### Nuprl Lemma : bool_bar_induction

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

Proof

Definitions occuring in Statement :  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` set: `{x:A| B[x]} ` function: `x:A ⟶ B[x]` universe: `Type`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` implies: `P `` Q` member: `t ∈ T` so_lambda: `λ2x.t[x]` so_apply: `x[s]` prop: `ℙ` 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` guard: `{T}` decidable: `Dec(P)` or: `P ∨ Q`
Lemmas referenced :  basic-bar-induction assert_wf list_wf decidable__assert all_wf append_wf cons_wf nil_wf nat_wf squash_wf exists_wf map_wf int_seg_wf subtype_rel_dep_function int_seg_subtype_nat false_wf upto_wf not_wf bool_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation lambdaFormation cut lemma_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality sqequalRule lambdaEquality applyEquality hypothesis independent_functionElimination dependent_functionElimination because_Cache functionEquality cumulativity natural_numberEquality setElimination rename independent_isectElimination independent_pairFormation setEquality universeEquality dependent_set_memberEquality unionElimination

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

Date html generated: 2016_05_14-PM-03_19_42
Last ObjectModification: 2015_12_26-PM-01_41_54

Theory : list_1

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