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_68c05002cd23_4 | b7a2843577be2bd3 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/ComplementClosedConsistentFinset.lean | ComplementClosedConsistentFinset | 4 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 12 | 1 | [
{
"theorem_name": "enum_subset",
"depth": 1,
"n_commands": 0,
"n_lines": 4,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n induction l with\n | nil => simp;\n | cons ψ qs ih => exact Set.Subset.trans ih $ by apply enum_subset_step;",
"n_chars":... | [
{
"name": "enum_subset_step",
"text": "lemma enum_subset_step {l : List (Formula α)} : (Φ[l]) ⊆ (Φ[(ψ :: l)]) := by\n simp [enum, next];\n split <;> simp;\n\n",
"fan_in": 1,
"n_lines": 5,
"n_chars": 123,
"n_subproofs": 0,
"n_tactics": 6,
"cyclomatic": 3,
"n_automation": 2,
... | [
{
"name": "enum_subset",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 6,
"n_chars": 176,
"n_subproofs": 0,
"n_tactics": 6,
"cyclomatic": 2,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 3,
"automation_only": false,
"max_nesting": 2... | 1 | module
public import Foundation.Modal.MaximalConsistentSet
public import Foundation.Modal.Formula.Complement
@[expose] public section
namespace LO.Modal
open LO.Entailment
variable {α : Type*} [DecidableEq α]
variable {S} [Entailment S (Formula α)]
variable {𝓢 : S}
namespace FormulaFinset
variable {Φ Φ₁ Φ₂ : Fo... | module
public import Foundation.Modal.MaximalConsistentSet
public import Foundation.Modal.Formula.Complement
@[expose] public section
namespace LO.Modal
open LO.Entailment
variable {α : Type*} [DecidableEq α]
variable {S} [Entailment S (Formula α)]
variable {𝓢 : S}
namespace FormulaFinset
variable {Φ Φ₁ Φ₂ : Fo... | @@ -99,10 +99,14 @@
@[simp] lemma enum_nil {Φ : FormulaFinset α} : (Φ[[]]) = Φ := by simp [enum]
+lemma enum_subset_step {l : List (Formula α)} : (Φ[l]) ⊆ (Φ[(ψ :: l)]) := by
+ simp [enum, next];
+ split <;> simp;
+
lemma enum_subset {l : List (Formula α)} : Φ ⊆ Φ[l] := by
induction l with
| nil => 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_68c05002cd23_5 | 663bca5edbf6be14 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/ComplementClosedConsistentFinset.lean | ComplementClosedConsistentFinset | 5 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 12 | 2 | [
{
"theorem_name": "unprovable_falsum",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " X.consistent",
"n_chars": 13,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_... | [
{
"name": "consistent",
"text": "lemma consistent (X : ComplementClosedConsistentFinset 𝓢 Ψ) : Consistent 𝓢 X := X.2.1\n\n",
"fan_in": 2,
"n_lines": 3,
"n_chars": 94,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": ... | [
{
"name": "unprovable_falsum",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 3,
"n_chars": 70,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
"max_nestin... | 1 | module
public import Foundation.Modal.MaximalConsistentSet
public import Foundation.Modal.Formula.Complement
@[expose] public section
namespace LO.Modal
open LO.Entailment
variable {α : Type*} [DecidableEq α]
variable {S} [Entailment S (Formula α)]
variable {𝓢 : S}
namespace FormulaFinset
variable {Φ Φ₁ Φ₂ : Fo... | module
public import Foundation.Modal.MaximalConsistentSet
public import Foundation.Modal.Formula.Complement
@[expose] public section
namespace LO.Modal
open LO.Entailment
variable {α : Type*} [DecidableEq α]
variable {S} [Entailment S (Formula α)]
variable {𝓢 : S}
namespace FormulaFinset
variable {Φ Φ₁ Φ₂ : Fo... | @@ -185,11 +185,13 @@
instance : Membership (Formula α) (ComplementClosedConsistentFinset 𝓢 Ψ) := ⟨λ X φ => φ ∈ X.1⟩
+lemma consistent (X : ComplementClosedConsistentFinset 𝓢 Ψ) : Consistent 𝓢 X := X.2.1
+
lemma closed (X : ComplementClosedConsistentFinset 𝓢 Ψ) : ComplementaryClosed X Ψ := X.2.2
variable {... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_68c05002cd23_6 | d75af685364bc31b | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/ComplementClosedConsistentFinset.lean | ComplementClosedConsistentFinset | 6 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 12 | 2 | [
{
"theorem_name": "mem_of_not_mem_compl",
"depth": 1,
"n_commands": 0,
"n_lines": 3,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n apply Not.imp_symm;\n exact mem_compl_of_not_mem hs;",
"n_chars": 58,
"n_subproofs": 0,
"n_tactics": 5,
... | [
{
"name": "mem_compl_of_not_mem",
"text": "lemma mem_compl_of_not_mem (hs : ψ ∈ Ψ) : ψ ∉ X → -ψ ∈ X := by\n intro h;\n rcases X.closed.either ψ (by assumption) with (h | h);\n . contradiction;\n . assumption;\n\n",
"fan_in": 2,
"n_lines": 7,
"n_chars": 180,
"n_subproofs": 0,
"n_tacti... | [
{
"name": "mem_of_not_mem_compl",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 3,
"n_lines": 5,
"n_chars": 131,
"n_subproofs": 0,
"n_tactics": 5,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 2,
"automation_only": false,
"max_ne... | 1 | module
public import Foundation.Modal.MaximalConsistentSet
public import Foundation.Modal.Formula.Complement
@[expose] public section
namespace LO.Modal
open LO.Entailment
variable {α : Type*} [DecidableEq α]
variable {S} [Entailment S (Formula α)]
variable {𝓢 : S}
namespace FormulaFinset
variable {Φ Φ₁ Φ₂ : Fo... | module
public import Foundation.Modal.MaximalConsistentSet
public import Foundation.Modal.Formula.Complement
@[expose] public section
namespace LO.Modal
open LO.Entailment
variable {α : Type*} [DecidableEq α]
variable {S} [Entailment S (Formula α)]
variable {𝓢 : S}
namespace FormulaFinset
variable {Φ Φ₁ Φ₂ : Fo... | @@ -193,8 +193,16 @@
@[simp] lemma unprovable_falsum : X.1 *⊬[𝓢] ⊥ := X.consistent
-lemma mem_of_not_mem_compl (hs : ψ ∈ Ψ) : -ψ ∉ X → ψ ∈ X := sorry
+lemma mem_compl_of_not_mem (hs : ψ ∈ Ψ) : ψ ∉ X → -ψ ∈ X := by
+ intro h;
+ rcases X.closed.either ψ (by assumption) with (h | h);
+ . contradiction;
+ . assum... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_68c05002cd23_7 | c9db63da27bd840c | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/ComplementClosedConsistentFinset.lean | ComplementClosedConsistentFinset | 7 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 12 | 3 | [
{
"theorem_name": "mem_verum",
"depth": 1,
"n_commands": 0,
"n_lines": 3,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n apply membership_iff h |>.mpr;\n exact verum!;",
"n_chars": 52,
"n_subproofs": 0,
"n_tactics": 5,
"cyclomatic": ... | [
{
"name": "membership_iff",
"text": "lemma membership_iff (hq_sub : ψ ∈ Ψ) : (ψ ∈ X) ↔ (X *⊢[𝓢] ψ) := by\n constructor;\n . intro h; exact Context.by_axm! h;\n . intro hp;\n suffices -ψ ∉ X by\n apply or_iff_not_imp_right.mp $ ?_;\n assumption;\n exact X.closed.either ψ hq_sub;\n by... | [
{
"name": "mem_verum",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 3,
"n_lines": 5,
"n_chars": 101,
"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.MaximalConsistentSet
public import Foundation.Modal.Formula.Complement
@[expose] public section
namespace LO.Modal
open LO.Entailment
variable {α : Type*} [DecidableEq α]
variable {S} [Entailment S (Formula α)]
variable {𝓢 : S}
namespace FormulaFinset
variable {Φ Φ₁ Φ₂ : Fo... | module
public import Foundation.Modal.MaximalConsistentSet
public import Foundation.Modal.Formula.Complement
@[expose] public section
namespace LO.Modal
open LO.Entailment
variable {α : Type*} [DecidableEq α]
variable {S} [Entailment S (Formula α)]
variable {𝓢 : S}
namespace FormulaFinset
variable {Φ Φ₁ Φ₂ : Fo... | @@ -209,8 +209,23 @@
noncomputable instance [Entailment.Consistent 𝓢] : Inhabited (ComplementClosedConsistentFinset 𝓢 Ψ) := ⟨lindenbaum (Φ := ∅) (Ψ := Ψ) (by simp) (FormulaFinset.empty_conisistent) |>.choose⟩
-lemma mem_verum (h : ⊤ ∈ Ψ) : ⊤ ∈ X := sorry
+lemma membership_iff (hq_sub : ψ ∈ Ψ) : (ψ ∈ X) ↔ (X *⊢[�... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_68c05002cd23_8 | d0bee287a68d84dc | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/ComplementClosedConsistentFinset.lean | ComplementClosedConsistentFinset | 8 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 12 | 2 | [
{
"theorem_name": "mem_of_not_mem_compl",
"depth": 1,
"n_commands": 0,
"n_lines": 3,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n apply Not.imp_symm;\n exact mem_compl_of_not_mem hs;",
"n_chars": 58,
"n_subproofs": 0,
"n_tactics": 5,
... | [
{
"name": "mem_compl_of_not_mem",
"text": "lemma mem_compl_of_not_mem (hs : ψ ∈ Ψ) : ψ ∉ X → -ψ ∈ X := by\n intro h;\n rcases X.closed.either ψ (by assumption) with (h | h);\n . contradiction;\n . assumption;\n\n",
"fan_in": 2,
"n_lines": 7,
"n_chars": 180,
"n_subproofs": 0,
"n_tacti... | [
{
"name": "iff_not_mem_compl",
"fan_in": 2,
"n_deps_direct": 3,
"n_deps_transitive": 7,
"n_lines": 29,
"n_chars": 1159,
"n_subproofs": 8,
"n_tactics": 50,
"cyclomatic": 3,
"n_automation": 9,
"n_rewrites": 0,
"n_structural": 6,
"automation_only": false,
"max_ne... | 4 | module
public import Foundation.Modal.MaximalConsistentSet
public import Foundation.Modal.Formula.Complement
@[expose] public section
namespace LO.Modal
open LO.Entailment
variable {α : Type*} [DecidableEq α]
variable {S} [Entailment S (Formula α)]
variable {𝓢 : S}
namespace FormulaFinset
variable {Φ Φ₁ Φ₂ : Fo... | module
public import Foundation.Modal.MaximalConsistentSet
public import Foundation.Modal.Formula.Complement
@[expose] public section
namespace LO.Modal
open LO.Entailment
variable {α : Type*} [DecidableEq α]
variable {S} [Entailment S (Formula α)]
variable {𝓢 : S}
namespace FormulaFinset
variable {Φ Φ₁ Φ₂ : Fo... | @@ -193,8 +193,16 @@
@[simp] lemma unprovable_falsum : X.1 *⊬[𝓢] ⊥ := X.consistent
-lemma mem_of_not_mem_compl (hs : ψ ∈ Ψ) : -ψ ∉ X → ψ ∈ X := sorry
+lemma mem_compl_of_not_mem (hs : ψ ∈ Ψ) : ψ ∉ X → -ψ ∈ X := by
+ intro h;
+ rcases X.closed.either ψ (by assumption) with (h | h);
+ . contradiction;
+ . assum... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_68c05002cd23_9 | a0f1a88ff27e406f | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/ComplementClosedConsistentFinset.lean | ComplementClosedConsistentFinset | 9 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 12 | 2 | [
{
"theorem_name": "mem_of_not_mem_compl",
"depth": 1,
"n_commands": 0,
"n_lines": 3,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n apply Not.imp_symm;\n exact mem_compl_of_not_mem hs;",
"n_chars": 58,
"n_subproofs": 0,
"n_tactics": 5,
... | [
{
"name": "mem_compl_of_not_mem",
"text": "lemma mem_compl_of_not_mem (hs : ψ ∈ Ψ) : ψ ∉ X → -ψ ∈ X := by\n intro h;\n rcases X.closed.either ψ (by assumption) with (h | h);\n . contradiction;\n . assumption;\n\n",
"fan_in": 2,
"n_lines": 7,
"n_chars": 180,
"n_subproofs": 0,
"n_tacti... | [
{
"name": "iff_mem_compl",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 8,
"n_lines": 3,
"n_chars": 134,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"max_nesting": ... | 5 | module
public import Foundation.Modal.MaximalConsistentSet
public import Foundation.Modal.Formula.Complement
@[expose] public section
namespace LO.Modal
open LO.Entailment
variable {α : Type*} [DecidableEq α]
variable {S} [Entailment S (Formula α)]
variable {𝓢 : S}
namespace FormulaFinset
variable {Φ Φ₁ Φ₂ : Fo... | module
public import Foundation.Modal.MaximalConsistentSet
public import Foundation.Modal.Formula.Complement
@[expose] public section
namespace LO.Modal
open LO.Entailment
variable {α : Type*} [DecidableEq α]
variable {S} [Entailment S (Formula α)]
variable {𝓢 : S}
namespace FormulaFinset
variable {Φ Φ₁ Φ₂ : Fo... | @@ -193,8 +193,16 @@
@[simp] lemma unprovable_falsum : X.1 *⊬[𝓢] ⊥ := X.consistent
-lemma mem_of_not_mem_compl (hs : ψ ∈ Ψ) : -ψ ∉ X → ψ ∈ X := sorry
+lemma mem_compl_of_not_mem (hs : ψ ∈ Ψ) : ψ ∉ X → -ψ ∈ X := by
+ intro h;
+ rcases X.closed.either ψ (by assumption) with (h | h);
+ . contradiction;
+ . assum... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_68c05002cd23_10 | e2924d286e9e485e | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/ComplementClosedConsistentFinset.lean | ComplementClosedConsistentFinset | 10 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 12 | 2 | [
{
"theorem_name": "mem_of_not_mem_compl",
"depth": 1,
"n_commands": 0,
"n_lines": 3,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n apply Not.imp_symm;\n exact mem_compl_of_not_mem hs;",
"n_chars": 58,
"n_subproofs": 0,
"n_tactics": 5,
... | [
{
"name": "mem_compl_of_not_mem",
"text": "lemma mem_compl_of_not_mem (hs : ψ ∈ Ψ) : ψ ∉ X → -ψ ∈ X := by\n intro h;\n rcases X.closed.either ψ (by assumption) with (h | h);\n . contradiction;\n . assumption;\n\n",
"fan_in": 2,
"n_lines": 7,
"n_chars": 180,
"n_subproofs": 0,
"n_tacti... | [
{
"name": "iff_mem_imp",
"fan_in": 1,
"n_deps_direct": 3,
"n_deps_transitive": 8,
"n_lines": 29,
"n_chars": 1196,
"n_subproofs": 0,
"n_tactics": 42,
"cyclomatic": 4,
"n_automation": 5,
"n_rewrites": 0,
"n_structural": 15,
"automation_only": false,
"max_nesting... | 5 | module
public import Foundation.Modal.MaximalConsistentSet
public import Foundation.Modal.Formula.Complement
@[expose] public section
namespace LO.Modal
open LO.Entailment
variable {α : Type*} [DecidableEq α]
variable {S} [Entailment S (Formula α)]
variable {𝓢 : S}
namespace FormulaFinset
variable {Φ Φ₁ Φ₂ : Fo... | module
public import Foundation.Modal.MaximalConsistentSet
public import Foundation.Modal.Formula.Complement
@[expose] public section
namespace LO.Modal
open LO.Entailment
variable {α : Type*} [DecidableEq α]
variable {S} [Entailment S (Formula α)]
variable {𝓢 : S}
namespace FormulaFinset
variable {Φ Φ₁ Φ₂ : Fo... | @@ -193,8 +193,16 @@
@[simp] lemma unprovable_falsum : X.1 *⊬[𝓢] ⊥ := X.consistent
-lemma mem_of_not_mem_compl (hs : ψ ∈ Ψ) : -ψ ∉ X → ψ ∈ X := sorry
+lemma mem_compl_of_not_mem (hs : ψ ∈ Ψ) : ψ ∉ X → -ψ ∈ X := by
+ intro h;
+ rcases X.closed.either ψ (by assumption) with (h | h);
+ . contradiction;
+ . assum... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_68c05002cd23_11 | 0c1678c5f7bd5279 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/ComplementClosedConsistentFinset.lean | ComplementClosedConsistentFinset | 11 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 12 | 1 | [
{
"theorem_name": "iff_not_mem_compl",
"depth": 1,
"n_commands": 0,
"n_lines": 27,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n constructor;\n . intro hq; replace hq := membership_iff hq_sub |>.mp hq;\n by_contra hnq;\n induction ψ using For... | [
{
"name": "mem_of_not_mem_compl",
"text": "lemma mem_of_not_mem_compl (hs : ψ ∈ Ψ) : -ψ ∉ X → ψ ∈ X := by\n apply Not.imp_symm;\n exact mem_compl_of_not_mem hs;\n\n",
"fan_in": 1,
"n_lines": 5,
"n_chars": 131,
"n_subproofs": 0,
"n_tactics": 5,
"cyclomatic": 1,
"n_automation": 0... | [
{
"name": "iff_not_mem_imp",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 9,
"n_lines": 8,
"n_chars": 257,
"n_subproofs": 0,
"n_tactics": 3,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"max_nesting"... | 6 | module
public import Foundation.Modal.MaximalConsistentSet
public import Foundation.Modal.Formula.Complement
@[expose] public section
namespace LO.Modal
open LO.Entailment
variable {α : Type*} [DecidableEq α]
variable {S} [Entailment S (Formula α)]
variable {𝓢 : S}
namespace FormulaFinset
variable {Φ Φ₁ Φ₂ : Fo... | module
public import Foundation.Modal.MaximalConsistentSet
public import Foundation.Modal.Formula.Complement
@[expose] public section
namespace LO.Modal
open LO.Entailment
variable {α : Type*} [DecidableEq α]
variable {S} [Entailment S (Formula α)]
variable {𝓢 : S}
namespace FormulaFinset
variable {Φ Φ₁ Φ₂ : Fo... | @@ -199,6 +199,10 @@
. contradiction;
. assumption;
+lemma mem_of_not_mem_compl (hs : ψ ∈ Ψ) : -ψ ∉ X → ψ ∈ X := by
+ apply Not.imp_symm;
+ exact mem_compl_of_not_mem hs;
+
lemma equality_def : X₁ = X₂ ↔ X₁.1 = X₂.1 := by
constructor;
. intro h; cases h; rfl;
@@ -256,7 +260,7 @@
have : ↑X *⊢[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_42331e6d7cb1_0 | 5a74a78b513c5d26 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Vorspiel/Rel/Basic.lean | Basic | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 6 | 1 | [
{
"theorem_name": "comp",
"depth": 1,
"n_commands": 0,
"n_lines": 23,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n constructor;\n . rintro ⟨z, hzx, hzy⟩;\n induction n generalizing x with\n | zero => simp_all;\n | succ n ih =>\n suff... | [
{
"name": "congr",
"text": "lemma congr (h : R.Iterate n x y) (he : n = m) : R.Iterate m x y := by\n subst he;\n exact h;\n\n",
"fan_in": 1,
"n_lines": 5,
"n_chars": 95,
"n_subproofs": 0,
"n_tactics": 5,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 1,
"n_structural... | [
{
"name": "comp",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 25,
"n_chars": 713,
"n_subproofs": 5,
"n_tactics": 41,
"cyclomatic": 8,
"n_automation": 5,
"n_rewrites": 2,
"n_structural": 16,
"automation_only": false,
"max_nesting": 6
}... | 1 | module
public import Mathlib.Data.PNat.Basic
public import Mathlib.Data.Rel
@[expose]
public section
namespace Rel
variable {α} {R : Rel α α} {x y z : α}
local infix:50 " ≺ " => R
def Iterate (R : Rel α α) : ℕ → Rel α α
| 0 => (· = ·)
| n + 1 => fun x y ↦ ∃ z, R x z ∧ R.Iterate n z y
namespace Iterate
@[sim... | module
public import Mathlib.Data.PNat.Basic
public import Mathlib.Data.Rel
@[expose]
public section
namespace Rel
variable {α} {R : Rel α α} {x y z : α}
local infix:50 " ≺ " => R
def Iterate (R : Rel α α) : ℕ → Rel α α
| 0 => (· = ·)
| n + 1 => fun x y ↦ ∃ z, R x z ∧ R.Iterate n z y
namespace Iterate
@[sim... | @@ -48,17 +48,27 @@
. apply ih.mpr;
use z;
+lemma congr (h : R.Iterate n x y) (he : n = m) : R.Iterate m x y := by
+ subst he;
+ exact h;
+
lemma comp : (∃ z, R.Iterate n x z ∧ R.Iterate m z y) ↔ R.Iterate (n + m) x y := by
constructor;
. rintro ⟨z, hzx, hzy⟩;
induction n generalizing x ... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_42331e6d7cb1_1 | 7bdeb4d2a0808cf1 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Vorspiel/Rel/Basic.lean | Basic | 1 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 6 | 3 | [
{
"theorem_name": "unwrap_of_trans_of_pos",
"depth": 1,
"n_commands": 0,
"n_lines": 4,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n have : ∃ m : ℕ+, n = m := ⟨⟨n, h⟩, by simp⟩\n rcases this with ⟨n, rfl⟩\n exact unwrap_of_trans Rxy",
"n_chars... | [
{
"name": "unwrap_of_trans",
"text": "lemma unwrap_of_trans {n : ℕ+} [IsTrans _ R] (Rxy : R.Iterate n x y) : R x y := by\n induction n using PNat.recOn generalizing x with\n | one => simpa using Rxy;\n | succ n ih =>\n obtain ⟨z, Rxz, Rzy⟩ := Rxy;\n exact IsTrans.trans _ _ _ Rxz (ih Rzy);\n\n",
... | [
{
"name": "unwrap_of_trans_of_pos",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 2,
"n_lines": 6,
"n_chars": 222,
"n_subproofs": 1,
"n_tactics": 4,
"cyclomatic": 2,
"n_automation": 2,
"n_rewrites": 0,
"n_structural": 2,
"automation_only": false,
"max_... | 1 | module
public import Mathlib.Data.PNat.Basic
public import Mathlib.Data.Rel
@[expose]
public section
namespace Rel
variable {α} {R : Rel α α} {x y z : α}
local infix:50 " ≺ " => R
def Iterate (R : Rel α α) : ℕ → Rel α α
| 0 => (· = ·)
| n + 1 => fun x y ↦ ∃ z, R x z ∧ R.Iterate n z y
namespace Iterate
@[sim... | module
public import Mathlib.Data.PNat.Basic
public import Mathlib.Data.Rel
@[expose]
public section
namespace Rel
variable {α} {R : Rel α α} {x y z : α}
local infix:50 " ≺ " => R
def Iterate (R : Rel α α) : ℕ → Rel α α
| 0 => (· = ·)
| n + 1 => fun x y ↦ ∃ z, R x z ∧ R.Iterate n z y
namespace Iterate
@[sim... | @@ -48,8 +48,18 @@
. apply ih.mpr;
use z;
-lemma unwrap_of_trans_of_pos {n : ℕ} (h : 0 < n) [IsTrans _ R] (Rxy : R.Iterate n x y) : R x y := sorry
+lemma unwrap_of_trans {n : ℕ+} [IsTrans _ R] (Rxy : R.Iterate n x y) : R x y := by
+ induction n using PNat.recOn generalizing x with
+ | one => simpa u... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_42331e6d7cb1_2 | 9f5eca3b0e26ca81 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Vorspiel/Rel/Basic.lean | Basic | 2 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 6 | 1 | [
{
"theorem_name": "constant_trans_of_pos",
"depth": 1,
"n_commands": 0,
"n_lines": 5,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n rcases (pos_succ_iff pos).mp Rxy with ⟨w, Rxw, hwy⟩\n have : R z w := IsTrans.trans _ _ _ Rzx Rxw\n have := succ_le... | [
{
"name": "succ_left",
"text": "lemma succ_left (Rxz : R x z) (Rzy : R.Iterate n z y) : R.Iterate (n + 1) x y := iff_succ.mp ⟨z, Rxz, Rzy⟩\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,
... | [
{
"name": "constant_trans_of_pos",
"fan_in": 0,
"n_deps_direct": 3,
"n_deps_transitive": 4,
"n_lines": 7,
"n_chars": 317,
"n_subproofs": 2,
"n_tactics": 5,
"cyclomatic": 2,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": false,
"max_n... | 2 | module
public import Mathlib.Data.PNat.Basic
public import Mathlib.Data.Rel
@[expose]
public section
namespace Rel
variable {α} {R : Rel α α} {x y z : α}
local infix:50 " ≺ " => R
def Iterate (R : Rel α α) : ℕ → Rel α α
| 0 => (· = ·)
| n + 1 => fun x y ↦ ∃ z, R x z ∧ R.Iterate n z y
namespace Iterate
@[sim... | module
public import Mathlib.Data.PNat.Basic
public import Mathlib.Data.Rel
@[expose]
public section
namespace Rel
variable {α} {R : Rel α α} {x y z : α}
local infix:50 " ≺ " => R
def Iterate (R : Rel α α) : ℕ → Rel α α
| 0 => (· = ·)
| n + 1 => fun x y ↦ ∃ z, R x z ∧ R.Iterate n z y
namespace Iterate
@[sim... | @@ -28,6 +28,8 @@
have : n.pred + 1 = n := Nat.sub_one_add_one_eq_of_pos pos
simpa only [this] using iff_succ (n := n.pred)
+lemma succ_left (Rxz : R x z) (Rzy : R.Iterate n z y) : R.Iterate (n + 1) x y := iff_succ.mp ⟨z, Rxz, Rzy⟩
+
@[simp]
lemma eq : Rel.Iterate (α := α) (· = ·) n = (· = ·) := by
inducti... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_42331e6d7cb1_3 | 4e4854ff733b5313 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Vorspiel/Rel/Basic.lean | Basic | 3 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 6 | 2 | [
{
"theorem_name": "exists_iterate",
"depth": 1,
"n_commands": 0,
"n_lines": 21,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n constructor;\n . intro h;\n induction h with\n | single h => use 1; simpa;\n | tail Rxy Ryz ih =>\n obtain ⟨... | [
{
"name": "forward",
"text": "lemma forward : (R.Iterate (n + 1) x y) ↔ ∃ z, R.Iterate n x z ∧ R z y := by\n induction n generalizing x y with\n | zero => simp_all;\n | succ n ih =>\n constructor;\n . rintro ⟨z, Rxz, Rzy⟩;\n obtain ⟨w, Rzw, Rwy⟩ := ih.mp Rzy;\n use w;\n constructor;\... | [
{
"name": "exists_iterate",
"fan_in": 2,
"n_deps_direct": 4,
"n_deps_transitive": 4,
"n_lines": 23,
"n_chars": 596,
"n_subproofs": 2,
"n_tactics": 37,
"cyclomatic": 5,
"n_automation": 3,
"n_rewrites": 0,
"n_structural": 16,
"automation_only": false,
"max_nesti... | 1 | module
public import Mathlib.Data.PNat.Basic
public import Mathlib.Data.Rel
@[expose]
public section
namespace Rel
variable {α} {R : Rel α α} {x y z : α}
local infix:50 " ≺ " => R
def Iterate (R : Rel α α) : ℕ → Rel α α
| 0 => (· = ·)
| n + 1 => fun x y ↦ ∃ z, R x z ∧ R.Iterate n z y
namespace Iterate
@[sim... | module
public import Mathlib.Data.PNat.Basic
public import Mathlib.Data.Rel
@[expose]
public section
namespace Rel
variable {α} {R : Rel α α} {x y z : α}
local infix:50 " ≺ " => R
def Iterate (R : Rel α α) : ℕ → Rel α α
| 0 => (· = ·)
| n + 1 => fun x y ↦ ∃ z, R x z ∧ R.Iterate n z y
namespace Iterate
@[sim... | @@ -30,6 +30,24 @@
| zero => rfl;
| succ n ih => simp [Iterate]; aesop
+lemma forward : (R.Iterate (n + 1) x y) ↔ ∃ z, R.Iterate n x z ∧ R z y := by
+ induction n generalizing x y with
+ | zero => simp_all;
+ | succ n ih =>
+ constructor;
+ . rintro ⟨z, Rxz, Rzy⟩;
+ obtain ⟨w, Rzw, Rwy⟩ := ih.mp R... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_42331e6d7cb1_4 | 47f2c2d3406abc0c | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Vorspiel/Rel/Basic.lean | Basic | 4 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 6 | 3 | [
{
"theorem_name": "unwrap_of_trans_of_pos",
"depth": 1,
"n_commands": 0,
"n_lines": 4,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n have : ∃ m : ℕ+, n = m := ⟨⟨n, h⟩, by simp⟩\n rcases this with ⟨n, rfl⟩\n exact unwrap_of_trans Rxy",
"n_chars... | [
{
"name": "unwrap_of_trans",
"text": "lemma unwrap_of_trans {n : ℕ+} [IsTrans _ R] (Rxy : R.Iterate n x y) : R x y := by\n induction n using PNat.recOn generalizing x with\n | one => simpa using Rxy;\n | succ n ih =>\n obtain ⟨z, Rxz, Rzy⟩ := Rxy;\n exact IsTrans.trans _ _ _ Rxz (ih Rzy);\n\n",
... | [
{
"name": "remove_iterate",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 2,
"n_lines": 4,
"n_chars": 127,
"n_subproofs": 0,
"n_tactics": 3,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": false,
"max_nesting"... | 1 | module
public import Mathlib.Data.PNat.Basic
public import Mathlib.Data.Rel
@[expose]
public section
namespace Rel
variable {α} {R : Rel α α} {x y z : α}
local infix:50 " ≺ " => R
def Iterate (R : Rel α α) : ℕ → Rel α α
| 0 => (· = ·)
| n + 1 => fun x y ↦ ∃ z, R x z ∧ R.Iterate n z y
namespace Iterate
@[sim... | module
public import Mathlib.Data.PNat.Basic
public import Mathlib.Data.Rel
@[expose]
public section
namespace Rel
variable {α} {R : Rel α α} {x y z : α}
local infix:50 " ≺ " => R
def Iterate (R : Rel α α) : ℕ → Rel α α
| 0 => (· = ·)
| n + 1 => fun x y ↦ ∃ z, R x z ∧ R.Iterate n z y
namespace Iterate
@[sim... | @@ -48,6 +48,13 @@
. apply ih.mpr;
use z;
+lemma unwrap_of_trans {n : ℕ+} [IsTrans _ R] (Rxy : R.Iterate n x y) : R x y := by
+ induction n using PNat.recOn generalizing x with
+ | one => simpa using Rxy;
+ | succ n ih =>
+ obtain ⟨z, Rxz, Rzy⟩ := Rxy;
+ exact IsTrans.trans _ _ _ Rxz (ih Rzy)... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_42331e6d7cb1_5 | 12fd8dd931d1be53 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Vorspiel/Rel/Basic.lean | Basic | 5 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 6 | 3 | [
{
"theorem_name": "unwrap_of_trans_of_pos",
"depth": 1,
"n_commands": 0,
"n_lines": 4,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n have : ∃ m : ℕ+, n = m := ⟨⟨n, h⟩, by simp⟩\n rcases this with ⟨n, rfl⟩\n exact unwrap_of_trans Rxy",
"n_chars... | [
{
"name": "unwrap_of_trans",
"text": "lemma unwrap_of_trans {n : ℕ+} [IsTrans _ R] (Rxy : R.Iterate n x y) : R x y := by\n induction n using PNat.recOn generalizing x with\n | one => simpa using Rxy;\n | succ n ih =>\n obtain ⟨z, Rxz, Rzy⟩ := Rxy;\n exact IsTrans.trans _ _ _ Rxz (ih Rzy);\n\n",
... | [
{
"name": "unwrap",
"fan_in": 1,
"n_deps_direct": 2,
"n_deps_transitive": 7,
"n_lines": 5,
"n_chars": 158,
"n_subproofs": 1,
"n_tactics": 5,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": false,
"max_nesting": 2
}
... | 2 | module
public import Mathlib.Data.PNat.Basic
public import Mathlib.Data.Rel
@[expose]
public section
namespace Rel
variable {α} {R : Rel α α} {x y z : α}
local infix:50 " ≺ " => R
def Iterate (R : Rel α α) : ℕ → Rel α α
| 0 => (· = ·)
| n + 1 => fun x y ↦ ∃ z, R x z ∧ R.Iterate n z y
namespace Iterate
@[sim... | module
public import Mathlib.Data.PNat.Basic
public import Mathlib.Data.Rel
@[expose]
public section
namespace Rel
variable {α} {R : Rel α α} {x y z : α}
local infix:50 " ≺ " => R
def Iterate (R : Rel α α) : ℕ → Rel α α
| 0 => (· = ·)
| n + 1 => fun x y ↦ ∃ z, R x z ∧ R.Iterate n z y
namespace Iterate
@[sim... | @@ -48,6 +48,13 @@
. apply ih.mpr;
use z;
+lemma unwrap_of_trans {n : ℕ+} [IsTrans _ R] (Rxy : R.Iterate n x y) : R x y := by
+ induction n using PNat.recOn generalizing x with
+ | one => simpa using Rxy;
+ | succ n ih =>
+ obtain ⟨z, Rxz, Rzy⟩ := Rxy;
+ exact IsTrans.trans _ _ _ Rxz (ih Rzy)... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_42331e6d7cb1_6 | a831ca2490b683a0 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Vorspiel/Rel/Basic.lean | Basic | 6 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 6 | 2 | [
{
"theorem_name": "remove_iterate",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n apply unwrap_of_refl_trans (n := n) Rxy;",
"n_chars": 46,
"n_subproofs": 0,
"n_tactics": 3,
"cyclomatic": 1,... | [
{
"name": "unwrap_of_refl_trans",
"text": "lemma unwrap_of_refl_trans {n : ℕ} [Std.Refl R] [IsTrans _ R] (Rxy : R.Iterate n x y) : R x y := by\n induction n generalizing x with\n | zero => subst Rxy; apply Std.Refl.refl;\n | succ n ih =>\n obtain ⟨z, Rxz, Rzy⟩ := Rxy;\n exact IsTrans.trans _ _ _ Rx... | [
{
"name": "remove_iterate",
"fan_in": 0,
"n_deps_direct": 4,
"n_deps_transitive": 4,
"n_lines": 4,
"n_chars": 133,
"n_subproofs": 0,
"n_tactics": 3,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": false,
"max_nesting"... | 2 | module
public import Mathlib.Data.PNat.Basic
public import Mathlib.Data.Rel
@[expose]
public section
namespace Rel
variable {α} {R : Rel α α} {x y z : α}
local infix:50 " ≺ " => R
def Iterate (R : Rel α α) : ℕ → Rel α α
| 0 => (· = ·)
| n + 1 => fun x y ↦ ∃ z, R x z ∧ R.Iterate n z y
namespace Iterate
@[sim... | module
public import Mathlib.Data.PNat.Basic
public import Mathlib.Data.Rel
@[expose]
public section
namespace Rel
variable {α} {R : Rel α α} {x y z : α}
local infix:50 " ≺ " => R
def Iterate (R : Rel α α) : ℕ → Rel α α
| 0 => (· = ·)
| n + 1 => fun x y ↦ ∃ z, R x z ∧ R.Iterate n z y
namespace Iterate
@[sim... | @@ -55,6 +55,13 @@
obtain ⟨z, Rxz, Rzy⟩ := Rxy;
exact IsTrans.trans _ _ _ Rxz (ih Rzy);
+lemma unwrap_of_refl_trans {n : ℕ} [Std.Refl R] [IsTrans _ R] (Rxy : R.Iterate n x y) : R x y := by
+ induction n generalizing x with
+ | zero => subst Rxy; apply Std.Refl.refl;
+ | succ n ih =>
+ obtain ⟨z, Rxz, ... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_42331e6d7cb1_7 | 19c8792c9c37abd3 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Vorspiel/Rel/Basic.lean | Basic | 7 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 6 | 2 | [
{
"theorem_name": "remove_iterate",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n apply unwrap_of_refl_trans (n := n) Rxy;",
"n_chars": 46,
"n_subproofs": 0,
"n_tactics": 3,
"cyclomatic": 1,... | [
{
"name": "unwrap_of_refl_trans",
"text": "lemma unwrap_of_refl_trans {n : ℕ} [Std.Refl R] [IsTrans _ R] (Rxy : R.Iterate n x y) : R x y := by\n induction n generalizing x with\n | zero => subst Rxy; apply Std.Refl.refl;\n | succ n ih =>\n obtain ⟨z, Rxz, Rzy⟩ := Rxy;\n exact IsTrans.trans _ _ _ Rx... | [
{
"name": "unwrap",
"fan_in": 1,
"n_deps_direct": 5,
"n_deps_transitive": 9,
"n_lines": 5,
"n_chars": 179,
"n_subproofs": 1,
"n_tactics": 5,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 2,
"automation_only": false,
"max_nesting": 2
}
... | 3 | module
public import Mathlib.Data.PNat.Basic
public import Mathlib.Data.Rel
@[expose]
public section
namespace Rel
variable {α} {R : Rel α α} {x y z : α}
local infix:50 " ≺ " => R
def Iterate (R : Rel α α) : ℕ → Rel α α
| 0 => (· = ·)
| n + 1 => fun x y ↦ ∃ z, R x z ∧ R.Iterate n z y
namespace Iterate
@[sim... | module
public import Mathlib.Data.PNat.Basic
public import Mathlib.Data.Rel
@[expose]
public section
namespace Rel
variable {α} {R : Rel α α} {x y z : α}
local infix:50 " ≺ " => R
def Iterate (R : Rel α α) : ℕ → Rel α α
| 0 => (· = ·)
| n + 1 => fun x y ↦ ∃ z, R x z ∧ R.Iterate n z y
namespace Iterate
@[sim... | @@ -55,6 +55,13 @@
obtain ⟨z, Rxz, Rzy⟩ := Rxy;
exact IsTrans.trans _ _ _ Rxz (ih Rzy);
+lemma unwrap_of_refl_trans {n : ℕ} [Std.Refl R] [IsTrans _ R] (Rxy : R.Iterate n x y) : R x y := by
+ induction n generalizing x with
+ | zero => subst Rxy; apply Std.Refl.refl;
+ | succ n ih =>
+ obtain ⟨z, Rxz, ... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_5ca9ddc3641a_0 | 9533666a0d41ce8f | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/SetTheory/TransitiveModel.lean | TransitiveModel | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 1 | 1 | [
{
"theorem_name": "ind",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": "\n wellFounded.induction x ind",
"n_chars": 30,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
... | [
{
"name": "wellFounded",
"text": "lemma wellFounded : WellFounded (α := U) (· ∈ ·) := Universe.wellFounded.onFun (f := Subtype.val)\n\n",
"fan_in": 1,
"n_lines": 3,
"n_chars": 104,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_s... | [
{
"name": "ind",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 5,
"n_chars": 135,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
"max_nesting": 2
}
] | 1 | module
public import Foundation.FirstOrder.SetTheory.Universe
@[expose] public section
namespace LO.FirstOrder.SetTheory
abbrev TransitiveModel.{u} := Set Universe.{u}
namespace TransitiveModel
variable {U : TransitiveModel}
@[elab_as_elim]
theorem ind {P : U → Prop} (ind : ∀ x, (∀ y ∈ x, P y) → P x) (x : U) : P ... | module
public import Foundation.FirstOrder.SetTheory.Universe
@[expose] public section
namespace LO.FirstOrder.SetTheory
abbrev TransitiveModel.{u} := Set Universe.{u}
namespace TransitiveModel
variable {U : TransitiveModel}
lemma wellFounded : WellFounded (α := U) (· ∈ ·) := Universe.wellFounded.onFun (f := Subt... | @@ -11,8 +11,11 @@
variable {U : TransitiveModel}
+lemma wellFounded : WellFounded (α := U) (· ∈ ·) := Universe.wellFounded.onFun (f := Subtype.val)
+
@[elab_as_elim]
-theorem ind {P : U → Prop} (ind : ∀ x, (∀ y ∈ x, P y) → P x) (x : U) : P x := sorry
+theorem ind {P : U → Prop} (ind : ∀ x, (∀ y ∈ x, P y) → P x) ... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_326141441e9a_0 | 6d79ce163c735f9e | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Propositional/Boolean/ZeroSubst.lean | ZeroSubst | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 1 | 1 | [
{
"theorem_name": "tautology_of_forall_zeroSubst",
"depth": 1,
"n_commands": 0,
"n_lines": 3,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n contrapose!;\n apply exists_neg_zeroSubst_of_not_tautology;",
"n_chars": 65,
"n_subproofs": 0,
"... | [
{
"name": "exists_neg_zeroSubst_of_not_tautology",
"text": "theorem exists_neg_zeroSubst_of_not_tautology (h : ¬φ.IsTautology)\n : ∃ s : ZeroSubstitution α, Formula.IsTautology (∼(φ⟦s.1⟧)) := by\n unfold Formula.IsTautology Valid at h;\n push Not at h;\n obtain ⟨v, hv⟩ := h;\n use vfSubst v;\n intro u... | [
{
"name": "tautology_of_forall_zeroSubst",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 5,
"n_chars": 190,
"n_subproofs": 0,
"n_tactics": 5,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": false,
... | 1 | module
public import Foundation.Propositional.Boolean.Basic
@[expose] public section
namespace LO.Propositional.Boolean
variable {v : Boolean.Valuation α} {φ ψ : Formula α}
open Classical
open Semantics (Valid)
open Formula (atom)
open Formula.Boolean
open Classical in
noncomputable def vfSubst (v : Valuation α) ... | module
public import Foundation.Propositional.Boolean.Basic
@[expose] public section
namespace LO.Propositional.Boolean
variable {v : Boolean.Valuation α} {φ ψ : Formula α}
open Classical
open Semantics (Valid)
open Formula (atom)
open Formula.Boolean
open Classical in
noncomputable def vfSubst (v : Valuation α) ... | @@ -19,8 +19,24 @@
by intro a; simp [Formula.subst_atom]; split <;> tauto;
⟩
-lemma tautology_of_forall_zeroSubst : (∀ s : ZeroSubstitution α, ¬(∼(φ⟦s.1⟧)).IsTautology) → φ.IsTautology := sorry
+theorem exists_neg_zeroSubst_of_not_tautology (h : ¬φ.IsTautology)
+ : ∃ s : ZeroSubstitution α, Formula.IsTautology... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_b5bc440d2dac_0 | d8fb5b5d63c61535 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Kripke/Cluster.lean | Cluster | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 8 | 2 | [
{
"theorem_name": "refl_rel_of_more_than_one",
"depth": 1,
"n_commands": 0,
"n_lines": 4,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n obtain ⟨c, rfl⟩ := Quotient.exists_rep C;\n apply refl_in_cluster_of_more_than_one h;\n tauto;",
"n_chars":... | [
{
"name": "refl_in_cluster_of_more_than_one",
"text": "lemma refl_in_cluster_of_more_than_one (h : ∃ x y, x ≠ y ∧ x ∈ C ∧ y ∈ C) : ∀ z, z ∈ C → z ≺ z := by\n obtain ⟨c, rfl⟩ := Quotient.exists_rep C;\n obtain ⟨x, y, hxy, hx, hy⟩ := h;\n intro z hz;\n simp only [mem_iff₂] at hx hy hz;\n rcases hx with r... | [
{
"name": "refl_rel_of_more_than_one",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 3,
"n_lines": 6,
"n_chars": 197,
"n_subproofs": 1,
"n_tactics": 7,
"cyclomatic": 1,
"n_automation": 2,
"n_rewrites": 0,
"n_structural": 2,
"automation_only": false,
"m... | 1 | module
public import Foundation.Modal.Kripke.Root
public import Mathlib.Data.Finite.Card
@[expose] public section
def IsEquiv.equivalence [IsEquiv α r] : Equivalence r where
refl := Std.Refl.refl
symm {_ _} := by apply Std.Symm.symm
trans {_ _ _} := by apply IsTrans.trans
namespace LO.Modal
namespace Kripke
... | module
public import Foundation.Modal.Kripke.Root
public import Mathlib.Data.Finite.Card
@[expose] public section
def IsEquiv.equivalence [IsEquiv α r] : Equivalence r where
refl := Std.Refl.refl
symm {_ _} := by apply Std.Symm.symm
trans {_ _ _} := by apply IsTrans.trans
namespace LO.Modal
namespace Kripke
... | @@ -169,8 +169,28 @@
apply mem_iff.mpr;
simpa using h;
-lemma refl_rel_of_more_than_one (h : ∃ x y, x ≠ y ∧ x ∈ C ∧ y ∈ C) : C ≼ C := sorry
+lemma refl_in_cluster_of_more_than_one (h : ∃ x y, x ≠ y ∧ x ∈ C ∧ y ∈ C) : ∀ z, z ∈ C → z ≺ z := by
+ obtain ⟨c, rfl⟩ := Quotient.exists_rep C;
+ obtain ⟨x, y, hxy,... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_b5bc440d2dac_1 | 1bc7b9c4f05ef9b7 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Kripke/Cluster.lean | Cluster | 1 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 8 | 2 | [
{
"theorem_name": "refl_rel_of_more_than_one",
"depth": 1,
"n_commands": 0,
"n_lines": 4,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n obtain ⟨c, rfl⟩ := Quotient.exists_rep C;\n apply refl_in_cluster_of_more_than_one h;\n tauto;",
"n_chars":... | [
{
"name": "refl_in_cluster_of_more_than_one",
"text": "lemma refl_in_cluster_of_more_than_one (h : ∃ x y, x ≠ y ∧ x ∈ C ∧ y ∈ C) : ∀ z, z ∈ C → z ≺ z := by\n obtain ⟨c, rfl⟩ := Quotient.exists_rep C;\n obtain ⟨x, y, hxy, hx, hy⟩ := h;\n intro z hz;\n simp only [mem_iff₂] at hx hy hz;\n rcases hx with r... | [
{
"name": "not_more_than_two_of_degenerate",
"fan_in": 2,
"n_deps_direct": 1,
"n_deps_transitive": 4,
"n_lines": 4,
"n_chars": 160,
"n_subproofs": 0,
"n_tactics": 3,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": false,
... | 2 | module
public import Foundation.Modal.Kripke.Root
public import Mathlib.Data.Finite.Card
@[expose] public section
def IsEquiv.equivalence [IsEquiv α r] : Equivalence r where
refl := Std.Refl.refl
symm {_ _} := by apply Std.Symm.symm
trans {_ _ _} := by apply IsTrans.trans
namespace LO.Modal
namespace Kripke
... | module
public import Foundation.Modal.Kripke.Root
public import Mathlib.Data.Finite.Card
@[expose] public section
def IsEquiv.equivalence [IsEquiv α r] : Equivalence r where
refl := Std.Refl.refl
symm {_ _} := by apply Std.Symm.symm
trans {_ _ _} := by apply IsTrans.trans
namespace LO.Modal
namespace Kripke
... | @@ -169,8 +169,28 @@
apply mem_iff.mpr;
simpa using h;
-lemma refl_rel_of_more_than_one (h : ∃ x y, x ≠ y ∧ x ∈ C ∧ y ∈ C) : C ≼ C := sorry
+lemma refl_in_cluster_of_more_than_one (h : ∃ x y, x ≠ y ∧ x ∈ C ∧ y ∈ C) : ∀ z, z ∈ C → z ≺ z := by
+ obtain ⟨c, rfl⟩ := Quotient.exists_rep C;
+ obtain ⟨x, y, hxy,... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_b5bc440d2dac_2 | cf64a84e25dc554e | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Kripke/Cluster.lean | Cluster | 2 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 8 | 1 | [
{
"theorem_name": "not_more_than_two_of_degenerate",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n apply not_imp_not.mpr $ refl_rel_of_more_than_one;",
"n_chars": 56,
"n_subproofs": 0,
"n_tactic... | [
{
"name": "refl_rel_of_more_than_one",
"text": "lemma refl_rel_of_more_than_one (h : ∃ x y, x ≠ y ∧ x ∈ C ∧ y ∈ C) : C ≼ C := by\n obtain ⟨c, rfl⟩ := Quotient.exists_rep C;\n apply refl_in_cluster_of_more_than_one h;\n tauto;\n\n",
"fan_in": 1,
"n_lines": 6,
"n_chars": 197,
"n_subproofs":... | [
{
"name": "degenerate_iff_exists_unique_irrefl_point",
"fan_in": 1,
"n_deps_direct": 2,
"n_deps_transitive": 5,
"n_lines": 20,
"n_chars": 611,
"n_subproofs": 2,
"n_tactics": 35,
"cyclomatic": 6,
"n_automation": 5,
"n_rewrites": 0,
"n_structural": 10,
"automation_o... | 3 | module
public import Foundation.Modal.Kripke.Root
public import Mathlib.Data.Finite.Card
@[expose] public section
def IsEquiv.equivalence [IsEquiv α r] : Equivalence r where
refl := Std.Refl.refl
symm {_ _} := by apply Std.Symm.symm
trans {_ _ _} := by apply IsTrans.trans
namespace LO.Modal
namespace Kripke
... | module
public import Foundation.Modal.Kripke.Root
public import Mathlib.Data.Finite.Card
@[expose] public section
def IsEquiv.equivalence [IsEquiv α r] : Equivalence r where
refl := Std.Refl.refl
symm {_ _} := by apply Std.Symm.symm
trans {_ _ _} := by apply IsTrans.trans
namespace LO.Modal
namespace Kripke
... | @@ -186,9 +186,15 @@
. exact _root_.trans Rcy Ryc;
. exact _root_.trans Rzc Rcz;
+lemma refl_rel_of_more_than_one (h : ∃ x y, x ≠ y ∧ x ∈ C ∧ y ∈ C) : C ≼ C := by
+ obtain ⟨c, rfl⟩ := Quotient.exists_rep C;
+ apply refl_in_cluster_of_more_than_one h;
+ tauto;
+
def degenerate (C : Cluster F) := ¬C ≼ C
-le... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_b5bc440d2dac_3 | 676e4d12e172121d | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Kripke/Cluster.lean | Cluster | 3 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 8 | 2 | [
{
"theorem_name": "degenerate_iff_exists_unique_irrefl_point",
"depth": 1,
"n_commands": 0,
"n_lines": 18,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n obtain ⟨c, rfl⟩ := Quotient.exists_rep C;\n constructor;\n . intro h;\n use c;\n constru... | [
{
"name": "not_more_than_two_of_degenerate",
"text": "lemma not_more_than_two_of_degenerate : C.degenerate → ¬∃ x y, x ≠ y ∧ x ∈ C ∧ y ∈ C := by\n apply not_imp_not.mpr $ refl_rel_of_more_than_one;\n\n",
"fan_in": 2,
"n_lines": 4,
"n_chars": 160,
"n_subproofs": 0,
"n_tactics": 3,
"c... | [
{
"name": "not_degenerate_of_simple",
"fan_in": 0,
"n_deps_direct": 2,
"n_deps_transitive": 6,
"n_lines": 13,
"n_chars": 428,
"n_subproofs": 4,
"n_tactics": 21,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 1,
"n_structural": 7,
"automation_only": false,
"... | 4 | module
public import Foundation.Modal.Kripke.Root
public import Mathlib.Data.Finite.Card
@[expose] public section
def IsEquiv.equivalence [IsEquiv α r] : Equivalence r where
refl := Std.Refl.refl
symm {_ _} := by apply Std.Symm.symm
trans {_ _ _} := by apply IsTrans.trans
namespace LO.Modal
namespace Kripke
... | module
public import Foundation.Modal.Kripke.Root
public import Mathlib.Data.Finite.Card
@[expose] public section
def IsEquiv.equivalence [IsEquiv α r] : Equivalence r where
refl := Std.Refl.refl
symm {_ _} := by apply Std.Symm.symm
trans {_ _ _} := by apply IsTrans.trans
namespace LO.Modal
namespace Kripke
... | @@ -193,8 +193,28 @@
def degenerate (C : Cluster F) := ¬C ≼ C
-lemma degenerate_iff_exists_unique_irrefl_point : C.degenerate ↔ (∃! x, x ∈ C ∧ ¬x ≺ x) := sorry
+lemma not_more_than_two_of_degenerate : C.degenerate → ¬∃ x y, x ≠ y ∧ x ∈ C ∧ y ∈ C := by
+ apply not_imp_not.mpr $ refl_rel_of_more_than_one;
+lemma ... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_b5bc440d2dac_4 | 2d0cf8dd636c2c75 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Kripke/Cluster.lean | Cluster | 4 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 8 | 2 | [
{
"theorem_name": "refl_in_simple",
"depth": 1,
"n_commands": 0,
"n_lines": 5,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n obtain ⟨y, ⟨hy, _⟩, _⟩ := h;\n rcases mem_same_cluster hx hy with rfl | ⟨Rxy, Ryx⟩;\n . assumption;\n . exact _root_.tran... | [
{
"name": "mem_same_cluster",
"text": "lemma mem_same_cluster (hx : x ∈ C) (hy : y ∈ C): y = x ∨ (y ≺ x ∧ x ≺ y) := by\n obtain ⟨c, rfl⟩ := Quotient.exists_rep C;\n replace hx := mem_iff₂.mp hx;\n replace hy := mem_iff₂.mp hy;\n rcases hx with rfl | ⟨Rcx, Rxc⟩ <;>\n rcases hy with rfl | ⟨Rcy, Ryc⟩;\n ... | [
{
"name": "refl_in_simple",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 3,
"n_lines": 7,
"n_chars": 214,
"n_subproofs": 1,
"n_tactics": 9,
"cyclomatic": 2,
"n_automation": 2,
"n_rewrites": 0,
"n_structural": 3,
"automation_only": false,
"max_nesting"... | 1 | module
public import Foundation.Modal.Kripke.Root
public import Mathlib.Data.Finite.Card
@[expose] public section
def IsEquiv.equivalence [IsEquiv α r] : Equivalence r where
refl := Std.Refl.refl
symm {_ _} := by apply Std.Symm.symm
trans {_ _ _} := by apply IsTrans.trans
namespace LO.Modal
namespace Kripke
... | module
public import Foundation.Modal.Kripke.Root
public import Mathlib.Data.Finite.Card
@[expose] public section
def IsEquiv.equivalence [IsEquiv α r] : Equivalence r where
refl := Std.Refl.refl
symm {_ _} := by apply Std.Symm.symm
trans {_ _ _} := by apply IsTrans.trans
namespace LO.Modal
namespace Kripke
... | @@ -169,11 +169,29 @@
apply mem_iff.mpr;
simpa using h;
+lemma mem_same_cluster (hx : x ∈ C) (hy : y ∈ C): y = x ∨ (y ≺ x ∧ x ≺ y) := by
+ obtain ⟨c, rfl⟩ := Quotient.exists_rep C;
+ replace hx := mem_iff₂.mp hx;
+ replace hy := mem_iff₂.mp hy;
+ rcases hx with rfl | ⟨Rcx, Rxc⟩ <;>
+ rcases hy with 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_b5bc440d2dac_5 | e36ed847e1a78c26 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Kripke/Cluster.lean | Cluster | 5 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 8 | 1 | [
{
"theorem_name": "not_more_than_two_of_degenerate",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n apply not_imp_not.mpr $ refl_rel_of_more_than_one;",
"n_chars": 56,
"n_subproofs": 0,
"n_tactic... | [
{
"name": "refl_rel_of_more_than_one",
"text": "lemma refl_rel_of_more_than_one (h : ∃ x y, x ≠ y ∧ x ∈ C ∧ y ∈ C) : C ≼ C := by\n obtain ⟨c, rfl⟩ := Quotient.exists_rep C;\n apply refl_in_cluster_of_more_than_one h;\n tauto;\n\n",
"fan_in": 1,
"n_lines": 6,
"n_chars": 197,
"n_subproofs":... | [
{
"name": "not_degenerate_of_proper",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 5,
"n_lines": 5,
"n_chars": 132,
"n_subproofs": 0,
"n_tactics": 5,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": false,
"ma... | 3 | module
public import Foundation.Modal.Kripke.Root
public import Mathlib.Data.Finite.Card
@[expose] public section
def IsEquiv.equivalence [IsEquiv α r] : Equivalence r where
refl := Std.Refl.refl
symm {_ _} := by apply Std.Symm.symm
trans {_ _ _} := by apply IsTrans.trans
namespace LO.Modal
namespace Kripke
... | module
public import Foundation.Modal.Kripke.Root
public import Mathlib.Data.Finite.Card
@[expose] public section
def IsEquiv.equivalence [IsEquiv α r] : Equivalence r where
refl := Std.Refl.refl
symm {_ _} := by apply Std.Symm.symm
trans {_ _ _} := by apply IsTrans.trans
namespace LO.Modal
namespace Kripke
... | @@ -186,9 +186,15 @@
. exact _root_.trans Rcy Ryc;
. exact _root_.trans Rzc Rcz;
+lemma refl_rel_of_more_than_one (h : ∃ x y, x ≠ y ∧ x ∈ C ∧ y ∈ C) : C ≼ C := by
+ obtain ⟨c, rfl⟩ := Quotient.exists_rep C;
+ apply refl_in_cluster_of_more_than_one h;
+ tauto;
+
def degenerate (C : Cluster F) := ¬C ≼ C
-le... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_b5bc440d2dac_6 | 0915db7f2def7260 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Kripke/Cluster.lean | Cluster | 6 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 8 | 2 | [
{
"theorem_name": "refl_in_simple",
"depth": 1,
"n_commands": 0,
"n_lines": 5,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n obtain ⟨y, ⟨hy, _⟩, _⟩ := h;\n rcases mem_same_cluster hx hy with rfl | ⟨Rxy, Ryx⟩;\n . assumption;\n . exact _root_.tran... | [
{
"name": "mem_same_cluster",
"text": "lemma mem_same_cluster (hx : x ∈ C) (hy : y ∈ C): y = x ∨ (y ≺ x ∧ x ≺ y) := by\n obtain ⟨c, rfl⟩ := Quotient.exists_rep C;\n replace hx := mem_iff₂.mp hx;\n replace hy := mem_iff₂.mp hy;\n rcases hx with rfl | ⟨Rcx, Rxc⟩ <;>\n rcases hy with rfl | ⟨Rcy, Ryc⟩;\n ... | [
{
"name": "refl_in_proper",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 3,
"n_lines": 9,
"n_chars": 314,
"n_subproofs": 1,
"n_tactics": 13,
"cyclomatic": 3,
"n_automation": 2,
"n_rewrites": 0,
"n_structural": 5,
"automation_only": false,
"max_nesting... | 1 | module
public import Foundation.Modal.Kripke.Root
public import Mathlib.Data.Finite.Card
@[expose] public section
def IsEquiv.equivalence [IsEquiv α r] : Equivalence r where
refl := Std.Refl.refl
symm {_ _} := by apply Std.Symm.symm
trans {_ _ _} := by apply IsTrans.trans
namespace LO.Modal
namespace Kripke
... | module
public import Foundation.Modal.Kripke.Root
public import Mathlib.Data.Finite.Card
@[expose] public section
def IsEquiv.equivalence [IsEquiv α r] : Equivalence r where
refl := Std.Refl.refl
symm {_ _} := by apply Std.Symm.symm
trans {_ _ _} := by apply IsTrans.trans
namespace LO.Modal
namespace Kripke
... | @@ -169,13 +169,33 @@
apply mem_iff.mpr;
simpa using h;
+lemma mem_same_cluster (hx : x ∈ C) (hy : y ∈ C): y = x ∨ (y ≺ x ∧ x ≺ y) := by
+ obtain ⟨c, rfl⟩ := Quotient.exists_rep C;
+ replace hx := mem_iff₂.mp hx;
+ replace hy := mem_iff₂.mp hy;
+ rcases hx with rfl | ⟨Rcx, Rxc⟩ <;>
+ rcases hy with 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_b5bc440d2dac_7 | df77bc348d7a1d77 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Kripke/Cluster.lean | Cluster | 7 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 8 | 1 | [
{
"theorem_name": "refl_of_mem_non_degenerate",
"depth": 1,
"n_commands": 0,
"n_lines": 4,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n rcases (either_simple_or_proper_of_non_degenerate h) with h | h;\n . apply refl_in_simple h hx;\n . apply refl... | [
{
"name": "refl_in_proper",
"text": "lemma refl_in_proper (h : C.proper) (hx : x ∈ C) : x ≺ x := by\n obtain ⟨y, z, hxy, hy, hz⟩ := h;\n rcases mem_same_cluster hx hy with rfl | ⟨Rxy, Ryx⟩;\n . rcases mem_same_cluster hy hz with rfl | ⟨Ryz, Rzy⟩;\n . contradiction;\n . exact _root_.trans Rzy Ryz;\n... | [
{
"name": "refl_of_mem_non_degenerate",
"fan_in": 0,
"n_deps_direct": 3,
"n_deps_transitive": 7,
"n_lines": 6,
"n_chars": 215,
"n_subproofs": 0,
"n_tactics": 7,
"cyclomatic": 2,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 3,
"automation_only": false,
"... | 4 | module
public import Foundation.Modal.Kripke.Root
public import Mathlib.Data.Finite.Card
@[expose] public section
def IsEquiv.equivalence [IsEquiv α r] : Equivalence r where
refl := Std.Refl.refl
symm {_ _} := by apply Std.Symm.symm
trans {_ _ _} := by apply IsTrans.trans
namespace LO.Modal
namespace Kripke
... | module
public import Foundation.Modal.Kripke.Root
public import Mathlib.Data.Finite.Card
@[expose] public section
def IsEquiv.equivalence [IsEquiv α r] : Equivalence r where
refl := Std.Refl.refl
symm {_ _} := by apply Std.Symm.symm
trans {_ _ _} := by apply IsTrans.trans
namespace LO.Modal
namespace Kripke
... | @@ -195,6 +195,14 @@
def proper (C : Cluster F) := ∃ x y, x ≠ y ∧ x ∈ C ∧ y ∈ C
+lemma refl_in_proper (h : C.proper) (hx : x ∈ C) : x ≺ x := by
+ obtain ⟨y, z, hxy, hy, hz⟩ := h;
+ rcases mem_same_cluster hx hy with rfl | ⟨Rxy, Ryx⟩;
+ . rcases mem_same_cluster hy hz with rfl | ⟨Ryz, Rzy⟩;
+ . contradiction;... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_b5bc440d2dac_8 | d91df36517f156ec | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Kripke/Cluster.lean | Cluster | 8 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 8 | 2 | [
{
"theorem_name": "refl_of_mem_non_degenerate",
"depth": 1,
"n_commands": 0,
"n_lines": 4,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n rcases (either_simple_or_proper_of_non_degenerate h) with h | h;\n . apply refl_in_simple h hx;\n . apply refl... | [
{
"name": "either_simple_or_proper_of_non_degenerate",
"text": "lemma either_simple_or_proper_of_non_degenerate (h : ¬C.degenerate) : C.simple ∨ C.proper := by\n obtain ⟨x, rfl⟩ := Quotient.exists_rep C;\n by_cases ex : ∃ y, x ≠ y ∧ (⟦x⟧ : Cluster F) = ⟦y⟧;\n . right;\n obtain ⟨y, nexy, h⟩ := ex;\n ... | [
{
"name": "degenerate_or_simple_or_proper",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 2,
"n_lines": 8,
"n_chars": 205,
"n_subproofs": 0,
"n_tactics": 11,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 2,
"automation_only": false,
... | 1 | module
public import Foundation.Modal.Kripke.Root
public import Mathlib.Data.Finite.Card
@[expose] public section
def IsEquiv.equivalence [IsEquiv α r] : Equivalence r where
refl := Std.Refl.refl
symm {_ _} := by apply Std.Symm.symm
trans {_ _ _} := by apply IsTrans.trans
namespace LO.Modal
namespace Kripke
... | module
public import Foundation.Modal.Kripke.Root
public import Mathlib.Data.Finite.Card
@[expose] public section
def IsEquiv.equivalence [IsEquiv α r] : Equivalence r where
refl := Std.Refl.refl
symm {_ _} := by apply Std.Symm.symm
trans {_ _ _} := by apply IsTrans.trans
namespace LO.Modal
namespace Kripke
... | @@ -175,11 +175,33 @@
def proper (C : Cluster F) := ∃ x y, x ≠ y ∧ x ∈ C ∧ y ∈ C
+lemma either_simple_or_proper_of_non_degenerate (h : ¬C.degenerate) : C.simple ∨ C.proper := by
+ obtain ⟨x, rfl⟩ := Quotient.exists_rep C;
+ by_cases ex : ∃ y, x ≠ y ∧ (⟦x⟧ : Cluster F) = ⟦y⟧;
+ . right;
+ obtain ⟨y, nexy, 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_d144cb8a619f_0 | 6c974e213aa827d5 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Exponential/Exp.lean | Exp | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 39 | 2 | [
{
"theorem_name": "ext_add_of_dvd_sq_left",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n rw [add_comm]; exact ext_add_of_dvd_sq_right pos h",
"n_chars": 56,
"n_subproofs": 0,
"n_tactics": 3,
... | [
{
"name": "ext_add_of_dvd_sq_right",
"text": "lemma ext_add_of_dvd_sq_right {u z₁ z₂ : V} (pos : 0 < u) (h : u^2 ∣ z₂) : ext u (z₁ + z₂) = ext u z₁ := by\n have : ∃ z', z₂ = z' * u * u := by rcases h with ⟨u', rfl⟩; exact ⟨u', by simp [mul_comm _ u', mul_assoc]; simp [sq]⟩\n rcases this with ⟨z₂, rfl⟩\n ... | [
{
"name": "ext_add_of_dvd_sq_left",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 4,
"n_chars": 175,
"n_subproofs": 0,
"n_tactics": 3,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 1,
"n_structural": 1,
"automation_only": false,
"max_... | 1 | module
public import Foundation.FirstOrder.Arithmetic.Exponential.PPow2
public import Mathlib.Algebra.Order.Ring.Basic
@[expose] public section
/-!
# Exponential function
This file provides a proof of the theorem states that
the graph of exponential function is definable by $\Sigma_0$-formula and
it's inductive prop... | module
public import Foundation.FirstOrder.Arithmetic.Exponential.PPow2
public import Mathlib.Algebra.Order.Ring.Basic
@[expose] public section
/-!
# Exponential function
This file provides a proof of the theorem states that
the graph of exponential function is definable by $\Sigma_0$-formula and
it's inductive prop... | @@ -40,8 +40,14 @@
@[simp] lemma ext_lt {u} (z : V) (pos : 0 < u) : ext u z < u := by simp [ext, pos]
-lemma ext_add_of_dvd_sq_left {u z₁ z₂ : V} (pos : 0 < u) (h : u^2 ∣ z₁) : ext u (z₁ + z₂) = ext u z₂ := sorry
+lemma ext_add_of_dvd_sq_right {u z₁ z₂ : V} (pos : 0 < u) (h : u^2 ∣ z₂) : ext u (z₁ + z₂) = ext u z₁... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_d144cb8a619f_1 | fdbfcdd8af48d421 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Exponential/Exp.lean | Exp | 1 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 39 | 1 | [
{
"theorem_name": "two_lt_four",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " lt_trans two_lt_three three_lt_four",
"n_chars": 36,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automa... | [
{
"name": "two_lt_three",
"text": "lemma two_lt_three : (2 : V) < 3 := by rw [←two_add_one_eq_three]; exact lt_add_one 2\n\n",
"fan_in": 1,
"n_lines": 3,
"n_chars": 89,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 1,
"n_structural": ... | [
{
"name": "two_lt_four",
"fan_in": 2,
"n_deps_direct": 2,
"n_deps_transitive": 2,
"n_lines": 3,
"n_chars": 72,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
"max_nesting": 1
... | 2 | module
public import Foundation.FirstOrder.Arithmetic.Exponential.PPow2
public import Mathlib.Algebra.Order.Ring.Basic
@[expose] public section
/-!
# Exponential function
This file provides a proof of the theorem states that
the graph of exponential function is definable by $\Sigma_0$-formula and
it's inductive prop... | module
public import Foundation.FirstOrder.Arithmetic.Exponential.PPow2
public import Mathlib.Algebra.Order.Ring.Basic
@[expose] public section
/-!
# Exponential function
This file provides a proof of the theorem states that
the graph of exponential function is definable by $\Sigma_0$-formula and
it's inductive prop... | @@ -128,9 +128,11 @@
def seqY₀ : V := 2 * 4
+lemma two_lt_three : (2 : V) < 3 := by rw [←two_add_one_eq_three]; exact lt_add_one 2
+
lemma three_lt_four : (3 : V) < 4 := by rw [←three_add_one_eq_four]; exact lt_add_one 3
-lemma two_lt_four : (2 : V) < 4 := sorry
+lemma two_lt_four : (2 : V) < 4 := lt_trans two_... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_d144cb8a619f_2 | 8dbda46ca6c63ef7 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Exponential/Exp.lean | Exp | 2 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 39 | 1 | [
{
"theorem_name": "two_lt_four",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " lt_trans two_lt_three three_lt_four",
"n_chars": 36,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automa... | [
{
"name": "two_lt_three",
"text": "lemma two_lt_three : (2 : V) < 3 := by rw [←two_add_one_eq_three]; exact lt_add_one 2\n\n",
"fan_in": 1,
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{
"name": "seq₀_zero_two",
"fan_in": 0,
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"max_nesting":... | 3 | module
public import Foundation.FirstOrder.Arithmetic.Exponential.PPow2
public import Mathlib.Algebra.Order.Ring.Basic
@[expose] public section
/-!
# Exponential function
This file provides a proof of the theorem states that
the graph of exponential function is definable by $\Sigma_0$-formula and
it's inductive prop... | module
public import Foundation.FirstOrder.Arithmetic.Exponential.PPow2
public import Mathlib.Algebra.Order.Ring.Basic
@[expose] public section
/-!
# Exponential function
This file provides a proof of the theorem states that
the graph of exponential function is definable by $\Sigma_0$-formula and
it's inductive prop... | @@ -128,9 +128,11 @@
def seqY₀ : V := 2 * 4
+lemma two_lt_three : (2 : V) < 3 := by rw [←two_add_one_eq_three]; exact lt_add_one 2
+
lemma three_lt_four : (3 : V) < 4 := by rw [←three_add_one_eq_four]; exact lt_add_one 3
-lemma two_lt_four : (2 : V) < 4 := sorry
+lemma two_lt_four : (2 : V) < 4 := lt_trans two_... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_d144cb8a619f_3 | a52aebeaa05ad015 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Exponential/Exp.lean | Exp | 3 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 39 | 2 | [
{
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"depth": 1,
"n_commands": 0,
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"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n rintro (⟨hx, rfl⟩ | ⟨X, _, Y, _, hseq₀, hseqₛ, i, hi, ne2, ppi, hXx, hYy⟩)\n · rcases hx with rfl; simp\n have hxsqi : 2 * x < ... | [
{
"name": "ext_append_last",
"text": "lemma ext_append_last (i X : V) {z} (hz : z < i) :\n ext i (append i X z) = z := by\n simp [ext, append, div_add_mul_self, show 0 < i from pos_of_gt hz, hz]\n\n",
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"c... | [
{
"name": "append_lt",
"fan_in": 2,
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... | 22 | module
public import Foundation.FirstOrder.Arithmetic.Exponential.PPow2
public import Mathlib.Algebra.Order.Ring.Basic
@[expose] public section
/-!
# Exponential function
This file provides a proof of the theorem states that
the graph of exponential function is definable by $\Sigma_0$-formula and
it's inductive prop... | module
public import Foundation.FirstOrder.Arithmetic.Exponential.PPow2
public import Mathlib.Algebra.Order.Ring.Basic
@[expose] public section
/-!
# Exponential function
This file provides a proof of the theorem states that
the graph of exponential function is definable by $\Sigma_0$-formula and
it's inductive prop... | @@ -155,6 +155,10 @@
_ < (1 + z) * i := by simpa [add_mul] using mod_lt _ (pos_of_gt hz)
_ ≤ i^2 := by simpa [sq, add_comm] using mul_le_mul_of_nonneg_right (lt_iff_succ_le.mp hz) (by simp)
+lemma ext_append_last (i X : V) {z} (hz : z < i) :
+ ext i (append i X z) = ... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_d144cb8a619f_4 | 912c423a3066c14a | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Exponential/Exp.lean | Exp | 4 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 39 | 3 | [
{
"theorem_name": "range_pow2",
"depth": 1,
"n_commands": 0,
"n_lines": 4,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n rcases h with (⟨rfl, rfl⟩ | ⟨X, bX, Y, bY, H₀, Hₛ, ⟨u, hu, ne2, ppu, rfl, rfl⟩⟩)\n · simp\n · exact pow2_ext_of_seq₀_of_seqₛ H... | [
{
"name": "pow2_ext_of_seq₀_of_seqₛ",
"text": "lemma pow2_ext_of_seq₀_of_seqₛ {y X Y : V} (h₀ : Exponential.Seq₀ X Y) (hₛ : Exponential.Seqₛ y X Y)\n {i} (ne2 : i ≠ 2) (hi : i ≤ y^2) (ppi : PPow2 i) : Pow2 (ext i Y) := by\n induction i using ISigma0.order_induction\n · definability\n case ind i IH =>\... | [
{
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... | 1 | module
public import Foundation.FirstOrder.Arithmetic.Exponential.PPow2
public import Mathlib.Algebra.Order.Ring.Basic
@[expose] public section
/-!
# Exponential function
This file provides a proof of the theorem states that
the graph of exponential function is definable by $\Sigma_0$-formula and
it's inductive prop... | module
public import Foundation.FirstOrder.Arithmetic.Exponential.PPow2
public import Mathlib.Algebra.Order.Ring.Basic
@[expose] public section
/-!
# Exponential function
This file provides a proof of the theorem states that
the graph of exponential function is definable by $\Sigma_0$-formula and
it's inductive prop... | @@ -196,10 +196,26 @@
by simpa [Seqₛ] using fun i hi ne2 ppi ↦ False.elim <| not_le.mpr (ppi.two_lt ne2) hi,
⟨4, by simp [two_pow_two_eq_four], by simp, by simp [ext, two_lt_four]⟩⟩
+lemma pow2_ext_of_seq₀_of_seqₛ {y X Y : V} (h₀ : Exponential.Seq₀ X Y) (hₛ : Exponential.Seqₛ y X Y)
+ {i} (ne2 : i ≠ 2) (... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_d144cb8a619f_5 | dd6a3b518a38788a | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Exponential/Exp.lean | Exp | 5 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 39 | 2 | [
{
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"proof_text": " by\n rintro (⟨hx, rfl⟩ | ⟨X, _, Y, _, hseq₀, hseqₛ, i, hi, ne2, ppi, hXx, hYy⟩)\n · rcases hx with rfl; simp\n have hxsqi : 2 * x < ... | [
{
"name": "append_lt",
"text": "lemma append_lt (i X : V) {z} (hz : z < i) : append i X z < i^2 := calc\n append i X z = (X % i) + z * i := rfl\n _ < (1 + z) * i := by simpa [add_mul] using mod_lt _ (pos_of_gt hz)\n _ ≤ i^2 := by simpa [sq, add_comm] using mul_le... | [
{
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"max_nesting... | 22 | module
public import Foundation.FirstOrder.Arithmetic.Exponential.PPow2
public import Mathlib.Algebra.Order.Ring.Basic
@[expose] public section
/-!
# Exponential function
This file provides a proof of the theorem states that
the graph of exponential function is definable by $\Sigma_0$-formula and
it's inductive prop... | module
public import Foundation.FirstOrder.Arithmetic.Exponential.PPow2
public import Mathlib.Algebra.Order.Ring.Basic
@[expose] public section
/-!
# Exponential function
This file provides a proof of the theorem states that
the graph of exponential function is definable by $\Sigma_0$-formula and
it's inductive prop... | @@ -150,6 +150,11 @@
noncomputable def append (i X z : V) : V := X % i + z * i
+lemma append_lt (i X : V) {z} (hz : z < i) : append i X z < i^2 := calc
+ append i X z = (X % i) + z * i := rfl
+ _ < (1 + z) * i := by simpa [add_mul] using mod_lt _ (pos_of_gt hz)
+ _ ≤ i^2 ... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_d144cb8a619f_6 | 86608d2762536898 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Exponential/Exp.lean | Exp | 6 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 39 | 2 | [
{
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"depth": 1,
"n_commands": 0,
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"method": "deleted-dep",
"proof_text": " by\n rintro (⟨hx, rfl⟩ | ⟨X, _, Y, _, hseq₀, hseqₛ, i, hi, ne2, ppi, hXx, hYy⟩)\n · rcases hx with rfl; simp\n have hxsqi : 2 * x < ... | [
{
"name": "append_lt",
"text": "lemma append_lt (i X : V) {z} (hz : z < i) : append i X z < i^2 := calc\n append i X z = (X % i) + z * i := rfl\n _ < (1 + z) * i := by simpa [add_mul] using mod_lt _ (pos_of_gt hz)\n _ ≤ i^2 := by simpa [sq, add_comm] using mul_le... | [
{
"name": "exponential_even_sq",
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"automation_only": false,
"max_ne... | 23 | module
public import Foundation.FirstOrder.Arithmetic.Exponential.PPow2
public import Mathlib.Algebra.Order.Ring.Basic
@[expose] public section
/-!
# Exponential function
This file provides a proof of the theorem states that
the graph of exponential function is definable by $\Sigma_0$-formula and
it's inductive prop... | module
public import Foundation.FirstOrder.Arithmetic.Exponential.PPow2
public import Mathlib.Algebra.Order.Ring.Basic
@[expose] public section
/-!
# Exponential function
This file provides a proof of the theorem states that
the graph of exponential function is definable by $\Sigma_0$-formula and
it's inductive prop... | @@ -150,6 +150,11 @@
noncomputable def append (i X z : V) : V := X % i + z * i
+lemma append_lt (i X : V) {z} (hz : z < i) : append i X z < i^2 := calc
+ append i X z = (X % i) + z * i := rfl
+ _ < (1 + z) * i := by simpa [add_mul] using mod_lt _ (pos_of_gt hz)
+ _ ≤ i^2 ... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_d144cb8a619f_7 | 396e05318814516f | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Exponential/Exp.lean | Exp | 7 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 39 | 2 | [
{
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"depth": 1,
"n_commands": 0,
"n_lines": 40,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n rintro (⟨hx, rfl⟩ | ⟨X, _, Y, _, hseq₀, hseqₛ, i, hi, ne2, ppi, hXx, hYy⟩)\n · rcases hx with rfl; simp\n have hxsqi : 2 * x < ... | [
{
"name": "append_lt",
"text": "lemma append_lt (i X : V) {z} (hz : z < i) : append i X z < i^2 := calc\n append i X z = (X % i) + z * i := rfl\n _ < (1 + z) * i := by simpa [add_mul] using mod_lt _ (pos_of_gt hz)\n _ ≤ i^2 := by simpa [sq, add_comm] using mul_le... | [
{
"name": "exponential_odd_two_mul_sq",
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"n_rewrites": 1,
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"automation_only": false,
... | 23 | module
public import Foundation.FirstOrder.Arithmetic.Exponential.PPow2
public import Mathlib.Algebra.Order.Ring.Basic
@[expose] public section
/-!
# Exponential function
This file provides a proof of the theorem states that
the graph of exponential function is definable by $\Sigma_0$-formula and
it's inductive prop... | module
public import Foundation.FirstOrder.Arithmetic.Exponential.PPow2
public import Mathlib.Algebra.Order.Ring.Basic
@[expose] public section
/-!
# Exponential function
This file provides a proof of the theorem states that
the graph of exponential function is definable by $\Sigma_0$-formula and
it's inductive prop... | @@ -150,6 +150,11 @@
noncomputable def append (i X z : V) : V := X % i + z * i
+lemma append_lt (i X : V) {z} (hz : z < i) : append i X z < i^2 := calc
+ append i X z = (X % i) + z * i := rfl
+ _ < (1 + z) * i := by simpa [add_mul] using mod_lt _ (pos_of_gt hz)
+ _ ≤ i^2 ... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_d144cb8a619f_8 | c9895ba9b2cb5f6d | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Exponential/Exp.lean | Exp | 8 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 39 | 3 | [
{
"theorem_name": "ext_le_ext_of_seq₀_of_seqₛ",
"depth": 1,
"n_commands": 0,
"n_lines": 24,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n induction i using ISigma0.order_induction\n · definability\n case ind i IH =>\n by_cases ne4 : i = 4\n ... | [
{
"name": "two_le_ext_of_seq₀_of_seqₛ",
"text": "lemma two_le_ext_of_seq₀_of_seqₛ {y X Y : V} (h₀ : Exponential.Seq₀ X Y) (hₛ : Exponential.Seqₛ y X Y)\n {i} (ne2 : i ≠ 2) (hi : i ≤ y^2) (ppi : PPow2 i) : 2 ≤ ext i Y := by\n induction i using ISigma0.order_induction\n · definability\n case ind i IH =>... | [
{
"name": "ext_le_ext_of_seq₀_of_seqₛ",
"fan_in": 1,
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... | 1 | module
public import Foundation.FirstOrder.Arithmetic.Exponential.PPow2
public import Mathlib.Algebra.Order.Ring.Basic
@[expose] public section
/-!
# Exponential function
This file provides a proof of the theorem states that
the graph of exponential function is definable by $\Sigma_0$-formula and
it's inductive prop... | module
public import Foundation.FirstOrder.Arithmetic.Exponential.PPow2
public import Mathlib.Algebra.Order.Ring.Basic
@[expose] public section
/-!
# Exponential function
This file provides a proof of the theorem states that
the graph of exponential function is definable by $\Sigma_0$-formula and
it's inductive prop... | @@ -424,6 +424,27 @@
lemma exponential_odd {x y : V} : Exponential (2 * x + 1) y ↔ ∃ y', y = 2 * y' ^ 2 ∧ Exponential x y' :=
⟨exponential_exists_sq_of_exponential_odd, by rintro ⟨y, rfl, h⟩; exact bit_one h⟩
+lemma two_le_ext_of_seq₀_of_seqₛ {y X Y : V} (h₀ : Exponential.Seq₀ X Y) (hₛ : Exponential.Seqₛ y X Y)
+... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_d144cb8a619f_9 | f9e7e7347a9d9e61 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Exponential/Exp.lean | Exp | 9 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 39 | 1 | [
{
"theorem_name": "zero_uniq",
"depth": 1,
"n_commands": 0,
"n_lines": 5,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n rcases h with (⟨_, rfl⟩ | ⟨X, _, Y, _, H₀, Hₛ, ⟨u, hu, ne2, ppu, hX, _⟩⟩)\n · rfl\n · have : 1 ≤ ext u X := one_le_ext_of_seq₀... | [
{
"name": "one_le_ext_of_seq₀_of_seqₛ",
"text": "lemma one_le_ext_of_seq₀_of_seqₛ {y X Y : V} (h₀ : Exponential.Seq₀ X Y) (hₛ : Exponential.Seqₛ y X Y)\n {i} (ne2 : i ≠ 2) (hi : i ≤ y^2) (ppi : PPow2 i) : 1 ≤ ext i X := by\n induction i using ISigma0.order_induction\n · definability\n case ind i IH =>... | [
{
"name": "zero_uniq",
"fan_in": 6,
"n_deps_direct": 1,
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"n_lines": 7,
"n_chars": 266,
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... | 1 | module
public import Foundation.FirstOrder.Arithmetic.Exponential.PPow2
public import Mathlib.Algebra.Order.Ring.Basic
@[expose] public section
/-!
# Exponential function
This file provides a proof of the theorem states that
the graph of exponential function is definable by $\Sigma_0$-formula and
it's inductive prop... | module
public import Foundation.FirstOrder.Arithmetic.Exponential.PPow2
public import Mathlib.Algebra.Order.Ring.Basic
@[expose] public section
/-!
# Exponential function
This file provides a proof of the theorem states that
the graph of exponential function is definable by $\Sigma_0$-formula and
it's inductive prop... | @@ -547,10 +547,27 @@
alias ⟨of_succ_two_mul, succ⟩ := exponential_succ_mul_two
+lemma one_le_ext_of_seq₀_of_seqₛ {y X Y : V} (h₀ : Exponential.Seq₀ X Y) (hₛ : Exponential.Seqₛ y X Y)
+ {i} (ne2 : i ≠ 2) (hi : i ≤ y^2) (ppi : PPow2 i) : 1 ≤ ext i X := by
+ induction i using ISigma0.order_induction
+ · definab... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
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"--cor... |
ablate_d144cb8a619f_10 | 00254cb5d2dc77f1 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Exponential/Exp.lean | Exp | 10 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 39 | 6 | [
{
"theorem_name": "zero_uniq_iff",
"depth": 1,
"n_commands": 0,
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"is_leaf": true,
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"method": "deleted-dep",
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"n_auto... | [
{
"name": "zero_uniq",
"text": "lemma zero_uniq {y : V} (h : Exponential 0 y) : y = 1 := by\n rcases h with (⟨_, rfl⟩ | ⟨X, _, Y, _, H₀, Hₛ, ⟨u, hu, ne2, ppu, hX, _⟩⟩)\n · rfl\n · have : 1 ≤ ext u X := one_le_ext_of_seq₀_of_seqₛ H₀ Hₛ ne2 hu ppu\n simp [hX] at this\n\n",
"fan_in": 6,
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{
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"max_nesting":... | 2 | module
public import Foundation.FirstOrder.Arithmetic.Exponential.PPow2
public import Mathlib.Algebra.Order.Ring.Basic
@[expose] public section
/-!
# Exponential function
This file provides a proof of the theorem states that
the graph of exponential function is definable by $\Sigma_0$-formula and
it's inductive prop... | module
public import Foundation.FirstOrder.Arithmetic.Exponential.PPow2
public import Mathlib.Algebra.Order.Ring.Basic
@[expose] public section
/-!
# Exponential function
This file provides a proof of the theorem states that
the graph of exponential function is definable by $\Sigma_0$-formula and
it's inductive prop... | @@ -563,8 +563,15 @@
· have : ext i X = 2 * ext (√i) X + 1 := by simpa [ppi.sq_sqrt_eq ne2] using hodd.1
simp [this]
-@[simp] lemma zero_uniq_iff {y : V} : Exponential 0 y ↔ y = 1 := sorry
+lemma zero_uniq {y : V} (h : Exponential 0 y) : y = 1 := by
+ rcases h with (⟨_, rfl⟩ | ⟨X, _, Y, _, H₀, Hₛ, ⟨u... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_d144cb8a619f_11 | c455b75dc79a9ee3 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Exponential/Exp.lean | Exp | 11 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 39 | 2 | [
{
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{
"name": "ext_append_last",
"text": "lemma ext_append_last (i X : V) {z} (hz : z < i) :\n ext i (append i X z) = z := by\n simp [ext, append, div_add_mul_self, show 0 < i from pos_of_gt hz, hz]\n\n",
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{
"name": "inj",
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... | 24 | module
public import Foundation.FirstOrder.Arithmetic.Exponential.PPow2
public import Mathlib.Algebra.Order.Ring.Basic
@[expose] public section
/-!
# Exponential function
This file provides a proof of the theorem states that
the graph of exponential function is definable by $\Sigma_0$-formula and
it's inductive prop... | module
public import Foundation.FirstOrder.Arithmetic.Exponential.PPow2
public import Mathlib.Algebra.Order.Ring.Basic
@[expose] public section
/-!
# Exponential function
This file provides a proof of the theorem states that
the graph of exponential function is definable by $\Sigma_0$-formula and
it's inductive prop... | @@ -155,6 +155,10 @@
_ < (1 + z) * i := by simpa [add_mul] using mod_lt _ (pos_of_gt hz)
_ ≤ i^2 := by simpa [sq, add_comm] using mul_le_mul_of_nonneg_right (lt_iff_succ_le.mp hz) (by simp)
+lemma ext_append_last (i X : V) {z} (hz : z < i) :
+ ext i (append i X z) = ... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
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ablate_d144cb8a619f_12 | 2e622bfafcb30bee | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Exponential/Exp.lean | Exp | 12 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 39 | 2 | [
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{
"name": "append_lt",
"text": "lemma append_lt (i X : V) {z} (hz : z < i) : append i X z < i^2 := calc\n append i X z = (X % i) + z * i := rfl\n _ < (1 + z) * i := by simpa [add_mul] using mod_lt _ (pos_of_gt hz)\n _ ≤ i^2 := by simpa [sq, add_comm] using mul_le... | [
{
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"max_nes... | 23 | module
public import Foundation.FirstOrder.Arithmetic.Exponential.PPow2
public import Mathlib.Algebra.Order.Ring.Basic
@[expose] public section
/-!
# Exponential function
This file provides a proof of the theorem states that
the graph of exponential function is definable by $\Sigma_0$-formula and
it's inductive prop... | module
public import Foundation.FirstOrder.Arithmetic.Exponential.PPow2
public import Mathlib.Algebra.Order.Ring.Basic
@[expose] public section
/-!
# Exponential function
This file provides a proof of the theorem states that
the graph of exponential function is definable by $\Sigma_0$-formula and
it's inductive prop... | @@ -150,6 +150,11 @@
noncomputable def append (i X z : V) : V := X % i + z * i
+lemma append_lt (i X : V) {z} (hz : z < i) : append i X z < i^2 := calc
+ append i X z = (X % i) + z * i := rfl
+ _ < (1 + z) * i := by simpa [add_mul] using mod_lt _ (pos_of_gt hz)
+ _ ≤ i^2 ... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_d144cb8a619f_13 | b4a51cebd98789dd | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Exponential/Exp.lean | Exp | 13 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 39 | 4 | [
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"method": "deleted-dep",
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"n_chars": 86,
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... | [
{
"name": "ext_append_of_lt",
"text": "lemma ext_append_of_lt {i j : V} (hi : PPow2 i) (hj : PPow2 j) (hij : i < j) (X z : V) :\n ext i (append j X z) = ext i X := by\n have : i^2 ∣ j := Pow2.dvd_of_le hi.pow2.sq hj.pow2 (PPow2.sq_le_of_lt hi hj hij)\n calc\n ext i (append j X z) = ext i ((X % j) + ... | [
{
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... | 28 | module
public import Foundation.FirstOrder.Arithmetic.Exponential.PPow2
public import Mathlib.Algebra.Order.Ring.Basic
@[expose] public section
/-!
# Exponential function
This file provides a proof of the theorem states that
the graph of exponential function is definable by $\Sigma_0$-formula and
it's inductive prop... | module
public import Foundation.FirstOrder.Arithmetic.Exponential.PPow2
public import Mathlib.Algebra.Order.Ring.Basic
@[expose] public section
/-!
# Exponential function
This file provides a proof of the theorem states that
the graph of exponential function is definable by $\Sigma_0$-formula and
it's inductive prop... | @@ -159,8 +159,19 @@
ext i (append i X z) = z := by
simp [ext, append, div_add_mul_self, show 0 < i from pos_of_gt hz, hz]
+lemma ext_append_of_lt {i j : V} (hi : PPow2 i) (hj : PPow2 j) (hij : i < j) (X z : V) :
+ ext i (append j X z) = ext i X := by
+ have : i^2 ∣ j := Pow2.dvd_of_le hi.pow2.sq hj.pow2 ... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_d144cb8a619f_14 | 6f0a2bc621ffbd5f | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Exponential/Exp.lean | Exp | 14 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 39 | 4 | [
{
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... | [
{
"name": "exp_of_exponential",
"text": "lemma exp_of_exponential {a b : V} (h : Exponential a b) : Exp.exp a = b :=\n Eq.symm <| exponential_graph.mpr h\n\n",
"fan_in": 4,
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{
"name": "exp_zero",
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}
... | 1 | module
public import Foundation.FirstOrder.Arithmetic.Exponential.PPow2
public import Mathlib.Algebra.Order.Ring.Basic
@[expose] public section
/-!
# Exponential function
This file provides a proof of the theorem states that
the graph of exponential function is definable by $\Sigma_0$-formula and
it's inductive prop... | module
public import Foundation.FirstOrder.Arithmetic.Exponential.PPow2
public import Mathlib.Algebra.Order.Ring.Basic
@[expose] public section
/-!
# Exponential function
This file provides a proof of the theorem states that
the graph of exponential function is definable by $\Sigma_0$-formula and
it's inductive prop... | @@ -690,12 +690,16 @@
instance exp_definable_deltaZero : 𝚺₀-Function₁ (Exp.exp : V → V) := exp_defined_deltaZero.to_definable
-@[simp] lemma exp_zero : Exp.exp (0 : V) = 1 := sorry
+lemma exp_of_exponential {a b : V} (h : Exponential a b) : Exp.exp a = b :=
+ Eq.symm <| exponential_graph.mpr h
-@[simp] lemma 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_d144cb8a619f_15 | 95271d9926671818 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Exponential/Exp.lean | Exp | 15 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 39 | 4 | [
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"depth": 1,
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"method": "deleted-dep",
"proof_text": " by\n rw [Seq₀, ext_append_of_lt, ext_append_of_lt] <;> try simp [ppi, hi]\n exact H",
"n_chars": 86,
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{
"name": "ext_append_of_lt",
"text": "lemma ext_append_of_lt {i j : V} (hi : PPow2 i) (hj : PPow2 j) (hij : i < j) (X z : V) :\n ext i (append j X z) = ext i X := by\n have : i^2 ∣ j := Pow2.dvd_of_le hi.pow2.sq hj.pow2 (PPow2.sq_le_of_lt hi hj hij)\n calc\n ext i (append j X z) = ext i ((X % j) + ... | [
{
"name": "exp_succ",
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... | 29 | module
public import Foundation.FirstOrder.Arithmetic.Exponential.PPow2
public import Mathlib.Algebra.Order.Ring.Basic
@[expose] public section
/-!
# Exponential function
This file provides a proof of the theorem states that
the graph of exponential function is definable by $\Sigma_0$-formula and
it's inductive prop... | module
public import Foundation.FirstOrder.Arithmetic.Exponential.PPow2
public import Mathlib.Algebra.Order.Ring.Basic
@[expose] public section
/-!
# Exponential function
This file provides a proof of the theorem states that
the graph of exponential function is definable by $\Sigma_0$-formula and
it's inductive prop... | @@ -159,8 +159,19 @@
ext i (append i X z) = z := by
simp [ext, append, div_add_mul_self, show 0 < i from pos_of_gt hz, hz]
+lemma ext_append_of_lt {i j : V} (hi : PPow2 i) (hj : PPow2 j) (hij : i < j) (X z : V) :
+ ext i (append j X z) = ext i X := by
+ have : i^2 ∣ j := Pow2.dvd_of_le hi.pow2.sq hj.pow2 ... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_d144cb8a619f_16 | 1ce6ce1be9f7e738 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Exponential/Exp.lean | Exp | 16 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 39 | 4 | [
{
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"depth": 1,
"n_commands": 0,
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"method": "deleted-dep",
"proof_text": " by\n rw [Seq₀, ext_append_of_lt, ext_append_of_lt] <;> try simp [ppi, hi]\n exact H",
"n_chars": 86,
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... | [
{
"name": "ext_append_of_lt",
"text": "lemma ext_append_of_lt {i j : V} (hi : PPow2 i) (hj : PPow2 j) (hij : i < j) (X z : V) :\n ext i (append j X z) = ext i X := by\n have : i^2 ∣ j := Pow2.dvd_of_le hi.pow2.sq hj.pow2 (PPow2.sq_le_of_lt hi hj hij)\n calc\n ext i (append j X z) = ext i ((X % j) + ... | [
{
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... | 29 | module
public import Foundation.FirstOrder.Arithmetic.Exponential.PPow2
public import Mathlib.Algebra.Order.Ring.Basic
@[expose] public section
/-!
# Exponential function
This file provides a proof of the theorem states that
the graph of exponential function is definable by $\Sigma_0$-formula and
it's inductive prop... | module
public import Foundation.FirstOrder.Arithmetic.Exponential.PPow2
public import Mathlib.Algebra.Order.Ring.Basic
@[expose] public section
/-!
# Exponential function
This file provides a proof of the theorem states that
the graph of exponential function is definable by $\Sigma_0$-formula and
it's inductive prop... | @@ -159,8 +159,19 @@
ext i (append i X z) = z := by
simp [ext, append, div_add_mul_self, show 0 < i from pos_of_gt hz, hz]
+lemma ext_append_of_lt {i j : V} (hi : PPow2 i) (hj : PPow2 j) (hij : i < j) (X z : V) :
+ ext i (append j X z) = ext i X := by
+ have : i^2 ∣ j := Pow2.dvd_of_le hi.pow2.sq hj.pow2 ... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_5c8e33b6b57f_0 | 5f8a6d5ef0e18bb6 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Incompleteness/ProvabilityAbstraction/Basic.lean | Basic | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 14 | 1 | [
{
"theorem_name": "bew_distribute_and'",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
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"method": "deleted-dep",
"proof_text": " λ h => bew_distribute_and ⨀ h",
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"n_auto... | [
{
"name": "bew_distribute_and",
"text": "lemma bew_distribute_and [𝔅.HBL2] [L₀.DecidableEq] : T₀ ⊢ 𝔅 (σ ⋏ τ) 🡒 𝔅 σ ⋏ 𝔅 τ := by\n have h₁ : T₀ ⊢ 𝔅 (σ ⋏ τ) 🡒 𝔅 σ := bew_distribute_imply $ D1 and₁!;\n have h₂ : T₀ ⊢ 𝔅 (σ ⋏ τ) 🡒 𝔅 τ := bew_distribute_imply $ D1 and₂!;\n cl_prover [h₁, h₂];\n\n",
... | [
{
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"max_nes... | 1 | module
public import Foundation.FirstOrder.Arithmetic.Basic
public import Foundation.Meta.ClProver
@[expose] public section
/-!
# Abstract incompleteness theorems and related results
-/
namespace LO
open LO.Entailment
namespace FirstOrder
variable {L₀ L : Language}
abbrev Language.ReferenceableBy (L L₀ : Languag... | module
public import Foundation.FirstOrder.Arithmetic.Basic
public import Foundation.Meta.ClProver
@[expose] public section
/-!
# Abstract incompleteness theorems and related results
-/
namespace LO
open LO.Entailment
namespace FirstOrder
variable {L₀ L : Language}
abbrev Language.ReferenceableBy (L L₀ : Languag... | @@ -118,8 +118,13 @@
instance [𝔅.HBL2] : 𝔅.Mono := ⟨λ h => bew_distribute_imply $ D1 h⟩
instance [𝔅.HBL2] : 𝔅.Ext := ⟨λ h => E!_intro (mono (K!_left h)) (mono (K!_right h))⟩
-lemma bew_distribute_and' [𝔅.HBL2] [L₀.DecidableEq] : T₀ ⊢ 𝔅 (σ ⋏ τ) → T₀ ⊢ 𝔅 σ ⋏ 𝔅 τ := sorry
+lemma bew_distribute_and [𝔅.HBL2] [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_5c8e33b6b57f_1 | af333ee90a914d75 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Incompleteness/ProvabilityAbstraction/Basic.lean | Basic | 1 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 14 | 3 | [
{
"theorem_name": "g",
"depth": 1,
"n_commands": 0,
"n_lines": 4,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n constructor\n . apply unprovable_gödel\n . apply unrefutable_gödel",
"n_chars": 74,
"n_subproofs": 0,
"n_tactics": 4,
"... | [
{
"name": "unprovable_g",
"text": "theorem unprovable_gödel : T ⊬ (gödel 𝔅) := by\n intro h;\n have h₁ : T ⊢ 𝔅 (gödel 𝔅) := WeakerThan.pbl $ D1 h;\n have h₂ : T ⊢ (gödel 𝔅) 🡘 ∼𝔅 (gödel 𝔅) := WeakerThan.pbl $ gödel_spec;\n have : T ⊢ ⊥ := by cl_prover [h₁, h₂, h];\n have : ¬Consistent T := not_co... | [
{
"name": "g",
"fan_in": 11,
"n_deps_direct": 8,
"n_deps_transitive": 4,
"n_lines": 6,
"n_chars": 150,
"n_subproofs": 0,
"n_tactics": 4,
"cyclomatic": 2,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 3,
"automation_only": false,
"max_nesting": 2
}
] | 2 | module
public import Foundation.FirstOrder.Arithmetic.Basic
public import Foundation.Meta.ClProver
@[expose] public section
/-!
# Abstract incompleteness theorems and related results
-/
namespace LO
open LO.Entailment
namespace FirstOrder
variable {L₀ L : Language}
abbrev Language.ReferenceableBy (L L₀ : Languag... | module
public import Foundation.FirstOrder.Arithmetic.Basic
public import Foundation.Meta.ClProver
@[expose] public section
/-!
# Abstract incompleteness theorems and related results
-/
namespace LO
open LO.Entailment
namespace FirstOrder
variable {L₀ L : Language}
abbrev Language.ReferenceableBy (L L₀ : Languag... | @@ -154,6 +154,14 @@
variable [L.DecidableEq]
variable [T₀ ⪯ T] [Consistent T]
+theorem unprovable_gödel : T ⊬ (gödel 𝔅) := by
+ intro h;
+ have h₁ : T ⊢ 𝔅 (gödel 𝔅) := WeakerThan.pbl $ D1 h;
+ have h₂ : T ⊢ (gödel 𝔅) 🡘 ∼𝔅 (gödel 𝔅) := WeakerThan.pbl $ gödel_spec;
+ have : T ⊢ ⊥ := by cl_prover [h₁, h₂, ... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_5c8e33b6b57f_2 | 450759e1af4f8d2d | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Incompleteness/ProvabilityAbstraction/Basic.lean | Basic | 2 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 14 | 1 | [
{
"theorem_name": "formalized_unprovable_g",
"depth": 1,
"n_commands": 0,
"n_lines": 6,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n suffices T₀ ⊢ ∼𝔅 ⊥ 🡒 ∼𝔅 𝐆 from this\n have h₁ : T₀ ⊢ 𝔅 𝐆 🡒 𝔅 (𝔅 𝐆) := D3\n have h₂ : T₀ ⊢ 𝔅 𝐆 🡒 𝔅 (... | [
{
"name": "mono'",
"text": "lemma mono' [𝔅.Mono] (h : T₀ ⊢ σ 🡒 τ) : T₀ ⊢ 𝔅 σ 🡒 𝔅 τ := 𝔅.mono $ WeakerThan.pbl h\n",
"fan_in": 1,
"n_lines": 2,
"n_chars": 114,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
... | [
{
"name": "formalized_unprovable_g",
"fan_in": 1,
"n_deps_direct": 2,
"n_deps_transitive": 2,
"n_lines": 9,
"n_chars": 495,
"n_subproofs": 4,
"n_tactics": 6,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
"max... | 1 | module
public import Foundation.FirstOrder.Arithmetic.Basic
public import Foundation.Meta.ClProver
@[expose] public section
/-!
# Abstract incompleteness theorems and related results
-/
namespace LO
open LO.Entailment
namespace FirstOrder
variable {L₀ L : Language}
abbrev Language.ReferenceableBy (L L₀ : Languag... | module
public import Foundation.FirstOrder.Arithmetic.Basic
public import Foundation.Meta.ClProver
@[expose] public section
/-!
# Abstract incompleteness theorems and related results
-/
namespace LO
open LO.Entailment
namespace FirstOrder
variable {L₀ L : Language}
abbrev Language.ReferenceableBy (L L₀ : Languag... | @@ -127,6 +127,7 @@
{𝔅 : Provability T₀ T}
{σ τ : Sentence L}
+lemma mono' [𝔅.Mono] (h : T₀ ⊢ σ 🡒 τ) : T₀ ⊢ 𝔅 σ 🡒 𝔅 τ := 𝔅.mono $ WeakerThan.pbl h
end
end Provability
@@ -190,7 +191,7 @@
theorem formalized_unprovable_gödel : T₀ ⊢ 𝔅.con 🡒 ∼𝔅 𝐆 := by
suffices T₀ ⊢ ∼𝔅 ⊥ 🡒 ∼𝔅 𝐆 from this
... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_5c8e33b6b57f_3 | 917ffd3f24408c65 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Incompleteness/ProvabilityAbstraction/Basic.lean | Basic | 3 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 14 | 1 | [
{
"theorem_name": "g",
"depth": 1,
"n_commands": 0,
"n_lines": 5,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n have h₁ : T₀ ⊢ ∼𝔅 𝐆 🡒 𝔅.con := formalized_consistent_of_existance_unprovable\n have h₂ : T₀ ⊢ 𝔅.con 🡒 ∼𝔅 𝐆 := formalized_unprova... | [
{
"name": "formalized_consistent_of_existance_unprovable",
"text": "lemma formalized_consistent_of_existance_unprovable [L.DecidableEq] : T₀ ⊢ ∼𝔅 σ 🡒 𝔅.con := contra! $ mdp! D2 $ D1 efq!\n\n",
"fan_in": 1,
"n_lines": 3,
"n_chars": 136,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic... | [
{
"name": "g",
"fan_in": 10,
"n_deps_direct": 6,
"n_deps_transitive": 5,
"n_lines": 7,
"n_chars": 334,
"n_subproofs": 3,
"n_tactics": 6,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
"max_nesting": 2
}
] | 3 | module
public import Foundation.FirstOrder.Arithmetic.Basic
public import Foundation.Meta.ClProver
@[expose] public section
/-!
# Abstract incompleteness theorems and related results
-/
namespace LO
open LO.Entailment
namespace FirstOrder
variable {L₀ L : Language}
abbrev Language.ReferenceableBy (L L₀ : Languag... | module
public import Foundation.FirstOrder.Arithmetic.Basic
public import Foundation.Meta.ClProver
@[expose] public section
/-!
# Abstract incompleteness theorems and related results
-/
namespace LO
open LO.Entailment
namespace FirstOrder
variable {L₀ L : Language}
abbrev Language.ReferenceableBy (L L₀ : Languag... | @@ -181,6 +181,8 @@
variable [𝔅.HBL]
omit [Diagonalization T₀] in
+lemma formalized_consistent_of_existance_unprovable [L.DecidableEq] : T₀ ⊢ ∼𝔅 σ 🡒 𝔅.con := contra! $ mdp! D2 $ D1 efq!
+
local notation "𝐆" => gödel 𝔅
variable [L.DecidableEq] [T₀ ⪯ T]
@@ -194,7 +196,7 @@
cl_prover [h₁, h₂, h₃]
theor... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_5c8e33b6b57f_4 | 2bbde93a1021c4d0 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Incompleteness/ProvabilityAbstraction/Basic.lean | Basic | 4 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 14 | 3 | [
{
"theorem_name": "g",
"depth": 1,
"n_commands": 0,
"n_lines": 4,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n constructor\n . apply unprovable_gödel\n . apply unrefutable_gödel",
"n_chars": 74,
"n_subproofs": 0,
"n_tactics": 4,
"... | [
{
"name": "unprovable_g",
"text": "theorem unprovable_gödel : T ⊬ (gödel 𝔅) := by\n intro h;\n have h₁ : T ⊢ 𝔅 (gödel 𝔅) := WeakerThan.pbl $ D1 h;\n have h₂ : T ⊢ (gödel 𝔅) 🡘 ∼𝔅 (gödel 𝔅) := WeakerThan.pbl $ gödel_spec;\n have : T ⊢ ⊥ := by cl_prover [h₁, h₂, h];\n have : ¬Consistent T := not_co... | [
{
"name": "con_unprovable",
"fan_in": 1,
"n_deps_direct": 2,
"n_deps_transitive": 3,
"n_lines": 7,
"n_chars": 204,
"n_subproofs": 2,
"n_tactics": 5,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 2,
"automation_only": false,
"max_nesting"... | 1 | module
public import Foundation.FirstOrder.Arithmetic.Basic
public import Foundation.Meta.ClProver
@[expose] public section
/-!
# Abstract incompleteness theorems and related results
-/
namespace LO
open LO.Entailment
namespace FirstOrder
variable {L₀ L : Language}
abbrev Language.ReferenceableBy (L L₀ : Languag... | module
public import Foundation.FirstOrder.Arithmetic.Basic
public import Foundation.Meta.ClProver
@[expose] public section
/-!
# Abstract incompleteness theorems and related results
-/
namespace LO
open LO.Entailment
namespace FirstOrder
variable {L₀ L : Language}
abbrev Language.ReferenceableBy (L L₀ : Languag... | @@ -154,6 +154,14 @@
variable [L.DecidableEq]
variable [T₀ ⪯ T] [Consistent T]
+theorem unprovable_gödel : T ⊬ (gödel 𝔅) := by
+ intro h;
+ have h₁ : T ⊢ 𝔅 (gödel 𝔅) := WeakerThan.pbl $ D1 h;
+ have h₂ : T ⊢ (gödel 𝔅) 🡘 ∼𝔅 (gödel 𝔅) := WeakerThan.pbl $ gödel_spec;
+ have : T ⊢ ⊥ := by cl_prover [h₁, h₂, ... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_5c8e33b6b57f_5 | cb1b7c9a95b74079 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Incompleteness/ProvabilityAbstraction/Basic.lean | Basic | 5 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 14 | 2 | [
{
"theorem_name": "g",
"depth": 1,
"n_commands": 0,
"n_lines": 4,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n constructor\n . apply unprovable_gödel\n . apply unrefutable_gödel",
"n_chars": 74,
"n_subproofs": 0,
"n_tactics": 4,
"... | [
{
"name": "unrefutable_g",
"text": "theorem unrefutable_gödel [𝔅.Kreisel] : T ⊬ ∼(gödel 𝔅) := by\n intro h₂;\n have h₁ : T ⊢ (gödel 𝔅) := WeakerThan.pbl $ 𝔅.KR $ by cl_prover [gödel_spec (T₀ := T₀), h₂];\n have : T ⊢ ⊥ := (N!_iff_CO!.mp $ WeakerThan.pbl $ h₂) ⨀ h₁;\n have : ¬Consistent T := not_cons... | [
{
"name": "con_unrefutable",
"fan_in": 2,
"n_deps_direct": 2,
"n_deps_transitive": 2,
"n_lines": 7,
"n_chars": 242,
"n_subproofs": 2,
"n_tactics": 6,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 2,
"automation_only": false,
"max_nesting... | 1 | module
public import Foundation.FirstOrder.Arithmetic.Basic
public import Foundation.Meta.ClProver
@[expose] public section
/-!
# Abstract incompleteness theorems and related results
-/
namespace LO
open LO.Entailment
namespace FirstOrder
variable {L₀ L : Language}
abbrev Language.ReferenceableBy (L L₀ : Languag... | module
public import Foundation.FirstOrder.Arithmetic.Basic
public import Foundation.Meta.ClProver
@[expose] public section
/-!
# Abstract incompleteness theorems and related results
-/
namespace LO
open LO.Entailment
namespace FirstOrder
variable {L₀ L : Language}
abbrev Language.ReferenceableBy (L L₀ : Languag... | @@ -162,10 +162,17 @@
have : ¬Consistent T := not_consistent_iff_inconsistent.mpr <| inconsistent_iff_provable_bot.mpr this;
contradiction
+theorem unrefutable_gödel [𝔅.Kreisel] : T ⊬ ∼(gödel 𝔅) := by
+ intro h₂;
+ have h₁ : T ⊢ (gödel 𝔅) := WeakerThan.pbl $ 𝔅.KR $ by cl_prover [gödel_spec (T₀ := T₀), h₂]... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_5c8e33b6b57f_6 | a58b50c60c27a04f | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Incompleteness/ProvabilityAbstraction/Basic.lean | Basic | 6 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 14 | 2 | [
{
"theorem_name": "g",
"depth": 1,
"n_commands": 0,
"n_lines": 4,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n constructor\n . apply unprovable_gödel\n . apply unrefutable_gödel",
"n_chars": 74,
"n_subproofs": 0,
"n_tactics": 4,
"... | [
{
"name": "unrefutable_g",
"text": "theorem unrefutable_gödel [𝔅.Kreisel] : T ⊬ ∼(gödel 𝔅) := by\n intro h₂;\n have h₁ : T ⊢ (gödel 𝔅) := WeakerThan.pbl $ 𝔅.KR $ by cl_prover [gödel_spec (T₀ := T₀), h₂];\n have : T ⊢ ⊥ := (N!_iff_CO!.mp $ WeakerThan.pbl $ h₂) ⨀ h₁;\n have : ¬Consistent T := not_cons... | [
{
"name": "con_independent",
"fan_in": 0,
"n_deps_direct": 2,
"n_deps_transitive": 6,
"n_lines": 6,
"n_chars": 151,
"n_subproofs": 0,
"n_tactics": 4,
"cyclomatic": 2,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 3,
"automation_only": false,
"max_nesting... | 4 | module
public import Foundation.FirstOrder.Arithmetic.Basic
public import Foundation.Meta.ClProver
@[expose] public section
/-!
# Abstract incompleteness theorems and related results
-/
namespace LO
open LO.Entailment
namespace FirstOrder
variable {L₀ L : Language}
abbrev Language.ReferenceableBy (L L₀ : Languag... | module
public import Foundation.FirstOrder.Arithmetic.Basic
public import Foundation.Meta.ClProver
@[expose] public section
/-!
# Abstract incompleteness theorems and related results
-/
namespace LO
open LO.Entailment
namespace FirstOrder
variable {L₀ L : Language}
abbrev Language.ReferenceableBy (L L₀ : Languag... | @@ -162,10 +162,17 @@
have : ¬Consistent T := not_consistent_iff_inconsistent.mpr <| inconsistent_iff_provable_bot.mpr this;
contradiction
+theorem unrefutable_gödel [𝔅.Kreisel] : T ⊬ ∼(gödel 𝔅) := by
+ intro h₂;
+ have h₁ : T ⊢ (gödel 𝔅) := WeakerThan.pbl $ 𝔅.KR $ by cl_prover [gödel_spec (T₀ := T₀), h₂]... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_5c8e33b6b57f_7 | 4604bf31c815fb2c | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Incompleteness/ProvabilityAbstraction/Basic.lean | Basic | 7 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 14 | 2 | [
{
"theorem_name": "kreisel_specAux₂",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " K!_right kreisel_spec",
"n_chars": 22,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,... | [
{
"name": "kreisel_spec",
"text": "lemma kreisel_spec : T₀ ⊢ (𝐊 σ) 🡘 (𝔅 (𝐊 σ) 🡒 σ) := by\n simpa [kreisel, Provability.pr, Rew.subst_comp_subst, ←TransitiveRewriting.comp_app] using diag “x. !𝔅.prov x → !σ”;\n\n",
"fan_in": 2,
"n_lines": 4,
"n_chars": 208,
"n_subproofs": 0,
"n_tac... | [
{
"name": "kreisel_specAux₂",
"fan_in": 2,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 3,
"n_chars": 110,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
"max_nestin... | 1 | module
public import Foundation.FirstOrder.Arithmetic.Basic
public import Foundation.Meta.ClProver
@[expose] public section
/-!
# Abstract incompleteness theorems and related results
-/
namespace LO
open LO.Entailment
namespace FirstOrder
variable {L₀ L : Language}
abbrev Language.ReferenceableBy (L L₀ : Languag... | module
public import Foundation.FirstOrder.Arithmetic.Basic
public import Foundation.Meta.ClProver
@[expose] public section
/-!
# Abstract incompleteness theorems and related results
-/
namespace LO
open LO.Entailment
namespace FirstOrder
variable {L₀ L : Language}
abbrev Language.ReferenceableBy (L L₀ : Languag... | @@ -211,8 +211,11 @@
local notation "𝐊" => kreisel 𝔅
-private lemma kreisel_specAux₂ : T₀ ⊢ (𝔅 (𝐊 σ) 🡒 σ) 🡒 (𝐊 σ) := sorry
+lemma kreisel_spec : T₀ ⊢ (𝐊 σ) 🡘 (𝔅 (𝐊 σ) 🡒 σ) := by
+ simpa [kreisel, Provability.pr, Rew.subst_comp_subst, ←TransitiveRewriting.comp_app] using diag “x. !𝔅.prov x → !σ”;
+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_5c8e33b6b57f_8 | 5e669007247c9620 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Incompleteness/ProvabilityAbstraction/Basic.lean | Basic | 8 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 14 | 2 | [
{
"theorem_name": "kreisel_specAux₂",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " K!_right kreisel_spec",
"n_chars": 22,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,... | [
{
"name": "kreisel_spec",
"text": "lemma kreisel_spec : T₀ ⊢ (𝐊 σ) 🡘 (𝔅 (𝐊 σ) 🡒 σ) := by\n simpa [kreisel, Provability.pr, Rew.subst_comp_subst, ←TransitiveRewriting.comp_app] using diag “x. !𝔅.prov x → !σ”;\n\n",
"fan_in": 2,
"n_lines": 4,
"n_chars": 208,
"n_subproofs": 0,
"n_tac... | [
{
"name": "kreisel_specAux₁",
"fan_in": 2,
"n_deps_direct": 2,
"n_deps_transitive": 2,
"n_lines": 4,
"n_chars": 201,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
"max_nestin... | 1 | module
public import Foundation.FirstOrder.Arithmetic.Basic
public import Foundation.Meta.ClProver
@[expose] public section
/-!
# Abstract incompleteness theorems and related results
-/
namespace LO
open LO.Entailment
namespace FirstOrder
variable {L₀ L : Language}
abbrev Language.ReferenceableBy (L L₀ : Languag... | module
public import Foundation.FirstOrder.Arithmetic.Basic
public import Foundation.Meta.ClProver
@[expose] public section
/-!
# Abstract incompleteness theorems and related results
-/
namespace LO
open LO.Entailment
namespace FirstOrder
variable {L₀ L : Language}
abbrev Language.ReferenceableBy (L L₀ : Languag... | @@ -211,9 +211,13 @@
local notation "𝐊" => kreisel 𝔅
+lemma kreisel_spec : T₀ ⊢ (𝐊 σ) 🡘 (𝔅 (𝐊 σ) 🡒 σ) := by
+ simpa [kreisel, Provability.pr, Rew.subst_comp_subst, ←TransitiveRewriting.comp_app] using diag “x. !𝔅.prov x → !σ”;
+
variable [𝔅.HBL]
-private lemma kreisel_specAux₁ [L.DecidableEq] [T₀ ⪯ 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_5c8e33b6b57f_9 | 7beeb0e333fc95a5 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Incompleteness/ProvabilityAbstraction/Basic.lean | Basic | 9 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 14 | 2 | [
{
"theorem_name": "l",
"depth": 1,
"n_commands": 0,
"n_lines": 4,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n have d₁ : T ⊢ 𝔅 (𝐊 σ) 🡒 σ := C!_trans (WeakerThan.pbl kreisel_specAux₁) H;\n have d₂ : T ⊢ 𝔅 (𝐊 σ) := WeakerThan.pbl $ D1 $ Wea... | [
{
"name": "kreisel_specAux₂",
"text": "private lemma kreisel_specAux₂ : T₀ ⊢ (𝔅 (𝐊 σ) 🡒 σ) 🡒 (𝐊 σ) := K!_right kreisel_spec\n\n",
"fan_in": 2,
"n_lines": 3,
"n_chars": 110,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_stru... | [
{
"name": "l",
"fan_in": 1,
"n_deps_direct": 3,
"n_deps_transitive": 4,
"n_lines": 6,
"n_chars": 292,
"n_subproofs": 2,
"n_tactics": 7,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": false,
"max_nesting": 2
}
] | 3 | module
public import Foundation.FirstOrder.Arithmetic.Basic
public import Foundation.Meta.ClProver
@[expose] public section
/-!
# Abstract incompleteness theorems and related results
-/
namespace LO
open LO.Entailment
namespace FirstOrder
variable {L₀ L : Language}
abbrev Language.ReferenceableBy (L L₀ : Languag... | module
public import Foundation.FirstOrder.Arithmetic.Basic
public import Foundation.Meta.ClProver
@[expose] public section
/-!
# Abstract incompleteness theorems and related results
-/
namespace LO
open LO.Entailment
namespace FirstOrder
variable {L₀ L : Language}
abbrev Language.ReferenceableBy (L L₀ : Languag... | @@ -214,6 +214,8 @@
lemma kreisel_spec : T₀ ⊢ (𝐊 σ) 🡘 (𝔅 (𝐊 σ) 🡒 σ) := by
simpa [kreisel, Provability.pr, Rew.subst_comp_subst, ←TransitiveRewriting.comp_app] using diag “x. !𝔅.prov x → !σ”;
+private lemma kreisel_specAux₂ : T₀ ⊢ (𝔅 (𝐊 σ) 🡒 σ) 🡒 (𝐊 σ) := K!_right kreisel_spec
+
variable [𝔅.HBL]
pr... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_5c8e33b6b57f_10 | 010591e1f9385f1d | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Incompleteness/ProvabilityAbstraction/Basic.lean | Basic | 10 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 14 | 2 | [
{
"theorem_name": "l",
"depth": 1,
"n_commands": 0,
"n_lines": 4,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n have d₁ : T ⊢ 𝔅 (𝐊 σ) 🡒 σ := C!_trans (WeakerThan.pbl kreisel_specAux₁) H;\n have d₂ : T ⊢ 𝔅 (𝐊 σ) := WeakerThan.pbl $ D1 $ Wea... | [
{
"name": "kreisel_specAux₂",
"text": "private lemma kreisel_specAux₂ : T₀ ⊢ (𝔅 (𝐊 σ) 🡒 σ) 🡒 (𝐊 σ) := K!_right kreisel_spec\n\n",
"fan_in": 2,
"n_lines": 3,
"n_chars": 110,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_stru... | [
{
"name": "formalized_l",
"fan_in": 0,
"n_deps_direct": 3,
"n_deps_transitive": 4,
"n_lines": 7,
"n_chars": 421,
"n_subproofs": 3,
"n_tactics": 9,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": false,
"max_nesting": ... | 3 | module
public import Foundation.FirstOrder.Arithmetic.Basic
public import Foundation.Meta.ClProver
@[expose] public section
/-!
# Abstract incompleteness theorems and related results
-/
namespace LO
open LO.Entailment
namespace FirstOrder
variable {L₀ L : Language}
abbrev Language.ReferenceableBy (L L₀ : Languag... | module
public import Foundation.FirstOrder.Arithmetic.Basic
public import Foundation.Meta.ClProver
@[expose] public section
/-!
# Abstract incompleteness theorems and related results
-/
namespace LO
open LO.Entailment
namespace FirstOrder
variable {L₀ L : Language}
abbrev Language.ReferenceableBy (L L₀ : Languag... | @@ -214,6 +214,8 @@
lemma kreisel_spec : T₀ ⊢ (𝐊 σ) 🡘 (𝔅 (𝐊 σ) 🡒 σ) := by
simpa [kreisel, Provability.pr, Rew.subst_comp_subst, ←TransitiveRewriting.comp_app] using diag “x. !𝔅.prov x → !σ”;
+private lemma kreisel_specAux₂ : T₀ ⊢ (𝔅 (𝐊 σ) 🡒 σ) 🡒 (𝐊 σ) := K!_right kreisel_spec
+
variable [𝔅.HBL]
pr... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_5c8e33b6b57f_11 | af75dc01d4a61882 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Incompleteness/ProvabilityAbstraction/Basic.lean | Basic | 11 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 14 | 2 | [
{
"theorem_name": "kreisel_specAux₂",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " K!_right kreisel_spec",
"n_chars": 22,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,... | [
{
"name": "kreisel_spec",
"text": "lemma kreisel_spec : T₀ ⊢ (𝐊 σ) 🡘 (𝔅 (𝐊 σ) 🡒 σ) := by\n simpa [kreisel, Provability.pr, Rew.subst_comp_subst, ←TransitiveRewriting.comp_app] using diag “x. !𝔅.prov x → !σ”;\n\n",
"fan_in": 2,
"n_lines": 4,
"n_chars": 208,
"n_subproofs": 0,
"n_tac... | [
{
"name": "formalized_unprovable_not_con",
"fan_in": 1,
"n_deps_direct": 2,
"n_deps_transitive": 8,
"n_lines": 7,
"n_chars": 263,
"n_subproofs": 2,
"n_tactics": 9,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
... | 5 | module
public import Foundation.FirstOrder.Arithmetic.Basic
public import Foundation.Meta.ClProver
@[expose] public section
/-!
# Abstract incompleteness theorems and related results
-/
namespace LO
open LO.Entailment
namespace FirstOrder
variable {L₀ L : Language}
abbrev Language.ReferenceableBy (L L₀ : Languag... | module
public import Foundation.FirstOrder.Arithmetic.Basic
public import Foundation.Meta.ClProver
@[expose] public section
/-!
# Abstract incompleteness theorems and related results
-/
namespace LO
open LO.Entailment
namespace FirstOrder
variable {L₀ L : Language}
abbrev Language.ReferenceableBy (L L₀ : Languag... | @@ -217,11 +217,15 @@
local notation "𝐊" => kreisel 𝔅
-private lemma kreisel_specAux₂ : T₀ ⊢ (𝔅 (𝐊 σ) 🡒 σ) 🡒 (𝐊 σ) := sorry
+lemma kreisel_spec : T₀ ⊢ (𝐊 σ) 🡘 (𝔅 (𝐊 σ) 🡒 σ) := by
+ simpa [kreisel, Provability.pr, Rew.subst_comp_subst, ←TransitiveRewriting.comp_app] using diag “x. !𝔅.prov x → !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_5c8e33b6b57f_12 | 042d5ad3d1a1340e | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Incompleteness/ProvabilityAbstraction/Basic.lean | Basic | 12 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 14 | 2 | [
{
"theorem_name": "l",
"depth": 1,
"n_commands": 0,
"n_lines": 4,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n have d₁ : T ⊢ 𝔅 (𝐊 σ) 🡒 σ := C!_trans (WeakerThan.pbl kreisel_specAux₁) H;\n have d₂ : T ⊢ 𝔅 (𝐊 σ) := WeakerThan.pbl $ D1 $ Wea... | [
{
"name": "kreisel_specAux₂",
"text": "private lemma kreisel_specAux₂ : T₀ ⊢ (𝔅 (𝐊 σ) 🡒 σ) 🡒 (𝐊 σ) := K!_right kreisel_spec\n\n",
"fan_in": 2,
"n_lines": 3,
"n_chars": 110,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_stru... | [
{
"name": "formalized_unrefutable_g",
"fan_in": 0,
"n_deps_direct": 5,
"n_deps_transitive": 9,
"n_lines": 12,
"n_chars": 422,
"n_subproofs": 2,
"n_tactics": 13,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
"... | 6 | module
public import Foundation.FirstOrder.Arithmetic.Basic
public import Foundation.Meta.ClProver
@[expose] public section
/-!
# Abstract incompleteness theorems and related results
-/
namespace LO
open LO.Entailment
namespace FirstOrder
variable {L₀ L : Language}
abbrev Language.ReferenceableBy (L L₀ : Languag... | module
public import Foundation.FirstOrder.Arithmetic.Basic
public import Foundation.Meta.ClProver
@[expose] public section
/-!
# Abstract incompleteness theorems and related results
-/
namespace LO
open LO.Entailment
namespace FirstOrder
variable {L₀ L : Language}
abbrev Language.ReferenceableBy (L L₀ : Languag... | @@ -220,6 +220,8 @@
lemma kreisel_spec : T₀ ⊢ (𝐊 σ) 🡘 (𝔅 (𝐊 σ) 🡒 σ) := by
simpa [kreisel, Provability.pr, Rew.subst_comp_subst, ←TransitiveRewriting.comp_app] using diag “x. !𝔅.prov x → !σ”;
+private lemma kreisel_specAux₂ : T₀ ⊢ (𝔅 (𝐊 σ) 🡒 σ) 🡒 (𝐊 σ) := K!_right kreisel_spec
+
variable [𝔅.HBL]
pr... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_5c8e33b6b57f_13 | 967829c636964c60 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Incompleteness/ProvabilityAbstraction/Basic.lean | Basic | 13 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 14 | 3 | [
{
"theorem_name": "g",
"depth": 1,
"n_commands": 0,
"n_lines": 4,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n constructor\n . apply unprovable_gödel\n . apply unrefutable_gödel",
"n_chars": 74,
"n_subproofs": 0,
"n_tactics": 4,
"... | [
{
"name": "unprovable_g",
"text": "theorem unprovable_gödel : T ⊬ (gödel 𝔅) := by\n intro h;\n have h₁ : T ⊢ 𝔅 (gödel 𝔅) := WeakerThan.pbl $ D1 h;\n have h₂ : T ⊢ (gödel 𝔅) 🡘 ∼𝔅 (gödel 𝔅) := WeakerThan.pbl $ gödel_spec;\n have : T ⊢ ⊥ := by cl_prover [h₁, h₂, h];\n have : ¬Consistent T := not_co... | [
{
"name": "rosser_independent",
"fan_in": 1,
"n_deps_direct": 2,
"n_deps_transitive": 4,
"n_lines": 6,
"n_chars": 156,
"n_subproofs": 0,
"n_tactics": 4,
"cyclomatic": 2,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 3,
"automation_only": false,
"max_nest... | 2 | module
public import Foundation.FirstOrder.Arithmetic.Basic
public import Foundation.Meta.ClProver
@[expose] public section
/-!
# Abstract incompleteness theorems and related results
-/
namespace LO
open LO.Entailment
namespace FirstOrder
variable {L₀ L : Language}
abbrev Language.ReferenceableBy (L L₀ : Languag... | module
public import Foundation.FirstOrder.Arithmetic.Basic
public import Foundation.Meta.ClProver
@[expose] public section
/-!
# Abstract incompleteness theorems and related results
-/
namespace LO
open LO.Entailment
namespace FirstOrder
variable {L₀ L : Language}
abbrev Language.ReferenceableBy (L L₀ : Languag... | @@ -154,6 +154,14 @@
variable [L.DecidableEq]
variable [T₀ ⪯ T] [Consistent T]
+theorem unprovable_gödel : T ⊬ (gödel 𝔅) := by
+ intro h;
+ have h₁ : T ⊢ 𝔅 (gödel 𝔅) := WeakerThan.pbl $ D1 h;
+ have h₂ : T ⊢ (gödel 𝔅) 🡘 ∼𝔅 (gödel 𝔅) := WeakerThan.pbl $ gödel_spec;
+ have : T ⊢ ⊥ := by cl_prover [h₁, h₂, ... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_13fe48712ba7_0 | 258b29f69f0d5cb1 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Bootstrapping/Syntax/Term/Coding.lean | Coding | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 7 | 1 | [
{
"theorem_name": "typed_quote_numeral_eq_numeral",
"depth": 1,
"n_commands": 0,
"n_lines": 12,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n match k with\n | 0 =>\n simp [Bootstrapping.Arithmetic.typedNumeral,\n Bootstrapping.Arith... | [
{
"name": "typed_quote_numeral_eq_numeral_one",
"text": "lemma typed_quote_numeral_eq_numeral_one :\n (⌜((1 : ℕ) : SyntacticSemiterm ℒₒᵣ n)⌝ : Bootstrapping.Semiterm V ℒₒᵣ n) = typedNumeral 1 := by\n simp [Bootstrapping.Arithmetic.typedNumeral,\n Bootstrapping.Arithmetic.one, Bootstrapping.Arithmetic... | [
{
"name": "typed_quote_numeral_eq_numeral",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 15,
"n_chars": 863,
"n_subproofs": 0,
"n_tactics": 12,
"cyclomatic": 2,
"n_automation": 6,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": false,... | 1 | module
public import Foundation.FirstOrder.Bootstrapping.Syntax.Term.Typed
public import Mathlib.Combinatorics.Colex
@[expose] public section
open Encodable LO FirstOrder Arithmetic PeanoMinus Bootstrapping
namespace LO.FirstOrder.Semiterm
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
variable {L : La... | module
public import Foundation.FirstOrder.Bootstrapping.Syntax.Term.Typed
public import Mathlib.Combinatorics.Colex
@[expose] public section
open Encodable LO FirstOrder Arithmetic PeanoMinus Bootstrapping
namespace LO.FirstOrder.Semiterm
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
variable {L : La... | @@ -53,6 +53,12 @@
@[simp] lemma typed_quote_mul (t u : SyntacticSemiterm ℒₒᵣ n) :
(⌜(‘!!t * !!u’ : SyntacticSemiterm ℒₒᵣ n)⌝ : Bootstrapping.Semiterm V ℒₒᵣ n) = ⌜t⌝ * ⌜u⌝ := rfl
+lemma typed_quote_numeral_eq_numeral_one :
+ (⌜((1 : ℕ) : SyntacticSemiterm ℒₒᵣ n)⌝ : Bootstrapping.Semiterm V ℒₒᵣ n) = typedNume... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_13fe48712ba7_1 | 05d22e5eacf0ba37 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Bootstrapping/Syntax/Term/Coding.lean | Coding | 1 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 7 | 1 | [
{
"theorem_name": "typed_quote_inj_iff",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " ⟨typed_quote_inj, by rintro rfl; rfl⟩",
"n_chars": 42,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 2,
... | [
{
"name": "typed_quote_inj",
"text": "lemma typed_quote_inj {t u : SyntacticSemiterm L n} : (⌜t⌝ : Bootstrapping.Semiterm V L n) = ⌜u⌝ → t = u :=\n match t, u with\n | #x, #y => by simp\n | &x, &y => by simp\n | func f₁ v₁, func f₂ v₂ => by\n simp only [typed_quote_fun... | [
{
"name": "typed_quote_inj_iff",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 2,
"n_lines": 4,
"n_chars": 177,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 2,
"n_automation": 2,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": false,
"max_nes... | 1 | module
public import Foundation.FirstOrder.Bootstrapping.Syntax.Term.Typed
public import Mathlib.Combinatorics.Colex
@[expose] public section
open Encodable LO FirstOrder Arithmetic PeanoMinus Bootstrapping
namespace LO.FirstOrder.Semiterm
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
variable {L : La... | module
public import Foundation.FirstOrder.Bootstrapping.Syntax.Term.Typed
public import Mathlib.Combinatorics.Colex
@[expose] public section
open Encodable LO FirstOrder Arithmetic PeanoMinus Bootstrapping
namespace LO.FirstOrder.Semiterm
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
variable {L : La... | @@ -73,8 +73,30 @@
_ = typedNumeral ↑(k + 1) + typedNumeral 1 := by simp [typed_quote_numeral_eq_numeral (k + 1), typed_quote_numeral_eq_numeral_one]
_ = typedNumeral (↑k + 1 + 1) := by simp
+lemma typed_quote_inj {t u : SyntacticSemiterm L n} : (⌜t⌝ : Bootstrapping.Semiterm V L n) = ⌜u⌝ → ... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_13fe48712ba7_2 | 2182b61d34798c29 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Bootstrapping/Syntax/Term/Coding.lean | Coding | 2 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 7 | 2 | [
{
"theorem_name": "quote_eq_encode",
"depth": 1,
"n_commands": 0,
"n_lines": 10,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n match t with\n | #x => simp [quote_def, encode_eq_toNat, toNat, qqBvar, coe_pair_eq_pair_coe]\n | &x => simp... | [
{
"name": "quote_eq_encode'_aux",
"text": "private lemma quote_eq_encode'_aux (v : Fin k → Semiterm L ℕ n)\n (H : ∀ i, (⌜v i⌝ : Bootstrapping.Semiterm V L n).val = encode ↑(v i)) :\n (SemitermVec.val fun i ↦ (⌜v i⌝ : Bootstrapping.Semiterm V L n)) = ↑(Matrix.vecToNat fun i ↦ encode (v i)) := by\n ind... | [
{
"name": "quote_eq_encode",
"fan_in": 5,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 12,
"n_chars": 639,
"n_subproofs": 1,
"n_tactics": 10,
"cyclomatic": 3,
"n_automation": 4,
"n_rewrites": 0,
"n_structural": 2,
"automation_only": false,
"max_nesti... | 1 | module
public import Foundation.FirstOrder.Bootstrapping.Syntax.Term.Typed
public import Mathlib.Combinatorics.Colex
@[expose] public section
open Encodable LO FirstOrder Arithmetic PeanoMinus Bootstrapping
namespace LO.FirstOrder.Semiterm
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
variable {L : La... | module
public import Foundation.FirstOrder.Bootstrapping.Syntax.Term.Typed
public import Mathlib.Combinatorics.Colex
@[expose] public section
open Encodable LO FirstOrder Arithmetic PeanoMinus Bootstrapping
namespace LO.FirstOrder.Semiterm
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
variable {L : La... | @@ -103,6 +103,20 @@
def quote_def (t : SyntacticSemiterm L n) : (⌜t⌝ : V) = (⌜t⌝ : Bootstrapping.Semiterm V L n).val := rfl
+private lemma quote_eq_encode'_aux (v : Fin k → Semiterm L ℕ n)
+ (H : ∀ i, (⌜v i⌝ : Bootstrapping.Semiterm V L n).val = encode ↑(v i)) :
+ (SemitermVec.val fun i ↦ (⌜v i⌝ : Bootstrap... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_13fe48712ba7_3 | c0c53572effbeb8b | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Bootstrapping/Syntax/Term/Coding.lean | Coding | 3 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 7 | 5 | [
{
"theorem_name": "quote_eq_encode'",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": "\n quote_eq_encode'_aux _ fun i ↦ quote_eq_encode (v i)",
"n_chars": 57,
"n_subproofs": 0,
"n_tactics": 1,
"cycl... | [
{
"name": "quote_eq_encode",
"text": "lemma quote_eq_encode (t : SyntacticSemiterm L n) : (⌜t⌝ : V) = ↑(encode t) := by\n match t with\n | #x => simp [quote_def, encode_eq_toNat, toNat, qqBvar, coe_pair_eq_pair_coe]\n | &x => simp [quote_def, encode_eq_toNat, toNat, qqFvar, coe_pair_eq_pair_c... | [
{
"name": "quote_eq_encode'",
"fan_in": 0,
"n_deps_direct": 2,
"n_deps_transitive": 2,
"n_lines": 5,
"n_chars": 240,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
"max_nestin... | 2 | module
public import Foundation.FirstOrder.Bootstrapping.Syntax.Term.Typed
public import Mathlib.Combinatorics.Colex
@[expose] public section
open Encodable LO FirstOrder Arithmetic PeanoMinus Bootstrapping
namespace LO.FirstOrder.Semiterm
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
variable {L : La... | module
public import Foundation.FirstOrder.Bootstrapping.Syntax.Term.Typed
public import Mathlib.Combinatorics.Colex
@[expose] public section
open Encodable LO FirstOrder Arithmetic PeanoMinus Bootstrapping
namespace LO.FirstOrder.Semiterm
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
variable {L : La... | @@ -117,8 +117,20 @@
· exact H 0
· exact ih (fun i ↦ v i.succ) (fun i ↦ by simpa using H i.succ)
+lemma quote_eq_encode (t : SyntacticSemiterm L n) : (⌜t⌝ : V) = ↑(encode t) := by
+ match t with
+ | #x => simp [quote_def, encode_eq_toNat, toNat, qqBvar, coe_pair_eq_pair_coe]
+ | &x => simp [q... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_13fe48712ba7_4 | 57f8af7e20219d30 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Bootstrapping/Syntax/Term/Coding.lean | Coding | 4 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 7 | 5 | [
{
"theorem_name": "quote_eq_encode'",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": "\n quote_eq_encode'_aux _ fun i ↦ quote_eq_encode (v i)",
"n_chars": 57,
"n_subproofs": 0,
"n_tactics": 1,
"cycl... | [
{
"name": "quote_eq_encode",
"text": "lemma quote_eq_encode (t : SyntacticSemiterm L n) : (⌜t⌝ : V) = ↑(encode t) := by\n match t with\n | #x => simp [quote_def, encode_eq_toNat, toNat, qqBvar, coe_pair_eq_pair_coe]\n | &x => simp [quote_def, encode_eq_toNat, toNat, qqFvar, coe_pair_eq_pair_c... | [
{
"name": "quote_eq_encode_standard",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 2,
"n_lines": 3,
"n_chars": 118,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"max... | 2 | module
public import Foundation.FirstOrder.Bootstrapping.Syntax.Term.Typed
public import Mathlib.Combinatorics.Colex
@[expose] public section
open Encodable LO FirstOrder Arithmetic PeanoMinus Bootstrapping
namespace LO.FirstOrder.Semiterm
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
variable {L : La... | module
public import Foundation.FirstOrder.Bootstrapping.Syntax.Term.Typed
public import Mathlib.Combinatorics.Colex
@[expose] public section
open Encodable LO FirstOrder Arithmetic PeanoMinus Bootstrapping
namespace LO.FirstOrder.Semiterm
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
variable {L : La... | @@ -117,8 +117,19 @@
· exact H 0
· exact ih (fun i ↦ v i.succ) (fun i ↦ by simpa using H i.succ)
-lemma quote_eq_encode_standard (t : SyntacticSemiterm L n) : (⌜t⌝ : ℕ) = encode t := sorry
+lemma quote_eq_encode (t : SyntacticSemiterm L n) : (⌜t⌝ : V) = ↑(encode t) := by
+ match t with
+ | #x => sim... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_13fe48712ba7_5 | 44728547fb17fce0 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Bootstrapping/Syntax/Term/Coding.lean | Coding | 5 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 7 | 5 | [
{
"theorem_name": "quote_eq_encode'",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": "\n quote_eq_encode'_aux _ fun i ↦ quote_eq_encode (v i)",
"n_chars": 57,
"n_subproofs": 0,
"n_tactics": 1,
"cycl... | [
{
"name": "quote_eq_encode",
"text": "lemma quote_eq_encode (t : SyntacticSemiterm L n) : (⌜t⌝ : V) = ↑(encode t) := by\n match t with\n | #x => simp [quote_def, encode_eq_toNat, toNat, qqBvar, coe_pair_eq_pair_coe]\n | &x => simp [quote_def, encode_eq_toNat, toNat, qqFvar, coe_pair_eq_pair_c... | [
{
"name": "coe_quote_eq_quote",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 2,
"n_lines": 4,
"n_chars": 122,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"max_nesti... | 2 | module
public import Foundation.FirstOrder.Bootstrapping.Syntax.Term.Typed
public import Mathlib.Combinatorics.Colex
@[expose] public section
open Encodable LO FirstOrder Arithmetic PeanoMinus Bootstrapping
namespace LO.FirstOrder.Semiterm
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
variable {L : La... | module
public import Foundation.FirstOrder.Bootstrapping.Syntax.Term.Typed
public import Mathlib.Combinatorics.Colex
@[expose] public section
open Encodable LO FirstOrder Arithmetic PeanoMinus Bootstrapping
namespace LO.FirstOrder.Semiterm
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
variable {L : La... | @@ -117,8 +117,20 @@
· exact H 0
· exact ih (fun i ↦ v i.succ) (fun i ↦ by simpa using H i.succ)
-lemma coe_quote_eq_quote (t : SyntacticSemiterm L n) : (↑(⌜t⌝ : ℕ) : V) = ⌜t⌝ := sorry
+lemma quote_eq_encode (t : SyntacticSemiterm L n) : (⌜t⌝ : V) = ↑(encode t) := by
+ match t with
+ | #x => simp [q... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_13fe48712ba7_6 | 29d45efe2a2d878e | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Bootstrapping/Syntax/Term/Coding.lean | Coding | 6 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 7 | 5 | [
{
"theorem_name": "quote_eq_encode'",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": "\n quote_eq_encode'_aux _ fun i ↦ quote_eq_encode (v i)",
"n_chars": 57,
"n_subproofs": 0,
"n_tactics": 1,
"cycl... | [
{
"name": "quote_eq_encode",
"text": "lemma quote_eq_encode (t : SyntacticSemiterm L n) : (⌜t⌝ : V) = ↑(encode t) := by\n match t with\n | #x => simp [quote_def, encode_eq_toNat, toNat, qqBvar, coe_pair_eq_pair_coe]\n | &x => simp [quote_def, encode_eq_toNat, toNat, qqFvar, coe_pair_eq_pair_c... | [
{
"name": "coe_quote_eq_quote'",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 3,
"n_lines": 5,
"n_chars": 190,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
"max_nes... | 3 | module
public import Foundation.FirstOrder.Bootstrapping.Syntax.Term.Typed
public import Mathlib.Combinatorics.Colex
@[expose] public section
open Encodable LO FirstOrder Arithmetic PeanoMinus Bootstrapping
namespace LO.FirstOrder.Semiterm
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
variable {L : La... | module
public import Foundation.FirstOrder.Bootstrapping.Syntax.Term.Typed
public import Mathlib.Combinatorics.Colex
@[expose] public section
open Encodable LO FirstOrder Arithmetic PeanoMinus Bootstrapping
namespace LO.FirstOrder.Semiterm
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
variable {L : La... | @@ -117,8 +117,20 @@
· exact H 0
· exact ih (fun i ↦ v i.succ) (fun i ↦ by simpa using H i.succ)
-lemma coe_quote_eq_quote (t : SyntacticSemiterm L n) : (↑(⌜t⌝ : ℕ) : V) = ⌜t⌝ := sorry
+lemma quote_eq_encode (t : SyntacticSemiterm L n) : (⌜t⌝ : V) = ↑(encode t) := by
+ match t with
+ | #x => simp [q... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_13fe48712ba7_7 | ae1426b6c9a928ee | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Bootstrapping/Syntax/Term/Coding.lean | Coding | 7 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 7 | 5 | [
{
"theorem_name": "quote_eq_encode'",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": "\n quote_eq_encode'_aux _ fun i ↦ quote_eq_encode (v i)",
"n_chars": 57,
"n_subproofs": 0,
"n_tactics": 1,
"cycl... | [
{
"name": "quote_eq_encode",
"text": "lemma quote_eq_encode (t : SyntacticSemiterm L n) : (⌜t⌝ : V) = ↑(encode t) := by\n match t with\n | #x => simp [quote_def, encode_eq_toNat, toNat, qqBvar, coe_pair_eq_pair_coe]\n | &x => simp [quote_def, encode_eq_toNat, toNat, qqFvar, coe_pair_eq_pair_c... | [
{
"name": "empty_quote_eq_encode",
"fan_in": 1,
"n_deps_direct": 2,
"n_deps_transitive": 3,
"n_lines": 3,
"n_chars": 132,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"max_ne... | 3 | module
public import Foundation.FirstOrder.Bootstrapping.Syntax.Term.Typed
public import Mathlib.Combinatorics.Colex
@[expose] public section
open Encodable LO FirstOrder Arithmetic PeanoMinus Bootstrapping
namespace LO.FirstOrder.Semiterm
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
variable {L : La... | module
public import Foundation.FirstOrder.Bootstrapping.Syntax.Term.Typed
public import Mathlib.Combinatorics.Colex
@[expose] public section
open Encodable LO FirstOrder Arithmetic PeanoMinus Bootstrapping
namespace LO.FirstOrder.Semiterm
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
variable {L : La... | @@ -117,6 +117,17 @@
· exact H 0
· exact ih (fun i ↦ v i.succ) (fun i ↦ by simpa using H i.succ)
+lemma quote_eq_encode (t : SyntacticSemiterm L n) : (⌜t⌝ : V) = ↑(encode t) := by
+ match t with
+ | #x => simp [quote_def, encode_eq_toNat, toNat, qqBvar, coe_pair_eq_pair_coe]
+ | &x => simp [q... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_98b8a872245a_0 | ff6e294ffef6d80e | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Propositional/FMT/Axiom/Ser.lean | Ser | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 2 | 1 | [
{
"theorem_name": "valid_axiomO_f",
"depth": 1,
"n_commands": 0,
"n_lines": 3,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n haveI := isDNFSerial_of_valid_axiomSer h;\n apply valid_axiomSer_of_isDNFSerial;",
"n_chars": 86,
"n_subproofs": 0,... | [
{
"name": "valid_axiomSer_of_isDNFSerial",
"text": "lemma valid_axiomSer_of_isDNFSerial [F.IsNTSerial] : F ⊧ ∼∼⊤ := by\n intro V x y Rxy h₁;\n obtain ⟨z, Ryz⟩ := F.nt_serial y;\n have : Formula.FMT.Forces (M := ⟨F, V⟩) z ⊥ := h₁ Ryz (by tauto);\n contradiction;\n\n",
"fan_in": 1,
"n_lines": 7,
... | [
{
"name": "valid_axiomO_f",
"fan_in": 0,
"n_deps_direct": 2,
"n_deps_transitive": 2,
"n_lines": 5,
"n_chars": 148,
"n_subproofs": 0,
"n_tactics": 5,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": false,
"max_nesting"... | 2 | module
public import Foundation.Propositional.FMT.Hilbert.VF
@[expose] public section
namespace LO.Propositional
namespace FMT
variable {F : FMT.Frame} {M : FMT.Model} {Γ : FormulaSet ℕ} {φ ψ χ : Formula ℕ}
namespace Frame
class IsNTSerial (F : Frame) : Prop where
nt_serial : ∀ x : F, ∃ y, x ≺[∼⊤] y
export IsN... | module
public import Foundation.Propositional.FMT.Hilbert.VF
@[expose] public section
namespace LO.Propositional
namespace FMT
variable {F : FMT.Frame} {M : FMT.Model} {Γ : FormulaSet ℕ} {φ ψ χ : Formula ℕ}
namespace Frame
class IsNTSerial (F : Frame) : Prop where
nt_serial : ∀ x : F, ∃ y, x ≺[∼⊤] y
export IsN... | @@ -18,12 +18,20 @@
end Frame
+lemma valid_axiomSer_of_isDNFSerial [F.IsNTSerial] : F ⊧ ∼∼⊤ := by
+ intro V x y Rxy h₁;
+ obtain ⟨z, Ryz⟩ := F.nt_serial y;
+ have : Formula.FMT.Forces (M := ⟨F, V⟩) z ⊥ := h₁ Ryz (by tauto);
+ contradiction;
+
lemma isDNFSerial_of_valid_axiomSer (h : F ⊧ ∼⊤) : F.IsNTSerial whe... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"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_0 | 84c29b984bea361a | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Exponential/Log.lean | Log | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 32 | 3 | [
{
"theorem_name": "log_exists_unique",
"depth": 1,
"n_commands": 0,
"n_lines": 4,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n by_cases hy : y = 0\n · rcases hy; simp\n · simp [hy, pos_iff_ne_zero.mpr hy, log_exists_unique_pos]",
"n_chars": 1... | [
{
"name": "log_exists_unique_pos",
"text": "lemma log_exists_unique_pos {y : V} (hy : 0 < y) : ∃! x, x < y ∧ ∃ y' ≤ y, Exponential x y' ∧ y < 2 * y' := by\n have : ∃ x < y, ∃ y' ≤ y, Exponential x y' ∧ y < 2 * y' := by\n induction y using sigma0_polynomial_induction\n · definability\n case zero =>... | [
{
"name": "log_exists_unique",
"fan_in": 3,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 6,
"n_chars": 244,
"n_subproofs": 0,
"n_tactics": 5,
"cyclomatic": 2,
"n_automation": 2,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": false,
"max_nesti... | 1 | 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 (y : V) : ∃! x, (y = 0 → x = 0) ∧ (0 < y → x ... | 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'... | @@ -15,10 +15,40 @@
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₀]
+lemma log_exists_unique_pos {y : V} (hy : 0 < y) : ∃! x, x < y ∧ ∃ y' ≤ y, Exponential x y' ∧ y < 2 * y' := by
+ have : ∃ x < y, ∃ y' ≤ y, Exponential x y' ∧ y < 2 * y' := by
+ induction y using sigma0_polynomial_induction
+ · definability
+ case zero => 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_7471c35686f5_1 | 4ec403f4953b7f28 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Exponential/Log.lean | Log | 1 | 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_le_self",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 3,
"n_lines": 6,
"n_chars": 158,
"n_subproofs": 0,
"n_tactics": 4,
"cyclomatic": 2,
"n_automation": 2,
"n_rewrites": 0,
"n_structural": 2,
"automation_only": false,
"max_nesting": 2... | 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'... | @@ -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_2 | 0e5829897f3f2749 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Exponential/Log.lean | Log | 2 | 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": "log_eq_of_pos",
"fan_in": 6,
"n_deps_direct": 3,
"n_deps_transitive": 4,
"n_lines": 4,
"n_chars": 248,
"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.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_3 | b17ecb21d0fb1ce2 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Exponential/Log.lean | Log | 3 | 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": "exponential_of_pow2",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 3,
"n_lines": 11,
"n_chars": 374,
"n_subproofs": 3,
"n_tactics": 9,
"cyclomatic": 2,
"n_automation": 2,
"n_rewrites": 0,
"n_structural": 2,
"automation_only": false,
"max_ne... | 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_4 | 23bd8ea64478d247 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Exponential/Log.lean | Log | 4 | 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_pow2_add_of_lt",
"fan_in": 3,
"n_deps_direct": 3,
"n_deps_transitive": 6,
"n_lines": 12,
"n_chars": 644,
"n_subproofs": 0,
"n_tactics": 10,
"cyclomatic": 4,
"n_automation": 3,
"n_rewrites": 0,
"n_structural": 5,
"automation_only": false,
"ma... | 5 | 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_5 | 758f56c9c1c8796e | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Exponential/Log.lean | Log | 5 | 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_pow2",
"fan_in": 2,
"n_deps_direct": 1,
"n_deps_transitive": 7,
"n_lines": 4,
"n_chars": 145,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"max_nesting": 2... | 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_6 | 055b2734a6c9407a | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Exponential/Log.lean | Log | 6 | 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_zero",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 3,
"n_chars": 83,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"max_nesting": 1
... | 1 | module
public import Foundation.FirstOrder.Arithmetic.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'... | @@ -116,10 +116,12 @@
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_7 | 305326996f40af06 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Exponential/Log.lean | Log | 7 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 32 | 14 | [
{
"theorem_name": "length_le",
"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 · simp [pos, length_of_pos, ←lt_iff_succ_le, log_lt_self_of_pos]",
... | [
{
"name": "length_of_pos",
"text": "lemma length_of_pos {a : V} (pos : 0 < a) : ‖a‖ = log a + 1 := by simp [length_eq_binaryLength, pos]\n\n",
"fan_in": 14,
"n_lines": 3,
"n_chars": 106,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
... | [
{
"name": "length_le",
"fan_in": 0,
"n_deps_direct": 2,
"n_deps_transitive": 5,
"n_lines": 6,
"n_chars": 182,
"n_subproofs": 0,
"n_tactics": 4,
"cyclomatic": 2,
"n_automation": 3,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": false,
"max_nesting": 2
... | 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'... | @@ -120,15 +120,17 @@
@[simp] lemma length_zero : ‖(0 : V)‖ = 0 := by simp [length_eq_binaryLength]
+lemma length_of_pos {a : V} (pos : 0 < a) : ‖a‖ = log a + 1 := by simp [length_eq_binaryLength, pos]
+
@[simp] lemma length_le (a : V) : ‖a‖ ≤ a := by
rcases Arithmetic.zero_le a with (rfl | pos)
· simp
- ·... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_7471c35686f5_8 | 84697d90f7621e17 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Exponential/Log.lean | Log | 8 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 32 | 14 | [
{
"theorem_name": "length_le",
"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 · simp [pos, length_of_pos, ←lt_iff_succ_le, log_lt_self_of_pos]",
... | [
{
"name": "length_of_pos",
"text": "lemma length_of_pos {a : V} (pos : 0 < a) : ‖a‖ = log a + 1 := by simp [length_eq_binaryLength, pos]\n\n",
"fan_in": 14,
"n_lines": 3,
"n_chars": 106,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
... | [
{
"name": "Exponential.length_eq",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 2,
"n_lines": 4,
"n_chars": 143,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"max_ne... | 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'... | @@ -120,15 +120,17 @@
@[simp] lemma length_zero : ‖(0 : V)‖ = 0 := by simp [length_eq_binaryLength]
+lemma length_of_pos {a : V} (pos : 0 < a) : ‖a‖ = log a + 1 := by simp [length_eq_binaryLength, pos]
+
@[simp] lemma length_le (a : V) : ‖a‖ ≤ a := by
rcases Arithmetic.zero_le a with (rfl | pos)
· simp
- ·... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_7471c35686f5_9 | 4c29ae093ac8eb9c | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Exponential/Log.lean | Log | 9 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 32 | 3 | [
{
"theorem_name": "log_mul_pow2",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n simpa using log_mul_pow2_add_of_lt pos pp pp.pos",
"n_chars": 54,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomati... | [
{
"name": "log_mul_pow2_add_of_lt",
"text": "lemma log_mul_pow2_add_of_lt {a p b : V} (pos : 0 < a) (pp : Pow2 p) (hb : b < p) : log (a * p + b) = log a + log p := by\n rcases log_pos pos with ⟨a', ha', Ha, ha⟩\n rcases log_pos pp.pos with ⟨p', hp', Hp, hp⟩\n exact log_eq_of_pos (lt_of_lt_of_le (mul_pos ... | [
{
"name": "length_mul_pow2_add_of_lt",
"fan_in": 1,
"n_deps_direct": 2,
"n_deps_transitive": 9,
"n_lines": 4,
"n_chars": 232,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"ma... | 8 | 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'... | @@ -106,6 +106,17 @@
_ < 2 * q := hp
simp at this
+lemma log_mul_pow2_add_of_lt {a p b : V} (pos : 0 < a) (pp : Pow2 p) (hb : b < p) : log (a * p + b) = log a + log p := by
+ rcases log_pos pos with ⟨a', ha', Ha, ha⟩
+ rcases log_pos pp.pos with ⟨p', hp', Hp, hp⟩
+ exact log_eq_of_pos (lt_of_lt_of_le (... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_7471c35686f5_10 | 2ca64bdbe0bd8c3a | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Exponential/Log.lean | Log | 10 | 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_pow2",
"fan_in": 1,
"n_deps_direct": 2,
"n_deps_transitive": 10,
"n_lines": 4,
"n_chars": 190,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"max_nesting... | 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'... | @@ -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_11 | 22d6a2e5c828b405 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Exponential/Log.lean | Log | 11 | 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": "exponential_log_le_self",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 3,
"n_lines": 5,
"n_chars": 172,
"n_subproofs": 0,
"n_tactics": 4,
"cyclomatic": 3,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 2,
"automation_only": false,
"max... | 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_12 | f5990b4fa2286a85 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Exponential/Log.lean | Log | 12 | 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_exponential_log_self",
"fan_in": 2,
"n_deps_direct": 1,
"n_deps_transitive": 3,
"n_lines": 7,
"n_chars": 232,
"n_subproofs": 0,
"n_tactics": 7,
"cyclomatic": 4,
"n_automation": 4,
"n_rewrites": 0,
"n_structural": 3,
"automation_only": false,
"max... | 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_13 | 9a7f1862ea269099 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Exponential/Log.lean | Log | 13 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 32 | 14 | [
{
"theorem_name": "length_le",
"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 · simp [pos, length_of_pos, ←lt_iff_succ_le, log_lt_self_of_pos]",
... | [
{
"name": "length_of_pos",
"text": "lemma length_of_pos {a : V} (pos : 0 < a) : ‖a‖ = log a + 1 := by simp [length_eq_binaryLength, pos]\n\n",
"fan_in": 14,
"n_lines": 3,
"n_chars": 106,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
... | [
{
"name": "lt_exp_len_self",
"fan_in": 1,
"n_deps_direct": 2,
"n_deps_transitive": 6,
"n_lines": 8,
"n_chars": 323,
"n_subproofs": 1,
"n_tactics": 7,
"cyclomatic": 3,
"n_automation": 5,
"n_rewrites": 0,
"n_structural": 3,
"automation_only": false,
"max_nesting... | 5 | 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'... | @@ -120,15 +120,17 @@
@[simp] lemma length_zero : ‖(0 : V)‖ = 0 := by simp [length_eq_binaryLength]
+lemma length_of_pos {a : V} (pos : 0 < a) : ‖a‖ = log a + 1 := by simp [length_eq_binaryLength, pos]
+
@[simp] lemma length_le (a : V) : ‖a‖ ≤ a := by
rcases Arithmetic.zero_le a with (rfl | pos)
· simp
- ·... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_7471c35686f5_14 | 50bc5829bd46071d | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Exponential/Log.lean | Log | 14 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 32 | 14 | [
{
"theorem_name": "length_le",
"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 · simp [pos, length_of_pos, ←lt_iff_succ_le, log_lt_self_of_pos]",
... | [
{
"name": "length_of_pos",
"text": "lemma length_of_pos {a : V} (pos : 0 < a) : ‖a‖ = log a + 1 := by simp [length_eq_binaryLength, pos]\n\n",
"fan_in": 14,
"n_lines": 3,
"n_chars": 106,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
... | [
{
"name": "le_iff_lt_length_of_exp",
"fan_in": 3,
"n_deps_direct": 2,
"n_deps_transitive": 7,
"n_lines": 6,
"n_chars": 266,
"n_subproofs": 0,
"n_tactics": 4,
"cyclomatic": 2,
"n_automation": 3,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": false,
"max... | 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'... | @@ -120,15 +120,17 @@
@[simp] lemma length_zero : ‖(0 : V)‖ = 0 := by simp [length_eq_binaryLength]
+lemma length_of_pos {a : V} (pos : 0 < a) : ‖a‖ = log a + 1 := by simp [length_eq_binaryLength, pos]
+
@[simp] lemma length_le (a : V) : ‖a‖ ≤ a := by
rcases Arithmetic.zero_le a with (rfl | pos)
· simp
- ·... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_7471c35686f5_15 | af5964e3b5c7428c | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Exponential/Log.lean | Log | 15 | 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": "Exponential.lt_iff_log_lt",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 5,
"n_lines": 4,
"n_chars": 167,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"ma... | 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_7471c35686f5_16 | 26f75c170906f7a5 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Exponential/Log.lean | Log | 16 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 32 | 14 | [
{
"theorem_name": "length_le",
"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 · simp [pos, length_of_pos, ←lt_iff_succ_le, log_lt_self_of_pos]",
... | [
{
"name": "length_of_pos",
"text": "lemma length_of_pos {a : V} (pos : 0 < a) : ‖a‖ = log a + 1 := by simp [length_eq_binaryLength, pos]\n\n",
"fan_in": 14,
"n_lines": 3,
"n_chars": 106,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
... | [
{
"name": "Exponential.lt_iff_len_le",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 8,
"n_lines": 3,
"n_chars": 154,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"ma... | 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'... | @@ -120,15 +120,17 @@
@[simp] lemma length_zero : ‖(0 : V)‖ = 0 := by simp [length_eq_binaryLength]
+lemma length_of_pos {a : V} (pos : 0 < a) : ‖a‖ = log a + 1 := by simp [length_eq_binaryLength, pos]
+
@[simp] lemma length_le (a : V) : ‖a‖ ≤ a := by
rcases Arithmetic.zero_le a with (rfl | pos)
· simp
- ·... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_7471c35686f5_17 | 32c169b7e31b12bc | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Exponential/Log.lean | Log | 17 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 32 | 14 | [
{
"theorem_name": "length_le",
"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 · simp [pos, length_of_pos, ←lt_iff_succ_le, log_lt_self_of_pos]",
... | [
{
"name": "length_of_pos",
"text": "lemma length_of_pos {a : V} (pos : 0 < a) : ‖a‖ = log a + 1 := by simp [length_eq_binaryLength, pos]\n\n",
"fan_in": 14,
"n_lines": 3,
"n_chars": 106,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
... | [
{
"name": "Exponential.le_of_lt_length",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 8,
"n_lines": 4,
"n_chars": 144,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
... | 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'... | @@ -120,15 +120,17 @@
@[simp] lemma length_zero : ‖(0 : V)‖ = 0 := by simp [length_eq_binaryLength]
+lemma length_of_pos {a : V} (pos : 0 < a) : ‖a‖ = log a + 1 := by simp [length_eq_binaryLength, pos]
+
@[simp] lemma length_le (a : V) : ‖a‖ ≤ a := by
rcases Arithmetic.zero_le a with (rfl | pos)
· simp
- ·... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_7471c35686f5_18 | cd49e363ac472378 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Exponential/Log.lean | Log | 18 | 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": "Exponential.le_log",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 5,
"n_lines": 3,
"n_chars": 124,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"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_7471c35686f5_19 | d42425cd56fc4b20 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Exponential/Log.lean | Log | 19 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 32 | 14 | [
{
"theorem_name": "length_le",
"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 · simp [pos, length_of_pos, ←lt_iff_succ_le, log_lt_self_of_pos]",
... | [
{
"name": "length_of_pos",
"text": "lemma length_of_pos {a : V} (pos : 0 < a) : ‖a‖ = log a + 1 := by simp [length_eq_binaryLength, pos]\n\n",
"fan_in": 14,
"n_lines": 3,
"n_chars": 106,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
... | [
{
"name": "Exponential.lt_length",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 8,
"n_lines": 3,
"n_chars": 118,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"max_ne... | 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'... | @@ -120,15 +120,17 @@
@[simp] lemma length_zero : ‖(0 : V)‖ = 0 := by simp [length_eq_binaryLength]
+lemma length_of_pos {a : V} (pos : 0 < a) : ‖a‖ = log a + 1 := by simp [length_eq_binaryLength, pos]
+
@[simp] lemma length_le (a : V) : ‖a‖ ≤ a := by
rcases Arithmetic.zero_le a with (rfl | pos)
· simp
- ·... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_7471c35686f5_20 | 933d716a19dacaf1 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Exponential/Log.lean | Log | 20 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 32 | 14 | [
{
"theorem_name": "length_le",
"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 · simp [pos, length_of_pos, ←lt_iff_succ_le, log_lt_self_of_pos]",
... | [
{
"name": "length_of_pos",
"text": "lemma length_of_pos {a : V} (pos : 0 < a) : ‖a‖ = log a + 1 := by simp [length_eq_binaryLength, pos]\n\n",
"fan_in": 14,
"n_lines": 3,
"n_chars": 106,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
... | [
{
"name": "lt_exponential_length",
"fan_in": 0,
"n_deps_direct": 2,
"n_deps_transitive": 6,
"n_lines": 8,
"n_chars": 329,
"n_subproofs": 1,
"n_tactics": 7,
"cyclomatic": 3,
"n_automation": 5,
"n_rewrites": 0,
"n_structural": 3,
"automation_only": false,
"max_n... | 5 | 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'... | @@ -120,15 +120,17 @@
@[simp] lemma length_zero : ‖(0 : V)‖ = 0 := by simp [length_eq_binaryLength]
+lemma length_of_pos {a : V} (pos : 0 < a) : ‖a‖ = log a + 1 := by simp [length_eq_binaryLength, pos]
+
@[simp] lemma length_le (a : V) : ‖a‖ ≤ a := by
rcases Arithmetic.zero_le a with (rfl | pos)
· simp
- ·... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_7471c35686f5_21 | 78270eed838aad94 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Exponential/Log.lean | Log | 21 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 32 | 14 | [
{
"theorem_name": "length_le",
"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 · simp [pos, length_of_pos, ←lt_iff_succ_le, log_lt_self_of_pos]",
... | [
{
"name": "length_of_pos",
"text": "lemma length_of_pos {a : V} (pos : 0 < a) : ‖a‖ = log a + 1 := by simp [length_eq_binaryLength, pos]\n\n",
"fan_in": 14,
"n_lines": 3,
"n_chars": 106,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
... | [
{
"name": "sq_len_le_three_mul",
"fan_in": 0,
"n_deps_direct": 2,
"n_deps_transitive": 11,
"n_lines": 25,
"n_chars": 1381,
"n_subproofs": 0,
"n_tactics": 23,
"cyclomatic": 3,
"n_automation": 14,
"n_rewrites": 2,
"n_structural": 4,
"automation_only": false,
"ma... | 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'... | @@ -120,15 +120,17 @@
@[simp] lemma length_zero : ‖(0 : V)‖ = 0 := by simp [length_eq_binaryLength]
+lemma length_of_pos {a : V} (pos : 0 < a) : ‖a‖ = log a + 1 := by simp [length_eq_binaryLength, pos]
+
@[simp] lemma length_le (a : V) : ‖a‖ ≤ a := by
rcases Arithmetic.zero_le a with (rfl | pos)
· simp
- ·... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_7471c35686f5_22 | c0f2dec29dd9a6bd | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Exponential/Log.lean | Log | 22 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 32 | 5 | [
{
"theorem_name": "le_iff_lt_length_of_exp",
"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 · simpa using pos_iff_ne_zero.mp H.range_pos\n simp [le_iff_le_log... | [
{
"name": "le_iff_le_log_of_exp",
"text": "lemma le_iff_le_log_of_exp {x y a : V} (H : Exponential x y) (pos : 0 < a) : y ≤ a ↔ x ≤ log a :=\n ⟨by rcases H.log_eq_of_exp; exact log_monotone,\n fun h ↦ by rcases log_pos pos with ⟨a', ha', Haa', _⟩; exact le_trans (Exponential.monotone_le H Haa' h) ha'⟩\n\... | [
{
"name": "exp_bexp_of_lt_iff",
"fan_in": 0,
"n_deps_direct": 2,
"n_deps_transitive": 13,
"n_lines": 8,
"n_chars": 262,
"n_subproofs": 2,
"n_tactics": 7,
"cyclomatic": 1,
"n_automation": 2,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": false,
"max_nes... | 11 | 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'... | @@ -170,6 +170,10 @@
have : 0 < b := pos_of_lt_length h
exact le_iff_lt_succ.mpr (by simpa [length_of_pos this] using h)
+lemma le_iff_le_log_of_exp {x y a : V} (H : Exponential x y) (pos : 0 < a) : y ≤ a ↔ x ≤ log a :=
+ ⟨by rcases H.log_eq_of_exp; exact log_monotone,
+ fun h ↦ by rcases log_pos pos with ⟨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_7471c35686f5_23 | a1e98117b592911f | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Exponential/Log.lean | Log | 23 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 32 | 3 | [
{
"theorem_name": "le_log_of_lt_length",
"depth": 1,
"n_commands": 0,
"n_lines": 3,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n have : 0 < b := pos_of_lt_length h\n exact le_iff_lt_succ.mpr (by simpa [length_of_pos this] using h)",
"n_chars":... | [
{
"name": "pos_of_lt_length",
"text": "lemma pos_of_lt_length {a b : V} (h : a < ‖b‖) : 0 < b := by\n by_contra A; rcases (show b = 0 from by simpa using A); simp_all\n\n",
"fan_in": 3,
"n_lines": 4,
"n_chars": 133,
"n_subproofs": 0,
"n_tactics": 4,
"cyclomatic": 2,
"n_automatio... | [
{
"name": "bexp_pow2",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 12,
"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": 1
... | 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'... | @@ -157,6 +157,9 @@
· simpa [length_of_pos posa, length_of_pos (lt_of_lt_of_le posa h)]
using log_monotone h
+lemma pos_of_lt_length {a b : V} (h : a < ‖b‖) : 0 < b := by
+ by_contra A; rcases (show b = 0 from by simpa using A); simp_all
+
@[simp] lemma length_pos_iff {a : V} : 0 < ‖a‖ ↔ 0 < a :=
⟨by int... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"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_24 | dd7e76ceb30b2c81 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Exponential/Log.lean | Log | 24 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 32 | 2 | [
{
"theorem_name": "brange_exists_unique",
"depth": 1,
"n_commands": 0,
"n_lines": 17,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n suffices ∀ x < ‖a‖, ∃ y ≤ a, Exponential x y by\n intro x hx; rcases this x hx with ⟨_, _, H⟩\n exact ExistsUni... | [
{
"name": "le_log_of_lt_length",
"text": "lemma le_log_of_lt_length {a b : V} (h : a < ‖b‖) : a ≤ log b := by\n have : 0 < b := pos_of_lt_length h\n exact le_iff_lt_succ.mpr (by simpa [length_of_pos this] using h)\n\n",
"fan_in": 2,
"n_lines": 5,
"n_chars": 179,
"n_subproofs": 1,
"n_ta... | [
{
"name": "lt_bexp_len",
"fan_in": 0,
"n_deps_direct": 2,
"n_deps_transitive": 14,
"n_lines": 3,
"n_chars": 112,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
"max_nesting": ... | 12 | 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'... | @@ -166,6 +166,10 @@
@[simp] lemma length_eq_zero_iff {a : V} : ‖a‖ = 0 ↔ a = 0 := not_iff_not.mp (by simp [←pos_iff_ne_zero])
+lemma le_log_of_lt_length {a b : V} (h : a < ‖b‖) : a ≤ log b := by
+ have : 0 < b := pos_of_lt_length h
+ exact le_iff_lt_succ.mpr (by simpa [length_of_pos this] using h)
+
lemma lt_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_7471c35686f5_25 | 72ebc5a3a78ac7a2 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Exponential/Log.lean | Log | 25 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 32 | 14 | [
{
"theorem_name": "length_le",
"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 · simp [pos, length_of_pos, ←lt_iff_succ_le, log_lt_self_of_pos]",
... | [
{
"name": "length_of_pos",
"text": "lemma length_of_pos {a : V} (pos : 0 < a) : ‖a‖ = log a + 1 := by simp [length_eq_binaryLength, pos]\n\n",
"fan_in": 14,
"n_lines": 3,
"n_chars": 106,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
... | [
{
"name": "log_bexp",
"fan_in": 1,
"n_deps_direct": 2,
"n_deps_transitive": 15,
"n_lines": 3,
"n_chars": 113,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
"max_nesting": 1
... | 13 | 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'... | @@ -120,15 +120,17 @@
@[simp] lemma length_zero : ‖(0 : V)‖ = 0 := by simp [length_eq_binaryLength]
+lemma length_of_pos {a : V} (pos : 0 < a) : ‖a‖ = log a + 1 := by simp [length_eq_binaryLength, pos]
+
@[simp] lemma length_le (a : V) : ‖a‖ ≤ a := by
rcases Arithmetic.zero_le a with (rfl | pos)
· simp
- ·... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_7471c35686f5_26 | 14171afa8578b608 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Exponential/Log.lean | Log | 26 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 32 | 14 | [
{
"theorem_name": "length_le",
"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 · simp [pos, length_of_pos, ←lt_iff_succ_le, log_lt_self_of_pos]",
... | [
{
"name": "length_of_pos",
"text": "lemma length_of_pos {a : V} (pos : 0 < a) : ‖a‖ = log a + 1 := by simp [length_eq_binaryLength, pos]\n\n",
"fan_in": 14,
"n_lines": 3,
"n_chars": 106,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
... | [
{
"name": "len_bexp",
"fan_in": 0,
"n_deps_direct": 3,
"n_deps_transitive": 17,
"n_lines": 3,
"n_chars": 119,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 1,
"n_structural": 0,
"automation_only": false,
"max_nesting": 1
... | 14 | 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'... | @@ -120,15 +120,17 @@
@[simp] lemma length_zero : ‖(0 : V)‖ = 0 := by simp [length_eq_binaryLength]
+lemma length_of_pos {a : V} (pos : 0 < a) : ‖a‖ = log a + 1 := by simp [length_eq_binaryLength, pos]
+
@[simp] lemma length_le (a : V) : ‖a‖ ≤ a := by
rcases Arithmetic.zero_le a with (rfl | pos)
· simp
- ·... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_7471c35686f5_27 | f150a5d2146f4900 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Exponential/Log.lean | Log | 27 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 32 | 3 | [
{
"theorem_name": "le_log_of_lt_length",
"depth": 1,
"n_commands": 0,
"n_lines": 3,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n have : 0 < b := pos_of_lt_length h\n exact le_iff_lt_succ.mpr (by simpa [length_of_pos this] using h)",
"n_chars":... | [
{
"name": "pos_of_lt_length",
"text": "lemma pos_of_lt_length {a b : V} (h : a < ‖b‖) : 0 < b := by\n by_contra A; rcases (show b = 0 from by simpa using A); simp_all\n\n",
"fan_in": 3,
"n_lines": 4,
"n_chars": 133,
"n_subproofs": 0,
"n_tactics": 4,
"cyclomatic": 2,
"n_automatio... | [
{
"name": "bexp_monotone",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 12,
"n_lines": 4,
"n_chars": 225,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
"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'... | @@ -157,6 +157,9 @@
· simpa [length_of_pos posa, length_of_pos (lt_of_lt_of_le posa h)]
using log_monotone h
+lemma pos_of_lt_length {a b : V} (h : a < ‖b‖) : 0 < b := by
+ by_contra A; rcases (show b = 0 from by simpa using A); simp_all
+
@[simp] lemma length_pos_iff {a : V} : 0 < ‖a‖ ↔ 0 < a :=
⟨by int... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"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_28 | 058f04b065067ddb | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Exponential/Log.lean | Log | 28 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 32 | 3 | [
{
"theorem_name": "le_log_of_lt_length",
"depth": 1,
"n_commands": 0,
"n_lines": 3,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n have : 0 < b := pos_of_lt_length h\n exact le_iff_lt_succ.mpr (by simpa [length_of_pos this] using h)",
"n_chars":... | [
{
"name": "pos_of_lt_length",
"text": "lemma pos_of_lt_length {a b : V} (h : a < ‖b‖) : 0 < b := by\n by_contra A; rcases (show b = 0 from by simpa using A); simp_all\n\n",
"fan_in": 3,
"n_lines": 4,
"n_chars": 133,
"n_subproofs": 0,
"n_tactics": 4,
"cyclomatic": 2,
"n_automatio... | [
{
"name": "bexp_monotone_le",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 12,
"n_lines": 4,
"n_chars": 235,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
"max_nesti... | 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'... | @@ -157,6 +157,9 @@
· simpa [length_of_pos posa, length_of_pos (lt_of_lt_of_le posa h)]
using log_monotone h
+lemma pos_of_lt_length {a b : V} (h : a < ‖b‖) : 0 < b := by
+ by_contra A; rcases (show b = 0 from by simpa using A); simp_all
+
@[simp] lemma length_pos_iff {a : V} : 0 < ‖a‖ ↔ 0 < a :=
⟨by int... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"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_29 | 3f7789408921a527 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Exponential/Log.lean | Log | 29 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 32 | 3 | [
{
"theorem_name": "le_log_of_lt_length",
"depth": 1,
"n_commands": 0,
"n_lines": 3,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n have : 0 < b := pos_of_lt_length h\n exact le_iff_lt_succ.mpr (by simpa [length_of_pos this] using h)",
"n_chars":... | [
{
"name": "pos_of_lt_length",
"text": "lemma pos_of_lt_length {a b : V} (h : a < ‖b‖) : 0 < b := by\n by_contra A; rcases (show b = 0 from by simpa using A); simp_all\n\n",
"fan_in": 3,
"n_lines": 4,
"n_chars": 133,
"n_subproofs": 0,
"n_tactics": 4,
"cyclomatic": 2,
"n_automatio... | [
{
"name": "bexp_add",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 12,
"n_lines": 5,
"n_chars": 252,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
"max_nesting": 2
... | 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'... | @@ -157,6 +157,9 @@
· simpa [length_of_pos posa, length_of_pos (lt_of_lt_of_le posa h)]
using log_monotone h
+lemma pos_of_lt_length {a b : V} (h : a < ‖b‖) : 0 < b := by
+ by_contra A; rcases (show b = 0 from by simpa using A); simp_all
+
@[simp] lemma length_pos_iff {a : V} : 0 < ‖a‖ ↔ 0 < a :=
⟨by int... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"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_30 | c5f7743a0291999f | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Exponential/Log.lean | Log | 30 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 32 | 5 | [
{
"theorem_name": "le_iff_lt_length_of_exp",
"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 · simpa using pos_iff_ne_zero.mp H.range_pos\n simp [le_iff_le_log... | [
{
"name": "le_iff_le_log_of_exp",
"text": "lemma le_iff_le_log_of_exp {x y a : V} (H : Exponential x y) (pos : 0 < a) : y ≤ a ↔ x ≤ log a :=\n ⟨by rcases H.log_eq_of_exp; exact log_monotone,\n fun h ↦ by rcases log_pos pos with ⟨a', ha', Haa', _⟩; exact le_trans (Exponential.monotone_le H Haa' h) ha'⟩\n\... | [
{
"name": "bexp_two_mul",
"fan_in": 0,
"n_deps_direct": 2,
"n_deps_transitive": 13,
"n_lines": 5,
"n_chars": 169,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
"max_nesting":... | 11 | 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'... | @@ -170,6 +170,10 @@
have : 0 < b := pos_of_lt_length h
exact le_iff_lt_succ.mpr (by simpa [length_of_pos this] using h)
+lemma le_iff_le_log_of_exp {x y a : V} (H : Exponential x y) (pos : 0 < a) : y ≤ a ↔ x ≤ log a :=
+ ⟨by rcases H.log_eq_of_exp; exact log_monotone,
+ fun h ↦ by rcases log_pos pos with ⟨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_7471c35686f5_31 | df409d3cc1f5fbd9 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Exponential/Log.lean | Log | 31 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 32 | 14 | [
{
"theorem_name": "length_le",
"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 · simp [pos, length_of_pos, ←lt_iff_succ_le, log_lt_self_of_pos]",
... | [
{
"name": "length_of_pos",
"text": "lemma length_of_pos {a : V} (pos : 0 < a) : ‖a‖ = log a + 1 := by simp [length_eq_binaryLength, pos]\n\n",
"fan_in": 14,
"n_lines": 3,
"n_chars": 106,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
... | [
{
"name": "fbit_eq_zero_of_le",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 12,
"n_lines": 3,
"n_chars": 113,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"max_nest... | 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'... | @@ -120,15 +120,17 @@
@[simp] lemma length_zero : ‖(0 : V)‖ = 0 := by simp [length_eq_binaryLength]
+lemma length_of_pos {a : V} (pos : 0 < a) : ‖a‖ = log a + 1 := by simp [length_eq_binaryLength, pos]
+
@[simp] lemma length_le (a : V) : ‖a‖ ≤ a := by
rcases Arithmetic.zero_le a with (rfl | pos)
· simp
- ·... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_7471c35686f5_32 | aa9781aa0799e280 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Exponential/Log.lean | Log | 32 | 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_log_le_self",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 3,
"n_lines": 6,
"n_chars": 173,
"n_subproofs": 0,
"n_tactics": 4,
"cyclomatic": 3,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 2,
"automation_only": false,
"max_nesting... | 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... |
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