### Nuprl Lemma : not_mem_remove1

`∀s:DSet. ∀a:|s|. ∀bs:|s| List.  ((¬↑(a ∈b bs)) `` ((bs \ a) = bs ∈ (|s| List)))`

Proof

Definitions occuring in Statement :  remove1: `as \ a` mem: `a ∈b as` list: `T List` assert: `↑b` all: `∀x:A. B[x]` not: `¬A` implies: `P `` Q` equal: `s = t ∈ T` dset: `DSet` set_car: `|p|`
Definitions unfolded in proof :  all: `∀x:A. B[x]` implies: `P `` Q` uall: `∀[x:A]. B[x]` member: `t ∈ T` nat: `ℕ` false: `False` ge: `i ≥ j ` uimplies: `b supposing a` not: `¬A` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` top: `Top` and: `P ∧ Q` prop: `ℙ` or: `P ∨ Q` assert: `↑b` ifthenelse: `if b then t else f fi ` bfalse: `ff` cons: `[a / b]` dset: `DSet` le: `A ≤ B` less_than': `less_than'(a;b)` colength: `colength(L)` nil: `[]` it: `⋅` guard: `{T}` so_lambda: `λ2x.t[x]` so_apply: `x[s]` sq_type: `SQType(T)` less_than: `a < b` squash: `↓T` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` decidable: `Dec(P)` subtype_rel: `A ⊆r B` infix_ap: `x f y` bool: `𝔹` unit: `Unit` btrue: `tt` uiff: `uiff(P;Q)` true: `True` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` bor: `p ∨bq`
Lemmas referenced :  nat_properties full-omega-unsat intformand_wf intformle_wf itermConstant_wf itermVar_wf intformless_wf istype-int int_formula_prop_and_lemma istype-void 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 list-cases mem_nil_lemma remove1_nil_lemma product_subtype_list colength-cons-not-zero colength_wf_list set_car_wf istype-false le_wf subtract-1-ge-0 subtype_base_sq intformeq_wf int_formula_prop_eq_lemma set_subtype_base int_subtype_base spread_cons_lemma decidable__equal_int subtract_wf intformnot_wf itermSubtract_wf itermAdd_wf int_formula_prop_not_lemma int_term_value_subtract_lemma int_term_value_add_lemma decidable__le mem_cons_lemma remove1_cons_lemma nat_wf not_wf assert_wf mem_wf list_wf dset_wf nil_wf false_wf set_eq_wf uiff_transitivity equal-wf-T-base bool_wf equal_wf eqtt_to_assert assert_of_dset_eq testxxx_lemma true_wf iff_transitivity bnot_wf iff_weakening_uiff eqff_to_assert assert_of_bnot cons_wf squash_wf istype-universe
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation_alt cut thin introduction extract_by_obid sqequalHypSubstitution isectElimination hypothesisEquality hypothesis setElimination rename intWeakElimination natural_numberEquality independent_isectElimination approximateComputation independent_functionElimination dependent_pairFormation_alt lambdaEquality_alt int_eqEquality dependent_functionElimination isect_memberEquality_alt voidElimination sqequalRule independent_pairFormation universeIsType axiomEquality functionIsTypeImplies inhabitedIsType because_Cache unionElimination promote_hyp hypothesis_subsumption productElimination equalityIsType1 dependent_set_memberEquality_alt instantiate equalityTransitivity equalitySymmetry applyLambdaEquality imageElimination equalityIsType4 baseApply closedConclusion baseClosed applyEquality intEquality equalityElimination universeEquality imageMemberEquality

Latex:
\mforall{}s:DSet.  \mforall{}a:|s|.  \mforall{}bs:|s|  List.    ((\mneg{}\muparrow{}(a  \mmember{}\msubb{}  bs))  {}\mRightarrow{}  ((bs  \mbackslash{}  a)  =  bs))

Date html generated: 2019_10_16-PM-01_03_48
Last ObjectModification: 2018_10_08-AM-11_14_45

Theory : list_2

Home Index