### Nuprl Lemma : assert-deq-all-disjoint

`∀[A:Type]. ∀[eq:EqDecider(A)]. ∀[ass:A List List]. ∀[bs:A List].`
`  uiff(↑deq-all-disjoint(eq;ass;bs);(∀as∈ass.l_disjoint(A;as;bs)))`

Proof

Definitions occuring in Statement :  deq-all-disjoint: `deq-all-disjoint(eq;ass;bs)` l_disjoint: `l_disjoint(T;l1;l2)` l_all: `(∀x∈L.P[x])` list: `T List` deq: `EqDecider(T)` assert: `↑b` uiff: `uiff(P;Q)` uall: `∀[x:A]. B[x]` universe: `Type`
Definitions unfolded in proof :  deq-all-disjoint: `deq-all-disjoint(eq;ass;bs)` uiff: `uiff(P;Q)` and: `P ∧ Q` uimplies: `b supposing a` member: `t ∈ T` l_all: `(∀x∈L.P[x])` all: `∀x:A. B[x]` l_disjoint: `l_disjoint(T;l1;l2)` not: `¬A` implies: `P `` Q` false: `False` uall: `∀[x:A]. B[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` prop: `ℙ` less_than: `a < b` squash: `↓T` so_lambda: `λ2x.t[x]` so_apply: `x[s]` subtype_rel: `A ⊆r B` iff: `P `⇐⇒` Q` rev_implies: `P `` Q`
Lemmas referenced :  assert-deq-disjoint l_all_functionality assert-bl-all iff_weakening_uiff iff_transitivity assert_witness deq_wf deq-all-disjoint_wf uiff_wf deq-disjoint_wf bl-all_wf assert_wf l_disjoint_wf l_all_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 length_wf int_seg_properties list_wf select_wf l_member_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity independent_pairFormation isect_memberFormation introduction cut hypothesis sqequalRule sqequalHypSubstitution lambdaEquality dependent_functionElimination thin hypothesisEquality voidElimination productEquality lemma_by_obid isectElimination cumulativity because_Cache setElimination rename independent_isectElimination natural_numberEquality productElimination unionElimination dependent_pairFormation int_eqEquality intEquality isect_memberEquality voidEquality computeAll imageElimination setEquality applyEquality universeEquality independent_pairEquality equalityTransitivity equalitySymmetry independent_functionElimination addLevel lambdaFormation

Latex:
\mforall{}[A:Type].  \mforall{}[eq:EqDecider(A)].  \mforall{}[ass:A  List  List].  \mforall{}[bs:A  List].
uiff(\muparrow{}deq-all-disjoint(eq;ass;bs);(\mforall{}as\mmember{}ass.l\_disjoint(A;as;bs)))

Date html generated: 2016_05_14-PM-03_24_08
Last ObjectModification: 2016_01_14-PM-11_22_44

Theory : decidable!equality

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