Isabelle/HOL Support Wiki

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===== rule (method) =====
''rule'' is a proof method. It applies a [[rule (calculus)|rule]] if possible.

==== Definition ====
Assume we have a [[subgoal]]

$\quad\bigwedge x_1 \dots x_k : [|\ A_1; \dots ; A_m\ |] \Longrightarrow C$

and we want to use ''rule'' with rule

$\quad[|\ P_1; \dots ; P_n\ |] \Longrightarrow Q$

Then, ''rule'' does the following:
  * [[unification|Unify]] $C$ and $Q$. The application fails if there is no unifier; otherwise, let $U$ be this unifier.
  * Remove the old subgoal and create one new subgoal \\ \\ $\quad\bigwedge x_1 \dots x_k : [|\ U(A_1); \dots ; U(A_m)\ |] \Longrightarrow U(P_k)$ \\ \\ for each $k = 1, \dots, n$.

==== Example ====
Assume we have a goal

$\quad[|\ A |] \Longrightarrow A \vee B$

Applying ''apply (rule disjI1)'' yields the new subgoal

$\quad [|\ A\ |] \Longrightarrow A$

which can obviously be solved by one application of ''[[assumption]]''. Note that ''apply (rule(1) disjI)'' is a shortcut for this and immediately solves the goal. 


==== Non-Example ====

==== Variants ====
=== rule_tac ===
With ''rule_tac'', you can force [[schematic variable|schematic variables]] in the used rule to take specific values. The extended syntax is:
  apply (rule_tac ident1="expr1" and ident2="expr1" and ... in rule)
This means that the variable ''?ident1'' is replaced by expression ''expr1'' and similarly for the others. Note that you have to leave out the question mark marking schematic variables. Find out which variables a rule uses with ''thm rule''.

=== rule(k) ===
Oftentimes, a rule application results in several subgoals that can directly be solved by ''[[assumption]]''; see [[#Example|above]] for an example. Instead of applying ''assumption'' by hand, you can apply ''rule(k)'' which forces Isabelle to apply ''assumption'' $k$ times after the rule application.

==== Relatives ====
  * ''[[drule]]''
  * ''[[erule]]''
  * ''[[frule]]''