Nuprl Lemma : bar_recursion_wf0

[T:Type]. ∀[R,A:n:ℕ ⟶ (ℕn ⟶ T) ⟶ ℙ]. ∀[d:∀n:ℕ. ∀s:ℕn ⟶ T.  Dec(R[n;s])].
[b:∀n:ℕ. ∀s:ℕn ⟶ T.  (R[n;s]  A[n;s])]. ∀[i:∀n:ℕ. ∀s:ℕn ⟶ T.  ((∀t:T. A[n 1;seq-append(n;1;s;λi.t)])  A[n;s])].
  ((∀alpha:ℕ ⟶ T. (↓∃m:ℕR[m;alpha]))  (bar_recursion(d;b;i;0;λm.eval in ⊥) ∈ A[0;λm.⊥]))


Proof




Definitions occuring in Statement :  bar_recursion: bar_recursion seq-append: seq-append(n;m;s1;s2) int_seg: {i..j-} nat: bottom: callbyvalue: callbyvalue decidable: Dec(P) uall: [x:A]. B[x] prop: so_apply: x[s1;s2] all: x:A. B[x] exists: x:A. B[x] squash: T implies:  Q member: t ∈ T lambda: λx.A[x] function: x:A ⟶ B[x] add: m natural_number: $n universe: Type
Definitions unfolded in proof :  label: ...$L... t guard: {T} lelt: i ≤ j < k int_seg: {i..j-} true: True top: Top subtract: m squash: T sq_stable: SqStable(P) uiff: uiff(P;Q) rev_implies:  Q iff: ⇐⇒ Q or: P ∨ Q decidable: Dec(P) exists: x:A. B[x] all: x:A. B[x] prop: not: ¬A false: False less_than': less_than'(a;b) and: P ∧ Q le: A ≤ B uimplies: supposing a nat: so_apply: x[s] subtype_rel: A ⊆B so_apply: x[s1;s2] so_lambda: λ2x.t[x] implies:  Q member: t ∈ T uall: [x:A]. B[x]
Lemmas referenced :  ext-eq_weakening subtype_rel_weakening less_than_irreflexivity less_than_transitivity1 le_reflexive decidable_wf seq-append_wf le_wf le-add-cancel add-zero add_functionality_wrt_le add-commutes add-swap add-associates minus-one-mul-top zero-add minus-one-mul minus-add condition-implies-le sq_stable__le not-le-2 decidable__le false_wf int_seg_subtype_nat int_seg_wf subtype_rel_dep_function exists_wf squash_wf nat_wf all_wf bar_recursion_wf1
Rules used in proof :  equalitySymmetry equalityTransitivity instantiate universeEquality minusEquality intEquality voidEquality isect_memberEquality imageElimination baseClosed imageMemberEquality productElimination voidElimination unionElimination dependent_functionElimination addEquality dependent_set_memberEquality independent_pairFormation independent_isectElimination rename setElimination natural_numberEquality functionExtensionality applyEquality because_Cache lambdaEquality sqequalRule cumulativity functionEquality independent_functionElimination lambdaFormation hypothesisEquality thin isectElimination sqequalHypSubstitution hypothesis isect_memberFormation sqequalReflexivity computationStep sqequalTransitivity sqequalSubstitution extract_by_obid introduction cut

Latex:
\mforall{}[T:Type].  \mforall{}[R,A:n:\mBbbN{}  {}\mrightarrow{}  (\mBbbN{}n  {}\mrightarrow{}  T)  {}\mrightarrow{}  \mBbbP{}].  \mforall{}[d:\mforall{}n:\mBbbN{}.  \mforall{}s:\mBbbN{}n  {}\mrightarrow{}  T.    Dec(R[n;s])].  \mforall{}[b:\mforall{}n:\mBbbN{}.  \mforall{}s:\mBbbN{}n  {}\mrightarrow{}  T.
                                                                                                                                                                        (R[n;s]
                                                                                                                                                                        {}\mRightarrow{}  A[n;s])].
\mforall{}[i:\mforall{}n:\mBbbN{}.  \mforall{}s:\mBbbN{}n  {}\mrightarrow{}  T.    ((\mforall{}t:T.  A[n  +  1;seq-append(n;1;s;\mlambda{}i.t)])  {}\mRightarrow{}  A[n;s])].
    ((\mforall{}alpha:\mBbbN{}  {}\mrightarrow{}  T.  (\mdownarrow{}\mexists{}m:\mBbbN{}.  R[m;alpha]))  {}\mRightarrow{}  (bar\_recursion(d;b;i;0;\mlambda{}m.eval  x  =  m  in  \mbot{})  \mmember{}  A[0;\mlambda{}m.\mbot{}]))



Date html generated: 2017_09_29-PM-05_47_35
Last ObjectModification: 2017_09_01-PM-11_45_59

Theory : bar-induction


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