renamed Thm.capply to Thm.apply, and Thm.cabs to Thm.lambda in conformance with similar operations in structure Term and Logic;
(* Title: HOL/Tools/SMT/z3_proof_literals.ML
Author: Sascha Boehme, TU Muenchen
Proof tools related to conjunctions and disjunctions.
*)
signature Z3_PROOF_LITERALS =
sig
(*literal table*)
type littab = thm Termtab.table
val make_littab: thm list -> littab
val insert_lit: thm -> littab -> littab
val delete_lit: thm -> littab -> littab
val lookup_lit: littab -> term -> thm option
val get_first_lit: (term -> bool) -> littab -> thm option
(*rules*)
val true_thm: thm
val rewrite_true: thm
(*properties*)
val is_conj: term -> bool
val is_disj: term -> bool
val exists_lit: bool -> (term -> bool) -> term -> bool
val negate: cterm -> cterm
(*proof tools*)
val explode: bool -> bool -> bool -> term list -> thm -> thm list
val join: bool -> littab -> term -> thm
val prove_conj_disj_eq: cterm -> thm
end
structure Z3_Proof_Literals: Z3_PROOF_LITERALS =
struct
(* literal table *)
type littab = thm Termtab.table
fun make_littab thms =
fold (Termtab.update o `SMT_Utils.prop_of) thms Termtab.empty
fun insert_lit thm = Termtab.update (`SMT_Utils.prop_of thm)
fun delete_lit thm = Termtab.delete (SMT_Utils.prop_of thm)
fun lookup_lit lits = Termtab.lookup lits
fun get_first_lit f =
Termtab.get_first (fn (t, thm) => if f t then SOME thm else NONE)
(* rules *)
val true_thm = @{lemma "~False" by simp}
val rewrite_true = @{lemma "True == ~ False" by simp}
(* properties and term operations *)
val is_neg = (fn @{const Not} $ _ => true | _ => false)
fun is_neg' f = (fn @{const Not} $ t => f t | _ => false)
val is_dneg = is_neg' is_neg
val is_conj = (fn @{const HOL.conj} $ _ $ _ => true | _ => false)
val is_disj = (fn @{const HOL.disj} $ _ $ _ => true | _ => false)
fun dest_disj_term' f = (fn
@{const Not} $ (@{const HOL.disj} $ t $ u) => SOME (f t, f u)
| _ => NONE)
val dest_conj_term = (fn @{const HOL.conj} $ t $ u => SOME (t, u) | _ => NONE)
val dest_disj_term =
dest_disj_term' (fn @{const Not} $ t => t | t => @{const Not} $ t)
fun exists_lit is_conj P =
let
val dest = if is_conj then dest_conj_term else dest_disj_term
fun exists t = P t orelse
(case dest t of
SOME (t1, t2) => exists t1 orelse exists t2
| NONE => false)
in exists end
val negate = Thm.apply (Thm.cterm_of @{theory} @{const Not})
(* proof tools *)
(** explosion of conjunctions and disjunctions **)
local
val precomp = Z3_Proof_Tools.precompose2
fun destc ct = Thm.dest_binop (Thm.dest_arg ct)
val dest_conj1 = precomp destc @{thm conjunct1}
val dest_conj2 = precomp destc @{thm conjunct2}
fun dest_conj_rules t =
dest_conj_term t |> Option.map (K (dest_conj1, dest_conj2))
fun destd f ct = f (Thm.dest_binop (Thm.dest_arg (Thm.dest_arg ct)))
val dn1 = apfst Thm.dest_arg and dn2 = apsnd Thm.dest_arg
val dest_disj1 = precomp (destd I) @{lemma "~(P | Q) ==> ~P" by fast}
val dest_disj2 = precomp (destd dn1) @{lemma "~(~P | Q) ==> P" by fast}
val dest_disj3 = precomp (destd I) @{lemma "~(P | Q) ==> ~Q" by fast}
val dest_disj4 = precomp (destd dn2) @{lemma "~(P | ~Q) ==> Q" by fast}
fun dest_disj_rules t =
(case dest_disj_term' is_neg t of
SOME (true, true) => SOME (dest_disj2, dest_disj4)
| SOME (true, false) => SOME (dest_disj2, dest_disj3)
| SOME (false, true) => SOME (dest_disj1, dest_disj4)
| SOME (false, false) => SOME (dest_disj1, dest_disj3)
| NONE => NONE)
fun destn ct = [Thm.dest_arg (Thm.dest_arg (Thm.dest_arg ct))]
val dneg_rule = Z3_Proof_Tools.precompose destn @{thm notnotD}
in
(*
explode a term into literals and collect all rules to be able to deduce
particular literals afterwards
*)
fun explode_term is_conj =
let
val dest = if is_conj then dest_conj_term else dest_disj_term
val dest_rules = if is_conj then dest_conj_rules else dest_disj_rules
fun add (t, rs) = Termtab.map_default (t, rs)
(fn rs' => if length rs' < length rs then rs' else rs)
fun explode1 rules t =
(case dest t of
SOME (t1, t2) =>
let val (rule1, rule2) = the (dest_rules t)
in
explode1 (rule1 :: rules) t1 #>
explode1 (rule2 :: rules) t2 #>
add (t, rev rules)
end
| NONE => add (t, rev rules))
fun explode0 (@{const Not} $ (@{const Not} $ t)) =
Termtab.make [(t, [dneg_rule])]
| explode0 t = explode1 [] t Termtab.empty
in explode0 end
(*
extract a literal by applying previously collected rules
*)
fun extract_lit thm rules = fold Z3_Proof_Tools.compose rules thm
(*
explode a theorem into its literals
*)
fun explode is_conj full keep_intermediate stop_lits =
let
val dest_rules = if is_conj then dest_conj_rules else dest_disj_rules
val tab = fold (Termtab.update o rpair ()) stop_lits Termtab.empty
fun explode1 thm =
if Termtab.defined tab (SMT_Utils.prop_of thm) then cons thm
else
(case dest_rules (SMT_Utils.prop_of thm) of
SOME (rule1, rule2) =>
explode2 rule1 thm #>
explode2 rule2 thm #>
keep_intermediate ? cons thm
| NONE => cons thm)
and explode2 dest_rule thm =
if full orelse
exists_lit is_conj (Termtab.defined tab) (SMT_Utils.prop_of thm)
then explode1 (Z3_Proof_Tools.compose dest_rule thm)
else cons (Z3_Proof_Tools.compose dest_rule thm)
fun explode0 thm =
if not is_conj andalso is_dneg (SMT_Utils.prop_of thm)
then [Z3_Proof_Tools.compose dneg_rule thm]
else explode1 thm []
in explode0 end
end
(** joining of literals to conjunctions or disjunctions **)
local
fun on_cprem i f thm = f (Thm.cprem_of thm i)
fun on_cprop f thm = f (Thm.cprop_of thm)
fun precomp2 f g thm = (on_cprem 1 f thm, on_cprem 2 g thm, f, g, thm)
fun comp2 (cv1, cv2, f, g, rule) thm1 thm2 =
Thm.instantiate ([], [(cv1, on_cprop f thm1), (cv2, on_cprop g thm2)]) rule
|> Z3_Proof_Tools.discharge thm1 |> Z3_Proof_Tools.discharge thm2
fun d1 ct = Thm.dest_arg ct and d2 ct = Thm.dest_arg (Thm.dest_arg ct)
val conj_rule = precomp2 d1 d1 @{thm conjI}
fun comp_conj ((_, thm1), (_, thm2)) = comp2 conj_rule thm1 thm2
val disj1 = precomp2 d2 d2 @{lemma "~P ==> ~Q ==> ~(P | Q)" by fast}
val disj2 = precomp2 d2 d1 @{lemma "~P ==> Q ==> ~(P | ~Q)" by fast}
val disj3 = precomp2 d1 d2 @{lemma "P ==> ~Q ==> ~(~P | Q)" by fast}
val disj4 = precomp2 d1 d1 @{lemma "P ==> Q ==> ~(~P | ~Q)" by fast}
fun comp_disj ((false, thm1), (false, thm2)) = comp2 disj1 thm1 thm2
| comp_disj ((false, thm1), (true, thm2)) = comp2 disj2 thm1 thm2
| comp_disj ((true, thm1), (false, thm2)) = comp2 disj3 thm1 thm2
| comp_disj ((true, thm1), (true, thm2)) = comp2 disj4 thm1 thm2
fun dest_conj (@{const HOL.conj} $ t $ u) = ((false, t), (false, u))
| dest_conj t = raise TERM ("dest_conj", [t])
val neg = (fn @{const Not} $ t => (true, t) | t => (false, @{const Not} $ t))
fun dest_disj (@{const Not} $ (@{const HOL.disj} $ t $ u)) = (neg t, neg u)
| dest_disj t = raise TERM ("dest_disj", [t])
val precomp = Z3_Proof_Tools.precompose
val dnegE = precomp (single o d2 o d1) @{thm notnotD}
val dnegI = precomp (single o d1) @{lemma "P ==> ~~P" by fast}
fun as_dneg f t = f (@{const Not} $ (@{const Not} $ t))
val precomp2 = Z3_Proof_Tools.precompose2
fun dni f = apsnd f o Thm.dest_binop o f o d1
val negIffE = precomp2 (dni d1) @{lemma "~(P = (~Q)) ==> Q = P" by fast}
val negIffI = precomp2 (dni I) @{lemma "P = Q ==> ~(Q = (~P))" by fast}
val iff_const = @{const HOL.eq (bool)}
fun as_negIff f (@{const HOL.eq (bool)} $ t $ u) =
f (@{const Not} $ (iff_const $ u $ (@{const Not} $ t)))
| as_negIff _ _ = NONE
in
fun join is_conj littab t =
let
val comp = if is_conj then comp_conj else comp_disj
val dest = if is_conj then dest_conj else dest_disj
val lookup = lookup_lit littab
fun lookup_rule t =
(case t of
@{const Not} $ (@{const Not} $ t) =>
(Z3_Proof_Tools.compose dnegI, lookup t)
| @{const Not} $ (@{const HOL.eq (bool)} $ t $ (@{const Not} $ u)) =>
(Z3_Proof_Tools.compose negIffI, lookup (iff_const $ u $ t))
| @{const Not} $ ((eq as Const (@{const_name HOL.eq}, _)) $ t $ u) =>
let fun rewr lit = lit COMP @{thm not_sym}
in (rewr, lookup (@{const Not} $ (eq $ u $ t))) end
| _ =>
(case as_dneg lookup t of
NONE => (Z3_Proof_Tools.compose negIffE, as_negIff lookup t)
| x => (Z3_Proof_Tools.compose dnegE, x)))
fun join1 (s, t) =
(case lookup t of
SOME lit => (s, lit)
| NONE =>
(case lookup_rule t of
(rewrite, SOME lit) => (s, rewrite lit)
| (_, NONE) => (s, comp (pairself join1 (dest t)))))
in snd (join1 (if is_conj then (false, t) else (true, t))) end
end
(** proving equality of conjunctions or disjunctions **)
fun iff_intro thm1 thm2 = thm2 COMP (thm1 COMP @{thm iffI})
local
val cp1 = @{lemma "(~P) = (~Q) ==> P = Q" by simp}
val cp2 = @{lemma "(~P) = Q ==> P = (~Q)" by fastforce}
val cp3 = @{lemma "P = (~Q) ==> (~P) = Q" by simp}
in
fun contrapos1 prove (ct, cu) = prove (negate ct, negate cu) COMP cp1
fun contrapos2 prove (ct, cu) = prove (negate ct, Thm.dest_arg cu) COMP cp2
fun contrapos3 prove (ct, cu) = prove (Thm.dest_arg ct, negate cu) COMP cp3
end
local
val contra_rule = @{lemma "P ==> ~P ==> False" by (rule notE)}
fun contra_left conj thm =
let
val rules = explode_term conj (SMT_Utils.prop_of thm)
fun contra_lits (t, rs) =
(case t of
@{const Not} $ u => Termtab.lookup rules u |> Option.map (pair rs)
| _ => NONE)
in
(case Termtab.lookup rules @{const False} of
SOME rs => extract_lit thm rs
| NONE =>
the (Termtab.get_first contra_lits rules)
|> pairself (extract_lit thm)
|> (fn (nlit, plit) => nlit COMP (plit COMP contra_rule)))
end
val falseE_v = Thm.dest_arg (Thm.dest_arg (Thm.cprop_of @{thm FalseE}))
fun contra_right ct = Thm.instantiate ([], [(falseE_v, ct)]) @{thm FalseE}
in
fun contradict conj ct =
iff_intro (Z3_Proof_Tools.under_assumption (contra_left conj) ct)
(contra_right ct)
end
local
fun prove_eq l r (cl, cr) =
let
fun explode' is_conj = explode is_conj true (l <> r) []
fun make_tab is_conj thm = make_littab (true_thm :: explode' is_conj thm)
fun prove is_conj ct tab = join is_conj tab (Thm.term_of ct)
val thm1 = Z3_Proof_Tools.under_assumption (prove r cr o make_tab l) cl
val thm2 = Z3_Proof_Tools.under_assumption (prove l cl o make_tab r) cr
in iff_intro thm1 thm2 end
datatype conj_disj = CONJ | DISJ | NCON | NDIS
fun kind_of t =
if is_conj t then SOME CONJ
else if is_disj t then SOME DISJ
else if is_neg' is_conj t then SOME NCON
else if is_neg' is_disj t then SOME NDIS
else NONE
in
fun prove_conj_disj_eq ct =
let val cp as (cl, cr) = Thm.dest_binop (Thm.dest_arg ct)
in
(case (kind_of (Thm.term_of cl), Thm.term_of cr) of
(SOME CONJ, @{const False}) => contradict true cl
| (SOME DISJ, @{const Not} $ @{const False}) =>
contrapos2 (contradict false o fst) cp
| (kl, _) =>
(case (kl, kind_of (Thm.term_of cr)) of
(SOME CONJ, SOME CONJ) => prove_eq true true cp
| (SOME CONJ, SOME NDIS) => prove_eq true false cp
| (SOME CONJ, _) => prove_eq true true cp
| (SOME DISJ, SOME DISJ) => contrapos1 (prove_eq false false) cp
| (SOME DISJ, SOME NCON) => contrapos2 (prove_eq false true) cp
| (SOME DISJ, _) => contrapos1 (prove_eq false false) cp
| (SOME NCON, SOME NCON) => contrapos1 (prove_eq true true) cp
| (SOME NCON, SOME DISJ) => contrapos3 (prove_eq true false) cp
| (SOME NCON, NONE) => contrapos3 (prove_eq true false) cp
| (SOME NDIS, SOME NDIS) => prove_eq false false cp
| (SOME NDIS, SOME CONJ) => prove_eq false true cp
| (SOME NDIS, NONE) => prove_eq false true cp
| _ => raise CTERM ("prove_conj_disj_eq", [ct])))
end
end
end