author  wenzelm 
Sat, 07 Apr 2012 16:41:59 +0200  
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parent 42814  5af15f1e2ef6 
child 51309  473303ef6e34 
permissions  rwrr 
17481  1 
(* Title: Sequents/ILL.thy 
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Author: Sara Kalvala and Valeria de Paiva 
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Copyright 1995 University of Cambridge 
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*) 
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17481  6 
theory ILL 
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imports Sequents 

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begin 

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consts 
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Trueprop :: "two_seqi" 
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tens :: "[o, o] => o" (infixr "><" 35) 
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limp :: "[o, o] => o" (infixr "o" 45) 

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liff :: "[o, o] => o" (infixr "oo" 45) 

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FShriek :: "o => o" ("! _" [100] 1000) 

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lconj :: "[o, o] => o" (infixr "&&" 35) 

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ldisj :: "[o, o] => o" (infixr "++" 35) 

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zero :: "o" ("0") 

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top :: "o" ("1") 

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eye :: "o" ("I") 

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aneg :: "o=>o" ("~_") 

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14765  25 
(* context manipulation *) 
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Context :: "two_seqi" 
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(* promotion rule *) 
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PromAux :: "three_seqi" 
14765  32 

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syntax 

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"_Trueprop" :: "single_seqe" ("((_)/  (_))" [6,6] 5) 
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"_Context" :: "two_seqe" ("((_)/ :=: (_))" [6,6] 5) 

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"_PromAux" :: "three_seqe" ("promaux {___}") 

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parse_translation {* 
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[(@{syntax_const "_Trueprop"}, single_tr @{const_syntax Trueprop}), 
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(@{syntax_const "_Context"}, two_seq_tr @{const_syntax Context}), 

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(@{syntax_const "_PromAux"}, three_seq_tr @{const_syntax PromAux})] 

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*} 

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print_translation {* 

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[(@{const_syntax Trueprop}, single_tr' @{syntax_const "_Trueprop"}), 
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(@{const_syntax Context}, two_seq_tr' @{syntax_const "_Context"}), 

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(@{const_syntax PromAux}, three_seq_tr' @{syntax_const "_PromAux"})] 

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*} 

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defs 
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liff_def: "P oo Q == (P o Q) >< (Q o P)" 
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aneg_def: "~A == A o 0" 
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axioms 
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identity: "P  P" 
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zerol: "$G, 0, $H  A" 
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(* RULES THAT DO NOT DIVIDE CONTEXT *) 
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derelict: "$F, A, $G  C ==> $F, !A, $G  C" 
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(* unfortunately, this one removes !A *) 
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contract: "$F, !A, !A, $G  C ==> $F, !A, $G  C" 
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weaken: "$F, $G  C ==> $G, !A, $F  C" 
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(* weak form of weakening, in practice just to clean context *) 
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(* weaken and contract not needed (CHECK) *) 
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promote2: "promaux{  $H  B} ==> $H  !B" 
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promote1: "promaux{!A, $G  $H  B} 

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==> promaux {$G  $H, !A  B}" 

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promote0: "$G  A ==> promaux {$G   A}" 

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tensl: "$H, A, B, $G  C ==> $H, A >< B, $G  C" 
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impr: "A, $F  B ==> $F  A o B" 
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conjr: "[ $F  A ; 
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$F  B ] 
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==> $F  (A && B)" 
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conjll: "$G, A, $H  C ==> $G, A && B, $H  C" 
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conjlr: "$G, B, $H  C ==> $G, A && B, $H  C" 
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disjrl: "$G  A ==> $G  A ++ B" 
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disjrr: "$G  B ==> $G  A ++ B" 

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disjl: "[ $G, A, $H  C ; 

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$G, B, $H  C ] 

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==> $G, A ++ B, $H  C" 

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(* RULES THAT DIVIDE CONTEXT *) 
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tensr: "[ $F, $J :=: $G; 
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$F  A ; 

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$J  B ] 

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==> $G  A >< B" 

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impl: "[ $G, $F :=: $J, $H ; 
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B, $F  C ; 

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$G  A ] 

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==> $J, A o B, $H  C" 

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cut: " [ $J1, $H1, $J2, $H3, $J3, $H2, $J4, $H4 :=: $F ; 
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$H1, $H2, $H3, $H4  A ; 

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$J1, $J2, A, $J3, $J4  B ] ==> $F  B" 

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(* CONTEXT RULES *) 
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context1: "$G :=: $G" 
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context2: "$F, $G :=: $H, !A, $G ==> $F, A, $G :=: $H, !A, $G" 

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context3: "$F, $G :=: $H, $J ==> $F, A, $G :=: $H, A, $J" 

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context4a: "$F :=: $H, $G ==> $F :=: $H, !A, $G" 

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context4b: "$F, $H :=: $G ==> $F, !A, $H :=: $G" 

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context5: "$F, $G :=: $H ==> $G, $F :=: $H" 

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ML {* 

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val lazy_cs = empty_pack 

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add_safes [@{thm tensl}, @{thm conjr}, @{thm disjl}, @{thm promote0}, 
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@{thm context2}, @{thm context3}] 

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add_unsafes [@{thm identity}, @{thm zerol}, @{thm conjll}, @{thm conjlr}, 

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@{thm disjrl}, @{thm disjrr}, @{thm impr}, @{thm tensr}, @{thm impl}, 

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@{thm derelict}, @{thm weaken}, @{thm promote1}, @{thm promote2}, 

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@{thm context1}, @{thm context4a}, @{thm context4b}]; 

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fun prom_tac n = 
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REPEAT (resolve_tac [@{thm promote0}, @{thm promote1}, @{thm promote2}] n) 

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*} 
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method_setup best_lazy = 

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{* Scan.succeed (K (SIMPLE_METHOD' (best_tac lazy_cs))) *} 
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"lazy classical reasoning" 
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lemma aux_impl: "$F, $G  A ==> $F, !(A o B), $G  B" 

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apply (rule derelict) 

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apply (rule impl) 

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apply (rule_tac [2] identity) 

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apply (rule context1) 

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apply assumption 

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done 

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lemma conj_lemma: " $F, !A, !B, $G  C ==> $F, !(A && B), $G  C" 

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apply (rule contract) 

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apply (rule_tac A = " (!A) >< (!B) " in cut) 

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apply (rule_tac [2] tensr) 

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prefer 3 

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apply (subgoal_tac "! (A && B)  !A") 

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apply assumption 

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apply best_lazy 

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prefer 3 

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apply (subgoal_tac "! (A && B)  !B") 

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apply assumption 

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apply best_lazy 

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apply (rule_tac [2] context1) 

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apply (rule_tac [2] tensl) 

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prefer 2 apply (assumption) 

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apply (rule context3) 

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apply (rule context3) 

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apply (rule context1) 

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done 

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lemma impr_contract: "!A, !A, $G  B ==> $G  (!A) o B" 

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apply (rule impr) 

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apply (rule contract) 

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apply assumption 

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done 

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lemma impr_contr_der: "A, !A, $G  B ==> $G  (!A) o B" 

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apply (rule impr) 

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apply (rule contract) 

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apply (rule derelict) 

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apply assumption 

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done 

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lemma contrad1: "$F, (!B) o 0, $G, !B, $H  A" 

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apply (rule impl) 

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apply (rule_tac [3] identity) 

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apply (rule context3) 

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apply (rule context1) 

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apply (rule zerol) 

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done 

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lemma contrad2: "$F, !B, $G, (!B) o 0, $H  A" 

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apply (rule impl) 

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apply (rule_tac [3] identity) 

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apply (rule context3) 

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apply (rule context1) 

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apply (rule zerol) 

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done 

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lemma ll_mp: "A o B, A  B" 

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apply (rule impl) 

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apply (rule_tac [2] identity) 

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apply (rule_tac [2] identity) 

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apply (rule context1) 

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done 

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lemma mp_rule1: "$F, B, $G, $H  C ==> $F, A, $G, A o B, $H  C" 

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apply (rule_tac A = "B" in cut) 

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apply (rule_tac [2] ll_mp) 

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prefer 2 apply (assumption) 

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apply (rule context3) 

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apply (rule context3) 

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apply (rule context1) 

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done 

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lemma mp_rule2: "$F, B, $G, $H  C ==> $F, A o B, $G, A, $H  C" 

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apply (rule_tac A = "B" in cut) 

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apply (rule_tac [2] ll_mp) 

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prefer 2 apply (assumption) 

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apply (rule context3) 

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apply (rule context3) 

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apply (rule context1) 

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done 

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lemma or_to_and: "!((!(A ++ B)) o 0)  !( ((!A) o 0) && ((!B) o 0))" 

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by best_lazy 

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lemma o_a_rule: "$F, !( ((!A) o 0) && ((!B) o 0)), $G  C ==> 

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$F, !((!(A ++ B)) o 0), $G  C" 

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apply (rule cut) 

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apply (rule_tac [2] or_to_and) 

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prefer 2 apply (assumption) 

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apply (rule context3) 

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apply (rule context1) 

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done 

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lemma conj_imp: "((!A) o C) ++ ((!B) o C)  (!(A && B)) o C" 

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apply (rule impr) 

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apply (rule conj_lemma) 

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apply (rule disjl) 

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apply (rule mp_rule1, best_lazy)+ 

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done 

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lemma not_imp: "!A, !((!B) o 0)  (!((!A) o B)) o 0" 

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by best_lazy 

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lemma a_not_a: "!A o (!A o 0)  !A o 0" 

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apply (rule impr) 

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apply (rule contract) 

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apply (rule impl) 

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apply (rule_tac [3] identity) 

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apply (rule context1) 

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apply best_lazy 

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done 

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lemma a_not_a_rule: "$J1, !A o 0, $J2  B ==> $J1, !A o (!A o 0), $J2  B" 

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apply (rule_tac A = "!A o 0" in cut) 

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apply (rule_tac [2] a_not_a) 

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prefer 2 apply (assumption) 

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apply best_lazy 

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done 

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ML {* 

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val safe_cs = lazy_cs add_safes [@{thm conj_lemma}, @{thm ll_mp}, @{thm contrad1}, 
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@{thm contrad2}, @{thm mp_rule1}, @{thm mp_rule2}, @{thm o_a_rule}, 

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@{thm a_not_a_rule}] 

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add_unsafes [@{thm aux_impl}]; 

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val power_cs = safe_cs add_unsafes [@{thm impr_contr_der}]; 
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*} 
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method_setup best_safe = 

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{* Scan.succeed (K (SIMPLE_METHOD' (best_tac safe_cs))) *} 
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method_setup best_power = 

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{* Scan.succeed (K (SIMPLE_METHOD' (best_tac power_cs))) *} 
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(* Some examples from Troelstra and van Dalen *) 

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lemma "!((!A) o ((!B) o 0))  (!(A && B)) o 0" 

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by best_safe 

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lemma "!((!(A && B)) o 0)  !((!A) o ((!B) o 0))" 

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by best_safe 

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lemma "!( (!((! ((!A) o B) ) o 0)) o 0)  

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(!A) o ( (! ((!B) o 0)) o 0)" 

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by best_safe 

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lemma "!( (!A) o ( (! ((!B) o 0)) o 0) )  

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(!((! ((!A) o B) ) o 0)) o 0" 

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by best_power 

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end 