### Nuprl Lemma : quicksort-int-last

`∀[L:ℤ List]. ∀n:ℤ. n ≤ last(quicksort-int(L)) supposing (n ∈ L)`

Proof

Definitions occuring in Statement :  quicksort-int: `quicksort-int(L)` last: `last(L)` l_member: `(x ∈ l)` list: `T List` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` le: `A ≤ B` all: `∀x:A. B[x]` int: `ℤ`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` all: `∀x:A. B[x]` uimplies: `b supposing a` iff: `P `⇐⇒` Q` and: `P ∧ Q` implies: `P `` Q` l_member: `(x ∈ l)` exists: `∃x:A. B[x]` cand: `A c∧ B` last: `last(L)` squash: `↓T` prop: `ℙ` subtype_rel: `A ⊆r B` nat: `ℕ` ge: `i ≥ j ` decidable: `Dec(P)` or: `P ∨ Q` less_than: `a < b` satisfiable_int_formula: `satisfiable_int_formula(fmla)` false: `False` not: `¬A` top: `Top` true: `True` guard: `{T}` rev_implies: `P `` Q` int_seg: `{i..j-}` lelt: `i ≤ j < k` le: `A ≤ B` so_lambda: `λ2x.t[x]` so_apply: `x[s]` uiff: `uiff(P;Q)`
Lemmas referenced :  quicksort-int-length not_wf assert_wf subtype_rel_set assert_elim assert_of_null nil_wf btrue_neq_bfalse member-implies-null-eq-bfalse top_wf subtype_rel_list null_wf3 equal_wf btrue_wf null_nil_lemma last_wf less_than'_wf lelt_wf quicksort-int-prop1 int_formula_prop_eq_lemma intformeq_wf less_than_wf decidable__equal_int iff_weakening_equal decidable__lt int_formula_prop_wf int_formula_prop_less_lemma int_term_value_var_lemma int_term_value_subtract_lemma int_term_value_constant_lemma int_formula_prop_le_lemma int_formula_prop_not_lemma int_formula_prop_and_lemma intformless_wf itermVar_wf itermSubtract_wf itermConstant_wf intformle_wf intformnot_wf intformand_wf satisfiable-full-omega-tt permutation_wf l_member_wf sorted-by_wf and_wf list_wf decidable__le nat_properties length_wf subtract_wf quicksort-int_wf select_wf true_wf squash_wf le_wf quicksort-int-member
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut lambdaFormation lemma_by_obid sqequalHypSubstitution dependent_functionElimination thin hypothesisEquality hypothesis productElimination independent_functionElimination applyEquality lambdaEquality imageElimination isectElimination equalityTransitivity equalitySymmetry intEquality because_Cache sqequalRule natural_numberEquality independent_isectElimination setElimination rename setEquality unionElimination dependent_pairFormation int_eqEquality isect_memberEquality voidElimination voidEquality independent_pairFormation computeAll imageMemberEquality baseClosed universeEquality dependent_set_memberEquality independent_pairEquality addLevel impliesFunctionality levelHypothesis axiomEquality

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

Date html generated: 2016_05_15-PM-04_30_47
Last ObjectModification: 2016_01_16-AM-11_15_58

Theory : general

Home Index