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_7471c35686f5_33 | f6c50f6882675e69 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Exponential/Log.lean | Log | 33 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 32 | 11 | [
{
"theorem_name": "log_eq_of_pos",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": "\n (log_exists_unique_pos pos).unique ⟨log_lt_self_of_pos pos, log_pos pos⟩ ⟨lt_of_lt_of_le H.lt hy', y', hy', H, hy⟩",
"n_char... | [
{
"name": "log_pos",
"text": "lemma log_pos {y : V} (pos : 0 < y) : ∃ y' ≤ y, Exponential (log y) y' ∧ y < 2 * y' :=\n ((Classical.choose!_spec (log_exists_unique y)).2 pos).2\n\n",
"fan_in": 11,
"n_lines": 4,
"n_chars": 153,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n... | [
{
"name": "lt_two_mul_exponential_log",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 3,
"n_lines": 6,
"n_chars": 186,
"n_subproofs": 0,
"n_tactics": 4,
"cyclomatic": 3,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 2,
"automation_only": false,
"... | 2 | module
public import Foundation.FirstOrder.Arithmetic.Exponential.Exp
@[expose] public section
/-!
# Logarithmic function
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section ISigma0
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₀]
lemma log_exists_unique_pos {y : V} (hy : 0 < y) : ∃! x, x < y ∧ ∃ y'... | module
public import Foundation.FirstOrder.Arithmetic.Exponential.Exp
@[expose] public section
/-!
# Logarithmic function
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section ISigma0
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₀]
lemma log_exists_unique_pos {y : V} (hy : 0 < y) : ∃! x, x < y ∧ ∃ y'... | @@ -55,6 +55,9 @@
@[simp] lemma log_zero : log (0 : V) = 0 :=
(Classical.choose!_spec (log_exists_unique (0 : V))).1 rfl
+lemma log_pos {y : V} (pos : 0 < y) : ∃ y' ≤ y, Exponential (log y) y' ∧ y < 2 * y' :=
+ ((Classical.choose!_spec (log_exists_unique y)).2 pos).2
+
lemma log_lt_self_of_pos {y : V} (pos : 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_7471c35686f5_34 | 7bc0eafe701f98c0 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Exponential/Log.lean | Log | 34 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 32 | 3 | [
{
"theorem_name": "log_le_self",
"depth": 1,
"n_commands": 0,
"n_lines": 4,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n rcases Arithmetic.zero_le a with (rfl | pos)\n · simp\n · exact le_of_lt <| log_lt_self_of_pos pos",
"n_chars": 106,
... | [
{
"name": "log_lt_self_of_pos",
"text": "lemma log_lt_self_of_pos {y : V} (pos : 0 < y) : log y < y :=\n ((Classical.choose!_spec (log_exists_unique y)).2 pos).1\n\n",
"fan_in": 3,
"n_lines": 4,
"n_chars": 122,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0... | [
{
"name": "log_mul_exp_add_of_lt",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 7,
"n_lines": 4,
"n_chars": 178,
"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 Foundation.FirstOrder.Arithmetic.Exponential.Exp
@[expose] public section
/-!
# Logarithmic function
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section ISigma0
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₀]
lemma log_exists_unique_pos {y : V} (hy : 0 < y) : ∃! x, x < y ∧ ∃ y'... | module
public import Foundation.FirstOrder.Arithmetic.Exponential.Exp
@[expose] public section
/-!
# Logarithmic function
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section ISigma0
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₀]
lemma log_exists_unique_pos {y : V} (hy : 0 < y) : ∃! x, x < y ∧ ∃ y'... | @@ -58,10 +58,13 @@
lemma log_pos {y : V} (pos : 0 < y) : ∃ y' ≤ y, Exponential (log y) y' ∧ y < 2 * y' :=
((Classical.choose!_spec (log_exists_unique y)).2 pos).2
+lemma log_lt_self_of_pos {y : V} (pos : 0 < y) : log y < y :=
+ ((Classical.choose!_spec (log_exists_unique y)).2 pos).1
+
@[simp] lemma log_le_sel... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_7471c35686f5_35 | 40bcbfb6e5629b59 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Exponential/Log.lean | Log | 35 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 32 | 6 | [
{
"theorem_name": "log_one",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " log_eq_of_pos (by simp) (y' := 1) (by simp) (by rfl) (by simp)",
"n_chars": 63,
"n_subproofs": 0,
"n_tactics": 1,
"cyclom... | [
{
"name": "log_eq_of_pos",
"text": "lemma log_eq_of_pos {x y : V} (pos : 0 < y) {y'} (H : Exponential x y') (hy' : y' ≤ y) (hy : y < 2 * y') : log y = x :=\n (log_exists_unique_pos pos).unique ⟨log_lt_self_of_pos pos, log_pos pos⟩ ⟨lt_of_lt_of_le H.lt hy', y', hy', H, hy⟩\n\n",
"fan_in": 6,
"n_line... | [
{
"name": "log_mul_exp",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 8,
"n_lines": 4,
"n_chars": 128,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"max_nesting": 2
... | 7 | module
public import Foundation.FirstOrder.Arithmetic.Exponential.Exp
@[expose] public section
/-!
# Logarithmic function
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section ISigma0
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₀]
lemma log_exists_unique_pos {y : V} (hy : 0 < y) : ∃! x, x < y ∧ ∃ y'... | module
public import Foundation.FirstOrder.Arithmetic.Exponential.Exp
@[expose] public section
/-!
# Logarithmic function
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section ISigma0
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₀]
lemma log_exists_unique_pos {y : V} (hy : 0 < y) : ∃! x, x < y ∧ ∃ y'... | @@ -77,16 +77,25 @@
instance : Bounded₁ (log : V → V) := ⟨#0, λ _ ↦ by simp⟩
-@[simp] lemma log_one : log (1 : V) = 0 := sorry
+lemma log_eq_of_pos {x y : V} (pos : 0 < y) {y'} (H : Exponential x y') (hy' : y' ≤ y) (hy : y < 2 * y') : log y = x :=
+ (log_exists_unique_pos pos).unique ⟨log_lt_self_of_pos pos, 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_7471c35686f5_36 | d075762b307ec822 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Exponential/Log.lean | Log | 36 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 32 | 3 | [
{
"theorem_name": "length_zero",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by simp [length_eq_binaryLength]",
"n_chars": 33,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automatio... | [
{
"name": "length_eq_binaryLength",
"text": "lemma length_eq_binaryLength (a : V) : ‖a‖ = if 0 < a then log a + 1 else 0 := rfl\n\n",
"fan_in": 3,
"n_lines": 3,
"n_chars": 88,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_struct... | [
{
"name": "length_mul_exp_add_of_lt",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 10,
"n_lines": 4,
"n_chars": 186,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"ma... | 9 | module
public import Foundation.FirstOrder.Arithmetic.Exponential.Exp
@[expose] public section
/-!
# Logarithmic function
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section ISigma0
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₀]
lemma log_exists_unique_pos {y : V} (hy : 0 < y) : ∃! x, x < y ∧ ∃ y'... | module
public import Foundation.FirstOrder.Arithmetic.Exponential.Exp
@[expose] public section
/-!
# Logarithmic function
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section ISigma0
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₀]
lemma log_exists_unique_pos {y : V} (hy : 0 < y) : ∃! x, x < y ∧ ∃ y'... | @@ -137,10 +137,12 @@
noncomputable scoped instance : Length V := ⟨binaryLength⟩
-@[simp] lemma length_zero : ‖(0 : V)‖ = 0 := sorry
+lemma length_eq_binaryLength (a : V) : ‖a‖ = if 0 < a then log a + 1 else 0 := rfl
-lemma length_of_pos {a : V} (pos : 0 < a) : ‖a‖ = log a + 1 := sorry
+@[simp] lemma length_zero... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_7471c35686f5_37 | 9d8037b2ceb01f0b | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Exponential/Log.lean | Log | 37 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 32 | 2 | [
{
"theorem_name": "length_mul_pow2",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n simp [length_of_pos, pos, pp.pos, log_mul_pow2 pos pp, add_right_comm (log a) (log p) 1]",
"n_chars": 94,
"n_subpro... | [
{
"name": "log_mul_pow2",
"text": "lemma log_mul_pow2 {a p : V} (pos : 0 < a) (pp : Pow2 p) : log (a * p) = log a + log p := by\n simpa using log_mul_pow2_add_of_lt pos pp pp.pos\n\n",
"fan_in": 2,
"n_lines": 4,
"n_chars": 145,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
... | [
{
"name": "length_mul_exp",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 11,
"n_lines": 4,
"n_chars": 136,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"max_nesting"... | 10 | module
public import Foundation.FirstOrder.Arithmetic.Exponential.Exp
@[expose] public section
/-!
# Logarithmic function
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section ISigma0
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₀]
lemma log_exists_unique_pos {y : V} (hy : 0 < y) : ∃! x, x < y ∧ ∃ y'... | module
public import Foundation.FirstOrder.Arithmetic.Exponential.Exp
@[expose] public section
/-!
# Logarithmic function
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section ISigma0
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₀]
lemma log_exists_unique_pos {y : V} (hy : 0 < y) : ∃! x, x < y ∧ ∃ y'... | @@ -117,6 +117,9 @@
_ = (a + 1) * p := by simp [add_mul]
_ ≤ 2 * (a' * p) := by simpa [←mul_assoc] using mul_le_mul_right (lt_iff_succ_le.mp ha))
+lemma log_mul_pow2 {a p : V} (pos : 0 < a) (pp : Pow2 p) : log (a * p) = log a + log p := by
+ simpa using log_mul_pow2_add_of_lt pos 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_7471c35686f5_38 | 42e21208b05156e1 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Exponential/Log.lean | Log | 38 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 32 | 11 | [
{
"theorem_name": "log_eq_of_pos",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": "\n (log_exists_unique_pos pos).unique ⟨log_lt_self_of_pos pos, log_pos pos⟩ ⟨lt_of_lt_of_le H.lt hy', y', hy', H, hy⟩",
"n_char... | [
{
"name": "log_pos",
"text": "lemma log_pos {y : V} (pos : 0 < y) : ∃ y' ≤ y, Exponential (log y) y' ∧ y < 2 * y' :=\n ((Classical.choose!_spec (log_exists_unique y)).2 pos).2\n\n",
"fan_in": 11,
"n_lines": 4,
"n_chars": 153,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n... | [
{
"name": "exp_le_iff_le_log",
"fan_in": 0,
"n_deps_direct": 2,
"n_deps_transitive": 5,
"n_lines": 4,
"n_chars": 199,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": false,
"max_nesti... | 4 | module
public import Foundation.FirstOrder.Arithmetic.Exponential.Exp
@[expose] public section
/-!
# Logarithmic function
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section ISigma0
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₀]
lemma log_exists_unique_pos {y : V} (hy : 0 < y) : ∃! x, x < y ∧ ∃ y'... | module
public import Foundation.FirstOrder.Arithmetic.Exponential.Exp
@[expose] public section
/-!
# Logarithmic function
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section ISigma0
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₀]
lemma log_exists_unique_pos {y : V} (hy : 0 < y) : ∃! x, x < y ∧ ∃ y'... | @@ -55,6 +55,9 @@
@[simp] lemma log_zero : log (0 : V) = 0 :=
(Classical.choose!_spec (log_exists_unique (0 : V))).1 rfl
+lemma log_pos {y : V} (pos : 0 < y) : ∃ y' ≤ y, Exponential (log y) y' ∧ y < 2 * y' :=
+ ((Classical.choose!_spec (log_exists_unique y)).2 pos).2
+
lemma log_lt_self_of_pos {y : V} (pos : 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_58cd21a20cd8_0 | b7b387d5b50f776d | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/TA/Nonstandard.lean | Nonstandard | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 2 | 1 | [
{
"theorem_name": "satisfiable_union_trueArithWithStarUnbounded",
"depth": 1,
"n_commands": 0,
"n_lines": 3,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": "\n (Compact.compact_cumulative trueArithWithStarUnbounded.cumulative).mpr\n satisfiable_trueArith... | [
{
"name": "satisfiable_trueArithWithStarUnbounded",
"text": "lemma satisfiable_trueArithWithStarUnbounded (c : ℕ) : Satisfiable (trueArithWithStarUnbounded c) := by\n letI : Structure Language.unit ℕ := modelStar c\n haveI : Structure.Zero ℒₒᵣ⋆ ℕ := ⟨rfl⟩\n haveI : Structure.One ℒₒᵣ⋆ ℕ := ⟨rfl⟩\n haveI ... | [
{
"name": "satisfiable_union_trueArithWithStarUnbounded",
"fan_in": 0,
"n_deps_direct": 2,
"n_deps_transitive": 2,
"n_lines": 6,
"n_chars": 227,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_o... | 2 | module
public import Foundation.FirstOrder.Arithmetic.TA.Basic
public import Foundation.FirstOrder.Arithmetic.PeanoMinus.Basic
@[expose] public section
namespace LO.FirstOrder.Arithmetic
abbrev withStar := Language.add ℒₒᵣ Language.unit
local notation "ℒₒᵣ⋆" => withStar
def starUnbounded (c : ℕ) : Theory ℒₒᵣ⋆ := S... | module
public import Foundation.FirstOrder.Arithmetic.TA.Basic
public import Foundation.FirstOrder.Arithmetic.PeanoMinus.Basic
@[expose] public section
namespace LO.FirstOrder.Arithmetic
abbrev withStar := Language.add ℒₒᵣ Language.unit
local notation "ℒₒᵣ⋆" => withStar
def starUnbounded (c : ℕ) : Theory ℒₒᵣ⋆ := S... | @@ -23,8 +23,24 @@
func := fun _ ⟨⟨⟩⟩ _ ↦ c
rel := fun _ r _ ↦ PEmpty.elim r
+lemma satisfiable_trueArithWithStarUnbounded (c : ℕ) : Satisfiable (trueArithWithStarUnbounded c) := by
+ letI : Structure Language.unit ℕ := modelStar c
+ haveI : Structure.Zero ℒₒᵣ⋆ ℕ := ⟨rfl⟩
+ haveI : Structure.One ℒₒᵣ⋆ ℕ := ⟨... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_4ac77af5e4b7_0 | 9e352f66707cc6cd | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Semantics/CoherenceSpace/StableFunction.lean | StableFunction | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 2 | 1 | [
{
"theorem_name": "union_clique",
"depth": 1,
"n_commands": 0,
"n_lines": 9,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n let u : Point α := ⟨a ∪ b, h⟩\n have directed : DirectedOn (· ≤ ·) {a, b, u} :=\n DirectedOn.of_terminal_elem (u := u) (by... | [
{
"name": "of_terminal_elem",
"text": "lemma of_terminal_elem (hu : u ∈ s) (h : ∀ x ∈ s, r x u) : DirectedOn r s := by\n intro x hx y hy\n exact ⟨u, hu, h x hx, h y hy⟩\n\n",
"fan_in": 1,
"n_lines": 5,
"n_chars": 141,
"n_subproofs": 0,
"n_tactics": 3,
"cyclomatic": 1,
"n_automa... | [
{
"name": "union_clique",
"fan_in": 0,
"n_deps_direct": 2,
"n_deps_transitive": 3,
"n_lines": 11,
"n_chars": 664,
"n_subproofs": 4,
"n_tactics": 9,
"cyclomatic": 1,
"n_automation": 5,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": false,
"max_nesting":... | 1 | module
public import Foundation.Semantics.CoherenceSpace.Basic
/-!
# Stable functions
Reference: Jean-Yves Girard, Paul Taylor, Yves Lafont, Proofs and Types [G.T.L89]
-/
@[expose] public section
namespace DirectedOn
variable {α : Type*} {r : α → α → Prop}
end DirectedOn
namespace LO
@[ext] structure StableFun... | module
public import Foundation.Semantics.CoherenceSpace.Basic
/-!
# Stable functions
Reference: Jean-Yves Girard, Paul Taylor, Yves Lafont, Proofs and Types [G.T.L89]
-/
@[expose] public section
namespace DirectedOn
variable {α : Type*} {r : α → α → Prop}
lemma of_terminal_elem (hu : u ∈ s) (h : ∀ x ∈ s, r x u)... | @@ -14,6 +14,10 @@
variable {α : Type*} {r : α → α → Prop}
+lemma of_terminal_elem (hu : u ∈ s) (h : ∀ x ∈ s, r x u) : DirectedOn r s := by
+ intro x hx y hy
+ exact ⟨u, hu, h x hx, h y hy⟩
+
end DirectedOn
namespace LO
@@ -46,7 +50,8 @@
lemma union_clique {f : α →ₛ β} {a b : Point α} (h : IsClique (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_4ac77af5e4b7_1 | 031c829026be6b55 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Semantics/CoherenceSpace/StableFunction.lean | StableFunction | 1 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 2 | 3 | [
{
"theorem_name": "id_comp",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " ext_func (by simp)",
"n_chars": 19,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 1,
"n_rewr... | [
{
"name": "ext_func",
"text": "lemma ext_func {f g : α →ₛ β} : (f : Point α → Point β) = g → f = g := DFunLike.coe_fn_eq.mp\n\n",
"fan_in": 3,
"n_lines": 3,
"n_chars": 106,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structura... | [
{
"name": "id_comp",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 3,
"n_chars": 100,
"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.Semantics.CoherenceSpace.Basic
/-!
# Stable functions
Reference: Jean-Yves Girard, Paul Taylor, Yves Lafont, Proofs and Types [G.T.L89]
-/
@[expose] public section
namespace DirectedOn
variable {α : Type*} {r : α → α → Prop}
lemma of_terminal_elem (hu : u ∈ s) (h : ∀ x ∈ s, r x u)... | module
public import Foundation.Semantics.CoherenceSpace.Basic
/-!
# Stable functions
Reference: Jean-Yves Girard, Paul Taylor, Yves Lafont, Proofs and Types [G.T.L89]
-/
@[expose] public section
namespace DirectedOn
variable {α : Type*} {r : α → α → Prop}
lemma of_terminal_elem (hu : u ∈ s) (h : ∀ x ∈ s, r x u)... | @@ -58,6 +58,8 @@
have : IsClique (↑(f a) ∪ ↑(f b) ∪ ↑(f u) : Set β) := by simp [←this]
refine this.of_subset (by simp)
+lemma ext_func {f g : α →ₛ β} : (f : Point α → Point β) = g → f = g := DFunLike.coe_fn_eq.mp
+
protected def id (α : Type*) [CoherenceSpace α] : α →ₛ α where
toFun := id
monotone' 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_4ac77af5e4b7_2 | 0dd56ca4a8c55095 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Semantics/CoherenceSpace/StableFunction.lean | StableFunction | 2 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 2 | 3 | [
{
"theorem_name": "id_comp",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " ext_func (by simp)",
"n_chars": 19,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 1,
"n_rewr... | [
{
"name": "ext_func",
"text": "lemma ext_func {f g : α →ₛ β} : (f : Point α → Point β) = g → f = g := DFunLike.coe_fn_eq.mp\n\n",
"fan_in": 3,
"n_lines": 3,
"n_chars": 106,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structura... | [
{
"name": "comp_assoc",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 4,
"n_chars": 147,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
"n_automation": 2,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"max_nesting": 1
... | 1 | module
public import Foundation.Semantics.CoherenceSpace.Basic
/-!
# Stable functions
Reference: Jean-Yves Girard, Paul Taylor, Yves Lafont, Proofs and Types [G.T.L89]
-/
@[expose] public section
namespace DirectedOn
variable {α : Type*} {r : α → α → Prop}
lemma of_terminal_elem (hu : u ∈ s) (h : ∀ x ∈ s, r x u)... | module
public import Foundation.Semantics.CoherenceSpace.Basic
/-!
# Stable functions
Reference: Jean-Yves Girard, Paul Taylor, Yves Lafont, Proofs and Types [G.T.L89]
-/
@[expose] public section
namespace DirectedOn
variable {α : Type*} {r : α → α → Prop}
lemma of_terminal_elem (hu : u ∈ s) (h : ∀ x ∈ s, r x u)... | @@ -58,6 +58,8 @@
have : IsClique (↑(f a) ∪ ↑(f b) ∪ ↑(f u) : Set β) := by simp [←this]
refine this.of_subset (by simp)
+lemma ext_func {f g : α →ₛ β} : (f : Point α → Point β) = g → f = g := DFunLike.coe_fn_eq.mp
+
protected def id (α : Type*) [CoherenceSpace α] : α →ₛ α where
toFun := id
monotone' 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_c5303c6282e4_0 | d1ddf8742340467f | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Kripke/Logic/KHen.lean | KHen | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 7 | 1 | [
{
"theorem_name": "valid_atomic_axiomL_iff_valid_atomic_axiomH",
"depth": 1,
"n_commands": 0,
"n_lines": 4,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n constructor;\n . exact valid_atomic_axiomHen_of_valid_atomic_axiomL;\n . exact valid_atomic_a... | [
{
"name": "valid_atomic_axiomHen_of_valid_atomic_axiomL",
"text": "lemma valid_atomic_axiomHen_of_valid_atomic_axiomL : F ⊧ (Axioms.L (atom a)) → F ⊧ (Axioms.Hen (atom a)) := by\n intro h V x hx;\n have : Satisfies ⟨F, V⟩ x (□(□a 🡒 a)) := by\n intro y Rxy;\n exact (Satisfies.and_def.mp $ @hx y Rxy)... | [
{
"name": "valid_atomic_axiomL_iff_valid_atomic_axiomH",
"fan_in": 1,
"n_deps_direct": 2,
"n_deps_transitive": 2,
"n_lines": 6,
"n_chars": 240,
"n_subproofs": 0,
"n_tactics": 7,
"cyclomatic": 2,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 3,
"automation_on... | 2 | module
public import Foundation.Modal.Kripke.AxiomL
public import Foundation.Modal.Kripke.Logic.K
@[expose] public section
namespace LO.Modal
open Entailment
open Formula
open Formula.Kripke
open Kripke
open System
namespace Kripke
variable {F : Kripke.Frame} {a : ℕ} {φ : Formula ℕ}
lemma valid_atomic_axiomL_of_... | module
public import Foundation.Modal.Kripke.AxiomL
public import Foundation.Modal.Kripke.Logic.K
@[expose] public section
namespace LO.Modal
open Entailment
open Formula
open Formula.Kripke
open Kripke
open System
namespace Kripke
variable {F : Kripke.Frame} {a : ℕ} {φ : Formula ℕ}
lemma valid_atomic_axiomHen_o... | @@ -17,6 +17,13 @@
variable {F : Kripke.Frame} {a : ℕ} {φ : Formula ℕ}
+lemma valid_atomic_axiomHen_of_valid_atomic_axiomL : F ⊧ (Axioms.L (atom a)) → F ⊧ (Axioms.Hen (atom a)) := by
+ intro h V x hx;
+ have : Satisfies ⟨F, V⟩ x (□(□a 🡒 a)) := by
+ intro y Rxy;
+ exact (Satisfies.and_def.mp $ @hx y Rxy) |... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_c5303c6282e4_1 | eb3f5e69c9dda92e | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Kripke/Logic/KHen.lean | KHen | 1 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 7 | 1 | [
{
"theorem_name": "valid_atomic_axiomL_iff_valid_atomic_axiomH",
"depth": 1,
"n_commands": 0,
"n_lines": 4,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n constructor;\n . exact valid_atomic_axiomHen_of_valid_atomic_axiomL;\n . exact valid_atomic_a... | [
{
"name": "valid_atomic_axiomHen_of_valid_atomic_axiomL",
"text": "lemma valid_atomic_axiomHen_of_valid_atomic_axiomL : F ⊧ (Axioms.L (atom a)) → F ⊧ (Axioms.Hen (atom a)) := by\n intro h V x hx;\n have : Satisfies ⟨F, V⟩ x (□(□a 🡒 a)) := by\n intro y Rxy;\n exact (Satisfies.and_def.mp $ @hx y Rxy)... | [
{
"name": "valid_atomic_axiomFour_of_valid_atomic_axiomH",
"fan_in": 1,
"n_deps_direct": 2,
"n_deps_transitive": 4,
"n_lines": 7,
"n_chars": 244,
"n_subproofs": 0,
"n_tactics": 7,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 2,
"automation_... | 4 | module
public import Foundation.Modal.Kripke.AxiomL
public import Foundation.Modal.Kripke.Logic.K
@[expose] public section
namespace LO.Modal
open Entailment
open Formula
open Formula.Kripke
open Kripke
open System
namespace Kripke
variable {F : Kripke.Frame} {a : ℕ} {φ : Formula ℕ}
lemma valid_atomic_axiomL_of_... | module
public import Foundation.Modal.Kripke.AxiomL
public import Foundation.Modal.Kripke.Logic.K
@[expose] public section
namespace LO.Modal
open Entailment
open Formula
open Formula.Kripke
open Kripke
open System
namespace Kripke
variable {F : Kripke.Frame} {a : ℕ} {φ : Formula ℕ}
lemma valid_atomic_axiomHen_o... | @@ -17,6 +17,13 @@
variable {F : Kripke.Frame} {a : ℕ} {φ : Formula ℕ}
+lemma valid_atomic_axiomHen_of_valid_atomic_axiomL : F ⊧ (Axioms.L (atom a)) → F ⊧ (Axioms.Hen (atom a)) := by
+ intro h V x hx;
+ have : Satisfies ⟨F, V⟩ x (□(□a 🡒 a)) := by
+ intro y Rxy;
+ exact (Satisfies.and_def.mp $ @hx y Rxy) |... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_c5303c6282e4_2 | c46635b4be072ebe | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Kripke/Logic/KHen.lean | KHen | 2 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 7 | 1 | [
{
"theorem_name": "unprovable_atomic_axiomFour",
"depth": 1,
"n_commands": 0,
"n_lines": 3,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n by_contra hC;\n exact cresswellModel.not_valid_axiomFour $ Kripke.valid_cresswellModel_of_provable hC 2♯;",
... | [
{
"name": "cresswellModel.not_valid_axiomFour",
"text": "lemma cresswellModel.not_valid_axiomFour : ¬(Satisfies cresswellModel 2♯ (Axioms.Four (atom 0))) := by\n apply Satisfies.imp_def.not.mpr;\n push Not;\n constructor;\n . intro x;\n match x with\n | n♯ =>\n intro h2n;\n suffices n ≠ ... | [
{
"name": "unprovable_atomic_axiomFour",
"fan_in": 1,
"n_deps_direct": 2,
"n_deps_transitive": 3,
"n_lines": 5,
"n_chars": 189,
"n_subproofs": 0,
"n_tactics": 5,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": false,
... | 2 | module
public import Foundation.Modal.Kripke.AxiomL
public import Foundation.Modal.Kripke.Logic.K
@[expose] public section
namespace LO.Modal
open Entailment
open Formula
open Formula.Kripke
open Kripke
open System
namespace Kripke
variable {F : Kripke.Frame} {a : ℕ} {φ : Formula ℕ}
abbrev cresswellFrame : Kripk... | module
public import Foundation.Modal.Kripke.AxiomL
public import Foundation.Modal.Kripke.Logic.K
@[expose] public section
namespace LO.Modal
open Entailment
open Formula
open Formula.Kripke
open Kripke
open System
namespace Kripke
variable {F : Kripke.Frame} {a : ℕ} {φ : Formula ℕ}
abbrev cresswellFrame : Kripk... | @@ -59,6 +59,30 @@
open cresswellFrame cresswellModel
+lemma cresswellModel.not_valid_axiomFour : ¬(Satisfies cresswellModel 2♯ (Axioms.Four (atom 0))) := by
+ apply Satisfies.imp_def.not.mpr;
+ push Not;
+ constructor;
+ . intro x;
+ match x with
+ | n♯ =>
+ intro h2n;
+ suffices n ≠ 0 by simp... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_c5303c6282e4_3 | 24563fade4d49080 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Kripke/Logic/KHen.lean | KHen | 3 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 7 | 1 | [
{
"theorem_name": "valid_atomic_axiomL_iff_valid_atomic_axiomH",
"depth": 1,
"n_commands": 0,
"n_lines": 4,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n constructor;\n . exact valid_atomic_axiomHen_of_valid_atomic_axiomL;\n . exact valid_atomic_a... | [
{
"name": "valid_atomic_axiomL_of_valid_atomic_axiomHen",
"text": "lemma valid_atomic_axiomL_of_valid_atomic_axiomHen : F ⊧ Axioms.Hen (atom a) → F ⊧ Axioms.L (atom a) := by\n intro hH V x hx;\n\n let V' : Valuation F := λ a w => ∀ n : ℕ, Satisfies ⟨F, V⟩ w (□^[n] a);\n\n have h₁ : Satisfies ⟨F, V'⟩ x (□... | [
{
"name": "Kripke.incomplete",
"fan_in": 0,
"n_deps_direct": 2,
"n_deps_transitive": 9,
"n_lines": 8,
"n_chars": 446,
"n_subproofs": 3,
"n_tactics": 11,
"cyclomatic": 2,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 2,
"automation_only": false,
"max_nest... | 7 | module
public import Foundation.Modal.Kripke.AxiomL
public import Foundation.Modal.Kripke.Logic.K
@[expose] public section
namespace LO.Modal
open Entailment
open Formula
open Formula.Kripke
open Kripke
open System
namespace Kripke
variable {F : Kripke.Frame} {a : ℕ} {φ : Formula ℕ}
lemma valid_atomic_axiomHen_o... | module
public import Foundation.Modal.Kripke.AxiomL
public import Foundation.Modal.Kripke.Logic.K
@[expose] public section
namespace LO.Modal
open Entailment
open Formula
open Formula.Kripke
open Kripke
open System
namespace Kripke
variable {F : Kripke.Frame} {a : ℕ} {φ : Formula ℕ}
lemma valid_atomic_axiomHen_o... | @@ -24,10 +24,41 @@
exact (Satisfies.and_def.mp $ @hx y Rxy) |>.1;
exact @h V x this;
+lemma valid_atomic_axiomL_of_valid_atomic_axiomHen : F ⊧ Axioms.Hen (atom a) → F ⊧ Axioms.L (atom a) := by
+ intro hH V x hx;
+
+ let V' : Valuation F := λ a w => ∀ n : ℕ, Satisfies ⟨F, V⟩ w (□^[n] a);
+
+ have h₁ : Sati... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_edfef76770c3_0 | f6d15a545c843287 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Incompleteness/RestrictedProvability.lean | RestrictedProvability | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 6 | 2 | [
{
"theorem_name": "iff_provable_restrictedG",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n apply Entailment.iff_of_E! provable_E_restrictedGödel_restrictedGödel';",
"n_chars": 79,
"n_subproofs": 0,... | [
{
"name": "provable_E_restrictedG",
"text": "private lemma provable_E_restrictedGödel_restrictedGödel' [𝗜𝚺₁ ⪯ U] : U ⊢ T.restrictedGödel e 🡘 T.restrictedGödel' e := by\n apply Entailment.E!_trans;\n . exact def_restrictedGödel;\n . exact Entailment.E!_symm $ def_restrictedGödel';\n\n",
"fan_in": 2... | [
{
"name": "iff_provable_restrictedG",
"fan_in": 2,
"n_deps_direct": 3,
"n_deps_transitive": 3,
"n_lines": 4,
"n_chars": 239,
"n_subproofs": 0,
"n_tactics": 3,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": false,
"ma... | 1 | module
public import Foundation.FirstOrder.Incompleteness.RosserProvability
@[expose] public section
/-!
# Provability with restricted proof size
Some results to consider provable predicate modified to state that "provable by proof whose Gödel number is less than `2^e`" (where `e` is an arbitary meta natural number)... | module
public import Foundation.FirstOrder.Incompleteness.RosserProvability
@[expose] public section
/-!
# Provability with restricted proof size
Some results to consider provable predicate modified to state that "provable by proof whose Gödel number is less than `2^e`" (where `e` is an arbitary meta natural number)... | @@ -50,10 +50,19 @@
private lemma def_restrictedGödel' [𝗜𝚺₁ ⪯ U] : U ⊢ T.restrictedGödel' e 🡘 (∼T.restrictedProvable e)/[⌜T.restrictedGödel e⌝] := by simp;
-private lemma iff_provable_restrictedGödel_provable_restrictedGödel' [𝗜𝚺₁ ⪯ U] : U ⊢ (T.restrictedGödel e) ↔ U ⊢ (T.restrictedGödel' e) := sorry
+private... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_edfef76770c3_2 | 5301e6a924e62494 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Incompleteness/RestrictedProvability.lean | RestrictedProvability | 2 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 6 | 2 | [
{
"theorem_name": "true_restrictedG",
"depth": 1,
"n_commands": 0,
"n_lines": 9,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n by_contra hC;\n obtain ⟨e, _, he⟩ := models_neg_restrictedGödel (e := e) |>.mp hC;\n apply hC;\n apply iff_true_restric... | [
{
"name": "iff_provable_restrictedG",
"text": "private lemma iff_provable_restrictedGödel_provable_restrictedGödel' [𝗜𝚺₁ ⪯ U] : U ⊢ (T.restrictedGödel e) ↔ U ⊢ (T.restrictedGödel' e) := by\n apply Entailment.iff_of_E! provable_E_restrictedGödel_restrictedGödel';\n\n",
"fan_in": 2,
"n_lines": 4,
... | [
{
"name": "true_restrictedG",
"fan_in": 2,
"n_deps_direct": 5,
"n_deps_transitive": 7,
"n_lines": 12,
"n_chars": 565,
"n_subproofs": 1,
"n_tactics": 17,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 6,
"automation_only": false,
"max_nest... | 5 | module
public import Foundation.FirstOrder.Incompleteness.RosserProvability
@[expose] public section
/-!
# Provability with restricted proof size
Some results to consider provable predicate modified to state that "provable by proof whose Gödel number is less than `2^e`" (where `e` is an arbitary meta natural number)... | module
public import Foundation.FirstOrder.Incompleteness.RosserProvability
@[expose] public section
/-!
# Provability with restricted proof size
Some results to consider provable predicate modified to state that "provable by proof whose Gödel number is less than `2^e`" (where `e` is an arbitary meta natural number)... | @@ -55,6 +55,9 @@
. exact def_restrictedGödel;
. exact Entailment.E!_symm $ def_restrictedGödel';
+private lemma iff_provable_restrictedGödel_provable_restrictedGödel' [𝗜𝚺₁ ⪯ U] : U ⊢ (T.restrictedGödel e) ↔ U ⊢ (T.restrictedGödel' e) := by
+ apply Entailment.iff_of_E! provable_E_restrictedGödel_restrictedGö... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_edfef76770c3_3 | 926d54814933a453 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Incompleteness/RestrictedProvability.lean | RestrictedProvability | 3 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 6 | 2 | [
{
"theorem_name": "true_restrictedG",
"depth": 1,
"n_commands": 0,
"n_lines": 9,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n by_contra hC;\n obtain ⟨e, _, he⟩ := models_neg_restrictedGödel (e := e) |>.mp hC;\n apply hC;\n apply iff_true_restric... | [
{
"name": "iff_true_restrictedG",
"text": "private lemma iff_true_restrictedGödel_true_restrictedGödel' : ℕ↓[ℒₒᵣ] ⊧ (T.restrictedGödel e) ↔ ℕ↓[ℒₒᵣ] ⊧ (T.restrictedGödel' e) := by\n apply Semantics.models_iff.mp;\n apply models_of_provable (T := 𝗜𝚺₁) inferInstance;\n apply provable_E_restrictedGödel_res... | [
{
"name": "provable_restrictedG",
"fan_in": 0,
"n_deps_direct": 5,
"n_deps_transitive": 8,
"n_lines": 7,
"n_chars": 336,
"n_subproofs": 0,
"n_tactics": 7,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 3,
"automation_only": false,
"max_ne... | 6 | module
public import Foundation.FirstOrder.Incompleteness.RosserProvability
@[expose] public section
/-!
# Provability with restricted proof size
Some results to consider provable predicate modified to state that "provable by proof whose Gödel number is less than `2^e`" (where `e` is an arbitary meta natural number)... | module
public import Foundation.FirstOrder.Incompleteness.RosserProvability
@[expose] public section
/-!
# Provability with restricted proof size
Some results to consider provable predicate modified to state that "provable by proof whose Gödel number is less than `2^e`" (where `e` is an arbitary meta natural number)... | @@ -58,6 +58,11 @@
private lemma iff_provable_restrictedGödel_provable_restrictedGödel' [𝗜𝚺₁ ⪯ U] : U ⊢ (T.restrictedGödel e) ↔ U ⊢ (T.restrictedGödel' e) := by
apply Entailment.iff_of_E! provable_E_restrictedGödel_restrictedGödel';
+private lemma iff_true_restrictedGödel_true_restrictedGödel' : ℕ↓[ℒₒᵣ] ⊧ (T.re... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_cf00234c7d93_0 | 160c20589c5b2e90 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/LinearLogic/FirstOrder/ClassicalEmbedding.lean | ClassicalEmbedding | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 1 | 2 | [
{
"theorem_name": "girard_negative_iff",
"depth": 1,
"n_commands": 0,
"n_lines": 7,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n constructor\n · contrapose\n intro h\n have : φ.girard.Positive := girard_positive (by simpa using h)\n grind... | [
{
"name": "girard_positive",
"text": "lemma girard_positive {φ : Semiformula L ξ n} (h : φ.Positive) : φ.girard.Positive := by\n have : (∼φ).Negative := by simpa\n simpa using girard_negative this\n\n",
"fan_in": 2,
"n_lines": 5,
"n_chars": 167,
"n_subproofs": 1,
"n_tactics": 3,
"c... | [
{
"name": "girard_negative_iff",
"fan_in": 0,
"n_deps_direct": 4,
"n_deps_transitive": 3,
"n_lines": 9,
"n_chars": 262,
"n_subproofs": 1,
"n_tactics": 8,
"cyclomatic": 2,
"n_automation": 2,
"n_rewrites": 0,
"n_structural": 4,
"automation_only": false,
"max_nes... | 1 | module
public import Foundation.Logic.Embedding
public import Foundation.FirstOrder.Polarity
public import Foundation.LinearLogic.FirstOrder.Calculus
/-! # Girard's embedding of classical logic into linear logic -/
@[expose] public section
namespace LO.FirstOrder
variable {L : Language}
/-! ## $\mathbf{LL}$ to $\... | module
public import Foundation.Logic.Embedding
public import Foundation.FirstOrder.Polarity
public import Foundation.LinearLogic.FirstOrder.Calculus
/-! # Girard's embedding of classical logic into linear logic -/
@[expose] public section
namespace LO.FirstOrder
variable {L : Language}
/-! ## $\mathbf{LL}$ to $\... | @@ -273,11 +273,15 @@
| true => simp [girard, hφ]
| false => simp [girard, hφ, girard_negative hφ]
+lemma girard_positive {φ : Semiformula L ξ n} (h : φ.Positive) : φ.girard.Positive := by
+ have : (∼φ).Negative := by simpa
+ simpa using girard_negative this
+
@[simp] lemma girard_negative_iff {φ : Semi... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_b92a3d9d578b_0 | 41d8e781759e26f7 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Bootstrapping/Syntax/Language.lean | Language | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 8 | 1 | [
{
"theorem_name": "codeIn_func_quote",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": "\n (isFunc_quote_quote (V := V)).mpr ⟨f, rfl⟩",
"n_chars": 49,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": ... | [
{
"name": "isFunc_quote_quote",
"text": "lemma isFunc_quote_quote {k x : ℕ} : L.IsFunc (V := V) k x ↔ ∃ f : L.Func k, Encodable.encode f = x :=\n have : V ⊧/![k, x] L.isFunc.val ↔ ℕ ⊧/![k, x] L.isFunc.val := by\n simpa [Matrix.comp_vecCons', Matrix.constant_eq_singleton]\n using models_iff_of_Sigma... | [
{
"name": "codeIn_func_quote",
"fan_in": 4,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 4,
"n_chars": 140,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"max_nestin... | 1 | module
public import Foundation.FirstOrder.Arithmetic.HFS
@[expose] public section
/-! # Internalized languages of first-order logic -/
namespace LO.FirstOrder.Arithmetic.Bootstrapping
variable {L : Language} [L.Encodable]
instance (k) : Semiterm.Operator.GödelNumber ℒₒᵣ (L.Func k) := ⟨fun f ↦ Semiterm.Operator.nu... | module
public import Foundation.FirstOrder.Arithmetic.HFS
@[expose] public section
/-! # Internalized languages of first-order logic -/
namespace LO.FirstOrder.Arithmetic.Bootstrapping
variable {L : Language} [L.Encodable]
instance (k) : Semiterm.Operator.GödelNumber ℒₒᵣ (L.Func k) := ⟨fun f ↦ Semiterm.Operator.nu... | @@ -70,13 +70,20 @@
omit [L.LORDefinable] in
lemma quote_rel_def (R : L.Rel k) : (⌜R⌝ : V) = ↑(Encodable.encode R) := rfl
+lemma isFunc_quote_quote {k x : ℕ} : L.IsFunc (V := V) k x ↔ ∃ f : L.Func k, Encodable.encode f = x :=
+ have : V ⊧/![k, x] L.isFunc.val ↔ ℕ ⊧/![k, x] L.isFunc.val := by
+ simpa [Matrix.com... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_b92a3d9d578b_1 | 24dba8dbe2d8637e | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Bootstrapping/Syntax/Language.lean | Language | 1 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 8 | 1 | [
{
"theorem_name": "codeIn_rel_quote",
"depth": 1,
"n_commands": 0,
"n_lines": 4,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": "\n (isRel_quote_quote (V := V)).mpr ⟨r, rfl⟩\n\nomit [L.LORDefinable]",
"n_chars": 71,
"n_subproofs": 0,
"n_tactics"... | [
{
"name": "isRel_quote_quote",
"text": "lemma isRel_quote_quote {k x : ℕ} : L.IsRel (V := V) k x ↔ ∃ R : L.Rel k, Encodable.encode R = x :=\n have : V ⊧/![k, x] L.isRel.val ↔ ℕ ⊧/![k, x] L.isRel.val := by\n simpa [Matrix.comp_vecCons', Matrix.constant_eq_singleton]\n using models_iff_of_Sigma0 (V :... | [
{
"name": "codeIn_rel_quote",
"fan_in": 2,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 6,
"n_chars": 159,
"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.Arithmetic.HFS
@[expose] public section
/-! # Internalized languages of first-order logic -/
namespace LO.FirstOrder.Arithmetic.Bootstrapping
variable {L : Language} [L.Encodable]
instance (k) : Semiterm.Operator.GödelNumber ℒₒᵣ (L.Func k) := ⟨fun f ↦ Semiterm.Operator.nu... | module
public import Foundation.FirstOrder.Arithmetic.HFS
@[expose] public section
/-! # Internalized languages of first-order logic -/
namespace LO.FirstOrder.Arithmetic.Bootstrapping
variable {L : Language} [L.Encodable]
instance (k) : Semiterm.Operator.GödelNumber ℒₒᵣ (L.Func k) := ⟨fun f ↦ Semiterm.Operator.nu... | @@ -76,11 +76,20 @@
using models_iff_of_Sigma0 (V := V) (σ := L.isFunc.val) (by simp) (e := ![k, x])
Iff.trans this <| Iff.trans L.iff_isFunc.symm <| by simp
+lemma isRel_quote_quote {k x : ℕ} : L.IsRel (V := V) k x ↔ ∃ R : L.Rel k, Encodable.encode R = x :=
+ have : V ⊧/![k, x] L.isRel.val ↔ ℕ ⊧/![k, x] 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_b92a3d9d578b_2 | 8e978e2b68bf146e | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Bootstrapping/Syntax/Language.lean | Language | 2 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 8 | 2 | [
{
"theorem_name": "quote_func_inj",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n simp [quote_func_def]",
"n_chars": 27,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
"n_automation"... | [
{
"name": "quote_func_def",
"text": "lemma quote_func_def (f : L.Func k) : (⌜f⌝ : V) = ↑(Encodable.encode f) := rfl\n\nomit [L.LORDefinable] in\n",
"fan_in": 2,
"n_lines": 4,
"n_chars": 111,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0... | [
{
"name": "quote_func_inj",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 4,
"n_chars": 137,
"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.Arithmetic.HFS
@[expose] public section
/-! # Internalized languages of first-order logic -/
namespace LO.FirstOrder.Arithmetic.Bootstrapping
variable {L : Language} [L.Encodable]
instance (k) : Semiterm.Operator.GödelNumber ℒₒᵣ (L.Func k) := ⟨fun f ↦ Semiterm.Operator.nu... | module
public import Foundation.FirstOrder.Arithmetic.HFS
@[expose] public section
/-! # Internalized languages of first-order logic -/
namespace LO.FirstOrder.Arithmetic.Bootstrapping
variable {L : Language} [L.Encodable]
instance (k) : Semiterm.Operator.GödelNumber ℒₒᵣ (L.Func k) := ⟨fun f ↦ Semiterm.Operator.nu... | @@ -65,6 +65,9 @@
instance gödelQuoteRel (k) : GödelQuote (L.Rel k) V := ⟨fun R ↦ ↑(Encodable.encode R)⟩
omit [L.LORDefinable] in
+lemma quote_func_def (f : L.Func k) : (⌜f⌝ : V) = ↑(Encodable.encode f) := rfl
+
+omit [L.LORDefinable] in
lemma quote_rel_def (R : L.Rel k) : (⌜R⌝ : V) = ↑(Encodable.encode R) := rfl
... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_b92a3d9d578b_4 | 3fc83a5509ac6893 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Bootstrapping/Syntax/Language.lean | Language | 4 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 8 | 1 | [
{
"theorem_name": "isFunc_iff_LOR",
"depth": 1,
"n_commands": 0,
"n_lines": 12,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n rw [isFunc_def (L := ℒₒᵣ), func_def_LOR]\n suffices\n k = 0 ∧ f = 0 ∨ k = 0 ∧ f = 1 ∨ k = 2 ∧ f = 0 ∨ k = 2 ∧ f = 1 ↔ _... | [
{
"name": "isFunc_def",
"text": "lemma isFunc_def (k f : V) : L.IsFunc k f ↔ V ⊧/![k, f] L.isFunc.val := by rfl\n\n",
"fan_in": 1,
"n_lines": 3,
"n_chars": 84,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"a... | [
{
"name": "isFunc_iff_LOR",
"fan_in": 0,
"n_deps_direct": 2,
"n_deps_transitive": 2,
"n_lines": 19,
"n_chars": 768,
"n_subproofs": 1,
"n_tactics": 12,
"cyclomatic": 1,
"n_automation": 5,
"n_rewrites": 2,
"n_structural": 3,
"automation_only": false,
"max_nestin... | 2 | module
public import Foundation.FirstOrder.Arithmetic.HFS
@[expose] public section
/-! # Internalized languages of first-order logic -/
namespace LO.FirstOrder.Arithmetic.Bootstrapping
variable {L : Language} [L.Encodable]
instance (k) : Semiterm.Operator.GödelNumber ℒₒᵣ (L.Func k) := ⟨fun f ↦ Semiterm.Operator.nu... | module
public import Foundation.FirstOrder.Arithmetic.HFS
@[expose] public section
/-! # Internalized languages of first-order logic -/
namespace LO.FirstOrder.Arithmetic.Bootstrapping
variable {L : Language} [L.Encodable]
instance (k) : Semiterm.Operator.GödelNumber ℒₒᵣ (L.Func k) := ⟨fun f ↦ Semiterm.Operator.nu... | @@ -36,6 +36,8 @@
variable {L}
+lemma isFunc_def (k f : V) : L.IsFunc k f ↔ V ⊧/![k, f] L.isFunc.val := by rfl
+
@[simp] lemma eval_func (v : Fin 2 → V) :
L.isFunc.val.Evalb v ↔ L.IsFunc (v 0) (v 1) := by simp [Language.IsFunc, ← Matrix.fun_eq_vec_two]
@@ -186,7 +188,18 @@
(k = 0 ∧ f = ⌜(Language.Zero... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_b92a3d9d578b_5 | e5320d7a573d231d | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Bootstrapping/Syntax/Language.lean | Language | 5 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 8 | 1 | [
{
"theorem_name": "isRel_iff_LOR",
"depth": 1,
"n_commands": 0,
"n_lines": 7,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n rw [isRel_def (L := ℒₒᵣ), rel_def_LOR]\n suffices\n k = 2 ∧ R = 0 ∨ k = 2 ∧ R = 1 ↔ _ by simpa\n apply iff_of_eq\n cong... | [
{
"name": "isRel_def",
"text": "lemma isRel_def (k R : V) : L.IsRel k R ↔ V ⊧/![k, R] L.isRel.val := by rfl\n\n",
"fan_in": 1,
"n_lines": 3,
"n_chars": 81,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"autom... | [
{
"name": "isRel_iff_LOR",
"fan_in": 0,
"n_deps_direct": 2,
"n_deps_transitive": 2,
"n_lines": 12,
"n_chars": 379,
"n_subproofs": 1,
"n_tactics": 7,
"cyclomatic": 1,
"n_automation": 2,
"n_rewrites": 2,
"n_structural": 2,
"automation_only": false,
"max_nesting"... | 2 | module
public import Foundation.FirstOrder.Arithmetic.HFS
@[expose] public section
/-! # Internalized languages of first-order logic -/
namespace LO.FirstOrder.Arithmetic.Bootstrapping
variable {L : Language} [L.Encodable]
instance (k) : Semiterm.Operator.GödelNumber ℒₒᵣ (L.Func k) := ⟨fun f ↦ Semiterm.Operator.nu... | module
public import Foundation.FirstOrder.Arithmetic.HFS
@[expose] public section
/-! # Internalized languages of first-order logic -/
namespace LO.FirstOrder.Arithmetic.Bootstrapping
variable {L : Language} [L.Encodable]
instance (k) : Semiterm.Operator.GödelNumber ℒₒᵣ (L.Func k) := ⟨fun f ↦ Semiterm.Operator.nu... | @@ -36,6 +36,8 @@
variable {L}
+lemma isRel_def (k R : V) : L.IsRel k R ↔ V ⊧/![k, R] L.isRel.val := by rfl
+
@[simp] lemma eval_func (v : Fin 2 → V) :
L.isFunc.val.Evalb v ↔ L.IsFunc (v 0) (v 1) := by simp [Language.IsFunc, ← Matrix.fun_eq_vec_two]
@@ -184,7 +186,13 @@
lemma isRel_iff_LOR {k R : V} :
... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_0083f62f8ed5_0 | 21d336ee091d4f9b | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Bootstrapping/DerivabilityCondition/D3.lean | D3 | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 5 | 1 | [
{
"theorem_name": "bold_sigma_one_complete",
"depth": 1,
"n_commands": 0,
"n_lines": 74,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n revert w\n apply sigma₁_induction' hp\n case hVerum => intro n; simp\n case hFalsum => intro n; simp\n case hE... | [
{
"name": "term_complete",
"text": "theorem term_complete {n : ℕ} (t : FirstOrder.ClosedSemiterm ℒₒᵣ n) (w : Fin n → V) :\n T.internalize V ⊢ (toNumVec w ⤕ ⌜t⌝) ≐ 𝕹 (t.valb w) :=\n match t with\n | #z => by simp\n | &x => Empty.elim x\n | .func Langua... | [
{
"name": "bold_sigma_one_complete",
"fan_in": 1,
"n_deps_direct": 2,
"n_deps_transitive": 2,
"n_lines": 77,
"n_chars": 4405,
"n_subproofs": 30,
"n_tactics": 76,
"cyclomatic": 3,
"n_automation": 24,
"n_rewrites": 0,
"n_structural": 25,
"automation_only": false,
... | 2 | module
public import Foundation.FirstOrder.Bootstrapping.DerivabilityCondition.PeanoMinus
@[expose] public section
/-!
# Hilbert-Bernays-Löb derivability condition $\mathbf{D3}$ and formalized $\Sigma_1$-completeness
-/
namespace LO.FirstOrder.Arithmetic.Bootstrapping.Arithmetic
open Classical
open LO.Entailment L... | module
public import Foundation.FirstOrder.Bootstrapping.DerivabilityCondition.PeanoMinus
@[expose] public section
/-!
# Hilbert-Bernays-Löb derivability condition $\mathbf{D3}$ and formalized $\Sigma_1$-completeness
-/
namespace LO.FirstOrder.Arithmetic.Bootstrapping.Arithmetic
open Classical
open LO.Entailment L... | @@ -38,10 +38,109 @@
variable (T)
+theorem term_complete {n : ℕ} (t : FirstOrder.ClosedSemiterm ℒₒᵣ n) (w : Fin n → V) :
+ T.internalize V ⊢ (toNumVec w ⤕ ⌜t⌝) ≐ 𝕹 (t.valb w) :=
+ match t with
+ | #z => by simp
+ | &x => Empty.elim x
+ | .func Language.Zero.... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_0083f62f8ed5_1 | 61a03078547c66bb | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Bootstrapping/DerivabilityCondition/D3.lean | D3 | 1 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 5 | 1 | [
{
"theorem_name": "sigma_one_provable_of_models",
"depth": 1,
"n_commands": 0,
"n_lines": 5,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n intro h\n have : T.internalize V ⊢ (toNumVec ![] ⤔ ⌜σ⌝) :=\n bold_sigma_one_complete T hσ (by simpa [model... | [
{
"name": "bold_sigma_one_complete",
"text": "theorem bold_sigma_one_complete {n} {φ : ArithmeticSemisentence n} (hp : Hierarchy 𝚺 1 φ) {w} :\n V ⊧/w φ → T.internalize V ⊢ (toNumVec w ⤔ ⌜φ⌝) := by\n revert w\n apply sigma₁_induction' hp\n case hVerum => intro n; simp\n case hFalsum => intro n; simp\... | [
{
"name": "sigma_one_provable_of_models",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 3,
"n_lines": 8,
"n_chars": 316,
"n_subproofs": 1,
"n_tactics": 5,
"cyclomatic": 1,
"n_automation": 2,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": false,
... | 3 | module
public import Foundation.FirstOrder.Bootstrapping.DerivabilityCondition.PeanoMinus
@[expose] public section
/-!
# Hilbert-Bernays-Löb derivability condition $\mathbf{D3}$ and formalized $\Sigma_1$-completeness
-/
namespace LO.FirstOrder.Arithmetic.Bootstrapping.Arithmetic
open Classical
open LO.Entailment L... | module
public import Foundation.FirstOrder.Bootstrapping.DerivabilityCondition.PeanoMinus
@[expose] public section
/-!
# Hilbert-Bernays-Löb derivability condition $\mathbf{D3}$ and formalized $\Sigma_1$-completeness
-/
namespace LO.FirstOrder.Arithmetic.Bootstrapping.Arithmetic
open Classical
open LO.Entailment L... | @@ -66,10 +66,87 @@
open FirstOrder.Arithmetic
+theorem bold_sigma_one_complete {n} {φ : ArithmeticSemisentence n} (hp : Hierarchy 𝚺 1 φ) {w} :
+ V ⊧/w φ → T.internalize V ⊢ (toNumVec w ⤔ ⌜φ⌝) := by
+ revert w
+ apply sigma₁_induction' hp
+ case hVerum => intro n; simp
+ case hFalsum => intro n; simp
+ ca... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_0083f62f8ed5_2 | b2dbd01301155de1 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Bootstrapping/DerivabilityCondition/D3.lean | D3 | 2 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 5 | 1 | [
{
"theorem_name": "sigma_one_complete",
"depth": 1,
"n_commands": 0,
"n_lines": 3,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " fun h ↦ by\n simpa [tprovable_iff_provable]\n using! Bootstrapping.Arithmetic.sigma_one_provable_of_models T hσ h",
"n... | [
{
"name": "sigma_one_provable_of_models",
"text": "theorem sigma_one_provable_of_models {σ : ArithmeticSentence} (hσ : Hierarchy 𝚺 1 σ) :\n V↓[ℒₒᵣ] ⊧ σ → T.internalize V ⊢ ⌜σ⌝ := by\n intro h\n have : T.internalize V ⊢ (toNumVec ![] ⤔ ⌜σ⌝) :=\n bold_sigma_one_complete T hσ (by simpa [models_iff] u... | [
{
"name": "sigma_one_complete",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 4,
"n_lines": 7,
"n_chars": 313,
"n_subproofs": 0,
"n_tactics": 3,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"max_nesti... | 4 | module
public import Foundation.FirstOrder.Bootstrapping.DerivabilityCondition.PeanoMinus
@[expose] public section
/-!
# Hilbert-Bernays-Löb derivability condition $\mathbf{D3}$ and formalized $\Sigma_1$-completeness
-/
namespace LO.FirstOrder.Arithmetic.Bootstrapping.Arithmetic
open Classical
open LO.Entailment L... | module
public import Foundation.FirstOrder.Bootstrapping.DerivabilityCondition.PeanoMinus
@[expose] public section
/-!
# Hilbert-Bernays-Löb derivability condition $\mathbf{D3}$ and formalized $\Sigma_1$-completeness
-/
namespace LO.FirstOrder.Arithmetic.Bootstrapping.Arithmetic
open Classical
open LO.Entailment L... | @@ -142,9 +142,18 @@
simpa [Semiformula.substs_substs, Matrix.map_map_comp']
exact ih hφ
+theorem sigma_one_provable_of_models {σ : ArithmeticSentence} (hσ : Hierarchy 𝚺 1 σ) :
+ V↓[ℒₒᵣ] ⊧ σ → T.internalize V ⊢ ⌜σ⌝ := by
+ intro h
+ have : T.internalize V ⊢ (toNumVec ![] ⤔ ⌜σ⌝) :=
+ bold_sigma_o... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_0083f62f8ed5_3 | de1fdbb744a87837 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Bootstrapping/DerivabilityCondition/D3.lean | D3 | 3 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 5 | 1 | [
{
"theorem_name": "provable_internalize",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n simpa [models_iff] using sigma_one_complete (V := V) (T := T) (σ := provabilityPred T σ) (by simp)",
"n_chars": 10... | [
{
"name": "sigma_one_complete",
"text": "/-- Hilbert–Bernays provability condition D3 -/\ntheorem sigma_one_complete {σ : ArithmeticSentence} (hσ : Hierarchy 𝚺 1 σ) :\n V↓[ℒₒᵣ] ⊧ σ → Provable T (⌜σ⌝ : V) := fun h ↦ by\n simpa [tprovable_iff_provable]\n using! Bootstrapping.Arithmetic.sigma_one_prova... | [
{
"name": "provable_internalize",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 5,
"n_lines": 5,
"n_chars": 245,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
"n_automation": 2,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"max_nes... | 5 | module
public import Foundation.FirstOrder.Bootstrapping.DerivabilityCondition.PeanoMinus
@[expose] public section
/-!
# Hilbert-Bernays-Löb derivability condition $\mathbf{D3}$ and formalized $\Sigma_1$-completeness
-/
namespace LO.FirstOrder.Arithmetic.Bootstrapping.Arithmetic
open Classical
open LO.Entailment L... | module
public import Foundation.FirstOrder.Bootstrapping.DerivabilityCondition.PeanoMinus
@[expose] public section
/-!
# Hilbert-Bernays-Löb derivability condition $\mathbf{D3}$ and formalized $\Sigma_1$-completeness
-/
namespace LO.FirstOrder.Arithmetic.Bootstrapping.Arithmetic
open Classical
open LO.Entailment L... | @@ -149,8 +149,15 @@
bold_sigma_one_complete T hσ (by simpa [models_iff] using h)
simpa using this
+/-- Hilbert–Bernays provability condition D3 -/
+theorem sigma_one_complete {σ : ArithmeticSentence} (hσ : Hierarchy 𝚺 1 σ) :
+ V↓[ℒₒᵣ] ⊧ σ → Provable T (⌜σ⌝ : V) := fun h ↦ by
+ simpa [tprovable_iff_prova... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_9dd7df63fb59_0 | 9d1cae97412efb14 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Q/Basic.lean | Basic | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 17 | 1 | [
{
"theorem_name": "exists_succ_of_ne_zero",
"depth": 1,
"n_commands": 0,
"n_lines": 3,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n have := zero_or_succ (a := a);\n tauto;",
"n_chars": 45,
"n_subproofs": 1,
"n_tactics": 5,
"cycloma... | [
{
"name": "zero_or_succ",
"text": "lemma zero_or_succ (a : M) : a = 0 ∨ ∃ b : M, a = b + 1 := by\n have := by simpa [models_iff] using Theory.models M _ RobinsonQ.zeroOrSucc\n exact this a\n\n",
"fan_in": 1,
"n_lines": 5,
"n_chars": 159,
"n_subproofs": 1,
"n_tactics": 3,
"cyclomati... | [
{
"name": "exists_succ_of_ne_zero",
"fan_in": 3,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 5,
"n_chars": 124,
"n_subproofs": 1,
"n_tactics": 5,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
"max_... | 1 | module
public import Foundation.FirstOrder.Arithmetic.R0.Basic
public import Mathlib.Data.ENat.Basic
@[expose] public section
/-!
# Robinson's theory $\mathsf{Q}$
-/
noncomputable section
namespace LO.FirstOrder.Arithmetic
inductive RobinsonQ : ArithmeticTheory
| equal : ∀ φ ∈ 𝗘𝗤 ℒₒᵣ, RobinsonQ φ
| succNeZer... | module
public import Foundation.FirstOrder.Arithmetic.R0.Basic
public import Mathlib.Data.ENat.Basic
@[expose] public section
/-!
# Robinson's theory $\mathsf{Q}$
-/
noncomputable section
namespace LO.FirstOrder.Arithmetic
inductive RobinsonQ : ArithmeticTheory
| equal : ∀ φ ∈ 𝗘𝗤 ℒₒᵣ, RobinsonQ φ
| succNeZer... | @@ -79,8 +79,14 @@
protected lemma mul_succ : ∀ a b : M, a * (b + 1) = a * b + a := by
simpa [models_iff] using Theory.models M _ RobinsonQ.mulSucc
-lemma exists_succ_of_ne_zero {a : M} (ha : a ≠ 0) : ∃ b : M, a = b + 1 := sorry
+lemma zero_or_succ (a : M) : a = 0 ∨ ∃ b : M, a = b + 1 := by
+ have := by simpa [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_9dd7df63fb59_1 | 6eb669721d540302 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Q/Basic.lean | Basic | 1 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 17 | 3 | [
{
"theorem_name": "exists_succ_of_ne_zero'",
"depth": 1,
"n_commands": 0,
"n_lines": 3,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n obtain ⟨b, rfl⟩ := exists_succ_of_ne_zero ha;\n use b;",
"n_chars": 64,
"n_subproofs": 1,
"n_tactics":... | [
{
"name": "exists_succ_of_ne_zero",
"text": "lemma exists_succ_of_ne_zero {a : M} (ha : a ≠ 0) : ∃ b : M, a = b + 1 := by\n have := zero_or_succ (a := a);\n tauto;\n\n",
"fan_in": 3,
"n_lines": 5,
"n_chars": 124,
"n_subproofs": 1,
"n_tactics": 5,
"cyclomatic": 1,
"n_automation"... | [
{
"name": "exists_succ_of_ne_zero'",
"fan_in": 2,
"n_deps_direct": 1,
"n_deps_transitive": 2,
"n_lines": 5,
"n_chars": 144,
"n_subproofs": 1,
"n_tactics": 5,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 2,
"automation_only": false,
"max... | 2 | module
public import Foundation.FirstOrder.Arithmetic.R0.Basic
public import Mathlib.Data.ENat.Basic
@[expose] public section
/-!
# Robinson's theory $\mathsf{Q}$
-/
noncomputable section
namespace LO.FirstOrder.Arithmetic
inductive RobinsonQ : ArithmeticTheory
| equal : ∀ φ ∈ 𝗘𝗤 ℒₒᵣ, RobinsonQ φ
| succNeZer... | module
public import Foundation.FirstOrder.Arithmetic.R0.Basic
public import Mathlib.Data.ENat.Basic
@[expose] public section
/-!
# Robinson's theory $\mathsf{Q}$
-/
noncomputable section
namespace LO.FirstOrder.Arithmetic
inductive RobinsonQ : ArithmeticTheory
| equal : ∀ φ ∈ 𝗘𝗤 ℒₒᵣ, RobinsonQ φ
| succNeZer... | @@ -83,8 +83,14 @@
have := by simpa [models_iff] using Theory.models M _ RobinsonQ.zeroOrSucc
exact this a
-lemma exists_succ_of_ne_zero' {a : M} (ha : a ≠ 0) : ∃ b : M, b + 1 = a := sorry
+lemma exists_succ_of_ne_zero {a : M} (ha : a ≠ 0) : ∃ b : M, a = b + 1 := by
+ have := zero_or_succ (a := a);
+ tauto;
... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_9dd7df63fb59_2 | 8e17abd2b11fd673 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Q/Basic.lean | Basic | 2 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 17 | 7 | [
{
"theorem_name": "zero_add_one",
"depth": 1,
"n_commands": 0,
"n_lines": 11,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n obtain ⟨a, ha⟩ := exists_succ_of_ne_zero' (M := M) one_ne_zero;\n convert ha;\n by_contra;\n obtain ⟨b, rfl⟩ := exists_suc... | [
{
"name": "add_succ",
"text": "protected lemma add_succ (a b : M) : a + (b + 1) = a + b + 1 := by\n have := by simpa [models_iff] using Theory.models M _ RobinsonQ.addSucc\n exact this a b\n\n",
"fan_in": 7,
"n_lines": 5,
"n_chars": 159,
"n_subproofs": 1,
"n_tactics": 3,
"cyclomati... | [
{
"name": "zero_add_one",
"fan_in": 1,
"n_deps_direct": 5,
"n_deps_transitive": 8,
"n_lines": 14,
"n_chars": 539,
"n_subproofs": 2,
"n_tactics": 20,
"cyclomatic": 1,
"n_automation": 2,
"n_rewrites": 3,
"n_structural": 5,
"automation_only": false,
"max_nesting"... | 5 | module
public import Foundation.FirstOrder.Arithmetic.R0.Basic
public import Mathlib.Data.ENat.Basic
@[expose] public section
/-!
# Robinson's theory $\mathsf{Q}$
-/
noncomputable section
namespace LO.FirstOrder.Arithmetic
inductive RobinsonQ : ArithmeticTheory
| equal : ∀ φ ∈ 𝗘𝗤 ℒₒᵣ, RobinsonQ φ
| succNeZer... | module
public import Foundation.FirstOrder.Arithmetic.R0.Basic
public import Mathlib.Data.ENat.Basic
@[expose] public section
/-!
# Robinson's theory $\mathsf{Q}$
-/
noncomputable section
namespace LO.FirstOrder.Arithmetic
inductive RobinsonQ : ArithmeticTheory
| equal : ∀ φ ∈ 𝗘𝗤 ℒₒᵣ, RobinsonQ φ
| succNeZer... | @@ -68,6 +68,10 @@
have := by simpa [models_iff] using Theory.models M _ RobinsonQ.addZero
exact this a
+protected lemma add_succ (a b : M) : a + (b + 1) = a + b + 1 := by
+ have := by simpa [models_iff] using Theory.models M _ RobinsonQ.addSucc
+ exact this a b
+
@[simp] protected lemma mul_zero (a : M) : 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_9dd7df63fb59_3 | aebba467b7a2c2c7 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Q/Basic.lean | Basic | 3 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 17 | 5 | [
{
"theorem_name": "one_mul_one",
"depth": 1,
"n_commands": 0,
"n_lines": 4,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " calc\n (1 : M) * 1 = 1 * (0 + 1) := by simp\n _ = 1 * 0 + 1 := by rw [Arithmetic.mul_succ]\n _ = 1 ... | [
{
"name": "mul_succ",
"text": "protected lemma mul_succ : ∀ a b : M, a * (b + 1) = a * b + a := by\n simpa [models_iff] using Theory.models M _ RobinsonQ.mulSucc\n\n",
"fan_in": 5,
"n_lines": 4,
"n_chars": 134,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
"n_automation": 1... | [
{
"name": "one_mul_one",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 7,
"n_chars": 190,
"n_subproofs": 0,
"n_tactics": 4,
"cyclomatic": 1,
"n_automation": 2,
"n_rewrites": 1,
"n_structural": 1,
"automation_only": false,
"max_nesting": 2... | 1 | module
public import Foundation.FirstOrder.Arithmetic.R0.Basic
public import Mathlib.Data.ENat.Basic
@[expose] public section
/-!
# Robinson's theory $\mathsf{Q}$
-/
noncomputable section
namespace LO.FirstOrder.Arithmetic
inductive RobinsonQ : ArithmeticTheory
| equal : ∀ φ ∈ 𝗘𝗤 ℒₒᵣ, RobinsonQ φ
| succNeZer... | module
public import Foundation.FirstOrder.Arithmetic.R0.Basic
public import Mathlib.Data.ENat.Basic
@[expose] public section
/-!
# Robinson's theory $\mathsf{Q}$
-/
noncomputable section
namespace LO.FirstOrder.Arithmetic
inductive RobinsonQ : ArithmeticTheory
| equal : ∀ φ ∈ 𝗘𝗤 ℒₒᵣ, RobinsonQ φ
| succNeZer... | @@ -76,6 +76,9 @@
have := by simpa [models_iff] using Theory.models M _ RobinsonQ.mulZero
exact this a
+protected lemma mul_succ : ∀ a b : M, a * (b + 1) = a * b + a := by
+ simpa [models_iff] using Theory.models M _ RobinsonQ.mulSucc
+
lemma zero_or_succ (a : M) : a = 0 ∨ ∃ b : M, a = b + 1 := by
have := ... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_9dd7df63fb59_4 | bad2d4c9c7835d7d | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Q/Basic.lean | Basic | 4 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 17 | 5 | [
{
"theorem_name": "not_le_zero",
"depth": 1,
"n_commands": 0,
"n_lines": 7,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n apply Arithmetic.lt_def.not.mpr;\n push Not;\n intro b;\n calc\n a + (b + 1) = (a + b) + 1 := Arithmetic.add_succ _ _\n ... | [
{
"name": "lt_def",
"text": "protected lemma lt_def {a b : M} : a < b ↔ ∃ c : M, a + (c + 1) = b := by\n have := by simpa [models_iff] using Theory.models M _ RobinsonQ.ltDef\n exact this a b\n\n",
"fan_in": 5,
"n_lines": 5,
"n_chars": 168,
"n_subproofs": 1,
"n_tactics": 3,
"cyclom... | [
{
"name": "lt_of_not_zero",
"fan_in": 1,
"n_deps_direct": 2,
"n_deps_transitive": 3,
"n_lines": 6,
"n_chars": 166,
"n_subproofs": 1,
"n_tactics": 7,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 3,
"automation_only": false,
"max_nesting"... | 3 | module
public import Foundation.FirstOrder.Arithmetic.R0.Basic
public import Mathlib.Data.ENat.Basic
@[expose] public section
/-!
# Robinson's theory $\mathsf{Q}$
-/
noncomputable section
namespace LO.FirstOrder.Arithmetic
inductive RobinsonQ : ArithmeticTheory
| equal : ∀ φ ∈ 𝗘𝗤 ℒₒᵣ, RobinsonQ φ
| succNeZer... | module
public import Foundation.FirstOrder.Arithmetic.R0.Basic
public import Mathlib.Data.ENat.Basic
@[expose] public section
/-!
# Robinson's theory $\mathsf{Q}$
-/
noncomputable section
namespace LO.FirstOrder.Arithmetic
inductive RobinsonQ : ArithmeticTheory
| equal : ∀ φ ∈ 𝗘𝗤 ℒₒᵣ, RobinsonQ φ
| succNeZer... | @@ -91,6 +91,10 @@
obtain ⟨b, rfl⟩ := exists_succ_of_ne_zero ha;
use b;
+protected lemma lt_def {a b : M} : a < b ↔ ∃ c : M, a + (c + 1) = b := by
+ have := by simpa [models_iff] using Theory.models M _ RobinsonQ.ltDef
+ exact this a b
+
@[simp]
lemma one_ne_zero : (1 : M) ≠ 0 := by
by_contra h;
@@ -144,... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_9dd7df63fb59_5 | 5ee7b959b32c2590 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Q/Basic.lean | Basic | 5 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 17 | 5 | [
{
"theorem_name": "one_mul_one",
"depth": 1,
"n_commands": 0,
"n_lines": 4,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " calc\n (1 : M) * 1 = 1 * (0 + 1) := by simp\n _ = 1 * 0 + 1 := by rw [Arithmetic.mul_succ]\n _ = 1 ... | [
{
"name": "mul_succ",
"text": "protected lemma mul_succ : ∀ a b : M, a * (b + 1) = a * b + a := by\n simpa [models_iff] using Theory.models M _ RobinsonQ.mulSucc\n\n",
"fan_in": 5,
"n_lines": 4,
"n_chars": 134,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
"n_automation": 1... | [
{
"name": "numeral_zero_mul",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 10,
"n_chars": 377,
"n_subproofs": 0,
"n_tactics": 8,
"cyclomatic": 2,
"n_automation": 4,
"n_rewrites": 1,
"n_structural": 1,
"automation_only": false,
"max_nesti... | 1 | module
public import Foundation.FirstOrder.Arithmetic.R0.Basic
public import Mathlib.Data.ENat.Basic
@[expose] public section
/-!
# Robinson's theory $\mathsf{Q}$
-/
noncomputable section
namespace LO.FirstOrder.Arithmetic
inductive RobinsonQ : ArithmeticTheory
| equal : ∀ φ ∈ 𝗘𝗤 ℒₒᵣ, RobinsonQ φ
| succNeZer... | module
public import Foundation.FirstOrder.Arithmetic.R0.Basic
public import Mathlib.Data.ENat.Basic
@[expose] public section
/-!
# Robinson's theory $\mathsf{Q}$
-/
noncomputable section
namespace LO.FirstOrder.Arithmetic
inductive RobinsonQ : ArithmeticTheory
| equal : ∀ φ ∈ 𝗘𝗤 ℒₒᵣ, RobinsonQ φ
| succNeZer... | @@ -76,6 +76,9 @@
have := by simpa [models_iff] using Theory.models M _ RobinsonQ.mulZero
exact this a
+protected lemma mul_succ : ∀ a b : M, a * (b + 1) = a * b + a := by
+ simpa [models_iff] using Theory.models M _ RobinsonQ.mulSucc
+
lemma zero_or_succ (a : M) : a = 0 ∨ ∃ b : M, a = b + 1 := by
have := ... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_9dd7df63fb59_6 | cd909d1be34c6401 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Q/Basic.lean | Basic | 6 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 17 | 4 | [
{
"theorem_name": "numeral_add",
"depth": 1,
"n_commands": 0,
"n_lines": 8,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n match m with\n | 0 => simp\n | 1 => simp [numeral_add_one]\n | m + 2 => calc\n (numeral n : M) + (numeral (m + 1)... | [
{
"name": "numeral_add_one",
"text": "lemma numeral_add_one (n : ℕ) : (numeral n : M) + 1 = numeral (n + 1) := by\n match n with\n | 0 => simp;\n | n + 1 => rfl\n\n",
"fan_in": 4,
"n_lines": 6,
"n_chars": 130,
"n_subproofs": 0,
"n_tactics": 5,
"cyclomatic": 2,
"n_automatio... | [
{
"name": "exists_numeral_of_ne_zero",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 13,
"n_chars": 515,
"n_subproofs": 1,
"n_tactics": 17,
"cyclomatic": 2,
"n_automation": 3,
"n_rewrites": 1,
"n_structural": 4,
"automation_only": false,
... | 1 | module
public import Foundation.FirstOrder.Arithmetic.R0.Basic
public import Mathlib.Data.ENat.Basic
@[expose] public section
/-!
# Robinson's theory $\mathsf{Q}$
-/
noncomputable section
namespace LO.FirstOrder.Arithmetic
inductive RobinsonQ : ArithmeticTheory
| equal : ∀ φ ∈ 𝗘𝗤 ℒₒᵣ, RobinsonQ φ
| succNeZer... | module
public import Foundation.FirstOrder.Arithmetic.R0.Basic
public import Mathlib.Data.ENat.Basic
@[expose] public section
/-!
# Robinson's theory $\mathsf{Q}$
-/
noncomputable section
namespace LO.FirstOrder.Arithmetic
inductive RobinsonQ : ArithmeticTheory
| equal : ∀ φ ∈ 𝗘𝗤 ℒₒᵣ, RobinsonQ φ
| succNeZer... | @@ -182,10 +182,15 @@
| 1 => simp
| n + 2 => simp [ORingStructure.numeral, Arithmetic.add_succ, numeral_zero_add (n + 1)]
+lemma numeral_add_one (n : ℕ) : (numeral n : M) + 1 = numeral (n + 1) := by
+ match n with
+ | 0 => simp;
+ | n + 1 => rfl
+
lemma numeral_add (n m : ℕ) : (numeral n : M) + nume... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_8afb1b320ec1_0 | 658e7bb347c95ff0 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Logic/Basic.lean | Basic | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 2 | 2 | [
{
"theorem_name": "weakerThan_of_subset",
"depth": 1,
"n_commands": 0,
"n_lines": 3,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n apply weakerThan_of_provable;\n grind;",
"n_chars": 44,
"n_subproofs": 0,
"n_tactics": 5,
"cyclomatic... | [
{
"name": "weakerThan_of_provable",
"text": "lemma weakerThan_of_provable (h : ∀ φ, L₁ ⊢ φ → L₂ ⊢ φ) : L₁ ⪯ L₂ := by\n constructor;\n simpa [Entailment.theory, forall_exists_index];\n\n",
"fan_in": 2,
"n_lines": 5,
"n_chars": 159,
"n_subproofs": 0,
"n_tactics": 5,
"cyclomatic": 2,
... | [
{
"name": "weakerThan_of_subset",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 5,
"n_chars": 111,
"n_subproofs": 0,
"n_tactics": 5,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": false,
"max_ne... | 1 | module
public import Foundation.Modal.Formula.Basic
public import Foundation.Modal.Entailment.Basic
public import Foundation.Meta.ClProver
@[expose] public section
namespace LO.Modal
open LO.Entailment
open Entailment
variable {α : Type*}
abbrev Logic (α) := Set (Modal.Formula α)
instance : Entailment (Logic α) ... | module
public import Foundation.Modal.Formula.Basic
public import Foundation.Modal.Entailment.Basic
public import Foundation.Meta.ClProver
@[expose] public section
namespace LO.Modal
open LO.Entailment
open Entailment
variable {α : Type*}
abbrev Logic (α) := Set (Modal.Formula α)
instance : Entailment (Logic α) ... | @@ -89,8 +89,14 @@
end
-lemma weakerThan_of_subset (h : L₁ ⊆ L₂) : L₁ ⪯ L₂ := sorry
+lemma weakerThan_of_provable (h : ∀ φ, L₁ ⊢ φ → L₂ ⊢ φ) : L₁ ⪯ L₂ := by
+ constructor;
+ simpa [Entailment.theory, forall_exists_index];
+lemma weakerThan_of_subset (h : L₁ ⊆ L₂) : L₁ ⪯ L₂ := by
+ apply weakerThan_of_provable... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_8afb1b320ec1_1 | 7ce36fbec08e1167 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Logic/Basic.lean | Basic | 1 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 2 | 2 | [
{
"theorem_name": "weakerThan_of_subset",
"depth": 1,
"n_commands": 0,
"n_lines": 3,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n apply weakerThan_of_provable;\n grind;",
"n_chars": 44,
"n_subproofs": 0,
"n_tactics": 5,
"cyclomatic... | [
{
"name": "weakerThan_of_provable",
"text": "lemma weakerThan_of_provable (h : ∀ φ, L₁ ⊢ φ → L₂ ⊢ φ) : L₁ ⪯ L₂ := by\n constructor;\n simpa [Entailment.theory, forall_exists_index];\n\n",
"fan_in": 2,
"n_lines": 5,
"n_chars": 159,
"n_subproofs": 0,
"n_tactics": 5,
"cyclomatic": 2,
... | [
{
"name": "equiv_of_provable",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 7,
"n_chars": 187,
"n_subproofs": 0,
"n_tactics": 9,
"cyclomatic": 3,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 3,
"automation_only": false,
"max_nesti... | 1 | module
public import Foundation.Modal.Formula.Basic
public import Foundation.Modal.Entailment.Basic
public import Foundation.Meta.ClProver
@[expose] public section
namespace LO.Modal
open LO.Entailment
open Entailment
variable {α : Type*}
abbrev Logic (α) := Set (Modal.Formula α)
instance : Entailment (Logic α) ... | module
public import Foundation.Modal.Formula.Basic
public import Foundation.Modal.Entailment.Basic
public import Foundation.Meta.ClProver
@[expose] public section
namespace LO.Modal
open LO.Entailment
open Entailment
variable {α : Type*}
abbrev Logic (α) := Set (Modal.Formula α)
instance : Entailment (Logic α) ... | @@ -89,12 +89,19 @@
end
-lemma weakerThan_of_subset (h : L₁ ⊆ L₂) : L₁ ⪯ L₂ := sorry
+lemma weakerThan_of_provable (h : ∀ φ, L₁ ⊢ φ → L₂ ⊢ φ) : L₁ ⪯ L₂ := by
+ constructor;
+ simpa [Entailment.theory, forall_exists_index];
+lemma weakerThan_of_subset (h : L₁ ⊆ L₂) : L₁ ⪯ L₂ := by
+ apply weakerThan_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_8afb1b320ec1_2 | 8bfe4a7c692046d3 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Logic/Basic.lean | Basic | 2 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 2 | 1 | [
{
"theorem_name": "fconj_subst",
"depth": 1,
"n_commands": 0,
"n_lines": 6,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n apply C!_trans ?_ $ lconj_subst (L := L) (X := X.toList) (s := s);\n apply right_Conj₂!_intro;\n intro φ hφ;\n apply left_Fc... | [
{
"name": "lconj_subst",
"text": "lemma lconj_subst {X : List (Formula α)} {s : Substitution α} : L ⊢ (X.map (·⟦s⟧)).conj₂ 🡒 X.conj₂⟦s⟧ := by\n induction X using List.induction_with_singleton with\n | hnil => simp;\n | hsingle => simp;\n | hcons φ X hφ ih =>\n suffices L ⊢ φ⟦s⟧ ⋏ ⋀(X.map (·⟦s⟧)) 🡒 ... | [
{
"name": "fconj_subst",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 8,
"n_chars": 281,
"n_subproofs": 0,
"n_tactics": 11,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 4,
"automation_only": false,
"max_nesting": ... | 1 | module
public import Foundation.Modal.Formula.Basic
public import Foundation.Modal.Entailment.Basic
public import Foundation.Meta.ClProver
@[expose] public section
namespace LO.Modal
open LO.Entailment
open Entailment
variable {α : Type*}
abbrev Logic (α) := Set (Modal.Formula α)
instance : Entailment (Logic α) ... | module
public import Foundation.Modal.Formula.Basic
public import Foundation.Modal.Entailment.Basic
public import Foundation.Meta.ClProver
@[expose] public section
namespace LO.Modal
open LO.Entailment
open Entailment
variable {α : Type*}
abbrev Logic (α) := Set (Modal.Formula α)
instance : Entailment (Logic α) ... | @@ -122,8 +122,22 @@
variable [DecidableEq α] [Entailment.Cl L]
-lemma fconj_subst {X : Finset (Formula α)} {s : Substitution α} : L ⊢ (X.image (·⟦s⟧)).conj 🡒 X.conj⟦s⟧ := sorry
+lemma lconj_subst {X : List (Formula α)} {s : Substitution α} : L ⊢ (X.map (·⟦s⟧)).conj₂ 🡒 X.conj₂⟦s⟧ := by
+ induction X using List.... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_cfd7804baf2b_0 | db498fb2eef54e85 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Boxdot/K4_S4.lean | K4_S4 | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 3 | 1 | [
{
"theorem_name": "provable_S4_of_provable_boxdotTranslated_K4",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n exact (K!_right provable_S4_iff_boxdotTranslated) ⨀ (WeakerThan.pbl h);",
"n_chars": 79,
... | [
{
"name": "provable_S4_iff_boxdotTranslated",
"text": "lemma provable_S4_iff_boxdotTranslated : Modal.S4 ⊢ φ 🡘 φᵇ := by\n induction φ with\n | hbox φ ihp => exact E!_trans (box_iff! ihp) iff_box_boxdot!;\n | himp φ ψ ihp ihq => exact ECC!_of_E!_of_E! ihp ihq;\n | _ => exact E!_id;\n\n",
"fan_in": 1... | [
{
"name": "provable_S4_of_provable_boxdotTranslated_K4",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 4,
"n_chars": 176,
"n_subproofs": 0,
"n_tactics": 3,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 1,
"automation_on... | 1 | module
public import Foundation.Modal.Boxdot.Basic
public import Foundation.Modal.Entailment.S4
public import Foundation.Modal.Kripke.Logic.S4
@[expose] public section
namespace LO.Modal
open LO.Entailment LO.Entailment.FiniteContext LO.Modal.Entailment
open Formula
lemma provable_S4_of_provable_boxdotTranslated_K... | module
public import Foundation.Modal.Boxdot.Basic
public import Foundation.Modal.Entailment.S4
public import Foundation.Modal.Kripke.Logic.S4
@[expose] public section
namespace LO.Modal
open LO.Entailment LO.Entailment.FiniteContext LO.Modal.Entailment
open Formula
lemma provable_S4_iff_boxdotTranslated : Modal.S... | @@ -11,8 +11,15 @@
open LO.Entailment LO.Entailment.FiniteContext LO.Modal.Entailment
open Formula
-lemma provable_S4_of_provable_boxdotTranslated_K4 (h : Modal.K4 ⊢ φᵇ) : Modal.S4 ⊢ φ := sorry
+lemma provable_S4_iff_boxdotTranslated : Modal.S4 ⊢ φ 🡘 φᵇ := by
+ induction φ with
+ | hbox φ ihp => exact E!_trans (... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_9d68f5a044f6_0 | f58837a036678d53 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Incompleteness/Dense.lean | Dense | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 2 | 1 | [
{
"theorem_name": "consistent_cons_of_unprovable",
"depth": 1,
"n_commands": 0,
"n_lines": 4,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n apply consistent_cons_of_unprovable_neg;\n contrapose! h;\n cl_prover [h];",
"n_chars": 80,
"n_subp... | [
{
"name": "consistent_cons_of_unprovable_neg",
"text": "lemma consistent_cons_of_unprovable_neg (h : 𝓢 ⊬ ∼φ) : Consistent (adjoin φ 𝓢) := by\n apply consistent_iff_exists_unprovable.mpr;\n use ⊥;\n apply deduction_iff.not.mpr;\n contrapose! h;\n cl_prover [h];\n\n",
"fan_in": 1,
"n_lines": 8,... | [
{
"name": "consistent_cons_of_unprovable",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 6,
"n_chars": 173,
"n_subproofs": 0,
"n_tactics": 7,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": false,
... | 1 | module
public import Foundation.FirstOrder.Incompleteness.GödelRosser
public import Foundation.Logic.LindenbaumAlgebra
@[expose] public section
namespace LO
namespace Entailment
variable {F S : Type*} [DecidableEq F] [LogicalConnective F] [Entailment S F] [AdjunctiveSet F S] [Deduction S]
{𝓢 : S} [Entailm... | module
public import Foundation.FirstOrder.Incompleteness.GödelRosser
public import Foundation.Logic.LindenbaumAlgebra
@[expose] public section
namespace LO
namespace Entailment
variable {F S : Type*} [DecidableEq F] [LogicalConnective F] [Entailment S F] [AdjunctiveSet F S] [Deduction S]
{𝓢 : S} [Entailm... | @@ -11,8 +11,18 @@
variable {F S : Type*} [DecidableEq F] [LogicalConnective F] [Entailment S F] [AdjunctiveSet F S] [Deduction S]
{𝓢 : S} [Entailment.Cl 𝓢]
-lemma consistent_cons_of_unprovable (h : 𝓢 ⊬ φ) : Consistent (adjoin (∼φ) 𝓢) := sorry
+lemma consistent_cons_of_unprovable_neg (h : 𝓢 ⊬ ∼φ) : Co... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_9d68f5a044f6_1 | 757a678a38920ddd | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Incompleteness/Dense.lean | Dense | 1 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 2 | 1 | [
{
"theorem_name": "FirstOrder.Arithmetic.dense",
"depth": 1,
"n_commands": 0,
"n_lines": 6,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " fun h ↦ by\n refine LindenbaumAlgebra.dense_of_finite_extend_incomplete T ?_ h\n intro σ con\n have : 𝗜𝚺₁ ⪯ T :=... | [
{
"name": "dense_of_finite_extend_incomplete",
"text": "lemma dense_of_finite_extend_incomplete\n (hE : ∀ φ : F, Consistent (adjoin φ 𝓢) → Incomplete (adjoin φ 𝓢))\n (h : φ < ψ) : ∃ ξ : LindenbaumAlgebra 𝓢, φ < ξ ∧ ξ < ψ := by\n obtain ⟨φ, rfl⟩ := Quotient.exists_rep φ;\n obtain ⟨ψ, rfl⟩ := Quoti... | [
{
"name": "FirstOrder.Arithmetic.dense",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 10,
"n_chars": 541,
"n_subproofs": 2,
"n_tactics": 6,
"cyclomatic": 1,
"n_automation": 2,
"n_rewrites": 0,
"n_structural": 2,
"automation_only": false,
... | 1 | module
public import Foundation.FirstOrder.Incompleteness.GödelRosser
public import Foundation.Logic.LindenbaumAlgebra
@[expose] public section
namespace LO
namespace Entailment
variable {F S : Type*} [DecidableEq F] [LogicalConnective F] [Entailment S F] [AdjunctiveSet F S] [Deduction S]
{𝓢 : S} [Entailm... | module
public import Foundation.FirstOrder.Incompleteness.GödelRosser
public import Foundation.Logic.LindenbaumAlgebra
@[expose] public section
namespace LO
namespace Entailment
variable {F S : Type*} [DecidableEq F] [LogicalConnective F] [Entailment S F] [AdjunctiveSet F S] [Deduction S]
{𝓢 : S} [Entailm... | @@ -20,13 +20,48 @@
variable {F S : Type*} [DecidableEq F] [LogicalConnective F] [Entailment S F] [AdjunctiveSet F S] [Deduction S]
(𝓢 : S) [Entailment.Cl 𝓢]
+lemma dense_of_finite_extend_incomplete
+ (hE : ∀ φ : F, Consistent (adjoin φ 𝓢) → Incomplete (adjoin φ 𝓢))
+ (h : φ < ψ) : ∃ ξ : Lindenba... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_4d9dc55be850_0 | d113a004b93c01e0 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Entailment/GL.lean | GL | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 3 | 1 | [
{
"theorem_name": "imply_box_box_of_imply_boxdot_plain!",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n exact imply_box_box_of_imply_boxdot_axiomT! $ imply_boxdot_axiomT_of_imply_boxdot_boxdot! $ imply_boxd... | [
{
"name": "imply_boxdot_boxdot_of_imply_boxdot_plain!",
"text": "lemma imply_boxdot_boxdot_of_imply_boxdot_plain! (h : 𝓢 ⊢ ⊡φ 🡒 ψ) : 𝓢 ⊢ ⊡φ 🡒 ⊡ψ := ⟨imply_boxdot_boxdot_of_imply_boxdot_plain h.some⟩\n\n\n",
"fan_in": 1,
"n_lines": 4,
"n_chars": 166,
"n_subproofs": 0,
"n_tactics": 1,
... | [
{
"name": "imply_box_box_of_imply_boxdot_plain!",
"fan_in": 0,
"n_deps_direct": 3,
"n_deps_transitive": 3,
"n_lines": 4,
"n_chars": 246,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": fa... | 3 | module
public import Foundation.Modal.Entailment.K4
@[expose] public section
namespace LO.Modal.Entailment
open LO.Entailment LO.Entailment.FiniteContext LO.Modal.Entailment
variable {S F : Type*} [BasicModalLogicalConnective F] [DecidableEq F] [Entailment S F]
variable {𝓢 : S} [Entailment.GL 𝓢]
def gödel2 : 𝓢... | module
public import Foundation.Modal.Entailment.K4
@[expose] public section
namespace LO.Modal.Entailment
open LO.Entailment LO.Entailment.FiniteContext LO.Modal.Entailment
variable {S F : Type*} [BasicModalLogicalConnective F] [DecidableEq F] [Entailment S F]
variable {𝓢 : S} [Entailment.GL 𝓢]
def gödel2 : 𝓢... | @@ -81,6 +81,8 @@
have : 𝓢 ⊢! □φ 🡒 □ψ := C_trans imply_Box_BoxBoxdot this;
have : 𝓢 ⊢! ⊡φ 🡒 □ψ := C_trans boxdotBox this;
exact right_K_intro h this;
+lemma imply_boxdot_boxdot_of_imply_boxdot_plain! (h : 𝓢 ⊢ ⊡φ 🡒 ψ) : 𝓢 ⊢ ⊡φ 🡒 ⊡ψ := ⟨imply_boxdot_boxdot_of_imply_boxdot_plain h.some⟩
+
def imply_boxdo... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_8cb510154e98_0 | b4f156d6b6be6ce7 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Propositional/Entailment/Cl/Basic.lean | Basic | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 2 | 1 | [
{
"theorem_name": "not_imply_prem''!",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": "\n deduct'! $ (contra! $ CCAN!) ⨀ (NA!_of_KNN! $ K!_intro (dni'! $ of'! hpq ⨀ (by_axm!)) (of'! hpnr ⨀ (by_axm!)))",
"n_char... | [
{
"name": "CCAN!",
"text": "lemma CCAN! : 𝓢 ⊢ (φ 🡒 ψ) 🡒 ∼φ ⋎ ψ := ⟨CCAN⟩\n\n\n",
"fan_in": 1,
"n_lines": 4,
"n_chars": 70,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
"max_n... | [
{
"name": "not_imply_prem''!",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 4,
"n_chars": 242,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
"max_nesti... | 1 | module
public import Foundation.Propositional.Entailment.AxiomDNE
public import Foundation.Propositional.Entailment.AxiomElimContra
public import Foundation.Propositional.Entailment.AxiomLEM
public import Foundation.Propositional.Entailment.LC
public import Foundation.Propositional.Entailment.AxiomPeirce
@[expose] pub... | module
public import Foundation.Propositional.Entailment.AxiomDNE
public import Foundation.Propositional.Entailment.AxiomElimContra
public import Foundation.Propositional.Entailment.AxiomLEM
public import Foundation.Propositional.Entailment.LC
public import Foundation.Propositional.Entailment.AxiomPeirce
@[expose] pub... | @@ -70,6 +70,8 @@
apply deduct';
apply AN_of_C;
exact FiniteContext.byAxm;
+lemma CCAN! : 𝓢 ⊢ (φ 🡒 ψ) 🡒 ∼φ ⋎ ψ := ⟨CCAN⟩
+
instance : HasAxiomEFQ 𝓢 where
efq {φ} := by
apply C_of_CNN;
@@ -85,7 +87,8 @@
instance : HasAxiomLEM 𝓢 := ⟨A_of_ANNNN $ AN_of_C dni⟩
-lemma not_imply_prem''! (hpq : 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_8cb510154e98_1 | 2eb6d268d00651ce | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Propositional/Entailment/Cl/Basic.lean | Basic | 1 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 2 | 1 | [
{
"theorem_name": "unprovable_iff_consistent_adjoin",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n simpa using provable_iff_inconsistent_adjoin.not",
"n_chars": 54,
"n_subproofs": 0,
"n_tactics... | [
{
"name": "provable_iff_inconsistent_adjoin",
"text": "lemma provable_iff_inconsistent_adjoin {φ : F} :\n 𝓢 ⊢ φ ↔ Inconsistent (adjoin (∼φ) 𝓢) := by\n constructor\n · intro h\n apply inconsistent_of_provable_of_unprovable (φ := φ)\n · exact Axiomatized.to_adjoin h\n · exact Axiomatized.adjoi... | [
{
"name": "unprovable_iff_consistent_adjoin",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 5,
"n_chars": 161,
"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.Propositional.Entailment.AxiomDNE
public import Foundation.Propositional.Entailment.AxiomElimContra
public import Foundation.Propositional.Entailment.AxiomLEM
public import Foundation.Propositional.Entailment.LC
public import Foundation.Propositional.Entailment.AxiomPeirce
@[expose] pub... | module
public import Foundation.Propositional.Entailment.AxiomDNE
public import Foundation.Propositional.Entailment.AxiomElimContra
public import Foundation.Propositional.Entailment.AxiomLEM
public import Foundation.Propositional.Entailment.LC
public import Foundation.Propositional.Entailment.AxiomPeirce
@[expose] pub... | @@ -119,8 +119,20 @@
variable [AdjunctiveSet F S] [Axiomatized S] [Deduction S] [∀ 𝓢 : S, Entailment.Cl 𝓢]
+lemma provable_iff_inconsistent_adjoin {φ : F} :
+ 𝓢 ⊢ φ ↔ Inconsistent (adjoin (∼φ) 𝓢) := by
+ constructor
+ · intro h
+ apply inconsistent_of_provable_of_unprovable (φ := φ)
+ · exact Axioma... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_403efce48305_0 | af4f07b6aac28964 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Kripke/Logic/KTMk.lean | KTMk | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 4 | 1 | [
{
"theorem_name": "model_infinitity_of_not_validate_axiomFour",
"depth": 1,
"n_commands": 0,
"n_lines": 5,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n contrapose!;\n intro h;\n apply validate_axiomFour_of_model_finitely hM;\n simpa using h;",
... | [
{
"name": "validate_axiomFour_of_model_finitely",
"text": "lemma validate_axiomFour_of_model_finitely {M : Kripke.Model} (hM : M ⊧* Modal.KTMk)\n : Finite M → ∀ φ : Formula ℕ, M ⊧ Axioms.Four φ := by\n contrapose!;\n rintro ⟨φ, hφ⟩;\n apply List.Nodup.infinite_of_infinite;\n have H : ∀ n : ℕ+, ∃ l : Li... | [
{
"name": "model_infinitity_of_not_validate_axiomFour",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 8,
"n_chars": 258,
"n_subproofs": 0,
"n_tactics": 9,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 2,
"automation_onl... | 1 | module
public import Foundation.Modal.Kripke.Logic.S4
public import Foundation.Modal.Kripke.AxiomMk
public import Foundation.Vorspiel.List.Chain
@[expose] public section
namespace LO.Modal
open Entailment
open Formula
open Kripke
open Modal.Kripke
namespace Kripke
variable {F : Kripke.Frame}
protected class Fram... | module
public import Foundation.Modal.Kripke.Logic.S4
public import Foundation.Modal.Kripke.AxiomMk
public import Foundation.Vorspiel.List.Chain
@[expose] public section
namespace LO.Modal
open Entailment
open Formula
open Kripke
open Modal.Kripke
namespace Kripke
variable {F : Kripke.Frame}
protected class Fram... | @@ -43,8 +43,91 @@
open Formula.Kripke
open Entailment
+lemma validate_axiomFour_of_model_finitely {M : Kripke.Model} (hM : M ⊧* Modal.KTMk)
+ : Finite M → ∀ φ : Formula ℕ, M ⊧ Axioms.Four φ := by
+ contrapose!;
+ rintro ⟨φ, hφ⟩;
+ apply List.Nodup.infinite_of_infinite;
+ have H : ∀ n : ℕ+, ∃ l : List M.World,... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_403efce48305_1 | b1b9655a49cb5a15 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Kripke/Logic/KTMk.lean | KTMk | 1 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 4 | 1 | [
{
"theorem_name": "exists_not_validate_axiomFour",
"depth": 1,
"n_commands": 0,
"n_lines": 3,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n use (.atom 0);\n exact not_imp_not.mpr transitive_of_validate_AxiomFour not_transitive;",
"n_chars": 93,... | [
{
"name": "not_transitive",
"text": "lemma not_transitive : ¬recessionFrame.IsTransitive := by\n by_contra h_trans;\n have := @Frame.trans recessionFrame _ 2 1 0;\n omega;\n\n",
"fan_in": 1,
"n_lines": 6,
"n_chars": 137,
"n_subproofs": 1,
"n_tactics": 7,
"cyclomatic": 1,
"n_au... | [
{
"name": "exists_not_validate_axiomFour",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 5,
"n_chars": 193,
"n_subproofs": 0,
"n_tactics": 5,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 2,
"automation_only": false,
... | 1 | module
public import Foundation.Modal.Kripke.Logic.S4
public import Foundation.Modal.Kripke.AxiomMk
public import Foundation.Vorspiel.List.Chain
@[expose] public section
namespace LO.Modal
open Entailment
open Formula
open Kripke
open Modal.Kripke
namespace Kripke
variable {F : Kripke.Frame}
protected class Fram... | module
public import Foundation.Modal.Kripke.Logic.S4
public import Foundation.Modal.Kripke.AxiomMk
public import Foundation.Vorspiel.List.Chain
@[expose] public section
namespace LO.Modal
open Entailment
open Formula
open Kripke
open Modal.Kripke
namespace Kripke
variable {F : Kripke.Frame}
protected class Fram... | @@ -142,8 +142,15 @@
use i + 1;
refine ⟨by omega, by omega, by simp_all; omega⟩;
-lemma exists_not_validate_axiomFour : ∃ φ : Formula ℕ, ¬recessionFrame ⊧ Axioms.Four φ := sorry
+lemma not_transitive : ¬recessionFrame.IsTransitive := by
+ by_contra h_trans;
+ have := @Frame.trans recessionFrame _ 2 1 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_403efce48305_2 | 3d89973b63772fb0 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Kripke/Logic/KTMk.lean | KTMk | 2 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 4 | 1 | [
{
"theorem_name": "exists_not_provable_axiomFour",
"depth": 1,
"n_commands": 0,
"n_lines": 10,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n obtain ⟨φ, hφ⟩ := recessionFrame.exists_not_validate_axiomFour;\n use! φ;\n apply Sound.not_provable_of_co... | [
{
"name": "exists_not_validate_axiomFour",
"text": "lemma exists_not_validate_axiomFour : ∃ φ : Formula ℕ, ¬recessionFrame ⊧ Axioms.Four φ := by\n use (.atom 0);\n exact not_imp_not.mpr transitive_of_validate_AxiomFour not_transitive;\n\n",
"fan_in": 1,
"n_lines": 5,
"n_chars": 193,
"n_sub... | [
{
"name": "exists_not_provable_axiomFour",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 2,
"n_lines": 12,
"n_chars": 414,
"n_subproofs": 1,
"n_tactics": 19,
"cyclomatic": 2,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 6,
"automation_only": false,
... | 2 | module
public import Foundation.Modal.Kripke.Logic.S4
public import Foundation.Modal.Kripke.AxiomMk
public import Foundation.Vorspiel.List.Chain
@[expose] public section
namespace LO.Modal
open Entailment
open Formula
open Kripke
open Modal.Kripke
namespace Kripke
variable {F : Kripke.Frame}
protected class Fram... | module
public import Foundation.Modal.Kripke.Logic.S4
public import Foundation.Modal.Kripke.AxiomMk
public import Foundation.Vorspiel.List.Chain
@[expose] public section
namespace LO.Modal
open Entailment
open Formula
open Kripke
open Modal.Kripke
namespace Kripke
variable {F : Kripke.Frame}
protected class Fram... | @@ -147,9 +147,22 @@
have := @Frame.trans recessionFrame _ 2 1 0;
omega;
+lemma exists_not_validate_axiomFour : ∃ φ : Formula ℕ, ¬recessionFrame ⊧ Axioms.Four φ := by
+ use (.atom 0);
+ exact not_imp_not.mpr transitive_of_validate_AxiomFour not_transitive;
+
end recessionFrame
-lemma exists_not_provable_ax... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_8ca4be547887_0 | 46b4f846a450c456 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Kripke/AxiomL.lean | AxiomL | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 1 | 1 | [
{
"theorem_name": "validate_AxiomL_of_finite_trans_irrefl",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " validate_AxiomL_of_trans_cwf",
"n_chars": 29,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomati... | [
{
"name": "validate_AxiomL_of_trans_cwf",
"text": "lemma validate_AxiomL_of_trans_cwf [F.IsTransitive] [F.IsConverseWellFounded] : F ⊧ (Axioms.L φ) := by\n rintro V w;\n apply Satisfies.imp_def.mpr;\n contrapose;\n intro h;\n obtain ⟨x, Rwx, h⟩ := by simpa using Satisfies.box_def.not.mp h;\n obtain ⟨m... | [
{
"name": "validate_AxiomL_of_finite_trans_irrefl",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 3,
"n_chars": 148,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": ... | 1 | module
public import Foundation.Modal.Kripke.Irreflexive
@[expose] public section
namespace LO.Modal
namespace Kripke
open Formula.Kripke
variable {F : Kripke.Frame}
protected abbrev Frame.IsConverseWellFounded (F : Frame) := _root_.IsConverseWellFounded _ F.Rel
lemma Frame.cwf [F.IsConverseWellFounded] : Conve... | module
public import Foundation.Modal.Kripke.Irreflexive
@[expose] public section
namespace LO.Modal
namespace Kripke
open Formula.Kripke
variable {F : Kripke.Frame}
protected abbrev Frame.IsConverseWellFounded (F : Frame) := _root_.IsConverseWellFounded _ F.Rel
lemma Frame.cwf [F.IsConverseWellFounded] : Conve... | @@ -20,8 +20,30 @@
instance [F.IsConverseWellFounded] : F.IsIrreflexive := F.cwf.irrefl.swap
-lemma validate_AxiomL_of_finite_trans_irrefl [F.IsFinite] [F.IsTransitive] [F.IsIrreflexive] : F ⊧ (Axioms.L φ) := sorry
+lemma validate_AxiomL_of_trans_cwf [F.IsTransitive] [F.IsConverseWellFounded] : F ⊧ (Axioms.L 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_063f2fb28aab_0 | 2d8e5a7b3d051d35 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Entailment/K.lean | K | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 16 | 1 | [
{
"theorem_name": "dia_congruence!",
"depth": 1,
"n_commands": 0,
"n_lines": 6,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n apply E!_replace (E!_symm $ dia_duality!) (E!_symm $ dia_duality!);\n apply neg_congruence!;\n apply box_congruence!;\n ... | [
{
"name": "neg_congruence!",
"text": "lemma neg_congruence! (h : 𝓢 ⊢ φ 🡘 ψ) : 𝓢 ⊢ ∼φ 🡘 ∼ψ := by\n apply E!_intro;\n . apply contra! $ C_of_E_mpr! h;\n . apply contra! $ C_of_E_mp! h;\n\n\n",
"fan_in": 1,
"n_lines": 7,
"n_chars": 171,
"n_subproofs": 0,
"n_tactics": 7,
"cyclomat... | [
{
"name": "dia_congruence!",
"fan_in": 0,
"n_deps_direct": 2,
"n_deps_transitive": 2,
"n_lines": 8,
"n_chars": 239,
"n_subproofs": 0,
"n_tactics": 11,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 5,
"automation_only": false,
"max_nestin... | 1 | module
public import Foundation.Modal.Entailment.E
@[expose] public section
namespace LO.Modal.Entailment
open LO.Entailment LO.Entailment.FiniteContext
variable {S F : Type*} [BasicModalLogicalConnective F] [Entailment S F]
variable {𝓢 : S} [Entailment.K 𝓢] {n : ℕ} {φ ψ ξ χ: F}
section E
variable [DecidableEq... | module
public import Foundation.Modal.Entailment.E
@[expose] public section
namespace LO.Modal.Entailment
open LO.Entailment LO.Entailment.FiniteContext
variable {S F : Type*} [BasicModalLogicalConnective F] [Entailment S F]
variable {𝓢 : S} [Entailment.K 𝓢] {n : ℕ} {φ ψ ξ χ: F}
section E
variable [DecidableEq... | @@ -16,6 +16,11 @@
variable [DecidableEq F]
-- TODO: move
+lemma neg_congruence! (h : 𝓢 ⊢ φ 🡘 ψ) : 𝓢 ⊢ ∼φ 🡘 ∼ψ := by
+ apply E!_intro;
+ . apply contra! $ C_of_E_mpr! h;
+ . apply contra! $ C_of_E_mp! h;
+
def box_regularity (h : 𝓢 ⊢! φ 🡒 ψ) : 𝓢 ⊢! □φ 🡒 □ψ := by
apply ?_ ⨀ nec h;
exact axiomK;
@@ ... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_063f2fb28aab_1 | 810aa24647f93402 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Entailment/K.lean | K | 1 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 16 | 4 | [
{
"theorem_name": "diaK''!",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " (diaK'! h₁) ⨀ h₂",
"n_chars": 23,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrit... | [
{
"name": "diaK'!",
"text": "lemma diaK'! (h : 𝓢 ⊢ φ 🡒 ψ) : 𝓢 ⊢ ◇φ 🡒 ◇ψ := ⟨diaK' h.some⟩\n\n",
"fan_in": 4,
"n_lines": 3,
"n_chars": 90,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": f... | [
{
"name": "diaK''!",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 3,
"n_chars": 111,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
"max_nesting": 1
}... | 1 | module
public import Foundation.Modal.Entailment.E
@[expose] public section
namespace LO.Modal.Entailment
open LO.Entailment LO.Entailment.FiniteContext
variable {S F : Type*} [BasicModalLogicalConnective F] [Entailment S F]
variable {𝓢 : S} [Entailment.K 𝓢] {n : ℕ} {φ ψ ξ χ: F}
section E
variable [DecidableEq... | module
public import Foundation.Modal.Entailment.E
@[expose] public section
namespace LO.Modal.Entailment
open LO.Entailment LO.Entailment.FiniteContext
variable {S F : Type*} [BasicModalLogicalConnective F] [Entailment S F]
variable {𝓢 : S} [Entailment.K 𝓢] {n : ℕ} {φ ψ ξ χ: F}
section E
variable [DecidableEq... | @@ -106,12 +106,16 @@
apply nec;
apply contra;
assumption;
-lemma diaK''! (h₁ : 𝓢 ⊢ φ 🡒 ψ) (h₂ : 𝓢 ⊢ ◇φ) : 𝓢 ⊢ ◇ψ := sorry
+lemma diaK'! (h : 𝓢 ⊢ φ 🡒 ψ) : 𝓢 ⊢ ◇φ 🡒 ◇ψ := ⟨diaK' h.some⟩
+lemma diaK''! (h₁ : 𝓢 ⊢ φ 🡒 ψ) (h₂ : 𝓢 ⊢ ◇φ) : 𝓢 ⊢ ◇ψ := (diaK'! h₁) ⨀ h₂
+
lemma CMultidiaMultidia_of_C (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_063f2fb28aab_2 | 2a6f0e4251e854c7 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Entailment/K.lean | K | 2 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 16 | 4 | [
{
"theorem_name": "diaK''!",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " (diaK'! h₁) ⨀ h₂",
"n_chars": 23,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrit... | [
{
"name": "diaK'!",
"text": "lemma diaK'! (h : 𝓢 ⊢ φ 🡒 ψ) : 𝓢 ⊢ ◇φ 🡒 ◇ψ := ⟨diaK' h.some⟩\n\n",
"fan_in": 4,
"n_lines": 3,
"n_chars": 90,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": f... | [
{
"name": "CMultidiaMultidia_of_C",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 10,
"n_chars": 209,
"n_subproofs": 0,
"n_tactics": 9,
"cyclomatic": 2,
"n_automation": 2,
"n_rewrites": 0,
"n_structural": 2,
"automation_only": false,
"max... | 1 | module
public import Foundation.Modal.Entailment.E
@[expose] public section
namespace LO.Modal.Entailment
open LO.Entailment LO.Entailment.FiniteContext
variable {S F : Type*} [BasicModalLogicalConnective F] [Entailment S F]
variable {𝓢 : S} [Entailment.K 𝓢] {n : ℕ} {φ ψ ξ χ: F}
section E
variable [DecidableEq... | module
public import Foundation.Modal.Entailment.E
@[expose] public section
namespace LO.Modal.Entailment
open LO.Entailment LO.Entailment.FiniteContext
variable {S F : Type*} [BasicModalLogicalConnective F] [Entailment S F]
variable {𝓢 : S} [Entailment.K 𝓢] {n : ℕ} {φ ψ ξ χ: F}
section E
variable [DecidableEq... | @@ -106,10 +106,14 @@
apply nec;
apply contra;
assumption;
+lemma diaK'! (h : 𝓢 ⊢ φ 🡒 ψ) : 𝓢 ⊢ ◇φ 🡒 ◇ψ := ⟨diaK' h.some⟩
+
lemma CMultidiaMultidia_of_C (h : 𝓢 ⊢ φ 🡒 ψ) : 𝓢 ⊢ ◇^[n]φ 🡒 ◇^[n]ψ := by
induction n with
| zero => simpa;
- | succ n ih => sorry
+ | succ n ih =>
+ simp only [Dia.diaI... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_063f2fb28aab_3 | 8794260a1dd77642 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Entailment/K.lean | K | 3 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 16 | 4 | [
{
"theorem_name": "diaK''!",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " (diaK'! h₁) ⨀ h₂",
"n_chars": 23,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrit... | [
{
"name": "diaK'!",
"text": "lemma diaK'! (h : 𝓢 ⊢ φ 🡒 ψ) : 𝓢 ⊢ ◇φ 🡒 ◇ψ := ⟨diaK' h.some⟩\n\n",
"fan_in": 4,
"n_lines": 3,
"n_chars": 90,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": f... | [
{
"name": "NDia_of_BoxN!",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 2,
"n_lines": 3,
"n_chars": 87,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
"max_nesting": ... | 1 | module
public import Foundation.Modal.Entailment.E
@[expose] public section
namespace LO.Modal.Entailment
open LO.Entailment LO.Entailment.FiniteContext
variable {S F : Type*} [BasicModalLogicalConnective F] [Entailment S F]
variable {𝓢 : S} [Entailment.K 𝓢] {n : ℕ} {φ ψ ξ χ: F}
section E
variable [DecidableEq... | module
public import Foundation.Modal.Entailment.E
@[expose] public section
namespace LO.Modal.Entailment
open LO.Entailment LO.Entailment.FiniteContext
variable {S F : Type*} [BasicModalLogicalConnective F] [Entailment S F]
variable {𝓢 : S} [Entailment.K 𝓢] {n : ℕ} {φ ψ ξ χ: F}
section E
variable [DecidableEq... | @@ -106,10 +106,14 @@
apply nec;
apply contra;
assumption;
+lemma diaK'! (h : 𝓢 ⊢ φ 🡒 ψ) : 𝓢 ⊢ ◇φ 🡒 ◇ψ := ⟨diaK' h.some⟩
+
lemma CMultidiaMultidia_of_C (h : 𝓢 ⊢ φ 🡒 ψ) : 𝓢 ⊢ ◇^[n]φ 🡒 ◇^[n]ψ := by
induction n with
| zero => simpa;
- | succ n ih => sorry
+ | succ n ih =>
+ simp only [Dia.diaI... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_063f2fb28aab_4 | 94cb8783d46d9b61 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Entailment/K.lean | K | 4 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 16 | 2 | [
{
"theorem_name": "box_duality'!",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " boxItr_duality'! (n := 1)",
"n_chars": 26,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0... | [
{
"name": "boxItr_duality'!",
"text": "lemma boxItr_duality'! : 𝓢 ⊢ □^[n]φ ↔ 𝓢 ⊢ ∼(◇^[n](∼φ)) := by\n constructor;\n . intro h; exact (K!_left boxItr_duality!) ⨀ h;\n . intro h; exact (K!_right boxItr_duality!) ⨀ h;\n\n",
"fan_in": 2,
"n_lines": 6,
"n_chars": 204,
"n_subproofs": 0,
... | [
{
"name": "box_duality'!",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 4,
"n_chars": 97,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
"max_nesting": ... | 1 | module
public import Foundation.Modal.Entailment.E
@[expose] public section
namespace LO.Modal.Entailment
open LO.Entailment LO.Entailment.FiniteContext
variable {S F : Type*} [BasicModalLogicalConnective F] [Entailment S F]
variable {𝓢 : S} [Entailment.K 𝓢] {n : ℕ} {φ ψ ξ χ: F}
section E
variable [DecidableEq... | module
public import Foundation.Modal.Entailment.E
@[expose] public section
namespace LO.Modal.Entailment
open LO.Entailment LO.Entailment.FiniteContext
variable {S F : Type*} [BasicModalLogicalConnective F] [Entailment S F]
variable {𝓢 : S} [Entailment.K 𝓢] {n : ℕ} {φ ψ ξ χ: F}
section E
variable [DecidableEq... | @@ -139,8 +139,13 @@
apply diaK'!;
simp;
-lemma box_duality'! : 𝓢 ⊢ □φ ↔ 𝓢 ⊢ ∼(◇(∼φ)) := sorry
+lemma boxItr_duality'! : 𝓢 ⊢ □^[n]φ ↔ 𝓢 ⊢ ∼(◇^[n](∼φ)) := by
+ constructor;
+ . intro h; exact (K!_left boxItr_duality!) ⨀ h;
+ . intro h; exact (K!_right boxItr_duality!) ⨀ h;
+lemma box_duality'! : 𝓢 ⊢ □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_063f2fb28aab_5 | 74d5d268d6cc4db4 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Entailment/K.lean | K | 5 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 16 | 3 | [
{
"theorem_name": "distribute_boxItr_conj!",
"depth": 1,
"n_commands": 0,
"n_lines": 17,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n induction Γ using List.induction_with_singleton with\n | hnil => simp;\n | hsingle => simp;\n | hcons φ Γ h ih ... | [
{
"name": "imply_boxItr_distribute'!",
"text": "lemma imply_boxItr_distribute'! (h : 𝓢 ⊢ φ 🡒 ψ) : 𝓢 ⊢ □^[n]φ 🡒 □^[n]ψ := ⟨implyMultiboxDistribute' h.some⟩\n\n",
"fan_in": 3,
"n_lines": 3,
"n_chars": 136,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
... | [
{
"name": "distribute_boxItr_conj!",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 20,
"n_chars": 842,
"n_subproofs": 4,
"n_tactics": 29,
"cyclomatic": 3,
"n_automation": 5,
"n_rewrites": 0,
"n_structural": 8,
"automation_only": false,
"m... | 1 | module
public import Foundation.Modal.Entailment.E
@[expose] public section
namespace LO.Modal.Entailment
open LO.Entailment LO.Entailment.FiniteContext
variable {S F : Type*} [BasicModalLogicalConnective F] [Entailment S F]
variable {𝓢 : S} [Entailment.K 𝓢] {n : ℕ} {φ ψ ξ χ: F}
section E
variable [DecidableEq... | module
public import Foundation.Modal.Entailment.E
@[expose] public section
namespace LO.Modal.Entailment
open LO.Entailment LO.Entailment.FiniteContext
variable {S F : Type*} [BasicModalLogicalConnective F] [Entailment S F]
variable {𝓢 : S} [Entailment.K 𝓢] {n : ℕ} {φ ψ ξ χ: F}
section E
variable [DecidableEq... | @@ -70,6 +70,8 @@
@[simp] lemma boxdotverum! : 𝓢 ⊢ (⊡⊤ : F) := ⟨boxdotverum⟩
def implyMultiboxDistribute' (h : 𝓢 ⊢! φ 🡒 ψ) : 𝓢 ⊢! □^[n]φ 🡒 □^[n]ψ := boxItr_axiomK' $ multinec h
+lemma imply_boxItr_distribute'! (h : 𝓢 ⊢ φ 🡒 ψ) : 𝓢 ⊢ □^[n]φ 🡒 □^[n]ψ := ⟨implyMultiboxDistribute' h.some⟩
+
def implyBoxDistrib... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_063f2fb28aab_6 | 8113825e1b63fc07 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Entailment/K.lean | K | 6 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 16 | 1 | [
{
"theorem_name": "boxItrConj'_iff!",
"depth": 1,
"n_commands": 0,
"n_lines": 13,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n induction Γ using List.induction_with_singleton with\n | hcons φ Γ h ih =>\n suffices 𝓢 ⊢ □^[n](φ ⋏ ⋀Γ) ↔ 𝓢 ⊢ □^[n]... | [
{
"name": "collect_boxItr_and'!",
"text": "lemma collect_boxItr_and'! (h : 𝓢 ⊢ □^[n]φ ⋏ □^[n]ψ) : 𝓢 ⊢ □^[n](φ ⋏ ψ) := ⟨collect_boxItr_and' h.some⟩\n\n",
"fan_in": 1,
"n_lines": 3,
"n_chars": 133,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewri... | [
{
"name": "boxItrConj'_iff!",
"fan_in": 2,
"n_deps_direct": 2,
"n_deps_transitive": 2,
"n_lines": 14,
"n_chars": 560,
"n_subproofs": 2,
"n_tactics": 23,
"cyclomatic": 5,
"n_automation": 2,
"n_rewrites": 0,
"n_structural": 8,
"automation_only": false,
"max_nest... | 2 | module
public import Foundation.Modal.Entailment.E
@[expose] public section
namespace LO.Modal.Entailment
open LO.Entailment LO.Entailment.FiniteContext
variable {S F : Type*} [BasicModalLogicalConnective F] [Entailment S F]
variable {𝓢 : S} [Entailment.K 𝓢] {n : ℕ} {φ ψ ξ χ: F}
section E
variable [DecidableEq... | module
public import Foundation.Modal.Entailment.E
@[expose] public section
namespace LO.Modal.Entailment
open LO.Entailment LO.Entailment.FiniteContext
variable {S F : Type*} [BasicModalLogicalConnective F] [Entailment S F]
variable {𝓢 : S} [Entailment.K 𝓢] {n : ℕ} {φ ψ ξ χ: F}
section E
variable [DecidableEq... | @@ -87,6 +87,8 @@
instance : Entailment.HasAxiomC 𝓢 := ⟨λ _ _ => collect_box_and⟩
def collect_boxItr_and' (h : 𝓢 ⊢! □^[n]φ ⋏ □^[n]ψ) : 𝓢 ⊢! □^[n](φ ⋏ ψ) := collect_boxItr_and ⨀ h
+lemma collect_boxItr_and'! (h : 𝓢 ⊢ □^[n]φ ⋏ □^[n]ψ) : 𝓢 ⊢ □^[n](φ ⋏ ψ) := ⟨collect_boxItr_and' h.some⟩
+
def collect_box_and' (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_063f2fb28aab_7 | 3cb36ccbfaeed312 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Entailment/K.lean | K | 7 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 16 | 1 | [
{
"theorem_name": "boxItrConj'_iff!",
"depth": 1,
"n_commands": 0,
"n_lines": 13,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n induction Γ using List.induction_with_singleton with\n | hcons φ Γ h ih =>\n suffices 𝓢 ⊢ □^[n](φ ⋏ ⋀Γ) ↔ 𝓢 ⊢ □^[n]... | [
{
"name": "distribute_boxItr_and'!",
"text": "lemma distribute_boxItr_and'! (d : 𝓢 ⊢ □^[n](φ ⋏ ψ)) : 𝓢 ⊢ □^[n]φ ⋏ □^[n]ψ := ⟨distribute_boxItr_and' d.some⟩\n\n",
"fan_in": 1,
"n_lines": 3,
"n_chars": 139,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
... | [
{
"name": "boxConj'_iff!",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 3,
"n_lines": 3,
"n_chars": 104,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
"max_nesting":... | 3 | module
public import Foundation.Modal.Entailment.E
@[expose] public section
namespace LO.Modal.Entailment
open LO.Entailment LO.Entailment.FiniteContext
variable {S F : Type*} [BasicModalLogicalConnective F] [Entailment S F]
variable {𝓢 : S} [Entailment.K 𝓢] {n : ℕ} {φ ψ ξ χ: F}
section E
variable [DecidableEq... | module
public import Foundation.Modal.Entailment.E
@[expose] public section
namespace LO.Modal.Entailment
open LO.Entailment LO.Entailment.FiniteContext
variable {S F : Type*} [BasicModalLogicalConnective F] [Entailment S F]
variable {𝓢 : S} [Entailment.K 𝓢] {n : ℕ} {φ ψ ξ χ: F}
section E
variable [DecidableEq... | @@ -160,6 +160,8 @@
@[simp] lemma distribute_box_and! : 𝓢 ⊢ □(φ ⋏ ψ) 🡒 □φ ⋏ □ψ := ⟨distribute_box_and⟩
def distribute_boxItr_and' (h : 𝓢 ⊢! □^[n](φ ⋏ ψ)) : 𝓢 ⊢! □^[n]φ ⋏ □^[n]ψ := distribute_boxItr_and ⨀ h
+lemma distribute_boxItr_and'! (d : 𝓢 ⊢ □^[n](φ ⋏ ψ)) : 𝓢 ⊢ □^[n]φ ⋏ □^[n]ψ := ⟨distribute_boxItr_and' d... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_063f2fb28aab_8 | 9c08186e5a34d11d | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Entailment/K.lean | K | 8 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 16 | 1 | [
{
"theorem_name": "boxItrConj'_iff!",
"depth": 1,
"n_commands": 0,
"n_lines": 13,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n induction Γ using List.induction_with_singleton with\n | hcons φ Γ h ih =>\n suffices 𝓢 ⊢ □^[n](φ ⋏ ⋀Γ) ↔ 𝓢 ⊢ □^[n]... | [
{
"name": "distribute_boxItr_and'!",
"text": "lemma distribute_boxItr_and'! (d : 𝓢 ⊢ □^[n](φ ⋏ ψ)) : 𝓢 ⊢ □^[n]φ ⋏ □^[n]ψ := ⟨distribute_boxItr_and' d.some⟩\n\n",
"fan_in": 1,
"n_lines": 3,
"n_chars": 139,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
... | [
{
"name": "boxItrconj_of_conjboxItr!",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 3,
"n_lines": 6,
"n_chars": 205,
"n_subproofs": 0,
"n_tactics": 7,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 3,
"automation_only": false,
"m... | 3 | module
public import Foundation.Modal.Entailment.E
@[expose] public section
namespace LO.Modal.Entailment
open LO.Entailment LO.Entailment.FiniteContext
variable {S F : Type*} [BasicModalLogicalConnective F] [Entailment S F]
variable {𝓢 : S} [Entailment.K 𝓢] {n : ℕ} {φ ψ ξ χ: F}
section E
variable [DecidableEq... | module
public import Foundation.Modal.Entailment.E
@[expose] public section
namespace LO.Modal.Entailment
open LO.Entailment LO.Entailment.FiniteContext
variable {S F : Type*} [BasicModalLogicalConnective F] [Entailment S F]
variable {𝓢 : S} [Entailment.K 𝓢] {n : ℕ} {φ ψ ξ χ: F}
section E
variable [DecidableEq... | @@ -160,6 +160,8 @@
@[simp] lemma distribute_box_and! : 𝓢 ⊢ □(φ ⋏ ψ) 🡒 □φ ⋏ □ψ := ⟨distribute_box_and⟩
def distribute_boxItr_and' (h : 𝓢 ⊢! □^[n](φ ⋏ ψ)) : 𝓢 ⊢! □^[n]φ ⋏ □^[n]ψ := distribute_boxItr_and ⨀ h
+lemma distribute_boxItr_and'! (d : 𝓢 ⊢ □^[n](φ ⋏ ψ)) : 𝓢 ⊢ □^[n]φ ⋏ □^[n]ψ := ⟨distribute_boxItr_and' d... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_063f2fb28aab_9 | 938be84a571b9138 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Entailment/K.lean | K | 9 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 16 | 1 | [
{
"theorem_name": "contextual_nec!",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": "\n provable_iff.mpr $ C!_trans collect_box_conj! $ imply_box_distribute'! $ provable_iff.mp h",
"n_chars": 93,
"n_subproo... | [
{
"name": "imply_box_distribute'!",
"text": "lemma imply_box_distribute'! (h : 𝓢 ⊢ φ 🡒 ψ) : 𝓢 ⊢ □φ 🡒 □ψ := ⟨implyBoxDistribute' h.some⟩\n\n",
"fan_in": 1,
"n_lines": 3,
"n_chars": 120,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
... | [
{
"name": "contextual_nec!",
"fan_in": 1,
"n_deps_direct": 2,
"n_deps_transitive": 5,
"n_lines": 4,
"n_chars": 168,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
"max_nesting... | 1 | module
public import Foundation.Modal.Entailment.E
@[expose] public section
namespace LO.Modal.Entailment
open LO.Entailment LO.Entailment.FiniteContext
variable {S F : Type*} [BasicModalLogicalConnective F] [Entailment S F]
variable {𝓢 : S} [Entailment.K 𝓢] {n : ℕ} {φ ψ ξ χ: F}
section E
variable [DecidableEq... | module
public import Foundation.Modal.Entailment.E
@[expose] public section
namespace LO.Modal.Entailment
open LO.Entailment LO.Entailment.FiniteContext
variable {S F : Type*} [BasicModalLogicalConnective F] [Entailment S F]
variable {𝓢 : S} [Entailment.K 𝓢] {n : ℕ} {φ ψ ξ χ: F}
section E
variable [DecidableEq... | @@ -73,6 +73,8 @@
lemma imply_boxItr_distribute'! (h : 𝓢 ⊢ φ 🡒 ψ) : 𝓢 ⊢ □^[n]φ 🡒 □^[n]ψ := ⟨implyMultiboxDistribute' h.some⟩
def implyBoxDistribute' (h : 𝓢 ⊢! φ 🡒 ψ) : 𝓢 ⊢! □φ 🡒 □ψ := implyMultiboxDistribute' (n := 1) h
+lemma imply_box_distribute'! (h : 𝓢 ⊢ φ 🡒 ψ) : 𝓢 ⊢ □φ 🡒 □ψ := ⟨implyBoxDistribute' ... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_063f2fb28aab_10 | d0f71a67ade107fb | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Entailment/K.lean | K | 10 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 16 | 4 | [
{
"theorem_name": "diaItr_or_inst₁!",
"depth": 1,
"n_commands": 0,
"n_lines": 12,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n induction n with\n | zero => simp;\n | succ n ih =>\n suffices 𝓢 ⊢ ◇◇^[n]φ 🡒 ◇◇^[n](φ ⋎ ψ) by simpa;\n apply C!... | [
{
"name": "nec!",
"text": "lemma nec! {Γ : Set F} (h : Γ *⊢[𝓢] φ) : (□'Γ) *⊢[𝓢] □φ := by\n apply Context.provable_iff.mpr;\n obtain ⟨Δ, hΔ₁, hΔ₂⟩ := Context.provable_iff.mp h;\n use (□'Δ);\n constructor;\n . grind;\n . exact contextual_nec! hΔ₂;\n\n",
"fan_in": 4,
"n_lines": 9,
"n_chars"... | [
{
"name": "diaItr_or_inst₁!",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 7,
"n_lines": 14,
"n_chars": 389,
"n_subproofs": 1,
"n_tactics": 21,
"cyclomatic": 2,
"n_automation": 2,
"n_rewrites": 0,
"n_structural": 8,
"automation_only": false,
"max_nest... | 3 | module
public import Foundation.Modal.Entailment.E
@[expose] public section
namespace LO.Modal.Entailment
open LO.Entailment LO.Entailment.FiniteContext
variable {S F : Type*} [BasicModalLogicalConnective F] [Entailment S F]
variable {𝓢 : S} [Entailment.K 𝓢] {n : ℕ} {φ ψ ξ χ: F}
section E
variable [DecidableEq... | module
public import Foundation.Modal.Entailment.E
@[expose] public section
namespace LO.Modal.Entailment
open LO.Entailment LO.Entailment.FiniteContext
variable {S F : Type*} [BasicModalLogicalConnective F] [Entailment S F]
variable {𝓢 : S} [Entailment.K 𝓢] {n : ℕ} {φ ψ ξ χ: F}
section E
variable [DecidableEq... | @@ -274,7 +274,15 @@
@[simp] lemma diaItr_or_inst₁! : 𝓢 ⊢ ◇^[n]φ 🡒 ◇^[n](φ ⋎ ψ) := by
induction n with
| zero => simp;
- | succ n ih => sorry
+ | succ n ih =>
+ suffices 𝓢 ⊢ ◇◇^[n]φ 🡒 ◇◇^[n](φ ⋎ ψ) by simpa;
+ apply C!_trans (K!_left dia_duality!);
+ apply C!_trans ?_ (K!_right dia_duality!);
+ ... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_063f2fb28aab_11 | a26faf3d94835bae | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Entailment/K.lean | K | 11 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 16 | 2 | [
{
"theorem_name": "box_duality'!",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " boxItr_duality'! (n := 1)",
"n_chars": 26,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0... | [
{
"name": "boxItr_duality'!",
"text": "lemma boxItr_duality'! : 𝓢 ⊢ □^[n]φ ↔ 𝓢 ⊢ ∼(◇^[n](∼φ)) := by\n constructor;\n . intro h; exact (K!_left boxItr_duality!) ⨀ h;\n . intro h; exact (K!_right boxItr_duality!) ⨀ h;\n\n",
"fan_in": 2,
"n_lines": 6,
"n_chars": 204,
"n_subproofs": 0,
... | [
{
"name": "not_dia_bot",
"fan_in": 1,
"n_deps_direct": 3,
"n_deps_transitive": 3,
"n_lines": 7,
"n_chars": 249,
"n_subproofs": 0,
"n_tactics": 5,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 2,
"automation_only": false,
"max_nesting": 2... | 1 | module
public import Foundation.Modal.Entailment.E
@[expose] public section
namespace LO.Modal.Entailment
open LO.Entailment LO.Entailment.FiniteContext
variable {S F : Type*} [BasicModalLogicalConnective F] [Entailment S F]
variable {𝓢 : S} [Entailment.K 𝓢] {n : ℕ} {φ ψ ξ χ: F}
section E
variable [DecidableEq... | module
public import Foundation.Modal.Entailment.E
@[expose] public section
namespace LO.Modal.Entailment
open LO.Entailment LO.Entailment.FiniteContext
variable {S F : Type*} [BasicModalLogicalConnective F] [Entailment S F]
variable {𝓢 : S} [Entailment.K 𝓢] {n : ℕ} {φ ψ ξ χ: F}
section E
variable [DecidableEq... | @@ -139,6 +139,11 @@
apply diaK'!;
simp;
+lemma boxItr_duality'! : 𝓢 ⊢ □^[n]φ ↔ 𝓢 ⊢ ∼(◇^[n](∼φ)) := by
+ constructor;
+ . intro h; exact (K!_left boxItr_duality!) ⨀ h;
+ . intro h; exact (K!_right boxItr_duality!) ⨀ h;
+
def box_dni' (h : 𝓢 ⊢! □φ) : 𝓢 ⊢! □(∼∼φ) := box_dni ⨀ h
@[simp] lemma negbox_dni! ... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_063f2fb28aab_12 | 8179ee14493794c2 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Entailment/K.lean | K | 12 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 16 | 3 | [
{
"theorem_name": "distribute_boxItr_conj!",
"depth": 1,
"n_commands": 0,
"n_lines": 17,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n induction Γ using List.induction_with_singleton with\n | hnil => simp;\n | hsingle => simp;\n | hcons φ Γ h ih ... | [
{
"name": "imply_boxItr_distribute'!",
"text": "lemma imply_boxItr_distribute'! (h : 𝓢 ⊢ φ 🡒 ψ) : 𝓢 ⊢ □^[n]φ 🡒 □^[n]ψ := ⟨implyMultiboxDistribute' h.some⟩\n\n",
"fan_in": 3,
"n_lines": 3,
"n_chars": 136,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
... | [
{
"name": "distribute_dia_and'!",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 4,
"n_lines": 3,
"n_chars": 114,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
"max_ne... | 1 | module
public import Foundation.Modal.Entailment.E
@[expose] public section
namespace LO.Modal.Entailment
open LO.Entailment LO.Entailment.FiniteContext
variable {S F : Type*} [BasicModalLogicalConnective F] [Entailment S F]
variable {𝓢 : S} [Entailment.K 𝓢] {n : ℕ} {φ ψ ξ χ: F}
section E
variable [DecidableEq... | module
public import Foundation.Modal.Entailment.E
@[expose] public section
namespace LO.Modal.Entailment
open LO.Entailment LO.Entailment.FiniteContext
variable {S F : Type*} [BasicModalLogicalConnective F] [Entailment S F]
variable {𝓢 : S} [Entailment.K 𝓢] {n : ℕ} {φ ψ ξ χ: F}
section E
variable [DecidableEq... | @@ -70,6 +70,8 @@
@[simp] lemma boxdotverum! : 𝓢 ⊢ (⊡⊤ : F) := ⟨boxdotverum⟩
def implyMultiboxDistribute' (h : 𝓢 ⊢! φ 🡒 ψ) : 𝓢 ⊢! □^[n]φ 🡒 □^[n]ψ := boxItr_axiomK' $ multinec h
+lemma imply_boxItr_distribute'! (h : 𝓢 ⊢ φ 🡒 ψ) : 𝓢 ⊢ □^[n]φ 🡒 □^[n]ψ := ⟨implyMultiboxDistribute' h.some⟩
+
def implyBoxDistrib... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_063f2fb28aab_13 | 1c9025b36e43a2b1 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Entailment/K.lean | K | 13 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 16 | 1 | [
{
"theorem_name": "distribute_diaItr_fdisj!",
"depth": 1,
"n_commands": 0,
"n_lines": 8,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n refine C!_replace ?_ ?_ (distribute_diaItr_disj! (n := n) (Γ := Γ.toList));\n . apply CMultidiaMultidia_of_C;\n ... | [
{
"name": "CMultidiaMultidia_of_C",
"text": "lemma CMultidiaMultidia_of_C (h : 𝓢 ⊢ φ 🡒 ψ) : 𝓢 ⊢ ◇^[n]φ 🡒 ◇^[n]ψ := by\n induction n with\n | zero => simpa;\n | succ n ih =>\n simp only [Dia.diaItr_succ];\n apply diaK'! ih;\n\n\n\n",
"fan_in": 1,
"n_lines": 10,
"n_chars": 209,
"n... | [
{
"name": "distribute_diaItr_fdisj!",
"fan_in": 1,
"n_deps_direct": 2,
"n_deps_transitive": 15,
"n_lines": 11,
"n_chars": 350,
"n_subproofs": 0,
"n_tactics": 14,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 6,
"automation_only": false,
... | 6 | module
public import Foundation.Modal.Entailment.E
@[expose] public section
namespace LO.Modal.Entailment
open LO.Entailment LO.Entailment.FiniteContext
variable {S F : Type*} [BasicModalLogicalConnective F] [Entailment S F]
variable {𝓢 : S} [Entailment.K 𝓢] {n : ℕ} {φ ψ ξ χ: F}
section E
variable [DecidableEq... | module
public import Foundation.Modal.Entailment.E
@[expose] public section
namespace LO.Modal.Entailment
open LO.Entailment LO.Entailment.FiniteContext
variable {S F : Type*} [BasicModalLogicalConnective F] [Entailment S F]
variable {𝓢 : S} [Entailment.K 𝓢] {n : ℕ} {φ ψ ξ χ: F}
section E
variable [DecidableEq... | @@ -108,6 +108,13 @@
assumption;
lemma diaK'! (h : 𝓢 ⊢ φ 🡒 ψ) : 𝓢 ⊢ ◇φ 🡒 ◇ψ := ⟨diaK' h.some⟩
+lemma CMultidiaMultidia_of_C (h : 𝓢 ⊢ φ 🡒 ψ) : 𝓢 ⊢ ◇^[n]φ 🡒 ◇^[n]ψ := by
+ induction n with
+ | zero => simpa;
+ | succ n ih =>
+ simp only [Dia.diaItr_succ];
+ apply diaK'! ih;
+
def diaIff' (h : 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_063f2fb28aab_14 | e70cc67dea18ebb2 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Entailment/K.lean | K | 14 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 16 | 1 | [
{
"theorem_name": "boxLe_lt_add!",
"depth": 1,
"n_commands": 0,
"n_lines": 6,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n induction m with\n | zero => simp;\n | succ m ih =>\n rw [(show n + (m + 1) = n + m + 1 by rfl)];\n apply C!_trans bo... | [
{
"name": "boxLe_lt_succ!",
"text": "lemma boxLe_lt_succ! : 𝓢 ⊢ (□^≤[n + 1] φ) 🡒 (□^≤[n] φ) := by\n apply CFconjFconj!_of_provable;\n intro ψ hψ;\n simp only [Finset.mem_image, Finset.mem_range] at hψ;\n obtain ⟨i, hi, rfl⟩ := hψ;\n apply Context.by_axm!\n simp only [Finset.coe_image, Finset.coe_ran... | [
{
"name": "boxLe_lt_add!",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 8,
"n_chars": 219,
"n_subproofs": 0,
"n_tactics": 9,
"cyclomatic": 2,
"n_automation": 2,
"n_rewrites": 2,
"n_structural": 2,
"automation_only": false,
"max_nesting":... | 1 | module
public import Foundation.Modal.Entailment.E
@[expose] public section
namespace LO.Modal.Entailment
open LO.Entailment LO.Entailment.FiniteContext
variable {S F : Type*} [BasicModalLogicalConnective F] [Entailment S F]
variable {𝓢 : S} [Entailment.K 𝓢] {n : ℕ} {φ ψ ξ χ: F}
section E
variable [DecidableEq... | module
public import Foundation.Modal.Entailment.E
@[expose] public section
namespace LO.Modal.Entailment
open LO.Entailment LO.Entailment.FiniteContext
variable {S F : Type*} [BasicModalLogicalConnective F] [Entailment S F]
variable {𝓢 : S} [Entailment.K 𝓢] {n : ℕ} {φ ψ ξ χ: F}
section E
variable [DecidableEq... | @@ -444,10 +444,24 @@
variable {n m : ℕ}
+lemma boxLe_lt_succ! : 𝓢 ⊢ (□^≤[n + 1] φ) 🡒 (□^≤[n] φ) := by
+ apply CFconjFconj!_of_provable;
+ intro ψ hψ;
+ simp only [Finset.mem_image, Finset.mem_range] at hψ;
+ obtain ⟨i, hi, rfl⟩ := hψ;
+ apply Context.by_axm!
+ simp only [Finset.coe_image, Finset.coe_range... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_063f2fb28aab_15 | 68d7b6addb3d5abf | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Entailment/K.lean | K | 15 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 16 | 1 | [
{
"theorem_name": "boxLe_lt_add!",
"depth": 1,
"n_commands": 0,
"n_lines": 6,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n induction m with\n | zero => simp;\n | succ m ih =>\n rw [(show n + (m + 1) = n + m + 1 by rfl)];\n apply C!_trans bo... | [
{
"name": "boxLe_lt_succ!",
"text": "lemma boxLe_lt_succ! : 𝓢 ⊢ (□^≤[n + 1] φ) 🡒 (□^≤[n] φ) := by\n apply CFconjFconj!_of_provable;\n intro ψ hψ;\n simp only [Finset.mem_image, Finset.mem_range] at hψ;\n obtain ⟨i, hi, rfl⟩ := hψ;\n apply Context.by_axm!\n simp only [Finset.coe_image, Finset.coe_ran... | [
{
"name": "boxLe_lt!",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 2,
"n_lines": 5,
"n_chars": 167,
"n_subproofs": 0,
"n_tactics": 5,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"max_nesting": 2
... | 2 | module
public import Foundation.Modal.Entailment.E
@[expose] public section
namespace LO.Modal.Entailment
open LO.Entailment LO.Entailment.FiniteContext
variable {S F : Type*} [BasicModalLogicalConnective F] [Entailment S F]
variable {𝓢 : S} [Entailment.K 𝓢] {n : ℕ} {φ ψ ξ χ: F}
section E
variable [DecidableEq... | module
public import Foundation.Modal.Entailment.E
@[expose] public section
namespace LO.Modal.Entailment
open LO.Entailment LO.Entailment.FiniteContext
variable {S F : Type*} [BasicModalLogicalConnective F] [Entailment S F]
variable {𝓢 : S} [Entailment.K 𝓢] {n : ℕ} {φ ψ ξ χ: F}
section E
variable [DecidableEq... | @@ -444,10 +444,24 @@
variable {n m : ℕ}
+lemma boxLe_lt_succ! : 𝓢 ⊢ (□^≤[n + 1] φ) 🡒 (□^≤[n] φ) := by
+ apply CFconjFconj!_of_provable;
+ intro ψ hψ;
+ simp only [Finset.mem_image, Finset.mem_range] at hψ;
+ obtain ⟨i, hi, rfl⟩ := hψ;
+ apply Context.by_axm!
+ simp only [Finset.coe_image, Finset.coe_range... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_063f2fb28aab_16 | c7b35909e618ee7c | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Entailment/K.lean | K | 16 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 16 | 1 | [
{
"theorem_name": "boxLe_lt!",
"depth": 1,
"n_commands": 0,
"n_lines": 3,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n convert boxLe_lt_add! (𝓢 := 𝓢) (n := m) (m := n - m) (φ := φ);\n omega;",
"n_chars": 85,
"n_subproofs": 0,
"n_tact... | [
{
"name": "boxLe_lt_add!",
"text": "lemma boxLe_lt_add! : 𝓢 ⊢ (□^≤[n + m] φ) 🡒 (□^≤[n] φ) := by\n induction m with\n | zero => simp;\n | succ m ih =>\n rw [(show n + (m + 1) = n + m + 1 by rfl)];\n apply C!_trans boxLe_lt_succ! ih;\n\n",
"fan_in": 1,
"n_lines": 8,
"n_chars": 219,
... | [
{
"name": "E_boxLe_succ!",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 3,
"n_lines": 20,
"n_chars": 830,
"n_subproofs": 1,
"n_tactics": 34,
"cyclomatic": 2,
"n_automation": 6,
"n_rewrites": 2,
"n_structural": 8,
"automation_only": false,
"max_nesting... | 3 | module
public import Foundation.Modal.Entailment.E
@[expose] public section
namespace LO.Modal.Entailment
open LO.Entailment LO.Entailment.FiniteContext
variable {S F : Type*} [BasicModalLogicalConnective F] [Entailment S F]
variable {𝓢 : S} [Entailment.K 𝓢] {n : ℕ} {φ ψ ξ χ: F}
section E
variable [DecidableEq... | module
public import Foundation.Modal.Entailment.E
@[expose] public section
namespace LO.Modal.Entailment
open LO.Entailment LO.Entailment.FiniteContext
variable {S F : Type*} [BasicModalLogicalConnective F] [Entailment S F]
variable {𝓢 : S} [Entailment.K 𝓢] {n : ℕ} {φ ψ ξ χ: F}
section E
variable [DecidableEq... | @@ -456,8 +456,17 @@
. omega;
. simp;
-lemma boxLe_lt! (h : n ≥ m) : 𝓢 ⊢ (□^≤[n] φ) 🡒 (□^≤[m] φ) := sorry
+lemma boxLe_lt_add! : 𝓢 ⊢ (□^≤[n + m] φ) 🡒 (□^≤[n] φ) := by
+ induction m with
+ | zero => simp;
+ | succ m ih =>
+ rw [(show n + (m + 1) = n + m + 1 by rfl)];
+ apply C!_trans boxLe_lt_succ! ... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_063f2fb28aab_17 | 3a22307e7e8013cb | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Entailment/K.lean | K | 17 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 16 | 1 | [
{
"theorem_name": "boxLe_lt_add!",
"depth": 1,
"n_commands": 0,
"n_lines": 6,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n induction m with\n | zero => simp;\n | succ m ih =>\n rw [(show n + (m + 1) = n + m + 1 by rfl)];\n apply C!_trans bo... | [
{
"name": "boxLe_lt_succ!",
"text": "lemma boxLe_lt_succ! : 𝓢 ⊢ (□^≤[n + 1] φ) 🡒 (□^≤[n] φ) := by\n apply CFconjFconj!_of_provable;\n intro ψ hψ;\n simp only [Finset.mem_image, Finset.mem_range] at hψ;\n obtain ⟨i, hi, rfl⟩ := hψ;\n apply Context.by_axm!\n simp only [Finset.coe_image, Finset.coe_ran... | [
{
"name": "boxLe_regularity!",
"fan_in": 1,
"n_deps_direct": 2,
"n_deps_transitive": 5,
"n_lines": 12,
"n_chars": 466,
"n_subproofs": 1,
"n_tactics": 14,
"cyclomatic": 2,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 5,
"automation_only": false,
"max_nes... | 5 | module
public import Foundation.Modal.Entailment.E
@[expose] public section
namespace LO.Modal.Entailment
open LO.Entailment LO.Entailment.FiniteContext
variable {S F : Type*} [BasicModalLogicalConnective F] [Entailment S F]
variable {𝓢 : S} [Entailment.K 𝓢] {n : ℕ} {φ ψ ξ χ: F}
section E
variable [DecidableEq... | module
public import Foundation.Modal.Entailment.E
@[expose] public section
namespace LO.Modal.Entailment
open LO.Entailment LO.Entailment.FiniteContext
variable {S F : Type*} [BasicModalLogicalConnective F] [Entailment S F]
variable {𝓢 : S} [Entailment.K 𝓢] {n : ℕ} {φ ψ ξ χ: F}
section E
variable [DecidableEq... | @@ -444,10 +444,24 @@
variable {n m : ℕ}
+lemma boxLe_lt_succ! : 𝓢 ⊢ (□^≤[n + 1] φ) 🡒 (□^≤[n] φ) := by
+ apply CFconjFconj!_of_provable;
+ intro ψ hψ;
+ simp only [Finset.mem_image, Finset.mem_range] at hψ;
+ obtain ⟨i, hi, rfl⟩ := hψ;
+ apply Context.by_axm!
+ simp only [Finset.coe_image, Finset.coe_range... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_063f2fb28aab_18 | fbce4fef4be9a863 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Entailment/K.lean | K | 18 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 16 | 1 | [
{
"theorem_name": "boxLe_lt_add!",
"depth": 1,
"n_commands": 0,
"n_lines": 6,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n induction m with\n | zero => simp;\n | succ m ih =>\n rw [(show n + (m + 1) = n + m + 1 by rfl)];\n apply C!_trans bo... | [
{
"name": "boxLe_lt_succ!",
"text": "lemma boxLe_lt_succ! : 𝓢 ⊢ (□^≤[n + 1] φ) 🡒 (□^≤[n] φ) := by\n apply CFconjFconj!_of_provable;\n intro ψ hψ;\n simp only [Finset.mem_image, Finset.mem_range] at hψ;\n obtain ⟨i, hi, rfl⟩ := hψ;\n apply Context.by_axm!\n simp only [Finset.coe_image, Finset.coe_ran... | [
{
"name": "boxLe_fconj_regularity_of_subset",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 6,
"n_lines": 5,
"n_chars": 199,
"n_subproofs": 0,
"n_tactics": 5,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 2,
"automation_only": false,... | 6 | module
public import Foundation.Modal.Entailment.E
@[expose] public section
namespace LO.Modal.Entailment
open LO.Entailment LO.Entailment.FiniteContext
variable {S F : Type*} [BasicModalLogicalConnective F] [Entailment S F]
variable {𝓢 : S} [Entailment.K 𝓢] {n : ℕ} {φ ψ ξ χ: F}
section E
variable [DecidableEq... | module
public import Foundation.Modal.Entailment.E
@[expose] public section
namespace LO.Modal.Entailment
open LO.Entailment LO.Entailment.FiniteContext
variable {S F : Type*} [BasicModalLogicalConnective F] [Entailment S F]
variable {𝓢 : S} [Entailment.K 𝓢] {n : ℕ} {φ ψ ξ χ: F}
section E
variable [DecidableEq... | @@ -444,10 +444,24 @@
variable {n m : ℕ}
+lemma boxLe_lt_succ! : 𝓢 ⊢ (□^≤[n + 1] φ) 🡒 (□^≤[n] φ) := by
+ apply CFconjFconj!_of_provable;
+ intro ψ hψ;
+ simp only [Finset.mem_image, Finset.mem_range] at hψ;
+ obtain ⟨i, hi, rfl⟩ := hψ;
+ apply Context.by_axm!
+ simp only [Finset.coe_image, Finset.coe_range... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_063f2fb28aab_19 | 722947d30225855a | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Entailment/K.lean | K | 19 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 16 | 4 | [
{
"theorem_name": "diaItr_or_inst₁!",
"depth": 1,
"n_commands": 0,
"n_lines": 12,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n induction n with\n | zero => simp;\n | succ n ih =>\n suffices 𝓢 ⊢ ◇◇^[n]φ 🡒 ◇◇^[n](φ ⋎ ψ) by simpa;\n apply C!... | [
{
"name": "nec!",
"text": "lemma nec! {Γ : Set F} (h : Γ *⊢[𝓢] φ) : (□'Γ) *⊢[𝓢] □φ := by\n apply Context.provable_iff.mpr;\n obtain ⟨Δ, hΔ₁, hΔ₂⟩ := Context.provable_iff.mp h;\n use (□'Δ);\n constructor;\n . grind;\n . exact contextual_nec! hΔ₂;\n\n",
"fan_in": 4,
"n_lines": 9,
"n_chars"... | [
{
"name": "box_regularity!",
"fan_in": 1,
"n_deps_direct": 4,
"n_deps_transitive": 8,
"n_lines": 6,
"n_chars": 239,
"n_subproofs": 2,
"n_tactics": 7,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": false,
"max_nesting... | 3 | module
public import Foundation.Modal.Entailment.E
@[expose] public section
namespace LO.Modal.Entailment
open LO.Entailment LO.Entailment.FiniteContext
variable {S F : Type*} [BasicModalLogicalConnective F] [Entailment S F]
variable {𝓢 : S} [Entailment.K 𝓢] {n : ℕ} {φ ψ ξ χ: F}
section E
variable [DecidableEq... | module
public import Foundation.Modal.Entailment.E
@[expose] public section
namespace LO.Modal.Entailment
open LO.Entailment LO.Entailment.FiniteContext
variable {S F : Type*} [BasicModalLogicalConnective F] [Entailment S F]
variable {𝓢 : S} [Entailment.K 𝓢] {n : ℕ} {φ ψ ξ χ: F}
section E
variable [DecidableEq... | @@ -276,7 +276,15 @@
@[simp] lemma diaItr_or_inst₁! : 𝓢 ⊢ ◇^[n]φ 🡒 ◇^[n](φ ⋎ ψ) := by
induction n with
| zero => simp;
- | succ n ih => sorry
+ | succ n ih =>
+ suffices 𝓢 ⊢ ◇◇^[n]φ 🡒 ◇◇^[n](φ ⋎ ψ) by simpa;
+ apply C!_trans (K!_left dia_duality!);
+ apply C!_trans ?_ (K!_right dia_duality!);
+ ... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_c4801069e298_0 | bb9003dcea4d41e5 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Vorspiel/Finset/Basic.lean | Basic | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 2 | 1 | [
{
"theorem_name": "mem_imageOfFinset_iff",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n simp [imageOfFinset, mem_rangeOfFinite_iff]",
"n_chars": 49,
"n_subproofs": 0,
"n_tactics": 2,
"cyclo... | [
{
"name": "mem_rangeOfFinite_iff",
"text": "lemma mem_rangeOfFinite_iff {ι : Sort v} [Finite ι] {f : ι → α} {a : α} :\n a ∈ rangeOfFinite f ↔ ∃ i : ι, f i = a := by simp [rangeOfFinite]\n\n",
"fan_in": 1,
"n_lines": 4,
"n_chars": 160,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic... | [
{
"name": "mem_imageOfFinset_iff",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 5,
"n_chars": 231,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"max_ne... | 1 | module
public import Mathlib.Data.Fin.VecNotation
public import Mathlib.Data.Finset.Preimage
public import Mathlib.Data.Finset.Sort
public import Mathlib.Data.Fintype.Sigma
public import Mathlib.Data.Fintype.Vector
public import Mathlib.Data.Set.Finite.Range
@[expose]
public section
namespace Finset
variable {α : T... | module
public import Mathlib.Data.Fin.VecNotation
public import Mathlib.Data.Finset.Preimage
public import Mathlib.Data.Finset.Sort
public import Mathlib.Data.Fintype.Sigma
public import Mathlib.Data.Fintype.Vector
public import Mathlib.Data.Set.Finite.Range
@[expose]
public section
namespace Finset
variable {α : T... | @@ -36,11 +36,15 @@
noncomputable def rangeOfFinite {ι : Sort v} [Finite ι] (f : ι → α) : Finset α :=
Set.Finite.toFinset (s := Set.range f) (Set.finite_range f)
+lemma mem_rangeOfFinite_iff {ι : Sort v} [Finite ι] {f : ι → α} {a : α} :
+ a ∈ rangeOfFinite f ↔ ∃ i : ι, f i = a := by simp [rangeOfFinite]
+
no... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_c4801069e298_1 | 174c1e27ee4fec42 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Vorspiel/Finset/Basic.lean | Basic | 1 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 2 | 1 | [
{
"theorem_name": "mem_imageOfFinset",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by simpa [mem_imageOfFinset_iff] using ⟨a, ha, rfl⟩",
"n_chars": 56,
"n_subproofs": 0,
"n_tactics": 1,
"cycloma... | [
{
"name": "mem_imageOfFinset_iff",
"text": "lemma mem_imageOfFinset_iff [DecidableEq β] {s : Finset α} {f : (a : α) → a ∈ s → β} {b : β} :\n b ∈ imageOfFinset s f ↔ ∃ (a : α) (ha : a ∈ s), f a ha = b := by\n simp [imageOfFinset, mem_rangeOfFinite_iff]\n\n",
"fan_in": 1,
"n_lines": 5,
"n_char... | [
{
"name": "mem_imageOfFinset",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 2,
"n_lines": 4,
"n_chars": 219,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 2,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"max_nestin... | 2 | module
public import Mathlib.Data.Fin.VecNotation
public import Mathlib.Data.Finset.Preimage
public import Mathlib.Data.Finset.Sort
public import Mathlib.Data.Fintype.Sigma
public import Mathlib.Data.Fintype.Vector
public import Mathlib.Data.Set.Finite.Range
@[expose]
public section
namespace Finset
variable {α : T... | module
public import Mathlib.Data.Fin.VecNotation
public import Mathlib.Data.Finset.Preimage
public import Mathlib.Data.Finset.Sort
public import Mathlib.Data.Fintype.Sigma
public import Mathlib.Data.Fintype.Vector
public import Mathlib.Data.Set.Finite.Range
@[expose]
public section
namespace Finset
variable {α : T... | @@ -42,8 +42,12 @@
noncomputable def imageOfFinset [DecidableEq β] (s : Finset α) (f : (a : α) → a ∈ s → β) : Finset β :=
Finset.biUnion s (rangeOfFinite $ f ·)
+lemma mem_imageOfFinset_iff [DecidableEq β] {s : Finset α} {f : (a : α) → a ∈ s → β} {b : β} :
+ b ∈ imageOfFinset s f ↔ ∃ (a : α) (ha : a ∈ s), f 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_62b2771b776f_0 | 2d988d5319e9e3e0 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Basic/Coding.lean | Coding | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 9 | 1 | [
{
"theorem_name": "encode_emb",
"depth": 1,
"n_commands": 0,
"n_lines": 6,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n simp only [encode_eq_toNat]\n induction t\n · simp [toNat]\n · contradiction\n · simp [Rew.func, toNat_func, *]",
"n_cha... | [
{
"name": "toNat_func",
"text": "lemma toNat_func {k} (f : L.Func k) (v : Fin k → Semiterm L ξ n) :\n toNat (func f v) = (Nat.pair 2 <| Nat.pair k <| Nat.pair (encode f) <| Matrix.vecToNat fun i ↦ toNat (v i)) + 1 := rfl\n\n",
"fan_in": 1,
"n_lines": 4,
"n_chars": 196,
"n_subproofs": 0,
... | [
{
"name": "encode_emb",
"fan_in": 0,
"n_deps_direct": 2,
"n_deps_transitive": 2,
"n_lines": 8,
"n_chars": 224,
"n_subproofs": 0,
"n_tactics": 6,
"cyclomatic": 2,
"n_automation": 3,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": false,
"max_nesting": 2
... | 1 | module
public import Foundation.FirstOrder.Basic.BinderNotation
public import Foundation.Vorspiel.Nat.Matrix
@[expose] public section
namespace LO.FirstOrder
variable {L : Language} [(k : ℕ) → Encodable (L.Func k)]
variable {ξ : Type*} [Encodable ξ]
open Encodable
namespace Semiterm
def toNat {n : ℕ} : Semiterm ... | module
public import Foundation.FirstOrder.Basic.BinderNotation
public import Foundation.Vorspiel.Nat.Matrix
@[expose] public section
namespace LO.FirstOrder
variable {L : Language} [(k : ℕ) → Encodable (L.Func k)]
variable {ξ : Type*} [Encodable ξ]
open Encodable
namespace Semiterm
def toNat {n : ℕ} : Semiterm ... | @@ -62,12 +62,15 @@
lemma encode_eq_toNat (t : Semiterm L ξ n) : encode t = toNat t := rfl
+lemma toNat_func {k} (f : L.Func k) (v : Fin k → Semiterm L ξ n) :
+ toNat (func f v) = (Nat.pair 2 <| Nat.pair k <| Nat.pair (encode f) <| Matrix.vecToNat fun i ↦ toNat (v i)) + 1 := rfl
+
@[simp] lemma encode_emb (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_62b2771b776f_1 | f607427476146e53 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Basic/Coding.lean | Coding | 1 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 9 | 1 | [
{
"theorem_name": "encode_emb",
"depth": 1,
"n_commands": 0,
"n_lines": 4,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n induction σ using rec' <;>\n simp [encode_rel, encode_nrel, encode_verum, encode_falsum, encode_and, encode_or, encode_all, e... | [
{
"name": "encode_nrel",
"text": "lemma encode_nrel {arity : ℕ} (R : L.Rel arity) (v : Fin arity → Semiterm L ξ n) :\n encode (Semiformula.nrel R v) = (Nat.pair 1 <| arity.pair <| (encode R).pair <| Matrix.vecToNat fun i ↦ encode (v i)) + 1 := rfl\n\n",
"fan_in": 1,
"n_lines": 4,
"n_chars": 2... | [
{
"name": "encode_emb",
"fan_in": 0,
"n_deps_direct": 12,
"n_deps_transitive": 11,
"n_lines": 7,
"n_chars": 291,
"n_subproofs": 0,
"n_tactics": 5,
"cyclomatic": 3,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": false,
"max_nesting": ... | 9 | module
public import Foundation.FirstOrder.Basic.BinderNotation
public import Foundation.Vorspiel.Nat.Matrix
@[expose] public section
namespace LO.FirstOrder
variable {L : Language} [(k : ℕ) → Encodable (L.Func k)]
variable {ξ : Type*} [Encodable ξ]
open Encodable
namespace Semiterm
def toNat {n : ℕ} : Semiterm ... | module
public import Foundation.FirstOrder.Basic.BinderNotation
public import Foundation.Vorspiel.Nat.Matrix
@[expose] public section
namespace LO.FirstOrder
variable {L : Language} [(k : ℕ) → Encodable (L.Func k)]
variable {ξ : Type*} [Encodable ξ]
open Encodable
namespace Semiterm
def toNat {n : ℕ} : Semiterm ... | @@ -171,6 +171,9 @@
lemma encode_rel {arity : ℕ} (R : L.Rel arity) (v : Fin arity → Semiterm L ξ n) :
encode (Semiformula.rel R v) = (Nat.pair 0 <| arity.pair <| (encode R).pair <| Matrix.vecToNat fun i ↦ encode (v i)) + 1 := rfl
+lemma encode_nrel {arity : ℕ} (R : L.Rel arity) (v : Fin arity → Semiterm L ξ n) ... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_676df67f1461_0 | e01a0bb334c7051d | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Vorspiel/AdjunctiveSet.lean | AdjunctiveSet | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 2 | 4 | [
{
"theorem_name": "subset_refl",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " subset_iff_set_subset_set.mpr (Set.Subset.refl _)",
"n_chars": 50,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,... | [
{
"name": "subset_iff_set_subset_set",
"text": "lemma subset_iff_set_subset_set {a b : α} : a ⊆ b ↔ set a ⊆ set b := by simp [subset_iff, set]\n\n",
"fan_in": 4,
"n_lines": 3,
"n_chars": 103,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": ... | [
{
"name": "subset_refl",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 3,
"n_chars": 105,
"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 Mathlib.Data.Set.Finite.Basic
@[expose] public section
class Adjoin (β : outParam Type*) (α : Type*) where
adjoin : β → α → α
export Adjoin (adjoin)
instance (α : Type*) : Adjoin α (Set α) := ⟨insert⟩
instance (α : Type*) : Adjoin α (List α) := ⟨List.cons⟩
instance (α : Type*) : Adjoin α (M... | module
public import Mathlib.Data.Set.Finite.Basic
@[expose] public section
class Adjoin (β : outParam Type*) (α : Type*) where
adjoin : β → α → α
export Adjoin (adjoin)
instance (α : Type*) : Adjoin α (Set α) := ⟨insert⟩
instance (α : Type*) : Adjoin α (List α) := ⟨List.cons⟩
instance (α : Type*) : Adjoin α (M... | @@ -51,10 +51,13 @@
@[simp] lemma mem_set_iff {x : β} {a : α} : x ∈ (set a : Set β) ↔ x ∈ a := by simp [set]
-@[simp, refl] lemma subset_refl (a : α) : a ⊆ a := sorry
+lemma subset_iff_set_subset_set {a b : α} : a ⊆ b ↔ set a ⊆ set b := by simp [subset_iff, set]
-@[trans] lemma subset_trans {a b c : α} (ha : 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_676df67f1461_1 | def3e1d25f1af082 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Vorspiel/AdjunctiveSet.lean | AdjunctiveSet | 1 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 2 | 4 | [
{
"theorem_name": "subset_refl",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " subset_iff_set_subset_set.mpr (Set.Subset.refl _)",
"n_chars": 50,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,... | [
{
"name": "subset_iff_set_subset_set",
"text": "lemma subset_iff_set_subset_set {a b : α} : a ⊆ b ↔ set a ⊆ set b := by simp [subset_iff, set]\n\n",
"fan_in": 4,
"n_lines": 3,
"n_chars": 103,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": ... | [
{
"name": "subset_antisymm",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 4,
"n_chars": 173,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
"max_nesting... | 1 | module
public import Mathlib.Data.Set.Finite.Basic
@[expose] public section
class Adjoin (β : outParam Type*) (α : Type*) where
adjoin : β → α → α
export Adjoin (adjoin)
instance (α : Type*) : Adjoin α (Set α) := ⟨insert⟩
instance (α : Type*) : Adjoin α (List α) := ⟨List.cons⟩
instance (α : Type*) : Adjoin α (M... | module
public import Mathlib.Data.Set.Finite.Basic
@[expose] public section
class Adjoin (β : outParam Type*) (α : Type*) where
adjoin : β → α → α
export Adjoin (adjoin)
instance (α : Type*) : Adjoin α (Set α) := ⟨insert⟩
instance (α : Type*) : Adjoin α (List α) := ⟨List.cons⟩
instance (α : Type*) : Adjoin α (M... | @@ -51,12 +51,16 @@
@[simp] lemma mem_set_iff {x : β} {a : α} : x ∈ (set a : Set β) ↔ x ∈ a := by simp [set]
-@[simp, refl] lemma subset_refl (a : α) : a ⊆ a := sorry
+lemma subset_iff_set_subset_set {a b : α} : a ⊆ b ↔ set a ⊆ set b := by simp [subset_iff, set]
-@[trans] lemma subset_trans {a b c : α} (ha : 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_41ea6949616d_0 | fe55bac37fc93d92 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Basic/Semantics/Elementary.lean | Elementary | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 5 | 1 | [
{
"theorem_name": "eval_hom_iff_of_open",
"depth": 1,
"n_commands": 0,
"n_lines": 5,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": "\n match φ with\n | rel r v | nrel r v => by simp [Function.comp_def, ←EmbeddingClass.rel Θ, HomClass.val_term]\n | ⊤ | ⊥ ... | [
{
"name": "val_term",
"text": "lemma val_term (e : Fin n → M₁) (ε : ξ → M₁) (t : Semiterm L ξ n) :\n φ (t.val e ε) = t.val (φ ∘ e) (φ ∘ ε) := by\n induction t <;> simp [*, HomClass.func, Function.comp_def]\n\n",
"fan_in": 1,
"n_lines": 5,
"n_chars": 198,
"n_subproofs": 0,
"n_tactics"... | [
{
"name": "eval_hom_iff_of_open",
"fan_in": 1,
"n_deps_direct": 2,
"n_deps_transitive": 3,
"n_lines": 8,
"n_chars": 432,
"n_subproofs": 0,
"n_tactics": 5,
"cyclomatic": 2,
"n_automation": 4,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
"max_ne... | 1 | module
public import Foundation.FirstOrder.Basic.Semantics.Semantics
@[expose] public section
namespace LO.FirstOrder
section
variable {L : Language}
variable {M : Type*} {M₁ : Type*} {M₂ : Type*} {M₃ : Type*}
variable [Nonempty M] [Nonempty M₁] [Nonempty M₂] [Nonempty M₃]
[s : Structure L M] [s₁ : Structure L M... | module
public import Foundation.FirstOrder.Basic.Semantics.Semantics
@[expose] public section
namespace LO.FirstOrder
section
variable {L : Language}
variable {M : Type*} {M₁ : Type*} {M₂ : Type*} {M₃ : Type*}
variable [Nonempty M] [Nonempty M₁] [Nonempty M₂] [Nonempty M₃]
[s : Structure L M] [s₁ : Structure L M... | @@ -83,6 +83,10 @@
protected lemma rel {k} (r : L.Rel k) (v : Fin k → M₁) :
s₁.rel r v → s₂.rel r (φ ∘ v) := map_rel φ r v
+lemma val_term (e : Fin n → M₁) (ε : ξ → M₁) (t : Semiterm L ξ n) :
+ φ (t.val e ε) = t.val (φ ∘ e) (φ ∘ ε) := by
+ induction t <;> simp [*, HomClass.func, Function.comp_def]
+
end Ho... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_41ea6949616d_1 | a476471582998b87 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Basic/Semantics/Elementary.lean | Elementary | 1 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 5 | 1 | [
{
"theorem_name": "eval_hom_iff_of_open",
"depth": 1,
"n_commands": 0,
"n_lines": 5,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": "\n match φ with\n | rel r v | nrel r v => by simp [Function.comp_def, ←EmbeddingClass.rel Θ, HomClass.val_term]\n | ⊤ | ⊥ ... | [
{
"name": "val_term",
"text": "lemma val_term (e : Fin n → M₁) (ε : ξ → M₁) (t : Semiterm L ξ n) :\n φ (t.val e ε) = t.val (φ ∘ e) (φ ∘ ε) := by\n induction t <;> simp [*, HomClass.func, Function.comp_def]\n\n",
"fan_in": 1,
"n_lines": 5,
"n_chars": 198,
"n_subproofs": 0,
"n_tactics"... | [
{
"name": "eval_hom_allClosure",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 4,
"n_lines": 6,
"n_chars": 272,
"n_subproofs": 0,
"n_tactics": 4,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 2,
"automation_only": false,
"max_nes... | 2 | module
public import Foundation.FirstOrder.Basic.Semantics.Semantics
@[expose] public section
namespace LO.FirstOrder
section
variable {L : Language}
variable {M : Type*} {M₁ : Type*} {M₂ : Type*} {M₃ : Type*}
variable [Nonempty M] [Nonempty M₁] [Nonempty M₂] [Nonempty M₃]
[s : Structure L M] [s₁ : Structure L M... | module
public import Foundation.FirstOrder.Basic.Semantics.Semantics
@[expose] public section
namespace LO.FirstOrder
section
variable {L : Language}
variable {M : Type*} {M₁ : Type*} {M₂ : Type*} {M₃ : Type*}
variable [Nonempty M] [Nonempty M₁] [Nonempty M₂] [Nonempty M₃]
[s : Structure L M] [s₁ : Structure L M... | @@ -83,6 +83,10 @@
protected lemma rel {k} (r : L.Rel k) (v : Fin k → M₁) :
s₁.rel r v → s₂.rel r (φ ∘ v) := map_rel φ r v
+lemma val_term (e : Fin n → M₁) (ε : ξ → M₁) (t : Semiterm L ξ n) :
+ φ (t.val e ε) = t.val (φ ∘ e) (φ ∘ ε) := by
+ induction t <;> simp [*, HomClass.func, Function.comp_def]
+
end Ho... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_41ea6949616d_2 | 990bd05a50093870 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Basic/Semantics/Elementary.lean | Elementary | 2 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 5 | 1 | [
{
"theorem_name": "modelsTheory'",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " modelsTheory.mpr h",
"n_chars": 19,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"... | [
{
"name": "modelsTheory",
"text": "lemma modelsTheory [h : M₁ ≡ₑ[L] M₂] {T : Theory L} :\n M₁↓[L] ⊧* T ↔ M₂↓[L] ⊧* T := by simp [models_theory_iff, h.models]\n\n",
"fan_in": 1,
"n_lines": 4,
"n_chars": 148,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 1,
... | [
{
"name": "modelsTheory'",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 4,
"n_chars": 128,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
"max_nesting":... | 1 | module
public import Foundation.FirstOrder.Basic.Semantics.Semantics
@[expose] public section
namespace LO.FirstOrder
section
variable {L : Language}
variable {M : Type*} {M₁ : Type*} {M₂ : Type*} {M₃ : Type*}
variable [Nonempty M] [Nonempty M₁] [Nonempty M₂] [Nonempty M₃]
[s : Structure L M] [s₁ : Structure L M... | module
public import Foundation.FirstOrder.Basic.Semantics.Semantics
@[expose] public section
namespace LO.FirstOrder
section
variable {L : Language}
variable {M : Type*} {M₁ : Type*} {M₂ : Type*} {M₃ : Type*}
variable [Nonempty M] [Nonempty M₁] [Nonempty M₂] [Nonempty M₃]
[s : Structure L M] [s₁ : Structure L M... | @@ -197,10 +197,13 @@
@[trans] lemma trans : (M₁ ≡ₑ[L] M₂) → (M₂ ≡ₑ[L] M₃) → (M₁ ≡ₑ[L] M₃) :=
fun h₁ h₂ ↦ ⟨Iff.trans h₁.models h₂.models⟩
+lemma modelsTheory [h : M₁ ≡ₑ[L] M₂] {T : Theory L} :
+ M₁↓[L] ⊧* T ↔ M₂↓[L] ⊧* T := by simp [models_theory_iff, h.models]
+
variable (M₁ M₂)
lemma modelsTheory' [M₁ ≡ₑ... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_e70fb2495c21_0 | 93d40e1c74c15fca | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Logic/D/Basic.lean | Basic | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 4 | 1 | [
{
"theorem_name": "D.fdisj_axiomDz",
"depth": 1,
"n_commands": 0,
"n_lines": 15,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n apply C!_replace ?_ ?_ $ D.ldisj_axiomDz (l := s.toList);\n . apply sumQuasiNormal.mem₁!;\n apply axiomK'!;\n apply... | [
{
"name": "D.ldisj_axiomDz",
"text": "lemma D.ldisj_axiomDz {l : List (Formula ℕ)} : Modal.D ⊢ □((□'l).disj) 🡒 (□'l).disj := by\n induction l with\n | nil => exact axiomP!;\n | cons φ l ih =>\n apply C!_replace ?_ ?_ (D.mem_axiomDz (φ := φ) (ψ := (□'l).disj));\n . apply sumQuasiNormal.mem₁!;\n ... | [
{
"name": "D.fdisj_axiomDz",
"fan_in": 2,
"n_deps_direct": 1,
"n_deps_transitive": 3,
"n_lines": 17,
"n_chars": 569,
"n_subproofs": 2,
"n_tactics": 29,
"cyclomatic": 2,
"n_automation": 4,
"n_rewrites": 0,
"n_structural": 12,
"automation_only": false,
"max_nest... | 1 | module
public import Foundation.Modal.Logic.GL.Independency
public import Foundation.Modal.Logic.S.Basic
public import Mathlib.Order.WellFoundedSet
@[expose] public section
namespace LO.Modal
open Formula (atom)
open Logic
protected abbrev D := sumQuasiNormal Modal.GL {∼□⊥, □(□(atom 0) ⋎ □(.atom 1)) 🡒 □(atom 0) ⋎... | module
public import Foundation.Modal.Logic.GL.Independency
public import Foundation.Modal.Logic.S.Basic
public import Mathlib.Order.WellFoundedSet
@[expose] public section
namespace LO.Modal
open Formula (atom)
open Logic
protected abbrev D := sumQuasiNormal Modal.GL {∼□⊥, □(□(atom 0) ⋎ □(.atom 1)) 🡒 □(atom 0) ⋎... | @@ -113,8 +113,36 @@
apply right_Disj!_intro;
assumption;
-lemma D.fdisj_axiomDz {s : Finset (Formula ℕ)} : Modal.D ⊢ □((□'s).disj) 🡒 (□'s).disj := sorry
+lemma D.ldisj_axiomDz {l : List (Formula ℕ)} : Modal.D ⊢ □((□'l).disj) 🡒 (□'l).disj := by
+ induction l with
+ | nil => exact axiomP!;
+ | cons φ l ih =... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_e70fb2495c21_1 | 411dd990d4728cd0 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Logic/D/Basic.lean | Basic | 1 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 4 | 1 | [
{
"theorem_name": "GL_D_TFAE",
"depth": 1,
"n_commands": 0,
"n_lines": 131,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n tfae_have 1 → 2 := by\n intro h M _ _ _ _ o;\n induction h using D.rec' with\n | mem_GL h =>\n apply So... | [
{
"name": "satisfies_box_of_satisfies_box_at_root",
"text": "lemma satisfies_box_of_satisfies_box_at_root [M.IsTransitive] (h : Satisfies _ (M.tailModel₀ o).root.1 (□φ)) {x : M.tailModel₀ o} : Satisfies _ x (□φ) := by\n intro y Rxy;\n apply h;\n dsimp [Frame.root, default];\n grind;\n\n",
"fan_in": ... | [
{
"name": "GL_D_TFAE",
"fan_in": 2,
"n_deps_direct": 3,
"n_deps_transitive": 6,
"n_lines": 139,
"n_chars": 6098,
"n_subproofs": 17,
"n_tactics": 216,
"cyclomatic": 13,
"n_automation": 31,
"n_rewrites": 5,
"n_structural": 60,
"automation_only": false,
"max_nest... | 3 | module
public import Foundation.Modal.Logic.GL.Independency
public import Foundation.Modal.Logic.S.Basic
public import Mathlib.Order.WellFoundedSet
@[expose] public section
namespace LO.Modal
open Formula (atom)
open Logic
protected abbrev D := sumQuasiNormal Modal.GL {∼□⊥, □(□(atom 0) ⋎ □(.atom 1)) 🡒 □(atom 0) ⋎... | module
public import Foundation.Modal.Logic.GL.Independency
public import Foundation.Modal.Logic.S.Basic
public import Mathlib.Order.WellFoundedSet
@[expose] public section
namespace LO.Modal
open Formula (atom)
open Logic
protected abbrev D := sumQuasiNormal Modal.GL {∼□⊥, □(□(atom 0) ⋎ □(.atom 1)) 🡒 □(atom 0) ⋎... | @@ -248,6 +248,12 @@
open Formula.Kripke
+lemma satisfies_box_of_satisfies_box_at_root [M.IsTransitive] (h : Satisfies _ (M.tailModel₀ o).root.1 (□φ)) {x : M.tailModel₀ o} : Satisfies _ x (□φ) := by
+ intro y Rxy;
+ apply h;
+ dsimp [Frame.root, default];
+ grind;
+
protected def pMorphism_extendRoot (M : Mod... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_e70fb2495c21_2 | 04db67c07277cd9e | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Logic/D/Basic.lean | Basic | 2 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 4 | 2 | [
{
"theorem_name": "iff_provable_D_provable_GL",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " GL_D_TFAE (φ := φ) |>.out 0 3",
"n_chars": 32,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
... | [
{
"name": "GL_D_TFAE",
"text": "theorem GL_D_TFAE :\n [\n Modal.D ⊢ φ,\n ∀ M : Kripke.Model, [M.IsFinite] → [M.IsTransitive] → [M.IsIrreflexive] → [M.IsRooted] → ∀ o, Satisfies _ (M.tailModel₀ o).root.1 φ,\n ∀ M : Kripke.Model, [M.IsFinite] → [M.IsTransitive] → [M.IsIrreflexive] → [M.IsRooted] → M... | [
{
"name": "iff_provable_D_provable_GL",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 7,
"n_lines": 3,
"n_chars": 132,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
"... | 4 | module
public import Foundation.Modal.Logic.GL.Independency
public import Foundation.Modal.Logic.S.Basic
public import Mathlib.Order.WellFoundedSet
@[expose] public section
namespace LO.Modal
open Formula (atom)
open Logic
protected abbrev D := sumQuasiNormal Modal.GL {∼□⊥, □(□(atom 0) ⋎ □(.atom 1)) 🡒 □(atom 0) ⋎... | module
public import Foundation.Modal.Logic.GL.Independency
public import Foundation.Modal.Logic.S.Basic
public import Mathlib.Order.WellFoundedSet
@[expose] public section
namespace LO.Modal
open Formula (atom)
open Logic
protected abbrev D := sumQuasiNormal Modal.GL {∼□⊥, □(□(atom 0) ⋎ □(.atom 1)) 🡒 □(atom 0) ⋎... | @@ -332,10 +332,164 @@
open Kripke.Model
open Formula.Kripke
-lemma iff_provable_D_provable_GL : Modal.D ⊢ φ ↔ Modal.GL ⊢ φ.dzSubformula.conj 🡒 φ := sorry
+theorem GL_D_TFAE :
+ [
+ Modal.D ⊢ φ,
+ ∀ M : Kripke.Model, [M.IsFinite] → [M.IsTransitive] → [M.IsIrreflexive] → [M.IsRooted] → ∀ o, Satisfies _ (M.ta... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_e70fb2495c21_3 | 8c5ee837d5bc9a21 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Logic/D/Basic.lean | Basic | 3 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 4 | 2 | [
{
"theorem_name": "iff_provable_D_provable_GL",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " GL_D_TFAE (φ := φ) |>.out 0 3",
"n_chars": 32,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
... | [
{
"name": "GL_D_TFAE",
"text": "theorem GL_D_TFAE :\n [\n Modal.D ⊢ φ,\n ∀ M : Kripke.Model, [M.IsFinite] → [M.IsTransitive] → [M.IsIrreflexive] → [M.IsRooted] → ∀ o, Satisfies _ (M.tailModel₀ o).root.1 φ,\n ∀ M : Kripke.Model, [M.IsFinite] → [M.IsTransitive] → [M.IsIrreflexive] → [M.IsRooted] → M... | [
{
"name": "D.unprovable_T",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 7,
"n_lines": 19,
"n_chars": 520,
"n_subproofs": 0,
"n_tactics": 26,
"cyclomatic": 3,
"n_automation": 4,
"n_rewrites": 0,
"n_structural": 6,
"automation_only": false,
"max_nestin... | 4 | module
public import Foundation.Modal.Logic.GL.Independency
public import Foundation.Modal.Logic.S.Basic
public import Mathlib.Order.WellFoundedSet
@[expose] public section
namespace LO.Modal
open Formula (atom)
open Logic
protected abbrev D := sumQuasiNormal Modal.GL {∼□⊥, □(□(atom 0) ⋎ □(.atom 1)) 🡒 □(atom 0) ⋎... | module
public import Foundation.Modal.Logic.GL.Independency
public import Foundation.Modal.Logic.S.Basic
public import Mathlib.Order.WellFoundedSet
@[expose] public section
namespace LO.Modal
open Formula (atom)
open Logic
protected abbrev D := sumQuasiNormal Modal.GL {∼□⊥, □(□(atom 0) ⋎ □(.atom 1)) 🡒 □(atom 0) ⋎... | @@ -332,8 +332,162 @@
open Kripke.Model
open Formula.Kripke
-lemma D.unprovable_T : Modal.D ⊬ (Axioms.T (.atom 0)) := sorry
+theorem GL_D_TFAE :
+ [
+ Modal.D ⊢ φ,
+ ∀ M : Kripke.Model, [M.IsFinite] → [M.IsTransitive] → [M.IsIrreflexive] → [M.IsRooted] → ∀ o, Satisfies _ (M.tailModel₀ o).root.1 φ,
+ ∀ 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_fe74d68e0680_0 | 486d2a4165ccabe9 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Vorspiel/Fintype.lean | Fintype | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 1 | 1 | [
{
"theorem_name": "elem_le_sup",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " Finset.le_sup (by simp)",
"n_chars": 24,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 1,
... | [
{
"name": "le_sup",
"text": "lemma le_sup {a : α} {f : ι → α} (i : ι) (le : a ≤ f i) : a ≤ sup f := le_trans le (elem_le_sup _ _)\n\n",
"fan_in": 1,
"n_lines": 3,
"n_chars": 112,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_str... | [
{
"name": "elem_le_sup",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 2,
"n_lines": 3,
"n_chars": 95,
"n_subproofs": 0,
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"cyclomatic": 1,
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"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"max_nesting": 1
... | 1 | module
public import Mathlib.Data.Fintype.Sigma
public import Mathlib.Data.Fintype.Vector
@[expose] public section
namespace Fintype
variable {ι : Type _} [Fintype ι]
section
variable {α : Type _} [SemilatticeSup α] [OrderBot α]
def sup (f : ι → α) : α := (Finset.univ : Finset ι).sup f
@[simp] lemma elem_le_sup ... | module
public import Mathlib.Data.Fintype.Sigma
public import Mathlib.Data.Fintype.Vector
@[expose] public section
namespace Fintype
variable {ι : Type _} [Fintype ι]
section
variable {α : Type _} [SemilatticeSup α] [OrderBot α]
def sup (f : ι → α) : α := (Finset.univ : Finset ι).sup f
@[simp] lemma elem_le_sup ... | @@ -14,8 +14,10 @@
def sup (f : ι → α) : α := (Finset.univ : Finset ι).sup f
-@[simp] lemma elem_le_sup (f : ι → α) (i : ι) : f i ≤ sup f := sorry
+@[simp] lemma elem_le_sup (f : ι → α) (i : ι) : f i ≤ sup f := Finset.le_sup (by simp)
+lemma le_sup {a : α} {f : ι → α} (i : ι) (le : a ≤ f i) : a ≤ sup f := le_tra... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_bc7e5cdeb2c6_0 | efcbc35ed8456748 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Ultraproduct.lean | Ultraproduct | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 5 | 1 | [
{
"theorem_name": "eval_Uprod",
"depth": 1,
"n_commands": 0,
"n_lines": 50,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n induction φ using rec'\n case hverum =>\n suffices Set.univ ∈ 𝓤 by simp [*]\n exact Filter.univ_mem\n case hfalsum =>\... | [
{
"name": "val_vecCons_val_eq",
"text": "lemma val_vecCons_val_eq {z : Uprod A 𝓤} {i : I} :\n (z.val i :> fun x ↦ (e x).val i) = (fun x ↦ ((z :> e) x).val i) := by\n simp [Matrix.comp_vecCons (Uprod.val · i), Function.comp_def]\n\n",
"fan_in": 1,
"n_lines": 5,
"n_chars": 198,
"n_subproo... | [
{
"name": "eval_Uprod",
"fan_in": 1,
"n_deps_direct": 2,
"n_deps_transitive": 2,
"n_lines": 53,
"n_chars": 2615,
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"cyclomatic": 6,
"n_automation": 10,
"n_rewrites": 2,
"n_structural": 21,
"automation_only": false,
"max_nesting... | 1 | module
public import Foundation.FirstOrder.Basic
public import Mathlib.Order.Filter.Ultrafilter.Basic
@[expose] public section
namespace LO
namespace FirstOrder
section
universe u v
variable {L : Language.{u}} {ξ : Type v}
{I : Type u} (A : I → Type u)
[s : (i : I) → FirstOrder.Structure L (A i)]
(𝓤 : Ultr... | module
public import Foundation.FirstOrder.Basic
public import Mathlib.Order.Filter.Ultrafilter.Basic
@[expose] public section
namespace LO
namespace FirstOrder
section
universe u v
variable {L : Language.{u}} {ξ : Type v}
{I : Type u} (A : I → Type u)
[s : (i : I) → FirstOrder.Structure L (A i)]
(𝓤 : Ultr... | @@ -56,6 +56,10 @@
variable {e : Fin n → Uprod A 𝓤} {ε : ξ → Uprod A 𝓤}
+lemma val_vecCons_val_eq {z : Uprod A 𝓤} {i : I} :
+ (z.val i :> fun x ↦ (e x).val i) = (fun x ↦ ((z :> e) x).val i) := by
+ simp [Matrix.comp_vecCons (Uprod.val · i), Function.comp_def]
+
lemma eval_Uprod [(i : I) → Nonempty (A 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_bc7e5cdeb2c6_1 | f126e6a99ad4dea5 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Ultraproduct.lean | Ultraproduct | 1 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 5 | 1 | [
{
"theorem_name": "val_Uprod",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n simp [Evalf, eval_Uprod, Matrix.empty_eq]",
"n_chars": 47,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
... | [
{
"name": "eval_Uprod",
"text": "lemma eval_Uprod [(i : I) → Nonempty (A i)] {φ : Semiformula L ξ n} :\n φ.Eval e ε ↔ {i | Eval (fun x ↦ (e x).val i) (fun x ↦ (ε x).val i) φ} ∈ 𝓤 := by\n induction φ using rec'\n case hverum =>\n suffices Set.univ ∈ 𝓤 by simp [*]\n exact Filter.univ_mem\n case ... | [
{
"name": "val_Uprod",
"fan_in": 2,
"n_deps_direct": 4,
"n_deps_transitive": 3,
"n_lines": 5,
"n_chars": 187,
"n_subproofs": 0,
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"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"max_nesting": 2
... | 2 | module
public import Foundation.FirstOrder.Basic
public import Mathlib.Order.Filter.Ultrafilter.Basic
@[expose] public section
namespace LO
namespace FirstOrder
section
universe u v
variable {L : Language.{u}} {ξ : Type v}
{I : Type u} (A : I → Type u)
[s : (i : I) → FirstOrder.Structure L (A i)]
(𝓤 : Ultr... | module
public import Foundation.FirstOrder.Basic
public import Mathlib.Order.Filter.Ultrafilter.Basic
@[expose] public section
namespace LO
namespace FirstOrder
section
universe u v
variable {L : Language.{u}} {ξ : Type v}
{I : Type u} (A : I → Type u)
[s : (i : I) → FirstOrder.Structure L (A i)]
(𝓤 : Ultr... | @@ -60,8 +60,61 @@
(z.val i :> fun x ↦ (e x).val i) = (fun x ↦ ((z :> e) x).val i) := by
simp [Matrix.comp_vecCons (Uprod.val · i), Function.comp_def]
+lemma eval_Uprod [(i : I) → Nonempty (A i)] {φ : Semiformula L ξ n} :
+ φ.Eval e ε ↔ {i | Eval (fun x ↦ (e x).val i) (fun x ↦ (ε x).val i) φ} ∈ 𝓤 := 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_bc7e5cdeb2c6_2 | 190ab16b5b10603a | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Ultraproduct.lean | Ultraproduct | 2 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 5 | 1 | [
{
"theorem_name": "compactness_aux",
"depth": 1,
"n_commands": 0,
"n_lines": 11,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n constructor\n · rintro h ⟨t, ht⟩; exact Semantics.Satisfiable.of_subset h ht\n · intro h\n have : ∀ i : FinSubtheory ... | [
{
"name": "ultrafilter_exists",
"text": "lemma ultrafilter_exists [(t : FinSubtheory T) → Nonempty (A t)]\n (H : ∀ (i : FinSubtheory T), (A i)↓[L] ⊧* (i.val : Theory L)) :\n ∃ 𝓤 : Ultrafilter (FinSubtheory T), Set.image (Sentence.domain A) T ⊆ 𝓤.sets :=\n Ultrafilter.exists_ultrafilter_of_finite_in... | [
{
"name": "compactness_aux",
"fan_in": 1,
"n_deps_direct": 2,
"n_deps_transitive": 3,
"n_lines": 14,
"n_chars": 787,
"n_subproofs": 3,
"n_tactics": 14,
"cyclomatic": 4,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 10,
"automation_only": false,
"max_nest... | 2 | module
public import Foundation.FirstOrder.Basic
public import Mathlib.Order.Filter.Ultrafilter.Basic
@[expose] public section
namespace LO
namespace FirstOrder
section
universe u v
variable {L : Language.{u}} {ξ : Type v}
{I : Type u} (A : I → Type u)
[s : (i : I) → FirstOrder.Structure L (A i)]
(𝓤 : Ultr... | module
public import Foundation.FirstOrder.Basic
public import Mathlib.Order.Filter.Ultrafilter.Basic
@[expose] public section
namespace LO
namespace FirstOrder
section
universe u v
variable {L : Language.{u}} {ξ : Type v}
{I : Type u} (A : I → Type u)
[s : (i : I) → FirstOrder.Structure L (A i)]
(𝓤 : Ultr... | @@ -137,6 +137,19 @@
instance : Nonempty (FinSubtheory T) := ⟨∅, by simp⟩
+lemma ultrafilter_exists [(t : FinSubtheory T) → Nonempty (A t)]
+ (H : ∀ (i : FinSubtheory T), (A i)↓[L] ⊧* (i.val : Theory L)) :
+ ∃ 𝓤 : Ultrafilter (FinSubtheory T), Set.image (Sentence.domain A) T ⊆ 𝓤.sets :=
+ Ultrafilter.exis... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_bc7e5cdeb2c6_3 | 8b7de51e7c0b9352 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Ultraproduct.lean | Ultraproduct | 3 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 5 | 1 | [
{
"theorem_name": "compact",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n rw [compactness_aux]; simp",
"n_chars": 32,
"n_subproofs": 0,
"n_tactics": 3,
"cyclomatic": 1,
"n_automation": ... | [
{
"name": "compactness_aux",
"text": "lemma compactness_aux :\n Satisfiable T ↔ ∀ i : FinSubtheory T, Satisfiable (i.val : Theory L) := by\n constructor\n · rintro h ⟨t, ht⟩; exact Semantics.Satisfiable.of_subset h ht\n · intro h\n have : ∀ i : FinSubtheory T, ∃ (M : Type u) (_ : Nonempty M) (_ : S... | [
{
"name": "compact",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 4,
"n_lines": 5,
"n_chars": 147,
"n_subproofs": 0,
"n_tactics": 3,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 1,
"n_structural": 0,
"automation_only": false,
"max_nesting": 2
}... | 3 | module
public import Foundation.FirstOrder.Basic
public import Mathlib.Order.Filter.Ultrafilter.Basic
@[expose] public section
namespace LO
namespace FirstOrder
section
universe u v
variable {L : Language.{u}} {ξ : Type v}
{I : Type u} (A : I → Type u)
[s : (i : I) → FirstOrder.Structure L (A i)]
(𝓤 : Ultr... | module
public import Foundation.FirstOrder.Basic
public import Mathlib.Order.Filter.Ultrafilter.Basic
@[expose] public section
namespace LO
namespace FirstOrder
section
universe u v
variable {L : Language.{u}} {ξ : Type v}
{I : Type u} (A : I → Type u)
[s : (i : I) → FirstOrder.Structure L (A i)]
(𝓤 : Ultr... | @@ -150,8 +150,22 @@
suffices ∀ i ∈ t, (A ⟨t, htT⟩)↓[L] ⊧ i by simpa [Sentence.domain] using this
intro i hi; exact (H ⟨t, htT⟩).models_set hi⟩)
+lemma compactness_aux :
+ Satisfiable T ↔ ∀ i : FinSubtheory T, Satisfiable (i.val : Theory L) := by
+ constructor
+ · rintro h ⟨t, ht⟩; exact Semantics.... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_7a83834db98b_0 | d49c07550a9261ad | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Incompleteness/RosserProvability.lean | RosserProvability | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 1 | 3 | [
{
"theorem_name": "rosser_quote₀",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n simpa [Sentence.quote_def] using rosser_quote",
"n_chars": 51,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic"... | [
{
"name": "rosser_quote",
"text": "lemma rosser_quote {φ : Proposition L} : T.RosserProvable (V := V) ⌜φ⌝ ↔ T.ProvabilityComparisonLE (V := V) ⌜φ⌝ ⌜∼φ⌝ := by\n simp [Theory.RosserProvable, Semiformula.quote_def]\n\n",
"fan_in": 3,
"n_lines": 4,
"n_chars": 198,
"n_subproofs": 0,
"n_tacti... | [
{
"name": "rosser_quote₀",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 4,
"n_chars": 192,
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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.Incompleteness.WitnessComparison
@[expose] public section
/-!
# Rosser's provability predicate
-/
namespace LO.FirstOrder.Arithmetic.Bootstrapping
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
variable {L : Language} [L.Encodable] [L.LORDefinable]
variable (... | module
public import Foundation.FirstOrder.Incompleteness.WitnessComparison
@[expose] public section
/-!
# Rosser's provability predicate
-/
namespace LO.FirstOrder.Arithmetic.Bootstrapping
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
variable {L : Language} [L.Encodable] [L.LORDefinable]
variable (... | @@ -35,10 +35,14 @@
variable {T}
-lemma rosser_quote₀ {φ : Sentence L} : T.RosserProvable (V := V) ⌜φ⌝ ↔ T.ProvabilityComparisonLE (V := V) ⌜φ⌝ ⌜∼φ⌝ := sorry
+lemma rosser_quote {φ : Proposition L} : T.RosserProvable (V := V) ⌜φ⌝ ↔ T.ProvabilityComparisonLE (V := V) ⌜φ⌝ ⌜∼φ⌝ := by
+ simp [Theory.RosserProvable, 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_7a83834db98b_1 | 2abff6c5a1e59423 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Incompleteness/RosserProvability.lean | RosserProvability | 1 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 1 | 3 | [
{
"theorem_name": "rosser_quote₀",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n simpa [Sentence.quote_def] using rosser_quote",
"n_chars": 51,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic"... | [
{
"name": "rosser_quote",
"text": "lemma rosser_quote {φ : Proposition L} : T.RosserProvable (V := V) ⌜φ⌝ ↔ T.ProvabilityComparisonLE (V := V) ⌜φ⌝ ⌜∼φ⌝ := by\n simp [Theory.RosserProvable, Semiformula.quote_def]\n\n",
"fan_in": 3,
"n_lines": 4,
"n_chars": 198,
"n_subproofs": 0,
"n_tacti... | [
{
"name": "rosser_quote_def",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 4,
"n_chars": 177,
"n_subproofs": 0,
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"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
"max_nestin... | 1 | module
public import Foundation.FirstOrder.Incompleteness.WitnessComparison
@[expose] public section
/-!
# Rosser's provability predicate
-/
namespace LO.FirstOrder.Arithmetic.Bootstrapping
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
variable {L : Language} [L.Encodable] [L.LORDefinable]
variable (... | module
public import Foundation.FirstOrder.Incompleteness.WitnessComparison
@[expose] public section
/-!
# Rosser's provability predicate
-/
namespace LO.FirstOrder.Arithmetic.Bootstrapping
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
variable {L : Language} [L.Encodable] [L.LORDefinable]
variable (... | @@ -35,13 +35,17 @@
variable {T}
-lemma rosser_quote₀ {φ : Sentence L} : T.RosserProvable (V := V) ⌜φ⌝ ↔ T.ProvabilityComparisonLE (V := V) ⌜φ⌝ ⌜∼φ⌝ := sorry
+lemma rosser_quote {φ : Proposition L} : T.RosserProvable (V := V) ⌜φ⌝ ↔ T.ProvabilityComparisonLE (V := V) ⌜φ⌝ ⌜∼φ⌝ := by
+ simp [Theory.RosserProvable, 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_330ab79e591d_0 | 5e7f53e7a130f4e6 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Entailment/DiaDuality.lean | DiaDuality | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 2 | 1 | [
{
"theorem_name": "iff_top_right!",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " iff_symm'! $ iff_top_left'! h",
"n_chars": 30,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automatio... | [
{
"name": "iff_top_left'!",
"text": "lemma iff_top_left'! : 𝓢 ⊢ φ → 𝓢 ⊢ φ 🡘 ⊤ := λ ⟨h⟩ => ⟨iff_top_left' h⟩\n\n",
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"a... | [
{
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"n_structural": 0,
"automation_only": false,
"max_nesting":... | 2 | module
public import Foundation.Modal.Entailment.Basic
@[expose] public section
namespace LO.Modal.Entailment
open LO.Entailment LO.Entailment.FiniteContext
variable {S F : Type*} [BasicModalLogicalConnective F] [Entailment S F]
variable {𝓢 : S} [Entailment.Cl 𝓢] [Entailment.HasDiaDuality 𝓢] {n : ℕ} {φ ψ ξ χ: F... | module
public import Foundation.Modal.Entailment.Basic
@[expose] public section
namespace LO.Modal.Entailment
open LO.Entailment LO.Entailment.FiniteContext
variable {S F : Type*} [BasicModalLogicalConnective F] [Entailment S F]
variable {𝓢 : S} [Entailment.Cl 𝓢] [Entailment.HasDiaDuality 𝓢] {n : ℕ} {φ ψ ξ χ: F... | @@ -20,12 +20,14 @@
. exact CV;
. exact C_of_conseq h;
+lemma iff_top_left'! : 𝓢 ⊢ φ → 𝓢 ⊢ φ 🡘 ⊤ := λ ⟨h⟩ => ⟨iff_top_left' h⟩
+
lemma iff_symm'! (h : 𝓢 ⊢ φ 🡘 ψ) : 𝓢 ⊢ ψ 🡘 φ := by
apply E!_intro;
. exact K!_right h;
. exact K!_left h;
-lemma iff_top_right! (h : 𝓢 ⊢ φ) : 𝓢 ⊢ ⊤ 🡘 φ := sorry
... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_6e7f31886414_0 | c1024b9aba6ff56b | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Vorspiel/Arithmetic.lean | Arithmetic | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 13 | 2 | [
{
"theorem_name": "and_eq_one",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by simp [and_eq, imp_false, Nat.pos_iff_ne_zero]",
"n_chars": 49,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
... | [
{
"name": "and_eq",
"text": "lemma and_eq (n m : ℕ) : and n m = if 0 < n ∧ 0 < m then 1 else 0 := by simp [mul_pos_iff, and, isLtNat]\n\n",
"fan_in": 2,
"n_lines": 3,
"n_chars": 110,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n... | [
{
"name": "and_eq_one",
"fan_in": 0,
"n_deps_direct": 4,
"n_deps_transitive": 8,
"n_lines": 3,
"n_chars": 116,
"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.Vorspiel.List.Basic
public import Mathlib.Computability.Halting
public import Mathlib.Computability.PartrecBasis
public import Mathlib.Logic.Godel.GodelBetaFunction
@[expose] public section
open Mathlib List.Vector Part
namespace Nat
def isEqNat (n m : ℕ) : ℕ := if n = m then 1 else... | module
public import Foundation.Vorspiel.List.Basic
public import Mathlib.Computability.Halting
public import Mathlib.Computability.PartrecBasis
public import Mathlib.Logic.Godel.GodelBetaFunction
@[expose] public section
open Mathlib List.Vector Part
namespace Nat
def isEqNat (n m : ℕ) : ℕ := if n = m then 1 else... | @@ -45,11 +45,13 @@
def or (n m : ℕ) : ℕ := isLtNat 0 (n + m)
-lemma and_eq_one (n m : ℕ) : and n m = 1 ↔ 0 < n ∧ 0 < m := sorry
+lemma and_eq (n m : ℕ) : and n m = if 0 < n ∧ 0 < m then 1 else 0 := by simp [mul_pos_iff, and, isLtNat]
+lemma and_eq_one (n m : ℕ) : and n m = 1 ↔ 0 < n ∧ 0 < m := by simp [and_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_6e7f31886414_1 | 5271ce067d9b7378 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Vorspiel/Arithmetic.lean | Arithmetic | 1 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 13 | 2 | [
{
"theorem_name": "and_eq_one",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by simp [and_eq, imp_false, Nat.pos_iff_ne_zero]",
"n_chars": 49,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
... | [
{
"name": "and_eq",
"text": "lemma and_eq (n m : ℕ) : and n m = if 0 < n ∧ 0 < m then 1 else 0 := by simp [mul_pos_iff, and, isLtNat]\n\n",
"fan_in": 2,
"n_lines": 3,
"n_chars": 110,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n... | [
{
"name": "and_pos_iff",
"fan_in": 1,
"n_deps_direct": 4,
"n_deps_transitive": 8,
"n_lines": 3,
"n_chars": 132,
"n_subproofs": 0,
"n_tactics": 3,
"cyclomatic": 2,
"n_automation": 2,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
"max_nesting": 1... | 1 | module
public import Foundation.Vorspiel.List.Basic
public import Mathlib.Computability.Halting
public import Mathlib.Computability.PartrecBasis
public import Mathlib.Logic.Godel.GodelBetaFunction
@[expose] public section
open Mathlib List.Vector Part
namespace Nat
def isEqNat (n m : ℕ) : ℕ := if n = m then 1 else... | module
public import Foundation.Vorspiel.List.Basic
public import Mathlib.Computability.Halting
public import Mathlib.Computability.PartrecBasis
public import Mathlib.Logic.Godel.GodelBetaFunction
@[expose] public section
open Mathlib List.Vector Part
namespace Nat
def isEqNat (n m : ℕ) : ℕ := if n = m then 1 else... | @@ -45,9 +45,11 @@
def or (n m : ℕ) : ℕ := isLtNat 0 (n + m)
+lemma and_eq (n m : ℕ) : and n m = if 0 < n ∧ 0 < m then 1 else 0 := by simp [mul_pos_iff, and, isLtNat]
+
lemma or_eq (n m : ℕ) : or n m = if 0 < n ∨ 0 < m then 1 else 0 := by simp [or, isLtNat]
-@[simp] lemma and_pos_iff (n m : ℕ) : 0 < and n m ↔ 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_6e7f31886414_2 | 55b325c5ddff0c14 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Vorspiel/Arithmetic.lean | Arithmetic | 2 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 13 | 2 | [
{
"theorem_name": "or_pos_iff",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by simp [or_eq]; by_cases 0 < n ∨ 0 < m <;> simp [*]",
"n_chars": 55,
"n_subproofs": 0,
"n_tactics": 3,
"cyclomatic": ... | [
{
"name": "or_eq",
"text": "lemma or_eq (n m : ℕ) : or n m = if 0 < n ∨ 0 < m then 1 else 0 := by simp [or, isLtNat]\n\n",
"fan_in": 2,
"n_lines": 3,
"n_chars": 94,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
... | [
{
"name": "or_pos_iff",
"fan_in": 1,
"n_deps_direct": 4,
"n_deps_transitive": 8,
"n_lines": 3,
"n_chars": 129,
"n_subproofs": 0,
"n_tactics": 3,
"cyclomatic": 2,
"n_automation": 2,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
"max_nesting": 1
... | 1 | module
public import Foundation.Vorspiel.List.Basic
public import Mathlib.Computability.Halting
public import Mathlib.Computability.PartrecBasis
public import Mathlib.Logic.Godel.GodelBetaFunction
@[expose] public section
open Mathlib List.Vector Part
namespace Nat
def isEqNat (n m : ℕ) : ℕ := if n = m then 1 else... | module
public import Foundation.Vorspiel.List.Basic
public import Mathlib.Computability.Halting
public import Mathlib.Computability.PartrecBasis
public import Mathlib.Logic.Godel.GodelBetaFunction
@[expose] public section
open Mathlib List.Vector Part
namespace Nat
def isEqNat (n m : ℕ) : ℕ := if n = m then 1 else... | @@ -47,9 +47,11 @@
lemma and_eq (n m : ℕ) : and n m = if 0 < n ∧ 0 < m then 1 else 0 := by simp [mul_pos_iff, and, isLtNat]
+lemma or_eq (n m : ℕ) : or n m = if 0 < n ∨ 0 < m then 1 else 0 := by simp [or, isLtNat]
+
@[simp] lemma and_pos_iff (n m : ℕ) : 0 < and n m ↔ 0 < n ∧ 0 < m := by simp [and_eq]; by_cases 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_6e7f31886414_3 | 3302e31da00bb6a7 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Vorspiel/Arithmetic.lean | Arithmetic | 3 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 13 | 1 | [
{
"theorem_name": "map",
"depth": 1,
"n_commands": 0,
"n_lines": 3,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": "\n (bind (Part.some $ f · ·) (hf.of_eq <| by simp) hg).of_eq <| by\n intro v; rcases Part.eq_none_or_eq_some (g v) with (_ | ⟨x, _⟩) <;> sim... | [
{
"name": "bind",
"text": "lemma bind (f : List.Vector ℕ n → ℕ →. ℕ) (hf : @ArithPart₁ (n + 1) fun v => f v.tail v.head) {g} (hg : @ArithPart₁ n g) :\n @ArithPart₁ n fun v ↦ (g v).bind (f v) :=\n (hf.comp (g :> fun i v ↦ v.get i) (fun i ↦ by\n cases i using Fin.cases\n · simp_all\n · solve_by_e... | [
{
"name": "map",
"fan_in": 0,
"n_deps_direct": 3,
"n_deps_transitive": 5,
"n_lines": 6,
"n_chars": 336,
"n_subproofs": 0,
"n_tactics": 4,
"cyclomatic": 3,
"n_automation": 2,
"n_rewrites": 0,
"n_structural": 2,
"automation_only": false,
"max_nesting": 2
}
] | 1 | module
public import Foundation.Vorspiel.List.Basic
public import Mathlib.Computability.Halting
public import Mathlib.Computability.PartrecBasis
public import Mathlib.Logic.Godel.GodelBetaFunction
@[expose] public section
open Mathlib List.Vector Part
namespace Nat
def isEqNat (n m : ℕ) : ℕ := if n = m then 1 else... | module
public import Foundation.Vorspiel.List.Basic
public import Mathlib.Computability.Halting
public import Mathlib.Computability.PartrecBasis
public import Mathlib.Logic.Godel.GodelBetaFunction
@[expose] public section
open Mathlib List.Vector Part
namespace Nat
def isEqNat (n m : ℕ) : ℕ := if n = m then 1 else... | @@ -96,8 +96,25 @@
lemma of_eq {n} {f g : List.Vector ℕ n →. ℕ} (hf : ArithPart₁ f) (H : ∀ i, f i = g i) : ArithPart₁ g :=
(funext H : f = g) ▸ hf
+lemma bind (f : List.Vector ℕ n → ℕ →. ℕ) (hf : @ArithPart₁ (n + 1) fun v => f v.tail v.head) {g} (hg : @ArithPart₁ n g) :
+ @ArithPart₁ n fun v ↦ (g v).bind (f 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_6e7f31886414_4 | 1420089ec46a401d | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Vorspiel/Arithmetic.lean | Arithmetic | 4 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 13 | 2 | [
{
"theorem_name": "or_pos_iff",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by simp [or_eq]; by_cases 0 < n ∨ 0 < m <;> simp [*]",
"n_chars": 55,
"n_subproofs": 0,
"n_tactics": 3,
"cyclomatic": ... | [
{
"name": "or_eq",
"text": "lemma or_eq (n m : ℕ) : or n m = if 0 < n ∨ 0 < m then 1 else 0 := by simp [or, isLtNat]\n\n",
"fan_in": 2,
"n_lines": 3,
"n_chars": 94,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
... | [
{
"name": "inv_fun",
"fan_in": 0,
"n_deps_direct": 9,
"n_deps_transitive": 19,
"n_lines": 9,
"n_chars": 597,
"n_subproofs": 1,
"n_tactics": 7,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 2,
"automation_only": false,
"max_nesting": 6
... | 2 | module
public import Foundation.Vorspiel.List.Basic
public import Mathlib.Computability.Halting
public import Mathlib.Computability.PartrecBasis
public import Mathlib.Logic.Godel.GodelBetaFunction
@[expose] public section
open Mathlib List.Vector Part
namespace Nat
def isEqNat (n m : ℕ) : ℕ := if n = m then 1 else... | module
public import Foundation.Vorspiel.List.Basic
public import Mathlib.Computability.Halting
public import Mathlib.Computability.PartrecBasis
public import Mathlib.Logic.Godel.GodelBetaFunction
@[expose] public section
open Mathlib List.Vector Part
namespace Nat
def isEqNat (n m : ℕ) : ℕ := if n = m then 1 else... | @@ -47,9 +47,11 @@
lemma and_eq (n m : ℕ) : and n m = if 0 < n ∧ 0 < m then 1 else 0 := by simp [mul_pos_iff, and, isLtNat]
+lemma or_eq (n m : ℕ) : or n m = if 0 < n ∨ 0 < m then 1 else 0 := by simp [or, isLtNat]
+
@[simp] lemma and_pos_iff (n m : ℕ) : 0 < and n m ↔ 0 < n ∧ 0 < m := by simp [and_eq]; by_cases 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_6e7f31886414_5 | c35377e9e989790e | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Vorspiel/Arithmetic.lean | Arithmetic | 5 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 13 | 3 | [
{
"theorem_name": "pair",
"depth": 1,
"n_commands": 0,
"n_lines": 6,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n have := if_pos (lt i j)\n ((add 0 1).comp₂ _ (mul j j) (proj i))\n ((add 0 1).comp₂ _ ((add 0 1).comp₂ _ (mul i i) (proj i)) (pr... | [
{
"name": "if_pos",
"text": "lemma if_pos {n} {f g h : List.Vector ℕ n → ℕ} (hf : Arithmetic₁ f) (hg : Arithmetic₁ g) (hh : Arithmetic₁ h) :\n Arithmetic₁ (fun v => if 0 < f v then g v else h v) := by\n have : Arithmetic₁ (fun v => (f v).pos * (g v) + (f v).inv * (h v)) :=\n (add 0 1).comp₂ _\n ... | [
{
"name": "pair",
"fan_in": 0,
"n_deps_direct": 7,
"n_deps_transitive": 13,
"n_lines": 8,
"n_chars": 295,
"n_subproofs": 1,
"n_tactics": 7,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 2,
"automation_only": false,
"max_nesting": 4
}
] | 1 | module
public import Foundation.Vorspiel.List.Basic
public import Mathlib.Computability.Halting
public import Mathlib.Computability.PartrecBasis
public import Mathlib.Logic.Godel.GodelBetaFunction
@[expose] public section
open Mathlib List.Vector Part
namespace Nat
def isEqNat (n m : ℕ) : ℕ := if n = m then 1 else... | module
public import Foundation.Vorspiel.List.Basic
public import Mathlib.Computability.Halting
public import Mathlib.Computability.PartrecBasis
public import Mathlib.Logic.Godel.GodelBetaFunction
@[expose] public section
open Mathlib List.Vector Part
namespace Nat
def isEqNat (n m : ℕ) : ℕ := if n = m then 1 else... | @@ -193,6 +193,15 @@
lemma le {n} (i j : Fin n) : @Arithmetic₁ n (fun v => isLeNat (v.get i) (v.get j)) :=
((or 0 1).comp₂ _ (lt i j) (equal i j)).of_eq <| by simp [Nat.or_eq, Nat.le_iff_lt_or_eq, isLeNat]
+lemma if_pos {n} {f g h : List.Vector ℕ n → ℕ} (hf : Arithmetic₁ f) (hg : Arithmetic₁ g) (hh : Arithmetic₁ ... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_6e7f31886414_6 | b11b0cf4c9a9077c | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Vorspiel/Arithmetic.lean | Arithmetic | 6 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 13 | 3 | [
{
"theorem_name": "pair",
"depth": 1,
"n_commands": 0,
"n_lines": 6,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n have := if_pos (lt i j)\n ((add 0 1).comp₂ _ (mul j j) (proj i))\n ((add 0 1).comp₂ _ ((add 0 1).comp₂ _ (mul i i) (proj i)) (pr... | [
{
"name": "if_pos",
"text": "lemma if_pos {n} {f g h : List.Vector ℕ n → ℕ} (hf : Arithmetic₁ f) (hg : Arithmetic₁ g) (hh : Arithmetic₁ h) :\n Arithmetic₁ (fun v => if 0 < f v then g v else h v) := by\n have : Arithmetic₁ (fun v => (f v).pos * (g v) + (f v).inv * (h v)) :=\n (add 0 1).comp₂ _\n ... | [
{
"name": "beta_unbeta_recSequence_zero",
"fan_in": 0,
"n_deps_direct": 3,
"n_deps_transitive": 37,
"n_lines": 5,
"n_chars": 276,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
"n_automation": 2,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
... | 10 | module
public import Foundation.Vorspiel.List.Basic
public import Mathlib.Computability.Halting
public import Mathlib.Computability.PartrecBasis
public import Mathlib.Logic.Godel.GodelBetaFunction
@[expose] public section
open Mathlib List.Vector Part
namespace Nat
def isEqNat (n m : ℕ) : ℕ := if n = m then 1 else... | module
public import Foundation.Vorspiel.List.Basic
public import Mathlib.Computability.Halting
public import Mathlib.Computability.PartrecBasis
public import Mathlib.Logic.Godel.GodelBetaFunction
@[expose] public section
open Mathlib List.Vector Part
namespace Nat
def isEqNat (n m : ℕ) : ℕ := if n = m then 1 else... | @@ -193,6 +193,15 @@
lemma le {n} (i j : Fin n) : @Arithmetic₁ n (fun v => isLeNat (v.get i) (v.get j)) :=
((or 0 1).comp₂ _ (lt i j) (equal i j)).of_eq <| by simp [Nat.or_eq, Nat.le_iff_lt_or_eq, isLeNat]
+lemma if_pos {n} {f g h : List.Vector ℕ n → ℕ} (hf : Arithmetic₁ f) (hg : Arithmetic₁ g) (hh : Arithmetic₁ ... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_6e7f31886414_7 | 43ab0a1ddbea9c9f | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Vorspiel/Arithmetic.lean | Arithmetic | 7 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 13 | 1 | [
{
"theorem_name": "_root_.Nat.ArithPart₁.of_partrec",
"depth": 1,
"n_commands": 0,
"n_lines": 5,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n induction hf\n case prim f hf => exact of_primrec hf\n case comp f g _ _ hf hg => exact hf.comp... | [
{
"name": "of_primrec",
"text": "lemma of_primrec {f : List.Vector ℕ n → ℕ} (hf : Primrec' f) : Arithmetic₁ f := by\n induction hf\n case zero => exact zero\n case succ => exact (@succ 1 0).of_eq (by simp)\n case get i => exact proj i\n case comp f g _ _ hf hg =... | [
{
"name": "_root_.Nat.ArithPart₁.of_partrec",
"fan_in": 0,
"n_deps_direct": 2,
"n_deps_transitive": 9,
"n_lines": 7,
"n_chars": 285,
"n_subproofs": 0,
"n_tactics": 5,
"cyclomatic": 2,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 4,
"automation_only": false,... | 1 | module
public import Foundation.Vorspiel.List.Basic
public import Mathlib.Computability.Halting
public import Mathlib.Computability.PartrecBasis
public import Mathlib.Logic.Godel.GodelBetaFunction
@[expose] public section
open Mathlib List.Vector Part
namespace Nat
def isEqNat (n m : ℕ) : ℕ := if n = m then 1 else... | module
public import Foundation.Vorspiel.List.Basic
public import Mathlib.Computability.Halting
public import Mathlib.Computability.PartrecBasis
public import Mathlib.Logic.Godel.GodelBetaFunction
@[expose] public section
open Mathlib List.Vector Part
namespace Nat
def isEqNat (n m : ℕ) : ℕ := if n = m then 1 else... | @@ -446,8 +446,20 @@
beta_unbeta_recSequence_zero f g v.head v.tail,
fun i hi => beta_unbeta_recSequence_succ f g v.head v.tail hi⟩
-lemma _root_.Nat.ArithPart₁.of_partrec {f : List.Vector ℕ n →. ℕ} (hf : Partrec' f) : ArithPart₁ f := sorry
+lemma of_primrec {f : List.Vector ℕ n → ℕ} (hf : Primrec' 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_7fe832d7440d_0 | 7460292f67cbee5a | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Definability/Definable.lean | Definable | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 15 | 1 | [
{
"theorem_name": "to_definable₀",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " Defined.to_definable (φ.ofZero ℌ) hP.of_zero",
"n_chars": 48,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
... | [
{
"name": "to_definable",
"text": "lemma to_definable (φ : ℌ.Semisentence k) (hP : Defined P φ) : ℌ.Definable P := ⟨φ.rew Rew.emb, by\n match ℌ with\n | 𝚺-[_] => intro; simp [hP.iff]\n | 𝚷-[_] => intro; simp [hP.iff]\n | 𝚫-[_] => exact ⟨\n fun v ↦ by rcases φ; simpa [HierarchySymbol.Semiformula.re... | [
{
"name": "to_definable₀",
"fan_in": 0,
"n_deps_direct": 2,
"n_deps_transitive": 5,
"n_lines": 4,
"n_chars": 146,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
"max_nesting":... | 1 | module
public import Foundation.FirstOrder.Arithmetic.Definability.Hierarchy
@[expose] public section
namespace LO.FirstOrder.Arithmetic.HierarchySymbol
variable (ξ : Type*) (n : ℕ)
open PeanoMinus
variable {V : Type*} [ORingStructure V]
abbrev IsDefinedBy (R : (Fin k → V) → Prop) : {ℌ : HierarchySymbol} → ℌ.Semi... | module
public import Foundation.FirstOrder.Arithmetic.Definability.Hierarchy
@[expose] public section
namespace LO.FirstOrder.Arithmetic.HierarchySymbol
variable (ξ : Type*) (n : ℕ)
open PeanoMinus
variable {V : Type*} [ORingStructure V]
abbrev IsDefinedBy (R : (Fin k → V) → Prop) : {ℌ : HierarchySymbol} → ℌ.Semi... | @@ -203,8 +203,16 @@
lemma of_iff {P Q : (Fin k → V) → Prop} (h : ∀ x, P x ↔ Q x) {φ : ℌ.Semisentence k} (H : Defined Q φ) : Defined P φ := by
rwa [show P = Q from by funext v; simp [h]]
+lemma to_definable (φ : ℌ.Semisentence k) (hP : Defined P φ) : ℌ.Definable P := ⟨φ.rew Rew.emb, by
+ match ℌ with
+ | 𝚺-[_]... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_7fe832d7440d_1 | 9c98287137962b31 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Definability/Definable.lean | Definable | 1 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 15 | 1 | [
{
"theorem_name": "not",
"depth": 1,
"n_commands": 0,
"n_lines": 5,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": "\n match Γ with\n | 𝚺 => h.notPi\n | 𝚷 => h.notSigma\n | 𝚫 => h.notDelta",
"n_chars": 82,
"n_subproofs": 0,
"n_tactics": 4,... | [
{
"name": "notPi",
"text": "lemma notPi (h : 𝚷-[m].Definable P) : 𝚺-[m].Definable fun x ↦ ¬P x := by\n rcases h with ⟨φ, h⟩\n exact Definable.mk' φ.negPi fun v ↦ by simp [h.iff]\n\n",
"fan_in": 1,
"n_lines": 5,
"n_chars": 168,
"n_subproofs": 0,
"n_tactics": 3,
"cyclomatic": 2,
... | [
{
"name": "not",
"fan_in": 0,
"n_deps_direct": 3,
"n_deps_transitive": 7,
"n_lines": 8,
"n_chars": 166,
"n_subproofs": 0,
"n_tactics": 4,
"cyclomatic": 2,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
"max_nesting": 2
}
] | 3 | module
public import Foundation.FirstOrder.Arithmetic.Definability.Hierarchy
@[expose] public section
namespace LO.FirstOrder.Arithmetic.HierarchySymbol
variable (ξ : Type*) (n : ℕ)
open PeanoMinus
variable {V : Type*} [ORingStructure V]
abbrev IsDefinedBy (R : (Fin k → V) → Prop) : {ℌ : HierarchySymbol} → ℌ.Semi... | module
public import Foundation.FirstOrder.Arithmetic.Definability.Hierarchy
@[expose] public section
namespace LO.FirstOrder.Arithmetic.HierarchySymbol
variable (ξ : Type*) (n : ℕ)
open PeanoMinus
variable {V : Type*} [ORingStructure V]
abbrev IsDefinedBy (R : (Fin k → V) → Prop) : {ℌ : HierarchySymbol} → ℌ.Semi... | @@ -358,6 +358,10 @@
lemma notSigma (h : 𝚺-[m].Definable P) : 𝚷-[m].Definable fun x ↦ ¬P x := by
rcases h with ⟨φ, h⟩; exact Definable.mk' φ.negSigma fun v ↦ by simp [h.iff]
+lemma notPi (h : 𝚷-[m].Definable P) : 𝚺-[m].Definable fun x ↦ ¬P x := by
+ rcases h with ⟨φ, h⟩
+ exact Definable.mk' φ.negPi fun 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_7fe832d7440d_2 | 2fba50eca022e213 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Definability/Definable.lean | Definable | 2 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 15 | 2 | [
{
"theorem_name": "not",
"depth": 1,
"n_commands": 0,
"n_lines": 5,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": "\n match Γ with\n | 𝚺 => h.notPi\n | 𝚷 => h.notSigma\n | 𝚫 => h.notDelta",
"n_chars": 82,
"n_subproofs": 0,
"n_tactics": 4,... | [
{
"name": "notDelta",
"text": "lemma notDelta (h : 𝚫-[m].Definable P) : 𝚫-[m].Definable fun x ↦ ¬P x := by\n rcases h with ⟨φ, h⟩\n exact Definable.mk' (∼φ) ⟨h.proper.neg, by intro v; simp [h.proper.eval_neg, h.iff]⟩\n\n",
"fan_in": 2,
"n_lines": 5,
"n_chars": 208,
"n_subproofs": 0,
... | [
{
"name": "impDelta",
"fan_in": 1,
"n_deps_direct": 3,
"n_deps_transitive": 7,
"n_lines": 4,
"n_chars": 185,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": false,
"max_nesting": 1
... | 1 | module
public import Foundation.FirstOrder.Arithmetic.Definability.Hierarchy
@[expose] public section
namespace LO.FirstOrder.Arithmetic.HierarchySymbol
variable (ξ : Type*) (n : ℕ)
open PeanoMinus
variable {V : Type*} [ORingStructure V]
abbrev IsDefinedBy (R : (Fin k → V) → Prop) : {ℌ : HierarchySymbol} → ℌ.Semi... | module
public import Foundation.FirstOrder.Arithmetic.Definability.Hierarchy
@[expose] public section
namespace LO.FirstOrder.Arithmetic.HierarchySymbol
variable (ξ : Type*) (n : ℕ)
open PeanoMinus
variable {V : Type*} [ORingStructure V]
abbrev IsDefinedBy (R : (Fin k → V) → Prop) : {ℌ : HierarchySymbol} → ℌ.Semi... | @@ -362,15 +362,19 @@
rcases h with ⟨φ, h⟩
exact Definable.mk' φ.negPi fun v ↦ by simp [h.iff]
+lemma notDelta (h : 𝚫-[m].Definable P) : 𝚫-[m].Definable fun x ↦ ¬P x := by
+ rcases h with ⟨φ, h⟩
+ exact Definable.mk' (∼φ) ⟨h.proper.neg, by intro v; simp [h.proper.eval_neg, h.iff]⟩
+
lemma not (h : Γ.alt-[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_7fe832d7440d_3 | c0bcb55fe76b0d9e | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Definability/Definable.lean | Definable | 3 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 15 | 1 | [
{
"theorem_name": "ball'",
"depth": 1,
"n_commands": 0,
"n_lines": 3,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n apply (ball h ‘!!t + 1’).of_iff\n intro v; simp [lt_succ_iff_le]",
"n_chars": 74,
"n_subproofs": 0,
"n_tactics": 4,
... | [
{
"name": "ball",
"text": "lemma ball {P : (Fin k → V) → V → Prop} (h : ℌ.Definable fun w ↦ P (w ·.succ) (w 0)) (t : ArithmeticSemiterm V k) :\n ℌ.Definable fun v ↦ ∀ x < t.val v id, P v x := by\n rcases h with ⟨φ, h⟩\n match ℌ with\n | 𝚺-[m] => exact ⟨HierarchySymbol.Semiformula.ball t φ, by intro v... | [
{
"name": "ball'",
"fan_in": 0,
"n_deps_direct": 2,
"n_deps_transitive": 5,
"n_lines": 6,
"n_chars": 297,
"n_subproofs": 0,
"n_tactics": 4,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 2,
"automation_only": false,
"max_nesting": 2
}
] | 1 | module
public import Foundation.FirstOrder.Arithmetic.Definability.Hierarchy
@[expose] public section
namespace LO.FirstOrder.Arithmetic.HierarchySymbol
variable (ξ : Type*) (n : ℕ)
open PeanoMinus
variable {V : Type*} [ORingStructure V]
abbrev IsDefinedBy (R : (Fin k → V) → Prop) : {ℌ : HierarchySymbol} → ℌ.Semi... | module
public import Foundation.FirstOrder.Arithmetic.Definability.Hierarchy
@[expose] public section
namespace LO.FirstOrder.Arithmetic.HierarchySymbol
variable (ξ : Type*) (n : ℕ)
open PeanoMinus
variable {V : Type*} [ORingStructure V]
abbrev IsDefinedBy (R : (Fin k → V) → Prop) : {ℌ : HierarchySymbol} → ℌ.Semi... | @@ -392,8 +392,18 @@
Γ-[m].Definable (fun v ↦ P v ↔ Q v) :=
.of_delta <| ((h₁.impDelta h₂).and (h₂.impDelta h₁)).of_iff <| by intro v; simp [iff_iff_implies_and_implies]
+lemma ball {P : (Fin k → V) → V → Prop} (h : ℌ.Definable fun w ↦ P (w ·.succ) (w 0)) (t : ArithmeticSemiterm V k) :
+ ℌ.Definable fun 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... |
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