reduce imported mathlib chunk

server/adam/Adam/Levels/SetTheory/L01_Univ.lean

@@ -1,7 +1,6 @@
 import Adam.Metadata
 
-import Mathlib.Init.Set
-import Mathlib.Tactic.Tauto
+import Adam.Options.MathlibPart
 
 set_option tactic.hygienic false
 set_option autoImplicit false

server/adam/Adam/Levels/SetTheory/L02_Empty.lean

@@ -1,7 +1,5 @@
 import Adam.Metadata
-
-import Mathlib.Init.Set
-import Mathlib.Tactic.Tauto
+import Adam.Options.MathlibPart
 
 set_option tactic.hygienic false
 

server/adam/Adam/Levels/SetTheory/L03_Subset.lean

@@ -1,7 +1,6 @@
 import Adam.Metadata
 
-import Mathlib.Init.Set
-import Mathlib.Tactic.Tauto
+import Adam.Options.MathlibPart
 
 set_option tactic.hygienic false
 

server/adam/Adam/Levels/SetTheory/L04_SubsetEmpty.lean

@@ -1,9 +1,7 @@
 import Adam.Metadata
 import Adam.Levels.SetTheory.L03_Subset
 
-import Mathlib.Init.Set
-import Mathlib.Tactic.Tauto
-import Mathlib
+import Adam.Options.MathlibPart
 
 set_option tactic.hygienic false
 

server/adam/Adam/Levels/SetTheory/L05_Empty.lean

@@ -1,10 +1,7 @@
 import Adam.Metadata
 import Adam.Levels.SetTheory.L04_SubsetEmpty
 
---import Mathlib.Data.Set.Basic
-import Mathlib.Init.Set
-import Mathlib.Tactic.Tauto
-import Mathlib.Tactic.PushNeg
+import Adam.Options.MathlibPart
 
 set_option tactic.hygienic false
 

server/adam/Adam/Levels/SetTheory/L06_Nonempty.lean

@@ -1,7 +1,7 @@
 import Adam.Metadata
 import Adam.Levels.SetTheory.L05_Empty
 
-import Mathlib.Data.Set.Basic
+import Adam.Options.MathlibPart
 
 set_option tactic.hygienic false
 

server/adam/Adam/Levels/SetTheory/L07_UnionInter.lean

@@ -1,6 +1,6 @@
 import Adam.Metadata
 
-import Mathlib.Data.Set.Basic
+import Adam.Options.MathlibPart
 
 set_option tactic.hygienic false
 

server/adam/Adam/Levels/SetTheory/L08_UnionInter.lean

@@ -1,6 +1,6 @@
 import Adam.Metadata
 
-import Mathlib.Data.Set.Basic
+import Adam.Options.MathlibPart
 
 set_option tactic.hygienic false
 

server/adam/Adam/Levels/SetTheory/L09_Complement.lean

@@ -1,6 +1,6 @@
 import Adam.Metadata
 
-import Mathlib.Data.Set.Basic
+import Adam.Options.MathlibPart
 
 Game "Adam"
 World "SetTheory"

server/adam/Adam/Levels/SetTheory/L10_Morgan.lean

@@ -1,6 +1,6 @@
 import Adam.Metadata
 
-import Mathlib.Data.Set.Basic
+import Adam.Options.MathlibPart
 
 Game "Adam"
 World "SetTheory"

server/adam/Adam/Levels/SetTheory/L11_SSubset.lean

@@ -1,5 +1,5 @@
 import Adam.Metadata
-import Mathlib.Data.Set.Basic
+import Adam.Options.MathlibPart
 
 Game "Adam"
 World "SetTheory"

server/adam/Adam/Levels/SetTheory/L12_Insert.lean

@@ -1,6 +1,6 @@
 import Adam.Metadata
 
-import Mathlib.Data.Set.Basic
+import Adam.Options.MathlibPart
 
 Game "Adam"
 World "SetTheory"

server/adam/Adam/Levels/SetTheory/L13_Insert.lean

@@ -1,6 +1,6 @@
 import Adam.Metadata
 
-import Mathlib.Data.Set.Basic
+import Adam.Options.MathlibPart
 
 Game "Adam"
 World "SetTheory"

server/adam/Adam/Levels/SetTheory/L14_SetOf.lean

@@ -1,8 +1,6 @@
 import Adam.Metadata
 
-import Mathlib.Data.Set.Basic
-import Mathlib.Algebra.Parity
-import Mathlib.Tactic.Ring
+import Adam.Options.MathlibPart
 
 Game "Adam"
 World "SetTheory2"

server/adam/Adam/Levels/SetTheory/L15_Powerset.lean

@@ -1,8 +1,6 @@
 import Adam.Metadata
 
-import Mathlib.Data.Set.Basic
-import Mathlib.Algebra.Parity
-import Mathlib.Tactic.Ring
+import Adam.Options.MathlibPart
 
 import Adam.ToBePorted
 

server/adam/Adam/Levels/SetTheory/L16_Disjoint.lean

@@ -1,8 +1,7 @@
 import Adam.Metadata
 
-import Mathlib
-import Mathlib.Algebra.Parity
-import Mathlib.Tactic.Ring
+
+import Adam.Options.MathlibPart
 
 Game "Adam"
 World "SetTheory2"

server/adam/Adam/Levels/SetTheory/L17_SetOf.lean

@@ -1,8 +1,7 @@
 import Adam.Metadata
 
-import Mathlib.Data.Set.Basic
-import Mathlib.Algebra.Parity
-import Mathlib
+import Adam.Options.MathlibPart
+
 
 Game "Adam"
 World "SetTheory2"

server/adam/Adam/Levels/SetTheory/L18_SetOf.lean

@@ -1,8 +1,7 @@
 import Adam.Metadata
 
-import Mathlib.Data.Set.Basic
-import Mathlib.Algebra.Parity
-import Mathlib
+import Adam.Options.MathlibPart
+
 
 Game "Adam"
 World "SetTheory2"

server/adam/Adam/Levels/SetTheory/L19_Subtype.lean

@@ -1,8 +1,6 @@
 import Adam.Metadata
 
-import Mathlib.Data.Set.Basic
-import Mathlib.Algebra.Parity
-import Mathlib.Tactic.Ring
+import Adam.Options.MathlibPart
 
 Game "Adam"
 World "SetTheory2"

server/adam/Adam/Levels/SetTheory/L20_UnionInter.lean

@@ -1,8 +1,6 @@
 import Adam.Metadata
 
-import Mathlib.Data.Set.Basic
-import Mathlib.Algebra.Parity
-import Mathlib
+import Adam.Options.MathlibPart
 
 Game "Adam"
 World "SetTheory2"

server/adam/Adam/Levels/SetTheory/L21_Summary.lean

@@ -1,8 +1,6 @@
 import Adam.Metadata
 
-import Mathlib.Data.Set.Basic
-import Mathlib.Algebra.Parity
-import Mathlib.Tactic.Ring
+import Adam.Options.MathlibPart
 
 Game "Adam"
 World "SetTheory"

server/adam/Adam/Levels/SetTheory/PowersetPlayground.lean

@@ -1,4 +1,4 @@
-import Mathlib
+
 
 open Set
 

server/adam/Adam/Levels/SetTheory/T01_Set.lean

@@ -1,296 +1,286 @@
-import Adam.Metadata
+-- import Adam.Metadata
 
-import Mathlib.Order.SymmDiff
-import Mathlib.Logic.Function.Iterate
-import Mathlib.Tactic.Use
-import Mathlib.Tactic.SolveByElim
-import Mathlib.Tactic.Tauto
-import Mathlib.Tactic.ByContra
-import Mathlib.Tactic.Lift
-import Mathlib.Data.Set.Basic
-import Mathlib.Data.Finset.Powerset
-import Mathlib.Tactic.LibrarySearch
-import Mathlib
+-- import Adam.Options.MathlibPart
 
-set_option tactic.hygienic false
+-- set_option tactic.hygienic false
 
-Game "Adam"
-World "SetTheory"
-Level 1
+-- Game "Adam"
+-- World "SetTheory"
+-- Level 1
 
-Title "Mengen"
+-- Title "Mengen"
 
-Introduction
-"
-In diesem Kapitel schauen wir uns Mengen an.
+-- Introduction
+-- "
+-- In diesem Kapitel schauen wir uns Mengen an.
 
-Zuerst ein ganz wichtiger Punkt: Alle Mengen in Lean sind homogen. Das heisst,
-alle Elemente in einer Menge haben den gleichen Typ.
+-- Zuerst ein ganz wichtiger Punkt: Alle Mengen in Lean sind homogen. Das heisst,
+-- alle Elemente in einer Menge haben den gleichen Typ.
 
-Zum Beispiel `{1, 4, 6}` ist eine Menge von natürlichen Zahlen. Aber man kann keine
-Menge `{(2 : ℕ), {3, 1}, \"e\", (1 : ℂ)}` definieren, da die Elemente unterschiedliche Typen haben.
+-- Zum Beispiel `{1, 4, 6}` ist eine Menge von natürlichen Zahlen. Aber man kann keine
+-- Menge `{(2 : ℕ), {3, 1}, \"e\", (1 : ℂ)}` definieren, da die Elemente unterschiedliche Typen haben.
 
-Für einen Typen `{X : Type*}` definiert damit also `set X` der Type aller Mengen mit Elementen aus
-`X`.  `set.univ` ist dann ganz `X` also Menge betrachtet, und es ist wichtig den Unterschied
-zu kennen: `(univ : set X)` und `(X : Typ*)` haben nicht den gleichen Typ und sind damit auch nicht
-austauschbar!
+-- Für einen Typen `{X : Type*}` definiert damit also `set X` der Type aller Mengen mit Elementen aus
+-- `X`.  `set.univ` ist dann ganz `X` also Menge betrachtet, und es ist wichtig den Unterschied
+-- zu kennen: `(univ : set X)` und `(X : Typ*)` haben nicht den gleichen Typ und sind damit auch nicht
+-- austauschbar!
 
-Am anderen Ende sitzt die leere Menge `(∅ : set X)` (`\\empty`). Bei `univ` und `∅` versucht Lean
-automatisch den Typ zu erraten, in exotischen Beispielen muss man wie oben diesen explizit angeben.
+-- Am anderen Ende sitzt die leere Menge `(∅ : set X)` (`\\empty`). Bei `univ` und `∅` versucht Lean
+-- automatisch den Typ zu erraten, in exotischen Beispielen muss man wie oben diesen explizit angeben.
 
-Als erste Operationen schauen wir uns `∪` (`\\union` oder `\\cup`), `∩`
-(`\\inter` oder `\\cap`) und `\\` (`\\\\` oder `\\ `)
+-- Als erste Operationen schauen wir uns `∪` (`\\union` oder `\\cup`), `∩`
+-- (`\\inter` oder `\\cap`) und `\\` (`\\\\` oder `\\ `)
 
-"
+-- "
 
-open Set
+-- open Set
 
-Statement
-    "Wenn $B$ wahr ist, dann ist die Implikation $A \\Rightarrow (A ∧ B)$ wahr."
-    {X : Type _} {A B : Set X} : univ \ (A ∩ B) ∪ ∅ = (univ \ A) ∪ (univ \ B) ∪ (A \ B) := by
-  rw [Set.diff_inter]
-  rw [Set.union_empty]
-  rw [Set.union_assoc]
-  rw [←Set.union_diff_distrib]
-  rw [Set.univ_union]
+-- Statement
+--     "Wenn $B$ wahr ist, dann ist die Implikation $A \\Rightarrow (A ∧ B)$ wahr."
+--     {X : Type _} {A B : Set X} : univ \ (A ∩ B) ∪ ∅ = (univ \ A) ∪ (univ \ B) ∪ (A \ B) := by
+--   rw [Set.diff_inter]
+--   rw [Set.union_empty]
+--   rw [Set.union_assoc]
+--   rw [←Set.union_diff_distrib]
+--   rw [Set.univ_union]
 
-NewTactic rw
+-- NewTactic rw
 
-example : 4 ∈ (univ : Set ℕ) := by
-  trivial
+-- example : 4 ∈ (univ : Set ℕ) := by
+--   trivial
 
-example (A : Set ℕ) : 4 ∉ (∅ : Set ℕ) := by
-  trivial
+-- example (A : Set ℕ) : 4 ∉ (∅ : Set ℕ) := by
+--   trivial
 
-example (A : Set ℕ) : A ⊆ univ := by
-  intro x h
-  trivial
+-- example (A : Set ℕ) : A ⊆ univ := by
+--   intro x h
+--   trivial
 
--- -- subset_empty_iff
--- example {s : Set α} : s ⊆ ∅ ↔ s = ∅ := by
+-- -- -- subset_empty_iff
+-- -- example {s : Set α} : s ⊆ ∅ ↔ s = ∅ := by
+-- --   constructor
+-- --   intro h
+-- --   rw [Subset.antisymm_iff]
+-- --   constructor
+-- --   assumption
+-- --   simp only [empty_subset]
+-- --   intro a
+-- --   rw [Subset.antisymm_iff] at a
+-- --   rcases a with ⟨h₁, h₂⟩
+-- --   assumption
+
+-- -- -- eq_empty_iff_forall_not_mem
+-- -- example {s : Set α} : s = ∅ ↔ ∀ x, x ∉ s := by
+-- --   rw [←subset_empty_iff]
+-- --   rfl
+
+-- -- -- nonempty_iff_ne_empty
+-- -- example (X : Type _) (s : Set X) : Set.Nonempty s ↔ s ≠ ∅ := by
+-- --   rw [Set.Nonempty]
+-- --   rw [ne_eq, eq_empty_iff_forall_not_mem]
+-- --   push_neg
+-- --   rfl
+
+
+-- -- example (A B : ℝ): A = B ↔ A ≤ B ∧ B ≤ A := by library_search
+
+-- namespace Finset
+
+-- theorem powerset_singleton {U : Type _} [DecidableEq U] (x : U) :
+--     Finset.powerset {x} = {∅, {x}} := by
+--   ext y
+--   rw [mem_powerset, subset_singleton_iff, mem_insert, mem_singleton]
+
+-- end Finset
+
+-- /- The powerset of a singleton contains only `∅` and the singleton. -/
+-- theorem powerset_singleton {U : Type _} (x : U) :
+--     𝒫 ({x} : Set U) = {∅, {x}} := by
+--   ext y
+--   rw [mem_powerset_iff, subset_singleton_iff_eq]
+--   rw [mem_insert_iff, mem_singleton_iff]
+
+
+-- lemma mem_powerset_insert_iff {U : Type _} (A S : Set U) (x : U) :
+--     S ∈ 𝒫 (insert x A) ↔ S \ {x} ∈ 𝒫 A := by
+--   rw [mem_powerset_iff, mem_powerset_iff, diff_singleton_subset_iff]
+
+-- -- lemma powerset_insert {U : Type _} (A : Set U) (x : U) :
+-- --     𝒫 (insert x A) = 𝒫 A ∪ ({B : Set U | B \ {x} ∈ 𝒫 A}) := by
+-- --   sorry
+
+-- theorem subset_insert_iff_of_not_mem {U : Type _ }{s t : Set U} (h : a ∉ s) : s ⊆ insert a t ↔ s ⊆ t := by
+--   rw [subset_insert_iff, erase_eq_of_not_mem h]
+
+-- lemma mem_powerset_insert_iff₂ {U : Type _} (A S : Set U) (x : U) :
+--     S ∈ 𝒫 (insert x A) ↔ S ∈ 𝒫 A ∨ ∃ B ∈ 𝒫 A , insert x B = S := by
+--   simp_rw [mem_powerset_iff]
 --   constructor
---   intro h
---   rw [Subset.antisymm_iff]
---   constructor
---   assumption
---   simp only [empty_subset]
---   intro a
---   rw [Subset.antisymm_iff] at a
---   rcases a with ⟨h₁, h₂⟩
---   assumption
-
--- -- eq_empty_iff_forall_not_mem
--- example {s : Set α} : s = ∅ ↔ ∀ x, x ∉ s := by
---   rw [←subset_empty_iff]
---   rfl
-
--- -- nonempty_iff_ne_empty
--- example (X : Type _) (s : Set X) : Set.Nonempty s ↔ s ≠ ∅ := by
---   rw [Set.Nonempty]
---   rw [ne_eq, eq_empty_iff_forall_not_mem]
---   push_neg
---   rfl
-
-
--- example (A B : ℝ): A = B ↔ A ≤ B ∧ B ≤ A := by library_search
-
-namespace Finset
-
-theorem powerset_singleton {U : Type _} [DecidableEq U] (x : U) :
-    Finset.powerset {x} = {∅, {x}} := by
-  ext y
-  rw [mem_powerset, subset_singleton_iff, mem_insert, mem_singleton]
-
-end Finset
-
-/- The powerset of a singleton contains only `∅` and the singleton. -/
-theorem powerset_singleton {U : Type _} (x : U) :
-    𝒫 ({x} : Set U) = {∅, {x}} := by
-  ext y
-  rw [mem_powerset_iff, subset_singleton_iff_eq]
-  rw [mem_insert_iff, mem_singleton_iff]
-
-
-lemma mem_powerset_insert_iff {U : Type _} (A S : Set U) (x : U) :
-    S ∈ 𝒫 (insert x A) ↔ S \ {x} ∈ 𝒫 A := by
-  rw [mem_powerset_iff, mem_powerset_iff, diff_singleton_subset_iff]
+--   · intro h
+--     by_cases hs : x ∈ S
+--     · sorry
+--     · left
+--       rw [Finset.subset_insert_iff_of_not_mem]
+--   · intro h
+--     rcases h with h | ⟨B, h₁, h₂⟩
+--     · exact le_trans h (subset_insert x A)
+--     · rw [←h₂]
+--       exact insert_subset_insert h₁
 
 -- lemma powerset_insert {U : Type _} (A : Set U) (x : U) :
---     𝒫 (insert x A) = 𝒫 A ∪ ({B : Set U | B \ {x} ∈ 𝒫 A}) := by
+--     𝒫 (insert x A) = A.powerset ∪ A.powerset.image (insert x) := by
+--   rw [Subset.antisymm_iff]
+--   constructor <;>
+--   intro B hB <;>
+--   simp_rw [mem_powerset_insert_iff₂, mem_union, mem_image] at hB ⊢ <;>
+--   assumption
+
+
+-- example : powerset {0, 1} = {∅, {0}, {1}, {0, 1}} := by
+--   simp_rw [powerset_insert, powerset_singleton, Finset.powerset_insert, Finset.powerset_singleton]
+--   simp only [image_insert_eq, union_insert, image_singleton, union_singleton]
+--   simp only [insert_comm, ←insert_emptyc_eq]
+
+-- example [DecidableEq ℕ] : Finset.powerset {0, 1} = {∅, {0}, {1}, {0, 1}} := by
+--   repeat' rw [@ Finset.powerset_insert ℕ]
+--   repeat' rw [@Finset.powerset_singleton ℕ]
+--   --simp only [image_insert_eq, union_insert, image_singleton, union_singleton]
+
+
+-- example : powerset {0, 1, 2, 4} =
+--     {∅, {0}, {1}, {0, 1}, {2}, {1, 2}, {0, 2}, {0, 1, 2},
+--     {4}, {0, 4}, {1, 4}, {0, 1, 4}, {2, 4}, {1, 2, 4}, {0, 2, 4}, {0, 1, 2, 4}} := by
+--   simp_rw [powerset_insert, powerset_singleton]
+--   simp_rw [image_insert_eq, image_singleton]
+
+
+
+-- example : Finset.powerset ({0, 1} : Finset ℕ) = {{0, 1}, ∅, {1}, {0}} := by
+--   have h : Finset.powerset ({0, 1} : Finset ℕ) = {∅, {0}, {1}, {0, 1}}
+--   rfl
+--   rw [h]
+--   simp only []
+
+
+-- example : ({∅, {0}, {1}, {0, 1}} : Finset (Finset ℕ)) = {∅, {1}, {0}, {0, 1}} := by
+--   simp only []
+
+
+
+-- lemma subset_of_subset_diff {U : Type _} (A B C : Set U) (h : A ⊆ B \ C) : A ⊆ B :=
+--   fun _ hx => mem_of_mem_diff (h hx)
+
+-- lemma subset_of_subset_diff' {U : Type _} (A B C : Set U) (h : A ⊆ B) : A \ C ⊆ B :=
 --   sorry
 
-theorem subset_insert_iff_of_not_mem {U : Type _ }{s t : Set U} (h : a ∉ s) : s ⊆ insert a t ↔ s ⊆ t := by
-  rw [subset_insert_iff, erase_eq_of_not_mem h]
+-- lemma mem_powerset' {U : Type _} {A B C : Set U}
+--     (h' : B ∈ 𝒫 C) (h : A ⊆ B) :
+--     A ∈ 𝒫 C := by
+--   rw [mem_powerset_iff] at h' ⊢
+--   exact le_trans h h'
 
-lemma mem_powerset_insert_iff₂ {U : Type _} (A S : Set U) (x : U) :
-    S ∈ 𝒫 (insert x A) ↔ S ∈ 𝒫 A ∨ ∃ B ∈ 𝒫 A , insert x B = S := by
-  simp_rw [mem_powerset_iff]
-  constructor
-  · intro h
-    by_cases hs : x ∈ S
-    · sorry
-    · left
-      rw [Finset.subset_insert_iff_of_not_mem]
-  · intro h
-    rcases h with h | ⟨B, h₁, h₂⟩
-    · exact le_trans h (subset_insert x A)
-    · rw [←h₂]
-      exact insert_subset_insert h₁
-
-lemma powerset_insert {U : Type _} (A : Set U) (x : U) :
-    𝒫 (insert x A) = A.powerset ∪ A.powerset.image (insert x) := by
-  rw [Subset.antisymm_iff]
-  constructor <;>
-  intro B hB <;>
-  simp_rw [mem_powerset_insert_iff₂, mem_union, mem_image] at hB ⊢ <;>
-  assumption
-
-
-example : powerset {0, 1} = {∅, {0}, {1}, {0, 1}} := by
-  simp_rw [powerset_insert, powerset_singleton, Finset.powerset_insert, Finset.powerset_singleton]
-  simp only [image_insert_eq, union_insert, image_singleton, union_singleton]
-  simp only [insert_comm, ←insert_emptyc_eq]
-
-example [DecidableEq ℕ] : Finset.powerset {0, 1} = {∅, {0}, {1}, {0, 1}} := by
-  repeat' rw [@ Finset.powerset_insert ℕ]
-  repeat' rw [@Finset.powerset_singleton ℕ]
-  --simp only [image_insert_eq, union_insert, image_singleton, union_singleton]
-
-
-example : powerset {0, 1, 2, 4} =
-    {∅, {0}, {1}, {0, 1}, {2}, {1, 2}, {0, 2}, {0, 1, 2},
-    {4}, {0, 4}, {1, 4}, {0, 1, 4}, {2, 4}, {1, 2, 4}, {0, 2, 4}, {0, 1, 2, 4}} := by
-  simp_rw [powerset_insert, powerset_singleton]
-  simp_rw [image_insert_eq, image_singleton]
+-- example (A B : Set ℕ) : A = B ↔ ∀ x, x ∈ A ↔ x ∈ B := by
+--   exact ext_iff
 
 
 
-example : Finset.powerset ({0, 1} : Finset ℕ) = {{0, 1}, ∅, {1}, {0}} := by
-  have h : Finset.powerset ({0, 1} : Finset ℕ) = {∅, {0}, {1}, {0, 1}}
-  rfl
-  rw [h]
-  simp only []
+-- lemma NO.powerset_insert {U : Type _} (A : Set U) (x : U) :
+--     𝒫 (insert x A) = 𝒫 A ∪ ({insert x B | B ∈ 𝒫 A}) := by
+--   ext y
+--   rw [mem_powerset_insert_iff]
+--   constructor
+--   · intro h
+--     rw [mem_union, mem_setOf]
+--     by_cases hx : x ∈ y
+--     · right
+--       use y \ {x}
+--       constructor
+--       · assumption
+--       · rw [insert_diff_singleton, insert_eq_of_mem hx]
+--     · left
+--       rw [diff_singleton_eq_self hx] at h
+--       assumption
+--   · intro h
+--     rw [mem_union, mem_setOf] at h
+--     rcases h with h | h
+--     · apply mem_powerset' h
+--       simp
+--     · rcases h with ⟨B, hB⟩
+--       rw [←hB.2]
+--       apply mem_powerset' hB.1
+--       simp
+
+-- -- lemma xxx {U : Type _} (x y : U):
+-- --     ({insert x B | B ∈ {∅, }}) = {({x} : Set U), {x, y}} := by
+-- --   sorry
+
+-- -- lemma hh {U : Type _} (A : Set U) (x : U) :
+-- --     A \ {x} ∈
+
+-- lemma diff_singleton_eq_iff {U : Type _} {A B : Set U} {x : U} :
+--     A \ {x} = B ↔ A = B ∨ A = insert x B := by sorry
+
+-- lemma x1 {U : Type _} (x : U) : insert x (∅ : Set U) = {x} := by sorry
+
+-- --set_option maxHeartbeats 20000
 
 
-example : ({∅, {0}, {1}, {0, 1}} : Finset (Finset ℕ)) = {∅, {1}, {0}, {0, 1}} := by
-  simp only []
+-- example {U : Type _} {x y : U} : ({x, y} : Set U) = {y, x} := by
+--   exact pair_comm x y
+
+-- example {U : Type _} {A : Set U} {x y : U} : insert x (insert y A) = insert y (insert x A) := by
+--   exact insert_comm x y A
+
+-- open Classical
+
+-- #check decide
+
+-- example : ({{0, 2}, {3, 5, 6}} : Set (Set ℕ)) = {{2, 0}, {5, 3, 6}} := by
+--   rw [Subset.antisymm_iff]
+--   constructor <;>
+--   intro A hA <;>
+--   simp_rw [mem_insert_iff, mem_singleton_iff] at *
+--   · rw [pair_comm 2 0, insert_comm 5 3]
+--     tauto
+--   · rw [pair_comm 0 2, insert_comm 3 5]
+--     tauto
+
+-- example (A : Set ℕ) : ({{0, 2}, A, {3, 5, 6}} : Set (Set ℕ)) = {{2, 0}, {5, 3, 6}, A} := by
+--   simp only [insert_comm, ←insert_emptyc_eq]
 
 
-
-lemma subset_of_subset_diff {U : Type _} (A B C : Set U) (h : A ⊆ B \ C) : A ⊆ B :=
-  fun _ hx => mem_of_mem_diff (h hx)
-
-lemma subset_of_subset_diff' {U : Type _} (A B C : Set U) (h : A ⊆ B) : A \ C ⊆ B :=
-  sorry
-
-lemma mem_powerset' {U : Type _} {A B C : Set U}
-    (h' : B ∈ 𝒫 C) (h : A ⊆ B) :
-    A ∈ 𝒫 C := by
-  rw [mem_powerset_iff] at h' ⊢
-  exact le_trans h h'
-
-example (A B : Set ℕ) : A = B ↔ ∀ x, x ∈ A ↔ x ∈ B := by
-  exact ext_iff
+-- example : ({{0, 2}, {3, 5, 6}} : Finset (Finset ℕ)) = {{2, 0}, {5, 3, 6}} := by
+--   simp only
 
 
-
-lemma NO.powerset_insert {U : Type _} (A : Set U) (x : U) :
-    𝒫 (insert x A) = 𝒫 A ∪ ({insert x B | B ∈ 𝒫 A}) := by
-  ext y
-  rw [mem_powerset_insert_iff]
-  constructor
-  · intro h
-    rw [mem_union, mem_setOf]
-    by_cases hx : x ∈ y
-    · right
-      use y \ {x}
-      constructor
-      · assumption
-      · rw [insert_diff_singleton, insert_eq_of_mem hx]
-    · left
-      rw [diff_singleton_eq_self hx] at h
-      assumption
-  · intro h
-    rw [mem_union, mem_setOf] at h
-    rcases h with h | h
-    · apply mem_powerset' h
-      simp
-    · rcases h with ⟨B, hB⟩
-      rw [←hB.2]
-      apply mem_powerset' hB.1
-      simp
-
--- lemma xxx {U : Type _} (x y : U):
---     ({insert x B | B ∈ {∅, }}) = {({x} : Set U), {x, y}} := by
---   sorry
-
--- lemma hh {U : Type _} (A : Set U) (x : U) :
---     A \ {x} ∈
-
-lemma diff_singleton_eq_iff {U : Type _} {A B : Set U} {x : U} :
-    A \ {x} = B ↔ A = B ∨ A = insert x B := by sorry
-
-lemma x1 {U : Type _} (x : U) : insert x (∅ : Set U) = {x} := by sorry
-
---set_option maxHeartbeats 20000
-
-
-example {U : Type _} {x y : U} : ({x, y} : Set U) = {y, x} := by
-  exact pair_comm x y
-
-example {U : Type _} {A : Set U} {x y : U} : insert x (insert y A) = insert y (insert x A) := by
-  exact insert_comm x y A
-
-open Classical
-
-#check decide
-
-example : ({{0, 2}, {3, 5, 6}} : Set (Set ℕ)) = {{2, 0}, {5, 3, 6}} := by
-  rw [Subset.antisymm_iff]
-  constructor <;>
-  intro A hA <;>
-  simp_rw [mem_insert_iff, mem_singleton_iff] at *
-  · rw [pair_comm 2 0, insert_comm 5 3]
-    tauto
-  · rw [pair_comm 0 2, insert_comm 3 5]
-    tauto
-
-example (A : Set ℕ) : ({{0, 2}, A, {3, 5, 6}} : Set (Set ℕ)) = {{2, 0}, {5, 3, 6}, A} := by
-  simp only [insert_comm, ←insert_emptyc_eq]
-
-
-example : ({{0, 2}, {3, 5, 6}} : Finset (Finset ℕ)) = {{2, 0}, {5, 3, 6}} := by
-  simp only
-
-
-  -- -- This works but does not scale well
-  -- ext x
-  -- simp_rw [powerset_insert, powerset_singleton]
-  -- simp_rw [mem_union, mem_setOf, mem_insert_iff, mem_singleton_iff]
-  -- simp_rw [diff_singleton_eq_iff, x1]
-  -- tauto
+--   -- -- This works but does not scale well
+--   -- ext x
+--   -- simp_rw [powerset_insert, powerset_singleton]
+--   -- simp_rw [mem_union, mem_setOf, mem_insert_iff, mem_singleton_iff]
+--   -- simp_rw [diff_singleton_eq_iff, x1]
+--   -- tauto
 
 
 
 
-  -- rw [Subset.antisymm_iff]
-  -- constructor
-  -- intro A hA
-  -- rw [mem_powerset_iff] at hA
-  -- simp_rw [mem_insert_iff, mem_singleton_iff] at *
+--   -- rw [Subset.antisymm_iff]
+--   -- constructor
+--   -- intro A hA
+--   -- rw [mem_powerset_iff] at hA
+--   -- simp_rw [mem_insert_iff, mem_singleton_iff] at *
 
 
-example : ({2, 3, 5} : Set ℕ) ⊆ {4, 2, 5, 7, 3} := by
-  intro a ha
-  simp_rw [Set.mem_insert_iff, Set.mem_singleton_iff] at *
-  tauto
+-- example : ({2, 3, 5} : Set ℕ) ⊆ {4, 2, 5, 7, 3} := by
+--   intro a ha
+--   simp_rw [Set.mem_insert_iff, Set.mem_singleton_iff] at *
+--   tauto
 
-example : ({0} : Set ℕ) ∪ {1, 2} = {0, 2, 1} := by
-  rw [Subset.antisymm_iff]
-  intro A hA
-  --rw [Set.mem_insert_iff]
+-- example : ({0} : Set ℕ) ∪ {1, 2} = {0, 2, 1} := by
+--   rw [Subset.antisymm_iff]
+--   intro A hA
+--   --rw [Set.mem_insert_iff]
 
 
 
--- Trick:
--- attribute [default_instance] Set.instSingletonSet
+-- -- Trick:
+-- -- attribute [default_instance] Set.instSingletonSet

server/adam/Adam/Levels/SetTheory/T04_xx.lean

@@ -1,96 +1,96 @@
-import Adam.Metadata
-
-import Mathlib
-import Duper.Tactic
+-- import Adam.Metadata
 
 
-import Mathlib.Order.Disjoint
+-- import Duper.Tactic
 
-set_option tactic.hygienic false
 
-Game "Adam"
-World "SetTheory"
-Level 4
+-- import Mathlib.Order.Disjoint
 
-Title "Mengen"
+-- set_option tactic.hygienic false
 
-Introduction
-"
-`∈ ∉ ⊆ ⊂ ⋃ ⋂`
-"
+-- Game "Adam"
+-- World "SetTheory"
+-- Level 4
 
-open Set
+-- Title "Mengen"
 
--- open_locale cardinal
+-- Introduction
+-- "
+-- `∈ ∉ ⊆ ⊂ ⋃ ⋂`
+-- "
 
-variables {X : Type _} (x : X) (A B : Set X)
+-- open Set
 
-#check A.Nonempty
-#check Nonempty A
-#check insert x A -- {x} ∪ A
--- #check disjoint A B -- needs Mathlib.Order.Disjoint
-#check A ∆ B
-#check Nontrivial A
+-- -- open_locale cardinal
 
-#check ({2} : Set ℕ)
+-- variables {X : Type _} (x : X) (A B : Set X)
 
-example : ({2} : Set ℕ) ⊆ {4, 2, 3} := by
-  simp only [mem_singleton_iff, mem_insert_iff, or_self, singleton_subset_iff, or_false, or_true]
+-- #check A.Nonempty
+-- #check Nonempty A
+-- #check insert x A -- {x} ∪ A
+-- -- #check disjoint A B -- needs Mathlib.Order.Disjoint
+-- #check A ∆ B
+-- #check Nontrivial A
 
-example : ({2, 3, 5} : Set ℕ) ⊆ {4, 2, 5, 7, 3} := by
-  intro n hn
-  simp only [mem_insert_iff, mem_singleton_iff] at *
-  tauto
+-- #check ({2} : Set ℕ)
 
-example : {2, 3, 5} ⊆ (univ : Set ℕ) := by
-  tauto
+-- example : ({2} : Set ℕ) ⊆ {4, 2, 3} := by
+--   simp only [mem_singleton_iff, mem_insert_iff, or_self, singleton_subset_iff, or_false, or_true]
 
-example : 3 ∈ ({4, 2, 5, 7, 3} : Set ℕ) := by
-  tauto
+-- example : ({2, 3, 5} : Set ℕ) ⊆ {4, 2, 5, 7, 3} := by
+--   intro n hn
+--   simp only [mem_insert_iff, mem_singleton_iff] at *
+--   tauto
 
-example : ({4, 9} : Set ℕ) = Set.insert 4 {9} := by
-  rfl
+-- example : {2, 3, 5} ⊆ (univ : Set ℕ) := by
+--   tauto
+
+-- example : 3 ∈ ({4, 2, 5, 7, 3} : Set ℕ) := by
+--   tauto
+
+-- example : ({4, 9} : Set ℕ) = Set.insert 4 {9} := by
+--   rfl
 
 
 
 
-#check Finset.card
+-- #check Finset.card
 
-variable (A : Finset ℕ)
+-- variable (A : Finset ℕ)
 
-#check A.card
+-- #check A.card
 
--- card_union_eq
-example (A B : Finset ℕ) (h : Disjoint A B) : (A ∪ B).card = A.card + B.card := by
-  -- Not a suitable proof for the course.
-  rw [← Finset.disjUnion_eq_union A B h, Finset.card_disjUnion _ _ _]
+-- -- card_union_eq
+-- example (A B : Finset ℕ) (h : Disjoint A B) : (A ∪ B).card = A.card + B.card := by
+--   -- Not a suitable proof for the course.
+--   rw [← Finset.disjUnion_eq_union A B h, Finset.card_disjUnion _ _ _]
 
-example : ¬Disjoint {n : ℤ | ∃ k, n = 2 * k} {3, 5, 6, 9, 11} := by
-  rw [not_disjoint_iff]
-  use 6
-  constructor
-  rw [mem_setOf_eq]
-  use 3
-  tauto
+-- example : ¬Disjoint {n : ℤ | ∃ k, n = 2 * k} {3, 5, 6, 9, 11} := by
+--   rw [not_disjoint_iff]
+--   use 6
+--   constructor
+--   rw [mem_setOf_eq]
+--   use 3
+--   tauto
 
-example : {n : ℕ | Even n ∨ Odd n} = univ := by
-  rw [setOf_or]
+-- example : {n : ℕ | Even n ∨ Odd n} = univ := by
+--   rw [setOf_or]
 
 
 
 
-#check Set.eq_of_mem_singleton
-#check {n : ℤ | ∃ k, n = 2 * k}
-#check {n : ℤ // ∃ k, n = 2 * k}
-#check ℤ
-variables (C : Set ℤ)
-#check {n ∈ C | ∃ (k : ℤ), n = 2 * k}
-#check {n : ℤ | n ∈ C ∧ ∃ (k : ℤ), n = 2 * k}
-#check {x ∈ A | x = x}
-#check {y | y ∈ A}
-#check setOf_and
-#check setOf_or
-#check Disjoint C univ
+-- #check Set.eq_of_mem_singleton
+-- #check {n : ℤ | ∃ k, n = 2 * k}
+-- #check {n : ℤ // ∃ k, n = 2 * k}
+-- #check ℤ
+-- variables (C : Set ℤ)
+-- #check {n ∈ C | ∃ (k : ℤ), n = 2 * k}
+-- #check {n : ℤ | n ∈ C ∧ ∃ (k : ℤ), n = 2 * k}
+-- #check {x ∈ A | x = x}
+-- #check {y | y ∈ A}
+-- #check setOf_and
+-- #check setOf_or
+-- #check Disjoint C univ
 
-example : {n ∈ C | ∃ (k : ℤ), n = 2 * k} = {n : ℤ | n ∈ C ∧ ∃ (k : ℤ), n = 2 * k} := by
-  rfl
+-- example : {n ∈ C | ∃ (k : ℤ), n = 2 * k} = {n : ℤ | n ∈ C ∧ ∃ (k : ℤ), n = 2 * k} := by
+--   rfl