### Nuprl Lemma : basic_bar_induction

`∀[T:Type]. ∀[R,A:n:ℕ ⟶ (ℕn ⟶ T) ⟶ ℙ].`
`  ((∀n:ℕ. ∀s:ℕn ⟶ T.  Dec(R[n;s]))`
`  `` (∀n:ℕ. ∀s:ℕn ⟶ T.  (R[n;s] `` A[n;s]))`
`  `` (∀n:ℕ. ∀s:ℕn ⟶ T.  ((∀t:T. A[n + 1;s++t]) `` A[n;s]))`
`  `` (∀alpha:ℕ ⟶ T. (↓∃m:ℕ. R[m;alpha]))`
`  `` (∀x:Top. A[0;x]))`

Proof

Definitions occuring in Statement :  seq-adjoin: `s++t` int_seg: `{i..j-}` nat: `ℕ` decidable: `Dec(P)` uall: `∀[x:A]. B[x]` top: `Top` prop: `ℙ` so_apply: `x[s1;s2]` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` squash: `↓T` implies: `P `` Q` function: `x:A ⟶ B[x]` add: `n + m` natural_number: `\$n` universe: `Type`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` implies: `P `` Q` all: `∀x:A. B[x]` member: `t ∈ T` int_seg: `{i..j-}` false: `False` lelt: `i ≤ j < k` and: `P ∧ Q` guard: `{T}` uimplies: `b supposing a` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` nat: `ℕ` le: `A ≤ B` less_than': `less_than'(a;b)` not: `¬A` prop: `ℙ` squash: `↓T` subtype_rel: `A ⊆r B` true: `True` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` so_lambda: `λ2x.t[x]` so_apply: `x[s]` decidable: `Dec(P)` or: `P ∨ Q` uiff: `uiff(P;Q)` sq_stable: `SqStable(P)` subtract: `n - m` top: `Top` exists: `∃x:A. B[x]` seq-append: `seq-append(n;m;s1;s2)` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` less_than: `a < b` bfalse: `ff` sq_type: `SQType(T)` bnot: `¬bb` ifthenelse: `if b then t else f fi ` assert: `↑b`
Lemmas referenced :  less_than_transitivity1 less_than_irreflexivity int_seg_wf bar_recursion_wf false_wf le_wf nat_wf subtype_rel-equal equal_wf iff_weakening_equal top_wf all_wf squash_wf exists_wf subtype_rel_dep_function int_seg_subtype_nat decidable__le not-le-2 sq_stable__le condition-implies-le minus-add minus-one-mul zero-add minus-one-mul-top add-associates add-swap add-commutes add_functionality_wrt_le add-zero le-add-cancel seq-adjoin_wf decidable_wf minus-zero seq-append_wf lt_int_wf bool_wf eqtt_to_assert assert_of_lt_int less_than_wf eqff_to_assert bool_cases_sqequal subtype_base_sq bool_subtype_base iff_transitivity assert_wf bnot_wf not_wf iff_weakening_uiff assert_of_bnot subtract_wf not-lt-2
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation lambdaFormation rename introduction cut functionExtensionality sqequalHypSubstitution setElimination thin productElimination hypothesis extract_by_obid isectElimination hypothesisEquality natural_numberEquality independent_isectElimination independent_functionElimination voidElimination because_Cache sqequalRule dependent_set_memberEquality independent_pairFormation equalityTransitivity equalitySymmetry imageElimination imageMemberEquality baseClosed functionEquality cumulativity applyEquality instantiate lambdaEquality addEquality dependent_functionElimination unionElimination isect_memberEquality voidEquality intEquality minusEquality universeEquality dependent_pairFormation hyp_replacement equalityElimination lessCases sqequalAxiom promote_hyp impliesFunctionality

Latex:
\mforall{}[T:Type].  \mforall{}[R,A:n:\mBbbN{}  {}\mrightarrow{}  (\mBbbN{}n  {}\mrightarrow{}  T)  {}\mrightarrow{}  \mBbbP{}].
((\mforall{}n:\mBbbN{}.  \mforall{}s:\mBbbN{}n  {}\mrightarrow{}  T.    Dec(R[n;s]))
{}\mRightarrow{}  (\mforall{}n:\mBbbN{}.  \mforall{}s:\mBbbN{}n  {}\mrightarrow{}  T.    (R[n;s]  {}\mRightarrow{}  A[n;s]))
{}\mRightarrow{}  (\mforall{}n:\mBbbN{}.  \mforall{}s:\mBbbN{}n  {}\mrightarrow{}  T.    ((\mforall{}t:T.  A[n  +  1;s++t])  {}\mRightarrow{}  A[n;s]))
{}\mRightarrow{}  (\mforall{}alpha:\mBbbN{}  {}\mrightarrow{}  T.  (\mdownarrow{}\mexists{}m:\mBbbN{}.  R[m;alpha]))
{}\mRightarrow{}  (\mforall{}x:Top.  A[0;x]))

Date html generated: 2017_04_14-AM-07_27_19
Last ObjectModification: 2017_02_27-PM-02_56_36

Theory : bar-induction

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