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Jon Eugster
2023-03-29 15:37:21 +02:00
parent 4d1d97a164
commit 4ae7e0eab6
9 changed files with 179 additions and 136 deletions
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server/adam/Adam/DefinitionDocs.lean
+87
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@@ -0,0 +1,87 @@
import GameServer.Commands
/-! ## Definitions -/
DefinitionDoc Even as "Even"
"
`even n` ist definiert als `∃ r, a = 2 * r`.
Die Definition kann man mit `unfold even at *` einsetzen.
## Eigenschaften
* Mathlib Doc: [#Even](https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Parity.html#Even)"
DefinitionDoc Odd as "Odd"
"
`odd n` ist definiert als `∃ r, a = 2 * r + 1`.
Die Definition kann man mit `unfold odd at *` einsetzen.
* Mathlib Doc: [Odd](https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Parity.html#Odd)"
DefinitionDoc Injective as "Injective"
"
`Injective f` ist definiert als
```
∀ a b, f a = f b → a = b
```
definiert.
* Mathlib Doc: [Injective](https://leanprover-community.github.io/mathlib4_docs/Mathlib/Init/Function.html#Function.Injective)"
DefinitionDoc Surjective as "Surjective"
"
`Surjective f` ist definiert als
```
∀ a, (∃ b, f a = b)
```
* Mathlib Doc: [Surjective](https://leanprover-community.github.io/mathlib4_docs/Mathlib/Init/Function.html#Function.Surjective)"
DefinitionDoc Bijective as "Bijective"
"
## Eigenschaften
* Mathlib Doc: [#Bijective](https://leanprover-community.github.io/mathlib4_docs/Mathlib/Init/Function.html#Function.Bijective)
"
DefinitionDoc LeftInverse as "LeftInverse"
"
## Eigenschaften
* Mathlib Doc: [#LeftInverse](https://leanprover-community.github.io/mathlib4_docs/Mathlib/Init/Function.html#Function.LeftInverse)
"
DefinitionDoc RightInverse as "RightInverse"
"
## Eigenschaften
* Mathlib Doc: [#RightInverse](https://leanprover-community.github.io/mathlib4_docs/Mathlib/Init/Logic.html#RightInverse)
"
DefinitionDoc StrictMono as "StrictMono"
"
`StrictMono f` ist definiert als
```
∀ a b, a < b → f a < f b
```
## Eigenschaften
* Mathlib Doc: [#StrictMono](https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Monotone/Basic.html#StrictMono)
"
DefinitionDoc Symbol.Subset as "⊆" "
Auf Mengen (`Set`) ist `A ⊆ B` als `∀x, x ∈ A → x ∈ B` implementiert.
"
DefinitionDoc Set.Nonempty as "Nonempty" "
`A.Nonemty` ist als `∃ x, x ∈ A` definiert.
"
server/adam/Adam/LemmaDocs.lean
+42 -90
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@@ -165,7 +165,7 @@ Jedes Element ist in `univ`, der Menge aller Elemente eines Typs `α`.
* Mathlib Doc: [#mem_univ](https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Set/Basic.html#Set.mem_univ)
"
LemmaDoc not_mem_empty as "not_mem_empty" in "Set"
LemmaDoc Set.not_mem_empty as "not_mem_empty" in "Set"
"
`Set.not_mem_empty {α : Type _} (x : α) : x ∉ ∅`
@@ -175,11 +175,21 @@ Kein Element ist in der leeren Menge.
* `simp`-Lemma: Nein
* Namespace: `Set`
* Minimal Import: `Mathlib.Data.Set.Basic`
* Mathlib Doc: [#not_mem_empty](https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Set/Basic.html#Set.not_mem_empty)
"
LemmaDoc empty_subset as "empty_subset" in "Set"
LemmaDoc Set.subset_empty_iff as "subset_empty_iff" in "Set"
"
`Set.subset_empty_iff.{u} {α : Type u} {s : Set α} : s ⊆ ∅ ↔ s = ∅`
## Eigenschaften
* `simp`-Lemma: Nein
* Namespace: `Set`
* Mathlib Doc: [#empty_subset](https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Set/Basic.html#Set.subset_empty_iff)
"
LemmaDoc Set.empty_subset as "empty_subset" in "Set"
"
`Set.empty_subset {α : Type u} (s : Set α) : ∅ ⊆ s`
@@ -187,11 +197,10 @@ LemmaDoc empty_subset as "empty_subset" in "Set"
* `simp`-Lemma: Ja
* Namespace: `Set`
* Minimal Import: `Mathlib.Data.Set.Basic`
* Mathlib Doc: [#empty_subset](https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Set/Basic.html#Set.empty_subset)
"
LemmaDoc Subset.antisymm as "Subset.antisymm" in "Set"
LemmaDoc Set.Subset.antisymm as "Subset.antisymm" in "Set"
"
`Set.Subset.antisymm {α : Type u} {a : Set α} {b : Set α} (h₁ : a ⊆ b) (h₂ : b ⊆ a) : a = b`
@@ -212,7 +221,7 @@ für die Iff-Version.
* Mathlib Doc: [#Subset.antisymm](https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Set/Basic.html#Set.Subset.antisymm)
"
LemmaDoc Subset.antisymm_iff as "Subset.antisymm_iff" in "Set"
LemmaDoc Set.Subset.antisymm_iff as "Subset.antisymm_iff" in "Set"
"
`Set.Subset.antisymm_iff {α : Type u} {a : Set α} {b : Set α} : a = b ↔ a ⊆ b ∧ b ⊆ a`
@@ -234,6 +243,17 @@ für eine verwandte Version.
* Mathlib Doc: [#Subset.antisymm_iff](https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Set/Basic.html#Set.Subset.antisymm_iff)
"
LemmaDoc Set.diff_inter as "union_assoc" in "Set"
""
LemmaDoc Set.union_assoc as "union_assoc" in "Set"
""
LemmaDoc Set.union_diff_distrib as "union_diff_distrib" in "Set"
""
LemmaDoc Set.univ_union as "univ_union" in "Set"
""
LemmaDoc Nat.prime_def_lt'' as "prime_def_lt''" in "Nat"
"
@@ -249,7 +269,6 @@ Die bekannte Definition einer Primmzahl in `ℕ`: Eine Zahl (`p ≥ 2`) mit gena
* Mathlib Doc: [#Nat.prime_def_lt''](https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Nat/Prime.html#Nat.prime_def_lt'')
"
LemmaDoc Finset.sum_add_distrib as "sum_add_distrib" in "Sum"
"
@@ -280,6 +299,22 @@ LemmaDoc Nat.succ_eq_add_one as "succ_eq_add_one" in "Sum"
* Mathlib Doc: [#succ_eq_add_one](https://leanprover-community.github.io/mathlib4_docs/Init/Data/Nat/Basic.html#Nat.succ_eq_add_one)
"
LemmaDoc ne_eq as "ne_eq" in "Logic"
"
## Eigenschaften
* Mathlib Doc: [#ne_eq](https://leanprover-community.github.io/mathlib4_docs/Init/SimpLemmas.html#ne_eq)
"
LemmaDoc Set.eq_empty_iff_forall_not_mem as "eq_empty_iff_forall_not_mem" in "Sum"
"
## Eigenschaften
* Mathlib Doc: [#eq_empty_iff_forall_not_mem](https://leanprover-community.github.io/mathlib4_docs/Mathlib/Data/Set/Basic.html#Set.eq_empty_iff_forall_not_mem)
"
LemmaDoc Nat.zero_eq as "zero_eq" in "Sum"
"
@@ -429,86 +464,3 @@ LemmaDoc Iff.symm as "Iff.symm" in "Logic"
* Mathlib Doc: [#Iff.symm](https://leanprover-community.github.io/mathlib4_docs/Init/Core.html#Iff.symm)
"
/-! ## Definitions -/
DefinitionDoc Even as "Even"
"
`even n` ist definiert als `∃ r, a = 2 * r`.
Die Definition kann man mit `unfold even at *` einsetzen.
## Eigenschaften
* Mathlib Doc: [#Even](https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Parity.html#Even)"
DefinitionDoc Odd as "Odd"
"
`odd n` ist definiert als `∃ r, a = 2 * r + 1`.
Die Definition kann man mit `unfold odd at *` einsetzen.
* Mathlib Doc: [Odd](https://leanprover-community.github.io/mathlib4_docs/Mathlib/Algebra/Parity.html#Odd)"
DefinitionDoc Injective as "Injective"
"
`Injective f` ist definiert als
```
∀ a b, f a = f b → a = b
```
definiert.
* Mathlib Doc: [Injective](https://leanprover-community.github.io/mathlib4_docs/Mathlib/Init/Function.html#Function.Injective)"
DefinitionDoc Surjective as "Surjective"
"
`Surjective f` ist definiert als
```
∀ a, (∃ b, f a = b)
```
* Mathlib Doc: [Surjective](https://leanprover-community.github.io/mathlib4_docs/Mathlib/Init/Function.html#Function.Surjective)"
DefinitionDoc Bijective as "Bijective"
"
## Eigenschaften
* Mathlib Doc: [#Bijective](https://leanprover-community.github.io/mathlib4_docs/Mathlib/Init/Function.html#Function.Bijective)
"
DefinitionDoc LeftInverse as "LeftInverse"
"
## Eigenschaften
* Mathlib Doc: [#LeftInverse](https://leanprover-community.github.io/mathlib4_docs/Mathlib/Init/Function.html#Function.LeftInverse)
"
DefinitionDoc RightInverse as "RightInverse"
"
## Eigenschaften
* Mathlib Doc: [#RightInverse](https://leanprover-community.github.io/mathlib4_docs/Mathlib/Init/Logic.html#RightInverse)
"
DefinitionDoc StrictMono as "StrictMono"
"
`StrictMono f` ist definiert als
```
∀ a b, a < b → f a < f b
```
## Eigenschaften
* Mathlib Doc: [#StrictMono](https://leanprover-community.github.io/mathlib4_docs/Mathlib/Order/Monotone/Basic.html#StrictMono)
"
DefinitionDoc Symbol.Subset as "⊆" "
Auf Mengen (`Set`) ist `A ⊆ B` als `∀x, x ∈ A → x ∈ B` implementiert.
"
server/adam/Adam/Levels/Implication/L06_Iff.lean
+2 -2
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@@ -29,8 +29,8 @@ Statement (A B : Prop) (mp : A → B) (mpr : B → A) : A ↔ B := by
Conclusion
"
**Robo**: Übrigens, bei `(h : A ∧ B)` haben die beiden Teile `h.left` und `h.right` geheissen,
hier bei `(h : A ↔ B)` heissen sie `h.mp` und `h.mpr`.
**Robo**: Übrigens, bei `(h : A ∧ B)` haben die beiden Teile `h.left` und `h.right` geheißen,
hier bei `(h : A ↔ B)` heißen sie `h.mp` und `h.mpr`.
**Du**: Also `h.mp` ist `A → B`? Wieso `mp`?
server/adam/Adam/Levels/SetTheory/L04_SubsetEmpty.lean
+9 -22
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@@ -29,21 +29,8 @@ Fast immer wenn man Gleichheiten von Mengen zeigen muss, will man diese in zwei
aufteilen.
"
namespace MySet
open Set Subset
-- Copied some lemmas from `Matlib.Data.Set.Basic` in order to not import the entire file.
theorem tmp {α : Type _} {s t : Set α} : s = t → s ⊆ t :=
fun h₁ _ h₂ => by rw [← h₁] ; exact h₂
theorem Subset.antisymm_iff {α : Type _} {a b : Set α} : a = b ↔ a ⊆ b ∧ b ⊆ a :=
⟨fun e => ⟨tmp e, tmp e.symm⟩, fun ⟨h₁, h₂⟩ => Set.ext fun _ => ⟨@h₁ _, @h₂ _⟩⟩
@[simp]
theorem empty_subset {α : Type _} (s : Set α) : ∅ ⊆ s :=
fun.
Statement subset_empty_iff {A : Type _} (s : Set A) :
s ⊆ ∅ ↔ s = ∅ := by
Hint "**Du**: Ja, die einzige Teilmenge der leeren Menge ist die leere Menge.
@@ -53,16 +40,16 @@ Statement subset_empty_iff {A : Type _} (s : Set A) :
Hint (hidden := true) "**Robo**: Fang doch einmal mit `constructor` an."
constructor
intro h
Hint "**Robo**: "
Hint "**Robo**: Gleichheit zwischen Mengen kann man zum Beispiel zeigen,
indem man `A ⊆ B` und `B ⊆ A` zeigt.
Dieser Schritt ist `apply Subset.antisymm`"
apply Subset.antisymm
assumption
simp only [empty_subset]
intro a
rw [Subset.antisymm_iff] at a
rcases a with ⟨h₁, h₂⟩
assumption
Hint "**Robo**: Hier ist das Lemma `empty_subset` hilfreich."
apply empty_subset
intro h
rw [h]
DisabledTactic tauto
NewLemma Subset.antisymm Subset.antisymm_iff empty_subset
end MySet
NewLemma Set.Subset.antisymm Set.Subset.antisymm_iff Set.empty_subset
server/adam/Adam/Levels/SetTheory/L05_Empty.lean
+4 -20
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@@ -12,39 +12,23 @@ Game "Adam"
World "SetTheory"
Level 5
Title "Nonempty"
Title "Empty"
Introduction
"
Das Gegenteil von `A = ∅` ist `A ≠ ∅`, aber in Lean wird der Ausdruck `A.Nonempty` bevorzugt.
Dieser ist dadurch existiert, dass in `A` ein Element existiert: `∃x, x ∈ A`.
Zeige dass die beiden Ausdrücke äquivalent sind:
Zeige folgendes Lemma, welches wir gleich brauchen werden:
"
namespace MySet
open Set
theorem subset_empty_iff {A : Type _} (s : Set A) : 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
Statement eq_empty_iff_forall_not_mem
""
{A : Type _} (s : Set A) :
s = ∅ ↔ ∀ x, x ∉ s := by
Hint "Das Lemma `subset_empty_iff` von letzter Aufgabe könnte hilfreich sein."
rw [←subset_empty_iff]
rfl -- This is quite a miracle :)
NewTactic constructor intro rw assumption rcases simp tauto trivial
end MySet
NewLemma Set.subset_empty_iff
server/adam/Adam/Levels/SetTheory/L06_Nonempty.lean
+6 -2
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@@ -25,9 +25,13 @@ Statement nonempty_iff_ne_empty
""
{A : Type _} (s : Set A) :
s.Nonempty ↔ s ≠ ∅ := by
rw [Set.Nonempty]
Hint "Am besten fängst du mit `unfold Set.Nonempty` an."
unfold Set.Nonempty
Hint "Mit `ne_eq` und `eq_empty_iff_forall_not_mem` kannst du hier weiterkommen."
rw [ne_eq, eq_empty_iff_forall_not_mem]
Hint (hidden := true) "`push_neg` kann hier helfen."
push_neg
rfl
NewTactic constructor intro rw assumption rcases simp tauto trivial
NewLemma ne_eq Set.eq_empty_iff_forall_not_mem
NewDefinition Set.Nonempty
server/adam/Adam/Levels/SetTheory/L08_UnionInter.lean
+1
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@@ -31,3 +31,4 @@ Statement
rw [univ_union]
NewTactic constructor intro rw assumption rcases simp tauto trivial
NewLemma Set.diff_inter Set.union_assoc Set.union_diff_distrib Set.univ_union
server/adam/Adam/Metadata.lean
+1
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@@ -1,4 +1,5 @@
import GameServer.Commands
import Adam.TacticDocs
import Adam.LemmaDocs
import Adam.DefinitionDocs
import Mathlib.Init.Data.Nat.Basic -- Imports the notation ℕ.