### Nuprl Lemma : permutation-when-no_repeats

`∀[T:Type]`
`  ∀sa,sb:T List.  ((∀x:T. ((x ∈ sa) `⇐⇒` (x ∈ sb))) `` no_repeats(T;sb) `` no_repeats(T;sa) `` permutation(T;sb;sa))`

Proof

Definitions occuring in Statement :  permutation: `permutation(T;L1;L2)` no_repeats: `no_repeats(T;l)` l_member: `(x ∈ l)` list: `T List` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` iff: `P `⇐⇒` Q` implies: `P `` Q` universe: `Type`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` implies: `P `` Q` member: `t ∈ T` int_seg: `{i..j-}` uimplies: `b supposing a` lelt: `i ≤ j < k` and: `P ∧ Q` le: `A ≤ B` less_than: `a < b` squash: `↓T` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` decidable: `Dec(P)` or: `P ∨ Q` not: `¬A` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` prop: `ℙ` l_member: `(x ∈ l)` cand: `A c∧ B` nat: `ℕ` ge: `i ≥ j ` subtype_rel: `A ⊆r B` guard: `{T}` pi1: `fst(t)` inject: `Inj(A;B;f)` so_lambda: `λ2x.t[x]` so_apply: `x[s]` no_repeats: `no_repeats(T;l)` less_than': `less_than'(a;b)` sq_type: `SQType(T)` true: `True` permutation: `permutation(T;L1;L2)`
Lemmas referenced :  select_wf int_seg_properties decidable__le full-omega-unsat intformand_wf intformnot_wf intformle_wf itermConstant_wf itermVar_wf istype-int 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 length_wf intformless_wf int_formula_prop_less_lemma select_member nat_properties istype-le istype-less_than int_seg_wf no_repeats_wf l_member_wf list_wf istype-universe non_neg_length length_wf_nat pigeon-hole decidable__equal_int_seg set_subtype_base lelt_wf int_subtype_base int_seg_subtype_nat istype-false intformeq_wf int_formula_prop_eq_lemma subtype_base_sq istype-nat equal_wf squash_wf true_wf subtype_rel_self iff_weakening_equal le_wf less_than_wf decidable__equal_int subtype_rel_function int_seg_subtype le_weakening inject_wf permute_list_wf list_extensionality permute_list_length permute_list_select
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation_alt lambdaFormation_alt cut hypothesis sqequalHypSubstitution dependent_functionElimination thin introduction extract_by_obid isectElimination hypothesisEquality setElimination rename because_Cache independent_isectElimination productElimination imageElimination sqequalRule natural_numberEquality unionElimination approximateComputation independent_functionElimination dependent_pairFormation_alt lambdaEquality_alt int_eqEquality Error :memTop,  independent_pairFormation universeIsType voidElimination dependent_set_memberEquality_alt productIsType equalityIstype functionIsType inhabitedIsType instantiate universeEquality promote_hyp applyEquality equalityTransitivity equalitySymmetry applyLambdaEquality functionExtensionality intEquality sqequalBase cumulativity imageMemberEquality baseClosed productEquality

Latex:
\mforall{}[T:Type]
\mforall{}sa,sb:T  List.
((\mforall{}x:T.  ((x  \mmember{}  sa)  \mLeftarrow{}{}\mRightarrow{}  (x  \mmember{}  sb)))
{}\mRightarrow{}  no\_repeats(T;sb)
{}\mRightarrow{}  no\_repeats(T;sa)
{}\mRightarrow{}  permutation(T;sb;sa))

Date html generated: 2020_05_19-PM-09_45_17
Last ObjectModification: 2019_12_31-PM-00_12_11

Theory : list_1

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