### Nuprl Lemma : bar-induction (dup of thm in list_1)

`∀[T:Type]. ∀[R,A:(T List) ⟶ ℙ].`
`  ((∀s:T List. Dec(R[s]))`
`  `` (∀s:T List. (R[s] `` A[s]))`
`  `` (∀s:T List. ((∀t:T. A[s @ [t]]) `` A[s]))`
`  `` (∀s:T List. ((∀alpha:ℕ ⟶ T. (↓∃n:ℕ. R[s @ map(alpha;upto(n))])) `` A[s])))`

Proof

Definitions occuring in Statement :  upto: `upto(n)` map: `map(f;as)` append: `as @ bs` cons: `[a / b]` nil: `[]` list: `T List` nat: `ℕ` decidable: `Dec(P)` uall: `∀[x:A]. B[x]` prop: `ℙ` so_apply: `x[s]` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` squash: `↓T` implies: `P `` Q` function: `x:A ⟶ B[x]` universe: `Type`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` implies: `P `` Q` all: `∀x:A. B[x]` member: `t ∈ T` so_apply: `x[s]` nat: `ℕ` so_apply: `x[s1;s2]` prop: `ℙ` so_lambda: `λ2x.t[x]` 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}` top: `Top` ge: `i ≥ j ` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` int_seg: `{i..j-}` lelt: `i ≤ j < k` squash: `↓T` true: `True` iff: `P `⇐⇒` Q` seq-adjoin: `s++t` seq-append: `seq-append(n;m;s1;s2)` less_than: `a < b` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` uiff: `uiff(P;Q)` ifthenelse: `if b then t else f fi ` label: `...\$L... t` rev_implies: `P `` Q` bfalse: `ff` sq_type: `SQType(T)` bnot: `¬bb` assert: `↑b` cand: `A c∧ B`
Lemmas referenced :  bar_induction list_wf map_wf int_seg_wf upto_wf nat_wf all_wf seq-adjoin_wf squash_wf exists_wf append_wf subtype_rel_dep_function int_seg_subtype_nat false_wf cons_wf nil_wf decidable_wf list_extensionality length-append map-length length_of_cons_lemma length_of_nil_lemma length_upto nat_properties decidable__le satisfiable-full-omega-tt intformand_wf intformnot_wf intformle_wf itermConstant_wf itermAdd_wf itermVar_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_add_lemma int_term_value_var_lemma int_formula_prop_wf le_wf select-map subtype_rel_list top_wf less_than_wf true_wf length_append iff_weakening_equal add_functionality_wrt_eq length_wf map_length_nat length-singleton lelt_wf length-map intformless_wf int_formula_prop_less_lemma lt_int_wf bool_wf eqtt_to_assert assert_of_lt_int equal_wf select_upto decidable__equal_int intformeq_wf int_formula_prop_eq_lemma decidable__lt eqff_to_assert bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot select-cons-hd subtract_wf itermSubtract_wf int_term_value_subtract_lemma select-append select-upto length_wf_nat select_wf int_seg_properties seq-append_wf add_nat_wf add-is-int-iff non_neg_length
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation lambdaFormation cut introduction extract_by_obid sqequalHypSubstitution isectElimination thin sqequalRule hypothesisEquality lambdaEquality applyEquality functionExtensionality cumulativity hypothesis natural_numberEquality setElimination rename because_Cache functionEquality independent_functionElimination dependent_functionElimination addEquality independent_isectElimination independent_pairFormation universeEquality hyp_replacement equalitySymmetry isect_memberEquality voidElimination voidEquality dependent_set_memberEquality unionElimination dependent_pairFormation int_eqEquality intEquality computeAll imageElimination equalityTransitivity productElimination imageMemberEquality baseClosed lessCases sqequalAxiom equalityElimination promote_hyp instantiate applyLambdaEquality pointwiseFunctionality baseApply closedConclusion

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

Date html generated: 2018_05_21-PM-10_17_46
Last ObjectModification: 2017_07_26-PM-06_36_26

Theory : bar!induction

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