src/FOL/ex/LocaleTest.thy

simplified get_thm(s): back to plain name argument;

(*  Title:      FOL/ex/LocaleTest.thy
    ID:         $Id$
    Author:     Clemens Ballarin
    Copyright (c) 2005 by Clemens Ballarin

Collection of regression tests for locales.
*)

header {* Test of Locale Interpretation *}

theory LocaleTest
imports FOL
begin

ML {* set Toplevel.debug *}
ML {* set show_hyps *}
ML {* set show_sorts *}

ML {*
  fun check_thm name = let
    val thy = the_context ();
    val thm = PureThy.get_thm thy name;
    val {prop, hyps, ...} = rep_thm thm;
    val prems = Logic.strip_imp_prems prop;
    val _ = if null hyps then ()
        else error ("Theorem " ^ quote name ^ " has meta hyps.\n" ^
          "Consistency check of locales package failed.");
    val _ = if null prems then ()
        else error ("Theorem " ^ quote name ^ " has premises.\n" ^
          "Consistency check of locales package failed.");
  in () end;
*}

section {* Context Elements and Locale Expressions *}

text {* Naming convention for global objects: prefixes L and l *}

subsection {* Renaming with Syntax *}

locale (open) LS = var mult +
  assumes "mult(x, y) = mult(y, x)"

print_locale LS

locale LS' = LS mult (infixl "**" 60)

print_locale LS'

locale LT = var mult (infixl "**" 60) +
  assumes "x ** y = y ** x"

locale LU = LT mult (infixl "**" 60) + LT add (infixl "++" 55) + var h +
  assumes hom: "h(x ** y) = h(x) ++ h(y)"


(* FIXME: graceful handling of type errors?
locale LY = LT mult (infixl "**" 60) + LT add (binder "++" 55) + var h +
  assumes "mult(x) == add"
*)


locale LV = LU _ add

locale LW = LU _ mult (infixl "**" 60)


subsection {* Constrains *}

locale LZ = fixes a (structure)
locale LZ' = LZ +
  constrains a :: "'a => 'b"
  assumes "a (x :: 'a) = a (y)"
print_locale LZ'


section {* Interpretation *}

text {* Naming convention for global objects: prefixes I and i *}

text {* interpretation input syntax *}

locale IL
locale IM = fixes a and b and c

interpretation test [simp]: IL + IM a b c [x y z] .

print_interps IL    (* output: test *)
print_interps IM    (* output: test *)

interpretation test [simp]: IL print_interps IM .

interpretation IL .

text {* Processing of locale expression *}

locale IA = fixes a assumes asm_A: "a = a"

locale (open) IB = fixes b assumes asm_B [simp]: "b = b"

locale IC = IA + IB + assumes asm_C: "c = c"
  (* TODO: independent type var in c, prohibit locale declaration *)

locale ID = IA + IB + fixes d defines def_D: "d == (a = b)"

theorem (in IA)
  includes ID
  shows True
  print_interps! IA
  print_interps! ID
  ..

theorem (in ID) True ..

typedecl i
arities i :: "term"


interpretation i1: IC ["X::i" "Y::i"] by unfold_locales auto

print_interps IA  (* output: i1 *)

(* possible accesses *)
thm i1.a.asm_A thm LocaleTest.i1.a.asm_A
thm i1.asm_A thm LocaleTest.i1.asm_A

ML {* check_thm "i1.a.asm_A" *}

(* without prefix *)

interpretation IC ["W::i" "Z::i"] .  (* subsumed by i1: IC *)
interpretation IC ["W::'a" "Z::i"] by unfold_locales auto
  (* subsumes i1: IA and i1: IC *)

print_interps IA  (* output: <no prefix>, i1 *)

(* possible accesses *)
thm asm_C thm a_b.asm_C thm LocaleTest.a_b.asm_C thm LocaleTest.a_b.asm_C

ML {* check_thm "asm_C" *}

interpretation i2: ID [X "Y::i" "Y = X"]
  by (simp add: eq_commute) unfold_locales

print_interps IA  (* output: <no prefix>, i1 *)
print_interps ID  (* output: i2 *)


interpretation i3: ID [X "Y::i"] by simp unfold_locales

(* duplicate: thm not added *)

(* thm i3.a.asm_A *)


print_interps IA  (* output: <no prefix>, i1 *)
print_interps IB  (* output: i1 *)
print_interps IC  (* output: <no prefix, i1 *)
print_interps ID  (* output: i2, i3 *)


interpretation i10: ID + ID a' b' d' [X "Y::i" _ u "v::i" _] .

corollary (in ID) th_x: True ..

(* possible accesses: for each registration *)

thm i2.th_x thm i3.th_x

ML {* check_thm "i2.th_x"; check_thm "i3.th_x" *}

lemma (in ID) th_y: "d == (a = b)" by (rule d_def)

thm i2.th_y thm i3.th_y

ML {* check_thm "i2.th_y"; check_thm "i3.th_y" *}

lemmas (in ID) th_z = th_y

thm i2.th_z

ML {* check_thm "i2.th_z" *}


subsection {* Interpretation in Proof Contexts *}

locale IF = fixes f assumes asm_F: "f & f --> f"

consts default :: "'a"

theorem True
proof -
  fix alpha::i and beta
  have alpha_A: "IA(alpha)" by unfold_locales
  interpret i5: IA [alpha] .  (* subsumed *)
  print_interps IA  (* output: <no prefix>, i1 *)
  interpret i6: IC [alpha beta] by unfold_locales auto
  print_interps IA   (* output: <no prefix>, i1 *)
  print_interps IC   (* output: <no prefix>, i1, i6 *)
  interpret i11: IF ["default = default"] by (fast intro: IF.intro)
  thm i11.asm_F [where 'a = i]     (* default has schematic type *)
qed rule

theorem (in IA) True
proof -
  print_interps! IA
  fix beta and gamma
  interpret i9: ID [a beta _]
    apply - apply assumption
    apply unfold_locales
    apply (rule refl) done
qed rule


(* Definition involving free variable *)

ML {* reset show_sorts *}

locale IE = fixes e defines e_def: "e(x) == x & x"
  notes e_def2 = e_def

lemma (in IE) True thm e_def by fast

interpretation i7: IE ["%x. x"] by simp

thm i7.e_def2 (* has no premise *)

ML {* check_thm "i7.e_def2" *}

locale IE' = fixes e defines e_def: "e == (%x. x & x)"
  notes e_def2 = e_def

interpretation i7': IE' ["(%x. x)"] by simp

thm i7'.e_def2

ML {* check_thm "i7'.e_def2" *}

(* Definition involving free variable in assm *)

locale (open) IG = fixes g assumes asm_G: "g --> x"
  notes asm_G2 = asm_G

interpretation i8: IG ["False"] by fast

thm i8.asm_G2

ML {* check_thm "i8.asm_G2" *}

text {* Locale without assumptions *}

locale IL1 = notes rev_conjI [intro] = conjI [THEN iffD1 [OF conj_commute]]

lemma "[| P; Q |] ==> P & Q"
proof -
  interpret my: IL1 .          txt {* No chained fact required. *}
  assume Q and P               txt {* order reversed *}
  then show "P & Q" ..         txt {* Applies @{thm my.rev_conjI}. *}
qed

locale IL11 = notes rev_conjI = conjI [THEN iffD1 [OF conj_commute]]

lemma "[| P; Q |] ==> P & Q"
proof -
  interpret [intro]: IL11 .     txt {* Attribute supplied at instantiation. *}
  assume Q and P
  then show "P & Q" ..
qed

subsection {* Simple locale with assumptions *}

consts ibin :: "[i, i] => i" (infixl "#" 60)

axioms i_assoc: "(x # y) # z = x # (y # z)"
  i_comm: "x # y = y # x"

locale IL2 =
  fixes OP (infixl "+" 60)
  assumes assoc: "(x + y) + z = x + (y + z)"
    and comm: "x + y = y + x"

lemma (in IL2) lcomm: "x + (y + z) = y + (x + z)"
proof -
  have "x + (y + z) = (x + y) + z" by (simp add: assoc)
  also have "... = (y + x) + z" by (simp add: comm)
  also have "... = y + (x + z)" by (simp add: assoc)
  finally show ?thesis .
qed

lemmas (in IL2) AC = comm assoc lcomm

lemma "(x::i) # y # z # w = y # x # w # z"
proof -
  interpret my: IL2 ["op #"] by (rule IL2.intro [of "op #", OF i_assoc i_comm])
  show ?thesis by (simp only: my.OP.AC)  (* or my.AC *)
qed

subsection {* Nested locale with assumptions *}

locale IL3 =
  fixes OP (infixl "+" 60)
  assumes assoc: "(x + y) + z = x + (y + z)"

locale IL4 = IL3 +
  assumes comm: "x + y = y + x"

lemma (in IL4) lcomm: "x + (y + z) = y + (x + z)"
proof -
  have "x + (y + z) = (x + y) + z" by (simp add: assoc)
  also have "... = (y + x) + z" by (simp add: comm)
  also have "... = y + (x + z)" by (simp add: assoc)
  finally show ?thesis .
qed

lemmas (in IL4) AC = comm assoc lcomm

lemma "(x::i) # y # z # w = y # x # w # z"
proof -
  interpret my: IL4 ["op #"]
    by (auto intro: IL4.intro IL3.intro IL4_axioms.intro i_assoc i_comm)
  show ?thesis by (simp only: my.OP.AC)  (* or simply AC *)
qed

text {* Locale with definition *}

text {* This example is admittedly not very creative :-) *}

locale IL5 = IL4 + var A +
  defines A_def: "A == True"

lemma (in IL5) lem: A
  by (unfold A_def) rule

lemma "IL5(op #) ==> True"
proof -
  assume "IL5(op #)"
  then interpret IL5 ["op #"] by (auto intro: IL5.axioms)
  show ?thesis by (rule lem)  (* lem instantiated to True *)
qed

text {* Interpretation in a context with target *}

lemma (in IL4)
  fixes A (infixl "$" 60)
  assumes A: "IL4(A)"
  shows "(x::i) $ y $ z $ w = y $ x $ w $ z"
proof -
  from A interpret A: IL4 ["A"] by (auto intro: IL4.axioms)
  show ?thesis by (simp only: A.OP.AC)
qed


section {* Interpretation in Locales *}

text {* Naming convention for global objects: prefixes R and r *}

locale (open) Rsemi = var prod (infixl "**" 65) +
  assumes assoc: "(x ** y) ** z = x ** (y ** z)"

locale (open) Rlgrp = Rsemi + var one + var inv +
  assumes lone: "one ** x = x"
    and linv: "inv(x) ** x = one"

lemma (in Rlgrp) lcancel:
  "x ** y = x ** z <-> y = z"
proof
  assume "x ** y = x ** z"
  then have "inv(x) ** x ** y = inv(x) ** x ** z" by (simp add: assoc)
  then show "y = z" by (simp add: lone linv)
qed simp

locale (open) Rrgrp = Rsemi + var one + var inv +
  assumes rone: "x ** one = x"
    and rinv: "x ** inv(x) = one"

lemma (in Rrgrp) rcancel:
  "y ** x = z ** x <-> y = z"
proof
  assume "y ** x = z ** x"
  then have "y ** (x ** inv(x)) = z ** (x ** inv(x))"
    by (simp add: assoc [symmetric])
  then show "y = z" by (simp add: rone rinv)
qed simp

interpretation Rlgrp < Rrgrp
  proof -
    {
      fix x
      have "inv(x) ** x ** one = inv(x) ** x" by (simp add: linv lone)
      then show "x ** one = x" by (simp add: assoc lcancel)
    }
    note rone = this
    {
      fix x
      have "inv(x) ** x ** inv(x) = inv(x) ** one"
	by (simp add: linv lone rone)
      then show "x ** inv(x) = one" by (simp add: assoc lcancel)
    }
  qed

(* effect on printed locale *)

print_locale! Rlgrp

(* use of derived theorem *)

lemma (in Rlgrp)
  "y ** x = z ** x <-> y = z"
  apply (rule rcancel)
  print_interps Rrgrp thm lcancel rcancel
  done

(* circular interpretation *)

interpretation Rrgrp < Rlgrp
  proof -
    {
      fix x
      have "one ** (x ** inv(x)) = x ** inv(x)" by (simp add: rinv rone)
      then show "one ** x = x" by (simp add: assoc [symmetric] rcancel)
    }
    note lone = this
    {
      fix x
      have "inv(x) ** (x ** inv(x)) = one ** inv(x)"
	by (simp add: rinv lone rone)
      then show "inv(x) ** x = one" by (simp add: assoc [symmetric] rcancel)
    }
  qed

(* effect on printed locale *)

print_locale! Rrgrp
print_locale! Rlgrp

(* locale with many parameters ---
   interpretations generate alternating group A5 *)

locale RA5 = var A + var B + var C + var D + var E +
  assumes eq: "A <-> B <-> C <-> D <-> E"

interpretation RA5 < RA5 _ _ D E C
print_facts
print_interps RA5
  using A_B_C_D_E.eq apply (blast intro: RA5.intro) done

interpretation RA5 < RA5 C _ E _ A
print_facts
print_interps RA5
  using A_B_C_D_E.eq apply (blast intro: RA5.intro) done

interpretation RA5 < RA5 B C A _ _
print_facts
print_interps RA5
  using A_B_C_D_E.eq apply (blast intro: RA5.intro) done

interpretation RA5 < RA5 _ C D B _ .
  (* Any even permutation of parameters is subsumed by the above. *)

(* circle of three locales, forward direction *)

locale (open) RA1 = var A + var B + assumes p: "A <-> B"
locale (open) RA2 = var A + var B + assumes q: "A & B | ~ A & ~ B"
locale (open) RA3 = var A + var B + assumes r: "(A --> B) & (B --> A)"

interpretation RA1 < RA2
  print_facts
  using p apply fast done
interpretation RA2 < RA3
  print_facts
  using q apply fast done
interpretation RA3 < RA1
  print_facts
  using r apply fast done

(* circle of three locales, backward direction *)

locale (open) RB1 = var A + var B + assumes p: "A <-> B"
locale (open) RB2 = var A + var B + assumes q: "A & B | ~ A & ~ B"
locale (open) RB3 = var A + var B + assumes r: "(A --> B) & (B --> A)"

interpretation RB1 < RB2
  print_facts
  using p apply fast done
interpretation RB3 < RB1
  print_facts
  using r apply fast done
interpretation RB2 < RB3
  print_facts
  using q apply fast done

lemma (in RB1) True
  print_facts
  ..


(* Group example revisited, with predicates *)

locale Rpsemi = var prod (infixl "**" 65) +
  assumes assoc: "(x ** y) ** z = x ** (y ** z)"

locale Rplgrp = Rpsemi + var one + var inv +
  assumes lone: "one ** x = x"
    and linv: "inv(x) ** x = one"

lemma (in Rplgrp) lcancel:
  "x ** y = x ** z <-> y = z"
proof
  assume "x ** y = x ** z"
  then have "inv(x) ** x ** y = inv(x) ** x ** z" by (simp add: assoc)
  then show "y = z" by (simp add: lone linv)
qed simp

locale Rprgrp = Rpsemi + var one + var inv +
  assumes rone: "x ** one = x"
    and rinv: "x ** inv(x) = one"

lemma (in Rprgrp) rcancel:
  "y ** x = z ** x <-> y = z"
proof
  assume "y ** x = z ** x"
  then have "y ** (x ** inv(x)) = z ** (x ** inv(x))"
    by (simp add: assoc [symmetric])
  then show "y = z" by (simp add: rone rinv)
qed simp

interpretation Rplgrp < Rprgrp
  proof unfold_locales
    {
      fix x
      have "inv(x) ** x ** one = inv(x) ** x" by (simp add: linv lone)
      then show "x ** one = x" by (simp add: assoc lcancel)
    }
    note rone = this
    {
      fix x
      have "inv(x) ** x ** inv(x) = inv(x) ** one"
	by (simp add: linv lone rone)
      then show "x ** inv(x) = one" by (simp add: assoc lcancel)
    }
  qed

(* effect on printed locale *)

print_locale! Rplgrp

(* use of derived theorem *)

lemma (in Rplgrp)
  "y ** x = z ** x <-> y = z"
  apply (rule rcancel)
  print_interps Rprgrp thm lcancel rcancel
  done

(* circular interpretation *)

interpretation Rprgrp < Rplgrp
  proof unfold_locales
    {
      fix x
      have "one ** (x ** inv(x)) = x ** inv(x)" by (simp add: rinv rone)
      then show "one ** x = x" by (simp add: assoc [symmetric] rcancel)
    }
    note lone = this
    {
      fix x
      have "inv(x) ** (x ** inv(x)) = one ** inv(x)"
	by (simp add: rinv lone rone)
      then show "inv(x) ** x = one" by (simp add: assoc [symmetric] rcancel)
    }
  qed

(* effect on printed locale *)

print_locale! Rprgrp
print_locale! Rplgrp

subsection {* Interaction of Interpretation in Theories and Locales:
  in Locale, then in Theory *}

consts
  rone :: i
  rinv :: "i => i"

axioms
  r_one : "rone # x = x"
  r_inv : "rinv(x) # x = rone"

interpretation Rbool: Rlgrp ["op #" "rone" "rinv"]
proof -
  fix x y z
  {
    show "(x # y) # z = x # (y # z)" by (rule i_assoc)
  next
    show "rone # x = x" by (rule r_one)
  next
    show "rinv(x) # x = rone" by (rule r_inv)
  }
qed

(* derived elements *)

print_interps Rrgrp
print_interps Rlgrp

lemma "y # x = z # x <-> y = z" by (rule Rbool.rcancel)

(* adding lemma to derived element *)

lemma (in Rrgrp) new_cancel:
  "b ** a = c ** a <-> b = c"
  by (rule rcancel)

thm Rbool.new_cancel (* additional prems discharged!! *)

ML {* check_thm "Rbool.new_cancel" *}

lemma "b # a = c # a <-> b = c" by (rule Rbool.new_cancel)


subsection {* Interaction of Interpretation in Theories and Locales:
  in Theory, then in Locale *}

(* Another copy of the group example *)

locale Rqsemi = var prod (infixl "**" 65) +
  assumes assoc: "(x ** y) ** z = x ** (y ** z)"

locale Rqlgrp = Rqsemi + var one + var inv +
  assumes lone: "one ** x = x"
    and linv: "inv(x) ** x = one"

lemma (in Rqlgrp) lcancel:
  "x ** y = x ** z <-> y = z"
proof
  assume "x ** y = x ** z"
  then have "inv(x) ** x ** y = inv(x) ** x ** z" by (simp add: assoc)
  then show "y = z" by (simp add: lone linv)
qed simp

locale Rqrgrp = Rqsemi + var one + var inv +
  assumes rone: "x ** one = x"
    and rinv: "x ** inv(x) = one"

lemma (in Rqrgrp) rcancel:
  "y ** x = z ** x <-> y = z"
proof
  assume "y ** x = z ** x"
  then have "y ** (x ** inv(x)) = z ** (x ** inv(x))"
    by (simp add: assoc [symmetric])
  then show "y = z" by (simp add: rone rinv)
qed simp

interpretation Rqrgrp < Rprgrp
  apply unfold_locales
  apply (rule assoc)
  apply (rule rone)
  apply (rule rinv)
  done

interpretation R2: Rqlgrp ["op #" "rone" "rinv"] 
  apply unfold_locales  (* FIXME: unfold_locales is too eager and shouldn't
                          solve this. *)
  done

print_interps Rqsemi
print_interps Rqlgrp
print_interps Rplgrp  (* no interpretations yet *)


interpretation Rqlgrp < Rqrgrp
  proof unfold_locales
    {
      fix x
      have "inv(x) ** x ** one = inv(x) ** x" by (simp add: linv lone)
      then show "x ** one = x" by (simp add: assoc lcancel)
    }
    note rone = this
    {
      fix x
      have "inv(x) ** x ** inv(x) = inv(x) ** one"
	by (simp add: linv lone rone)
      then show "x ** inv(x) = one" by (simp add: assoc lcancel)
    }
  qed

print_interps! Rqrgrp
print_interps! Rpsemi  (* witness must not have meta hyps *)
print_interps! Rprgrp  (* witness must not have meta hyps *)
print_interps! Rplgrp  (* witness must not have meta hyps *)
thm R2.rcancel
thm R2.lcancel

ML {* check_thm "R2.rcancel"; check_thm "R2.lcancel" *}


subsection {* Generation of Witness Theorems for Transitive Interpretations *}

locale Rtriv = var x +
  assumes x: "x = x"

locale Rtriv2 = var x + var y +
  assumes x: "x = x" and y: "y = y"

interpretation Rtriv2 < Rtriv x
  apply unfold_locales
  apply (rule x)
  done

interpretation Rtriv2 < Rtriv y
  apply unfold_locales
  apply (rule y)
  done

print_locale Rtriv2

locale Rtriv3 = var x + var y + var z +
  assumes x: "x = x" and y: "y = y" and z: "z = z"

interpretation Rtriv3 < Rtriv2 x y
  apply unfold_locales
  apply (rule x)
  apply (rule y)
  done

print_locale Rtriv3

interpretation Rtriv3 < Rtriv2 x z
  apply unfold_locales
  apply (rule x_y_z.x)
  apply (rule z)
  done

ML {* set show_types *}

print_locale Rtriv3


subsection {* Normalisation Replaces Assumed Element by Derived Element *}

typedecl ('a, 'b) pair
arities pair :: ("term", "term") "term"

consts
  pair :: "['a, 'b] => ('a, 'b) pair"
  fst :: "('a, 'b) pair => 'a"
  snd :: "('a, 'b) pair => 'b"

axioms
  fst [simp]: "fst(pair(x, y)) = x"
  snd [simp]: "snd(pair(x, y)) = y"

locale Rpair = var prod (infixl "**" 65) + var prodP (infixl "***" 65) +
  defines P_def: "x *** y == pair(fst(x) ** fst(y), snd(x) ** snd(y))"

locale Rpair_semi = Rpair + Rpsemi

interpretation Rpair_semi < Rpsemi prodP (infixl "***" 65)
proof (rule Rpsemi.intro)
  fix x y z
  show "(x *** y) *** z = x *** (y *** z)"
    apply (simp only: P_def) apply (simp add: assoc) (* FIXME: unfold P_def fails *)
    done
qed

locale Rsemi_rev = Rpsemi + var rprod (infixl "++" 65) +
  defines r_def: "x ++ y == y ** x"

lemma (in Rsemi_rev) r_assoc:
  "(x ++ y) ++ z = x ++ (y ++ z)"
  by (simp add: r_def assoc)

lemma (in Rpair_semi)
  includes Rsemi_rev prodP (infixl "***" 65) rprodP (infixl "+++" 65)
  constrains prod :: "['a, 'a] => 'a"
    and rprodP :: "[('a, 'a) pair, ('a, 'a) pair] => ('a, 'a) pair"
  shows "(x +++ y) +++ z = x +++ (y +++ z)"
  apply (rule r_assoc) done


subsection {* Import of Locales with Predicates as Interpretation *}

locale Ra =
  assumes Ra: "True"

locale Rb = Ra +
  assumes Rb: "True"

locale Rc = Rb +
  assumes Rc: "True"

print_locale! Rc


section {* Interpretation of Defined Concepts *}

text {* Naming convention for global objects: prefixes D and d *}


subsection {* Simple examples *}

locale Da = fixes a :: o
  assumes true: a

text {* In the following examples, @{term "~ a"} is the defined concept. *}

lemma (in Da) not_false: "~ a <-> False"
  apply simp apply (rule true) done

interpretation D1: Da ["True"]
  where "~ True == False"
  apply -
  apply unfold_locales [1] apply rule
  by simp

thm D1.not_false
lemma "False <-> False" apply (rule D1.not_false) done

interpretation D2: Da ["x | ~ x"]
  where "~ (x | ~ x) <-> ~ x & x"
  apply -
  apply unfold_locales [1] apply fast
  by simp

thm D2.not_false
lemma "~ x & x <-> False" apply (rule D2.not_false) done

print_interps! Da

(* Subscriptions of interpretations *)

lemma (in Da) not_false2: "~a <-> False"
  apply simp apply (rule true) done

thm D1.not_false2 D2.not_false2
lemma "False <-> False" apply (rule D1.not_false2) done
lemma "~x & x <-> False" apply (rule D2.not_false2) done

(* Unfolding in attributes *)

locale Db = Da +
  fixes b :: o
  assumes a_iff_b: "~a <-> b"

lemmas (in Db) not_false_b = not_false [unfolded a_iff_b]

interpretation D2: Db ["x | ~ x" "~ (x <-> x)"]
  apply unfold_locales apply fast done

thm D2.not_false_b
lemma "~(x <-> x) <-> False" apply (rule D2.not_false_b) done

(* Subscription and attributes *)

lemmas (in Db) not_false_b2 = not_false [unfolded a_iff_b]

thm D2.not_false_b2
lemma "~(x <-> x) <-> False" apply (rule D2.not_false_b2) done

end