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_c9e9d75d2f5b_2 | 69b8c4d06fdd76f6 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Syntax/Predicate/Rew.lean | Rew | 2 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 14 | 1 | [
{
"theorem_name": "shift_Injective",
"depth": 1,
"n_commands": 0,
"n_lines": 3,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": "\n Function.LeftInverse.injective (g := map id Nat.pred)\n (by intros φ; simp only [← comp_app]; apply eq_id_of_eq <;> simp [c... | [
{
"name": "eq_id_of_eq",
"text": "lemma eq_id_of_eq {ω : Rew L ξ n ξ n} (hb : ∀ x, ω #x = #x) (he : ∀ x, ω &x = &x) (t) : ω t = t := by\n have : ω = Rew.id := by ext <;> simp [*]\n simp [this]\n\n",
"fan_in": 1,
"n_lines": 5,
"n_chars": 171,
"n_subproofs": 1,
"n_tactics": 4,
"cyclo... | [
{
"name": "shift_Injective",
"fan_in": 0,
"n_deps_direct": 2,
"n_deps_transitive": 4,
"n_lines": 5,
"n_chars": 200,
"n_subproofs": 0,
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"cyclomatic": 2,
"n_automation": 2,
"n_rewrites": 0,
"n_structural": 2,
"automation_only": false,
"max_nesting... | 1 | module
public import Foundation.Syntax.Predicate.Quantifier
public import Foundation.Syntax.Predicate.Term
public import Foundation.Vorspiel.Function
/-!
# Rewriting
term/formula morphisms such as Rewritings, substitutions, and embs are handled by the structure `LO.FirstOrder.Rew`.
- `LO.FirstOrder.Rew.rewrite f` is... | module
public import Foundation.Syntax.Predicate.Quantifier
public import Foundation.Syntax.Predicate.Term
public import Foundation.Vorspiel.Function
/-!
# Rewriting
term/formula morphisms such as Rewritings, substitutions, and embs are handled by the structure `LO.FirstOrder.Rew`.
- `LO.FirstOrder.Rew.rewrite f` is... | @@ -110,6 +110,10 @@
protected def q (ω : Rew L ξ₁ n₁ ξ₂ n₂) : Rew L ξ₁ (n₁ + 1) ξ₂ (n₂ + 1) :=
bind (#0 :> bShift ∘ ω ∘ bvar) (bShift ∘ ω ∘ fvar)
+lemma eq_id_of_eq {ω : Rew L ξ n ξ n} (hb : ∀ x, ω #x = #x) (he : ∀ x, ω &x = &x) (t) : ω t = t := by
+ have : ω = Rew.id := by ext <;> simp [*]
+ simp [this]
+
de... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_c9e9d75d2f5b_3 | 4c7ab61121978235 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Syntax/Predicate/Rew.lean | Rew | 3 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 14 | 2 | [
{
"theorem_name": "fixitr_bvar",
"depth": 1,
"n_commands": 0,
"n_lines": 5,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n induction m\n · simp [*]\n case succ m ih =>\n rw [fixitr_succ, comp_app, ih]; simp [Fin.castSucc_castAdd]",
"n_chars"... | [
{
"name": "fixitr_succ",
"text": "lemma fixitr_succ (m) :\n fixitr (L := L) n (m + 1) = Rew.fix.comp (fixitr n m) := by\n simp [fixitr]\n\n",
"fan_in": 2,
"n_lines": 5,
"n_chars": 105,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
... | [
{
"name": "fixitr_bvar",
"fan_in": 0,
"n_deps_direct": 2,
"n_deps_transitive": 2,
"n_lines": 7,
"n_chars": 222,
"n_subproofs": 0,
"n_tactics": 6,
"cyclomatic": 2,
"n_automation": 2,
"n_rewrites": 1,
"n_structural": 1,
"automation_only": false,
"max_nesting": 4... | 1 | module
public import Foundation.Syntax.Predicate.Quantifier
public import Foundation.Syntax.Predicate.Term
public import Foundation.Vorspiel.Function
/-!
# Rewriting
term/formula morphisms such as Rewritings, substitutions, and embs are handled by the structure `LO.FirstOrder.Rew`.
- `LO.FirstOrder.Rew.rewrite f` is... | module
public import Foundation.Syntax.Predicate.Quantifier
public import Foundation.Syntax.Predicate.Term
public import Foundation.Vorspiel.Function
/-!
# Rewriting
term/formula morphisms such as Rewritings, substitutions, and embs are handled by the structure `LO.FirstOrder.Rew`.
- `LO.FirstOrder.Rew.rewrite f` is... | @@ -493,8 +493,16 @@
@[simp] lemma fixitr_zero :
fixitr (L := L) n 0 = Rew.id := by simp [fixitr]
-@[simp] lemma fixitr_bvar (n m) (x : Fin n) : fixitr n m (#x : SyntacticSemiterm L n) = #(x.castAdd m) := sorry
+lemma fixitr_succ (m) :
+ fixitr (L := L) n (m + 1) = Rew.fix.comp (fixitr n m) := by
+ simp [fi... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_c9e9d75d2f5b_4 | d07c8125a01a780e | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Syntax/Predicate/Rew.lean | Rew | 4 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 14 | 1 | [
{
"theorem_name": "subst_positive",
"depth": 1,
"n_commands": 0,
"n_lines": 3,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n simpa [Semiterm.Positive, subst_bv]\n using ⟨fun H i hi x hx ↦ H x i hi hx, fun H x i hi hx ↦ H i hi x hx⟩",
"n_chars... | [
{
"name": "subst_bv",
"text": "lemma subst_bv (t : Semiterm L ξ n) (v : Fin n → Semiterm L ξ m) :\n (Rew.subst v t).bv = t.bv.biUnion (fun i ↦ (v i).bv) := by\n induction t <;> simp [Rew.func, Semiterm.bv_func, Finset.biUnion_biUnion, *]\n\n",
"fan_in": 1,
"n_lines": 5,
"n_chars": 216,
"... | [
{
"name": "subst_positive",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 2,
"n_lines": 6,
"n_chars": 281,
"n_subproofs": 0,
"n_tactics": 3,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"max_nesting":... | 1 | module
public import Foundation.Syntax.Predicate.Quantifier
public import Foundation.Syntax.Predicate.Term
public import Foundation.Vorspiel.Function
/-!
# Rewriting
term/formula morphisms such as Rewritings, substitutions, and embs are handled by the structure `LO.FirstOrder.Rew`.
- `LO.FirstOrder.Rew.rewrite f` is... | module
public import Foundation.Syntax.Predicate.Quantifier
public import Foundation.Syntax.Predicate.Term
public import Foundation.Vorspiel.Function
/-!
# Rewriting
term/formula morphisms such as Rewritings, substitutions, and embs are handled by the structure `LO.FirstOrder.Rew`.
- `LO.FirstOrder.Rew.rewrite f` is... | @@ -505,8 +505,14 @@
end Syntactic
+lemma subst_bv (t : Semiterm L ξ n) (v : Fin n → Semiterm L ξ m) :
+ (Rew.subst v t).bv = t.bv.biUnion (fun i ↦ (v i).bv) := by
+ induction t <;> simp [Rew.func, Semiterm.bv_func, Finset.biUnion_biUnion, *]
+
@[simp] lemma subst_positive (t : Semiterm L ξ n) (v : Fin n → 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_c9e9d75d2f5b_5 | 46e705597199720c | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Syntax/Predicate/Rew.lean | Rew | 5 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 14 | 1 | [
{
"theorem_name": "embSubsts_positive",
"depth": 1,
"n_commands": 0,
"n_lines": 3,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n simpa [Semiterm.Positive, embSubsts_bv]\n using ⟨fun H i hi x hx ↦ H x i hi hx, fun H x i hi hx ↦ H i hi x hx⟩",
... | [
{
"name": "embSubsts_bv",
"text": "lemma embSubsts_bv (t : ClosedSemiterm L n) (v : Fin n → Semiterm L ξ m) :\n (Rew.embSubsts v t).bv = t.bv.biUnion (fun i ↦ (v i).bv) := by\n induction t\n · simp\n · contradiction\n · simp [Rew.func, Semiterm.bv_func, Finset.biUnion_biUnion, *]\n\n",
"fan_in": ... | [
{
"name": "embSubsts_positive",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 2,
"n_lines": 6,
"n_chars": 296,
"n_subproofs": 0,
"n_tactics": 3,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"max_nesti... | 1 | module
public import Foundation.Syntax.Predicate.Quantifier
public import Foundation.Syntax.Predicate.Term
public import Foundation.Vorspiel.Function
/-!
# Rewriting
term/formula morphisms such as Rewritings, substitutions, and embs are handled by the structure `LO.FirstOrder.Rew`.
- `LO.FirstOrder.Rew.rewrite f` is... | module
public import Foundation.Syntax.Predicate.Quantifier
public import Foundation.Syntax.Predicate.Term
public import Foundation.Vorspiel.Function
/-!
# Rewriting
term/formula morphisms such as Rewritings, substitutions, and embs are handled by the structure `LO.FirstOrder.Rew`.
- `LO.FirstOrder.Rew.rewrite f` is... | @@ -514,8 +514,17 @@
simpa [Semiterm.Positive, subst_bv]
using ⟨fun H i hi x hx ↦ H x i hi hx, fun H x i hi hx ↦ H i hi x hx⟩
+lemma embSubsts_bv (t : ClosedSemiterm L n) (v : Fin n → Semiterm L ξ m) :
+ (Rew.embSubsts v t).bv = t.bv.biUnion (fun i ↦ (v i).bv) := by
+ induction t
+ · simp
+ · contradict... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_c9e9d75d2f5b_6 | 5e28db3b3ca07354 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Syntax/Predicate/Rew.lean | Rew | 6 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 14 | 1 | [
{
"theorem_name": "emb_bShift_term",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n simp [←comp_app, emb_comp_bShift_comm]",
"n_chars": 46,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
... | [
{
"name": "emb_comp_bShift_comm",
"text": "lemma emb_comp_bShift_comm {o : Type v₁} [IsEmpty o] :\n Rew.bShift.comp (Rew.emb : Rew L o n ξ n) = Rew.emb.comp Rew.bShift := by\n ext x\n · simp [comp_app]\n · exact IsEmpty.elim (by assumption) x\n\n",
"fan_in": 1,
"n_lines": 7,
"n_chars": 208... | [
{
"name": "emb_bShift_term",
"fan_in": 0,
"n_deps_direct": 2,
"n_deps_transitive": 4,
"n_lines": 5,
"n_chars": 193,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"max_nesting"... | 1 | module
public import Foundation.Syntax.Predicate.Quantifier
public import Foundation.Syntax.Predicate.Term
public import Foundation.Vorspiel.Function
/-!
# Rewriting
term/formula morphisms such as Rewritings, substitutions, and embs are handled by the structure `LO.FirstOrder.Rew`.
- `LO.FirstOrder.Rew.rewrite f` is... | module
public import Foundation.Syntax.Predicate.Quantifier
public import Foundation.Syntax.Predicate.Term
public import Foundation.Vorspiel.Function
/-!
# Rewriting
term/formula morphisms such as Rewritings, substitutions, and embs are handled by the structure `LO.FirstOrder.Rew`.
- `LO.FirstOrder.Rew.rewrite f` is... | @@ -529,8 +529,15 @@
@[simp] lemma bshift_positive (t : Semiterm L ξ n) : Positive (Rew.bShift t) := by
induction t <;> simp
+lemma emb_comp_bShift_comm {o : Type v₁} [IsEmpty o] :
+ Rew.bShift.comp (Rew.emb : Rew L o n ξ n) = Rew.emb.comp Rew.bShift := by
+ ext x
+ · simp [comp_app]
+ · exact IsEmpty.elim ... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_c9e9d75d2f5b_7 | 6a6277500f9e7552 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Syntax/Predicate/Rew.lean | Rew | 7 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 14 | 1 | [
{
"theorem_name": "lMap_bind",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n induction t <;> simp [*, -lMap_func, lMap_func', -Rew.func, Rew.func']",
"n_chars": 76,
"n_subproofs": 0,
"n_tactics"... | [
{
"name": "func'",
"text": "lemma func' {k} (f : L.Func k) (v : Fin k → Semiterm L ξ₁ n₁) :\n ω (func f v) = func f fun i ↦ ω (v i) := ω.func'' f v\n\n",
"fan_in": 1,
"n_lines": 4,
"n_chars": 135,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rew... | [
{
"name": "lMap_bind",
"fan_in": 3,
"n_deps_direct": 2,
"n_deps_transitive": 2,
"n_lines": 5,
"n_chars": 263,
"n_subproofs": 0,
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"cyclomatic": 3,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": false,
"max_nesting": 2
... | 1 | module
public import Foundation.Syntax.Predicate.Quantifier
public import Foundation.Syntax.Predicate.Term
public import Foundation.Vorspiel.Function
/-!
# Rewriting
term/formula morphisms such as Rewritings, substitutions, and embs are handled by the structure `LO.FirstOrder.Rew`.
- `LO.FirstOrder.Rew.rewrite f` is... | module
public import Foundation.Syntax.Predicate.Quantifier
public import Foundation.Syntax.Predicate.Term
public import Foundation.Vorspiel.Function
/-!
# Rewriting
term/formula morphisms such as Rewritings, substitutions, and embs are handled by the structure `LO.FirstOrder.Rew`.
- `LO.FirstOrder.Rew.rewrite f` is... | @@ -45,6 +45,9 @@
@[simp] protected lemma func {k} (f : L.Func k) (v : Fin k → Semiterm L ξ₁ n₁) :
ω (func f v) = func f (ω ∘ v) := ω.func'' f v
+lemma func' {k} (f : L.Func k) (v : Fin k → Semiterm L ξ₁ n₁) :
+ ω (func f v) = func f fun i ↦ ω (v i) := ω.func'' f v
+
@[ext] lemma ext (ω₁ ω₂ : Rew L ξ₁ n₁ ξ₂... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_c9e9d75d2f5b_8 | d5bb4f0593ac58c3 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Syntax/Predicate/Rew.lean | Rew | 8 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 14 | 3 | [
{
"theorem_name": "lMap_map",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n simp [map, lMap_bind, Function.comp_def]",
"n_chars": 46,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
"... | [
{
"name": "lMap_bind",
"text": "lemma lMap_bind (b : Fin n₁ → Semiterm L₁ ξ₂ n₂) (e : ξ₁ → Semiterm L₁ ξ₂ n₂) (t) :\n lMap Φ (bind b e t) = bind (lMap Φ ∘ b) (lMap Φ ∘ e) (t.lMap Φ) := by\n induction t <;> simp [*, -lMap_func, lMap_func', -Rew.func, Rew.func']\n\n",
"fan_in": 3,
"n_lines": 5,
... | [
{
"name": "lMap_map",
"fan_in": 2,
"n_deps_direct": 1,
"n_deps_transitive": 3,
"n_lines": 5,
"n_chars": 167,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"max_nesting": 2
}... | 2 | module
public import Foundation.Syntax.Predicate.Quantifier
public import Foundation.Syntax.Predicate.Term
public import Foundation.Vorspiel.Function
/-!
# Rewriting
term/formula morphisms such as Rewritings, substitutions, and embs are handled by the structure `LO.FirstOrder.Rew`.
- `LO.FirstOrder.Rew.rewrite f` is... | module
public import Foundation.Syntax.Predicate.Quantifier
public import Foundation.Syntax.Predicate.Term
public import Foundation.Vorspiel.Function
/-!
# Rewriting
term/formula morphisms such as Rewritings, substitutions, and embs are handled by the structure `LO.FirstOrder.Rew`.
- `LO.FirstOrder.Rew.rewrite f` is... | @@ -555,8 +555,13 @@
variable (Φ : L₁ →ᵥ L₂)
open Rew
+lemma lMap_bind (b : Fin n₁ → Semiterm L₁ ξ₂ n₂) (e : ξ₁ → Semiterm L₁ ξ₂ n₂) (t) :
+ lMap Φ (bind b e t) = bind (lMap Φ ∘ b) (lMap Φ ∘ e) (t.lMap Φ) := by
+ induction t <;> simp [*, -lMap_func, lMap_func', -Rew.func, Rew.func']
+
lemma lMap_map (b : Fin n... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_c9e9d75d2f5b_9 | f57a3717cef3d701 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Syntax/Predicate/Rew.lean | Rew | 9 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 14 | 3 | [
{
"theorem_name": "lMap_map",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n simp [map, lMap_bind, Function.comp_def]",
"n_chars": 46,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
"... | [
{
"name": "lMap_bind",
"text": "lemma lMap_bind (b : Fin n₁ → Semiterm L₁ ξ₂ n₂) (e : ξ₁ → Semiterm L₁ ξ₂ n₂) (t) :\n lMap Φ (bind b e t) = bind (lMap Φ ∘ b) (lMap Φ ∘ e) (t.lMap Φ) := by\n induction t <;> simp [*, -lMap_func, lMap_func', -Rew.func, Rew.func']\n\n",
"fan_in": 3,
"n_lines": 5,
... | [
{
"name": "lMap_bShift",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 4,
"n_lines": 4,
"n_chars": 121,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"max_nesting": 2
... | 3 | module
public import Foundation.Syntax.Predicate.Quantifier
public import Foundation.Syntax.Predicate.Term
public import Foundation.Vorspiel.Function
/-!
# Rewriting
term/formula morphisms such as Rewritings, substitutions, and embs are handled by the structure `LO.FirstOrder.Rew`.
- `LO.FirstOrder.Rew.rewrite f` is... | module
public import Foundation.Syntax.Predicate.Quantifier
public import Foundation.Syntax.Predicate.Term
public import Foundation.Vorspiel.Function
/-!
# Rewriting
term/formula morphisms such as Rewritings, substitutions, and embs are handled by the structure `LO.FirstOrder.Rew`.
- `LO.FirstOrder.Rew.rewrite f` is... | @@ -555,8 +555,13 @@
variable (Φ : L₁ →ᵥ L₂)
open Rew
+lemma lMap_bind (b : Fin n₁ → Semiterm L₁ ξ₂ n₂) (e : ξ₁ → Semiterm L₁ ξ₂ n₂) (t) :
+ lMap Φ (bind b e t) = bind (lMap Φ ∘ b) (lMap Φ ∘ e) (t.lMap Φ) := by
+ induction t <;> simp [*, -lMap_func, lMap_func', -Rew.func, Rew.func']
+
lemma lMap_map (b : Fin n... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_c9e9d75d2f5b_10 | 566a7f4b4ac7569b | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Syntax/Predicate/Rew.lean | Rew | 10 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 14 | 3 | [
{
"theorem_name": "lMap_map",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n simp [map, lMap_bind, Function.comp_def]",
"n_chars": 46,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
"... | [
{
"name": "lMap_bind",
"text": "lemma lMap_bind (b : Fin n₁ → Semiterm L₁ ξ₂ n₂) (e : ξ₁ → Semiterm L₁ ξ₂ n₂) (t) :\n lMap Φ (bind b e t) = bind (lMap Φ ∘ b) (lMap Φ ∘ e) (t.lMap Φ) := by\n induction t <;> simp [*, -lMap_func, lMap_func', -Rew.func, Rew.func']\n\n",
"fan_in": 3,
"n_lines": 5,
... | [
{
"name": "lMap_shift",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 4,
"n_lines": 4,
"n_chars": 120,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"max_nesting": 2
... | 3 | module
public import Foundation.Syntax.Predicate.Quantifier
public import Foundation.Syntax.Predicate.Term
public import Foundation.Vorspiel.Function
/-!
# Rewriting
term/formula morphisms such as Rewritings, substitutions, and embs are handled by the structure `LO.FirstOrder.Rew`.
- `LO.FirstOrder.Rew.rewrite f` is... | module
public import Foundation.Syntax.Predicate.Quantifier
public import Foundation.Syntax.Predicate.Term
public import Foundation.Vorspiel.Function
/-!
# Rewriting
term/formula morphisms such as Rewritings, substitutions, and embs are handled by the structure `LO.FirstOrder.Rew`.
- `LO.FirstOrder.Rew.rewrite f` is... | @@ -555,8 +555,13 @@
variable (Φ : L₁ →ᵥ L₂)
open Rew
+lemma lMap_bind (b : Fin n₁ → Semiterm L₁ ξ₂ n₂) (e : ξ₁ → Semiterm L₁ ξ₂ n₂) (t) :
+ lMap Φ (bind b e t) = bind (lMap Φ ∘ b) (lMap Φ ∘ e) (t.lMap Φ) := by
+ induction t <;> simp [*, -lMap_func, lMap_func', -Rew.func, Rew.func']
+
lemma lMap_map (b : Fin n... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_c9e9d75d2f5b_11 | 2576605c9c53f2d3 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Syntax/Predicate/Rew.lean | Rew | 11 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 14 | 3 | [
{
"theorem_name": "lMap_map",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n simp [map, lMap_bind, Function.comp_def]",
"n_chars": 46,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
"... | [
{
"name": "lMap_bind",
"text": "lemma lMap_bind (b : Fin n₁ → Semiterm L₁ ξ₂ n₂) (e : ξ₁ → Semiterm L₁ ξ₂ n₂) (t) :\n lMap Φ (bind b e t) = bind (lMap Φ ∘ b) (lMap Φ ∘ e) (t.lMap Φ) := by\n induction t <;> simp [*, -lMap_func, lMap_func', -Rew.func, Rew.func']\n\n",
"fan_in": 3,
"n_lines": 5,
... | [
{
"name": "lMap_free",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 3,
"n_lines": 5,
"n_chars": 207,
"n_subproofs": 0,
"n_tactics": 4,
"cyclomatic": 1,
"n_automation": 3,
"n_rewrites": 1,
"n_structural": 2,
"automation_only": false,
"max_nesting": 2
... | 2 | module
public import Foundation.Syntax.Predicate.Quantifier
public import Foundation.Syntax.Predicate.Term
public import Foundation.Vorspiel.Function
/-!
# Rewriting
term/formula morphisms such as Rewritings, substitutions, and embs are handled by the structure `LO.FirstOrder.Rew`.
- `LO.FirstOrder.Rew.rewrite f` is... | module
public import Foundation.Syntax.Predicate.Quantifier
public import Foundation.Syntax.Predicate.Term
public import Foundation.Vorspiel.Function
/-!
# Rewriting
term/formula morphisms such as Rewritings, substitutions, and embs are handled by the structure `LO.FirstOrder.Rew`.
- `LO.FirstOrder.Rew.rewrite f` is... | @@ -555,8 +555,14 @@
variable (Φ : L₁ →ᵥ L₂)
open Rew
-lemma lMap_free (t : SyntacticSemiterm L₁ (n + 1)) : (free t).lMap Φ = free (t.lMap Φ) := sorry
+lemma lMap_bind (b : Fin n₁ → Semiterm L₁ ξ₂ n₂) (e : ξ₁ → Semiterm L₁ ξ₂ n₂) (t) :
+ lMap Φ (bind b e t) = bind (lMap Φ ∘ b) (lMap Φ ∘ e) (t.lMap Φ) := by
+ in... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_c9e9d75d2f5b_12 | 1e8d43c845512e4a | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Syntax/Predicate/Rew.lean | Rew | 12 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 14 | 3 | [
{
"theorem_name": "lMap_map",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n simp [map, lMap_bind, Function.comp_def]",
"n_chars": 46,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
"... | [
{
"name": "lMap_bind",
"text": "lemma lMap_bind (b : Fin n₁ → Semiterm L₁ ξ₂ n₂) (e : ξ₁ → Semiterm L₁ ξ₂ n₂) (t) :\n lMap Φ (bind b e t) = bind (lMap Φ ∘ b) (lMap Φ ∘ e) (t.lMap Φ) := by\n induction t <;> simp [*, -lMap_func, lMap_func', -Rew.func, Rew.func']\n\n",
"fan_in": 3,
"n_lines": 5,
... | [
{
"name": "lMap_fix",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 3,
"n_lines": 5,
"n_chars": 154,
"n_subproofs": 0,
"n_tactics": 6,
"cyclomatic": 3,
"n_automation": 2,
"n_rewrites": 1,
"n_structural": 2,
"automation_only": false,
"max_nesting": 2
... | 2 | module
public import Foundation.Syntax.Predicate.Quantifier
public import Foundation.Syntax.Predicate.Term
public import Foundation.Vorspiel.Function
/-!
# Rewriting
term/formula morphisms such as Rewritings, substitutions, and embs are handled by the structure `LO.FirstOrder.Rew`.
- `LO.FirstOrder.Rew.rewrite f` is... | module
public import Foundation.Syntax.Predicate.Quantifier
public import Foundation.Syntax.Predicate.Term
public import Foundation.Vorspiel.Function
/-!
# Rewriting
term/formula morphisms such as Rewritings, substitutions, and embs are handled by the structure `LO.FirstOrder.Rew`.
- `LO.FirstOrder.Rew.rewrite f` is... | @@ -555,8 +555,14 @@
variable (Φ : L₁ →ᵥ L₂)
open Rew
-lemma lMap_fix (t : SyntacticSemiterm L₁ n) : (fix t).lMap Φ = fix (t.lMap Φ) := sorry
+lemma lMap_bind (b : Fin n₁ → Semiterm L₁ ξ₂ n₂) (e : ξ₁ → Semiterm L₁ ξ₂ n₂) (t) :
+ lMap Φ (bind b e t) = bind (lMap Φ ∘ b) (lMap Φ ∘ e) (t.lMap Φ) := by
+ induction 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_c9e9d75d2f5b_13 | c9307e552f17e68b | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Syntax/Predicate/Rew.lean | Rew | 13 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 14 | 8 | [
{
"theorem_name": "free_rewrite_eq",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n simpa [← comp_app] using smul_ext' <| by ext x <;> simp [Rew.comp_app]",
"n_chars": 78,
"n_subproofs": 0,
"n_ta... | [
{
"name": "smul_ext'",
"text": "lemma smul_ext' {ω₁ ω₂ : Rew L ξ n₁ ζ n₂} (h : ω₁ = ω₂) {φ : F n₁} : ω₁ ▹ φ = ω₂ ▹ φ := by rw [h]\n\n",
"fan_in": 8,
"n_lines": 3,
"n_chars": 132,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 1,
"n_str... | [
{
"name": "free_rewrite_eq",
"fan_in": 0,
"n_deps_direct": 3,
"n_deps_transitive": 4,
"n_lines": 6,
"n_chars": 273,
"n_subproofs": 0,
"n_tactics": 3,
"cyclomatic": 2,
"n_automation": 2,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": false,
"max_nesting... | 1 | module
public import Foundation.Syntax.Predicate.Quantifier
public import Foundation.Syntax.Predicate.Term
public import Foundation.Vorspiel.Function
/-!
# Rewriting
term/formula morphisms such as Rewritings, substitutions, and embs are handled by the structure `LO.FirstOrder.Rew`.
- `LO.FirstOrder.Rew.rewrite f` is... | module
public import Foundation.Syntax.Predicate.Quantifier
public import Foundation.Syntax.Predicate.Term
public import Foundation.Vorspiel.Function
/-!
# Rewriting
term/formula morphisms such as Rewritings, substitutions, and embs are handled by the structure `LO.FirstOrder.Rew`.
- `LO.FirstOrder.Rew.rewrite f` is... | @@ -598,6 +598,8 @@
infixr:73 " ▹ " => app
+lemma smul_ext' {ω₁ ω₂ : Rew L ξ n₁ ζ n₂} (h : ω₁ = ω₂) {φ : F n₁} : ω₁ ▹ φ = ω₂ ▹ φ := by rw [h]
+
@[simp] lemma smul_ball (ω : Rew L ξ n₁ ζ n₂) (φ ψ : F (n₁ + 1)) : ω ▹ (∀⁰[φ] ψ) = ∀⁰[ω.q ▹ φ] (ω.q ▹ ψ) := by simp [ball]
@[simp] lemma smul_bexs (ω : Rew L ξ n₁ ζ n₂... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_c9e9d75d2f5b_14 | 81e762a5d8b55e23 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Syntax/Predicate/Rew.lean | Rew | 14 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 14 | 8 | [
{
"theorem_name": "free_rewrite_eq",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n simpa [← comp_app] using smul_ext' <| by ext x <;> simp [Rew.comp_app]",
"n_chars": 78,
"n_subproofs": 0,
"n_ta... | [
{
"name": "smul_ext'",
"text": "lemma smul_ext' {ω₁ ω₂ : Rew L ξ n₁ ζ n₂} (h : ω₁ = ω₂) {φ : F n₁} : ω₁ ▹ φ = ω₂ ▹ φ := by rw [h]\n\n",
"fan_in": 8,
"n_lines": 3,
"n_chars": 132,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 1,
"n_str... | [
{
"name": "shift_rewrite_eq",
"fan_in": 0,
"n_deps_direct": 3,
"n_deps_transitive": 4,
"n_lines": 5,
"n_chars": 246,
"n_subproofs": 0,
"n_tactics": 3,
"cyclomatic": 2,
"n_automation": 2,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": false,
"max_nestin... | 1 | module
public import Foundation.Syntax.Predicate.Quantifier
public import Foundation.Syntax.Predicate.Term
public import Foundation.Vorspiel.Function
/-!
# Rewriting
term/formula morphisms such as Rewritings, substitutions, and embs are handled by the structure `LO.FirstOrder.Rew`.
- `LO.FirstOrder.Rew.rewrite f` is... | module
public import Foundation.Syntax.Predicate.Quantifier
public import Foundation.Syntax.Predicate.Term
public import Foundation.Vorspiel.Function
/-!
# Rewriting
term/formula morphisms such as Rewritings, substitutions, and embs are handled by the structure `LO.FirstOrder.Rew`.
- `LO.FirstOrder.Rew.rewrite f` is... | @@ -598,6 +598,8 @@
infixr:73 " ▹ " => app
+lemma smul_ext' {ω₁ ω₂ : Rew L ξ n₁ ζ n₂} (h : ω₁ = ω₂) {φ : F n₁} : ω₁ ▹ φ = ω₂ ▹ φ := by rw [h]
+
@[simp] lemma smul_ball (ω : Rew L ξ n₁ ζ n₂) (φ ψ : F (n₁ + 1)) : ω ▹ (∀⁰[φ] ψ) = ∀⁰[ω.q ▹ φ] (ω.q ▹ ψ) := by simp [ball]
@[simp] lemma smul_bexs (ω : Rew L ξ n₁ ζ n₂... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_c9e9d75d2f5b_15 | 920154e765488d38 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Syntax/Predicate/Rew.lean | Rew | 15 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 14 | 8 | [
{
"theorem_name": "free_rewrite_eq",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n simpa [← comp_app] using smul_ext' <| by ext x <;> simp [Rew.comp_app]",
"n_chars": 78,
"n_subproofs": 0,
"n_ta... | [
{
"name": "smul_ext'",
"text": "lemma smul_ext' {ω₁ ω₂ : Rew L ξ n₁ ζ n₂} (h : ω₁ = ω₂) {φ : F n₁} : ω₁ ▹ φ = ω₂ ▹ φ := by rw [h]\n\n",
"fan_in": 8,
"n_lines": 3,
"n_chars": 132,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 1,
"n_str... | [
{
"name": "rewrite_subst_eq",
"fan_in": 0,
"n_deps_direct": 3,
"n_deps_transitive": 4,
"n_lines": 5,
"n_chars": 240,
"n_subproofs": 0,
"n_tactics": 3,
"cyclomatic": 2,
"n_automation": 2,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": false,
"max_nestin... | 1 | module
public import Foundation.Syntax.Predicate.Quantifier
public import Foundation.Syntax.Predicate.Term
public import Foundation.Vorspiel.Function
/-!
# Rewriting
term/formula morphisms such as Rewritings, substitutions, and embs are handled by the structure `LO.FirstOrder.Rew`.
- `LO.FirstOrder.Rew.rewrite f` is... | module
public import Foundation.Syntax.Predicate.Quantifier
public import Foundation.Syntax.Predicate.Term
public import Foundation.Vorspiel.Function
/-!
# Rewriting
term/formula morphisms such as Rewritings, substitutions, and embs are handled by the structure `LO.FirstOrder.Rew`.
- `LO.FirstOrder.Rew.rewrite f` is... | @@ -598,6 +598,8 @@
infixr:73 " ▹ " => app
+lemma smul_ext' {ω₁ ω₂ : Rew L ξ n₁ ζ n₂} (h : ω₁ = ω₂) {φ : F n₁} : ω₁ ▹ φ = ω₂ ▹ φ := by rw [h]
+
@[simp] lemma smul_ball (ω : Rew L ξ n₁ ζ n₂) (φ ψ : F (n₁ + 1)) : ω ▹ (∀⁰[φ] ψ) = ∀⁰[ω.q ▹ φ] (ω.q ▹ ψ) := by simp [ball]
@[simp] lemma smul_bexs (ω : Rew L ξ n₁ ζ n₂... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_c9e9d75d2f5b_16 | 0cac471914a373d8 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Syntax/Predicate/Rew.lean | Rew | 16 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 14 | 8 | [
{
"theorem_name": "free_rewrite_eq",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n simpa [← comp_app] using smul_ext' <| by ext x <;> simp [Rew.comp_app]",
"n_chars": 78,
"n_subproofs": 0,
"n_ta... | [
{
"name": "smul_ext'",
"text": "lemma smul_ext' {ω₁ ω₂ : Rew L ξ n₁ ζ n₂} (h : ω₁ = ω₂) {φ : F n₁} : ω₁ ▹ φ = ω₂ ▹ φ := by rw [h]\n\n",
"fan_in": 8,
"n_lines": 3,
"n_chars": 132,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 1,
"n_str... | [
{
"name": "free_subst_nil",
"fan_in": 0,
"n_deps_direct": 6,
"n_deps_transitive": 7,
"n_lines": 5,
"n_chars": 255,
"n_subproofs": 0,
"n_tactics": 5,
"cyclomatic": 2,
"n_automation": 2,
"n_rewrites": 0,
"n_structural": 2,
"automation_only": false,
"max_nesting"... | 1 | module
public import Foundation.Syntax.Predicate.Quantifier
public import Foundation.Syntax.Predicate.Term
public import Foundation.Vorspiel.Function
/-!
# Rewriting
term/formula morphisms such as Rewritings, substitutions, and embs are handled by the structure `LO.FirstOrder.Rew`.
- `LO.FirstOrder.Rew.rewrite f` is... | module
public import Foundation.Syntax.Predicate.Quantifier
public import Foundation.Syntax.Predicate.Term
public import Foundation.Vorspiel.Function
/-!
# Rewriting
term/formula morphisms such as Rewritings, substitutions, and embs are handled by the structure `LO.FirstOrder.Rew`.
- `LO.FirstOrder.Rew.rewrite f` is... | @@ -598,6 +598,8 @@
infixr:73 " ▹ " => app
+lemma smul_ext' {ω₁ ω₂ : Rew L ξ n₁ ζ n₂} (h : ω₁ = ω₂) {φ : F n₁} : ω₁ ▹ φ = ω₂ ▹ φ := by rw [h]
+
@[simp] lemma smul_ball (ω : Rew L ξ n₁ ζ n₂) (φ ψ : F (n₁ + 1)) : ω ▹ (∀⁰[φ] ψ) = ∀⁰[ω.q ▹ φ] (ω.q ▹ ψ) := by simp [ball]
@[simp] lemma smul_bexs (ω : Rew L ξ n₁ ζ n₂... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_c9e9d75d2f5b_17 | 13780297de645188 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Syntax/Predicate/Rew.lean | Rew | 17 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 14 | 8 | [
{
"theorem_name": "free_rewrite_eq",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n simpa [← comp_app] using smul_ext' <| by ext x <;> simp [Rew.comp_app]",
"n_chars": 78,
"n_subproofs": 0,
"n_ta... | [
{
"name": "smul_ext'",
"text": "lemma smul_ext' {ω₁ ω₂ : Rew L ξ n₁ ζ n₂} (h : ω₁ = ω₂) {φ : F n₁} : ω₁ ▹ φ = ω₂ ▹ φ := by rw [h]\n\n",
"fan_in": 8,
"n_lines": 3,
"n_chars": 132,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 1,
"n_str... | [
{
"name": "rewrite_subst_nil",
"fan_in": 0,
"n_deps_direct": 4,
"n_deps_transitive": 5,
"n_lines": 9,
"n_chars": 336,
"n_subproofs": 0,
"n_tactics": 5,
"cyclomatic": 1,
"n_automation": 2,
"n_rewrites": 0,
"n_structural": 2,
"automation_only": false,
"max_nesti... | 2 | module
public import Foundation.Syntax.Predicate.Quantifier
public import Foundation.Syntax.Predicate.Term
public import Foundation.Vorspiel.Function
/-!
# Rewriting
term/formula morphisms such as Rewritings, substitutions, and embs are handled by the structure `LO.FirstOrder.Rew`.
- `LO.FirstOrder.Rew.rewrite f` is... | module
public import Foundation.Syntax.Predicate.Quantifier
public import Foundation.Syntax.Predicate.Term
public import Foundation.Vorspiel.Function
/-!
# Rewriting
term/formula morphisms such as Rewritings, substitutions, and embs are handled by the structure `LO.FirstOrder.Rew`.
- `LO.FirstOrder.Rew.rewrite f` is... | @@ -604,6 +604,8 @@
infixr:73 " ▹ " => app
+lemma smul_ext' {ω₁ ω₂ : Rew L ξ n₁ ζ n₂} (h : ω₁ = ω₂) {φ : F n₁} : ω₁ ▹ φ = ω₂ ▹ φ := by rw [h]
+
@[simp] lemma smul_ball (ω : Rew L ξ n₁ ζ n₂) (φ ψ : F (n₁ + 1)) : ω ▹ (∀⁰[φ] ψ) = ∀⁰[ω.q ▹ φ] (ω.q ▹ ψ) := by simp [ball]
@[simp] lemma smul_bexs (ω : Rew L ξ n₁ ζ n₂... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_c9e9d75d2f5b_18 | 079c48ba40a73d17 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Syntax/Predicate/Rew.lean | Rew | 18 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 14 | 8 | [
{
"theorem_name": "free_rewrite_eq",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n simpa [← comp_app] using smul_ext' <| by ext x <;> simp [Rew.comp_app]",
"n_chars": 78,
"n_subproofs": 0,
"n_ta... | [
{
"name": "smul_ext'",
"text": "lemma smul_ext' {ω₁ ω₂ : Rew L ξ n₁ ζ n₂} (h : ω₁ = ω₂) {φ : F n₁} : ω₁ ▹ φ = ω₂ ▹ φ := by rw [h]\n\n",
"fan_in": 8,
"n_lines": 3,
"n_chars": 132,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 1,
"n_str... | [
{
"name": "rewrite_free_eq_subst",
"fan_in": 0,
"n_deps_direct": 3,
"n_deps_transitive": 4,
"n_lines": 5,
"n_chars": 204,
"n_subproofs": 0,
"n_tactics": 3,
"cyclomatic": 2,
"n_automation": 2,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": false,
"max_n... | 1 | module
public import Foundation.Syntax.Predicate.Quantifier
public import Foundation.Syntax.Predicate.Term
public import Foundation.Vorspiel.Function
/-!
# Rewriting
term/formula morphisms such as Rewritings, substitutions, and embs are handled by the structure `LO.FirstOrder.Rew`.
- `LO.FirstOrder.Rew.rewrite f` is... | module
public import Foundation.Syntax.Predicate.Quantifier
public import Foundation.Syntax.Predicate.Term
public import Foundation.Vorspiel.Function
/-!
# Rewriting
term/formula morphisms such as Rewritings, substitutions, and embs are handled by the structure `LO.FirstOrder.Rew`.
- `LO.FirstOrder.Rew.rewrite f` is... | @@ -598,6 +598,8 @@
infixr:73 " ▹ " => app
+lemma smul_ext' {ω₁ ω₂ : Rew L ξ n₁ ζ n₂} (h : ω₁ = ω₂) {φ : F n₁} : ω₁ ▹ φ = ω₂ ▹ φ := by rw [h]
+
@[simp] lemma smul_ball (ω : Rew L ξ n₁ ζ n₂) (φ ψ : F (n₁ + 1)) : ω ▹ (∀⁰[φ] ψ) = ∀⁰[ω.q ▹ φ] (ω.q ▹ ψ) := by simp [ball]
@[simp] lemma smul_bexs (ω : Rew L ξ n₁ ζ n₂... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_c9e9d75d2f5b_19 | b3243347382794ad | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Syntax/Predicate/Rew.lean | Rew | 19 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 14 | 2 | [
{
"theorem_name": "fixitr_bvar",
"depth": 1,
"n_commands": 0,
"n_lines": 5,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n induction m\n · simp [*]\n case succ m ih =>\n rw [fixitr_succ, comp_app, ih]; simp [Fin.castSucc_castAdd]",
"n_chars"... | [
{
"name": "fixitr_succ",
"text": "lemma fixitr_succ (m) :\n fixitr (L := L) n (m + 1) = Rew.fix.comp (fixitr n m) := by\n simp [fixitr]\n\n",
"fan_in": 2,
"n_lines": 5,
"n_chars": 105,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
... | [
{
"name": "allClosure_fixitr",
"fan_in": 0,
"n_deps_direct": 3,
"n_deps_transitive": 3,
"n_lines": 4,
"n_chars": 185,
"n_subproofs": 0,
"n_tactics": 3,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"max_nestin... | 2 | module
public import Foundation.Syntax.Predicate.Quantifier
public import Foundation.Syntax.Predicate.Term
public import Foundation.Vorspiel.Function
/-!
# Rewriting
term/formula morphisms such as Rewritings, substitutions, and embs are handled by the structure `LO.FirstOrder.Rew`.
- `LO.FirstOrder.Rew.rewrite f` is... | module
public import Foundation.Syntax.Predicate.Quantifier
public import Foundation.Syntax.Predicate.Term
public import Foundation.Vorspiel.Function
/-!
# Rewriting
term/formula morphisms such as Rewritings, substitutions, and embs are handled by the structure `LO.FirstOrder.Rew`.
- `LO.FirstOrder.Rew.rewrite f` is... | @@ -493,8 +493,16 @@
@[simp] lemma fixitr_zero :
fixitr (L := L) n 0 = Rew.id := by simp [fixitr]
-@[simp] lemma fixitr_bvar (n m) (x : Fin n) : fixitr n m (#x : SyntacticSemiterm L n) = #(x.castAdd m) := sorry
+lemma fixitr_succ (m) :
+ fixitr (L := L) n (m + 1) = Rew.fix.comp (fixitr n m) := by
+ simp [fi... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_c9e9d75d2f5b_20 | 0a1ee0006301bd5b | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Syntax/Predicate/Rew.lean | Rew | 20 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 14 | 1 | [
{
"theorem_name": "emb_subst_eq_subst_coe₁",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n simpa [Matrix.constant_eq_singleton] using emb_subst_eq_subst_emb φ ![t]",
"n_chars": 79,
"n_subproofs": 0,... | [
{
"name": "emb_subst_eq_subst_emb",
"text": "/-- `coe_subst_eq_subst_coe` -/\nlemma emb_subst_eq_subst_emb\n [Rewriting L ο O ο O] [Rewriting L ο O ξ F] [Rewriting L ξ F ξ F]\n [TransitiveRewriting L ο O ο O ξ F]\n [TransitiveRewriting L ο O ξ F ξ F]\n (φ : O k) (v : Fin k → Semiterm L ο n) :\n ... | [
{
"name": "emb_subst_eq_subst_coe₁",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 4,
"n_lines": 10,
"n_chars": 435,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"max... | 1 | module
public import Foundation.Syntax.Predicate.Quantifier
public import Foundation.Syntax.Predicate.Term
public import Foundation.Vorspiel.Function
/-!
# Rewriting
term/formula morphisms such as Rewritings, substitutions, and embs are handled by the structure `LO.FirstOrder.Rew`.
- `LO.FirstOrder.Rew.rewrite f` is... | module
public import Foundation.Syntax.Predicate.Quantifier
public import Foundation.Syntax.Predicate.Term
public import Foundation.Vorspiel.Function
/-!
# Rewriting
term/formula morphisms such as Rewritings, substitutions, and embs are handled by the structure `LO.FirstOrder.Rew`.
- `LO.FirstOrder.Rew.rewrite f` is... | @@ -750,13 +750,28 @@
congr 2
simp
+/-- `coe_subst_eq_subst_coe` -/
+lemma emb_subst_eq_subst_emb
+ [Rewriting L ο O ο O] [Rewriting L ο O ξ F] [Rewriting L ξ F ξ F]
+ [TransitiveRewriting L ο O ο O ξ F]
+ [TransitiveRewriting L ο O ξ F ξ F]
+ (φ : O k) (v : Fin k → Semiterm L ο n) :
+ (emb (ξ :=... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_44a7008b72c9_0 | 6e330f7426f4872c | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Boxdot/Jerabek.lean | Jerabek | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 5 | 1 | [
{
"theorem_name": "jerabek_SBDP",
"depth": 1,
"n_commands": 0,
"n_lines": 148,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n intro L _;\n contrapose!;\n intro hL;\n obtain ⟨φ, hφL, hφL₀⟩ := Set.not_subset.mp hL;\n apply Set.not_subset.mpr;\n\n ... | [
{
"name": "jerabek_SBDP.lemma₃",
"text": "private lemma jerabek_SBDP.lemma₃ : L ⊢ (□^[n]Γ.conj)ᵇ 🡒 □^≤[n](Γ.image (·ᵇ)).conj := by\n apply normal_provable_of_K_provable;\n apply Complete.complete (𝓜 := Kripke.FrameClass.K);\n intro F hF V x h;\n apply Satisfies.fconj_def.mpr;\n simp only [Finset.mem_... | [
{
"name": "jerabek_SBDP.lemma₁",
"fan_in": 1,
"n_deps_direct": 2,
"n_deps_transitive": 4,
"n_lines": 18,
"n_chars": 537,
"n_subproofs": 0,
"n_tactics": 31,
"cyclomatic": 3,
"n_automation": 2,
"n_rewrites": 0,
"n_structural": 10,
"automation_only": false,
"max_... | 2 | module
/-
Jeřábek's proof of boxdot conjecture.
References:
- E. Jeřábek - "Cluster Expansion and the Boxdot Conjecture": https://arxiv.org/abs/1308.0994
-/
public import Foundation.Modal.Boxdot.Basic
public import Foundation.Modal.Kripke.Logic.S5
public import Foundation.Modal.Kripke.Logic.S4McK
public import ... | module
/-
Jeřábek's proof of boxdot conjecture.
References:
- E. Jeřábek - "Cluster Expansion and the Boxdot Conjecture": https://arxiv.org/abs/1308.0994
-/
public import Foundation.Modal.Boxdot.Basic
public import Foundation.Modal.Kripke.Logic.S5
public import Foundation.Modal.Kripke.Logic.S4McK
public import ... | @@ -174,6 +174,25 @@
apply normal_provable_of_K_provable;
exact lemma₁;
+private lemma jerabek_SBDP.lemma₃ : L ⊢ (□^[n]Γ.conj)ᵇ 🡒 □^≤[n](Γ.image (·ᵇ)).conj := by
+ apply normal_provable_of_K_provable;
+ apply Complete.complete (𝓜 := Kripke.FrameClass.K);
+ intro F hF V x h;
+ apply Satisfies.fconj_def.mpr... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_44a7008b72c9_1 | d09890d1d51ab51b | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Boxdot/Jerabek.lean | Jerabek | 1 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 5 | 1 | [
{
"theorem_name": "jerabek_SBDP",
"depth": 1,
"n_commands": 0,
"n_lines": 148,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n intro L _;\n contrapose!;\n intro hL;\n obtain ⟨φ, hφL, hφL₀⟩ := Set.not_subset.mp hL;\n apply Set.not_subset.mpr;\n\n ... | [
{
"name": "jerabek_SBDP.lemma₂",
"text": "private lemma jerabek_SBDP.lemma₂ : L ⊢ (flag (.atom p) b) ⋏ □φᵇ 🡒 ⊡((flag (.atom p) !b) 🡒 φᵇ) := by\n apply normal_provable_of_K_provable;\n exact lemma₁;\n\n",
"fan_in": 1,
"n_lines": 5,
"n_chars": 181,
"n_subproofs": 0,
"n_tactics": 5,
... | [
{
"name": "jerabek_SBDP.lemma₃",
"fan_in": 1,
"n_deps_direct": 2,
"n_deps_transitive": 4,
"n_lines": 20,
"n_chars": 932,
"n_subproofs": 0,
"n_tactics": 35,
"cyclomatic": 1,
"n_automation": 4,
"n_rewrites": 0,
"n_structural": 12,
"automation_only": false,
"max_... | 1 | module
/-
Jeřábek's proof of boxdot conjecture.
References:
- E. Jeřábek - "Cluster Expansion and the Boxdot Conjecture": https://arxiv.org/abs/1308.0994
-/
public import Foundation.Modal.Boxdot.Basic
public import Foundation.Modal.Kripke.Logic.S5
public import Foundation.Modal.Kripke.Logic.S4McK
public import ... | module
/-
Jeřábek's proof of boxdot conjecture.
References:
- E. Jeřábek - "Cluster Expansion and the Boxdot Conjecture": https://arxiv.org/abs/1308.0994
-/
public import Foundation.Modal.Boxdot.Basic
public import Foundation.Modal.Kripke.Logic.S5
public import Foundation.Modal.Kripke.Logic.S4McK
public import ... | @@ -170,6 +170,10 @@
apply hx;
assumption;
+private lemma jerabek_SBDP.lemma₂ : L ⊢ (flag (.atom p) b) ⋏ □φᵇ 🡒 ⊡((flag (.atom p) !b) 🡒 φᵇ) := by
+ apply normal_provable_of_K_provable;
+ exact lemma₁;
+
private lemma jerabek_SBDP.lemma₃ : L ⊢ (□^[n]Γ.conj)ᵇ 🡒 □^≤[n](Γ.image (·ᵇ)).conj := by
apply no... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_44a7008b72c9_2 | c126a7875fecfb00 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Boxdot/Jerabek.lean | Jerabek | 2 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 5 | 1 | [
{
"theorem_name": "jerabek_SBDP",
"depth": 1,
"n_commands": 0,
"n_lines": 148,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n intro L _;\n contrapose!;\n intro hL;\n obtain ⟨φ, hφL, hφL₀⟩ := Set.not_subset.mp hL;\n apply Set.not_subset.mpr;\n\n ... | [
{
"name": "jerabek_SBDP.lemma₃",
"text": "private lemma jerabek_SBDP.lemma₃ : L ⊢ (□^[n]Γ.conj)ᵇ 🡒 □^≤[n](Γ.image (·ᵇ)).conj := by\n apply normal_provable_of_K_provable;\n apply Complete.complete (𝓜 := Kripke.FrameClass.K);\n intro F hF V x h;\n apply Satisfies.fconj_def.mpr;\n simp only [Finset.mem_... | [
{
"name": "jerabek_BDP",
"fan_in": 6,
"n_deps_direct": 2,
"n_deps_transitive": 5,
"n_lines": 10,
"n_chars": 366,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
"max_nesting": ... | 3 | module
/-
Jeřábek's proof of boxdot conjecture.
References:
- E. Jeřábek - "Cluster Expansion and the Boxdot Conjecture": https://arxiv.org/abs/1308.0994
-/
public import Foundation.Modal.Boxdot.Basic
public import Foundation.Modal.Kripke.Logic.S5
public import Foundation.Modal.Kripke.Logic.S4McK
public import ... | module
/-
Jeřábek's proof of boxdot conjecture.
References:
- E. Jeřábek - "Cluster Expansion and the Boxdot Conjecture": https://arxiv.org/abs/1308.0994
-/
public import Foundation.Modal.Boxdot.Basic
public import Foundation.Modal.Kripke.Logic.S5
public import Foundation.Modal.Kripke.Logic.S4McK
public import ... | @@ -181,6 +181,25 @@
apply normal_provable_of_K_provable;
exact lemma₁;
+private lemma jerabek_SBDP.lemma₃ : L ⊢ (□^[n]Γ.conj)ᵇ 🡒 □^≤[n](Γ.image (·ᵇ)).conj := by
+ apply normal_provable_of_K_provable;
+ apply Complete.complete (𝓜 := Kripke.FrameClass.K);
+ intro F hF V x h;
+ apply Satisfies.fconj_def.mpr... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_44a7008b72c9_3 | 29873c5da97f080d | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Boxdot/Jerabek.lean | Jerabek | 3 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 5 | 6 | [
{
"theorem_name": "KT.BDP",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " jerabek_BDP Modal.KT Kripke.FrameClass.KT\nalias boxdot_conjecture := KT.BDP",
"n_chars": 76,
"n_subproofs": 0,
"n_tactics": 2... | [
{
"name": "jerabek_BDP",
"text": "/--\n Every Logic `L₀` which is `Modal.KT ⪯ L₀` and sound and complete to frame class `C` satisfies Jeřábek's assumption, has boxdot property.\n-/\ntheorem jerabek_BDP\n (L₀ : Logic ℕ) [Modal.KT ⪯ L₀]\n (C : Kripke.FrameClass) [C.JerabekAssumption]\n [Sound L₀ C] [Compl... | [
{
"name": "KT.BDP",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 6,
"n_lines": 5,
"n_chars": 196,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
"max_nesting": 1
}
... | 4 | module
/-
Jeřábek's proof of boxdot conjecture.
References:
- E. Jeřábek - "Cluster Expansion and the Boxdot Conjecture": https://arxiv.org/abs/1308.0994
-/
public import Foundation.Modal.Boxdot.Basic
public import Foundation.Modal.Kripke.Logic.S5
public import Foundation.Modal.Kripke.Logic.S4McK
public import ... | module
/-
Jeřábek's proof of boxdot conjecture.
References:
- E. Jeřábek - "Cluster Expansion and the Boxdot Conjecture": https://arxiv.org/abs/1308.0994
-/
public import Foundation.Modal.Boxdot.Basic
public import Foundation.Modal.Kripke.Logic.S5
public import Foundation.Modal.Kripke.Logic.S4McK
public import ... | @@ -358,23 +358,33 @@
apply ValidOnModel.multinec n this;
. exact H2 φ (by grind) x _ |>.not.mpr hF;
+/--
+ Every Logic `L₀` which is `Modal.KT ⪯ L₀` and sound and complete to frame class `C` satisfies Jeřábek's assumption, has boxdot property.
+-/
+theorem jerabek_BDP
+ (L₀ : Logic ℕ) [Modal.KT ⪯ 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_811dc6ccbcea_0 | 62f642b1f07bfd26 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Completeness/CountableSublanguage.lean | CountableSublanguage | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 3 | 1 | [
{
"theorem_name": "eval_lMap",
"depth": 1,
"n_commands": 0,
"n_lines": 15,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n induction φ using Semiformula.rec'\n case hrel k r v =>\n simp only [Semiformula.lMap_rel, eval_rel, val_lMap s₁ Φ bv fv, Fu... | [
{
"name": "val_lMap",
"text": "lemma val_lMap\n (Φ : L₁ →ᵥ L₂) [Φ.Injective]\n (bv : Fin n → M) (fv : ξ → M)\n (t : Semiterm L₁ ξ n) :\n (t.lMap Φ).val (s := s₁.extendStructure Φ) bv fv = t.val bv fv := by\n induction t\n case func k f v ih =>\n simpa [ih, Function.comp_def] using extendStr... | [
{
"name": "eval_lMap",
"fan_in": 1,
"n_deps_direct": 2,
"n_deps_transitive": 3,
"n_lines": 20,
"n_chars": 752,
"n_subproofs": 0,
"n_tactics": 15,
"cyclomatic": 2,
"n_automation": 9,
"n_rewrites": 0,
"n_structural": 2,
"automation_only": false,
"max_nesting": 6... | 1 | module
public import Foundation.FirstOrder.Basic
public import Mathlib.Logic.Equiv.List
@[expose] public section
namespace LO
namespace FirstOrder
variable {L : Language.{u}}
/-! ### Finite sublanguage -/
namespace Language
def sublanguage (L : Language) (pFunc : ∀ k, L.Func k → Prop) (pRel : ∀ k, L.Rel k → Prop... | module
public import Foundation.FirstOrder.Basic
public import Mathlib.Logic.Equiv.List
@[expose] public section
namespace LO
namespace FirstOrder
variable {L : Language.{u}}
/-! ### Finite sublanguage -/
namespace Language
def sublanguage (L : Language) (pFunc : ∀ k, L.Func k → Prop) (pRel : ∀ k, L.Rel k → Prop... | @@ -191,12 +191,37 @@
(s₁.extendStructure Φ).rel (Φ.rel r₁) v ↔ s₁.rel r₁ v := by
refine ⟨by intros h; rcases h with ⟨r₁', bv, h⟩; rcases hΦ.rel k bv with rfl; exact h, by intros h; refine ⟨r₁, rfl, h⟩⟩
+lemma val_lMap
+ (Φ : L₁ →ᵥ L₂) [Φ.Injective]
+ (bv : Fin n → M) (fv : ξ → M)
+ (t : Semiterm L₁ ... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_811dc6ccbcea_1 | 722e9fb759376ea1 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Completeness/CountableSublanguage.lean | CountableSublanguage | 1 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 3 | 1 | [
{
"theorem_name": "models_lMap",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n simp [Semantics.Models, eval_lMap s₁ Φ]",
"n_chars": 48,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
... | [
{
"name": "eval_lMap",
"text": "lemma eval_lMap (Φ : L₁ →ᵥ L₂) [Φ.Injective]\n (bv : Fin n → M) (fv : ξ → M)\n {φ : Semiformula L₁ ξ n} :\n (lMap Φ φ).Eval (s := s₁.extendStructure Φ) bv fv ↔ φ.Eval bv fv := by\n induction φ using Semiformula.rec'\n case hrel k r v =>\n simp only [Semiformula.... | [
{
"name": "models_lMap",
"fan_in": 2,
"n_deps_direct": 1,
"n_deps_transitive": 4,
"n_lines": 5,
"n_chars": 209,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"max_nesting": 2
... | 2 | module
public import Foundation.FirstOrder.Basic
public import Mathlib.Logic.Equiv.List
@[expose] public section
namespace LO
namespace FirstOrder
variable {L : Language.{u}}
/-! ### Finite sublanguage -/
namespace Language
def sublanguage (L : Language) (pFunc : ∀ k, L.Func k → Prop) (pRel : ∀ k, L.Rel k → Prop... | module
public import Foundation.FirstOrder.Basic
public import Mathlib.Logic.Equiv.List
@[expose] public section
namespace LO
namespace FirstOrder
variable {L : Language.{u}}
/-! ### Finite sublanguage -/
namespace Language
def sublanguage (L : Language) (pFunc : ∀ k, L.Func k → Prop) (pRel : ∀ k, L.Rel k → Prop... | @@ -204,8 +204,28 @@
open Semiformula
+lemma eval_lMap (Φ : L₁ →ᵥ L₂) [Φ.Injective]
+ (bv : Fin n → M) (fv : ξ → M)
+ {φ : Semiformula L₁ ξ n} :
+ (lMap Φ φ).Eval (s := s₁.extendStructure Φ) bv fv ↔ φ.Eval bv fv := by
+ induction φ using Semiformula.rec'
+ case hrel k r v =>
+ simp only [Semiformula.... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_811dc6ccbcea_2 | 8064dd9e531fede1 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Completeness/CountableSublanguage.lean | CountableSublanguage | 2 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 3 | 2 | [
{
"theorem_name": "lMap_models_lMap_iff",
"depth": 1,
"n_commands": 0,
"n_lines": 9,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n constructor\n · intro h s₁ hs₁\n refine (Structure.extendStructure.models_lMap s₁.struc Φ φ).mp <| h ?_\n suffi... | [
{
"name": "models_lMap",
"text": "lemma models_lMap (Φ : L₁ →ᵥ L₂) [Φ.Injective] (φ : Sentence L₁) :\n (s₁.extendStructure Φ).toStruc ⊧ φ.lMap Φ ↔ s₁.toStruc ⊧ φ := by\n simp [Semantics.Models, eval_lMap s₁ Φ]\n\n",
"fan_in": 2,
"n_lines": 5,
"n_chars": 209,
"n_subproofs": 0,
"n_tact... | [
{
"name": "lMap_models_lMap_iff",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 5,
"n_lines": 12,
"n_chars": 574,
"n_subproofs": 1,
"n_tactics": 9,
"cyclomatic": 2,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 6,
"automation_only": false,
"max_n... | 3 | module
public import Foundation.FirstOrder.Basic
public import Mathlib.Logic.Equiv.List
@[expose] public section
namespace LO
namespace FirstOrder
variable {L : Language.{u}}
/-! ### Finite sublanguage -/
namespace Language
def sublanguage (L : Language) (pFunc : ∀ k, L.Func k → Prop) (pRel : ∀ k, L.Rel k → Prop... | module
public import Foundation.FirstOrder.Basic
public import Mathlib.Logic.Equiv.List
@[expose] public section
namespace LO
namespace FirstOrder
variable {L : Language.{u}}
/-! ### Finite sublanguage -/
namespace Language
def sublanguage (L : Language) (pFunc : ∀ k, L.Func k → Prop) (pRel : ∀ k, L.Rel k → Prop... | @@ -223,6 +223,10 @@
case _ => simp [*]
case _ => simp [*]
+lemma models_lMap (Φ : L₁ →ᵥ L₂) [Φ.Injective] (φ : Sentence L₁) :
+ (s₁.extendStructure Φ).toStruc ⊧ φ.lMap Φ ↔ s₁.toStruc ⊧ φ := by
+ simp [Semantics.Models, eval_lMap s₁ Φ]
+
end extendStructure
end Structure
@@ -234,7 +238,12 @@
lemma lMap... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_811dc6ccbcea_3 | 97e6502e3a566a2b | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Completeness/CountableSublanguage.lean | CountableSublanguage | 3 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 3 | 2 | [
{
"theorem_name": "lMap_models_lMap_iff",
"depth": 1,
"n_commands": 0,
"n_lines": 9,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n constructor\n · intro h s₁ hs₁\n refine (Structure.extendStructure.models_lMap s₁.struc Φ φ).mp <| h ?_\n suffi... | [
{
"name": "models_lMap",
"text": "lemma models_lMap (Φ : L₁ →ᵥ L₂) [Φ.Injective] (φ : Sentence L₁) :\n (s₁.extendStructure Φ).toStruc ⊧ φ.lMap Φ ↔ s₁.toStruc ⊧ φ := by\n simp [Semantics.Models, eval_lMap s₁ Φ]\n\n",
"fan_in": 2,
"n_lines": 5,
"n_chars": 209,
"n_subproofs": 0,
"n_tact... | [
{
"name": "satisfiable_lMap",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 5,
"n_lines": 9,
"n_chars": 438,
"n_subproofs": 1,
"n_tactics": 6,
"cyclomatic": 2,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 2,
"automation_only": false,
"max_nestin... | 3 | module
public import Foundation.FirstOrder.Basic
public import Mathlib.Logic.Equiv.List
@[expose] public section
namespace LO
namespace FirstOrder
variable {L : Language.{u}}
/-! ### Finite sublanguage -/
namespace Language
def sublanguage (L : Language) (pFunc : ∀ k, L.Func k → Prop) (pRel : ∀ k, L.Rel k → Prop... | module
public import Foundation.FirstOrder.Basic
public import Mathlib.Logic.Equiv.List
@[expose] public section
namespace LO
namespace FirstOrder
variable {L : Language.{u}}
/-! ### Finite sublanguage -/
namespace Language
def sublanguage (L : Language) (pFunc : ∀ k, L.Func k → Prop) (pRel : ∀ k, L.Rel k → Prop... | @@ -223,6 +223,10 @@
case _ => simp [*]
case _ => simp [*]
+lemma models_lMap (Φ : L₁ →ᵥ L₂) [Φ.Injective] (φ : Sentence L₁) :
+ (s₁.extendStructure Φ).toStruc ⊧ φ.lMap Φ ↔ s₁.toStruc ⊧ φ := by
+ simp [Semantics.Models, eval_lMap s₁ Φ]
+
end extendStructure
end Structure
@@ -232,7 +236,12 @@
variable {... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_3fb12358fbd9_0 | 7a34ab7efe31b2bf | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Bootstrapping/Syntax/Term/Functions.lean | Functions | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 6 | 3 | [
{
"theorem_name": "termSubstVec_cons₁",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n rw [show (1 : V) = 0 + 1 by simp, termSubstVec_cons] <;> simp [*]",
"n_chars": 72,
"n_subproofs": 0,
"n_tac... | [
{
"name": "termSubstVec_cons",
"text": "lemma termSubstVec_cons {k t ts : V} (ht : IsUTerm L t) (hts : IsUTermVec L k ts) :\n termSubstVec L (k + 1) w (t ∷ ts) = termSubst L w t ∷ termSubstVec L k w ts :=\n construction.resultVec_cons L ![w] hts ht\n\n",
"fan_in": 3,
"n_lines": 5,
"n_chars":... | [
{
"name": "termSubstVec_cons₁",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 5,
"n_chars": 189,
"n_subproofs": 0,
"n_tactics": 3,
"cyclomatic": 2,
"n_automation": 2,
"n_rewrites": 2,
"n_structural": 0,
"automation_only": false,
"max_nest... | 1 | module
public import Foundation.FirstOrder.Bootstrapping.Syntax.Term.Basic
@[expose] public section
namespace LO.FirstOrder.Arithmetic.Bootstrapping
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
section
variable {L : Language} [L.Encodable] [L.LORDefinable]
namespace TermSubst
def blueprint : Langua... | module
public import Foundation.FirstOrder.Bootstrapping.Syntax.Term.Basic
@[expose] public section
namespace LO.FirstOrder.Arithmetic.Bootstrapping
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
section
variable {L : Language} [L.Encodable] [L.LORDefinable]
namespace TermSubst
def blueprint : Langua... | @@ -85,11 +85,17 @@
@[simp] lemma termSubstVec_nil (w : V) : termSubstVec L 0 w 0 = 0 := construction.resultVec_nil L _
+lemma termSubstVec_cons {k t ts : V} (ht : IsUTerm L t) (hts : IsUTermVec L k ts) :
+ termSubstVec L (k + 1) w (t ∷ ts) = termSubst L w t ∷ termSubstVec L k w ts :=
+ construction.resultVec_... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_3fb12358fbd9_1 | 795fe84aaa0fd3c6 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Bootstrapping/Syntax/Term/Functions.lean | Functions | 1 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 6 | 2 | [
{
"theorem_name": "termShiftVec_cons₁",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n rw [show (1 : V) = 0 + 1 by simp, termShiftVec_cons] <;> simp [*]",
"n_chars": 72,
"n_subproofs": 0,
"n_tac... | [
{
"name": "termShiftVec_cons",
"text": "lemma termShiftVec_cons {k t ts : V} (ht : IsUTerm L t) (hts : IsUTermVec L k ts) :\n termShiftVec L (k + 1) (t ∷ ts) = termShift L t ∷ termShiftVec L k ts :=\n construction.resultVec_cons L ![] hts ht\n\n",
"fan_in": 2,
"n_lines": 5,
"n_chars": 209,
... | [
{
"name": "termShiftVec_cons₁",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 5,
"n_chars": 206,
"n_subproofs": 0,
"n_tactics": 3,
"cyclomatic": 2,
"n_automation": 2,
"n_rewrites": 2,
"n_structural": 0,
"automation_only": false,
"max_nest... | 1 | module
public import Foundation.FirstOrder.Bootstrapping.Syntax.Term.Basic
@[expose] public section
namespace LO.FirstOrder.Arithmetic.Bootstrapping
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
section
variable {L : Language} [L.Encodable] [L.LORDefinable]
namespace TermSubst
def blueprint : Langua... | module
public import Foundation.FirstOrder.Bootstrapping.Syntax.Term.Basic
@[expose] public section
namespace LO.FirstOrder.Arithmetic.Bootstrapping
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
section
variable {L : Language} [L.Encodable] [L.LORDefinable]
namespace TermSubst
def blueprint : Langua... | @@ -198,11 +198,17 @@
@[simp] lemma termShiftVec_nil : termShiftVec (V := V) L 0 0 = 0 := construction.resultVec_nil L ![]
+lemma termShiftVec_cons {k t ts : V} (ht : IsUTerm L t) (hts : IsUTermVec L k ts) :
+ termShiftVec L (k + 1) (t ∷ ts) = termShift L t ∷ termShiftVec L k ts :=
+ construction.resultVec_con... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_3fb12358fbd9_2 | 6ab8e886743375d0 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Bootstrapping/Syntax/Term/Functions.lean | Functions | 2 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 6 | 2 | [
{
"theorem_name": "termBShiftVec_cons₁",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n rw [show (1 : V) = 0 + 1 by simp, termBShiftVec_cons] <;> simp [*]",
"n_chars": 73,
"n_subproofs": 0,
"n_t... | [
{
"name": "termBShiftVec_cons",
"text": "lemma termBShiftVec_cons {k t ts : V} (ht : IsUTerm L t) (hts : IsUTermVec L k ts) :\n termBShiftVec L (k + 1) (t ∷ ts) = termBShift L t ∷ termBShiftVec L k ts :=\n construction.resultVec_cons L ![] hts ht\n\n",
"fan_in": 2,
"n_lines": 5,
"n_chars": 2... | [
{
"name": "termBShiftVec_cons₁",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 5,
"n_chars": 210,
"n_subproofs": 0,
"n_tactics": 3,
"cyclomatic": 2,
"n_automation": 2,
"n_rewrites": 2,
"n_structural": 0,
"automation_only": false,
"max_nes... | 1 | module
public import Foundation.FirstOrder.Bootstrapping.Syntax.Term.Basic
@[expose] public section
namespace LO.FirstOrder.Arithmetic.Bootstrapping
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
section
variable {L : Language} [L.Encodable] [L.LORDefinable]
namespace TermSubst
def blueprint : Langua... | module
public import Foundation.FirstOrder.Bootstrapping.Syntax.Term.Basic
@[expose] public section
namespace LO.FirstOrder.Arithmetic.Bootstrapping
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
section
variable {L : Language} [L.Encodable] [L.LORDefinable]
namespace TermSubst
def blueprint : Langua... | @@ -333,11 +333,17 @@
@[simp] lemma termBShiftVec_nil : termBShiftVec (V := V) L 0 0 = 0 :=
construction.resultVec_nil L ![]
+lemma termBShiftVec_cons {k t ts : V} (ht : IsUTerm L t) (hts : IsUTermVec L k ts) :
+ termBShiftVec L (k + 1) (t ∷ ts) = termBShift L t ∷ termBShiftVec L k ts :=
+ construction.result... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_3fb12358fbd9_3 | 59fdff78497f0d9a | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Bootstrapping/Syntax/Term/Functions.lean | Functions | 3 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 6 | 2 | [
{
"theorem_name": "substs_qVec_bShift",
"depth": 1,
"n_commands": 0,
"n_lines": 3,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n rcases hw.lh\n simp [qVec, substs_cons_bShift ht, bShift_substs ht hw]",
"n_chars": 76,
"n_subproofs": 0,
"... | [
{
"name": "bShift_substs",
"text": "lemma bShift_substs {n m w t : V} (ht : IsSemiterm L n t) (hw : IsSemitermVec L n m w) :\n termBShift L (termSubst L w t) = termSubst L (termBShiftVec L n w) t := by\n apply IsSemiterm.induction 𝚺 ?_ ?_ ?_ ?_ t ht\n · definability\n · intro z hz; simp [nth_termBShi... | [
{
"name": "substs_qVec_bShift",
"fan_in": 0,
"n_deps_direct": 2,
"n_deps_transitive": 8,
"n_lines": 6,
"n_chars": 251,
"n_subproofs": 0,
"n_tactics": 3,
"cyclomatic": 2,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": false,
"max_nest... | 2 | module
public import Foundation.FirstOrder.Bootstrapping.Syntax.Term.Basic
@[expose] public section
namespace LO.FirstOrder.Arithmetic.Bootstrapping
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
section
variable {L : Language} [L.Encodable] [L.LORDefinable]
namespace TermSubst
def blueprint : Langua... | module
public import Foundation.FirstOrder.Bootstrapping.Syntax.Term.Basic
@[expose] public section
namespace LO.FirstOrder.Arithmetic.Bootstrapping
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
section
variable {L : Language} [L.Encodable] [L.LORDefinable]
namespace TermSubst
def blueprint : Langua... | @@ -474,8 +474,27 @@
nth_termBShiftVec hv.isUTerm hi,
ih i hi]
+lemma bShift_substs {n m w t : V} (ht : IsSemiterm L n t) (hw : IsSemitermVec L n m w) :
+ termBShift L (termSubst L w t) = termSubst L (termBShiftVec L n w) t := by
+ apply IsSemiterm.induction 𝚺 ?_ ?_ ?_ ?_ t ht
+ · definability
+ ... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_3fb12358fbd9_4 | a1094232fad7ee5f | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Bootstrapping/Syntax/Term/Functions.lean | Functions | 4 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 6 | 2 | [
{
"theorem_name": "substs_qVec_bShift",
"depth": 1,
"n_commands": 0,
"n_lines": 3,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n rcases hw.lh\n simp [qVec, substs_cons_bShift ht, bShift_substs ht hw]",
"n_chars": 76,
"n_subproofs": 0,
"... | [
{
"name": "bShift_substs",
"text": "lemma bShift_substs {n m w t : V} (ht : IsSemiterm L n t) (hw : IsSemitermVec L n m w) :\n termBShift L (termSubst L w t) = termSubst L (termBShiftVec L n w) t := by\n apply IsSemiterm.induction 𝚺 ?_ ?_ ?_ ?_ t ht\n · definability\n · intro z hz; simp [nth_termBShi... | [
{
"name": "termSubstVec_qVec_qVec",
"fan_in": 0,
"n_deps_direct": 7,
"n_deps_transitive": 10,
"n_lines": 21,
"n_chars": 873,
"n_subproofs": 1,
"n_tactics": 20,
"cyclomatic": 5,
"n_automation": 6,
"n_rewrites": 5,
"n_structural": 7,
"automation_only": false,
"m... | 3 | module
public import Foundation.FirstOrder.Bootstrapping.Syntax.Term.Basic
@[expose] public section
namespace LO.FirstOrder.Arithmetic.Bootstrapping
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
section
variable {L : Language} [L.Encodable] [L.LORDefinable]
namespace TermSubst
def blueprint : Langua... | module
public import Foundation.FirstOrder.Bootstrapping.Syntax.Term.Basic
@[expose] public section
namespace LO.FirstOrder.Arithmetic.Bootstrapping
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
section
variable {L : Language} [L.Encodable] [L.LORDefinable]
namespace TermSubst
def blueprint : Langua... | @@ -474,6 +474,23 @@
nth_termBShiftVec hv.isUTerm hi,
ih i hi]
+lemma bShift_substs {n m w t : V} (ht : IsSemiterm L n t) (hw : IsSemitermVec L n m w) :
+ termBShift L (termSubst L w t) = termSubst L (termBShiftVec L n w) t := by
+ apply IsSemiterm.induction 𝚺 ?_ ?_ ?_ ?_ t ht
+ · definability
+ ... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_3fb12358fbd9_5 | e4c55024fd6db439 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Bootstrapping/Syntax/Term/Functions.lean | Functions | 5 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 6 | 1 | [
{
"theorem_name": "termShift_qVec",
"depth": 1,
"n_commands": 0,
"n_lines": 15,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n apply nth_ext' (n + 1)\n (by rw [len_termShiftVec hw.qVec.isUTerm])\n (by rw [len_qVec hw.termShiftVec.isUTerm])\n i... | [
{
"name": "termBShift_termShift",
"text": "lemma termBShift_termShift {t : V} (ht : IsSemiterm L n t) : termBShift L (termShift L t) = termShift L (termBShift L t) := by\n apply IsSemiterm.induction 𝚺 ?_ ?_ ?_ ?_ t ht\n · definability\n · intro z hz; simp\n · intro x; simp\n · intro k f v hkf hv ih\n ... | [
{
"name": "termShift_qVec",
"fan_in": 0,
"n_deps_direct": 5,
"n_deps_transitive": 8,
"n_lines": 18,
"n_chars": 697,
"n_subproofs": 0,
"n_tactics": 15,
"cyclomatic": 3,
"n_automation": 8,
"n_rewrites": 5,
"n_structural": 4,
"automation_only": false,
"max_nestin... | 1 | module
public import Foundation.FirstOrder.Bootstrapping.Syntax.Term.Basic
@[expose] public section
namespace LO.FirstOrder.Arithmetic.Bootstrapping
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
section
variable {L : Language} [L.Encodable] [L.LORDefinable]
namespace TermSubst
def blueprint : Langua... | module
public import Foundation.FirstOrder.Bootstrapping.Syntax.Term.Basic
@[expose] public section
namespace LO.FirstOrder.Arithmetic.Bootstrapping
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
section
variable {L : Language} [L.Encodable] [L.LORDefinable]
namespace TermSubst
def blueprint : Langua... | @@ -363,6 +363,23 @@
⟨by simp [hv.isUTerm], fun i hi ↦ by
rw [nth_termBShiftVec hv.isUTerm hi]; exact (hv.nth hi).termBShift⟩
+lemma termBShift_termShift {t : V} (ht : IsSemiterm L n t) : termBShift L (termShift L t) = termShift L (termBShift L t) := by
+ apply IsSemiterm.induction 𝚺 ?_ ?_ ?_ ?_ t ht
+ · d... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_77474d4ead5f_0 | 47d1c2a61fdbb6d2 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Basic/Syntax/Formula.lean | Formula | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 6 | 1 | [
{
"theorem_name": "qr_iff",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": "\n by simp [iff_eq, total_of]",
"n_chars": 29,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 1,
... | [
{
"name": "iff_eq",
"text": "lemma iff_eq (φ ψ : Semiformula L ξ n) : φ 🡘 ψ = (∼φ ⋎ ψ) ⋏ (∼ψ ⋎ φ) := rfl\n\n",
"fan_in": 1,
"n_lines": 3,
"n_chars": 99,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automat... | [
{
"name": "qr_iff",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 4,
"n_chars": 119,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"max_nesting": 2
}
] | 1 | module
public import Foundation.Syntax.Predicate.Term
public import Foundation.Syntax.Predicate.Quantifier
public import Mathlib.Data.Nat.Cast.Order.Basic
@[expose] public section
/-!
# Formulas of first-order logic
This file defines the formulas of first-order logic.
`φ : Semiformula L ξ n` is a (semi-)formula of... | module
public import Foundation.Syntax.Predicate.Term
public import Foundation.Syntax.Predicate.Quantifier
public import Mathlib.Data.Nat.Cast.Order.Basic
@[expose] public section
/-!
# Formulas of first-order logic
This file defines the formulas of first-order logic.
`φ : Semiformula L ξ n` is a (semi-)formula of... | @@ -128,6 +128,8 @@
lemma imp_eq (φ ψ : Semiformula L ξ n) : φ 🡒 ψ = ∼φ ⋎ ψ := rfl
+lemma iff_eq (φ ψ : Semiformula L ξ n) : φ 🡘 ψ = (∼φ ⋎ ψ) ⋏ (∼ψ ⋎ φ) := rfl
+
@[simp] lemma neg_ball (φ ψ : Semiformula L ξ (n + 1)) : ∼(∀⁰[φ] ψ) = ∃⁰[φ] ∼ψ := by
simp [ball, bexs, imp_eq]
@@ -329,7 +331,8 @@
@[simp] lemma... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_77474d4ead5f_1 | 83c9f0cf0bf40c61 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Basic/Syntax/Formula.lean | Formula | 1 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 6 | 3 | [
{
"theorem_name": "freeVariables_not",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n induction φ using rec' <;> simp [*, freeVariables_rel, freeVariables_nrel]",
"n_chars": 81,
"n_subproofs": 0,
... | [
{
"name": "freeVariables_rel",
"text": "lemma freeVariables_rel {k} (r : L.Rel k) (v : Fin k → Semiterm L ξ n) : (rel r v).freeVariables = .biUnion .univ fun i ↦ (v i).freeVariables := rfl\n\n",
"fan_in": 3,
"n_lines": 3,
"n_chars": 155,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic"... | [
{
"name": "freeVariables_not",
"fan_in": 0,
"n_deps_direct": 2,
"n_deps_transitive": 2,
"n_lines": 4,
"n_chars": 186,
"n_subproofs": 0,
"n_tactics": 3,
"cyclomatic": 3,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": false,
"max_nesti... | 2 | module
public import Foundation.Syntax.Predicate.Term
public import Foundation.Syntax.Predicate.Quantifier
public import Mathlib.Data.Nat.Cast.Order.Basic
@[expose] public section
/-!
# Formulas of first-order logic
This file defines the formulas of first-order logic.
`φ : Semiformula L ξ n` is a (semi-)formula of... | module
public import Foundation.Syntax.Predicate.Term
public import Foundation.Syntax.Predicate.Quantifier
public import Mathlib.Data.Nat.Cast.Order.Basic
@[expose] public section
/-!
# Formulas of first-order logic
This file defines the formulas of first-order logic.
`φ : Semiformula L ξ n` is a (semi-)formula of... | @@ -385,6 +385,8 @@
| ∀⁰ φ => freeVariables φ
| ∃⁰ φ => freeVariables φ
+lemma freeVariables_rel {k} (r : L.Rel k) (v : Fin k → Semiterm L ξ n) : (rel r v).freeVariables = .biUnion .univ fun i ↦ (v i).freeVariables := rfl
+
lemma freeVariables_nrel {k} (r : L.Rel k) (v : Fin k → Semiterm L ξ n) : (nrel... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_77474d4ead5f_2 | 463326b472d05b1d | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Basic/Syntax/Formula.lean | Formula | 2 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 6 | 3 | [
{
"theorem_name": "freeVariables_not",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n induction φ using rec' <;> simp [*, freeVariables_rel, freeVariables_nrel]",
"n_chars": 81,
"n_subproofs": 0,
... | [
{
"name": "freeVariables_nrel",
"text": "lemma freeVariables_nrel {k} (r : L.Rel k) (v : Fin k → Semiterm L ξ n) : (nrel r v).freeVariables = .biUnion .univ fun i ↦ (v i).freeVariables := rfl\n\n",
"fan_in": 3,
"n_lines": 3,
"n_chars": 157,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomat... | [
{
"name": "fvar?_nrel",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 4,
"n_chars": 167,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"max_nesting": 1
... | 1 | module
public import Foundation.Syntax.Predicate.Term
public import Foundation.Syntax.Predicate.Quantifier
public import Mathlib.Data.Nat.Cast.Order.Basic
@[expose] public section
/-!
# Formulas of first-order logic
This file defines the formulas of first-order logic.
`φ : Semiformula L ξ n` is a (semi-)formula of... | module
public import Foundation.Syntax.Predicate.Term
public import Foundation.Syntax.Predicate.Quantifier
public import Mathlib.Data.Nat.Cast.Order.Basic
@[expose] public section
/-!
# Formulas of first-order logic
This file defines the formulas of first-order logic.
`φ : Semiformula L ξ n` is a (semi-)formula of... | @@ -387,6 +387,8 @@
lemma freeVariables_rel {k} (r : L.Rel k) (v : Fin k → Semiterm L ξ n) : (rel r v).freeVariables = .biUnion .univ fun i ↦ (v i).freeVariables := rfl
+lemma freeVariables_nrel {k} (r : L.Rel k) (v : Fin k → Semiterm L ξ n) : (nrel r v).freeVariables = .biUnion .univ fun i ↦ (v i).freeVariables :... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_77474d4ead5f_3 | 895585c311268c14 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Basic/Syntax/Formula.lean | Formula | 3 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 6 | 1 | [
{
"theorem_name": "not_fvar?_of_lt_fvSup",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": "\n fun hm ↦ (lt_self_iff_false _).mp (lt_of_le_of_lt h <| lt_fvSup_of_fvar? hm)",
"n_chars": 81,
"n_subproofs": 0,
... | [
{
"name": "lt_fvSup_of_fvar?",
"text": "lemma lt_fvSup_of_fvar? {φ : Semiproposition L n} : φ.FVar? m → m < φ.fvSup := by\n unfold fvSup FVar?\n intro hm\n have : ∃ s : ℕ, φ.freeVariables.max = s := Finset.max_of_mem hm\n rcases this with ⟨s, hs⟩\n have : m ≤ s := by\n have : (m : WithBot ℕ) ≤ ↑s :=... | [
{
"name": "not_fvar?_of_lt_fvSup",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 4,
"n_chars": 176,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
"max_n... | 1 | module
public import Foundation.Syntax.Predicate.Term
public import Foundation.Syntax.Predicate.Quantifier
public import Mathlib.Data.Nat.Cast.Order.Basic
@[expose] public section
/-!
# Formulas of first-order logic
This file defines the formulas of first-order logic.
`φ : Semiformula L ξ n` is a (semi-)formula of... | module
public import Foundation.Syntax.Predicate.Term
public import Foundation.Syntax.Predicate.Quantifier
public import Mathlib.Data.Nat.Cast.Order.Basic
@[expose] public section
/-!
# Formulas of first-order logic
This file defines the formulas of first-order logic.
`φ : Semiformula L ξ n` is a (semi-)formula of... | @@ -436,8 +436,19 @@
def fvSup (φ : Semiproposition L n) : ℕ := (φ.freeVariables.max).recBotCoe 0 .succ
-lemma not_fvar?_of_lt_fvSup (φ : Semiproposition L n) (h : φ.fvSup ≤ m) : ¬φ.FVar? m := sorry
+lemma lt_fvSup_of_fvar? {φ : Semiproposition L n} : φ.FVar? m → m < φ.fvSup := by
+ unfold fvSup FVar?
+ intro hm... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_77474d4ead5f_4 | 340da534a2dda704 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Basic/Syntax/Formula.lean | Formula | 4 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 6 | 2 | [
{
"theorem_name": "ne_or_left",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " ne_of_ne_complexity (by simp)",
"n_chars": 30,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": ... | [
{
"name": "ne_of_ne_complexity",
"text": "lemma ne_of_ne_complexity {φ ψ : Semiformula L ξ n} (h : φ.complexity ≠ ψ.complexity) : φ ≠ ψ :=\n by rintro rfl; contradiction\n\n",
"fan_in": 2,
"n_lines": 4,
"n_chars": 140,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 2,
"n_automa... | [
{
"name": "ne_or_left",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 3,
"n_chars": 107,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"max_nesting": 1
... | 1 | module
public import Foundation.Syntax.Predicate.Term
public import Foundation.Syntax.Predicate.Quantifier
public import Mathlib.Data.Nat.Cast.Order.Basic
@[expose] public section
/-!
# Formulas of first-order logic
This file defines the formulas of first-order logic.
`φ : Semiformula L ξ n` is a (semi-)formula of... | module
public import Foundation.Syntax.Predicate.Term
public import Foundation.Syntax.Predicate.Quantifier
public import Mathlib.Data.Nat.Cast.Order.Basic
@[expose] public section
/-!
# Formulas of first-order logic
This file defines the formulas of first-order logic.
`φ : Semiformula L ξ n` is a (semi-)formula of... | @@ -460,10 +460,13 @@
end
-@[simp] lemma ne_or_left (φ ψ : Semiformula L ξ n) : φ ≠ φ ⋎ ψ := sorry
+lemma ne_of_ne_complexity {φ ψ : Semiformula L ξ n} (h : φ.complexity ≠ ψ.complexity) : φ ≠ ψ :=
+ by rintro rfl; contradiction
-@[simp] lemma ne_or_right (φ ψ : Semiformula L ξ n) : ψ ≠ φ ⋎ ψ := sorry
+@[simp] 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_df5598afe937_0 | c779ec867b0d27d4 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Syntax/Predicate/Relational.lean | Relational | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 2 | 2 | [
{
"theorem_name": "fvar_of_relational",
"depth": 1,
"n_commands": 0,
"n_lines": 4,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n rcases bvar_or_fvar_of_relational t with (⟨x, rfl⟩ | ⟨x, rfl⟩)\n · exact finZeroElim (α := fun _ ↦ _) x\n · exact ⟨x, ... | [
{
"name": "bvar_or_fvar_of_relational",
"text": "lemma bvar_or_fvar_of_relational (t : Semiterm L ξ n) : (∃ x, t = #x) ∨ (∃ x, t = &x) :=\n match t with\n | #x => by simp\n | &x => by simp\n | .func f _ => Language.Relational.func_empty _ |>.elim' f\n\n",
"fan_in": 2,
"n_lines": 7,... | [
{
"name": "fvar_of_relational",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 6,
"n_chars": 206,
"n_subproofs": 0,
"n_tactics": 4,
"cyclomatic": 2,
"n_automation": 3,
"n_rewrites": 0,
"n_structural": 3,
"automation_only": false,
"max_nest... | 1 | module
public import Foundation.Syntax.Predicate.Rew
@[expose] public section
namespace LO.FirstOrder
namespace Semiterm
variable {L : Language} [L.Relational]
lemma fvar_of_relational (t : Term L ξ) : ∃ x, t = &x := sorry
variable {M : Type*} (bv : Fin n → M) (fv : ξ → M)
def relationalVal : Semiterm L ξ n → M... | module
public import Foundation.Syntax.Predicate.Rew
@[expose] public section
namespace LO.FirstOrder
namespace Semiterm
variable {L : Language} [L.Relational]
lemma bvar_or_fvar_of_relational (t : Semiterm L ξ n) : (∃ x, t = #x) ∨ (∃ x, t = &x) :=
match t with
| #x => by simp
| &x => by simp
... | @@ -10,8 +10,17 @@
variable {L : Language} [L.Relational]
-lemma fvar_of_relational (t : Term L ξ) : ∃ x, t = &x := sorry
+lemma bvar_or_fvar_of_relational (t : Semiterm L ξ n) : (∃ x, t = #x) ∨ (∃ x, t = &x) :=
+ match t with
+ | #x => by simp
+ | &x => by simp
+ | .func f _ => Language.Relation... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_df5598afe937_1 | 46e23d3a963544a5 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Syntax/Predicate/Relational.lean | Relational | 1 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 2 | 2 | [
{
"theorem_name": "fvar_of_relational",
"depth": 1,
"n_commands": 0,
"n_lines": 4,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n rcases bvar_or_fvar_of_relational t with (⟨x, rfl⟩ | ⟨x, rfl⟩)\n · exact finZeroElim (α := fun _ ↦ _) x\n · exact ⟨x, ... | [
{
"name": "bvar_or_fvar_of_relational",
"text": "lemma bvar_or_fvar_of_relational (t : Semiterm L ξ n) : (∃ x, t = #x) ∨ (∃ x, t = &x) :=\n match t with\n | #x => by simp\n | &x => by simp\n | .func f _ => Language.Relational.func_empty _ |>.elim' f\n\n",
"fan_in": 2,
"n_lines": 7,... | [
{
"name": "relationalVal_rew",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 5,
"n_chars": 344,
"n_subproofs": 0,
"n_tactics": 3,
"cyclomatic": 3,
"n_automation": 3,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": false,
"max_nesti... | 1 | module
public import Foundation.Syntax.Predicate.Rew
@[expose] public section
namespace LO.FirstOrder
namespace Semiterm
variable {L : Language} [L.Relational]
variable {M : Type*} (bv : Fin n → M) (fv : ξ → M)
def relationalVal : Semiterm L ξ n → M
| #x => bv x
| &x => fv x
| .func f _ => La... | module
public import Foundation.Syntax.Predicate.Rew
@[expose] public section
namespace LO.FirstOrder
namespace Semiterm
variable {L : Language} [L.Relational]
lemma bvar_or_fvar_of_relational (t : Semiterm L ξ n) : (∃ x, t = #x) ∨ (∃ x, t = &x) :=
match t with
| #x => by simp
| &x => by simp
... | @@ -10,6 +10,12 @@
variable {L : Language} [L.Relational]
+lemma bvar_or_fvar_of_relational (t : Semiterm L ξ n) : (∃ x, t = #x) ∨ (∃ x, t = &x) :=
+ match t with
+ | #x => by simp
+ | &x => by simp
+ | .func f _ => Language.Relational.func_empty _ |>.elim' f
+
variable {M : Type*} (bv : Fin n →... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_df5598afe937_2 | 047b8f7ed22f2398 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Syntax/Predicate/Relational.lean | Relational | 2 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 2 | 1 | [
{
"theorem_name": "relationalVal_bShift",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n simp [relationalVal_rew, Function.comp_def]",
"n_chars": 49,
"n_subproofs": 0,
"n_tactics": 2,
"cyclom... | [
{
"name": "relationalVal_rew",
"text": "lemma relationalVal_rew {bv : Fin n₂ → M} {fv : ξ₂ → M} (ω : Rew L ξ₁ n₁ ξ₂ n₂) (t : Semiterm L ξ₁ n₁) :\n relationalVal bv fv (ω t) = relationalVal (relationalVal bv fv ∘ ω ∘ bvar) (relationalVal bv fv ∘ ω ∘ fvar) t := by\n rcases bvar_or_fvar_of_relational t wit... | [
{
"name": "relationalVal_bShift",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 2,
"n_lines": 5,
"n_chars": 190,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"max_nes... | 2 | module
public import Foundation.Syntax.Predicate.Rew
@[expose] public section
namespace LO.FirstOrder
namespace Semiterm
variable {L : Language} [L.Relational]
lemma bvar_or_fvar_of_relational (t : Semiterm L ξ n) : (∃ x, t = #x) ∨ (∃ x, t = &x) :=
match t with
| #x => by simp
| &x => by simp
... | module
public import Foundation.Syntax.Predicate.Rew
@[expose] public section
namespace LO.FirstOrder
namespace Semiterm
variable {L : Language} [L.Relational]
lemma bvar_or_fvar_of_relational (t : Semiterm L ξ n) : (∃ x, t = #x) ∨ (∃ x, t = &x) :=
match t with
| #x => by simp
| &x => by simp
... | @@ -29,8 +29,13 @@
@[simp] lemma relationalVal_fvar : (&x : Semiterm L ξ n).relationalVal bv fv = fv x := rfl
+lemma relationalVal_rew {bv : Fin n₂ → M} {fv : ξ₂ → M} (ω : Rew L ξ₁ n₁ ξ₂ n₂) (t : Semiterm L ξ₁ n₁) :
+ relationalVal bv fv (ω t) = relationalVal (relationalVal bv fv ∘ ω ∘ bvar) (relationalVal bv 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_0b526a48560d_0 | c5d361f969b8972a | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Entailment/K4.lean | K4 | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 2 | 1 | [
{
"theorem_name": "Context.boxItr_2_in_context_to_box",
"depth": 1,
"n_commands": 0,
"n_lines": 16,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n apply Context.provable_iff_finset.mpr;\n obtain ⟨Δ, hΔ₁, hΔ₂⟩ := Context.provable_iff_finset.mp h;\n ... | [
{
"name": "Context.boxItr_2_in_context_to_box_finset",
"text": "lemma Context.boxItr_2_in_context_to_box_finset {Γ : Finset F} (h : ↑(□^[2]'Γ) *⊢[𝓢] φ) : ↑(□'Γ) *⊢[𝓢] φ := by\n apply FConj_DT.mp;\n refine C!_trans ?_ $ FConj_DT.mpr h;\n apply CFconjFconj!_of_provable;\n intro ξ hξ;\n obtain ⟨ξ, h, rf... | [
{
"name": "Context.boxItr_2_in_context_to_box",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 18,
"n_chars": 780,
"n_subproofs": 5,
"n_tactics": 31,
"cyclomatic": 2,
"n_automation": 3,
"n_rewrites": 0,
"n_structural": 12,
"automation_only": f... | 1 | module
public import Foundation.Modal.Entailment.K
public import Foundation.Modal.Entailment.AxiomGeach
@[expose] public section
namespace LO.Modal.Entailment
open LO.Entailment Entailment.FiniteContext
variable {S F : Type*} [BasicModalLogicalConnective F] [DecidableEq F] [Entailment S F]
variable {𝓢 : S} [Entai... | module
public import Foundation.Modal.Entailment.K
public import Foundation.Modal.Entailment.AxiomGeach
@[expose] public section
namespace LO.Modal.Entailment
open LO.Entailment Entailment.FiniteContext
variable {S F : Type*} [BasicModalLogicalConnective F] [DecidableEq F] [Entailment S F]
variable {𝓢 : S} [Entai... | @@ -44,6 +44,16 @@
def boxdotAxiomFour : 𝓢 ⊢! ⊡φ 🡒 ⊡⊡φ := K_left iff_Boxdot_BoxdotBoxdot
@[simp] lemma boxdot_axiomFour! : 𝓢 ⊢ ⊡φ 🡒 ⊡⊡φ := ⟨boxdotAxiomFour⟩
+lemma Context.boxItr_2_in_context_to_box_finset {Γ : Finset F} (h : ↑(□^[2]'Γ) *⊢[𝓢] φ) : ↑(□'Γ) *⊢[𝓢] φ := by
+ apply FConj_DT.mp;
+ refine C!_trans ... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_0b526a48560d_1 | c442d976de1d8f61 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/Entailment/K4.lean | K4 | 1 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 2 | 1 | [
{
"theorem_name": "Context.boxbox_in_context_to_box",
"depth": 1,
"n_commands": 0,
"n_lines": 5,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n apply Context.boxItr_2_in_context_to_box;\n suffices (□'□'Γ) = (□^[2]'Γ) by rwa [this] at h;\n ext;\n s... | [
{
"name": "Context.boxItr_2_in_context_to_box",
"text": "lemma Context.boxItr_2_in_context_to_box [InjectiveBox F] {Γ : Set F} (h : (□^[2]'Γ) *⊢[𝓢] φ) : (□'Γ) *⊢[𝓢] φ := by\n apply Context.provable_iff_finset.mpr;\n obtain ⟨Δ, hΔ₁, hΔ₂⟩ := Context.provable_iff_finset.mp h;\n use □'(□⁻¹^[2]'Δ);\n const... | [
{
"name": "Context.boxbox_in_context_to_box",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 2,
"n_lines": 7,
"n_chars": 268,
"n_subproofs": 1,
"n_tactics": 9,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 2,
"automation_only": false,... | 2 | module
public import Foundation.Modal.Entailment.K
public import Foundation.Modal.Entailment.AxiomGeach
@[expose] public section
namespace LO.Modal.Entailment
open LO.Entailment Entailment.FiniteContext
variable {S F : Type*} [BasicModalLogicalConnective F] [DecidableEq F] [Entailment S F]
variable {𝓢 : S} [Entai... | module
public import Foundation.Modal.Entailment.K
public import Foundation.Modal.Entailment.AxiomGeach
@[expose] public section
namespace LO.Modal.Entailment
open LO.Entailment Entailment.FiniteContext
variable {S F : Type*} [BasicModalLogicalConnective F] [DecidableEq F] [Entailment S F]
variable {𝓢 : S} [Entai... | @@ -54,8 +54,29 @@
apply Context.by_axm!
grind;
-lemma Context.boxbox_in_context_to_box [InjectiveBox F] {Γ : Set F} (h : (□'□'Γ) *⊢[𝓢] φ) : (□'Γ) *⊢[𝓢] φ := sorry
+lemma Context.boxItr_2_in_context_to_box [InjectiveBox F] {Γ : Set F} (h : (□^[2]'Γ) *⊢[𝓢] φ) : (□'Γ) *⊢[𝓢] φ := by
+ apply Context.provable_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_ed41caf2641c_1 | aa405fcff47bce32 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Bootstrapping/Syntax/Proof/Basic.lean | Basic | 1 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 11 | 6 | [
{
"theorem_name": "IsFormulaSet.setShift",
"depth": 1,
"n_commands": 0,
"n_lines": 3,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n simp only [IsFormulaSet, mem_setShift_iff, forall_exists_index, and_imp]\n rintro _ p hp rfl; exact (h p hp).shift",... | [
{
"name": "mem_setShift_iff",
"text": "lemma mem_setShift_iff {s y : V} : y ∈ setShift L s ↔ ∃ x ∈ s, y = shift L x :=\n Classical.choose!_spec (setShift_existsUnique L s) y\n\n",
"fan_in": 6,
"n_lines": 4,
"n_chars": 144,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_au... | [
{
"name": "case_iff",
"fan_in": 0,
"n_deps_direct": 28,
"n_deps_transitive": 20,
"n_lines": 18,
"n_chars": 1209,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
"max_nesting": ... | 2 | module
public import Foundation.FirstOrder.Bootstrapping.Syntax.Theory
@[expose] public section
namespace LO
open FirstOrder Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
variable {L : Language} [L.Encodable] [L.LORDefinable]
variable {T U : Theory L} [T.Δ₁] [U.Δ₁]
namespace FirstOrder.Ar... | module
public import Foundation.FirstOrder.Bootstrapping.Syntax.Theory
@[expose] public section
namespace LO
open FirstOrder Arithmetic
variable {V : Type*} [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗜𝚺₁]
variable {L : Language} [L.Encodable] [L.LORDefinable]
variable {T U : Theory L} [T.Δ₁] [U.Δ₁]
namespace FirstOrder.Ar... | @@ -79,19 +79,34 @@
section setShift
-lemma IsFormulaSet.setShift {s : V} (h : IsFormulaSet L s) : IsFormulaSet L (setShift L s) := sorry
+lemma mem_setShift_iff {s y : V} : y ∈ setShift L s ↔ ∃ x ∈ s, y = shift L x :=
+ Classical.choose!_spec (setShift_existsUnique L s) y
-lemma shift_mem_setShift {p s : V} (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_48c8c271a9a9_0 | f4177975d1c4cfb8 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Propositional/Entailment/Corsi/VF.lean | VF | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 11 | 2 | [
{
"theorem_name": "CA_replace_left",
"depth": 1,
"n_commands": 0,
"n_lines": 4,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n apply CA_replace_both;\n . assumption;\n . simp;",
"n_chars": 54,
"n_subproofs": 0,
"n_tactics": 7,
"cycl... | [
{
"name": "CA_replace_both",
"text": "lemma CA_replace_both (h₁ : 𝓢 ⊢ φ 🡒 φ') (h₂ : 𝓢 ⊢ ψ 🡒 ψ') : 𝓢 ⊢ φ ⋎ ψ 🡒 φ' ⋎ ψ' := by\n apply ruleD;\n . apply C_trans h₁; simp;\n . apply C_trans h₂; simp;\n\n",
"fan_in": 2,
"n_lines": 6,
"n_chars": 202,
"n_subproofs": 0,
"n_tactics": 9,
... | [
{
"name": "CA_replace_left",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 6,
"n_chars": 144,
"n_subproofs": 0,
"n_tactics": 7,
"cyclomatic": 1,
"n_automation": 2,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": false,
"max_nesting... | 1 | module
public import Foundation.Propositional.Entailment.Corsi.Basic
public import Foundation.Logic.Disjunctive
@[expose] public section
namespace LO.Propositional
namespace Entailment
variable {S F : Type*} [LogicalConnective F] [Entailment S F]
variable {𝓢 : S} {φ ψ χ : F}
protected class VF (𝓢 : S) extends
... | module
public import Foundation.Propositional.Entailment.Corsi.Basic
public import Foundation.Logic.Disjunctive
@[expose] public section
namespace LO.Propositional
namespace Entailment
variable {S F : Type*} [LogicalConnective F] [Entailment S F]
variable {𝓢 : S} {φ ψ χ : F}
protected class VF (𝓢 : S) extends
... | @@ -48,8 +48,16 @@
contrapose! hφ;
exact efq ⨀ hφ;
-lemma CA_replace_left (h : 𝓢 ⊢ φ' 🡒 φ) : 𝓢 ⊢ φ' ⋎ ψ 🡒 φ ⋎ ψ := sorry
+lemma CA_replace_both (h₁ : 𝓢 ⊢ φ 🡒 φ') (h₂ : 𝓢 ⊢ ψ 🡒 ψ') : 𝓢 ⊢ φ ⋎ ψ 🡒 φ' ⋎ ψ' := by
+ apply ruleD;
+ . apply C_trans h₁; simp;
+ . apply C_trans h₂; simp;
+lemma CA_replace_... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_48c8c271a9a9_1 | b4d049b0cef19062 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Propositional/Entailment/Corsi/VF.lean | VF | 1 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 11 | 2 | [
{
"theorem_name": "CA_replace_left",
"depth": 1,
"n_commands": 0,
"n_lines": 4,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n apply CA_replace_both;\n . assumption;\n . simp;",
"n_chars": 54,
"n_subproofs": 0,
"n_tactics": 7,
"cycl... | [
{
"name": "CA_replace_both",
"text": "lemma CA_replace_both (h₁ : 𝓢 ⊢ φ 🡒 φ') (h₂ : 𝓢 ⊢ ψ 🡒 ψ') : 𝓢 ⊢ φ ⋎ ψ 🡒 φ' ⋎ ψ' := by\n apply ruleD;\n . apply C_trans h₁; simp;\n . apply C_trans h₂; simp;\n\n",
"fan_in": 2,
"n_lines": 6,
"n_chars": 202,
"n_subproofs": 0,
"n_tactics": 9,
... | [
{
"name": "CA_replace_right",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 7,
"n_chars": 146,
"n_subproofs": 0,
"n_tactics": 7,
"cyclomatic": 1,
"n_automation": 2,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": false,
"max_nestin... | 1 | module
public import Foundation.Propositional.Entailment.Corsi.Basic
public import Foundation.Logic.Disjunctive
@[expose] public section
namespace LO.Propositional
namespace Entailment
variable {S F : Type*} [LogicalConnective F] [Entailment S F]
variable {𝓢 : S} {φ ψ χ : F}
protected class VF (𝓢 : S) extends
... | module
public import Foundation.Propositional.Entailment.Corsi.Basic
public import Foundation.Logic.Disjunctive
@[expose] public section
namespace LO.Propositional
namespace Entailment
variable {S F : Type*} [LogicalConnective F] [Entailment S F]
variable {𝓢 : S} {φ ψ χ : F}
protected class VF (𝓢 : S) extends
... | @@ -48,8 +48,16 @@
contrapose! hφ;
exact efq ⨀ hφ;
-lemma CA_replace_right (h : 𝓢 ⊢ ψ 🡒 ψ') : 𝓢 ⊢ φ ⋎ ψ 🡒 φ ⋎ ψ' := sorry
+lemma CA_replace_both (h₁ : 𝓢 ⊢ φ 🡒 φ') (h₂ : 𝓢 ⊢ ψ 🡒 ψ') : 𝓢 ⊢ φ ⋎ ψ 🡒 φ' ⋎ ψ' := by
+ apply ruleD;
+ . apply C_trans h₁; simp;
+ . apply C_trans h₂; simp;
+lemma CA_replace... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_48c8c271a9a9_2 | e45732e77791f8f1 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Propositional/Entailment/Corsi/VF.lean | VF | 2 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 11 | 2 | [
{
"theorem_name": "ruleC_fconj",
"depth": 1,
"n_commands": 0,
"n_lines": 5,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n apply ruleC_lconj₂;\n intro γ hγ;\n apply h γ;\n simpa using hγ;",
"n_chars": 76,
"n_subproofs": 0,
"n_tactics":... | [
{
"name": "ruleC_lconj₂",
"text": "lemma ruleC_lconj₂ {Γ : List F} (h : ∀ γ ∈ Γ, 𝓢 ⊢ φ 🡒 γ) : 𝓢 ⊢ φ 🡒 Γ.conj₂ := by\n induction Γ using List.induction_with_singleton with\n | hnil => apply af; simp;\n | hsingle ψ => apply h; simp;\n | hcons ψ Δ he ih =>\n simp only [List.conj₂_cons_nonempty he];\... | [
{
"name": "ruleC_fconj",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 7,
"n_chars": 182,
"n_subproofs": 0,
"n_tactics": 9,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 3,
"automation_only": false,
"max_nesting": 2... | 1 | module
public import Foundation.Propositional.Entailment.Corsi.Basic
public import Foundation.Logic.Disjunctive
@[expose] public section
namespace LO.Propositional
namespace Entailment
variable {S F : Type*} [LogicalConnective F] [Entailment S F]
variable {𝓢 : S} {φ ψ χ : F}
protected class VF (𝓢 : S) extends
... | module
public import Foundation.Propositional.Entailment.Corsi.Basic
public import Foundation.Logic.Disjunctive
@[expose] public section
namespace LO.Propositional
namespace Entailment
variable {S F : Type*} [LogicalConnective F] [Entailment S F]
variable {𝓢 : S} {φ ψ χ : F}
protected class VF (𝓢 : S) extends
... | @@ -58,8 +58,23 @@
@[simp, grind .]
lemma conjconj {Γ : Finset F} : 𝓢 ⊢ (Γ.conj) 🡒 Γ.toList.conj₂ := by simp [Finset.conj];
-lemma ruleC_fconj {Γ : Finset F} (h : ∀ γ ∈ Γ, 𝓢 ⊢ φ 🡒 γ) : 𝓢 ⊢ φ 🡒 Γ.conj := sorry
+lemma ruleC_lconj₂ {Γ : List F} (h : ∀ γ ∈ Γ, 𝓢 ⊢ φ 🡒 γ) : 𝓢 ⊢ φ 🡒 Γ.conj₂ := by
+ induction Γ ... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_48c8c271a9a9_3 | 7b3db63a330d8e36 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Propositional/Entailment/Corsi/VF.lean | VF | 3 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 11 | 2 | [
{
"theorem_name": "ruleC_fconj",
"depth": 1,
"n_commands": 0,
"n_lines": 5,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n apply ruleC_lconj₂;\n intro γ hγ;\n apply h γ;\n simpa using hγ;",
"n_chars": 76,
"n_subproofs": 0,
"n_tactics":... | [
{
"name": "ruleC_lconj₂",
"text": "lemma ruleC_lconj₂ {Γ : List F} (h : ∀ γ ∈ Γ, 𝓢 ⊢ φ 🡒 γ) : 𝓢 ⊢ φ 🡒 Γ.conj₂ := by\n induction Γ using List.induction_with_singleton with\n | hnil => apply af; simp;\n | hsingle ψ => apply h; simp;\n | hcons ψ Δ he ih =>\n simp only [List.conj₂_cons_nonempty he];\... | [
{
"name": "ruleC_fconj'",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 7,
"n_chars": 176,
"n_subproofs": 0,
"n_tactics": 5,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": false,
"max_nesting": ... | 1 | module
public import Foundation.Propositional.Entailment.Corsi.Basic
public import Foundation.Logic.Disjunctive
@[expose] public section
namespace LO.Propositional
namespace Entailment
variable {S F : Type*} [LogicalConnective F] [Entailment S F]
variable {𝓢 : S} {φ ψ χ : F}
protected class VF (𝓢 : S) extends
... | module
public import Foundation.Propositional.Entailment.Corsi.Basic
public import Foundation.Logic.Disjunctive
@[expose] public section
namespace LO.Propositional
namespace Entailment
variable {S F : Type*} [LogicalConnective F] [Entailment S F]
variable {𝓢 : S} {φ ψ χ : F}
protected class VF (𝓢 : S) extends
... | @@ -58,8 +58,21 @@
@[simp, grind .]
lemma conjconj {Γ : Finset F} : 𝓢 ⊢ (Γ.conj) 🡒 Γ.toList.conj₂ := by simp [Finset.conj];
-lemma ruleC_fconj' {Γ : Finset ι} (Φ : ι → F) (h : ∀ i ∈ Γ, 𝓢 ⊢ φ 🡒 Φ i) : 𝓢 ⊢ φ 🡒 (⩕ i ∈ Γ, Φ i) := sorry
+lemma ruleC_lconj₂ {Γ : List F} (h : ∀ γ ∈ Γ, 𝓢 ⊢ φ 🡒 γ) : 𝓢 ⊢ φ 🡒 Γ.conj... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_48c8c271a9a9_4 | 296a584ab5ca1328 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Propositional/Entailment/Corsi/VF.lean | VF | 4 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 11 | 4 | [
{
"theorem_name": "mem_fconj",
"depth": 1,
"n_commands": 0,
"n_lines": 3,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n apply mem_lconj₂;\n simpa using h;",
"n_chars": 42,
"n_subproofs": 0,
"n_tactics": 5,
"cyclomatic": 1,
"n_au... | [
{
"name": "mem_lconj₂",
"text": "lemma mem_lconj₂ {Γ : List F} (h : φ ∈ Γ) : 𝓢 ⊢ ⋀Γ 🡒 φ := by\n induction Γ using List.induction_with_singleton with\n | hcons ψ Δ he ih =>\n simp [List.conj₂_cons_nonempty he];\n simp at h;\n rcases h with rfl | h;\n . simp;\n . apply C_trans ?_ $ ih h;\n ... | [
{
"name": "mem_fconj",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 5,
"n_chars": 121,
"n_subproofs": 0,
"n_tactics": 5,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": false,
"max_nesting": 2
... | 1 | module
public import Foundation.Propositional.Entailment.Corsi.Basic
public import Foundation.Logic.Disjunctive
@[expose] public section
namespace LO.Propositional
namespace Entailment
variable {S F : Type*} [LogicalConnective F] [Entailment S F]
variable {𝓢 : S} {φ ψ χ : F}
protected class VF (𝓢 : S) extends
... | module
public import Foundation.Propositional.Entailment.Corsi.Basic
public import Foundation.Logic.Disjunctive
@[expose] public section
namespace LO.Propositional
namespace Entailment
variable {S F : Type*} [LogicalConnective F] [Entailment S F]
variable {𝓢 : S} {φ ψ χ : F}
protected class VF (𝓢 : S) extends
... | @@ -58,8 +58,21 @@
@[simp, grind .]
lemma conjconj {Γ : Finset F} : 𝓢 ⊢ (Γ.conj) 🡒 Γ.toList.conj₂ := by simp [Finset.conj];
-lemma mem_fconj {Γ : Finset F} (h : φ ∈ Γ) : 𝓢 ⊢ Γ.conj 🡒 φ := sorry
+lemma mem_lconj₂ {Γ : List F} (h : φ ∈ Γ) : 𝓢 ⊢ ⋀Γ 🡒 φ := by
+ induction Γ using List.induction_with_singleton 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_48c8c271a9a9_5 | 3f3cc71ef01d6256 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Propositional/Entailment/Corsi/VF.lean | VF | 5 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 11 | 4 | [
{
"theorem_name": "mem_fconj",
"depth": 1,
"n_commands": 0,
"n_lines": 3,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n apply mem_lconj₂;\n simpa using h;",
"n_chars": 42,
"n_subproofs": 0,
"n_tactics": 5,
"cyclomatic": 1,
"n_au... | [
{
"name": "mem_lconj₂",
"text": "lemma mem_lconj₂ {Γ : List F} (h : φ ∈ Γ) : 𝓢 ⊢ ⋀Γ 🡒 φ := by\n induction Γ using List.induction_with_singleton with\n | hcons ψ Δ he ih =>\n simp [List.conj₂_cons_nonempty he];\n simp at h;\n rcases h with rfl | h;\n . simp;\n . apply C_trans ?_ $ ih h;\n ... | [
{
"name": "mem_fconj'",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 6,
"n_chars": 161,
"n_subproofs": 0,
"n_tactics": 5,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": false,
"max_nesting": 2
... | 1 | module
public import Foundation.Propositional.Entailment.Corsi.Basic
public import Foundation.Logic.Disjunctive
@[expose] public section
namespace LO.Propositional
namespace Entailment
variable {S F : Type*} [LogicalConnective F] [Entailment S F]
variable {𝓢 : S} {φ ψ χ : F}
protected class VF (𝓢 : S) extends
... | module
public import Foundation.Propositional.Entailment.Corsi.Basic
public import Foundation.Logic.Disjunctive
@[expose] public section
namespace LO.Propositional
namespace Entailment
variable {S F : Type*} [LogicalConnective F] [Entailment S F]
variable {𝓢 : S} {φ ψ χ : F}
protected class VF (𝓢 : S) extends
... | @@ -58,8 +58,21 @@
@[simp, grind .]
lemma conjconj {Γ : Finset F} : 𝓢 ⊢ (Γ.conj) 🡒 Γ.toList.conj₂ := by simp [Finset.conj];
-lemma mem_fconj' {Γ : Finset ι} (Φ : ι → F) (hΦ : ∃ i ∈ Γ, Φ i = ψ) : 𝓢 ⊢ (⩕ i ∈ Γ, Φ i) 🡒 ψ := sorry
+lemma mem_lconj₂ {Γ : List F} (h : φ ∈ Γ) : 𝓢 ⊢ ⋀Γ 🡒 φ := by
+ induction Γ 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_48c8c271a9a9_6 | 3fe9cba4ace67cb8 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Propositional/Entailment/Corsi/VF.lean | VF | 6 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 11 | 3 | [
{
"theorem_name": "LConj₂Conj₂_of_subset",
"depth": 1,
"n_commands": 0,
"n_lines": 4,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n apply LConj₂Conj₂_of_provable;\n intro δ hδ;\n apply mem_lconj₂ $ h δ hδ;",
"n_chars": 89,
"n_subproofs": 0... | [
{
"name": "LConj₂Conj₂_of_provable",
"text": "lemma LConj₂Conj₂_of_provable {Δ : List F} (h : ∀ δ ∈ Δ, 𝓢 ⊢ γ 🡒 δ) : 𝓢 ⊢ γ 🡒 Δ.conj₂ := by\n induction Δ using List.induction_with_singleton with\n | hnil => apply af; simp;\n | hsingle φ =>\n apply h;\n simp;\n | hcons ψ Δ he ih =>\n simp only... | [
{
"name": "LConj₂Conj₂_of_subset",
"fan_in": 1,
"n_deps_direct": 2,
"n_deps_transitive": 2,
"n_lines": 6,
"n_chars": 218,
"n_subproofs": 0,
"n_tactics": 7,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 3,
"automation_only": false,
"max_n... | 2 | module
public import Foundation.Propositional.Entailment.Corsi.Basic
public import Foundation.Logic.Disjunctive
@[expose] public section
namespace LO.Propositional
namespace Entailment
variable {S F : Type*} [LogicalConnective F] [Entailment S F]
variable {𝓢 : S} {φ ψ χ : F}
protected class VF (𝓢 : S) extends
... | module
public import Foundation.Propositional.Entailment.Corsi.Basic
public import Foundation.Logic.Disjunctive
@[expose] public section
namespace LO.Propositional
namespace Entailment
variable {S F : Type*} [LogicalConnective F] [Entailment S F]
variable {𝓢 : S} {φ ψ χ : F}
protected class VF (𝓢 : S) extends
... | @@ -58,6 +58,19 @@
@[simp, grind .]
lemma conjconj {Γ : Finset F} : 𝓢 ⊢ (Γ.conj) 🡒 Γ.toList.conj₂ := by simp [Finset.conj];
+lemma LConj₂Conj₂_of_provable {Δ : List F} (h : ∀ δ ∈ Δ, 𝓢 ⊢ γ 🡒 δ) : 𝓢 ⊢ γ 🡒 Δ.conj₂ := by
+ induction Δ using List.induction_with_singleton with
+ | hnil => apply af; simp;
+ | hsi... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_48c8c271a9a9_7 | 6bde42cd1cb9282c | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Propositional/Entailment/Corsi/VF.lean | VF | 7 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 11 | 3 | [
{
"theorem_name": "LConj₂Conj₂_of_subset",
"depth": 1,
"n_commands": 0,
"n_lines": 4,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n apply LConj₂Conj₂_of_provable;\n intro δ hδ;\n apply mem_lconj₂ $ h δ hδ;",
"n_chars": 89,
"n_subproofs": 0... | [
{
"name": "LConj₂Conj₂_of_provable",
"text": "lemma LConj₂Conj₂_of_provable {Δ : List F} (h : ∀ δ ∈ Δ, 𝓢 ⊢ γ 🡒 δ) : 𝓢 ⊢ γ 🡒 Δ.conj₂ := by\n induction Δ using List.induction_with_singleton with\n | hnil => apply af; simp;\n | hsingle φ =>\n apply h;\n simp;\n | hcons ψ Δ he ih =>\n simp only... | [
{
"name": "CFConjFConj_of_subset",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 3,
"n_lines": 5,
"n_chars": 146,
"n_subproofs": 0,
"n_tactics": 5,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": false,
"max_n... | 3 | module
public import Foundation.Propositional.Entailment.Corsi.Basic
public import Foundation.Logic.Disjunctive
@[expose] public section
namespace LO.Propositional
namespace Entailment
variable {S F : Type*} [LogicalConnective F] [Entailment S F]
variable {𝓢 : S} {φ ψ χ : F}
protected class VF (𝓢 : S) extends
... | module
public import Foundation.Propositional.Entailment.Corsi.Basic
public import Foundation.Logic.Disjunctive
@[expose] public section
namespace LO.Propositional
namespace Entailment
variable {S F : Type*} [LogicalConnective F] [Entailment S F]
variable {𝓢 : S} {φ ψ χ : F}
protected class VF (𝓢 : S) extends
... | @@ -58,6 +58,19 @@
@[simp, grind .]
lemma conjconj {Γ : Finset F} : 𝓢 ⊢ (Γ.conj) 🡒 Γ.toList.conj₂ := by simp [Finset.conj];
+lemma LConj₂Conj₂_of_provable {Δ : List F} (h : ∀ δ ∈ Δ, 𝓢 ⊢ γ 🡒 δ) : 𝓢 ⊢ γ 🡒 Δ.conj₂ := by
+ induction Δ using List.induction_with_singleton with
+ | hnil => apply af; simp;
+ | hsi... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_48c8c271a9a9_8 | 238f6713349d1686 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Propositional/Entailment/Corsi/VF.lean | VF | 8 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 11 | 3 | [
{
"theorem_name": "LConj₂Conj₂_of_subset",
"depth": 1,
"n_commands": 0,
"n_lines": 4,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n apply LConj₂Conj₂_of_provable;\n intro δ hδ;\n apply mem_lconj₂ $ h δ hδ;",
"n_chars": 89,
"n_subproofs": 0... | [
{
"name": "LConj₂Conj₂_of_provable",
"text": "lemma LConj₂Conj₂_of_provable {Δ : List F} (h : ∀ δ ∈ Δ, 𝓢 ⊢ γ 🡒 δ) : 𝓢 ⊢ γ 🡒 Δ.conj₂ := by\n induction Δ using List.induction_with_singleton with\n | hnil => apply af; simp;\n | hsingle φ =>\n apply h;\n simp;\n | hcons ψ Δ he ih =>\n simp only... | [
{
"name": "FConj₂_of_LConj",
"fan_in": 1,
"n_deps_direct": 2,
"n_deps_transitive": 2,
"n_lines": 7,
"n_chars": 183,
"n_subproofs": 0,
"n_tactics": 9,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 3,
"automation_only": false,
"max_nesting... | 2 | module
public import Foundation.Propositional.Entailment.Corsi.Basic
public import Foundation.Logic.Disjunctive
@[expose] public section
namespace LO.Propositional
namespace Entailment
variable {S F : Type*} [LogicalConnective F] [Entailment S F]
variable {𝓢 : S} {φ ψ χ : F}
protected class VF (𝓢 : S) extends
... | module
public import Foundation.Propositional.Entailment.Corsi.Basic
public import Foundation.Logic.Disjunctive
@[expose] public section
namespace LO.Propositional
namespace Entailment
variable {S F : Type*} [LogicalConnective F] [Entailment S F]
variable {𝓢 : S} {φ ψ χ : F}
protected class VF (𝓢 : S) extends
... | @@ -58,6 +58,19 @@
@[simp, grind .]
lemma conjconj {Γ : Finset F} : 𝓢 ⊢ (Γ.conj) 🡒 Γ.toList.conj₂ := by simp [Finset.conj];
+lemma LConj₂Conj₂_of_provable {Δ : List F} (h : ∀ δ ∈ Δ, 𝓢 ⊢ γ 🡒 δ) : 𝓢 ⊢ γ 🡒 Δ.conj₂ := by
+ induction Δ using List.induction_with_singleton with
+ | hnil => apply af; simp;
+ | hsi... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_48c8c271a9a9_9 | 0c09d5e1c4606ab6 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Propositional/Entailment/Corsi/VF.lean | VF | 9 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 11 | 1 | [
{
"theorem_name": "insert_FConj",
"depth": 1,
"n_commands": 0,
"n_lines": 9,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n apply C_replace_both $ insert_LConj;\n . show 𝓢 ⊢ φ ⋏ Γ.conj 🡒 φ ⋏ ⋀Γ.toList;\n apply CKK_right_replace;\n simp;\n .... | [
{
"name": "insert_LConj",
"text": "lemma insert_LConj {Γ : List F} : 𝓢 ⊢ φ ⋏ Γ.conj₂ 🡒 (φ :: Γ).conj₂ := by\n match Γ with\n | [] => simp [List.conj₂];\n | γ :: Γ => apply ruleC andElimL andElimR;\n\n",
"fan_in": 1,
"n_lines": 6,
"n_chars": 187,
"n_subproofs": 0,
"n_tactics": 6,
... | [
{
"name": "insert_FConj",
"fan_in": 0,
"n_deps_direct": 4,
"n_deps_transitive": 6,
"n_lines": 11,
"n_chars": 414,
"n_subproofs": 0,
"n_tactics": 17,
"cyclomatic": 1,
"n_automation": 2,
"n_rewrites": 4,
"n_structural": 4,
"automation_only": false,
"max_nesting"... | 5 | module
public import Foundation.Propositional.Entailment.Corsi.Basic
public import Foundation.Logic.Disjunctive
@[expose] public section
namespace LO.Propositional
namespace Entailment
variable {S F : Type*} [LogicalConnective F] [Entailment S F]
variable {𝓢 : S} {φ ψ χ : F}
protected class VF (𝓢 : S) extends
... | module
public import Foundation.Propositional.Entailment.Corsi.Basic
public import Foundation.Logic.Disjunctive
@[expose] public section
namespace LO.Propositional
namespace Entailment
variable {S F : Type*} [LogicalConnective F] [Entailment S F]
variable {𝓢 : S} {φ ψ χ : F}
protected class VF (𝓢 : S) extends
... | @@ -60,6 +60,11 @@
. apply C_trans ?_ h;
simp;
+lemma insert_LConj {Γ : List F} : 𝓢 ⊢ φ ⋏ Γ.conj₂ 🡒 (φ :: Γ).conj₂ := by
+ match Γ with
+ | [] => simp [List.conj₂];
+ | γ :: Γ => apply ruleC andElimL andElimR;
+
@[simp, grind .]
lemma conjconj {Γ : Finset F} : 𝓢 ⊢ (Γ.conj) 🡒 Γ.toList.conj₂ := by simp... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_48c8c271a9a9_10 | 546e50c9057754f9 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Propositional/Entailment/Corsi/VF.lean | VF | 10 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 11 | 3 | [
{
"theorem_name": "LConj₂Conj₂_of_subset",
"depth": 1,
"n_commands": 0,
"n_lines": 4,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n apply LConj₂Conj₂_of_provable;\n intro δ hδ;\n apply mem_lconj₂ $ h δ hδ;",
"n_chars": 89,
"n_subproofs": 0... | [
{
"name": "LConj₂Conj₂_of_provable",
"text": "lemma LConj₂Conj₂_of_provable {Δ : List F} (h : ∀ δ ∈ Δ, 𝓢 ⊢ γ 🡒 δ) : 𝓢 ⊢ γ 🡒 Δ.conj₂ := by\n induction Δ using List.induction_with_singleton with\n | hnil => apply af; simp;\n | hsingle φ =>\n apply h;\n simp;\n | hcons ψ Δ he ih =>\n simp only... | [
{
"name": "CFConjFConj_of_provable",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 9,
"n_chars": 228,
"n_subproofs": 0,
"n_tactics": 9,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 3,
"automation_only": false,
"max... | 1 | module
public import Foundation.Propositional.Entailment.Corsi.Basic
public import Foundation.Logic.Disjunctive
@[expose] public section
namespace LO.Propositional
namespace Entailment
variable {S F : Type*} [LogicalConnective F] [Entailment S F]
variable {𝓢 : S} {φ ψ χ : F}
protected class VF (𝓢 : S) extends
... | module
public import Foundation.Propositional.Entailment.Corsi.Basic
public import Foundation.Logic.Disjunctive
@[expose] public section
namespace LO.Propositional
namespace Entailment
variable {S F : Type*} [LogicalConnective F] [Entailment S F]
variable {𝓢 : S} {φ ψ χ : F}
protected class VF (𝓢 : S) extends
... | @@ -58,9 +58,26 @@
@[simp, grind .]
lemma conjconj {Γ : Finset F} : 𝓢 ⊢ (Γ.conj) 🡒 Γ.toList.conj₂ := by simp [Finset.conj];
+lemma LConj₂Conj₂_of_provable {Δ : List F} (h : ∀ δ ∈ Δ, 𝓢 ⊢ γ 🡒 δ) : 𝓢 ⊢ γ 🡒 Δ.conj₂ := by
+ induction Δ using List.induction_with_singleton with
+ | hnil => apply af; simp;
+ | hsi... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_efb3a2afc8cf_0 | b6adf1945ac57bb2 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Definability/Absoluteness.lean | Absoluteness | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 9 | 3 | [
{
"theorem_name": "shigmaZero_absolute",
"depth": 1,
"n_commands": 0,
"n_lines": 3,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": "\n ⟨by simpa [nat_modelsWithParam_iff_models_substs, modelsWithParam_iff_models_substs] using nat_extention_sigmaOne V (by si... | [
{
"name": "nat_modelsWithParam_iff_models_substs",
"text": "lemma nat_modelsWithParam_iff_models_substs {v : Fin k → ℕ} {φ : ArithmeticSemisentence k} :\n φ.Evalb v ↔ ℕ↓[ℒₒᵣ] ⊧ (φ ⇜ (fun i ↦ Semiterm.Operator.numeral ℒₒᵣ (v i))) := by\n simp [models_iff, Function.comp_def, Matrix.empty_eq]\n\n",
"fa... | [
{
"name": "shigmaZero_absolute",
"fan_in": 2,
"n_deps_direct": 2,
"n_deps_transitive": 2,
"n_lines": 6,
"n_chars": 407,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
"n_automation": 4,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"max_nest... | 2 | module
public import Foundation.FirstOrder.Arithmetic.Definability.BoundedDefinable
@[expose] public section
namespace LO.FirstOrder.Arithmetic
open PeanoMinus R0
variable (V : Type*) [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗣𝗔⁻]
lemma modelsWithParam_iff_models_substs {v : Fin k → ℕ} {φ : ArithmeticSemisentence k} :
... | module
public import Foundation.FirstOrder.Arithmetic.Definability.BoundedDefinable
@[expose] public section
namespace LO.FirstOrder.Arithmetic
open PeanoMinus R0
lemma nat_modelsWithParam_iff_models_substs {v : Fin k → ℕ} {φ : ArithmeticSemisentence k} :
φ.Evalb v ↔ ℕ↓[ℒₒᵣ] ⊧ (φ ⇜ (fun i ↦ Semiterm.Operator.nu... | @@ -7,6 +7,10 @@
open PeanoMinus R0
+lemma nat_modelsWithParam_iff_models_substs {v : Fin k → ℕ} {φ : ArithmeticSemisentence k} :
+ φ.Evalb v ↔ ℕ↓[ℒₒᵣ] ⊧ (φ ⇜ (fun i ↦ Semiterm.Operator.numeral ℒₒᵣ (v i))) := by
+ simp [models_iff, Function.comp_def, Matrix.empty_eq]
+
variable (V : Type*) [ORingStructure 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_efb3a2afc8cf_1 | ea97e75018563e79 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Definability/Absoluteness.lean | Absoluteness | 1 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 9 | 3 | [
{
"theorem_name": "shigmaZero_absolute",
"depth": 1,
"n_commands": 0,
"n_lines": 3,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": "\n ⟨by simpa [nat_modelsWithParam_iff_models_substs, modelsWithParam_iff_models_substs] using nat_extention_sigmaOne V (by si... | [
{
"name": "nat_modelsWithParam_iff_models_substs",
"text": "lemma nat_modelsWithParam_iff_models_substs {v : Fin k → ℕ} {φ : ArithmeticSemisentence k} :\n φ.Evalb v ↔ ℕ↓[ℒₒᵣ] ⊧ (φ ⇜ (fun i ↦ Semiterm.Operator.numeral ℒₒᵣ (v i))) := by\n simp [models_iff, Function.comp_def, Matrix.empty_eq]\n\n",
"fa... | [
{
"name": "DefinedFunction.shigmaZero_absolute_func",
"fan_in": 0,
"n_deps_direct": 2,
"n_deps_transitive": 4,
"n_lines": 6,
"n_chars": 369,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only"... | 2 | module
public import Foundation.FirstOrder.Arithmetic.Definability.BoundedDefinable
@[expose] public section
namespace LO.FirstOrder.Arithmetic
open PeanoMinus R0
variable (V : Type*) [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗣𝗔⁻]
lemma modelsWithParam_iff_models_substs {v : Fin k → ℕ} {φ : ArithmeticSemisentence k} :
... | module
public import Foundation.FirstOrder.Arithmetic.Definability.BoundedDefinable
@[expose] public section
namespace LO.FirstOrder.Arithmetic
open PeanoMinus R0
lemma nat_modelsWithParam_iff_models_substs {v : Fin k → ℕ} {φ : ArithmeticSemisentence k} :
φ.Evalb v ↔ ℕ↓[ℒₒᵣ] ⊧ (φ ⇜ (fun i ↦ Semiterm.Operator.nu... | @@ -7,6 +7,10 @@
open PeanoMinus R0
+lemma nat_modelsWithParam_iff_models_substs {v : Fin k → ℕ} {φ : ArithmeticSemisentence k} :
+ φ.Evalb v ↔ ℕ↓[ℒₒᵣ] ⊧ (φ ⇜ (fun i ↦ Semiterm.Operator.numeral ℒₒᵣ (v i))) := by
+ simp [models_iff, Function.comp_def, Matrix.empty_eq]
+
variable (V : Type*) [ORingStructure 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_efb3a2afc8cf_2 | d4f299b3ee4497ad | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Definability/Absoluteness.lean | Absoluteness | 2 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 9 | 3 | [
{
"theorem_name": "shigmaZero_absolute",
"depth": 1,
"n_commands": 0,
"n_lines": 3,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": "\n ⟨by simpa [nat_modelsWithParam_iff_models_substs, modelsWithParam_iff_models_substs] using nat_extention_sigmaOne V (by si... | [
{
"name": "nat_modelsWithParam_iff_models_substs",
"text": "lemma nat_modelsWithParam_iff_models_substs {v : Fin k → ℕ} {φ : ArithmeticSemisentence k} :\n φ.Evalb v ↔ ℕ↓[ℒₒᵣ] ⊧ (φ ⇜ (fun i ↦ Semiterm.Operator.numeral ℒₒᵣ (v i))) := by\n simp [models_iff, Function.comp_def, Matrix.empty_eq]\n\n",
"fa... | [
{
"name": "sigmaOne_upward_absolute",
"fan_in": 1,
"n_deps_direct": 2,
"n_deps_transitive": 2,
"n_lines": 6,
"n_chars": 282,
"n_subproofs": 0,
"n_tactics": 3,
"cyclomatic": 1,
"n_automation": 2,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"max... | 2 | module
public import Foundation.FirstOrder.Arithmetic.Definability.BoundedDefinable
@[expose] public section
namespace LO.FirstOrder.Arithmetic
open PeanoMinus R0
variable (V : Type*) [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗣𝗔⁻]
lemma modelsWithParam_iff_models_substs {v : Fin k → ℕ} {φ : ArithmeticSemisentence k} :
... | module
public import Foundation.FirstOrder.Arithmetic.Definability.BoundedDefinable
@[expose] public section
namespace LO.FirstOrder.Arithmetic
open PeanoMinus R0
lemma nat_modelsWithParam_iff_models_substs {v : Fin k → ℕ} {φ : ArithmeticSemisentence k} :
φ.Evalb v ↔ ℕ↓[ℒₒᵣ] ⊧ (φ ⇜ (fun i ↦ Semiterm.Operator.nu... | @@ -7,6 +7,10 @@
open PeanoMinus R0
+lemma nat_modelsWithParam_iff_models_substs {v : Fin k → ℕ} {φ : ArithmeticSemisentence k} :
+ φ.Evalb v ↔ ℕ↓[ℒₒᵣ] ⊧ (φ ⇜ (fun i ↦ Semiterm.Operator.numeral ℒₒᵣ (v i))) := by
+ simp [models_iff, Function.comp_def, Matrix.empty_eq]
+
variable (V : Type*) [ORingStructure 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_efb3a2afc8cf_3 | da64742491fec9f2 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Definability/Absoluteness.lean | Absoluteness | 3 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 9 | 3 | [
{
"theorem_name": "shigmaZero_absolute",
"depth": 1,
"n_commands": 0,
"n_lines": 3,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": "\n ⟨by simpa [nat_modelsWithParam_iff_models_substs, modelsWithParam_iff_models_substs] using nat_extention_sigmaOne V (by si... | [
{
"name": "nat_modelsWithParam_iff_models_substs",
"text": "lemma nat_modelsWithParam_iff_models_substs {v : Fin k → ℕ} {φ : ArithmeticSemisentence k} :\n φ.Evalb v ↔ ℕ↓[ℒₒᵣ] ⊧ (φ ⇜ (fun i ↦ Semiterm.Operator.numeral ℒₒᵣ (v i))) := by\n simp [models_iff, Function.comp_def, Matrix.empty_eq]\n\n",
"fa... | [
{
"name": "piOne_downward_absolute",
"fan_in": 1,
"n_deps_direct": 2,
"n_deps_transitive": 2,
"n_lines": 6,
"n_chars": 278,
"n_subproofs": 0,
"n_tactics": 3,
"cyclomatic": 1,
"n_automation": 2,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"max_... | 2 | module
public import Foundation.FirstOrder.Arithmetic.Definability.BoundedDefinable
@[expose] public section
namespace LO.FirstOrder.Arithmetic
open PeanoMinus R0
variable (V : Type*) [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗣𝗔⁻]
lemma modelsWithParam_iff_models_substs {v : Fin k → ℕ} {φ : ArithmeticSemisentence k} :
... | module
public import Foundation.FirstOrder.Arithmetic.Definability.BoundedDefinable
@[expose] public section
namespace LO.FirstOrder.Arithmetic
open PeanoMinus R0
lemma nat_modelsWithParam_iff_models_substs {v : Fin k → ℕ} {φ : ArithmeticSemisentence k} :
φ.Evalb v ↔ ℕ↓[ℒₒᵣ] ⊧ (φ ⇜ (fun i ↦ Semiterm.Operator.nu... | @@ -7,6 +7,10 @@
open PeanoMinus R0
+lemma nat_modelsWithParam_iff_models_substs {v : Fin k → ℕ} {φ : ArithmeticSemisentence k} :
+ φ.Evalb v ↔ ℕ↓[ℒₒᵣ] ⊧ (φ ⇜ (fun i ↦ Semiterm.Operator.numeral ℒₒᵣ (v i))) := by
+ simp [models_iff, Function.comp_def, Matrix.empty_eq]
+
variable (V : Type*) [ORingStructure 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_efb3a2afc8cf_4 | 197473780bf1f8fa | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Definability/Absoluteness.lean | Absoluteness | 4 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 9 | 3 | [
{
"theorem_name": "shigmaZero_absolute",
"depth": 1,
"n_commands": 0,
"n_lines": 3,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": "\n ⟨by simpa [nat_modelsWithParam_iff_models_substs, modelsWithParam_iff_models_substs] using nat_extention_sigmaOne V (by si... | [
{
"name": "modelsWithParam_iff_models_substs",
"text": "lemma modelsWithParam_iff_models_substs {v : Fin k → ℕ} {φ : ArithmeticSemisentence k} :\n φ.Evalb (M := V) (Nat.cast ∘ v) ↔ V↓[ℒₒᵣ] ⊧ (φ ⇜ (fun i ↦ Semiterm.Operator.numeral ℒₒᵣ (v i))) := by\n simp [models_iff, Function.comp_def, Matrix.empty_eq,... | [
{
"name": "deltaOne_absolute",
"fan_in": 1,
"n_deps_direct": 2,
"n_deps_transitive": 4,
"n_lines": 7,
"n_chars": 389,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
"n_automation": 2,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"max_nestin... | 4 | module
public import Foundation.FirstOrder.Arithmetic.Definability.BoundedDefinable
@[expose] public section
namespace LO.FirstOrder.Arithmetic
open PeanoMinus R0
lemma nat_modelsWithParam_iff_models_substs {v : Fin k → ℕ} {φ : ArithmeticSemisentence k} :
φ.Evalb v ↔ ℕ↓[ℒₒᵣ] ⊧ (φ ⇜ (fun i ↦ Semiterm.Operator.nu... | module
public import Foundation.FirstOrder.Arithmetic.Definability.BoundedDefinable
@[expose] public section
namespace LO.FirstOrder.Arithmetic
open PeanoMinus R0
lemma nat_modelsWithParam_iff_models_substs {v : Fin k → ℕ} {φ : ArithmeticSemisentence k} :
φ.Evalb v ↔ ℕ↓[ℒₒᵣ] ⊧ (φ ⇜ (fun i ↦ Semiterm.Operator.nu... | @@ -13,11 +13,19 @@
variable (V : Type*) [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗣𝗔⁻]
+lemma modelsWithParam_iff_models_substs {v : Fin k → ℕ} {φ : ArithmeticSemisentence k} :
+ φ.Evalb (M := V) (Nat.cast ∘ v) ↔ V↓[ℒₒᵣ] ⊧ (φ ⇜ (fun i ↦ Semiterm.Operator.numeral ℒₒᵣ (v i))) := by
+ simp [models_iff, Function.comp_def,... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_efb3a2afc8cf_5 | da8976941895f601 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Definability/Absoluteness.lean | Absoluteness | 5 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 9 | 3 | [
{
"theorem_name": "shigmaZero_absolute",
"depth": 1,
"n_commands": 0,
"n_lines": 3,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": "\n ⟨by simpa [nat_modelsWithParam_iff_models_substs, modelsWithParam_iff_models_substs] using nat_extention_sigmaOne V (by si... | [
{
"name": "modelsWithParam_iff_models_substs",
"text": "lemma modelsWithParam_iff_models_substs {v : Fin k → ℕ} {φ : ArithmeticSemisentence k} :\n φ.Evalb (M := V) (Nat.cast ∘ v) ↔ V↓[ℒₒᵣ] ⊧ (φ ⇜ (fun i ↦ Semiterm.Operator.numeral ℒₒᵣ (v i))) := by\n simp [models_iff, Function.comp_def, Matrix.empty_eq,... | [
{
"name": "Defined.shigmaOne_absolute",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 5,
"n_lines": 6,
"n_chars": 311,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"m... | 5 | module
public import Foundation.FirstOrder.Arithmetic.Definability.BoundedDefinable
@[expose] public section
namespace LO.FirstOrder.Arithmetic
open PeanoMinus R0
lemma nat_modelsWithParam_iff_models_substs {v : Fin k → ℕ} {φ : ArithmeticSemisentence k} :
φ.Evalb v ↔ ℕ↓[ℒₒᵣ] ⊧ (φ ⇜ (fun i ↦ Semiterm.Operator.nu... | module
public import Foundation.FirstOrder.Arithmetic.Definability.BoundedDefinable
@[expose] public section
namespace LO.FirstOrder.Arithmetic
open PeanoMinus R0
lemma nat_modelsWithParam_iff_models_substs {v : Fin k → ℕ} {φ : ArithmeticSemisentence k} :
φ.Evalb v ↔ ℕ↓[ℒₒᵣ] ⊧ (φ ⇜ (fun i ↦ Semiterm.Operator.nu... | @@ -13,11 +13,19 @@
variable (V : Type*) [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗣𝗔⁻]
+lemma modelsWithParam_iff_models_substs {v : Fin k → ℕ} {φ : ArithmeticSemisentence k} :
+ φ.Evalb (M := V) (Nat.cast ∘ v) ↔ V↓[ℒₒᵣ] ⊧ (φ ⇜ (fun i ↦ Semiterm.Operator.numeral ℒₒᵣ (v i))) := by
+ simp [models_iff, Function.comp_def,... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_efb3a2afc8cf_6 | 89010e3ed0da0c89 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Definability/Absoluteness.lean | Absoluteness | 6 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 9 | 3 | [
{
"theorem_name": "shigmaZero_absolute",
"depth": 1,
"n_commands": 0,
"n_lines": 3,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": "\n ⟨by simpa [nat_modelsWithParam_iff_models_substs, modelsWithParam_iff_models_substs] using nat_extention_sigmaOne V (by si... | [
{
"name": "modelsWithParam_iff_models_substs",
"text": "lemma modelsWithParam_iff_models_substs {v : Fin k → ℕ} {φ : ArithmeticSemisentence k} :\n φ.Evalb (M := V) (Nat.cast ∘ v) ↔ V↓[ℒₒᵣ] ⊧ (φ ⇜ (fun i ↦ Semiterm.Operator.numeral ℒₒᵣ (v i))) := by\n simp [models_iff, Function.comp_def, Matrix.empty_eq,... | [
{
"name": "DefinedFunction.shigmaOne_absolute_func",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 6,
"n_lines": 6,
"n_chars": 391,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only":... | 6 | module
public import Foundation.FirstOrder.Arithmetic.Definability.BoundedDefinable
@[expose] public section
namespace LO.FirstOrder.Arithmetic
open PeanoMinus R0
lemma nat_modelsWithParam_iff_models_substs {v : Fin k → ℕ} {φ : ArithmeticSemisentence k} :
φ.Evalb v ↔ ℕ↓[ℒₒᵣ] ⊧ (φ ⇜ (fun i ↦ Semiterm.Operator.nu... | module
public import Foundation.FirstOrder.Arithmetic.Definability.BoundedDefinable
@[expose] public section
namespace LO.FirstOrder.Arithmetic
open PeanoMinus R0
lemma nat_modelsWithParam_iff_models_substs {v : Fin k → ℕ} {φ : ArithmeticSemisentence k} :
φ.Evalb v ↔ ℕ↓[ℒₒᵣ] ⊧ (φ ⇜ (fun i ↦ Semiterm.Operator.nu... | @@ -13,11 +13,19 @@
variable (V : Type*) [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗣𝗔⁻]
+lemma modelsWithParam_iff_models_substs {v : Fin k → ℕ} {φ : ArithmeticSemisentence k} :
+ φ.Evalb (M := V) (Nat.cast ∘ v) ↔ V↓[ℒₒᵣ] ⊧ (φ ⇜ (fun i ↦ Semiterm.Operator.numeral ℒₒᵣ (v i))) := by
+ simp [models_iff, Function.comp_def,... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_efb3a2afc8cf_7 | 786e43ff171ec829 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Definability/Absoluteness.lean | Absoluteness | 7 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 9 | 2 | [
{
"theorem_name": "models_iff_provable_of_Sigma0_param",
"depth": 1,
"n_commands": 0,
"n_lines": 6,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n calc\n V ⊧/(Nat.cast ∘ e) σ ↔ ℕ ⊧/e σ := by\n simp [models_iff_of_Sigma0 hσ]\n _ ... | [
{
"name": "sigma_one_completeness_iff_param",
"text": "theorem sigma_one_completeness_iff_param {σ : ArithmeticSemisentence n} (hσ : Hierarchy 𝚺 1 σ) {e : Fin n → ℕ} :\n ℕ ⊧/e σ ↔ T ⊢ (σ ⇜ fun x ↦ Semiterm.Operator.numeral ℒₒᵣ (e x)) := Iff.trans\n (by simp [models_iff, Semiformula.eval_substs, Functio... | [
{
"name": "models_iff_provable_of_Sigma0_param",
"fan_in": 0,
"n_deps_direct": 2,
"n_deps_transitive": 2,
"n_lines": 9,
"n_chars": 534,
"n_subproofs": 0,
"n_tactics": 6,
"cyclomatic": 1,
"n_automation": 2,
"n_rewrites": 0,
"n_structural": 2,
"automation_only": fal... | 2 | module
public import Foundation.FirstOrder.Arithmetic.Definability.BoundedDefinable
@[expose] public section
namespace LO.FirstOrder.Arithmetic
open PeanoMinus R0
variable (V : Type*) [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗣𝗔⁻]
variable {V}
lemma models_iff_of_Sigma0 {σ : ArithmeticSemisentence n} (hσ : Hierarchy 𝚺 0 ... | module
public import Foundation.FirstOrder.Arithmetic.Definability.BoundedDefinable
@[expose] public section
namespace LO.FirstOrder.Arithmetic
open PeanoMinus R0
variable (V : Type*) [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗣𝗔⁻]
variable {V}
lemma models_iff_of_Sigma0 {σ : ArithmeticSemisentence n} (hσ : Hierarchy 𝚺 0 ... | @@ -26,8 +26,18 @@
noncomputable instance : 𝗥₀ ⪯ T := Entailment.WeakerThan.trans (𝓣 := 𝗣𝗔⁻) inferInstance inferInstance
+theorem sigma_one_completeness_iff_param {σ : ArithmeticSemisentence n} (hσ : Hierarchy 𝚺 1 σ) {e : Fin n → ℕ} :
+ ℕ ⊧/e σ ↔ T ⊢ (σ ⇜ fun x ↦ Semiterm.Operator.numeral ℒₒᵣ (e x)) := 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_efb3a2afc8cf_8 | 06204a81cd4aa0e3 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/Definability/Absoluteness.lean | Absoluteness | 8 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 9 | 2 | [
{
"theorem_name": "models_iff_provable_of_Sigma0_param",
"depth": 1,
"n_commands": 0,
"n_lines": 6,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n calc\n V ⊧/(Nat.cast ∘ e) σ ↔ ℕ ⊧/e σ := by\n simp [models_iff_of_Sigma0 hσ]\n _ ... | [
{
"name": "sigma_one_completeness_iff_param",
"text": "theorem sigma_one_completeness_iff_param {σ : ArithmeticSemisentence n} (hσ : Hierarchy 𝚺 1 σ) {e : Fin n → ℕ} :\n ℕ ⊧/e σ ↔ T ⊢ (σ ⇜ fun x ↦ Semiterm.Operator.numeral ℒₒᵣ (e x)) := Iff.trans\n (by simp [models_iff, Semiformula.eval_substs, Functio... | [
{
"name": "models_iff_provable_of_Delta1_param",
"fan_in": 0,
"n_deps_direct": 2,
"n_deps_transitive": 2,
"n_lines": 13,
"n_chars": 825,
"n_subproofs": 0,
"n_tactics": 10,
"cyclomatic": 1,
"n_automation": 4,
"n_rewrites": 0,
"n_structural": 2,
"automation_only": f... | 2 | module
public import Foundation.FirstOrder.Arithmetic.Definability.BoundedDefinable
@[expose] public section
namespace LO.FirstOrder.Arithmetic
open PeanoMinus R0
variable (V : Type*) [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗣𝗔⁻]
variable {V}
lemma models_iff_of_Delta1 {σ : 𝚫₁.Semisentence n} (hσ : σ.ProperOn ℕ) (hσV : ... | module
public import Foundation.FirstOrder.Arithmetic.Definability.BoundedDefinable
@[expose] public section
namespace LO.FirstOrder.Arithmetic
open PeanoMinus R0
variable (V : Type*) [ORingStructure V] [V↓[ℒₒᵣ] ⊧* 𝗣𝗔⁻]
variable {V}
lemma models_iff_of_Delta1 {σ : 𝚫₁.Semisentence n} (hσ : σ.ProperOn ℕ) (hσV : ... | @@ -26,8 +26,22 @@
noncomputable instance : 𝗥₀ ⪯ T := Entailment.WeakerThan.trans (𝓣 := 𝗣𝗔⁻) inferInstance inferInstance
+theorem sigma_one_completeness_iff_param {σ : ArithmeticSemisentence n} (hσ : Hierarchy 𝚺 1 σ) {e : Fin n → ℕ} :
+ ℕ ⊧/e σ ↔ T ⊢ (σ ⇜ fun x ↦ Semiterm.Operator.numeral ℒₒᵣ (e x)) := 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_7d8da48f1248_0 | 832bc2b6d524c515 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Incompleteness/Tarski.lean | Tarski | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 1 | 1 | [
{
"theorem_name": "undefinability_of_truth",
"depth": 1,
"n_commands": 0,
"n_lines": 8,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n have := not_exists_tarski_predicate (T := 𝗧𝗔);\n contrapose! this;\n obtain ⟨τ, hτ⟩ := this;\n use τ;\n intro... | [
{
"name": "not_exists_tarski_predicate",
"text": "/-- There is no predicate `τ`, s.t. for any sentence `σ`, `σ` is provable in `T` iff `τ/[⌜σ⌝]` is so. -/\nlemma not_exists_tarski_predicate : ¬∃ τ : ArithmeticSemisentence 1, ∀ σ, T ⊢ σ 🡘 τ/[⌜σ⌝] := by\n rintro ⟨τ, hτ⟩;\n apply Entailment.Consistent.not_b... | [
{
"name": "undefinability_of_truth",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 11,
"n_chars": 401,
"n_subproofs": 2,
"n_tactics": 15,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 4,
"automation_only": false,
"m... | 1 | module
public import Foundation.FirstOrder.Bootstrapping.FixedPoint
public import Foundation.Meta.ClProver
@[expose] public section
namespace LO.FirstOrder.Arithmetic
variable {T : ArithmeticTheory} [𝗜𝚺₁ ⪯ T] [Entailment.Consistent T]
/-- Tarski's Undefinability of Truth Theorem. -/
theorem undefinability_of_trut... | module
public import Foundation.FirstOrder.Bootstrapping.FixedPoint
public import Foundation.Meta.ClProver
@[expose] public section
namespace LO.FirstOrder.Arithmetic
variable {T : ArithmeticTheory} [𝗜𝚺₁ ⪯ T] [Entailment.Consistent T]
/-- There is no predicate `τ`, s.t. for any sentence `σ`, `σ` is provable in `T... | @@ -8,8 +8,24 @@
variable {T : ArithmeticTheory} [𝗜𝚺₁ ⪯ T] [Entailment.Consistent T]
+/-- There is no predicate `τ`, s.t. for any sentence `σ`, `σ` is provable in `T` iff `τ/[⌜σ⌝]` is so. -/
+lemma not_exists_tarski_predicate : ¬∃ τ : ArithmeticSemisentence 1, ∀ σ, T ⊢ σ 🡘 τ/[⌜σ⌝] := by
+ rintro ⟨τ, hτ⟩;
+ ap... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_b8017aa3a967_0 | 6cd286b242534e2e | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/IOpen/Basic.lean | Basic | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 35 | 2 | [
{
"theorem_name": "div_exists_unique_pos",
"depth": 1,
"n_commands": 0,
"n_lines": 20,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n have : ∃ u, b * u ≤ a ∧ a < b * (u + 1) := by\n have : a < b * (a + 1) → ∃ u, b * u ≤ a ∧ a < b * (u + 1) := by\n... | [
{
"name": "least_number",
"text": "lemma least_number {P : V → Prop}\n (hP : ∃ φ : ArithmeticSemiformula V 1, φ.Open ∧ ∀ x, P x ↔ φ.Eval ![x] id)\n (zero : P 0) {a} (counterex : ¬P a) : ∃ x, P x ∧ ¬P (x + 1) := by\n by_contra A\n have : ∀ x, P x := by\n intro x; induction x using succ_induction\n... | [
{
"name": "div_exists_unique_pos",
"fan_in": 2,
"n_deps_direct": 1,
"n_deps_transitive": 2,
"n_lines": 22,
"n_chars": 946,
"n_subproofs": 5,
"n_tactics": 21,
"cyclomatic": 2,
"n_automation": 5,
"n_rewrites": 0,
"n_structural": 8,
"automation_only": false,
"max... | 1 | module
public import Foundation.FirstOrder.Arithmetic.Schemata
@[expose] public section
/-!
# Basic properties of theory $\mathsf{IOpen}$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section IOpen
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝗢𝗽𝗲𝗻]
instance : V↓[ℒₒᵣ] ⊧* 𝗣𝗔⁻ :=
have : V↓[ℒₒᵣ] ⊧... | module
public import Foundation.FirstOrder.Arithmetic.Schemata
@[expose] public section
/-!
# Basic properties of theory $\mathsf{IOpen}$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section IOpen
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝗢𝗽𝗲𝗻]
instance : V↓[ℒₒᵣ] ⊧* 𝗣𝗔⁻ :=
have : V↓[ℒₒᵣ] ⊧... | @@ -39,11 +39,28 @@
simp [Semiformula.enumarateFVar_idxOfFVar (Semiformula.mem_fvarList_iff_fvar?.mpr hz)]))
zero succ
+lemma least_number {P : V → Prop}
+ (hP : ∃ φ : ArithmeticSemiformula V 1, φ.Open ∧ ∀ x, P x ↔ φ.Eval ![x] id)
+ (zero : P 0) {a} (counterex : ¬P a) : ∃ x, P x ∧ ¬P (x + 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_b8017aa3a967_1 | 8f117bd9ee38a469 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/IOpen/Basic.lean | Basic | 1 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 35 | 2 | [
{
"theorem_name": "div_exists_unique",
"depth": 1,
"n_commands": 0,
"n_lines": 4,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n have : 0 ≤ b := by exact Arithmetic.zero_le b\n rcases this with (rfl | pos) <;> simp [*]\n · simpa [pos_iff_ne_zero.mp... | [
{
"name": "div_exists_unique_pos",
"text": "lemma div_exists_unique_pos (a : V) {b} (pos : 0 < b) : ∃! u, b * u ≤ a ∧ a < b * (u + 1) := by\n have : ∃ u, b * u ≤ a ∧ a < b * (u + 1) := by\n have : a < b * (a + 1) → ∃ u, b * u ≤ a ∧ a < b * (u + 1) := by\n simpa using least_number (P := fun u ↦ b * ... | [
{
"name": "eq_mul_div_add_of_pos",
"fan_in": 3,
"n_deps_direct": 2,
"n_deps_transitive": 6,
"n_lines": 13,
"n_chars": 494,
"n_subproofs": 3,
"n_tactics": 11,
"cyclomatic": 1,
"n_automation": 4,
"n_rewrites": 0,
"n_structural": 2,
"automation_only": false,
"max... | 4 | module
public import Foundation.FirstOrder.Arithmetic.Schemata
@[expose] public section
/-!
# Basic properties of theory $\mathsf{IOpen}$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section IOpen
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝗢𝗽𝗲𝗻]
instance : V↓[ℒₒᵣ] ⊧* 𝗣𝗔⁻ :=
have : V↓[ℒₒᵣ] ⊧... | module
public import Foundation.FirstOrder.Arithmetic.Schemata
@[expose] public section
/-!
# Basic properties of theory $\mathsf{IOpen}$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section IOpen
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝗢𝗽𝗲𝗻]
instance : V↓[ℒₒᵣ] ⊧* 𝗣𝗔⁻ :=
have : V↓[ℒₒᵣ] ⊧... | @@ -57,10 +57,31 @@
section div
+lemma div_exists_unique_pos (a : V) {b} (pos : 0 < b) : ∃! u, b * u ≤ a ∧ a < b * (u + 1) := by
+ have : ∃ u, b * u ≤ a ∧ a < b * (u + 1) := by
+ have : a < b * (a + 1) → ∃ u, b * u ≤ a ∧ a < b * (u + 1) := by
+ simpa using least_number (P := fun u ↦ b * u ≤ a) ⟨“x. &b * x... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_b8017aa3a967_2 | eb8a424ace0a850f | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/IOpen/Basic.lean | Basic | 2 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 35 | 3 | [
{
"theorem_name": "eq_mul_div_add_of_pos",
"depth": 1,
"n_commands": 0,
"n_lines": 11,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n let r := a - b * (a / b)\n have e : a = b * (a / b) + r := by simp [r, add_tsub_self_of_le (mul_div_le_pos a hb)]\n... | [
{
"name": "mul_div_le_pos",
"text": "lemma mul_div_le_pos (a : V) (h : 0 < b) : b * (a / b) ≤ a := ((Classical.choose!_spec (div_exists_unique a b)).1 h).1\n\n",
"fan_in": 3,
"n_lines": 3,
"n_chars": 122,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"... | [
{
"name": "div_spec_of_pos'",
"fan_in": 2,
"n_deps_direct": 1,
"n_deps_transitive": 7,
"n_lines": 4,
"n_chars": 134,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"max_nesting... | 5 | module
public import Foundation.FirstOrder.Arithmetic.Schemata
@[expose] public section
/-!
# Basic properties of theory $\mathsf{IOpen}$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section IOpen
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝗢𝗽𝗲𝗻]
instance : V↓[ℒₒᵣ] ⊧* 𝗣𝗔⁻ :=
have : V↓[ℒₒᵣ] ⊧... | module
public import Foundation.FirstOrder.Arithmetic.Schemata
@[expose] public section
/-!
# Basic properties of theory $\mathsf{IOpen}$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section IOpen
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝗢𝗽𝗲𝗻]
instance : V↓[ℒₒᵣ] ⊧* 𝗣𝗔⁻ :=
have : V↓[ℒₒᵣ] ⊧... | @@ -85,11 +85,13 @@
noncomputable scoped instance : Div V := ⟨fun a b ↦ Classical.choose! (div_exists_unique a b)⟩
+lemma mul_div_le_pos (a : V) (h : 0 < b) : b * (a / b) ≤ a := ((Classical.choose!_spec (div_exists_unique a b)).1 h).1
+
lemma lt_mul_div_succ (a : V) (h : 0 < b) : a < b * (a / b + 1) := ((Classica... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_b8017aa3a967_3 | c7ddc9569ee33297 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/IOpen/Basic.lean | Basic | 3 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 35 | 3 | [
{
"theorem_name": "eq_mul_div_add_of_pos",
"depth": 1,
"n_commands": 0,
"n_lines": 11,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n let r := a - b * (a / b)\n have e : a = b * (a / b) + r := by simp [r, add_tsub_self_of_le (mul_div_le_pos a hb)]\n... | [
{
"name": "mul_div_le_pos",
"text": "lemma mul_div_le_pos (a : V) (h : 0 < b) : b * (a / b) ≤ a := ((Classical.choose!_spec (div_exists_unique a b)).1 h).1\n\n",
"fan_in": 3,
"n_lines": 3,
"n_chars": 122,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"... | [
{
"name": "div_mul_add'",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 8,
"n_lines": 3,
"n_chars": 117,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"max_nesting": 1... | 6 | module
public import Foundation.FirstOrder.Arithmetic.Schemata
@[expose] public section
/-!
# Basic properties of theory $\mathsf{IOpen}$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section IOpen
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝗢𝗽𝗲𝗻]
instance : V↓[ℒₒᵣ] ⊧* 𝗣𝗔⁻ :=
have : V↓[ℒₒᵣ] ⊧... | module
public import Foundation.FirstOrder.Arithmetic.Schemata
@[expose] public section
/-!
# Basic properties of theory $\mathsf{IOpen}$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section IOpen
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝗢𝗽𝗲𝗻]
instance : V↓[ℒₒᵣ] ⊧* 𝗣𝗔⁻ :=
have : V↓[ℒₒᵣ] ⊧... | @@ -85,11 +85,13 @@
noncomputable scoped instance : Div V := ⟨fun a b ↦ Classical.choose! (div_exists_unique a b)⟩
+lemma mul_div_le_pos (a : V) (h : 0 < b) : b * (a / b) ≤ a := ((Classical.choose!_spec (div_exists_unique a b)).1 h).1
+
lemma lt_mul_div_succ (a : V) (h : 0 < b) : a < b * (a / b + 1) := ((Classica... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_b8017aa3a967_4 | 9e2c05d05e0131c8 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/IOpen/Basic.lean | Basic | 4 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 35 | 3 | [
{
"theorem_name": "eq_mul_div_add_of_pos",
"depth": 1,
"n_commands": 0,
"n_lines": 11,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n let r := a - b * (a / b)\n have e : a = b * (a / b) + r := by simp [r, add_tsub_self_of_le (mul_div_le_pos a hb)]\n... | [
{
"name": "mul_div_le_pos",
"text": "lemma mul_div_le_pos (a : V) (h : 0 < b) : b * (a / b) ≤ a := ((Classical.choose!_spec (div_exists_unique a b)).1 h).1\n\n",
"fan_in": 3,
"n_lines": 3,
"n_chars": 122,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"... | [
{
"name": "div_mul",
"fan_in": 1,
"n_deps_direct": 3,
"n_deps_transitive": 7,
"n_lines": 14,
"n_chars": 594,
"n_subproofs": 0,
"n_tactics": 12,
"cyclomatic": 3,
"n_automation": 6,
"n_rewrites": 0,
"n_structural": 5,
"automation_only": false,
"max_nesting": 10
... | 5 | module
public import Foundation.FirstOrder.Arithmetic.Schemata
@[expose] public section
/-!
# Basic properties of theory $\mathsf{IOpen}$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section IOpen
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝗢𝗽𝗲𝗻]
instance : V↓[ℒₒᵣ] ⊧* 𝗣𝗔⁻ :=
have : V↓[ℒₒᵣ] ⊧... | module
public import Foundation.FirstOrder.Arithmetic.Schemata
@[expose] public section
/-!
# Basic properties of theory $\mathsf{IOpen}$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section IOpen
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝗢𝗽𝗲𝗻]
instance : V↓[ℒₒᵣ] ⊧* 𝗣𝗔⁻ :=
have : V↓[ℒₒᵣ] ⊧... | @@ -85,11 +85,13 @@
noncomputable scoped instance : Div V := ⟨fun a b ↦ Classical.choose! (div_exists_unique a b)⟩
+lemma mul_div_le_pos (a : V) (h : 0 < b) : b * (a / b) ≤ a := ((Classical.choose!_spec (div_exists_unique a b)).1 h).1
+
lemma lt_mul_div_succ (a : V) (h : 0 < b) : a < b * (a / b + 1) := ((Classica... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_b8017aa3a967_5 | c041dfe062824911 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/IOpen/Basic.lean | Basic | 5 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 35 | 4 | [
{
"theorem_name": "eq_mul_div_add_of_pos",
"depth": 1,
"n_commands": 0,
"n_lines": 11,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n let r := a - b * (a / b)\n have e : a = b * (a / b) + r := by simp [r, add_tsub_self_of_le (mul_div_le_pos a hb)]\n... | [
{
"name": "lt_mul_div_succ",
"text": "lemma lt_mul_div_succ (a : V) (h : 0 < b) : a < b * (a / b + 1) := ((Classical.choose!_spec (div_exists_unique a b)).1 h).2\n\n",
"fan_in": 4,
"n_lines": 3,
"n_chars": 125,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,... | [
{
"name": "div_add_mul_self",
"fan_in": 3,
"n_deps_direct": 2,
"n_deps_transitive": 10,
"n_lines": 5,
"n_chars": 218,
"n_subproofs": 0,
"n_tactics": 3,
"cyclomatic": 2,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": false,
"max_nesti... | 8 | module
public import Foundation.FirstOrder.Arithmetic.Schemata
@[expose] public section
/-!
# Basic properties of theory $\mathsf{IOpen}$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section IOpen
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝗢𝗽𝗲𝗻]
instance : V↓[ℒₒᵣ] ⊧* 𝗣𝗔⁻ :=
have : V↓[ℒₒᵣ] ⊧... | module
public import Foundation.FirstOrder.Arithmetic.Schemata
@[expose] public section
/-!
# Basic properties of theory $\mathsf{IOpen}$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section IOpen
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝗢𝗽𝗲𝗻]
instance : V↓[ℒₒᵣ] ⊧* 𝗣𝗔⁻ :=
have : V↓[ℒₒᵣ] ⊧... | @@ -87,10 +87,19 @@
lemma mul_div_le_pos (a : V) (h : 0 < b) : b * (a / b) ≤ a := ((Classical.choose!_spec (div_exists_unique a b)).1 h).1
+lemma lt_mul_div_succ (a : V) (h : 0 < b) : a < b * (a / b + 1) := ((Classical.choose!_spec (div_exists_unique a b)).1 h).2
+
lemma eq_mul_div_add_of_pos (a : V) {b} (hb : 0 ... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_b8017aa3a967_6 | e81c06d79bcb84db | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/IOpen/Basic.lean | Basic | 6 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 35 | 4 | [
{
"theorem_name": "eq_mul_div_add_of_pos",
"depth": 1,
"n_commands": 0,
"n_lines": 11,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n let r := a - b * (a / b)\n have e : a = b * (a / b) + r := by simp [r, add_tsub_self_of_le (mul_div_le_pos a hb)]\n... | [
{
"name": "lt_mul_div_succ",
"text": "lemma lt_mul_div_succ (a : V) (h : 0 < b) : a < b * (a / b + 1) := ((Classical.choose!_spec (div_exists_unique a b)).1 h).2\n\n",
"fan_in": 4,
"n_lines": 3,
"n_chars": 125,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,... | [
{
"name": "div_add_mul_self'",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 11,
"n_lines": 4,
"n_chars": 139,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"max_nesti... | 9 | module
public import Foundation.FirstOrder.Arithmetic.Schemata
@[expose] public section
/-!
# Basic properties of theory $\mathsf{IOpen}$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section IOpen
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝗢𝗽𝗲𝗻]
instance : V↓[ℒₒᵣ] ⊧* 𝗣𝗔⁻ :=
have : V↓[ℒₒᵣ] ⊧... | module
public import Foundation.FirstOrder.Arithmetic.Schemata
@[expose] public section
/-!
# Basic properties of theory $\mathsf{IOpen}$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section IOpen
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝗢𝗽𝗲𝗻]
instance : V↓[ℒₒᵣ] ⊧* 𝗣𝗔⁻ :=
have : V↓[ℒₒᵣ] ⊧... | @@ -87,10 +87,19 @@
lemma mul_div_le_pos (a : V) (h : 0 < b) : b * (a / b) ≤ a := ((Classical.choose!_spec (div_exists_unique a b)).1 h).1
+lemma lt_mul_div_succ (a : V) (h : 0 < b) : a < b * (a / b + 1) := ((Classical.choose!_spec (div_exists_unique a b)).1 h).2
+
lemma eq_mul_div_add_of_pos (a : V) {b} (hb : 0 ... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_b8017aa3a967_7 | e48a79a4ffe1f133 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/IOpen/Basic.lean | Basic | 7 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 35 | 2 | [
{
"theorem_name": "div_exists_unique",
"depth": 1,
"n_commands": 0,
"n_lines": 4,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n have : 0 ≤ b := by exact Arithmetic.zero_le b\n rcases this with (rfl | pos) <;> simp [*]\n · simpa [pos_iff_ne_zero.mp... | [
{
"name": "div_exists_unique_pos",
"text": "lemma div_exists_unique_pos (a : V) {b} (pos : 0 < b) : ∃! u, b * u ≤ a ∧ a < b * (u + 1) := by\n have : ∃ u, b * u ≤ a ∧ a < b * (u + 1) := by\n have : a < b * (a + 1) → ∃ u, b * u ≤ a ∧ a < b * (u + 1) := by\n simpa using least_number (P := fun u ↦ b * ... | [
{
"name": "le_two_mul_div_two_add_one",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 5,
"n_lines": 5,
"n_chars": 238,
"n_subproofs": 1,
"n_tactics": 3,
"cyclomatic": 1,
"n_automation": 2,
"n_rewrites": 1,
"n_structural": 1,
"automation_only": false,
"... | 3 | module
public import Foundation.FirstOrder.Arithmetic.Schemata
@[expose] public section
/-!
# Basic properties of theory $\mathsf{IOpen}$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section IOpen
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝗢𝗽𝗲𝗻]
instance : V↓[ℒₒᵣ] ⊧* 𝗣𝗔⁻ :=
have : V↓[ℒₒᵣ] ⊧... | module
public import Foundation.FirstOrder.Arithmetic.Schemata
@[expose] public section
/-!
# Basic properties of theory $\mathsf{IOpen}$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section IOpen
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝗢𝗽𝗲𝗻]
instance : V↓[ℒₒᵣ] ⊧* 𝗣𝗔⁻ :=
have : V↓[ℒₒᵣ] ⊧... | @@ -57,10 +57,31 @@
section div
+lemma div_exists_unique_pos (a : V) {b} (pos : 0 < b) : ∃! u, b * u ≤ a ∧ a < b * (u + 1) := by
+ have : ∃ u, b * u ≤ a ∧ a < b * (u + 1) := by
+ have : a < b * (a + 1) → ∃ u, b * u ≤ a ∧ a < b * (u + 1) := by
+ simpa using least_number (P := fun u ↦ b * u ≤ a) ⟨“x. &b * x... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_b8017aa3a967_8 | f8f0b5e47ae10a6d | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/IOpen/Basic.lean | Basic | 8 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 35 | 3 | [
{
"theorem_name": "eq_mul_div_add_of_pos",
"depth": 1,
"n_commands": 0,
"n_lines": 11,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n let r := a - b * (a / b)\n have e : a = b * (a / b) + r := by simp [r, add_tsub_self_of_le (mul_div_le_pos a hb)]\n... | [
{
"name": "mul_div_le_pos",
"text": "lemma mul_div_le_pos (a : V) (h : 0 < b) : b * (a / b) ≤ a := ((Classical.choose!_spec (div_exists_unique a b)).1 h).1\n\n",
"fan_in": 3,
"n_lines": 3,
"n_chars": 122,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"... | [
{
"name": "div_monotone",
"fan_in": 0,
"n_deps_direct": 2,
"n_deps_transitive": 9,
"n_lines": 13,
"n_chars": 420,
"n_subproofs": 2,
"n_tactics": 11,
"cyclomatic": 2,
"n_automation": 4,
"n_rewrites": 0,
"n_structural": 2,
"automation_only": false,
"max_nesting"... | 6 | module
public import Foundation.FirstOrder.Arithmetic.Schemata
@[expose] public section
/-!
# Basic properties of theory $\mathsf{IOpen}$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section IOpen
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝗢𝗽𝗲𝗻]
instance : V↓[ℒₒᵣ] ⊧* 𝗣𝗔⁻ :=
have : V↓[ℒₒᵣ] ⊧... | module
public import Foundation.FirstOrder.Arithmetic.Schemata
@[expose] public section
/-!
# Basic properties of theory $\mathsf{IOpen}$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section IOpen
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝗢𝗽𝗲𝗻]
instance : V↓[ℒₒᵣ] ⊧* 𝗣𝗔⁻ :=
have : V↓[ℒₒᵣ] ⊧... | @@ -85,11 +85,13 @@
noncomputable scoped instance : Div V := ⟨fun a b ↦ Classical.choose! (div_exists_unique a b)⟩
+lemma mul_div_le_pos (a : V) (h : 0 < b) : b * (a / b) ≤ a := ((Classical.choose!_spec (div_exists_unique a b)).1 h).1
+
lemma lt_mul_div_succ (a : V) (h : 0 < b) : a < b * (a / b + 1) := ((Classica... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_b8017aa3a967_9 | 1ecc513dbb32e82f | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/IOpen/Basic.lean | Basic | 9 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 35 | 3 | [
{
"theorem_name": "eq_mul_div_add_of_pos",
"depth": 1,
"n_commands": 0,
"n_lines": 11,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n let r := a - b * (a / b)\n have e : a = b * (a / b) + r := by simp [r, add_tsub_self_of_le (mul_div_le_pos a hb)]\n... | [
{
"name": "mul_div_le_pos",
"text": "lemma mul_div_le_pos (a : V) (h : 0 < b) : b * (a / b) ≤ a := ((Classical.choose!_spec (div_exists_unique a b)).1 h).1\n\n",
"fan_in": 3,
"n_lines": 3,
"n_chars": 122,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"... | [
{
"name": "div_cancel_left",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 8,
"n_lines": 3,
"n_chars": 105,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"max_nesting"... | 6 | module
public import Foundation.FirstOrder.Arithmetic.Schemata
@[expose] public section
/-!
# Basic properties of theory $\mathsf{IOpen}$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section IOpen
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝗢𝗽𝗲𝗻]
instance : V↓[ℒₒᵣ] ⊧* 𝗣𝗔⁻ :=
have : V↓[ℒₒᵣ] ⊧... | module
public import Foundation.FirstOrder.Arithmetic.Schemata
@[expose] public section
/-!
# Basic properties of theory $\mathsf{IOpen}$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section IOpen
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝗢𝗽𝗲𝗻]
instance : V↓[ℒₒᵣ] ⊧* 𝗣𝗔⁻ :=
have : V↓[ℒₒᵣ] ⊧... | @@ -85,11 +85,13 @@
noncomputable scoped instance : Div V := ⟨fun a b ↦ Classical.choose! (div_exists_unique a b)⟩
+lemma mul_div_le_pos (a : V) (h : 0 < b) : b * (a / b) ≤ a := ((Classical.choose!_spec (div_exists_unique a b)).1 h).1
+
lemma lt_mul_div_succ (a : V) (h : 0 < b) : a < b * (a / b + 1) := ((Classica... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_b8017aa3a967_10 | 3ee1f23b05365e57 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/IOpen/Basic.lean | Basic | 10 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 35 | 3 | [
{
"theorem_name": "eq_mul_div_add_of_pos",
"depth": 1,
"n_commands": 0,
"n_lines": 11,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n let r := a - b * (a / b)\n have e : a = b * (a / b) + r := by simp [r, add_tsub_self_of_le (mul_div_le_pos a hb)]\n... | [
{
"name": "mul_div_le_pos",
"text": "lemma mul_div_le_pos (a : V) (h : 0 < b) : b * (a / b) ≤ a := ((Classical.choose!_spec (div_exists_unique a b)).1 h).1\n\n",
"fan_in": 3,
"n_lines": 3,
"n_chars": 122,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"... | [
{
"name": "div_cancel_right",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 9,
"n_lines": 3,
"n_chars": 127,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"max_nesting... | 7 | module
public import Foundation.FirstOrder.Arithmetic.Schemata
@[expose] public section
/-!
# Basic properties of theory $\mathsf{IOpen}$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section IOpen
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝗢𝗽𝗲𝗻]
instance : V↓[ℒₒᵣ] ⊧* 𝗣𝗔⁻ :=
have : V↓[ℒₒᵣ] ⊧... | module
public import Foundation.FirstOrder.Arithmetic.Schemata
@[expose] public section
/-!
# Basic properties of theory $\mathsf{IOpen}$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section IOpen
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝗢𝗽𝗲𝗻]
instance : V↓[ℒₒᵣ] ⊧* 𝗣𝗔⁻ :=
have : V↓[ℒₒᵣ] ⊧... | @@ -85,11 +85,13 @@
noncomputable scoped instance : Div V := ⟨fun a b ↦ Classical.choose! (div_exists_unique a b)⟩
+lemma mul_div_le_pos (a : V) (h : 0 < b) : b * (a / b) ≤ a := ((Classical.choose!_spec (div_exists_unique a b)).1 h).1
+
lemma lt_mul_div_succ (a : V) (h : 0 < b) : a < b * (a / b + 1) := ((Classica... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_b8017aa3a967_11 | 8f996a0cbf830757 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/IOpen/Basic.lean | Basic | 11 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 35 | 8 | [
{
"theorem_name": "rem_graph",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n simp [mod_def]",
"n_chars": 20,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
"n_automation": 1,
"n_... | [
{
"name": "mod_def",
"text": "lemma mod_def (a b : V) : a % b = a - b * (a / b) := rfl\n\n",
"fan_in": 8,
"n_lines": 3,
"n_chars": 58,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
... | [
{
"name": "rem_graph",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 4,
"n_chars": 111,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"max_nesting": 2
... | 1 | module
public import Foundation.FirstOrder.Arithmetic.Schemata
@[expose] public section
/-!
# Basic properties of theory $\mathsf{IOpen}$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section IOpen
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝗢𝗽𝗲𝗻]
instance : V↓[ℒₒᵣ] ⊧* 𝗣𝗔⁻ :=
have : V↓[ℒₒᵣ] ⊧... | module
public import Foundation.FirstOrder.Arithmetic.Schemata
@[expose] public section
/-!
# Basic properties of theory $\mathsf{IOpen}$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section IOpen
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝗢𝗽𝗲𝗻]
instance : V↓[ℒₒᵣ] ⊧* 𝗣𝗔⁻ :=
have : V↓[ℒₒᵣ] ⊧... | @@ -206,10 +206,13 @@
noncomputable scoped instance : Mod V := ⟨rem⟩
+lemma mod_def (a b : V) : a % b = a - b * (a / b) := rfl
+
def _root_.LO.FirstOrder.Arithmetic.remDef : 𝚺₀.Semisentence 3 :=
.mkSigma “c a b. ∃ d <⁺ a, !divDef.val d a b ∧ !subDef.val c a (b * d)”
-lemma rem_graph (a b c : V) : a = b % c ... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_b8017aa3a967_12 | e04fde92b81004a8 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/IOpen/Basic.lean | Basic | 12 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 35 | 3 | [
{
"theorem_name": "div_spec_of_pos'",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n simpa [mul_comm] using eq_mul_div_add_of_pos a h",
"n_chars": 54,
"n_subproofs": 0,
"n_tactics": 2,
"cyclo... | [
{
"name": "eq_mul_div_add_of_pos",
"text": "lemma eq_mul_div_add_of_pos (a : V) {b} (hb : 0 < b) : ∃ r < b, a = b * (a / b) + r := by\n let r := a - b * (a / b)\n have e : a = b * (a / b) + r := by simp [r, add_tsub_self_of_le (mul_div_le_pos a hb)]\n exact ⟨r, by\n by_contra A\n have hyv : b ≤ r :... | [
{
"name": "mod_mul_add",
"fan_in": 1,
"n_deps_direct": 2,
"n_deps_transitive": 13,
"n_lines": 4,
"n_chars": 193,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 1,
"n_structural": 0,
"automation_only": false,
"max_nesting": ... | 11 | module
public import Foundation.FirstOrder.Arithmetic.Schemata
@[expose] public section
/-!
# Basic properties of theory $\mathsf{IOpen}$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section IOpen
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝗢𝗽𝗲𝗻]
instance : V↓[ℒₒᵣ] ⊧* 𝗣𝗔⁻ :=
have : V↓[ℒₒᵣ] ⊧... | module
public import Foundation.FirstOrder.Arithmetic.Schemata
@[expose] public section
/-!
# Basic properties of theory $\mathsf{IOpen}$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section IOpen
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝗢𝗽𝗲𝗻]
instance : V↓[ℒₒᵣ] ⊧* 𝗣𝗔⁻ :=
have : V↓[ℒₒᵣ] ⊧... | @@ -89,6 +89,18 @@
lemma lt_mul_div_succ (a : V) (h : 0 < b) : a < b * (a / b + 1) := ((Classical.choose!_spec (div_exists_unique a b)).1 h).2
+lemma eq_mul_div_add_of_pos (a : V) {b} (hb : 0 < b) : ∃ r < b, a = b * (a / b) + r := by
+ let r := a - b * (a / b)
+ have e : a = b * (a / b) + r := by simp [r, add_ts... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_b8017aa3a967_13 | 343e906c2d26e67f | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/IOpen/Basic.lean | Basic | 13 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 35 | 3 | [
{
"theorem_name": "eq_mul_div_add_of_pos",
"depth": 1,
"n_commands": 0,
"n_lines": 11,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n let r := a - b * (a / b)\n have e : a = b * (a / b) + r := by simp [r, add_tsub_self_of_le (mul_div_le_pos a hb)]\n... | [
{
"name": "mul_div_le_pos",
"text": "lemma mul_div_le_pos (a : V) (h : 0 < b) : b * (a / b) ≤ a := ((Classical.choose!_spec (div_exists_unique a b)).1 h).1\n\n",
"fan_in": 3,
"n_lines": 3,
"n_chars": 122,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"... | [
{
"name": "mod_eq_zero_iff_dvd",
"fan_in": 1,
"n_deps_direct": 3,
"n_deps_transitive": 10,
"n_lines": 7,
"n_chars": 246,
"n_subproofs": 0,
"n_tactics": 7,
"cyclomatic": 2,
"n_automation": 2,
"n_rewrites": 0,
"n_structural": 4,
"automation_only": false,
"max_ne... | 7 | module
public import Foundation.FirstOrder.Arithmetic.Schemata
@[expose] public section
/-!
# Basic properties of theory $\mathsf{IOpen}$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section IOpen
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝗢𝗽𝗲𝗻]
instance : V↓[ℒₒᵣ] ⊧* 𝗣𝗔⁻ :=
have : V↓[ℒₒᵣ] ⊧... | module
public import Foundation.FirstOrder.Arithmetic.Schemata
@[expose] public section
/-!
# Basic properties of theory $\mathsf{IOpen}$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section IOpen
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝗢𝗽𝗲𝗻]
instance : V↓[ℒₒᵣ] ⊧* 𝗣𝗔⁻ :=
have : V↓[ℒₒᵣ] ⊧... | @@ -85,11 +85,13 @@
noncomputable scoped instance : Div V := ⟨fun a b ↦ Classical.choose! (div_exists_unique a b)⟩
+lemma mul_div_le_pos (a : V) (h : 0 < b) : b * (a / b) ≤ a := ((Classical.choose!_spec (div_exists_unique a b)).1 h).1
+
lemma lt_mul_div_succ (a : V) (h : 0 < b) : a < b * (a / b + 1) := ((Classica... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_b8017aa3a967_14 | 66ba3c48ff0e19a0 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/IOpen/Basic.lean | Basic | 14 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 35 | 3 | [
{
"theorem_name": "eq_mul_div_add_of_pos",
"depth": 1,
"n_commands": 0,
"n_lines": 11,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n let r := a - b * (a / b)\n have e : a = b * (a / b) + r := by simp [r, add_tsub_self_of_le (mul_div_le_pos a hb)]\n... | [
{
"name": "mul_div_le_pos",
"text": "lemma mul_div_le_pos (a : V) (h : 0 < b) : b * (a / b) ≤ a := ((Classical.choose!_spec (div_exists_unique a b)).1 h).1\n\n",
"fan_in": 3,
"n_lines": 3,
"n_chars": 122,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"... | [
{
"name": "mod_add",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 9,
"n_lines": 6,
"n_chars": 384,
"n_subproofs": 0,
"n_tactics": 4,
"cyclomatic": 1,
"n_automation": 3,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": false,
"max_nesting": 2
}... | 6 | module
public import Foundation.FirstOrder.Arithmetic.Schemata
@[expose] public section
/-!
# Basic properties of theory $\mathsf{IOpen}$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section IOpen
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝗢𝗽𝗲𝗻]
instance : V↓[ℒₒᵣ] ⊧* 𝗣𝗔⁻ :=
have : V↓[ℒₒᵣ] ⊧... | module
public import Foundation.FirstOrder.Arithmetic.Schemata
@[expose] public section
/-!
# Basic properties of theory $\mathsf{IOpen}$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section IOpen
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝗢𝗽𝗲𝗻]
instance : V↓[ℒₒᵣ] ⊧* 𝗣𝗔⁻ :=
have : V↓[ℒₒᵣ] ⊧... | @@ -85,11 +85,13 @@
noncomputable scoped instance : Div V := ⟨fun a b ↦ Classical.choose! (div_exists_unique a b)⟩
+lemma mul_div_le_pos (a : V) (h : 0 < b) : b * (a / b) ≤ a := ((Classical.choose!_spec (div_exists_unique a b)).1 h).1
+
lemma lt_mul_div_succ (a : V) (h : 0 < b) : a < b * (a / b + 1) := ((Classica... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_b8017aa3a967_15 | f761c8b9c044be26 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/IOpen/Basic.lean | Basic | 15 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 35 | 3 | [
{
"theorem_name": "eq_mul_div_add_of_pos",
"depth": 1,
"n_commands": 0,
"n_lines": 11,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n let r := a - b * (a / b)\n have e : a = b * (a / b) + r := by simp [r, add_tsub_self_of_le (mul_div_le_pos a hb)]\n... | [
{
"name": "mul_div_le_pos",
"text": "lemma mul_div_le_pos (a : V) (h : 0 < b) : b * (a / b) ≤ a := ((Classical.choose!_spec (div_exists_unique a b)).1 h).1\n\n",
"fan_in": 3,
"n_lines": 3,
"n_chars": 122,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"... | [
{
"name": "mod_mul",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 9,
"n_lines": 5,
"n_chars": 330,
"n_subproofs": 0,
"n_tactics": 3,
"cyclomatic": 1,
"n_automation": 2,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": false,
"max_nesting": 2
}... | 6 | module
public import Foundation.FirstOrder.Arithmetic.Schemata
@[expose] public section
/-!
# Basic properties of theory $\mathsf{IOpen}$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section IOpen
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝗢𝗽𝗲𝗻]
instance : V↓[ℒₒᵣ] ⊧* 𝗣𝗔⁻ :=
have : V↓[ℒₒᵣ] ⊧... | module
public import Foundation.FirstOrder.Arithmetic.Schemata
@[expose] public section
/-!
# Basic properties of theory $\mathsf{IOpen}$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section IOpen
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝗢𝗽𝗲𝗻]
instance : V↓[ℒₒᵣ] ⊧* 𝗣𝗔⁻ :=
have : V↓[ℒₒᵣ] ⊧... | @@ -85,11 +85,13 @@
noncomputable scoped instance : Div V := ⟨fun a b ↦ Classical.choose! (div_exists_unique a b)⟩
+lemma mul_div_le_pos (a : V) (h : 0 < b) : b * (a / b) ≤ a := ((Classical.choose!_spec (div_exists_unique a b)).1 h).1
+
lemma lt_mul_div_succ (a : V) (h : 0 < b) : a < b * (a / b + 1) := ((Classica... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_b8017aa3a967_16 | 3ef86f136f814811 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/IOpen/Basic.lean | Basic | 16 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 35 | 3 | [
{
"theorem_name": "div_spec_of_pos'",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n simpa [mul_comm] using eq_mul_div_add_of_pos a h",
"n_chars": 54,
"n_subproofs": 0,
"n_tactics": 2,
"cyclo... | [
{
"name": "eq_mul_div_add_of_pos",
"text": "lemma eq_mul_div_add_of_pos (a : V) {b} (hb : 0 < b) : ∃ r < b, a = b * (a / b) + r := by\n let r := a - b * (a / b)\n have e : a = b * (a / b) + r := by simp [r, add_tsub_self_of_le (mul_div_le_pos a hb)]\n exact ⟨r, by\n by_contra A\n have hyv : b ≤ r :... | [
{
"name": "two_dvd_mul",
"fan_in": 1,
"n_deps_direct": 4,
"n_deps_transitive": 19,
"n_lines": 15,
"n_chars": 694,
"n_subproofs": 7,
"n_tactics": 14,
"cyclomatic": 1,
"n_automation": 8,
"n_rewrites": 2,
"n_structural": 1,
"automation_only": false,
"max_nesting"... | 15 | module
public import Foundation.FirstOrder.Arithmetic.Schemata
@[expose] public section
/-!
# Basic properties of theory $\mathsf{IOpen}$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section IOpen
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝗢𝗽𝗲𝗻]
instance : V↓[ℒₒᵣ] ⊧* 𝗣𝗔⁻ :=
have : V↓[ℒₒᵣ] ⊧... | module
public import Foundation.FirstOrder.Arithmetic.Schemata
@[expose] public section
/-!
# Basic properties of theory $\mathsf{IOpen}$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section IOpen
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝗢𝗽𝗲𝗻]
instance : V↓[ℒₒᵣ] ⊧* 𝗣𝗔⁻ :=
have : V↓[ℒₒᵣ] ⊧... | @@ -89,6 +89,18 @@
lemma lt_mul_div_succ (a : V) (h : 0 < b) : a < b * (a / b + 1) := ((Classical.choose!_spec (div_exists_unique a b)).1 h).2
+lemma eq_mul_div_add_of_pos (a : V) {b} (hb : 0 < b) : ∃ r < b, a = b * (a / b) + r := by
+ let r := a - b * (a / b)
+ have e : a = b * (a / b) + r := by simp [r, add_ts... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_b8017aa3a967_17 | 6925121b6c939920 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/IOpen/Basic.lean | Basic | 17 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 35 | 3 | [
{
"theorem_name": "div_spec_of_pos'",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n simpa [mul_comm] using eq_mul_div_add_of_pos a h",
"n_chars": 54,
"n_subproofs": 0,
"n_tactics": 2,
"cyclo... | [
{
"name": "eq_mul_div_add_of_pos",
"text": "lemma eq_mul_div_add_of_pos (a : V) {b} (hb : 0 < b) : ∃ r < b, a = b * (a / b) + r := by\n let r := a - b * (a / b)\n have e : a = b * (a / b) + r := by simp [r, add_tsub_self_of_le (mul_div_le_pos a hb)]\n exact ⟨r, by\n by_contra A\n have hyv : b ≤ r :... | [
{
"name": "even_or_odd",
"fan_in": 0,
"n_deps_direct": 2,
"n_deps_transitive": 16,
"n_lines": 6,
"n_chars": 213,
"n_subproofs": 1,
"n_tactics": 5,
"cyclomatic": 3,
"n_automation": 2,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": false,
"max_nesting": ... | 12 | module
public import Foundation.FirstOrder.Arithmetic.Schemata
@[expose] public section
/-!
# Basic properties of theory $\mathsf{IOpen}$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section IOpen
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝗢𝗽𝗲𝗻]
instance : V↓[ℒₒᵣ] ⊧* 𝗣𝗔⁻ :=
have : V↓[ℒₒᵣ] ⊧... | module
public import Foundation.FirstOrder.Arithmetic.Schemata
@[expose] public section
/-!
# Basic properties of theory $\mathsf{IOpen}$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section IOpen
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝗢𝗽𝗲𝗻]
instance : V↓[ℒₒᵣ] ⊧* 𝗣𝗔⁻ :=
have : V↓[ℒₒᵣ] ⊧... | @@ -89,6 +89,18 @@
lemma lt_mul_div_succ (a : V) (h : 0 < b) : a < b * (a / b + 1) := ((Classical.choose!_spec (div_exists_unique a b)).1 h).2
+lemma eq_mul_div_add_of_pos (a : V) {b} (hb : 0 < b) : ∃ r < b, a = b * (a / b) + r := by
+ let r := a - b * (a / b)
+ have e : a = b * (a / b) + r := by simp [r, add_ts... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_b8017aa3a967_18 | 60ee9416362f1af4 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/IOpen/Basic.lean | Basic | 18 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 35 | 2 | [
{
"theorem_name": "div_add_mul_self",
"depth": 1,
"n_commands": 0,
"n_lines": 3,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n rcases div_spec_of_pos' a pos with ⟨r, hr, ex⟩\n simpa [add_mul, add_right_comm, ← ex] using div_mul_add (a / b + c) _ hr... | [
{
"name": "div_spec_of_pos'",
"text": "lemma div_spec_of_pos' (a : V) (h : 0 < b) : ∃ v < b, a = (a / b) * b + v := by\n simpa [mul_comm] using eq_mul_div_add_of_pos a h\n\n",
"fan_in": 2,
"n_lines": 4,
"n_chars": 134,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
"n_automa... | [
{
"name": "two_prime",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 20,
"n_lines": 5,
"n_chars": 120,
"n_subproofs": 0,
"n_tactics": 2,
"cyclomatic": 1,
"n_automation": 2,
"n_rewrites": 0,
"n_structural": 2,
"automation_only": false,
"max_nesting": 1
... | 16 | module
public import Foundation.FirstOrder.Arithmetic.Schemata
@[expose] public section
/-!
# Basic properties of theory $\mathsf{IOpen}$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section IOpen
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝗢𝗽𝗲𝗻]
instance : V↓[ℒₒᵣ] ⊧* 𝗣𝗔⁻ :=
have : V↓[ℒₒᵣ] ⊧... | module
public import Foundation.FirstOrder.Arithmetic.Schemata
@[expose] public section
/-!
# Basic properties of theory $\mathsf{IOpen}$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section IOpen
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝗢𝗽𝗲𝗻]
instance : V↓[ℒₒᵣ] ⊧* 𝗣𝗔⁻ :=
have : V↓[ℒₒᵣ] ⊧... | @@ -111,6 +111,9 @@
instance div_defined : 𝚺₀-Function₂[V] HDiv.hDiv via divDef := .mk fun v ↦ by simp [div_graph, divDef]
+lemma div_spec_of_pos' (a : V) (h : 0 < b) : ∃ v < b, a = (a / b) * b + v := by
+ simpa [mul_comm] using eq_mul_div_add_of_pos a h
+
lemma div_eq_of {b : V} (hb : b * c ≤ a) (ha : a < b * ... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_b8017aa3a967_19 | b9f8ef6cc3b54344 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/IOpen/Basic.lean | Basic | 19 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 35 | 4 | [
{
"theorem_name": "sqrt_eq_of_le_of_lt",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": "\n Eq.symm <| eq_sqrt x a ⟨le, lt⟩",
"n_chars": 38,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n... | [
{
"name": "eq_sqrt",
"text": "lemma eq_sqrt (x a : V) : x * x ≤ a ∧ a < (x + 1) * (x + 1) → x = √a := Classical.choose_uniq (sqrt_exists_unique a)\n\n",
"fan_in": 4,
"n_lines": 3,
"n_chars": 126,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrite... | [
{
"name": "sqrt_eq_of_le_of_le",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 5,
"n_lines": 4,
"n_chars": 177,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"max_nest... | 3 | module
public import Foundation.FirstOrder.Arithmetic.Schemata
@[expose] public section
/-!
# Basic properties of theory $\mathsf{IOpen}$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section IOpen
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝗢𝗽𝗲𝗻]
instance : V↓[ℒₒᵣ] ⊧* 𝗣𝗔⁻ :=
have : V↓[ℒₒᵣ] ⊧... | module
public import Foundation.FirstOrder.Arithmetic.Schemata
@[expose] public section
/-!
# Basic properties of theory $\mathsf{IOpen}$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section IOpen
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝗢𝗽𝗲𝗻]
instance : V↓[ℒₒᵣ] ⊧* 𝗣𝗔⁻ :=
have : V↓[ℒₒᵣ] ⊧... | @@ -336,8 +336,11 @@
instance sqrt_definable : 𝚺₀-Function₁[V] sqrt := sqrt_defined.to_definable
-lemma sqrt_eq_of_le_of_lt {x a : V} (le : x * x ≤ a) (lt : a < (x + 1) * (x + 1)) : √a = x := sorry
+lemma eq_sqrt (x a : V) : x * x ≤ a ∧ a < (x + 1) * (x + 1) → x = √a := Classical.choose_uniq (sqrt_exists_unique a... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_b8017aa3a967_20 | 45c9cdad2a7cbb64 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/IOpen/Basic.lean | Basic | 20 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 35 | 2 | [
{
"theorem_name": "div_exists_unique_pos",
"depth": 1,
"n_commands": 0,
"n_lines": 20,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n have : ∃ u, b * u ≤ a ∧ a < b * (u + 1) := by\n have : a < b * (a + 1) → ∃ u, b * u ≤ a ∧ a < b * (u + 1) := by\n... | [
{
"name": "least_number",
"text": "lemma least_number {P : V → Prop}\n (hP : ∃ φ : ArithmeticSemiformula V 1, φ.Open ∧ ∀ x, P x ↔ φ.Eval ![x] id)\n (zero : P 0) {a} (counterex : ¬P a) : ∃ x, P x ∧ ¬P (x + 1) := by\n by_contra A\n have : ∀ x, P x := by\n intro x; induction x using succ_induction\n... | [
{
"name": "sqrt_three",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 4,
"n_lines": 4,
"n_chars": 142,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"max_nesting": 2
... | 2 | module
public import Foundation.FirstOrder.Arithmetic.Schemata
@[expose] public section
/-!
# Basic properties of theory $\mathsf{IOpen}$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section IOpen
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝗢𝗽𝗲𝗻]
instance : V↓[ℒₒᵣ] ⊧* 𝗣𝗔⁻ :=
have : V↓[ℒₒᵣ] ⊧... | module
public import Foundation.FirstOrder.Arithmetic.Schemata
@[expose] public section
/-!
# Basic properties of theory $\mathsf{IOpen}$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section IOpen
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝗢𝗽𝗲𝗻]
instance : V↓[ℒₒᵣ] ⊧* 𝗣𝗔⁻ :=
have : V↓[ℒₒᵣ] ⊧... | @@ -39,11 +39,28 @@
simp [Semiformula.enumarateFVar_idxOfFVar (Semiformula.mem_fvarList_iff_fvar?.mpr hz)]))
zero succ
+lemma least_number {P : V → Prop}
+ (hP : ∃ φ : ArithmeticSemiformula V 1, φ.Open ∧ ∀ x, P x ↔ φ.Eval ![x] id)
+ (zero : P 0) {a} (counterex : ¬P a) : ∃ x, P x ∧ ¬P (x + 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_b8017aa3a967_21 | 145a486991bc254f | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/IOpen/Basic.lean | Basic | 21 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 35 | 2 | [
{
"theorem_name": "div_exists_unique_pos",
"depth": 1,
"n_commands": 0,
"n_lines": 20,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n have : ∃ u, b * u ≤ a ∧ a < b * (u + 1) := by\n have : a < b * (a + 1) → ∃ u, b * u ≤ a ∧ a < b * (u + 1) := by\n... | [
{
"name": "least_number",
"text": "lemma least_number {P : V → Prop}\n (hP : ∃ φ : ArithmeticSemiformula V 1, φ.Open ∧ ∀ x, P x ↔ φ.Eval ![x] id)\n (zero : P 0) {a} (counterex : ¬P a) : ∃ x, P x ∧ ¬P (x + 1) := by\n by_contra A\n have : ∀ x, P x := by\n intro x; induction x using succ_induction\n... | [
{
"name": "sqrt_lt_self_of_one_lt",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 4,
"n_lines": 7,
"n_chars": 260,
"n_subproofs": 2,
"n_tactics": 5,
"cyclomatic": 1,
"n_automation": 2,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": false,
"max_... | 1 | module
public import Foundation.FirstOrder.Arithmetic.Schemata
@[expose] public section
/-!
# Basic properties of theory $\mathsf{IOpen}$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section IOpen
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝗢𝗽𝗲𝗻]
instance : V↓[ℒₒᵣ] ⊧* 𝗣𝗔⁻ :=
have : V↓[ℒₒᵣ] ⊧... | module
public import Foundation.FirstOrder.Arithmetic.Schemata
@[expose] public section
/-!
# Basic properties of theory $\mathsf{IOpen}$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section IOpen
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝗢𝗽𝗲𝗻]
instance : V↓[ℒₒᵣ] ⊧* 𝗣𝗔⁻ :=
have : V↓[ℒₒᵣ] ⊧... | @@ -39,11 +39,28 @@
simp [Semiformula.enumarateFVar_idxOfFVar (Semiformula.mem_fvarList_iff_fvar?.mpr hz)]))
zero succ
+lemma least_number {P : V → Prop}
+ (hP : ∃ φ : ArithmeticSemiformula V 1, φ.Open ∧ ∀ x, P x ↔ φ.Eval ![x] id)
+ (zero : P 0) {a} (counterex : ¬P a) : ∃ x, P x ∧ ¬P (x + 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_b8017aa3a967_22 | f88aa2a2514b0efa | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/IOpen/Basic.lean | Basic | 22 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 35 | 1 | [
{
"theorem_name": "sq_lt_of_lt_sqrt",
"depth": 1,
"n_commands": 0,
"n_lines": 3,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n intro h; by_contra A\n exact not_le.mpr h (sqrt_le_of_le_sq $ show b ≤ a^2 from by simpa using A)",
"n_chars": 105,
... | [
{
"name": "sqrt_le_of_le_sq",
"text": "lemma sqrt_le_of_le_sq {a b : V} : a ≤ b^2 → √a ≤ b := by\n intro h; by_contra A\n have : a < a := calc\n a ≤ b^2 := h\n _ < (√a)^2 := sq_lt_sq.mpr (by simpa using A)\n _ ≤ a := by simp\n simp_all\n\n",
"fan_in": 1,
"n_lines": 9,
"n_char... | [
{
"name": "sq_lt_of_lt_sqrt",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 5,
"n_chars": 165,
"n_subproofs": 0,
"n_tactics": 4,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 1,
"n_structural": 2,
"automation_only": false,
"max_nestin... | 1 | module
public import Foundation.FirstOrder.Arithmetic.Schemata
@[expose] public section
/-!
# Basic properties of theory $\mathsf{IOpen}$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section IOpen
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝗢𝗽𝗲𝗻]
instance : V↓[ℒₒᵣ] ⊧* 𝗣𝗔⁻ :=
have : V↓[ℒₒᵣ] ⊧... | module
public import Foundation.FirstOrder.Arithmetic.Schemata
@[expose] public section
/-!
# Basic properties of theory $\mathsf{IOpen}$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section IOpen
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝗢𝗽𝗲𝗻]
instance : V↓[ℒₒᵣ] ⊧* 𝗣𝗔⁻ :=
have : V↓[ℒₒᵣ] ⊧... | @@ -381,8 +381,18 @@
instance : Bounded₁ ((√·) : V → V) := ⟨#0, by intro v; simp⟩
-lemma sq_lt_of_lt_sqrt {a b : V} : a < √b → a^2 < b := sorry
+lemma sqrt_le_of_le_sq {a b : V} : a ≤ b^2 → √a ≤ b := by
+ intro h; by_contra A
+ have : a < a := calc
+ a ≤ b^2 := h
+ _ < (√a)^2 := sq_lt_sq.mpr (by simpa u... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_b8017aa3a967_23 | 7d92a1f9c16bbc53 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/IOpen/Basic.lean | Basic | 23 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 35 | 2 | [
{
"theorem_name": "pair_le_pair_left",
"depth": 1,
"n_commands": 0,
"n_lines": 4,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n rcases h with (rfl | lt)\n · simp\n · exact le_of_lt (pair_lt_pair_left lt b)",
"n_chars": 85,
"n_subproofs": 0... | [
{
"name": "pair_lt_pair_left",
"text": "lemma pair_lt_pair_left {a₁ a₂ : V} (h : a₁ < a₂) (b) : ⟪a₁, b⟫ < ⟪a₂, b⟫ := by\n by_cases h₁ : a₁ < b\n · simp only [pair, h₁, ↓reduceIte]\n by_cases h₂ : a₂ < b\n · simp [h₂, h]\n · suffices b * b + a₁ < a₂ * a₂ + a₂ + b by simpa [pair, h₂, h]\n calc... | [
{
"name": "pair_le_pair_left",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 6,
"n_chars": 187,
"n_subproofs": 0,
"n_tactics": 4,
"cyclomatic": 2,
"n_automation": 2,
"n_rewrites": 0,
"n_structural": 2,
"automation_only": false,
"max_nesti... | 1 | module
public import Foundation.FirstOrder.Arithmetic.Schemata
@[expose] public section
/-!
# Basic properties of theory $\mathsf{IOpen}$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section IOpen
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝗢𝗽𝗲𝗻]
instance : V↓[ℒₒᵣ] ⊧* 𝗣𝗔⁻ :=
have : V↓[ℒₒᵣ] ⊧... | module
public import Foundation.FirstOrder.Arithmetic.Schemata
@[expose] public section
/-!
# Basic properties of theory $\mathsf{IOpen}$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section IOpen
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝗢𝗽𝗲𝗻]
instance : V↓[ℒₒᵣ] ⊧* 𝗣𝗔⁻ :=
have : V↓[ℒₒᵣ] ⊧... | @@ -502,10 +502,24 @@
instance pi₂_definable : 𝚺₀-Function₁ (pi₂ : V → V) := pi₂_defined.to_definable₀
+lemma pair_lt_pair_left {a₁ a₂ : V} (h : a₁ < a₂) (b) : ⟪a₁, b⟫ < ⟪a₂, b⟫ := by
+ by_cases h₁ : a₁ < b
+ · simp only [pair, h₁, ↓reduceIte]
+ by_cases h₂ : a₂ < b
+ · simp [h₂, h]
+ · suffices b * b ... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_b8017aa3a967_24 | cbdb7ba2276f157f | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/IOpen/Basic.lean | Basic | 24 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 35 | 2 | [
{
"theorem_name": "pair_le_pair_right",
"depth": 1,
"n_commands": 0,
"n_lines": 4,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n rcases h with (rfl | lt)\n · simp\n · exact le_of_lt (pair_lt_pair_right a lt)",
"n_chars": 86,
"n_subproofs":... | [
{
"name": "pair_lt_pair_right",
"text": "lemma pair_lt_pair_right (a : V) {b₁ b₂} (h : b₁ < b₂) : ⟪a, b₁⟫ < ⟪a, b₂⟫ := by\n by_cases h₁ : a < b₁\n · simpa [pair, h₁, lt_trans h₁ h, ←sq] using h\n · by_cases h₂ : a < b₂\n · suffices a * a + a + b₁ < b₂ * b₂ + a by simpa [pair, h₁, h₂, h]\n calc\n ... | [
{
"name": "pair_le_pair_right",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 6,
"n_chars": 189,
"n_subproofs": 0,
"n_tactics": 4,
"cyclomatic": 2,
"n_automation": 2,
"n_rewrites": 0,
"n_structural": 2,
"automation_only": false,
"max_nest... | 1 | module
public import Foundation.FirstOrder.Arithmetic.Schemata
@[expose] public section
/-!
# Basic properties of theory $\mathsf{IOpen}$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section IOpen
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝗢𝗽𝗲𝗻]
instance : V↓[ℒₒᵣ] ⊧* 𝗣𝗔⁻ :=
have : V↓[ℒₒᵣ] ⊧... | module
public import Foundation.FirstOrder.Arithmetic.Schemata
@[expose] public section
/-!
# Basic properties of theory $\mathsf{IOpen}$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section IOpen
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝗢𝗽𝗲𝗻]
instance : V↓[ℒₒᵣ] ⊧* 𝗣𝗔⁻ :=
have : V↓[ℒₒᵣ] ⊧... | @@ -502,10 +502,21 @@
instance pi₂_definable : 𝚺₀-Function₁ (pi₂ : V → V) := pi₂_defined.to_definable₀
+lemma pair_lt_pair_right (a : V) {b₁ b₂} (h : b₁ < b₂) : ⟪a, b₁⟫ < ⟪a, b₂⟫ := by
+ by_cases h₁ : a < b₁
+ · simpa [pair, h₁, lt_trans h₁ h, ←sq] using h
+ · by_cases h₂ : a < b₂
+ · suffices a * a + a + b... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_b8017aa3a967_25 | 1c17fe8d30117f40 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/IOpen/Basic.lean | Basic | 25 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 35 | 2 | [
{
"theorem_name": "pair_le_pair_left",
"depth": 1,
"n_commands": 0,
"n_lines": 4,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n rcases h with (rfl | lt)\n · simp\n · exact le_of_lt (pair_lt_pair_left lt b)",
"n_chars": 85,
"n_subproofs": 0... | [
{
"name": "pair_lt_pair_left",
"text": "lemma pair_lt_pair_left {a₁ a₂ : V} (h : a₁ < a₂) (b) : ⟪a₁, b⟫ < ⟪a₂, b⟫ := by\n by_cases h₁ : a₁ < b\n · simp only [pair, h₁, ↓reduceIte]\n by_cases h₂ : a₂ < b\n · simp [h₂, h]\n · suffices b * b + a₁ < a₂ * a₂ + a₂ + b by simpa [pair, h₂, h]\n calc... | [
{
"name": "pair_le_pair",
"fan_in": 0,
"n_deps_direct": 2,
"n_deps_transitive": 4,
"n_lines": 6,
"n_chars": 273,
"n_subproofs": 0,
"n_tactics": 3,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": false,
"max_nesting": ... | 4 | module
public import Foundation.FirstOrder.Arithmetic.Schemata
@[expose] public section
/-!
# Basic properties of theory $\mathsf{IOpen}$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section IOpen
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝗢𝗽𝗲𝗻]
instance : V↓[ℒₒᵣ] ⊧* 𝗣𝗔⁻ :=
have : V↓[ℒₒᵣ] ⊧... | module
public import Foundation.FirstOrder.Arithmetic.Schemata
@[expose] public section
/-!
# Basic properties of theory $\mathsf{IOpen}$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section IOpen
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝗢𝗽𝗲𝗻]
instance : V↓[ℒₒᵣ] ⊧* 𝗣𝗔⁻ :=
have : V↓[ℒₒᵣ] ⊧... | @@ -502,10 +502,24 @@
instance pi₂_definable : 𝚺₀-Function₁ (pi₂ : V → V) := pi₂_defined.to_definable₀
+lemma pair_lt_pair_left {a₁ a₂ : V} (h : a₁ < a₂) (b) : ⟪a₁, b⟫ < ⟪a₂, b⟫ := by
+ by_cases h₁ : a₁ < b
+ · simp only [pair, h₁, ↓reduceIte]
+ by_cases h₂ : a₂ < b
+ · simp [h₂, h]
+ · suffices b * b ... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_b8017aa3a967_26 | 6c331e8c6b19a1d2 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/IOpen/Basic.lean | Basic | 26 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 35 | 2 | [
{
"theorem_name": "pair_le_pair_left",
"depth": 1,
"n_commands": 0,
"n_lines": 4,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n rcases h with (rfl | lt)\n · simp\n · exact le_of_lt (pair_lt_pair_left lt b)",
"n_chars": 85,
"n_subproofs": 0... | [
{
"name": "pair_lt_pair_left",
"text": "lemma pair_lt_pair_left {a₁ a₂ : V} (h : a₁ < a₂) (b) : ⟪a₁, b⟫ < ⟪a₂, b⟫ := by\n by_cases h₁ : a₁ < b\n · simp only [pair, h₁, ↓reduceIte]\n by_cases h₂ : a₂ < b\n · simp [h₂, h]\n · suffices b * b + a₁ < a₂ * a₂ + a₂ + b by simpa [pair, h₂, h]\n calc... | [
{
"name": "pair_lt_pair",
"fan_in": 0,
"n_deps_direct": 2,
"n_deps_transitive": 2,
"n_lines": 6,
"n_chars": 263,
"n_subproofs": 0,
"n_tactics": 3,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": false,
"max_nesting": ... | 2 | module
public import Foundation.FirstOrder.Arithmetic.Schemata
@[expose] public section
/-!
# Basic properties of theory $\mathsf{IOpen}$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section IOpen
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝗢𝗽𝗲𝗻]
instance : V↓[ℒₒᵣ] ⊧* 𝗣𝗔⁻ :=
have : V↓[ℒₒᵣ] ⊧... | module
public import Foundation.FirstOrder.Arithmetic.Schemata
@[expose] public section
/-!
# Basic properties of theory $\mathsf{IOpen}$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section IOpen
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝗢𝗽𝗲𝗻]
instance : V↓[ℒₒᵣ] ⊧* 𝗣𝗔⁻ :=
have : V↓[ℒₒᵣ] ⊧... | @@ -502,6 +502,20 @@
instance pi₂_definable : 𝚺₀-Function₁ (pi₂ : V → V) := pi₂_defined.to_definable₀
+lemma pair_lt_pair_left {a₁ a₂ : V} (h : a₁ < a₂) (b) : ⟪a₁, b⟫ < ⟪a₂, b⟫ := by
+ by_cases h₁ : a₁ < b
+ · simp only [pair, h₁, ↓reduceIte]
+ by_cases h₂ : a₂ < b
+ · simp [h₂, h]
+ · suffices b * b +... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_b8017aa3a967_27 | 4a435bdf32cb7cee | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/IOpen/Basic.lean | Basic | 27 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 35 | 1 | [
{
"theorem_name": "coe_pair_eq_pair_coe",
"depth": 1,
"n_commands": 0,
"n_lines": 1,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by simp [←nat_pair_eq, nat_cast_pair]",
"n_chars": 40,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
... | [
{
"name": "nat_cast_pair",
"text": "lemma nat_cast_pair (n m : ℕ) : (⟪n, m⟫ : ℕ) = ⟪(↑n : V), (↑m : V)⟫ := by simp [pair]\n\n",
"fan_in": 1,
"n_lines": 3,
"n_chars": 103,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural"... | [
{
"name": "coe_pair_eq_pair_coe",
"fan_in": 0,
"n_deps_direct": 2,
"n_deps_transitive": 2,
"n_lines": 3,
"n_chars": 135,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"max_nes... | 2 | module
public import Foundation.FirstOrder.Arithmetic.Schemata
@[expose] public section
/-!
# Basic properties of theory $\mathsf{IOpen}$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section IOpen
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝗢𝗽𝗲𝗻]
instance : V↓[ℒₒᵣ] ⊧* 𝗣𝗔⁻ :=
have : V↓[ℒₒᵣ] ⊧... | module
public import Foundation.FirstOrder.Arithmetic.Schemata
@[expose] public section
/-!
# Basic properties of theory $\mathsf{IOpen}$
-/
namespace LO.FirstOrder.Arithmetic
variable {V : Type*} [ORingStructure V]
section IOpen
variable [V↓[ℒₒᵣ] ⊧* 𝗜𝗢𝗽𝗲𝗻]
instance : V↓[ℒₒᵣ] ⊧* 𝗣𝗔⁻ :=
have : V↓[ℒₒᵣ] ⊧... | @@ -584,9 +584,11 @@
end
+lemma nat_cast_pair (n m : ℕ) : (⟪n, m⟫ : ℕ) = ⟪(↑n : V), (↑m : V)⟫ := by simp [pair]
+
lemma nat_pair_eq (m n : ℕ) : ⟪n, m⟫ = Nat.pair n m := by simp [pair, Nat.pair]
-lemma coe_pair_eq_pair_coe (n m : ℕ) : (Nat.pair n m : V) = ⟪(↑n : V), (↑m : V)⟫ := sorry
+lemma coe_pair_eq_pair_coe... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_7384fdb77b52_0 | 92b777c80be11d05 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/R0/Representation.lean | Representation | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 5 | 1 | [
{
"theorem_name": "code_sigma_one",
"depth": 1,
"n_commands": 0,
"n_lines": 2,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": "\n Hierarchy.rew _ (codeAux_sigma_one c)",
"n_chars": 40,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"... | [
{
"name": "codeAux_sigma_one",
"text": "private lemma codeAux_sigma_one {k} (c : Nat.ArithPart₁.Code k) : Hierarchy 𝚺 1 (codeAux c) := by\n induction c\n case zero => simp [codeAux]\n case one => simp [codeAux]\n case add => simp [codeAux]\n case mul => simp [codeAux]\n case lt => simp [codeAux, Matr... | [
{
"name": "code_sigma_one",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 4,
"n_chars": 135,
"n_subproofs": 0,
"n_tactics": 1,
"cyclomatic": 1,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": false,
"max_nesting"... | 1 | module
public import Foundation.FirstOrder.Arithmetic.R0.Basic
public import Foundation.Vorspiel.Arithmetic
public import Foundation.Vorspiel.Computability
@[expose] public section
open Encodable Denumerable
namespace LO.FirstOrder.Arithmetic
open Mathlib Encodable Semiterm.Operator.GödelNumber
section
lemma term... | module
public import Foundation.FirstOrder.Arithmetic.R0.Basic
public import Foundation.Vorspiel.Arithmetic
public import Foundation.Vorspiel.Computability
@[expose] public section
open Encodable Denumerable
namespace LO.FirstOrder.Arithmetic
open Mathlib Encodable Semiterm.Operator.GödelNumber
section
lemma term... | @@ -93,8 +93,22 @@
end model
-/
-@[simp] lemma code_sigma_one {k} (c : Nat.ArithPart₁.Code k) : Hierarchy 𝚺 1 (code c) := sorry
+private lemma codeAux_sigma_one {k} (c : Nat.ArithPart₁.Code k) : Hierarchy 𝚺 1 (codeAux c) := by
+ induction c
+ case zero => simp [codeAux]
+ case one => simp [codeAux]
+ case add... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_7384fdb77b52_1 | c8244c59cdbcffdc | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/R0/Representation.lean | Representation | 1 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 5 | 1 | [
{
"theorem_name": "models_code",
"depth": 1,
"n_commands": 0,
"n_lines": 3,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n simpa [code, models_iff, Semiformula.eval_rew, Matrix.empty_eq, Function.comp_def,\n Matrix.comp_vecCons', ←Part.eq_some_iff... | [
{
"name": "models_codeAux",
"text": "private lemma models_codeAux {c : Code k} {f : List.Vector ℕ k →. ℕ} (hc : c.eval f) (y : ℕ) (v : Fin k → ℕ) :\n (codeAux c).Evalf (y :> v) ↔ f (List.Vector.ofFn v) = Part.some y := by\n induction hc generalizing y\n case zero =>\n have : (0 : Part ℕ) = Part.some... | [
{
"name": "models_code",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 6,
"n_chars": 340,
"n_subproofs": 0,
"n_tactics": 3,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"max_nesting": 4
... | 1 | module
public import Foundation.FirstOrder.Arithmetic.R0.Basic
public import Foundation.Vorspiel.Arithmetic
public import Foundation.Vorspiel.Computability
@[expose] public section
open Encodable Denumerable
namespace LO.FirstOrder.Arithmetic
open Mathlib Encodable Semiterm.Operator.GödelNumber
section
lemma term... | module
public import Foundation.FirstOrder.Arithmetic.R0.Basic
public import Foundation.Vorspiel.Arithmetic
public import Foundation.Vorspiel.Computability
@[expose] public section
open Encodable Denumerable
namespace LO.FirstOrder.Arithmetic
open Mathlib Encodable Semiterm.Operator.GödelNumber
section
lemma term... | @@ -111,8 +111,55 @@
@[simp] lemma natCast_nat' (n : ℕ) : Nat.cast n = n := by rfl
+private lemma models_codeAux {c : Code k} {f : List.Vector ℕ k →. ℕ} (hc : c.eval f) (y : ℕ) (v : Fin k → ℕ) :
+ (codeAux c).Evalf (y :> v) ↔ f (List.Vector.ofFn v) = Part.some y := by
+ induction hc generalizing y
+ case zero... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_7384fdb77b52_2 | 4ea2faf2abcca950 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/R0/Representation.lean | Representation | 2 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 5 | 1 | [
{
"theorem_name": "models_code",
"depth": 1,
"n_commands": 0,
"n_lines": 3,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n simpa [code, models_iff, Semiformula.eval_rew, Matrix.empty_eq, Function.comp_def,\n Matrix.comp_vecCons', ←Part.eq_some_iff... | [
{
"name": "models_codeAux",
"text": "private lemma models_codeAux {c : Code k} {f : List.Vector ℕ k →. ℕ} (hc : c.eval f) (y : ℕ) (v : Fin k → ℕ) :\n (codeAux c).Evalf (y :> v) ↔ f (List.Vector.ofFn v) = Part.some y := by\n induction hc generalizing y\n case zero =>\n have : (0 : Part ℕ) = Part.some... | [
{
"name": "codeOfPartrec'_spec",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 2,
"n_lines": 8,
"n_chars": 430,
"n_subproofs": 1,
"n_tactics": 5,
"cyclomatic": 2,
"n_automation": 0,
"n_rewrites": 0,
"n_structural": 3,
"automation_only": false,
"max_nes... | 2 | module
public import Foundation.FirstOrder.Arithmetic.R0.Basic
public import Foundation.Vorspiel.Arithmetic
public import Foundation.Vorspiel.Computability
@[expose] public section
open Encodable Denumerable
namespace LO.FirstOrder.Arithmetic
open Mathlib Encodable Semiterm.Operator.GödelNumber
section
lemma term... | module
public import Foundation.FirstOrder.Arithmetic.R0.Basic
public import Foundation.Vorspiel.Arithmetic
public import Foundation.Vorspiel.Computability
@[expose] public section
open Encodable Denumerable
namespace LO.FirstOrder.Arithmetic
open Mathlib Encodable Semiterm.Operator.GödelNumber
section
lemma term... | @@ -111,8 +111,55 @@
@[simp] lemma natCast_nat' (n : ℕ) : Nat.cast n = n := by rfl
+private lemma models_codeAux {c : Code k} {f : List.Vector ℕ k →. ℕ} (hc : c.eval f) (y : ℕ) (v : Fin k → ℕ) :
+ (codeAux c).Evalf (y :> v) ↔ f (List.Vector.ofFn v) = Part.some y := by
+ induction hc generalizing y
+ case zero... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_7384fdb77b52_3 | fde806728f901230 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/R0/Representation.lean | Representation | 3 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 5 | 1 | [
{
"theorem_name": "models_code",
"depth": 1,
"n_commands": 0,
"n_lines": 3,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n simpa [code, models_iff, Semiformula.eval_rew, Matrix.empty_eq, Function.comp_def,\n Matrix.comp_vecCons', ←Part.eq_some_iff... | [
{
"name": "models_codeAux",
"text": "private lemma models_codeAux {c : Code k} {f : List.Vector ℕ k →. ℕ} (hc : c.eval f) (y : ℕ) (v : Fin k → ℕ) :\n (codeAux c).Evalf (y :> v) ↔ f (List.Vector.ofFn v) = Part.some y := by\n induction hc generalizing y\n case zero =>\n have : (0 : Part ℕ) = Part.some... | [
{
"name": "codeOfREPred_spec",
"fan_in": 1,
"n_deps_direct": 1,
"n_deps_transitive": 3,
"n_lines": 10,
"n_chars": 716,
"n_subproofs": 2,
"n_tactics": 7,
"cyclomatic": 1,
"n_automation": 2,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": false,
"max_nest... | 3 | module
public import Foundation.FirstOrder.Arithmetic.R0.Basic
public import Foundation.Vorspiel.Arithmetic
public import Foundation.Vorspiel.Computability
@[expose] public section
open Encodable Denumerable
namespace LO.FirstOrder.Arithmetic
open Mathlib Encodable Semiterm.Operator.GödelNumber
section
lemma term... | module
public import Foundation.FirstOrder.Arithmetic.R0.Basic
public import Foundation.Vorspiel.Arithmetic
public import Foundation.Vorspiel.Computability
@[expose] public section
open Encodable Denumerable
namespace LO.FirstOrder.Arithmetic
open Mathlib Encodable Semiterm.Operator.GödelNumber
section
lemma term... | @@ -111,8 +111,55 @@
@[simp] lemma natCast_nat' (n : ℕ) : Nat.cast n = n := by rfl
+private lemma models_codeAux {c : Code k} {f : List.Vector ℕ k →. ℕ} (hc : c.eval f) (y : ℕ) (v : Fin k → ℕ) :
+ (codeAux c).Evalf (y :> v) ↔ f (List.Vector.ofFn v) = Part.some y := by
+ induction hc generalizing y
+ case zero... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_7384fdb77b52_4 | 607152ff8c0fb329 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/FirstOrder/Arithmetic/R0/Representation.lean | Representation | 4 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 5 | 1 | [
{
"theorem_name": "models_code",
"depth": 1,
"n_commands": 0,
"n_lines": 3,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n simpa [code, models_iff, Semiformula.eval_rew, Matrix.empty_eq, Function.comp_def,\n Matrix.comp_vecCons', ←Part.eq_some_iff... | [
{
"name": "models_codeAux",
"text": "private lemma models_codeAux {c : Code k} {f : List.Vector ℕ k →. ℕ} (hc : c.eval f) (y : ℕ) (v : Fin k → ℕ) :\n (codeAux c).Evalf (y :> v) ↔ f (List.Vector.ofFn v) = Part.some y := by\n induction hc generalizing y\n case zero =>\n have : (0 : Part ℕ) = Part.some... | [
{
"name": "re_complete",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 4,
"n_lines": 7,
"n_chars": 373,
"n_subproofs": 0,
"n_tactics": 3,
"cyclomatic": 1,
"n_automation": 2,
"n_rewrites": 0,
"n_structural": 0,
"automation_only": true,
"max_nesting": 2
... | 4 | module
public import Foundation.FirstOrder.Arithmetic.R0.Basic
public import Foundation.Vorspiel.Arithmetic
public import Foundation.Vorspiel.Computability
@[expose] public section
open Encodable Denumerable
namespace LO.FirstOrder.Arithmetic
open Mathlib Encodable Semiterm.Operator.GödelNumber
section
lemma term... | module
public import Foundation.FirstOrder.Arithmetic.R0.Basic
public import Foundation.Vorspiel.Arithmetic
public import Foundation.Vorspiel.Computability
@[expose] public section
open Encodable Denumerable
namespace LO.FirstOrder.Arithmetic
open Mathlib Encodable Semiterm.Operator.GödelNumber
section
lemma term... | @@ -111,8 +111,55 @@
@[simp] lemma natCast_nat' (n : ℕ) : Nat.cast n = n := by rfl
+private lemma models_codeAux {c : Code k} {f : List.Vector ℕ k →. ℕ} (hc : c.eval f) (y : ℕ) (v : Fin k → ℕ) :
+ (codeAux c).Evalf (y :> v) ↔ f (List.Vector.ofFn v) = Part.some y := by
+ induction hc generalizing y
+ case zero... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_68c05002cd23_0 | 97654655444ae051 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/ComplementClosedConsistentFinset.lean | ComplementClosedConsistentFinset | 0 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 12 | 4 | [
{
"theorem_name": "empty_conisistent",
"depth": 1,
"n_commands": 0,
"n_lines": 4,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n apply iff_theory_consistent_formulae_consistent.mp;\n simp only [Finset.coe_empty];\n apply FormulaSet.emptyset_consist... | [
{
"name": "iff_theory_consistent_formulae_consistent",
"text": "lemma iff_theory_consistent_formulae_consistent {Φ : FormulaFinset α} : FormulaSet.Consistent 𝓢 Φ ↔ FormulaFinset.Consistent 𝓢 Φ := by\n simp [Consistent, FormulaSet.Consistent]\n\nomit [DecidableEq α] in\n",
"fan_in": 4,
"n_lines": ... | [
{
"name": "empty_conisistent",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 7,
"n_chars": 230,
"n_subproofs": 0,
"n_tactics": 7,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 2,
"automation_only": false,
"max_nesti... | 1 | module
public import Foundation.Modal.MaximalConsistentSet
public import Foundation.Modal.Formula.Complement
@[expose] public section
namespace LO.Modal
open LO.Entailment
variable {α : Type*} [DecidableEq α]
variable {S} [Entailment S (Formula α)]
variable {𝓢 : S}
namespace FormulaFinset
variable {Φ Φ₁ Φ₂ : Fo... | module
public import Foundation.Modal.MaximalConsistentSet
public import Foundation.Modal.Formula.Complement
@[expose] public section
namespace LO.Modal
open LO.Entailment
variable {α : Type*} [DecidableEq α]
variable {S} [Entailment S (Formula α)]
variable {𝓢 : S}
namespace FormulaFinset
variable {Φ Φ₁ Φ₂ : Fo... | @@ -22,6 +22,10 @@
abbrev Inconsistent (𝓢 : S) (Φ : FormulaFinset α) : Prop := ¬(Consistent 𝓢 Φ)
omit [DecidableEq α] in
+lemma iff_theory_consistent_formulae_consistent {Φ : FormulaFinset α} : FormulaSet.Consistent 𝓢 Φ ↔ FormulaFinset.Consistent 𝓢 Φ := by
+ simp [Consistent, FormulaSet.Consistent]
+
+omit [De... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
ablate_68c05002cd23_3 | 032000043747fc14 | lean | lean | github.com/FormalizedFormalLogic/Foundation | 87d4dd68835a6c1eb8448b9c392d9ca51fe08d63 | Foundation/Modal/ComplementClosedConsistentFinset.lean | ComplementClosedConsistentFinset | 3 | lemma_delete | null | null | false | 0.5 | 1 | 1 | false | 0 | inf | 0 | inf | 42 | 12 | 4 | [
{
"theorem_name": "empty_conisistent",
"depth": 1,
"n_commands": 0,
"n_lines": 4,
"is_leaf": true,
"centrality": 0,
"method": "deleted-dep",
"proof_text": " by\n apply iff_theory_consistent_formulae_consistent.mp;\n simp only [Finset.coe_empty];\n apply FormulaSet.emptyset_consist... | [
{
"name": "iff_theory_consistent_formulae_consistent",
"text": "lemma iff_theory_consistent_formulae_consistent {Φ : FormulaFinset α} : FormulaSet.Consistent 𝓢 Φ ↔ FormulaFinset.Consistent 𝓢 Φ := by\n simp [Consistent, FormulaSet.Consistent]\n\nomit [DecidableEq α] in\n",
"fan_in": 4,
"n_lines": ... | [
{
"name": "intro_union_consistent",
"fan_in": 0,
"n_deps_direct": 1,
"n_deps_transitive": 1,
"n_lines": 26,
"n_chars": 1130,
"n_subproofs": 0,
"n_tactics": 5,
"cyclomatic": 1,
"n_automation": 1,
"n_rewrites": 0,
"n_structural": 1,
"automation_only": false,
"ma... | 1 | module
public import Foundation.Modal.MaximalConsistentSet
public import Foundation.Modal.Formula.Complement
@[expose] public section
namespace LO.Modal
open LO.Entailment
variable {α : Type*} [DecidableEq α]
variable {S} [Entailment S (Formula α)]
variable {𝓢 : S}
namespace FormulaFinset
variable {Φ Φ₁ Φ₂ : Fo... | module
public import Foundation.Modal.MaximalConsistentSet
public import Foundation.Modal.Formula.Complement
@[expose] public section
namespace LO.Modal
open LO.Entailment
variable {α : Type*} [DecidableEq α]
variable {S} [Entailment S (Formula α)]
variable {𝓢 : S}
namespace FormulaFinset
variable {Φ Φ₁ Φ₂ : Fo... | @@ -22,6 +22,10 @@
abbrev Inconsistent (𝓢 : S) (Φ : FormulaFinset α) : Prop := ¬(Consistent 𝓢 Φ)
omit [DecidableEq α] in
+lemma iff_theory_consistent_formulae_consistent {Φ : FormulaFinset α} : FormulaSet.Consistent 𝓢 Φ ↔ FormulaFinset.Consistent 𝓢 Φ := by
+ simp [Consistent, FormulaSet.Consistent]
+
+omit [De... | {
"repo": "Foundation",
"git_repo": "github.com/FormalizedFormalLogic/Foundation",
"revision": "87d4dd68835a6c1eb8448b9c392d9ca51fe08d63",
"lean_toolchain": "leanprover/lean4:v4.31.0",
"proof_assistant": "lean",
"ablator": "ablators/lean (corollary-delete-lemmas-leaves-all)",
"ablator_flags": [
"--cor... |
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