Nuprl Lemma : remove-combine-implies-member2

`∀[T:Type]. ∀cmp:T ⟶ ℤ. ∀x:T. ∀l:T List.  ((¬((cmp x) = 0 ∈ ℤ)) `` (x ∈ l) `` (x ∈ remove-combine(cmp;l)))`

Proof

Definitions occuring in Statement :  remove-combine: `remove-combine(cmp;l)` l_member: `(x ∈ l)` list: `T List` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` not: `¬A` implies: `P `` Q` apply: `f a` function: `x:A ⟶ B[x]` natural_number: `\$n` int: `ℤ` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` member: `t ∈ T` so_lambda: `λ2x.t[x]` implies: `P `` Q` prop: `ℙ` so_apply: `x[s]` false: `False` iff: `P `⇐⇒` Q` and: `P ∧ Q` rev_implies: `P `` Q` top: `Top` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` uiff: `uiff(P;Q)` uimplies: `b supposing a` ifthenelse: `if b then t else f fi ` or: `P ∨ Q` not: `¬A` bfalse: `ff` exists: `∃x:A. B[x]` sq_type: `SQType(T)` guard: `{T}` bnot: `¬bb` assert: `↑b`
Lemmas referenced :  list_induction not_wf equal-wf-T-base l_member_wf remove-combine_wf list_wf false_wf nil_member remove-combine-nil nil_wf eq_int_wf bool_wf eqtt_to_assert assert_of_eq_int and_wf equal_wf or_wf eqff_to_assert bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot neg_assert_of_eq_int lt_int_wf assert_of_lt_int cons_member less_than_wf remove-combine-cons cons_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation lambdaFormation cut thin introduction extract_by_obid sqequalHypSubstitution isectElimination because_Cache sqequalRule lambdaEquality functionEquality intEquality applyEquality functionExtensionality hypothesisEquality cumulativity baseClosed hypothesis dependent_functionElimination independent_functionElimination voidElimination addLevel impliesFunctionality productElimination isect_memberEquality voidEquality rename natural_numberEquality unionElimination equalityElimination equalityTransitivity equalitySymmetry independent_isectElimination dependent_set_memberEquality independent_pairFormation applyLambdaEquality setElimination dependent_pairFormation promote_hyp instantiate inlFormation inrFormation universeEquality

Latex:
\mforall{}[T:Type]
\mforall{}cmp:T  {}\mrightarrow{}  \mBbbZ{}.  \mforall{}x:T.  \mforall{}l:T  List.    ((\mneg{}((cmp  x)  =  0))  {}\mRightarrow{}  (x  \mmember{}  l)  {}\mRightarrow{}  (x  \mmember{}  remove-combine(cmp;l)))

Date html generated: 2017_04_17-AM-08_30_28
Last ObjectModification: 2017_02_27-PM-04_51_54

Theory : list_1

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