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 |
|---|---|---|---|---|---|---|
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 <$> f n) fun x => g n | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | rcases hf with ⟨cf, rfl⟩ | /-- A function is partial recursive if and only if there is a code implementing it. -/
theorem exists_code {f : ℕ →. ℕ} : Nat.Partrec f ↔ ∃ c : Code, eval c = f :=
⟨fun h => by
induction h
case zero => exact ⟨zero, rfl⟩
case succ => exact ⟨succ, rfl⟩
case left => exact ⟨left, rfl⟩
case right => ex... | Mathlib.Computability.PartrecCode.698_0.A3c3Aev6SyIRjCJ | /-- A function is partial recursive if and only if there is a code implementing it. -/
theorem exists_code {f : ℕ →. ℕ} : Nat.Partrec f ↔ ∃ c : Code, eval c = f | Mathlib_Computability_PartrecCode |
case intro
f g : ℕ →. ℕ
pg : Partrec g
hg : ∃ c, eval c = g
cf : Code
pf : Partrec (eval cf)
⊢ ∃ c, eval c = fun n => Seq.seq (Nat.pair <$> eval cf 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... | rcases hg with ⟨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 |
case intro.intro
f : ℕ →. ℕ
cf : Code
pf : Partrec (eval cf)
cg : Code
pg : Partrec (eval cg)
⊢ ∃ c, eval c = fun n => Seq.seq (Nat.pair <$> eval cf n) fun x => eval cg 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... | 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 |
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 >>= f✝
case prec
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 =
unpaired fun a n =>
Nat.rec... | /-
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 comp f g pf pg hf hg =>
rcases hf with ⟨cf, rfl⟩; rcases hg with ⟨cg, rfl⟩
exact ⟨comp 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 => g n >>= 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... | case comp f g pf pg hf hg =>
rcases hf with ⟨cf, rfl⟩; rcases hg with ⟨cg, rfl⟩
exact ⟨comp 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 => g n >>= 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... | rcases hf with ⟨cf, 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 |
case intro
f g : ℕ →. ℕ
pg : Partrec g
hg : ∃ c, eval c = g
cf : Code
pf : Partrec (eval cf)
⊢ ∃ c, eval c = fun n => g n >>= eval cf | /-
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... | rcases hg with ⟨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 |
case intro.intro
f : ℕ →. ℕ
cf : Code
pf : Partrec (eval cf)
cg : Code
pg : Partrec (eval cg)
⊢ ∃ c, eval c = fun n => eval cg n >>= eval cf | /-
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 ⟨comp 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 |
case prec
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 =
unpaired fun a n =>
Nat.rec (f✝ a)
(fun y IH => do
let i ← IH
g✝ (Nat.pair a (Nat.pair y i)))
n
case rfind
f f✝ : ℕ →. ℕ
a✝ : Partr... | /-
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 prec f g pf pg hf hg =>
rcases hf with ⟨cf, rfl⟩; rcases hg with ⟨cg, rfl⟩
exact ⟨prec 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 =
unpaired fun a n =>
Nat.rec (f a)
(fun y IH => do
let i ← IH
g (Nat.pair a (Nat.pair y i)))
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 prec f g pf pg hf hg =>
rcases hf with ⟨cf, rfl⟩; rcases hg with ⟨cg, rfl⟩
exact ⟨prec 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 =
unpaired fun a n =>
Nat.rec (f a)
(fun y IH => do
let i ← IH
g (Nat.pair a (Nat.pair y i)))
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... | rcases hf with ⟨cf, 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 |
case intro
f g : ℕ →. ℕ
pg : Partrec g
hg : ∃ c, eval c = g
cf : Code
pf : Partrec (eval cf)
⊢ ∃ c,
eval c =
unpaired fun a n =>
Nat.rec (eval cf a)
(fun y IH => do
let i ← IH
g (Nat.pair a (Nat.pair y i)))
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... | rcases hg with ⟨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 |
case intro.intro
f : ℕ →. ℕ
cf : Code
pf : Partrec (eval cf)
cg : Code
pg : Partrec (eval cg)
⊢ ∃ c,
eval c =
unpaired fun a n =>
Nat.rec (eval cf a)
(fun y IH => do
let i ← IH
eval cg (Nat.pair a (Nat.pair y i)))
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... | exact ⟨prec 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 |
case rfind
f f✝ : ℕ →. ℕ
a✝ : Partrec f✝
a_ih✝ : ∃ c, eval c = f✝
⊢ ∃ c, eval c = fun a => Nat.rfind fun n => (fun m => decide (m = 0)) <$> f✝ (Nat.pair a 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 rfind f pf hf =>
rcases hf with ⟨cf, rfl⟩
refine' ⟨comp (rfind' cf) (pair Code.id zero), _⟩
simp [eval, Seq.seq, pure, PFun.pure, Part.map_id'] | /-- 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 : ℕ →. ℕ
pf : Partrec f
hf : ∃ c, eval c = f
⊢ ∃ c, eval c = fun a => Nat.rfind fun n => (fun m => decide (m = 0)) <$> f (Nat.pair a n) | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | case rfind f pf hf =>
rcases hf with ⟨cf, rfl⟩
refine' ⟨comp (rfind' cf) (pair Code.id zero), _⟩
simp [eval, Seq.seq, pure, PFun.pure, Part.map_id'] | /-- A function is partial recursive if and only if there is a code implementing it. -/
theorem exists_code {f : ℕ →. ℕ} : Nat.Partrec f ↔ ∃ c : Code, eval c = f :=
⟨fun h => by
induction h
case zero => exact ⟨zero, rfl⟩
case succ => exact ⟨succ, rfl⟩
case left => exact ⟨left, rfl⟩
case right => ex... | Mathlib.Computability.PartrecCode.698_0.A3c3Aev6SyIRjCJ | /-- A function is partial recursive if and only if there is a code implementing it. -/
theorem exists_code {f : ℕ →. ℕ} : Nat.Partrec f ↔ ∃ c : Code, eval c = f | Mathlib_Computability_PartrecCode |
f✝ f : ℕ →. ℕ
pf : Partrec f
hf : ∃ c, eval c = f
⊢ ∃ c, eval c = fun a => Nat.rfind fun n => (fun m => decide (m = 0)) <$> f (Nat.pair a 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... | rcases hf with ⟨cf, 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 |
case intro
f : ℕ →. ℕ
cf : Code
pf : Partrec (eval cf)
⊢ ∃ c, eval c = fun a => Nat.rfind fun n => (fun m => decide (m = 0)) <$> eval cf (Nat.pair a n) | /-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
/-!
# G... | refine' ⟨comp (rfind' cf) (pair Code.id zero), _⟩ | /-- A function is partial recursive if and only if there is a code implementing it. -/
theorem exists_code {f : ℕ →. ℕ} : Nat.Partrec f ↔ ∃ c : Code, eval c = f :=
⟨fun h => by
induction h
case zero => exact ⟨zero, rfl⟩
case succ => exact ⟨succ, rfl⟩
case left => exact ⟨left, rfl⟩
case right => ex... | Mathlib.Computability.PartrecCode.698_0.A3c3Aev6SyIRjCJ | /-- A function is partial recursive if and only if there is a code implementing it. -/
theorem exists_code {f : ℕ →. ℕ} : Nat.Partrec f ↔ ∃ c : Code, eval c = f | Mathlib_Computability_PartrecCode |
case intro
f : ℕ →. ℕ
cf : Code
pf : Partrec (eval cf)
⊢ eval (comp (rfind' cf) (pair Code.id zero)) = fun a =>
Nat.rfind fun n => (fun m => decide (m = 0)) <$> eval cf (Nat.pair a 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 [eval, Seq.seq, pure, PFun.pure, Part.map_id'] | /-- 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 : ℕ →. ℕ
h : ∃ c, eval c = f
⊢ Partrec 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... | rcases h with ⟨c, 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 |
case intro
c : Code
⊢ Partrec (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... | induction c | /-- 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 |
case intro.zero
⊢ Partrec (eval zero)
case intro.succ
⊢ Partrec (eval succ)
case intro.left
⊢ Partrec (eval left)
case intro.right
⊢ Partrec (eval right)
case intro.pair
a✝¹ a✝ : Code
a_ih✝¹ : Partrec (eval a✝¹)
a_ih✝ : Partrec (eval a✝)
⊢ Partrec (eval (pair a✝¹ a✝))
case intro.comp
a✝¹ a✝ : Code
a_ih✝¹ : Partrec (eva... | /-
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 Nat.Partrec.zero | /-- 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 |
⊢ Partrec (eval 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... | case zero => exact Nat.Partrec.zero | /-- 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 |
⊢ Partrec (eval 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... | exact Nat.Partrec.zero | /-- 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 |
case intro.succ
⊢ Partrec (eval succ)
case intro.left
⊢ Partrec (eval left)
case intro.right
⊢ Partrec (eval right)
case intro.pair
a✝¹ a✝ : Code
a_ih✝¹ : Partrec (eval a✝¹)
a_ih✝ : Partrec (eval a✝)
⊢ Partrec (eval (pair a✝¹ a✝))
case intro.comp
a✝¹ a✝ : Code
a_ih✝¹ : Partrec (eval a✝¹)
a_ih✝ : Partrec (eval a✝)
⊢ Par... | /-
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 Nat.Partrec.succ | /-- 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 |
⊢ Partrec (eval 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 Nat.Partrec.succ | /-- 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 |
⊢ Partrec (eval 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 Nat.Partrec.succ | /-- 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 |
case intro.left
⊢ Partrec (eval left)
case intro.right
⊢ Partrec (eval right)
case intro.pair
a✝¹ a✝ : Code
a_ih✝¹ : Partrec (eval a✝¹)
a_ih✝ : Partrec (eval a✝)
⊢ Partrec (eval (pair a✝¹ a✝))
case intro.comp
a✝¹ a✝ : Code
a_ih✝¹ : Partrec (eval a✝¹)
a_ih✝ : Partrec (eval a✝)
⊢ Partrec (eval (comp a✝¹ a✝))
case intro.p... | /-
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 Nat.Partrec.left | /-- 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 |
⊢ Partrec (eval left) | /-
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 Nat.Partrec.left | /-- 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 |
⊢ Partrec (eval left) | /-
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.Partrec.left | /-- 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 |
case intro.right
⊢ Partrec (eval right)
case intro.pair
a✝¹ a✝ : Code
a_ih✝¹ : Partrec (eval a✝¹)
a_ih✝ : Partrec (eval a✝)
⊢ Partrec (eval (pair a✝¹ a✝))
case intro.comp
a✝¹ a✝ : Code
a_ih✝¹ : Partrec (eval a✝¹)
a_ih✝ : Partrec (eval a✝)
⊢ Partrec (eval (comp a✝¹ a✝))
case intro.prec
a✝¹ a✝ : Code
a_ih✝¹ : Partrec (ev... | /-
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 Nat.Partrec.right | /-- 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 |
⊢ Partrec (eval right) | /-
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 Nat.Partrec.right | /-- 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 |
⊢ Partrec (eval right) | /-
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.Partrec.right | /-- 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 |
case intro.pair
a✝¹ a✝ : Code
a_ih✝¹ : Partrec (eval a✝¹)
a_ih✝ : Partrec (eval a✝)
⊢ Partrec (eval (pair a✝¹ a✝))
case intro.comp
a✝¹ a✝ : Code
a_ih✝¹ : Partrec (eval a✝¹)
a_ih✝ : Partrec (eval a✝)
⊢ Partrec (eval (comp a✝¹ a✝))
case intro.prec
a✝¹ a✝ : Code
a_ih✝¹ : Partrec (eval a✝¹)
a_ih✝ : Partrec (eval a✝)
⊢ Part... | /-
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 cf cg pf pg => exact pf.pair pg | /-- 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 |
cf cg : Code
pf : Partrec (eval cf)
pg : Partrec (eval cg)
⊢ Partrec (eval (pair cf cg)) | /-
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 cf cg pf pg => exact pf.pair pg | /-- 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 |
cf cg : Code
pf : Partrec (eval cf)
pg : Partrec (eval cg)
⊢ Partrec (eval (pair cf cg)) | /-
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 pf.pair pg | /-- 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 |
case intro.comp
a✝¹ a✝ : Code
a_ih✝¹ : Partrec (eval a✝¹)
a_ih✝ : Partrec (eval a✝)
⊢ Partrec (eval (comp a✝¹ a✝))
case intro.prec
a✝¹ a✝ : Code
a_ih✝¹ : Partrec (eval a✝¹)
a_ih✝ : Partrec (eval a✝)
⊢ Partrec (eval (prec a✝¹ a✝))
case intro.rfind' a✝ : Code a_ih✝ : Partrec (eval a✝) ⊢ Partrec (eval (rfind' 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... | case comp cf cg pf pg => exact pf.comp pg | /-- 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 |
cf cg : Code
pf : Partrec (eval cf)
pg : Partrec (eval cg)
⊢ Partrec (eval (comp cf cg)) | /-
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 comp cf cg pf pg => exact pf.comp pg | /-- 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 |
cf cg : Code
pf : Partrec (eval cf)
pg : Partrec (eval cg)
⊢ Partrec (eval (comp cf cg)) | /-
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 pf.comp pg | /-- 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 |
case intro.prec
a✝¹ a✝ : Code
a_ih✝¹ : Partrec (eval a✝¹)
a_ih✝ : Partrec (eval a✝)
⊢ Partrec (eval (prec a✝¹ a✝))
case intro.rfind' a✝ : Code a_ih✝ : Partrec (eval a✝) ⊢ Partrec (eval (rfind' 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... | case prec cf cg pf pg => exact pf.prec pg | /-- 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 |
cf cg : Code
pf : Partrec (eval cf)
pg : Partrec (eval cg)
⊢ Partrec (eval (prec cf cg)) | /-
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 prec cf cg pf pg => exact pf.prec pg | /-- 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 |
cf cg : Code
pf : Partrec (eval cf)
pg : Partrec (eval cg)
⊢ Partrec (eval (prec cf cg)) | /-
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 pf.prec pg | /-- 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 |
case intro.rfind'
a✝ : Code
a_ih✝ : Partrec (eval a✝)
⊢ Partrec (eval (rfind' 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... | case rfind' cf pf => exact pf.rfind' | /-- 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 |
cf : Code
pf : Partrec (eval cf)
⊢ Partrec (eval (rfind' cf)) | /-
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 rfind' cf pf => exact pf.rfind' | /-- 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 |
cf : Code
pf : Partrec (eval cf)
⊢ Partrec (eval (rfind' cf)) | /-
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 pf.rfind' | /-- 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 |
k : ℕ
cf cg : Code
⊢ (invImage (fun a => PSigma.casesOn a fun k snd => (k, snd)) Prod.instWellFoundedRelationProd).1
{ fst := k + 1, snd := cf } { fst := Nat.succ k, snd := pair cf cg } | /-
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... | { decreasing_with simp (config := { arith := true }) [Zero.zero]; done } | /-- A modified evaluation for the code which returns an `Option ℕ` instead of a `Part ℕ`. To avoid
undecidability, `evaln` takes a parameter `k` and fails if it encounters a number ≥ k in the course
of its execution. Other than this, the semantics are the same as in `Nat.Partrec.Code.eval`.
-/
def evaln : ℕ → Code → ℕ ... | Mathlib.Computability.PartrecCode.732_0.A3c3Aev6SyIRjCJ | /-- A modified evaluation for the code which returns an `Option ℕ` instead of a `Part ℕ`. To avoid
undecidability, `evaln` takes a parameter `k` and fails if it encounters a number ≥ k in the course
of its execution. Other than this, the semantics are the same as in `Nat.Partrec.Code.eval`.
-/
def evaln : ℕ → Code → ℕ ... | Mathlib_Computability_PartrecCode |
k : ℕ
cf cg : Code
⊢ (invImage (fun a => PSigma.casesOn a fun k snd => (k, snd)) Prod.instWellFoundedRelationProd).1
{ fst := k + 1, snd := cf } { fst := Nat.succ k, snd := pair cf cg } | /-
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... | decreasing_with simp (config := { arith := true }) [Zero.zero]; done | /-- A modified evaluation for the code which returns an `Option ℕ` instead of a `Part ℕ`. To avoid
undecidability, `evaln` takes a parameter `k` and fails if it encounters a number ≥ k in the course
of its execution. Other than this, the semantics are the same as in `Nat.Partrec.Code.eval`.
-/
def evaln : ℕ → Code → ℕ ... | Mathlib.Computability.PartrecCode.732_0.A3c3Aev6SyIRjCJ | /-- A modified evaluation for the code which returns an `Option ℕ` instead of a `Part ℕ`. To avoid
undecidability, `evaln` takes a parameter `k` and fails if it encounters a number ≥ k in the course
of its execution. Other than this, the semantics are the same as in `Nat.Partrec.Code.eval`.
-/
def evaln : ℕ → Code → ℕ ... | Mathlib_Computability_PartrecCode |
case h
k : ℕ
cf cg : Code
⊢ sizeOf cf < sizeOf (pair cf cg) | /-
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 := { arith := true }) [Zero.zero] | /-- A modified evaluation for the code which returns an `Option ℕ` instead of a `Part ℕ`. To avoid
undecidability, `evaln` takes a parameter `k` and fails if it encounters a number ≥ k in the course
of its execution. Other than this, the semantics are the same as in `Nat.Partrec.Code.eval`.
-/
def evaln : ℕ → Code → ℕ ... | Mathlib.Computability.PartrecCode.732_0.A3c3Aev6SyIRjCJ | /-- A modified evaluation for the code which returns an `Option ℕ` instead of a `Part ℕ`. To avoid
undecidability, `evaln` takes a parameter `k` and fails if it encounters a number ≥ k in the course
of its execution. Other than this, the semantics are the same as in `Nat.Partrec.Code.eval`.
-/
def evaln : ℕ → Code → ℕ ... | Mathlib_Computability_PartrecCode |
/-
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... | done | /-- A modified evaluation for the code which returns an `Option ℕ` instead of a `Part ℕ`. To avoid
undecidability, `evaln` takes a parameter `k` and fails if it encounters a number ≥ k in the course
of its execution. Other than this, the semantics are the same as in `Nat.Partrec.Code.eval`.
-/
def evaln : ℕ → Code → ℕ ... | Mathlib.Computability.PartrecCode.732_0.A3c3Aev6SyIRjCJ | /-- A modified evaluation for the code which returns an `Option ℕ` instead of a `Part ℕ`. To avoid
undecidability, `evaln` takes a parameter `k` and fails if it encounters a number ≥ k in the course
of its execution. Other than this, the semantics are the same as in `Nat.Partrec.Code.eval`.
-/
def evaln : ℕ → Code → ℕ ... | Mathlib_Computability_PartrecCode | |
k : ℕ
cf cg : Code
⊢ (invImage (fun a => PSigma.casesOn a fun k snd => (k, snd)) Prod.instWellFoundedRelationProd).1
{ fst := k + 1, snd := cg } { fst := Nat.succ k, snd := pair cf cg } | /-
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... | { decreasing_with simp (config := { arith := true }) [Zero.zero]; done } | /-- A modified evaluation for the code which returns an `Option ℕ` instead of a `Part ℕ`. To avoid
undecidability, `evaln` takes a parameter `k` and fails if it encounters a number ≥ k in the course
of its execution. Other than this, the semantics are the same as in `Nat.Partrec.Code.eval`.
-/
def evaln : ℕ → Code → ℕ ... | Mathlib.Computability.PartrecCode.732_0.A3c3Aev6SyIRjCJ | /-- A modified evaluation for the code which returns an `Option ℕ` instead of a `Part ℕ`. To avoid
undecidability, `evaln` takes a parameter `k` and fails if it encounters a number ≥ k in the course
of its execution. Other than this, the semantics are the same as in `Nat.Partrec.Code.eval`.
-/
def evaln : ℕ → Code → ℕ ... | Mathlib_Computability_PartrecCode |
k : ℕ
cf cg : Code
⊢ (invImage (fun a => PSigma.casesOn a fun k snd => (k, snd)) Prod.instWellFoundedRelationProd).1
{ fst := k + 1, snd := cg } { fst := Nat.succ k, snd := pair cf cg } | /-
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... | decreasing_with simp (config := { arith := true }) [Zero.zero]; done | /-- A modified evaluation for the code which returns an `Option ℕ` instead of a `Part ℕ`. To avoid
undecidability, `evaln` takes a parameter `k` and fails if it encounters a number ≥ k in the course
of its execution. Other than this, the semantics are the same as in `Nat.Partrec.Code.eval`.
-/
def evaln : ℕ → Code → ℕ ... | Mathlib.Computability.PartrecCode.732_0.A3c3Aev6SyIRjCJ | /-- A modified evaluation for the code which returns an `Option ℕ` instead of a `Part ℕ`. To avoid
undecidability, `evaln` takes a parameter `k` and fails if it encounters a number ≥ k in the course
of its execution. Other than this, the semantics are the same as in `Nat.Partrec.Code.eval`.
-/
def evaln : ℕ → Code → ℕ ... | Mathlib_Computability_PartrecCode |
case h
k : ℕ
cf cg : Code
⊢ sizeOf cg < sizeOf (pair cf cg) | /-
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 := { arith := true }) [Zero.zero] | /-- A modified evaluation for the code which returns an `Option ℕ` instead of a `Part ℕ`. To avoid
undecidability, `evaln` takes a parameter `k` and fails if it encounters a number ≥ k in the course
of its execution. Other than this, the semantics are the same as in `Nat.Partrec.Code.eval`.
-/
def evaln : ℕ → Code → ℕ ... | Mathlib.Computability.PartrecCode.732_0.A3c3Aev6SyIRjCJ | /-- A modified evaluation for the code which returns an `Option ℕ` instead of a `Part ℕ`. To avoid
undecidability, `evaln` takes a parameter `k` and fails if it encounters a number ≥ k in the course
of its execution. Other than this, the semantics are the same as in `Nat.Partrec.Code.eval`.
-/
def evaln : ℕ → Code → ℕ ... | Mathlib_Computability_PartrecCode |
/-
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... | done | /-- A modified evaluation for the code which returns an `Option ℕ` instead of a `Part ℕ`. To avoid
undecidability, `evaln` takes a parameter `k` and fails if it encounters a number ≥ k in the course
of its execution. Other than this, the semantics are the same as in `Nat.Partrec.Code.eval`.
-/
def evaln : ℕ → Code → ℕ ... | Mathlib.Computability.PartrecCode.732_0.A3c3Aev6SyIRjCJ | /-- A modified evaluation for the code which returns an `Option ℕ` instead of a `Part ℕ`. To avoid
undecidability, `evaln` takes a parameter `k` and fails if it encounters a number ≥ k in the course
of its execution. Other than this, the semantics are the same as in `Nat.Partrec.Code.eval`.
-/
def evaln : ℕ → Code → ℕ ... | Mathlib_Computability_PartrecCode | |
k : ℕ
cf cg : Code
⊢ (invImage (fun a => PSigma.casesOn a fun k snd => (k, snd)) Prod.instWellFoundedRelationProd).1
{ fst := k + 1, snd := cg } { fst := Nat.succ k, snd := comp cf cg } | /-
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... | { decreasing_with simp (config := { arith := true }) [Zero.zero]; done } | /-- A modified evaluation for the code which returns an `Option ℕ` instead of a `Part ℕ`. To avoid
undecidability, `evaln` takes a parameter `k` and fails if it encounters a number ≥ k in the course
of its execution. Other than this, the semantics are the same as in `Nat.Partrec.Code.eval`.
-/
def evaln : ℕ → Code → ℕ ... | Mathlib.Computability.PartrecCode.732_0.A3c3Aev6SyIRjCJ | /-- A modified evaluation for the code which returns an `Option ℕ` instead of a `Part ℕ`. To avoid
undecidability, `evaln` takes a parameter `k` and fails if it encounters a number ≥ k in the course
of its execution. Other than this, the semantics are the same as in `Nat.Partrec.Code.eval`.
-/
def evaln : ℕ → Code → ℕ ... | Mathlib_Computability_PartrecCode |
k : ℕ
cf cg : Code
⊢ (invImage (fun a => PSigma.casesOn a fun k snd => (k, snd)) Prod.instWellFoundedRelationProd).1
{ fst := k + 1, snd := cg } { fst := Nat.succ k, snd := comp cf cg } | /-
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... | decreasing_with simp (config := { arith := true }) [Zero.zero]; done | /-- A modified evaluation for the code which returns an `Option ℕ` instead of a `Part ℕ`. To avoid
undecidability, `evaln` takes a parameter `k` and fails if it encounters a number ≥ k in the course
of its execution. Other than this, the semantics are the same as in `Nat.Partrec.Code.eval`.
-/
def evaln : ℕ → Code → ℕ ... | Mathlib.Computability.PartrecCode.732_0.A3c3Aev6SyIRjCJ | /-- A modified evaluation for the code which returns an `Option ℕ` instead of a `Part ℕ`. To avoid
undecidability, `evaln` takes a parameter `k` and fails if it encounters a number ≥ k in the course
of its execution. Other than this, the semantics are the same as in `Nat.Partrec.Code.eval`.
-/
def evaln : ℕ → Code → ℕ ... | Mathlib_Computability_PartrecCode |
case h
k : ℕ
cf cg : Code
⊢ sizeOf cg < sizeOf (comp cf cg) | /-
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 := { arith := true }) [Zero.zero] | /-- A modified evaluation for the code which returns an `Option ℕ` instead of a `Part ℕ`. To avoid
undecidability, `evaln` takes a parameter `k` and fails if it encounters a number ≥ k in the course
of its execution. Other than this, the semantics are the same as in `Nat.Partrec.Code.eval`.
-/
def evaln : ℕ → Code → ℕ ... | Mathlib.Computability.PartrecCode.732_0.A3c3Aev6SyIRjCJ | /-- A modified evaluation for the code which returns an `Option ℕ` instead of a `Part ℕ`. To avoid
undecidability, `evaln` takes a parameter `k` and fails if it encounters a number ≥ k in the course
of its execution. Other than this, the semantics are the same as in `Nat.Partrec.Code.eval`.
-/
def evaln : ℕ → Code → ℕ ... | Mathlib_Computability_PartrecCode |
/-
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... | done | /-- A modified evaluation for the code which returns an `Option ℕ` instead of a `Part ℕ`. To avoid
undecidability, `evaln` takes a parameter `k` and fails if it encounters a number ≥ k in the course
of its execution. Other than this, the semantics are the same as in `Nat.Partrec.Code.eval`.
-/
def evaln : ℕ → Code → ℕ ... | Mathlib.Computability.PartrecCode.732_0.A3c3Aev6SyIRjCJ | /-- A modified evaluation for the code which returns an `Option ℕ` instead of a `Part ℕ`. To avoid
undecidability, `evaln` takes a parameter `k` and fails if it encounters a number ≥ k in the course
of its execution. Other than this, the semantics are the same as in `Nat.Partrec.Code.eval`.
-/
def evaln : ℕ → Code → ℕ ... | Mathlib_Computability_PartrecCode | |
k : ℕ
cf cg : Code
⊢ (invImage (fun a => PSigma.casesOn a fun k snd => (k, snd)) Prod.instWellFoundedRelationProd).1
{ fst := k + 1, snd := cf } { fst := Nat.succ k, snd := comp cf cg } | /-
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... | { decreasing_with simp (config := { arith := true }) [Zero.zero]; done } | /-- A modified evaluation for the code which returns an `Option ℕ` instead of a `Part ℕ`. To avoid
undecidability, `evaln` takes a parameter `k` and fails if it encounters a number ≥ k in the course
of its execution. Other than this, the semantics are the same as in `Nat.Partrec.Code.eval`.
-/
def evaln : ℕ → Code → ℕ ... | Mathlib.Computability.PartrecCode.732_0.A3c3Aev6SyIRjCJ | /-- A modified evaluation for the code which returns an `Option ℕ` instead of a `Part ℕ`. To avoid
undecidability, `evaln` takes a parameter `k` and fails if it encounters a number ≥ k in the course
of its execution. Other than this, the semantics are the same as in `Nat.Partrec.Code.eval`.
-/
def evaln : ℕ → Code → ℕ ... | Mathlib_Computability_PartrecCode |
k : ℕ
cf cg : Code
⊢ (invImage (fun a => PSigma.casesOn a fun k snd => (k, snd)) Prod.instWellFoundedRelationProd).1
{ fst := k + 1, snd := cf } { fst := Nat.succ k, snd := comp cf cg } | /-
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... | decreasing_with simp (config := { arith := true }) [Zero.zero]; done | /-- A modified evaluation for the code which returns an `Option ℕ` instead of a `Part ℕ`. To avoid
undecidability, `evaln` takes a parameter `k` and fails if it encounters a number ≥ k in the course
of its execution. Other than this, the semantics are the same as in `Nat.Partrec.Code.eval`.
-/
def evaln : ℕ → Code → ℕ ... | Mathlib.Computability.PartrecCode.732_0.A3c3Aev6SyIRjCJ | /-- A modified evaluation for the code which returns an `Option ℕ` instead of a `Part ℕ`. To avoid
undecidability, `evaln` takes a parameter `k` and fails if it encounters a number ≥ k in the course
of its execution. Other than this, the semantics are the same as in `Nat.Partrec.Code.eval`.
-/
def evaln : ℕ → Code → ℕ ... | Mathlib_Computability_PartrecCode |
case h
k : ℕ
cf cg : Code
⊢ sizeOf cf < sizeOf (comp cf cg) | /-
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 := { arith := true }) [Zero.zero] | /-- A modified evaluation for the code which returns an `Option ℕ` instead of a `Part ℕ`. To avoid
undecidability, `evaln` takes a parameter `k` and fails if it encounters a number ≥ k in the course
of its execution. Other than this, the semantics are the same as in `Nat.Partrec.Code.eval`.
-/
def evaln : ℕ → Code → ℕ ... | Mathlib.Computability.PartrecCode.732_0.A3c3Aev6SyIRjCJ | /-- A modified evaluation for the code which returns an `Option ℕ` instead of a `Part ℕ`. To avoid
undecidability, `evaln` takes a parameter `k` and fails if it encounters a number ≥ k in the course
of its execution. Other than this, the semantics are the same as in `Nat.Partrec.Code.eval`.
-/
def evaln : ℕ → Code → ℕ ... | Mathlib_Computability_PartrecCode |
/-
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... | done | /-- A modified evaluation for the code which returns an `Option ℕ` instead of a `Part ℕ`. To avoid
undecidability, `evaln` takes a parameter `k` and fails if it encounters a number ≥ k in the course
of its execution. Other than this, the semantics are the same as in `Nat.Partrec.Code.eval`.
-/
def evaln : ℕ → Code → ℕ ... | Mathlib.Computability.PartrecCode.732_0.A3c3Aev6SyIRjCJ | /-- A modified evaluation for the code which returns an `Option ℕ` instead of a `Part ℕ`. To avoid
undecidability, `evaln` takes a parameter `k` and fails if it encounters a number ≥ k in the course
of its execution. Other than this, the semantics are the same as in `Nat.Partrec.Code.eval`.
-/
def evaln : ℕ → Code → ℕ ... | Mathlib_Computability_PartrecCode | |
k : ℕ
cf cg : Code
⊢ (invImage (fun a => PSigma.casesOn a fun k snd => (k, snd)) Prod.instWellFoundedRelationProd).1
{ fst := k + 1, snd := cf } { fst := Nat.succ k, snd := prec cf cg } | /-
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... | { decreasing_with simp (config := { arith := true }) [Zero.zero]; done } | /-- A modified evaluation for the code which returns an `Option ℕ` instead of a `Part ℕ`. To avoid
undecidability, `evaln` takes a parameter `k` and fails if it encounters a number ≥ k in the course
of its execution. Other than this, the semantics are the same as in `Nat.Partrec.Code.eval`.
-/
def evaln : ℕ → Code → ℕ ... | Mathlib.Computability.PartrecCode.732_0.A3c3Aev6SyIRjCJ | /-- A modified evaluation for the code which returns an `Option ℕ` instead of a `Part ℕ`. To avoid
undecidability, `evaln` takes a parameter `k` and fails if it encounters a number ≥ k in the course
of its execution. Other than this, the semantics are the same as in `Nat.Partrec.Code.eval`.
-/
def evaln : ℕ → Code → ℕ ... | Mathlib_Computability_PartrecCode |
k : ℕ
cf cg : Code
⊢ (invImage (fun a => PSigma.casesOn a fun k snd => (k, snd)) Prod.instWellFoundedRelationProd).1
{ fst := k + 1, snd := cf } { fst := Nat.succ k, snd := prec cf cg } | /-
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... | decreasing_with simp (config := { arith := true }) [Zero.zero]; done | /-- A modified evaluation for the code which returns an `Option ℕ` instead of a `Part ℕ`. To avoid
undecidability, `evaln` takes a parameter `k` and fails if it encounters a number ≥ k in the course
of its execution. Other than this, the semantics are the same as in `Nat.Partrec.Code.eval`.
-/
def evaln : ℕ → Code → ℕ ... | Mathlib.Computability.PartrecCode.732_0.A3c3Aev6SyIRjCJ | /-- A modified evaluation for the code which returns an `Option ℕ` instead of a `Part ℕ`. To avoid
undecidability, `evaln` takes a parameter `k` and fails if it encounters a number ≥ k in the course
of its execution. Other than this, the semantics are the same as in `Nat.Partrec.Code.eval`.
-/
def evaln : ℕ → Code → ℕ ... | Mathlib_Computability_PartrecCode |
case h
k : ℕ
cf cg : Code
⊢ sizeOf cf < sizeOf (prec cf cg) | /-
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 := { arith := true }) [Zero.zero] | /-- A modified evaluation for the code which returns an `Option ℕ` instead of a `Part ℕ`. To avoid
undecidability, `evaln` takes a parameter `k` and fails if it encounters a number ≥ k in the course
of its execution. Other than this, the semantics are the same as in `Nat.Partrec.Code.eval`.
-/
def evaln : ℕ → Code → ℕ ... | Mathlib.Computability.PartrecCode.732_0.A3c3Aev6SyIRjCJ | /-- A modified evaluation for the code which returns an `Option ℕ` instead of a `Part ℕ`. To avoid
undecidability, `evaln` takes a parameter `k` and fails if it encounters a number ≥ k in the course
of its execution. Other than this, the semantics are the same as in `Nat.Partrec.Code.eval`.
-/
def evaln : ℕ → Code → ℕ ... | Mathlib_Computability_PartrecCode |
/-
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... | done | /-- A modified evaluation for the code which returns an `Option ℕ` instead of a `Part ℕ`. To avoid
undecidability, `evaln` takes a parameter `k` and fails if it encounters a number ≥ k in the course
of its execution. Other than this, the semantics are the same as in `Nat.Partrec.Code.eval`.
-/
def evaln : ℕ → Code → ℕ ... | Mathlib.Computability.PartrecCode.732_0.A3c3Aev6SyIRjCJ | /-- A modified evaluation for the code which returns an `Option ℕ` instead of a `Part ℕ`. To avoid
undecidability, `evaln` takes a parameter `k` and fails if it encounters a number ≥ k in the course
of its execution. Other than this, the semantics are the same as in `Nat.Partrec.Code.eval`.
-/
def evaln : ℕ → Code → ℕ ... | Mathlib_Computability_PartrecCode | |
k : ℕ
cf cg : Code
⊢ (invImage (fun a => PSigma.casesOn a fun k snd => (k, snd)) Prod.instWellFoundedRelationProd).1
{ fst := k, snd := prec cf cg } { fst := Nat.succ k, snd := prec cf cg } | /-
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... | { decreasing_with simp (config := { arith := true }) [Zero.zero]; done } | /-- A modified evaluation for the code which returns an `Option ℕ` instead of a `Part ℕ`. To avoid
undecidability, `evaln` takes a parameter `k` and fails if it encounters a number ≥ k in the course
of its execution. Other than this, the semantics are the same as in `Nat.Partrec.Code.eval`.
-/
def evaln : ℕ → Code → ℕ ... | Mathlib.Computability.PartrecCode.732_0.A3c3Aev6SyIRjCJ | /-- A modified evaluation for the code which returns an `Option ℕ` instead of a `Part ℕ`. To avoid
undecidability, `evaln` takes a parameter `k` and fails if it encounters a number ≥ k in the course
of its execution. Other than this, the semantics are the same as in `Nat.Partrec.Code.eval`.
-/
def evaln : ℕ → Code → ℕ ... | Mathlib_Computability_PartrecCode |
k : ℕ
cf cg : Code
⊢ (invImage (fun a => PSigma.casesOn a fun k snd => (k, snd)) Prod.instWellFoundedRelationProd).1
{ fst := k, snd := prec cf cg } { fst := Nat.succ k, snd := prec cf cg } | /-
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... | decreasing_with simp (config := { arith := true }) [Zero.zero]; done | /-- A modified evaluation for the code which returns an `Option ℕ` instead of a `Part ℕ`. To avoid
undecidability, `evaln` takes a parameter `k` and fails if it encounters a number ≥ k in the course
of its execution. Other than this, the semantics are the same as in `Nat.Partrec.Code.eval`.
-/
def evaln : ℕ → Code → ℕ ... | Mathlib.Computability.PartrecCode.732_0.A3c3Aev6SyIRjCJ | /-- A modified evaluation for the code which returns an `Option ℕ` instead of a `Part ℕ`. To avoid
undecidability, `evaln` takes a parameter `k` and fails if it encounters a number ≥ k in the course
of its execution. Other than this, the semantics are the same as in `Nat.Partrec.Code.eval`.
-/
def evaln : ℕ → Code → ℕ ... | Mathlib_Computability_PartrecCode |
case h
k : ℕ
cf cg : Code
⊢ k < Nat.succ k | /-
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 := { arith := true }) [Zero.zero] | /-- A modified evaluation for the code which returns an `Option ℕ` instead of a `Part ℕ`. To avoid
undecidability, `evaln` takes a parameter `k` and fails if it encounters a number ≥ k in the course
of its execution. Other than this, the semantics are the same as in `Nat.Partrec.Code.eval`.
-/
def evaln : ℕ → Code → ℕ ... | Mathlib.Computability.PartrecCode.732_0.A3c3Aev6SyIRjCJ | /-- A modified evaluation for the code which returns an `Option ℕ` instead of a `Part ℕ`. To avoid
undecidability, `evaln` takes a parameter `k` and fails if it encounters a number ≥ k in the course
of its execution. Other than this, the semantics are the same as in `Nat.Partrec.Code.eval`.
-/
def evaln : ℕ → Code → ℕ ... | Mathlib_Computability_PartrecCode |
/-
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... | done | /-- A modified evaluation for the code which returns an `Option ℕ` instead of a `Part ℕ`. To avoid
undecidability, `evaln` takes a parameter `k` and fails if it encounters a number ≥ k in the course
of its execution. Other than this, the semantics are the same as in `Nat.Partrec.Code.eval`.
-/
def evaln : ℕ → Code → ℕ ... | Mathlib.Computability.PartrecCode.732_0.A3c3Aev6SyIRjCJ | /-- A modified evaluation for the code which returns an `Option ℕ` instead of a `Part ℕ`. To avoid
undecidability, `evaln` takes a parameter `k` and fails if it encounters a number ≥ k in the course
of its execution. Other than this, the semantics are the same as in `Nat.Partrec.Code.eval`.
-/
def evaln : ℕ → Code → ℕ ... | Mathlib_Computability_PartrecCode | |
k : ℕ
cf cg : Code
⊢ (invImage (fun a => PSigma.casesOn a fun k snd => (k, snd)) Prod.instWellFoundedRelationProd).1
{ fst := k + 1, snd := cg } { fst := Nat.succ k, snd := prec cf cg } | /-
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... | { decreasing_with simp (config := { arith := true }) [Zero.zero]; done } | /-- A modified evaluation for the code which returns an `Option ℕ` instead of a `Part ℕ`. To avoid
undecidability, `evaln` takes a parameter `k` and fails if it encounters a number ≥ k in the course
of its execution. Other than this, the semantics are the same as in `Nat.Partrec.Code.eval`.
-/
def evaln : ℕ → Code → ℕ ... | Mathlib.Computability.PartrecCode.732_0.A3c3Aev6SyIRjCJ | /-- A modified evaluation for the code which returns an `Option ℕ` instead of a `Part ℕ`. To avoid
undecidability, `evaln` takes a parameter `k` and fails if it encounters a number ≥ k in the course
of its execution. Other than this, the semantics are the same as in `Nat.Partrec.Code.eval`.
-/
def evaln : ℕ → Code → ℕ ... | Mathlib_Computability_PartrecCode |
k : ℕ
cf cg : Code
⊢ (invImage (fun a => PSigma.casesOn a fun k snd => (k, snd)) Prod.instWellFoundedRelationProd).1
{ fst := k + 1, snd := cg } { fst := Nat.succ k, snd := prec cf cg } | /-
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... | decreasing_with simp (config := { arith := true }) [Zero.zero]; done | /-- A modified evaluation for the code which returns an `Option ℕ` instead of a `Part ℕ`. To avoid
undecidability, `evaln` takes a parameter `k` and fails if it encounters a number ≥ k in the course
of its execution. Other than this, the semantics are the same as in `Nat.Partrec.Code.eval`.
-/
def evaln : ℕ → Code → ℕ ... | Mathlib.Computability.PartrecCode.732_0.A3c3Aev6SyIRjCJ | /-- A modified evaluation for the code which returns an `Option ℕ` instead of a `Part ℕ`. To avoid
undecidability, `evaln` takes a parameter `k` and fails if it encounters a number ≥ k in the course
of its execution. Other than this, the semantics are the same as in `Nat.Partrec.Code.eval`.
-/
def evaln : ℕ → Code → ℕ ... | Mathlib_Computability_PartrecCode |
case h
k : ℕ
cf cg : Code
⊢ sizeOf cg < sizeOf (prec cf cg) | /-
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 := { arith := true }) [Zero.zero] | /-- A modified evaluation for the code which returns an `Option ℕ` instead of a `Part ℕ`. To avoid
undecidability, `evaln` takes a parameter `k` and fails if it encounters a number ≥ k in the course
of its execution. Other than this, the semantics are the same as in `Nat.Partrec.Code.eval`.
-/
def evaln : ℕ → Code → ℕ ... | Mathlib.Computability.PartrecCode.732_0.A3c3Aev6SyIRjCJ | /-- A modified evaluation for the code which returns an `Option ℕ` instead of a `Part ℕ`. To avoid
undecidability, `evaln` takes a parameter `k` and fails if it encounters a number ≥ k in the course
of its execution. Other than this, the semantics are the same as in `Nat.Partrec.Code.eval`.
-/
def evaln : ℕ → Code → ℕ ... | Mathlib_Computability_PartrecCode |
/-
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... | done | /-- A modified evaluation for the code which returns an `Option ℕ` instead of a `Part ℕ`. To avoid
undecidability, `evaln` takes a parameter `k` and fails if it encounters a number ≥ k in the course
of its execution. Other than this, the semantics are the same as in `Nat.Partrec.Code.eval`.
-/
def evaln : ℕ → Code → ℕ ... | Mathlib.Computability.PartrecCode.732_0.A3c3Aev6SyIRjCJ | /-- A modified evaluation for the code which returns an `Option ℕ` instead of a `Part ℕ`. To avoid
undecidability, `evaln` takes a parameter `k` and fails if it encounters a number ≥ k in the course
of its execution. Other than this, the semantics are the same as in `Nat.Partrec.Code.eval`.
-/
def evaln : ℕ → Code → ℕ ... | Mathlib_Computability_PartrecCode | |
k : ℕ
cf : Code
⊢ (invImage (fun a => PSigma.casesOn a fun k snd => (k, snd)) Prod.instWellFoundedRelationProd).1
{ fst := k + 1, snd := cf } { fst := Nat.succ k, snd := rfind' cf } | /-
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... | { decreasing_with simp (config := { arith := true }) [Zero.zero]; done } | /-- A modified evaluation for the code which returns an `Option ℕ` instead of a `Part ℕ`. To avoid
undecidability, `evaln` takes a parameter `k` and fails if it encounters a number ≥ k in the course
of its execution. Other than this, the semantics are the same as in `Nat.Partrec.Code.eval`.
-/
def evaln : ℕ → Code → ℕ ... | Mathlib.Computability.PartrecCode.732_0.A3c3Aev6SyIRjCJ | /-- A modified evaluation for the code which returns an `Option ℕ` instead of a `Part ℕ`. To avoid
undecidability, `evaln` takes a parameter `k` and fails if it encounters a number ≥ k in the course
of its execution. Other than this, the semantics are the same as in `Nat.Partrec.Code.eval`.
-/
def evaln : ℕ → Code → ℕ ... | Mathlib_Computability_PartrecCode |
k : ℕ
cf : Code
⊢ (invImage (fun a => PSigma.casesOn a fun k snd => (k, snd)) Prod.instWellFoundedRelationProd).1
{ fst := k + 1, snd := cf } { fst := Nat.succ k, snd := rfind' cf } | /-
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... | decreasing_with simp (config := { arith := true }) [Zero.zero]; done | /-- A modified evaluation for the code which returns an `Option ℕ` instead of a `Part ℕ`. To avoid
undecidability, `evaln` takes a parameter `k` and fails if it encounters a number ≥ k in the course
of its execution. Other than this, the semantics are the same as in `Nat.Partrec.Code.eval`.
-/
def evaln : ℕ → Code → ℕ ... | Mathlib.Computability.PartrecCode.732_0.A3c3Aev6SyIRjCJ | /-- A modified evaluation for the code which returns an `Option ℕ` instead of a `Part ℕ`. To avoid
undecidability, `evaln` takes a parameter `k` and fails if it encounters a number ≥ k in the course
of its execution. Other than this, the semantics are the same as in `Nat.Partrec.Code.eval`.
-/
def evaln : ℕ → Code → ℕ ... | Mathlib_Computability_PartrecCode |
case h
k : ℕ
cf : Code
⊢ sizeOf cf < sizeOf (rfind' cf) | /-
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 := { arith := true }) [Zero.zero] | /-- A modified evaluation for the code which returns an `Option ℕ` instead of a `Part ℕ`. To avoid
undecidability, `evaln` takes a parameter `k` and fails if it encounters a number ≥ k in the course
of its execution. Other than this, the semantics are the same as in `Nat.Partrec.Code.eval`.
-/
def evaln : ℕ → Code → ℕ ... | Mathlib.Computability.PartrecCode.732_0.A3c3Aev6SyIRjCJ | /-- A modified evaluation for the code which returns an `Option ℕ` instead of a `Part ℕ`. To avoid
undecidability, `evaln` takes a parameter `k` and fails if it encounters a number ≥ k in the course
of its execution. Other than this, the semantics are the same as in `Nat.Partrec.Code.eval`.
-/
def evaln : ℕ → Code → ℕ ... | Mathlib_Computability_PartrecCode |
/-
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... | done | /-- A modified evaluation for the code which returns an `Option ℕ` instead of a `Part ℕ`. To avoid
undecidability, `evaln` takes a parameter `k` and fails if it encounters a number ≥ k in the course
of its execution. Other than this, the semantics are the same as in `Nat.Partrec.Code.eval`.
-/
def evaln : ℕ → Code → ℕ ... | Mathlib.Computability.PartrecCode.732_0.A3c3Aev6SyIRjCJ | /-- A modified evaluation for the code which returns an `Option ℕ` instead of a `Part ℕ`. To avoid
undecidability, `evaln` takes a parameter `k` and fails if it encounters a number ≥ k in the course
of its execution. Other than this, the semantics are the same as in `Nat.Partrec.Code.eval`.
-/
def evaln : ℕ → Code → ℕ ... | Mathlib_Computability_PartrecCode | |
k : ℕ
cf : Code
⊢ (invImage (fun a => PSigma.casesOn a fun k snd => (k, snd)) Prod.instWellFoundedRelationProd).1
{ fst := k, snd := rfind' cf } { fst := Nat.succ k, snd := rfind' cf } | /-
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... | { decreasing_with simp (config := { arith := true }) [Zero.zero]; done } | /-- A modified evaluation for the code which returns an `Option ℕ` instead of a `Part ℕ`. To avoid
undecidability, `evaln` takes a parameter `k` and fails if it encounters a number ≥ k in the course
of its execution. Other than this, the semantics are the same as in `Nat.Partrec.Code.eval`.
-/
def evaln : ℕ → Code → ℕ ... | Mathlib.Computability.PartrecCode.732_0.A3c3Aev6SyIRjCJ | /-- A modified evaluation for the code which returns an `Option ℕ` instead of a `Part ℕ`. To avoid
undecidability, `evaln` takes a parameter `k` and fails if it encounters a number ≥ k in the course
of its execution. Other than this, the semantics are the same as in `Nat.Partrec.Code.eval`.
-/
def evaln : ℕ → Code → ℕ ... | Mathlib_Computability_PartrecCode |
k : ℕ
cf : Code
⊢ (invImage (fun a => PSigma.casesOn a fun k snd => (k, snd)) Prod.instWellFoundedRelationProd).1
{ fst := k, snd := rfind' cf } { fst := Nat.succ k, snd := rfind' cf } | /-
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... | decreasing_with simp (config := { arith := true }) [Zero.zero]; done | /-- A modified evaluation for the code which returns an `Option ℕ` instead of a `Part ℕ`. To avoid
undecidability, `evaln` takes a parameter `k` and fails if it encounters a number ≥ k in the course
of its execution. Other than this, the semantics are the same as in `Nat.Partrec.Code.eval`.
-/
def evaln : ℕ → Code → ℕ ... | Mathlib.Computability.PartrecCode.732_0.A3c3Aev6SyIRjCJ | /-- A modified evaluation for the code which returns an `Option ℕ` instead of a `Part ℕ`. To avoid
undecidability, `evaln` takes a parameter `k` and fails if it encounters a number ≥ k in the course
of its execution. Other than this, the semantics are the same as in `Nat.Partrec.Code.eval`.
-/
def evaln : ℕ → Code → ℕ ... | Mathlib_Computability_PartrecCode |
case h
k : ℕ
cf : Code
⊢ k < Nat.succ k | /-
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 := { arith := true }) [Zero.zero] | /-- A modified evaluation for the code which returns an `Option ℕ` instead of a `Part ℕ`. To avoid
undecidability, `evaln` takes a parameter `k` and fails if it encounters a number ≥ k in the course
of its execution. Other than this, the semantics are the same as in `Nat.Partrec.Code.eval`.
-/
def evaln : ℕ → Code → ℕ ... | Mathlib.Computability.PartrecCode.732_0.A3c3Aev6SyIRjCJ | /-- A modified evaluation for the code which returns an `Option ℕ` instead of a `Part ℕ`. To avoid
undecidability, `evaln` takes a parameter `k` and fails if it encounters a number ≥ k in the course
of its execution. Other than this, the semantics are the same as in `Nat.Partrec.Code.eval`.
-/
def evaln : ℕ → Code → ℕ ... | Mathlib_Computability_PartrecCode |
/-
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... | done | /-- A modified evaluation for the code which returns an `Option ℕ` instead of a `Part ℕ`. To avoid
undecidability, `evaln` takes a parameter `k` and fails if it encounters a number ≥ k in the course
of its execution. Other than this, the semantics are the same as in `Nat.Partrec.Code.eval`.
-/
def evaln : ℕ → Code → ℕ ... | Mathlib.Computability.PartrecCode.732_0.A3c3Aev6SyIRjCJ | /-- A modified evaluation for the code which returns an `Option ℕ` instead of a `Part ℕ`. To avoid
undecidability, `evaln` takes a parameter `k` and fails if it encounters a number ≥ k in the course
of its execution. Other than this, the semantics are the same as in `Nat.Partrec.Code.eval`.
-/
def evaln : ℕ → Code → ℕ ... | Mathlib_Computability_PartrecCode | |
c : Code
n x : ℕ
h : x ∈ evaln 0 c n
⊢ n < 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... | simp [evaln] at h | theorem evaln_bound : ∀ {k c n x}, x ∈ evaln k c n → n < k
| 0, c, n, x, h => by | Mathlib.Computability.PartrecCode.775_0.A3c3Aev6SyIRjCJ | theorem evaln_bound : ∀ {k c n x}, x ∈ evaln k c n → n < k
| 0, c, n, x, h => by simp [evaln] at h
| k + 1, c, n, x, h => by
suffices ∀ {o : Option ℕ}, x ∈ do { guard (n ≤ k); o } → n < k + 1 by
cases c <;> rw [evaln] at h <;> exact this h
simpa [Bind.bind] using Nat.lt_succ_of_le | Mathlib_Computability_PartrecCode |
k : ℕ
c : Code
n x : ℕ
h : x ∈ evaln (k + 1) c n
⊢ n < k + 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... | suffices ∀ {o : Option ℕ}, x ∈ do { guard (n ≤ k); o } → n < k + 1 by
cases c <;> rw [evaln] at h <;> exact this h | theorem evaln_bound : ∀ {k c n x}, x ∈ evaln k c n → n < k
| 0, c, n, x, h => by simp [evaln] at h
| k + 1, c, n, x, h => by
| Mathlib.Computability.PartrecCode.775_0.A3c3Aev6SyIRjCJ | theorem evaln_bound : ∀ {k c n x}, x ∈ evaln k c n → n < k
| 0, c, n, x, h => by simp [evaln] at h
| k + 1, c, n, x, h => by
suffices ∀ {o : Option ℕ}, x ∈ do { guard (n ≤ k); o } → n < k + 1 by
cases c <;> rw [evaln] at h <;> exact this h
simpa [Bind.bind] using Nat.lt_succ_of_le | Mathlib_Computability_PartrecCode |
k : ℕ
c : Code
n x : ℕ
h : x ∈ evaln (k + 1) c n
this :
∀ {o : Option ℕ},
(x ∈ do
guard (n ≤ k)
o) →
n < k + 1
⊢ n < k + 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... | cases c | theorem evaln_bound : ∀ {k c n x}, x ∈ evaln k c n → n < k
| 0, c, n, x, h => by simp [evaln] at h
| k + 1, c, n, x, h => by
suffices ∀ {o : Option ℕ}, x ∈ do { guard (n ≤ k); o } → n < k + 1 by
| Mathlib.Computability.PartrecCode.775_0.A3c3Aev6SyIRjCJ | theorem evaln_bound : ∀ {k c n x}, x ∈ evaln k c n → n < k
| 0, c, n, x, h => by simp [evaln] at h
| k + 1, c, n, x, h => by
suffices ∀ {o : Option ℕ}, x ∈ do { guard (n ≤ k); o } → n < k + 1 by
cases c <;> rw [evaln] at h <;> exact this h
simpa [Bind.bind] using Nat.lt_succ_of_le | Mathlib_Computability_PartrecCode |
case zero
k n x : ℕ
this :
∀ {o : Option ℕ},
(x ∈ do
guard (n ≤ k)
o) →
n < k + 1
h : x ∈ evaln (k + 1) zero n
⊢ n < k + 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... | rw [evaln] at h | theorem evaln_bound : ∀ {k c n x}, x ∈ evaln k c n → n < k
| 0, c, n, x, h => by simp [evaln] at h
| k + 1, c, n, x, h => by
suffices ∀ {o : Option ℕ}, x ∈ do { guard (n ≤ k); o } → n < k + 1 by
cases c <;> | Mathlib.Computability.PartrecCode.775_0.A3c3Aev6SyIRjCJ | theorem evaln_bound : ∀ {k c n x}, x ∈ evaln k c n → n < k
| 0, c, n, x, h => by simp [evaln] at h
| k + 1, c, n, x, h => by
suffices ∀ {o : Option ℕ}, x ∈ do { guard (n ≤ k); o } → n < k + 1 by
cases c <;> rw [evaln] at h <;> exact this h
simpa [Bind.bind] using Nat.lt_succ_of_le | Mathlib_Computability_PartrecCode |
case succ
k n x : ℕ
this :
∀ {o : Option ℕ},
(x ∈ do
guard (n ≤ k)
o) →
n < k + 1
h : x ∈ evaln (k + 1) succ n
⊢ n < k + 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... | rw [evaln] at h | theorem evaln_bound : ∀ {k c n x}, x ∈ evaln k c n → n < k
| 0, c, n, x, h => by simp [evaln] at h
| k + 1, c, n, x, h => by
suffices ∀ {o : Option ℕ}, x ∈ do { guard (n ≤ k); o } → n < k + 1 by
cases c <;> | Mathlib.Computability.PartrecCode.775_0.A3c3Aev6SyIRjCJ | theorem evaln_bound : ∀ {k c n x}, x ∈ evaln k c n → n < k
| 0, c, n, x, h => by simp [evaln] at h
| k + 1, c, n, x, h => by
suffices ∀ {o : Option ℕ}, x ∈ do { guard (n ≤ k); o } → n < k + 1 by
cases c <;> rw [evaln] at h <;> exact this h
simpa [Bind.bind] using Nat.lt_succ_of_le | Mathlib_Computability_PartrecCode |
case left
k n x : ℕ
this :
∀ {o : Option ℕ},
(x ∈ do
guard (n ≤ k)
o) →
n < k + 1
h : x ∈ evaln (k + 1) left n
⊢ n < k + 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... | rw [evaln] at h | theorem evaln_bound : ∀ {k c n x}, x ∈ evaln k c n → n < k
| 0, c, n, x, h => by simp [evaln] at h
| k + 1, c, n, x, h => by
suffices ∀ {o : Option ℕ}, x ∈ do { guard (n ≤ k); o } → n < k + 1 by
cases c <;> | Mathlib.Computability.PartrecCode.775_0.A3c3Aev6SyIRjCJ | theorem evaln_bound : ∀ {k c n x}, x ∈ evaln k c n → n < k
| 0, c, n, x, h => by simp [evaln] at h
| k + 1, c, n, x, h => by
suffices ∀ {o : Option ℕ}, x ∈ do { guard (n ≤ k); o } → n < k + 1 by
cases c <;> rw [evaln] at h <;> exact this h
simpa [Bind.bind] using Nat.lt_succ_of_le | Mathlib_Computability_PartrecCode |
case right
k n x : ℕ
this :
∀ {o : Option ℕ},
(x ∈ do
guard (n ≤ k)
o) →
n < k + 1
h : x ∈ evaln (k + 1) right n
⊢ n < k + 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... | rw [evaln] at h | theorem evaln_bound : ∀ {k c n x}, x ∈ evaln k c n → n < k
| 0, c, n, x, h => by simp [evaln] at h
| k + 1, c, n, x, h => by
suffices ∀ {o : Option ℕ}, x ∈ do { guard (n ≤ k); o } → n < k + 1 by
cases c <;> | Mathlib.Computability.PartrecCode.775_0.A3c3Aev6SyIRjCJ | theorem evaln_bound : ∀ {k c n x}, x ∈ evaln k c n → n < k
| 0, c, n, x, h => by simp [evaln] at h
| k + 1, c, n, x, h => by
suffices ∀ {o : Option ℕ}, x ∈ do { guard (n ≤ k); o } → n < k + 1 by
cases c <;> rw [evaln] at h <;> exact this h
simpa [Bind.bind] using Nat.lt_succ_of_le | Mathlib_Computability_PartrecCode |
case pair
k n x : ℕ
this :
∀ {o : Option ℕ},
(x ∈ do
guard (n ≤ k)
o) →
n < k + 1
a✝¹ a✝ : Code
h : x ∈ evaln (k + 1) (pair a✝¹ a✝) n
⊢ n < k + 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... | rw [evaln] at h | theorem evaln_bound : ∀ {k c n x}, x ∈ evaln k c n → n < k
| 0, c, n, x, h => by simp [evaln] at h
| k + 1, c, n, x, h => by
suffices ∀ {o : Option ℕ}, x ∈ do { guard (n ≤ k); o } → n < k + 1 by
cases c <;> | Mathlib.Computability.PartrecCode.775_0.A3c3Aev6SyIRjCJ | theorem evaln_bound : ∀ {k c n x}, x ∈ evaln k c n → n < k
| 0, c, n, x, h => by simp [evaln] at h
| k + 1, c, n, x, h => by
suffices ∀ {o : Option ℕ}, x ∈ do { guard (n ≤ k); o } → n < k + 1 by
cases c <;> rw [evaln] at h <;> exact this h
simpa [Bind.bind] using Nat.lt_succ_of_le | Mathlib_Computability_PartrecCode |
case comp
k n x : ℕ
this :
∀ {o : Option ℕ},
(x ∈ do
guard (n ≤ k)
o) →
n < k + 1
a✝¹ a✝ : Code
h : x ∈ evaln (k + 1) (comp a✝¹ a✝) n
⊢ n < k + 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... | rw [evaln] at h | theorem evaln_bound : ∀ {k c n x}, x ∈ evaln k c n → n < k
| 0, c, n, x, h => by simp [evaln] at h
| k + 1, c, n, x, h => by
suffices ∀ {o : Option ℕ}, x ∈ do { guard (n ≤ k); o } → n < k + 1 by
cases c <;> | Mathlib.Computability.PartrecCode.775_0.A3c3Aev6SyIRjCJ | theorem evaln_bound : ∀ {k c n x}, x ∈ evaln k c n → n < k
| 0, c, n, x, h => by simp [evaln] at h
| k + 1, c, n, x, h => by
suffices ∀ {o : Option ℕ}, x ∈ do { guard (n ≤ k); o } → n < k + 1 by
cases c <;> rw [evaln] at h <;> exact this h
simpa [Bind.bind] using Nat.lt_succ_of_le | Mathlib_Computability_PartrecCode |
case prec
k n x : ℕ
this :
∀ {o : Option ℕ},
(x ∈ do
guard (n ≤ k)
o) →
n < k + 1
a✝¹ a✝ : Code
h : x ∈ evaln (k + 1) (prec a✝¹ a✝) n
⊢ n < k + 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... | rw [evaln] at h | theorem evaln_bound : ∀ {k c n x}, x ∈ evaln k c n → n < k
| 0, c, n, x, h => by simp [evaln] at h
| k + 1, c, n, x, h => by
suffices ∀ {o : Option ℕ}, x ∈ do { guard (n ≤ k); o } → n < k + 1 by
cases c <;> | Mathlib.Computability.PartrecCode.775_0.A3c3Aev6SyIRjCJ | theorem evaln_bound : ∀ {k c n x}, x ∈ evaln k c n → n < k
| 0, c, n, x, h => by simp [evaln] at h
| k + 1, c, n, x, h => by
suffices ∀ {o : Option ℕ}, x ∈ do { guard (n ≤ k); o } → n < k + 1 by
cases c <;> rw [evaln] at h <;> exact this h
simpa [Bind.bind] using Nat.lt_succ_of_le | Mathlib_Computability_PartrecCode |
case rfind'
k n x : ℕ
this :
∀ {o : Option ℕ},
(x ∈ do
guard (n ≤ k)
o) →
n < k + 1
a✝ : Code
h : x ∈ evaln (k + 1) (rfind' a✝) n
⊢ n < k + 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... | rw [evaln] at h | theorem evaln_bound : ∀ {k c n x}, x ∈ evaln k c n → n < k
| 0, c, n, x, h => by simp [evaln] at h
| k + 1, c, n, x, h => by
suffices ∀ {o : Option ℕ}, x ∈ do { guard (n ≤ k); o } → n < k + 1 by
cases c <;> | Mathlib.Computability.PartrecCode.775_0.A3c3Aev6SyIRjCJ | theorem evaln_bound : ∀ {k c n x}, x ∈ evaln k c n → n < k
| 0, c, n, x, h => by simp [evaln] at h
| k + 1, c, n, x, h => by
suffices ∀ {o : Option ℕ}, x ∈ do { guard (n ≤ k); o } → n < k + 1 by
cases c <;> rw [evaln] at h <;> exact this h
simpa [Bind.bind] using Nat.lt_succ_of_le | Mathlib_Computability_PartrecCode |
case zero
k n x : ℕ
this :
∀ {o : Option ℕ},
(x ∈ do
guard (n ≤ k)
o) →
n < k + 1
h :
x ∈
(fun n => do
guard (n ≤ k)
pure 0)
n
⊢ n < k + 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 this h | theorem evaln_bound : ∀ {k c n x}, x ∈ evaln k c n → n < k
| 0, c, n, x, h => by simp [evaln] at h
| k + 1, c, n, x, h => by
suffices ∀ {o : Option ℕ}, x ∈ do { guard (n ≤ k); o } → n < k + 1 by
cases c <;> rw [evaln] at h <;> | Mathlib.Computability.PartrecCode.775_0.A3c3Aev6SyIRjCJ | theorem evaln_bound : ∀ {k c n x}, x ∈ evaln k c n → n < k
| 0, c, n, x, h => by simp [evaln] at h
| k + 1, c, n, x, h => by
suffices ∀ {o : Option ℕ}, x ∈ do { guard (n ≤ k); o } → n < k + 1 by
cases c <;> rw [evaln] at h <;> exact this h
simpa [Bind.bind] using Nat.lt_succ_of_le | Mathlib_Computability_PartrecCode |
case succ
k n x : ℕ
this :
∀ {o : Option ℕ},
(x ∈ do
guard (n ≤ k)
o) →
n < k + 1
h :
x ∈
(fun n => do
guard (n ≤ k)
pure (Nat.succ n))
n
⊢ n < k + 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 this h | theorem evaln_bound : ∀ {k c n x}, x ∈ evaln k c n → n < k
| 0, c, n, x, h => by simp [evaln] at h
| k + 1, c, n, x, h => by
suffices ∀ {o : Option ℕ}, x ∈ do { guard (n ≤ k); o } → n < k + 1 by
cases c <;> rw [evaln] at h <;> | Mathlib.Computability.PartrecCode.775_0.A3c3Aev6SyIRjCJ | theorem evaln_bound : ∀ {k c n x}, x ∈ evaln k c n → n < k
| 0, c, n, x, h => by simp [evaln] at h
| k + 1, c, n, x, h => by
suffices ∀ {o : Option ℕ}, x ∈ do { guard (n ≤ k); o } → n < k + 1 by
cases c <;> rw [evaln] at h <;> exact this h
simpa [Bind.bind] using Nat.lt_succ_of_le | Mathlib_Computability_PartrecCode |
case left
k n x : ℕ
this :
∀ {o : Option ℕ},
(x ∈ do
guard (n ≤ k)
o) →
n < k + 1
h :
x ∈
(fun n => do
guard (n ≤ k)
pure (unpair n).1)
n
⊢ n < k + 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 this h | theorem evaln_bound : ∀ {k c n x}, x ∈ evaln k c n → n < k
| 0, c, n, x, h => by simp [evaln] at h
| k + 1, c, n, x, h => by
suffices ∀ {o : Option ℕ}, x ∈ do { guard (n ≤ k); o } → n < k + 1 by
cases c <;> rw [evaln] at h <;> | Mathlib.Computability.PartrecCode.775_0.A3c3Aev6SyIRjCJ | theorem evaln_bound : ∀ {k c n x}, x ∈ evaln k c n → n < k
| 0, c, n, x, h => by simp [evaln] at h
| k + 1, c, n, x, h => by
suffices ∀ {o : Option ℕ}, x ∈ do { guard (n ≤ k); o } → n < k + 1 by
cases c <;> rw [evaln] at h <;> exact this h
simpa [Bind.bind] using Nat.lt_succ_of_le | Mathlib_Computability_PartrecCode |
case right
k n x : ℕ
this :
∀ {o : Option ℕ},
(x ∈ do
guard (n ≤ k)
o) →
n < k + 1
h :
x ∈
(fun n => do
guard (n ≤ k)
pure (unpair n).2)
n
⊢ n < k + 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 this h | theorem evaln_bound : ∀ {k c n x}, x ∈ evaln k c n → n < k
| 0, c, n, x, h => by simp [evaln] at h
| k + 1, c, n, x, h => by
suffices ∀ {o : Option ℕ}, x ∈ do { guard (n ≤ k); o } → n < k + 1 by
cases c <;> rw [evaln] at h <;> | Mathlib.Computability.PartrecCode.775_0.A3c3Aev6SyIRjCJ | theorem evaln_bound : ∀ {k c n x}, x ∈ evaln k c n → n < k
| 0, c, n, x, h => by simp [evaln] at h
| k + 1, c, n, x, h => by
suffices ∀ {o : Option ℕ}, x ∈ do { guard (n ≤ k); o } → n < k + 1 by
cases c <;> rw [evaln] at h <;> exact this h
simpa [Bind.bind] using Nat.lt_succ_of_le | Mathlib_Computability_PartrecCode |
case pair
k n x : ℕ
this :
∀ {o : Option ℕ},
(x ∈ do
guard (n ≤ k)
o) →
n < k + 1
a✝¹ a✝ : Code
h :
x ∈
(fun n => do
guard (n ≤ k)
Seq.seq (Nat.pair <$> evaln (k + 1) a✝¹ n) fun x => evaln (k + 1) a✝ n)
n
⊢ n < k + 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 this h | theorem evaln_bound : ∀ {k c n x}, x ∈ evaln k c n → n < k
| 0, c, n, x, h => by simp [evaln] at h
| k + 1, c, n, x, h => by
suffices ∀ {o : Option ℕ}, x ∈ do { guard (n ≤ k); o } → n < k + 1 by
cases c <;> rw [evaln] at h <;> | Mathlib.Computability.PartrecCode.775_0.A3c3Aev6SyIRjCJ | theorem evaln_bound : ∀ {k c n x}, x ∈ evaln k c n → n < k
| 0, c, n, x, h => by simp [evaln] at h
| k + 1, c, n, x, h => by
suffices ∀ {o : Option ℕ}, x ∈ do { guard (n ≤ k); o } → n < k + 1 by
cases c <;> rw [evaln] at h <;> exact this h
simpa [Bind.bind] using Nat.lt_succ_of_le | Mathlib_Computability_PartrecCode |
case comp
k n x : ℕ
this :
∀ {o : Option ℕ},
(x ∈ do
guard (n ≤ k)
o) →
n < k + 1
a✝¹ a✝ : Code
h :
x ∈
(fun n => do
guard (n ≤ k)
let x ← evaln (k + 1) a✝ n
evaln (k + 1) a✝¹ x)
n
⊢ n < k + 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 this h | theorem evaln_bound : ∀ {k c n x}, x ∈ evaln k c n → n < k
| 0, c, n, x, h => by simp [evaln] at h
| k + 1, c, n, x, h => by
suffices ∀ {o : Option ℕ}, x ∈ do { guard (n ≤ k); o } → n < k + 1 by
cases c <;> rw [evaln] at h <;> | Mathlib.Computability.PartrecCode.775_0.A3c3Aev6SyIRjCJ | theorem evaln_bound : ∀ {k c n x}, x ∈ evaln k c n → n < k
| 0, c, n, x, h => by simp [evaln] at h
| k + 1, c, n, x, h => by
suffices ∀ {o : Option ℕ}, x ∈ do { guard (n ≤ k); o } → n < k + 1 by
cases c <;> rw [evaln] at h <;> exact this h
simpa [Bind.bind] using Nat.lt_succ_of_le | Mathlib_Computability_PartrecCode |
case prec
k n x : ℕ
this :
∀ {o : Option ℕ},
(x ∈ do
guard (n ≤ k)
o) →
n < k + 1
a✝¹ a✝ : Code
h :
x ∈
(fun n => do
guard (n ≤ k)
unpaired
(fun a n =>
Nat.casesOn n (evaln (k + 1) a✝¹ a) fun y => do
let i ← evaln k (prec a✝¹ 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... | exact this h | theorem evaln_bound : ∀ {k c n x}, x ∈ evaln k c n → n < k
| 0, c, n, x, h => by simp [evaln] at h
| k + 1, c, n, x, h => by
suffices ∀ {o : Option ℕ}, x ∈ do { guard (n ≤ k); o } → n < k + 1 by
cases c <;> rw [evaln] at h <;> | Mathlib.Computability.PartrecCode.775_0.A3c3Aev6SyIRjCJ | theorem evaln_bound : ∀ {k c n x}, x ∈ evaln k c n → n < k
| 0, c, n, x, h => by simp [evaln] at h
| k + 1, c, n, x, h => by
suffices ∀ {o : Option ℕ}, x ∈ do { guard (n ≤ k); o } → n < k + 1 by
cases c <;> rw [evaln] at h <;> exact this h
simpa [Bind.bind] using Nat.lt_succ_of_le | Mathlib_Computability_PartrecCode |
case rfind'
k n x : ℕ
this :
∀ {o : Option ℕ},
(x ∈ do
guard (n ≤ k)
o) →
n < k + 1
a✝ : Code
h :
x ∈
(fun n => do
guard (n ≤ k)
unpaired
(fun a m => do
let x ← evaln (k + 1) a✝ (Nat.pair a m)
if x = 0 then pure m else evaln k (rf... | /-
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 this h | theorem evaln_bound : ∀ {k c n x}, x ∈ evaln k c n → n < k
| 0, c, n, x, h => by simp [evaln] at h
| k + 1, c, n, x, h => by
suffices ∀ {o : Option ℕ}, x ∈ do { guard (n ≤ k); o } → n < k + 1 by
cases c <;> rw [evaln] at h <;> | Mathlib.Computability.PartrecCode.775_0.A3c3Aev6SyIRjCJ | theorem evaln_bound : ∀ {k c n x}, x ∈ evaln k c n → n < k
| 0, c, n, x, h => by simp [evaln] at h
| k + 1, c, n, x, h => by
suffices ∀ {o : Option ℕ}, x ∈ do { guard (n ≤ k); o } → n < k + 1 by
cases c <;> rw [evaln] at h <;> exact this h
simpa [Bind.bind] using Nat.lt_succ_of_le | Mathlib_Computability_PartrecCode |
k : ℕ
c : Code
n x : ℕ
h : x ∈ evaln (k + 1) c n
⊢ ∀ {o : Option ℕ},
(x ∈ do
guard (n ≤ k)
o) →
n < k + 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... | simpa [Bind.bind] using Nat.lt_succ_of_le | theorem evaln_bound : ∀ {k c n x}, x ∈ evaln k c n → n < k
| 0, c, n, x, h => by simp [evaln] at h
| k + 1, c, n, x, h => by
suffices ∀ {o : Option ℕ}, x ∈ do { guard (n ≤ k); o } → n < k + 1 by
cases c <;> rw [evaln] at h <;> exact this h
| Mathlib.Computability.PartrecCode.775_0.A3c3Aev6SyIRjCJ | theorem evaln_bound : ∀ {k c n x}, x ∈ evaln k c n → n < k
| 0, c, n, x, h => by simp [evaln] at h
| k + 1, c, n, x, h => by
suffices ∀ {o : Option ℕ}, x ∈ do { guard (n ≤ k); o } → n < k + 1 by
cases c <;> rw [evaln] at h <;> exact this h
simpa [Bind.bind] using Nat.lt_succ_of_le | Mathlib_Computability_PartrecCode |
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