### Nuprl Lemma : remove-combine_wf

`∀T:Type. ∀cmp:T ⟶ ℤ. ∀l:T List.  (remove-combine(cmp;l) ∈ T List)`

Proof

Definitions occuring in Statement :  remove-combine: `remove-combine(cmp;l)` list: `T List` all: `∀x:A. B[x]` member: `t ∈ T` function: `x:A ⟶ B[x]` int: `ℤ` universe: `Type`
Definitions unfolded in proof :  all: `∀x:A. B[x]` uall: `∀[x:A]. B[x]` member: `t ∈ T` 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` remove-combine: `remove-combine(cmp;l)` so_lambda: `so_lambda(x,y,z.t[x; y; z])` so_apply: `x[s1;s2;s3]` 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)` has-value: `(a)↓` bool: `𝔹` unit: `Unit` btrue: `tt` ifthenelse: `if b then t else f fi ` bfalse: `ff` uiff: `uiff(P;Q)` bnot: `¬bb` assert: `↑b`
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 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 list_ind_cons_lemma value-type-has-value int-value-type eq_int_wf bool_wf eqff_to_assert bool_cases_sqequal bool_subtype_base assert-bnot neg_assert_of_eq_int lt_int_wf eqtt_to_assert assert_of_lt_int cons_wf list_ind_wf list_wf ifthenelse_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation cut thin introduction extract_by_obid sqequalHypSubstitution isectElimination hypothesisEquality hypothesis setElimination rename sqequalRule intWeakElimination natural_numberEquality independent_isectElimination dependent_pairFormation lambdaEquality int_eqEquality intEquality dependent_functionElimination isect_memberEquality voidElimination voidEquality independent_pairFormation computeAll independent_functionElimination axiomEquality equalityTransitivity equalitySymmetry cumulativity applyEquality because_Cache unionElimination promote_hyp hypothesis_subsumption productElimination applyLambdaEquality dependent_set_memberEquality addEquality baseClosed instantiate imageElimination callbyvalueReduce functionExtensionality equalityElimination functionEquality universeEquality

Latex:
\mforall{}T:Type.  \mforall{}cmp:T  {}\mrightarrow{}  \mBbbZ{}.  \mforall{}l:T  List.    (remove-combine(cmp;l)  \mmember{}  T  List)

Date html generated: 2017_04_17-AM-08_30_02
Last ObjectModification: 2017_02_27-PM-04_51_44

Theory : list_1

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