### Nuprl Lemma : not-isl-priority-select

`∀[T:Type]. ∀[as:T List]. ∀[f,g:T ⟶ 𝔹].  uiff(¬↑isl(priority-select(f;g;as));(∀a∈as.(¬↑(f a)) ∧ (¬↑(g a))))`

Proof

Definitions occuring in Statement :  priority-select: `priority-select(f;g;as)` l_all: `(∀x∈L.P[x])` list: `T List` assert: `↑b` isl: `isl(x)` bool: `𝔹` uiff: `uiff(P;Q)` uall: `∀[x:A]. B[x]` not: `¬A` and: `P ∧ Q` apply: `f a` function: `x:A ⟶ B[x]` universe: `Type`
Definitions unfolded in proof :  member: `t ∈ T` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` implies: `P `` Q` uiff: `uiff(P;Q)` and: `P ∧ Q` uimplies: `b supposing a` prop: `ℙ` not: `¬A` false: `False` isl: `isl(x)` rev_implies: `P `` Q` so_lambda: `λ2x.t[x]` so_apply: `x[s]` iff: `P `⇐⇒` Q` l_all: `(∀x∈L.P[x])` int_seg: `{i..j-}` guard: `{T}` lelt: `i ≤ j < k` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` top: `Top` less_than: `a < b` squash: `↓T` subtype_rel: `A ⊆r B` assert: `↑b` ifthenelse: `if b then t else f fi ` btrue: `tt` true: `True` bfalse: `ff`
Lemmas referenced :  priority-select_wf bool_wf unit_wf2 not_wf assert_wf isl_wf assert_elim and_wf equal_wf bfalse_wf btrue_neq_bfalse equal-wf-T-base iff_weakening_uiff l_all_wf2 l_member_wf priority-select-inr select_wf int_seg_properties length_wf decidable__le satisfiable-full-omega-tt intformand_wf intformnot_wf intformle_wf itermConstant_wf itermVar_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_wf decidable__lt intformless_wf int_formula_prop_less_lemma int_seg_wf uiff_wf list_wf equal-unit it_wf
Rules used in proof :  cut sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity introduction extract_by_obid sqequalHypSubstitution isectElimination thin cumulativity hypothesisEquality functionExtensionality applyEquality hypothesis unionEquality lambdaFormation independent_pairFormation isect_memberFormation addLevel independent_isectElimination levelHypothesis dependent_set_memberEquality equalityTransitivity equalitySymmetry applyLambdaEquality setElimination rename productElimination sqequalRule independent_functionElimination voidElimination lambdaEquality dependent_functionElimination because_Cache productEquality setEquality baseClosed independent_pairEquality natural_numberEquality unionElimination dependent_pairFormation int_eqEquality intEquality isect_memberEquality voidEquality computeAll imageElimination universeEquality functionEquality inrEquality

Latex:
\mforall{}[T:Type].  \mforall{}[as:T  List].  \mforall{}[f,g:T  {}\mrightarrow{}  \mBbbB{}].
uiff(\mneg{}\muparrow{}isl(priority-select(f;g;as));(\mforall{}a\mmember{}as.(\mneg{}\muparrow{}(f  a))  \mwedge{}  (\mneg{}\muparrow{}(g  a))))

Date html generated: 2018_05_21-PM-06_49_50
Last ObjectModification: 2017_07_26-PM-04_56_55

Theory : general

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