src/Sequents/LK.thy
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(*  Title:      LK/LK.ML
Author:     Lawrence C Paulson, Cambridge University Computer Laboratory

Axiom to express monotonicity (a variant of the deduction theorem).  Makes the
link between |- and ==>, needed for instance to prove imp_cong.

Axiom left_cong allows the simplifier to use left-side formulas.  Ideally it
should be derived from lower-level axioms.

CANNOT be added to LK0.thy because modal logic is built upon it, and
various modal rules would become inconsistent.
*)

theory LK
imports LK0
uses ("simpdata.ML")
begin

axiomatization where
monotonic:  "(\$H |- P ==> \$H |- Q) ==> \$H, P |- Q" and

left_cong:  "[| P == P';  |- P' ==> (\$H |- \$F) == (\$H' |- \$F') |]
==> (P, \$H |- \$F) == (P', \$H' |- \$F')"

subsection {* Rewrite rules *}

lemma conj_simps:
"|- P & True <-> P"
"|- True & P <-> P"
"|- P & False <-> False"
"|- False & P <-> False"
"|- P & P <-> P"
"|- P & P & Q <-> P & Q"
"|- P & ~P <-> False"
"|- ~P & P <-> False"
"|- (P & Q) & R <-> P & (Q & R)"
apply (tactic {* ALLGOALS (fast_tac (LK_pack add_safes @{thms subst})) *})
done

lemma disj_simps:
"|- P | True <-> True"
"|- True | P <-> True"
"|- P | False <-> P"
"|- False | P <-> P"
"|- P | P <-> P"
"|- P | P | Q <-> P | Q"
"|- (P | Q) | R <-> P | (Q | R)"
apply (tactic {* ALLGOALS (fast_tac (LK_pack add_safes @{thms subst})) *})
done

lemma not_simps:
"|- ~ False <-> True"
"|- ~ True <-> False"
apply (tactic {* ALLGOALS (fast_tac (LK_pack add_safes @{thms subst})) *})
done

lemma imp_simps:
"|- (P --> False) <-> ~P"
"|- (P --> True) <-> True"
"|- (False --> P) <-> True"
"|- (True --> P) <-> P"
"|- (P --> P) <-> True"
"|- (P --> ~P) <-> ~P"
apply (tactic {* ALLGOALS (fast_tac (LK_pack add_safes @{thms subst})) *})
done

lemma iff_simps:
"|- (True <-> P) <-> P"
"|- (P <-> True) <-> P"
"|- (P <-> P) <-> True"
"|- (False <-> P) <-> ~P"
"|- (P <-> False) <-> ~P"
apply (tactic {* ALLGOALS (fast_tac (LK_pack add_safes @{thms subst})) *})
done

lemma quant_simps:
"!!P. |- (ALL x. P) <-> P"
"!!P. |- (ALL x. x=t --> P(x)) <-> P(t)"
"!!P. |- (ALL x. t=x --> P(x)) <-> P(t)"
"!!P. |- (EX x. P) <-> P"
"!!P. |- (EX x. x=t & P(x)) <-> P(t)"
"!!P. |- (EX x. t=x & P(x)) <-> P(t)"
apply (tactic {* ALLGOALS (fast_tac (LK_pack add_safes @{thms subst})) *})
done

subsection {* Miniscoping: pushing quantifiers in *}

text {*
We do NOT distribute of ALL over &, or dually that of EX over |
Baaz and Leitsch, On Skolemization and Proof Complexity (1994)
show that this step can increase proof length!
*}

text {*existential miniscoping*}
lemma ex_simps:
"!!P Q. |- (EX x. P(x) & Q) <-> (EX x. P(x)) & Q"
"!!P Q. |- (EX x. P & Q(x)) <-> P & (EX x. Q(x))"
"!!P Q. |- (EX x. P(x) | Q) <-> (EX x. P(x)) | Q"
"!!P Q. |- (EX x. P | Q(x)) <-> P | (EX x. Q(x))"
"!!P Q. |- (EX x. P(x) --> Q) <-> (ALL x. P(x)) --> Q"
"!!P Q. |- (EX x. P --> Q(x)) <-> P --> (EX x. Q(x))"
apply (tactic {* ALLGOALS (fast_tac (LK_pack add_safes @{thms subst})) *})
done

text {*universal miniscoping*}
lemma all_simps:
"!!P Q. |- (ALL x. P(x) & Q) <-> (ALL x. P(x)) & Q"
"!!P Q. |- (ALL x. P & Q(x)) <-> P & (ALL x. Q(x))"
"!!P Q. |- (ALL x. P(x) --> Q) <-> (EX x. P(x)) --> Q"
"!!P Q. |- (ALL x. P --> Q(x)) <-> P --> (ALL x. Q(x))"
"!!P Q. |- (ALL x. P(x) | Q) <-> (ALL x. P(x)) | Q"
"!!P Q. |- (ALL x. P | Q(x)) <-> P | (ALL x. Q(x))"
apply (tactic {* ALLGOALS (fast_tac (LK_pack add_safes @{thms subst})) *})
done

text {*These are NOT supplied by default!*}
lemma distrib_simps:
"|- P & (Q | R) <-> P&Q | P&R"
"|- (Q | R) & P <-> Q&P | R&P"
"|- (P | Q --> R) <-> (P --> R) & (Q --> R)"
apply (tactic {* ALLGOALS (fast_tac (LK_pack add_safes @{thms subst})) *})
done

lemma P_iff_F: "|- ~P ==> |- (P <-> False)"
apply (erule thinR [THEN cut])
apply (tactic {* fast_tac LK_pack 1 *})
done

lemmas iff_reflection_F = P_iff_F [THEN iff_reflection, standard]

lemma P_iff_T: "|- P ==> |- (P <-> True)"
apply (erule thinR [THEN cut])
apply (tactic {* fast_tac LK_pack 1 *})
done

lemmas iff_reflection_T = P_iff_T [THEN iff_reflection, standard]

lemma LK_extra_simps:
"|- P | ~P"
"|- ~P | P"
"|- ~ ~ P <-> P"
"|- (~P --> P) <-> P"
"|- (~P <-> ~Q) <-> (P<->Q)"
apply (tactic {* ALLGOALS (fast_tac (LK_pack add_safes @{thms subst})) *})
done

subsection {* Named rewrite rules *}

lemma conj_commute: "|- P&Q <-> Q&P"
and conj_left_commute: "|- P&(Q&R) <-> Q&(P&R)"
apply (tactic {* ALLGOALS (fast_tac (LK_pack add_safes @{thms subst})) *})
done

lemmas conj_comms = conj_commute conj_left_commute

lemma disj_commute: "|- P|Q <-> Q|P"
and disj_left_commute: "|- P|(Q|R) <-> Q|(P|R)"
apply (tactic {* ALLGOALS (fast_tac (LK_pack add_safes @{thms subst})) *})
done

lemmas disj_comms = disj_commute disj_left_commute

lemma conj_disj_distribL: "|- P&(Q|R) <-> (P&Q | P&R)"
and conj_disj_distribR: "|- (P|Q)&R <-> (P&R | Q&R)"

and disj_conj_distribL: "|- P|(Q&R) <-> (P|Q) & (P|R)"
and disj_conj_distribR: "|- (P&Q)|R <-> (P|R) & (Q|R)"

and imp_conj_distrib: "|- (P --> (Q&R)) <-> (P-->Q) & (P-->R)"
and imp_conj: "|- ((P&Q)-->R)   <-> (P --> (Q --> R))"
and imp_disj: "|- (P|Q --> R)   <-> (P-->R) & (Q-->R)"

and imp_disj1: "|- (P-->Q) | R <-> (P-->Q | R)"
and imp_disj2: "|- Q | (P-->R) <-> (P-->Q | R)"

and de_Morgan_disj: "|- (~(P | Q)) <-> (~P & ~Q)"
and de_Morgan_conj: "|- (~(P & Q)) <-> (~P | ~Q)"

and not_iff: "|- ~(P <-> Q) <-> (P <-> ~Q)"
apply (tactic {* ALLGOALS (fast_tac (LK_pack add_safes @{thms subst})) *})
done

lemma imp_cong:
assumes p1: "|- P <-> P'"
and p2: "|- P' ==> |- Q <-> Q'"
shows "|- (P-->Q) <-> (P'-->Q')"
apply (tactic {* lemma_tac @{thm p1} 1 *})
apply (tactic {* safe_tac LK_pack 1 *})
apply (tactic {*
REPEAT (rtac @{thm cut} 1 THEN
DEPTH_SOLVE_1
(resolve_tac [@{thm thinL}, @{thm thinR}, @{thm p2} COMP @{thm monotonic}] 1) THEN
safe_tac LK_pack 1) *})
done

lemma conj_cong:
assumes p1: "|- P <-> P'"
and p2: "|- P' ==> |- Q <-> Q'"
shows "|- (P&Q) <-> (P'&Q')"
apply (tactic {* lemma_tac @{thm p1} 1 *})
apply (tactic {* safe_tac LK_pack 1 *})
apply (tactic {*
REPEAT (rtac @{thm cut} 1 THEN
DEPTH_SOLVE_1
(resolve_tac [@{thm thinL}, @{thm thinR}, @{thm p2} COMP @{thm monotonic}] 1) THEN
safe_tac LK_pack 1) *})
done

lemma eq_sym_conv: "|- (x=y) <-> (y=x)"
apply (tactic {* fast_tac (LK_pack add_safes @{thms subst}) 1 *})
done

use "simpdata.ML"

text {* To create substition rules *}

lemma eq_imp_subst: "|- a=b ==> \$H, A(a), \$G |- \$E, A(b), \$F"
apply (tactic {* asm_simp_tac LK_basic_ss 1 *})
done

lemma split_if: "|- P(if Q then x else y) <-> ((Q --> P(x)) & (~Q --> P(y)))"
apply (rule_tac P = Q in cut)
apply (tactic {* simp_tac (@{simpset} addsimps @{thms if_P}) 2 *})
apply (rule_tac P = "~Q" in cut)
apply (tactic {* simp_tac (@{simpset} addsimps @{thms if_not_P}) 2 *})
apply (tactic {* fast_tac LK_pack 1 *})
done

lemma if_cancel: "|- (if P then x else x) = x"
apply (tactic {* lemma_tac @{thm split_if} 1 *})
apply (tactic {* fast_tac LK_pack 1 *})
done

lemma if_eq_cancel: "|- (if x=y then y else x) = x"
apply (tactic {* lemma_tac @{thm split_if} 1 *})
apply (tactic {* safe_tac LK_pack 1 *})
apply (rule symL)
apply (rule basic)
done

end
```