Nuprl Lemma : sorted-seq-iff

∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ].
  ∀s:sequence(T). (Trans(T;x,y.R[x;y]) ⇒ (sorted-seq(x,y.R[x;y];s) ⇐⇒ ∀i:ℕ||s|| - 1. R[s[i];s[i + 1]]))


Proof



Definitions occuring in Statement :  trans: Trans(T;x,y.E[x; y]),  sorted-seq: sorted-seq(x,y.R[x; y];s),  seq-item: s[i],  seq-len: ||s||,  sequence: sequence(T),  int_seg: {i..j-},  uall: ∀[x:A]. B[x],  prop: ℙ,  so_apply: x[s1;s2],  all: ∀x:A. B[x],  iff: P ⇐⇒ Q,  implies: P ⇒ Q,  function: x:A ⟶ B[x],  subtract: n - m,  add: n + m,  natural_number: $n,  universe: Type
Definitions unfolded in proof :  uall: ∀[x:A]. B[x],  all: ∀x:A. B[x],  implies: P ⇒ Q,  iff: P ⇐⇒ Q,  and: P ∧ Q,  sorted-seq: sorted-seq(x,y.R[x; y];s),  member: t ∈ T,  int_seg: {i..j-},  lelt: i ≤ j < k,  less_than: a < b,  squash: ↓T,  cand: A c∧ B,  le: A ≤ B,  subtype_rel: A ⊆r B,  nat: ℕ,  decidable: Dec(P),  or: P ∨ Q,  not: ¬A,  rev_implies: P ⇐ Q,  false: False,  uiff: uiff(P;Q),  uimplies: b supposing a,  subtract: n - m,  less_than': less_than'(a;b),  true: True,  so_lambda: λ2x y.t[x; y],  so_apply: x[s1;s2],  prop: ℙ,  exists: ∃x:A. B[x],  so_lambda: λ2x.t[x],  so_apply: x[s],  guard: {T},  sq_stable: SqStable(P),  sq_type: SQType(T),  ge: i ≥ j ,  nat_plus: ℕ+,  top: Top,  trans: Trans(T;x,y.E[x; y])
Lemmas referenced :  decidable__lt,  seq-len_wf,  istype-false,  not-lt-2,  less-iff-le,  condition-implies-le,  add-associates,  minus-add,  minus-one-mul,  add-swap,  minus-one-mul-top,  add-commutes,  add_functionality_wrt_le,  le-add-cancel2,  istype-le,  istype-less_than,  add-member-int_seg2,  decidable__le,  subtract_wf,  not-le-2,  zero-add,  add-zero,  add-mul-special,  zero-mul,  le-add-cancel,  int_seg_wf,  sorted-seq_wf,  seq-item_wf,  trans_wf,  sequence_wf,  istype-universe,  le_weakening2,  istype-sqequal,  set_subtype_base,  le_wf,  int_subtype_base,  add-is-int-iff,  istype-int,  primrec-wf2,  all_wf,  less_than_wf,  istype-nat,  subtract_nat_wf,  sq_stable__le,  subtype_base_sq,  minus-zero,  minus-minus,  le_reflexive,  one-mul,  two-mul,  mul-distributes-right,  mul-associates,  omega-shadow,  mul-swap,  mul-distributes,  mul-commutes,  le-add-cancel-alt,  nat_properties,  int_seg_properties,  lelt_wf,  istype-void
Rules used in proof :  sqequalSubstitution,  sqequalTransitivity,  computationStep,  sqequalReflexivity,  isect_memberFormation_alt,  lambdaFormation_alt,  independent_pairFormation,  sqequalHypSubstitution,  cut,  hypothesis,  dependent_functionElimination,  thin,  setElimination,  rename,  dependent_set_memberEquality_alt,  hypothesisEquality,  productElimination,  imageElimination,  introduction,  extract_by_obid,  isectElimination,  applyEquality,  lambdaEquality_alt,  inhabitedIsType,  equalityTransitivity,  equalitySymmetry,  sqequalRule,  unionElimination,  voidElimination,  independent_functionElimination,  because_Cache,  independent_isectElimination,  addEquality,  natural_numberEquality,  minusEquality,  Error :memTop,  productIsType,  closedConclusion,  multiplyEquality,  universeIsType,  functionIsType,  universeEquality,  instantiate,  dependent_pairFormation_alt,  intEquality,  equalityIstype,  promote_hyp,  baseApply,  baseClosed,  setIsType,  functionEquality,  imageMemberEquality,  cumulativity,  isect_memberEquality_alt
Latex:
\mforall{}[T:Type].  \mforall{}[R:T  {}\mrightarrow{}  T  {}\mrightarrow{}  \mBbbP{}].
    \mforall{}s:sequence(T)
        (Trans(T;x,y.R[x;y])  {}\mRightarrow{}  (sorted-seq(x,y.R[x;y];s)  \mLeftarrow{}{}\mRightarrow{}  \mforall{}i:\mBbbN{}||s||  -  1.  R[s[i];s[i  +  1]]))


Date html generated: 2020_05_19-PM-09_36_26
Last ObjectModification: 2020_02_07-AM-09_44_18

Theory : rel_1


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