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_df8bcbceb234_2 | 04dfe094d9cabae6 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/PeanoMinus/Functions.lean | Functions | 2 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 10 | 2 | [
{
"theorem_name": "zero_sub",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " sub_spec_of_le (by simp)",
"n_chars": 25,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 1,
... | [
{
"name": "sub_spec_of_le",
"text": "lemma sub_spec_of_le (h : a ≤ b) : a - b = 0 := by\n rcases lt_or_eq_of_le h with (lt | rfl) <;> simp [sub_spec_of_lt, *]\n\n",
"fan_in": 2,
"n_lines": 4,
"n_chars": 125,
"n_subproofs": 0,
"n_tactics": 3,
"cyclomatic": 3,
"n_automation": 2,
... | [
{
"name": "zero_sub",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 3,
"n_lines": 3,
"n_chars": 72,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"max_nesting": 1
}
... | 1 | module
public import Foundation.FirstOrder.Arithmetic.Definability
public import Mathlib.Algebra.Prime.Lemmas
public import Foundation.Vorspiel.ExistsUnique
@[expose] public section
/-!
# Functions and relations defined in $\mathsf{PA^-}$
This file provides functions and relations defined in $\mathsf{PA^-}
-/
name... | module
public import Foundation.FirstOrder.Arithmetic.Definability
public import Mathlib.Algebra.Prime.Lemmas
public import Foundation.Vorspiel.ExistsUnique
@[expose] public section
/-!
# Functions and relations defined in $\mathsf{PA^-}$
This file provides functions and relations defined in $\mathsf{PA^-}
-/
name... | @@ -60,6 +60,9 @@
@[simp] lemma sub_self (a : V) : a - a = 0 :=
add_eq_left.mp (sub_spec_of_ge (a := a) (b := a) (by rfl)).symm
+lemma sub_spec_of_le (h : a ≤ b) : a - b = 0 := by
+ rcases lt_or_eq_of_le h with (lt | rfl) <;> simp [sub_spec_of_lt, *]
+
lemma sub_add_self_of_le (h : b ≤ a) : a - b + b = a := by ... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_df8bcbceb234_3 | ce8e02867d8088d0 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/PeanoMinus/Functions.lean | Functions | 3 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 10 | 6 | [
{
"theorem_name": "sub_le_self",
"depth": 1,
"n_commands": 0,
"n_lines": 5,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n have : b ≤ a ∨ a < b := le_or_gt b a\n rcases this with (hxy | hxy) <;> simp\n · simpa [← sub_spec_of_ge hxy] using show a - ... | [
{
"name": "sub_spec_of_ge",
"text": "lemma sub_spec_of_ge (h : a ≥ b) : a = b + (a - b) := (Classical.choose!_spec (sub_existsUnique a b)).1 h\n\n",
"fan_in": 6,
"n_lines": 3,
"n_chars": 109,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": ... | [
{
"name": "sub_sub",
"fan_in": 1,
"n_deps_direct": 4,
"n_deps_transitive": 6,
"n_lines": 16,
"n_chars": 658,
"n_subproofs": 2,
"n_tactics": 14,
"cyclomatic": 1,
"n_automation": 5,
"n_rewrites": 3,
"n_structural": 3,
"automation_only": false,
"max_nesting": 10
... | 3 | module
public import Foundation.FirstOrder.Arithmetic.Definability
public import Mathlib.Algebra.Prime.Lemmas
public import Foundation.Vorspiel.ExistsUnique
@[expose] public section
/-!
# Functions and relations defined in $\mathsf{PA^-}$
This file provides functions and relations defined in $\mathsf{PA^-}
-/
name... | module
public import Foundation.FirstOrder.Arithmetic.Definability
public import Mathlib.Algebra.Prime.Lemmas
public import Foundation.Vorspiel.ExistsUnique
@[expose] public section
/-!
# Functions and relations defined in $\mathsf{PA^-}$
This file provides functions and relations defined in $\mathsf{PA^-}
-/
name... | @@ -34,6 +34,8 @@
noncomputable scoped instance : Sub V := ⟨sub⟩
+lemma sub_spec_of_ge (h : a ≥ b) : a = b + (a - b) := (Classical.choose!_spec (sub_existsUnique a b)).1 h
+
lemma sub_spec_of_lt (h : a < b) : a - b = 0 := (Classical.choose!_spec (sub_existsUnique a b)).2 h
lemma sub_eq_iff : c = a - b ↔ ((a ≥ ... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_df8bcbceb234_4 | c71436b6b81dd980 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/PeanoMinus/Functions.lean | Functions | 4 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 10 | 6 | [
{
"theorem_name": "sub_le_self",
"depth": 1,
"n_commands": 0,
"n_lines": 5,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n have : b ≤ a ∨ a < b := le_or_gt b a\n rcases this with (hxy | hxy) <;> simp\n · simpa [← sub_spec_of_ge hxy] using show a - ... | [
{
"name": "sub_spec_of_ge",
"text": "lemma sub_spec_of_ge (h : a ≥ b) : a = b + (a - b) := (Classical.choose!_spec (sub_existsUnique a b)).1 h\n\n",
"fan_in": 6,
"n_lines": 3,
"n_chars": 109,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": ... | [
{
"name": "sub_mul",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 3,
"n_lines": 8,
"n_chars": 340,
"n_subproofs": 1,
"n_tactics": 6,
"cyclomatic": 1,
"n_automation": 4,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": false,
"max_nesting": 4
}... | 1 | module
public import Foundation.FirstOrder.Arithmetic.Definability
public import Mathlib.Algebra.Prime.Lemmas
public import Foundation.Vorspiel.ExistsUnique
@[expose] public section
/-!
# Functions and relations defined in $\mathsf{PA^-}$
This file provides functions and relations defined in $\mathsf{PA^-}
-/
name... | module
public import Foundation.FirstOrder.Arithmetic.Definability
public import Mathlib.Algebra.Prime.Lemmas
public import Foundation.Vorspiel.ExistsUnique
@[expose] public section
/-!
# Functions and relations defined in $\mathsf{PA^-}$
This file provides functions and relations defined in $\mathsf{PA^-}
-/
name... | @@ -34,6 +34,8 @@
noncomputable scoped instance : Sub V := ⟨sub⟩
+lemma sub_spec_of_ge (h : a ≥ b) : a = b + (a - b) := (Classical.choose!_spec (sub_existsUnique a b)).1 h
+
lemma sub_spec_of_lt (h : a < b) : a - b = 0 := (Classical.choose!_spec (sub_existsUnique a b)).2 h
lemma sub_eq_iff : c = a - b ↔ ((a ≥ ... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_df8bcbceb234_5 | 3a442f96dde24c9e | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/PeanoMinus/Functions.lean | Functions | 5 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 10 | 6 | [
{
"theorem_name": "sub_le_self",
"depth": 1,
"n_commands": 0,
"n_lines": 5,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n have : b ≤ a ∨ a < b := le_or_gt b a\n rcases this with (hxy | hxy) <;> simp\n · simpa [← sub_spec_of_ge hxy] using show a - ... | [
{
"name": "sub_spec_of_ge",
"text": "lemma sub_spec_of_ge (h : a ≥ b) : a = b + (a - b) := (Classical.choose!_spec (sub_existsUnique a b)).1 h\n\n",
"fan_in": 6,
"n_lines": 3,
"n_chars": 109,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": ... | [
{
"name": "mul_sub",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 4,
"n_lines": 3,
"n_chars": 95,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"max_nesting": 1
}
] | 2 | module
public import Foundation.FirstOrder.Arithmetic.Definability
public import Mathlib.Algebra.Prime.Lemmas
public import Foundation.Vorspiel.ExistsUnique
@[expose] public section
/-!
# Functions and relations defined in $\mathsf{PA^-}$
This file provides functions and relations defined in $\mathsf{PA^-}
-/
name... | module
public import Foundation.FirstOrder.Arithmetic.Definability
public import Mathlib.Algebra.Prime.Lemmas
public import Foundation.Vorspiel.ExistsUnique
@[expose] public section
/-!
# Functions and relations defined in $\mathsf{PA^-}$
This file provides functions and relations defined in $\mathsf{PA^-}
-/
name... | @@ -34,6 +34,8 @@
noncomputable scoped instance : Sub V := ⟨sub⟩
+lemma sub_spec_of_ge (h : a ≥ b) : a = b + (a - b) := (Classical.choose!_spec (sub_existsUnique a b)).1 h
+
lemma sub_spec_of_lt (h : a < b) : a - b = 0 := (Classical.choose!_spec (sub_existsUnique a b)).2 h
lemma sub_eq_iff : c = a - b ↔ ((a ≥ ... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_df8bcbceb234_6 | 8848303d16db277a | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/PeanoMinus/Functions.lean | Functions | 6 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 10 | 6 | [
{
"theorem_name": "sub_le_self",
"depth": 1,
"n_commands": 0,
"n_lines": 5,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n have : b ≤ a ∨ a < b := le_or_gt b a\n rcases this with (hxy | hxy) <;> simp\n · simpa [← sub_spec_of_ge hxy] using show a - ... | [
{
"name": "sub_spec_of_ge",
"text": "lemma sub_spec_of_ge (h : a ≥ b) : a = b + (a - b) := (Classical.choose!_spec (sub_existsUnique a b)).1 h\n\n",
"fan_in": 6,
"n_lines": 3,
"n_chars": 109,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": ... | [
{
"name": "sub_succ_add_succ",
"fan_in": 0,
"n_deps_direct": 3,
"n_deps_transitive": 9,
"n_lines": 7,
"n_chars": 402,
"n_subproofs": 0,
"n_tactics": 5,
"cyclomatic": 1,
"n_automation": 3,
"n_rewrites": 1,
"n_structural": 1,
"automation_only": false,
"max_nesti... | 5 | module
public import Foundation.FirstOrder.Arithmetic.Definability
public import Mathlib.Algebra.Prime.Lemmas
public import Foundation.Vorspiel.ExistsUnique
@[expose] public section
/-!
# Functions and relations defined in $\mathsf{PA^-}$
This file provides functions and relations defined in $\mathsf{PA^-}
-/
name... | module
public import Foundation.FirstOrder.Arithmetic.Definability
public import Mathlib.Algebra.Prime.Lemmas
public import Foundation.Vorspiel.ExistsUnique
@[expose] public section
/-!
# Functions and relations defined in $\mathsf{PA^-}$
This file provides functions and relations defined in $\mathsf{PA^-}
-/
name... | @@ -34,6 +34,8 @@
noncomputable scoped instance : Sub V := ⟨sub⟩
+lemma sub_spec_of_ge (h : a ≥ b) : a = b + (a - b) := (Classical.choose!_spec (sub_existsUnique a b)).1 h
+
lemma sub_spec_of_lt (h : a < b) : a - b = 0 := (Classical.choose!_spec (sub_existsUnique a b)).2 h
lemma sub_eq_iff : c = a - b ↔ ((a ≥ ... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_df8bcbceb234_7 | a7d53f2c15858d6d | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/PeanoMinus/Functions.lean | Functions | 7 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 10 | 6 | [
{
"theorem_name": "sub_le_self",
"depth": 1,
"n_commands": 0,
"n_lines": 5,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n have : b ≤ a ∨ a < b := le_or_gt b a\n rcases this with (hxy | hxy) <;> simp\n · simpa [← sub_spec_of_ge hxy] using show a - ... | [
{
"name": "sub_spec_of_ge",
"text": "lemma sub_spec_of_ge (h : a ≥ b) : a = b + (a - b) := (Classical.choose!_spec (sub_existsUnique a b)).1 h\n\n",
"fan_in": 6,
"n_lines": 3,
"n_chars": 109,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": ... | [
{
"name": "le_sub_one_of_lt",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 3,
"n_lines": 5,
"n_chars": 172,
"n_subproofs": 1,
"n_tactics": 3,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
"max_nestin... | 1 | module
public import Foundation.FirstOrder.Arithmetic.Definability
public import Mathlib.Algebra.Prime.Lemmas
public import Foundation.Vorspiel.ExistsUnique
@[expose] public section
/-!
# Functions and relations defined in $\mathsf{PA^-}$
This file provides functions and relations defined in $\mathsf{PA^-}
-/
name... | module
public import Foundation.FirstOrder.Arithmetic.Definability
public import Mathlib.Algebra.Prime.Lemmas
public import Foundation.Vorspiel.ExistsUnique
@[expose] public section
/-!
# Functions and relations defined in $\mathsf{PA^-}$
This file provides functions and relations defined in $\mathsf{PA^-}
-/
name... | @@ -34,6 +34,8 @@
noncomputable scoped instance : Sub V := ⟨sub⟩
+lemma sub_spec_of_ge (h : a ≥ b) : a = b + (a - b) := (Classical.choose!_spec (sub_existsUnique a b)).1 h
+
lemma sub_spec_of_lt (h : a < b) : a - b = 0 := (Classical.choose!_spec (sub_existsUnique a b)).2 h
lemma sub_eq_iff : c = a - b ↔ ((a ≥ ... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_df8bcbceb234_8 | ed5ee79f91da5e9b | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/PeanoMinus/Functions.lean | Functions | 8 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 10 | 2 | [
{
"theorem_name": "le_mul_self_of_pos_right",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n simpa [mul_comm a b] using le_mul_self_of_pos_left hy",
"n_chars": 59,
"n_subproofs": 0,
"n_tactics": ... | [
{
"name": "le_mul_self_of_pos_left",
"text": "lemma le_mul_self_of_pos_left (hy : 0 < b) : a ≤ b * a := by\n have : 1 * a ≤ b * a := mul_le_mul_of_nonneg_right (one_le_of_zero_lt b hy) (by simp)\n simpa using this\n\n",
"fan_in": 2,
"n_lines": 5,
"n_chars": 173,
"n_subproofs": 1,
"n_ta... | [
{
"name": "le_mul_self_of_pos_right",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 4,
"n_chars": 121,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"max... | 1 | module
public import Foundation.FirstOrder.Arithmetic.Definability
public import Mathlib.Algebra.Prime.Lemmas
public import Foundation.Vorspiel.ExistsUnique
@[expose] public section
/-!
# Functions and relations defined in $\mathsf{PA^-}$
This file provides functions and relations defined in $\mathsf{PA^-}
-/
name... | module
public import Foundation.FirstOrder.Arithmetic.Definability
public import Mathlib.Algebra.Prime.Lemmas
public import Foundation.Vorspiel.ExistsUnique
@[expose] public section
/-!
# Functions and relations defined in $\mathsf{PA^-}$
This file provides functions and relations defined in $\mathsf{PA^-}
-/
name... | @@ -104,15 +104,20 @@
section Dvd
-lemma le_mul_self_of_pos_right (hy : 0 < b) : a ≤ a * b := sorry
+lemma le_mul_self_of_pos_left (hy : 0 < b) : a ≤ b * a := by
+ have : 1 * a ≤ b * a := mul_le_mul_of_nonneg_right (one_le_of_zero_lt b hy) (by simp)
+ simpa using this
+lemma le_mul_self_of_pos_right (hy : 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_df8bcbceb234_9 | dfa504cd67109636 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/PeanoMinus/Functions.lean | Functions | 9 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 10 | 1 | [
{
"theorem_name": "le_of_dvd",
"depth": 1,
"n_commands": 0,
"n_lines": 4,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n rintro ⟨c, rfl⟩\n exact le_mul_self_of_pos_right\n (pos_iff_ne_zero.mpr (show c ≠ 0 from by rintro rfl; simp at h))",
"n_... | [
{
"name": "le_mul_self_of_pos_right",
"text": "lemma le_mul_self_of_pos_right (hy : 0 < b) : a ≤ a * b := by\n simpa [mul_comm a b] using le_mul_self_of_pos_left hy\n\n",
"fan_in": 1,
"n_lines": 4,
"n_chars": 121,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
"n_automation"... | [
{
"name": "not_dvd_of_lt",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 3,
"n_lines": 4,
"n_chars": 115,
"n_subproofs": 0,
"n_tactics": 3,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 2,
"automation_only": false,
"max_nesting":... | 3 | module
public import Foundation.FirstOrder.Arithmetic.Definability
public import Mathlib.Algebra.Prime.Lemmas
public import Foundation.Vorspiel.ExistsUnique
@[expose] public section
/-!
# Functions and relations defined in $\mathsf{PA^-}$
This file provides functions and relations defined in $\mathsf{PA^-}
-/
name... | module
public import Foundation.FirstOrder.Arithmetic.Definability
public import Mathlib.Algebra.Prime.Lemmas
public import Foundation.Vorspiel.ExistsUnique
@[expose] public section
/-!
# Functions and relations defined in $\mathsf{PA^-}$
This file provides functions and relations defined in $\mathsf{PA^-}
-/
name... | @@ -108,6 +108,9 @@
have : 1 * a ≤ b * a := mul_le_mul_of_nonneg_right (one_le_of_zero_lt b hy) (by simp)
simpa using this
+lemma le_mul_self_of_pos_right (hy : 0 < b) : a ≤ a * b := by
+ simpa [mul_comm a b] using le_mul_self_of_pos_left hy
+
open Classical
lemma dvd_iff_bounded {a b : V} : a ∣ b ↔ ∃ c ≤ ... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_df8bcbceb234_10 | 2f37fd1093e2a65d | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/PeanoMinus/Functions.lean | Functions | 10 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 10 | 1 | [
{
"theorem_name": "le_of_dvd",
"depth": 1,
"n_commands": 0,
"n_lines": 4,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n rintro ⟨c, rfl⟩\n exact le_mul_self_of_pos_right\n (pos_iff_ne_zero.mpr (show c ≠ 0 from by rintro rfl; simp at h))",
"n_... | [
{
"name": "le_mul_self_of_pos_right",
"text": "lemma le_mul_self_of_pos_right (hy : 0 < b) : a ≤ a * b := by\n simpa [mul_comm a b] using le_mul_self_of_pos_left hy\n\n",
"fan_in": 1,
"n_lines": 4,
"n_chars": 121,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
"n_automation"... | [
{
"name": "dvd_antisymm",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 3,
"n_lines": 9,
"n_chars": 370,
"n_subproofs": 0,
"n_tactics": 7,
"cyclomatic": 3,
"n_automation": 6,
"n_rewrites": 4,
"n_structural": 4,
"automation_only": false,
"max_nesting": ... | 3 | module
public import Foundation.FirstOrder.Arithmetic.Definability
public import Mathlib.Algebra.Prime.Lemmas
public import Foundation.Vorspiel.ExistsUnique
@[expose] public section
/-!
# Functions and relations defined in $\mathsf{PA^-}$
This file provides functions and relations defined in $\mathsf{PA^-}
-/
name... | module
public import Foundation.FirstOrder.Arithmetic.Definability
public import Mathlib.Algebra.Prime.Lemmas
public import Foundation.Vorspiel.ExistsUnique
@[expose] public section
/-!
# Functions and relations defined in $\mathsf{PA^-}$
This file provides functions and relations defined in $\mathsf{PA^-}
-/
name... | @@ -108,6 +108,9 @@
have : 1 * a ≤ b * a := mul_le_mul_of_nonneg_right (one_le_of_zero_lt b hy) (by simp)
simpa using this
+lemma le_mul_self_of_pos_right (hy : 0 < b) : a ≤ a * b := by
+ simpa [mul_comm a b] using le_mul_self_of_pos_left hy
+
open Classical
lemma dvd_iff_bounded {a b : V} : a ∣ b ↔ ∃ c ≤ ... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_df8bcbceb234_11 | 06badc74ec7b53ef | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/PeanoMinus/Functions.lean | Functions | 11 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 10 | 2 | [
{
"theorem_name": "not_dvd_of_lt",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n intro hb h; exact not_le.mpr hb (le_of_dvd pos h)",
"n_chars": 55,
"n_subproofs": 0,
"n_tactics": 3,
"cycloma... | [
{
"name": "le_of_dvd",
"text": "lemma le_of_dvd (h : 0 < b) : a ∣ b → a ≤ b := by\n rintro ⟨c, rfl⟩\n exact le_mul_self_of_pos_right\n (pos_iff_ne_zero.mpr (show c ≠ 0 from by rintro rfl; simp at h))\n\n",
"fan_in": 2,
"n_lines": 6,
"n_chars": 183,
"n_subproofs": 0,
"n_tactics": 5,
... | [
{
"name": "unit_iff_eq_one",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 6,
"n_lines": 6,
"n_chars": 163,
"n_subproofs": 0,
"n_tactics": 3,
"cyclomatic": 3,
"n_automation": 4,
"n_rewrites": 0,
"n_structural": 2,
"automation_only": false,
"max_nesting... | 4 | module
public import Foundation.FirstOrder.Arithmetic.Definability
public import Mathlib.Algebra.Prime.Lemmas
public import Foundation.Vorspiel.ExistsUnique
@[expose] public section
/-!
# Functions and relations defined in $\mathsf{PA^-}$
This file provides functions and relations defined in $\mathsf{PA^-}
-/
name... | module
public import Foundation.FirstOrder.Arithmetic.Definability
public import Mathlib.Algebra.Prime.Lemmas
public import Foundation.Vorspiel.ExistsUnique
@[expose] public section
/-!
# Functions and relations defined in $\mathsf{PA^-}$
This file provides functions and relations defined in $\mathsf{PA^-}
-/
name... | @@ -138,13 +138,18 @@
end Dvd
+lemma le_of_dvd (h : 0 < b) : a ∣ b → a ≤ b := by
+ rintro ⟨c, rfl⟩
+ exact le_mul_self_of_pos_right
+ (pos_iff_ne_zero.mpr (show c ≠ 0 from by rintro rfl; simp at h))
+
lemma dvd_antisymm : a ∣ b → b ∣ a → a = b := by
intro hx hy
rcases show a = 0 ∨ 0 < a from eq_zero_or... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_ad52969947f9_21 | 4ee1dc6913edd7c6 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Tableau.lean | Tableau | 21 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 48 | 3 | [
{
"theorem_name": "iff_not_mem₁_mem₂",
"depth": 1,
"n_commands": 0,
"n_lines": 4,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n constructor;\n . apply mem₂_of_not_mem₁ hMax;\n . apply Set.disjoint_right.mp $ disjoint_of_consistent hCon |>.1;",
... | [
{
"name": "disjoint_of_consistent",
"text": "lemma disjoint_of_consistent (hCon : t.Consistent 𝓢) : t.Disjoint := by\n constructor;\n . by_contra hC;\n obtain ⟨T, hT₁, hT₂, hT⟩ := by simpa [Disjoint] using hC;\n obtain ⟨φ, hφ⟩ := Set.nonempty_def.mp $ Set.nonempty_iff_ne_empty.mpr hT;\n apply hC... | [
{
"name": "iff_mem₂_fdisj",
"fan_in": 0,
"n_deps_direct": 2,
"n_deps_transitive": 16,
"n_lines": 15,
"n_chars": 434,
"n_subproofs": 0,
"n_tactics": 24,
"cyclomatic": 2,
"n_automation": 4,
"n_rewrites": 0,
"n_structural": 8,
"automation_only": false,
"max_nesti... | 11 | module
public import Foundation.Modal.Formula.Basic
public import Foundation.Modal.Entailment.K
public import Foundation.Vorspiel.Set.Basic
@[expose] public section
namespace LO.Modal
open LO.Entailment LO.Modal.Entailment
variable {α : Type*}
variable {S} [Entailment S (Formula α)]
variable {𝓢 : S}
def Tableau ... | module
public import Foundation.Modal.Formula.Basic
public import Foundation.Modal.Entailment.K
public import Foundation.Vorspiel.Set.Basic
@[expose] public section
namespace LO.Modal
open LO.Entailment LO.Modal.Entailment
variable {α : Type*}
variable {S} [Entailment S (Formula α)]
variable {𝓢 : S}
def Tableau ... | @@ -41,6 +41,18 @@
variable [Entailment.Cl 𝓢]
+lemma disjoint_of_consistent (hCon : t.Consistent 𝓢) : t.Disjoint := by
+ constructor;
+ . by_contra hC;
+ obtain ⟨T, hT₁, hT₂, hT⟩ := by simpa [Disjoint] using hC;
+ obtain ⟨φ, hφ⟩ := Set.nonempty_def.mp $ Set.nonempty_iff_ne_empty.mpr hT;
+ apply hCon (... | {
"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_76211d00003b_0 | 11d825ab72c9bfc2 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Formula/NNFormula.lean | NNFormula | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 1 | 1 | [
{
"theorem_name": "negneg",
"depth": 1,
"n_commands": 0,
"n_lines": 15,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n induction φ with\n | and φ ψ ihφ ihψ =>\n simp only [←neg_eq, neg, and.injEq];\n exact ⟨ihφ, ihψ⟩;\n | or φ ψ ihφ ihψ =>\n ... | [
{
"name": "neg_eq",
"text": "lemma neg_eq : neg φ = ∼φ := rfl\n\n",
"fan_in": 1,
"n_lines": 3,
"n_chars": 38,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"max_nesting": 1
}
] | [
{
"name": "negneg",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 18,
"n_chars": 433,
"n_subproofs": 0,
"n_tactics": 24,
"cyclomatic": 2,
"n_automation": 5,
"n_rewrites": 0,
"n_structural": 5,
"automation_only": false,
"max_nesting": 4
... | 1 | module
public import Foundation.Modal.Formula.Basic
@[expose] public section
namespace LO.Modal
variable {α : Type u}
inductive NNFormula (α : Type u) : Type u where
| atom : α → NNFormula α
| natom : α → NNFormula α
| falsum : NNFormula α
| verum : NNFormula α
| or : NNFormula α → NNFormula α → ... | module
public import Foundation.Modal.Formula.Basic
@[expose] public section
namespace LO.Modal
variable {α : Type u}
inductive NNFormula (α : Type u) : Type u where
| atom : α → NNFormula α
| natom : α → NNFormula α
| falsum : NNFormula α
| verum : NNFormula α
| or : NNFormula α → NNFormula α → ... | @@ -57,6 +57,8 @@
@[simp] lemma neg_inj (φ ψ : Formula α) : ∼φ = ∼ψ ↔ φ = ψ := by simp [NegAbbrev.neg];
+lemma neg_eq : neg φ = ∼φ := rfl
+
@[simp] lemma neg_atom (a : α) : ∼atom a = natom a := by rfl
@[simp] lemma neg_natom (a : α) : ∼natom a = atom a := by rfl
@@ -64,10 +66,18 @@
@[grind =]
lemma negneg : ... | {
"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_ef9f7cda8c07_0 | d1f5a8e0f558b697 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Logic/SumQuasiNormal.lean | SumQuasiNormal | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 9 | 1 | [
{
"theorem_name": "iff_provable_finite_provable",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " ⟨finite_provable_of_provable h, provable_of_finite_provable⟩",
"n_chars": 65,
"n_subproofs": 0,
"n_tacti... | [
{
"name": "finite_provable_of_provable",
"text": "lemma finite_provable_of_provable [DecidableEq α] [L₁.Substitution] (h : ∀ ξ ∈ L₂, ∀ s : Substitution _, ξ⟦s⟧ ∈ L₂) :\n sumQuasiNormal L₁ L₂ ⊢ φ → ∃ X : Finset _, (↑X ⊆ L₂) ∧ L₁ ⊢ X.conj 🡒 φ := by\n intro h;\n induction h using sumQuasiNormal.rec! with\n... | [
{
"name": "iff_provable_finite_provable",
"fan_in": 1,
"n_deps_direct": 2,
"n_deps_transitive": 5,
"n_lines": 4,
"n_chars": 310,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
... | 2 | module
public import Foundation.Modal.Logic.SumNormal
@[expose] public section
namespace LO.Modal
namespace Logic
variable {L L₁ L₂ : Logic α} {φ ψ : Formula α} {s : Substitution α}
inductive sumQuasiNormal (L₁ L₂ : Logic α) : Logic α
| mem₁ {φ : Formula α} : L₁ ⊢ φ → sumQuasiNormal L₁ L₂ φ
| mem₂ {φ : For... | module
public import Foundation.Modal.Logic.SumNormal
@[expose] public section
namespace LO.Modal
namespace Logic
variable {L L₁ L₂ : Logic α} {φ ψ : Formula α} {s : Substitution α}
inductive sumQuasiNormal (L₁ L₂ : Logic α) : Logic α
| mem₁ {φ : Formula α} : L₁ ⊢ φ → sumQuasiNormal L₁ L₂ φ
| mem₂ {φ : For... | @@ -143,8 +143,42 @@
apply iff_provable.mpr;
apply hX₂ hχ;
+lemma finite_provable_of_provable [DecidableEq α] [L₁.Substitution] (h : ∀ ξ ∈ L₂, ∀ s : Substitution _, ξ⟦s⟧ ∈ L₂) :
+ sumQuasiNormal L₁ L₂ ⊢ φ → ∃ X : Finset _, (↑X ⊆ L₂) ∧ L₁ ⊢ X.conj 🡒 φ := by
+ intro h;
+ induction h using sumQuasiNormal.rec! ... | {
"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_ef9f7cda8c07_1 | b1f7a7ea47d4ce9c | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Logic/SumQuasiNormal.lean | SumQuasiNormal | 1 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 9 | 1 | [
{
"theorem_name": "iff_provable_finite_provable_letterless",
"depth": 1,
"n_commands": 0,
"n_lines": 3,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n apply iff_provable_finite_provable;\n grind;",
"n_chars": 50,
"n_subproofs": 0,
"n_tac... | [
{
"name": "iff_provable_finite_provable",
"text": "lemma iff_provable_finite_provable [DecidableEq α] [L₁.Substitution] (h : ∀ ξ ∈ L₂, ∀ s : Substitution _, ξ⟦s⟧ ∈ L₂) :\n sumQuasiNormal L₁ L₂ ⊢ φ ↔ ∃ X : Finset _, (↑X ⊆ L₂) ∧ L₁ ⊢ X.conj 🡒 φ := ⟨finite_provable_of_provable h, provable_of_finite_provable... | [
{
"name": "iff_provable_finite_provable_letterless",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 6,
"n_lines": 6,
"n_chars": 286,
"n_subproofs": 0,
"n_tactics": 5,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 1,
"automation_only":... | 3 | module
public import Foundation.Modal.Logic.SumNormal
@[expose] public section
namespace LO.Modal
namespace Logic
variable {L L₁ L₂ : Logic α} {φ ψ : Formula α} {s : Substitution α}
inductive sumQuasiNormal (L₁ L₂ : Logic α) : Logic α
| mem₁ {φ : Formula α} : L₁ ⊢ φ → sumQuasiNormal L₁ L₂ φ
| mem₂ {φ : For... | module
public import Foundation.Modal.Logic.SumNormal
@[expose] public section
namespace LO.Modal
namespace Logic
variable {L L₁ L₂ : Logic α} {φ ψ : Formula α} {s : Substitution α}
inductive sumQuasiNormal (L₁ L₂ : Logic α) : Logic α
| mem₁ {φ : Formula α} : L₁ ⊢ φ → sumQuasiNormal L₁ L₂ φ
| mem₂ {φ : For... | @@ -177,8 +177,13 @@
. apply C!_trans ?_ (Logic.subst s hφ);
exact fconj_subst;
+lemma iff_provable_finite_provable [DecidableEq α] [L₁.Substitution] (h : ∀ ξ ∈ L₂, ∀ s : Substitution _, ξ⟦s⟧ ∈ L₂) :
+ sumQuasiNormal L₁ L₂ ⊢ φ ↔ ∃ X : Finset _, (↑X ⊆ L₂) ∧ L₁ ⊢ X.conj 🡒 φ := ⟨finite_provable_of_provabl... | {
"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_ef9f7cda8c07_2 | 3cb4ad2ff9c7433e | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Logic/SumQuasiNormal.lean | SumQuasiNormal | 2 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 9 | 1 | [
{
"theorem_name": "sum_sum_eq_sum_union",
"depth": 1,
"n_commands": 0,
"n_lines": 17,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n apply Set.Subset.antisymm;\n . apply covered;\n . apply covered;\n . grind;\n . trans (L.sumQuasiNormal... | [
{
"name": "covered",
"text": "/-- If `L₁` and `L₂` are already subset `L₃`, then `L₁ + L₂` is a subset of `L₃`. -/\nlemma covered [Entailment.ModusPonens L₃] [Logic.Substitution L₃] (hX : L₁ ⊆ L₃) (h : L₂ ⊆ L₃) : sumQuasiNormal L₁ L₂ ⊆ L₃ := by\n intro A hA;\n rw [←Logic.iff_provable] at hA ⊢;\n inductio... | [
{
"name": "sum_sum_eq_sum_union",
"fan_in": 0,
"n_deps_direct": 2,
"n_deps_transitive": 5,
"n_lines": 21,
"n_chars": 540,
"n_subproofs": 0,
"n_tactics": 35,
"cyclomatic": 4,
"n_automation": 8,
"n_rewrites": 0,
"n_structural": 6,
"automation_only": false,
"max_... | 2 | module
public import Foundation.Modal.Logic.SumNormal
@[expose] public section
namespace LO.Modal
namespace Logic
variable {L L₁ L₂ : Logic α} {φ ψ : Formula α} {s : Substitution α}
inductive sumQuasiNormal (L₁ L₂ : Logic α) : Logic α
| mem₁ {φ : Formula α} : L₁ ⊢ φ → sumQuasiNormal L₁ L₂ φ
| mem₂ {φ : For... | module
public import Foundation.Modal.Logic.SumNormal
@[expose] public section
namespace LO.Modal
namespace Logic
variable {L L₁ L₂ : Logic α} {φ ψ : Formula α} {s : Substitution α}
inductive sumQuasiNormal (L₁ L₂ : Logic α) : Logic α
| mem₁ {φ : Formula α} : L₁ ⊢ φ → sumQuasiNormal L₁ L₂ φ
| mem₂ {φ : For... | @@ -150,11 +150,34 @@
. intro h;
exact Entailment.WeakerThan.pbl h;
+/-- If `L₁` and `L₂` are already subset `L₃`, then `L₁ + L₂` is a subset of `L₃`. -/
+lemma covered [Entailment.ModusPonens L₃] [Logic.Substitution L₃] (hX : L₁ ⊆ L₃) (h : L₂ ⊆ L₃) : sumQuasiNormal L₁ L₂ ⊆ L₃ := by
+ intro A hA;
+ rw [←Log... | {
"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_ef9f7cda8c07_3 | bc74d884287eee98 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Logic/SumQuasiNormal.lean | SumQuasiNormal | 3 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 9 | 1 | [
{
"theorem_name": "iff_provable_sumQuasiNormal'_provable_sumQuasiNormal",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n rw [eq_sumQuasiNormal_sumQuasiNormal'];",
"n_chars": 45,
"n_subproofs": 0,
... | [
{
"name": "eq_sumQuasiNormal_sumQuasiNormal'",
"text": "lemma eq_sumQuasiNormal_sumQuasiNormal' : Logic.sumQuasiNormal L₁ L₂ = Logic.sumQuasiNormal' L₁ L₂ := by\n ext φ;\n suffices (Logic.sumQuasiNormal L₁ L₂ ⊢ φ) ↔ (Logic.sumQuasiNormal' L₁ L₂ ⊢ φ) by grind;\n constructor;\n . intro h;\n induction h... | [
{
"name": "iff_provable_sumQuasiNormal'_provable_sumQuasiNormal",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 4,
"n_lines": 5,
"n_chars": 194,
"n_subproofs": 0,
"n_tactics": 3,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 1,
"n_structural": 0,
"auto... | 1 | module
public import Foundation.Modal.Logic.SumNormal
@[expose] public section
namespace LO.Modal
namespace Logic
variable {L L₁ L₂ : Logic α} {φ ψ : Formula α} {s : Substitution α}
inductive sumQuasiNormal (L₁ L₂ : Logic α) : Logic α
| mem₁ {φ : Formula α} : L₁ ⊢ φ → sumQuasiNormal L₁ L₂ φ
| mem₂ {φ : For... | module
public import Foundation.Modal.Logic.SumNormal
@[expose] public section
namespace LO.Modal
namespace Logic
variable {L L₁ L₂ : Logic α} {φ ψ : Formula α} {s : Substitution α}
inductive sumQuasiNormal (L₁ L₂ : Logic α) : Logic α
| mem₁ {φ : Formula α} : L₁ ⊢ φ → sumQuasiNormal L₁ L₂ φ
| mem₂ {φ : For... | @@ -244,8 +244,23 @@
end sumQuasiNormal'
+lemma eq_sumQuasiNormal_sumQuasiNormal' : Logic.sumQuasiNormal L₁ L₂ = Logic.sumQuasiNormal' L₁ L₂ := by
+ ext φ;
+ suffices (Logic.sumQuasiNormal L₁ L₂ ⊢ φ) ↔ (Logic.sumQuasiNormal' L₁ L₂ ⊢ φ) by grind;
+ constructor;
+ . intro h;
+ induction h using Logic.sumQuasi... | {
"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_ef9f7cda8c07_4 | 7457dd06410973dc | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Logic/SumQuasiNormal.lean | SumQuasiNormal | 4 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 9 | 1 | [
{
"theorem_name": "iff_provable_sumQuasiNormal'_provable_sumQuasiNormal",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n rw [eq_sumQuasiNormal_sumQuasiNormal'];",
"n_chars": 45,
"n_subproofs": 0,
... | [
{
"name": "eq_sumQuasiNormal_sumQuasiNormal'",
"text": "lemma eq_sumQuasiNormal_sumQuasiNormal' : Logic.sumQuasiNormal L₁ L₂ = Logic.sumQuasiNormal' L₁ L₂ := by\n ext φ;\n suffices (Logic.sumQuasiNormal L₁ L₂ ⊢ φ) ↔ (Logic.sumQuasiNormal' L₁ L₂ ⊢ φ) by grind;\n constructor;\n . intro h;\n induction h... | [
{
"name": "sumQuasiNormal.rec!_omitSubst",
"fan_in": 2,
"n_deps_direct": 8,
"n_deps_transitive": 5,
"n_lines": 17,
"n_chars": 899,
"n_subproofs": 0,
"n_tactics": 10,
"cyclomatic": 2,
"n_automation": 4,
"n_rewrites": 0,
"n_structural": 3,
"automation_only": false,
... | 1 | module
public import Foundation.Modal.Logic.SumNormal
@[expose] public section
namespace LO.Modal
namespace Logic
variable {L L₁ L₂ : Logic α} {φ ψ : Formula α} {s : Substitution α}
inductive sumQuasiNormal (L₁ L₂ : Logic α) : Logic α
| mem₁ {φ : Formula α} : L₁ ⊢ φ → sumQuasiNormal L₁ L₂ φ
| mem₂ {φ : For... | module
public import Foundation.Modal.Logic.SumNormal
@[expose] public section
namespace LO.Modal
namespace Logic
variable {L L₁ L₂ : Logic α} {φ ψ : Formula α} {s : Substitution α}
inductive sumQuasiNormal (L₁ L₂ : Logic α) : Logic α
| mem₁ {φ : Formula α} : L₁ ⊢ φ → sumQuasiNormal L₁ L₂ φ
| mem₂ {φ : For... | @@ -244,8 +244,23 @@
end sumQuasiNormal'
+lemma eq_sumQuasiNormal_sumQuasiNormal' : Logic.sumQuasiNormal L₁ L₂ = Logic.sumQuasiNormal' L₁ L₂ := by
+ ext φ;
+ suffices (Logic.sumQuasiNormal L₁ L₂ ⊢ φ) ↔ (Logic.sumQuasiNormal' L₁ L₂ ⊢ φ) by grind;
+ constructor;
+ . intro h;
+ induction h using Logic.sumQuasi... | {
"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_ef9f7cda8c07_5 | 56a9b1fb1a5a1ab7 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Logic/SumQuasiNormal.lean | SumQuasiNormal | 5 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 9 | 2 | [
{
"theorem_name": "sumQuasiNormal.rec!_omitSubst₁",
"depth": 1,
"n_commands": 0,
"n_lines": 7,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n apply sumQuasiNormal.rec!_omitSubst;\n . intro φ s h;\n apply mem₁;\n grind;\n . assumption;\n . as... | [
{
"name": "sumQuasiNormal.rec!_omitSubst",
"text": "lemma sumQuasiNormal.rec!_omitSubst\n {motive : (φ : Formula α) → ((sumQuasiNormal L₁ L₂) ⊢ φ) → Sort}\n (mem₁ : ∀ {φ}, ∀ s, (h : L₁ ⊢ φ) → motive (φ⟦s⟧) (Logic.subst s $ mem₁! h))\n (mem₂ : ∀ {φ}, ∀ s, (h : L₂ ⊢ φ) → motive (φ⟦s⟧) (Logic.subst s $ me... | [
{
"name": "sumQuasiNormal.rec!_omitSubst₁",
"fan_in": 1,
"n_deps_direct": 7,
"n_deps_transitive": 6,
"n_lines": 17,
"n_chars": 749,
"n_subproofs": 0,
"n_tactics": 13,
"cyclomatic": 1,
"n_automation": 3,
"n_rewrites": 0,
"n_structural": 3,
"automation_only": false,... | 2 | module
public import Foundation.Modal.Logic.SumNormal
@[expose] public section
namespace LO.Modal
namespace Logic
variable {L L₁ L₂ : Logic α} {φ ψ : Formula α} {s : Substitution α}
inductive sumQuasiNormal (L₁ L₂ : Logic α) : Logic α
| mem₁ {φ : Formula α} : L₁ ⊢ φ → sumQuasiNormal L₁ L₂ φ
| mem₂ {φ : For... | module
public import Foundation.Modal.Logic.SumNormal
@[expose] public section
namespace LO.Modal
namespace Logic
variable {L L₁ L₂ : Logic α} {φ ψ : Formula α} {s : Substitution α}
inductive sumQuasiNormal (L₁ L₂ : Logic α) : Logic α
| mem₁ {φ : Formula α} : L₁ ⊢ φ → sumQuasiNormal L₁ L₂ φ
| mem₂ {φ : For... | @@ -262,6 +262,21 @@
lemma iff_provable_sumQuasiNormal'_provable_sumQuasiNormal : (sumQuasiNormal' L₁ L₂ ⊢ φ) ↔ (sumQuasiNormal L₁ L₂ ⊢ φ) := by
rw [eq_sumQuasiNormal_sumQuasiNormal'];
+lemma sumQuasiNormal.rec!_omitSubst
+ {motive : (φ : Formula α) → ((sumQuasiNormal L₁ L₂) ⊢ φ) → Sort}
+ (mem₁ : ∀ {φ}, ∀ 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_ef9f7cda8c07_6 | deba5186db0c9b8e | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Logic/SumQuasiNormal.lean | SumQuasiNormal | 6 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 9 | 1 | [
{
"theorem_name": "sumQuasiNormal.rec!_omitSubst₂",
"depth": 1,
"n_commands": 0,
"n_lines": 3,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n simp_all only [Logic.sumQuasiNormal.symm (L₁ := L₁) (L₂ := L₂)]\n apply sumQuasiNormal.rec!_omitSubst₁ <;> ... | [
{
"name": "sumQuasiNormal.rec!_omitSubst₁",
"text": "lemma sumQuasiNormal.rec!_omitSubst₁ (hL₁ : L₁.Substitution)\n {motive : (φ : Formula α) → ((sumQuasiNormal L₁ L₂) ⊢ φ) → Sort}\n (mem₁ : ∀ {φ}, (h : L₁ ⊢ φ) → motive φ (mem₁! h))\n (mem₂ : ∀ {φ}, ∀ s, (h : L₂ ⊢ φ) → motive (φ⟦s⟧) (Logic.subst s $ me... | [
{
"name": "sumQuasiNormal.rec!_omitSubst₂",
"fan_in": 0,
"n_deps_direct": 8,
"n_deps_transitive": 8,
"n_lines": 13,
"n_chars": 762,
"n_subproofs": 0,
"n_tactics": 5,
"cyclomatic": 2,
"n_automation": 2,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": false,
... | 3 | module
public import Foundation.Modal.Logic.SumNormal
@[expose] public section
namespace LO.Modal
namespace Logic
variable {L L₁ L₂ : Logic α} {φ ψ : Formula α} {s : Substitution α}
inductive sumQuasiNormal (L₁ L₂ : Logic α) : Logic α
| mem₁ {φ : Formula α} : L₁ ⊢ φ → sumQuasiNormal L₁ L₂ φ
| mem₂ {φ : For... | module
public import Foundation.Modal.Logic.SumNormal
@[expose] public section
namespace LO.Modal
namespace Logic
variable {L L₁ L₂ : Logic α} {φ ψ : Formula α} {s : Substitution α}
inductive sumQuasiNormal (L₁ L₂ : Logic α) : Logic α
| mem₁ {φ : Formula α} : L₁ ⊢ φ → sumQuasiNormal L₁ L₂ φ
| mem₂ {φ : For... | @@ -280,6 +280,22 @@
def substitution_of_letterless (L_letterless : FormulaSet.Letterless L) : L.Substitution where
subst s hφ := by grind;
+lemma sumQuasiNormal.rec!_omitSubst₁ (hL₁ : L₁.Substitution)
+ {motive : (φ : Formula α) → ((sumQuasiNormal L₁ L₂) ⊢ φ) → Sort}
+ (mem₁ : ∀ {φ}, (h : L₁ ⊢ φ) → motive 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_ef9f7cda8c07_7 | 7a32c6b2c3acbeec | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Logic/SumQuasiNormal.lean | SumQuasiNormal | 7 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 9 | 2 | [
{
"theorem_name": "sumQuasiNormal.rec!_omitSubst₁",
"depth": 1,
"n_commands": 0,
"n_lines": 7,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n apply sumQuasiNormal.rec!_omitSubst;\n . intro φ s h;\n apply mem₁;\n grind;\n . assumption;\n . as... | [
{
"name": "sumQuasiNormal.rec!_omitSubst",
"text": "lemma sumQuasiNormal.rec!_omitSubst\n {motive : (φ : Formula α) → ((sumQuasiNormal L₁ L₂) ⊢ φ) → Sort}\n (mem₁ : ∀ {φ}, ∀ s, (h : L₁ ⊢ φ) → motive (φ⟦s⟧) (Logic.subst s $ mem₁! h))\n (mem₂ : ∀ {φ}, ∀ s, (h : L₂ ⊢ φ) → motive (φ⟦s⟧) (Logic.subst s $ me... | [
{
"name": "sumQuasiNormal.rec!_omitSubst_strong",
"fan_in": 1,
"n_deps_direct": 7,
"n_deps_transitive": 6,
"n_lines": 15,
"n_chars": 764,
"n_subproofs": 0,
"n_tactics": 13,
"cyclomatic": 1,
"n_automation": 3,
"n_rewrites": 0,
"n_structural": 5,
"automation_only": ... | 2 | module
public import Foundation.Modal.Logic.SumNormal
@[expose] public section
namespace LO.Modal
namespace Logic
variable {L L₁ L₂ : Logic α} {φ ψ : Formula α} {s : Substitution α}
inductive sumQuasiNormal (L₁ L₂ : Logic α) : Logic α
| mem₁ {φ : Formula α} : L₁ ⊢ φ → sumQuasiNormal L₁ L₂ φ
| mem₂ {φ : For... | module
public import Foundation.Modal.Logic.SumNormal
@[expose] public section
namespace LO.Modal
namespace Logic
variable {L L₁ L₂ : Logic α} {φ ψ : Formula α} {s : Substitution α}
inductive sumQuasiNormal (L₁ L₂ : Logic α) : Logic α
| mem₁ {φ : Formula α} : L₁ ⊢ φ → sumQuasiNormal L₁ L₂ φ
| mem₂ {φ : For... | @@ -262,6 +262,21 @@
lemma iff_provable_sumQuasiNormal'_provable_sumQuasiNormal : (sumQuasiNormal' L₁ L₂ ⊢ φ) ↔ (sumQuasiNormal L₁ L₂ ⊢ φ) := by
rw [eq_sumQuasiNormal_sumQuasiNormal'];
+lemma sumQuasiNormal.rec!_omitSubst
+ {motive : (φ : Formula α) → ((sumQuasiNormal L₁ L₂) ⊢ φ) → Sort}
+ (mem₁ : ∀ {φ}, ∀ 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_2eb7e9c1e9b4_5 | 51f5a9ea64ae2a6e | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/HFS/Seq.lean | Seq | 5 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 16 | 1 | [
{
"theorem_name": "Seq.cases_iff",
"depth": 1,
"n_commands": 0,
"n_lines": 25,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " ⟨fun h ↦ by\n by_cases hs : lh s = 0\n · left\n simpa [hs] using h.domain_eq\n · right\n let i := lh s - 1\n have hi :... | [
{
"name": "ne_zero_iff_one_le",
"text": "/-- TODO: move to Lemmata.lean-/\nlemma ne_zero_iff_one_le {a : V} : a ≠ 0 ↔ 1 ≤ a := Iff.trans pos_iff_ne_zero.symm (pos_iff_one_le (a := a))\n\n",
"fan_in": 1,
"n_lines": 4,
"n_chars": 149,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
... | [
{
"name": "Seq.cases_iff",
"fan_in": 0,
"n_deps_direct": 2,
"n_deps_transitive": 2,
"n_lines": 27,
"n_chars": 1153,
"n_subproofs": 6,
"n_tactics": 25,
"cyclomatic": 3,
"n_automation": 11,
"n_rewrites": 0,
"n_structural": 4,
"automation_only": false,
"max_nesti... | 2 | module
public import Foundation.FirstOrder.Arithmetic.HFS.Basic
@[expose] public section
/-!
# Sequence
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
def Seq (s : V) : Prop := IsMapping s ∧ ∃ l, domain s = under l
def Seq.isMapping {s : V} (h : Seq s) : IsMappi... | module
public import Foundation.FirstOrder.Arithmetic.HFS.Basic
@[expose] public section
/-!
# Sequence
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
def Seq (s : V) : Prop := IsMapping s ∧ ∃ l, domain s = under l
def Seq.isMapping {s : V} (h : Seq s) : IsMappi... | @@ -239,6 +239,9 @@
exact ⟨ha₁a₂, H₁.eq_of_eq_of_subset H₂ hs₁s₂ this⟩,
by rintro ⟨rfl, rfl⟩; rfl⟩
+/-- TODO: move to Lemmata.lean-/
+lemma ne_zero_iff_one_le {a : V} : a ≠ 0 ↔ 1 ≤ a := Iff.trans pos_iff_ne_zero.symm (pos_iff_one_le (a := a))
+
lemma Seq.cases_iff {s : V} : Seq s ↔ s = ∅ ∨ ∃ x s', Seq 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_ff5646339880_0 | b76d87ab862e5d38 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Logic/LindenbaumAlgebra.lean | LindenbaumAlgebra | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 3 | 1 | [
{
"theorem_name": "provable_iff_eq_top",
"depth": 1,
"n_commands": 0,
"n_lines": 3,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " calc\n _ ↔ ProvablyEquivalent 𝓢 φ ⊤ := by rw [provable_iff_provablyEquivalent_verum]\n _ ↔ _ := by rw [top_def, Quotient.e... | [
{
"name": "provable_iff_provablyEquivalent_verum",
"text": "lemma provable_iff_provablyEquivalent_verum [Entailment.Minimal 𝓢] {φ : F} : 𝓢 ⊢ φ ↔ φ ≡ ⊤ :=\n ⟨fun h ↦ E!_intro CV! (C!_of_conseq! h), fun h ↦ (K!_right h) ⨀ verum!⟩\n\n",
"fan_in": 1,
"n_lines": 4,
"n_chars": 196,
"n_subproofs... | [
{
"name": "provable_iff_eq_top",
"fan_in": 1,
"n_deps_direct": 2,
"n_deps_transitive": 2,
"n_lines": 5,
"n_chars": 240,
"n_subproofs": 0,
"n_tactics": 5,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 2,
"n_structural": 1,
"automation_only": false,
"max_nes... | 1 | module
public import Foundation.Propositional.Entailment.Cl.Basic
@[expose] public section
namespace LO
variable {F S : Type*} [LogicalConnective F] [Entailment S F]
namespace Entailment
variable (𝓢 : S)
def ProvablyEquivalent (φ ψ : F) : Prop := 𝓢 ⊢ φ 🡘 ψ
local infix:45 " ≡ " => ProvablyEquivalent 𝓢
prote... | module
public import Foundation.Propositional.Entailment.Cl.Basic
@[expose] public section
namespace LO
variable {F S : Type*} [LogicalConnective F] [Entailment S F]
namespace Entailment
variable (𝓢 : S)
def ProvablyEquivalent (φ ψ : F) : Prop := 𝓢 ⊢ φ 🡘 ψ
local infix:45 " ≡ " => ProvablyEquivalent 𝓢
prote... | @@ -24,6 +24,9 @@
protected lemma ProvablyEquivalent.trans [Entailment.Minimal 𝓢] {φ ψ χ : F} : φ ≡ ψ → ψ ≡ χ → φ ≡ χ := E!_trans
+lemma provable_iff_provablyEquivalent_verum [Entailment.Minimal 𝓢] {φ : F} : 𝓢 ⊢ φ ↔ φ ≡ ⊤ :=
+ ⟨fun h ↦ E!_intro CV! (C!_of_conseq! h), fun h ↦ (K!_right h) ⨀ verum!⟩
+
variable ... | {
"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_ff5646339880_1 | ef317abbd39598f5 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Logic/LindenbaumAlgebra.lean | LindenbaumAlgebra | 1 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 3 | 1 | [
{
"theorem_name": "inconsistent_iff_trivial",
"depth": 1,
"n_commands": 0,
"n_lines": 7,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n simp only [Inconsistent, provable_iff_eq_top]\n constructor\n · intro h φ;\n induction φ using Quotient.ind\n... | [
{
"name": "provable_iff_eq_top",
"text": "lemma provable_iff_eq_top {φ : F} : 𝓢 ⊢ φ ↔ (⟦φ⟧ : LindenbaumAlgebra 𝓢) = ⊤ := calc\n _ ↔ ProvablyEquivalent 𝓢 φ ⊤ := by rw [provable_iff_provablyEquivalent_verum]\n _ ↔ _ := by rw [top_def, Quotient.eq]; rfl;\n\n",
"fan_in": 1,
"n_lines": 5,
"n_cha... | [
{
"name": "inconsistent_iff_trivial",
"fan_in": 1,
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"n_lines": 9,
"n_chars": 259,
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"n_rewrites": 0,
"n_structural": 4,
"automation_only": false,
"ma... | 2 | module
public import Foundation.Propositional.Entailment.Cl.Basic
@[expose] public section
namespace LO
variable {F S : Type*} [LogicalConnective F] [Entailment S F]
namespace Entailment
variable (𝓢 : S)
def ProvablyEquivalent (φ ψ : F) : Prop := 𝓢 ⊢ φ 🡘 ψ
local infix:45 " ≡ " => ProvablyEquivalent 𝓢
prote... | module
public import Foundation.Propositional.Entailment.Cl.Basic
@[expose] public section
namespace LO
variable {F S : Type*} [LogicalConnective F] [Entailment S F]
namespace Entailment
variable (𝓢 : S)
def ProvablyEquivalent (φ ψ : F) : Prop := 𝓢 ⊢ φ 🡘 ψ
local infix:45 " ≡ " => ProvablyEquivalent 𝓢
prote... | @@ -136,8 +136,18 @@
variable {𝓢}
-lemma inconsistent_iff_trivial : Inconsistent 𝓢 ↔ (∀ φ : LindenbaumAlgebra 𝓢, φ = ⊤) := sorry
+lemma provable_iff_eq_top {φ : F} : 𝓢 ⊢ φ ↔ (⟦φ⟧ : LindenbaumAlgebra 𝓢) = ⊤ := calc
+ _ ↔ ProvablyEquivalent 𝓢 φ ⊤ := by rw [provable_iff_provablyEquivalent_verum]
+ _ ↔ _ := by... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_ff5646339880_2 | 59f9f6ae513c7e7a | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Logic/LindenbaumAlgebra.lean | LindenbaumAlgebra | 2 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 3 | 1 | [
{
"theorem_name": "consistent_iff_nontrivial",
"depth": 1,
"n_commands": 0,
"n_lines": 6,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n apply not_iff_not.mp\n simp only [not_consistent_iff_inconsistent, inconsistent_iff_trivial, nontrivial_iff, ne_... | [
{
"name": "inconsistent_iff_trivial",
"text": "lemma inconsistent_iff_trivial : Inconsistent 𝓢 ↔ (∀ φ : LindenbaumAlgebra 𝓢, φ = ⊤) := by\n simp only [Inconsistent, provable_iff_eq_top]\n constructor\n · intro h φ;\n induction φ using Quotient.ind\n simp [h]\n · intro h f; simp [h]\n\n",
"fa... | [
{
"name": "consistent_iff_nontrivial",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 4,
"n_lines": 8,
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"automation_only": false,
"m... | 3 | module
public import Foundation.Propositional.Entailment.Cl.Basic
@[expose] public section
namespace LO
variable {F S : Type*} [LogicalConnective F] [Entailment S F]
namespace Entailment
variable (𝓢 : S)
def ProvablyEquivalent (φ ψ : F) : Prop := 𝓢 ⊢ φ 🡘 ψ
local infix:45 " ≡ " => ProvablyEquivalent 𝓢
prote... | module
public import Foundation.Propositional.Entailment.Cl.Basic
@[expose] public section
namespace LO
variable {F S : Type*} [LogicalConnective F] [Entailment S F]
namespace Entailment
variable (𝓢 : S)
def ProvablyEquivalent (φ ψ : F) : Prop := 𝓢 ⊢ φ 🡘 ψ
local infix:45 " ≡ " => ProvablyEquivalent 𝓢
prote... | @@ -140,8 +140,21 @@
_ ↔ ProvablyEquivalent 𝓢 φ ⊤ := by rw [provable_iff_provablyEquivalent_verum]
_ ↔ _ := by rw [top_def, Quotient.eq]; rfl;
-lemma consistent_iff_nontrivial : Consistent 𝓢 ↔ Nontrivial (LindenbaumAlgebra 𝓢) := sorry
+lemma inconsistent_iff_trivial : Inconsistent 𝓢 ↔ (∀ φ : LindenbaumAlgeb... | {
"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_9b46659a3dc2_0 | f8754abb69ce9d95 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Kripke/Basic.lean | Basic | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 1 | 1 | [
{
"theorem_name": "finite_colimit_domain",
"depth": 1,
"n_commands": 0,
"n_lines": 6,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n have : ∀ i, ∃ p ∈ F, p ⊩↓ ↑(v i) := fun i ↦ (v i).mem_filter\n choose p hp using this\n have : ∃ q ∈ F, ∀ i, q ≤ p ... | [
{
"name": "finite_colimit",
"text": "lemma finite_colimit [Fintype ι] (p : ι → W) (hp : ∀ i, p i ∈ F) : ∃ q ∈ F, ∀ i, q ≤ p i :=\n DirectedOn.fintype_colimit isTrans_ge (Order.PFilter.nonempty F) F.directed p hp\n\n",
"fan_in": 1,
"n_lines": 4,
"n_chars": 192,
"n_subproofs": 0,
"n_tacti... | [
{
"name": "finite_colimit_domain",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 9,
"n_chars": 400,
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"n_tactics": 6,
"cyclomatic": 2,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 2,
"automation_only": false,
"max_n... | 1 | module
public import Foundation.FirstOrder.Basic
public import Foundation.Syntax.Predicate.Relational
public import Foundation.Logic.ForcingRelation
public import Foundation.Vorspiel.Order.Dense
@[expose] public section
namespace LO.FirstOrder
/-- Kripke model for relational first-order language -/
class KripkeModel... | module
public import Foundation.FirstOrder.Basic
public import Foundation.Syntax.Predicate.Relational
public import Foundation.Logic.ForcingRelation
public import Foundation.Vorspiel.Order.Dense
@[expose] public section
namespace LO.FirstOrder
/-- Kripke model for relational first-order language -/
class KripkeModel... | @@ -67,11 +67,14 @@
instance : CoeOut F.Model C := ⟨fun x ↦ x.val⟩
+lemma finite_colimit [Fintype ι] (p : ι → W) (hp : ∀ i, p i ∈ F) : ∃ q ∈ F, ∀ i, q ≤ p i :=
+ DirectedOn.fintype_colimit isTrans_ge (Order.PFilter.nonempty F) F.directed p hp
+
lemma finite_colimit_domain [Fintype ι] (v : ι → F.Model) :
∃ q... | {
"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_a105c606a367_0 | 04bcce53dbace89c | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Propositional/Boolean/Basic.lean | Basic | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 1 | 1 | [
{
"theorem_name": "subst_isTautology",
"depth": 1,
"n_commands": 0,
"n_lines": 4,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n intro s v;\n apply Formula.Boolean.iff_subst_self s |>.mp;\n apply h;",
"n_chars": 75,
"n_subproofs": 0,
"n... | [
{
"name": "iff_subst_self",
"text": "lemma iff_subst_self (s) :\n ((λ a => val v ((.atom a)⟦s⟧)) : Valuation α) ⊧ φ ↔ v ⊧ (φ⟦s⟧) := by\n induction φ with\n | hatom a => simp [val, models_iff_val];\n | hfalsum => simp;\n | himp φ ψ ihφ ihψ =>\n constructor;\n . intro hφψ hφ;\n apply ihψ.mp;\n... | [
{
"name": "subst_isTautology",
"fan_in": 0,
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"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 3,
"automation_only": false,
"max_nesti... | 1 | module
public import Foundation.Propositional.Formula.Basic
public import Foundation.Logic.Semantics
@[expose] public section
namespace LO.Propositional
variable {α : Type*}
abbrev Boolean.Valuation (α : Type*) := α → Prop
namespace Formula.Boolean
open Propositional.Boolean (Valuation)
def val (v : Valuation α... | module
public import Foundation.Propositional.Formula.Basic
public import Foundation.Logic.Semantics
@[expose] public section
namespace LO.Propositional
variable {α : Type*}
abbrev Boolean.Valuation (α : Type*) := α → Prop
namespace Formula.Boolean
open Propositional.Boolean (Valuation)
def val (v : Valuation α... | @@ -38,6 +38,42 @@
@[simp] protected lemma models_atom : v ⊧ (.atom a) ↔ v a := iff_of_eq rfl
+lemma iff_subst_self (s) :
+ ((λ a => val v ((.atom a)⟦s⟧)) : Valuation α) ⊧ φ ↔ v ⊧ (φ⟦s⟧) := by
+ induction φ with
+ | hatom a => simp [val, models_iff_val];
+ | hfalsum => simp;
+ | himp φ ψ ihφ ihψ =>
+ const... | {
"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_fa2767d51d56_0 | 36fe27a2c4ffa2cd | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Propositional/Logic/Superintuitionistic.lean | Superintuitionistic | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 2 | 1 | [
{
"theorem_name": "trivial_of_mem_bot",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " L.mdp (L.subset_Int Entailment.efq!) h",
"n_chars": 39,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
... | [
{
"name": "subset_Int",
"text": "lemma subset_Int (h : φ ∈ Propositional.Int) : φ ∈ L.logic := L.subset_L h\n\n",
"fan_in": 1,
"n_lines": 3,
"n_chars": 82,
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"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"autom... | [
{
"name": "trivial_of_mem_bot",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 2,
"n_chars": 120,
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"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
"max_nest... | 1 | module
public import Foundation.Propositional.Hilbert.Minimal.Basic
@[expose] public section
namespace LO.Propositional
abbrev SuperintuitionisticLogic (α) := Logic.ExtensionOf (Propositional.Int : Logic α)
namespace SuperintuitionisticLogic
variable {L : SuperintuitionisticLogic α} {φ ψ : Formula α}
lemma trivi... | module
public import Foundation.Propositional.Hilbert.Minimal.Basic
@[expose] public section
namespace LO.Propositional
abbrev SuperintuitionisticLogic (α) := Logic.ExtensionOf (Propositional.Int : Logic α)
namespace SuperintuitionisticLogic
variable {L : SuperintuitionisticLogic α} {φ ψ : Formula α}
lemma subse... | @@ -12,7 +12,9 @@
variable {L : SuperintuitionisticLogic α} {φ ψ : Formula α}
-lemma trivial_of_mem_bot (h : ⊥ ∈ L.toLogic) : ∀ {φ}, φ ∈ L.toLogic := sorry
+lemma subset_Int (h : φ ∈ Propositional.Int) : φ ∈ L.logic := L.subset_L h
+
+lemma trivial_of_mem_bot (h : ⊥ ∈ L.toLogic) : ∀ {φ}, φ ∈ L.toLogic := L.mdp (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_fa2767d51d56_1 | 1299c4c29ea9b730 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Propositional/Logic/Superintuitionistic.lean | Superintuitionistic | 1 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 2 | 1 | [
{
"theorem_name": "isTrivial_of_mem_bot",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " ⟨Set.eq_univ_iff_forall.mpr $ @trivial_of_mem_bot α L h⟩",
"n_chars": 62,
"n_subproofs": 0,
"n_tactics": 1,
... | [
{
"name": "trivial_of_mem_bot",
"text": "lemma trivial_of_mem_bot (h : ⊥ ∈ L.toLogic) : ∀ {φ}, φ ∈ L.toLogic := L.mdp (L.subset_Int Entailment.efq!) h\n",
"fan_in": 1,
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"n_chars": 120,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewri... | [
{
"name": "isTrivial_of_mem_bot",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 2,
"n_lines": 3,
"n_chars": 131,
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"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
"max_ne... | 2 | module
public import Foundation.Propositional.Hilbert.Minimal.Basic
@[expose] public section
namespace LO.Propositional
abbrev SuperintuitionisticLogic (α) := Logic.ExtensionOf (Propositional.Int : Logic α)
namespace SuperintuitionisticLogic
variable {L : SuperintuitionisticLogic α} {φ ψ : Formula α}
lemma subse... | module
public import Foundation.Propositional.Hilbert.Minimal.Basic
@[expose] public section
namespace LO.Propositional
abbrev SuperintuitionisticLogic (α) := Logic.ExtensionOf (Propositional.Int : Logic α)
namespace SuperintuitionisticLogic
variable {L : SuperintuitionisticLogic α} {φ ψ : Formula α}
lemma subse... | @@ -14,7 +14,8 @@
lemma subset_Int (h : φ ∈ Propositional.Int) : φ ∈ L.logic := L.subset_L h
-lemma isTrivial_of_mem_bot (h : ⊥ ∈ L.toLogic) : L.IsTrivial := sorry
+lemma trivial_of_mem_bot (h : ⊥ ∈ L.toLogic) : ∀ {φ}, φ ∈ L.toLogic := L.mdp (L.subset_Int Entailment.efq!) h
+lemma isTrivial_of_mem_bot (h : ⊥ ∈ L.t... | {
"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_a75c31a2da29_0 | 10e5ff4b72300ab3 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Basic/Model.lean | Model | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 2 | 1 | [
{
"theorem_name": "Semiformula.eval_uLift",
"depth": 1,
"n_commands": 0,
"n_lines": 3,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n induction φ using Semiformula.rec' <;>\n simp [*, Semiterm.val_uLift, Matrix.comp_vecCons', Function.comp_def]",
... | [
{
"name": "Semiterm.val_uLift",
"text": "lemma Semiterm.val_uLift {e : Fin n → ULift.{v'} M} {f : ξ → ULift.{v'} M} {t : Semiterm L ξ n} :\n Semiterm.val e f t = ⟨Semiterm.val (ULift.down ∘ e) (ULift.down ∘ f) t⟩ := by\n induction t <;> simp [*, Function.comp_def]\n\n",
"fan_in": 1,
"n_lines": 5... | [
{
"name": "Semiformula.eval_uLift",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 6,
"n_chars": 301,
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"cyclomatic": 3,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": false,
"max_... | 1 | module
public import Foundation.FirstOrder.Basic.Operator
public import Foundation.FirstOrder.Basic.Semantics.Elementary
@[expose] public section
namespace LO
namespace FirstOrder
namespace Structure
structure Model (L : Language) (M : Type*) where
intro : M
namespace Model
variable [Structure L M]
def equiv (... | module
public import Foundation.FirstOrder.Basic.Operator
public import Foundation.FirstOrder.Basic.Semantics.Elementary
@[expose] public section
namespace LO
namespace FirstOrder
namespace Structure
structure Model (L : Language) (M : Type*) where
intro : M
namespace Model
variable [Structure L M]
def equiv (... | @@ -171,8 +171,14 @@
@[simp] lemma Structure.rel_uLift {k} (r : L.Rel k) (v : Fin k → ULift.{v'} M) :
Structure.rel r v = Structure.rel r (ULift.down ∘ v) := rfl
+lemma Semiterm.val_uLift {e : Fin n → ULift.{v'} M} {f : ξ → ULift.{v'} M} {t : Semiterm L ξ n} :
+ Semiterm.val e f t = ⟨Semiterm.val (ULift.down... | {
"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_a75c31a2da29_1 | 05b2ee147c855edf | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Basic/Model.lean | Model | 1 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 2 | 1 | [
{
"theorem_name": "uLift_elementaryEquiv",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " ⟨by\n intro σ; simp [models_iff, Semiformula.eval_uLift, Matrix.empty_eq, Empty.eq_elim]⟩",
"n_chars": 95,
"n_subp... | [
{
"name": "Semiformula.eval_uLift",
"text": "lemma Semiformula.eval_uLift {e : Fin n → ULift.{v'} M} {f : ξ → ULift.{v'} M} {φ : Semiformula L ξ n} :\n φ.Eval e f ↔ φ.Eval (ULift.down ∘ e) (ULift.down ∘ f) := by\n induction φ using Semiformula.rec' <;>\n simp [*, Semiterm.val_uLift, Matrix.comp_vecCo... | [
{
"name": "uLift_elementaryEquiv",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 2,
"n_lines": 4,
"n_chars": 167,
"n_subproofs": 0,
"n_tactics": 3,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": false,
"max_n... | 2 | module
public import Foundation.FirstOrder.Basic.Operator
public import Foundation.FirstOrder.Basic.Semantics.Elementary
@[expose] public section
namespace LO
namespace FirstOrder
namespace Structure
structure Model (L : Language) (M : Type*) where
intro : M
namespace Model
variable [Structure L M]
def equiv (... | module
public import Foundation.FirstOrder.Basic.Operator
public import Foundation.FirstOrder.Basic.Semantics.Elementary
@[expose] public section
namespace LO
namespace FirstOrder
namespace Structure
structure Model (L : Language) (M : Type*) where
intro : M
namespace Model
variable [Structure L M]
def equiv (... | @@ -175,9 +175,15 @@
Semiterm.val e f t = ⟨Semiterm.val (ULift.down ∘ e) (ULift.down ∘ f) t⟩ := by
induction t <;> simp [*, Function.comp_def]
+lemma Semiformula.eval_uLift {e : Fin n → ULift.{v'} M} {f : ξ → ULift.{v'} M} {φ : Semiformula L ξ n} :
+ φ.Eval e f ↔ φ.Eval (ULift.down ∘ e) (ULift.down ∘ f) :=... | {
"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_36d55c101c52_0 | e22c5cecda19c25e | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Vorspiel/Order/Ideal.lean | Ideal | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 8 | 1 | [
{
"theorem_name": "mem_iSup_iff",
"depth": 1,
"n_commands": 0,
"n_lines": 26,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n revert x\n let K : Ideal P := {\n carrier := { x | ∃ u : Finset ι, x ∈ u.sup I }\n lower' x y hyx := by\n rintro ... | [
{
"name": "iSup_def",
"text": "lemma iSup_def (I : ι → Ideal P) : ⨆ i, I i = sInf (upperBounds (Set.range I)) := rfl\n\n",
"fan_in": 1,
"n_lines": 3,
"n_chars": 92,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
... | [
{
"name": "mem_iSup_iff",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 29,
"n_chars": 1064,
"n_subproofs": 5,
"n_tactics": 26,
"cyclomatic": 4,
"n_automation": 11,
"n_rewrites": 0,
"n_structural": 11,
"automation_only": false,
"max_nesti... | 1 |
module
public import Mathlib.Order.Ideal
public import Mathlib.Order.PFilter
public import Mathlib.Data.Finset.Lattice.Basic
public import Mathlib.Order.PrimeIdeal
@[expose] public section
namespace Order
namespace Ideal
variable {P : Type*}
section semilatticeSup
variable [SemilatticeSup P] [OrderBot P]
@[sim... |
module
public import Mathlib.Order.Ideal
public import Mathlib.Order.PFilter
public import Mathlib.Data.Finset.Lattice.Basic
public import Mathlib.Order.PrimeIdeal
@[expose] public section
namespace Order
namespace Ideal
variable {P : Type*}
section semilatticeSup
variable [SemilatticeSup P] [OrderBot P]
lemma... | @@ -18,6 +18,8 @@
variable [SemilatticeSup P] [OrderBot P]
+lemma iSup_def (I : ι → Ideal P) : ⨆ i, I i = sInf (upperBounds (Set.range I)) := rfl
+
@[simp] lemma mem_bot {x : P} : x ∈ (⊥ : Ideal P) ↔ x = ⊥ := le_bot_iff
lemma mem_iSup_iff [DecidableEq ι] {I : ι → Ideal P} {x : P} :
@@ -38,7 +40,12 @@
have m... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_36d55c101c52_1 | 888eface17d97b5a | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Vorspiel/Order/Ideal.lean | Ideal | 1 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 8 | 1 | [
{
"theorem_name": "top_not_mem",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": "\n iff_top_not_mem.mp inferInstance",
"n_chars": 35,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automa... | [
{
"name": "iff_top_not_mem",
"text": "lemma iff_top_not_mem [OrderTop P] {I : Ideal P} : I.IsProper ↔ ⊤ ∉ I := by\n constructor\n · intro h ht\n have : (I : Set P) = Set.univ := by\n ext x\n suffices x ∈ I by simpa\n exact I.lower (show x ≤ ⊤ by simp) ht\n exact h.ne_univ this\n · in... | [
{
"name": "top_not_mem",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 4,
"n_chars": 117,
"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 Mathlib.Order.Ideal
public import Mathlib.Order.PFilter
public import Mathlib.Data.Finset.Lattice.Basic
public import Mathlib.Order.PrimeIdeal
@[expose] public section
namespace Order
namespace Ideal
variable {P : Type*}
section semilatticeSup
variable [SemilatticeSup P] [OrderBot P]
@[sim... |
module
public import Mathlib.Order.Ideal
public import Mathlib.Order.PFilter
public import Mathlib.Data.Finset.Lattice.Basic
public import Mathlib.Order.PrimeIdeal
@[expose] public section
namespace Order
namespace Ideal
variable {P : Type*}
section semilatticeSup
variable [SemilatticeSup P] [OrderBot P]
@[sim... | @@ -34,8 +34,24 @@
variable {P : Type*} [SemilatticeSup P]
-@[simp] lemma top_not_mem [OrderTop P] (I : Ideal P) [I.IsProper] : ⊤ ∉ I := sorry
+lemma iff_top_not_mem [OrderTop P] {I : Ideal P} : I.IsProper ↔ ⊤ ∉ I := by
+ constructor
+ · intro h ht
+ have : (I : Set P) = Set.univ := by
+ ext x
+ suf... | {
"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_36d55c101c52_2 | d6d673c686afe4b1 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Vorspiel/Order/Ideal.lean | Ideal | 2 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 8 | 3 | [
{
"theorem_name": "mem_or_compl_mem_F",
"depth": 1,
"n_commands": 0,
"n_lines": 7,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n by_contra!\n have hx : x ∈ IF.I := by simpa [not_mem_F_iff_mem_I] using this.1\n have hxc : xᶜ ∈ IF.I := by simpa [not... | [
{
"name": "not_mem_F_iff_mem_I",
"text": "lemma not_mem_F_iff_mem_I {x : P} :\n x ∉ IF.F ↔ x ∈ IF.I := by\n have : x ∈ IF.I ∨ x ∈ IF.F := by simpa using Set.ext_iff.mp IF.I_union_F x\n have : x ∈ IF.I → x ∉ IF.F := @Set.disjoint_left.mp IF.disjoint x\n tauto\n\n",
"fan_in": 3,
"n_lines": 7,
... | [
{
"name": "mem_or_compl_mem_F",
"fan_in": 1,
"n_deps_direct": 2,
"n_deps_transitive": 3,
"n_lines": 9,
"n_chars": 368,
"n_subproofs": 4,
"n_tactics": 7,
"cyclomatic": 1,
"n_automation": 3,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
"max_nest... | 2 |
module
public import Mathlib.Order.Ideal
public import Mathlib.Order.PFilter
public import Mathlib.Data.Finset.Lattice.Basic
public import Mathlib.Order.PrimeIdeal
@[expose] public section
namespace Order
namespace Ideal
variable {P : Type*}
section semilatticeSup
variable [SemilatticeSup P] [OrderBot P]
@[sim... |
module
public import Mathlib.Order.Ideal
public import Mathlib.Order.PFilter
public import Mathlib.Data.Finset.Lattice.Basic
public import Mathlib.Order.PrimeIdeal
@[expose] public section
namespace Order
namespace Ideal
variable {P : Type*}
section semilatticeSup
variable [SemilatticeSup P] [OrderBot P]
@[sim... | @@ -66,6 +66,12 @@
variable [Preorder P] (IF : PrimePair P)
+lemma not_mem_F_iff_mem_I {x : P} :
+ x ∉ IF.F ↔ x ∈ IF.I := by
+ have : x ∈ IF.I ∨ x ∈ IF.F := by simpa using Set.ext_iff.mp IF.I_union_F x
+ have : x ∈ IF.I → x ∉ IF.F := @Set.disjoint_left.mp IF.disjoint x
+ tauto
+
lemma not_mem_I_iff_mem_F {x... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_36d55c101c52_3 | 7d40a26aa6e82819 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Vorspiel/Order/Ideal.lean | Ideal | 3 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 8 | 1 | [
{
"theorem_name": "compl_mem_I_iff_mem_F",
"depth": 1,
"n_commands": 0,
"n_lines": 6,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n constructor\n · have : x ∈ IF.F ∨ xᶜ ∉ IF.I := by simpa [not_mem_I_iff_mem_F] using mem_or_compl_mem_F x\n tauto\... | [
{
"name": "mem_or_compl_mem_F",
"text": "lemma mem_or_compl_mem_F (x : P) : x ∈ IF.F ∨ xᶜ ∈ IF.F := by\n by_contra!\n have hx : x ∈ IF.I := by simpa [not_mem_F_iff_mem_I] using this.1\n have hxc : xᶜ ∈ IF.I := by simpa [not_mem_F_iff_mem_I] using this.2\n have : ⊤ ∈ IF.I := by simpa using Order.Ideal.su... | [
{
"name": "compl_mem_I_iff_mem_F",
"fan_in": 2,
"n_deps_direct": 4,
"n_deps_transitive": 6,
"n_lines": 9,
"n_chars": 308,
"n_subproofs": 2,
"n_tactics": 6,
"cyclomatic": 2,
"n_automation": 4,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": false,
"max_n... | 5 |
module
public import Mathlib.Order.Ideal
public import Mathlib.Order.PFilter
public import Mathlib.Data.Finset.Lattice.Basic
public import Mathlib.Order.PrimeIdeal
@[expose] public section
namespace Order
namespace Ideal
variable {P : Type*}
section semilatticeSup
variable [SemilatticeSup P] [OrderBot P]
@[sim... |
module
public import Mathlib.Order.Ideal
public import Mathlib.Order.PFilter
public import Mathlib.Data.Finset.Lattice.Basic
public import Mathlib.Order.PrimeIdeal
@[expose] public section
namespace Order
namespace Ideal
variable {P : Type*}
section semilatticeSup
variable [SemilatticeSup P] [OrderBot P]
@[sim... | @@ -91,10 +91,19 @@
lemma mem_or_compl_mem_I (x : P) : x ∈ IF.I ∨ xᶜ ∈ IF.I :=
IF.I_isPrime.mem_or_compl_mem
+lemma mem_or_compl_mem_F (x : P) : x ∈ IF.F ∨ xᶜ ∈ IF.F := by
+ by_contra!
+ have hx : x ∈ IF.I := by simpa [not_mem_F_iff_mem_I] using this.1
+ have hxc : xᶜ ∈ IF.I := by simpa [not_mem_F_iff_mem_I] u... | {
"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_36d55c101c52_4 | 8e8225bd7b42b3e7 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Vorspiel/Order/Ideal.lean | Ideal | 4 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 8 | 2 | [
{
"theorem_name": "compl_mem_F_iff_mem_I",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n simpa using (compl_mem_I_iff_mem_F (x := xᶜ)).symm",
"n_chars": 58,
"n_subproofs": 0,
"n_tactics": 2,
... | [
{
"name": "compl_mem_I_iff_mem_F",
"text": "lemma compl_mem_I_iff_mem_F :\n xᶜ ∈ IF.I ↔ x ∈ IF.F := by\n constructor\n · have : x ∈ IF.F ∨ xᶜ ∉ IF.I := by simpa [not_mem_I_iff_mem_F] using mem_or_compl_mem_F x\n tauto\n · have : x ∉ IF.F ∨ xᶜ ∈ IF.I := by simpa [not_mem_F_iff_mem_I] using mem_or_co... | [
{
"name": "compl_mem_F_iff_mem_I",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 7,
"n_lines": 5,
"n_chars": 125,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"max_ne... | 6 |
module
public import Mathlib.Order.Ideal
public import Mathlib.Order.PFilter
public import Mathlib.Data.Finset.Lattice.Basic
public import Mathlib.Order.PrimeIdeal
@[expose] public section
namespace Order
namespace Ideal
variable {P : Type*}
section semilatticeSup
variable [SemilatticeSup P] [OrderBot P]
@[sim... |
module
public import Mathlib.Order.Ideal
public import Mathlib.Order.PFilter
public import Mathlib.Data.Finset.Lattice.Basic
public import Mathlib.Order.PrimeIdeal
@[expose] public section
namespace Order
namespace Ideal
variable {P : Type*}
section semilatticeSup
variable [SemilatticeSup P] [OrderBot P]
@[sim... | @@ -99,8 +99,17 @@
have : ⊤ ∉ IF.I := Ideal.IsProper.top_not_mem IF.I
contradiction
+lemma compl_mem_I_iff_mem_F :
+ xᶜ ∈ IF.I ↔ x ∈ IF.F := by
+ constructor
+ · have : x ∈ IF.F ∨ xᶜ ∉ IF.I := by simpa [not_mem_I_iff_mem_F] using mem_or_compl_mem_F x
+ tauto
+ · have : x ∉ IF.F ∨ xᶜ ∈ IF.I := 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_36d55c101c52_5 | 9486525f9c637ca8 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Vorspiel/Order/Ideal.lean | Ideal | 5 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 8 | 1 | [
{
"theorem_name": "himp_mem_F_iff",
"depth": 1,
"n_commands": 0,
"n_lines": 3,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n simp [himp_eq, compl_mem_F_iff_mem_I, ←not_mem_F_iff_mem_I]\n tauto",
"n_chars": 75,
"n_subproofs": 0,
"n_tacti... | [
{
"name": "compl_mem_F_iff_mem_I",
"text": "lemma compl_mem_F_iff_mem_I :\n xᶜ ∈ IF.F ↔ x ∈ IF.I := by\n simpa using (compl_mem_I_iff_mem_F (x := xᶜ)).symm\n\n",
"fan_in": 1,
"n_lines": 5,
"n_chars": 125,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
"n_automation": 1,
... | [
{
"name": "himp_mem_F_iff",
"fan_in": 0,
"n_deps_direct": 2,
"n_deps_transitive": 8,
"n_lines": 6,
"n_chars": 173,
"n_subproofs": 0,
"n_tactics": 3,
"cyclomatic": 1,
"n_automation": 2,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"max_nesting":... | 7 |
module
public import Mathlib.Order.Ideal
public import Mathlib.Order.PFilter
public import Mathlib.Data.Finset.Lattice.Basic
public import Mathlib.Order.PrimeIdeal
@[expose] public section
namespace Order
namespace Ideal
variable {P : Type*}
section semilatticeSup
variable [SemilatticeSup P] [OrderBot P]
@[sim... |
module
public import Mathlib.Order.Ideal
public import Mathlib.Order.PFilter
public import Mathlib.Data.Finset.Lattice.Basic
public import Mathlib.Order.PrimeIdeal
@[expose] public section
namespace Order
namespace Ideal
variable {P : Type*}
section semilatticeSup
variable [SemilatticeSup P] [OrderBot P]
@[sim... | @@ -107,6 +107,10 @@
· have : x ∉ IF.F ∨ xᶜ ∈ IF.I := by simpa [not_mem_F_iff_mem_I] using mem_or_compl_mem_I x
tauto
+lemma compl_mem_F_iff_mem_I :
+ xᶜ ∈ IF.F ↔ x ∈ IF.I := by
+ simpa using (compl_mem_I_iff_mem_F (x := xᶜ)).symm
+
@[simp] lemma inf_mem_I_iff {x y : P} :
x ⊓ y ∈ IF.I ↔ x ∈ IF.I ∨ 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_22178f2c1b69_0 | 135f791a37f54ed7 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Kripke/Logic/K4n.lean | K4n | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 3 | 1 | [
{
"theorem_name": "succ_strictlyWeakerThan",
"depth": 1,
"n_commands": 0,
"n_lines": 34,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n constructor;\n . apply Hilbert.Normal.weakerThan_of_provable_axioms;\n rintro φ (rfl | rfl);\n . simp;\n ... | [
{
"name": "counterframe.iff_rel_from_zero",
"text": "lemma counterframe.iff_rel_from_zero {m : ℕ} {x : counterframe n} : (0 : counterframe n) ≺^[m] x ↔ x.1 = min m (n + 1) := by\n induction m generalizing x with\n | zero =>\n simp;\n tauto;\n | succ m ih =>\n constructor;\n . intro R0x;\n ... | [
{
"name": "succ_strictlyWeakerThan",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 36,
"n_chars": 1339,
"n_subproofs": 1,
"n_tactics": 66,
"cyclomatic": 8,
"n_automation": 10,
"n_rewrites": 0,
"n_structural": 22,
"automation_only": false,
... | 1 | module
public import Foundation.Modal.Kripke.AxiomFourN
public import Foundation.Modal.Kripke.Logic.KTc
@[expose] public section
namespace LO.Modal
open Kripke
open Modal.Kripke
namespace Kripke
abbrev Frame.IsK4n (F : Frame) (n) := F.IsWeakTransitive n
protected abbrev FrameClass.K4n (n : ℕ) : FrameClass := { F... | module
public import Foundation.Modal.Kripke.AxiomFourN
public import Foundation.Modal.Kripke.Logic.KTc
@[expose] public section
namespace LO.Modal
open Kripke
open Modal.Kripke
namespace Kripke
abbrev Frame.IsK4n (F : Frame) (n) := F.IsWeakTransitive n
protected abbrev FrameClass.K4n (n : ℕ) : FrameClass := { F... | @@ -53,6 +53,29 @@
abbrev counterframe.last : counterframe n |>.World := ⟨n + 1, by omega⟩
+lemma counterframe.iff_rel_from_zero {m : ℕ} {x : counterframe n} : (0 : counterframe n) ≺^[m] x ↔ x.1 = min m (n + 1) := by
+ induction m generalizing x with
+ | zero =>
+ simp;
+ tauto;
+ | succ m ih =>
+ 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_22178f2c1b69_1 | 65a27a5dd85b205c | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Kripke/Logic/K4n.lean | K4n | 1 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 3 | 1 | [
{
"theorem_name": "add_strictlyWeakerThan",
"depth": 1,
"n_commands": 0,
"n_lines": 7,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n induction m with\n | one => apply succ_strictlyWeakerThan;\n | succ m ih =>\n trans Modal.K4n (n + m);\n . a... | [
{
"name": "succ_strictlyWeakerThan",
"text": "lemma succ_strictlyWeakerThan : Modal.K4n (n + 1) ⪱ Modal.K4n n := by\n constructor;\n . apply Hilbert.Normal.weakerThan_of_provable_axioms;\n rintro φ (rfl | rfl);\n . simp;\n . suffices Modal.K4n n ⊢ □□^[n](.atom 0) 🡒 □□^[(n + 1)](.atom 0) by simpa... | [
{
"name": "add_strictlyWeakerThan",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 2,
"n_lines": 9,
"n_chars": 243,
"n_subproofs": 0,
"n_tactics": 11,
"cyclomatic": 2,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 4,
"automation_only": false,
"max... | 2 | module
public import Foundation.Modal.Kripke.AxiomFourN
public import Foundation.Modal.Kripke.Logic.KTc
@[expose] public section
namespace LO.Modal
open Kripke
open Modal.Kripke
namespace Kripke
abbrev Frame.IsK4n (F : Frame) (n) := F.IsWeakTransitive n
protected abbrev FrameClass.K4n (n : ℕ) : FrameClass := { F... | module
public import Foundation.Modal.Kripke.AxiomFourN
public import Foundation.Modal.Kripke.Logic.KTc
@[expose] public section
namespace LO.Modal
open Kripke
open Modal.Kripke
namespace Kripke
abbrev Frame.IsK4n (F : Frame) (n) := F.IsWeakTransitive n
protected abbrev FrameClass.K4n (n : ℕ) : FrameClass := { F... | @@ -133,12 +133,47 @@
. apply counterframe.iff_rel_from.mpr; simp;
. simp [Semantics.Models, Satisfies, M];
+lemma succ_strictlyWeakerThan : Modal.K4n (n + 1) ⪱ Modal.K4n n := by
+ constructor;
+ . apply Hilbert.Normal.weakerThan_of_provable_axioms;
+ rintro φ (rfl | rfl);
+ . 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_22178f2c1b69_2 | 940d52da7528649b | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Kripke/Logic/K4n.lean | K4n | 2 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 3 | 1 | [
{
"theorem_name": "strictlyWeakerThan_of_lt",
"depth": 1,
"n_commands": 0,
"n_lines": 4,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n convert add_strictlyWeakerThan (n := n) (m := ⟨m - n, by omega⟩);\n simp;\n omega;",
"n_chars": 92,
"n_s... | [
{
"name": "add_strictlyWeakerThan",
"text": "lemma add_strictlyWeakerThan {m : ℕ+} : Modal.K4n (n + m) ⪱ Modal.K4n n := by\n induction m with\n | one => apply succ_strictlyWeakerThan;\n | succ m ih =>\n trans Modal.K4n (n + m);\n . apply succ_strictlyWeakerThan;\n . apply ih;\n\n",
"fan_in":... | [
{
"name": "strictlyWeakerThan_of_lt",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 3,
"n_lines": 6,
"n_chars": 171,
"n_subproofs": 0,
"n_tactics": 7,
"cyclomatic": 1,
"n_automation": 3,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"max... | 3 | module
public import Foundation.Modal.Kripke.AxiomFourN
public import Foundation.Modal.Kripke.Logic.KTc
@[expose] public section
namespace LO.Modal
open Kripke
open Modal.Kripke
namespace Kripke
abbrev Frame.IsK4n (F : Frame) (n) := F.IsWeakTransitive n
protected abbrev FrameClass.K4n (n : ℕ) : FrameClass := { F... | module
public import Foundation.Modal.Kripke.AxiomFourN
public import Foundation.Modal.Kripke.Logic.KTc
@[expose] public section
namespace LO.Modal
open Kripke
open Modal.Kripke
namespace Kripke
abbrev Frame.IsK4n (F : Frame) (n) := F.IsWeakTransitive n
protected abbrev FrameClass.K4n (n : ℕ) : FrameClass := { F... | @@ -168,8 +168,19 @@
simp;
. simp [Semantics.Models, Satisfies, M];
-lemma strictlyWeakerThan_of_lt (hnm : n < m) : Modal.K4n m ⪱ Modal.K4n n := sorry
+lemma add_strictlyWeakerThan {m : ℕ+} : Modal.K4n (n + m) ⪱ Modal.K4n n := by
+ induction m with
+ | one => apply succ_strictlyWeakerThan;
+... | {
"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_7d27477f81aa_0 | 980e0be1a6861e0e | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Propositional/FMT/Completeness.lean | Completeness | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 7 | 1 | [
{
"theorem_name": "either_consistent_insert",
"depth": 1,
"n_commands": 0,
"n_lines": 45,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n contrapose! H_consis;\n apply iff_not_consistent.mpr;\n\n obtain ⟨h₁, h₂⟩ := H_consis;\n\n let Γ₀ : Formula ℕ ... | [
{
"name": "iff_not_consistent",
"text": "lemma iff_not_consistent : ¬(H.Consistent 𝓢) ↔ 𝓢 ⊢ (⩕ x ∈ H.1, ↑x) 🡒 (⩖ x ∈ H.2, ↑x) := by simp [Consistent];\n\n",
"fan_in": 1,
"n_lines": 3,
"n_chars": 137,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
"n_automation": 1,
"n_... | [
{
"name": "either_consistent_insert",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 50,
"n_chars": 1544,
"n_subproofs": 1,
"n_tactics": 75,
"cyclomatic": 3,
"n_automation": 4,
"n_rewrites": 4,
"n_structural": 22,
"automation_only": false,
... | 1 | module
public import Foundation.Propositional.FMT.Hilbert.Basic
public import Foundation.Propositional.Hilbert.WF_VF
@[expose] public section
namespace LO.Propositional
open Entailment.Corsi
variable {S} [Entailment S (Formula ℕ)]
{𝓢 : S}
abbrev SubformulaOf (φ : Formula ℕ) : Type := { ψ // ψ ∈ φ.subfor... | module
public import Foundation.Propositional.FMT.Hilbert.Basic
public import Foundation.Propositional.Hilbert.WF_VF
@[expose] public section
namespace LO.Propositional
open Entailment.Corsi
variable {S} [Entailment S (Formula ℕ)]
{𝓢 : S}
abbrev SubformulaOf (φ : Formula ℕ) : Type := { ψ // ψ ∈ φ.subfor... | @@ -26,6 +26,8 @@
def Consistent (𝓢 : S) (H : HintikkaPair φ) : Prop := 𝓢 ⊬ (⩕ x ∈ H.1, ↑x) 🡒 (⩖ x ∈ H.2, ↑x)
lemma iff_consistent : H.Consistent 𝓢 ↔ (𝓢 ⊬ (⩕ x ∈ H.1, ↑x) 🡒 (⩖ x ∈ H.2, ↑x)) := by simp [Consistent];
+lemma iff_not_consistent : ¬(H.Consistent 𝓢) ↔ 𝓢 ⊢ (⩕ x ∈ H.1, ↑x) 🡒 (⩖ x ∈ H.2, ↑x) := by ... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_7d27477f81aa_2 | 247a32a5bbbd6582 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Propositional/FMT/Completeness.lean | Completeness | 2 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 7 | 2 | [
{
"theorem_name": "enum_monotone₁",
"depth": 1,
"n_commands": 0,
"n_lines": 7,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n induction Γ with\n | nil => simp [enum];\n | cons ψ Γ ih =>\n trans (enum 𝓢 H Γ).1;\n . exact ih;\n . apply next... | [
{
"name": "next_monotone₁",
"text": "lemma next_monotone₁ : H.1 ⊆ (next 𝓢 ψ H).1 := by\n simp [next, insert₁, insert₂];\n split <;> grind;\n\n",
"fan_in": 2,
"n_lines": 5,
"n_chars": 115,
"n_subproofs": 0,
"n_tactics": 6,
"cyclomatic": 3,
"n_automation": 2,
"n_rewrites": 0... | [
{
"name": "enum_monotone₁",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 9,
"n_chars": 229,
"n_subproofs": 0,
"n_tactics": 10,
"cyclomatic": 2,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 3,
"automation_only": false,
"max_nesting... | 1 | module
public import Foundation.Propositional.FMT.Hilbert.Basic
public import Foundation.Propositional.Hilbert.WF_VF
@[expose] public section
namespace LO.Propositional
open Entailment.Corsi
variable {S} [Entailment S (Formula ℕ)]
{𝓢 : S}
abbrev SubformulaOf (φ : Formula ℕ) : Type := { ψ // ψ ∈ φ.subfor... | module
public import Foundation.Propositional.FMT.Hilbert.Basic
public import Foundation.Propositional.Hilbert.WF_VF
@[expose] public section
namespace LO.Propositional
open Entailment.Corsi
variable {S} [Entailment S (Formula ℕ)]
{𝓢 : S}
abbrev SubformulaOf (φ : Formula ℕ) : Type := { ψ // ψ ∈ φ.subfor... | @@ -110,6 +110,10 @@
. dsimp [next, h];
grind;
+lemma next_monotone₁ : H.1 ⊆ (next 𝓢 ψ H).1 := by
+ simp [next, insert₁, insert₂];
+ split <;> grind;
+
lemma next_monotone₂ : H.2 ⊆ (next 𝓢 ψ H).2 := by
simp [next, insert₁, insert₂];
split <;> grind;
@@ -134,7 +138,7 @@
| cons ψ Γ ih =>
tran... | {
"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_7d27477f81aa_3 | 427af857914993d5 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Propositional/FMT/Completeness.lean | Completeness | 3 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 7 | 2 | [
{
"theorem_name": "enum_monotone₂",
"depth": 1,
"n_commands": 0,
"n_lines": 7,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n induction Γ with\n | nil => simp [enum];\n | cons ψ Γ ih =>\n trans (enum 𝓢 H Γ).2;\n . exact ih;\n . apply next... | [
{
"name": "next_monotone₂",
"text": "lemma next_monotone₂ : H.2 ⊆ (next 𝓢 ψ H).2 := by\n simp [next, insert₁, insert₂];\n split <;> grind;\n\n",
"fan_in": 2,
"n_lines": 5,
"n_chars": 115,
"n_subproofs": 0,
"n_tactics": 6,
"cyclomatic": 3,
"n_automation": 2,
"n_rewrites": 0... | [
{
"name": "enum_monotone₂",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 9,
"n_chars": 229,
"n_subproofs": 0,
"n_tactics": 10,
"cyclomatic": 2,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 3,
"automation_only": false,
"max_nesting... | 1 | module
public import Foundation.Propositional.FMT.Hilbert.Basic
public import Foundation.Propositional.Hilbert.WF_VF
@[expose] public section
namespace LO.Propositional
open Entailment.Corsi
variable {S} [Entailment S (Formula ℕ)]
{𝓢 : S}
abbrev SubformulaOf (φ : Formula ℕ) : Type := { ψ // ψ ∈ φ.subfor... | module
public import Foundation.Propositional.FMT.Hilbert.Basic
public import Foundation.Propositional.Hilbert.WF_VF
@[expose] public section
namespace LO.Propositional
open Entailment.Corsi
variable {S} [Entailment S (Formula ℕ)]
{𝓢 : S}
abbrev SubformulaOf (φ : Formula ℕ) : Type := { ψ // ψ ∈ φ.subfor... | @@ -114,6 +114,10 @@
simp [next, insert₁, insert₂];
split <;> grind;
+lemma next_monotone₂ : H.2 ⊆ (next 𝓢 ψ H).2 := by
+ simp [next, insert₁, insert₂];
+ split <;> grind;
+
lemma next_either_mem (ψ) : ψ ∈ (next 𝓢 ψ H).1 ∨ ψ ∈ (next 𝓢 ψ H).2 := by
simp [next, insert₁, insert₂];
split <;> grind;
@@ -... | {
"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_7d27477f81aa_4 | c4d3b0bbfa65d032 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Propositional/FMT/Completeness.lean | Completeness | 4 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 7 | 1 | [
{
"theorem_name": "enum_of_mem",
"depth": 1,
"n_commands": 0,
"n_lines": 12,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n induction Γ with\n | nil => simp at hψ;\n | cons χ Δ ih =>\n simp only [List.mem_cons] at hψ;\n rcases hψ with rfl | h... | [
{
"name": "next_either_mem",
"text": "lemma next_either_mem (ψ) : ψ ∈ (next 𝓢 ψ H).1 ∨ ψ ∈ (next 𝓢 ψ H).2 := by\n simp [next, insert₁, insert₂];\n split <;> grind;\n\n",
"fan_in": 1,
"n_lines": 5,
"n_chars": 148,
"n_subproofs": 0,
"n_tactics": 6,
"cyclomatic": 3,
"n_automatio... | [
{
"name": "sat_saturated",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 4,
"n_lines": 7,
"n_chars": 150,
"n_subproofs": 0,
"n_tactics": 9,
"cyclomatic": 1,
"n_automation": 2,
"n_rewrites": 0,
"n_structural": 2,
"automation_only": false,
"max_nesting":... | 4 | module
public import Foundation.Propositional.FMT.Hilbert.Basic
public import Foundation.Propositional.Hilbert.WF_VF
@[expose] public section
namespace LO.Propositional
open Entailment.Corsi
variable {S} [Entailment S (Formula ℕ)]
{𝓢 : S}
abbrev SubformulaOf (φ : Formula ℕ) : Type := { ψ // ψ ∈ φ.subfor... | module
public import Foundation.Propositional.FMT.Hilbert.Basic
public import Foundation.Propositional.Hilbert.WF_VF
@[expose] public section
namespace LO.Propositional
open Entailment.Corsi
variable {S} [Entailment S (Formula ℕ)]
{𝓢 : S}
abbrev SubformulaOf (φ : Formula ℕ) : Type := { ψ // ψ ∈ φ.subfor... | @@ -118,6 +118,10 @@
simp [next, insert₁, insert₂];
split <;> grind;
+lemma next_either_mem (ψ) : ψ ∈ (next 𝓢 ψ H).1 ∨ ψ ∈ (next 𝓢 ψ H).2 := by
+ simp [next, insert₁, insert₂];
+ split <;> grind;
+
noncomputable def enum (𝓢 : S) (H : HintikkaPair φ) : List (SubformulaOf φ) → HintikkaPair 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_7d27477f81aa_5 | 791731b057f4e7df | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Propositional/FMT/Completeness.lean | Completeness | 5 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 7 | 2 | [
{
"theorem_name": "iff_mem_and",
"depth": 1,
"n_commands": 0,
"n_lines": 18,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n constructor;\n . rintro h;\n constructor;\n . apply imp_closed ?_ ?_ andElimL h <;> grind;\n . apply imp_closed ?_ ?... | [
{
"name": "imp_closed",
"text": "lemma imp_closed (hSψ : ψ ∈ φ.subformulas) (hSχ : χ ∈ φ.subformulas) : 𝓢 ⊢ ψ 🡒 χ → ⟨ψ, hSψ⟩ ∈ H.1.1 → ⟨χ, hSχ⟩ ∈ H.1.1 := by\n rintro h₁ hφ;\n by_contra hψ;\n replace hψ := iff_mem₂_not_mem₁.mpr hψ;\n apply H.consistent;\n apply C_replace_both h₁;\n . apply mem_fconj... | [
{
"name": "iff_mem_and",
"fan_in": 0,
"n_deps_direct": 3,
"n_deps_transitive": 6,
"n_lines": 20,
"n_chars": 670,
"n_subproofs": 0,
"n_tactics": 37,
"cyclomatic": 8,
"n_automation": 4,
"n_rewrites": 1,
"n_structural": 12,
"automation_only": false,
"max_nesting"... | 1 | module
public import Foundation.Propositional.FMT.Hilbert.Basic
public import Foundation.Propositional.Hilbert.WF_VF
@[expose] public section
namespace LO.Propositional
open Entailment.Corsi
variable {S} [Entailment S (Formula ℕ)]
{𝓢 : S}
abbrev SubformulaOf (φ : Formula ℕ) : Type := { ψ // ψ ∈ φ.subfor... | module
public import Foundation.Propositional.FMT.Hilbert.Basic
public import Foundation.Propositional.Hilbert.WF_VF
@[expose] public section
namespace LO.Propositional
open Entailment.Corsi
variable {S} [Entailment S (Formula ℕ)]
{𝓢 : S}
abbrev SubformulaOf (φ : Formula ℕ) : Type := { ψ // ψ ∈ φ.subfor... | @@ -225,6 +225,15 @@
apply not_mem_both hC.symm;
. apply HintikkaPair.mem₂_of_not_mem₁_of_saturated H.saturated;
+lemma imp_closed (hSψ : ψ ∈ φ.subformulas) (hSχ : χ ∈ φ.subformulas) : 𝓢 ⊢ ψ 🡒 χ → ⟨ψ, hSψ⟩ ∈ H.1.1 → ⟨χ, hSχ⟩ ∈ H.1.1 := by
+ rintro h₁ hφ;
+ by_contra hψ;
+ replace hψ := iff_mem₂_not_mem₁.... | {
"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_a03ceaa2dbfc_0 | ca0cd87570a369ef | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Vorspiel/List/Basic.lean | Basic | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 6 | 6 | [
{
"theorem_name": "mem_of_mem_remove",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n rw [mem_remove_iff] at h; exact h.1",
"n_chars": 41,
"n_subproofs": 0,
"n_tactics": 3,
"cyclomatic": 1,
... | [
{
"name": "mem_remove_iff",
"text": "lemma mem_remove_iff {l : List α} : b ∈ l.remove a ↔ b ∈ l ∧ b ≠ a := by\n simp [List.remove]\n\n",
"fan_in": 6,
"n_lines": 4,
"n_chars": 106,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_s... | [
{
"name": "mem_of_mem_remove",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 4,
"n_chars": 127,
"n_subproofs": 0,
"n_tactics": 3,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 1,
"n_structural": 1,
"automation_only": false,
"max_nesti... | 1 | module
public import Mathlib.Data.Fintype.Sigma
public import Mathlib.Data.Fintype.Vector
public import Mathlib.Data.List.GetD
public import Foundation.Vorspiel.Matrix
@[expose]
public section
/-- Compatibility shim: mathlib v4.31 removed the named `Finset.toSet` that used to back the
`Finset → Set` coercion (the coe... | module
public import Mathlib.Data.Fintype.Sigma
public import Mathlib.Data.Fintype.Vector
public import Mathlib.Data.List.GetD
public import Foundation.Vorspiel.Matrix
@[expose]
public section
/-- Compatibility shim: mathlib v4.31 removed the named `Finset.toSet` that used to back the
`Finset → Set` coercion (the coe... | @@ -67,16 +67,24 @@
@[simp]
lemma remove_singleton_of_ne {φ ψ : α} (h : φ ≠ ψ) : [φ].remove ψ = [φ] := by simp_all [List.remove];
-lemma mem_of_mem_remove {a b : α} {l : List α} (h : b ∈ l.remove a) : b ∈ l := sorry
+lemma mem_remove_iff {l : List α} : b ∈ l.remove a ↔ b ∈ l ∧ b ≠ a := by
+ simp [List.remove]
+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_a03ceaa2dbfc_1 | f7fc8aa1ac7ff63f | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Vorspiel/List/Basic.lean | Basic | 1 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 6 | 6 | [
{
"theorem_name": "mem_of_mem_remove",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n rw [mem_remove_iff] at h; exact h.1",
"n_chars": 41,
"n_subproofs": 0,
"n_tactics": 3,
"cyclomatic": 1,
... | [
{
"name": "mem_remove_iff",
"text": "lemma mem_remove_iff {l : List α} : b ∈ l.remove a ↔ b ∈ l ∧ b ≠ a := by\n simp [List.remove]\n\n",
"fan_in": 6,
"n_lines": 4,
"n_chars": 106,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_s... | [
{
"name": "remove_subset",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 6,
"n_chars": 152,
"n_subproofs": 0,
"n_tactics": 4,
"cyclomatic": 1,
"n_automation": 2,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": false,
"max_nesting":... | 1 | module
public import Mathlib.Data.Fintype.Sigma
public import Mathlib.Data.Fintype.Vector
public import Mathlib.Data.List.GetD
public import Foundation.Vorspiel.Matrix
@[expose]
public section
/-- Compatibility shim: mathlib v4.31 removed the named `Finset.toSet` that used to back the
`Finset → Set` coercion (the coe... | module
public import Mathlib.Data.Fintype.Sigma
public import Mathlib.Data.Fintype.Vector
public import Mathlib.Data.List.GetD
public import Foundation.Vorspiel.Matrix
@[expose]
public section
/-- Compatibility shim: mathlib v4.31 removed the named `Finset.toSet` that used to back the
`Finset → Set` coercion (the coe... | @@ -67,14 +67,21 @@
@[simp]
lemma remove_singleton_of_ne {φ ψ : α} (h : φ ≠ ψ) : [φ].remove ψ = [φ] := by simp_all [List.remove];
+lemma mem_remove_iff {l : List α} : b ∈ l.remove a ↔ b ∈ l ∧ b ≠ a := by
+ simp [List.remove]
+
@[simp] lemma remove_cons_self (l : List α) (a) :
(a :: l).remove a = l.remove 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_a03ceaa2dbfc_2 | f4fe42c60c9b3e7e | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Vorspiel/List/Basic.lean | Basic | 2 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 6 | 6 | [
{
"theorem_name": "mem_of_mem_remove",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n rw [mem_remove_iff] at h; exact h.1",
"n_chars": 41,
"n_subproofs": 0,
"n_tactics": 3,
"cyclomatic": 1,
... | [
{
"name": "mem_remove_iff",
"text": "lemma mem_remove_iff {l : List α} : b ∈ l.remove a ↔ b ∈ l ∧ b ≠ a := by\n simp [List.remove]\n\n",
"fan_in": 6,
"n_lines": 4,
"n_chars": 106,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_s... | [
{
"name": "remove_subset_remove",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 7,
"n_chars": 220,
"n_subproofs": 0,
"n_tactics": 4,
"cyclomatic": 1,
"n_automation": 3,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": false,
"max_ne... | 1 | module
public import Mathlib.Data.Fintype.Sigma
public import Mathlib.Data.Fintype.Vector
public import Mathlib.Data.List.GetD
public import Foundation.Vorspiel.Matrix
@[expose]
public section
/-- Compatibility shim: mathlib v4.31 removed the named `Finset.toSet` that used to back the
`Finset → Set` coercion (the coe... | module
public import Mathlib.Data.Fintype.Sigma
public import Mathlib.Data.Fintype.Vector
public import Mathlib.Data.List.GetD
public import Foundation.Vorspiel.Matrix
@[expose]
public section
/-- Compatibility shim: mathlib v4.31 removed the named `Finset.toSet` that used to back the
`Finset → Set` coercion (the coe... | @@ -67,17 +67,27 @@
@[simp]
lemma remove_singleton_of_ne {φ ψ : α} (h : φ ≠ ψ) : [φ].remove ψ = [φ] := by simp_all [List.remove];
+lemma mem_remove_iff {l : List α} : b ∈ l.remove a ↔ b ∈ l ∧ b ≠ a := by
+ simp [List.remove]
+
@[simp] lemma remove_cons_self (l : List α) (a) :
(a :: l).remove a = l.remove 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_a03ceaa2dbfc_3 | e12e49e450899fac | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Vorspiel/List/Basic.lean | Basic | 3 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 6 | 6 | [
{
"theorem_name": "mem_of_mem_remove",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n rw [mem_remove_iff] at h; exact h.1",
"n_chars": 41,
"n_subproofs": 0,
"n_tactics": 3,
"cyclomatic": 1,
... | [
{
"name": "mem_remove_iff",
"text": "lemma mem_remove_iff {l : List α} : b ∈ l.remove a ↔ b ∈ l ∧ b ≠ a := by\n simp [List.remove]\n\n",
"fan_in": 6,
"n_lines": 4,
"n_chars": 106,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_s... | [
{
"name": "remove_cons_subset_cons_remove",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 7,
"n_chars": 210,
"n_subproofs": 0,
"n_tactics": 5,
"cyclomatic": 3,
"n_automation": 3,
"n_rewrites": 0,
"n_structural": 2,
"automation_only": false,
... | 1 | module
public import Mathlib.Data.Fintype.Sigma
public import Mathlib.Data.Fintype.Vector
public import Mathlib.Data.List.GetD
public import Foundation.Vorspiel.Matrix
@[expose]
public section
/-- Compatibility shim: mathlib v4.31 removed the named `Finset.toSet` that used to back the
`Finset → Set` coercion (the coe... | module
public import Mathlib.Data.Fintype.Sigma
public import Mathlib.Data.Fintype.Vector
public import Mathlib.Data.List.GetD
public import Foundation.Vorspiel.Matrix
@[expose]
public section
/-- Compatibility shim: mathlib v4.31 removed the named `Finset.toSet` that used to back the
`Finset → Set` coercion (the coe... | @@ -67,17 +67,27 @@
@[simp]
lemma remove_singleton_of_ne {φ ψ : α} (h : φ ≠ ψ) : [φ].remove ψ = [φ] := by simp_all [List.remove];
+lemma mem_remove_iff {l : List α} : b ∈ l.remove a ↔ b ∈ l ∧ b ≠ a := by
+ simp [List.remove]
+
@[simp] lemma remove_cons_self (l : List α) (a) :
(a :: l).remove a = l.remove 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_a03ceaa2dbfc_4 | cfeafbb772297d5c | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Vorspiel/List/Basic.lean | Basic | 4 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 6 | 6 | [
{
"theorem_name": "mem_of_mem_remove",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n rw [mem_remove_iff] at h; exact h.1",
"n_chars": 41,
"n_subproofs": 0,
"n_tactics": 3,
"cyclomatic": 1,
... | [
{
"name": "mem_remove_iff",
"text": "lemma mem_remove_iff {l : List α} : b ∈ l.remove a ↔ b ∈ l ∧ b ≠ a := by\n simp [List.remove]\n\n",
"fan_in": 6,
"n_lines": 4,
"n_chars": 106,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_s... | [
{
"name": "remove_map_substet_map_remove",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 8,
"n_chars": 344,
"n_subproofs": 0,
"n_tactics": 7,
"cyclomatic": 2,
"n_automation": 4,
"n_rewrites": 0,
"n_structural": 4,
"automation_only": false,
... | 1 | module
public import Mathlib.Data.Fintype.Sigma
public import Mathlib.Data.Fintype.Vector
public import Mathlib.Data.List.GetD
public import Foundation.Vorspiel.Matrix
@[expose]
public section
/-- Compatibility shim: mathlib v4.31 removed the named `Finset.toSet` that used to back the
`Finset → Set` coercion (the coe... | module
public import Mathlib.Data.Fintype.Sigma
public import Mathlib.Data.Fintype.Vector
public import Mathlib.Data.List.GetD
public import Foundation.Vorspiel.Matrix
@[expose]
public section
/-- Compatibility shim: mathlib v4.31 removed the named `Finset.toSet` that used to back the
`Finset → Set` coercion (the coe... | @@ -67,17 +67,28 @@
@[simp]
lemma remove_singleton_of_ne {φ ψ : α} (h : φ ≠ ψ) : [φ].remove ψ = [φ] := by simp_all [List.remove];
+lemma mem_remove_iff {l : List α} : b ∈ l.remove a ↔ b ∈ l ∧ b ≠ a := by
+ simp [List.remove]
+
@[simp] lemma remove_cons_self (l : List α) (a) :
(a :: l).remove a = l.remove 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_a03ceaa2dbfc_5 | 9a4548197182c68c | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Vorspiel/List/Basic.lean | Basic | 5 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 6 | 1 | [
{
"theorem_name": "exists_of_not_suffix",
"depth": 1,
"n_commands": 0,
"n_lines": 9,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": "\n match l₁ with\n | [] => by simp\n | a :: l₁ => by\n intro h\n by_cases h₁₂ : l₁ <:+ l₂\n · exact ⟨l₁, a, ... | [
{
"name": "suffix_cons_of",
"text": "lemma suffix_cons_of {l₁ l₂ : List α} {a} : l₁ <:+ l₂ → l₁ <:+ a :: l₂ := fun h ↦ suffix_cons_iff.mpr <| Or.inr h\n\n",
"fan_in": 1,
"n_lines": 3,
"n_chars": 132,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rew... | [
{
"name": "exists_of_not_suffix",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 11,
"n_chars": 426,
"n_subproofs": 0,
"n_tactics": 8,
"cyclomatic": 3,
"n_automation": 2,
"n_rewrites": 0,
"n_structural": 4,
"automation_only": false,
"max_n... | 1 | module
public import Mathlib.Data.Fintype.Sigma
public import Mathlib.Data.Fintype.Vector
public import Mathlib.Data.List.GetD
public import Foundation.Vorspiel.Matrix
@[expose]
public section
/-- Compatibility shim: mathlib v4.31 removed the named `Finset.toSet` that used to back the
`Finset → Set` coercion (the coe... | module
public import Mathlib.Data.Fintype.Sigma
public import Mathlib.Data.Fintype.Vector
public import Mathlib.Data.List.GetD
public import Foundation.Vorspiel.Matrix
@[expose]
public section
/-- Compatibility shim: mathlib v4.31 removed the named `Finset.toSet` that used to back the
`Finset → Set` coercion (the coe... | @@ -121,6 +121,8 @@
section suffix
+lemma suffix_cons_of {l₁ l₂ : List α} {a} : l₁ <:+ l₂ → l₁ <:+ a :: l₂ := fun h ↦ suffix_cons_iff.mpr <| Or.inr h
+
lemma exists_of_not_suffix (l₁ l₂ : List α) : ¬l₁ <:+ l₂ → ∃ l a, a :: l <:+ l₁ ∧ l <:+ l₂ ∧ ¬a :: l <:+ l₂ :=
match l₁ with
| [] => by simp
@@ -128,7 ... | {
"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_a03ceaa2dbfc_6 | e0991fcb69e27263 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Vorspiel/List/Basic.lean | Basic | 6 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 6 | 1 | [
{
"theorem_name": "exists_of_not_suffix",
"depth": 1,
"n_commands": 0,
"n_lines": 9,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": "\n match l₁ with\n | [] => by simp\n | a :: l₁ => by\n intro h\n by_cases h₁₂ : l₁ <:+ l₂\n · exact ⟨l₁, a, ... | [
{
"name": "suffix_cons_of",
"text": "lemma suffix_cons_of {l₁ l₂ : List α} {a} : l₁ <:+ l₂ → l₁ <:+ a :: l₂ := fun h ↦ suffix_cons_iff.mpr <| Or.inr h\n\n",
"fan_in": 1,
"n_lines": 3,
"n_chars": 132,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rew... | [
{
"name": "suffix_trichotomy",
"fan_in": 0,
"n_deps_direct": 2,
"n_deps_transitive": 3,
"n_lines": 11,
"n_chars": 502,
"n_subproofs": 2,
"n_tactics": 10,
"cyclomatic": 5,
"n_automation": 2,
"n_rewrites": 0,
"n_structural": 6,
"automation_only": false,
"max_nes... | 3 | module
public import Mathlib.Data.Fintype.Sigma
public import Mathlib.Data.Fintype.Vector
public import Mathlib.Data.List.GetD
public import Foundation.Vorspiel.Matrix
@[expose]
public section
/-- Compatibility shim: mathlib v4.31 removed the named `Finset.toSet` that used to back the
`Finset → Set` coercion (the coe... | module
public import Mathlib.Data.Fintype.Sigma
public import Mathlib.Data.Fintype.Vector
public import Mathlib.Data.List.GetD
public import Foundation.Vorspiel.Matrix
@[expose]
public section
/-- Compatibility shim: mathlib v4.31 removed the named `Finset.toSet` that used to back the
`Finset → Set` coercion (the coe... | @@ -123,6 +123,8 @@
lemma suffix_of_cons_suffix {l₁ l₂ : List α} {a} : a :: l₁ <:+ l₂ → l₁ <:+ l₂ := by rintro ⟨l₂, rfl⟩; exact ⟨l₂ ++ [a], by simp⟩
+lemma suffix_cons_of {l₁ l₂ : List α} {a} : l₁ <:+ l₂ → l₁ <:+ a :: l₂ := fun h ↦ suffix_cons_iff.mpr <| Or.inr h
+
lemma exists_of_not_suffix (l₁ l₂ : List α) : ¬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_a03ceaa2dbfc_7 | d56dc87a27c2c8fe | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Vorspiel/List/Basic.lean | Basic | 7 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 6 | 1 | [
{
"theorem_name": "iff_nil_forall",
"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 subst h;\n tauto;\n . contrapose!;\n rintro h;\n obtain ⟨a, ha⟩ := exists_of_not_... | [
{
"name": "exists_of_not_nil",
"text": "lemma exists_of_not_nil (hl : l ≠ []) : ∃ a, a ∈ l := by\n match l with\n | [] => tauto;\n | a :: l => use a; simp only [mem_cons, true_or];\n\n",
"fan_in": 1,
"n_lines": 6,
"n_chars": 148,
"n_subproofs": 0,
"n_tactics": 7,
"cyclomatic": 2,
... | [
{
"name": "iff_nil_forall",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 12,
"n_chars": 219,
"n_subproofs": 1,
"n_tactics": 19,
"cyclomatic": 3,
"n_automation": 2,
"n_rewrites": 1,
"n_structural": 5,
"automation_only": false,
"max_nestin... | 1 | module
public import Mathlib.Data.Fintype.Sigma
public import Mathlib.Data.Fintype.Vector
public import Mathlib.Data.List.GetD
public import Foundation.Vorspiel.Matrix
@[expose]
public section
/-- Compatibility shim: mathlib v4.31 removed the named `Finset.toSet` that used to back the
`Finset → Set` coercion (the coe... | module
public import Mathlib.Data.Fintype.Sigma
public import Mathlib.Data.Fintype.Vector
public import Mathlib.Data.List.GetD
public import Foundation.Vorspiel.Matrix
@[expose]
public section
/-- Compatibility shim: mathlib v4.31 removed the named `Finset.toSet` that used to back the
`Finset → Set` coercion (the coe... | @@ -129,12 +129,21 @@
end Vector
+lemma exists_of_not_nil (hl : l ≠ []) : ∃ a, a ∈ l := by
+ match l with
+ | [] => tauto;
+ | a :: l => use a; simp only [mem_cons, true_or];
+
lemma iff_nil_forall : (l = []) ↔ (∀ a ∈ l, a ∈ []) := by
constructor;
. intro h;
subst h;
tauto;
- . sorry
+ . cont... | {
"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_61c6521971bb_0 | 1d5f4e29d6b8728f | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Kripke/NNFormula.lean | NNFormula | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 2 | 2 | [
{
"theorem_name": "neg_def",
"depth": 1,
"n_commands": 0,
"n_lines": 45,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n induction φ using NNFormula.rec' generalizing x with\n | hOr φ ψ ihφ ihψ =>\n simp only [DeMorgan.or, Satisfies.or_def, not_or... | [
{
"name": "dia_def",
"text": "protected lemma dia_def : x ⊧ ◇φ ↔ ∃ y, x ≺ y ∧ y ⊧ φ := by simp [Satisfies.iff_models, Satisfies];\n\n",
"fan_in": 2,
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"n_str... | [
{
"name": "neg_def",
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"n_deps_transitive": 5,
"n_lines": 47,
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"n_structural": 34,
"automation_only": false,
"max_nesting": ... | 2 | module
public import Foundation.Modal.Formula.NNFormula
public import Foundation.Modal.Kripke.Basic
@[expose] public section
namespace LO.Modal
open Entailment
variable {φ ψ : NNFormula ℕ}
namespace NNFormula.Kripke
def Satisfies (M : Kripke.Model) (x : M.World) : NNFormula ℕ → Prop
| atom a => M a x
| nat... | module
public import Foundation.Modal.Formula.NNFormula
public import Foundation.Modal.Kripke.Basic
@[expose] public section
namespace LO.Modal
open Entailment
variable {φ ψ : NNFormula ℕ}
namespace NNFormula.Kripke
def Satisfies (M : Kripke.Model) (x : M.World) : NNFormula ℕ → Prop
| atom a => M a x
| nat... | @@ -47,6 +47,8 @@
protected lemma box_def : x ⊧ □φ ↔ ∀ y, x ≺ y → y ⊧ φ := by simp [Satisfies.iff_models, Satisfies];
+protected lemma dia_def : x ⊧ ◇φ ↔ ∃ y, x ≺ y ∧ y ⊧ φ := by simp [Satisfies.iff_models, Satisfies];
+
protected lemma neg_def : x ⊧ ∼φ ↔ ¬x ⊧ φ := by
induction φ using NNFormula.rec' generaliz... | {
"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_61c6521971bb_1 | d62fcab121b3fa73 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Kripke/NNFormula.lean | NNFormula | 1 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 2 | 2 | [
{
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"depth": 1,
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"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n induction φ using NNFormula.rec' generalizing x with\n | hOr φ ψ ihφ ihψ =>\n simp only [DeMorgan.or, Satisfies.or_def, not_or... | [
{
"name": "dia_def",
"text": "protected lemma dia_def : x ⊧ ◇φ ↔ ∃ y, x ≺ y ∧ y ⊧ φ := by simp [Satisfies.iff_models, Satisfies];\n\n",
"fan_in": 2,
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"n_str... | [
{
"name": "Satisfies.toFormula",
"fan_in": 2,
"n_deps_direct": 3,
"n_deps_transitive": 4,
"n_lines": 54,
"n_chars": 1563,
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"cyclomatic": 17,
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"n_structural": 43,
"automation_only": false,
"m... | 1 | module
public import Foundation.Modal.Formula.NNFormula
public import Foundation.Modal.Kripke.Basic
@[expose] public section
namespace LO.Modal
open Entailment
variable {φ ψ : NNFormula ℕ}
namespace NNFormula.Kripke
def Satisfies (M : Kripke.Model) (x : M.World) : NNFormula ℕ → Prop
| atom a => M a x
| nat... | module
public import Foundation.Modal.Formula.NNFormula
public import Foundation.Modal.Kripke.Basic
@[expose] public section
namespace LO.Modal
open Entailment
variable {φ ψ : NNFormula ℕ}
namespace NNFormula.Kripke
def Satisfies (M : Kripke.Model) (x : M.World) : NNFormula ℕ → Prop
| atom a => M a x
| nat... | @@ -47,6 +47,8 @@
protected lemma box_def : x ⊧ □φ ↔ ∀ y, x ≺ y → y ⊧ φ := by simp [Satisfies.iff_models, Satisfies];
+protected lemma dia_def : x ⊧ ◇φ ↔ ∃ y, x ≺ y ∧ y ⊧ φ := by simp [Satisfies.iff_models, Satisfies];
+
protected lemma neg_def : x ⊧ ∼φ ↔ ¬x ⊧ φ := by
induction φ using NNFormula.rec' generaliz... | {
"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_61c6521971bb_2 | 11259c3246fc6198 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Kripke/NNFormula.lean | NNFormula | 2 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 2 | 2 | [
{
"theorem_name": "neg_def",
"depth": 1,
"n_commands": 0,
"n_lines": 45,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n induction φ using NNFormula.rec' generalizing x with\n | hOr φ ψ ihφ ihψ =>\n simp only [DeMorgan.or, Satisfies.or_def, not_or... | [
{
"name": "dia_def",
"text": "protected lemma dia_def : x ⊧ ◇φ ↔ ∃ y, x ≺ y ∧ y ⊧ φ := by simp [Satisfies.iff_models, Satisfies];\n\n",
"fan_in": 2,
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"n_str... | [
{
"name": "Satisfies.toNNFormula",
"fan_in": 2,
"n_deps_direct": 1,
"n_deps_transitive": 7,
"n_lines": 27,
"n_chars": 725,
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"cyclomatic": 5,
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"n_rewrites": 0,
"n_structural": 21,
"automation_only": false,
"ma... | 2 | module
public import Foundation.Modal.Formula.NNFormula
public import Foundation.Modal.Kripke.Basic
@[expose] public section
namespace LO.Modal
open Entailment
variable {φ ψ : NNFormula ℕ}
namespace NNFormula.Kripke
def Satisfies (M : Kripke.Model) (x : M.World) : NNFormula ℕ → Prop
| atom a => M a x
| nat... | module
public import Foundation.Modal.Formula.NNFormula
public import Foundation.Modal.Kripke.Basic
@[expose] public section
namespace LO.Modal
open Entailment
variable {φ ψ : NNFormula ℕ}
namespace NNFormula.Kripke
def Satisfies (M : Kripke.Model) (x : M.World) : NNFormula ℕ → Prop
| atom a => M a x
| nat... | @@ -47,6 +47,8 @@
protected lemma box_def : x ⊧ □φ ↔ ∀ y, x ≺ y → y ⊧ φ := by simp [Satisfies.iff_models, Satisfies];
+protected lemma dia_def : x ⊧ ◇φ ↔ ∃ y, x ≺ y ∧ y ⊧ φ := by simp [Satisfies.iff_models, Satisfies];
+
protected lemma neg_def : x ⊧ ∼φ ↔ ¬x ⊧ φ := by
induction φ using NNFormula.rec' generaliz... | {
"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_02bda5101c4a_0 | 6cedd49284776393 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Semantics/Algebra/Modal/Basic.lean | Basic | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 2 | 1 | [
{
"theorem_name": "box_axiomK",
"depth": 1,
"n_commands": 0,
"n_lines": 3,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n rw [himp_eq_top_iff];\n exact box_imp_le_box_imp_box;",
"n_chars": 59,
"n_subproofs": 0,
"n_tactics": 5,
"cyclo... | [
{
"name": "box_imp_le_box_imp_box",
"text": "lemma box_imp_le_box_imp_box : □(a ⇨ b) ≤ (□a ⇨ □b) := by\n suffices □(a ⇨ b) ⊓ □a ≤ □b by simpa;\n calc\n □(a ⇨ b) ⊓ □a ≤ □(a ⇨ b) ⊓ □a ⊓ □b := by simp [←box_meet];\n _ ≤ □b := by simp;\n\n",
"fan_in": 1,
"n_lines": 7,... | [
{
"name": "box_axiomK",
"fan_in": 0,
"n_deps_direct": 1,
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"max_nesting": 2
... | 1 | module
public import Foundation.Logic.LindenbaumAlgebra
@[expose] public section
namespace LO
/-- Algebra corresponding to `Modal.K` -/
class ModalAlgebra (α : Type*) extends BooleanAlgebra α, Box α, Dia α where
box_top : □(⊤ : α) = ⊤
box_meet (a b : α) : □(a ⊓ b) = □a ⊓ □b
dual_dia {a : α} : (◇a) = (□aᶜ)ᶜ
n... | module
public import Foundation.Logic.LindenbaumAlgebra
@[expose] public section
namespace LO
/-- Algebra corresponding to `Modal.K` -/
class ModalAlgebra (α : Type*) extends BooleanAlgebra α, Box α, Dia α where
box_top : □(⊤ : α) = ⊤
box_meet (a b : α) : □(a ⊓ b) = □a ⊓ □b
dual_dia {a : α} : (◇a) = (□aᶜ)ᶜ
n... | @@ -27,8 +27,16 @@
attribute [simp, grind .] box_top
@[simp, grind .] lemma dia_bot : ◇(⊥ : α) = ⊥ := by simp [dual_dia];
-lemma box_axiomK : □(a ⇨ b) ⇨ (□a ⇨ □b) = ⊤ := sorry
+lemma box_imp_le_box_imp_box : □(a ⇨ b) ≤ (□a ⇨ □b) := by
+ suffices □(a ⇨ b) ⊓ □a ≤ □b by simpa;
+ calc
+ □(a ⇨ b) ⊓ □a ≤ □(a ⇨ 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_02bda5101c4a_1 | b5b1dcf536855f0d | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Semantics/Algebra/Modal/Basic.lean | Basic | 1 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 2 | 1 | [
{
"theorem_name": "dia_monotone",
"depth": 1,
"n_commands": 0,
"n_lines": 3,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " calc\n ◇a ≤ ◇(a ⊔ (b \\ a)) := by simp [dia_or]\n _ = ◇b := by simp [sup_sdiff_cancel_right h]",
"n_chars": 117,
... | [
{
"name": "dia_or",
"text": "lemma dia_or : ◇(a ⊔ b) = ◇a ⊔ ◇b := by\n simp [dual_dia, compl_sup, box_meet];\n\n",
"fan_in": 1,
"n_lines": 4,
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"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"aut... | [
{
"name": "dia_monotone",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 6,
"n_chars": 182,
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"cyclomatic": 1,
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"n_rewrites": 0,
"n_structural": 1,
"automation_only": false,
"max_nesting": ... | 1 | module
public import Foundation.Logic.LindenbaumAlgebra
@[expose] public section
namespace LO
/-- Algebra corresponding to `Modal.K` -/
class ModalAlgebra (α : Type*) extends BooleanAlgebra α, Box α, Dia α where
box_top : □(⊤ : α) = ⊤
box_meet (a b : α) : □(a ⊓ b) = □a ⊓ □b
dual_dia {a : α} : (◇a) = (□aᶜ)ᶜ
n... | module
public import Foundation.Logic.LindenbaumAlgebra
@[expose] public section
namespace LO
/-- Algebra corresponding to `Modal.K` -/
class ModalAlgebra (α : Type*) extends BooleanAlgebra α, Box α, Dia α where
box_top : □(⊤ : α) = ⊤
box_meet (a b : α) : □(a ⊓ b) = □a ⊓ □b
dual_dia {a : α} : (◇a) = (□aᶜ)ᶜ
n... | @@ -27,8 +27,13 @@
attribute [simp, grind .] box_top
@[simp, grind .] lemma dia_bot : ◇(⊥ : α) = ⊥ := by simp [dual_dia];
+lemma dia_or : ◇(a ⊔ b) = ◇a ⊔ ◇b := by
+ simp [dual_dia, compl_sup, box_meet];
+
@[grind <-]
-lemma dia_monotone (h : a ≤ b) : ◇a ≤ ◇b := sorry
+lemma dia_monotone (h : a ≤ b) : ◇a ≤ ◇b := c... | {
"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_b742e608c839_0 | 999d9f63783f49f8 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/HFS/PRF.lean | PRF | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 4 | 1 | [
{
"theorem_name": "cseq_defined",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " .mk fun v ↦ by\n simp [Blueprint.cseqDef, cseq_iff, c.zero_defined.iff, c.succ_defined.iff]",
"n_chars": 94,
"n_subproofs":... | [
{
"name": "cseq_iff",
"text": "private lemma cseq_iff (s : V) : c.CSeq v s ↔\n Seq s\n ∧ (∃ z < s, z = c.zero v ∧ ⟪0, z⟫ ∈ s)\n ∧ (∀ i < 2 * s,\n (∃ l ≤ 2 * s, l = lh s ∧ i + 1 < l) → ∀ z < s, ⟪i, z⟫ ∈ s → ∃ u < s, u = c.succ v i z ∧ ⟪i + 1, u⟫ ∈ s) :=\n ⟨by rintro ⟨Hs, hz, hs⟩\n exact ⟨... | [
{
"name": "cseq_defined",
"fan_in": 1,
"n_deps_direct": 4,
"n_deps_transitive": 3,
"n_lines": 4,
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"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"max_nesting": 2... | 1 | module
public import Foundation.FirstOrder.Arithmetic.HFS.Seq
@[expose] public section
/-!
# Primitive Recursive Functions in $\mathsf{I} \Sigma_1$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
namespace PR
structure Blueprint (k : ℕ) where
zero : 𝚺₁.Semisen... | module
public import Foundation.FirstOrder.Arithmetic.HFS.Seq
@[expose] public section
/-!
# Primitive Recursive Functions in $\mathsf{I} \Sigma_1$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
namespace PR
structure Blueprint (k : ℕ) where
zero : 𝚺₁.Semisen... | @@ -48,8 +48,26 @@
def CSeq (s : V) : Prop := Seq s ∧ ⟪0, c.zero v⟫ ∈ s ∧ ∀ i < lh s - 1, ∀ z, ⟪i, z⟫ ∈ s → ⟪i + 1, c.succ v i z⟫ ∈ s
-lemma cseq_defined : 𝚺₁.Defined (fun v ↦ c.CSeq (v ·.succ) (v 0) : (Fin (k + 1) → V) → Prop) p.cseqDef := sorry
+private lemma cseq_iff (s : V) : c.CSeq v s ↔
+ Seq 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_b742e608c839_1 | f7adb7d9af57e530 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/HFS/PRF.lean | PRF | 1 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 4 | 1 | [
{
"theorem_name": "cseq_defined_iff",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " c.cseq_defined.iff",
"n_chars": 19,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
... | [
{
"name": "cseq_defined",
"text": "lemma cseq_defined : 𝚺₁.Defined (fun v ↦ c.CSeq (v ·.succ) (v 0) : (Fin (k + 1) → V) → Prop) p.cseqDef := .mk fun v ↦ by\n simp [Blueprint.cseqDef, cseq_iff, c.zero_defined.iff, c.succ_defined.iff]\n\n",
"fan_in": 1,
"n_lines": 4,
"n_chars": 213,
"n_subpr... | [
{
"name": "cseq_defined_iff",
"fan_in": 0,
"n_deps_direct": 4,
"n_deps_transitive": 4,
"n_lines": 4,
"n_chars": 135,
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"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
"max_nestin... | 2 | module
public import Foundation.FirstOrder.Arithmetic.HFS.Seq
@[expose] public section
/-!
# Primitive Recursive Functions in $\mathsf{I} \Sigma_1$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
namespace PR
structure Blueprint (k : ℕ) where
zero : 𝚺₁.Semisen... | module
public import Foundation.FirstOrder.Arithmetic.HFS.Seq
@[expose] public section
/-!
# Primitive Recursive Functions in $\mathsf{I} \Sigma_1$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
namespace PR
structure Blueprint (k : ℕ) where
zero : 𝚺₁.Semisen... | @@ -65,8 +65,11 @@
⟨lh s, by simp, rfl, by simpa [lt_tsub_iff_right] using hi⟩ z (lt_of_mem_rng hiz) hiz with ⟨_, _, rfl, h⟩
exact h⟩⟩
+lemma cseq_defined : 𝚺₁.Defined (fun v ↦ c.CSeq (v ·.succ) (v 0) : (Fin (k + 1) → V) → Prop) p.cseqDef := .mk fun v ↦ by
+ simp [Blueprint.cseqDef, cseq_iff, c.... | {
"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_b742e608c839_2 | 7ecc51d825dd071b | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/HFS/PRF.lean | PRF | 2 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 4 | 2 | [
{
"theorem_name": "cSeq_result_existsUnique",
"depth": 1,
"n_commands": 0,
"n_lines": 7,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n rcases CSeq.exists c v l with ⟨s, Hs, h⟩\n have : ∃ z, ⟪l, z⟫ ∈ s := Hs.seq.exists (show l < lh s from by simp [←... | [
{
"name": "unique",
"text": "lemma unique {s₁ s₂ : V} (H₁ : c.CSeq v s₁) (H₂ : c.CSeq v s₂) (h₁₂ : lh s₁ ≤ lh s₂) {i} (hi : i < lh s₁) {z₁ z₂} :\n ⟪i, z₁⟫ ∈ s₁ → ⟪i, z₂⟫ ∈ s₂ → z₁ = z₂ := by\n revert z₁ z₂\n suffices ∀ z₁ < s₁, ∀ z₂ < s₂, ⟪i, z₁⟫ ∈ s₁ → ⟪i, z₂⟫ ∈ s₂ → z₁ = z₂\n by intro z₁ z₂ hz₁ hz₂;... | [
{
"name": "cSeq_result_existsUnique",
"fan_in": 2,
"n_deps_direct": 2,
"n_deps_transitive": 4,
"n_lines": 9,
"n_chars": 468,
"n_subproofs": 1,
"n_tactics": 7,
"cyclomatic": 4,
"n_automation": 3,
"n_rewrites": 2,
"n_structural": 5,
"automation_only": false,
"ma... | 1 | module
public import Foundation.FirstOrder.Arithmetic.HFS.Seq
@[expose] public section
/-!
# Primitive Recursive Functions in $\mathsf{I} \Sigma_1$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
namespace PR
structure Blueprint (k : ℕ) where
zero : 𝚺₁.Semisen... | module
public import Foundation.FirstOrder.Arithmetic.HFS.Seq
@[expose] public section
/-!
# Primitive Recursive Functions in $\mathsf{I} \Sigma_1$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
namespace PR
structure Blueprint (k : ℕ) where
zero : 𝚺₁.Semisen... | @@ -83,6 +83,33 @@
lemma succ (h : c.CSeq v s) : ∀ i < lh s - 1, ∀ z, ⟪i, z⟫ ∈ s → ⟪i + 1, c.succ v i z⟫ ∈ s := h.2.2
+lemma unique {s₁ s₂ : V} (H₁ : c.CSeq v s₁) (H₂ : c.CSeq v s₂) (h₁₂ : lh s₁ ≤ lh s₂) {i} (hi : i < lh s₁) {z₁ z₂} :
+ ⟪i, z₁⟫ ∈ s₁ → ⟪i, z₂⟫ ∈ s₂ → z₁ = z₂ := by
+ revert z₁ z₂
+ suffices ∀ 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_b742e608c839_3 | e78e2f962a9b4739 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/HFS/PRF.lean | PRF | 3 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 4 | 3 | [
{
"theorem_name": "result_zero",
"depth": 1,
"n_commands": 0,
"n_lines": 3,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n rcases c.result_spec v 0 with ⟨s, Hs, _, h0⟩\n exact Hs.seq.isMapping.uniq h0 Hs.zero",
"n_chars": 95,
"n_subproofs": ... | [
{
"name": "result_spec",
"text": "lemma result_spec (u : V) : ∃ s, c.CSeq v s ∧ u + 1 = lh s ∧ ⟪u, c.result v u⟫ ∈ s :=\n Classical.choose!_spec (c.cSeq_result_existsUnique v u)\n\n",
"fan_in": 3,
"n_lines": 4,
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"n_subproofs": 0,
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"... | [
{
"name": "result_zero",
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"n_deps_direct": 4,
"n_deps_transitive": 6,
"n_lines": 5,
"n_chars": 153,
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"cyclomatic": 2,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 2,
"automation_only": false,
"max_nesting": 2... | 2 | module
public import Foundation.FirstOrder.Arithmetic.HFS.Seq
@[expose] public section
/-!
# Primitive Recursive Functions in $\mathsf{I} \Sigma_1$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
namespace PR
structure Blueprint (k : ℕ) where
zero : 𝚺₁.Semisen... | module
public import Foundation.FirstOrder.Arithmetic.HFS.Seq
@[expose] public section
/-!
# Primitive Recursive Functions in $\mathsf{I} \Sigma_1$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
namespace PR
structure Blueprint (k : ℕ) where
zero : 𝚺₁.Semisen... | @@ -161,13 +161,29 @@
noncomputable def result (u : V) : V := Classical.choose! (c.cSeq_result_existsUnique v u)
-@[simp] theorem result_zero : c.result v 0 = c.zero v := sorry
+lemma result_spec (u : V) : ∃ s, c.CSeq v s ∧ u + 1 = lh s ∧ ⟪u, c.result v u⟫ ∈ s :=
+ Classical.choose!_spec (c.cSeq_result_existsUniq... | {
"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_315da6e556da_0 | 55a1a7816716f09d | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Vorspiel/Matrix.lean | Matrix | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 4 | 2 | [
{
"theorem_name": "comp_vecCons'",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": "\n comp_vecCons f a s",
"n_chars": 21,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
... | [
{
"name": "comp_vecCons",
"text": "lemma comp_vecCons (f : α → β) (a : α) (s : Fin n → α) :\n (fun x ↦ f <| (a :> s) x) = f a :> f ∘ s :=\n funext (fun i => Fin.cases (by simp) (by simp) i)\n\n",
"fan_in": 2,
"n_lines": 5,
"n_chars": 170,
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"cyclom... | [
{
"name": "comp_vecCons'",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 5,
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"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
"max_nesting":... | 1 | module
public import Mathlib.Data.Fin.VecNotation
public import Foundation.Vorspiel.Nat.Basic
public import Foundation.Vorspiel.Fin.Basic
@[expose] public section
namespace Matrix
open Fin
section
variable {n : ℕ} {α : Type u}
infixr:70 " :> " => vecCons
@[simp] lemma vecCons_zero :
(a :> s) 0 = a := by simp
... | module
public import Mathlib.Data.Fin.VecNotation
public import Foundation.Vorspiel.Nat.Basic
public import Foundation.Vorspiel.Fin.Basic
@[expose] public section
namespace Matrix
open Fin
section
variable {n : ℕ} {α : Type u}
infixr:70 " :> " => vecCons
@[simp] lemma vecCons_zero :
(a :> s) 0 = a := by simp
... | @@ -92,10 +92,16 @@
haveI : Decidable (v ∘ Fin.succ = w ∘ Fin.succ) := decVec _ _ (by intros i; simpa using d _)
refine instDecidableAnd
+lemma comp_vecCons (f : α → β) (a : α) (s : Fin n → α) :
+ (fun x ↦ f <| (a :> s) x) = f a :> f ∘ s :=
+ funext (fun i => Fin.cases (by simp) (by simp) i)
+
lemm... | {
"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_315da6e556da_1 | b3ce629f91b6cf3c | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Vorspiel/Matrix.lean | Matrix | 1 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 4 | 2 | [
{
"theorem_name": "comp_vecCons'",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": "\n comp_vecCons f a s",
"n_chars": 21,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
... | [
{
"name": "comp_vecCons",
"text": "lemma comp_vecCons (f : α → β) (a : α) (s : Fin n → α) :\n (fun x ↦ f <| (a :> s) x) = f a :> f ∘ s :=\n funext (fun i => Fin.cases (by simp) (by simp) i)\n\n",
"fan_in": 2,
"n_lines": 5,
"n_chars": 170,
"n_subproofs": 0,
"n_tactics": 1,
"cyclom... | [
{
"name": "comp_vecCons''",
"fan_in": 4,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 4,
"n_chars": 124,
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"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
"max_nesting"... | 1 | module
public import Mathlib.Data.Fin.VecNotation
public import Foundation.Vorspiel.Nat.Basic
public import Foundation.Vorspiel.Fin.Basic
@[expose] public section
namespace Matrix
open Fin
section
variable {n : ℕ} {α : Type u}
infixr:70 " :> " => vecCons
@[simp] lemma vecCons_zero :
(a :> s) 0 = a := by simp
... | module
public import Mathlib.Data.Fin.VecNotation
public import Foundation.Vorspiel.Nat.Basic
public import Foundation.Vorspiel.Fin.Basic
@[expose] public section
namespace Matrix
open Fin
section
variable {n : ℕ} {α : Type u}
infixr:70 " :> " => vecCons
@[simp] lemma vecCons_zero :
(a :> s) 0 = a := by simp
... | @@ -92,8 +92,13 @@
haveI : Decidable (v ∘ Fin.succ = w ∘ Fin.succ) := decVec _ _ (by intros i; simpa using d _)
refine instDecidableAnd
-lemma comp_vecCons'' (f : α → β) (a : α) (s : Fin n → α) : f ∘ (a :> s) = f a :> f ∘ s := sorry
+lemma comp_vecCons (f : α → β) (a : α) (s : Fin n → α) :
+ (fun x ↦... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_315da6e556da_2 | 683f5ac725e959c2 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Vorspiel/Matrix.lean | Matrix | 2 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 4 | 4 | [
{
"theorem_name": "comp₁",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by simp [comp_vecCons'']",
"n_chars": 25,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 1,
"n_... | [
{
"name": "comp_vecCons''",
"text": "lemma comp_vecCons'' (f : α → β) (a : α) (s : Fin n → α) : f ∘ (a :> s) = f a :> f ∘ s :=\n comp_vecCons f a s\n\n",
"fan_in": 4,
"n_lines": 4,
"n_chars": 124,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewri... | [
{
"name": "comp₁",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 2,
"n_lines": 3,
"n_chars": 82,
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"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"max_nesting": 1
}
] | 2 | module
public import Mathlib.Data.Fin.VecNotation
public import Foundation.Vorspiel.Nat.Basic
public import Foundation.Vorspiel.Fin.Basic
@[expose] public section
namespace Matrix
open Fin
section
variable {n : ℕ} {α : Type u}
infixr:70 " :> " => vecCons
@[simp] lemma vecCons_zero :
(a :> s) 0 = a := by simp
... | module
public import Mathlib.Data.Fin.VecNotation
public import Foundation.Vorspiel.Nat.Basic
public import Foundation.Vorspiel.Fin.Basic
@[expose] public section
namespace Matrix
open Fin
section
variable {n : ℕ} {α : Type u}
infixr:70 " :> " => vecCons
@[simp] lemma vecCons_zero :
(a :> s) 0 = a := by simp
... | @@ -96,15 +96,18 @@
(fun x ↦ f <| (a :> s) x) = f a :> f ∘ s :=
funext (fun i => Fin.cases (by simp) (by simp) i)
+lemma comp_vecCons'' (f : α → β) (a : α) (s : Fin n → α) : f ∘ (a :> s) = f a :> f ∘ s :=
+ comp_vecCons f a s
+
@[simp] lemma comp₀ : f ∘ (![] : Fin 0 → α) = ![] := by simp [Matrix.empty_eq]
... | {
"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_315da6e556da_3 | 302d92c53b6f8947 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Vorspiel/Matrix.lean | Matrix | 3 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 4 | 1 | [
{
"theorem_name": "fun_eq_vec_four'",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n rw [fun_eq_vec_four v]; simp",
"n_chars": 34,
"n_subproofs": 0,
"n_tactics": 3,
"cyclomatic": 1,
"n_au... | [
{
"name": "fun_eq_vec_four",
"text": "lemma fun_eq_vec_four (v : Fin 4 → α) : v = ![v 0, v 1, v 2, v 3] := by\n funext x\n repeat cases' x using Fin.cases with x <;> simp\n\n",
"fan_in": 1,
"n_lines": 5,
"n_chars": 137,
"n_subproofs": 0,
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"n_auto... | [
{
"name": "fun_eq_vec_four'",
"fan_in": 0,
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"n_lines": 4,
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"cyclomatic": 1,
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"n_structural": 0,
"automation_only": false,
"max_nestin... | 1 | module
public import Mathlib.Data.Fin.VecNotation
public import Foundation.Vorspiel.Nat.Basic
public import Foundation.Vorspiel.Fin.Basic
@[expose] public section
namespace Matrix
open Fin
section
variable {n : ℕ} {α : Type u}
infixr:70 " :> " => vecCons
@[simp] lemma vecCons_zero :
(a :> s) 0 = a := by simp
... | module
public import Mathlib.Data.Fin.VecNotation
public import Foundation.Vorspiel.Nat.Basic
public import Foundation.Vorspiel.Fin.Basic
@[expose] public section
namespace Matrix
open Fin
section
variable {n : ℕ} {α : Type u}
infixr:70 " :> " => vecCons
@[simp] lemma vecCons_zero :
(a :> s) 0 = a := by simp
... | @@ -115,8 +115,13 @@
@[simp] lemma vecTail_comp (f : α → β) (v : Fin (n + 1) → α) : vecTail (f ∘ v) = f ∘ (vecTail v) := by
simp [vecTail, Function.comp_assoc]
-lemma fun_eq_vec_four' (f : α → β) (v : Fin 4 → α) : f ∘ v = ![f (v 0), f (v 1), f (v 2), f (v 3)] := sorry
+lemma fun_eq_vec_four (v : Fin 4 → α) : 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_315da6e556da_4 | 053ae44cd930ca80 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Vorspiel/Matrix.lean | Matrix | 4 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 4 | 1 | [
{
"theorem_name": "getM_some",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " getM_pure v",
"n_chars": 12,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites"... | [
{
"name": "getM_pure",
"text": "lemma getM_pure [LawfulMonad m] {n} {β : Fin n → Type u} (v : (i : Fin n) → β i) :\n getM (fun i => (pure (v i) : m (β i))) = pure v := by\n induction' n with n ih\n · unfold getM; congr; funext x; exact x.elim0\n · simp only [getM, map_pure, ih, seq_pure]\n exact co... | [
{
"name": "getM_some",
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"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 4,
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"max_nesting": 1
... | 1 | module
public import Mathlib.Data.Fin.VecNotation
public import Foundation.Vorspiel.Nat.Basic
public import Foundation.Vorspiel.Fin.Basic
@[expose] public section
namespace Matrix
open Fin
section
variable {n : ℕ} {α : Type u}
infixr:70 " :> " => vecCons
@[simp] lemma vecCons_zero :
(a :> s) 0 = a := by simp
... | module
public import Mathlib.Data.Fin.VecNotation
public import Foundation.Vorspiel.Nat.Basic
public import Foundation.Vorspiel.Fin.Basic
@[expose] public section
namespace Matrix
open Fin
section
variable {n : ℕ} {α : Type u}
infixr:70 " :> " => vecCons
@[simp] lemma vecCons_zero :
(a :> s) 0 = a := by simp
... | @@ -155,8 +155,15 @@
| 0, _, _ => pure finZeroElim
| _ + 1, _, f => Fin.cases <$> f 0 <*> getM (f ·.succ)
+lemma getM_pure [LawfulMonad m] {n} {β : Fin n → Type u} (v : (i : Fin n) → β i) :
+ getM (fun i => (pure (v i) : m (β i))) = pure v := by
+ induction' n with n ih
+ · unfold getM; congr; funext x... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_ad814b86861d_0 | 290ba7d2938becbf | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Kripke/Preservation.lean | Preservation | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 4 | 1 | [
{
"theorem_name": "toFun_rel_iterate_toFun_iff_of_inj",
"depth": 1,
"n_commands": 0,
"n_lines": 6,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": "\n ⟨ fun h ↦ by\n rcases f.back_iterate h with ⟨z, he, hz⟩\n have : z = y := inj he\n simpa [th... | [
{
"name": "back_iterate",
"text": "lemma back_iterate {w v} :\n f w ≺^[n] v → ∃ u, f u = v ∧ w ≺^[n] u := by\n match n with\n | 0 => simp\n | n + 1 =>\n intro h\n have : ∃ z, f w ≺ z ∧ z ≺^[n] v := by simpa using h\n rcases this with ⟨z, rfwz, hz⟩\n rcases f.back rfwz with ⟨z, rfl, rwz⟩\n ... | [
{
"name": "toFun_rel_iterate_toFun_iff_of_inj",
"fan_in": 0,
"n_deps_direct": 2,
"n_deps_transitive": 2,
"n_lines": 9,
"n_chars": 277,
"n_subproofs": 1,
"n_tactics": 5,
"cyclomatic": 2,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": fals... | 2 | module
public import Foundation.Modal.Kripke.Closure
@[expose] public section
namespace LO.Modal
namespace Kripke
open Formula.Kripke
section Bisimulation
structure Model.Bisimulation (M₁ M₂ : Kripke.Model) where
toRel : M₁.World → M₂.World → Prop
atomic {x₁ : M₁.World} {x₂ : M₂.World} {a : ℕ} : toRel x₁ x₂ ... | module
public import Foundation.Modal.Kripke.Closure
@[expose] public section
namespace LO.Modal
namespace Kripke
open Formula.Kripke
section Bisimulation
structure Model.Bisimulation (M₁ M₂ : Kripke.Model) where
toRel : M₁.World → M₂.World → Prop
atomic {x₁ : M₁.World} {x₂ : M₂.World} {a : ℕ} : toRel x₁ x₂ ... | @@ -131,8 +131,25 @@
rcases this with ⟨z, rxz, hz⟩
exact Rel.Iterate.succ_left (f.forth rxz) (forth_iterate hz)
+lemma back_iterate {w v} :
+ f w ≺^[n] v → ∃ u, f u = v ∧ w ≺^[n] u := by
+ match n with
+ | 0 => simp
+ | n + 1 =>
+ intro h
+ have : ∃ z, f w ≺ z ∧ z ≺^[n] v := by simpa using 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_ad814b86861d_1 | a992f09c630c8e52 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Kripke/Preservation.lean | Preservation | 1 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 4 | 1 | [
{
"theorem_name": "modal_equivalence",
"depth": 1,
"n_commands": 0,
"n_lines": 3,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n apply modal_equivalent_of_bisimilar $ Model.PseudoEpimorphism.bisimulation f;\n simp [Model.PseudoEpimorphism.bisimulati... | [
{
"name": "modal_equivalent_of_bisimilar",
"text": "lemma modal_equivalent_of_bisimilar (Bi : M₁ ⇄ M₂) (bisx : Bi x₁ x₂) : x₁ ↭ x₂ := by\n intro φ;\n induction φ generalizing x₁ x₂ with\n | hatom a => exact Bi.atomic bisx;\n | himp φ ψ ihp ihq =>\n constructor;\n . intro hpq hp;\n exact ihq b... | [
{
"name": "modal_equivalence",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 5,
"n_chars": 212,
"n_subproofs": 0,
"n_tactics": 5,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": false,
"max_nesti... | 1 | module
public import Foundation.Modal.Kripke.Closure
@[expose] public section
namespace LO.Modal
namespace Kripke
open Formula.Kripke
section Bisimulation
structure Model.Bisimulation (M₁ M₂ : Kripke.Model) where
toRel : M₁.World → M₂.World → Prop
atomic {x₁ : M₁.World} {x₂ : M₂.World} {a : ℕ} : toRel x₁ x₂ ... | module
public import Foundation.Modal.Kripke.Closure
@[expose] public section
namespace LO.Modal
namespace Kripke
open Formula.Kripke
section Bisimulation
structure Model.Bisimulation (M₁ M₂ : Kripke.Model) where
toRel : M₁.World → M₂.World → Prop
atomic {x₁ : M₁.World} {x₂ : M₂.World} {a : ℕ} : toRel x₁ x₂ ... | @@ -44,6 +44,26 @@
def ModalEquivalent {M₁ M₂ : Model} (w₁ : M₁.World) (w₂ : M₂.World) : Prop := ∀ {φ}, w₁ ⊧ φ ↔ w₂ ⊧ φ
infix:50 " ↭ " => ModalEquivalent
+lemma modal_equivalent_of_bisimilar (Bi : M₁ ⇄ M₂) (bisx : Bi x₁ x₂) : x₁ ↭ x₂ := by
+ intro φ;
+ induction φ generalizing x₁ x₂ with
+ | hatom a => exact Bi.... | {
"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_ad814b86861d_2 | a0bcdce3d1cd37f6 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Kripke/Preservation.lean | Preservation | 2 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 4 | 1 | [
{
"theorem_name": "modal_equivalence",
"depth": 1,
"n_commands": 0,
"n_lines": 3,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n apply modal_equivalent_of_bisimilar $ Model.PseudoEpimorphism.bisimulation f;\n simp [Model.PseudoEpimorphism.bisimulati... | [
{
"name": "modal_equivalent_of_bisimilar",
"text": "lemma modal_equivalent_of_bisimilar (Bi : M₁ ⇄ M₂) (bisx : Bi x₁ x₂) : x₁ ↭ x₂ := by\n intro φ;\n induction φ generalizing x₁ x₂ with\n | hatom a => exact Bi.atomic bisx;\n | himp φ ψ ihp ihq =>\n constructor;\n . intro hpq hp;\n exact ihq b... | [
{
"name": "validOnFrame_of_surjective_pseudoMorphism",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 2,
"n_lines": 7,
"n_chars": 370,
"n_subproofs": 1,
"n_tactics": 9,
"cyclomatic": 1,
"n_automation": 2,
"n_rewrites": 0,
"n_structural": 3,
"automation_only... | 1 | module
public import Foundation.Modal.Kripke.Closure
@[expose] public section
namespace LO.Modal
namespace Kripke
open Formula.Kripke
section Bisimulation
structure Model.Bisimulation (M₁ M₂ : Kripke.Model) where
toRel : M₁.World → M₂.World → Prop
atomic {x₁ : M₁.World} {x₂ : M₂.World} {a : ℕ} : toRel x₁ x₂ ... | module
public import Foundation.Modal.Kripke.Closure
@[expose] public section
namespace LO.Modal
namespace Kripke
open Formula.Kripke
section Bisimulation
structure Model.Bisimulation (M₁ M₂ : Kripke.Model) where
toRel : M₁.World → M₂.World → Prop
atomic {x₁ : M₁.World} {x₂ : M₂.World} {a : ℕ} : toRel x₁ x₂ ... | @@ -44,6 +44,26 @@
def ModalEquivalent {M₁ M₂ : Model} (w₁ : M₁.World) (w₂ : M₂.World) : Prop := ∀ {φ}, w₁ ⊧ φ ↔ w₂ ⊧ φ
infix:50 " ↭ " => ModalEquivalent
+lemma modal_equivalent_of_bisimilar (Bi : M₁ ⇄ M₂) (bisx : Bi x₁ x₂) : x₁ ↭ x₂ := by
+ intro φ;
+ induction φ generalizing x₁ x₂ with
+ | hatom a => exact Bi.... | {
"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_ad814b86861d_3 | 5dbba7d49d558b4c | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Kripke/Preservation.lean | Preservation | 3 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 4 | 1 | [
{
"theorem_name": "theory_ValidOnFrame_of_surjective_pseudoMorphism",
"depth": 1,
"n_commands": 0,
"n_lines": 4,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n simp only [Semantics.modelsSet_iff];\n intro h φ hp;\n exact validOnFrame_of_surjective_... | [
{
"name": "validOnFrame_of_surjective_pseudoMorphism",
"text": "lemma validOnFrame_of_surjective_pseudoMorphism (f : F₁ →ₚ F₂) (f_surjective : Function.Surjective f) : F₁ ⊧ φ → F₂ ⊧ φ := by\n intro h V₂ u;\n obtain ⟨x, rfl⟩ := f_surjective u;\n refine (Model.PseudoEpimorphism.ofAtomic (M₁ := ⟨F₁, λ a w =... | [
{
"name": "theory_ValidOnFrame_of_surjective_pseudoMorphism",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 3,
"n_lines": 6,
"n_chars": 283,
"n_subproofs": 0,
"n_tactics": 7,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 2,
"automati... | 2 | module
public import Foundation.Modal.Kripke.Closure
@[expose] public section
namespace LO.Modal
namespace Kripke
open Formula.Kripke
section Bisimulation
structure Model.Bisimulation (M₁ M₂ : Kripke.Model) where
toRel : M₁.World → M₂.World → Prop
atomic {x₁ : M₁.World} {x₂ : M₂.World} {a : ℕ} : toRel x₁ x₂ ... | module
public import Foundation.Modal.Kripke.Closure
@[expose] public section
namespace LO.Modal
namespace Kripke
open Formula.Kripke
section Bisimulation
structure Model.Bisimulation (M₁ M₂ : Kripke.Model) where
toRel : M₁.World → M₂.World → Prop
atomic {x₁ : M₁.World} {x₂ : M₂.World} {a : ℕ} : toRel x₁ x₂ ... | @@ -182,8 +182,17 @@
variable {F₁ F₂ : Kripke.Frame} {M₁ M₂ : Kripke.Model} {φ : Formula ℕ} {T : FormulaSet ℕ}
-lemma theory_ValidOnFrame_of_surjective_pseudoMorphism (f : F₁ →ₚ F₂) (f_surjective : Function.Surjective f) : F₁ ⊧* T → F₂ ⊧* T := sorry
+lemma validOnFrame_of_surjective_pseudoMorphism (f : F₁ →ₚ F₂) (... | {
"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_d2e9778766d4_0 | 8be12adaf991eb6f | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/SetTheory/Function.lean | Function | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 19 | 1 | [
{
"theorem_name": "mem_domain_iff",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n simpa [domain] using fun _ ↦ mem_sUnion_sUnion_of_kpair_mem_left",
"n_chars": 73,
"n_subproofs": 0,
"n_tactics"... | [
{
"name": "mem_sUnion_sUnion_of_kpair_mem_left",
"text": "lemma mem_sUnion_sUnion_of_kpair_mem_left {x y R : V} (h : ⟨x, y⟩ₖ ∈ R) : x ∈ ⋃ˢ ⋃ˢ R := by\n simp only [mem_sUnion_iff]\n refine ⟨{x, y}, ⟨⟨x, y⟩ₖ, h, by simp [kpair]⟩, by simp⟩\n\n",
"fan_in": 1,
"n_lines": 5,
"n_chars": 210,
"n_s... | [
{
"name": "mem_domain_iff",
"fan_in": 9,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 4,
"n_chars": 156,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"max_nesting":... | 1 | module
public import Foundation.FirstOrder.SetTheory.Z
@[expose] public section
/-!
# Basic definitions and lemmata for relations and functions
-/
namespace LO.FirstOrder.SetTheory
variable {V : Type*} [SetStructure V] [Nonempty V] [V↓[ℒₛₑₜ] ⊧* 𝗭]
/-! ### Relations -/
noncomputable def domain (R : V) : V := {x ∈... | module
public import Foundation.FirstOrder.SetTheory.Z
@[expose] public section
/-!
# Basic definitions and lemmata for relations and functions
-/
namespace LO.FirstOrder.SetTheory
variable {V : Type*} [SetStructure V] [Nonempty V] [V↓[ℒₛₑₜ] ⊧* 𝗭]
/-! ### Relations -/
noncomputable def domain (R : V) : V := {x ∈... | @@ -19,8 +19,13 @@
section domain
-lemma mem_domain_iff {R x : V} : x ∈ domain R ↔ ∃ y, ⟨x, y⟩ₖ ∈ R := sorry
+lemma mem_sUnion_sUnion_of_kpair_mem_left {x y R : V} (h : ⟨x, y⟩ₖ ∈ R) : x ∈ ⋃ˢ ⋃ˢ R := by
+ simp only [mem_sUnion_iff]
+ refine ⟨{x, y}, ⟨⟨x, y⟩ₖ, h, by simp [kpair]⟩, by simp⟩
+lemma mem_domain_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_d2e9778766d4_1 | 021cbd9d349d6cc8 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/SetTheory/Function.lean | Function | 1 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 19 | 1 | [
{
"theorem_name": "mem_domain_iff",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n simpa [domain] using fun _ ↦ mem_sUnion_sUnion_of_kpair_mem_left",
"n_chars": 73,
"n_subproofs": 0,
"n_tactics"... | [
{
"name": "mem_sUnion_sUnion_of_kpair_mem_left",
"text": "lemma mem_sUnion_sUnion_of_kpair_mem_left {x y R : V} (h : ⟨x, y⟩ₖ ∈ R) : x ∈ ⋃ˢ ⋃ˢ R := by\n simp only [mem_sUnion_iff]\n refine ⟨{x, y}, ⟨⟨x, y⟩ₖ, h, by simp [kpair]⟩, by simp⟩\n\n",
"fan_in": 1,
"n_lines": 5,
"n_chars": 210,
"n_s... | [
{
"name": "domain_subset_of_subset_prod",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 2,
"n_lines": 8,
"n_chars": 279,
"n_subproofs": 2,
"n_tactics": 6,
"cyclomatic": 2,
"n_automation": 2,
"n_rewrites": 0,
"n_structural": 3,
"automation_only": false,
... | 1 | module
public import Foundation.FirstOrder.SetTheory.Z
@[expose] public section
/-!
# Basic definitions and lemmata for relations and functions
-/
namespace LO.FirstOrder.SetTheory
variable {V : Type*} [SetStructure V] [Nonempty V] [V↓[ℒₛₑₜ] ⊧* 𝗭]
/-! ### Relations -/
noncomputable def domain (R : V) : V := {x ∈... | module
public import Foundation.FirstOrder.SetTheory.Z
@[expose] public section
/-!
# Basic definitions and lemmata for relations and functions
-/
namespace LO.FirstOrder.SetTheory
variable {V : Type*} [SetStructure V] [Nonempty V] [V↓[ℒₛₑₜ] ⊧* 𝗭]
/-! ### Relations -/
noncomputable def domain (R : V) : V := {x ∈... | @@ -19,8 +19,13 @@
section domain
-lemma mem_domain_iff {R x : V} : x ∈ domain R ↔ ∃ y, ⟨x, y⟩ₖ ∈ R := sorry
+lemma mem_sUnion_sUnion_of_kpair_mem_left {x y R : V} (h : ⟨x, y⟩ₖ ∈ R) : x ∈ ⋃ˢ ⋃ˢ R := by
+ simp only [mem_sUnion_iff]
+ refine ⟨{x, y}, ⟨⟨x, y⟩ₖ, h, by simp [kpair]⟩, by simp⟩
+lemma mem_domain_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_d2e9778766d4_2 | e93d78f77cbb7056 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/SetTheory/Function.lean | Function | 2 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 19 | 1 | [
{
"theorem_name": "mem_domain_iff",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n simpa [domain] using fun _ ↦ mem_sUnion_sUnion_of_kpair_mem_left",
"n_chars": 73,
"n_subproofs": 0,
"n_tactics"... | [
{
"name": "mem_sUnion_sUnion_of_kpair_mem_left",
"text": "lemma mem_sUnion_sUnion_of_kpair_mem_left {x y R : V} (h : ⟨x, y⟩ₖ ∈ R) : x ∈ ⋃ˢ ⋃ˢ R := by\n simp only [mem_sUnion_iff]\n refine ⟨{x, y}, ⟨⟨x, y⟩ₖ, h, by simp [kpair]⟩, by simp⟩\n\n",
"fan_in": 1,
"n_lines": 5,
"n_chars": 210,
"n_s... | [
{
"name": "domain_inter_subset",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 2,
"n_lines": 4,
"n_chars": 166,
"n_subproofs": 0,
"n_tactics": 4,
"cyclomatic": 1,
"n_automation": 2,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": false,
"max_nes... | 1 | module
public import Foundation.FirstOrder.SetTheory.Z
@[expose] public section
/-!
# Basic definitions and lemmata for relations and functions
-/
namespace LO.FirstOrder.SetTheory
variable {V : Type*} [SetStructure V] [Nonempty V] [V↓[ℒₛₑₜ] ⊧* 𝗭]
/-! ### Relations -/
noncomputable def domain (R : V) : V := {x ∈... | module
public import Foundation.FirstOrder.SetTheory.Z
@[expose] public section
/-!
# Basic definitions and lemmata for relations and functions
-/
namespace LO.FirstOrder.SetTheory
variable {V : Type*} [SetStructure V] [Nonempty V] [V↓[ℒₛₑₜ] ⊧* 𝗭]
/-! ### Relations -/
noncomputable def domain (R : V) : V := {x ∈... | @@ -19,8 +19,13 @@
section domain
-lemma mem_domain_iff {R x : V} : x ∈ domain R ↔ ∃ y, ⟨x, y⟩ₖ ∈ R := sorry
+lemma mem_sUnion_sUnion_of_kpair_mem_left {x y R : V} (h : ⟨x, y⟩ₖ ∈ R) : x ∈ ⋃ˢ ⋃ˢ R := by
+ simp only [mem_sUnion_iff]
+ refine ⟨{x, y}, ⟨⟨x, y⟩ₖ, h, by simp [kpair]⟩, by simp⟩
+lemma mem_domain_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_d2e9778766d4_3 | fa37da9056862c0d | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/SetTheory/Function.lean | Function | 3 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 19 | 1 | [
{
"theorem_name": "mem_range_iff",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n simpa [range] using fun _ ↦ mem_sUnion_sUnion_of_kpair_mem_right",
"n_chars": 72,
"n_subproofs": 0,
"n_tactics": ... | [
{
"name": "mem_sUnion_sUnion_of_kpair_mem_right",
"text": "lemma mem_sUnion_sUnion_of_kpair_mem_right {x y R : V} (h : ⟨x, y⟩ₖ ∈ R) : y ∈ ⋃ˢ ⋃ˢ R := by\n simp only [mem_sUnion_iff]\n refine ⟨{x, y}, ⟨⟨x, y⟩ₖ, h, by simp [kpair]⟩, by simp⟩\n\n",
"fan_in": 1,
"n_lines": 5,
"n_chars": 211,
"n... | [
{
"name": "mem_range_iff",
"fan_in": 11,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 4,
"n_chars": 153,
"n_subproofs": 0,
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"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"max_nesting":... | 1 | module
public import Foundation.FirstOrder.SetTheory.Z
@[expose] public section
/-!
# Basic definitions and lemmata for relations and functions
-/
namespace LO.FirstOrder.SetTheory
variable {V : Type*} [SetStructure V] [Nonempty V] [V↓[ℒₛₑₜ] ⊧* 𝗭]
/-! ### Relations -/
noncomputable def domain (R : V) : V := {x ∈... | module
public import Foundation.FirstOrder.SetTheory.Z
@[expose] public section
/-!
# Basic definitions and lemmata for relations and functions
-/
namespace LO.FirstOrder.SetTheory
variable {V : Type*} [SetStructure V] [Nonempty V] [V↓[ℒₛₑₜ] ⊧* 𝗭]
/-! ### Relations -/
noncomputable def domain (R : V) : V := {x ∈... | @@ -54,8 +54,13 @@
section range
-lemma mem_range_iff {R y : V} : y ∈ range R ↔ ∃ x, ⟨x, y⟩ₖ ∈ R := sorry
+lemma mem_sUnion_sUnion_of_kpair_mem_right {x y R : V} (h : ⟨x, y⟩ₖ ∈ R) : y ∈ ⋃ˢ ⋃ˢ R := by
+ simp only [mem_sUnion_iff]
+ refine ⟨{x, y}, ⟨⟨x, y⟩ₖ, h, by simp [kpair]⟩, by simp⟩
+lemma mem_range_iff {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_d2e9778766d4_4 | 3d43eb88755587c3 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/SetTheory/Function.lean | Function | 4 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 19 | 1 | [
{
"theorem_name": "mem_range_iff",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n simpa [range] using fun _ ↦ mem_sUnion_sUnion_of_kpair_mem_right",
"n_chars": 72,
"n_subproofs": 0,
"n_tactics": ... | [
{
"name": "mem_sUnion_sUnion_of_kpair_mem_right",
"text": "lemma mem_sUnion_sUnion_of_kpair_mem_right {x y R : V} (h : ⟨x, y⟩ₖ ∈ R) : y ∈ ⋃ˢ ⋃ˢ R := by\n simp only [mem_sUnion_iff]\n refine ⟨{x, y}, ⟨⟨x, y⟩ₖ, h, by simp [kpair]⟩, by simp⟩\n\n",
"fan_in": 1,
"n_lines": 5,
"n_chars": 211,
"n... | [
{
"name": "range_subset_of_subset_prod",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 2,
"n_lines": 8,
"n_chars": 276,
"n_subproofs": 2,
"n_tactics": 6,
"cyclomatic": 2,
"n_automation": 2,
"n_rewrites": 0,
"n_structural": 3,
"automation_only": false,
... | 1 | module
public import Foundation.FirstOrder.SetTheory.Z
@[expose] public section
/-!
# Basic definitions and lemmata for relations and functions
-/
namespace LO.FirstOrder.SetTheory
variable {V : Type*} [SetStructure V] [Nonempty V] [V↓[ℒₛₑₜ] ⊧* 𝗭]
/-! ### Relations -/
noncomputable def domain (R : V) : V := {x ∈... | module
public import Foundation.FirstOrder.SetTheory.Z
@[expose] public section
/-!
# Basic definitions and lemmata for relations and functions
-/
namespace LO.FirstOrder.SetTheory
variable {V : Type*} [SetStructure V] [Nonempty V] [V↓[ℒₛₑₜ] ⊧* 𝗭]
/-! ### Relations -/
noncomputable def domain (R : V) : V := {x ∈... | @@ -54,8 +54,13 @@
section range
-lemma mem_range_iff {R y : V} : y ∈ range R ↔ ∃ x, ⟨x, y⟩ₖ ∈ R := sorry
+lemma mem_sUnion_sUnion_of_kpair_mem_right {x y R : V} (h : ⟨x, y⟩ₖ ∈ R) : y ∈ ⋃ˢ ⋃ˢ R := by
+ simp only [mem_sUnion_iff]
+ refine ⟨{x, y}, ⟨⟨x, y⟩ₖ, h, by simp [kpair]⟩, by simp⟩
+lemma mem_range_iff {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_d2e9778766d4_5 | a17a5ee83655a08e | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/SetTheory/Function.lean | Function | 5 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 19 | 1 | [
{
"theorem_name": "mem_range_iff",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n simpa [range] using fun _ ↦ mem_sUnion_sUnion_of_kpair_mem_right",
"n_chars": 72,
"n_subproofs": 0,
"n_tactics": ... | [
{
"name": "mem_sUnion_sUnion_of_kpair_mem_right",
"text": "lemma mem_sUnion_sUnion_of_kpair_mem_right {x y R : V} (h : ⟨x, y⟩ₖ ∈ R) : y ∈ ⋃ˢ ⋃ˢ R := by\n simp only [mem_sUnion_iff]\n refine ⟨{x, y}, ⟨⟨x, y⟩ₖ, h, by simp [kpair]⟩, by simp⟩\n\n",
"fan_in": 1,
"n_lines": 5,
"n_chars": 211,
"n... | [
{
"name": "range_inter_subset",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 2,
"n_lines": 4,
"n_chars": 161,
"n_subproofs": 0,
"n_tactics": 4,
"cyclomatic": 1,
"n_automation": 2,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": false,
"max_nest... | 1 | module
public import Foundation.FirstOrder.SetTheory.Z
@[expose] public section
/-!
# Basic definitions and lemmata for relations and functions
-/
namespace LO.FirstOrder.SetTheory
variable {V : Type*} [SetStructure V] [Nonempty V] [V↓[ℒₛₑₜ] ⊧* 𝗭]
/-! ### Relations -/
noncomputable def domain (R : V) : V := {x ∈... | module
public import Foundation.FirstOrder.SetTheory.Z
@[expose] public section
/-!
# Basic definitions and lemmata for relations and functions
-/
namespace LO.FirstOrder.SetTheory
variable {V : Type*} [SetStructure V] [Nonempty V] [V↓[ℒₛₑₜ] ⊧* 𝗭]
/-! ### Relations -/
noncomputable def domain (R : V) : V := {x ∈... | @@ -54,8 +54,13 @@
section range
-lemma mem_range_iff {R y : V} : y ∈ range R ↔ ∃ x, ⟨x, y⟩ₖ ∈ R := sorry
+lemma mem_sUnion_sUnion_of_kpair_mem_right {x y R : V} (h : ⟨x, y⟩ₖ ∈ R) : y ∈ ⋃ˢ ⋃ˢ R := by
+ simp only [mem_sUnion_iff]
+ refine ⟨{x, y}, ⟨⟨x, y⟩ₖ, h, by simp [kpair]⟩, by simp⟩
+lemma mem_range_iff {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_d2e9778766d4_6 | 8a8bdcf929164349 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/SetTheory/Function.lean | Function | 6 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 19 | 1 | [
{
"theorem_name": "mem_function_iff",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by simp [function, function_def]",
"n_chars": 33,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_auto... | [
{
"name": "function_def",
"text": "lemma function_def {Y X : V} : Y ^ X = function Y X := rfl\n\n",
"fan_in": 1,
"n_lines": 3,
"n_chars": 60,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": t... | [
{
"name": "mem_function_iff",
"fan_in": 5,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 3,
"n_chars": 150,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"max_nesting... | 1 | module
public import Foundation.FirstOrder.SetTheory.Z
@[expose] public section
/-!
# Basic definitions and lemmata for relations and functions
-/
namespace LO.FirstOrder.SetTheory
variable {V : Type*} [SetStructure V] [Nonempty V] [V↓[ℒₛₑₜ] ⊧* 𝗭]
/-! ### Relations -/
noncomputable def domain (R : V) : V := {x ∈... | module
public import Foundation.FirstOrder.SetTheory.Z
@[expose] public section
/-!
# Basic definitions and lemmata for relations and functions
-/
namespace LO.FirstOrder.SetTheory
variable {V : Type*} [SetStructure V] [Nonempty V] [V↓[ℒₛₑₜ] ⊧* 𝗭]
/-! ### Relations -/
noncomputable def domain (R : V) : V := {x ∈... | @@ -93,8 +93,10 @@
noncomputable instance : Pow V V := ⟨fun Y X ↦ function Y X⟩
-lemma mem_function_iff {f Y X : V} : f ∈ Y ^ X ↔ f ⊆ X ×ˢ Y ∧ ∀ x ∈ X, ∃! y, ⟨x, y⟩ₖ ∈ f := sorry
+lemma function_def {Y X : V} : Y ^ X = function Y X := rfl
+lemma mem_function_iff {f Y X : V} : f ∈ Y ^ X ↔ f ⊆ X ×ˢ Y ∧ ∀ x ∈ X, ∃!... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_d2e9778766d4_7 | 5befece85f9023ad | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/SetTheory/Function.lean | Function | 7 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 19 | 1 | [
{
"theorem_name": "mem_function_iff",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by simp [function, function_def]",
"n_chars": 33,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_auto... | [
{
"name": "function_def",
"text": "lemma function_def {Y X : V} : Y ^ X = function Y X := rfl\n\n",
"fan_in": 1,
"n_lines": 3,
"n_chars": 60,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": t... | [
{
"name": "function_subset_power_prod",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 3,
"n_lines": 3,
"n_chars": 133,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"m... | 1 | module
public import Foundation.FirstOrder.SetTheory.Z
@[expose] public section
/-!
# Basic definitions and lemmata for relations and functions
-/
namespace LO.FirstOrder.SetTheory
variable {V : Type*} [SetStructure V] [Nonempty V] [V↓[ℒₛₑₜ] ⊧* 𝗭]
/-! ### Relations -/
noncomputable def domain (R : V) : V := {x ∈... | module
public import Foundation.FirstOrder.SetTheory.Z
@[expose] public section
/-!
# Basic definitions and lemmata for relations and functions
-/
namespace LO.FirstOrder.SetTheory
variable {V : Type*} [SetStructure V] [Nonempty V] [V↓[ℒₛₑₜ] ⊧* 𝗭]
/-! ### Relations -/
noncomputable def domain (R : V) : V := {x ∈... | @@ -93,8 +93,10 @@
noncomputable instance : Pow V V := ⟨fun Y X ↦ function Y X⟩
-lemma mem_function_iff {f Y X : V} : f ∈ Y ^ X ↔ f ⊆ X ×ˢ Y ∧ ∀ x ∈ X, ∃! y, ⟨x, y⟩ₖ ∈ f := sorry
+lemma function_def {Y X : V} : Y ^ X = function Y X := rfl
+lemma mem_function_iff {f Y X : V} : f ∈ Y ^ X ↔ f ⊆ X ×ˢ Y ∧ ∀ x ∈ X, ∃!... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_d2e9778766d4_8 | 08be4dc7e268e452 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/SetTheory/Function.lean | Function | 8 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 19 | 5 | [
{
"theorem_name": "exists_of_mem_function",
"depth": 1,
"n_commands": 0,
"n_lines": 5,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n intro x hx\n rcases (exists_unique_of_mem_function h x hx).exists with ⟨y, hy⟩\n have : x ∈ X ∧ y ∈ Y := mem_of_me... | [
{
"name": "mem_of_mem_functions",
"text": "lemma mem_of_mem_functions {f X Y : V} (h : f ∈ Y ^ X) (hx : ⟨x, y⟩ₖ ∈ f) : x ∈ X ∧ y ∈ Y := by\n simpa using subset_prod_of_mem_function h _ hx\n\n",
"fan_in": 5,
"n_lines": 4,
"n_chars": 162,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic"... | [
{
"name": "exists_of_mem_function",
"fan_in": 5,
"n_deps_direct": 2,
"n_deps_transitive": 5,
"n_lines": 7,
"n_chars": 285,
"n_subproofs": 1,
"n_tactics": 5,
"cyclomatic": 2,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 4,
"automation_only": false,
"max_... | 3 | module
public import Foundation.FirstOrder.SetTheory.Z
@[expose] public section
/-!
# Basic definitions and lemmata for relations and functions
-/
namespace LO.FirstOrder.SetTheory
variable {V : Type*} [SetStructure V] [Nonempty V] [V↓[ℒₛₑₜ] ⊧* 𝗭]
/-! ### Relations -/
noncomputable def domain (R : V) : V := {x ∈... | module
public import Foundation.FirstOrder.SetTheory.Z
@[expose] public section
/-!
# Basic definitions and lemmata for relations and functions
-/
namespace LO.FirstOrder.SetTheory
variable {V : Type*} [SetStructure V] [Nonempty V] [V↓[ℒₛₑₜ] ⊧* 𝗭]
/-! ### Relations -/
noncomputable def domain (R : V) : V := {x ∈... | @@ -109,12 +109,15 @@
lemma subset_prod_of_mem_function {f X Y : V} (h : f ∈ Y ^ X) : f ⊆ X ×ˢ Y := mem_function_iff.mp h |>.1
+lemma mem_of_mem_functions {f X Y : V} (h : f ∈ Y ^ X) (hx : ⟨x, y⟩ₖ ∈ f) : x ∈ X ∧ y ∈ Y := by
+ simpa using subset_prod_of_mem_function h _ hx
+
lemma exists_unique_of_mem_function {f... | {
"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_d2e9778766d4_9 | 9a5a5341f313fbaa | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/SetTheory/Function.lean | Function | 9 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 19 | 1 | [
{
"theorem_name": "mem_range_iff",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n simpa [range] using fun _ ↦ mem_sUnion_sUnion_of_kpair_mem_right",
"n_chars": 72,
"n_subproofs": 0,
"n_tactics": ... | [
{
"name": "mem_sUnion_sUnion_of_kpair_mem_right",
"text": "lemma mem_sUnion_sUnion_of_kpair_mem_right {x y R : V} (h : ⟨x, y⟩ₖ ∈ R) : y ∈ ⋃ˢ ⋃ˢ R := by\n simp only [mem_sUnion_iff]\n refine ⟨{x, y}, ⟨⟨x, y⟩ₖ, h, by simp [kpair]⟩, by simp⟩\n\n",
"fan_in": 1,
"n_lines": 5,
"n_chars": 211,
"n... | [
{
"name": "range_subset_of_mem_function",
"fan_in": 0,
"n_deps_direct": 2,
"n_deps_transitive": 6,
"n_lines": 8,
"n_chars": 280,
"n_subproofs": 2,
"n_tactics": 6,
"cyclomatic": 2,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 3,
"automation_only": false,
... | 3 | module
public import Foundation.FirstOrder.SetTheory.Z
@[expose] public section
/-!
# Basic definitions and lemmata for relations and functions
-/
namespace LO.FirstOrder.SetTheory
variable {V : Type*} [SetStructure V] [Nonempty V] [V↓[ℒₛₑₜ] ⊧* 𝗭]
/-! ### Relations -/
noncomputable def domain (R : V) : V := {x ∈... | module
public import Foundation.FirstOrder.SetTheory.Z
@[expose] public section
/-!
# Basic definitions and lemmata for relations and functions
-/
namespace LO.FirstOrder.SetTheory
variable {V : Type*} [SetStructure V] [Nonempty V] [V↓[ℒₛₑₜ] ⊧* 𝗭]
/-! ### Relations -/
noncomputable def domain (R : V) : V := {x ∈... | @@ -54,8 +54,13 @@
section range
-lemma mem_range_iff {R y : V} : y ∈ range R ↔ ∃ x, ⟨x, y⟩ₖ ∈ R := sorry
+lemma mem_sUnion_sUnion_of_kpair_mem_right {x y R : V} (h : ⟨x, y⟩ₖ ∈ R) : y ∈ ⋃ˢ ⋃ˢ R := by
+ simp only [mem_sUnion_iff]
+ refine ⟨{x, y}, ⟨⟨x, y⟩ₖ, h, by simp [kpair]⟩, by simp⟩
+lemma mem_range_iff {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_d2e9778766d4_10 | caa63e183f4a7232 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/SetTheory/Function.lean | Function | 10 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 19 | 1 | [
{
"theorem_name": "mem_function_iff",
"depth": 1,
"n_commands": 0,
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"proof_text": " by simp [function, function_def]",
"n_chars": 33,
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"n_auto... | [
{
"name": "function_def",
"text": "lemma function_def {Y X : V} : Y ^ X = function Y X := rfl\n\n",
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{
"name": "function_subset_function_of_subset",
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"automation_only": fal... | 10 | module
public import Foundation.FirstOrder.SetTheory.Z
@[expose] public section
/-!
# Basic definitions and lemmata for relations and functions
-/
namespace LO.FirstOrder.SetTheory
variable {V : Type*} [SetStructure V] [Nonempty V] [V↓[ℒₛₑₜ] ⊧* 𝗭]
/-! ### Relations -/
noncomputable def domain (R : V) : V := {x ∈... | module
public import Foundation.FirstOrder.SetTheory.Z
@[expose] public section
/-!
# Basic definitions and lemmata for relations and functions
-/
namespace LO.FirstOrder.SetTheory
variable {V : Type*} [SetStructure V] [Nonempty V] [V↓[ℒₛₑₜ] ⊧* 𝗭]
/-! ### Relations -/
noncomputable def domain (R : V) : V := {x ∈... | @@ -93,8 +93,10 @@
noncomputable instance : Pow V V := ⟨fun Y X ↦ function Y X⟩
-lemma mem_function_iff {f Y X : V} : f ∈ Y ^ X ↔ f ⊆ X ×ˢ Y ∧ ∀ x ∈ X, ∃! y, ⟨x, y⟩ₖ ∈ f := sorry
+lemma function_def {Y X : V} : Y ^ X = function Y X := rfl
+lemma mem_function_iff {f Y X : V} : f ∈ Y ^ X ↔ f ⊆ X ×ˢ Y ∧ ∀ x ∈ X, ∃!... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_d2e9778766d4_11 | 5f01b7aa72949b40 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/SetTheory/Function.lean | Function | 11 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 19 | 1 | [
{
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{
"name": "function_def",
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{
"name": "function_eq_of_subset",
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"ma... | 9 | module
public import Foundation.FirstOrder.SetTheory.Z
@[expose] public section
/-!
# Basic definitions and lemmata for relations and functions
-/
namespace LO.FirstOrder.SetTheory
variable {V : Type*} [SetStructure V] [Nonempty V] [V↓[ℒₛₑₜ] ⊧* 𝗭]
/-! ### Relations -/
noncomputable def domain (R : V) : V := {x ∈... | module
public import Foundation.FirstOrder.SetTheory.Z
@[expose] public section
/-!
# Basic definitions and lemmata for relations and functions
-/
namespace LO.FirstOrder.SetTheory
variable {V : Type*} [SetStructure V] [Nonempty V] [V↓[ℒₛₑₜ] ⊧* 𝗭]
/-! ### Relations -/
noncomputable def domain (R : V) : V := {x ∈... | @@ -93,8 +93,10 @@
noncomputable instance : Pow V V := ⟨fun Y X ↦ function Y X⟩
-lemma mem_function_iff {f Y X : V} : f ∈ Y ^ X ↔ f ⊆ X ×ˢ Y ∧ ∀ x ∈ X, ∃! y, ⟨x, y⟩ₖ ∈ f := sorry
+lemma function_def {Y X : V} : Y ^ X = function Y X := rfl
+lemma mem_function_iff {f Y X : V} : f ∈ Y ^ X ↔ f ⊆ X ×ˢ Y ∧ ∀ x ∈ X, ∃!... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_d2e9778766d4_12 | d5a89a4a2a52cfae | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/SetTheory/Function.lean | Function | 12 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 19 | 1 | [
{
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{
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{
"name": "function_ext",
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"max_nesting":... | 10 | module
public import Foundation.FirstOrder.SetTheory.Z
@[expose] public section
/-!
# Basic definitions and lemmata for relations and functions
-/
namespace LO.FirstOrder.SetTheory
variable {V : Type*} [SetStructure V] [Nonempty V] [V↓[ℒₛₑₜ] ⊧* 𝗭]
/-! ### Relations -/
noncomputable def domain (R : V) : V := {x ∈... | module
public import Foundation.FirstOrder.SetTheory.Z
@[expose] public section
/-!
# Basic definitions and lemmata for relations and functions
-/
namespace LO.FirstOrder.SetTheory
variable {V : Type*} [SetStructure V] [Nonempty V] [V↓[ℒₛₑₜ] ⊧* 𝗭]
/-! ### Relations -/
noncomputable def domain (R : V) : V := {x ∈... | @@ -93,8 +93,10 @@
noncomputable instance : Pow V V := ⟨fun Y X ↦ function Y X⟩
-lemma mem_function_iff {f Y X : V} : f ∈ Y ^ X ↔ f ⊆ X ×ˢ Y ∧ ∀ x ∈ X, ∃! y, ⟨x, y⟩ₖ ∈ f := sorry
+lemma function_def {Y X : V} : Y ^ X = function Y X := rfl
+lemma mem_function_iff {f Y X : V} : f ∈ Y ^ X ↔ f ⊆ X ×ˢ Y ∧ ∀ x ∈ X, ∃!... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_d2e9778766d4_13 | 95316b3b6fb5a084 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/SetTheory/Function.lean | Function | 13 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 19 | 1 | [
{
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"n_chars": 33,
"n_subproofs": 0,
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"n_auto... | [
{
"name": "function_def",
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{
"name": "value_mem_range",
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"automation_only": false,
"max_nest... | 9 | module
public import Foundation.FirstOrder.SetTheory.Z
@[expose] public section
/-!
# Basic definitions and lemmata for relations and functions
-/
namespace LO.FirstOrder.SetTheory
variable {V : Type*} [SetStructure V] [Nonempty V] [V↓[ℒₛₑₜ] ⊧* 𝗭]
/-! ### Relations -/
noncomputable def domain (R : V) : V := {x ∈... | module
public import Foundation.FirstOrder.SetTheory.Z
@[expose] public section
/-!
# Basic definitions and lemmata for relations and functions
-/
namespace LO.FirstOrder.SetTheory
variable {V : Type*} [SetStructure V] [Nonempty V] [V↓[ℒₛₑₜ] ⊧* 𝗭]
/-! ### Relations -/
noncomputable def domain (R : V) : V := {x ∈... | @@ -93,8 +93,10 @@
noncomputable instance : Pow V V := ⟨fun Y X ↦ function Y X⟩
-lemma mem_function_iff {f Y X : V} : f ∈ Y ^ X ↔ f ⊆ X ×ˢ Y ∧ ∀ x ∈ X, ∃! y, ⟨x, y⟩ₖ ∈ f := sorry
+lemma function_def {Y X : V} : Y ^ X = function Y X := rfl
+lemma mem_function_iff {f Y X : V} : f ∈ Y ^ X ↔ f ⊆ X ×ˢ Y ∧ ∀ x ∈ X, ∃!... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_d2e9778766d4_14 | 6759e1ed5bf28d30 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/SetTheory/Function.lean | Function | 14 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 19 | 1 | [
{
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"depth": 1,
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"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n simpa [domain] using fun _ ↦ mem_sUnion_sUnion_of_kpair_mem_left",
"n_chars": 73,
"n_subproofs": 0,
"n_tactics"... | [
{
"name": "mem_sUnion_sUnion_of_kpair_mem_left",
"text": "lemma mem_sUnion_sUnion_of_kpair_mem_left {x y R : V} (h : ⟨x, y⟩ₖ ∈ R) : x ∈ ⋃ˢ ⋃ˢ R := by\n simp only [mem_sUnion_iff]\n refine ⟨{x, y}, ⟨⟨x, y⟩ₖ, h, by simp [kpair]⟩, by simp⟩\n\n",
"fan_in": 1,
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"n_s... | [
{
"name": "domain_restrict_eq",
"fan_in": 0,
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"n_rewrites": 0,
"n_structural": 6,
"automation_only": false,
"max_ne... | 2 | module
public import Foundation.FirstOrder.SetTheory.Z
@[expose] public section
/-!
# Basic definitions and lemmata for relations and functions
-/
namespace LO.FirstOrder.SetTheory
variable {V : Type*} [SetStructure V] [Nonempty V] [V↓[ℒₛₑₜ] ⊧* 𝗭]
/-! ### Relations -/
noncomputable def domain (R : V) : V := {x ∈... | module
public import Foundation.FirstOrder.SetTheory.Z
@[expose] public section
/-!
# Basic definitions and lemmata for relations and functions
-/
namespace LO.FirstOrder.SetTheory
variable {V : Type*} [SetStructure V] [Nonempty V] [V↓[ℒₛₑₜ] ⊧* 𝗭]
/-! ### Relations -/
noncomputable def domain (R : V) : V := {x ∈... | @@ -19,8 +19,13 @@
section domain
-lemma mem_domain_iff {R x : V} : x ∈ domain R ↔ ∃ y, ⟨x, y⟩ₖ ∈ R := sorry
+lemma mem_sUnion_sUnion_of_kpair_mem_left {x y R : V} (h : ⟨x, y⟩ₖ ∈ R) : x ∈ ⋃ˢ ⋃ˢ R := by
+ simp only [mem_sUnion_iff]
+ refine ⟨{x, y}, ⟨⟨x, y⟩ₖ, h, by simp [kpair]⟩, by simp⟩
+lemma mem_domain_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_d2e9778766d4_15 | 1ffd0ce781a9cf6b | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/SetTheory/Function.lean | Function | 15 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 19 | 1 | [
{
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"depth": 1,
"n_commands": 0,
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"n_chars": 33,
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"n_auto... | [
{
"name": "function_def",
"text": "lemma function_def {Y X : V} : Y ^ X = function Y X := rfl\n\n",
"fan_in": 1,
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"automation_only": t... | [
{
"name": "cardLT_power",
"fan_in": 0,
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"n_lines": 36,
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"n_rewrites": 2,
"n_structural": 16,
"automation_only": false,
"max_nest... | 9 | module
public import Foundation.FirstOrder.SetTheory.Z
@[expose] public section
/-!
# Basic definitions and lemmata for relations and functions
-/
namespace LO.FirstOrder.SetTheory
variable {V : Type*} [SetStructure V] [Nonempty V] [V↓[ℒₛₑₜ] ⊧* 𝗭]
/-! ### Relations -/
noncomputable def domain (R : V) : V := {x ∈... | module
public import Foundation.FirstOrder.SetTheory.Z
@[expose] public section
/-!
# Basic definitions and lemmata for relations and functions
-/
namespace LO.FirstOrder.SetTheory
variable {V : Type*} [SetStructure V] [Nonempty V] [V↓[ℒₛₑₜ] ⊧* 𝗭]
/-! ### Relations -/
noncomputable def domain (R : V) : V := {x ∈... | @@ -93,8 +93,10 @@
noncomputable instance : Pow V V := ⟨fun Y X ↦ function Y X⟩
-lemma mem_function_iff {f Y X : V} : f ∈ Y ^ X ↔ f ⊆ X ×ˢ Y ∧ ∀ x ∈ X, ∃! y, ⟨x, y⟩ₖ ∈ f := sorry
+lemma function_def {Y X : V} : Y ^ X = function Y X := rfl
+lemma mem_function_iff {f Y X : V} : f ∈ Y ^ X ↔ f ⊆ X ×ˢ Y ∧ ∀ x ∈ X, ∃!... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_d2e9778766d4_16 | c889dec8cfcf6c04 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/SetTheory/Function.lean | Function | 16 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 19 | 1 | [
{
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"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by simp [function, function_def]",
"n_chars": 33,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_auto... | [
{
"name": "function_def",
"text": "lemma function_def {Y X : V} : Y ^ X = function Y X := rfl\n\n",
"fan_in": 1,
"n_lines": 3,
"n_chars": 60,
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"n_automation": 1,
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"automation_only": t... | [
{
"name": "two_pow_cardEQ_power",
"fan_in": 0,
"n_deps_direct": 4,
"n_deps_transitive": 20,
"n_lines": 76,
"n_chars": 3761,
"n_subproofs": 14,
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"cyclomatic": 7,
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"n_rewrites": 2,
"n_structural": 33,
"automation_only": false,
... | 11 | module
public import Foundation.FirstOrder.SetTheory.Z
@[expose] public section
/-!
# Basic definitions and lemmata for relations and functions
-/
namespace LO.FirstOrder.SetTheory
variable {V : Type*} [SetStructure V] [Nonempty V] [V↓[ℒₛₑₜ] ⊧* 𝗭]
/-! ### Relations -/
noncomputable def domain (R : V) : V := {x ∈... | module
public import Foundation.FirstOrder.SetTheory.Z
@[expose] public section
/-!
# Basic definitions and lemmata for relations and functions
-/
namespace LO.FirstOrder.SetTheory
variable {V : Type*} [SetStructure V] [Nonempty V] [V↓[ℒₛₑₜ] ⊧* 𝗭]
/-! ### Relations -/
noncomputable def domain (R : V) : V := {x ∈... | @@ -93,8 +93,10 @@
noncomputable instance : Pow V V := ⟨fun Y X ↦ function Y X⟩
-lemma mem_function_iff {f Y X : V} : f ∈ Y ^ X ↔ f ⊆ X ×ˢ Y ∧ ∀ x ∈ X, ∃! y, ⟨x, y⟩ₖ ∈ f := sorry
+lemma function_def {Y X : V} : Y ^ X = function Y X := rfl
+lemma mem_function_iff {f Y X : V} : f ∈ Y ^ X ↔ f ⊆ X ×ˢ Y ∧ ∀ x ∈ X, ∃!... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_2db400535676_0 | da2c9fa958ed9719 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Kripke/Filtration.lean | Filtration | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 1 | 3 | [
{
"theorem_name": "isFinite",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " FilterEqvQuotient.finite T_finite",
"n_chars": 34,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation"... | [
{
"name": "finite",
"text": "lemma finite (T_finite : T.Finite) : Finite (FilterEqvQuotient M T) := by\n have : Finite (𝒫 T) := Set.Finite.powerset T_finite\n let f : FilterEqvQuotient M T → 𝒫 T :=\n λ (X : FilterEqvQuotient M T) => Quotient.lift (λ x => ⟨{ φ ∈ T | x ⊧ φ }, (by simp_all)⟩) (by\n ... | [
{
"name": "isFinite",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 2,
"n_chars": 134,
"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.Modal.Kripke.Root
@[expose] public section
universe u v
namespace LO.Modal
namespace Kripke
open FormulaSet.IsSubformulaClosed
open Formula (atom)
open Formula.Kripke
def filterEquiv (M : Kripke.Model) (T : FormulaSet ℕ) [T.IsSubformulaClosed] (x y : M.World) := ∀ φ, (_ : φ ∈ T) →... | module
public import Foundation.Modal.Kripke.Root
@[expose] public section
universe u v
namespace LO.Modal
namespace Kripke
open FormulaSet.IsSubformulaClosed
open Formula (atom)
open Formula.Kripke
def filterEquiv (M : Kripke.Model) (T : FormulaSet ℕ) [T.IsSubformulaClosed] (x y : M.World) := ∀ φ, (_ : φ ∈ T) →... | @@ -38,6 +38,33 @@
apply @Quotient.eq_iff_equiv.mp h;
assumption;
+lemma finite (T_finite : T.Finite) : Finite (FilterEqvQuotient M T) := by
+ have : Finite (𝒫 T) := Set.Finite.powerset T_finite
+ let f : FilterEqvQuotient M T → 𝒫 T :=
+ λ (X : FilterEqvQuotient M T) => Quotient.lift (λ x => ⟨{ φ ∈ T | x... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_28d45ef51d92_0 | 6c90ef6b9f2da0fa | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Incompleteness/Jeroslow.lean | Jeroslow | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 2 | 1 | [
{
"theorem_name": "provable_E_jeroslow_jeroslow'",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " Entailment.E!_trans def_jeroslow def_jeroslow'",
"n_chars": 47,
"n_subproofs": 0,
"n_tactics": 1,
"... | [
{
"name": "def_jeroslow",
"text": "lemma def_jeroslow [𝗜𝚺₁ ⪯ U] : U ⊢ T.jeroslow 🡘 (T.refutable)/[⌜T.jeroslow⌝] := diagonal _\n\n",
"fan_in": 1,
"n_lines": 3,
"n_chars": 111,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_stru... | [
{
"name": "provable_E_jeroslow_jeroslow'",
"fan_in": 1,
"n_deps_direct": 2,
"n_deps_transitive": 2,
"n_lines": 3,
"n_chars": 151,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
... | 2 | module
public import Foundation.FirstOrder.Incompleteness.RosserProvability
public import Foundation.FirstOrder.Incompleteness.ProvabilityAbstraction.Refutability
/-!
# Jeroslow's Second Incompleteness Theorem
Jeroslow's formulation of the second incompleteness theorem
states that the sentence represents _formalized... | module
public import Foundation.FirstOrder.Incompleteness.RosserProvability
public import Foundation.FirstOrder.Incompleteness.ProvabilityAbstraction.Refutability
/-!
# Jeroslow's Second Incompleteness Theorem
Jeroslow's formulation of the second incompleteness theorem
states that the sentence represents _formalized... | @@ -69,9 +69,11 @@
private lemma jeroslow'_sigmaOne : Hierarchy 𝚺 1 (T.jeroslow') := by definability;
+lemma def_jeroslow [𝗜𝚺₁ ⪯ U] : U ⊢ T.jeroslow 🡘 (T.refutable)/[⌜T.jeroslow⌝] := diagonal _
+
private lemma def_jeroslow' [𝗜𝚺₁ ⪯ U] : U ⊢ T.jeroslow' 🡘 (T.refutable)/[⌜T.jeroslow⌝] := by simp;
-private 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_28d45ef51d92_1 | 620d9b91dd55861c | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Incompleteness/Jeroslow.lean | Jeroslow | 1 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 2 | 1 | [
{
"theorem_name": "provable_E_jeroslow_jeroslow'",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " Entailment.E!_trans def_jeroslow def_jeroslow'",
"n_chars": 47,
"n_subproofs": 0,
"n_tactics": 1,
"... | [
{
"name": "def_jeroslow",
"text": "lemma def_jeroslow [𝗜𝚺₁ ⪯ U] : U ⊢ T.jeroslow 🡘 (T.refutable)/[⌜T.jeroslow⌝] := diagonal _\n\n",
"fan_in": 1,
"n_lines": 3,
"n_chars": 111,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_stru... | [
{
"name": "iff_provable_jeroslow_provable_jeroslow'",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 3,
"n_lines": 4,
"n_chars": 187,
"n_subproofs": 0,
"n_tactics": 3,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 1,
"automation_only"... | 2 | module
public import Foundation.FirstOrder.Incompleteness.RosserProvability
public import Foundation.FirstOrder.Incompleteness.ProvabilityAbstraction.Refutability
/-!
# Jeroslow's Second Incompleteness Theorem
Jeroslow's formulation of the second incompleteness theorem
states that the sentence represents _formalized... | module
public import Foundation.FirstOrder.Incompleteness.RosserProvability
public import Foundation.FirstOrder.Incompleteness.ProvabilityAbstraction.Refutability
/-!
# Jeroslow's Second Incompleteness Theorem
Jeroslow's formulation of the second incompleteness theorem
states that the sentence represents _formalized... | @@ -69,9 +69,11 @@
private lemma jeroslow'_sigmaOne : Hierarchy 𝚺 1 (T.jeroslow') := by definability;
+lemma def_jeroslow [𝗜𝚺₁ ⪯ U] : U ⊢ T.jeroslow 🡘 (T.refutable)/[⌜T.jeroslow⌝] := diagonal _
+
private lemma def_jeroslow' [𝗜𝚺₁ ⪯ U] : U ⊢ T.jeroslow' 🡘 (T.refutable)/[⌜T.jeroslow⌝] := by simp;
-private 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_6f4bb7b3d2dc_0 | 951f43c4f426781a | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Induction.lean | Induction | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 2 | 3 | [
{
"theorem_name": "bounded_all_sigma1_order_induction'",
"depth": 1,
"n_commands": 0,
"n_lines": 3,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": "\n have : 𝚺₁-Function₂ (fun _ ↦ f) := DefinableFunction₁.comp (by simp)\n bounded_all_sigma1_order_inductio... | [
{
"name": "bounded_all_sigma1_order_induction",
"text": "theorem bounded_all_sigma1_order_induction {f : V → V → V} (hf : 𝚺₁-Function₂ f) {P : V → V → Prop} (hP : 𝚺₁-Relation P)\n (ind : ∀ x y, (∀ x' < x, ∀ y' ≤ f x y, P x' y') → P x y) : ∀ x y, P x y := by\n have maxf : ∀ x y, ∃ m, ∀ x' ≤ x, ∀ y' ≤ y... | [
{
"name": "bounded_all_sigma1_order_induction'",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 6,
"n_chars": 355,
"n_subproofs": 1,
"n_tactics": 2,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": fal... | 1 | module
public import Foundation.FirstOrder.Arithmetic.HFS
@[expose] public section
/-!
# Various induction-related principles in $\mathsf{I}\Sigma_n$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section ISigma1
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
@[elab_as_elim] lemma sigma1_pos_succ_ind... | module
public import Foundation.FirstOrder.Arithmetic.HFS
@[expose] public section
/-!
# Various induction-related principles in $\mathsf{I}\Sigma_n$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section ISigma1
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
@[elab_as_elim] lemma sigma1_pos_succ_ind... | @@ -32,8 +32,98 @@
open HierarchySymbol
+theorem bounded_all_sigma1_order_induction {f : V → V → V} (hf : 𝚺₁-Function₂ f) {P : V → V → Prop} (hP : 𝚺₁-Relation P)
+ (ind : ∀ x y, (∀ x' < x, ∀ y' ≤ f x y, P x' y') → P x y) : ∀ x y, P x y := by
+ have maxf : ∀ x y, ∃ m, ∀ x' ≤ x, ∀ y' ≤ y, f x' y' ≤ m := by
+ ... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_6f4bb7b3d2dc_1 | 1fd95df2e7f15ca0 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Induction.lean | Induction | 1 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 2 | 3 | [
{
"theorem_name": "bounded_all_sigma1_order_induction'",
"depth": 1,
"n_commands": 0,
"n_lines": 3,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": "\n have : 𝚺₁-Function₂ (fun _ ↦ f) := DefinableFunction₁.comp (by simp)\n bounded_all_sigma1_order_inductio... | [
{
"name": "bounded_all_sigma1_order_induction",
"text": "theorem bounded_all_sigma1_order_induction {f : V → V → V} (hf : 𝚺₁-Function₂ f) {P : V → V → Prop} (hP : 𝚺₁-Relation P)\n (ind : ∀ x y, (∀ x' < x, ∀ y' ≤ f x y, P x' y') → P x y) : ∀ x y, P x y := by\n have maxf : ∀ x y, ∃ m, ∀ x' ≤ x, ∀ y' ≤ y... | [
{
"name": "bounded_all_sigma1_order_induction₂",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 25,
"n_chars": 1277,
"n_subproofs": 2,
"n_tactics": 20,
"cyclomatic": 1,
"n_automation": 3,
"n_rewrites": 0,
"n_structural": 9,
"automation_only": ... | 1 | module
public import Foundation.FirstOrder.Arithmetic.HFS
@[expose] public section
/-!
# Various induction-related principles in $\mathsf{I}\Sigma_n$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section ISigma1
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
@[elab_as_elim] lemma sigma1_pos_succ_ind... | module
public import Foundation.FirstOrder.Arithmetic.HFS
@[expose] public section
/-!
# Various induction-related principles in $\mathsf{I}\Sigma_n$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section ISigma1
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
@[elab_as_elim] lemma sigma1_pos_succ_ind... | @@ -32,10 +32,117 @@
open HierarchySymbol
+theorem bounded_all_sigma1_order_induction {f : V → V → V} (hf : 𝚺₁-Function₂ f) {P : V → V → Prop} (hP : 𝚺₁-Relation P)
+ (ind : ∀ x y, (∀ x' < x, ∀ y' ≤ f x y, P x' y') → P x y) : ∀ x y, P x y := by
+ have maxf : ∀ x y, ∃ m, ∀ x' ≤ x, ∀ y' ≤ y, f x' y' ≤ m := by
+... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_6f4bb7b3d2dc_2 | 48246ed7b9bf3d21 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Induction.lean | Induction | 2 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 2 | 3 | [
{
"theorem_name": "bounded_all_sigma1_order_induction'",
"depth": 1,
"n_commands": 0,
"n_lines": 3,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": "\n have : 𝚺₁-Function₂ (fun _ ↦ f) := DefinableFunction₁.comp (by simp)\n bounded_all_sigma1_order_inductio... | [
{
"name": "bounded_all_sigma1_order_induction",
"text": "theorem bounded_all_sigma1_order_induction {f : V → V → V} (hf : 𝚺₁-Function₂ f) {P : V → V → Prop} (hP : 𝚺₁-Relation P)\n (ind : ∀ x y, (∀ x' < x, ∀ y' ≤ f x y, P x' y') → P x y) : ∀ x y, P x y := by\n have maxf : ∀ x y, ∃ m, ∀ x' ≤ x, ∀ y' ≤ y... | [
{
"name": "bounded_all_sigma1_order_induction₃",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 40,
"n_chars": 2207,
"n_subproofs": 3,
"n_tactics": 35,
"cyclomatic": 1,
"n_automation": 3,
"n_rewrites": 0,
"n_structural": 7,
"automation_only": ... | 1 | module
public import Foundation.FirstOrder.Arithmetic.HFS
@[expose] public section
/-!
# Various induction-related principles in $\mathsf{I}\Sigma_n$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section ISigma1
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
@[elab_as_elim] lemma sigma1_pos_succ_ind... | module
public import Foundation.FirstOrder.Arithmetic.HFS
@[expose] public section
/-!
# Various induction-related principles in $\mathsf{I}\Sigma_n$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section ISigma1
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
@[elab_as_elim] lemma sigma1_pos_succ_ind... | @@ -32,10 +32,132 @@
open HierarchySymbol
+theorem bounded_all_sigma1_order_induction {f : V → V → V} (hf : 𝚺₁-Function₂ f) {P : V → V → Prop} (hP : 𝚺₁-Relation P)
+ (ind : ∀ x y, (∀ x' < x, ∀ y' ≤ f x y, P x' y') → P x y) : ∀ x y, P x y := by
+ have maxf : ∀ x y, ∃ m, ∀ x' ≤ x, ∀ y' ≤ y, f x' y' ≤ m := by
+... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_6f4bb7b3d2dc_3 | 456b172977d51c39 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Induction.lean | Induction | 3 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 2 | 1 | [
{
"theorem_name": "measured_bounded_sigma1_order_induction",
"depth": 1,
"n_commands": 0,
"n_lines": 7,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n let Q : V → V → Prop := fun k x ↦ m x ≤ k → P x\n have hQ : 𝚺₁-Relation Q := by unfold Q; definab... | [
{
"name": "bounded_all_sigma1_order_induction'",
"text": "lemma bounded_all_sigma1_order_induction' {f : V → V} (hf : 𝚺₁-Function₁ f) {P : V → V → Prop} (hP : 𝚺₁-Relation P)\n (ind : ∀ x y, (∀ x' < x, ∀ y' ≤ f y, P x' y') → P x y) : ∀ x y, P x y :=\n have : 𝚺₁-Function₂ (fun _ ↦ f) := DefinableFuncti... | [
{
"name": "measured_bounded_sigma1_order_induction",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 2,
"n_lines": 11,
"n_chars": 580,
"n_subproofs": 2,
"n_tactics": 8,
"cyclomatic": 1,
"n_automation": 2,
"n_rewrites": 1,
"n_structural": 2,
"automation_only"... | 2 | module
public import Foundation.FirstOrder.Arithmetic.HFS
@[expose] public section
/-!
# Various induction-related principles in $\mathsf{I}\Sigma_n$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section ISigma1
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
@[elab_as_elim] lemma sigma1_pos_succ_ind... | module
public import Foundation.FirstOrder.Arithmetic.HFS
@[expose] public section
/-!
# Various induction-related principles in $\mathsf{I}\Sigma_n$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section ISigma1
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
@[elab_as_elim] lemma sigma1_pos_succ_ind... | @@ -120,12 +120,18 @@
exact ih m₁ (by simp [m₁]) (by simp [m₁]) x'' (lt_succ_iff_le.mp (lt_of_lt_of_le hx'' hx')) y'' (le_trans hy'' this)
exact this x (by rfl) y (lt_of_mem_rng hW₀) (by simpa using hW₀) x (by rfl) y (by rfl)
+lemma bounded_all_sigma1_order_induction' {f : V → V} (hf : 𝚺₁-Function₁ f) {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_20b4e5ef02af_0 | 353981b3b11f6371 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Exponential/Bit.lean | Bit | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 23 | 9 | [
{
"theorem_name": "lt_of_mem",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " lt_of_lt_of_le (lt_exp i) (exp_le_of_mem h)",
"n_chars": 44,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_... | [
{
"name": "exp_le_of_mem",
"text": "lemma exp_le_of_mem {i a : V} (h : i ∈ a) : Exp.exp i ≤ a := LenBit.le h\n\n",
"fan_in": 9,
"n_lines": 3,
"n_chars": 78,
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"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"auto... | [
{
"name": "lt_of_mem",
"fan_in": 4,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 3,
"n_chars": 96,
"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.Log
/-!
# $\mathrm{Bit}$ predicate
-/
@[expose] public section
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
section model
def Bit (i a : V) : Prop := LenBit (Exp.exp i) a
instanc... | module
public import Foundation.FirstOrder.Arithmetic.Exponential.Log
/-!
# $\mathrm{Bit}$ predicate
-/
@[expose] public section
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
section model
def Bit (i a : V) : Prop := LenBit (Exp.exp i) a
instanc... | @@ -41,8 +41,10 @@
lemma mem_iff_bit {i a : V} : i ∈ a ↔ Bit i a := iff_of_eq rfl
-lemma lt_of_mem {i a : V} (h : i ∈ a) : i < a := sorry
+lemma exp_le_of_mem {i a : V} (h : i ∈ a) : Exp.exp i ≤ a := LenBit.le h
+lemma lt_of_mem {i a : V} (h : i ∈ a) : i < a := lt_of_lt_of_le (lt_exp i) (exp_le_of_mem h)
+
@[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_20b4e5ef02af_1 | 877010cb67e10a04 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Exponential/Bit.lean | Bit | 1 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 23 | 9 | [
{
"theorem_name": "lt_of_mem",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " lt_of_lt_of_le (lt_exp i) (exp_le_of_mem h)",
"n_chars": 44,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_... | [
{
"name": "exp_le_of_mem",
"text": "lemma exp_le_of_mem {i a : V} (h : i ∈ a) : Exp.exp i ≤ a := LenBit.le h\n\n",
"fan_in": 9,
"n_lines": 3,
"n_chars": 78,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"auto... | [
{
"name": "not_mem_of_lt_exp",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 3,
"n_chars": 138,
"n_subproofs": 1,
"n_tactics": 2,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
"max_nesti... | 1 | module
public import Foundation.FirstOrder.Arithmetic.Exponential.Log
/-!
# $\mathrm{Bit}$ predicate
-/
@[expose] public section
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
section model
def Bit (i a : V) : Prop := LenBit (Exp.exp i) a
instanc... | module
public import Foundation.FirstOrder.Arithmetic.Exponential.Log
/-!
# $\mathrm{Bit}$ predicate
-/
@[expose] public section
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
section model
def Bit (i a : V) : Prop := LenBit (Exp.exp i) a
instanc... | @@ -41,10 +41,12 @@
lemma mem_iff_bit {i a : V} : i ∈ a ↔ Bit i a := iff_of_eq rfl
-lemma lt_of_mem {i a : V} (h : i ∈ a) : i < a := sorry
+lemma exp_le_of_mem {i a : V} (h : i ∈ a) : Exp.exp i ≤ a := LenBit.le h
-lemma not_mem_of_lt_exp {i a : V} (h : a < Exp.exp i) : i ∉ a := sorry
+lemma lt_of_mem {i a : V} (... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_20b4e5ef02af_2 | 314ae93b748a38d4 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Exponential/Bit.lean | Bit | 2 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 23 | 10 | [
{
"theorem_name": "mem_iff_mul_exp_add_exp_add",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n simpa [mem_iff_bit, exp_succ] using! lenbit_iff_add_mul (exp_pow2 i) (a := a)",
"n_chars": 83,
"n_subpr... | [
{
"name": "mem_iff_bit",
"text": "lemma mem_iff_bit {i a : V} : i ∈ a ↔ Bit i a := iff_of_eq rfl\n\n",
"fan_in": 10,
"n_lines": 3,
"n_chars": 68,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only... | [
{
"name": "mem_iff_mul_exp_add_exp_add",
"fan_in": 2,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 4,
"n_chars": 211,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"... | 1 | module
public import Foundation.FirstOrder.Arithmetic.Exponential.Log
/-!
# $\mathrm{Bit}$ predicate
-/
@[expose] public section
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
section model
def Bit (i a : V) : Prop := LenBit (Exp.exp i) a
instanc... | module
public import Foundation.FirstOrder.Arithmetic.Exponential.Log
/-!
# $\mathrm{Bit}$ predicate
-/
@[expose] public section
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
section model
def Bit (i a : V) : Prop := LenBit (Exp.exp i) a
instanc... | @@ -39,6 +39,8 @@
instance mem_definable'' (ℌ : HierarchySymbol) : ℌ-Relation[V] Membership.mem := by
simpa using (mem_definable' ℌ).retraction ![1, 0]
+lemma mem_iff_bit {i a : V} : i ∈ a ↔ Bit i a := iff_of_eq rfl
+
lemma exp_le_of_mem {i a : V} (h : i ∈ a) : Exp.exp i ≤ a := LenBit.le h
lemma lt_of_mem {i ... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_20b4e5ef02af_3 | 10527633649477ed | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Exponential/Bit.lean | Bit | 3 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 23 | 10 | [
{
"theorem_name": "mem_iff_mul_exp_add_exp_add",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n simpa [mem_iff_bit, exp_succ] using! lenbit_iff_add_mul (exp_pow2 i) (a := a)",
"n_chars": 83,
"n_subpr... | [
{
"name": "mem_iff_bit",
"text": "lemma mem_iff_bit {i a : V} : i ∈ a ↔ Bit i a := iff_of_eq rfl\n\n",
"fan_in": 10,
"n_lines": 3,
"n_chars": 68,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only... | [
{
"name": "not_mem_iff_mul_exp_add",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 4,
"n_chars": 199,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"max_... | 1 | module
public import Foundation.FirstOrder.Arithmetic.Exponential.Log
/-!
# $\mathrm{Bit}$ predicate
-/
@[expose] public section
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
section model
def Bit (i a : V) : Prop := LenBit (Exp.exp i) a
instanc... | module
public import Foundation.FirstOrder.Arithmetic.Exponential.Log
/-!
# $\mathrm{Bit}$ predicate
-/
@[expose] public section
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
section model
def Bit (i a : V) : Prop := LenBit (Exp.exp i) a
instanc... | @@ -39,6 +39,8 @@
instance mem_definable'' (ℌ : HierarchySymbol) : ℌ-Relation[V] Membership.mem := by
simpa using (mem_definable' ℌ).retraction ![1, 0]
+lemma mem_iff_bit {i a : V} : i ∈ a ↔ Bit i a := iff_of_eq rfl
+
lemma exp_le_of_mem {i a : V} (h : i ∈ a) : Exp.exp i ≤ a := LenBit.le h
lemma lt_of_mem {i ... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_20b4e5ef02af_4 | e9a8ba807666682f | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Exponential/Bit.lean | Bit | 4 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 23 | 2 | [
{
"theorem_name": "singleton_injective",
"depth": 1,
"n_commands": 0,
"n_lines": 5,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n constructor\n · intro h\n simpa [singleton_def] using congr_arg log h\n · rintro rfl; rfl",
"n_chars": 99,
... | [
{
"name": "singleton_def",
"text": "lemma singleton_def (a : V) : {a} = Exp.exp a := rfl\n\n",
"fan_in": 2,
"n_lines": 3,
"n_chars": 54,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
... | [
{
"name": "singleton_injective",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 7,
"n_chars": 175,
"n_subproofs": 0,
"n_tactics": 6,
"cyclomatic": 3,
"n_automation": 3,
"n_rewrites": 0,
"n_structural": 3,
"automation_only": false,
"max_nes... | 1 | module
public import Foundation.FirstOrder.Arithmetic.Exponential.Log
/-!
# $\mathrm{Bit}$ predicate
-/
@[expose] public section
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
section model
def Bit (i a : V) : Prop := LenBit (Exp.exp i) a
instanc... | module
public import Foundation.FirstOrder.Arithmetic.Exponential.Log
/-!
# $\mathrm{Bit}$ predicate
-/
@[expose] public section
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
section model
def Bit (i a : V) : Prop := LenBit (Exp.exp i) a
instanc... | @@ -200,9 +200,12 @@
noncomputable scoped instance : Singleton V V := ⟨fun a ↦ Exp.exp a⟩
+lemma singleton_def (a : V) : {a} = Exp.exp a := rfl
+
@[simp] lemma singleton_injective (a b : V) : ({a} : V) = {b} ↔ a = b := by
constructor
- · sorry
+ · intro h
+ simpa [singleton_def] using congr_arg log 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_20b4e5ef02af_5 | 85af9c4b4a91b0d5 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Exponential/Bit.lean | Bit | 5 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 23 | 2 | [
{
"theorem_name": "mem_bitInsert_iff",
"depth": 1,
"n_commands": 0,
"n_lines": 5,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n by_cases h : j ∈ a <;> simp [h, insert_eq, bitInsert]\n · by_cases e : i = j <;> simp [h, e]\n · simpa [exp_inj.eq_iff]... | [
{
"name": "insert_eq",
"text": "lemma insert_eq {i a : V} : insert i a = bitInsert i a := rfl\n\n",
"fan_in": 2,
"n_lines": 3,
"n_chars": 63,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": t... | [
{
"name": "mem_bitInsert_iff",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 8,
"n_chars": 297,
"n_subproofs": 0,
"n_tactics": 7,
"cyclomatic": 3,
"n_automation": 3,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
"max_nesti... | 1 | module
public import Foundation.FirstOrder.Arithmetic.Exponential.Log
/-!
# $\mathrm{Bit}$ predicate
-/
@[expose] public section
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
section model
def Bit (i a : V) : Prop := LenBit (Exp.exp i) a
instanc... | module
public import Foundation.FirstOrder.Arithmetic.Exponential.Log
/-!
# $\mathrm{Bit}$ predicate
-/
@[expose] public section
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
section model
def Bit (i a : V) : Prop := LenBit (Exp.exp i) a
instanc... | @@ -220,13 +220,19 @@
noncomputable scoped instance : Insert V V := ⟨bitInsert⟩
+lemma insert_eq {i a : V} : insert i a = bitInsert i a := rfl
+
lemma singleton_eq_insert (i : V) : ({i} : V) = insert i ∅ := by simp [singleton_def, insert, bitInsert, emptyset_def]
instance : LawfulSingleton V V where
insert_... | {
"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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