### Nuprl Lemma : sorted-by-strict-no_repeats

`∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ]. ∀[L:T List].  (no_repeats(T;L)) supposing (sorted-by(R;L) and (∀a:T. (¬(R a a))))`

Proof

Definitions occuring in Statement :  sorted-by: `sorted-by(R;L)` no_repeats: `no_repeats(T;l)` list: `T List` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` prop: `ℙ` all: `∀x:A. B[x]` not: `¬A` apply: `f a` function: `x:A ⟶ B[x]` universe: `Type`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` no_repeats: `no_repeats(T;l)` not: `¬A` implies: `P `` Q` false: `False` prop: `ℙ` nat: `ℕ` ge: `i ≥ j ` all: `∀x:A. B[x]` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` top: `Top` and: `P ∧ Q` subtype_rel: `A ⊆r B` so_lambda: `λ2x.t[x]` so_apply: `x[s]` guard: `{T}` int_seg: `{i..j-}` lelt: `i ≤ j < k` le: `A ≤ B` sorted-by: `sorted-by(R;L)`
Lemmas referenced :  equal_wf select_wf nat_properties decidable__le satisfiable-full-omega-tt 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 not_wf nat_wf less_than_wf length_wf no_repeats_witness sorted-by_wf subtype_rel_dep_function l_member_wf subtype_rel_self set_wf all_wf list_wf decidable__lt lelt_wf decidable__equal_int intformeq_wf intformless_wf int_formula_prop_eq_lemma int_formula_prop_less_lemma le_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut lambdaFormation thin sqequalHypSubstitution independent_functionElimination voidElimination because_Cache hypothesis extract_by_obid isectElimination setElimination rename independent_isectElimination hypothesisEquality dependent_functionElimination natural_numberEquality unionElimination dependent_pairFormation lambdaEquality int_eqEquality intEquality isect_memberEquality voidEquality sqequalRule independent_pairFormation computeAll equalityTransitivity equalitySymmetry cumulativity applyEquality instantiate functionEquality universeEquality setEquality functionExtensionality hyp_replacement applyLambdaEquality dependent_set_memberEquality productElimination

Latex:
\mforall{}[T:Type].  \mforall{}[R:T  {}\mrightarrow{}  T  {}\mrightarrow{}  \mBbbP{}].  \mforall{}[L:T  List].
(no\_repeats(T;L))  supposing  (sorted-by(R;L)  and  (\mforall{}a:T.  (\mneg{}(R  a  a))))

Date html generated: 2017_04_17-AM-07_43_46
Last ObjectModification: 2017_02_27-PM-04_16_18

Theory : list_1

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