### Nuprl Lemma : comparison-sort_wf

`∀[T:Type]. ∀[cmp:comparison(T)].`
`  ∀L:T List. (comparison-sort(cmp;L) ∈ {srtd:T List| sorted-by(λx,y. (0 ≤ (cmp x y));srtd)} ) supposing valueall-type(T)`

Proof

Definitions occuring in Statement :  comparison-sort: `comparison-sort(cmp;L)` comparison: `comparison(T)` sorted-by: `sorted-by(R;L)` list: `T List` valueall-type: `valueall-type(T)` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` le: `A ≤ B` all: `∀x:A. B[x]` member: `t ∈ T` set: `{x:A| B[x]} ` apply: `f a` lambda: `λx.A[x]` natural_number: `\$n` universe: `Type`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` all: `∀x:A. B[x]` comparison-sort: `comparison-sort(cmp;L)` prop: `ℙ` comparison: `comparison(T)` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` so_lambda: `λ2x.t[x]` so_apply: `x[s]` sorted-by: `sorted-by(R;L)` select: `L[n]` nil: `[]` it: `⋅` top: `Top` guard: `{T}` int_seg: `{i..j-}` lelt: `i ≤ j < k` and: `P ∧ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` implies: `P `` Q` not: `¬A` sq_stable: `SqStable(P)` squash: `↓T`
Lemmas referenced :  sq_stable__le insert-no-combine-sorted-by insert-no-combine_wf int_seg_wf int_formula_prop_wf int_term_value_constant_lemma int_formula_prop_le_lemma int_term_value_var_lemma int_formula_prop_less_lemma int_formula_prop_and_lemma itermConstant_wf intformle_wf itermVar_wf intformless_wf intformand_wf satisfiable-full-omega-tt int_seg_properties base_wf stuck-spread length_of_nil_lemma nil_wf comparison_wf valueall-type_wf list-valueall-type set-valueall-type le_wf l_member_wf sorted-by_wf list_wf eager-accum_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut lambdaFormation sqequalRule lemma_by_obid sqequalHypSubstitution isectElimination thin cumulativity hypothesisEquality setEquality because_Cache hypothesis lambdaEquality setElimination rename natural_numberEquality applyEquality dependent_functionElimination independent_isectElimination axiomEquality equalityTransitivity equalitySymmetry isect_memberEquality universeEquality dependent_set_memberEquality baseClosed voidElimination voidEquality productElimination dependent_pairFormation int_eqEquality intEquality independent_pairFormation computeAll independent_functionElimination imageMemberEquality imageElimination

Latex:
\mforall{}[T:Type].  \mforall{}[cmp:comparison(T)].
\mforall{}L:T  List.  (comparison-sort(cmp;L)  \mmember{}  \{srtd:T  List|  sorted-by(\mlambda{}x,y.  (0  \mleq{}  (cmp  x  y));srtd)\}  )  suppos\000Cing  valueall-type(T)

Date html generated: 2016_05_14-PM-02_43_59
Last ObjectModification: 2016_01_15-AM-07_37_55

Theory : list_1

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