Impressum Examples.thy Sprache: Isabelle

(*  Title:      HOL/Eisbach/Examples.thy
    Author:     Daniel Matichuk, NICTA/UNSW
*)

section \<open>Basic Eisbach examples\<close>

theory Examples
imports Main Eisbach_Tools
begin

subsection \<open>Basic methods\<close>

method my_intros = (rule conjI | rule impI)

lemma "P \ Q \ Z \ X"
  apply my_intros+
  oops

method my_intros' uses intros = (rule conjI | rule impI | rule intros)

lemma "P \ Q \ Z \ X"
  apply (my_intros' intros: disjI1)+
  oops

method my_spec for x :: 'a = (drule spec[where x="x"])

lemma "\x. P x \ P x"
  apply (my_spec x)
  apply assumption
  done

subsection \<open>Focusing and matching\<close>

method match_test =
  (match premises in U: "P x \ Q x" for P Q x \
    \<open>print_term P,
     print_term Q,
     print_term x,
     print_fact U\<close>)

lemma "\x. P x \ Q x \ A x \ B x \ R x y \ True"
  apply match_test  \<comment> \<open>Valid match, but not quite what we were expecting..\<close>
  back  \<comment> \<open>Can backtrack over matches, exploring all bindings\<close>
  back
  back
  back
  back
  back  \<comment> \<open>Found the other conjunction\<close>
  back
  back
  back
  oops

text \<open>Use matching to avoid "improper" methods\<close>

lemma focus_test:
  shows "\x. \x. P x \ P x"
  apply (my_spec "x :: 'a", assumption)?  \<comment> \<open>Wrong x\<close>
  apply (match conclusion in "P x" for x \<Rightarrow> \<open>my_spec x, assumption\<close>)
  done

text \<open>Matches are exclusive. Backtracking will not occur past a match\<close>

method match_test' =
  (match conclusion in
    "P \ Q" for P Q \
      \<open>print_term P, print_term Q, rule conjI[where P="P" and Q="Q"]; assumption\<close>
    \<bar> "H" for H \<Rightarrow> \<open>print_term H\<close>)

text \<open>Solves goal\<close>
lemma "P \ Q \ P \ Q"
  apply match_test'
  done

text \<open>Fall-through case never taken\<close>
lemma "P \ Q"
  apply match_test'?
  oops

lemma "P"
  apply match_test'
  oops

method my_spec_guess =
  (match conclusion in "P (x :: 'a)" for P x \<Rightarrow>
    \<open>drule spec[where x=x],
     print_term P,
     print_term x\<close>)

lemma "\x. P (x :: nat) \ Q (x :: nat)"
  apply my_spec_guess
  oops

method my_spec_guess2 =
  (match premises in U[thin]:"\x. P x \ Q x" and U':"P x" for P Q x \
    \<open>insert spec[where x=x, OF U],
     print_term P,
     print_term Q\<close>)

lemma "\x. P x \ Q x \ Q x \ Q x"
  apply my_spec_guess2?  \<comment> \<open>Fails. Note that both "P"s must match\<close>
  oops

lemma "\x. P x \ Q x \ P x \ Q x"
  apply my_spec_guess2
  apply (erule mp)
  apply assumption
  done

subsection \<open>Higher-order methods\<close>

method higher_order_example for x methods meth =
  (cases x, meth, meth)

lemma
  assumes A: "x = Some a"
  shows "the x = a"
  by (higher_order_example x \<open>simp add: A\<close>)

subsection \<open>Recursion\<close>

method recursion_example for x :: bool =
  (print_term x,
     match (x) in "A \ B" for A B \
    \<open>print_term A,
     print_term B,
     recursion_example A,
     recursion_example B | -\<close>)

lemma "P"
  apply (recursion_example "(A \ D) \ (B \ C)")
  oops

subsection \<open>Solves combinator\<close>

lemma "A \ B \ A \ B"
  apply (solves \<open>rule conjI\<close>)?  \<comment> \<open>Doesn't solve the goal!\<close>
  apply (solves \<open>rule conjI, assumption, assumption\<close>)
  done

subsection \<open>Demo\<close>

named_theorems intros and elims and subst

method prop_solver declares intros elims subst =
  (assumption |
    rule intros | erule elims |
    subst subst | subst (asm) subst |
    (erule notE; solves prop_solver))+

lemmas [intros] =
  conjI
  impI
  disjCI
  iffI
  notI
lemmas [elims] =
  impCE
  conjE
  disjE

lemma "((A \ B) \ (A \ C) \ (B \ C)) \ C"
  apply prop_solver
  done

method guess_all =
  (match premises in U[thin]:"\x. P (x :: 'a)" for P \
    \<open>match premises in "?H (y :: 'a)" for y \<Rightarrow>
       \<open>rule allE[where P = P and x = y, OF U]\<close>
   | match conclusion in "?H (y :: 'a)" for y \<Rightarrow>
       \<open>rule allE[where P = P and x = y, OF U]\<close>\<close>)

lemma "(\x. P x \ Q x) \ P y \ Q y"
  apply guess_all
  apply prop_solver
  done

lemma "(\x. P x \ Q x) \ P z \ P y \ Q y"
  apply (solves \<open>guess_all, prop_solver\<close>)  \<comment> \<open>Try it without solve\<close>
  done

method guess_ex =
  (match conclusion in
    "\x. P (x :: 'a)" for P \
      \<open>match premises in "?H (x :: 'a)" for x \<Rightarrow>
              \<open>rule exI[where x=x]\<close>\<close>)

lemma "P x \ \x. P x"
  apply guess_ex
  apply prop_solver
  done

method fol_solver =
  ((guess_ex | guess_all | prop_solver); solves fol_solver)

declare
  allI [intros]
  exE [elims]
  ex_simps [subst]
  all_simps [subst]

lemma "(\x. P x) \ (\x. Q x) \ (\x. P x \ Q x)"
  and  "\x. P x \ (\x. P x)"
  and "(\x. \y. R x y) \ (\y. \x. R x y)"
  by fol_solver+

text \<open>
  Eisbach_Tools provides the catch method, which catches run-time method
  errors. In this example the OF attribute throws an error when it can't
  compose H with A, forcing H to be re-bound to different members of imps
  until it succeeds.
\<close>

lemma
  assumes imps:  "A \ B" "A \ C" "B \ D"
  assumes A: "A"
  shows "B \ C"
  apply (rule conjI)
  apply ((match imps in H:"_ \ _" \ \catch \rule H[OF A], print_fact H\ \print_fact H, fail\\)+)
  done

text \<open>
  Eisbach_Tools provides the curry and uncurry attributes. This is useful
  when the number of premises of a thm isn't known statically. The pattern
  \<^term>\<open>P \<Longrightarrow> Q\<close> matches P against the major premise of a thm, and Q is the
  rest of the premises with the conclusion. If we first uncurry, then \<^term>\<open>P \<Longrightarrow> Q\<close> will match P with the conjunction of all the premises, and Q with
  the final conclusion of the rule.
\<close>

lemma
  assumes imps: "A \ B \ C" "D \ C" "E \ D \ A"
  shows "(A \ B \ C) \ (D \ C)"
    by (match imps[uncurry] in H[curry]:"_ \ C" (cut, multi) \
                    \<open>match H in "E \<Longrightarrow> _" \<Rightarrow> fail
                                      \<bar> _ \<Rightarrow> \<open>simp add: H\<close>\<close>)

end

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