairEquality axiomEquality

Latex:
\mforall{}[L:\mBbbZ{}  List].  \mforall{}[i:\mBbbN{}||L||].  \mforall{}[j:\mBbbN{}i].    (quicksort-int(L)[j]  \mleq{}  quicksort-int(L)[i])

Date html generated: 2016_05_15-PM-04_30_09
Last ObjectModification: 2016_01_16-AM-11_15_38

Theory : general

Home Index