### Nuprl Lemma : remove-repeats-append-sq

`∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[L1,L2:T List].`
`  (remove-repeats(eq;L1 @ L2) ~ remove-repeats(eq;L1) @ filter(λx.(¬bx ∈b L1);remove-repeats(eq;L2)))`

Proof

Definitions occuring in Statement :  remove-repeats: `remove-repeats(eq;L)` deq-member: `x ∈b L` filter: `filter(P;l)` append: `as @ bs` list: `T List` deq: `EqDecider(T)` bnot: `¬bb` uall: `∀[x:A]. B[x]` lambda: `λx.A[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` append: `as @ bs` so_lambda: `so_lambda(x,y,z.t[x; y; z])` so_apply: `x[s1;s2;s3]` bnot: `¬bb` ifthenelse: `if b then t else f fi ` bfalse: `ff` remove-repeats: `remove-repeats(eq;L)` 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)` btrue: `tt` bool: `𝔹` unit: `Unit` uiff: `uiff(P;Q)` iff: `P `⇐⇒` Q` band: `p ∧b q` deq: `EqDecider(T)` eqof: `eqof(d)` assert: `↑b` bor: `p ∨bq` 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 equal-wf-T-base nat_wf colength_wf_list less_than_transitivity1 less_than_irreflexivity list-cases list_ind_nil_lemma deq_member_nil_lemma 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 list_ind_cons_lemma deq_member_cons_lemma list_wf deq_wf remove-repeats_wf filter_nil_lemma filter_cons_lemma filter_append_sq filter-filter deq-member_wf bool_wf eqtt_to_assert assert-deq-member safe-assert-deq testxxx_lemma eqff_to_assert bool_cases_sqequal bool_subtype_base assert-bnot l_member_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut thin lambdaFormation 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 because_Cache cumulativity applyEquality unionElimination promote_hyp hypothesis_subsumption productElimination equalityTransitivity equalitySymmetry applyLambdaEquality dependent_set_memberEquality addEquality baseClosed instantiate imageElimination universeEquality equalityElimination

Latex:
\mforall{}[T:Type].  \mforall{}[eq:EqDecider(T)].  \mforall{}[L1,L2:T  List].
(remove-repeats(eq;L1  @  L2)  \msim{}  remove-repeats(eq;L1)
@  filter(\mlambda{}x.(\mneg{}\msubb{}x  \mmember{}\msubb{}  L1);remove-repeats(eq;L2)))

Date html generated: 2017_04_17-AM-09_10_21
Last ObjectModification: 2017_02_27-PM-05_18_45

Theory : decidable!equality

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