task_id stringlengths 21 23 | challenge_id stringlengths 16 16 | proof_assistant stringclasses 1
value | session stringclasses 1
value | repo stringclasses 57
values | revision stringclasses 57
values | file_path stringlengths 13 92 | theory stringlengths 1 48 | variant int64 0 132 | challenge_type stringclasses 1
value | difficulty null | count null | by_centrality bool 1
class | ablation_prob float64 0.5 0.5 | min_depth int64 1 1 | max_depth int64 1 1 | leaves_only bool 1
class | min_size int64 0 0 | max_size stringclasses 1
value | min_centrality int64 0 0 | max_centrality stringclasses 1
value | seed int64 42 42 | n_proofs int64 1 173 | n_ablated int64 1 60 | holes_filled listlengths 1 60 | deleted_lemmas listlengths 1 1 | corollaries listlengths 1 1 | closure_size int64 1 172 | challenge_file_content stringlengths 133 350k | solution_file_content stringlengths 394 351k | solution_diff stringlengths 243 134k | manifest dict |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
ablate_e5cd8def0acd_3 | 6166196e20126869 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Bootstrapping/FixedPoint.lean | FixedPoint | 3 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 7 | 1 | [
{
"theorem_name": "multidiagonal",
"depth": 1,
"n_commands": 0,
"n_lines": 14,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": "\n haveI : 𝗘𝗤 _ ⪯ T := Entailment.WeakerThan.trans (𝓣 := 𝗜𝚺₁) inferInstance inferInstance\n complete.{0} T _ fun (V : Type) ... | [
{
"name": "substNumerals_app_quote_quote",
"text": "lemma substNumerals_app_quote_quote (σ : ArithmeticSemisentence k) (π : Fin k → ArithmeticSemisentence k) :\n substNumerals (⌜σ⌝ : V) (fun i ↦ ⌜π i⌝) = ⌜((Rew.subst (fun i ↦ ⌜π i⌝)) ▹ σ : ArithmeticSentence)⌝ := by\n simpa [Sentence.coe_quote_eq_quote]... | [
{
"name": "exclusiveMultidiagonal",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 3,
"n_lines": 7,
"n_chars": 472,
"n_subproofs": 1,
"n_tactics": 4,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": false,
"max_... | 3 | module
public import Foundation.FirstOrder.Arithmetic.Basic.Model
public import Foundation.FirstOrder.Bootstrapping.Syntax
@[expose] public section
open Classical
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
namespace Bootstrapping.Arithmetic
noncomputable def subs... | module
public import Foundation.FirstOrder.Arithmetic.Basic.Model
public import Foundation.FirstOrder.Bootstrapping.Syntax
@[expose] public section
open Classical
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
namespace Bootstrapping.Arithmetic
noncomputable def subs... | @@ -22,6 +22,10 @@
Rewriting.emb_subst_eq_subst_emb]
rfl
+lemma substNumerals_app_quote_quote (σ : ArithmeticSemisentence k) (π : Fin k → ArithmeticSemisentence k) :
+ substNumerals (⌜σ⌝ : V) (fun i ↦ ⌜π i⌝) = ⌜((Rew.subst (fun i ↦ ⌜π i⌝)) ▹ σ : ArithmeticSentence)⌝ := by
+ simpa [Sentence.coe_quote_eq_qu... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_e5cd8def0acd_4 | 0b754ffe26da682d | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Bootstrapping/FixedPoint.lean | FixedPoint | 4 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 7 | 1 | [
{
"theorem_name": "exclusiveMultifixedpoint_pi",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n apply multifixedpoint_pi; simp [h]",
"n_chars": 40,
"n_subproofs": 0,
"n_tactics": 3,
"cyclomat... | [
{
"name": "multifixedpoint_pi",
"text": "lemma multifixedpoint_pi {θ : Fin k → ArithmeticSemisentence k} (h : ∀ i, Hierarchy 𝚷 (m + 1) (θ i)) :\n Hierarchy 𝚷 (m + 1) (multifixedpoint θ i) := by\n simpa [multifixedpoint, multidiag, h] using fun _ ↦ Hierarchy.mono (s := 1) (by simp) (by simp)\n\n",
... | [
{
"name": "exclusiveMultifixedpoint_pi",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 5,
"n_chars": 224,
"n_subproofs": 0,
"n_tactics": 3,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": false,
... | 1 | module
public import Foundation.FirstOrder.Arithmetic.Basic.Model
public import Foundation.FirstOrder.Bootstrapping.Syntax
@[expose] public section
open Classical
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
namespace Bootstrapping.Arithmetic
noncomputable def subs... | module
public import Foundation.FirstOrder.Arithmetic.Basic.Model
public import Foundation.FirstOrder.Bootstrapping.Syntax
@[expose] public section
open Classical
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
namespace Bootstrapping.Arithmetic
noncomputable def subs... | @@ -121,8 +121,13 @@
simp [multidiag, Fin.val_inj]
· rintro rfl; rfl
+lemma multifixedpoint_pi {θ : Fin k → ArithmeticSemisentence k} (h : ∀ i, Hierarchy 𝚷 (m + 1) (θ i)) :
+ Hierarchy 𝚷 (m + 1) (multifixedpoint θ i) := by
+ simpa [multifixedpoint, multidiag, h] using fun _ ↦ Hierarchy.mono (s := 1) (by... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_e5cd8def0acd_5 | ea88891c17808875 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Bootstrapping/FixedPoint.lean | FixedPoint | 5 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 7 | 1 | [
{
"theorem_name": "parameterized_diagonal",
"depth": 1,
"n_commands": 0,
"n_lines": 17,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": "\n haveI : 𝗘𝗤 _ ⪯ T := Entailment.WeakerThan.trans (𝓣 := 𝗜𝚺₁) inferInstance inferInstance\n complete.{0} T _ fun (V... | [
{
"name": "substNumeralParams_app_quote",
"text": "lemma substNumeralParams_app_quote (σ τ : ArithmeticSemisentence (k + 1)) :\n (substNumeralParams k ⌜σ⌝ ⌜τ⌝ : V) = ⌜((Rew.subst (⌜τ⌝ :> fun i : Fin k ↦ #i)) ▹ σ : ArithmeticSemisentence k)⌝ := by\n simp [substNumeralParams, Sentence.quote_def, Semiformu... | [
{
"name": "parameterized_diagonal",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 20,
"n_chars": 1366,
"n_subproofs": 2,
"n_tactics": 16,
"cyclomatic": 1,
"n_automation": 5,
"n_rewrites": 0,
"n_structural": 2,
"automation_only": false,
"m... | 1 | module
public import Foundation.FirstOrder.Arithmetic.Basic.Model
public import Foundation.FirstOrder.Bootstrapping.Syntax
@[expose] public section
open Classical
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
namespace Bootstrapping.Arithmetic
noncomputable def subs... | module
public import Foundation.FirstOrder.Arithmetic.Basic.Model
public import Foundation.FirstOrder.Bootstrapping.Syntax
@[expose] public section
open Classical
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
namespace Bootstrapping.Arithmetic
noncomputable def subs... | @@ -18,6 +18,12 @@
noncomputable def substNumeralParams (k : ℕ) (φ x : V) : V := subst ℒₒᵣ (matrixToVec (numeral x :> fun i : Fin k ↦ qqBvar i)) φ
+lemma substNumeralParams_app_quote (σ τ : ArithmeticSemisentence (k + 1)) :
+ (substNumeralParams k ⌜σ⌝ ⌜τ⌝ : V) = ⌜((Rew.subst (⌜τ⌝ :> fun i : Fin k ↦ #i)) ▹ σ : A... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_e5cd8def0acd_6 | 79e2282fb9f39909 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Bootstrapping/FixedPoint.lean | FixedPoint | 6 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 7 | 1 | [
{
"theorem_name": "parameterized_diagonal",
"depth": 1,
"n_commands": 0,
"n_lines": 17,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": "\n haveI : 𝗘𝗤 _ ⪯ T := Entailment.WeakerThan.trans (𝓣 := 𝗜𝚺₁) inferInstance inferInstance\n complete.{0} T _ fun (V... | [
{
"name": "substNumeralParams_app_quote",
"text": "lemma substNumeralParams_app_quote (σ τ : ArithmeticSemisentence (k + 1)) :\n (substNumeralParams k ⌜σ⌝ ⌜τ⌝ : V) = ⌜((Rew.subst (⌜τ⌝ :> fun i : Fin k ↦ #i)) ▹ σ : ArithmeticSemisentence k)⌝ := by\n simp [substNumeralParams, Sentence.quote_def, Semiformu... | [
{
"name": "parameterized_diagonal₁",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 2,
"n_lines": 6,
"n_chars": 284,
"n_subproofs": 0,
"n_tactics": 3,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"max_... | 2 | module
public import Foundation.FirstOrder.Arithmetic.Basic.Model
public import Foundation.FirstOrder.Bootstrapping.Syntax
@[expose] public section
open Classical
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
namespace Bootstrapping.Arithmetic
noncomputable def subs... | module
public import Foundation.FirstOrder.Arithmetic.Basic.Model
public import Foundation.FirstOrder.Bootstrapping.Syntax
@[expose] public section
open Classical
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
namespace Bootstrapping.Arithmetic
noncomputable def subs... | @@ -18,6 +18,12 @@
noncomputable def substNumeralParams (k : ℕ) (φ x : V) : V := subst ℒₒᵣ (matrixToVec (numeral x :> fun i : Fin k ↦ qqBvar i)) φ
+lemma substNumeralParams_app_quote (σ τ : ArithmeticSemisentence (k + 1)) :
+ (substNumeralParams k ⌜σ⌝ ⌜τ⌝ : V) = ⌜((Rew.subst (⌜τ⌝ :> fun i : Fin k ↦ #i)) ▹ σ : A... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_e698833437c2_0 | c524a346affecc43 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Propositional/Kripke/Completeness.lean | Completeness | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 1 | 1 | [
{
"theorem_name": "truthlemma",
"depth": 1,
"n_commands": 0,
"n_lines": 7,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n induction φ generalizing t with\n | hatom => tauto;\n | hfalsum => simp only [Semantics.Bot.models_falsum, not_mem₁_falsum];\n... | [
{
"name": "truthlemma.himp",
"text": "private lemma truthlemma.himp\n (ihp : ∀ {t : (Kripke.canonicalModel 𝓢).World}, t ⊧ φ ↔ φ ∈ t.1.1)\n (ihq : ∀ {t : (Kripke.canonicalModel 𝓢).World}, t ⊧ ψ ↔ ψ ∈ t.1.1)\n : t ⊧ φ 🡒 ψ ↔ φ 🡒 ψ ∈ t.1.1 := by\n constructor;\n . contrapose;\n intro h;\n replace... | [
{
"name": "truthlemma",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 2,
"n_lines": 9,
"n_chars": 395,
"n_subproofs": 0,
"n_tactics": 12,
"cyclomatic": 2,
"n_automation": 4,
"n_rewrites": 0,
"n_structural": 2,
"automation_only": false,
"max_nesting": 2... | 1 | module
public import Foundation.Propositional.Kripke.Basic
public import Foundation.Propositional.ConsistentTableau
@[expose] public section
namespace LO.Propositional
variable {S} [Entailment S (Formula ℕ)]
variable {𝓢 : S} [Entailment.Consistent 𝓢] [Entailment.Int 𝓢]
variable {t t₁ t₂ : SaturatedConsistentTabl... | module
public import Foundation.Propositional.Kripke.Basic
public import Foundation.Propositional.ConsistentTableau
@[expose] public section
namespace LO.Propositional
variable {S} [Entailment S (Formula ℕ)]
variable {𝓢 : S} [Entailment.Consistent 𝓢] [Entailment.Int 𝓢]
variable {t t₁ t₂ : SaturatedConsistentTabl... | @@ -50,11 +50,61 @@
variable {t : (Kripke.canonicalModel 𝓢).World}
+private lemma truthlemma.himp
+ (ihp : ∀ {t : (Kripke.canonicalModel 𝓢).World}, t ⊧ φ ↔ φ ∈ t.1.1)
+ (ihq : ∀ {t : (Kripke.canonicalModel 𝓢).World}, t ⊧ ψ ↔ ψ ∈ t.1.1)
+ : t ⊧ φ 🡒 ψ ↔ φ 🡒 ψ ∈ t.1.1 := by
+ constructor;
+ . contrapose;
+ ... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_70ca7e247e94_0 | f21127b1fdd76117 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Completeness/CounterModel.lean | CounterModel | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 9 | 1 | [
{
"theorem_name": "GenericForces.neg",
"depth": 1,
"n_commands": 0,
"n_lines": 10,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n suffices p ⊫ ∼φ → p ⊫ φ → False by\n have := GenericForces.em p φ\n grind\n rintro ⟨q₀, hG₀, h₀⟩\n rintro ⟨q₁, h... | [
{
"name": "GenericForces.em",
"text": "lemma GenericForces.em (p : ℙ⁻) (φ : Proposition K) : p ⊫ φ ∨ p ⊫ ∼φ := by\n have : ∃ q ∈ genericFilter p, q ∈ decidablePoints φ :=\n (genericFilter_isGeneric p).isGeneric (decidablePoints φ) (by simp)\n rcases this with ⟨q, hqG, hq⟩\n have : q ⊩ᶜ φ ∨ q ⊩ᶜ ∼φ := ... | [
{
"name": "GenericForces.neg",
"fan_in": 3,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 12,
"n_chars": 475,
"n_subproofs": 4,
"n_tactics": 10,
"cyclomatic": 4,
"n_automation": 2,
"n_rewrites": 0,
"n_structural": 3,
"automation_only": false,
"max_nes... | 1 | module
public import Foundation.FirstOrder.Completeness.CanonicalModel
public import Foundation.FirstOrder.Completeness.CountableSublanguage
public import Foundation.FirstOrder.Ultraproduct
public import Foundation.Vorspiel.Order.Dense
public import Mathlib.Logic.Equiv.List
public import Mathlib.Logic.Encodable.Basic
... | module
public import Foundation.FirstOrder.Completeness.CanonicalModel
public import Foundation.FirstOrder.Completeness.CountableSublanguage
public import Foundation.FirstOrder.Ultraproduct
public import Foundation.Vorspiel.Order.Dense
public import Mathlib.Logic.Equiv.List
public import Mathlib.Logic.Encodable.Basic
... | @@ -80,8 +80,26 @@
local infix: 60 " ⊫ " => GenericForces
-@[simp] lemma GenericForces.neg {p : ℙ⁻} {φ : Proposition K} : p ⊫ ∼φ ↔ ¬p ⊫ φ := sorry
+lemma GenericForces.em (p : ℙ⁻) (φ : Proposition K) : p ⊫ φ ∨ p ⊫ ∼φ := by
+ have : ∃ q ∈ genericFilter p, q ∈ decidablePoints φ :=
+ (genericFilter_isGeneric p).i... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_70ca7e247e94_1 | 64b568aa1207a8c0 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Completeness/CounterModel.lean | CounterModel | 1 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 9 | 1 | [
{
"theorem_name": "GenericForces.exs",
"depth": 1,
"n_commands": 0,
"n_lines": 8,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n constructor\n · exact GenericForces.henkin\n · rintro ⟨t, q, hqG, h⟩\n refine ⟨q, hqG, ?_⟩\n suffices ∀ r ≤ q, ∃ ... | [
{
"name": "GenericForces.henkin",
"text": "lemma GenericForces.henkin {p : ℙ⁻} {φ : Semiproposition K 1} : p ⊫ ∃⁰ φ → ∃ t, p ⊫ φ/[t] := by\n have : ∃ q ∈ genericFilter p, ∀ r ≤ q, r ⊩ᶜ ∃⁰ φ → ∃ t, r ⊩ᶜ φ/[t] :=\n (genericFilter_isGeneric p).isGeneric (henkinPoints φ) (by simp)\n rcases this with ⟨q, hq... | [
{
"name": "GenericForces.exs",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 10,
"n_chars": 333,
"n_subproofs": 1,
"n_tactics": 8,
"cyclomatic": 3,
"n_automation": 2,
"n_rewrites": 0,
"n_structural": 6,
"automation_only": false,
"max_nest... | 1 | module
public import Foundation.FirstOrder.Completeness.CanonicalModel
public import Foundation.FirstOrder.Completeness.CountableSublanguage
public import Foundation.FirstOrder.Ultraproduct
public import Foundation.Vorspiel.Order.Dense
public import Mathlib.Logic.Equiv.List
public import Mathlib.Logic.Encodable.Basic
... | module
public import Foundation.FirstOrder.Completeness.CanonicalModel
public import Foundation.FirstOrder.Completeness.CountableSublanguage
public import Foundation.FirstOrder.Ultraproduct
public import Foundation.Vorspiel.Order.Dense
public import Mathlib.Logic.Equiv.List
public import Mathlib.Logic.Encodable.Basic
... | @@ -110,9 +110,19 @@
p ⊫ .nrel R v ↔ p ⊫ ∼(.rel R v) := by simp
_ ↔ ¬p ⊫ .rel R v := by rw [GenericForces.neg]
+lemma GenericForces.henkin {p : ℙ⁻} {φ : Semiproposition K 1} : p ⊫ ∃⁰ φ → ∃ t, p ⊫ φ/[t] := by
+ have : ∃ q ∈ genericFilter p, ∀ r ≤ q, r ⊩ᶜ ∃⁰ φ → ∃ t, r ⊩ᶜ φ/[t] :=
+ (genericFilter_isG... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_70ca7e247e94_2 | c161de2e19fc2658 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Completeness/CounterModel.lean | CounterModel | 2 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 9 | 1 | [
{
"theorem_name": "refl",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": "\n (forcing_lemma φ).mpr ⟨ConsistentSequent.ofUnprovable φ h, by simp, by simpa using IsWeaklyForced.refl φ h⟩",
"n_chars": 117,
"n_... | [
{
"name": "forcing_lemma",
"text": "lemma forcing_lemma (φ : Semiformula K ξ n) {fv : ξ → 𝔗} {bv : Fin n → 𝔗} :\n φ.Eval (s := termModelOf p) bv fv ↔ p ⊫ Rew.bind bv fv ▹ φ :=\n have e (t : 𝔗) (φ : Semiformula K ξ (n + 1)) : ((Rew.bind bv fv).q ▹ φ)/[t] = Rew.bind (t :> bv) fv ▹ φ := by\n unfold R... | [
{
"name": "refl",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 5,
"n_chars": 258,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 2,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"max_nesting": 2
}
] | 1 | module
public import Foundation.FirstOrder.Completeness.CanonicalModel
public import Foundation.FirstOrder.Completeness.CountableSublanguage
public import Foundation.FirstOrder.Ultraproduct
public import Foundation.Vorspiel.Order.Dense
public import Mathlib.Logic.Equiv.List
public import Mathlib.Logic.Encodable.Basic
... | module
public import Foundation.FirstOrder.Completeness.CanonicalModel
public import Foundation.FirstOrder.Completeness.CountableSublanguage
public import Foundation.FirstOrder.Ultraproduct
public import Foundation.Vorspiel.Order.Dense
public import Mathlib.Logic.Equiv.List
public import Mathlib.Logic.Encodable.Basic
... | @@ -168,8 +168,22 @@
t.val (s := termModelOf p) bv fv = Rew.bind bv fv t := by
induction t <;> simp [*, Function.comp_def]
+lemma forcing_lemma (φ : Semiformula K ξ n) {fv : ξ → 𝔗} {bv : Fin n → 𝔗} :
+ φ.Eval (s := termModelOf p) bv fv ↔ p ⊫ Rew.bind bv fv ▹ φ :=
+ have e (t : 𝔗) (φ : Semiformula K ξ (... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_70ca7e247e94_3 | e32a4e4958f7288c | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Completeness/CounterModel.lean | CounterModel | 3 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 9 | 1 | [
{
"theorem_name": "satisfiable_of_irrefutable",
"depth": 1,
"n_commands": 0,
"n_lines": 11,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n let K := σ.sublanguage\n let π : Sentence K := σ.toSubLanguageSelf\n have : 𝐋𝐊¹ ⊬ ∼(π : Proposition K) := f... | [
{
"name": "refl",
"text": "lemma refl (φ : Proposition K) (h : 𝐋𝐊¹ ⊬ ∼φ) :\n φ.Evalf (s := termModelOf (ConsistentSequent.ofUnprovable φ h)) (&·) :=\n (forcing_lemma φ).mpr ⟨ConsistentSequent.ofUnprovable φ h, by simp, by simpa using IsWeaklyForced.refl φ h⟩\n\n",
"fan_in": 1,
"n_lines": 5,
... | [
{
"name": "Proof.complete",
"fan_in": 3,
"n_deps_direct": 1,
"n_deps_transitive": 5,
"n_lines": 12,
"n_chars": 499,
"n_subproofs": 3,
"n_tactics": 8,
"cyclomatic": 2,
"n_automation": 2,
"n_rewrites": 0,
"n_structural": 2,
"automation_only": false,
"max_nesting... | 5 | module
public import Foundation.FirstOrder.Completeness.CanonicalModel
public import Foundation.FirstOrder.Completeness.CountableSublanguage
public import Foundation.FirstOrder.Ultraproduct
public import Foundation.Vorspiel.Order.Dense
public import Mathlib.Logic.Equiv.List
public import Mathlib.Logic.Encodable.Basic
... | module
public import Foundation.FirstOrder.Completeness.CanonicalModel
public import Foundation.FirstOrder.Completeness.CountableSublanguage
public import Foundation.FirstOrder.Ultraproduct
public import Foundation.Vorspiel.Order.Dense
public import Mathlib.Logic.Equiv.List
public import Mathlib.Logic.Encodable.Basic
... | @@ -181,6 +181,10 @@
| φ ⋏ ψ | φ ⋎ ψ => by simp [forcing_lemma φ, forcing_lemma ψ]
| ∀⁰ φ | ∃⁰ φ => by simp [e, forcing_lemma φ]
+lemma refl (φ : Proposition K) (h : 𝐋𝐊¹ ⊬ ∼φ) :
+ φ.Evalf (s := termModelOf (ConsistentSequent.ofUnprovable φ h)) (&·) :=
+ (forcing_lemma φ).mpr ⟨ConsistentSequent.ofUnprovabl... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_70ca7e247e94_4 | 3e3a853683b67ea3 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Completeness/CounterModel.lean | CounterModel | 4 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 9 | 1 | [
{
"theorem_name": "satisfiable_of_irrefutable",
"depth": 1,
"n_commands": 0,
"n_lines": 11,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n let K := σ.sublanguage\n let π : Sentence K := σ.toSubLanguageSelf\n have : 𝐋𝐊¹ ⊬ ∼(π : Proposition K) := f... | [
{
"name": "refl",
"text": "lemma refl (φ : Proposition K) (h : 𝐋𝐊¹ ⊬ ∼φ) :\n φ.Evalf (s := termModelOf (ConsistentSequent.ofUnprovable φ h)) (&·) :=\n (forcing_lemma φ).mpr ⟨ConsistentSequent.ofUnprovable φ h, by simp, by simpa using IsWeaklyForced.refl φ h⟩\n\n",
"fan_in": 1,
"n_lines": 5,
... | [
{
"name": "Proof.small_complete",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 6,
"n_lines": 3,
"n_chars": 72,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
"max_nes... | 5 | module
public import Foundation.FirstOrder.Completeness.CanonicalModel
public import Foundation.FirstOrder.Completeness.CountableSublanguage
public import Foundation.FirstOrder.Ultraproduct
public import Foundation.Vorspiel.Order.Dense
public import Mathlib.Logic.Equiv.List
public import Mathlib.Logic.Encodable.Basic
... | module
public import Foundation.FirstOrder.Completeness.CanonicalModel
public import Foundation.FirstOrder.Completeness.CountableSublanguage
public import Foundation.FirstOrder.Ultraproduct
public import Foundation.Vorspiel.Order.Dense
public import Mathlib.Logic.Equiv.List
public import Mathlib.Logic.Encodable.Basic
... | @@ -181,6 +181,10 @@
| φ ⋏ ψ | φ ⋎ ψ => by simp [forcing_lemma φ, forcing_lemma ψ]
| ∀⁰ φ | ∃⁰ φ => by simp [e, forcing_lemma φ]
+lemma refl (φ : Proposition K) (h : 𝐋𝐊¹ ⊬ ∼φ) :
+ φ.Evalf (s := termModelOf (ConsistentSequent.ofUnprovable φ h)) (&·) :=
+ (forcing_lemma φ).mpr ⟨ConsistentSequent.ofUnprovabl... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_70ca7e247e94_5 | 1b577087364455bd | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Completeness/CounterModel.lean | CounterModel | 5 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 9 | 1 | [
{
"theorem_name": "satisfiable_of_irrefutable",
"depth": 1,
"n_commands": 0,
"n_lines": 11,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n let K := σ.sublanguage\n let π : Sentence K := σ.toSubLanguageSelf\n have : 𝐋𝐊¹ ⊬ ∼(π : Proposition K) := f... | [
{
"name": "refl",
"text": "lemma refl (φ : Proposition K) (h : 𝐋𝐊¹ ⊬ ∼φ) :\n φ.Evalf (s := termModelOf (ConsistentSequent.ofUnprovable φ h)) (&·) :=\n (forcing_lemma φ).mpr ⟨ConsistentSequent.ofUnprovable φ h, by simp, by simpa using IsWeaklyForced.refl φ h⟩\n\n",
"fan_in": 1,
"n_lines": 5,
... | [
{
"name": "Proof.complete_iff",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 6,
"n_lines": 3,
"n_chars": 101,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
"max_nest... | 5 | module
public import Foundation.FirstOrder.Completeness.CanonicalModel
public import Foundation.FirstOrder.Completeness.CountableSublanguage
public import Foundation.FirstOrder.Ultraproduct
public import Foundation.Vorspiel.Order.Dense
public import Mathlib.Logic.Equiv.List
public import Mathlib.Logic.Encodable.Basic
... | module
public import Foundation.FirstOrder.Completeness.CanonicalModel
public import Foundation.FirstOrder.Completeness.CountableSublanguage
public import Foundation.FirstOrder.Ultraproduct
public import Foundation.Vorspiel.Order.Dense
public import Mathlib.Logic.Equiv.List
public import Mathlib.Logic.Encodable.Basic
... | @@ -181,6 +181,10 @@
| φ ⋏ ψ | φ ⋎ ψ => by simp [forcing_lemma φ, forcing_lemma ψ]
| ∀⁰ φ | ∃⁰ φ => by simp [e, forcing_lemma φ]
+lemma refl (φ : Proposition K) (h : 𝐋𝐊¹ ⊬ ∼φ) :
+ φ.Evalf (s := termModelOf (ConsistentSequent.ofUnprovable φ h)) (&·) :=
+ (forcing_lemma φ).mpr ⟨ConsistentSequent.ofUnprovabl... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_70ca7e247e94_6 | cf1a3f32fb604c4b | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Completeness/CounterModel.lean | CounterModel | 6 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 9 | 1 | [
{
"theorem_name": "satisfiable_of_irrefutable",
"depth": 1,
"n_commands": 0,
"n_lines": 11,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n let K := σ.sublanguage\n let π : Sentence K := σ.toSubLanguageSelf\n have : 𝐋𝐊¹ ⊬ ∼(π : Proposition K) := f... | [
{
"name": "refl",
"text": "lemma refl (φ : Proposition K) (h : 𝐋𝐊¹ ⊬ ∼φ) :\n φ.Evalf (s := termModelOf (ConsistentSequent.ofUnprovable φ h)) (&·) :=\n (forcing_lemma φ).mpr ⟨ConsistentSequent.ofUnprovable φ h, by simp, by simpa using IsWeaklyForced.refl φ h⟩\n\n",
"fan_in": 1,
"n_lines": 5,
... | [
{
"name": "satisfiable_iff_satisfiable",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 5,
"n_lines": 4,
"n_chars": 183,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"... | 5 | module
public import Foundation.FirstOrder.Completeness.CanonicalModel
public import Foundation.FirstOrder.Completeness.CountableSublanguage
public import Foundation.FirstOrder.Ultraproduct
public import Foundation.Vorspiel.Order.Dense
public import Mathlib.Logic.Equiv.List
public import Mathlib.Logic.Encodable.Basic
... | module
public import Foundation.FirstOrder.Completeness.CanonicalModel
public import Foundation.FirstOrder.Completeness.CountableSublanguage
public import Foundation.FirstOrder.Ultraproduct
public import Foundation.Vorspiel.Order.Dense
public import Mathlib.Logic.Equiv.List
public import Mathlib.Logic.Encodable.Basic
... | @@ -181,6 +181,10 @@
| φ ⋏ ψ | φ ⋎ ψ => by simp [forcing_lemma φ, forcing_lemma ψ]
| ∀⁰ φ | ∃⁰ φ => by simp [e, forcing_lemma φ]
+lemma refl (φ : Proposition K) (h : 𝐋𝐊¹ ⊬ ∼φ) :
+ φ.Evalf (s := termModelOf (ConsistentSequent.ofUnprovable φ h)) (&·) :=
+ (forcing_lemma φ).mpr ⟨ConsistentSequent.ofUnprovabl... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_70ca7e247e94_7 | 75ac3361dbccc857 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Completeness/CounterModel.lean | CounterModel | 7 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 9 | 1 | [
{
"theorem_name": "satisfiable_of_irrefutable",
"depth": 1,
"n_commands": 0,
"n_lines": 11,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n let K := σ.sublanguage\n let π : Sentence K := σ.toSubLanguageSelf\n have : 𝐋𝐊¹ ⊬ ∼(π : Proposition K) := f... | [
{
"name": "refl",
"text": "lemma refl (φ : Proposition K) (h : 𝐋𝐊¹ ⊬ ∼φ) :\n φ.Evalf (s := termModelOf (ConsistentSequent.ofUnprovable φ h)) (&·) :=\n (forcing_lemma φ).mpr ⟨ConsistentSequent.ofUnprovable φ h, by simp, by simpa using IsWeaklyForced.refl φ h⟩\n\n",
"fan_in": 1,
"n_lines": 5,
... | [
{
"name": "consequence_iff_consequence",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 6,
"n_lines": 4,
"n_chars": 159,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"... | 6 | module
public import Foundation.FirstOrder.Completeness.CanonicalModel
public import Foundation.FirstOrder.Completeness.CountableSublanguage
public import Foundation.FirstOrder.Ultraproduct
public import Foundation.Vorspiel.Order.Dense
public import Mathlib.Logic.Equiv.List
public import Mathlib.Logic.Encodable.Basic
... | module
public import Foundation.FirstOrder.Completeness.CanonicalModel
public import Foundation.FirstOrder.Completeness.CountableSublanguage
public import Foundation.FirstOrder.Ultraproduct
public import Foundation.Vorspiel.Order.Dense
public import Mathlib.Logic.Equiv.List
public import Mathlib.Logic.Encodable.Basic
... | @@ -181,6 +181,10 @@
| φ ⋏ ψ | φ ⋎ ψ => by simp [forcing_lemma φ, forcing_lemma ψ]
| ∀⁰ φ | ∃⁰ φ => by simp [e, forcing_lemma φ]
+lemma refl (φ : Proposition K) (h : 𝐋𝐊¹ ⊬ ∼φ) :
+ φ.Evalf (s := termModelOf (ConsistentSequent.ofUnprovable φ h)) (&·) :=
+ (forcing_lemma φ).mpr ⟨ConsistentSequent.ofUnprovabl... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_70ca7e247e94_8 | 602f6bf22cb063b4 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Completeness/CounterModel.lean | CounterModel | 8 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 9 | 1 | [
{
"theorem_name": "satisfiable_of_irrefutable",
"depth": 1,
"n_commands": 0,
"n_lines": 11,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n let K := σ.sublanguage\n let π : Sentence K := σ.toSubLanguageSelf\n have : 𝐋𝐊¹ ⊬ ∼(π : Proposition K) := f... | [
{
"name": "refl",
"text": "lemma refl (φ : Proposition K) (h : 𝐋𝐊¹ ⊬ ∼φ) :\n φ.Evalf (s := termModelOf (ConsistentSequent.ofUnprovable φ h)) (&·) :=\n (forcing_lemma φ).mpr ⟨ConsistentSequent.ofUnprovable φ h, by simp, by simpa using IsWeaklyForced.refl φ h⟩\n\n",
"fan_in": 1,
"n_lines": 5,
... | [
{
"name": "Theory.Proof.complete_on_eq_models",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 7,
"n_lines": 14,
"n_chars": 516,
"n_subproofs": 1,
"n_tactics": 4,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": fal... | 7 | module
public import Foundation.FirstOrder.Completeness.CanonicalModel
public import Foundation.FirstOrder.Completeness.CountableSublanguage
public import Foundation.FirstOrder.Ultraproduct
public import Foundation.Vorspiel.Order.Dense
public import Mathlib.Logic.Equiv.List
public import Mathlib.Logic.Encodable.Basic
... | module
public import Foundation.FirstOrder.Completeness.CanonicalModel
public import Foundation.FirstOrder.Completeness.CountableSublanguage
public import Foundation.FirstOrder.Ultraproduct
public import Foundation.Vorspiel.Order.Dense
public import Mathlib.Logic.Equiv.List
public import Mathlib.Logic.Encodable.Basic
... | @@ -181,6 +181,10 @@
| φ ⋏ ψ | φ ⋎ ψ => by simp [forcing_lemma φ, forcing_lemma ψ]
| ∀⁰ φ | ∃⁰ φ => by simp [e, forcing_lemma φ]
+lemma refl (φ : Proposition K) (h : 𝐋𝐊¹ ⊬ ∼φ) :
+ φ.Evalf (s := termModelOf (ConsistentSequent.ofUnprovable φ h)) (&·) :=
+ (forcing_lemma φ).mpr ⟨ConsistentSequent.ofUnprovabl... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_5b363eaa7d16_0 | bbf9a26c22706691 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Formula/Complement.lean | Complement | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 2 | 1 | [
{
"theorem_name": "complementary_mem_box",
"depth": 1,
"n_commands": 0,
"n_lines": 7,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n intro h;\n rcases (mem_of h) with (h | ⟨ψ, hq, eq⟩);\n . assumption;\n . replace eq := Formula.complement.resort_b... | [
{
"name": "resort_box",
"text": "lemma resort_box (h : -φ = □ψ) : φ = ∼□ψ := by\n dsimp [complement] at h;\n split at h;\n . subst h; rfl;\n . contradiction;\n\n",
"fan_in": 1,
"n_lines": 7,
"n_chars": 136,
"n_subproofs": 0,
"n_tactics": 10,
"cyclomatic": 2,
"n_automation": 1... | [
{
"name": "complementary_mem_box",
"fan_in": 0,
"n_deps_direct": 2,
"n_deps_transitive": 2,
"n_lines": 10,
"n_chars": 278,
"n_subproofs": 0,
"n_tactics": 13,
"cyclomatic": 2,
"n_automation": 1,
"n_rewrites": 1,
"n_structural": 3,
"automation_only": false,
"max... | 2 | module
public import Foundation.Modal.Formula.Basic
public import Foundation.Propositional.Entailment.Cl.Basic
@[expose] public section
namespace LO.Modal
namespace Formula
@[grind]
def complement : Formula α → Formula α
| ∼φ => φ
| φ => ∼φ
prefix:80 "-" => complement
namespace complement
variable {φ ψ : Fo... | module
public import Foundation.Modal.Formula.Basic
public import Foundation.Propositional.Entailment.Cl.Basic
@[expose] public section
namespace LO.Modal
namespace Formula
@[grind]
def complement : Formula α → Formula α
| ∼φ => φ
| φ => ∼φ
prefix:80 "-" => complement
namespace complement
variable {φ ψ : Fo... | @@ -33,6 +33,12 @@
subst hq;
apply neg_def;
+lemma resort_box (h : -φ = □ψ) : φ = ∼□ψ := by
+ dsimp [complement] at h;
+ split at h;
+ . subst h; rfl;
+ . contradiction;
+
end complement
end Formula
@@ -52,7 +58,9 @@
intro h;
rcases (mem_of h) with (h | ⟨ψ, hq, eq⟩);
. assumption;
- . sorry
+ ... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_64a93373aba9_0 | 88330658b9b0208f | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Vorspiel/ENat.lean | ENat | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 1 | 1 | [
{
"theorem_name": "exists_of_find_le",
"depth": 1,
"n_commands": 0,
"n_lines": 3,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n by_contra A\n exact Std.Irrefl.irrefl _ <| lt_of_le_of_lt h <| lt_find P n (by simpa using A)",
"n_chars": 99,
"... | [
{
"name": "lt_find",
"text": "theorem lt_find (n : ℕ) (h : ∀ m ≤ n, ¬P m) : (n : ℕ∞) < find P := by\n by_cases h : ∃ x : ℕ, P x\n · simpa [find, h]\n · simp [find, h]\n\n",
"fan_in": 1,
"n_lines": 6,
"n_chars": 155,
"n_subproofs": 0,
"n_tactics": 4,
"cyclomatic": 1,
"n_automat... | [
{
"name": "exists_of_find_le",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 5,
"n_chars": 186,
"n_subproofs": 0,
"n_tactics": 3,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": false,
"max_nesti... | 1 | module
public import Mathlib.Data.ENat.Basic
@[expose]
public section
namespace ENat
open Classical
noncomputable def find (P : ℕ → Prop) : ℕ∞ := if h : ∃ x : ℕ, P x then Nat.find h else ⊤
variable (P : ℕ → Prop)
theorem exists_of_find_le (n : ℕ) (h : find P ≤ (n : ENat)) : ∃ m ≤ n, P m := sorry
end ENat
end
| module
public import Mathlib.Data.ENat.Basic
@[expose]
public section
namespace ENat
open Classical
noncomputable def find (P : ℕ → Prop) : ℕ∞ := if h : ∃ x : ℕ, P x then Nat.find h else ⊤
variable (P : ℕ → Prop)
theorem lt_find (n : ℕ) (h : ∀ m ≤ n, ¬P m) : (n : ℕ∞) < find P := by
by_cases h : ∃ x : ℕ, P x
... | @@ -13,8 +13,15 @@
variable (P : ℕ → Prop)
-theorem exists_of_find_le (n : ℕ) (h : find P ≤ (n : ENat)) : ∃ m ≤ n, P m := sorry
+theorem lt_find (n : ℕ) (h : ∀ m ≤ n, ¬P m) : (n : ℕ∞) < find P := by
+ by_cases h : ∃ x : ℕ, P x
+ · simpa [find, h]
+ · simp [find, h]
+theorem exists_of_find_le (n : ℕ) (h : find... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_85dfb68b9467_0 | b23c9248e676735e | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Basic/Soundness.lean | Soundness | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 2 | 1 | [
{
"theorem_name": "Proof.sound",
"depth": 1,
"n_commands": 0,
"n_lines": 3,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " fun b s hS ↦ by\n simpa [struc_models_iff_models (s := s), models_iff]\n using Proof.sound_proposition b hS",
"n_chars": 112,... | [
{
"name": "Proof.sound_proposition",
"text": "theorem Proof.sound_proposition {M : Type*} [s : Structure L M] [Nonempty M] :\n T ⊢ φ → M↓[L] ⊧* T → φ.Realize M := fun b H ↦ by\n rcases Proof.provable_iff.mp b with ⟨Γ, hΓ, ⟨b⟩⟩\n let f : ℕ → M := fun _ ↦ Nonempty.some inferInstance\n have : φ.Realize M... | [
{
"name": "Proof.sound",
"fan_in": 7,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 7,
"n_chars": 246,
"n_subproofs": 0,
"n_tactics": 3,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"max_nesting": 4
... | 1 | module
public import Foundation.FirstOrder.Basic.Semantics.Semantics
public import Foundation.FirstOrder.Basic.Calculus
@[expose] public section
/-! # Soundness theorem for first-order classical logic -/
namespace LO.FirstOrder
open Semiformula
variable {L : Language}
namespace Derivation
lemma sound {M : Type*} ... | module
public import Foundation.FirstOrder.Basic.Semantics.Semantics
public import Foundation.FirstOrder.Basic.Calculus
@[expose] public section
/-! # Soundness theorem for first-order classical logic -/
namespace LO.FirstOrder
open Semiformula
variable {L : Language}
namespace Derivation
lemma sound {M : Type*} ... | @@ -73,9 +73,28 @@
namespace Theory
+theorem Proof.sound_proposition {M : Type*} [s : Structure L M] [Nonempty M] :
+ T ⊢ φ → M↓[L] ⊧* T → φ.Realize M := fun b H ↦ by
+ rcases Proof.provable_iff.mp b with ⟨Γ, hΓ, ⟨b⟩⟩
+ let f : ℕ → M := fun _ ↦ Nonempty.some inferInstance
+ have : φ.Realize M ∨ ∃ ψ, ∼ψ ∈ Seq... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_85dfb68b9467_1 | 7713fb2ec40f2fb9 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Basic/Soundness.lean | Soundness | 1 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 2 | 1 | [
{
"theorem_name": "Proof.sound",
"depth": 1,
"n_commands": 0,
"n_lines": 3,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " fun b s hS ↦ by\n simpa [struc_models_iff_models (s := s), models_iff]\n using Proof.sound_proposition b hS",
"n_chars": 112,... | [
{
"name": "Proof.sound_proposition",
"text": "theorem Proof.sound_proposition {M : Type*} [s : Structure L M] [Nonempty M] :\n T ⊢ φ → M↓[L] ⊧* T → φ.Realize M := fun b H ↦ by\n rcases Proof.provable_iff.mp b with ⟨Γ, hΓ, ⟨b⟩⟩\n let f : ℕ → M := fun _ ↦ Nonempty.some inferInstance\n have : φ.Realize M... | [
{
"name": "Proof.sound_small",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 2,
"n_lines": 3,
"n_chars": 66,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
"max_nestin... | 1 | module
public import Foundation.FirstOrder.Basic.Semantics.Semantics
public import Foundation.FirstOrder.Basic.Calculus
@[expose] public section
/-! # Soundness theorem for first-order classical logic -/
namespace LO.FirstOrder
open Semiformula
variable {L : Language}
namespace Derivation
lemma sound {M : Type*} ... | module
public import Foundation.FirstOrder.Basic.Semantics.Semantics
public import Foundation.FirstOrder.Basic.Calculus
@[expose] public section
/-! # Soundness theorem for first-order classical logic -/
namespace LO.FirstOrder
open Semiformula
variable {L : Language}
namespace Derivation
lemma sound {M : Type*} ... | @@ -73,9 +73,28 @@
namespace Theory
+theorem Proof.sound_proposition {M : Type*} [s : Structure L M] [Nonempty M] :
+ T ⊢ φ → M↓[L] ⊧* T → φ.Realize M := fun b H ↦ by
+ rcases Proof.provable_iff.mp b with ⟨Γ, hΓ, ⟨b⟩⟩
+ let f : ℕ → M := fun _ ↦ Nonempty.some inferInstance
+ have : φ.Realize M ∨ ∃ ψ, ∼ψ ∈ Seq... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_85dfb68b9467_2 | 3f244dedb924625a | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Basic/Soundness.lean | Soundness | 2 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 2 | 1 | [
{
"theorem_name": "consistent_of_model",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " Theory.consistent_of_satisfiable ⟨M↓[L], hM⟩",
"n_chars": 51,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic":... | [
{
"name": "consistent_of_satisfiable",
"text": "lemma consistent_of_satisfiable (h : Semantics.Satisfiable (Struc.{v, u} L) T) : Entailment.Consistent T :=\n Sound.consistent_of_satisfiable h\n\n",
"fan_in": 1,
"n_lines": 4,
"n_chars": 145,
"n_subproofs": 0,
"n_tactics": 1,
"cycloma... | [
{
"name": "consistent_of_model",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 4,
"n_chars": 133,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
"max_nes... | 1 | module
public import Foundation.FirstOrder.Basic.Semantics.Semantics
public import Foundation.FirstOrder.Basic.Calculus
@[expose] public section
/-! # Soundness theorem for first-order classical logic -/
namespace LO.FirstOrder
open Semiformula
variable {L : Language}
namespace Derivation
lemma sound {M : Type*} ... | module
public import Foundation.FirstOrder.Basic.Semantics.Semantics
public import Foundation.FirstOrder.Basic.Calculus
@[expose] public section
/-! # Soundness theorem for first-order classical logic -/
namespace LO.FirstOrder
open Semiformula
variable {L : Language}
namespace Derivation
lemma sound {M : Type*} ... | @@ -98,6 +98,9 @@
instance (T : Theory L) : Sound T (Semantics.models (Struc.{v, u} L) T) := ⟨Theory.Proof.sound⟩
+lemma consistent_of_satisfiable (h : Semantics.Satisfiable (Struc.{v, u} L) T) : Entailment.Consistent T :=
+ Sound.consistent_of_satisfiable h
+
end Theory
section model
@@ -105,7 +108,7 @@
var... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_b12decf9ef43_0 | 2f214172b1f2a716 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Kripke/Intuitionistic.lean | Intuitionistic | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 1 | 3 | [
{
"theorem_name": "free",
"depth": 1,
"n_commands": 0,
"n_lines": 4,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n have : (fun i ↦ Semiterm.relationalVal (L := L) bv (x :>ₙ fv) (Rew.free #i)) = (bv <: x) := by\n ext i; cases i using Fin.lastCases... | [
{
"name": "rew",
"text": "lemma rew {bv : Fin n₂ → C} {fv : ξ₂ → C} {ω : Rew L ξ₁ n₁ ξ₂ n₂} {φ : Semiformulaᵢ L ξ₁ n₁} :\n w ⊩[bv|fv] (ω ▹ φ) ↔\n w ⊩[fun x ↦ (ω #x).relationalVal bv fv|fun x ↦ (ω &x).relationalVal bv fv] φ := by\n induction φ using Semiformulaᵢ.rec' generalizing n₂ w\n case hRel k R... | [
{
"name": "free",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 2,
"n_lines": 7,
"n_chars": 351,
"n_subproofs": 1,
"n_tactics": 6,
"cyclomatic": 3,
"n_automation": 2,
"n_rewrites": 0,
"n_structural": 2,
"automation_only": false,
"max_nesting": 4
}
] | 1 | module
public import Foundation.FirstOrder.Intuitionistic.Deduction
public import Foundation.FirstOrder.Kripke.Basic
@[expose] public section
/-! # Kripke semantics for intuitionistic first-order logic -/
namespace LO.FirstOrder
variable {L : Language} [L.Relational]
namespace KripkeModel
variable {W : Type*} [Pr... | module
public import Foundation.FirstOrder.Intuitionistic.Deduction
public import Foundation.FirstOrder.Kripke.Basic
@[expose] public section
/-! # Kripke semantics for intuitionistic first-order logic -/
namespace LO.FirstOrder
variable {L : Language} [L.Relational]
namespace KripkeModel
variable {W : Type*} [Pr... | @@ -76,11 +76,37 @@
| [φ] => by simp
| φ :: ψ :: Γ => by simp [disj (Γ := ψ :: Γ)]
+lemma rew {bv : Fin n₂ → C} {fv : ξ₂ → C} {ω : Rew L ξ₁ n₁ ξ₂ n₂} {φ : Semiformulaᵢ L ξ₁ n₁} :
+ w ⊩[bv|fv] (ω ▹ φ) ↔
+ w ⊩[fun x ↦ (ω #x).relationalVal bv fv|fun x ↦ (ω &x).relationalVal bv fv] φ := by
+ inductio... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_03e25fe3d076_0 | 3bde751903395ba7 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/ModalCompanion/Standard/LC.lean | LC | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 3 | 2 | [
{
"theorem_name": "modalCompanion_S4Point3",
"depth": 1,
"n_commands": 0,
"n_lines": 10,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n apply modalCompanion_via_kripkeSemantics (P := λ F => IsPiecewiseStronglyConnected F);\n . apply provable_S4Point... | [
{
"name": "provable_S4Point3_of_mem",
"text": "lemma provable_S4Point3_of_mem : φ ∈ Propositional.LC → Modal.S4Point3 ⊢ φᵍ := by\n apply Propositional.Hilbert.provable_gödelTranslated_of_mem_logic;\n rintro _ ⟨a, (⟨_, rfl⟩ | ⟨_, _, rfl⟩), rfl⟩ <;> simp;\n\n",
"fan_in": 2,
"n_lines": 5,
"n_char... | [
{
"name": "modalCompanion_S4Point3",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 12,
"n_chars": 440,
"n_subproofs": 0,
"n_tactics": 19,
"cyclomatic": 2,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 8,
"automation_only": false,
"m... | 1 | module
public import Foundation.Logic.Embedding
public import Foundation.Modal.Boxdot.GLPoint3_GrzPoint3
public import Foundation.Modal.Kripke.Logic.GrzPoint3
public import Foundation.Modal.ModalCompanion.Standard.Basic
public import Foundation.Propositional.Kripke.Hilbert.LC
@[expose] public section
namespace LO
o... | module
public import Foundation.Logic.Embedding
public import Foundation.Modal.Boxdot.GLPoint3_GrzPoint3
public import Foundation.Modal.Kripke.Logic.GrzPoint3
public import Foundation.Modal.ModalCompanion.Standard.Basic
public import Foundation.Propositional.Kripke.Hilbert.LC
@[expose] public section
namespace LO
o... | @@ -39,9 +39,13 @@
namespace Propositional.LC
+lemma provable_S4Point3_of_mem : φ ∈ Propositional.LC → Modal.S4Point3 ⊢ φᵍ := by
+ apply Propositional.Hilbert.provable_gödelTranslated_of_mem_logic;
+ rintro _ ⟨a, (⟨_, rfl⟩ | ⟨_, _, rfl⟩), rfl⟩ <;> simp;
+
theorem modalCompanion_S4Point3 : ModalCompanion Proposi... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_03e25fe3d076_1 | aaf2109688445f40 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/ModalCompanion/Standard/LC.lean | LC | 1 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 3 | 2 | [
{
"theorem_name": "modalCompanion_S4Point3",
"depth": 1,
"n_commands": 0,
"n_lines": 10,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n apply modalCompanion_via_kripkeSemantics (P := λ F => IsPiecewiseStronglyConnected F);\n . apply provable_S4Point... | [
{
"name": "provable_S4Point3_of_mem",
"text": "lemma provable_S4Point3_of_mem : φ ∈ Propositional.LC → Modal.S4Point3 ⊢ φᵍ := by\n apply Propositional.Hilbert.provable_gödelTranslated_of_mem_logic;\n rintro _ ⟨a, (⟨_, rfl⟩ | ⟨_, _, rfl⟩), rfl⟩ <;> simp;\n\n",
"fan_in": 2,
"n_lines": 5,
"n_char... | [
{
"name": "provable_GrzPoint3_of_mem",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 3,
"n_chars": 146,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
"m... | 1 | module
public import Foundation.Logic.Embedding
public import Foundation.Modal.Boxdot.GLPoint3_GrzPoint3
public import Foundation.Modal.Kripke.Logic.GrzPoint3
public import Foundation.Modal.ModalCompanion.Standard.Basic
public import Foundation.Propositional.Kripke.Hilbert.LC
@[expose] public section
namespace LO
o... | module
public import Foundation.Logic.Embedding
public import Foundation.Modal.Boxdot.GLPoint3_GrzPoint3
public import Foundation.Modal.Kripke.Logic.GrzPoint3
public import Foundation.Modal.ModalCompanion.Standard.Basic
public import Foundation.Propositional.Kripke.Hilbert.LC
@[expose] public section
namespace LO
o... | @@ -39,8 +39,12 @@
namespace Propositional.LC
-lemma provable_GrzPoint3_of_mem : φ ∈ Propositional.LC → Modal.GrzPoint3 ⊢ φᵍ := sorry
+lemma provable_S4Point3_of_mem : φ ∈ Propositional.LC → Modal.S4Point3 ⊢ φᵍ := by
+ apply Propositional.Hilbert.provable_gödelTranslated_of_mem_logic;
+ rintro _ ⟨a, (⟨_, rfl⟩ | ... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_03e25fe3d076_2 | 5b53fa084af5404e | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/ModalCompanion/Standard/LC.lean | LC | 2 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 3 | 2 | [
{
"theorem_name": "modalCompanion_S4Point3",
"depth": 1,
"n_commands": 0,
"n_lines": 10,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n apply modalCompanion_via_kripkeSemantics (P := λ F => IsPiecewiseStronglyConnected F);\n . apply provable_S4Point... | [
{
"name": "provable_S4Point3_of_mem",
"text": "lemma provable_S4Point3_of_mem : φ ∈ Propositional.LC → Modal.S4Point3 ⊢ φᵍ := by\n apply Propositional.Hilbert.provable_gödelTranslated_of_mem_logic;\n rintro _ ⟨a, (⟨_, rfl⟩ | ⟨_, _, rfl⟩), rfl⟩ <;> simp;\n\n",
"fan_in": 2,
"n_lines": 5,
"n_char... | [
{
"name": "modalCompanion_GrzPoint3",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 2,
"n_lines": 12,
"n_chars": 491,
"n_subproofs": 0,
"n_tactics": 19,
"cyclomatic": 2,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 8,
"automation_only": false,
"... | 2 | module
public import Foundation.Logic.Embedding
public import Foundation.Modal.Boxdot.GLPoint3_GrzPoint3
public import Foundation.Modal.Kripke.Logic.GrzPoint3
public import Foundation.Modal.ModalCompanion.Standard.Basic
public import Foundation.Propositional.Kripke.Hilbert.LC
@[expose] public section
namespace LO
o... | module
public import Foundation.Logic.Embedding
public import Foundation.Modal.Boxdot.GLPoint3_GrzPoint3
public import Foundation.Modal.Kripke.Logic.GrzPoint3
public import Foundation.Modal.ModalCompanion.Standard.Basic
public import Foundation.Propositional.Kripke.Hilbert.LC
@[expose] public section
namespace LO
o... | @@ -39,8 +39,12 @@
namespace Propositional.LC
-lemma provable_GrzPoint3_of_mem : φ ∈ Propositional.LC → Modal.GrzPoint3 ⊢ φᵍ := sorry
+lemma provable_S4Point3_of_mem : φ ∈ Propositional.LC → Modal.S4Point3 ⊢ φᵍ := by
+ apply Propositional.Hilbert.provable_gödelTranslated_of_mem_logic;
+ rintro _ ⟨a, (⟨_, rfl⟩ | ... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_03e25fe3d076_3 | 62dd3f9b6b83ae76 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/ModalCompanion/Standard/LC.lean | LC | 3 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 3 | 2 | [
{
"theorem_name": "modalCompanion_S4Point3",
"depth": 1,
"n_commands": 0,
"n_lines": 10,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n apply modalCompanion_via_kripkeSemantics (P := λ F => IsPiecewiseStronglyConnected F);\n . apply provable_S4Point... | [
{
"name": "provable_S4Point3_of_mem",
"text": "lemma provable_S4Point3_of_mem : φ ∈ Propositional.LC → Modal.S4Point3 ⊢ φᵍ := by\n apply Propositional.Hilbert.provable_gödelTranslated_of_mem_logic;\n rintro _ ⟨a, (⟨_, rfl⟩ | ⟨_, _, rfl⟩), rfl⟩ <;> simp;\n\n",
"fan_in": 2,
"n_lines": 5,
"n_char... | [
{
"name": "boxdotModalCompanion_LC",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 3,
"n_lines": 4,
"n_chars": 190,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
"max... | 3 | module
public import Foundation.Logic.Embedding
public import Foundation.Modal.Boxdot.GLPoint3_GrzPoint3
public import Foundation.Modal.Kripke.Logic.GrzPoint3
public import Foundation.Modal.ModalCompanion.Standard.Basic
public import Foundation.Propositional.Kripke.Hilbert.LC
@[expose] public section
namespace LO
o... | module
public import Foundation.Logic.Embedding
public import Foundation.Modal.Boxdot.GLPoint3_GrzPoint3
public import Foundation.Modal.Kripke.Logic.GrzPoint3
public import Foundation.Modal.ModalCompanion.Standard.Basic
public import Foundation.Propositional.Kripke.Hilbert.LC
@[expose] public section
namespace LO
o... | @@ -39,8 +39,12 @@
namespace Propositional.LC
-lemma provable_GrzPoint3_of_mem : φ ∈ Propositional.LC → Modal.GrzPoint3 ⊢ φᵍ := sorry
+lemma provable_S4Point3_of_mem : φ ∈ Propositional.LC → Modal.S4Point3 ⊢ φᵍ := by
+ apply Propositional.Hilbert.provable_gödelTranslated_of_mem_logic;
+ rintro _ ⟨a, (⟨_, rfl⟩ | ... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_fe3fe2f56391_0 | b7e84d714b10617b | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Propositional/Kripke/Basic.lean | Basic | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 3 | 4 | [
{
"theorem_name": "formula_hereditary_not",
"depth": 1,
"n_commands": 0,
"n_lines": 3,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n contrapose!;\n exact formula_hereditary hw;",
"n_chars": 49,
"n_subproofs": 0,
"n_tactics": 5,
"cyc... | [
{
"name": "formula_hereditary",
"text": "lemma formula_hereditary\n (hw : w ≺ w') : w ⊧ φ → w' ⊧ φ := by\n induction φ with\n | hatom => apply M.Val.hereditary hw;\n | himp =>\n intro hpq v hv;\n exact hpq $ M.trans hw hv;\n | hor => simp_all; tauto;\n | _ => simp_all;\n\n",
"fan_in": 4,
... | [
{
"name": "formula_hereditary_not",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 2,
"n_lines": 5,
"n_chars": 127,
"n_subproofs": 0,
"n_tactics": 5,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": false,
"max_... | 1 | module
public import Foundation.Propositional.Formula.Basic
public import Foundation.Propositional.Entailment.Cl.Basic
public import Foundation.Vorspiel.Rel.Basic
@[expose] public section
namespace LO.Propositional
namespace Kripke
structure Frame where
World : Type
Rel : Rel World World
[world_nonempty : Non... | module
public import Foundation.Propositional.Formula.Basic
public import Foundation.Propositional.Entailment.Cl.Basic
public import Foundation.Vorspiel.Rel.Basic
@[expose] public section
namespace LO.Propositional
namespace Kripke
structure Frame where
World : Type
Rel : Rel World World
[world_nonempty : Non... | @@ -108,8 +108,20 @@
instance : Semantics.Or M.World where
models_or := by simp [Satisfies];
-lemma formula_hereditary_not (hw : w ≺ w') : ¬w' ⊧ φ → ¬w ⊧ φ := sorry
+lemma formula_hereditary
+ (hw : w ≺ w') : w ⊧ φ → w' ⊧ φ := by
+ induction φ with
+ | hatom => apply M.Val.hereditary hw;
+ | himp =>
+ intr... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_fe3fe2f56391_2 | f68cbc6f923f8cc7 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Propositional/Kripke/Basic.lean | Basic | 2 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 3 | 4 | [
{
"theorem_name": "formula_hereditary_not",
"depth": 1,
"n_commands": 0,
"n_lines": 3,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n contrapose!;\n exact formula_hereditary hw;",
"n_chars": 49,
"n_subproofs": 0,
"n_tactics": 5,
"cyc... | [
{
"name": "formula_hereditary",
"text": "lemma formula_hereditary\n (hw : w ≺ w') : w ⊧ φ → w' ⊧ φ := by\n induction φ with\n | hatom => apply M.Val.hereditary hw;\n | himp =>\n intro hpq v hv;\n exact hpq $ M.trans hw hv;\n | hor => simp_all; tauto;\n | _ => simp_all;\n\n",
"fan_in": 4,
... | [
{
"name": "andInst₃",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 2,
"n_lines": 6,
"n_chars": 179,
"n_subproofs": 0,
"n_tactics": 7,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 2,
"automation_only": false,
"max_nesting": 2
... | 1 | module
public import Foundation.Propositional.Formula.Basic
public import Foundation.Propositional.Entailment.Cl.Basic
public import Foundation.Vorspiel.Rel.Basic
@[expose] public section
namespace LO.Propositional
namespace Kripke
structure Frame where
World : Type
Rel : Rel World World
[world_nonempty : Non... | module
public import Foundation.Propositional.Formula.Basic
public import Foundation.Propositional.Entailment.Cl.Basic
public import Foundation.Vorspiel.Rel.Basic
@[expose] public section
namespace LO.Propositional
namespace Kripke
structure Frame where
World : Type
Rel : Rel World World
[world_nonempty : Non... | @@ -108,6 +108,16 @@
instance : Semantics.Or M.World where
models_or := by simp [Satisfies];
+lemma formula_hereditary
+ (hw : w ≺ w') : w ⊧ φ → w' ⊧ φ := by
+ induction φ with
+ | hatom => apply M.Val.hereditary hw;
+ | himp =>
+ intro hpq v hv;
+ exact hpq $ M.trans hw hv;
+ | hor => simp_all; tauto;... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_fe3fe2f56391_3 | ca95a99b85ac91f3 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Propositional/Kripke/Basic.lean | Basic | 3 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 3 | 1 | [
{
"theorem_name": "iff_not_exists_model_world",
"depth": 1,
"n_commands": 0,
"n_lines": 10,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n constructor;\n . intro h;\n obtain ⟨V, x, h⟩ := iff_not_exists_valuation_world.mp h;\n use ⟨F, V⟩, x;\n ... | [
{
"name": "iff_not_exists_valuation_world",
"text": "lemma iff_not_exists_valuation_world : (¬F ⊧ φ) ↔ (∃ V : Kripke.Valuation F, ∃ x : (⟨F, V⟩ : Kripke.Model).World, ¬Satisfies _ x φ) := by\n simp [ValidOnFrame, ValidOnModel, Semantics.Models];\n\nalias ⟨exists_valuation_world_of_not, not_of_exists_valuat... | [
{
"name": "iff_not_exists_model_world",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 12,
"n_chars": 450,
"n_subproofs": 1,
"n_tactics": 16,
"cyclomatic": 3,
"n_automation": 2,
"n_rewrites": 0,
"n_structural": 6,
"automation_only": false,
... | 1 | module
public import Foundation.Propositional.Formula.Basic
public import Foundation.Propositional.Entailment.Cl.Basic
public import Foundation.Vorspiel.Rel.Basic
@[expose] public section
namespace LO.Propositional
namespace Kripke
structure Frame where
World : Type
Rel : Rel World World
[world_nonempty : Non... | module
public import Foundation.Propositional.Formula.Basic
public import Foundation.Propositional.Entailment.Cl.Basic
public import Foundation.Vorspiel.Rel.Basic
@[expose] public section
namespace LO.Propositional
namespace Kripke
structure Frame where
World : Type
Rel : Rel World World
[world_nonempty : Non... | @@ -191,10 +191,19 @@
exact ⟨(λ _ _ => True), by tauto⟩;
instance : Semantics.Bot (Frame) := ⟨λ _ => ValidOnFrame.bot⟩
+lemma iff_not_exists_valuation_world : (¬F ⊧ φ) ↔ (∃ V : Kripke.Valuation F, ∃ x : (⟨F, V⟩ : Kripke.Model).World, ¬Satisfies _ x φ) := by
+ simp [ValidOnFrame, ValidOnModel, Semantics.Models];
... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_832a19d088f8_0 | b21a7f78dae4882a | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Incompleteness/ProvabilityAbstraction/Refutability.lean | Refutability | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 4 | 1 | [
{
"theorem_name": "jeroslow_def'",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " WeakerThan.pbl $ jeroslow_def",
"n_chars": 30,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation... | [
{
"name": "jeroslow_def",
"text": "lemma jeroslow_def : T₀ ⊢ jeroslow 𝔚 🡘 𝔚 (jeroslow 𝔚) := Diagonalization.diag _\n\n",
"fan_in": 1,
"n_lines": 3,
"n_chars": 97,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,... | [
{
"name": "jeroslow_def'",
"fan_in": 3,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 4,
"n_chars": 116,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
"max_nesting":... | 1 | module
public import Foundation.FirstOrder.Incompleteness.RosserProvability
@[expose] public section
namespace LO.FirstOrder
namespace ProvabilityAbstraction
open LO.Entailment FirstOrder Diagonalization Provability
variable {L₀ L : Language}
structure Refutability [L.ReferenceableBy L₀] (T₀ : Theory L₀) (T : Th... | module
public import Foundation.FirstOrder.Incompleteness.RosserProvability
@[expose] public section
namespace LO.FirstOrder
namespace ProvabilityAbstraction
open LO.Entailment FirstOrder Diagonalization Provability
variable {L₀ L : Language}
structure Refutability [L.ReferenceableBy L₀] (T₀ : Theory L₀) (T : Th... | @@ -43,8 +43,10 @@
/-- This sentence is refutable. -/
def jeroslow (𝔚 : Refutability T₀ T) : Sentence L := fixedpoint T₀ 𝔚.refu
-lemma jeroslow_def' [T₀ ⪯ T] : T ⊢ jeroslow 𝔚 🡘 𝔚 (jeroslow 𝔚) := sorry
+lemma jeroslow_def : T₀ ⊢ jeroslow 𝔚 🡘 𝔚 (jeroslow 𝔚) := Diagonalization.diag _
+lemma jeroslow_def' [... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_832a19d088f8_1 | 04ef32a7b54b4734 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Incompleteness/ProvabilityAbstraction/Refutability.lean | Refutability | 1 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 4 | 1 | [
{
"theorem_name": "jeroslow_def'",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " WeakerThan.pbl $ jeroslow_def",
"n_chars": 30,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation... | [
{
"name": "jeroslow_def",
"text": "lemma jeroslow_def : T₀ ⊢ jeroslow 𝔚 🡘 𝔚 (jeroslow 𝔚) := Diagonalization.diag _\n\n",
"fan_in": 1,
"n_lines": 3,
"n_chars": 97,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,... | [
{
"name": "unprovable_jeroslow",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 2,
"n_lines": 8,
"n_chars": 352,
"n_subproofs": 1,
"n_tactics": 11,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 2,
"automation_only": false,
"max_ne... | 2 | module
public import Foundation.FirstOrder.Incompleteness.RosserProvability
@[expose] public section
namespace LO.FirstOrder
namespace ProvabilityAbstraction
open LO.Entailment FirstOrder Diagonalization Provability
variable {L₀ L : Language}
structure Refutability [L.ReferenceableBy L₀] (T₀ : Theory L₀) (T : Th... | module
public import Foundation.FirstOrder.Incompleteness.RosserProvability
@[expose] public section
namespace LO.FirstOrder
namespace ProvabilityAbstraction
open LO.Entailment FirstOrder Diagonalization Provability
variable {L₀ L : Language}
structure Refutability [L.ReferenceableBy L₀] (T₀ : Theory L₀) (T : Th... | @@ -43,8 +43,10 @@
/-- This sentence is refutable. -/
def jeroslow (𝔚 : Refutability T₀ T) : Sentence L := fixedpoint T₀ 𝔚.refu
-lemma jeroslow_def' [T₀ ⪯ T] : T ⊢ jeroslow 𝔚 🡘 𝔚 (jeroslow 𝔚) := sorry
+lemma jeroslow_def : T₀ ⊢ jeroslow 𝔚 🡘 𝔚 (jeroslow 𝔚) := Diagonalization.diag _
+lemma jeroslow_def' [... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_832a19d088f8_2 | f8e7529acec814c3 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Incompleteness/ProvabilityAbstraction/Refutability.lean | Refutability | 2 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 4 | 1 | [
{
"theorem_name": "jeroslow_def'",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " WeakerThan.pbl $ jeroslow_def",
"n_chars": 30,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation... | [
{
"name": "jeroslow_def",
"text": "lemma jeroslow_def : T₀ ⊢ jeroslow 𝔚 🡘 𝔚 (jeroslow 𝔚) := Diagonalization.diag _\n\n",
"fan_in": 1,
"n_lines": 3,
"n_chars": 97,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,... | [
{
"name": "jeroslow_not_safe",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 2,
"n_lines": 6,
"n_chars": 292,
"n_subproofs": 2,
"n_tactics": 7,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
"max_nesti... | 2 | module
public import Foundation.FirstOrder.Incompleteness.RosserProvability
@[expose] public section
namespace LO.FirstOrder
namespace ProvabilityAbstraction
open LO.Entailment FirstOrder Diagonalization Provability
variable {L₀ L : Language}
structure Refutability [L.ReferenceableBy L₀] (T₀ : Theory L₀) (T : Th... | module
public import Foundation.FirstOrder.Incompleteness.RosserProvability
@[expose] public section
namespace LO.FirstOrder
namespace ProvabilityAbstraction
open LO.Entailment FirstOrder Diagonalization Provability
variable {L₀ L : Language}
structure Refutability [L.ReferenceableBy L₀] (T₀ : Theory L₀) (T : Th... | @@ -43,8 +43,10 @@
/-- This sentence is refutable. -/
def jeroslow (𝔚 : Refutability T₀ T) : Sentence L := fixedpoint T₀ 𝔚.refu
-lemma jeroslow_def' [T₀ ⪯ T] : T ⊢ jeroslow 𝔚 🡘 𝔚 (jeroslow 𝔚) := sorry
+lemma jeroslow_def : T₀ ⊢ jeroslow 𝔚 🡘 𝔚 (jeroslow 𝔚) := Diagonalization.diag _
+lemma jeroslow_def' [... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_832a19d088f8_3 | cff89509c54c0ace | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Incompleteness/ProvabilityAbstraction/Refutability.lean | Refutability | 3 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 4 | 1 | [
{
"theorem_name": "unprovable_flon",
"depth": 1,
"n_commands": 0,
"n_lines": 10,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n contrapose! consis;\n replace consis : T ⊢ ∀⁰ safe 𝔅 𝔚 := by simpa [flon] using consis;\n have h₁ : T ⊢ ∼(𝔅 𝐉 ⋏ 𝔚 �... | [
{
"name": "R1'",
"text": "lemma R1' {L : Language} [L.ReferenceableBy L] {T₀ T : Theory L}\n {𝔚 : Refutability T₀ T} {σ : Sentence L} [T₀ ⪯ T] : T ⊢ ∼σ → T ⊢ 𝔚 σ := fun h ↦\n WeakerThan.pbl $ R1 h\n\n",
"fan_in": 1,
"n_lines": 5,
"n_chars": 199,
"n_subproofs": 0,
"n_tactics": 2,
... | [
{
"name": "unprovable_flon",
"fan_in": 0,
"n_deps_direct": 3,
"n_deps_transitive": 5,
"n_lines": 16,
"n_chars": 859,
"n_subproofs": 6,
"n_tactics": 19,
"cyclomatic": 1,
"n_automation": 2,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": false,
"max_nesti... | 4 | module
public import Foundation.FirstOrder.Incompleteness.RosserProvability
@[expose] public section
namespace LO.FirstOrder
namespace ProvabilityAbstraction
open LO.Entailment FirstOrder Diagonalization Provability
variable {L₀ L : Language}
structure Refutability [L.ReferenceableBy L₀] (T₀ : Theory L₀) (T : Th... | module
public import Foundation.FirstOrder.Incompleteness.RosserProvability
@[expose] public section
namespace LO.FirstOrder
namespace ProvabilityAbstraction
open LO.Entailment FirstOrder Diagonalization Provability
variable {L₀ L : Language}
structure Refutability [L.ReferenceableBy L₀] (T₀ : Theory L₀) (T : Th... | @@ -33,6 +33,10 @@
lemma R1 {𝔚 : Refutability T₀ T} {σ : Sentence L} : T ⊢ ∼σ → T₀ ⊢ 𝔚 σ := fun h ↦ 𝔚.refu_def h
+lemma R1' {L : Language} [L.ReferenceableBy L] {T₀ T : Theory L}
+ {𝔚 : Refutability T₀ T} {σ : Sentence L} [T₀ ⪯ T] : T ⊢ ∼σ → T ⊢ 𝔚 σ := fun h ↦
+ WeakerThan.pbl $ R1 h
+
end
section
@@ -1... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_a15271e54cd0_0 | 244b0e110bda3c03 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Logic/Embedding.lean | Embedding | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 1 | 1 | [
{
"theorem_name": "trans",
"depth": 1,
"n_commands": 0,
"n_lines": 5,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n rcases fun_exists 𝓢₁ 𝓢₂ with ⟨f₁₂, h₁₂⟩\n rcases fun_exists 𝓢₂ 𝓢₃ with ⟨f₂₃, h₂₃⟩\n refine ⟨f₂₃ ∘ f₁₂, ?_⟩\n intro φ; ... | [
{
"name": "fun_exists",
"text": "lemma fun_exists (𝓢₁ : S₁) (𝓢₂ : S₂) [FaithfullyEmbeddable 𝓢₁ 𝓢₂] :\n ∃ f : F₁ → F₂, IsFaithfulEmbedding 𝓢₁ 𝓢₂ f := FaithfullyEmbeddable.prop\n\n",
"fan_in": 1,
"n_lines": 4,
"n_chars": 187,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
... | [
{
"name": "trans",
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"n_deps_direct": 1,
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"n_lines": 9,
"n_chars": 440,
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"automation_only": false,
"max_nesting": 4
}
] | 1 | module
public import Foundation.Propositional.Entailment.Cl.Basic
@[expose] public section
/-! # Faithful embeddings among logical systems -/
namespace LO.Entailment
variable {F₁ F₂ F₃ : Type*} {S₁ S₂ S₃ : Type*} [Entailment S₁ F₁] [Entailment S₂ F₂] [Entailment S₃ F₃]
def IsFaithfulEmbedding (𝓢₁ : S₁) (𝓢₂ : S₂... | module
public import Foundation.Propositional.Entailment.Cl.Basic
@[expose] public section
/-! # Faithful embeddings among logical systems -/
namespace LO.Entailment
variable {F₁ F₂ F₃ : Type*} {S₁ S₂ S₃ : Type*} [Entailment S₁ F₁] [Entailment S₂ F₂] [Entailment S₃ F₃]
def IsFaithfulEmbedding (𝓢₁ : S₁) (𝓢₂ : S₂... | @@ -17,12 +17,19 @@
namespace FaithfullyEmbeddable
+lemma fun_exists (𝓢₁ : S₁) (𝓢₂ : S₂) [FaithfullyEmbeddable 𝓢₁ 𝓢₂] :
+ ∃ f : F₁ → F₂, IsFaithfulEmbedding 𝓢₁ 𝓢₂ f := FaithfullyEmbeddable.prop
+
@[refl] protected instance refl (𝓢₁ : S₁) : FaithfullyEmbeddable 𝓢₁ 𝓢₁ where
prop := ⟨id, by intros _; ... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_e249934e738b_0 | 5f4be342e4a5fbe3 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Omega1/Nuon.lean | Nuon | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 32 | 3 | [
{
"theorem_name": "ext_graph_aux",
"depth": 1,
"n_commands": 0,
"n_lines": 10,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n rcases show ‖S‖ ≤ i * ‖L‖ ∨ i * ‖L‖ < ‖S‖ from le_or_gt _ _ with (le | lt)\n · simp [ext_eq_zero_of_lt le, le, not_lt.mpr l... | [
{
"name": "ext_eq_zero_of_lt",
"text": "lemma ext_eq_zero_of_lt {L S i : V} (h : ‖S‖ ≤ i * ‖L‖) : S{L}[i] = 0 := by simp [ext, bexp_eq_zero_of_le h]\n\n",
"fan_in": 3,
"n_lines": 3,
"n_chars": 120,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 1,
"n_rewri... | [
{
"name": "ext_graph_aux",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 12,
"n_chars": 714,
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"cyclomatic": 6,
"n_automation": 7,
"n_rewrites": 1,
"n_structural": 7,
"automation_only": false,
"max_nesting... | 1 | module
public import Foundation.FirstOrder.Arithmetic.Omega1.Basic
@[expose] public section
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₀ ∪ 𝝮₁]
namespace Nuon
@[simp] lemma llen_lt_len_smash_len (K : V) : ‖‖K‖‖ < ‖K ⨳ ‖K‖‖ := by
suffices ‖‖K‖‖ ≤ ‖K‖ * ‖‖K‖‖ by simp... | module
public import Foundation.FirstOrder.Arithmetic.Omega1.Basic
@[expose] public section
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₀ ∪ 𝝮₁]
namespace Nuon
@[simp] lemma llen_lt_len_smash_len (K : V) : ‖‖K‖‖ < ‖K ⨳ ‖K‖‖ := by
suffices ‖‖K‖‖ ≤ ‖K‖ * ‖‖K‖‖ by simp... | @@ -22,11 +22,13 @@
local notation S "{" L "}[" i "]" => ext L S i
+lemma ext_eq_zero_of_lt {L S i : V} (h : ‖S‖ ≤ i * ‖L‖) : S{L}[i] = 0 := by simp [ext, bexp_eq_zero_of_le h]
+
@[simp] lemma ext_le_self (L S i : V) : S{L}[i] ≤ S := le_trans (mod_le _ _) (by simp)
lemma ext_graph_aux (z S L i : V) : z = S{L}[... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_e249934e738b_1 | c1240387cf86a696 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Omega1/Nuon.lean | Nuon | 1 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 32 | 1 | [
{
"theorem_name": "ext_graph",
"depth": 1,
"n_commands": 0,
"n_lines": 12,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n rw [ext_graph_aux]\n rcases show ‖S‖ ≤ i * ‖L‖ ∨ i * ‖L‖ < ‖S‖ from le_or_gt _ _ with (le | lt)\n · simp [le, not_lt.mpr le]\n... | [
{
"name": "ext_graph_aux",
"text": "lemma ext_graph_aux (z S L i : V) : z = S{L}[i] ↔ (‖S‖ ≤ i * ‖L‖ → z = 0) ∧ (i * ‖L‖ < ‖S‖ → ∃ b ≤ S, Exponential (i * ‖L‖) b ∧ z = S / b % (L ⨳ 1)) := by\n rcases show ‖S‖ ≤ i * ‖L‖ ∨ i * ‖L‖ < ‖S‖ from le_or_gt _ _ with (le | lt)\n · simp [ext_eq_zero_of_lt le, le, no... | [
{
"name": "ext_graph",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 2,
"n_lines": 18,
"n_chars": 960,
"n_subproofs": 1,
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"cyclomatic": 5,
"n_automation": 7,
"n_rewrites": 3,
"n_structural": 6,
"automation_only": false,
"max_nesting": 6... | 2 | module
public import Foundation.FirstOrder.Arithmetic.Omega1.Basic
@[expose] public section
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₀ ∪ 𝝮₁]
namespace Nuon
@[simp] lemma llen_lt_len_smash_len (K : V) : ‖‖K‖‖ < ‖K ⨳ ‖K‖‖ := by
suffices ‖‖K‖‖ ≤ ‖K‖ * ‖‖K‖‖ by simp... | module
public import Foundation.FirstOrder.Arithmetic.Omega1.Basic
@[expose] public section
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₀ ∪ 𝝮₁]
namespace Nuon
@[simp] lemma llen_lt_len_smash_len (K : V) : ‖‖K‖‖ < ‖K ⨳ ‖K‖‖ := by
suffices ‖‖K‖‖ ≤ ‖K‖ * ‖‖K‖‖ by simp... | @@ -26,11 +26,33 @@
@[simp] lemma ext_le_self (L S i : V) : S{L}[i] ≤ S := le_trans (mod_le _ _) (by simp)
+lemma ext_graph_aux (z S L i : V) : z = S{L}[i] ↔ (‖S‖ ≤ i * ‖L‖ → z = 0) ∧ (i * ‖L‖ < ‖S‖ → ∃ b ≤ S, Exponential (i * ‖L‖) b ∧ z = S / b % (L ⨳ 1)) := by
+ rcases show ‖S‖ ≤ i * ‖L‖ ∨ i * ‖L‖ < ‖S‖ from le... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_e249934e738b_2 | bdc6ee62150dea3e | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Omega1/Nuon.lean | Nuon | 2 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 32 | 1 | [
{
"theorem_name": "ext_add₁_pow2",
"depth": 1,
"n_commands": 0,
"n_lines": 19,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n rcases Arithmetic.zero_le S₂ with (rfl | pos₂)\n · simp\n have lt_len : i * ‖L‖ < ‖S₁ + S₂ * p‖ := calc\n i * ‖L‖ ≤... | [
{
"name": "ext_eq_of_ge",
"text": "lemma ext_eq_of_ge {L S S' i : V} (h : S ≤ S') : S / bexp S' (i * ‖L‖) % (L ⨳ 1) = S{L}[i] := by\n rcases show i * ‖L‖ < ‖S‖ ∨ ‖S‖ ≤ i * ‖L‖ from lt_or_ge (i * ‖L‖) ‖S‖ with (lt | le)\n · unfold ext; congr 2; exact bexp_eq_of_lt_length (lt_of_lt_of_le lt $ length_monoton... | [
{
"name": "ext_add₁_pow2",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 2,
"n_lines": 22,
"n_chars": 1621,
"n_subproofs": 3,
"n_tactics": 19,
"cyclomatic": 3,
"n_automation": 10,
"n_rewrites": 1,
"n_structural": 5,
"automation_only": false,
"max_nesti... | 2 | module
public import Foundation.FirstOrder.Arithmetic.Omega1.Basic
@[expose] public section
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₀ ∪ 𝝮₁]
namespace Nuon
@[simp] lemma llen_lt_len_smash_len (K : V) : ‖‖K‖‖ < ‖K ⨳ ‖K‖‖ := by
suffices ‖‖K‖‖ ≤ ‖K‖ * ‖‖K‖‖ by simp... | module
public import Foundation.FirstOrder.Arithmetic.Omega1.Basic
@[expose] public section
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₀ ∪ 𝝮₁]
namespace Nuon
@[simp] lemma llen_lt_len_smash_len (K : V) : ‖‖K‖‖ < ‖K ⨳ ‖K‖‖ := by
suffices ‖‖K‖‖ ≤ ‖K‖ * ‖‖K‖‖ by simp... | @@ -85,6 +85,15 @@
have : bexp S 0 = 1 := (exp_bexp_of_lt (show 0 < ‖S‖ from by simp [pos])).zero_uniq
simp [this, lt_smash_one_iff.mpr h]
+lemma ext_eq_of_ge {L S S' i : V} (h : S ≤ S') : S / bexp S' (i * ‖L‖) % (L ⨳ 1) = S{L}[i] := by
+ rcases show i * ‖L‖ < ‖S‖ ∨ ‖S‖ ≤ i * ‖L‖ from lt_or_ge (i * ‖L‖) ‖S... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_e249934e738b_3 | a122845cb1124b63 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Omega1/Nuon.lean | Nuon | 3 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 32 | 6 | [
{
"theorem_name": "ext_eq_smash_of_le",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": "\n ext_eq_of_gt (mul_len_lt_len_smash h)",
"n_chars": 40,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
... | [
{
"name": "mul_len_lt_len_smash",
"text": "lemma mul_len_lt_len_smash {i I L : V} (hi : i ≤ ‖I‖) : i * ‖L‖ < ‖I ⨳ L‖ := by\n simpa [length_smash, lt_succ_iff_le] using _root_.mul_le_mul_left hi ‖L‖\n\n",
"fan_in": 6,
"n_lines": 4,
"n_chars": 176,
"n_subproofs": 0,
"n_tactics": 2,
"c... | [
{
"name": "len_append",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 8,
"n_chars": 776,
"n_subproofs": 0,
"n_tactics": 6,
"cyclomatic": 1,
"n_automation": 2,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": false,
"max_nesting": 86... | 1 | module
public import Foundation.FirstOrder.Arithmetic.Omega1.Basic
@[expose] public section
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₀ ∪ 𝝮₁]
namespace Nuon
@[simp] lemma llen_lt_len_smash_len (K : V) : ‖‖K‖‖ < ‖K ⨳ ‖K‖‖ := by
suffices ‖‖K‖‖ ≤ ‖K‖ * ‖‖K‖‖ by simp... | module
public import Foundation.FirstOrder.Arithmetic.Omega1.Basic
@[expose] public section
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₀ ∪ 𝝮₁]
namespace Nuon
@[simp] lemma llen_lt_len_smash_len (K : V) : ‖‖K‖‖ < ‖K ⨳ ‖K‖‖ := by
suffices ‖‖K‖‖ ≤ ‖K‖ * ‖‖K‖‖ by simp... | @@ -15,6 +15,9 @@
· simp [←hK]
· exact le_mul_of_pos_left pos
+lemma mul_len_lt_len_smash {i I L : V} (hi : i ≤ ‖I‖) : i * ‖L‖ < ‖I ⨳ L‖ := by
+ simpa [length_smash, lt_succ_iff_le] using _root_.mul_le_mul_left hi ‖L‖
+
noncomputable def ext (L S i : V) : V := S / bexp S (i * ‖L‖) % (L ⨳ 1)
local notation ... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_e249934e738b_4 | a2f7448f99045c82 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Omega1/Nuon.lean | Nuon | 4 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 32 | 1 | [
{
"theorem_name": "append_lt_smash",
"depth": 1,
"n_commands": 0,
"n_lines": 8,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n rcases Arithmetic.zero_le X with (rfl | pos)\n · simpa [append_nil]\n using lt_of_lt_of_le (mod_lt _ (bexp_pos $ mul_... | [
{
"name": "append_nil",
"text": "lemma append_nil (I L S i : V) : append I L S i 0 = S % bexp (I ⨳ L) (i * ‖L‖) := by simp [append]\n\n",
"fan_in": 1,
"n_lines": 3,
"n_chars": 106,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_s... | [
{
"name": "append_lt_smash",
"fan_in": 1,
"n_deps_direct": 3,
"n_deps_transitive": 3,
"n_lines": 10,
"n_chars": 605,
"n_subproofs": 1,
"n_tactics": 8,
"cyclomatic": 2,
"n_automation": 5,
"n_rewrites": 0,
"n_structural": 2,
"automation_only": false,
"max_nestin... | 3 | module
public import Foundation.FirstOrder.Arithmetic.Omega1.Basic
@[expose] public section
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₀ ∪ 𝝮₁]
namespace Nuon
@[simp] lemma llen_lt_len_smash_len (K : V) : ‖‖K‖‖ < ‖K ⨳ ‖K‖‖ := by
suffices ‖‖K‖‖ ≤ ‖K‖ * ‖‖K‖‖ by simp... | module
public import Foundation.FirstOrder.Arithmetic.Omega1.Basic
@[expose] public section
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₀ ∪ 𝝮₁]
namespace Nuon
@[simp] lemma llen_lt_len_smash_len (K : V) : ‖‖K‖‖ < ‖K ⨳ ‖K‖‖ := by
suffices ‖‖K‖‖ ≤ ‖K‖ * ‖‖K‖‖ by simp... | @@ -143,6 +143,8 @@
noncomputable def append (I L S i X : V) : V := S % bexp (I ⨳ L) (i * ‖L‖) + X * bexp (I ⨳ L) (i * ‖L‖)
+lemma append_nil (I L S i : V) : append I L S i 0 = S % bexp (I ⨳ L) (i * ‖L‖) := by simp [append]
+
lemma len_append (I L S : V) {i X} (hi : i ≤ ‖I‖) (hX : 0 < X) : ‖append I L S i X‖ = ‖X... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_e249934e738b_5 | c851f25703575df8 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Omega1/Nuon.lean | Nuon | 5 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 32 | 1 | [
{
"theorem_name": "append_lt_smash",
"depth": 1,
"n_commands": 0,
"n_lines": 8,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n rcases Arithmetic.zero_le X with (rfl | pos)\n · simpa [append_nil]\n using lt_of_lt_of_le (mod_lt _ (bexp_pos $ mul_... | [
{
"name": "len_append",
"text": "lemma len_append (I L S : V) {i X} (hi : i ≤ ‖I‖) (hX : 0 < X) : ‖append I L S i X‖ = ‖X‖ + i * ‖L‖ := calc\n ‖append I L S i X‖ = ‖X * bexp (I ⨳ L) (i * ‖L‖) + S % bexp (I ⨳ L) (i * ‖L‖)‖ := by simp [append, add_comm]\n _ = ‖X‖ + log (bexp (I ⨳ L) (i * ‖L... | [
{
"name": "append_lt_sq_smash",
"fan_in": 2,
"n_deps_direct": 1,
"n_deps_transitive": 4,
"n_lines": 10,
"n_chars": 687,
"n_subproofs": 0,
"n_tactics": 8,
"cyclomatic": 2,
"n_automation": 6,
"n_rewrites": 0,
"n_structural": 3,
"automation_only": false,
"max_nes... | 4 | module
public import Foundation.FirstOrder.Arithmetic.Omega1.Basic
@[expose] public section
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₀ ∪ 𝝮₁]
namespace Nuon
@[simp] lemma llen_lt_len_smash_len (K : V) : ‖‖K‖‖ < ‖K ⨳ ‖K‖‖ := by
suffices ‖‖K‖‖ ≤ ‖K‖ * ‖‖K‖‖ by simp... | module
public import Foundation.FirstOrder.Arithmetic.Omega1.Basic
@[expose] public section
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₀ ∪ 𝝮₁]
namespace Nuon
@[simp] lemma llen_lt_len_smash_len (K : V) : ‖‖K‖‖ < ‖K ⨳ ‖K‖‖ := by
suffices ‖‖K‖‖ ≤ ‖K‖ * ‖‖K‖‖ by simp... | @@ -145,11 +145,21 @@
lemma append_nil (I L S i : V) : append I L S i 0 = S % bexp (I ⨳ L) (i * ‖L‖) := by simp [append]
+lemma len_append (I L S : V) {i X} (hi : i ≤ ‖I‖) (hX : 0 < X) : ‖append I L S i X‖ = ‖X‖ + i * ‖L‖ := calc
+ ‖append I L S i X‖ = ‖X * bexp (I ⨳ L) (i * ‖L‖) + S % bexp (I ⨳ L) (i * ‖L‖)‖ := ... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_e249934e738b_6 | cc8f6a78765f813a | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Omega1/Nuon.lean | Nuon | 6 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 32 | 2 | [
{
"theorem_name": "ext_append_last",
"depth": 1,
"n_commands": 0,
"n_lines": 6,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " calc\n (append I L S i X){L}[i] = (S % bexp (I ⨳ L) (i * ‖L‖) + X * bexp (I ⨳ L) (i * ‖L‖)){L}[i + 0] := by simp [append]\n _ ... | [
{
"name": "ext_zero_eq_self_of_le",
"text": "lemma ext_zero_eq_self_of_le {L S : V} (h : ‖S‖ ≤ ‖L‖) : S{L}[0] = S := by\n rcases Arithmetic.zero_le S with (rfl | pos)\n · simp [ext]\n · simp [ext]\n have : bexp S 0 = 1 := (exp_bexp_of_lt (show 0 < ‖S‖ from by simp [pos])).zero_uniq\n simp [this, lt... | [
{
"name": "ext_append_last",
"fan_in": 2,
"n_deps_direct": 3,
"n_deps_transitive": 6,
"n_lines": 8,
"n_chars": 835,
"n_subproofs": 0,
"n_tactics": 6,
"cyclomatic": 1,
"n_automation": 2,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": false,
"max_nesting... | 6 | module
public import Foundation.FirstOrder.Arithmetic.Omega1.Basic
@[expose] public section
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₀ ∪ 𝝮₁]
namespace Nuon
@[simp] lemma llen_lt_len_smash_len (K : V) : ‖‖K‖‖ < ‖K ⨳ ‖K‖‖ := by
suffices ‖‖K‖‖ ≤ ‖K‖ * ‖‖K‖‖ by simp... | module
public import Foundation.FirstOrder.Arithmetic.Omega1.Basic
@[expose] public section
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₀ ∪ 𝝮₁]
namespace Nuon
@[simp] lemma llen_lt_len_smash_len (K : V) : ‖‖K‖‖ < ‖K ⨳ ‖K‖‖ := by
suffices ‖‖K‖‖ ≤ ‖K‖ * ‖‖K‖‖ by simp... | @@ -78,6 +78,13 @@
@[simp] lemma ext_zero (L i : V) : 0{L}[i] = 0 := by simp [ext]
+lemma ext_zero_eq_self_of_le {L S : V} (h : ‖S‖ ≤ ‖L‖) : S{L}[0] = S := by
+ rcases Arithmetic.zero_le S with (rfl | pos)
+ · simp [ext]
+ · simp [ext]
+ have : bexp S 0 = 1 := (exp_bexp_of_lt (show 0 < ‖S‖ from by simp [pos]... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_e249934e738b_7 | 8bbfdd05ec5e8518 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Omega1/Nuon.lean | Nuon | 7 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 32 | 2 | [
{
"theorem_name": "Segment.succ",
"depth": 1,
"n_commands": 0,
"n_lines": 24,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n rcases H with ⟨S, _, H, rfl, rfl⟩\n let S' := append I L S (intv + 1) (S{L}[intv] + fbit A (start + intv))\n have le_len_L ... | [
{
"name": "append_lt_sq_smash",
"text": "lemma append_lt_sq_smash (I L S : V) {i X} (hi : i ≤ ‖I‖) (hX : ‖X‖ ≤ ‖L‖) (Ipos : 0 < I) : append I L S i X < (I ⨳ L)^2 := by\n rcases hi with (rfl | hi)\n · calc\n append I L S ‖I‖ X = S % I ⨳ L + X * I ⨳ L := by simp [append, bexp_eq_smash]\n _ ... | [
{
"name": "Segment.succ",
"fan_in": 11,
"n_deps_direct": 3,
"n_deps_transitive": 15,
"n_lines": 27,
"n_chars": 1927,
"n_subproofs": 2,
"n_tactics": 28,
"cyclomatic": 3,
"n_automation": 12,
"n_rewrites": 3,
"n_structural": 9,
"automation_only": false,
"max_nest... | 15 | module
public import Foundation.FirstOrder.Arithmetic.Omega1.Basic
@[expose] public section
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₀ ∪ 𝝮₁]
namespace Nuon
@[simp] lemma llen_lt_len_smash_len (K : V) : ‖‖K‖‖ < ‖K ⨳ ‖K‖‖ := by
suffices ‖‖K‖‖ ≤ ‖K‖ * ‖‖K‖‖ by simp... | module
public import Foundation.FirstOrder.Arithmetic.Omega1.Basic
@[expose] public section
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₀ ∪ 𝝮₁]
namespace Nuon
@[simp] lemma llen_lt_len_smash_len (K : V) : ‖‖K‖‖ < ‖K ⨳ ‖K‖‖ := by
suffices ‖‖K‖‖ ≤ ‖K‖ * ‖‖K‖‖ by simp... | @@ -161,6 +161,15 @@
‖X‖ + i * ‖L‖ ≤ (i + 1) * ‖L‖ := by simp [add_mul, add_comm (i * ‖L‖), hX]
_ ≤ ‖I‖ * ‖L‖ := mul_le_mul_right (succ_le_iff_lt.mpr hi)
+lemma append_lt_sq_smash (I L S : V) {i X} (hi : i ≤ ‖I‖) (hX : ‖X‖ ≤ ‖L‖) (Ipos : 0 < I) : append I L S i X < (I ⨳ L)^2 := by
+ rca... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_e249934e738b_8 | cd2e945b193a75b6 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Omega1/Nuon.lean | Nuon | 8 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 32 | 1 | [
{
"theorem_name": "SeriesSegment.succ",
"depth": 1,
"n_commands": 0,
"n_lines": 31,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n have hhk : (k + 1)/‖I‖ ≤ ‖I‖ := by simpa using div_monotone (succ_le_iff_lt.mpr hk) ‖I‖\n have hnk : n ≤ k := H.le\n ... | [
{
"name": "div_mod_succ",
"text": "lemma div_mod_succ (a b : V) : ((a + 1) / b = a / b + 1 ∧ (a + 1) % b = 0 ∧ a % b + 1 = b) ∨ ((a + 1) / b = a / b ∧ (a + 1) % b = a % b + 1) := by\n rcases Arithmetic.zero_le b with (rfl | pos)\n · simp\n have : a % b + 1 ≤ b := lt_iff_succ_le.mp <| mod_lt a (by simp [p... | [
{
"name": "SeriesSegment.succ",
"fan_in": 10,
"n_deps_direct": 2,
"n_deps_transitive": 3,
"n_lines": 35,
"n_chars": 2253,
"n_subproofs": 9,
"n_tactics": 33,
"cyclomatic": 3,
"n_automation": 18,
"n_rewrites": 1,
"n_structural": 9,
"automation_only": false,
"max... | 3 | module
public import Foundation.FirstOrder.Arithmetic.Omega1.Basic
@[expose] public section
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₀ ∪ 𝝮₁]
namespace Nuon
@[simp] lemma llen_lt_len_smash_len (K : V) : ‖‖K‖‖ < ‖K ⨳ ‖K‖‖ := by
suffices ‖‖K‖‖ ≤ ‖K‖ * ‖‖K‖‖ by simp... | module
public import Foundation.FirstOrder.Arithmetic.Omega1.Basic
@[expose] public section
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₀ ∪ 𝝮₁]
namespace Nuon
@[simp] lemma llen_lt_len_smash_len (K : V) : ‖‖K‖‖ < ‖K ⨳ ‖K‖‖ := by
suffices ‖‖K‖‖ ≤ ‖K‖ * ‖‖K‖‖ by simp... | @@ -339,9 +339,57 @@
Eq.trans (ext_append_lt I L T (succ_le_iff_lt.mpr hiter) (by simp)) Tₛ,
hTlast ⟩
+lemma div_mod_succ (a b : V) : ((a + 1) / b = a / b + 1 ∧ (a + 1) % b = 0 ∧ a % b + 1 = b) ∨ ((a + 1) / b = a / b ∧ (a + 1) % b = a % b + 1) := by
+ rcases Arithmetic.zero_le b with (rfl | pos)
+ · simp
... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_e249934e738b_9 | 783e7be891072e51 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Omega1/Nuon.lean | Nuon | 9 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 32 | 2 | [
{
"theorem_name": "polyI_le",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by simp [len_polyI pos]",
"n_chars": 24,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 1,
"... | [
{
"name": "len_polyI",
"text": "lemma len_polyI {A : V} (pos : 0 < A) : ‖polyI A‖ = √‖A‖ + 1 :=\n len_bexp (show √‖A‖ < ‖2 * A‖ from by simp [length_two_mul_of_pos pos, lt_succ_iff_le])\n\n",
"fan_in": 2,
"n_lines": 4,
"n_chars": 175,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": ... | [
{
"name": "polyI_le",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 3,
"n_chars": 95,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"max_nesting": 1
}
... | 1 | module
public import Foundation.FirstOrder.Arithmetic.Omega1.Basic
@[expose] public section
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₀ ∪ 𝝮₁]
namespace Nuon
@[simp] lemma llen_lt_len_smash_len (K : V) : ‖‖K‖‖ < ‖K ⨳ ‖K‖‖ := by
suffices ‖‖K‖‖ ≤ ‖K‖ * ‖‖K‖‖ by simp... | module
public import Foundation.FirstOrder.Arithmetic.Omega1.Basic
@[expose] public section
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₀ ∪ 𝝮₁]
namespace Nuon
@[simp] lemma llen_lt_len_smash_len (K : V) : ‖‖K‖‖ < ‖K ⨳ ‖K‖‖ := by
suffices ‖‖K‖‖ ≤ ‖K‖ * ‖‖K‖‖ by simp... | @@ -408,8 +408,11 @@
def polyU (A : V) : V := (2 * A + 1) ^ 128
-lemma polyI_le {A : V} (pos : 0 < A) : ‖A‖ < ‖polyI A‖ ^ 2 := sorry
+lemma len_polyI {A : V} (pos : 0 < A) : ‖polyI A‖ = √‖A‖ + 1 :=
+ len_bexp (show √‖A‖ < ‖2 * A‖ from by simp [length_two_mul_of_pos pos, lt_succ_iff_le])
+lemma polyI_le {A : V} ... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_e249934e738b_10 | e13164db8af6cc94 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Omega1/Nuon.lean | Nuon | 10 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 32 | 1 | [
{
"theorem_name": "four_mul_eq_two_mul_two_mul",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by simp [←two_add_two_eq_four, add_mul, two_mul]",
"n_chars": 51,
"n_subproofs": 0,
"n_tactics": 1,
"... | [
{
"name": "two_add_two_eq_four",
"text": "lemma two_add_two_eq_four : 2 + 2 = (4 : V) := by simp [←three_add_one_eq_four, ←two_add_one_eq_three, ←one_add_one_eq_two, add_assoc]\n\n",
"fan_in": 1,
"n_lines": 3,
"n_chars": 142,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_... | [
{
"name": "four_mul_eq_two_mul_two_mul",
"fan_in": 2,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 3,
"n_chars": 119,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"... | 1 | module
public import Foundation.FirstOrder.Arithmetic.Omega1.Basic
@[expose] public section
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₀ ∪ 𝝮₁]
namespace Nuon
@[simp] lemma llen_lt_len_smash_len (K : V) : ‖‖K‖‖ < ‖K ⨳ ‖K‖‖ := by
suffices ‖‖K‖‖ ≤ ‖K‖ * ‖‖K‖‖ by simp... | module
public import Foundation.FirstOrder.Arithmetic.Omega1.Basic
@[expose] public section
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₀ ∪ 𝝮₁]
namespace Nuon
@[simp] lemma llen_lt_len_smash_len (K : V) : ‖‖K‖‖ < ‖K ⨳ ‖K‖‖ := by
suffices ‖‖K‖‖ ≤ ‖K‖ * ‖‖K‖‖ by simp... | @@ -413,8 +413,10 @@
lemma polyI_le {A : V} (pos : 0 < A) : ‖A‖ < ‖polyI A‖ ^ 2 := by simp [len_polyI pos]
-lemma four_mul_eq_two_mul_two_mul (a : V) : 4 * a = 2 * (2 * a) := sorry
+lemma two_add_two_eq_four : 2 + 2 = (4 : V) := by simp [←three_add_one_eq_four, ←two_add_one_eq_three, ←one_add_one_eq_two, add_assoc... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_e249934e738b_11 | 8e1aa7261bf89519 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Omega1/Nuon.lean | Nuon | 11 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 32 | 2 | [
{
"theorem_name": "four_mul_smash_self",
"depth": 1,
"n_commands": 0,
"n_lines": 7,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " calc\n (4 * a) ⨳ (4 * a) ≤ ((4 * a) ⨳ (2 * a)) ^ 2 := by simp [four_mul_eq_two_mul_two_mul, smash_two_mul_le_sq_smash]\n _ ... | [
{
"name": "four_mul_eq_two_mul_two_mul",
"text": "lemma four_mul_eq_two_mul_two_mul (a : V) : 4 * a = 2 * (2 * a) := by simp [←two_add_two_eq_four, add_mul, two_mul]\n\n",
"fan_in": 2,
"n_lines": 3,
"n_chars": 119,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation"... | [
{
"name": "four_mul_smash_self",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 2,
"n_lines": 9,
"n_chars": 704,
"n_subproofs": 0,
"n_tactics": 7,
"cyclomatic": 1,
"n_automation": 5,
"n_rewrites": 1,
"n_structural": 1,
"automation_only": false,
"max_nes... | 2 | module
public import Foundation.FirstOrder.Arithmetic.Omega1.Basic
@[expose] public section
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₀ ∪ 𝝮₁]
namespace Nuon
@[simp] lemma llen_lt_len_smash_len (K : V) : ‖‖K‖‖ < ‖K ⨳ ‖K‖‖ := by
suffices ‖‖K‖‖ ≤ ‖K‖ * ‖‖K‖‖ by simp... | module
public import Foundation.FirstOrder.Arithmetic.Omega1.Basic
@[expose] public section
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₀ ∪ 𝝮₁]
namespace Nuon
@[simp] lemma llen_lt_len_smash_len (K : V) : ‖‖K‖‖ < ‖K ⨳ ‖K‖‖ := by
suffices ‖‖K‖‖ ≤ ‖K‖ * ‖‖K‖‖ by simp... | @@ -415,9 +415,17 @@
lemma two_add_two_eq_four : 2 + 2 = (4 : V) := by simp [←three_add_one_eq_four, ←two_add_one_eq_three, ←one_add_one_eq_two, add_assoc]
+lemma four_mul_eq_two_mul_two_mul (a : V) : 4 * a = 2 * (2 * a) := by simp [←two_add_two_eq_four, add_mul, two_mul]
+
@[simp] lemma two_mul_sqrt_le_self (a :... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_e249934e738b_12 | ad9a33a2ada1e807 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Omega1/Nuon.lean | Nuon | 12 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 32 | 1 | [
{
"theorem_name": "polyI_smash_self_polybounded",
"depth": 1,
"n_commands": 0,
"n_lines": 16,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " calc\n (polyI A) ⨳ (polyI A) = bexp ((polyI A) ⨳ (polyI A)) ((√‖A‖ + 1) ^ 2) := Eq.symm <| by simpa [sq, len_polyI... | [
{
"name": "bexp_four_mul",
"text": "lemma bexp_four_mul {a a' x : V} (hx : 4 * x < ‖a‖) (hx' : x < ‖a'‖) :\n bexp a (4 * x) = (bexp a' x) ^ 4 := by\n rw [four_mul_eq_two_mul_two_mul, bexp_two_mul (a' := a), bexp_two_mul (a := a), pow_four_eq_sq_sq]\n · exact lt_of_le_of_lt (by simp [four_mul_eq_two_mul... | [
{
"name": "polyI_smash_polyL_polybounded",
"fan_in": 1,
"n_deps_direct": 3,
"n_deps_transitive": 7,
"n_lines": 8,
"n_chars": 738,
"n_subproofs": 0,
"n_tactics": 6,
"cyclomatic": 1,
"n_automation": 7,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": false,
... | 6 | module
public import Foundation.FirstOrder.Arithmetic.Omega1.Basic
@[expose] public section
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₀ ∪ 𝝮₁]
namespace Nuon
@[simp] lemma llen_lt_len_smash_len (K : V) : ‖‖K‖‖ < ‖K ⨳ ‖K‖‖ := by
suffices ‖‖K‖‖ ≤ ‖K‖ * ‖‖K‖‖ by simp... | module
public import Foundation.FirstOrder.Arithmetic.Omega1.Basic
@[expose] public section
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₀ ∪ 𝝮₁]
namespace Nuon
@[simp] lemma llen_lt_len_smash_len (K : V) : ‖‖K‖‖ < ‖K ⨳ ‖K‖‖ := by
suffices ‖‖K‖‖ ≤ ‖K‖ * ‖‖K‖‖ by simp... | @@ -434,8 +434,31 @@
@[simp] lemma pow_four_le_pow_four {a b : V} : a ^ 4 ≤ b ^ 4 ↔ a ≤ b := by simp [pow_four_eq_sq_sq]
-lemma polyI_smash_self_polybounded {A : V} (pos : 0 < A) : (polyI A) ⨳ (polyI A) ≤ (2 * A + 1) ^ 4 := sorry
+lemma bexp_four_mul {a a' x : V} (hx : 4 * x < ‖a‖) (hx' : x < ‖a'‖) :
+ bexp a (... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_e249934e738b_13 | 0841c6fa251259e6 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Omega1/Nuon.lean | Nuon | 13 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 32 | 1 | [
{
"theorem_name": "polyI_smash_polyL_polybounded",
"depth": 1,
"n_commands": 0,
"n_lines": 6,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " calc\n (polyI A) ⨳ (polyL A) ≤ (polyI A) ⨳ (3 * polyI A) := smash_monotone (by rfl) (by simp [polyL, sq_le... | [
{
"name": "polyI_smash_self_polybounded",
"text": "lemma polyI_smash_self_polybounded {A : V} (pos : 0 < A) : (polyI A) ⨳ (polyI A) ≤ (2 * A + 1) ^ 4 := calc\n (polyI A) ⨳ (polyI A) = bexp ((polyI A) ⨳ (polyI A)) ((√‖A‖ + 1) ^ 2) := Eq.symm <| by simpa [sq, len_polyI pos] using bexp_eq_smash (polyI A) (pol... | [
{
"name": "sq_polyI_smash_polyL_polybounded",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 8,
"n_lines": 5,
"n_chars": 315,
"n_subproofs": 0,
"n_tactics": 3,
"cyclomatic": 1,
"n_automation": 2,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": false,... | 7 | module
public import Foundation.FirstOrder.Arithmetic.Omega1.Basic
@[expose] public section
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₀ ∪ 𝝮₁]
namespace Nuon
@[simp] lemma llen_lt_len_smash_len (K : V) : ‖‖K‖‖ < ‖K ⨳ ‖K‖‖ := by
suffices ‖‖K‖‖ ≤ ‖K‖ * ‖‖K‖‖ by simp... | module
public import Foundation.FirstOrder.Arithmetic.Omega1.Basic
@[expose] public section
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₀ ∪ 𝝮₁]
namespace Nuon
@[simp] lemma llen_lt_len_smash_len (K : V) : ‖‖K‖‖ < ‖K ⨳ ‖K‖‖ := by
suffices ‖‖K‖‖ ≤ ‖K‖ * ‖‖K‖‖ by simp... | @@ -442,8 +442,30 @@
· simpa [four_mul_eq_two_mul_two_mul] using hx
· exact lt_of_le_of_lt (by simp [four_mul_eq_two_mul_two_mul]) hx
-lemma polyI_smash_polyL_polybounded {A : V} (pos : 0 < A) : (polyI A) ⨳ (polyL A) ≤ (2 * A + 1) ^ 64 := sorry
+lemma polyI_smash_self_polybounded {A : V} (pos : 0 < A) : (polyI ... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_e249934e738b_14 | 19eff12043b27401 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Omega1/Nuon.lean | Nuon | 14 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 32 | 3 | [
{
"theorem_name": "NuonAux.succ_elim",
"depth": 1,
"n_commands": 0,
"n_lines": 4,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n rcases NuonAux.exists hk with ⟨n', H'⟩\n rcases H.uniq (H'.succ hk)\n exact ⟨n', rfl, H'⟩",
"n_chars": 103,
"n_... | [
{
"name": "NuonAux.exists",
"text": "lemma NuonAux.exists {k : V} (hk : k ≤ ‖A‖) : ∃ n, NuonAux A k n := by\n suffices ∃ n ≤ k, NuonAux A k n by\n rcases this with ⟨n, _, h⟩; exact ⟨n, h⟩\n revert hk\n induction k using ISigma0.succ_induction\n · definability\n case zero =>\n intro _; exact ⟨0, b... | [
{
"name": "NuonAux.succ_elim",
"fan_in": 2,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 6,
"n_chars": 235,
"n_subproofs": 0,
"n_tactics": 4,
"cyclomatic": 3,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 3,
"automation_only": false,
"max_nesti... | 1 | module
public import Foundation.FirstOrder.Arithmetic.Omega1.Basic
@[expose] public section
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₀ ∪ 𝝮₁]
namespace Nuon
@[simp] lemma llen_lt_len_smash_len (K : V) : ‖‖K‖‖ < ‖K ⨳ ‖K‖‖ := by
suffices ‖‖K‖‖ ≤ ‖K‖ * ‖‖K‖‖ by simp... | module
public import Foundation.FirstOrder.Arithmetic.Omega1.Basic
@[expose] public section
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₀ ∪ 𝝮₁]
namespace Nuon
@[simp] lemma llen_lt_len_smash_len (K : V) : ‖‖K‖‖ < ‖K ⨳ ‖K‖‖ := by
suffices ‖‖K‖‖ ≤ ‖K‖ * ‖‖K‖‖ by simp... | @@ -560,8 +560,24 @@
· rcases show n = 0 from H.uniq (NuonAux.zero k); simp
exact SeriesSegment.succ (sq_polyI_smash_polyL_polybounded pos) (by simp [polyL]) (lt_of_le_of_lt hk $ polyI_le pos) H
-lemma NuonAux.succ_elim {A k : V} (hk : k ≤ ‖A‖) (H : NuonAux A (k + 1) n) : ∃ n', n = n' + fbit A k ∧ NuonAux A k n... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_e249934e738b_15 | 4400b7c01b9af423 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Omega1/Nuon.lean | Nuon | 15 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 32 | 1 | [
{
"theorem_name": "NuonAux.succ",
"depth": 1,
"n_commands": 0,
"n_lines": 4,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n rcases Arithmetic.zero_le A with (rfl | pos)\n · rcases show n = 0 from H.uniq (NuonAux.zero k); simp\n exact SeriesSegment.... | [
{
"name": "sq_polyI_smash_polyL_polybounded",
"text": "lemma sq_polyI_smash_polyL_polybounded {A : V} (pos : 0 < A) : ((polyI A) ⨳ (polyL A)) ^ 2 ≤ polyU A := calc\n ((polyI A) ⨳ (polyL A)) ^ 2 ≤ ((2 * A + 1) ^ 64) ^ 2 := by simp [polyI_smash_polyL_polybounded pos]\n _ = polyU A ... | [
{
"name": "NuonAux.succ_iff",
"fan_in": 0,
"n_deps_direct": 2,
"n_deps_transitive": 18,
"n_lines": 9,
"n_chars": 296,
"n_subproofs": 0,
"n_tactics": 7,
"cyclomatic": 4,
"n_automation": 1,
"n_rewrites": 1,
"n_structural": 6,
"automation_only": false,
"max_nesti... | 13 | module
public import Foundation.FirstOrder.Arithmetic.Omega1.Basic
@[expose] public section
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₀ ∪ 𝝮₁]
namespace Nuon
@[simp] lemma llen_lt_len_smash_len (K : V) : ‖‖K‖‖ < ‖K ⨳ ‖K‖‖ := by
suffices ‖‖K‖‖ ≤ ‖K‖ * ‖‖K‖‖ by simp... | module
public import Foundation.FirstOrder.Arithmetic.Omega1.Basic
@[expose] public section
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₀ ∪ 𝝮₁]
namespace Nuon
@[simp] lemma llen_lt_len_smash_len (K : V) : ‖‖K‖‖ < ‖K ⨳ ‖K‖‖ := by
suffices ‖‖K‖‖ ≤ ‖K‖ * ‖‖K‖‖ by simp... | @@ -466,6 +466,10 @@
_ ≤ ((2 * A + 1) ^ 4) ^ (4 * 4) := by simp only [pow_mul, pow_four_le_pow_four, polyI_smash_self_polybounded pos]
_ = (2 * A + 1) ^ 64 := by simp [←pow_mul]
+lemma sq_polyI_smash_polyL_polybounded {A : V} (pos : 0 < A) : ((polyI A) ⨳ (po... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_e249934e738b_16 | 1a6c0961559b0d2d | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Omega1/Nuon.lean | Nuon | 16 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 32 | 3 | [
{
"theorem_name": "NuonAux.succ_elim",
"depth": 1,
"n_commands": 0,
"n_lines": 4,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n rcases NuonAux.exists hk with ⟨n', H'⟩\n rcases H.uniq (H'.succ hk)\n exact ⟨n', rfl, H'⟩",
"n_chars": 103,
"n_... | [
{
"name": "NuonAux.exists",
"text": "lemma NuonAux.exists {k : V} (hk : k ≤ ‖A‖) : ∃ n, NuonAux A k n := by\n suffices ∃ n ≤ k, NuonAux A k n by\n rcases this with ⟨n, _, h⟩; exact ⟨n, h⟩\n revert hk\n induction k using ISigma0.succ_induction\n · definability\n case zero =>\n intro _; exact ⟨0, b... | [
{
"name": "nuon_bit0",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 3,
"n_lines": 7,
"n_chars": 253,
"n_subproofs": 1,
"n_tactics": 5,
"cyclomatic": 2,
"n_automation": 4,
"n_rewrites": 0,
"n_structural": 2,
"automation_only": false,
"max_nesting": 4
... | 1 | module
public import Foundation.FirstOrder.Arithmetic.Omega1.Basic
@[expose] public section
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₀ ∪ 𝝮₁]
namespace Nuon
@[simp] lemma llen_lt_len_smash_len (K : V) : ‖‖K‖‖ < ‖K ⨳ ‖K‖‖ := by
suffices ‖‖K‖‖ ≤ ‖K‖ * ‖‖K‖‖ by simp... | module
public import Foundation.FirstOrder.Arithmetic.Omega1.Basic
@[expose] public section
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₀ ∪ 𝝮₁]
namespace Nuon
@[simp] lemma llen_lt_len_smash_len (K : V) : ‖‖K‖‖ < ‖K ⨳ ‖K‖‖ := by
suffices ‖‖K‖‖ ≤ ‖K‖ * ‖‖K‖‖ by simp... | @@ -560,8 +560,24 @@
· rcases show n = 0 from H.uniq (NuonAux.zero k); simp
exact SeriesSegment.succ (sq_polyI_smash_polyL_polybounded pos) (by simp [polyL]) (lt_of_le_of_lt hk $ polyI_le pos) H
-lemma NuonAux.succ_elim {A k : V} (hk : k ≤ ‖A‖) (H : NuonAux A (k + 1) n) : ∃ n', n = n' + fbit A k ∧ NuonAux A k n... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_e249934e738b_17 | e81dcac2149227cd | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Omega1/Nuon.lean | Nuon | 17 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 32 | 3 | [
{
"theorem_name": "NuonAux.succ_elim",
"depth": 1,
"n_commands": 0,
"n_lines": 4,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n rcases NuonAux.exists hk with ⟨n', H'⟩\n rcases H.uniq (H'.succ hk)\n exact ⟨n', rfl, H'⟩",
"n_chars": 103,
"n_... | [
{
"name": "NuonAux.exists",
"text": "lemma NuonAux.exists {k : V} (hk : k ≤ ‖A‖) : ∃ n, NuonAux A k n := by\n suffices ∃ n ≤ k, NuonAux A k n by\n rcases this with ⟨n, _, h⟩; exact ⟨n, h⟩\n revert hk\n induction k using ISigma0.succ_induction\n · definability\n case zero =>\n intro _; exact ⟨0, b... | [
{
"name": "nuon_bit1",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 3,
"n_lines": 5,
"n_chars": 212,
"n_subproofs": 1,
"n_tactics": 3,
"cyclomatic": 1,
"n_automation": 2,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": false,
"max_nesting": 2
... | 1 | module
public import Foundation.FirstOrder.Arithmetic.Omega1.Basic
@[expose] public section
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₀ ∪ 𝝮₁]
namespace Nuon
@[simp] lemma llen_lt_len_smash_len (K : V) : ‖‖K‖‖ < ‖K ⨳ ‖K‖‖ := by
suffices ‖‖K‖‖ ≤ ‖K‖ * ‖‖K‖‖ by simp... | module
public import Foundation.FirstOrder.Arithmetic.Omega1.Basic
@[expose] public section
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₀ ∪ 𝝮₁]
namespace Nuon
@[simp] lemma llen_lt_len_smash_len (K : V) : ‖‖K‖‖ < ‖K ⨳ ‖K‖‖ := by
suffices ‖‖K‖‖ ≤ ‖K‖ * ‖‖K‖‖ by simp... | @@ -560,8 +560,24 @@
· rcases show n = 0 from H.uniq (NuonAux.zero k); simp
exact SeriesSegment.succ (sq_polyI_smash_polyL_polybounded pos) (by simp [polyL]) (lt_of_le_of_lt hk $ polyI_le pos) H
-lemma NuonAux.succ_elim {A k : V} (hk : k ≤ ‖A‖) (H : NuonAux A (k + 1) n) : ∃ n', n = n' + fbit A k ∧ NuonAux A k n... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_e249934e738b_18 | 3e512d4274a9c95b | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Omega1/Nuon.lean | Nuon | 18 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 32 | 3 | [
{
"theorem_name": "nuon_bit0",
"depth": 1,
"n_commands": 0,
"n_lines": 5,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n rcases Arithmetic.zero_le a with (rfl | pos)\n · simp\n · have : Nuon (2 * a) (nuon a) := by simpa [Nuon, length_two_mul_of_pos... | [
{
"name": "Nuon.nuon_eq",
"text": "lemma Nuon.nuon_eq {a b : V} (h : Nuon a b) : nuon a = b := (nuon_nuon a).uniq h\n\n",
"fan_in": 3,
"n_lines": 3,
"n_chars": 82,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
... | [
{
"name": "nuon_zero",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 4,
"n_lines": 3,
"n_chars": 77,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"max_nesting": 1
}... | 2 | module
public import Foundation.FirstOrder.Arithmetic.Omega1.Basic
@[expose] public section
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₀ ∪ 𝝮₁]
namespace Nuon
@[simp] lemma llen_lt_len_smash_len (K : V) : ‖‖K‖‖ < ‖K ⨳ ‖K‖‖ := by
suffices ‖‖K‖‖ ≤ ‖K‖ * ‖‖K‖‖ by simp... | module
public import Foundation.FirstOrder.Arithmetic.Omega1.Basic
@[expose] public section
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₀ ∪ 𝝮₁]
namespace Nuon
@[simp] lemma llen_lt_len_smash_len (K : V) : ‖‖K‖‖ < ‖K ⨳ ‖K‖‖ := by
suffices ‖‖K‖‖ ≤ ‖K‖ * ‖‖K‖‖ by simp... | @@ -607,9 +607,11 @@
@[simp] lemma nuon_nuon (a : V) : Nuon a (nuon a) := Classical.choose!_spec (Nuon.exists_unique a)
+lemma Nuon.nuon_eq {a b : V} (h : Nuon a b) : nuon a = b := (nuon_nuon a).uniq h
+
lemma Nuon.nuon_eq_iff {a b : V} : b = nuon a ↔ Nuon a b := Classical.choose!_eq_iff_right (Nuon.exists_unique... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_6ec1955dec30_0 | 1cf872435a3d1df7 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Bootstrapping/Syntax/Formula/Basic.lean | Basic | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 26 | 2 | [
{
"theorem_name": "graph_exists",
"depth": 1,
"n_commands": 0,
"n_lines": 27,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n haveI : 𝚺₁-Function₁ c.allChanges := c.allChanges_defined.to_definable\n haveI : 𝚺₁-Function₁ c.exsChanges := c.exChanges_... | [
{
"name": "graph_all",
"text": "lemma graph_all {p₁ r₁ : V} (hp₁ : IsUFormula L p₁) (h₁ : c.Graph L (c.allChanges param) p₁ r₁) :\n c.Graph L param (^∀ p₁) (c.all param p₁ r₁) :=\n (Graph.case_iff).mpr ⟨by simp [hp₁], Or.inr <| Or.inr <| Or.inr <| Or.inr <| Or.inr <| Or.inr <| Or.inl ⟨p₁, r₁, h₁, rfl, r... | [
{
"name": "graph_exists",
"fan_in": 1,
"n_deps_direct": 10,
"n_deps_transitive": 12,
"n_lines": 29,
"n_chars": 1728,
"n_subproofs": 1,
"n_tactics": 27,
"cyclomatic": 8,
"n_automation": 20,
"n_rewrites": 0,
"n_structural": 17,
"automation_only": false,
"max_nes... | 8 | module
public import Foundation.FirstOrder.Bootstrapping.Syntax.Term.Basic
public import Foundation.FirstOrder.Arithmetic.Induction
@[expose] public section
namespace LO.FirstOrder.Arithmetic.Bootstrapping
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
variable {L : Language} [L.Encodable] [L.LORDefinab... | module
public import Foundation.FirstOrder.Bootstrapping.Syntax.Term.Basic
public import Foundation.FirstOrder.Arithmetic.Induction
@[expose] public section
namespace LO.FirstOrder.Arithmetic.Bootstrapping
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
variable {L : Language} [L.Encodable] [L.LORDefinab... | @@ -869,6 +869,10 @@
· simp [qqOr, qqAll] at H
· simp [qqOr, qqExs] at H
+lemma graph_all {p₁ r₁ : V} (hp₁ : IsUFormula L p₁) (h₁ : c.Graph L (c.allChanges param) p₁ r₁) :
+ c.Graph L param (^∀ p₁) (c.all param p₁ r₁) :=
+ (Graph.case_iff).mpr ⟨by simp [hp₁], Or.inr <| Or.inr <| Or.inr <| Or.inr <| Or.inr <... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_6ec1955dec30_1 | dfa0cd547c81203a | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Bootstrapping/Syntax/Formula/Basic.lean | Basic | 1 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 26 | 1 | [
{
"theorem_name": "graph_unique",
"depth": 1,
"n_commands": 0,
"n_lines": 33,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n apply IsUFormula.ISigma1.pi1_succ_induction (P := fun p ↦ ∀ {param r r'}, c.Graph L param p r → c.Graph L param p r' → r = r'... | [
{
"name": "graph_all_inv",
"text": "lemma graph_all_inv {p₁ r : V} :\n c.Graph L param (^∀ p₁) r → ∃ r₁, c.Graph L (c.allChanges param) p₁ r₁ ∧ r = c.all param p₁ r₁ := by\n intro h\n rcases Graph.case_iff.mp h with ⟨_, (⟨_, _, _, H, _⟩ | ⟨_, _, _, H, _⟩ | ⟨H, _⟩ | ⟨H, _⟩ |\n ⟨_, _, _, _, _, _, H, _... | [
{
"name": "graph_unique",
"fan_in": 1,
"n_deps_direct": 8,
"n_deps_transitive": 12,
"n_lines": 35,
"n_chars": 1457,
"n_subproofs": 0,
"n_tactics": 39,
"cyclomatic": 15,
"n_automation": 16,
"n_rewrites": 0,
"n_structural": 21,
"automation_only": false,
"max_nes... | 8 | module
public import Foundation.FirstOrder.Bootstrapping.Syntax.Term.Basic
public import Foundation.FirstOrder.Arithmetic.Induction
@[expose] public section
namespace LO.FirstOrder.Arithmetic.Bootstrapping
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
variable {L : Language} [L.Encodable] [L.LORDefinab... | module
public import Foundation.FirstOrder.Bootstrapping.Syntax.Term.Basic
public import Foundation.FirstOrder.Arithmetic.Induction
@[expose] public section
namespace LO.FirstOrder.Arithmetic.Bootstrapping
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
variable {L : Language} [L.Encodable] [L.LORDefinab... | @@ -873,6 +873,21 @@
c.Graph L param (^∀ p₁) (c.all param p₁ r₁) :=
(Graph.case_iff).mpr ⟨by simp [hp₁], Or.inr <| Or.inr <| Or.inr <| Or.inr <| Or.inr <| Or.inr <| Or.inl ⟨p₁, r₁, h₁, rfl, rfl⟩⟩
+lemma graph_all_inv {p₁ r : V} :
+ c.Graph L param (^∀ p₁) r → ∃ r₁, c.Graph L (c.allChanges param) p₁ r₁ ∧ r ... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_6ec1955dec30_2 | 92863d8e99381615 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Bootstrapping/Syntax/Formula/Basic.lean | Basic | 2 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 26 | 2 | [
{
"theorem_name": "graph_exists",
"depth": 1,
"n_commands": 0,
"n_lines": 27,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n haveI : 𝚺₁-Function₁ c.allChanges := c.allChanges_defined.to_definable\n haveI : 𝚺₁-Function₁ c.exsChanges := c.exChanges_... | [
{
"name": "graph_ex",
"text": "lemma graph_ex {p₁ r₁ : V} (hp₁ : IsUFormula L p₁) (h₁ : c.Graph L (c.exsChanges param) p₁ r₁) :\n c.Graph L param (^∃ p₁) (c.exs param p₁ r₁) :=\n (Graph.case_iff).mpr ⟨by simp [hp₁], Or.inr <| Or.inr <| Or.inr <| Or.inr <| Or.inr <| Or.inr <| Or.inr ⟨p₁, r₁, h₁, rfl, rfl... | [
{
"name": "exists_unique",
"fan_in": 1,
"n_deps_direct": 2,
"n_deps_transitive": 22,
"n_lines": 5,
"n_chars": 211,
"n_subproofs": 0,
"n_tactics": 3,
"cyclomatic": 2,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 2,
"automation_only": false,
"max_nesting"... | 18 | module
public import Foundation.FirstOrder.Bootstrapping.Syntax.Term.Basic
public import Foundation.FirstOrder.Arithmetic.Induction
@[expose] public section
namespace LO.FirstOrder.Arithmetic.Bootstrapping
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
variable {L : Language} [L.Encodable] [L.LORDefinab... | module
public import Foundation.FirstOrder.Bootstrapping.Syntax.Term.Basic
public import Foundation.FirstOrder.Arithmetic.Induction
@[expose] public section
namespace LO.FirstOrder.Arithmetic.Bootstrapping
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
variable {L : Language} [L.Encodable] [L.LORDefinab... | @@ -888,6 +888,10 @@
exact ⟨_, by assumption, rfl⟩
· simp [qqAll, qqExs] at H
+lemma graph_ex {p₁ r₁ : V} (hp₁ : IsUFormula L p₁) (h₁ : c.Graph L (c.exsChanges param) p₁ r₁) :
+ c.Graph L param (^∃ p₁) (c.exs param p₁ r₁) :=
+ (Graph.case_iff).mpr ⟨by simp [hp₁], Or.inr <| Or.inr <| Or.inr <| Or.inr <| Or... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_6ec1955dec30_3 | 826186304c8d2b5b | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Bootstrapping/Syntax/Formula/Basic.lean | Basic | 3 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 26 | 1 | [
{
"theorem_name": "graph_unique",
"depth": 1,
"n_commands": 0,
"n_lines": 33,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n apply IsUFormula.ISigma1.pi1_succ_induction (P := fun p ↦ ∀ {param r r'}, c.Graph L param p r → c.Graph L param p r' → r = r'... | [
{
"name": "graph_ex_inv",
"text": "lemma graph_ex_inv {p₁ r : V} :\n c.Graph L param (^∃ p₁) r → ∃ r₁, c.Graph L (c.exsChanges param) p₁ r₁ ∧ r = c.exs param p₁ r₁ := by\n intro h\n rcases Graph.case_iff.mp h with ⟨_, (⟨_, _, _, H, _⟩ | ⟨_, _, _, H, _⟩ | ⟨H, _⟩ | ⟨H, _⟩ |\n ⟨_, _, _, _, _, _, H, _⟩ ... | [
{
"name": "exists_unique_all",
"fan_in": 3,
"n_deps_direct": 1,
"n_deps_transitive": 23,
"n_lines": 4,
"n_chars": 183,
"n_subproofs": 0,
"n_tactics": 3,
"cyclomatic": 2,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
"max_nest... | 19 | module
public import Foundation.FirstOrder.Bootstrapping.Syntax.Term.Basic
public import Foundation.FirstOrder.Arithmetic.Induction
@[expose] public section
namespace LO.FirstOrder.Arithmetic.Bootstrapping
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
variable {L : Language} [L.Encodable] [L.LORDefinab... | module
public import Foundation.FirstOrder.Bootstrapping.Syntax.Term.Basic
public import Foundation.FirstOrder.Arithmetic.Induction
@[expose] public section
namespace LO.FirstOrder.Arithmetic.Bootstrapping
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
variable {L : Language} [L.Encodable] [L.LORDefinab... | @@ -892,6 +892,21 @@
c.Graph L param (^∃ p₁) (c.exs param p₁ r₁) :=
(Graph.case_iff).mpr ⟨by simp [hp₁], Or.inr <| Or.inr <| Or.inr <| Or.inr <| Or.inr <| Or.inr <| Or.inr ⟨p₁, r₁, h₁, rfl, rfl⟩⟩
+lemma graph_ex_inv {p₁ r : V} :
+ c.Graph L param (^∃ p₁) r → ∃ r₁, c.Graph L (c.exsChanges param) p₁ r₁ ∧ r =... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_6ec1955dec30_4 | a5c28e22c131f9b9 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Bootstrapping/Syntax/Formula/Basic.lean | Basic | 4 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 26 | 1 | [
{
"theorem_name": "graph_unique",
"depth": 1,
"n_commands": 0,
"n_lines": 33,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n apply IsUFormula.ISigma1.pi1_succ_induction (P := fun p ↦ ∀ {param r r'}, c.Graph L param p r → c.Graph L param p r' → r = r'... | [
{
"name": "graph_ex_inv",
"text": "lemma graph_ex_inv {p₁ r : V} :\n c.Graph L param (^∃ p₁) r → ∃ r₁, c.Graph L (c.exsChanges param) p₁ r₁ ∧ r = c.exs param p₁ r₁ := by\n intro h\n rcases Graph.case_iff.mp h with ⟨_, (⟨_, _, _, H, _⟩ | ⟨_, _, _, H, _⟩ | ⟨H, _⟩ | ⟨H, _⟩ |\n ⟨_, _, _, _, _, _, H, _⟩ ... | [
{
"name": "result_prop_not",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 24,
"n_lines": 4,
"n_chars": 148,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
"max_nestin... | 19 | module
public import Foundation.FirstOrder.Bootstrapping.Syntax.Term.Basic
public import Foundation.FirstOrder.Arithmetic.Induction
@[expose] public section
namespace LO.FirstOrder.Arithmetic.Bootstrapping
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
variable {L : Language} [L.Encodable] [L.LORDefinab... | module
public import Foundation.FirstOrder.Bootstrapping.Syntax.Term.Basic
public import Foundation.FirstOrder.Arithmetic.Induction
@[expose] public section
namespace LO.FirstOrder.Arithmetic.Bootstrapping
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
variable {L : Language} [L.Encodable] [L.LORDefinab... | @@ -892,6 +892,21 @@
c.Graph L param (^∃ p₁) (c.exs param p₁ r₁) :=
(Graph.case_iff).mpr ⟨by simp [hp₁], Or.inr <| Or.inr <| Or.inr <| Or.inr <| Or.inr <| Or.inr <| Or.inr ⟨p₁, r₁, h₁, rfl, rfl⟩⟩
+lemma graph_ex_inv {p₁ r : V} :
+ c.Graph L param (^∃ p₁) r → ∃ r₁, c.Graph L (c.exsChanges param) p₁ r₁ ∧ r =... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_6ec1955dec30_5 | 87cf9a8757c5a811 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Bootstrapping/Syntax/Formula/Basic.lean | Basic | 5 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 26 | 8 | [
{
"theorem_name": "result_rel",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": "\n c.result_eq_of_graph (c.graph_rel hR hv)",
"n_chars": 43,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n... | [
{
"name": "result_eq_of_graph",
"text": "lemma result_eq_of_graph {p r} (h : c.Graph L param p r) : c.result L param p = r := Eq.symm <|\n Classical.choose_uniq (c.exists_unique_all L param p) (by simp [c.graph_dom_uformula h, h])\n\n",
"fan_in": 8,
"n_lines": 4,
"n_chars": 191,
"n_subproof... | [
{
"name": "result_rel",
"fan_in": 0,
"n_deps_direct": 2,
"n_deps_transitive": 26,
"n_lines": 5,
"n_chars": 179,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
"max_nesting": 2... | 21 | module
public import Foundation.FirstOrder.Bootstrapping.Syntax.Term.Basic
public import Foundation.FirstOrder.Arithmetic.Induction
@[expose] public section
namespace LO.FirstOrder.Arithmetic.Bootstrapping
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
variable {L : Language} [L.Encodable] [L.LORDefinab... | module
public import Foundation.FirstOrder.Bootstrapping.Syntax.Term.Basic
public import Foundation.FirstOrder.Arithmetic.Induction
@[expose] public section
namespace LO.FirstOrder.Arithmetic.Bootstrapping
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
variable {L : Language} [L.Encodable] [L.LORDefinab... | @@ -989,29 +989,38 @@
variable {param}
+lemma result_eq_of_graph {p r} (h : c.Graph L param p r) : c.result L param p = r := Eq.symm <|
+ Classical.choose_uniq (c.exists_unique_all L param p) (by simp [c.graph_dom_uformula h, h])
+
@[simp] lemma result_rel {k R v} (hR : L.IsRel k R) (hv : IsUTermVec L k v) :
- ... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_6ec1955dec30_6 | a2669424aa3d245d | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Bootstrapping/Syntax/Formula/Basic.lean | Basic | 6 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 26 | 8 | [
{
"theorem_name": "result_rel",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": "\n c.result_eq_of_graph (c.graph_rel hR hv)",
"n_chars": 43,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n... | [
{
"name": "result_eq_of_graph",
"text": "lemma result_eq_of_graph {p r} (h : c.Graph L param p r) : c.result L param p = r := Eq.symm <|\n Classical.choose_uniq (c.exists_unique_all L param p) (by simp [c.graph_dom_uformula h, h])\n\n",
"fan_in": 8,
"n_lines": 4,
"n_chars": 191,
"n_subproof... | [
{
"name": "uformula_result_induction",
"fan_in": 0,
"n_deps_direct": 6,
"n_deps_transitive": 31,
"n_lines": 46,
"n_chars": 3200,
"n_subproofs": 1,
"n_tactics": 29,
"cyclomatic": 2,
"n_automation": 28,
"n_rewrites": 0,
"n_structural": 5,
"automation_only": false,
... | 22 | module
public import Foundation.FirstOrder.Bootstrapping.Syntax.Term.Basic
public import Foundation.FirstOrder.Arithmetic.Induction
@[expose] public section
namespace LO.FirstOrder.Arithmetic.Bootstrapping
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
variable {L : Language} [L.Encodable] [L.LORDefinab... | module
public import Foundation.FirstOrder.Bootstrapping.Syntax.Term.Basic
public import Foundation.FirstOrder.Arithmetic.Induction
@[expose] public section
namespace LO.FirstOrder.Arithmetic.Bootstrapping
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
variable {L : Language} [L.Encodable] [L.LORDefinab... | @@ -989,29 +989,38 @@
variable {param}
+lemma result_eq_of_graph {p r} (h : c.Graph L param p r) : c.result L param p = r := Eq.symm <|
+ Classical.choose_uniq (c.exists_unique_all L param p) (by simp [c.graph_dom_uformula h, h])
+
@[simp] lemma result_rel {k R v} (hR : L.IsRel k R) (hv : IsUTermVec L k v) :
- ... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_6ec1955dec30_7 | 29ba9f59592aab18 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Bootstrapping/Syntax/Formula/Basic.lean | Basic | 7 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 26 | 1 | [
{
"theorem_name": "exists_unique",
"depth": 1,
"n_commands": 0,
"n_lines": 3,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n rcases c.graph_exists param hp with ⟨r, hr⟩\n exact ExistsUnique.intro r hr (fun r' hr' ↦ c.graph_unique hp hr' hr)",
"n... | [
{
"name": "graph_exists",
"text": "lemma graph_exists {p : V} : IsUFormula L p → ∃ y, c.Graph L param p y := by\n haveI : 𝚺₁-Function₁ c.allChanges := c.allChanges_defined.to_definable\n haveI : 𝚺₁-Function₁ c.exsChanges := c.exChanges_defined.to_definable\n let f : V → V → V := fun _ param ↦ Max.max p... | [
{
"name": "bv_eq_of_not_isUFormula",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 25,
"n_lines": 3,
"n_chars": 115,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
"ma... | 20 | module
public import Foundation.FirstOrder.Bootstrapping.Syntax.Term.Basic
public import Foundation.FirstOrder.Arithmetic.Induction
@[expose] public section
namespace LO.FirstOrder.Arithmetic.Bootstrapping
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
variable {L : Language} [L.Encodable] [L.LORDefinab... | module
public import Foundation.FirstOrder.Bootstrapping.Syntax.Term.Basic
public import Foundation.FirstOrder.Arithmetic.Induction
@[expose] public section
namespace LO.FirstOrder.Arithmetic.Bootstrapping
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
variable {L : Language} [L.Encodable] [L.LORDefinab... | @@ -909,6 +909,34 @@
variable (param)
+lemma graph_exists {p : V} : IsUFormula L p → ∃ y, c.Graph L param p y := by
+ haveI : 𝚺₁-Function₁ c.allChanges := c.allChanges_defined.to_definable
+ haveI : 𝚺₁-Function₁ c.exsChanges := c.exChanges_defined.to_definable
+ let f : V → V → V := fun _ param ↦ Max.max para... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_6ec1955dec30_8 | d81210ce6b4e4bb0 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Bootstrapping/Syntax/Formula/Basic.lean | Basic | 8 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 26 | 1 | [
{
"theorem_name": "IsSemiformula.induction1",
"depth": 1,
"n_commands": 0,
"n_lines": 5,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": "\n match Γ with\n | 𝚺 => IsSemiformula.sigma1_structural_induction hP hrel hnrel hverum hfalsum hand hor hall hexs\n ... | [
{
"name": "IsSemiformula.pi1_structural_induction",
"text": "lemma IsSemiformula.pi1_structural_induction {P : V → V → Prop} (hP : 𝚷₁-Relation P)\n (hrel : ∀ n k r v, L.IsRel k r → IsSemitermVec L k n v → P n (^rel k r v))\n (hnrel : ∀ n k r v, L.IsRel k r → IsSemitermVec L k n v → P n (^nrel k r v))... | [
{
"name": "IsSemiformula.induction1",
"fan_in": 2,
"n_deps_direct": 3,
"n_deps_transitive": 5,
"n_lines": 17,
"n_chars": 1069,
"n_subproofs": 0,
"n_tactics": 4,
"cyclomatic": 2,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
"... | 2 | module
public import Foundation.FirstOrder.Bootstrapping.Syntax.Term.Basic
public import Foundation.FirstOrder.Arithmetic.Induction
@[expose] public section
namespace LO.FirstOrder.Arithmetic.Bootstrapping
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
variable {L : Language} [L.Encodable] [L.LORDefinab... | module
public import Foundation.FirstOrder.Bootstrapping.Syntax.Term.Basic
public import Foundation.FirstOrder.Arithmetic.Induction
@[expose] public section
namespace LO.FirstOrder.Arithmetic.Bootstrapping
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
variable {L : Language} [L.Encodable] [L.LORDefinab... | @@ -1308,6 +1308,40 @@
· apply hall _ _ h₁ (ih p₁ (by simp) (n + 1) (by simp) h₁)
· apply hexs _ _ h₁ (ih p₁ (by simp) (n + 1) (by simp) h₁)
+lemma IsSemiformula.pi1_structural_induction {P : V → V → Prop} (hP : 𝚷₁-Relation P)
+ (hrel : ∀ n k r v, L.IsRel k r → IsSemitermVec L k n v → P n (^rel k r v))
+ ... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_baba430b5bd9_0 | da2bbcaec5c2031c | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/R0/Basic.lean | Basic | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 9 | 1 | [
{
"theorem_name": "numeral_inj_iff",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": "\n ⟨by contrapose; exact numeral_ne_numeral_of_ne, by rintro rfl; rfl⟩",
"n_chars": 74,
"n_subproofs": 0,
"n_tactics"... | [
{
"name": "numeral_ne_numeral_of_ne",
"text": "lemma numeral_ne_numeral_of_ne {n m : ℕ} (h : n ≠ m) : (numeral n : M) ≠ numeral m := by\n simpa [models_iff] using Theory.models M _ (R0.Ω₃ n m h)\n\n",
"fan_in": 1,
"n_lines": 4,
"n_chars": 158,
"n_subproofs": 0,
"n_tactics": 2,
"cycl... | [
{
"name": "numeral_inj_iff",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 4,
"n_chars": 160,
"n_subproofs": 0,
"n_tactics": 3,
"cyclomatic": 2,
"n_automation": 2,
"n_rewrites": 0,
"n_structural": 2,
"automation_only": false,
"max_nesting... | 1 | module
public import Foundation.FirstOrder.Arithmetic.Basic
@[expose] public section
/-!
# Cobham's theory $\mathsf{R_0}$
-/
noncomputable section
namespace LO.FirstOrder.Arithmetic
inductive R0 : ArithmeticTheory
| equal : ∀ φ ∈ 𝗘𝗤 ℒₒᵣ, R0 φ
| Ω₁ (n m : ℕ) : R0 “↑n + ↑m = ↑(n + m)”
| Ω₂ (n m : ℕ) : R0 “↑... | module
public import Foundation.FirstOrder.Arithmetic.Basic
@[expose] public section
/-!
# Cobham's theory $\mathsf{R_0}$
-/
noncomputable section
namespace LO.FirstOrder.Arithmetic
inductive R0 : ArithmeticTheory
| equal : ∀ φ ∈ 𝗘𝗤 ℒₒᵣ, R0 φ
| Ω₁ (n m : ℕ) : R0 “↑n + ↑m = ↑(n + m)”
| Ω₂ (n m : ℕ) : R0 “↑... | @@ -48,6 +48,9 @@
lemma numeral_mul_numeral (n m : ℕ) : (numeral n : M) * numeral m = numeral (n * m) := by
simpa [models_iff] using Theory.models M _ (R0.Ω₂ n m)
+lemma numeral_ne_numeral_of_ne {n m : ℕ} (h : n ≠ m) : (numeral n : M) ≠ numeral m := by
+ simpa [models_iff] using Theory.models M _ (R0.Ω₃ n m h)
+... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_baba430b5bd9_1 | 5352bb6aec6b347e | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/R0/Basic.lean | Basic | 1 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 9 | 2 | [
{
"theorem_name": "numeral_lt_numeral_iff",
"depth": 1,
"n_commands": 0,
"n_lines": 7,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": "\n ⟨by contrapose\n intro h H\n rcases lt_numeral_iff.mp H with ⟨i, hi⟩\n rcases numeral_inj_iff.mp hi\n ... | [
{
"name": "lt_numeral_iff",
"text": "lemma lt_numeral_iff {x : M} {n : ℕ} : x < numeral n ↔ ∃ i : Fin n, x = numeral i := by\n have := by simpa [models_iff] using Theory.models M _ (R0.Ω₄ n)\n constructor\n · intro hx\n rcases (this x).mp hx with ⟨i, hi, rfl⟩\n exact ⟨⟨i, hi⟩, by simp⟩\n · rintro ... | [
{
"name": "numeral_lt_numeral_iff",
"fan_in": 0,
"n_deps_direct": 2,
"n_deps_transitive": 3,
"n_lines": 9,
"n_chars": 345,
"n_subproofs": 0,
"n_tactics": 6,
"cyclomatic": 3,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 4,
"automation_only": false,
"max_... | 2 | module
public import Foundation.FirstOrder.Arithmetic.Basic
@[expose] public section
/-!
# Cobham's theory $\mathsf{R_0}$
-/
noncomputable section
namespace LO.FirstOrder.Arithmetic
inductive R0 : ArithmeticTheory
| equal : ∀ φ ∈ 𝗘𝗤 ℒₒᵣ, R0 φ
| Ω₁ (n m : ℕ) : R0 “↑n + ↑m = ↑(n + m)”
| Ω₂ (n m : ℕ) : R0 “↑... | module
public import Foundation.FirstOrder.Arithmetic.Basic
@[expose] public section
/-!
# Cobham's theory $\mathsf{R_0}$
-/
noncomputable section
namespace LO.FirstOrder.Arithmetic
inductive R0 : ArithmeticTheory
| equal : ∀ φ ∈ 𝗘𝗤 ℒₒᵣ, R0 φ
| Ω₁ (n m : ℕ) : R0 “↑n + ↑m = ↑(n + m)”
| Ω₂ (n m : ℕ) : R0 “↑... | @@ -51,10 +51,25 @@
lemma numeral_ne_numeral_of_ne {n m : ℕ} (h : n ≠ m) : (numeral n : M) ≠ numeral m := by
simpa [models_iff] using Theory.models M _ (R0.Ω₃ n m h)
+lemma lt_numeral_iff {x : M} {n : ℕ} : x < numeral n ↔ ∃ i : Fin n, x = numeral i := by
+ have := by simpa [models_iff] using Theory.models M _ (R... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_baba430b5bd9_2 | 5fd390f20118ec56 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/R0/Basic.lean | Basic | 2 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 9 | 1 | [
{
"theorem_name": "val_numeral",
"depth": 1,
"n_commands": 0,
"n_lines": 8,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": "\n match t with\n | #_ => by simp\n | &_ => by simp\n | .func Language.Zero.zero _... | [
{
"name": "numeral_mul_numeral",
"text": "lemma numeral_mul_numeral (n m : ℕ) : (numeral n : M) * numeral m = numeral (n * m) := by\n simpa [models_iff] using Theory.models M _ (R0.Ω₂ n m)\n\n",
"fan_in": 1,
"n_lines": 4,
"n_chars": 153,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic... | [
{
"name": "val_numeral",
"fan_in": 1,
"n_deps_direct": 2,
"n_deps_transitive": 2,
"n_lines": 11,
"n_chars": 654,
"n_subproofs": 0,
"n_tactics": 7,
"cyclomatic": 2,
"n_automation": 6,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
"max_nesting": ... | 2 | module
public import Foundation.FirstOrder.Arithmetic.Basic
@[expose] public section
/-!
# Cobham's theory $\mathsf{R_0}$
-/
noncomputable section
namespace LO.FirstOrder.Arithmetic
inductive R0 : ArithmeticTheory
| equal : ∀ φ ∈ 𝗘𝗤 ℒₒᵣ, R0 φ
| Ω₁ (n m : ℕ) : R0 “↑n + ↑m = ↑(n + m)”
| Ω₂ (n m : ℕ) : R0 “↑... | module
public import Foundation.FirstOrder.Arithmetic.Basic
@[expose] public section
/-!
# Cobham's theory $\mathsf{R_0}$
-/
noncomputable section
namespace LO.FirstOrder.Arithmetic
inductive R0 : ArithmeticTheory
| equal : ∀ φ ∈ 𝗘𝗤 ℒₒᵣ, R0 φ
| Ω₁ (n m : ℕ) : R0 “↑n + ↑m = ↑(n + m)”
| Ω₂ (n m : ℕ) : R0 “↑... | @@ -45,6 +45,9 @@
lemma numeral_add_numeral (n m : ℕ) : (numeral n : M) + numeral m = numeral (n + m) := by
simpa [models_iff] using Theory.models M _ (R0.Ω₁ n m)
+lemma numeral_mul_numeral (n m : ℕ) : (numeral n : M) * numeral m = numeral (n * m) := by
+ simpa [models_iff] using Theory.models M _ (R0.Ω₂ n m)
+
... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_baba430b5bd9_3 | 8477101cf56a12a5 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/R0/Basic.lean | Basic | 3 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 9 | 1 | [
{
"theorem_name": "bold_sigma_one_completeness",
"depth": 1,
"n_commands": 0,
"n_lines": 24,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n revert bv\n apply sigma₁_induction' hp\n case hVerum => simp\n case hFalsum => simp\n case hEQ => intro n ... | [
{
"name": "val_numeral",
"text": "lemma val_numeral {n ξ} (bv : Fin n → ℕ) (fv : ξ → ℕ) (t : ArithmeticSemiterm ξ n) :\n t.val (M := M) (numeral ∘ bv) (numeral ∘ fv) = numeral (t.val bv fv) :=\n match t with\n | #_ => by simp\n | &_ => by simp\n | .func... | [
{
"name": "R0.model_complete",
"fan_in": 2,
"n_deps_direct": 1,
"n_deps_transitive": 5,
"n_lines": 7,
"n_chars": 317,
"n_subproofs": 1,
"n_tactics": 4,
"cyclomatic": 1,
"n_automation": 2,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": false,
"max_nesti... | 5 | module
public import Foundation.FirstOrder.Arithmetic.Basic
@[expose] public section
/-!
# Cobham's theory $\mathsf{R_0}$
-/
noncomputable section
namespace LO.FirstOrder.Arithmetic
inductive R0 : ArithmeticTheory
| equal : ∀ φ ∈ 𝗘𝗤 ℒₒᵣ, R0 φ
| Ω₁ (n m : ℕ) : R0 “↑n + ↑m = ↑(n + m)”
| Ω₂ (n m : ℕ) : R0 “↑... | module
public import Foundation.FirstOrder.Arithmetic.Basic
@[expose] public section
/-!
# Cobham's theory $\mathsf{R_0}$
-/
noncomputable section
namespace LO.FirstOrder.Arithmetic
inductive R0 : ArithmeticTheory
| equal : ∀ φ ∈ 𝗘𝗤 ℒₒᵣ, R0 φ
| Ω₁ (n m : ℕ) : R0 “↑n + ↑m = ↑(n + m)”
| Ω₂ (n m : ℕ) : R0 “↑... | @@ -73,8 +73,41 @@
open Hierarchy
+lemma val_numeral {n ξ} (bv : Fin n → ℕ) (fv : ξ → ℕ) (t : ArithmeticSemiterm ξ n) :
+ t.val (M := M) (numeral ∘ bv) (numeral ∘ fv) = numeral (t.val bv fv) :=
+ match t with
+ | #_ => by simp
+ | &_ => by simp
+ | .func Langu... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_baba430b5bd9_4 | 99c55316fcc26aa9 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/R0/Basic.lean | Basic | 4 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 9 | 2 | [
{
"theorem_name": "R0.model_complete",
"depth": 1,
"n_commands": 0,
"n_lines": 4,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n suffices σ.Evalb (M := ℕ) ![] → σ.Evalb (M := M) ![] by simpa [models_iff]\n intro h\n simpa [Matrix.empty_eq, Empty.eq... | [
{
"name": "bold_sigma_one_completeness",
"text": "lemma bold_sigma_one_completeness {n} {φ : ArithmeticSemiformula ξ n} (hp : Hierarchy 𝚺 1 φ) {bv : Fin n → ℕ} {fv : ξ → ℕ} :\n φ.Eval bv fv → φ.Eval (M := M) (numeral ∘ bv) (numeral ∘ fv) := by\n revert bv\n apply sigma₁_induction' hp\n case hVerum =>... | [
{
"name": "nat_extention_sigmaOne",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 6,
"n_lines": 5,
"n_chars": 223,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"max_n... | 6 | module
public import Foundation.FirstOrder.Arithmetic.Basic
@[expose] public section
/-!
# Cobham's theory $\mathsf{R_0}$
-/
noncomputable section
namespace LO.FirstOrder.Arithmetic
inductive R0 : ArithmeticTheory
| equal : ∀ φ ∈ 𝗘𝗤 ℒₒᵣ, R0 φ
| Ω₁ (n m : ℕ) : R0 “↑n + ↑m = ↑(n + m)”
| Ω₂ (n m : ℕ) : R0 “↑... | module
public import Foundation.FirstOrder.Arithmetic.Basic
@[expose] public section
/-!
# Cobham's theory $\mathsf{R_0}$
-/
noncomputable section
namespace LO.FirstOrder.Arithmetic
inductive R0 : ArithmeticTheory
| equal : ∀ φ ∈ 𝗘𝗤 ℒₒᵣ, R0 φ
| Ω₁ (n m : ℕ) : R0 “↑n + ↑m = ↑(n + m)”
| Ω₂ (n m : ℕ) : R0 “↑... | @@ -83,8 +83,37 @@
| .func Language.Add.add v => by simp [Semiterm.val_func, val_numeral _ _ (v 0), val_numeral _ _ (v 1), numeral_add_numeral]
| .func Language.Mul.mul v => by simp [Semiterm.val_func, val_numeral _ _ (v 0), val_numeral _ _ (v 1), numeral_mul_numeral]
+lemma bold_sigma_one_completeness {n} ... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_baba430b5bd9_5 | adac14ac30c61048 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/R0/Basic.lean | Basic | 5 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 9 | 2 | [
{
"theorem_name": "R0.model_complete",
"depth": 1,
"n_commands": 0,
"n_lines": 4,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n suffices σ.Evalb (M := ℕ) ![] → σ.Evalb (M := M) ![] by simpa [models_iff]\n intro h\n simpa [Matrix.empty_eq, Empty.eq... | [
{
"name": "bold_sigma_one_completeness",
"text": "lemma bold_sigma_one_completeness {n} {φ : ArithmeticSemiformula ξ n} (hp : Hierarchy 𝚺 1 φ) {bv : Fin n → ℕ} {fv : ξ → ℕ} :\n φ.Eval bv fv → φ.Eval (M := M) (numeral ∘ bv) (numeral ∘ fv) := by\n revert bv\n apply sigma₁_induction' hp\n case hVerum =>... | [
{
"name": "nat_extention_piOne",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 7,
"n_lines": 6,
"n_chars": 232,
"n_subproofs": 0,
"n_tactics": 3,
"cyclomatic": 1,
"n_automation": 2,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"max_nest... | 7 | module
public import Foundation.FirstOrder.Arithmetic.Basic
@[expose] public section
/-!
# Cobham's theory $\mathsf{R_0}$
-/
noncomputable section
namespace LO.FirstOrder.Arithmetic
inductive R0 : ArithmeticTheory
| equal : ∀ φ ∈ 𝗘𝗤 ℒₒᵣ, R0 φ
| Ω₁ (n m : ℕ) : R0 “↑n + ↑m = ↑(n + m)”
| Ω₂ (n m : ℕ) : R0 “↑... | module
public import Foundation.FirstOrder.Arithmetic.Basic
@[expose] public section
/-!
# Cobham's theory $\mathsf{R_0}$
-/
noncomputable section
namespace LO.FirstOrder.Arithmetic
inductive R0 : ArithmeticTheory
| equal : ∀ φ ∈ 𝗘𝗤 ℒₒᵣ, R0 φ
| Ω₁ (n m : ℕ) : R0 “↑n + ↑m = ↑(n + m)”
| Ω₂ (n m : ℕ) : R0 “↑... | @@ -83,8 +83,37 @@
| .func Language.Add.add v => by simp [Semiterm.val_func, val_numeral _ _ (v 0), val_numeral _ _ (v 1), numeral_add_numeral]
| .func Language.Mul.mul v => by simp [Semiterm.val_func, val_numeral _ _ (v 0), val_numeral _ _ (v 1), numeral_mul_numeral]
+lemma bold_sigma_one_completeness {n} ... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_baba430b5bd9_6 | 5b622e3778c0b621 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/R0/Basic.lean | Basic | 6 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 9 | 1 | [
{
"theorem_name": "bold_sigma_one_completeness",
"depth": 1,
"n_commands": 0,
"n_lines": 24,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n revert bv\n apply sigma₁_induction' hp\n case hVerum => simp\n case hFalsum => simp\n case hEQ => intro n ... | [
{
"name": "val_numeral",
"text": "lemma val_numeral {n ξ} (bv : Fin n → ℕ) (fv : ξ → ℕ) (t : ArithmeticSemiterm ξ n) :\n t.val (M := M) (numeral ∘ bv) (numeral ∘ fv) = numeral (t.val bv fv) :=\n match t with\n | #_ => by simp\n | &_ => by simp\n | .func... | [
{
"name": "bold_sigma_one_completeness'",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 5,
"n_lines": 5,
"n_chars": 303,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
... | 5 | module
public import Foundation.FirstOrder.Arithmetic.Basic
@[expose] public section
/-!
# Cobham's theory $\mathsf{R_0}$
-/
noncomputable section
namespace LO.FirstOrder.Arithmetic
inductive R0 : ArithmeticTheory
| equal : ∀ φ ∈ 𝗘𝗤 ℒₒᵣ, R0 φ
| Ω₁ (n m : ℕ) : R0 “↑n + ↑m = ↑(n + m)”
| Ω₂ (n m : ℕ) : R0 “↑... | module
public import Foundation.FirstOrder.Arithmetic.Basic
@[expose] public section
/-!
# Cobham's theory $\mathsf{R_0}$
-/
noncomputable section
namespace LO.FirstOrder.Arithmetic
inductive R0 : ArithmeticTheory
| equal : ∀ φ ∈ 𝗘𝗤 ℒₒᵣ, R0 φ
| Ω₁ (n m : ℕ) : R0 “↑n + ↑m = ↑(n + m)”
| Ω₂ (n m : ℕ) : R0 “↑... | @@ -73,8 +73,41 @@
open Hierarchy
+lemma val_numeral {n ξ} (bv : Fin n → ℕ) (fv : ξ → ℕ) (t : ArithmeticSemiterm ξ n) :
+ t.val (M := M) (numeral ∘ bv) (numeral ∘ fv) = numeral (t.val bv fv) :=
+ match t with
+ | #_ => by simp
+ | &_ => by simp
+ | .func Langu... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_baba430b5bd9_7 | 3f86398f6676a78e | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/R0/Basic.lean | Basic | 7 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 9 | 2 | [
{
"theorem_name": "R0.model_complete",
"depth": 1,
"n_commands": 0,
"n_lines": 4,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n suffices σ.Evalb (M := ℕ) ![] → σ.Evalb (M := M) ![] by simpa [models_iff]\n intro h\n simpa [Matrix.empty_eq, Empty.eq... | [
{
"name": "bold_sigma_one_completeness",
"text": "lemma bold_sigma_one_completeness {n} {φ : ArithmeticSemiformula ξ n} (hp : Hierarchy 𝚺 1 φ) {bv : Fin n → ℕ} {fv : ξ → ℕ} :\n φ.Eval bv fv → φ.Eval (M := M) (numeral ∘ bv) (numeral ∘ fv) := by\n revert bv\n apply sigma₁_induction' hp\n case hVerum =>... | [
{
"name": "sigma_one_completeness",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 6,
"n_lines": 8,
"n_chars": 424,
"n_subproofs": 0,
"n_tactics": 5,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": false,
"max_... | 6 | module
public import Foundation.FirstOrder.Arithmetic.Basic
@[expose] public section
/-!
# Cobham's theory $\mathsf{R_0}$
-/
noncomputable section
namespace LO.FirstOrder.Arithmetic
inductive R0 : ArithmeticTheory
| equal : ∀ φ ∈ 𝗘𝗤 ℒₒᵣ, R0 φ
| Ω₁ (n m : ℕ) : R0 “↑n + ↑m = ↑(n + m)”
| Ω₂ (n m : ℕ) : R0 “↑... | module
public import Foundation.FirstOrder.Arithmetic.Basic
@[expose] public section
/-!
# Cobham's theory $\mathsf{R_0}$
-/
noncomputable section
namespace LO.FirstOrder.Arithmetic
inductive R0 : ArithmeticTheory
| equal : ∀ φ ∈ 𝗘𝗤 ℒₒᵣ, R0 φ
| Ω₁ (n m : ℕ) : R0 “↑n + ↑m = ↑(n + m)”
| Ω₂ (n m : ℕ) : R0 “↑... | @@ -83,8 +83,37 @@
| .func Language.Add.add v => by simp [Semiterm.val_func, val_numeral _ _ (v 0), val_numeral _ _ (v 1), numeral_add_numeral]
| .func Language.Mul.mul v => by simp [Semiterm.val_func, val_numeral _ _ (v 0), val_numeral _ _ (v 1), numeral_mul_numeral]
+lemma bold_sigma_one_completeness {n} ... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_baba430b5bd9_8 | 9fa618d505e06743 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/R0/Basic.lean | Basic | 8 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 9 | 1 | [
{
"theorem_name": "unprovable_addZero",
"depth": 1,
"n_commands": 0,
"n_lines": 3,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": "\n unprovable_of_countermodel _ (M := Countermodel.OmegaAddOne) <| by\n simpa [notModels_iff] using Countermodel.OmegaAddOn... | [
{
"name": "exists_add_zero_ne_self",
"text": "lemma exists_add_zero_ne_self : ∃ x : OmegaAddOne, x + 0 ≠ x :=\n ⟨⊤, by simp⟩\n\n",
"fan_in": 1,
"n_lines": 4,
"n_chars": 90,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structur... | [
{
"name": "unprovable_addZero",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 5,
"n_chars": 216,
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"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"max_nesti... | 1 | module
public import Foundation.FirstOrder.Arithmetic.Basic
@[expose] public section
/-!
# Cobham's theory $\mathsf{R_0}$
-/
noncomputable section
namespace LO.FirstOrder.Arithmetic
inductive R0 : ArithmeticTheory
| equal : ∀ φ ∈ 𝗘𝗤 ℒₒᵣ, R0 φ
| Ω₁ (n m : ℕ) : R0 “↑n + ↑m = ↑(n + m)”
| Ω₂ (n m : ℕ) : R0 “↑... | module
public import Foundation.FirstOrder.Arithmetic.Basic
@[expose] public section
/-!
# Cobham's theory $\mathsf{R_0}$
-/
noncomputable section
namespace LO.FirstOrder.Arithmetic
inductive R0 : ArithmeticTheory
| equal : ∀ φ ∈ 𝗘𝗤 ℒₒᵣ, R0 φ
| Ω₁ (n m : ℕ) : R0 “↑n + ↑m = ↑(n + m)”
| Ω₂ (n m : ℕ) : R0 “↑... | @@ -192,6 +192,9 @@
@[simp] lemma top_add_zero : (⊤ : OmegaAddOne) + 0 = 0 := by rfl
+lemma exists_add_zero_ne_self : ∃ x : OmegaAddOne, x + 0 ≠ x :=
+ ⟨⊤, by simp⟩
+
@[simp] lemma numeral_eq (n : ℕ) : (ORingStructure.numeral n : OmegaAddOne) = n :=
match n with
| 0 => rfl
@@ -222,7 +225,9 @@
end Co... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_e70c06bb2780_0 | e6a5faf8f2529ef2 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Vorspiel/Order/Dense.lean | Dense | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 2 | 1 | [
{
"theorem_name": "fintype_colimit",
"depth": 1,
"n_commands": 0,
"n_lines": 4,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n let f : Fin (Fintype.card ι) → α := fun x ↦ v ((Fintype.equivFin ι).symm x)\n rcases DirectedOn.vector_le tr hs h f (by in... | [
{
"name": "vector_le",
"text": "private lemma vector_le (tr : IsTrans α r) {s : Set α} (hs : s.Nonempty) (h : DirectedOn r s) (v : Fin n → α) (hv : ∀ i, v i ∈ s) :\n ∃ z ∈ s, ∀ i, r (v i) z :=\n match n with\n | 0 => by\n rcases hs with ⟨x, hx⟩; exact ⟨x, hx, fun i ↦ IsEmpty.elim inferInstance i... | [
{
"name": "fintype_colimit",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 8,
"n_chars": 446,
"n_subproofs": 0,
"n_tactics": 5,
"cyclomatic": 2,
"n_automation": 2,
"n_rewrites": 0,
"n_structural": 3,
"automation_only": false,
"max_nesting... | 1 | module
public import Mathlib.Order.PFilter
public import Mathlib.Data.Set.Countable
@[expose] public section
namespace Nat
end Nat
namespace DirectedOn
variable {α : Type*} {r : α → α → Prop}
lemma fintype_colimit [Fintype ι] (tr : IsTrans α r)
{s : Set α} (hs : s.Nonempty) (h : DirectedOn r s) (v : ι → α) (... | module
public import Mathlib.Order.PFilter
public import Mathlib.Data.Set.Countable
@[expose] public section
namespace Nat
end Nat
namespace DirectedOn
variable {α : Type*} {r : α → α → Prop}
private lemma vector_le (tr : IsTrans α r) {s : Set α} (hs : s.Nonempty) (h : DirectedOn r s) (v : Fin n → α) (hv : ∀ i, ... | @@ -13,9 +13,29 @@
variable {α : Type*} {r : α → α → Prop}
+private lemma vector_le (tr : IsTrans α r) {s : Set α} (hs : s.Nonempty) (h : DirectedOn r s) (v : Fin n → α) (hv : ∀ i, v i ∈ s) :
+ ∃ z ∈ s, ∀ i, r (v i) z :=
+ match n with
+ | 0 => by
+ rcases hs with ⟨x, hx⟩; exact ⟨x, hx, fun i ↦ IsEmpty... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_e70c06bb2780_1 | 9eb50caa8b4445a1 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Vorspiel/Order/Dense.lean | Dense | 1 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 2 | 1 | [
{
"theorem_name": "exists_genericFilter_of_countable",
"depth": 1,
"n_commands": 0,
"n_lines": 21,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n by_cases emp : 𝓓.Nonempty\n case neg =>\n exact ⟨principal a, by simp [Set.not_nonempty_iff_eq_empt... | [
{
"name": "monotone_of_succ_monotone",
"text": "lemma monotone_of_succ_monotone {r : ℕ → ℕ → Prop} (rfx : Reflexive r) (tr : IsTrans ℕ r)\n (succ : ∀ n, r n (n + 1)) : n ≤ m → r n m := by\n revert n m\n suffices ∀ n d, r n (n + d) by\n intro n m hnm\n have := this n (m - n)\n grind\n intro n ... | [
{
"name": "exists_genericFilter_of_countable",
"fan_in": 0,
"n_deps_direct": 2,
"n_deps_transitive": 2,
"n_lines": 25,
"n_chars": 1039,
"n_subproofs": 3,
"n_tactics": 21,
"cyclomatic": 3,
"n_automation": 8,
"n_rewrites": 1,
"n_structural": 8,
"automation_only": fa... | 1 | module
public import Mathlib.Order.PFilter
public import Mathlib.Data.Set.Countable
@[expose] public section
namespace Nat
end Nat
namespace DirectedOn
variable {α : Type*} {r : α → α → Prop}
end DirectedOn
namespace Order
variable {α : Type*} [Preorder α]
/-! ### Compatibility and incompatibility -/
def IsC... | module
public import Mathlib.Order.PFilter
public import Mathlib.Data.Set.Countable
@[expose] public section
namespace Nat
lemma monotone_of_succ_monotone {r : ℕ → ℕ → Prop} (rfx : Reflexive r) (tr : IsTrans ℕ r)
(succ : ∀ n, r n (n + 1)) : n ≤ m → r n m := by
revert n m
suffices ∀ n d, r n (n + d) by
i... | @@ -7,6 +7,19 @@
namespace Nat
+lemma monotone_of_succ_monotone {r : ℕ → ℕ → Prop} (rfx : Reflexive r) (tr : IsTrans ℕ r)
+ (succ : ∀ n, r n (n + 1)) : n ≤ m → r n m := by
+ revert n m
+ suffices ∀ n d, r n (n + d) by
+ intro n m hnm
+ have := this n (m - n)
+ grind
+ intro n d
+ induction d
+ cas... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_f1b4189b7433_0 | 8183195a4ade31d0 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Incompleteness/First.lean | First | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 2 | 1 | [
{
"theorem_name": "incomplete",
"depth": 1,
"n_commands": 0,
"n_lines": 28,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n have con : Consistent T := inferInstance\n let D : ℕ → Prop := fun φ : ℕ ↦\n IsSemiformula ℒₒᵣ 1 φ ∧ Provable T (neg ℒₒᵣ <|... | [
{
"name": "re_iff_sigma1",
"text": "lemma re_iff_sigma1 {P : ℕ → Prop} : REPred P ↔ 𝚺₁-Predicate P := by\n constructor\n · intro h\n refine ⟨.mkSigma (codeOfREPred P) (by simp [codeOfREPred, codeOfPartrec']), ?_⟩\n intro v\n simpa [←Matrix.fun_eq_vec_one] using codeOfREPred_spec h (x := v 0)\n ... | [
{
"name": "incomplete",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 32,
"n_chars": 1596,
"n_subproofs": 6,
"n_tactics": 28,
"cyclomatic": 1,
"n_automation": 5,
"n_rewrites": 2,
"n_structural": 7,
"automation_only": false,
"max_nesting":... | 1 | module
public import Foundation.FirstOrder.Incompleteness.StandardProvability
public import Foundation.FirstOrder.Arithmetic.R0.Representation
@[expose] public section
/-!
# Gödel's first incompleteness theorem for arithmetic theories stronger than $\mathsf{R_0}$
-/
namespace LO.FirstOrder.Arithmetic
open LO.Entail... | module
public import Foundation.FirstOrder.Incompleteness.StandardProvability
public import Foundation.FirstOrder.Arithmetic.R0.Representation
@[expose] public section
/-!
# Gödel's first incompleteness theorem for arithmetic theories stronger than $\mathsf{R_0}$
-/
namespace LO.FirstOrder.Arithmetic
lemma re_iff_s... | @@ -10,6 +10,18 @@
namespace LO.FirstOrder.Arithmetic
+lemma re_iff_sigma1 {P : ℕ → Prop} : REPred P ↔ 𝚺₁-Predicate P := by
+ constructor
+ · intro h
+ refine ⟨.mkSigma (codeOfREPred P) (by simp [codeOfREPred, codeOfPartrec']), ?_⟩
+ intro v
+ simpa [←Matrix.fun_eq_vec_one] using codeOfREPred_spec h (x... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_f1b4189b7433_1 | 7467d72488225903 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Incompleteness/First.lean | First | 1 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 2 | 1 | [
{
"theorem_name": "exists_true_but_unprovable_sentence",
"depth": 1,
"n_commands": 0,
"n_lines": 5,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n obtain ⟨δ, hδ⟩ := incomplete_def.mp $ Arithmetic.incomplete T;\n by_cases ℕ↓[ℒₒᵣ] ⊧ δ\n . exact ⟨δ, b... | [
{
"name": "incomplete",
"text": "/-- Gödel's first incompleteness theorem-/\ntheorem incomplete (T : ArithmeticTheory) [T.Δ₁] [𝗥₀ ⪯ T] [T.SoundOnHierarchy 𝚺 1] :\n Incomplete T := by\n have con : Consistent T := inferInstance\n let D : ℕ → Prop := fun φ : ℕ ↦\n IsSemiformula ℒₒᵣ 1 φ ∧ Provable T (... | [
{
"name": "exists_true_but_unprovable_sentence",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 2,
"n_lines": 9,
"n_chars": 392,
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"n_tactics": 6,
"cyclomatic": 1,
"n_automation": 2,
"n_rewrites": 0,
"n_structural": 3,
"automation_only": fal... | 2 | module
public import Foundation.FirstOrder.Incompleteness.StandardProvability
public import Foundation.FirstOrder.Arithmetic.R0.Representation
@[expose] public section
/-!
# Gödel's first incompleteness theorem for arithmetic theories stronger than $\mathsf{R_0}$
-/
namespace LO.FirstOrder.Arithmetic
lemma re_iff_s... | module
public import Foundation.FirstOrder.Incompleteness.StandardProvability
public import Foundation.FirstOrder.Arithmetic.R0.Representation
@[expose] public section
/-!
# Gödel's first incompleteness theorem for arithmetic theories stronger than $\mathsf{R_0}$
-/
namespace LO.FirstOrder.Arithmetic
lemma re_iff_s... | @@ -24,9 +24,44 @@
open LO.Entailment Bootstrapping Bootstrapping.Arithmetic
+/-- Gödel's first incompleteness theorem-/
+theorem incomplete (T : ArithmeticTheory) [T.Δ₁] [𝗥₀ ⪯ T] [T.SoundOnHierarchy 𝚺 1] :
+ Incomplete T := by
+ have con : Consistent T := inferInstance
+ let D : ℕ → Prop := fun φ : ℕ ↦
+ ... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_f74726e1fd3c_0 | 527ae7bec556a193 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/SecondOrder/Derivation.lean | Derivation | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 1 | 1 | [
{
"theorem_name": "schema_theory_def",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by simp [provable_def, Schema.theory]",
"n_chars": 38,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"... | [
{
"name": "provable_def",
"text": "lemma provable_def {φ : Sentence L} : T ⊢ φ ↔ φ ∈ T :=\n ⟨fun h ↦ PLift.down h.some, fun h ↦ ⟨⟨h⟩⟩⟩\n\n",
"fan_in": 1,
"n_lines": 4,
"n_chars": 126,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
... | [
{
"name": "schema_theory_def",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 4,
"n_chars": 154,
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"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"max_nestin... | 1 | module
public import Foundation.SecondOrder.Syntax.Rew
@[expose] public section
/-!
# Second-order one-sided $\mathbf{LK}$
-/
namespace LO.SecondOrder
open FirstOrder
variable {L : Language}
abbrev Sequent (L : Language) := List (Proposition L)
namespace Sequent
def shift₀ (Γ : Sequent L) : Sequent L := Γ.map ... | module
public import Foundation.SecondOrder.Syntax.Rew
@[expose] public section
/-!
# Second-order one-sided $\mathbf{LK}$
-/
namespace LO.SecondOrder
open FirstOrder
variable {L : Language}
abbrev Sequent (L : Language) := List (Proposition L)
namespace Sequent
def shift₀ (Γ : Sequent L) : Sequent L := Γ.map ... | @@ -94,8 +94,11 @@
variable {T : Theory L}
+lemma provable_def {φ : Sentence L} : T ⊢ φ ↔ φ ∈ T :=
+ ⟨fun h ↦ PLift.down h.some, fun h ↦ ⟨⟨h⟩⟩⟩
+
@[simp] lemma schema_theory_def {𝓢 : Schema L} {φ : Sentence L} :
- 𝓢.theory ⊢ φ ↔ 𝓢 ⊢ ↑φ := sorry
+ 𝓢.theory ⊢ φ ↔ 𝓢 ⊢ ↑φ := by simp [provable_def, Schema.... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_23153f3e9ec3_0 | 0adcd828285b1907 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Incompleteness/Yablo.lean | Yablo | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 10 | 1 | [
{
"theorem_name": "yablo_diagonal_modeled",
"depth": 1,
"n_commands": 0,
"n_lines": 4,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n have : V↓[ℒₒᵣ] ⊧ ∀⁰ (T.yablo 🡘 ↑(T.yabloSystem)/[⌜T.yablo⌝, #0]) := models_of_provable (T := 𝗜𝚺₁) (by assumption)... | [
{
"name": "yablo_diagonal",
"text": "lemma yablo_diagonal : U ⊢ ∀⁰ (T.yablo 🡘 (T.yabloSystem)/[⌜T.yablo⌝, #0]) := parameterized_diagonal₁ _\n\n",
"fan_in": 1,
"n_lines": 3,
"n_chars": 119,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,... | [
{
"name": "yablo_diagonal_modeled",
"fan_in": 3,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 6,
"n_chars": 447,
"n_subproofs": 2,
"n_tactics": 7,
"cyclomatic": 1,
"n_automation": 2,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": false,
"max_... | 1 | module
public import Foundation.FirstOrder.Incompleteness.StandardProvability
@[expose] public section
/-
Formalizing Yablo's Paradox.
*References*
- C. Cieśliński, R. Urbaniak, "Gödelizing the Yablo Sequence"
-/
namespace LO.FirstOrder.Arithmetic
open ORingStructure
variable {M : Type*} [ORingStructure M] [... | module
public import Foundation.FirstOrder.Incompleteness.StandardProvability
@[expose] public section
/-
Formalizing Yablo's Paradox.
*References*
- C. Cieśliński, R. Urbaniak, "Gödelizing the Yablo Sequence"
-/
namespace LO.FirstOrder.Arithmetic
open ORingStructure
variable {M : Type*} [ORingStructure M] [... | @@ -58,8 +58,10 @@
variable {V : Type} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
variable {U : ArithmeticTheory} [𝗜𝚺₁ ⪯ U]
+lemma yablo_diagonal : U ⊢ ∀⁰ (T.yablo 🡘 (T.yabloSystem)/[⌜T.yablo⌝, #0]) := parameterized_diagonal₁ _
+
lemma yablo_diagonal_modeled (n : V) : V ⊧/![n] (T.yablo) ↔ ∀ m, n < m → ¬Provable T (s... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_23153f3e9ec3_1 | d43b8154fd0443d7 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Incompleteness/Yablo.lean | Yablo | 1 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 10 | 3 | [
{
"theorem_name": "yablo_diagonal_neg_modeled",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n simpa using yablo_diagonal_modeled n |>.not;",
"n_chars": 50,
"n_subproofs": 0,
"n_tactics": 3,
... | [
{
"name": "yablo_diagonal_modeled",
"text": "lemma yablo_diagonal_modeled (n : V) : V ⊧/![n] (T.yablo) ↔ ∀ m, n < m → ¬Provable T (substNumeral ⌜T.yablo⌝ m) := by\n have : V↓[ℒₒᵣ] ⊧ ∀⁰ (T.yablo 🡘 ↑(T.yabloSystem)/[⌜T.yablo⌝, #0]) := models_of_provable (T := 𝗜𝚺₁) (by assumption) $ yablo_diagonal;\n have... | [
{
"name": "yablo_diagonal_neg_modeled",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 2,
"n_lines": 4,
"n_chars": 183,
"n_subproofs": 0,
"n_tactics": 3,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"m... | 2 | module
public import Foundation.FirstOrder.Incompleteness.StandardProvability
@[expose] public section
/-
Formalizing Yablo's Paradox.
*References*
- C. Cieśliński, R. Urbaniak, "Gödelizing the Yablo Sequence"
-/
namespace LO.FirstOrder.Arithmetic
open ORingStructure
variable {M : Type*} [ORingStructure M] [... | module
public import Foundation.FirstOrder.Incompleteness.StandardProvability
@[expose] public section
/-
Formalizing Yablo's Paradox.
*References*
- C. Cieśliński, R. Urbaniak, "Gödelizing the Yablo Sequence"
-/
namespace LO.FirstOrder.Arithmetic
open ORingStructure
variable {M : Type*} [ORingStructure M] [... | @@ -60,8 +60,14 @@
lemma yablo_diagonal : U ⊢ ∀⁰ (T.yablo 🡘 (T.yabloSystem)/[⌜T.yablo⌝, #0]) := parameterized_diagonal₁ _
-lemma yablo_diagonal_neg_modeled (n : V) : ¬V ⊧/![n] (T.yablo) ↔ ∃ m, n < m ∧ Provable T (substNumeral ⌜T.yablo⌝ m) := sorry
+lemma yablo_diagonal_modeled (n : V) : V ⊧/![n] (T.yablo) ↔ ∀ m, ... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_23153f3e9ec3_2 | 7ecdecd8d4e33edb | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Incompleteness/Yablo.lean | Yablo | 2 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 10 | 3 | [
{
"theorem_name": "yablo_diagonal_neg_modeled",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n simpa using yablo_diagonal_modeled n |>.not;",
"n_chars": 50,
"n_subproofs": 0,
"n_tactics": 3,
... | [
{
"name": "yablo_diagonal_modeled",
"text": "lemma yablo_diagonal_modeled (n : V) : V ⊧/![n] (T.yablo) ↔ ∀ m, n < m → ¬Provable T (substNumeral ⌜T.yablo⌝ m) := by\n have : V↓[ℒₒᵣ] ⊧ ∀⁰ (T.yablo 🡘 ↑(T.yabloSystem)/[⌜T.yablo⌝, #0]) := models_of_provable (T := 𝗜𝚺₁) (by assumption) $ yablo_diagonal;\n have... | [
{
"name": "iff_yablo_provable",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 2,
"n_lines": 10,
"n_chars": 720,
"n_subproofs": 1,
"n_tactics": 16,
"cyclomatic": 3,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 4,
"automation_only": false,
"max_ne... | 2 | module
public import Foundation.FirstOrder.Incompleteness.StandardProvability
@[expose] public section
/-
Formalizing Yablo's Paradox.
*References*
- C. Cieśliński, R. Urbaniak, "Gödelizing the Yablo Sequence"
-/
namespace LO.FirstOrder.Arithmetic
open ORingStructure
variable {M : Type*} [ORingStructure M] [... | module
public import Foundation.FirstOrder.Incompleteness.StandardProvability
@[expose] public section
/-
Formalizing Yablo's Paradox.
*References*
- C. Cieśliński, R. Urbaniak, "Gödelizing the Yablo Sequence"
-/
namespace LO.FirstOrder.Arithmetic
open ORingStructure
variable {M : Type*} [ORingStructure M] [... | @@ -60,8 +60,20 @@
lemma yablo_diagonal : U ⊢ ∀⁰ (T.yablo 🡘 (T.yabloSystem)/[⌜T.yablo⌝, #0]) := parameterized_diagonal₁ _
-lemma iff_yablo_provable (n : ℕ) : U ⊢ T.yabloPred n ↔ U ⊢ “∀ m, ↑n < m → ∀ nσ, !ssnum nσ ⌜T.yablo⌝ m → ¬!(provable T) (nσ)” := sorry
+lemma yablo_diagonal_modeled (n : V) : V ⊧/![n] (T.yablo... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_23153f3e9ec3_3 | 9e2630d03e638859 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Incompleteness/Yablo.lean | Yablo | 3 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 10 | 3 | [
{
"theorem_name": "yablo_diagonal_neg_modeled",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n simpa using yablo_diagonal_modeled n |>.not;",
"n_chars": 50,
"n_subproofs": 0,
"n_tactics": 3,
... | [
{
"name": "yablo_diagonal_modeled",
"text": "lemma yablo_diagonal_modeled (n : V) : V ⊧/![n] (T.yablo) ↔ ∀ m, n < m → ¬Provable T (substNumeral ⌜T.yablo⌝ m) := by\n have : V↓[ℒₒᵣ] ⊧ ∀⁰ (T.yablo 🡘 ↑(T.yabloSystem)/[⌜T.yablo⌝, #0]) := models_of_provable (T := 𝗜𝚺₁) (by assumption) $ yablo_diagonal;\n have... | [
{
"name": "iff_neg_yablo_provable",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 3,
"n_lines": 10,
"n_chars": 760,
"n_subproofs": 1,
"n_tactics": 16,
"cyclomatic": 3,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 4,
"automation_only": false,
"ma... | 3 | module
public import Foundation.FirstOrder.Incompleteness.StandardProvability
@[expose] public section
/-
Formalizing Yablo's Paradox.
*References*
- C. Cieśliński, R. Urbaniak, "Gödelizing the Yablo Sequence"
-/
namespace LO.FirstOrder.Arithmetic
open ORingStructure
variable {M : Type*} [ORingStructure M] [... | module
public import Foundation.FirstOrder.Incompleteness.StandardProvability
@[expose] public section
/-
Formalizing Yablo's Paradox.
*References*
- C. Cieśliński, R. Urbaniak, "Gödelizing the Yablo Sequence"
-/
namespace LO.FirstOrder.Arithmetic
open ORingStructure
variable {M : Type*} [ORingStructure M] [... | @@ -60,8 +60,14 @@
lemma yablo_diagonal : U ⊢ ∀⁰ (T.yablo 🡘 (T.yabloSystem)/[⌜T.yablo⌝, #0]) := parameterized_diagonal₁ _
-lemma yablo_diagonal_neg_modeled (n : V) : ¬V ⊧/![n] (T.yablo) ↔ ∃ m, n < m ∧ Provable T (substNumeral ⌜T.yablo⌝ m) := sorry
+lemma yablo_diagonal_modeled (n : V) : V ⊧/![n] (T.yablo) ↔ ∀ m, ... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_23153f3e9ec3_4 | 065c23c147f54f74 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Incompleteness/Yablo.lean | Yablo | 4 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 10 | 3 | [
{
"theorem_name": "yablo_diagonal_neg_modeled",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n simpa using yablo_diagonal_modeled n |>.not;",
"n_chars": 50,
"n_subproofs": 0,
"n_tactics": 3,
... | [
{
"name": "yablo_diagonal_modeled",
"text": "lemma yablo_diagonal_modeled (n : V) : V ⊧/![n] (T.yablo) ↔ ∀ m, n < m → ¬Provable T (substNumeral ⌜T.yablo⌝ m) := by\n have : V↓[ℒₒᵣ] ⊧ ∀⁰ (T.yablo 🡘 ↑(T.yabloSystem)/[⌜T.yablo⌝, #0]) := models_of_provable (T := 𝗜𝚺₁) (by assumption) $ yablo_diagonal;\n have... | [
{
"name": "provable_greater_yablo",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 2,
"n_lines": 14,
"n_chars": 716,
"n_subproofs": 2,
"n_tactics": 15,
"cyclomatic": 1,
"n_automation": 3,
"n_rewrites": 0,
"n_structural": 6,
"automation_only": false,
"ma... | 2 | module
public import Foundation.FirstOrder.Incompleteness.StandardProvability
@[expose] public section
/-
Formalizing Yablo's Paradox.
*References*
- C. Cieśliński, R. Urbaniak, "Gödelizing the Yablo Sequence"
-/
namespace LO.FirstOrder.Arithmetic
open ORingStructure
variable {M : Type*} [ORingStructure M] [... | module
public import Foundation.FirstOrder.Incompleteness.StandardProvability
@[expose] public section
/-
Formalizing Yablo's Paradox.
*References*
- C. Cieśliński, R. Urbaniak, "Gödelizing the Yablo Sequence"
-/
namespace LO.FirstOrder.Arithmetic
open ORingStructure
variable {M : Type*} [ORingStructure M] [... | @@ -60,6 +60,11 @@
lemma yablo_diagonal : U ⊢ ∀⁰ (T.yablo 🡘 (T.yabloSystem)/[⌜T.yablo⌝, #0]) := parameterized_diagonal₁ _
+lemma yablo_diagonal_modeled (n : V) : V ⊧/![n] (T.yablo) ↔ ∀ m, n < m → ¬Provable T (substNumeral ⌜T.yablo⌝ m) := by
+ have : V↓[ℒₒᵣ] ⊧ ∀⁰ (T.yablo 🡘 ↑(T.yabloSystem)/[⌜T.yablo⌝, #0]) := m... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_23153f3e9ec3_5 | 9ea3e6fd227bdc1b | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Incompleteness/Yablo.lean | Yablo | 5 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 10 | 3 | [
{
"theorem_name": "yablo_diagonal_neg_modeled",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n simpa using yablo_diagonal_modeled n |>.not;",
"n_chars": 50,
"n_subproofs": 0,
"n_tactics": 3,
... | [
{
"name": "yablo_diagonal_modeled",
"text": "lemma yablo_diagonal_modeled (n : V) : V ⊧/![n] (T.yablo) ↔ ∀ m, n < m → ¬Provable T (substNumeral ⌜T.yablo⌝ m) := by\n have : V↓[ℒₒᵣ] ⊧ ∀⁰ (T.yablo 🡘 ↑(T.yabloSystem)/[⌜T.yablo⌝, #0]) := models_of_provable (T := 𝗜𝚺₁) (by assumption) $ yablo_diagonal;\n have... | [
{
"name": "yablo_unprovable",
"fan_in": 2,
"n_deps_direct": 3,
"n_deps_transitive": 5,
"n_lines": 21,
"n_chars": 1204,
"n_subproofs": 5,
"n_tactics": 32,
"cyclomatic": 1,
"n_automation": 5,
"n_rewrites": 1,
"n_structural": 6,
"automation_only": false,
"max_nes... | 5 | module
public import Foundation.FirstOrder.Incompleteness.StandardProvability
@[expose] public section
/-
Formalizing Yablo's Paradox.
*References*
- C. Cieśliński, R. Urbaniak, "Gödelizing the Yablo Sequence"
-/
namespace LO.FirstOrder.Arithmetic
open ORingStructure
variable {M : Type*} [ORingStructure M] [... | module
public import Foundation.FirstOrder.Incompleteness.StandardProvability
@[expose] public section
/-
Formalizing Yablo's Paradox.
*References*
- C. Cieśliński, R. Urbaniak, "Gödelizing the Yablo Sequence"
-/
namespace LO.FirstOrder.Arithmetic
open ORingStructure
variable {M : Type*} [ORingStructure M] [... | @@ -67,8 +67,20 @@
lemma yablo_diagonal : U ⊢ ∀⁰ (T.yablo 🡘 (T.yabloSystem)/[⌜T.yablo⌝, #0]) := parameterized_diagonal₁ _
-lemma iff_yablo_provable (n : ℕ) : U ⊢ T.yabloPred n ↔ U ⊢ “∀ m, ↑n < m → ∀ nσ, !ssnum nσ ⌜T.yablo⌝ m → ¬!(provable T) (nσ)” := sorry
+lemma yablo_diagonal_modeled (n : V) : V ⊧/![n] (T.yablo... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_23153f3e9ec3_6 | a643a5348399062a | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Incompleteness/Yablo.lean | Yablo | 6 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 10 | 1 | [
{
"theorem_name": "iff_neg_yablo_provable",
"depth": 1,
"n_commands": 0,
"n_lines": 8,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n suffices U ⊢ ∼T.yablo/[n] 🡘 “∃ m, ↑n < m ∧ ∃ nσ, !ssnum nσ ⌜T.yablo⌝ m ∧ !(provable T) (nσ)” by\n constructor <;... | [
{
"name": "yablo_diagonal_neg_modeled",
"text": "lemma yablo_diagonal_neg_modeled (n : V) : ¬V ⊧/![n] (T.yablo) ↔ ∃ m, n < m ∧ Provable T (substNumeral ⌜T.yablo⌝ m) := by\n simpa using yablo_diagonal_modeled n |>.not;\n\n",
"fan_in": 1,
"n_lines": 4,
"n_chars": 183,
"n_subproofs": 0,
"n... | [
{
"name": "yablo_unrefutable",
"fan_in": 1,
"n_deps_direct": 3,
"n_deps_transitive": 9,
"n_lines": 8,
"n_chars": 437,
"n_subproofs": 1,
"n_tactics": 11,
"cyclomatic": 1,
"n_automation": 2,
"n_rewrites": 1,
"n_structural": 2,
"automation_only": false,
"max_nest... | 9 | module
public import Foundation.FirstOrder.Incompleteness.StandardProvability
@[expose] public section
/-
Formalizing Yablo's Paradox.
*References*
- C. Cieśliński, R. Urbaniak, "Gödelizing the Yablo Sequence"
-/
namespace LO.FirstOrder.Arithmetic
open ORingStructure
variable {M : Type*} [ORingStructure M] [... | module
public import Foundation.FirstOrder.Incompleteness.StandardProvability
@[expose] public section
/-
Formalizing Yablo's Paradox.
*References*
- C. Cieśliński, R. Urbaniak, "Gödelizing the Yablo Sequence"
-/
namespace LO.FirstOrder.Arithmetic
open ORingStructure
variable {M : Type*} [ORingStructure M] [... | @@ -79,6 +79,9 @@
have : ∀ (n : V), V ⊧/![n] (T.yablo) ↔ T.YabloSystem ⌜T.yablo⌝ n := by simpa [models_iff, Matrix.comp_vecCons'] using this;
apply this;
+lemma yablo_diagonal_neg_modeled (n : V) : ¬V ⊧/![n] (T.yablo) ↔ ∃ m, n < m ∧ Provable T (substNumeral ⌜T.yablo⌝ m) := by
+ simpa using yablo_diagonal_model... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_23153f3e9ec3_7 | e117e35800dae9d1 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Incompleteness/Yablo.lean | Yablo | 7 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 10 | 1 | [
{
"theorem_name": "yablo_unprovable",
"depth": 1,
"n_commands": 0,
"n_lines": 19,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n by_contra! hC;\n have H₁ : T ⊢ provabilityPred T (T.yabloPred (n + 1)) := by\n apply Entailment.WeakerThan.pbl $ prov... | [
{
"name": "iff_yablo_provable",
"text": "lemma iff_yablo_provable (n : ℕ) : U ⊢ T.yabloPred n ↔ U ⊢ “∀ m, ↑n < m → ∀ nσ, !ssnum nσ ⌜T.yablo⌝ m → ¬!(provable T) (nσ)” := by\n suffices U ⊢ T.yablo/[n] 🡘 “∀ m, ↑n < m → ∀ nσ, !ssnum nσ ⌜T.yablo⌝ m → ¬!(provable T) (nσ)” by\n constructor <;> . intro h; cl_p... | [
{
"name": "yablo_independent",
"fan_in": 0,
"n_deps_direct": 2,
"n_deps_transitive": 10,
"n_lines": 3,
"n_chars": 143,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
"max_nest... | 10 | module
public import Foundation.FirstOrder.Incompleteness.StandardProvability
@[expose] public section
/-
Formalizing Yablo's Paradox.
*References*
- C. Cieśliński, R. Urbaniak, "Gödelizing the Yablo Sequence"
-/
namespace LO.FirstOrder.Arithmetic
open ORingStructure
variable {M : Type*} [ORingStructure M] [... | module
public import Foundation.FirstOrder.Incompleteness.StandardProvability
@[expose] public section
/-
Formalizing Yablo's Paradox.
*References*
- C. Cieśliński, R. Urbaniak, "Gödelizing the Yablo Sequence"
-/
namespace LO.FirstOrder.Arithmetic
open ORingStructure
variable {M : Type*} [ORingStructure M] [... | @@ -82,6 +82,15 @@
lemma yablo_diagonal_neg_modeled (n : V) : ¬V ⊧/![n] (T.yablo) ↔ ∃ m, n < m ∧ Provable T (substNumeral ⌜T.yablo⌝ m) := by
simpa using yablo_diagonal_modeled n |>.not;
+lemma iff_yablo_provable (n : ℕ) : U ⊢ T.yabloPred n ↔ U ⊢ “∀ m, ↑n < m → ∀ nσ, !ssnum nσ ⌜T.yablo⌝ m → ¬!(provable T) (nσ)” :=... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_321cf358e627_0 | 931c1bd67ccb6970 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Vorspiel/List/Perm.lean | Perm | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 2 | 1 | [
{
"theorem_name": "perm_normalize",
"depth": 1,
"n_commands": 0,
"n_lines": 10,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": "\n match l with\n | [] => by simp\n | b :: l => by\n by_cases h : a = b\n · simp [h, List.replicate, perm_normalize l... | [
{
"name": "count_def",
"text": "lemma count_def (a b : α) (l : List α) : count a (b :: l) = if a = b then count a l + 1 else count a l := by\n simp [count]; grind\n\n",
"fan_in": 1,
"n_lines": 4,
"n_chars": 134,
"n_subproofs": 0,
"n_tactics": 3,
"cyclomatic": 1,
"n_automation": ... | [
{
"name": "perm_normalize",
"fan_in": 0,
"n_deps_direct": 2,
"n_deps_transitive": 2,
"n_lines": 12,
"n_chars": 513,
"n_subproofs": 1,
"n_tactics": 9,
"cyclomatic": 2,
"n_automation": 4,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": false,
"max_nesting... | 2 | module
public import Foundation.Vorspiel.List.Basic
@[expose]
public section
namespace List
variable {α : Type*}
inductive CompSubset : List α → List α → Type _
| refl (l) : CompSubset l l
| perm : CompSubset l₁ l₂ → l₂ ~ l₃ → CompSubset l₁ l₃
| add (a : α) :
CompSubset l₁ l₂ → CompSubset l₁ (a :: l₂)
... | module
public import Foundation.Vorspiel.List.Basic
@[expose]
public section
namespace List
variable {α : Type*}
inductive CompSubset : List α → List α → Type _
| refl (l) : CompSubset l l
| perm : CompSubset l₁ l₂ → l₂ ~ l₃ → CompSubset l₁ l₃
| add (a : α) :
CompSubset l₁ l₂ → CompSubset l₁ (a :: l₂)
... | @@ -22,13 +22,19 @@
lemma remove_def (a b : α) (l : List α) : remove a (b :: l) = if a = b then remove a l else b :: remove a l := by
simp [remove, List.filter]; grind
+lemma count_def (a b : α) (l : List α) : count a (b :: l) = if a = b then count a l + 1 else count a l := by
+ simp [count]; grind
+
lemma perm... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_886b91094ddb_0 | 82242f7d933e345c | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Kripke/AxiomPoint4.lean | AxiomPoint4 | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 1 | 1 | [
{
"theorem_name": "validate_axiomPoint4_of_satisfiesSobocinskiCondition",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": "\n validate_axiomPoint4_of_sobocinskiCondition Frame.SatisfiesSobocinskiCondition.sobocinski... | [
{
"name": "validate_axiomPoint4_of_sobocinskiCondition",
"text": "private lemma validate_axiomPoint4_of_sobocinskiCondition : (∀ ⦃x y z : F⦄, x ≠ y → x ≺ y → x ≺ z → z ≺ y) → F ⊧ (Axioms.Point4 (.atom 0)) := by\n contrapose!;\n intro h;\n obtain ⟨V, x, h⟩ := ValidOnFrame.exists_valuation_world_of_not h;\... | [
{
"name": "validate_axiomPoint4_of_satisfiesSobocinskiCondition",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 4,
"n_chars": 222,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"auto... | 1 | module
public import Foundation.Modal.Kripke.AxiomGeach
public import Foundation.Modal.Kripke.AxiomPoint3
@[expose] public section
namespace LO.Modal
namespace Kripke
variable {F : Kripke.Frame}
namespace Frame
class SatisfiesSobocinskiCondition (F : Kripke.Frame) where
sobocinski : ∀ ⦃x y z : F⦄, x ≠ y → x ≺ ... | module
public import Foundation.Modal.Kripke.AxiomGeach
public import Foundation.Modal.Kripke.AxiomPoint3
@[expose] public section
namespace LO.Modal
namespace Kripke
variable {F : Kripke.Frame}
namespace Frame
class SatisfiesSobocinskiCondition (F : Kripke.Frame) where
sobocinski : ∀ ⦃x y z : F⦄, x ≠ y → x ≺ ... | @@ -41,8 +41,30 @@
variable {F : Kripke.Frame}
-lemma validate_axiomPoint4_of_satisfiesSobocinskiCondition [F.SatisfiesSobocinskiCondition] : F ⊧ (Axioms.Point4 (.atom 0)) := sorry
+private lemma validate_axiomPoint4_of_sobocinskiCondition : (∀ ⦃x y z : F⦄, x ≠ y → x ≺ y → x ≺ z → z ≺ y) → F ⊧ (Axioms.Point4 (.ato... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_746f3aa76f97_0 | e221929b495721e2 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Logic/GLPlusBoxBot/Basic.lean | Basic | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 3 | 1 | [
{
"theorem_name": "Logic.GLPlusBoxBot.weakerThan_succ",
"depth": 1,
"n_commands": 0,
"n_lines": 13,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n apply weakerThan_iff.mpr;\n intro φ;\n suffices Modal.GL ⊢ □^[(n + 1)]⊥ 🡒 φ → Modal.GL ⊢ □^[n]⊥ 🡒 φ... | [
{
"name": "iff_provable_GLPlusBoxBot_provable_GL",
"text": "lemma iff_provable_GLPlusBoxBot_provable_GL {n : ℕ} {φ} : Modal.GLPlusBoxBot n ⊢ φ ↔ Modal.GL ⊢ (□^[n]⊥) 🡒 φ := by\n constructor;\n . intro h;\n induction h using sumQuasiNormal.rec! with\n | mem₁ h => cl_prover [h]\n | mem₂ h => simp_a... | [
{
"name": "Logic.GLPlusBoxBot.weakerThan_succ",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 15,
"n_chars": 568,
"n_subproofs": 2,
"n_tactics": 20,
"cyclomatic": 1,
"n_automation": 3,
"n_rewrites": 1,
"n_structural": 5,
"automation_only": fa... | 1 | module
public import Foundation.Modal.Logic.SumQuasiNormal
public import Foundation.Modal.Maximal.Unprovability
public import Mathlib.Data.ENat.Basic
public import Foundation.Modal.Kripke.Logic.GL.Completeness
@[expose] public section
namespace LO.Modal
open Logic
variable {n : ℕ∞}
protected def GLPlusBoxBot (n :... | module
public import Foundation.Modal.Logic.SumQuasiNormal
public import Foundation.Modal.Maximal.Unprovability
public import Mathlib.Data.ENat.Basic
public import Foundation.Modal.Kripke.Logic.GL.Completeness
@[expose] public section
namespace LO.Modal
open Logic
variable {n : ℕ∞}
protected def GLPlusBoxBot (n :... | @@ -31,13 +31,37 @@
apply Logic.sumQuasiNormal.mem₂!;
tauto;
+lemma iff_provable_GLPlusBoxBot_provable_GL {n : ℕ} {φ} : Modal.GLPlusBoxBot n ⊢ φ ↔ Modal.GL ⊢ (□^[n]⊥) 🡒 φ := by
+ constructor;
+ . intro h;
+ induction h using sumQuasiNormal.rec! with
+ | mem₁ h => cl_prover [h]
+ | mem₂ h => simp_all... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_746f3aa76f97_1 | 14137fb8aa1dfe58 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Logic/GLPlusBoxBot/Basic.lean | Basic | 1 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 3 | 1 | [
{
"theorem_name": "Logic.GLPlusBoxBot.weakerThan_add",
"depth": 1,
"n_commands": 0,
"n_lines": 7,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n induction k with\n | zero => simp [Modal.GLPlusBoxBot];\n | succ k ih =>\n trans (Modal.GLPlusBoxBot... | [
{
"name": "Logic.GLPlusBoxBot.weakerThan_succ",
"text": "lemma Logic.GLPlusBoxBot.weakerThan_succ {n : ℕ} : (Modal.GLPlusBoxBot (n + 1)) ⪯ (Modal.GLPlusBoxBot n) := by\n apply weakerThan_iff.mpr;\n intro φ;\n suffices Modal.GL ⊢ □^[(n + 1)]⊥ 🡒 φ → Modal.GL ⊢ □^[n]⊥ 🡒 φ by\n simpa [\n (show Moda... | [
{
"name": "Logic.GLPlusBoxBot.weakerThan_add",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 2,
"n_lines": 9,
"n_chars": 298,
"n_subproofs": 0,
"n_tactics": 11,
"cyclomatic": 2,
"n_automation": 2,
"n_rewrites": 0,
"n_structural": 2,
"automation_only": fals... | 2 | module
public import Foundation.Modal.Logic.SumQuasiNormal
public import Foundation.Modal.Maximal.Unprovability
public import Mathlib.Data.ENat.Basic
public import Foundation.Modal.Kripke.Logic.GL.Completeness
@[expose] public section
namespace LO.Modal
open Logic
variable {n : ℕ∞}
protected def GLPlusBoxBot (n :... | module
public import Foundation.Modal.Logic.SumQuasiNormal
public import Foundation.Modal.Maximal.Unprovability
public import Mathlib.Data.ENat.Basic
public import Foundation.Modal.Kripke.Logic.GL.Completeness
@[expose] public section
namespace LO.Modal
open Logic
variable {n : ℕ∞}
protected def GLPlusBoxBot (n :... | @@ -49,12 +49,26 @@
open LO.Entailment LO.Modal.Entailment
+lemma Logic.GLPlusBoxBot.weakerThan_succ {n : ℕ} : (Modal.GLPlusBoxBot (n + 1)) ⪯ (Modal.GLPlusBoxBot n) := by
+ apply weakerThan_iff.mpr;
+ intro φ;
+ suffices Modal.GL ⊢ □^[(n + 1)]⊥ 🡒 φ → Modal.GL ⊢ □^[n]⊥ 🡒 φ by
+ simpa [
+ (show Modal.GL... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_746f3aa76f97_2 | 9ad6366f67da67da | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Logic/GLPlusBoxBot/Basic.lean | Basic | 2 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 3 | 1 | [
{
"theorem_name": "Logic.GLPlusBoxBot.weakerThan_lt",
"depth": 1,
"n_commands": 0,
"n_lines": 3,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n rw [(show m = n + (m - n) by omega)];\n apply Logic.GLPlusBoxBot.weakerThan_add;",
"n_chars": 86,
... | [
{
"name": "Logic.GLPlusBoxBot.weakerThan_add",
"text": "lemma Logic.GLPlusBoxBot.weakerThan_add {n k : ℕ} : (Modal.GLPlusBoxBot (n + k)) ⪯ (Modal.GLPlusBoxBot n) := by\n induction k with\n | zero => simp [Modal.GLPlusBoxBot];\n | succ k ih =>\n trans (Modal.GLPlusBoxBot (n + k));\n . apply Logic.GL... | [
{
"name": "Logic.GLPlusBoxBot.weakerThan_lt",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 3,
"n_lines": 5,
"n_chars": 207,
"n_subproofs": 0,
"n_tactics": 5,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 2,
"n_structural": 1,
"automation_only": false,... | 3 | module
public import Foundation.Modal.Logic.SumQuasiNormal
public import Foundation.Modal.Maximal.Unprovability
public import Mathlib.Data.ENat.Basic
public import Foundation.Modal.Kripke.Logic.GL.Completeness
@[expose] public section
namespace LO.Modal
open Logic
variable {n : ℕ∞}
protected def GLPlusBoxBot (n :... | module
public import Foundation.Modal.Logic.SumQuasiNormal
public import Foundation.Modal.Maximal.Unprovability
public import Mathlib.Data.ENat.Basic
public import Foundation.Modal.Kripke.Logic.GL.Completeness
@[expose] public section
namespace LO.Modal
open Logic
variable {n : ℕ∞}
protected def GLPlusBoxBot (n :... | @@ -63,8 +63,18 @@
apply imply_boxItr_distribute'!;
cl_prover;
-lemma Logic.GLPlusBoxBot.weakerThan_lt {n m : ℕ} (hmn : m > n) : (Modal.GLPlusBoxBot m) ⪯ (Modal.GLPlusBoxBot n) := sorry
+lemma Logic.GLPlusBoxBot.weakerThan_add {n k : ℕ} : (Modal.GLPlusBoxBot (n + k)) ⪯ (Modal.GLPlusBoxBot n) := by
+ induction ... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_99c0762192a9_0 | 533d61f6c7e23d0f | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Definability/Hierarchy.lean | Hierarchy | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 1 | 8 | [
{
"theorem_name": "ProperOn.iff'",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by simp [←h.iff, val_sigma]",
"n_chars": 30,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation":... | [
{
"name": "val_sigma",
"text": "lemma val_sigma (φ : 𝚫-[m].Semiformula ξ n) : φ.sigma.val = φ.val := by rcases φ; simp\n\n",
"fan_in": 8,
"n_lines": 3,
"n_chars": 96,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 2,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 1... | [
{
"name": "ProperOn.iff'",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 5,
"n_chars": 170,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"max_nesting": ... | 1 | module
public import Foundation.FirstOrder.Arithmetic.PeanoMinus.Basic
@[expose] public section
/-!
# Arithmetical Formula Sorted by Arithmetical Hierarchy
This file defines the $\Sigma_n / \Pi_n / \Delta_n$ formulas of arithmetic of first-order logic.
- `𝚺-[m].Semiformula ξ n` is a `ArithmeticSemiformula ξ n` wh... | module
public import Foundation.FirstOrder.Arithmetic.PeanoMinus.Basic
@[expose] public section
/-!
# Arithmetical Formula Sorted by Arithmetical Hierarchy
This file defines the $\Sigma_n / \Pi_n / \Delta_n$ formulas of arithmetic of first-order logic.
- `𝚺-[m].Semiformula ξ n` is a `ArithmeticSemiformula ξ n` wh... | @@ -106,6 +106,8 @@
@[simp] lemma pi_mkDelta (φ : 𝚺-[m].Semiformula ξ n) (ψ : 𝚷-[m].Semiformula ξ n) : (mkDelta φ ψ).pi = ψ := rfl
+lemma val_sigma (φ : 𝚫-[m].Semiformula ξ n) : φ.sigma.val = φ.val := by rcases φ; simp
+
def mkPolarity (φ : ArithmeticSemiformula ξ n) : (Γ : Polarity) → Hierarchy Γ m φ → Γ-[m].... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_99c0762192a9_1 | 43b581c95fbbda9b | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Definability/Hierarchy.lean | Hierarchy | 1 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 1 | 8 | [
{
"theorem_name": "ProperOn.iff'",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by simp [←h.iff, val_sigma]",
"n_chars": 30,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation":... | [
{
"name": "val_sigma",
"text": "lemma val_sigma (φ : 𝚫-[m].Semiformula ξ n) : φ.sigma.val = φ.val := by rcases φ; simp\n\n",
"fan_in": 8,
"n_lines": 3,
"n_chars": 96,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 2,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 1... | [
{
"name": "val_and",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 5,
"n_chars": 225,
"n_subproofs": 1,
"n_tactics": 5,
"cyclomatic": 4,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 2,
"automation_only": false,
"max_nesting": 2
}... | 1 | module
public import Foundation.FirstOrder.Arithmetic.PeanoMinus.Basic
@[expose] public section
/-!
# Arithmetical Formula Sorted by Arithmetical Hierarchy
This file defines the $\Sigma_n / \Pi_n / \Delta_n$ formulas of arithmetic of first-order logic.
- `𝚺-[m].Semiformula ξ n` is a `ArithmeticSemiformula ξ n` wh... | module
public import Foundation.FirstOrder.Arithmetic.PeanoMinus.Basic
@[expose] public section
/-!
# Arithmetical Formula Sorted by Arithmetical Hierarchy
This file defines the $\Sigma_n / \Pi_n / \Delta_n$ formulas of arithmetic of first-order logic.
- `𝚺-[m].Semiformula ξ n` is a `ArithmeticSemiformula ξ n` wh... | @@ -106,6 +106,8 @@
@[simp] lemma pi_mkDelta (φ : 𝚺-[m].Semiformula ξ n) (ψ : 𝚷-[m].Semiformula ξ n) : (mkDelta φ ψ).pi = ψ := rfl
+lemma val_sigma (φ : 𝚫-[m].Semiformula ξ n) : φ.sigma.val = φ.val := by rcases φ; simp
+
def mkPolarity (φ : ArithmeticSemiformula ξ n) : (Γ : Polarity) → Hierarchy Γ m φ → Γ-[m].... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_99c0762192a9_2 | 1298e63a64ff8adc | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Definability/Hierarchy.lean | Hierarchy | 2 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 1 | 8 | [
{
"theorem_name": "ProperOn.iff'",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by simp [←h.iff, val_sigma]",
"n_chars": 30,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation":... | [
{
"name": "val_sigma",
"text": "lemma val_sigma (φ : 𝚫-[m].Semiformula ξ n) : φ.sigma.val = φ.val := by rcases φ; simp\n\n",
"fan_in": 8,
"n_lines": 3,
"n_chars": 96,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 2,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 1... | [
{
"name": "ProperOn.eval_neg",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 5,
"n_chars": 178,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"max_nestin... | 1 | module
public import Foundation.FirstOrder.Arithmetic.PeanoMinus.Basic
@[expose] public section
/-!
# Arithmetical Formula Sorted by Arithmetical Hierarchy
This file defines the $\Sigma_n / \Pi_n / \Delta_n$ formulas of arithmetic of first-order logic.
- `𝚺-[m].Semiformula ξ n` is a `ArithmeticSemiformula ξ n` wh... | module
public import Foundation.FirstOrder.Arithmetic.PeanoMinus.Basic
@[expose] public section
/-!
# Arithmetical Formula Sorted by Arithmetical Hierarchy
This file defines the $\Sigma_n / \Pi_n / \Delta_n$ formulas of arithmetic of first-order logic.
- `𝚺-[m].Semiformula ξ n` is a `ArithmeticSemiformula ξ n` wh... | @@ -106,6 +106,8 @@
@[simp] lemma pi_mkDelta (φ : 𝚺-[m].Semiformula ξ n) (ψ : 𝚷-[m].Semiformula ξ n) : (mkDelta φ ψ).pi = ψ := rfl
+lemma val_sigma (φ : 𝚫-[m].Semiformula ξ n) : φ.sigma.val = φ.val := by rcases φ; simp
+
def mkPolarity (φ : ArithmeticSemiformula ξ n) : (Γ : Polarity) → Hierarchy Γ m φ → Γ-[m].... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_a1402c3b36f3_0 | f1cafc5adb44d42d | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Kripke/Logic/GL/MDP.lean | MDP | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 4 | 1 | [
{
"theorem_name": "MDP_Aux",
"depth": 1,
"n_commands": 0,
"n_lines": 62,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n obtain ⟨Δ, sΓ, hΓ⟩ := Context.provable_iff_boxed.mp h;\n\n have : Modal.GL ⊢ ⋀(□'Δ) 🡒 (□φ₁ ⋎ □φ₂) := FiniteContext.provable_iff.... | [
{
"name": "modal_equivalence_original_world₂",
"text": "lemma modal_equivalence_original_world₂ {x : M₂.World} : ModalEquivalent (M₁ := M₂) (M₂ := (mdpCounterexmpleModel M₁ M₂)) x (↑x) := by\n apply Kripke.Model.PseudoEpimorphism.modal_equivalence $ pMorphism₂ M₁ M₂;\n\n",
"fan_in": 1,
"n_lines": 4... | [
{
"name": "MDP_Aux",
"fan_in": 1,
"n_deps_direct": 3,
"n_deps_transitive": 3,
"n_lines": 64,
"n_chars": 3038,
"n_subproofs": 17,
"n_tactics": 103,
"cyclomatic": 6,
"n_automation": 6,
"n_rewrites": 1,
"n_structural": 22,
"automation_only": false,
"max_nesting":... | 3 | module
public import Foundation.Modal.Kripke.Logic.GL.Unnecessitation
@[expose] public section
namespace LO.Modal
open LO.Entailment LO.Entailment.FiniteContext LO.Modal.Entailment
open Kripke
open Formula.Kripke
namespace GL
namespace Kripke
abbrev mdpCounterexmpleFrame (F₁ F₂ : Frame) : Frame where
World := ... | module
public import Foundation.Modal.Kripke.Logic.GL.Unnecessitation
@[expose] public section
namespace LO.Modal
open LO.Entailment LO.Entailment.FiniteContext LO.Modal.Entailment
open Kripke
open Formula.Kripke
namespace GL
namespace Kripke
abbrev mdpCounterexmpleFrame (F₁ F₂ : Frame) : Frame where
World := ... | @@ -106,6 +106,9 @@
lemma modal_equivalence_original_world₁ {x : M₁.World} : ModalEquivalent (M₁ := M₁) (M₂ := (mdpCounterexmpleModel M₁ M₂)) x (↑x) := by
apply Kripke.Model.PseudoEpimorphism.modal_equivalence $ pMorphism₁ M₁ M₂;
+lemma modal_equivalence_original_world₂ {x : M₂.World} : ModalEquivalent (M₁ := M₂)... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_a1402c3b36f3_1 | faf65cbc3447363f | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Kripke/Logic/GL/MDP.lean | MDP | 1 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 4 | 1 | [
{
"theorem_name": "modal_disjunctive",
"depth": 1,
"n_commands": 0,
"n_lines": 6,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n have : ∅ *⊢[Modal.GL] □φ₁ ∨ ∅ *⊢[Modal.GL] □φ₂ := by simpa [Set.LO.boxItr, Set.LO.preboxItr] using MDP_Aux (X := ∅) (φ₁ :... | [
{
"name": "MDP_Aux",
"text": "lemma MDP_Aux {X : Set _} (h : (□'X) *⊢[Modal.GL] □φ₁ ⋎ □φ₂) : (□'X) *⊢[Modal.GL] □φ₁ ∨ (□'X) *⊢[Modal.GL] □φ₂ := by\n obtain ⟨Δ, sΓ, hΓ⟩ := Context.provable_iff_boxed.mp h;\n\n have : Modal.GL ⊢ ⋀(□'Δ) 🡒 (□φ₁ ⋎ □φ₂) := FiniteContext.provable_iff.mp hΓ;\n have : Modal.GL ⊢ ... | [
{
"name": "modal_disjunctive",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 4,
"n_lines": 7,
"n_chars": 398,
"n_subproofs": 2,
"n_tactics": 10,
"cyclomatic": 3,
"n_automation": 2,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": false,
"max_nest... | 4 | module
public import Foundation.Modal.Kripke.Logic.GL.Unnecessitation
@[expose] public section
namespace LO.Modal
open LO.Entailment LO.Entailment.FiniteContext LO.Modal.Entailment
open Kripke
open Formula.Kripke
namespace GL
namespace Kripke
abbrev mdpCounterexmpleFrame (F₁ F₂ : Frame) : Frame where
World := ... | module
public import Foundation.Modal.Kripke.Logic.GL.Unnecessitation
@[expose] public section
namespace LO.Modal
open LO.Entailment LO.Entailment.FiniteContext LO.Modal.Entailment
open Kripke
open Formula.Kripke
namespace GL
namespace Kripke
abbrev mdpCounterexmpleFrame (F₁ F₂ : Frame) : Frame where
World := ... | @@ -113,8 +113,71 @@
end Kripke
+lemma MDP_Aux {X : Set _} (h : (□'X) *⊢[Modal.GL] □φ₁ ⋎ □φ₂) : (□'X) *⊢[Modal.GL] □φ₁ ∨ (□'X) *⊢[Modal.GL] □φ₂ := by
+ obtain ⟨Δ, sΓ, hΓ⟩ := Context.provable_iff_boxed.mp h;
+
+ have : Modal.GL ⊢ ⋀(□'Δ) 🡒 (□φ₁ ⋎ □φ₂) := FiniteContext.provable_iff.mp hΓ;
+ have : Modal.GL ⊢ □⋀Δ ... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_cc04283e781c_0 | 4a22b853f47933dd | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Skolemization/Hull.lean | Hull | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 3 | 1 | [
{
"theorem_name": "closed",
"depth": 1,
"n_commands": 0,
"n_lines": 7,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n choose u hu using fun i ↦ mem_iff.mp (hv i)\n let t : Term L.skolemFunction₁ s := .func φ.skolem₁ u\n refine ⟨t.val ![] (↑), by si... | [
{
"name": "mem_iff",
"text": "lemma mem_iff :\n x ∈ SkolemHull L s ↔ ∃ t : Term L.skolemFunction₁ s, t.val ![] (↑) = x := by\n simp [SkolemHull]\n\n",
"fan_in": 1,
"n_lines": 5,
"n_chars": 128,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
"n_automation": 1,
"n_rewri... | [
{
"name": "closed",
"fan_in": 3,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 11,
"n_chars": 510,
"n_subproofs": 1,
"n_tactics": 7,
"cyclomatic": 1,
"n_automation": 2,
"n_rewrites": 0,
"n_structural": 2,
"automation_only": false,
"max_nesting": 4
}... | 1 | module
public import Foundation.FirstOrder.Basic
public import Mathlib.SetTheory.Cardinal.Basic
@[expose] public section
/-! # Skolem hull -/
namespace LO.FirstOrder
/-- Skolem function of rank 1 -/
def Language.skolemFunction₁ (L : Language) : Language where
Func k := Semisentence L (k + 1)
Rel _ := PEmpty
ab... | module
public import Foundation.FirstOrder.Basic
public import Mathlib.SetTheory.Cardinal.Basic
@[expose] public section
/-! # Skolem hull -/
namespace LO.FirstOrder
/-- Skolem function of rank 1 -/
def Language.skolemFunction₁ (L : Language) : Language where
Func k := Semisentence L (k + 1)
Rel _ := PEmpty
ab... | @@ -51,11 +51,21 @@
variable {s : Set M}
+lemma mem_iff :
+ x ∈ SkolemHull L s ↔ ∃ t : Term L.skolemFunction₁ s, t.val ![] (↑) = x := by
+ simp [SkolemHull]
+
@[simp] lemma val_mem (t : Term L.skolemFunction₁ s) : t.val ![] (↑) ∈ SkolemHull L s := by simp [SkolemHull]
lemma closed {v : Fin k → M} (hv : ∀ i... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_cc04283e781c_1 | 2afc7c7660610c5f | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Skolemization/Hull.lean | Hull | 1 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 3 | 3 | [
{
"theorem_name": "closed_func",
"depth": 1,
"n_commands": 0,
"n_lines": 7,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n have : ∃ z ∈ SkolemHull L s, “#0 = !!(Semiterm.func f fun i ↦ #i.succ)”.Evalb (z :> v) :=\n closed hv (φ := “#0 = !!(Semiter... | [
{
"name": "closed",
"text": "lemma closed {v : Fin k → M} (hv : ∀ i, v i ∈ SkolemHull L s)\n {φ : Semisentence L (k + 1)} (H : ∃ z, φ.Evalb (z :> v)) :\n ∃ z ∈ SkolemHull L s, φ.Evalb (z :> v) := by\n choose u hu using fun i ↦ mem_iff.mp (hv i)\n let t : Term L.skolemFunction₁ s := .func φ.skolem₁ u... | [
{
"name": "closed_func",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 2,
"n_lines": 10,
"n_chars": 497,
"n_subproofs": 2,
"n_tactics": 9,
"cyclomatic": 3,
"n_automation": 4,
"n_rewrites": 0,
"n_structural": 2,
"automation_only": false,
"max_nesting": ... | 2 | module
public import Foundation.FirstOrder.Basic
public import Mathlib.SetTheory.Cardinal.Basic
@[expose] public section
/-! # Skolem hull -/
namespace LO.FirstOrder
/-- Skolem function of rank 1 -/
def Language.skolemFunction₁ (L : Language) : Language where
Func k := Semisentence L (k + 1)
Rel _ := PEmpty
ab... | module
public import Foundation.FirstOrder.Basic
public import Mathlib.SetTheory.Cardinal.Basic
@[expose] public section
/-! # Skolem hull -/
namespace LO.FirstOrder
/-- Skolem function of rank 1 -/
def Language.skolemFunction₁ (L : Language) : Language where
Func k := Semisentence L (k + 1)
Rel _ := PEmpty
ab... | @@ -57,11 +57,23 @@
@[simp] lemma val_mem (t : Term L.skolemFunction₁ s) : t.val ![] (↑) ∈ SkolemHull L s := by simp [SkolemHull]
+lemma closed {v : Fin k → M} (hv : ∀ i, v i ∈ SkolemHull L s)
+ {φ : Semisentence L (k + 1)} (H : ∃ z, φ.Evalb (z :> v)) :
+ ∃ z ∈ SkolemHull L s, φ.Evalb (z :> v) := by
+ choos... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_cc04283e781c_2 | 2414144d857c7284 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Skolemization/Hull.lean | Hull | 2 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 3 | 2 | [
{
"theorem_name": "set_countable",
"depth": 1,
"n_commands": 0,
"n_lines": 4,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n have : Countable s := hs\n have : Countable (Term L.skolemFunction₁ s) := Semiterm.countable\n exact Set.countable_range _"... | [
{
"name": "countable",
"text": "lemma countable (hs : s.Countable) : Countable (SkolemHull L s) := set_countable hs\n\n",
"fan_in": 2,
"n_lines": 3,
"n_chars": 85,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
... | [
{
"name": "set_countable",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 2,
"n_lines": 6,
"n_chars": 202,
"n_subproofs": 2,
"n_tactics": 4,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": false,
"max_nesting":... | 1 | module
public import Foundation.FirstOrder.Basic
public import Mathlib.SetTheory.Cardinal.Basic
@[expose] public section
/-! # Skolem hull -/
namespace LO.FirstOrder
/-- Skolem function of rank 1 -/
def Language.skolemFunction₁ (L : Language) : Language where
Func k := Semisentence L (k + 1)
Rel _ := PEmpty
ab... | module
public import Foundation.FirstOrder.Basic
public import Mathlib.SetTheory.Cardinal.Basic
@[expose] public section
/-! # Skolem hull -/
namespace LO.FirstOrder
/-- Skolem function of rank 1 -/
def Language.skolemFunction₁ (L : Language) : Language where
Func k := Semisentence L (k + 1)
Rel _ := PEmpty
ab... | @@ -165,9 +165,11 @@
lemma set_countable (hs : s.Countable) : (SkolemHull L s).Countable := by
have : Countable s := hs
- have : Countable (Term L.skolemFunction₁ s) := sorry
+ have : Countable (Term L.skolemFunction₁ s) := Semiterm.countable
exact Set.countable_range _
+lemma countable (hs : s.Countable) ... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_cc04283e781c_3 | be76f6f2c00cf200 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Skolemization/Hull.lean | Hull | 3 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 3 | 2 | [
{
"theorem_name": "set_countable",
"depth": 1,
"n_commands": 0,
"n_lines": 4,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n have : Countable s := hs\n have : Countable (Term L.skolemFunction₁ s) := Semiterm.countable\n exact Set.countable_range _"... | [
{
"name": "countable",
"text": "lemma countable (hs : s.Countable) : Countable (SkolemHull L s) := set_countable hs\n\n",
"fan_in": 2,
"n_lines": 3,
"n_chars": 85,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
... | [
{
"name": "card_le_aleph0",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 2,
"n_lines": 6,
"n_chars": 227,
"n_subproofs": 1,
"n_tactics": 2,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
"max_nesting"... | 1 | module
public import Foundation.FirstOrder.Basic
public import Mathlib.SetTheory.Cardinal.Basic
@[expose] public section
/-! # Skolem hull -/
namespace LO.FirstOrder
/-- Skolem function of rank 1 -/
def Language.skolemFunction₁ (L : Language) : Language where
Func k := Semisentence L (k + 1)
Rel _ := PEmpty
ab... | module
public import Foundation.FirstOrder.Basic
public import Mathlib.SetTheory.Cardinal.Basic
@[expose] public section
/-! # Skolem hull -/
namespace LO.FirstOrder
/-- Skolem function of rank 1 -/
def Language.skolemFunction₁ (L : Language) : Language where
Func k := Semisentence L (k + 1)
Rel _ := PEmpty
ab... | @@ -165,14 +165,16 @@
lemma set_countable (hs : s.Countable) : (SkolemHull L s).Countable := by
have : Countable s := hs
- have : Countable (Term L.skolemFunction₁ s) := sorry
+ have : Countable (Term L.skolemFunction₁ s) := Semiterm.countable
exact Set.countable_range _
+lemma countable (hs : s.Countable)... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_ada4b7170845_0 | 4fa0e4e01daa1bae | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/NegationTranslation/GoedelGentzen.lean | GoedelGentzen | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 3 | 2 | [
{
"theorem_name": "neg_doubleNegation'",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by simpa using neg_doubleNegation (∼φ)",
"n_chars": 42,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
... | [
{
"name": "neg_doubleNegation",
"text": "lemma neg_doubleNegation (φ : Proposition L) : Λ ⊢ ∼φᴺ 🡘 (∼φ)ᴺ := ⟨negDoubleNegation φ⟩\n\n",
"fan_in": 2,
"n_lines": 3,
"n_chars": 111,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_str... | [
{
"name": "neg_doubleNegation'",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 3,
"n_chars": 127,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"max_nest... | 1 | module
public import Foundation.FirstOrder.Intuitionistic.Deduction
public import Foundation.Meta.IntProver
@[expose] public section
namespace LO.FirstOrder
namespace Semiformula
def doubleNegation {n} : Semiformula L ξ n → Semiformulaᵢ L ξ n
| rel r v => ∼∼(.rel r v)
| nrel r v => ∼(.rel r v)
| ⊤ => ... | module
public import Foundation.FirstOrder.Intuitionistic.Deduction
public import Foundation.Meta.IntProver
@[expose] public section
namespace LO.FirstOrder
namespace Semiformula
def doubleNegation {n} : Semiformula L ξ n → Semiformulaᵢ L ξ n
| rel r v => ∼∼(.rel r v)
| nrel r v => ∼(.rel r v)
| ⊤ => ... | @@ -114,8 +114,10 @@
this
termination_by φ => φ.complexity
-lemma neg_doubleNegation' (φ : Proposition L) : Λ ⊢ ∼(∼φ)ᴺ 🡘 φᴺ := sorry
+lemma neg_doubleNegation (φ : Proposition L) : Λ ⊢ ∼φᴺ 🡘 (∼φ)ᴺ := ⟨negDoubleNegation φ⟩
+lemma neg_doubleNegation' (φ : Proposition L) : Λ ⊢ ∼(∼φ)ᴺ 🡘 φᴺ := by simpa using ... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_ada4b7170845_1 | b19a62c0c375383f | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/NegationTranslation/GoedelGentzen.lean | GoedelGentzen | 1 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 3 | 1 | [
{
"theorem_name": "imply_doubleNegation",
"depth": 1,
"n_commands": 0,
"n_lines": 18,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n suffices Λ ⊢ (φᴺ 🡒 ψᴺ) 🡘 ∼(∼(∼φ)ᴺ ⋏ ∼ψᴺ) by simpa [Semiformula.doubleNegation_imply]\n have hφ₀ : Λ ⊢ ∼(∼φ)ᴺ 🡘 φᴺ... | [
{
"name": "doubleNegation_imply",
"text": "lemma doubleNegation_imply (φ ψ : Semiformula L ξ n) : (φ 🡒 ψ)ᴺ = ∼(∼(∼φ)ᴺ ⋏ ∼ψᴺ) := by simp [imp_eq]\n\n",
"fan_in": 1,
"n_lines": 3,
"n_chars": 129,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites... | [
{
"name": "imply_doubleNegation",
"fan_in": 0,
"n_deps_direct": 2,
"n_deps_transitive": 2,
"n_lines": 20,
"n_chars": 1060,
"n_subproofs": 6,
"n_tactics": 18,
"cyclomatic": 1,
"n_automation": 3,
"n_rewrites": 0,
"n_structural": 9,
"automation_only": false,
"max... | 2 | module
public import Foundation.FirstOrder.Intuitionistic.Deduction
public import Foundation.Meta.IntProver
@[expose] public section
namespace LO.FirstOrder
namespace Semiformula
def doubleNegation {n} : Semiformula L ξ n → Semiformulaᵢ L ξ n
| rel r v => ∼∼(.rel r v)
| nrel r v => ∼(.rel r v)
| ⊤ => ... | module
public import Foundation.FirstOrder.Intuitionistic.Deduction
public import Foundation.Meta.IntProver
@[expose] public section
namespace LO.FirstOrder
namespace Semiformula
def doubleNegation {n} : Semiformula L ξ n → Semiformulaᵢ L ξ n
| rel r v => ∼∼(.rel r v)
| nrel r v => ∼(.rel r v)
| ⊤ => ... | @@ -36,6 +36,8 @@
@[simp] lemma doubleNegation_ex (φ : Semiformula L ξ (n + 1)) : (∃⁰ φ)ᴺ = ∼(∀⁰ ∼φᴺ) := rfl
+lemma doubleNegation_imply (φ ψ : Semiformula L ξ n) : (φ 🡒 ψ)ᴺ = ∼(∼(∼φ)ᴺ ⋏ ∼ψᴺ) := by simp [imp_eq]
+
@[simp] lemma doubleNegation_isNegative (φ : Semiformula L ξ n) : φᴺ.IsNegative := by
induction ... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_df8bcbceb234_0 | cc566eb07d5e8465 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/PeanoMinus/Functions.lean | Functions | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 10 | 6 | [
{
"theorem_name": "sub_le_self",
"depth": 1,
"n_commands": 0,
"n_lines": 5,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n have : b ≤ a ∨ a < b := le_or_gt b a\n rcases this with (hxy | hxy) <;> simp\n · simpa [← sub_spec_of_ge hxy] using show a - ... | [
{
"name": "sub_spec_of_ge",
"text": "lemma sub_spec_of_ge (h : a ≥ b) : a = b + (a - b) := (Classical.choose!_spec (sub_existsUnique a b)).1 h\n\n",
"fan_in": 6,
"n_lines": 3,
"n_chars": 109,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": ... | [
{
"name": "sub_le_self",
"fan_in": 0,
"n_deps_direct": 2,
"n_deps_transitive": 3,
"n_lines": 7,
"n_chars": 257,
"n_subproofs": 1,
"n_tactics": 6,
"cyclomatic": 3,
"n_automation": 3,
"n_rewrites": 1,
"n_structural": 1,
"automation_only": false,
"max_nesting": 2... | 1 | module
public import Foundation.FirstOrder.Arithmetic.Definability
public import Mathlib.Algebra.Prime.Lemmas
public import Foundation.Vorspiel.ExistsUnique
@[expose] public section
/-!
# Functions and relations defined in $\mathsf{PA^-}$
This file provides functions and relations defined in $\mathsf{PA^-}
-/
name... | module
public import Foundation.FirstOrder.Arithmetic.Definability
public import Mathlib.Algebra.Prime.Lemmas
public import Foundation.Vorspiel.ExistsUnique
@[expose] public section
/-!
# Functions and relations defined in $\mathsf{PA^-}$
This file provides functions and relations defined in $\mathsf{PA^-}
-/
name... | @@ -34,6 +34,8 @@
noncomputable scoped instance : Sub V := ⟨sub⟩
+lemma sub_spec_of_ge (h : a ≥ b) : a = b + (a - b) := (Classical.choose!_spec (sub_existsUnique a b)).1 h
+
lemma sub_spec_of_lt (h : a < b) : a - b = 0 := (Classical.choose!_spec (sub_existsUnique a b)).2 h
lemma sub_eq_iff : c = a - b ↔ ((a ≥ ... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_df8bcbceb234_1 | 3bd2f10a0378ca10 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/PeanoMinus/Functions.lean | Functions | 1 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 10 | 6 | [
{
"theorem_name": "sub_le_self",
"depth": 1,
"n_commands": 0,
"n_lines": 5,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n have : b ≤ a ∨ a < b := le_or_gt b a\n rcases this with (hxy | hxy) <;> simp\n · simpa [← sub_spec_of_ge hxy] using show a - ... | [
{
"name": "sub_spec_of_ge",
"text": "lemma sub_spec_of_ge (h : a ≥ b) : a = b + (a - b) := (Classical.choose!_spec (sub_existsUnique a b)).1 h\n\n",
"fan_in": 6,
"n_lines": 3,
"n_chars": 109,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": ... | [
{
"name": "add_tsub_self_of_le",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 2,
"n_lines": 3,
"n_chars": 94,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": false,
"max_nest... | 1 | module
public import Foundation.FirstOrder.Arithmetic.Definability
public import Mathlib.Algebra.Prime.Lemmas
public import Foundation.Vorspiel.ExistsUnique
@[expose] public section
/-!
# Functions and relations defined in $\mathsf{PA^-}$
This file provides functions and relations defined in $\mathsf{PA^-}
-/
name... | module
public import Foundation.FirstOrder.Arithmetic.Definability
public import Mathlib.Algebra.Prime.Lemmas
public import Foundation.Vorspiel.ExistsUnique
@[expose] public section
/-!
# Functions and relations defined in $\mathsf{PA^-}$
This file provides functions and relations defined in $\mathsf{PA^-}
-/
name... | @@ -34,6 +34,8 @@
noncomputable scoped instance : Sub V := ⟨sub⟩
+lemma sub_spec_of_ge (h : a ≥ b) : a = b + (a - b) := (Classical.choose!_spec (sub_existsUnique a b)).1 h
+
lemma sub_spec_of_lt (h : a < b) : a - b = 0 := (Classical.choose!_spec (sub_existsUnique a b)).2 h
lemma sub_eq_iff : c = a - b ↔ ((a ≥ ... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
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