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* fix: delete all posts * chore: Pull-Request [blog-7-12-2023] from Obsidian (#41) * PUSH NOTE : 수학 공부에 대한 고찰.md * PUSH NOTE : Rules of Inference with Coq.md * PUSH NOTE : 블로그 이주 이야기.md * PUSH ATTACHMENT : blog-logo.png * PUSH ATTACHMENT : github-publisher.png * PUSH NOTE : 10. StatefulSets - Deploying Replicated Stateful Applications.md * PUSH ATTACHMENT : k8s-10.jpeg * PUSH NOTE : 09. Deployments - Updating Applications Declaratively.md * PUSH ATTACHMENT : k8s-09.jpeg * PUSH NOTE : 08. Accessing Pod Metadata and Other Resources from Applications.md * PUSH ATTACHMENT : k8s-08.jpeg * PUSH NOTE : 07. ConfigMaps and Secrets - Configuring Applications.md * PUSH ATTACHMENT : k8s-07.jpeg * PUSH NOTE : 06. Volumes - Attaching Disk Storage to Containers.md * PUSH ATTACHMENT : k8s-06.jpeg * PUSH NOTE : 05. Services - Enabling Clients to Discover and Talk to Pods.md * PUSH ATTACHMENT : k8s-05.jpeg * PUSH NOTE : 04. Replication and Other Controllers - Deploying Managed Pods.md * PUSH ATTACHMENT : k8s-04.jpeg * PUSH NOTE : 03. Pods - Running Containers in Kubernetes.md * PUSH ATTACHMENT : k8s-03.jpeg * PUSH NOTE : 02. First Steps with Docker and Kubernetes.md * PUSH ATTACHMENT : k8s-02.jpeg * PUSH NOTE : 01. Introducing Kubernetes.md * PUSH ATTACHMENT : k8s-01.jpeg * DELETE FILE : _posts/Development/Kubernetes/2021-02-28-01-introducing-kubernetes.md * DELETE FILE : _posts/Development/Kubernetes/2021-03-07-02-first-steps-with-docker-and-kubernetes.md * DELETE FILE : _posts/Development/Kubernetes/2021-03-21-04-replication-and-other-controllers.md * DELETE FILE : _posts/Development/Kubernetes/2021-04-18-08-accessing-pod-metadata-and-other-resources-from-applications.md * DELETE FILE : _posts/Mathematics/Coq/2023-07-08-rules-of-inference-with-coq.md
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---
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share: true
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toc: true
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math: true
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categories: [Mathematics, Coq]
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tags: [math, coq, proof-verification]
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title: "Rules of Inference with Coq"
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date: "2023-07-08"
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github_title: "2023-07-08-rules-of-inference"
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---
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This is a list of proofs with Coq, for each rule of inference stated in [List of Rules of Inference (Wikipedia)](https://en.wikipedia.org/wiki/List_of_rules_of_inference)
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Some rules that need [the law of excluded middle](https://en.wikipedia.org/wiki/Law_of_excluded_middle) are at the end of the section.
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## Rules for Negation
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```coq
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Lemma reductio_ad_absurdum : forall P Q : Prop,
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(P -> Q) -> (P -> ~Q) -> ~P.
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Proof.
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intros. intros HP.
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apply H0 in HP as HNQ.
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apply H in HP as HQ.
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contradiction.
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Qed.
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Lemma ex_contradictione_quodlibet : forall P Q : Prop,
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P -> ~P -> Q.
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Proof. intros. contradiction. Qed.
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Lemma double_negation_introduction : forall P : Prop,
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P -> ~~P.
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Proof. auto. Qed.
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```
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## Rules for Conditionals
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```coq
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Lemma modus_ponens : forall P Q : Prop,
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(P -> Q) -> P -> Q.
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Proof. auto. Qed.
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Lemma modus_tollens : forall P Q : Prop,
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(P -> Q) -> ~Q -> ~P.
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Proof. auto. Qed.
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```
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## Rules for Conjunctions
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```coq
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Lemma conjuction_introduction : forall P Q : Prop,
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P -> Q -> P /\ Q.
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Proof. auto. Qed.
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Lemma conjunction_elimination_left : forall P Q : Prop,
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P /\ Q -> P.
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Proof. intros P Q [HP HQ]; auto. Qed.
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Lemma conjunction_elimination_right : forall P Q : Prop,
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P /\ Q -> Q.
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Proof. intros P Q [HP HQ]; auto. Qed.
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Lemma conjunction_commutative : forall P Q : Prop,
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P /\ Q -> Q /\ P.
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Proof.
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intros. destruct H; split; auto.
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Qed.
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Lemma conjunction_associative : forall P Q R : Prop,
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(P /\ Q) /\ R -> P /\ (Q /\ R).
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Proof.
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intros. destruct H as [H H1]; destruct H; split; auto.
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Qed.
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```
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## Rules for Disjunctions
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```coq
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Lemma disjunction_introduction_left : forall P Q : Prop,
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P -> P \/ Q.
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Proof. auto. Qed.
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Lemma disjunction_introduction_right : forall P Q : Prop,
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Q -> P \/ Q.
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Proof. auto. Qed.
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Lemma disjunction_elimination : forall P Q R : Prop,
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(P -> R) -> (Q -> R) -> (P \/ Q) -> R.
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Proof.
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intros. destruct H1; auto.
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Qed.
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Lemma disjunctive_syllogism_left : forall P Q : Prop,
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(P \/ Q) -> ~P -> Q.
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Proof.
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intros. destruct H; auto. contradiction.
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Qed.
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Lemma disjunctive_syllogism_right : forall P Q : Prop,
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(P \/ Q) -> ~Q -> P.
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Proof.
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intros. destruct H; auto. contradiction.
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Qed.
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Lemma constructive_dilemma : forall P Q R S : Prop,
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(P -> R) -> (Q -> S) -> (P \/ Q) -> (R \/ S).
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Proof.
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intros. destruct H1; auto.
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Qed.
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Lemma disjunction_commutative : forall P Q : Prop,
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P \/ Q -> Q \/ P.
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Proof. intros. destruct H; auto. Qed.
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Lemma disunction_associative : forall P Q R : Prop,
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(P \/ Q) \/ R -> P \/ (Q \/ R).
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Proof.
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intros. destruct H as [H | H]; auto.
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destruct H; auto.
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Qed.
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```
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## Rules for Biconditionals
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```coq
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Lemma biconditional_introduction : forall P Q : Prop,
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(P -> Q) -> (Q -> P) -> (P <-> Q).
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Proof. split; auto. Qed.
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Lemma biconditional_elimination_left_mp : forall P Q : Prop,
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(P <-> Q) -> P -> Q.
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Proof.
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intros. destruct H; auto.
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Qed.
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Lemma biconditional_elimination_right_mp : forall P Q : Prop,
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(P <-> Q) -> Q -> P.
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Proof.
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intros. destruct H; auto.
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Qed.
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Lemma biconditional_elimination_left_mt : forall P Q : Prop,
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(P <-> Q) -> ~P -> ~Q.
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Proof.
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intros. destruct H; auto.
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Qed.
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Lemma biconditional_elimination_right_mt : forall P Q : Prop,
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(P <-> Q) -> ~Q -> ~P.
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Proof.
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intros. destruct H; auto.
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Qed.
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Lemma biconditional_elimination_disjunction : forall P Q : Prop,
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(P <-> Q) -> (P \/ Q) -> (P /\ Q).
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Proof.
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intros. destruct H, H0; auto.
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Qed.
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Lemma biconditional_elimination_disjunction_not : forall P Q : Prop,
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(P <-> Q) -> (~P \/ ~Q) -> (~P /\ ~Q).
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Proof.
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intros. destruct H, H0; auto.
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Qed.
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```
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## Other Rules
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```coq
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Lemma exportation : forall P Q R : Prop,
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(P /\ Q) -> R <-> (P -> Q -> R).
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Proof.
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split; auto.
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destruct H; auto.
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Qed.
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Lemma distributive_disjunction : forall P Q R : Prop,
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P \/ (Q /\ R) <-> (P \/ Q) /\ (P \/ R).
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Proof.
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split; intros.
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- destruct H as [H | [H1 H2]]; split; auto.
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- destruct H as [H1 H2]; destruct H1, H2; auto.
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Qed.
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Lemma distributive_conjunction : forall P Q R : Prop,
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P /\ (Q \/ R) <-> (P /\ Q) \/ (P /\ R).
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Proof.
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split; intros.
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- destruct H as [H [H1 | H1]]; auto.
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- destruct H as [ [H1 H2](H1%20H2.md); auto.
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Qed.
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Lemma material_implication_converse : forall P Q : Prop,
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(~P \/ Q) -> (P -> Q).
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Proof.
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intros. destruct H; auto. contradiction.
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Qed.
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Lemma resolution : forall P Q R : Prop,
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(P \/ Q) -> (~P \/ R) -> (Q \/ R).
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Proof.
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intros. destruct H, H0; auto. contradiction.
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Qed.
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```
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## Rules that require Excluded Middle
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We declare the law of excluded middle as an axiom.
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```coq
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Axiom excluded_middle : forall P : Prop, P \/ ~P.
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Lemma double_negation_elimination : forall P : Prop,
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~~P -> P.
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Proof.
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intros. destruct (excluded_middle P); auto. contradiction.
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Qed.
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Lemma material_implication : forall P Q : Prop,
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(P -> Q) -> (~P \/ Q).
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Proof.
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intros. destruct (excluded_middle P); auto.
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Qed.
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Lemma reductio_ad_absurdum_neg : forall P Q : Prop,
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(~P -> Q) -> (~P -> ~Q) -> P.
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Proof.
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intros. destruct (excluded_middle P); auto.
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apply H in H1 as HQ.
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apply H0 in H1 as HNQ.
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contradiction.
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Qed.
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```
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---
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I was supposed to be reading the [source](https://github.com/snu-sf/promising-seq-coq) for the paper [Sequential Reasoning for Optimizing Compilers Under Weak Memory Concurrency](https://dl.acm.org/doi/abs/10.1145/3519939.3523718) but I got carried away...
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