20 appNil: "app(Nil,ys,ys)" |
20 appNil: "app(Nil,ys,ys)" |
21 appCons: "app(xs,ys,zs) ==> app(x:xs, ys, x:zs)" |
21 appCons: "app(xs,ys,zs) ==> app(x:xs, ys, x:zs)" |
22 revNil: "rev(Nil,Nil)" |
22 revNil: "rev(Nil,Nil)" |
23 revCons: "[| rev(xs,ys); app(ys, x:Nil, zs) |] ==> rev(x:xs, zs)" |
23 revCons: "[| rev(xs,ys); app(ys, x:Nil, zs) |] ==> rev(x:xs, zs)" |
24 |
24 |
25 lemma "app(a:b:c:Nil, d:e:Nil, ?x)" |
25 schematic_lemma "app(a:b:c:Nil, d:e:Nil, ?x)" |
26 apply (rule appNil appCons) |
26 apply (rule appNil appCons) |
27 apply (rule appNil appCons) |
27 apply (rule appNil appCons) |
28 apply (rule appNil appCons) |
28 apply (rule appNil appCons) |
29 apply (rule appNil appCons) |
29 apply (rule appNil appCons) |
30 done |
30 done |
31 |
31 |
32 lemma "app(?x, c:d:Nil, a:b:c:d:Nil)" |
32 schematic_lemma "app(?x, c:d:Nil, a:b:c:d:Nil)" |
33 apply (rule appNil appCons)+ |
33 apply (rule appNil appCons)+ |
34 done |
34 done |
35 |
35 |
36 lemma "app(?x, ?y, a:b:c:d:Nil)" |
36 schematic_lemma "app(?x, ?y, a:b:c:d:Nil)" |
37 apply (rule appNil appCons)+ |
37 apply (rule appNil appCons)+ |
38 back |
38 back |
39 back |
39 back |
40 back |
40 back |
41 back |
41 back |
44 (*app([x1,...,xn], y, ?z) requires (n+1) inferences*) |
44 (*app([x1,...,xn], y, ?z) requires (n+1) inferences*) |
45 (*rev([x1,...,xn], ?y) requires (n+1)(n+2)/2 inferences*) |
45 (*rev([x1,...,xn], ?y) requires (n+1)(n+2)/2 inferences*) |
46 |
46 |
47 lemmas rules = appNil appCons revNil revCons |
47 lemmas rules = appNil appCons revNil revCons |
48 |
48 |
49 lemma "rev(a:b:c:d:Nil, ?x)" |
49 schematic_lemma "rev(a:b:c:d:Nil, ?x)" |
50 apply (rule rules)+ |
50 apply (rule rules)+ |
51 done |
51 done |
52 |
52 |
53 lemma "rev(a:b:c:d:e:f:g:h:i:j:k:l:m:n:Nil, ?w)" |
53 schematic_lemma "rev(a:b:c:d:e:f:g:h:i:j:k:l:m:n:Nil, ?w)" |
54 apply (rule rules)+ |
54 apply (rule rules)+ |
55 done |
55 done |
56 |
56 |
57 lemma "rev(?x, a:b:c:Nil)" |
57 schematic_lemma "rev(?x, a:b:c:Nil)" |
58 apply (rule rules)+ -- {* does not solve it directly! *} |
58 apply (rule rules)+ -- {* does not solve it directly! *} |
59 back |
59 back |
60 back |
60 back |
61 done |
61 done |
62 |
62 |
63 (*backtracking version*) |
63 (*backtracking version*) |
64 ML {* |
64 ML {* |
65 val prolog_tac = DEPTH_FIRST (has_fewer_prems 1) (resolve_tac (@{thms rules}) 1) |
65 val prolog_tac = DEPTH_FIRST (has_fewer_prems 1) (resolve_tac (@{thms rules}) 1) |
66 *} |
66 *} |
67 |
67 |
68 lemma "rev(?x, a:b:c:Nil)" |
68 schematic_lemma "rev(?x, a:b:c:Nil)" |
69 apply (tactic prolog_tac) |
69 apply (tactic prolog_tac) |
70 done |
70 done |
71 |
71 |
72 lemma "rev(a:?x:c:?y:Nil, d:?z:b:?u)" |
72 schematic_lemma "rev(a:?x:c:?y:Nil, d:?z:b:?u)" |
73 apply (tactic prolog_tac) |
73 apply (tactic prolog_tac) |
74 done |
74 done |
75 |
75 |
76 (*rev([a..p], ?w) requires 153 inferences *) |
76 (*rev([a..p], ?w) requires 153 inferences *) |
77 lemma "rev(a:b:c:d:e:f:g:h:i:j:k:l:m:n:o:p:Nil, ?w)" |
77 schematic_lemma "rev(a:b:c:d:e:f:g:h:i:j:k:l:m:n:o:p:Nil, ?w)" |
78 apply (tactic {* DEPTH_SOLVE (resolve_tac ([@{thm refl}, @{thm conjI}] @ @{thms rules}) 1) *}) |
78 apply (tactic {* DEPTH_SOLVE (resolve_tac ([@{thm refl}, @{thm conjI}] @ @{thms rules}) 1) *}) |
79 done |
79 done |
80 |
80 |
81 (*?x has 16, ?y has 32; rev(?y,?w) requires 561 (rather large) inferences |
81 (*?x has 16, ?y has 32; rev(?y,?w) requires 561 (rather large) inferences |
82 total inferences = 2 + 1 + 17 + 561 = 581*) |
82 total inferences = 2 + 1 + 17 + 561 = 581*) |
83 lemma "a:b:c:d:e:f:g:h:i:j:k:l:m:n:o:p:Nil = ?x & app(?x,?x,?y) & rev(?y,?w)" |
83 schematic_lemma "a:b:c:d:e:f:g:h:i:j:k:l:m:n:o:p:Nil = ?x & app(?x,?x,?y) & rev(?y,?w)" |
84 apply (tactic {* DEPTH_SOLVE (resolve_tac ([@{thm refl}, @{thm conjI}] @ @{thms rules}) 1) *}) |
84 apply (tactic {* DEPTH_SOLVE (resolve_tac ([@{thm refl}, @{thm conjI}] @ @{thms rules}) 1) *}) |
85 done |
85 done |
86 |
86 |
87 end |
87 end |