### Nuprl Lemma : quicksort-int-iseg

`∀[L,L':ℤ List]. ∀[n:ℤ].  ∀[x:ℤ]. x ≤ n supposing (x ∈ L') supposing L' @ [n] ≤ quicksort-int(L)`

Proof

Definitions occuring in Statement :  quicksort-int: `quicksort-int(L)` iseg: `l1 ≤ l2` l_member: `(x ∈ l)` append: `as @ bs` cons: `[a / b]` nil: `[]` list: `T List` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` le: `A ≤ B` int: `ℤ`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` l_member: `(x ∈ l)` exists: `∃x:A. B[x]` cand: `A c∧ B` iseg: `l1 ≤ l2` int_seg: `{i..j-}` nat: `ℕ` lelt: `i ≤ j < k` and: `P ∧ Q` ge: `i ≥ j ` all: `∀x:A. B[x]` decidable: `Dec(P)` or: `P ∨ Q` less_than: `a < b` squash: `↓T` not: `¬A` implies: `P `` Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` false: `False` top: `Top` prop: `ℙ` subtype_rel: `A ⊆r B` so_lambda: `λ2x.t[x]` so_apply: `x[s]` sq_type: `SQType(T)` guard: `{T}` true: `True` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` le: `A ≤ B` select: `L[n]` cons: `[a / b]`
Lemmas referenced :  subtract_wf length_wf nat_properties decidable__le full-omega-unsat intformand_wf intformnot_wf intformle_wf itermConstant_wf itermSubtract_wf itermVar_wf intformless_wf istype-int int_formula_prop_and_lemma istype-void int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_subtract_lemma int_term_value_var_lemma int_formula_prop_less_lemma int_formula_prop_wf decidable__lt le_wf less_than_wf decidable__equal_int intformeq_wf int_formula_prop_eq_lemma set_subtype_base int_subtype_base lelt_wf list_subtype_base subtype_base_sq squash_wf true_wf subtype_rel_self iff_weakening_equal le_witness_for_triv l_member_wf iseg_wf append_wf cons_wf nil_wf quicksort-int_wf list_wf select_wf int_seg_properties length-append length_of_cons_lemma length_of_nil_lemma non_neg_length itermAdd_wf int_term_value_add_lemma equal_wf istype-universe select_append_front select_append_back quicksort-int-length quicksort-int-prop1 length_nil length_cons length_append subtype_rel_list top_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation_alt introduction cut sqequalHypSubstitution productElimination thin dependent_pairFormation_alt dependent_set_memberEquality_alt extract_by_obid isectElimination intEquality hypothesisEquality hypothesis natural_numberEquality setElimination rename independent_pairFormation dependent_functionElimination unionElimination imageElimination independent_isectElimination approximateComputation independent_functionElimination lambdaEquality_alt int_eqEquality isect_memberEquality_alt voidElimination sqequalRule universeIsType because_Cache productIsType equalityIsType4 inhabitedIsType baseApply closedConclusion baseClosed applyEquality promote_hyp instantiate cumulativity equalityTransitivity equalitySymmetry imageMemberEquality universeEquality hyp_replacement applyLambdaEquality addEquality productEquality voidEquality

Latex:
\mforall{}[L,L':\mBbbZ{}  List].  \mforall{}[n:\mBbbZ{}].    \mforall{}[x:\mBbbZ{}].  x  \mleq{}  n  supposing  (x  \mmember{}  L')  supposing  L'  @  [n]  \mleq{}  quicksort-int(L)

Date html generated: 2019_10_15-AM-11_13_26
Last ObjectModification: 2018_10_10-PM-02_08_55

Theory : general

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