state stringlengths 0 159k | srcUpToTactic stringlengths 387 167k | nextTactic stringlengths 3 9k | declUpToTactic stringlengths 22 11.5k | declId stringlengths 38 95 | decl stringlengths 16 1.89k | file_tag stringlengths 17 73 |
|---|---|---|---|---|---|---|
case succ.succ.succ.succ
α : Type u_1
σ : Type u_2
inst✝¹ : Primcodable α
inst✝ : Primcodable σ
c : α → Code
hc : Primrec c
z : α → σ
hz : Primrec z
s : α → σ
hs : Primrec s
l : α → σ
hl : Primrec l
r : α → σ
hr : Primrec r
pr : α → Code → Code → σ → σ → σ
hpr : Primrec fun a => pr a.1 a.2.1 a.2.2.1 a.2.2.2.1 a.2.2.2.2... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | have hm : m < n + 4 := by
simp only [div2_val]
exact
lt_of_le_of_lt (le_trans (Nat.div_le_self _ _) (Nat.div_le_self _ _))
(Nat.succ_le_succ (Nat.le_add_right _ _)) | /-- Recursion on `Nat.Partrec.Code` is primitive recursive. -/
theorem rec_prim {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Primrec c) {z : α → σ}
(hz : Primrec z) {s : α → σ} (hs : Primrec s) {l : α → σ} (hl : Primrec l) {r : α → σ}
(hr : Primrec r) {pr : α → Code → Code → σ → σ → σ}
(hpr : ... | Mathlib.Computability.PartrecCode.385_0.A3c3Aev6SyIRjCJ | /-- Recursion on `Nat.Partrec.Code` is primitive recursive. -/
theorem rec_prim {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Primrec c) {z : α → σ}
(hz : Primrec z) {s : α → σ} (hs : Primrec s) {l : α → σ} (hl : Primrec l) {r : α → σ}
(hr : Primrec r) {pr : α → Code → Code → σ → σ → σ}
(hpr : ... | Mathlib_Computability_PartrecCode |
α : Type u_1
σ : Type u_2
inst✝¹ : Primcodable α
inst✝ : Primcodable σ
c : α → Code
hc : Primrec c
z : α → σ
hz : Primrec z
s : α → σ
hs : Primrec s
l : α → σ
hl : Primrec l
r : α → σ
hr : Primrec r
pr : α → Code → Code → σ → σ → σ
hpr : Primrec fun a => pr a.1 a.2.1 a.2.2.1 a.2.2.2.1 a.2.2.2.2
co : α → Code → Code → σ... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | simp only [div2_val] | /-- Recursion on `Nat.Partrec.Code` is primitive recursive. -/
theorem rec_prim {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Primrec c) {z : α → σ}
(hz : Primrec z) {s : α → σ} (hs : Primrec s) {l : α → σ} (hl : Primrec l) {r : α → σ}
(hr : Primrec r) {pr : α → Code → Code → σ → σ → σ}
(hpr : ... | Mathlib.Computability.PartrecCode.385_0.A3c3Aev6SyIRjCJ | /-- Recursion on `Nat.Partrec.Code` is primitive recursive. -/
theorem rec_prim {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Primrec c) {z : α → σ}
(hz : Primrec z) {s : α → σ} (hs : Primrec s) {l : α → σ} (hl : Primrec l) {r : α → σ}
(hr : Primrec r) {pr : α → Code → Code → σ → σ → σ}
(hpr : ... | Mathlib_Computability_PartrecCode |
α : Type u_1
σ : Type u_2
inst✝¹ : Primcodable α
inst✝ : Primcodable σ
c : α → Code
hc : Primrec c
z : α → σ
hz : Primrec z
s : α → σ
hs : Primrec s
l : α → σ
hl : Primrec l
r : α → σ
hr : Primrec r
pr : α → Code → Code → σ → σ → σ
hpr : Primrec fun a => pr a.1 a.2.1 a.2.2.1 a.2.2.2.1 a.2.2.2.2
co : α → Code → Code → σ... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | exact
lt_of_le_of_lt (le_trans (Nat.div_le_self _ _) (Nat.div_le_self _ _))
(Nat.succ_le_succ (Nat.le_add_right _ _)) | /-- Recursion on `Nat.Partrec.Code` is primitive recursive. -/
theorem rec_prim {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Primrec c) {z : α → σ}
(hz : Primrec z) {s : α → σ} (hs : Primrec s) {l : α → σ} (hl : Primrec l) {r : α → σ}
(hr : Primrec r) {pr : α → Code → Code → σ → σ → σ}
(hpr : ... | Mathlib.Computability.PartrecCode.385_0.A3c3Aev6SyIRjCJ | /-- Recursion on `Nat.Partrec.Code` is primitive recursive. -/
theorem rec_prim {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Primrec c) {z : α → σ}
(hz : Primrec z) {s : α → σ} (hs : Primrec s) {l : α → σ} (hl : Primrec l) {r : α → σ}
(hr : Primrec r) {pr : α → Code → Code → σ → σ → σ}
(hpr : ... | Mathlib_Computability_PartrecCode |
case succ.succ.succ.succ
α : Type u_1
σ : Type u_2
inst✝¹ : Primcodable α
inst✝ : Primcodable σ
c : α → Code
hc : Primrec c
z : α → σ
hz : Primrec z
s : α → σ
hs : Primrec s
l : α → σ
hl : Primrec l
r : α → σ
hr : Primrec r
pr : α → Code → Code → σ → σ → σ
hpr : Primrec fun a => pr a.1 a.2.1 a.2.2.1 a.2.2.2.1 a.2.2.2.2... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | have m1 : m.unpair.1 < n + 4 := lt_of_le_of_lt m.unpair_left_le hm | /-- Recursion on `Nat.Partrec.Code` is primitive recursive. -/
theorem rec_prim {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Primrec c) {z : α → σ}
(hz : Primrec z) {s : α → σ} (hs : Primrec s) {l : α → σ} (hl : Primrec l) {r : α → σ}
(hr : Primrec r) {pr : α → Code → Code → σ → σ → σ}
(hpr : ... | Mathlib.Computability.PartrecCode.385_0.A3c3Aev6SyIRjCJ | /-- Recursion on `Nat.Partrec.Code` is primitive recursive. -/
theorem rec_prim {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Primrec c) {z : α → σ}
(hz : Primrec z) {s : α → σ} (hs : Primrec s) {l : α → σ} (hl : Primrec l) {r : α → σ}
(hr : Primrec r) {pr : α → Code → Code → σ → σ → σ}
(hpr : ... | Mathlib_Computability_PartrecCode |
case succ.succ.succ.succ
α : Type u_1
σ : Type u_2
inst✝¹ : Primcodable α
inst✝ : Primcodable σ
c : α → Code
hc : Primrec c
z : α → σ
hz : Primrec z
s : α → σ
hs : Primrec s
l : α → σ
hl : Primrec l
r : α → σ
hr : Primrec r
pr : α → Code → Code → σ → σ → σ
hpr : Primrec fun a => pr a.1 a.2.1 a.2.2.1 a.2.2.2.1 a.2.2.2.2... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | have m2 : m.unpair.2 < n + 4 := lt_of_le_of_lt m.unpair_right_le hm | /-- Recursion on `Nat.Partrec.Code` is primitive recursive. -/
theorem rec_prim {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Primrec c) {z : α → σ}
(hz : Primrec z) {s : α → σ} (hs : Primrec s) {l : α → σ} (hl : Primrec l) {r : α → σ}
(hr : Primrec r) {pr : α → Code → Code → σ → σ → σ}
(hpr : ... | Mathlib.Computability.PartrecCode.385_0.A3c3Aev6SyIRjCJ | /-- Recursion on `Nat.Partrec.Code` is primitive recursive. -/
theorem rec_prim {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Primrec c) {z : α → σ}
(hz : Primrec z) {s : α → σ} (hs : Primrec s) {l : α → σ} (hl : Primrec l) {r : α → σ}
(hr : Primrec r) {pr : α → Code → Code → σ → σ → σ}
(hpr : ... | Mathlib_Computability_PartrecCode |
case succ.succ.succ.succ
α : Type u_1
σ : Type u_2
inst✝¹ : Primcodable α
inst✝ : Primcodable σ
c : α → Code
hc : Primrec c
z : α → σ
hz : Primrec z
s : α → σ
hs : Primrec s
l : α → σ
hl : Primrec l
r : α → σ
hr : Primrec r
pr : α → Code → Code → σ → σ → σ
hpr : Primrec fun a => pr a.1 a.2.1 a.2.2.1 a.2.2.2.1 a.2.2.2.2... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | simp [List.get?_map, List.get?_range, hm, m1, m2] | /-- Recursion on `Nat.Partrec.Code` is primitive recursive. -/
theorem rec_prim {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Primrec c) {z : α → σ}
(hz : Primrec z) {s : α → σ} (hs : Primrec s) {l : α → σ} (hl : Primrec l) {r : α → σ}
(hr : Primrec r) {pr : α → Code → Code → σ → σ → σ}
(hpr : ... | Mathlib.Computability.PartrecCode.385_0.A3c3Aev6SyIRjCJ | /-- Recursion on `Nat.Partrec.Code` is primitive recursive. -/
theorem rec_prim {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Primrec c) {z : α → σ}
(hz : Primrec z) {s : α → σ} (hs : Primrec s) {l : α → σ} (hl : Primrec l) {r : α → σ}
(hr : Primrec r) {pr : α → Code → Code → σ → σ → σ}
(hpr : ... | Mathlib_Computability_PartrecCode |
case succ.succ.succ.succ
α : Type u_1
σ : Type u_2
inst✝¹ : Primcodable α
inst✝ : Primcodable σ
c : α → Code
hc : Primrec c
z : α → σ
hz : Primrec z
s : α → σ
hs : Primrec s
l : α → σ
hl : Primrec l
r : α → σ
hr : Primrec r
pr : α → Code → Code → σ → σ → σ
hpr : Primrec fun a => pr a.1 a.2.1 a.2.2.1 a.2.2.2.1 a.2.2.2.2... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | rw [show ofNat Code (n + 4) = ofNatCode (n + 4) from rfl] | /-- Recursion on `Nat.Partrec.Code` is primitive recursive. -/
theorem rec_prim {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Primrec c) {z : α → σ}
(hz : Primrec z) {s : α → σ} (hs : Primrec s) {l : α → σ} (hl : Primrec l) {r : α → σ}
(hr : Primrec r) {pr : α → Code → Code → σ → σ → σ}
(hpr : ... | Mathlib.Computability.PartrecCode.385_0.A3c3Aev6SyIRjCJ | /-- Recursion on `Nat.Partrec.Code` is primitive recursive. -/
theorem rec_prim {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Primrec c) {z : α → σ}
(hz : Primrec z) {s : α → σ} (hs : Primrec s) {l : α → σ} (hl : Primrec l) {r : α → σ}
(hr : Primrec r) {pr : α → Code → Code → σ → σ → σ}
(hpr : ... | Mathlib_Computability_PartrecCode |
case succ.succ.succ.succ
α : Type u_1
σ : Type u_2
inst✝¹ : Primcodable α
inst✝ : Primcodable σ
c : α → Code
hc : Primrec c
z : α → σ
hz : Primrec z
s : α → σ
hs : Primrec s
l : α → σ
hl : Primrec l
r : α → σ
hr : Primrec r
pr : α → Code → Code → σ → σ → σ
hpr : Primrec fun a => pr a.1 a.2.1 a.2.2.1 a.2.2.2.1 a.2.2.2.2... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | simp [ofNatCode] | /-- Recursion on `Nat.Partrec.Code` is primitive recursive. -/
theorem rec_prim {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Primrec c) {z : α → σ}
(hz : Primrec z) {s : α → σ} (hs : Primrec s) {l : α → σ} (hl : Primrec l) {r : α → σ}
(hr : Primrec r) {pr : α → Code → Code → σ → σ → σ}
(hpr : ... | Mathlib.Computability.PartrecCode.385_0.A3c3Aev6SyIRjCJ | /-- Recursion on `Nat.Partrec.Code` is primitive recursive. -/
theorem rec_prim {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Primrec c) {z : α → σ}
(hz : Primrec z) {s : α → σ} (hs : Primrec s) {l : α → σ} (hl : Primrec l) {r : α → σ}
(hr : Primrec r) {pr : α → Code → Code → σ → σ → σ}
(hpr : ... | Mathlib_Computability_PartrecCode |
case succ.succ.succ.succ
α : Type u_1
σ : Type u_2
inst✝¹ : Primcodable α
inst✝ : Primcodable σ
c : α → Code
hc : Primrec c
z : α → σ
hz : Primrec z
s : α → σ
hs : Primrec s
l : α → σ
hl : Primrec l
r : α → σ
hr : Primrec r
pr : α → Code → Code → σ → σ → σ
hpr : Primrec fun a => pr a.1 a.2.1 a.2.2.1 a.2.2.2.1 a.2.2.2.2... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | cases n.bodd | /-- Recursion on `Nat.Partrec.Code` is primitive recursive. -/
theorem rec_prim {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Primrec c) {z : α → σ}
(hz : Primrec z) {s : α → σ} (hs : Primrec s) {l : α → σ} (hl : Primrec l) {r : α → σ}
(hr : Primrec r) {pr : α → Code → Code → σ → σ → σ}
(hpr : ... | Mathlib.Computability.PartrecCode.385_0.A3c3Aev6SyIRjCJ | /-- Recursion on `Nat.Partrec.Code` is primitive recursive. -/
theorem rec_prim {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Primrec c) {z : α → σ}
(hz : Primrec z) {s : α → σ} (hs : Primrec s) {l : α → σ} (hl : Primrec l) {r : α → σ}
(hr : Primrec r) {pr : α → Code → Code → σ → σ → σ}
(hpr : ... | Mathlib_Computability_PartrecCode |
case succ.succ.succ.succ.false
α : Type u_1
σ : Type u_2
inst✝¹ : Primcodable α
inst✝ : Primcodable σ
c : α → Code
hc : Primrec c
z : α → σ
hz : Primrec z
s : α → σ
hs : Primrec s
l : α → σ
hl : Primrec l
r : α → σ
hr : Primrec r
pr : α → Code → Code → σ → σ → σ
hpr : Primrec fun a => pr a.1 a.2.1 a.2.2.1 a.2.2.2.1 a.2... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | cases n.div2.bodd | /-- Recursion on `Nat.Partrec.Code` is primitive recursive. -/
theorem rec_prim {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Primrec c) {z : α → σ}
(hz : Primrec z) {s : α → σ} (hs : Primrec s) {l : α → σ} (hl : Primrec l) {r : α → σ}
(hr : Primrec r) {pr : α → Code → Code → σ → σ → σ}
(hpr : ... | Mathlib.Computability.PartrecCode.385_0.A3c3Aev6SyIRjCJ | /-- Recursion on `Nat.Partrec.Code` is primitive recursive. -/
theorem rec_prim {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Primrec c) {z : α → σ}
(hz : Primrec z) {s : α → σ} (hs : Primrec s) {l : α → σ} (hl : Primrec l) {r : α → σ}
(hr : Primrec r) {pr : α → Code → Code → σ → σ → σ}
(hpr : ... | Mathlib_Computability_PartrecCode |
case succ.succ.succ.succ.true
α : Type u_1
σ : Type u_2
inst✝¹ : Primcodable α
inst✝ : Primcodable σ
c : α → Code
hc : Primrec c
z : α → σ
hz : Primrec z
s : α → σ
hs : Primrec s
l : α → σ
hl : Primrec l
r : α → σ
hr : Primrec r
pr : α → Code → Code → σ → σ → σ
hpr : Primrec fun a => pr a.1 a.2.1 a.2.2.1 a.2.2.2.1 a.2.... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | cases n.div2.bodd | /-- Recursion on `Nat.Partrec.Code` is primitive recursive. -/
theorem rec_prim {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Primrec c) {z : α → σ}
(hz : Primrec z) {s : α → σ} (hs : Primrec s) {l : α → σ} (hl : Primrec l) {r : α → σ}
(hr : Primrec r) {pr : α → Code → Code → σ → σ → σ}
(hpr : ... | Mathlib.Computability.PartrecCode.385_0.A3c3Aev6SyIRjCJ | /-- Recursion on `Nat.Partrec.Code` is primitive recursive. -/
theorem rec_prim {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Primrec c) {z : α → σ}
(hz : Primrec z) {s : α → σ} (hs : Primrec s) {l : α → σ} (hl : Primrec l) {r : α → σ}
(hr : Primrec r) {pr : α → Code → Code → σ → σ → σ}
(hpr : ... | Mathlib_Computability_PartrecCode |
case succ.succ.succ.succ.false.false
α : Type u_1
σ : Type u_2
inst✝¹ : Primcodable α
inst✝ : Primcodable σ
c : α → Code
hc : Primrec c
z : α → σ
hz : Primrec z
s : α → σ
hs : Primrec s
l : α → σ
hl : Primrec l
r : α → σ
hr : Primrec r
pr : α → Code → Code → σ → σ → σ
hpr : Primrec fun a => pr a.1 a.2.1 a.2.2.1 a.2.2.2... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | rfl | /-- Recursion on `Nat.Partrec.Code` is primitive recursive. -/
theorem rec_prim {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Primrec c) {z : α → σ}
(hz : Primrec z) {s : α → σ} (hs : Primrec s) {l : α → σ} (hl : Primrec l) {r : α → σ}
(hr : Primrec r) {pr : α → Code → Code → σ → σ → σ}
(hpr : ... | Mathlib.Computability.PartrecCode.385_0.A3c3Aev6SyIRjCJ | /-- Recursion on `Nat.Partrec.Code` is primitive recursive. -/
theorem rec_prim {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Primrec c) {z : α → σ}
(hz : Primrec z) {s : α → σ} (hs : Primrec s) {l : α → σ} (hl : Primrec l) {r : α → σ}
(hr : Primrec r) {pr : α → Code → Code → σ → σ → σ}
(hpr : ... | Mathlib_Computability_PartrecCode |
case succ.succ.succ.succ.false.true
α : Type u_1
σ : Type u_2
inst✝¹ : Primcodable α
inst✝ : Primcodable σ
c : α → Code
hc : Primrec c
z : α → σ
hz : Primrec z
s : α → σ
hs : Primrec s
l : α → σ
hl : Primrec l
r : α → σ
hr : Primrec r
pr : α → Code → Code → σ → σ → σ
hpr : Primrec fun a => pr a.1 a.2.1 a.2.2.1 a.2.2.2.... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | rfl | /-- Recursion on `Nat.Partrec.Code` is primitive recursive. -/
theorem rec_prim {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Primrec c) {z : α → σ}
(hz : Primrec z) {s : α → σ} (hs : Primrec s) {l : α → σ} (hl : Primrec l) {r : α → σ}
(hr : Primrec r) {pr : α → Code → Code → σ → σ → σ}
(hpr : ... | Mathlib.Computability.PartrecCode.385_0.A3c3Aev6SyIRjCJ | /-- Recursion on `Nat.Partrec.Code` is primitive recursive. -/
theorem rec_prim {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Primrec c) {z : α → σ}
(hz : Primrec z) {s : α → σ} (hs : Primrec s) {l : α → σ} (hl : Primrec l) {r : α → σ}
(hr : Primrec r) {pr : α → Code → Code → σ → σ → σ}
(hpr : ... | Mathlib_Computability_PartrecCode |
case succ.succ.succ.succ.true.false
α : Type u_1
σ : Type u_2
inst✝¹ : Primcodable α
inst✝ : Primcodable σ
c : α → Code
hc : Primrec c
z : α → σ
hz : Primrec z
s : α → σ
hs : Primrec s
l : α → σ
hl : Primrec l
r : α → σ
hr : Primrec r
pr : α → Code → Code → σ → σ → σ
hpr : Primrec fun a => pr a.1 a.2.1 a.2.2.1 a.2.2.2.... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | rfl | /-- Recursion on `Nat.Partrec.Code` is primitive recursive. -/
theorem rec_prim {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Primrec c) {z : α → σ}
(hz : Primrec z) {s : α → σ} (hs : Primrec s) {l : α → σ} (hl : Primrec l) {r : α → σ}
(hr : Primrec r) {pr : α → Code → Code → σ → σ → σ}
(hpr : ... | Mathlib.Computability.PartrecCode.385_0.A3c3Aev6SyIRjCJ | /-- Recursion on `Nat.Partrec.Code` is primitive recursive. -/
theorem rec_prim {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Primrec c) {z : α → σ}
(hz : Primrec z) {s : α → σ} (hs : Primrec s) {l : α → σ} (hl : Primrec l) {r : α → σ}
(hr : Primrec r) {pr : α → Code → Code → σ → σ → σ}
(hpr : ... | Mathlib_Computability_PartrecCode |
case succ.succ.succ.succ.true.true
α : Type u_1
σ : Type u_2
inst✝¹ : Primcodable α
inst✝ : Primcodable σ
c : α → Code
hc : Primrec c
z : α → σ
hz : Primrec z
s : α → σ
hs : Primrec s
l : α → σ
hl : Primrec l
r : α → σ
hr : Primrec r
pr : α → Code → Code → σ → σ → σ
hpr : Primrec fun a => pr a.1 a.2.1 a.2.2.1 a.2.2.2.1... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | rfl | /-- Recursion on `Nat.Partrec.Code` is primitive recursive. -/
theorem rec_prim {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Primrec c) {z : α → σ}
(hz : Primrec z) {s : α → σ} (hs : Primrec s) {l : α → σ} (hl : Primrec l) {r : α → σ}
(hr : Primrec r) {pr : α → Code → Code → σ → σ → σ}
(hpr : ... | Mathlib.Computability.PartrecCode.385_0.A3c3Aev6SyIRjCJ | /-- Recursion on `Nat.Partrec.Code` is primitive recursive. -/
theorem rec_prim {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Primrec c) {z : α → σ}
(hz : Primrec z) {s : α → σ} (hs : Primrec s) {l : α → σ} (hl : Primrec l) {r : α → σ}
(hr : Primrec r) {pr : α → Code → Code → σ → σ → σ}
(hpr : ... | Mathlib_Computability_PartrecCode |
α : Type u_1
σ : Type u_2
inst✝¹ : Primcodable α
inst✝ : Primcodable σ
c : α → Code
hc : Computable c
z : α → σ
hz : Computable z
s : α → σ
hs : Computable s
l : α → σ
hl : Computable l
r : α → σ
hr : Computable r
pr : α → Code × Code × σ × σ → σ
hpr : Computable₂ pr
co : α → Code × Code × σ × σ → σ
hco : Computable₂ c... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | intros _ _ _ _ F | /-- Recursion on `Nat.Partrec.Code` is computable. -/
theorem rec_computable {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Computable c)
{z : α → σ} (hz : Computable z) {s : α → σ} (hs : Computable s) {l : α → σ} (hl : Computable l)
{r : α → σ} (hr : Computable r) {pr : α → Code × Code × σ × σ → σ}... | Mathlib.Computability.PartrecCode.498_0.A3c3Aev6SyIRjCJ | /-- Recursion on `Nat.Partrec.Code` is computable. -/
theorem rec_computable {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Computable c)
{z : α → σ} (hz : Computable z) {s : α → σ} (hs : Computable s) {l : α → σ} (hl : Computable l)
{r : α → σ} (hr : Computable r) {pr : α → Code × Code × σ × σ → σ}... | Mathlib_Computability_PartrecCode |
α : Type u_1
σ : Type u_2
inst✝¹ : Primcodable α
inst✝ : Primcodable σ
c : α → Code
hc : Computable c
z : α → σ
hz : Computable z
s : α → σ
hs : Computable s
l : α → σ
hl : Computable l
r : α → σ
hr : Computable r
pr : α → Code × Code × σ × σ → σ
hpr : Computable₂ pr
co : α → Code × Code × σ × σ → σ
hco : Computable₂ c... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | let G₁ : (α × List σ) × ℕ × ℕ → Option σ := fun p =>
let a := p.1.1
let IH := p.1.2
let n := p.2.1
let m := p.2.2
(IH.get? m).bind fun s =>
(IH.get? m.unpair.1).bind fun s₁ =>
(IH.get? m.unpair.2).map fun s₂ =>
cond n.bodd
(cond n.div2.bodd (rf a (ofNat Code m, s)... | /-- Recursion on `Nat.Partrec.Code` is computable. -/
theorem rec_computable {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Computable c)
{z : α → σ} (hz : Computable z) {s : α → σ} (hs : Computable s) {l : α → σ} (hl : Computable l)
{r : α → σ} (hr : Computable r) {pr : α → Code × Code × σ × σ → σ}... | Mathlib.Computability.PartrecCode.498_0.A3c3Aev6SyIRjCJ | /-- Recursion on `Nat.Partrec.Code` is computable. -/
theorem rec_computable {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Computable c)
{z : α → σ} (hz : Computable z) {s : α → σ} (hs : Computable s) {l : α → σ} (hl : Computable l)
{r : α → σ} (hr : Computable r) {pr : α → Code × Code × σ × σ → σ}... | Mathlib_Computability_PartrecCode |
α : Type u_1
σ : Type u_2
inst✝¹ : Primcodable α
inst✝ : Primcodable σ
c : α → Code
hc : Computable c
z : α → σ
hz : Computable z
s : α → σ
hs : Computable s
l : α → σ
hl : Computable l
r : α → σ
hr : Computable r
pr : α → Code × Code × σ × σ → σ
hpr : Computable₂ pr
co : α → Code × Code × σ × σ → σ
hco : Computable₂ c... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | have : Computable G₁ := by
refine' option_bind (list_get?.comp (snd.comp fst) (snd.comp snd)) _
unfold Computable₂
refine'
option_bind
((list_get?.comp (snd.comp fst)
(fst.comp <| Computable.unpair.comp (snd.comp snd))).comp fst) _
unfold Computable₂
refine'
option_map
... | /-- Recursion on `Nat.Partrec.Code` is computable. -/
theorem rec_computable {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Computable c)
{z : α → σ} (hz : Computable z) {s : α → σ} (hs : Computable s) {l : α → σ} (hl : Computable l)
{r : α → σ} (hr : Computable r) {pr : α → Code × Code × σ × σ → σ}... | Mathlib.Computability.PartrecCode.498_0.A3c3Aev6SyIRjCJ | /-- Recursion on `Nat.Partrec.Code` is computable. -/
theorem rec_computable {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Computable c)
{z : α → σ} (hz : Computable z) {s : α → σ} (hs : Computable s) {l : α → σ} (hl : Computable l)
{r : α → σ} (hr : Computable r) {pr : α → Code × Code × σ × σ → σ}... | Mathlib_Computability_PartrecCode |
α : Type u_1
σ : Type u_2
inst✝¹ : Primcodable α
inst✝ : Primcodable σ
c : α → Code
hc : Computable c
z : α → σ
hz : Computable z
s : α → σ
hs : Computable s
l : α → σ
hl : Computable l
r : α → σ
hr : Computable r
pr : α → Code × Code × σ × σ → σ
hpr : Computable₂ pr
co : α → Code × Code × σ × σ → σ
hco : Computable₂ c... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | refine' option_bind (list_get?.comp (snd.comp fst) (snd.comp snd)) _ | /-- Recursion on `Nat.Partrec.Code` is computable. -/
theorem rec_computable {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Computable c)
{z : α → σ} (hz : Computable z) {s : α → σ} (hs : Computable s) {l : α → σ} (hl : Computable l)
{r : α → σ} (hr : Computable r) {pr : α → Code × Code × σ × σ → σ}... | Mathlib.Computability.PartrecCode.498_0.A3c3Aev6SyIRjCJ | /-- Recursion on `Nat.Partrec.Code` is computable. -/
theorem rec_computable {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Computable c)
{z : α → σ} (hz : Computable z) {s : α → σ} (hs : Computable s) {l : α → σ} (hl : Computable l)
{r : α → σ} (hr : Computable r) {pr : α → Code × Code × σ × σ → σ}... | Mathlib_Computability_PartrecCode |
α : Type u_1
σ : Type u_2
inst✝¹ : Primcodable α
inst✝ : Primcodable σ
c : α → Code
hc : Computable c
z : α → σ
hz : Computable z
s : α → σ
hs : Computable s
l : α → σ
hl : Computable l
r : α → σ
hr : Computable r
pr : α → Code × Code × σ × σ → σ
hpr : Computable₂ pr
co : α → Code × Code × σ × σ → σ
hco : Computable₂ c... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | unfold Computable₂ | /-- Recursion on `Nat.Partrec.Code` is computable. -/
theorem rec_computable {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Computable c)
{z : α → σ} (hz : Computable z) {s : α → σ} (hs : Computable s) {l : α → σ} (hl : Computable l)
{r : α → σ} (hr : Computable r) {pr : α → Code × Code × σ × σ → σ}... | Mathlib.Computability.PartrecCode.498_0.A3c3Aev6SyIRjCJ | /-- Recursion on `Nat.Partrec.Code` is computable. -/
theorem rec_computable {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Computable c)
{z : α → σ} (hz : Computable z) {s : α → σ} (hs : Computable s) {l : α → σ} (hl : Computable l)
{r : α → σ} (hr : Computable r) {pr : α → Code × Code × σ × σ → σ}... | Mathlib_Computability_PartrecCode |
α : Type u_1
σ : Type u_2
inst✝¹ : Primcodable α
inst✝ : Primcodable σ
c : α → Code
hc : Computable c
z : α → σ
hz : Computable z
s : α → σ
hs : Computable s
l : α → σ
hl : Computable l
r : α → σ
hr : Computable r
pr : α → Code × Code × σ × σ → σ
hpr : Computable₂ pr
co : α → Code × Code × σ × σ → σ
hco : Computable₂ c... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | refine'
option_bind
((list_get?.comp (snd.comp fst)
(fst.comp <| Computable.unpair.comp (snd.comp snd))).comp fst) _ | /-- Recursion on `Nat.Partrec.Code` is computable. -/
theorem rec_computable {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Computable c)
{z : α → σ} (hz : Computable z) {s : α → σ} (hs : Computable s) {l : α → σ} (hl : Computable l)
{r : α → σ} (hr : Computable r) {pr : α → Code × Code × σ × σ → σ}... | Mathlib.Computability.PartrecCode.498_0.A3c3Aev6SyIRjCJ | /-- Recursion on `Nat.Partrec.Code` is computable. -/
theorem rec_computable {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Computable c)
{z : α → σ} (hz : Computable z) {s : α → σ} (hs : Computable s) {l : α → σ} (hl : Computable l)
{r : α → σ} (hr : Computable r) {pr : α → Code × Code × σ × σ → σ}... | Mathlib_Computability_PartrecCode |
α : Type u_1
σ : Type u_2
inst✝¹ : Primcodable α
inst✝ : Primcodable σ
c : α → Code
hc : Computable c
z : α → σ
hz : Computable z
s : α → σ
hs : Computable s
l : α → σ
hl : Computable l
r : α → σ
hr : Computable r
pr : α → Code × Code × σ × σ → σ
hpr : Computable₂ pr
co : α → Code × Code × σ × σ → σ
hco : Computable₂ c... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | unfold Computable₂ | /-- Recursion on `Nat.Partrec.Code` is computable. -/
theorem rec_computable {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Computable c)
{z : α → σ} (hz : Computable z) {s : α → σ} (hs : Computable s) {l : α → σ} (hl : Computable l)
{r : α → σ} (hr : Computable r) {pr : α → Code × Code × σ × σ → σ}... | Mathlib.Computability.PartrecCode.498_0.A3c3Aev6SyIRjCJ | /-- Recursion on `Nat.Partrec.Code` is computable. -/
theorem rec_computable {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Computable c)
{z : α → σ} (hz : Computable z) {s : α → σ} (hs : Computable s) {l : α → σ} (hl : Computable l)
{r : α → σ} (hr : Computable r) {pr : α → Code × Code × σ × σ → σ}... | Mathlib_Computability_PartrecCode |
α : Type u_1
σ : Type u_2
inst✝¹ : Primcodable α
inst✝ : Primcodable σ
c : α → Code
hc : Computable c
z : α → σ
hz : Computable z
s : α → σ
hs : Computable s
l : α → σ
hl : Computable l
r : α → σ
hr : Computable r
pr : α → Code × Code × σ × σ → σ
hpr : Computable₂ pr
co : α → Code × Code × σ × σ → σ
hco : Computable₂ c... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | refine'
option_map
((list_get?.comp (snd.comp fst)
(snd.comp <| Computable.unpair.comp (snd.comp snd))).comp <| fst.comp fst) _ | /-- Recursion on `Nat.Partrec.Code` is computable. -/
theorem rec_computable {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Computable c)
{z : α → σ} (hz : Computable z) {s : α → σ} (hs : Computable s) {l : α → σ} (hl : Computable l)
{r : α → σ} (hr : Computable r) {pr : α → Code × Code × σ × σ → σ}... | Mathlib.Computability.PartrecCode.498_0.A3c3Aev6SyIRjCJ | /-- Recursion on `Nat.Partrec.Code` is computable. -/
theorem rec_computable {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Computable c)
{z : α → σ} (hz : Computable z) {s : α → σ} (hs : Computable s) {l : α → σ} (hl : Computable l)
{r : α → σ} (hr : Computable r) {pr : α → Code × Code × σ × σ → σ}... | Mathlib_Computability_PartrecCode |
α : Type u_1
σ : Type u_2
inst✝¹ : Primcodable α
inst✝ : Primcodable σ
c : α → Code
hc : Computable c
z : α → σ
hz : Computable z
s : α → σ
hs : Computable s
l : α → σ
hl : Computable l
r : α → σ
hr : Computable r
pr : α → Code × Code × σ × σ → σ
hpr : Computable₂ pr
co : α → Code × Code × σ × σ → σ
hco : Computable₂ c... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | have a : Computable (fun p : ((((α × List σ) × ℕ × ℕ) × σ) × σ) × σ => p.1.1.1.1.1) :=
fst.comp (fst.comp <| fst.comp <| fst.comp fst) | /-- Recursion on `Nat.Partrec.Code` is computable. -/
theorem rec_computable {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Computable c)
{z : α → σ} (hz : Computable z) {s : α → σ} (hs : Computable s) {l : α → σ} (hl : Computable l)
{r : α → σ} (hr : Computable r) {pr : α → Code × Code × σ × σ → σ}... | Mathlib.Computability.PartrecCode.498_0.A3c3Aev6SyIRjCJ | /-- Recursion on `Nat.Partrec.Code` is computable. -/
theorem rec_computable {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Computable c)
{z : α → σ} (hz : Computable z) {s : α → σ} (hs : Computable s) {l : α → σ} (hl : Computable l)
{r : α → σ} (hr : Computable r) {pr : α → Code × Code × σ × σ → σ}... | Mathlib_Computability_PartrecCode |
α : Type u_1
σ : Type u_2
inst✝¹ : Primcodable α
inst✝ : Primcodable σ
c : α → Code
hc : Computable c
z : α → σ
hz : Computable z
s : α → σ
hs : Computable s
l : α → σ
hl : Computable l
r : α → σ
hr : Computable r
pr : α → Code × Code × σ × σ → σ
hpr : Computable₂ pr
co : α → Code × Code × σ × σ → σ
hco : Computable₂ c... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | have n : Computable (fun p : ((((α × List σ) × ℕ × ℕ) × σ) × σ) × σ => p.1.1.1.2.1) :=
fst.comp (snd.comp <| fst.comp <| fst.comp fst) | /-- Recursion on `Nat.Partrec.Code` is computable. -/
theorem rec_computable {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Computable c)
{z : α → σ} (hz : Computable z) {s : α → σ} (hs : Computable s) {l : α → σ} (hl : Computable l)
{r : α → σ} (hr : Computable r) {pr : α → Code × Code × σ × σ → σ}... | Mathlib.Computability.PartrecCode.498_0.A3c3Aev6SyIRjCJ | /-- Recursion on `Nat.Partrec.Code` is computable. -/
theorem rec_computable {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Computable c)
{z : α → σ} (hz : Computable z) {s : α → σ} (hs : Computable s) {l : α → σ} (hl : Computable l)
{r : α → σ} (hr : Computable r) {pr : α → Code × Code × σ × σ → σ}... | Mathlib_Computability_PartrecCode |
α : Type u_1
σ : Type u_2
inst✝¹ : Primcodable α
inst✝ : Primcodable σ
c : α → Code
hc : Computable c
z : α → σ
hz : Computable z
s : α → σ
hs : Computable s
l : α → σ
hl : Computable l
r : α → σ
hr : Computable r
pr : α → Code × Code × σ × σ → σ
hpr : Computable₂ pr
co : α → Code × Code × σ × σ → σ
hco : Computable₂ c... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | have m : Computable (fun p : ((((α × List σ) × ℕ × ℕ) × σ) × σ) × σ => p.1.1.1.2.2) :=
snd.comp (snd.comp <| fst.comp <| fst.comp fst) | /-- Recursion on `Nat.Partrec.Code` is computable. -/
theorem rec_computable {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Computable c)
{z : α → σ} (hz : Computable z) {s : α → σ} (hs : Computable s) {l : α → σ} (hl : Computable l)
{r : α → σ} (hr : Computable r) {pr : α → Code × Code × σ × σ → σ}... | Mathlib.Computability.PartrecCode.498_0.A3c3Aev6SyIRjCJ | /-- Recursion on `Nat.Partrec.Code` is computable. -/
theorem rec_computable {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Computable c)
{z : α → σ} (hz : Computable z) {s : α → σ} (hs : Computable s) {l : α → σ} (hl : Computable l)
{r : α → σ} (hr : Computable r) {pr : α → Code × Code × σ × σ → σ}... | Mathlib_Computability_PartrecCode |
α : Type u_1
σ : Type u_2
inst✝¹ : Primcodable α
inst✝ : Primcodable σ
c : α → Code
hc : Computable c
z : α → σ
hz : Computable z
s : α → σ
hs : Computable s
l : α → σ
hl : Computable l
r : α → σ
hr : Computable r
pr : α → Code × Code × σ × σ → σ
hpr : Computable₂ pr
co : α → Code × Code × σ × σ → σ
hco : Computable₂ c... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | have m₁ := fst.comp (Computable.unpair.comp m) | /-- Recursion on `Nat.Partrec.Code` is computable. -/
theorem rec_computable {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Computable c)
{z : α → σ} (hz : Computable z) {s : α → σ} (hs : Computable s) {l : α → σ} (hl : Computable l)
{r : α → σ} (hr : Computable r) {pr : α → Code × Code × σ × σ → σ}... | Mathlib.Computability.PartrecCode.498_0.A3c3Aev6SyIRjCJ | /-- Recursion on `Nat.Partrec.Code` is computable. -/
theorem rec_computable {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Computable c)
{z : α → σ} (hz : Computable z) {s : α → σ} (hs : Computable s) {l : α → σ} (hl : Computable l)
{r : α → σ} (hr : Computable r) {pr : α → Code × Code × σ × σ → σ}... | Mathlib_Computability_PartrecCode |
α : Type u_1
σ : Type u_2
inst✝¹ : Primcodable α
inst✝ : Primcodable σ
c : α → Code
hc : Computable c
z : α → σ
hz : Computable z
s : α → σ
hs : Computable s
l : α → σ
hl : Computable l
r : α → σ
hr : Computable r
pr : α → Code × Code × σ × σ → σ
hpr : Computable₂ pr
co : α → Code × Code × σ × σ → σ
hco : Computable₂ c... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | have m₂ := snd.comp (Computable.unpair.comp m) | /-- Recursion on `Nat.Partrec.Code` is computable. -/
theorem rec_computable {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Computable c)
{z : α → σ} (hz : Computable z) {s : α → σ} (hs : Computable s) {l : α → σ} (hl : Computable l)
{r : α → σ} (hr : Computable r) {pr : α → Code × Code × σ × σ → σ}... | Mathlib.Computability.PartrecCode.498_0.A3c3Aev6SyIRjCJ | /-- Recursion on `Nat.Partrec.Code` is computable. -/
theorem rec_computable {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Computable c)
{z : α → σ} (hz : Computable z) {s : α → σ} (hs : Computable s) {l : α → σ} (hl : Computable l)
{r : α → σ} (hr : Computable r) {pr : α → Code × Code × σ × σ → σ}... | Mathlib_Computability_PartrecCode |
α : Type u_1
σ : Type u_2
inst✝¹ : Primcodable α
inst✝ : Primcodable σ
c : α → Code
hc : Computable c
z : α → σ
hz : Computable z
s : α → σ
hs : Computable s
l : α → σ
hl : Computable l
r : α → σ
hr : Computable r
pr : α → Code × Code × σ × σ → σ
hpr : Computable₂ pr
co : α → Code × Code × σ × σ → σ
hco : Computable₂ c... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | have s : Computable (fun p : ((((α × List σ) × ℕ × ℕ) × σ) × σ) × σ => p.1.1.2) :=
snd.comp (fst.comp fst) | /-- Recursion on `Nat.Partrec.Code` is computable. -/
theorem rec_computable {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Computable c)
{z : α → σ} (hz : Computable z) {s : α → σ} (hs : Computable s) {l : α → σ} (hl : Computable l)
{r : α → σ} (hr : Computable r) {pr : α → Code × Code × σ × σ → σ}... | Mathlib.Computability.PartrecCode.498_0.A3c3Aev6SyIRjCJ | /-- Recursion on `Nat.Partrec.Code` is computable. -/
theorem rec_computable {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Computable c)
{z : α → σ} (hz : Computable z) {s : α → σ} (hs : Computable s) {l : α → σ} (hl : Computable l)
{r : α → σ} (hr : Computable r) {pr : α → Code × Code × σ × σ → σ}... | Mathlib_Computability_PartrecCode |
α : Type u_1
σ : Type u_2
inst✝¹ : Primcodable α
inst✝ : Primcodable σ
c : α → Code
hc : Computable c
z : α → σ
hz : Computable z
s✝ : α → σ
hs : Computable s✝
l : α → σ
hl : Computable l
r : α → σ
hr : Computable r
pr : α → Code × Code × σ × σ → σ
hpr : Computable₂ pr
co : α → Code × Code × σ × σ → σ
hco : Computable₂... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | have s₁ : Computable (fun p : ((((α × List σ) × ℕ × ℕ) × σ) × σ) × σ => p.1.2) :=
snd.comp fst | /-- Recursion on `Nat.Partrec.Code` is computable. -/
theorem rec_computable {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Computable c)
{z : α → σ} (hz : Computable z) {s : α → σ} (hs : Computable s) {l : α → σ} (hl : Computable l)
{r : α → σ} (hr : Computable r) {pr : α → Code × Code × σ × σ → σ}... | Mathlib.Computability.PartrecCode.498_0.A3c3Aev6SyIRjCJ | /-- Recursion on `Nat.Partrec.Code` is computable. -/
theorem rec_computable {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Computable c)
{z : α → σ} (hz : Computable z) {s : α → σ} (hs : Computable s) {l : α → σ} (hl : Computable l)
{r : α → σ} (hr : Computable r) {pr : α → Code × Code × σ × σ → σ}... | Mathlib_Computability_PartrecCode |
α : Type u_1
σ : Type u_2
inst✝¹ : Primcodable α
inst✝ : Primcodable σ
c : α → Code
hc : Computable c
z : α → σ
hz : Computable z
s✝ : α → σ
hs : Computable s✝
l : α → σ
hl : Computable l
r : α → σ
hr : Computable r
pr : α → Code × Code × σ × σ → σ
hpr : Computable₂ pr
co : α → Code × Code × σ × σ → σ
hco : Computable₂... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | have s₂ : Computable (fun p : ((((α × List σ) × ℕ × ℕ) × σ) × σ) × σ => p.2) :=
snd | /-- Recursion on `Nat.Partrec.Code` is computable. -/
theorem rec_computable {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Computable c)
{z : α → σ} (hz : Computable z) {s : α → σ} (hs : Computable s) {l : α → σ} (hl : Computable l)
{r : α → σ} (hr : Computable r) {pr : α → Code × Code × σ × σ → σ}... | Mathlib.Computability.PartrecCode.498_0.A3c3Aev6SyIRjCJ | /-- Recursion on `Nat.Partrec.Code` is computable. -/
theorem rec_computable {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Computable c)
{z : α → σ} (hz : Computable z) {s : α → σ} (hs : Computable s) {l : α → σ} (hl : Computable l)
{r : α → σ} (hr : Computable r) {pr : α → Code × Code × σ × σ → σ}... | Mathlib_Computability_PartrecCode |
α : Type u_1
σ : Type u_2
inst✝¹ : Primcodable α
inst✝ : Primcodable σ
c : α → Code
hc : Computable c
z : α → σ
hz : Computable z
s✝ : α → σ
hs : Computable s✝
l : α → σ
hl : Computable l
r : α → σ
hr : Computable r
pr : α → Code × Code × σ × σ → σ
hpr : Computable₂ pr
co : α → Code × Code × σ × σ → σ
hco : Computable₂... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | exact
(nat_bodd.comp n).cond
((nat_bodd.comp <| nat_div2.comp n).cond
(hrf.comp a (((Computable.ofNat Code).comp m).pair s))
(hpc.comp a
(((Computable.ofNat Code).comp m₁).pair <|
((Computable.ofNat Code).comp m₂).pair <| s₁.pair s₂)))
(Computable.cond... | /-- Recursion on `Nat.Partrec.Code` is computable. -/
theorem rec_computable {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Computable c)
{z : α → σ} (hz : Computable z) {s : α → σ} (hs : Computable s) {l : α → σ} (hl : Computable l)
{r : α → σ} (hr : Computable r) {pr : α → Code × Code × σ × σ → σ}... | Mathlib.Computability.PartrecCode.498_0.A3c3Aev6SyIRjCJ | /-- Recursion on `Nat.Partrec.Code` is computable. -/
theorem rec_computable {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Computable c)
{z : α → σ} (hz : Computable z) {s : α → σ} (hs : Computable s) {l : α → σ} (hl : Computable l)
{r : α → σ} (hr : Computable r) {pr : α → Code × Code × σ × σ → σ}... | Mathlib_Computability_PartrecCode |
α : Type u_1
σ : Type u_2
inst✝¹ : Primcodable α
inst✝ : Primcodable σ
c : α → Code
hc : Computable c
z : α → σ
hz : Computable z
s : α → σ
hs : Computable s
l : α → σ
hl : Computable l
r : α → σ
hr : Computable r
pr : α → Code × Code × σ × σ → σ
hpr : Computable₂ pr
co : α → Code × Code × σ × σ → σ
hco : Computable₂ c... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | let G : α → List σ → Option σ := fun a IH =>
IH.length.casesOn (some (z a)) fun n =>
n.casesOn (some (s a)) fun n =>
n.casesOn (some (l a)) fun n =>
n.casesOn (some (r a)) fun n => G₁ ((a, IH), n, n.div2.div2) | /-- Recursion on `Nat.Partrec.Code` is computable. -/
theorem rec_computable {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Computable c)
{z : α → σ} (hz : Computable z) {s : α → σ} (hs : Computable s) {l : α → σ} (hl : Computable l)
{r : α → σ} (hr : Computable r) {pr : α → Code × Code × σ × σ → σ}... | Mathlib.Computability.PartrecCode.498_0.A3c3Aev6SyIRjCJ | /-- Recursion on `Nat.Partrec.Code` is computable. -/
theorem rec_computable {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Computable c)
{z : α → σ} (hz : Computable z) {s : α → σ} (hs : Computable s) {l : α → σ} (hl : Computable l)
{r : α → σ} (hr : Computable r) {pr : α → Code × Code × σ × σ → σ}... | Mathlib_Computability_PartrecCode |
α : Type u_1
σ : Type u_2
inst✝¹ : Primcodable α
inst✝ : Primcodable σ
c : α → Code
hc : Computable c
z : α → σ
hz : Computable z
s : α → σ
hs : Computable s
l : α → σ
hl : Computable l
r : α → σ
hr : Computable r
pr : α → Code × Code × σ × σ → σ
hpr : Computable₂ pr
co : α → Code × Code × σ × σ → σ
hco : Computable₂ c... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | have : Computable₂ G :=
Computable.nat_casesOn (list_length.comp snd) (option_some_iff.2 (hz.comp fst)) <|
Computable.nat_casesOn snd (option_some_iff.2 (hs.comp (fst.comp fst))) <|
Computable.nat_casesOn snd (option_some_iff.2 (hl.comp (fst.comp <| fst.comp fst))) <|
Computable.nat_casesOn ... | /-- Recursion on `Nat.Partrec.Code` is computable. -/
theorem rec_computable {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Computable c)
{z : α → σ} (hz : Computable z) {s : α → σ} (hs : Computable s) {l : α → σ} (hl : Computable l)
{r : α → σ} (hr : Computable r) {pr : α → Code × Code × σ × σ → σ}... | Mathlib.Computability.PartrecCode.498_0.A3c3Aev6SyIRjCJ | /-- Recursion on `Nat.Partrec.Code` is computable. -/
theorem rec_computable {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Computable c)
{z : α → σ} (hz : Computable z) {s : α → σ} (hs : Computable s) {l : α → σ} (hl : Computable l)
{r : α → σ} (hr : Computable r) {pr : α → Code × Code × σ × σ → σ}... | Mathlib_Computability_PartrecCode |
α : Type u_1
σ : Type u_2
inst✝¹ : Primcodable α
inst✝ : Primcodable σ
c : α → Code
hc : Computable c
z : α → σ
hz : Computable z
s : α → σ
hs : Computable s
l : α → σ
hl : Computable l
r : α → σ
hr : Computable r
pr : α → Code × Code × σ × σ → σ
hpr : Computable₂ pr
co : α → Code × Code × σ × σ → σ
hco : Computable₂ c... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | refine'
((nat_strong_rec (fun a n => F a (ofNat Code n)) this.to₂ fun a n => _).comp Computable.id <|
encode_iff.2 hc).of_eq fun a => by simp | /-- Recursion on `Nat.Partrec.Code` is computable. -/
theorem rec_computable {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Computable c)
{z : α → σ} (hz : Computable z) {s : α → σ} (hs : Computable s) {l : α → σ} (hl : Computable l)
{r : α → σ} (hr : Computable r) {pr : α → Code × Code × σ × σ → σ}... | Mathlib.Computability.PartrecCode.498_0.A3c3Aev6SyIRjCJ | /-- Recursion on `Nat.Partrec.Code` is computable. -/
theorem rec_computable {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Computable c)
{z : α → σ} (hz : Computable z) {s : α → σ} (hs : Computable s) {l : α → σ} (hl : Computable l)
{r : α → σ} (hr : Computable r) {pr : α → Code × Code × σ × σ → σ}... | Mathlib_Computability_PartrecCode |
α : Type u_1
σ : Type u_2
inst✝¹ : Primcodable α
inst✝ : Primcodable σ
c : α → Code
hc : Computable c
z : α → σ
hz : Computable z
s : α → σ
hs : Computable s
l : α → σ
hl : Computable l
r : α → σ
hr : Computable r
pr : α → Code × Code × σ × σ → σ
hpr : Computable₂ pr
co : α → Code × Code × σ × σ → σ
hco : Computable₂ c... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | simp | /-- Recursion on `Nat.Partrec.Code` is computable. -/
theorem rec_computable {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Computable c)
{z : α → σ} (hz : Computable z) {s : α → σ} (hs : Computable s) {l : α → σ} (hl : Computable l)
{r : α → σ} (hr : Computable r) {pr : α → Code × Code × σ × σ → σ}... | Mathlib.Computability.PartrecCode.498_0.A3c3Aev6SyIRjCJ | /-- Recursion on `Nat.Partrec.Code` is computable. -/
theorem rec_computable {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Computable c)
{z : α → σ} (hz : Computable z) {s : α → σ} (hs : Computable s) {l : α → σ} (hl : Computable l)
{r : α → σ} (hr : Computable r) {pr : α → Code × Code × σ × σ → σ}... | Mathlib_Computability_PartrecCode |
α : Type u_1
σ : Type u_2
inst✝¹ : Primcodable α
inst✝ : Primcodable σ
c : α → Code
hc : Computable c
z : α → σ
hz : Computable z
s : α → σ
hs : Computable s
l : α → σ
hl : Computable l
r : α → σ
hr : Computable r
pr : α → Code × Code × σ × σ → σ
hpr : Computable₂ pr
co : α → Code × Code × σ × σ → σ
hco : Computable₂ c... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | simp (config := { zeta := false }) | /-- Recursion on `Nat.Partrec.Code` is computable. -/
theorem rec_computable {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Computable c)
{z : α → σ} (hz : Computable z) {s : α → σ} (hs : Computable s) {l : α → σ} (hl : Computable l)
{r : α → σ} (hr : Computable r) {pr : α → Code × Code × σ × σ → σ}... | Mathlib.Computability.PartrecCode.498_0.A3c3Aev6SyIRjCJ | /-- Recursion on `Nat.Partrec.Code` is computable. -/
theorem rec_computable {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Computable c)
{z : α → σ} (hz : Computable z) {s : α → σ} (hs : Computable s) {l : α → σ} (hl : Computable l)
{r : α → σ} (hr : Computable r) {pr : α → Code × Code × σ × σ → σ}... | Mathlib_Computability_PartrecCode |
α : Type u_1
σ : Type u_2
inst✝¹ : Primcodable α
inst✝ : Primcodable σ
c : α → Code
hc : Computable c
z : α → σ
hz : Computable z
s : α → σ
hs : Computable s
l : α → σ
hl : Computable l
r : α → σ
hr : Computable r
pr : α → Code × Code × σ × σ → σ
hpr : Computable₂ pr
co : α → Code × Code × σ × σ → σ
hco : Computable₂ c... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | iterate 4 cases' n with n; · simp (config := { zeta := false }) [ofNatCode_eq, ofNatCode]; rfl | /-- Recursion on `Nat.Partrec.Code` is computable. -/
theorem rec_computable {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Computable c)
{z : α → σ} (hz : Computable z) {s : α → σ} (hs : Computable s) {l : α → σ} (hl : Computable l)
{r : α → σ} (hr : Computable r) {pr : α → Code × Code × σ × σ → σ}... | Mathlib.Computability.PartrecCode.498_0.A3c3Aev6SyIRjCJ | /-- Recursion on `Nat.Partrec.Code` is computable. -/
theorem rec_computable {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Computable c)
{z : α → σ} (hz : Computable z) {s : α → σ} (hs : Computable s) {l : α → σ} (hl : Computable l)
{r : α → σ} (hr : Computable r) {pr : α → Code × Code × σ × σ → σ}... | Mathlib_Computability_PartrecCode |
α : Type u_1
σ : Type u_2
inst✝¹ : Primcodable α
inst✝ : Primcodable σ
c : α → Code
hc : Computable c
z : α → σ
hz : Computable z
s : α → σ
hs : Computable s
l : α → σ
hl : Computable l
r : α → σ
hr : Computable r
pr : α → Code × Code × σ × σ → σ
hpr : Computable₂ pr
co : α → Code × Code × σ × σ → σ
hco : Computable₂ c... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | cases' n with n | /-- Recursion on `Nat.Partrec.Code` is computable. -/
theorem rec_computable {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Computable c)
{z : α → σ} (hz : Computable z) {s : α → σ} (hs : Computable s) {l : α → σ} (hl : Computable l)
{r : α → σ} (hr : Computable r) {pr : α → Code × Code × σ × σ → σ}... | Mathlib.Computability.PartrecCode.498_0.A3c3Aev6SyIRjCJ | /-- Recursion on `Nat.Partrec.Code` is computable. -/
theorem rec_computable {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Computable c)
{z : α → σ} (hz : Computable z) {s : α → σ} (hs : Computable s) {l : α → σ} (hl : Computable l)
{r : α → σ} (hr : Computable r) {pr : α → Code × Code × σ × σ → σ}... | Mathlib_Computability_PartrecCode |
case zero
α : Type u_1
σ : Type u_2
inst✝¹ : Primcodable α
inst✝ : Primcodable σ
c : α → Code
hc : Computable c
z : α → σ
hz : Computable z
s : α → σ
hs : Computable s
l : α → σ
hl : Computable l
r : α → σ
hr : Computable r
pr : α → Code × Code × σ × σ → σ
hpr : Computable₂ pr
co : α → Code × Code × σ × σ → σ
hco : Com... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | simp (config := { zeta := false }) [ofNatCode_eq, ofNatCode] | /-- Recursion on `Nat.Partrec.Code` is computable. -/
theorem rec_computable {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Computable c)
{z : α → σ} (hz : Computable z) {s : α → σ} (hs : Computable s) {l : α → σ} (hl : Computable l)
{r : α → σ} (hr : Computable r) {pr : α → Code × Code × σ × σ → σ}... | Mathlib.Computability.PartrecCode.498_0.A3c3Aev6SyIRjCJ | /-- Recursion on `Nat.Partrec.Code` is computable. -/
theorem rec_computable {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Computable c)
{z : α → σ} (hz : Computable z) {s : α → σ} (hs : Computable s) {l : α → σ} (hl : Computable l)
{r : α → σ} (hr : Computable r) {pr : α → Code × Code × σ × σ → σ}... | Mathlib_Computability_PartrecCode |
case zero
α : Type u_1
σ : Type u_2
inst✝¹ : Primcodable α
inst✝ : Primcodable σ
c : α → Code
hc : Computable c
z : α → σ
hz : Computable z
s : α → σ
hs : Computable s
l : α → σ
hl : Computable l
r : α → σ
hr : Computable r
pr : α → Code × Code × σ × σ → σ
hpr : Computable₂ pr
co : α → Code × Code × σ × σ → σ
hco : Com... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | rfl | /-- Recursion on `Nat.Partrec.Code` is computable. -/
theorem rec_computable {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Computable c)
{z : α → σ} (hz : Computable z) {s : α → σ} (hs : Computable s) {l : α → σ} (hl : Computable l)
{r : α → σ} (hr : Computable r) {pr : α → Code × Code × σ × σ → σ}... | Mathlib.Computability.PartrecCode.498_0.A3c3Aev6SyIRjCJ | /-- Recursion on `Nat.Partrec.Code` is computable. -/
theorem rec_computable {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Computable c)
{z : α → σ} (hz : Computable z) {s : α → σ} (hs : Computable s) {l : α → σ} (hl : Computable l)
{r : α → σ} (hr : Computable r) {pr : α → Code × Code × σ × σ → σ}... | Mathlib_Computability_PartrecCode |
case succ
α : Type u_1
σ : Type u_2
inst✝¹ : Primcodable α
inst✝ : Primcodable σ
c : α → Code
hc : Computable c
z : α → σ
hz : Computable z
s : α → σ
hs : Computable s
l : α → σ
hl : Computable l
r : α → σ
hr : Computable r
pr : α → Code × Code × σ × σ → σ
hpr : Computable₂ pr
co : α → Code × Code × σ × σ → σ
hco : Com... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | cases' n with n | /-- Recursion on `Nat.Partrec.Code` is computable. -/
theorem rec_computable {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Computable c)
{z : α → σ} (hz : Computable z) {s : α → σ} (hs : Computable s) {l : α → σ} (hl : Computable l)
{r : α → σ} (hr : Computable r) {pr : α → Code × Code × σ × σ → σ}... | Mathlib.Computability.PartrecCode.498_0.A3c3Aev6SyIRjCJ | /-- Recursion on `Nat.Partrec.Code` is computable. -/
theorem rec_computable {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Computable c)
{z : α → σ} (hz : Computable z) {s : α → σ} (hs : Computable s) {l : α → σ} (hl : Computable l)
{r : α → σ} (hr : Computable r) {pr : α → Code × Code × σ × σ → σ}... | Mathlib_Computability_PartrecCode |
case succ.zero
α : Type u_1
σ : Type u_2
inst✝¹ : Primcodable α
inst✝ : Primcodable σ
c : α → Code
hc : Computable c
z : α → σ
hz : Computable z
s : α → σ
hs : Computable s
l : α → σ
hl : Computable l
r : α → σ
hr : Computable r
pr : α → Code × Code × σ × σ → σ
hpr : Computable₂ pr
co : α → Code × Code × σ × σ → σ
hco ... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | simp (config := { zeta := false }) [ofNatCode_eq, ofNatCode] | /-- Recursion on `Nat.Partrec.Code` is computable. -/
theorem rec_computable {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Computable c)
{z : α → σ} (hz : Computable z) {s : α → σ} (hs : Computable s) {l : α → σ} (hl : Computable l)
{r : α → σ} (hr : Computable r) {pr : α → Code × Code × σ × σ → σ}... | Mathlib.Computability.PartrecCode.498_0.A3c3Aev6SyIRjCJ | /-- Recursion on `Nat.Partrec.Code` is computable. -/
theorem rec_computable {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Computable c)
{z : α → σ} (hz : Computable z) {s : α → σ} (hs : Computable s) {l : α → σ} (hl : Computable l)
{r : α → σ} (hr : Computable r) {pr : α → Code × Code × σ × σ → σ}... | Mathlib_Computability_PartrecCode |
case succ.zero
α : Type u_1
σ : Type u_2
inst✝¹ : Primcodable α
inst✝ : Primcodable σ
c : α → Code
hc : Computable c
z : α → σ
hz : Computable z
s : α → σ
hs : Computable s
l : α → σ
hl : Computable l
r : α → σ
hr : Computable r
pr : α → Code × Code × σ × σ → σ
hpr : Computable₂ pr
co : α → Code × Code × σ × σ → σ
hco ... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | rfl | /-- Recursion on `Nat.Partrec.Code` is computable. -/
theorem rec_computable {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Computable c)
{z : α → σ} (hz : Computable z) {s : α → σ} (hs : Computable s) {l : α → σ} (hl : Computable l)
{r : α → σ} (hr : Computable r) {pr : α → Code × Code × σ × σ → σ}... | Mathlib.Computability.PartrecCode.498_0.A3c3Aev6SyIRjCJ | /-- Recursion on `Nat.Partrec.Code` is computable. -/
theorem rec_computable {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Computable c)
{z : α → σ} (hz : Computable z) {s : α → σ} (hs : Computable s) {l : α → σ} (hl : Computable l)
{r : α → σ} (hr : Computable r) {pr : α → Code × Code × σ × σ → σ}... | Mathlib_Computability_PartrecCode |
case succ.succ
α : Type u_1
σ : Type u_2
inst✝¹ : Primcodable α
inst✝ : Primcodable σ
c : α → Code
hc : Computable c
z : α → σ
hz : Computable z
s : α → σ
hs : Computable s
l : α → σ
hl : Computable l
r : α → σ
hr : Computable r
pr : α → Code × Code × σ × σ → σ
hpr : Computable₂ pr
co : α → Code × Code × σ × σ → σ
hco ... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | cases' n with n | /-- Recursion on `Nat.Partrec.Code` is computable. -/
theorem rec_computable {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Computable c)
{z : α → σ} (hz : Computable z) {s : α → σ} (hs : Computable s) {l : α → σ} (hl : Computable l)
{r : α → σ} (hr : Computable r) {pr : α → Code × Code × σ × σ → σ}... | Mathlib.Computability.PartrecCode.498_0.A3c3Aev6SyIRjCJ | /-- Recursion on `Nat.Partrec.Code` is computable. -/
theorem rec_computable {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Computable c)
{z : α → σ} (hz : Computable z) {s : α → σ} (hs : Computable s) {l : α → σ} (hl : Computable l)
{r : α → σ} (hr : Computable r) {pr : α → Code × Code × σ × σ → σ}... | Mathlib_Computability_PartrecCode |
case succ.succ.zero
α : Type u_1
σ : Type u_2
inst✝¹ : Primcodable α
inst✝ : Primcodable σ
c : α → Code
hc : Computable c
z : α → σ
hz : Computable z
s : α → σ
hs : Computable s
l : α → σ
hl : Computable l
r : α → σ
hr : Computable r
pr : α → Code × Code × σ × σ → σ
hpr : Computable₂ pr
co : α → Code × Code × σ × σ → σ... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | simp (config := { zeta := false }) [ofNatCode_eq, ofNatCode] | /-- Recursion on `Nat.Partrec.Code` is computable. -/
theorem rec_computable {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Computable c)
{z : α → σ} (hz : Computable z) {s : α → σ} (hs : Computable s) {l : α → σ} (hl : Computable l)
{r : α → σ} (hr : Computable r) {pr : α → Code × Code × σ × σ → σ}... | Mathlib.Computability.PartrecCode.498_0.A3c3Aev6SyIRjCJ | /-- Recursion on `Nat.Partrec.Code` is computable. -/
theorem rec_computable {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Computable c)
{z : α → σ} (hz : Computable z) {s : α → σ} (hs : Computable s) {l : α → σ} (hl : Computable l)
{r : α → σ} (hr : Computable r) {pr : α → Code × Code × σ × σ → σ}... | Mathlib_Computability_PartrecCode |
case succ.succ.zero
α : Type u_1
σ : Type u_2
inst✝¹ : Primcodable α
inst✝ : Primcodable σ
c : α → Code
hc : Computable c
z : α → σ
hz : Computable z
s : α → σ
hs : Computable s
l : α → σ
hl : Computable l
r : α → σ
hr : Computable r
pr : α → Code × Code × σ × σ → σ
hpr : Computable₂ pr
co : α → Code × Code × σ × σ → σ... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | rfl | /-- Recursion on `Nat.Partrec.Code` is computable. -/
theorem rec_computable {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Computable c)
{z : α → σ} (hz : Computable z) {s : α → σ} (hs : Computable s) {l : α → σ} (hl : Computable l)
{r : α → σ} (hr : Computable r) {pr : α → Code × Code × σ × σ → σ}... | Mathlib.Computability.PartrecCode.498_0.A3c3Aev6SyIRjCJ | /-- Recursion on `Nat.Partrec.Code` is computable. -/
theorem rec_computable {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Computable c)
{z : α → σ} (hz : Computable z) {s : α → σ} (hs : Computable s) {l : α → σ} (hl : Computable l)
{r : α → σ} (hr : Computable r) {pr : α → Code × Code × σ × σ → σ}... | Mathlib_Computability_PartrecCode |
case succ.succ.succ
α : Type u_1
σ : Type u_2
inst✝¹ : Primcodable α
inst✝ : Primcodable σ
c : α → Code
hc : Computable c
z : α → σ
hz : Computable z
s : α → σ
hs : Computable s
l : α → σ
hl : Computable l
r : α → σ
hr : Computable r
pr : α → Code × Code × σ × σ → σ
hpr : Computable₂ pr
co : α → Code × Code × σ × σ → σ... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | cases' n with n | /-- Recursion on `Nat.Partrec.Code` is computable. -/
theorem rec_computable {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Computable c)
{z : α → σ} (hz : Computable z) {s : α → σ} (hs : Computable s) {l : α → σ} (hl : Computable l)
{r : α → σ} (hr : Computable r) {pr : α → Code × Code × σ × σ → σ}... | Mathlib.Computability.PartrecCode.498_0.A3c3Aev6SyIRjCJ | /-- Recursion on `Nat.Partrec.Code` is computable. -/
theorem rec_computable {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Computable c)
{z : α → σ} (hz : Computable z) {s : α → σ} (hs : Computable s) {l : α → σ} (hl : Computable l)
{r : α → σ} (hr : Computable r) {pr : α → Code × Code × σ × σ → σ}... | Mathlib_Computability_PartrecCode |
case succ.succ.succ.zero
α : Type u_1
σ : Type u_2
inst✝¹ : Primcodable α
inst✝ : Primcodable σ
c : α → Code
hc : Computable c
z : α → σ
hz : Computable z
s : α → σ
hs : Computable s
l : α → σ
hl : Computable l
r : α → σ
hr : Computable r
pr : α → Code × Code × σ × σ → σ
hpr : Computable₂ pr
co : α → Code × Code × σ × ... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | simp (config := { zeta := false }) [ofNatCode_eq, ofNatCode] | /-- Recursion on `Nat.Partrec.Code` is computable. -/
theorem rec_computable {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Computable c)
{z : α → σ} (hz : Computable z) {s : α → σ} (hs : Computable s) {l : α → σ} (hl : Computable l)
{r : α → σ} (hr : Computable r) {pr : α → Code × Code × σ × σ → σ}... | Mathlib.Computability.PartrecCode.498_0.A3c3Aev6SyIRjCJ | /-- Recursion on `Nat.Partrec.Code` is computable. -/
theorem rec_computable {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Computable c)
{z : α → σ} (hz : Computable z) {s : α → σ} (hs : Computable s) {l : α → σ} (hl : Computable l)
{r : α → σ} (hr : Computable r) {pr : α → Code × Code × σ × σ → σ}... | Mathlib_Computability_PartrecCode |
case succ.succ.succ.zero
α : Type u_1
σ : Type u_2
inst✝¹ : Primcodable α
inst✝ : Primcodable σ
c : α → Code
hc : Computable c
z : α → σ
hz : Computable z
s : α → σ
hs : Computable s
l : α → σ
hl : Computable l
r : α → σ
hr : Computable r
pr : α → Code × Code × σ × σ → σ
hpr : Computable₂ pr
co : α → Code × Code × σ × ... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | rfl | /-- Recursion on `Nat.Partrec.Code` is computable. -/
theorem rec_computable {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Computable c)
{z : α → σ} (hz : Computable z) {s : α → σ} (hs : Computable s) {l : α → σ} (hl : Computable l)
{r : α → σ} (hr : Computable r) {pr : α → Code × Code × σ × σ → σ}... | Mathlib.Computability.PartrecCode.498_0.A3c3Aev6SyIRjCJ | /-- Recursion on `Nat.Partrec.Code` is computable. -/
theorem rec_computable {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Computable c)
{z : α → σ} (hz : Computable z) {s : α → σ} (hs : Computable s) {l : α → σ} (hl : Computable l)
{r : α → σ} (hr : Computable r) {pr : α → Code × Code × σ × σ → σ}... | Mathlib_Computability_PartrecCode |
case succ.succ.succ.succ
α : Type u_1
σ : Type u_2
inst✝¹ : Primcodable α
inst✝ : Primcodable σ
c : α → Code
hc : Computable c
z : α → σ
hz : Computable z
s : α → σ
hs : Computable s
l : α → σ
hl : Computable l
r : α → σ
hr : Computable r
pr : α → Code × Code × σ × σ → σ
hpr : Computable₂ pr
co : α → Code × Code × σ × ... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | simp only [] | /-- Recursion on `Nat.Partrec.Code` is computable. -/
theorem rec_computable {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Computable c)
{z : α → σ} (hz : Computable z) {s : α → σ} (hs : Computable s) {l : α → σ} (hl : Computable l)
{r : α → σ} (hr : Computable r) {pr : α → Code × Code × σ × σ → σ}... | Mathlib.Computability.PartrecCode.498_0.A3c3Aev6SyIRjCJ | /-- Recursion on `Nat.Partrec.Code` is computable. -/
theorem rec_computable {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Computable c)
{z : α → σ} (hz : Computable z) {s : α → σ} (hs : Computable s) {l : α → σ} (hl : Computable l)
{r : α → σ} (hr : Computable r) {pr : α → Code × Code × σ × σ → σ}... | Mathlib_Computability_PartrecCode |
case succ.succ.succ.succ
α : Type u_1
σ : Type u_2
inst✝¹ : Primcodable α
inst✝ : Primcodable σ
c : α → Code
hc : Computable c
z : α → σ
hz : Computable z
s : α → σ
hs : Computable s
l : α → σ
hl : Computable l
r : α → σ
hr : Computable r
pr : α → Code × Code × σ × σ → σ
hpr : Computable₂ pr
co : α → Code × Code × σ × ... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | rw [List.length_map, List.length_range] | /-- Recursion on `Nat.Partrec.Code` is computable. -/
theorem rec_computable {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Computable c)
{z : α → σ} (hz : Computable z) {s : α → σ} (hs : Computable s) {l : α → σ} (hl : Computable l)
{r : α → σ} (hr : Computable r) {pr : α → Code × Code × σ × σ → σ}... | Mathlib.Computability.PartrecCode.498_0.A3c3Aev6SyIRjCJ | /-- Recursion on `Nat.Partrec.Code` is computable. -/
theorem rec_computable {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Computable c)
{z : α → σ} (hz : Computable z) {s : α → σ} (hs : Computable s) {l : α → σ} (hl : Computable l)
{r : α → σ} (hr : Computable r) {pr : α → Code × Code × σ × σ → σ}... | Mathlib_Computability_PartrecCode |
case succ.succ.succ.succ
α : Type u_1
σ : Type u_2
inst✝¹ : Primcodable α
inst✝ : Primcodable σ
c : α → Code
hc : Computable c
z : α → σ
hz : Computable z
s : α → σ
hs : Computable s
l : α → σ
hl : Computable l
r : α → σ
hr : Computable r
pr : α → Code × Code × σ × σ → σ
hpr : Computable₂ pr
co : α → Code × Code × σ × ... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | let m := n.div2.div2 | /-- Recursion on `Nat.Partrec.Code` is computable. -/
theorem rec_computable {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Computable c)
{z : α → σ} (hz : Computable z) {s : α → σ} (hs : Computable s) {l : α → σ} (hl : Computable l)
{r : α → σ} (hr : Computable r) {pr : α → Code × Code × σ × σ → σ}... | Mathlib.Computability.PartrecCode.498_0.A3c3Aev6SyIRjCJ | /-- Recursion on `Nat.Partrec.Code` is computable. -/
theorem rec_computable {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Computable c)
{z : α → σ} (hz : Computable z) {s : α → σ} (hs : Computable s) {l : α → σ} (hl : Computable l)
{r : α → σ} (hr : Computable r) {pr : α → Code × Code × σ × σ → σ}... | Mathlib_Computability_PartrecCode |
case succ.succ.succ.succ
α : Type u_1
σ : Type u_2
inst✝¹ : Primcodable α
inst✝ : Primcodable σ
c : α → Code
hc : Computable c
z : α → σ
hz : Computable z
s : α → σ
hs : Computable s
l : α → σ
hl : Computable l
r : α → σ
hr : Computable r
pr : α → Code × Code × σ × σ → σ
hpr : Computable₂ pr
co : α → Code × Code × σ × ... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | show
G₁ ((a, (List.range (n + 4)).map fun n => F a (ofNat Code n)), n, m) =
some (F a (ofNat Code (n + 4))) | /-- Recursion on `Nat.Partrec.Code` is computable. -/
theorem rec_computable {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Computable c)
{z : α → σ} (hz : Computable z) {s : α → σ} (hs : Computable s) {l : α → σ} (hl : Computable l)
{r : α → σ} (hr : Computable r) {pr : α → Code × Code × σ × σ → σ}... | Mathlib.Computability.PartrecCode.498_0.A3c3Aev6SyIRjCJ | /-- Recursion on `Nat.Partrec.Code` is computable. -/
theorem rec_computable {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Computable c)
{z : α → σ} (hz : Computable z) {s : α → σ} (hs : Computable s) {l : α → σ} (hl : Computable l)
{r : α → σ} (hr : Computable r) {pr : α → Code × Code × σ × σ → σ}... | Mathlib_Computability_PartrecCode |
case succ.succ.succ.succ
α : Type u_1
σ : Type u_2
inst✝¹ : Primcodable α
inst✝ : Primcodable σ
c : α → Code
hc : Computable c
z : α → σ
hz : Computable z
s : α → σ
hs : Computable s
l : α → σ
hl : Computable l
r : α → σ
hr : Computable r
pr : α → Code × Code × σ × σ → σ
hpr : Computable₂ pr
co : α → Code × Code × σ × ... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | have hm : m < n + 4 := by
simp only [div2_val]
exact
lt_of_le_of_lt (le_trans (Nat.div_le_self _ _) (Nat.div_le_self _ _))
(Nat.succ_le_succ (Nat.le_add_right _ _)) | /-- Recursion on `Nat.Partrec.Code` is computable. -/
theorem rec_computable {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Computable c)
{z : α → σ} (hz : Computable z) {s : α → σ} (hs : Computable s) {l : α → σ} (hl : Computable l)
{r : α → σ} (hr : Computable r) {pr : α → Code × Code × σ × σ → σ}... | Mathlib.Computability.PartrecCode.498_0.A3c3Aev6SyIRjCJ | /-- Recursion on `Nat.Partrec.Code` is computable. -/
theorem rec_computable {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Computable c)
{z : α → σ} (hz : Computable z) {s : α → σ} (hs : Computable s) {l : α → σ} (hl : Computable l)
{r : α → σ} (hr : Computable r) {pr : α → Code × Code × σ × σ → σ}... | Mathlib_Computability_PartrecCode |
α : Type u_1
σ : Type u_2
inst✝¹ : Primcodable α
inst✝ : Primcodable σ
c : α → Code
hc : Computable c
z : α → σ
hz : Computable z
s : α → σ
hs : Computable s
l : α → σ
hl : Computable l
r : α → σ
hr : Computable r
pr : α → Code × Code × σ × σ → σ
hpr : Computable₂ pr
co : α → Code × Code × σ × σ → σ
hco : Computable₂ c... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | simp only [div2_val] | /-- Recursion on `Nat.Partrec.Code` is computable. -/
theorem rec_computable {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Computable c)
{z : α → σ} (hz : Computable z) {s : α → σ} (hs : Computable s) {l : α → σ} (hl : Computable l)
{r : α → σ} (hr : Computable r) {pr : α → Code × Code × σ × σ → σ}... | Mathlib.Computability.PartrecCode.498_0.A3c3Aev6SyIRjCJ | /-- Recursion on `Nat.Partrec.Code` is computable. -/
theorem rec_computable {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Computable c)
{z : α → σ} (hz : Computable z) {s : α → σ} (hs : Computable s) {l : α → σ} (hl : Computable l)
{r : α → σ} (hr : Computable r) {pr : α → Code × Code × σ × σ → σ}... | Mathlib_Computability_PartrecCode |
α : Type u_1
σ : Type u_2
inst✝¹ : Primcodable α
inst✝ : Primcodable σ
c : α → Code
hc : Computable c
z : α → σ
hz : Computable z
s : α → σ
hs : Computable s
l : α → σ
hl : Computable l
r : α → σ
hr : Computable r
pr : α → Code × Code × σ × σ → σ
hpr : Computable₂ pr
co : α → Code × Code × σ × σ → σ
hco : Computable₂ c... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | exact
lt_of_le_of_lt (le_trans (Nat.div_le_self _ _) (Nat.div_le_self _ _))
(Nat.succ_le_succ (Nat.le_add_right _ _)) | /-- Recursion on `Nat.Partrec.Code` is computable. -/
theorem rec_computable {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Computable c)
{z : α → σ} (hz : Computable z) {s : α → σ} (hs : Computable s) {l : α → σ} (hl : Computable l)
{r : α → σ} (hr : Computable r) {pr : α → Code × Code × σ × σ → σ}... | Mathlib.Computability.PartrecCode.498_0.A3c3Aev6SyIRjCJ | /-- Recursion on `Nat.Partrec.Code` is computable. -/
theorem rec_computable {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Computable c)
{z : α → σ} (hz : Computable z) {s : α → σ} (hs : Computable s) {l : α → σ} (hl : Computable l)
{r : α → σ} (hr : Computable r) {pr : α → Code × Code × σ × σ → σ}... | Mathlib_Computability_PartrecCode |
case succ.succ.succ.succ
α : Type u_1
σ : Type u_2
inst✝¹ : Primcodable α
inst✝ : Primcodable σ
c : α → Code
hc : Computable c
z : α → σ
hz : Computable z
s : α → σ
hs : Computable s
l : α → σ
hl : Computable l
r : α → σ
hr : Computable r
pr : α → Code × Code × σ × σ → σ
hpr : Computable₂ pr
co : α → Code × Code × σ × ... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | have m1 : m.unpair.1 < n + 4 := lt_of_le_of_lt m.unpair_left_le hm | /-- Recursion on `Nat.Partrec.Code` is computable. -/
theorem rec_computable {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Computable c)
{z : α → σ} (hz : Computable z) {s : α → σ} (hs : Computable s) {l : α → σ} (hl : Computable l)
{r : α → σ} (hr : Computable r) {pr : α → Code × Code × σ × σ → σ}... | Mathlib.Computability.PartrecCode.498_0.A3c3Aev6SyIRjCJ | /-- Recursion on `Nat.Partrec.Code` is computable. -/
theorem rec_computable {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Computable c)
{z : α → σ} (hz : Computable z) {s : α → σ} (hs : Computable s) {l : α → σ} (hl : Computable l)
{r : α → σ} (hr : Computable r) {pr : α → Code × Code × σ × σ → σ}... | Mathlib_Computability_PartrecCode |
case succ.succ.succ.succ
α : Type u_1
σ : Type u_2
inst✝¹ : Primcodable α
inst✝ : Primcodable σ
c : α → Code
hc : Computable c
z : α → σ
hz : Computable z
s : α → σ
hs : Computable s
l : α → σ
hl : Computable l
r : α → σ
hr : Computable r
pr : α → Code × Code × σ × σ → σ
hpr : Computable₂ pr
co : α → Code × Code × σ × ... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | have m2 : m.unpair.2 < n + 4 := lt_of_le_of_lt m.unpair_right_le hm | /-- Recursion on `Nat.Partrec.Code` is computable. -/
theorem rec_computable {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Computable c)
{z : α → σ} (hz : Computable z) {s : α → σ} (hs : Computable s) {l : α → σ} (hl : Computable l)
{r : α → σ} (hr : Computable r) {pr : α → Code × Code × σ × σ → σ}... | Mathlib.Computability.PartrecCode.498_0.A3c3Aev6SyIRjCJ | /-- Recursion on `Nat.Partrec.Code` is computable. -/
theorem rec_computable {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Computable c)
{z : α → σ} (hz : Computable z) {s : α → σ} (hs : Computable s) {l : α → σ} (hl : Computable l)
{r : α → σ} (hr : Computable r) {pr : α → Code × Code × σ × σ → σ}... | Mathlib_Computability_PartrecCode |
case succ.succ.succ.succ
α : Type u_1
σ : Type u_2
inst✝¹ : Primcodable α
inst✝ : Primcodable σ
c : α → Code
hc : Computable c
z : α → σ
hz : Computable z
s : α → σ
hs : Computable s
l : α → σ
hl : Computable l
r : α → σ
hr : Computable r
pr : α → Code × Code × σ × σ → σ
hpr : Computable₂ pr
co : α → Code × Code × σ × ... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | simp [List.get?_map, List.get?_range, hm, m1, m2] | /-- Recursion on `Nat.Partrec.Code` is computable. -/
theorem rec_computable {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Computable c)
{z : α → σ} (hz : Computable z) {s : α → σ} (hs : Computable s) {l : α → σ} (hl : Computable l)
{r : α → σ} (hr : Computable r) {pr : α → Code × Code × σ × σ → σ}... | Mathlib.Computability.PartrecCode.498_0.A3c3Aev6SyIRjCJ | /-- Recursion on `Nat.Partrec.Code` is computable. -/
theorem rec_computable {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Computable c)
{z : α → σ} (hz : Computable z) {s : α → σ} (hs : Computable s) {l : α → σ} (hl : Computable l)
{r : α → σ} (hr : Computable r) {pr : α → Code × Code × σ × σ → σ}... | Mathlib_Computability_PartrecCode |
case succ.succ.succ.succ
α : Type u_1
σ : Type u_2
inst✝¹ : Primcodable α
inst✝ : Primcodable σ
c : α → Code
hc : Computable c
z : α → σ
hz : Computable z
s : α → σ
hs : Computable s
l : α → σ
hl : Computable l
r : α → σ
hr : Computable r
pr : α → Code × Code × σ × σ → σ
hpr : Computable₂ pr
co : α → Code × Code × σ × ... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | rw [show ofNat Code (n + 4) = ofNatCode (n + 4) from rfl] | /-- Recursion on `Nat.Partrec.Code` is computable. -/
theorem rec_computable {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Computable c)
{z : α → σ} (hz : Computable z) {s : α → σ} (hs : Computable s) {l : α → σ} (hl : Computable l)
{r : α → σ} (hr : Computable r) {pr : α → Code × Code × σ × σ → σ}... | Mathlib.Computability.PartrecCode.498_0.A3c3Aev6SyIRjCJ | /-- Recursion on `Nat.Partrec.Code` is computable. -/
theorem rec_computable {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Computable c)
{z : α → σ} (hz : Computable z) {s : α → σ} (hs : Computable s) {l : α → σ} (hl : Computable l)
{r : α → σ} (hr : Computable r) {pr : α → Code × Code × σ × σ → σ}... | Mathlib_Computability_PartrecCode |
case succ.succ.succ.succ
α : Type u_1
σ : Type u_2
inst✝¹ : Primcodable α
inst✝ : Primcodable σ
c : α → Code
hc : Computable c
z : α → σ
hz : Computable z
s : α → σ
hs : Computable s
l : α → σ
hl : Computable l
r : α → σ
hr : Computable r
pr : α → Code × Code × σ × σ → σ
hpr : Computable₂ pr
co : α → Code × Code × σ × ... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | simp [ofNatCode] | /-- Recursion on `Nat.Partrec.Code` is computable. -/
theorem rec_computable {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Computable c)
{z : α → σ} (hz : Computable z) {s : α → σ} (hs : Computable s) {l : α → σ} (hl : Computable l)
{r : α → σ} (hr : Computable r) {pr : α → Code × Code × σ × σ → σ}... | Mathlib.Computability.PartrecCode.498_0.A3c3Aev6SyIRjCJ | /-- Recursion on `Nat.Partrec.Code` is computable. -/
theorem rec_computable {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Computable c)
{z : α → σ} (hz : Computable z) {s : α → σ} (hs : Computable s) {l : α → σ} (hl : Computable l)
{r : α → σ} (hr : Computable r) {pr : α → Code × Code × σ × σ → σ}... | Mathlib_Computability_PartrecCode |
case succ.succ.succ.succ
α : Type u_1
σ : Type u_2
inst✝¹ : Primcodable α
inst✝ : Primcodable σ
c : α → Code
hc : Computable c
z : α → σ
hz : Computable z
s : α → σ
hs : Computable s
l : α → σ
hl : Computable l
r : α → σ
hr : Computable r
pr : α → Code × Code × σ × σ → σ
hpr : Computable₂ pr
co : α → Code × Code × σ × ... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | cases n.bodd | /-- Recursion on `Nat.Partrec.Code` is computable. -/
theorem rec_computable {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Computable c)
{z : α → σ} (hz : Computable z) {s : α → σ} (hs : Computable s) {l : α → σ} (hl : Computable l)
{r : α → σ} (hr : Computable r) {pr : α → Code × Code × σ × σ → σ}... | Mathlib.Computability.PartrecCode.498_0.A3c3Aev6SyIRjCJ | /-- Recursion on `Nat.Partrec.Code` is computable. -/
theorem rec_computable {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Computable c)
{z : α → σ} (hz : Computable z) {s : α → σ} (hs : Computable s) {l : α → σ} (hl : Computable l)
{r : α → σ} (hr : Computable r) {pr : α → Code × Code × σ × σ → σ}... | Mathlib_Computability_PartrecCode |
case succ.succ.succ.succ.false
α : Type u_1
σ : Type u_2
inst✝¹ : Primcodable α
inst✝ : Primcodable σ
c : α → Code
hc : Computable c
z : α → σ
hz : Computable z
s : α → σ
hs : Computable s
l : α → σ
hl : Computable l
r : α → σ
hr : Computable r
pr : α → Code × Code × σ × σ → σ
hpr : Computable₂ pr
co : α → Code × Code ... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | cases n.div2.bodd | /-- Recursion on `Nat.Partrec.Code` is computable. -/
theorem rec_computable {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Computable c)
{z : α → σ} (hz : Computable z) {s : α → σ} (hs : Computable s) {l : α → σ} (hl : Computable l)
{r : α → σ} (hr : Computable r) {pr : α → Code × Code × σ × σ → σ}... | Mathlib.Computability.PartrecCode.498_0.A3c3Aev6SyIRjCJ | /-- Recursion on `Nat.Partrec.Code` is computable. -/
theorem rec_computable {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Computable c)
{z : α → σ} (hz : Computable z) {s : α → σ} (hs : Computable s) {l : α → σ} (hl : Computable l)
{r : α → σ} (hr : Computable r) {pr : α → Code × Code × σ × σ → σ}... | Mathlib_Computability_PartrecCode |
case succ.succ.succ.succ.true
α : Type u_1
σ : Type u_2
inst✝¹ : Primcodable α
inst✝ : Primcodable σ
c : α → Code
hc : Computable c
z : α → σ
hz : Computable z
s : α → σ
hs : Computable s
l : α → σ
hl : Computable l
r : α → σ
hr : Computable r
pr : α → Code × Code × σ × σ → σ
hpr : Computable₂ pr
co : α → Code × Code ×... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | cases n.div2.bodd | /-- Recursion on `Nat.Partrec.Code` is computable. -/
theorem rec_computable {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Computable c)
{z : α → σ} (hz : Computable z) {s : α → σ} (hs : Computable s) {l : α → σ} (hl : Computable l)
{r : α → σ} (hr : Computable r) {pr : α → Code × Code × σ × σ → σ}... | Mathlib.Computability.PartrecCode.498_0.A3c3Aev6SyIRjCJ | /-- Recursion on `Nat.Partrec.Code` is computable. -/
theorem rec_computable {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Computable c)
{z : α → σ} (hz : Computable z) {s : α → σ} (hs : Computable s) {l : α → σ} (hl : Computable l)
{r : α → σ} (hr : Computable r) {pr : α → Code × Code × σ × σ → σ}... | Mathlib_Computability_PartrecCode |
case succ.succ.succ.succ.false.false
α : Type u_1
σ : Type u_2
inst✝¹ : Primcodable α
inst✝ : Primcodable σ
c : α → Code
hc : Computable c
z : α → σ
hz : Computable z
s : α → σ
hs : Computable s
l : α → σ
hl : Computable l
r : α → σ
hr : Computable r
pr : α → Code × Code × σ × σ → σ
hpr : Computable₂ pr
co : α → Code ×... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | rfl | /-- Recursion on `Nat.Partrec.Code` is computable. -/
theorem rec_computable {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Computable c)
{z : α → σ} (hz : Computable z) {s : α → σ} (hs : Computable s) {l : α → σ} (hl : Computable l)
{r : α → σ} (hr : Computable r) {pr : α → Code × Code × σ × σ → σ}... | Mathlib.Computability.PartrecCode.498_0.A3c3Aev6SyIRjCJ | /-- Recursion on `Nat.Partrec.Code` is computable. -/
theorem rec_computable {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Computable c)
{z : α → σ} (hz : Computable z) {s : α → σ} (hs : Computable s) {l : α → σ} (hl : Computable l)
{r : α → σ} (hr : Computable r) {pr : α → Code × Code × σ × σ → σ}... | Mathlib_Computability_PartrecCode |
case succ.succ.succ.succ.false.true
α : Type u_1
σ : Type u_2
inst✝¹ : Primcodable α
inst✝ : Primcodable σ
c : α → Code
hc : Computable c
z : α → σ
hz : Computable z
s : α → σ
hs : Computable s
l : α → σ
hl : Computable l
r : α → σ
hr : Computable r
pr : α → Code × Code × σ × σ → σ
hpr : Computable₂ pr
co : α → Code × ... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | rfl | /-- Recursion on `Nat.Partrec.Code` is computable. -/
theorem rec_computable {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Computable c)
{z : α → σ} (hz : Computable z) {s : α → σ} (hs : Computable s) {l : α → σ} (hl : Computable l)
{r : α → σ} (hr : Computable r) {pr : α → Code × Code × σ × σ → σ}... | Mathlib.Computability.PartrecCode.498_0.A3c3Aev6SyIRjCJ | /-- Recursion on `Nat.Partrec.Code` is computable. -/
theorem rec_computable {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Computable c)
{z : α → σ} (hz : Computable z) {s : α → σ} (hs : Computable s) {l : α → σ} (hl : Computable l)
{r : α → σ} (hr : Computable r) {pr : α → Code × Code × σ × σ → σ}... | Mathlib_Computability_PartrecCode |
case succ.succ.succ.succ.true.false
α : Type u_1
σ : Type u_2
inst✝¹ : Primcodable α
inst✝ : Primcodable σ
c : α → Code
hc : Computable c
z : α → σ
hz : Computable z
s : α → σ
hs : Computable s
l : α → σ
hl : Computable l
r : α → σ
hr : Computable r
pr : α → Code × Code × σ × σ → σ
hpr : Computable₂ pr
co : α → Code × ... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | rfl | /-- Recursion on `Nat.Partrec.Code` is computable. -/
theorem rec_computable {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Computable c)
{z : α → σ} (hz : Computable z) {s : α → σ} (hs : Computable s) {l : α → σ} (hl : Computable l)
{r : α → σ} (hr : Computable r) {pr : α → Code × Code × σ × σ → σ}... | Mathlib.Computability.PartrecCode.498_0.A3c3Aev6SyIRjCJ | /-- Recursion on `Nat.Partrec.Code` is computable. -/
theorem rec_computable {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Computable c)
{z : α → σ} (hz : Computable z) {s : α → σ} (hs : Computable s) {l : α → σ} (hl : Computable l)
{r : α → σ} (hr : Computable r) {pr : α → Code × Code × σ × σ → σ}... | Mathlib_Computability_PartrecCode |
case succ.succ.succ.succ.true.true
α : Type u_1
σ : Type u_2
inst✝¹ : Primcodable α
inst✝ : Primcodable σ
c : α → Code
hc : Computable c
z : α → σ
hz : Computable z
s : α → σ
hs : Computable s
l : α → σ
hl : Computable l
r : α → σ
hr : Computable r
pr : α → Code × Code × σ × σ → σ
hpr : Computable₂ pr
co : α → Code × C... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | rfl | /-- Recursion on `Nat.Partrec.Code` is computable. -/
theorem rec_computable {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Computable c)
{z : α → σ} (hz : Computable z) {s : α → σ} (hs : Computable s) {l : α → σ} (hl : Computable l)
{r : α → σ} (hr : Computable r) {pr : α → Code × Code × σ × σ → σ}... | Mathlib.Computability.PartrecCode.498_0.A3c3Aev6SyIRjCJ | /-- Recursion on `Nat.Partrec.Code` is computable. -/
theorem rec_computable {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Computable c)
{z : α → σ} (hz : Computable z) {s : α → σ} (hs : Computable s) {l : α → σ} (hl : Computable l)
{r : α → σ} (hr : Computable r) {pr : α → Code × Code × σ × σ → σ}... | Mathlib_Computability_PartrecCode |
cf cg : Code
a : ℕ
⊢ eval (prec cf cg) (Nat.pair a 0) = eval cf a | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | rw [eval, Nat.unpaired, Nat.unpair_pair] | /-- Helper lemma for the evaluation of `prec` in the base case. -/
@[simp]
theorem eval_prec_zero (cf cg : Code) (a : ℕ) : eval (prec cf cg) (Nat.pair a 0) = eval cf a := by
| Mathlib.Computability.PartrecCode.637_0.A3c3Aev6SyIRjCJ | /-- Helper lemma for the evaluation of `prec` in the base case. -/
@[simp]
theorem eval_prec_zero (cf cg : Code) (a : ℕ) : eval (prec cf cg) (Nat.pair a 0) = eval cf a | Mathlib_Computability_PartrecCode |
cf cg : Code
a : ℕ
⊢ Nat.rec (eval cf (a, 0).1)
(fun y IH => do
let i ← IH
eval cg (Nat.pair (a, 0).1 (Nat.pair y i)))
(a, 0).2 =
eval cf a | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | simp (config := { Lean.Meta.Simp.neutralConfig with proj := true }) only [] | /-- Helper lemma for the evaluation of `prec` in the base case. -/
@[simp]
theorem eval_prec_zero (cf cg : Code) (a : ℕ) : eval (prec cf cg) (Nat.pair a 0) = eval cf a := by
rw [eval, Nat.unpaired, Nat.unpair_pair]
| Mathlib.Computability.PartrecCode.637_0.A3c3Aev6SyIRjCJ | /-- Helper lemma for the evaluation of `prec` in the base case. -/
@[simp]
theorem eval_prec_zero (cf cg : Code) (a : ℕ) : eval (prec cf cg) (Nat.pair a 0) = eval cf a | Mathlib_Computability_PartrecCode |
cf cg : Code
a : ℕ
⊢ Nat.rec (eval cf a)
(fun y IH => do
let i ← IH
eval cg (Nat.pair a (Nat.pair y i)))
0 =
eval cf a | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | rw [Nat.rec_zero] | /-- Helper lemma for the evaluation of `prec` in the base case. -/
@[simp]
theorem eval_prec_zero (cf cg : Code) (a : ℕ) : eval (prec cf cg) (Nat.pair a 0) = eval cf a := by
rw [eval, Nat.unpaired, Nat.unpair_pair]
simp (config := { Lean.Meta.Simp.neutralConfig with proj := true }) only []
| Mathlib.Computability.PartrecCode.637_0.A3c3Aev6SyIRjCJ | /-- Helper lemma for the evaluation of `prec` in the base case. -/
@[simp]
theorem eval_prec_zero (cf cg : Code) (a : ℕ) : eval (prec cf cg) (Nat.pair a 0) = eval cf a | Mathlib_Computability_PartrecCode |
cf cg : Code
a k : ℕ
⊢ eval (prec cf cg) (Nat.pair a (Nat.succ k)) = do
let ih ← eval (prec cf cg) (Nat.pair a k)
eval cg (Nat.pair a (Nat.pair k ih)) | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | rw [eval, Nat.unpaired, Part.bind_eq_bind, Nat.unpair_pair] | /-- Helper lemma for the evaluation of `prec` in the recursive case. -/
theorem eval_prec_succ (cf cg : Code) (a k : ℕ) :
eval (prec cf cg) (Nat.pair a (Nat.succ k)) =
do {let ih ← eval (prec cf cg) (Nat.pair a k); eval cg (Nat.pair a (Nat.pair k ih))} := by
| Mathlib.Computability.PartrecCode.645_0.A3c3Aev6SyIRjCJ | /-- Helper lemma for the evaluation of `prec` in the recursive case. -/
theorem eval_prec_succ (cf cg : Code) (a k : ℕ) :
eval (prec cf cg) (Nat.pair a (Nat.succ k)) =
do {let ih ← eval (prec cf cg) (Nat.pair a k); eval cg (Nat.pair a (Nat.pair k ih))} | Mathlib_Computability_PartrecCode |
cf cg : Code
a k : ℕ
⊢ Nat.rec (eval cf (a, Nat.succ k).1)
(fun y IH => do
let i ← IH
eval cg (Nat.pair (a, Nat.succ k).1 (Nat.pair y i)))
(a, Nat.succ k).2 =
Part.bind
(unpaired
(fun a n =>
Nat.rec (eval cf a)
(fun y IH => do
let i ← IH
... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | simp | /-- Helper lemma for the evaluation of `prec` in the recursive case. -/
theorem eval_prec_succ (cf cg : Code) (a k : ℕ) :
eval (prec cf cg) (Nat.pair a (Nat.succ k)) =
do {let ih ← eval (prec cf cg) (Nat.pair a k); eval cg (Nat.pair a (Nat.pair k ih))} := by
rw [eval, Nat.unpaired, Part.bind_eq_bind, Nat.un... | Mathlib.Computability.PartrecCode.645_0.A3c3Aev6SyIRjCJ | /-- Helper lemma for the evaluation of `prec` in the recursive case. -/
theorem eval_prec_succ (cf cg : Code) (a k : ℕ) :
eval (prec cf cg) (Nat.pair a (Nat.succ k)) =
do {let ih ← eval (prec cf cg) (Nat.pair a k); eval cg (Nat.pair a (Nat.pair k ih))} | Mathlib_Computability_PartrecCode |
n m : ℕ
⊢ eval (Code.const (n + 1)) m = Part.some (n + 1) | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | simp! [eval_const n m] | @[simp]
theorem eval_const : ∀ n m, eval (Code.const n) m = Part.some n
| 0, m => rfl
| n + 1, m => by | Mathlib.Computability.PartrecCode.656_0.A3c3Aev6SyIRjCJ | @[simp]
theorem eval_const : ∀ n m, eval (Code.const n) m = Part.some n
| 0, m => rfl
| n + 1, m => by simp! [eval_const n m] | Mathlib_Computability_PartrecCode |
n : ℕ
⊢ eval Code.id n = Part.some n | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | simp! [Seq.seq] | @[simp]
theorem eval_id (n) : eval Code.id n = Part.some n := by | Mathlib.Computability.PartrecCode.662_0.A3c3Aev6SyIRjCJ | @[simp]
theorem eval_id (n) : eval Code.id n = Part.some n | Mathlib_Computability_PartrecCode |
c : Code
n x : ℕ
⊢ eval (curry c n) x = eval c (Nat.pair n x) | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | simp! [Seq.seq] | @[simp]
theorem eval_curry (c n x) : eval (curry c n) x = eval c (Nat.pair n x) := by | Mathlib.Computability.PartrecCode.666_0.A3c3Aev6SyIRjCJ | @[simp]
theorem eval_curry (c n x) : eval (curry c n) x = eval c (Nat.pair n x) | Mathlib_Computability_PartrecCode |
n : ℕ
⊢ (fun b => comp succ (n, b).2)^[id n] zero = Code.const n | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | simp | theorem const_prim : Primrec Code.const :=
(_root_.Primrec.id.nat_iterate (_root_.Primrec.const zero)
(comp_prim.comp (_root_.Primrec.const succ) Primrec.snd).to₂).of_eq
fun n => by | Mathlib.Computability.PartrecCode.670_0.A3c3Aev6SyIRjCJ | theorem const_prim : Primrec Code.const | Mathlib_Computability_PartrecCode |
n : ℕ
⊢ (fun b => comp succ b)^[n] zero = Code.const n | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | induction n | theorem const_prim : Primrec Code.const :=
(_root_.Primrec.id.nat_iterate (_root_.Primrec.const zero)
(comp_prim.comp (_root_.Primrec.const succ) Primrec.snd).to₂).of_eq
fun n => by simp; | Mathlib.Computability.PartrecCode.670_0.A3c3Aev6SyIRjCJ | theorem const_prim : Primrec Code.const | Mathlib_Computability_PartrecCode |
case zero
⊢ (fun b => comp succ b)^[Nat.zero] zero = Code.const Nat.zero | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | simp [*, Code.const, Function.iterate_succ', -Function.iterate_succ] | theorem const_prim : Primrec Code.const :=
(_root_.Primrec.id.nat_iterate (_root_.Primrec.const zero)
(comp_prim.comp (_root_.Primrec.const succ) Primrec.snd).to₂).of_eq
fun n => by simp; induction n <;>
| Mathlib.Computability.PartrecCode.670_0.A3c3Aev6SyIRjCJ | theorem const_prim : Primrec Code.const | Mathlib_Computability_PartrecCode |
case succ
n✝ : ℕ
n_ih✝ : (fun b => comp succ b)^[n✝] zero = Code.const n✝
⊢ (fun b => comp succ b)^[Nat.succ n✝] zero = Code.const (Nat.succ n✝) | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | simp [*, Code.const, Function.iterate_succ', -Function.iterate_succ] | theorem const_prim : Primrec Code.const :=
(_root_.Primrec.id.nat_iterate (_root_.Primrec.const zero)
(comp_prim.comp (_root_.Primrec.const succ) Primrec.snd).to₂).of_eq
fun n => by simp; induction n <;>
| Mathlib.Computability.PartrecCode.670_0.A3c3Aev6SyIRjCJ | theorem const_prim : Primrec Code.const | Mathlib_Computability_PartrecCode |
c₁ c₂ : Code
n₁ n₂ : ℕ
h : curry c₁ n₁ = curry c₂ n₂
⊢ c₁ = c₂ | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | injection h | theorem curry_inj {c₁ c₂ n₁ n₂} (h : curry c₁ n₁ = curry c₂ n₂) : c₁ = c₂ ∧ n₁ = n₂ :=
⟨by | Mathlib.Computability.PartrecCode.682_0.A3c3Aev6SyIRjCJ | theorem curry_inj {c₁ c₂ n₁ n₂} (h : curry c₁ n₁ = curry c₂ n₂) : c₁ = c₂ ∧ n₁ = n₂ | Mathlib_Computability_PartrecCode |
c₁ c₂ : Code
n₁ n₂ : ℕ
h : curry c₁ n₁ = curry c₂ n₂
⊢ n₁ = n₂ | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | injection h with h₁ h₂ | theorem curry_inj {c₁ c₂ n₁ n₂} (h : curry c₁ n₁ = curry c₂ n₂) : c₁ = c₂ ∧ n₁ = n₂ :=
⟨by injection h, by
| Mathlib.Computability.PartrecCode.682_0.A3c3Aev6SyIRjCJ | theorem curry_inj {c₁ c₂ n₁ n₂} (h : curry c₁ n₁ = curry c₂ n₂) : c₁ = c₂ ∧ n₁ = n₂ | Mathlib_Computability_PartrecCode |
c₁ c₂ : Code
n₁ n₂ : ℕ
h₁ : c₁ = c₂
h₂ : pair (Code.const n₁) Code.id = pair (Code.const n₂) Code.id
⊢ n₁ = n₂ | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | injection h₂ with h₃ h₄ | theorem curry_inj {c₁ c₂ n₁ n₂} (h : curry c₁ n₁ = curry c₂ n₂) : c₁ = c₂ ∧ n₁ = n₂ :=
⟨by injection h, by
injection h with h₁ h₂
| Mathlib.Computability.PartrecCode.682_0.A3c3Aev6SyIRjCJ | theorem curry_inj {c₁ c₂ n₁ n₂} (h : curry c₁ n₁ = curry c₂ n₂) : c₁ = c₂ ∧ n₁ = n₂ | Mathlib_Computability_PartrecCode |
c₁ c₂ : Code
n₁ n₂ : ℕ
h₁ : c₁ = c₂
h₃ : Code.const n₁ = Code.const n₂
h₄ : Code.id = Code.id
⊢ n₁ = n₂ | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | exact const_inj h₃ | theorem curry_inj {c₁ c₂ n₁ n₂} (h : curry c₁ n₁ = curry c₂ n₂) : c₁ = c₂ ∧ n₁ = n₂ :=
⟨by injection h, by
injection h with h₁ h₂
injection h₂ with h₃ h₄
| Mathlib.Computability.PartrecCode.682_0.A3c3Aev6SyIRjCJ | theorem curry_inj {c₁ c₂ n₁ n₂} (h : curry c₁ n₁ = curry c₂ n₂) : c₁ = c₂ ∧ n₁ = n₂ | Mathlib_Computability_PartrecCode |
f : ℕ →. ℕ
h : Partrec f
⊢ ∃ c, eval c = f | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | induction h | /-- A function is partial recursive if and only if there is a code implementing it. -/
theorem exists_code {f : ℕ →. ℕ} : Nat.Partrec f ↔ ∃ c : Code, eval c = f :=
⟨fun h => by
| Mathlib.Computability.PartrecCode.698_0.A3c3Aev6SyIRjCJ | /-- A function is partial recursive if and only if there is a code implementing it. -/
theorem exists_code {f : ℕ →. ℕ} : Nat.Partrec f ↔ ∃ c : Code, eval c = f | Mathlib_Computability_PartrecCode |
case zero
f : ℕ →. ℕ
⊢ ∃ c, eval c = pure 0
case succ
f : ℕ →. ℕ
⊢ ∃ c, eval c = ↑Nat.succ
case left
f : ℕ →. ℕ
⊢ ∃ c, eval c = ↑fun n => (unpair n).1
case right
f : ℕ →. ℕ
⊢ ∃ c, eval c = ↑fun n => (unpair n).2
case pair
f f✝ g✝ : ℕ →. ℕ
a✝¹ : Partrec f✝
a✝ : Partrec g✝
a_ih✝¹ : ∃ c, eval c = f✝
a_ih✝ : ∃ c, eval c = ... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | case zero => exact ⟨zero, rfl⟩ | /-- A function is partial recursive if and only if there is a code implementing it. -/
theorem exists_code {f : ℕ →. ℕ} : Nat.Partrec f ↔ ∃ c : Code, eval c = f :=
⟨fun h => by
induction h
| Mathlib.Computability.PartrecCode.698_0.A3c3Aev6SyIRjCJ | /-- A function is partial recursive if and only if there is a code implementing it. -/
theorem exists_code {f : ℕ →. ℕ} : Nat.Partrec f ↔ ∃ c : Code, eval c = f | Mathlib_Computability_PartrecCode |
f : ℕ →. ℕ
⊢ ∃ c, eval c = pure 0 | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | case zero => exact ⟨zero, rfl⟩ | /-- A function is partial recursive if and only if there is a code implementing it. -/
theorem exists_code {f : ℕ →. ℕ} : Nat.Partrec f ↔ ∃ c : Code, eval c = f :=
⟨fun h => by
induction h
| Mathlib.Computability.PartrecCode.698_0.A3c3Aev6SyIRjCJ | /-- A function is partial recursive if and only if there is a code implementing it. -/
theorem exists_code {f : ℕ →. ℕ} : Nat.Partrec f ↔ ∃ c : Code, eval c = f | Mathlib_Computability_PartrecCode |
f : ℕ →. ℕ
⊢ ∃ c, eval c = pure 0 | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | exact ⟨zero, rfl⟩ | /-- A function is partial recursive if and only if there is a code implementing it. -/
theorem exists_code {f : ℕ →. ℕ} : Nat.Partrec f ↔ ∃ c : Code, eval c = f :=
⟨fun h => by
induction h
case zero => | Mathlib.Computability.PartrecCode.698_0.A3c3Aev6SyIRjCJ | /-- A function is partial recursive if and only if there is a code implementing it. -/
theorem exists_code {f : ℕ →. ℕ} : Nat.Partrec f ↔ ∃ c : Code, eval c = f | Mathlib_Computability_PartrecCode |
case succ
f : ℕ →. ℕ
⊢ ∃ c, eval c = ↑Nat.succ
case left
f : ℕ →. ℕ
⊢ ∃ c, eval c = ↑fun n => (unpair n).1
case right
f : ℕ →. ℕ
⊢ ∃ c, eval c = ↑fun n => (unpair n).2
case pair
f f✝ g✝ : ℕ →. ℕ
a✝¹ : Partrec f✝
a✝ : Partrec g✝
a_ih✝¹ : ∃ c, eval c = f✝
a_ih✝ : ∃ c, eval c = g✝
⊢ ∃ c, eval c = fun n => Seq.seq (Nat.pai... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | case succ => exact ⟨succ, rfl⟩ | /-- A function is partial recursive if and only if there is a code implementing it. -/
theorem exists_code {f : ℕ →. ℕ} : Nat.Partrec f ↔ ∃ c : Code, eval c = f :=
⟨fun h => by
induction h
case zero => exact ⟨zero, rfl⟩
| Mathlib.Computability.PartrecCode.698_0.A3c3Aev6SyIRjCJ | /-- A function is partial recursive if and only if there is a code implementing it. -/
theorem exists_code {f : ℕ →. ℕ} : Nat.Partrec f ↔ ∃ c : Code, eval c = f | Mathlib_Computability_PartrecCode |
f : ℕ →. ℕ
⊢ ∃ c, eval c = ↑Nat.succ | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | case succ => exact ⟨succ, rfl⟩ | /-- A function is partial recursive if and only if there is a code implementing it. -/
theorem exists_code {f : ℕ →. ℕ} : Nat.Partrec f ↔ ∃ c : Code, eval c = f :=
⟨fun h => by
induction h
case zero => exact ⟨zero, rfl⟩
| Mathlib.Computability.PartrecCode.698_0.A3c3Aev6SyIRjCJ | /-- A function is partial recursive if and only if there is a code implementing it. -/
theorem exists_code {f : ℕ →. ℕ} : Nat.Partrec f ↔ ∃ c : Code, eval c = f | Mathlib_Computability_PartrecCode |
f : ℕ →. ℕ
⊢ ∃ c, eval c = ↑Nat.succ | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | exact ⟨succ, rfl⟩ | /-- A function is partial recursive if and only if there is a code implementing it. -/
theorem exists_code {f : ℕ →. ℕ} : Nat.Partrec f ↔ ∃ c : Code, eval c = f :=
⟨fun h => by
induction h
case zero => exact ⟨zero, rfl⟩
case succ => | Mathlib.Computability.PartrecCode.698_0.A3c3Aev6SyIRjCJ | /-- A function is partial recursive if and only if there is a code implementing it. -/
theorem exists_code {f : ℕ →. ℕ} : Nat.Partrec f ↔ ∃ c : Code, eval c = f | Mathlib_Computability_PartrecCode |
case left
f : ℕ →. ℕ
⊢ ∃ c, eval c = ↑fun n => (unpair n).1
case right
f : ℕ →. ℕ
⊢ ∃ c, eval c = ↑fun n => (unpair n).2
case pair
f f✝ g✝ : ℕ →. ℕ
a✝¹ : Partrec f✝
a✝ : Partrec g✝
a_ih✝¹ : ∃ c, eval c = f✝
a_ih✝ : ∃ c, eval c = g✝
⊢ ∃ c, eval c = fun n => Seq.seq (Nat.pair <$> f✝ n) fun x => g✝ n
case comp
f f✝ g✝ : ℕ... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | case left => exact ⟨left, rfl⟩ | /-- A function is partial recursive if and only if there is a code implementing it. -/
theorem exists_code {f : ℕ →. ℕ} : Nat.Partrec f ↔ ∃ c : Code, eval c = f :=
⟨fun h => by
induction h
case zero => exact ⟨zero, rfl⟩
case succ => exact ⟨succ, rfl⟩
| Mathlib.Computability.PartrecCode.698_0.A3c3Aev6SyIRjCJ | /-- A function is partial recursive if and only if there is a code implementing it. -/
theorem exists_code {f : ℕ →. ℕ} : Nat.Partrec f ↔ ∃ c : Code, eval c = f | Mathlib_Computability_PartrecCode |
f : ℕ →. ℕ
⊢ ∃ c, eval c = ↑fun n => (unpair n).1 | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | case left => exact ⟨left, rfl⟩ | /-- A function is partial recursive if and only if there is a code implementing it. -/
theorem exists_code {f : ℕ →. ℕ} : Nat.Partrec f ↔ ∃ c : Code, eval c = f :=
⟨fun h => by
induction h
case zero => exact ⟨zero, rfl⟩
case succ => exact ⟨succ, rfl⟩
| Mathlib.Computability.PartrecCode.698_0.A3c3Aev6SyIRjCJ | /-- A function is partial recursive if and only if there is a code implementing it. -/
theorem exists_code {f : ℕ →. ℕ} : Nat.Partrec f ↔ ∃ c : Code, eval c = f | Mathlib_Computability_PartrecCode |
f : ℕ →. ℕ
⊢ ∃ c, eval c = ↑fun n => (unpair n).1 | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | exact ⟨left, rfl⟩ | /-- A function is partial recursive if and only if there is a code implementing it. -/
theorem exists_code {f : ℕ →. ℕ} : Nat.Partrec f ↔ ∃ c : Code, eval c = f :=
⟨fun h => by
induction h
case zero => exact ⟨zero, rfl⟩
case succ => exact ⟨succ, rfl⟩
case left => | Mathlib.Computability.PartrecCode.698_0.A3c3Aev6SyIRjCJ | /-- A function is partial recursive if and only if there is a code implementing it. -/
theorem exists_code {f : ℕ →. ℕ} : Nat.Partrec f ↔ ∃ c : Code, eval c = f | Mathlib_Computability_PartrecCode |
case right
f : ℕ →. ℕ
⊢ ∃ c, eval c = ↑fun n => (unpair n).2
case pair
f f✝ g✝ : ℕ →. ℕ
a✝¹ : Partrec f✝
a✝ : Partrec g✝
a_ih✝¹ : ∃ c, eval c = f✝
a_ih✝ : ∃ c, eval c = g✝
⊢ ∃ c, eval c = fun n => Seq.seq (Nat.pair <$> f✝ n) fun x => g✝ n
case comp
f f✝ g✝ : ℕ →. ℕ
a✝¹ : Partrec f✝
a✝ : Partrec g✝
a_ih✝¹ : ∃ c, eval c ... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | case right => exact ⟨right, rfl⟩ | /-- A function is partial recursive if and only if there is a code implementing it. -/
theorem exists_code {f : ℕ →. ℕ} : Nat.Partrec f ↔ ∃ c : Code, eval c = f :=
⟨fun h => by
induction h
case zero => exact ⟨zero, rfl⟩
case succ => exact ⟨succ, rfl⟩
case left => exact ⟨left, rfl⟩
| Mathlib.Computability.PartrecCode.698_0.A3c3Aev6SyIRjCJ | /-- A function is partial recursive if and only if there is a code implementing it. -/
theorem exists_code {f : ℕ →. ℕ} : Nat.Partrec f ↔ ∃ c : Code, eval c = f | Mathlib_Computability_PartrecCode |
f : ℕ →. ℕ
⊢ ∃ c, eval c = ↑fun n => (unpair n).2 | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | case right => exact ⟨right, rfl⟩ | /-- A function is partial recursive if and only if there is a code implementing it. -/
theorem exists_code {f : ℕ →. ℕ} : Nat.Partrec f ↔ ∃ c : Code, eval c = f :=
⟨fun h => by
induction h
case zero => exact ⟨zero, rfl⟩
case succ => exact ⟨succ, rfl⟩
case left => exact ⟨left, rfl⟩
| Mathlib.Computability.PartrecCode.698_0.A3c3Aev6SyIRjCJ | /-- A function is partial recursive if and only if there is a code implementing it. -/
theorem exists_code {f : ℕ →. ℕ} : Nat.Partrec f ↔ ∃ c : Code, eval c = f | Mathlib_Computability_PartrecCode |
f : ℕ →. ℕ
⊢ ∃ c, eval c = ↑fun n => (unpair n).2 | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | exact ⟨right, rfl⟩ | /-- A function is partial recursive if and only if there is a code implementing it. -/
theorem exists_code {f : ℕ →. ℕ} : Nat.Partrec f ↔ ∃ c : Code, eval c = f :=
⟨fun h => by
induction h
case zero => exact ⟨zero, rfl⟩
case succ => exact ⟨succ, rfl⟩
case left => exact ⟨left, rfl⟩
case right => | Mathlib.Computability.PartrecCode.698_0.A3c3Aev6SyIRjCJ | /-- A function is partial recursive if and only if there is a code implementing it. -/
theorem exists_code {f : ℕ →. ℕ} : Nat.Partrec f ↔ ∃ c : Code, eval c = f | Mathlib_Computability_PartrecCode |
case pair
f f✝ g✝ : ℕ →. ℕ
a✝¹ : Partrec f✝
a✝ : Partrec g✝
a_ih✝¹ : ∃ c, eval c = f✝
a_ih✝ : ∃ c, eval c = g✝
⊢ ∃ c, eval c = fun n => Seq.seq (Nat.pair <$> f✝ n) fun x => g✝ n
case comp
f f✝ g✝ : ℕ →. ℕ
a✝¹ : Partrec f✝
a✝ : Partrec g✝
a_ih✝¹ : ∃ c, eval c = f✝
a_ih✝ : ∃ c, eval c = g✝
⊢ ∃ c, eval c = fun n => g✝ n >... | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | case pair f g pf pg hf hg =>
rcases hf with ⟨cf, rfl⟩; rcases hg with ⟨cg, rfl⟩
exact ⟨pair cf cg, rfl⟩ | /-- A function is partial recursive if and only if there is a code implementing it. -/
theorem exists_code {f : ℕ →. ℕ} : Nat.Partrec f ↔ ∃ c : Code, eval c = f :=
⟨fun h => by
induction h
case zero => exact ⟨zero, rfl⟩
case succ => exact ⟨succ, rfl⟩
case left => exact ⟨left, rfl⟩
case right => ex... | Mathlib.Computability.PartrecCode.698_0.A3c3Aev6SyIRjCJ | /-- A function is partial recursive if and only if there is a code implementing it. -/
theorem exists_code {f : ℕ →. ℕ} : Nat.Partrec f ↔ ∃ c : Code, eval c = f | Mathlib_Computability_PartrecCode |
f✝ f g : ℕ →. ℕ
pf : Partrec f
pg : Partrec g
hf : ∃ c, eval c = f
hg : ∃ c, eval c = g
⊢ ∃ c, eval c = fun n => Seq.seq (Nat.pair <$> f n) fun x => g n | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | case pair f g pf pg hf hg =>
rcases hf with ⟨cf, rfl⟩; rcases hg with ⟨cg, rfl⟩
exact ⟨pair cf cg, rfl⟩ | /-- A function is partial recursive if and only if there is a code implementing it. -/
theorem exists_code {f : ℕ →. ℕ} : Nat.Partrec f ↔ ∃ c : Code, eval c = f :=
⟨fun h => by
induction h
case zero => exact ⟨zero, rfl⟩
case succ => exact ⟨succ, rfl⟩
case left => exact ⟨left, rfl⟩
case right => ex... | Mathlib.Computability.PartrecCode.698_0.A3c3Aev6SyIRjCJ | /-- A function is partial recursive if and only if there is a code implementing it. -/
theorem exists_code {f : ℕ →. ℕ} : Nat.Partrec f ↔ ∃ c : Code, eval c = f | Mathlib_Computability_PartrecCode |
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