mirror of
https://github.com/calofmijuck/blog.git
synced 2025-12-06 14:53:50 +00:00
Sungchan Yi
6960edd3d4
* PUSH NOTE : 05. Lebesgue Integration.md * PUSH NOTE : 04. Measurable Functions.md * PUSH NOTE : 03. Measure Spaces.md * PUSH NOTE : 02. Construction of Measure.md * PUSH NOTE : Rules of Inference with Coq.md * PUSH NOTE : 9. Public Key Encryption.md * PUSH NOTE : 7. Key Exchange.md * PUSH NOTE : 6. Hash Functions.md * PUSH NOTE : 5. CCA-Security and Authenticated Encryption.md * PUSH NOTE : 2. PRFs, PRPs and Block Ciphers.md * PUSH NOTE : 14. Secure Multiparty Computation.md * PUSH NOTE : 07. Public Key Cryptography.md * PUSH NOTE : 06. RSA and ElGamal Encryption.md * PUSH NOTE : 05. Modular Arithmetic (2).md * PUSH NOTE : 03. Symmetric Key Cryptography (2).md * PUSH NOTE : 02. Symmetric Key Cryptography (1).md * DELETE FILE : _posts/Lecture Notes/Modern Cryptography/2023-10-19-public-key-encryption.md * DELETE FILE : _posts/lecture-notes/modern-cryptography/2023-10-09-public-key-cryptography.md
244 lines
5.4 KiB
Markdown
244 lines
5.4 KiB
Markdown
---
|
|
share: true
|
|
toc: true
|
|
math: true
|
|
categories:
|
|
- Mathematics
|
|
- Coq
|
|
tags:
|
|
- math
|
|
- coq
|
|
- proof-verification
|
|
title: Rules of Inference with Coq
|
|
date: 2023-07-08
|
|
github_title: 2023-07-08-rules-of-inference
|
|
---
|
|
|
|
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)
|
|
|
|
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.
|
|
|
|
## Rules for Negation
|
|
|
|
```coq
|
|
Lemma reductio_ad_absurdum : forall P Q : Prop,
|
|
(P -> Q) -> (P -> ~Q) -> ~P.
|
|
Proof.
|
|
intros. intros HP.
|
|
apply H0 in HP as HNQ.
|
|
apply H in HP as HQ.
|
|
contradiction.
|
|
Qed.
|
|
|
|
Lemma ex_contradictione_quodlibet : forall P Q : Prop,
|
|
P -> ~P -> Q.
|
|
Proof. intros. contradiction. Qed.
|
|
|
|
Lemma double_negation_introduction : forall P : Prop,
|
|
P -> ~~P.
|
|
Proof. auto. Qed.
|
|
```
|
|
|
|
## Rules for Conditionals
|
|
|
|
```coq
|
|
Lemma modus_ponens : forall P Q : Prop,
|
|
(P -> Q) -> P -> Q.
|
|
Proof. auto. Qed.
|
|
|
|
Lemma modus_tollens : forall P Q : Prop,
|
|
(P -> Q) -> ~Q -> ~P.
|
|
Proof. auto. Qed.
|
|
```
|
|
|
|
## Rules for Conjunctions
|
|
|
|
```coq
|
|
Lemma conjuction_introduction : forall P Q : Prop,
|
|
P -> Q -> P /\ Q.
|
|
Proof. auto. Qed.
|
|
|
|
Lemma conjunction_elimination_left : forall P Q : Prop,
|
|
P /\ Q -> P.
|
|
Proof. intros P Q [HP HQ]; auto. Qed.
|
|
|
|
Lemma conjunction_elimination_right : forall P Q : Prop,
|
|
P /\ Q -> Q.
|
|
Proof. intros P Q [HP HQ]; auto. Qed.
|
|
|
|
Lemma conjunction_commutative : forall P Q : Prop,
|
|
P /\ Q -> Q /\ P.
|
|
Proof.
|
|
intros. destruct H; split; auto.
|
|
Qed.
|
|
|
|
Lemma conjunction_associative : forall P Q R : Prop,
|
|
(P /\ Q) /\ R -> P /\ (Q /\ R).
|
|
Proof.
|
|
intros. destruct H as [H H1]; destruct H; split; auto.
|
|
Qed.
|
|
```
|
|
|
|
## Rules for Disjunctions
|
|
|
|
```coq
|
|
Lemma disjunction_introduction_left : forall P Q : Prop,
|
|
P -> P \/ Q.
|
|
Proof. auto. Qed.
|
|
|
|
Lemma disjunction_introduction_right : forall P Q : Prop,
|
|
Q -> P \/ Q.
|
|
Proof. auto. Qed.
|
|
|
|
Lemma disjunction_elimination : forall P Q R : Prop,
|
|
(P -> R) -> (Q -> R) -> (P \/ Q) -> R.
|
|
Proof.
|
|
intros. destruct H1; auto.
|
|
Qed.
|
|
|
|
Lemma disjunctive_syllogism_left : forall P Q : Prop,
|
|
(P \/ Q) -> ~P -> Q.
|
|
Proof.
|
|
intros. destruct H; auto. contradiction.
|
|
Qed.
|
|
|
|
Lemma disjunctive_syllogism_right : forall P Q : Prop,
|
|
(P \/ Q) -> ~Q -> P.
|
|
Proof.
|
|
intros. destruct H; auto. contradiction.
|
|
Qed.
|
|
|
|
Lemma constructive_dilemma : forall P Q R S : Prop,
|
|
(P -> R) -> (Q -> S) -> (P \/ Q) -> (R \/ S).
|
|
Proof.
|
|
intros. destruct H1; auto.
|
|
Qed.
|
|
|
|
Lemma disjunction_commutative : forall P Q : Prop,
|
|
P \/ Q -> Q \/ P.
|
|
Proof. intros. destruct H; auto. Qed.
|
|
|
|
Lemma disunction_associative : forall P Q R : Prop,
|
|
(P \/ Q) \/ R -> P \/ (Q \/ R).
|
|
Proof.
|
|
intros. destruct H as [H | H]; auto.
|
|
destruct H; auto.
|
|
Qed.
|
|
```
|
|
|
|
## Rules for Biconditionals
|
|
|
|
```coq
|
|
Lemma biconditional_introduction : forall P Q : Prop,
|
|
(P -> Q) -> (Q -> P) -> (P <-> Q).
|
|
Proof. split; auto. Qed.
|
|
|
|
Lemma biconditional_elimination_left_mp : forall P Q : Prop,
|
|
(P <-> Q) -> P -> Q.
|
|
Proof.
|
|
intros. destruct H; auto.
|
|
Qed.
|
|
|
|
Lemma biconditional_elimination_right_mp : forall P Q : Prop,
|
|
(P <-> Q) -> Q -> P.
|
|
Proof.
|
|
intros. destruct H; auto.
|
|
Qed.
|
|
|
|
Lemma biconditional_elimination_left_mt : forall P Q : Prop,
|
|
(P <-> Q) -> ~P -> ~Q.
|
|
Proof.
|
|
intros. destruct H; auto.
|
|
Qed.
|
|
|
|
Lemma biconditional_elimination_right_mt : forall P Q : Prop,
|
|
(P <-> Q) -> ~Q -> ~P.
|
|
Proof.
|
|
intros. destruct H; auto.
|
|
Qed.
|
|
|
|
Lemma biconditional_elimination_disjunction : forall P Q : Prop,
|
|
(P <-> Q) -> (P \/ Q) -> (P /\ Q).
|
|
Proof.
|
|
intros. destruct H, H0; auto.
|
|
Qed.
|
|
|
|
Lemma biconditional_elimination_disjunction_not : forall P Q : Prop,
|
|
(P <-> Q) -> (~P \/ ~Q) -> (~P /\ ~Q).
|
|
Proof.
|
|
intros. destruct H, H0; auto.
|
|
Qed.
|
|
```
|
|
|
|
## Other Rules
|
|
|
|
```coq
|
|
Lemma exportation : forall P Q R : Prop,
|
|
(P /\ Q) -> R <-> (P -> Q -> R).
|
|
Proof.
|
|
split; auto.
|
|
destruct H; auto.
|
|
Qed.
|
|
|
|
Lemma distributive_disjunction : forall P Q R : Prop,
|
|
P \/ (Q /\ R) <-> (P \/ Q) /\ (P \/ R).
|
|
Proof.
|
|
split; intros.
|
|
- destruct H as [H | [H1 H2]]; split; auto.
|
|
- destruct H as [H1 H2]; destruct H1, H2; auto.
|
|
Qed.
|
|
|
|
Lemma distributive_conjunction : forall P Q R : Prop,
|
|
P /\ (Q \/ R) <-> (P /\ Q) \/ (P /\ R).
|
|
Proof.
|
|
split; intros.
|
|
- destruct H as [H [H1 | H1]]; auto.
|
|
- destruct H as [[H1 H2] | [H1 H2]]; auto.
|
|
Qed.
|
|
|
|
Lemma material_implication_converse : forall P Q : Prop,
|
|
(~P \/ Q) -> (P -> Q).
|
|
Proof.
|
|
intros. destruct H; auto. contradiction.
|
|
Qed.
|
|
|
|
Lemma resolution : forall P Q R : Prop,
|
|
(P \/ Q) -> (~P \/ R) -> (Q \/ R).
|
|
Proof.
|
|
intros. destruct H, H0; auto. contradiction.
|
|
Qed.
|
|
```
|
|
|
|
## Rules that require Excluded Middle
|
|
|
|
We declare the law of excluded middle as an axiom.
|
|
|
|
```coq
|
|
Axiom excluded_middle : forall P : Prop, P \/ ~P.
|
|
|
|
Lemma double_negation_elimination : forall P : Prop,
|
|
~~P -> P.
|
|
Proof.
|
|
intros. destruct (excluded_middle P); auto. contradiction.
|
|
Qed.
|
|
|
|
Lemma material_implication : forall P Q : Prop,
|
|
(P -> Q) -> (~P \/ Q).
|
|
Proof.
|
|
intros. destruct (excluded_middle P); auto.
|
|
Qed.
|
|
|
|
Lemma reductio_ad_absurdum_neg : forall P Q : Prop,
|
|
(~P -> Q) -> (~P -> ~Q) -> P.
|
|
Proof.
|
|
intros. destruct (excluded_middle P); auto.
|
|
apply H in H1 as HQ.
|
|
apply H0 in H1 as HNQ.
|
|
contradiction.
|
|
Qed.
|
|
```
|
|
|
|
---
|
|
|
|
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...
|