Nuprl Lemma : l-ordered-decomp2

∀[T:Type]. ∀[R:T ⟶ T ⟶ 𝔹]. ∀[x:T].
  (∀[L:T List]
     (L = (filter(λy.R[y;x];L) @ [x / filter(λy.R[x;y];L)]) ∈ (T List)) supposing 
        (l-ordered(T;x,y.↑R[x;y];L) and 
        (x ∈ L))) supposing 
     (Trans(T;x,y.↑R[x;y]) and 
     Irrefl(T;x,y.↑R[x;y]) and 
     StAntiSym(T;x,y.↑R[x;y]))


Proof



Definitions occuring in Statement :  l-ordered: l-ordered(T;x,y.R[x; y];L),  l_member: (x ∈ l),  filter: filter(P;l),  append: as @ bs,  cons: [a / b],  list: T List,  irrefl: Irrefl(T;x,y.E[x; y]),  st_anti_sym: StAntiSym(T;x,y.R[x; y]),  trans: Trans(T;x,y.E[x; y]),  assert: ↑b,  bool: 𝔹,  uimplies: b supposing a,  uall: ∀[x:A]. B[x],  so_apply: x[s1;s2],  lambda: λx.A[x],  function: x:A ⟶ B[x],  universe: Type,  equal: s = t ∈ T
Definitions unfolded in proof :  uall: ∀[x:A]. B[x],  member: t ∈ T,  uimplies: b supposing a,  all: ∀x:A. B[x],  nat: ℕ,  implies: P ⇒ Q,  false: False,  ge: i ≥ j ,  not: ¬A,  satisfiable_int_formula: satisfiable_int_formula(fmla),  exists: ∃x:A. B[x],  and: P ∧ Q,  prop: ℙ,  or: P ∨ Q,  append: as @ bs,  so_lambda: so_lambda3,  so_apply: x[s1;s2;s3],  iff: P ⇐⇒ Q,  so_lambda: λ2x y.t[x; y],  so_apply: x[s1;s2],  true: True,  cons: [a / b],  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,  decidable: Dec(P),  subtype_rel: A ⊆r B,  bool: 𝔹,  unit: Unit,  btrue: tt,  uiff: uiff(P;Q),  ifthenelse: if b then t else f fi ,  irrefl: Irrefl(T;x,y.E[x; y]),  st_anti_sym: StAntiSym(T;x,y.R[x; y]),  bfalse: ff,  bnot: ¬bb,  assert: ↑b,  rev_implies: P ⇐ Q,  cand: A c∧ B,  rev_uimplies: rev_uimplies(P;Q),  trans: Trans(T;x,y.E[x; y])
Lemmas referenced :  nat_properties,  full-omega-unsat,  intformand_wf,  intformle_wf,  itermConstant_wf,  itermVar_wf,  intformless_wf,  istype-int,  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,  istype-less_than,  list-cases,  filter_nil_lemma,  list_ind_nil_lemma,  istype-void,  nil_member,  l-ordered-nil-true,  assert_wf,  l-ordered_wf,  nil_wf,  l_member_wf,  product_subtype_list,  colength-cons-not-zero,  colength_wf_list,  istype-le,  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,  le_wf,  filter_cons_lemma,  eqtt_to_assert,  list_ind_cons_lemma,  eqff_to_assert,  bool_cases_sqequal,  bool_wf,  bool_subtype_base,  assert-bnot,  assert_elim,  not_assert_elim,  btrue_neq_bfalse,  cons_member,  l-ordered-cons,  cons_wf,  istype-nat,  list_wf,  trans_wf,  irrefl_wf,  st_anti_sym_wf,  istype-universe,  iff_imp_equal_bool,  btrue_wf,  istype-true,  equal_wf,  iff_weakening_equal,  squash_wf,  true_wf,  filter_is_nil,  l_all_iff,  not_wf,  istype-assert,  filter_trivial,  assert_functionality_wrt_uiff,  subtype_rel_self
Rules used in proof :  sqequalSubstitution,  sqequalTransitivity,  computationStep,  sqequalReflexivity,  isect_memberFormation_alt,  introduction,  cut,  thin,  lambdaFormation_alt,  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,  Error :memTop,  sqequalRule,  independent_pairFormation,  universeIsType,  voidElimination,  isect_memberEquality_alt,  axiomEquality,  isectIsTypeImplies,  inhabitedIsType,  functionIsTypeImplies,  unionElimination,  productElimination,  because_Cache,  applyEquality,  promote_hyp,  hypothesis_subsumption,  equalityIstype,  dependent_set_memberEquality_alt,  instantiate,  equalityTransitivity,  equalitySymmetry,  applyLambdaEquality,  imageElimination,  baseApply,  closedConclusion,  baseClosed,  intEquality,  sqequalBase,  equalityElimination,  cumulativity,  functionIsType,  universeEquality,  imageMemberEquality,  hyp_replacement,  productIsType,  setIsType,  functionEquality
Latex:
\mforall{}[T:Type].  \mforall{}[R:T  {}\mrightarrow{}  T  {}\mrightarrow{}  \mBbbB{}].  \mforall{}[x:T].
    (\mforall{}[L:T  List]
          (L  =  (filter(\mlambda{}y.R[y;x];L)  @  [x  /  filter(\mlambda{}y.R[x;y];L)]))  supposing 
                (l-ordered(T;x,y.\muparrow{}R[x;y];L)  and 
                (x  \mmember{}  L)))  supposing 
          (Trans(T;x,y.\muparrow{}R[x;y])  and 
          Irrefl(T;x,y.\muparrow{}R[x;y])  and 
          StAntiSym(T;x,y.\muparrow{}R[x;y]))


Date html generated: 2020_05_20-AM-08_09_24
Last ObjectModification: 2020_01_25-PM-11_57_54

Theory : general


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