Nuprl Lemma : member-s-insert

`∀[T:Type]. ∀x:T. ∀L:T List. ∀z:T.  ((z ∈ s-insert(x;L)) `⇐⇒` (z = x ∈ T) ∨ (z ∈ L)) supposing T ⊆r ℤ`

Proof

Definitions occuring in Statement :  s-insert: `s-insert(x;l)` l_member: `(x ∈ l)` list: `T List` uimplies: `b supposing a` subtype_rel: `A ⊆r B` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` iff: `P `⇐⇒` Q` or: `P ∨ Q` int: `ℤ` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  bfalse: `ff` ifthenelse: `if b then t else f fi ` uiff: `uiff(P;Q)` btrue: `tt` it: `⋅` unit: `Unit` bool: `𝔹` guard: `{T}` sq_type: `SQType(T)` false: `False` not: `¬A` rev_implies: `P `` Q` and: `P ∧ Q` iff: `P `⇐⇒` Q` so_apply: `x[s1;s2;s3]` top: `Top` so_lambda: `so_lambda(x,y,z.t[x; y; z])` s-insert: `s-insert(x;l)` implies: `P `` Q` so_apply: `x[s]` or: `P ∨ Q` prop: `ℙ` so_lambda: `λ2x.t[x]` all: `∀x:A. B[x]` subtype_rel: `A ⊆r B` member: `t ∈ T` uimplies: `b supposing a` uall: `∀[x:A]. B[x]`
Lemmas referenced :  assert_of_le_int bnot_of_lt_int assert_functionality_wrt_uiff assert_of_lt_int assert_of_bnot eqff_to_assert iff_weakening_uiff iff_transitivity assert_of_eq_int eqtt_to_assert uiff_transitivity le_wf le_int_wf less_than_wf lt_int_wf istype-assert istype-int not_wf bnot_wf cons_member int_subtype_base subtype_base_sq assert_wf bool_wf equal-wf-T-base eq_int_wf istype-universe subtype_rel_wf list_ind_cons_lemma cons_wf member_singleton btrue_neq_bfalse member-implies-null-eq-bfalse btrue_wf null_nil_lemma nil_wf istype-void list_ind_nil_lemma list_wf equal_wf s-insert_wf l_member_wf iff_wf list_induction
Rules used in proof :  equalityElimination cumulativity Error :inrFormation_alt,  baseClosed applyEquality universeEquality instantiate intEquality Error :productIsType,  Error :functionIsType,  promote_hyp productElimination Error :unionIsType,  equalitySymmetry equalityTransitivity because_Cache unionElimination Error :inhabitedIsType,  Error :equalityIstype,  Error :inlFormation_alt,  independent_pairFormation voidElimination Error :isect_memberEquality_alt,  dependent_functionElimination independent_functionElimination Error :universeIsType,  unionEquality independent_isectElimination functionEquality Error :lambdaEquality_alt,  hypothesisEquality isectElimination sqequalHypSubstitution extract_by_obid Error :lambdaFormation_alt,  rename thin hypothesis axiomEquality sqequalRule introduction cut Error :isect_memberFormation_alt,  sqequalReflexivity computationStep sqequalTransitivity sqequalSubstitution

Latex:
\mforall{}[T:Type].  \mforall{}x:T.  \mforall{}L:T  List.  \mforall{}z:T.    ((z  \mmember{}  s-insert(x;L))  \mLeftarrow{}{}\mRightarrow{}  (z  =  x)  \mvee{}  (z  \mmember{}  L))  supposing  T  \msubseteq{}r  \mBbbZ{}

Date html generated: 2019_06_20-PM-00_42_20
Last ObjectModification: 2019_06_19-AM-10_43_11

Theory : list_0

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