### Nuprl Lemma : l-ordered-reorder

`∀[A:Type]`
`  ∀R:A ⟶ A ⟶ 𝔹. ∀L:A List.`
`    (Trans(A;x,y.↑R[x;y])`
`    `` (∀x∈L.(∀y∈L.(¬↑R[x;y]) `` (↑R[y;x])))`
`    `` (∃L':A List. (l-ordered(A;x,y.↑R[x;y];L') ∧ permutation(A;L;L'))))`

Proof

Definitions occuring in Statement :  l-ordered: `l-ordered(T;x,y.R[x; y];L)` permutation: `permutation(T;L1;L2)` l_all: `(∀x∈L.P[x])` list: `T List` trans: `Trans(T;x,y.E[x; y])` assert: `↑b` bool: `𝔹` uall: `∀[x:A]. B[x]` so_apply: `x[s1;s2]` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` not: `¬A` implies: `P `` Q` and: `P ∧ Q` function: `x:A ⟶ B[x]` universe: `Type`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` member: `t ∈ T` so_lambda: `λ2x.t[x]` implies: `P `` Q` prop: `ℙ` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` so_apply: `x[s]` and: `P ∧ Q` exists: `∃x:A. B[x]` cand: `A c∧ B` true: `True` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` subtype_rel: `A ⊆r B` guard: `{T}` or: `P ∨ Q` uimplies: `b supposing a` uiff: `uiff(P;Q)` sq_type: `SQType(T)` assert: `↑b` ifthenelse: `if b then t else f fi ` btrue: `tt` trans: `Trans(T;x,y.E[x; y])` squash: `↓T` top: `Top`
Lemmas referenced :  list_induction trans_wf assert_wf l_all_wf2 l_member_wf not_wf exists_wf list_wf l-ordered_wf permutation_wf nil_wf l-ordered-nil-true permutation-nil-iff true_wf equal-wf-base l_all_iff cons_wf cons_member equal_wf assert_witness all_wf l-ordered-decomp append_wf filter_wf5 bnot_wf bool_wf l-ordered-append l-ordered-filter l-ordered-cons member_filter_2 assert_of_bnot member-permutation and_wf assert_elim subtype_base_sq bool_subtype_base permutation-cons squash_wf iff_weakening_equal length_wf length-append
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation lambdaFormation cut thin introduction extract_by_obid sqequalHypSubstitution isectElimination hypothesisEquality sqequalRule lambdaEquality functionEquality cumulativity applyEquality functionExtensionality because_Cache hypothesis setElimination rename setEquality dependent_functionElimination productEquality independent_functionElimination dependent_pairFormation natural_numberEquality independent_pairFormation addLevel productElimination baseClosed voidEquality voidElimination allFunctionality promote_hyp inrFormation impliesFunctionality independent_isectElimination universeEquality inlFormation unionElimination dependent_set_memberEquality applyLambdaEquality equalityTransitivity equalitySymmetry instantiate imageElimination imageMemberEquality isect_memberEquality

Latex:
\mforall{}[A:Type]
\mforall{}R:A  {}\mrightarrow{}  A  {}\mrightarrow{}  \mBbbB{}.  \mforall{}L:A  List.
(Trans(A;x,y.\muparrow{}R[x;y])
{}\mRightarrow{}  (\mforall{}x\mmember{}L.(\mforall{}y\mmember{}L.(\mneg{}\muparrow{}R[x;y])  {}\mRightarrow{}  (\muparrow{}R[y;x])))
{}\mRightarrow{}  (\mexists{}L':A  List.  (l-ordered(A;x,y.\muparrow{}R[x;y];L')  \mwedge{}  permutation(A;L;L'))))

Date html generated: 2018_05_21-PM-07_39_58
Last ObjectModification: 2017_07_26-PM-05_14_08

Theory : general

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