### Nuprl Lemma : no_repeats-same-length-l_contains

`∀[T:Type]. ∀as,bs:T List.  (no_repeats(T;as) `` (||as|| = ||bs|| ∈ ℤ) `` as ⊆ bs `` bs ⊆ as)`

Proof

Definitions occuring in Statement :  l_contains: `A ⊆ B` no_repeats: `no_repeats(T;l)` length: `||as||` list: `T List` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` implies: `P `` Q` int: `ℤ` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` implies: `P `` Q` member: `t ∈ T` prop: `ℙ` l_contains: `A ⊆ B` l_member: `(x ∈ l)` l_all: `(∀x∈L.P[x])` exists: `∃x:A. B[x]` nat: `ℕ` int_seg: `{i..j-}` lelt: `i ≤ j < k` and: `P ∧ Q` le: `A ≤ B` cand: `A c∧ B` uimplies: `b supposing a` guard: `{T}` ge: `i ≥ j ` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` false: `False` not: `¬A` top: `Top` so_lambda: `λ2x.t[x]` so_apply: `x[s]` subtype_rel: `A ⊆r B` pi1: `fst(t)` inject: `Inj(A;B;f)` squash: `↓T` true: `True` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` no_repeats: `no_repeats(T;l)` less_than': `less_than'(a;b)` surject: `Surj(A;B;f)` less_than: `a < b` sq_type: `SQType(T)`
Lemmas referenced :  l_contains_wf equal_wf length_wf no_repeats_wf list_wf lelt_wf select_wf int_seg_properties 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 decidable__lt intformless_wf int_formula_prop_less_lemma intformeq_wf int_formula_prop_eq_lemma int_seg_wf exists_wf all_wf non_neg_length length_wf_nat and_wf less_than_wf injection-is-surjection squash_wf true_wf iff_weakening_equal le_wf decidable__equal_int_seg int_seg_subtype_nat false_wf nat_wf subtype_base_sq set_subtype_base int_subtype_base l_member_wf select_member
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation lambdaFormation cut introduction extract_by_obid sqequalHypSubstitution isectElimination thin cumulativity hypothesisEquality hypothesis intEquality universeEquality sqequalRule dependent_functionElimination productElimination dependent_pairFormation setElimination rename dependent_set_memberEquality independent_pairFormation equalityTransitivity equalitySymmetry because_Cache independent_isectElimination natural_numberEquality unionElimination lambdaEquality int_eqEquality isect_memberEquality voidElimination voidEquality computeAll promote_hyp applyEquality functionExtensionality applyLambdaEquality independent_functionElimination hyp_replacement imageElimination imageMemberEquality baseClosed productEquality instantiate

Latex:
\mforall{}[T:Type].  \mforall{}as,bs:T  List.    (no\_repeats(T;as)  {}\mRightarrow{}  (||as||  =  ||bs||)  {}\mRightarrow{}  as  \msubseteq{}  bs  {}\mRightarrow{}  bs  \msubseteq{}  as)

Date html generated: 2017_04_17-AM-07_29_53
Last ObjectModification: 2017_02_27-PM-04_07_56

Theory : list_1

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