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_7fe832d7440d_4 | 4e81aae3f4ca858a | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Definability/Definable.lean | Definable | 4 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 15 | 1 | [
{
"theorem_name": "bexs'",
"depth": 1,
"n_commands": 0,
"n_lines": 3,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n apply (bexs h ‘!!t + 1’).of_iff\n intro v; simp [lt_succ_iff_le]",
"n_chars": 74,
"n_subproofs": 0,
"n_tactics": 4,
... | [
{
"name": "bexs",
"text": "lemma bexs {P : (Fin k → V) → V → Prop} (h : ℌ.Definable fun w ↦ P (w ·.succ) (w 0)) (t : ArithmeticSemiterm V k) :\n ℌ.Definable fun v ↦ ∃ x < t.val v id, P v x := by\n rcases h with ⟨φ, h⟩\n match ℌ with\n | 𝚺-[m] => exact ⟨HierarchySymbol.Semiformula.bexs t φ, by intro v... | [
{
"name": "bexs'",
"fan_in": 0,
"n_deps_direct": 2,
"n_deps_transitive": 5,
"n_lines": 6,
"n_chars": 297,
"n_subproofs": 0,
"n_tactics": 4,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 2,
"automation_only": false,
"max_nesting": 2
}
] | 1 | module
public import Foundation.FirstOrder.Arithmetic.Definability.Hierarchy
@[expose] public section
namespace LO.FirstOrder.Arithmetic.HierarchySymbol
variable (ξ : Type*) (n : ℕ)
open PeanoMinus
variable {V : Type*} [ORingStructure V]
abbrev IsDefinedBy (R : (Fin k → V) → Prop) : {ℌ : HierarchySymbol} → ℌ.Semi... | module
public import Foundation.FirstOrder.Arithmetic.Definability.Hierarchy
@[expose] public section
namespace LO.FirstOrder.Arithmetic.HierarchySymbol
variable (ξ : Type*) (n : ℕ)
open PeanoMinus
variable {V : Type*} [ORingStructure V]
abbrev IsDefinedBy (R : (Fin k → V) → Prop) : {ℌ : HierarchySymbol} → ℌ.Semi... | @@ -392,8 +392,18 @@
Γ-[m].Definable (fun v ↦ P v ↔ Q v) :=
.of_delta <| ((h₁.impDelta h₂).and (h₂.impDelta h₁)).of_iff <| by intro v; simp [iff_iff_implies_and_implies]
+lemma bexs {P : (Fin k → V) → V → Prop} (h : ℌ.Definable fun w ↦ P (w ·.succ) (w 0)) (t : ArithmeticSemiterm V k) :
+ ℌ.Definable fun v ... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_7fe832d7440d_5 | 48465028ab4efe54 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Definability/Definable.lean | Definable | 5 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 15 | 4 | [
{
"theorem_name": "of_sigma",
"depth": 1,
"n_commands": 0,
"n_lines": 9,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n cases' m with m\n · exact of_zero h\n apply of_sigma_of_pi\n · exact h\n · have : 𝚷-[m + 1].Definable fun v ↦ ∀ y, y = f (v ·... | [
{
"name": "of_sigma_of_pi",
"text": "lemma of_sigma_of_pi (hσ : 𝚺-[m].Definable P) (hπ : 𝚷-[m].Definable P) : Γ-[m].Definable P :=\n match Γ with\n | 𝚺 => hσ\n | 𝚷 => hπ\n | 𝚫 => by\n rcases hσ with ⟨φ, hp⟩; rcases hπ with ⟨ψ, hq⟩\n exact ⟨.mkDelta φ ψ, by intro v; simp [hp.df.iff, hq.df.iff]... | [
{
"name": "of_sigma",
"fan_in": 2,
"n_deps_direct": 6,
"n_deps_transitive": 10,
"n_lines": 11,
"n_chars": 420,
"n_subproofs": 1,
"n_tactics": 9,
"cyclomatic": 1,
"n_automation": 3,
"n_rewrites": 0,
"n_structural": 4,
"automation_only": false,
"max_nesting": 6
... | 2 | module
public import Foundation.FirstOrder.Arithmetic.Definability.Hierarchy
@[expose] public section
namespace LO.FirstOrder.Arithmetic.HierarchySymbol
variable (ξ : Type*) (n : ℕ)
open PeanoMinus
variable {V : Type*} [ORingStructure V]
abbrev IsDefinedBy (R : (Fin k → V) → Prop) : {ℌ : HierarchySymbol} → ℌ.Semi... | module
public import Foundation.FirstOrder.Arithmetic.Definability.Hierarchy
@[expose] public section
namespace LO.FirstOrder.Arithmetic.HierarchySymbol
variable (ξ : Type*) (n : ℕ)
open PeanoMinus
variable {V : Type*} [ORingStructure V]
abbrev IsDefinedBy (R : (Fin k → V) → Prop) : {ℌ : HierarchySymbol} → ℌ.Semi... | @@ -309,6 +309,14 @@
instance [𝚫-[m].Definable P] (Γ) : Γ-[m].Definable P := of_delta inferInstance
+lemma of_sigma_of_pi (hσ : 𝚺-[m].Definable P) (hπ : 𝚷-[m].Definable P) : Γ-[m].Definable P :=
+ match Γ with
+ | 𝚺 => hσ
+ | 𝚷 => hπ
+ | 𝚫 => by
+ rcases hσ with ⟨φ, hp⟩; rcases hπ with ⟨ψ, hq⟩
+ ex... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_7fe832d7440d_6 | c198063b3cff0147 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Definability/Definable.lean | Definable | 6 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 15 | 2 | [
{
"theorem_name": "substitution_sigma",
"depth": 1,
"n_commands": 0,
"n_lines": 14,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n have : 𝚺-[m+1].Definable fun z ↦ ∃ ys : Fin k → V, (∀ i, ys i = f i z) ∧ P ys := by\n apply exsVec; apply and\n ... | [
{
"name": "fintype_all",
"text": "lemma fintype_all [Fintype ι] {P : ι → (Fin k → V) → Prop}\n (h : ∀ i, ℌ.Definable fun w : Fin k → V ↦ P i w) :\n ℌ.Definable fun v : Fin k → V ↦ ∀ i, P i v := by\n simpa using fconj Finset.univ h\n\n",
"fan_in": 2,
"n_lines": 6,
"n_chars": 226,
"n_su... | [
{
"name": "substitution_sigma",
"fan_in": 1,
"n_deps_direct": 6,
"n_deps_transitive": 15,
"n_lines": 17,
"n_chars": 718,
"n_subproofs": 2,
"n_tactics": 17,
"cyclomatic": 4,
"n_automation": 2,
"n_rewrites": 0,
"n_structural": 14,
"automation_only": false,
"max_... | 5 | module
public import Foundation.FirstOrder.Arithmetic.Definability.Hierarchy
@[expose] public section
namespace LO.FirstOrder.Arithmetic.HierarchySymbol
variable (ξ : Type*) (n : ℕ)
open PeanoMinus
variable {V : Type*} [ORingStructure V]
abbrev IsDefinedBy (R : (Fin k → V) → Prop) : {ℌ : HierarchySymbol} → ℌ.Semi... | module
public import Foundation.FirstOrder.Arithmetic.Definability.Hierarchy
@[expose] public section
namespace LO.FirstOrder.Arithmetic.HierarchySymbol
variable (ξ : Type*) (n : ℕ)
open PeanoMinus
variable {V : Type*} [ORingStructure V]
abbrev IsDefinedBy (R : (Fin k → V) → Prop) : {ℌ : HierarchySymbol} → ℌ.Semi... | @@ -422,6 +422,11 @@
lemma fdisj (s : Finset ι) {R : ι → (Fin k → V) → Prop} (h : ∀ i, ℌ.Definable (R i)) :
ℌ.Definable fun x ↦ ∃ i ∈ s, R i x := by simpa using disj₂ s.toList h
+lemma fintype_all [Fintype ι] {P : ι → (Fin k → V) → Prop}
+ (h : ∀ i, ℌ.Definable fun w : Fin k → V ↦ P i w) :
+ ℌ.Definable f... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_7fe832d7440d_7 | 0990b8db88c87940 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Definability/Definable.lean | Definable | 7 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 15 | 1 | [
{
"theorem_name": "of_sigmaOne",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " Definable.of_deltaOne (graph_delta h)",
"n_chars": 38,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_auto... | [
{
"name": "of_deltaOne",
"text": "lemma of_deltaOne {Γ m} (h : 𝚫₁.Definable P) : Γ-[m+1].Definable P := by\n rcases h with ⟨φ, h⟩\n apply Definable.mk' (φ.ofDeltaOne Γ m)\n match Γ with\n | 𝚺 => intro _; simp [HierarchySymbol.Semiformula.ofDeltaOne, h.iff, HierarchySymbol.Semiformula.val_sigma]\n | �... | [
{
"name": "of_sigmaOne",
"fan_in": 0,
"n_deps_direct": 2,
"n_deps_transitive": 6,
"n_lines": 4,
"n_chars": 138,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
"max_nesting": 1... | 1 | module
public import Foundation.FirstOrder.Arithmetic.Definability.Hierarchy
@[expose] public section
namespace LO.FirstOrder.Arithmetic.HierarchySymbol
variable (ξ : Type*) (n : ℕ)
open PeanoMinus
variable {V : Type*} [ORingStructure V]
abbrev IsDefinedBy (R : (Fin k → V) → Prop) : {ℌ : HierarchySymbol} → ℌ.Semi... | module
public import Foundation.FirstOrder.Arithmetic.Definability.Hierarchy
@[expose] public section
namespace LO.FirstOrder.Arithmetic.HierarchySymbol
variable (ξ : Type*) (n : ℕ)
open PeanoMinus
variable {V : Type*} [ORingStructure V]
abbrev IsDefinedBy (R : (Fin k → V) → Prop) : {ℌ : HierarchySymbol} → ℌ.Semi... | @@ -300,6 +300,15 @@
instance [𝚺₀.Definable P] (ℌ : HierarchySymbol) : ℌ.Definable P := of_zero (Γ' := 𝚺) inferInstance
+lemma of_deltaOne {Γ m} (h : 𝚫₁.Definable P) : Γ-[m+1].Definable P := by
+ rcases h with ⟨φ, h⟩
+ apply Definable.mk' (φ.ofDeltaOne Γ m)
+ match Γ with
+ | 𝚺 => intro _; simp [HierarchyS... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_7fe832d7440d_8 | e85f7890bdd3575c | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Definability/Definable.lean | Definable | 8 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 15 | 4 | [
{
"theorem_name": "of_sigma",
"depth": 1,
"n_commands": 0,
"n_lines": 9,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n cases' m with m\n · exact of_zero h\n apply of_sigma_of_pi\n · exact h\n · have : 𝚷-[m + 1].Definable fun v ↦ ∀ y, y = f (v ·... | [
{
"name": "of_sigma_of_pi",
"text": "lemma of_sigma_of_pi (hσ : 𝚺-[m].Definable P) (hπ : 𝚷-[m].Definable P) : Γ-[m].Definable P :=\n match Γ with\n | 𝚺 => hσ\n | 𝚷 => hπ\n | 𝚫 => by\n rcases hσ with ⟨φ, hp⟩; rcases hπ with ⟨ψ, hq⟩\n exact ⟨.mkDelta φ ψ, by intro v; simp [hp.df.iff, hq.df.iff]... | [
{
"name": "ball_lt'",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 5,
"n_lines": 5,
"n_chars": 267,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
"max_nesting": 1
... | 1 | module
public import Foundation.FirstOrder.Arithmetic.Definability.Hierarchy
@[expose] public section
namespace LO.FirstOrder.Arithmetic.HierarchySymbol
variable (ξ : Type*) (n : ℕ)
open PeanoMinus
variable {V : Type*} [ORingStructure V]
abbrev IsDefinedBy (R : (Fin k → V) → Prop) : {ℌ : HierarchySymbol} → ℌ.Semi... | module
public import Foundation.FirstOrder.Arithmetic.Definability.Hierarchy
@[expose] public section
namespace LO.FirstOrder.Arithmetic.HierarchySymbol
variable (ξ : Type*) (n : ℕ)
open PeanoMinus
variable {V : Type*} [ORingStructure V]
abbrev IsDefinedBy (R : (Fin k → V) → Prop) : {ℌ : HierarchySymbol} → ℌ.Semi... | @@ -309,6 +309,14 @@
instance [𝚫-[m].Definable P] (Γ) : Γ-[m].Definable P := of_delta inferInstance
+lemma of_sigma_of_pi (hσ : 𝚺-[m].Definable P) (hπ : 𝚷-[m].Definable P) : Γ-[m].Definable P :=
+ match Γ with
+ | 𝚺 => hσ
+ | 𝚷 => hπ
+ | 𝚫 => by
+ rcases hσ with ⟨φ, hp⟩; rcases hπ with ⟨ψ, hq⟩
+ ex... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_7fe832d7440d_9 | b9890babb589283c | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Definability/Definable.lean | Definable | 9 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 15 | 2 | [
{
"theorem_name": "substitution_sigma",
"depth": 1,
"n_commands": 0,
"n_lines": 14,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n have : 𝚺-[m+1].Definable fun z ↦ ∃ ys : Fin k → V, (∀ i, ys i = f i z) ∧ P ys := by\n apply exsVec; apply and\n ... | [
{
"name": "fintype_all",
"text": "lemma fintype_all [Fintype ι] {P : ι → (Fin k → V) → Prop}\n (h : ∀ i, ℌ.Definable fun w : Fin k → V ↦ P i w) :\n ℌ.Definable fun v : Fin k → V ↦ ∀ i, P i v := by\n simpa using fconj Finset.univ h\n\n",
"fan_in": 2,
"n_lines": 6,
"n_chars": 226,
"n_su... | [
{
"name": "ball_le'",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 23,
"n_lines": 5,
"n_chars": 304,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
"max_nesting": 1
... | 8 | module
public import Foundation.FirstOrder.Arithmetic.Definability.Hierarchy
@[expose] public section
namespace LO.FirstOrder.Arithmetic.HierarchySymbol
variable (ξ : Type*) (n : ℕ)
open PeanoMinus
variable {V : Type*} [ORingStructure V]
abbrev IsDefinedBy (R : (Fin k → V) → Prop) : {ℌ : HierarchySymbol} → ℌ.Semi... | module
public import Foundation.FirstOrder.Arithmetic.Definability.Hierarchy
@[expose] public section
namespace LO.FirstOrder.Arithmetic.HierarchySymbol
variable (ξ : Type*) (n : ℕ)
open PeanoMinus
variable {V : Type*} [ORingStructure V]
abbrev IsDefinedBy (R : (Fin k → V) → Prop) : {ℌ : HierarchySymbol} → ℌ.Semi... | @@ -422,6 +422,11 @@
lemma fdisj (s : Finset ι) {R : ι → (Fin k → V) → Prop} (h : ∀ i, ℌ.Definable (R i)) :
ℌ.Definable fun x ↦ ∃ i ∈ s, R i x := by simpa using disj₂ s.toList h
+lemma fintype_all [Fintype ι] {P : ι → (Fin k → V) → Prop}
+ (h : ∀ i, ℌ.Definable fun w : Fin k → V ↦ P i w) :
+ ℌ.Definable 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_b2f33299bc1e_0 | 4087d733cd4f704d | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Kripke/AxiomGrz.lean | AxiomGrz | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 6 | 1 | [
{
"theorem_name": "validate_AxiomGrz_of_finite_strict_preorder",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " validate_AxiomGrz_of_refl_trans_wcwf",
"n_chars": 37,
"n_subproofs": 0,
"n_tactics": 1,
... | [
{
"name": "validate_AxiomGrz_of_refl_trans_wcwf",
"text": "lemma validate_AxiomGrz_of_refl_trans_wcwf [F.IsReflexive] [F.IsTransitive] [F.IsWeaklyConverseWellFounded] : F ⊧ (Axioms.Grz (.atom 0)) := by\n intro V;\n let M : Model := ⟨F, V⟩;\n let X := { x | Satisfies M x (□(□((.atom 0) 🡒 □(.atom 0)) 🡒 (... | [
{
"name": "validate_AxiomGrz_of_finite_strict_preorder",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 3,
"n_chars": 154,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_on... | 1 | module
public import Foundation.Modal.Kripke.Logic.K
public import Foundation.Vorspiel.Rel.WCWF
@[expose] public section
namespace LO.Modal
namespace Kripke
open Entailment
open Kripke
open Formula (atom)
open Formula.Kripke
open Rel (IrreflGen)
variable {F : Kripke.Frame}
protected abbrev Frame.IsWeaklyConverse... | module
public import Foundation.Modal.Kripke.Logic.K
public import Foundation.Vorspiel.Rel.WCWF
@[expose] public section
namespace LO.Modal
namespace Kripke
open Entailment
open Kripke
open Formula (atom)
open Formula.Kripke
open Rel (IrreflGen)
variable {F : Kripke.Frame}
protected abbrev Frame.IsWeaklyConverse... | @@ -23,8 +23,62 @@
instance [F.IsFinite] [F.IsTransitive] [F.IsAntisymmetric] : F.IsWeaklyConverseWellFounded := ⟨IsWeaklyConverseWellFounded.wcwf⟩
-lemma validate_AxiomGrz_of_finite_strict_preorder [F.IsFinite] [F.IsPartialOrder] : F ⊧ (Axioms.Grz (.atom 0)) := sorry
+lemma validate_AxiomGrz_of_refl_trans_wcwf [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_b2f33299bc1e_1 | 936815529681abd0 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Kripke/AxiomGrz.lean | AxiomGrz | 1 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 6 | 2 | [
{
"theorem_name": "validate_AxiomT_of_validate_AxiomGrz",
"depth": 1,
"n_commands": 0,
"n_lines": 3,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n intro V x hx;\n exact Satisfies.and_def.mp (validate_AxiomT_AxiomFour_of_validate_Grz h V x hx) |>.1;... | [
{
"name": "validate_AxiomT_AxiomFour_of_validate_Grz",
"text": "lemma validate_AxiomT_AxiomFour_of_validate_Grz (h : F ⊧ Axioms.Grz (.atom 0)) : F ⊧ □(.atom 0) 🡒 ((.atom 0) ⋏ □□(.atom 0)) := by\n let ψ : Formula _ := (.atom 0) ⋏ (□(.atom 0) 🡒 □□(.atom 0));\n intro V x;\n simp only [Axioms.Grz, ValidOnF... | [
{
"name": "validate_AxiomT_of_validate_AxiomGrz",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 5,
"n_chars": 216,
"n_subproofs": 0,
"n_tactics": 5,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 2,
"automation_only": fa... | 1 | module
public import Foundation.Modal.Kripke.Logic.K
public import Foundation.Vorspiel.Rel.WCWF
@[expose] public section
namespace LO.Modal
namespace Kripke
open Entailment
open Kripke
open Formula (atom)
open Formula.Kripke
open Rel (IrreflGen)
variable {F : Kripke.Frame}
protected abbrev Frame.IsWeaklyConverse... | module
public import Foundation.Modal.Kripke.Logic.K
public import Foundation.Vorspiel.Rel.WCWF
@[expose] public section
namespace LO.Modal
namespace Kripke
open Entailment
open Kripke
open Formula (atom)
open Formula.Kripke
open Rel (IrreflGen)
variable {F : Kripke.Frame}
protected abbrev Frame.IsWeaklyConverse... | @@ -23,8 +23,26 @@
instance [F.IsFinite] [F.IsTransitive] [F.IsAntisymmetric] : F.IsWeaklyConverseWellFounded := ⟨IsWeaklyConverseWellFounded.wcwf⟩
-lemma validate_AxiomT_of_validate_AxiomGrz (h : F ⊧ Axioms.Grz (.atom 0)) : F ⊧ (Axioms.T (.atom 0)) := sorry
+lemma validate_AxiomT_AxiomFour_of_validate_Grz (h : 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_b2f33299bc1e_2 | fa0c2710ea5b7b66 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Kripke/AxiomGrz.lean | AxiomGrz | 2 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 6 | 2 | [
{
"theorem_name": "validate_AxiomT_of_validate_AxiomGrz",
"depth": 1,
"n_commands": 0,
"n_lines": 3,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n intro V x hx;\n exact Satisfies.and_def.mp (validate_AxiomT_AxiomFour_of_validate_Grz h V x hx) |>.1;... | [
{
"name": "validate_AxiomT_AxiomFour_of_validate_Grz",
"text": "lemma validate_AxiomT_AxiomFour_of_validate_Grz (h : F ⊧ Axioms.Grz (.atom 0)) : F ⊧ □(.atom 0) 🡒 ((.atom 0) ⋏ □□(.atom 0)) := by\n let ψ : Formula _ := (.atom 0) ⋏ (□(.atom 0) 🡒 □□(.atom 0));\n intro V x;\n simp only [Axioms.Grz, ValidOnF... | [
{
"name": "validate_AxiomFour_of_validate_AxiomGrz",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 5,
"n_chars": 223,
"n_subproofs": 0,
"n_tactics": 5,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 2,
"automation_only":... | 1 | module
public import Foundation.Modal.Kripke.Logic.K
public import Foundation.Vorspiel.Rel.WCWF
@[expose] public section
namespace LO.Modal
namespace Kripke
open Entailment
open Kripke
open Formula (atom)
open Formula.Kripke
open Rel (IrreflGen)
variable {F : Kripke.Frame}
protected abbrev Frame.IsWeaklyConverse... | module
public import Foundation.Modal.Kripke.Logic.K
public import Foundation.Vorspiel.Rel.WCWF
@[expose] public section
namespace LO.Modal
namespace Kripke
open Entailment
open Kripke
open Formula (atom)
open Formula.Kripke
open Rel (IrreflGen)
variable {F : Kripke.Frame}
protected abbrev Frame.IsWeaklyConverse... | @@ -23,8 +23,26 @@
instance [F.IsFinite] [F.IsTransitive] [F.IsAntisymmetric] : F.IsWeaklyConverseWellFounded := ⟨IsWeaklyConverseWellFounded.wcwf⟩
-lemma validate_AxiomFour_of_validate_AxiomGrz (h : F ⊧ Axioms.Grz (.atom 0)) : F ⊧ (Axioms.Four (.atom 0)) := sorry
+lemma validate_AxiomT_AxiomFour_of_validate_Grz ... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_b2f33299bc1e_3 | fcd305d6e174d1d9 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Kripke/AxiomGrz.lean | AxiomGrz | 3 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 6 | 2 | [
{
"theorem_name": "validate_AxiomT_of_validate_AxiomGrz",
"depth": 1,
"n_commands": 0,
"n_lines": 3,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n intro V x hx;\n exact Satisfies.and_def.mp (validate_AxiomT_AxiomFour_of_validate_Grz h V x hx) |>.1;... | [
{
"name": "validate_AxiomT_AxiomFour_of_validate_Grz",
"text": "lemma validate_AxiomT_AxiomFour_of_validate_Grz (h : F ⊧ Axioms.Grz (.atom 0)) : F ⊧ □(.atom 0) 🡒 ((.atom 0) ⋏ □□(.atom 0)) := by\n let ψ : Formula _ := (.atom 0) ⋏ (□(.atom 0) 🡒 □□(.atom 0));\n intro V x;\n simp only [Axioms.Grz, ValidOnF... | [
{
"name": "reflexive_of_validate_AxiomGrz",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 2,
"n_lines": 5,
"n_chars": 179,
"n_subproofs": 0,
"n_tactics": 5,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 2,
"automation_only": false,
... | 2 | module
public import Foundation.Modal.Kripke.Logic.K
public import Foundation.Vorspiel.Rel.WCWF
@[expose] public section
namespace LO.Modal
namespace Kripke
open Entailment
open Kripke
open Formula (atom)
open Formula.Kripke
open Rel (IrreflGen)
variable {F : Kripke.Frame}
protected abbrev Frame.IsWeaklyConverse... | module
public import Foundation.Modal.Kripke.Logic.K
public import Foundation.Vorspiel.Rel.WCWF
@[expose] public section
namespace LO.Modal
namespace Kripke
open Entailment
open Kripke
open Formula (atom)
open Formula.Kripke
open Rel (IrreflGen)
variable {F : Kripke.Frame}
protected abbrev Frame.IsWeaklyConverse... | @@ -23,8 +23,26 @@
instance [F.IsFinite] [F.IsTransitive] [F.IsAntisymmetric] : F.IsWeaklyConverseWellFounded := ⟨IsWeaklyConverseWellFounded.wcwf⟩
-lemma validate_AxiomT_of_validate_AxiomGrz (h : F ⊧ Axioms.Grz (.atom 0)) : F ⊧ (Axioms.T (.atom 0)) := sorry
+lemma validate_AxiomT_AxiomFour_of_validate_Grz (h : 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_b2f33299bc1e_4 | 1cec44b5c3272f52 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Kripke/AxiomGrz.lean | AxiomGrz | 4 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 6 | 2 | [
{
"theorem_name": "validate_AxiomT_of_validate_AxiomGrz",
"depth": 1,
"n_commands": 0,
"n_lines": 3,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n intro V x hx;\n exact Satisfies.and_def.mp (validate_AxiomT_AxiomFour_of_validate_Grz h V x hx) |>.1;... | [
{
"name": "validate_AxiomT_AxiomFour_of_validate_Grz",
"text": "lemma validate_AxiomT_AxiomFour_of_validate_Grz (h : F ⊧ Axioms.Grz (.atom 0)) : F ⊧ □(.atom 0) 🡒 ((.atom 0) ⋏ □□(.atom 0)) := by\n let ψ : Formula _ := (.atom 0) ⋏ (□(.atom 0) 🡒 □□(.atom 0));\n intro V x;\n simp only [Axioms.Grz, ValidOnF... | [
{
"name": "transitive_of_validate_AxiomGrz",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 2,
"n_lines": 5,
"n_chars": 188,
"n_subproofs": 0,
"n_tactics": 5,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 2,
"automation_only": false,
... | 2 | module
public import Foundation.Modal.Kripke.Logic.K
public import Foundation.Vorspiel.Rel.WCWF
@[expose] public section
namespace LO.Modal
namespace Kripke
open Entailment
open Kripke
open Formula (atom)
open Formula.Kripke
open Rel (IrreflGen)
variable {F : Kripke.Frame}
protected abbrev Frame.IsWeaklyConverse... | module
public import Foundation.Modal.Kripke.Logic.K
public import Foundation.Vorspiel.Rel.WCWF
@[expose] public section
namespace LO.Modal
namespace Kripke
open Entailment
open Kripke
open Formula (atom)
open Formula.Kripke
open Rel (IrreflGen)
variable {F : Kripke.Frame}
protected abbrev Frame.IsWeaklyConverse... | @@ -23,8 +23,26 @@
instance [F.IsFinite] [F.IsTransitive] [F.IsAntisymmetric] : F.IsWeaklyConverseWellFounded := ⟨IsWeaklyConverseWellFounded.wcwf⟩
-lemma validate_AxiomFour_of_validate_AxiomGrz (h : F ⊧ Axioms.Grz (.atom 0)) : F ⊧ (Axioms.Four (.atom 0)) := sorry
+lemma validate_AxiomT_AxiomFour_of_validate_Grz ... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_b2f33299bc1e_5 | 56614b88457c28bf | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Kripke/AxiomGrz.lean | AxiomGrz | 5 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 6 | 1 | [
{
"theorem_name": "transitive_of_validate_AxiomGrz",
"depth": 1,
"n_commands": 0,
"n_lines": 3,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n apply transitive_of_validate_AxiomFour;\n exact validate_AxiomFour_of_validate_AxiomGrz h;",
"n_chars"... | [
{
"name": "validate_AxiomFour_of_validate_AxiomGrz",
"text": "lemma validate_AxiomFour_of_validate_AxiomGrz (h : F ⊧ Axioms.Grz (.atom 0)) : F ⊧ (Axioms.Four (.atom 0)) := by\n intro V x hx;\n exact Satisfies.and_def.mp (validate_AxiomT_AxiomFour_of_validate_Grz h V x hx) |>.2;\n\n",
"fan_in": 1,
... | [
{
"name": "WCWF_of_validate_AxiomGrz",
"fan_in": 0,
"n_deps_direct": 2,
"n_deps_transitive": 5,
"n_lines": 104,
"n_chars": 3811,
"n_subproofs": 11,
"n_tactics": 116,
"cyclomatic": 7,
"n_automation": 15,
"n_rewrites": 1,
"n_structural": 32,
"automation_only": false... | 5 | module
public import Foundation.Modal.Kripke.Logic.K
public import Foundation.Vorspiel.Rel.WCWF
@[expose] public section
namespace LO.Modal
namespace Kripke
open Entailment
open Kripke
open Formula (atom)
open Formula.Kripke
open Rel (IrreflGen)
variable {F : Kripke.Frame}
protected abbrev Frame.IsWeaklyConverse... | module
public import Foundation.Modal.Kripke.Logic.K
public import Foundation.Vorspiel.Rel.WCWF
@[expose] public section
namespace LO.Modal
namespace Kripke
open Entailment
open Kripke
open Formula (atom)
open Formula.Kripke
open Rel (IrreflGen)
variable {F : Kripke.Frame}
protected abbrev Frame.IsWeaklyConverse... | @@ -43,11 +43,17 @@
intro V x hx;
exact Satisfies.and_def.mp (validate_AxiomT_AxiomFour_of_validate_Grz h V x hx) |>.1;
+lemma validate_AxiomFour_of_validate_AxiomGrz (h : F ⊧ Axioms.Grz (.atom 0)) : F ⊧ (Axioms.Four (.atom 0)) := by
+ intro V x hx;
+ exact Satisfies.and_def.mp (validate_AxiomT_AxiomFour_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_ac257c79ad72_0 | b2e7b0a13929ee47 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Logic/Disjunctive.lean | Disjunctive | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 1 | 1 | [
{
"theorem_name": "iff_complete_disjunctive",
"depth": 1,
"n_commands": 0,
"n_lines": 12,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n constructor;\n . intro hComp;\n apply iff_disjunctive.mpr;\n intro φ ψ hpq;\n rcases (hComp.con φ) with... | [
{
"name": "iff_disjunctive",
"text": "lemma iff_disjunctive {𝓢 : S} : (Disjunctive 𝓢) ↔ ∀ {φ ψ}, 𝓢 ⊢ φ ⋎ ψ → 𝓢 ⊢ φ ∨ 𝓢 ⊢ ψ := by\n constructor;\n . apply Disjunctive.disjunctive;\n . exact λ d ↦ ⟨d⟩;\n\n",
"fan_in": 1,
"n_lines": 6,
"n_chars": 208,
"n_subproofs": 0,
"n_tactics":... | [
{
"name": "iff_complete_disjunctive",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 14,
"n_chars": 545,
"n_subproofs": 0,
"n_tactics": 24,
"cyclomatic": 3,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 9,
"automation_only": false,
"... | 1 | module
public import Foundation.Propositional.Entailment.Cl.Basic
@[expose] public section
namespace LO.Entailment
variable {F : Type*} [LogicalConnective F]
variable {S : Type*} [Entailment S F]
class Disjunctive (𝓢 : S) : Prop where
disjunctive : ∀ {φ ψ}, 𝓢 ⊢ φ ⋎ ψ → 𝓢 ⊢ φ ∨ 𝓢 ⊢ ψ
alias disjunctive := Dis... | module
public import Foundation.Propositional.Entailment.Cl.Basic
@[expose] public section
namespace LO.Entailment
variable {F : Type*} [LogicalConnective F]
variable {S : Type*} [Entailment S F]
class Disjunctive (𝓢 : S) : Prop where
disjunctive : ∀ {φ ψ}, 𝓢 ⊢ φ ⋎ ψ → 𝓢 ⊢ φ ∨ 𝓢 ⊢ ψ
alias disjunctive := Dis... | @@ -14,12 +14,22 @@
alias disjunctive := Disjunctive.disjunctive
+lemma iff_disjunctive {𝓢 : S} : (Disjunctive 𝓢) ↔ ∀ {φ ψ}, 𝓢 ⊢ φ ⋎ ψ → 𝓢 ⊢ φ ∨ 𝓢 ⊢ ψ := by
+ constructor;
+ . apply Disjunctive.disjunctive;
+ . exact λ d ↦ ⟨d⟩;
+
lemma iff_complete_disjunctive [DecidableEq F] {𝓢 : S} [Entailment.Cl 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_2a3f669a01ec_0 | b774b148305e580a | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Logic/Entailment.lean | Entailment | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 20 | 5 | [
{
"theorem_name": "not_weakerThan_iff",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by simp [weakerThan_iff, Unprovable];",
"n_chars": 38,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
... | [
{
"name": "weakerThan_iff",
"text": "lemma weakerThan_iff : 𝓢 ⪯ 𝓣 ↔ (∀ {φ}, 𝓢 ⊢ φ → 𝓣 ⊢ φ) :=\n ⟨fun h _ hf ↦ h.subset hf, fun h ↦ ⟨fun _ hf ↦ h hf⟩⟩\n\n",
"fan_in": 5,
"n_lines": 4,
"n_chars": 155,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n... | [
{
"name": "not_weakerThan_iff",
"fan_in": 2,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 3,
"n_chars": 127,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"max_nesti... | 1 | module
public import Foundation.Logic.Semantics
public import Foundation.Vorspiel.AdjunctiveSet
/-!
# Basic definitions and properties of proof system related notions
This file defines a characterization of the system/proof/provability/calculus of formulae.
Also defines soundness and completeness.
## Main Definitio... | module
public import Foundation.Logic.Semantics
public import Foundation.Vorspiel.AdjunctiveSet
/-!
# Basic definitions and properties of proof system related notions
This file defines a characterization of the system/proof/provability/calculus of formulae.
Also defines soundness and completeness.
## Main Definitio... | @@ -116,8 +116,11 @@
instance : Trans (α := S) (β := T) (γ := U) (· ⪯ ·) (· ⪯ ·) (· ⪯ ·) where
trans := WeakerThan.trans
-lemma not_weakerThan_iff : ¬𝓢 ⪯ 𝓣 ↔ (∃ φ, 𝓢 ⊢ φ ∧ 𝓣 ⊬ φ) := sorry
+lemma weakerThan_iff : 𝓢 ⪯ 𝓣 ↔ (∀ {φ}, 𝓢 ⊢ φ → 𝓣 ⊢ φ) :=
+ ⟨fun h _ hf ↦ h.subset hf, fun h ↦ ⟨fun _ hf ↦ h hf⟩⟩
+... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_2a3f669a01ec_1 | 302822321cc68e42 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Logic/Entailment.lean | Entailment | 1 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 20 | 5 | [
{
"theorem_name": "not_weakerThan_iff",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by simp [weakerThan_iff, Unprovable];",
"n_chars": 38,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
... | [
{
"name": "weakerThan_iff",
"text": "lemma weakerThan_iff : 𝓢 ⪯ 𝓣 ↔ (∀ {φ}, 𝓢 ⊢ φ → 𝓣 ⊢ φ) :=\n ⟨fun h _ hf ↦ h.subset hf, fun h ↦ ⟨fun _ hf ↦ h hf⟩⟩\n\n",
"fan_in": 5,
"n_lines": 4,
"n_chars": 155,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n... | [
{
"name": "strictlyWeakerThan_iff",
"fan_in": 0,
"n_deps_direct": 2,
"n_deps_transitive": 2,
"n_lines": 8,
"n_chars": 408,
"n_subproofs": 0,
"n_tactics": 7,
"cyclomatic": 5,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 7,
"automation_only": false,
"max_... | 2 | module
public import Foundation.Logic.Semantics
public import Foundation.Vorspiel.AdjunctiveSet
/-!
# Basic definitions and properties of proof system related notions
This file defines a characterization of the system/proof/provability/calculus of formulae.
Also defines soundness and completeness.
## Main Definitio... | module
public import Foundation.Logic.Semantics
public import Foundation.Vorspiel.AdjunctiveSet
/-!
# Basic definitions and properties of proof system related notions
This file defines a characterization of the system/proof/provability/calculus of formulae.
Also defines soundness and completeness.
## Main Definitio... | @@ -116,12 +116,17 @@
instance : Trans (α := S) (β := T) (γ := U) (· ⪯ ·) (· ⪯ ·) (· ⪯ ·) where
trans := WeakerThan.trans
-lemma not_weakerThan_iff : ¬𝓢 ⪯ 𝓣 ↔ (∃ φ, 𝓢 ⊢ φ ∧ 𝓣 ⊬ φ) := sorry
+lemma weakerThan_iff : 𝓢 ⪯ 𝓣 ↔ (∀ {φ}, 𝓢 ⊢ φ → 𝓣 ⊢ φ) :=
+ ⟨fun h _ hf ↦ h.subset hf, fun h ↦ ⟨fun _ hf ↦ h hf⟩⟩
... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_2a3f669a01ec_2 | b072277d4d691a13 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Logic/Entailment.lean | Entailment | 2 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 20 | 5 | [
{
"theorem_name": "not_weakerThan_iff",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by simp [weakerThan_iff, Unprovable];",
"n_chars": 38,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
... | [
{
"name": "weakerThan_iff",
"text": "lemma weakerThan_iff : 𝓢 ⪯ 𝓣 ↔ (∀ {φ}, 𝓢 ⊢ φ → 𝓣 ⊢ φ) :=\n ⟨fun h _ hf ↦ h.subset hf, fun h ↦ ⟨fun _ hf ↦ h hf⟩⟩\n\n",
"fan_in": 5,
"n_lines": 4,
"n_chars": 155,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n... | [
{
"name": "StrictlyWeakerThan.trans",
"fan_in": 4,
"n_deps_direct": 2,
"n_deps_transitive": 4,
"n_lines": 3,
"n_chars": 143,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
"ma... | 1 | module
public import Foundation.Logic.Semantics
public import Foundation.Vorspiel.AdjunctiveSet
/-!
# Basic definitions and properties of proof system related notions
This file defines a characterization of the system/proof/provability/calculus of formulae.
Also defines soundness and completeness.
## Main Definitio... | module
public import Foundation.Logic.Semantics
public import Foundation.Vorspiel.AdjunctiveSet
/-!
# Basic definitions and properties of proof system related notions
This file defines a characterization of the system/proof/provability/calculus of formulae.
Also defines soundness and completeness.
## Main Definitio... | @@ -116,6 +116,9 @@
instance : Trans (α := S) (β := T) (γ := U) (· ⪯ ·) (· ⪯ ·) (· ⪯ ·) where
trans := WeakerThan.trans
+lemma weakerThan_iff : 𝓢 ⪯ 𝓣 ↔ (∀ {φ}, 𝓢 ⊢ φ → 𝓣 ⊢ φ) :=
+ ⟨fun h _ hf ↦ h.subset hf, fun h ↦ ⟨fun _ hf ↦ h hf⟩⟩
+
lemma swt_of_swt_of_wt : 𝓢 ⪱ 𝓣 → 𝓣 ⪯ 𝓤 → 𝓢 ⪱ 𝓤 := by
rintro ⟨h₁... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_2a3f669a01ec_3 | 3a38742ac09538d4 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Logic/Entailment.lean | Entailment | 3 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 20 | 5 | [
{
"theorem_name": "not_weakerThan_iff",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by simp [weakerThan_iff, Unprovable];",
"n_chars": 38,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
... | [
{
"name": "weakerThan_iff",
"text": "lemma weakerThan_iff : 𝓢 ⪯ 𝓣 ↔ (∀ {φ}, 𝓢 ⊢ φ → 𝓣 ⊢ φ) :=\n ⟨fun h _ hf ↦ h.subset hf, fun h ↦ ⟨fun _ hf ↦ h hf⟩⟩\n\n",
"fan_in": 5,
"n_lines": 4,
"n_chars": 155,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n... | [
{
"name": "Incomparable.of_unprovable",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 2,
"n_lines": 9,
"n_chars": 252,
"n_subproofs": 0,
"n_tactics": 7,
"cyclomatic": 3,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 2,
"automation_only": false,
"... | 2 | module
public import Foundation.Logic.Semantics
public import Foundation.Vorspiel.AdjunctiveSet
/-!
# Basic definitions and properties of proof system related notions
This file defines a characterization of the system/proof/provability/calculus of formulae.
Also defines soundness and completeness.
## Main Definitio... | module
public import Foundation.Logic.Semantics
public import Foundation.Vorspiel.AdjunctiveSet
/-!
# Basic definitions and properties of proof system related notions
This file defines a characterization of the system/proof/provability/calculus of formulae.
Also defines soundness and completeness.
## Main Definitio... | @@ -116,8 +116,11 @@
instance : Trans (α := S) (β := T) (γ := U) (· ⪯ ·) (· ⪯ ·) (· ⪯ ·) where
trans := WeakerThan.trans
-lemma not_weakerThan_iff : ¬𝓢 ⪯ 𝓣 ↔ (∃ φ, 𝓢 ⊢ φ ∧ 𝓣 ⊬ φ) := sorry
+lemma weakerThan_iff : 𝓢 ⪯ 𝓣 ↔ (∀ {φ}, 𝓢 ⊢ φ → 𝓣 ⊢ φ) :=
+ ⟨fun h _ hf ↦ h.subset hf, fun h ↦ ⟨fun _ hf ↦ h hf⟩⟩
+... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_2a3f669a01ec_4 | ddf01d7872042964 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Logic/Entailment.lean | Entailment | 4 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 20 | 2 | [
{
"theorem_name": "consistent_iff_exists_unprovable",
"depth": 1,
"n_commands": 0,
"n_lines": 4,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n simp [←not_inconsistent_iff_consistent, inconsistent_def]\n\nalias ⟨Consistent.exists_unprovable, _⟩ := co... | [
{
"name": "inconsistent_def",
"text": "lemma inconsistent_def {𝓢 : S} :\n Inconsistent 𝓢 ↔ ∀ φ, 𝓢 ⊢ φ := by simp [Inconsistent]\n\n",
"fan_in": 2,
"n_lines": 4,
"n_chars": 109,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n... | [
{
"name": "consistent_iff_exists_unprovable",
"fan_in": 0,
"n_deps_direct": 2,
"n_deps_transitive": 2,
"n_lines": 7,
"n_chars": 246,
"n_subproofs": 0,
"n_tactics": 3,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
... | 1 | module
public import Foundation.Logic.Semantics
public import Foundation.Vorspiel.AdjunctiveSet
/-!
# Basic definitions and properties of proof system related notions
This file defines a characterization of the system/proof/provability/calculus of formulae.
Also defines soundness and completeness.
## Main Definitio... | module
public import Foundation.Logic.Semantics
public import Foundation.Vorspiel.AdjunctiveSet
/-!
# Basic definitions and properties of proof system related notions
This file defines a characterization of the system/proof/provability/calculus of formulae.
Also defines soundness and completeness.
## Main Definitio... | @@ -205,6 +205,9 @@
class Consistent (𝓢 : S) : Prop where
not_inconsistent : ¬Inconsistent 𝓢
+lemma inconsistent_def {𝓢 : S} :
+ Inconsistent 𝓢 ↔ ∀ φ, 𝓢 ⊢ φ := by simp [Inconsistent]
+
@[simp] lemma not_inconsistent_iff_consistent {𝓢 : S} :
¬Inconsistent 𝓢 ↔ Consistent 𝓢 :=
⟨fun h ↦ ⟨h⟩, by ri... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_2a3f669a01ec_5 | 5fb05c00fcd40aed | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Logic/Entailment.lean | Entailment | 5 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 20 | 2 | [
{
"theorem_name": "consistent_iff_exists_unprovable",
"depth": 1,
"n_commands": 0,
"n_lines": 4,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n simp [←not_inconsistent_iff_consistent, inconsistent_def]\n\nalias ⟨Consistent.exists_unprovable, _⟩ := co... | [
{
"name": "inconsistent_def",
"text": "lemma inconsistent_def {𝓢 : S} :\n Inconsistent 𝓢 ↔ ∀ φ, 𝓢 ⊢ φ := by simp [Inconsistent]\n\n",
"fan_in": 2,
"n_lines": 4,
"n_chars": 109,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n... | [
{
"name": "inconsistent_iff_theory_eq_univ",
"fan_in": 0,
"n_deps_direct": 2,
"n_deps_transitive": 2,
"n_lines": 6,
"n_chars": 226,
"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.Logic.Semantics
public import Foundation.Vorspiel.AdjunctiveSet
/-!
# Basic definitions and properties of proof system related notions
This file defines a characterization of the system/proof/provability/calculus of formulae.
Also defines soundness and completeness.
## Main Definitio... | module
public import Foundation.Logic.Semantics
public import Foundation.Vorspiel.AdjunctiveSet
/-!
# Basic definitions and properties of proof system related notions
This file defines a characterization of the system/proof/provability/calculus of formulae.
Also defines soundness and completeness.
## Main Definitio... | @@ -205,6 +205,9 @@
class Consistent (𝓢 : S) : Prop where
not_inconsistent : ¬Inconsistent 𝓢
+lemma inconsistent_def {𝓢 : S} :
+ Inconsistent 𝓢 ↔ ∀ φ, 𝓢 ⊢ φ := by simp [Inconsistent]
+
@[simp] lemma not_inconsistent_iff_consistent {𝓢 : S} :
¬Inconsistent 𝓢 ↔ Consistent 𝓢 :=
⟨fun h ↦ ⟨h⟩, by ri... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_2a3f669a01ec_6 | 8dc7cafd01f8679c | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Logic/Entailment.lean | Entailment | 6 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 20 | 1 | [
{
"theorem_name": "consistent_iff_unprovable_bot",
"depth": 1,
"n_commands": 0,
"n_lines": 4,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n simp [inconsistent_iff_provable_bot, ←not_inconsistent_iff_consistent]\n\nalias ⟨Consistent.not_bot, _⟩ := co... | [
{
"name": "inconsistent_iff_provable_bot",
"text": "lemma inconsistent_iff_provable_bot {𝓢 : S} :\n Inconsistent 𝓢 ↔ 𝓢 ⊢ ⊥ := ⟨fun h ↦ h ⊥, fun h φ ↦ DeductiveExplosion.dexp! h φ⟩\n\nalias ⟨_, inconsistent_of_provable⟩ := inconsistent_iff_provable_bot\n\n",
"fan_in": 1,
"n_lines": 6,
"n_ch... | [
{
"name": "consistent_iff_unprovable_bot",
"fan_in": 0,
"n_deps_direct": 2,
"n_deps_transitive": 2,
"n_lines": 7,
"n_chars": 236,
"n_subproofs": 0,
"n_tactics": 3,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
... | 1 | module
public import Foundation.Logic.Semantics
public import Foundation.Vorspiel.AdjunctiveSet
/-!
# Basic definitions and properties of proof system related notions
This file defines a characterization of the system/proof/provability/calculus of formulae.
Also defines soundness and completeness.
## Main Definitio... | module
public import Foundation.Logic.Semantics
public import Foundation.Vorspiel.AdjunctiveSet
/-!
# Basic definitions and properties of proof system related notions
This file defines a characterization of the system/proof/provability/calculus of formulae.
Also defines soundness and completeness.
## Main Definitio... | @@ -239,9 +239,17 @@
def DeductiveExplosion.dexp! {𝓢 : S} (h : 𝓢 ⊢ ⊥) (φ : F) : 𝓢 ⊢ φ := by
rcases h with ⟨b⟩; exact ⟨dexp b φ⟩
+lemma inconsistent_iff_provable_bot {𝓢 : S} :
+ Inconsistent 𝓢 ↔ 𝓢 ⊢ ⊥ := ⟨fun h ↦ h ⊥, fun h φ ↦ DeductiveExplosion.dexp! h φ⟩
+
+alias ⟨_, inconsistent_of_provable⟩ := incons... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_2a3f669a01ec_7 | c5d263979caccbb9 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Logic/Entailment.lean | Entailment | 7 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 20 | 1 | [
{
"theorem_name": "not_complete_iff_incomplete",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n simp [complete_def, incomplete_def, Independent, not_or]",
"n_chars": 62,
"n_subproofs": 0,
"n_tact... | [
{
"name": "complete_def",
"text": "lemma complete_def : Complete 𝓢 ↔ ∀ φ, 𝓢 ⊢ φ ∨ 𝓢 ⊢ ∼φ :=\n ⟨fun h ↦ h.con, Complete.mk⟩\n\n",
"fan_in": 1,
"n_lines": 4,
"n_chars": 119,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_struct... | [
{
"name": "not_complete_iff_incomplete",
"fan_in": 1,
"n_deps_direct": 2,
"n_deps_transitive": 2,
"n_lines": 4,
"n_chars": 146,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"... | 2 | module
public import Foundation.Logic.Semantics
public import Foundation.Vorspiel.AdjunctiveSet
/-!
# Basic definitions and properties of proof system related notions
This file defines a characterization of the system/proof/provability/calculus of formulae.
Also defines soundness and completeness.
## Main Definitio... | module
public import Foundation.Logic.Semantics
public import Foundation.Vorspiel.AdjunctiveSet
/-!
# Basic definitions and properties of proof system related notions
This file defines a characterization of the system/proof/provability/calculus of formulae.
Also defines soundness and completeness.
## Main Definitio... | @@ -260,10 +260,14 @@
variable {𝓢}
+lemma complete_def : Complete 𝓢 ↔ ∀ φ, 𝓢 ⊢ φ ∨ 𝓢 ⊢ ∼φ :=
+ ⟨fun h ↦ h.con, Complete.mk⟩
+
lemma incomplete_def : Incomplete 𝓢 ↔ ∃ φ, Independent 𝓢 φ :=
⟨fun h ↦ h.indep, Incomplete.mk⟩
-@[simp] lemma not_complete_iff_incomplete : ¬Complete 𝓢 ↔ Incomplete 𝓢 := sorr... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_2a3f669a01ec_8 | 55eeec77809d45b8 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Logic/Entailment.lean | Entailment | 8 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 20 | 5 | [
{
"theorem_name": "not_weakerThan_iff",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by simp [weakerThan_iff, Unprovable];",
"n_chars": 38,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
... | [
{
"name": "weakerThan_iff",
"text": "lemma weakerThan_iff : 𝓢 ⪯ 𝓣 ↔ (∀ {φ}, 𝓢 ⊢ φ → 𝓣 ⊢ φ) :=\n ⟨fun h _ hf ↦ h.subset hf, fun h ↦ ⟨fun _ hf ↦ h hf⟩⟩\n\n",
"fan_in": 5,
"n_lines": 4,
"n_chars": 155,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n... | [
{
"name": "le_of_subset",
"fan_in": 2,
"n_deps_direct": 1,
"n_deps_transitive": 2,
"n_lines": 3,
"n_chars": 113,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 2,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 2,
"automation_only": false,
"max_nesting": ... | 1 | module
public import Foundation.Logic.Semantics
public import Foundation.Vorspiel.AdjunctiveSet
/-!
# Basic definitions and properties of proof system related notions
This file defines a characterization of the system/proof/provability/calculus of formulae.
Also defines soundness and completeness.
## Main Definitio... | module
public import Foundation.Logic.Semantics
public import Foundation.Vorspiel.AdjunctiveSet
/-!
# Basic definitions and properties of proof system related notions
This file defines a characterization of the system/proof/provability/calculus of formulae.
Also defines soundness and completeness.
## Main Definitio... | @@ -116,6 +116,9 @@
instance : Trans (α := S) (β := T) (γ := U) (· ⪯ ·) (· ⪯ ·) (· ⪯ ·) where
trans := WeakerThan.trans
+lemma weakerThan_iff : 𝓢 ⪯ 𝓣 ↔ (∀ {φ}, 𝓢 ⊢ φ → 𝓣 ⊢ φ) :=
+ ⟨fun h _ hf ↦ h.subset hf, fun h ↦ ⟨fun _ hf ↦ h hf⟩⟩
+
lemma swt_of_swt_of_wt : 𝓢 ⪱ 𝓣 → 𝓣 ⪯ 𝓤 → 𝓢 ⪱ 𝓤 := by
rintro ⟨h₁... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_2a3f669a01ec_9 | 7917c40c27ce3a62 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Logic/Entailment.lean | Entailment | 9 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 20 | 5 | [
{
"theorem_name": "not_weakerThan_iff",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by simp [weakerThan_iff, Unprovable];",
"n_chars": 38,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
... | [
{
"name": "weakerThan_iff",
"text": "lemma weakerThan_iff : 𝓢 ⪯ 𝓣 ↔ (∀ {φ}, 𝓢 ⊢ φ → 𝓣 ⊢ φ) :=\n ⟨fun h _ hf ↦ h.subset hf, fun h ↦ ⟨fun _ hf ↦ h hf⟩⟩\n\n",
"fan_in": 5,
"n_lines": 4,
"n_chars": 155,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n... | [
{
"name": "Consistent.of_subset",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 3,
"n_lines": 4,
"n_chars": 154,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
"max_ne... | 2 | module
public import Foundation.Logic.Semantics
public import Foundation.Vorspiel.AdjunctiveSet
/-!
# Basic definitions and properties of proof system related notions
This file defines a characterization of the system/proof/provability/calculus of formulae.
Also defines soundness and completeness.
## Main Definitio... | module
public import Foundation.Logic.Semantics
public import Foundation.Vorspiel.AdjunctiveSet
/-!
# Basic definitions and properties of proof system related notions
This file defines a characterization of the system/proof/provability/calculus of formulae.
Also defines soundness and completeness.
## Main Definitio... | @@ -116,6 +116,9 @@
instance : Trans (α := S) (β := T) (γ := U) (· ⪯ ·) (· ⪯ ·) (· ⪯ ·) where
trans := WeakerThan.trans
+lemma weakerThan_iff : 𝓢 ⪯ 𝓣 ↔ (∀ {φ}, 𝓢 ⊢ φ → 𝓣 ⊢ φ) :=
+ ⟨fun h _ hf ↦ h.subset hf, fun h ↦ ⟨fun _ hf ↦ h hf⟩⟩
+
lemma swt_of_swt_of_wt : 𝓢 ⪱ 𝓣 → 𝓣 ⪯ 𝓤 → 𝓢 ⪱ 𝓤 := by
rintro ⟨h₁... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_2a3f669a01ec_10 | 3c73281c34f96a76 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Logic/Entailment.lean | Entailment | 10 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 20 | 5 | [
{
"theorem_name": "not_weakerThan_iff",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by simp [weakerThan_iff, Unprovable];",
"n_chars": 38,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
... | [
{
"name": "weakerThan_iff",
"text": "lemma weakerThan_iff : 𝓢 ⪯ 𝓣 ↔ (∀ {φ}, 𝓢 ⊢ φ → 𝓣 ⊢ φ) :=\n ⟨fun h _ hf ↦ h.subset hf, fun h ↦ ⟨fun _ hf ↦ h hf⟩⟩\n\n",
"fan_in": 5,
"n_lines": 4,
"n_chars": 155,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n... | [
{
"name": "Inconsistent.of_supset",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 3,
"n_lines": 4,
"n_chars": 160,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
"max_... | 2 | module
public import Foundation.Logic.Semantics
public import Foundation.Vorspiel.AdjunctiveSet
/-!
# Basic definitions and properties of proof system related notions
This file defines a characterization of the system/proof/provability/calculus of formulae.
Also defines soundness and completeness.
## Main Definitio... | module
public import Foundation.Logic.Semantics
public import Foundation.Vorspiel.AdjunctiveSet
/-!
# Basic definitions and properties of proof system related notions
This file defines a characterization of the system/proof/provability/calculus of formulae.
Also defines soundness and completeness.
## Main Definitio... | @@ -116,6 +116,9 @@
instance : Trans (α := S) (β := T) (γ := U) (· ⪯ ·) (· ⪯ ·) (· ⪯ ·) where
trans := WeakerThan.trans
+lemma weakerThan_iff : 𝓢 ⪯ 𝓣 ↔ (∀ {φ}, 𝓢 ⊢ φ → 𝓣 ⊢ φ) :=
+ ⟨fun h _ hf ↦ h.subset hf, fun h ↦ ⟨fun _ hf ↦ h hf⟩⟩
+
lemma swt_of_swt_of_wt : 𝓢 ⪱ 𝓣 → 𝓣 ⪯ 𝓤 → 𝓢 ⪱ 𝓤 := by
rintro ⟨h₁... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_2a3f669a01ec_14 | a456bfa3ad117d54 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Logic/Entailment.lean | Entailment | 14 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 20 | 2 | [
{
"theorem_name": "consistent_of_model",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": "\n consistent_of_meaningful (𝓜 := 𝓜) inferInstance",
"n_chars": 56,
"n_subproofs": 0,
"n_tactics": 1,
"cycl... | [
{
"name": "consistent_of_meaningful",
"text": "lemma consistent_of_meaningful : Semantics.Meaningful 𝓜 → Entailment.Consistent 𝓢 :=\n fun H ↦ ⟨fun h ↦ by rcases H with ⟨φ, hf⟩; exact hf (Sound.sound (h φ))⟩\n\n",
"fan_in": 2,
"n_lines": 4,
"n_chars": 183,
"n_subproofs": 0,
"n_tactics"... | [
{
"name": "consistent_of_model",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 4,
"n_chars": 184,
"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.Semantics
public import Foundation.Vorspiel.AdjunctiveSet
/-!
# Basic definitions and properties of proof system related notions
This file defines a characterization of the system/proof/provability/calculus of formulae.
Also defines soundness and completeness.
## Main Definitio... | module
public import Foundation.Logic.Semantics
public import Foundation.Vorspiel.AdjunctiveSet
/-!
# Basic definitions and properties of proof system related notions
This file defines a characterization of the system/proof/provability/calculus of formulae.
Also defines soundness and completeness.
## Main Definitio... | @@ -414,8 +414,12 @@
variable {𝓢 𝓣 : S} {𝓜 𝓝 : M} [Sound 𝓢 𝓜] [Sound 𝓣 𝓝]
-lemma consistent_of_model [LogicalConnective F] [Semantics.Bot M] (𝓜 : M) [Sound 𝓢 𝓜] : Entailment.Consistent 𝓢 := sorry
+lemma consistent_of_meaningful : Semantics.Meaningful 𝓜 → Entailment.Consistent 𝓢 :=
+ fun H ↦ ⟨fun 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_2a3f669a01ec_15 | 9a838df80d2b488d | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Logic/Entailment.lean | Entailment | 15 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 20 | 2 | [
{
"theorem_name": "consistent_of_model",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": "\n consistent_of_meaningful (𝓜 := 𝓜) inferInstance",
"n_chars": 56,
"n_subproofs": 0,
"n_tactics": 1,
"cycl... | [
{
"name": "consistent_of_meaningful",
"text": "lemma consistent_of_meaningful : Semantics.Meaningful 𝓜 → Entailment.Consistent 𝓢 :=\n fun H ↦ ⟨fun h ↦ by rcases H with ⟨φ, hf⟩; exact hf (Sound.sound (h φ))⟩\n\n",
"fan_in": 2,
"n_lines": 4,
"n_chars": 183,
"n_subproofs": 0,
"n_tactics"... | [
{
"name": "consistent_of_satisfiable",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 4,
"n_chars": 243,
"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.Logic.Semantics
public import Foundation.Vorspiel.AdjunctiveSet
/-!
# Basic definitions and properties of proof system related notions
This file defines a characterization of the system/proof/provability/calculus of formulae.
Also defines soundness and completeness.
## Main Definitio... | module
public import Foundation.Logic.Semantics
public import Foundation.Vorspiel.AdjunctiveSet
/-!
# Basic definitions and properties of proof system related notions
This file defines a characterization of the system/proof/provability/calculus of formulae.
Also defines soundness and completeness.
## Main Definitio... | @@ -414,13 +414,17 @@
variable {𝓢 𝓣 : S} {𝓜 𝓝 : M} [Sound 𝓢 𝓜] [Sound 𝓣 𝓝]
+lemma consistent_of_meaningful : Semantics.Meaningful 𝓜 → Entailment.Consistent 𝓢 :=
+ fun H ↦ ⟨fun h ↦ by rcases H with ⟨φ, hf⟩; exact hf (Sound.sound (h φ))⟩
+
end
section
variable {𝓢 : S} {T : Set F} [Sound 𝓢 (Semant... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_2a3f669a01ec_16 | d2098a7ef4804ff8 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Logic/Entailment.lean | Entailment | 16 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 20 | 1 | [
{
"theorem_name": "satisfiable_of_consistent",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": "\n fun H ↦ Semantics.meaningful_iff_satisfiableSet.mpr (meaningful_of_consistent H)",
"n_chars": 85,
"n_subproo... | [
{
"name": "meaningful_of_consistent",
"text": "lemma meaningful_of_consistent : Entailment.Consistent 𝓢 → Semantics.Meaningful 𝓜 := by\n contrapose\n suffices (∀ (φ : F), 𝓜 ⊧ φ) → Entailment.Inconsistent 𝓢 by\n simpa [Semantics.not_meaningful_iff, Entailment.not_consistent_iff_inconsistent]\n exac... | [
{
"name": "satisfiable_of_consistent",
"fan_in": 2,
"n_deps_direct": 1,
"n_deps_transitive": 3,
"n_lines": 5,
"n_chars": 184,
"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.Logic.Semantics
public import Foundation.Vorspiel.AdjunctiveSet
/-!
# Basic definitions and properties of proof system related notions
This file defines a characterization of the system/proof/provability/calculus of formulae.
Also defines soundness and completeness.
## Main Definitio... | module
public import Foundation.Logic.Semantics
public import Foundation.Vorspiel.AdjunctiveSet
/-!
# Basic definitions and properties of proof system related notions
This file defines a characterization of the system/proof/provability/calculus of formulae.
Also defines soundness and completeness.
## Main Definitio... | @@ -430,6 +430,12 @@
variable {𝓢 : S} {𝓜 : M} [Complete 𝓢 𝓜]
+lemma meaningful_of_consistent : Entailment.Consistent 𝓢 → Semantics.Meaningful 𝓜 := by
+ contrapose
+ suffices (∀ (φ : F), 𝓜 ⊧ φ) → Entailment.Inconsistent 𝓢 by
+ simpa [Semantics.not_meaningful_iff, Entailment.not_consistent_iff_inconsist... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_2a3f669a01ec_17 | ff511a848d83a9e0 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Logic/Entailment.lean | Entailment | 17 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 20 | 1 | [
{
"theorem_name": "satisfiable_of_consistent",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": "\n fun H ↦ Semantics.meaningful_iff_satisfiableSet.mpr (meaningful_of_consistent H)",
"n_chars": 85,
"n_subproo... | [
{
"name": "meaningful_of_consistent",
"text": "lemma meaningful_of_consistent : Entailment.Consistent 𝓢 → Semantics.Meaningful 𝓜 := by\n contrapose\n suffices (∀ (φ : F), 𝓜 ⊧ φ) → Entailment.Inconsistent 𝓢 by\n simpa [Semantics.not_meaningful_iff, Entailment.not_consistent_iff_inconsistent]\n exac... | [
{
"name": "inconsistent_of_unsatisfiable",
"fan_in": 0,
"n_deps_direct": 2,
"n_deps_transitive": 4,
"n_lines": 5,
"n_chars": 210,
"n_subproofs": 0,
"n_tactics": 3,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
... | 2 | module
public import Foundation.Logic.Semantics
public import Foundation.Vorspiel.AdjunctiveSet
/-!
# Basic definitions and properties of proof system related notions
This file defines a characterization of the system/proof/provability/calculus of formulae.
Also defines soundness and completeness.
## Main Definitio... | module
public import Foundation.Logic.Semantics
public import Foundation.Vorspiel.AdjunctiveSet
/-!
# Basic definitions and properties of proof system related notions
This file defines a characterization of the system/proof/provability/calculus of formulae.
Also defines soundness and completeness.
## Main Definitio... | @@ -430,6 +430,12 @@
variable {𝓢 : S} {𝓜 : M} [Complete 𝓢 𝓜]
+lemma meaningful_of_consistent : Entailment.Consistent 𝓢 → Semantics.Meaningful 𝓜 := by
+ contrapose
+ suffices (∀ (φ : F), 𝓜 ⊧ φ) → Entailment.Inconsistent 𝓢 by
+ simpa [Semantics.not_meaningful_iff, Entailment.not_consistent_iff_inconsist... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_2a3f669a01ec_18 | 95e2b563c00733bc | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Logic/Entailment.lean | Entailment | 18 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 20 | 2 | [
{
"theorem_name": "consistent_of_model",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": "\n consistent_of_meaningful (𝓜 := 𝓜) inferInstance",
"n_chars": 56,
"n_subproofs": 0,
"n_tactics": 1,
"cycl... | [
{
"name": "consistent_of_meaningful",
"text": "lemma consistent_of_meaningful : Semantics.Meaningful 𝓜 → Entailment.Consistent 𝓢 :=\n fun H ↦ ⟨fun h ↦ by rcases H with ⟨φ, hf⟩; exact hf (Sound.sound (h φ))⟩\n\n",
"fan_in": 2,
"n_lines": 4,
"n_chars": 183,
"n_subproofs": 0,
"n_tactics"... | [
{
"name": "consistent_iff_satisfiable",
"fan_in": 0,
"n_deps_direct": 2,
"n_deps_transitive": 6,
"n_lines": 4,
"n_chars": 199,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
"... | 4 | module
public import Foundation.Logic.Semantics
public import Foundation.Vorspiel.AdjunctiveSet
/-!
# Basic definitions and properties of proof system related notions
This file defines a characterization of the system/proof/provability/calculus of formulae.
Also defines soundness and completeness.
## Main Definitio... | module
public import Foundation.Logic.Semantics
public import Foundation.Vorspiel.AdjunctiveSet
/-!
# Basic definitions and properties of proof system related notions
This file defines a characterization of the system/proof/provability/calculus of formulae.
Also defines soundness and completeness.
## Main Definitio... | @@ -414,13 +414,17 @@
variable {𝓢 𝓣 : S} {𝓜 𝓝 : M} [Sound 𝓢 𝓜] [Sound 𝓣 𝓝]
+lemma consistent_of_meaningful : Semantics.Meaningful 𝓜 → Entailment.Consistent 𝓢 :=
+ fun H ↦ ⟨fun h ↦ by rcases H with ⟨φ, hf⟩; exact hf (Sound.sound (h φ))⟩
+
end
section
variable {𝓢 : S} {T : Set F} [Sound 𝓢 (Semant... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_2a3f669a01ec_19 | 01953efa78425a31 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Logic/Entailment.lean | Entailment | 19 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 20 | 5 | [
{
"theorem_name": "not_weakerThan_iff",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by simp [weakerThan_iff, Unprovable];",
"n_chars": 38,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
... | [
{
"name": "weakerThan_iff",
"text": "lemma weakerThan_iff : 𝓢 ⪯ 𝓣 ↔ (∀ {φ}, 𝓢 ⊢ φ → 𝓣 ⊢ φ) :=\n ⟨fun h _ hf ↦ h.subset hf, fun h ↦ ⟨fun _ hf ↦ h hf⟩⟩\n\n",
"fan_in": 5,
"n_lines": 4,
"n_chars": 155,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n... | [
{
"name": "weakerthan_of_models",
"fan_in": 0,
"n_deps_direct": 3,
"n_deps_transitive": 3,
"n_lines": 6,
"n_chars": 318,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
"max_ne... | 3 | module
public import Foundation.Logic.Semantics
public import Foundation.Vorspiel.AdjunctiveSet
/-!
# Basic definitions and properties of proof system related notions
This file defines a characterization of the system/proof/provability/calculus of formulae.
Also defines soundness and completeness.
## Main Definitio... | module
public import Foundation.Logic.Semantics
public import Foundation.Vorspiel.AdjunctiveSet
/-!
# Basic definitions and properties of proof system related notions
This file defines a characterization of the system/proof/provability/calculus of formulae.
Also defines soundness and completeness.
## Main Definitio... | @@ -116,6 +116,9 @@
instance : Trans (α := S) (β := T) (γ := U) (· ⪯ ·) (· ⪯ ·) (· ⪯ ·) where
trans := WeakerThan.trans
+lemma weakerThan_iff : 𝓢 ⪯ 𝓣 ↔ (∀ {φ}, 𝓢 ⊢ φ → 𝓣 ⊢ φ) :=
+ ⟨fun h _ hf ↦ h.subset hf, fun h ↦ ⟨fun _ hf ↦ h hf⟩⟩
+
lemma swt_of_swt_of_wt : 𝓢 ⪱ 𝓣 → 𝓣 ⪯ 𝓤 → 𝓢 ⪱ 𝓤 := by
rintro ⟨h₁... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_2a3f669a01ec_20 | 305d633b16e82956 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Logic/Entailment.lean | Entailment | 20 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 20 | 1 | [
{
"theorem_name": "consistent",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n contrapose; simpa using inconsistent",
"n_chars": 42,
"n_subproofs": 0,
"n_tactics": 3,
"cyclomatic": 1,
"n_... | [
{
"name": "inconsistent",
"text": "lemma inconsistent {𝓢 : S} : Inconsistent 𝓢 → Inconsistent (pullback 𝓢 f) := by\n simp_all [Inconsistent]\n\n",
"fan_in": 1,
"n_lines": 4,
"n_chars": 118,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites"... | [
{
"name": "consistent",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 4,
"n_chars": 125,
"n_subproofs": 0,
"n_tactics": 3,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"max_nesting": 2
... | 1 | module
public import Foundation.Logic.Semantics
public import Foundation.Vorspiel.AdjunctiveSet
/-!
# Basic definitions and properties of proof system related notions
This file defines a characterization of the system/proof/provability/calculus of formulae.
Also defines soundness and completeness.
## Main Definitio... | module
public import Foundation.Logic.Semantics
public import Foundation.Vorspiel.AdjunctiveSet
/-!
# Basic definitions and properties of proof system related notions
This file defines a characterization of the system/proof/provability/calculus of formulae.
Also defines soundness and completeness.
## Main Definitio... | @@ -483,8 +483,12 @@
lemma weakerThan (𝓢 𝓣 : S) (h : 𝓢 ⪯ 𝓣) : pullback 𝓢 f ⪯ pullback 𝓣 f := by
simp_all [Entailment.weakerThan_iff]
-lemma consistent {𝓢 : S} : Consistent (pullback 𝓢 f) → Consistent 𝓢 := sorry
+lemma inconsistent {𝓢 : S} : Inconsistent 𝓢 → Inconsistent (pullback 𝓢 f) := by
+ simp_al... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_e8992ae7c059_0 | 066d5df1ade43f16 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Vorspiel/Computability.lean | Computability | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 3 | 4 | [
{
"theorem_name": "projection",
"depth": 1,
"n_commands": 0,
"n_lines": 52,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n obtain ⟨cf, rfl⟩ := Code.exists_code.1 hf\n let F : ℕ → ℕ → Option ℕ := fun m n ↦ Nat.rec .none (fun x ih ↦ ih.casesOn (cf.eva... | [
{
"name": "comp",
"text": "protected lemma comp {f : α → β} (hf : Computable f) {p : β → Prop} (hp : REPred p) : REPred fun x ↦ p (f x) := by\n rcases REPred.iff.mp hp with ⟨p, pp, rfl⟩\n exact REPred.iff'.mpr ⟨_, pp.comp hf, by intro x; simp⟩\n\n",
"fan_in": 4,
"n_lines": 5,
"n_chars": 235,
... | [
{
"name": "projection",
"fan_in": 2,
"n_deps_direct": 2,
"n_deps_transitive": 4,
"n_lines": 55,
"n_chars": 3061,
"n_subproofs": 11,
"n_tactics": 55,
"cyclomatic": 15,
"n_automation": 11,
"n_rewrites": 0,
"n_structural": 29,
"automation_only": false,
"max_nesti... | 3 | module
public import Mathlib.Computability.Halting
public import Foundation.Vorspiel.Part
@[expose]
public section
namespace Nat.Partrec
open Part _root_.Primrec
lemma projection {f : ℕ →. ℕ} (hf : Nat.Partrec f) (unif : ∀ {m n₁ n₂ a₁ a₂ : ℕ}, a₁ ∈ f (m.pair n₁) → a₂ ∈ f (m.pair n₂) → a₁ = a₂) :
∃ g : ℕ →. ℕ, ... | module
public import Mathlib.Computability.Halting
public import Foundation.Vorspiel.Part
@[expose]
public section
namespace Nat.Partrec
open Part _root_.Primrec
lemma projection {f : ℕ →. ℕ} (hf : Nat.Partrec f) (unif : ∀ {m n₁ n₂ a₁ a₂ : ℕ}, a₁ ∈ f (m.pair n₁) → a₂ ∈ f (m.pair n₂) → a₁ = a₂) :
∃ g : ℕ →. ℕ, ... | @@ -14,7 +14,10 @@
∃ g : ℕ →. ℕ, Nat.Partrec g ∧ (∀ a m, a ∈ g m ↔ ∃ z, a ∈ f (m.pair z)) := by
obtain ⟨cf, rfl⟩ := Code.exists_code.1 hf
let F : ℕ → ℕ → Option ℕ := fun m n ↦ Nat.rec .none (fun x ih ↦ ih.casesOn (cf.evaln n (m.pair x)) .some) n
- have : Primrec₂ F := sorry
+ have : Primrec₂ F := .to₂ <| 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_e8992ae7c059_1 | 98fd9791c21ef1ca | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Vorspiel/Computability.lean | Computability | 1 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 3 | 4 | [
{
"theorem_name": "projection",
"depth": 1,
"n_commands": 0,
"n_lines": 52,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n obtain ⟨cf, rfl⟩ := Code.exists_code.1 hf\n let F : ℕ → ℕ → Option ℕ := fun m n ↦ Nat.rec .none (fun x ih ↦ ih.casesOn (cf.eva... | [
{
"name": "comp",
"text": "protected lemma comp {f : α → β} (hf : Computable f) {p : β → Prop} (hp : REPred p) : REPred fun x ↦ p (f x) := by\n rcases REPred.iff.mp hp with ⟨p, pp, rfl⟩\n exact REPred.iff'.mpr ⟨_, pp.comp hf, by intro x; simp⟩\n\n",
"fan_in": 4,
"n_lines": 5,
"n_chars": 235,
... | [
{
"name": "and",
"fan_in": 2,
"n_deps_direct": 3,
"n_deps_transitive": 4,
"n_lines": 8,
"n_chars": 439,
"n_subproofs": 1,
"n_tactics": 7,
"cyclomatic": 3,
"n_automation": 3,
"n_rewrites": 0,
"n_structural": 4,
"automation_only": false,
"max_nesting": 2
}
] | 3 | module
public import Mathlib.Computability.Halting
public import Foundation.Vorspiel.Part
@[expose]
public section
namespace Nat.Partrec
open Part _root_.Primrec
end Nat.Partrec
namespace Partrec
variable {α β γ : Type*} [Primcodable α] [Primcodable β] [Primcodable γ]
end Partrec
namespace REPred
variable {α ... | module
public import Mathlib.Computability.Halting
public import Foundation.Vorspiel.Part
@[expose]
public section
namespace Nat.Partrec
open Part _root_.Primrec
end Nat.Partrec
namespace Partrec
variable {α β γ : Type*} [Primcodable α] [Primcodable β] [Primcodable γ]
end Partrec
namespace REPred
variable {α ... | @@ -38,7 +38,7 @@
rcases REPred.iff.mp hp with ⟨f, hf, rfl⟩
rcases REPred.iff.mp hq with ⟨g, hg, rfl⟩
let h : α →. Unit := fun x ↦ (f x).bind fun _ ↦ (g x).map fun _ ↦ ()
- have : Partrec h := sorry
+ have : Partrec h := Partrec.bind hf <| Partrec.to₂ <| Partrec.map (hg.comp Computable.fst) (Computable.const... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_e8992ae7c059_2 | 76bd3447084b53b8 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Vorspiel/Computability.lean | Computability | 2 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 3 | 4 | [
{
"theorem_name": "and",
"depth": 1,
"n_commands": 0,
"n_lines": 6,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n rcases REPred.iff.mp hp with ⟨f, hf, rfl⟩\n rcases REPred.iff.mp hq with ⟨g, hg, rfl⟩\n let h : α →. Unit := fun x ↦ (f x).bind fun _... | [
{
"name": "iff",
"text": "lemma iff : REPred p ↔ ∃ f : α →. Unit, Partrec f ∧ p = fun x ↦ (f x).Dom :=\n ⟨fun h ↦ ⟨_, h, by ext x; simp [Part.assert]⟩, by rintro ⟨f, hf, rfl⟩; exact hf.dom_re⟩\n\n",
"fan_in": 4,
"n_lines": 4,
"n_chars": 193,
"n_subproofs": 0,
"n_tactics": 3,
"cyclom... | [
{
"name": "or",
"fan_in": 2,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 7,
"n_chars": 314,
"n_subproofs": 0,
"n_tactics": 7,
"cyclomatic": 4,
"n_automation": 4,
"n_rewrites": 0,
"n_structural": 6,
"automation_only": false,
"max_nesting": 2
}
] | 1 | module
public import Mathlib.Computability.Halting
public import Foundation.Vorspiel.Part
@[expose]
public section
namespace Nat.Partrec
open Part _root_.Primrec
end Nat.Partrec
namespace Partrec
variable {α β γ : Type*} [Primcodable α] [Primcodable β] [Primcodable γ]
end Partrec
namespace REPred
variable {α ... | module
public import Mathlib.Computability.Halting
public import Foundation.Vorspiel.Part
@[expose]
public section
namespace Nat.Partrec
open Part _root_.Primrec
end Nat.Partrec
namespace Partrec
variable {α β γ : Type*} [Primcodable α] [Primcodable β] [Primcodable γ]
end Partrec
namespace REPred
variable {α ... | @@ -28,14 +28,28 @@
· simpa [h] using Partrec.some.dom_re
· simpa [h] using (Partrec.none (α := α) (σ := α)).dom_re
+lemma iff : REPred p ↔ ∃ f : α →. Unit, Partrec f ∧ p = fun x ↦ (f x).Dom :=
+ ⟨fun h ↦ ⟨_, h, by ext x; simp [Part.assert]⟩, by rintro ⟨f, hf, rfl⟩; exact hf.dom_re⟩
+
lemma iff' : REPred 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_e8992ae7c059_3 | 8a4b02e229732ece | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Vorspiel/Computability.lean | Computability | 3 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 3 | 1 | [
{
"theorem_name": "comp",
"depth": 1,
"n_commands": 0,
"n_lines": 3,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n rcases REPred.iff.mp hp with ⟨p, pp, rfl⟩\n exact REPred.iff'.mpr ⟨_, pp.comp hf, by intro x; simp⟩",
"n_chars": 113,
"n_subp... | [
{
"name": "iff'",
"text": "lemma iff' : REPred p ↔ ∃ f : α →. Unit, Partrec f ∧ ∀ x, p x ↔ (f x).Dom :=\n ⟨fun h ↦ ⟨_, h, by intro x; simp [Part.assert]⟩, by rintro ⟨f, hf, H⟩; exact hf.dom_re.of_eq (by simp [H])⟩\n\n",
"fan_in": 1,
"n_lines": 4,
"n_chars": 215,
"n_subproofs": 0,
"n_tac... | [
{
"name": "comp",
"fan_in": 4,
"n_deps_direct": 2,
"n_deps_transitive": 2,
"n_lines": 5,
"n_chars": 235,
"n_subproofs": 0,
"n_tactics": 4,
"cyclomatic": 2,
"n_automation": 2,
"n_rewrites": 0,
"n_structural": 3,
"automation_only": false,
"max_nesting": 2
}
] | 2 | module
public import Mathlib.Computability.Halting
public import Foundation.Vorspiel.Part
@[expose]
public section
namespace Nat.Partrec
open Part _root_.Primrec
end Nat.Partrec
namespace Partrec
variable {α β γ : Type*} [Primcodable α] [Primcodable β] [Primcodable γ]
end Partrec
namespace REPred
variable {α ... | module
public import Mathlib.Computability.Halting
public import Foundation.Vorspiel.Part
@[expose]
public section
namespace Nat.Partrec
open Part _root_.Primrec
end Nat.Partrec
namespace Partrec
variable {α β γ : Type*} [Primcodable α] [Primcodable β] [Primcodable γ]
end Partrec
namespace REPred
variable {α ... | @@ -31,8 +31,13 @@
lemma iff : REPred p ↔ ∃ f : α →. Unit, Partrec f ∧ p = fun x ↦ (f x).Dom :=
⟨fun h ↦ ⟨_, h, by ext x; simp [Part.assert]⟩, by rintro ⟨f, hf, rfl⟩; exact hf.dom_re⟩
-protected lemma comp {f : α → β} (hf : Computable f) {p : β → Prop} (hp : REPred p) : REPred fun x ↦ p (f x) := sorry
+lemma iff'... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_d1d343cc0a9a_0 | 8ac36bedc82245d0 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Propositional/ConsistentTableau.lean | ConsistentTableau | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 40 | 1 | [
{
"theorem_name": "iff_not_consistent_insert₁",
"depth": 1,
"n_commands": 0,
"n_lines": 4,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n constructor;\n . contrapose!; apply iff_consistent_insert₁.mpr;\n . contrapose!; apply iff_consistent_insert₁.... | [
{
"name": "iff_consistent_insert₁",
"text": "lemma iff_consistent_insert₁\n : Tableau.Consistent 𝓢 ((insert φ T), U) ↔ ∀ {Γ Δ : Finset (Formula α)}, (↑Γ ⊆ T) → (↑Δ ⊆ U) → 𝓢 ⊬ φ ⋏ Γ.conj 🡒 Δ.disj := by\n constructor;\n . intro h Γ Δ hΓ hΔ;\n by_contra hC;\n apply h (Γ := insert φ Γ) (Δ := Δ) ?_ h... | [
{
"name": "iff_not_consistent_insert₁",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 6,
"n_chars": 319,
"n_subproofs": 0,
"n_tactics": 9,
"cyclomatic": 2,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 3,
"automation_only": false,
"... | 1 | module
public import Foundation.Propositional.Formula.Basic
public import Foundation.Propositional.Entailment.Cl.Basic
public import Foundation.Vorspiel.Set.Basic
@[expose] public section
namespace LO.Propositional
open Entailment FiniteContext
open Formula
variable {α : Type*}
variable {S} [Entailment S (Formula ... | module
public import Foundation.Propositional.Formula.Basic
public import Foundation.Propositional.Entailment.Cl.Basic
public import Foundation.Vorspiel.Set.Basic
@[expose] public section
namespace LO.Propositional
open Entailment FiniteContext
open Formula
variable {α : Type*}
variable {S} [Entailment S (Formula ... | @@ -50,10 +50,27 @@
variable [DecidableEq α]
+lemma iff_consistent_insert₁
+ : Tableau.Consistent 𝓢 ((insert φ T), U) ↔ ∀ {Γ Δ : Finset (Formula α)}, (↑Γ ⊆ T) → (↑Δ ⊆ U) → 𝓢 ⊬ φ ⋏ Γ.conj 🡒 Δ.disj := by
+ constructor;
+ . intro h Γ Δ hΓ hΔ;
+ by_contra hC;
+ apply h (Γ := insert φ Γ) (Δ := Δ) ?_ hΔ;
+ ... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_d1d343cc0a9a_1 | 8f8ecdc0f382a24c | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Propositional/ConsistentTableau.lean | ConsistentTableau | 1 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 40 | 1 | [
{
"theorem_name": "iff_not_consistent_insert₂",
"depth": 1,
"n_commands": 0,
"n_lines": 4,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n constructor;\n . contrapose!; apply iff_consistent_insert₂.mpr;\n . contrapose!; apply iff_consistent_insert₂.... | [
{
"name": "iff_consistent_insert₂",
"text": "lemma iff_consistent_insert₂ : Tableau.Consistent 𝓢 (T, (insert φ U)) ↔ ∀ {Γ Δ : Finset (Formula α)}, (↑Γ ⊆ T) → (↑Δ ⊆ U) → 𝓢 ⊬ Γ.conj 🡒 φ ⋎ Δ.disj := by\n constructor;\n . intro h Γ Δ hΓ hΔ;\n by_contra hC;\n apply h (Γ := Γ) (Δ := insert φ Δ) hΓ ?_;\... | [
{
"name": "iff_not_consistent_insert₂",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 6,
"n_chars": 319,
"n_subproofs": 0,
"n_tactics": 9,
"cyclomatic": 2,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 3,
"automation_only": false,
"... | 1 | module
public import Foundation.Propositional.Formula.Basic
public import Foundation.Propositional.Entailment.Cl.Basic
public import Foundation.Vorspiel.Set.Basic
@[expose] public section
namespace LO.Propositional
open Entailment FiniteContext
open Formula
variable {α : Type*}
variable {S} [Entailment S (Formula ... | module
public import Foundation.Propositional.Formula.Basic
public import Foundation.Propositional.Entailment.Cl.Basic
public import Foundation.Vorspiel.Set.Basic
@[expose] public section
namespace LO.Propositional
open Entailment FiniteContext
open Formula
variable {α : Type*}
variable {S} [Entailment S (Formula ... | @@ -72,10 +72,28 @@
. contrapose!; apply iff_consistent_insert₁.mpr;
. contrapose!; apply iff_consistent_insert₁.mp;
+lemma iff_consistent_insert₂ : Tableau.Consistent 𝓢 (T, (insert φ U)) ↔ ∀ {Γ Δ : Finset (Formula α)}, (↑Γ ⊆ T) → (↑Δ ⊆ U) → 𝓢 ⊬ Γ.conj 🡒 φ ⋎ Δ.disj := by
+ constructor;
+ . intro h Γ Δ hΓ 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_d1d343cc0a9a_2 | dd8b042c9a8eb441 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Propositional/ConsistentTableau.lean | ConsistentTableau | 2 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 40 | 1 | [
{
"theorem_name": "consistent_either",
"depth": 1,
"n_commands": 0,
"n_lines": 13,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n by_contra hC;\n push Not at hC;\n have ⟨hC₁, hC₂⟩ := hC;\n\n obtain ⟨Γ₁, Δ₁, hΓ₁, hΔ₁, h₁⟩ := iff_not_consistent_inse... | [
{
"name": "iff_not_consistent_insert₂",
"text": "lemma iff_not_consistent_insert₂ : ¬Tableau.Consistent 𝓢 (T, (insert φ U)) ↔ ∃ Γ Δ : Finset (Formula α), (↑Γ ⊆ T) ∧ (↑Δ ⊆ U) ∧ 𝓢 ⊢ Γ.conj 🡒 φ ⋎ Δ.disj := by\n constructor;\n . contrapose!; apply iff_consistent_insert₂.mpr;\n . contrapose!; apply iff_con... | [
{
"name": "consistent_either",
"fan_in": 1,
"n_deps_direct": 2,
"n_deps_transitive": 4,
"n_lines": 17,
"n_chars": 920,
"n_subproofs": 3,
"n_tactics": 23,
"cyclomatic": 1,
"n_automation": 6,
"n_rewrites": 0,
"n_structural": 4,
"automation_only": false,
"max_nes... | 4 | module
public import Foundation.Propositional.Formula.Basic
public import Foundation.Propositional.Entailment.Cl.Basic
public import Foundation.Vorspiel.Set.Basic
@[expose] public section
namespace LO.Propositional
open Entailment FiniteContext
open Formula
variable {α : Type*}
variable {S} [Entailment S (Formula ... | module
public import Foundation.Propositional.Formula.Basic
public import Foundation.Propositional.Entailment.Cl.Basic
public import Foundation.Vorspiel.Set.Basic
@[expose] public section
namespace LO.Propositional
open Entailment FiniteContext
open Formula
variable {α : Type*}
variable {S} [Entailment S (Formula ... | @@ -90,12 +90,29 @@
simp only [Finset.mem_toList, Finset.mem_insert, Finset.mem_erase, ne_eq];
tauto;
+lemma iff_not_consistent_insert₂ : ¬Tableau.Consistent 𝓢 (T, (insert φ U)) ↔ ∃ Γ Δ : Finset (Formula α), (↑Γ ⊆ T) ∧ (↑Δ ⊆ U) ∧ 𝓢 ⊢ Γ.conj 🡒 φ ⋎ Δ.disj := by
+ constructor;
+ . contrapose!; apply 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_d1d343cc0a9a_3 | 391bbf1dd8bf8b8a | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Propositional/ConsistentTableau.lean | ConsistentTableau | 3 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 40 | 1 | [
{
"theorem_name": "not_mem₁_iff_mem₂",
"depth": 1,
"n_commands": 0,
"n_lines": 4,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n constructor;\n . apply mem₂_of_not_mem₁ hMat;\n . apply Set.disjoint_right.mp $ disjoint_of_consistent hCon;",
"n_c... | [
{
"name": "mem₂_of_not_mem₁",
"text": "lemma mem₂_of_not_mem₁ (hMat : Saturated t) : φ ∉ t.1 → φ ∈ t.2 := by\n intro h;\n cases (hMat φ) with\n | inl h' => exact absurd h' h;\n | inr _ => assumption;\n\n",
"fan_in": 1,
"n_lines": 7,
"n_chars": 175,
"n_subproofs": 0,
"n_tactics": 8,
... | [
{
"name": "not_mem₁_iff_mem₂",
"fan_in": 2,
"n_deps_direct": 2,
"n_deps_transitive": 2,
"n_lines": 6,
"n_chars": 226,
"n_subproofs": 0,
"n_tactics": 7,
"cyclomatic": 2,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 3,
"automation_only": false,
"max_nesti... | 2 | module
public import Foundation.Propositional.Formula.Basic
public import Foundation.Propositional.Entailment.Cl.Basic
public import Foundation.Vorspiel.Set.Basic
@[expose] public section
namespace LO.Propositional
open Entailment FiniteContext
open Formula
variable {α : Type*}
variable {S} [Entailment S (Formula ... | module
public import Foundation.Propositional.Formula.Basic
public import Foundation.Propositional.Entailment.Cl.Basic
public import Foundation.Vorspiel.Set.Basic
@[expose] public section
namespace LO.Propositional
open Entailment FiniteContext
open Formula
variable {α : Type*}
variable {S} [Entailment S (Formula ... | @@ -126,6 +126,12 @@
variable [Entailment.Int 𝓢]
variable {t : Tableau α}
+lemma mem₂_of_not_mem₁ (hMat : Saturated t) : φ ∉ t.1 → φ ∈ t.2 := by
+ intro h;
+ cases (hMat φ) with
+ | inl h' => exact absurd h' h;
+ | inr _ => assumption;
+
lemma mem₁_of_not_mem₂ (hMat : Saturated t) : φ ∉ t.2 → φ ∈ t.1 := 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_d1d343cc0a9a_4 | 94d5a560290db117 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Propositional/ConsistentTableau.lean | ConsistentTableau | 4 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 40 | 1 | [
{
"theorem_name": "not_mem₂_iff_mem₁",
"depth": 1,
"n_commands": 0,
"n_lines": 4,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n constructor;\n . apply mem₁_of_not_mem₂ hMat;\n . apply Set.disjoint_left.mp $ disjoint_of_consistent hCon;",
"n_ch... | [
{
"name": "mem₁_of_not_mem₂",
"text": "lemma mem₁_of_not_mem₂ (hMat : Saturated t) : φ ∉ t.2 → φ ∈ t.1 := by\n intro h;\n cases (hMat φ) with\n | inl _ => assumption;\n | inr h' => exact absurd h' h;\n\n",
"fan_in": 1,
"n_lines": 7,
"n_chars": 175,
"n_subproofs": 0,
"n_tactics": 8,
... | [
{
"name": "not_mem₂_iff_mem₁",
"fan_in": 2,
"n_deps_direct": 2,
"n_deps_transitive": 2,
"n_lines": 6,
"n_chars": 225,
"n_subproofs": 0,
"n_tactics": 7,
"cyclomatic": 2,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 3,
"automation_only": false,
"max_nesti... | 2 | module
public import Foundation.Propositional.Formula.Basic
public import Foundation.Propositional.Entailment.Cl.Basic
public import Foundation.Vorspiel.Set.Basic
@[expose] public section
namespace LO.Propositional
open Entailment FiniteContext
open Formula
variable {α : Type*}
variable {S} [Entailment S (Formula ... | module
public import Foundation.Propositional.Formula.Basic
public import Foundation.Propositional.Entailment.Cl.Basic
public import Foundation.Vorspiel.Set.Basic
@[expose] public section
namespace LO.Propositional
open Entailment FiniteContext
open Formula
variable {α : Type*}
variable {S} [Entailment S (Formula ... | @@ -132,6 +132,12 @@
| inl h' => exact absurd h' h;
| inr _ => assumption;
+lemma mem₁_of_not_mem₂ (hMat : Saturated t) : φ ∉ t.2 → φ ∈ t.1 := by
+ intro h;
+ cases (hMat φ) with
+ | inl _ => assumption;
+ | inr h' => exact absurd h' h;
+
lemma not_mem₁_iff_mem₂ (hCon : t.Consistent 𝓢) (hMat : Saturated 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_d1d343cc0a9a_5 | 7a2cf3fc9ea9646d | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Propositional/ConsistentTableau.lean | ConsistentTableau | 5 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 40 | 2 | [
{
"theorem_name": "saturated_duality",
"depth": 1,
"n_commands": 0,
"n_lines": 18,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n constructor;\n . intro h;\n apply Set.eq_of_subset_of_subset;\n . intro φ hp;\n apply not_mem₁_iff_mem₂ ct₂ ... | [
{
"name": "not_mem₁_iff_mem₂",
"text": "lemma not_mem₁_iff_mem₂ (hCon : t.Consistent 𝓢) (hMat : Saturated t) : φ ∉ t.1 ↔ φ ∈ t.2 := by\n constructor;\n . apply mem₂_of_not_mem₁ hMat;\n . apply Set.disjoint_right.mp $ disjoint_of_consistent hCon;\n\n",
"fan_in": 2,
"n_lines": 6,
"n_chars": 22... | [
{
"name": "saturated_duality",
"fan_in": 3,
"n_deps_direct": 2,
"n_deps_transitive": 5,
"n_lines": 24,
"n_chars": 885,
"n_subproofs": 0,
"n_tactics": 39,
"cyclomatic": 2,
"n_automation": 0,
"n_rewrites": 4,
"n_structural": 17,
"automation_only": false,
"max_ne... | 5 | module
public import Foundation.Propositional.Formula.Basic
public import Foundation.Propositional.Entailment.Cl.Basic
public import Foundation.Vorspiel.Set.Basic
@[expose] public section
namespace LO.Propositional
open Entailment FiniteContext
open Formula
variable {α : Type*}
variable {S} [Entailment S (Formula ... | module
public import Foundation.Propositional.Formula.Basic
public import Foundation.Propositional.Entailment.Cl.Basic
public import Foundation.Vorspiel.Set.Basic
@[expose] public section
namespace LO.Propositional
open Entailment FiniteContext
open Formula
variable {α : Type*}
variable {S} [Entailment S (Formula ... | @@ -138,6 +138,11 @@
| inl _ => assumption;
| inr h' => exact absurd h' h;
+lemma not_mem₁_iff_mem₂ (hCon : t.Consistent 𝓢) (hMat : Saturated t) : φ ∉ t.1 ↔ φ ∈ t.2 := by
+ constructor;
+ . apply mem₂_of_not_mem₁ hMat;
+ . apply Set.disjoint_right.mp $ disjoint_of_consistent hCon;
+
lemma not_mem₂_iff_mem₁... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_d1d343cc0a9a_6 | c8da18564a0c9312 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Propositional/ConsistentTableau.lean | ConsistentTableau | 6 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 40 | 1 | [
{
"theorem_name": "next_parametericConsistent",
"depth": 1,
"n_commands": 0,
"n_lines": 7,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n dsimp [lindenbaum_next];\n split;\n . simpa;\n . rcases (consistent_either consistent φ) with (h | h);\n .... | [
{
"name": "consistent_either",
"text": "lemma consistent_either (hCon : t.Consistent 𝓢) (φ : Formula α) : Tableau.Consistent 𝓢 ((insert φ t.1), t.2) ∨ Tableau.Consistent 𝓢 (t.1, (insert φ t.2)) := by\n by_contra hC;\n push Not at hC;\n have ⟨hC₁, hC₂⟩ := hC;\n\n obtain ⟨Γ₁, Δ₁, hΓ₁, hΔ₁, h₁⟩ := iff_n... | [
{
"name": "lindenbaum_next_indexed_parametricConsistent_succ",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 7,
"n_lines": 9,
"n_chars": 284,
"n_subproofs": 0,
"n_tactics": 13,
"cyclomatic": 2,
"n_automation": 2,
"n_rewrites": 0,
"n_structural": 2,
"automa... | 6 | module
public import Foundation.Propositional.Formula.Basic
public import Foundation.Propositional.Entailment.Cl.Basic
public import Foundation.Vorspiel.Set.Basic
@[expose] public section
namespace LO.Propositional
open Entailment FiniteContext
open Formula
variable {α : Type*}
variable {S} [Entailment S (Formula ... | module
public import Foundation.Propositional.Formula.Basic
public import Foundation.Propositional.Entailment.Cl.Basic
public import Foundation.Vorspiel.Set.Basic
@[expose] public section
namespace LO.Propositional
open Entailment FiniteContext
open Formula
variable {α : Type*}
variable {S} [Entailment S (Formula ... | @@ -99,6 +99,22 @@
variable {t : Tableau α}
+lemma consistent_either (hCon : t.Consistent 𝓢) (φ : Formula α) : Tableau.Consistent 𝓢 ((insert φ t.1), t.2) ∨ Tableau.Consistent 𝓢 (t.1, (insert φ t.2)) := by
+ by_contra hC;
+ push Not at hC;
+ have ⟨hC₁, hC₂⟩ := hC;
+
+ obtain ⟨Γ₁, Δ₁, hΓ₁, hΔ₁, h₁⟩ := iff_not... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_d1d343cc0a9a_7 | 6504b6e2d9e0755e | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Propositional/ConsistentTableau.lean | ConsistentTableau | 7 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 40 | 1 | [
{
"theorem_name": "lindenbaum_next_indexed_parametricConsistent_succ",
"depth": 1,
"n_commands": 0,
"n_lines": 7,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n dsimp [lindenbaum_next_indexed];\n split;\n . intro h;\n apply next_parametericConsi... | [
{
"name": "next_parametericConsistent",
"text": "lemma next_parametericConsistent [Entailment.Int 𝓢] (consistent : t.Consistent 𝓢) (φ : Formula α) : (t.lindenbaum_next 𝓢 φ).Consistent 𝓢 := by\n dsimp [lindenbaum_next];\n split;\n . simpa;\n . rcases (consistent_either consistent φ) with (h | h);\n ... | [
{
"name": "lindenbaum_next_indexed_parametricConsistent",
"fan_in": 1,
"n_deps_direct": 4,
"n_deps_transitive": 8,
"n_lines": 6,
"n_chars": 270,
"n_subproofs": 0,
"n_tactics": 7,
"cyclomatic": 2,
"n_automation": 2,
"n_rewrites": 0,
"n_structural": 2,
"automation_o... | 7 | module
public import Foundation.Propositional.Formula.Basic
public import Foundation.Propositional.Entailment.Cl.Basic
public import Foundation.Vorspiel.Set.Basic
@[expose] public section
namespace LO.Propositional
open Entailment FiniteContext
open Formula
variable {α : Type*}
variable {S} [Entailment S (Formula ... | module
public import Foundation.Propositional.Formula.Basic
public import Foundation.Propositional.Entailment.Cl.Basic
public import Foundation.Vorspiel.Set.Basic
@[expose] public section
namespace LO.Propositional
open Entailment FiniteContext
open Formula
variable {α : Type*}
variable {S} [Entailment S (Formula ... | @@ -185,12 +185,22 @@
variable {𝓢}
+lemma next_parametericConsistent [Entailment.Int 𝓢] (consistent : t.Consistent 𝓢) (φ : Formula α) : (t.lindenbaum_next 𝓢 φ).Consistent 𝓢 := by
+ dsimp [lindenbaum_next];
+ split;
+ . simpa;
+ . rcases (consistent_either consistent φ) with (h | h);
+ . contradiction;
... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_d1d343cc0a9a_8 | 4599f928306704ba | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Propositional/ConsistentTableau.lean | ConsistentTableau | 8 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 40 | 2 | [
{
"theorem_name": "exists_list_lindenbaum_index₁",
"depth": 1,
"n_commands": 0,
"n_lines": 15,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n induction Γ with\n | nil => simp;\n | cons φ Γ ih =>\n simp_all only [List.coe_toFinset, List.toFinset_... | [
{
"name": "lindenbaum_next_indexed_subset₁_of_lt",
"text": "lemma lindenbaum_next_indexed_subset₁_of_lt (h : m ≤ n) : t[m].1 ⊆ t[n].1 := by\n induction h with\n | refl => simp;\n | step h ih =>\n simp [lindenbaum_next_indexed, lindenbaum_next];\n split;\n . split <;> tauto;\n . tauto;\n\n",
... | [
{
"name": "exists_list_lindenbaum_index₁",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 17,
"n_chars": 657,
"n_subproofs": 2,
"n_tactics": 27,
"cyclomatic": 3,
"n_automation": 5,
"n_rewrites": 0,
"n_structural": 8,
"automation_only": false,
... | 1 | module
public import Foundation.Propositional.Formula.Basic
public import Foundation.Propositional.Entailment.Cl.Basic
public import Foundation.Vorspiel.Set.Basic
@[expose] public section
namespace LO.Propositional
open Entailment FiniteContext
open Formula
variable {α : Type*}
variable {S} [Entailment S (Formula ... | module
public import Foundation.Propositional.Formula.Basic
public import Foundation.Propositional.Entailment.Cl.Basic
public import Foundation.Vorspiel.Set.Basic
@[expose] public section
namespace LO.Propositional
open Entailment FiniteContext
open Formula
variable {α : Type*}
variable {S} [Entailment S (Formula ... | @@ -216,6 +216,15 @@
variable {m n : ℕ}
+lemma lindenbaum_next_indexed_subset₁_of_lt (h : m ≤ n) : t[m].1 ⊆ t[n].1 := by
+ induction h with
+ | refl => simp;
+ | step h ih =>
+ simp [lindenbaum_next_indexed, lindenbaum_next];
+ split;
+ . split <;> tauto;
+ . tauto;
+
lemma lindenbaum_next_indexed_... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_d1d343cc0a9a_9 | dd9460cdff7783f7 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Propositional/ConsistentTableau.lean | ConsistentTableau | 9 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 40 | 1 | [
{
"theorem_name": "exists_finset_lindenbaum_index₁",
"depth": 1,
"n_commands": 0,
"n_lines": 6,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n obtain ⟨m, hΓ⟩ := exists_list_lindenbaum_index₁ (Γ := Γ.toList) (t := t) (by simpa);\n use m;\n intro φ h... | [
{
"name": "exists_list_lindenbaum_index₁",
"text": "lemma exists_list_lindenbaum_index₁ {Γ : List _} (hΓ : ↑Γ.toFinset ⊆ ⋃ i, t[i].1): ∃ m, ∀ φ ∈ Γ, φ ∈ t[m].1 := by\n induction Γ with\n | nil => simp;\n | cons φ Γ ih =>\n simp_all only [List.coe_toFinset, List.toFinset_cons, Finset.coe_insert, List.m... | [
{
"name": "exists_finset_lindenbaum_index₁",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 2,
"n_lines": 8,
"n_chars": 286,
"n_subproofs": 1,
"n_tactics": 11,
"cyclomatic": 1,
"n_automation": 2,
"n_rewrites": 0,
"n_structural": 4,
"automation_only": false,... | 2 | module
public import Foundation.Propositional.Formula.Basic
public import Foundation.Propositional.Entailment.Cl.Basic
public import Foundation.Vorspiel.Set.Basic
@[expose] public section
namespace LO.Propositional
open Entailment FiniteContext
open Formula
variable {α : Type*}
variable {S} [Entailment S (Formula ... | module
public import Foundation.Propositional.Formula.Basic
public import Foundation.Propositional.Entailment.Cl.Basic
public import Foundation.Vorspiel.Set.Basic
@[expose] public section
namespace LO.Propositional
open Entailment FiniteContext
open Formula
variable {α : Type*}
variable {S} [Entailment S (Formula ... | @@ -234,8 +234,29 @@
. split <;> tauto;
. tauto;
-lemma exists_finset_lindenbaum_index₁ {Γ : Finset _} (hΓ : (SetLike.coe Γ) ⊆ ⋃ i, t[i].1): ∃ m, ∀ φ ∈ Γ, φ ∈ t[m].1 := sorry
+lemma exists_list_lindenbaum_index₁ {Γ : List _} (hΓ : ↑Γ.toFinset ⊆ ⋃ i, t[i].1): ∃ m, ∀ φ ∈ Γ, φ ∈ t[m].1 := by
+ induction Γ wit... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_d1d343cc0a9a_10 | 45b618ad56c53f1e | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Propositional/ConsistentTableau.lean | ConsistentTableau | 10 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 40 | 2 | [
{
"theorem_name": "exists_list_lindenbaum_index₂",
"depth": 1,
"n_commands": 0,
"n_lines": 15,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n induction Δ with\n | nil => simp;\n | cons φ Δ ih =>\n simp_all only [List.coe_toFinset, List.toFinset_... | [
{
"name": "lindenbaum_next_indexed_subset₂_of_lt",
"text": "lemma lindenbaum_next_indexed_subset₂_of_lt (h : m ≤ n) : t[m].2 ⊆ t[n].2 := by\n induction h with\n | refl => simp;\n | step h ih =>\n simp [lindenbaum_next_indexed, lindenbaum_next];\n split;\n . split <;> tauto;\n . tauto;\n\n",
... | [
{
"name": "exists_list_lindenbaum_index₂",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 17,
"n_chars": 657,
"n_subproofs": 2,
"n_tactics": 26,
"cyclomatic": 3,
"n_automation": 5,
"n_rewrites": 0,
"n_structural": 8,
"automation_only": false,
... | 1 | module
public import Foundation.Propositional.Formula.Basic
public import Foundation.Propositional.Entailment.Cl.Basic
public import Foundation.Vorspiel.Set.Basic
@[expose] public section
namespace LO.Propositional
open Entailment FiniteContext
open Formula
variable {α : Type*}
variable {S} [Entailment S (Formula ... | module
public import Foundation.Propositional.Formula.Basic
public import Foundation.Propositional.Entailment.Cl.Basic
public import Foundation.Vorspiel.Set.Basic
@[expose] public section
namespace LO.Propositional
open Entailment FiniteContext
open Formula
variable {α : Type*}
variable {S} [Entailment S (Formula ... | @@ -225,6 +225,15 @@
. split <;> tauto;
. tauto;
+lemma lindenbaum_next_indexed_subset₂_of_lt (h : m ≤ n) : t[m].2 ⊆ t[n].2 := by
+ induction h with
+ | refl => simp;
+ | step h ih =>
+ simp [lindenbaum_next_indexed, lindenbaum_next];
+ split;
+ . split <;> tauto;
+ . tauto;
+
lemma exists_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_d1d343cc0a9a_11 | a63fda96173296ac | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Propositional/ConsistentTableau.lean | ConsistentTableau | 11 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 40 | 1 | [
{
"theorem_name": "exists_finset_lindenbaum_index₂",
"depth": 1,
"n_commands": 0,
"n_lines": 6,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n obtain ⟨m, hΔ⟩ := exists_list_lindenbaum_index₂ (Δ := Δ.toList) (𝓢 := 𝓢) (t := t) (by simpa);\n use m;\n... | [
{
"name": "exists_list_lindenbaum_index₂",
"text": "lemma exists_list_lindenbaum_index₂ {Δ : List _} (hΔ : ↑Δ.toFinset ⊆ ⋃ i, t[i].2) : ∃ n, ∀ φ ∈ Δ, φ ∈ t[n].2 := by\n induction Δ with\n | nil => simp;\n | cons φ Δ ih =>\n simp_all only [List.coe_toFinset, List.toFinset_cons, Finset.coe_insert, List.... | [
{
"name": "exists_finset_lindenbaum_index₂",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 2,
"n_lines": 8,
"n_chars": 302,
"n_subproofs": 1,
"n_tactics": 11,
"cyclomatic": 1,
"n_automation": 2,
"n_rewrites": 0,
"n_structural": 4,
"automation_only": false,... | 2 | module
public import Foundation.Propositional.Formula.Basic
public import Foundation.Propositional.Entailment.Cl.Basic
public import Foundation.Vorspiel.Set.Basic
@[expose] public section
namespace LO.Propositional
open Entailment FiniteContext
open Formula
variable {α : Type*}
variable {S} [Entailment S (Formula ... | module
public import Foundation.Propositional.Formula.Basic
public import Foundation.Propositional.Entailment.Cl.Basic
public import Foundation.Vorspiel.Set.Basic
@[expose] public section
namespace LO.Propositional
open Entailment FiniteContext
open Formula
variable {α : Type*}
variable {S} [Entailment S (Formula ... | @@ -257,8 +257,29 @@
apply hΓ;
simpa;
-lemma exists_finset_lindenbaum_index₂ {Δ : Finset _} (hΓ : (SetLike.coe Δ) ⊆ ⋃ i, t[i].2) : ∃ n, ∀ φ ∈ Δ, φ ∈ t[n].2 := sorry
+lemma exists_list_lindenbaum_index₂ {Δ : List _} (hΔ : ↑Δ.toFinset ⊆ ⋃ i, t[i].2) : ∃ n, ∀ φ ∈ Δ, φ ∈ t[n].2 := by
+ induction Δ with
+ | nil =>... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_d1d343cc0a9a_12 | 936b3fbed3e87b42 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Propositional/ConsistentTableau.lean | ConsistentTableau | 12 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 40 | 1 | [
{
"theorem_name": "exists_parametricConsistent_saturated_tableau",
"depth": 1,
"n_commands": 0,
"n_lines": 20,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n use t∞;\n refine ⟨?subset, ?consistent, ?saturated⟩;\n case subset => constructor <;> appl... | [
{
"name": "exists_finset_lindenbaum_index₂",
"text": "lemma exists_finset_lindenbaum_index₂ {Δ : Finset _} (hΓ : (SetLike.coe Δ) ⊆ ⋃ i, t[i].2) : ∃ n, ∀ φ ∈ Δ, φ ∈ t[n].2 := by\n obtain ⟨m, hΔ⟩ := exists_list_lindenbaum_index₂ (Δ := Δ.toList) (𝓢 := 𝓢) (t := t) (by simpa);\n use m;\n intro φ hφ;\n appl... | [
{
"name": "exists_parametricConsistent_saturated_tableau",
"fan_in": 0,
"n_deps_direct": 8,
"n_deps_transitive": 17,
"n_lines": 22,
"n_chars": 1211,
"n_subproofs": 2,
"n_tactics": 42,
"cyclomatic": 5,
"n_automation": 4,
"n_rewrites": 1,
"n_structural": 19,
"automa... | 16 | module
public import Foundation.Propositional.Formula.Basic
public import Foundation.Propositional.Entailment.Cl.Basic
public import Foundation.Vorspiel.Set.Basic
@[expose] public section
namespace LO.Propositional
open Entailment FiniteContext
open Formula
variable {α : Type*}
variable {S} [Entailment S (Formula ... | module
public import Foundation.Propositional.Formula.Basic
public import Foundation.Propositional.Entailment.Cl.Basic
public import Foundation.Vorspiel.Set.Basic
@[expose] public section
namespace LO.Propositional
open Entailment FiniteContext
open Formula
variable {α : Type*}
variable {S} [Entailment S (Formula ... | @@ -273,8 +273,34 @@
. intro ψ hq;
exact lindenbaum_next_indexed_subset₂_of_lt (by simp) $ hn ψ hq;
-lemma exists_parametricConsistent_saturated_tableau [Entailment.Int 𝓢] (hCon : t.Consistent 𝓢) : ∃ u, t ⊆ u ∧ (Tableau.Consistent 𝓢 u) ∧ (Saturated u) := sorry
+lemma exists_finset_lindenbaum_index₂ {Δ ... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_d1d343cc0a9a_13 | 9c803849672b0358 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Propositional/ConsistentTableau.lean | ConsistentTableau | 13 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 40 | 3 | [
{
"theorem_name": "not_mem₁_iff_mem₂",
"depth": 1,
"n_commands": 0,
"n_lines": 4,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n constructor;\n . apply mem₂_of_not_mem₁ hMat;\n . apply Set.disjoint_right.mp $ disjoint_of_consistent hCon;",
"n_c... | [
{
"name": "disjoint_of_consistent",
"text": "lemma disjoint_of_consistent (hCon : t.Consistent 𝓢) : Disjoint t.1 t.2 := by\n by_contra h;\n obtain ⟨T, hp₁, hp₂, hp⟩ := by simpa [Disjoint] using h;\n obtain ⟨φ, hp⟩ := Set.nonempty_def.mp $ Set.nonempty_iff_ne_empty.mpr hp;\n have : 𝓢 ⊬ (Finset.conj {φ}... | [
{
"name": "disjoint",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 3,
"n_chars": 83,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
"max_nesting": 1
}... | 1 | module
public import Foundation.Propositional.Formula.Basic
public import Foundation.Propositional.Entailment.Cl.Basic
public import Foundation.Vorspiel.Set.Basic
@[expose] public section
namespace LO.Propositional
open Entailment FiniteContext
open Formula
variable {α : Type*}
variable {S} [Entailment S (Formula ... | module
public import Foundation.Propositional.Formula.Basic
public import Foundation.Propositional.Entailment.Cl.Basic
public import Foundation.Vorspiel.Set.Basic
@[expose] public section
namespace LO.Propositional
open Entailment FiniteContext
open Formula
variable {α : Type*}
variable {S} [Entailment S (Formula ... | @@ -37,6 +37,17 @@
variable [Entailment.Int 𝓢]
+lemma disjoint_of_consistent (hCon : t.Consistent 𝓢) : Disjoint t.1 t.2 := by
+ by_contra h;
+ obtain ⟨T, hp₁, hp₂, hp⟩ := by simpa [Disjoint] using h;
+ obtain ⟨φ, hp⟩ := Set.nonempty_def.mp $ Set.nonempty_iff_ne_empty.mpr hp;
+ have : 𝓢 ⊬ (Finset.conj {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_d1d343cc0a9a_14 | f17aa18d0baf1fca | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Propositional/ConsistentTableau.lean | ConsistentTableau | 14 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 40 | 5 | [
{
"theorem_name": "exists_parametricConsistent_saturated_tableau",
"depth": 1,
"n_commands": 0,
"n_lines": 20,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n use t∞;\n refine ⟨?subset, ?consistent, ?saturated⟩;\n case subset => constructor <;> appl... | [
{
"name": "saturated",
"text": "lemma saturated (t : SaturatedConsistentTableau 𝓢) : Saturated t.1 := t.2.1\n\n",
"fan_in": 5,
"n_lines": 3,
"n_chars": 80,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"auto... | [
{
"name": "iff_not_mem₁_mem₂",
"fan_in": 8,
"n_deps_direct": 3,
"n_deps_transitive": 5,
"n_lines": 3,
"n_chars": 119,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
"max_nesti... | 4 | module
public import Foundation.Propositional.Formula.Basic
public import Foundation.Propositional.Entailment.Cl.Basic
public import Foundation.Vorspiel.Set.Basic
@[expose] public section
namespace LO.Propositional
open Entailment FiniteContext
open Formula
variable {α : Type*}
variable {S} [Entailment S (Formula ... | module
public import Foundation.Propositional.Formula.Basic
public import Foundation.Propositional.Entailment.Cl.Basic
public import Foundation.Vorspiel.Set.Basic
@[expose] public section
namespace LO.Propositional
open Entailment FiniteContext
open Formula
variable {α : Type*}
variable {S} [Entailment S (Formula ... | @@ -280,7 +280,26 @@
apply hΔ;
simpa;
-lemma exists_parametricConsistent_saturated_tableau [Entailment.Int 𝓢] (hCon : t.Consistent 𝓢) : ∃ u, t ⊆ u ∧ (Tableau.Consistent 𝓢 u) ∧ (Saturated u) := sorry
+lemma exists_parametricConsistent_saturated_tableau [Entailment.Int 𝓢] (hCon : t.Consistent 𝓢) : ∃ u, t ⊆ u... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_d1d343cc0a9a_15 | 2aabbe37ad2eebd4 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Propositional/ConsistentTableau.lean | ConsistentTableau | 15 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 40 | 5 | [
{
"theorem_name": "exists_parametricConsistent_saturated_tableau",
"depth": 1,
"n_commands": 0,
"n_lines": 20,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n use t∞;\n refine ⟨?subset, ?consistent, ?saturated⟩;\n case subset => constructor <;> appl... | [
{
"name": "saturated",
"text": "lemma saturated (t : SaturatedConsistentTableau 𝓢) : Saturated t.1 := t.2.1\n\n",
"fan_in": 5,
"n_lines": 3,
"n_chars": 80,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"auto... | [
{
"name": "saturated_duality",
"fan_in": 2,
"n_deps_direct": 6,
"n_deps_transitive": 8,
"n_lines": 3,
"n_chars": 161,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
"max_nesti... | 6 | module
public import Foundation.Propositional.Formula.Basic
public import Foundation.Propositional.Entailment.Cl.Basic
public import Foundation.Vorspiel.Set.Basic
@[expose] public section
namespace LO.Propositional
open Entailment FiniteContext
open Formula
variable {α : Type*}
variable {S} [Entailment S (Formula ... | module
public import Foundation.Propositional.Formula.Basic
public import Foundation.Propositional.Entailment.Cl.Basic
public import Foundation.Vorspiel.Set.Basic
@[expose] public section
namespace LO.Propositional
open Entailment FiniteContext
open Formula
variable {α : Type*}
variable {S} [Entailment S (Formula ... | @@ -303,7 +303,26 @@
apply hΔ;
simpa;
-lemma exists_parametricConsistent_saturated_tableau [Entailment.Int 𝓢] (hCon : t.Consistent 𝓢) : ∃ u, t ⊆ u ∧ (Tableau.Consistent 𝓢 u) ∧ (Saturated u) := sorry
+lemma exists_parametricConsistent_saturated_tableau [Entailment.Int 𝓢] (hCon : t.Consistent 𝓢) : ∃ u, t ⊆ u... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_d1d343cc0a9a_16 | e898bded95403d54 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Propositional/ConsistentTableau.lean | ConsistentTableau | 16 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 40 | 5 | [
{
"theorem_name": "exists_parametricConsistent_saturated_tableau",
"depth": 1,
"n_commands": 0,
"n_lines": 20,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n use t∞;\n refine ⟨?subset, ?consistent, ?saturated⟩;\n case subset => constructor <;> appl... | [
{
"name": "saturated",
"text": "lemma saturated (t : SaturatedConsistentTableau 𝓢) : Saturated t.1 := t.2.1\n\n",
"fan_in": 5,
"n_lines": 3,
"n_chars": 80,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"auto... | [
{
"name": "equality_of₁",
"fan_in": 1,
"n_deps_direct": 4,
"n_deps_transitive": 9,
"n_lines": 7,
"n_chars": 295,
"n_subproofs": 1,
"n_tactics": 8,
"cyclomatic": 1,
"n_automation": 2,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": false,
"max_nesting": ... | 6 | module
public import Foundation.Propositional.Formula.Basic
public import Foundation.Propositional.Entailment.Cl.Basic
public import Foundation.Vorspiel.Set.Basic
@[expose] public section
namespace LO.Propositional
open Entailment FiniteContext
open Formula
variable {α : Type*}
variable {S} [Entailment S (Formula ... | module
public import Foundation.Propositional.Formula.Basic
public import Foundation.Propositional.Entailment.Cl.Basic
public import Foundation.Vorspiel.Set.Basic
@[expose] public section
namespace LO.Propositional
open Entailment FiniteContext
open Formula
variable {α : Type*}
variable {S} [Entailment S (Formula ... | @@ -303,7 +303,26 @@
apply hΔ;
simpa;
-lemma exists_parametricConsistent_saturated_tableau [Entailment.Int 𝓢] (hCon : t.Consistent 𝓢) : ∃ u, t ⊆ u ∧ (Tableau.Consistent 𝓢 u) ∧ (Saturated u) := sorry
+lemma exists_parametricConsistent_saturated_tableau [Entailment.Int 𝓢] (hCon : t.Consistent 𝓢) : ∃ u, t ⊆ u... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_d1d343cc0a9a_17 | 01e99ec63bcd9a2d | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Propositional/ConsistentTableau.lean | ConsistentTableau | 17 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 40 | 5 | [
{
"theorem_name": "exists_parametricConsistent_saturated_tableau",
"depth": 1,
"n_commands": 0,
"n_lines": 20,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n use t∞;\n refine ⟨?subset, ?consistent, ?saturated⟩;\n case subset => constructor <;> appl... | [
{
"name": "saturated",
"text": "lemma saturated (t : SaturatedConsistentTableau 𝓢) : Saturated t.1 := t.2.1\n\n",
"fan_in": 5,
"n_lines": 3,
"n_chars": 80,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"auto... | [
{
"name": "equality_of₂",
"fan_in": 0,
"n_deps_direct": 2,
"n_deps_transitive": 10,
"n_lines": 3,
"n_chars": 112,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
"max_nesting":... | 7 | module
public import Foundation.Propositional.Formula.Basic
public import Foundation.Propositional.Entailment.Cl.Basic
public import Foundation.Vorspiel.Set.Basic
@[expose] public section
namespace LO.Propositional
open Entailment FiniteContext
open Formula
variable {α : Type*}
variable {S} [Entailment S (Formula ... | module
public import Foundation.Propositional.Formula.Basic
public import Foundation.Propositional.Entailment.Cl.Basic
public import Foundation.Vorspiel.Set.Basic
@[expose] public section
namespace LO.Propositional
open Entailment FiniteContext
open Formula
variable {α : Type*}
variable {S} [Entailment S (Formula ... | @@ -303,7 +303,26 @@
apply hΔ;
simpa;
-lemma exists_parametricConsistent_saturated_tableau [Entailment.Int 𝓢] (hCon : t.Consistent 𝓢) : ∃ u, t ⊆ u ∧ (Tableau.Consistent 𝓢 u) ∧ (Saturated u) := sorry
+lemma exists_parametricConsistent_saturated_tableau [Entailment.Int 𝓢] (hCon : t.Consistent 𝓢) : ∃ u, t ⊆ u... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_d1d343cc0a9a_18 | fbc8de60aeb7cbdc | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Propositional/ConsistentTableau.lean | ConsistentTableau | 18 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 40 | 14 | [
{
"theorem_name": "iff_provable_include₁",
"depth": 1,
"n_commands": 0,
"n_lines": 30,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n constructor;\n . intro h t hT;\n by_contra hφ;\n replace hφ := iff_not_mem₁_mem₂.mp hφ;\n obtain ⟨Γ, hΓ₁, ... | [
{
"name": "iff_not_mem₂_mem₁",
"text": "lemma iff_not_mem₂_mem₁ : φ ∉ t.1.2 ↔ φ ∈ t.1.1 := Tableau.not_mem₂_iff_mem₁ t.consistent t.saturated\n\n",
"fan_in": 14,
"n_lines": 3,
"n_chars": 119,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": ... | [
{
"name": "iff_provable_include₁",
"fan_in": 2,
"n_deps_direct": 4,
"n_deps_transitive": 10,
"n_lines": 32,
"n_chars": 1137,
"n_subproofs": 2,
"n_tactics": 58,
"cyclomatic": 3,
"n_automation": 9,
"n_rewrites": 2,
"n_structural": 15,
"automation_only": false,
"... | 8 | module
public import Foundation.Propositional.Formula.Basic
public import Foundation.Propositional.Entailment.Cl.Basic
public import Foundation.Vorspiel.Set.Basic
@[expose] public section
namespace LO.Propositional
open Entailment FiniteContext
open Formula
variable {α : Type*}
variable {S} [Entailment S (Formula ... | module
public import Foundation.Propositional.Formula.Basic
public import Foundation.Propositional.Entailment.Cl.Basic
public import Foundation.Vorspiel.Set.Basic
@[expose] public section
namespace LO.Propositional
open Entailment FiniteContext
open Formula
variable {α : Type*}
variable {S} [Entailment S (Formula ... | @@ -333,6 +333,8 @@
lemma iff_not_mem₁_mem₂ : φ ∉ t.1.1 ↔ φ ∈ t.1.2 := Tableau.not_mem₁_iff_mem₂ t.consistent t.saturated
+lemma iff_not_mem₂_mem₁ : φ ∉ t.1.2 ↔ φ ∈ t.1.1 := Tableau.not_mem₂_iff_mem₁ t.consistent t.saturated
+
section
variable [DecidableEq α] [Encodable α]
@@ -351,13 +353,32 @@
apply hΓ... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_d1d343cc0a9a_19 | dc5bee24ca009bba | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Propositional/ConsistentTableau.lean | ConsistentTableau | 19 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 40 | 14 | [
{
"theorem_name": "iff_provable_include₁",
"depth": 1,
"n_commands": 0,
"n_lines": 30,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n constructor;\n . intro h t hT;\n by_contra hφ;\n replace hφ := iff_not_mem₁_mem₂.mp hφ;\n obtain ⟨Γ, hΓ₁, ... | [
{
"name": "iff_not_mem₂_mem₁",
"text": "lemma iff_not_mem₂_mem₁ : φ ∉ t.1.2 ↔ φ ∈ t.1.1 := Tableau.not_mem₂_iff_mem₁ t.consistent t.saturated\n\n",
"fan_in": 14,
"n_lines": 3,
"n_chars": 119,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": ... | [
{
"name": "iff_provable_include₁_finset",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 11,
"n_lines": 3,
"n_chars": 194,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
... | 9 | module
public import Foundation.Propositional.Formula.Basic
public import Foundation.Propositional.Entailment.Cl.Basic
public import Foundation.Vorspiel.Set.Basic
@[expose] public section
namespace LO.Propositional
open Entailment FiniteContext
open Formula
variable {α : Type*}
variable {S} [Entailment S (Formula ... | module
public import Foundation.Propositional.Formula.Basic
public import Foundation.Propositional.Entailment.Cl.Basic
public import Foundation.Vorspiel.Set.Basic
@[expose] public section
namespace LO.Propositional
open Entailment FiniteContext
open Formula
variable {α : Type*}
variable {S} [Entailment S (Formula ... | @@ -333,6 +333,8 @@
lemma iff_not_mem₁_mem₂ : φ ∉ t.1.1 ↔ φ ∈ t.1.2 := Tableau.not_mem₁_iff_mem₂ t.consistent t.saturated
+lemma iff_not_mem₂_mem₁ : φ ∉ t.1.2 ↔ φ ∈ t.1.1 := Tableau.not_mem₂_iff_mem₁ t.consistent t.saturated
+
section
variable [DecidableEq α] [Encodable α]
@@ -351,7 +353,22 @@
apply hΓ₁... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_d1d343cc0a9a_20 | bb42c6f08b84b11e | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Propositional/ConsistentTableau.lean | ConsistentTableau | 20 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 40 | 14 | [
{
"theorem_name": "iff_provable_include₁",
"depth": 1,
"n_commands": 0,
"n_lines": 30,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n constructor;\n . intro h t hT;\n by_contra hφ;\n replace hφ := iff_not_mem₁_mem₂.mp hφ;\n obtain ⟨Γ, hΓ₁, ... | [
{
"name": "iff_not_mem₂_mem₁",
"text": "lemma iff_not_mem₂_mem₁ : φ ∉ t.1.2 ↔ φ ∈ t.1.1 := Tableau.not_mem₂_iff_mem₁ t.consistent t.saturated\n\n",
"fan_in": 14,
"n_lines": 3,
"n_chars": 119,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": ... | [
{
"name": "iff_provable_mem₁",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 11,
"n_lines": 10,
"n_chars": 309,
"n_subproofs": 0,
"n_tactics": 15,
"cyclomatic": 2,
"n_automation": 2,
"n_rewrites": 0,
"n_structural": 6,
"automation_only": false,
"max_ne... | 9 | module
public import Foundation.Propositional.Formula.Basic
public import Foundation.Propositional.Entailment.Cl.Basic
public import Foundation.Vorspiel.Set.Basic
@[expose] public section
namespace LO.Propositional
open Entailment FiniteContext
open Formula
variable {α : Type*}
variable {S} [Entailment S (Formula ... | module
public import Foundation.Propositional.Formula.Basic
public import Foundation.Propositional.Entailment.Cl.Basic
public import Foundation.Vorspiel.Set.Basic
@[expose] public section
namespace LO.Propositional
open Entailment FiniteContext
open Formula
variable {α : Type*}
variable {S} [Entailment S (Formula ... | @@ -333,6 +333,8 @@
lemma iff_not_mem₁_mem₂ : φ ∉ t.1.1 ↔ φ ∈ t.1.2 := Tableau.not_mem₁_iff_mem₂ t.consistent t.saturated
+lemma iff_not_mem₂_mem₁ : φ ∉ t.1.2 ↔ φ ∈ t.1.1 := Tableau.not_mem₂_iff_mem₁ t.consistent t.saturated
+
section
variable [DecidableEq α] [Encodable α]
@@ -351,7 +353,22 @@
apply hΓ₁... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_d1d343cc0a9a_21 | 9abe889894c67275 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Propositional/ConsistentTableau.lean | ConsistentTableau | 21 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 40 | 14 | [
{
"theorem_name": "iff_provable_include₁",
"depth": 1,
"n_commands": 0,
"n_lines": 30,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n constructor;\n . intro h t hT;\n by_contra hφ;\n replace hφ := iff_not_mem₁_mem₂.mp hφ;\n obtain ⟨Γ, hΓ₁, ... | [
{
"name": "iff_not_mem₂_mem₁",
"text": "lemma iff_not_mem₂_mem₁ : φ ∉ t.1.2 ↔ φ ∈ t.1.1 := Tableau.not_mem₂_iff_mem₁ t.consistent t.saturated\n\n",
"fan_in": 14,
"n_lines": 3,
"n_chars": 119,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": ... | [
{
"name": "mdp_mem₁_provable",
"fan_in": 7,
"n_deps_direct": 2,
"n_deps_transitive": 6,
"n_lines": 7,
"n_chars": 246,
"n_subproofs": 0,
"n_tactics": 9,
"cyclomatic": 1,
"n_automation": 3,
"n_rewrites": 0,
"n_structural": 3,
"automation_only": false,
"max_nesti... | 5 | module
public import Foundation.Propositional.Formula.Basic
public import Foundation.Propositional.Entailment.Cl.Basic
public import Foundation.Vorspiel.Set.Basic
@[expose] public section
namespace LO.Propositional
open Entailment FiniteContext
open Formula
variable {α : Type*}
variable {S} [Entailment S (Formula ... | module
public import Foundation.Propositional.Formula.Basic
public import Foundation.Propositional.Entailment.Cl.Basic
public import Foundation.Vorspiel.Set.Basic
@[expose] public section
namespace LO.Propositional
open Entailment FiniteContext
open Formula
variable {α : Type*}
variable {S} [Entailment S (Formula ... | @@ -333,6 +333,8 @@
lemma iff_not_mem₁_mem₂ : φ ∉ t.1.1 ↔ φ ∈ t.1.2 := Tableau.not_mem₁_iff_mem₂ t.consistent t.saturated
+lemma iff_not_mem₂_mem₁ : φ ∉ t.1.2 ↔ φ ∈ t.1.1 := Tableau.not_mem₂_iff_mem₁ t.consistent t.saturated
+
section
variable [DecidableEq α] [Encodable α]
@@ -341,9 +343,17 @@
section Satur... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_d1d343cc0a9a_22 | 03368880f2abd6e9 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Propositional/ConsistentTableau.lean | ConsistentTableau | 22 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 40 | 14 | [
{
"theorem_name": "iff_provable_include₁",
"depth": 1,
"n_commands": 0,
"n_lines": 30,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n constructor;\n . intro h t hT;\n by_contra hφ;\n replace hφ := iff_not_mem₁_mem₂.mp hφ;\n obtain ⟨Γ, hΓ₁, ... | [
{
"name": "iff_not_mem₂_mem₁",
"text": "lemma iff_not_mem₂_mem₁ : φ ∉ t.1.2 ↔ φ ∈ t.1.1 := Tableau.not_mem₂_iff_mem₁ t.consistent t.saturated\n\n",
"fan_in": 14,
"n_lines": 3,
"n_chars": 119,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": ... | [
{
"name": "mdp_mem₂_provable",
"fan_in": 1,
"n_deps_direct": 2,
"n_deps_transitive": 7,
"n_lines": 6,
"n_chars": 227,
"n_subproofs": 1,
"n_tactics": 7,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
"max_nesti... | 6 | module
public import Foundation.Propositional.Formula.Basic
public import Foundation.Propositional.Entailment.Cl.Basic
public import Foundation.Vorspiel.Set.Basic
@[expose] public section
namespace LO.Propositional
open Entailment FiniteContext
open Formula
variable {α : Type*}
variable {S} [Entailment S (Formula ... | module
public import Foundation.Propositional.Formula.Basic
public import Foundation.Propositional.Entailment.Cl.Basic
public import Foundation.Vorspiel.Set.Basic
@[expose] public section
namespace LO.Propositional
open Entailment FiniteContext
open Formula
variable {α : Type*}
variable {S} [Entailment S (Formula ... | @@ -333,6 +333,8 @@
lemma iff_not_mem₁_mem₂ : φ ∉ t.1.1 ↔ φ ∈ t.1.2 := Tableau.not_mem₁_iff_mem₂ t.consistent t.saturated
+lemma iff_not_mem₂_mem₁ : φ ∉ t.1.2 ↔ φ ∈ t.1.1 := Tableau.not_mem₂_iff_mem₁ t.consistent t.saturated
+
section
variable [DecidableEq α] [Encodable α]
@@ -341,11 +343,22 @@
section Satu... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_d1d343cc0a9a_23 | 9a852a2dfec6da5d | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Propositional/ConsistentTableau.lean | ConsistentTableau | 23 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 40 | 14 | [
{
"theorem_name": "iff_provable_include₁",
"depth": 1,
"n_commands": 0,
"n_lines": 30,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n constructor;\n . intro h t hT;\n by_contra hφ;\n replace hφ := iff_not_mem₁_mem₂.mp hφ;\n obtain ⟨Γ, hΓ₁, ... | [
{
"name": "iff_not_mem₂_mem₁",
"text": "lemma iff_not_mem₂_mem₁ : φ ∉ t.1.2 ↔ φ ∈ t.1.1 := Tableau.not_mem₂_iff_mem₁ t.consistent t.saturated\n\n",
"fan_in": 14,
"n_lines": 3,
"n_chars": 119,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": ... | [
{
"name": "mem₁_verum",
"fan_in": 1,
"n_deps_direct": 2,
"n_deps_transitive": 6,
"n_lines": 7,
"n_chars": 192,
"n_subproofs": 0,
"n_tactics": 9,
"cyclomatic": 1,
"n_automation": 4,
"n_rewrites": 0,
"n_structural": 2,
"automation_only": false,
"max_nesting": 2
... | 5 | module
public import Foundation.Propositional.Formula.Basic
public import Foundation.Propositional.Entailment.Cl.Basic
public import Foundation.Vorspiel.Set.Basic
@[expose] public section
namespace LO.Propositional
open Entailment FiniteContext
open Formula
variable {α : Type*}
variable {S} [Entailment S (Formula ... | module
public import Foundation.Propositional.Formula.Basic
public import Foundation.Propositional.Entailment.Cl.Basic
public import Foundation.Vorspiel.Set.Basic
@[expose] public section
namespace LO.Propositional
open Entailment FiniteContext
open Formula
variable {α : Type*}
variable {S} [Entailment S (Formula ... | @@ -333,6 +333,8 @@
lemma iff_not_mem₁_mem₂ : φ ∉ t.1.1 ↔ φ ∈ t.1.2 := Tableau.not_mem₁_iff_mem₂ t.consistent t.saturated
+lemma iff_not_mem₂_mem₁ : φ ∉ t.1.2 ↔ φ ∈ t.1.1 := Tableau.not_mem₂_iff_mem₁ t.consistent t.saturated
+
section
variable [DecidableEq α] [Encodable α]
@@ -341,7 +343,11 @@
section Satur... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_d1d343cc0a9a_24 | 3741a8b3b10f9fe5 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Propositional/ConsistentTableau.lean | ConsistentTableau | 24 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 40 | 8 | [
{
"theorem_name": "iff_provable_include₁",
"depth": 1,
"n_commands": 0,
"n_lines": 30,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n constructor;\n . intro h t hT;\n by_contra hφ;\n replace hφ := iff_not_mem₁_mem₂.mp hφ;\n obtain ⟨Γ, hΓ₁, ... | [
{
"name": "iff_not_mem₁_mem₂",
"text": "lemma iff_not_mem₁_mem₂ : φ ∉ t.1.1 ↔ φ ∈ t.1.2 := Tableau.not_mem₁_iff_mem₂ t.consistent t.saturated\n\n",
"fan_in": 8,
"n_lines": 3,
"n_chars": 119,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0... | [
{
"name": "mem₂_falsum",
"fan_in": 0,
"n_deps_direct": 2,
"n_deps_transitive": 7,
"n_lines": 3,
"n_chars": 91,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
"max_nesting": 1
... | 5 | module
public import Foundation.Propositional.Formula.Basic
public import Foundation.Propositional.Entailment.Cl.Basic
public import Foundation.Vorspiel.Set.Basic
@[expose] public section
namespace LO.Propositional
open Entailment FiniteContext
open Formula
variable {α : Type*}
variable {S} [Entailment S (Formula ... | module
public import Foundation.Propositional.Formula.Basic
public import Foundation.Propositional.Entailment.Cl.Basic
public import Foundation.Vorspiel.Set.Basic
@[expose] public section
namespace LO.Propositional
open Entailment FiniteContext
open Formula
variable {α : Type*}
variable {S} [Entailment S (Formula ... | @@ -331,6 +331,8 @@
@[simp] lemma disjoint : Disjoint t.1.1 t.1.2 := t.1.disjoint_of_consistent t.2.2
+lemma iff_not_mem₁_mem₂ : φ ∉ t.1.1 ↔ φ ∈ t.1.2 := Tableau.not_mem₁_iff_mem₂ t.consistent t.saturated
+
lemma iff_not_mem₂_mem₁ : φ ∉ t.1.2 ↔ φ ∈ t.1.1 := Tableau.not_mem₂_iff_mem₁ t.consistent t.saturated
se... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_d1d343cc0a9a_25 | 31f71d7ca3e415be | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Propositional/ConsistentTableau.lean | ConsistentTableau | 25 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 40 | 14 | [
{
"theorem_name": "iff_provable_include₁",
"depth": 1,
"n_commands": 0,
"n_lines": 30,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n constructor;\n . intro h t hT;\n by_contra hφ;\n replace hφ := iff_not_mem₁_mem₂.mp hφ;\n obtain ⟨Γ, hΓ₁, ... | [
{
"name": "iff_not_mem₂_mem₁",
"text": "lemma iff_not_mem₂_mem₁ : φ ∉ t.1.2 ↔ φ ∈ t.1.1 := Tableau.not_mem₂_iff_mem₁ t.consistent t.saturated\n\n",
"fan_in": 14,
"n_lines": 3,
"n_chars": 119,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": ... | [
{
"name": "mem₁_of_provable",
"fan_in": 0,
"n_deps_direct": 2,
"n_deps_transitive": 8,
"n_lines": 5,
"n_chars": 134,
"n_subproofs": 0,
"n_tactics": 5,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 2,
"automation_only": false,
"max_nestin... | 6 | module
public import Foundation.Propositional.Formula.Basic
public import Foundation.Propositional.Entailment.Cl.Basic
public import Foundation.Vorspiel.Set.Basic
@[expose] public section
namespace LO.Propositional
open Entailment FiniteContext
open Formula
variable {α : Type*}
variable {S} [Entailment S (Formula ... | module
public import Foundation.Propositional.Formula.Basic
public import Foundation.Propositional.Entailment.Cl.Basic
public import Foundation.Vorspiel.Set.Basic
@[expose] public section
namespace LO.Propositional
open Entailment FiniteContext
open Formula
variable {α : Type*}
variable {S} [Entailment S (Formula ... | @@ -333,6 +333,8 @@
lemma iff_not_mem₁_mem₂ : φ ∉ t.1.1 ↔ φ ∈ t.1.2 := Tableau.not_mem₁_iff_mem₂ t.consistent t.saturated
+lemma iff_not_mem₂_mem₁ : φ ∉ t.1.2 ↔ φ ∈ t.1.1 := Tableau.not_mem₂_iff_mem₁ t.consistent t.saturated
+
section
variable [DecidableEq α] [Encodable α]
@@ -341,9 +343,17 @@
section Satur... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_d1d343cc0a9a_26 | 3dd8b95185890930 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Propositional/ConsistentTableau.lean | ConsistentTableau | 26 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 40 | 14 | [
{
"theorem_name": "iff_provable_include₁",
"depth": 1,
"n_commands": 0,
"n_lines": 30,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n constructor;\n . intro h t hT;\n by_contra hφ;\n replace hφ := iff_not_mem₁_mem₂.mp hφ;\n obtain ⟨Γ, hΓ₁, ... | [
{
"name": "iff_not_mem₂_mem₁",
"text": "lemma iff_not_mem₂_mem₁ : φ ∉ t.1.2 ↔ φ ∈ t.1.1 := Tableau.not_mem₂_iff_mem₁ t.consistent t.saturated\n\n",
"fan_in": 14,
"n_lines": 3,
"n_chars": 119,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": ... | [
{
"name": "mdp_mem₁",
"fan_in": 2,
"n_deps_direct": 2,
"n_deps_transitive": 6,
"n_lines": 10,
"n_chars": 399,
"n_subproofs": 0,
"n_tactics": 16,
"cyclomatic": 2,
"n_automation": 4,
"n_rewrites": 0,
"n_structural": 4,
"automation_only": false,
"max_nesting": 4
... | 5 | module
public import Foundation.Propositional.Formula.Basic
public import Foundation.Propositional.Entailment.Cl.Basic
public import Foundation.Vorspiel.Set.Basic
@[expose] public section
namespace LO.Propositional
open Entailment FiniteContext
open Formula
variable {α : Type*}
variable {S} [Entailment S (Formula ... | module
public import Foundation.Propositional.Formula.Basic
public import Foundation.Propositional.Entailment.Cl.Basic
public import Foundation.Vorspiel.Set.Basic
@[expose] public section
namespace LO.Propositional
open Entailment FiniteContext
open Formula
variable {α : Type*}
variable {S} [Entailment S (Formula ... | @@ -333,6 +333,8 @@
lemma iff_not_mem₁_mem₂ : φ ∉ t.1.1 ↔ φ ∈ t.1.2 := Tableau.not_mem₁_iff_mem₂ t.consistent t.saturated
+lemma iff_not_mem₂_mem₁ : φ ∉ t.1.2 ↔ φ ∈ t.1.1 := Tableau.not_mem₂_iff_mem₁ t.consistent t.saturated
+
section
variable [DecidableEq α] [Encodable α]
@@ -341,7 +343,11 @@
section Satur... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_d1d343cc0a9a_27 | c5e517cf8037f553 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Propositional/ConsistentTableau.lean | ConsistentTableau | 27 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 40 | 14 | [
{
"theorem_name": "iff_provable_include₁",
"depth": 1,
"n_commands": 0,
"n_lines": 30,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n constructor;\n . intro h t hT;\n by_contra hφ;\n replace hφ := iff_not_mem₁_mem₂.mp hφ;\n obtain ⟨Γ, hΓ₁, ... | [
{
"name": "iff_not_mem₂_mem₁",
"text": "lemma iff_not_mem₂_mem₁ : φ ∉ t.1.2 ↔ φ ∈ t.1.1 := Tableau.not_mem₂_iff_mem₁ t.consistent t.saturated\n\n",
"fan_in": 14,
"n_lines": 3,
"n_chars": 119,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": ... | [
{
"name": "iff_mem₁_and",
"fan_in": 2,
"n_deps_direct": 3,
"n_deps_transitive": 7,
"n_lines": 13,
"n_chars": 519,
"n_subproofs": 0,
"n_tactics": 23,
"cyclomatic": 6,
"n_automation": 5,
"n_rewrites": 0,
"n_structural": 9,
"automation_only": false,
"max_nesting"... | 6 | module
public import Foundation.Propositional.Formula.Basic
public import Foundation.Propositional.Entailment.Cl.Basic
public import Foundation.Vorspiel.Set.Basic
@[expose] public section
namespace LO.Propositional
open Entailment FiniteContext
open Formula
variable {α : Type*}
variable {S} [Entailment S (Formula ... | module
public import Foundation.Propositional.Formula.Basic
public import Foundation.Propositional.Entailment.Cl.Basic
public import Foundation.Vorspiel.Set.Basic
@[expose] public section
namespace LO.Propositional
open Entailment FiniteContext
open Formula
variable {α : Type*}
variable {S} [Entailment S (Formula ... | @@ -333,6 +333,8 @@
lemma iff_not_mem₁_mem₂ : φ ∉ t.1.1 ↔ φ ∈ t.1.2 := Tableau.not_mem₁_iff_mem₂ t.consistent t.saturated
+lemma iff_not_mem₂_mem₁ : φ ∉ t.1.2 ↔ φ ∈ t.1.1 := Tableau.not_mem₂_iff_mem₁ t.consistent t.saturated
+
section
variable [DecidableEq α] [Encodable α]
@@ -341,9 +343,17 @@
section Satur... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_d1d343cc0a9a_28 | 393d77c34727ecfc | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Propositional/ConsistentTableau.lean | ConsistentTableau | 28 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 40 | 7 | [
{
"theorem_name": "mdp_mem₂_provable",
"depth": 1,
"n_commands": 0,
"n_lines": 4,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n by_contra hq₂;\n have := iff_not_mem₂_mem₁.mpr $ mdp_mem₁_provable h $ iff_not_mem₂_mem₁.mp hq₂;\n contradiction;",
... | [
{
"name": "mdp_mem₁_provable",
"text": "lemma mdp_mem₁_provable (h : 𝓢 ⊢ φ 🡒 ψ) (hp₁ : φ ∈ t.1.1) : ψ ∈ t.1.1 := by\n apply iff_not_mem₂_mem₁.mp;\n by_contra hq₂;\n apply by simpa using t.consistent (Γ := {φ}) (Δ := {ψ}) (by simpa) (by simpa);\n exact h;\n\n",
"fan_in": 7,
"n_lines": 7,
"n... | [
{
"name": "iff_mem₁_conj₂",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 8,
"n_lines": 14,
"n_chars": 495,
"n_subproofs": 0,
"n_tactics": 21,
"cyclomatic": 4,
"n_automation": 4,
"n_rewrites": 0,
"n_structural": 7,
"automation_only": false,
"max_nestin... | 7 | module
public import Foundation.Propositional.Formula.Basic
public import Foundation.Propositional.Entailment.Cl.Basic
public import Foundation.Vorspiel.Set.Basic
@[expose] public section
namespace LO.Propositional
open Entailment FiniteContext
open Formula
variable {α : Type*}
variable {S} [Entailment S (Formula ... | module
public import Foundation.Propositional.Formula.Basic
public import Foundation.Propositional.Entailment.Cl.Basic
public import Foundation.Vorspiel.Set.Basic
@[expose] public section
namespace LO.Propositional
open Entailment FiniteContext
open Formula
variable {α : Type*}
variable {S} [Entailment S (Formula ... | @@ -343,6 +343,12 @@
section Saturated
+lemma mdp_mem₁_provable (h : 𝓢 ⊢ φ 🡒 ψ) (hp₁ : φ ∈ t.1.1) : ψ ∈ t.1.1 := by
+ apply iff_not_mem₂_mem₁.mp;
+ by_contra hq₂;
+ apply by simpa using t.consistent (Γ := {φ}) (Δ := {ψ}) (by simpa) (by simpa);
+ exact h;
+
@[simp] lemma mem₁_verum : ⊤ ∈ t.1.1 := by
apply... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_d1d343cc0a9a_29 | 77c14c78d7be01f8 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Propositional/ConsistentTableau.lean | ConsistentTableau | 29 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 40 | 7 | [
{
"theorem_name": "mdp_mem₂_provable",
"depth": 1,
"n_commands": 0,
"n_lines": 4,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n by_contra hq₂;\n have := iff_not_mem₂_mem₁.mpr $ mdp_mem₁_provable h $ iff_not_mem₂_mem₁.mp hq₂;\n contradiction;",
... | [
{
"name": "mdp_mem₁_provable",
"text": "lemma mdp_mem₁_provable (h : 𝓢 ⊢ φ 🡒 ψ) (hp₁ : φ ∈ t.1.1) : ψ ∈ t.1.1 := by\n apply iff_not_mem₂_mem₁.mp;\n by_contra hq₂;\n apply by simpa using t.consistent (Γ := {φ}) (Δ := {ψ}) (by simpa) (by simpa);\n exact h;\n\n",
"fan_in": 7,
"n_lines": 7,
"n... | [
{
"name": "iff_mem₁_fconj",
"fan_in": 0,
"n_deps_direct": 2,
"n_deps_transitive": 9,
"n_lines": 14,
"n_chars": 436,
"n_subproofs": 0,
"n_tactics": 23,
"cyclomatic": 2,
"n_automation": 4,
"n_rewrites": 0,
"n_structural": 8,
"automation_only": false,
"max_nestin... | 8 | module
public import Foundation.Propositional.Formula.Basic
public import Foundation.Propositional.Entailment.Cl.Basic
public import Foundation.Vorspiel.Set.Basic
@[expose] public section
namespace LO.Propositional
open Entailment FiniteContext
open Formula
variable {α : Type*}
variable {S} [Entailment S (Formula ... | module
public import Foundation.Propositional.Formula.Basic
public import Foundation.Propositional.Entailment.Cl.Basic
public import Foundation.Vorspiel.Set.Basic
@[expose] public section
namespace LO.Propositional
open Entailment FiniteContext
open Formula
variable {α : Type*}
variable {S} [Entailment S (Formula ... | @@ -343,6 +343,12 @@
section Saturated
+lemma mdp_mem₁_provable (h : 𝓢 ⊢ φ 🡒 ψ) (hp₁ : φ ∈ t.1.1) : ψ ∈ t.1.1 := by
+ apply iff_not_mem₂_mem₁.mp;
+ by_contra hq₂;
+ apply by simpa using t.consistent (Γ := {φ}) (Δ := {ψ}) (by simpa) (by simpa);
+ exact h;
+
@[simp] lemma mem₁_verum : ⊤ ∈ t.1.1 := by
apply... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_d1d343cc0a9a_30 | a98460a876b3b588 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Propositional/ConsistentTableau.lean | ConsistentTableau | 30 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 40 | 8 | [
{
"theorem_name": "iff_provable_include₁",
"depth": 1,
"n_commands": 0,
"n_lines": 30,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n constructor;\n . intro h t hT;\n by_contra hφ;\n replace hφ := iff_not_mem₁_mem₂.mp hφ;\n obtain ⟨Γ, hΓ₁, ... | [
{
"name": "iff_not_mem₁_mem₂",
"text": "lemma iff_not_mem₁_mem₂ : φ ∉ t.1.1 ↔ φ ∈ t.1.2 := Tableau.not_mem₁_iff_mem₂ t.consistent t.saturated\n\n",
"fan_in": 8,
"n_lines": 3,
"n_chars": 119,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0... | [
{
"name": "of_mem₁_or",
"fan_in": 1,
"n_deps_direct": 2,
"n_deps_transitive": 6,
"n_lines": 12,
"n_chars": 485,
"n_subproofs": 0,
"n_tactics": 21,
"cyclomatic": 3,
"n_automation": 3,
"n_rewrites": 0,
"n_structural": 7,
"automation_only": false,
"max_nesting": ... | 5 | module
public import Foundation.Propositional.Formula.Basic
public import Foundation.Propositional.Entailment.Cl.Basic
public import Foundation.Vorspiel.Set.Basic
@[expose] public section
namespace LO.Propositional
open Entailment FiniteContext
open Formula
variable {α : Type*}
variable {S} [Entailment S (Formula ... | module
public import Foundation.Propositional.Formula.Basic
public import Foundation.Propositional.Entailment.Cl.Basic
public import Foundation.Vorspiel.Set.Basic
@[expose] public section
namespace LO.Propositional
open Entailment FiniteContext
open Formula
variable {α : Type*}
variable {S} [Entailment S (Formula ... | @@ -331,6 +331,8 @@
@[simp] lemma disjoint : Disjoint t.1.1 t.1.2 := t.1.disjoint_of_consistent t.2.2
+lemma iff_not_mem₁_mem₂ : φ ∉ t.1.1 ↔ φ ∈ t.1.2 := Tableau.not_mem₁_iff_mem₂ t.consistent t.saturated
+
lemma iff_not_mem₂_mem₁ : φ ∉ t.1.2 ↔ φ ∈ t.1.1 := Tableau.not_mem₂_iff_mem₁ t.consistent t.saturated
se... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_d1d343cc0a9a_31 | cdf83a7c250a05b2 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Propositional/ConsistentTableau.lean | ConsistentTableau | 31 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 40 | 14 | [
{
"theorem_name": "iff_provable_include₁",
"depth": 1,
"n_commands": 0,
"n_lines": 30,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n constructor;\n . intro h t hT;\n by_contra hφ;\n replace hφ := iff_not_mem₁_mem₂.mp hφ;\n obtain ⟨Γ, hΓ₁, ... | [
{
"name": "iff_not_mem₂_mem₁",
"text": "lemma iff_not_mem₂_mem₁ : φ ∉ t.1.2 ↔ φ ∈ t.1.1 := Tableau.not_mem₂_iff_mem₁ t.consistent t.saturated\n\n",
"fan_in": 14,
"n_lines": 3,
"n_chars": 119,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": ... | [
{
"name": "of_mem₂_or",
"fan_in": 1,
"n_deps_direct": 2,
"n_deps_transitive": 7,
"n_lines": 10,
"n_chars": 415,
"n_subproofs": 1,
"n_tactics": 15,
"cyclomatic": 2,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 5,
"automation_only": false,
"max_nesting": ... | 6 | module
public import Foundation.Propositional.Formula.Basic
public import Foundation.Propositional.Entailment.Cl.Basic
public import Foundation.Vorspiel.Set.Basic
@[expose] public section
namespace LO.Propositional
open Entailment FiniteContext
open Formula
variable {α : Type*}
variable {S} [Entailment S (Formula ... | module
public import Foundation.Propositional.Formula.Basic
public import Foundation.Propositional.Entailment.Cl.Basic
public import Foundation.Vorspiel.Set.Basic
@[expose] public section
namespace LO.Propositional
open Entailment FiniteContext
open Formula
variable {α : Type*}
variable {S} [Entailment S (Formula ... | @@ -333,6 +333,8 @@
lemma iff_not_mem₁_mem₂ : φ ∉ t.1.1 ↔ φ ∈ t.1.2 := Tableau.not_mem₁_iff_mem₂ t.consistent t.saturated
+lemma iff_not_mem₂_mem₁ : φ ∉ t.1.2 ↔ φ ∈ t.1.1 := Tableau.not_mem₂_iff_mem₁ t.consistent t.saturated
+
section
variable [DecidableEq α] [Encodable α]
@@ -341,9 +343,17 @@
section Satur... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_d1d343cc0a9a_32 | 241a71a66cf9919c | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Propositional/ConsistentTableau.lean | ConsistentTableau | 32 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 40 | 14 | [
{
"theorem_name": "iff_provable_include₁",
"depth": 1,
"n_commands": 0,
"n_lines": 30,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n constructor;\n . intro h t hT;\n by_contra hφ;\n replace hφ := iff_not_mem₁_mem₂.mp hφ;\n obtain ⟨Γ, hΓ₁, ... | [
{
"name": "iff_not_mem₂_mem₁",
"text": "lemma iff_not_mem₂_mem₁ : φ ∉ t.1.2 ↔ φ ∈ t.1.1 := Tableau.not_mem₂_iff_mem₁ t.consistent t.saturated\n\n",
"fan_in": 14,
"n_lines": 3,
"n_chars": 119,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": ... | [
{
"name": "iff_mem₁_or",
"fan_in": 2,
"n_deps_direct": 2,
"n_deps_transitive": 11,
"n_lines": 9,
"n_chars": 271,
"n_subproofs": 0,
"n_tactics": 12,
"cyclomatic": 3,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 6,
"automation_only": false,
"max_nesting":... | 10 | module
public import Foundation.Propositional.Formula.Basic
public import Foundation.Propositional.Entailment.Cl.Basic
public import Foundation.Vorspiel.Set.Basic
@[expose] public section
namespace LO.Propositional
open Entailment FiniteContext
open Formula
variable {α : Type*}
variable {S} [Entailment S (Formula ... | module
public import Foundation.Propositional.Formula.Basic
public import Foundation.Propositional.Entailment.Cl.Basic
public import Foundation.Vorspiel.Set.Basic
@[expose] public section
namespace LO.Propositional
open Entailment FiniteContext
open Formula
variable {α : Type*}
variable {S} [Entailment S (Formula ... | @@ -333,6 +333,8 @@
lemma iff_not_mem₁_mem₂ : φ ∉ t.1.1 ↔ φ ∈ t.1.2 := Tableau.not_mem₁_iff_mem₂ t.consistent t.saturated
+lemma iff_not_mem₂_mem₁ : φ ∉ t.1.2 ↔ φ ∈ t.1.1 := Tableau.not_mem₂_iff_mem₁ t.consistent t.saturated
+
section
variable [DecidableEq α] [Encodable α]
@@ -341,9 +343,17 @@
section Satur... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_d1d343cc0a9a_33 | 05aa1d74f9994851 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Propositional/ConsistentTableau.lean | ConsistentTableau | 33 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 40 | 14 | [
{
"theorem_name": "iff_provable_include₁",
"depth": 1,
"n_commands": 0,
"n_lines": 30,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n constructor;\n . intro h t hT;\n by_contra hφ;\n replace hφ := iff_not_mem₁_mem₂.mp hφ;\n obtain ⟨Γ, hΓ₁, ... | [
{
"name": "iff_not_mem₂_mem₁",
"text": "lemma iff_not_mem₂_mem₁ : φ ∉ t.1.2 ↔ φ ∈ t.1.1 := Tableau.not_mem₂_iff_mem₁ t.consistent t.saturated\n\n",
"fan_in": 14,
"n_lines": 3,
"n_chars": 119,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": ... | [
{
"name": "iff_mem₂_or",
"fan_in": 1,
"n_deps_direct": 3,
"n_deps_transitive": 13,
"n_lines": 11,
"n_chars": 372,
"n_subproofs": 1,
"n_tactics": 18,
"cyclomatic": 3,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 5,
"automation_only": false,
"max_nesting"... | 12 | module
public import Foundation.Propositional.Formula.Basic
public import Foundation.Propositional.Entailment.Cl.Basic
public import Foundation.Vorspiel.Set.Basic
@[expose] public section
namespace LO.Propositional
open Entailment FiniteContext
open Formula
variable {α : Type*}
variable {S} [Entailment S (Formula ... | module
public import Foundation.Propositional.Formula.Basic
public import Foundation.Propositional.Entailment.Cl.Basic
public import Foundation.Vorspiel.Set.Basic
@[expose] public section
namespace LO.Propositional
open Entailment FiniteContext
open Formula
variable {α : Type*}
variable {S} [Entailment S (Formula ... | @@ -333,6 +333,8 @@
lemma iff_not_mem₁_mem₂ : φ ∉ t.1.1 ↔ φ ∈ t.1.2 := Tableau.not_mem₁_iff_mem₂ t.consistent t.saturated
+lemma iff_not_mem₂_mem₁ : φ ∉ t.1.2 ↔ φ ∈ t.1.1 := Tableau.not_mem₂_iff_mem₁ t.consistent t.saturated
+
section
variable [DecidableEq α] [Encodable α]
@@ -341,9 +343,17 @@
section Satur... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_d1d343cc0a9a_34 | 691b4c79812542d5 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Propositional/ConsistentTableau.lean | ConsistentTableau | 34 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 40 | 7 | [
{
"theorem_name": "mdp_mem₂_provable",
"depth": 1,
"n_commands": 0,
"n_lines": 4,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n by_contra hq₂;\n have := iff_not_mem₂_mem₁.mpr $ mdp_mem₁_provable h $ iff_not_mem₂_mem₁.mp hq₂;\n contradiction;",
... | [
{
"name": "mdp_mem₁_provable",
"text": "lemma mdp_mem₁_provable (h : 𝓢 ⊢ φ 🡒 ψ) (hp₁ : φ ∈ t.1.1) : ψ ∈ t.1.1 := by\n apply iff_not_mem₂_mem₁.mp;\n by_contra hq₂;\n apply by simpa using t.consistent (Γ := {φ}) (Δ := {ψ}) (by simpa) (by simpa);\n exact h;\n\n",
"fan_in": 7,
"n_lines": 7,
"n... | [
{
"name": "iff_mem₂_disj",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 14,
"n_lines": 14,
"n_chars": 489,
"n_subproofs": 0,
"n_tactics": 21,
"cyclomatic": 4,
"n_automation": 4,
"n_rewrites": 0,
"n_structural": 7,
"automation_only": false,
"max_nestin... | 13 | module
public import Foundation.Propositional.Formula.Basic
public import Foundation.Propositional.Entailment.Cl.Basic
public import Foundation.Vorspiel.Set.Basic
@[expose] public section
namespace LO.Propositional
open Entailment FiniteContext
open Formula
variable {α : Type*}
variable {S} [Entailment S (Formula ... | module
public import Foundation.Propositional.Formula.Basic
public import Foundation.Propositional.Entailment.Cl.Basic
public import Foundation.Vorspiel.Set.Basic
@[expose] public section
namespace LO.Propositional
open Entailment FiniteContext
open Formula
variable {α : Type*}
variable {S} [Entailment S (Formula ... | @@ -343,6 +343,12 @@
section Saturated
+lemma mdp_mem₁_provable (h : 𝓢 ⊢ φ 🡒 ψ) (hp₁ : φ ∈ t.1.1) : ψ ∈ t.1.1 := by
+ apply iff_not_mem₂_mem₁.mp;
+ by_contra hq₂;
+ apply by simpa using t.consistent (Γ := {φ}) (Δ := {ψ}) (by simpa) (by simpa);
+ exact h;
+
@[simp] lemma mem₁_verum : ⊤ ∈ t.1.1 := by
apply... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_d1d343cc0a9a_35 | 38a0663395763655 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Propositional/ConsistentTableau.lean | ConsistentTableau | 35 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 40 | 7 | [
{
"theorem_name": "mdp_mem₂_provable",
"depth": 1,
"n_commands": 0,
"n_lines": 4,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n by_contra hq₂;\n have := iff_not_mem₂_mem₁.mpr $ mdp_mem₁_provable h $ iff_not_mem₂_mem₁.mp hq₂;\n contradiction;",
... | [
{
"name": "mdp_mem₁_provable",
"text": "lemma mdp_mem₁_provable (h : 𝓢 ⊢ φ 🡒 ψ) (hp₁ : φ ∈ t.1.1) : ψ ∈ t.1.1 := by\n apply iff_not_mem₂_mem₁.mp;\n by_contra hq₂;\n apply by simpa using t.consistent (Γ := {φ}) (Δ := {ψ}) (by simpa) (by simpa);\n exact h;\n\n",
"fan_in": 7,
"n_lines": 7,
"n... | [
{
"name": "iff_mem₂_fdisj",
"fan_in": 0,
"n_deps_direct": 2,
"n_deps_transitive": 16,
"n_lines": 7,
"n_chars": 319,
"n_subproofs": 0,
"n_tactics": 10,
"cyclomatic": 3,
"n_automation": 3,
"n_rewrites": 1,
"n_structural": 4,
"automation_only": false,
"max_nestin... | 15 | module
public import Foundation.Propositional.Formula.Basic
public import Foundation.Propositional.Entailment.Cl.Basic
public import Foundation.Vorspiel.Set.Basic
@[expose] public section
namespace LO.Propositional
open Entailment FiniteContext
open Formula
variable {α : Type*}
variable {S} [Entailment S (Formula ... | module
public import Foundation.Propositional.Formula.Basic
public import Foundation.Propositional.Entailment.Cl.Basic
public import Foundation.Vorspiel.Set.Basic
@[expose] public section
namespace LO.Propositional
open Entailment FiniteContext
open Formula
variable {α : Type*}
variable {S} [Entailment S (Formula ... | @@ -343,8 +343,17 @@
section Saturated
-lemma mdp_mem₂_provable (h : 𝓢 ⊢ φ 🡒 ψ) (hp₁ : ψ ∈ t.1.2) : φ ∈ t.1.2 := sorry
+lemma mdp_mem₁_provable (h : 𝓢 ⊢ φ 🡒 ψ) (hp₁ : φ ∈ t.1.1) : ψ ∈ t.1.1 := by
+ apply iff_not_mem₂_mem₁.mp;
+ by_contra hq₂;
+ apply by simpa using t.consistent (Γ := {φ}) (Δ := {ψ}) (by sim... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_d1d343cc0a9a_36 | 35064ed4ba728b06 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Propositional/ConsistentTableau.lean | ConsistentTableau | 36 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 40 | 14 | [
{
"theorem_name": "iff_provable_include₁",
"depth": 1,
"n_commands": 0,
"n_lines": 30,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n constructor;\n . intro h t hT;\n by_contra hφ;\n replace hφ := iff_not_mem₁_mem₂.mp hφ;\n obtain ⟨Γ, hΓ₁, ... | [
{
"name": "iff_not_mem₂_mem₁",
"text": "lemma iff_not_mem₂_mem₁ : φ ∉ t.1.2 ↔ φ ∈ t.1.1 := Tableau.not_mem₂_iff_mem₁ t.consistent t.saturated\n\n",
"fan_in": 14,
"n_lines": 3,
"n_chars": 119,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": ... | [
{
"name": "of_mem₁_imp",
"fan_in": 4,
"n_deps_direct": 2,
"n_deps_transitive": 7,
"n_lines": 7,
"n_chars": 208,
"n_subproofs": 0,
"n_tactics": 8,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 2,
"automation_only": false,
"max_nesting": 2... | 6 | module
public import Foundation.Propositional.Formula.Basic
public import Foundation.Propositional.Entailment.Cl.Basic
public import Foundation.Vorspiel.Set.Basic
@[expose] public section
namespace LO.Propositional
open Entailment FiniteContext
open Formula
variable {α : Type*}
variable {S} [Entailment S (Formula ... | module
public import Foundation.Propositional.Formula.Basic
public import Foundation.Propositional.Entailment.Cl.Basic
public import Foundation.Vorspiel.Set.Basic
@[expose] public section
namespace LO.Propositional
open Entailment FiniteContext
open Formula
variable {α : Type*}
variable {S} [Entailment S (Formula ... | @@ -333,6 +333,8 @@
lemma iff_not_mem₁_mem₂ : φ ∉ t.1.1 ↔ φ ∈ t.1.2 := Tableau.not_mem₁_iff_mem₂ t.consistent t.saturated
+lemma iff_not_mem₂_mem₁ : φ ∉ t.1.2 ↔ φ ∈ t.1.1 := Tableau.not_mem₂_iff_mem₁ t.consistent t.saturated
+
section
variable [DecidableEq α] [Encodable α]
@@ -341,7 +343,11 @@
section Satur... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_d1d343cc0a9a_37 | 92f54b6fcea03645 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Propositional/ConsistentTableau.lean | ConsistentTableau | 37 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 40 | 14 | [
{
"theorem_name": "iff_provable_include₁",
"depth": 1,
"n_commands": 0,
"n_lines": 30,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n constructor;\n . intro h t hT;\n by_contra hφ;\n replace hφ := iff_not_mem₁_mem₂.mp hφ;\n obtain ⟨Γ, hΓ₁, ... | [
{
"name": "iff_not_mem₂_mem₁",
"text": "lemma iff_not_mem₂_mem₁ : φ ∉ t.1.2 ↔ φ ∈ t.1.1 := Tableau.not_mem₂_iff_mem₁ t.consistent t.saturated\n\n",
"fan_in": 14,
"n_lines": 3,
"n_chars": 119,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": ... | [
{
"name": "of_mem₁_imp'",
"fan_in": 0,
"n_deps_direct": 2,
"n_deps_transitive": 8,
"n_lines": 6,
"n_chars": 210,
"n_subproofs": 0,
"n_tactics": 7,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 3,
"automation_only": false,
"max_nesting": ... | 7 | module
public import Foundation.Propositional.Formula.Basic
public import Foundation.Propositional.Entailment.Cl.Basic
public import Foundation.Vorspiel.Set.Basic
@[expose] public section
namespace LO.Propositional
open Entailment FiniteContext
open Formula
variable {α : Type*}
variable {S} [Entailment S (Formula ... | module
public import Foundation.Propositional.Formula.Basic
public import Foundation.Propositional.Entailment.Cl.Basic
public import Foundation.Vorspiel.Set.Basic
@[expose] public section
namespace LO.Propositional
open Entailment FiniteContext
open Formula
variable {α : Type*}
variable {S} [Entailment S (Formula ... | @@ -333,6 +333,8 @@
lemma iff_not_mem₁_mem₂ : φ ∉ t.1.1 ↔ φ ∈ t.1.2 := Tableau.not_mem₁_iff_mem₂ t.consistent t.saturated
+lemma iff_not_mem₂_mem₁ : φ ∉ t.1.2 ↔ φ ∈ t.1.1 := Tableau.not_mem₂_iff_mem₁ t.consistent t.saturated
+
section
variable [DecidableEq α] [Encodable α]
@@ -341,7 +343,11 @@
section Satur... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_d1d343cc0a9a_38 | 480d6e1be028e129 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Propositional/ConsistentTableau.lean | ConsistentTableau | 38 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 40 | 14 | [
{
"theorem_name": "iff_provable_include₁",
"depth": 1,
"n_commands": 0,
"n_lines": 30,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n constructor;\n . intro h t hT;\n by_contra hφ;\n replace hφ := iff_not_mem₁_mem₂.mp hφ;\n obtain ⟨Γ, hΓ₁, ... | [
{
"name": "iff_not_mem₂_mem₁",
"text": "lemma iff_not_mem₂_mem₁ : φ ∉ t.1.2 ↔ φ ∈ t.1.1 := Tableau.not_mem₂_iff_mem₁ t.consistent t.saturated\n\n",
"fan_in": 14,
"n_lines": 3,
"n_chars": 119,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": ... | [
{
"name": "of_mem₁_neg",
"fan_in": 1,
"n_deps_direct": 2,
"n_deps_transitive": 9,
"n_lines": 7,
"n_chars": 210,
"n_subproofs": 0,
"n_tactics": 9,
"cyclomatic": 2,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 2,
"automation_only": false,
"max_nesting": 4... | 7 | module
public import Foundation.Propositional.Formula.Basic
public import Foundation.Propositional.Entailment.Cl.Basic
public import Foundation.Vorspiel.Set.Basic
@[expose] public section
namespace LO.Propositional
open Entailment FiniteContext
open Formula
variable {α : Type*}
variable {S} [Entailment S (Formula ... | module
public import Foundation.Propositional.Formula.Basic
public import Foundation.Propositional.Entailment.Cl.Basic
public import Foundation.Vorspiel.Set.Basic
@[expose] public section
namespace LO.Propositional
open Entailment FiniteContext
open Formula
variable {α : Type*}
variable {S} [Entailment S (Formula ... | @@ -333,6 +333,8 @@
lemma iff_not_mem₁_mem₂ : φ ∉ t.1.1 ↔ φ ∈ t.1.2 := Tableau.not_mem₁_iff_mem₂ t.consistent t.saturated
+lemma iff_not_mem₂_mem₁ : φ ∉ t.1.2 ↔ φ ∈ t.1.1 := Tableau.not_mem₂_iff_mem₁ t.consistent t.saturated
+
section
variable [DecidableEq α] [Encodable α]
@@ -341,7 +343,11 @@
section Satur... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_d1d343cc0a9a_42 | 5525873b2387f6c9 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Propositional/ConsistentTableau.lean | ConsistentTableau | 42 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 40 | 7 | [
{
"theorem_name": "mdp_mem₂_provable",
"depth": 1,
"n_commands": 0,
"n_lines": 4,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n by_contra hq₂;\n have := iff_not_mem₂_mem₁.mpr $ mdp_mem₁_provable h $ iff_not_mem₂_mem₁.mp hq₂;\n contradiction;",
... | [
{
"name": "mdp_mem₁_provable",
"text": "lemma mdp_mem₁_provable (h : 𝓢 ⊢ φ 🡒 ψ) (hp₁ : φ ∈ t.1.1) : ψ ∈ t.1.1 := by\n apply iff_not_mem₂_mem₁.mp;\n by_contra hq₂;\n apply by simpa using t.consistent (Γ := {φ}) (Δ := {ψ}) (by simpa) (by simpa);\n exact h;\n\n",
"fan_in": 7,
"n_lines": 7,
"n... | [
{
"name": "iff_mem₂_imp",
"fan_in": 0,
"n_deps_direct": 4,
"n_deps_transitive": 18,
"n_lines": 11,
"n_chars": 410,
"n_subproofs": 1,
"n_tactics": 18,
"cyclomatic": 3,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 5,
"automation_only": false,
"max_nesting... | 16 | module
public import Foundation.Propositional.Formula.Basic
public import Foundation.Propositional.Entailment.Cl.Basic
public import Foundation.Vorspiel.Set.Basic
@[expose] public section
namespace LO.Propositional
open Entailment FiniteContext
open Formula
variable {α : Type*}
variable {S} [Entailment S (Formula ... | module
public import Foundation.Propositional.Formula.Basic
public import Foundation.Propositional.Entailment.Cl.Basic
public import Foundation.Vorspiel.Set.Basic
@[expose] public section
namespace LO.Propositional
open Entailment FiniteContext
open Formula
variable {α : Type*}
variable {S} [Entailment S (Formula ... | @@ -383,6 +383,12 @@
section Saturated
+lemma mdp_mem₁_provable (h : 𝓢 ⊢ φ 🡒 ψ) (hp₁ : φ ∈ t.1.1) : ψ ∈ t.1.1 := by
+ apply iff_not_mem₂_mem₁.mp;
+ by_contra hq₂;
+ apply by simpa using t.consistent (Γ := {φ}) (Δ := {ψ}) (by simpa) (by simpa);
+ exact h;
+
@[simp] lemma mem₁_verum : ⊤ ∈ t.1.1 := by
apply... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_d1d343cc0a9a_43 | 8dabe808a4cb1c7b | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Propositional/ConsistentTableau.lean | ConsistentTableau | 43 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 40 | 7 | [
{
"theorem_name": "mdp_mem₂_provable",
"depth": 1,
"n_commands": 0,
"n_lines": 4,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n by_contra hq₂;\n have := iff_not_mem₂_mem₁.mpr $ mdp_mem₁_provable h $ iff_not_mem₂_mem₁.mp hq₂;\n contradiction;",
... | [
{
"name": "mdp_mem₁_provable",
"text": "lemma mdp_mem₁_provable (h : 𝓢 ⊢ φ 🡒 ψ) (hp₁ : φ ∈ t.1.1) : ψ ∈ t.1.1 := by\n apply iff_not_mem₂_mem₁.mp;\n by_contra hq₂;\n apply by simpa using t.consistent (Γ := {φ}) (Δ := {ψ}) (by simpa) (by simpa);\n exact h;\n\n",
"fan_in": 7,
"n_lines": 7,
"n... | [
{
"name": "not_mem₁_neg_of_mem₁",
"fan_in": 1,
"n_deps_direct": 3,
"n_deps_transitive": 9,
"n_lines": 9,
"n_chars": 282,
"n_subproofs": 3,
"n_tactics": 12,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": false,
"max_n... | 7 | module
public import Foundation.Propositional.Formula.Basic
public import Foundation.Propositional.Entailment.Cl.Basic
public import Foundation.Vorspiel.Set.Basic
@[expose] public section
namespace LO.Propositional
open Entailment FiniteContext
open Formula
variable {α : Type*}
variable {S} [Entailment S (Formula ... | module
public import Foundation.Propositional.Formula.Basic
public import Foundation.Propositional.Entailment.Cl.Basic
public import Foundation.Vorspiel.Set.Basic
@[expose] public section
namespace LO.Propositional
open Entailment FiniteContext
open Formula
variable {α : Type*}
variable {S} [Entailment S (Formula ... | @@ -343,6 +343,12 @@
section Saturated
+lemma mdp_mem₁_provable (h : 𝓢 ⊢ φ 🡒 ψ) (hp₁ : φ ∈ t.1.1) : ψ ∈ t.1.1 := by
+ apply iff_not_mem₂_mem₁.mp;
+ by_contra hq₂;
+ apply by simpa using t.consistent (Γ := {φ}) (Δ := {ψ}) (by simpa) (by simpa);
+ exact h;
+
@[simp] lemma mem₁_verum : ⊤ ∈ t.1.1 := by
apply... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_d1d343cc0a9a_44 | f0fd1ae6ba83c8cd | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Propositional/ConsistentTableau.lean | ConsistentTableau | 44 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 40 | 14 | [
{
"theorem_name": "iff_provable_include₁",
"depth": 1,
"n_commands": 0,
"n_lines": 30,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n constructor;\n . intro h t hT;\n by_contra hφ;\n replace hφ := iff_not_mem₁_mem₂.mp hφ;\n obtain ⟨Γ, hΓ₁, ... | [
{
"name": "iff_not_mem₂_mem₁",
"text": "lemma iff_not_mem₂_mem₁ : φ ∉ t.1.2 ↔ φ ∈ t.1.1 := Tableau.not_mem₂_iff_mem₁ t.consistent t.saturated\n\n",
"fan_in": 14,
"n_lines": 3,
"n_chars": 119,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": ... | [
{
"name": "mem₂_neg_of_mem₁",
"fan_in": 0,
"n_deps_direct": 2,
"n_deps_transitive": 13,
"n_lines": 5,
"n_chars": 186,
"n_subproofs": 0,
"n_tactics": 5,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 2,
"automation_only": false,
"max_nesti... | 11 | module
public import Foundation.Propositional.Formula.Basic
public import Foundation.Propositional.Entailment.Cl.Basic
public import Foundation.Vorspiel.Set.Basic
@[expose] public section
namespace LO.Propositional
open Entailment FiniteContext
open Formula
variable {α : Type*}
variable {S} [Entailment S (Formula ... | module
public import Foundation.Propositional.Formula.Basic
public import Foundation.Propositional.Entailment.Cl.Basic
public import Foundation.Vorspiel.Set.Basic
@[expose] public section
namespace LO.Propositional
open Entailment FiniteContext
open Formula
variable {α : Type*}
variable {S} [Entailment S (Formula ... | @@ -333,6 +333,8 @@
lemma iff_not_mem₁_mem₂ : φ ∉ t.1.1 ↔ φ ∈ t.1.2 := Tableau.not_mem₁_iff_mem₂ t.consistent t.saturated
+lemma iff_not_mem₂_mem₁ : φ ∉ t.1.2 ↔ φ ∈ t.1.1 := Tableau.not_mem₂_iff_mem₁ t.consistent t.saturated
+
section
variable [DecidableEq α] [Encodable α]
@@ -341,9 +343,17 @@
section Satur... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_d1d343cc0a9a_45 | f43ab945b7afef7f | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Propositional/ConsistentTableau.lean | ConsistentTableau | 45 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 40 | 14 | [
{
"theorem_name": "iff_provable_include₁",
"depth": 1,
"n_commands": 0,
"n_lines": 30,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n constructor;\n . intro h t hT;\n by_contra hφ;\n replace hφ := iff_not_mem₁_mem₂.mp hφ;\n obtain ⟨Γ, hΓ₁, ... | [
{
"name": "iff_not_mem₂_mem₁",
"text": "lemma iff_not_mem₂_mem₁ : φ ∉ t.1.2 ↔ φ ∈ t.1.1 := Tableau.not_mem₂_iff_mem₁ t.consistent t.saturated\n\n",
"fan_in": 14,
"n_lines": 3,
"n_chars": 119,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": ... | [
{
"name": "mdp₁_mem",
"fan_in": 0,
"n_deps_direct": 2,
"n_deps_transitive": 6,
"n_lines": 10,
"n_chars": 420,
"n_subproofs": 0,
"n_tactics": 17,
"cyclomatic": 4,
"n_automation": 4,
"n_rewrites": 0,
"n_structural": 5,
"automation_only": false,
"max_nesting": 4
... | 5 | module
public import Foundation.Propositional.Formula.Basic
public import Foundation.Propositional.Entailment.Cl.Basic
public import Foundation.Vorspiel.Set.Basic
@[expose] public section
namespace LO.Propositional
open Entailment FiniteContext
open Formula
variable {α : Type*}
variable {S} [Entailment S (Formula ... | module
public import Foundation.Propositional.Formula.Basic
public import Foundation.Propositional.Entailment.Cl.Basic
public import Foundation.Vorspiel.Set.Basic
@[expose] public section
namespace LO.Propositional
open Entailment FiniteContext
open Formula
variable {α : Type*}
variable {S} [Entailment S (Formula ... | @@ -333,6 +333,8 @@
lemma iff_not_mem₁_mem₂ : φ ∉ t.1.1 ↔ φ ∈ t.1.2 := Tableau.not_mem₁_iff_mem₂ t.consistent t.saturated
+lemma iff_not_mem₂_mem₁ : φ ∉ t.1.2 ↔ φ ∈ t.1.1 := Tableau.not_mem₂_iff_mem₁ t.consistent t.saturated
+
section
variable [DecidableEq α] [Encodable α]
@@ -341,7 +343,11 @@
section Satur... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_d2d9ab791de4_0 | c78973444728f464 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Vorspiel/Quotient.lean | Quotient | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 1 | 2 | [
{
"theorem_name": "liftVec_mk₁",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " liftVec_mk f h ![a]",
"n_chars": 20,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n... | [
{
"name": "liftVec_mk",
"text": "lemma liftVec_mk {n} (f : (Fin n → α) → β) (h) (v : Fin n → α) :\n liftVec f h (Quotient.mk s ∘ v) = f v := by\n induction' n with n ih <;> simp [liftVec, empty_eq]\n simpa using! ih (fun v' => f (vecHead v :> v'))\n (fun v₁ v₂ hv => h (vecHead v :> v₁) (vecHead v :>... | [
{
"name": "liftVec_mk₁",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 4,
"n_chars": 139,
"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.Vorspiel.Fin.Matrix
@[expose] public section
namespace Quotient
open Matrix
variable {α : Type u} [s : Setoid α] {β : Sort v}
@[elab_as_elim]
lemma inductionOnVec {φ : (Fin n → Quotient s) → Prop} (v : Fin n → Quotient s)
(h : ∀ v : Fin n → α, φ (fun i => Quotient.mk s (v i))) : φ ... | module
public import Foundation.Vorspiel.Fin.Matrix
@[expose] public section
namespace Quotient
open Matrix
variable {α : Type u} [s : Setoid α] {β : Sort v}
@[elab_as_elim]
lemma inductionOnVec {φ : (Fin n → Quotient s) → Prop} (v : Fin n → Quotient s)
(h : ∀ v : Fin n → α, φ (fun i => Quotient.mk s (v i))) : φ ... | @@ -27,11 +27,17 @@
@[simp] lemma liftVec_zero (f : (Fin 0 → α) → β) (h) (v : Fin 0 → Quotient s) : liftVec f h v = f ![] := rfl
+lemma liftVec_mk {n} (f : (Fin n → α) → β) (h) (v : Fin n → α) :
+ liftVec f h (Quotient.mk s ∘ v) = f v := by
+ induction' n with n ih <;> simp [liftVec, empty_eq]
+ simpa using! ... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_05d280bf781a_0 | e4bfbd546535436e | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/ModalCompanion/Standard/KC.lean | KC | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 2 | 2 | [
{
"theorem_name": "modalCompanion_S4Point2",
"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 F => IsPiecewiseStronglyConvergent F);\n . apply provable_S4Poin... | [
{
"name": "provable_S4Point2_of_mem",
"text": "lemma provable_S4Point2_of_mem : φ ∈ Propositional.KC → Modal.S4Point2 ⊢ φᵍ := by\n apply Propositional.Hilbert.provable_gödelTranslated_of_mem_logic;\n rintro _ ⟨a, (⟨_, rfl⟩ | ⟨_, rfl⟩), rfl⟩ <;> simp;\n\n",
"fan_in": 2,
"n_lines": 5,
"n_chars":... | [
{
"name": "modalCompanion_S4Point2",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 12,
"n_chars": 441,
"n_subproofs": 0,
"n_tactics": 19,
"cyclomatic": 2,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 8,
"automation_only": false,
"m... | 1 | module
public import Foundation.Logic.Embedding
public import Foundation.Modal.Boxdot.GL_Grz
public import Foundation.Modal.Kripke.Logic.GrzPoint2
public import Foundation.Modal.ModalCompanion.Standard.Basic
public import Foundation.Propositional.Kripke.Hilbert.KC
@[expose] public section
namespace LO
open LO.Entai... | module
public import Foundation.Logic.Embedding
public import Foundation.Modal.Boxdot.GL_Grz
public import Foundation.Modal.Kripke.Logic.GrzPoint2
public import Foundation.Modal.ModalCompanion.Standard.Basic
public import Foundation.Propositional.Kripke.Hilbert.KC
@[expose] public section
namespace LO
open LO.Entai... | @@ -43,9 +43,13 @@
namespace Propositional.KC
+lemma provable_S4Point2_of_mem : φ ∈ Propositional.KC → Modal.S4Point2 ⊢ φᵍ := by
+ apply Propositional.Hilbert.provable_gödelTranslated_of_mem_logic;
+ rintro _ ⟨a, (⟨_, rfl⟩ | ⟨_, rfl⟩), rfl⟩ <;> simp;
+
theorem modalCompanion_S4Point2 : ModalCompanion Propositio... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_05d280bf781a_1 | 0035bdb4b1fb9e12 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/ModalCompanion/Standard/KC.lean | KC | 1 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 2 | 2 | [
{
"theorem_name": "modalCompanion_S4Point2",
"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 F => IsPiecewiseStronglyConvergent F);\n . apply provable_S4Poin... | [
{
"name": "provable_S4Point2_of_mem",
"text": "lemma provable_S4Point2_of_mem : φ ∈ Propositional.KC → Modal.S4Point2 ⊢ φᵍ := by\n apply Propositional.Hilbert.provable_gödelTranslated_of_mem_logic;\n rintro _ ⟨a, (⟨_, rfl⟩ | ⟨_, rfl⟩), rfl⟩ <;> simp;\n\n",
"fan_in": 2,
"n_lines": 5,
"n_chars":... | [
{
"name": "provable_GrzPoint2_of_mem",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 3,
"n_chars": 146,
"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.Logic.Embedding
public import Foundation.Modal.Boxdot.GL_Grz
public import Foundation.Modal.Kripke.Logic.GrzPoint2
public import Foundation.Modal.ModalCompanion.Standard.Basic
public import Foundation.Propositional.Kripke.Hilbert.KC
@[expose] public section
namespace LO
open LO.Entai... | module
public import Foundation.Logic.Embedding
public import Foundation.Modal.Boxdot.GL_Grz
public import Foundation.Modal.Kripke.Logic.GrzPoint2
public import Foundation.Modal.ModalCompanion.Standard.Basic
public import Foundation.Propositional.Kripke.Hilbert.KC
@[expose] public section
namespace LO
open LO.Entai... | @@ -43,8 +43,12 @@
namespace Propositional.KC
-lemma provable_GrzPoint2_of_mem : φ ∈ Propositional.KC → Modal.GrzPoint2 ⊢ φᵍ := sorry
+lemma provable_S4Point2_of_mem : φ ∈ Propositional.KC → Modal.S4Point2 ⊢ φᵍ := by
+ apply Propositional.Hilbert.provable_gödelTranslated_of_mem_logic;
+ rintro _ ⟨a, (⟨_, 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_05d280bf781a_2 | b0095d5a8f08e066 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/ModalCompanion/Standard/KC.lean | KC | 2 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 2 | 2 | [
{
"theorem_name": "modalCompanion_S4Point2",
"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 F => IsPiecewiseStronglyConvergent F);\n . apply provable_S4Poin... | [
{
"name": "provable_S4Point2_of_mem",
"text": "lemma provable_S4Point2_of_mem : φ ∈ Propositional.KC → Modal.S4Point2 ⊢ φᵍ := by\n apply Propositional.Hilbert.provable_gödelTranslated_of_mem_logic;\n rintro _ ⟨a, (⟨_, rfl⟩ | ⟨_, rfl⟩), rfl⟩ <;> simp;\n\n",
"fan_in": 2,
"n_lines": 5,
"n_chars":... | [
{
"name": "modalCompanion_GrzPoint2",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 2,
"n_lines": 12,
"n_chars": 487,
"n_subproofs": 0,
"n_tactics": 19,
"cyclomatic": 2,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 8,
"automation_only": false,
"... | 2 | module
public import Foundation.Logic.Embedding
public import Foundation.Modal.Boxdot.GL_Grz
public import Foundation.Modal.Kripke.Logic.GrzPoint2
public import Foundation.Modal.ModalCompanion.Standard.Basic
public import Foundation.Propositional.Kripke.Hilbert.KC
@[expose] public section
namespace LO
open LO.Entai... | module
public import Foundation.Logic.Embedding
public import Foundation.Modal.Boxdot.GL_Grz
public import Foundation.Modal.Kripke.Logic.GrzPoint2
public import Foundation.Modal.ModalCompanion.Standard.Basic
public import Foundation.Propositional.Kripke.Hilbert.KC
@[expose] public section
namespace LO
open LO.Entai... | @@ -43,8 +43,12 @@
namespace Propositional.KC
-lemma provable_GrzPoint2_of_mem : φ ∈ Propositional.KC → Modal.GrzPoint2 ⊢ φᵍ := sorry
+lemma provable_S4Point2_of_mem : φ ∈ Propositional.KC → Modal.S4Point2 ⊢ φᵍ := by
+ apply Propositional.Hilbert.provable_gödelTranslated_of_mem_logic;
+ rintro _ ⟨a, (⟨_, 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_79dd459e53bb_0 | 0b1c5617cdf8116a | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/VanBentham/StandardTranslation.lean | StandardTranslation | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 1 | 1 | [
{
"theorem_name": "correspondence_validOnModel",
"depth": 1,
"n_commands": 0,
"n_lines": 5,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n suffices M ⊧ φ ↔ ∀ x : M.World, (φ¹).Evalb ![x] by simpa [FirstOrder.models_iff];\n constructor;\n . intro h ... | [
{
"name": "correspondence_satisfies",
"text": "/-- BdRV Prop 2.47 (i) -/\nlemma correspondence_satisfies : x ⊧ φ ↔ φ¹.Evalb ![x] := by\n induction φ using NNFormula.rec' generalizing x with\n | hBox φ ihφ =>\n suffices x ⊧ □φ ↔ ∀ y, x ≺ y → (φ¹).Evalb ![y] by\n simpa [standardTranslation]\n con... | [
{
"name": "correspondence_validOnModel",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 8,
"n_chars": 335,
"n_subproofs": 1,
"n_tactics": 11,
"cyclomatic": 2,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 5,
"automation_only": false,
... | 1 | module
public import Foundation.FirstOrder.Basic
public import Foundation.Modal.Kripke.NNFormula
@[expose] public section
namespace LO.FirstOrder
namespace Language
namespace Frame
inductive Rel : ℕ → Type _
/-- propositional variable -/
| pred : ℕ → Rel 1
/-- binary relation -/
| lt : Rel 2
end Frame
/... | module
public import Foundation.FirstOrder.Basic
public import Foundation.Modal.Kripke.NNFormula
@[expose] public section
namespace LO.FirstOrder
namespace Language
namespace Frame
inductive Rel : ℕ → Type _
/-- propositional variable -/
| pred : ℕ → Rel 1
/-- binary relation -/
| lt : Rel 2
end Frame
/... | @@ -86,12 +86,39 @@
@[simp] lemma lt_iff_rel : (@Operator.LT.lt 𝓛𝓕 _).val ![x, y] ↔ x ≺ y := by rfl
+/-- BdRV Prop 2.47 (i) -/
+lemma correspondence_satisfies : x ⊧ φ ↔ φ¹.Evalb ![x] := by
+ induction φ using NNFormula.rec' generalizing x with
+ | hBox φ ihφ =>
+ suffices x ⊧ □φ ↔ ∀ y, x ≺ y → (φ¹).Evalb ![... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_e630f1fe68e4_0 | 41218753bea4dadc | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Basic/Model.lean | Model | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 4 | 1 | [
{
"theorem_name": "complete",
"depth": 1,
"n_commands": 0,
"n_lines": 3,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " complete_aux T φ fun M _ s _ _ ↦ by\n rcases standardModel_unique M s\n exact H M",
"n_chars": 85,
"n_subproofs": 0,
"n_ta... | [
{
"name": "standardModel_unique",
"text": "lemma standardModel_unique (s : Structure ℒₒᵣ M)\n [hZero : Structure.Zero ℒₒᵣ M] [hOne : Structure.One ℒₒᵣ M] [hAdd : Structure.Add ℒₒᵣ M] [hMul : Structure.Mul ℒₒᵣ M]\n [hEq : Structure.Eq ℒₒᵣ M] [hLT : Structure.LT ℒₒᵣ M] : s = standardModel M :=\n standa... | [
{
"name": "complete_aux",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 4,
"n_lines": 12,
"n_chars": 574,
"n_subproofs": 0,
"n_tactics": 3,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
"max_nesting":... | 3 | module
public import Foundation.FirstOrder.Arithmetic.Basic.Misc
@[expose] public section
namespace LO.FirstOrder.Arithmetic
private lemma complete_aux (T : ArithmeticTheory) [𝗘𝗤 ℒₒᵣ ⪯ T] (φ : ArithmeticSentence)
(H : ∀ (M : Type w)
[ORingStructure M]
[Structure ℒₒᵣ M]
[Structu... | module
public import Foundation.FirstOrder.Arithmetic.Basic.Misc
@[expose] public section
namespace LO.FirstOrder.Arithmetic
private lemma complete_aux (T : ArithmeticTheory) [𝗘𝗤 ℒₒᵣ ⪯ T] (φ : ArithmeticSentence)
(H : ∀ (M : Type w)
[ORingStructure M]
[Structure ℒₒᵣ M]
[Structu... | @@ -67,11 +67,18 @@
| _, Language.Eq.eq => by simp
| _, Language.LT.lt => by simp)
+lemma standardModel_unique (s : Structure ℒₒᵣ M)
+ [hZero : Structure.Zero ℒₒᵣ M] [hOne : Structure.One ℒₒᵣ M] [hAdd : Structure.Add ℒₒᵣ M] [hMul : Structure.Mul ℒₒᵣ M]
+ [hEq : Structure.Eq ℒₒᵣ M] [hLT : Structure.LT ... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_e630f1fe68e4_1 | a1ef904451091a20 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Basic/Model.lean | Model | 1 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 4 | 1 | [
{
"theorem_name": "standardModel_unique",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": "\n standardModel_unique' M s hZero hOne hAdd hMul hEq hLT",
"n_chars": 57,
"n_subproofs": 0,
"n_tactics": 1,
... | [
{
"name": "standardModel_unique'",
"text": "lemma standardModel_unique' (s : Structure ℒₒᵣ M)\n (hZero : Structure.Zero ℒₒᵣ M) (hOne : Structure.One ℒₒᵣ M) (hAdd : Structure.Add ℒₒᵣ M) (hMul : Structure.Mul ℒₒᵣ M)\n (hEq : Structure.Eq ℒₒᵣ M) (hLT : Structure.LT ℒₒᵣ M) : s = standardModel M := Structu... | [
{
"name": "standardModel_unique",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 6,
"n_chars": 354,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
"max_ne... | 1 | module
public import Foundation.FirstOrder.Arithmetic.Basic.Misc
@[expose] public section
namespace LO.FirstOrder.Arithmetic
open Language
section semantics
variable (M : Type*) [ORingStructure M]
instance standardModel : Structure ℒₒᵣ M where
func := fun _ f ↦
match f with
| ORing.Func.zero => fun _ ↦ ... | module
public import Foundation.FirstOrder.Arithmetic.Basic.Misc
@[expose] public section
namespace LO.FirstOrder.Arithmetic
open Language
section semantics
variable (M : Type*) [ORingStructure M]
instance standardModel : Structure ℒₒᵣ M where
func := fun _ f ↦
match f with
| ORing.Func.zero => fun _ ↦ ... | @@ -42,9 +42,24 @@
instance : ORing ℒₒᵣ := ORing.mk
+lemma standardModel_unique' (s : Structure ℒₒᵣ M)
+ (hZero : Structure.Zero ℒₒᵣ M) (hOne : Structure.One ℒₒᵣ M) (hAdd : Structure.Add ℒₒᵣ M) (hMul : Structure.Mul ℒₒᵣ M)
+ (hEq : Structure.Eq ℒₒᵣ M) (hLT : Structure.LT ℒₒᵣ M) : s = standardModel M := Struc... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_e630f1fe68e4_2 | 8a466100a601289d | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Basic/Model.lean | Model | 2 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 4 | 1 | [
{
"theorem_name": "standardModel_unique",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": "\n standardModel_unique' M s hZero hOne hAdd hMul hEq hLT",
"n_chars": 57,
"n_subproofs": 0,
"n_tactics": 1,
... | [
{
"name": "standardModel_unique'",
"text": "lemma standardModel_unique' (s : Structure ℒₒᵣ M)\n (hZero : Structure.Zero ℒₒᵣ M) (hOne : Structure.One ℒₒᵣ M) (hAdd : Structure.Add ℒₒᵣ M) (hMul : Structure.Mul ℒₒᵣ M)\n (hEq : Structure.Eq ℒₒᵣ M) (hLT : Structure.LT ℒₒᵣ M) : s = standardModel M := Structu... | [
{
"name": "complete",
"fan_in": 2,
"n_deps_direct": 2,
"n_deps_transitive": 4,
"n_lines": 7,
"n_chars": 311,
"n_subproofs": 0,
"n_tactics": 3,
"cyclomatic": 2,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 2,
"automation_only": false,
"max_nesting": 2
... | 3 | module
public import Foundation.FirstOrder.Arithmetic.Basic.Misc
@[expose] public section
namespace LO.FirstOrder.Arithmetic
private lemma complete_aux (T : ArithmeticTheory) [𝗘𝗤 ℒₒᵣ ⪯ T] (φ : ArithmeticSentence)
(H : ∀ (M : Type w)
[ORingStructure M]
[Structure ℒₒᵣ M]
[Structu... | module
public import Foundation.FirstOrder.Arithmetic.Basic.Misc
@[expose] public section
namespace LO.FirstOrder.Arithmetic
private lemma complete_aux (T : ArithmeticTheory) [𝗘𝗤 ℒₒᵣ ⪯ T] (φ : ArithmeticSentence)
(H : ∀ (M : Type w)
[ORingStructure M]
[Structure ℒₒᵣ M]
[Structu... | @@ -53,9 +53,24 @@
instance : ORing ℒₒᵣ := ORing.mk
+lemma standardModel_unique' (s : Structure ℒₒᵣ M)
+ (hZero : Structure.Zero ℒₒᵣ M) (hOne : Structure.One ℒₒᵣ M) (hAdd : Structure.Add ℒₒᵣ M) (hMul : Structure.Mul ℒₒᵣ M)
+ (hEq : Structure.Eq ℒₒᵣ M) (hLT : Structure.LT ℒₒᵣ M) : s = standardModel M := Struc... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_e630f1fe68e4_3 | 2d4fa73d1de17978 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Basic/Model.lean | Model | 3 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 4 | 1 | [
{
"theorem_name": "complete",
"depth": 1,
"n_commands": 0,
"n_lines": 3,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " complete_aux T φ fun M _ s _ _ ↦ by\n rcases standardModel_unique M s\n exact H M",
"n_chars": 85,
"n_subproofs": 0,
"n_ta... | [
{
"name": "standardModel_unique",
"text": "lemma standardModel_unique (s : Structure ℒₒᵣ M)\n [hZero : Structure.Zero ℒₒᵣ M] [hOne : Structure.One ℒₒᵣ M] [hAdd : Structure.Add ℒₒᵣ M] [hMul : Structure.Mul ℒₒᵣ M]\n [hEq : Structure.Eq ℒₒᵣ M] [hLT : Structure.LT ℒₒᵣ M] : s = standardModel M :=\n standa... | [
{
"name": "weakerThan_of_models",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 4,
"n_lines": 8,
"n_chars": 316,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
"max_ne... | 4 | module
public import Foundation.FirstOrder.Arithmetic.Basic.Misc
@[expose] public section
namespace LO.FirstOrder.Arithmetic
private lemma complete_aux (T : ArithmeticTheory) [𝗘𝗤 ℒₒᵣ ⪯ T] (φ : ArithmeticSentence)
(H : ∀ (M : Type w)
[ORingStructure M]
[Structure ℒₒᵣ M]
[Structu... | module
public import Foundation.FirstOrder.Arithmetic.Basic.Misc
@[expose] public section
namespace LO.FirstOrder.Arithmetic
private lemma complete_aux (T : ArithmeticTheory) [𝗘𝗤 ℒₒᵣ ⪯ T] (φ : ArithmeticSentence)
(H : ∀ (M : Type w)
[ORingStructure M]
[Structure ℒₒᵣ M]
[Structu... | @@ -67,11 +67,18 @@
| _, Language.Eq.eq => by simp
| _, Language.LT.lt => by simp)
+lemma standardModel_unique (s : Structure ℒₒᵣ M)
+ [hZero : Structure.Zero ℒₒᵣ M] [hOne : Structure.One ℒₒᵣ M] [hAdd : Structure.Add ℒₒᵣ M] [hMul : Structure.Mul ℒₒᵣ M]
+ [hEq : Structure.Eq ℒₒᵣ M] [hLT : Structure.LT ... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_6db9c1fef89d_0 | 2203ee412acbf321 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Kripke/Logic/GL/Completeness.lean | Completeness | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 2 | 2 | [
{
"theorem_name": "iff_unprovable_exists_finite_rooted_model",
"depth": 1,
"n_commands": 0,
"n_lines": 5,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n apply Iff.not_left;\n apply Iff.trans $ finite_completeness_TFAE (φ := φ) |>.out 0 3;\n push No... | [
{
"name": "finite_completeness_TFAE",
"text": "theorem finite_completeness_TFAE : [\n Modal.GL ⊢ φ,\n FrameClass.finite_GL ⊧ φ,\n ∀ F : Kripke.Frame, [F.IsFinite] → [F.IsTransitive] → [F.IsIrreflexive] → [F.IsRooted] → F ⊧ φ,\n ∀ M : Kripke.Model, [M.IsFinite] → [M.IsTransitive] → [M.IsIrreflexive] → [M... | [
{
"name": "iff_unprovable_exists_finite_rooted_model",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 7,
"n_chars": 310,
"n_subproofs": 0,
"n_tactics": 9,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 2,
"automation_only... | 1 | module
public import Foundation.Modal.ComplementClosedConsistentFinset
public import Foundation.Modal.Kripke.Logic.GL.Soundness
public import Foundation.Modal.Kripke.Logic.K4
@[expose] public section
namespace LO.Modal
open Kripke
open LO.Entailment LO.Entailment.FiniteContext LO.Modal.Entailment
open Formula
open ... | module
public import Foundation.Modal.ComplementClosedConsistentFinset
public import Foundation.Modal.Kripke.Logic.GL.Soundness
public import Foundation.Modal.Kripke.Logic.K4
@[expose] public section
namespace LO.Modal
open Kripke
open LO.Entailment LO.Entailment.FiniteContext LO.Modal.Entailment
open Formula
open ... | @@ -212,8 +212,33 @@
all_goals grind;
⟩
-lemma iff_unprovable_exists_finite_rooted_model : Modal.GL ⊬ φ ↔ ∃ M : Model, ∃ _ : M.IsFinite, ∃ _ : M.IsTransitive, ∃ _ : M.IsIrreflexive, ∃ _ : M.IsRooted, ¬M.root.1 ⊧ φ := sorry
+theorem finite_completeness_TFAE : [
+ Modal.GL ⊢ φ,
+ FrameClass.finite_GL ⊧ 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_6db9c1fef89d_1 | 72972f3ff5db0cb8 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Kripke/Logic/GL/Completeness.lean | Completeness | 1 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 2 | 2 | [
{
"theorem_name": "iff_unprovable_exists_finite_rooted_model",
"depth": 1,
"n_commands": 0,
"n_lines": 5,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n apply Iff.not_left;\n apply Iff.trans $ finite_completeness_TFAE (φ := φ) |>.out 0 3;\n push No... | [
{
"name": "finite_completeness_TFAE",
"text": "theorem finite_completeness_TFAE : [\n Modal.GL ⊢ φ,\n FrameClass.finite_GL ⊧ φ,\n ∀ F : Kripke.Frame, [F.IsFinite] → [F.IsTransitive] → [F.IsIrreflexive] → [F.IsRooted] → F ⊧ φ,\n ∀ M : Kripke.Model, [M.IsFinite] → [M.IsTransitive] → [M.IsIrreflexive] → [M... | [
{
"name": "fintype_completeness_TFAE",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 21,
"n_chars": 672,
"n_subproofs": 2,
"n_tactics": 26,
"cyclomatic": 2,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 8,
"automation_only": false,
... | 1 | module
public import Foundation.Modal.ComplementClosedConsistentFinset
public import Foundation.Modal.Kripke.Logic.GL.Soundness
public import Foundation.Modal.Kripke.Logic.K4
@[expose] public section
namespace LO.Modal
open Kripke
open LO.Entailment LO.Entailment.FiniteContext LO.Modal.Entailment
open Formula
open ... | module
public import Foundation.Modal.ComplementClosedConsistentFinset
public import Foundation.Modal.Kripke.Logic.GL.Soundness
public import Foundation.Modal.Kripke.Logic.K4
@[expose] public section
namespace LO.Modal
open Kripke
open LO.Entailment LO.Entailment.FiniteContext LO.Modal.Entailment
open Formula
open ... | @@ -212,11 +212,46 @@
all_goals grind;
⟩
+theorem finite_completeness_TFAE : [
+ Modal.GL ⊢ φ,
+ FrameClass.finite_GL ⊧ φ,
+ ∀ F : Kripke.Frame, [F.IsFinite] → [F.IsTransitive] → [F.IsIrreflexive] → [F.IsRooted] → F ⊧ φ,
+ ∀ M : Kripke.Model, [M.IsFinite] → [M.IsTransitive] → [M.IsIrreflexive] → [M.IsRoot... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_6db9c1fef89d_2 | 92c6e5d0a5cb2a6f | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Kripke/Logic/GL/Completeness.lean | Completeness | 2 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 2 | 1 | [
{
"theorem_name": "iff_unprovable_exists_fintype_rooted_model",
"depth": 1,
"n_commands": 0,
"n_lines": 5,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n apply Iff.not_left;\n apply Iff.trans $ fintype_completeness_TFAE (φ := φ) |>.out 0 2;\n push ... | [
{
"name": "fintype_completeness_TFAE",
"text": "theorem fintype_completeness_TFAE : [\n Modal.GL ⊢ φ,\n ∀ F : Kripke.Frame, [Fintype F] → [F.IsTransitive] → [F.IsIrreflexive] → [F.IsRooted] → F ⊧ φ,\n ∀ M : Kripke.Model, [Fintype M] → [M.IsTransitive] → [M.IsIrreflexive] → [M.IsRooted] → M.root.1 ⊧ φ,\n]... | [
{
"name": "iff_unprovable_exists_fintype_rooted_model",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 2,
"n_lines": 7,
"n_chars": 311,
"n_subproofs": 0,
"n_tactics": 9,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 2,
"automation_onl... | 2 | module
public import Foundation.Modal.ComplementClosedConsistentFinset
public import Foundation.Modal.Kripke.Logic.GL.Soundness
public import Foundation.Modal.Kripke.Logic.K4
@[expose] public section
namespace LO.Modal
open Kripke
open LO.Entailment LO.Entailment.FiniteContext LO.Modal.Entailment
open Formula
open ... | module
public import Foundation.Modal.ComplementClosedConsistentFinset
public import Foundation.Modal.Kripke.Logic.GL.Soundness
public import Foundation.Modal.Kripke.Logic.K4
@[expose] public section
namespace LO.Modal
open Kripke
open LO.Entailment LO.Entailment.FiniteContext LO.Modal.Entailment
open Formula
open ... | @@ -233,8 +233,32 @@
exact Model.pointGenerate.pMorphism M x |>.modal_equivalence ⟨x, by tauto⟩ |>.mp $ H (M↾x);
tfae_finish;
-lemma iff_unprovable_exists_fintype_rooted_model : Modal.GL ⊬ φ ↔ ∃ M : Model, ∃ _ : Fintype M, ∃ _ : M.IsTransitive, ∃ _ : M.IsIrreflexive, ∃ _ : M.IsRooted, ¬M.root.1 ⊧ φ := sorry
+... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_cc99a6eb1ab1_0 | 54da9242e4fbbf28 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Bootstrapping/DerivabilityCondition/PeanoMinus.lean | PeanoMinus | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 6 | 1 | [
{
"theorem_name": "numeral_add",
"depth": 1,
"n_commands": 0,
"n_lines": 30,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n induction m using sigma1_pos_succ_induction\n · simp only [tprovable_iff_provable, val_equals, val_add, val_numeral]\n def... | [
{
"name": "term_add_assoc",
"text": "lemma term_add_assoc (t₁ t₂ t₃ : Term V ℒₒᵣ) :\n T.internalize V ⊢ t₁ + (t₂ + t₃) ≐ (t₁ + t₂) + t₃ := by\n have : T ⊢ “∀ x y z, x + (y + z) = (x + y) + z” :=\n complete.{0} T _ fun (M : Type) _ (hM : M↓[ℒₒᵣ] ⊧* T) ↦ by\n have : M↓[ℒₒᵣ] ⊧* 𝗣𝗔⁻ := models_of_s... | [
{
"name": "numeral_add",
"fan_in": 2,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 33,
"n_chars": 1677,
"n_subproofs": 10,
"n_tactics": 30,
"cyclomatic": 3,
"n_automation": 11,
"n_rewrites": 0,
"n_structural": 2,
"automation_only": false,
"max_nestin... | 1 | module
public import Foundation.FirstOrder.Bootstrapping.DerivabilityCondition.EquationalTheory
@[expose] public section
/-!
# Bootstrapping theory $\mathsf{PA}^-$, $\mathsf{R_0}$ in $\mathsf{I}\Sigma_1$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗣𝗔⁻]
end LO.FirstOr... | module
public import Foundation.FirstOrder.Bootstrapping.DerivabilityCondition.EquationalTheory
@[expose] public section
/-!
# Bootstrapping theory $\mathsf{PA}^-$, $\mathsf{R_0}$ in $\mathsf{I}\Sigma_1$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗣𝗔⁻]
end LO.FirstOr... | @@ -35,6 +35,16 @@
have : 𝗣𝗔⁻ ⪯ T := inferInstance
WeakerThan.trans inferInstance this
+lemma term_add_assoc (t₁ t₂ t₃ : Term V ℒₒᵣ) :
+ T.internalize V ⊢ t₁ + (t₂ + t₃) ≐ (t₁ + t₂) + t₃ := by
+ have : T ⊢ “∀ x y z, x + (y + z) = (x + y) + z” :=
+ complete.{0} T _ fun (M : Type) _ (hM : M↓[ℒₒᵣ] ⊧* 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_cc99a6eb1ab1_1 | 19e0c758ed35248b | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Bootstrapping/DerivabilityCondition/PeanoMinus.lean | PeanoMinus | 1 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 6 | 2 | [
{
"theorem_name": "numeral_mul",
"depth": 1,
"n_commands": 0,
"n_lines": 37,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n induction m using sigma1_pos_succ_induction\n · simp only [tprovable_iff_provable, val_equals, val_mul, val_numeral]\n def... | [
{
"name": "numeral_add",
"text": "lemma numeral_add (n m : V) :\n T.internalize V ⊢ (𝕹 n + 𝕹 m) ≐ 𝕹 (n + m) := by\n induction m using sigma1_pos_succ_induction\n · simp only [tprovable_iff_provable, val_equals, val_add, val_numeral]\n definability\n case zero =>\n have : T.internalize V ⊢ ∀⁰ ... | [
{
"name": "numeral_mul",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 2,
"n_lines": 40,
"n_chars": 2127,
"n_subproofs": 15,
"n_tactics": 37,
"cyclomatic": 2,
"n_automation": 11,
"n_rewrites": 0,
"n_structural": 2,
"automation_only": false,
"max_nestin... | 2 | module
public import Foundation.FirstOrder.Bootstrapping.DerivabilityCondition.EquationalTheory
@[expose] public section
/-!
# Bootstrapping theory $\mathsf{PA}^-$, $\mathsf{R_0}$ in $\mathsf{I}\Sigma_1$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗣𝗔⁻]
end LO.FirstOr... | module
public import Foundation.FirstOrder.Bootstrapping.DerivabilityCondition.EquationalTheory
@[expose] public section
/-!
# Bootstrapping theory $\mathsf{PA}^-$, $\mathsf{R_0}$ in $\mathsf{I}\Sigma_1$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗣𝗔⁻]
end LO.FirstOr... | @@ -45,6 +45,38 @@
simpa using internal_provable_of_outer_provable (V := V) this
simpa using TProof.specialize₃! this t₃ t₂ t₁
+lemma numeral_add (n m : V) :
+ T.internalize V ⊢ (𝕹 n + 𝕹 m) ≐ 𝕹 (n + m) := by
+ induction m using sigma1_pos_succ_induction
+ · simp only [tprovable_iff_provable, val_equal... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_cc99a6eb1ab1_2 | f04129ce32e67280 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Bootstrapping/DerivabilityCondition/PeanoMinus.lean | PeanoMinus | 2 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 6 | 1 | [
{
"theorem_name": "numeral_add",
"depth": 1,
"n_commands": 0,
"n_lines": 30,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n induction m using sigma1_pos_succ_induction\n · simp only [tprovable_iff_provable, val_equals, val_add, val_numeral]\n def... | [
{
"name": "term_add_assoc",
"text": "lemma term_add_assoc (t₁ t₂ t₃ : Term V ℒₒᵣ) :\n T.internalize V ⊢ t₁ + (t₂ + t₃) ≐ (t₁ + t₂) + t₃ := by\n have : T ⊢ “∀ x y z, x + (y + z) = (x + y) + z” :=\n complete.{0} T _ fun (M : Type) _ (hM : M↓[ℒₒᵣ] ⊧* T) ↦ by\n have : M↓[ℒₒᵣ] ⊧* 𝗣𝗔⁻ := models_of_s... | [
{
"name": "numeral_lt",
"fan_in": 3,
"n_deps_direct": 2,
"n_deps_transitive": 3,
"n_lines": 22,
"n_chars": 1298,
"n_subproofs": 9,
"n_tactics": 19,
"cyclomatic": 1,
"n_automation": 5,
"n_rewrites": 0,
"n_structural": 2,
"automation_only": false,
"max_nesting":... | 3 | module
public import Foundation.FirstOrder.Bootstrapping.DerivabilityCondition.EquationalTheory
@[expose] public section
/-!
# Bootstrapping theory $\mathsf{PA}^-$, $\mathsf{R_0}$ in $\mathsf{I}\Sigma_1$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗣𝗔⁻]
lemma lt_add_s... | module
public import Foundation.FirstOrder.Bootstrapping.DerivabilityCondition.EquationalTheory
@[expose] public section
/-!
# Bootstrapping theory $\mathsf{PA}^-$, $\mathsf{R_0}$ in $\mathsf{I}\Sigma_1$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗣𝗔⁻]
lemma lt_add_s... | @@ -37,6 +37,16 @@
have : 𝗣𝗔⁻ ⪯ T := inferInstance
WeakerThan.trans inferInstance this
+lemma term_add_assoc (t₁ t₂ t₃ : Term V ℒₒᵣ) :
+ T.internalize V ⊢ t₁ + (t₂ + t₃) ≐ (t₁ + t₂) + t₃ := by
+ have : T ⊢ “∀ x y z, x + (y + z) = (x + y) + z” :=
+ complete.{0} T _ fun (M : Type) _ (hM : M↓[ℒₒᵣ] ⊧* 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_cc99a6eb1ab1_3 | 0c7a2be30d3ea37a | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Bootstrapping/DerivabilityCondition/PeanoMinus.lean | PeanoMinus | 3 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 6 | 2 | [
{
"theorem_name": "numeral_mul",
"depth": 1,
"n_commands": 0,
"n_lines": 37,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n induction m using sigma1_pos_succ_induction\n · simp only [tprovable_iff_provable, val_equals, val_mul, val_numeral]\n def... | [
{
"name": "numeral_add",
"text": "lemma numeral_add (n m : V) :\n T.internalize V ⊢ (𝕹 n + 𝕹 m) ≐ 𝕹 (n + m) := by\n induction m using sigma1_pos_succ_induction\n · simp only [tprovable_iff_provable, val_equals, val_add, val_numeral]\n definability\n case zero =>\n have : T.internalize V ⊢ ∀⁰ ... | [
{
"name": "numeral_ne",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 4,
"n_lines": 16,
"n_chars": 827,
"n_subproofs": 6,
"n_tactics": 13,
"cyclomatic": 1,
"n_automation": 4,
"n_rewrites": 0,
"n_structural": 3,
"automation_only": false,
"max_nesting": ... | 4 | module
public import Foundation.FirstOrder.Bootstrapping.DerivabilityCondition.EquationalTheory
@[expose] public section
/-!
# Bootstrapping theory $\mathsf{PA}^-$, $\mathsf{R_0}$ in $\mathsf{I}\Sigma_1$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗣𝗔⁻]
lemma lt_add_s... | module
public import Foundation.FirstOrder.Bootstrapping.DerivabilityCondition.EquationalTheory
@[expose] public section
/-!
# Bootstrapping theory $\mathsf{PA}^-$, $\mathsf{R_0}$ in $\mathsf{I}\Sigma_1$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗣𝗔⁻]
lemma lt_add_s... | @@ -47,6 +47,38 @@
simpa using internal_provable_of_outer_provable (V := V) this
simpa using TProof.specialize₃! this t₃ t₂ t₁
+lemma numeral_add (n m : V) :
+ T.internalize V ⊢ (𝕹 n + 𝕹 m) ≐ 𝕹 (n + m) := by
+ induction m using sigma1_pos_succ_induction
+ · simp only [tprovable_iff_provable, val_equal... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_cc99a6eb1ab1_4 | 602d0b109cc51564 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Bootstrapping/DerivabilityCondition/PeanoMinus.lean | PeanoMinus | 4 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 6 | 2 | [
{
"theorem_name": "numeral_mul",
"depth": 1,
"n_commands": 0,
"n_lines": 37,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n induction m using sigma1_pos_succ_induction\n · simp only [tprovable_iff_provable, val_equals, val_mul, val_numeral]\n def... | [
{
"name": "numeral_add",
"text": "lemma numeral_add (n m : V) :\n T.internalize V ⊢ (𝕹 n + 𝕹 m) ≐ 𝕹 (n + m) := by\n induction m using sigma1_pos_succ_induction\n · simp only [tprovable_iff_provable, val_equals, val_add, val_numeral]\n definability\n case zero =>\n have : T.internalize V ⊢ ∀⁰ ... | [
{
"name": "numeral_nlt",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 4,
"n_lines": 20,
"n_chars": 1081,
"n_subproofs": 8,
"n_tactics": 17,
"cyclomatic": 2,
"n_automation": 7,
"n_rewrites": 0,
"n_structural": 2,
"automation_only": false,
"max_nesting"... | 4 | module
public import Foundation.FirstOrder.Bootstrapping.DerivabilityCondition.EquationalTheory
@[expose] public section
/-!
# Bootstrapping theory $\mathsf{PA}^-$, $\mathsf{R_0}$ in $\mathsf{I}\Sigma_1$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗣𝗔⁻]
lemma lt_add_s... | module
public import Foundation.FirstOrder.Bootstrapping.DerivabilityCondition.EquationalTheory
@[expose] public section
/-!
# Bootstrapping theory $\mathsf{PA}^-$, $\mathsf{R_0}$ in $\mathsf{I}\Sigma_1$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗣𝗔⁻]
lemma lt_add_s... | @@ -47,6 +47,38 @@
simpa using internal_provable_of_outer_provable (V := V) this
simpa using TProof.specialize₃! this t₃ t₂ t₁
+lemma numeral_add (n m : V) :
+ T.internalize V ⊢ (𝕹 n + 𝕹 m) ≐ 𝕹 (n + m) := by
+ induction m using sigma1_pos_succ_induction
+ · simp only [tprovable_iff_provable, val_equal... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_cc99a6eb1ab1_5 | 2b8fca14e575980d | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Bootstrapping/DerivabilityCondition/PeanoMinus.lean | PeanoMinus | 5 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 6 | 1 | [
{
"theorem_name": "lt_iff_substItrDisj",
"depth": 1,
"n_commands": 0,
"n_lines": 39,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n induction m using sigma1_pos_succ_induction\n · simp only [Nat.succ_eq_add_one, Nat.reduceAdd, Fin.isValue, tprovable... | [
{
"name": "lt_succ_iff_eq_or_succ",
"text": "lemma lt_succ_iff_eq_or_succ {a b : V} : a < b + 1 ↔ a = b ∨ a < b := by\n simp [lt_succ_iff_le, le_iff_eq_or_lt]\n\n",
"fan_in": 1,
"n_lines": 4,
"n_chars": 119,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
"n_automation": 1,
... | [
{
"name": "lt_iff_substItrDisj",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 42,
"n_chars": 2485,
"n_subproofs": 13,
"n_tactics": 39,
"cyclomatic": 2,
"n_automation": 13,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": false,
"ma... | 1 | module
public import Foundation.FirstOrder.Bootstrapping.DerivabilityCondition.EquationalTheory
@[expose] public section
/-!
# Bootstrapping theory $\mathsf{PA}^-$, $\mathsf{R_0}$ in $\mathsf{I}\Sigma_1$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗣𝗔⁻]
end LO.FirstOr... | module
public import Foundation.FirstOrder.Bootstrapping.DerivabilityCondition.EquationalTheory
@[expose] public section
/-!
# Bootstrapping theory $\mathsf{PA}^-$, $\mathsf{R_0}$ in $\mathsf{I}\Sigma_1$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗣𝗔⁻]
lemma lt_succ_... | @@ -11,6 +11,9 @@
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗣𝗔⁻]
+lemma lt_succ_iff_eq_or_succ {a b : V} : a < b + 1 ↔ a = b ∨ a < b := by
+ simp [lt_succ_iff_le, le_iff_eq_or_lt]
+
end LO.FirstOrder.Arithmetic
namespace LO.FirstOrder.Arithmetic.Bootstrapping.Arithmetic
@@ -68,7 +71,10 @@
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_cc99a6eb1ab1_6 | 5a71ebdc14a25b3f | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Bootstrapping/DerivabilityCondition/PeanoMinus.lean | PeanoMinus | 6 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 6 | 1 | [
{
"theorem_name": "lt_iff_substItrDisj",
"depth": 1,
"n_commands": 0,
"n_lines": 39,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n induction m using sigma1_pos_succ_induction\n · simp only [Nat.succ_eq_add_one, Nat.reduceAdd, Fin.isValue, tprovable... | [
{
"name": "lt_succ_iff_eq_or_succ",
"text": "lemma lt_succ_iff_eq_or_succ {a b : V} : a < b + 1 ↔ a = b ∨ a < b := by\n simp [lt_succ_iff_le, le_iff_eq_or_lt]\n\n",
"fan_in": 1,
"n_lines": 4,
"n_chars": 119,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
"n_automation": 1,
... | [
{
"name": "ball_intro",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 2,
"n_lines": 19,
"n_chars": 1071,
"n_subproofs": 6,
"n_tactics": 15,
"cyclomatic": 1,
"n_automation": 3,
"n_rewrites": 6,
"n_structural": 3,
"automation_only": false,
"max_nesting":... | 2 | module
public import Foundation.FirstOrder.Bootstrapping.DerivabilityCondition.EquationalTheory
@[expose] public section
/-!
# Bootstrapping theory $\mathsf{PA}^-$, $\mathsf{R_0}$ in $\mathsf{I}\Sigma_1$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗣𝗔⁻]
end LO.FirstOr... | module
public import Foundation.FirstOrder.Bootstrapping.DerivabilityCondition.EquationalTheory
@[expose] public section
/-!
# Bootstrapping theory $\mathsf{PA}^-$, $\mathsf{R_0}$ in $\mathsf{I}\Sigma_1$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗣𝗔⁻]
lemma lt_succ_... | @@ -11,6 +11,9 @@
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗣𝗔⁻]
+lemma lt_succ_iff_eq_or_succ {a b : V} : a < b + 1 ↔ a = b ∨ a < b := by
+ simp [lt_succ_iff_le, le_iff_eq_or_lt]
+
end LO.FirstOrder.Arithmetic
namespace LO.FirstOrder.Arithmetic.Bootstrapping.Arithmetic
@@ -68,7 +71,10 @@
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_cc99a6eb1ab1_7 | c4fb003e8ef2194e | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Bootstrapping/DerivabilityCondition/PeanoMinus.lean | PeanoMinus | 7 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 6 | 2 | [
{
"theorem_name": "numeral_mul",
"depth": 1,
"n_commands": 0,
"n_lines": 37,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n induction m using sigma1_pos_succ_induction\n · simp only [tprovable_iff_provable, val_equals, val_mul, val_numeral]\n def... | [
{
"name": "numeral_add",
"text": "lemma numeral_add (n m : V) :\n T.internalize V ⊢ (𝕹 n + 𝕹 m) ≐ 𝕹 (n + m) := by\n induction m using sigma1_pos_succ_induction\n · simp only [tprovable_iff_provable, val_equals, val_add, val_numeral]\n definability\n case zero =>\n have : T.internalize V ⊢ ∀⁰ ... | [
{
"name": "bexs_intro",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 4,
"n_lines": 10,
"n_chars": 342,
"n_subproofs": 1,
"n_tactics": 6,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 4,
"automation_only": false,
"max_nesting": 2... | 4 | module
public import Foundation.FirstOrder.Bootstrapping.DerivabilityCondition.EquationalTheory
@[expose] public section
/-!
# Bootstrapping theory $\mathsf{PA}^-$, $\mathsf{R_0}$ in $\mathsf{I}\Sigma_1$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗣𝗔⁻]
lemma lt_add_s... | module
public import Foundation.FirstOrder.Bootstrapping.DerivabilityCondition.EquationalTheory
@[expose] public section
/-!
# Bootstrapping theory $\mathsf{PA}^-$, $\mathsf{R_0}$ in $\mathsf{I}\Sigma_1$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗣𝗔⁻]
lemma lt_add_s... | @@ -47,6 +47,38 @@
simpa using internal_provable_of_outer_provable (V := V) this
simpa using TProof.specialize₃! this t₃ t₂ t₁
+lemma numeral_add (n m : V) :
+ T.internalize V ⊢ (𝕹 n + 𝕹 m) ≐ 𝕹 (n + m) := by
+ induction m using sigma1_pos_succ_induction
+ · simp only [tprovable_iff_provable, val_equal... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"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_0 | 84a9dd4ef56dd985 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Logic/GL/Independency.lean | Independency | 0 | 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_independency",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 5,
"n_chars": 124,
"n_subproofs": 0,
"n_tactics": 5,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": false,
"max... | 1 | 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 ⋏ ∼□(... | @@ -22,8 +22,18 @@
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 : ∼∼⊥ ∈ Pr... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_e48307bb12d9_1 | 8d7fd2b04b513f95 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Logic/GL/Independency.lean | Independency | 1 | 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_independency_of_consistency",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 16,
"n_chars": 506,
"n_subproofs": 0,
"n_tactics": 11,
"cyclomatic": 2,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 4,
"automation_o... | 1 | 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 ⋏ ∼□(... | @@ -22,10 +22,18 @@
variable {n : ℕ} {φ : Formula ℕ}
+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 : ∼∼⊥ ∈ Propositional.Cl := GL.provable_verTranslated_Cl this;
+ have := Hil... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/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.