### Nuprl Lemma : member-insert-no-combine

`∀T:Type. ∀cmp:comparison(T). ∀x,z:T. ∀v:T List.  ((z ∈ insert-no-combine(cmp;x;v)) `⇐⇒` (z ∈ [x / v]))`

Proof

Definitions occuring in Statement :  insert-no-combine: `insert-no-combine(cmp;x;l)` comparison: `comparison(T)` l_member: `(x ∈ l)` cons: `[a / b]` list: `T List` all: `∀x:A. B[x]` iff: `P `⇐⇒` Q` universe: `Type`
Definitions unfolded in proof :  all: `∀x:A. B[x]` uall: `∀[x:A]. B[x]` member: `t ∈ T` so_lambda: `λ2x.t[x]` so_apply: `x[s]` implies: `P `` Q` insert-no-combine: `insert-no-combine(cmp;x;l)` so_lambda: `so_lambda(x,y,z.t[x; y; z])` top: `Top` so_apply: `x[s1;s2;s3]` prop: `ℙ` iff: `P `⇐⇒` Q` and: `P ∧ Q` rev_implies: `P `` Q` comparison: `comparison(T)` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` uiff: `uiff(P;Q)` uimplies: `b supposing a` ifthenelse: `if b then t else f fi ` bfalse: `ff` exists: `∃x:A. B[x]` or: `P ∨ Q` sq_type: `SQType(T)` guard: `{T}` bnot: `¬bb` assert: `↑b` false: `False` not: `¬A`
Lemmas referenced :  list_induction iff_wf l_member_wf insert-no-combine_wf cons_wf list_wf list_ind_nil_lemma list_ind_cons_lemma comparison_wf nil_wf le_int_wf bool_wf eqtt_to_assert assert_of_le_int eqff_to_assert equal_wf bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot le_wf cons_member or_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation cut thin introduction extract_by_obid sqequalHypSubstitution isectElimination because_Cache sqequalRule lambdaEquality cumulativity hypothesisEquality hypothesis independent_functionElimination dependent_functionElimination isect_memberEquality voidElimination voidEquality rename universeEquality independent_pairFormation natural_numberEquality applyEquality setElimination unionElimination equalityElimination equalityTransitivity equalitySymmetry productElimination independent_isectElimination dependent_pairFormation promote_hyp instantiate addLevel orFunctionality inrFormation inlFormation

Latex:
\mforall{}T:Type.  \mforall{}cmp:comparison(T).  \mforall{}x,z:T.  \mforall{}v:T  List.
((z  \mmember{}  insert-no-combine(cmp;x;v))  \mLeftarrow{}{}\mRightarrow{}  (z  \mmember{}  [x  /  v]))

Date html generated: 2017_04_17-AM-08_30_57
Last ObjectModification: 2017_02_27-PM-04_52_03

Theory : list_1

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