### Nuprl Lemma : l_contains-cons

`∀[T:Type]`
`  ∀u:T. ∀v,bs:T List.`
`    ([u / v] ⊆ bs `⇐⇒` ∃cs,ds:T List. ((bs = (cs @ [u / ds]) ∈ (T List)) ∧ v ⊆ cs @ ds)) supposing `
`       (no_repeats(T;bs) and `
`       no_repeats(T;[u / v]))`

Proof

Definitions occuring in Statement :  l_contains: `A ⊆ B` no_repeats: `no_repeats(T;l)` append: `as @ bs` cons: `[a / b]` list: `T List` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` iff: `P `⇐⇒` Q` and: `P ∧ Q` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` uimplies: `b supposing a` member: `t ∈ T` implies: `P `` Q` iff: `P `⇐⇒` Q` and: `P ∧ Q` prop: `ℙ` rev_implies: `P `` Q` so_lambda: `λ2x.t[x]` top: `Top` so_apply: `x[s]` l_contains: `A ⊆ B` l_all: `(∀x∈L.P[x])` int_seg: `{i..j-}` lelt: `i ≤ j < k` le: `A ≤ B` less_than': `less_than'(a;b)` false: `False` not: `¬A` nat_plus: `ℕ+` less_than: `a < b` squash: `↓T` true: `True` guard: `{T}` decidable: `Dec(P)` or: `P ∨ Q` uiff: `uiff(P;Q)` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` append: `as @ bs` so_lambda: `so_lambda(x,y,z.t[x; y; z])` so_apply: `x[s1;s2;s3]` cand: `A c∧ B` sq_type: `SQType(T)` select: `L[n]` cons: `[a / b]` ge: `i ≥ j `
Lemmas referenced :  no_repeats_witness cons_wf l_contains_wf exists_wf list_wf equal_wf append_wf length_wf length-append no_repeats_wf length_of_cons_lemma false_wf add_nat_plus length_wf_nat less_than_wf nat_plus_wf nat_plus_properties decidable__lt add-is-int-iff satisfiable-full-omega-tt intformand_wf intformnot_wf intformless_wf itermConstant_wf itermVar_wf itermAdd_wf intformeq_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_less_lemma int_term_value_constant_lemma int_term_value_var_lemma int_term_value_add_lemma int_formula_prop_eq_lemma int_formula_prop_wf lelt_wf select-cons-hd l_member_decomp list_ind_cons_lemma list_ind_nil_lemma l_all_iff l_member_wf cons_member nat_wf member_append or_wf no_repeats_cons and_wf select_wf int_seg_properties decidable__le intformle_wf int_formula_prop_le_lemma decidable__equal_int subtype_base_sq int_subtype_base int_seg_wf all_wf select-cons-tl subtract_wf itermSubtract_wf int_term_value_subtract_lemma non_neg_length
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation lambdaFormation cut introduction extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality cumulativity hypothesis independent_functionElimination rename because_Cache independent_pairFormation sqequalRule lambdaEquality productEquality applyLambdaEquality isect_memberEquality voidElimination voidEquality universeEquality dependent_functionElimination dependent_set_memberEquality natural_numberEquality imageMemberEquality baseClosed equalityTransitivity equalitySymmetry setElimination unionElimination pointwiseFunctionality promote_hyp baseApply closedConclusion productElimination independent_isectElimination dependent_pairFormation int_eqEquality intEquality computeAll addEquality setEquality inrFormation hyp_replacement addLevel orFunctionality inlFormation levelHypothesis allFunctionality imageElimination instantiate allLevelFunctionality

Latex:
\mforall{}[T:Type]
\mforall{}u:T.  \mforall{}v,bs:T  List.
([u  /  v]  \msubseteq{}  bs  \mLeftarrow{}{}\mRightarrow{}  \mexists{}cs,ds:T  List.  ((bs  =  (cs  @  [u  /  ds]))  \mwedge{}  v  \msubseteq{}  cs  @  ds))  supposing
(no\_repeats(T;bs)  and
no\_repeats(T;[u  /  v]))

Date html generated: 2017_04_17-AM-07_29_37
Last ObjectModification: 2017_02_27-PM-04_07_51

Theory : list_1

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