
1 (* Title: Pure/deriv.ML 

2 ID: $Id$ 

3 Author: Lawrence C Paulson, Cambridge University Computer Laboratory 

4 Copyright 1996 University of Cambridge 

5 

6 Derivations (proof objects) and functions for examining them 

7 *) 

8 

9 signature DERIV = 

10 sig 

11 (*Objectlevel rules*) 

12 datatype orule = Subgoal of cterm 

13  Asm of int 

14  Res of deriv 

15  Equal of deriv 

16  Thm of theory * string 

17  Other of deriv; 

18 

19 val size : deriv > int 

20 val drop : 'a mtree * int > 'a mtree 

21 val linear : deriv > deriv list 

22 val tree : deriv > orule mtree 

23 end; 

24 

25 structure Deriv : DERIV = 

26 struct 

27 

28 fun size (Join(Theorem _, _)) = 1 

29  size (Join(_, ders)) = foldl op+ (1, map size ders); 

30 

31 (*Conversion to linear format. Children of a node are the LIST of inferences 

32 justifying ONE of the premises*) 

33 fun rev_deriv (Join (rl, [])) = [Join(rl,[])] 

34  rev_deriv (Join (Theorem arg, _)) = [Join(Theorem arg, [])] 

35  rev_deriv (Join (Assumption arg, [der])) = 

36 Join(Assumption arg,[]) :: rev_deriv der 

37  rev_deriv (Join (Bicompose arg, [rder, sder])) = 

38 Join (Bicompose arg, linear rder) :: rev_deriv sder 

39  rev_deriv (Join (_, [der])) = rev_deriv der 

40  rev_deriv (Join (rl, der::ders)) = (*catchall case; doubtful?*) 

41 Join(rl, flat (map linear ders)) :: rev_deriv der 

42 and linear der = rev (rev_deriv der); 

43 

44 

45 (*** Conversion of objectlevel proof trees ***) 

46 

47 (*Objectlevel rules*) 

48 datatype orule = Subgoal of cterm 

49  Asm of int 

50  Res of deriv 

51  Equal of deriv 

52  Thm of theory * string 

53  Other of deriv; 

54 

55 (*At position i, splice in value x, removing ngoal elements*) 

56 fun splice (i,x,ngoal,prfs) = 

57 let val prfs0 = take(i1,prfs) 

58 and prfs1 = drop(i1,prfs) 

59 val prfs2 = Join (x, take(ngoal, prfs1)) :: drop(ngoal, prfs1) 

60 in prfs0 @ prfs2 end; 

61 

62 (*Deletes trivial uses of Equal_elim; hides derivations of Theorems*) 

63 fun simp_deriv (Join (Equal_elim, [Join (Rewrite_cterm _, []), der])) = 

64 simp_deriv der 

65  simp_deriv (Join (Equal_elim, [Join (Reflexive _, []), der])) = 

66 simp_deriv der 

67  simp_deriv (Join (rule as Theorem arg, [_])) = Join (rule, []) 

68  simp_deriv (Join (rule, ders)) = Join (rule, map simp_deriv ders); 

69 

70 (*Proof term is an equality: first premise of equal_elim. 

71 Attempt to decode proof terms made by Drule.goals_conv. 

72 Subgoal numbers are returned; they are wrong if original subgoal 

73 had flexflex pairs! 

74 NEGATIVE i means "could affect all subgoals starting from i"*) 

75 fun scan_equals (i, Join (Combination, 

76 [Join (Combination, [_, der1]), der2])) = 

77 (case der1 of (*ignore trivial cases*) 

78 Join (Reflexive _, _) => scan_equals (i+1, der2) 

79  Join (Rewrite_cterm _, []) => scan_equals (i+1, der2) 

80  Join (Rewrite_cterm _, _) => (i,der1) :: scan_equals (i+1, der2) 

81  _ (*impossible in gconv*) => []) 

82  scan_equals (i, Join (Reflexive _, [])) = [] 

83  scan_equals (i, Join (Rewrite_cterm _, [])) = [] 

84 (*Anything else could affect ALL following goals*) 

85  scan_equals (i, der) = [(~i,der)]; 

86 

87 (*Record uses of equality reasoning on 1 or more subgoals*) 

88 fun update_equals ((i,der), prfs) = 

89 if i>0 then splice (i, Equal (simp_deriv der), 1, prfs) 

90 else take (~i1, prfs) @ 

91 map (fn prf => Join (Equal (simp_deriv der), [prf])) 

92 (drop (~i1, prfs)); 

93 

94 fun delift (Join (Lift_rule _, [der])) = der 

95  delift der = der; 

96 

97 (*Conversion to an objectlevel proof tree. 

98 Uses embedded Lift_rules to "annotate" the proof tree with subgoals; 

99  assumes that Lift_rule never occurs except with resolution 

100  may contain Vars that, in fact, are instantiated in that step*) 

101 fun tree_aux (Join (Trivial ct, []), prfs) = Join(Subgoal ct, prfs) 

102  tree_aux (Join (Assumption(i,_), [der]), prfs) = 

103 tree_aux (der, splice (i, Asm i, 0, prfs)) 

104  tree_aux (Join (Equal_elim, [der1,der2]), prfs) = 

105 tree_aux (der2, foldr update_equals (scan_equals (1, der1), prfs)) 

106  tree_aux (Join (Bicompose (match,true,i,ngoal,env), ders), prfs) = 

107 (*change eresolve_tac to proof by assumption*) 

108 tree_aux (Join (Assumption(i, Some env), 

109 [Join (Bicompose (match,false,i,ngoal,env), ders)]), 

110 prfs) 

111  tree_aux (Join (Lift_rule (ct,i), [der]), prfs) = 

112 tree_aux (der, splice (i, Subgoal ct, 1, prfs)) 

113  tree_aux (Join (Bicompose arg, 

114 [Join (Instantiate _, [rder]), sder]), prfs) = 

115 (*Ignore Instantiate*) 

116 tree_aux (Join (Bicompose arg, [rder, sder]), prfs) 

117  tree_aux (Join (Bicompose arg, 

118 [Join (Lift_rule larg, [rder]), sder]), prfs) = 

119 (*Move Lift_rule: to make a Subgoal on the result*) 

120 tree_aux (Join (Bicompose arg, [rder, 

121 Join(Lift_rule larg, [sder])]), prfs) 

122  tree_aux (Join (Bicompose (match,ef,i,ngoal,env), 

123 [Join (Bicompose (match',ef',i',ngoal',env'), 

124 [der1,der2]), 

125 der3]), prfs) = 

126 (*associate resolutions to the right*) 

127 tree_aux (Join (Bicompose (match', ef', i'+i1, ngoal', env'), 

128 [delift der1, (*This Lift_rule would be wrong!*) 

129 Join (Bicompose (match, ef, i, ngoalngoal'+1, env), 

130 [der2, der3])]), prfs) 

131  tree_aux (Join (Bicompose (arg as (_,_,i,ngoal,_)), 

132 [rder, sder]), prfs) = 

133 (*resolution with basic rule/assumption  we hope!*) 

134 tree_aux (sder, splice (i, Res (simp_deriv rder), ngoal, prfs)) 

135  tree_aux (Join (Theorem arg, _), prfs) = Join(Thm arg, prfs) 

136  tree_aux (Join (_, [der]), prfs) = tree_aux (der,prfs) 

137  tree_aux (der, prfs) = Join(Other (simp_deriv der), prfs); 

138 

139 

140 fun tree der = tree_aux (der,[]); 

141 

142 (*Currently declared at end, to avoid conflicting with library's drop 

143 Can put it after "size" once we switch to List.drop*) 

144 fun drop (der,0) = der 

145  drop (Join (_, der::_), n) = drop (der, n1); 

146 

147 end; 

148 

149 

150 (*We do NOT open this structure*) 