rule is a proof method. It applies a rule if possible.
Assume we have a subgoal
and we want to use
rule with rule
rule does the following:
Assume we have a goal
apply (rule disjI1) yields the new subgoal
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.
rule_tac, you can force 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
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
rule(k) which forces Isabelle to apply
assumption times after the rule application.