### Nuprl Lemma : bbar-recursion_wf

`∀[T:Type]. ∀[R:(T List) ⟶ 𝔹]. ∀[A:(T List) ⟶ ℙ]. ∀[b:∀s:{s:T List| ↑R[s]} . A[s]]. ∀[i:∀s:{s:T List| ¬↑R[s]} `
`                                                                                          ((∀t:T. A[s @ [t]]) `` A[s])].`
`∀[s:T List].`
`  ((∀alpha:ℕ ⟶ T. (↓∃n:ℕ. (↑R[s @ map(alpha;upto(n))]))) `` (bbar-recursion(R;b;i;s) ∈ A[s]))`

Proof

Definitions occuring in Statement :  bbar-recursion: bbar-recursion upto: `upto(n)` map: `map(f;as)` append: `as @ bs` cons: `[a / b]` nil: `[]` list: `T List` nat: `ℕ` assert: `↑b` bool: `𝔹` uall: `∀[x:A]. B[x]` prop: `ℙ` so_apply: `x[s]` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` not: `¬A` squash: `↓T` implies: `P `` Q` member: `t ∈ T` set: `{x:A| B[x]} ` function: `x:A ⟶ B[x]` universe: `Type`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` implies: `P `` Q` prop: `ℙ` so_lambda: `λ2x.t[x]` so_apply: `x[s]` nat: `ℕ` subtype_rel: `A ⊆r B` uimplies: `b supposing a` le: `A ≤ B` and: `P ∧ Q` less_than': `less_than'(a;b)` false: `False` not: `¬A` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` top: `Top` decidable: `Dec(P)` or: `P ∨ Q` guard: `{T}` bbar-recursion: bbar-recursion bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` uiff: `uiff(P;Q)` ifthenelse: `if b then t else f fi ` bfalse: `ff` sq_type: `SQType(T)` bnot: `¬bb` assert: `↑b`
Lemmas referenced :  all_wf nat_wf squash_wf exists_wf assert_wf list_wf append_wf map_wf int_seg_wf subtype_rel_dep_function int_seg_subtype_nat false_wf upto_wf not_wf cons_wf nil_wf bool_wf subtype_rel_list top_wf append-nil decidable__assert eqtt_to_assert eqff_to_assert equal_wf bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot append_assoc
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut lambdaFormation sqequalRule sqequalHypSubstitution hypothesis extract_by_obid isectElimination thin functionEquality cumulativity hypothesisEquality lambdaEquality because_Cache applyEquality functionExtensionality natural_numberEquality setElimination rename independent_isectElimination independent_pairFormation dependent_functionElimination axiomEquality equalityTransitivity equalitySymmetry isect_memberEquality setEquality universeEquality voidElimination voidEquality barInduction unionElimination inlFormation inrFormation equalityElimination productElimination dependent_set_memberEquality dependent_pairFormation promote_hyp instantiate independent_functionElimination

Latex:
\mforall{}[T:Type].  \mforall{}[R:(T  List)  {}\mrightarrow{}  \mBbbB{}].  \mforall{}[A:(T  List)  {}\mrightarrow{}  \mBbbP{}].  \mforall{}[b:\mforall{}s:\{s:T  List|  \muparrow{}R[s]\}  .  A[s]].
\mforall{}[i:\mforall{}s:\{s:T  List|  \mneg{}\muparrow{}R[s]\}  .  ((\mforall{}t:T.  A[s  @  [t]])  {}\mRightarrow{}  A[s])].  \mforall{}[s:T  List].
((\mforall{}alpha:\mBbbN{}  {}\mrightarrow{}  T.  (\mdownarrow{}\mexists{}n:\mBbbN{}.  (\muparrow{}R[s  @  map(alpha;upto(n))])))  {}\mRightarrow{}  (bbar-recursion(R;b;i;s)  \mmember{}  A[s]))

Date html generated: 2018_05_21-PM-10_17_40
Last ObjectModification: 2017_07_26-PM-06_36_26

Theory : bar!induction

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