Theory Meson

(*  Title:      HOL/Meson.thy
    Author:     Lawrence C. Paulson, Cambridge University Computer Laboratory
    Author:     Tobias Nipkow, TU Muenchen
    Author:     Jasmin Blanchette, TU Muenchen
    Copyright   2001  University of Cambridge

section ‹MESON Proof Method›

theory Meson
imports Nat

subsection ‹Negation Normal Form›

text ‹de Morgan laws›

lemma not_conjD: "¬(PQ)  ¬P  ¬Q"
  and not_disjD: "¬(PQ)  ¬P  ¬Q"
  and not_notD: "¬¬P  P"
  and not_allD: "P. ¬(x. P(x))  x. ¬P(x)"
  and not_exD: "P. ¬(x. P(x))  x. ¬P(x)"
  by fast+

text ‹Removal of ⟶› and ⟷› (positive and negative occurrences)›

lemma imp_to_disjD: "PQ  ¬P  Q"
  and not_impD: "¬(PQ)  P  ¬Q"
  and iff_to_disjD: "P=Q  (¬P  Q)  (¬Q  P)"
  and not_iffD: "¬(P=Q)  (P  Q)  (¬P  ¬Q)"
    ― ‹Much more efficient than prop(P  ¬Q)  (Q  ¬P) for computing CNF›
  and not_refl_disj_D: "x  x  P  P"
  by fast+

subsection ‹Pulling out the existential quantifiers›

text ‹Conjunction›

lemma conj_exD1: "P Q. (x. P(x))  Q  x. P(x)  Q"
  and conj_exD2: "P Q. P  (x. Q(x))  x. P  Q(x)"
  by fast+

text ‹Disjunction›

lemma disj_exD: "P Q. (x. P(x))  (x. Q(x))  x. P(x)  Q(x)"
  ― ‹DO NOT USE with forall-Skolemization: makes fewer schematic variables!!›
  ― ‹With ex-Skolemization, makes fewer Skolem constants›
  and disj_exD1: "P Q. (x. P(x))  Q  x. P(x)  Q"
  and disj_exD2: "P Q. P  (x. Q(x))  x. P  Q(x)"
  by fast+

lemma disj_assoc: "(PQ)R  P(QR)"
  and disj_comm: "PQ  QP"
  and disj_FalseD1: "FalseP  P"
  and disj_FalseD2: "PFalse  P"
  by fast+

text‹Generation of contrapositives›

text‹Inserts negated disjunct after removing the negation; P is a literal.
  Model elimination requires assuming the negation of every attempted subgoal,
  hence the negated disjuncts.›
lemma make_neg_rule: "¬PQ  ((¬PP)  Q)"
by blast

text‹Version for Plaisted's "Postive refinement" of the Meson procedure›
lemma make_refined_neg_rule: "¬PQ  (P  Q)"
by blast

texttermP should be a literal›
lemma make_pos_rule: "PQ  ((P¬P)  Q)"
by blast

text‹Versions of make_neg_rule› and make_pos_rule› that don't
insert new assumptions, for ordinary resolution.›

lemmas make_neg_rule' = make_refined_neg_rule

lemma make_pos_rule': "PQ; ¬P  Q"
by blast

text‹Generation of a goal clause -- put away the final literal›

lemma make_neg_goal: "¬P  ((¬PP)  False)"
by blast

lemma make_pos_goal: "P  ((P¬P)  False)"
by blast

subsection ‹Lemmas for Forward Proof›

text‹There is a similarity to congruence rules. They are also useful in ordinary proofs.›

(*NOTE: could handle conjunctions (faster?) by
    nf(th RS conjunct2) RS (nf(th RS conjunct1) RS conjI) *)
lemma conj_forward: "P'Q';  P'  P;  Q'  Q   PQ"
by blast

lemma disj_forward: "P'Q';  P'  P;  Q'  Q   PQ"
by blast

lemma imp_forward: "P'  Q';  P  P';  Q'  Q   P  Q"
by blast

lemma imp_forward2: "P'  Q';  P  P';  P'  Q'  Q   P  Q"
  by blast

(*Version of @{text disj_forward} for removal of duplicate literals*)
lemma disj_forward2: " P'Q';  P'  P;  Q'; PFalse  Q  PQ"
apply blast 

lemma all_forward: "[| x. P'(x);  !!x. P'(x) ==> P(x) |] ==> x. P(x)"
by blast

lemma ex_forward: "[| x. P'(x);  !!x. P'(x) ==> P(x) |] ==> x. P(x)"
by blast

subsection ‹Clausification helper›

lemma TruepropI: "P  Q  Trueprop P  Trueprop Q"
by simp

lemma ext_cong_neq: "F g  F h  F g  F h  (x. g x  h x)"
apply (erule contrapos_np)
apply clarsimp
apply (rule cong[where f = F])
by auto

text‹Combinator translation helpers›

definition COMBI :: "'a  'a" where

definition COMBK :: "'a  'b  'a" where

definition COMBB :: "('b => 'c)  ('a => 'b)  'a  'c" where
"COMBB P Q R = P (Q R)"

definition COMBC :: "('a  'b  'c)  'b  'a  'c" where

definition COMBS :: "('a  'b  'c)  ('a  'b)  'a  'c" where
"COMBS P Q R = P R (Q R)"

lemma abs_S: "λx. (f x) (g x)  COMBS f g"
apply (rule eq_reflection)
apply (rule ext) 
apply (simp add: COMBS_def) 

lemma abs_I: "λx. x  COMBI"
apply (rule eq_reflection)
apply (rule ext) 
apply (simp add: COMBI_def) 

lemma abs_K: "λx. y  COMBK y"
apply (rule eq_reflection)
apply (rule ext) 
apply (simp add: COMBK_def) 

lemma abs_B: "λx. a (g x)  COMBB a g"
apply (rule eq_reflection)
apply (rule ext) 
apply (simp add: COMBB_def) 

lemma abs_C: "λx. (f x) b  COMBC f b"
apply (rule eq_reflection)
apply (rule ext) 
apply (simp add: COMBC_def) 

subsection ‹Skolemization helpers›

definition skolem :: "'a  'a" where
"skolem = (λx. x)"

lemma skolem_COMBK_iff: "P  skolem (COMBK P (i::nat))"
unfolding skolem_def COMBK_def by (rule refl)

lemmas skolem_COMBK_I = iffD1 [OF skolem_COMBK_iff]

subsection ‹Meson package›

ML_file ‹Tools/Meson/meson.ML›
ML_file ‹Tools/Meson/meson_clausify.ML›
ML_file ‹Tools/Meson/meson_tactic.ML›

hide_const (open) COMBI COMBK COMBB COMBC COMBS skolem
hide_fact (open) not_conjD not_disjD not_notD not_allD not_exD imp_to_disjD
    not_impD iff_to_disjD not_iffD not_refl_disj_D conj_exD1 conj_exD2 disj_exD
    disj_exD1 disj_exD2 disj_assoc disj_comm disj_FalseD1 disj_FalseD2 TruepropI
    ext_cong_neq COMBI_def COMBK_def COMBB_def COMBC_def COMBS_def abs_I abs_K
    abs_B abs_C abs_S skolem_def skolem_COMBK_iff skolem_COMBK_I