# HG changeset patch
# User wenzelm
# Date 1006796037 -3600
# Node ID b344486c33e2800fe4fd8bd55708d11fd82bb5a0
# Parent  2ce7b42b0a64c19b994ba1247a08750ba020a26e
tuned;
meson lemmas from Tools/meson.ML;

diff -r 2ce7b42b0a64 -r b344486c33e2 src/HOL/Hilbert_Choice.thy
--- a/src/HOL/Hilbert_Choice.thy	Mon Nov 26 18:33:21 2001 +0100
+++ b/src/HOL/Hilbert_Choice.thy	Mon Nov 26 18:33:57 2001 +0100
@@ -9,199 +9,247 @@
 theory Hilbert_Choice = NatArith
 files ("Hilbert_Choice_lemmas.ML") ("meson_lemmas.ML") ("Tools/meson.ML"):
 
+
+subsection {* Hilbert's epsilon *}
+
 consts
   Eps           :: "('a => bool) => 'a"
 
-
 syntax (input)
-  "_Eps"        :: "[pttrn, bool] => 'a"                 ("(3\<epsilon>_./ _)" [0, 10] 10)
-
+  "_Eps"        :: "[pttrn, bool] => 'a"    ("(3\<epsilon>_./ _)" [0, 10] 10)
 syntax (HOL)
-  "_Eps"        :: "[pttrn, bool] => 'a"                 ("(3@ _./ _)" [0, 10] 10)
-
+  "_Eps"        :: "[pttrn, bool] => 'a"    ("(3@ _./ _)" [0, 10] 10)
 syntax
-  "_Eps"        :: "[pttrn, bool] => 'a"                 ("(3SOME _./ _)" [0, 10] 10)
-
+  "_Eps"        :: "[pttrn, bool] => 'a"    ("(3SOME _./ _)" [0, 10] 10)
 translations
-  "SOME x. P"             == "Eps (%x. P)"
+  "SOME x. P" == "Eps (%x. P)"
 
-axioms  
-  someI:        "P (x::'a) ==> P (SOME x. P x)"
+axioms
+  someI: "P (x::'a) ==> P (SOME x. P x)"
 
 
-(*used in TFL*)
-lemma tfl_some: "\<forall>P x. P x --> P (Eps P)"
-  by (blast intro: someI)
-
+constdefs
+  inv :: "('a => 'b) => ('b => 'a)"
+  "inv(f :: 'a => 'b) == %y. SOME x. f x = y"
 
-constdefs  
-  inv :: "('a => 'b) => ('b => 'a)"
-    "inv(f::'a=>'b) == % y. @x. f(x)=y"
-
-  Inv :: "['a set, 'a => 'b] => ('b => 'a)"
-    "Inv A f == (% x. (@ y. y : A & f y = x))"
+  Inv :: "'a set => ('a => 'b) => ('b => 'a)"
+  "Inv A f == %x. SOME y. y : A & f y = x"
 
 
 use "Hilbert_Choice_lemmas.ML"
 
+lemma tfl_some: "\<forall>P x. P x --> P (Eps P)"
+  -- {* dynamically-scoped fact for TFL *}
+  by (blast intro: someI)
 
-(** Least value operator **)
+
+subsection {* Least value operator *}
 
 constdefs
-  LeastM   :: "['a => 'b::ord, 'a => bool] => 'a"
-              "LeastM m P == @x. P x & (ALL y. P y --> m x <= m y)"
+  LeastM :: "['a => 'b::ord, 'a => bool] => 'a"
+  "LeastM m P == SOME x. P x & (ALL y. P y --> m x <= m y)"
 
 syntax
- "@LeastM" :: "[pttrn, 'a=>'b::ord, bool] => 'a" ("LEAST _ WRT _. _" [0,4,10]10)
-
+  "_LeastM" :: "[pttrn, 'a => 'b::ord, bool] => 'a"    ("LEAST _ WRT _. _" [0, 4, 10] 10)
 translations
-                "LEAST x WRT m. P" == "LeastM m (%x. P)"
+  "LEAST x WRT m. P" == "LeastM m (%x. P)"
 
 lemma LeastMI2:
-  "[| P x; !!y. P y ==> m x <= m y;
-           !!x. [| P x; \<forall>y. P y --> m x \<le> m y |] ==> Q x |]
-   ==> Q (LeastM m P)";
-apply (unfold LeastM_def)
-apply (rule someI2_ex)
-apply  blast
-apply blast
-done
+  "P x ==> (!!y. P y ==> m x <= m y)
+    ==> (!!x. P x ==> \<forall>y. P y --> m x \<le> m y ==> Q x)
+    ==> Q (LeastM m P)"
+  apply (unfold LeastM_def)
+  apply (rule someI2_ex)
+   apply blast
+  apply blast
+  done
 
 lemma LeastM_equality:
- "[| P k; !!x. P x ==> m k <= m x |] ==> m (LEAST x WRT m. P x) = 
-     (m k::'a::order)";
-apply (rule LeastMI2)
-apply   assumption
-apply  blast
-apply (blast intro!: order_antisym) 
-done
+  "P k ==> (!!x. P x ==> m k <= m x)
+    ==> m (LEAST x WRT m. P x) = (m k::'a::order)"
+  apply (rule LeastMI2)
+    apply assumption
+   apply blast
+  apply (blast intro!: order_antisym)
+  done
 
 lemma wf_linord_ex_has_least:
-     "[|wf r;  ALL x y. ((x,y):r^+) = ((y,x)~:r^*);  P k|]  \
-\     ==> EX x. P x & (!y. P y --> (m x,m y):r^*)" 
-apply (drule wf_trancl [THEN wf_eq_minimal [THEN iffD1]])
-apply (drule_tac x = "m`Collect P" in spec)
-apply force
-done
+  "wf r ==> ALL x y. ((x,y):r^+) = ((y,x)~:r^*) ==> P k
+    ==> EX x. P x & (!y. P y --> (m x,m y):r^*)"
+  apply (drule wf_trancl [THEN wf_eq_minimal [THEN iffD1]])
+  apply (drule_tac x = "m`Collect P" in spec)
+  apply force
+  done
 
-(* successor of obsolete nonempty_has_least *)
 lemma ex_has_least_nat:
-     "P k ==> EX x. P x & (ALL y. P y --> m x <= (m y::nat))"
-apply (simp only: pred_nat_trancl_eq_le [symmetric])
-apply (rule wf_pred_nat [THEN wf_linord_ex_has_least])
-apply (simp (no_asm) add: less_eq not_le_iff_less pred_nat_trancl_eq_le)
-apply assumption
-done
+    "P k ==> EX x. P x & (ALL y. P y --> m x <= (m y::nat))"
+  apply (simp only: pred_nat_trancl_eq_le [symmetric])
+  apply (rule wf_pred_nat [THEN wf_linord_ex_has_least])
+   apply (simp add: less_eq not_le_iff_less pred_nat_trancl_eq_le)
+  apply assumption
+  done
 
-lemma LeastM_nat_lemma: 
-  "P k ==> P (LeastM m P) & (ALL y. P y --> m (LeastM m P) <= (m y::nat))"
-apply (unfold LeastM_def)
-apply (rule someI_ex)
-apply (erule ex_has_least_nat)
-done
+lemma LeastM_nat_lemma:
+    "P k ==> P (LeastM m P) & (ALL y. P y --> m (LeastM m P) <= (m y::nat))"
+  apply (unfold LeastM_def)
+  apply (rule someI_ex)
+  apply (erule ex_has_least_nat)
+  done
 
 lemmas LeastM_natI = LeastM_nat_lemma [THEN conjunct1, standard]
 
 lemma LeastM_nat_le: "P x ==> m (LeastM m P) <= (m x::nat)"
-apply (rule LeastM_nat_lemma [THEN conjunct2, THEN spec, THEN mp])
-apply assumption
-apply assumption
-done
+  apply (rule LeastM_nat_lemma [THEN conjunct2, THEN spec, THEN mp])
+   apply assumption
+  apply assumption
+  done
 
 
-(** Greatest value operator **)
+subsection {* Greatest value operator *}
 
 constdefs
-  GreatestM   :: "['a => 'b::ord, 'a => bool] => 'a"
-              "GreatestM m P == @x. P x & (ALL y. P y --> m y <= m x)"
-  
-  Greatest    :: "('a::ord => bool) => 'a"         (binder "GREATEST " 10)
-              "Greatest     == GreatestM (%x. x)"
+  GreatestM :: "['a => 'b::ord, 'a => bool] => 'a"
+  "GreatestM m P == SOME x. P x & (ALL y. P y --> m y <= m x)"
+
+  Greatest :: "('a::ord => bool) => 'a"    (binder "GREATEST " 10)
+  "Greatest == GreatestM (%x. x)"
 
 syntax
- "@GreatestM" :: "[pttrn, 'a=>'b::ord, bool] => 'a"
-                                        ("GREATEST _ WRT _. _" [0,4,10]10)
+  "_GreatestM" :: "[pttrn, 'a=>'b::ord, bool] => 'a"
+      ("GREATEST _ WRT _. _" [0, 4, 10] 10)
 
 translations
-              "GREATEST x WRT m. P" == "GreatestM m (%x. P)"
+  "GREATEST x WRT m. P" == "GreatestM m (%x. P)"
 
 lemma GreatestMI2:
-     "[| P x;
-	 !!y. P y ==> m y <= m x;
-         !!x. [| P x; \<forall>y. P y --> m y \<le> m x |] ==> Q x |]
-      ==> Q (GreatestM m P)";
-apply (unfold GreatestM_def)
-apply (rule someI2_ex)
-apply  blast
-apply blast
-done
+  "P x ==> (!!y. P y ==> m y <= m x)
+    ==> (!!x. P x ==> \<forall>y. P y --> m y \<le> m x ==> Q x)
+    ==> Q (GreatestM m P)"
+  apply (unfold GreatestM_def)
+  apply (rule someI2_ex)
+   apply blast
+  apply blast
+  done
 
 lemma GreatestM_equality:
- "[| P k;  !!x. P x ==> m x <= m k |]
-  ==> m (GREATEST x WRT m. P x) = (m k::'a::order)";
-apply (rule_tac m=m in GreatestMI2)
-apply   assumption
-apply  blast
-apply (blast intro!: order_antisym) 
-done
+ "P k ==> (!!x. P x ==> m x <= m k)
+    ==> m (GREATEST x WRT m. P x) = (m k::'a::order)"
+  apply (rule_tac m = m in GreatestMI2)
+    apply assumption
+   apply blast
+  apply (blast intro!: order_antisym)
+  done
 
 lemma Greatest_equality:
-  "[| P (k::'a::order); !!x. P x ==> x <= k |] ==> (GREATEST x. P x) = k";
-apply (unfold Greatest_def)
-apply (erule GreatestM_equality)
-apply blast
-done
+  "P (k::'a::order) ==> (!!x. P x ==> x <= k) ==> (GREATEST x. P x) = k"
+  apply (unfold Greatest_def)
+  apply (erule GreatestM_equality)
+  apply blast
+  done
 
 lemma ex_has_greatest_nat_lemma:
-     "[|P k;  ALL x. P x --> (EX y. P y & ~ ((m y::nat) <= m x))|]  
-      ==> EX y. P y & ~ (m y < m k + n)"
-apply (induct_tac "n")
-apply force
-(*ind step*)
-apply (force simp add: le_Suc_eq)
-done
+  "P k ==> ALL x. P x --> (EX y. P y & ~ ((m y::nat) <= m x))
+    ==> EX y. P y & ~ (m y < m k + n)"
+  apply (induct_tac n)
+   apply force
+  apply (force simp add: le_Suc_eq)
+  done
 
-lemma ex_has_greatest_nat: "[|P k;  ! y. P y --> m y < b|]  
-      ==> ? x. P x & (! y. P y --> (m y::nat) <= m x)"
-apply (rule ccontr)
-apply (cut_tac P = "P" and n = "b - m k" in ex_has_greatest_nat_lemma)
-apply (subgoal_tac [3] "m k <= b")
-apply auto
-done
+lemma ex_has_greatest_nat:
+  "P k ==> ALL y. P y --> m y < b
+    ==> EX x. P x & (ALL y. P y --> (m y::nat) <= m x)"
+  apply (rule ccontr)
+  apply (cut_tac P = P and n = "b - m k" in ex_has_greatest_nat_lemma)
+    apply (subgoal_tac [3] "m k <= b")
+     apply auto
+  done
 
-lemma GreatestM_nat_lemma: 
-     "[|P k;  ! y. P y --> m y < b|]  
-      ==> P (GreatestM m P) & (!y. P y --> (m y::nat) <= m (GreatestM m P))"
-apply (unfold GreatestM_def)
-apply (rule someI_ex)
-apply (erule ex_has_greatest_nat)
-apply assumption
-done
+lemma GreatestM_nat_lemma:
+  "P k ==> ALL y. P y --> m y < b
+    ==> P (GreatestM m P) & (ALL y. P y --> (m y::nat) <= m (GreatestM m P))"
+  apply (unfold GreatestM_def)
+  apply (rule someI_ex)
+  apply (erule ex_has_greatest_nat)
+  apply assumption
+  done
 
 lemmas GreatestM_natI = GreatestM_nat_lemma [THEN conjunct1, standard]
 
-lemma GreatestM_nat_le: "[|P x;  ! y. P y --> m y < b|]  
-      ==> (m x::nat) <= m (GreatestM m P)"
-apply (blast dest: GreatestM_nat_lemma [THEN conjunct2, THEN spec]) 
-done
+lemma GreatestM_nat_le:
+  "P x ==> ALL y. P y --> m y < b
+    ==> (m x::nat) <= m (GreatestM m P)"
+  apply (blast dest: GreatestM_nat_lemma [THEN conjunct2, THEN spec])
+  done
+
+
+text {* \medskip Specialization to @{text GREATEST}. *}
+
+lemma GreatestI: "P (k::nat) ==> ALL y. P y --> y < b ==> P (GREATEST x. P x)"
+  apply (unfold Greatest_def)
+  apply (rule GreatestM_natI)
+   apply auto
+  done
 
-(** Specialization to GREATEST **)
+lemma Greatest_le:
+    "P x ==> ALL y. P y --> y < b ==> (x::nat) <= (GREATEST x. P x)"
+  apply (unfold Greatest_def)
+  apply (rule GreatestM_nat_le)
+   apply auto
+  done
+
+
+subsection {* The Meson proof procedure *}
 
-lemma GreatestI: 
-     "[|P (k::nat);  ! y. P y --> y < b|] ==> P (GREATEST x. P x)"
+subsubsection {* Negation Normal Form *}
+
+text {* de Morgan laws *}
+
+lemma meson_not_conjD: "~(P&Q) ==> ~P | ~Q"
+  and meson_not_disjD: "~(P|Q) ==> ~P & ~Q"
+  and meson_not_notD: "~~P ==> P"
+  and meson_not_allD: "!!P. ~(ALL x. P(x)) ==> EX x. ~P(x)"
+  and meson_not_exD: "!!P. ~(EX x. P(x)) ==> ALL x. ~P(x)"
+  by fast+
 
-apply (unfold Greatest_def)
-apply (rule GreatestM_natI)
-apply auto
-done
+text {* Removal of @{text "-->"} and @{text "<->"} (positive and
+negative occurrences) *}
+
+lemma meson_imp_to_disjD: "P-->Q ==> ~P | Q"
+  and meson_not_impD: "~(P-->Q) ==> P & ~Q"
+  and meson_iff_to_disjD: "P=Q ==> (~P | Q) & (~Q | P)"
+  and meson_not_iffD: "~(P=Q) ==> (P | Q) & (~P | ~Q)"
+    -- {* Much more efficient than @{prop "(P & ~Q) | (Q & ~P)"} for computing CNF *}
+  by fast+
+
+
+subsubsection {* Pulling out the existential quantifiers *}
+
+text {* Conjunction *}
+
+lemma meson_conj_exD1: "!!P Q. (EX x. P(x)) & Q ==> EX x. P(x) & Q"
+  and meson_conj_exD2: "!!P Q. P & (EX x. Q(x)) ==> EX x. P & Q(x)"
+  by fast+
+
 
-lemma Greatest_le: 
-     "[|P x;  ! y. P y --> y < b|] ==> (x::nat) <= (GREATEST x. P x)"
-apply (unfold Greatest_def)
-apply (rule GreatestM_nat_le)
-apply auto
-done
+text {* Disjunction *}
+
+lemma meson_disj_exD: "!!P Q. (EX x. P(x)) | (EX x. Q(x)) ==> EX x. P(x) | Q(x)"
+  -- {* DO NOT USE with forall-Skolemization: makes fewer schematic variables!! *}
+  -- {* With ex-Skolemization, makes fewer Skolem constants *}
+  and meson_disj_exD1: "!!P Q. (EX x. P(x)) | Q ==> EX x. P(x) | Q"
+  and meson_disj_exD2: "!!P Q. P | (EX x. Q(x)) ==> EX x. P | Q(x)"
+  by fast+
+
 
+subsubsection {* Generating clauses for the Meson Proof Procedure *}
+
+text {* Disjunctions *}
+
+lemma meson_disj_assoc: "(P|Q)|R ==> P|(Q|R)"
+  and meson_disj_comm: "P|Q ==> Q|P"
+  and meson_disj_FalseD1: "False|P ==> P"
+  and meson_disj_FalseD2: "P|False ==> P"
+  by fast+
 
 use "meson_lemmas.ML"
 use "Tools/meson.ML"