### Nuprl Lemma : no-repeats-list-to-set

`∀[T:Type]. ∀eq:EqDecider(T). ∀L:T List.  list-to-set(eq;L) ~ L supposing no_repeats(T;L)`

Proof

Definitions occuring in Statement :  list-to-set: `list-to-set(eq;L)` no_repeats: `no_repeats(T;l)` list: `T List` deq: `EqDecider(T)` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` universe: `Type` sqequal: `s ~ t`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` all: `∀x:A. B[x]` nat: `ℕ` implies: `P `` Q` false: `False` ge: `i ≥ j ` uimplies: `b supposing a` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` not: `¬A` top: `Top` and: `P ∧ Q` prop: `ℙ` subtype_rel: `A ⊆r B` guard: `{T}` or: `P ∨ Q` cons: `[a / b]` colength: `colength(L)` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` decidable: `Dec(P)` nil: `[]` it: `⋅` so_lambda: `λ2x.t[x]` so_apply: `x[s]` sq_type: `SQType(T)` less_than: `a < b` squash: `↓T` less_than': `less_than'(a;b)` uiff: `uiff(P;Q)` bool: `𝔹` unit: `Unit` btrue: `tt` iff: `P `⇐⇒` Q` ifthenelse: `if b then t else f fi ` bfalse: `ff` bnot: `¬bb` assert: `↑b` rev_implies: `P `` Q`
Lemmas referenced :  nat_properties satisfiable-full-omega-tt intformand_wf intformle_wf itermConstant_wf itermVar_wf intformless_wf int_formula_prop_and_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_less_lemma int_formula_prop_wf ge_wf less_than_wf no_repeats_wf equal-wf-T-base nat_wf colength_wf_list less_than_transitivity1 less_than_irreflexivity list-cases list_to_set_nil_lemma nil_wf product_subtype_list spread_cons_lemma intformeq_wf itermAdd_wf int_formula_prop_eq_lemma int_term_value_add_lemma decidable__le intformnot_wf int_formula_prop_not_lemma le_wf equal_wf subtract_wf itermSubtract_wf int_term_value_subtract_lemma subtype_base_sq set_subtype_base int_subtype_base decidable__equal_int no_repeats_cons list-to-set-cons deq-member_wf list-to-set_wf bool_wf eqtt_to_assert assert-deq-member eqff_to_assert bool_cases_sqequal bool_subtype_base assert-bnot l_member_wf cons_wf list_wf deq_wf member-list-to-set
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut lambdaFormation thin extract_by_obid sqequalHypSubstitution isectElimination hypothesisEquality hypothesis setElimination rename intWeakElimination natural_numberEquality independent_isectElimination dependent_pairFormation lambdaEquality int_eqEquality intEquality dependent_functionElimination isect_memberEquality voidElimination voidEquality sqequalRule independent_pairFormation computeAll independent_functionElimination sqequalAxiom cumulativity equalityTransitivity equalitySymmetry applyEquality because_Cache unionElimination promote_hyp hypothesis_subsumption productElimination applyLambdaEquality dependent_set_memberEquality addEquality baseClosed instantiate imageElimination equalityElimination universeEquality

Latex:
\mforall{}[T:Type].  \mforall{}eq:EqDecider(T).  \mforall{}L:T  List.    list-to-set(eq;L)  \msim{}  L  supposing  no\_repeats(T;L)

Date html generated: 2017_04_17-AM-09_10_04
Last ObjectModification: 2017_02_27-PM-05_18_03

Theory : decidable!equality

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