Theory Examples_FOL

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

section ‹Basic Eisbach examples (in FOL)›

theory Examples_FOL
imports FOL Eisbach_Old_Appl_Syntax
begin


subsection ‹Basic methods›

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 ‹Demo›

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 ⇒
    ‹match premises in "?H (y :: 'a)" for y ⇒
       ‹rule allE[where P = P and x = y, OF U]›
   | match conclusion in "?H (y :: 'a)" for y ⇒
       ‹rule allE[where P = P and x = y, OF U]››)

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 ‹guess_all, prop_solver›)  ― ‹Try it without solve›
  done

method guess_ex =
  (match conclusion in
    "∃x. P (x :: 'a)" for P ⇒
      ‹match premises in "?H (x :: 'a)" for x ⇒
              ‹rule exI[where x=x]››)

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+

end