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_e48307bb12d9_2 | 2436a96a552ea679 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Logic/GL/Independency.lean | Independency | 2 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 3 | 3 | [
{
"theorem_name": "unprovable_independency",
"depth": 1,
"n_commands": 0,
"n_lines": 3,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n by_contra hC;\n exact unprovable_notbox $ K!_left hC;",
"n_chars": 59,
"n_subproofs": 0,
"n_tactics": ... | [
{
"name": "unprovable_notbox",
"text": "lemma unprovable_notbox : Modal.GL ⊬ ∼□φ := by\n by_contra hC;\n have : Modal.GL ⊢ ∼□φ 🡒 ∼□⊥ := contra! (imply_box_distribute'! efq!)\n have : Modal.GL ⊢ ∼□⊥ := this ⨀ hC;\n have : ∼∼⊥ ∈ Propositional.Cl := GL.provable_verTranslated_Cl this;\n have := Hilbert.Cl... | [
{
"name": "unprovable_higherIndependency_of_consistency",
"fan_in": 1,
"n_deps_direct": 2,
"n_deps_transitive": 2,
"n_lines": 6,
"n_chars": 210,
"n_subproofs": 0,
"n_tactics": 6,
"cyclomatic": 2,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 3,
"automation_o... | 2 | module
public import Foundation.Modal.Maximal.Unprovability
public import Foundation.Modal.Kripke.Logic.GL.MDP
@[expose] public section
namespace LO.Modal
open LO.Entailment LO.Entailment.FiniteContext LO.Modal.Entailment
open Propositional
variable [DecidableEq α]
abbrev independency (φ : Formula α) := ∼□φ ⋏ ∼□(... | module
public import Foundation.Modal.Maximal.Unprovability
public import Foundation.Modal.Kripke.Logic.GL.MDP
@[expose] public section
namespace LO.Modal
open LO.Entailment LO.Entailment.FiniteContext LO.Modal.Entailment
open Propositional
variable [DecidableEq α]
abbrev independency (φ : Formula α) := ∼□φ ⋏ ∼□(... | @@ -22,11 +22,21 @@
variable {n : ℕ} {φ : Formula ℕ}
-lemma unprovable_independency : Modal.GL ⊬ independency φ := sorry
+lemma unprovable_notbox : Modal.GL ⊬ ∼□φ := by
+ by_contra hC;
+ have : Modal.GL ⊢ ∼□φ 🡒 ∼□⊥ := contra! (imply_box_distribute'! efq!)
+ have : Modal.GL ⊢ ∼□⊥ := this ⨀ hC;
+ have : ∼∼⊥ ∈ 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_e48307bb12d9_3 | 94728199619b24ab | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Logic/GL/Independency.lean | Independency | 3 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 3 | 3 | [
{
"theorem_name": "unprovable_independency",
"depth": 1,
"n_commands": 0,
"n_lines": 3,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n by_contra hC;\n exact unprovable_notbox $ K!_left hC;",
"n_chars": 59,
"n_subproofs": 0,
"n_tactics": ... | [
{
"name": "unprovable_notbox",
"text": "lemma unprovable_notbox : Modal.GL ⊬ ∼□φ := by\n by_contra hC;\n have : Modal.GL ⊢ ∼□φ 🡒 ∼□⊥ := contra! (imply_box_distribute'! efq!)\n have : Modal.GL ⊢ ∼□⊥ := this ⨀ hC;\n have : ∼∼⊥ ∈ Propositional.Cl := GL.provable_verTranslated_Cl this;\n have := Hilbert.Cl... | [
{
"name": "unprovable_not_higherIndependency_of_consistency",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 3,
"n_lines": 21,
"n_chars": 705,
"n_subproofs": 0,
"n_tactics": 20,
"cyclomatic": 3,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 7,
"automa... | 3 | module
public import Foundation.Modal.Maximal.Unprovability
public import Foundation.Modal.Kripke.Logic.GL.MDP
@[expose] public section
namespace LO.Modal
open LO.Entailment LO.Entailment.FiniteContext LO.Modal.Entailment
open Propositional
variable [DecidableEq α]
abbrev independency (φ : Formula α) := ∼□φ ⋏ ∼□(... | module
public import Foundation.Modal.Maximal.Unprovability
public import Foundation.Modal.Kripke.Logic.GL.MDP
@[expose] public section
namespace LO.Modal
open LO.Entailment LO.Entailment.FiniteContext LO.Modal.Entailment
open Propositional
variable [DecidableEq α]
abbrev independency (φ : Formula α) := ∼□φ ⋏ ∼□(... | @@ -22,11 +22,21 @@
variable {n : ℕ} {φ : Formula ℕ}
-lemma unprovable_independency : Modal.GL ⊬ independency φ := sorry
+lemma unprovable_notbox : Modal.GL ⊬ ∼□φ := by
+ by_contra hC;
+ have : Modal.GL ⊢ ∼□φ 🡒 ∼□⊥ := contra! (imply_box_distribute'! efq!)
+ have : Modal.GL ⊢ ∼□⊥ := this ⨀ hC;
+ have : ∼∼⊥ ∈ 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_e7b1e1450b99_0 | 9d65264c5d737603 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Kripke/ExtendRoot.lean | ExtendRoot | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 7 | 2 | [
{
"theorem_name": "eq_root_or_eq_original",
"depth": 1,
"n_commands": 0,
"n_lines": 4,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n rcases eq_extend_or_eq_original x with (⟨i, hi, rfl⟩ | ⟨x₀, rfl⟩);\n . left; simp [Frame.root, default, extend];\n ... | [
{
"name": "eq_extend_or_eq_original",
"text": "lemma eq_extend_or_eq_original (x : F.extendRoot n)\n : (∃ i : Fin n, x = extend i) ∨ (∃ x₀ : F, x = x₀) := by\n match x with\n | .inl i => left; use i;\n | .inr x => grind;\n\n\n",
"fan_in": 2,
"n_lines": 8,
"n_chars": 186,
"n_subproofs": 0... | [
{
"name": "eq_root_or_eq_original",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 6,
"n_chars": 247,
"n_subproofs": 0,
"n_tactics": 8,
"cyclomatic": 2,
"n_automation": 4,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": false,
"max_... | 1 | module
public import Foundation.Modal.Boxdot.Basic
public import Foundation.Modal.Kripke.Tree
public import Foundation.Modal.Kripke.AxiomL
public import Foundation.Vorspiel.Finset.Card
public import Mathlib.Data.Finite.Sum
@[expose] public section
namespace LO.Modal
open Formula.Kripke
namespace Kripke
abbrev Fr... | module
public import Foundation.Modal.Boxdot.Basic
public import Foundation.Modal.Kripke.Tree
public import Foundation.Modal.Kripke.AxiomL
public import Foundation.Vorspiel.Finset.Card
public import Mathlib.Data.Finite.Sum
@[expose] public section
namespace LO.Modal
open Formula.Kripke
namespace Kripke
abbrev Fr... | @@ -87,9 +87,18 @@
lemma not_eq_extendRoot_root_of_rel_original_root [F.IsIrreflexive] (x : F.extendRoot n) (h : (extendRoot F n).root ≺ x) : x ≠ (extendRoot F n).root := by
grind;
+lemma eq_extend_or_eq_original (x : F.extendRoot n)
+ : (∃ i : Fin n, x = extend i) ∨ (∃ x₀ : F, x = x₀) := by
+ match x 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_e7b1e1450b99_1 | 30288d780448fd12 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Kripke/ExtendRoot.lean | ExtendRoot | 1 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 7 | 2 | [
{
"theorem_name": "eq_root_or_eq_original",
"depth": 1,
"n_commands": 0,
"n_lines": 4,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n rcases eq_extend_or_eq_original x with (⟨i, hi, rfl⟩ | ⟨x₀, rfl⟩);\n . left; simp [Frame.root, default, extend];\n ... | [
{
"name": "eq_extend_or_eq_original",
"text": "lemma eq_extend_or_eq_original (x : F.extendRoot n)\n : (∃ i : Fin n, x = extend i) ∨ (∃ x₀ : F, x = x₀) := by\n match x with\n | .inl i => left; use i;\n | .inr x => grind;\n\n\n",
"fan_in": 2,
"n_lines": 8,
"n_chars": 186,
"n_subproofs": 0... | [
{
"name": "eq_original_of_rel_extendRoot_root₁",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 7,
"n_chars": 274,
"n_subproofs": 0,
"n_tactics": 8,
"cyclomatic": 2,
"n_automation": 4,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": fal... | 1 | module
public import Foundation.Modal.Boxdot.Basic
public import Foundation.Modal.Kripke.Tree
public import Foundation.Modal.Kripke.AxiomL
public import Foundation.Vorspiel.Finset.Card
public import Mathlib.Data.Finite.Sum
@[expose] public section
namespace LO.Modal
open Formula.Kripke
namespace Kripke
abbrev Fr... | module
public import Foundation.Modal.Boxdot.Basic
public import Foundation.Modal.Kripke.Tree
public import Foundation.Modal.Kripke.AxiomL
public import Foundation.Vorspiel.Finset.Card
public import Mathlib.Data.Finite.Sum
@[expose] public section
namespace LO.Modal
open Formula.Kripke
namespace Kripke
abbrev Fr... | @@ -87,10 +87,19 @@
lemma not_eq_extendRoot_root_of_rel_original_root [F.IsIrreflexive] (x : F.extendRoot n) (h : (extendRoot F n).root ≺ x) : x ≠ (extendRoot F n).root := by
grind;
+lemma eq_extend_or_eq_original (x : F.extendRoot n)
+ : (∃ i : Fin n, x = extend i) ∨ (∃ x₀ : F, x = x₀) := by
+ match x 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_e7b1e1450b99_2 | c5f6de644c98287b | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Kripke/ExtendRoot.lean | ExtendRoot | 2 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 7 | 1 | [
{
"theorem_name": "eq_original_of_neq_extendRoot_root₁",
"depth": 1,
"n_commands": 0,
"n_lines": 3,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n apply eq_original_of_rel_extendRoot_root₁;\n grind;",
"n_chars": 59,
"n_subproofs": 0,
"n_... | [
{
"name": "eq_original_of_rel_extendRoot_root₁",
"text": "lemma eq_original_of_rel_extendRoot_root₁ [F.IsIrreflexive] (x : F.extendRoot 1) (h : (extendRoot F 1).root ≺ x)\n : ∃ x₀ : F, x = x₀ := by\n rcases eq_extend_or_eq_original x with (⟨i, hi, rfl⟩ | _);\n . simp [Frame.root, default] at h; grind;\n ... | [
{
"name": "eq_original_of_neq_extendRoot_root₁",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 2,
"n_lines": 6,
"n_chars": 207,
"n_subproofs": 0,
"n_tactics": 5,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": fal... | 2 | module
public import Foundation.Modal.Boxdot.Basic
public import Foundation.Modal.Kripke.Tree
public import Foundation.Modal.Kripke.AxiomL
public import Foundation.Vorspiel.Finset.Card
public import Mathlib.Data.Finite.Sum
@[expose] public section
namespace LO.Modal
open Formula.Kripke
namespace Kripke
abbrev Fr... | module
public import Foundation.Modal.Boxdot.Basic
public import Foundation.Modal.Kripke.Tree
public import Foundation.Modal.Kripke.AxiomL
public import Foundation.Vorspiel.Finset.Card
public import Mathlib.Data.Finite.Sum
@[expose] public section
namespace LO.Modal
open Formula.Kripke
namespace Kripke
abbrev Fr... | @@ -95,8 +95,16 @@
section
+lemma eq_original_of_rel_extendRoot_root₁ [F.IsIrreflexive] (x : F.extendRoot 1) (h : (extendRoot F 1).root ≺ x)
+ : ∃ x₀ : F, x = x₀ := by
+ rcases eq_extend_or_eq_original x with (⟨i, hi, rfl⟩ | _);
+ . simp [Frame.root, default] at h; grind;
+ . simp_all;
+
lemma eq_original_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_e7b1e1450b99_3 | 800f664609b51804 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Kripke/ExtendRoot.lean | ExtendRoot | 3 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 7 | 1 | [
{
"theorem_name": "inr_satisfies_iff",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " modal_equivalence_original_world",
"n_chars": 33,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_aut... | [
{
"name": "modal_equivalence_original_world",
"text": "lemma modal_equivalence_original_world : (embed x : M.extendRoot n) ↭ x :=\n Model.PseudoEpimorphism.modal_equivalence pMorphism _ |>.symm\n\n",
"fan_in": 1,
"n_lines": 4,
"n_chars": 142,
"n_subproofs": 0,
"n_tactics": 1,
"cyclo... | [
{
"name": "inr_satisfies_iff",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 4,
"n_chars": 124,
"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.Modal.Boxdot.Basic
public import Foundation.Modal.Kripke.Tree
public import Foundation.Modal.Kripke.AxiomL
public import Foundation.Vorspiel.Finset.Card
public import Mathlib.Data.Finite.Sum
@[expose] public section
namespace LO.Modal
open Formula.Kripke
namespace Kripke
abbrev Fr... | module
public import Foundation.Modal.Boxdot.Basic
public import Foundation.Modal.Kripke.Tree
public import Foundation.Modal.Kripke.AxiomL
public import Foundation.Vorspiel.Finset.Card
public import Mathlib.Data.Finite.Sum
@[expose] public section
namespace LO.Modal
open Formula.Kripke
namespace Kripke
abbrev Fr... | @@ -109,8 +109,11 @@
def pMorphism : M →ₚ M.extendRoot n := PseudoEpimorphism.ofAtomic Frame.extendRoot.pMorphism $ by grind;
+lemma modal_equivalence_original_world : (embed x : M.extendRoot n) ↭ x :=
+ Model.PseudoEpimorphism.modal_equivalence pMorphism _ |>.symm
+
@[simp]
-lemma inr_satisfies_iff : Satisfies ... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_e7b1e1450b99_5 | 257f8ddcc0dbe241 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Kripke/ExtendRoot.lean | ExtendRoot | 5 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 7 | 1 | [
{
"theorem_name": "validates_axiomT_set_in_irrefl_trans_chain",
"depth": 1,
"n_commands": 0,
"n_lines": 55,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n haveI : Fintype M.World := Fintype.ofFinite _;\n let t₁ : Finset M.World := { x | x ∈ l ∧ ∃ A ... | [
{
"name": "atmost_one_validates_axiomT_in_irrefl_trans_chain",
"text": "lemma atmost_one_validates_axiomT_in_irrefl_trans_chain\n (l_chain : List.IsChain (· ≺ ·) l) :\n haveI : Fintype M.World := Fintype.ofFinite _;\n Finset.card { x | x ∈ l ∧ ¬x ⊧ (□A 🡒 A) } ≤ 1 := by\n apply Nat.le_one_iff_eq_z... | [
{
"name": "validates_axiomT_set_in_irrefl_trans_chain",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 2,
"n_lines": 61,
"n_chars": 2268,
"n_subproofs": 8,
"n_tactics": 99,
"cyclomatic": 4,
"n_automation": 10,
"n_rewrites": 1,
"n_structural": 28,
"automatio... | 2 | module
public import Foundation.Modal.Boxdot.Basic
public import Foundation.Modal.Kripke.Tree
public import Foundation.Modal.Kripke.AxiomL
public import Foundation.Vorspiel.Finset.Card
public import Mathlib.Data.Finite.Sum
@[expose] public section
namespace LO.Modal
open Formula.Kripke
namespace Kripke
abbrev Fr... | module
public import Foundation.Modal.Boxdot.Basic
public import Foundation.Modal.Kripke.Tree
public import Foundation.Modal.Kripke.AxiomL
public import Foundation.Vorspiel.Finset.Card
public import Mathlib.Data.Finite.Sum
@[expose] public section
namespace LO.Modal
open Formula.Kripke
namespace Kripke
abbrev Fr... | @@ -177,6 +177,20 @@
. have : x ⊧ A := hy₁ x Ryx; contradiction;
. have : y ⊧ A := hx₁ y Rxy; contradiction;
+lemma atmost_one_validates_axiomT_in_irrefl_trans_chain
+ (l_chain : List.IsChain (· ≺ ·) l) :
+ haveI : Fintype M.World := Fintype.ofFinite _;
+ Finset.card { x | x ∈ l ∧ ¬x ⊧ (□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_6c78f2e69e34_0 | 2198bb8c114ff2ae | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/HFS/Basic.lean | Basic | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 27 | 1 | [
{
"theorem_name": "sUnion_le",
"depth": 1,
"n_commands": 0,
"n_lines": 4,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n rcases Arithmetic.zero_le a with (rfl | pos)\n · simp [←emptyset_def]\n · exact le_of_lt (sUnion_lt_of_pos pos)",
"n_chars"... | [
{
"name": "sUnion_lt_of_pos",
"text": "lemma sUnion_lt_of_pos {a : V} (ha : 0 < a) : ⋃ʰᶠ a < a :=\n lt_of_lt_log ha <| by\n suffices ∀ i x, x ∈ a → i ∈ x → i < log a by simpa\n intro i x hx hi\n exact lt_of_lt_of_le (lt_of_mem hi) (le_log_of_mem hx)\n\n",
"fan_in": 1,
"n_lines": 7,
"n_... | [
{
"name": "sUnion_le",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 6,
"n_chars": 176,
"n_subproofs": 0,
"n_tactics": 4,
"cyclomatic": 2,
"n_automation": 2,
"n_rewrites": 0,
"n_structural": 2,
"automation_only": false,
"max_nesting": 2
... | 1 | module
public import Foundation.FirstOrder.Arithmetic.Exponential
@[expose] public section
/-!
# Hereditary Finite Set Theory in $\mathsf{I}\Sigma_1$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
@[simp] lemma susbset_insert (x a : V) : a ⊆ insert x a := by intr... | module
public import Foundation.FirstOrder.Arithmetic.Exponential
@[expose] public section
/-!
# Hereditary Finite Set Theory in $\mathsf{I}\Sigma_1$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
@[simp] lemma susbset_insert (x a : V) : a ⊆ insert x a := by intr... | @@ -51,10 +51,16 @@
@[simp] lemma sUnion_empty : (⋃ʰᶠ ∅ : V) = ∅ := mem_ext (by simp)
+lemma sUnion_lt_of_pos {a : V} (ha : 0 < a) : ⋃ʰᶠ a < a :=
+ lt_of_lt_log ha <| by
+ suffices ∀ i x, x ∈ a → i ∈ x → i < log a by simpa
+ intro i x hx hi
+ exact lt_of_lt_of_le (lt_of_mem hi) (le_log_of_mem hx)
+
@[si... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_6c78f2e69e34_1 | 36dd1698a2410031 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/HFS/Basic.lean | Basic | 1 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 27 | 2 | [
{
"theorem_name": "mem_sInter_iff_empty",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " mem_ext (by simp [mem_sInter_iff])",
"n_chars": 35,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"... | [
{
"name": "mem_sInter_iff",
"text": "lemma mem_sInter_iff {x s : V} : x ∈ ⋂ʰᶠ s ↔ s ≠ ∅ ∧ ∀ t ∈ s, x ∈ t := Classical.choose!_spec (sInter_exists_unique s) x\n\n",
"fan_in": 2,
"n_lines": 3,
"n_chars": 143,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
... | [
{
"name": "mem_sInter_iff_empty",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 2,
"n_lines": 3,
"n_chars": 101,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"max_nes... | 1 | module
public import Foundation.FirstOrder.Arithmetic.Exponential
@[expose] public section
/-!
# Hereditary Finite Set Theory in $\mathsf{I}\Sigma_1$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
@[simp] lemma susbset_insert (x a : V) : a ⊆ insert x a := by intr... | module
public import Foundation.FirstOrder.Arithmetic.Exponential
@[expose] public section
/-!
# Hereditary Finite Set Theory in $\mathsf{I}\Sigma_1$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
@[simp] lemma susbset_insert (x a : V) : a ⊆ insert x a := by intr... | @@ -158,10 +158,12 @@
prefix:80 "⋂ʰᶠ " => sInter
-@[simp] lemma mem_sInter_iff_empty : ⋂ʰᶠ (∅ : V) = ∅ := sorry
+lemma mem_sInter_iff {x s : V} : x ∈ ⋂ʰᶠ s ↔ s ≠ ∅ ∧ ∀ t ∈ s, x ∈ t := Classical.choose!_spec (sInter_exists_unique s) x
-lemma mem_sInter_iff_of_pos {x s : V} (h : s ≠ ∅) : x ∈ ⋂ʰᶠ s ↔ ∀ t ∈ s, x ∈ t... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_6c78f2e69e34_2 | c54d356975282557 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/HFS/Basic.lean | Basic | 2 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 27 | 3 | [
{
"theorem_name": "mem_product_iff'",
"depth": 1,
"n_commands": 0,
"n_lines": 5,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n suffices (∃ y ∈ a, ∃ z ∈ b, x = ⟪y, z⟫) ↔ π₁ x ∈ a ∧ π₂ x ∈ b by simpa [mem_product_iff]\n constructor\n · rintro ⟨y, hy... | [
{
"name": "mem_product_iff",
"text": "lemma mem_product_iff {x a b : V} : x ∈ a ×ʰᶠ b ↔ ∃ y ∈ a, ∃ z ∈ b, x = ⟪y, z⟫ := Classical.choose!_spec (product_exists_unique a b) x\n\n",
"fan_in": 3,
"n_lines": 3,
"n_chars": 156,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_auto... | [
{
"name": "mem_product_iff'",
"fan_in": 2,
"n_deps_direct": 1,
"n_deps_transitive": 2,
"n_lines": 7,
"n_chars": 355,
"n_subproofs": 1,
"n_tactics": 7,
"cyclomatic": 4,
"n_automation": 4,
"n_rewrites": 0,
"n_structural": 4,
"automation_only": false,
"max_nestin... | 1 | module
public import Foundation.FirstOrder.Arithmetic.Exponential
@[expose] public section
/-!
# Hereditary Finite Set Theory in $\mathsf{I}\Sigma_1$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
@[simp] lemma susbset_insert (x a : V) : a ⊆ insert x a := by intr... | module
public import Foundation.FirstOrder.Arithmetic.Exponential
@[expose] public section
/-!
# Hereditary Finite Set Theory in $\mathsf{I}\Sigma_1$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
@[simp] lemma susbset_insert (x a : V) : a ⊆ insert x a := by intr... | @@ -194,11 +194,26 @@
infixl:60 " ×ʰᶠ " => product
-lemma mem_product_iff' {x a b : V} : x ∈ a ×ʰᶠ b ↔ π₁ x ∈ a ∧ π₂ x ∈ b := sorry
+lemma mem_product_iff {x a b : V} : x ∈ a ×ʰᶠ b ↔ ∃ y ∈ a, ∃ z ∈ b, x = ⟪y, z⟫ := Classical.choose!_spec (product_exists_unique a b) x
+lemma mem_product_iff' {x a b : V} : x ∈ 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_6c78f2e69e34_3 | 2e662c7b3907b669 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/HFS/Basic.lean | Basic | 3 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 27 | 3 | [
{
"theorem_name": "mem_product_iff'",
"depth": 1,
"n_commands": 0,
"n_lines": 5,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n suffices (∃ y ∈ a, ∃ z ∈ b, x = ⟪y, z⟫) ↔ π₁ x ∈ a ∧ π₂ x ∈ b by simpa [mem_product_iff]\n constructor\n · rintro ⟨y, hy... | [
{
"name": "mem_product_iff",
"text": "lemma mem_product_iff {x a b : V} : x ∈ a ×ʰᶠ b ↔ ∃ y ∈ a, ∃ z ∈ b, x = ⟪y, z⟫ := Classical.choose!_spec (product_exists_unique a b) x\n\n",
"fan_in": 3,
"n_lines": 3,
"n_chars": 156,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_auto... | [
{
"name": "pair_mem_product",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 2,
"n_lines": 4,
"n_chars": 148,
"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.Exponential
@[expose] public section
/-!
# Hereditary Finite Set Theory in $\mathsf{I}\Sigma_1$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
@[simp] lemma susbset_insert (x a : V) : a ⊆ insert x a := by intr... | module
public import Foundation.FirstOrder.Arithmetic.Exponential
@[expose] public section
/-!
# Hereditary Finite Set Theory in $\mathsf{I}\Sigma_1$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
@[simp] lemma susbset_insert (x a : V) : a ⊆ insert x a := by intr... | @@ -194,13 +194,29 @@
infixl:60 " ×ʰᶠ " => product
-lemma mem_product_iff' {x a b : V} : x ∈ a ×ʰᶠ b ↔ π₁ x ∈ a ∧ π₂ x ∈ b := sorry
+lemma mem_product_iff {x a b : V} : x ∈ a ×ʰᶠ b ↔ ∃ y ∈ a, ∃ z ∈ b, x = ⟪y, z⟫ := Classical.choose!_spec (product_exists_unique a b) x
+lemma mem_product_iff' {x a b : V} : x ∈ 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_6c78f2e69e34_4 | 64104931d1d9d8ef | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/HFS/Basic.lean | Basic | 4 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 27 | 12 | [
{
"theorem_name": "domain_graph",
"depth": 1,
"n_commands": 0,
"n_lines": 14,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n constructor\n · rintro rfl x _\n suffices (∃ y, ⟪x, y⟫ ∈ s) ↔ ∃ y < s, ⟪x, y⟫ ∈ s by simpa [mem_domain_iff]\n exact ⟨b... | [
{
"name": "mem_domain_iff",
"text": "lemma mem_domain_iff {x s : V} : x ∈ domain s ↔ ∃ y, ⟪x, y⟫ ∈ s := Classical.choose!_spec (domain_exists_unique s) x\n\n",
"fan_in": 12,
"n_lines": 3,
"n_chars": 130,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n... | [
{
"name": "domain_graph",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 2,
"n_lines": 16,
"n_chars": 886,
"n_subproofs": 2,
"n_tactics": 19,
"cyclomatic": 8,
"n_automation": 7,
"n_rewrites": 0,
"n_structural": 16,
"automation_only": false,
"max_nesting... | 1 | module
public import Foundation.FirstOrder.Arithmetic.Exponential
@[expose] public section
/-!
# Hereditary Finite Set Theory in $\mathsf{I}\Sigma_1$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
@[simp] lemma susbset_insert (x a : V) : a ⊆ insert x a := by intr... | module
public import Foundation.FirstOrder.Arithmetic.Exponential
@[expose] public section
/-!
# Hereditary Finite Set Theory in $\mathsf{I}\Sigma_1$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
@[simp] lemma susbset_insert (x a : V) : a ⊆ insert x a := by intr... | @@ -242,10 +242,22 @@
noncomputable def domain (s : V) : V := Classical.choose! (domain_exists_unique s)
+lemma mem_domain_iff {x s : V} : x ∈ domain s ↔ ∃ y, ⟪x, y⟫ ∈ s := Classical.choose!_spec (domain_exists_unique s) x
+
private lemma domain_graph {u s : V} : u = domain s ↔ ∀ x < u + s, (x ∈ u ↔ ∃ y < s, ∃ z ... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_6c78f2e69e34_5 | c233cc12386e459b | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/HFS/Basic.lean | Basic | 5 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 27 | 2 | [
{
"theorem_name": "domain_insert",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by simp [insert_eq_union_singleton]",
"n_chars": 36,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_auto... | [
{
"name": "insert_eq_union_singleton",
"text": "lemma insert_eq_union_singleton (a s : V) : insert a s = {a} ∪ s := mem_ext (fun x ↦ by simp)\n\n",
"fan_in": 2,
"n_lines": 3,
"n_chars": 99,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,... | [
{
"name": "domain_insert",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 3,
"n_chars": 133,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"max_nesting": ... | 1 | module
public import Foundation.FirstOrder.Arithmetic.Exponential
@[expose] public section
/-!
# Hereditary Finite Set Theory in $\mathsf{I}\Sigma_1$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
@[simp] lemma susbset_insert (x a : V) : a ⊆ insert x a := by intr... | module
public import Foundation.FirstOrder.Arithmetic.Exponential
@[expose] public section
/-!
# Hereditary Finite Set Theory in $\mathsf{I}\Sigma_1$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
@[simp] lemma susbset_insert (x a : V) : a ⊆ insert x a := by intr... | @@ -113,6 +113,8 @@
instance union_definable' (ℌ : HierarchySymbol) : ℌ-Function₂ ((· ∪ ·) : V → V → V) := union_definable.of_zero
+lemma insert_eq_union_singleton (a s : V) : insert a s = {a} ∪ s := mem_ext (fun x ↦ by simp)
+
@[simp] lemma union_polybound (a b : V) : a ∪ b ≤ 2 * (a + b) := by
suffices ∀ i, i... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_6c78f2e69e34_7 | 47505f3f19977b03 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/HFS/Basic.lean | Basic | 7 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 27 | 3 | [
{
"theorem_name": "range_graph",
"depth": 1,
"n_commands": 0,
"n_lines": 14,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": "\n ⟨by rintro rfl y _\n suffices (∃ x, ⟪x, y⟫ ∈ s) ↔ ∃ x < s, ⟪x, y⟫ ∈ s by simpa [mem_range_iff]\n exact ⟨by rintro ⟨x, h... | [
{
"name": "mem_range_iff",
"text": "lemma mem_range_iff {y s : V} : y ∈ range s ↔ ∃ x, ⟪x, y⟫ ∈ s := Classical.choose!_spec (range_exists_unique s) y\n\n",
"fan_in": 3,
"n_lines": 3,
"n_chars": 127,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewr... | [
{
"name": "range_empty",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 2,
"n_lines": 3,
"n_chars": 88,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"max_nesting": 1
... | 1 | module
public import Foundation.FirstOrder.Arithmetic.Exponential
@[expose] public section
/-!
# Hereditary Finite Set Theory in $\mathsf{I}\Sigma_1$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
@[simp] lemma susbset_insert (x a : V) : a ⊆ insert x a := by intr... | module
public import Foundation.FirstOrder.Arithmetic.Exponential
@[expose] public section
/-!
# Hereditary Finite Set Theory in $\mathsf{I}\Sigma_1$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
@[simp] lemma susbset_insert (x a : V) : a ⊆ insert x a := by intr... | @@ -331,8 +331,23 @@
noncomputable def range (s : V) : V := Classical.choose! (range_exists_unique s)
-private lemma range_graph {s' s : V} : s' = range s ↔ ∀ y < s' + s, (y ∈ s' ↔ ∃ x < s, ∃ z ∈ s, z = ⟪x, y⟫) := sorry
+lemma mem_range_iff {y s : V} : y ∈ range s ↔ ∃ x, ⟪x, y⟫ ∈ s := Classical.choose!_spec (range... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_6c78f2e69e34_8 | 6c344ceae814f9b9 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/HFS/Basic.lean | Basic | 8 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 27 | 1 | [
{
"theorem_name": "Disjoint.not_of_mem",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n simpa [Disjoint.iff, not_or] using ⟨x, hs, ht⟩",
"n_chars": 56,
"n_subproofs": 0,
"n_tactics": 2,
"cycl... | [
{
"name": "Disjoint.iff",
"text": "lemma Disjoint.iff {s t : V} : Disjoint s t ↔ ∀ x, x ∉ s ∨ x ∉ t := by simp [Disjoint, isempty_iff, imp_iff_not_or]\n\n",
"fan_in": 1,
"n_lines": 3,
"n_chars": 127,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 1,
"n_rew... | [
{
"name": "Disjoint.not_of_mem",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 4,
"n_chars": 145,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"max_nest... | 1 | module
public import Foundation.FirstOrder.Arithmetic.Exponential
@[expose] public section
/-!
# Hereditary Finite Set Theory in $\mathsf{I}\Sigma_1$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
@[simp] lemma susbset_insert (x a : V) : a ⊆ insert x a := by intr... | module
public import Foundation.FirstOrder.Arithmetic.Exponential
@[expose] public section
/-!
# Hereditary Finite Set Theory in $\mathsf{I}\Sigma_1$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
@[simp] lemma susbset_insert (x a : V) : a ⊆ insert x a := by intr... | @@ -367,8 +367,11 @@
def Disjoint (s t : V) : Prop := s ∩ t = ∅
-lemma Disjoint.not_of_mem {s t x : V} (hs : x ∈ s) (ht : x ∈ t) : ¬Disjoint s t := sorry
+lemma Disjoint.iff {s t : V} : Disjoint s t ↔ ∀ x, x ∉ s ∨ x ∉ t := by simp [Disjoint, isempty_iff, imp_iff_not_or]
+lemma Disjoint.not_of_mem {s t x : V} (hs... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_6c78f2e69e34_9 | 0d5e685dcafb72f8 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/HFS/Basic.lean | Basic | 9 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 27 | 2 | [
{
"theorem_name": "Disjoint.symm",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by simpa [Disjoint, inter_comm t s] using h",
"n_chars": 44,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
... | [
{
"name": "inter_comm",
"text": "lemma inter_comm (a b : V) : a ∩ b = b ∩ a := mem_ext (by simp [and_comm])\n\n",
"fan_in": 2,
"n_lines": 3,
"n_chars": 80,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"autom... | [
{
"name": "Disjoint.symm",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 3,
"n_chars": 112,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"max_nesting": ... | 1 | module
public import Foundation.FirstOrder.Arithmetic.Exponential
@[expose] public section
/-!
# Hereditary Finite Set Theory in $\mathsf{I}\Sigma_1$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
@[simp] lemma susbset_insert (x a : V) : a ⊆ insert x a := by intr... | module
public import Foundation.FirstOrder.Arithmetic.Exponential
@[expose] public section
/-!
# Hereditary Finite Set Theory in $\mathsf{I}\Sigma_1$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
@[simp] lemma susbset_insert (x a : V) : a ⊆ insert x a := by intr... | @@ -175,6 +175,8 @@
@[simp] lemma mem_inter_iff {a b c : V} : a ∈ b ∩ c ↔ a ∈ b ∧ a ∈ c := by
simp [Inter.inter, inter, mem_sInter_iff_of_pos (s := {b, c}) (nonempty_iff.mpr ⟨b, by simp⟩)]
+lemma inter_comm (a b : V) : a ∩ b = b ∩ a := mem_ext (by simp [and_comm])
+
end inter
section product
@@ -370,7 +372,7 ... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_6c78f2e69e34_10 | b1f2a63017252a11 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/HFS/Basic.lean | Basic | 10 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 27 | 3 | [
{
"theorem_name": "pair_mem_restr_iff",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by simp [mem_restr_iff]",
"n_chars": 24,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation"... | [
{
"name": "mem_restr_iff",
"text": "lemma mem_restr_iff {x f s : V} : x ∈ f ↾ s ↔ x ∈ f ∧ π₁ x ∈ s := Classical.choose!_spec (restr_exists_unique f s) x\n\n",
"fan_in": 3,
"n_lines": 3,
"n_chars": 133,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_r... | [
{
"name": "pair_mem_restr_iff",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 2,
"n_lines": 3,
"n_chars": 133,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"max_nesti... | 1 | module
public import Foundation.FirstOrder.Arithmetic.Exponential
@[expose] public section
/-!
# Hereditary Finite Set Theory in $\mathsf{I}\Sigma_1$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
@[simp] lemma susbset_insert (x a : V) : a ⊆ insert x a := by intr... | module
public import Foundation.FirstOrder.Arithmetic.Exponential
@[expose] public section
/-!
# Hereditary Finite Set Theory in $\mathsf{I}\Sigma_1$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
@[simp] lemma susbset_insert (x a : V) : a ⊆ insert x a := by intr... | @@ -469,12 +469,14 @@
scoped infix:80 " ↾ " => restr
-@[simp] lemma pair_mem_restr_iff {x y f s : V} : ⟪x, y⟫ ∈ f ↾ s ↔ ⟪x, y⟫ ∈ f ∧ x ∈ s := sorry
+lemma mem_restr_iff {x f s : V} : x ∈ f ↾ s ↔ x ∈ f ∧ π₁ x ∈ s := Classical.choose!_spec (restr_exists_unique f s) x
-@[simp] lemma restr_empty (f : V) : f ↾ ∅ = ∅ ... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_6c78f2e69e34_11 | 99e4e0cae2d91415 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/HFS/Basic.lean | Basic | 11 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 27 | 12 | [
{
"theorem_name": "domain_graph",
"depth": 1,
"n_commands": 0,
"n_lines": 14,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n constructor\n · rintro rfl x _\n suffices (∃ y, ⟪x, y⟫ ∈ s) ↔ ∃ y < s, ⟪x, y⟫ ∈ s by simpa [mem_domain_iff]\n exact ⟨b... | [
{
"name": "mem_domain_iff",
"text": "lemma mem_domain_iff {x s : V} : x ∈ domain s ↔ ∃ y, ⟪x, y⟫ ∈ s := Classical.choose!_spec (domain_exists_unique s) x\n\n",
"fan_in": 12,
"n_lines": 3,
"n_chars": 130,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n... | [
{
"name": "domain_restr",
"fan_in": 1,
"n_deps_direct": 3,
"n_deps_transitive": 9,
"n_lines": 4,
"n_chars": 159,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"max_nesting": 2... | 4 | module
public import Foundation.FirstOrder.Arithmetic.Exponential
@[expose] public section
/-!
# Hereditary Finite Set Theory in $\mathsf{I}\Sigma_1$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
@[simp] lemma susbset_insert (x a : V) : a ⊆ insert x a := by intr... | module
public import Foundation.FirstOrder.Arithmetic.Exponential
@[expose] public section
/-!
# Hereditary Finite Set Theory in $\mathsf{I}\Sigma_1$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
@[simp] lemma susbset_insert (x a : V) : a ⊆ insert x a := by intr... | @@ -242,10 +242,22 @@
noncomputable def domain (s : V) : V := Classical.choose! (domain_exists_unique s)
+lemma mem_domain_iff {x s : V} : x ∈ domain s ↔ ∃ y, ⟪x, y⟫ ∈ s := Classical.choose!_spec (domain_exists_unique s) x
+
private lemma domain_graph {u s : V} : u = domain s ↔ ∀ x < u + s, (x ∈ u ↔ ∃ y < s, ∃ z ... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_6c78f2e69e34_12 | c598ab8a2ec4bf9b | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/HFS/Basic.lean | Basic | 12 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 27 | 12 | [
{
"theorem_name": "domain_graph",
"depth": 1,
"n_commands": 0,
"n_lines": 14,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n constructor\n · rintro rfl x _\n suffices (∃ y, ⟪x, y⟫ ∈ s) ↔ ∃ y < s, ⟪x, y⟫ ∈ s by simpa [mem_domain_iff]\n exact ⟨b... | [
{
"name": "mem_domain_iff",
"text": "lemma mem_domain_iff {x s : V} : x ∈ domain s ↔ ∃ y, ⟪x, y⟫ ∈ s := Classical.choose!_spec (domain_exists_unique s) x\n\n",
"fan_in": 12,
"n_lines": 3,
"n_chars": 130,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n... | [
{
"name": "domain_restr_of_subset_domain",
"fan_in": 0,
"n_deps_direct": 3,
"n_deps_transitive": 12,
"n_lines": 4,
"n_chars": 158,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
... | 7 | module
public import Foundation.FirstOrder.Arithmetic.Exponential
@[expose] public section
/-!
# Hereditary Finite Set Theory in $\mathsf{I}\Sigma_1$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
@[simp] lemma susbset_insert (x a : V) : a ⊆ insert x a := by intr... | module
public import Foundation.FirstOrder.Arithmetic.Exponential
@[expose] public section
/-!
# Hereditary Finite Set Theory in $\mathsf{I}\Sigma_1$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
@[simp] lemma susbset_insert (x a : V) : a ⊆ insert x a := by intr... | @@ -245,10 +245,22 @@
noncomputable def domain (s : V) : V := Classical.choose! (domain_exists_unique s)
+lemma mem_domain_iff {x s : V} : x ∈ domain s ↔ ∃ y, ⟪x, y⟫ ∈ s := Classical.choose!_spec (domain_exists_unique s) x
+
private lemma domain_graph {u s : V} : u = domain s ↔ ∀ x < u + s, (x ∈ u ↔ ∃ y < s, ∃ z ... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_6c78f2e69e34_13 | 7a998fca09cd86bd | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/HFS/Basic.lean | Basic | 13 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 27 | 2 | [
{
"theorem_name": "insert_induction_sigmaOne",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": "\n insert_induction hP hempty hinsert",
"n_chars": 37,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": ... | [
{
"name": "insert_induction",
"text": "theorem insert_induction {P : V → Prop} (hP : Γ-[1]-Predicate P)\n (hempty : P ∅) (hinsert : ∀ a s, a ∉ s → P s → P (insert a s)) : ∀ s, P s :=\n InductionOnHierarchy.order_induction_sigma Γ 1 hP <| by\n intro s IH\n rcases eq_empty_or_nonempty s with (rfl | ... | [
{
"name": "insert_induction_sigmaOne",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 6,
"n_chars": 223,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
"m... | 1 | module
public import Foundation.FirstOrder.Arithmetic.Exponential
@[expose] public section
/-!
# Hereditary Finite Set Theory in $\mathsf{I}\Sigma_1$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
@[simp] lemma susbset_insert (x a : V) : a ⊆ insert x a := by intr... | module
public import Foundation.FirstOrder.Arithmetic.Exponential
@[expose] public section
/-!
# Hereditary Finite Set Theory in $\mathsf{I}\Sigma_1$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
@[simp] lemma susbset_insert (x a : V) : a ⊆ insert x a := by intr... | @@ -483,13 +483,24 @@
end restriction
+theorem insert_induction {P : V → Prop} (hP : Γ-[1]-Predicate P)
+ (hempty : P ∅) (hinsert : ∀ a s, a ∉ s → P s → P (insert a s)) : ∀ s, P s :=
+ InductionOnHierarchy.order_induction_sigma Γ 1 hP <| by
+ intro s IH
+ rcases eq_empty_or_nonempty s with (rfl | ⟨x, hx⟩... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_6c78f2e69e34_14 | 5296cd4458182e6a | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/HFS/Basic.lean | Basic | 14 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 27 | 12 | [
{
"theorem_name": "domain_graph",
"depth": 1,
"n_commands": 0,
"n_lines": 14,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n constructor\n · rintro rfl x _\n suffices (∃ y, ⟪x, y⟫ ∈ s) ↔ ∃ y < s, ⟪x, y⟫ ∈ s by simpa [mem_domain_iff]\n exact ⟨b... | [
{
"name": "mem_domain_iff",
"text": "lemma mem_domain_iff {x s : V} : x ∈ domain s ↔ ∃ y, ⟪x, y⟫ ∈ s := Classical.choose!_spec (domain_exists_unique s) x\n\n",
"fan_in": 12,
"n_lines": 3,
"n_chars": 130,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n... | [
{
"name": "sigma₁_replacement₂",
"fan_in": 0,
"n_deps_direct": 2,
"n_deps_transitive": 14,
"n_lines": 14,
"n_chars": 713,
"n_subproofs": 1,
"n_tactics": 14,
"cyclomatic": 4,
"n_automation": 6,
"n_rewrites": 0,
"n_structural": 10,
"automation_only": false,
"max... | 8 | module
public import Foundation.FirstOrder.Arithmetic.Exponential
@[expose] public section
/-!
# Hereditary Finite Set Theory in $\mathsf{I}\Sigma_1$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
@[simp] lemma susbset_insert (x a : V) : a ⊆ insert x a := by intr... | module
public import Foundation.FirstOrder.Arithmetic.Exponential
@[expose] public section
/-!
# Hereditary Finite Set Theory in $\mathsf{I}\Sigma_1$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
@[simp] lemma susbset_insert (x a : V) : a ⊆ insert x a := by intr... | @@ -242,10 +242,22 @@
noncomputable def domain (s : V) : V := Classical.choose! (domain_exists_unique s)
+lemma mem_domain_iff {x s : V} : x ∈ domain s ↔ ∃ y, ⟪x, y⟫ ∈ s := Classical.choose!_spec (domain_exists_unique s) x
+
private lemma domain_graph {u s : V} : u = domain s ↔ ∀ x < u + s, (x ∈ u ↔ ∃ y < s, ∃ z ... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_6c78f2e69e34_15 | 293e1762233ad388 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/HFS/Basic.lean | Basic | 15 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 27 | 12 | [
{
"theorem_name": "domain_graph",
"depth": 1,
"n_commands": 0,
"n_lines": 14,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n constructor\n · rintro rfl x _\n suffices (∃ y, ⟪x, y⟫ ∈ s) ↔ ∃ y < s, ⟪x, y⟫ ∈ s by simpa [mem_domain_iff]\n exact ⟨b... | [
{
"name": "mem_domain_iff",
"text": "lemma mem_domain_iff {x s : V} : x ∈ domain s ↔ ∃ y, ⟪x, y⟫ ∈ s := Classical.choose!_spec (domain_exists_unique s) x\n\n",
"fan_in": 12,
"n_lines": 3,
"n_chars": 130,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n... | [
{
"name": "app_mem_hfsImage",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 12,
"n_lines": 4,
"n_chars": 113,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"max_nestin... | 7 | module
public import Foundation.FirstOrder.Arithmetic.Exponential
@[expose] public section
/-!
# Hereditary Finite Set Theory in $\mathsf{I}\Sigma_1$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
@[simp] lemma susbset_insert (x a : V) : a ⊆ insert x a := by intr... | module
public import Foundation.FirstOrder.Arithmetic.Exponential
@[expose] public section
/-!
# Hereditary Finite Set Theory in $\mathsf{I}\Sigma_1$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
@[simp] lemma susbset_insert (x a : V) : a ⊆ insert x a := by intr... | @@ -242,10 +242,22 @@
noncomputable def domain (s : V) : V := Classical.choose! (domain_exists_unique s)
+lemma mem_domain_iff {x s : V} : x ∈ domain s ↔ ∃ y, ⟪x, y⟫ ∈ s := Classical.choose!_spec (domain_exists_unique s) x
+
private lemma domain_graph {u s : V} : u = domain s ↔ ∀ x < u + s, (x ∈ u ↔ ∃ y < s, ∃ z ... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_6c78f2e69e34_16 | 5c9f8ff392f7ec0b | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/HFS/Basic.lean | Basic | 16 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 27 | 12 | [
{
"theorem_name": "domain_graph",
"depth": 1,
"n_commands": 0,
"n_lines": 14,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n constructor\n · rintro rfl x _\n suffices (∃ y, ⟪x, y⟫ ∈ s) ↔ ∃ y < s, ⟪x, y⟫ ∈ s by simpa [mem_domain_iff]\n exact ⟨b... | [
{
"name": "mem_domain_iff",
"text": "lemma mem_domain_iff {x s : V} : x ∈ domain s ↔ ∃ y, ⟪x, y⟫ ∈ s := Classical.choose!_spec (domain_exists_unique s) x\n\n",
"fan_in": 12,
"n_lines": 3,
"n_chars": 130,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n... | [
{
"name": "hfsImage_subset_of_subset",
"fan_in": 0,
"n_deps_direct": 2,
"n_deps_transitive": 13,
"n_lines": 6,
"n_chars": 194,
"n_subproofs": 0,
"n_tactics": 4,
"cyclomatic": 2,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 3,
"automation_only": false,
"... | 8 | module
public import Foundation.FirstOrder.Arithmetic.Exponential
@[expose] public section
/-!
# Hereditary Finite Set Theory in $\mathsf{I}\Sigma_1$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
@[simp] lemma susbset_insert (x a : V) : a ⊆ insert x a := by intr... | module
public import Foundation.FirstOrder.Arithmetic.Exponential
@[expose] public section
/-!
# Hereditary Finite Set Theory in $\mathsf{I}\Sigma_1$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
@[simp] lemma susbset_insert (x a : V) : a ⊆ insert x a := by intr... | @@ -242,10 +242,22 @@
noncomputable def domain (s : V) : V := Classical.choose! (domain_exists_unique s)
+lemma mem_domain_iff {x s : V} : x ∈ domain s ↔ ∃ y, ⟪x, y⟫ ∈ s := Classical.choose!_spec (domain_exists_unique s) x
+
private lemma domain_graph {u s : V} : u = domain s ↔ ∀ x < u + s, (x ∈ u ↔ ∃ y < s, ∃ z ... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_84eb14c25090_0 | 787d18c5765af1b7 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Schemata.lean | Schemata | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 4 | 1 | [
{
"theorem_name": "induction_eval",
"depth": 1,
"n_commands": 0,
"n_lines": 5,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n have : V↓[ℒₒᵣ] ⊧ .univCl (succInd φ) :=\n Theory.models (T := InductionScheme _ C) V (by simpa using mem_InductionScheme_... | [
{
"name": "mem_InductionScheme_of_mem",
"text": "lemma mem_InductionScheme_of_mem {φ : ArithmeticSemiformula ℕ 1} (hp : C φ) :\n .univCl (succInd φ) ∈ InductionScheme ℒₒᵣ C := by\n simpa [InductionScheme] using ⟨φ, hp, rfl⟩\n\n",
"fan_in": 1,
"n_lines": 5,
"n_chars": 196,
"n_subproofs": ... | [
{
"name": "induction_eval",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 10,
"n_chars": 484,
"n_subproofs": 1,
"n_tactics": 5,
"cyclomatic": 1,
"n_automation": 2,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
"max_nesting... | 1 | module
public import Foundation.FirstOrder.Arithmetic.PeanoMinus.Functions
public import Foundation.FirstOrder.Arithmetic.TA.Basic
@[expose] public section
/-!
# Induction schemata of Arithmetic
-/
namespace LO.FirstOrder.Arithmetic
section axioms
variable {L : Language} [L.ORing] {ξ : Type*} [DecidableEq ξ]
def... | module
public import Foundation.FirstOrder.Arithmetic.PeanoMinus.Functions
public import Foundation.FirstOrder.Arithmetic.TA.Basic
@[expose] public section
/-!
# Induction schemata of Arithmetic
-/
namespace LO.FirstOrder.Arithmetic
section axioms
variable {L : Language} [L.ORing] {ξ : Type*} [DecidableEq ξ]
def... | @@ -83,6 +83,10 @@
instance : 𝗜𝚺i ⪯ 𝗣𝗔 :=
Entailment.WeakerThan.ofSubset <| Set.union_subset_union_right _ <| InductionScheme_subset (by intros; trivial)
+lemma mem_InductionScheme_of_mem {φ : ArithmeticSemiformula ℕ 1} (hp : C φ) :
+ .univCl (succInd φ) ∈ InductionScheme ℒₒᵣ C := by
+ simpa [InductionSc... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_84eb14c25090_1 | 0747e163056beb91 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Schemata.lean | Schemata | 1 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 4 | 1 | [
{
"theorem_name": "succ_induction",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n rcases hP with ⟨e, φ, Cp, hp⟩; simpa [←hp] using induction_eval (V := V) Cp e",
"n_chars": 90,
"n_subproofs": 0,
... | [
{
"name": "induction_eval",
"text": "private lemma induction_eval {φ : ArithmeticSemiformula ℕ 1} (hp : C φ) (v : ℕ → V) :\n φ.Eval ![0] v →\n (∀ x, φ.Eval ![x] v → φ.Eval ![x + 1] v) →\n ∀ x, φ.Eval ![x] v := by\n have : V↓[ℒₒᵣ] ⊧ .univCl (succInd φ) :=\n Theory.models (T := InductionScheme _ ... | [
{
"name": "succ_induction",
"fan_in": 8,
"n_deps_direct": 1,
"n_deps_transitive": 2,
"n_lines": 7,
"n_chars": 311,
"n_subproofs": 0,
"n_tactics": 3,
"cyclomatic": 2,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": false,
"max_nesting"... | 2 | module
public import Foundation.FirstOrder.Arithmetic.PeanoMinus.Functions
public import Foundation.FirstOrder.Arithmetic.TA.Basic
@[expose] public section
/-!
# Induction schemata of Arithmetic
-/
namespace LO.FirstOrder.Arithmetic
section axioms
variable {L : Language} [L.ORing] {ξ : Type*} [DecidableEq ξ]
def... | module
public import Foundation.FirstOrder.Arithmetic.PeanoMinus.Functions
public import Foundation.FirstOrder.Arithmetic.TA.Basic
@[expose] public section
/-!
# Induction schemata of Arithmetic
-/
namespace LO.FirstOrder.Arithmetic
section axioms
variable {L : Language} [L.ORing] {ξ : Type*} [DecidableEq ξ]
def... | @@ -103,10 +103,20 @@
variable {C : ArithmeticSemiformula ℕ 1 → Prop} [V↓[ℒₒᵣ] ⊧* InductionScheme ℒₒᵣ C]
+private lemma induction_eval {φ : ArithmeticSemiformula ℕ 1} (hp : C φ) (v : ℕ → V) :
+ φ.Eval ![0] v →
+ (∀ x, φ.Eval ![x] v → φ.Eval ![x + 1] v) →
+ ∀ x, φ.Eval ![x] v := by
+ have : V↓[ℒₒᵣ] ⊧ .uni... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_84eb14c25090_2 | 7af488a8f88d5c2d | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Schemata.lean | Schemata | 2 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 4 | 1 | [
{
"theorem_name": "succ_induction",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n rcases hP with ⟨e, φ, Cp, hp⟩; simpa [←hp] using induction_eval (V := V) Cp e",
"n_chars": 90,
"n_subproofs": 0,
... | [
{
"name": "induction_eval",
"text": "private lemma induction_eval {φ : ArithmeticSemiformula ℕ 1} (hp : C φ) (v : ℕ → V) :\n φ.Eval ![0] v →\n (∀ x, φ.Eval ![x] v → φ.Eval ![x + 1] v) →\n ∀ x, φ.Eval ![x] v := by\n have : V↓[ℒₒᵣ] ⊧ .univCl (succInd φ) :=\n Theory.models (T := InductionScheme _ ... | [
{
"name": "least_number",
"fan_in": 2,
"n_deps_direct": 1,
"n_deps_transitive": 3,
"n_lines": 27,
"n_chars": 1075,
"n_subproofs": 6,
"n_tactics": 24,
"cyclomatic": 4,
"n_automation": 5,
"n_rewrites": 0,
"n_structural": 14,
"automation_only": false,
"max_nestin... | 2 | module
public import Foundation.FirstOrder.Arithmetic.PeanoMinus.Functions
public import Foundation.FirstOrder.Arithmetic.TA.Basic
@[expose] public section
/-!
# Induction schemata of Arithmetic
-/
namespace LO.FirstOrder.Arithmetic
section axioms
variable {L : Language} [L.ORing] {ξ : Type*} [DecidableEq ξ]
def... | module
public import Foundation.FirstOrder.Arithmetic.PeanoMinus.Functions
public import Foundation.FirstOrder.Arithmetic.TA.Basic
@[expose] public section
/-!
# Induction schemata of Arithmetic
-/
namespace LO.FirstOrder.Arithmetic
section axioms
variable {L : Language} [L.ORing] {ξ : Type*} [DecidableEq ξ]
def... | @@ -103,10 +103,20 @@
variable {C : ArithmeticSemiformula ℕ 1 → Prop} [V↓[ℒₒᵣ] ⊧* InductionScheme ℒₒᵣ C]
+private lemma induction_eval {φ : ArithmeticSemiformula ℕ 1} (hp : C φ) (v : ℕ → V) :
+ φ.Eval ![0] v →
+ (∀ x, φ.Eval ![x] v → φ.Eval ![x + 1] v) →
+ ∀ x, φ.Eval ![x] v := by
+ have : V↓[ℒₒᵣ] ⊧ .uni... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_84eb14c25090_3 | 5548df9a6679dd12 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Schemata.lean | Schemata | 3 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 4 | 1 | [
{
"theorem_name": "succ_induction",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n rcases hP with ⟨e, φ, Cp, hp⟩; simpa [←hp] using induction_eval (V := V) Cp e",
"n_chars": 90,
"n_subproofs": 0,
... | [
{
"name": "induction_eval",
"text": "private lemma induction_eval {φ : ArithmeticSemiformula ℕ 1} (hp : C φ) (v : ℕ → V) :\n φ.Eval ![0] v →\n (∀ x, φ.Eval ![x] v → φ.Eval ![x + 1] v) →\n ∀ x, φ.Eval ![x] v := by\n have : V↓[ℒₒᵣ] ⊧ .univCl (succInd φ) :=\n Theory.models (T := InductionScheme _ ... | [
{
"name": "least_number_sigma",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 4,
"n_lines": 10,
"n_chars": 342,
"n_subproofs": 0,
"n_tactics": 6,
"cyclomatic": 2,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
"max_nes... | 2 | module
public import Foundation.FirstOrder.Arithmetic.PeanoMinus.Functions
public import Foundation.FirstOrder.Arithmetic.TA.Basic
@[expose] public section
/-!
# Induction schemata of Arithmetic
-/
namespace LO.FirstOrder.Arithmetic
section axioms
variable {L : Language} [L.ORing] {ξ : Type*} [DecidableEq ξ]
def... | module
public import Foundation.FirstOrder.Arithmetic.PeanoMinus.Functions
public import Foundation.FirstOrder.Arithmetic.TA.Basic
@[expose] public section
/-!
# Induction schemata of Arithmetic
-/
namespace LO.FirstOrder.Arithmetic
section axioms
variable {L : Language} [L.ORing] {ξ : Type*} [DecidableEq ξ]
def... | @@ -103,10 +103,20 @@
variable {C : ArithmeticSemiformula ℕ 1 → Prop} [V↓[ℒₒᵣ] ⊧* InductionScheme ℒₒᵣ C]
+private lemma induction_eval {φ : ArithmeticSemiformula ℕ 1} (hp : C φ) (v : ℕ → V) :
+ φ.Eval ![0] v →
+ (∀ x, φ.Eval ![x] v → φ.Eval ![x + 1] v) →
+ ∀ x, φ.Eval ![x] v := by
+ have : V↓[ℒₒᵣ] ⊧ .uni... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_92d3b5c52d13_0 | cea28bf87ba127ee | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Basic/Definability.lean | Definability | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 5 | 1 | [
{
"theorem_name": "fintype_all",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n simpa using fconj Finset.univ h",
"n_chars": 37,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
"n_auto... | [
{
"name": "fconj",
"text": "lemma fconj {P : ι → (Fin k → M) → Prop} (s : Finset ι)\n (h : ∀ i, L.Definable fun w : Fin k → M ↦ P i w) : L.Definable fun v : Fin k → M ↦ ∀ i ∈ s, P i v := by\n have : ∀ i, ∃ φ, IsDefinedByWithParam (P i) φ := fun i ↦ (h i).definable\n rcases Classical.axiomOfChoice this ... | [
{
"name": "fintype_all",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 5,
"n_chars": 218,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"max_nesting": 2
... | 1 | module
public import Foundation.FirstOrder.Basic.Semantics.Elementary
public import Foundation.FirstOrder.Basic.BinderNotation
public import Foundation.FirstOrder.Basic.AesopInit
public import Foundation.Vorspiel.Finset.Card
public import Foundation.Vorspiel.Graph
@[expose] public section
/-!
# Relations and functio... | module
public import Foundation.FirstOrder.Basic.Semantics.Elementary
public import Foundation.FirstOrder.Basic.BinderNotation
public import Foundation.FirstOrder.Basic.AesopInit
public import Foundation.Vorspiel.Finset.Card
public import Foundation.Vorspiel.Graph
@[expose] public section
/-!
# Relations and functio... | @@ -246,8 +246,15 @@
instance mem [L.Mem] [Membership M M] [Structure.Mem L M] : L-relation[M] Membership.mem := ⟨“x y. y ∈ x”, fun _ ↦ by simp⟩
+lemma fconj {P : ι → (Fin k → M) → Prop} (s : Finset ι)
+ (h : ∀ i, L.Definable fun w : Fin k → M ↦ P i w) : L.Definable fun v : Fin k → M ↦ ∀ i ∈ s, P i v := by
+ 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_92d3b5c52d13_1 | d9371906700288b1 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Basic/Definability.lean | Definability | 1 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 5 | 1 | [
{
"theorem_name": "fintype_exs",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n simpa using fdisj Finset.univ h",
"n_chars": 37,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
"n_auto... | [
{
"name": "fdisj",
"text": "lemma fdisj {P : ι → (Fin k → M) → Prop} (s : Finset ι)\n (h : ∀ i, L.Definable fun w : Fin k → M ↦ P i w) : L.Definable fun v : Fin k → M ↦ ∃ i ∈ s, P i v := by\n have : ∀ i, ∃ φ, IsDefinedByWithParam (P i) φ := fun i ↦ (h i).definable\n rcases Classical.axiomOfChoice this ... | [
{
"name": "fintype_exs",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 5,
"n_chars": 218,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"max_nesting": 2
... | 1 | module
public import Foundation.FirstOrder.Basic.Semantics.Elementary
public import Foundation.FirstOrder.Basic.BinderNotation
public import Foundation.FirstOrder.Basic.AesopInit
public import Foundation.Vorspiel.Finset.Card
public import Foundation.Vorspiel.Graph
@[expose] public section
/-!
# Relations and functio... | module
public import Foundation.FirstOrder.Basic.Semantics.Elementary
public import Foundation.FirstOrder.Basic.BinderNotation
public import Foundation.FirstOrder.Basic.AesopInit
public import Foundation.Vorspiel.Finset.Card
public import Foundation.Vorspiel.Graph
@[expose] public section
/-!
# Relations and functio... | @@ -252,12 +252,19 @@
rcases Classical.axiomOfChoice this with ⟨φ, H⟩
exact ⟨⩕ i ∈ s, φ i, fun v ↦ by simp [fun i ↦ (H i).iff]⟩
+lemma fdisj {P : ι → (Fin k → M) → Prop} (s : Finset ι)
+ (h : ∀ i, L.Definable fun w : Fin k → M ↦ P i w) : L.Definable fun v : Fin k → M ↦ ∃ i ∈ s, P i v := by
+ have : ∀ i, ∃ φ... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_92d3b5c52d13_2 | 483736184dfbf501 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Basic/Definability.lean | Definability | 2 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 5 | 4 | [
{
"theorem_name": "exsVec",
"depth": 1,
"n_commands": 0,
"n_lines": 19,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n induction l generalizing k\n case zero => simpa [Matrix.empty_eq] using h\n case succ l ih =>\n suffices L.Definable fun v : F... | [
{
"name": "retraction",
"text": "lemma retraction (h : L.Definable P) {n} (f : Fin k → Fin n) :\n L.Definable fun v ↦ P (fun i ↦ v (f i)) := by\n rcases h with ⟨φ, hφ⟩\n exact ⟨(Rew.subst fun i ↦ #(f i)) ▹ φ, fun v ↦ by simp [←hφ.iff, Function.comp_def]⟩\n\n",
"fan_in": 4,
"n_lines": 6,
"n_... | [
{
"name": "exsVec",
"fan_in": 1,
"n_deps_direct": 3,
"n_deps_transitive": 3,
"n_lines": 23,
"n_chars": 1190,
"n_subproofs": 1,
"n_tactics": 34,
"cyclomatic": 5,
"n_automation": 9,
"n_rewrites": 4,
"n_structural": 18,
"automation_only": false,
"max_nesting": 8
... | 1 | module
public import Foundation.FirstOrder.Basic.Semantics.Elementary
public import Foundation.FirstOrder.Basic.BinderNotation
public import Foundation.FirstOrder.Basic.AesopInit
public import Foundation.Vorspiel.Finset.Card
public import Foundation.Vorspiel.Graph
@[expose] public section
/-!
# Relations and functio... | module
public import Foundation.FirstOrder.Basic.Semantics.Elementary
public import Foundation.FirstOrder.Basic.BinderNotation
public import Foundation.FirstOrder.Basic.AesopInit
public import Foundation.Vorspiel.Finset.Card
public import Foundation.Vorspiel.Graph
@[expose] public section
/-!
# Relations and functio... | @@ -256,17 +256,41 @@
(h : ∀ i, L.Definable fun w : Fin k → M ↦ P i w) : L.Definable fun v : Fin k → M ↦ ∀ i, P i v := by
simpa using fconj Finset.univ h
+lemma retraction (h : L.Definable P) {n} (f : Fin k → Fin n) :
+ L.Definable fun v ↦ P (fun i ↦ v (f i)) := by
+ rcases h with ⟨φ, hφ⟩
+ exact ⟨(Rew.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_92d3b5c52d13_3 | 27710516b205030d | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Basic/Definability.lean | Definability | 3 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 5 | 4 | [
{
"theorem_name": "exsVec",
"depth": 1,
"n_commands": 0,
"n_lines": 19,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n induction l generalizing k\n case zero => simpa [Matrix.empty_eq] using h\n case succ l ih =>\n suffices L.Definable fun v : F... | [
{
"name": "retraction",
"text": "lemma retraction (h : L.Definable P) {n} (f : Fin k → Fin n) :\n L.Definable fun v ↦ P (fun i ↦ v (f i)) := by\n rcases h with ⟨φ, hφ⟩\n exact ⟨(Rew.subst fun i ↦ #(f i)) ▹ φ, fun v ↦ by simp [←hφ.iff, Function.comp_def]⟩\n\n",
"fan_in": 4,
"n_lines": 6,
"n_... | [
{
"name": "allVec",
"fan_in": 0,
"n_deps_direct": 3,
"n_deps_transitive": 3,
"n_lines": 23,
"n_chars": 1149,
"n_subproofs": 1,
"n_tactics": 34,
"cyclomatic": 3,
"n_automation": 9,
"n_rewrites": 4,
"n_structural": 17,
"automation_only": false,
"max_nesting": 8
... | 1 | module
public import Foundation.FirstOrder.Basic.Semantics.Elementary
public import Foundation.FirstOrder.Basic.BinderNotation
public import Foundation.FirstOrder.Basic.AesopInit
public import Foundation.Vorspiel.Finset.Card
public import Foundation.Vorspiel.Graph
@[expose] public section
/-!
# Relations and functio... | module
public import Foundation.FirstOrder.Basic.Semantics.Elementary
public import Foundation.FirstOrder.Basic.BinderNotation
public import Foundation.FirstOrder.Basic.AesopInit
public import Foundation.Vorspiel.Finset.Card
public import Foundation.Vorspiel.Graph
@[expose] public section
/-!
# Relations and functio... | @@ -256,21 +256,63 @@
(h : ∀ i, L.Definable fun w : Fin k → M ↦ P i w) : L.Definable fun v : Fin k → M ↦ ∀ i, P i v := by
simpa using fconj Finset.univ h
+lemma retraction (h : L.Definable P) {n} (f : Fin k → Fin n) :
+ L.Definable fun v ↦ P (fun i ↦ v (f i)) := by
+ rcases h with ⟨φ, hφ⟩
+ exact ⟨(Rew.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_d1f65fd8eab6_0 | 021be97da5f10d1a | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/ModalCompanion/Standard/Int.lean | Int | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 4 | 2 | [
{
"theorem_name": "modalCompanion_S4",
"depth": 1,
"n_commands": 0,
"n_lines": 10,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n apply modalCompanion_via_kripkeSemantics (P := λ _ => True);\n . apply provable_S4_of_mem;\n . intro φ;\n apply Hil... | [
{
"name": "provable_S4_of_mem",
"text": "lemma provable_S4_of_mem : φ ∈ Propositional.Int → Modal.S4 ⊢ φᵍ := by\n apply Propositional.Hilbert.provable_gödelTranslated_of_mem_logic;\n rintro _ ⟨φ, ⟨_, rfl⟩, rfl⟩;\n simp;\n\n",
"fan_in": 2,
"n_lines": 6,
"n_chars": 200,
"n_subproofs": 0,
... | [
{
"name": "modalCompanion_S4",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 12,
"n_chars": 387,
"n_subproofs": 0,
"n_tactics": 19,
"cyclomatic": 2,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 8,
"automation_only": false,
"max_nes... | 1 | module
public import Foundation.Logic.Embedding
public import Foundation.Modal.Boxdot.GL_Grz
public import Foundation.Modal.ModalCompanion.Standard.Basic
public import Foundation.Propositional.Kripke.Hilbert.Int.Basic
public import Foundation.Propositional.Hilbert.Minimal.Glivenko
@[expose] public section
namespace ... | module
public import Foundation.Logic.Embedding
public import Foundation.Modal.Boxdot.GL_Grz
public import Foundation.Modal.ModalCompanion.Standard.Basic
public import Foundation.Propositional.Kripke.Hilbert.Int.Basic
public import Foundation.Propositional.Hilbert.Minimal.Glivenko
@[expose] public section
namespace ... | @@ -19,9 +19,14 @@
namespace Propositional.Int
+lemma provable_S4_of_mem : φ ∈ Propositional.Int → Modal.S4 ⊢ φᵍ := by
+ apply Propositional.Hilbert.provable_gödelTranslated_of_mem_logic;
+ rintro _ ⟨φ, ⟨_, rfl⟩, rfl⟩;
+ simp;
+
theorem modalCompanion_S4 : ModalCompanion Propositional.Int Modal.S4 := by
app... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_d1f65fd8eab6_1 | eb98f8bfc3ce1595 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/ModalCompanion/Standard/Int.lean | Int | 1 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 4 | 2 | [
{
"theorem_name": "modalCompanion_S4",
"depth": 1,
"n_commands": 0,
"n_lines": 10,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n apply modalCompanion_via_kripkeSemantics (P := λ _ => True);\n . apply provable_S4_of_mem;\n . intro φ;\n apply Hil... | [
{
"name": "provable_S4_of_mem",
"text": "lemma provable_S4_of_mem : φ ∈ Propositional.Int → Modal.S4 ⊢ φᵍ := by\n apply Propositional.Hilbert.provable_gödelTranslated_of_mem_logic;\n rintro _ ⟨φ, ⟨_, rfl⟩, rfl⟩;\n simp;\n\n",
"fan_in": 2,
"n_lines": 6,
"n_chars": 200,
"n_subproofs": 0,
... | [
{
"name": "provable_Grz_of_mem",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 3,
"n_chars": 129,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
"max_nes... | 1 | module
public import Foundation.Logic.Embedding
public import Foundation.Modal.Boxdot.GL_Grz
public import Foundation.Modal.ModalCompanion.Standard.Basic
public import Foundation.Propositional.Kripke.Hilbert.Int.Basic
public import Foundation.Propositional.Hilbert.Minimal.Glivenko
@[expose] public section
namespace ... | module
public import Foundation.Logic.Embedding
public import Foundation.Modal.Boxdot.GL_Grz
public import Foundation.Modal.ModalCompanion.Standard.Basic
public import Foundation.Propositional.Kripke.Hilbert.Int.Basic
public import Foundation.Propositional.Hilbert.Minimal.Glivenko
@[expose] public section
namespace ... | @@ -19,8 +19,13 @@
namespace Propositional.Int
-lemma provable_Grz_of_mem : φ ∈ Propositional.Int → Modal.Grz ⊢ φᵍ := sorry
+lemma provable_S4_of_mem : φ ∈ Propositional.Int → Modal.S4 ⊢ φᵍ := by
+ apply Propositional.Hilbert.provable_gödelTranslated_of_mem_logic;
+ rintro _ ⟨φ, ⟨_, rfl⟩, rfl⟩;
+ simp;
+lemma... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_d1f65fd8eab6_2 | b6aff23639254cd8 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/ModalCompanion/Standard/Int.lean | Int | 2 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 4 | 2 | [
{
"theorem_name": "modalCompanion_S4",
"depth": 1,
"n_commands": 0,
"n_lines": 10,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n apply modalCompanion_via_kripkeSemantics (P := λ _ => True);\n . apply provable_S4_of_mem;\n . intro φ;\n apply Hil... | [
{
"name": "provable_S4_of_mem",
"text": "lemma provable_S4_of_mem : φ ∈ Propositional.Int → Modal.S4 ⊢ φᵍ := by\n apply Propositional.Hilbert.provable_gödelTranslated_of_mem_logic;\n rintro _ ⟨φ, ⟨_, rfl⟩, rfl⟩;\n simp;\n\n",
"fan_in": 2,
"n_lines": 6,
"n_chars": 200,
"n_subproofs": 0,
... | [
{
"name": "modalCompanion_Grz",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 2,
"n_lines": 12,
"n_chars": 414,
"n_subproofs": 0,
"n_tactics": 19,
"cyclomatic": 2,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 8,
"automation_only": false,
"max_ne... | 2 | module
public import Foundation.Logic.Embedding
public import Foundation.Modal.Boxdot.GL_Grz
public import Foundation.Modal.ModalCompanion.Standard.Basic
public import Foundation.Propositional.Kripke.Hilbert.Int.Basic
public import Foundation.Propositional.Hilbert.Minimal.Glivenko
@[expose] public section
namespace ... | module
public import Foundation.Logic.Embedding
public import Foundation.Modal.Boxdot.GL_Grz
public import Foundation.Modal.ModalCompanion.Standard.Basic
public import Foundation.Propositional.Kripke.Hilbert.Int.Basic
public import Foundation.Propositional.Hilbert.Minimal.Glivenko
@[expose] public section
namespace ... | @@ -19,8 +19,13 @@
namespace Propositional.Int
-lemma provable_Grz_of_mem : φ ∈ Propositional.Int → Modal.Grz ⊢ φᵍ := sorry
+lemma provable_S4_of_mem : φ ∈ Propositional.Int → Modal.S4 ⊢ φᵍ := by
+ apply Propositional.Hilbert.provable_gödelTranslated_of_mem_logic;
+ rintro _ ⟨φ, ⟨_, rfl⟩, rfl⟩;
+ simp;
+lemma... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_d1f65fd8eab6_3 | e4ab3848a9314ddf | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/ModalCompanion/Standard/Int.lean | Int | 3 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 4 | 2 | [
{
"theorem_name": "modalCompanion_S4",
"depth": 1,
"n_commands": 0,
"n_lines": 10,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n apply modalCompanion_via_kripkeSemantics (P := λ _ => True);\n . apply provable_S4_of_mem;\n . intro φ;\n apply Hil... | [
{
"name": "provable_S4_of_mem",
"text": "lemma provable_S4_of_mem : φ ∈ Propositional.Int → Modal.S4 ⊢ φᵍ := by\n apply Propositional.Hilbert.provable_gödelTranslated_of_mem_logic;\n rintro _ ⟨φ, ⟨_, rfl⟩, rfl⟩;\n simp;\n\n",
"fan_in": 2,
"n_lines": 6,
"n_chars": 200,
"n_subproofs": 0,
... | [
{
"name": "boxdotModalCompanion_GL",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 3,
"n_lines": 9,
"n_chars": 328,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
"max... | 3 | module
public import Foundation.Logic.Embedding
public import Foundation.Modal.Boxdot.GL_Grz
public import Foundation.Modal.ModalCompanion.Standard.Basic
public import Foundation.Propositional.Kripke.Hilbert.Int.Basic
public import Foundation.Propositional.Hilbert.Minimal.Glivenko
@[expose] public section
namespace ... | module
public import Foundation.Logic.Embedding
public import Foundation.Modal.Boxdot.GL_Grz
public import Foundation.Modal.ModalCompanion.Standard.Basic
public import Foundation.Propositional.Kripke.Hilbert.Int.Basic
public import Foundation.Propositional.Hilbert.Minimal.Glivenko
@[expose] public section
namespace ... | @@ -19,8 +19,13 @@
namespace Propositional.Int
-lemma provable_Grz_of_mem : φ ∈ Propositional.Int → Modal.Grz ⊢ φᵍ := sorry
+lemma provable_S4_of_mem : φ ∈ Propositional.Int → Modal.S4 ⊢ φᵍ := by
+ apply Propositional.Hilbert.provable_gödelTranslated_of_mem_logic;
+ rintro _ ⟨φ, ⟨_, rfl⟩, rfl⟩;
+ simp;
+lemma... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_d1f65fd8eab6_4 | b8a1c3adc4fba8d7 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/ModalCompanion/Standard/Int.lean | Int | 4 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 4 | 2 | [
{
"theorem_name": "modalCompanion_S4",
"depth": 1,
"n_commands": 0,
"n_lines": 10,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n apply modalCompanion_via_kripkeSemantics (P := λ _ => True);\n . apply provable_S4_of_mem;\n . intro φ;\n apply Hil... | [
{
"name": "provable_S4_of_mem",
"text": "lemma provable_S4_of_mem : φ ∈ Propositional.Int → Modal.S4 ⊢ φᵍ := by\n apply Propositional.Hilbert.provable_gödelTranslated_of_mem_logic;\n rintro _ ⟨φ, ⟨_, rfl⟩, rfl⟩;\n simp;\n\n",
"fan_in": 2,
"n_lines": 6,
"n_chars": 200,
"n_subproofs": 0,
... | [
{
"name": "iff_mem_S4_dia",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 2,
"n_lines": 8,
"n_chars": 287,
"n_subproofs": 0,
"n_tactics": 11,
"cyclomatic": 2,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 5,
"automation_only": false,
"max_nesting... | 2 | module
public import Foundation.Logic.Embedding
public import Foundation.Modal.Boxdot.GL_Grz
public import Foundation.Modal.ModalCompanion.Standard.Basic
public import Foundation.Propositional.Kripke.Hilbert.Int.Basic
public import Foundation.Propositional.Hilbert.Minimal.Glivenko
@[expose] public section
namespace ... | module
public import Foundation.Logic.Embedding
public import Foundation.Modal.Boxdot.GL_Grz
public import Foundation.Modal.ModalCompanion.Standard.Basic
public import Foundation.Propositional.Kripke.Hilbert.Int.Basic
public import Foundation.Propositional.Hilbert.Minimal.Glivenko
@[expose] public section
namespace ... | @@ -19,9 +19,14 @@
namespace Propositional.Int
+lemma provable_S4_of_mem : φ ∈ Propositional.Int → Modal.S4 ⊢ φᵍ := by
+ apply Propositional.Hilbert.provable_gödelTranslated_of_mem_logic;
+ rintro _ ⟨φ, ⟨_, rfl⟩, rfl⟩;
+ simp;
+
theorem modalCompanion_S4 : ModalCompanion Propositional.Int Modal.S4 := by
app... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_48f6f5d268e6_0 | 5b3e4464a3d26a95 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Basic/Eq.lean | Eq | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 10 | 1 | [
{
"theorem_name": "EqAxiom.finite",
"depth": 1,
"n_commands": 0,
"n_lines": 5,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n haveI : Fintype ((k : ℕ) × L.Func k) := Language.Finite.func\n haveI : Fintype ((k : ℕ) × L.Rel k) := Language.Finite.rel\n... | [
{
"name": "Eq.defeq",
"text": "lemma Eq.defeq :\n 𝗘𝗤 L = {Eq.refl L, Eq.symm L, Eq.trans L}\n ∪ Set.range (fun f : (k : ℕ) × L.Func k ↦ Eq.funcExt f.2)\n ∪ Set.range (fun f : (k : ℕ) × L.Rel k ↦ Eq.relExt f.2) := by\n ext φ; constructor\n · rintro ⟨⟩\n case refl => simp\n case symm =>... | [
{
"name": "EqAxiom.finite",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 7,
"n_chars": 245,
"n_subproofs": 0,
"n_tactics": 5,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 1,
"n_structural": 0,
"automation_only": false,
"max_nesting"... | 1 | module
public import Foundation.FirstOrder.Basic.BinderNotation
public import Foundation.FirstOrder.Basic.Semantics.Elementary
public import Foundation.FirstOrder.Basic.Soundness
public import Foundation.Vorspiel.Quotient
public import Foundation.Vorspiel.Finset.Card
@[expose] public section
namespace Matrix
variabl... | module
public import Foundation.FirstOrder.Basic.BinderNotation
public import Foundation.FirstOrder.Basic.Semantics.Elementary
public import Foundation.FirstOrder.Basic.Soundness
public import Foundation.Vorspiel.Quotient
public import Foundation.Vorspiel.Finset.Card
@[expose] public section
namespace Matrix
variabl... | @@ -60,8 +60,32 @@
variable {L}
-@[simp] lemma EqAxiom.finite [L.Finite] : Set.Finite (𝗘𝗤 L) := sorry
+lemma Eq.defeq :
+ 𝗘𝗤 L = {Eq.refl L, Eq.symm L, Eq.trans L}
+ ∪ Set.range (fun f : (k : ℕ) × L.Func k ↦ Eq.funcExt f.2)
+ ∪ Set.range (fun f : (k : ℕ) × L.Rel k ↦ Eq.relExt f.2) := by
+ ext φ; ... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_48f6f5d268e6_1 | 94f3a5923ccf0cae | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Basic/Eq.lean | Eq | 1 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 10 | 3 | [
{
"theorem_name": "consequence_iff_eq",
"depth": 1,
"n_commands": 0,
"n_lines": 9,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n simp only [consequence_iff, Nonempty.forall]\n constructor\n · intro h M x s _ hM; exact h M x hM\n · intro h M x s h... | [
{
"name": "elementaryEquiv",
"text": "lemma elementaryEquiv : QuotEq L M ≡ₑ[L] M := ⟨models_iff⟩\n\n",
"fan_in": 3,
"n_lines": 3,
"n_chars": 68,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only"... | [
{
"name": "eqv_refl",
"fan_in": 1,
"n_deps_direct": 2,
"n_deps_transitive": 7,
"n_lines": 6,
"n_chars": 219,
"n_subproofs": 2,
"n_tactics": 4,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": false,
"max_nesting": 2
... | 4 | module
public import Foundation.FirstOrder.Basic.BinderNotation
public import Foundation.FirstOrder.Basic.Semantics.Elementary
public import Foundation.FirstOrder.Basic.Soundness
public import Foundation.Vorspiel.Quotient
public import Foundation.Vorspiel.Finset.Card
@[expose] public section
namespace Matrix
variabl... | module
public import Foundation.FirstOrder.Basic.BinderNotation
public import Foundation.FirstOrder.Basic.Semantics.Elementary
public import Foundation.FirstOrder.Basic.Soundness
public import Foundation.Vorspiel.Quotient
public import Foundation.Vorspiel.Finset.Card
@[expose] public section
namespace Matrix
variabl... | @@ -226,6 +226,8 @@
variable (L M)
+lemma elementaryEquiv : QuotEq L M ≡ₑ[L] M := ⟨models_iff⟩
+
variable {L M}
set_option backward.isDefEq.respectTransparency false in
@@ -261,7 +263,7 @@
noncomputable instance : Structure.Eq L (ModelOfSatEq sat) := Structure.Eq.QuotEq.structureEq
lemma models : (ModelOfS... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_48f6f5d268e6_2 | 09dfd5136425af2b | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Basic/Eq.lean | Eq | 2 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 10 | 1 | [
{
"theorem_name": "val_mk",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n induction t <;> simp [*, funk_mk, Function.comp_def]",
"n_chars": 58,
"n_subproofs": 0,
"n_tactics": 3,
"cyclomatic"... | [
{
"name": "funk_mk",
"text": "lemma funk_mk {k} (f : L.Func k) (v : Fin k → M) : Structure.func (M := QuotEq L M) f (⟦v ·⟧) = ⟦Structure.func f v⟧ :=\n Quotient.liftVec_mk (s := eqvSetoid L M) _ _ _\n\n",
"fan_in": 1,
"n_lines": 4,
"n_chars": 181,
"n_subproofs": 0,
"n_tactics": 1,
"... | [
{
"name": "val_mk",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 5,
"n_chars": 183,
"n_subproofs": 0,
"n_tactics": 3,
"cyclomatic": 3,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": false,
"max_nesting": 2
}
... | 1 | module
public import Foundation.FirstOrder.Basic.BinderNotation
public import Foundation.FirstOrder.Basic.Semantics.Elementary
public import Foundation.FirstOrder.Basic.Soundness
public import Foundation.Vorspiel.Quotient
public import Foundation.Vorspiel.Finset.Card
@[expose] public section
namespace Matrix
variabl... | module
public import Foundation.FirstOrder.Basic.BinderNotation
public import Foundation.FirstOrder.Basic.Semantics.Elementary
public import Foundation.FirstOrder.Basic.Soundness
public import Foundation.Vorspiel.Quotient
public import Foundation.Vorspiel.Finset.Card
@[expose] public section
namespace Matrix
variabl... | @@ -187,11 +187,15 @@
func := QuotEq.func
rel := QuotEq.Rel
+lemma funk_mk {k} (f : L.Func k) (v : Fin k → M) : Structure.func (M := QuotEq L M) f (⟦v ·⟧) = ⟦Structure.func f v⟧ :=
+ Quotient.liftVec_mk (s := eqvSetoid L M) _ _ _
+
lemma rel_mk {k} (r : L.Rel k) (v : Fin k → M) : Structure.rel (M := QuotEq 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_48f6f5d268e6_3 | 8a5559dfd9177e50 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Basic/Eq.lean | Eq | 3 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 10 | 1 | [
{
"theorem_name": "eval_mk",
"depth": 1,
"n_commands": 0,
"n_lines": 21,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n induction φ using Semiformula.rec'\n case hall n φ ih =>\n constructor\n · intro h a; exact (ih (bv := a :> bv)).mp (by sim... | [
{
"name": "val_mk",
"text": "lemma val_mk {bv fv} (t : Semiterm L ξ n) :\n t.val (M := QuotEq L M) (⟦bv ·⟧) (⟦fv ·⟧) = ⟦t.val bv fv⟧ := by\n induction t <;> simp [*, funk_mk, Function.comp_def]\n\n",
"fan_in": 1,
"n_lines": 5,
"n_chars": 183,
"n_subproofs": 0,
"n_tactics": 3,
"cy... | [
{
"name": "eval_mk",
"fan_in": 2,
"n_deps_direct": 2,
"n_deps_transitive": 3,
"n_lines": 24,
"n_chars": 974,
"n_subproofs": 2,
"n_tactics": 30,
"cyclomatic": 4,
"n_automation": 10,
"n_rewrites": 0,
"n_structural": 14,
"automation_only": false,
"max_nesting": 6... | 3 | module
public import Foundation.FirstOrder.Basic.BinderNotation
public import Foundation.FirstOrder.Basic.Semantics.Elementary
public import Foundation.FirstOrder.Basic.Soundness
public import Foundation.Vorspiel.Quotient
public import Foundation.Vorspiel.Finset.Card
@[expose] public section
namespace Matrix
variabl... | module
public import Foundation.FirstOrder.Basic.BinderNotation
public import Foundation.FirstOrder.Basic.Semantics.Elementary
public import Foundation.FirstOrder.Basic.Soundness
public import Foundation.Vorspiel.Quotient
public import Foundation.Vorspiel.Finset.Card
@[expose] public section
namespace Matrix
variabl... | @@ -193,8 +193,32 @@
lemma rel_mk {k} (r : L.Rel k) (v : Fin k → M) : Structure.rel (M := QuotEq L M) r (⟦v ·⟧) ↔ Structure.rel r v :=
of_eq <| Quotient.liftVec_mk (s := eqvSetoid L M) _ _ _
+lemma val_mk {bv fv} (t : Semiterm L ξ n) :
+ t.val (M := QuotEq L M) (⟦bv ·⟧) (⟦fv ·⟧) = ⟦t.val bv fv⟧ := by
+ induct... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_48f6f5d268e6_4 | aa25efc95ea89eeb | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Basic/Eq.lean | Eq | 4 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 10 | 1 | [
{
"theorem_name": "eval_mk",
"depth": 1,
"n_commands": 0,
"n_lines": 21,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n induction φ using Semiformula.rec'\n case hall n φ ih =>\n constructor\n · intro h a; exact (ih (bv := a :> bv)).mp (by sim... | [
{
"name": "val_mk",
"text": "lemma val_mk {bv fv} (t : Semiterm L ξ n) :\n t.val (M := QuotEq L M) (⟦bv ·⟧) (⟦fv ·⟧) = ⟦t.val bv fv⟧ := by\n induction t <;> simp [*, funk_mk, Function.comp_def]\n\n",
"fan_in": 1,
"n_lines": 5,
"n_chars": 183,
"n_subproofs": 0,
"n_tactics": 3,
"cy... | [
{
"name": "evalf_mk",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 4,
"n_lines": 6,
"n_chars": 208,
"n_subproofs": 1,
"n_tactics": 4,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": false,
"max_nesting": 2
... | 3 | module
public import Foundation.FirstOrder.Basic.BinderNotation
public import Foundation.FirstOrder.Basic.Semantics.Elementary
public import Foundation.FirstOrder.Basic.Soundness
public import Foundation.Vorspiel.Quotient
public import Foundation.Vorspiel.Finset.Card
@[expose] public section
namespace Matrix
variabl... | module
public import Foundation.FirstOrder.Basic.BinderNotation
public import Foundation.FirstOrder.Basic.Semantics.Elementary
public import Foundation.FirstOrder.Basic.Soundness
public import Foundation.Vorspiel.Quotient
public import Foundation.Vorspiel.Finset.Card
@[expose] public section
namespace Matrix
variabl... | @@ -193,8 +193,32 @@
lemma rel_mk {k} (r : L.Rel k) (v : Fin k → M) : Structure.rel (M := QuotEq L M) r (⟦v ·⟧) ↔ Structure.rel r v :=
of_eq <| Quotient.liftVec_mk (s := eqvSetoid L M) _ _ _
+lemma val_mk {bv fv} (t : Semiterm L ξ n) :
+ t.val (M := QuotEq L M) (⟦bv ·⟧) (⟦fv ·⟧) = ⟦t.val bv fv⟧ := by
+ induct... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_48f6f5d268e6_5 | 901a770a4f30b749 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Basic/Eq.lean | Eq | 5 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 10 | 3 | [
{
"theorem_name": "consequence_iff_eq",
"depth": 1,
"n_commands": 0,
"n_lines": 9,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n simp only [consequence_iff, Nonempty.forall]\n constructor\n · intro h M x s _ hM; exact h M x hM\n · intro h M x s h... | [
{
"name": "elementaryEquiv",
"text": "lemma elementaryEquiv : QuotEq L M ≡ₑ[L] M := ⟨models_iff⟩\n\n",
"fan_in": 3,
"n_lines": 3,
"n_chars": 68,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only"... | [
{
"name": "consequence_iff_eq",
"fan_in": 1,
"n_deps_direct": 2,
"n_deps_transitive": 7,
"n_lines": 12,
"n_chars": 617,
"n_subproofs": 2,
"n_tactics": 10,
"cyclomatic": 2,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 5,
"automation_only": false,
"max_ne... | 4 | module
public import Foundation.FirstOrder.Basic.BinderNotation
public import Foundation.FirstOrder.Basic.Semantics.Elementary
public import Foundation.FirstOrder.Basic.Soundness
public import Foundation.Vorspiel.Quotient
public import Foundation.Vorspiel.Finset.Card
@[expose] public section
namespace Matrix
variabl... | module
public import Foundation.FirstOrder.Basic.BinderNotation
public import Foundation.FirstOrder.Basic.Semantics.Elementary
public import Foundation.FirstOrder.Basic.Soundness
public import Foundation.Vorspiel.Quotient
public import Foundation.Vorspiel.Finset.Card
@[expose] public section
namespace Matrix
variabl... | @@ -226,6 +226,8 @@
variable (L M)
+lemma elementaryEquiv : QuotEq L M ≡ₑ[L] M := ⟨models_iff⟩
+
variable {L M}
set_option backward.isDefEq.respectTransparency false in
@@ -252,7 +254,7 @@
· intro h M x s hM
haveI : Nonempty M := ⟨x⟩
have H : M↓[L] ⊧* 𝗘𝗤 L := models_of_subtheory hM
- have e :... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_48f6f5d268e6_6 | bd1833c788ec03f2 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Basic/Eq.lean | Eq | 6 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 10 | 1 | [
{
"theorem_name": "consequence_iff_eq'",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n rw [consequence_iff_eq]",
"n_chars": 29,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
"n_auto... | [
{
"name": "consequence_iff_eq",
"text": "lemma consequence_iff_eq {T : Theory L} [𝗘𝗤 L ⪯ T] {σ : Sentence L} :\n T ⊨[Struc.{v, u} L] σ ↔ (∀ (M : Type v) [Nonempty M] [Structure L M] [Structure.Eq L M], M↓[L] ⊧* T → M↓[L] ⊧ σ) := by\n simp only [consequence_iff, Nonempty.forall]\n constructor\n · int... | [
{
"name": "consequence_iff_eq'",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 8,
"n_lines": 5,
"n_chars": 246,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 1,
"n_structural": 0,
"automation_only": false,
"max_nes... | 5 | module
public import Foundation.FirstOrder.Basic.BinderNotation
public import Foundation.FirstOrder.Basic.Semantics.Elementary
public import Foundation.FirstOrder.Basic.Soundness
public import Foundation.Vorspiel.Quotient
public import Foundation.Vorspiel.Finset.Card
@[expose] public section
namespace Matrix
variabl... | module
public import Foundation.FirstOrder.Basic.BinderNotation
public import Foundation.FirstOrder.Basic.Semantics.Elementary
public import Foundation.FirstOrder.Basic.Soundness
public import Foundation.Vorspiel.Quotient
public import Foundation.Vorspiel.Finset.Card
@[expose] public section
namespace Matrix
variabl... | @@ -246,8 +246,20 @@
end Structure
+lemma consequence_iff_eq {T : Theory L} [𝗘𝗤 L ⪯ T] {σ : Sentence L} :
+ T ⊨[Struc.{v, u} L] σ ↔ (∀ (M : Type v) [Nonempty M] [Structure L M] [Structure.Eq L M], M↓[L] ⊧* T → M↓[L] ⊧ σ) := by
+ simp only [consequence_iff, Nonempty.forall]
+ constructor
+ · intro h M x s _... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_48f6f5d268e6_7 | 36d14e4ab0788e71 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Basic/Eq.lean | Eq | 7 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 10 | 3 | [
{
"theorem_name": "consequence_iff_eq",
"depth": 1,
"n_commands": 0,
"n_lines": 9,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n simp only [consequence_iff, Nonempty.forall]\n constructor\n · intro h M x s _ hM; exact h M x hM\n · intro h M x s h... | [
{
"name": "elementaryEquiv",
"text": "lemma elementaryEquiv : QuotEq L M ≡ₑ[L] M := ⟨models_iff⟩\n\n",
"fan_in": 3,
"n_lines": 3,
"n_chars": 68,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only"... | [
{
"name": "satisfiable_iff_eq",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 6,
"n_lines": 12,
"n_chars": 664,
"n_subproofs": 2,
"n_tactics": 11,
"cyclomatic": 2,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 5,
"automation_only": false,
"max_ne... | 4 | module
public import Foundation.FirstOrder.Basic.BinderNotation
public import Foundation.FirstOrder.Basic.Semantics.Elementary
public import Foundation.FirstOrder.Basic.Soundness
public import Foundation.Vorspiel.Quotient
public import Foundation.Vorspiel.Finset.Card
@[expose] public section
namespace Matrix
variabl... | module
public import Foundation.FirstOrder.Basic.BinderNotation
public import Foundation.FirstOrder.Basic.Semantics.Elementary
public import Foundation.FirstOrder.Basic.Soundness
public import Foundation.Vorspiel.Quotient
public import Foundation.Vorspiel.Finset.Card
@[expose] public section
namespace Matrix
variabl... | @@ -226,6 +226,8 @@
variable (L M)
+lemma elementaryEquiv : QuotEq L M ≡ₑ[L] M := ⟨models_iff⟩
+
variable {L M}
set_option backward.isDefEq.respectTransparency false in
@@ -251,7 +253,7 @@
· intro ⟨M, x, s, hM⟩;
haveI : Nonempty M := ⟨x⟩
have H : M↓[L] ⊧* 𝗘𝗤 L := models_of_subtheory hM
- have... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_8fa40e2ed44a_0 | 73c1ff30f6be53f2 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Propositional/Formula/Basic.lean | Basic | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 1 | 1 | [
{
"theorem_name": "mem_neg",
"depth": 1,
"n_commands": 0,
"n_lines": 3,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n rw [neg_def] at h;\n grind;",
"n_chars": 33,
"n_subproofs": 0,
"n_tactics": 5,
"cyclomatic": 1,
"n_automation"... | [
{
"name": "neg_def",
"text": "lemma neg_def : ∼φ = φ 🡒 ⊥ := rfl\n",
"fan_in": 1,
"n_lines": 2,
"n_chars": 43,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"max_nesting": 1
}
] | [
{
"name": "mem_neg",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 6,
"n_chars": 158,
"n_subproofs": 0,
"n_tactics": 5,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 1,
"n_structural": 0,
"automation_only": false,
"max_nesting": 2
}... | 1 | module
public import Foundation.Logic.LogicSymbol
public import Mathlib.Logic.Encodable.Basic
@[expose]
public section
namespace LO.Propositional
variable {α : Type*}
inductive Formula (α : Type u) : Type u
| atom : α → Formula α
| falsum : Formula α
| and : Formula α → Formula α → Formula α
| or ... | module
public import Foundation.Logic.LogicSymbol
public import Mathlib.Logic.Encodable.Basic
@[expose]
public section
namespace LO.Propositional
variable {α : Type*}
inductive Formula (α : Type u) : Type u
| atom : α → Formula α
| falsum : Formula α
| and : Formula α → Formula α → Formula α
| or ... | @@ -63,6 +63,7 @@
lemma and_inj : φ₁ ⋏ φ₂ = ψ₁ ⋏ ψ₂ ↔ φ₁ = ψ₁ ∧ φ₂ = ψ₂ := by simp [Wedge.wedge]
lemma or_inj : φ₁ ⋎ φ₂ = ψ₁ ⋎ ψ₂ ↔ φ₁ = ψ₁ ∧ φ₂ = ψ₂ := by simp [Vee.vee]
lemma imp_inj : φ₁ 🡒 φ₂ = ψ₁ 🡒 ψ₂ ↔ φ₁ = ψ₁ ∧ φ₂ = ψ₂ := by simp [Arrow.arrow]
+lemma neg_def : ∼φ = φ 🡒 ⊥ := rfl
@[grind]
def complexity : F... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_022c0ad287ca_0 | 76c4133bac74aa48 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Kripke/Completeness.lean | Completeness | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 10 | 2 | [
{
"theorem_name": "truthlemma₁",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " truthlemma.1",
"n_chars": 13,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrit... | [
{
"name": "truthlemma",
"text": "lemma truthlemma : ((φ ∈ t.1.1) ↔ t ⊧ φ) ∧ ((φ ∈ t.1.2) ↔ ¬t ⊧ φ) := by\n induction φ generalizing t with\n | hatom =>\n simp_all only [Semantics.Models, Satisfies, true_and];\n exact iff_not_mem₁_mem₂.symm;\n | hfalsum => simp [Semantics.Models, Satisfies];\n | hi... | [
{
"name": "truthlemma₁",
"fan_in": 5,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 3,
"n_chars": 67,
"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.Tableau
public import Foundation.Modal.Kripke.Basic
@[expose] public section
namespace LO.Modal
open Entailment
open Formula
open Kripke
open MaximalConsistentTableau
variable {S} [Entailment S (Formula ℕ)]
variable {𝓢 : S} [Entailment.Consistent 𝓢] [Entailment.K 𝓢]
namesp... | module
public import Foundation.Modal.Tableau
public import Foundation.Modal.Kripke.Basic
@[expose] public section
namespace LO.Modal
open Entailment
open Formula
open Kripke
open MaximalConsistentTableau
variable {S} [Entailment S (Formula ℕ)]
variable {𝓢 : S} [Entailment.Consistent 𝓢] [Entailment.K 𝓢]
namesp... | @@ -37,10 +37,77 @@
variable {φ ψ : Formula ℕ}
variable {t : (canonicalModel 𝓢).World}
-lemma truthlemma₁ : (φ ∈ t.1.1) ↔ t ⊧ φ := sorry
+lemma truthlemma : ((φ ∈ t.1.1) ↔ t ⊧ φ) ∧ ((φ ∈ t.1.2) ↔ ¬t ⊧ φ) := by
+ induction φ generalizing t with
+ | hatom =>
+ simp_all only [Semantics.Models, Satisfies, true_an... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_022c0ad287ca_1 | 114130b0be341ccc | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Kripke/Completeness.lean | Completeness | 1 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 10 | 5 | [
{
"theorem_name": "iff_valid_on_canonicalModel_deducible",
"depth": 1,
"n_commands": 0,
"n_lines": 15,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n constructor;\n . contrapose;\n intro h;\n have : Tableau.Consistent 𝓢 (∅, {φ}) := by\n ... | [
{
"name": "truthlemma₁",
"text": "lemma truthlemma₁ : (φ ∈ t.1.1) ↔ t ⊧ φ := truthlemma.1\n\n",
"fan_in": 5,
"n_lines": 3,
"n_chars": 67,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false... | [
{
"name": "iff_valid_on_canonicalModel_deducible",
"fan_in": 0,
"n_deps_direct": 2,
"n_deps_transitive": 3,
"n_lines": 17,
"n_chars": 552,
"n_subproofs": 2,
"n_tactics": 28,
"cyclomatic": 2,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 11,
"automation_only"... | 3 | module
public import Foundation.Modal.Tableau
public import Foundation.Modal.Kripke.Basic
@[expose] public section
namespace LO.Modal
open Entailment
open Formula
open Kripke
open MaximalConsistentTableau
variable {S} [Entailment S (Formula ℕ)]
variable {𝓢 : S} [Entailment.Consistent 𝓢] [Entailment.K 𝓢]
namesp... | module
public import Foundation.Modal.Tableau
public import Foundation.Modal.Kripke.Basic
@[expose] public section
namespace LO.Modal
open Entailment
open Formula
open Kripke
open MaximalConsistentTableau
variable {S} [Entailment S (Formula ℕ)]
variable {𝓢 : S} [Entailment.Consistent 𝓢] [Entailment.K 𝓢]
namesp... | @@ -104,6 +104,8 @@
. exact Rtt';
. exact ihφ.2.mpr ht';
+lemma truthlemma₁ : (φ ∈ t.1.1) ↔ t ⊧ φ := truthlemma.1
+
lemma truthlemma₂ : (φ ∈ t.1.2) ↔ ¬t ⊧ φ := truthlemma.2
lemma iff_valid_on_canonicalModel_deducible : (canonicalModel 𝓢) ⊧ φ ↔ 𝓢 ⊢ φ := by
@@ -119,7 +121,8 @@
apply truthle... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_022c0ad287ca_2 | f348e7fc14eb344b | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Kripke/Completeness.lean | Completeness | 2 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 10 | 5 | [
{
"theorem_name": "iff_valid_on_canonicalModel_deducible",
"depth": 1,
"n_commands": 0,
"n_lines": 15,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n constructor;\n . contrapose;\n intro h;\n have : Tableau.Consistent 𝓢 (∅, {φ}) := by\n ... | [
{
"name": "truthlemma₁",
"text": "lemma truthlemma₁ : (φ ∈ t.1.1) ↔ t ⊧ φ := truthlemma.1\n\n",
"fan_in": 5,
"n_lines": 3,
"n_chars": 67,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false... | [
{
"name": "def_rel_box_satisfies",
"fan_in": 0,
"n_deps_direct": 2,
"n_deps_transitive": 3,
"n_lines": 10,
"n_chars": 318,
"n_subproofs": 0,
"n_tactics": 14,
"cyclomatic": 2,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 7,
"automation_only": false,
"max... | 3 | module
public import Foundation.Modal.Tableau
public import Foundation.Modal.Kripke.Basic
@[expose] public section
namespace LO.Modal
open Entailment
open Formula
open Kripke
open MaximalConsistentTableau
variable {S} [Entailment S (Formula ℕ)]
variable {𝓢 : S} [Entailment.Consistent 𝓢] [Entailment.K 𝓢]
namesp... | module
public import Foundation.Modal.Tableau
public import Foundation.Modal.Kripke.Basic
@[expose] public section
namespace LO.Modal
open Entailment
open Formula
open Kripke
open MaximalConsistentTableau
variable {S} [Entailment S (Formula ℕ)]
variable {𝓢 : S} [Entailment.Consistent 𝓢] [Entailment.K 𝓢]
namesp... | @@ -104,6 +104,8 @@
. exact Rtt';
. exact ihφ.2.mpr ht';
+lemma truthlemma₁ : (φ ∈ t.1.1) ↔ t ⊧ φ := truthlemma.1
+
lemma truthlemma₂ : (φ ∈ t.1.2) ↔ ¬t ⊧ φ := truthlemma.2
lemma iff_valid_on_canonicalModel_deducible : (canonicalModel 𝓢) ⊧ φ ↔ 𝓢 ⊢ φ := by
@@ -119,7 +121,8 @@
apply truthle... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_022c0ad287ca_3 | 71882138549b44d9 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Kripke/Completeness.lean | Completeness | 3 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 10 | 5 | [
{
"theorem_name": "iff_valid_on_canonicalModel_deducible",
"depth": 1,
"n_commands": 0,
"n_lines": 15,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n constructor;\n . contrapose;\n intro h;\n have : Tableau.Consistent 𝓢 (∅, {φ}) := by\n ... | [
{
"name": "truthlemma₁",
"text": "lemma truthlemma₁ : (φ ∈ t.1.1) ↔ t ⊧ φ := truthlemma.1\n\n",
"fan_in": 5,
"n_lines": 3,
"n_chars": 67,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false... | [
{
"name": "def_multirel_boxItr_satisfies",
"fan_in": 2,
"n_deps_direct": 2,
"n_deps_transitive": 3,
"n_lines": 53,
"n_chars": 2155,
"n_subproofs": 13,
"n_tactics": 96,
"cyclomatic": 5,
"n_automation": 6,
"n_rewrites": 1,
"n_structural": 35,
"automation_only": fals... | 3 | module
public import Foundation.Modal.Tableau
public import Foundation.Modal.Kripke.Basic
@[expose] public section
namespace LO.Modal
open Entailment
open Formula
open Kripke
open MaximalConsistentTableau
variable {S} [Entailment S (Formula ℕ)]
variable {𝓢 : S} [Entailment.Consistent 𝓢] [Entailment.K 𝓢]
namesp... | module
public import Foundation.Modal.Tableau
public import Foundation.Modal.Kripke.Basic
@[expose] public section
namespace LO.Modal
open Entailment
open Formula
open Kripke
open MaximalConsistentTableau
variable {S} [Entailment S (Formula ℕ)]
variable {𝓢 : S} [Entailment.Consistent 𝓢] [Entailment.K 𝓢]
namesp... | @@ -104,6 +104,8 @@
. exact Rtt';
. exact ihφ.2.mpr ht';
+lemma truthlemma₁ : (φ ∈ t.1.1) ↔ t ⊧ φ := truthlemma.1
+
lemma truthlemma₂ : (φ ∈ t.1.2) ↔ ¬t ⊧ φ := truthlemma.2
lemma iff_valid_on_canonicalModel_deducible : (canonicalModel 𝓢) ⊧ φ ↔ 𝓢 ⊢ φ := by
@@ -119,7 +121,8 @@
apply truthle... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_022c0ad287ca_4 | 8fa586923ebfc631 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Kripke/Completeness.lean | Completeness | 4 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 10 | 5 | [
{
"theorem_name": "iff_valid_on_canonicalModel_deducible",
"depth": 1,
"n_commands": 0,
"n_lines": 15,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n constructor;\n . contrapose;\n intro h;\n have : Tableau.Consistent 𝓢 (∅, {φ}) := by\n ... | [
{
"name": "truthlemma₁",
"text": "lemma truthlemma₁ : (φ ∈ t.1.1) ↔ t ⊧ φ := truthlemma.1\n\n",
"fan_in": 5,
"n_lines": 3,
"n_chars": 67,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false... | [
{
"name": "def_multirel_boxItr_mem₁",
"fan_in": 1,
"n_deps_direct": 2,
"n_deps_transitive": 4,
"n_lines": 6,
"n_chars": 293,
"n_subproofs": 0,
"n_tactics": 4,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
"ma... | 4 | module
public import Foundation.Modal.Tableau
public import Foundation.Modal.Kripke.Basic
@[expose] public section
namespace LO.Modal
open Entailment
open Formula
open Kripke
open MaximalConsistentTableau
variable {S} [Entailment S (Formula ℕ)]
variable {𝓢 : S} [Entailment.Consistent 𝓢] [Entailment.K 𝓢]
namesp... | module
public import Foundation.Modal.Tableau
public import Foundation.Modal.Kripke.Basic
@[expose] public section
namespace LO.Modal
open Entailment
open Formula
open Kripke
open MaximalConsistentTableau
variable {S} [Entailment S (Formula ℕ)]
variable {𝓢 : S} [Entailment.Consistent 𝓢] [Entailment.K 𝓢]
namesp... | @@ -104,6 +104,8 @@
. exact Rtt';
. exact ihφ.2.mpr ht';
+lemma truthlemma₁ : (φ ∈ t.1.1) ↔ t ⊧ φ := truthlemma.1
+
lemma truthlemma₂ : (φ ∈ t.1.2) ↔ ¬t ⊧ φ := truthlemma.2
lemma iff_valid_on_canonicalModel_deducible : (canonicalModel 𝓢) ⊧ φ ↔ 𝓢 ⊢ φ := by
@@ -119,7 +121,8 @@
apply truthle... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_022c0ad287ca_5 | 73ca6645b6c9fe40 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Kripke/Completeness.lean | Completeness | 5 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 10 | 2 | [
{
"theorem_name": "iff_valid_on_canonicalModel_deducible",
"depth": 1,
"n_commands": 0,
"n_lines": 15,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n constructor;\n . contrapose;\n intro h;\n have : Tableau.Consistent 𝓢 (∅, {φ}) := by\n ... | [
{
"name": "truthlemma₂",
"text": "lemma truthlemma₂ : (φ ∈ t.1.2) ↔ ¬t ⊧ φ := truthlemma.2\n\n",
"fan_in": 2,
"n_lines": 3,
"n_chars": 69,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": fals... | [
{
"name": "def_multirel_boxItr_mem₂",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 5,
"n_lines": 19,
"n_chars": 452,
"n_subproofs": 0,
"n_tactics": 33,
"cyclomatic": 2,
"n_automation": 2,
"n_rewrites": 0,
"n_structural": 12,
"automation_only": false,
... | 5 | module
public import Foundation.Modal.Tableau
public import Foundation.Modal.Kripke.Basic
@[expose] public section
namespace LO.Modal
open Entailment
open Formula
open Kripke
open MaximalConsistentTableau
variable {S} [Entailment S (Formula ℕ)]
variable {𝓢 : S} [Entailment.Consistent 𝓢] [Entailment.K 𝓢]
namesp... | module
public import Foundation.Modal.Tableau
public import Foundation.Modal.Kripke.Basic
@[expose] public section
namespace LO.Modal
open Entailment
open Formula
open Kripke
open MaximalConsistentTableau
variable {S} [Entailment S (Formula ℕ)]
variable {𝓢 : S} [Entailment.Consistent 𝓢] [Entailment.K 𝓢]
namesp... | @@ -106,9 +106,21 @@
lemma truthlemma₁ : (φ ∈ t.1.1) ↔ t ⊧ φ := truthlemma.1
+lemma truthlemma₂ : (φ ∈ t.1.2) ↔ ¬t ⊧ φ := truthlemma.2
+
lemma iff_valid_on_canonicalModel_deducible : (canonicalModel 𝓢) ⊧ φ ↔ 𝓢 ⊢ φ := by
constructor;
- . sorry
+ . contrapose;
+ intro h;
+ have : Tableau.Consistent 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_022c0ad287ca_6 | 99d456f3b239e03c | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Kripke/Completeness.lean | Completeness | 6 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 10 | 2 | [
{
"theorem_name": "iff_valid_on_canonicalModel_deducible",
"depth": 1,
"n_commands": 0,
"n_lines": 15,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n constructor;\n . contrapose;\n intro h;\n have : Tableau.Consistent 𝓢 (∅, {φ}) := by\n ... | [
{
"name": "truthlemma₂",
"text": "lemma truthlemma₂ : (φ ∈ t.1.2) ↔ ¬t ⊧ φ := truthlemma.2\n\n",
"fan_in": 2,
"n_lines": 3,
"n_chars": 69,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": fals... | [
{
"name": "def_rel_box_mem₂",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 6,
"n_lines": 4,
"n_chars": 127,
"n_subproofs": 0,
"n_tactics": 3,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"max_nesting... | 6 | module
public import Foundation.Modal.Tableau
public import Foundation.Modal.Kripke.Basic
@[expose] public section
namespace LO.Modal
open Entailment
open Formula
open Kripke
open MaximalConsistentTableau
variable {S} [Entailment S (Formula ℕ)]
variable {𝓢 : S} [Entailment.Consistent 𝓢] [Entailment.K 𝓢]
namesp... | module
public import Foundation.Modal.Tableau
public import Foundation.Modal.Kripke.Basic
@[expose] public section
namespace LO.Modal
open Entailment
open Formula
open Kripke
open MaximalConsistentTableau
variable {S} [Entailment S (Formula ℕ)]
variable {𝓢 : S} [Entailment.Consistent 𝓢] [Entailment.K 𝓢]
namesp... | @@ -106,9 +106,21 @@
lemma truthlemma₁ : (φ ∈ t.1.1) ↔ t ⊧ φ := truthlemma.1
+lemma truthlemma₂ : (φ ∈ t.1.2) ↔ ¬t ⊧ φ := truthlemma.2
+
lemma iff_valid_on_canonicalModel_deducible : (canonicalModel 𝓢) ⊧ φ ↔ 𝓢 ⊢ φ := by
constructor;
- . sorry
+ . contrapose;
+ intro h;
+ have : Tableau.Consistent 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_022c0ad287ca_7 | b2865b2cbdf45f3d | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Kripke/Completeness.lean | Completeness | 7 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 10 | 5 | [
{
"theorem_name": "iff_valid_on_canonicalModel_deducible",
"depth": 1,
"n_commands": 0,
"n_lines": 15,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n constructor;\n . contrapose;\n intro h;\n have : Tableau.Consistent 𝓢 (∅, {φ}) := by\n ... | [
{
"name": "truthlemma₁",
"text": "lemma truthlemma₁ : (φ ∈ t.1.1) ↔ t ⊧ φ := truthlemma.1\n\n",
"fan_in": 5,
"n_lines": 3,
"n_chars": 67,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false... | [
{
"name": "def_multirel_diaItr_satisfies",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 4,
"n_lines": 19,
"n_chars": 559,
"n_subproofs": 1,
"n_tactics": 33,
"cyclomatic": 2,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 13,
"automation_only": false,... | 4 | module
public import Foundation.Modal.Tableau
public import Foundation.Modal.Kripke.Basic
@[expose] public section
namespace LO.Modal
open Entailment
open Formula
open Kripke
open MaximalConsistentTableau
variable {S} [Entailment S (Formula ℕ)]
variable {𝓢 : S} [Entailment.Consistent 𝓢] [Entailment.K 𝓢]
namesp... | module
public import Foundation.Modal.Tableau
public import Foundation.Modal.Kripke.Basic
@[expose] public section
namespace LO.Modal
open Entailment
open Formula
open Kripke
open MaximalConsistentTableau
variable {S} [Entailment S (Formula ℕ)]
variable {𝓢 : S} [Entailment.Consistent 𝓢] [Entailment.K 𝓢]
namesp... | @@ -104,6 +104,8 @@
. exact Rtt';
. exact ihφ.2.mpr ht';
+lemma truthlemma₁ : (φ ∈ t.1.1) ↔ t ⊧ φ := truthlemma.1
+
lemma truthlemma₂ : (φ ∈ t.1.2) ↔ ¬t ⊧ φ := truthlemma.2
lemma iff_valid_on_canonicalModel_deducible : (canonicalModel 𝓢) ⊧ φ ↔ 𝓢 ⊢ φ := by
@@ -119,7 +121,8 @@
apply truthle... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_eb7e41cfa91c_0 | 7d0a183e26232791 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Intuitionistic/Formula.lean | Formula | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 2 | 1 | [
{
"theorem_name": "complexity_neg",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by simp [neg_def]",
"n_chars": 18,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 1,
"... | [
{
"name": "neg_def",
"text": "lemma neg_def (φ : Semiformulaᵢ L ξ n) : ∼φ = φ 🡒 ⊥ := rfl\n\n",
"fan_in": 1,
"n_lines": 3,
"n_chars": 73,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,... | [
{
"name": "complexity_neg",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 3,
"n_chars": 121,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"max_nesting":... | 1 | module
public import Foundation.FirstOrder.Basic
@[expose] public section
/-!
# Formulas of intuitionistic first-order logic
This file defines the formulas of first-order logic.
`φ : Semiformulaᵢ L ξ n` is a (semi-)formula of language `L` with bounded variables of `Fin n` and free variables of `ξ`.
The quantificati... | module
public import Foundation.FirstOrder.Basic
@[expose] public section
/-!
# Formulas of intuitionistic first-order logic
This file defines the formulas of first-order logic.
`φ : Semiformulaᵢ L ξ n` is a (semi-)formula of language `L` with bounded variables of `Fin n` and free variables of `ξ`.
The quantificati... | @@ -52,6 +52,8 @@
top := verum
tilde := neg
+lemma neg_def (φ : Semiformulaᵢ L ξ n) : ∼φ = φ 🡒 ⊥ := rfl
+
lemma verum_def : (⊤ : Semiformulaᵢ L ξ n) = ⊥ 🡒 ⊥ := rfl
instance : Quantifier (Semiformulaᵢ L ξ) where
@@ -135,7 +137,7 @@
@[simp] lemma complexity_exs (φ : Semiformulaᵢ L ξ (n + 1)) : complexity (... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_eb7e41cfa91c_1 | 5fc79adf6434ebe6 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Intuitionistic/Formula.lean | Formula | 1 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 2 | 1 | [
{
"theorem_name": "verum",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by simp [verum_def]",
"n_chars": 20,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 1,
"n_rewri... | [
{
"name": "verum_def",
"text": "lemma verum_def : (⊤ : Semiformulaᵢ L ξ n) = ⊥ 🡒 ⊥ := rfl\n\n",
"fan_in": 1,
"n_lines": 3,
"n_chars": 71,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true... | [
{
"name": "verum",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 3,
"n_chars": 87,
"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.Basic
@[expose] public section
/-!
# Formulas of intuitionistic first-order logic
This file defines the formulas of first-order logic.
`φ : Semiformulaᵢ L ξ n` is a (semi-)formula of language `L` with bounded variables of `Fin n` and free variables of `ξ`.
The quantificati... | module
public import Foundation.FirstOrder.Basic
@[expose] public section
/-!
# Formulas of intuitionistic first-order logic
This file defines the formulas of first-order logic.
`φ : Semiformulaᵢ L ξ n` is a (semi-)formula of language `L` with bounded variables of `Fin n` and free variables of `ξ`.
The quantificati... | @@ -54,6 +54,8 @@
lemma neg_def (φ : Semiformulaᵢ L ξ n) : ∼φ = φ 🡒 ⊥ := rfl
+lemma verum_def : (⊤ : Semiformulaᵢ L ξ n) = ⊥ 🡒 ⊥ := rfl
+
instance : Quantifier (Semiformulaᵢ L ξ) where
all := all
exs := exs
@@ -252,7 +254,7 @@
@[simp] lemma all_iff {φ : Semiformulaᵢ L ξ (n + 1)} : (∀⁰ φ).IsNegative ↔ φ.I... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_79e2f8da8e39_0 | 6cc3374843867db1 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Propositional/Kripke/Hilbert/Int/DP.lean | DP | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 2 | 1 | [
{
"theorem_name": "disjunctive",
"depth": 1,
"n_commands": 0,
"n_lines": 22,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n contrapose!;\n rintro ⟨hnφ, hnψ⟩;\n\n obtain ⟨M₁, w₁, hM₁, hφ⟩ := iff_not_validOnFrameClass_exists_model_world.mp $ instKrip... | [
{
"name": "satisfies_right_on_counterexampleDPModel",
"text": "lemma satisfies_right_on_counterexampleDPModel :\n w ⊧ φ ↔ (Satisfies (counterexampleDPModel M₁ M₂ w₁ w₂) (Sum.inr $ Sum.inr w) φ) := by\n induction φ generalizing w with\n | himp φ ψ ihp ihq =>\n constructor;\n . intro h X hWX hp;\n ... | [
{
"name": "disjunctive",
"fan_in": 0,
"n_deps_direct": 2,
"n_deps_transitive": 2,
"n_lines": 24,
"n_chars": 1170,
"n_subproofs": 2,
"n_tactics": 39,
"cyclomatic": 4,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 15,
"automation_only": false,
"max_nesting... | 2 | module
public import Foundation.Propositional.Kripke.Hilbert.Int.Basic
@[expose] public section
namespace LO.Propositional.Hilbert.Int
open Kripke
open Formula.Kripke
variable {M₁ : Kripke.Model} {M₂ : Kripke.Model} {φ ψ : Formula ℕ}
abbrev counterexampleDPModel (M₁ : Kripke.Model) (M₂ : Kripke.Model) (w₁ : M₁.Wo... | module
public import Foundation.Propositional.Kripke.Hilbert.Int.Basic
@[expose] public section
namespace LO.Propositional.Hilbert.Int
open Kripke
open Formula.Kripke
variable {M₁ : Kripke.Model} {M₂ : Kripke.Model} {φ ψ : Formula ℕ}
abbrev counterexampleDPModel (M₁ : Kripke.Model) (M₂ : Kripke.Model) (w₁ : M₁.Wo... | @@ -70,6 +70,23 @@
apply @ihq v |>.mpr $ h (by simpa) $ ihp.mp hp;
| _ => simp_all [counterexampleDPModel, Satisfies.iff_models, Satisfies];
+lemma satisfies_right_on_counterexampleDPModel :
+ w ⊧ φ ↔ (Satisfies (counterexampleDPModel M₁ M₂ w₁ w₂) (Sum.inr $ Sum.inr w) φ) := by
+ induction φ generalizing ... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_10609d3a3ca0_0 | eb43d3ee12f89374 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Propositional/Kripke/Preservation.lean | Preservation | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 1 | 1 | [
{
"theorem_name": "modal_equivalence",
"depth": 1,
"n_commands": 0,
"n_lines": 3,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n apply modal_equivalent_of_bisimilar $ Model.PseudoEpimorphism.bisimulation f;\n simp [Model.PseudoEpimorphism.bisimulati... | [
{
"name": "modal_equivalent_of_bisimilar",
"text": "lemma modal_equivalent_of_bisimilar (Bi : M₁ ⇄ M₂) (bisx : Bi x₁ x₂) : x₁ ↭ x₂ := by\n intro φ;\n induction φ generalizing x₁ x₂ with\n | hatom a => exact Bi.atomic bisx;\n | hand φ ψ ihφ ihψ =>\n constructor;\n . rintro ⟨hφ, hψ⟩;\n construc... | [
{
"name": "modal_equivalence",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 5,
"n_chars": 212,
"n_subproofs": 0,
"n_tactics": 5,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": false,
"max_nesti... | 1 | module
public import Foundation.Propositional.Kripke.Basic
@[expose] public section
namespace LO.Propositional
namespace Kripke
open Formula.Kripke
section Bisimulation
structure Model.Bisimulation (M₁ M₂ : Kripke.Model) where
toRel : M₁.World → M₂.World → Prop
atomic {x₁ : M₁.World} {x₂ : M₂.World} {a : ℕ} ... | module
public import Foundation.Propositional.Kripke.Basic
@[expose] public section
namespace LO.Propositional
namespace Kripke
open Formula.Kripke
section Bisimulation
structure Model.Bisimulation (M₁ M₂ : Kripke.Model) where
toRel : M₁.World → M₂.World → Prop
atomic {x₁ : M₁.World} {x₂ : M₂.World} {a : ℕ} ... | @@ -44,6 +44,38 @@
def ModalEquivalent {M₁ M₂ : Model} (w₁ : M₁.World) (w₂ : M₂.World) : Prop := ∀ {φ}, w₁ ⊧ φ ↔ w₂ ⊧ φ
infix:50 " ↭ " => ModalEquivalent
+lemma modal_equivalent_of_bisimilar (Bi : M₁ ⇄ M₂) (bisx : Bi x₁ x₂) : x₁ ↭ x₂ := by
+ intro φ;
+ induction φ generalizing x₁ x₂ with
+ | hatom a => exact Bi.... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_d36854a307c3_0 | a82708472f66c506 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Hilbert/NNFormula.lean | NNFormula | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 2 | 1 | [
{
"theorem_name": "exists_iff",
"depth": 1,
"n_commands": 0,
"n_lines": 22,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n induction φ with\n | hatom a => use (.atom a); simp;\n | hfalsum => use ⊥; simp;\n | himp φ ψ ihφ ihψ =>\n obtain ⟨φ', hφ... | [
{
"name": "iff_neg",
"text": "lemma iff_neg {φ : NNFormula _} : Modal.K ⊢ ∼(φ.toFormula) 🡘 (∼φ).toFormula := by\n induction φ using NNFormula.rec' with\n | hAtom a => simp;\n | hNatom a => apply K!_intro <;> simp;\n | hVerum => simp;\n | hFalsum => simp [Formula.eq_verum_neg_falsum];\n | hAnd φ ψ ihφ... | [
{
"name": "exists_iff",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 24,
"n_chars": 820,
"n_subproofs": 3,
"n_tactics": 42,
"cyclomatic": 2,
"n_automation": 3,
"n_rewrites": 1,
"n_structural": 19,
"automation_only": false,
"max_nesting":... | 1 | module
public import Foundation.Modal.Formula.NNFormula
public import Foundation.Modal.Hilbert.Normal.Basic
@[expose] public section
namespace LO.Modal.Hilbert
namespace NNFormula
open
NNFormula
LO.Entailment
LO.Entailment.FiniteContext
LO.Modal.Entailment
lemma exists_iff {φ} : ∃ ψ : NNFormula _, Modal.K... | module
public import Foundation.Modal.Formula.NNFormula
public import Foundation.Modal.Hilbert.Normal.Basic
@[expose] public section
namespace LO.Modal.Hilbert
namespace NNFormula
open
NNFormula
LO.Entailment
LO.Entailment.FiniteContext
LO.Modal.Entailment
lemma iff_neg {φ : NNFormula _} : Modal.K ⊢ ∼(φ.t... | @@ -15,6 +15,61 @@
LO.Entailment.FiniteContext
LO.Modal.Entailment
+lemma iff_neg {φ : NNFormula _} : Modal.K ⊢ ∼(φ.toFormula) 🡘 (∼φ).toFormula := by
+ induction φ using NNFormula.rec' with
+ | hAtom a => simp;
+ | hNatom a => apply K!_intro <;> simp;
+ | hVerum => simp;
+ | hFalsum => simp [Formula.eq_ve... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_0132ea924d07_0 | 9449d0f0323061c4 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/SetTheory/LoewenheimSkolem.lean | LoewenheimSkolem | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 1 | 1 | [
{
"theorem_name": "hull_models_iff",
"depth": 1,
"n_commands": 0,
"n_lines": 7,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n have :\n φ.Evalb (s := Structure.SkolemHull.str (standardStructure V) s) b ↔\n V ⊧/(b ·) φ :=\n Structure.Skol... | [
{
"name": "str_eq",
"text": "lemma str_eq : Structure.SkolemHull.str (standardStructure V) s = standardStructure (Hull s) := by\n have : (Structure.SkolemHull.str (standardStructure V) s).Eq ℒₛₑₜ (Hull s) := Structure.SkolemHull.eq\n have : (Structure.SkolemHull.str (standardStructure V) s).Mem ℒₛₑₜ (Hull... | [
{
"name": "hull_models_iff",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 10,
"n_chars": 342,
"n_subproofs": 1,
"n_tactics": 7,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 1,
"n_structural": 1,
"automation_only": false,
"max_nestin... | 1 | module
public import Foundation.FirstOrder.SetTheory.Basic
public import Foundation.FirstOrder.Skolemization.Hull
@[expose] public section
/-!
# Downward Löwenheim-Skolem theorem for models of set theory
-/
namespace LO.FirstOrder.SetTheory
variable {V : Type*} [SetStructure V] [Nonempty V] (s : Set V)
def Hull : ... | module
public import Foundation.FirstOrder.SetTheory.Basic
public import Foundation.FirstOrder.Skolemization.Hull
@[expose] public section
/-!
# Downward Löwenheim-Skolem theorem for models of set theory
-/
namespace LO.FirstOrder.SetTheory
variable {V : Type*} [SetStructure V] [Nonempty V] (s : Set V)
def Hull : ... | @@ -24,10 +24,21 @@
@[simp] lemma mk_mem_mk_iff {x y : V} {hx hy} : (⟨x, hx⟩ : Hull s) ∈ (⟨y, hy⟩ : Hull s) ↔ x ∈ y := by rfl
+lemma str_eq : Structure.SkolemHull.str (standardStructure V) s = standardStructure (Hull s) := by
+ have : (Structure.SkolemHull.str (standardStructure V) s).Eq ℒₛₑₜ (Hull s) := Structur... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_5e92081d42e0_0 | 90738bd576db6eb7 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Incompleteness/Halting.lean | Halting | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 3 | 1 | [
{
"theorem_name": "incomplete_of_REPred_not_ComputablePred_Nat",
"depth": 1,
"n_commands": 0,
"n_lines": 5,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n obtain ⟨φ, a, hφ₁, hφ₂⟩ := incomplete_of_REPred_not_ComputablePred_Nat' T hRE hC;\n apply inco... | [
{
"name": "incomplete_of_REPred_not_ComputablePred_Nat'",
"text": "lemma incomplete_of_REPred_not_ComputablePred_Nat' {P : ℕ → Prop} (hRE : REPred P) (hC : ¬ComputablePred P) :\n ∃ φ : ArithmeticSemisentence 1, ∃ a : ℕ, T ⊬ φ/[a] ∧ T ⊬ ∼φ/[a] := by\n let φ := codeOfREPred P;\n use φ;\n have hP : P = { n... | [
{
"name": "incomplete_of_REPred_not_ComputablePred_Nat",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 11,
"n_chars": 417,
"n_subproofs": 1,
"n_tactics": 10,
"cyclomatic": 3,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 4,
"automation_... | 1 | module
public import Foundation.FirstOrder.Incompleteness.First
@[expose] public section
namespace LO.FirstOrder.Arithmetic
variable (T : ArithmeticTheory) [T.Δ₁] [𝗜𝚺₁ ⪯ T] [T.SoundOnHierarchy 𝚺 1]
open LO.Entailment
open Classical
/--
If r.e. but not recursive predicate `P` on `ℕ` exists, then implies incom... | module
public import Foundation.FirstOrder.Incompleteness.First
@[expose] public section
namespace LO.FirstOrder.Arithmetic
variable (T : ArithmeticTheory) [T.Δ₁] [𝗜𝚺₁ ⪯ T] [T.SoundOnHierarchy 𝚺 1]
open LO.Entailment
open Classical
lemma incomplete_of_REPred_not_ComputablePred_Nat' {P : ℕ → Prop} (hRE : REPred... | @@ -11,11 +11,50 @@
open Classical
+lemma incomplete_of_REPred_not_ComputablePred_Nat' {P : ℕ → Prop} (hRE : REPred P) (hC : ¬ComputablePred P) :
+ ∃ φ : ArithmeticSemisentence 1, ∃ a : ℕ, T ⊬ φ/[a] ∧ T ⊬ ∼φ/[a] := by
+ let φ := codeOfREPred P;
+ use φ;
+ have hP : P = { n : ℕ | T ⊢ φ/[n] } := Set.ext fun x ↦ ... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_5e92081d42e0_1 | f45a9c41e9d8195e | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Incompleteness/Halting.lean | Halting | 1 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 3 | 1 | [
{
"theorem_name": "incomplete_of_REPred_not_ComputablePred_Nat",
"depth": 1,
"n_commands": 0,
"n_lines": 5,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n obtain ⟨φ, a, hφ₁, hφ₂⟩ := incomplete_of_REPred_not_ComputablePred_Nat' T hRE hC;\n apply inco... | [
{
"name": "incomplete_of_REPred_not_ComputablePred_Nat'",
"text": "lemma incomplete_of_REPred_not_ComputablePred_Nat' {P : ℕ → Prop} (hRE : REPred P) (hC : ¬ComputablePred P) :\n ∃ φ : ArithmeticSemisentence 1, ∃ a : ℕ, T ⊬ φ/[a] ∧ T ⊬ ∼φ/[a] := by\n let φ := codeOfREPred P;\n use φ;\n have hP : P = { n... | [
{
"name": "incomplete_of_REPred_not_ComputablePred",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 2,
"n_lines": 11,
"n_chars": 508,
"n_subproofs": 0,
"n_tactics": 5,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 3,
"automation_only"... | 2 | module
public import Foundation.FirstOrder.Incompleteness.First
@[expose] public section
namespace LO.FirstOrder.Arithmetic
variable (T : ArithmeticTheory) [T.Δ₁] [𝗜𝚺₁ ⪯ T] [T.SoundOnHierarchy 𝚺 1]
open LO.Entailment
open Classical
/--
If r.e. but not recursive predicate `P` on `ℕ` exists, then implies incom... | module
public import Foundation.FirstOrder.Incompleteness.First
@[expose] public section
namespace LO.FirstOrder.Arithmetic
variable (T : ArithmeticTheory) [T.Δ₁] [𝗜𝚺₁ ⪯ T] [T.SoundOnHierarchy 𝚺 1]
open LO.Entailment
open Classical
lemma incomplete_of_REPred_not_ComputablePred_Nat' {P : ℕ → Prop} (hRE : REPred... | @@ -11,11 +11,50 @@
open Classical
+lemma incomplete_of_REPred_not_ComputablePred_Nat' {P : ℕ → Prop} (hRE : REPred P) (hC : ¬ComputablePred P) :
+ ∃ φ : ArithmeticSemisentence 1, ∃ a : ℕ, T ⊬ φ/[a] ∧ T ⊬ ∼φ/[a] := by
+ let φ := codeOfREPred P;
+ use φ;
+ have hP : P = { n : ℕ | T ⊢ φ/[n] } := Set.ext fun x ↦ ... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_5e92081d42e0_2 | e88fdb79d679054f | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Incompleteness/Halting.lean | Halting | 2 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 3 | 1 | [
{
"theorem_name": "incomplete_of_REPred_not_ComputablePred",
"depth": 1,
"n_commands": 0,
"n_lines": 5,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n apply incomplete_of_REPred_not_ComputablePred_Nat T\n (P := fun n : ℕ ↦ (Encodable.decode (α := ... | [
{
"name": "incomplete_of_REPred_not_ComputablePred_Nat",
"text": "/--\n If r.e. but not recursive predicate `P` on `ℕ` exists, then implies incompleteness.\n-/\nlemma incomplete_of_REPred_not_ComputablePred_Nat\n {P : ℕ → Prop} (hRE : REPred P) (hC : ¬ComputablePred P) : Entailment.Incomplete T := by\n ... | [
{
"name": "incomplete_of_halting_problem",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 3,
"n_lines": 9,
"n_chars": 299,
"n_subproofs": 0,
"n_tactics": 3,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
... | 3 | module
public import Foundation.FirstOrder.Incompleteness.First
@[expose] public section
namespace LO.FirstOrder.Arithmetic
variable (T : ArithmeticTheory) [T.Δ₁] [𝗜𝚺₁ ⪯ T] [T.SoundOnHierarchy 𝚺 1]
open LO.Entailment
open Classical
lemma incomplete_of_REPred_not_ComputablePred_Nat' {P : ℕ → Prop} (hRE : REPred... | module
public import Foundation.FirstOrder.Incompleteness.First
@[expose] public section
namespace LO.FirstOrder.Arithmetic
variable (T : ArithmeticTheory) [T.Δ₁] [𝗜𝚺₁ ⪯ T] [T.SoundOnHierarchy 𝚺 1]
open LO.Entailment
open Classical
lemma incomplete_of_REPred_not_ComputablePred_Nat' {P : ℕ → Prop} (hRE : REPred... | @@ -47,6 +47,16 @@
cl_prover [hd₁, hd₂];
/--
+ If r.e. but not recursive predicate `P` on `ℕ` exists, then implies incompleteness.
+-/
+lemma incomplete_of_REPred_not_ComputablePred_Nat
+ {P : ℕ → Prop} (hRE : REPred P) (hC : ¬ComputablePred P) : Entailment.Incomplete T := by
+ obtain ⟨φ, a, hφ₁, hφ₂⟩ := i... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_9d02ddf619ed_0 | d50fae97511cad4f | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Propositional/Formula/NNFormula.lean | NNFormula | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 1 | 1 | [
{
"theorem_name": "neg_neg'",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " neg_neg φ",
"n_chars": 11,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0... | [
{
"name": "neg_neg",
"text": "lemma neg_neg (φ : NNFormula α) : neg (neg φ) = φ :=\n by induction φ <;> simp [*, neg]\n\n",
"fan_in": 1,
"n_lines": 4,
"n_chars": 94,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 3,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 1,... | [
{
"name": "neg_neg'",
"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.Logic.LogicSymbol
@[expose] public section
namespace LO.Propositional
inductive NNFormula (α : Type u) : Type u where
| verum : NNFormula α
| falsum : NNFormula α
| atom : α → NNFormula α
| natom : α → NNFormula α
| and : NNFormula α → NNFormula α → NNFormula α
| o... | module
public import Foundation.Logic.LogicSymbol
@[expose] public section
namespace LO.Propositional
inductive NNFormula (α : Type u) : Type u where
| verum : NNFormula α
| falsum : NNFormula α
| atom : α → NNFormula α
| natom : α → NNFormula α
| and : NNFormula α → NNFormula α → NNFormula α
| o... | @@ -26,6 +26,9 @@
| and φ ψ => or (neg φ) (neg ψ)
| or φ ψ => and (neg φ) (neg ψ)
+lemma neg_neg (φ : NNFormula α) : neg (neg φ) = φ :=
+ by induction φ <;> simp [*, neg]
+
instance : LogicalConnective (NNFormula α) where
tilde := neg
arrow := fun φ ψ => or (neg φ) ψ
@@ -64,7 +67,7 @@
@[simp] lemma ... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_863502fb112d_0 | 3244cbef74a9d2dd | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Vorspiel/ExistsUnique.lean | ExistsUnique | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 4 | 2 | [
{
"theorem_name": "choose_uniq",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " h.unique hx (choose!_spec h)",
"n_chars": 29,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": ... | [
{
"name": "choose!_spec",
"text": "lemma choose!_spec (h : ∃! x, p x) : p (choose! h) := choose_spec h.exists\n\n",
"fan_in": 2,
"n_lines": 3,
"n_chars": 78,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"aut... | [
{
"name": "choose_uniq",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 3,
"n_chars": 96,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
"max_nesting": 1
... | 1 | module
public import Mathlib.Logic.IsEmpty
@[expose]
public section
namespace Classical
variable {α : Sort*} {p : α → Prop} {r : α → α → Prop}
noncomputable def choose! (h : ∃! x, p x) : α := choose h.exists
lemma choose_uniq (h : ∃! x, p x) (hx : p x) : x = choose! h := sorry
@[simp] lemma choose!_eq_iff_right (... | module
public import Mathlib.Logic.IsEmpty
@[expose]
public section
namespace Classical
variable {α : Sort*} {p : α → Prop} {r : α → α → Prop}
noncomputable def choose! (h : ∃! x, p x) : α := choose h.exists
lemma choose!_spec (h : ∃! x, p x) : p (choose! h) := choose_spec h.exists
lemma choose_uniq (h : ∃! x, p ... | @@ -10,10 +10,12 @@
noncomputable def choose! (h : ∃! x, p x) : α := choose h.exists
-lemma choose_uniq (h : ∃! x, p x) (hx : p x) : x = choose! h := sorry
+lemma choose!_spec (h : ∃! x, p x) : p (choose! h) := choose_spec h.exists
+lemma choose_uniq (h : ∃! x, p x) (hx : p x) : x = choose! h := h.unique hx (cho... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_863502fb112d_1 | b2921d579a05ea9f | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Vorspiel/ExistsUnique.lean | ExistsUnique | 1 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 4 | 2 | [
{
"theorem_name": "extendedChoose!_uniq",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": "\n (h x hpx).unique hrx (extendedchoose!_spec h default hpx)",
"n_chars": 60,
"n_subproofs": 0,
"n_tactics": 1,
... | [
{
"name": "extendedchoose!_spec",
"text": "lemma extendedchoose!_spec (h : ∀ x, p x → ∃! y, r x y) (default : α) (hx : p x) :\n r x (extendedChoose! h default x) :=\n choose_spec (exitsUnique_extend h default x).exists |>.1 hx\n\n",
"fan_in": 2,
"n_lines": 5,
"n_chars": 194,
"n_subproofs... | [
{
"name": "extendedChoose!_uniq",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 2,
"n_lines": 5,
"n_chars": 205,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
"max_ne... | 1 | module
public import Mathlib.Logic.IsEmpty
@[expose]
public section
namespace Classical
variable {α : Sort*} {p : α → Prop} {r : α → α → Prop}
noncomputable def choose! (h : ∃! x, p x) : α := choose h.exists
lemma choose!_spec (h : ∃! x, p x) : p (choose! h) := choose_spec h.exists
lemma choose_uniq (h : ∃! x, p ... | module
public import Mathlib.Logic.IsEmpty
@[expose]
public section
namespace Classical
variable {α : Sort*} {p : α → Prop} {r : α → α → Prop}
noncomputable def choose! (h : ∃! x, p x) : α := choose h.exists
lemma choose!_spec (h : ∃! x, p x) : p (choose! h) := choose_spec h.exists
lemma choose_uniq (h : ∃! x, p ... | @@ -29,8 +29,13 @@
noncomputable def extendedChoose! (h : ∀ x, p x → ∃! y, r x y)
(default : α) (x : α) : α := choose (exitsUnique_extend h default x).exists
+lemma extendedchoose!_spec (h : ∀ x, p x → ∃! y, r x y) (default : α) (hx : p x) :
+ r x (extendedChoose! h default x) :=
+ choose_spec (exitsUnique_... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_863502fb112d_2 | abcdc87aab951871 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Vorspiel/ExistsUnique.lean | ExistsUnique | 2 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 4 | 2 | [
{
"theorem_name": "extendedChoose!_uniq",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": "\n (h x hpx).unique hrx (extendedchoose!_spec h default hpx)",
"n_chars": 60,
"n_subproofs": 0,
"n_tactics": 1,
... | [
{
"name": "extendedchoose!_spec",
"text": "lemma extendedchoose!_spec (h : ∀ x, p x → ∃! y, r x y) (default : α) (hx : p x) :\n r x (extendedChoose! h default x) :=\n choose_spec (exitsUnique_extend h default x).exists |>.1 hx\n\n",
"fan_in": 2,
"n_lines": 5,
"n_chars": 194,
"n_subproofs... | [
{
"name": "extendedChoose!_eq_iff",
"fan_in": 0,
"n_deps_direct": 2,
"n_deps_transitive": 3,
"n_lines": 5,
"n_chars": 243,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 2,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 2,
"automation_only": false,
"max_... | 2 | module
public import Mathlib.Logic.IsEmpty
@[expose]
public section
namespace Classical
variable {α : Sort*} {p : α → Prop} {r : α → α → Prop}
noncomputable def choose! (h : ∃! x, p x) : α := choose h.exists
lemma choose!_spec (h : ∃! x, p x) : p (choose! h) := choose_spec h.exists
lemma choose_uniq (h : ∃! x, p ... | module
public import Mathlib.Logic.IsEmpty
@[expose]
public section
namespace Classical
variable {α : Sort*} {p : α → Prop} {r : α → α → Prop}
noncomputable def choose! (h : ∃! x, p x) : α := choose h.exists
lemma choose!_spec (h : ∃! x, p x) : p (choose! h) := choose_spec h.exists
lemma choose_uniq (h : ∃! x, p ... | @@ -29,12 +29,17 @@
noncomputable def extendedChoose! (h : ∀ x, p x → ∃! y, r x y)
(default : α) (x : α) : α := choose (exitsUnique_extend h default x).exists
+lemma extendedchoose!_spec (h : ∀ x, p x → ∃! y, r x y) (default : α) (hx : p x) :
+ r x (extendedChoose! h default x) :=
+ choose_spec (exitsUnique... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_1e0995290f4a_0 | 794181b78e6a61d0 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Formula/Basic.lean | Basic | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 3 | 2 | [
{
"theorem_name": "le_max_atoms_freshAtom",
"depth": 1,
"n_commands": 0,
"n_lines": 9,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n simp only [freshAtom, Finset.max'_lt_iff];\n intro a ha;\n split;\n . have := le_max_atoms_of_mem_atoms ha;\n ... | [
{
"name": "le_max_atoms_of_mem_atoms",
"text": "lemma le_max_atoms_of_mem_atoms {a : ℕ} (ha : a ∈ φ.atoms) : a ≤ φ.atoms.max' (⟨a, ha⟩) := by\n induction φ with\n | hfalsum => simp [atoms] at ha;\n | hatom b => simp [atoms] at ha ⊢; omega;\n | hbox φ ihφ => apply ihφ; simpa [atoms] using ha;\n | himp φ... | [
{
"name": "le_max_atoms_freshAtom",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 11,
"n_chars": 298,
"n_subproofs": 2,
"n_tactics": 17,
"cyclomatic": 2,
"n_automation": 2,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": false,
"ma... | 1 | module
public import Foundation.Modal.LogicSymbol
public import Foundation.Propositional.Formula.Basic
@[expose] public section
namespace LO
namespace Modal
@[grind]
inductive Formula (α : Type*) where
| atom : α → Formula α
| falsum : Formula α
| imp : Formula α → Formula α → Formula α
| box : For... | module
public import Foundation.Modal.LogicSymbol
public import Foundation.Propositional.Formula.Basic
@[expose] public section
namespace LO
namespace Modal
@[grind]
inductive Formula (α : Type*) where
| atom : α → Formula α
| falsum : Formula α
| imp : Formula α → Formula α → Formula α
| box : For... | @@ -467,11 +467,26 @@
def freshAtom (φ : Formula ℕ) : ℕ := if h : φ.atoms.Nonempty then φ.atoms.max' h + 1 else 0
+lemma le_max_atoms_of_mem_atoms {a : ℕ} (ha : a ∈ φ.atoms) : a ≤ φ.atoms.max' (⟨a, ha⟩) := by
+ induction φ with
+ | hfalsum => simp [atoms] at ha;
+ | hatom b => simp [atoms] at ha ⊢; omega;
+ | ... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_1e0995290f4a_1 | a40fa3bb372c82d4 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Formula/Basic.lean | Basic | 1 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 3 | 1 | [
{
"theorem_name": "ne_freshAtom_of_mem_subformulas",
"depth": 1,
"n_commands": 0,
"n_lines": 4,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n by_contra hC; subst hC;\n apply Nat.not_lt.mpr $ le_max_atoms_of_mem_atoms $ iff_mem_atoms_mem_subformula.... | [
{
"name": "le_max_atoms_freshAtom",
"text": "lemma le_max_atoms_freshAtom (h : φ.atoms.Nonempty) : Finset.max' φ.atoms h < φ.freshAtom := by\n simp only [freshAtom, Finset.max'_lt_iff];\n intro a ha;\n split;\n . have := le_max_atoms_of_mem_atoms ha;\n omega;\n . exfalso;\n have : φ.atoms.Nonempt... | [
{
"name": "ne_freshAtom_of_mem_subformulas",
"fan_in": 0,
"n_deps_direct": 3,
"n_deps_transitive": 3,
"n_lines": 6,
"n_chars": 245,
"n_subproofs": 0,
"n_tactics": 8,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 1,
"n_structural": 2,
"automation_only": false,
... | 3 | module
public import Foundation.Modal.LogicSymbol
public import Foundation.Propositional.Formula.Basic
@[expose] public section
namespace LO
namespace Modal
@[grind]
inductive Formula (α : Type*) where
| atom : α → Formula α
| falsum : Formula α
| imp : Formula α → Formula α → Formula α
| box : For... | module
public import Foundation.Modal.LogicSymbol
public import Foundation.Propositional.Formula.Basic
@[expose] public section
namespace LO
namespace Modal
@[grind]
inductive Formula (α : Type*) where
| atom : α → Formula α
| falsum : Formula α
| imp : Formula α → Formula α → Formula α
| box : For... | @@ -484,8 +484,21 @@
. simp [atoms, Finset.max'_union hφ ⟨_, hψ⟩, ihψ hψ];
. simp [atoms, Finset.not_nonempty_iff_eq_empty.mp hφ, ihψ hψ];
-lemma ne_freshAtom_of_mem_subformulas (h : atom a ∈ φ.subformulas) : φ.freshAtom ≠ a := sorry
+lemma le_max_atoms_freshAtom (h : φ.atoms.Nonempty) : Finset.max' φ.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_6ac59065e7dd_0 | 242800755d504cde | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Propositional/Kripke/AxiomLEM.lean | AxiomLEM | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 2 | 1 | [
{
"theorem_name": "validate_axiomLEM_of_isEuclidean",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " validate_axiomLEM_of_isSymmetric",
"n_chars": 33,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic"... | [
{
"name": "validate_axiomLEM_of_isSymmetric",
"text": "lemma validate_axiomLEM_of_isSymmetric [F.IsSymmetric] : F ⊧ (Axioms.LEM φ) := by\n have := F.symm;\n revert this;\n contrapose!;\n intro h;\n\n obtain ⟨V, x, h⟩ := ValidOnFrame.exists_valuation_world_of_not h;\n unfold Satisfies at h;\n push Not... | [
{
"name": "validate_axiomLEM_of_isEuclidean",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 3,
"n_chars": 116,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,... | 1 | module
public import Foundation.Propositional.Kripke.Completeness
@[expose] public section
namespace LO.Propositional
open Kripke
open Formula.Kripke
namespace Kripke
protected abbrev Frame.IsSymmetric (F : Frame) := _root_.Std.Symm F.Rel
lemma Frame.symm {F : Frame} [F.IsSymmetric] : ∀ ⦃x y : F⦄, x ≺ y → y ≺ x :... | module
public import Foundation.Propositional.Kripke.Completeness
@[expose] public section
namespace LO.Propositional
open Kripke
open Formula.Kripke
namespace Kripke
protected abbrev Frame.IsSymmetric (F : Frame) := _root_.Std.Symm F.Rel
lemma Frame.symm {F : Frame} [F.IsSymmetric] : ∀ ⦃x y : F⦄, x ≺ y → y ≺ x :... | @@ -19,8 +19,29 @@
variable {F : Kripke.Frame}
-lemma validate_axiomLEM_of_isEuclidean [F.IsEuclidean] : F ⊧ (Axioms.LEM φ) := sorry
+lemma validate_axiomLEM_of_isSymmetric [F.IsSymmetric] : F ⊧ (Axioms.LEM φ) := by
+ have := F.symm;
+ revert this;
+ contrapose!;
+ intro h;
+ obtain ⟨V, x, h⟩ := ValidOnFram... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_64c4494ce03b_0 | 6e7aa02cc633f832 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Logic/GLPoint3OplusBoxBot/Basic.lean | Basic | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 6 | 1 | [
{
"theorem_name": "GLPoint3OplusBoxBot.weakerThan_lt",
"depth": 1,
"n_commands": 0,
"n_lines": 3,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n rw [(show m = n + (m - n) by omega)];\n apply GLPoint3OplusBoxBot.weakerThan_add;",
"n_chars": 87,
... | [
{
"name": "GLPoint3OplusBoxBot.weakerThan_add",
"text": "lemma GLPoint3OplusBoxBot.weakerThan_add {n k : ℕ} : (Modal.GLPoint3OplusBoxBot (n + k)) ⪯ (Modal.GLPoint3OplusBoxBot n) := by\n induction k with\n | zero => simp [Modal.GLPoint3OplusBoxBot];\n | succ k ih =>\n trans (Modal.GLPoint3OplusBoxBot (... | [
{
"name": "GLPoint3OplusBoxBot.weakerThan_lt",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 5,
"n_chars": 223,
"n_subproofs": 0,
"n_tactics": 5,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 2,
"n_structural": 1,
"automation_only": false... | 1 | module
public import Foundation.Modal.Logic.SumNormal
public import Foundation.Modal.Maximal.Unprovability
public import Mathlib.Data.ENat.Basic
public import Foundation.Modal.Kripke.Logic.GL.Completeness
public import Foundation.Modal.Kripke.Logic.Ver
public import Foundation.Modal.Logic.Global
@[expose] public sect... | module
public import Foundation.Modal.Logic.SumNormal
public import Foundation.Modal.Maximal.Unprovability
public import Mathlib.Data.ENat.Basic
public import Foundation.Modal.Kripke.Logic.GL.Completeness
public import Foundation.Modal.Kripke.Logic.Ver
public import Foundation.Modal.Logic.Global
@[expose] public sect... | @@ -64,8 +64,18 @@
| subst ih => apply Logic.subst _ ih;
| nec ih => apply nec! ih;
-lemma GLPoint3OplusBoxBot.weakerThan_lt {n m : ℕ} (hmn : m > n) : (Modal.GLPoint3OplusBoxBot m) ⪯ (Modal.GLPoint3OplusBoxBot n) := sorry
+lemma GLPoint3OplusBoxBot.weakerThan_add {n k : ℕ} : (Modal.GLPoint3OplusBoxBot (n + 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_64c4494ce03b_2 | ffb9b4784fe6b9fb | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Logic/GLPoint3OplusBoxBot/Basic.lean | Basic | 2 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 6 | 1 | [
{
"theorem_name": "eq_GLPoint3OplusBoxBot_2_GLPoint2",
"depth": 1,
"n_commands": 0,
"n_lines": 24,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n ext φ;\n constructor;\n . simp only [←iff_provable];\n intro h;\n induction h using sumNormal.re... | [
{
"name": "GLPoint3OplusBoxBot.provable_weakPoint2_in_2",
"text": "lemma GLPoint3OplusBoxBot.provable_weakPoint2_in_2 : Modal.GLPoint3OplusBoxBot 2 ⊢ Axioms.WeakPoint2 (.atom 0) (.atom 1) := by\n suffices Modal.GLPoint3OplusBoxBot 2 ⊢ Axioms.CD (.atom 0) by\n apply C!_trans (Logic.subst (λ _ => (□(.atom... | [
{
"name": "eq_GLPoint3OplusBoxBot_2_GLPoint2",
"fan_in": 0,
"n_deps_direct": 2,
"n_deps_transitive": 4,
"n_lines": 26,
"n_chars": 980,
"n_subproofs": 1,
"n_tactics": 45,
"cyclomatic": 5,
"n_automation": 8,
"n_rewrites": 3,
"n_structural": 14,
"automation_only": fa... | 3 | module
public import Foundation.Modal.Logic.SumNormal
public import Foundation.Modal.Maximal.Unprovability
public import Mathlib.Data.ENat.Basic
public import Foundation.Modal.Kripke.Logic.GL.Completeness
public import Foundation.Modal.Kripke.Logic.Ver
public import Foundation.Modal.Logic.Global
@[expose] public sect... | module
public import Foundation.Modal.Logic.SumNormal
public import Foundation.Modal.Maximal.Unprovability
public import Mathlib.Data.ENat.Basic
public import Foundation.Modal.Kripke.Logic.GL.Completeness
public import Foundation.Modal.Kripke.Logic.Ver
public import Foundation.Modal.Logic.Global
@[expose] public sect... | @@ -151,6 +151,56 @@
instance : (Modal.GLPoint3OplusBoxBot 1) ≊ Modal.Ver := by rw [eq_GLPoint3OplusBoxBot_1_Ver];
+lemma GLPoint3OplusBoxBot.provable_weakPoint2_in_2 : Modal.GLPoint3OplusBoxBot 2 ⊢ Axioms.WeakPoint2 (.atom 0) (.atom 1) := by
+ suffices Modal.GLPoint3OplusBoxBot 2 ⊢ Axioms.CD (.atom 0) by
+ ap... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_50283ea28128_0 | aab5996485b14df3 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Vorspiel/Rel/CWF.lean | CWF | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 8 | 1 | [
{
"theorem_name": "ConverseWellFounded.has_max",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n apply ConverseWellFounded.iff_has_max.mp h;",
"n_chars": 49,
"n_subproofs": 0,
"n_tactics": 3,
... | [
{
"name": "ConverseWellFounded.iff_has_max",
"text": "lemma ConverseWellFounded.iff_has_max : ConverseWellFounded R ↔ (∀ (s : Set α), Set.Nonempty s → ∃ m ∈ s, ∀ x ∈ s, ¬(R m x)) := by\n simp [ConverseWellFounded, WellFounded.wellFounded_iff_has_min, flip]\n\n",
"fan_in": 1,
"n_lines": 4,
"n_ch... | [
{
"name": "ConverseWellFounded.has_max",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 4,
"n_chars": 190,
"n_subproofs": 0,
"n_tactics": 3,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": false,
... | 1 | module
public import Mathlib.Data.Fintype.Card
public import Mathlib.Data.Finset.Lattice.Fold
public import Foundation.Vorspiel.Rel.Connected
@[expose]
public section
section
abbrev ConverseWellFounded {α} (rel : Rel α α) := WellFounded $ flip rel
class IsConverseWellFounded (α) (rel : Rel α α) : Prop where cwf : ... | module
public import Mathlib.Data.Fintype.Card
public import Mathlib.Data.Finset.Lattice.Fold
public import Foundation.Vorspiel.Rel.Connected
@[expose]
public section
section
abbrev ConverseWellFounded {α} (rel : Rel α α) := WellFounded $ flip rel
class IsConverseWellFounded (α) (rel : Rel α α) : Prop where cwf : ... | @@ -19,8 +19,12 @@
variable {α} {R : Rel α α}
-lemma ConverseWellFounded.has_max (h : ConverseWellFounded R) : ∀ (s : Set α), Set.Nonempty s → ∃ m ∈ s, ∀ x ∈ s, ¬(R m x) := sorry
+lemma ConverseWellFounded.iff_has_max : ConverseWellFounded R ↔ (∀ (s : Set α), Set.Nonempty s → ∃ m ∈ s, ∀ x ∈ s, ¬(R m x)) := by
+ 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_50283ea28128_1 | 18fe4295db16b140 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Vorspiel/Rel/CWF.lean | CWF | 1 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 8 | 5 | [
{
"theorem_name": "cwfHeight_gt_of",
"depth": 1,
"n_commands": 0,
"n_lines": 3,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " fun h ↦ calc\n cwfHeight R a = Finset.sup {x : α | R a x} fun b ↦ cwfHeight R b + 1 := cwfHeight_eq a\n _ ≥ cwfHe... | [
{
"name": "cwfHeight_eq",
"text": "lemma cwfHeight_eq (a : α) :\n cwfHeight R a = Finset.sup {x : α | R a x} (fun b ↦ cwfHeight R b + 1) := by\n have h : cwfHeight R a = Finset.univ.sup fun b : {y : α // R a y} ↦ cwfHeight R b + 1 :=\n WellFounded.fix_eq _ _ a\n suffices\n Finset.univ.sup (fun b ... | [
{
"name": "cwfHeight_gt_of",
"fan_in": 3,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 6,
"n_chars": 291,
"n_subproofs": 0,
"n_tactics": 3,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": false,
"max_nesting... | 1 | module
public import Mathlib.Data.Fintype.Card
public import Mathlib.Data.Finset.Lattice.Fold
public import Foundation.Vorspiel.Rel.Connected
@[expose]
public section
section
abbrev ConverseWellFounded {α} (rel : Rel α α) := WellFounded $ flip rel
class IsConverseWellFounded (α) (rel : Rel α α) : Prop where cwf : ... | module
public import Mathlib.Data.Fintype.Card
public import Mathlib.Data.Finset.Lattice.Fold
public import Foundation.Vorspiel.Rel.Connected
@[expose]
public section
section
abbrev ConverseWellFounded {α} (rel : Rel α α) := WellFounded $ flip rel
class IsConverseWellFounded (α) (rel : Rel α α) : Prop where cwf : ... | @@ -46,8 +46,26 @@
open Classical
+lemma cwfHeight_eq (a : α) :
+ cwfHeight R a = Finset.sup {x : α | R a x} (fun b ↦ cwfHeight R b + 1) := by
+ have h : cwfHeight R a = Finset.univ.sup fun b : {y : α // R a y} ↦ cwfHeight R b + 1 :=
+ WellFounded.fix_eq _ _ a
+ suffices
+ Finset.univ.sup (fun b : {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_50283ea28128_2 | 5bab1b8c9d145836 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Vorspiel/Rel/CWF.lean | CWF | 2 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 8 | 5 | [
{
"theorem_name": "cwfHeight_gt_of",
"depth": 1,
"n_commands": 0,
"n_lines": 3,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " fun h ↦ calc\n cwfHeight R a = Finset.sup {x : α | R a x} fun b ↦ cwfHeight R b + 1 := cwfHeight_eq a\n _ ≥ cwfHe... | [
{
"name": "cwfHeight_eq",
"text": "lemma cwfHeight_eq (a : α) :\n cwfHeight R a = Finset.sup {x : α | R a x} (fun b ↦ cwfHeight R b + 1) := by\n have h : cwfHeight R a = Finset.univ.sup fun b : {y : α // R a y} ↦ cwfHeight R b + 1 :=\n WellFounded.fix_eq _ _ a\n suffices\n Finset.univ.sup (fun b ... | [
{
"name": "cwfHeight_eq_zero_iff",
"fan_in": 1,
"n_deps_direct": 2,
"n_deps_transitive": 2,
"n_lines": 13,
"n_chars": 491,
"n_subproofs": 1,
"n_tactics": 10,
"cyclomatic": 2,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 6,
"automation_only": false,
"max... | 2 | module
public import Mathlib.Data.Fintype.Card
public import Mathlib.Data.Finset.Lattice.Fold
public import Foundation.Vorspiel.Rel.Connected
@[expose]
public section
section
abbrev ConverseWellFounded {α} (rel : Rel α α) := WellFounded $ flip rel
class IsConverseWellFounded (α) (rel : Rel α α) : Prop where cwf : ... | module
public import Mathlib.Data.Fintype.Card
public import Mathlib.Data.Finset.Lattice.Fold
public import Foundation.Vorspiel.Rel.Connected
@[expose]
public section
section
abbrev ConverseWellFounded {α} (rel : Rel α α) := WellFounded $ flip rel
class IsConverseWellFounded (α) (rel : Rel α α) : Prop where cwf : ... | @@ -46,8 +46,26 @@
open Classical
+lemma cwfHeight_eq (a : α) :
+ cwfHeight R a = Finset.sup {x : α | R a x} (fun b ↦ cwfHeight R b + 1) := by
+ have h : cwfHeight R a = Finset.univ.sup fun b : {y : α // R a y} ↦ cwfHeight R b + 1 :=
+ WellFounded.fix_eq _ _ a
+ suffices
+ Finset.univ.sup (fun b : {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_50283ea28128_3 | 906c26d0c94bea88 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Vorspiel/Rel/CWF.lean | CWF | 3 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 8 | 5 | [
{
"theorem_name": "cwfHeight_gt_of",
"depth": 1,
"n_commands": 0,
"n_lines": 3,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " fun h ↦ calc\n cwfHeight R a = Finset.sup {x : α | R a x} fun b ↦ cwfHeight R b + 1 := cwfHeight_eq a\n _ ≥ cwfHe... | [
{
"name": "cwfHeight_eq",
"text": "lemma cwfHeight_eq (a : α) :\n cwfHeight R a = Finset.sup {x : α | R a x} (fun b ↦ cwfHeight R b + 1) := by\n have h : cwfHeight R a = Finset.univ.sup fun b : {y : α // R a y} ↦ cwfHeight R b + 1 :=\n WellFounded.fix_eq _ _ a\n suffices\n Finset.univ.sup (fun b ... | [
{
"name": "cwfHeight_le",
"fan_in": 2,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 8,
"n_chars": 191,
"n_subproofs": 0,
"n_tactics": 5,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 1,
"n_structural": 3,
"automation_only": false,
"max_nesting": ... | 1 | module
public import Mathlib.Data.Fintype.Card
public import Mathlib.Data.Finset.Lattice.Fold
public import Foundation.Vorspiel.Rel.Connected
@[expose]
public section
section
abbrev ConverseWellFounded {α} (rel : Rel α α) := WellFounded $ flip rel
class IsConverseWellFounded (α) (rel : Rel α α) : Prop where cwf : ... | module
public import Mathlib.Data.Fintype.Card
public import Mathlib.Data.Finset.Lattice.Fold
public import Foundation.Vorspiel.Rel.Connected
@[expose]
public section
section
abbrev ConverseWellFounded {α} (rel : Rel α α) := WellFounded $ flip rel
class IsConverseWellFounded (α) (rel : Rel α α) : Prop where cwf : ... | @@ -46,8 +46,28 @@
open Classical
+lemma cwfHeight_eq (a : α) :
+ cwfHeight R a = Finset.sup {x : α | R a x} (fun b ↦ cwfHeight R b + 1) := by
+ have h : cwfHeight R a = Finset.univ.sup fun b : {y : α // R a y} ↦ cwfHeight R b + 1 :=
+ WellFounded.fix_eq _ _ a
+ suffices
+ Finset.univ.sup (fun b : {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_50283ea28128_4 | c0172c4e1373d1a3 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Vorspiel/Rel/CWF.lean | CWF | 4 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 8 | 5 | [
{
"theorem_name": "cwfHeight_gt_of",
"depth": 1,
"n_commands": 0,
"n_lines": 3,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " fun h ↦ calc\n cwfHeight R a = Finset.sup {x : α | R a x} fun b ↦ cwfHeight R b + 1 := cwfHeight_eq a\n _ ≥ cwfHe... | [
{
"name": "cwfHeight_eq",
"text": "lemma cwfHeight_eq (a : α) :\n cwfHeight R a = Finset.sup {x : α | R a x} (fun b ↦ cwfHeight R b + 1) := by\n have h : cwfHeight R a = Finset.univ.sup fun b : {y : α // R a y} ↦ cwfHeight R b + 1 :=\n WellFounded.fix_eq _ _ a\n suffices\n Finset.univ.sup (fun b ... | [
{
"name": "lt_cwfHeight",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 9,
"n_chars": 334,
"n_subproofs": 1,
"n_tactics": 7,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 1,
"n_structural": 3,
"automation_only": false,
"max_nesting": ... | 1 | module
public import Mathlib.Data.Fintype.Card
public import Mathlib.Data.Finset.Lattice.Fold
public import Foundation.Vorspiel.Rel.Connected
@[expose]
public section
section
abbrev ConverseWellFounded {α} (rel : Rel α α) := WellFounded $ flip rel
class IsConverseWellFounded (α) (rel : Rel α α) : Prop where cwf : ... | module
public import Mathlib.Data.Fintype.Card
public import Mathlib.Data.Finset.Lattice.Fold
public import Foundation.Vorspiel.Rel.Connected
@[expose]
public section
section
abbrev ConverseWellFounded {α} (rel : Rel α α) := WellFounded $ flip rel
class IsConverseWellFounded (α) (rel : Rel α α) : Prop where cwf : ... | @@ -46,8 +46,28 @@
open Classical
+lemma cwfHeight_eq (a : α) :
+ cwfHeight R a = Finset.sup {x : α | R a x} (fun b ↦ cwfHeight R b + 1) := by
+ have h : cwfHeight R a = Finset.univ.sup fun b : {y : α // R a y} ↦ cwfHeight R b + 1 :=
+ WellFounded.fix_eq _ _ a
+ suffices
+ Finset.univ.sup (fun b : {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_50283ea28128_5 | 0b8daa5d45acb73e | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Vorspiel/Rel/CWF.lean | CWF | 5 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 8 | 5 | [
{
"theorem_name": "cwfHeight_gt_of",
"depth": 1,
"n_commands": 0,
"n_lines": 3,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " fun h ↦ calc\n cwfHeight R a = Finset.sup {x : α | R a x} fun b ↦ cwfHeight R b + 1 := cwfHeight_eq a\n _ ≥ cwfHe... | [
{
"name": "cwfHeight_eq",
"text": "lemma cwfHeight_eq (a : α) :\n cwfHeight R a = Finset.sup {x : α | R a x} (fun b ↦ cwfHeight R b + 1) := by\n have h : cwfHeight R a = Finset.univ.sup fun b : {y : α // R a y} ↦ cwfHeight R b + 1 :=\n WellFounded.fix_eq _ _ a\n suffices\n Finset.univ.sup (fun b ... | [
{
"name": "cwfHeight_eq_of_lt_of_le",
"fan_in": 1,
"n_deps_direct": 2,
"n_deps_transitive": 3,
"n_lines": 11,
"n_chars": 449,
"n_subproofs": 2,
"n_tactics": 8,
"cyclomatic": 3,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 5,
"automation_only": false,
"m... | 3 | module
public import Mathlib.Data.Fintype.Card
public import Mathlib.Data.Finset.Lattice.Fold
public import Foundation.Vorspiel.Rel.Connected
@[expose]
public section
section
abbrev ConverseWellFounded {α} (rel : Rel α α) := WellFounded $ flip rel
class IsConverseWellFounded (α) (rel : Rel α α) : Prop where cwf : ... | module
public import Mathlib.Data.Fintype.Card
public import Mathlib.Data.Finset.Lattice.Fold
public import Foundation.Vorspiel.Rel.Connected
@[expose]
public section
section
abbrev ConverseWellFounded {α} (rel : Rel α α) := WellFounded $ flip rel
class IsConverseWellFounded (α) (rel : Rel α α) : Prop where cwf : ... | @@ -46,11 +46,35 @@
open Classical
+lemma cwfHeight_eq (a : α) :
+ cwfHeight R a = Finset.sup {x : α | R a x} (fun b ↦ cwfHeight R b + 1) := by
+ have h : cwfHeight R a = Finset.univ.sup fun b : {y : α // R a y} ↦ cwfHeight R b + 1 :=
+ WellFounded.fix_eq _ _ a
+ suffices
+ Finset.univ.sup (fun b : {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_50283ea28128_6 | c6323390a92a9a37 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Vorspiel/Rel/CWF.lean | CWF | 6 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 8 | 5 | [
{
"theorem_name": "cwfHeight_gt_of",
"depth": 1,
"n_commands": 0,
"n_lines": 3,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " fun h ↦ calc\n cwfHeight R a = Finset.sup {x : α | R a x} fun b ↦ cwfHeight R b + 1 := cwfHeight_eq a\n _ ≥ cwfHe... | [
{
"name": "cwfHeight_eq",
"text": "lemma cwfHeight_eq (a : α) :\n cwfHeight R a = Finset.sup {x : α | R a x} (fun b ↦ cwfHeight R b + 1) := by\n have h : cwfHeight R a = Finset.univ.sup fun b : {y : α // R a y} ↦ cwfHeight R b + 1 :=\n WellFounded.fix_eq _ _ a\n suffices\n Finset.univ.sup (fun b ... | [
{
"name": "cwfHeight_eq_succ",
"fan_in": 1,
"n_deps_direct": 2,
"n_deps_transitive": 3,
"n_lines": 10,
"n_chars": 461,
"n_subproofs": 3,
"n_tactics": 7,
"cyclomatic": 1,
"n_automation": 4,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
"max_nest... | 3 | module
public import Mathlib.Data.Fintype.Card
public import Mathlib.Data.Finset.Lattice.Fold
public import Foundation.Vorspiel.Rel.Connected
@[expose]
public section
section
abbrev ConverseWellFounded {α} (rel : Rel α α) := WellFounded $ flip rel
class IsConverseWellFounded (α) (rel : Rel α α) : Prop where cwf : ... | module
public import Mathlib.Data.Fintype.Card
public import Mathlib.Data.Finset.Lattice.Fold
public import Foundation.Vorspiel.Rel.Connected
@[expose]
public section
section
abbrev ConverseWellFounded {α} (rel : Rel α α) := WellFounded $ flip rel
class IsConverseWellFounded (α) (rel : Rel α α) : Prop where cwf : ... | @@ -46,8 +46,26 @@
open Classical
+lemma cwfHeight_eq (a : α) :
+ cwfHeight R a = Finset.sup {x : α | R a x} (fun b ↦ cwfHeight R b + 1) := by
+ have h : cwfHeight R a = Finset.univ.sup fun b : {y : α // R a y} ↦ cwfHeight R b + 1 :=
+ WellFounded.fix_eq _ _ a
+ suffices
+ Finset.univ.sup (fun b : {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_50283ea28128_7 | d423fd47923b9df5 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Vorspiel/Rel/CWF.lean | CWF | 7 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 8 | 5 | [
{
"theorem_name": "cwfHeight_gt_of",
"depth": 1,
"n_commands": 0,
"n_lines": 3,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " fun h ↦ calc\n cwfHeight R a = Finset.sup {x : α | R a x} fun b ↦ cwfHeight R b + 1 := cwfHeight_eq a\n _ ≥ cwfHe... | [
{
"name": "cwfHeight_eq",
"text": "lemma cwfHeight_eq (a : α) :\n cwfHeight R a = Finset.sup {x : α | R a x} (fun b ↦ cwfHeight R b + 1) := by\n have h : cwfHeight R a = Finset.univ.sup fun b : {y : α // R a y} ↦ cwfHeight R b + 1 :=\n WellFounded.fix_eq _ _ a\n suffices\n Finset.univ.sup (fun b ... | [
{
"name": "cwfHeight_eq_succ_cwfHeight",
"fan_in": 0,
"n_deps_direct": 2,
"n_deps_transitive": 5,
"n_lines": 11,
"n_chars": 361,
"n_subproofs": 1,
"n_tactics": 8,
"cyclomatic": 2,
"n_automation": 2,
"n_rewrites": 0,
"n_structural": 5,
"automation_only": false,
... | 5 | module
public import Mathlib.Data.Fintype.Card
public import Mathlib.Data.Finset.Lattice.Fold
public import Foundation.Vorspiel.Rel.Connected
@[expose]
public section
section
abbrev ConverseWellFounded {α} (rel : Rel α α) := WellFounded $ flip rel
class IsConverseWellFounded (α) (rel : Rel α α) : Prop where cwf : ... | module
public import Mathlib.Data.Fintype.Card
public import Mathlib.Data.Finset.Lattice.Fold
public import Foundation.Vorspiel.Rel.Connected
@[expose]
public section
section
abbrev ConverseWellFounded {α} (rel : Rel α α) := WellFounded $ flip rel
class IsConverseWellFounded (α) (rel : Rel α α) : Prop where cwf : ... | @@ -46,14 +46,40 @@
open Classical
+lemma cwfHeight_eq (a : α) :
+ cwfHeight R a = Finset.sup {x : α | R a x} (fun b ↦ cwfHeight R b + 1) := by
+ have h : cwfHeight R a = Finset.univ.sup fun b : {y : α // R a y} ↦ cwfHeight R b + 1 :=
+ WellFounded.fix_eq _ _ a
+ suffices
+ Finset.univ.sup (fun b : {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_50283ea28128_8 | d48b73a0f215bd22 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Vorspiel/Rel/CWF.lean | CWF | 8 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 8 | 5 | [
{
"theorem_name": "cwfHeight_gt_of",
"depth": 1,
"n_commands": 0,
"n_lines": 3,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " fun h ↦ calc\n cwfHeight R a = Finset.sup {x : α | R a x} fun b ↦ cwfHeight R b + 1 := cwfHeight_eq a\n _ ≥ cwfHe... | [
{
"name": "cwfHeight_eq",
"text": "lemma cwfHeight_eq (a : α) :\n cwfHeight R a = Finset.sup {x : α | R a x} (fun b ↦ cwfHeight R b + 1) := by\n have h : cwfHeight R a = Finset.univ.sup fun b : {y : α // R a y} ↦ cwfHeight R b + 1 :=\n WellFounded.fix_eq _ _ a\n suffices\n Finset.univ.sup (fun b ... | [
{
"name": "cwfHeight_lt",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 4,
"n_lines": 19,
"n_chars": 796,
"n_subproofs": 3,
"n_tactics": 16,
"cyclomatic": 6,
"n_automation": 8,
"n_rewrites": 0,
"n_structural": 11,
"automation_only": false,
"max_nesting... | 4 | module
public import Mathlib.Data.Fintype.Card
public import Mathlib.Data.Finset.Lattice.Fold
public import Foundation.Vorspiel.Rel.Connected
@[expose]
public section
section
abbrev ConverseWellFounded {α} (rel : Rel α α) := WellFounded $ flip rel
class IsConverseWellFounded (α) (rel : Rel α α) : Prop where cwf : ... | module
public import Mathlib.Data.Fintype.Card
public import Mathlib.Data.Finset.Lattice.Fold
public import Foundation.Vorspiel.Rel.Connected
@[expose]
public section
section
abbrev ConverseWellFounded {α} (rel : Rel α α) := WellFounded $ flip rel
class IsConverseWellFounded (α) (rel : Rel α α) : Prop where cwf : ... | @@ -46,8 +46,26 @@
open Classical
+lemma cwfHeight_eq (a : α) :
+ cwfHeight R a = Finset.sup {x : α | R a x} (fun b ↦ cwfHeight R b + 1) := by
+ have h : cwfHeight R a = Finset.univ.sup fun b : {y : α // R a y} ↦ cwfHeight R b + 1 :=
+ WellFounded.fix_eq _ _ a
+ suffices
+ Finset.univ.sup (fun b : {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_50283ea28128_9 | e95aedfc6940952e | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Vorspiel/Rel/CWF.lean | CWF | 9 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 8 | 5 | [
{
"theorem_name": "cwfHeight_gt_of",
"depth": 1,
"n_commands": 0,
"n_lines": 3,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " fun h ↦ calc\n cwfHeight R a = Finset.sup {x : α | R a x} fun b ↦ cwfHeight R b + 1 := cwfHeight_eq a\n _ ≥ cwfHe... | [
{
"name": "cwfHeight_eq",
"text": "lemma cwfHeight_eq (a : α) :\n cwfHeight R a = Finset.sup {x : α | R a x} (fun b ↦ cwfHeight R b + 1) := by\n have h : cwfHeight R a = Finset.univ.sup fun b : {y : α // R a y} ↦ cwfHeight R b + 1 :=\n WellFounded.fix_eq _ _ a\n suffices\n Finset.univ.sup (fun b ... | [
{
"name": "cwfHeight_congr",
"fan_in": 0,
"n_deps_direct": 2,
"n_deps_transitive": 3,
"n_lines": 20,
"n_chars": 746,
"n_subproofs": 2,
"n_tactics": 14,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 2,
"n_structural": 10,
"automation_only": false,
"max_nest... | 3 | module
public import Mathlib.Data.Fintype.Card
public import Mathlib.Data.Finset.Lattice.Fold
public import Foundation.Vorspiel.Rel.Connected
@[expose]
public section
section
abbrev ConverseWellFounded {α} (rel : Rel α α) := WellFounded $ flip rel
class IsConverseWellFounded (α) (rel : Rel α α) : Prop where cwf : ... | module
public import Mathlib.Data.Fintype.Card
public import Mathlib.Data.Finset.Lattice.Fold
public import Foundation.Vorspiel.Rel.Connected
@[expose]
public section
section
abbrev ConverseWellFounded {α} (rel : Rel α α) := WellFounded $ flip rel
class IsConverseWellFounded (α) (rel : Rel α α) : Prop where cwf : ... | @@ -46,11 +46,33 @@
open Classical
+lemma cwfHeight_eq (a : α) :
+ cwfHeight R a = Finset.sup {x : α | R a x} (fun b ↦ cwfHeight R b + 1) := by
+ have h : cwfHeight R a = Finset.univ.sup fun b : {y : α // R a y} ↦ cwfHeight R b + 1 :=
+ WellFounded.fix_eq _ _ a
+ suffices
+ Finset.univ.sup (fun b : {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_980f7e998f41_0 | 7d123f7489da0522 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Vorspiel/List/Chain.lean | Chain | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 5 | 2 | [
{
"theorem_name": "finRange.le_isChain_map_strictMono_succ",
"depth": 1,
"n_commands": 0,
"n_lines": 3,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n apply List.isChain_map_of_isChain f f_mono;\n apply finRange.le_isChain_succ;",
"n_chars": 83,... | [
{
"name": "finRange.le_isChain_succ",
"text": "lemma finRange.le_isChain_succ : List.IsChain (· < ·) (List.finRange (n + 1)) := by\n rw [finRange_succ];\n induction n with\n | zero => simp [finRange]\n | succ n ih =>\n rw [List.finRange_succ, List.map]\n apply List.isChain_append_cons_cons (α := F... | [
{
"name": "finRange.le_isChain_map_strictMono_succ",
"fan_in": 2,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 6,
"n_chars": 246,
"n_subproofs": 0,
"n_tactics": 5,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 2,
"automation_only":... | 1 | module
public import Mathlib.Data.Fintype.List
public import Foundation.Vorspiel.Fin.Basic
@[expose]
public section
namespace List
variable {α}
variable {l : List α} {x y : α}
section
variable [DecidableEq α]
def finIdxOf (l : List α) (hx : x ∈ l) : Fin l.length := ⟨l.idxOf x, idxOf_lt_length_of_mem hx⟩
@[simp,... | module
public import Mathlib.Data.Fintype.List
public import Foundation.Vorspiel.Fin.Basic
@[expose]
public section
namespace List
variable {α}
variable {l : List α} {x y : α}
section
variable [DecidableEq α]
def finIdxOf (l : List α) (hx : x ∈ l) : Fin l.length := ⟨l.idxOf x, idxOf_lt_length_of_mem hx⟩
@[simp,... | @@ -59,8 +59,28 @@
| 0 => simp;
| n + 1 => apply range.lt_isChain_map_strictAnti_succ f f_anti;
+lemma finRange.le_isChain_succ : List.IsChain (· < ·) (List.finRange (n + 1)) := by
+ rw [finRange_succ];
+ induction n with
+ | zero => simp [finRange]
+ | succ n ih =>
+ rw [List.finRange_succ, List.map]
+ ... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_980f7e998f41_1 | ef76e7a8adf8b08e | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Vorspiel/List/Chain.lean | Chain | 1 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 5 | 1 | [
{
"theorem_name": "chains_finite",
"depth": 1,
"n_commands": 0,
"n_lines": 4,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n apply @Finite.of_injective { l : List α // l.IsChain R } { l : List α // l.Nodup } _ ?f;\n case f => intro ⟨l, hl⟩; refine ⟨... | [
{
"name": "nodup_of_trans_irreflex",
"text": "lemma nodup_of_trans_irreflex [Std.Irrefl R] (h_chain : l.IsChain R) : l.Nodup := by\n by_contra hC;\n replace ⟨d, hC⟩ := List.exists_duplicate_iff_not_nodup.mpr hC;\n have := List.duplicate_iff_sublist.mp hC;\n have := @List.IsChain.sublist α R [d, d] l ⟨by... | [
{
"name": "chains_finite",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 7,
"n_chars": 336,
"n_subproofs": 0,
"n_tactics": 8,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 3,
"automation_only": false,
"max_nesting":... | 1 | module
public import Mathlib.Data.Fintype.List
public import Foundation.Vorspiel.Fin.Basic
@[expose]
public section
namespace List
variable {α}
variable {l : List α} {x y : α}
section
variable [DecidableEq α]
def finIdxOf (l : List α) (hx : x ∈ l) : Fin l.length := ⟨l.idxOf x, idxOf_lt_length_of_mem hx⟩
@[simp,... | module
public import Mathlib.Data.Fintype.List
public import Foundation.Vorspiel.Fin.Basic
@[expose]
public section
namespace List
variable {α}
variable {l : List α} {x y : α}
section
variable [DecidableEq α]
def finIdxOf (l : List α) (hx : x ∈ l) : Fin l.length := ⟨l.idxOf x, idxOf_lt_length_of_mem hx⟩
@[simp,... | @@ -132,11 +132,22 @@
lemma connected_of_trans [DecidableEq α] (h : List.IsChain R l) (hx : x ∈ l) (hy : y ∈ l) (exy : x ≠ y) : R x y ∨ R y x := by
grind;
+lemma nodup_of_trans_irreflex [Std.Irrefl R] (h_chain : l.IsChain R) : l.Nodup := by
+ by_contra hC;
+ replace ⟨d, hC⟩ := List.exists_duplicate_iff_not_nodu... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_980f7e998f41_2 | 14a9c45c5800f7ae | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Vorspiel/List/Chain.lean | Chain | 2 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 5 | 1 | [
{
"theorem_name": "isChain_concat_of_not_nil",
"depth": 1,
"n_commands": 0,
"n_lines": 10,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n apply Iff.trans List.isChain_concat;\n suffices (∀ x ∈ l.getLast?, R x a) ↔ R (l.getLast hl) a by tauto;\n con... | [
{
"name": "isChain_concat",
"text": "lemma isChain_concat : List.IsChain R (l.concat a) ↔ List.IsChain R l ∧ ∀ x ∈ l.getLast?, R x a := by\n rw [List.concat_eq_append]\n constructor;\n . intro h;\n simpa using List.isChain_append.mp h;\n . rintro ⟨h₁, h₂⟩;\n apply List.isChain_append.mpr;\n re... | [
{
"name": "isChain_concat_of_not_nil",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 12,
"n_chars": 437,
"n_subproofs": 1,
"n_tactics": 19,
"cyclomatic": 3,
"n_automation": 2,
"n_rewrites": 0,
"n_structural": 6,
"automation_only": false,
... | 1 | module
public import Mathlib.Data.Fintype.List
public import Foundation.Vorspiel.Fin.Basic
@[expose]
public section
namespace List
variable {α}
variable {l : List α} {x y : α}
section
variable [DecidableEq α]
def finIdxOf (l : List α) (hx : x ∈ l) : Fin l.length := ⟨l.idxOf x, idxOf_lt_length_of_mem hx⟩
@[simp,... | module
public import Mathlib.Data.Fintype.List
public import Foundation.Vorspiel.Fin.Basic
@[expose]
public section
namespace List
variable {α}
variable {l : List α} {x y : α}
section
variable [DecidableEq α]
def finIdxOf (l : List α) (hx : x ∈ l) : Fin l.length := ⟨l.idxOf x, idxOf_lt_length_of_mem hx⟩
@[simp,... | @@ -136,8 +136,30 @@
instance finiteNodupList [DecidableEq α] [Finite α] : Finite { l : List α // l.Nodup } := @fintypeNodupList α (Fintype.ofFinite α) |>.finite
-lemma isChain_concat_of_not_nil (hl : l ≠ []) : List.IsChain R (l.concat a) ↔ List.IsChain R l ∧ R (l.getLast hl) a := sorry
+lemma isChain_concat : Li... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_980f7e998f41_3 | 1953d926f4855f8b | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Vorspiel/List/Chain.lean | Chain | 3 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 5 | 1 | [
{
"theorem_name": "rel_head_of_isChain_preorder",
"depth": 1,
"n_commands": 0,
"n_lines": 6,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n intro x hx;\n by_cases e : x = l.head lh;\n . subst e;\n apply refl;\n . apply rel_head_of_isChain_trans... | [
{
"name": "rel_head_of_isChain_trans",
"text": "lemma rel_head_of_isChain_trans [IsTrans _ R] (h : List.IsChain R l) (lh : l ≠ []) : ∀ x ∈ l, x ≠ l.head lh → R (l.head lh) x := by\n intro x hx₁ hx₂;\n rw [List.head_eq_getElem_zero, ←getElem_finIdxOf hx₁];\n apply List.IsChain.of_lt h;\n apply lt_of_le_o... | [
{
"name": "rel_head_of_isChain_preorder",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 3,
"n_lines": 8,
"n_chars": 259,
"n_subproofs": 0,
"n_tactics": 12,
"cyclomatic": 2,
"n_automation": 1,
"n_rewrites": 1,
"n_structural": 3,
"automation_only": false,
... | 1 | module
public import Mathlib.Data.Fintype.List
public import Foundation.Vorspiel.Fin.Basic
@[expose]
public section
namespace List
variable {α}
variable {l : List α} {x y : α}
section
variable [DecidableEq α]
def finIdxOf (l : List α) (hx : x ∈ l) : Fin l.length := ⟨l.idxOf x, idxOf_lt_length_of_mem hx⟩
@[simp,... | module
public import Mathlib.Data.Fintype.List
public import Foundation.Vorspiel.Fin.Basic
@[expose]
public section
namespace List
variable {α}
variable {l : List α} {x y : α}
section
variable [DecidableEq α]
def finIdxOf (l : List α) (hx : x ∈ l) : Fin l.length := ⟨l.idxOf x, idxOf_lt_length_of_mem hx⟩
@[simp,... | @@ -140,12 +140,20 @@
variable [DecidableEq α] {R : α → α → Prop}
+lemma rel_head_of_isChain_trans [IsTrans _ R] (h : List.IsChain R l) (lh : l ≠ []) : ∀ x ∈ l, x ≠ l.head lh → R (l.head lh) x := by
+ intro x hx₁ hx₂;
+ rw [List.head_eq_getElem_zero, ←getElem_finIdxOf hx₁];
+ apply List.IsChain.of_lt h;
+ appl... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_980f7e998f41_4 | d3a104ef71ccbe23 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Vorspiel/List/Chain.lean | Chain | 4 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 5 | 1 | [
{
"theorem_name": "rel_getLast_of_isChain_preorder",
"depth": 1,
"n_commands": 0,
"n_lines": 6,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n intro x hx;\n by_cases e : x = l.getLast lh;\n . subst e;\n apply refl;\n . apply rel_getLast_of_isCh... | [
{
"name": "rel_getLast_of_isChain_trans",
"text": "lemma rel_getLast_of_isChain_trans [IsTrans _ R] (h : List.IsChain R l) (lh : l ≠ []) : ∀ x ∈ l, x ≠ l.getLast lh → R x (l.getLast lh) := by\n intro x hx₁ hx₂;\n rw [←getElem_finIdxOf hx₁, List.getLast_eq_getElem];\n apply List.IsChain.of_lt h;\n apply ... | [
{
"name": "rel_getLast_of_isChain_preorder",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 3,
"n_lines": 8,
"n_chars": 271,
"n_subproofs": 0,
"n_tactics": 12,
"cyclomatic": 2,
"n_automation": 1,
"n_rewrites": 1,
"n_structural": 3,
"automation_only": false,... | 1 | module
public import Mathlib.Data.Fintype.List
public import Foundation.Vorspiel.Fin.Basic
@[expose]
public section
namespace List
variable {α}
variable {l : List α} {x y : α}
section
variable [DecidableEq α]
def finIdxOf (l : List α) (hx : x ∈ l) : Fin l.length := ⟨l.idxOf x, idxOf_lt_length_of_mem hx⟩
@[simp,... | module
public import Mathlib.Data.Fintype.List
public import Foundation.Vorspiel.Fin.Basic
@[expose]
public section
namespace List
variable {α}
variable {l : List α} {x y : α}
section
variable [DecidableEq α]
def finIdxOf (l : List α) (hx : x ∈ l) : Fin l.length := ⟨l.idxOf x, idxOf_lt_length_of_mem hx⟩
@[simp,... | @@ -140,12 +140,20 @@
variable [DecidableEq α] {R : α → α → Prop}
+lemma rel_getLast_of_isChain_trans [IsTrans _ R] (h : List.IsChain R l) (lh : l ≠ []) : ∀ x ∈ l, x ≠ l.getLast lh → R x (l.getLast lh) := by
+ intro x hx₁ hx₂;
+ rw [←getElem_finIdxOf hx₁, List.getLast_eq_getElem];
+ apply List.IsChain.of_lt 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_a12adc433eb6_2 | 903ea0f208b13d67 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Bootstrapping/Syntax/Proof/Coding.lean | Coding | 2 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 5 | 2 | [
{
"theorem_name": "setShift_typed_quote",
"depth": 1,
"n_commands": 0,
"n_lines": 3,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n apply Sequent.ext'\n simp [←setShift_quote]; rfl",
"n_chars": 56,
"n_subproofs": 0,
"n_tactics": 4,
"... | [
{
"name": "setShift_quote",
"text": "lemma setShift_quote (Γ : Finset (Proposition L)) :\n setShift L (⌜Γ⌝ : V) = ⌜Finset.image Rewriting.shift Γ⌝ := by\n apply mem_ext\n intro x; simp only [mem_setShift_iff]\n constructor\n · rintro ⟨x, hx, rfl⟩\n rcases Sequent.mem_quote hx with ⟨p, _, rfl⟩\n ... | [
{
"name": "setShift_typed_quote",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 5,
"n_lines": 6,
"n_chars": 246,
"n_subproofs": 0,
"n_tactics": 4,
"cyclomatic": 1,
"n_automation": 2,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": false,
"max_ne... | 2 | module
public import Foundation.FirstOrder.Bootstrapping.Syntax.Proof.Typed
@[expose] public section
namespace LO.FirstOrder
open Arithmetic Bootstrapping
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
variable {L : Language} [L.DecidableEq] [L.Encodable] [L.LORDefinable]
variable {T : Theory L} [T.Δ... | module
public import Foundation.FirstOrder.Bootstrapping.Syntax.Proof.Typed
@[expose] public section
namespace LO.FirstOrder
open Arithmetic Bootstrapping
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
variable {L : Language} [L.DecidableEq] [L.Encodable] [L.LORDefinable]
variable {T : Theory L} [T.Δ... | @@ -74,6 +74,20 @@
Sequent.mem_quote_iff.mp <| h <| Sequent.mem_quote_iff.mpr hp,
fun h x hx ↦ by rcases Sequent.mem_quote hx with ⟨φ, hφ, rfl⟩; simp [h hφ]⟩
+lemma setShift_quote (Γ : Finset (Proposition L)) :
+ setShift L (⌜Γ⌝ : V) = ⌜Finset.image Rewriting.shift Γ⌝ := by
+ apply mem_ext
+ intro x; s... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_a12adc433eb6_3 | 69ae851015db7424 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Bootstrapping/Syntax/Proof/Coding.lean | Coding | 3 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 5 | 1 | [
{
"theorem_name": "coe_quote_proof_eq",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n simp [quote_proof_def, Derivation2.coe_quote_eq]",
"n_chars": 54,
"n_subproofs": 0,
"n_tactics": 2,
"cyc... | [
{
"name": "quote_proof_def",
"text": "lemma quote_proof_def {φ : Sentence L} (b : T ⊢! φ) : (⌜b⌝ : V) = ⌜b.toProof2⌝ := rfl\n\n",
"fan_in": 1,
"n_lines": 3,
"n_chars": 99,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural... | [
{
"name": "coe_quote_proof_eq",
"fan_in": 0,
"n_deps_direct": 2,
"n_deps_transitive": 2,
"n_lines": 4,
"n_chars": 136,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"max_nesti... | 2 | module
public import Foundation.FirstOrder.Bootstrapping.Syntax.Proof.Typed
@[expose] public section
namespace LO.FirstOrder
open Arithmetic Bootstrapping
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
variable {L : Language} [L.DecidableEq] [L.Encodable] [L.LORDefinable]
variable {T : Theory L} [T.Δ... | module
public import Foundation.FirstOrder.Bootstrapping.Syntax.Proof.Typed
@[expose] public section
namespace LO.FirstOrder
open Arithmetic Bootstrapping
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
variable {L : Language} [L.DecidableEq] [L.Encodable] [L.LORDefinable]
variable {T : Theory L} [T.Δ... | @@ -214,6 +214,8 @@
noncomputable instance (φ : Sentence L) : GödelQuote (T ⊢! φ) V := ⟨fun b ↦
⌜b.toProof2⌝⟩
+lemma quote_proof_def {φ : Sentence L} (b : T ⊢! φ) : (⌜b⌝ : V) = ⌜b.toProof2⌝ := rfl
+
@[simp] lemma derivation_of_quote_derivation {Γ : Sequent L} (b : ⊢ᴸᴷ¹ Γ) : DerivationOf T (⌜b⌝ : V) ⌜Γ.toFinset⌝ ... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_692985ff5a4d_0 | 2ad0c34c8b25ae05 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Bootstrapping/Syntax/Term/Basic.lean | Basic | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 15 | 3 | [
{
"theorem_name": "graph_exists",
"depth": 1,
"n_commands": 0,
"n_lines": 11,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n apply IsUTerm.induction 𝚺 (P := fun t ↦ ∃ y, c.Graph L param t y) (by definability)\n case hbvar =>\n intro z; exact ⟨c.... | [
{
"name": "graph_fvar_iff",
"text": "lemma graph_fvar_iff (x) :\n c.Graph L param ^&x y ↔ y = c.fvar param x := by\n constructor\n · intro H\n rcases Graph.case_iff.mp H with ⟨_, (⟨_, h, _⟩ | ⟨_, h, rfl⟩ | ⟨_, _, _, _, _, h, _⟩)⟩\n · simp [qqFvar, qqBvar] at h\n · rcases (by simpa using h); rf... | [
{
"name": "graph_exists",
"fan_in": 1,
"n_deps_direct": 7,
"n_deps_transitive": 10,
"n_lines": 13,
"n_chars": 612,
"n_subproofs": 1,
"n_tactics": 13,
"cyclomatic": 2,
"n_automation": 2,
"n_rewrites": 0,
"n_structural": 8,
"automation_only": false,
"max_nesting... | 3 | module
public import Foundation.FirstOrder.Bootstrapping.Syntax.Language
@[expose] public section
namespace LO.FirstOrder.Arithmetic.Bootstrapping
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
variable {L : Language} [L.Encodable] [L.LORDefinable]
section term
noncomputable def qqBvar (z : V) : V := ... | module
public import Foundation.FirstOrder.Bootstrapping.Syntax.Language
@[expose] public section
namespace LO.FirstOrder.Arithmetic.Bootstrapping
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
variable {L : Language} [L.Encodable] [L.LORDefinable]
section term
noncomputable def qqBvar (z : V) : V := ... | @@ -411,6 +411,16 @@
· simp [qqBvar, qqFunc] at h
· rintro rfl; exact Graph.case_iff.mpr ⟨by simp, Or.inl ⟨z, by simp⟩⟩
+lemma graph_fvar_iff (x) :
+ c.Graph L param ^&x y ↔ y = c.fvar param x := by
+ constructor
+ · intro H
+ rcases Graph.case_iff.mp H with ⟨_, (⟨_, h, _⟩ | ⟨_, h, rfl⟩ | ⟨_, _, _, _,... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_692985ff5a4d_1 | 78b7e9c2db0896fb | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Bootstrapping/Syntax/Term/Basic.lean | Basic | 1 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 15 | 1 | [
{
"theorem_name": "graph_unique",
"depth": 1,
"n_commands": 0,
"n_lines": 17,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n revert y₁ y₂\n suffices IsUTerm L t → ∀ y₁ y₂, c.Graph L param t y₁ → c.Graph L param t y₂ → y₁ = y₂\n by intro u₁ u₂ h₁ h₂... | [
{
"name": "graph_func_inv",
"text": "lemma graph_func_inv {k f v y} :\n c.Graph L param (^func k f v) y → ∃ w,\n (k = len w ∧ ∀ i < k, c.Graph L param v.[i] w.[i]) ∧\n y = c.func param k f v w := by\n intro H\n rcases Graph.case_iff.mp H with ⟨_, (⟨_, h, _⟩ | ⟨_, h, rfl⟩ | ⟨k', f', v', w, hw,... | [
{
"name": "graph_existsUnique",
"fan_in": 1,
"n_deps_direct": 2,
"n_deps_transitive": 14,
"n_lines": 5,
"n_chars": 210,
"n_subproofs": 0,
"n_tactics": 3,
"cyclomatic": 2,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 2,
"automation_only": false,
"max_nes... | 7 | module
public import Foundation.FirstOrder.Bootstrapping.Syntax.Language
@[expose] public section
namespace LO.FirstOrder.Arithmetic.Bootstrapping
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
variable {L : Language} [L.Encodable] [L.LORDefinable]
section term
noncomputable def qqBvar (z : V) : V := ... | module
public import Foundation.FirstOrder.Bootstrapping.Syntax.Language
@[expose] public section
namespace LO.FirstOrder.Arithmetic.Bootstrapping
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
variable {L : Language} [L.Encodable] [L.LORDefinable]
section term
noncomputable def qqBvar (z : V) : V := ... | @@ -426,6 +426,17 @@
c.Graph L param (^func k f v) (c.func param k f v w) := by
exact Graph.case_iff.mpr ⟨by simp [hkr, hv], Or.inr <| Or.inr ⟨k, f, v, w, ⟨hkw, hw⟩, by simp⟩⟩
+lemma graph_func_inv {k f v y} :
+ c.Graph L param (^func k f v) y → ∃ w,
+ (k = len w ∧ ∀ i < k, c.Graph L param v.[i] w.[i]... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_692985ff5a4d_2 | 7d75451932c576dc | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Bootstrapping/Syntax/Term/Basic.lean | Basic | 2 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 15 | 1 | [
{
"theorem_name": "graph_existsUnique",
"depth": 1,
"n_commands": 0,
"n_lines": 3,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n rcases graph_exists param ht with ⟨y, hy⟩\n exact ExistsUnique.intro y hy (fun y' h' ↦ graph_unique param h' hy)",
... | [
{
"name": "graph_exists",
"text": "lemma graph_exists {t : V} : IsUTerm L t → ∃ y, c.Graph L param t y := by\n apply IsUTerm.induction 𝚺 (P := fun t ↦ ∃ y, c.Graph L param t y) (by definability)\n case hbvar =>\n intro z; exact ⟨c.bvar param z, by simp [c.graph_bvar_iff]⟩\n case hfvar =>\n intro x... | [
{
"name": "graph_existsUnique_total",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 15,
"n_lines": 5,
"n_chars": 200,
"n_subproofs": 0,
"n_tactics": 3,
"cyclomatic": 2,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
"m... | 8 | module
public import Foundation.FirstOrder.Bootstrapping.Syntax.Language
@[expose] public section
namespace LO.FirstOrder.Arithmetic.Bootstrapping
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
variable {L : Language} [L.Encodable] [L.LORDefinable]
section term
noncomputable def qqBvar (z : V) : V := ... | module
public import Foundation.FirstOrder.Bootstrapping.Syntax.Language
@[expose] public section
namespace LO.FirstOrder.Arithmetic.Bootstrapping
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
variable {L : Language} [L.Encodable] [L.LORDefinable]
section term
noncomputable def qqBvar (z : V) : V := ... | @@ -439,6 +439,18 @@
variable {c} (param n)
+lemma graph_exists {t : V} : IsUTerm L t → ∃ y, c.Graph L param t y := by
+ apply IsUTerm.induction 𝚺 (P := fun t ↦ ∃ y, c.Graph L param t y) (by definability)
+ case hbvar =>
+ intro z; exact ⟨c.bvar param z, by simp [c.graph_bvar_iff]⟩
+ case hfvar =>
+ intr... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_692985ff5a4d_3 | 8aeefe640846048c | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Bootstrapping/Syntax/Term/Basic.lean | Basic | 3 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 15 | 1 | [
{
"theorem_name": "graph_existsUnique_total",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n by_cases ht : IsUTerm L t <;> simp [ht, c.graph_existsUnique L param]",
"n_chars": 75,
"n_subproofs": 0,
... | [
{
"name": "graph_existsUnique",
"text": "lemma graph_existsUnique {t : V} (ht : IsUTerm L t) : ∃! y, c.Graph L param t y := by\n rcases graph_exists param ht with ⟨y, hy⟩\n exact ExistsUnique.intro y hy (fun y' h' ↦ graph_unique param h' hy)\n\n",
"fan_in": 1,
"n_lines": 5,
"n_chars": 210,
... | [
{
"name": "result_bvar",
"fan_in": 0,
"n_deps_direct": 3,
"n_deps_transitive": 17,
"n_lines": 4,
"n_chars": 138,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 2,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"max_nesting": 2... | 9 | module
public import Foundation.FirstOrder.Bootstrapping.Syntax.Language
@[expose] public section
namespace LO.FirstOrder.Arithmetic.Bootstrapping
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
variable {L : Language} [L.Encodable] [L.LORDefinable]
section term
noncomputable def qqBvar (z : V) : V := ... | module
public import Foundation.FirstOrder.Bootstrapping.Syntax.Language
@[expose] public section
namespace LO.FirstOrder.Arithmetic.Bootstrapping
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
variable {L : Language} [L.Encodable] [L.LORDefinable]
section term
noncomputable def qqBvar (z : V) : V := ... | @@ -471,8 +471,13 @@
variable (L c)
+lemma graph_existsUnique {t : V} (ht : IsUTerm L t) : ∃! y, c.Graph L param t y := by
+ rcases graph_exists param ht with ⟨y, hy⟩
+ exact ExistsUnique.intro y hy (fun y' h' ↦ graph_unique param h' hy)
+
lemma graph_existsUnique_total (t : V) : ∃! y,
- (IsUTerm L t → c.Gra... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_692985ff5a4d_4 | d26e4b48e2ff93ed | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Bootstrapping/Syntax/Term/Basic.lean | Basic | 4 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 15 | 2 | [
{
"theorem_name": "graph_existsUnique",
"depth": 1,
"n_commands": 0,
"n_lines": 3,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n rcases graph_exists param ht with ⟨y, hy⟩\n exact ExistsUnique.intro y hy (fun y' h' ↦ graph_unique param h' hy)",
... | [
{
"name": "graph_unique",
"text": "lemma graph_unique {t y₁ y₂ : V} : c.Graph L param t y₁ → c.Graph L param t y₂ → y₁ = y₂ := by\n revert y₁ y₂\n suffices IsUTerm L t → ∀ y₁ y₂, c.Graph L param t y₁ → c.Graph L param t y₂ → y₁ = y₂\n by intro u₁ u₂ h₁ h₂; exact this (c.graph_dom_isUTerm h₁) u₁ u₂ h₁ h₂\... | [
{
"name": "graph_existsUnique_vec_total",
"fan_in": 3,
"n_deps_direct": 1,
"n_deps_transitive": 14,
"n_lines": 6,
"n_chars": 256,
"n_subproofs": 0,
"n_tactics": 3,
"cyclomatic": 2,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
... | 6 | module
public import Foundation.FirstOrder.Bootstrapping.Syntax.Language
@[expose] public section
namespace LO.FirstOrder.Arithmetic.Bootstrapping
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
variable {L : Language} [L.Encodable] [L.LORDefinable]
section term
noncomputable def qqBvar (z : V) : V := ... | module
public import Foundation.FirstOrder.Bootstrapping.Syntax.Language
@[expose] public section
namespace LO.FirstOrder.Arithmetic.Bootstrapping
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
variable {L : Language} [L.Encodable] [L.LORDefinable]
section term
noncomputable def qqBvar (z : V) : V := ... | @@ -451,9 +451,29 @@
rcases this with ⟨w, hwk, hvw⟩
exact ⟨c.func param k f v w, c.graph_func hkf hv (Eq.symm hwk) hvw⟩
+lemma graph_unique {t y₁ y₂ : V} : c.Graph L param t y₁ → c.Graph L param t y₂ → y₁ = y₂ := by
+ revert y₁ y₂
+ suffices IsUTerm L t → ∀ y₁ y₂, c.Graph L param t y₁ → c.Graph L param t ... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_692985ff5a4d_5 | a14a2aa0a480c098 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Bootstrapping/Syntax/Term/Basic.lean | Basic | 5 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 15 | 4 | [
{
"theorem_name": "result_bvar",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": "\n c.result_eq_of_graph (by simp) (by simp [c.graph_bvar_iff])",
"n_chars": 62,
"n_subproofs": 0,
"n_tactics": 1,
"cy... | [
{
"name": "result_eq_of_graph",
"text": "lemma result_eq_of_graph {t y} (ht : IsUTerm L t) (h : c.Graph L param t y) :\n c.result L param t = y := Eq.symm <| Classical.choose_uniq (c.graph_existsUnique_total L param t) (by simp [h, ht])\n\n",
"fan_in": 4,
"n_lines": 4,
"n_chars": 199,
"n_... | [
{
"name": "nth_resultVec",
"fan_in": 3,
"n_deps_direct": 3,
"n_deps_transitive": 21,
"n_lines": 5,
"n_chars": 218,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
"max_nesting"... | 11 | module
public import Foundation.FirstOrder.Bootstrapping.Syntax.Language
@[expose] public section
namespace LO.FirstOrder.Arithmetic.Bootstrapping
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
variable {L : Language} [L.Encodable] [L.LORDefinable]
section term
noncomputable def qqBvar (z : V) : V := ... | module
public import Foundation.FirstOrder.Bootstrapping.Syntax.Language
@[expose] public section
namespace LO.FirstOrder.Arithmetic.Bootstrapping
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
variable {L : Language} [L.Encodable] [L.LORDefinable]
section term
noncomputable def qqBvar (z : V) : V := ... | @@ -489,13 +489,20 @@
variable {L c param}
-@[simp] lemma result_bvar (z) : c.result L param (^#z) = c.bvar param z := sorry
+lemma result_eq_of_graph {t y} (ht : IsUTerm L t) (h : c.Graph L param t y) :
+ c.result L param t = y := Eq.symm <| Classical.choose_uniq (c.graph_existsUnique_total L param t) (by simp... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_692985ff5a4d_6 | 1ea6369416dd279f | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Bootstrapping/Syntax/Term/Basic.lean | Basic | 6 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 15 | 4 | [
{
"theorem_name": "result_bvar",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": "\n c.result_eq_of_graph (by simp) (by simp [c.graph_bvar_iff])",
"n_chars": 62,
"n_subproofs": 0,
"n_tactics": 1,
"cy... | [
{
"name": "result_eq_of_graph",
"text": "lemma result_eq_of_graph {t y} (ht : IsUTerm L t) (h : c.Graph L param t y) :\n c.result L param t = y := Eq.symm <| Classical.choose_uniq (c.graph_existsUnique_total L param t) (by simp [h, ht])\n\n",
"fan_in": 4,
"n_lines": 4,
"n_chars": 199,
"n_... | [
{
"name": "resultVec_cons",
"fan_in": 1,
"n_deps_direct": 3,
"n_deps_transitive": 25,
"n_lines": 11,
"n_chars": 482,
"n_subproofs": 1,
"n_tactics": 7,
"cyclomatic": 2,
"n_automation": 7,
"n_rewrites": 1,
"n_structural": 2,
"automation_only": false,
"max_nestin... | 12 | module
public import Foundation.FirstOrder.Bootstrapping.Syntax.Language
@[expose] public section
namespace LO.FirstOrder.Arithmetic.Bootstrapping
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
variable {L : Language} [L.Encodable] [L.LORDefinable]
section term
noncomputable def qqBvar (z : V) : V := ... | module
public import Foundation.FirstOrder.Bootstrapping.Syntax.Language
@[expose] public section
namespace LO.FirstOrder.Arithmetic.Bootstrapping
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
variable {L : Language} [L.Encodable] [L.LORDefinable]
section term
noncomputable def qqBvar (z : V) : V := ... | @@ -489,13 +489,20 @@
variable {L c param}
-@[simp] lemma result_bvar (z) : c.result L param (^#z) = c.bvar param z := sorry
+lemma result_eq_of_graph {t y} (ht : IsUTerm L t) (h : c.Graph L param t y) :
+ c.result L param t = y := Eq.symm <| Classical.choose_uniq (c.graph_existsUnique_total L param t) (by simp... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_692985ff5a4d_7 | aa5035975ba8a3af | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Bootstrapping/Syntax/Term/Basic.lean | Basic | 7 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 15 | 2 | [
{
"theorem_name": "graph_existsUnique",
"depth": 1,
"n_commands": 0,
"n_lines": 3,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n rcases graph_exists param ht with ⟨y, hy⟩\n exact ExistsUnique.intro y hy (fun y' h' ↦ graph_unique param h' hy)",
... | [
{
"name": "graph_unique",
"text": "lemma graph_unique {t y₁ y₂ : V} : c.Graph L param t y₁ → c.Graph L param t y₂ → y₁ = y₂ := by\n revert y₁ y₂\n suffices IsUTerm L t → ∀ y₁ y₂, c.Graph L param t y₁ → c.Graph L param t y₂ → y₁ = y₂\n by intro u₁ u₂ h₁ h₂; exact this (c.graph_dom_isUTerm h₁) u₁ u₂ h₁ h₂\... | [
{
"name": "eval_resultVec",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 19,
"n_lines": 4,
"n_chars": 170,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
"max_nesting... | 7 | module
public import Foundation.FirstOrder.Bootstrapping.Syntax.Language
@[expose] public section
namespace LO.FirstOrder.Arithmetic.Bootstrapping
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
variable {L : Language} [L.Encodable] [L.LORDefinable]
section term
noncomputable def qqBvar (z : V) : V := ... | module
public import Foundation.FirstOrder.Bootstrapping.Syntax.Language
@[expose] public section
namespace LO.FirstOrder.Arithmetic.Bootstrapping
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
variable {L : Language} [L.Encodable] [L.LORDefinable]
section term
noncomputable def qqBvar (z : V) : V := ... | @@ -451,9 +451,29 @@
rcases this with ⟨w, hwk, hvw⟩
exact ⟨c.func param k f v w, c.graph_func hkf hv (Eq.symm hwk) hvw⟩
+lemma graph_unique {t y₁ y₂ : V} : c.Graph L param t y₁ → c.Graph L param t y₂ → y₁ = y₂ := by
+ revert y₁ y₂
+ suffices IsUTerm L t → ∀ y₁ y₂, c.Graph L param t y₁ → c.Graph L param t ... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_692985ff5a4d_8 | 24e237de55fee2bb | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Bootstrapping/Syntax/Term/Basic.lean | Basic | 8 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 15 | 4 | [
{
"theorem_name": "result_bvar",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": "\n c.result_eq_of_graph (by simp) (by simp [c.graph_bvar_iff])",
"n_chars": 62,
"n_subproofs": 0,
"n_tactics": 1,
"cy... | [
{
"name": "result_eq_of_graph",
"text": "lemma result_eq_of_graph {t y} (ht : IsUTerm L t) (h : c.Graph L param t y) :\n c.result L param t = y := Eq.symm <| Classical.choose_uniq (c.graph_existsUnique_total L param t) (by simp [h, ht])\n\n",
"fan_in": 4,
"n_lines": 4,
"n_chars": 199,
"n_... | [
{
"name": "termBVVec_cons",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 26,
"n_lines": 5,
"n_chars": 197,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
"max_nesting... | 13 | module
public import Foundation.FirstOrder.Bootstrapping.Syntax.Language
@[expose] public section
namespace LO.FirstOrder.Arithmetic.Bootstrapping
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
variable {L : Language} [L.Encodable] [L.LORDefinable]
section term
noncomputable def qqBvar (z : V) : V := ... | module
public import Foundation.FirstOrder.Bootstrapping.Syntax.Language
@[expose] public section
namespace LO.FirstOrder.Arithmetic.Bootstrapping
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
variable {L : Language} [L.Encodable] [L.LORDefinable]
section term
noncomputable def qqBvar (z : V) : V := ... | @@ -489,13 +489,20 @@
variable {L c param}
-@[simp] lemma result_bvar (z) : c.result L param (^#z) = c.bvar param z := sorry
+lemma result_eq_of_graph {t y} (ht : IsUTerm L t) (h : c.Graph L param t y) :
+ c.result L param t = y := Eq.symm <| Classical.choose_uniq (c.graph_existsUnique_total L param t) (by simp... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_e5cd8def0acd_0 | b2b10278a635e177 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Bootstrapping/FixedPoint.lean | FixedPoint | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 7 | 1 | [
{
"theorem_name": "substNumerals_app_quote_quote",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n simpa [Sentence.coe_quote_eq_quote] using substNumerals_app_quote (V := V) σ (fun i ↦ ⌜π i⌝)",
"n_chars":... | [
{
"name": "substNumerals_app_quote",
"text": "lemma substNumerals_app_quote (σ : ArithmeticSemisentence k) (v : Fin k → ℕ) :\n (substNumerals ⌜σ⌝ (v ·) : V) = ⌜((Rew.subst (fun i ↦ ↑(v i))) ▹ σ : ArithmeticSentence)⌝ := by\n simp [substNumerals, Sentence.quote_def, Semiformula.quote_def,\n Rewriting.... | [
{
"name": "substNumerals_app_quote_quote",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 5,
"n_chars": 351,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
... | 1 | module
public import Foundation.FirstOrder.Arithmetic.Basic.Model
public import Foundation.FirstOrder.Bootstrapping.Syntax
@[expose] public section
open Classical
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
namespace Bootstrapping.Arithmetic
noncomputable def subs... | module
public import Foundation.FirstOrder.Arithmetic.Basic.Model
public import Foundation.FirstOrder.Bootstrapping.Syntax
@[expose] public section
open Classical
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
namespace Bootstrapping.Arithmetic
noncomputable def subs... | @@ -16,8 +16,15 @@
noncomputable def substNumerals (φ : V) (v : Fin k → V) : V := subst ℒₒᵣ (matrixToVec (fun i ↦ numeral (v i))) φ
+lemma substNumerals_app_quote (σ : ArithmeticSemisentence k) (v : Fin k → ℕ) :
+ (substNumerals ⌜σ⌝ (v ·) : V) = ⌜((Rew.subst (fun i ↦ ↑(v i))) ▹ σ : ArithmeticSentence)⌝ := by
+ ... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_e5cd8def0acd_1 | 4477717e16de852d | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Bootstrapping/FixedPoint.lean | FixedPoint | 1 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 7 | 1 | [
{
"theorem_name": "diagonal",
"depth": 1,
"n_commands": 0,
"n_lines": 14,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": "\n haveI : 𝗘𝗤 _ ⪯ T := Entailment.WeakerThan.trans (𝓣 := 𝗜𝚺₁) inferInstance inferInstance\n complete.{0} T _ fun (V : Type) _ _ ↦... | [
{
"name": "substNumeral_app_quote",
"text": "lemma substNumeral_app_quote (σ π : ArithmeticSemisentence 1) :\n substNumeral ⌜σ⌝ (⌜π⌝ : V) = ⌜(σ/[⌜π⌝] : ArithmeticSentence)⌝ := by\n simp [substNumeral, Sentence.quote_def, Semiformula.quote_def,\n Rewriting.emb_subst_eq_subst_coe₁]\n\n",
"fan_in": ... | [
{
"name": "diagonal",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 17,
"n_chars": 873,
"n_subproofs": 2,
"n_tactics": 13,
"cyclomatic": 1,
"n_automation": 5,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": false,
"max_nesting": 6
... | 1 | module
public import Foundation.FirstOrder.Arithmetic.Basic.Model
public import Foundation.FirstOrder.Bootstrapping.Syntax
@[expose] public section
open Classical
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
namespace Bootstrapping.Arithmetic
noncomputable def subs... | module
public import Foundation.FirstOrder.Arithmetic.Basic.Model
public import Foundation.FirstOrder.Bootstrapping.Syntax
@[expose] public section
open Classical
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
namespace Bootstrapping.Arithmetic
noncomputable def subs... | @@ -14,6 +14,11 @@
noncomputable def substNumeral (φ x : V) : V := subst ℒₒᵣ ?[numeral x] φ
+lemma substNumeral_app_quote (σ π : ArithmeticSemisentence 1) :
+ substNumeral ⌜σ⌝ (⌜π⌝ : V) = ⌜(σ/[⌜π⌝] : ArithmeticSentence)⌝ := by
+ simp [substNumeral, Sentence.quote_def, Semiformula.quote_def,
+ Rewriting.emb_... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_e5cd8def0acd_2 | b05fc64503bcf254 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Bootstrapping/FixedPoint.lean | FixedPoint | 2 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 7 | 1 | [
{
"theorem_name": "substNumerals_app_quote_quote",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n simpa [Sentence.coe_quote_eq_quote] using substNumerals_app_quote (V := V) σ (fun i ↦ ⌜π i⌝)",
"n_chars":... | [
{
"name": "substNumerals_app_quote",
"text": "lemma substNumerals_app_quote (σ : ArithmeticSemisentence k) (v : Fin k → ℕ) :\n (substNumerals ⌜σ⌝ (v ·) : V) = ⌜((Rew.subst (fun i ↦ ↑(v i))) ▹ σ : ArithmeticSentence)⌝ := by\n simp [substNumerals, Sentence.quote_def, Semiformula.quote_def,\n Rewriting.... | [
{
"name": "multidiagonal",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 2,
"n_lines": 17,
"n_chars": 1161,
"n_subproofs": 2,
"n_tactics": 14,
"cyclomatic": 1,
"n_automation": 5,
"n_rewrites": 0,
"n_structural": 2,
"automation_only": false,
"max_nestin... | 2 | module
public import Foundation.FirstOrder.Arithmetic.Basic.Model
public import Foundation.FirstOrder.Bootstrapping.Syntax
@[expose] public section
open Classical
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
namespace Bootstrapping.Arithmetic
noncomputable def subs... | module
public import Foundation.FirstOrder.Arithmetic.Basic.Model
public import Foundation.FirstOrder.Bootstrapping.Syntax
@[expose] public section
open Classical
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
namespace Bootstrapping.Arithmetic
noncomputable def subs... | @@ -16,8 +16,15 @@
noncomputable def substNumerals (φ : V) (v : Fin k → V) : V := subst ℒₒᵣ (matrixToVec (fun i ↦ numeral (v i))) φ
+lemma substNumerals_app_quote (σ : ArithmeticSemisentence k) (v : Fin k → ℕ) :
+ (substNumerals ⌜σ⌝ (v ·) : V) = ⌜((Rew.subst (fun i ↦ ↑(v i))) ▹ σ : ArithmeticSentence)⌝ := by
+ ... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.