Rule of Sequent Introduction

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Theorem

Let the statements $P_1, P_2, \ldots, P_n$ be conclusions in a proof, on various assumptions.

Let $P_1, P_2, \ldots, P_n \vdash Q$ be a substitution instance of a sequent for which we already have a proof.



Then we may introduce, at any stage of a proof (citing SI), either:

The conclusion $Q$ of the sequent already proved

or:

A substitution instance of such a conclusion, together with a reference to the sequent that is being cited.


This conclusion depend upon the pool of assumptions upon which $P_1, P_2, \ldots, P_n \vdash Q$ rests.


This is called the rule of sequent introduction.


Proof

By hypothesis and substitution instance we have a proof, using primitive rules, of:

$P_1, P_2, \ldots, P_n \vdash Q$

By the Extended Rule of Implication, we have:

$\vdash P_1 \implies \left({P_2 \implies \left({P_3 \implies \left({\ldots \implies \left({P_n \implies Q}\right) \ldots }\right)}\right)}\right)$

$\blacksquare$


Also known as

This rule is also known as the rule of replacement.


Also see

  • Rule of Theorem Introduction, which is a direct corollary of this. Thus we can convert any sequent into a theorem so as to use results already calculated in order to prove further results.


Technical Note

When invoking Rule of Sequent Introduction in a tableau proof, use the {{SequentIntro}} template:

{{SequentIntro|line|pool|statement|depends|sequent}}

where:

line is the number of the line on the tableau proof where Rule of Sequent Introduction is to be invoked
pool is the pool of assumptions (comma-separated list)
statement is the statement of logic that is to be displayed in the Formula column, without the $ ... $ delimiters
depends is the line (or lines) of the tableau proof upon which this line directly depends
sequent is the link to the sequent in question that will be displayed in the Notes column.


Sources