### Nuprl Lemma : gen-bar-rec

`∀P:n:ℕ ⟶ (ℕn ⟶ ℕ) ⟶ ℙ`
`  ((∀n:ℕ. ∀s:ℕn ⟶ ℕ.  ((∀m:ℕ. P[n + 1;s.m@n]) `` P[n;s])) `` (∀f:ℕ ⟶ ℕ. ⇃(∃n:ℕ. ∀m:{n...}. P[m;f])) `` ⇃(P[0;λx.⊥]))`

Proof

Definitions occuring in Statement :  quotient: `x,y:A//B[x; y]` seq-add: `s.x@n` int_upper: `{i...}` int_seg: `{i..j-}` nat: `ℕ` bottom: `⊥` prop: `ℙ` so_apply: `x[s1;s2]` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` implies: `P `` Q` true: `True` lambda: `λx.A[x]` function: `x:A ⟶ B[x]` add: `n + m` natural_number: `\$n`
Definitions unfolded in proof :  or: `P ∨ Q` decidable: `Dec(P)` ge: `i ≥ j ` top: `Top` exists: `∃x:A. B[x]` satisfiable_int_formula: `satisfiable_int_formula(fmla)` lelt: `i ≤ j < k` guard: `{T}` int_seg: `{i..j-}` prop: `ℙ` not: `¬A` false: `False` less_than': `less_than'(a;b)` and: `P ∧ Q` le: `A ≤ B` uimplies: `b supposing a` int_upper: `{i...}` so_apply: `x[s]` subtype_rel: `A ⊆r B` so_apply: `x[s1;s2]` so_lambda: `λ2x.t[x]` nat: `ℕ` uall: `∀[x:A]. B[x]` member: `t ∈ T` so_lambda: `λ2x y.t[x; y]` implies: `P `` Q` all: `∀x:A. B[x]` squash: `↓T` isl: `isl(x)` true: `True` btrue: `tt` ifthenelse: `if b then t else f fi ` assert: `↑b` bfalse: `ff` sq_type: `SQType(T)` uiff: `uiff(P;Q)` rev_implies: `P `` Q` iff: `P `⇐⇒` Q` spector-bar-rec: `spector-bar-rec(Y;G;H;n;s)` bnot: `¬bb` it: `⋅` unit: `Unit` bool: `𝔹` ext2Baire: `ext2Baire(n;f;d)` seq-add: `s.x@n`
Lemmas referenced :  seq-add_wf int_term_value_add_lemma int_formula_prop_not_lemma itermAdd_wf intformnot_wf decidable__le nat_properties equiv_rel_true true_wf quotient_wf int_formula_prop_wf int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_less_lemma int_formula_prop_and_lemma intformle_wf itermConstant_wf itermVar_wf intformless_wf intformand_wf satisfiable-full-omega-tt int_seg_properties isl_wf assert_wf equal_wf unit_wf2 le_wf ext2Baire_wf exists_wf implies-quotient-true subtype_rel_self false_wf int_seg_subtype_nat int_seg_wf nat_wf subtype_rel_dep_function int_upper_subtype_nat int_upper_wf all_wf strong-continuity-rel-unique-pair decidable__equal_int btrue_wf bfalse_wf seq-normalize_wf seq-normalize-equal iff_wf assert_of_le_int le_int_wf iff_imp_equal_bool bool_subtype_base bool_wf subtype_base_sq less_than_wf assert-bnot bool_cases_sqequal eqff_to_assert assert_of_lt_int eqtt_to_assert lt_int_wf
Rules used in proof :  cumulativity universeEquality unionElimination addEquality independent_functionElimination computeAll voidEquality voidElimination isect_memberEquality intEquality int_eqEquality dependent_pairFormation productElimination dependent_pairEquality inlEquality dependent_set_memberEquality productEquality unionEquality functionEquality independent_pairFormation independent_isectElimination natural_numberEquality hypothesisEquality functionExtensionality applyEquality hypothesis because_Cache setElimination isectElimination lambdaEquality sqequalRule thin dependent_functionElimination sqequalHypSubstitution extract_by_obid introduction cut rename lambdaFormation sqequalReflexivity computationStep sqequalTransitivity sqequalSubstitution applyLambdaEquality hyp_replacement equalitySymmetry equalityTransitivity imageElimination SquashedBarInduction Error :inhabitedIsType,  Error :lambdaFormation_alt,  Error :equalityIstype,  baseClosed imageMemberEquality impliesFunctionality addLevel instantiate promote_hyp equalityElimination

Latex:
\mforall{}P:n:\mBbbN{}  {}\mrightarrow{}  (\mBbbN{}n  {}\mrightarrow{}  \mBbbN{})  {}\mrightarrow{}  \mBbbP{}
((\mforall{}n:\mBbbN{}.  \mforall{}s:\mBbbN{}n  {}\mrightarrow{}  \mBbbN{}.    ((\mforall{}m:\mBbbN{}.  P[n  +  1;s.m@n])  {}\mRightarrow{}  P[n;s]))
{}\mRightarrow{}  (\mforall{}f:\mBbbN{}  {}\mrightarrow{}  \mBbbN{}.  \00D9(\mexists{}n:\mBbbN{}.  \mforall{}m:\{n...\}.  P[m;f]))
{}\mRightarrow{}  \00D9(P[0;\mlambda{}x.\mbot{}]))

Date html generated: 2019_06_20-PM-03_07_08
Last ObjectModification: 2019_01_15-PM-03_07_44

Theory : continuity

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