├── .gitignore ├── images ├── Logo1.png ├── Menu.png ├── format1.png ├── tac-left.png ├── formation1.png ├── formation2.png ├── hello-coq1.png ├── hello-coq2.png ├── hello-coq3.png ├── hello-coq4.png ├── hello-coq5.png ├── tac-apply.png ├── tac-exact.png ├── tac-right.png ├── tac-simpl1.png ├── tac-simpl2.png ├── tac-simpl3.png ├── tac-split.png ├── coq-response.png ├── logicalSymbol.png ├── tac-destruct1.png ├── tac-destruct2.png ├── tac-intros1.png ├── tac-intros2.png ├── tac-intros3.png ├── tac-rewrite1.png ├── tac-rewrite2.png ├── tac-rewrite3.png ├── two-proccess.png ├── two-subgoals.png ├── tac-assumption.png └── tac-reflexivity.png ├── .ipynb_checkpoints ├── 1-Logic-Copy1-checkpoint.ipynb ├── 1-Logic-Copy2-checkpoint.ipynb ├── Logic-checkpoint.ipynb ├── 1−Logic-checkpoint.ipynb ├── Equal−Ex-checkpoint.ipynb ├── Equal-Ex-checkpoint.ipynb ├── Induction-Ex-checkpoint.ipynb ├── Logic-Ex-checkpoint.ipynb ├── 1_basic-checkpoint.ipynb ├── 0-Contents-checkpoint.ipynb ├── Introduction-checkpoint.ipynb ├── 0-Introduction-checkpoint.ipynb ├── Equal-Ex-Work-checkpoint.ipynb ├── Equal-Ex-Answer-checkpoint.ipynb ├── Induction-Ex-Work-checkpoint.ipynb └── Logic-Ex-Work-checkpoint.ipynb ├── README.md ├── 0-Contents.ipynb └── 0-Introduction.ipynb /.gitignore: -------------------------------------------------------------------------------- 1 | 2 | .DS_Store 3 | -------------------------------------------------------------------------------- /images/Logo1.png: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/maruyama097/coq-tutorial/HEAD/images/Logo1.png -------------------------------------------------------------------------------- /images/Menu.png: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/maruyama097/coq-tutorial/HEAD/images/Menu.png -------------------------------------------------------------------------------- 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"execution_count": null, 28 | "metadata": { 29 | "coq_kernel_metadata": { 30 | "auto_roll_back": true 31 | } 32 | }, 33 | "outputs": [], 34 | "source": [] 35 | } 36 | ], 37 | "metadata": { 38 | "kernelspec": { 39 | "display_name": "Coq", 40 | "language": "coq", 41 | "name": "coq" 42 | }, 43 | "language_info": { 44 | "file_extension": ".v", 45 | "mimetype": "text/x-coq", 46 | "name": "coq", 47 | "version": "8.9.1" 48 | } 49 | }, 50 | "nbformat": 4, 51 | "nbformat_minor": 2 52 | } 53 | -------------------------------------------------------------------------------- /.ipynb_checkpoints/1-Logic-Copy2-checkpoint.ipynb: -------------------------------------------------------------------------------- 1 | { 2 | "cells": [ 3 | { 4 | "cell_type": "markdown", 5 | "metadata": { 6 | "coq_kernel_metadata": { 7 | "auto_roll_back": true 8 | } 9 | }, 10 | "source": [ 11 | "" 12 | ] 13 | }, 14 | { 15 | "cell_type": "markdown", 16 | "metadata": { 17 | "coq_kernel_metadata": { 18 | "auto_roll_back": true 19 | } 20 | }, 21 | "source": [ 22 | "# 等式の証明" 23 | ] 24 | }, 25 | { 26 | "cell_type": "code", 27 | "execution_count": null, 28 | "metadata": { 29 | "coq_kernel_metadata": { 30 | "auto_roll_back": true 31 | } 32 | }, 33 | "outputs": [], 34 | "source": [] 35 | } 36 | ], 37 | "metadata": { 38 | "kernelspec": { 39 | "display_name": "Coq", 40 | "language": "coq", 41 | "name": "coq" 42 | }, 43 | "language_info": { 44 | "file_extension": ".v", 45 | "mimetype": "text/x-coq", 46 | "name": "coq", 47 | "version": "8.9.1" 48 | } 49 | }, 50 | "nbformat": 4, 51 | "nbformat_minor": 2 52 | } 53 | -------------------------------------------------------------------------------- /README.md: -------------------------------------------------------------------------------- 1 | # このチュートリアルについて 2 | 3 | このチュートリアルは、coq-jupyter の形式で記述されています。Coqとcoq-jupyter をインストールしていれば、ドキュメントの内部でCoqのコマンドが実行できます。 4 | 5 | 全体の構成については[目次](0-Contents.ipynb)をご覧ください。（ファイル名は、0-Contents.ipynbです。） 6 | git clone https://github.com/maruyama097/coq-tutorial でダウンロードできます。 7 | 8 | Coqとcoq-jupyterのインストールなしでも、jupyter のnotebookとしてだけなら、GitHubへのアクセスだけでも、画像を含めてドキュメントの多くの部分を読むことは可能です。 9 | - ただ、文字の一部が欠けていたり、HTMLのタグがそのまま表示されることがあります。 10 | - もちろん、Coqのコマンドは実行できません。 11 | 12 | Coqとcoq-jupyterのインストールをお勧めします。 13 | 14 | # Coqとcoq-jupyterのインストールについて 15 | 16 | Mac とWindowsでは、インストールのやり方が異なりますので、ご注意ください。 17 | 18 | ## Macの場合 19 | 20 | Anacondaを既にインストールしている場合と、そうでない場合とで、別々のインストールの仕方を紹介します。 21 | 22 | ### Anacondaがインストール済の場合 23 | 24 | 次のコマンドで、Coqとcoq-jupyterのインストールができます。 25 | ``` 26 | conda config --add channels conda-forge 27 | conda create -n coq coq-jupyter 28 | conda activate coq 29 | ``` 30 | ### Anacondaを使わずにインストールしたい場合 31 | 32 | 次の資料を参考にしてください。 33 | 34 | #### 「macOSにbrewでcoq_jupyterをインストールする」 　http://bit.ly/33vHeX3 35 | 36 | ## Windowsの場合 37 | 38 | Windowsの場合には、WSLを使って Coqとcoq-jupyter をインストールします。 39 | 40 | 次の資料を参考にしてください。 41 | 42 | #### 「WSLにcoq-jupyterをインストールする」　http://bit.ly/33nw8TB 43 | 44 | 平原さん、情報提供ありがとうございました！ -------------------------------------------------------------------------------- /.ipynb_checkpoints/Logic-checkpoint.ipynb: -------------------------------------------------------------------------------- 1 | { 2 | "cells": [ 3 | { 4 | "cell_type": "markdown", 5 | "metadata": { 6 | "coq_kernel_metadata": { 7 | "auto_roll_back": true 8 | } 9 | }, 10 | "source": [ 11 | "# 論理式の証明" 12 | ] 13 | }, 14 | { 15 | "cell_type": "markdown", 16 | "metadata": { 17 | "coq_kernel_metadata": { 18 | "auto_roll_back": true 19 | } 20 | }, 21 | "source": [ 22 | "## 演習問題\n", 23 | "#### 演習問題 1-1 \n", 24 | "\n", 25 | "次の定理を証明せよ。\n", 26 | " Theorem my_first_proof : (forall A : Prop, A -> A).\n", 27 | "
\n", 28 | "[作業用ページで証明をして見る](./samples/work/ex01.ipynb)　/ [ヒントを見る](./samples/hints/ex01.ipynb ) / [解答を見る](./samples/answers/ex01.ipynb) / [演習問題に戻る](../../Logic-Ex.ipynb ) " 29 | ] 30 | }, 31 | { 32 | "cell_type": "code", 33 | "execution_count": null, 34 | "metadata": { 35 | "collapsed": true, 36 | "coq_kernel_metadata": { 37 | "auto_roll_back": true 38 | } 39 | }, 40 | "outputs": [], 41 | "source": [] 42 | } 43 | ], 44 | "metadata": { 45 | "kernelspec": { 46 | "display_name": "Coq", 47 | "language": "coq", 48 | "name": "coq" 49 | }, 50 | "language_info": { 51 | "file_extension": ".v", 52 | "mimetype": "text/x-coq", 53 | "name": "coq", 54 | "version": "8.9.1" 55 | } 56 | }, 57 | "nbformat": 4, 58 | "nbformat_minor": 2 59 | } 60 | -------------------------------------------------------------------------------- /.ipynb_checkpoints/1−Logic-checkpoint.ipynb: -------------------------------------------------------------------------------- 1 | { 2 | "cells": [ 3 | { 4 | "cell_type": "markdown", 5 | "metadata": { 6 | "coq_kernel_metadata": { 7 | "auto_roll_back": true 8 | } 9 | }, 10 | "source": [ 11 | "# 論理式の証明" 12 | ] 13 | }, 14 | { 15 | "cell_type": "code", 16 | "execution_count": null, 17 | "metadata": { 18 | "collapsed": true, 19 | "coq_kernel_metadata": { 20 | "auto_roll_back": true 21 | } 22 | }, 23 | "outputs": [], 24 | "source": [] 25 | }, 26 | { 27 | "cell_type": "markdown", 28 | "metadata": { 29 | "coq_kernel_metadata": { 30 | "auto_roll_back": true 31 | } 32 | }, 33 | "source": [ 34 | "## 演習問題\n", 35 | "\n", 36 | "#### 演習問題 1-1 \n", 37 | "次の定理を証明せよ。\n", 38 | " Theorem my_first_proof : (forall A : Prop, A -> A).\n", 39 | "
\n", 40 | "[作業用ページで証明をして見る](./samples/work/ex01.ipynb)　/ [ヒントを見る](./samples/hints/ex01.ipynb ) / [解答を見る](./samples/answers/ex01.ipynb) " 41 | ] 42 | }, 43 | { 44 | "cell_type": "code", 45 | "execution_count": null, 46 | "metadata": { 47 | "collapsed": true, 48 | "coq_kernel_metadata": { 49 | "auto_roll_back": true 50 | } 51 | }, 52 | "outputs": [], 53 | "source": [] 54 | } 55 | ], 56 | "metadata": { 57 | "kernelspec": { 58 | "display_name": "Coq", 59 | "language": "coq", 60 | "name": "coq" 61 | }, 62 | "language_info": { 63 | "file_extension": ".v", 64 | "mimetype": "text/x-coq", 65 | "name": "coq", 66 | "version": "8.9.1" 67 | } 68 | }, 69 | "nbformat": 4, 70 | "nbformat_minor": 2 71 | } 72 | -------------------------------------------------------------------------------- /.ipynb_checkpoints/Equal−Ex-checkpoint.ipynb: -------------------------------------------------------------------------------- 1 | { 2 | "cells": [ 3 | { 4 | "cell_type": "markdown", 5 | "metadata": { 6 | "coq_kernel_metadata": { 7 | "auto_roll_back": true 8 | } 9 | }, 10 | "source": [ 11 | "# 等式の証明 演習問題" 12 | ] 13 | }, 14 | { 15 | "cell_type": "markdown", 16 | "metadata": { 17 | "coq_kernel_metadata": { 18 | "auto_roll_back": true 19 | } 20 | }, 21 | "source": [ 22 | "## 演習問題 2-1\n", 23 | "　次の定理を証明せよ。\n", 24 | " Theorem plus_O_n : (forall n : nat , 0+n = n ).\n", 25 | "
\n", 26 | "[作業用ページで証明をして見る](./samples/work/ex10.ipynb)　/ [ヒントを見る](./samples/hints/ex10.ipynb ) / [解答を見る](./samples/answers/ex10.ipynb) " 27 | ] 28 | }, 29 | { 30 | "cell_type": "markdown", 31 | "metadata": { 32 | "coq_kernel_metadata": { 33 | "auto_roll_back": true 34 | } 35 | }, 36 | "source": [ 37 | "## 演習問題 2-2\n", 38 | "　次の定理を証明せよ。\n", 39 | " Theorem plus_1_n : (forall n : nat , 1+n = S n ).\n", 40 | "
\n", 41 | "[作業用ページで証明をして見る](./samples/work/ex11.ipynb)　/ [ヒントを見る](./samples/hints/ex11.ipynb ) / [解答を見る](./samples/answers/ex11.ipynb) " 42 | ] 43 | }, 44 | { 45 | "cell_type": "markdown", 46 | "metadata": { 47 | "coq_kernel_metadata": { 48 | "auto_roll_back": true 49 | } 50 | }, 51 | "source": [ 52 | "## 演習問題 2-3\n", 53 | "　次の定理を証明せよ。\n", 54 | " Theorem mult_0_n : (forall n : nat , 0 * n = 0 ).\n", 55 | "
\n", 56 | "[作業用ページで証明をして見る](./samples/work/ex12.ipynb)　/ [ヒントを見る](./samples/hints/ex12.ipynb ) / [解答を見る](./samples/answers/ex12.ipynb) " 57 | ] 58 | }, 59 | { 60 | "cell_type": "markdown", 61 | "metadata": { 62 | "coq_kernel_metadata": { 63 | "auto_roll_back": true 64 | } 65 | }, 66 | "source": [ 67 | "## 演習問題 2-4\n", 68 | "　次の定理を証明せよ。\n", 69 | " Theorem plus_id_nm : (forall n m : nat , n = m -> n+n = m + m ).\n", 70 | "
\n", 71 | "[作業用ページで証明をして見る](./samples/work/ex13.ipynb)　/ [ヒントを見る](./samples/hints/ex13.ipynb ) / [解答を見る](./samples/answers/ex13.ipynb) " 72 | ] 73 | }, 74 | { 75 | "cell_type": "markdown", 76 | "metadata": { 77 | "coq_kernel_metadata": { 78 | "auto_roll_back": true 79 | } 80 | }, 81 | "source": [ 82 | "## 演習問題 2-5\n", 83 | "　次の定理を証明せよ。\n", 84 | " Theorem plus_id : (forall n m o : nat , n = m -> m = o -> n + m = m + o ).\n", 85 | "
\n", 86 | "[作業用ページで証明をして見る](./samples/work/ex14.ipynb)　/ [ヒントを見る](./samples/hints/ex14.ipynb ) / [解答を見る](./samples/answers/ex14.ipynb) " 87 | ] 88 | }, 89 | { 90 | "cell_type": "markdown", 91 | "metadata": { 92 | "coq_kernel_metadata": { 93 | "auto_roll_back": true 94 | } 95 | }, 96 | "source": [ 97 | "## 演習問題 2-6\n", 98 | "　次の定理を証明せよ。\n", 99 | " Theorem mult_0_plus : (forall n m : nat , ( 0 + n ) \\* m = n \\* m ).\n", 100 | "
\n", 101 | "[作業用ページで証明をして見る](./samples/work/ex15.ipynb)　/ [ヒントを見る](./samples/hints/ex15.ipynb ) / [解答を見る](./samples/answers/ex15.ipynb) " 102 | ] 103 | }, 104 | { 105 | "cell_type": "markdown", 106 | "metadata": { 107 | "coq_kernel_metadata": { 108 | "auto_roll_back": true 109 | } 110 | }, 111 | "source": [ 112 | "## 演習問題 2-7\n", 113 | "　次の定理を証明せよ。\n", 114 | " Theorem mult_S_1 : (forall n m : nat, m = S n -> m \\* (1 + n) = m \\* m).\n", 115 | "
\n", 116 | "[作業用ページで証明をして見る](./samples/work/ex16.ipynb)　/ [ヒントを見る](./samples/hints/ex16.ipynb ) / [解答を見る](./samples/answers/ex16.ipynb) " 117 | ] 118 | } 119 | ], 120 | "metadata": { 121 | "kernelspec": { 122 | "display_name": "Coq", 123 | "language": "coq", 124 | "name": "coq" 125 | }, 126 | "language_info": { 127 | "file_extension": ".v", 128 | "mimetype": "text/x-coq", 129 | "name": "coq", 130 | "version": "8.9.1" 131 | } 132 | }, 133 | "nbformat": 4, 134 | "nbformat_minor": 2 135 | } 136 | -------------------------------------------------------------------------------- /.ipynb_checkpoints/Equal-Ex-checkpoint.ipynb: -------------------------------------------------------------------------------- 1 | { 2 | "cells": [ 3 | { 4 | "cell_type": "markdown", 5 | "metadata": { 6 | "coq_kernel_metadata": { 7 | "auto_roll_back": true 8 | } 9 | }, 10 | "source": [ 11 | "# 演習問題 2\n", 12 | "\n", 13 | "[演習問題 2-1](./Equal-Ex.ipynb#ex21) [演習問題 2-2](./Equal-Ex.ipynb#ex22) [演習問題 2-3](./Equal-Ex.ipynb#ex23) [演習問題 2-4](./Equal-Ex.ipynb#ex24) [演習問題 2-5](./Equal-Ex.ipynb#ex25) [演習問題 2-6](./Equal-Ex.ipynb#ex26) [演習問題 2-7](./Equal-Ex.ipynb#ex27) " 14 | ] 15 | }, 16 | { 17 | "cell_type": "markdown", 18 | "metadata": { 19 | "coq_kernel_metadata": { 20 | "auto_roll_back": true 21 | } 22 | }, 23 | "source": [ 24 | "\n", 25 | "## 演習問題 2-1\n", 26 | "　次の定理を証明せよ。\n", 27 | "
　Theorem plus_O_n : (forall n : nat , 0+n = n ).\n", 28 | "\n", 29 | "#### [作業用ページで証明をしてみる](./Equal-Ex-Work.ipynb#ex21) / [解答を見る](./Equal-Ex-Answer.ipynb#ex21) \n", 30 | "### ヒント\n", 31 | "- 冒頭の forall を intros で削除する。\n", 32 | "- A -> A の最初のAを intros で仮説に入れる。\n", 33 | "- そうすると、サブゴールが仮説の一つと一致していることがわかるので、assumption または、exact を使う。" 34 | ] 35 | }, 36 | { 37 | "cell_type": "markdown", 38 | "metadata": { 39 | "coq_kernel_metadata": { 40 | "auto_roll_back": true 41 | } 42 | }, 43 | "source": [ 44 | "\n", 45 | "## 演習問題 2-2\n", 46 | "　次の定理を証明せよ。\n", 47 | "
Theorem plus_1_n : (forall n : nat , 1+n = S n ).\n", 48 | "\n", 49 | "#### [作業用ページで証明をしてみる](./Equal-Ex-Work.ipynb#ex22) / [解答を見る](./Equal-Ex-Answer.ipynb#ex22) \n", 50 | "### ヒント\n", 51 | "- intros で forall を削除する。\n", 52 | "- simpl. で、サブゴールの計算式を簡単にする。\n", 53 | "- 左項と右項が等しかったら reflexivity を使う。" 54 | ] 55 | }, 56 | { 57 | "cell_type": "markdown", 58 | "metadata": { 59 | "coq_kernel_metadata": { 60 | "auto_roll_back": true 61 | } 62 | }, 63 | "source": [ 64 | "\n", 65 | "## 演習問題 2-3\n", 66 | "　次の定理を証明せよ。\n", 67 | "
Theorem mult_0_n : (forall n : nat , 0 * n = 0 ).\n", 68 | "\n", 69 | "#### [作業用ページで証明をしてみる](./Equal-Ex-Work.ipynb#ex23) / [解答を見る](./Equal-Ex-Answer.ipynb#ex23) \n", 70 | "### ヒント\n", 71 | "- intros で forall を削除する。\n", 72 | "- simpl. で、サブゴールの計算式を簡単にする。\n", 73 | "- 左項と右項が等しかったら reflexivity を使う。" 74 | ] 75 | }, 76 | { 77 | "cell_type": "markdown", 78 | "metadata": { 79 | "coq_kernel_metadata": { 80 | "auto_roll_back": true 81 | } 82 | }, 83 | "source": [ 84 | "\n", 85 | "## 演習問題 2-4\n", 86 | "　次の定理を証明せよ。\n", 87 | "
Theorem plus_id_nm : (forall n m : nat , n = m -> n+n = m + m ).\n", 88 | "\n", 89 | "#### [作業用ページで証明をしてみる](./Equal-Ex-Work.ipynb#ex24) / [解答を見る](./Equal-Ex-Answer.ipynb#ex24) \n", 90 | "### ヒント\n", 91 | "- intros で forall を消す。\n", 92 | "- サブゴールの前提部分を intros で仮定に移す。\n", 93 | "- サブゴールの左項を、等式Hを使って rewriite で書き換える。\n", 94 | "- サブゴールの左項と右項が等しいので、reflexivity を使う。" 95 | ] 96 | }, 97 | { 98 | "cell_type": "markdown", 99 | "metadata": { 100 | "coq_kernel_metadata": { 101 | "auto_roll_back": true 102 | } 103 | }, 104 | "source": [ 105 | "\n", 106 | "## 演習問題 2-5\n", 107 | "　次の定理を証明せよ。\n", 108 | "
Theorem plus_id : (forall n m o : nat , n = m -> m = o -> n + m = m + o ).\n", 109 | "\n", 110 | "#### [作業用ページで証明をしてみる](./Equal-Ex-Work.ipynb#ex25) / [解答を見る](./Equal-Ex-Answer.ipynb#ex25) \n", 111 | "### ヒント\n", 112 | "- intros で forall を削除 \n", 113 | "- intoros で、前提部分を仮説に移す。\n", 114 | "- 等式　n_eq_m　の左辺 を使って、rewriteでサブゴールを書き換える。\n", 115 | "- 等式　m_eq_o の右辺を使って、rewriteでサブゴールのを書き換える。\n", 116 | "- サブゴールの等式の左辺と右辺が等しくなるので reflexivity を使う。" 117 | ] 118 | }, 119 | { 120 | "cell_type": "markdown", 121 | "metadata": { 122 | "coq_kernel_metadata": { 123 | "auto_roll_back": true 124 | } 125 | }, 126 | "source": [ 127 | "\n", 128 | "## 演習問題 2-6\n", 129 | "　次の定理を証明せよ。\n", 130 | "
Theorem mult_0_plus : (forall n m : nat , ( 0 + n ) \\* m = n \\* m ).\n", 131 | "\n", 132 | "#### [作業用ページで証明をしてみる](./Equal-Ex-Work.ipynb#ex26) / [解答を見る](./Equal-Ex-Answer.ipynb#ex26) \n", 133 | "### ヒント\n", 134 | "- intros で forall を消す。\n", 135 | "- simpl.　で、サブゴール中の計算式を簡単なものにする。\n", 136 | "- 左辺と右辺が等しいので、reflexivity を使う。" 137 | ] 138 | }, 139 | { 140 | "cell_type": "markdown", 141 | "metadata": { 142 | "coq_kernel_metadata": { 143 | "auto_roll_back": true 144 | } 145 | }, 146 | "source": [ 147 | "\n", 148 | "## 演習問題 2-7\n", 149 | "　次の定理を証明せよ。\n", 150 | "
Theorem mult_S_1 : (forall n m : nat, m = S n -> m \\* (1 + n) = m \\* m).\n", 151 | "\n", 152 | "#### [作業用ページで証明をしてみる](./Equal-Ex-Work.ipynb#ex27) / [解答を見る](./Equal-Ex-Answer.ipynb#ex27) \n", 153 | "### ヒント\n", 154 | "- intros で　forall を消し、前提を仮説に移す。\n", 155 | "- サブゴールの計算式を simpl で簡単にする。\n", 156 | "- rewriteで、サブゴールを m_eq_Sn を使って書き換える。\n", 157 | "- サブゴールの左辺と右辺は等しい。" 158 | ] 159 | }, 160 | { 161 | "cell_type": "code", 162 | "execution_count": null, 163 | "metadata": { 164 | "coq_kernel_metadata": { 165 | "auto_roll_back": true 166 | } 167 | }, 168 | "outputs": [], 169 | "source": [] 170 | } 171 | ], 172 | "metadata": { 173 | "kernelspec": { 174 | "display_name": "Coq", 175 | "language": "coq", 176 | "name": "coq" 177 | }, 178 | "language_info": { 179 | "file_extension": ".v", 180 | "mimetype": "text/x-coq", 181 | "name": "coq", 182 | "version": "8.9.1" 183 | } 184 | }, 185 | "nbformat": 4, 186 | "nbformat_minor": 2 187 | } 188 | -------------------------------------------------------------------------------- /.ipynb_checkpoints/Induction-Ex-checkpoint.ipynb: -------------------------------------------------------------------------------- 1 | { 2 | "cells": [ 3 | { 4 | "cell_type": "markdown", 5 | "metadata": { 6 | "coq_kernel_metadata": { 7 | "auto_roll_back": true 8 | } 9 | }, 10 | "source": [ 11 | "# 演習問題 3\n", 12 | "\n", 13 | "[演習問題 3-1](./Induction-Ex.ipynb#ex11) [演習問題 3-2](./Induction-Ex.ipynb#ex12) [演習問題 3-3](./Induction-Ex.ipynb#ex13) [演習問題 3-4](./Induction-Ex.ipynb#ex14) [演習問題 3-5](./Induction-Ex.ipynb#ex15) [演習問題 3-6](./Induction-Ex.ipynb#ex16) [演習問題 3-7](./Induction-Ex.ipynb#ex17) [演習問題 3-8](./Induction-Ex.ipynb#ex18) " 14 | ] 15 | }, 16 | { 17 | "cell_type": "markdown", 18 | "metadata": { 19 | "coq_kernel_metadata": { 20 | "auto_roll_back": true 21 | } 22 | }, 23 | "source": [ 24 | "\n", 25 | "## 演習問題 3-1\n", 26 | "　次の定理を証明せよ。\n", 27 | "
　Theorem plus_n_O : (forall n : nat, n + 0 = n).\n", 28 | "\n", 29 | "[作業用ページで証明をしてみる](./Induction-Ex-Work.ipynb#ex31)\n", 30 | "\n", 31 | "### ヒント\n", 32 | "- intros で　forall の削除 \n", 33 | "- tactic induction n を使って、nについての帰納法を利用する。\n", 34 | "- サブゴールは、n=0の場合とそうでない場合の、二つに分かれる。\n", 35 | "- 一つ目のサブゴールについて。simplで簡単な形にする。\n", 36 | "- 一つ目のサブゴールについて。reflexivityを使う。\n", 37 | "- 一つ目のゴールは消える。\n", 38 | "- IHn の左項を使って、サブゴールを rewrite で書き換える。\n", 39 | "- 左項と右項が等しいので、reflexivityを使う。\n", 40 | "\n", 41 | "### [解答を見る](./Induction-Ex-Answer.ipynb#ex31) " 42 | ] 43 | }, 44 | { 45 | "cell_type": "markdown", 46 | "metadata": { 47 | "coq_kernel_metadata": { 48 | "auto_roll_back": true 49 | } 50 | }, 51 | "source": [ 52 | "\n", 53 | "## 演習問題 3-2\n", 54 | "　次の定理を証明せよ。\n", 55 | "
　Theorem minus_diag : (forall n : nat, minus n n = 0).\n", 56 | "\n", 57 | "[作業用ページで証明をしてみる](./Induction-Ex-Work.ipynb#ex32)\n", 58 | "\n", 59 | "### ヒント\n", 60 | "- ntros でforall を消す。\n", 61 | "- induction n で、nについての minus の帰納法の定義を使う。\n", 62 | "- 最初のサブゴールを simpl で簡単な形にする。\n", 63 | "- reflexivityで、最初のサブゴールは消える。\n", 64 | "- simpl で残りのサブゴールを簡単な形にする。\n", 65 | "- 仮定の一つが、サブゴールと一致している。\n", 66 | "\n", 67 | "### [解答を見る](./Induction-Ex-Answer.ipynb#ex32) " 68 | ] 69 | }, 70 | { 71 | "cell_type": "markdown", 72 | "metadata": { 73 | "coq_kernel_metadata": { 74 | "auto_roll_back": true 75 | } 76 | }, 77 | "source": [ 78 | "\n", 79 | "## 演習問題 3-3\n", 80 | "　次の定理を証明せよ。\n", 81 | "
　Theorem mult_0_r : (forall n : nat, n * 0 = 0).\n", 82 | "\n", 83 | "[作業用ページで証明をしてみる](./Induction-Ex-Work.ipynb#ex33)\n", 84 | "\n", 85 | "### ヒント\n", 86 | "\n", 87 | "\n", 88 | "### [解答を見る](./Induction-Ex-Answer.ipynb#ex33) " 89 | ] 90 | }, 91 | { 92 | "cell_type": "markdown", 93 | "metadata": { 94 | "coq_kernel_metadata": { 95 | "auto_roll_back": true 96 | } 97 | }, 98 | "source": [ 99 | "\n", 100 | "## 演習問題 3-4\n", 101 | "　次の定理を証明せよ。\n", 102 | "
　Theorem plus_n_Sm : (forall n m : nat, S (n + m) = n + (S m) ).\n", 103 | "\n", 104 | "[作業用ページで証明をしてみる](./Induction-Ex-Work.ipynb#ex34)\n", 105 | "\n", 106 | "### ヒント\n", 107 | "\n", 108 | "\n", 109 | "### [解答を見る](./Induction-Ex-Answer.ipynb#ex34) " 110 | ] 111 | }, 112 | { 113 | "cell_type": "markdown", 114 | "metadata": { 115 | "coq_kernel_metadata": { 116 | "auto_roll_back": true 117 | } 118 | }, 119 | "source": [ 120 | "\n", 121 | "## 演習問題 3-5\n", 122 | "　次の定理を証明せよ。\n", 123 | "
　Theorem plus_comm : (forall n m : nat, n + m = m + n ).\n", 124 | "\n", 125 | "[作業用ページで証明をしてみる](./Induction-Ex-Work.ipynb#ex35)\n", 126 | "\n", 127 | "### ヒント\n", 128 | "\n", 129 | "\n", 130 | "### [解答を見る](./Induction-Ex-Answer.ipynb#ex35) " 131 | ] 132 | }, 133 | { 134 | "cell_type": "markdown", 135 | "metadata": { 136 | "coq_kernel_metadata": { 137 | "auto_roll_back": true 138 | } 139 | }, 140 | "source": [ 141 | "\n", 142 | "## 演習問題 3-6\n", 143 | "　次の定理を証明せよ。\n", 144 | "
　Theorem plus_assoc : (forall n m o : nat, n + (m +o) = (n + m) + o ).\n", 145 | "\n", 146 | "[作業用ページで証明をしてみる](./Induction-Ex-Work.ipynb#ex36)\n", 147 | "\n", 148 | "### ヒント\n", 149 | "\n", 150 | "\n", 151 | "### [解答を見る](./Induction-Ex-Answer.ipynb#ex36) " 152 | ] 153 | }, 154 | { 155 | "cell_type": "markdown", 156 | "metadata": { 157 | "coq_kernel_metadata": { 158 | "auto_roll_back": true 159 | } 160 | }, 161 | "source": [ 162 | "\n", 163 | "## 演習問題 3-7\n", 164 | "　次の定理を証明せよ。\n", 165 | "
　Theorem mul_succ_r : forall n m : nat, n \\* S m = n \\* m + n ).\n", 166 | "\n", 167 | "[作業用ページで証明をしてみる](./Induction-Ex-Work.ipynb#ex37)\n", 168 | "\n", 169 | "### ヒント\n", 170 | "\n", 171 | "\n", 172 | "### [解答を見る](./Induction-Ex-Answer.ipynb#ex37) " 173 | ] 174 | }, 175 | { 176 | "cell_type": "markdown", 177 | "metadata": { 178 | "coq_kernel_metadata": { 179 | "auto_roll_back": true 180 | } 181 | }, 182 | "source": [ 183 | "\n", 184 | "## 演習問題 3-8\n", 185 | "　次の定理を証明せよ。\n", 186 | "
　Theorem mult_comm : (forall n m : nat, n \\* m = m \\* n ).\n", 187 | "\n", 188 | "[作業用ページで証明をしてみる](./Induction-Ex-Work.ipynb#ex38)\n", 189 | "\n", 190 | "### ヒント\n", 191 | "\n", 192 | "\n", 193 | "### [解答を見る](./Induction-Ex-Answer.ipynb#ex38) " 194 | ] 195 | } 196 | ], 197 | "metadata": { 198 | "kernelspec": { 199 | "display_name": "Coq", 200 | "language": "coq", 201 | "name": "coq" 202 | }, 203 | "language_info": { 204 | "file_extension": ".v", 205 | "mimetype": "text/x-coq", 206 | "name": "coq", 207 | "version": "8.9.1" 208 | } 209 | }, 210 | "nbformat": 4, 211 | "nbformat_minor": 2 212 | } 213 | -------------------------------------------------------------------------------- /.ipynb_checkpoints/Logic-Ex-checkpoint.ipynb: -------------------------------------------------------------------------------- 1 | { 2 | "cells": [ 3 | { 4 | "cell_type": "markdown", 5 | "metadata": { 6 | "coq_kernel_metadata": { 7 | "auto_roll_back": true 8 | } 9 | }, 10 | "source": [ 11 | "# 演習問題 1\n", 12 | "\n", 13 | "[演習問題 1-1](./Logic-Ex.ipynb#ex11) [演習問題 1-2](./Logic-Ex.ipynb#ex12) [演習問題 1-3](./Logic-Ex.ipynb#ex13) [演習問題 1-4](./Logic-Ex.ipynb#ex14) [演習問題 1-5](./Logic-Ex.ipynb#ex15) [演習問題 1-6](./Logic-Ex.ipynb#ex16) [演習問題 1-7](./Logic-Ex.ipynb#ex17) [演習問題 1-8](./Logic-Ex.ipynb#ex18) [演習問題 1-9](./Logic-Ex.ipynb#ex19) " 14 | ] 15 | }, 16 | { 17 | "cell_type": "markdown", 18 | "metadata": { 19 | "coq_kernel_metadata": { 20 | "auto_roll_back": true 21 | } 22 | }, 23 | "source": [ 24 | "\n", 25 | "## 演習問題 1-1 \n", 26 | "次の定理を証明せよ。\n", 27 | "
Theorem my_first_proof : (forall A : Prop, A -> A).\n", 28 | "\n", 29 | "#### [作業用ページで証明をしてみる](./Logic-Ex-Work.ipynb#ex11) / [解答を見る](./Logic-Ex-Answer.ipynb#ex11) \n", 30 | "### ヒント\n", 31 | "- 冒頭の forall を intros で削除する。\n", 32 | "- A -> A の最初のAを intros で仮説に入れる。\n", 33 | "- そうすると、サブゴールが仮説の一つと一致していることがわかるので、assumption または、exact を使う。" 34 | ] 35 | }, 36 | { 37 | "cell_type": "markdown", 38 | "metadata": { 39 | "collapsed": true, 40 | "coq_kernel_metadata": { 41 | "auto_roll_back": true 42 | } 43 | }, 44 | "source": [ 45 | "\n", 46 | "## 演習問題 1-2\n", 47 | "次の定理を証明せよ。\n", 48 | "
Theorem and1 : (forall A B: Prop, A /\\ B -> A).\n", 49 | "\n", 50 | "#### [作業用ページで証明をしてみる](./Logic-Ex-Work.ipynb#ex12) / [解答を見る](./Logic-Ex-Answer.ipynb#ex12) \n", 51 | "### ヒント\n", 52 | "- intros. で forall を削除し、`A /\\ B -> A` の前提部分の`A /\\ B`を仮説に移す。\n", 53 | "- 仮説中の`A /\\ B`を、destruct で分解する。\n", 54 | "- 仮説の一つがサブゴールに一致している。" 55 | ] 56 | }, 57 | { 58 | "cell_type": "markdown", 59 | "metadata": { 60 | "coq_kernel_metadata": { 61 | "auto_roll_back": true 62 | } 63 | }, 64 | "source": [ 65 | "\n", 66 | "## 演習問題 1-3\n", 67 | "次の定理を証明せよ。\n", 68 | "
Theorem and1 : (forall A B: Prop, A /\\ B -> B).\n", 69 | "\n", 70 | "#### [作業用ページで証明をしてみる](./Logic-Ex-Work.ipynb#ex13) / [解答を見る](./Logic-Ex-Answer.ipynb#ex13) \n", 71 | "### ヒント\n", 72 | "- intros. で forall を削除し、`A /\\ B -> B` の前提部分の`A /\\ B`を仮説に移す。\n", 73 | "- 仮説中の`A /\\ B`を、destruct で分解する。\n", 74 | "- 仮説の一つがサブゴールに一致している。" 75 | ] 76 | }, 77 | { 78 | "cell_type": "markdown", 79 | "metadata": { 80 | "coq_kernel_metadata": { 81 | "auto_roll_back": true 82 | } 83 | }, 84 | "source": [ 85 | "\n", 86 | "## 演習問題 1-4\n", 87 | "次の定理を証明せよ。\n", 88 | "
Theorem or1 : (forall A B: Prop, A -> A \\\\/ B).\n", 89 | "\n", 90 | "#### [作業用ページで証明をしてみる](./Logic-Ex-Work.ipynb#ex14) / [解答を見る](./Logic-Ex-Answer.ipynb#ex14) \n", 91 | "### ヒント\n", 92 | "- intros. でforallを消す。\n", 93 | "- left. でAを仮説に取り込む。\n", 94 | "- サブゴールと仮説が一致している。" 95 | ] 96 | }, 97 | { 98 | "cell_type": "markdown", 99 | "metadata": { 100 | "coq_kernel_metadata": { 101 | "auto_roll_back": true 102 | } 103 | }, 104 | "source": [ 105 | "\n", 106 | "## 演習問題 1-5\n", 107 | "次の定理を証明せよ。\n", 108 | "
Theorem or2 : (forall A B: Prop, B -> A \\\\/ B).\n", 109 | "\n", 110 | "#### [作業用ページで証明をしてみる](./Logic-Ex-Work.ipynb#ex15) / [解答を見る](./Logic-Ex-Answer.ipynb#ex15) \n", 111 | "### ヒント\n", 112 | "- intros. でforallを消す。\n", 113 | "- right. でBを仮説に取り込む。\n", 114 | "- サブゴールと仮説が一致している。" 115 | ] 116 | }, 117 | { 118 | "cell_type": "markdown", 119 | "metadata": { 120 | "coq_kernel_metadata": { 121 | "auto_roll_back": true 122 | } 123 | }, 124 | "source": [ 125 | "\n", 126 | "## 演習問題 1-6\n", 127 | "次の定理を証明せよ。\n", 128 | "
Theorem modus_ponens : (forall A B: Prop, A -> (A -> B) ->B).\n", 129 | "\n", 130 | "#### [作業用ページで証明をしてみる](./Logic-Ex-Work.ipynb#ex16) / [解答を見る](./Logic-Ex-Answer.ipynb#ex16) \n", 131 | "### ヒント\n", 132 | "- intros. で先頭のforall を削除する。\n", 133 | "- 再び intros で、AとA -> Bを仮説に移す。\n", 134 | "- 仮説A -> Bを仮説（この場合はA）に適用して、サブゴールをBに変える。applyを使う。\n", 135 | "- 仮説とサブゴールが一致する。" 136 | ] 137 | }, 138 | { 139 | "cell_type": "markdown", 140 | "metadata": { 141 | "coq_kernel_metadata": { 142 | "auto_roll_back": true 143 | } 144 | }, 145 | "source": [ 146 | "\n", 147 | "## 演習問題 1-7\n", 148 | "次の定理を証明せよ。\n", 149 | "
Theorem modus_ponens2 : (forall A B C: Prop, A -> (A -> B) ->(B ->C) -> C).\n", 150 | "\n", 151 | "#### [作業用ページで証明をしてみる](./Logic-Ex-Work.ipynb#ex17) / [解答を見る](./Logic-Ex-Answer.ipynb#ex17) \n", 152 | "### ヒント\n", 153 | "- intros でforallを削除する。\n", 154 | "- intros で、全ての -> の前提部分を仮説に移す。この時、適当なラベルをつけておくと分かりやすくなる。この例の場合なら、intros Proof_of_A A_implies_B B_implies_C とか。\n", 155 | "- apply B_implies_C で、サブゴールをBに変える。\n", 156 | "- apply A_implies_B で、サブゴールをAに変える。\n", 157 | "- 仮説とサブゴールが一致している。" 158 | ] 159 | }, 160 | { 161 | "cell_type": "markdown", 162 | "metadata": { 163 | "coq_kernel_metadata": { 164 | "auto_roll_back": true 165 | } 166 | }, 167 | "source": [ 168 | "\n", 169 | "## 演習問題 1-8\n", 170 | "次の定理を証明せよ。\n", 171 | "
Theorem and_comm : (forall A B: Prop, A /\\ B -> B /\\ A ).\n", 172 | "\n", 173 | "#### [作業用ページで証明をしてみる](./Logic-Ex-Work.ipynb#ex18) / [解答を見る](./Logic-Ex-Answer.ipynb#ex18) \n", 174 | "### ヒント\n", 175 | "- intros で、forall を削除し、\"A /\\ B\" にH というラベルをつける 。\n", 176 | "- \"H: A/\\ B\" を destruct で分解し、A, B を仮説に追加する。　\n", 177 | "- サブゴール \"A\\/ B\" を split で、二つのサブゴールに分割する。　\n", 178 | "- 一つ目のサブゴールは仮説の一つと同じなので解ける。サブゴールは一つになる。\n", 179 | "- 残ったサブゴールも仮説の一つと一緒なので、解ける。\n", 180 | "- サブゴールは残っていないので証明終了。" 181 | ] 182 | }, 183 | { 184 | "cell_type": "markdown", 185 | "metadata": { 186 | "coq_kernel_metadata": { 187 | "auto_roll_back": true 188 | } 189 | }, 190 | "source": [ 191 | "\n", 192 | "## 演習問題 1-9\n", 193 | "次の定理を証明せよ。\n", 194 | "
Theorem or_comm : (forall A B: Prop, A \\\\/ B -> B \\\\/ A ) .\n", 195 | "\n", 196 | "#### [作業用ページで証明をしてみる](./Logic-Ex-Work.ipynb#ex19) / [解答を見る](./Logic-Ex-Answer.ipynb#ex19) \n", 197 | "### ヒント\n", 198 | "- intros で\"A \\/ B\" を仮定に移し、A_or_B というラベルをつける。\n", 199 | "- A\\/B をdestructすると、Aが仮定に残り、サブゴールが二つに分割される。\n", 200 | "- 一つ目のサブゴールから、right でB\\/A の右の項Aをサブゴールに残す。 \n", 201 | "- サブゴールと仮定が一致している。 サブゴールは一つになる。\n", 202 | "- 一つ目のサブゴールから、left でB\\/A の左の項Bをサブゴールに残す。 " 203 | ] 204 | } 205 | ], 206 | "metadata": { 207 | "kernelspec": { 208 | "display_name": "Coq", 209 | "language": "coq", 210 | "name": "coq" 211 | }, 212 | "language_info": { 213 | "file_extension": ".v", 214 | "mimetype": "text/x-coq", 215 | "name": "coq", 216 | "version": "8.9.1" 217 | } 218 | }, 219 | "nbformat": 4, 220 | "nbformat_minor": 2 221 | } 222 | -------------------------------------------------------------------------------- /.ipynb_checkpoints/1_basic-checkpoint.ipynb: -------------------------------------------------------------------------------- 1 | { 2 | "cells": [ 3 | { 4 | "cell_type": "markdown", 5 | "metadata": {}, 6 | "source": [ 7 | "\"Note: Trusted Notebook\" width=\"700 px\" align=\"center\">" 8 | ] 9 | }, 10 | { 11 | "cell_type": "markdown", 12 | "metadata": {}, 13 | "source": [ 14 | "# 第一章　量子情報理論の三つの基本原理 \n", 15 | "\n", 16 | "**本章の内容**　　　　　[全体目次](./Contents.ipynb)\n", 17 | "\n", 18 | "- [量子情報理論の三つの基本原理](#postulate )\n", 19 | " - [重ね合わせの原理](#superposition )\n", 20 | " - [観測の原理](#measurement)\n", 21 | " - [ユニタリ発展の原理](#unitary)\n", 22 | " \n", 23 | "- [数学的準備１ ](#math1)\n", 24 | "\n", 25 | "\n", 26 | "### 量子情報理論には、次の三つの基本原理がある。\n", 27 | "\n", 28 | "1. #### 重ね合わせの原理\n", 29 | "1. #### 観測の原理\n", 30 | "1. #### ユニタリ発展の原理\n", 31 | "\n", 32 | "\n", 33 | "## 1. 重ね合わせの原理 \n", 34 | "### qubitの状態は、互いに直交する二つのベクトルの和として、列ベクトルで表される。\n", 35 | "\n", 36 | "### x-y平面との類似で考える\n", 37 | "- それは、x-y平面上の任意の点を表すベクトルが、x軸方向のベクトルとy軸方向のベクトルの和として表されるのと同じである。\n", 38 | "- x-y平面で、二つのx軸方向の長さ1のベクトル(1,0) をx、y軸方向の長さ1のベクトル(0,1)をyとすると、原点から点P(m,n)に向かうベクトルPは、P=mx+ny=m(1,0)+n(0,1) と表すことができる。\n", 39 | "- この時、mをPの「x成分」、nをPの「y成分」という。\n", 40 | "- また、二つの単位ベクトル x(1,0), y(0,1)を、**「基底」ベクトル**と呼ぶ。\n", 41 | "\"Note: Trusted Notebook\" width=\"550 px\" align=\"left\">\n", 42 | "　　" 43 | ] 44 | }, 45 | { 46 | "cell_type": "markdown", 47 | "metadata": {}, 48 | "source": [ 49 | "### qubitの場合の重ね合わせ\n", 50 | "- x-y平面の場合ベクトルの成分は実数だが、qubitの重ね合わせの場合には、その成分は **複素数** となる。qubitの状態は、二つの複素数のペアで定義される。\n", 51 | "- x-y平面の場合、基底 x(1,0), y(0,1)は、行ベクトルとして表現されるが、qubitの基底は、$\\begin{pmatrix} 1 \\\\ 0 \\\\\\end{pmatrix}$ と$\\begin{pmatrix} 0 \\\\ 1 \\\\\\end{pmatrix}$ と、**列ベクトル** で表現される。\n", 52 | "- x-y平面の成分(m,n)を持つ任意のベクトルが、二つの基底ベクトル x(1,0), y(0,1)を用いて、(m,n)=m(1,0)+n(0,1)と表すことができるように、qubitの成分を$\\begin{pmatrix} \\alpha \\\\ \\beta \\\\\\end{pmatrix}$とする時、$\\begin{pmatrix} \\alpha \\\\ \\beta \\\\\\end{pmatrix}= \\alpha \\begin{pmatrix} 1 \\\\ 0 \\\\\\end{pmatrix} + \\beta \\begin{pmatrix} 0 \\\\ 1 \\\\\\end{pmatrix}$と表すことができる。\n", 53 | "- ただし、qubitの成分 $\\alpha, \\beta$は、次の条件を満たす。$|\\alpha|^2+|\\beta|^2=1$\n", 54 | "\n", 55 | "### ket記法\n", 56 | "- Aが列ベクトルの時、それを$|A\\rangle$と表す。これを **ket記法** と呼ぶ。\n", 57 | "- qubitの基底である二つの列ベクトルを、$|0\\rangle, |1\\rangle$と表す。すなわち、$|0\\rangle=\\begin{pmatrix} 1 \\\\ 0 \\\\\\end{pmatrix}, |1\\rangle=\\begin{pmatrix} 0 \\\\ 1 \\\\\\end{pmatrix}$とする。\n", 58 | "- この時、任意のqubitの状態$|\\psi\\rangleは、|\\psi\\rangle=\\alpha |0\\rangle+\\beta|1\\rangle 　（ただし|\\alpha|^2+|\\beta|^2=1）$と表される。\n", 59 | "\n", 60 | "### 一般化\n", 61 | "- qubitの状態は、二つの複素数を成分とする列ベクトルで表現されるが、任意の量子状態$|\\psi\\rangle$は、n個の基底$|0\\rangle, |1\\rangle, |2\\rangle, \\cdots, |n-1\\rangle$を持つn個の複素数$\\alpha_0, \\alpha_1, \\alpha_{n-1}$を成分とする列ベクトルで表現される。$|\\psi\\rangle=\\alpha_0|0\\rangle+\\alpha_1|1\\rangle+\\alpha|2\\rangle+ \\cdots + \\alpha_{n-1}|n-1\\rangle$\n", 62 | "- この時、各成分の絶対値の二乗を全て加えた値は、１に等しくなければならない。$|\\alpha_0|^2+|\\alpha_1|^2+|\\alpha|_2|^2+\\cdots + |\\alpha_{n-1}|^2=1$\n", 63 | "\n", 64 | "\"Note: Trusted Notebook\" width=\"600 px\" align=\"left\">" 65 | ] 66 | }, 67 | { 68 | "cell_type": "markdown", 69 | "metadata": {}, 70 | "source": [ 71 | "## 2. 観測の原理 \n", 72 | "### qubitを観測すると、重ね合わせの状態は失われて、古典bit 0か1が観測される\n", 73 | "一般に、n個の基底からなる量子の状態を観測すると、重ね合わせの状態は失われて、その基底の一つが観測される。\n", 74 | "\n", 75 | "### qubit $|\\psi\\rangle=\\alpha |0\\rangle+\\beta|1\\rangle$を観測した時、$|0\\rangle$ (古典bit 0に対応)が観測される確率は$|\\alpha|^2$、$|1\\rangle$ (古典bit 1に対応)が観測される確率は$|\\beta|^2$に等しい\n", 76 | "一般に、先の量子の状態 $|\\psi\\rangle$を観測した時、重ね合わせの状態が失われて、基底$|i\\rangle$が観測される確率は、$|\\alpha_i|^2$に等しい。\n", 77 | "$|\\alpha_0|^2+|\\alpha_1|^2+|\\alpha|_2|^2+\\cdots + |\\alpha_{n-1}|^2=1$という条件は、観測によって、いずれかの基底が観測されるという条件と等しい。\n", 78 | "\n", 79 | "\"Note: Trusted Notebook\" width=\"550 px\" align=\"left\">\n", 80 | "\"Note: Trusted Notebook\" width=\"550 px\" align=\"left\">" 81 | ] 82 | }, 83 | { 84 | "cell_type": "markdown", 85 | "metadata": {}, 86 | "source": [ 87 | "## 3. ユニタリ発展の原理 \n", 88 | "### 量子の状態$|\\psi_0\\rangle$が$|\\psi_1\\rangle$に変化したとする。この時、$|\\psi_1\\rangle=U|\\psi_1\\rangle$となるユニタリ行列$U$が存在する\n", 89 | "\"Note: Trusted Notebook\" width=\"550 px\" align=\"left\">\n" 90 | ] 91 | }, 92 | { 93 | "cell_type": "markdown", 94 | "metadata": {}, 95 | "source": [ 96 | "\n", 97 | "\n", 98 | "# 数学的準備 1 \n", 99 | "\n", 100 | "\"Note: Trusted Notebook\" width=\"550 px\" align=\"left\">\n", 101 | "\"Note: Trusted Notebook\" width=\"550 px\" align=\"left\">\n", 102 | "\n", 103 | "\"Note: Trusted Notebook\" width=\"550 px\" align=\"left\">\n", 104 | "\"Note: Trusted Notebook\" width=\"550 px\" align=\"left\">\n", 105 | "\"Note: Trusted Notebook\" width=\"550 px\" align=\"left\">\n", 106 | "\"Note: Trusted Notebook\" width=\"550 px\" align=\"left\">\n", 107 | "\"Note: Trusted Notebook\" width=\"550 px\" align=\"left\">\n", 108 | "\"Note: Trusted Notebook\" width=\"550 px\" align=\"left\">\n", 109 | "\"Note: Trusted Notebook\" width=\"550 px\" align=\"left\">\n", 110 | "\"Note: Trusted Notebook\" width=\"550 px\" align=\"left\">\n" 111 | ] 112 | }, 113 | { 114 | "cell_type": "markdown", 115 | "metadata": {}, 116 | "source": [ 117 | "\n", 118 | "### 　　　　　　　　[前の章へ](./0_index.ipynb)　[全体目次](./Contents.ipynb)　 [次の章へ](./2_circuit.ipynb)" 119 | ] 120 | }, 121 | { 122 | "cell_type": "code", 123 | "execution_count": null, 124 | "metadata": {}, 125 | "outputs": [], 126 | "source": [] 127 | } 128 | ], 129 | "metadata": { 130 | "anaconda-cloud": {}, 131 | "kernelspec": { 132 | "display_name": "Python 3", 133 | "language": "python", 134 | "name": "python3" 135 | }, 136 | "language_info": { 137 | "codemirror_mode": { 138 | "name": "ipython", 139 | "version": 3 140 | }, 141 | "file_extension": ".py", 142 | "mimetype": "text/x-python", 143 | "name": "python", 144 | "nbconvert_exporter": "python", 145 | "pygments_lexer": "ipython3", 146 | "version": "3.6.8" 147 | } 148 | }, 149 | "nbformat": 4, 150 | "nbformat_minor": 1 151 | } 152 | -------------------------------------------------------------------------------- /0-Contents.ipynb: -------------------------------------------------------------------------------- 1 | { 2 | "cells": [ 3 | { 4 | "cell_type": "markdown", 5 | "metadata": { 6 | "coq_kernel_metadata": { 7 | "auto_roll_back": true 8 | } 9 | }, 10 | "source": [ 11 | "" 12 | ] 13 | }, 14 | { 15 | "cell_type": "markdown", 16 | "metadata": { 17 | "coq_kernel_metadata": { 18 | "auto_roll_back": true 19 | } 20 | }, 21 | "source": [ 22 | "# 目次 " 23 | ] 24 | }, 25 | { 26 | "cell_type": "markdown", 27 | "metadata": { 28 | "coq_kernel_metadata": { 29 | "auto_roll_back": true 30 | } 31 | }, 32 | "source": [ 33 | "# [Hello Coq!](./0-Introduction.ipynb)\n", 34 | "\n", 35 | "- [Coqとのはじめての対話](./0-Introduction.ipynb#sec1)\n", 36 | "- [Coqとのはじめての対話をふりかえる。](./0-Introduction.ipynb#sec2)\n", 37 | " - [人間は、Coqに何を伝えたか？](./0-Introduction.ipynb#sec21)\n", 38 | " - [Coqは、反応として人間に何を伝えたか？](./0-Introduction.ipynb#sec22)\n", 39 | "\n" 40 | ] 41 | }, 42 | { 43 | "cell_type": "markdown", 44 | "metadata": { 45 | "coq_kernel_metadata": { 46 | "auto_roll_back": true 47 | } 48 | }, 49 | "source": [ 50 | "# [論理式の証明](./1-Logic.ipynb)\n", 51 | "- [論理式はどのような形をしているか？](./1-Logic.ipynb#sec1)\n", 52 | " - [命題の構成ルール -- Formation Rule](./1-Logic.ipynb#sec11)\n", 53 | " - [命題の分解ルール -- 部分式への分解](./1-Logic.ipynb#sec12)\n", 54 | " - [証明のサブゴールと命題の部分式](./1-Logic.ipynb#sec13)\n", 55 | "- [Coqが返す「証明の状態」が持つ情報](./1-Logic.ipynb#sec2)\n", 56 | " - [「証明の状態」＝「仮説部分」＋「サブゴール」](./1-Logic.ipynb#sec21)\n", 57 | " - [証明が進むと「仮説部分」も「サブゴール」も部分式に置き換わります](./1-Logic.ipynb#sec22)\n", 58 | "- [tacticの選択](./1-Logic.ipynb#sec3)\n", 59 | " - [サブゴールの論理式の形で、選択すべきtactic](./1-Logic.ipynb#sec31)\n", 60 | " - [仮説部の論理式の形で、選択すべきtactic](./1-Logic.ipynb#sec32)\n", 61 | " - [仮説部の論理式の一つとサブゴールが一致する時](./1-Logic.ipynb#sec33)\n", 62 | "- [論理式の証明で利用できるtacticsの働きを見る](./1-Logic.ipynb#sec4)\n", 63 | " - [intros](./1-Logic.ipynb#sec41)\n", 64 | " - [introsの もう一つの使い方](./1-Logic.ipynb#sec42)\n", 65 | " - [split](./1-Logic.ipynb#sec43)\n", 66 | " - [left または right](./1-Logic.ipynb#sec44)\n", 67 | " - [destruct](./1-Logic.ipynb#sec45)\n", 68 | " - [apply](./1-Logic.ipynb#sec46)\n", 69 | " - [exact](./1-Logic.ipynb#sec47)\n", 70 | " - [assumption](./1-Logic.ipynb#sec48)\n", 71 | " - [trivial](./1-Logic.ipynb#sec49)\n", 72 | "- [論理式の証明の自動化　tauto](./1-Logic.ipynb#sec5)\n", 73 | " - [tautoで証明できる命題](./1-Logic.ipynb#sec51)\n", 74 | " - [tautoで証明できない命題](./1-Logic.ipynb#sec52)\n", 75 | "\n" 76 | ] 77 | }, 78 | { 79 | "cell_type": "markdown", 80 | "metadata": { 81 | "coq_kernel_metadata": { 82 | "auto_roll_back": true 83 | } 84 | }, 85 | "source": [ 86 | "# [演習問題 1](./1-logic-answer.ipynb)\n", 87 | "- [演習問題 1-1](./1-logic-answer.ipynb#ex11) [解答](./1-logic-answer.ipynb#ans11) 　A -> A\n", 88 | "- [演習問題 1-2](./1-logic-answer.ipynb#ex12) [解答](./1-logic-answer.ipynb#ans12) 　A /\\ B -> A\n", 89 | "- [演習問題 1-3](./1-logic-answer.ipynb#ex13) [解答](./1-logic-answer.ipynb#ans13) 　A /\\ B -> B\n", 90 | "- [演習問題 1-4](./1-logic-answer.ipynb#ex14) [解答](./1-logic-answer.ipynb#ans14) 　A -> A \\\\/ B\n", 91 | "- [演習問題 1-5](./1-logic-answer.ipynb#ex15) [解答](./1-logic-answer.ipynb#ans15) 　B -> A \\\\/ B\n", 92 | "- [演習問題 1-6](./1-logic-answer.ipynb#ex16) [解答](./1-logic-answer.ipynb#ans16) 　A -> (A -> B) -> B\n", 93 | "- [演習問題 1-7](./1-logic-answer.ipynb#ex17) [解答](./1-logic-answer.ipynb#ans17) 　A -> (A -> B) -> (B ->C) -> C\n", 94 | "- [演習問題 1-8](./1-logic-answer.ipynb#ex18) [解答](./1-logic-answer.ipynb#ans18) 　A /\\ B -> B /\\ A\n", 95 | "- [演習問題 1-9](./1-logic-answer.ipynb#ex19) [解答](./1-logic-answer.ipynb#ans19) 　A \\\\/ B -> B \\\\/ A\n", 96 | "\n" 97 | ] 98 | }, 99 | { 100 | "cell_type": "markdown", 101 | "metadata": { 102 | "coq_kernel_metadata": { 103 | "auto_roll_back": true 104 | } 105 | }, 106 | "source": [ 107 | "# [関数と等式を含んだ命題の証明](./2-Equality.ipynb)\n", 108 | "- [Coqでの関数と等式を含んだ命題の証明](./2-Equality.ipynb#sec1)\n", 109 | "- [等式の性質](./2-Equality.ipynb#sec2)\n", 110 | " - [Coqでの等式の証明の戦略](./2-Equality.ipynb#sec21)\n", 111 | "- [Coqでの関数](./2-Equality.ipynb#sec3)\n", 112 | " - [Coqでの関数の定義](./2-Equality.ipynb#sec31)\n", 113 | " - [Coqでの関数の型](./2-Equality.ipynb#sec32)\n", 114 | "- [簡単な関数の例](./2-Equality.ipynb#sec4)\n", 115 | " - [関数 negb:　引数に与えられたブール値の否定を返す関数](./2-Equality.ipynb#sec41)\n", 116 | " - [関数 eqb 　引数に与えられた二つのブール値が等しいかを返す関数](./2-Equality.ipynb#sec42)\n", 117 | "- [再帰的な関数定義の例](./2-Equality.ipynb#sec5)\n", 118 | " - [二つの自然数の和を返す関数 plus](./2-Equality.ipynb#sec51)\n", 119 | " - [二つの自然数の積を返す関数 mult](./2-Equality.ipynb#sec52)\n", 120 | "- [等式を含む命題の証明で利用できるtacticsについて](./2-Equality.ipynb#sec6)\n", 121 | "- [reflexivity tactics](./2-Equality.ipynb#sec605)\n", 122 | "- [simpl tactic](./2-Equality.ipynb#sec7)\n", 123 | " - [1 + n = S n で式を単純にする](./2-Equality.ipynb#sec71)\n", 124 | " - [0 + n = n で式を単純にする](./2-Equality.ipynb#sec72)\n", 125 | " - [0 * n = 0 で式を単純にする](./2-Equality.ipynb#sec73)\n", 126 | "- [rewrite tactic](./2-Equality.ipynb#sec8)\n", 127 | " - [rewrite -> H の例](./2-Equality.ipynb#sec81)\n", 128 | " - [もう一つ rewrite -> H の例](./2-Equality.ipynb#sec82)\n", 129 | " - [rewrite <- H の例](./2-Equality.ipynb#sec83)\n", 130 | " " 131 | ] 132 | }, 133 | { 134 | "cell_type": "markdown", 135 | "metadata": { 136 | "coq_kernel_metadata": { 137 | "auto_roll_back": true 138 | } 139 | }, 140 | "source": [ 141 | "# [演習問題 2](./2-equality-answer.ipynb)\n", 142 | "- [演習問題 2-1](./2-equality-answer.ipynb#ex21) [解答](./2-equality-answer.ipynb#ans21) 　0 + n = n\n", 143 | "- [演習問題 2-2](./2-equality-answer.ipynb#ex22) [解答](./2-equality-answer.ipynb#ans22) 　1 + n = S n\n", 144 | "- [演習問題 2-3](./2-equality-answer.ipynb#ex23) [解答](./2-equality-answer.ipynb#ans23) 　0 * n = 0\n", 145 | "- [演習問題 2-4](./2-equality-answer.ipynb#ex24) [解答](./2-equality-answer.ipynb#ans24) 　n = m -> n+n = m + m\n", 146 | "- [演習問題 2-5](./2-equality-answer.ipynb#ex25) [解答](./2-equality-answer.ipynb#ans25) 　n = m -> m = o -> n + m = m + o\n", 147 | "- [演習問題 2-6](./2-equality-answer.ipynb#ex26) [解答](./2-equality-answer.ipynb#ans26) 　( 0 + n ) * m = n * m\n", 148 | "- [演習問題 2-7](./2-equality-answer.ipynb#ex27) [解答](./2-equality-answer.ipynb#ans27) 　m = S n -> m * (1 + n) = m * m" 149 | ] 150 | }, 151 | { 152 | "cell_type": "markdown", 153 | "metadata": { 154 | "coq_kernel_metadata": { 155 | "auto_roll_back": true 156 | } 157 | }, 158 | "source": [ 159 | "# [帰納的データ型と帰納法を用いた証明](./3-Induction.ipynb)\n", 160 | "- [Coq での「型」について](./3-Induction.ipynb#sec1)\n", 161 | "- [列挙によるデータ型の定義](./3-Induction.ipynb#sec2)\n", 162 | " - [型 bool ](./3-Induction.ipynb#sec21)\n", 163 | " - [型 rgb](./3-Induction.ipynb#sec22)\n", 164 | " - [定義済みの型から、新しい型を定義する](./3-Induction.ipynb#sec23)\n", 165 | "- [帰納的データ型](./3-Induction.ipynb#sec3)\n", 166 | " - [Coqでの自然数の定義](./3-Induction.ipynb#sec31)\n", 167 | "- [t : T という表記の意味](./3-Induction.ipynb#sec32)\n", 168 | "- [帰納法を使った命題の証明](./3-Induction.ipynb#sec4)\n", 169 | " - [数学的帰納法について](./3-Induction.ipynb#sec41)\n", 170 | " - [帰納法を使った証明の例 1： n + 0 = n](./3-Induction.ipynb#sec42)\n", 171 | " - [帰納法を使った証明の例 2： S (n + m) = n + (S m)](./3-Induction.ipynb#sec43)\n", 172 | "- [帰納法を使う場合の証明の流れ](./3-Induction.ipynb#sec5)\n", 173 | " - [tactic induction](./3-Induction.ipynb#sec51)\n", 174 | " - [induction nを用いた証明の留意点](./3-Induction.ipynb#sec52)\n", 175 | "- [Coqのライブラリーの利用](./3-Induction.ipynb#sec6)\n", 176 | " - [Coqでは、多数の証明済みの命題がライブラリーとして提供されています。](./3-Induction.ipynb#sec61)\n", 177 | " - [Coq ライブラリーを利用した証明の例 : n * m = m * n](./3-Induction.ipynb#sec62)\n", 178 | "\n" 179 | ] 180 | }, 181 | { 182 | "cell_type": "markdown", 183 | "metadata": { 184 | "coq_kernel_metadata": { 185 | "auto_roll_back": true 186 | } 187 | }, 188 | "source": [ 189 | "# [演習問題 3](./3-induction-answer.ipynb)\n", 190 | "- [演習問題 3-1](./3-induction-answer.ipynb#ex31) [解答](./3-induction-answer.ipynb#ans31) 　n + 0 = n\n", 191 | "- [演習問題 3-2](./3-induction-answer.ipynb#ex32) [解答](./3-induction-answer.ipynb#ans32) 　minus n n = 0\n", 192 | "- [演習問題 3-3](./3-induction-answer.ipynb#ex33) [解答](./3-induction-answer.ipynb#ans33) 　n * 0 = 0\n", 193 | "- [演習問題 3-4](./3-induction-answer.ipynb#ex34) [解答](./3-induction-answer.ipynb#ans34) 　S (n + m) = n + (S m)\n", 194 | "- [演習問題 3-5](./3-induction-answer.ipynb#ex35) [解答](./3-induction-answer.ipynb#ans35) 　n + m = m + n\n", 195 | "- [演習問題 3-6](./3-induction-answer.ipynb#ex36) [解答](./3-induction-answer.ipynb#ans36) 　n + (m +o) = (n + m) + o \n", 196 | "- [演習問題 3-7](./3-induction-answer.ipynb#ex37) [解答](./3-induction-answer.ipynb#ans37) 　n * S m = n * m + n\n", 197 | "- [演習問題 3-8](./3-induction-answer.ipynb#ex38) [解答](./3-induction-answer.ipynb#ans38) 　n * m = m * n\n" 198 | ] 199 | } 200 | ], 201 | "metadata": { 202 | "kernelspec": { 203 | "display_name": "Coq", 204 | "language": "coq", 205 | "name": "coq" 206 | }, 207 | "language_info": { 208 | "file_extension": ".v", 209 | "mimetype": "text/x-coq", 210 | "name": "coq", 211 | "version": "8.9.1" 212 | } 213 | }, 214 | "nbformat": 4, 215 | "nbformat_minor": 2 216 | } 217 | -------------------------------------------------------------------------------- /.ipynb_checkpoints/0-Contents-checkpoint.ipynb: -------------------------------------------------------------------------------- 1 | { 2 | "cells": [ 3 | { 4 | "cell_type": "markdown", 5 | "metadata": { 6 | "coq_kernel_metadata": { 7 | "auto_roll_back": true 8 | } 9 | }, 10 | "source": [ 11 | "" 12 | ] 13 | }, 14 | { 15 | "cell_type": "markdown", 16 | "metadata": { 17 | "coq_kernel_metadata": { 18 | "auto_roll_back": true 19 | } 20 | }, 21 | "source": [ 22 | "# 目次 " 23 | ] 24 | }, 25 | { 26 | "cell_type": "markdown", 27 | "metadata": { 28 | "coq_kernel_metadata": { 29 | "auto_roll_back": true 30 | } 31 | }, 32 | "source": [ 33 | "# [Hello Coq!](./0-Introduction.ipynb)\n", 34 | "\n", 35 | "- [Coqとのはじめての対話](./0-Introduction.ipynb#sec1)\n", 36 | "- [Coqとのはじめての対話をふりかえる。](./0-Introduction.ipynb#sec2)\n", 37 | " - [人間は、Coqに何を伝えたか？](./0-Introduction.ipynb#sec21)\n", 38 | " - [Coqは、反応として人間に何を伝えたか？](./0-Introduction.ipynb#sec22)\n", 39 | "\n" 40 | ] 41 | }, 42 | { 43 | "cell_type": "markdown", 44 | "metadata": { 45 | "coq_kernel_metadata": { 46 | "auto_roll_back": true 47 | } 48 | }, 49 | "source": [ 50 | "# [論理式の証明](./1-Logic.ipynb)\n", 51 | "- [論理式はどのような形をしているか？](./1-Logic.ipynb#sec1)\n", 52 | " - [命題の構成ルール -- Formation Rule](./1-Logic.ipynb#sec11)\n", 53 | " - [命題の分解ルール -- 部分式への分解](./1-Logic.ipynb#sec12)\n", 54 | " - [証明のサブゴールと命題の部分式](./1-Logic.ipynb#sec13)\n", 55 | "- [Coqが返す「証明の状態」が持つ情報](./1-Logic.ipynb#sec2)\n", 56 | " - [「証明の状態」＝「仮説部分」＋「サブゴール」](./1-Logic.ipynb#sec21)\n", 57 | " - [証明が進むと「仮説部分」も「サブゴール」も部分式に置き換わります](./1-Logic.ipynb#sec22)\n", 58 | "- [tacticの選択](./1-Logic.ipynb#sec3)\n", 59 | " - [サブゴールの論理式の形で、選択すべきtactic](./1-Logic.ipynb#sec31)\n", 60 | " - [仮説部の論理式の形で、選択すべきtactic](./1-Logic.ipynb#sec32)\n", 61 | " - [仮説部の論理式の一つとサブゴールが一致する時](./1-Logic.ipynb#sec33)\n", 62 | "- [論理式の証明で利用できるtacticsの働きを見る](./1-Logic.ipynb#sec4)\n", 63 | " - [intros](./1-Logic.ipynb#sec41)\n", 64 | " - [introsの もう一つの使い方](./1-Logic.ipynb#sec42)\n", 65 | " - [split](./1-Logic.ipynb#sec43)\n", 66 | " - [left または right](./1-Logic.ipynb#sec44)\n", 67 | " - [destruct](./1-Logic.ipynb#sec45)\n", 68 | " - [apply](./1-Logic.ipynb#sec46)\n", 69 | " - [exact](./1-Logic.ipynb#sec47)\n", 70 | " - [assumption](./1-Logic.ipynb#sec48)\n", 71 | " - [trivial](./1-Logic.ipynb#sec49)\n", 72 | "- [論理式の証明の自動化　tauto](./1-Logic.ipynb#sec5)\n", 73 | " - [tautoで証明できる命題](./1-Logic.ipynb#sec51)\n", 74 | " - [tautoで証明できない命題](./1-Logic.ipynb#sec52)\n", 75 | "\n" 76 | ] 77 | }, 78 | { 79 | "cell_type": "markdown", 80 | "metadata": { 81 | "coq_kernel_metadata": { 82 | "auto_roll_back": true 83 | } 84 | }, 85 | "source": [ 86 | "# [演習問題 1](./1-logic-answer.ipynb)\n", 87 | "- [演習問題 1-1](./1-logic-answer.ipynb#ex11) [解答](./1-logic-answer.ipynb#ans11) 　A -> A\n", 88 | "- [演習問題 1-2](./1-logic-answer.ipynb#ex12) [解答](./1-logic-answer.ipynb#ans12) 　A /\\ B -> A\n", 89 | "- [演習問題 1-3](./1-logic-answer.ipynb#ex13) [解答](./1-logic-answer.ipynb#ans13) 　A /\\ B -> B\n", 90 | "- [演習問題 1-4](./1-logic-answer.ipynb#ex14) [解答](./1-logic-answer.ipynb#ans14) 　A -> A \\\\/ B\n", 91 | "- [演習問題 1-5](./1-logic-answer.ipynb#ex15) [解答](./1-logic-answer.ipynb#ans15) 　B -> A \\\\/ B\n", 92 | "- [演習問題 1-6](./1-logic-answer.ipynb#ex16) [解答](./1-logic-answer.ipynb#ans16) 　A -> (A -> B) -> B\n", 93 | "- [演習問題 1-7](./1-logic-answer.ipynb#ex17) [解答](./1-logic-answer.ipynb#ans17) 　A -> (A -> B) -> (B ->C) -> C\n", 94 | "- [演習問題 1-8](./1-logic-answer.ipynb#ex18) [解答](./1-logic-answer.ipynb#ans18) 　A /\\ B -> B /\\ A\n", 95 | "- [演習問題 1-9](./1-logic-answer.ipynb#ex19) [解答](./1-logic-answer.ipynb#ans19) 　A \\\\/ B -> B \\\\/ A\n", 96 | "\n" 97 | ] 98 | }, 99 | { 100 | "cell_type": "markdown", 101 | "metadata": { 102 | "coq_kernel_metadata": { 103 | "auto_roll_back": true 104 | } 105 | }, 106 | "source": [ 107 | "# [関数と等式を含んだ命題の証明](./2-Equality.ipynb)\n", 108 | "- [Coqでの関数と等式を含んだ命題の証明](./2-Equality.ipynb#sec1)\n", 109 | "- [等式の性質](./2-Equality.ipynb#sec2)\n", 110 | " - [Coqでの等式の証明の戦略](./2-Equality.ipynb#sec21)\n", 111 | "- [Coqでの関数](./2-Equality.ipynb#sec3)\n", 112 | " - [Coqでの関数の定義](./2-Equality.ipynb#sec31)\n", 113 | " - [Coqでの関数の型](./2-Equality.ipynb#sec32)\n", 114 | "- [簡単な関数の例](./2-Equality.ipynb#sec4)\n", 115 | " - [関数 negb:　引数に与えられたブール値の否定を返す関数](./2-Equality.ipynb#sec41)\n", 116 | " - [関数 eqb 　引数に与えられた二つのブール値が等しいかを返す関数](./2-Equality.ipynb#sec42)\n", 117 | "- [再帰的な関数定義の例](./2-Equality.ipynb#sec5)\n", 118 | " - [二つの自然数の和を返す関数 plus](./2-Equality.ipynb#sec51)\n", 119 | " - [二つの自然数の積を返す関数 mult](./2-Equality.ipynb#sec52)\n", 120 | "- [等式を含む命題の証明で利用できるtacticsについて](./2-Equality.ipynb#sec6)\n", 121 | "- [reflexivity tactics](./2-Equality.ipynb#sec605)\n", 122 | "- [simpl tactic](./2-Equality.ipynb#sec7)\n", 123 | " - [1 + n = S n で式を単純にする](./2-Equality.ipynb#sec71)\n", 124 | " - [0 + n = n で式を単純にする](./2-Equality.ipynb#sec72)\n", 125 | " - [0 * n = 0 で式を単純にする](./2-Equality.ipynb#sec73)\n", 126 | "- [rewrite tactic](./2-Equality.ipynb#sec8)\n", 127 | " - [rewrite -> H の例](./2-Equality.ipynb#sec81)\n", 128 | " - [もう一つ rewrite -> H の例](./2-Equality.ipynb#sec82)\n", 129 | " - [rewrite <- H の例](./2-Equality.ipynb#sec83)\n", 130 | " " 131 | ] 132 | }, 133 | { 134 | "cell_type": "markdown", 135 | "metadata": { 136 | "coq_kernel_metadata": { 137 | "auto_roll_back": true 138 | } 139 | }, 140 | "source": [ 141 | "# [演習問題 2](./2-equality-answer.ipynb)\n", 142 | "- [演習問題 2-1](./2-equality-answer.ipynb#ex21) [解答](./2-equality-answer.ipynb#ans21) 　0 + n = n\n", 143 | "- [演習問題 2-2](./2-equality-answer.ipynb#ex22) [解答](./2-equality-answer.ipynb#ans22) 　1 + n = S n\n", 144 | "- [演習問題 2-3](./2-equality-answer.ipynb#ex23) [解答](./2-equality-answer.ipynb#ans23) 　0 * n = 0\n", 145 | "- [演習問題 2-4](./2-equality-answer.ipynb#ex24) [解答](./2-equality-answer.ipynb#ans24) 　n = m -> n+n = m + m\n", 146 | "- [演習問題 2-5](./2-equality-answer.ipynb#ex25) [解答](./2-equality-answer.ipynb#ans25) 　n = m -> m = o -> n + m = m + o\n", 147 | "- [演習問題 2-6](./2-equality-answer.ipynb#ex26) [解答](./2-equality-answer.ipynb#ans26) 　( 0 + n ) * m = n * m\n", 148 | "- [演習問題 2-7](./2-equality-answer.ipynb#ex27) [解答](./2-equality-answer.ipynb#ans27) 　m = S n -> m * (1 + n) = m * m" 149 | ] 150 | }, 151 | { 152 | "cell_type": "markdown", 153 | "metadata": { 154 | "coq_kernel_metadata": { 155 | "auto_roll_back": true 156 | } 157 | }, 158 | "source": [ 159 | "# [帰納的データ型と帰納法を用いた証明](./3-Induction.ipynb)\n", 160 | "- [Coq での「型」について](./3-Induction.ipynb#sec1)\n", 161 | "- [列挙によるデータ型の定義](./3-Induction.ipynb#sec2)\n", 162 | " - [型 bool ](./3-Induction.ipynb#sec21)\n", 163 | " - [型 rgb](./3-Induction.ipynb#sec22)\n", 164 | " - [定義済みの型から、新しい型を定義する](./3-Induction.ipynb#sec23)\n", 165 | "- [帰納的データ型](./3-Induction.ipynb#sec3)\n", 166 | " - [Coqでの自然数の定義](./3-Induction.ipynb#sec31)\n", 167 | "- [t : T という表記の意味](./3-Induction.ipynb#sec32)\n", 168 | "- [帰納法を使った命題の証明](./3-Induction.ipynb#sec4)\n", 169 | " - [数学的帰納法について](./3-Induction.ipynb#sec41)n * 0 = 0\n", 170 | " - [帰納法を使った証明の例 1： n + 0 = n](./3-Induction.ipynb#sec42)\n", 171 | " - [帰納法を使った証明の例 2： S (n + m) = n + (S m)](./3-Induction.ipynb#sec43)\n", 172 | "- [帰納法を使う場合の証明の流れ](./3-Induction.ipynb#sec5)\n", 173 | " - [tactic induction](./3-Induction.ipynb#sec51)\n", 174 | " - [induction nを用いた証明の留意点](./3-Induction.ipynb#sec52)\n", 175 | "- [Coqのライブラリーの利用](./3-Induction.ipynb#sec6)\n", 176 | " - [Coqでは、多数の証明済みの命題がライブラリーとして提供されています。](./3-Induction.ipynb#sec61)\n", 177 | " - [Coq ライブラリーを利用した証明の例 : n * m = m * n](./3-Induction.ipynb#sec62)\n", 178 | "\n" 179 | ] 180 | }, 181 | { 182 | "cell_type": "markdown", 183 | "metadata": { 184 | "coq_kernel_metadata": { 185 | "auto_roll_back": true 186 | } 187 | }, 188 | "source": [ 189 | "# [演習問題 3](./3-induction-answer.ipynb)\n", 190 | "- [演習問題 3-1](./3-induction-answer.ipynb#ex31) [解答](./3-induction-answer.ipynb#ans31) 　n + 0 = n\n", 191 | "- [演習問題 3-2](./3-induction-answer.ipynb#ex32) [解答](./3-induction-answer.ipynb#ans32) 　minus n n = 0\n", 192 | "- [演習問題 3-3](./3-induction-answer.ipynb#ex33) [解答](./3-induction-answer.ipynb#ans33) 　n * 0 = 0\n", 193 | "- [演習問題 3-4](./3-induction-answer.ipynb#ex34) [解答](./3-induction-answer.ipynb#ans34) 　S (n + m) = n + (S m)\n", 194 | "- [演習問題 3-5](./3-induction-answer.ipynb#ex35) [解答](./3-induction-answer.ipynb#ans35) 　n + m = m + n\n", 195 | "- [演習問題 3-6](./3-induction-answer.ipynb#ex36) [解答](./3-induction-answer.ipynb#ans36) 　n + (m +o) = (n + m) + o \n", 196 | "- [演習問題 3-7](./3-induction-answer.ipynb#ex37) [解答](./3-induction-answer.ipynb#ans37) 　n * S m = n * m + n\n", 197 | "- [演習問題 3-8](./3-induction-answer.ipynb#ex38) [解答](./3-induction-answer.ipynb#ans38) 　n * m = m * n\n" 198 | ] 199 | } 200 | ], 201 | "metadata": { 202 | "kernelspec": { 203 | "display_name": "Coq", 204 | "language": "coq", 205 | "name": "coq" 206 | }, 207 | "language_info": { 208 | "file_extension": ".v", 209 | "mimetype": "text/x-coq", 210 | "name": "coq", 211 | "version": "8.9.1" 212 | } 213 | }, 214 | "nbformat": 4, 215 | "nbformat_minor": 2 216 | } 217 | -------------------------------------------------------------------------------- /.ipynb_checkpoints/Introduction-checkpoint.ipynb: -------------------------------------------------------------------------------- 1 | { 2 | "cells": [ 3 | { 4 | "cell_type": "markdown", 5 | "metadata": { 6 | "coq_kernel_metadata": { 7 | "auto_roll_back": true 8 | } 9 | }, 10 | "source": [ 11 | "# MaruLabo " 12 | ] 13 | }, 14 | { 15 | "cell_type": "markdown", 16 | "metadata": { 17 | "coq_kernel_metadata": { 18 | "auto_roll_back": true 19 | } 20 | }, 21 | "source": [ 22 | "# はじめての証明" 23 | ] 24 | }, 25 | { 26 | "cell_type": "markdown", 27 | "metadata": { 28 | "coq_kernel_metadata": { 29 | "auto_roll_back": true 30 | } 31 | }, 32 | "source": [ 33 | "次のセルを実行してみてください。次のような出力があるはずです。\n", 34 | "" 35 | ] 36 | }, 37 | { 38 | "cell_type": "markdown", 39 | "metadata": { 40 | "coq_kernel_metadata": { 41 | "auto_roll_back": true 42 | } 43 | }, 44 | "source": [ 45 | "
\n", 46 | "\n", 47 | "
" 48 | ] 49 | }, 50 | { 51 | "cell_type": "code", 52 | "execution_count": 1, 53 | "metadata": { 54 | "coq_kernel_metadata": { 55 | "auto_roll_back": true, 56 | "cell_id": "7774a2227f494ad18622054a5fda738d", 57 | "execution_id": "9B0735FF845E4B73820D47975CE978CF" 58 | } 59 | }, 60 | "outputs": [ 61 | { 62 | "data": { 63 | "text/html": [ 64 | "\n", 65 | "
\n", 66 | " 
Proving: Hello_Coq\n",
 67 |        "\n",
 68 |        "1 subgoal\n",
 69 |        "\n",
 70 |        "1/1 -----------\n",
 71 |        "forall A : Prop, A -> A
\n", 72 | "
\n", 73 | "\n", 74 | "
\n", 75 | " \n", 76 | " Cell evaluated.\n", 77 | "
\n", 78 | "\n", 79 | "
\n", 80 | " \n", 81 | " Cell rolled back.\n", 82 | "
\n", 83 | "\n", 84 | "
\n", 85 | " \n", 89 | "\n", 90 | "
\n", 91 | " \n", 92 | " Auto rollback\n", 93 | "
\n", 94 | "
\n" 95 | ], 96 | "text/plain": [ 97 | "Proving: Hello_Coq\n", 98 | "\n", 99 | "1 subgoal\n", 100 | "\n", 101 | "1/1 -----------\n", 102 | "forall A : Prop, A -> A" 103 | ] 104 | }, 105 | "execution_count": 2, 106 | "metadata": { 107 | "coq_kernel_evaluated": true, 108 | "coq_kernel_execution_id": "9B0735FF845E4B73820D47975CE978CF", 109 | "coq_kernel_rolled_back": false 110 | }, 111 | "output_type": "execute_result" 112 | } 113 | ], 114 | "source": [ 115 | "Theorem Hello_Coq : (forall A : Prop, A -> A)." 116 | ] 117 | }, 118 | { 119 | "cell_type": "markdown", 120 | "metadata": { 121 | "coq_kernel_metadata": { 122 | "auto_roll_back": true 123 | } 124 | }, 125 | "source": [ 126 | "今度は、次のセルを実行してみてください。次のような出力があるはずです。\n", 127 | "" 128 | ] 129 | }, 130 | { 131 | "cell_type": "code", 132 | "execution_count": 2, 133 | "metadata": { 134 | "coq_kernel_metadata": { 135 | "auto_roll_back": true, 136 | "cell_id": "107c5f7ec1354ef492e28a159d5d9d9c", 137 | "execution_id": "DAD112078A37407E8EE32A0E3FF9DA77" 138 | }, 139 | "scrolled": false 140 | }, 141 | "outputs": [ 142 | { 143 | "data": { 144 | "text/html": [ 145 | "\n", 146 | "
\n", 147 | " 
Proving: Hello_Coq\n",
148 |        "\n",
149 |        "1 subgoal\n",
150 |        "\n",
151 |        "1/1 -----------\n",
152 |        "forall A : Prop, A -> A
\n", 153 | "
\n", 154 | "\n", 155 | "
\n", 156 | " \n", 157 | " Cell evaluated.\n", 158 | "
\n", 159 | "\n", 160 | "
\n", 161 | " \n", 162 | " Cell rolled back.\n", 163 | "
\n", 164 | "\n", 165 | "
\n", 166 | " \n", 170 | "\n", 171 | "
\n", 172 | " \n", 173 | " Auto rollback\n", 174 | "
\n", 175 | "
\n" 176 | ], 177 | "text/plain": [ 178 | "Proving: Hello_Coq\n", 179 | "\n", 180 | "1 subgoal\n", 181 | "\n", 182 | "1/1 -----------\n", 183 | "forall A : Prop, A -> A" 184 | ] 185 | }, 186 | "execution_count": 3, 187 | "metadata": { 188 | "coq_kernel_evaluated": true, 189 | "coq_kernel_execution_id": "DAD112078A37407E8EE32A0E3FF9DA77", 190 | "coq_kernel_rolled_back": false 191 | }, 192 | "output_type": "execute_result" 193 | } 194 | ], 195 | "source": [ 196 | "Proof." 197 | ] 198 | }, 199 | { 200 | "cell_type": "markdown", 201 | "metadata": { 202 | "coq_kernel_metadata": { 203 | "auto_roll_back": true 204 | } 205 | }, 206 | "source": [ 207 | "今度は、次のセルを実行してみてください。次のような出力があるはずです。その意味を説明します。\n", 208 | "" 209 | ] 210 | }, 211 | { 212 | "cell_type": "code", 213 | "execution_count": 3, 214 | "metadata": { 215 | "coq_kernel_metadata": { 216 | "auto_roll_back": true, 217 | "cell_id": "456f37c34dee4c5680781cf89cc9a1ce", 218 | "execution_id": "C2AF206FEA7042F4816B1E717626F4CD" 219 | }, 220 | "scrolled": true 221 | }, 222 | "outputs": [ 223 | { 224 | "data": { 225 | "text/html": [ 226 | "\n", 227 | "
\n", 228 | " 
Proving: Hello_Coq\n",
229 |        "\n",
230 |        "1 subgoal\n",
231 |        "\n",
232 |        "A : Prop\n",
233 |        "H : A\n",
234 |        "\n",
235 |        "1/1 -----------\n",
236 |        "A
\n", 237 | "
\n", 238 | "\n", 239 | "
\n", 240 | " \n", 241 | " Cell evaluated.\n", 242 | "
\n", 243 | "\n", 244 | "
\n", 245 | " \n", 246 | " Cell rolled back.\n", 247 | "
\n", 248 | "\n", 249 | "
\n", 250 | " \n", 254 | "\n", 255 | "
\n", 256 | " \n", 257 | " Auto rollback\n", 258 | "
\n", 259 | "
\n" 260 | ], 261 | "text/plain": [ 262 | "Proving: Hello_Coq\n", 263 | "\n", 264 | "1 subgoal\n", 265 | "\n", 266 | "A : Prop\n", 267 | "H : A\n", 268 | "\n", 269 | "1/1 -----------\n", 270 | "A" 271 | ] 272 | }, 273 | "execution_count": 4, 274 | "metadata": { 275 | "coq_kernel_evaluated": true, 276 | "coq_kernel_execution_id": "C2AF206FEA7042F4816B1E717626F4CD", 277 | "coq_kernel_rolled_back": false 278 | }, 279 | "output_type": "execute_result" 280 | } 281 | ], 282 | "source": [ 283 | "intros." 284 | ] 285 | }, 286 | { 287 | "cell_type": "markdown", 288 | "metadata": { 289 | "coq_kernel_metadata": { 290 | "auto_roll_back": true 291 | } 292 | }, 293 | "source": [ 294 | "今度は、次のセルを実行してみてください。次のような出力があるはずです。\n", 295 | "" 296 | ] 297 | }, 298 | { 299 | "cell_type": "code", 300 | "execution_count": 4, 301 | "metadata": { 302 | "coq_kernel_metadata": { 303 | "auto_roll_back": true, 304 | "cell_id": "3f4dfe31f672462fa18cde95827a8d8b", 305 | "execution_id": "5236AD7A7C9F41838CE5B3A9F37F2C4E" 306 | }, 307 | "scrolled": true 308 | }, 309 | "outputs": [ 310 | { 311 | "data": { 312 | "text/html": [ 313 | "\n", 314 | "
\n", 315 | " 
Proving: Hello_Coq\n",
316 |        "\n",
317 |        "No more subgoals
\n", 318 | "
\n", 319 | "\n", 320 | "
\n", 321 | " \n", 322 | " Cell evaluated.\n", 323 | "
\n", 324 | "\n", 325 | "
\n", 326 | " \n", 327 | " Cell rolled back.\n", 328 | "
\n", 329 | "\n", 330 | "
\n", 331 | " \n", 335 | "\n", 336 | "
\n", 337 | " \n", 338 | " Auto rollback\n", 339 | "
\n", 340 | "
\n" 341 | ], 342 | "text/plain": [ 343 | "Proving: Hello_Coq\n", 344 | "\n", 345 | "No more subgoals" 346 | ] 347 | }, 348 | "execution_count": 5, 349 | "metadata": { 350 | "coq_kernel_evaluated": true, 351 | "coq_kernel_execution_id": "5236AD7A7C9F41838CE5B3A9F37F2C4E", 352 | "coq_kernel_rolled_back": false 353 | }, 354 | "output_type": "execute_result" 355 | } 356 | ], 357 | "source": [ 358 | "exact H." 359 | ] 360 | }, 361 | { 362 | "cell_type": "markdown", 363 | "metadata": { 364 | "coq_kernel_metadata": { 365 | "auto_roll_back": true 366 | } 367 | }, 368 | "source": [ 369 | "今度は、次のセルを実行してみてください。次のような出力があるはずです。\n", 370 | "" 371 | ] 372 | }, 373 | { 374 | "cell_type": "code", 375 | "execution_count": 5, 376 | "metadata": { 377 | "coq_kernel_metadata": { 378 | "auto_roll_back": true, 379 | "cell_id": "6b1e72f14e3a4bb8bc2d0300c92ae95f", 380 | "execution_id": "C5AF1A1C5CCB4A0C9B8B439056C556F1" 381 | } 382 | }, 383 | "outputs": [ 384 | { 385 | "data": { 386 | "text/html": [ 387 | "\n", 388 | "
\n", 389 | " 
\n",
390 |        "
\n",
391 |        "\n",
392 |        "
\n",
393 |        "    \n",
394 |        "    Cell evaluated.\n",
395 |        "
\n",
396 |        "\n",
397 |        "
\n",
398 |        "    \n",
399 |        "    Cell rolled back.\n",
400 |        "
\n",
401 |        "\n",
402 |        "
\n",
403 |        "    \n",
407 |        "\n",
408 |        "    
\n",
409 |        "      \n",
410 |        "      Auto rollback\n",
411 |        "    
\n",
412 |        "
\n"
413 |       ],
414 |       "text/plain": []
415 |      },
416 |      "execution_count": 6,
417 |      "metadata": {
418 |       "coq_kernel_evaluated": true,
419 |       "coq_kernel_execution_id": "C5AF1A1C5CCB4A0C9B8B439056C556F1",
420 |       "coq_kernel_rolled_back": false
421 |      },
422 |      "output_type": "execute_result"
423 |     }
424 |    ],
425 |    "source": [
426 |     "Qed."
427 |    ]
428 |   },
429 |   {
430 |    "cell_type": "markdown",
431 |    "metadata": {
432 |     "collapsed": true,
433 |     "coq_kernel_metadata": {
434 |      "auto_roll_back": true
435 |     }
436 |    },
437 |    "source": []
438 |   }
439 |  ],
440 |  "metadata": {
441 |   "kernelspec": {
442 |    "display_name": "Coq",
443 |    "language": "coq",
444 |    "name": "coq"
445 |   },
446 |   "language_info": {
447 |    "file_extension": ".v",
448 |    "mimetype": "text/x-coq",
449 |    "name": "coq",
450 |    "version": "8.9.1"
451 |   }
452 |  },
453 |  "nbformat": 4,
454 |  "nbformat_minor": 2
455 | }
456 | 


--------------------------------------------------------------------------------
/0-Introduction.ipynb:
--------------------------------------------------------------------------------
  1 | {
  2 |  "cells": [
  3 |   {
  4 |    "cell_type": "markdown",
  5 |    "metadata": {
  6 |     "coq_kernel_metadata": {
  7 |      "auto_roll_back": true
  8 |     }
  9 |    },
 10 |    "source": [
 11 |     ""
 12 |    ]
 13 |   },
 14 |   {
 15 |    "cell_type": "markdown",
 16 |    "metadata": {
 17 |     "coq_kernel_metadata": {
 18 |      "auto_roll_back": true
 19 |     }
 20 |    },
 21 |    "source": [
 22 |     "　　　　[全体目次](./0-Contents.ipynb)　　[次の章へ](./1-Logic.ipynb)"
 23 |    ]
 24 |   },
 25 |   {
 26 |    "cell_type": "markdown",
 27 |    "metadata": {
 28 |     "coq_kernel_metadata": {
 29 |      "auto_roll_back": true
 30 |     }
 31 |    },
 32 |    "source": [
 33 |     "# Hello Coq !"
 34 |    ]
 35 |   },
 36 |   {
 37 |    "cell_type": "markdown",
 38 |    "metadata": {
 39 |     "coq_kernel_metadata": {
 40 |      "auto_roll_back": true
 41 |     }
 42 |    },
 43 |    "source": [
 44 |     "\n",
 45 |     "# Coqとのはじめての対話\n",
 46 |     "\n",
 47 |     "Coqは、対話型の証明支援システムです。対話するのは人間と機械（Coq）です。\n",
 48 |     "\n",
 49 |     "人間と機械の対話でどのように証明が進んでいくかについては、おいおい説明するとして、ここではまず、Coqとのはじめての対話を経験して見ましょう。簡単な証明をしてみます。\n",
 50 |     "\n",
 51 |     "次の In [1]: で始まる空白の行に、次のブルーの文字列を入力して実行して見てください。"
 52 |    ]
 53 |   },
 54 |   {
 55 |    "cell_type": "markdown",
 56 |    "metadata": {
 57 |     "coq_kernel_metadata": {
 58 |      "auto_roll_back": true
 59 |     }
 60 |    },
 61 |    "source": [
 62 |     "
\n",
 63 |     "    ハンズオン環境 coq-jupyter -- 入力の実行\n",
 64 |     "

\n",
 65 |     "    今回のハンズオンで利用している環境は、jupyter notebookからCoqを呼び出せるようにした coq-jupyter というものです。操作法はjupyterと同じです。\n",
 66 |     " jupyterでは、操作の単位をセルと呼んでいます。 セルに入力したの字列を実行するには、セルにカーソルを合わせて、 画面上部の Runボタンを押します。\n",
 67 |     "
   \n",
 68 |     "\n"
 69 |    ]
 70 |   },
 71 |   {
 72 |    "cell_type": "markdown",
 73 |    "metadata": {
 74 |     "coq_kernel_metadata": {
 75 |      "auto_roll_back": true
 76 |     }
 77 |    },
 78 |    "source": [
 79 |     "
\n",
 80 |     "Shift-RET（シフトキーを押しながらRETキーを押す）でも、セルの実行ができます。\n",
 81 |     "
 \n",
 82 |     "\n",
 83 |     "最初に入力して実行するの、この文字列です。この文字列で一番大事なのは、冒頭の \"Theorem\" です。この文字列は、Coqに証明すべき式と、証明すべき式の名前を伝えています。こうした文字列を Coqコマンドと呼ぶことにしましょう。"
 84 |    ]
 85 |   },
 86 |   {
 87 |    "cell_type": "markdown",
 88 |    "metadata": {
 89 |     "coq_kernel_metadata": {
 90 |      "auto_roll_back": true
 91 |     }
 92 |    },
 93 |    "source": [
 94 |     "Theorem Hello_Coq : (forall A : Prop, A -> A ).\n",
 95 |     "\n",
 96 |     "文字列の入力では、コロンやコンマやピリオド、\"->\"(マイナス記号'-'＋'>' )に注意してください。\n",
 97 |     ""
 98 |    ]
 99 |   },
100 |   {
101 |    "cell_type": "code",
102 |    "execution_count": null,
103 |    "metadata": {
104 |     "coq_kernel_metadata": {
105 |      "auto_roll_back": true,
106 |      "cell_id": "ecf27221f8c64117aa9fd72f870003df",
107 |      "execution_id": "a4e06ef9b93e4d0d8237490a7eec840c"
108 |     },
109 |     "scrolled": true
110 |    },
111 |    "outputs": [],
112 |    "source": []
113 |   },
114 |   {
115 |    "cell_type": "markdown",
116 |    "metadata": {
117 |     "coq_kernel_metadata": {
118 |      "auto_roll_back": true
119 |     }
120 |    },
121 |    "source": [
122 |     "入力がおわったら、先のセルを実行してみてください。Coqから次のような反応があるはずです。\n",
123 |     "ここで初めてCoqに証明したい定理の情報を伝えました。Coqが返した情報の意味は、次のようなものです。\n",
124 |     "\n",
125 |     "\n"
126 |    ]
127 |   },
128 |   {
129 |    "cell_type": "markdown",
130 |    "metadata": {
131 |     "coq_kernel_metadata": {
132 |      "auto_roll_back": true
133 |     }
134 |    },
135 |    "source": [
136 |     "今度は、次の文字列を入力して実行してください。（最後の、ピリオドを忘れずに）\n",
137 |     "

\n",
138 |     "Proof.\n",
139 |     "
"
140 |    ]
141 |   },
142 |   {
143 |    "cell_type": "code",
144 |    "execution_count": null,
145 |    "metadata": {
146 |     "coq_kernel_metadata": {
147 |      "auto_roll_back": true,
148 |      "cell_id": "84d24fe8d5bd418f909b9d922ad3f4bf",
149 |      "execution_id": "9B0735FF845E4B73820D47975CE978CF"
150 |     },
151 |     "scrolled": true
152 |    },
153 |    "outputs": [],
154 |    "source": []
155 |   },
156 |   {
157 |    "cell_type": "markdown",
158 |    "metadata": {
159 |     "coq_kernel_metadata": {
160 |      "auto_roll_back": true
161 |     }
162 |    },
163 |    "source": [
164 |     "次のような反応がcoqからあるはずです。基本的には、先の反応と同じです。\n",
165 |     "coqコマンドの \"Proof.\"は、これから証明が始まることを宣言しています。\n",
166 |     "\n",
167 |     "ここでは、先に説明していなかった下二行の説明をしています。これらの出力は、coq-jupyterからのものです。\n",
168 |     "\"Cell evaluated\"がでていれば、証明は、順調に進んでいると思って構いません。\n",
169 |     "\n",
170 |     "\n",
171 |     ""
172 |    ]
173 |   },
174 |   {
175 |    "cell_type": "markdown",
176 |    "metadata": {
177 |     "coq_kernel_metadata": {
178 |      "auto_roll_back": true
179 |     }
180 |    },
181 |    "source": [
182 |     "今度は、次の文字列を入力して実行してください。（最後の、ピリオドを忘れずに）\n",
183 |     "

\n",
184 |     "intros.\n",
185 |     "
"
186 |    ]
187 |   },
188 |   {
189 |    "cell_type": "code",
190 |    "execution_count": null,
191 |    "metadata": {
192 |     "coq_kernel_metadata": {
193 |      "auto_roll_back": true,
194 |      "cell_id": "2b72a7e19d34461982cfd26a9dad43de",
195 |      "execution_id": "DAD112078A37407E8EE32A0E3FF9DA77"
196 |     },
197 |     "scrolled": false
198 |    },
199 |    "outputs": [],
200 |    "source": []
201 |   },
202 |   {
203 |    "cell_type": "markdown",
204 |    "metadata": {
205 |     "coq_kernel_metadata": {
206 |      "auto_roll_back": true
207 |     }
208 |    },
209 |    "source": [
210 |     "次のような反応があるはずです。その意味を説明しています。\n",
211 |     ""
212 |    ]
213 |   },
214 |   {
215 |    "cell_type": "markdown",
216 |    "metadata": {
217 |     "coq_kernel_metadata": {
218 |      "auto_roll_back": true
219 |     }
220 |    },
221 |    "source": [
222 |     "今度は、次の文字列を入力して実行してください。（最後の、ピリオドを忘れずに）\n",
223 |     "

\n",
224 |     "exact H.\n",
225 |     "
"
226 |    ]
227 |   },
228 |   {
229 |    "cell_type": "code",
230 |    "execution_count": null,
231 |    "metadata": {
232 |     "coq_kernel_metadata": {
233 |      "auto_roll_back": true
234 |     }
235 |    },
236 |    "outputs": [],
237 |    "source": []
238 |   },
239 |   {
240 |    "cell_type": "markdown",
241 |    "metadata": {
242 |     "coq_kernel_metadata": {
243 |      "auto_roll_back": true
244 |     }
245 |    },
246 |    "source": [
247 |     "次のような反応があるはずです。\n",
248 |     "\n",
249 |     "\n",
250 |     ""
251 |    ]
252 |   },
253 |   {
254 |    "cell_type": "markdown",
255 |    "metadata": {
256 |     "coq_kernel_metadata": {
257 |      "auto_roll_back": true
258 |     }
259 |    },
260 |    "source": [
261 |     "今度は、次の文字列を入力して実行してください。（最後の、ピリオドを忘れずに）\n",
262 |     "

\n",
263 |     "Qed..\n",
264 |     "
"
265 |    ]
266 |   },
267 |   {
268 |    "cell_type": "code",
269 |    "execution_count": null,
270 |    "metadata": {
271 |     "coq_kernel_metadata": {
272 |      "auto_roll_back": true,
273 |      "cell_id": "868b4dae6c1c456abad8bd188cf0d134",
274 |      "execution_id": "5236AD7A7C9F41838CE5B3A9F37F2C4E"
275 |     },
276 |     "scrolled": true
277 |    },
278 |    "outputs": [],
279 |    "source": []
280 |   },
281 |   {
282 |    "cell_type": "markdown",
283 |    "metadata": {
284 |     "coq_kernel_metadata": {
285 |      "auto_roll_back": true
286 |     }
287 |    },
288 |    "source": [
289 |     "次のような反応ががあるはずです。\n",
290 |     "\n",
291 |     ""
292 |    ]
293 |   },
294 |   {
295 |    "cell_type": "markdown",
296 |    "metadata": {
297 |     "coq_kernel_metadata": {
298 |      "auto_roll_back": true
299 |     }
300 |    },
301 |    "source": [
302 |     "これで、定理 Hello_Coq の証明は、終了です。\n",
303 |     "\n",
304 |     "\n",
305 |     "\n",
306 |     "# Coqとのはじめての対話をふりかえる。\n",
307 |     "\n",
308 |     "Coqとのはじめての対話を、人間の側とCoqの側の、それぞれから振り返ってみましょう。\n",
309 |     "\n",
310 |     "\n",
311 |     "## 人間は、Coqに何を伝えたか？\n",
312 |     "\n",
313 |     "人間が、Coqに伝えた一連の命令を抜き出してみましょう。次のようになります。"
314 |    ]
315 |   },
316 |   {
317 |    "cell_type": "code",
318 |    "execution_count": 1,
319 |    "metadata": {
320 |     "coq_kernel_metadata": {
321 |      "auto_roll_back": true,
322 |      "cell_id": "f6945826b13e445788c25fbd1254ca52",
323 |      "execution_id": "548394fdcbcd42faa4958c4cadd30d99"
324 |     },
325 |     "scrolled": true
326 |    },
327 |    "outputs": [
328 |     {
329 |      "data": {
330 |       "text/html": [
331 |        "\n",
332 |        "
\n",
333 |        "    
\n",
334 |        "
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335 |        "\n",
336 |        "
\n",
337 |        "    \n",
338 |        "    Cell evaluated.\n",
339 |        "
\n",
340 |        "\n",
341 |        "
\n",
342 |        "    \n",
343 |        "    Cell rolled back.\n",
344 |        "
\n",
345 |        "\n",
346 |        "
\n",
347 |        "    \n",
351 |        "\n",
352 |        "    
\n",
353 |        "      \n",
354 |        "      Auto rollback\n",
355 |        "    
\n",
356 |        "
\n"
357 |       ],
358 |       "text/plain": []
359 |      },
360 |      "execution_count": 6,
361 |      "metadata": {
362 |       "coq_kernel_evaluated": true,
363 |       "coq_kernel_execution_id": "548394fdcbcd42faa4958c4cadd30d99",
364 |       "coq_kernel_rolled_back": false
365 |      },
366 |      "output_type": "execute_result"
367 |     }
368 |    ],
369 |    "source": [
370 |     "Theorem Hello_Coq' : (forall A : Prop, A -> A ).\n",
371 |     "Proof.\n",
372 |     "  intros.\n",
373 |     "  exact H.\n",
374 |     "Qed."
375 |    ]
376 |   },
377 |   {
378 |    "cell_type": "markdown",
379 |    "metadata": {
380 |     "coq_kernel_metadata": {
381 |      "auto_roll_back": true
382 |     }
383 |    },
384 |    "source": [
385 |     "これだけ見ていると、これが定理Hello_Coq  (forall A : Prop, A -> A )の証明とは、とても思えないと思います。\n",
386 |     "逆に、定理Hello_Coq が与えられた時、上のような証明をすぐに思いつくことは、ほとんどの人にはできないと思います。\n",
387 |     "\n",
388 |     "それには理由があります。Coqの証明は、人間とCoqの「対話」を通じて進行します。人間は、Coqの反応を見て、証明を進めるために次にCoqに何を命ずるかを考えます。そのやり取りの過程が省略されて、その結果だけを抜き出しても、その命令を選択した意図は見えてきません。\n",
389 |     "\n",
390 |     "### 証明を進めるために、人間がCoqに与える命令を tactic と呼びます。\n",
391 |     "\n",
392 |     "tactic というのは「戦略」という意味ですね。Coqの反応を見て、証明を進めるために、どのような攻め方をするのかと考えて出てきた結論が、あるtacticを選択するということです。なぜ、あるtactic が選ばれたかは、tacticの選択の直前のCoqの反応を見てみないとわかりません。この例では、introsやexact H がtacticです。\n",
393 |     "\n",
394 |     "一方、人間の入力のうち、Theorem や Proof やQed は、証明の進め方やCoqの反応とは直接は関係がありません。これらは、tactics とは別のグループのCoqの命令群に族します。これらのCoqコマンドは、大文字で始まっています。tactics は小文字から始まりますので、区別は容易です。\n",
395 |     "\n",
396 |     "### 完成した証明を、「対話」なしに、人間はCoqに与えることができます。\n",
397 |     "\n",
398 |     "では、ある証明が正しいことを示すために、人間は、Coqとの「対話」を毎回繰り返す必要があるのでしょうか？　そうでは、ありません。先のセルを実行してみてください。\"Cell evaluated.\"が出て、証明がすぐに終わることが確認できます。\n",
399 |     "\n",
400 |     "人間とCoqが「対話」していったん完成した証明は、人間はtactic ごとのCoqの反応を見ることなく、それを丸ごとCoqに与えることで証明を実行することができます。これは、証明の完成のためにCoqと「対話」した相手が誰であろうと構わないという点で、実践的には、とても重要なことです。我々は、第三者の行なった証明を、利用することができるのです。ただし、これは、人間はとCoqとの「対話」を通じて証明を作っていく過程とは区別されるべき、「証明の検証」 ともいうべきものです。\n",
401 |     "\n",
402 |     "上のセルのことを、「これだけ見ていると、これが定理Hello_Coq  (forall A : Prop, A -> A )の証明とは、とても思えない」と、先ほど書きましたが、それは人間にとってのこと。Coqは、これを正しく証明として認識し、それが正しい証明かチェックできます。\n",
403 |     "\n",
404 |     "\n",
405 |     "\n",
406 |     "\n"
407 |    ]
408 |   },
409 |   {
410 |    "cell_type": "markdown",
411 |    "metadata": {
412 |     "coq_kernel_metadata": {
413 |      "auto_roll_back": true
414 |     }
415 |    },
416 |    "source": [
417 |     "
\n",
418 |     " 先に実行したセルの中では、定理の名前が、元のHello_CoqからHello_Coq'に変更されています（最後にダッシュ＝プライムが追加されている）それは、元の名前の定理が、このページの演習の中で、すでに「証明済」になっていて、再度の証明の実行が開始できないからです。\n",
419 |     " 
\n",
420 |     "\n",
421 |     "\n",
422 |     "## Coqは、反応として人間に何を伝えたか？\n",
423 |     "\n",
424 |     "ここでは、先の例での人間が intros と打ち込んだ時のCoqの反応を、改めて、見ることにしましょう。\n",
425 |     "次のような情報が含まれていることがわかります。\n",
426 |     "\n",
427 |     "- 証明中の定理の名前\n",
428 |     "- 証明すべきサブゴールの数\n",
429 |     "- 証明の状態を表す大事な情報\n",
430 |     "( この情報は、横線の上下で大きく二つに分かれています。)\n",
431 |     "  - サブゴールを証明するにあたって前提として利用できる仮説\n",
432 |     "  - 証明すべきサブゴール\n",
433 |     "  \n",
434 |     " \n"
435 |    ]
436 |   },
437 |   {
438 |    "cell_type": "code",
439 |    "execution_count": null,
440 |    "metadata": {
441 |     "coq_kernel_metadata": {
442 |      "auto_roll_back": true
443 |     }
444 |    },
445 |    "outputs": [],
446 |    "source": []
447 |   },
448 |   {
449 |    "cell_type": "code",
450 |    "execution_count": null,
451 |    "metadata": {
452 |     "coq_kernel_metadata": {
453 |      "auto_roll_back": true
454 |     }
455 |    },
456 |    "outputs": [],
457 |    "source": []
458 |   },
459 |   {
460 |    "cell_type": "markdown",
461 |    "metadata": {
462 |     "coq_kernel_metadata": {
463 |      "auto_roll_back": true
464 |     }
465 |    },
466 |    "source": [
467 |     "### Coqの反応には、当面の証明で集中すべきサブゴールが含まれています。\n",
468 |     "\n",
469 |     "Coqが返す反応で一番大事なのは、その時点での「証明の状態」を表す情報です。その状態は、直前に人間が投入した tactic コマンドによって変化します。tactics は、まさに、証明の状態を変化させるために、人間が利用するコマンドです。あるtactics コマンドが、どのように証明の状態を変化させるかについては、次章以降で説明したいと思います。ここでは、それについては触れません。ただ、ここでは、「証明の状態」の情報のうち、「サブゴール」といわれるものだけに注目しましょう。\n",
470 |     "\n",
471 |     "Coqの反応の中に含まれるサブゴールというのは、「当面は、この問題の証明に集中しようよ」という、Coqから人間へのサジェスションです。Coqから人間への指示と思っても構いません。人間とCoqは、大きな問題を複数のより簡単な問題に分割して、一つづつ分割された問題を解こうとします。こうして分割された問題の一つが、サブゴールです。ですので、ある問題を解くために必要なサブゴールの数は、一つとは限りません。\n",
472 |     "\n",
473 |     "次の例は、destruct というtactic を実行した結果、二つのサブゴールを持つ証明の状態が生まれたことを示しています。\n",
474 |     " \n"
475 |    ]
476 |   },
477 |   {
478 |    "cell_type": "markdown",
479 |    "metadata": {
480 |     "coq_kernel_metadata": {
481 |      "auto_roll_back": true
482 |     }
483 |    },
484 |    "source": [
485 |     "### Coqでの証明の最終目標は、サブゴールをなくすことです。\n",
486 |     "\n",
487 |     "先に述べたように、Coqは、元の問題をより簡単な問題（サブゴール）に分解して、それをすべて解こうとします。すべてのサブゴールが解かれたとき、元の問題の証明が終わります。別の言い方をすれば、Coqでの証明の最終的な目標は、証明されていないサブゴールの数をゼロにすることです。\n",
488 |     "\n",
489 |     "# 次章の目標\n",
490 |     "\n",
491 |     "次の章は、Coqでの論理式の証明を取り上げます。この章では詳しく触れなかった、あるtactic が証明の状態をどのように変化させるのかを学びます。"
492 |    ]
493 |   },
494 |   {
495 |    "cell_type": "markdown",
496 |    "metadata": {
497 |     "coq_kernel_metadata": {
498 |      "auto_roll_back": true
499 |     }
500 |    },
501 |    "source": [
502 |     "　　　　[全体目次](./0-Contents.ipynb) 　[この章の演習問題へ](./1-logic-answer.ipynb)　　[前の章へ](./0-Introduction.ipynb)　　[次の章へ](./2-Equality.ipynb)"
503 |    ]
504 |   }
505 |  ],
506 |  "metadata": {
507 |   "kernelspec": {
508 |    "display_name": "Coq",
509 |    "language": "coq",
510 |    "name": "coq"
511 |   },
512 |   "language_info": {
513 |    "file_extension": ".v",
514 |    "mimetype": "text/x-coq",
515 |    "name": "coq",
516 |    "version": "8.9.1"
517 |   }
518 |  },
519 |  "nbformat": 4,
520 |  "nbformat_minor": 2
521 | }
522 | 


--------------------------------------------------------------------------------
/.ipynb_checkpoints/0-Introduction-checkpoint.ipynb:
--------------------------------------------------------------------------------
  1 | {
  2 |  "cells": [
  3 |   {
  4 |    "cell_type": "markdown",
  5 |    "metadata": {
  6 |     "coq_kernel_metadata": {
  7 |      "auto_roll_back": true
  8 |     }
  9 |    },
 10 |    "source": [
 11 |     ""
 12 |    ]
 13 |   },
 14 |   {
 15 |    "cell_type": "markdown",
 16 |    "metadata": {
 17 |     "coq_kernel_metadata": {
 18 |      "auto_roll_back": true
 19 |     }
 20 |    },
 21 |    "source": [
 22 |     "　　　　[全体目次](./0-Contents.ipynb)　　[次の章へ](./1-Logic.ipynb)"
 23 |    ]
 24 |   },
 25 |   {
 26 |    "cell_type": "markdown",
 27 |    "metadata": {
 28 |     "coq_kernel_metadata": {
 29 |      "auto_roll_back": true
 30 |     }
 31 |    },
 32 |    "source": [
 33 |     "# Hello Coq !"
 34 |    ]
 35 |   },
 36 |   {
 37 |    "cell_type": "markdown",
 38 |    "metadata": {
 39 |     "coq_kernel_metadata": {
 40 |      "auto_roll_back": true
 41 |     }
 42 |    },
 43 |    "source": [
 44 |     "\n",
 45 |     "# Coqとのはじめての対話\n",
 46 |     "\n",
 47 |     "Coqは、対話型の証明支援システムです。対話するのは人間と機械（Coq）です。\n",
 48 |     "\n",
 49 |     "人間と機械の対話でどのように証明が進んでいくかについては、おいおい説明するとして、ここではまず、Coqとのはじめての対話を経験して見ましょう。簡単な証明をしてみます。\n",
 50 |     "\n",
 51 |     "次の In [1]: で始まる空白の行に、次のブルーの文字列を入力して実行して見てください。"
 52 |    ]
 53 |   },
 54 |   {
 55 |    "cell_type": "markdown",
 56 |    "metadata": {
 57 |     "coq_kernel_metadata": {
 58 |      "auto_roll_back": true
 59 |     }
 60 |    },
 61 |    "source": [
 62 |     "
\n",
 63 |     "    ハンズオン環境 coq-jupyter -- 入力の実行\n",
 64 |     "

\n",
 65 |     "    今回のハンズオンで利用している環境は、jupyter notebookからCoqを呼び出せるようにした coq-jupyter というものです。操作法はjupyterと同じです。\n",
 66 |     " jupyterでは、操作の単位をセルと呼んでいます。 セルに入力したの字列を実行するには、セルにカーソルを合わせて、 画面上部の Runボタンを押します。\n",
 67 |     "
   \n",
 68 |     "\n"
 69 |    ]
 70 |   },
 71 |   {
 72 |    "cell_type": "markdown",
 73 |    "metadata": {
 74 |     "coq_kernel_metadata": {
 75 |      "auto_roll_back": true
 76 |     }
 77 |    },
 78 |    "source": [
 79 |     "
\n",
 80 |     "Shift-RET（シフトキーを押しながらRETキーを押す）でも、セルの実行ができます。\n",
 81 |     "
 \n",
 82 |     "\n",
 83 |     "最初に入力して実行するの、この文字列です。この文字列で一番大事なのは、冒頭の \"Theorem\" です。この文字列は、Coqに証明すべき式と、証明すべき式の名前を伝えています。こうした文字列を Coqコマンドと呼ぶことにしましょう。"
 84 |    ]
 85 |   },
 86 |   {
 87 |    "cell_type": "markdown",
 88 |    "metadata": {
 89 |     "coq_kernel_metadata": {
 90 |      "auto_roll_back": true
 91 |     }
 92 |    },
 93 |    "source": [
 94 |     "Theorem Hello_Coq : (forall A : Prop, A -> A ).\n",
 95 |     "\n",
 96 |     "文字列の入力では、コロンやコンマやピリオド、\"->\"(マイナス記号'-'＋'>' )に注意してください。\n",
 97 |     ""
 98 |    ]
 99 |   },
100 |   {
101 |    "cell_type": "code",
102 |    "execution_count": null,
103 |    "metadata": {
104 |     "coq_kernel_metadata": {
105 |      "auto_roll_back": true,
106 |      "cell_id": "ecf27221f8c64117aa9fd72f870003df",
107 |      "execution_id": "a4e06ef9b93e4d0d8237490a7eec840c"
108 |     },
109 |     "scrolled": true
110 |    },
111 |    "outputs": [],
112 |    "source": []
113 |   },
114 |   {
115 |    "cell_type": "markdown",
116 |    "metadata": {
117 |     "coq_kernel_metadata": {
118 |      "auto_roll_back": true
119 |     }
120 |    },
121 |    "source": [
122 |     "入力がおわったら、先のセルを実行してみてください。Coqから次のような反応があるはずです。\n",
123 |     "ここで初めてCoqに証明したい定理の情報を伝えました。Coqが返した情報の意味は、次のようなものです。\n",
124 |     "\n",
125 |     "\n"
126 |    ]
127 |   },
128 |   {
129 |    "cell_type": "markdown",
130 |    "metadata": {
131 |     "coq_kernel_metadata": {
132 |      "auto_roll_back": true
133 |     }
134 |    },
135 |    "source": [
136 |     "今度は、次の文字列を入力して実行してください。（最後の、ピリオドを忘れずに）\n",
137 |     "

\n",
138 |     "Proof.\n",
139 |     "
"
140 |    ]
141 |   },
142 |   {
143 |    "cell_type": "code",
144 |    "execution_count": null,
145 |    "metadata": {
146 |     "coq_kernel_metadata": {
147 |      "auto_roll_back": true,
148 |      "cell_id": "84d24fe8d5bd418f909b9d922ad3f4bf",
149 |      "execution_id": "9B0735FF845E4B73820D47975CE978CF"
150 |     },
151 |     "scrolled": true
152 |    },
153 |    "outputs": [],
154 |    "source": []
155 |   },
156 |   {
157 |    "cell_type": "markdown",
158 |    "metadata": {
159 |     "coq_kernel_metadata": {
160 |      "auto_roll_back": true
161 |     }
162 |    },
163 |    "source": [
164 |     "次のような反応がcoqからあるはずです。基本的には、先の反応と同じです。\n",
165 |     "coqコマンドの \"Proof.\"は、これから証明が始まることを宣言しています。\n",
166 |     "\n",
167 |     "ここでは、先に説明していなかった下二行の説明をしています。これらの出力は、coq-jupyterからのものです。\n",
168 |     "\"Cell evaluated\"がでていれば、証明は、順調に進んでいると思って構いません。\n",
169 |     "\n",
170 |     "\n",
171 |     ""
172 |    ]
173 |   },
174 |   {
175 |    "cell_type": "markdown",
176 |    "metadata": {
177 |     "coq_kernel_metadata": {
178 |      "auto_roll_back": true
179 |     }
180 |    },
181 |    "source": [
182 |     "今度は、次の文字列を入力して実行してください。（最後の、ピリオドを忘れずに）\n",
183 |     "

\n",
184 |     "intros.\n",
185 |     "
"
186 |    ]
187 |   },
188 |   {
189 |    "cell_type": "code",
190 |    "execution_count": null,
191 |    "metadata": {
192 |     "coq_kernel_metadata": {
193 |      "auto_roll_back": true,
194 |      "cell_id": "2b72a7e19d34461982cfd26a9dad43de",
195 |      "execution_id": "DAD112078A37407E8EE32A0E3FF9DA77"
196 |     },
197 |     "scrolled": false
198 |    },
199 |    "outputs": [],
200 |    "source": []
201 |   },
202 |   {
203 |    "cell_type": "markdown",
204 |    "metadata": {
205 |     "coq_kernel_metadata": {
206 |      "auto_roll_back": true
207 |     }
208 |    },
209 |    "source": [
210 |     "次のような反応があるはずです。その意味を説明しています。\n",
211 |     ""
212 |    ]
213 |   },
214 |   {
215 |    "cell_type": "markdown",
216 |    "metadata": {
217 |     "coq_kernel_metadata": {
218 |      "auto_roll_back": true
219 |     }
220 |    },
221 |    "source": [
222 |     "今度は、次の文字列を入力して実行してください。（最後の、ピリオドを忘れずに）\n",
223 |     "

\n",
224 |     "exact H.\n",
225 |     "
"
226 |    ]
227 |   },
228 |   {
229 |    "cell_type": "code",
230 |    "execution_count": null,
231 |    "metadata": {
232 |     "coq_kernel_metadata": {
233 |      "auto_roll_back": true
234 |     }
235 |    },
236 |    "outputs": [],
237 |    "source": []
238 |   },
239 |   {
240 |    "cell_type": "markdown",
241 |    "metadata": {
242 |     "coq_kernel_metadata": {
243 |      "auto_roll_back": true
244 |     }
245 |    },
246 |    "source": [
247 |     "次のような反応があるはずです。\n",
248 |     "\n",
249 |     "\n",
250 |     ""
251 |    ]
252 |   },
253 |   {
254 |    "cell_type": "markdown",
255 |    "metadata": {
256 |     "coq_kernel_metadata": {
257 |      "auto_roll_back": true
258 |     }
259 |    },
260 |    "source": [
261 |     "今度は、次の文字列を入力して実行してください。（最後の、ピリオドを忘れずに）\n",
262 |     "

\n",
263 |     "Qed..\n",
264 |     "
"
265 |    ]
266 |   },
267 |   {
268 |    "cell_type": "code",
269 |    "execution_count": null,
270 |    "metadata": {
271 |     "coq_kernel_metadata": {
272 |      "auto_roll_back": true,
273 |      "cell_id": "868b4dae6c1c456abad8bd188cf0d134",
274 |      "execution_id": "5236AD7A7C9F41838CE5B3A9F37F2C4E"
275 |     },
276 |     "scrolled": true
277 |    },
278 |    "outputs": [],
279 |    "source": []
280 |   },
281 |   {
282 |    "cell_type": "markdown",
283 |    "metadata": {
284 |     "coq_kernel_metadata": {
285 |      "auto_roll_back": true
286 |     }
287 |    },
288 |    "source": [
289 |     "次のような反応ががあるはずです。\n",
290 |     "\n",
291 |     ""
292 |    ]
293 |   },
294 |   {
295 |    "cell_type": "markdown",
296 |    "metadata": {
297 |     "coq_kernel_metadata": {
298 |      "auto_roll_back": true
299 |     }
300 |    },
301 |    "source": [
302 |     "これで、定理 Hello_Coq の証明は、終了です。\n",
303 |     "\n",
304 |     "\n",
305 |     "\n",
306 |     "# Coqとのはじめての対話をふりかえる。\n",
307 |     "\n",
308 |     "Coqとのはじめての対話を、人間の側とCoqの側の、それぞれから振り返ってみましょう。\n",
309 |     "\n",
310 |     "\n",
311 |     "## 人間は、Coqに何を伝えたか？\n",
312 |     "\n",
313 |     "人間が、Coqに伝えた一連の命令を抜き出してみましょう。次のようになります。"
314 |    ]
315 |   },
316 |   {
317 |    "cell_type": "code",
318 |    "execution_count": 1,
319 |    "metadata": {
320 |     "coq_kernel_metadata": {
321 |      "auto_roll_back": true,
322 |      "cell_id": "f6945826b13e445788c25fbd1254ca52",
323 |      "execution_id": "548394fdcbcd42faa4958c4cadd30d99"
324 |     },
325 |     "scrolled": true
326 |    },
327 |    "outputs": [
328 |     {
329 |      "data": {
330 |       "text/html": [
331 |        "\n",
332 |        "
\n",
333 |        "    
\n",
334 |        "
\n",
335 |        "\n",
336 |        "
\n",
337 |        "    \n",
338 |        "    Cell evaluated.\n",
339 |        "
\n",
340 |        "\n",
341 |        "
\n",
342 |        "    \n",
343 |        "    Cell rolled back.\n",
344 |        "
\n",
345 |        "\n",
346 |        "
\n",
347 |        "    \n",
351 |        "\n",
352 |        "    
\n",
353 |        "      \n",
354 |        "      Auto rollback\n",
355 |        "    
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356 |        "
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357 |       ],
358 |       "text/plain": []
359 |      },
360 |      "execution_count": 6,
361 |      "metadata": {
362 |       "coq_kernel_evaluated": true,
363 |       "coq_kernel_execution_id": "548394fdcbcd42faa4958c4cadd30d99",
364 |       "coq_kernel_rolled_back": false
365 |      },
366 |      "output_type": "execute_result"
367 |     }
368 |    ],
369 |    "source": [
370 |     "Theorem Hello_Coq' : (forall A : Prop, A -> A ).\n",
371 |     "Proof.\n",
372 |     "  intros.\n",
373 |     "  exact H.\n",
374 |     "Qed."
375 |    ]
376 |   },
377 |   {
378 |    "cell_type": "markdown",
379 |    "metadata": {
380 |     "coq_kernel_metadata": {
381 |      "auto_roll_back": true
382 |     }
383 |    },
384 |    "source": [
385 |     "これだけ見ていると、これが定理Hello_Coq  (forall A : Prop, A -> A )の証明とは、とても思えないと思います。\n",
386 |     "逆に、定理Hello_Coq が与えられた時、上のような証明をすぐに思いつくことは、ほとんどの人にはできないと思います。\n",
387 |     "\n",
388 |     "それには理由があります。Coqの証明は、人間とCoqの「対話」を通じて進行します。人間は、Coqの反応を見て、証明を進めるために次にCoqに何を命ずるかを考えます。そのやり取りの過程が省略されて、その結果だけを抜き出しても、その命令を選択した意図は見えてきません。\n",
389 |     "\n",
390 |     "### 証明を進めるために、人間がCoqに与える命令を tactic と呼びます。\n",
391 |     "\n",
392 |     "tactic というのは「戦略」という意味ですね。Coqの反応を見て、証明を進めるために、どのような攻め方をするのかと考えて出てきた結論が、あるtacticを選択するということです。なぜ、あるtactic が選ばれたかは、tacticの選択の直前のCoqの反応を見てみないとわかりません。この例では、introsやexact H がtacticです。\n",
393 |     "\n",
394 |     "一方、人間の入力のうち、Theorem や Proof やQed は、証明の進め方やCoqの反応とは直接は関係がありません。これらは、tactics とは別のグループのCoqの命令群に族します。これらのCoqコマンドは、大文字で始まっています。tactics は小文字から始まりますので、区別は容易です。\n",
395 |     "\n",
396 |     "### 完成した証明を、「対話」なしに、人間はCoqに与えることができます。\n",
397 |     "\n",
398 |     "では、ある証明が正しいことを示すために、人間は、Coqとの「対話」を毎回繰り返す必要があるのでしょうか？　そうでは、ありません。先のセルを実行してみてください。\"Cell evaluated.\"が出て、証明がすぐに終わることが確認できます。\n",
399 |     "\n",
400 |     "人間とCoqが「対話」していったん完成した証明は、人間はtactic ごとのCoqの反応を見ることなく、それを丸ごとCoqに与えることで証明を実行することができます。これは、証明の完成のためにCoqと「対話」した相手が誰であろうと構わないという点で、実践的には、とても重要なことです。我々は、第三者の行なった証明を、利用することができるのです。ただし、これは、人間はとCoqとの「対話」を通じて証明を作っていく過程とは区別されるべき、「証明の検証」 ともいうべきものです。\n",
401 |     "\n",
402 |     "上のセルのことを、「これだけ見ていると、これが定理Hello_Coq  (forall A : Prop, A -> A )の証明とは、とても思えない」と、先ほど書きましたが、それは人間にとってのこと。Coqは、これを正しく証明として認識し、それが正しい証明かチェックできます。\n",
403 |     "\n",
404 |     "\n",
405 |     "\n",
406 |     "\n"
407 |    ]
408 |   },
409 |   {
410 |    "cell_type": "markdown",
411 |    "metadata": {
412 |     "coq_kernel_metadata": {
413 |      "auto_roll_back": true
414 |     }
415 |    },
416 |    "source": [
417 |     "
\n",
418 |     " 先に実行したセルの中では、定理の名前が、元のHello_CoqからHello_Coq'に変更されています（最後にダッシュ＝プライムが追加されている）それは、元の名前の定理が、このページの演習の中で、すでに「証明済」になっていて、再度の証明の実行が開始できないからです。\n",
419 |     " 
\n",
420 |     "\n",
421 |     "\n",
422 |     "## Coqは、反応として人間に何を伝えたか？\n",
423 |     "\n",
424 |     "ここでは、先の例での人間が intros と打ち込んだ時のCoqの反応を、改めて、見ることにしましょう。\n",
425 |     "次のような情報が含まれていることがわかります。\n",
426 |     "\n",
427 |     "- 証明中の定理の名前\n",
428 |     "- 証明すべきサブゴールの数\n",
429 |     "- 証明の状態を表す大事な情報\n",
430 |     "( この情報は、横線の上下で大きく二つに分かれています。)\n",
431 |     "  - サブゴールを証明するにあたって前提として利用できる仮説\n",
432 |     "  - 証明すべきサブゴール\n",
433 |     "  \n",
434 |     " \n"
435 |    ]
436 |   },
437 |   {
438 |    "cell_type": "code",
439 |    "execution_count": null,
440 |    "metadata": {
441 |     "coq_kernel_metadata": {
442 |      "auto_roll_back": true
443 |     }
444 |    },
445 |    "outputs": [],
446 |    "source": []
447 |   },
448 |   {
449 |    "cell_type": "code",
450 |    "execution_count": null,
451 |    "metadata": {
452 |     "coq_kernel_metadata": {
453 |      "auto_roll_back": true
454 |     }
455 |    },
456 |    "outputs": [],
457 |    "source": []
458 |   },
459 |   {
460 |    "cell_type": "markdown",
461 |    "metadata": {
462 |     "coq_kernel_metadata": {
463 |      "auto_roll_back": true
464 |     }
465 |    },
466 |    "source": [
467 |     "### Coqの反応には、当面の証明で集中すべきサブゴールが含まれています。\n",
468 |     "\n",
469 |     "Coqが返す反応で一番大事なのは、その時点での「証明の状態」を表す情報です。その状態は、直前に人間が投入した tactic コマンドによって変化します。tactics は、まさに、証明の状態を変化させるために、人間が利用するコマンドです。あるtactics コマンドが、どのように証明の状態を変化させるかについては、次章以降で説明したいと思います。ここでは、それについては触れません。ただ、ここでは、「証明の状態」の情報のうち、「サブゴール」といわれるものだけに注目しましょう。\n",
470 |     "\n",
471 |     "Coqの反応の中に含まれるサブゴールというのは、「当面は、この問題の証明に集中しようよ」という、Coqから人間へのサジェスションです。Coqから人間への指示と思っても構いません。人間とCoqは、大きな問題を複数のより簡単な問題に分割して、一つづつ分割された問題を解こうとします。こうして分割された問題の一つが、サブゴールです。ですので、ある問題を解くために必要なサブゴールの数は、一つとは限りません。\n",
472 |     "\n",
473 |     "次の例は、destruct というtactic を実行した結果、二つのサブゴールを持つ証明の状態が生まれたことを示しています。\n",
474 |     " \n"
475 |    ]
476 |   },
477 |   {
478 |    "cell_type": "markdown",
479 |    "metadata": {
480 |     "coq_kernel_metadata": {
481 |      "auto_roll_back": true
482 |     }
483 |    },
484 |    "source": [
485 |     "### Coqでの証明の最終目標は、サブゴールをなくすことです。\n",
486 |     "\n",
487 |     "先に述べたように、Coqは、元の問題をより簡単な問題（サブゴール）に分解して、それをすべて解こうとします。すべてのサブゴールが解かれたとき、元の問題の証明が終わります。別の言い方をすれば、Coqでの証明の最終的な目標は、証明されていないサブゴールの数をゼロにすることです。\n",
488 |     "\n",
489 |     "# 次章の目標\n",
490 |     "\n",
491 |     "次の章は、Coqでの論理式の証明を取り上げます。この章では詳しく触れなかった、あるtactic が証明の状態をどのように変化させるのかを学びます。"
492 |    ]
493 |   },
494 |   {
495 |    "cell_type": "markdown",
496 |    "metadata": {
497 |     "coq_kernel_metadata": {
498 |      "auto_roll_back": true
499 |     }
500 |    },
501 |    "source": [
502 |     "　　　　[全体目次](./0-Contents.ipynb) 　[この章の演習問題へ](./1-logic-answer.ipynb)　　[前の章へ](./0-Introduction.ipynb)　　[次の章へ](./2-Equality.ipynb)"
503 |    ]
504 |   }
505 |  ],
506 |  "metadata": {
507 |   "kernelspec": {
508 |    "display_name": "Coq",
509 |    "language": "coq",
510 |    "name": "coq"
511 |   },
512 |   "language_info": {
513 |    "file_extension": ".v",
514 |    "mimetype": "text/x-coq",
515 |    "name": "coq",
516 |    "version": "8.9.1"
517 |   }
518 |  },
519 |  "nbformat": 4,
520 |  "nbformat_minor": 2
521 | }
522 | 


--------------------------------------------------------------------------------
/.ipynb_checkpoints/Equal-Ex-Work-checkpoint.ipynb:
--------------------------------------------------------------------------------
  1 | {
  2 |  "cells": [
  3 |   {
  4 |    "cell_type": "markdown",
  5 |    "metadata": {
  6 |     "coq_kernel_metadata": {
  7 |      "auto_roll_back": true
  8 |     }
  9 |    },
 10 |    "source": [
 11 |     "# 演習問題 2  作業用ページ\n",
 12 |     "\n",
 13 |     "[演習問題 2-1](./Equal-Ex-Work.ipynb#ex21) [演習問題 2-2](./Equal-Ex-Work.ipynb#ex22) [演習問題 2-3](./Equal-Ex-Work.ipynb#ex23) [演習問題 2-4](./Equal-Ex-Work.ipynb#ex24) [演習問題 2-5](./Equal-Ex-Work.ipynb#ex25) [演習問題 2-6](./Equal-Ex-Work.ipynb#ex26) [演習問題 2-7](./Equal-Ex-Work.ipynb#ex27) "
 14 |    ]
 15 |   },
 16 |   {
 17 |    "cell_type": "markdown",
 18 |    "metadata": {
 19 |     "coq_kernel_metadata": {
 20 |      "auto_roll_back": true
 21 |     }
 22 |    },
 23 |    "source": [
 24 |     "\n",
 25 |     "## 演習問題 2-1\n",
 26 |     "　次の定理を証明せよ。\n",
 27 |     " 
　Theorem plus_O_n : (forall n : nat , 0+n = n ).\n",
 28 |     "\n",
 29 |     "### ヒント\n",
 30 |     "- 冒頭の forall を intros で削除する。\n",
 31 |     "- A -> A の最初のAを intros で仮説に入れる。\n",
 32 |     "- そうすると、サブゴールが仮説の一つと一致していることがわかるので、assumption  または、exact  を使う。\n",
 33 |     "\n",
 34 |     "###  [解答を見る](./Equal-Ex-Answer.ipynb#ex21) "
 35 |    ]
 36 |   },
 37 |   {
 38 |    "cell_type": "code",
 39 |    "execution_count": null,
 40 |    "metadata": {
 41 |     "coq_kernel_metadata": {
 42 |      "auto_roll_back": true
 43 |     }
 44 |    },
 45 |    "outputs": [],
 46 |    "source": [
 47 |     "Theorem plus_O_n : (forall n : nat , 0+n = n ).\n",
 48 |     "Proof."
 49 |    ]
 50 |   },
 51 |   {
 52 |    "cell_type": "code",
 53 |    "execution_count": null,
 54 |    "metadata": {
 55 |     "coq_kernel_metadata": {
 56 |      "auto_roll_back": true
 57 |     }
 58 |    },
 59 |    "outputs": [],
 60 |    "source": []
 61 |   },
 62 |   {
 63 |    "cell_type": "code",
 64 |    "execution_count": null,
 65 |    "metadata": {
 66 |     "coq_kernel_metadata": {
 67 |      "auto_roll_back": true
 68 |     }
 69 |    },
 70 |    "outputs": [],
 71 |    "source": []
 72 |   },
 73 |   {
 74 |    "cell_type": "code",
 75 |    "execution_count": null,
 76 |    "metadata": {
 77 |     "coq_kernel_metadata": {
 78 |      "auto_roll_back": true
 79 |     }
 80 |    },
 81 |    "outputs": [],
 82 |    "source": []
 83 |   },
 84 |   {
 85 |    "cell_type": "code",
 86 |    "execution_count": null,
 87 |    "metadata": {
 88 |     "coq_kernel_metadata": {
 89 |      "auto_roll_back": true
 90 |     }
 91 |    },
 92 |    "outputs": [],
 93 |    "source": []
 94 |   },
 95 |   {
 96 |    "cell_type": "code",
 97 |    "execution_count": null,
 98 |    "metadata": {
 99 |     "coq_kernel_metadata": {
100 |      "auto_roll_back": true
101 |     }
102 |    },
103 |    "outputs": [],
104 |    "source": []
105 |   },
106 |   {
107 |    "cell_type": "code",
108 |    "execution_count": null,
109 |    "metadata": {
110 |     "coq_kernel_metadata": {
111 |      "auto_roll_back": true
112 |     }
113 |    },
114 |    "outputs": [],
115 |    "source": [
116 |     "Admitted. (* 最後にここを Qed. に変える。*)"
117 |    ]
118 |   },
119 |   {
120 |    "cell_type": "markdown",
121 |    "metadata": {
122 |     "coq_kernel_metadata": {
123 |      "auto_roll_back": true
124 |     }
125 |    },
126 |    "source": [
127 |     "\n",
128 |     "## 演習問題 2-2\n",
129 |     "　次の定理を証明せよ。\n",
130 |     " 
 Theorem plus_1_n : (forall n : nat , 1+n = S n ).\n",
131 |     "\n",
132 |     "### ヒント\n",
133 |     "- intros で forall を削除する。\n",
134 |     "- simpl. で、サブゴールの計算式を簡単にする。\n",
135 |     "- 左項と右項が等しかったら reflexivity を使う。\n",
136 |     "\n",
137 |     "###  [解答を見る](./Equal-Ex-Answer.ipynb#ex22) "
138 |    ]
139 |   },
140 |   {
141 |    "cell_type": "code",
142 |    "execution_count": null,
143 |    "metadata": {
144 |     "coq_kernel_metadata": {
145 |      "auto_roll_back": true
146 |     }
147 |    },
148 |    "outputs": [],
149 |    "source": [
150 |     "Theorem plus_1_n : (forall n : nat , 1+n = S n ).\n",
151 |     "Proof."
152 |    ]
153 |   },
154 |   {
155 |    "cell_type": "code",
156 |    "execution_count": null,
157 |    "metadata": {
158 |     "coq_kernel_metadata": {
159 |      "auto_roll_back": true
160 |     }
161 |    },
162 |    "outputs": [],
163 |    "source": []
164 |   },
165 |   {
166 |    "cell_type": "code",
167 |    "execution_count": null,
168 |    "metadata": {
169 |     "coq_kernel_metadata": {
170 |      "auto_roll_back": true
171 |     }
172 |    },
173 |    "outputs": [],
174 |    "source": []
175 |   },
176 |   {
177 |    "cell_type": "code",
178 |    "execution_count": null,
179 |    "metadata": {
180 |     "coq_kernel_metadata": {
181 |      "auto_roll_back": true
182 |     }
183 |    },
184 |    "outputs": [],
185 |    "source": []
186 |   },
187 |   {
188 |    "cell_type": "code",
189 |    "execution_count": null,
190 |    "metadata": {
191 |     "coq_kernel_metadata": {
192 |      "auto_roll_back": true
193 |     }
194 |    },
195 |    "outputs": [],
196 |    "source": []
197 |   },
198 |   {
199 |    "cell_type": "code",
200 |    "execution_count": null,
201 |    "metadata": {
202 |     "coq_kernel_metadata": {
203 |      "auto_roll_back": true
204 |     }
205 |    },
206 |    "outputs": [],
207 |    "source": []
208 |   },
209 |   {
210 |    "cell_type": "code",
211 |    "execution_count": null,
212 |    "metadata": {
213 |     "coq_kernel_metadata": {
214 |      "auto_roll_back": true
215 |     }
216 |    },
217 |    "outputs": [],
218 |    "source": []
219 |   },
220 |   {
221 |    "cell_type": "code",
222 |    "execution_count": null,
223 |    "metadata": {
224 |     "coq_kernel_metadata": {
225 |      "auto_roll_back": true
226 |     }
227 |    },
228 |    "outputs": [],
229 |    "source": []
230 |   },
231 |   {
232 |    "cell_type": "code",
233 |    "execution_count": null,
234 |    "metadata": {
235 |     "coq_kernel_metadata": {
236 |      "auto_roll_back": true
237 |     }
238 |    },
239 |    "outputs": [],
240 |    "source": [
241 |     "Admitted. (* 最後にここを Qed. に変える。*)"
242 |    ]
243 |   },
244 |   {
245 |    "cell_type": "markdown",
246 |    "metadata": {
247 |     "coq_kernel_metadata": {
248 |      "auto_roll_back": true
249 |     }
250 |    },
251 |    "source": [
252 |     "\n",
253 |     "## 演習問題 2-3\n",
254 |     "　次の定理を証明せよ。\n",
255 |     " 
 Theorem mult_0_n : (forall n : nat , 0 * n =  0 ).\n",
256 |     "\n",
257 |     "### ヒント\n",
258 |     "- intros で forall を削除する。\n",
259 |     "- simpl. で、サブゴールの計算式を簡単にする。\n",
260 |     "- 左項と右項が等しかったら reflexivity を使う。\n",
261 |     "\n",
262 |     "###  [解答を見る](./Equal-Ex-Answer.ipynb#ex23) "
263 |    ]
264 |   },
265 |   {
266 |    "cell_type": "markdown",
267 |    "metadata": {
268 |     "coq_kernel_metadata": {
269 |      "auto_roll_back": true
270 |     }
271 |    },
272 |    "source": [
273 |     "\n",
274 |     "## 演習問題 2-4\n",
275 |     "　次の定理を証明せよ。\n",
276 |     " 
 Theorem plus_id_nm : (forall n m : nat , n = m -> n+n = m + m ).\n",
277 |     "\n",
278 |     "### ヒント\n",
279 |     "- intros で forall を消す。\n",
280 |     "- サブゴールの前提部分を intros で仮定に移す。\n",
281 |     "- サブゴールの左項を、等式Hを使って rewriite で書き換える。\n",
282 |     "- サブゴールの左項と右項が等しいので、reflexivity を使う。\n",
283 |     "\n",
284 |     "###  [解答を見る](./Equal-Ex-Answer.ipynb#ex24) "
285 |    ]
286 |   },
287 |   {
288 |    "cell_type": "code",
289 |    "execution_count": null,
290 |    "metadata": {
291 |     "coq_kernel_metadata": {
292 |      "auto_roll_back": true
293 |     }
294 |    },
295 |    "outputs": [],
296 |    "source": [
297 |     "Theorem plus_id_nm : (forall n m : nat , n = m -> n+n = m + m ).\n",
298 |     "Proof."
299 |    ]
300 |   },
301 |   {
302 |    "cell_type": "code",
303 |    "execution_count": null,
304 |    "metadata": {
305 |     "coq_kernel_metadata": {
306 |      "auto_roll_back": true
307 |     }
308 |    },
309 |    "outputs": [],
310 |    "source": []
311 |   },
312 |   {
313 |    "cell_type": "code",
314 |    "execution_count": null,
315 |    "metadata": {
316 |     "coq_kernel_metadata": {
317 |      "auto_roll_back": true
318 |     }
319 |    },
320 |    "outputs": [],
321 |    "source": []
322 |   },
323 |   {
324 |    "cell_type": "code",
325 |    "execution_count": null,
326 |    "metadata": {
327 |     "coq_kernel_metadata": {
328 |      "auto_roll_back": true
329 |     }
330 |    },
331 |    "outputs": [],
332 |    "source": []
333 |   },
334 |   {
335 |    "cell_type": "code",
336 |    "execution_count": null,
337 |    "metadata": {
338 |     "coq_kernel_metadata": {
339 |      "auto_roll_back": true
340 |     }
341 |    },
342 |    "outputs": [],
343 |    "source": []
344 |   },
345 |   {
346 |    "cell_type": "code",
347 |    "execution_count": null,
348 |    "metadata": {
349 |     "coq_kernel_metadata": {
350 |      "auto_roll_back": true
351 |     }
352 |    },
353 |    "outputs": [],
354 |    "source": []
355 |   },
356 |   {
357 |    "cell_type": "code",
358 |    "execution_count": null,
359 |    "metadata": {
360 |     "coq_kernel_metadata": {
361 |      "auto_roll_back": true
362 |     }
363 |    },
364 |    "outputs": [],
365 |    "source": [
366 |     "Admitted. (* 最後にここを Qed. に変える。*)"
367 |    ]
368 |   },
369 |   {
370 |    "cell_type": "markdown",
371 |    "metadata": {
372 |     "coq_kernel_metadata": {
373 |      "auto_roll_back": true
374 |     }
375 |    },
376 |    "source": [
377 |     "\n",
378 |     "## 演習問題 2-5\n",
379 |     "　次の定理を証明せよ。\n",
380 |     "  
 Theorem plus_id : (forall n m o : nat , n = m -> m = o -> n + m = m + o ).\n",
381 |     "\n",
382 |     "### ヒント\n",
383 |     "- intros で forall を削除 \n",
384 |     "- intoros で、前提部分を仮説に移す。\n",
385 |     "- 等式　n_eq_m　の左辺 を使って、rewriteでサブゴールを書き換える。\n",
386 |     "- 等式　m_eq_o の右辺を使って、rewriteでサブゴールのを書き換える。\n",
387 |     "- サブゴールの等式の左辺と右辺が等しくなるので reflexivity を使う。\n",
388 |     "\n",
389 |     "###  [解答を見る](./Equal-Ex-Answer.ipynb#ex25) "
390 |    ]
391 |   },
392 |   {
393 |    "cell_type": "code",
394 |    "execution_count": null,
395 |    "metadata": {
396 |     "coq_kernel_metadata": {
397 |      "auto_roll_back": true
398 |     }
399 |    },
400 |    "outputs": [],
401 |    "source": [
402 |     "Theorem plus_id : (forall n m o : nat , n = m -> m = o -> n + m = m + o ).\n",
403 |     "Proof."
404 |    ]
405 |   },
406 |   {
407 |    "cell_type": "code",
408 |    "execution_count": null,
409 |    "metadata": {
410 |     "coq_kernel_metadata": {
411 |      "auto_roll_back": true
412 |     }
413 |    },
414 |    "outputs": [],
415 |    "source": []
416 |   },
417 |   {
418 |    "cell_type": "code",
419 |    "execution_count": null,
420 |    "metadata": {
421 |     "coq_kernel_metadata": {
422 |      "auto_roll_back": true
423 |     }
424 |    },
425 |    "outputs": [],
426 |    "source": []
427 |   },
428 |   {
429 |    "cell_type": "code",
430 |    "execution_count": null,
431 |    "metadata": {
432 |     "coq_kernel_metadata": {
433 |      "auto_roll_back": true
434 |     }
435 |    },
436 |    "outputs": [],
437 |    "source": []
438 |   },
439 |   {
440 |    "cell_type": "code",
441 |    "execution_count": null,
442 |    "metadata": {
443 |     "coq_kernel_metadata": {
444 |      "auto_roll_back": true
445 |     }
446 |    },
447 |    "outputs": [],
448 |    "source": []
449 |   },
450 |   {
451 |    "cell_type": "code",
452 |    "execution_count": null,
453 |    "metadata": {
454 |     "coq_kernel_metadata": {
455 |      "auto_roll_back": true
456 |     }
457 |    },
458 |    "outputs": [],
459 |    "source": []
460 |   },
461 |   {
462 |    "cell_type": "code",
463 |    "execution_count": null,
464 |    "metadata": {
465 |     "coq_kernel_metadata": {
466 |      "auto_roll_back": true
467 |     }
468 |    },
469 |    "outputs": [],
470 |    "source": []
471 |   },
472 |   {
473 |    "cell_type": "code",
474 |    "execution_count": null,
475 |    "metadata": {
476 |     "coq_kernel_metadata": {
477 |      "auto_roll_back": true
478 |     }
479 |    },
480 |    "outputs": [],
481 |    "source": []
482 |   },
483 |   {
484 |    "cell_type": "code",
485 |    "execution_count": null,
486 |    "metadata": {
487 |     "coq_kernel_metadata": {
488 |      "auto_roll_back": true
489 |     }
490 |    },
491 |    "outputs": [],
492 |    "source": []
493 |   },
494 |   {
495 |    "cell_type": "code",
496 |    "execution_count": null,
497 |    "metadata": {
498 |     "coq_kernel_metadata": {
499 |      "auto_roll_back": true
500 |     }
501 |    },
502 |    "outputs": [],
503 |    "source": []
504 |   },
505 |   {
506 |    "cell_type": "code",
507 |    "execution_count": null,
508 |    "metadata": {
509 |     "coq_kernel_metadata": {
510 |      "auto_roll_back": true
511 |     }
512 |    },
513 |    "outputs": [],
514 |    "source": [
515 |     "Admitted. (* 最後にここを Qed. に変える。*)"
516 |    ]
517 |   },
518 |   {
519 |    "cell_type": "markdown",
520 |    "metadata": {
521 |     "coq_kernel_metadata": {
522 |      "auto_roll_back": true
523 |     }
524 |    },
525 |    "source": [
526 |     "\n",
527 |     "## 演習問題 2-6\n",
528 |     "　次の定理を証明せよ。\n",
529 |     " 
 Theorem mult_0_plus : (forall n m : nat , ( 0 + n ) \\* m =  n \\* m ).\n",
530 |     "\n",
531 |     "### ヒント\n",
532 |     "- intros で forall を消す。\n",
533 |     "- simpl.　で、サブゴール中の計算式を簡単なものにする。\n",
534 |     "- 左辺と右辺が等しいので、reflexivity を使う。\n",
535 |     "\n",
536 |     "###  [解答を見る](./Equal-Ex-Answer.ipynb#ex26) "
537 |    ]
538 |   },
539 |   {
540 |    "cell_type": "code",
541 |    "execution_count": null,
542 |    "metadata": {
543 |     "coq_kernel_metadata": {
544 |      "auto_roll_back": true
545 |     }
546 |    },
547 |    "outputs": [],
548 |    "source": [
549 |     "Theorem mult_0_plus : (forall n m : nat , ( 0 + n ) * m =  n * m )."
550 |    ]
551 |   },
552 |   {
553 |    "cell_type": "code",
554 |    "execution_count": null,
555 |    "metadata": {
556 |     "coq_kernel_metadata": {
557 |      "auto_roll_back": true
558 |     }
559 |    },
560 |    "outputs": [],
561 |    "source": []
562 |   },
563 |   {
564 |    "cell_type": "code",
565 |    "execution_count": null,
566 |    "metadata": {
567 |     "coq_kernel_metadata": {
568 |      "auto_roll_back": true
569 |     }
570 |    },
571 |    "outputs": [],
572 |    "source": []
573 |   },
574 |   {
575 |    "cell_type": "code",
576 |    "execution_count": null,
577 |    "metadata": {
578 |     "coq_kernel_metadata": {
579 |      "auto_roll_back": true
580 |     }
581 |    },
582 |    "outputs": [],
583 |    "source": []
584 |   },
585 |   {
586 |    "cell_type": "code",
587 |    "execution_count": null,
588 |    "metadata": {
589 |     "coq_kernel_metadata": {
590 |      "auto_roll_back": true
591 |     }
592 |    },
593 |    "outputs": [],
594 |    "source": []
595 |   },
596 |   {
597 |    "cell_type": "code",
598 |    "execution_count": null,
599 |    "metadata": {
600 |     "coq_kernel_metadata": {
601 |      "auto_roll_back": true
602 |     }
603 |    },
604 |    "outputs": [],
605 |    "source": []
606 |   },
607 |   {
608 |    "cell_type": "code",
609 |    "execution_count": null,
610 |    "metadata": {
611 |     "coq_kernel_metadata": {
612 |      "auto_roll_back": true
613 |     }
614 |    },
615 |    "outputs": [],
616 |    "source": []
617 |   },
618 |   {
619 |    "cell_type": "code",
620 |    "execution_count": null,
621 |    "metadata": {
622 |     "coq_kernel_metadata": {
623 |      "auto_roll_back": true
624 |     }
625 |    },
626 |    "outputs": [],
627 |    "source": []
628 |   },
629 |   {
630 |    "cell_type": "code",
631 |    "execution_count": null,
632 |    "metadata": {
633 |     "coq_kernel_metadata": {
634 |      "auto_roll_back": true
635 |     }
636 |    },
637 |    "outputs": [],
638 |    "source": [
639 |     "Admitted. (* 最後にここを Qed. に変える。*)"
640 |    ]
641 |   },
642 |   {
643 |    "cell_type": "markdown",
644 |    "metadata": {
645 |     "coq_kernel_metadata": {
646 |      "auto_roll_back": true
647 |     }
648 |    },
649 |    "source": [
650 |     "\n",
651 |     "## 演習問題 2-7\n",
652 |     "　次の定理を証明せよ。\n",
653 |     " 
 Theorem mult_S_1 : (forall n m : nat, m = S n -> m \\* (1 + n) = m \\* m).\n",
654 |     "\n",
655 |     "### ヒント\n",
656 |     "- intros で　forall を消し、前提を仮説に移す。\n",
657 |     "- サブゴールの計算式を simpl で簡単にする。\n",
658 |     "- rewriteで、サブゴールを m_eq_Sn を使って書き換える。\n",
659 |     "- サブゴールの左辺と右辺は等しい。\n",
660 |     "\n",
661 |     "###  [解答を見る](./Equal-Ex-Answer.ipynb#ex27) "
662 |    ]
663 |   },
664 |   {
665 |    "cell_type": "code",
666 |    "execution_count": null,
667 |    "metadata": {
668 |     "coq_kernel_metadata": {
669 |      "auto_roll_back": true
670 |     }
671 |    },
672 |    "outputs": [],
673 |    "source": [
674 |     "Theorem mult_S_1 : (forall n m : nat, m = S n -> m * (1 + n) = m * m).\n",
675 |     "Proof."
676 |    ]
677 |   },
678 |   {
679 |    "cell_type": "code",
680 |    "execution_count": null,
681 |    "metadata": {
682 |     "coq_kernel_metadata": {
683 |      "auto_roll_back": true
684 |     }
685 |    },
686 |    "outputs": [],
687 |    "source": []
688 |   },
689 |   {
690 |    "cell_type": "code",
691 |    "execution_count": null,
692 |    "metadata": {
693 |     "coq_kernel_metadata": {
694 |      "auto_roll_back": true
695 |     }
696 |    },
697 |    "outputs": [],
698 |    "source": []
699 |   },
700 |   {
701 |    "cell_type": "code",
702 |    "execution_count": null,
703 |    "metadata": {
704 |     "coq_kernel_metadata": {
705 |      "auto_roll_back": true
706 |     }
707 |    },
708 |    "outputs": [],
709 |    "source": []
710 |   },
711 |   {
712 |    "cell_type": "code",
713 |    "execution_count": null,
714 |    "metadata": {
715 |     "coq_kernel_metadata": {
716 |      "auto_roll_back": true
717 |     }
718 |    },
719 |    "outputs": [],
720 |    "source": []
721 |   },
722 |   {
723 |    "cell_type": "code",
724 |    "execution_count": null,
725 |    "metadata": {
726 |     "coq_kernel_metadata": {
727 |      "auto_roll_back": true
728 |     }
729 |    },
730 |    "outputs": [],
731 |    "source": []
732 |   },
733 |   {
734 |    "cell_type": "code",
735 |    "execution_count": null,
736 |    "metadata": {
737 |     "coq_kernel_metadata": {
738 |      "auto_roll_back": true
739 |     }
740 |    },
741 |    "outputs": [],
742 |    "source": []
743 |   },
744 |   {
745 |    "cell_type": "code",
746 |    "execution_count": null,
747 |    "metadata": {
748 |     "coq_kernel_metadata": {
749 |      "auto_roll_back": true
750 |     }
751 |    },
752 |    "outputs": [],
753 |    "source": []
754 |   },
755 |   {
756 |    "cell_type": "code",
757 |    "execution_count": null,
758 |    "metadata": {
759 |     "coq_kernel_metadata": {
760 |      "auto_roll_back": true
761 |     }
762 |    },
763 |    "outputs": [],
764 |    "source": [
765 |     "Admitted. (* 最後にここを Qed. に変える。*)"
766 |    ]
767 |   }
768 |  ],
769 |  "metadata": {
770 |   "kernelspec": {
771 |    "display_name": "Coq",
772 |    "language": "coq",
773 |    "name": "coq"
774 |   },
775 |   "language_info": {
776 |    "file_extension": ".v",
777 |    "mimetype": "text/x-coq",
778 |    "name": "coq",
779 |    "version": "8.9.1"
780 |   }
781 |  },
782 |  "nbformat": 4,
783 |  "nbformat_minor": 2
784 | }
785 | 


--------------------------------------------------------------------------------
/.ipynb_checkpoints/Equal-Ex-Answer-checkpoint.ipynb:
--------------------------------------------------------------------------------
  1 | {
  2 |  "cells": [
  3 |   {
  4 |    "cell_type": "markdown",
  5 |    "metadata": {
  6 |     "coq_kernel_metadata": {
  7 |      "auto_roll_back": true
  8 |     }
  9 |    },
 10 |    "source": [
 11 |     "# 演習問題 2 解答\n",
 12 |     "\n",
 13 |     "[演習問題 2-1](./Equal-Ex-Answer.ipynb#ex21) [演習問題 2-2](./Equal-Ex-Answer.ipynb#ex22) [演習問題 2-3](./Equal-Ex-Answer.ipynb#ex23) [演習問題 2-4](./Equal-Ex-Answer.ipynb#ex24) [演習問題 2-5](./Equal-Ex-Answer.ipynb#ex25) [演習問題 2-6](./Equal-Ex-Answer.ipynb#ex26) [演習問題 2-7](./Equal-Ex-Answer.ipynb#ex27) "
 14 |    ]
 15 |   },
 16 |   {
 17 |    "cell_type": "markdown",
 18 |    "metadata": {
 19 |     "coq_kernel_metadata": {
 20 |      "auto_roll_back": true
 21 |     }
 22 |    },
 23 |    "source": [
 24 |     "\n",
 25 |     "## 演習問題 2-1\n",
 26 |     "　次の定理を証明せよ。\n",
 27 |     " 
　Theorem plus_O_n : (forall n : nat , 0+n = n ).\n",
 28 |     "\n",
 29 |     "[作業用ページで証明をして見る](./Equal-Ex-Work.ipynb#ex21)　/ [演習問題に戻る](./Equal-Ex.ipynb#ex21 ) \n",
 30 |     "\n",
 31 |     "## 解答\n",
 32 |     "```\n",
 33 |     "Theorem  plus_O_n : (forall n : nat , 0+n = n ).\n",
 34 |     "Proof.\n",
 35 |     "  intro n.\n",
 36 |     "  simpl.\n",
 37 |     "  reflexivity.\n",
 38 |     "Qed.  \n",
 39 |     "```\n",
 40 |     "\n",
 41 |     "### 下のCellを実行して確かめよ。"
 42 |    ]
 43 |   },
 44 |   {
 45 |    "cell_type": "code",
 46 |    "execution_count": null,
 47 |    "metadata": {
 48 |     "coq_kernel_metadata": {
 49 |      "auto_roll_back": true
 50 |     }
 51 |    },
 52 |    "outputs": [],
 53 |    "source": [
 54 |     "Theorem  plus_O_n : (forall n : nat , 0+n = n ).\n",
 55 |     "Proof.\n",
 56 |     "  intro n.\n",
 57 |     "  simpl.\n",
 58 |     "  reflexivity.\n",
 59 |     "Qed.  "
 60 |    ]
 61 |   },
 62 |   {
 63 |    "cell_type": "markdown",
 64 |    "metadata": {
 65 |     "coq_kernel_metadata": {
 66 |      "auto_roll_back": true
 67 |     }
 68 |    },
 69 |    "source": [
 70 |     "### 次のCell を一つづつ実行して、各tacticsの働きを確認せよ。"
 71 |    ]
 72 |   },
 73 |   {
 74 |    "cell_type": "code",
 75 |    "execution_count": null,
 76 |    "metadata": {
 77 |     "coq_kernel_metadata": {
 78 |      "auto_roll_back": true
 79 |     }
 80 |    },
 81 |    "outputs": [],
 82 |    "source": [
 83 |     "Theorem  plus_O_n' : (forall n : nat , 0+n = n ).\n",
 84 |     "Proof."
 85 |    ]
 86 |   },
 87 |   {
 88 |    "cell_type": "code",
 89 |    "execution_count": null,
 90 |    "metadata": {
 91 |     "coq_kernel_metadata": {
 92 |      "auto_roll_back": true
 93 |     }
 94 |    },
 95 |    "outputs": [],
 96 |    "source": [
 97 |     "  intro n."
 98 |    ]
 99 |   },
100 |   {
101 |    "cell_type": "code",
102 |    "execution_count": null,
103 |    "metadata": {
104 |     "coq_kernel_metadata": {
105 |      "auto_roll_back": true
106 |     }
107 |    },
108 |    "outputs": [],
109 |    "source": [
110 |     "  simpl."
111 |    ]
112 |   },
113 |   {
114 |    "cell_type": "code",
115 |    "execution_count": null,
116 |    "metadata": {
117 |     "coq_kernel_metadata": {
118 |      "auto_roll_back": true
119 |     }
120 |    },
121 |    "outputs": [],
122 |    "source": [
123 |     "  reflexivity."
124 |    ]
125 |   },
126 |   {
127 |    "cell_type": "code",
128 |    "execution_count": null,
129 |    "metadata": {
130 |     "coq_kernel_metadata": {
131 |      "auto_roll_back": true
132 |     }
133 |    },
134 |    "outputs": [],
135 |    "source": [
136 |     "Qed.  "
137 |    ]
138 |   },
139 |   {
140 |    "cell_type": "markdown",
141 |    "metadata": {
142 |     "coq_kernel_metadata": {
143 |      "auto_roll_back": true
144 |     }
145 |    },
146 |    "source": [
147 |     "\n",
148 |     "## 演習問題 2-2\n",
149 |     "　次の定理を証明せよ。\n",
150 |     " 
 Theorem plus_1_n : (forall n : nat , 1+n = S n ).\n",
151 |     "\n",
152 |     "[作業用ページで証明をして見る](./Equal-Ex-Work.ipynb#ex22)　/ [演習問題に戻る](./Equal-Ex.ipynb#ex22 ) \n",
153 |     "\n",
154 |     "## 解答\n",
155 |     "```\n",
156 |     "Theorem  plus_1_n : (forall n : nat , 1+n = S n ).\n",
157 |     "Proof.\n",
158 |     "  intros n.\n",
159 |     "  simpl.\n",
160 |     "  reflexivity.\n",
161 |     "Qed.\n",
162 |     "```\n",
163 |     "\n",
164 |     "### 下のCellを実行して確かめよ。"
165 |    ]
166 |   },
167 |   {
168 |    "cell_type": "code",
169 |    "execution_count": null,
170 |    "metadata": {
171 |     "coq_kernel_metadata": {
172 |      "auto_roll_back": true
173 |     }
174 |    },
175 |    "outputs": [],
176 |    "source": [
177 |     "Theorem  plus_1_n : (forall n : nat , 1+n = S n ).\n",
178 |     "Proof.\n",
179 |     "  intros n.\n",
180 |     "  simpl.\n",
181 |     "  reflexivity.\n",
182 |     "Qed."
183 |    ]
184 |   },
185 |   {
186 |    "cell_type": "markdown",
187 |    "metadata": {
188 |     "coq_kernel_metadata": {
189 |      "auto_roll_back": true
190 |     }
191 |    },
192 |    "source": [
193 |     "### 次のCell を一つづつ実行して、各tacticsの働きを確認せよ。"
194 |    ]
195 |   },
196 |   {
197 |    "cell_type": "code",
198 |    "execution_count": null,
199 |    "metadata": {
200 |     "coq_kernel_metadata": {
201 |      "auto_roll_back": true
202 |     }
203 |    },
204 |    "outputs": [],
205 |    "source": [
206 |     "Theorem  plus_1_n' : (forall n : nat , 1+n = S n ).\n",
207 |     "Proof."
208 |    ]
209 |   },
210 |   {
211 |    "cell_type": "code",
212 |    "execution_count": null,
213 |    "metadata": {
214 |     "coq_kernel_metadata": {
215 |      "auto_roll_back": true
216 |     }
217 |    },
218 |    "outputs": [],
219 |    "source": [
220 |     "  intros n."
221 |    ]
222 |   },
223 |   {
224 |    "cell_type": "code",
225 |    "execution_count": null,
226 |    "metadata": {
227 |     "coq_kernel_metadata": {
228 |      "auto_roll_back": true
229 |     }
230 |    },
231 |    "outputs": [],
232 |    "source": [
233 |     "  simpl."
234 |    ]
235 |   },
236 |   {
237 |    "cell_type": "code",
238 |    "execution_count": null,
239 |    "metadata": {
240 |     "coq_kernel_metadata": {
241 |      "auto_roll_back": true
242 |     }
243 |    },
244 |    "outputs": [],
245 |    "source": [
246 |     "  reflexivity."
247 |    ]
248 |   },
249 |   {
250 |    "cell_type": "code",
251 |    "execution_count": null,
252 |    "metadata": {
253 |     "coq_kernel_metadata": {
254 |      "auto_roll_back": true
255 |     }
256 |    },
257 |    "outputs": [],
258 |    "source": [
259 |     "Qed."
260 |    ]
261 |   },
262 |   {
263 |    "cell_type": "markdown",
264 |    "metadata": {
265 |     "coq_kernel_metadata": {
266 |      "auto_roll_back": true
267 |     }
268 |    },
269 |    "source": [
270 |     "\n",
271 |     "## 演習問題 2-3\n",
272 |     "　次の定理を証明せよ。\n",
273 |     " 
 Theorem mult_0_n : (forall n : nat , 0 * n =  0 ).\n",
274 |     "\n",
275 |     "[作業用ページで証明をして見る](./Equal-Ex-Work.ipynb#ex23)　/ [演習問題に戻る](./Equal-Ex.ipynb#ex23 ) \n",
276 |     "\n",
277 |     "## 解答\n",
278 |     "```\n",
279 |     "Theorem  mult_0_n : (forall n : nat , 0 * n =  0 ).\n",
280 |     "Proof.\n",
281 |     "  intros n.\n",
282 |     "  simpl.\n",
283 |     "  reflexivity.\n",
284 |     "Qed.\n",
285 |     "```\n",
286 |     "\n",
287 |     "### 下のCellを実行して確かめよ。"
288 |    ]
289 |   },
290 |   {
291 |    "cell_type": "code",
292 |    "execution_count": null,
293 |    "metadata": {
294 |     "coq_kernel_metadata": {
295 |      "auto_roll_back": true
296 |     }
297 |    },
298 |    "outputs": [],
299 |    "source": [
300 |     "Theorem  mult_0_n : (forall n : nat , 0 * n =  0 ).\n",
301 |     "Proof.\n",
302 |     "  intros n.\n",
303 |     "  simpl.\n",
304 |     "  reflexivity.\n",
305 |     "Qed."
306 |    ]
307 |   },
308 |   {
309 |    "cell_type": "markdown",
310 |    "metadata": {
311 |     "coq_kernel_metadata": {
312 |      "auto_roll_back": true
313 |     }
314 |    },
315 |    "source": [
316 |     "### 次のCell を一つづつ実行して、各tacticsの働きを確認せよ。"
317 |    ]
318 |   },
319 |   {
320 |    "cell_type": "code",
321 |    "execution_count": null,
322 |    "metadata": {
323 |     "coq_kernel_metadata": {
324 |      "auto_roll_back": true
325 |     }
326 |    },
327 |    "outputs": [],
328 |    "source": [
329 |     "Theorem  mult_0_n' : (forall n : nat , 0 * n =  0 ).\n",
330 |     "Proof."
331 |    ]
332 |   },
333 |   {
334 |    "cell_type": "code",
335 |    "execution_count": null,
336 |    "metadata": {
337 |     "coq_kernel_metadata": {
338 |      "auto_roll_back": true
339 |     }
340 |    },
341 |    "outputs": [],
342 |    "source": [
343 |     "  intros n."
344 |    ]
345 |   },
346 |   {
347 |    "cell_type": "code",
348 |    "execution_count": null,
349 |    "metadata": {
350 |     "coq_kernel_metadata": {
351 |      "auto_roll_back": true
352 |     }
353 |    },
354 |    "outputs": [],
355 |    "source": [
356 |     "  simpl."
357 |    ]
358 |   },
359 |   {
360 |    "cell_type": "code",
361 |    "execution_count": null,
362 |    "metadata": {
363 |     "coq_kernel_metadata": {
364 |      "auto_roll_back": true
365 |     }
366 |    },
367 |    "outputs": [],
368 |    "source": [
369 |     "  reflexivity."
370 |    ]
371 |   },
372 |   {
373 |    "cell_type": "code",
374 |    "execution_count": null,
375 |    "metadata": {
376 |     "coq_kernel_metadata": {
377 |      "auto_roll_back": true
378 |     }
379 |    },
380 |    "outputs": [],
381 |    "source": [
382 |     "Qed."
383 |    ]
384 |   },
385 |   {
386 |    "cell_type": "markdown",
387 |    "metadata": {
388 |     "coq_kernel_metadata": {
389 |      "auto_roll_back": true
390 |     }
391 |    },
392 |    "source": [
393 |     "\n",
394 |     "## 演習問題 2-4\n",
395 |     "　次の定理を証明せよ。\n",
396 |     " 
 Theorem plus_id_nm : (forall n m : nat , n = m -> n+n = m + m ).\n",
397 |     "\n",
398 |     "[作業用ページで証明をして見る](./Equal-Ex-Work.ipynb#ex21)　/ [演習問題に戻る](./Equal-Ex.ipynb#ex21 ) \n",
399 |     "\n",
400 |     "## 解答\n",
401 |     "```\n",
402 |     "Theorem  plus_id_nm : (forall n m : nat , n = m -> n+n = m + m ).\n",
403 |     "Proof.\n",
404 |     "  intros n m.\n",
405 |     "  intros H.\n",
406 |     "  rewrite -> H.\n",
407 |     "  reflexivity.\n",
408 |     "Qed.\n",
409 |     "```\n",
410 |     "\n",
411 |     "### 下のCellを実行して確かめよ。"
412 |    ]
413 |   },
414 |   {
415 |    "cell_type": "code",
416 |    "execution_count": null,
417 |    "metadata": {
418 |     "coq_kernel_metadata": {
419 |      "auto_roll_back": true
420 |     }
421 |    },
422 |    "outputs": [],
423 |    "source": [
424 |     "Theorem  plus_id_nm : (forall n m : nat , n = m -> n+n = m + m ).\n",
425 |     "Proof.\n",
426 |     "  intros n m.\n",
427 |     "  intros H.\n",
428 |     "  rewrite -> H.\n",
429 |     "  reflexivity.\n",
430 |     "Qed."
431 |    ]
432 |   },
433 |   {
434 |    "cell_type": "markdown",
435 |    "metadata": {
436 |     "coq_kernel_metadata": {
437 |      "auto_roll_back": true
438 |     }
439 |    },
440 |    "source": [
441 |     "### 次のCell を一つづつ実行して、各tacticsの働きを確認せよ。"
442 |    ]
443 |   },
444 |   {
445 |    "cell_type": "code",
446 |    "execution_count": null,
447 |    "metadata": {
448 |     "coq_kernel_metadata": {
449 |      "auto_roll_back": true
450 |     }
451 |    },
452 |    "outputs": [],
453 |    "source": [
454 |     "Theorem  plus_id_nm' : (forall n m : nat , n = m -> n+n = m + m ).\n",
455 |     "Proof."
456 |    ]
457 |   },
458 |   {
459 |    "cell_type": "code",
460 |    "execution_count": null,
461 |    "metadata": {
462 |     "coq_kernel_metadata": {
463 |      "auto_roll_back": true
464 |     }
465 |    },
466 |    "outputs": [],
467 |    "source": [
468 |     "  intros n m."
469 |    ]
470 |   },
471 |   {
472 |    "cell_type": "code",
473 |    "execution_count": null,
474 |    "metadata": {
475 |     "coq_kernel_metadata": {
476 |      "auto_roll_back": true
477 |     }
478 |    },
479 |    "outputs": [],
480 |    "source": [
481 |     "  intros H."
482 |    ]
483 |   },
484 |   {
485 |    "cell_type": "code",
486 |    "execution_count": null,
487 |    "metadata": {
488 |     "coq_kernel_metadata": {
489 |      "auto_roll_back": true
490 |     }
491 |    },
492 |    "outputs": [],
493 |    "source": [
494 |     "  rewrite -> H."
495 |    ]
496 |   },
497 |   {
498 |    "cell_type": "code",
499 |    "execution_count": null,
500 |    "metadata": {
501 |     "coq_kernel_metadata": {
502 |      "auto_roll_back": true
503 |     }
504 |    },
505 |    "outputs": [],
506 |    "source": [
507 |     "  reflexivity."
508 |    ]
509 |   },
510 |   {
511 |    "cell_type": "code",
512 |    "execution_count": null,
513 |    "metadata": {
514 |     "coq_kernel_metadata": {
515 |      "auto_roll_back": true
516 |     }
517 |    },
518 |    "outputs": [],
519 |    "source": [
520 |     "Qed."
521 |    ]
522 |   },
523 |   {
524 |    "cell_type": "markdown",
525 |    "metadata": {
526 |     "coq_kernel_metadata": {
527 |      "auto_roll_back": true
528 |     }
529 |    },
530 |    "source": [
531 |     "\n",
532 |     "## 演習問題 2-5\n",
533 |     "　次の定理を証明せよ。\n",
534 |     "  
 Theorem plus_id : (forall n m o : nat , n = m -> m = o -> n + m = m + o ).\n",
535 |     "\n",
536 |     "[作業用ページで証明をして見る](./Equal-Ex-Work.ipynb#ex25)　/ [演習問題に戻る](./Equal-Ex.ipynb#ex25 ) \n",
537 |     "\n",
538 |     "## 解答\n",
539 |     "```\n",
540 |     "Theorem  plus_id : (forall n m o : nat , n = m -> m = o -> n + m = m + o ).\n",
541 |     "Proof.\n",
542 |     "  intros n m o.\n",
543 |     "  intros n_eq_m m_eq_o.\n",
544 |     "  rewrite -> n_eq_m.\n",
545 |     "  rewrite <- m_eq_o.\n",
546 |     "  reflexivity.\n",
547 |     "Qed.\n",
548 |     "```\n",
549 |     "\n",
550 |     "### 下のCellを実行して確かめよ。"
551 |    ]
552 |   },
553 |   {
554 |    "cell_type": "code",
555 |    "execution_count": null,
556 |    "metadata": {
557 |     "coq_kernel_metadata": {
558 |      "auto_roll_back": true
559 |     }
560 |    },
561 |    "outputs": [],
562 |    "source": [
563 |     "Theorem  plus_id : (forall n m o : nat , n = m -> m = o -> n + m = m + o ).\n",
564 |     "Proof.\n",
565 |     "  intros n m o.\n",
566 |     "  intros n_eq_m m_eq_o.\n",
567 |     "  rewrite -> n_eq_m.\n",
568 |     "  rewrite <- m_eq_o.\n",
569 |     "  reflexivity.\n",
570 |     "Qed."
571 |    ]
572 |   },
573 |   {
574 |    "cell_type": "markdown",
575 |    "metadata": {
576 |     "coq_kernel_metadata": {
577 |      "auto_roll_back": true
578 |     }
579 |    },
580 |    "source": [
581 |     "### 次のCell を一つづつ実行して、各tacticsの働きを確認せよ。"
582 |    ]
583 |   },
584 |   {
585 |    "cell_type": "code",
586 |    "execution_count": null,
587 |    "metadata": {
588 |     "coq_kernel_metadata": {
589 |      "auto_roll_back": true
590 |     }
591 |    },
592 |    "outputs": [],
593 |    "source": [
594 |     "Theorem  plus_id' : (forall n m o : nat , n = m -> m = o -> n + m = m + o ).\n",
595 |     "Proof."
596 |    ]
597 |   },
598 |   {
599 |    "cell_type": "code",
600 |    "execution_count": null,
601 |    "metadata": {
602 |     "coq_kernel_metadata": {
603 |      "auto_roll_back": true
604 |     }
605 |    },
606 |    "outputs": [],
607 |    "source": [
608 |     "  intros n m o."
609 |    ]
610 |   },
611 |   {
612 |    "cell_type": "code",
613 |    "execution_count": null,
614 |    "metadata": {
615 |     "coq_kernel_metadata": {
616 |      "auto_roll_back": true
617 |     }
618 |    },
619 |    "outputs": [],
620 |    "source": [
621 |     "  intros n_eq_m m_eq_o."
622 |    ]
623 |   },
624 |   {
625 |    "cell_type": "code",
626 |    "execution_count": null,
627 |    "metadata": {
628 |     "coq_kernel_metadata": {
629 |      "auto_roll_back": true
630 |     }
631 |    },
632 |    "outputs": [],
633 |    "source": [
634 |     "  rewrite -> n_eq_m."
635 |    ]
636 |   },
637 |   {
638 |    "cell_type": "code",
639 |    "execution_count": null,
640 |    "metadata": {
641 |     "coq_kernel_metadata": {
642 |      "auto_roll_back": true
643 |     }
644 |    },
645 |    "outputs": [],
646 |    "source": [
647 |     "  rewrite <- m_eq_o."
648 |    ]
649 |   },
650 |   {
651 |    "cell_type": "code",
652 |    "execution_count": null,
653 |    "metadata": {
654 |     "coq_kernel_metadata": {
655 |      "auto_roll_back": true
656 |     }
657 |    },
658 |    "outputs": [],
659 |    "source": [
660 |     "  reflexivity."
661 |    ]
662 |   },
663 |   {
664 |    "cell_type": "code",
665 |    "execution_count": null,
666 |    "metadata": {
667 |     "coq_kernel_metadata": {
668 |      "auto_roll_back": true
669 |     }
670 |    },
671 |    "outputs": [],
672 |    "source": [
673 |     "Qed."
674 |    ]
675 |   },
676 |   {
677 |    "cell_type": "markdown",
678 |    "metadata": {
679 |     "coq_kernel_metadata": {
680 |      "auto_roll_back": true
681 |     }
682 |    },
683 |    "source": [
684 |     "\n",
685 |     "## 演習問題 2-6\n",
686 |     "　次の定理を証明せよ。\n",
687 |     " 
 Theorem mult_0_plus : (forall n m : nat , ( 0 + n ) \\* m =  n \\* m ).\n",
688 |     "\n",
689 |     "[作業用ページで証明をして見る](./Equal-Ex-Work.ipynb#ex26)　/ [演習問題に戻る](./Equal-Ex.ipynb#ex26 ) \n",
690 |     "\n",
691 |     "## 解答\n",
692 |     "```\n",
693 |     "Theorem  mult_0_plus : (forall n m : nat , ( 0 + n ) * m =  n * m ).\n",
694 |     "Proof.\n",
695 |     "  intros.\n",
696 |     "  simpl.\n",
697 |     "  reflexivity.\n",
698 |     "Qed.\n",
699 |     "```\n",
700 |     "\n",
701 |     "### 下のCellを実行して確かめよ。"
702 |    ]
703 |   },
704 |   {
705 |    "cell_type": "code",
706 |    "execution_count": null,
707 |    "metadata": {
708 |     "coq_kernel_metadata": {
709 |      "auto_roll_back": true
710 |     }
711 |    },
712 |    "outputs": [],
713 |    "source": [
714 |     "Theorem  mult_0_plus : (forall n m : nat , ( 0 + n ) * m =  n * m ).\n",
715 |     "Proof.\n",
716 |     "  intros.\n",
717 |     "  simpl.\n",
718 |     "  reflexivity.\n",
719 |     "Qed."
720 |    ]
721 |   },
722 |   {
723 |    "cell_type": "markdown",
724 |    "metadata": {
725 |     "coq_kernel_metadata": {
726 |      "auto_roll_back": true
727 |     }
728 |    },
729 |    "source": [
730 |     "### 次のCell を一つづつ実行して、各tacticsの働きを確認せよ。"
731 |    ]
732 |   },
733 |   {
734 |    "cell_type": "code",
735 |    "execution_count": null,
736 |    "metadata": {
737 |     "coq_kernel_metadata": {
738 |      "auto_roll_back": true
739 |     }
740 |    },
741 |    "outputs": [],
742 |    "source": [
743 |     "Theorem  mult_0_plus' : (forall n m : nat , ( 0 + n ) * m =  n * m ).\n",
744 |     "Proof."
745 |    ]
746 |   },
747 |   {
748 |    "cell_type": "code",
749 |    "execution_count": null,
750 |    "metadata": {
751 |     "coq_kernel_metadata": {
752 |      "auto_roll_back": true
753 |     }
754 |    },
755 |    "outputs": [],
756 |    "source": [
757 |     "  intros."
758 |    ]
759 |   },
760 |   {
761 |    "cell_type": "code",
762 |    "execution_count": null,
763 |    "metadata": {
764 |     "coq_kernel_metadata": {
765 |      "auto_roll_back": true
766 |     }
767 |    },
768 |    "outputs": [],
769 |    "source": [
770 |     "  simpl."
771 |    ]
772 |   },
773 |   {
774 |    "cell_type": "code",
775 |    "execution_count": null,
776 |    "metadata": {
777 |     "coq_kernel_metadata": {
778 |      "auto_roll_back": true
779 |     }
780 |    },
781 |    "outputs": [],
782 |    "source": [
783 |     "  reflexivity."
784 |    ]
785 |   },
786 |   {
787 |    "cell_type": "code",
788 |    "execution_count": null,
789 |    "metadata": {
790 |     "coq_kernel_metadata": {
791 |      "auto_roll_back": true
792 |     }
793 |    },
794 |    "outputs": [],
795 |    "source": [
796 |     "Qed."
797 |    ]
798 |   },
799 |   {
800 |    "cell_type": "markdown",
801 |    "metadata": {
802 |     "coq_kernel_metadata": {
803 |      "auto_roll_back": true
804 |     }
805 |    },
806 |    "source": [
807 |     "\n",
808 |     "## 演習問題 2-7\n",
809 |     "　次の定理を証明せよ。\n",
810 |     " 
 Theorem mult_S_1 : (forall n m : nat, m = S n -> m \\* (1 + n) = m \\* m).\n",
811 |     "\n",
812 |     "[作業用ページで証明をして見る](./Equal-Ex-Work.ipynb#ex27)　/ [演習問題に戻る](./Equal-Ex.ipynb#ex27 ) \n",
813 |     "\n",
814 |     "## 解答\n",
815 |     "```\n",
816 |     "Theorem  mult_S_1 : (forall n m : nat, m = S n -> m * (1 + n) = m * m).\n",
817 |     "Proof.\n",
818 |     "  intros n m m_eq_Sn.\n",
819 |     "  simpl.\n",
820 |     "  rewrite -> m_eq_Sn.\n",
821 |     "  reflexivity.\n",
822 |     "Qed.\n",
823 |     "```\n",
824 |     "\n",
825 |     "### 下のCellを実行して確かめよ。"
826 |    ]
827 |   },
828 |   {
829 |    "cell_type": "code",
830 |    "execution_count": null,
831 |    "metadata": {
832 |     "coq_kernel_metadata": {
833 |      "auto_roll_back": true
834 |     }
835 |    },
836 |    "outputs": [],
837 |    "source": [
838 |     "Theorem  mult_S_1 : (forall n m : nat, m = S n -> m * (1 + n) = m * m).\n",
839 |     "Proof.\n",
840 |     "  intros n m m_eq_Sn.\n",
841 |     "  simpl.\n",
842 |     "  rewrite -> m_eq_Sn.\n",
843 |     "  reflexivity.\n",
844 |     "Qed."
845 |    ]
846 |   },
847 |   {
848 |    "cell_type": "markdown",
849 |    "metadata": {
850 |     "coq_kernel_metadata": {
851 |      "auto_roll_back": true
852 |     }
853 |    },
854 |    "source": [
855 |     "### 次のCell を一つづつ実行して、各tacticsの働きを確認せよ。"
856 |    ]
857 |   },
858 |   {
859 |    "cell_type": "code",
860 |    "execution_count": null,
861 |    "metadata": {
862 |     "coq_kernel_metadata": {
863 |      "auto_roll_back": true
864 |     }
865 |    },
866 |    "outputs": [],
867 |    "source": [
868 |     "Theorem  mult_S_1' : (forall n m : nat, m = S n -> m * (1 + n) = m * m).\n",
869 |     "Proof."
870 |    ]
871 |   },
872 |   {
873 |    "cell_type": "code",
874 |    "execution_count": null,
875 |    "metadata": {
876 |     "coq_kernel_metadata": {
877 |      "auto_roll_back": true
878 |     }
879 |    },
880 |    "outputs": [],
881 |    "source": [
882 |     "  intros n m m_eq_Sn."
883 |    ]
884 |   },
885 |   {
886 |    "cell_type": "code",
887 |    "execution_count": null,
888 |    "metadata": {
889 |     "coq_kernel_metadata": {
890 |      "auto_roll_back": true
891 |     }
892 |    },
893 |    "outputs": [],
894 |    "source": [
895 |     "  simpl."
896 |    ]
897 |   },
898 |   {
899 |    "cell_type": "code",
900 |    "execution_count": null,
901 |    "metadata": {
902 |     "coq_kernel_metadata": {
903 |      "auto_roll_back": true
904 |     }
905 |    },
906 |    "outputs": [],
907 |    "source": [
908 |     "  rewrite -> m_eq_Sn.\n",
909 |     " "
910 |    ]
911 |   },
912 |   {
913 |    "cell_type": "code",
914 |    "execution_count": null,
915 |    "metadata": {
916 |     "coq_kernel_metadata": {
917 |      "auto_roll_back": true
918 |     }
919 |    },
920 |    "outputs": [],
921 |    "source": [
922 |     " reflexivity."
923 |    ]
924 |   },
925 |   {
926 |    "cell_type": "code",
927 |    "execution_count": null,
928 |    "metadata": {
929 |     "coq_kernel_metadata": {
930 |      "auto_roll_back": true
931 |     }
932 |    },
933 |    "outputs": [],
934 |    "source": [
935 |     "Qed"
936 |    ]
937 |   }
938 |  ],
939 |  "metadata": {
940 |   "kernelspec": {
941 |    "display_name": "Coq",
942 |    "language": "coq",
943 |    "name": "coq"
944 |   },
945 |   "language_info": {
946 |    "file_extension": ".v",
947 |    "mimetype": "text/x-coq",
948 |    "name": "coq",
949 |    "version": "8.9.1"
950 |   }
951 |  },
952 |  "nbformat": 4,
953 |  "nbformat_minor": 2
954 | }
955 | 


--------------------------------------------------------------------------------
/.ipynb_checkpoints/Induction-Ex-Work-checkpoint.ipynb:
--------------------------------------------------------------------------------
   1 | {
   2 |  "cells": [
   3 |   {
   4 |    "cell_type": "markdown",
   5 |    "metadata": {
   6 |     "coq_kernel_metadata": {
   7 |      "auto_roll_back": true
   8 |     }
   9 |    },
  10 |    "source": [
  11 |     "# 演習問題 3　作業用ページ\n",
  12 |     "\n",
  13 |     "[演習問題 3-1](./Induction-Ex.ipynb#ex11) [演習問題 3-2](./Induction-Ex.ipynb#ex12) [演習問題 3-3](./Induction-Ex.ipynb#ex13) [演習問題 3-4](./Induction-Ex.ipynb#ex14) [演習問題 3-5](./Induction-Ex.ipynb#ex15) [演習問題 3-6](./Induction-Ex.ipynb#ex16) [演習問題 3-7](./Induction-Ex.ipynb#ex17) [演習問題 3-8](./Induction-Ex.ipynb#ex18) "
  14 |    ]
  15 |   },
  16 |   {
  17 |    "cell_type": "markdown",
  18 |    "metadata": {
  19 |     "coq_kernel_metadata": {
  20 |      "auto_roll_back": true
  21 |     }
  22 |    },
  23 |    "source": [
  24 |     "\n",
  25 |     "## 演習問題 3-1\n",
  26 |     "　次の定理を証明せよ。\n",
  27 |     " 
　Theorem plus_n_O : (forall n : nat, n + 0 = n).\n",
  28 |     "\n",
  29 |     "### 下記の証明を完成させよ。\n",
  30 |     "#### [ヒント: 解答のコメントを読む](./Induction-Ex-Answer.ipynb#hint31) /  [解答を見る](./Induction-Ex-Answer.ipynb#ex31) "
  31 |    ]
  32 |   },
  33 |   {
  34 |    "cell_type": "code",
  35 |    "execution_count": null,
  36 |    "metadata": {
  37 |     "coq_kernel_metadata": {
  38 |      "auto_roll_back": true
  39 |     }
  40 |    },
  41 |    "outputs": [],
  42 |    "source": [
  43 |     "Theorem plus_n_O : (forall n : nat, n + 0 = n)."
  44 |    ]
  45 |   },
  46 |   {
  47 |    "cell_type": "code",
  48 |    "execution_count": null,
  49 |    "metadata": {
  50 |     "coq_kernel_metadata": {
  51 |      "auto_roll_back": true
  52 |     }
  53 |    },
  54 |    "outputs": [],
  55 |    "source": []
  56 |   },
  57 |   {
  58 |    "cell_type": "code",
  59 |    "execution_count": null,
  60 |    "metadata": {
  61 |     "coq_kernel_metadata": {
  62 |      "auto_roll_back": true
  63 |     }
  64 |    },
  65 |    "outputs": [],
  66 |    "source": []
  67 |   },
  68 |   {
  69 |    "cell_type": "code",
  70 |    "execution_count": null,
  71 |    "metadata": {
  72 |     "coq_kernel_metadata": {
  73 |      "auto_roll_back": true
  74 |     }
  75 |    },
  76 |    "outputs": [],
  77 |    "source": []
  78 |   },
  79 |   {
  80 |    "cell_type": "code",
  81 |    "execution_count": null,
  82 |    "metadata": {
  83 |     "coq_kernel_metadata": {
  84 |      "auto_roll_back": true
  85 |     }
  86 |    },
  87 |    "outputs": [],
  88 |    "source": []
  89 |   },
  90 |   {
  91 |    "cell_type": "code",
  92 |    "execution_count": null,
  93 |    "metadata": {
  94 |     "coq_kernel_metadata": {
  95 |      "auto_roll_back": true
  96 |     }
  97 |    },
  98 |    "outputs": [],
  99 |    "source": []
 100 |   },
 101 |   {
 102 |    "cell_type": "code",
 103 |    "execution_count": null,
 104 |    "metadata": {
 105 |     "coq_kernel_metadata": {
 106 |      "auto_roll_back": true
 107 |     }
 108 |    },
 109 |    "outputs": [],
 110 |    "source": []
 111 |   },
 112 |   {
 113 |    "cell_type": "code",
 114 |    "execution_count": null,
 115 |    "metadata": {
 116 |     "coq_kernel_metadata": {
 117 |      "auto_roll_back": true
 118 |     }
 119 |    },
 120 |    "outputs": [],
 121 |    "source": []
 122 |   },
 123 |   {
 124 |    "cell_type": "code",
 125 |    "execution_count": null,
 126 |    "metadata": {
 127 |     "coq_kernel_metadata": {
 128 |      "auto_roll_back": true
 129 |     }
 130 |    },
 131 |    "outputs": [],
 132 |    "source": []
 133 |   },
 134 |   {
 135 |    "cell_type": "code",
 136 |    "execution_count": null,
 137 |    "metadata": {
 138 |     "coq_kernel_metadata": {
 139 |      "auto_roll_back": true
 140 |     }
 141 |    },
 142 |    "outputs": [],
 143 |    "source": [
 144 |     "Admitted. (* 最後にここを Qed. に変える。*)"
 145 |    ]
 146 |   },
 147 |   {
 148 |    "cell_type": "markdown",
 149 |    "metadata": {
 150 |     "coq_kernel_metadata": {
 151 |      "auto_roll_back": true
 152 |     }
 153 |    },
 154 |    "source": [
 155 |     "\n",
 156 |     "## 演習問題 3-2\n",
 157 |     "　次の定理を証明せよ。\n",
 158 |     " 
　Theorem minus_diag : (forall n : nat, minus n n = 0).\n",
 159 |     "\n",
 160 |     "### 下記の証明を完成させよ。\n",
 161 |     "#### [ヒント: 解答のコメントを読む](./Induction-Ex-Answer.ipynb#hint32) /  [解答を見る](./Induction-Ex-Answer.ipynb#ex32) "
 162 |    ]
 163 |   },
 164 |   {
 165 |    "cell_type": "code",
 166 |    "execution_count": null,
 167 |    "metadata": {
 168 |     "coq_kernel_metadata": {
 169 |      "auto_roll_back": true
 170 |     }
 171 |    },
 172 |    "outputs": [],
 173 |    "source": [
 174 |     "Theorem minus_diag : (forall n : nat, minus n n = 0)."
 175 |    ]
 176 |   },
 177 |   {
 178 |    "cell_type": "code",
 179 |    "execution_count": null,
 180 |    "metadata": {
 181 |     "coq_kernel_metadata": {
 182 |      "auto_roll_back": true
 183 |     }
 184 |    },
 185 |    "outputs": [],
 186 |    "source": []
 187 |   },
 188 |   {
 189 |    "cell_type": "code",
 190 |    "execution_count": null,
 191 |    "metadata": {
 192 |     "coq_kernel_metadata": {
 193 |      "auto_roll_back": true
 194 |     }
 195 |    },
 196 |    "outputs": [],
 197 |    "source": []
 198 |   },
 199 |   {
 200 |    "cell_type": "code",
 201 |    "execution_count": null,
 202 |    "metadata": {
 203 |     "coq_kernel_metadata": {
 204 |      "auto_roll_back": true
 205 |     }
 206 |    },
 207 |    "outputs": [],
 208 |    "source": []
 209 |   },
 210 |   {
 211 |    "cell_type": "code",
 212 |    "execution_count": null,
 213 |    "metadata": {
 214 |     "coq_kernel_metadata": {
 215 |      "auto_roll_back": true
 216 |     }
 217 |    },
 218 |    "outputs": [],
 219 |    "source": []
 220 |   },
 221 |   {
 222 |    "cell_type": "code",
 223 |    "execution_count": null,
 224 |    "metadata": {
 225 |     "coq_kernel_metadata": {
 226 |      "auto_roll_back": true
 227 |     }
 228 |    },
 229 |    "outputs": [],
 230 |    "source": []
 231 |   },
 232 |   {
 233 |    "cell_type": "code",
 234 |    "execution_count": null,
 235 |    "metadata": {
 236 |     "coq_kernel_metadata": {
 237 |      "auto_roll_back": true
 238 |     }
 239 |    },
 240 |    "outputs": [],
 241 |    "source": []
 242 |   },
 243 |   {
 244 |    "cell_type": "code",
 245 |    "execution_count": null,
 246 |    "metadata": {
 247 |     "coq_kernel_metadata": {
 248 |      "auto_roll_back": true
 249 |     }
 250 |    },
 251 |    "outputs": [],
 252 |    "source": []
 253 |   },
 254 |   {
 255 |    "cell_type": "code",
 256 |    "execution_count": null,
 257 |    "metadata": {
 258 |     "coq_kernel_metadata": {
 259 |      "auto_roll_back": true
 260 |     }
 261 |    },
 262 |    "outputs": [],
 263 |    "source": []
 264 |   },
 265 |   {
 266 |    "cell_type": "code",
 267 |    "execution_count": null,
 268 |    "metadata": {
 269 |     "coq_kernel_metadata": {
 270 |      "auto_roll_back": true
 271 |     }
 272 |    },
 273 |    "outputs": [],
 274 |    "source": [
 275 |     "Admitted. (* 最後にここを Qed. に変える。*)"
 276 |    ]
 277 |   },
 278 |   {
 279 |    "cell_type": "markdown",
 280 |    "metadata": {
 281 |     "coq_kernel_metadata": {
 282 |      "auto_roll_back": true
 283 |     }
 284 |    },
 285 |    "source": [
 286 |     "\n",
 287 |     "## 演習問題 3-3\n",
 288 |     "　次の定理を証明せよ。\n",
 289 |     " 
　Theorem mult_0_r : (forall n : nat,  n * 0 = 0).\n",
 290 |     "\n",
 291 |     "### 下記の証明を完成させよ。\n",
 292 |     "#### [ヒント: 解答のコメントを読む](./Induction-Ex-Answer.ipynb#hint33) /  [解答を見る](./Induction-Ex-Answer.ipynb#ex33) "
 293 |    ]
 294 |   },
 295 |   {
 296 |    "cell_type": "code",
 297 |    "execution_count": null,
 298 |    "metadata": {
 299 |     "coq_kernel_metadata": {
 300 |      "auto_roll_back": true
 301 |     }
 302 |    },
 303 |    "outputs": [],
 304 |    "source": [
 305 |     "Theorem mult_0_r : (forall n : nat,  n * 0 = 0)."
 306 |    ]
 307 |   },
 308 |   {
 309 |    "cell_type": "code",
 310 |    "execution_count": null,
 311 |    "metadata": {
 312 |     "coq_kernel_metadata": {
 313 |      "auto_roll_back": true
 314 |     }
 315 |    },
 316 |    "outputs": [],
 317 |    "source": []
 318 |   },
 319 |   {
 320 |    "cell_type": "code",
 321 |    "execution_count": null,
 322 |    "metadata": {
 323 |     "coq_kernel_metadata": {
 324 |      "auto_roll_back": true
 325 |     }
 326 |    },
 327 |    "outputs": [],
 328 |    "source": []
 329 |   },
 330 |   {
 331 |    "cell_type": "code",
 332 |    "execution_count": null,
 333 |    "metadata": {
 334 |     "coq_kernel_metadata": {
 335 |      "auto_roll_back": true
 336 |     }
 337 |    },
 338 |    "outputs": [],
 339 |    "source": []
 340 |   },
 341 |   {
 342 |    "cell_type": "code",
 343 |    "execution_count": null,
 344 |    "metadata": {
 345 |     "coq_kernel_metadata": {
 346 |      "auto_roll_back": true
 347 |     }
 348 |    },
 349 |    "outputs": [],
 350 |    "source": []
 351 |   },
 352 |   {
 353 |    "cell_type": "code",
 354 |    "execution_count": null,
 355 |    "metadata": {
 356 |     "coq_kernel_metadata": {
 357 |      "auto_roll_back": true
 358 |     }
 359 |    },
 360 |    "outputs": [],
 361 |    "source": []
 362 |   },
 363 |   {
 364 |    "cell_type": "code",
 365 |    "execution_count": null,
 366 |    "metadata": {
 367 |     "coq_kernel_metadata": {
 368 |      "auto_roll_back": true
 369 |     }
 370 |    },
 371 |    "outputs": [],
 372 |    "source": []
 373 |   },
 374 |   {
 375 |    "cell_type": "code",
 376 |    "execution_count": null,
 377 |    "metadata": {
 378 |     "coq_kernel_metadata": {
 379 |      "auto_roll_back": true
 380 |     }
 381 |    },
 382 |    "outputs": [],
 383 |    "source": []
 384 |   },
 385 |   {
 386 |    "cell_type": "code",
 387 |    "execution_count": null,
 388 |    "metadata": {
 389 |     "coq_kernel_metadata": {
 390 |      "auto_roll_back": true
 391 |     }
 392 |    },
 393 |    "outputs": [],
 394 |    "source": []
 395 |   },
 396 |   {
 397 |    "cell_type": "code",
 398 |    "execution_count": null,
 399 |    "metadata": {
 400 |     "coq_kernel_metadata": {
 401 |      "auto_roll_back": true
 402 |     }
 403 |    },
 404 |    "outputs": [],
 405 |    "source": []
 406 |   },
 407 |   {
 408 |    "cell_type": "code",
 409 |    "execution_count": null,
 410 |    "metadata": {
 411 |     "coq_kernel_metadata": {
 412 |      "auto_roll_back": true
 413 |     }
 414 |    },
 415 |    "outputs": [],
 416 |    "source": []
 417 |   },
 418 |   {
 419 |    "cell_type": "code",
 420 |    "execution_count": null,
 421 |    "metadata": {
 422 |     "coq_kernel_metadata": {
 423 |      "auto_roll_back": true
 424 |     }
 425 |    },
 426 |    "outputs": [],
 427 |    "source": [
 428 |     "Admitted. (* 最後にここを Qed. に変える。*)"
 429 |    ]
 430 |   },
 431 |   {
 432 |    "cell_type": "markdown",
 433 |    "metadata": {
 434 |     "coq_kernel_metadata": {
 435 |      "auto_roll_back": true
 436 |     }
 437 |    },
 438 |    "source": [
 439 |     "\n",
 440 |     "## 演習問題 3-4\n",
 441 |     "　次の定理を証明せよ。\n",
 442 |     " 
　Theorem plus_n_Sm : (forall n m : nat, S (n + m) = n + (S m) ).\n",
 443 |     "\n",
 444 |     "### 下記の証明を完成させよ。\n",
 445 |     "#### [ヒント: 解答のコメントを読む](./Induction-Ex-Answer.ipynb#hint34) /  [解答を見る](./Induction-Ex-Answer.ipynb#ex34) "
 446 |    ]
 447 |   },
 448 |   {
 449 |    "cell_type": "code",
 450 |    "execution_count": null,
 451 |    "metadata": {
 452 |     "coq_kernel_metadata": {
 453 |      "auto_roll_back": true
 454 |     }
 455 |    },
 456 |    "outputs": [],
 457 |    "source": [
 458 |     "Theorem plus_n_Sm : (forall n m : nat, S (n + m) = n + (S m) )."
 459 |    ]
 460 |   },
 461 |   {
 462 |    "cell_type": "code",
 463 |    "execution_count": null,
 464 |    "metadata": {
 465 |     "coq_kernel_metadata": {
 466 |      "auto_roll_back": true
 467 |     }
 468 |    },
 469 |    "outputs": [],
 470 |    "source": []
 471 |   },
 472 |   {
 473 |    "cell_type": "code",
 474 |    "execution_count": null,
 475 |    "metadata": {
 476 |     "coq_kernel_metadata": {
 477 |      "auto_roll_back": true
 478 |     }
 479 |    },
 480 |    "outputs": [],
 481 |    "source": []
 482 |   },
 483 |   {
 484 |    "cell_type": "code",
 485 |    "execution_count": null,
 486 |    "metadata": {
 487 |     "coq_kernel_metadata": {
 488 |      "auto_roll_back": true
 489 |     }
 490 |    },
 491 |    "outputs": [],
 492 |    "source": []
 493 |   },
 494 |   {
 495 |    "cell_type": "code",
 496 |    "execution_count": null,
 497 |    "metadata": {
 498 |     "coq_kernel_metadata": {
 499 |      "auto_roll_back": true
 500 |     }
 501 |    },
 502 |    "outputs": [],
 503 |    "source": []
 504 |   },
 505 |   {
 506 |    "cell_type": "code",
 507 |    "execution_count": null,
 508 |    "metadata": {
 509 |     "coq_kernel_metadata": {
 510 |      "auto_roll_back": true
 511 |     }
 512 |    },
 513 |    "outputs": [],
 514 |    "source": []
 515 |   },
 516 |   {
 517 |    "cell_type": "code",
 518 |    "execution_count": null,
 519 |    "metadata": {
 520 |     "coq_kernel_metadata": {
 521 |      "auto_roll_back": true
 522 |     }
 523 |    },
 524 |    "outputs": [],
 525 |    "source": []
 526 |   },
 527 |   {
 528 |    "cell_type": "code",
 529 |    "execution_count": null,
 530 |    "metadata": {
 531 |     "coq_kernel_metadata": {
 532 |      "auto_roll_back": true
 533 |     }
 534 |    },
 535 |    "outputs": [],
 536 |    "source": []
 537 |   },
 538 |   {
 539 |    "cell_type": "code",
 540 |    "execution_count": null,
 541 |    "metadata": {
 542 |     "coq_kernel_metadata": {
 543 |      "auto_roll_back": true
 544 |     }
 545 |    },
 546 |    "outputs": [],
 547 |    "source": [
 548 |     "Admitted. (* 最後にここを Qed. に変える。*)"
 549 |    ]
 550 |   },
 551 |   {
 552 |    "cell_type": "markdown",
 553 |    "metadata": {
 554 |     "coq_kernel_metadata": {
 555 |      "auto_roll_back": true
 556 |     }
 557 |    },
 558 |    "source": [
 559 |     "\n",
 560 |     "## 演習問題 3-5\n",
 561 |     "　次の定理を証明せよ。\n",
 562 |     " 
　Theorem plus_comm : (forall n m : nat, n + m = m + n ).\n",
 563 |     "\n",
 564 |     "### 下記の証明を完成させよ。\n",
 565 |     "#### [ヒント: 解答のコメントを読む](./Induction-Ex-Answer.ipynb#hint35) /  [解答を見る](./Induction-Ex-Answer.ipynb#ex35) "
 566 |    ]
 567 |   },
 568 |   {
 569 |    "cell_type": "code",
 570 |    "execution_count": null,
 571 |    "metadata": {
 572 |     "coq_kernel_metadata": {
 573 |      "auto_roll_back": true
 574 |     }
 575 |    },
 576 |    "outputs": [],
 577 |    "source": [
 578 |     "Theorem plus_comm : (forall n m : nat, n + m = m + n )."
 579 |    ]
 580 |   },
 581 |   {
 582 |    "cell_type": "code",
 583 |    "execution_count": null,
 584 |    "metadata": {
 585 |     "coq_kernel_metadata": {
 586 |      "auto_roll_back": true
 587 |     }
 588 |    },
 589 |    "outputs": [],
 590 |    "source": []
 591 |   },
 592 |   {
 593 |    "cell_type": "code",
 594 |    "execution_count": null,
 595 |    "metadata": {
 596 |     "coq_kernel_metadata": {
 597 |      "auto_roll_back": true
 598 |     }
 599 |    },
 600 |    "outputs": [],
 601 |    "source": []
 602 |   },
 603 |   {
 604 |    "cell_type": "code",
 605 |    "execution_count": null,
 606 |    "metadata": {
 607 |     "coq_kernel_metadata": {
 608 |      "auto_roll_back": true
 609 |     }
 610 |    },
 611 |    "outputs": [],
 612 |    "source": []
 613 |   },
 614 |   {
 615 |    "cell_type": "code",
 616 |    "execution_count": null,
 617 |    "metadata": {
 618 |     "coq_kernel_metadata": {
 619 |      "auto_roll_back": true
 620 |     }
 621 |    },
 622 |    "outputs": [],
 623 |    "source": []
 624 |   },
 625 |   {
 626 |    "cell_type": "code",
 627 |    "execution_count": null,
 628 |    "metadata": {
 629 |     "coq_kernel_metadata": {
 630 |      "auto_roll_back": true
 631 |     }
 632 |    },
 633 |    "outputs": [],
 634 |    "source": []
 635 |   },
 636 |   {
 637 |    "cell_type": "code",
 638 |    "execution_count": null,
 639 |    "metadata": {
 640 |     "coq_kernel_metadata": {
 641 |      "auto_roll_back": true
 642 |     }
 643 |    },
 644 |    "outputs": [],
 645 |    "source": []
 646 |   },
 647 |   {
 648 |    "cell_type": "code",
 649 |    "execution_count": null,
 650 |    "metadata": {
 651 |     "coq_kernel_metadata": {
 652 |      "auto_roll_back": true
 653 |     }
 654 |    },
 655 |    "outputs": [],
 656 |    "source": []
 657 |   },
 658 |   {
 659 |    "cell_type": "code",
 660 |    "execution_count": null,
 661 |    "metadata": {
 662 |     "coq_kernel_metadata": {
 663 |      "auto_roll_back": true
 664 |     }
 665 |    },
 666 |    "outputs": [],
 667 |    "source": []
 668 |   },
 669 |   {
 670 |    "cell_type": "code",
 671 |    "execution_count": null,
 672 |    "metadata": {
 673 |     "coq_kernel_metadata": {
 674 |      "auto_roll_back": true
 675 |     }
 676 |    },
 677 |    "outputs": [],
 678 |    "source": []
 679 |   },
 680 |   {
 681 |    "cell_type": "code",
 682 |    "execution_count": null,
 683 |    "metadata": {
 684 |     "coq_kernel_metadata": {
 685 |      "auto_roll_back": true
 686 |     }
 687 |    },
 688 |    "outputs": [],
 689 |    "source": []
 690 |   },
 691 |   {
 692 |    "cell_type": "code",
 693 |    "execution_count": null,
 694 |    "metadata": {
 695 |     "coq_kernel_metadata": {
 696 |      "auto_roll_back": true
 697 |     }
 698 |    },
 699 |    "outputs": [],
 700 |    "source": [
 701 |     "Admitted. (* 最後にここを Qed. に変える。*)"
 702 |    ]
 703 |   },
 704 |   {
 705 |    "cell_type": "markdown",
 706 |    "metadata": {
 707 |     "coq_kernel_metadata": {
 708 |      "auto_roll_back": true
 709 |     }
 710 |    },
 711 |    "source": [
 712 |     "\n",
 713 |     "## 演習問題 3-6\n",
 714 |     "　次の定理を証明せよ。\n",
 715 |     " 
　Theorem plus_assoc : (forall n m o : nat, n + (m +o)  = (n + m) + o ).\n",
 716 |     "\n",
 717 |     "### 下記の証明を完成させよ。\n",
 718 |     "#### [ヒント: 解答のコメントを読む](./Induction-Ex-Answer.ipynb#hint36) /  [解答を見る](./Induction-Ex-Answer.ipynb#ex36) "
 719 |    ]
 720 |   },
 721 |   {
 722 |    "cell_type": "code",
 723 |    "execution_count": null,
 724 |    "metadata": {
 725 |     "coq_kernel_metadata": {
 726 |      "auto_roll_back": true
 727 |     }
 728 |    },
 729 |    "outputs": [],
 730 |    "source": [
 731 |     "Theorem plus_assoc : (forall n m o : nat, n + (m +o)  = (n + m) + o )."
 732 |    ]
 733 |   },
 734 |   {
 735 |    "cell_type": "code",
 736 |    "execution_count": null,
 737 |    "metadata": {
 738 |     "coq_kernel_metadata": {
 739 |      "auto_roll_back": true
 740 |     }
 741 |    },
 742 |    "outputs": [],
 743 |    "source": []
 744 |   },
 745 |   {
 746 |    "cell_type": "code",
 747 |    "execution_count": null,
 748 |    "metadata": {
 749 |     "coq_kernel_metadata": {
 750 |      "auto_roll_back": true
 751 |     }
 752 |    },
 753 |    "outputs": [],
 754 |    "source": []
 755 |   },
 756 |   {
 757 |    "cell_type": "code",
 758 |    "execution_count": null,
 759 |    "metadata": {
 760 |     "coq_kernel_metadata": {
 761 |      "auto_roll_back": true
 762 |     }
 763 |    },
 764 |    "outputs": [],
 765 |    "source": []
 766 |   },
 767 |   {
 768 |    "cell_type": "code",
 769 |    "execution_count": null,
 770 |    "metadata": {
 771 |     "coq_kernel_metadata": {
 772 |      "auto_roll_back": true
 773 |     }
 774 |    },
 775 |    "outputs": [],
 776 |    "source": []
 777 |   },
 778 |   {
 779 |    "cell_type": "code",
 780 |    "execution_count": null,
 781 |    "metadata": {
 782 |     "coq_kernel_metadata": {
 783 |      "auto_roll_back": true
 784 |     }
 785 |    },
 786 |    "outputs": [],
 787 |    "source": []
 788 |   },
 789 |   {
 790 |    "cell_type": "code",
 791 |    "execution_count": null,
 792 |    "metadata": {
 793 |     "coq_kernel_metadata": {
 794 |      "auto_roll_back": true
 795 |     }
 796 |    },
 797 |    "outputs": [],
 798 |    "source": []
 799 |   },
 800 |   {
 801 |    "cell_type": "code",
 802 |    "execution_count": null,
 803 |    "metadata": {
 804 |     "coq_kernel_metadata": {
 805 |      "auto_roll_back": true
 806 |     }
 807 |    },
 808 |    "outputs": [],
 809 |    "source": []
 810 |   },
 811 |   {
 812 |    "cell_type": "code",
 813 |    "execution_count": null,
 814 |    "metadata": {
 815 |     "coq_kernel_metadata": {
 816 |      "auto_roll_back": true
 817 |     }
 818 |    },
 819 |    "outputs": [],
 820 |    "source": []
 821 |   },
 822 |   {
 823 |    "cell_type": "code",
 824 |    "execution_count": null,
 825 |    "metadata": {
 826 |     "coq_kernel_metadata": {
 827 |      "auto_roll_back": true
 828 |     }
 829 |    },
 830 |    "outputs": [],
 831 |    "source": []
 832 |   },
 833 |   {
 834 |    "cell_type": "code",
 835 |    "execution_count": null,
 836 |    "metadata": {
 837 |     "coq_kernel_metadata": {
 838 |      "auto_roll_back": true
 839 |     }
 840 |    },
 841 |    "outputs": [],
 842 |    "source": [
 843 |     "Admitted. (* 最後にここを Qed. に変える。*)"
 844 |    ]
 845 |   },
 846 |   {
 847 |    "cell_type": "markdown",
 848 |    "metadata": {
 849 |     "coq_kernel_metadata": {
 850 |      "auto_roll_back": true
 851 |     }
 852 |    },
 853 |    "source": [
 854 |     "\n",
 855 |     "## 演習問題 3-7\n",
 856 |     "　次の定理を証明せよ。\n",
 857 |     " 
　Theorem mul_succ_r : forall n m : nat,  n \\* S m = n \\* m + n  ).\n",
 858 |     "\n",
 859 |     "### 下記の証明を完成させよ。\n",
 860 |     "#### [ヒント: 解答のコメントを読む](./Induction-Ex-Answer.ipynb#hint37) /  [解答を見る](./Induction-Ex-Answer.ipynb#ex37) "
 861 |    ]
 862 |   },
 863 |   {
 864 |    "cell_type": "code",
 865 |    "execution_count": null,
 866 |    "metadata": {
 867 |     "coq_kernel_metadata": {
 868 |      "auto_roll_back": true
 869 |     }
 870 |    },
 871 |    "outputs": [],
 872 |    "source": [
 873 |     "Theorem mul_succ_r : forall n m : nat,  n * S m = n * m + n  )."
 874 |    ]
 875 |   },
 876 |   {
 877 |    "cell_type": "code",
 878 |    "execution_count": null,
 879 |    "metadata": {
 880 |     "coq_kernel_metadata": {
 881 |      "auto_roll_back": true
 882 |     }
 883 |    },
 884 |    "outputs": [],
 885 |    "source": []
 886 |   },
 887 |   {
 888 |    "cell_type": "code",
 889 |    "execution_count": null,
 890 |    "metadata": {
 891 |     "coq_kernel_metadata": {
 892 |      "auto_roll_back": true
 893 |     }
 894 |    },
 895 |    "outputs": [],
 896 |    "source": []
 897 |   },
 898 |   {
 899 |    "cell_type": "code",
 900 |    "execution_count": null,
 901 |    "metadata": {
 902 |     "coq_kernel_metadata": {
 903 |      "auto_roll_back": true
 904 |     }
 905 |    },
 906 |    "outputs": [],
 907 |    "source": []
 908 |   },
 909 |   {
 910 |    "cell_type": "code",
 911 |    "execution_count": null,
 912 |    "metadata": {
 913 |     "coq_kernel_metadata": {
 914 |      "auto_roll_back": true
 915 |     }
 916 |    },
 917 |    "outputs": [],
 918 |    "source": []
 919 |   },
 920 |   {
 921 |    "cell_type": "code",
 922 |    "execution_count": null,
 923 |    "metadata": {
 924 |     "coq_kernel_metadata": {
 925 |      "auto_roll_back": true
 926 |     }
 927 |    },
 928 |    "outputs": [],
 929 |    "source": []
 930 |   },
 931 |   {
 932 |    "cell_type": "code",
 933 |    "execution_count": null,
 934 |    "metadata": {
 935 |     "coq_kernel_metadata": {
 936 |      "auto_roll_back": true
 937 |     }
 938 |    },
 939 |    "outputs": [],
 940 |    "source": []
 941 |   },
 942 |   {
 943 |    "cell_type": "code",
 944 |    "execution_count": null,
 945 |    "metadata": {
 946 |     "coq_kernel_metadata": {
 947 |      "auto_roll_back": true
 948 |     }
 949 |    },
 950 |    "outputs": [],
 951 |    "source": []
 952 |   },
 953 |   {
 954 |    "cell_type": "code",
 955 |    "execution_count": null,
 956 |    "metadata": {
 957 |     "coq_kernel_metadata": {
 958 |      "auto_roll_back": true
 959 |     }
 960 |    },
 961 |    "outputs": [],
 962 |    "source": [
 963 |     "Admitted. (* 最後にここを Qed. に変える。*)"
 964 |    ]
 965 |   },
 966 |   {
 967 |    "cell_type": "markdown",
 968 |    "metadata": {
 969 |     "coq_kernel_metadata": {
 970 |      "auto_roll_back": true
 971 |     }
 972 |    },
 973 |    "source": [
 974 |     "\n",
 975 |     "## 演習問題 3-8\n",
 976 |     "　次の定理を証明せよ。\n",
 977 |     " 
　Theorem mult_comm : (forall n m : nat,  n \\* m = m \\* n ).\n",
 978 |     "\n",
 979 |     "### 下記の証明を完成させよ。\n",
 980 |     "#### [ヒント: 解答のコメントを読む](./Induction-Ex-Answer.ipynb#hint38) /  [解答を見る](./Induction-Ex-Answer.ipynb#ex38) "
 981 |    ]
 982 |   },
 983 |   {
 984 |    "cell_type": "code",
 985 |    "execution_count": null,
 986 |    "metadata": {
 987 |     "coq_kernel_metadata": {
 988 |      "auto_roll_back": true
 989 |     }
 990 |    },
 991 |    "outputs": [],
 992 |    "source": [
 993 |     "Theorem mult_comm : (forall n m : nat,  n * m = m * n )."
 994 |    ]
 995 |   },
 996 |   {
 997 |    "cell_type": "code",
 998 |    "execution_count": null,
 999 |    "metadata": {
1000 |     "coq_kernel_metadata": {
1001 |      "auto_roll_back": true
1002 |     }
1003 |    },
1004 |    "outputs": [],
1005 |    "source": []
1006 |   },
1007 |   {
1008 |    "cell_type": "code",
1009 |    "execution_count": null,
1010 |    "metadata": {
1011 |     "coq_kernel_metadata": {
1012 |      "auto_roll_back": true
1013 |     }
1014 |    },
1015 |    "outputs": [],
1016 |    "source": []
1017 |   },
1018 |   {
1019 |    "cell_type": "code",
1020 |    "execution_count": null,
1021 |    "metadata": {
1022 |     "coq_kernel_metadata": {
1023 |      "auto_roll_back": true
1024 |     }
1025 |    },
1026 |    "outputs": [],
1027 |    "source": []
1028 |   },
1029 |   {
1030 |    "cell_type": "code",
1031 |    "execution_count": null,
1032 |    "metadata": {
1033 |     "coq_kernel_metadata": {
1034 |      "auto_roll_back": true
1035 |     }
1036 |    },
1037 |    "outputs": [],
1038 |    "source": []
1039 |   },
1040 |   {
1041 |    "cell_type": "code",
1042 |    "execution_count": null,
1043 |    "metadata": {
1044 |     "coq_kernel_metadata": {
1045 |      "auto_roll_back": true
1046 |     }
1047 |    },
1048 |    "outputs": [],
1049 |    "source": []
1050 |   },
1051 |   {
1052 |    "cell_type": "code",
1053 |    "execution_count": null,
1054 |    "metadata": {
1055 |     "coq_kernel_metadata": {
1056 |      "auto_roll_back": true
1057 |     }
1058 |    },
1059 |    "outputs": [],
1060 |    "source": []
1061 |   },
1062 |   {
1063 |    "cell_type": "code",
1064 |    "execution_count": null,
1065 |    "metadata": {
1066 |     "coq_kernel_metadata": {
1067 |      "auto_roll_back": true
1068 |     }
1069 |    },
1070 |    "outputs": [],
1071 |    "source": []
1072 |   },
1073 |   {
1074 |    "cell_type": "code",
1075 |    "execution_count": null,
1076 |    "metadata": {
1077 |     "coq_kernel_metadata": {
1078 |      "auto_roll_back": true
1079 |     }
1080 |    },
1081 |    "outputs": [],
1082 |    "source": [
1083 |     "Admitted. (* 最後にここを Qed. に変える。*)"
1084 |    ]
1085 |   }
1086 |  ],
1087 |  "metadata": {
1088 |   "kernelspec": {
1089 |    "display_name": "Coq",
1090 |    "language": "coq",
1091 |    "name": "coq"
1092 |   },
1093 |   "language_info": {
1094 |    "file_extension": ".v",
1095 |    "mimetype": "text/x-coq",
1096 |    "name": "coq",
1097 |    "version": "8.9.1"
1098 |   }
1099 |  },
1100 |  "nbformat": 4,
1101 |  "nbformat_minor": 2
1102 | }
1103 | 


--------------------------------------------------------------------------------
/.ipynb_checkpoints/Logic-Ex-Work-checkpoint.ipynb:
--------------------------------------------------------------------------------
   1 | {
   2 |  "cells": [
   3 |   {
   4 |    "cell_type": "markdown",
   5 |    "metadata": {
   6 |     "coq_kernel_metadata": {
   7 |      "auto_roll_back": true
   8 |     }
   9 |    },
  10 |    "source": [
  11 |     "# 演習問題 1 作業用ページ\n",
  12 |     "\n",
  13 |     "[演習問題 1-1](./Logic-Ex-Work.ipynb#ex11) [演習問題 1-2](./Logic-Ex-Work.ipynb#ex12) [演習問題 1-3](./Logic-Ex-Work-Work.ipynb#ex13) [演習問題 1-4](./Logic-Ex-Work-Work.ipynb#ex14) [演習問題 1-5](./Logic-Ex-Work.ipynb#ex15) [演習問題 1-6](./Logic-Ex-Work.ipynb#ex16) [演習問題 1-7](./Logic-Ex-Work.ipynb#ex17) [演習問題 1-8](./Logic-Ex-Work.ipynb#ex18) [演習問題 1-9](./Logic-Ex-Work.ipynb#ex19) "
  14 |    ]
  15 |   },
  16 |   {
  17 |    "cell_type": "markdown",
  18 |    "metadata": {
  19 |     "coq_kernel_metadata": {
  20 |      "auto_roll_back": true
  21 |     }
  22 |    },
  23 |    "source": [
  24 |     "\n",
  25 |     "## 演習問題 1-1 \n",
  26 |     "次の定理を証明せよ。\n",
  27 |     " 
 Theorem my_first_proof : (forall A : Prop, A -> A).\n",
  28 |     "\n",
  29 |     "### ヒント\n",
  30 |     "- 冒頭の forall を intros で削除する。（Rule 1）\n",
  31 |     "- A -> A の最初のAを intros で仮説に入れる。(Rule 2)\n",
  32 |     "- そうすると、サブゴールが仮説の一つと一致していることがわかるので、assumption (Rule 3) または、exact (Rule 4) を使う。\n",
  33 |     "\n",
  34 |     "###  [解答を見る](./Logic-Ex-Answer.ipynb#ex11) "
  35 |    ]
  36 |   },
  37 |   {
  38 |    "cell_type": "code",
  39 |    "execution_count": null,
  40 |    "metadata": {
  41 |     "coq_kernel_metadata": {
  42 |      "auto_roll_back": true
  43 |     }
  44 |    },
  45 |    "outputs": [],
  46 |    "source": [
  47 |     "Theorem  my_first_proof : (forall A : Prop, A -> A).\n",
  48 |     "Proof."
  49 |    ]
  50 |   },
  51 |   {
  52 |    "cell_type": "code",
  53 |    "execution_count": null,
  54 |    "metadata": {
  55 |     "coq_kernel_metadata": {
  56 |      "auto_roll_back": true
  57 |     }
  58 |    },
  59 |    "outputs": [],
  60 |    "source": []
  61 |   },
  62 |   {
  63 |    "cell_type": "code",
  64 |    "execution_count": null,
  65 |    "metadata": {
  66 |     "coq_kernel_metadata": {
  67 |      "auto_roll_back": true
  68 |     }
  69 |    },
  70 |    "outputs": [],
  71 |    "source": []
  72 |   },
  73 |   {
  74 |    "cell_type": "code",
  75 |    "execution_count": null,
  76 |    "metadata": {
  77 |     "coq_kernel_metadata": {
  78 |      "auto_roll_back": true
  79 |     }
  80 |    },
  81 |    "outputs": [],
  82 |    "source": []
  83 |   },
  84 |   {
  85 |    "cell_type": "code",
  86 |    "execution_count": null,
  87 |    "metadata": {
  88 |     "coq_kernel_metadata": {
  89 |      "auto_roll_back": true
  90 |     }
  91 |    },
  92 |    "outputs": [],
  93 |    "source": []
  94 |   },
  95 |   {
  96 |    "cell_type": "code",
  97 |    "execution_count": null,
  98 |    "metadata": {
  99 |     "coq_kernel_metadata": {
 100 |      "auto_roll_back": true
 101 |     }
 102 |    },
 103 |    "outputs": [],
 104 |    "source": []
 105 |   },
 106 |   {
 107 |    "cell_type": "code",
 108 |    "execution_count": null,
 109 |    "metadata": {
 110 |     "coq_kernel_metadata": {
 111 |      "auto_roll_back": true
 112 |     }
 113 |    },
 114 |    "outputs": [],
 115 |    "source": [
 116 |     "Admitted. (* 最後にここを Qed. に変える。*)"
 117 |    ]
 118 |   },
 119 |   {
 120 |    "cell_type": "markdown",
 121 |    "metadata": {
 122 |     "collapsed": true,
 123 |     "coq_kernel_metadata": {
 124 |      "auto_roll_back": true
 125 |     }
 126 |    },
 127 |    "source": [
 128 |     "\n",
 129 |     "## 演習問題 1-2\n",
 130 |     "次の定理を証明せよ。\n",
 131 |     " 
 Theorem and1 : (forall A  B: Prop, A /\\ B -> A).\n",
 132 |     "\n",
 133 |     "### ヒント\n",
 134 |     "- intros. で forall を削除し、`A /\\ B -> A` の前提部分の`A /\\ B`を仮説に移す。（Rule 1,2）\n",
 135 |     "- 仮説中の`A /\\ B`を、destruct で分解する。(Rule 6)\n",
 136 |     "- 仮説の一つがサブゴールに一致している。（Rule 3,4,5）\n",
 137 |     "\n",
 138 |     "###  [解答を見る](./Logic-Ex-Answer.ipynb#ex12) "
 139 |    ]
 140 |   },
 141 |   {
 142 |    "cell_type": "code",
 143 |    "execution_count": null,
 144 |    "metadata": {
 145 |     "coq_kernel_metadata": {
 146 |      "auto_roll_back": true
 147 |     }
 148 |    },
 149 |    "outputs": [],
 150 |    "source": [
 151 |     "Theorem and1 : (forall A  B: Prop, A /\\ B -> A)."
 152 |    ]
 153 |   },
 154 |   {
 155 |    "cell_type": "code",
 156 |    "execution_count": null,
 157 |    "metadata": {
 158 |     "coq_kernel_metadata": {
 159 |      "auto_roll_back": true
 160 |     }
 161 |    },
 162 |    "outputs": [],
 163 |    "source": []
 164 |   },
 165 |   {
 166 |    "cell_type": "code",
 167 |    "execution_count": null,
 168 |    "metadata": {
 169 |     "coq_kernel_metadata": {
 170 |      "auto_roll_back": true
 171 |     }
 172 |    },
 173 |    "outputs": [],
 174 |    "source": []
 175 |   },
 176 |   {
 177 |    "cell_type": "code",
 178 |    "execution_count": null,
 179 |    "metadata": {
 180 |     "coq_kernel_metadata": {
 181 |      "auto_roll_back": true
 182 |     }
 183 |    },
 184 |    "outputs": [],
 185 |    "source": []
 186 |   },
 187 |   {
 188 |    "cell_type": "code",
 189 |    "execution_count": null,
 190 |    "metadata": {
 191 |     "coq_kernel_metadata": {
 192 |      "auto_roll_back": true
 193 |     }
 194 |    },
 195 |    "outputs": [],
 196 |    "source": []
 197 |   },
 198 |   {
 199 |    "cell_type": "code",
 200 |    "execution_count": null,
 201 |    "metadata": {
 202 |     "coq_kernel_metadata": {
 203 |      "auto_roll_back": true
 204 |     }
 205 |    },
 206 |    "outputs": [],
 207 |    "source": []
 208 |   },
 209 |   {
 210 |    "cell_type": "code",
 211 |    "execution_count": null,
 212 |    "metadata": {
 213 |     "coq_kernel_metadata": {
 214 |      "auto_roll_back": true
 215 |     }
 216 |    },
 217 |    "outputs": [],
 218 |    "source": []
 219 |   },
 220 |   {
 221 |    "cell_type": "code",
 222 |    "execution_count": null,
 223 |    "metadata": {
 224 |     "coq_kernel_metadata": {
 225 |      "auto_roll_back": true
 226 |     }
 227 |    },
 228 |    "outputs": [],
 229 |    "source": []
 230 |   },
 231 |   {
 232 |    "cell_type": "code",
 233 |    "execution_count": null,
 234 |    "metadata": {
 235 |     "coq_kernel_metadata": {
 236 |      "auto_roll_back": true
 237 |     }
 238 |    },
 239 |    "outputs": [],
 240 |    "source": [
 241 |     "Admitted. (* 最後にここを Qed. に変える。*)"
 242 |    ]
 243 |   },
 244 |   {
 245 |    "cell_type": "markdown",
 246 |    "metadata": {
 247 |     "coq_kernel_metadata": {
 248 |      "auto_roll_back": true
 249 |     }
 250 |    },
 251 |    "source": [
 252 |     "\n",
 253 |     "## 演習問題 1-3\n",
 254 |     "次の定理を証明せよ。\n",
 255 |     " 
 Theorem and1 : (forall A  B: Prop, A /\\ B -> B).\n",
 256 |     "\n",
 257 |     "### ヒント\n",
 258 |     "- intros. で forall を削除し、`A /\\ B -> B` の前提部分の`A /\\ B`を仮説に移す。（Rule 1,2）\n",
 259 |     "- 仮説中の`A /\\ B`を、destruct で分解する。(Rule 6)\n",
 260 |     "- 仮説の一つがサブゴールに一致している。（Rule 3,4,5）\n",
 261 |     "\n",
 262 |     "###  [解答を見る](./Logic-Ex-Answer.ipynb#ex13) "
 263 |    ]
 264 |   },
 265 |   {
 266 |    "cell_type": "code",
 267 |    "execution_count": null,
 268 |    "metadata": {
 269 |     "coq_kernel_metadata": {
 270 |      "auto_roll_back": true
 271 |     }
 272 |    },
 273 |    "outputs": [],
 274 |    "source": [
 275 |     " Theorem and1 : (forall A  B: Prop, A /\\ B -> B)."
 276 |    ]
 277 |   },
 278 |   {
 279 |    "cell_type": "code",
 280 |    "execution_count": null,
 281 |    "metadata": {
 282 |     "coq_kernel_metadata": {
 283 |      "auto_roll_back": true
 284 |     }
 285 |    },
 286 |    "outputs": [],
 287 |    "source": []
 288 |   },
 289 |   {
 290 |    "cell_type": "code",
 291 |    "execution_count": null,
 292 |    "metadata": {
 293 |     "coq_kernel_metadata": {
 294 |      "auto_roll_back": true
 295 |     }
 296 |    },
 297 |    "outputs": [],
 298 |    "source": []
 299 |   },
 300 |   {
 301 |    "cell_type": "code",
 302 |    "execution_count": null,
 303 |    "metadata": {
 304 |     "coq_kernel_metadata": {
 305 |      "auto_roll_back": true
 306 |     }
 307 |    },
 308 |    "outputs": [],
 309 |    "source": []
 310 |   },
 311 |   {
 312 |    "cell_type": "code",
 313 |    "execution_count": null,
 314 |    "metadata": {
 315 |     "coq_kernel_metadata": {
 316 |      "auto_roll_back": true
 317 |     }
 318 |    },
 319 |    "outputs": [],
 320 |    "source": []
 321 |   },
 322 |   {
 323 |    "cell_type": "code",
 324 |    "execution_count": null,
 325 |    "metadata": {
 326 |     "coq_kernel_metadata": {
 327 |      "auto_roll_back": true
 328 |     }
 329 |    },
 330 |    "outputs": [],
 331 |    "source": []
 332 |   },
 333 |   {
 334 |    "cell_type": "code",
 335 |    "execution_count": null,
 336 |    "metadata": {
 337 |     "coq_kernel_metadata": {
 338 |      "auto_roll_back": true
 339 |     }
 340 |    },
 341 |    "outputs": [],
 342 |    "source": []
 343 |   },
 344 |   {
 345 |    "cell_type": "code",
 346 |    "execution_count": null,
 347 |    "metadata": {
 348 |     "coq_kernel_metadata": {
 349 |      "auto_roll_back": true
 350 |     }
 351 |    },
 352 |    "outputs": [],
 353 |    "source": [
 354 |     "Admitted. (* 最後にここを Qed. に変える。*)"
 355 |    ]
 356 |   },
 357 |   {
 358 |    "cell_type": "markdown",
 359 |    "metadata": {
 360 |     "coq_kernel_metadata": {
 361 |      "auto_roll_back": true
 362 |     }
 363 |    },
 364 |    "source": [
 365 |     "\n",
 366 |     "## 演習問題 1-4\n",
 367 |     "次の定理を証明せよ。\n",
 368 |     " 
 Theorem or1 : (forall A  B: Prop, A -> A \\\\/ B).\n",
 369 |     "\n",
 370 |     "### ヒント\n",
 371 |     "- intros. でforallを消す。(Rule 1)\n",
 372 |     "- left. でAを仮説に取り込む。(Rule 7)\n",
 373 |     "- サブゴールと仮説が一致している。(Rule 3,4,5)\n",
 374 |     "\n",
 375 |     "###  [解答を見る](./Logic-Ex-Answer.ipynb#ex14) "
 376 |    ]
 377 |   },
 378 |   {
 379 |    "cell_type": "code",
 380 |    "execution_count": null,
 381 |    "metadata": {
 382 |     "coq_kernel_metadata": {
 383 |      "auto_roll_back": true
 384 |     }
 385 |    },
 386 |    "outputs": [],
 387 |    "source": [
 388 |     "Theorem or1 : (forall A  B: Prop, A -> A \\/ B)."
 389 |    ]
 390 |   },
 391 |   {
 392 |    "cell_type": "code",
 393 |    "execution_count": null,
 394 |    "metadata": {
 395 |     "coq_kernel_metadata": {
 396 |      "auto_roll_back": true
 397 |     }
 398 |    },
 399 |    "outputs": [],
 400 |    "source": []
 401 |   },
 402 |   {
 403 |    "cell_type": "code",
 404 |    "execution_count": null,
 405 |    "metadata": {
 406 |     "coq_kernel_metadata": {
 407 |      "auto_roll_back": true
 408 |     }
 409 |    },
 410 |    "outputs": [],
 411 |    "source": []
 412 |   },
 413 |   {
 414 |    "cell_type": "code",
 415 |    "execution_count": null,
 416 |    "metadata": {
 417 |     "coq_kernel_metadata": {
 418 |      "auto_roll_back": true
 419 |     }
 420 |    },
 421 |    "outputs": [],
 422 |    "source": []
 423 |   },
 424 |   {
 425 |    "cell_type": "code",
 426 |    "execution_count": null,
 427 |    "metadata": {
 428 |     "coq_kernel_metadata": {
 429 |      "auto_roll_back": true
 430 |     }
 431 |    },
 432 |    "outputs": [],
 433 |    "source": []
 434 |   },
 435 |   {
 436 |    "cell_type": "code",
 437 |    "execution_count": null,
 438 |    "metadata": {
 439 |     "coq_kernel_metadata": {
 440 |      "auto_roll_back": true
 441 |     }
 442 |    },
 443 |    "outputs": [],
 444 |    "source": []
 445 |   },
 446 |   {
 447 |    "cell_type": "code",
 448 |    "execution_count": null,
 449 |    "metadata": {
 450 |     "coq_kernel_metadata": {
 451 |      "auto_roll_back": true
 452 |     }
 453 |    },
 454 |    "outputs": [],
 455 |    "source": []
 456 |   },
 457 |   {
 458 |    "cell_type": "code",
 459 |    "execution_count": null,
 460 |    "metadata": {
 461 |     "coq_kernel_metadata": {
 462 |      "auto_roll_back": true
 463 |     }
 464 |    },
 465 |    "outputs": [],
 466 |    "source": []
 467 |   },
 468 |   {
 469 |    "cell_type": "code",
 470 |    "execution_count": null,
 471 |    "metadata": {
 472 |     "coq_kernel_metadata": {
 473 |      "auto_roll_back": true
 474 |     }
 475 |    },
 476 |    "outputs": [],
 477 |    "source": [
 478 |     "Admitted. (* 最後にここを Qed. に変える。*)"
 479 |    ]
 480 |   },
 481 |   {
 482 |    "cell_type": "markdown",
 483 |    "metadata": {
 484 |     "coq_kernel_metadata": {
 485 |      "auto_roll_back": true
 486 |     }
 487 |    },
 488 |    "source": [
 489 |     "\n",
 490 |     "## 演習問題 1-5\n",
 491 |     "次の定理を証明せよ。\n",
 492 |     " 
 Theorem or2 : (forall A  B: Prop, B -> A \\\\/ B).\n",
 493 |     "\n",
 494 |     "### ヒント\n",
 495 |     "- intros. でforallを消す。(Rule 1)\n",
 496 |     "- right. でBを仮説に取り込む。(Rule 7)\n",
 497 |     "- サブゴールと仮説が一致している。(Rule 3,4,5)\n",
 498 |     "\n",
 499 |     "###  [解答を見る](./Logic-Ex-Answer.ipynb#ex15) "
 500 |    ]
 501 |   },
 502 |   {
 503 |    "cell_type": "code",
 504 |    "execution_count": null,
 505 |    "metadata": {
 506 |     "coq_kernel_metadata": {
 507 |      "auto_roll_back": true
 508 |     }
 509 |    },
 510 |    "outputs": [],
 511 |    "source": [
 512 |     "Theorem or2 : (forall A  B: Prop, B -> A \\/ B)."
 513 |    ]
 514 |   },
 515 |   {
 516 |    "cell_type": "code",
 517 |    "execution_count": null,
 518 |    "metadata": {
 519 |     "coq_kernel_metadata": {
 520 |      "auto_roll_back": true
 521 |     }
 522 |    },
 523 |    "outputs": [],
 524 |    "source": []
 525 |   },
 526 |   {
 527 |    "cell_type": "code",
 528 |    "execution_count": null,
 529 |    "metadata": {
 530 |     "coq_kernel_metadata": {
 531 |      "auto_roll_back": true
 532 |     }
 533 |    },
 534 |    "outputs": [],
 535 |    "source": []
 536 |   },
 537 |   {
 538 |    "cell_type": "code",
 539 |    "execution_count": null,
 540 |    "metadata": {
 541 |     "coq_kernel_metadata": {
 542 |      "auto_roll_back": true
 543 |     }
 544 |    },
 545 |    "outputs": [],
 546 |    "source": []
 547 |   },
 548 |   {
 549 |    "cell_type": "code",
 550 |    "execution_count": null,
 551 |    "metadata": {
 552 |     "coq_kernel_metadata": {
 553 |      "auto_roll_back": true
 554 |     }
 555 |    },
 556 |    "outputs": [],
 557 |    "source": []
 558 |   },
 559 |   {
 560 |    "cell_type": "code",
 561 |    "execution_count": null,
 562 |    "metadata": {
 563 |     "coq_kernel_metadata": {
 564 |      "auto_roll_back": true
 565 |     }
 566 |    },
 567 |    "outputs": [],
 568 |    "source": []
 569 |   },
 570 |   {
 571 |    "cell_type": "code",
 572 |    "execution_count": null,
 573 |    "metadata": {
 574 |     "coq_kernel_metadata": {
 575 |      "auto_roll_back": true
 576 |     }
 577 |    },
 578 |    "outputs": [],
 579 |    "source": []
 580 |   },
 581 |   {
 582 |    "cell_type": "code",
 583 |    "execution_count": null,
 584 |    "metadata": {
 585 |     "coq_kernel_metadata": {
 586 |      "auto_roll_back": true
 587 |     }
 588 |    },
 589 |    "outputs": [],
 590 |    "source": [
 591 |     "Admitted. (* 最後にここを Qed. に変える。*)"
 592 |    ]
 593 |   },
 594 |   {
 595 |    "cell_type": "markdown",
 596 |    "metadata": {
 597 |     "coq_kernel_metadata": {
 598 |      "auto_roll_back": true
 599 |     }
 600 |    },
 601 |    "source": [
 602 |     "\n",
 603 |     "## 演習問題 1-6\n",
 604 |     "次の定理を証明せよ。\n",
 605 |     " 
 Theorem modus_ponens : (forall A  B: Prop, A -> (A -> B) ->B).\n",
 606 |     "\n",
 607 |     "### ヒント\n",
 608 |     "- intros. で先頭のforall を削除する。(Rule1)\n",
 609 |     "- 再び intros で、AとA -> Bを仮説に移す。(Rule 2) \n",
 610 |     "- 仮説A -> Bを仮説（この場合はA）に適用して、サブゴールをBに変える。applyを使う。(Rule 9)\n",
 611 |     "- 仮説とサブゴールが一致する。(Rule 3,4,5)\n",
 612 |     "\n",
 613 |     "###  [解答を見る](./Logic-Ex-Answer.ipynb#ex16) "
 614 |    ]
 615 |   },
 616 |   {
 617 |    "cell_type": "code",
 618 |    "execution_count": null,
 619 |    "metadata": {
 620 |     "coq_kernel_metadata": {
 621 |      "auto_roll_back": true
 622 |     }
 623 |    },
 624 |    "outputs": [],
 625 |    "source": [
 626 |     "Theorem modus_ponens : (forall A  B: Prop, A -> (A -> B) ->B)"
 627 |    ]
 628 |   },
 629 |   {
 630 |    "cell_type": "code",
 631 |    "execution_count": null,
 632 |    "metadata": {
 633 |     "coq_kernel_metadata": {
 634 |      "auto_roll_back": true
 635 |     }
 636 |    },
 637 |    "outputs": [],
 638 |    "source": []
 639 |   },
 640 |   {
 641 |    "cell_type": "code",
 642 |    "execution_count": null,
 643 |    "metadata": {
 644 |     "coq_kernel_metadata": {
 645 |      "auto_roll_back": true
 646 |     }
 647 |    },
 648 |    "outputs": [],
 649 |    "source": []
 650 |   },
 651 |   {
 652 |    "cell_type": "code",
 653 |    "execution_count": null,
 654 |    "metadata": {
 655 |     "coq_kernel_metadata": {
 656 |      "auto_roll_back": true
 657 |     }
 658 |    },
 659 |    "outputs": [],
 660 |    "source": []
 661 |   },
 662 |   {
 663 |    "cell_type": "code",
 664 |    "execution_count": null,
 665 |    "metadata": {
 666 |     "coq_kernel_metadata": {
 667 |      "auto_roll_back": true
 668 |     }
 669 |    },
 670 |    "outputs": [],
 671 |    "source": []
 672 |   },
 673 |   {
 674 |    "cell_type": "code",
 675 |    "execution_count": null,
 676 |    "metadata": {
 677 |     "coq_kernel_metadata": {
 678 |      "auto_roll_back": true
 679 |     }
 680 |    },
 681 |    "outputs": [],
 682 |    "source": []
 683 |   },
 684 |   {
 685 |    "cell_type": "code",
 686 |    "execution_count": null,
 687 |    "metadata": {
 688 |     "coq_kernel_metadata": {
 689 |      "auto_roll_back": true
 690 |     }
 691 |    },
 692 |    "outputs": [],
 693 |    "source": []
 694 |   },
 695 |   {
 696 |    "cell_type": "code",
 697 |    "execution_count": null,
 698 |    "metadata": {
 699 |     "coq_kernel_metadata": {
 700 |      "auto_roll_back": true
 701 |     }
 702 |    },
 703 |    "outputs": [],
 704 |    "source": [
 705 |     "Admitted. (* 最後にここを Qed. に変える。*)"
 706 |    ]
 707 |   },
 708 |   {
 709 |    "cell_type": "markdown",
 710 |    "metadata": {
 711 |     "coq_kernel_metadata": {
 712 |      "auto_roll_back": true
 713 |     }
 714 |    },
 715 |    "source": [
 716 |     "\n",
 717 |     "## 演習問題 1-7\n",
 718 |     "次の定理を証明せよ。\n",
 719 |     " 
 Theorem modus_ponens2 : (forall A  B C: Prop, A -> (A -> B) ->(B ->C) -> C).\n",
 720 |     "\n",
 721 |     "### ヒント\n",
 722 |     "- intros でforallを削除する。\n",
 723 |     "- intros で、全ての -> の前提部分を仮説に移す。この時、適当なラベルをつけておくと分かりやすくなる。この例の場合なら、intros Proof_of_A A_implies_B B_implies_C とか。\n",
 724 |     "- apply B_implies_C で、サブゴールをBに変える。\n",
 725 |     "- apply A_implies_B で、サブゴールをAに変える。\n",
 726 |     "- 仮説とサブゴールが一致している。\n",
 727 |     "\n",
 728 |     "###  [解答を見る](./Logic-Ex-Answer.ipynb#ex17) "
 729 |    ]
 730 |   },
 731 |   {
 732 |    "cell_type": "code",
 733 |    "execution_count": null,
 734 |    "metadata": {
 735 |     "coq_kernel_metadata": {
 736 |      "auto_roll_back": true
 737 |     }
 738 |    },
 739 |    "outputs": [],
 740 |    "source": [
 741 |     "Theorem modus_ponens2 : (forall A  B C: Prop, A -> (A -> B) ->(B ->C) -> C)."
 742 |    ]
 743 |   },
 744 |   {
 745 |    "cell_type": "code",
 746 |    "execution_count": null,
 747 |    "metadata": {
 748 |     "coq_kernel_metadata": {
 749 |      "auto_roll_back": true
 750 |     }
 751 |    },
 752 |    "outputs": [],
 753 |    "source": []
 754 |   },
 755 |   {
 756 |    "cell_type": "code",
 757 |    "execution_count": null,
 758 |    "metadata": {
 759 |     "coq_kernel_metadata": {
 760 |      "auto_roll_back": true
 761 |     }
 762 |    },
 763 |    "outputs": [],
 764 |    "source": []
 765 |   },
 766 |   {
 767 |    "cell_type": "code",
 768 |    "execution_count": null,
 769 |    "metadata": {
 770 |     "coq_kernel_metadata": {
 771 |      "auto_roll_back": true
 772 |     }
 773 |    },
 774 |    "outputs": [],
 775 |    "source": []
 776 |   },
 777 |   {
 778 |    "cell_type": "code",
 779 |    "execution_count": null,
 780 |    "metadata": {
 781 |     "coq_kernel_metadata": {
 782 |      "auto_roll_back": true
 783 |     }
 784 |    },
 785 |    "outputs": [],
 786 |    "source": []
 787 |   },
 788 |   {
 789 |    "cell_type": "code",
 790 |    "execution_count": null,
 791 |    "metadata": {
 792 |     "coq_kernel_metadata": {
 793 |      "auto_roll_back": true
 794 |     }
 795 |    },
 796 |    "outputs": [],
 797 |    "source": []
 798 |   },
 799 |   {
 800 |    "cell_type": "code",
 801 |    "execution_count": null,
 802 |    "metadata": {
 803 |     "coq_kernel_metadata": {
 804 |      "auto_roll_back": true
 805 |     }
 806 |    },
 807 |    "outputs": [],
 808 |    "source": []
 809 |   },
 810 |   {
 811 |    "cell_type": "code",
 812 |    "execution_count": null,
 813 |    "metadata": {
 814 |     "coq_kernel_metadata": {
 815 |      "auto_roll_back": true
 816 |     }
 817 |    },
 818 |    "outputs": [],
 819 |    "source": []
 820 |   },
 821 |   {
 822 |    "cell_type": "code",
 823 |    "execution_count": null,
 824 |    "metadata": {
 825 |     "coq_kernel_metadata": {
 826 |      "auto_roll_back": true
 827 |     }
 828 |    },
 829 |    "outputs": [],
 830 |    "source": [
 831 |     "Admitted. (* 最後にここを Qed. に変える。*)"
 832 |    ]
 833 |   },
 834 |   {
 835 |    "cell_type": "markdown",
 836 |    "metadata": {
 837 |     "coq_kernel_metadata": {
 838 |      "auto_roll_back": true
 839 |     }
 840 |    },
 841 |    "source": [
 842 |     "\n",
 843 |     "## 演習問題 1-8\n",
 844 |     "次の定理を証明せよ。\n",
 845 |     " 
 Theorem and_comm : (forall A  B: Prop, A /\\ B -> B /\\ A ).\n",
 846 |     "\n",
 847 |     "### ヒント\n",
 848 |     "- intros で、forall を削除し、\"A /\\ B\" にH というラベルをつける 。(Rule 1)\n",
 849 |     "- \"H: A/\\ B\" を destruct で分解し、A, B を仮説に追加する。　(Rule 6)\n",
 850 |     "- サブゴール \"A\\/ B\" を split で、二つのサブゴールに分割する。　(Rule 8)\n",
 851 |     "- 一つ目のサブゴールは仮説の一つと同じなので解ける。サブゴールは一つになる。(Rule 2,3,4)\n",
 852 |     "- 残ったサブゴールも仮説の一つと一緒なので、解ける。(Rule 2,3,4)\n",
 853 |     "- サブゴールは残っていないので証明終了。\n",
 854 |     "\n",
 855 |     "###  [解答を見る](./Logic-Ex-Answer.ipynb#ex18) "
 856 |    ]
 857 |   },
 858 |   {
 859 |    "cell_type": "code",
 860 |    "execution_count": null,
 861 |    "metadata": {
 862 |     "coq_kernel_metadata": {
 863 |      "auto_roll_back": true
 864 |     }
 865 |    },
 866 |    "outputs": [],
 867 |    "source": [
 868 |     "Theorem and_comm : (forall A  B: Prop, A /\\ B -> B /\\ A )."
 869 |    ]
 870 |   },
 871 |   {
 872 |    "cell_type": "code",
 873 |    "execution_count": null,
 874 |    "metadata": {
 875 |     "coq_kernel_metadata": {
 876 |      "auto_roll_back": true
 877 |     }
 878 |    },
 879 |    "outputs": [],
 880 |    "source": []
 881 |   },
 882 |   {
 883 |    "cell_type": "code",
 884 |    "execution_count": null,
 885 |    "metadata": {
 886 |     "coq_kernel_metadata": {
 887 |      "auto_roll_back": true
 888 |     }
 889 |    },
 890 |    "outputs": [],
 891 |    "source": []
 892 |   },
 893 |   {
 894 |    "cell_type": "code",
 895 |    "execution_count": null,
 896 |    "metadata": {
 897 |     "coq_kernel_metadata": {
 898 |      "auto_roll_back": true
 899 |     }
 900 |    },
 901 |    "outputs": [],
 902 |    "source": []
 903 |   },
 904 |   {
 905 |    "cell_type": "code",
 906 |    "execution_count": null,
 907 |    "metadata": {
 908 |     "coq_kernel_metadata": {
 909 |      "auto_roll_back": true
 910 |     }
 911 |    },
 912 |    "outputs": [],
 913 |    "source": []
 914 |   },
 915 |   {
 916 |    "cell_type": "code",
 917 |    "execution_count": null,
 918 |    "metadata": {
 919 |     "coq_kernel_metadata": {
 920 |      "auto_roll_back": true
 921 |     }
 922 |    },
 923 |    "outputs": [],
 924 |    "source": []
 925 |   },
 926 |   {
 927 |    "cell_type": "code",
 928 |    "execution_count": null,
 929 |    "metadata": {
 930 |     "coq_kernel_metadata": {
 931 |      "auto_roll_back": true
 932 |     }
 933 |    },
 934 |    "outputs": [],
 935 |    "source": []
 936 |   },
 937 |   {
 938 |    "cell_type": "code",
 939 |    "execution_count": null,
 940 |    "metadata": {
 941 |     "coq_kernel_metadata": {
 942 |      "auto_roll_back": true
 943 |     }
 944 |    },
 945 |    "outputs": [],
 946 |    "source": []
 947 |   },
 948 |   {
 949 |    "cell_type": "code",
 950 |    "execution_count": null,
 951 |    "metadata": {
 952 |     "coq_kernel_metadata": {
 953 |      "auto_roll_back": true
 954 |     }
 955 |    },
 956 |    "outputs": [],
 957 |    "source": [
 958 |     "Admitted. (* 最後にここを Qed. に変える。*)"
 959 |    ]
 960 |   },
 961 |   {
 962 |    "cell_type": "markdown",
 963 |    "metadata": {
 964 |     "coq_kernel_metadata": {
 965 |      "auto_roll_back": true
 966 |     }
 967 |    },
 968 |    "source": [
 969 |     "\n",
 970 |     "## 演習問題 1-9\n",
 971 |     "次の定理を証明せよ。\n",
 972 |     " 
 Theorem or_comm : (forall A B: Prop, A \\\\/ B -> B \\\\/ A ) .\n",
 973 |     "\n",
 974 |     "### ヒント\n",
 975 |     "-  intros で\"A \\/ B\" を仮定に移し、A_or_B というラベルをつける。\n",
 976 |     "-  A\\/B をdestructすると、Aが仮定に残り、サブゴールが二つに分割される。\n",
 977 |     "-  一つ目のサブゴールから、right でB\\/A の右の項Aをサブゴールに残す。 \n",
 978 |     "-  サブゴールと仮定が一致している。 サブゴールは一つになる。\n",
 979 |     "-  一つ目のサブゴールから、left でB\\/A の左の項Bをサブゴールに残す。 \n",
 980 |     "\n",
 981 |     "###  [解答を見る](./Logic-Ex-Answer.ipynb#ex) "
 982 |    ]
 983 |   },
 984 |   {
 985 |    "cell_type": "code",
 986 |    "execution_count": null,
 987 |    "metadata": {
 988 |     "coq_kernel_metadata": {
 989 |      "auto_roll_back": true
 990 |     }
 991 |    },
 992 |    "outputs": [],
 993 |    "source": [
 994 |     "Theorem or_comm : (forall A B: Prop, A \\/ B -> B \\/ A ) ."
 995 |    ]
 996 |   },
 997 |   {
 998 |    "cell_type": "code",
 999 |    "execution_count": null,
1000 |    "metadata": {
1001 |     "coq_kernel_metadata": {
1002 |      "auto_roll_back": true
1003 |     }
1004 |    },
1005 |    "outputs": [],
1006 |    "source": []
1007 |   },
1008 |   {
1009 |    "cell_type": "code",
1010 |    "execution_count": null,
1011 |    "metadata": {
1012 |     "coq_kernel_metadata": {
1013 |      "auto_roll_back": true
1014 |     }
1015 |    },
1016 |    "outputs": [],
1017 |    "source": []
1018 |   },
1019 |   {
1020 |    "cell_type": "code",
1021 |    "execution_count": null,
1022 |    "metadata": {
1023 |     "coq_kernel_metadata": {
1024 |      "auto_roll_back": true
1025 |     }
1026 |    },
1027 |    "outputs": [],
1028 |    "source": []
1029 |   },
1030 |   {
1031 |    "cell_type": "code",
1032 |    "execution_count": null,
1033 |    "metadata": {
1034 |     "coq_kernel_metadata": {
1035 |      "auto_roll_back": true
1036 |     }
1037 |    },
1038 |    "outputs": [],
1039 |    "source": []
1040 |   },
1041 |   {
1042 |    "cell_type": "code",
1043 |    "execution_count": null,
1044 |    "metadata": {
1045 |     "coq_kernel_metadata": {
1046 |      "auto_roll_back": true
1047 |     }
1048 |    },
1049 |    "outputs": [],
1050 |    "source": []
1051 |   },
1052 |   {
1053 |    "cell_type": "code",
1054 |    "execution_count": null,
1055 |    "metadata": {
1056 |     "coq_kernel_metadata": {
1057 |      "auto_roll_back": true
1058 |     }
1059 |    },
1060 |    "outputs": [],
1061 |    "source": []
1062 |   },
1063 |   {
1064 |    "cell_type": "code",
1065 |    "execution_count": null,
1066 |    "metadata": {
1067 |     "coq_kernel_metadata": {
1068 |      "auto_roll_back": true
1069 |     }
1070 |    },
1071 |    "outputs": [],
1072 |    "source": []
1073 |   },
1074 |   {
1075 |    "cell_type": "code",
1076 |    "execution_count": null,
1077 |    "metadata": {
1078 |     "coq_kernel_metadata": {
1079 |      "auto_roll_back": true
1080 |     }
1081 |    },
1082 |    "outputs": [],
1083 |    "source": [
1084 |     "Admitted. (* 最後にここを Qed. に変える。*)"
1085 |    ]
1086 |   }
1087 |  ],
1088 |  "metadata": {
1089 |   "kernelspec": {
1090 |    "display_name": "Coq",
1091 |    "language": "coq",
1092 |    "name": "coq"
1093 |   },
1094 |   "language_info": {
1095 |    "file_extension": ".v",
1096 |    "mimetype": "text/x-coq",
1097 |    "name": "coq",
1098 |    "version": "8.9.1"
1099 |   }
1100 |  },
1101 |  "nbformat": 4,
1102 |  "nbformat_minor": 2
1103 | }
1104 | 


--------------------------------------------------------------------------------