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16 changes: 14 additions & 2 deletions Incompleteness/Arith/First.lean
Original file line number Diff line number Diff line change
@@ -1,5 +1,17 @@
import Incompleteness.Arith.D1


namespace List.Vector

variable {α : Type*}

lemma cons_get {x : α} : (x ::ᵥ List.Vector.nil).get = ![x] := by
ext i;
simp;

end List.Vector


namespace LO.FirstOrder

namespace Arith
Expand All @@ -13,8 +25,8 @@ lemma re_iff_sigma1 {P : ℕ → Prop} : RePred P ↔ 𝚺₁-Predicate P := by
intro v; symm; simp; simpa [←Matrix.constant_eq_singleton'] using codeOfRePred_spec h (x := v 0)⟩
· rintro ⟨φ, hφ⟩
have := (sigma1_re id (φ.sigma_prop)).comp
(f := fun x : ℕ ↦ x ::ᵥ Mathlib.Vector.nil) (Primrec.to_comp <| Primrec.vector_cons.comp .id (.const _))
exact this.of_eq <| by intro x; symm; simpa [Mathlib.Vector.cons_get] using hφ ![x]
(f := fun x : ℕ ↦ x ::ᵥ List.Vector.nil) (Primrec.to_comp <| Primrec.vector_cons.comp .id (.const _))
exact this.of_eq <| by intro x; symm; simpa [List.Vector.cons_get] using hφ ![x];

variable (T : Theory ℒₒᵣ) [𝐑₀ ≼ T] [Sigma1Sound T] [T.Delta1Definable]

Expand Down
30 changes: 15 additions & 15 deletions lake-manifest.json
Original file line number Diff line number Diff line change
Expand Up @@ -5,7 +5,7 @@
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Expand All @@ -15,7 +15,7 @@
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Expand All @@ -25,7 +25,7 @@
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Expand All @@ -35,17 +35,17 @@
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Expand All @@ -55,27 +55,27 @@
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Expand All @@ -85,17 +85,17 @@
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Expand All @@ -105,7 +105,7 @@
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Expand Down
2 changes: 1 addition & 1 deletion lean-toolchain
Original file line number Diff line number Diff line change
@@ -1 +1 @@
leanprover/lean4:v4.14.0-rc2
leanprover/lean4:v4.15.0-rc1
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