### Nuprl Lemma : bl-rev-exists-sq

`∀[T:Type]. ∀[L:T List]. ∀[P:{x:T| (x ∈ L)}  ⟶ 𝔹].  ((∃x∈rev(L).P[x])_b ~ (∃x∈L.P[x])_b)`

Proof

Definitions occuring in Statement :  bl-rev-exists: `(∃x∈rev(L).P[x])_b` bl-exists: `(∃x∈L.P[x])_b` l_member: `(x ∈ l)` list: `T List` bool: `𝔹` uall: `∀[x:A]. B[x]` so_apply: `x[s]` set: `{x:A| B[x]} ` function: `x:A ⟶ B[x]` universe: `Type` sqequal: `s ~ t`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` all: `∀x:A. B[x]` nat: `ℕ` implies: `P `` Q` false: `False` ge: `i ≥ j ` uimplies: `b supposing a` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` not: `¬A` top: `Top` and: `P ∧ Q` prop: `ℙ` subtype_rel: `A ⊆r B` guard: `{T}` or: `P ∨ Q` bl-exists: `(∃x∈L.P[x])_b` bl-rev-exists: `(∃x∈rev(L).P[x])_b` so_lambda: `so_lambda(x,y,z.t[x; y; z])` so_apply: `x[s1;s2;s3]` cons: `[a / b]` colength: `colength(L)` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` decidable: `Dec(P)` nil: `[]` it: `⋅` so_lambda: `λ2x.t[x]` so_apply: `x[s]` sq_type: `SQType(T)` less_than: `a < b` squash: `↓T` less_than': `less_than'(a;b)` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` bool: `𝔹` unit: `Unit` btrue: `tt` uiff: `uiff(P;Q)` bfalse: `ff` bnot: `¬bb` ifthenelse: `if b then t else f fi ` assert: `↑b`
Lemmas referenced :  nat_properties satisfiable-full-omega-tt intformand_wf intformle_wf itermConstant_wf itermVar_wf intformless_wf int_formula_prop_and_lemma 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 l_member_wf bool_wf equal-wf-T-base nat_wf colength_wf_list less_than_transitivity1 less_than_irreflexivity list-cases list_ind_nil_lemma reduce_nil_lemma nil_wf product_subtype_list spread_cons_lemma intformeq_wf itermAdd_wf int_formula_prop_eq_lemma int_term_value_add_lemma decidable__le intformnot_wf int_formula_prop_not_lemma le_wf equal_wf subtract_wf itermSubtract_wf int_term_value_subtract_lemma subtype_base_sq set_subtype_base int_subtype_base decidable__equal_int list_ind_cons_lemma reduce_cons_lemma cons_wf list_wf cons_member bl-exists_wf eqtt_to_assert assert-bl-exists bor-btrue eqff_to_assert bool_cases_sqequal bool_subtype_base assert-bnot l_exists_wf assert_wf bor-bfalse
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut thin lambdaFormation extract_by_obid sqequalHypSubstitution isectElimination hypothesisEquality hypothesis setElimination rename intWeakElimination natural_numberEquality independent_isectElimination dependent_pairFormation lambdaEquality int_eqEquality intEquality dependent_functionElimination isect_memberEquality voidElimination voidEquality sqequalRule independent_pairFormation computeAll independent_functionElimination sqequalAxiom functionEquality setEquality cumulativity applyEquality because_Cache unionElimination promote_hyp hypothesis_subsumption productElimination equalityTransitivity equalitySymmetry applyLambdaEquality dependent_set_memberEquality addEquality baseClosed instantiate imageElimination universeEquality functionExtensionality inrFormation equalityElimination inlFormation

Latex:
\mforall{}[T:Type].  \mforall{}[L:T  List].  \mforall{}[P:\{x:T|  (x  \mmember{}  L)\}    {}\mrightarrow{}  \mBbbB{}].    ((\mexists{}x\mmember{}rev(L).P[x])\_b  \msim{}  (\mexists{}x\mmember{}L.P[x])\_b)

Date html generated: 2018_05_21-PM-06_46_03
Last ObjectModification: 2017_07_26-PM-04_55_52

Theory : general

Home Index