author | blanchet |
Fri, 01 Aug 2014 19:44:18 +0200 | |
changeset 57762 | 1649841f3b38 |
parent 57747 | 816f96fff418 |
permissions | -rw-r--r-- |
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(* Title: HOL/Tools/SMT2/z3_new_proof.ML |
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Author: Sascha Boehme, TU Muenchen |
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Z3 proofs: parsing and abstract syntax tree. |
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*) |
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signature Z3_NEW_PROOF = |
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sig |
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(*proof rules*) |
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datatype z3_rule = |
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True_Axiom | Asserted | Goal | Modus_Ponens | Reflexivity | Symmetry | Transitivity | |
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Transitivity_Star | Monotonicity | Quant_Intro | Distributivity | And_Elim | Not_Or_Elim | |
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Rewrite | Rewrite_Star | Pull_Quant | Pull_Quant_Star | Push_Quant | Elim_Unused_Vars | |
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Dest_Eq_Res | Quant_Inst | Hypothesis | Lemma | Unit_Resolution | Iff_True | Iff_False | |
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Commutativity | Def_Axiom | Intro_Def | Apply_Def | Iff_Oeq | Nnf_Pos | Nnf_Neg | Nnf_Star | |
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Cnf_Star | Skolemize | Modus_Ponens_Oeq | Th_Lemma of string |
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val is_assumption: z3_rule -> bool |
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val string_of_rule: z3_rule -> string |
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(*proofs*) |
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datatype z3_step = Z3_Step of { |
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id: int, |
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rule: z3_rule, |
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prems: int list, |
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concl: term, |
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fixes: string list, |
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is_fix_step: bool} |
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(*proof parser*) |
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val parse: typ Symtab.table -> term Symtab.table -> string list -> |
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Proof.context -> z3_step list * Proof.context |
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end; |
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structure Z3_New_Proof: Z3_NEW_PROOF = |
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struct |
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open SMTLIB2_Proof |
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(* proof rules *) |
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datatype z3_rule = |
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True_Axiom | Asserted | Goal | Modus_Ponens | Reflexivity | Symmetry | Transitivity | |
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Transitivity_Star | Monotonicity | Quant_Intro | Distributivity | And_Elim | Not_Or_Elim | |
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Rewrite | Rewrite_Star | Pull_Quant | Pull_Quant_Star | Push_Quant | Elim_Unused_Vars | |
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Dest_Eq_Res | Quant_Inst | Hypothesis | Lemma | Unit_Resolution | Iff_True | Iff_False | |
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Commutativity | Def_Axiom | Intro_Def | Apply_Def | Iff_Oeq | Nnf_Pos | Nnf_Neg | Nnf_Star | |
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Cnf_Star | Skolemize | Modus_Ponens_Oeq | Th_Lemma of string |
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(* some proof rules include further information that is currently dropped by the parser *) |
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val rule_names = Symtab.make [ |
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("true-axiom", True_Axiom), |
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("asserted", Asserted), |
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("goal", Goal), |
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("mp", Modus_Ponens), |
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("refl", Reflexivity), |
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("symm", Symmetry), |
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("trans", Transitivity), |
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("trans*", Transitivity_Star), |
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("monotonicity", Monotonicity), |
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("quant-intro", Quant_Intro), |
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("distributivity", Distributivity), |
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("and-elim", And_Elim), |
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("not-or-elim", Not_Or_Elim), |
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("rewrite", Rewrite), |
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("rewrite*", Rewrite_Star), |
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("pull-quant", Pull_Quant), |
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("pull-quant*", Pull_Quant_Star), |
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("push-quant", Push_Quant), |
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("elim-unused", Elim_Unused_Vars), |
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("der", Dest_Eq_Res), |
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("quant-inst", Quant_Inst), |
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("hypothesis", Hypothesis), |
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("lemma", Lemma), |
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("unit-resolution", Unit_Resolution), |
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("iff-true", Iff_True), |
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("iff-false", Iff_False), |
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("commutativity", Commutativity), |
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("def-axiom", Def_Axiom), |
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("intro-def", Intro_Def), |
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("apply-def", Apply_Def), |
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("iff~", Iff_Oeq), |
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("nnf-pos", Nnf_Pos), |
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("nnf-neg", Nnf_Neg), |
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("nnf*", Nnf_Star), |
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("cnf*", Cnf_Star), |
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("sk", Skolemize), |
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("mp~", Modus_Ponens_Oeq)] |
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fun is_assumption Asserted = true |
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| is_assumption Goal = true |
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| is_assumption Hypothesis = true |
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| is_assumption Intro_Def = true |
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| is_assumption Skolemize = true |
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| is_assumption _ = false |
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fun rule_of_string name = |
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(case Symtab.lookup rule_names name of |
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SOME rule => rule |
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| NONE => error ("unknown Z3 proof rule " ^ quote name)) |
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fun string_of_rule (Th_Lemma kind) = "th-lemma" ^ (if kind = "" then "" else " " ^ kind) |
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| string_of_rule r = |
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let fun eq_rule (s, r') = if r = r' then SOME s else NONE |
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in the (Symtab.get_first eq_rule rule_names) end |
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(* proofs *) |
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datatype z3_node = Z3_Node of { |
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id: int, |
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rule: z3_rule, |
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prems: z3_node list, |
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concl: term, |
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bounds: string list} |
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fun mk_node id rule prems concl bounds = |
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Z3_Node {id = id, rule = rule, prems = prems, concl = concl, bounds = bounds} |
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datatype z3_step = Z3_Step of { |
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id: int, |
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rule: z3_rule, |
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prems: int list, |
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concl: term, |
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fixes: string list, |
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is_fix_step: bool} |
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fun mk_step id rule prems concl fixes is_fix_step = |
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Z3_Step {id = id, rule = rule, prems = prems, concl = concl, fixes = fixes, |
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is_fix_step = is_fix_step} |
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(* proof parser *) |
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fun rule_of (SMTLIB2.Sym name) = rule_of_string name |
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| rule_of (SMTLIB2.S (SMTLIB2.Sym "_" :: SMTLIB2.Sym name :: args)) = |
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(case (name, args) of |
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("th-lemma", SMTLIB2.Sym kind :: _) => Th_Lemma kind |
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| _ => rule_of_string name) |
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| rule_of r = raise SMTLIB2_PARSE ("bad Z3 proof rule format", r) |
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142 |
|
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fun node_of p cx = |
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(case p of |
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SMTLIB2.Sym name => |
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(case lookup_binding cx name of |
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Proof node => (node, cx) |
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| Tree p' => |
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cx |
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|> node_of p' |
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|-> (fn node => pair node o update_binding (name, Proof node)) |
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| _ => raise SMTLIB2_PARSE ("bad Z3 proof format", p)) |
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| SMTLIB2.S [SMTLIB2.Sym "let", SMTLIB2.S bindings, p] => |
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with_bindings (map dest_binding bindings) (node_of p) cx |
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| SMTLIB2.S (name :: parts) => |
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let |
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val (ps, p) = split_last parts |
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val r = rule_of name |
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159 |
in |
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cx |
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|> fold_map node_of ps |
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||>> `(with_fresh_names (term_of p)) |
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||>> next_id |
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|>> (fn ((prems, (t, ns)), id) => mk_node id r prems t ns) |
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165 |
end |
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| _ => raise SMTLIB2_PARSE ("bad Z3 proof format", p)) |
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167 |
|
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fun dest_name (SMTLIB2.Sym name) = name |
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| dest_name t = raise SMTLIB2_PARSE ("bad name", t) |
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|
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fun dest_seq (SMTLIB2.S ts) = ts |
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| dest_seq t = raise SMTLIB2_PARSE ("bad Z3 proof format", t) |
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173 |
|
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fun parse' (SMTLIB2.S (SMTLIB2.Sym "set-logic" :: _) :: ts) cx = parse' ts cx |
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| parse' (SMTLIB2.S [SMTLIB2.Sym "declare-fun", n, tys, ty] :: ts) cx = |
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let |
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val name = dest_name n |
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val Ts = map (type_of cx) (dest_seq tys) |
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val T = type_of cx ty |
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in parse' ts (declare_fun name (Ts ---> T) cx) end |
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| parse' (SMTLIB2.S [SMTLIB2.Sym "proof", p] :: _) cx = node_of p cx |
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| parse' ts _ = raise SMTLIB2_PARSE ("bad Z3 proof declarations", SMTLIB2.S ts) |
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183 |
|
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fun parse_proof typs funs lines ctxt = |
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185 |
let |
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val ts = dest_seq (SMTLIB2.parse lines) |
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val (node, cx) = parse' ts (empty_context ctxt typs funs) |
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188 |
in (node, ctxt_of cx) end |
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handle SMTLIB2.PARSE (l, msg) => error ("parsing error at line " ^ string_of_int l ^ ": " ^ msg) |
190 |
| SMTLIB2_PARSE (msg, t) => error (msg ^ ": " ^ SMTLIB2.str_of t) |
|
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191 |
|
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192 |
|
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(* handling of bound variables *) |
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194 |
|
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195 |
fun subst_of tyenv = |
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let fun add (ix, (S, T)) = cons (TVar (ix, S), T) |
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in Vartab.fold add tyenv [] end |
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198 |
|
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199 |
fun substTs_same subst = |
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let val applyT = Same.function (AList.lookup (op =) subst) |
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in Term_Subst.map_atypsT_same applyT end |
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202 |
|
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203 |
fun subst_types ctxt env bounds t = |
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204 |
let |
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val match = Sign.typ_match (Proof_Context.theory_of ctxt) |
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206 |
|
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207 |
val t' = singleton (Variable.polymorphic ctxt) t |
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val patTs = map snd (Term.strip_qnt_vars @{const_name Pure.all} t') |
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val objTs = map (the o Symtab.lookup env) bounds |
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val subst = subst_of (fold match (patTs ~~ objTs) Vartab.empty) |
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211 |
in Same.commit (Term_Subst.map_types_same (substTs_same subst)) t' end |
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212 |
|
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213 |
fun eq_quant (@{const_name HOL.All}, _) (@{const_name HOL.All}, _) = true |
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| eq_quant (@{const_name HOL.Ex}, _) (@{const_name HOL.Ex}, _) = true |
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215 |
| eq_quant _ _ = false |
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216 |
|
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217 |
fun opp_quant (@{const_name HOL.All}, _) (@{const_name HOL.Ex}, _) = true |
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| opp_quant (@{const_name HOL.Ex}, _) (@{const_name HOL.All}, _) = true |
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| opp_quant _ _ = false |
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220 |
|
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221 |
fun with_quant pred i (Const q1 $ Abs (_, T1, t1), Const q2 $ Abs (_, T2, t2)) = |
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if pred q1 q2 andalso T1 = T2 then |
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223 |
let val t = Var (("", i), T1) |
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in SOME (pairself Term.subst_bound ((t, t1), (t, t2))) end |
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225 |
else NONE |
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226 |
| with_quant _ _ _ = NONE |
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227 |
|
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228 |
fun dest_quant_pair i (@{term HOL.Not} $ t1, t2) = |
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229 |
Option.map (apfst HOLogic.mk_not) (with_quant opp_quant i (t1, t2)) |
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230 |
| dest_quant_pair i (t1, t2) = with_quant eq_quant i (t1, t2) |
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231 |
|
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232 |
fun dest_quant i t = |
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(case dest_quant_pair i (HOLogic.dest_eq (HOLogic.dest_Trueprop t)) of |
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234 |
SOME (t1, t2) => HOLogic.mk_Trueprop (HOLogic.mk_eq (t1, t2)) |
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235 |
| NONE => raise TERM ("lift_quant", [t])) |
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236 |
|
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237 |
fun match_types ctxt pat obj = |
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238 |
(Vartab.empty, Vartab.empty) |
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239 |
|> Pattern.first_order_match (Proof_Context.theory_of ctxt) (pat, obj) |
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240 |
|
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241 |
fun strip_match ctxt pat i obj = |
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242 |
(case try (match_types ctxt pat) obj of |
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243 |
SOME (tyenv, _) => subst_of tyenv |
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244 |
| NONE => strip_match ctxt pat (i + 1) (dest_quant i obj)) |
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245 |
|
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fun dest_all i (Const (@{const_name Pure.all}, _) $ (a as Abs (_, T, _))) = |
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247 |
dest_all (i + 1) (Term.betapply (a, Var (("", i), T))) |
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248 |
| dest_all i t = (i, t) |
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249 |
|
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250 |
fun dest_alls t = dest_all (Term.maxidx_of_term t + 1) t |
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251 |
|
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fun match_rule ctxt env (Z3_Node {bounds = bs', concl = t', ...}) bs t = |
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let |
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val t'' = singleton (Variable.polymorphic ctxt) t' |
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val (i, obj) = dest_alls (subst_types ctxt env bs t) |
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in |
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(case try (strip_match ctxt (snd (dest_alls t'')) i) obj of |
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NONE => NONE |
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| SOME subst => |
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let |
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val applyT = Same.commit (substTs_same subst) |
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val patTs = map snd (Term.strip_qnt_vars @{const_name Pure.all} t'') |
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in SOME (Symtab.make (bs' ~~ map applyT patTs)) end) |
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end |
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|
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|
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(* linearizing proofs and resolving types of bound variables *) |
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|
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fun has_step (tab, _) = Inttab.defined tab |
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|
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fun add_step id rule bounds concl is_fix_step ids (tab, sts) = |
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let val step = mk_step id rule ids concl bounds is_fix_step |
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in (id, (Inttab.update (id, ()) tab, step :: sts)) end |
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|
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fun is_fix_rule rule prems = |
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member (op =) [Quant_Intro, Nnf_Pos, Nnf_Neg] rule andalso length prems = 1 |
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|
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fun lin_proof ctxt env (Z3_Node {id, rule, prems, concl, bounds}) steps = |
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if has_step steps id then (id, steps) |
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else |
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let |
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val t = subst_types ctxt env bounds concl |
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val add = add_step id rule bounds t |
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fun rec_apply e b = fold_map (lin_proof ctxt e) prems #-> add b |
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in |
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if is_fix_rule rule prems then |
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(case match_rule ctxt env (hd prems) bounds t of |
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NONE => rec_apply env false steps |
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| SOME env' => rec_apply env' true steps) |
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else rec_apply env false steps |
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end |
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|
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fun linearize ctxt node = |
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rev (snd (snd (lin_proof ctxt Symtab.empty node (Inttab.empty, [])))) |
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|
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|
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(* overall proof parser *) |
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|
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fun parse typs funs lines ctxt = |
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let val (node, ctxt') = parse_proof typs funs lines ctxt |
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in (linearize ctxt' node, ctxt') end |
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|
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end; |