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_20b4e5ef02af_6 | 20769a3e02a2073e | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Exponential/Bit.lean | Bit | 6 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 23 | 2 | [
{
"theorem_name": "singleton_injective",
"depth": 1,
"n_commands": 0,
"n_lines": 5,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n constructor\n · intro h\n simpa [singleton_def] using congr_arg log h\n · rintro rfl; rfl",
"n_chars": 99,
... | [
{
"name": "singleton_def",
"text": "lemma singleton_def (a : V) : {a} = Exp.exp a := rfl\n\n",
"fan_in": 2,
"n_lines": 3,
"n_chars": 54,
"n_subproofs": 0,
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"n_automation": 1,
"n_rewrites": 0,
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"automation_only": true,
... | [
{
"name": "one_eq_singleton",
"fan_in": 0,
"n_deps_direct": 2,
"n_deps_transitive": 3,
"n_lines": 3,
"n_chars": 108,
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"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"max_nesting... | 2 | module
public import Foundation.FirstOrder.Arithmetic.Exponential.Log
/-!
# $\mathrm{Bit}$ predicate
-/
@[expose] public section
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
section model
def Bit (i a : V) : Prop := LenBit (Exp.exp i) a
instanc... | module
public import Foundation.FirstOrder.Arithmetic.Exponential.Log
/-!
# $\mathrm{Bit}$ predicate
-/
@[expose] public section
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
section model
def Bit (i a : V) : Prop := LenBit (Exp.exp i) a
instanc... | @@ -200,9 +200,12 @@
noncomputable scoped instance : Singleton V V := ⟨fun a ↦ Exp.exp a⟩
+lemma singleton_def (a : V) : {a} = Exp.exp a := rfl
+
@[simp] lemma singleton_injective (a b : V) : ({a} : V) = {b} ↔ a = b := by
constructor
- · sorry
+ · intro h
+ simpa [singleton_def] using congr_arg log h
·... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_20b4e5ef02af_7 | c0941f3eb216c3c7 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Exponential/Bit.lean | Bit | 7 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 23 | 9 | [
{
"theorem_name": "lt_of_mem",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " lt_of_lt_of_le (lt_exp i) (exp_le_of_mem h)",
"n_chars": 44,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_... | [
{
"name": "exp_le_of_mem",
"text": "lemma exp_le_of_mem {i a : V} (h : i ∈ a) : Exp.exp i ≤ a := LenBit.le h\n\n",
"fan_in": 9,
"n_lines": 3,
"n_chars": 78,
"n_subproofs": 0,
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"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
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"auto... | [
{
"name": "bitRemove_lt_of_mem",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 4,
"n_chars": 135,
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"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"max_nest... | 1 | module
public import Foundation.FirstOrder.Arithmetic.Exponential.Log
/-!
# $\mathrm{Bit}$ predicate
-/
@[expose] public section
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
section model
def Bit (i a : V) : Prop := LenBit (Exp.exp i) a
instanc... | module
public import Foundation.FirstOrder.Arithmetic.Exponential.Log
/-!
# $\mathrm{Bit}$ predicate
-/
@[expose] public section
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
section model
def Bit (i a : V) : Prop := LenBit (Exp.exp i) a
instanc... | @@ -41,8 +41,10 @@
lemma mem_iff_bit {i a : V} : i ∈ a ↔ Bit i a := iff_of_eq rfl
-lemma lt_of_mem {i a : V} (h : i ∈ a) : i < a := sorry
+lemma exp_le_of_mem {i a : V} (h : i ∈ a) : Exp.exp i ≤ a := LenBit.le h
+lemma lt_of_mem {i a : V} (h : i ∈ a) : i < a := lt_of_lt_of_le (lt_exp i) (exp_le_of_mem h)
+
@[de... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_20b4e5ef02af_8 | cd2440a466c9be77 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Exponential/Bit.lean | Bit | 8 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 23 | 2 | [
{
"theorem_name": "mem_bitInsert_iff",
"depth": 1,
"n_commands": 0,
"n_lines": 5,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n by_cases h : j ∈ a <;> simp [h, insert_eq, bitInsert]\n · by_cases e : i = j <;> simp [h, e]\n · simpa [exp_inj.eq_iff]... | [
{
"name": "insert_eq",
"text": "lemma insert_eq {i a : V} : insert i a = bitInsert i a := rfl\n\n",
"fan_in": 2,
"n_lines": 3,
"n_chars": 63,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": t... | [
{
"name": "insert_eq_self_of_mem",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 4,
"n_chars": 109,
"n_subproofs": 0,
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"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"max_ne... | 1 | module
public import Foundation.FirstOrder.Arithmetic.Exponential.Log
/-!
# $\mathrm{Bit}$ predicate
-/
@[expose] public section
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
section model
def Bit (i a : V) : Prop := LenBit (Exp.exp i) a
instanc... | module
public import Foundation.FirstOrder.Arithmetic.Exponential.Log
/-!
# $\mathrm{Bit}$ predicate
-/
@[expose] public section
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
section model
def Bit (i a : V) : Prop := LenBit (Exp.exp i) a
instanc... | @@ -220,13 +220,19 @@
noncomputable scoped instance : Insert V V := ⟨bitInsert⟩
+lemma insert_eq {i a : V} : insert i a = bitInsert i a := rfl
+
lemma singleton_eq_insert (i : V) : ({i} : V) = insert i ∅ := by simp [singleton_def, insert, bitInsert, emptyset_def]
instance : LawfulSingleton V V where
insert_... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_20b4e5ef02af_9 | 7f691e949617603e | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Exponential/Bit.lean | Bit | 9 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 23 | 9 | [
{
"theorem_name": "lt_of_mem",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " lt_of_lt_of_le (lt_exp i) (exp_le_of_mem h)",
"n_chars": 44,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_... | [
{
"name": "exp_le_of_mem",
"text": "lemma exp_le_of_mem {i a : V} (h : i ∈ a) : Exp.exp i ≤ a := LenBit.le h\n\n",
"fan_in": 9,
"n_lines": 3,
"n_chars": 78,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"auto... | [
{
"name": "le_log_of_mem",
"fan_in": 1,
"n_deps_direct": 2,
"n_deps_transitive": 2,
"n_lines": 3,
"n_chars": 123,
"n_subproofs": 0,
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"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
"max_nesting":... | 2 | module
public import Foundation.FirstOrder.Arithmetic.Exponential.Log
/-!
# $\mathrm{Bit}$ predicate
-/
@[expose] public section
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
section model
def Bit (i a : V) : Prop := LenBit (Exp.exp i) a
instanc... | module
public import Foundation.FirstOrder.Arithmetic.Exponential.Log
/-!
# $\mathrm{Bit}$ predicate
-/
@[expose] public section
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
section model
def Bit (i a : V) : Prop := LenBit (Exp.exp i) a
instanc... | @@ -41,8 +41,10 @@
lemma mem_iff_bit {i a : V} : i ∈ a ↔ Bit i a := iff_of_eq rfl
-lemma lt_of_mem {i a : V} (h : i ∈ a) : i < a := sorry
+lemma exp_le_of_mem {i a : V} (h : i ∈ a) : Exp.exp i ≤ a := LenBit.le h
+lemma lt_of_mem {i a : V} (h : i ∈ a) : i < a := lt_of_lt_of_le (lt_exp i) (exp_le_of_mem h)
+
@[de... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_20b4e5ef02af_10 | 3b86f5745ebb7745 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Exponential/Bit.lean | Bit | 10 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 23 | 9 | [
{
"theorem_name": "lt_of_mem",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " lt_of_lt_of_le (lt_exp i) (exp_le_of_mem h)",
"n_chars": 44,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_... | [
{
"name": "exp_le_of_mem",
"text": "lemma exp_le_of_mem {i a : V} (h : i ∈ a) : Exp.exp i ≤ a := LenBit.le h\n\n",
"fan_in": 9,
"n_lines": 3,
"n_chars": 78,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"auto... | [
{
"name": "lt_length_of_mem",
"fan_in": 0,
"n_deps_direct": 2,
"n_deps_transitive": 3,
"n_lines": 4,
"n_chars": 153,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"max_nesting... | 3 | module
public import Foundation.FirstOrder.Arithmetic.Exponential.Log
/-!
# $\mathrm{Bit}$ predicate
-/
@[expose] public section
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
section model
def Bit (i a : V) : Prop := LenBit (Exp.exp i) a
instanc... | module
public import Foundation.FirstOrder.Arithmetic.Exponential.Log
/-!
# $\mathrm{Bit}$ predicate
-/
@[expose] public section
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
section model
def Bit (i a : V) : Prop := LenBit (Exp.exp i) a
instanc... | @@ -41,8 +41,10 @@
lemma mem_iff_bit {i a : V} : i ∈ a ↔ Bit i a := iff_of_eq rfl
-lemma lt_of_mem {i a : V} (h : i ∈ a) : i < a := sorry
+lemma exp_le_of_mem {i a : V} (h : i ∈ a) : Exp.exp i ≤ a := LenBit.le h
+lemma lt_of_mem {i a : V} (h : i ∈ a) : i < a := lt_of_lt_of_le (lt_exp i) (exp_le_of_mem h)
+
@[de... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_20b4e5ef02af_11 | 98f4452b14dd56ab | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Exponential/Bit.lean | Bit | 11 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 23 | 9 | [
{
"theorem_name": "lt_of_mem",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " lt_of_lt_of_le (lt_exp i) (exp_le_of_mem h)",
"n_chars": 44,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_... | [
{
"name": "exp_le_of_mem",
"text": "lemma exp_le_of_mem {i a : V} (h : i ∈ a) : Exp.exp i ≤ a := LenBit.le h\n\n",
"fan_in": 9,
"n_lines": 3,
"n_chars": 78,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"auto... | [
{
"name": "lt_exp_iff",
"fan_in": 0,
"n_deps_direct": 2,
"n_deps_transitive": 4,
"n_lines": 8,
"n_chars": 367,
"n_subproofs": 2,
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"cyclomatic": 1,
"n_automation": 2,
"n_rewrites": 0,
"n_structural": 2,
"automation_only": false,
"max_nesting": 6
... | 4 | module
public import Foundation.FirstOrder.Arithmetic.Exponential.Log
/-!
# $\mathrm{Bit}$ predicate
-/
@[expose] public section
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
section model
def Bit (i a : V) : Prop := LenBit (Exp.exp i) a
instanc... | module
public import Foundation.FirstOrder.Arithmetic.Exponential.Log
/-!
# $\mathrm{Bit}$ predicate
-/
@[expose] public section
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
section model
def Bit (i a : V) : Prop := LenBit (Exp.exp i) a
instanc... | @@ -41,8 +41,10 @@
lemma mem_iff_bit {i a : V} : i ∈ a ↔ Bit i a := iff_of_eq rfl
-lemma lt_of_mem {i a : V} (h : i ∈ a) : i < a := sorry
+lemma exp_le_of_mem {i a : V} (h : i ∈ a) : Exp.exp i ≤ a := LenBit.le h
+lemma lt_of_mem {i a : V} (h : i ∈ a) : i < a := lt_of_lt_of_le (lt_exp i) (exp_le_of_mem h)
+
@[de... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_20b4e5ef02af_12 | d420a999f1c01b89 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Exponential/Bit.lean | Bit | 12 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 23 | 2 | [
{
"theorem_name": "log_mem_of_pos",
"depth": 1,
"n_commands": 0,
"n_lines": 5,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": "\n mem_iff_mul_exp_add_exp_add.mpr\n ⟨0, a - Exp.exp (log a),\n (tsub_lt_iff_left (exp_log_le_self h)).mpr (by rw [←two_mu... | [
{
"name": "mem_iff_mul_exp_add_exp_add",
"text": "lemma mem_iff_mul_exp_add_exp_add {i a : V} : i ∈ a ↔ ∃ k, ∃ r < Exp.exp i, a = k * Exp.exp (i + 1) + Exp.exp i + r := by\n simpa [mem_iff_bit, exp_succ] using! lenbit_iff_add_mul (exp_pow2 i) (a := a)\n\n",
"fan_in": 2,
"n_lines": 4,
"n_chars":... | [
{
"name": "eq_zero_of_subset_zero",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 3,
"n_lines": 6,
"n_chars": 171,
"n_subproofs": 1,
"n_tactics": 5,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": false,
"max_... | 3 | module
public import Foundation.FirstOrder.Arithmetic.Exponential.Log
/-!
# $\mathrm{Bit}$ predicate
-/
@[expose] public section
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
section model
def Bit (i a : V) : Prop := LenBit (Exp.exp i) a
instanc... | module
public import Foundation.FirstOrder.Arithmetic.Exponential.Log
/-!
# $\mathrm{Bit}$ predicate
-/
@[expose] public section
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
section model
def Bit (i a : V) : Prop := LenBit (Exp.exp i) a
instanc... | @@ -180,6 +180,9 @@
section model
+lemma mem_iff_mul_exp_add_exp_add {i a : V} : i ∈ a ↔ ∃ k, ∃ r < Exp.exp i, a = k * Exp.exp (i + 1) + Exp.exp i + r := by
+ simpa [mem_iff_bit, exp_succ] using! lenbit_iff_add_mul (exp_pow2 i) (a := a)
+
section empty
scoped instance : EmptyCollection V := ⟨0⟩
@@ -272,7 +275... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_20b4e5ef02af_13 | 0210bd7bf540afca | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Exponential/Bit.lean | Bit | 13 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 23 | 3 | [
{
"theorem_name": "lt_exp_iff",
"depth": 1,
"n_commands": 0,
"n_lines": 6,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": "\n ⟨fun h j hj ↦ exp_monotone.mp <| lt_of_le_of_lt (exp_le_of_mem hj) h,\n by suffices Exp.exp i ≤ a → ∃ x ∈ a, i ≤ x by contrapose;... | [
{
"name": "log_mem_of_pos",
"text": "lemma log_mem_of_pos {a : V} (h : 0 < a) : log a ∈ a :=\n mem_iff_mul_exp_add_exp_add.mpr\n ⟨0, a - Exp.exp (log a),\n (tsub_lt_iff_left (exp_log_le_self h)).mpr (by rw [←two_mul]; exact lt_two_mul_exponential_log h),\n by simpa using Eq.symm <| add_tsub_se... | [
{
"name": "le_of_subset",
"fan_in": 1,
"n_deps_direct": 3,
"n_deps_transitive": 7,
"n_lines": 15,
"n_chars": 643,
"n_subproofs": 4,
"n_tactics": 15,
"cyclomatic": 2,
"n_automation": 6,
"n_rewrites": 0,
"n_structural": 3,
"automation_only": false,
"max_nesting"... | 7 | module
public import Foundation.FirstOrder.Arithmetic.Exponential.Log
/-!
# $\mathrm{Bit}$ predicate
-/
@[expose] public section
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
section model
def Bit (i a : V) : Prop := LenBit (Exp.exp i) a
instanc... | module
public import Foundation.FirstOrder.Arithmetic.Exponential.Log
/-!
# $\mathrm{Bit}$ predicate
-/
@[expose] public section
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
section model
def Bit (i a : V) : Prop := LenBit (Exp.exp i) a
instanc... | @@ -275,6 +275,12 @@
@[simp] lemma mem_insert (i a : V) : i ∈ insert i a := by simp
+lemma log_mem_of_pos {a : V} (h : 0 < a) : log a ∈ a :=
+ mem_iff_mul_exp_add_exp_add.mpr
+ ⟨0, a - Exp.exp (log a),
+ (tsub_lt_iff_left (exp_log_le_self h)).mpr (by rw [←two_mul]; exact lt_two_mul_exponential_log h),
+ ... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_20b4e5ef02af_14 | 46cb5c5ea206646b | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Exponential/Bit.lean | Bit | 14 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 23 | 1 | [
{
"theorem_name": "mem_ext",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": "\n le_antisymm (le_of_subset fun i hi ↦ (h i).mp hi) (le_of_subset fun i hi ↦ (h i).mpr hi)",
"n_chars": 95,
"n_subproofs": 0,
... | [
{
"name": "le_of_subset",
"text": "lemma le_of_subset {a b : V} (h : a ⊆ b) : a ≤ b := by\n induction b using pi1_polynomial_induction generalizing a\n · definability\n case zero =>\n simp [eq_zero_of_subset_zero h]\n case even b _ IH =>\n have IH : a / 2 ≤ b := IH (by simpa using subset_div_two h... | [
{
"name": "mem_ext_iff",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 9,
"n_lines": 3,
"n_chars": 106,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 2,
"n_automation": 2,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": false,
"max_nesting": 1... | 9 | module
public import Foundation.FirstOrder.Arithmetic.Exponential.Log
/-!
# $\mathrm{Bit}$ predicate
-/
@[expose] public section
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
section model
def Bit (i a : V) : Prop := LenBit (Exp.exp i) a
instanc... | module
public import Foundation.FirstOrder.Arithmetic.Exponential.Log
/-!
# $\mathrm{Bit}$ predicate
-/
@[expose] public section
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
section model
def Bit (i a : V) : Prop := LenBit (Exp.exp i) a
instanc... | @@ -374,8 +374,23 @@
@[simp] lemma succ_mem_two_mul_succ_iff {i a : V} : i + 1 ∈ 2 * a + 1 ↔ i ∈ a := by
simp [mem_iff_bit, Bit, LenBit, exp_succ, Arithmetic.div_mul]
-lemma mem_ext {a b : V} (h : ∀ i, i ∈ a ↔ i ∈ b) : a = b := sorry
+lemma le_of_subset {a b : V} (h : a ⊆ b) : a ≤ b := by
+ induction b using pi1... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_20b4e5ef02af_15 | e622e00706153403 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Exponential/Bit.lean | Bit | 15 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 23 | 1 | [
{
"theorem_name": "mem_ext",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": "\n le_antisymm (le_of_subset fun i hi ↦ (h i).mp hi) (le_of_subset fun i hi ↦ (h i).mpr hi)",
"n_chars": 95,
"n_subproofs": 0,
... | [
{
"name": "le_of_subset",
"text": "lemma le_of_subset {a b : V} (h : a ⊆ b) : a ≤ b := by\n induction b using pi1_polynomial_induction generalizing a\n · definability\n case zero =>\n simp [eq_zero_of_subset_zero h]\n case even b _ IH =>\n have IH : a / 2 ≤ b := IH (by simpa using subset_div_two h... | [
{
"name": "nonempty_of_pos",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 9,
"n_lines": 6,
"n_chars": 144,
"n_subproofs": 1,
"n_tactics": 4,
"cyclomatic": 1,
"n_automation": 2,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
"max_nesting... | 9 | module
public import Foundation.FirstOrder.Arithmetic.Exponential.Log
/-!
# $\mathrm{Bit}$ predicate
-/
@[expose] public section
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
section model
def Bit (i a : V) : Prop := LenBit (Exp.exp i) a
instanc... | module
public import Foundation.FirstOrder.Arithmetic.Exponential.Log
/-!
# $\mathrm{Bit}$ predicate
-/
@[expose] public section
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
section model
def Bit (i a : V) : Prop := LenBit (Exp.exp i) a
instanc... | @@ -374,8 +374,23 @@
@[simp] lemma succ_mem_two_mul_succ_iff {i a : V} : i + 1 ∈ 2 * a + 1 ↔ i ∈ a := by
simp [mem_iff_bit, Bit, LenBit, exp_succ, Arithmetic.div_mul]
-lemma mem_ext {a b : V} (h : ∀ i, i ∈ a ↔ i ∈ b) : a = b := sorry
+lemma le_of_subset {a b : V} (h : a ⊆ b) : a ≤ b := by
+ induction b using pi1... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_20b4e5ef02af_16 | def7c22706d37f66 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Exponential/Bit.lean | Bit | 16 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 23 | 1 | [
{
"theorem_name": "mem_ext",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": "\n le_antisymm (le_of_subset fun i hi ↦ (h i).mp hi) (le_of_subset fun i hi ↦ (h i).mpr hi)",
"n_chars": 95,
"n_subproofs": 0,
... | [
{
"name": "le_of_subset",
"text": "lemma le_of_subset {a b : V} (h : a ⊆ b) : a ≤ b := by\n induction b using pi1_polynomial_induction generalizing a\n · definability\n case zero =>\n simp [eq_zero_of_subset_zero h]\n case even b _ IH =>\n have IH : a / 2 ≤ b := IH (by simpa using subset_div_two h... | [
{
"name": "eq_empty_or_nonempty",
"fan_in": 1,
"n_deps_direct": 2,
"n_deps_transitive": 11,
"n_lines": 6,
"n_chars": 181,
"n_subproofs": 0,
"n_tactics": 5,
"cyclomatic": 2,
"n_automation": 2,
"n_rewrites": 0,
"n_structural": 2,
"automation_only": false,
"max_n... | 11 | module
public import Foundation.FirstOrder.Arithmetic.Exponential.Log
/-!
# $\mathrm{Bit}$ predicate
-/
@[expose] public section
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
section model
def Bit (i a : V) : Prop := LenBit (Exp.exp i) a
instanc... | module
public import Foundation.FirstOrder.Arithmetic.Exponential.Log
/-!
# $\mathrm{Bit}$ predicate
-/
@[expose] public section
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
section model
def Bit (i a : V) : Prop := LenBit (Exp.exp i) a
instanc... | @@ -374,8 +374,23 @@
@[simp] lemma succ_mem_two_mul_succ_iff {i a : V} : i + 1 ∈ 2 * a + 1 ↔ i ∈ a := by
simp [mem_iff_bit, Bit, LenBit, exp_succ, Arithmetic.div_mul]
-lemma mem_ext {a b : V} (h : ∀ i, i ∈ a ↔ i ∈ b) : a = b := sorry
+lemma le_of_subset {a b : V} (h : a ⊆ b) : a ≤ b := by
+ induction b using pi1... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_20b4e5ef02af_17 | b2bcc311dc1cfd3f | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Exponential/Bit.lean | Bit | 17 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 23 | 1 | [
{
"theorem_name": "mem_ext",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": "\n le_antisymm (le_of_subset fun i hi ↦ (h i).mp hi) (le_of_subset fun i hi ↦ (h i).mpr hi)",
"n_chars": 95,
"n_subproofs": 0,
... | [
{
"name": "le_of_subset",
"text": "lemma le_of_subset {a b : V} (h : a ⊆ b) : a ≤ b := by\n induction b using pi1_polynomial_induction generalizing a\n · definability\n case zero =>\n simp [eq_zero_of_subset_zero h]\n case even b _ IH =>\n have IH : a / 2 ≤ b := IH (by simpa using subset_div_two h... | [
{
"name": "nonempty_iff",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 12,
"n_lines": 6,
"n_chars": 183,
"n_subproofs": 0,
"n_tactics": 5,
"cyclomatic": 3,
"n_automation": 5,
"n_rewrites": 1,
"n_structural": 2,
"automation_only": false,
"max_nesting":... | 12 | module
public import Foundation.FirstOrder.Arithmetic.Exponential.Log
/-!
# $\mathrm{Bit}$ predicate
-/
@[expose] public section
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
section model
def Bit (i a : V) : Prop := LenBit (Exp.exp i) a
instanc... | module
public import Foundation.FirstOrder.Arithmetic.Exponential.Log
/-!
# $\mathrm{Bit}$ predicate
-/
@[expose] public section
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
section model
def Bit (i a : V) : Prop := LenBit (Exp.exp i) a
instanc... | @@ -374,8 +374,23 @@
@[simp] lemma succ_mem_two_mul_succ_iff {i a : V} : i + 1 ∈ 2 * a + 1 ↔ i ∈ a := by
simp [mem_iff_bit, Bit, LenBit, exp_succ, Arithmetic.div_mul]
-lemma mem_ext {a b : V} (h : ∀ i, i ∈ a ↔ i ∈ b) : a = b := sorry
+lemma le_of_subset {a b : V} (h : a ⊆ b) : a ≤ b := by
+ induction b using pi1... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_20b4e5ef02af_18 | 54dac807c8fae731 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Exponential/Bit.lean | Bit | 18 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 23 | 8 | [
{
"theorem_name": "mem_ext_iff",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " ⟨by rintro rfl; simp, mem_ext⟩",
"n_chars": 35,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 2,
"n_automation"... | [
{
"name": "mem_ext",
"text": "lemma mem_ext {a b : V} (h : ∀ i, i ∈ a ↔ i ∈ b) : a = b :=\n le_antisymm (le_of_subset fun i hi ↦ (h i).mp hi) (le_of_subset fun i hi ↦ (h i).mpr hi)\n\n",
"fan_in": 8,
"n_lines": 4,
"n_chars": 164,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
... | [
{
"name": "isempty_iff",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 13,
"n_lines": 4,
"n_chars": 116,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"max_nesting": 2... | 13 | module
public import Foundation.FirstOrder.Arithmetic.Exponential.Log
/-!
# $\mathrm{Bit}$ predicate
-/
@[expose] public section
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
section model
def Bit (i a : V) : Prop := LenBit (Exp.exp i) a
instanc... | module
public import Foundation.FirstOrder.Arithmetic.Exponential.Log
/-!
# $\mathrm{Bit}$ predicate
-/
@[expose] public section
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
section model
def Bit (i a : V) : Prop := LenBit (Exp.exp i) a
instanc... | @@ -388,9 +388,12 @@
have IH : a / 2 ≤ b := IH (by simpa [div_mul_add' b 2 one_lt_two] using subset_div_two h)
exact le_trans (le_two_mul_div_two_add_one a) (by simpa using IH)
+lemma mem_ext {a b : V} (h : ∀ i, i ∈ a ↔ i ∈ b) : a = b :=
+ le_antisymm (le_of_subset fun i hi ↦ (h i).mp hi) (le_of_subset fun... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_20b4e5ef02af_19 | 920a2d712763d6ae | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Exponential/Bit.lean | Bit | 19 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 23 | 2 | [
{
"theorem_name": "log_mem_of_pos",
"depth": 1,
"n_commands": 0,
"n_lines": 5,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": "\n mem_iff_mul_exp_add_exp_add.mpr\n ⟨0, a - Exp.exp (log a),\n (tsub_lt_iff_left (exp_log_le_self h)).mpr (by rw [←two_mu... | [
{
"name": "mem_iff_mul_exp_add_exp_add",
"text": "lemma mem_iff_mul_exp_add_exp_add {i a : V} : i ∈ a ↔ ∃ k, ∃ r < Exp.exp i, a = k * Exp.exp (i + 1) + Exp.exp i + r := by\n simpa [mem_iff_bit, exp_succ] using! lenbit_iff_add_mul (exp_pow2 i) (a := a)\n\n",
"fan_in": 2,
"n_lines": 4,
"n_chars":... | [
{
"name": "lt_of_lt_log",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 3,
"n_lines": 7,
"n_chars": 267,
"n_subproofs": 0,
"n_tactics": 5,
"cyclomatic": 2,
"n_automation": 2,
"n_rewrites": 1,
"n_structural": 3,
"automation_only": false,
"max_nesting": ... | 3 | module
public import Foundation.FirstOrder.Arithmetic.Exponential.Log
/-!
# $\mathrm{Bit}$ predicate
-/
@[expose] public section
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
section model
def Bit (i a : V) : Prop := LenBit (Exp.exp i) a
instanc... | module
public import Foundation.FirstOrder.Arithmetic.Exponential.Log
/-!
# $\mathrm{Bit}$ predicate
-/
@[expose] public section
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
section model
def Bit (i a : V) : Prop := LenBit (Exp.exp i) a
instanc... | @@ -180,6 +180,9 @@
section model
+lemma mem_iff_mul_exp_add_exp_add {i a : V} : i ∈ a ↔ ∃ k, ∃ r < Exp.exp i, a = k * Exp.exp (i + 1) + Exp.exp i + r := by
+ simpa [mem_iff_bit, exp_succ] using! lenbit_iff_add_mul (exp_pow2 i) (a := a)
+
section empty
scoped instance : EmptyCollection V := ⟨0⟩
@@ -272,7 +275... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_20b4e5ef02af_20 | 4fce8df6ea029f66 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Exponential/Bit.lean | Bit | 20 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 23 | 1 | [
{
"theorem_name": "le_of_subset",
"depth": 1,
"n_commands": 0,
"n_lines": 13,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n induction b using pi1_polynomial_induction generalizing a\n · definability\n case zero =>\n simp [eq_zero_of_subset_zero... | [
{
"name": "eq_zero_of_subset_zero",
"text": "lemma eq_zero_of_subset_zero {a : V} : a ⊆ 0 → a = 0 := by\n intro h; by_contra A\n have : log a ∈ (0 : V) := h (log_mem_of_pos (pos_iff_ne_zero.mpr A))\n simp_all\n\n",
"fan_in": 1,
"n_lines": 6,
"n_chars": 171,
"n_subproofs": 1,
"n_tactic... | [
{
"name": "under_zero",
"fan_in": 0,
"n_deps_direct": 2,
"n_deps_transitive": 12,
"n_lines": 3,
"n_chars": 85,
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"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"max_nesting": 1
... | 11 | module
public import Foundation.FirstOrder.Arithmetic.Exponential.Log
/-!
# $\mathrm{Bit}$ predicate
-/
@[expose] public section
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
section model
def Bit (i a : V) : Prop := LenBit (Exp.exp i) a
instanc... | module
public import Foundation.FirstOrder.Arithmetic.Exponential.Log
/-!
# $\mathrm{Bit}$ predicate
-/
@[expose] public section
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
section model
def Bit (i a : V) : Prop := LenBit (Exp.exp i) a
instanc... | @@ -352,6 +352,11 @@
instance under_definable' (Γ) : Γ-Function₁[V] under := under_definable.of_zero
+lemma eq_zero_of_subset_zero {a : V} : a ⊆ 0 → a = 0 := by
+ intro h; by_contra A
+ have : log a ∈ (0 : V) := h (log_mem_of_pos (pos_iff_ne_zero.mpr A))
+ simp_all
+
lemma subset_div_two {a b : V} : a ⊆ b → a ... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_20b4e5ef02af_21 | 4184bf1e6d912619 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Exponential/Bit.lean | Bit | 21 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 23 | 1 | [
{
"theorem_name": "mem_ext",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": "\n le_antisymm (le_of_subset fun i hi ↦ (h i).mp hi) (le_of_subset fun i hi ↦ (h i).mpr hi)",
"n_chars": 95,
"n_subproofs": 0,
... | [
{
"name": "le_of_subset",
"text": "lemma le_of_subset {a b : V} (h : a ⊆ b) : a ≤ b := by\n induction b using pi1_polynomial_induction generalizing a\n · definability\n case zero =>\n simp [eq_zero_of_subset_zero h]\n case even b _ IH =>\n have IH : a / 2 ≤ b := IH (by simpa using subset_div_two h... | [
{
"name": "insert_remove",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 9,
"n_lines": 8,
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"cyclomatic": 4,
"n_automation": 4,
"n_rewrites": 0,
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"automation_only": false,
"max_nesting":... | 9 | module
public import Foundation.FirstOrder.Arithmetic.Exponential.Log
/-!
# $\mathrm{Bit}$ predicate
-/
@[expose] public section
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
section model
def Bit (i a : V) : Prop := LenBit (Exp.exp i) a
instanc... | module
public import Foundation.FirstOrder.Arithmetic.Exponential.Log
/-!
# $\mathrm{Bit}$ predicate
-/
@[expose] public section
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
section model
def Bit (i a : V) : Prop := LenBit (Exp.exp i) a
instanc... | @@ -374,8 +374,23 @@
@[simp] lemma succ_mem_two_mul_succ_iff {i a : V} : i + 1 ∈ 2 * a + 1 ↔ i ∈ a := by
simp [mem_iff_bit, Bit, LenBit, exp_succ, Arithmetic.div_mul]
-lemma mem_ext {a b : V} (h : ∀ i, i ∈ a ↔ i ∈ b) : a = b := sorry
+lemma le_of_subset {a b : V} (h : a ⊆ b) : a ≤ b := by
+ induction b using pi1... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_20b4e5ef02af_22 | 3852ccc4fb6cc2d3 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Exponential/Bit.lean | Bit | 22 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 23 | 9 | [
{
"theorem_name": "lt_of_mem",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " lt_of_lt_of_le (lt_exp i) (exp_le_of_mem h)",
"n_chars": 44,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_... | [
{
"name": "exp_le_of_mem",
"text": "lemma exp_le_of_mem {i a : V} (h : i ∈ a) : Exp.exp i ≤ a := LenBit.le h\n\n",
"fan_in": 9,
"n_lines": 3,
"n_chars": 78,
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"n_automation": 0,
"n_rewrites": 0,
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"auto... | [
{
"name": "finset_comprehension_aux",
"fan_in": 1,
"n_deps_direct": 2,
"n_deps_transitive": 6,
"n_lines": 30,
"n_chars": 1600,
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"n_tactics": 28,
"cyclomatic": 6,
"n_automation": 7,
"n_rewrites": 2,
"n_structural": 11,
"automation_only": false,
... | 5 | module
public import Foundation.FirstOrder.Arithmetic.Exponential.Log
/-!
# $\mathrm{Bit}$ predicate
-/
@[expose] public section
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
section model
def Bit (i a : V) : Prop := LenBit (Exp.exp i) a
instanc... | module
public import Foundation.FirstOrder.Arithmetic.Exponential.Log
/-!
# $\mathrm{Bit}$ predicate
-/
@[expose] public section
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
section model
def Bit (i a : V) : Prop := LenBit (Exp.exp i) a
instanc... | @@ -41,8 +41,10 @@
lemma mem_iff_bit {i a : V} : i ∈ a ↔ Bit i a := iff_of_eq rfl
-lemma lt_of_mem {i a : V} (h : i ∈ a) : i < a := sorry
+lemma exp_le_of_mem {i a : V} (h : i ∈ a) : Exp.exp i ≤ a := LenBit.le h
+lemma lt_of_mem {i a : V} (h : i ∈ a) : i < a := lt_of_lt_of_le (lt_exp i) (exp_le_of_mem h)
+
@[de... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_20b4e5ef02af_23 | ddcdcc454525c632 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Exponential/Bit.lean | Bit | 23 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 23 | 9 | [
{
"theorem_name": "lt_of_mem",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " lt_of_lt_of_le (lt_exp i) (exp_le_of_mem h)",
"n_chars": 44,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_... | [
{
"name": "exp_le_of_mem",
"text": "lemma exp_le_of_mem {i a : V} (h : i ∈ a) : Exp.exp i ≤ a := LenBit.le h\n\n",
"fan_in": 9,
"n_lines": 3,
"n_chars": 78,
"n_subproofs": 0,
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"n_automation": 0,
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"auto... | [
{
"name": "finset_comprehension",
"fan_in": 2,
"n_deps_direct": 1,
"n_deps_transitive": 7,
"n_lines": 9,
"n_chars": 401,
"n_subproofs": 0,
"n_tactics": 6,
"cyclomatic": 2,
"n_automation": 0,
"n_rewrites": 1,
"n_structural": 0,
"automation_only": false,
"max_ne... | 6 | module
public import Foundation.FirstOrder.Arithmetic.Exponential.Log
/-!
# $\mathrm{Bit}$ predicate
-/
@[expose] public section
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
section model
def Bit (i a : V) : Prop := LenBit (Exp.exp i) a
instanc... | module
public import Foundation.FirstOrder.Arithmetic.Exponential.Log
/-!
# $\mathrm{Bit}$ predicate
-/
@[expose] public section
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
section model
def Bit (i a : V) : Prop := LenBit (Exp.exp i) a
instanc... | @@ -41,8 +41,10 @@
lemma mem_iff_bit {i a : V} : i ∈ a ↔ Bit i a := iff_of_eq rfl
-lemma lt_of_mem {i a : V} (h : i ∈ a) : i < a := sorry
+lemma exp_le_of_mem {i a : V} (h : i ∈ a) : Exp.exp i ≤ a := LenBit.le h
+lemma lt_of_mem {i a : V} (h : i ∈ a) : i < a := lt_of_lt_of_le (lt_exp i) (exp_le_of_mem h)
+
@[de... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_20b4e5ef02af_24 | 034c230a6a2ea52f | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Exponential/Bit.lean | Bit | 24 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 23 | 1 | [
{
"theorem_name": "le_of_subset",
"depth": 1,
"n_commands": 0,
"n_lines": 13,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n induction b using pi1_polynomial_induction generalizing a\n · definability\n case zero =>\n simp [eq_zero_of_subset_zero... | [
{
"name": "zero_mem_iff",
"text": "lemma zero_mem_iff {a : V} : 0 ∉ a ↔ 2 ∣ a := by simp [mem_iff_bit, Bit, LenBit]\n\n",
"fan_in": 1,
"n_lines": 3,
"n_chars": 88,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
... | [
{
"name": "finset_comprehension_exists_unique",
"fan_in": 0,
"n_deps_direct": 3,
"n_deps_transitive": 15,
"n_lines": 18,
"n_chars": 806,
"n_subproofs": 2,
"n_tactics": 15,
"cyclomatic": 3,
"n_automation": 0,
"n_rewrites": 2,
"n_structural": 10,
"automation_only": ... | 14 | module
public import Foundation.FirstOrder.Arithmetic.Exponential.Log
/-!
# $\mathrm{Bit}$ predicate
-/
@[expose] public section
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
section model
def Bit (i a : V) : Prop := LenBit (Exp.exp i) a
instanc... | module
public import Foundation.FirstOrder.Arithmetic.Exponential.Log
/-!
# $\mathrm{Bit}$ predicate
-/
@[expose] public section
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
section model
def Bit (i a : V) : Prop := LenBit (Exp.exp i) a
instanc... | @@ -365,6 +365,8 @@
have : i + 1 ∈ a := succ_mem_iff_mem_div_two.mpr hi
exact succ_mem_iff_mem_div_two.mp <| ss this
+lemma zero_mem_iff {a : V} : 0 ∉ a ↔ 2 ∣ a := by simp [mem_iff_bit, Bit, LenBit]
+
@[simp] lemma zero_not_mem (a : V) : 0 ∉ 2 * a := by simp [mem_iff_bit, Bit, LenBit]
@[simp] lemma zero_mem... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_20b4e5ef02af_25 | a35792f37b5d503b | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Exponential/Bit.lean | Bit | 25 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 23 | 9 | [
{
"theorem_name": "lt_of_mem",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " lt_of_lt_of_le (lt_exp i) (exp_le_of_mem h)",
"n_chars": 44,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_... | [
{
"name": "exp_le_of_mem",
"text": "lemma exp_le_of_mem {i a : V} (h : i ∈ a) : Exp.exp i ≤ a := LenBit.le h\n\n",
"fan_in": 9,
"n_lines": 3,
"n_chars": 78,
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"auto... | [
{
"name": "finset_comprehension₁",
"fan_in": 2,
"n_deps_direct": 1,
"n_deps_transitive": 8,
"n_lines": 5,
"n_chars": 167,
"n_subproofs": 0,
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"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
"max_n... | 7 | module
public import Foundation.FirstOrder.Arithmetic.Exponential.Log
/-!
# $\mathrm{Bit}$ predicate
-/
@[expose] public section
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
section model
def Bit (i a : V) : Prop := LenBit (Exp.exp i) a
instanc... | module
public import Foundation.FirstOrder.Arithmetic.Exponential.Log
/-!
# $\mathrm{Bit}$ predicate
-/
@[expose] public section
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
section model
def Bit (i a : V) : Prop := LenBit (Exp.exp i) a
instanc... | @@ -41,8 +41,10 @@
lemma mem_iff_bit {i a : V} : i ∈ a ↔ Bit i a := iff_of_eq rfl
-lemma lt_of_mem {i a : V} (h : i ∈ a) : i < a := sorry
+lemma exp_le_of_mem {i a : V} (h : i ∈ a) : Exp.exp i ≤ a := LenBit.le h
+lemma lt_of_mem {i a : V} (h : i ∈ a) : i < a := lt_of_lt_of_le (lt_exp i) (exp_le_of_mem h)
+
@[de... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_20b4e5ef02af_26 | 43ec6522d44cbbe1 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Exponential/Bit.lean | Bit | 26 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 23 | 1 | [
{
"theorem_name": "mem_ext",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": "\n le_antisymm (le_of_subset fun i hi ↦ (h i).mp hi) (le_of_subset fun i hi ↦ (h i).mpr hi)",
"n_chars": 95,
"n_subproofs": 0,
... | [
{
"name": "le_of_subset",
"text": "lemma le_of_subset {a b : V} (h : a ⊆ b) : a ≤ b := by\n induction b using pi1_polynomial_induction generalizing a\n · definability\n case zero =>\n simp [eq_zero_of_subset_zero h]\n case even b _ IH =>\n have IH : a / 2 ≤ b := IH (by simpa using subset_div_two h... | [
{
"name": "finset_comprehension₁!",
"fan_in": 0,
"n_deps_direct": 3,
"n_deps_transitive": 16,
"n_lines": 17,
"n_chars": 640,
"n_subproofs": 2,
"n_tactics": 14,
"cyclomatic": 4,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 10,
"automation_only": false,
"... | 15 | module
public import Foundation.FirstOrder.Arithmetic.Exponential.Log
/-!
# $\mathrm{Bit}$ predicate
-/
@[expose] public section
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
section model
def Bit (i a : V) : Prop := LenBit (Exp.exp i) a
instanc... | module
public import Foundation.FirstOrder.Arithmetic.Exponential.Log
/-!
# $\mathrm{Bit}$ predicate
-/
@[expose] public section
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
section model
def Bit (i a : V) : Prop := LenBit (Exp.exp i) a
instanc... | @@ -377,8 +377,23 @@
@[simp] lemma succ_mem_two_mul_succ_iff {i a : V} : i + 1 ∈ 2 * a + 1 ↔ i ∈ a := by
simp [mem_iff_bit, Bit, LenBit, exp_succ, Arithmetic.div_mul]
-lemma mem_ext {a b : V} (h : ∀ i, i ∈ a ↔ i ∈ b) : a = b := sorry
+lemma le_of_subset {a b : V} (h : a ⊆ b) : a ≤ b := by
+ induction b using pi1... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_20b4e5ef02af_27 | fb68d7f31720a41e | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Exponential/Bit.lean | Bit | 27 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 23 | 1 | [
{
"theorem_name": "mem_ext",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": "\n le_antisymm (le_of_subset fun i hi ↦ (h i).mp hi) (le_of_subset fun i hi ↦ (h i).mpr hi)",
"n_chars": 95,
"n_subproofs": 0,
... | [
{
"name": "le_of_subset",
"text": "lemma le_of_subset {a b : V} (h : a ⊆ b) : a ≤ b := by\n induction b using pi1_polynomial_induction generalizing a\n · definability\n case zero =>\n simp [eq_zero_of_subset_zero h]\n case even b _ IH =>\n have IH : a / 2 ≤ b := IH (by simpa using subset_div_two h... | [
{
"name": "finite_comprehension₁!",
"fan_in": 0,
"n_deps_direct": 3,
"n_deps_transitive": 16,
"n_lines": 11,
"n_chars": 515,
"n_subproofs": 1,
"n_tactics": 8,
"cyclomatic": 3,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 3,
"automation_only": false,
"ma... | 15 | module
public import Foundation.FirstOrder.Arithmetic.Exponential.Log
/-!
# $\mathrm{Bit}$ predicate
-/
@[expose] public section
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
section model
def Bit (i a : V) : Prop := LenBit (Exp.exp i) a
instanc... | module
public import Foundation.FirstOrder.Arithmetic.Exponential.Log
/-!
# $\mathrm{Bit}$ predicate
-/
@[expose] public section
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
section model
def Bit (i a : V) : Prop := LenBit (Exp.exp i) a
instanc... | @@ -377,8 +377,23 @@
@[simp] lemma succ_mem_two_mul_succ_iff {i a : V} : i + 1 ∈ 2 * a + 1 ↔ i ∈ a := by
simp [mem_iff_bit, Bit, LenBit, exp_succ, Arithmetic.div_mul]
-lemma mem_ext {a b : V} (h : ∀ i, i ∈ a ↔ i ∈ b) : a = b := sorry
+lemma le_of_subset {a b : V} (h : a ⊆ b) : a ≤ b := by
+ induction b using pi1... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_5e672913debe_0 | 6a39a5c140b1a4b0 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Vorspiel/Fin/Basic.lean | Basic | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 1 | 5 | [
{
"theorem_name": "one_pos''",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " pos_of_coe_ne_zero (Nat.succ_ne_zero 0)",
"n_chars": 40,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_auto... | [
{
"name": "pos_of_coe_ne_zero",
"text": "lemma pos_of_coe_ne_zero {i : Fin (n + 1)} (h : (i : ℕ) ≠ 0) : 0 < i := Nat.pos_of_ne_zero h\n\n",
"fan_in": 5,
"n_lines": 3,
"n_chars": 98,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_... | [
{
"name": "one_pos''",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 3,
"n_chars": 92,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
"max_nesting": 1
... | 1 | module
public import Mathlib.Algebra.GroupWithZero.Nat
public import Mathlib.Data.Fintype.Pigeonhole
public import Mathlib.Tactic.Cases
public import Mathlib.Tactic.TautoSet
@[expose]
public section
@[simp, grind .]
lemma Nat.sub_one_lt' [NeZero n] : n - 1 < n := sub_one_lt $ NeZero.ne n
namespace Fin
variable {n ... | module
public import Mathlib.Algebra.GroupWithZero.Nat
public import Mathlib.Data.Fintype.Pigeonhole
public import Mathlib.Tactic.Cases
public import Mathlib.Tactic.TautoSet
@[expose]
public section
@[simp, grind .]
lemma Nat.sub_one_lt' [NeZero n] : n - 1 < n := sub_one_lt $ NeZero.ne n
namespace Fin
variable {n ... | @@ -42,16 +42,18 @@
section
-@[simp] lemma one_pos'' : (0 : Fin (n + 2)) < 1 := sorry
+lemma pos_of_coe_ne_zero {i : Fin (n + 1)} (h : (i : ℕ) ≠ 0) : 0 < i := Nat.pos_of_ne_zero h
-@[simp] lemma two_pos : (0 : Fin (n + 3)) < 2 := sorry
+@[simp] lemma one_pos'' : (0 : Fin (n + 2)) < 1 := pos_of_coe_ne_zero (Nat.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_0e86813d27fd_0 | 9ec1645f9a6abd96 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Maximal/Unprovability.lean | Unprovability | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 2 | 1 | [
{
"theorem_name": "unprovable_AxiomL",
"depth": 1,
"n_commands": 0,
"n_lines": 5,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n apply not_imp_not.mpr provable_trivTranslated_Cl;\n apply Cl.not_provable_of_exists_valuation;\n use (λ _ => False);\n ... | [
{
"name": "provable_trivTranslated_Cl",
"text": "lemma provable_trivTranslated_Cl : Modal.K4 ⊢ φ → φᵀ.toPropFormula ∈ Propositional.Cl := by\n intro h;\n apply Triv.iff_provable_Cl.mp;\n apply WeakerThan.pbl h;\n\n",
"fan_in": 1,
"n_lines": 6,
"n_chars": 173,
"n_subproofs": 0,
"n_tact... | [
{
"name": "unprovable_AxiomL",
"fan_in": 0,
"n_deps_direct": 4,
"n_deps_transitive": 2,
"n_lines": 7,
"n_chars": 196,
"n_subproofs": 0,
"n_tactics": 9,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 3,
"automation_only": false,
"max_nesti... | 1 | module
public import Foundation.Modal.Maximal.Basic
@[expose] public section
namespace LO.Modal
variable {φ : Formula ℕ}
open LO.Entailment LO.Entailment.FiniteContext LO.Modal.Entailment
open Propositional
open Formula
open Hilbert
open Formula
namespace Triv
lemma unprovable_AxiomL : Modal.Triv ⊬ (Axioms.L (.a... | module
public import Foundation.Modal.Maximal.Basic
@[expose] public section
namespace LO.Modal
variable {φ : Formula ℕ}
open LO.Entailment LO.Entailment.FiniteContext LO.Modal.Entailment
open Propositional
open Formula
open Hilbert
open Formula
namespace Triv
lemma unprovable_AxiomL : Modal.Triv ⊬ (Axioms.L (.a... | @@ -30,8 +30,17 @@
namespace K4
-lemma unprovable_AxiomL : Modal.K4 ⊬ (Axioms.L (.atom a)) := sorry
+lemma provable_trivTranslated_Cl : Modal.K4 ⊢ φ → φᵀ.toPropFormula ∈ Propositional.Cl := by
+ intro h;
+ apply Triv.iff_provable_Cl.mp;
+ apply WeakerThan.pbl h;
+lemma unprovable_AxiomL : Modal.K4 ⊬ (Axioms.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_0e86813d27fd_1 | 26432ad9f3b08dd1 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Maximal/Unprovability.lean | Unprovability | 1 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 2 | 1 | [
{
"theorem_name": "unprovable_AxiomT",
"depth": 1,
"n_commands": 0,
"n_lines": 5,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n apply not_imp_not.mpr provable_verTranslated_Cl;\n apply Cl.not_provable_of_exists_valuation;\n use (λ _ => False);\n ... | [
{
"name": "provable_verTranslated_Cl",
"text": "lemma provable_verTranslated_Cl : Modal.GL ⊢ φ → φⱽ.toPropFormula ∈ Propositional.Cl := by\n rw [←Propositional.Hilbert.iff_mem_logic_provable];\n intro h;\n induction h using Hilbert.Normal.rec! with\n | axm _ a =>\n rcases a with (rfl | rfl | rfl)... | [
{
"name": "unprovable_AxiomT",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 8,
"n_chars": 212,
"n_subproofs": 0,
"n_tactics": 9,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 3,
"automation_only": false,
"max_nesti... | 1 | module
public import Foundation.Modal.Maximal.Basic
@[expose] public section
namespace LO.Modal
variable {φ : Formula ℕ}
open LO.Entailment LO.Entailment.FiniteContext LO.Modal.Entailment
open Propositional
open Formula
open Hilbert
open Formula
namespace Triv
lemma unprovable_AxiomL : Modal.Triv ⊬ (Axioms.L (.a... | module
public import Foundation.Modal.Maximal.Basic
@[expose] public section
namespace LO.Modal
variable {φ : Formula ℕ}
open LO.Entailment LO.Entailment.FiniteContext LO.Modal.Entailment
open Propositional
open Formula
open Hilbert
open Formula
namespace Triv
lemma unprovable_AxiomL : Modal.Triv ⊬ (Axioms.L (.a... | @@ -55,8 +55,23 @@
namespace GL
+lemma provable_verTranslated_Cl : Modal.GL ⊢ φ → φⱽ.toPropFormula ∈ Propositional.Cl := by
+ rw [←Propositional.Hilbert.iff_mem_logic_provable];
+ intro h;
+ induction h using Hilbert.Normal.rec! with
+ | axm _ a =>
+ rcases a with (rfl | rfl | rfl) <;> simp [verTranslat... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_0e86813d27fd_2 | 10b7cbb878059623 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Maximal/Unprovability.lean | Unprovability | 2 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 2 | 1 | [
{
"theorem_name": "unprovable_AxiomT",
"depth": 1,
"n_commands": 0,
"n_lines": 5,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n apply not_imp_not.mpr provable_verTranslated_Cl;\n apply Cl.not_provable_of_exists_valuation;\n use (λ _ => False);\n ... | [
{
"name": "provable_verTranslated_Cl",
"text": "lemma provable_verTranslated_Cl : Modal.GL ⊢ φ → φⱽ.toPropFormula ∈ Propositional.Cl := by\n rw [←Propositional.Hilbert.iff_mem_logic_provable];\n intro h;\n induction h using Hilbert.Normal.rec! with\n | axm _ a =>\n rcases a with (rfl | rfl | rfl)... | [
{
"name": "not_S4_weakerThan_GL",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 2,
"n_lines": 9,
"n_chars": 203,
"n_subproofs": 0,
"n_tactics": 11,
"cyclomatic": 2,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 5,
"automation_only": false,
"max_n... | 1 | module
public import Foundation.Modal.Maximal.Basic
@[expose] public section
namespace LO.Modal
variable {φ : Formula ℕ}
open LO.Entailment LO.Entailment.FiniteContext LO.Modal.Entailment
open Propositional
open Formula
open Hilbert
open Formula
namespace Triv
lemma unprovable_AxiomL : Modal.Triv ⊬ (Axioms.L (.a... | module
public import Foundation.Modal.Maximal.Basic
@[expose] public section
namespace LO.Modal
variable {φ : Formula ℕ}
open LO.Entailment LO.Entailment.FiniteContext LO.Modal.Entailment
open Propositional
open Formula
open Hilbert
open Formula
namespace Triv
lemma unprovable_AxiomL : Modal.Triv ⊬ (Axioms.L (.a... | @@ -55,8 +55,23 @@
namespace GL
+lemma provable_verTranslated_Cl : Modal.GL ⊢ φ → φⱽ.toPropFormula ∈ Propositional.Cl := by
+ rw [←Propositional.Hilbert.iff_mem_logic_provable];
+ intro h;
+ induction h using Hilbert.Normal.rec! with
+ | axm _ a =>
+ rcases a with (rfl | rfl | rfl) <;> simp [verTranslat... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_d24895e9760d_0 | 752c8d6ebded5fd9 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Interpretation.lean | Interpretation | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 9 | 2 | [
{
"theorem_name": "func_existsUnique",
"depth": 1,
"n_commands": 0,
"n_lines": 8,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n have : ∃! y, π.Dom y ∧ M ⊧/(y :> fun i ↦ v i) (π.func f) :=\n func_existsUnique_on_dom (π := π) f (fun i ↦ (v i : M)) ... | [
{
"name": "func_existsUnique_on_dom",
"text": "lemma func_existsUnique_on_dom {k} (f : L₂.Func k) :\n ∀ (v : Fin k → M), (∀ i, π.Dom (v i)) → ∃! y, π.Dom y ∧ (π.func f).Evalb (y :> v) := by\n simpa [Dom, models_iff, eval_substs, Matrix.constant_eq_singleton] using\n models_of_provable (M := M) inferI... | [
{
"name": "func_existsUnique",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 11,
"n_chars": 507,
"n_subproofs": 1,
"n_tactics": 8,
"cyclomatic": 3,
"n_automation": 3,
"n_rewrites": 1,
"n_structural": 3,
"automation_only": false,
"max_nest... | 1 | module
public import Foundation.FirstOrder.Completeness.CounterModel
public import Foundation.Vorspiel.ExistsUnique
@[expose] public section
/-!
# (Direct) Interpretation
-/
namespace LO.FirstOrder
@[ext]
structure DirectTranslation {L₁ : Language} [L₁.Eq] (T : Theory L₁) [𝗘𝗤 _ ⪯ T] (L₂ : Language) [L₂.Eq] where
... | module
public import Foundation.FirstOrder.Completeness.CounterModel
public import Foundation.Vorspiel.ExistsUnique
@[expose] public section
/-!
# (Direct) Interpretation
-/
namespace LO.FirstOrder
@[ext]
structure DirectTranslation {L₁ : Language} [L₁.Eq] (T : Theory L₁) [𝗘𝗤 _ ⪯ T] (L₂ : Language) [L₂.Eq] where
... | @@ -163,9 +163,15 @@
variable [Nonempty M] [M↓[L₁] ⊧* T] [Structure.Eq L₁ M]
+lemma func_existsUnique_on_dom {k} (f : L₂.Func k) :
+ ∀ (v : Fin k → M), (∀ i, π.Dom (v i)) → ∃! y, π.Dom y ∧ (π.func f).Evalb (y :> v) := by
+ simpa [Dom, models_iff, eval_substs, Matrix.constant_eq_singleton] using
+ models_of_... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_d24895e9760d_1 | 433706f0ea59e29e | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Interpretation.lean | Interpretation | 1 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 9 | 4 | [
{
"theorem_name": "eq_iff",
"depth": 1,
"n_commands": 0,
"n_lines": 3,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n simp [Model.rel_iff]\n simp [Matrix.fun_eq_vec_two (v := fun i ↦ (v i : M))]",
"n_chars": 84,
"n_subproofs": 0,
"n_tact... | [
{
"name": "rel_iff",
"text": "lemma rel_iff {k} (r : L₂.Rel k) (v : Fin k → π.Model M) :\n Structure.rel r v ↔ (π.rel r).Evalb (fun i ↦ (v i : M)) := iff_of_eq rfl\n\n",
"fan_in": 4,
"n_lines": 4,
"n_chars": 147,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation... | [
{
"name": "eq_iff",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 6,
"n_chars": 202,
"n_subproofs": 0,
"n_tactics": 3,
"cyclomatic": 1,
"n_automation": 2,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"max_nesting": 2
}
] | 1 | module
public import Foundation.FirstOrder.Completeness.CounterModel
public import Foundation.Vorspiel.ExistsUnique
@[expose] public section
/-!
# (Direct) Interpretation
-/
namespace LO.FirstOrder
@[ext]
structure DirectTranslation {L₁ : Language} [L₁.Eq] (T : Theory L₁) [𝗘𝗤 _ ⪯ T] (L₂ : Language) [L₂.Eq] where
... | module
public import Foundation.FirstOrder.Completeness.CounterModel
public import Foundation.Vorspiel.ExistsUnique
@[expose] public section
/-!
# (Direct) Interpretation
-/
namespace LO.FirstOrder
@[ext]
structure DirectTranslation {L₁ : Language} [L₁.Eq] (T : Theory L₁) [𝗘𝗤 _ ⪯ T] (L₂ : Language) [L₂.Eq] where
... | @@ -186,6 +186,9 @@
variable {π M}
+lemma rel_iff {k} (r : L₂.Rel k) (v : Fin k → π.Model M) :
+ Structure.rel r v ↔ (π.rel r).Evalb (fun i ↦ (v i : M)) := iff_of_eq rfl
+
@[simp] lemma eq_iff' {a b : π.Model M} :
M ⊧/![a, b] (π.rel Language.Eq.eq) ↔ a = b := by
have : M↓[L₁] ⊧* T := inferInstance
@@ -... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_7371a5fa1e89_0 | 945afaeed5343965 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Bootstrapping/Syntax/Formula/Coding.lean | Coding | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 2 | 1 | [
{
"theorem_name": "typed_quote_inj",
"depth": 1,
"n_commands": 0,
"n_lines": 49,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": "\n match φ₁, φ₂ with\n | rel R₁ v₁, rel R₂ v₂ => by\n simp only [typed_quote_rel, Bootstrapping.Semiformula.rel, Semiformula... | [
{
"name": "ne_iff_val_ne",
"text": "lemma ne_iff_val_ne (φ ψ : Bootstrapping.Semiformula V L n) : φ ≠ ψ ↔ φ.val ≠ ψ.val := Iff.ne Semiformula.ext_iff\n\n",
"fan_in": 1,
"n_lines": 3,
"n_chars": 127,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewr... | [
{
"name": "typed_quote_inj",
"fan_in": 1,
"n_deps_direct": 3,
"n_deps_transitive": 3,
"n_lines": 51,
"n_chars": 2997,
"n_subproofs": 2,
"n_tactics": 48,
"cyclomatic": 6,
"n_automation": 17,
"n_rewrites": 0,
"n_structural": 16,
"automation_only": false,
"max_ne... | 1 | module
public import Foundation.FirstOrder.Bootstrapping.Syntax.Formula.Typed
public import Foundation.FirstOrder.Bootstrapping.Syntax.Term.Coding
@[expose] public section
open Encodable LO FirstOrder Arithmetic Bootstrapping
namespace LO
class LCWQIsoGödelQuote (α β : ℕ → Type*) [LCWQ α] [LCWQ β] where
gq : ∀ n,... | module
public import Foundation.FirstOrder.Bootstrapping.Syntax.Formula.Typed
public import Foundation.FirstOrder.Bootstrapping.Syntax.Term.Coding
@[expose] public section
open Encodable LO FirstOrder Arithmetic Bootstrapping
namespace LO
class LCWQIsoGödelQuote (α β : ℕ → Type*) [LCWQ α] [LCWQ β] where
gq : ∀ n,... | @@ -134,6 +134,8 @@
@[simp] lemma typed_quote_nlt (t u : SyntacticSemiterm ℒₒᵣ n) :
(⌜(“!!t ≮ !!u” : ArithmeticSemiproposition n)⌝ : Bootstrapping.Semiformula V ℒₒᵣ n) = (⌜t⌝ ≮' ⌜u⌝) := rfl
+lemma ne_iff_val_ne (φ ψ : Bootstrapping.Semiformula V L n) : φ ≠ ψ ↔ φ.val ≠ ψ.val := Iff.ne Semiformula.ext_iff
+
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_1c61a30da128_0 | 8ea2a78dc52bdc91 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Modality/S5.lean | S5 | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 4 | 2 | [
{
"theorem_name": "modal_reduction_3",
"depth": 1,
"n_commands": 0,
"n_lines": 40,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n intro m hm;\n rcases split_left₁' hm with ⟨m₂, hm₂, (rfl | rfl | rfl)⟩;\n . obtain ⟨m', _, _⟩ := modal_reduction_2 m₂ ... | [
{
"name": "modal_reduction_2",
"text": "lemma modal_reduction_2 : ModalReduction Modal.S5 2 S5.modalities := by\n apply ModalReduction.of_allOfSize;\n intro m hm;\n simp only [\n allOfSize, Finset.image_singleton, Finset.union_assoc, Finset.mem_union,\n Finset.mem_image, Finset.mem_singleton, exist... | [
{
"name": "modal_reduction_3",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 42,
"n_chars": 1492,
"n_subproofs": 3,
"n_tactics": 79,
"cyclomatic": 13,
"n_automation": 31,
"n_rewrites": 0,
"n_structural": 21,
"automation_only": false,
"max... | 1 | module
public import Foundation.Modal.Modality.Basic
public import Foundation.Modal.Kripke.Logic.S5
@[expose] public section
namespace LO.Modal
namespace S5
set_option linter.style.multiGoal false
variable {m m₁ m₂ : Modality} {n}
open Entailment
open Modality
open Formula
open Kripke
open Modalities
protected ... | module
public import Foundation.Modal.Modality.Basic
public import Foundation.Modal.Kripke.Logic.S5
@[expose] public section
namespace LO.Modal
namespace S5
set_option linter.style.multiGoal false
variable {m m₁ m₂ : Modality} {n}
open Entailment
open Modality
open Formula
open Kripke
open Modalities
protected ... | @@ -54,6 +54,24 @@
. apply translation_of_axiomInstance (a := 0); simp;
. apply translation_of_axiomInstance (a := 0); simp;
+lemma modal_reduction_2 : ModalReduction Modal.S5 2 S5.modalities := by
+ apply ModalReduction.of_allOfSize;
+ intro m hm;
+ simp only [
+ allOfSize, Finset.image_singleton, Finset... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_1c61a30da128_1 | 80bbcb88ffd0a7d1 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Modality/S5.lean | S5 | 1 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 4 | 1 | [
{
"theorem_name": "modal_reduction",
"depth": 1,
"n_commands": 0,
"n_lines": 7,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " ModalReduction.forall_of_reducible_to_max (by simp) $ by\n intro n hn;\n match n with\n | 0 => exact modal_reduction_0;\n | 1... | [
{
"name": "modal_reduction_0",
"text": "lemma modal_reduction_0 : ModalReduction Modal.S5 0 Modal.S5.modalities := ModalReduction.reducible_0_of_mem $ by simp;\n\n",
"fan_in": 1,
"n_lines": 3,
"n_chars": 121,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
"n_automation": 1,
... | [
{
"name": "modal_reduction",
"fan_in": 0,
"n_deps_direct": 4,
"n_deps_transitive": 4,
"n_lines": 9,
"n_chars": 299,
"n_subproofs": 0,
"n_tactics": 12,
"cyclomatic": 2,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 5,
"automation_only": false,
"max_nestin... | 4 | module
public import Foundation.Modal.Modality.Basic
public import Foundation.Modal.Kripke.Logic.S5
@[expose] public section
namespace LO.Modal
namespace S5
set_option linter.style.multiGoal false
variable {m m₁ m₂ : Modality} {n}
open Entailment
open Modality
open Formula
open Kripke
open Modalities
protected ... | module
public import Foundation.Modal.Modality.Basic
public import Foundation.Modal.Kripke.Logic.S5
@[expose] public section
namespace LO.Modal
namespace S5
set_option linter.style.multiGoal false
variable {m m₁ m₂ : Modality} {n}
open Entailment
open Modality
open Formula
open Kripke
open Modalities
protected ... | @@ -21,6 +21,8 @@
protected abbrev modalities : Modalities := {-, ∼, □, ◇, ∼□, ∼◇}
+lemma modal_reduction_0 : ModalReduction Modal.S5 0 Modal.S5.modalities := ModalReduction.reducible_0_of_mem $ by simp;
+
lemma modal_reduction_1 : ModalReduction Modal.S5 1 Modal.S5.modalities := ModalReduction.reducible_1_of_mem... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_76e907bea2f3_0 | 0d813a71ae9d12f3 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Omega1/Basic.lean | Basic | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 7 | 2 | [
{
"theorem_name": "omega1_of_ISigma1",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " models_Omega1_iff.mpr (fun x ↦ Exponential.range_exists (‖x‖^2))",
"n_chars": 71,
"n_subproofs": 0,
"n_tactics": 1,... | [
{
"name": "models_Omega1_iff",
"text": "lemma models_Omega1_iff [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₀] : V↓[ℒₒᵣ] ⊧ Omega1.omega1 ↔ ∀ x : V, ∃ y, Exponential (‖x‖^2) y := by\n simp [models_iff, Omega1.omega1, sq]\n\n",
"fan_in": 2,
"n_lines": 4,
"n_chars": 189,
"n_subproofs": 0,
"n_tactics": 2,
"cycloma... | [
{
"name": "omega1_of_ISigma1",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 3,
"n_chars": 170,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
"max_nesti... | 1 | module
public import Foundation.FirstOrder.Arithmetic.Exponential
@[expose] public section
/-!
# Theory $\mathsf{I}\Sigma_0 + \Omega_1$
-/
namespace LO.FirstOrder.Arithmetic
/-- ∀ x, ∃ y, 2^{|x|^2} = y-/
def _root_.LO.Omega1.omega1 : ArithmeticSentence := “∀ x, ∃ y, ∃ l <⁺ x, !lengthDef l x ∧ !exponentialDef (l * ... | module
public import Foundation.FirstOrder.Arithmetic.Exponential
@[expose] public section
/-!
# Theory $\mathsf{I}\Sigma_0 + \Omega_1$
-/
namespace LO.FirstOrder.Arithmetic
/-- ∀ x, ∃ y, 2^{|x|^2} = y-/
def _root_.LO.Omega1.omega1 : ArithmeticSentence := “∀ x, ∃ y, ∃ l <⁺ x, !lengthDef l x ∧ !exponentialDef (l * ... | @@ -25,8 +25,11 @@
variable {V : Type*} [ORingStructure V]
-lemma omega1_of_ISigma1 [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁] : V↓[ℒₒᵣ] ⊧ Omega1.omega1 := sorry
+lemma models_Omega1_iff [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₀] : V↓[ℒₒᵣ] ⊧ Omega1.omega1 ↔ ∀ x : V, ∃ y, Exponential (‖x‖^2) y := by
+ simp [models_iff, Omega1.omega1, sq]
+lemma omega1_of_ISig... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_76e907bea2f3_1 | a75591fa52c1ba8d | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Omega1/Basic.lean | Basic | 1 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 7 | 2 | [
{
"theorem_name": "exists_unique_exponential_sq_length",
"depth": 1,
"n_commands": 0,
"n_lines": 3,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n rcases exists_exponential_sq_length x with ⟨y, h⟩\n exact ExistsUnique.intro y h (fun y' h' ↦ h'.uniq ... | [
{
"name": "exists_exponential_sq_length",
"text": "lemma exists_exponential_sq_length (x : V) : ∃ y, Exponential (‖x‖^2) y :=\n models_Omega1_iff.mp (Theory.models _ _ Omega1.omega) x\n\n",
"fan_in": 2,
"n_lines": 4,
"n_chars": 140,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,... | [
{
"name": "exists_unique_exponential_sq_length",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 2,
"n_lines": 5,
"n_chars": 206,
"n_subproofs": 0,
"n_tactics": 3,
"cyclomatic": 2,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 2,
"automation_only": fal... | 2 | module
public import Foundation.FirstOrder.Arithmetic.Exponential
@[expose] public section
/-!
# Theory $\mathsf{I}\Sigma_0 + \Omega_1$
-/
namespace LO.FirstOrder.Arithmetic
/-- ∀ x, ∃ y, 2^{|x|^2} = y-/
def _root_.LO.Omega1.omega1 : ArithmeticSentence := “∀ x, ∃ y, ∃ l <⁺ x, !lengthDef l x ∧ !exponentialDef (l * ... | module
public import Foundation.FirstOrder.Arithmetic.Exponential
@[expose] public section
/-!
# Theory $\mathsf{I}\Sigma_0 + \Omega_1$
-/
namespace LO.FirstOrder.Arithmetic
/-- ∀ x, ∃ y, 2^{|x|^2} = y-/
def _root_.LO.Omega1.omega1 : ArithmeticSentence := “∀ x, ∃ y, ∃ l <⁺ x, !lengthDef l x ∧ !exponentialDef (l * ... | @@ -40,10 +40,22 @@
instance : V↓[ℒₒᵣ] ⊧* 𝝮₁ := ModelsTheory.of_add_right V 𝗜𝚺₀ 𝝮₁
-lemma exists_unique_exponential_sq_length (x : V) : ∃! y, Exponential (‖x‖^2) y := sorry
+lemma exists_exponential_sq_length (x : V) : ∃ y, Exponential (‖x‖^2) y :=
+ models_Omega1_iff.mp (Theory.models _ _ Omega1.omega) 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_76e907bea2f3_2 | 2c4c466d233cdb69 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Omega1/Basic.lean | Basic | 2 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 7 | 2 | [
{
"theorem_name": "exists_unique_exponential_sq_length",
"depth": 1,
"n_commands": 0,
"n_lines": 3,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n rcases exists_exponential_sq_length x with ⟨y, h⟩\n exact ExistsUnique.intro y h (fun y' h' ↦ h'.uniq ... | [
{
"name": "exists_exponential_sq_length",
"text": "lemma exists_exponential_sq_length (x : V) : ∃ y, Exponential (‖x‖^2) y :=\n models_Omega1_iff.mp (Theory.models _ _ Omega1.omega) x\n\n",
"fan_in": 2,
"n_lines": 4,
"n_chars": 140,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,... | [
{
"name": "smash_exists_unique",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 2,
"n_lines": 10,
"n_chars": 566,
"n_subproofs": 2,
"n_tactics": 8,
"cyclomatic": 2,
"n_automation": 2,
"n_rewrites": 0,
"n_structural": 2,
"automation_only": false,
"max_ne... | 2 | module
public import Foundation.FirstOrder.Arithmetic.Exponential
@[expose] public section
/-!
# Theory $\mathsf{I}\Sigma_0 + \Omega_1$
-/
namespace LO.FirstOrder.Arithmetic
/-- ∀ x, ∃ y, 2^{|x|^2} = y-/
def _root_.LO.Omega1.omega1 : ArithmeticSentence := “∀ x, ∃ y, ∃ l <⁺ x, !lengthDef l x ∧ !exponentialDef (l * ... | module
public import Foundation.FirstOrder.Arithmetic.Exponential
@[expose] public section
/-!
# Theory $\mathsf{I}\Sigma_0 + \Omega_1$
-/
namespace LO.FirstOrder.Arithmetic
/-- ∀ x, ∃ y, 2^{|x|^2} = y-/
def _root_.LO.Omega1.omega1 : ArithmeticSentence := “∀ x, ∃ y, ∃ l <⁺ x, !lengthDef l x ∧ !exponentialDef (l * ... | @@ -40,8 +40,18 @@
instance : V↓[ℒₒᵣ] ⊧* 𝝮₁ := ModelsTheory.of_add_right V 𝗜𝚺₀ 𝝮₁
-lemma smash_exists_unique (x y : V) : ∃! z, Exponential (‖x‖ * ‖y‖) z := sorry
+lemma exists_exponential_sq_length (x : V) : ∃ y, Exponential (‖x‖^2) y :=
+ models_Omega1_iff.mp (Theory.models _ _ Omega1.omega) x
+lemma smash... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_76e907bea2f3_3 | 65b989fffdf8f98e | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Omega1/Basic.lean | Basic | 3 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 7 | 2 | [
{
"theorem_name": "exists_unique_exponential_sq_length",
"depth": 1,
"n_commands": 0,
"n_lines": 3,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n rcases exists_exponential_sq_length x with ⟨y, h⟩\n exact ExistsUnique.intro y h (fun y' h' ↦ h'.uniq ... | [
{
"name": "exists_exponential_sq_length",
"text": "lemma exists_exponential_sq_length (x : V) : ∃ y, Exponential (‖x‖^2) y :=\n models_Omega1_iff.mp (Theory.models _ _ Omega1.omega) x\n\n",
"fan_in": 2,
"n_lines": 4,
"n_chars": 140,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,... | [
{
"name": "exponential_smash_one",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 4,
"n_lines": 3,
"n_chars": 109,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"max_ne... | 2 | module
public import Foundation.FirstOrder.Arithmetic.Exponential
@[expose] public section
/-!
# Theory $\mathsf{I}\Sigma_0 + \Omega_1$
-/
namespace LO.FirstOrder.Arithmetic
/-- ∀ x, ∃ y, 2^{|x|^2} = y-/
def _root_.LO.Omega1.omega1 : ArithmeticSentence := “∀ x, ∃ y, ∃ l <⁺ x, !lengthDef l x ∧ !exponentialDef (l * ... | module
public import Foundation.FirstOrder.Arithmetic.Exponential
@[expose] public section
/-!
# Theory $\mathsf{I}\Sigma_0 + \Omega_1$
-/
namespace LO.FirstOrder.Arithmetic
/-- ∀ x, ∃ y, 2^{|x|^2} = y-/
def _root_.LO.Omega1.omega1 : ArithmeticSentence := “∀ x, ∃ y, ∃ l <⁺ x, !lengthDef l x ∧ !exponentialDef (l * ... | @@ -40,8 +40,18 @@
instance : V↓[ℒₒᵣ] ⊧* 𝝮₁ := ModelsTheory.of_add_right V 𝗜𝚺₀ 𝝮₁
-lemma smash_exists_unique (x y : V) : ∃! z, Exponential (‖x‖ * ‖y‖) z := sorry
+lemma exists_exponential_sq_length (x : V) : ∃ y, Exponential (‖x‖^2) y :=
+ models_Omega1_iff.mp (Theory.models _ _ Omega1.omega) x
+lemma smash... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_76e907bea2f3_4 | 72314f02e9cde50f | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Omega1/Basic.lean | Basic | 4 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 7 | 2 | [
{
"theorem_name": "exists_unique_exponential_sq_length",
"depth": 1,
"n_commands": 0,
"n_lines": 3,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n rcases exists_exponential_sq_length x with ⟨y, h⟩\n exact ExistsUnique.intro y h (fun y' h' ↦ h'.uniq ... | [
{
"name": "exists_exponential_sq_length",
"text": "lemma exists_exponential_sq_length (x : V) : ∃ y, Exponential (‖x‖^2) y :=\n models_Omega1_iff.mp (Theory.models _ _ Omega1.omega) x\n\n",
"fan_in": 2,
"n_lines": 4,
"n_chars": 140,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,... | [
{
"name": "length_smash",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 4,
"n_lines": 3,
"n_chars": 107,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
"max_nesting": ... | 2 | module
public import Foundation.FirstOrder.Arithmetic.Exponential
@[expose] public section
/-!
# Theory $\mathsf{I}\Sigma_0 + \Omega_1$
-/
namespace LO.FirstOrder.Arithmetic
/-- ∀ x, ∃ y, 2^{|x|^2} = y-/
def _root_.LO.Omega1.omega1 : ArithmeticSentence := “∀ x, ∃ y, ∃ l <⁺ x, !lengthDef l x ∧ !exponentialDef (l * ... | module
public import Foundation.FirstOrder.Arithmetic.Exponential
@[expose] public section
/-!
# Theory $\mathsf{I}\Sigma_0 + \Omega_1$
-/
namespace LO.FirstOrder.Arithmetic
/-- ∀ x, ∃ y, 2^{|x|^2} = y-/
def _root_.LO.Omega1.omega1 : ArithmeticSentence := “∀ x, ∃ y, ∃ l <⁺ x, !lengthDef l x ∧ !exponentialDef (l * ... | @@ -40,8 +40,18 @@
instance : V↓[ℒₒᵣ] ⊧* 𝝮₁ := ModelsTheory.of_add_right V 𝗜𝚺₀ 𝝮₁
-lemma smash_exists_unique (x y : V) : ∃! z, Exponential (‖x‖ * ‖y‖) z := sorry
+lemma exists_exponential_sq_length (x : V) : ∃ y, Exponential (‖x‖^2) y :=
+ models_Omega1_iff.mp (Theory.models _ _ Omega1.omega) x
+lemma smash... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_76e907bea2f3_5 | 03b36e0b17b0719e | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Omega1/Basic.lean | Basic | 5 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 7 | 2 | [
{
"theorem_name": "exists_unique_exponential_sq_length",
"depth": 1,
"n_commands": 0,
"n_lines": 3,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n rcases exists_exponential_sq_length x with ⟨y, h⟩\n exact ExistsUnique.intro y h (fun y' h' ↦ h'.uniq ... | [
{
"name": "exists_exponential_sq_length",
"text": "lemma exists_exponential_sq_length (x : V) : ∃ y, Exponential (‖x‖^2) y :=\n models_Omega1_iff.mp (Theory.models _ _ Omega1.omega) x\n\n",
"fan_in": 2,
"n_lines": 4,
"n_chars": 140,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,... | [
{
"name": "smash_comm",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 4,
"n_lines": 3,
"n_chars": 130,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"max_nesting": 1
... | 2 | module
public import Foundation.FirstOrder.Arithmetic.Exponential
@[expose] public section
/-!
# Theory $\mathsf{I}\Sigma_0 + \Omega_1$
-/
namespace LO.FirstOrder.Arithmetic
/-- ∀ x, ∃ y, 2^{|x|^2} = y-/
def _root_.LO.Omega1.omega1 : ArithmeticSentence := “∀ x, ∃ y, ∃ l <⁺ x, !lengthDef l x ∧ !exponentialDef (l * ... | module
public import Foundation.FirstOrder.Arithmetic.Exponential
@[expose] public section
/-!
# Theory $\mathsf{I}\Sigma_0 + \Omega_1$
-/
namespace LO.FirstOrder.Arithmetic
/-- ∀ x, ∃ y, 2^{|x|^2} = y-/
def _root_.LO.Omega1.omega1 : ArithmeticSentence := “∀ x, ∃ y, ∃ l <⁺ x, !lengthDef l x ∧ !exponentialDef (l * ... | @@ -40,8 +40,18 @@
instance : V↓[ℒₒᵣ] ⊧* 𝝮₁ := ModelsTheory.of_add_right V 𝗜𝚺₀ 𝝮₁
-lemma smash_exists_unique (x y : V) : ∃! z, Exponential (‖x‖ * ‖y‖) z := sorry
+lemma exists_exponential_sq_length (x : V) : ∃ y, Exponential (‖x‖^2) y :=
+ models_Omega1_iff.mp (Theory.models _ _ Omega1.omega) x
+lemma smash... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_76e907bea2f3_6 | 5acf61d515483551 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Omega1/Basic.lean | Basic | 6 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 7 | 2 | [
{
"theorem_name": "exists_unique_exponential_sq_length",
"depth": 1,
"n_commands": 0,
"n_lines": 3,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n rcases exists_exponential_sq_length x with ⟨y, h⟩\n exact ExistsUnique.intro y h (fun y' h' ↦ h'.uniq ... | [
{
"name": "exists_exponential_sq_length",
"text": "lemma exists_exponential_sq_length (x : V) : ∃ y, Exponential (‖x‖^2) y :=\n models_Omega1_iff.mp (Theory.models _ _ Omega1.omega) x\n\n",
"fan_in": 2,
"n_lines": 4,
"n_chars": 140,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,... | [
{
"name": "lt_smash_iff",
"fan_in": 2,
"n_deps_direct": 1,
"n_deps_transitive": 4,
"n_lines": 3,
"n_chars": 121,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
"max_nesting": ... | 2 | module
public import Foundation.FirstOrder.Arithmetic.Exponential
@[expose] public section
/-!
# Theory $\mathsf{I}\Sigma_0 + \Omega_1$
-/
namespace LO.FirstOrder.Arithmetic
/-- ∀ x, ∃ y, 2^{|x|^2} = y-/
def _root_.LO.Omega1.omega1 : ArithmeticSentence := “∀ x, ∃ y, ∃ l <⁺ x, !lengthDef l x ∧ !exponentialDef (l * ... | module
public import Foundation.FirstOrder.Arithmetic.Exponential
@[expose] public section
/-!
# Theory $\mathsf{I}\Sigma_0 + \Omega_1$
-/
namespace LO.FirstOrder.Arithmetic
/-- ∀ x, ∃ y, 2^{|x|^2} = y-/
def _root_.LO.Omega1.omega1 : ArithmeticSentence := “∀ x, ∃ y, ∃ l <⁺ x, !lengthDef l x ∧ !exponentialDef (l * ... | @@ -40,8 +40,18 @@
instance : V↓[ℒₒᵣ] ⊧* 𝝮₁ := ModelsTheory.of_add_right V 𝗜𝚺₀ 𝝮₁
-lemma smash_exists_unique (x y : V) : ∃! z, Exponential (‖x‖ * ‖y‖) z := sorry
+lemma exists_exponential_sq_length (x : V) : ∃ y, Exponential (‖x‖^2) y :=
+ models_Omega1_iff.mp (Theory.models _ _ Omega1.omega) x
+lemma smash... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_76e907bea2f3_7 | 0aec82fe1f0da847 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Omega1/Basic.lean | Basic | 7 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 7 | 2 | [
{
"theorem_name": "smash_le_iff",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": "\n not_iff_not.mp <| by simp [lt_smash_iff]",
"n_chars": 43,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
... | [
{
"name": "lt_smash_iff",
"text": "lemma lt_smash_iff {a b c : V} : a < b ⨳ c ↔ ‖a‖ ≤ ‖b‖ * ‖c‖ := (exponential_smash b c).lt_iff_len_le\n\n",
"fan_in": 2,
"n_lines": 3,
"n_chars": 121,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
... | [
{
"name": "smash_le_iff",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 5,
"n_lines": 4,
"n_chars": 126,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"max_nesting": 2... | 3 | module
public import Foundation.FirstOrder.Arithmetic.Exponential
@[expose] public section
/-!
# Theory $\mathsf{I}\Sigma_0 + \Omega_1$
-/
namespace LO.FirstOrder.Arithmetic
/-- ∀ x, ∃ y, 2^{|x|^2} = y-/
def _root_.LO.Omega1.omega1 : ArithmeticSentence := “∀ x, ∃ y, ∃ l <⁺ x, !lengthDef l x ∧ !exponentialDef (l * ... | module
public import Foundation.FirstOrder.Arithmetic.Exponential
@[expose] public section
/-!
# Theory $\mathsf{I}\Sigma_0 + \Omega_1$
-/
namespace LO.FirstOrder.Arithmetic
/-- ∀ x, ∃ y, 2^{|x|^2} = y-/
def _root_.LO.Omega1.omega1 : ArithmeticSentence := “∀ x, ∃ y, ∃ l <⁺ x, !lengthDef l x ∧ !exponentialDef (l * ... | @@ -91,8 +91,11 @@
simpa using length_mul_pow2_add_of_lt pos (show Pow2 2 from by simp) one_lt_two
simp [this])
-lemma smash_le_iff {a b c : V} : b ⨳ c ≤ a ↔ ‖b‖ * ‖c‖ < ‖a‖ := sorry
+lemma lt_smash_iff {a b c : V} : a < b ⨳ c ↔ ‖a‖ ≤ ‖b‖ * ‖c‖ := (exponential_smash b c).lt_iff_len_le
+lemma smash_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_76e907bea2f3_8 | 17597b0c3f303356 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Omega1/Basic.lean | Basic | 8 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 7 | 2 | [
{
"theorem_name": "smash_le_iff",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": "\n not_iff_not.mp <| by simp [lt_smash_iff]",
"n_chars": 43,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
... | [
{
"name": "lt_smash_iff",
"text": "lemma lt_smash_iff {a b c : V} : a < b ⨳ c ↔ ‖a‖ ≤ ‖b‖ * ‖c‖ := (exponential_smash b c).lt_iff_len_le\n\n",
"fan_in": 2,
"n_lines": 3,
"n_chars": 121,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
... | [
{
"name": "lt_smash_one_iff",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 5,
"n_lines": 3,
"n_chars": 130,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"max_nesting... | 3 | module
public import Foundation.FirstOrder.Arithmetic.Exponential
@[expose] public section
/-!
# Theory $\mathsf{I}\Sigma_0 + \Omega_1$
-/
namespace LO.FirstOrder.Arithmetic
/-- ∀ x, ∃ y, 2^{|x|^2} = y-/
def _root_.LO.Omega1.omega1 : ArithmeticSentence := “∀ x, ∃ y, ∃ l <⁺ x, !lengthDef l x ∧ !exponentialDef (l * ... | module
public import Foundation.FirstOrder.Arithmetic.Exponential
@[expose] public section
/-!
# Theory $\mathsf{I}\Sigma_0 + \Omega_1$
-/
namespace LO.FirstOrder.Arithmetic
/-- ∀ x, ∃ y, 2^{|x|^2} = y-/
def _root_.LO.Omega1.omega1 : ArithmeticSentence := “∀ x, ∃ y, ∃ l <⁺ x, !lengthDef l x ∧ !exponentialDef (l * ... | @@ -91,8 +91,10 @@
simpa using length_mul_pow2_add_of_lt pos (show Pow2 2 from by simp) one_lt_two
simp [this])
-lemma lt_smash_one_iff {a b : V} : a < b ⨳ 1 ↔ ‖a‖ ≤ ‖b‖ := sorry
+lemma lt_smash_iff {a b c : V} : a < b ⨳ c ↔ ‖a‖ ≤ ‖b‖ * ‖c‖ := (exponential_smash b c).lt_iff_len_le
+lemma lt_smash_on... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_76e907bea2f3_9 | 9898bd001e4b5162 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Omega1/Basic.lean | Basic | 9 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 7 | 2 | [
{
"theorem_name": "exists_unique_exponential_sq_length",
"depth": 1,
"n_commands": 0,
"n_lines": 3,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n rcases exists_exponential_sq_length x with ⟨y, h⟩\n exact ExistsUnique.intro y h (fun y' h' ↦ h'.uniq ... | [
{
"name": "exists_exponential_sq_length",
"text": "lemma exists_exponential_sq_length (x : V) : ∃ y, Exponential (‖x‖^2) y :=\n models_Omega1_iff.mp (Theory.models _ _ Omega1.omega) x\n\n",
"fan_in": 2,
"n_lines": 4,
"n_chars": 140,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,... | [
{
"name": "smash_monotone",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 4,
"n_lines": 4,
"n_chars": 284,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 2,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"max_nesting":... | 2 | module
public import Foundation.FirstOrder.Arithmetic.Exponential
@[expose] public section
/-!
# Theory $\mathsf{I}\Sigma_0 + \Omega_1$
-/
namespace LO.FirstOrder.Arithmetic
/-- ∀ x, ∃ y, 2^{|x|^2} = y-/
def _root_.LO.Omega1.omega1 : ArithmeticSentence := “∀ x, ∃ y, ∃ l <⁺ x, !lengthDef l x ∧ !exponentialDef (l * ... | module
public import Foundation.FirstOrder.Arithmetic.Exponential
@[expose] public section
/-!
# Theory $\mathsf{I}\Sigma_0 + \Omega_1$
-/
namespace LO.FirstOrder.Arithmetic
/-- ∀ x, ∃ y, 2^{|x|^2} = y-/
def _root_.LO.Omega1.omega1 : ArithmeticSentence := “∀ x, ∃ y, ∃ l <⁺ x, !lengthDef l x ∧ !exponentialDef (l * ... | @@ -40,8 +40,18 @@
instance : V↓[ℒₒᵣ] ⊧* 𝝮₁ := ModelsTheory.of_add_right V 𝗜𝚺₀ 𝝮₁
-lemma smash_exists_unique (x y : V) : ∃! z, Exponential (‖x‖ * ‖y‖) z := sorry
+lemma exists_exponential_sq_length (x : V) : ∃ y, Exponential (‖x‖^2) y :=
+ models_Omega1_iff.mp (Theory.models _ _ Omega1.omega) x
+lemma smash... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_76e907bea2f3_10 | d4ebf1640b7eb8a3 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Omega1/Basic.lean | Basic | 10 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 7 | 2 | [
{
"theorem_name": "exists_unique_exponential_sq_length",
"depth": 1,
"n_commands": 0,
"n_lines": 3,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n rcases exists_exponential_sq_length x with ⟨y, h⟩\n exact ExistsUnique.intro y h (fun y' h' ↦ h'.uniq ... | [
{
"name": "exists_exponential_sq_length",
"text": "lemma exists_exponential_sq_length (x : V) : ∃ y, Exponential (‖x‖^2) y :=\n models_Omega1_iff.mp (Theory.models _ _ Omega1.omega) x\n\n",
"fan_in": 2,
"n_lines": 4,
"n_chars": 140,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,... | [
{
"name": "bexp_eq_smash",
"fan_in": 0,
"n_deps_direct": 2,
"n_deps_transitive": 5,
"n_lines": 3,
"n_chars": 145,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"max_nesting": ... | 3 | module
public import Foundation.FirstOrder.Arithmetic.Exponential
@[expose] public section
/-!
# Theory $\mathsf{I}\Sigma_0 + \Omega_1$
-/
namespace LO.FirstOrder.Arithmetic
/-- ∀ x, ∃ y, 2^{|x|^2} = y-/
def _root_.LO.Omega1.omega1 : ArithmeticSentence := “∀ x, ∃ y, ∃ l <⁺ x, !lengthDef l x ∧ !exponentialDef (l * ... | module
public import Foundation.FirstOrder.Arithmetic.Exponential
@[expose] public section
/-!
# Theory $\mathsf{I}\Sigma_0 + \Omega_1$
-/
namespace LO.FirstOrder.Arithmetic
/-- ∀ x, ∃ y, 2^{|x|^2} = y-/
def _root_.LO.Omega1.omega1 : ArithmeticSentence := “∀ x, ∃ y, ∃ l <⁺ x, !lengthDef l x ∧ !exponentialDef (l * ... | @@ -40,8 +40,18 @@
instance : V↓[ℒₒᵣ] ⊧* 𝝮₁ := ModelsTheory.of_add_right V 𝗜𝚺₀ 𝝮₁
-lemma smash_exists_unique (x y : V) : ∃! z, Exponential (‖x‖ * ‖y‖) z := sorry
+lemma exists_exponential_sq_length (x : V) : ∃ y, Exponential (‖x‖^2) y :=
+ models_Omega1_iff.mp (Theory.models _ _ Omega1.omega) x
+lemma smash... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_76e907bea2f3_11 | 1007f668a82ece7a | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Omega1/Basic.lean | Basic | 11 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 7 | 2 | [
{
"theorem_name": "exists_unique_exponential_sq_length",
"depth": 1,
"n_commands": 0,
"n_lines": 3,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n rcases exists_exponential_sq_length x with ⟨y, h⟩\n exact ExistsUnique.intro y h (fun y' h' ↦ h'.uniq ... | [
{
"name": "exists_exponential_sq_length",
"text": "lemma exists_exponential_sq_length (x : V) : ∃ y, Exponential (‖x‖^2) y :=\n models_Omega1_iff.mp (Theory.models _ _ Omega1.omega) x\n\n",
"fan_in": 2,
"n_lines": 4,
"n_chars": 140,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,... | [
{
"name": "smash_two_mul",
"fan_in": 1,
"n_deps_direct": 2,
"n_deps_transitive": 5,
"n_lines": 7,
"n_chars": 412,
"n_subproofs": 2,
"n_tactics": 5,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": false,
"max_nesting":... | 3 | module
public import Foundation.FirstOrder.Arithmetic.Exponential
@[expose] public section
/-!
# Theory $\mathsf{I}\Sigma_0 + \Omega_1$
-/
namespace LO.FirstOrder.Arithmetic
/-- ∀ x, ∃ y, 2^{|x|^2} = y-/
def _root_.LO.Omega1.omega1 : ArithmeticSentence := “∀ x, ∃ y, ∃ l <⁺ x, !lengthDef l x ∧ !exponentialDef (l * ... | module
public import Foundation.FirstOrder.Arithmetic.Exponential
@[expose] public section
/-!
# Theory $\mathsf{I}\Sigma_0 + \Omega_1$
-/
namespace LO.FirstOrder.Arithmetic
/-- ∀ x, ∃ y, 2^{|x|^2} = y-/
def _root_.LO.Omega1.omega1 : ArithmeticSentence := “∀ x, ∃ y, ∃ l <⁺ x, !lengthDef l x ∧ !exponentialDef (l * ... | @@ -40,8 +40,18 @@
instance : V↓[ℒₒᵣ] ⊧* 𝝮₁ := ModelsTheory.of_add_right V 𝗜𝚺₀ 𝝮₁
-lemma smash_exists_unique (x y : V) : ∃! z, Exponential (‖x‖ * ‖y‖) z := sorry
+lemma exists_exponential_sq_length (x : V) : ∃ y, Exponential (‖x‖^2) y :=
+ models_Omega1_iff.mp (Theory.models _ _ Omega1.omega) x
+lemma smash... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_76e907bea2f3_12 | 90129cfdedff03ad | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Omega1/Basic.lean | Basic | 12 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 7 | 1 | [
{
"theorem_name": "smash_two_mul_le_sq_smash",
"depth": 1,
"n_commands": 0,
"n_lines": 5,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n rcases Arithmetic.zero_le b with (rfl | pos)\n · simp\n · simpa [smash_two_mul a pos, sq]\n using smash_mono... | [
{
"name": "smash_monotone",
"text": "lemma smash_monotone {a₁ a₂ b₁ b₂ : V} (h₁ : a₁ ≤ b₁) (h₂ : a₂ ≤ b₂) : a₁ ⨳ a₂ ≤ b₁ ⨳ b₂ :=\n (exponential_smash a₁ a₂).monotone_le (exponential_smash b₁ b₂) (mul_le_mul (length_monotone h₁) (length_monotone h₂) (by simp) (by simp))\n\n",
"fan_in": 1,
"n_lines":... | [
{
"name": "smash_two_mul_le_sq_smash",
"fan_in": 0,
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"cyclomatic": 2,
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"n_rewrites": 0,
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"automation_only": false,
"m... | 5 | module
public import Foundation.FirstOrder.Arithmetic.Exponential
@[expose] public section
/-!
# Theory $\mathsf{I}\Sigma_0 + \Omega_1$
-/
namespace LO.FirstOrder.Arithmetic
/-- ∀ x, ∃ y, 2^{|x|^2} = y-/
def _root_.LO.Omega1.omega1 : ArithmeticSentence := “∀ x, ∃ y, ∃ l <⁺ x, !lengthDef l x ∧ !exponentialDef (l * ... | module
public import Foundation.FirstOrder.Arithmetic.Exponential
@[expose] public section
/-!
# Theory $\mathsf{I}\Sigma_0 + \Omega_1$
-/
namespace LO.FirstOrder.Arithmetic
/-- ∀ x, ∃ y, 2^{|x|^2} = y-/
def _root_.LO.Omega1.omega1 : ArithmeticSentence := “∀ x, ∃ y, ∃ l <⁺ x, !lengthDef l x ∧ !exponentialDef (l * ... | @@ -93,6 +93,9 @@
simpa using length_mul_pow2_add_of_lt pos (show Pow2 2 from by simp) one_lt_two
simp [this])
+lemma smash_monotone {a₁ a₂ b₁ b₂ : V} (h₁ : a₁ ≤ b₁) (h₂ : a₂ ≤ b₂) : a₁ ⨳ a₂ ≤ b₁ ⨳ b₂ :=
+ (exponential_smash a₁ a₂).monotone_le (exponential_smash b₁ b₂) (mul_le_mul (length_monotone h₁... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_d6c8e1aee028_0 | 4681a0b416e1a351 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Propositional/Kripke2/Axiom/Ser.lean | Ser | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 1 | 1 | [
{
"theorem_name": "valid_axiomSer_of_isSerial",
"depth": 1,
"n_commands": 0,
"n_lines": 9,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n intro V x y Rxy h;\n contrapose! h;\n apply Satisfies.not_def_neg.mpr;\n obtain ⟨z, Ryz⟩ := F.serial y;\n us... | [
{
"name": "serial",
"text": "lemma serial [F.IsSerial] : ∀ x : F, ∃ y, x ≺ y := IsSerial.serial\n\n",
"fan_in": 1,
"n_lines": 3,
"n_chars": 74,
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"automation_only":... | [
{
"name": "valid_axiomSer_of_isSerial",
"fan_in": 0,
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"n_deps_transitive": 1,
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"cyclomatic": 2,
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"n_structural": 6,
"automation_only": false,
... | 1 | module
public import Foundation.Propositional.Kripke2.Basic
public import Foundation.Vorspiel.Rel.Serial
@[expose] public section
namespace LO.Propositional
open Kripke2
open Formula.Kripke2
namespace Kripke2
variable {F : Kripke2.Frame}
namespace Frame
protected abbrev IsSerial (F : Kripke2.Frame) := _root_.Is... | module
public import Foundation.Propositional.Kripke2.Basic
public import Foundation.Vorspiel.Rel.Serial
@[expose] public section
namespace LO.Propositional
open Kripke2
open Formula.Kripke2
namespace Kripke2
variable {F : Kripke2.Frame}
namespace Frame
protected abbrev IsSerial (F : Kripke2.Frame) := _root_.Is... | @@ -17,10 +17,20 @@
namespace Frame
protected abbrev IsSerial (F : Kripke2.Frame) := _root_.IsSerial F.Rel
+lemma serial [F.IsSerial] : ∀ x : F, ∃ y, x ≺ y := IsSerial.serial
+
end Frame
@[simp high, grind .]
-lemma valid_axiomSer_of_isSerial [F.IsSerial] : F ⊧ Axioms.Ser := sorry
+lemma valid_axiomSer_of_isSer... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_ba8c1955d116_0 | bb937d4674a83507 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Vorspiel/List/ChainI.lean | ChainI | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 7 | 2 | [
{
"theorem_name": "cons_cons_iff",
"depth": 1,
"n_commands": 0,
"n_lines": 8,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n constructor\n · rintro ⟨⟩\n case cons d' hR hC =>\n rcases cons_iff.mp hC with ⟨rfl, hC⟩\n simp_all\n · rintro... | [
{
"name": "cons_iff",
"text": "lemma cons_iff : ChainI R a b (c :: l) ↔ a = c ∧ ChainI R c b (c :: l) := by\n constructor\n · rintro (_ | _)\n · simp\n case cons a' hR h =>\n simp [h.cons hR]\n · rintro ⟨rfl, _⟩\n assumption\n\n",
"fan_in": 2,
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"n... | [
{
"name": "cons_cons_iff",
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"n_rewrites": 0,
"n_structural": 5,
"automation_only": false,
"max_nesting"... | 1 | module
public import Mathlib.Data.Fintype.List
public import Foundation.Vorspiel.List.Basic
@[expose]
public section
namespace List
/--
```
ChainI R x y [a, b, c, d] ↔ x = a ∧ R a b ∧ R b c ∧ R c d ∧ d = y
```
-/
inductive ChainI {α : Type*} (R : α → α → Prop) : α → α → List α → Prop
| singleton (a : α) : Chai... | module
public import Mathlib.Data.Fintype.List
public import Foundation.Vorspiel.List.Basic
@[expose]
public section
namespace List
/--
```
ChainI R x y [a, b, c, d] ↔ x = a ∧ R a b ∧ R b c ∧ R c d ∧ d = y
```
-/
inductive ChainI {α : Type*} (R : α → α → Prop) : α → α → List α → Prop
| singleton (a : α) : Chai... | @@ -30,10 +30,22 @@
attribute [simp] ChainI.singleton
+lemma cons_iff : ChainI R a b (c :: l) ↔ a = c ∧ ChainI R c b (c :: l) := by
+ constructor
+ · rintro (_ | _)
+ · simp
+ case cons a' hR h =>
+ simp [h.cons hR]
+ · rintro ⟨rfl, _⟩
+ assumption
+
lemma cons_cons_iff :
ChainI R a b (c :: d ... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_ba8c1955d116_1 | ef09b81045fab165 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Vorspiel/List/ChainI.lean | ChainI | 1 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 7 | 1 | [
{
"theorem_name": "nodup",
"depth": 1,
"n_commands": 0,
"n_lines": 10,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": "\n match l with\n | [] => by simp\n | a' :: l => by\n rintro (_ | _)\n case singleton => simp\n case cons a' Raa' h =>\n ... | [
{
"name": "not_mem_of_rel",
"text": "lemma not_mem_of_rel (IR : Std.Irrefl R) (TR : IsTrans α R) {a b x : α} {l : List α} : ChainI R a b l → R x a → x ∉ l := by\n match l with\n | [] => simp\n | a' :: l =>\n rintro (_ | _)\n case singleton => simp; intro hR; rintro rfl; exact IR.irrefl _ hR\n ... | [
{
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"n_automation": 3,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": false,
"max_nesting": 6
}
... | 1 | module
public import Mathlib.Data.Fintype.List
public import Foundation.Vorspiel.List.Basic
@[expose]
public section
namespace List
/--
```
ChainI R x y [a, b, c, d] ↔ x = a ∧ R a b ∧ R b c ∧ R c d ∧ d = y
```
-/
inductive ChainI {α : Type*} (R : α → α → Prop) : α → α → List α → Prop
| singleton (a : α) : Chai... | module
public import Mathlib.Data.Fintype.List
public import Foundation.Vorspiel.List.Basic
@[expose]
public section
namespace List
/--
```
ChainI R x y [a, b, c, d] ↔ x = a ∧ R a b ∧ R b c ∧ R c d ∧ d = y
```
-/
inductive ChainI {α : Type*} (R : α → α → Prop) : α → α → List α → Prop
| singleton (a : α) : Chai... | @@ -30,6 +30,20 @@
attribute [simp] ChainI.singleton
+lemma not_mem_of_rel (IR : Std.Irrefl R) (TR : IsTrans α R) {a b x : α} {l : List α} : ChainI R a b l → R x a → x ∉ l := by
+ match l with
+ | [] => simp
+ | a' :: l =>
+ rintro (_ | _)
+ case singleton => simp; intro hR; rintro rfl; exact IR.irre... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_ba8c1955d116_2 | fc029d5c3315d65b | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Vorspiel/List/ChainI.lean | ChainI | 2 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 7 | 1 | [
{
"theorem_name": "nodup",
"depth": 1,
"n_commands": 0,
"n_lines": 10,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": "\n match l with\n | [] => by simp\n | a' :: l => by\n rintro (_ | _)\n case singleton => simp\n case cons a' Raa' h =>\n ... | [
{
"name": "not_mem_of_rel",
"text": "lemma not_mem_of_rel (IR : Std.Irrefl R) (TR : IsTrans α R) {a b x : α} {l : List α} : ChainI R a b l → R x a → x ∉ l := by\n match l with\n | [] => simp\n | a' :: l =>\n rintro (_ | _)\n case singleton => simp; intro hR; rintro rfl; exact IR.irrefl _ hR\n ... | [
{
"name": "finite_of_irreflexive_of_transitive",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 2,
"n_lines": 8,
"n_chars": 436,
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"n_tactics": 6,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 2,
"automation_only": fal... | 2 | module
public import Mathlib.Data.Fintype.List
public import Foundation.Vorspiel.List.Basic
@[expose]
public section
namespace List
/--
```
ChainI R x y [a, b, c, d] ↔ x = a ∧ R a b ∧ R b c ∧ R c d ∧ d = y
```
-/
inductive ChainI {α : Type*} (R : α → α → Prop) : α → α → List α → Prop
| singleton (a : α) : Chai... | module
public import Mathlib.Data.Fintype.List
public import Foundation.Vorspiel.List.Basic
@[expose]
public section
namespace List
/--
```
ChainI R x y [a, b, c, d] ↔ x = a ∧ R a b ∧ R b c ∧ R c d ∧ d = y
```
-/
inductive ChainI {α : Type*} (R : α → α → Prop) : α → α → List α → Prop
| singleton (a : α) : Chai... | @@ -30,6 +30,20 @@
attribute [simp] ChainI.singleton
+lemma not_mem_of_rel (IR : Std.Irrefl R) (TR : IsTrans α R) {a b x : α} {l : List α} : ChainI R a b l → R x a → x ∉ l := by
+ match l with
+ | [] => simp
+ | a' :: l =>
+ rintro (_ | _)
+ case singleton => simp; intro hR; rintro rfl; exact IR.irre... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_ba8c1955d116_3 | 705478589aa1d562 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Vorspiel/List/ChainI.lean | ChainI | 3 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 7 | 2 | [
{
"theorem_name": "prec_exists_of_ne",
"depth": 1,
"n_commands": 0,
"n_lines": 10,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n intro _\n match l with\n | [] => rcases h\n | [b'] => rcases h <;> simp_all\n | b' :: c :: l' =>... | [
{
"name": "cons_eq",
"text": "lemma cons_eq {l} : ChainI R a b (a' :: l) → a = a' := by\n rintro ⟨⟩ <;> simp\n\n",
"fan_in": 2,
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"aut... | [
{
"name": "prec_exists_of_ne",
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"automation_only": false,
"max_nes... | 1 | module
public import Mathlib.Data.Fintype.List
public import Foundation.Vorspiel.List.Basic
@[expose]
public section
namespace List
/--
```
ChainI R x y [a, b, c, d] ↔ x = a ∧ R a b ∧ R b c ∧ R c d ∧ d = y
```
-/
inductive ChainI {α : Type*} (R : α → α → Prop) : α → α → List α → Prop
| singleton (a : α) : Chai... | module
public import Mathlib.Data.Fintype.List
public import Foundation.Vorspiel.List.Basic
@[expose]
public section
namespace List
/--
```
ChainI R x y [a, b, c, d] ↔ x = a ∧ R a b ∧ R b c ∧ R c d ∧ d = y
```
-/
inductive ChainI {α : Type*} (R : α → α → Prop) : α → α → List α → Prop
| singleton (a : α) : Chai... | @@ -30,13 +30,20 @@
attribute [simp] ChainI.singleton
+lemma cons_eq {l} : ChainI R a b (a' :: l) → a = a' := by
+ rintro ⟨⟩ <;> simp
+
lemma prec_exists_of_ne {l} (h : ChainI R a b l) :
a ≠ b → ∃ l' c, R a c ∧ l = a :: c :: l' ∧ ChainI R c b (c :: l') := by
intro _
match l with
| [] => ... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_ba8c1955d116_4 | 5eb658b2b7de3fca | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Vorspiel/List/ChainI.lean | ChainI | 4 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 7 | 2 | [
{
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"depth": 1,
"n_commands": 0,
"n_lines": 10,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n intro _\n match l with\n | [] => rcases h\n | [b'] => rcases h <;> simp_all\n | b' :: c :: l' =>... | [
{
"name": "cons_eq",
"text": "lemma cons_eq {l} : ChainI R a b (a' :: l) → a = a' := by\n rintro ⟨⟩ <;> simp\n\n",
"fan_in": 2,
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"aut... | [
{
"name": "tail_exists",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 11,
"n_chars": 280,
"n_subproofs": 0,
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"cyclomatic": 7,
"n_automation": 2,
"n_rewrites": 1,
"n_structural": 5,
"automation_only": false,
"max_nesting":... | 1 | module
public import Mathlib.Data.Fintype.List
public import Foundation.Vorspiel.List.Basic
@[expose]
public section
namespace List
/--
```
ChainI R x y [a, b, c, d] ↔ x = a ∧ R a b ∧ R b c ∧ R c d ∧ d = y
```
-/
inductive ChainI {α : Type*} (R : α → α → Prop) : α → α → List α → Prop
| singleton (a : α) : Chai... | module
public import Mathlib.Data.Fintype.List
public import Foundation.Vorspiel.List.Basic
@[expose]
public section
namespace List
/--
```
ChainI R x y [a, b, c, d] ↔ x = a ∧ R a b ∧ R b c ∧ R c d ∧ d = y
```
-/
inductive ChainI {α : Type*} (R : α → α → Prop) : α → α → List α → Prop
| singleton (a : α) : Chai... | @@ -30,11 +30,18 @@
attribute [simp] ChainI.singleton
+lemma cons_eq {l} : ChainI R a b (a' :: l) → a = a' := by
+ rintro ⟨⟩ <;> simp
+
lemma tail_exists (h : ChainI R a b l) : ∃ l', l = a :: l' := by
match l with
| [] => rcases h
| [b'] => rcases h <;> simp_all
- | b' :: c :: 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_ba8c1955d116_5 | c2036d3b3fe990c1 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Vorspiel/List/ChainI.lean | ChainI | 5 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 7 | 2 | [
{
"theorem_name": "cons_cons_iff",
"depth": 1,
"n_commands": 0,
"n_lines": 8,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n constructor\n · rintro ⟨⟩\n case cons d' hR hC =>\n rcases cons_iff.mp hC with ⟨rfl, hC⟩\n simp_all\n · rintro... | [
{
"name": "cons_iff",
"text": "lemma cons_iff : ChainI R a b (c :: l) ↔ a = c ∧ ChainI R c b (c :: l) := by\n constructor\n · rintro (_ | _)\n · simp\n case cons a' hR h =>\n simp [h.cons hR]\n · rintro ⟨rfl, _⟩\n assumption\n\n",
"fan_in": 2,
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"n... | [
{
"name": "append_singleton_append_iff",
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"n_lines": 11,
"n_chars": 559,
"n_subproofs": 1,
"n_tactics": 8,
"cyclomatic": 2,
"n_automation": 3,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
... | 2 | module
public import Mathlib.Data.Fintype.List
public import Foundation.Vorspiel.List.Basic
@[expose]
public section
namespace List
/--
```
ChainI R x y [a, b, c, d] ↔ x = a ∧ R a b ∧ R b c ∧ R c d ∧ d = y
```
-/
inductive ChainI {α : Type*} (R : α → α → Prop) : α → α → List α → Prop
| singleton (a : α) : Chai... | module
public import Mathlib.Data.Fintype.List
public import Foundation.Vorspiel.List.Basic
@[expose]
public section
namespace List
/--
```
ChainI R x y [a, b, c, d] ↔ x = a ∧ R a b ∧ R b c ∧ R c d ∧ d = y
```
-/
inductive ChainI {α : Type*} (R : α → α → Prop) : α → α → List α → Prop
| singleton (a : α) : Chai... | @@ -30,17 +30,29 @@
attribute [simp] ChainI.singleton
+lemma cons_iff : ChainI R a b (c :: l) ↔ a = c ∧ ChainI R c b (c :: l) := by
+ constructor
+ · rintro (_ | _)
+ · simp
+ case cons a' hR h =>
+ simp [h.cons hR]
+ · rintro ⟨rfl, _⟩
+ assumption
+
lemma cons_cons_iff :
ChainI R a b (c :: d ... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_ba8c1955d116_6 | d94e20bd882732d5 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Vorspiel/List/ChainI.lean | ChainI | 6 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 7 | 2 | [
{
"theorem_name": "cons_cons_iff",
"depth": 1,
"n_commands": 0,
"n_lines": 8,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n constructor\n · rintro ⟨⟩\n case cons d' hR hC =>\n rcases cons_iff.mp hC with ⟨rfl, hC⟩\n simp_all\n · rintro... | [
{
"name": "cons_iff",
"text": "lemma cons_iff : ChainI R a b (c :: l) ↔ a = c ∧ ChainI R c b (c :: l) := by\n constructor\n · rintro (_ | _)\n · simp\n case cons a' hR h =>\n simp [h.cons hR]\n · rintro ⟨rfl, _⟩\n assumption\n\n",
"fan_in": 2,
"n_lines": 10,
"n_chars": 214,
"n... | [
{
"name": "rel_of_infix",
"fan_in": 0,
"n_deps_direct": 2,
"n_deps_transitive": 3,
"n_lines": 8,
"n_chars": 282,
"n_subproofs": 1,
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"cyclomatic": 2,
"n_automation": 2,
"n_rewrites": 0,
"n_structural": 3,
"automation_only": false,
"max_nesting": ... | 3 | module
public import Mathlib.Data.Fintype.List
public import Foundation.Vorspiel.List.Basic
@[expose]
public section
namespace List
/--
```
ChainI R x y [a, b, c, d] ↔ x = a ∧ R a b ∧ R b c ∧ R c d ∧ d = y
```
-/
inductive ChainI {α : Type*} (R : α → α → Prop) : α → α → List α → Prop
| singleton (a : α) : Chai... | module
public import Mathlib.Data.Fintype.List
public import Foundation.Vorspiel.List.Basic
@[expose]
public section
namespace List
/--
```
ChainI R x y [a, b, c, d] ↔ x = a ∧ R a b ∧ R b c ∧ R c d ∧ d = y
```
-/
inductive ChainI {α : Type*} (R : α → α → Prop) : α → α → List α → Prop
| singleton (a : α) : Chai... | @@ -30,17 +30,29 @@
attribute [simp] ChainI.singleton
+lemma cons_iff : ChainI R a b (c :: l) ↔ a = c ∧ ChainI R c b (c :: l) := by
+ constructor
+ · rintro (_ | _)
+ · simp
+ case cons a' hR h =>
+ simp [h.cons hR]
+ · rintro ⟨rfl, _⟩
+ assumption
+
lemma cons_cons_iff :
ChainI R a b (c :: d ... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_ba8c1955d116_7 | 42a37f6e621876ce | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Vorspiel/List/ChainI.lean | ChainI | 7 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 7 | 2 | [
{
"theorem_name": "prec_exists_of_ne",
"depth": 1,
"n_commands": 0,
"n_lines": 10,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n intro _\n match l with\n | [] => rcases h\n | [b'] => rcases h <;> simp_all\n | b' :: c :: l' =>... | [
{
"name": "cons_eq",
"text": "lemma cons_eq {l} : ChainI R a b (a' :: l) → a = a' := by\n rintro ⟨⟩ <;> simp\n\n",
"fan_in": 2,
"n_lines": 4,
"n_chars": 86,
"n_subproofs": 0,
"n_tactics": 3,
"cyclomatic": 3,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 1,
"aut... | [
{
"name": "infix_of_suffix_of",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 2,
"n_lines": 6,
"n_chars": 230,
"n_subproofs": 0,
"n_tactics": 4,
"cyclomatic": 2,
"n_automation": 2,
"n_rewrites": 0,
"n_structural": 3,
"automation_only": false,
"max_nest... | 2 | module
public import Mathlib.Data.Fintype.List
public import Foundation.Vorspiel.List.Basic
@[expose]
public section
namespace List
/--
```
ChainI R x y [a, b, c, d] ↔ x = a ∧ R a b ∧ R b c ∧ R c d ∧ d = y
```
-/
inductive ChainI {α : Type*} (R : α → α → Prop) : α → α → List α → Prop
| singleton (a : α) : Chai... | module
public import Mathlib.Data.Fintype.List
public import Foundation.Vorspiel.List.Basic
@[expose]
public section
namespace List
/--
```
ChainI R x y [a, b, c, d] ↔ x = a ∧ R a b ∧ R b c ∧ R c d ∧ d = y
```
-/
inductive ChainI {α : Type*} (R : α → α → Prop) : α → α → List α → Prop
| singleton (a : α) : Chai... | @@ -30,11 +30,18 @@
attribute [simp] ChainI.singleton
+lemma cons_eq {l} : ChainI R a b (a' :: l) → a = a' := by
+ rintro ⟨⟩ <;> simp
+
lemma tail_exists (h : ChainI R a b l) : ∃ l', l = a :: l' := by
match l with
| [] => rcases h
| [b'] => rcases h <;> simp_all
- | b' :: c :: 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_8952d0b07064_0 | 26827decdbb5e92a | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Exponential/Pow2.lean | Pow2 | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 32 | 2 | [
{
"theorem_name": "pow2_mul_four",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n simp [←two_mul_two_eq_four, mul_assoc, pow2_mul_two]",
"n_chars": 60,
"n_subproofs": 0,
"n_tactics": 2,
"cycl... | [
{
"name": "pow2_mul_two",
"text": "lemma pow2_mul_two {a : V} : Pow2 (2 * a) ↔ Pow2 a :=\n ⟨by intro H\n have : ∀ r ≤ a, 1 < r → r ∣ a → 2 ∣ r := by\n intro r hr ltr dvd\n exact H.dvd (show r ≤ 2 * a from le_trans hr (le_mul_of_one_le_left (by simp) one_le_two)) ltr (Dvd.dvd.mul_left dvd... | [
{
"name": "pow2_mul_four",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 4,
"n_chars": 118,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"max_nesting": ... | 1 | module
public import Foundation.FirstOrder.Arithmetic.IOpen.Basic
@[expose] public section
/-!
# On $\mathrm{Pow2}(x)$ and $\mathrm{BitLen}(x, y)$
$\mathrm{Pow2}(n)$ is a property that holds iff $n = 2^i$ for some $i$.
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section IOpen
va... | module
public import Foundation.FirstOrder.Arithmetic.IOpen.Basic
@[expose] public section
/-!
# On $\mathrm{Pow2}(x)$ and $\mathrm{BitLen}(x, y)$
$\mathrm{Pow2}(n)$ is a property that holds iff $n = 2^i$ for some $i$.
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section IOpen
va... | @@ -52,15 +52,31 @@
exact ⟨this, fun r hr ltr hb ↦
ha.dvd (show r ≤ a from le_trans hr (le_of_dvd ha.pos h)) ltr (dvd_trans hb h)⟩
-lemma pow2_mul_four {a : V} : Pow2 (4 * a) ↔ Pow2 a := sorry
+lemma pow2_mul_two {a : V} : Pow2 (2 * a) ↔ Pow2 a :=
+ ⟨by intro H
+ have : ∀ r ≤ a, 1 < r → r ∣ a → 2 ∣ 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_8952d0b07064_1 | 64346fe700e0541c | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Exponential/Pow2.lean | Pow2 | 1 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 32 | 4 | [
{
"theorem_name": "pow2_two",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " Pow2.elim.mpr (Or.inr ⟨1, by simp⟩)",
"n_chars": 40,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automatio... | [
{
"name": "Pow2.elim",
"text": "lemma Pow2.elim {p : V} : Pow2 p ↔ p = 1 ∨ ∃ q, p = 2 * q ∧ Pow2 q :=\n ⟨by intro H\n by_cases hp : 1 < p\n · have : 2 ∣ p := H.two_dvd hp\n rcases this with ⟨q, rfl⟩\n right; exact ⟨q, rfl, pow2_mul_two.mp H⟩\n · have : p = 1 := le_antisymm (by ... | [
{
"name": "pow2_two",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 3,
"n_lines": 3,
"n_chars": 82,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"max_nesting": 1
}
... | 2 | module
public import Foundation.FirstOrder.Arithmetic.IOpen.Basic
@[expose] public section
/-!
# On $\mathrm{Pow2}(x)$ and $\mathrm{BitLen}(x, y)$
$\mathrm{Pow2}(n)$ is a property that holds iff $n = 2^i$ for some $i$.
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section IOpen
va... | module
public import Foundation.FirstOrder.Arithmetic.IOpen.Basic
@[expose] public section
/-!
# On $\mathrm{Pow2}(x)$ and $\mathrm{BitLen}(x, y)$
$\mathrm{Pow2}(n)$ is a property that holds iff $n = 2^i$ for some $i$.
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section IOpen
va... | @@ -68,11 +68,21 @@
lemma pow2_mul_four {a : V} : Pow2 (4 * a) ↔ Pow2 a := by
simp [←two_mul_two_eq_four, mul_assoc, pow2_mul_two]
-@[simp] lemma pow2_two : Pow2 (2 : V) := sorry
+lemma Pow2.elim {p : V} : Pow2 p ↔ p = 1 ∨ ∃ q, p = 2 * q ∧ Pow2 q :=
+ ⟨by intro H
+ by_cases hp : 1 < p
+ · have : 2 ∣ 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_8952d0b07064_2 | 03f617bb60ff1dca | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Exponential/Pow2.lean | Pow2 | 2 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 32 | 4 | [
{
"theorem_name": "pow2_two",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " Pow2.elim.mpr (Or.inr ⟨1, by simp⟩)",
"n_chars": 40,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automatio... | [
{
"name": "Pow2.elim",
"text": "lemma Pow2.elim {p : V} : Pow2 p ↔ p = 1 ∨ ∃ q, p = 2 * q ∧ Pow2 q :=\n ⟨by intro H\n by_cases hp : 1 < p\n · have : 2 ∣ p := H.two_dvd hp\n rcases this with ⟨q, rfl⟩\n right; exact ⟨q, rfl, pow2_mul_two.mp H⟩\n · have : p = 1 := le_antisymm (by ... | [
{
"name": "Pow2.div_two",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 3,
"n_lines": 6,
"n_chars": 155,
"n_subproofs": 0,
"n_tactics": 4,
"cyclomatic": 2,
"n_automation": 4,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": false,
"max_nesting": ... | 2 | module
public import Foundation.FirstOrder.Arithmetic.IOpen.Basic
@[expose] public section
/-!
# On $\mathrm{Pow2}(x)$ and $\mathrm{BitLen}(x, y)$
$\mathrm{Pow2}(n)$ is a property that holds iff $n = 2^i$ for some $i$.
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section IOpen
va... | module
public import Foundation.FirstOrder.Arithmetic.IOpen.Basic
@[expose] public section
/-!
# On $\mathrm{Pow2}(x)$ and $\mathrm{BitLen}(x, y)$
$\mathrm{Pow2}(n)$ is a property that holds iff $n = 2^i$ for some $i$.
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section IOpen
va... | @@ -68,13 +68,26 @@
lemma pow2_mul_four {a : V} : Pow2 (4 * a) ↔ Pow2 a := by
simp [←two_mul_two_eq_four, mul_assoc, pow2_mul_two]
-@[simp] lemma pow2_two : Pow2 (2 : V) := sorry
+lemma Pow2.elim {p : V} : Pow2 p ↔ p = 1 ∨ ∃ q, p = 2 * q ∧ Pow2 q :=
+ ⟨by intro H
+ by_cases hp : 1 < p
+ · have : 2 ∣ 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_8952d0b07064_3 | 1d0357ac251b55e2 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Exponential/Pow2.lean | Pow2 | 3 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 32 | 4 | [
{
"theorem_name": "pow2_two",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " Pow2.elim.mpr (Or.inr ⟨1, by simp⟩)",
"n_chars": 40,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automatio... | [
{
"name": "Pow2.elim",
"text": "lemma Pow2.elim {p : V} : Pow2 p ↔ p = 1 ∨ ∃ q, p = 2 * q ∧ Pow2 q :=\n ⟨by intro H\n by_cases hp : 1 < p\n · have : 2 ∣ p := H.two_dvd hp\n rcases this with ⟨q, rfl⟩\n right; exact ⟨q, rfl, pow2_mul_two.mp H⟩\n · have : p = 1 := le_antisymm (by ... | [
{
"name": "Pow2.two_mul_div_two",
"fan_in": 3,
"n_deps_direct": 1,
"n_deps_transitive": 3,
"n_lines": 6,
"n_chars": 164,
"n_subproofs": 0,
"n_tactics": 4,
"cyclomatic": 2,
"n_automation": 4,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": false,
"max_ne... | 2 | module
public import Foundation.FirstOrder.Arithmetic.IOpen.Basic
@[expose] public section
/-!
# On $\mathrm{Pow2}(x)$ and $\mathrm{BitLen}(x, y)$
$\mathrm{Pow2}(n)$ is a property that holds iff $n = 2^i$ for some $i$.
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section IOpen
va... | module
public import Foundation.FirstOrder.Arithmetic.IOpen.Basic
@[expose] public section
/-!
# On $\mathrm{Pow2}(x)$ and $\mathrm{BitLen}(x, y)$
$\mathrm{Pow2}(n)$ is a property that holds iff $n = 2^i$ for some $i$.
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section IOpen
va... | @@ -68,13 +68,26 @@
lemma pow2_mul_four {a : V} : Pow2 (4 * a) ↔ Pow2 a := by
simp [←two_mul_two_eq_four, mul_assoc, pow2_mul_two]
-@[simp] lemma pow2_two : Pow2 (2 : V) := sorry
+lemma Pow2.elim {p : V} : Pow2 p ↔ p = 1 ∨ ∃ q, p = 2 * q ∧ Pow2 q :=
+ ⟨by intro H
+ by_cases hp : 1 < p
+ · have : 2 ∣ 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_8952d0b07064_4 | 1340a2e3412c24e7 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Exponential/Pow2.lean | Pow2 | 4 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 32 | 7 | [
{
"theorem_name": "LenBit.self",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by simp [LenBit.iff_rem, pos]",
"n_chars": 30,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation":... | [
{
"name": "LenBit.iff_rem",
"text": "lemma LenBit.iff_rem {i a : V} : LenBit i a ↔ (a / i) % 2 = 1 := by\n simp only [LenBit]\n rcases mod_two (a / i) with (h | h) <;> simp [h, ←mod_eq_zero_iff_dvd]\n\n",
"fan_in": 7,
"n_lines": 5,
"n_chars": 167,
"n_subproofs": 0,
"n_tactics": 4,
... | [
{
"name": "LenBit.self",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 3,
"n_chars": 95,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"max_nesting": 1
... | 1 | module
public import Foundation.FirstOrder.Arithmetic.IOpen.Basic
@[expose] public section
/-!
# On $\mathrm{Pow2}(x)$ and $\mathrm{BitLen}(x, y)$
$\mathrm{Pow2}(n)$ is a property that holds iff $n = 2^i$ for some $i$.
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section IOpen
va... | module
public import Foundation.FirstOrder.Arithmetic.IOpen.Basic
@[expose] public section
/-!
# On $\mathrm{Pow2}(x)$ and $\mathrm{BitLen}(x, y)$
$\mathrm{Pow2}(n)$ is a property that holds iff $n = 2^i$ for some $i$.
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section IOpen
va... | @@ -106,10 +106,27 @@
@[simp] lemma LenBit.on_zero (a : V) : ¬LenBit a 0 := by simp [LenBit]
-@[simp] lemma LenBit.self {a : V} (pos : 0 < a) : LenBit a a := sorry
+lemma LenBit.iff_rem {i a : V} : LenBit i a ↔ (a / i) % 2 = 1 := by
+ simp only [LenBit]
+ rcases mod_two (a / i) with (h | h) <;> simp [h, ←mod_eq_... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_8952d0b07064_5 | cfaf3619c26f4a54 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Exponential/Pow2.lean | Pow2 | 5 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 32 | 7 | [
{
"theorem_name": "LenBit.self",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by simp [LenBit.iff_rem, pos]",
"n_chars": 30,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation":... | [
{
"name": "LenBit.iff_rem",
"text": "lemma LenBit.iff_rem {i a : V} : LenBit i a ↔ (a / i) % 2 = 1 := by\n simp only [LenBit]\n rcases mod_two (a / i) with (h | h) <;> simp [h, ←mod_eq_zero_iff_dvd]\n\n",
"fan_in": 7,
"n_lines": 5,
"n_chars": 167,
"n_subproofs": 0,
"n_tactics": 4,
... | [
{
"name": "LenBit.add",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 12,
"n_chars": 522,
"n_subproofs": 2,
"n_tactics": 10,
"cyclomatic": 3,
"n_automation": 4,
"n_rewrites": 1,
"n_structural": 3,
"automation_only": false,
"max_nesting": ... | 1 | module
public import Foundation.FirstOrder.Arithmetic.IOpen.Basic
@[expose] public section
/-!
# On $\mathrm{Pow2}(x)$ and $\mathrm{BitLen}(x, y)$
$\mathrm{Pow2}(n)$ is a property that holds iff $n = 2^i$ for some $i$.
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section IOpen
va... | module
public import Foundation.FirstOrder.Arithmetic.IOpen.Basic
@[expose] public section
/-!
# On $\mathrm{Pow2}(x)$ and $\mathrm{BitLen}(x, y)$
$\mathrm{Pow2}(n)$ is a property that holds iff $n = 2^i$ for some $i$.
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section IOpen
va... | @@ -106,13 +106,39 @@
@[simp] lemma LenBit.on_zero (a : V) : ¬LenBit a 0 := by simp [LenBit]
-@[simp] lemma LenBit.self {a : V} (pos : 0 < a) : LenBit a a := sorry
+lemma LenBit.iff_rem {i a : V} : LenBit i a ↔ (a / i) % 2 = 1 := by
+ simp only [LenBit]
+ rcases mod_two (a / i) with (h | h) <;> simp [h, ←mod_eq_... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_8952d0b07064_6 | 915e8038bde19205 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Exponential/Pow2.lean | Pow2 | 6 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 32 | 7 | [
{
"theorem_name": "LenBit.self",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by simp [LenBit.iff_rem, pos]",
"n_chars": 30,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation":... | [
{
"name": "LenBit.iff_rem",
"text": "lemma LenBit.iff_rem {i a : V} : LenBit i a ↔ (a / i) % 2 = 1 := by\n simp only [LenBit]\n rcases mod_two (a / i) with (h | h) <;> simp [h, ←mod_eq_zero_iff_dvd]\n\n",
"fan_in": 7,
"n_lines": 5,
"n_chars": 167,
"n_subproofs": 0,
"n_tactics": 4,
... | [
{
"name": "LenBit.add_self",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 5,
"n_chars": 166,
"n_subproofs": 1,
"n_tactics": 3,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": false,
"max_nesting... | 1 | module
public import Foundation.FirstOrder.Arithmetic.IOpen.Basic
@[expose] public section
/-!
# On $\mathrm{Pow2}(x)$ and $\mathrm{BitLen}(x, y)$
$\mathrm{Pow2}(n)$ is a property that holds iff $n = 2^i$ for some $i$.
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section IOpen
va... | module
public import Foundation.FirstOrder.Arithmetic.IOpen.Basic
@[expose] public section
/-!
# On $\mathrm{Pow2}(x)$ and $\mathrm{BitLen}(x, y)$
$\mathrm{Pow2}(n)$ is a property that holds iff $n = 2^i$ for some $i$.
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section IOpen
va... | @@ -106,13 +106,32 @@
@[simp] lemma LenBit.on_zero (a : V) : ¬LenBit a 0 := by simp [LenBit]
-@[simp] lemma LenBit.self {a : V} (pos : 0 < a) : LenBit a a := sorry
+lemma LenBit.iff_rem {i a : V} : LenBit i a ↔ (a / i) % 2 = 1 := by
+ simp only [LenBit]
+ rcases mod_two (a / i) with (h | h) <;> simp [h, ←mod_eq_... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_8952d0b07064_10 | 9839a42540e86365 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Exponential/Pow2.lean | Pow2 | 10 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 32 | 4 | [
{
"theorem_name": "pow2_two",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " Pow2.elim.mpr (Or.inr ⟨1, by simp⟩)",
"n_chars": 40,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automatio... | [
{
"name": "Pow2.elim",
"text": "lemma Pow2.elim {p : V} : Pow2 p ↔ p = 1 ∨ ∃ q, p = 2 * q ∧ Pow2 q :=\n ⟨by intro H\n by_cases hp : 1 < p\n · have : 2 ∣ p := H.two_dvd hp\n rcases this with ⟨q, rfl⟩\n right; exact ⟨q, rfl, pow2_mul_two.mp H⟩\n · have : p = 1 := le_antisymm (by ... | [
{
"name": "dvd_of_le",
"fan_in": 4,
"n_deps_direct": 1,
"n_deps_transitive": 4,
"n_lines": 19,
"n_chars": 931,
"n_subproofs": 5,
"n_tactics": 21,
"cyclomatic": 7,
"n_automation": 14,
"n_rewrites": 0,
"n_structural": 10,
"automation_only": false,
"max_nesting":... | 3 | module
public import Foundation.FirstOrder.Arithmetic.IOpen.Basic
@[expose] public section
/-!
# On $\mathrm{Pow2}(x)$ and $\mathrm{BitLen}(x, y)$
$\mathrm{Pow2}(n)$ is a property that holds iff $n = 2^i$ for some $i$.
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section IOpen
va... | module
public import Foundation.FirstOrder.Arithmetic.IOpen.Basic
@[expose] public section
/-!
# On $\mathrm{Pow2}(x)$ and $\mathrm{BitLen}(x, y)$
$\mathrm{Pow2}(n)$ is a property that holds iff $n = 2^i$ for some $i$.
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section IOpen
va... | @@ -68,11 +68,21 @@
lemma pow2_mul_four {a : V} : Pow2 (4 * a) ↔ Pow2 a := by
simp [←two_mul_two_eq_four, mul_assoc, pow2_mul_two]
-@[simp] lemma pow2_two : Pow2 (2 : V) := sorry
+lemma Pow2.elim {p : V} : Pow2 p ↔ p = 1 ∨ ∃ q, p = 2 * q ∧ Pow2 q :=
+ ⟨by intro H
+ by_cases hp : 1 < p
+ · have : 2 ∣ 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_8952d0b07064_11 | b81b5e8c7c7c117b | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Exponential/Pow2.lean | Pow2 | 11 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 32 | 4 | [
{
"theorem_name": "pow2_two",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " Pow2.elim.mpr (Or.inr ⟨1, by simp⟩)",
"n_chars": 40,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automatio... | [
{
"name": "Pow2.elim",
"text": "lemma Pow2.elim {p : V} : Pow2 p ↔ p = 1 ∨ ∃ q, p = 2 * q ∧ Pow2 q :=\n ⟨by intro H\n by_cases hp : 1 < p\n · have : 2 ∣ p := H.two_dvd hp\n rcases this with ⟨q, rfl⟩\n right; exact ⟨q, rfl, pow2_mul_two.mp H⟩\n · have : p = 1 := le_antisymm (by ... | [
{
"name": "le_iff_dvd",
"fan_in": 2,
"n_deps_direct": 1,
"n_deps_transitive": 5,
"n_lines": 4,
"n_chars": 128,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
"max_nesting": 2
... | 4 | module
public import Foundation.FirstOrder.Arithmetic.IOpen.Basic
@[expose] public section
/-!
# On $\mathrm{Pow2}(x)$ and $\mathrm{BitLen}(x, y)$
$\mathrm{Pow2}(n)$ is a property that holds iff $n = 2^i$ for some $i$.
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section IOpen
va... | module
public import Foundation.FirstOrder.Arithmetic.IOpen.Basic
@[expose] public section
/-!
# On $\mathrm{Pow2}(x)$ and $\mathrm{BitLen}(x, y)$
$\mathrm{Pow2}(n)$ is a property that holds iff $n = 2^i$ for some $i$.
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section IOpen
va... | @@ -68,11 +68,21 @@
lemma pow2_mul_four {a : V} : Pow2 (4 * a) ↔ Pow2 a := by
simp [←two_mul_two_eq_four, mul_assoc, pow2_mul_two]
-@[simp] lemma pow2_two : Pow2 (2 : V) := sorry
+lemma Pow2.elim {p : V} : Pow2 p ↔ p = 1 ∨ ∃ q, p = 2 * q ∧ Pow2 q :=
+ ⟨by intro H
+ by_cases hp : 1 < p
+ · have : 2 ∣ 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_8952d0b07064_12 | e311173c7e3c7855 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Exponential/Pow2.lean | Pow2 | 12 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 32 | 4 | [
{
"theorem_name": "pow2_two",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " Pow2.elim.mpr (Or.inr ⟨1, by simp⟩)",
"n_chars": 40,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automatio... | [
{
"name": "Pow2.elim",
"text": "lemma Pow2.elim {p : V} : Pow2 p ↔ p = 1 ∨ ∃ q, p = 2 * q ∧ Pow2 q :=\n ⟨by intro H\n by_cases hp : 1 < p\n · have : 2 ∣ p := H.two_dvd hp\n rcases this with ⟨q, rfl⟩\n right; exact ⟨q, rfl, pow2_mul_two.mp H⟩\n · have : p = 1 := le_antisymm (by ... | [
{
"name": "le_iff_lt_two",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 5,
"n_lines": 19,
"n_chars": 838,
"n_subproofs": 5,
"n_tactics": 19,
"cyclomatic": 6,
"n_automation": 11,
"n_rewrites": 1,
"n_structural": 9,
"automation_only": false,
"max_nestin... | 4 | module
public import Foundation.FirstOrder.Arithmetic.IOpen.Basic
@[expose] public section
/-!
# On $\mathrm{Pow2}(x)$ and $\mathrm{BitLen}(x, y)$
$\mathrm{Pow2}(n)$ is a property that holds iff $n = 2^i$ for some $i$.
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section IOpen
va... | module
public import Foundation.FirstOrder.Arithmetic.IOpen.Basic
@[expose] public section
/-!
# On $\mathrm{Pow2}(x)$ and $\mathrm{BitLen}(x, y)$
$\mathrm{Pow2}(n)$ is a property that holds iff $n = 2^i$ for some $i$.
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section IOpen
va... | @@ -75,16 +75,29 @@
lemma pow2_mul_four {a : V} : Pow2 (4 * a) ↔ Pow2 a := by
simp [←two_mul_two_eq_four, mul_assoc, pow2_mul_two]
-@[simp] lemma pow2_two : Pow2 (2 : V) := sorry
+lemma Pow2.elim {p : V} : Pow2 p ↔ p = 1 ∨ ∃ q, p = 2 * q ∧ Pow2 q :=
+ ⟨by intro H
+ by_cases hp : 1 < p
+ · have : 2 ∣ 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_8952d0b07064_13 | 10da4bdb64628e52 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Exponential/Pow2.lean | Pow2 | 13 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 32 | 4 | [
{
"theorem_name": "pow2_two",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " Pow2.elim.mpr (Or.inr ⟨1, by simp⟩)",
"n_chars": 40,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automatio... | [
{
"name": "Pow2.elim",
"text": "lemma Pow2.elim {p : V} : Pow2 p ↔ p = 1 ∨ ∃ q, p = 2 * q ∧ Pow2 q :=\n ⟨by intro H\n by_cases hp : 1 < p\n · have : 2 ∣ p := H.two_dvd hp\n rcases this with ⟨q, rfl⟩\n right; exact ⟨q, rfl, pow2_mul_two.mp H⟩\n · have : p = 1 := le_antisymm (by ... | [
{
"name": "lt_iff_two_mul_le",
"fan_in": 4,
"n_deps_direct": 1,
"n_deps_transitive": 6,
"n_lines": 6,
"n_chars": 228,
"n_subproofs": 0,
"n_tactics": 5,
"cyclomatic": 1,
"n_automation": 2,
"n_rewrites": 1,
"n_structural": 0,
"automation_only": false,
"max_nesti... | 5 | module
public import Foundation.FirstOrder.Arithmetic.IOpen.Basic
@[expose] public section
/-!
# On $\mathrm{Pow2}(x)$ and $\mathrm{BitLen}(x, y)$
$\mathrm{Pow2}(n)$ is a property that holds iff $n = 2^i$ for some $i$.
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section IOpen
va... | module
public import Foundation.FirstOrder.Arithmetic.IOpen.Basic
@[expose] public section
/-!
# On $\mathrm{Pow2}(x)$ and $\mathrm{BitLen}(x, y)$
$\mathrm{Pow2}(n)$ is a property that holds iff $n = 2^i$ for some $i$.
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section IOpen
va... | @@ -75,18 +75,34 @@
lemma pow2_mul_four {a : V} : Pow2 (4 * a) ↔ Pow2 a := by
simp [←two_mul_two_eq_four, mul_assoc, pow2_mul_two]
-@[simp] lemma pow2_two : Pow2 (2 : V) := sorry
+lemma Pow2.elim {p : V} : Pow2 p ↔ p = 1 ∨ ∃ q, p = 2 * q ∧ Pow2 q :=
+ ⟨by intro H
+ by_cases hp : 1 < p
+ · have : 2 ∣ 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_8952d0b07064_14 | d064dbdd2e42f90f | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Exponential/Pow2.lean | Pow2 | 14 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 32 | 4 | [
{
"theorem_name": "pow2_two",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " Pow2.elim.mpr (Or.inr ⟨1, by simp⟩)",
"n_chars": 40,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automatio... | [
{
"name": "Pow2.elim",
"text": "lemma Pow2.elim {p : V} : Pow2 p ↔ p = 1 ∨ ∃ q, p = 2 * q ∧ Pow2 q :=\n ⟨by intro H\n by_cases hp : 1 < p\n · have : 2 ∣ p := H.two_dvd hp\n rcases this with ⟨q, rfl⟩\n right; exact ⟨q, rfl, pow2_mul_two.mp H⟩\n · have : p = 1 := le_antisymm (by ... | [
{
"name": "sq_or_dsq",
"fan_in": 0,
"n_deps_direct": 2,
"n_deps_transitive": 5,
"n_lines": 16,
"n_chars": 745,
"n_subproofs": 2,
"n_tactics": 16,
"cyclomatic": 5,
"n_automation": 11,
"n_rewrites": 0,
"n_structural": 8,
"automation_only": false,
"max_nesting": ... | 3 | module
public import Foundation.FirstOrder.Arithmetic.IOpen.Basic
@[expose] public section
/-!
# On $\mathrm{Pow2}(x)$ and $\mathrm{BitLen}(x, y)$
$\mathrm{Pow2}(n)$ is a property that holds iff $n = 2^i$ for some $i$.
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section IOpen
va... | module
public import Foundation.FirstOrder.Arithmetic.IOpen.Basic
@[expose] public section
/-!
# On $\mathrm{Pow2}(x)$ and $\mathrm{BitLen}(x, y)$
$\mathrm{Pow2}(n)$ is a property that holds iff $n = 2^i$ for some $i$.
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section IOpen
va... | @@ -68,11 +68,21 @@
lemma pow2_mul_four {a : V} : Pow2 (4 * a) ↔ Pow2 a := by
simp [←two_mul_two_eq_four, mul_assoc, pow2_mul_two]
-@[simp] lemma pow2_two : Pow2 (2 : V) := sorry
+lemma Pow2.elim {p : V} : Pow2 p ↔ p = 1 ∨ ∃ q, p = 2 * q ∧ Pow2 q :=
+ ⟨by intro H
+ by_cases hp : 1 < p
+ · have : 2 ∣ 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_8952d0b07064_15 | badf30dd03dfc375 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Exponential/Pow2.lean | Pow2 | 15 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 32 | 4 | [
{
"theorem_name": "pow2_two",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " Pow2.elim.mpr (Or.inr ⟨1, by simp⟩)",
"n_chars": 40,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automatio... | [
{
"name": "Pow2.elim",
"text": "lemma Pow2.elim {p : V} : Pow2 p ↔ p = 1 ∨ ∃ q, p = 2 * q ∧ Pow2 q :=\n ⟨by intro H\n by_cases hp : 1 < p\n · have : 2 ∣ p := H.two_dvd hp\n rcases this with ⟨q, rfl⟩\n right; exact ⟨q, rfl, pow2_mul_two.mp H⟩\n · have : p = 1 := le_antisymm (by ... | [
{
"name": "mul_add_lt_of_mul_lt_of_pos",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 5,
"n_lines": 13,
"n_chars": 582,
"n_subproofs": 2,
"n_tactics": 10,
"cyclomatic": 3,
"n_automation": 6,
"n_rewrites": 0,
"n_structural": 3,
"automation_only": false,
... | 4 | module
public import Foundation.FirstOrder.Arithmetic.IOpen.Basic
@[expose] public section
/-!
# On $\mathrm{Pow2}(x)$ and $\mathrm{BitLen}(x, y)$
$\mathrm{Pow2}(n)$ is a property that holds iff $n = 2^i$ for some $i$.
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section IOpen
va... | module
public import Foundation.FirstOrder.Arithmetic.IOpen.Basic
@[expose] public section
/-!
# On $\mathrm{Pow2}(x)$ and $\mathrm{BitLen}(x, y)$
$\mathrm{Pow2}(n)$ is a property that holds iff $n = 2^i$ for some $i$.
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section IOpen
va... | @@ -68,11 +68,21 @@
lemma pow2_mul_four {a : V} : Pow2 (4 * a) ↔ Pow2 a := by
simp [←two_mul_two_eq_four, mul_assoc, pow2_mul_two]
-@[simp] lemma pow2_two : Pow2 (2 : V) := sorry
+lemma Pow2.elim {p : V} : Pow2 p ↔ p = 1 ∨ ∃ q, p = 2 * q ∧ Pow2 q :=
+ ⟨by intro H
+ by_cases hp : 1 < p
+ · have : 2 ∣ 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_8952d0b07064_16 | ef478df342745058 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Exponential/Pow2.lean | Pow2 | 16 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 32 | 7 | [
{
"theorem_name": "LenBit.self",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by simp [LenBit.iff_rem, pos]",
"n_chars": 30,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation":... | [
{
"name": "LenBit.iff_rem",
"text": "lemma LenBit.iff_rem {i a : V} : LenBit i a ↔ (a / i) % 2 = 1 := by\n simp only [LenBit]\n rcases mod_two (a / i) with (h | h) <;> simp [h, ←mod_eq_zero_iff_dvd]\n\n",
"fan_in": 7,
"n_lines": 5,
"n_chars": 167,
"n_subproofs": 0,
"n_tactics": 4,
... | [
{
"name": "LenBit.mod_pow2",
"fan_in": 0,
"n_deps_direct": 3,
"n_deps_transitive": 10,
"n_lines": 4,
"n_chars": 202,
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"cyclomatic": 2,
"n_automation": 1,
"n_rewrites": 1,
"n_structural": 0,
"automation_only": false,
"max_nestin... | 9 | module
public import Foundation.FirstOrder.Arithmetic.IOpen.Basic
@[expose] public section
/-!
# On $\mathrm{Pow2}(x)$ and $\mathrm{BitLen}(x, y)$
$\mathrm{Pow2}(n)$ is a property that holds iff $n = 2^i$ for some $i$.
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section IOpen
va... | module
public import Foundation.FirstOrder.Arithmetic.IOpen.Basic
@[expose] public section
/-!
# On $\mathrm{Pow2}(x)$ and $\mathrm{BitLen}(x, y)$
$\mathrm{Pow2}(n)$ is a property that holds iff $n = 2^i$ for some $i$.
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section IOpen
va... | @@ -126,10 +126,27 @@
@[simp] lemma LenBit.on_zero (a : V) : ¬LenBit a 0 := by simp [LenBit]
-@[simp] lemma LenBit.self {a : V} (pos : 0 < a) : LenBit a a := sorry
+lemma LenBit.iff_rem {i a : V} : LenBit i a ↔ (a / i) % 2 = 1 := by
+ simp only [LenBit]
+ rcases mod_two (a / i) with (h | h) <;> simp [h, ←mod_eq_... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_8952d0b07064_17 | 624491583e0ab4dc | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Exponential/Pow2.lean | Pow2 | 17 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 32 | 7 | [
{
"theorem_name": "LenBit.self",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by simp [LenBit.iff_rem, pos]",
"n_chars": 30,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation":... | [
{
"name": "LenBit.iff_rem",
"text": "lemma LenBit.iff_rem {i a : V} : LenBit i a ↔ (a / i) % 2 = 1 := by\n simp only [LenBit]\n rcases mod_two (a / i) with (h | h) <;> simp [h, ←mod_eq_zero_iff_dvd]\n\n",
"fan_in": 7,
"n_lines": 5,
"n_chars": 167,
"n_subproofs": 0,
"n_tactics": 4,
... | [
{
"name": "LenBit.add_pow2",
"fan_in": 2,
"n_deps_direct": 3,
"n_deps_transitive": 10,
"n_lines": 4,
"n_chars": 202,
"n_subproofs": 0,
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"cyclomatic": 2,
"n_automation": 1,
"n_rewrites": 1,
"n_structural": 0,
"automation_only": false,
"max_nestin... | 9 | module
public import Foundation.FirstOrder.Arithmetic.IOpen.Basic
@[expose] public section
/-!
# On $\mathrm{Pow2}(x)$ and $\mathrm{BitLen}(x, y)$
$\mathrm{Pow2}(n)$ is a property that holds iff $n = 2^i$ for some $i$.
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section IOpen
va... | module
public import Foundation.FirstOrder.Arithmetic.IOpen.Basic
@[expose] public section
/-!
# On $\mathrm{Pow2}(x)$ and $\mathrm{BitLen}(x, y)$
$\mathrm{Pow2}(n)$ is a property that holds iff $n = 2^i$ for some $i$.
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section IOpen
va... | @@ -126,13 +126,39 @@
@[simp] lemma LenBit.on_zero (a : V) : ¬LenBit a 0 := by simp [LenBit]
-@[simp] lemma LenBit.self {a : V} (pos : 0 < a) : LenBit a a := sorry
+lemma LenBit.iff_rem {i a : V} : LenBit i a ↔ (a / i) % 2 = 1 := by
+ simp only [LenBit]
+ rcases mod_two (a / i) with (h | h) <;> simp [h, ←mod_eq_... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_8952d0b07064_18 | 7c9962c6641175d0 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Exponential/Pow2.lean | Pow2 | 18 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 32 | 7 | [
{
"theorem_name": "LenBit.self",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by simp [LenBit.iff_rem, pos]",
"n_chars": 30,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation":... | [
{
"name": "LenBit.iff_rem",
"text": "lemma LenBit.iff_rem {i a : V} : LenBit i a ↔ (a / i) % 2 = 1 := by\n simp only [LenBit]\n rcases mod_two (a / i) with (h | h) <;> simp [h, ←mod_eq_zero_iff_dvd]\n\n",
"fan_in": 7,
"n_lines": 5,
"n_chars": 167,
"n_subproofs": 0,
"n_tactics": 4,
... | [
{
"name": "LenBit.add_pow2_iff_of_lt",
"fan_in": 0,
"n_deps_direct": 4,
"n_deps_transitive": 13,
"n_lines": 10,
"n_chars": 528,
"n_subproofs": 1,
"n_tactics": 8,
"cyclomatic": 2,
"n_automation": 5,
"n_rewrites": 2,
"n_structural": 2,
"automation_only": false,
... | 12 | module
public import Foundation.FirstOrder.Arithmetic.IOpen.Basic
@[expose] public section
/-!
# On $\mathrm{Pow2}(x)$ and $\mathrm{BitLen}(x, y)$
$\mathrm{Pow2}(n)$ is a property that holds iff $n = 2^i$ for some $i$.
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section IOpen
va... | module
public import Foundation.FirstOrder.Arithmetic.IOpen.Basic
@[expose] public section
/-!
# On $\mathrm{Pow2}(x)$ and $\mathrm{BitLen}(x, y)$
$\mathrm{Pow2}(n)$ is a property that holds iff $n = 2^i$ for some $i$.
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section IOpen
va... | @@ -129,15 +129,43 @@
@[simp] lemma LenBit.on_zero (a : V) : ¬LenBit a 0 := by simp [LenBit]
-@[simp] lemma LenBit.self {a : V} (pos : 0 < a) : LenBit a a := sorry
+lemma LenBit.iff_rem {i a : V} : LenBit i a ↔ (a / i) % 2 = 1 := by
+ simp only [LenBit]
+ rcases mod_two (a / i) with (h | h) <;> simp [h, ←mod_eq_... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_8952d0b07064_19 | 0b40c95165bfc7e7 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Exponential/Pow2.lean | Pow2 | 19 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 32 | 7 | [
{
"theorem_name": "LenBit.self",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by simp [LenBit.iff_rem, pos]",
"n_chars": 30,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation":... | [
{
"name": "LenBit.iff_rem",
"text": "lemma LenBit.iff_rem {i a : V} : LenBit i a ↔ (a / i) % 2 = 1 := by\n simp only [LenBit]\n rcases mod_two (a / i) with (h | h) <;> simp [h, ←mod_eq_zero_iff_dvd]\n\n",
"fan_in": 7,
"n_lines": 5,
"n_chars": 167,
"n_subproofs": 0,
"n_tactics": 4,
... | [
{
"name": "lenbit_iff_add_mul",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 14,
"n_chars": 740,
"n_subproofs": 2,
"n_tactics": 11,
"cyclomatic": 3,
"n_automation": 6,
"n_rewrites": 0,
"n_structural": 5,
"automation_only": false,
"max_ne... | 1 | module
public import Foundation.FirstOrder.Arithmetic.IOpen.Basic
@[expose] public section
/-!
# On $\mathrm{Pow2}(x)$ and $\mathrm{BitLen}(x, y)$
$\mathrm{Pow2}(n)$ is a property that holds iff $n = 2^i$ for some $i$.
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section IOpen
va... | module
public import Foundation.FirstOrder.Arithmetic.IOpen.Basic
@[expose] public section
/-!
# On $\mathrm{Pow2}(x)$ and $\mathrm{BitLen}(x, y)$
$\mathrm{Pow2}(n)$ is a property that holds iff $n = 2^i$ for some $i$.
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section IOpen
va... | @@ -106,10 +106,27 @@
@[simp] lemma LenBit.on_zero (a : V) : ¬LenBit a 0 := by simp [LenBit]
-@[simp] lemma LenBit.self {a : V} (pos : 0 < a) : LenBit a a := sorry
+lemma LenBit.iff_rem {i a : V} : LenBit i a ↔ (a / i) % 2 = 1 := by
+ simp only [LenBit]
+ rcases mod_two (a / i) with (h | h) <;> simp [h, ←mod_eq_... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_8952d0b07064_20 | 1569a0ab1e950309 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Exponential/Pow2.lean | Pow2 | 20 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 32 | 1 | [
{
"theorem_name": "not_lenbit_iff_add_mul",
"depth": 1,
"n_commands": 0,
"n_lines": 11,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n constructor\n · intro h\n have : 2 * ((a / i) / 2) = a / i := by simpa [not_lenbit_iff_rem.mp h] using div_add_... | [
{
"name": "not_lenbit_iff_rem",
"text": "lemma not_lenbit_iff_rem {i a : V} : ¬LenBit i a ↔ (a / i) % 2 = 0 := by\n simp [LenBit, ←mod_eq_zero_iff_dvd]\n\n",
"fan_in": 1,
"n_lines": 4,
"n_chars": 117,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
"n_automation": 1,
"n_r... | [
{
"name": "not_lenbit_iff_add_mul",
"fan_in": 2,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 14,
"n_chars": 710,
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"cyclomatic": 3,
"n_automation": 6,
"n_rewrites": 0,
"n_structural": 5,
"automation_only": false,
"ma... | 1 | module
public import Foundation.FirstOrder.Arithmetic.IOpen.Basic
@[expose] public section
/-!
# On $\mathrm{Pow2}(x)$ and $\mathrm{BitLen}(x, y)$
$\mathrm{Pow2}(n)$ is a property that holds iff $n = 2^i$ for some $i$.
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section IOpen
va... | module
public import Foundation.FirstOrder.Arithmetic.IOpen.Basic
@[expose] public section
/-!
# On $\mathrm{Pow2}(x)$ and $\mathrm{BitLen}(x, y)$
$\mathrm{Pow2}(n)$ is a property that holds iff $n = 2^i$ for some $i$.
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section IOpen
va... | @@ -110,6 +110,9 @@
simp only [LenBit]
rcases mod_two (a / i) with (h | h) <;> simp [h, ←mod_eq_zero_iff_dvd]
+lemma not_lenbit_iff_rem {i a : V} : ¬LenBit i a ↔ (a / i) % 2 = 0 := by
+ simp [LenBit, ←mod_eq_zero_iff_dvd]
+
@[simp] lemma LenBit.self {a : V} (pos : 0 < a) : LenBit a a := by simp [LenBit.iff_re... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_8952d0b07064_21 | e670b42cc984b07b | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Exponential/Pow2.lean | Pow2 | 21 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 32 | 7 | [
{
"theorem_name": "LenBit.self",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by simp [LenBit.iff_rem, pos]",
"n_chars": 30,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation":... | [
{
"name": "LenBit.iff_rem",
"text": "lemma LenBit.iff_rem {i a : V} : LenBit i a ↔ (a / i) % 2 = 1 := by\n simp only [LenBit]\n rcases mod_two (a / i) with (h | h) <;> simp [h, ←mod_eq_zero_iff_dvd]\n\n",
"fan_in": 7,
"n_lines": 5,
"n_chars": 167,
"n_subproofs": 0,
"n_tactics": 4,
... | [
{
"name": "lenbit_mul_add",
"fan_in": 1,
"n_deps_direct": 3,
"n_deps_transitive": 10,
"n_lines": 23,
"n_chars": 1069,
"n_subproofs": 4,
"n_tactics": 20,
"cyclomatic": 3,
"n_automation": 10,
"n_rewrites": 0,
"n_structural": 4,
"automation_only": false,
"max_nes... | 9 | module
public import Foundation.FirstOrder.Arithmetic.IOpen.Basic
@[expose] public section
/-!
# On $\mathrm{Pow2}(x)$ and $\mathrm{BitLen}(x, y)$
$\mathrm{Pow2}(n)$ is a property that holds iff $n = 2^i$ for some $i$.
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section IOpen
va... | module
public import Foundation.FirstOrder.Arithmetic.IOpen.Basic
@[expose] public section
/-!
# On $\mathrm{Pow2}(x)$ and $\mathrm{BitLen}(x, y)$
$\mathrm{Pow2}(n)$ is a property that holds iff $n = 2^i$ for some $i$.
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section IOpen
va... | @@ -106,12 +106,29 @@
@[simp] lemma LenBit.on_zero (a : V) : ¬LenBit a 0 := by simp [LenBit]
+lemma LenBit.iff_rem {i a : V} : LenBit i a ↔ (a / i) % 2 = 1 := by
+ simp only [LenBit]
+ rcases mod_two (a / i) with (h | h) <;> simp [h, ←mod_eq_zero_iff_dvd]
+
lemma not_lenbit_iff_rem {i a : V} : ¬LenBit i a ↔ (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_8952d0b07064_23 | 1448d6bc3d3f3a83 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Exponential/Pow2.lean | Pow2 | 23 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 32 | 1 | [
{
"theorem_name": "not_lenbit_iff_add_mul",
"depth": 1,
"n_commands": 0,
"n_lines": 11,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n constructor\n · intro h\n have : 2 * ((a / i) / 2) = a / i := by simpa [not_lenbit_iff_rem.mp h] using div_add_... | [
{
"name": "not_lenbit_iff_rem",
"text": "lemma not_lenbit_iff_rem {i a : V} : ¬LenBit i a ↔ (a / i) % 2 = 0 := by\n simp [LenBit, ←mod_eq_zero_iff_dvd]\n\n",
"fan_in": 1,
"n_lines": 4,
"n_chars": 117,
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"n_tactics": 2,
"cyclomatic": 1,
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"n_r... | [
{
"name": "lenbit_sub_pow2_iff_of_lenbit",
"fan_in": 0,
"n_deps_direct": 2,
"n_deps_transitive": 20,
"n_lines": 11,
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... | 19 | module
public import Foundation.FirstOrder.Arithmetic.IOpen.Basic
@[expose] public section
/-!
# On $\mathrm{Pow2}(x)$ and $\mathrm{BitLen}(x, y)$
$\mathrm{Pow2}(n)$ is a property that holds iff $n = 2^i$ for some $i$.
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section IOpen
va... | module
public import Foundation.FirstOrder.Arithmetic.IOpen.Basic
@[expose] public section
/-!
# On $\mathrm{Pow2}(x)$ and $\mathrm{BitLen}(x, y)$
$\mathrm{Pow2}(n)$ is a property that holds iff $n = 2^i$ for some $i$.
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section IOpen
va... | @@ -130,6 +130,9 @@
simp only [LenBit]
rcases mod_two (a / i) with (h | h) <;> simp [h, ←mod_eq_zero_iff_dvd]
+lemma not_lenbit_iff_rem {i a : V} : ¬LenBit i a ↔ (a / i) % 2 = 0 := by
+ simp [LenBit, ←mod_eq_zero_iff_dvd]
+
@[simp] lemma LenBit.self {a : V} (pos : 0 < a) : LenBit a a := by simp [LenBit.iff_re... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_01fc1f3fd842_0 | 74baa03e09bfe71b | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Syntax/Predicate/Term.lean | Term | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 2 | 1 | [
{
"theorem_name": "complexity_func_lt",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n simpa [complexity_func, Nat.lt_add_one_iff] using Finset.le_sup (f := fun i ↦ complexity (v i)) (by simp)",
"n_chars... | [
{
"name": "complexity_func",
"text": "lemma complexity_func {k} (f : L.Func k) (v : Fin k → Semiterm L ξ n) : (func f v).complexity = Finset.sup Finset.univ (fun i ↦ complexity (v i)) + 1 := rfl\n\n",
"fan_in": 1,
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"n_chars": 163,
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"cyclo... | [
{
"name": "complexity_func_lt",
"fan_in": 0,
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"automation_only": true,
"max_nesti... | 1 | module
public import Foundation.Syntax.Predicate.Language
public import Foundation.Vorspiel.String
@[expose] public section
/-!
# Terms of first-order logic
This file defines the terms of first-order logic.
The bounded variables are denoted by `#x` for `x : Fin n`, and free variables are denoted by `&x` for `x : ξ... | module
public import Foundation.Syntax.Predicate.Language
public import Foundation.Vorspiel.String
@[expose] public section
/-!
# Terms of first-order logic
This file defines the terms of first-order logic.
The bounded variables are denoted by `#x` for `x : Fin n`, and free variables are denoted by `&x` for `x : ξ... | @@ -99,8 +99,11 @@
@[simp] lemma complexity_fvar (x : ξ) : (&x : Semiterm L ξ n).complexity = 0 := rfl
+lemma complexity_func {k} (f : L.Func k) (v : Fin k → Semiterm L ξ n) : (func f v).complexity = Finset.sup Finset.univ (fun i ↦ complexity (v i)) + 1 := rfl
+
@[simp] lemma complexity_func_lt {k} (f : L.Func k)... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_01fc1f3fd842_1 | 2169cd10bc71bcf4 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Syntax/Predicate/Term.lean | Term | 1 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 2 | 2 | [
{
"theorem_name": "fvar?_func",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by simp [FVar?, freeVariables_func]",
"n_chars": 36,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automat... | [
{
"name": "freeVariables_func",
"text": "lemma freeVariables_func {k} (f : L.Func k) (v : Fin k → Semiterm L ξ n) :\n (func f v).freeVariables = .biUnion .univ fun i ↦ (v i).freeVariables := rfl\n\n",
"fan_in": 2,
"n_lines": 4,
"n_chars": 162,
"n_subproofs": 0,
"n_tactics": 1,
"cy... | [
{
"name": "fvar?_func",
"fan_in": 0,
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"n_deps_transitive": 2,
"n_lines": 4,
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"automation_only": true,
"max_nesting": 1
... | 1 | module
public import Foundation.Syntax.Predicate.Language
public import Foundation.Vorspiel.String
@[expose] public section
/-!
# Terms of first-order logic
This file defines the terms of first-order logic.
The bounded variables are denoted by `#x` for `x : Fin n`, and free variables are denoted by `&x` for `x : ξ... | module
public import Foundation.Syntax.Predicate.Language
public import Foundation.Vorspiel.String
@[expose] public section
/-!
# Terms of first-order logic
This file defines the terms of first-order logic.
The bounded variables are denoted by `#x` for `x : Fin n`, and free variables are denoted by `&x` for `x : ξ... | @@ -151,6 +151,9 @@
@[simp] lemma freeVariables_fvar : (&x : Semiterm L ξ n).freeVariables = {x} := rfl
+lemma freeVariables_func {k} (f : L.Func k) (v : Fin k → Semiterm L ξ n) :
+ (func f v).freeVariables = .biUnion .univ fun i ↦ (v i).freeVariables := rfl
+
@[simp] lemma freeVariables_constant (f : L.Func 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_eb0310953f4a_0 | 68966ee5aa121e95 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/HFS/Vec.lean | Vec | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 21 | 2 | [
{
"theorem_name": "nil_or_adjoin",
"depth": 1,
"n_commands": 0,
"n_lines": 4,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n rcases zero_or_succ z with (rfl | ⟨z, rfl⟩)\n · left; rfl\n · right; exact ⟨π₁ z, π₂ z, by simp [succ_eq_adjoin]⟩",
"n_... | [
{
"name": "succ_eq_adjoin",
"text": "lemma succ_eq_adjoin (x : V) : x + 1 = π₁ x ∷ π₂ x := by simp [adjoin_def]\n\n",
"fan_in": 2,
"n_lines": 3,
"n_chars": 84,
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"a... | [
{
"name": "nil_or_adjoin",
"fan_in": 5,
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"n_deps_transitive": 2,
"n_lines": 6,
"n_chars": 200,
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"cyclomatic": 2,
"n_automation": 4,
"n_rewrites": 0,
"n_structural": 2,
"automation_only": false,
"max_nesting":... | 1 | module
public import Foundation.FirstOrder.Arithmetic.HFS.Fixpoint
@[expose] public section
/-!
# Vec
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
section adjoin
noncomputable instance : Adjoin V V := ⟨(⟪·, ·⟫ + 1)⟩
scoped infixr:67 " ∷ " => adjoin
syntax "?... | module
public import Foundation.FirstOrder.Arithmetic.HFS.Fixpoint
@[expose] public section
/-!
# Vec
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
section adjoin
noncomputable instance : Adjoin V V := ⟨(⟪·, ·⟫ + 1)⟩
scoped infixr:67 " ∷ " => adjoin
syntax "?... | @@ -38,6 +38,8 @@
@[simp] lemma sndIdx_adjoin (x v : V) : sndIdx (x ∷ v) = v := by simp [adjoin_def, sndIdx]
+lemma succ_eq_adjoin (x : V) : x + 1 = π₁ x ∷ π₂ x := by simp [adjoin_def]
+
@[simp] lemma lt_adjoin (x v : V) : x < x ∷ v := by simp [adjoin_def, lt_succ_iff_le]
@[simp] lemma lt_adjoin' (x v : V) : v... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_eb0310953f4a_1 | 04d337b6f9777ddc | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/HFS/Vec.lean | Vec | 1 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 21 | 2 | [
{
"theorem_name": "graph_unique",
"depth": 1,
"n_commands": 0,
"n_lines": 9,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n induction i using ISigma1.pi1_succ_induction generalizing v x₁ x₂\n · definability\n case zero =>\n simp only [graph_zero... | [
{
"name": "graph_zero",
"text": "lemma graph_zero {v x : V} :\n Graph ⟪v, 0, x⟫ ↔ x = fstIdx v := by\n constructor\n · intro h\n rcases graph_case.mp h with (⟨w, h⟩ | ⟨w, i, x, h, _⟩)\n · rcases show v = w ∧ x = fstIdx w by simpa using h with ⟨rfl, rfl, rfl⟩; rfl\n · simp at h\n · rintro rfl;... | [
{
"name": "graph_unique",
"fan_in": 4,
"n_deps_direct": 2,
"n_deps_transitive": 3,
"n_lines": 11,
"n_chars": 330,
"n_subproofs": 0,
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"n_rewrites": 0,
"n_structural": 3,
"automation_only": false,
"max_nesting"... | 2 | module
public import Foundation.FirstOrder.Arithmetic.HFS.Fixpoint
@[expose] public section
/-!
# Vec
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
section adjoin
noncomputable instance : Adjoin V V := ⟨(⟪·, ·⟫ + 1)⟩
scoped infixr:67 " ∷ " => adjoin
syntax "?... | module
public import Foundation.FirstOrder.Arithmetic.HFS.Fixpoint
@[expose] public section
/-!
# Vec
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
section adjoin
noncomputable instance : Adjoin V V := ⟨(⟪·, ·⟫ + 1)⟩
scoped infixr:67 " ∷ " => adjoin
syntax "?... | @@ -161,6 +161,15 @@
(∃ v, pr = ⟪v, 0, fstIdx v⟫) ∨ (∃ v i x, pr = ⟪v, i + 1, x⟫ ∧ Graph ⟪sndIdx v, i, x⟫) :=
adjointruction.case
+lemma graph_zero {v x : V} :
+ Graph ⟪v, 0, x⟫ ↔ x = fstIdx v := by
+ constructor
+ · intro h
+ rcases graph_case.mp h with (⟨w, h⟩ | ⟨w, i, x, h, _⟩)
+ · rcases show v... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_eb0310953f4a_2 | aae4eece93dde6e0 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/HFS/Vec.lean | Vec | 2 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 21 | 2 | [
{
"theorem_name": "nil_or_adjoin",
"depth": 1,
"n_commands": 0,
"n_lines": 4,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n rcases zero_or_succ z with (rfl | ⟨z, rfl⟩)\n · left; rfl\n · right; exact ⟨π₁ z, π₂ z, by simp [succ_eq_adjoin]⟩",
"n_... | [
{
"name": "succ_eq_adjoin",
"text": "lemma succ_eq_adjoin (x : V) : x + 1 = π₁ x ∷ π₂ x := by simp [adjoin_def]\n\n",
"fan_in": 2,
"n_lines": 3,
"n_chars": 84,
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"cyclomatic": 1,
"n_automation": 1,
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"a... | [
{
"name": "nth_lt_len",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 5,
"n_lines": 10,
"n_chars": 304,
"n_subproofs": 0,
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"cyclomatic": 3,
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"n_rewrites": 0,
"n_structural": 2,
"automation_only": false,
"max_nesting": 4... | 3 | module
public import Foundation.FirstOrder.Arithmetic.HFS.Fixpoint
@[expose] public section
/-!
# Vec
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
section adjoin
noncomputable instance : Adjoin V V := ⟨(⟪·, ·⟫ + 1)⟩
scoped infixr:67 " ∷ " => adjoin
syntax "?... | module
public import Foundation.FirstOrder.Arithmetic.HFS.Fixpoint
@[expose] public section
/-!
# Vec
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
section adjoin
noncomputable instance : Adjoin V V := ⟨(⟪·, ·⟫ + 1)⟩
scoped infixr:67 " ∷ " => adjoin
syntax "?... | @@ -38,6 +38,8 @@
@[simp] lemma sndIdx_adjoin (x v : V) : sndIdx (x ∷ v) = v := by simp [adjoin_def, sndIdx]
+lemma succ_eq_adjoin (x : V) : x + 1 = π₁ x ∷ π₂ x := by simp [adjoin_def]
+
@[simp] lemma lt_adjoin (x v : V) : x < x ∷ v := by simp [adjoin_def, lt_succ_iff_le]
@[simp] lemma lt_adjoin' (x v : V) : v... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_eb0310953f4a_3 | 769d3683f868ffa4 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/HFS/Vec.lean | Vec | 3 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 21 | 2 | [
{
"theorem_name": "nil_or_adjoin",
"depth": 1,
"n_commands": 0,
"n_lines": 4,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n rcases zero_or_succ z with (rfl | ⟨z, rfl⟩)\n · left; rfl\n · right; exact ⟨π₁ z, π₂ z, by simp [succ_eq_adjoin]⟩",
"n_... | [
{
"name": "succ_eq_adjoin",
"text": "lemma succ_eq_adjoin (x : V) : x + 1 = π₁ x ∷ π₂ x := by simp [adjoin_def]\n\n",
"fan_in": 2,
"n_lines": 3,
"n_chars": 84,
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"cyclomatic": 1,
"n_automation": 1,
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"a... | [
{
"name": "nth_ext",
"fan_in": 3,
"n_deps_direct": 3,
"n_deps_transitive": 6,
"n_lines": 13,
"n_chars": 591,
"n_subproofs": 2,
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"cyclomatic": 3,
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"n_rewrites": 0,
"n_structural": 4,
"automation_only": false,
"max_nesting": 4
... | 3 | module
public import Foundation.FirstOrder.Arithmetic.HFS.Fixpoint
@[expose] public section
/-!
# Vec
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
section adjoin
noncomputable instance : Adjoin V V := ⟨(⟪·, ·⟫ + 1)⟩
scoped infixr:67 " ∷ " => adjoin
syntax "?... | module
public import Foundation.FirstOrder.Arithmetic.HFS.Fixpoint
@[expose] public section
/-!
# Vec
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
section adjoin
noncomputable instance : Adjoin V V := ⟨(⟪·, ·⟫ + 1)⟩
scoped infixr:67 " ∷ " => adjoin
syntax "?... | @@ -38,6 +38,8 @@
@[simp] lemma sndIdx_adjoin (x v : V) : sndIdx (x ∷ v) = v := by simp [adjoin_def, sndIdx]
+lemma succ_eq_adjoin (x : V) : x + 1 = π₁ x ∷ π₂ x := by simp [adjoin_def]
+
@[simp] lemma lt_adjoin (x v : V) : x < x ∷ v := by simp [adjoin_def, lt_succ_iff_le]
@[simp] lemma lt_adjoin' (x v : V) : v... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_eb0310953f4a_4 | 121d9abb1b6a4530 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/HFS/Vec.lean | Vec | 4 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 21 | 2 | [
{
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"depth": 1,
"n_commands": 0,
"n_lines": 4,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n rcases zero_or_succ z with (rfl | ⟨z, rfl⟩)\n · left; rfl\n · right; exact ⟨π₁ z, π₂ z, by simp [succ_eq_adjoin]⟩",
"n_... | [
{
"name": "succ_eq_adjoin",
"text": "lemma succ_eq_adjoin (x : V) : x + 1 = π₁ x ∷ π₂ x := by simp [adjoin_def]\n\n",
"fan_in": 2,
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"a... | [
{
"name": "nth_ext'",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 7,
"n_lines": 4,
"n_chars": 206,
"n_subproofs": 0,
"n_tactics": 3,
"cyclomatic": 2,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 2,
"automation_only": false,
"max_nesting": 2
... | 4 | module
public import Foundation.FirstOrder.Arithmetic.HFS.Fixpoint
@[expose] public section
/-!
# Vec
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
section adjoin
noncomputable instance : Adjoin V V := ⟨(⟪·, ·⟫ + 1)⟩
scoped infixr:67 " ∷ " => adjoin
syntax "?... | module
public import Foundation.FirstOrder.Arithmetic.HFS.Fixpoint
@[expose] public section
/-!
# Vec
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
section adjoin
noncomputable instance : Adjoin V V := ⟨(⟪·, ·⟫ + 1)⟩
scoped infixr:67 " ∷ " => adjoin
syntax "?... | @@ -38,6 +38,8 @@
@[simp] lemma sndIdx_adjoin (x v : V) : sndIdx (x ∷ v) = v := by simp [adjoin_def, sndIdx]
+lemma succ_eq_adjoin (x : V) : x + 1 = π₁ x ∷ π₂ x := by simp [adjoin_def]
+
@[simp] lemma lt_adjoin (x v : V) : x < x ∷ v := by simp [adjoin_def, lt_succ_iff_le]
@[simp] lemma lt_adjoin' (x v : V) : v... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_eb0310953f4a_5 | 561148da7bdbb84f | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/HFS/Vec.lean | Vec | 5 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 21 | 2 | [
{
"theorem_name": "nil_or_adjoin",
"depth": 1,
"n_commands": 0,
"n_lines": 4,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n rcases zero_or_succ z with (rfl | ⟨z, rfl⟩)\n · left; rfl\n · right; exact ⟨π₁ z, π₂ z, by simp [succ_eq_adjoin]⟩",
"n_... | [
{
"name": "succ_eq_adjoin",
"text": "lemma succ_eq_adjoin (x : V) : x + 1 = π₁ x ∷ π₂ x := by simp [adjoin_def]\n\n",
"fan_in": 2,
"n_lines": 3,
"n_chars": 84,
"n_subproofs": 0,
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"n_automation": 1,
"n_rewrites": 0,
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"a... | [
{
"name": "le_of_nth_le_nth",
"fan_in": 1,
"n_deps_direct": 3,
"n_deps_transitive": 6,
"n_lines": 12,
"n_chars": 566,
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"cyclomatic": 3,
"n_automation": 9,
"n_rewrites": 0,
"n_structural": 4,
"automation_only": false,
"max_nest... | 4 | module
public import Foundation.FirstOrder.Arithmetic.HFS.Fixpoint
@[expose] public section
/-!
# Vec
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
section adjoin
noncomputable instance : Adjoin V V := ⟨(⟪·, ·⟫ + 1)⟩
scoped infixr:67 " ∷ " => adjoin
syntax "?... | module
public import Foundation.FirstOrder.Arithmetic.HFS.Fixpoint
@[expose] public section
/-!
# Vec
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
section adjoin
noncomputable instance : Adjoin V V := ⟨(⟪·, ·⟫ + 1)⟩
scoped infixr:67 " ∷ " => adjoin
syntax "?... | @@ -38,6 +38,8 @@
@[simp] lemma sndIdx_adjoin (x v : V) : sndIdx (x ∷ v) = v := by simp [adjoin_def, sndIdx]
+lemma succ_eq_adjoin (x : V) : x + 1 = π₁ x ∷ π₂ x := by simp [adjoin_def]
+
@[simp] lemma lt_adjoin (x v : V) : x < x ∷ v := by simp [adjoin_def, lt_succ_iff_le]
@[simp] lemma lt_adjoin' (x v : V) : v... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_eb0310953f4a_6 | 95fc87f404242483 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/HFS/Vec.lean | Vec | 6 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 21 | 2 | [
{
"theorem_name": "nil_or_adjoin",
"depth": 1,
"n_commands": 0,
"n_lines": 4,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n rcases zero_or_succ z with (rfl | ⟨z, rfl⟩)\n · left; rfl\n · right; exact ⟨π₁ z, π₂ z, by simp [succ_eq_adjoin]⟩",
"n_... | [
{
"name": "succ_eq_adjoin",
"text": "lemma succ_eq_adjoin (x : V) : x + 1 = π₁ x ∷ π₂ x := by simp [adjoin_def]\n\n",
"fan_in": 2,
"n_lines": 3,
"n_chars": 84,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"a... | [
{
"name": "nth_lt_self",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 5,
"n_lines": 10,
"n_chars": 327,
"n_subproofs": 0,
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"cyclomatic": 3,
"n_automation": 7,
"n_rewrites": 0,
"n_structural": 2,
"automation_only": false,
"max_nesting": ... | 3 | module
public import Foundation.FirstOrder.Arithmetic.HFS.Fixpoint
@[expose] public section
/-!
# Vec
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
section adjoin
noncomputable instance : Adjoin V V := ⟨(⟪·, ·⟫ + 1)⟩
scoped infixr:67 " ∷ " => adjoin
syntax "?... | module
public import Foundation.FirstOrder.Arithmetic.HFS.Fixpoint
@[expose] public section
/-!
# Vec
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
section adjoin
noncomputable instance : Adjoin V V := ⟨(⟪·, ·⟫ + 1)⟩
scoped infixr:67 " ∷ " => adjoin
syntax "?... | @@ -38,6 +38,8 @@
@[simp] lemma sndIdx_adjoin (x v : V) : sndIdx (x ∷ v) = v := by simp [adjoin_def, sndIdx]
+lemma succ_eq_adjoin (x : V) : x + 1 = π₁ x ∷ π₂ x := by simp [adjoin_def]
+
@[simp] lemma lt_adjoin (x v : V) : x < x ∷ v := by simp [adjoin_def, lt_succ_iff_le]
@[simp] lemma lt_adjoin' (x v : V) : v... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_eb0310953f4a_7 | 29673e1c4377a239 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/HFS/Vec.lean | Vec | 7 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 21 | 2 | [
{
"theorem_name": "nil_or_adjoin",
"depth": 1,
"n_commands": 0,
"n_lines": 4,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n rcases zero_or_succ z with (rfl | ⟨z, rfl⟩)\n · left; rfl\n · right; exact ⟨π₁ z, π₂ z, by simp [succ_eq_adjoin]⟩",
"n_... | [
{
"name": "succ_eq_adjoin",
"text": "lemma succ_eq_adjoin (x : V) : x + 1 = π₁ x ∷ π₂ x := by simp [adjoin_def]\n\n",
"fan_in": 2,
"n_lines": 3,
"n_chars": 84,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"a... | [
{
"name": "eq_singleton_iff_len_eq_one",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 7,
"n_lines": 6,
"n_chars": 195,
"n_subproofs": 0,
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"cyclomatic": 3,
"n_automation": 4,
"n_rewrites": 0,
"n_structural": 4,
"automation_only": false,
... | 4 | module
public import Foundation.FirstOrder.Arithmetic.HFS.Fixpoint
@[expose] public section
/-!
# Vec
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
section adjoin
noncomputable instance : Adjoin V V := ⟨(⟪·, ·⟫ + 1)⟩
scoped infixr:67 " ∷ " => adjoin
syntax "?... | module
public import Foundation.FirstOrder.Arithmetic.HFS.Fixpoint
@[expose] public section
/-!
# Vec
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
section adjoin
noncomputable instance : Adjoin V V := ⟨(⟪·, ·⟫ + 1)⟩
scoped infixr:67 " ∷ " => adjoin
syntax "?... | @@ -38,6 +38,8 @@
@[simp] lemma sndIdx_adjoin (x v : V) : sndIdx (x ∷ v) = v := by simp [adjoin_def, sndIdx]
+lemma succ_eq_adjoin (x : V) : x + 1 = π₁ x ∷ π₂ x := by simp [adjoin_def]
+
@[simp] lemma lt_adjoin (x v : V) : x < x ∷ v := by simp [adjoin_def, lt_succ_iff_le]
@[simp] lemma lt_adjoin' (x v : V) : v... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_eb0310953f4a_8 | 88151d8b6be86024 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/HFS/Vec.lean | Vec | 8 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 21 | 2 | [
{
"theorem_name": "nil_or_adjoin",
"depth": 1,
"n_commands": 0,
"n_lines": 4,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n rcases zero_or_succ z with (rfl | ⟨z, rfl⟩)\n · left; rfl\n · right; exact ⟨π₁ z, π₂ z, by simp [succ_eq_adjoin]⟩",
"n_... | [
{
"name": "succ_eq_adjoin",
"text": "lemma succ_eq_adjoin (x : V) : x + 1 = π₁ x ∷ π₂ x := by simp [adjoin_def]\n\n",
"fan_in": 2,
"n_lines": 3,
"n_chars": 84,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"a... | [
{
"name": "eq_doubleton_of_len_eq_two",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 7,
"n_lines": 14,
"n_chars": 333,
"n_subproofs": 0,
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"cyclomatic": 3,
"n_automation": 4,
"n_rewrites": 0,
"n_structural": 4,
"automation_only": false,
... | 4 | module
public import Foundation.FirstOrder.Arithmetic.HFS.Fixpoint
@[expose] public section
/-!
# Vec
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
section adjoin
noncomputable instance : Adjoin V V := ⟨(⟪·, ·⟫ + 1)⟩
scoped infixr:67 " ∷ " => adjoin
syntax "?... | module
public import Foundation.FirstOrder.Arithmetic.HFS.Fixpoint
@[expose] public section
/-!
# Vec
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
section adjoin
noncomputable instance : Adjoin V V := ⟨(⟪·, ·⟫ + 1)⟩
scoped infixr:67 " ∷ " => adjoin
syntax "?... | @@ -38,6 +38,8 @@
@[simp] lemma sndIdx_adjoin (x v : V) : sndIdx (x ∷ v) = v := by simp [adjoin_def, sndIdx]
+lemma succ_eq_adjoin (x : V) : x + 1 = π₁ x ∷ π₂ x := by simp [adjoin_def]
+
@[simp] lemma lt_adjoin (x v : V) : x < x ∷ v := by simp [adjoin_def, lt_succ_iff_le]
@[simp] lemma lt_adjoin' (x v : V) : v... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_eb0310953f4a_9 | 17f3148cb5b8630f | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/HFS/Vec.lean | Vec | 9 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 21 | 2 | [
{
"theorem_name": "nil_or_adjoin",
"depth": 1,
"n_commands": 0,
"n_lines": 4,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n rcases zero_or_succ z with (rfl | ⟨z, rfl⟩)\n · left; rfl\n · right; exact ⟨π₁ z, π₂ z, by simp [succ_eq_adjoin]⟩",
"n_... | [
{
"name": "succ_eq_adjoin",
"text": "lemma succ_eq_adjoin (x : V) : x + 1 = π₁ x ∷ π₂ x := by simp [adjoin_def]\n\n",
"fan_in": 2,
"n_lines": 3,
"n_chars": 84,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"a... | [
{
"name": "nth_le_listMax",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 5,
"n_lines": 10,
"n_chars": 306,
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"n_rewrites": 0,
"n_structural": 2,
"automation_only": false,
"max_nesting... | 3 | module
public import Foundation.FirstOrder.Arithmetic.HFS.Fixpoint
@[expose] public section
/-!
# Vec
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
section adjoin
noncomputable instance : Adjoin V V := ⟨(⟪·, ·⟫ + 1)⟩
scoped infixr:67 " ∷ " => adjoin
syntax "?... | module
public import Foundation.FirstOrder.Arithmetic.HFS.Fixpoint
@[expose] public section
/-!
# Vec
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
section adjoin
noncomputable instance : Adjoin V V := ⟨(⟪·, ·⟫ + 1)⟩
scoped infixr:67 " ∷ " => adjoin
syntax "?... | @@ -38,6 +38,8 @@
@[simp] lemma sndIdx_adjoin (x v : V) : sndIdx (x ∷ v) = v := by simp [adjoin_def, sndIdx]
+lemma succ_eq_adjoin (x : V) : x + 1 = π₁ x ∷ π₂ x := by simp [adjoin_def]
+
@[simp] lemma lt_adjoin (x v : V) : x < x ∷ v := by simp [adjoin_def, lt_succ_iff_le]
@[simp] lemma lt_adjoin' (x v : V) : v... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_eb0310953f4a_10 | 598774922aa90530 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/HFS/Vec.lean | Vec | 10 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 21 | 2 | [
{
"theorem_name": "nil_or_adjoin",
"depth": 1,
"n_commands": 0,
"n_lines": 4,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n rcases zero_or_succ z with (rfl | ⟨z, rfl⟩)\n · left; rfl\n · right; exact ⟨π₁ z, π₂ z, by simp [succ_eq_adjoin]⟩",
"n_... | [
{
"name": "succ_eq_adjoin",
"text": "lemma succ_eq_adjoin (x : V) : x + 1 = π₁ x ∷ π₂ x := by simp [adjoin_def]\n\n",
"fan_in": 2,
"n_lines": 3,
"n_chars": 84,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"a... | [
{
"name": "listMaxss_le",
"fan_in": 1,
"n_deps_direct": 2,
"n_deps_transitive": 6,
"n_lines": 11,
"n_chars": 365,
"n_subproofs": 0,
"n_tactics": 9,
"cyclomatic": 3,
"n_automation": 6,
"n_rewrites": 0,
"n_structural": 3,
"automation_only": false,
"max_nesting":... | 3 | module
public import Foundation.FirstOrder.Arithmetic.HFS.Fixpoint
@[expose] public section
/-!
# Vec
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
section adjoin
noncomputable instance : Adjoin V V := ⟨(⟪·, ·⟫ + 1)⟩
scoped infixr:67 " ∷ " => adjoin
syntax "?... | module
public import Foundation.FirstOrder.Arithmetic.HFS.Fixpoint
@[expose] public section
/-!
# Vec
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
section adjoin
noncomputable instance : Adjoin V V := ⟨(⟪·, ·⟫ + 1)⟩
scoped infixr:67 " ∷ " => adjoin
syntax "?... | @@ -38,6 +38,8 @@
@[simp] lemma sndIdx_adjoin (x v : V) : sndIdx (x ∷ v) = v := by simp [adjoin_def, sndIdx]
+lemma succ_eq_adjoin (x : V) : x + 1 = π₁ x ∷ π₂ x := by simp [adjoin_def]
+
@[simp] lemma lt_adjoin (x v : V) : x < x ∷ v := by simp [adjoin_def, lt_succ_iff_le]
@[simp] lemma lt_adjoin' (x v : V) : v... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_eb0310953f4a_11 | f6776510df61dbb0 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/HFS/Vec.lean | Vec | 11 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 21 | 2 | [
{
"theorem_name": "nil_or_adjoin",
"depth": 1,
"n_commands": 0,
"n_lines": 4,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n rcases zero_or_succ z with (rfl | ⟨z, rfl⟩)\n · left; rfl\n · right; exact ⟨π₁ z, π₂ z, by simp [succ_eq_adjoin]⟩",
"n_... | [
{
"name": "succ_eq_adjoin",
"text": "lemma succ_eq_adjoin (x : V) : x + 1 = π₁ x ∷ π₂ x := by simp [adjoin_def]\n\n",
"fan_in": 2,
"n_lines": 3,
"n_chars": 84,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"a... | [
{
"name": "listMaxss_le_iff",
"fan_in": 0,
"n_deps_direct": 2,
"n_deps_transitive": 8,
"n_lines": 15,
"n_chars": 409,
"n_subproofs": 0,
"n_tactics": 5,
"cyclomatic": 2,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 4,
"automation_only": false,
"max_nesti... | 5 | module
public import Foundation.FirstOrder.Arithmetic.HFS.Fixpoint
@[expose] public section
/-!
# Vec
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
section adjoin
noncomputable instance : Adjoin V V := ⟨(⟪·, ·⟫ + 1)⟩
scoped infixr:67 " ∷ " => adjoin
syntax "?... | module
public import Foundation.FirstOrder.Arithmetic.HFS.Fixpoint
@[expose] public section
/-!
# Vec
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
section adjoin
noncomputable instance : Adjoin V V := ⟨(⟪·, ·⟫ + 1)⟩
scoped infixr:67 " ∷ " => adjoin
syntax "?... | @@ -38,6 +38,8 @@
@[simp] lemma sndIdx_adjoin (x v : V) : sndIdx (x ∷ v) = v := by simp [adjoin_def, sndIdx]
+lemma succ_eq_adjoin (x : V) : x + 1 = π₁ x ∷ π₂ x := by simp [adjoin_def]
+
@[simp] lemma lt_adjoin (x v : V) : x < x ∷ v := by simp [adjoin_def, lt_succ_iff_le]
@[simp] lemma lt_adjoin' (x v : V) : v... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_eb0310953f4a_12 | a5fa27e24b7d89a9 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/HFS/Vec.lean | Vec | 12 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 21 | 2 | [
{
"theorem_name": "nil_or_adjoin",
"depth": 1,
"n_commands": 0,
"n_lines": 4,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n rcases zero_or_succ z with (rfl | ⟨z, rfl⟩)\n · left; rfl\n · right; exact ⟨π₁ z, π₂ z, by simp [succ_eq_adjoin]⟩",
"n_... | [
{
"name": "succ_eq_adjoin",
"text": "lemma succ_eq_adjoin (x : V) : x + 1 = π₁ x ∷ π₂ x := by simp [adjoin_def]\n\n",
"fan_in": 2,
"n_lines": 3,
"n_chars": 84,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"a... | [
{
"name": "len_takeLast",
"fan_in": 0,
"n_deps_direct": 2,
"n_deps_transitive": 6,
"n_lines": 14,
"n_chars": 486,
"n_subproofs": 1,
"n_tactics": 14,
"cyclomatic": 4,
"n_automation": 6,
"n_rewrites": 0,
"n_structural": 3,
"automation_only": false,
"max_nesting"... | 4 | module
public import Foundation.FirstOrder.Arithmetic.HFS.Fixpoint
@[expose] public section
/-!
# Vec
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
section adjoin
noncomputable instance : Adjoin V V := ⟨(⟪·, ·⟫ + 1)⟩
scoped infixr:67 " ∷ " => adjoin
syntax "?... | module
public import Foundation.FirstOrder.Arithmetic.HFS.Fixpoint
@[expose] public section
/-!
# Vec
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
section adjoin
noncomputable instance : Adjoin V V := ⟨(⟪·, ·⟫ + 1)⟩
scoped infixr:67 " ∷ " => adjoin
syntax "?... | @@ -38,6 +38,8 @@
@[simp] lemma sndIdx_adjoin (x v : V) : sndIdx (x ∷ v) = v := by simp [adjoin_def, sndIdx]
+lemma succ_eq_adjoin (x : V) : x + 1 = π₁ x ∷ π₂ x := by simp [adjoin_def]
+
@[simp] lemma lt_adjoin (x v : V) : x < x ∷ v := by simp [adjoin_def, lt_succ_iff_le]
@[simp] lemma lt_adjoin' (x v : V) : v... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_eb0310953f4a_13 | 5ba173f124bb010a | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/HFS/Vec.lean | Vec | 13 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 21 | 2 | [
{
"theorem_name": "nil_or_adjoin",
"depth": 1,
"n_commands": 0,
"n_lines": 4,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n rcases zero_or_succ z with (rfl | ⟨z, rfl⟩)\n · left; rfl\n · right; exact ⟨π₁ z, π₂ z, by simp [succ_eq_adjoin]⟩",
"n_... | [
{
"name": "succ_eq_adjoin",
"text": "lemma succ_eq_adjoin (x : V) : x + 1 = π₁ x ∷ π₂ x := by simp [adjoin_def]\n\n",
"fan_in": 2,
"n_lines": 3,
"n_chars": 84,
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"cyclomatic": 1,
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"a... | [
{
"name": "takeLast_succ_of_lt",
"fan_in": 0,
"n_deps_direct": 2,
"n_deps_transitive": 6,
"n_lines": 12,
"n_chars": 644,
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"n_rewrites": 2,
"n_structural": 2,
"automation_only": false,
"max_n... | 4 | module
public import Foundation.FirstOrder.Arithmetic.HFS.Fixpoint
@[expose] public section
/-!
# Vec
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
section adjoin
noncomputable instance : Adjoin V V := ⟨(⟪·, ·⟫ + 1)⟩
scoped infixr:67 " ∷ " => adjoin
syntax "?... | module
public import Foundation.FirstOrder.Arithmetic.HFS.Fixpoint
@[expose] public section
/-!
# Vec
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
section adjoin
noncomputable instance : Adjoin V V := ⟨(⟪·, ·⟫ + 1)⟩
scoped infixr:67 " ∷ " => adjoin
syntax "?... | @@ -38,6 +38,8 @@
@[simp] lemma sndIdx_adjoin (x v : V) : sndIdx (x ∷ v) = v := by simp [adjoin_def, sndIdx]
+lemma succ_eq_adjoin (x : V) : x + 1 = π₁ x ∷ π₂ x := by simp [adjoin_def]
+
@[simp] lemma lt_adjoin (x v : V) : x < x ∷ v := by simp [adjoin_def, lt_succ_iff_le]
@[simp] lemma lt_adjoin' (x v : V) : v... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_eb0310953f4a_14 | eb56befde345fbb8 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/HFS/Vec.lean | Vec | 14 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 21 | 2 | [
{
"theorem_name": "nil_or_adjoin",
"depth": 1,
"n_commands": 0,
"n_lines": 4,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n rcases zero_or_succ z with (rfl | ⟨z, rfl⟩)\n · left; rfl\n · right; exact ⟨π₁ z, π₂ z, by simp [succ_eq_adjoin]⟩",
"n_... | [
{
"name": "succ_eq_adjoin",
"text": "lemma succ_eq_adjoin (x : V) : x + 1 = π₁ x ∷ π₂ x := by simp [adjoin_def]\n\n",
"fan_in": 2,
"n_lines": 3,
"n_chars": 84,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"a... | [
{
"name": "concat_nth_lt",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 5,
"n_lines": 10,
"n_chars": 325,
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"cyclomatic": 3,
"n_automation": 6,
"n_rewrites": 0,
"n_structural": 2,
"automation_only": false,
"max_nesting"... | 3 | module
public import Foundation.FirstOrder.Arithmetic.HFS.Fixpoint
@[expose] public section
/-!
# Vec
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
section adjoin
noncomputable instance : Adjoin V V := ⟨(⟪·, ·⟫ + 1)⟩
scoped infixr:67 " ∷ " => adjoin
syntax "?... | module
public import Foundation.FirstOrder.Arithmetic.HFS.Fixpoint
@[expose] public section
/-!
# Vec
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
section adjoin
noncomputable instance : Adjoin V V := ⟨(⟪·, ·⟫ + 1)⟩
scoped infixr:67 " ∷ " => adjoin
syntax "?... | @@ -38,6 +38,8 @@
@[simp] lemma sndIdx_adjoin (x v : V) : sndIdx (x ∷ v) = v := by simp [adjoin_def, sndIdx]
+lemma succ_eq_adjoin (x : V) : x + 1 = π₁ x ∷ π₂ x := by simp [adjoin_def]
+
@[simp] lemma lt_adjoin (x v : V) : x < x ∷ v := by simp [adjoin_def, lt_succ_iff_le]
@[simp] lemma lt_adjoin' (x v : V) : v... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_eb0310953f4a_15 | d34ae3ac2b77533d | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/HFS/Vec.lean | Vec | 15 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 21 | 1 | [
{
"theorem_name": "memVec_adjoin_iff",
"depth": 1,
"n_commands": 0,
"n_lines": 10,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n constructor\n · rintro ⟨i, h, rfl⟩\n rcases zero_or_succ i with (rfl | ⟨i, rfl⟩)\n · simp\n · right; simpa usi... | [
{
"name": "nth_mem_memVec",
"text": "lemma nth_mem_memVec {i v : V} (h : i < len v) : v.[i] ∈ᵥ v := ⟨i, by simp [h]⟩\n\n",
"fan_in": 1,
"n_lines": 3,
"n_chars": 89,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
... | [
{
"name": "memVec_adjoin_iff",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 12,
"n_chars": 373,
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"cyclomatic": 6,
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"n_rewrites": 0,
"n_structural": 6,
"automation_only": false,
"max_ne... | 1 | module
public import Foundation.FirstOrder.Arithmetic.HFS.Fixpoint
@[expose] public section
/-!
# Vec
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
section adjoin
noncomputable instance : Adjoin V V := ⟨(⟪·, ·⟫ + 1)⟩
scoped infixr:67 " ∷ " => adjoin
syntax "?... | module
public import Foundation.FirstOrder.Arithmetic.HFS.Fixpoint
@[expose] public section
/-!
# Vec
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
section adjoin
noncomputable instance : Adjoin V V := ⟨(⟪·, ·⟫ + 1)⟩
scoped infixr:67 " ∷ " => adjoin
syntax "?... | @@ -734,6 +734,8 @@
@[simp] lemma not_memVec_empty (x : V) : ¬x ∈ᵥ 0 := by rintro ⟨i, h, _⟩; simp at h
+lemma nth_mem_memVec {i v : V} (h : i < len v) : v.[i] ∈ᵥ v := ⟨i, by simp [h]⟩
+
@[simp] lemma memVec_insert_fst {x v : V} : x ∈ᵥ x ∷ v := ⟨0, by simp⟩
@[simp] lemma memVec_adjoin_iff {x y v : V} : x ∈ᵥ 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_eb0310953f4a_16 | 6d94707d3c63764a | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/HFS/Vec.lean | Vec | 16 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 21 | 2 | [
{
"theorem_name": "nil_or_adjoin",
"depth": 1,
"n_commands": 0,
"n_lines": 4,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n rcases zero_or_succ z with (rfl | ⟨z, rfl⟩)\n · left; rfl\n · right; exact ⟨π₁ z, π₂ z, by simp [succ_eq_adjoin]⟩",
"n_... | [
{
"name": "succ_eq_adjoin",
"text": "lemma succ_eq_adjoin (x : V) : x + 1 = π₁ x ∷ π₂ x := by simp [adjoin_def]\n\n",
"fan_in": 2,
"n_lines": 3,
"n_chars": 84,
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"cyclomatic": 1,
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"a... | [
{
"name": "len_repeatVec_of_nth_le",
"fan_in": 0,
"n_deps_direct": 2,
"n_deps_transitive": 8,
"n_lines": 4,
"n_chars": 194,
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"cyclomatic": 1,
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"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"max_... | 6 | module
public import Foundation.FirstOrder.Arithmetic.HFS.Fixpoint
@[expose] public section
/-!
# Vec
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
section adjoin
noncomputable instance : Adjoin V V := ⟨(⟪·, ·⟫ + 1)⟩
scoped infixr:67 " ∷ " => adjoin
syntax "?... | module
public import Foundation.FirstOrder.Arithmetic.HFS.Fixpoint
@[expose] public section
/-!
# Vec
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
section adjoin
noncomputable instance : Adjoin V V := ⟨(⟪·, ·⟫ + 1)⟩
scoped infixr:67 " ∷ " => adjoin
syntax "?... | @@ -38,6 +38,8 @@
@[simp] lemma sndIdx_adjoin (x v : V) : sndIdx (x ∷ v) = v := by simp [adjoin_def, sndIdx]
+lemma succ_eq_adjoin (x : V) : x + 1 = π₁ x ∷ π₂ x := by simp [adjoin_def]
+
@[simp] lemma lt_adjoin (x v : V) : x < x ∷ v := by simp [adjoin_def, lt_succ_iff_le]
@[simp] lemma lt_adjoin' (x v : V) : v... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_8587c672228d_0 | 9c1b713026a73ab8 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Bootstrapping/DerivabilityCondition/D1.lean | D1 | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 3 | 1 | [
{
"theorem_name": "internalize_provability",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " fun h ↦ by\n simpa using! derivable_quote (V := V) (provable_iff_derivable2.mp h).some",
"n_chars": 89,
"n_subpr... | [
{
"name": "derivable_quote",
"text": "lemma derivable_quote {Γ : Finset (Proposition L)} (d : T ⟹₂ Γ) : Derivable T (⌜Γ⌝ : V) :=\n ⟨⌜d⌝, by simpa [Semiformula.quote_def] using! (⌜d⌝ : Theory.internalize V T ⊢!ᵈᵉʳ ⌜Γ⌝).derivationOf⟩\n\n",
"fan_in": 1,
"n_lines": 4,
"n_chars": 230,
"n_subproo... | [
{
"name": "internalize_provability",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 5,
"n_chars": 221,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"max_... | 1 | module
public import Foundation.FirstOrder.Bootstrapping.Syntax
@[expose] public section
/-!
# Hilbert-Bernays-Löb derivability condition $\mathbf{D1}$ and soundness of internal provability.
-/
namespace LO.FirstOrder.Arithmetic.Bootstrapping
open Classical FirstOrder
variable {V : Type*} [ORingStructure V] [V↓[ℒₒ... | module
public import Foundation.FirstOrder.Bootstrapping.Syntax
@[expose] public section
/-!
# Hilbert-Bernays-Löb derivability condition $\mathbf{D1}$ and soundness of internal provability.
-/
namespace LO.FirstOrder.Arithmetic.Bootstrapping
open Classical FirstOrder
variable {V : Type*} [ORingStructure V] [V↓[ℒₒ... | @@ -17,8 +17,12 @@
variable {T : Theory L} [T.Δ₁]
+lemma derivable_quote {Γ : Finset (Proposition L)} (d : T ⟹₂ Γ) : Derivable T (⌜Γ⌝ : V) :=
+ ⟨⌜d⌝, by simpa [Semiformula.quote_def] using! (⌜d⌝ : Theory.internalize V T ⊢!ᵈᵉʳ ⌜Γ⌝).derivationOf⟩
+
/-- Hilbert–Bernays provability condition D1 -/
-theorem internali... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_97495911a3b3_0 | 8e61e5cbc15763e8 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Kripke/Basic.lean | Basic | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 14 | 2 | [
{
"theorem_name": "box_dn",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " boxItr_dn (n := 1)",
"n_chars": 19,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewri... | [
{
"name": "boxItr_dn",
"text": "lemma boxItr_dn : x ⊧ □^[n](∼∼φ) ↔ x ⊧ □^[n]φ := by\n induction n generalizing x with\n | zero => simp [Semantics.NotModels];\n | succ n ih =>\n suffices x ⊧ (□□^[n](∼∼φ)) ↔ x ⊧ (□□^[n]φ) by simpa;\n constructor;\n . intro h y Rxy;\n exact ih.mp $ (h y Rxy);\... | [
{
"name": "box_dn",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 3,
"n_chars": 74,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
"max_nesting": 1
}
] | 1 | module
public import Foundation.Modal.Logic.Basic
public import Foundation.Vorspiel.Rel.Basic
@[expose] public section
namespace LO.Modal
open Entailment
namespace Kripke
structure Frame where
World : Type
Rel : Rel World World
[world_nonempty : Nonempty World]
attribute [simp] Frame.world_nonempty
instanc... | module
public import Foundation.Modal.Logic.Basic
public import Foundation.Vorspiel.Rel.Basic
@[expose] public section
namespace LO.Modal
open Entailment
namespace Kripke
structure Frame where
World : Type
Rel : Rel World World
[world_nonempty : Nonempty World]
attribute [simp] Frame.world_nonempty
instanc... | @@ -114,8 +114,19 @@
@[simp] lemma negneg_def : x ⊧ ∼∼φ ↔ x ⊧ φ := by simp [Semantics.NotModels];
-lemma box_dn : x ⊧ □(∼∼φ) ↔ x ⊧ □φ := sorry
+lemma boxItr_dn : x ⊧ □^[n](∼∼φ) ↔ x ⊧ □^[n]φ := by
+ induction n generalizing x with
+ | zero => simp [Semantics.NotModels];
+ | succ n ih =>
+ suffices x ⊧ (□□^[n]... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_97495911a3b3_1 | b5d4b7fa56eedad0 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Kripke/Basic.lean | Basic | 1 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 14 | 3 | [
{
"theorem_name": "diaItr_dn",
"depth": 1,
"n_commands": 0,
"n_lines": 20,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n induction n generalizing x with\n | zero => simp [Semantics.NotModels];\n | succ n ih =>\n suffices x ⊧ (◇◇^[n](∼∼φ)) ↔ x ⊧... | [
{
"name": "dia_def",
"text": "protected lemma dia_def : x ⊧ ◇φ ↔ ∃ y, x ≺ y ∧ y ⊧ φ := by simp [Satisfies];\n",
"fan_in": 3,
"n_lines": 2,
"n_chars": 94,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automat... | [
{
"name": "diaItr_dn",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 22,
"n_chars": 615,
"n_subproofs": 3,
"n_tactics": 37,
"cyclomatic": 5,
"n_automation": 2,
"n_rewrites": 0,
"n_structural": 16,
"automation_only": false,
"max_nesting": ... | 1 | module
public import Foundation.Modal.Logic.Basic
public import Foundation.Vorspiel.Rel.Basic
@[expose] public section
namespace LO.Modal
open Entailment
namespace Kripke
structure Frame where
World : Type
Rel : Rel World World
[world_nonempty : Nonempty World]
attribute [simp] Frame.world_nonempty
instanc... | module
public import Foundation.Modal.Logic.Basic
public import Foundation.Vorspiel.Rel.Basic
@[expose] public section
namespace LO.Modal
open Entailment
namespace Kripke
structure Frame where
World : Type
Rel : Rel World World
[world_nonempty : Nonempty World]
attribute [simp] Frame.world_nonempty
instanc... | @@ -104,6 +104,7 @@
protected lemma top_def : x ⊧ ⊤ := by simp [Satisfies];
+protected lemma dia_def : x ⊧ ◇φ ↔ ∃ y, x ≺ y ∧ y ⊧ φ := by simp [Satisfies];
protected instance : Semantics.Tarski (M.World) where
models_verum := λ _ => Satisfies.top_def;
models_falsum := λ _ => Satisfies.bot_def;
@@ -120,8 +121... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_97495911a3b3_2 | 07beca18c4efc7c2 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Kripke/Basic.lean | Basic | 2 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 14 | 3 | [
{
"theorem_name": "diaItr_dn",
"depth": 1,
"n_commands": 0,
"n_lines": 20,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n induction n generalizing x with\n | zero => simp [Semantics.NotModels];\n | succ n ih =>\n suffices x ⊧ (◇◇^[n](∼∼φ)) ↔ x ⊧... | [
{
"name": "dia_def",
"text": "protected lemma dia_def : x ⊧ ◇φ ↔ ∃ y, x ≺ y ∧ y ⊧ φ := by simp [Satisfies];\n",
"fan_in": 3,
"n_lines": 2,
"n_chars": 94,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automat... | [
{
"name": "dia_dn",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 2,
"n_lines": 3,
"n_chars": 74,
"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.Modal.Logic.Basic
public import Foundation.Vorspiel.Rel.Basic
@[expose] public section
namespace LO.Modal
open Entailment
namespace Kripke
structure Frame where
World : Type
Rel : Rel World World
[world_nonempty : Nonempty World]
attribute [simp] Frame.world_nonempty
instanc... | module
public import Foundation.Modal.Logic.Basic
public import Foundation.Vorspiel.Rel.Basic
@[expose] public section
namespace LO.Modal
open Entailment
namespace Kripke
structure Frame where
World : Type
Rel : Rel World World
[world_nonempty : Nonempty World]
attribute [simp] Frame.world_nonempty
instanc... | @@ -104,6 +104,7 @@
protected lemma top_def : x ⊧ ⊤ := by simp [Satisfies];
+protected lemma dia_def : x ⊧ ◇φ ↔ ∃ y, x ≺ y ∧ y ⊧ φ := by simp [Satisfies];
protected instance : Semantics.Tarski (M.World) where
models_verum := λ _ => Satisfies.top_def;
models_falsum := λ _ => Satisfies.bot_def;
@@ -120,8 +121... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_97495911a3b3_3 | fe92d79a894095b2 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Kripke/Basic.lean | Basic | 3 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 14 | 3 | [
{
"theorem_name": "boxItr_def",
"depth": 1,
"n_commands": 0,
"n_lines": 13,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n induction n generalizing x with\n | zero => simp;\n | succ n ih =>\n constructor;\n . rintro h y ⟨z, Rxz, Rzy⟩;\n ... | [
{
"name": "box_def",
"text": "protected lemma box_def : x ⊧ □φ ↔ ∀ y, x ≺ y → y ⊧ φ := by simp [Satisfies];\n",
"fan_in": 3,
"n_lines": 2,
"n_chars": 94,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automat... | [
{
"name": "boxItr_def",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 15,
"n_chars": 525,
"n_subproofs": 1,
"n_tactics": 23,
"cyclomatic": 4,
"n_automation": 3,
"n_rewrites": 0,
"n_structural": 8,
"automation_only": false,
"max_nesting": ... | 1 | module
public import Foundation.Modal.Logic.Basic
public import Foundation.Vorspiel.Rel.Basic
@[expose] public section
namespace LO.Modal
open Entailment
namespace Kripke
structure Frame where
World : Type
Rel : Rel World World
[world_nonempty : Nonempty World]
attribute [simp] Frame.world_nonempty
instanc... | module
public import Foundation.Modal.Logic.Basic
public import Foundation.Vorspiel.Rel.Basic
@[expose] public section
namespace LO.Modal
open Entailment
namespace Kripke
structure Frame where
World : Type
Rel : Rel World World
[world_nonempty : Nonempty World]
attribute [simp] Frame.world_nonempty
instanc... | @@ -104,6 +104,7 @@
protected lemma top_def : x ⊧ ⊤ := by simp [Satisfies];
+protected lemma box_def : x ⊧ □φ ↔ ∀ y, x ≺ y → y ⊧ φ := by simp [Satisfies];
protected instance : Semantics.Tarski (M.World) where
models_verum := λ _ => Satisfies.top_def;
models_falsum := λ _ => Satisfies.bot_def;
@@ -120,7 +121... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_97495911a3b3_4 | b408e28ce8f83fbf | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Kripke/Basic.lean | Basic | 4 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 14 | 3 | [
{
"theorem_name": "diaItr_dn",
"depth": 1,
"n_commands": 0,
"n_lines": 20,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n induction n generalizing x with\n | zero => simp [Semantics.NotModels];\n | succ n ih =>\n suffices x ⊧ (◇◇^[n](∼∼φ)) ↔ x ⊧... | [
{
"name": "dia_def",
"text": "protected lemma dia_def : x ⊧ ◇φ ↔ ∃ y, x ≺ y ∧ y ⊧ φ := by simp [Satisfies];\n",
"fan_in": 3,
"n_lines": 2,
"n_chars": 94,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automat... | [
{
"name": "diaItr_def",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 23,
"n_chars": 619,
"n_subproofs": 3,
"n_tactics": 39,
"cyclomatic": 6,
"n_automation": 5,
"n_rewrites": 0,
"n_structural": 14,
"automation_only": false,
"max_nesting":... | 1 | module
public import Foundation.Modal.Logic.Basic
public import Foundation.Vorspiel.Rel.Basic
@[expose] public section
namespace LO.Modal
open Entailment
namespace Kripke
structure Frame where
World : Type
Rel : Rel World World
[world_nonempty : Nonempty World]
attribute [simp] Frame.world_nonempty
instanc... | module
public import Foundation.Modal.Logic.Basic
public import Foundation.Vorspiel.Rel.Basic
@[expose] public section
namespace LO.Modal
open Entailment
namespace Kripke
structure Frame where
World : Type
Rel : Rel World World
[world_nonempty : Nonempty World]
attribute [simp] Frame.world_nonempty
instanc... | @@ -104,6 +104,7 @@
protected lemma top_def : x ⊧ ⊤ := by simp [Satisfies];
+protected lemma dia_def : x ⊧ ◇φ ↔ ∃ y, x ≺ y ∧ y ⊧ φ := by simp [Satisfies];
protected instance : Semantics.Tarski (M.World) where
models_verum := λ _ => Satisfies.top_def;
models_falsum := λ _ => Satisfies.bot_def;
@@ -119,8 +120... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_97495911a3b3_5 | 21f0cdd90787422f | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Kripke/Basic.lean | Basic | 5 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 14 | 3 | [
{
"theorem_name": "boxItr_def",
"depth": 1,
"n_commands": 0,
"n_lines": 13,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n induction n generalizing x with\n | zero => simp;\n | succ n ih =>\n constructor;\n . rintro h y ⟨z, Rxz, Rzy⟩;\n ... | [
{
"name": "box_def",
"text": "protected lemma box_def : x ⊧ □φ ↔ ∀ y, x ≺ y → y ⊧ φ := by simp [Satisfies];\n",
"fan_in": 3,
"n_lines": 2,
"n_chars": 94,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automat... | [
{
"name": "intro_boxItr_semiequiv",
"fan_in": 2,
"n_deps_direct": 2,
"n_deps_transitive": 2,
"n_lines": 14,
"n_chars": 474,
"n_subproofs": 0,
"n_tactics": 21,
"cyclomatic": 3,
"n_automation": 2,
"n_rewrites": 0,
"n_structural": 8,
"automation_only": false,
"ma... | 2 | module
public import Foundation.Modal.Logic.Basic
public import Foundation.Vorspiel.Rel.Basic
@[expose] public section
namespace LO.Modal
open Entailment
namespace Kripke
structure Frame where
World : Type
Rel : Rel World World
[world_nonempty : Nonempty World]
attribute [simp] Frame.world_nonempty
instanc... | module
public import Foundation.Modal.Logic.Basic
public import Foundation.Vorspiel.Rel.Basic
@[expose] public section
namespace LO.Modal
open Entailment
namespace Kripke
structure Frame where
World : Type
Rel : Rel World World
[world_nonempty : Nonempty World]
attribute [simp] Frame.world_nonempty
instanc... | @@ -104,6 +104,7 @@
protected lemma top_def : x ⊧ ⊤ := by simp [Satisfies];
+protected lemma box_def : x ⊧ □φ ↔ ∀ y, x ≺ y → y ⊧ φ := by simp [Satisfies];
protected instance : Semantics.Tarski (M.World) where
models_verum := λ _ => Satisfies.top_def;
models_falsum := λ _ => Satisfies.bot_def;
@@ -120,7 +121... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_97495911a3b3_6 | 85eaa7f5cf9fdcd0 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Kripke/Basic.lean | Basic | 6 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 14 | 3 | [
{
"theorem_name": "boxItr_def",
"depth": 1,
"n_commands": 0,
"n_lines": 13,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n induction n generalizing x with\n | zero => simp;\n | succ n ih =>\n constructor;\n . rintro h y ⟨z, Rxz, Rzy⟩;\n ... | [
{
"name": "box_def",
"text": "protected lemma box_def : x ⊧ □φ ↔ ∀ y, x ≺ y → y ⊧ φ := by simp [Satisfies];\n",
"fan_in": 3,
"n_lines": 2,
"n_chars": 94,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automat... | [
{
"name": "intro_box_semiequiv",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 3,
"n_lines": 5,
"n_chars": 160,
"n_subproofs": 0,
"n_tactics": 5,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": false,
"max_nes... | 3 | module
public import Foundation.Modal.Logic.Basic
public import Foundation.Vorspiel.Rel.Basic
@[expose] public section
namespace LO.Modal
open Entailment
namespace Kripke
structure Frame where
World : Type
Rel : Rel World World
[world_nonempty : Nonempty World]
attribute [simp] Frame.world_nonempty
instanc... | module
public import Foundation.Modal.Logic.Basic
public import Foundation.Vorspiel.Rel.Basic
@[expose] public section
namespace LO.Modal
open Entailment
namespace Kripke
structure Frame where
World : Type
Rel : Rel World World
[world_nonempty : Nonempty World]
attribute [simp] Frame.world_nonempty
instanc... | @@ -104,6 +104,7 @@
protected lemma top_def : x ⊧ ⊤ := by simp [Satisfies];
+protected lemma box_def : x ⊧ □φ ↔ ∀ y, x ≺ y → y ⊧ φ := by simp [Satisfies];
protected instance : Semantics.Tarski (M.World) where
models_verum := λ _ => Satisfies.top_def;
models_falsum := λ _ => Satisfies.bot_def;
@@ -120,7 +121... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
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