1593
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(* Title: Pure/deriv.ML
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ID: $Id$
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Author: Lawrence C Paulson, Cambridge University Computer Laboratory
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Copyright 1996 University of Cambridge
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Derivations (proof objects) and functions for examining them
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*)
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signature DERIV =
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sig
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(*Object-level rules*)
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datatype orule = Subgoal of cterm
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| Asm of int
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| Res of deriv
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| Equal of deriv
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| Thm of theory * string
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| Other of deriv;
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val size : deriv -> int
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val drop : 'a mtree * int -> 'a mtree
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val linear : deriv -> deriv list
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val tree : deriv -> orule mtree
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end;
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structure Deriv : DERIV =
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struct
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fun size (Join(Theorem _, _)) = 1
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| size (Join(_, ders)) = foldl op+ (1, map size ders);
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(*Conversion to linear format. Children of a node are the LIST of inferences
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justifying ONE of the premises*)
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fun rev_deriv (Join (rl, [])) = [Join(rl,[])]
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| rev_deriv (Join (Theorem arg, _)) = [Join(Theorem arg, [])]
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| rev_deriv (Join (Assumption arg, [der])) =
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Join(Assumption arg,[]) :: rev_deriv der
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| rev_deriv (Join (Bicompose arg, [rder, sder])) =
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Join (Bicompose arg, linear rder) :: rev_deriv sder
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| rev_deriv (Join (_, [der])) = rev_deriv der
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| rev_deriv (Join (rl, der::ders)) = (*catch-all case; doubtful?*)
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Join(rl, flat (map linear ders)) :: rev_deriv der
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and linear der = rev (rev_deriv der);
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(*** Conversion of object-level proof trees ***)
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(*Object-level rules*)
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datatype orule = Subgoal of cterm
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| Asm of int
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| Res of deriv
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| Equal of deriv
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| Thm of theory * string
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| Other of deriv;
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(*At position i, splice in value x, removing ngoal elements*)
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fun splice (i,x,ngoal,prfs) =
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let val prfs0 = take(i-1,prfs)
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and prfs1 = drop(i-1,prfs)
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val prfs2 = Join (x, take(ngoal, prfs1)) :: drop(ngoal, prfs1)
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in prfs0 @ prfs2 end;
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(*Deletes trivial uses of Equal_elim; hides derivations of Theorems*)
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fun simp_deriv (Join (Equal_elim, [Join (Rewrite_cterm _, []), der])) =
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simp_deriv der
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| simp_deriv (Join (Equal_elim, [Join (Reflexive _, []), der])) =
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simp_deriv der
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| simp_deriv (Join (rule as Theorem arg, [_])) = Join (rule, [])
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| simp_deriv (Join (rule, ders)) = Join (rule, map simp_deriv ders);
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(*Proof term is an equality: first premise of equal_elim.
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Attempt to decode proof terms made by Drule.goals_conv.
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Subgoal numbers are returned; they are wrong if original subgoal
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had flexflex pairs!
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NEGATIVE i means "could affect all subgoals starting from i"*)
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fun scan_equals (i, Join (Combination,
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[Join (Combination, [_, der1]), der2])) =
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(case der1 of (*ignore trivial cases*)
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Join (Reflexive _, _) => scan_equals (i+1, der2)
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| Join (Rewrite_cterm _, []) => scan_equals (i+1, der2)
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| Join (Rewrite_cterm _, _) => (i,der1) :: scan_equals (i+1, der2)
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| _ (*impossible in gconv*) => [])
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| scan_equals (i, Join (Reflexive _, [])) = []
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| scan_equals (i, Join (Rewrite_cterm _, [])) = []
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(*Anything else could affect ALL following goals*)
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| scan_equals (i, der) = [(~i,der)];
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(*Record uses of equality reasoning on 1 or more subgoals*)
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fun update_equals ((i,der), prfs) =
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if i>0 then splice (i, Equal (simp_deriv der), 1, prfs)
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else take (~i-1, prfs) @
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map (fn prf => Join (Equal (simp_deriv der), [prf]))
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(drop (~i-1, prfs));
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fun delift (Join (Lift_rule _, [der])) = der
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| delift der = der;
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(*Conversion to an object-level proof tree.
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Uses embedded Lift_rules to "annotate" the proof tree with subgoals;
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-- assumes that Lift_rule never occurs except with resolution
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-- may contain Vars that, in fact, are instantiated in that step*)
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fun tree_aux (Join (Trivial ct, []), prfs) = Join(Subgoal ct, prfs)
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| tree_aux (Join (Assumption(i,_), [der]), prfs) =
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tree_aux (der, splice (i, Asm i, 0, prfs))
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| tree_aux (Join (Equal_elim, [der1,der2]), prfs) =
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tree_aux (der2, foldr update_equals (scan_equals (1, der1), prfs))
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| tree_aux (Join (Bicompose (match,true,i,ngoal,env), ders), prfs) =
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(*change eresolve_tac to proof by assumption*)
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tree_aux (Join (Assumption(i, Some env),
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[Join (Bicompose (match,false,i,ngoal,env), ders)]),
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prfs)
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| tree_aux (Join (Lift_rule (ct,i), [der]), prfs) =
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tree_aux (der, splice (i, Subgoal ct, 1, prfs))
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| tree_aux (Join (Bicompose arg,
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[Join (Instantiate _, [rder]), sder]), prfs) =
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(*Ignore Instantiate*)
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tree_aux (Join (Bicompose arg, [rder, sder]), prfs)
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| tree_aux (Join (Bicompose arg,
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[Join (Lift_rule larg, [rder]), sder]), prfs) =
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(*Move Lift_rule: to make a Subgoal on the result*)
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tree_aux (Join (Bicompose arg, [rder,
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Join(Lift_rule larg, [sder])]), prfs)
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| tree_aux (Join (Bicompose (match,ef,i,ngoal,env),
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[Join (Bicompose (match',ef',i',ngoal',env'),
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[der1,der2]),
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der3]), prfs) =
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(*associate resolutions to the right*)
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tree_aux (Join (Bicompose (match', ef', i'+i-1, ngoal', env'),
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[delift der1, (*This Lift_rule would be wrong!*)
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Join (Bicompose (match, ef, i, ngoal-ngoal'+1, env),
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[der2, der3])]), prfs)
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| tree_aux (Join (Bicompose (arg as (_,_,i,ngoal,_)),
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[rder, sder]), prfs) =
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(*resolution with basic rule/assumption -- we hope!*)
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tree_aux (sder, splice (i, Res (simp_deriv rder), ngoal, prfs))
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| tree_aux (Join (Theorem arg, _), prfs) = Join(Thm arg, prfs)
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| tree_aux (Join (_, [der]), prfs) = tree_aux (der,prfs)
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| tree_aux (der, prfs) = Join(Other (simp_deriv der), prfs);
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fun tree der = tree_aux (der,[]);
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(*Currently declared at end, to avoid conflicting with library's drop
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Can put it after "size" once we switch to List.drop*)
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fun drop (der,0) = der
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| drop (Join (_, der::_), n) = drop (der, n-1);
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end;
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(*We do NOT open this structure*)
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