# Isabelle/HOL Support Wiki

reference:erule

## erule

erule is a proof method. It applies a 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 erule with rule

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

Then, erule does the following:

• Unify $Q$ with $C$ and $P_1$ with $A_j$ simultaneously for some $j$. The application fails if there is no unifier for any $j$; otherwise, let $U$ be this unifier.
• Remove the old subgoal and create new ones
$\quad\bigwedge x_1 \dots x_k : [|\ U(A_1); \dots ; U(A_{j-1}); U(A_{j+1}); \dots ; U(A_m)\ |] \Longrightarrow U(P_k)$
for each $k = 2, \dots, n$.

### Example

Assume we have the goal

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

Applying apply (erule conjE) yields the new goal

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

which can be solved by Assumption. Note that apply (erule(1) conjE) is a shortcut for this and immediately solves the goal.

### Variants

#### erule_tac

With erule_tac, you can force schematic variables in the used rule to take specific values. The extended syntax is:

apply (erule_tac ident1="expr1" and ident2="expr1" and ... in rule)

This means that the variables ?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.

#### erule(k)

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

### Relatives

• Rule
• drule
• frule 