### Nuprl Lemma : not-assert-bl-all

`∀[T:Type]. ∀L:T List. ∀P:{x:T| (x ∈ L)}  ⟶ 𝔹.  (¬↑(∀x∈L.P[x])_b `⇐⇒` (∃x∈L. ¬↑P[x]))`

Proof

Definitions occuring in Statement :  bl-all: `(∀x∈L.P[x])_b` l_exists: `(∃x∈L. P[x])` l_member: `(x ∈ l)` list: `T List` assert: `↑b` bool: `𝔹` uall: `∀[x:A]. B[x]` so_apply: `x[s]` all: `∀x:A. B[x]` iff: `P `⇐⇒` Q` not: `¬A` set: `{x:A| B[x]} ` function: `x:A ⟶ B[x]` universe: `Type`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` iff: `P `⇐⇒` Q` and: `P ∧ Q` implies: `P `` Q` member: `t ∈ T` prop: `ℙ` so_lambda: `λ2x.t[x]` so_apply: `x[s]` rev_implies: `P `` Q` not: `¬A` false: `False` decidable: `Dec(P)` or: `P ∨ Q` uiff: `uiff(P;Q)` rev_uimplies: `rev_uimplies(P;Q)` uimplies: `b supposing a` l_all: `(∀x∈L.P[x])` guard: `{T}` int_seg: `{i..j-}` lelt: `i ≤ j < k` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` top: `Top` less_than: `a < b` squash: `↓T` l_exists: `(∃x∈L. P[x])`
Lemmas referenced :  length_wf int_seg_wf int_formula_prop_less_lemma intformless_wf decidable__lt int_formula_prop_wf int_term_value_var_lemma int_term_value_constant_lemma int_formula_prop_le_lemma int_formula_prop_not_lemma int_formula_prop_and_lemma itermVar_wf itermConstant_wf intformle_wf intformnot_wf intformand_wf satisfiable-full-omega-tt decidable__le int_seg_properties select_wf assert-bl-all set_wf decidable__assert decidable__not list-subtype decidable__l_exists list_wf bool_wf l_exists_wf l_member_wf bl-all_wf assert_wf not_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation lambdaFormation independent_pairFormation cut lemma_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality sqequalRule lambdaEquality applyEquality setEquality hypothesis independent_functionElimination voidElimination functionEquality universeEquality dependent_functionElimination cumulativity equalityTransitivity equalitySymmetry unionElimination productElimination independent_isectElimination because_Cache setElimination rename natural_numberEquality dependent_pairFormation int_eqEquality intEquality isect_memberEquality voidEquality computeAll imageElimination

Latex:
\mforall{}[T:Type].  \mforall{}L:T  List.  \mforall{}P:\{x:T|  (x  \mmember{}  L)\}    {}\mrightarrow{}  \mBbbB{}.    (\mneg{}\muparrow{}(\mforall{}x\mmember{}L.P[x])\_b  \mLeftarrow{}{}\mRightarrow{}  (\mexists{}x\mmember{}L.  \mneg{}\muparrow{}P[x]))

Date html generated: 2016_05_14-PM-02_10_53
Last ObjectModification: 2016_01_15-AM-08_00_23

Theory : list_1

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