1 (* Title: LK/LK0 |
1 (* Title: LK/LK0.thy |
2 ID: $Id$ |
2 ID: $Id$ |
3 Author: Lawrence C Paulson, Cambridge University Computer Laboratory |
3 Author: Lawrence C Paulson, Cambridge University Computer Laboratory |
4 Copyright 1993 University of Cambridge |
4 Copyright 1993 University of Cambridge |
5 |
5 |
6 Classical First-Order Sequent Calculus |
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7 |
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8 There may be printing problems if a seqent is in expanded normal form |
6 There may be printing problems if a seqent is in expanded normal form |
9 (eta-expanded, beta-contracted) |
7 (eta-expanded, beta-contracted) |
10 *) |
8 *) |
11 |
9 |
12 LK0 = Sequents + |
10 header {* Classical First-Order Sequent Calculus *} |
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11 |
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12 theory LK0 |
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13 imports Sequents |
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14 begin |
13 |
15 |
14 global |
16 global |
15 |
17 |
16 classes term |
18 classes "term" |
17 default term |
19 defaultsort "term" |
18 |
20 |
19 consts |
21 consts |
20 |
22 |
21 Trueprop :: "two_seqi" |
23 Trueprop :: "two_seqi" |
22 |
24 |
23 True,False :: o |
25 True :: o |
24 "=" :: ['a,'a] => o (infixl 50) |
26 False :: o |
25 Not :: o => o ("~ _" [40] 40) |
27 "=" :: "['a,'a] => o" (infixl 50) |
26 "&" :: [o,o] => o (infixr 35) |
28 Not :: "o => o" ("~ _" [40] 40) |
27 "|" :: [o,o] => o (infixr 30) |
29 "&" :: "[o,o] => o" (infixr 35) |
28 "-->","<->" :: [o,o] => o (infixr 25) |
30 "|" :: "[o,o] => o" (infixr 30) |
29 The :: ('a => o) => 'a (binder "THE " 10) |
31 "-->" :: "[o,o] => o" (infixr 25) |
30 All :: ('a => o) => o (binder "ALL " 10) |
32 "<->" :: "[o,o] => o" (infixr 25) |
31 Ex :: ('a => o) => o (binder "EX " 10) |
33 The :: "('a => o) => 'a" (binder "THE " 10) |
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34 All :: "('a => o) => o" (binder "ALL " 10) |
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35 Ex :: "('a => o) => o" (binder "EX " 10) |
32 |
36 |
33 syntax |
37 syntax |
34 "@Trueprop" :: "two_seqe" ("((_)/ |- (_))" [6,6] 5) |
38 "@Trueprop" :: "two_seqe" ("((_)/ |- (_))" [6,6] 5) |
35 "_not_equal" :: ['a, 'a] => o (infixl "~=" 50) |
39 "_not_equal" :: "['a, 'a] => o" (infixl "~=" 50) |
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40 |
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41 parse_translation {* [("@Trueprop", two_seq_tr "Trueprop")] *} |
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42 print_translation {* [("Trueprop", two_seq_tr' "@Trueprop")] *} |
36 |
43 |
37 translations |
44 translations |
38 "x ~= y" == "~ (x = y)" |
45 "x ~= y" == "~ (x = y)" |
39 |
46 |
40 syntax (xsymbols) |
47 syntax (xsymbols) |
41 Not :: o => o ("\\<not> _" [40] 40) |
48 Not :: "o => o" ("\<not> _" [40] 40) |
42 "op &" :: [o, o] => o (infixr "\\<and>" 35) |
49 "op &" :: "[o, o] => o" (infixr "\<and>" 35) |
43 "op |" :: [o, o] => o (infixr "\\<or>" 30) |
50 "op |" :: "[o, o] => o" (infixr "\<or>" 30) |
44 "op -->" :: [o, o] => o (infixr "\\<longrightarrow>" 25) |
51 "op -->" :: "[o, o] => o" (infixr "\<longrightarrow>" 25) |
45 "op <->" :: [o, o] => o (infixr "\\<longleftrightarrow>" 25) |
52 "op <->" :: "[o, o] => o" (infixr "\<longleftrightarrow>" 25) |
46 "ALL " :: [idts, o] => o ("(3\\<forall>_./ _)" [0, 10] 10) |
53 "ALL " :: "[idts, o] => o" ("(3\<forall>_./ _)" [0, 10] 10) |
47 "EX " :: [idts, o] => o ("(3\\<exists>_./ _)" [0, 10] 10) |
54 "EX " :: "[idts, o] => o" ("(3\<exists>_./ _)" [0, 10] 10) |
48 "EX! " :: [idts, o] => o ("(3\\<exists>!_./ _)" [0, 10] 10) |
55 "EX! " :: "[idts, o] => o" ("(3\<exists>!_./ _)" [0, 10] 10) |
49 "_not_equal" :: ['a, 'a] => o (infixl "\\<noteq>" 50) |
56 "_not_equal" :: "['a, 'a] => o" (infixl "\<noteq>" 50) |
50 |
57 |
51 syntax (HTML output) |
58 syntax (HTML output) |
52 Not :: o => o ("\\<not> _" [40] 40) |
59 Not :: "o => o" ("\<not> _" [40] 40) |
53 "op &" :: [o, o] => o (infixr "\\<and>" 35) |
60 "op &" :: "[o, o] => o" (infixr "\<and>" 35) |
54 "op |" :: [o, o] => o (infixr "\\<or>" 30) |
61 "op |" :: "[o, o] => o" (infixr "\<or>" 30) |
55 "ALL " :: [idts, o] => o ("(3\\<forall>_./ _)" [0, 10] 10) |
62 "ALL " :: "[idts, o] => o" ("(3\<forall>_./ _)" [0, 10] 10) |
56 "EX " :: [idts, o] => o ("(3\\<exists>_./ _)" [0, 10] 10) |
63 "EX " :: "[idts, o] => o" ("(3\<exists>_./ _)" [0, 10] 10) |
57 "EX! " :: [idts, o] => o ("(3\\<exists>!_./ _)" [0, 10] 10) |
64 "EX! " :: "[idts, o] => o" ("(3\<exists>!_./ _)" [0, 10] 10) |
58 "_not_equal" :: ['a, 'a] => o (infixl "\\<noteq>" 50) |
65 "_not_equal" :: "['a, 'a] => o" (infixl "\<noteq>" 50) |
59 |
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60 |
66 |
61 local |
67 local |
62 |
68 |
63 rules |
69 axioms |
64 |
70 |
65 (*Structural rules: contraction, thinning, exchange [Soren Heilmann] *) |
71 (*Structural rules: contraction, thinning, exchange [Soren Heilmann] *) |
66 |
72 |
67 contRS "$H |- $E, $S, $S, $F ==> $H |- $E, $S, $F" |
73 contRS: "$H |- $E, $S, $S, $F ==> $H |- $E, $S, $F" |
68 contLS "$H, $S, $S, $G |- $E ==> $H, $S, $G |- $E" |
74 contLS: "$H, $S, $S, $G |- $E ==> $H, $S, $G |- $E" |
69 |
75 |
70 thinRS "$H |- $E, $F ==> $H |- $E, $S, $F" |
76 thinRS: "$H |- $E, $F ==> $H |- $E, $S, $F" |
71 thinLS "$H, $G |- $E ==> $H, $S, $G |- $E" |
77 thinLS: "$H, $G |- $E ==> $H, $S, $G |- $E" |
72 |
78 |
73 exchRS "$H |- $E, $R, $S, $F ==> $H |- $E, $S, $R, $F" |
79 exchRS: "$H |- $E, $R, $S, $F ==> $H |- $E, $S, $R, $F" |
74 exchLS "$H, $R, $S, $G |- $E ==> $H, $S, $R, $G |- $E" |
80 exchLS: "$H, $R, $S, $G |- $E ==> $H, $S, $R, $G |- $E" |
75 |
81 |
76 cut "[| $H |- $E, P; $H, P |- $E |] ==> $H |- $E" |
82 cut: "[| $H |- $E, P; $H, P |- $E |] ==> $H |- $E" |
77 |
83 |
78 (*Propositional rules*) |
84 (*Propositional rules*) |
79 |
85 |
80 basic "$H, P, $G |- $E, P, $F" |
86 basic: "$H, P, $G |- $E, P, $F" |
81 |
87 |
82 conjR "[| $H|- $E, P, $F; $H|- $E, Q, $F |] ==> $H|- $E, P&Q, $F" |
88 conjR: "[| $H|- $E, P, $F; $H|- $E, Q, $F |] ==> $H|- $E, P&Q, $F" |
83 conjL "$H, P, Q, $G |- $E ==> $H, P & Q, $G |- $E" |
89 conjL: "$H, P, Q, $G |- $E ==> $H, P & Q, $G |- $E" |
84 |
90 |
85 disjR "$H |- $E, P, Q, $F ==> $H |- $E, P|Q, $F" |
91 disjR: "$H |- $E, P, Q, $F ==> $H |- $E, P|Q, $F" |
86 disjL "[| $H, P, $G |- $E; $H, Q, $G |- $E |] ==> $H, P|Q, $G |- $E" |
92 disjL: "[| $H, P, $G |- $E; $H, Q, $G |- $E |] ==> $H, P|Q, $G |- $E" |
87 |
93 |
88 impR "$H, P |- $E, Q, $F ==> $H |- $E, P-->Q, $F" |
94 impR: "$H, P |- $E, Q, $F ==> $H |- $E, P-->Q, $F" |
89 impL "[| $H,$G |- $E,P; $H, Q, $G |- $E |] ==> $H, P-->Q, $G |- $E" |
95 impL: "[| $H,$G |- $E,P; $H, Q, $G |- $E |] ==> $H, P-->Q, $G |- $E" |
90 |
96 |
91 notR "$H, P |- $E, $F ==> $H |- $E, ~P, $F" |
97 notR: "$H, P |- $E, $F ==> $H |- $E, ~P, $F" |
92 notL "$H, $G |- $E, P ==> $H, ~P, $G |- $E" |
98 notL: "$H, $G |- $E, P ==> $H, ~P, $G |- $E" |
93 |
99 |
94 FalseL "$H, False, $G |- $E" |
100 FalseL: "$H, False, $G |- $E" |
95 |
101 |
96 True_def "True == False-->False" |
102 True_def: "True == False-->False" |
97 iff_def "P<->Q == (P-->Q) & (Q-->P)" |
103 iff_def: "P<->Q == (P-->Q) & (Q-->P)" |
98 |
104 |
99 (*Quantifiers*) |
105 (*Quantifiers*) |
100 |
106 |
101 allR "(!!x.$H |- $E, P(x), $F) ==> $H |- $E, ALL x. P(x), $F" |
107 allR: "(!!x.$H |- $E, P(x), $F) ==> $H |- $E, ALL x. P(x), $F" |
102 allL "$H, P(x), $G, ALL x. P(x) |- $E ==> $H, ALL x. P(x), $G |- $E" |
108 allL: "$H, P(x), $G, ALL x. P(x) |- $E ==> $H, ALL x. P(x), $G |- $E" |
103 |
109 |
104 exR "$H |- $E, P(x), $F, EX x. P(x) ==> $H |- $E, EX x. P(x), $F" |
110 exR: "$H |- $E, P(x), $F, EX x. P(x) ==> $H |- $E, EX x. P(x), $F" |
105 exL "(!!x.$H, P(x), $G |- $E) ==> $H, EX x. P(x), $G |- $E" |
111 exL: "(!!x.$H, P(x), $G |- $E) ==> $H, EX x. P(x), $G |- $E" |
106 |
112 |
107 (*Equality*) |
113 (*Equality*) |
108 |
114 |
109 refl "$H |- $E, a=a, $F" |
115 refl: "$H |- $E, a=a, $F" |
110 subst "$H(a), $G(a) |- $E(a) ==> $H(b), a=b, $G(b) |- $E(b)" |
116 subst: "$H(a), $G(a) |- $E(a) ==> $H(b), a=b, $G(b) |- $E(b)" |
111 |
117 |
112 (* Reflection *) |
118 (* Reflection *) |
113 |
119 |
114 eq_reflection "|- x=y ==> (x==y)" |
120 eq_reflection: "|- x=y ==> (x==y)" |
115 iff_reflection "|- P<->Q ==> (P==Q)" |
121 iff_reflection: "|- P<->Q ==> (P==Q)" |
116 |
122 |
117 (*Descriptions*) |
123 (*Descriptions*) |
118 |
124 |
119 The "[| $H |- $E, P(a), $F; !!x.$H, P(x) |- $E, x=a, $F |] ==> |
125 The: "[| $H |- $E, P(a), $F; !!x.$H, P(x) |- $E, x=a, $F |] ==> |
120 $H |- $E, P(THE x. P(x)), $F" |
126 $H |- $E, P(THE x. P(x)), $F" |
121 |
127 |
122 constdefs |
128 constdefs |
123 If :: [o, 'a, 'a] => 'a ("(if (_)/ then (_)/ else (_))" 10) |
129 If :: "[o, 'a, 'a] => 'a" ("(if (_)/ then (_)/ else (_))" 10) |
124 "If(P,x,y) == THE z::'a. (P --> z=x) & (~P --> z=y)" |
130 "If(P,x,y) == THE z::'a. (P --> z=x) & (~P --> z=y)" |
125 |
131 |
126 setup |
132 setup |
127 prover_setup |
133 prover_setup |
128 |
134 |
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135 ML {* use_legacy_bindings (the_context ()) *} |
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136 |
129 end |
137 end |
130 |
138 |
131 ML |
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132 |
139 |
133 |
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134 val parse_translation = [("@Trueprop",Sequents.two_seq_tr "Trueprop")]; |
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135 val print_translation = [("Trueprop",Sequents.two_seq_tr' "@Trueprop")]; |
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136 |
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