### Nuprl Lemma : mutual-primitive-recursion

`∀[A,B:Type].`
`  ∀f0:A. ∀g0:B. ∀F:ℕ ⟶ A ⟶ B ⟶ A. ∀G:ℕ ⟶ A ⟶ B ⟶ B.`
`    ∃f:ℕ ⟶ A`
`     ∃g:ℕ ⟶ B`
`      (((f 0) = f0 ∈ A)`
`      ∧ ((g 0) = g0 ∈ B)`
`      ∧ (∀s:ℕ+. (((f s) = F[s - 1;f (s - 1);g (s - 1)] ∈ A) ∧ ((g s) = G[s - 1;f (s - 1);g (s - 1)] ∈ B))))`

Proof

Definitions occuring in Statement :  nat_plus: `ℕ+` nat: `ℕ` uall: `∀[x:A]. B[x]` so_apply: `x[s1;s2;s3]` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` and: `P ∧ Q` apply: `f a` function: `x:A ⟶ B[x]` subtract: `n - m` natural_number: `\$n` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` member: `t ∈ T` so_apply: `x[s1;s2;s3]` subtype_rel: `A ⊆r B` nat: `ℕ` uimplies: `b supposing a` le: `A ≤ B` and: `P ∧ Q` less_than': `less_than'(a;b)` false: `False` not: `¬A` implies: `P `` Q` prop: `ℙ` top: `Top` so_lambda: `λ2x.t[x]` so_apply: `x[s]` pi1: `fst(t)` pi2: `snd(t)` cand: `A c∧ B` nat_plus: `ℕ+` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` uiff: `uiff(P;Q)` ifthenelse: `if b then t else f fi ` bfalse: `ff` iff: `P `⇐⇒` Q` rev_implies: `P `` Q`
Lemmas referenced :  primrec_wf int_seg_subtype_nat false_wf int_seg_wf pi1_wf_top equal_wf nat_wf pi2_wf primrec0_lemma primrec-unroll nat_plus_wf and_wf le_wf all_wf nat_plus_subtype_nat subtract_wf nat_plus_properties decidable__le satisfiable-full-omega-tt intformand_wf intformnot_wf intformle_wf itermConstant_wf itermSubtract_wf itermVar_wf intformless_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_subtract_lemma int_term_value_var_lemma int_formula_prop_less_lemma int_formula_prop_wf eq_int_wf bool_wf equal-wf-T-base assert_wf intformeq_wf int_formula_prop_eq_lemma bnot_wf not_wf uiff_transitivity eqtt_to_assert assert_of_eq_int iff_transitivity iff_weakening_uiff eqff_to_assert assert_of_bnot exists_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation lambdaFormation dependent_pairFormation lambdaEquality cut introduction extract_by_obid sqequalHypSubstitution isectElimination thin because_Cache productElimination sqequalRule independent_pairEquality applyEquality hypothesisEquality natural_numberEquality setElimination rename hypothesis independent_isectElimination independent_pairFormation functionExtensionality productEquality cumulativity isect_memberEquality voidElimination voidEquality equalityTransitivity equalitySymmetry dependent_functionElimination independent_functionElimination axiomEquality dependent_set_memberEquality unionElimination int_eqEquality intEquality computeAll functionEquality universeEquality baseClosed equalityElimination impliesFunctionality

Latex:
\mforall{}[A,B:Type].
\mforall{}f0:A.  \mforall{}g0:B.  \mforall{}F:\mBbbN{}  {}\mrightarrow{}  A  {}\mrightarrow{}  B  {}\mrightarrow{}  A.  \mforall{}G:\mBbbN{}  {}\mrightarrow{}  A  {}\mrightarrow{}  B  {}\mrightarrow{}  B.
\mexists{}f:\mBbbN{}  {}\mrightarrow{}  A
\mexists{}g:\mBbbN{}  {}\mrightarrow{}  B
(((f  0)  =  f0)
\mwedge{}  ((g  0)  =  g0)
\mwedge{}  (\mforall{}s:\mBbbN{}\msupplus{}.  (((f  s)  =  F[s  -  1;f  (s  -  1);g  (s  -  1)])  \mwedge{}  ((g  s)  =  G[s  -  1;f  (s  -  1);g  (s  -  1)]))))

Date html generated: 2018_05_21-PM-07_42_41
Last ObjectModification: 2017_07_26-PM-05_20_44

Theory : general

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