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_97495911a3b3_7 | 73c02e20c8261541 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Kripke/Basic.lean | Basic | 7 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 14 | 3 | [
{
"theorem_name": "boxItr_def",
"depth": 1,
"n_commands": 0,
"n_lines": 13,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n induction n generalizing x with\n | zero => simp;\n | succ n ih =>\n constructor;\n . rintro h y ⟨z, Rxz, Rzy⟩;\n ... | [
{
"name": "box_def",
"text": "protected lemma box_def : x ⊧ □φ ↔ ∀ y, x ≺ y → y ⊧ φ := by simp [Satisfies];\n",
"fan_in": 3,
"n_lines": 2,
"n_chars": 94,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automat... | [
{
"name": "intro_diaItr_semiequiv",
"fan_in": 2,
"n_deps_direct": 2,
"n_deps_transitive": 5,
"n_lines": 15,
"n_chars": 396,
"n_subproofs": 0,
"n_tactics": 23,
"cyclomatic": 2,
"n_automation": 2,
"n_rewrites": 0,
"n_structural": 9,
"automation_only": false,
"ma... | 5 | module
public import Foundation.Modal.Logic.Basic
public import Foundation.Vorspiel.Rel.Basic
@[expose] public section
namespace LO.Modal
open Entailment
namespace Kripke
structure Frame where
World : Type
Rel : Rel World World
[world_nonempty : Nonempty World]
attribute [simp] Frame.world_nonempty
instanc... | module
public import Foundation.Modal.Logic.Basic
public import Foundation.Vorspiel.Rel.Basic
@[expose] public section
namespace LO.Modal
open Entailment
namespace Kripke
structure Frame where
World : Type
Rel : Rel World World
[world_nonempty : Nonempty World]
attribute [simp] Frame.world_nonempty
instanc... | @@ -104,6 +104,7 @@
protected lemma top_def : x ⊧ ⊤ := by simp [Satisfies];
+protected lemma box_def : x ⊧ □φ ↔ ∀ y, x ≺ y → y ⊧ φ := by simp [Satisfies];
protected instance : Semantics.Tarski (M.World) where
models_verum := λ _ => Satisfies.top_def;
models_falsum := λ _ => Satisfies.bot_def;
@@ -120,7 +121... | {
"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_97495911a3b3_8 | 656f199d4757609c | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Kripke/Basic.lean | Basic | 8 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 14 | 3 | [
{
"theorem_name": "boxItr_def",
"depth": 1,
"n_commands": 0,
"n_lines": 13,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n induction n generalizing x with\n | zero => simp;\n | succ n ih =>\n constructor;\n . rintro h y ⟨z, Rxz, Rzy⟩;\n ... | [
{
"name": "box_def",
"text": "protected lemma box_def : x ⊧ □φ ↔ ∀ y, x ≺ y → y ⊧ φ := by simp [Satisfies];\n",
"fan_in": 3,
"n_lines": 2,
"n_chars": 94,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automat... | [
{
"name": "intro_dia_semiequiv",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 6,
"n_lines": 5,
"n_chars": 160,
"n_subproofs": 0,
"n_tactics": 5,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": false,
"max_nes... | 6 | module
public import Foundation.Modal.Logic.Basic
public import Foundation.Vorspiel.Rel.Basic
@[expose] public section
namespace LO.Modal
open Entailment
namespace Kripke
structure Frame where
World : Type
Rel : Rel World World
[world_nonempty : Nonempty World]
attribute [simp] Frame.world_nonempty
instanc... | module
public import Foundation.Modal.Logic.Basic
public import Foundation.Vorspiel.Rel.Basic
@[expose] public section
namespace LO.Modal
open Entailment
namespace Kripke
structure Frame where
World : Type
Rel : Rel World World
[world_nonempty : Nonempty World]
attribute [simp] Frame.world_nonempty
instanc... | @@ -104,6 +104,7 @@
protected lemma top_def : x ⊧ ⊤ := by simp [Satisfies];
+protected lemma box_def : x ⊧ □φ ↔ ∀ y, x ≺ y → y ⊧ φ := by simp [Satisfies];
protected instance : Semantics.Tarski (M.World) where
models_verum := λ _ => Satisfies.top_def;
models_falsum := λ _ => Satisfies.bot_def;
@@ -120,7 +121... | {
"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_97495911a3b3_9 | 0057253ee9674f8b | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Kripke/Basic.lean | Basic | 9 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 14 | 2 | [
{
"theorem_name": "intro_diaItr_semiequiv",
"depth": 1,
"n_commands": 0,
"n_lines": 13,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n induction n generalizing x with\n | zero => simp_all;\n | succ n ih =>\n simp only [Dia.diaItr_succ];\n app... | [
{
"name": "intro_neg_semiequiv",
"text": "lemma intro_neg_semiequiv (h : x ⊧ φ → x ⊧ ψ) : x ⊧ ∼ψ → x ⊧ ∼φ := by\n contrapose;\n simp_all;\n\n",
"fan_in": 2,
"n_lines": 5,
"n_chars": 117,
"n_subproofs": 0,
"n_tactics": 5,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
... | [
{
"name": "intro_negEquiv",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 6,
"n_chars": 178,
"n_subproofs": 0,
"n_tactics": 7,
"cyclomatic": 2,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 3,
"automation_only": false,
"max_nesting"... | 1 | module
public import Foundation.Modal.Logic.Basic
public import Foundation.Vorspiel.Rel.Basic
@[expose] public section
namespace LO.Modal
open Entailment
namespace Kripke
structure Frame where
World : Type
Rel : Rel World World
[world_nonempty : Nonempty World]
attribute [simp] Frame.world_nonempty
instanc... | module
public import Foundation.Modal.Logic.Basic
public import Foundation.Vorspiel.Rel.Basic
@[expose] public section
namespace LO.Modal
open Entailment
namespace Kripke
structure Frame where
World : Type
Rel : Rel World World
[world_nonempty : Nonempty World]
attribute [simp] Frame.world_nonempty
instanc... | @@ -114,10 +114,14 @@
@[simp] lemma negneg_def : x ⊧ ∼∼φ ↔ x ⊧ φ := by simp [Semantics.NotModels];
+lemma intro_neg_semiequiv (h : x ⊧ φ → x ⊧ ψ) : x ⊧ ∼ψ → x ⊧ ∼φ := by
+ contrapose;
+ simp_all;
+
lemma intro_negEquiv (h : x ⊧ φ ↔ x ⊧ ψ) : x ⊧ ∼φ ↔ x ⊧ ∼ψ := by
constructor;
- . sorry
- . sorry
+ . apply ... | {
"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_97495911a3b3_10 | 42307f0de9e62a6c | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Kripke/Basic.lean | Basic | 10 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 14 | 3 | [
{
"theorem_name": "boxItr_def",
"depth": 1,
"n_commands": 0,
"n_lines": 13,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n induction n generalizing x with\n | zero => simp;\n | succ n ih =>\n constructor;\n . rintro h y ⟨z, Rxz, Rzy⟩;\n ... | [
{
"name": "box_def",
"text": "protected lemma box_def : x ⊧ □φ ↔ ∀ y, x ≺ y → y ⊧ φ := by simp [Satisfies];\n",
"fan_in": 3,
"n_lines": 2,
"n_chars": 94,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automat... | [
{
"name": "intro_boxItr_equiv",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 3,
"n_lines": 6,
"n_chars": 275,
"n_subproofs": 0,
"n_tactics": 11,
"cyclomatic": 2,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 7,
"automation_only": false,
"max_nes... | 3 | module
public import Foundation.Modal.Logic.Basic
public import Foundation.Vorspiel.Rel.Basic
@[expose] public section
namespace LO.Modal
open Entailment
namespace Kripke
structure Frame where
World : Type
Rel : Rel World World
[world_nonempty : Nonempty World]
attribute [simp] Frame.world_nonempty
instanc... | module
public import Foundation.Modal.Logic.Basic
public import Foundation.Vorspiel.Rel.Basic
@[expose] public section
namespace LO.Modal
open Entailment
namespace Kripke
structure Frame where
World : Type
Rel : Rel World World
[world_nonempty : Nonempty World]
attribute [simp] Frame.world_nonempty
instanc... | @@ -104,6 +104,7 @@
protected lemma top_def : x ⊧ ⊤ := by simp [Satisfies];
+protected lemma box_def : x ⊧ □φ ↔ ∀ y, x ≺ y → y ⊧ φ := by simp [Satisfies];
protected instance : Semantics.Tarski (M.World) where
models_verum := λ _ => Satisfies.top_def;
models_falsum := λ _ => Satisfies.bot_def;
@@ -120,7 +121... | {
"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_97495911a3b3_11 | 2e34b6309b269a38 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Kripke/Basic.lean | Basic | 11 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 14 | 3 | [
{
"theorem_name": "boxItr_def",
"depth": 1,
"n_commands": 0,
"n_lines": 13,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n induction n generalizing x with\n | zero => simp;\n | succ n ih =>\n constructor;\n . rintro h y ⟨z, Rxz, Rzy⟩;\n ... | [
{
"name": "box_def",
"text": "protected lemma box_def : x ⊧ □φ ↔ ∀ y, x ≺ y → y ⊧ φ := by simp [Satisfies];\n",
"fan_in": 3,
"n_lines": 2,
"n_chars": 94,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automat... | [
{
"name": "intro_box_equiv",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 4,
"n_lines": 5,
"n_chars": 154,
"n_subproofs": 0,
"n_tactics": 5,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": false,
"max_nesting... | 4 | module
public import Foundation.Modal.Logic.Basic
public import Foundation.Vorspiel.Rel.Basic
@[expose] public section
namespace LO.Modal
open Entailment
namespace Kripke
structure Frame where
World : Type
Rel : Rel World World
[world_nonempty : Nonempty World]
attribute [simp] Frame.world_nonempty
instanc... | module
public import Foundation.Modal.Logic.Basic
public import Foundation.Vorspiel.Rel.Basic
@[expose] public section
namespace LO.Modal
open Entailment
namespace Kripke
structure Frame where
World : Type
Rel : Rel World World
[world_nonempty : Nonempty World]
attribute [simp] Frame.world_nonempty
instanc... | @@ -104,6 +104,7 @@
protected lemma top_def : x ⊧ ⊤ := by simp [Satisfies];
+protected lemma box_def : x ⊧ □φ ↔ ∀ y, x ≺ y → y ⊧ φ := by simp [Satisfies];
protected instance : Semantics.Tarski (M.World) where
models_verum := λ _ => Satisfies.top_def;
models_falsum := λ _ => Satisfies.bot_def;
@@ -120,7 +121... | {
"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_97495911a3b3_12 | cf53e1d3fef86a0e | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Kripke/Basic.lean | Basic | 12 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 14 | 3 | [
{
"theorem_name": "boxItr_def",
"depth": 1,
"n_commands": 0,
"n_lines": 13,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n induction n generalizing x with\n | zero => simp;\n | succ n ih =>\n constructor;\n . rintro h y ⟨z, Rxz, Rzy⟩;\n ... | [
{
"name": "box_def",
"text": "protected lemma box_def : x ⊧ □φ ↔ ∀ y, x ≺ y → y ⊧ φ := by simp [Satisfies];\n",
"fan_in": 3,
"n_lines": 2,
"n_chars": 94,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automat... | [
{
"name": "intro_diaItr_equiv",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 6,
"n_lines": 6,
"n_chars": 275,
"n_subproofs": 0,
"n_tactics": 11,
"cyclomatic": 2,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 7,
"automation_only": false,
"max_nes... | 6 | module
public import Foundation.Modal.Logic.Basic
public import Foundation.Vorspiel.Rel.Basic
@[expose] public section
namespace LO.Modal
open Entailment
namespace Kripke
structure Frame where
World : Type
Rel : Rel World World
[world_nonempty : Nonempty World]
attribute [simp] Frame.world_nonempty
instanc... | module
public import Foundation.Modal.Logic.Basic
public import Foundation.Vorspiel.Rel.Basic
@[expose] public section
namespace LO.Modal
open Entailment
namespace Kripke
structure Frame where
World : Type
Rel : Rel World World
[world_nonempty : Nonempty World]
attribute [simp] Frame.world_nonempty
instanc... | @@ -104,6 +104,7 @@
protected lemma top_def : x ⊧ ⊤ := by simp [Satisfies];
+protected lemma box_def : x ⊧ □φ ↔ ∀ y, x ≺ y → y ⊧ φ := by simp [Satisfies];
protected instance : Semantics.Tarski (M.World) where
models_verum := λ _ => Satisfies.top_def;
models_falsum := λ _ => Satisfies.bot_def;
@@ -120,7 +121... | {
"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_97495911a3b3_13 | 423cd367759abc67 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Kripke/Basic.lean | Basic | 13 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 14 | 3 | [
{
"theorem_name": "boxItr_def",
"depth": 1,
"n_commands": 0,
"n_lines": 13,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n induction n generalizing x with\n | zero => simp;\n | succ n ih =>\n constructor;\n . rintro h y ⟨z, Rxz, Rzy⟩;\n ... | [
{
"name": "box_def",
"text": "protected lemma box_def : x ⊧ □φ ↔ ∀ y, x ≺ y → y ⊧ φ := by simp [Satisfies];\n",
"fan_in": 3,
"n_lines": 2,
"n_chars": 94,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automat... | [
{
"name": "intro_dia_equiv",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 7,
"n_lines": 5,
"n_chars": 154,
"n_subproofs": 0,
"n_tactics": 5,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": false,
"max_nesting... | 7 | module
public import Foundation.Modal.Logic.Basic
public import Foundation.Vorspiel.Rel.Basic
@[expose] public section
namespace LO.Modal
open Entailment
namespace Kripke
structure Frame where
World : Type
Rel : Rel World World
[world_nonempty : Nonempty World]
attribute [simp] Frame.world_nonempty
instanc... | module
public import Foundation.Modal.Logic.Basic
public import Foundation.Vorspiel.Rel.Basic
@[expose] public section
namespace LO.Modal
open Entailment
namespace Kripke
structure Frame where
World : Type
Rel : Rel World World
[world_nonempty : Nonempty World]
attribute [simp] Frame.world_nonempty
instanc... | @@ -104,6 +104,7 @@
protected lemma top_def : x ⊧ ⊤ := by simp [Satisfies];
+protected lemma box_def : x ⊧ □φ ↔ ∀ y, x ≺ y → y ⊧ φ := by simp [Satisfies];
protected instance : Semantics.Tarski (M.World) where
models_verum := λ _ => Satisfies.top_def;
models_falsum := λ _ => Satisfies.bot_def;
@@ -120,7 +121... | {
"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_97495911a3b3_14 | 709b5d396d777a9f | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Kripke/Basic.lean | Basic | 14 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 14 | 3 | [
{
"theorem_name": "boxItr_def",
"depth": 1,
"n_commands": 0,
"n_lines": 13,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n induction n generalizing x with\n | zero => simp;\n | succ n ih =>\n constructor;\n . rintro h y ⟨z, Rxz, Rzy⟩;\n ... | [
{
"name": "box_def",
"text": "protected lemma box_def : x ⊧ □φ ↔ ∀ y, x ≺ y → y ⊧ φ := by simp [Satisfies];\n",
"fan_in": 3,
"n_lines": 2,
"n_chars": 94,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automat... | [
{
"name": "diaItr_dual",
"fan_in": 1,
"n_deps_direct": 3,
"n_deps_transitive": 3,
"n_lines": 31,
"n_chars": 922,
"n_subproofs": 5,
"n_tactics": 55,
"cyclomatic": 5,
"n_automation": 5,
"n_rewrites": 0,
"n_structural": 19,
"automation_only": false,
"max_nesting"... | 2 | module
public import Foundation.Modal.Logic.Basic
public import Foundation.Vorspiel.Rel.Basic
@[expose] public section
namespace LO.Modal
open Entailment
namespace Kripke
structure Frame where
World : Type
Rel : Rel World World
[world_nonempty : Nonempty World]
attribute [simp] Frame.world_nonempty
instanc... | module
public import Foundation.Modal.Logic.Basic
public import Foundation.Vorspiel.Rel.Basic
@[expose] public section
namespace LO.Modal
open Entailment
namespace Kripke
structure Frame where
World : Type
Rel : Rel World World
[world_nonempty : Nonempty World]
attribute [simp] Frame.world_nonempty
instanc... | @@ -104,6 +104,7 @@
protected lemma top_def : x ⊧ ⊤ := by simp [Satisfies];
+protected lemma box_def : x ⊧ □φ ↔ ∀ y, x ≺ y → y ⊧ φ := by simp [Satisfies];
protected lemma dia_def : x ⊧ ◇φ ↔ ∃ y, x ≺ y ∧ y ⊧ φ := by simp [Satisfies];
protected instance : Semantics.Tarski (M.World) where
models_verum := λ _ => ... | {
"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_97495911a3b3_15 | 5642fb8dc212911e | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Kripke/Basic.lean | Basic | 15 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 14 | 3 | [
{
"theorem_name": "boxItr_def",
"depth": 1,
"n_commands": 0,
"n_lines": 13,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n induction n generalizing x with\n | zero => simp;\n | succ n ih =>\n constructor;\n . rintro h y ⟨z, Rxz, Rzy⟩;\n ... | [
{
"name": "box_def",
"text": "protected lemma box_def : x ⊧ □φ ↔ ∀ y, x ≺ y → y ⊧ φ := by simp [Satisfies];\n",
"fan_in": 3,
"n_lines": 2,
"n_chars": 94,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automat... | [
{
"name": "boxItr_dual",
"fan_in": 0,
"n_deps_direct": 4,
"n_deps_transitive": 6,
"n_lines": 20,
"n_chars": 443,
"n_subproofs": 0,
"n_tactics": 29,
"cyclomatic": 2,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 10,
"automation_only": false,
"max_nesting"... | 4 | module
public import Foundation.Modal.Logic.Basic
public import Foundation.Vorspiel.Rel.Basic
@[expose] public section
namespace LO.Modal
open Entailment
namespace Kripke
structure Frame where
World : Type
Rel : Rel World World
[world_nonempty : Nonempty World]
attribute [simp] Frame.world_nonempty
instanc... | module
public import Foundation.Modal.Logic.Basic
public import Foundation.Vorspiel.Rel.Basic
@[expose] public section
namespace LO.Modal
open Entailment
namespace Kripke
structure Frame where
World : Type
Rel : Rel World World
[world_nonempty : Nonempty World]
attribute [simp] Frame.world_nonempty
instanc... | @@ -104,6 +104,7 @@
protected lemma top_def : x ⊧ ⊤ := by simp [Satisfies];
+protected lemma box_def : x ⊧ □φ ↔ ∀ y, x ≺ y → y ⊧ φ := by simp [Satisfies];
protected lemma dia_def : x ⊧ ◇φ ↔ ∃ y, x ≺ y ∧ y ⊧ φ := by simp [Satisfies];
protected instance : Semantics.Tarski (M.World) where
models_verum := λ _ => ... | {
"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_97495911a3b3_16 | 30c42e3a1d84dbe0 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Kripke/Basic.lean | Basic | 16 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 14 | 2 | [
{
"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.Modal.Logic.Basic
public import Foundation.Vorspiel.Rel.Basic
@[expose] public section
namespace LO.Modal
open Entailment
namespace Kripke
structure Frame where
World : Type
Rel : Rel World World
[world_nonempty : Nonempty World]
attribute [simp] Frame.world_nonempty
instanc... | module
public import Foundation.Modal.Logic.Basic
public import Foundation.Vorspiel.Rel.Basic
@[expose] public section
namespace LO.Modal
open Entailment
namespace Kripke
structure Frame where
World : Type
Rel : Rel World World
[world_nonempty : Nonempty World]
attribute [simp] Frame.world_nonempty
instanc... | @@ -158,10 +158,19 @@
instance : Semantics.Bot (Kripke.Frame) where
models_falsum _ := ValidOnFrame.bot_def
+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];
+
+alias ⟨e... | {
"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_97495911a3b3_17 | e288716542552c7b | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Kripke/Basic.lean | Basic | 17 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 14 | 1 | [
{
"theorem_name": "subst",
"depth": 1,
"n_commands": 0,
"n_lines": 6,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n by_contra hC;\n replace hC := iff_not_exists_valuation_world.mp hC;\n obtain ⟨V, ⟨x, hx⟩⟩ := hC;\n apply Satisfies.iff_subst_self ... | [
{
"name": "iff_subst_self",
"text": "lemma iff_subst_self {x : F.World} (s : Substitution ℕ) :\n letI U : Kripke.Valuation F := λ a w => Satisfies ⟨F, V⟩ w ((atom a)⟦s⟧);\n Satisfies ⟨F, U⟩ x φ ↔ Satisfies ⟨F, V⟩ x (φ⟦s⟧) := by\n induction φ generalizing x with\n | hatom a => simp [Satisfies];\n | hfal... | [
{
"name": "subst",
"fan_in": 0,
"n_deps_direct": 2,
"n_deps_transitive": 2,
"n_lines": 8,
"n_chars": 283,
"n_subproofs": 1,
"n_tactics": 11,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 3,
"automation_only": false,
"max_nesting": 2
}
... | 2 | module
public import Foundation.Modal.Logic.Basic
public import Foundation.Vorspiel.Rel.Basic
@[expose] public section
namespace LO.Modal
open Entailment
namespace Kripke
structure Frame where
World : Type
Rel : Rel World World
[world_nonempty : Nonempty World]
attribute [simp] Frame.world_nonempty
instanc... | module
public import Foundation.Modal.Logic.Basic
public import Foundation.Vorspiel.Rel.Basic
@[expose] public section
namespace LO.Modal
open Entailment
namespace Kripke
structure Frame where
World : Type
Rel : Rel World World
[world_nonempty : Nonempty World]
attribute [simp] Frame.world_nonempty
instanc... | @@ -114,6 +114,33 @@
@[simp] lemma negneg_def : x ⊧ ∼∼φ ↔ x ⊧ φ := by simp [Semantics.NotModels];
+lemma iff_subst_self {x : F.World} (s : Substitution ℕ) :
+ letI U : Kripke.Valuation F := λ a w => Satisfies ⟨F, V⟩ w ((atom a)⟦s⟧);
+ Satisfies ⟨F, U⟩ x φ ↔ Satisfies ⟨F, V⟩ x (φ⟦s⟧) := by
+ induction φ generali... | {
"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_1684411f74cd_0 | c73f45e2d17d5ea5 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Semantics/Algebra/Modal/Magari.lean | Magari | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 1 | 1 | [
{
"theorem_name": "dia_trans",
"depth": 1,
"n_commands": 0,
"n_lines": 7,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " calc\n ◇◇a ≤ ◇a ⊔ ◇◇a := by simp\n _ = ◇(a ⊔ ◇a) := by simp [dia_or]\n _ = ◇((a ⊔ ◇a) ⊓ (... | [
{
"name": "dia_diag",
"text": "lemma dia_diag : ◇(a ⊓ (◇a)ᶜ) = ◇a := by\n suffices □(aᶜ ⊔ (□aᶜ)ᶜ) = □aᶜ by simpa [ModalAlgebra.dual_dia];\n calc\n _ = □(□aᶜ ⇨ aᶜ) := by rw [BooleanAlgebra.himp_eq]\n _ = _ := by rw [box_diag];\n\n",
"fan_in": 1,
"n_lines": 7,
"n_chars": 242,
"n_... | [
{
"name": "dia_trans",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 10,
"n_chars": 580,
"n_subproofs": 0,
"n_tactics": 7,
"cyclomatic": 1,
"n_automation": 5,
"n_rewrites": 1,
"n_structural": 1,
"automation_only": false,
"max_nesting": 2
... | 1 | module
public import Foundation.Semantics.Algebra.Modal.Basic
@[expose] public section
namespace LO
class MagariAlgebra (α : Type*) extends ModalAlgebra α where
box_diag {a : α} : □(□a ⇨ a) = □a
namespace MagariAlgebra
open ModalAlgebra
variable {α : Type u} [MagariAlgebra α]
variable {a b : α}
@[simp]
lemma ... | module
public import Foundation.Semantics.Algebra.Modal.Basic
@[expose] public section
namespace LO
class MagariAlgebra (α : Type*) extends ModalAlgebra α where
box_diag {a : α} : □(□a ⇨ a) = □a
namespace MagariAlgebra
open ModalAlgebra
variable {α : Type u} [MagariAlgebra α]
variable {a b : α}
lemma dia_diag... | @@ -16,8 +16,20 @@
variable {α : Type u} [MagariAlgebra α]
variable {a b : α}
+lemma dia_diag : ◇(a ⊓ (◇a)ᶜ) = ◇a := by
+ suffices □(aᶜ ⊔ (□aᶜ)ᶜ) = □aᶜ by simpa [ModalAlgebra.dual_dia];
+ calc
+ _ = □(□aᶜ ⇨ aᶜ) := by rw [BooleanAlgebra.himp_eq]
+ _ = _ := by rw [box_diag];
+
@[simp]
-lemma dia_trans ... | {
"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_e0a56683dea4_0 | 7db08436e7f8b74f | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Kripke/Logic/GrzPoint2.lean | GrzPoint2 | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 1 | 1 | [
{
"theorem_name": "not_Grz_of_not_GrzPoint2",
"depth": 1,
"n_commands": 0,
"n_lines": 4,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n have := Context.provable_iff.not.mp $ not_imp_not.mpr GrzPoint2_of_Grz h;\n push Not at this;\n convert this ((φ... | [
{
"name": "GrzPoint2_of_Grz",
"text": "lemma GrzPoint2_of_Grz (h : (φ.atoms.image (λ a => Axioms.Point2 (.atom a))).toSet *⊢[Modal.Grz] φ) : Modal.GrzPoint2 ⊢ φ := by\n obtain ⟨Γ, hΓ₁, hΓ₂⟩ := Context.provable_iff.mp h;\n simp only [Finset.coe_image, Set.mem_image, Finset.mem_coe] at hΓ₁;\n replace hΓ₂ :... | [
{
"name": "not_Grz_of_not_GrzPoint2",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 6,
"n_chars": 327,
"n_subproofs": 1,
"n_tactics": 7,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
"ma... | 1 | module
public import Foundation.Modal.Kripke.Logic.Grz.Completeness
public import Foundation.Modal.Kripke.Logic.S4Point2McK
public import Mathlib.Data.Finite.Sum
@[expose] public section
namespace LO.Modal
open Entailment
open Entailment.Context
open Formula
open Formula.Kripke
open Modal.Kripke
open Kripke
sectio... | module
public import Foundation.Modal.Kripke.Logic.Grz.Completeness
public import Foundation.Modal.Kripke.Logic.S4Point2McK
public import Mathlib.Data.Finite.Sum
@[expose] public section
namespace LO.Modal
open Entailment
open Entailment.Context
open Formula
open Formula.Kripke
open Modal.Kripke
open Kripke
sectio... | @@ -61,8 +61,21 @@
instance : Modal.Grz ⪯ Modal.GrzPoint2 := Hilbert.Normal.weakerThan_of_subset_axioms $ by grind;
-lemma not_Grz_of_not_GrzPoint2 (h : Modal.GrzPoint2 ⊬ φ) : (φ.atoms.image (λ a => Axioms.Point2 (.atom a))).toList ⊬[Modal.Grz] φ := sorry
+lemma GrzPoint2_of_Grz (h : (φ.atoms.image (λ a => Axioms.... | {
"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_c83194f0ef4d_0 | 393fc1110634dbc4 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Vorspiel/Order/LowerSet.lean | LowerSet | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 2 | 1 | [
{
"theorem_name": "mem_dual_iff",
"depth": 1,
"n_commands": 0,
"n_lines": 10,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n suffices (∃ i, (∀ y ∈ i, ∀ z ∈ u, y ⟂ z) ∧ x ∈ i) ↔ ∀ y ∈ u, x ⟂ y by simpa [Compl.compl, inf_eq_bot_iff_incompatible]\n con... | [
{
"name": "inf_eq_bot_iff_incompatible",
"text": "lemma inf_eq_bot_iff_incompatible {u v : LowerSet α} : u ⊓ v = ⊥ ↔ (∀ x ∈ u, ∀ y ∈ v, x ⟂ y) := by\n suffices (u ∩ v : Set α) = ∅ ↔ ∀ x ∈ u, ∀ y ∈ v, x ⟂ y by simpa [LowerSet.ext_iff]\n constructor\n · rintro e x hx y hy ⟨z, hzx, hzy⟩\n have hzu : z ∈ ... | [
{
"name": "mem_dual_iff",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 12,
"n_chars": 503,
"n_subproofs": 1,
"n_tactics": 10,
"cyclomatic": 3,
"n_automation": 4,
"n_rewrites": 0,
"n_structural": 7,
"automation_only": false,
"max_nesting"... | 1 | module
public import Mathlib.Topology.Order.UpperLowerSetTopology
public import Mathlib.Topology.Sets.Opens
public import Mathlib.Order.Heyting.Regular
public import Foundation.Vorspiel.Order.Regular
public import Foundation.Vorspiel.Order.Dense
@[expose] public section
variable {α : Type*} [Preorder α]
namespace L... | module
public import Mathlib.Topology.Order.UpperLowerSetTopology
public import Mathlib.Topology.Sets.Opens
public import Mathlib.Order.Heyting.Regular
public import Foundation.Vorspiel.Order.Regular
public import Foundation.Vorspiel.Order.Dense
@[expose] public section
variable {α : Type*} [Preorder α]
namespace L... | @@ -16,8 +16,32 @@
instance : HeytingAlgebra (LowerSet α) := inferInstance
-lemma mem_dual_iff {u : LowerSet α} : x ∈ uᶜ ↔ ∀ y ∈ u, x ⟂ y := sorry
+lemma inf_eq_bot_iff_incompatible {u v : LowerSet α} : u ⊓ v = ⊥ ↔ (∀ x ∈ u, ∀ y ∈ v, x ⟂ y) := by
+ suffices (u ∩ v : Set α) = ∅ ↔ ∀ x ∈ u, ∀ y ∈ v, x ⟂ y by simpa [... | {
"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_c83194f0ef4d_1 | 15d3a76a3144fe36 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Vorspiel/Order/LowerSet.lean | LowerSet | 1 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 2 | 1 | [
{
"theorem_name": "mem_dual_iff",
"depth": 1,
"n_commands": 0,
"n_lines": 10,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n suffices (∃ i, (∀ y ∈ i, ∀ z ∈ u, y ⟂ z) ∧ x ∈ i) ↔ ∀ y ∈ u, x ⟂ y by simpa [Compl.compl, inf_eq_bot_iff_incompatible]\n con... | [
{
"name": "inf_eq_bot_iff_incompatible",
"text": "lemma inf_eq_bot_iff_incompatible {u v : LowerSet α} : u ⊓ v = ⊥ ↔ (∀ x ∈ u, ∀ y ∈ v, x ⟂ y) := by\n suffices (u ∩ v : Set α) = ∅ ↔ ∀ x ∈ u, ∀ y ∈ v, x ⟂ y by simpa [LowerSet.ext_iff]\n constructor\n · rintro e x hx y hy ⟨z, hzx, hzy⟩\n have hzu : z ∈ ... | [
{
"name": "coe_dual",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 2,
"n_lines": 4,
"n_chars": 105,
"n_subproofs": 0,
"n_tactics": 3,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": false,
"max_nesting": 2
... | 2 | module
public import Mathlib.Topology.Order.UpperLowerSetTopology
public import Mathlib.Topology.Sets.Opens
public import Mathlib.Order.Heyting.Regular
public import Foundation.Vorspiel.Order.Regular
public import Foundation.Vorspiel.Order.Dense
@[expose] public section
variable {α : Type*} [Preorder α]
namespace L... | module
public import Mathlib.Topology.Order.UpperLowerSetTopology
public import Mathlib.Topology.Sets.Opens
public import Mathlib.Order.Heyting.Regular
public import Foundation.Vorspiel.Order.Regular
public import Foundation.Vorspiel.Order.Dense
@[expose] public section
variable {α : Type*} [Preorder α]
namespace L... | @@ -16,8 +16,32 @@
instance : HeytingAlgebra (LowerSet α) := inferInstance
-lemma mem_dual_iff {u : LowerSet α} : x ∈ uᶜ ↔ ∀ y ∈ u, x ⟂ y := sorry
+lemma inf_eq_bot_iff_incompatible {u v : LowerSet α} : u ⊓ v = ⊥ ↔ (∀ x ∈ u, ∀ y ∈ v, x ⟂ y) := by
+ suffices (u ∩ v : Set α) = ∅ ↔ ∀ x ∈ u, ∀ y ∈ v, x ⟂ y by simpa [... | {
"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_235f99e6add7_0 | 74be48498228c4e7 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Propositional/Hilbert/Minimal/Basic.lean | Basic | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 5 | 1 | [
{
"theorem_name": "weakerThan_of_le",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " Entailment.weakerThan_iff.mpr $ of_le h",
"n_chars": 40,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
... | [
{
"name": "of_le",
"text": "lemma of_le (h : H₁.schema ⊆ H₂.schema) : H₁ ⊢ φ → H₂ ⊢ φ := λ ⟨hφ⟩ => ⟨ofLE h hφ⟩\n\n",
"fan_in": 1,
"n_lines": 3,
"n_chars": 113,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"a... | [
{
"name": "weakerThan_of_le",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 4,
"n_chars": 129,
"n_subproofs": 0,
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"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
"max_nestin... | 1 | module
public import Foundation.Propositional.Entailment.Int.DNE_of_LEM
public import Foundation.Propositional.Logic.Basic
public import Foundation.Propositional.Formula.Basic
public import Foundation.Propositional.Entailment.KreiselPutnam
public import Foundation.Propositional.Entailment.Scott
public import Foundatio... | module
public import Foundation.Propositional.Entailment.Int.DNE_of_LEM
public import Foundation.Propositional.Logic.Basic
public import Foundation.Propositional.Formula.Basic
public import Foundation.Propositional.Entailment.KreiselPutnam
public import Foundation.Propositional.Entailment.Scott
public import Foundatio... | @@ -99,8 +99,10 @@
| orIntroR => orIntroR
| orElim => orElim
+lemma of_le (h : H₁.schema ⊆ H₂.schema) : H₁ ⊢ φ → H₂ ⊢ φ := λ ⟨hφ⟩ => ⟨ofLE h hφ⟩
+
@[grind <=]
-lemma weakerThan_of_le (h : H₁.schema ⊆ H₂.schema) : H₁ ⪯ H₂ := sorry
+lemma weakerThan_of_le (h : H₁.schema ⊆ H₂.schema) : H₁ ⪯ H₂ := Entailment.weake... | {
"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_235f99e6add7_1 | e409659b6810cc96 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Propositional/Hilbert/Minimal/Basic.lean | Basic | 1 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 5 | 1 | [
{
"theorem_name": "weakerThan_of_provable_schema",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " Entailment.weakerThan_iff.mpr $ of_proof_schema h",
"n_chars": 50,
"n_subproofs": 0,
"n_tactics": 1,
... | [
{
"name": "of_proof_schema",
"text": "lemma of_proof_schema (h : H₂ ⊢* H₁.schema) : H₁ ⊢ φ → H₂ ⊢ φ := λ ⟨hφ⟩ => ⟨ofProofSchema (fun _ hφ ↦ (h hφ).get) hφ⟩\n\n",
"fan_in": 1,
"n_lines": 3,
"n_chars": 152,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"... | [
{
"name": "weakerThan_of_provable_schema",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 3,
"n_chars": 134,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
... | 1 | module
public import Foundation.Propositional.Entailment.Int.DNE_of_LEM
public import Foundation.Propositional.Logic.Basic
public import Foundation.Propositional.Formula.Basic
public import Foundation.Propositional.Entailment.KreiselPutnam
public import Foundation.Propositional.Entailment.Scott
public import Foundatio... | module
public import Foundation.Propositional.Entailment.Int.DNE_of_LEM
public import Foundation.Propositional.Logic.Basic
public import Foundation.Propositional.Formula.Basic
public import Foundation.Propositional.Entailment.KreiselPutnam
public import Foundation.Propositional.Entailment.Scott
public import Foundatio... | @@ -132,8 +132,10 @@
| orIntroR => orIntroR
| orElim => orElim
-lemma weakerThan_of_provable_schema (h : H₂ ⊢* H₁.schema) : H₁ ⪯ H₂ := sorry
+lemma of_proof_schema (h : H₂ ⊢* H₁.schema) : H₁ ⊢ φ → H₂ ⊢ φ := λ ⟨hφ⟩ => ⟨ofProofSchema (fun _ hφ ↦ (h hφ).get) hφ⟩
+lemma weakerThan_of_provable_schema (h : H₂ ⊢* H₁... | {
"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_235f99e6add7_2 | 3d0e4af5f2d27bc1 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Propositional/Hilbert/Minimal/Basic.lean | Basic | 2 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 5 | 1 | [
{
"theorem_name": "mem_logic_of_provable",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " mem_logic_of_proof h.get",
"n_chars": 25,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automat... | [
{
"name": "mem_logic_of_proof",
"text": "lemma mem_logic_of_proof (h : H ⊢! φ) : φ ∈ H.logic := ⟨h⟩\n\n",
"fan_in": 1,
"n_lines": 3,
"n_chars": 70,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_on... | [
{
"name": "mem_logic_of_provable",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 2,
"n_chars": 88,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
"max_ne... | 1 | module
public import Foundation.Propositional.Entailment.Int.DNE_of_LEM
public import Foundation.Propositional.Logic.Basic
public import Foundation.Propositional.Formula.Basic
public import Foundation.Propositional.Entailment.KreiselPutnam
public import Foundation.Propositional.Entailment.Scott
public import Foundatio... | module
public import Foundation.Propositional.Entailment.Int.DNE_of_LEM
public import Foundation.Propositional.Logic.Basic
public import Foundation.Propositional.Formula.Basic
public import Foundation.Propositional.Entailment.KreiselPutnam
public import Foundation.Propositional.Entailment.Scott
public import Foundatio... | @@ -167,7 +167,9 @@
variable {H : Hilbert α}
-lemma mem_logic_of_provable (h : H ⊢ φ) : φ ∈ H.logic := sorry
+lemma mem_logic_of_proof (h : H ⊢! φ) : φ ∈ H.logic := ⟨h⟩
+
+lemma mem_logic_of_provable (h : H ⊢ φ) : φ ∈ H.logic := mem_logic_of_proof h.get
lemma provable_of_mem_logic (h : φ ∈ H.logic) : H ⊢ φ := 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_235f99e6add7_3 | 1fda7b25905eb9d2 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Propositional/Hilbert/Minimal/Basic.lean | Basic | 3 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 5 | 1 | [
{
"theorem_name": "iff_mem_logic_provable",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " ⟨mem_logic_of_provable, provable_of_mem_logic⟩",
"n_chars": 51,
"n_subproofs": 0,
"n_tactics": 1,
"cycloma... | [
{
"name": "mem_logic_of_provable",
"text": "lemma mem_logic_of_provable (h : H ⊢ φ) : φ ∈ H.logic := mem_logic_of_proof h.get\n",
"fan_in": 1,
"n_lines": 2,
"n_chars": 88,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural... | [
{
"name": "iff_mem_logic_provable",
"fan_in": 0,
"n_deps_direct": 2,
"n_deps_transitive": 3,
"n_lines": 4,
"n_chars": 125,
"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.Propositional.Entailment.Int.DNE_of_LEM
public import Foundation.Propositional.Logic.Basic
public import Foundation.Propositional.Formula.Basic
public import Foundation.Propositional.Entailment.KreiselPutnam
public import Foundation.Propositional.Entailment.Scott
public import Foundatio... | module
public import Foundation.Propositional.Entailment.Int.DNE_of_LEM
public import Foundation.Propositional.Logic.Basic
public import Foundation.Propositional.Formula.Basic
public import Foundation.Propositional.Entailment.KreiselPutnam
public import Foundation.Propositional.Entailment.Scott
public import Foundatio... | @@ -169,10 +169,11 @@
lemma mem_logic_of_proof (h : H ⊢! φ) : φ ∈ H.logic := ⟨h⟩
+lemma mem_logic_of_provable (h : H ⊢ φ) : φ ∈ H.logic := mem_logic_of_proof h.get
lemma provable_of_mem_logic (h : φ ∈ H.logic) : H ⊢ φ := h
@[grind =]
-lemma iff_mem_logic_provable : H ⊢ φ ↔ φ ∈ H.logic := sorry
+lemma iff_mem_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_5234c2258d5f_0 | 93bddf34f6309b91 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Kripke/AxiomGeach.lean | AxiomGeach | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 2 | 7 | [
{
"theorem_name": "validate_AxiomT_of_reflexive",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " validate_axiomGeach_of_isGeachConvergent ⟨0, 0, 1, 0⟩",
"n_chars": 58,
"n_subproofs": 0,
"n_tactics": 1,... | [
{
"name": "validate_axiomGeach_of_isGeachConvergent",
"text": "lemma validate_axiomGeach_of_isGeachConvergent (g) [F.IsGeachConvergent g] : F ⊧ (Axioms.Geach g (.atom 0)) := by\n rintro V x h;\n apply Satisfies.boxItr_def.mpr;\n obtain ⟨y, Rxy, hbp⟩ := Satisfies.diaItr_def.mp h;\n intro z Rxz;\n apply ... | [
{
"name": "validate_AxiomT_of_reflexive",
"fan_in": 0,
"n_deps_direct": 4,
"n_deps_transitive": 2,
"n_lines": 2,
"n_chars": 148,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
... | 1 | module
public import Foundation.Modal.Kripke.Completeness
public import Foundation.Vorspiel.Rel.Coreflexive
@[expose] public section
namespace LO.Modal
namespace Kripke
variable {F : Frame}
namespace Frame
class IsGeachConvergent (F : Frame) (g : Axioms.Geach.Taple) where
gconv : ∀ ⦃x y z : F⦄, x ≺^[g.i] y → x... | module
public import Foundation.Modal.Kripke.Completeness
public import Foundation.Vorspiel.Rel.Coreflexive
@[expose] public section
namespace LO.Modal
namespace Kripke
variable {F : Frame}
namespace Frame
class IsGeachConvergent (F : Frame) (g : Axioms.Geach.Taple) where
gconv : ∀ ⦃x y z : F⦄, x ≺^[g.i] y → x... | @@ -171,7 +171,19 @@
open Formula (atom)
open Formula.Kripke
-lemma validate_AxiomT_of_reflexive [refl : F.IsReflexive] : F ⊧ (Axioms.T (.atom 0)) := sorry
+lemma validate_axiomGeach_of_isGeachConvergent (g) [F.IsGeachConvergent g] : F ⊧ (Axioms.Geach g (.atom 0)) := by
+ rintro V x h;
+ apply Satisfies.boxItr_de... | {
"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_5234c2258d5f_1 | c4acfd3427fedd07 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Kripke/AxiomGeach.lean | AxiomGeach | 1 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 2 | 7 | [
{
"theorem_name": "validate_AxiomT_of_reflexive",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " validate_axiomGeach_of_isGeachConvergent ⟨0, 0, 1, 0⟩",
"n_chars": 58,
"n_subproofs": 0,
"n_tactics": 1,... | [
{
"name": "validate_axiomGeach_of_isGeachConvergent",
"text": "lemma validate_axiomGeach_of_isGeachConvergent (g) [F.IsGeachConvergent g] : F ⊧ (Axioms.Geach g (.atom 0)) := by\n rintro V x h;\n apply Satisfies.boxItr_def.mpr;\n obtain ⟨y, Rxy, hbp⟩ := Satisfies.diaItr_def.mp h;\n intro z Rxz;\n apply ... | [
{
"name": "validate_AxiomD_of_serial",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 2,
"n_chars": 141,
"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.Modal.Kripke.Completeness
public import Foundation.Vorspiel.Rel.Coreflexive
@[expose] public section
namespace LO.Modal
namespace Kripke
variable {F : Frame}
namespace Frame
class IsGeachConvergent (F : Frame) (g : Axioms.Geach.Taple) where
gconv : ∀ ⦃x y z : F⦄, x ≺^[g.i] y → x... | module
public import Foundation.Modal.Kripke.Completeness
public import Foundation.Vorspiel.Rel.Coreflexive
@[expose] public section
namespace LO.Modal
namespace Kripke
variable {F : Frame}
namespace Frame
class IsGeachConvergent (F : Frame) (g : Axioms.Geach.Taple) where
gconv : ∀ ⦃x y z : F⦄, x ≺^[g.i] y → x... | @@ -171,7 +171,19 @@
open Formula (atom)
open Formula.Kripke
-lemma validate_AxiomD_of_serial [ser : F.IsSerial] : F ⊧ (Axioms.D (.atom 0)) := sorry
+lemma validate_axiomGeach_of_isGeachConvergent (g) [F.IsGeachConvergent g] : F ⊧ (Axioms.Geach g (.atom 0)) := by
+ rintro V x h;
+ apply Satisfies.boxItr_def.mpr;
... | {
"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_5234c2258d5f_2 | 6b3b6be02b48b518 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Kripke/AxiomGeach.lean | AxiomGeach | 2 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 2 | 7 | [
{
"theorem_name": "validate_AxiomT_of_reflexive",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " validate_axiomGeach_of_isGeachConvergent ⟨0, 0, 1, 0⟩",
"n_chars": 58,
"n_subproofs": 0,
"n_tactics": 1,... | [
{
"name": "validate_axiomGeach_of_isGeachConvergent",
"text": "lemma validate_axiomGeach_of_isGeachConvergent (g) [F.IsGeachConvergent g] : F ⊧ (Axioms.Geach g (.atom 0)) := by\n rintro V x h;\n apply Satisfies.boxItr_def.mpr;\n obtain ⟨y, Rxy, hbp⟩ := Satisfies.diaItr_def.mp h;\n intro z Rxz;\n apply ... | [
{
"name": "validate_AxiomB_of_symmetric",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 2,
"n_chars": 147,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
... | 1 | module
public import Foundation.Modal.Kripke.Completeness
public import Foundation.Vorspiel.Rel.Coreflexive
@[expose] public section
namespace LO.Modal
namespace Kripke
variable {F : Frame}
namespace Frame
class IsGeachConvergent (F : Frame) (g : Axioms.Geach.Taple) where
gconv : ∀ ⦃x y z : F⦄, x ≺^[g.i] y → x... | module
public import Foundation.Modal.Kripke.Completeness
public import Foundation.Vorspiel.Rel.Coreflexive
@[expose] public section
namespace LO.Modal
namespace Kripke
variable {F : Frame}
namespace Frame
class IsGeachConvergent (F : Frame) (g : Axioms.Geach.Taple) where
gconv : ∀ ⦃x y z : F⦄, x ≺^[g.i] y → x... | @@ -171,7 +171,19 @@
open Formula (atom)
open Formula.Kripke
-lemma validate_AxiomB_of_symmetric [sym : F.IsSymmetric] : F ⊧ (Axioms.B (.atom 0)) := sorry
+lemma validate_axiomGeach_of_isGeachConvergent (g) [F.IsGeachConvergent g] : F ⊧ (Axioms.Geach g (.atom 0)) := by
+ rintro V x h;
+ apply Satisfies.boxItr_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_5234c2258d5f_3 | 17cd0a429423431c | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Kripke/AxiomGeach.lean | AxiomGeach | 3 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 2 | 7 | [
{
"theorem_name": "validate_AxiomT_of_reflexive",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " validate_axiomGeach_of_isGeachConvergent ⟨0, 0, 1, 0⟩",
"n_chars": 58,
"n_subproofs": 0,
"n_tactics": 1,... | [
{
"name": "validate_axiomGeach_of_isGeachConvergent",
"text": "lemma validate_axiomGeach_of_isGeachConvergent (g) [F.IsGeachConvergent g] : F ⊧ (Axioms.Geach g (.atom 0)) := by\n rintro V x h;\n apply Satisfies.boxItr_def.mpr;\n obtain ⟨y, Rxy, hbp⟩ := Satisfies.diaItr_def.mp h;\n intro z Rxz;\n apply ... | [
{
"name": "validate_AxiomFour_of_transitive",
"fan_in": 0,
"n_deps_direct": 4,
"n_deps_transitive": 2,
"n_lines": 2,
"n_chars": 157,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,... | 1 | module
public import Foundation.Modal.Kripke.Completeness
public import Foundation.Vorspiel.Rel.Coreflexive
@[expose] public section
namespace LO.Modal
namespace Kripke
variable {F : Frame}
namespace Frame
class IsGeachConvergent (F : Frame) (g : Axioms.Geach.Taple) where
gconv : ∀ ⦃x y z : F⦄, x ≺^[g.i] y → x... | module
public import Foundation.Modal.Kripke.Completeness
public import Foundation.Vorspiel.Rel.Coreflexive
@[expose] public section
namespace LO.Modal
namespace Kripke
variable {F : Frame}
namespace Frame
class IsGeachConvergent (F : Frame) (g : Axioms.Geach.Taple) where
gconv : ∀ ⦃x y z : F⦄, x ≺^[g.i] y → x... | @@ -171,7 +171,19 @@
open Formula (atom)
open Formula.Kripke
-lemma validate_AxiomFour_of_transitive [trans : F.IsTransitive] : F ⊧ (Axioms.Four (.atom 0)) := sorry
+lemma validate_axiomGeach_of_isGeachConvergent (g) [F.IsGeachConvergent g] : F ⊧ (Axioms.Geach g (.atom 0)) := by
+ rintro V x h;
+ apply Satisfies.... | {
"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_5234c2258d5f_4 | 3953c7118311c23f | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Kripke/AxiomGeach.lean | AxiomGeach | 4 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 2 | 7 | [
{
"theorem_name": "validate_AxiomT_of_reflexive",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " validate_axiomGeach_of_isGeachConvergent ⟨0, 0, 1, 0⟩",
"n_chars": 58,
"n_subproofs": 0,
"n_tactics": 1,... | [
{
"name": "validate_axiomGeach_of_isGeachConvergent",
"text": "lemma validate_axiomGeach_of_isGeachConvergent (g) [F.IsGeachConvergent g] : F ⊧ (Axioms.Geach g (.atom 0)) := by\n rintro V x h;\n apply Satisfies.boxItr_def.mpr;\n obtain ⟨y, Rxy, hbp⟩ := Satisfies.diaItr_def.mp h;\n intro z Rxz;\n apply ... | [
{
"name": "validate_AxiomFive_of_euclidean",
"fan_in": 0,
"n_deps_direct": 4,
"n_deps_transitive": 2,
"n_lines": 2,
"n_chars": 154,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
... | 1 | module
public import Foundation.Modal.Kripke.Completeness
public import Foundation.Vorspiel.Rel.Coreflexive
@[expose] public section
namespace LO.Modal
namespace Kripke
variable {F : Frame}
namespace Frame
class IsGeachConvergent (F : Frame) (g : Axioms.Geach.Taple) where
gconv : ∀ ⦃x y z : F⦄, x ≺^[g.i] y → x... | module
public import Foundation.Modal.Kripke.Completeness
public import Foundation.Vorspiel.Rel.Coreflexive
@[expose] public section
namespace LO.Modal
namespace Kripke
variable {F : Frame}
namespace Frame
class IsGeachConvergent (F : Frame) (g : Axioms.Geach.Taple) where
gconv : ∀ ⦃x y z : F⦄, x ≺^[g.i] y → x... | @@ -173,7 +173,19 @@
open Formula (atom)
open Formula.Kripke
-lemma validate_AxiomFive_of_euclidean [eucl : F.IsEuclidean] : F ⊧ (Axioms.Five (.atom 0)) := sorry
+lemma validate_axiomGeach_of_isGeachConvergent (g) [F.IsGeachConvergent g] : F ⊧ (Axioms.Geach g (.atom 0)) := by
+ rintro V x h;
+ apply Satisfies.box... | {
"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_5234c2258d5f_5 | 9d8b2052049da000 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Kripke/AxiomGeach.lean | AxiomGeach | 5 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 2 | 7 | [
{
"theorem_name": "validate_AxiomT_of_reflexive",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " validate_axiomGeach_of_isGeachConvergent ⟨0, 0, 1, 0⟩",
"n_chars": 58,
"n_subproofs": 0,
"n_tactics": 1,... | [
{
"name": "validate_axiomGeach_of_isGeachConvergent",
"text": "lemma validate_axiomGeach_of_isGeachConvergent (g) [F.IsGeachConvergent g] : F ⊧ (Axioms.Geach g (.atom 0)) := by\n rintro V x h;\n apply Satisfies.boxItr_def.mpr;\n obtain ⟨y, Rxy, hbp⟩ := Satisfies.diaItr_def.mp h;\n intro z Rxz;\n apply ... | [
{
"name": "validate_AxiomPoint2_of_confluent",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 2,
"n_chars": 176,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false... | 1 | module
public import Foundation.Modal.Kripke.Completeness
public import Foundation.Vorspiel.Rel.Coreflexive
@[expose] public section
namespace LO.Modal
namespace Kripke
variable {F : Frame}
namespace Frame
class IsGeachConvergent (F : Frame) (g : Axioms.Geach.Taple) where
gconv : ∀ ⦃x y z : F⦄, x ≺^[g.i] y → x... | module
public import Foundation.Modal.Kripke.Completeness
public import Foundation.Vorspiel.Rel.Coreflexive
@[expose] public section
namespace LO.Modal
namespace Kripke
variable {F : Frame}
namespace Frame
class IsGeachConvergent (F : Frame) (g : Axioms.Geach.Taple) where
gconv : ∀ ⦃x y z : F⦄, x ≺^[g.i] y → x... | @@ -171,7 +171,19 @@
open Formula (atom)
open Formula.Kripke
-lemma validate_AxiomPoint2_of_confluent [conf : F.IsPiecewiseStronglyConvergent] : F ⊧ (Axioms.Point2 (.atom 0)) := sorry
+lemma validate_axiomGeach_of_isGeachConvergent (g) [F.IsGeachConvergent g] : F ⊧ (Axioms.Geach g (.atom 0)) := by
+ rintro V x 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_5234c2258d5f_6 | de4a3b6cf31f801d | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Kripke/AxiomGeach.lean | AxiomGeach | 6 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 2 | 7 | [
{
"theorem_name": "validate_AxiomT_of_reflexive",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " validate_axiomGeach_of_isGeachConvergent ⟨0, 0, 1, 0⟩",
"n_chars": 58,
"n_subproofs": 0,
"n_tactics": 1,... | [
{
"name": "validate_axiomGeach_of_isGeachConvergent",
"text": "lemma validate_axiomGeach_of_isGeachConvergent (g) [F.IsGeachConvergent g] : F ⊧ (Axioms.Geach g (.atom 0)) := by\n rintro V x h;\n apply Satisfies.boxItr_def.mpr;\n obtain ⟨y, Rxy, hbp⟩ := Satisfies.diaItr_def.mp h;\n intro z Rxz;\n apply ... | [
{
"name": "validate_AxiomTc_of_coreflexive",
"fan_in": 0,
"n_deps_direct": 4,
"n_deps_transitive": 2,
"n_lines": 3,
"n_chars": 157,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
... | 1 | module
public import Foundation.Modal.Kripke.Completeness
public import Foundation.Vorspiel.Rel.Coreflexive
@[expose] public section
namespace LO.Modal
namespace Kripke
variable {F : Frame}
namespace Frame
class IsGeachConvergent (F : Frame) (g : Axioms.Geach.Taple) where
gconv : ∀ ⦃x y z : F⦄, x ≺^[g.i] y → x... | module
public import Foundation.Modal.Kripke.Completeness
public import Foundation.Vorspiel.Rel.Coreflexive
@[expose] public section
namespace LO.Modal
namespace Kripke
variable {F : Frame}
namespace Frame
class IsGeachConvergent (F : Frame) (g : Axioms.Geach.Taple) where
gconv : ∀ ⦃x y z : F⦄, x ≺^[g.i] y → x... | @@ -171,8 +171,20 @@
open Formula (atom)
open Formula.Kripke
-lemma validate_AxiomTc_of_coreflexive [corefl : F.IsCoreflexive] : F ⊧ (Axioms.Tc (.atom 0)) := sorry
+lemma validate_axiomGeach_of_isGeachConvergent (g) [F.IsGeachConvergent g] : F ⊧ (Axioms.Geach g (.atom 0)) := by
+ rintro V x h;
+ apply Satisfies.b... | {
"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_5234c2258d5f_7 | 31daadaff71fa4a5 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Kripke/AxiomGeach.lean | AxiomGeach | 7 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 2 | 7 | [
{
"theorem_name": "reflexive_of_validate_AxiomT",
"depth": 1,
"n_commands": 0,
"n_lines": 4,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n suffices F.IsGeachConvergent ⟨0, 0, 1, 0⟩ by infer_instance;\n apply isGeachConvergent_of_validate_axiomGeach... | [
{
"name": "isGeachConvergent_of_validate_axiomGeach",
"text": "lemma isGeachConvergent_of_validate_axiomGeach {g} (h : F ⊧ (Axioms.Geach g (.atom 0))) : F.IsGeachConvergent g := ⟨by\n rintro x y z Rxy Rxz;\n let V : Kripke.Valuation F := λ _ v => y ≺^[g.m] v;\n have : Satisfies ⟨F, V⟩ x (□^[g.j](◇^[g.n](... | [
{
"name": "reflexive_of_validate_AxiomT",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 6,
"n_chars": 217,
"n_subproofs": 1,
"n_tactics": 7,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": false,
... | 1 | module
public import Foundation.Modal.Kripke.Completeness
public import Foundation.Vorspiel.Rel.Coreflexive
@[expose] public section
namespace LO.Modal
namespace Kripke
variable {F : Frame}
namespace Frame
class IsGeachConvergent (F : Frame) (g : Axioms.Geach.Taple) where
gconv : ∀ ⦃x y z : F⦄, x ≺^[g.i] y → x... | module
public import Foundation.Modal.Kripke.Completeness
public import Foundation.Vorspiel.Rel.Coreflexive
@[expose] public section
namespace LO.Modal
namespace Kripke
variable {F : Frame}
namespace Frame
class IsGeachConvergent (F : Frame) (g : Axioms.Geach.Taple) where
gconv : ∀ ⦃x y z : F⦄, x ≺^[g.i] y → x... | @@ -171,8 +171,26 @@
open Formula (atom)
open Formula.Kripke
-lemma reflexive_of_validate_AxiomT (h : F ⊧ (Axioms.T (.atom 0))) : F.IsReflexive := sorry
+lemma isGeachConvergent_of_validate_axiomGeach {g} (h : F ⊧ (Axioms.Geach g (.atom 0))) : F.IsGeachConvergent g := ⟨by
+ rintro x y z Rxy Rxz;
+ let V : Kripke.... | {
"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_5234c2258d5f_8 | 8b18c0de385b18ae | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Kripke/AxiomGeach.lean | AxiomGeach | 8 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 2 | 7 | [
{
"theorem_name": "reflexive_of_validate_AxiomT",
"depth": 1,
"n_commands": 0,
"n_lines": 4,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n suffices F.IsGeachConvergent ⟨0, 0, 1, 0⟩ by infer_instance;\n apply isGeachConvergent_of_validate_axiomGeach... | [
{
"name": "isGeachConvergent_of_validate_axiomGeach",
"text": "lemma isGeachConvergent_of_validate_axiomGeach {g} (h : F ⊧ (Axioms.Geach g (.atom 0))) : F.IsGeachConvergent g := ⟨by\n rintro x y z Rxy Rxz;\n let V : Kripke.Valuation F := λ _ v => y ≺^[g.m] v;\n have : Satisfies ⟨F, V⟩ x (□^[g.j](◇^[g.n](... | [
{
"name": "transitive_of_validate_AxiomFour",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 6,
"n_chars": 225,
"n_subproofs": 1,
"n_tactics": 7,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": false,... | 1 | module
public import Foundation.Modal.Kripke.Completeness
public import Foundation.Vorspiel.Rel.Coreflexive
@[expose] public section
namespace LO.Modal
namespace Kripke
variable {F : Frame}
namespace Frame
class IsGeachConvergent (F : Frame) (g : Axioms.Geach.Taple) where
gconv : ∀ ⦃x y z : F⦄, x ≺^[g.i] y → x... | module
public import Foundation.Modal.Kripke.Completeness
public import Foundation.Vorspiel.Rel.Coreflexive
@[expose] public section
namespace LO.Modal
namespace Kripke
variable {F : Frame}
namespace Frame
class IsGeachConvergent (F : Frame) (g : Axioms.Geach.Taple) where
gconv : ∀ ⦃x y z : F⦄, x ≺^[g.i] y → x... | @@ -171,8 +171,26 @@
open Formula (atom)
open Formula.Kripke
-lemma transitive_of_validate_AxiomFour (h : F ⊧ (Axioms.Four (.atom 0))) : F.IsTransitive := sorry
+lemma isGeachConvergent_of_validate_axiomGeach {g} (h : F ⊧ (Axioms.Geach g (.atom 0))) : F.IsGeachConvergent g := ⟨by
+ rintro x y z Rxy Rxz;
+ let 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_5234c2258d5f_9 | a5dc7d8b7a5d14f6 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Kripke/AxiomGeach.lean | AxiomGeach | 9 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 2 | 7 | [
{
"theorem_name": "reflexive_of_validate_AxiomT",
"depth": 1,
"n_commands": 0,
"n_lines": 4,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n suffices F.IsGeachConvergent ⟨0, 0, 1, 0⟩ by infer_instance;\n apply isGeachConvergent_of_validate_axiomGeach... | [
{
"name": "isGeachConvergent_of_validate_axiomGeach",
"text": "lemma isGeachConvergent_of_validate_axiomGeach {g} (h : F ⊧ (Axioms.Geach g (.atom 0))) : F.IsGeachConvergent g := ⟨by\n rintro x y z Rxy Rxz;\n let V : Kripke.Valuation F := λ _ v => y ≺^[g.m] v;\n have : Satisfies ⟨F, V⟩ x (□^[g.j](◇^[g.n](... | [
{
"name": "euclidean_of_validate_AxiomFive",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 6,
"n_chars": 223,
"n_subproofs": 1,
"n_tactics": 7,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": false,
... | 1 | module
public import Foundation.Modal.Kripke.Completeness
public import Foundation.Vorspiel.Rel.Coreflexive
@[expose] public section
namespace LO.Modal
namespace Kripke
variable {F : Frame}
namespace Frame
class IsGeachConvergent (F : Frame) (g : Axioms.Geach.Taple) where
gconv : ∀ ⦃x y z : F⦄, x ≺^[g.i] y → x... | module
public import Foundation.Modal.Kripke.Completeness
public import Foundation.Vorspiel.Rel.Coreflexive
@[expose] public section
namespace LO.Modal
namespace Kripke
variable {F : Frame}
namespace Frame
class IsGeachConvergent (F : Frame) (g : Axioms.Geach.Taple) where
gconv : ∀ ⦃x y z : F⦄, x ≺^[g.i] y → x... | @@ -171,8 +171,26 @@
open Formula (atom)
open Formula.Kripke
-lemma euclidean_of_validate_AxiomFive (h : F ⊧ (Axioms.Five (.atom 0))) : F.IsEuclidean := sorry
+lemma isGeachConvergent_of_validate_axiomGeach {g} (h : F ⊧ (Axioms.Geach g (.atom 0))) : F.IsGeachConvergent g := ⟨by
+ rintro x y z Rxy Rxz;
+ let V : K... | {
"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_5234c2258d5f_10 | d82279f3fd5e212b | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Kripke/AxiomGeach.lean | AxiomGeach | 10 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 2 | 7 | [
{
"theorem_name": "reflexive_of_validate_AxiomT",
"depth": 1,
"n_commands": 0,
"n_lines": 4,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n suffices F.IsGeachConvergent ⟨0, 0, 1, 0⟩ by infer_instance;\n apply isGeachConvergent_of_validate_axiomGeach... | [
{
"name": "isGeachConvergent_of_validate_axiomGeach",
"text": "lemma isGeachConvergent_of_validate_axiomGeach {g} (h : F ⊧ (Axioms.Geach g (.atom 0))) : F.IsGeachConvergent g := ⟨by\n rintro x y z Rxy Rxz;\n let V : Kripke.Valuation F := λ _ v => y ≺^[g.m] v;\n have : Satisfies ⟨F, V⟩ x (□^[g.j](◇^[g.n](... | [
{
"name": "symmetric_of_validate_AxiomB",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 6,
"n_chars": 217,
"n_subproofs": 1,
"n_tactics": 7,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": false,
... | 1 | module
public import Foundation.Modal.Kripke.Completeness
public import Foundation.Vorspiel.Rel.Coreflexive
@[expose] public section
namespace LO.Modal
namespace Kripke
variable {F : Frame}
namespace Frame
class IsGeachConvergent (F : Frame) (g : Axioms.Geach.Taple) where
gconv : ∀ ⦃x y z : F⦄, x ≺^[g.i] y → x... | module
public import Foundation.Modal.Kripke.Completeness
public import Foundation.Vorspiel.Rel.Coreflexive
@[expose] public section
namespace LO.Modal
namespace Kripke
variable {F : Frame}
namespace Frame
class IsGeachConvergent (F : Frame) (g : Axioms.Geach.Taple) where
gconv : ∀ ⦃x y z : F⦄, x ≺^[g.i] y → x... | @@ -171,8 +171,26 @@
open Formula (atom)
open Formula.Kripke
-lemma symmetric_of_validate_AxiomB (h : F ⊧ (Axioms.B (.atom 0))) : F.IsSymmetric := sorry
+lemma isGeachConvergent_of_validate_axiomGeach {g} (h : F ⊧ (Axioms.Geach g (.atom 0))) : F.IsGeachConvergent g := ⟨by
+ rintro x y z Rxy Rxz;
+ let V : Kripke.... | {
"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_5234c2258d5f_11 | 36f85ab971320b28 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Kripke/AxiomGeach.lean | AxiomGeach | 11 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 2 | 7 | [
{
"theorem_name": "reflexive_of_validate_AxiomT",
"depth": 1,
"n_commands": 0,
"n_lines": 4,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n suffices F.IsGeachConvergent ⟨0, 0, 1, 0⟩ by infer_instance;\n apply isGeachConvergent_of_validate_axiomGeach... | [
{
"name": "isGeachConvergent_of_validate_axiomGeach",
"text": "lemma isGeachConvergent_of_validate_axiomGeach {g} (h : F ⊧ (Axioms.Geach g (.atom 0))) : F.IsGeachConvergent g := ⟨by\n rintro x y z Rxy Rxz;\n let V : Kripke.Valuation F := λ _ v => y ≺^[g.m] v;\n have : Satisfies ⟨F, V⟩ x (□^[g.j](◇^[g.n](... | [
{
"name": "serial_of_validate_AxiomD",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 6,
"n_chars": 211,
"n_subproofs": 1,
"n_tactics": 7,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": false,
"m... | 1 | module
public import Foundation.Modal.Kripke.Completeness
public import Foundation.Vorspiel.Rel.Coreflexive
@[expose] public section
namespace LO.Modal
namespace Kripke
variable {F : Frame}
namespace Frame
class IsGeachConvergent (F : Frame) (g : Axioms.Geach.Taple) where
gconv : ∀ ⦃x y z : F⦄, x ≺^[g.i] y → x... | module
public import Foundation.Modal.Kripke.Completeness
public import Foundation.Vorspiel.Rel.Coreflexive
@[expose] public section
namespace LO.Modal
namespace Kripke
variable {F : Frame}
namespace Frame
class IsGeachConvergent (F : Frame) (g : Axioms.Geach.Taple) where
gconv : ∀ ⦃x y z : F⦄, x ≺^[g.i] y → x... | @@ -171,8 +171,26 @@
open Formula (atom)
open Formula.Kripke
-lemma serial_of_validate_AxiomD (h : F ⊧ (Axioms.D (.atom 0))) : F.IsSerial := sorry
+lemma isGeachConvergent_of_validate_axiomGeach {g} (h : F ⊧ (Axioms.Geach g (.atom 0))) : F.IsGeachConvergent g := ⟨by
+ rintro x y z Rxy Rxz;
+ let V : Kripke.Valuat... | {
"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_5234c2258d5f_12 | 38712260ecbbcf05 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Kripke/AxiomGeach.lean | AxiomGeach | 12 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 2 | 7 | [
{
"theorem_name": "reflexive_of_validate_AxiomT",
"depth": 1,
"n_commands": 0,
"n_lines": 4,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n suffices F.IsGeachConvergent ⟨0, 0, 1, 0⟩ by infer_instance;\n apply isGeachConvergent_of_validate_axiomGeach... | [
{
"name": "isGeachConvergent_of_validate_axiomGeach",
"text": "lemma isGeachConvergent_of_validate_axiomGeach {g} (h : F ⊧ (Axioms.Geach g (.atom 0))) : F.IsGeachConvergent g := ⟨by\n rintro x y z Rxy Rxz;\n let V : Kripke.Valuation F := λ _ v => y ≺^[g.m] v;\n have : Satisfies ⟨F, V⟩ x (□^[g.j](◇^[g.n](... | [
{
"name": "coreflexive_of_validate_AxiomTc",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 6,
"n_chars": 223,
"n_subproofs": 1,
"n_tactics": 7,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": false,
... | 1 | module
public import Foundation.Modal.Kripke.Completeness
public import Foundation.Vorspiel.Rel.Coreflexive
@[expose] public section
namespace LO.Modal
namespace Kripke
variable {F : Frame}
namespace Frame
class IsGeachConvergent (F : Frame) (g : Axioms.Geach.Taple) where
gconv : ∀ ⦃x y z : F⦄, x ≺^[g.i] y → x... | module
public import Foundation.Modal.Kripke.Completeness
public import Foundation.Vorspiel.Rel.Coreflexive
@[expose] public section
namespace LO.Modal
namespace Kripke
variable {F : Frame}
namespace Frame
class IsGeachConvergent (F : Frame) (g : Axioms.Geach.Taple) where
gconv : ∀ ⦃x y z : F⦄, x ≺^[g.i] y → x... | @@ -171,8 +171,26 @@
open Formula (atom)
open Formula.Kripke
-lemma coreflexive_of_validate_AxiomTc (h : F ⊧ (Axioms.Tc (.atom 0))) : F.IsCoreflexive := sorry
+lemma isGeachConvergent_of_validate_axiomGeach {g} (h : F ⊧ (Axioms.Geach g (.atom 0))) : F.IsGeachConvergent g := ⟨by
+ rintro x y z Rxy Rxz;
+ let V : K... | {
"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_5234c2258d5f_13 | 9414a5b97573d8a2 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Kripke/AxiomGeach.lean | AxiomGeach | 13 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 2 | 7 | [
{
"theorem_name": "reflexive_of_validate_AxiomT",
"depth": 1,
"n_commands": 0,
"n_lines": 4,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n suffices F.IsGeachConvergent ⟨0, 0, 1, 0⟩ by infer_instance;\n apply isGeachConvergent_of_validate_axiomGeach... | [
{
"name": "isGeachConvergent_of_validate_axiomGeach",
"text": "lemma isGeachConvergent_of_validate_axiomGeach {g} (h : F ⊧ (Axioms.Geach g (.atom 0))) : F.IsGeachConvergent g := ⟨by\n rintro x y z Rxy Rxz;\n let V : Kripke.Valuation F := λ _ v => y ≺^[g.m] v;\n have : Satisfies ⟨F, V⟩ x (□^[g.j](◇^[g.n](... | [
{
"name": "confluent_of_validate_AxiomPoint2",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 6,
"n_chars": 245,
"n_subproofs": 1,
"n_tactics": 7,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": false... | 1 | module
public import Foundation.Modal.Kripke.Completeness
public import Foundation.Vorspiel.Rel.Coreflexive
@[expose] public section
namespace LO.Modal
namespace Kripke
variable {F : Frame}
namespace Frame
class IsGeachConvergent (F : Frame) (g : Axioms.Geach.Taple) where
gconv : ∀ ⦃x y z : F⦄, x ≺^[g.i] y → x... | module
public import Foundation.Modal.Kripke.Completeness
public import Foundation.Vorspiel.Rel.Coreflexive
@[expose] public section
namespace LO.Modal
namespace Kripke
variable {F : Frame}
namespace Frame
class IsGeachConvergent (F : Frame) (g : Axioms.Geach.Taple) where
gconv : ∀ ⦃x y z : F⦄, x ≺^[g.i] y → x... | @@ -171,8 +171,26 @@
open Formula (atom)
open Formula.Kripke
-lemma confluent_of_validate_AxiomPoint2 (h : F ⊧ (Axioms.Point2 (.atom 0))) : F.IsPiecewiseStronglyConvergent := sorry
+lemma isGeachConvergent_of_validate_axiomGeach {g} (h : F ⊧ (Axioms.Geach g (.atom 0))) : F.IsGeachConvergent g := ⟨by
+ rintro x y z... | {
"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_1c0bfe828b43_0 | b5c68f4c9d54cd07 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Basic/Syntax/Rew.lean | Rew | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 9 | 1 | [
{
"theorem_name": "eq_ball_iff",
"depth": 1,
"n_commands": 0,
"n_lines": 4,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n simp only [ball, eq_all_iff, eq_neg_iff, exists_and_left]; constructor\n · rintro ⟨φ', ⟨φ₁, rfl, φ₂, rfl, rfl⟩, rfl⟩; exact ⟨φ... | [
{
"name": "eq_all_iff",
"text": "lemma eq_all_iff {φ : Semiformula L ξ₁ n₁} {ψ : Semiformula L ξ₂ (n₂ + 1)} :\n ω ▹ φ = ∀⁰ ψ ↔ ∃ φ' : Semiformula L ξ₁ (n₁ + 1), ω.q ▹ φ' = ψ ∧ φ = ∀⁰ φ' := by\n cases φ using Semiformula.rec' <;> simp\n\n",
"fan_in": 1,
"n_lines": 5,
"n_chars": 249,
"n_su... | [
{
"name": "eq_ball_iff",
"fan_in": 0,
"n_deps_direct": 2,
"n_deps_transitive": 3,
"n_lines": 7,
"n_chars": 505,
"n_subproofs": 0,
"n_tactics": 7,
"cyclomatic": 4,
"n_automation": 12,
"n_rewrites": 0,
"n_structural": 4,
"automation_only": false,
"max_nesting": ... | 1 | module
public import Foundation.Vorspiel.Finset.Basic
public import Foundation.Syntax.Predicate.Rew
public import Foundation.FirstOrder.Basic.Syntax.Formula
@[expose] public section
/-!
# Rewriting Entailment
term/formula morphisms such as Rewritings, substitutions, and embs are handled by the structure `LO.FirstOr... | module
public import Foundation.Vorspiel.Finset.Basic
public import Foundation.Syntax.Predicate.Rew
public import Foundation.FirstOrder.Basic.Syntax.Formula
@[expose] public section
/-!
# Rewriting Entailment
term/formula morphisms such as Rewritings, substitutions, and embs are handled by the structure `LO.FirstOr... | @@ -175,6 +175,10 @@
ω ▹ φ = ψ₁ ⋎ ψ₂ ↔ ∃ φ₁ φ₂ : Semiformula L ξ₁ n₁, ω ▹ φ₁ = ψ₁ ∧ ω ▹ φ₂ = ψ₂ ∧ φ = φ₁ ⋎ φ₂ := by
cases φ using Semiformula.rec' <;> simp
+lemma eq_all_iff {φ : Semiformula L ξ₁ n₁} {ψ : Semiformula L ξ₂ (n₂ + 1)} :
+ ω ▹ φ = ∀⁰ ψ ↔ ∃ φ' : Semiformula L ξ₁ (n₁ + 1), ω.q ▹ φ' = ψ ∧ φ = ∀⁰ ... | {
"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_1c0bfe828b43_1 | 9e3ec961634cac5e | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Basic/Syntax/Rew.lean | Rew | 1 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 9 | 1 | [
{
"theorem_name": "eq_bexs_iff",
"depth": 1,
"n_commands": 0,
"n_lines": 4,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n simp only [bexs, eq_exs_iff, eq_and_iff, exists_and_left]; constructor\n · rintro ⟨φ', ⟨φ₁, rfl, φ₂, rfl, rfl⟩, rfl⟩; exact ⟨φ... | [
{
"name": "eq_exs_iff",
"text": "lemma eq_exs_iff {φ : Semiformula L ξ₁ n₁} {ψ : Semiformula L ξ₂ (n₂ + 1)} :\n ω ▹ φ = ∃⁰ ψ ↔ ∃ φ' : Semiformula L ξ₁ (n₁ + 1), ω.q ▹ φ' = ψ ∧ φ = ∃⁰ φ' := by\n cases φ using Semiformula.rec' <;> simp\n\n",
"fan_in": 1,
"n_lines": 5,
"n_chars": 249,
"n_su... | [
{
"name": "eq_bexs_iff",
"fan_in": 0,
"n_deps_direct": 2,
"n_deps_transitive": 2,
"n_lines": 7,
"n_chars": 505,
"n_subproofs": 0,
"n_tactics": 7,
"cyclomatic": 4,
"n_automation": 12,
"n_rewrites": 0,
"n_structural": 4,
"automation_only": false,
"max_nesting": ... | 1 | module
public import Foundation.Vorspiel.Finset.Basic
public import Foundation.Syntax.Predicate.Rew
public import Foundation.FirstOrder.Basic.Syntax.Formula
@[expose] public section
/-!
# Rewriting Entailment
term/formula morphisms such as Rewritings, substitutions, and embs are handled by the structure `LO.FirstOr... | module
public import Foundation.Vorspiel.Finset.Basic
public import Foundation.Syntax.Predicate.Rew
public import Foundation.FirstOrder.Basic.Syntax.Formula
@[expose] public section
/-!
# Rewriting Entailment
term/formula morphisms such as Rewritings, substitutions, and embs are handled by the structure `LO.FirstOr... | @@ -175,6 +175,10 @@
ω ▹ φ = ψ₁ ⋎ ψ₂ ↔ ∃ φ₁ φ₂ : Semiformula L ξ₁ n₁, ω ▹ φ₁ = ψ₁ ∧ ω ▹ φ₂ = ψ₂ ∧ φ = φ₁ ⋎ φ₂ := by
cases φ using Semiformula.rec' <;> simp
+lemma eq_exs_iff {φ : Semiformula L ξ₁ n₁} {ψ : Semiformula L ξ₂ (n₂ + 1)} :
+ ω ▹ φ = ∃⁰ ψ ↔ ∃ φ' : Semiformula L ξ₁ (n₁ + 1), ω.q ▹ φ' = ψ ∧ φ = ∃⁰ ... | {
"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_1c0bfe828b43_2 | 8680e5eae8bfffb2 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Basic/Syntax/Rew.lean | Rew | 2 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 9 | 2 | [
{
"theorem_name": "freeVariables_emb",
"depth": 1,
"n_commands": 0,
"n_lines": 7,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n ext x\n suffices x ∉ (Rewriting.emb φ : Semiformula L ξ n).freeVariables by simpa using this\n intro hx\n rcases fvar?... | [
{
"name": "fvar?_rew",
"text": "lemma fvar?_rew [DecidableEq ξ₁] [DecidableEq ξ₂]\n {φ : Semiformula L ξ₁ n₁} {ω : Rew L ξ₁ n₁ ξ₂ n₂}\n {x} (h : (ω ▹ φ).FVar? x) :\n (∃ i : Fin n₁, (ω #i).FVar? x) ∨ (∃ z : ξ₁, φ.FVar? z ∧ (ω &z).FVar? x) := by\n induction φ using rec' generalizing n₂\n case hveru... | [
{
"name": "freeVariables_emb",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 3,
"n_lines": 10,
"n_chars": 394,
"n_subproofs": 1,
"n_tactics": 7,
"cyclomatic": 2,
"n_automation": 2,
"n_rewrites": 0,
"n_structural": 4,
"automation_only": false,
"max_nest... | 1 | module
public import Foundation.Vorspiel.Finset.Basic
public import Foundation.Syntax.Predicate.Rew
public import Foundation.FirstOrder.Basic.Syntax.Formula
@[expose] public section
/-!
# Rewriting Entailment
term/formula morphisms such as Rewritings, substitutions, and embs are handled by the structure `LO.FirstOr... | module
public import Foundation.Vorspiel.Finset.Basic
public import Foundation.Syntax.Predicate.Rew
public import Foundation.FirstOrder.Basic.Syntax.Formula
@[expose] public section
/-!
# Rewriting Entailment
term/formula morphisms such as Rewritings, substitutions, and embs are handled by the structure `LO.FirstOr... | @@ -208,8 +208,72 @@
| ∃⁰ φ => ex φ (formulaRec verum falsum rel nrel and or all ex (Rewriting.free φ))
termination_by φ => φ.complexity
+lemma fvar?_rew [DecidableEq ξ₁] [DecidableEq ξ₂]
+ {φ : Semiformula L ξ₁ n₁} {ω : Rew L ξ₁ n₁ ξ₂ n₂}
+ {x} (h : (ω ▹ φ).FVar? x) :
+ (∃ i : Fin n₁, (ω #i).FVar? ... | {
"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_1c0bfe828b43_3 | ae885167d5ad7f90 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Basic/Syntax/Rew.lean | Rew | 3 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 9 | 3 | [
{
"theorem_name": "rew_eq_of_funEqOn₀",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": "\n rew_eq_of_funEqOn (fun x ↦ Fin.elim0 x) hf",
"n_chars": 47,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic":... | [
{
"name": "rew_eq_of_funEqOn",
"text": "lemma rew_eq_of_funEqOn [DecidableEq ξ₁] {ω₁ ω₂ : Rew L ξ₁ n₁ ξ₂ n₂} {φ : Semiformula L ξ₁ n₁}\n (hb : ∀ x, ω₁ #x = ω₂ #x) (hf : Function.funEqOn φ.FVar? (ω₁ ∘ Semiterm.fvar) (ω₂ ∘ Semiterm.fvar)) :\n ω₁ ▹ φ = ω₂ ▹ φ := by\n induction φ using rec' generalizing n₂... | [
{
"name": "rew_eq_of_funEqOn₀",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 3,
"n_lines": 5,
"n_chars": 289,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
"max_nest... | 1 | module
public import Foundation.Vorspiel.Finset.Basic
public import Foundation.Syntax.Predicate.Rew
public import Foundation.FirstOrder.Basic.Syntax.Formula
@[expose] public section
/-!
# Rewriting Entailment
term/formula morphisms such as Rewritings, substitutions, and embs are handled by the structure `LO.FirstOr... | module
public import Foundation.Vorspiel.Finset.Basic
public import Foundation.Syntax.Predicate.Rew
public import Foundation.FirstOrder.Basic.Syntax.Formula
@[expose] public section
/-!
# Rewriting Entailment
term/formula morphisms such as Rewritings, substitutions, and embs are handled by the structure `LO.FirstOr... | @@ -275,12 +275,46 @@
· simp at hi
· exact IsEmpty.elim inferInstance z
+lemma rew_eq_of_funEqOn [DecidableEq ξ₁] {ω₁ ω₂ : Rew L ξ₁ n₁ ξ₂ n₂} {φ : Semiformula L ξ₁ n₁}
+ (hb : ∀ x, ω₁ #x = ω₂ #x) (hf : Function.funEqOn φ.FVar? (ω₁ ∘ Semiterm.fvar) (ω₂ ∘ Semiterm.fvar)) :
+ ω₁ ▹ φ = ω₂ ▹ φ := 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_1c0bfe828b43_4 | 67d895fd58fbb228 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Basic/Syntax/Rew.lean | Rew | 4 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 9 | 3 | [
{
"theorem_name": "rew_eq_of_funEqOn₀",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": "\n rew_eq_of_funEqOn (fun x ↦ Fin.elim0 x) hf",
"n_chars": 47,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic":... | [
{
"name": "rew_eq_of_funEqOn",
"text": "lemma rew_eq_of_funEqOn [DecidableEq ξ₁] {ω₁ ω₂ : Rew L ξ₁ n₁ ξ₂ n₂} {φ : Semiformula L ξ₁ n₁}\n (hb : ∀ x, ω₁ #x = ω₂ #x) (hf : Function.funEqOn φ.FVar? (ω₁ ∘ Semiterm.fvar) (ω₂ ∘ Semiterm.fvar)) :\n ω₁ ▹ φ = ω₂ ▹ φ := by\n induction φ using rec' generalizing n₂... | [
{
"name": "rew_eq_self_of",
"fan_in": 2,
"n_deps_direct": 1,
"n_deps_transitive": 3,
"n_lines": 9,
"n_chars": 352,
"n_subproofs": 1,
"n_tactics": 7,
"cyclomatic": 1,
"n_automation": 2,
"n_rewrites": 0,
"n_structural": 3,
"automation_only": false,
"max_nesting"... | 1 | module
public import Foundation.Vorspiel.Finset.Basic
public import Foundation.Syntax.Predicate.Rew
public import Foundation.FirstOrder.Basic.Syntax.Formula
@[expose] public section
/-!
# Rewriting Entailment
term/formula morphisms such as Rewritings, substitutions, and embs are handled by the structure `LO.FirstOr... | module
public import Foundation.Vorspiel.Finset.Basic
public import Foundation.Syntax.Predicate.Rew
public import Foundation.FirstOrder.Basic.Syntax.Formula
@[expose] public section
/-!
# Rewriting Entailment
term/formula morphisms such as Rewritings, substitutions, and embs are handled by the structure `LO.FirstOr... | @@ -275,9 +275,42 @@
· simp at hi
· exact IsEmpty.elim inferInstance z
+lemma rew_eq_of_funEqOn [DecidableEq ξ₁] {ω₁ ω₂ : Rew L ξ₁ n₁ ξ₂ n₂} {φ : Semiformula L ξ₁ n₁}
+ (hb : ∀ x, ω₁ #x = ω₂ #x) (hf : Function.funEqOn φ.FVar? (ω₁ ∘ Semiterm.fvar) (ω₂ ∘ Semiterm.fvar)) :
+ ω₁ ▹ φ = ω₂ ▹ φ := 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_1c0bfe828b43_5 | 87ec241f919e0483 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Basic/Syntax/Rew.lean | Rew | 5 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 9 | 2 | [
{
"theorem_name": "senetnce_univCl'_eq_self",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " univCl'_eq_self_of _ (by simp)",
"n_chars": 31,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"... | [
{
"name": "univCl'_eq_self_of",
"text": "lemma univCl'_eq_self_of (φ : Proposition L) (h : φ.freeVariables = ∅) : φ.univCl' = φ := by\n have : φ.fvSup = 0 := by simp [fvSup, h]\n simp only [univCl']; rw [this]; simp\n\n",
"fan_in": 2,
"n_lines": 5,
"n_chars": 183,
"n_subproofs": 1,
"n_... | [
{
"name": "senetnce_univCl'_eq_self",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 4,
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"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"max... | 1 | module
public import Foundation.Vorspiel.Finset.Basic
public import Foundation.Syntax.Predicate.Rew
public import Foundation.FirstOrder.Basic.Syntax.Formula
@[expose] public section
/-!
# Rewriting Entailment
term/formula morphisms such as Rewritings, substitutions, and embs are handled by the structure `LO.FirstOr... | module
public import Foundation.Vorspiel.Finset.Basic
public import Foundation.Syntax.Predicate.Rew
public import Foundation.FirstOrder.Basic.Syntax.Formula
@[expose] public section
/-!
# Rewriting Entailment
term/formula morphisms such as Rewritings, substitutions, and embs are handled by the structure `LO.FirstOr... | @@ -349,8 +349,12 @@
@[simp] lemma rew_univCl' (φ : Proposition L) (ω : SyntacticRew L 0 0) :
ω ▹ φ.univCl' = φ.univCl' := rew_eq_self_of (by simp) (by simp [univCl', not_fvar?_fixitr_fvSup])
+lemma univCl'_eq_self_of (φ : Proposition L) (h : φ.freeVariables = ∅) : φ.univCl' = φ := by
+ have : φ.fvSup = 0 := b... | {
"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_1c0bfe828b43_6 | 8d0b7c28f47c0de9 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Basic/Syntax/Rew.lean | Rew | 6 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 9 | 7 | [
{
"theorem_name": "lMap_map",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " lMap_bind _ _ _",
"n_chars": 16,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrit... | [
{
"name": "lMap_bind",
"text": "lemma lMap_bind (b : Fin n₁ → Semiterm L₁ ξ₂ n₂) (e : ξ₁ → Semiterm L₁ ξ₂ n₂) (φ : Semiformula L₁ ξ₁ n₁) :\n lMap Φ (Rew.bind b e ▹ φ) = Rew.bind (Semiterm.lMap Φ ∘ b) (Semiterm.lMap Φ ∘ e) ▹ (lMap Φ φ) := by\n induction φ using rec' generalizing ξ₂ n₂ <;>\n simp [*, lMa... | [
{
"name": "lMap_map",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 4,
"n_chars": 208,
"n_subproofs": 0,
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"cyclomatic": 1,
"n_automation": 0,
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"n_structural": 0,
"automation_only": false,
"max_nesting": 1
... | 1 | module
public import Foundation.Vorspiel.Finset.Basic
public import Foundation.Syntax.Predicate.Rew
public import Foundation.FirstOrder.Basic.Syntax.Formula
@[expose] public section
/-!
# Rewriting Entailment
term/formula morphisms such as Rewritings, substitutions, and embs are handled by the structure `LO.FirstOr... | module
public import Foundation.Vorspiel.Finset.Basic
public import Foundation.Syntax.Predicate.Rew
public import Foundation.FirstOrder.Basic.Syntax.Formula
@[expose] public section
/-!
# Rewriting Entailment
term/formula morphisms such as Rewritings, substitutions, and embs are handled by the structure `LO.FirstOr... | @@ -425,8 +425,13 @@
variable {L : Language.{u}} {L₁ : Language.{u₁}} {L₂ : Language.{u₂}} {L₃ : Language.{u₃}} {ξ : Type*} {Φ : L₁ →ᵥ L₂}
+lemma lMap_bind (b : Fin n₁ → Semiterm L₁ ξ₂ n₂) (e : ξ₁ → Semiterm L₁ ξ₂ n₂) (φ : Semiformula L₁ ξ₁ n₁) :
+ lMap Φ (Rew.bind b e ▹ φ) = Rew.bind (Semiterm.lMap Φ ∘ b) (Sem... | {
"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_1c0bfe828b43_7 | a68ceaabed23343f | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Basic/Syntax/Rew.lean | Rew | 7 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 9 | 7 | [
{
"theorem_name": "lMap_map",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " lMap_bind _ _ _",
"n_chars": 16,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrit... | [
{
"name": "lMap_bind",
"text": "lemma lMap_bind (b : Fin n₁ → Semiterm L₁ ξ₂ n₂) (e : ξ₁ → Semiterm L₁ ξ₂ n₂) (φ : Semiformula L₁ ξ₁ n₁) :\n lMap Φ (Rew.bind b e ▹ φ) = Rew.bind (Semiterm.lMap Φ ∘ b) (Semiterm.lMap Φ ∘ e) ▹ (lMap Φ φ) := by\n induction φ using rec' generalizing ξ₂ n₂ <;>\n simp [*, lMa... | [
{
"name": "lMap_rewrite",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 5,
"n_chars": 238,
"n_subproofs": 0,
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"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"max_nesting": 2... | 1 | module
public import Foundation.Vorspiel.Finset.Basic
public import Foundation.Syntax.Predicate.Rew
public import Foundation.FirstOrder.Basic.Syntax.Formula
@[expose] public section
/-!
# Rewriting Entailment
term/formula morphisms such as Rewritings, substitutions, and embs are handled by the structure `LO.FirstOr... | module
public import Foundation.Vorspiel.Finset.Basic
public import Foundation.Syntax.Predicate.Rew
public import Foundation.FirstOrder.Basic.Syntax.Formula
@[expose] public section
/-!
# Rewriting Entailment
term/formula morphisms such as Rewritings, substitutions, and embs are handled by the structure `LO.FirstOr... | @@ -425,8 +425,14 @@
variable {L : Language.{u}} {L₁ : Language.{u₁}} {L₂ : Language.{u₂}} {L₃ : Language.{u₃}} {ξ : Type*} {Φ : L₁ →ᵥ L₂}
+lemma lMap_bind (b : Fin n₁ → Semiterm L₁ ξ₂ n₂) (e : ξ₁ → Semiterm L₁ ξ₂ n₂) (φ : Semiformula L₁ ξ₁ n₁) :
+ lMap Φ (Rew.bind b e ▹ φ) = Rew.bind (Semiterm.lMap Φ ∘ b) (Sem... | {
"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_1c0bfe828b43_8 | a580dcfa6cc39a2c | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Basic/Syntax/Rew.lean | Rew | 8 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 9 | 7 | [
{
"theorem_name": "lMap_map",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " lMap_bind _ _ _",
"n_chars": 16,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrit... | [
{
"name": "lMap_bind",
"text": "lemma lMap_bind (b : Fin n₁ → Semiterm L₁ ξ₂ n₂) (e : ξ₁ → Semiterm L₁ ξ₂ n₂) (φ : Semiformula L₁ ξ₁ n₁) :\n lMap Φ (Rew.bind b e ▹ φ) = Rew.bind (Semiterm.lMap Φ ∘ b) (Semiterm.lMap Φ ∘ e) ▹ (lMap Φ φ) := by\n induction φ using rec' generalizing ξ₂ n₂ <;>\n simp [*, lMa... | [
{
"name": "lMap_subst",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 4,
"n_chars": 168,
"n_subproofs": 0,
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"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
"max_nesting": 1
... | 1 | module
public import Foundation.Vorspiel.Finset.Basic
public import Foundation.Syntax.Predicate.Rew
public import Foundation.FirstOrder.Basic.Syntax.Formula
@[expose] public section
/-!
# Rewriting Entailment
term/formula morphisms such as Rewritings, substitutions, and embs are handled by the structure `LO.FirstOr... | module
public import Foundation.Vorspiel.Finset.Basic
public import Foundation.Syntax.Predicate.Rew
public import Foundation.FirstOrder.Basic.Syntax.Formula
@[expose] public section
/-!
# Rewriting Entailment
term/formula morphisms such as Rewritings, substitutions, and embs are handled by the structure `LO.FirstOr... | @@ -425,8 +425,13 @@
variable {L : Language.{u}} {L₁ : Language.{u₁}} {L₂ : Language.{u₂}} {L₃ : Language.{u₃}} {ξ : Type*} {Φ : L₁ →ᵥ L₂}
+lemma lMap_bind (b : Fin n₁ → Semiterm L₁ ξ₂ n₂) (e : ξ₁ → Semiterm L₁ ξ₂ n₂) (φ : Semiformula L₁ ξ₁ n₁) :
+ lMap Φ (Rew.bind b e ▹ φ) = Rew.bind (Semiterm.lMap Φ ∘ b) (Sem... | {
"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_1c0bfe828b43_9 | 7d0205808d319e1e | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Basic/Syntax/Rew.lean | Rew | 9 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 9 | 7 | [
{
"theorem_name": "lMap_map",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " lMap_bind _ _ _",
"n_chars": 16,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrit... | [
{
"name": "lMap_bind",
"text": "lemma lMap_bind (b : Fin n₁ → Semiterm L₁ ξ₂ n₂) (e : ξ₁ → Semiterm L₁ ξ₂ n₂) (φ : Semiformula L₁ ξ₁ n₁) :\n lMap Φ (Rew.bind b e ▹ φ) = Rew.bind (Semiterm.lMap Φ ∘ b) (Semiterm.lMap Φ ∘ e) ▹ (lMap Φ φ) := by\n induction φ using rec' generalizing ξ₂ n₂ <;>\n simp [*, lMa... | [
{
"name": "lMap_shift",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 3,
"n_chars": 139,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
"max_nesting": 1
... | 1 | module
public import Foundation.Vorspiel.Finset.Basic
public import Foundation.Syntax.Predicate.Rew
public import Foundation.FirstOrder.Basic.Syntax.Formula
@[expose] public section
/-!
# Rewriting Entailment
term/formula morphisms such as Rewritings, substitutions, and embs are handled by the structure `LO.FirstOr... | module
public import Foundation.Vorspiel.Finset.Basic
public import Foundation.Syntax.Predicate.Rew
public import Foundation.FirstOrder.Basic.Syntax.Formula
@[expose] public section
/-!
# Rewriting Entailment
term/formula morphisms such as Rewritings, substitutions, and embs are handled by the structure `LO.FirstOr... | @@ -425,8 +425,13 @@
variable {L : Language.{u}} {L₁ : Language.{u₁}} {L₂ : Language.{u₂}} {L₃ : Language.{u₃}} {ξ : Type*} {Φ : L₁ →ᵥ L₂}
-lemma lMap_shift (φ : Semiproposition L₁ n) : lMap Φ (@Rew.shift L₁ n ▹ φ) = @Rew.shift L₂ n ▹ lMap Φ φ := sorry
+lemma lMap_bind (b : Fin n₁ → Semiterm L₁ ξ₂ n₂) (e : ξ₁ → Se... | {
"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_1c0bfe828b43_10 | a595fc20e4ec97ec | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Basic/Syntax/Rew.lean | Rew | 10 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 9 | 7 | [
{
"theorem_name": "lMap_map",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " lMap_bind _ _ _",
"n_chars": 16,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrit... | [
{
"name": "lMap_bind",
"text": "lemma lMap_bind (b : Fin n₁ → Semiterm L₁ ξ₂ n₂) (e : ξ₁ → Semiterm L₁ ξ₂ n₂) (φ : Semiformula L₁ ξ₁ n₁) :\n lMap Φ (Rew.bind b e ▹ φ) = Rew.bind (Semiterm.lMap Φ ∘ b) (Semiterm.lMap Φ ∘ e) ▹ (lMap Φ φ) := by\n induction φ using rec' generalizing ξ₂ n₂ <;>\n simp [*, lMa... | [
{
"name": "lMap_free",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 4,
"n_chars": 202,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"max_nesting": 2
... | 1 | module
public import Foundation.Vorspiel.Finset.Basic
public import Foundation.Syntax.Predicate.Rew
public import Foundation.FirstOrder.Basic.Syntax.Formula
@[expose] public section
/-!
# Rewriting Entailment
term/formula morphisms such as Rewritings, substitutions, and embs are handled by the structure `LO.FirstOr... | module
public import Foundation.Vorspiel.Finset.Basic
public import Foundation.Syntax.Predicate.Rew
public import Foundation.FirstOrder.Basic.Syntax.Formula
@[expose] public section
/-!
# Rewriting Entailment
term/formula morphisms such as Rewritings, substitutions, and embs are handled by the structure `LO.FirstOr... | @@ -425,8 +425,14 @@
variable {L : Language.{u}} {L₁ : Language.{u₁}} {L₂ : Language.{u₂}} {L₃ : Language.{u₃}} {ξ : Type*} {Φ : L₁ →ᵥ L₂}
-lemma lMap_free (φ : Semiproposition L₁ (n + 1)) : lMap Φ (@Rew.free L₁ n ▹ φ) = @Rew.free L₂ n ▹ lMap Φ φ := sorry
+lemma lMap_bind (b : Fin n₁ → Semiterm L₁ ξ₂ n₂) (e : ξ₁ →... | {
"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_1c0bfe828b43_11 | 31f24012152d71ee | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Basic/Syntax/Rew.lean | Rew | 11 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 9 | 7 | [
{
"theorem_name": "lMap_map",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " lMap_bind _ _ _",
"n_chars": 16,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrit... | [
{
"name": "lMap_bind",
"text": "lemma lMap_bind (b : Fin n₁ → Semiterm L₁ ξ₂ n₂) (e : ξ₁ → Semiterm L₁ ξ₂ n₂) (φ : Semiformula L₁ ξ₁ n₁) :\n lMap Φ (Rew.bind b e ▹ φ) = Rew.bind (Semiterm.lMap Φ ∘ b) (Semiterm.lMap Φ ∘ e) ▹ (lMap Φ φ) := by\n induction φ using rec' generalizing ξ₂ n₂ <;>\n simp [*, lMa... | [
{
"name": "lMap_fix",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 5,
"n_chars": 232,
"n_subproofs": 0,
"n_tactics": 6,
"cyclomatic": 3,
"n_automation": 2,
"n_rewrites": 1,
"n_structural": 2,
"automation_only": false,
"max_nesting": 2
... | 1 | module
public import Foundation.Vorspiel.Finset.Basic
public import Foundation.Syntax.Predicate.Rew
public import Foundation.FirstOrder.Basic.Syntax.Formula
@[expose] public section
/-!
# Rewriting Entailment
term/formula morphisms such as Rewritings, substitutions, and embs are handled by the structure `LO.FirstOr... | module
public import Foundation.Vorspiel.Finset.Basic
public import Foundation.Syntax.Predicate.Rew
public import Foundation.FirstOrder.Basic.Syntax.Formula
@[expose] public section
/-!
# Rewriting Entailment
term/formula morphisms such as Rewritings, substitutions, and embs are handled by the structure `LO.FirstOr... | @@ -425,8 +425,15 @@
variable {L : Language.{u}} {L₁ : Language.{u₁}} {L₂ : Language.{u₂}} {L₃ : Language.{u₃}} {ξ : Type*} {Φ : L₁ →ᵥ L₂}
-lemma lMap_fix (φ : Semiproposition L₁ n) : lMap Φ (@Rew.fix L₁ n ▹ φ) = @Rew.fix L₂ n ▹ lMap Φ φ := sorry
+lemma lMap_bind (b : Fin n₁ → Semiterm L₁ ξ₂ n₂) (e : ξ₁ → Semiterm... | {
"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_1c0bfe828b43_12 | b04c363edbaea4da | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Basic/Syntax/Rew.lean | Rew | 12 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 9 | 7 | [
{
"theorem_name": "lMap_map",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " lMap_bind _ _ _",
"n_chars": 16,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrit... | [
{
"name": "lMap_bind",
"text": "lemma lMap_bind (b : Fin n₁ → Semiterm L₁ ξ₂ n₂) (e : ξ₁ → Semiterm L₁ ξ₂ n₂) (φ : Semiformula L₁ ξ₁ n₁) :\n lMap Φ (Rew.bind b e ▹ φ) = Rew.bind (Semiterm.lMap Φ ∘ b) (Semiterm.lMap Φ ∘ e) ▹ (lMap Φ φ) := by\n induction φ using rec' generalizing ξ₂ n₂ <;>\n simp [*, lMa... | [
{
"name": "lMap_emb",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 4,
"n_chars": 179,
"n_subproofs": 0,
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"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
"max_nesting": 1
... | 1 | module
public import Foundation.Vorspiel.Finset.Basic
public import Foundation.Syntax.Predicate.Rew
public import Foundation.FirstOrder.Basic.Syntax.Formula
@[expose] public section
/-!
# Rewriting Entailment
term/formula morphisms such as Rewritings, substitutions, and embs are handled by the structure `LO.FirstOr... | module
public import Foundation.Vorspiel.Finset.Basic
public import Foundation.Syntax.Predicate.Rew
public import Foundation.FirstOrder.Basic.Syntax.Formula
@[expose] public section
/-!
# Rewriting Entailment
term/formula morphisms such as Rewritings, substitutions, and embs are handled by the structure `LO.FirstOr... | @@ -425,8 +425,13 @@
variable {L : Language.{u}} {L₁ : Language.{u₁}} {L₂ : Language.{u₂}} {L₃ : Language.{u₃}} {ξ : Type*} {Φ : L₁ →ᵥ L₂}
+lemma lMap_bind (b : Fin n₁ → Semiterm L₁ ξ₂ n₂) (e : ξ₁ → Semiterm L₁ ξ₂ n₂) (φ : Semiformula L₁ ξ₁ n₁) :
+ lMap Φ (Rew.bind b e ▹ φ) = Rew.bind (Semiterm.lMap Φ ∘ b) (Sem... | {
"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_1c0bfe828b43_13 | f65ac0186faaf0dd | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Basic/Syntax/Rew.lean | Rew | 13 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 9 | 7 | [
{
"theorem_name": "lMap_map",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " lMap_bind _ _ _",
"n_chars": 16,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrit... | [
{
"name": "lMap_bind",
"text": "lemma lMap_bind (b : Fin n₁ → Semiterm L₁ ξ₂ n₂) (e : ξ₁ → Semiterm L₁ ξ₂ n₂) (φ : Semiformula L₁ ξ₁ n₁) :\n lMap Φ (Rew.bind b e ▹ φ) = Rew.bind (Semiterm.lMap Φ ∘ b) (Semiterm.lMap Φ ∘ e) ▹ (lMap Φ φ) := by\n induction φ using rec' generalizing ξ₂ n₂ <;>\n simp [*, lMa... | [
{
"name": "lMap_rewriteMap",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 2,
"n_lines": 5,
"n_chars": 260,
"n_subproofs": 0,
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"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"max_nesting"... | 2 | module
public import Foundation.Vorspiel.Finset.Basic
public import Foundation.Syntax.Predicate.Rew
public import Foundation.FirstOrder.Basic.Syntax.Formula
@[expose] public section
/-!
# Rewriting Entailment
term/formula morphisms such as Rewritings, substitutions, and embs are handled by the structure `LO.FirstOr... | module
public import Foundation.Vorspiel.Finset.Basic
public import Foundation.Syntax.Predicate.Rew
public import Foundation.FirstOrder.Basic.Syntax.Formula
@[expose] public section
/-!
# Rewriting Entailment
term/formula morphisms such as Rewritings, substitutions, and embs are handled by the structure `LO.FirstOr... | @@ -425,8 +425,14 @@
variable {L : Language.{u}} {L₁ : Language.{u₁}} {L₂ : Language.{u₂}} {L₃ : Language.{u₃}} {ξ : Type*} {Φ : L₁ →ᵥ L₂}
+lemma lMap_bind (b : Fin n₁ → Semiterm L₁ ξ₂ n₂) (e : ξ₁ → Semiterm L₁ ξ₂ n₂) (φ : Semiformula L₁ ξ₁ n₁) :
+ lMap Φ (Rew.bind b e ▹ φ) = Rew.bind (Semiterm.lMap Φ ∘ b) (Sem... | {
"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_4d5fa2c55d0a_0 | 7ca2669dc1cc444e | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Bootstrapping/Syntax/Term/Typed.lean | Typed | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 3 | 1 | [
{
"theorem_name": "func_inj_iff",
"depth": 1,
"n_commands": 0,
"n_lines": 4,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n simp only [func, Semiterm.ext_iff, qqFunc_inj, quote_func_inj, true_and, and_congr_right_iff]\n rintro rfl\n symm; exact Sem... | [
{
"name": "SemitermVec.val_inj",
"text": "lemma SemitermVec.val_inj (v₁ v₂ : SemitermVec V L k n) : v₁ = v₂ ↔ v₁.val = v₂.val := by\n induction k\n · simp [Matrix.empty_eq]\n case succ k ih =>\n simp [← Semiterm.ext_iff, ←ih, Matrix.eq_iff_eq_vecHead_of_eq_vecTail]\n\n",
"fan_in": 1,
"... | [
{
"name": "func_inj_iff",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 6,
"n_chars": 310,
"n_subproofs": 0,
"n_tactics": 5,
"cyclomatic": 2,
"n_automation": 2,
"n_rewrites": 0,
"n_structural": 2,
"automation_only": false,
"max_nesting": ... | 1 | module
public import Foundation.FirstOrder.Bootstrapping.Syntax.Term.Functions
@[expose] public section
/-!
# Typed Formalized IsSemiterm/Term
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
noncomputable def matrixToVec (v : Fin k → V) : V := Matrix.foldr (fun t ... | module
public import Foundation.FirstOrder.Bootstrapping.Syntax.Term.Functions
@[expose] public section
/-!
# Typed Formalized IsSemiterm/Term
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
noncomputable def matrixToVec (v : Fin k → V) : V := Matrix.foldr (fun t ... | @@ -76,6 +76,12 @@
@[simp] lemma SemitermVec.val_succ (v : SemitermVec V L (k + 1) n) :
SemitermVec.val (v : SemitermVec V L (k + 1) n) = (Matrix.vecHead v).val ∷ SemitermVec.val (Matrix.vecTail v) := by rfl
+lemma SemitermVec.val_inj (v₁ v₂ : SemitermVec V L k n) : v₁ = v₂ ↔ v₁.val = v₂.val := 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_41fd8a024681_0 | 4696bcfadc7c55a1 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Kripke/Algebra.lean | Algebra | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 2 | 1 | [
{
"theorem_name": "algebraic_validOnFrame_iff",
"depth": 1,
"n_commands": 0,
"n_lines": 21,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n constructor;\n . intro H V;\n apply le_antisymm;\n . apply algebraic_validOnFrame_imp.1;\n intro V ... | [
{
"name": "algebraic_validOnFrame_imp",
"text": "lemma algebraic_validOnFrame_imp : F ⊧ φ 🡒 ψ ↔ (∀ V : ℕ → F⁺, (V ⊩ φ) ≤ (V ⊩ ψ)) := by\n constructor;\n . intro H V x h;\n have := @H V x;\n dsimp [Semantics.Models, Satisfies] at this;\n grind;\n . intro H V x h;\n have := @H V x;\n grind;... | [
{
"name": "algebraic_validOnFrame_iff",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 23,
"n_chars": 641,
"n_subproofs": 0,
"n_tactics": 41,
"cyclomatic": 3,
"n_automation": 2,
"n_rewrites": 0,
"n_structural": 18,
"automation_only": false,
... | 1 | module
public import Foundation.Modal.Algebra.Basic
@[expose] public section
namespace LO.Modal
namespace Kripke
namespace Frame
abbrev dual (F : Kripke.Frame) := Set F.World
postfix:max "⁺" => Frame.dual
variable {F : Kripke.Frame} {A B : F⁺} {x : F.World}
def box (F : Kripke.Frame) : F⁺ → F⁺ := λ A => { x | ∀... | module
public import Foundation.Modal.Algebra.Basic
@[expose] public section
namespace LO.Modal
namespace Kripke
namespace Frame
abbrev dual (F : Kripke.Frame) := Set F.World
postfix:max "⁺" => Frame.dual
variable {F : Kripke.Frame} {A B : F⁺} {x : F.World}
def box (F : Kripke.Frame) : F⁺ → F⁺ := λ A => { x | ∀... | @@ -50,17 +50,37 @@
| himp φ ψ ihφ ihψ => simp [Satisfies, ihφ x, ihψ x];
| hbox φ ihφ => simp [Satisfies, ihφ, Frame.mem_def_box];
+lemma algebraic_validOnFrame_imp : F ⊧ φ 🡒 ψ ↔ (∀ V : ℕ → F⁺, (V ⊩ φ) ≤ (V ⊩ ψ)) := by
+ constructor;
+ . intro H V x h;
+ have := @H V x;
+ dsimp [Semantics.Models, Sati... | {
"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_41fd8a024681_1 | f254eaeb846133a2 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Kripke/Algebra.lean | Algebra | 1 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 2 | 1 | [
{
"theorem_name": "algebraic_validOnFrame",
"depth": 1,
"n_commands": 0,
"n_lines": 8,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n have := algebraic_validOnFrame_iff (F := F) (φ := φ) (ψ := ⊤);\n simp_all only [\n ValidOnFrame.models_iff, Form... | [
{
"name": "algebraic_validOnFrame_iff",
"text": "lemma algebraic_validOnFrame_iff : F ⊧ φ 🡘 ψ ↔ (∀ V : ℕ → F⁺, (V ⊩ φ) = (V ⊩ ψ)) := by\n constructor;\n . intro H V;\n apply le_antisymm;\n . apply algebraic_validOnFrame_imp.1;\n intro V x;\n apply (Satisfies.iff_def.mp $ H V x).1;\n . ... | [
{
"name": "algebraic_validOnFrame",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 2,
"n_lines": 10,
"n_chars": 391,
"n_subproofs": 1,
"n_tactics": 13,
"cyclomatic": 3,
"n_automation": 2,
"n_rewrites": 0,
"n_structural": 2,
"automation_only": false,
"ma... | 2 | module
public import Foundation.Modal.Algebra.Basic
@[expose] public section
namespace LO.Modal
namespace Kripke
namespace Frame
abbrev dual (F : Kripke.Frame) := Set F.World
postfix:max "⁺" => Frame.dual
variable {F : Kripke.Frame} {A B : F⁺} {x : F.World}
def box (F : Kripke.Frame) : F⁺ → F⁺ := λ A => { x | ∀... | module
public import Foundation.Modal.Algebra.Basic
@[expose] public section
namespace LO.Modal
namespace Kripke
namespace Frame
abbrev dual (F : Kripke.Frame) := Set F.World
postfix:max "⁺" => Frame.dual
variable {F : Kripke.Frame} {A B : F⁺} {x : F.World}
def box (F : Kripke.Frame) : F⁺ → F⁺ := λ A => { x | ∀... | @@ -60,8 +60,37 @@
have := @H V x;
grind;
-lemma algebraic_validOnFrame : F ⊧ φ ↔ (∀ V : ℕ → F⁺, (V ⊩ φ) = ⊤) := sorry
+lemma algebraic_validOnFrame_iff : F ⊧ φ 🡘 ψ ↔ (∀ V : ℕ → F⁺, (V ⊩ φ) = (V ⊩ ψ)) := by
+ constructor;
+ . intro H V;
+ apply le_antisymm;
+ . apply algebraic_validOnFrame_imp.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_33d9bdc3496f_0 | 33dab3a98645e67b | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Kripke/Root.lean | Root | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 2 | 1 | [
{
"theorem_name": "connected'",
"depth": 1,
"n_commands": 0,
"n_lines": 6,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n rintro x y nexy;\n rcases F.connected x y with (Rxy | rfl | Ryx);\n . tauto;\n . contradiction;\n . tauto;",
"n_chars": ... | [
{
"name": "connected",
"text": "lemma connected [F.IsConnected] : ∀ x y : F.World, x ≺ y ∨ x = y ∨ y ≺ x := by grind [Std.Trichotomous.trichotomous (r := F.Rel)]\n",
"fan_in": 1,
"n_lines": 2,
"n_chars": 140,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 1,
... | [
{
"name": "connected'",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 8,
"n_chars": 202,
"n_subproofs": 0,
"n_tactics": 11,
"cyclomatic": 3,
"n_automation": 3,
"n_rewrites": 0,
"n_structural": 2,
"automation_only": false,
"max_nesting": 2... | 1 | module
public import Foundation.Modal.Kripke.Preservation
public import Foundation.Modal.Kripke.Asymmetric
public import Foundation.Modal.Kripke.AxiomWeakPoint2
public import Foundation.Modal.Kripke.AxiomPoint3
@[expose] public section
namespace LO.Modal
namespace Kripke
namespace Frame
variable {F : Kripke.Frame... | module
public import Foundation.Modal.Kripke.Preservation
public import Foundation.Modal.Kripke.Asymmetric
public import Foundation.Modal.Kripke.AxiomWeakPoint2
public import Foundation.Modal.Kripke.AxiomPoint3
@[expose] public section
namespace LO.Modal
namespace Kripke
namespace Frame
variable {F : Kripke.Frame... | @@ -21,7 +21,13 @@
protected abbrev IsStronglyConvergent (F : Frame) := _root_.IsStronglyConvergent F.Rel
protected abbrev IsStronglyConnected (F : Frame) := _root_.Std.Total F.Rel
protected abbrev IsConnected (F : Frame) := Std.Trichotomous F.Rel
-lemma connected' [F.IsConnected] : ∀ x y : F.World, x ≠ y → x ≺ y ∨ ... | {
"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_33d9bdc3496f_1 | e98a3ba04ff85d61 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Kripke/Root.lean | Root | 1 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 2 | 1 | [
{
"theorem_name": "modal_equivalent_at_root",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by apply modal_equivalent",
"n_chars": 26,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_aut... | [
{
"name": "modal_equivalent",
"text": "lemma modal_equivalent (r : M.World) (x : M↾r) : ModalEquivalent (M₁ := M↾r) (M₂ := M) x x.1\n := by apply Model.pointGenerate.pMorphism M r |>.modal_equivalence;\n\n",
"fan_in": 1,
"n_lines": 4,
"n_chars": 172,
"n_subproofs": 0,
"n_tactics": 2,
... | [
{
"name": "modal_equivalent_at_root",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 4,
"n_chars": 143,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": false,
"ma... | 1 | module
public import Foundation.Modal.Kripke.Preservation
public import Foundation.Modal.Kripke.Asymmetric
public import Foundation.Modal.Kripke.AxiomWeakPoint2
public import Foundation.Modal.Kripke.AxiomPoint3
@[expose] public section
namespace LO.Modal
namespace Kripke
namespace Frame
variable {F : Kripke.Frame... | module
public import Foundation.Modal.Kripke.Preservation
public import Foundation.Modal.Kripke.Asymmetric
public import Foundation.Modal.Kripke.AxiomWeakPoint2
public import Foundation.Modal.Kripke.AxiomPoint3
@[expose] public section
namespace LO.Modal
namespace Kripke
namespace Frame
variable {F : Kripke.Frame... | @@ -190,8 +190,11 @@
variable [M.IsTransitive]
+lemma modal_equivalent (r : M.World) (x : M↾r) : ModalEquivalent (M₁ := M↾r) (M₂ := M) x x.1
+ := by apply Model.pointGenerate.pMorphism M r |>.modal_equivalence;
+
lemma modal_equivalent_at_root (r : M.World) : ModalEquivalent (M₁ := M↾r) (M₂ := M) ⟨r, by grind⟩ 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_884b6d6ae831_0 | 7a50612535be8df5 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/SetTheory/Ordinal.lean | Ordinal | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 12 | 2 | [
{
"theorem_name": "ssubset_iff",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": "\n ⟨mem_of_ssubset, fun hαβ ↦ ⟨hβ.transitive _ hαβ, ne_of_mem hαβ⟩⟩",
"n_chars": 84,
"n_subproofs": 0,
"n_tactics": 1,
... | [
{
"name": "mem_of_ssubset",
"text": "lemma mem_of_ssubset [hα : IsOrdinal α] [hβ : IsOrdinal β] : α ⊊ β → α ∈ β := by\n intro ss\n have : ∃ γ, (γ ∈ β ∧ γ ∉ α) ∧ ∀ δ ∈ β, δ ∉ α → δ ∉ γ := by\n have : IsNonempty (β \\ α) := (isNonempty_sdiff_of_ssubset ss)\n simpa using foundation (β \\ α)\n rcases t... | [
{
"name": "ssubset_iff",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 4,
"n_chars": 185,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
"max_nesting": 2... | 1 | module
public import Foundation.FirstOrder.SetTheory.Z
@[expose] public section
/-!
# Ordinals and transitive sets
reference: Ralf Schindler, "Set Theory, Exploring Independence and Truth" [Sch14]
-/
namespace LO.FirstOrder.SetTheory
variable {V : Type*} [SetStructure V] [Nonempty V] [V↓[ℒₛₑₜ] ⊧* 𝗭]
/-! ### Tran... | module
public import Foundation.FirstOrder.SetTheory.Z
@[expose] public section
/-!
# Ordinals and transitive sets
reference: Ralf Schindler, "Set Theory, Exploring Independence and Truth" [Sch14]
-/
namespace LO.FirstOrder.SetTheory
variable {V : Type*} [SetStructure V] [Nonempty V] [V↓[ℒₛₑₜ] ⊧* 𝗭]
/-! ### Tran... | @@ -148,14 +148,42 @@
case zero => simp [zero_def]
case succ => intro α hα H; exact H.succ
-@[grind =] lemma ssubset_iff [hα : IsOrdinal α] [hβ : IsOrdinal β] : α ⊊ β ↔ α ∈ β := sorry
+lemma mem_of_ssubset [hα : IsOrdinal α] [hβ : IsOrdinal β] : α ⊊ β → α ∈ β := by
+ intro ss
+ have : ∃ γ, (γ ∈ β ∧ γ ∉ α) ∧ ∀... | {
"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_884b6d6ae831_1 | 4ae9ad860942b1c7 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/SetTheory/Ordinal.lean | Ordinal | 1 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 12 | 1 | [
{
"theorem_name": "subset_or_supset",
"depth": 1,
"n_commands": 0,
"n_lines": 49,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n by_contra! Aαβ\n let C : V := {α' ∈ succ α ; ∃ β, IsOrdinal β ∧ ¬α' ⊆ β ∧ ¬β ⊆ α'}\n have hαC : α ∈ C := by\n simp o... | [
{
"name": "union",
"text": "lemma union {x y : V} [hx : IsTransitive x] [hy : IsTransitive y] : IsTransitive (x ∪ y) := ⟨by\n simp only [mem_union_iff]\n rintro z (hzx | hzy)\n · exact subset_trans (hx.transitive z hzx) (by simp)\n · exact subset_trans (hy.transitive z hzy) (by simp)⟩\n\n",
"fan_in"... | [
{
"name": "subset_or_supset",
"fan_in": 1,
"n_deps_direct": 5,
"n_deps_transitive": 6,
"n_lines": 51,
"n_chars": 2669,
"n_subproofs": 21,
"n_tactics": 50,
"cyclomatic": 7,
"n_automation": 21,
"n_rewrites": 3,
"n_structural": 13,
"automation_only": false,
"max_... | 2 | module
public import Foundation.FirstOrder.SetTheory.Z
@[expose] public section
/-!
# Ordinals and transitive sets
reference: Ralf Schindler, "Set Theory, Exploring Independence and Truth" [Sch14]
-/
namespace LO.FirstOrder.SetTheory
variable {V : Type*} [SetStructure V] [Nonempty V] [V↓[ℒₛₑₜ] ⊧* 𝗭]
/-! ### Tran... | module
public import Foundation.FirstOrder.SetTheory.Z
@[expose] public section
/-!
# Ordinals and transitive sets
reference: Ralf Schindler, "Set Theory, Exploring Independence and Truth" [Sch14]
-/
namespace LO.FirstOrder.SetTheory
variable {V : Type*} [SetStructure V] [Nonempty V] [V↓[ℒₛₑₜ] ⊧* 𝗭]
/-! ### Tran... | @@ -49,6 +49,12 @@
intro x hx ih
exact ih.succ
+lemma union {x y : V} [hx : IsTransitive x] [hy : IsTransitive y] : IsTransitive (x ∪ y) := ⟨by
+ simp only [mem_union_iff]
+ rintro z (hzx | hzy)
+ · exact subset_trans (hx.transitive z hzx) (by simp)
+ · exact subset_trans (hy.transitive z hzy) (by 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_884b6d6ae831_2 | c64e12e96b1b597d | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/SetTheory/Ordinal.lean | Ordinal | 2 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 12 | 2 | [
{
"theorem_name": "ssubset_iff",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": "\n ⟨mem_of_ssubset, fun hαβ ↦ ⟨hβ.transitive _ hαβ, ne_of_mem hαβ⟩⟩",
"n_chars": 84,
"n_subproofs": 0,
"n_tactics": 1,
... | [
{
"name": "mem_of_ssubset",
"text": "lemma mem_of_ssubset [hα : IsOrdinal α] [hβ : IsOrdinal β] : α ⊊ β → α ∈ β := by\n intro ss\n have : ∃ γ, (γ ∈ β ∧ γ ∉ α) ∧ ∀ δ ∈ β, δ ∉ α → δ ∉ γ := by\n have : IsNonempty (β \\ α) := (isNonempty_sdiff_of_ssubset ss)\n simpa using foundation (β \\ α)\n rcases t... | [
{
"name": "sUnion",
"fan_in": 0,
"n_deps_direct": 6,
"n_deps_transitive": 10,
"n_lines": 9,
"n_chars": 391,
"n_subproofs": 2,
"n_tactics": 6,
"cyclomatic": 2,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 2,
"automation_only": false,
"max_nesting": 6
}... | 3 | module
public import Foundation.FirstOrder.SetTheory.Z
@[expose] public section
/-!
# Ordinals and transitive sets
reference: Ralf Schindler, "Set Theory, Exploring Independence and Truth" [Sch14]
-/
namespace LO.FirstOrder.SetTheory
variable {V : Type*} [SetStructure V] [Nonempty V] [V↓[ℒₛₑₜ] ⊧* 𝗭]
/-! ### Tran... | module
public import Foundation.FirstOrder.SetTheory.Z
@[expose] public section
/-!
# Ordinals and transitive sets
reference: Ralf Schindler, "Set Theory, Exploring Independence and Truth" [Sch14]
-/
namespace LO.FirstOrder.SetTheory
variable {V : Type*} [SetStructure V] [Nonempty V] [V↓[ℒₛₑₜ] ⊧* 𝗭]
/-! ### Tran... | @@ -154,14 +154,42 @@
case zero => simp [zero_def]
case succ => intro α hα H; exact H.succ
-@[grind =] lemma ssubset_iff [hα : IsOrdinal α] [hβ : IsOrdinal β] : α ⊊ β ↔ α ∈ β := sorry
+lemma mem_of_ssubset [hα : IsOrdinal α] [hβ : IsOrdinal β] : α ⊊ β → α ∈ β := by
+ intro ss
+ have : ∃ γ, (γ ∈ β ∧ γ ∉ α) ∧ ∀... | {
"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_884b6d6ae831_3 | 09213fd70ec0d284 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/SetTheory/Ordinal.lean | Ordinal | 3 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 12 | 2 | [
{
"theorem_name": "ssubset_iff",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": "\n ⟨mem_of_ssubset, fun hαβ ↦ ⟨hβ.transitive _ hαβ, ne_of_mem hαβ⟩⟩",
"n_chars": 84,
"n_subproofs": 0,
"n_tactics": 1,
... | [
{
"name": "mem_of_ssubset",
"text": "lemma mem_of_ssubset [hα : IsOrdinal α] [hβ : IsOrdinal β] : α ⊊ β → α ∈ β := by\n intro ss\n have : ∃ γ, (γ ∈ β ∧ γ ∉ α) ∧ ∀ δ ∈ β, δ ∉ α → δ ∉ γ := by\n have : IsNonempty (β \\ α) := (isNonempty_sdiff_of_ssubset ss)\n simpa using foundation (β \\ α)\n rcases t... | [
{
"name": "sInter_mem",
"fan_in": 1,
"n_deps_direct": 2,
"n_deps_transitive": 3,
"n_lines": 12,
"n_chars": 448,
"n_subproofs": 4,
"n_tactics": 10,
"cyclomatic": 2,
"n_automation": 3,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": false,
"max_nesting": ... | 1 | module
public import Foundation.FirstOrder.SetTheory.Z
@[expose] public section
/-!
# Ordinals and transitive sets
reference: Ralf Schindler, "Set Theory, Exploring Independence and Truth" [Sch14]
-/
namespace LO.FirstOrder.SetTheory
variable {V : Type*} [SetStructure V] [Nonempty V] [V↓[ℒₛₑₜ] ⊧* 𝗭]
/-! ### Tran... | module
public import Foundation.FirstOrder.SetTheory.Z
@[expose] public section
/-!
# Ordinals and transitive sets
reference: Ralf Schindler, "Set Theory, Exploring Independence and Truth" [Sch14]
-/
namespace LO.FirstOrder.SetTheory
variable {V : Type*} [SetStructure V] [Nonempty V] [V↓[ℒₛₑₜ] ⊧* 𝗭]
/-! ### Tran... | @@ -148,14 +148,42 @@
case zero => simp [zero_def]
case succ => intro α hα H; exact H.succ
-@[grind =] lemma ssubset_iff [hα : IsOrdinal α] [hβ : IsOrdinal β] : α ⊊ β ↔ α ∈ β := sorry
+lemma mem_of_ssubset [hα : IsOrdinal α] [hβ : IsOrdinal β] : α ⊊ β → α ∈ β := by
+ intro ss
+ have : ∃ γ, (γ ∈ β ∧ γ ∉ α) ∧ ∀... | {
"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_884b6d6ae831_4 | 08c789b89735687c | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/SetTheory/Ordinal.lean | Ordinal | 4 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 12 | 2 | [
{
"theorem_name": "ssubset_iff",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": "\n ⟨mem_of_ssubset, fun hαβ ↦ ⟨hβ.transitive _ hαβ, ne_of_mem hαβ⟩⟩",
"n_chars": 84,
"n_subproofs": 0,
"n_tactics": 1,
... | [
{
"name": "mem_of_ssubset",
"text": "lemma mem_of_ssubset [hα : IsOrdinal α] [hβ : IsOrdinal β] : α ⊊ β → α ∈ β := by\n intro ss\n have : ∃ γ, (γ ∈ β ∧ γ ∉ α) ∧ ∀ δ ∈ β, δ ∉ α → δ ∉ γ := by\n have : IsNonempty (β \\ α) := (isNonempty_sdiff_of_ssubset ss)\n simpa using foundation (β \\ α)\n rcases t... | [
{
"name": "empty_mem_iff_nonempty",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 2,
"n_lines": 8,
"n_chars": 249,
"n_subproofs": 1,
"n_tactics": 6,
"cyclomatic": 2,
"n_automation": 4,
"n_rewrites": 1,
"n_structural": 2,
"automation_only": false,
"max_... | 1 | module
public import Foundation.FirstOrder.SetTheory.Z
@[expose] public section
/-!
# Ordinals and transitive sets
reference: Ralf Schindler, "Set Theory, Exploring Independence and Truth" [Sch14]
-/
namespace LO.FirstOrder.SetTheory
variable {V : Type*} [SetStructure V] [Nonempty V] [V↓[ℒₛₑₜ] ⊧* 𝗭]
/-! ### Tran... | module
public import Foundation.FirstOrder.SetTheory.Z
@[expose] public section
/-!
# Ordinals and transitive sets
reference: Ralf Schindler, "Set Theory, Exploring Independence and Truth" [Sch14]
-/
namespace LO.FirstOrder.SetTheory
variable {V : Type*} [SetStructure V] [Nonempty V] [V↓[ℒₛₑₜ] ⊧* 𝗭]
/-! ### Tran... | @@ -148,14 +148,42 @@
case zero => simp [zero_def]
case succ => intro α hα H; exact H.succ
-@[grind =] lemma ssubset_iff [hα : IsOrdinal α] [hβ : IsOrdinal β] : α ⊊ β ↔ α ∈ β := sorry
+lemma mem_of_ssubset [hα : IsOrdinal α] [hβ : IsOrdinal β] : α ⊊ β → α ∈ β := by
+ intro ss
+ have : ∃ γ, (γ ∈ β ∧ γ ∉ α) ∧ ∀... | {
"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_884b6d6ae831_5 | 1bb0ef0b94b7070b | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/SetTheory/Ordinal.lean | Ordinal | 5 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 12 | 2 | [
{
"theorem_name": "ssubset_iff",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": "\n ⟨mem_of_ssubset, fun hαβ ↦ ⟨hβ.transitive _ hαβ, ne_of_mem hαβ⟩⟩",
"n_chars": 84,
"n_subproofs": 0,
"n_tactics": 1,
... | [
{
"name": "mem_of_ssubset",
"text": "lemma mem_of_ssubset [hα : IsOrdinal α] [hβ : IsOrdinal β] : α ⊊ β → α ∈ β := by\n intro ss\n have : ∃ γ, (γ ∈ β ∧ γ ∉ α) ∧ ∀ δ ∈ β, δ ∉ α → δ ∉ γ := by\n have : IsNonempty (β \\ α) := (isNonempty_sdiff_of_ssubset ss)\n simpa using foundation (β \\ α)\n rcases t... | [
{
"name": "pos_iff_nonempty",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 3,
"n_lines": 3,
"n_chars": 93,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
"max_nesting... | 2 | module
public import Foundation.FirstOrder.SetTheory.Z
@[expose] public section
/-!
# Ordinals and transitive sets
reference: Ralf Schindler, "Set Theory, Exploring Independence and Truth" [Sch14]
-/
namespace LO.FirstOrder.SetTheory
variable {V : Type*} [SetStructure V] [Nonempty V] [V↓[ℒₛₑₜ] ⊧* 𝗭]
/-! ### Tran... | module
public import Foundation.FirstOrder.SetTheory.Z
@[expose] public section
/-!
# Ordinals and transitive sets
reference: Ralf Schindler, "Set Theory, Exploring Independence and Truth" [Sch14]
-/
namespace LO.FirstOrder.SetTheory
variable {V : Type*} [SetStructure V] [Nonempty V] [V↓[ℒₛₑₜ] ⊧* 𝗭]
/-! ### Tran... | @@ -148,14 +148,42 @@
case zero => simp [zero_def]
case succ => intro α hα H; exact H.succ
-@[grind =] lemma ssubset_iff [hα : IsOrdinal α] [hβ : IsOrdinal β] : α ⊊ β ↔ α ∈ β := sorry
+lemma mem_of_ssubset [hα : IsOrdinal α] [hβ : IsOrdinal β] : α ⊊ β → α ∈ β := by
+ intro ss
+ have : ∃ γ, (γ ∈ β ∧ γ ∉ α) ∧ ∀... | {
"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_884b6d6ae831_6 | e345efc1ef001214 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/SetTheory/Ordinal.lean | Ordinal | 6 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 12 | 2 | [
{
"theorem_name": "lt_succ",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " lt_def.mpr <| by simp",
"n_chars": 22,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 1,
"n_r... | [
{
"name": "lt_def",
"text": "lemma lt_def : α < β ↔ α.val ∈ β.val := by rfl\n\nomit [Nonempty V] [V↓[ℒₛₑₜ] ⊧* 𝗭] in\n",
"fan_in": 2,
"n_lines": 4,
"n_chars": 108,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
... | [
{
"name": "lt_succ",
"fan_in": 0,
"n_deps_direct": 2,
"n_deps_transitive": 2,
"n_lines": 3,
"n_chars": 80,
"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.SetTheory.Z
@[expose] public section
/-!
# Ordinals and transitive sets
reference: Ralf Schindler, "Set Theory, Exploring Independence and Truth" [Sch14]
-/
namespace LO.FirstOrder.SetTheory
variable {V : Type*} [SetStructure V] [Nonempty V] [V↓[ℒₛₑₜ] ⊧* 𝗭]
/-! ### Tran... | module
public import Foundation.FirstOrder.SetTheory.Z
@[expose] public section
/-!
# Ordinals and transitive sets
reference: Ralf Schindler, "Set Theory, Exploring Independence and Truth" [Sch14]
-/
namespace LO.FirstOrder.SetTheory
variable {V : Type*} [SetStructure V] [Nonempty V] [V↓[ℒₛₑₜ] ⊧* 𝗭]
/-! ### Tran... | @@ -307,6 +307,9 @@
instance : LE (Ordinal V) := ⟨fun α β ↦ α.val ⊆ β.val⟩
omit [Nonempty V] [V↓[ℒₛₑₜ] ⊧* 𝗭] in
+lemma lt_def : α < β ↔ α.val ∈ β.val := by rfl
+
+omit [Nonempty V] [V↓[ℒₛₑₜ] ⊧* 𝗭] in
lemma le_def : α ≤ β ↔ α.val ⊆ β.val := by rfl
instance : IsOrdinal α.val := α.ordinal
@@ -331,7 +334,7 @@
... | {
"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_884b6d6ae831_7 | 3f76f2dcd9bb12df | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/SetTheory/Ordinal.lean | Ordinal | 7 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 12 | 2 | [
{
"theorem_name": "ssubset_iff",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": "\n ⟨mem_of_ssubset, fun hαβ ↦ ⟨hβ.transitive _ hαβ, ne_of_mem hαβ⟩⟩",
"n_chars": 84,
"n_subproofs": 0,
"n_tactics": 1,
... | [
{
"name": "mem_of_ssubset",
"text": "lemma mem_of_ssubset [hα : IsOrdinal α] [hβ : IsOrdinal β] : α ⊊ β → α ∈ β := by\n intro ss\n have : ∃ γ, (γ ∈ β ∧ γ ∉ α) ∧ ∀ δ ∈ β, δ ∉ α → δ ∉ γ := by\n have : IsNonempty (β \\ α) := (isNonempty_sdiff_of_ssubset ss)\n simpa using foundation (β \\ α)\n rcases t... | [
{
"name": "minimal_prop_of_exists_aux",
"fan_in": 3,
"n_deps_direct": 4,
"n_deps_transitive": 7,
"n_lines": 17,
"n_chars": 762,
"n_subproofs": 5,
"n_tactics": 15,
"cyclomatic": 2,
"n_automation": 6,
"n_rewrites": 0,
"n_structural": 6,
"automation_only": false,
... | 3 | module
public import Foundation.FirstOrder.SetTheory.Z
@[expose] public section
/-!
# Ordinals and transitive sets
reference: Ralf Schindler, "Set Theory, Exploring Independence and Truth" [Sch14]
-/
namespace LO.FirstOrder.SetTheory
variable {V : Type*} [SetStructure V] [Nonempty V] [V↓[ℒₛₑₜ] ⊧* 𝗭]
/-! ### Tran... | module
public import Foundation.FirstOrder.SetTheory.Z
@[expose] public section
/-!
# Ordinals and transitive sets
reference: Ralf Schindler, "Set Theory, Exploring Independence and Truth" [Sch14]
-/
namespace LO.FirstOrder.SetTheory
variable {V : Type*} [SetStructure V] [Nonempty V] [V↓[ℒₛₑₜ] ⊧* 𝗭]
/-! ### Tran... | @@ -148,14 +148,42 @@
case zero => simp [zero_def]
case succ => intro α hα H; exact H.succ
-@[grind =] lemma ssubset_iff [hα : IsOrdinal α] [hβ : IsOrdinal β] : α ⊊ β ↔ α ∈ β := sorry
+lemma mem_of_ssubset [hα : IsOrdinal α] [hβ : IsOrdinal β] : α ⊊ β → α ∈ β := by
+ intro ss
+ have : ∃ γ, (γ ∈ β ∧ γ ∉ α) ∧ ∀... | {
"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_884b6d6ae831_8 | 235ebdffd46ee3b4 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/SetTheory/Ordinal.lean | Ordinal | 8 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 12 | 1 | [
{
"theorem_name": "minimal_prop_of_exists_aux",
"depth": 1,
"n_commands": 0,
"n_lines": 14,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n let X := {x ∈ ↑α ; P x}\n have : IsNonempty X := by\n rcases H with ⟨β, hαβ, Pβ⟩\n exact ⟨β, by simp [X,... | [
{
"name": "sInter_mem",
"text": "lemma sInter_mem {X : V} [IsNonempty X] (h : ∀ α ∈ X, IsOrdinal α) : ⋂ˢ X ∈ X := by\n by_contra! hXX\n have : IsOrdinal (⋂ˢ X) := IsOrdinal.sInter h\n have : ⋂ˢ X ∈ ⋂ˢ X := mem_sInter_iff_of_nonempty.mpr fun α hαX ↦ by\n have : IsOrdinal α := h α hαX\n have : ⋂ˢ X ⊆... | [
{
"name": "minimal_lt_of_exists",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 8,
"n_lines": 3,
"n_chars": 108,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
"max_ne... | 4 | module
public import Foundation.FirstOrder.SetTheory.Z
@[expose] public section
/-!
# Ordinals and transitive sets
reference: Ralf Schindler, "Set Theory, Exploring Independence and Truth" [Sch14]
-/
namespace LO.FirstOrder.SetTheory
variable {V : Type*} [SetStructure V] [Nonempty V] [V↓[ℒₛₑₜ] ⊧* 𝗭]
/-! ### Tran... | module
public import Foundation.FirstOrder.SetTheory.Z
@[expose] public section
/-!
# Ordinals and transitive sets
reference: Ralf Schindler, "Set Theory, Exploring Independence and Truth" [Sch14]
-/
namespace LO.FirstOrder.SetTheory
variable {V : Type*} [SetStructure V] [Nonempty V] [V↓[ℒₛₑₜ] ⊧* 𝗭]
/-! ### Tran... | @@ -273,6 +273,17 @@
have : α ∈ α₀ := this α₀ hα₀X
exact of_mem this
+lemma sInter_mem {X : V} [IsNonempty X] (h : ∀ α ∈ X, IsOrdinal α) : ⋂ˢ X ∈ X := by
+ by_contra! hXX
+ have : IsOrdinal (⋂ˢ X) := IsOrdinal.sInter h
+ have : ⋂ˢ X ∈ ⋂ˢ X := mem_sInter_iff_of_nonempty.mpr fun α hαX ↦ by
+ have : IsOr... | {
"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_884b6d6ae831_9 | 999c38930ccb19b3 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/SetTheory/Ordinal.lean | Ordinal | 9 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 12 | 1 | [
{
"theorem_name": "minimal_prop_of_exists_aux",
"depth": 1,
"n_commands": 0,
"n_lines": 14,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n let X := {x ∈ ↑α ; P x}\n have : IsNonempty X := by\n rcases H with ⟨β, hαβ, Pβ⟩\n exact ⟨β, by simp [X,... | [
{
"name": "sInter_mem",
"text": "lemma sInter_mem {X : V} [IsNonempty X] (h : ∀ α ∈ X, IsOrdinal α) : ⋂ˢ X ∈ X := by\n by_contra! hXX\n have : IsOrdinal (⋂ˢ X) := IsOrdinal.sInter h\n have : ⋂ˢ X ∈ ⋂ˢ X := mem_sInter_iff_of_nonempty.mpr fun α hαX ↦ by\n have : IsOrdinal α := h α hαX\n have : ⋂ˢ X ⊆... | [
{
"name": "minimal_prop_of_exists",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 8,
"n_lines": 3,
"n_chars": 111,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
"max_... | 4 | module
public import Foundation.FirstOrder.SetTheory.Z
@[expose] public section
/-!
# Ordinals and transitive sets
reference: Ralf Schindler, "Set Theory, Exploring Independence and Truth" [Sch14]
-/
namespace LO.FirstOrder.SetTheory
variable {V : Type*} [SetStructure V] [Nonempty V] [V↓[ℒₛₑₜ] ⊧* 𝗭]
/-! ### Tran... | module
public import Foundation.FirstOrder.SetTheory.Z
@[expose] public section
/-!
# Ordinals and transitive sets
reference: Ralf Schindler, "Set Theory, Exploring Independence and Truth" [Sch14]
-/
namespace LO.FirstOrder.SetTheory
variable {V : Type*} [SetStructure V] [Nonempty V] [V↓[ℒₛₑₜ] ⊧* 𝗭]
/-! ### Tran... | @@ -273,6 +273,17 @@
have : α ∈ α₀ := this α₀ hα₀X
exact of_mem this
+lemma sInter_mem {X : V} [IsNonempty X] (h : ∀ α ∈ X, IsOrdinal α) : ⋂ˢ X ∈ X := by
+ by_contra! hXX
+ have : IsOrdinal (⋂ˢ X) := IsOrdinal.sInter h
+ have : ⋂ˢ X ∈ ⋂ˢ X := mem_sInter_iff_of_nonempty.mpr fun α hαX ↦ by
+ have : IsOr... | {
"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_884b6d6ae831_10 | aebc4e8d005cdaee | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/SetTheory/Ordinal.lean | Ordinal | 10 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 12 | 1 | [
{
"theorem_name": "minimal_prop_of_exists_aux",
"depth": 1,
"n_commands": 0,
"n_lines": 14,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n let X := {x ∈ ↑α ; P x}\n have : IsNonempty X := by\n rcases H with ⟨β, hαβ, Pβ⟩\n exact ⟨β, by simp [X,... | [
{
"name": "sInter_mem",
"text": "lemma sInter_mem {X : V} [IsNonempty X] (h : ∀ α ∈ X, IsOrdinal α) : ⋂ˢ X ∈ X := by\n by_contra! hXX\n have : IsOrdinal (⋂ˢ X) := IsOrdinal.sInter h\n have : ⋂ˢ X ∈ ⋂ˢ X := mem_sInter_iff_of_nonempty.mpr fun α hαX ↦ by\n have : IsOrdinal α := h α hαX\n have : ⋂ˢ X ⊆... | [
{
"name": "minimal_le_of_exists_aux",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 8,
"n_lines": 3,
"n_chars": 138,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
"ma... | 4 | module
public import Foundation.FirstOrder.SetTheory.Z
@[expose] public section
/-!
# Ordinals and transitive sets
reference: Ralf Schindler, "Set Theory, Exploring Independence and Truth" [Sch14]
-/
namespace LO.FirstOrder.SetTheory
variable {V : Type*} [SetStructure V] [Nonempty V] [V↓[ℒₛₑₜ] ⊧* 𝗭]
/-! ### Tran... | module
public import Foundation.FirstOrder.SetTheory.Z
@[expose] public section
/-!
# Ordinals and transitive sets
reference: Ralf Schindler, "Set Theory, Exploring Independence and Truth" [Sch14]
-/
namespace LO.FirstOrder.SetTheory
variable {V : Type*} [SetStructure V] [Nonempty V] [V↓[ℒₛₑₜ] ⊧* 𝗭]
/-! ### Tran... | @@ -273,6 +273,17 @@
have : α ∈ α₀ := this α₀ hα₀X
exact of_mem this
+lemma sInter_mem {X : V} [IsNonempty X] (h : ∀ α ∈ X, IsOrdinal α) : ⋂ˢ X ∈ X := by
+ by_contra! hXX
+ have : IsOrdinal (⋂ˢ X) := IsOrdinal.sInter h
+ have : ⋂ˢ X ∈ ⋂ˢ X := mem_sInter_iff_of_nonempty.mpr fun α hαX ↦ by
+ have : IsOr... | {
"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_884b6d6ae831_11 | ef533414f4a0407c | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/SetTheory/Ordinal.lean | Ordinal | 11 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 12 | 2 | [
{
"theorem_name": "lt_succ",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " lt_def.mpr <| by simp",
"n_chars": 22,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 1,
"n_r... | [
{
"name": "lt_def",
"text": "lemma lt_def : α < β ↔ α.val ∈ β.val := by rfl\n\nomit [Nonempty V] [V↓[ℒₛₑₜ] ⊧* 𝗭] in\n",
"fan_in": 2,
"n_lines": 4,
"n_chars": 108,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
... | [
{
"name": "minimal_le_of_exists",
"fan_in": 1,
"n_deps_direct": 2,
"n_deps_transitive": 10,
"n_lines": 9,
"n_chars": 354,
"n_subproofs": 0,
"n_tactics": 7,
"cyclomatic": 2,
"n_automation": 0,
"n_rewrites": 1,
"n_structural": 4,
"automation_only": false,
"max_n... | 6 | module
public import Foundation.FirstOrder.SetTheory.Z
@[expose] public section
/-!
# Ordinals and transitive sets
reference: Ralf Schindler, "Set Theory, Exploring Independence and Truth" [Sch14]
-/
namespace LO.FirstOrder.SetTheory
variable {V : Type*} [SetStructure V] [Nonempty V] [V↓[ℒₛₑₜ] ⊧* 𝗭]
/-! ### Tran... | module
public import Foundation.FirstOrder.SetTheory.Z
@[expose] public section
/-!
# Ordinals and transitive sets
reference: Ralf Schindler, "Set Theory, Exploring Independence and Truth" [Sch14]
-/
namespace LO.FirstOrder.SetTheory
variable {V : Type*} [SetStructure V] [Nonempty V] [V↓[ℒₛₑₜ] ⊧* 𝗭]
/-! ### Tran... | @@ -318,6 +318,9 @@
instance : LE (Ordinal V) := ⟨fun α β ↦ α.val ⊆ β.val⟩
omit [Nonempty V] [V↓[ℒₛₑₜ] ⊧* 𝗭] in
+lemma lt_def : α < β ↔ α.val ∈ β.val := by rfl
+
+omit [Nonempty V] [V↓[ℒₛₑₜ] ⊧* 𝗭] in
lemma le_def : α ≤ β ↔ α.val ⊆ β.val := by rfl
instance : IsOrdinal α.val := α.ordinal
@@ -342,7 +345,7 @@
... | {
"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_884b6d6ae831_12 | 5b78a15a975d663b | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/SetTheory/Ordinal.lean | Ordinal | 12 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 12 | 2 | [
{
"theorem_name": "lt_succ",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " lt_def.mpr <| by simp",
"n_chars": 22,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 1,
"n_r... | [
{
"name": "lt_def",
"text": "lemma lt_def : α < β ↔ α.val ∈ β.val := by rfl\n\nomit [Nonempty V] [V↓[ℒₛₑₜ] ⊧* 𝗭] in\n",
"fan_in": 2,
"n_lines": 4,
"n_chars": 108,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
... | [
{
"name": "exists_minimal",
"fan_in": 1,
"n_deps_direct": 3,
"n_deps_transitive": 13,
"n_lines": 7,
"n_chars": 377,
"n_subproofs": 1,
"n_tactics": 4,
"cyclomatic": 2,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 2,
"automation_only": false,
"max_nesting... | 8 | module
public import Foundation.FirstOrder.SetTheory.Z
@[expose] public section
/-!
# Ordinals and transitive sets
reference: Ralf Schindler, "Set Theory, Exploring Independence and Truth" [Sch14]
-/
namespace LO.FirstOrder.SetTheory
variable {V : Type*} [SetStructure V] [Nonempty V] [V↓[ℒₛₑₜ] ⊧* 𝗭]
/-! ### Tran... | module
public import Foundation.FirstOrder.SetTheory.Z
@[expose] public section
/-!
# Ordinals and transitive sets
reference: Ralf Schindler, "Set Theory, Exploring Independence and Truth" [Sch14]
-/
namespace LO.FirstOrder.SetTheory
variable {V : Type*} [SetStructure V] [Nonempty V] [V↓[ℒₛₑₜ] ⊧* 𝗭]
/-! ### Tran... | @@ -318,6 +318,9 @@
instance : LE (Ordinal V) := ⟨fun α β ↦ α.val ⊆ β.val⟩
omit [Nonempty V] [V↓[ℒₛₑₜ] ⊧* 𝗭] in
+lemma lt_def : α < β ↔ α.val ∈ β.val := by rfl
+
+omit [Nonempty V] [V↓[ℒₛₑₜ] ⊧* 𝗭] in
lemma le_def : α ≤ β ↔ α.val ⊆ β.val := by rfl
instance : IsOrdinal α.val := α.ordinal
@@ -342,7 +345,7 @@
... | {
"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_884b6d6ae831_13 | 71e3368b8a59f2a7 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/SetTheory/Ordinal.lean | Ordinal | 13 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 12 | 2 | [
{
"theorem_name": "lt_succ",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " lt_def.mpr <| by simp",
"n_chars": 22,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 1,
"n_r... | [
{
"name": "lt_def",
"text": "lemma lt_def : α < β ↔ α.val ∈ β.val := by rfl\n\nomit [Nonempty V] [V↓[ℒₛₑₜ] ⊧* 𝗭] in\n",
"fan_in": 2,
"n_lines": 4,
"n_chars": 108,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
... | [
{
"name": "transfinite_induction",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 14,
"n_lines": 15,
"n_chars": 549,
"n_subproofs": 3,
"n_tactics": 10,
"cyclomatic": 2,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 2,
"automation_only": false,
"ma... | 9 | module
public import Foundation.FirstOrder.SetTheory.Z
@[expose] public section
/-!
# Ordinals and transitive sets
reference: Ralf Schindler, "Set Theory, Exploring Independence and Truth" [Sch14]
-/
namespace LO.FirstOrder.SetTheory
variable {V : Type*} [SetStructure V] [Nonempty V] [V↓[ℒₛₑₜ] ⊧* 𝗭]
/-! ### Tran... | module
public import Foundation.FirstOrder.SetTheory.Z
@[expose] public section
/-!
# Ordinals and transitive sets
reference: Ralf Schindler, "Set Theory, Exploring Independence and Truth" [Sch14]
-/
namespace LO.FirstOrder.SetTheory
variable {V : Type*} [SetStructure V] [Nonempty V] [V↓[ℒₛₑₜ] ⊧* 𝗭]
/-! ### Tran... | @@ -318,6 +318,9 @@
instance : LE (Ordinal V) := ⟨fun α β ↦ α.val ⊆ β.val⟩
omit [Nonempty V] [V↓[ℒₛₑₜ] ⊧* 𝗭] in
+lemma lt_def : α < β ↔ α.val ∈ β.val := by rfl
+
+omit [Nonempty V] [V↓[ℒₛₑₜ] ⊧* 𝗭] in
lemma le_def : α ≤ β ↔ α.val ⊆ β.val := by rfl
instance : IsOrdinal α.val := α.ordinal
@@ -342,7 +345,7 @@
... | {
"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_51ba405263db_0 | 09cb4d1a9294f47a | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Vorspiel/Rel/WCWF.lean | WCWF | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 2 | 1 | [
{
"theorem_name": "weaklyConverseWellFounded_of_finite_trans_antisymm",
"depth": 1,
"n_commands": 0,
"n_lines": 20,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n simp only [AntiSymmetric, ConverseWellFounded.iff_has_max];\n contrapose!;\n ri... | [
{
"name": "Finite.exists_ne_map_eq_of_infinite_lt",
"text": "lemma Finite.exists_ne_map_eq_of_infinite_lt {α β} [LinearOrder α] [Infinite α] [Finite β] (f : α → β)\n : ∃ x y : α, (x < y) ∧ f x = f y\n := by\n obtain ⟨i, j, hij, e⟩ := Finite.exists_ne_map_eq_of_infinite f;\n rcases lt_trichotomy i j ... | [
{
"name": "weaklyConverseWellFounded_of_finite_trans_antisymm",
"fan_in": 0,
"n_deps_direct": 2,
"n_deps_transitive": 2,
"n_lines": 23,
"n_chars": 836,
"n_subproofs": 6,
"n_tactics": 32,
"cyclomatic": 3,
"n_automation": 6,
"n_rewrites": 0,
"n_structural": 9,
"auto... | 2 | module
public import Foundation.Vorspiel.Rel.CWF
public import Mathlib.Data.Fintype.Pigeonhole
@[expose]
public section
section
abbrev WeaklyConverseWellFounded {α} (rel : Rel α α) := ConverseWellFounded (rel.IrreflGen)
@[mk_iff]
class IsWeaklyConverseWellFounded (α) (rel : Rel α α) where wcwf : WeaklyConverseWell... | module
public import Foundation.Vorspiel.Rel.CWF
public import Mathlib.Data.Fintype.Pigeonhole
@[expose]
public section
section
abbrev WeaklyConverseWellFounded {α} (rel : Rel α α) := ConverseWellFounded (rel.IrreflGen)
@[mk_iff]
class IsWeaklyConverseWellFounded (α) (rel : Rel α α) where wcwf : WeaklyConverseWell... | @@ -31,6 +31,15 @@
| zero => simpa;
| succ n ih => simp only [Function.iterate_succ']; apply hfs _ ih;
+lemma Finite.exists_ne_map_eq_of_infinite_lt {α β} [LinearOrder α] [Infinite α] [Finite β] (f : α → β)
+ : ∃ x y : α, (x < y) ∧ f x = f y
+ := by
+ obtain ⟨i, j, hij, e⟩ := Finite.exists_ne_map_eq_of_inf... | {
"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_635b0d983ced_0 | f10c1a045917caff | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Basic/Hierarchy.lean | Hierarchy | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 3 | 3 | [
{
"theorem_name": "zero_iff",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by rcases Γ <;> rcases Γ' <;> simp [pi_zero_iff_sigma_zero]",
"n_chars": 62,
"n_subproofs": 0,
"n_tactics": 3,
"cyclomat... | [
{
"name": "pi_zero_iff_sigma_zero",
"text": "lemma pi_zero_iff_sigma_zero {φ : Semiformula L ξ n} : Hierarchy 𝚷 0 φ ↔ Hierarchy 𝚺 0 φ := ⟨zero_eq_alt, zero_eq_alt⟩\n\n",
"fan_in": 3,
"n_lines": 3,
"n_chars": 136,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation"... | [
{
"name": "zero_iff",
"fan_in": 2,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 3,
"n_chars": 159,
"n_subproofs": 0,
"n_tactics": 3,
"cyclomatic": 5,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 2,
"automation_only": false,
"max_nesting": 1
... | 1 | module
public import Foundation.FirstOrder.Arithmetic.Basic.Model
@[expose] public section
namespace LO.FirstOrder.Arithmetic
variable {L : Language} [L.LT]
inductive Hierarchy : Polarity → ℕ → {n : ℕ} → Semiformula L ξ n → Prop
| verum (Γ s n) : Hierarchy Γ s (⊤ : Semiformula L ξ n)
| falsum (Γ s n) : Hierarch... | module
public import Foundation.FirstOrder.Arithmetic.Basic.Model
@[expose] public section
namespace LO.FirstOrder.Arithmetic
variable {L : Language} [L.LT]
inductive Hierarchy : Polarity → ℕ → {n : ℕ} → Semiformula L ξ n → Prop
| verum (Γ s n) : Hierarchy Γ s (⊤ : Semiformula L ξ n)
| falsum (Γ s n) : Hierarch... | @@ -70,10 +70,12 @@
case ball pos _ ih => exact ball pos (ih hz)
case bexs pos _ ih => exact bexs pos (ih hz)
-lemma zero_iff {Γ Γ'} {φ : Semiformula L ξ n} : Hierarchy Γ 0 φ ↔ Hierarchy Γ' 0 φ := sorry
+lemma pi_zero_iff_sigma_zero {φ : Semiformula L ξ n} : Hierarchy 𝚷 0 φ ↔ Hierarchy 𝚺 0 φ := ⟨zero_eq_alt, ... | {
"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_635b0d983ced_1 | 2c8c47eb88c43384 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Basic/Hierarchy.lean | Hierarchy | 1 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 3 | 2 | [
{
"theorem_name": "zero_iff_delta_zero",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n simpa [DeltaZero, pi_zero_iff_sigma_zero] using zero_iff",
"n_chars": 62,
"n_subproofs": 0,
"n_tactics": 2,... | [
{
"name": "zero_iff",
"text": "lemma zero_iff {Γ Γ'} {φ : Semiformula L ξ n} : Hierarchy Γ 0 φ ↔ Hierarchy Γ' 0 φ := by rcases Γ <;> rcases Γ' <;> simp [pi_zero_iff_sigma_zero]\n\n",
"fan_in": 2,
"n_lines": 3,
"n_chars": 159,
"n_subproofs": 0,
"n_tactics": 3,
"cyclomatic": 5,
"n_... | [
{
"name": "zero_iff_delta_zero",
"fan_in": 0,
"n_deps_direct": 2,
"n_deps_transitive": 2,
"n_lines": 4,
"n_chars": 160,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"max_nest... | 2 | module
public import Foundation.FirstOrder.Arithmetic.Basic.Model
@[expose] public section
namespace LO.FirstOrder.Arithmetic
variable {L : Language} [L.LT]
inductive Hierarchy : Polarity → ℕ → {n : ℕ} → Semiformula L ξ n → Prop
| verum (Γ s n) : Hierarchy Γ s (⊤ : Semiformula L ξ n)
| falsum (Γ s n) : Hierarch... | module
public import Foundation.FirstOrder.Arithmetic.Basic.Model
@[expose] public section
namespace LO.FirstOrder.Arithmetic
variable {L : Language} [L.LT]
inductive Hierarchy : Polarity → ℕ → {n : ℕ} → Semiformula L ξ n → Prop
| verum (Γ s n) : Hierarchy Γ s (⊤ : Semiformula L ξ n)
| falsum (Γ s n) : Hierarch... | @@ -72,8 +72,11 @@
lemma pi_zero_iff_sigma_zero {φ : Semiformula L ξ n} : Hierarchy 𝚷 0 φ ↔ Hierarchy 𝚺 0 φ := ⟨zero_eq_alt, zero_eq_alt⟩
-lemma zero_iff_delta_zero {Γ} {φ : Semiformula L ξ n} : Hierarchy Γ 0 φ ↔ DeltaZero φ := sorry
+lemma zero_iff {Γ Γ'} {φ : Semiformula L ξ n} : Hierarchy Γ 0 φ ↔ Hierarchy Γ'... | {
"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_635b0d983ced_2 | e731069a87da96c9 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Basic/Hierarchy.lean | Hierarchy | 2 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 3 | 3 | [
{
"theorem_name": "zero_iff",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by rcases Γ <;> rcases Γ' <;> simp [pi_zero_iff_sigma_zero]",
"n_chars": 62,
"n_subproofs": 0,
"n_tactics": 3,
"cyclomat... | [
{
"name": "pi_zero_iff_sigma_zero",
"text": "lemma pi_zero_iff_sigma_zero {φ : Semiformula L ξ n} : Hierarchy 𝚷 0 φ ↔ Hierarchy 𝚺 0 φ := ⟨zero_eq_alt, zero_eq_alt⟩\n\n",
"fan_in": 3,
"n_lines": 3,
"n_chars": 136,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation"... | [
{
"name": "alt_zero_iff_zero",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 3,
"n_chars": 155,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 3,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": false,
"max_nesti... | 1 | module
public import Foundation.FirstOrder.Arithmetic.Basic.Model
@[expose] public section
namespace LO.FirstOrder.Arithmetic
variable {L : Language} [L.LT]
inductive Hierarchy : Polarity → ℕ → {n : ℕ} → Semiformula L ξ n → Prop
| verum (Γ s n) : Hierarchy Γ s (⊤ : Semiformula L ξ n)
| falsum (Γ s n) : Hierarch... | module
public import Foundation.FirstOrder.Arithmetic.Basic.Model
@[expose] public section
namespace LO.FirstOrder.Arithmetic
variable {L : Language} [L.LT]
inductive Hierarchy : Polarity → ℕ → {n : ℕ} → Semiformula L ξ n → Prop
| verum (Γ s n) : Hierarchy Γ s (⊤ : Semiformula L ξ n)
| falsum (Γ s n) : Hierarch... | @@ -70,8 +70,10 @@
case ball pos _ ih => exact ball pos (ih hz)
case bexs pos _ ih => exact bexs pos (ih hz)
-@[simp] lemma alt_zero_iff_zero {φ : Semiformula L ξ n} : Hierarchy Γ.alt 0 φ ↔ Hierarchy Γ 0 φ := sorry
+lemma pi_zero_iff_sigma_zero {φ : Semiformula L ξ n} : Hierarchy 𝚷 0 φ ↔ Hierarchy 𝚺 0 φ := ⟨z... | {
"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_635b0d983ced_3 | da2e7618652dbcee | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Basic/Hierarchy.lean | Hierarchy | 3 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 3 | 2 | [
{
"theorem_name": "mono",
"depth": 1,
"n_commands": 0,
"n_lines": 4,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n rcases Nat.lt_or_eq_of_le h with (lt | rfl)\n · exact hp.strict_mono Γ lt\n · assumption",
"n_chars": 97,
"n_subproofs": 0,
... | [
{
"name": "strict_mono",
"text": "lemma strict_mono {Γ s} {φ : Semiformula L ξ n} (hp : Hierarchy Γ s φ) (Γ') {s'} (h : s < s') : Hierarchy Γ' s' φ := by\n have : ∀ d, Hierarchy Γ' (s + d + 1) φ := by\n intro d\n induction' d with s ih\n · simpa using hp.accum Γ'\n · simpa only [Nat.add_succ, a... | [
{
"name": "mono",
"fan_in": 0,
"n_deps_direct": 2,
"n_deps_transitive": 2,
"n_lines": 6,
"n_chars": 217,
"n_subproofs": 0,
"n_tactics": 4,
"cyclomatic": 2,
"n_automation": 2,
"n_rewrites": 0,
"n_structural": 2,
"automation_only": false,
"max_nesting": 2
}
] | 1 | module
public import Foundation.FirstOrder.Arithmetic.Basic.Model
@[expose] public section
namespace LO.FirstOrder.Arithmetic
variable {L : Language} [L.LT]
inductive Hierarchy : Polarity → ℕ → {n : ℕ} → Semiformula L ξ n → Prop
| verum (Γ s n) : Hierarchy Γ s (⊤ : Semiformula L ξ n)
| falsum (Γ s n) : Hierarch... | module
public import Foundation.FirstOrder.Arithmetic.Basic.Model
@[expose] public section
namespace LO.FirstOrder.Arithmetic
variable {L : Language} [L.LT]
inductive Hierarchy : Polarity → ℕ → {n : ℕ} → Semiformula L ξ n → Prop
| verum (Γ s n) : Hierarchy Γ s (⊤ : Semiformula L ξ n)
| falsum (Γ s n) : Hierarch... | @@ -108,9 +108,17 @@
· exact ((hp.accum 𝚺).accum 𝚺).exs
· exact (hp.accum 𝚺).dummy_pi
+lemma strict_mono {Γ s} {φ : Semiformula L ξ n} (hp : Hierarchy Γ s φ) (Γ') {s'} (h : s < s') : Hierarchy Γ' s' φ := by
+ have : ∀ d, Hierarchy Γ' (s + d + 1) φ := by
+ intro d
+ induction' d with s ih
+ · si... | {
"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_635b0d983ced_4 | cbb89a1688c9a8ba | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Basic/Hierarchy.lean | Hierarchy | 4 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 3 | 2 | [
{
"theorem_name": "mono",
"depth": 1,
"n_commands": 0,
"n_lines": 4,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n rcases Nat.lt_or_eq_of_le h with (lt | rfl)\n · exact hp.strict_mono Γ lt\n · assumption",
"n_chars": 97,
"n_subproofs": 0,
... | [
{
"name": "strict_mono",
"text": "lemma strict_mono {Γ s} {φ : Semiformula L ξ n} (hp : Hierarchy Γ s φ) (Γ') {s'} (h : s < s') : Hierarchy Γ' s' φ := by\n have : ∀ d, Hierarchy Γ' (s + d + 1) φ := by\n intro d\n induction' d with s ih\n · simpa using hp.accum Γ'\n · simpa only [Nat.add_succ, a... | [
{
"name": "of_zero",
"fan_in": 0,
"n_deps_direct": 2,
"n_deps_transitive": 3,
"n_lines": 6,
"n_chars": 228,
"n_subproofs": 0,
"n_tactics": 4,
"cyclomatic": 2,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 3,
"automation_only": false,
"max_nesting": 2
}... | 3 | module
public import Foundation.FirstOrder.Arithmetic.Basic.Model
@[expose] public section
namespace LO.FirstOrder.Arithmetic
variable {L : Language} [L.LT]
inductive Hierarchy : Polarity → ℕ → {n : ℕ} → Semiformula L ξ n → Prop
| verum (Γ s n) : Hierarchy Γ s (⊤ : Semiformula L ξ n)
| falsum (Γ s n) : Hierarch... | module
public import Foundation.FirstOrder.Arithmetic.Basic.Model
@[expose] public section
namespace LO.FirstOrder.Arithmetic
variable {L : Language} [L.LT]
inductive Hierarchy : Polarity → ℕ → {n : ℕ} → Semiformula L ξ n → Prop
| verum (Γ s n) : Hierarchy Γ s (⊤ : Semiformula L ξ n)
| falsum (Γ s n) : Hierarch... | @@ -110,10 +110,18 @@
· exact ((hp.accum 𝚺).accum 𝚺).exs
· exact (hp.accum 𝚺).dummy_pi
+lemma strict_mono {Γ s} {φ : Semiformula L ξ n} (hp : Hierarchy Γ s φ) (Γ') {s'} (h : s < s') : Hierarchy Γ' s' φ := by
+ have : ∀ d, Hierarchy Γ' (s + d + 1) φ := by
+ intro d
+ induction' d with s ih
+ · 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_352cf89c8d51_0 | 7c97b869e752de4c | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Incompleteness/ProvabilityAbstraction/Height.lean | Height | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 4 | 3 | [
{
"theorem_name": "height_lt_pos_of_boxBot",
"depth": 1,
"n_commands": 0,
"n_lines": 8,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n have e : n.pred.succ = n := Eq.symm <| (Nat.sub_eq_iff_eq_add pos).mp rfl\n have : T₀ ⊢ 𝔅 (𝔅^[n.pred] ⊥) := by r... | [
{
"name": "height_le_of_boxBot",
"text": "lemma height_le_of_boxBot {n : ℕ} (h : T ⊢ 𝔅^[n] ⊥) : 𝔅.height ≤ n :=\n ENat.find_le (T ⊢ 𝔅^[·] ⊥) n h\n\n",
"fan_in": 3,
"n_lines": 4,
"n_chars": 126,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewri... | [
{
"name": "height_lt_pos_of_boxBot",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 11,
"n_chars": 530,
"n_subproofs": 4,
"n_tactics": 9,
"cyclomatic": 1,
"n_automation": 2,
"n_rewrites": 1,
"n_structural": 2,
"automation_only": false,
"ma... | 1 | module
public import Foundation.FirstOrder.Incompleteness.Examples
public import Foundation.Vorspiel.ENat
@[expose] public section
namespace LO.FirstOrder
variable {L : Language} [L.ReferenceableBy L] {T₀ T : Theory L}
open ProvabilityAbstraction
open Classical
namespace ProvabilityAbstraction
variable {𝔅 : Prov... | module
public import Foundation.FirstOrder.Incompleteness.Examples
public import Foundation.Vorspiel.ENat
@[expose] public section
namespace LO.FirstOrder
variable {L : Language} [L.ReferenceableBy L] {T₀ T : Theory L}
open ProvabilityAbstraction
open Classical
namespace ProvabilityAbstraction
variable {𝔅 : Prov... | @@ -31,11 +31,14 @@
lemma height_eq_top_iff : 𝔅.height = ⊤ ↔ ∀ n, T ⊬ 𝔅^[n] ⊥ := ENat.find_eq_top_iff _
+lemma height_le_of_boxBot {n : ℕ} (h : T ⊢ 𝔅^[n] ⊥) : 𝔅.height ≤ n :=
+ ENat.find_le (T ⊢ 𝔅^[·] ⊥) n h
+
lemma height_lt_pos_of_boxBot (hSound : ∀ {σ}, T₀ ⊢ 𝔅 σ → T ⊢ σ)
{n : ℕ} (pos : 0 < n) (h : T₀... | {
"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_352cf89c8d51_1 | 7c44b484c2f958ea | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Incompleteness/ProvabilityAbstraction/Height.lean | Height | 1 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 4 | 1 | [
{
"theorem_name": "height_le_iff_boxBot",
"depth": 1,
"n_commands": 0,
"n_lines": 7,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n constructor\n · intro h\n have : ∃ m ≤ n, T ⊢ (↑𝔅)^[m] ⊥ := ENat.exists_of_find_le _ n h\n rcases this with ⟨m... | [
{
"name": "boxBot_monotone",
"text": "lemma boxBot_monotone [T₀ ⪯ T] [𝔅.HBL] : n ≤ m → T ⊢ 𝔅^[n] ⊥ 🡒 𝔅^[m] ⊥ := by\n revert m\n suffices ∀ k, T ⊢ 𝔅^[n] ⊥ 🡒 𝔅^[n + k] ⊥ by\n intro m hnm\n simpa [Nat.add_sub_of_le hnm] using this (m - n)\n intro k\n induction k\n case zero => simp\n case su... | [
{
"name": "height_le_iff_boxBot",
"fan_in": 0,
"n_deps_direct": 2,
"n_deps_transitive": 2,
"n_lines": 10,
"n_chars": 328,
"n_subproofs": 1,
"n_tactics": 7,
"cyclomatic": 3,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 5,
"automation_only": false,
"max_n... | 2 | module
public import Foundation.FirstOrder.Incompleteness.Examples
public import Foundation.Vorspiel.ENat
@[expose] public section
namespace LO.FirstOrder
variable {L : Language} [L.ReferenceableBy L] {T₀ T : Theory L}
open ProvabilityAbstraction
open Classical
namespace ProvabilityAbstraction
variable {𝔅 : Prov... | module
public import Foundation.FirstOrder.Incompleteness.Examples
public import Foundation.Vorspiel.ENat
@[expose] public section
namespace LO.FirstOrder
variable {L : Language} [L.ReferenceableBy L] {T₀ T : Theory L}
open ProvabilityAbstraction
open Classical
namespace ProvabilityAbstraction
variable {𝔅 : Prov... | @@ -21,6 +21,25 @@
lemma neg_iterated_prov (φ : Sentence L) : ∼(𝔅^[n] φ) = 𝔅.dia^[n] (∼φ) := by
induction n generalizing φ <;> simp [Provability.dia, *]
+lemma boxBot_monotone [T₀ ⪯ T] [𝔅.HBL] : n ≤ m → T ⊢ 𝔅^[n] ⊥ 🡒 𝔅^[m] ⊥ := by
+ revert m
+ suffices ∀ k, T ⊢ 𝔅^[n] ⊥ 🡒 𝔅^[n + k] ⊥ by
+ intro m hnm... | {
"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_352cf89c8d51_2 | 64ff9abe300fca98 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Incompleteness/ProvabilityAbstraction/Height.lean | Height | 2 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 4 | 1 | [
{
"theorem_name": "height_eq_top_of_sound_and_consistent",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": "\n height_eq_top_iff.mpr iIncon_unprovable_of_sigma1_sound",
"n_chars": 58,
"n_subproofs": 0,
"... | [
{
"name": "height_eq_top_iff",
"text": "lemma height_eq_top_iff : 𝔅.height = ⊤ ↔ ∀ n, T ⊬ 𝔅^[n] ⊥ := ENat.find_eq_top_iff _\n\n",
"fan_in": 1,
"n_lines": 3,
"n_chars": 101,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structu... | [
{
"name": "height_eq_top_of_sound_and_consistent",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 4,
"n_chars": 167,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": f... | 1 | module
public import Foundation.FirstOrder.Incompleteness.Examples
public import Foundation.Vorspiel.ENat
@[expose] public section
namespace LO.FirstOrder
variable {L : Language} [L.ReferenceableBy L] {T₀ T : Theory L}
open ProvabilityAbstraction
open Classical
namespace ProvabilityAbstraction
variable {𝔅 : Prov... | module
public import Foundation.FirstOrder.Incompleteness.Examples
public import Foundation.Vorspiel.ENat
@[expose] public section
namespace LO.FirstOrder
variable {L : Language} [L.ReferenceableBy L] {T₀ T : Theory L}
open ProvabilityAbstraction
open Classical
namespace ProvabilityAbstraction
variable {𝔅 : Prov... | @@ -29,10 +29,13 @@
namespace Provability
+lemma height_eq_top_iff : 𝔅.height = ⊤ ↔ ∀ n, T ⊬ 𝔅^[n] ⊥ := ENat.find_eq_top_iff _
+
lemma height_le_of_boxBot {n : ℕ} (h : T ⊢ 𝔅^[n] ⊥) : 𝔅.height ≤ n :=
ENat.find_le (T ⊢ 𝔅^[·] ⊥) n h
-lemma height_eq_top_of_sound_and_consistent [𝔅.Kreisel] [Entailment.Cons... | {
"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_352cf89c8d51_3 | fbfccd4f94b1a109 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Incompleteness/ProvabilityAbstraction/Height.lean | Height | 3 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 4 | 3 | [
{
"theorem_name": "height_lt_pos_of_boxBot",
"depth": 1,
"n_commands": 0,
"n_lines": 8,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n have e : n.pred.succ = n := Eq.symm <| (Nat.sub_eq_iff_eq_add pos).mp rfl\n have : T₀ ⊢ 𝔅 (𝔅^[n.pred] ⊥) := by r... | [
{
"name": "height_le_of_boxBot",
"text": "lemma height_le_of_boxBot {n : ℕ} (h : T ⊢ 𝔅^[n] ⊥) : 𝔅.height ≤ n :=\n ENat.find_le (T ⊢ 𝔅^[·] ⊥) n h\n\n",
"fan_in": 3,
"n_lines": 4,
"n_chars": 126,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewri... | [
{
"name": "height_eq_zero_of_inconsistent",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 6,
"n_chars": 218,
"n_subproofs": 1,
"n_tactics": 3,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
... | 1 | module
public import Foundation.FirstOrder.Incompleteness.Examples
public import Foundation.Vorspiel.ENat
@[expose] public section
namespace LO.FirstOrder
variable {L : Language} [L.ReferenceableBy L] {T₀ T : Theory L}
open ProvabilityAbstraction
open Classical
namespace ProvabilityAbstraction
variable {𝔅 : Prov... | module
public import Foundation.FirstOrder.Incompleteness.Examples
public import Foundation.Vorspiel.ENat
@[expose] public section
namespace LO.FirstOrder
variable {L : Language} [L.ReferenceableBy L] {T₀ T : Theory L}
open ProvabilityAbstraction
open Classical
namespace ProvabilityAbstraction
variable {𝔅 : Prov... | @@ -31,11 +31,16 @@
lemma height_eq_top_iff : 𝔅.height = ⊤ ↔ ∀ n, T ⊬ 𝔅^[n] ⊥ := ENat.find_eq_top_iff _
+lemma height_le_of_boxBot {n : ℕ} (h : T ⊢ 𝔅^[n] ⊥) : 𝔅.height ≤ n :=
+ ENat.find_le (T ⊢ 𝔅^[·] ⊥) n h
+
lemma height_eq_top_of_sound_and_consistent [𝔅.Kreisel] [Entailment.Consistent T] : 𝔅.height = ⊤... | {
"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_51ee21c851ca_0 | bd733d2b0131880e | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Exponential/PPow2.lean | PPow2 | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 11 | 6 | [
{
"theorem_name": "sqrt",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " ((hm.lenbit_iff hi.le pi (hm.two_lt hi ne2)).mp hi).2",
"n_chars": 54,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
... | [
{
"name": "lenbit_iff",
"text": "lemma lenbit_iff (hm : SPPow2 m) {i : V} (hi : i ≤ m) (pi : Pow2 i) (lt2 : 2 < i) :\n LenBit i m ↔ (√i)^2 = i ∧ LenBit (√i) m := hm.2.2 i hi pi lt2\n\n",
"fan_in": 6,
"n_lines": 4,
"n_chars": 161,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
... | [
{
"name": "sqrt",
"fan_in": 7,
"n_deps_direct": 2,
"n_deps_transitive": 3,
"n_lines": 4,
"n_chars": 163,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
"max_nesting": 1
}
] | 1 | module
public import Foundation.FirstOrder.Arithmetic.Exponential.Pow2
@[expose] public section
/-!
# On $\mathrm{PPow2}(x)$
$\mathrm{PPow2}(n)$ is a property that holds iff $n = 2^{2^i}$ for some $i$.
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₀]
def SPPow2 (m :... | module
public import Foundation.FirstOrder.Arithmetic.Exponential.Pow2
@[expose] public section
/-!
# On $\mathrm{PPow2}(x)$
$\mathrm{PPow2}(n)$ is a property that holds iff $n = 2^{2^i}$ for some $i$.
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₀]
def SPPow2 (m :... | @@ -42,6 +42,9 @@
lemma lenbit_two (hm : SPPow2 m) : LenBit 2 m := hm.2.1
+lemma lenbit_iff (hm : SPPow2 m) {i : V} (hi : i ≤ m) (pi : Pow2 i) (lt2 : 2 < i) :
+ LenBit i m ↔ (√i)^2 = i ∧ LenBit (√i) m := hm.2.2 i hi pi lt2
+
lemma one_lt (hm : SPPow2 m) {i : V} (hi : LenBit i m) : 1 < i := by
by_contra 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_51ee21c851ca_1 | d88e5829ddce2d72 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Exponential/PPow2.lean | PPow2 | 1 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 11 | 4 | [
{
"theorem_name": "of_sqrt",
"depth": 1,
"n_commands": 0,
"n_lines": 8,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n by_cases ne1 : i = 1\n · rcases ne1; simpa using hi\n · have ne2 : i ≠ 2 := by\n rintro rfl; simp [sqrt_two] at hsqi\n ha... | [
{
"name": "pos",
"text": "lemma pos {i : V} (ppi : PPow2 i) : 0 < i := ppi.pow2.pos\n\n",
"fan_in": 4,
"n_lines": 3,
"n_chars": 59,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
... | [
{
"name": "of_sqrt",
"fan_in": 1,
"n_deps_direct": 2,
"n_deps_transitive": 3,
"n_lines": 11,
"n_chars": 459,
"n_subproofs": 2,
"n_tactics": 10,
"cyclomatic": 3,
"n_automation": 3,
"n_rewrites": 0,
"n_structural": 3,
"automation_only": false,
"max_nesting": 6
... | 3 | module
public import Foundation.FirstOrder.Arithmetic.Exponential.Pow2
@[expose] public section
/-!
# On $\mathrm{PPow2}(x)$
$\mathrm{PPow2}(n)$ is a property that holds iff $n = 2^{2^i}$ for some $i$.
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₀]
def SPPow2 (m :... | module
public import Foundation.FirstOrder.Arithmetic.Exponential.Pow2
@[expose] public section
/-!
# On $\mathrm{PPow2}(x)$
$\mathrm{PPow2}(n)$ is a property that holds iff $n = 2^{2^i}$ for some $i$.
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₀]
def SPPow2 (m :... | @@ -66,7 +66,8 @@
· rcases ne1; simpa using hi
· have ne2 : i ≠ 2 := by
rintro rfl; simp [sqrt_two] at hsqi
- have : 2 < i := sorry
+ have : 2 < i := lt_of_le_of_ne
+ (one_lt_iff_two_le.mp <| lt_of_le_of_ne (pos_iff_one_le.mp pi.pos) <| Ne.symm ne1) (Ne.symm ne2)
exact (hm.lenbit_iff him pi... | {
"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_51ee21c851ca_2 | e52f2a9dc7d1cdf5 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Exponential/PPow2.lean | PPow2 | 2 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 11 | 3 | [
{
"theorem_name": "sq_le_of_lt",
"depth": 1,
"n_commands": 0,
"n_lines": 26,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n intro hij\n suffices ∀ i < j, Pow2 i → Pow2 j → LenBit i m → LenBit j m → i^2 ≤ j from this i hij pi pj hi hj\n clear i pi h... | [
{
"name": "four_le",
"text": "lemma four_le {i : V} (hi : PPow2 i) (ne : i ≠ 2) : 4 ≤ i := by\n by_contra A\n have : i ≤ 3 := by simpa [←three_add_one_eq_four, ←le_iff_lt_succ] using A\n rcases le_three_iff_eq_zero_or_one_or_two_or_three.mp this with (rfl | rfl | rfl | rfl) <;> simp at ne hi\n · have : ... | [
{
"name": "sq_le_of_lt",
"fan_in": 3,
"n_deps_direct": 6,
"n_deps_transitive": 17,
"n_lines": 28,
"n_chars": 1330,
"n_subproofs": 5,
"n_tactics": 26,
"cyclomatic": 4,
"n_automation": 8,
"n_rewrites": 1,
"n_structural": 7,
"automation_only": false,
"max_nesting... | 11 | module
public import Foundation.FirstOrder.Arithmetic.Exponential.Pow2
@[expose] public section
/-!
# On $\mathrm{PPow2}(x)$
$\mathrm{PPow2}(n)$ is a property that holds iff $n = 2^{2^i}$ for some $i$.
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₀]
def SPPow2 (m :... | module
public import Foundation.FirstOrder.Arithmetic.Exponential.Pow2
@[expose] public section
/-!
# On $\mathrm{PPow2}(x)$
$\mathrm{PPow2}(n)$ is a property that holds iff $n = 2^{2^i}$ for some $i$.
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₀]
def SPPow2 (m :... | @@ -94,7 +94,8 @@
have : 2 ≤ i := one_lt_iff_two_le.mp (hm.one_lt hi)
exact False.elim ((not_lt.mpr this) hij)
· by_cases ine2 : i = 2
- · sorry
+ · rcases ine2 with rfl
+ simpa [sq, two_mul_two_eq_four] using pj.four_le hij
· have : √i < √j := by
by_contra 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_51ee21c851ca_3 | 01eb729eafafac72 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Exponential/PPow2.lean | PPow2 | 3 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 11 | 4 | [
{
"theorem_name": "pos",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " ppi.pow2.pos",
"n_chars": 13,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
... | [
{
"name": "pow2",
"text": "lemma pow2 {i : V} (h : PPow2 i) : Pow2 i := h.1\n\n",
"fan_in": 4,
"n_lines": 3,
"n_chars": 50,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
"max_nes... | [
{
"name": "pos",
"fan_in": 4,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 3,
"n_chars": 59,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
"max_nesting": 1
}
] | 1 | module
public import Foundation.FirstOrder.Arithmetic.Exponential.Pow2
@[expose] public section
/-!
# On $\mathrm{PPow2}(x)$
$\mathrm{PPow2}(n)$ is a property that holds iff $n = 2^{2^i}$ for some $i$.
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₀]
def SPPow2 (m :... | module
public import Foundation.FirstOrder.Arithmetic.Exponential.Pow2
@[expose] public section
/-!
# On $\mathrm{PPow2}(x)$
$\mathrm{PPow2}(n)$ is a property that holds iff $n = 2^{2^i}$ for some $i$.
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₀]
def SPPow2 (m :... | @@ -124,8 +124,10 @@
namespace PPow2
-lemma pos {i : V} (ppi : PPow2 i) : 0 < i := sorry
+lemma pow2 {i : V} (h : PPow2 i) : Pow2 i := h.1
+lemma pos {i : V} (ppi : PPow2 i) : 0 < i := ppi.pow2.pos
+
lemma one_lt {i : V} (ppi : PPow2 i) : 1 < i := by
rcases ppi with ⟨_, m, _, sppm, lb⟩; exact sppm.one_lt lb
... | {
"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_51ee21c851ca_4 | 68ce488f45395c21 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Exponential/PPow2.lean | PPow2 | 4 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 11 | 3 | [
{
"theorem_name": "sq_le_of_lt",
"depth": 1,
"n_commands": 0,
"n_lines": 26,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n intro hij\n suffices ∀ i < j, Pow2 i → Pow2 j → LenBit i m → LenBit j m → i^2 ≤ j from this i hij pi pj hi hj\n clear i pi h... | [
{
"name": "four_le",
"text": "lemma four_le {i : V} (hi : PPow2 i) (ne : i ≠ 2) : 4 ≤ i := by\n by_contra A\n have : i ≤ 3 := by simpa [←three_add_one_eq_four, ←le_iff_lt_succ] using A\n rcases le_three_iff_eq_zero_or_one_or_two_or_three.mp this with (rfl | rfl | rfl | rfl) <;> simp at ne hi\n · have : ... | [
{
"name": "elim'",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 17,
"n_lines": 7,
"n_chars": 276,
"n_subproofs": 1,
"n_tactics": 6,
"cyclomatic": 3,
"n_automation": 6,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": false,
"max_nesting": 4
}
... | 11 | module
public import Foundation.FirstOrder.Arithmetic.Exponential.Pow2
@[expose] public section
/-!
# On $\mathrm{PPow2}(x)$
$\mathrm{PPow2}(n)$ is a property that holds iff $n = 2^{2^i}$ for some $i$.
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₀]
def SPPow2 (m :... | module
public import Foundation.FirstOrder.Arithmetic.Exponential.Pow2
@[expose] public section
/-!
# On $\mathrm{PPow2}(x)$
$\mathrm{PPow2}(n)$ is a property that holds iff $n = 2^{2^i}$ for some $i$.
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₀]
def SPPow2 (m :... | @@ -94,7 +94,8 @@
have : 2 ≤ i := one_lt_iff_two_le.mp (hm.one_lt hi)
exact False.elim ((not_lt.mpr this) hij)
· by_cases ine2 : i = 2
- · sorry
+ · rcases ine2 with rfl
+ simpa [sq, two_mul_two_eq_four] using pj.four_le hij
· have : √i < √j := by
by_contra 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_51ee21c851ca_5 | 0d79fd38f6cd7215 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Exponential/PPow2.lean | PPow2 | 5 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 11 | 6 | [
{
"theorem_name": "sqrt",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " ((hm.lenbit_iff hi.le pi (hm.two_lt hi ne2)).mp hi).2",
"n_chars": 54,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
... | [
{
"name": "lenbit_iff",
"text": "lemma lenbit_iff (hm : SPPow2 m) {i : V} (hi : i ≤ m) (pi : Pow2 i) (lt2 : 2 < i) :\n LenBit i m ↔ (√i)^2 = i ∧ LenBit (√i) m := hm.2.2 i hi pi lt2\n\n",
"fan_in": 6,
"n_lines": 4,
"n_chars": 161,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
... | [
{
"name": "not_three",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 4,
"n_lines": 4,
"n_chars": 94,
"n_subproofs": 0,
"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.Exponential.Pow2
@[expose] public section
/-!
# On $\mathrm{PPow2}(x)$
$\mathrm{PPow2}(n)$ is a property that holds iff $n = 2^{2^i}$ for some $i$.
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₀]
def SPPow2 (m :... | module
public import Foundation.FirstOrder.Arithmetic.Exponential.Pow2
@[expose] public section
/-!
# On $\mathrm{PPow2}(x)$
$\mathrm{PPow2}(n)$ is a property that holds iff $n = 2^{2^i}$ for some $i$.
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₀]
def SPPow2 (m :... | @@ -42,6 +42,9 @@
lemma lenbit_two (hm : SPPow2 m) : LenBit 2 m := hm.2.1
+lemma lenbit_iff (hm : SPPow2 m) {i : V} (hi : i ≤ m) (pi : Pow2 i) (lt2 : 2 < i) :
+ LenBit i m ↔ (√i)^2 = i ∧ LenBit (√i) m := hm.2.2 i hi pi lt2
+
lemma one_lt (hm : SPPow2 m) {i : V} (hi : LenBit i m) : 1 < i := by
by_contra 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_51ee21c851ca_6 | 57cafd6993ed8c8a | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Exponential/PPow2.lean | PPow2 | 6 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 11 | 3 | [
{
"theorem_name": "sq_le_of_lt",
"depth": 1,
"n_commands": 0,
"n_lines": 26,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n intro hij\n suffices ∀ i < j, Pow2 i → Pow2 j → LenBit i m → LenBit j m → i^2 ≤ j from this i hij pi pj hi hj\n clear i pi h... | [
{
"name": "four_le",
"text": "lemma four_le {i : V} (hi : PPow2 i) (ne : i ≠ 2) : 4 ≤ i := by\n by_contra A\n have : i ≤ 3 := by simpa [←three_add_one_eq_four, ←le_iff_lt_succ] using A\n rcases le_three_iff_eq_zero_or_one_or_two_or_three.mp this with (rfl | rfl | rfl | rfl) <;> simp at ne hi\n · have : ... | [
{
"name": "four_lt",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 5,
"n_lines": 4,
"n_chars": 126,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
"max_nesting": 2
}... | 2 | module
public import Foundation.FirstOrder.Arithmetic.Exponential.Pow2
@[expose] public section
/-!
# On $\mathrm{PPow2}(x)$
$\mathrm{PPow2}(n)$ is a property that holds iff $n = 2^{2^i}$ for some $i$.
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₀]
def SPPow2 (m :... | module
public import Foundation.FirstOrder.Arithmetic.Exponential.Pow2
@[expose] public section
/-!
# On $\mathrm{PPow2}(x)$
$\mathrm{PPow2}(n)$ is a property that holds iff $n = 2^{2^i}$ for some $i$.
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₀]
def SPPow2 (m :... | @@ -94,7 +94,8 @@
have : 2 ≤ i := one_lt_iff_two_le.mp (hm.one_lt hi)
exact False.elim ((not_lt.mpr this) hij)
· by_cases ine2 : i = 2
- · sorry
+ · rcases ine2 with rfl
+ simpa [sq, two_mul_two_eq_four] using pj.four_le hij
· have : √i < √j := by
by_contra 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_51ee21c851ca_7 | 2a7b4bbeb2621063 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Exponential/PPow2.lean | PPow2 | 7 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 11 | 2 | [
{
"theorem_name": "sq",
"depth": 1,
"n_commands": 0,
"n_lines": 40,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n rcases ppi.exists_spp with ⟨m, hm, sppm, hi⟩\n have sppm' : SPPow2 (m + i^2) := by\n refine ⟨?_, ?_, ?_⟩\n · rw [LenBit.add_pow2... | [
{
"name": "two_le",
"text": "lemma two_le {i : V} (hi : PPow2 i) : 2 ≤ i := by\n simp [←one_add_one_eq_two, ←lt_iff_succ_le, hi.one_lt]\n\n",
"fan_in": 2,
"n_lines": 4,
"n_chars": 114,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
... | [
{
"name": "sq_ne_two",
"fan_in": 0,
"n_deps_direct": 2,
"n_deps_transitive": 2,
"n_lines": 5,
"n_chars": 167,
"n_subproofs": 1,
"n_tactics": 4,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 2,
"automation_only": false,
"max_nesting": 2
... | 1 | module
public import Foundation.FirstOrder.Arithmetic.Exponential.Pow2
@[expose] public section
/-!
# On $\mathrm{PPow2}(x)$
$\mathrm{PPow2}(n)$ is a property that holds iff $n = 2^{2^i}$ for some $i$.
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₀]
def SPPow2 (m :... | module
public import Foundation.FirstOrder.Arithmetic.Exponential.Pow2
@[expose] public section
/-!
# On $\mathrm{PPow2}(x)$
$\mathrm{PPow2}(n)$ is a property that holds iff $n = 2^{2^i}$ for some $i$.
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₀]
def SPPow2 (m :... | @@ -168,7 +168,7 @@
refine ⟨?_, ?_, ?_⟩
· rw [LenBit.add_pow2] <;> simp [ppi.pow2, sppm.not_lenbit_one, sppm.one_lt hi]
· have : i ≠ 1 := by rintro rfl; exact sppm.not_lenbit_one hi
- have : 2 < i^2 := sorry
+ have : 2 < i^2 := lt_of_le_of_ne (ppi.pow2.sq.two_le <| by simp [this]) (by 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_51ee21c851ca_8 | 3001d43e164a50e2 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Exponential/PPow2.lean | PPow2 | 8 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 11 | 6 | [
{
"theorem_name": "sqrt",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " ((hm.lenbit_iff hi.le pi (hm.two_lt hi ne2)).mp hi).2",
"n_chars": 54,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
... | [
{
"name": "lenbit_iff",
"text": "lemma lenbit_iff (hm : SPPow2 m) {i : V} (hi : i ≤ m) (pi : Pow2 i) (lt2 : 2 < i) :\n LenBit i m ↔ (√i)^2 = i ∧ LenBit (√i) m := hm.2.2 i hi pi lt2\n\n",
"fan_in": 6,
"n_lines": 4,
"n_chars": 161,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
... | [
{
"name": "sqrt_ne_two",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 4,
"n_lines": 6,
"n_chars": 206,
"n_subproofs": 1,
"n_tactics": 4,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": false,
"max_nesting": 2... | 1 | module
public import Foundation.FirstOrder.Arithmetic.Exponential.Pow2
@[expose] public section
/-!
# On $\mathrm{PPow2}(x)$
$\mathrm{PPow2}(n)$ is a property that holds iff $n = 2^{2^i}$ for some $i$.
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₀]
def SPPow2 (m :... | module
public import Foundation.FirstOrder.Arithmetic.Exponential.Pow2
@[expose] public section
/-!
# On $\mathrm{PPow2}(x)$
$\mathrm{PPow2}(n)$ is a property that holds iff $n = 2^{2^i}$ for some $i$.
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₀]
def SPPow2 (m :... | @@ -42,6 +42,9 @@
lemma lenbit_two (hm : SPPow2 m) : LenBit 2 m := hm.2.1
+lemma lenbit_iff (hm : SPPow2 m) {i : V} (hi : i ≤ m) (pi : Pow2 i) (lt2 : 2 < i) :
+ LenBit i m ↔ (√i)^2 = i ∧ LenBit (√i) m := hm.2.2 i hi pi lt2
+
lemma one_lt (hm : SPPow2 m) {i : V} (hi : LenBit i m) : 1 < i := by
by_contra 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_51ee21c851ca_9 | b03191e09d672733 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Exponential/PPow2.lean | PPow2 | 9 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 11 | 3 | [
{
"theorem_name": "sq_le_of_lt",
"depth": 1,
"n_commands": 0,
"n_lines": 26,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n intro hij\n suffices ∀ i < j, Pow2 i → Pow2 j → LenBit i m → LenBit j m → i^2 ≤ j from this i hij pi pj hi hj\n clear i pi h... | [
{
"name": "four_le",
"text": "lemma four_le {i : V} (hi : PPow2 i) (ne : i ≠ 2) : 4 ≤ i := by\n by_contra A\n have : i ≤ 3 := by simpa [←three_add_one_eq_four, ←le_iff_lt_succ] using A\n rcases le_three_iff_eq_zero_or_one_or_two_or_three.mp this with (rfl | rfl | rfl | rfl) <;> simp at ne hi\n · have : ... | [
{
"name": "sq_uniq",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 17,
"n_lines": 12,
"n_chars": 404,
"n_subproofs": 1,
"n_tactics": 9,
"cyclomatic": 1,
"n_automation": 2,
"n_rewrites": 0,
"n_structural": 2,
"automation_only": false,
"max_nesting": 4
... | 11 | module
public import Foundation.FirstOrder.Arithmetic.Exponential.Pow2
@[expose] public section
/-!
# On $\mathrm{PPow2}(x)$
$\mathrm{PPow2}(n)$ is a property that holds iff $n = 2^{2^i}$ for some $i$.
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₀]
def SPPow2 (m :... | module
public import Foundation.FirstOrder.Arithmetic.Exponential.Pow2
@[expose] public section
/-!
# On $\mathrm{PPow2}(x)$
$\mathrm{PPow2}(n)$ is a property that holds iff $n = 2^{2^i}$ for some $i$.
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₀]
def SPPow2 (m :... | @@ -94,7 +94,8 @@
have : 2 ≤ i := one_lt_iff_two_le.mp (hm.one_lt hi)
exact False.elim ((not_lt.mpr this) hij)
· by_cases ine2 : i = 2
- · sorry
+ · rcases ine2 with rfl
+ simpa [sq, two_mul_two_eq_four] using pj.four_le hij
· have : √i < √j := by
by_contra 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_51ee21c851ca_10 | 00befb644af7c563 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Exponential/PPow2.lean | PPow2 | 10 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 11 | 3 | [
{
"theorem_name": "sq_le_of_lt",
"depth": 1,
"n_commands": 0,
"n_lines": 26,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n intro hij\n suffices ∀ i < j, Pow2 i → Pow2 j → LenBit i m → LenBit j m → i^2 ≤ j from this i hij pi pj hi hj\n clear i pi h... | [
{
"name": "four_le",
"text": "lemma four_le {i : V} (hi : PPow2 i) (ne : i ≠ 2) : 4 ≤ i := by\n by_contra A\n have : i ≤ 3 := by simpa [←three_add_one_eq_four, ←le_iff_lt_succ] using A\n rcases le_three_iff_eq_zero_or_one_or_two_or_three.mp this with (rfl | rfl | rfl | rfl) <;> simp at ne hi\n · have : ... | [
{
"name": "two_mul_sq_uniq",
"fan_in": 0,
"n_deps_direct": 4,
"n_deps_transitive": 17,
"n_lines": 16,
"n_chars": 762,
"n_subproofs": 3,
"n_tactics": 13,
"cyclomatic": 1,
"n_automation": 6,
"n_rewrites": 0,
"n_structural": 2,
"automation_only": false,
"max_nest... | 11 | module
public import Foundation.FirstOrder.Arithmetic.Exponential.Pow2
@[expose] public section
/-!
# On $\mathrm{PPow2}(x)$
$\mathrm{PPow2}(n)$ is a property that holds iff $n = 2^{2^i}$ for some $i$.
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₀]
def SPPow2 (m :... | module
public import Foundation.FirstOrder.Arithmetic.Exponential.Pow2
@[expose] public section
/-!
# On $\mathrm{PPow2}(x)$
$\mathrm{PPow2}(n)$ is a property that holds iff $n = 2^{2^i}$ for some $i$.
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₀]
def SPPow2 (m :... | @@ -94,7 +94,8 @@
have : 2 ≤ i := one_lt_iff_two_le.mp (hm.one_lt hi)
exact False.elim ((not_lt.mpr this) hij)
· by_cases ine2 : i = 2
- · sorry
+ · rcases ine2 with rfl
+ simpa [sq, two_mul_two_eq_four] using pj.four_le hij
· have : √i < √j := by
by_contra 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_3ff5dee7e6da_0 | 91930176d89e4184 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Semantics/CoherenceSpace/Basic.lean | Basic | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 3 | 2 | [
{
"theorem_name": "mk_strictCoherence_mk_iff",
"depth": 1,
"n_commands": 0,
"n_lines": 3,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n simp [StrictCoherence.iff_coherence_ne,\n StrictIncoherence.iff_incoherence_ne, mk_coherence_mk_iff]",
"n_... | [
{
"name": "StrictCoherence.iff_coherence_ne",
"text": "lemma StrictCoherence.iff_coherence_ne (x y : α) : x ⌢ y ↔ x ⁐ y ∧ x ≠ y := by\n simp [StrictCoherence, Incoherence]\n\n",
"fan_in": 2,
"n_lines": 4,
"n_chars": 129,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
"n_auto... | [
{
"name": "mk_strictCoherence_mk_iff",
"fan_in": 0,
"n_deps_direct": 3,
"n_deps_transitive": 3,
"n_lines": 6,
"n_chars": 213,
"n_subproofs": 0,
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"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"ma... | 2 | module
public import Foundation.Vorspiel.Nat.Matrix
public import Foundation.Vorspiel.NotationClass
/-!
# Coherence space for denotational semantics of logics
Reference: Jean-Yves Girard, Paul Taylor, Yves Lafont, Proofs and Types [G.T.L89]
-/
@[expose] public section
namespace LO
/-- A coherence space is a set e... | module
public import Foundation.Vorspiel.Nat.Matrix
public import Foundation.Vorspiel.NotationClass
/-!
# Coherence space for denotational semantics of logics
Reference: Jean-Yves Girard, Paul Taylor, Yves Lafont, Proofs and Types [G.T.L89]
-/
@[expose] public section
namespace LO
/-- A coherence space is a set e... | @@ -71,6 +71,9 @@
infix:40 " ⌢ " => StrictCoherence
+lemma StrictCoherence.iff_coherence_ne (x y : α) : x ⌢ y ↔ x ⁐ y ∧ x ≠ y := by
+ simp [StrictCoherence, Incoherence]
+
lemma StrictCoherence.symm {x y : α} : x ⌢ y → y ⌢ x := by
simp [StrictCoherence]; grind
@@ -226,7 +229,9 @@
exact Coherence.mk 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_3ff5dee7e6da_1 | 2a8dc8b49e94b51d | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Semantics/CoherenceSpace/Basic.lean | Basic | 1 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 3 | 1 | [
{
"theorem_name": "arrowPar_strictCoherence_iff",
"depth": 1,
"n_commands": 0,
"n_lines": 3,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n simp [StrictCoherence.iff_coherence_ne, arrowPar_coherence_iff]\n grind",
"n_chars": 77,
"n_subproofs... | [
{
"name": "arrowPar_coherence_iff",
"text": "lemma arrowPar_coherence_iff (f g : (i : ι) → ρ i) :\n f ⁐ g ↔ f = g ∨ ∃ i, f i ⌢ g i := by\n constructor\n · rintro (h | ⟨i, h⟩) <;> grind\n · rintro (rfl | ⟨i, h⟩)\n · exact ArrowParCoherence.refl _\n · exact ArrowParCoherence.pointwise i h\n\n",
... | [
{
"name": "arrowPar_strictCoherence_iff",
"fan_in": 0,
"n_deps_direct": 2,
"n_deps_transitive": 2,
"n_lines": 6,
"n_chars": 179,
"n_subproofs": 0,
"n_tactics": 3,
"cyclomatic": 1,
"n_automation": 2,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
... | 2 | module
public import Foundation.Vorspiel.Nat.Matrix
public import Foundation.Vorspiel.NotationClass
/-!
# Coherence space for denotational semantics of logics
Reference: Jean-Yves Girard, Paul Taylor, Yves Lafont, Proofs and Types [G.T.L89]
-/
@[expose] public section
namespace LO
/-- A coherence space is a set e... | module
public import Foundation.Vorspiel.Nat.Matrix
public import Foundation.Vorspiel.NotationClass
/-!
# Coherence space for denotational semantics of logics
Reference: Jean-Yves Girard, Paul Taylor, Yves Lafont, Proofs and Types [G.T.L89]
-/
@[expose] public section
namespace LO
/-- A coherence space is a set e... | @@ -331,8 +331,18 @@
· exact ArrowParCoherence.refl _
· exact ArrowParCoherence.pointwise _ (symm h)
+lemma arrowPar_coherence_iff (f g : (i : ι) → ρ i) :
+ f ⁐ g ↔ f = g ∨ ∃ i, f i ⌢ g i := by
+ constructor
+ · rintro (h | ⟨i, h⟩) <;> grind
+ · rintro (rfl | ⟨i, h⟩)
+ · exact ArrowParCoherence.ref... | {
"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_c095e1feb02c_0 | 29ea59a9dc354644 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Bootstrapping/Syntax/Formula/Functions.lean | Functions | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 10 | 2 | [
{
"theorem_name": "IsUFormula.neg_iff",
"depth": 1,
"n_commands": 0,
"n_lines": 6,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n constructor\n · intro h; by_contra hp\n have Hp : IsUFormula L p := by by_contra hp; simp [neg_not_uformula hp] at h... | [
{
"name": "neg_not_uformula",
"text": "lemma neg_not_uformula {x : V} (h : ¬IsUFormula L x) :\n neg L x = 0 := construction.result_prop_not _ h\n\n",
"fan_in": 2,
"n_lines": 4,
"n_chars": 109,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrite... | [
{
"name": "IsUFormula.neg_iff",
"fan_in": 1,
"n_deps_direct": 2,
"n_deps_transitive": 2,
"n_lines": 8,
"n_chars": 272,
"n_subproofs": 1,
"n_tactics": 8,
"cyclomatic": 2,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 3,
"automation_only": false,
"max_nest... | 1 | module
public import Foundation.FirstOrder.Bootstrapping.Syntax.Formula.Basic
public import Foundation.FirstOrder.Bootstrapping.Syntax.Term.Functions
@[expose] public section
namespace LO.FirstOrder.Arithmetic.Bootstrapping
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
variable {L : Language} [L.Encoda... | module
public import Foundation.FirstOrder.Bootstrapping.Syntax.Formula.Basic
public import Foundation.FirstOrder.Bootstrapping.Syntax.Term.Functions
@[expose] public section
namespace LO.FirstOrder.Arithmetic.Bootstrapping
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
variable {L : Language} [L.Encoda... | @@ -97,6 +97,9 @@
@[simp] lemma neg_ex {p : V} (hp : IsUFormula L p) :
neg L (^∃ p) = ^∀ (neg L p) := by simp [neg, hp, construction]
+lemma neg_not_uformula {x : V} (h : ¬IsUFormula L x) :
+ neg L x = 0 := construction.result_prop_not _ h
+
lemma IsUFormula.neg {p : V} : IsUFormula L p → IsUFormula L (neg ... | {
"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_c095e1feb02c_1 | 045cdbccf2ece010 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Bootstrapping/Syntax/Formula/Functions.lean | Functions | 1 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 10 | 2 | [
{
"theorem_name": "IsUFormula.shift_iff",
"depth": 1,
"n_commands": 0,
"n_lines": 6,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n constructor\n · intro h; by_contra hp\n have Hp : IsUFormula L p := by by_contra hp; simp [shift_not_uformula hp] ... | [
{
"name": "shift_not_uformula",
"text": "lemma shift_not_uformula {x : V} (h : ¬IsUFormula L x) :\n shift L x = 0 := (construction L).result_prop_not _ h\n\n",
"fan_in": 2,
"n_lines": 4,
"n_chars": 117,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
... | [
{
"name": "IsUFormula.shift_iff",
"fan_in": 0,
"n_deps_direct": 2,
"n_deps_transitive": 2,
"n_lines": 8,
"n_chars": 280,
"n_subproofs": 1,
"n_tactics": 8,
"cyclomatic": 2,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 3,
"automation_only": false,
"max_ne... | 1 | module
public import Foundation.FirstOrder.Bootstrapping.Syntax.Formula.Basic
public import Foundation.FirstOrder.Bootstrapping.Syntax.Term.Functions
@[expose] public section
namespace LO.FirstOrder.Arithmetic.Bootstrapping
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
variable {L : Language} [L.Encoda... | module
public import Foundation.FirstOrder.Bootstrapping.Syntax.Formula.Basic
public import Foundation.FirstOrder.Bootstrapping.Syntax.Term.Functions
@[expose] public section
namespace LO.FirstOrder.Arithmetic.Bootstrapping
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
variable {L : Language} [L.Encoda... | @@ -312,6 +312,9 @@
@[simp] lemma shift_exs {p : V} (hp : IsUFormula L p) :
shift L (^∃ p) = ^∃ (shift L p) := by simp [shift, hp, construction]
+lemma shift_not_uformula {x : V} (h : ¬IsUFormula L x) :
+ shift L x = 0 := (construction L).result_prop_not _ h
+
lemma IsUFormula.shift {p : V} : IsUFormula L p... | {
"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_c095e1feb02c_2 | 486ec5bc33581558 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Bootstrapping/Syntax/Formula/Functions.lean | Functions | 2 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 10 | 1 | [
{
"theorem_name": "subst_eq_self₁",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " subst_eq_self hp (by simp) (by simp)",
"n_chars": 37,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_au... | [
{
"name": "subst_eq_self",
"text": "lemma subst_eq_self {n w : V} (hp : IsSemiformula L n p) (hw : IsSemitermVec L n n w) (H : ∀ i < n, w.[i] = ^#i) :\n subst L w p = p := by\n revert w\n apply IsSemiformula.pi1_structural_induction ?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_ hp\n · definability\n · intro n k R v hR ... | [
{
"name": "subst_eq_self₁",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 3,
"n_lines": 4,
"n_chars": 123,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 2,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"max_nesting":... | 1 | module
public import Foundation.FirstOrder.Bootstrapping.Syntax.Formula.Basic
public import Foundation.FirstOrder.Bootstrapping.Syntax.Term.Functions
@[expose] public section
namespace LO.FirstOrder.Arithmetic.Bootstrapping
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
variable {L : Language} [L.Encoda... | module
public import Foundation.FirstOrder.Bootstrapping.Syntax.Formula.Basic
public import Foundation.FirstOrder.Bootstrapping.Syntax.Term.Functions
@[expose] public section
namespace LO.FirstOrder.Arithmetic.Bootstrapping
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
variable {L : Language} [L.Encoda... | @@ -513,8 +513,50 @@
· simpa [h₁.isUFormula] using ih p₁ (by simp) (qVec L w) (by simp [fw]) (n + 1) (by simp [fn]) (m + 1) (by simp [fm]) h₁ hw.qVec
· simpa [h₁.isUFormula] using ih p₁ (by simp) (qVec L w) (by simp [fw]) (n + 1) (by simp [fn]) (m + 1) (by simp [fm]) h₁ hw.qVec
+lemma subst_eq_self {n w : V} (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_c095e1feb02c_3 | 11e2a2f2e7a995bb | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Bootstrapping/Syntax/Formula/Functions.lean | Functions | 3 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 10 | 2 | [
{
"theorem_name": "fomulaComplexity_substs1",
"depth": 1,
"n_commands": 0,
"n_lines": 3,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n unfold substs1\n rw [fomulaComplexity_substs hp (IsSemitermVec.singleton.mpr ht)]",
"n_chars": 87,
"n_sub... | [
{
"name": "fomulaComplexity_substs",
"text": "lemma fomulaComplexity_substs {p : V} (hp : IsSemiformula L n p) :\n IsSemitermVec L n m w → formulaComplexity L (subst L w p) = formulaComplexity L p := by\n revert m w\n apply IsSemiformula.pi1_structural_induction ?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_ hp\n · defin... | [
{
"name": "fomulaComplexity_substs1",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 15,
"n_lines": 7,
"n_chars": 248,
"n_subproofs": 0,
"n_tactics": 3,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 2,
"n_structural": 0,
"automation_only": false,
"m... | 1 | module
public import Foundation.FirstOrder.Bootstrapping.Syntax.Formula.Basic
public import Foundation.FirstOrder.Bootstrapping.Syntax.Term.Functions
@[expose] public section
namespace LO.FirstOrder.Arithmetic.Bootstrapping
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
variable {L : Language} [L.Encoda... | module
public import Foundation.FirstOrder.Bootstrapping.Syntax.Formula.Basic
public import Foundation.FirstOrder.Bootstrapping.Syntax.Term.Functions
@[expose] public section
namespace LO.FirstOrder.Arithmetic.Bootstrapping
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
variable {L : Language} [L.Encoda... | @@ -693,8 +693,44 @@
· intro p hp ihp; simp [hp, ihp]
· intro p hp ihp; simp [hp, ihp]
+lemma fomulaComplexity_substs {p : V} (hp : IsSemiformula L n p) :
+ IsSemitermVec L n m w → formulaComplexity L (subst L w p) = formulaComplexity L p := by
+ revert m w
+ apply IsSemiformula.pi1_structural_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_c095e1feb02c_4 | b47fb5ef1363a4b5 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Bootstrapping/Syntax/Formula/Functions.lean | Functions | 4 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 10 | 2 | [
{
"theorem_name": "fomulaComplexity_substs1",
"depth": 1,
"n_commands": 0,
"n_lines": 3,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n unfold substs1\n rw [fomulaComplexity_substs hp (IsSemitermVec.singleton.mpr ht)]",
"n_chars": 87,
"n_sub... | [
{
"name": "fomulaComplexity_substs",
"text": "lemma fomulaComplexity_substs {p : V} (hp : IsSemiformula L n p) :\n IsSemitermVec L n m w → formulaComplexity L (subst L w p) = formulaComplexity L p := by\n revert m w\n apply IsSemiformula.pi1_structural_induction ?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_ hp\n · defin... | [
{
"name": "fomulaComplexity_free1",
"fan_in": 2,
"n_deps_direct": 2,
"n_deps_transitive": 16,
"n_lines": 11,
"n_chars": 421,
"n_subproofs": 1,
"n_tactics": 8,
"cyclomatic": 1,
"n_automation": 3,
"n_rewrites": 3,
"n_structural": 3,
"automation_only": false,
"ma... | 1 | module
public import Foundation.FirstOrder.Bootstrapping.Syntax.Formula.Basic
public import Foundation.FirstOrder.Bootstrapping.Syntax.Term.Functions
@[expose] public section
namespace LO.FirstOrder.Arithmetic.Bootstrapping
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
variable {L : Language} [L.Encoda... | module
public import Foundation.FirstOrder.Bootstrapping.Syntax.Formula.Basic
public import Foundation.FirstOrder.Bootstrapping.Syntax.Term.Functions
@[expose] public section
namespace LO.FirstOrder.Arithmetic.Bootstrapping
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
variable {L : Language} [L.Encoda... | @@ -693,8 +693,44 @@
· intro p hp ihp; simp [hp, ihp]
· intro p hp ihp; simp [hp, ihp]
+lemma fomulaComplexity_substs {p : V} (hp : IsSemiformula L n p) :
+ IsSemitermVec L n m w → formulaComplexity L (subst L w p) = formulaComplexity L p := by
+ revert m w
+ apply IsSemiformula.pi1_structural_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_c095e1feb02c_5 | e80314e63bfa4863 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Bootstrapping/Syntax/Formula/Functions.lean | Functions | 5 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 10 | 2 | [
{
"theorem_name": "fomulaComplexity_substs1",
"depth": 1,
"n_commands": 0,
"n_lines": 3,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n unfold substs1\n rw [fomulaComplexity_substs hp (IsSemitermVec.singleton.mpr ht)]",
"n_chars": 87,
"n_sub... | [
{
"name": "fomulaComplexity_substs",
"text": "lemma fomulaComplexity_substs {p : V} (hp : IsSemiformula L n p) :\n IsSemitermVec L n m w → formulaComplexity L (subst L w p) = formulaComplexity L p := by\n revert m w\n apply IsSemiformula.pi1_structural_induction ?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_ hp\n · defin... | [
{
"name": "IsFormula.sigma1_structural_induction₂",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 17,
"n_lines": 47,
"n_chars": 2619,
"n_subproofs": 10,
"n_tactics": 49,
"cyclomatic": 2,
"n_automation": 28,
"n_rewrites": 1,
"n_structural": 15,
"automation_... | 2 | module
public import Foundation.FirstOrder.Bootstrapping.Syntax.Formula.Basic
public import Foundation.FirstOrder.Bootstrapping.Syntax.Term.Functions
@[expose] public section
namespace LO.FirstOrder.Arithmetic.Bootstrapping
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
variable {L : Language} [L.Encoda... | module
public import Foundation.FirstOrder.Bootstrapping.Syntax.Formula.Basic
public import Foundation.FirstOrder.Bootstrapping.Syntax.Term.Functions
@[expose] public section
namespace LO.FirstOrder.Arithmetic.Bootstrapping
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
variable {L : Language} [L.Encoda... | @@ -693,8 +693,44 @@
· intro p hp ihp; simp [hp, ihp]
· intro p hp ihp; simp [hp, ihp]
+lemma fomulaComplexity_substs {p : V} (hp : IsSemiformula L n p) :
+ IsSemitermVec L n m w → formulaComplexity L (subst L w p) = formulaComplexity L p := by
+ revert m w
+ apply IsSemiformula.pi1_structural_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_c095e1feb02c_6 | 41ee731e7509331d | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Bootstrapping/Syntax/Formula/Functions.lean | Functions | 6 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 10 | 2 | [
{
"theorem_name": "fomulaComplexity_substs1",
"depth": 1,
"n_commands": 0,
"n_lines": 3,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n unfold substs1\n rw [fomulaComplexity_substs hp (IsSemitermVec.singleton.mpr ht)]",
"n_chars": 87,
"n_sub... | [
{
"name": "fomulaComplexity_substs",
"text": "lemma fomulaComplexity_substs {p : V} (hp : IsSemiformula L n p) :\n IsSemitermVec L n m w → formulaComplexity L (subst L w p) = formulaComplexity L p := by\n revert m w\n apply IsSemiformula.pi1_structural_induction ?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_ ?_ hp\n · defin... | [
{
"name": "IsFormula.sigma1_structural_induction₂_ss",
"fan_in": 0,
"n_deps_direct": 2,
"n_deps_transitive": 17,
"n_lines": 99,
"n_chars": 4847,
"n_subproofs": 10,
"n_tactics": 55,
"cyclomatic": 2,
"n_automation": 28,
"n_rewrites": 3,
"n_structural": 15,
"automati... | 2 | module
public import Foundation.FirstOrder.Bootstrapping.Syntax.Formula.Basic
public import Foundation.FirstOrder.Bootstrapping.Syntax.Term.Functions
@[expose] public section
namespace LO.FirstOrder.Arithmetic.Bootstrapping
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
variable {L : Language} [L.Encoda... | module
public import Foundation.FirstOrder.Bootstrapping.Syntax.Formula.Basic
public import Foundation.FirstOrder.Bootstrapping.Syntax.Term.Functions
@[expose] public section
namespace LO.FirstOrder.Arithmetic.Bootstrapping
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
variable {L : Language} [L.Encoda... | @@ -693,8 +693,44 @@
· intro p hp ihp; simp [hp, ihp]
· intro p hp ihp; simp [hp, ihp]
+lemma fomulaComplexity_substs {p : V} (hp : IsSemiformula L n p) :
+ IsSemitermVec L n m w → formulaComplexity L (subst L w p) = formulaComplexity L p := by
+ revert m w
+ apply IsSemiformula.pi1_structural_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_2e0312786b76_0 | c88873b1b0226df9 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Bootstrapping/DerivabilityCondition/EquationalTheory.lean | EquationalTheory | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 16 | 2 | [
{
"theorem_name": "term_replace_aux",
"depth": 1,
"n_commands": 0,
"n_lines": 97,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n apply IsSemiterm.sigma1_induction\n · definability\n case hfunc =>\n intro k F v hF hv ih\n rcases isFunc_iff_LOR... | [
{
"name": "vec2_eq",
"text": "lemma vec2_eq {v : V} (h : len v = 2) : ?[v.[0], v.[1]] = v :=\n nth_ext' 2 (by simp [one_add_one_eq_two]) h (by\n intro i hi\n have : i = 0 ∨ i = 1 := le_one_iff_eq_zero_or_one.mp (lt_two_iff_le_one.mp hi)\n rcases this with (rfl | rfl) <;> simp)\n\n",
"fan_in": ... | [
{
"name": "term_replace_aux",
"fan_in": 1,
"n_deps_direct": 4,
"n_deps_transitive": 4,
"n_lines": 101,
"n_chars": 6269,
"n_subproofs": 37,
"n_tactics": 102,
"cyclomatic": 5,
"n_automation": 48,
"n_rewrites": 36,
"n_structural": 22,
"automation_only": false,
"m... | 3 | module
public import Foundation.Meta.ClProver
public import Foundation.FirstOrder.Arithmetic.Basic.Model
public import Foundation.FirstOrder.Bootstrapping.DerivabilityCondition.D1
@[expose] public section
/-!
# Bootstrapping theory of equality
-/
namespace LO.FirstOrder.Arithmetic.Bootstrapping
open Classical Entai... | module
public import Foundation.Meta.ClProver
public import Foundation.FirstOrder.Arithmetic.Basic.Model
public import Foundation.FirstOrder.Bootstrapping.DerivabilityCondition.D1
@[expose] public section
/-!
# Bootstrapping theory of equality
-/
namespace LO.FirstOrder.Arithmetic.Bootstrapping
open Classical Entai... | @@ -97,6 +97,12 @@
have := by simpa using internal_provable_of_outer_provable this (V := V)
simpa using TProof.specialize₄! this u₂ u₁ t₂ t₁
+lemma vec2_eq {v : V} (h : len v = 2) : ?[v.[0], v.[1]] = v :=
+ nth_ext' 2 (by simp [one_add_one_eq_two]) h (by
+ intro i hi
+ have : i = 0 ∨ i = 1 := le_one_iff_... | {
"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_2e0312786b76_1 | 34c990b9221e1307 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Bootstrapping/DerivabilityCondition/EquationalTheory.lean | EquationalTheory | 1 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 16 | 4 | [
{
"theorem_name": "replace_eq",
"depth": 1,
"n_commands": 0,
"n_lines": 24,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n suffices\n T.internalize V ⊢\n (&'1 ≐ &'0) 🡒\n (t⇞⇞.subst ![&'1] ≐ u⇞⇞.subst ![&'1]) 🡒\n (t⇞⇞.subst... | [
{
"name": "term_replace'",
"text": "lemma term_replace' (t : Semiterm V ℒₒᵣ 1) (u₁ u₂ : Term V ℒₒᵣ) :\n T.internalize V ⊢ (u₁ ≐ u₂) 🡒 (t.subst ![u₁] ≐ t.subst ![u₂]) := by\n have := TProof.specialize₂! (term_replace T t) u₂ u₁\n simpa [Semiterm.substs_substs] using this\n\n",
"fan_in": 4,
"n_l... | [
{
"name": "replace_eq",
"fan_in": 1,
"n_deps_direct": 2,
"n_deps_transitive": 8,
"n_lines": 27,
"n_chars": 1250,
"n_subproofs": 5,
"n_tactics": 24,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 18,
"n_structural": 5,
"automation_only": false,
"max_nesting"... | 7 | module
public import Foundation.Meta.ClProver
public import Foundation.FirstOrder.Arithmetic.Basic.Model
public import Foundation.FirstOrder.Bootstrapping.DerivabilityCondition.D1
@[expose] public section
/-!
# Bootstrapping theory of equality
-/
namespace LO.FirstOrder.Arithmetic.Bootstrapping
open Classical Entai... | module
public import Foundation.Meta.ClProver
public import Foundation.FirstOrder.Arithmetic.Basic.Model
public import Foundation.FirstOrder.Bootstrapping.DerivabilityCondition.D1
@[expose] public section
/-!
# Bootstrapping theory of equality
-/
namespace LO.FirstOrder.Arithmetic.Bootstrapping
open Classical Entai... | @@ -208,6 +208,11 @@
apply (tprovable_iff_provable (T := T)).mpr
simpa using term_replace_aux T t.val
+lemma term_replace' (t : Semiterm V ℒₒᵣ 1) (u₁ u₂ : Term V ℒₒᵣ) :
+ T.internalize V ⊢ (u₁ ≐ u₂) 🡒 (t.subst ![u₁] ≐ t.subst ![u₂]) := by
+ have := TProof.specialize₂! (term_replace T t) u₂ u₁
+ simpa [Sem... | {
"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_2f17a1389c2e_0 | 2e3520aebbef6dfd | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Completeness/CanonicalModel.lean | CanonicalModel | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 3 | 2 | [
{
"theorem_name": "dn_neg_iff",
"depth": 1,
"n_commands": 0,
"n_lines": 3,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n have := by simpa using (sound_minimal (Derivation.neg_doubleNegation φ) p)\n exact (this p (by simp)).symm",
"n_chars": 113... | [
{
"name": "sound_minimal",
"text": "lemma sound_minimal {φ : Propositionᵢ L} : 𝗠𝗶𝗻¹ ⊢ φ → ℙ⁻ ∀⊩ φ := by\n rintro ⟨d⟩ p; exact ⟨Forces.sound d p.val⟩\n\n",
"fan_in": 2,
"n_lines": 4,
"n_chars": 149,
"n_subproofs": 0,
"n_tactics": 3,
"cyclomatic": 2,
"n_automation": 0,
"n_r... | [
{
"name": "dn_neg_iff",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 5,
"n_chars": 204,
"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.Hauptsatz
public import Foundation.Logic.ForcingRelation
@[expose] public section
/-!
# Canonical model for classical first-order logic
Main reference: Jeremy Avigad, Algebraic proofs of cut elimination [Avi01]
-/
namespace LO.FirstOrder.Derivation.Canonical
variable {L... | module
public import Foundation.FirstOrder.Hauptsatz
public import Foundation.Logic.ForcingRelation
@[expose] public section
/-!
# Canonical model for classical first-order logic
Main reference: Jeremy Avigad, Algebraic proofs of cut elimination [Avi01]
-/
namespace LO.FirstOrder.Derivation.Canonical
variable {L... | @@ -136,6 +136,9 @@
not _ := not
monotone hφ _ hpq := hφ.monotone hpq
+lemma sound_minimal {φ : Propositionᵢ L} : 𝗠𝗶𝗻¹ ⊢ φ → ℙ⁻ ∀⊩ φ := by
+ rintro ⟨d⟩ p; exact ⟨Forces.sound d p.val⟩
+
end IsForced
namespace IsWeaklyForced
@@ -144,7 +147,9 @@
lemma iff_isForced {φ : Proposition L} {p : ℙ⁻} : p ⊩ᶜ φ ... | {
"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_2f17a1389c2e_1 | 31c8a10c601e54e9 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Completeness/CanonicalModel.lean | CanonicalModel | 1 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 3 | 6 | [
{
"theorem_name": "verum",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by simp [iff_isForced, IsForced.not]",
"n_chars": 37,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation"... | [
{
"name": "iff_isForced",
"text": "lemma iff_isForced {φ : Proposition L} {p : ℙ⁻} : p ⊩ᶜ φ ↔ p ⊩ φᴺ := by rfl\n\n",
"fan_in": 6,
"n_lines": 3,
"n_chars": 94,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"au... | [
{
"name": "verum",
"fan_in": 0,
"n_deps_direct": 2,
"n_deps_transitive": 3,
"n_lines": 3,
"n_chars": 89,
"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.Hauptsatz
public import Foundation.Logic.ForcingRelation
@[expose] public section
/-!
# Canonical model for classical first-order logic
Main reference: Jeremy Avigad, Algebraic proofs of cut elimination [Avi01]
-/
namespace LO.FirstOrder.Derivation.Canonical
variable {L... | module
public import Foundation.FirstOrder.Hauptsatz
public import Foundation.Logic.ForcingRelation
@[expose] public section
/-!
# Canonical model for classical first-order logic
Main reference: Jeremy Avigad, Algebraic proofs of cut elimination [Avi01]
-/
namespace LO.FirstOrder.Derivation.Canonical
variable {L... | @@ -145,24 +145,30 @@
open IsForced
+lemma iff_isForced {φ : Proposition L} {p : ℙ⁻} : p ⊩ᶜ φ ↔ p ⊩ φᴺ := by rfl
+
lemma dn_neg_iff {φ : Proposition L} {p : ℙ⁻} : p ⊩ᶜ ∼φ ↔ p ⊩ ∼φᴺ := by
have := by simpa using (sound_minimal (Derivation.neg_doubleNegation φ) p)
exact (this p (by simp)).symm
-@[simp] lemma... | {
"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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