### Nuprl Lemma : l_before_swap

`∀[T:Type]`
`  ∀L:T List. ∀i:ℕ||L|| - 1. ∀a,b:T.`
`    (a before b ∈ swap(L;i;i + 1) `` (a before b ∈ L ∨ ((a = L[i + 1] ∈ T) ∧ (b = L[i] ∈ T))))`

Proof

Definitions occuring in Statement :  swap: `swap(L;i;j)` l_before: `x before y ∈ l` select: `L[n]` length: `||as||` list: `T List` int_seg: `{i..j-}` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` implies: `P `` Q` or: `P ∨ Q` and: `P ∧ Q` subtract: `n - m` add: `n + m` natural_number: `\$n` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` implies: `P `` Q` l_before: `x before y ∈ l` sublist: `L1 ⊆ L2` exists: `∃x:A. B[x]` and: `P ∧ Q` member: `t ∈ T` top: `Top` int_seg: `{i..j-}` lelt: `i ≤ j < k` prop: `ℙ` uiff: `uiff(P;Q)` uimplies: `b supposing a` squash: `↓T` so_lambda: `λ2x.t[x]` guard: `{T}` decidable: `Dec(P)` or: `P ∨ Q` not: `¬A` satisfiable_int_formula: `satisfiable_int_formula(fmla)` false: `False` ge: `i ≥ j ` subtype_rel: `A ⊆r B` le: `A ≤ B` less_than': `less_than'(a;b)` true: `True` so_apply: `x[s]` iff: `P `⇐⇒` Q` less_than: `a < b` select: `L[n]` cons: `[a / b]` subtract: `n - m` flip: `(i, j)` nat: `ℕ` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` ifthenelse: `if b then t else f fi ` bfalse: `ff` sq_type: `SQType(T)` bnot: `¬bb` assert: `↑b` cand: `A c∧ B` nequal: `a ≠ b ∈ T `
Lemmas referenced :  length_of_cons_lemma length_of_nil_lemma swap_length lelt_wf length_wf add-member-int_seg2 subtract_wf all_wf squash_wf true_wf int_seg_wf equal_wf select_wf cons_wf nil_wf int_seg_properties decidable__le full-omega-unsat intformand_wf intformnot_wf intformle_wf itermConstant_wf itermVar_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_wf non_neg_length swap_wf add-member-int_seg1 decidable__lt intformless_wf itermAdd_wf int_formula_prop_less_lemma int_term_value_add_lemma swap_select int_seg_subtype false_wf le_weakening subtype_rel_self iff_weakening_equal flip_wf length_wf_nat or_wf sublist_wf itermSubtract_wf int_term_value_subtract_lemma l_before_wf subtract-is-int-iff list_wf sublist_pair increasing_implies le_wf eq_int_wf bool_wf eqtt_to_assert assert_of_eq_int intformeq_wf int_formula_prop_eq_lemma eqff_to_assert bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot neg_assert_of_eq_int not_wf less_than_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation lambdaFormation sqequalHypSubstitution productElimination thin sqequalRule cut introduction extract_by_obid dependent_functionElimination isect_memberEquality voidElimination voidEquality hypothesis isectElimination because_Cache hypothesisEquality setElimination rename dependent_set_memberEquality independent_pairFormation natural_numberEquality cumulativity independent_isectElimination applyEquality lambdaEquality imageElimination equalityTransitivity equalitySymmetry functionEquality universeEquality unionElimination approximateComputation independent_functionElimination dependent_pairFormation int_eqEquality intEquality addEquality imageMemberEquality baseClosed instantiate functionExtensionality hyp_replacement applyLambdaEquality productEquality pointwiseFunctionality promote_hyp baseApply closedConclusion inlFormation inrFormation equalityElimination independent_pairEquality axiomEquality

Latex:
\mforall{}[T:Type]
\mforall{}L:T  List.  \mforall{}i:\mBbbN{}||L||  -  1.  \mforall{}a,b:T.
(a  before  b  \mmember{}  swap(L;i;i  +  1)  {}\mRightarrow{}  (a  before  b  \mmember{}  L  \mvee{}  ((a  =  L[i  +  1])  \mwedge{}  (b  =  L[i]))))

Date html generated: 2018_05_21-PM-06_21_15
Last ObjectModification: 2018_05_19-PM-05_35_18

Theory : list!

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