From 38503bde0d58e80bd6c39152fb7d2f226d8ac3c9 Mon Sep 17 00:00:00 2001 From: Andrew Reynolds Date: Thu, 3 Oct 2013 09:52:24 -0500 Subject: [PATCH] Adding example proof signatures for LFSC. --- proofs/signatures/example.plf | 116 +++++++++++++ proofs/signatures/sat.plf | 127 ++++++++++++++ proofs/signatures/smt.plf | 313 ++++++++++++++++++++++++++++++++++ proofs/signatures/th_base.plf | 107 ++++++++++++ 4 files changed, 663 insertions(+) create mode 100755 proofs/signatures/example.plf create mode 100755 proofs/signatures/sat.plf create mode 100755 proofs/signatures/smt.plf create mode 100755 proofs/signatures/th_base.plf diff --git a/proofs/signatures/example.plf b/proofs/signatures/example.plf new file mode 100755 index 000000000..5df1f31c3 --- /dev/null +++ b/proofs/signatures/example.plf @@ -0,0 +1,116 @@ +; to check, run : lfsc sat.plf smt.plf th_base.plf example.plf + +; -------------------------------------------------------------------------------- +; input : +; ( a = b ) +; ( b = f(c) ) +; ( a != f(c) V a != b ) + +; theory lemma (by transitivity) : +; ( a != b V b != f(c) V a = f(c) ) + + +; With the theory lemma, the input is unsatisfiable. +; -------------------------------------------------------------------------------- + + + +(check +(% s sort +(% a (term s) +(% b (term s) +(% c (term s) +(% f (term (arrow s s)) + +; -------------------- declaration of input formula ----------------------------------- + +(% A1 (th_holds (= s a b)) +(% A2 (th_holds (= s b (apply _ _ f c))) +(% A3 (th_holds (or (not (= s a (apply _ _ f c))) (not (= s a b)))) + + +; ------------------- specify that the following is a proof of the empty clause ----------------- + +(: (holds cln) + +; ---------- use atoms (a1, a2, a3) to map theory literals to boolean literals (v1, v2, v3) ------ + +(decl_atom (= s a b) (\ v1 (\ a1 +(decl_atom (= s b (apply _ _ f c)) (\ v2 (\ a2 +(decl_atom (= s a (apply _ _ f c)) (\ v3 (\ a3 + + +; --------------------------- proof of theory lemma --------------------------------------------- + +(satlem _ _ +(ast _ _ _ a1 (\ l1 +(ast _ _ _ a2 (\ l2 +(asf _ _ _ a3 (\ l3 +(clausify_false + +; this should be a proof of l1 ^ l2 ^ ~l3 => false +; this is done by theory proof rules (currently just use "trust") + + trust + +))))))) (\ CT +; CT is the clause ( ~v1 V ~v2 V v3 ) + +; -------------------- clausification of input formulas ----------------------------------------- + +(satlem _ _ +(asf _ _ _ a1 (\ l1 +(clausify_false + +; input formula A1 is ( ~l1 ) +; the following should be a proof of l1 ^ ( ~l1 ) => false +; this is done by natural deduction rules + + (contra _ A1 l1) + +))) (\ C1 +; C1 is the clause ( v1 ) + + +(satlem _ _ +(asf _ _ _ a2 (\ l2 +(clausify_false + +; input formula A2 is ( ~l2 ) +; the following should be a proof of l2 ^ ( ~l2 ) => false +; this is done by natural deduction rules + + (contra _ A2 l2) + +))) (\ C2 +; C2 is the clause ( v2 ) + + +(satlem _ _ +(ast _ _ _ a3 (\ l3 +(ast _ _ _ a1 (\ l1 +(clausify_false + +; input formula A3 is ( ~a3 V ~a1 ) +; the following should be a proof of a3 ^ a1 ^ ( ~a3 V ~a1 ) => false +; this is done by natural deduction rules + + (contra _ l1 (or_elim_1 _ _ (not_not_intro _ l3) A3)) + +))))) (\ C3 +; C3 is the clause ( ~v3 V ~v1 ) + + + +; -------------------- resolution proof ------------------------------------------------------------ + +(satlem_simplify _ _ _ + +(R _ _ C1 +(R _ _ C2 +(R _ _ CT C3 v3) v2) v1) + +(\ x x)) + +)))))))))))))))))))))))))) +) \ No newline at end of file diff --git a/proofs/signatures/sat.plf b/proofs/signatures/sat.plf new file mode 100755 index 000000000..09255f612 --- /dev/null +++ b/proofs/signatures/sat.plf @@ -0,0 +1,127 @@ +(declare bool type) +(declare tt bool) +(declare ff bool) + +(declare var type) + +(declare lit type) +(declare pos (! x var lit)) +(declare neg (! x var lit)) + +(declare clause type) +(declare cln clause) +(declare clc (! x lit (! c clause clause))) + +; constructs for general clauses for R, Q, satlem + +(declare concat (! c1 clause (! c2 clause clause))) +(declare clr (! l lit (! c clause clause))) + +; code to check resolutions + +(program append ((c1 clause) (c2 clause)) clause + (match c1 (cln c2) ((clc l c1') (clc l (append c1' c2))))) + +; we use marks as follows: +; -- mark 1 to record if we are supposed to remove a positive occurrence of the variable. +; -- mark 2 to record if we are supposed to remove a negative occurrence of the variable. +; -- mark 3 if we did indeed remove the variable positively +; -- mark 4 if we did indeed remove the variable negatively +(program simplify_clause ((c clause)) clause + (match c + (cln cln) + ((clc l c1) + (match l + ; Set mark 1 on v if it is not set, to indicate we should remove it. + ; After processing the rest of the clause, set mark 3 if we were already + ; supposed to remove v (so if mark 1 was set when we began). Clear mark3 + ; if we were not supposed to be removing v when we began this call. + ((pos v) + (let m (ifmarked v tt (do (markvar v) ff)) + (let c' (simplify_clause c1) + (match m + (tt (do (ifmarked3 v v (markvar3 v)) c')) + (ff (do (ifmarked3 v (markvar3 v) v) (markvar v) (clc l c'))))))) + ; the same as the code for tt, but using different marks. + ((neg v) + (let m (ifmarked2 v tt (do (markvar2 v) ff)) + (let c' (simplify_clause c1) + (match m + (tt (do (ifmarked4 v v (markvar4 v)) c')) + (ff (do (ifmarked4 v (markvar4 v) v) (markvar2 v) (clc l c'))))))))) + ((concat c1 c2) (append (simplify_clause c1) (simplify_clause c2))) + ((clr l c1) + (match l + ; set mark 1 to indicate we should remove v, and fail if + ; mark 3 is not set after processing the rest of the clause + ; (we will set mark 3 if we remove a positive occurrence of v). + ((pos v) + (let m (ifmarked v tt (do (markvar v) ff)) + (let m3 (ifmarked3 v (do (markvar3 v) tt) ff) + (let c' (simplify_clause c1) + (ifmarked3 v (do (match m3 (tt v) (ff (markvar3 v))) + (match m (tt v) (ff (markvar v))) c') + (fail clause)))))) + ; same as the tt case, but with different marks. + ((neg v) + (let m2 (ifmarked2 v tt (do (markvar2 v) ff)) + (let m4 (ifmarked4 v (do (markvar4 v) tt) ff) + (let c' (simplify_clause c1) + (ifmarked4 v (do (match m4 (tt v) (ff (markvar4 v))) + (match m2 (tt v) (ff (markvar2 v))) c') + (fail clause)))))) + )))) + + +; resolution proofs + +(declare holds (! c clause type)) + +(declare R (! c1 clause (! c2 clause + (! u1 (holds c1) + (! u2 (holds c2) + (! n var + (holds (concat (clr (pos n) c1) + (clr (neg n) c2))))))))) + +(declare Q (! c1 clause (! c2 clause + (! u1 (holds c1) + (! u2 (holds c2) + (! n var + (holds (concat (clr (neg n) c1) + (clr (pos n) c2))))))))) + +(declare satlem_simplify + (! c1 clause + (! c2 clause + (! c3 clause + (! u1 (holds c1) + (! r (^ (simplify_clause c1) c2) + (! u2 (! x (holds c2) (holds c3)) + (holds c3)))))))) + +(declare satlem + (! c clause + (! c2 clause + (! u (holds c) + (! u2 (! v (holds c) (holds c2)) + (holds c2)))))) + +; A little example to demonstrate simplify_clause. +; It can handle nested clr's of both polarities, +; and correctly cleans up marks when it leaves a +; clr or clc scope. Uncomment and run with +; --show-runs to see it in action. +; +; (check +; (% v1 var +; (% u1 (holds (concat (clr (neg v1) (clr (pos v1) (clc (pos v1) (clr (pos v1) (clc (pos v1) (clc (neg v1) cln)))))) +; (clc (pos v1) (clc (pos v1) cln)))) +; (satlem _ _ _ u1 (\ x x)))))) + + +;(check +; (% v1 var +; (% u1 (holds (clr (neg v1) (concat (clc (neg v1) cln) +; (clr (neg v1) (clc (neg v1) cln))))) +; (satlem _ _ _ u1 (\ x x)))))) \ No newline at end of file diff --git a/proofs/signatures/smt.plf b/proofs/signatures/smt.plf new file mode 100755 index 000000000..75bfc442f --- /dev/null +++ b/proofs/signatures/smt.plf @@ -0,0 +1,313 @@ +;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; +; +; SMT syntax and semantics (not theory-specific) +; +;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; + +(declare formula type) +(declare th_holds (! f formula type)) + +; constants +(declare true formula) +(declare false formula) + +; logical connectives +(declare not (! f formula formula)) +(declare and (! f1 formula (! f2 formula formula))) +(declare or (! f1 formula (! f2 formula formula))) +(declare impl (! f1 formula (! f2 formula formula))) +(declare iff (! f1 formula (! f2 formula formula))) +(declare xor (! f1 formula (! f2 formula formula))) +(declare ifte (! b formula (! f1 formula (! f2 formula formula)))) + +; terms +(declare sort type) ; sort in the theory +(declare arrow (! s1 sort (! s2 sort sort))) ; function constructor + +(declare term (! t sort type)) ; declared terms in formula + +(declare apply (! s1 sort + (! s2 sort + (! t1 (term (arrow s1 s2)) + (! t2 (term s1) + (term s2)))))) + +(declare ite (! s sort + (! f formula + (! t1 (term s) + (! t2 (term s) + (term s)))))) + +; let/flet +(declare let (! s sort + (! t (term s) + (! f (! v (term s) formula) + formula)))) +(declare flet (! f1 formula + (! f2 (! v formula formula) + formula))) + +; predicates +(declare = (! s sort + (! x (term s) + (! y (term s) + formula)))) + +; To avoid duplicating some of the rules (e.g., cong), we will view +; applications of predicates as terms of sort "Bool". +; Such terms can be injected as atomic formulas using "p_app". + +(declare Bool sort) ; the special sort for predicates +(declare p_app (! x (term Bool) formula)) ; propositional application of term + + +;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; +; Examples + +; an example of "(p1 or p2(0)) and t1=t2(1)" +;(! p1 (term Bool) +;(! p2 (term (arrow Int Bool)) +;(! t1 (term Int) +;(! t2 (term (arrow Int Int)) +;(! F (th_holds (and (or (p_app p1) (p_app (apply _ _ p2 0))) +; (= _ t1 (apply _ _ t2 1)))) +; ... + +; another example of "p3(a,b)" +;(! a (term Int) +;(! b (term Int) +;(! p3 (term (arrow Int (arrow Int Bool))) ; arrow is right assoc. +;(! F (th_holds (p_app (apply _ _ (apply _ _ p3 a) b))) ; apply is left assoc. +; ... + + +;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; +; +; Natural deduction rules : used for CNF +; +;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; + + +;; contradiction + +(declare contra + (! f formula + (! r1 (th_holds f) + (! r2 (th_holds (not f)) + (th_holds false))))) + +;; not not + +(declare not_not_intro + (! f formula + (! u (th_holds f) + (th_holds (not (not f)))))) + +(declare not_not_elim + (! f formula + (! u (th_holds (not (not f))) + (th_holds f)))) + +;; or elimination + +(declare or_elim_1 + (! f1 formula + (! f2 formula + (! u1 (th_holds (not f1)) + (! u2 (th_holds (or f1 f2)) + (th_holds f2)))))) + +(declare or_elim_2 + (! f1 formula + (! f2 formula + (! u1 (th_holds (not f2)) + (! u2 (th_holds (or f1 f2)) + (th_holds f1)))))) + +;; and elimination + +(declare and_elim_1 + (! f1 formula + (! f2 formula + (! u (th_holds (and f1 f2)) + (th_holds f1))))) + +(declare and_elim_2 + (! f1 formula + (! f2 formula + (! u (th_holds (and f1 f2)) + (th_holds f2))))) + +;; not impl elimination + +(declare not_impl_elim_1 + (! f1 formula + (! f2 formula + (! u (th_holds (not (impl f1 f2))) + (th_holds f1))))) + +(declare not_impl_elim_2 + (! f1 formula + (! f2 formula + (! u (th_holds (not (impl f1 f2))) + (th_holds (not f2)))))) + +;; impl elimination + +(declare impl_intro (! f1 formula + (! f2 formula + (! i1 (! u (th_holds f1) + (th_holds f2)) + (th_holds (impl f1 f2)))))) + +(declare impl_elim + (! f1 formula + (! f2 formula + (! u1 (th_holds f1) + (! u2 (th_holds (impl f1 f2)) + (th_holds f2)))))) + +;; iff elimination + +(declare iff_elim_1 + (! f1 formula + (! f2 formula + (! u1 (th_holds (iff f1 f2)) + (th_holds (impl f1 f2)))))) + +(declare iff_elim_2 + (! f1 formula + (! f2 formula + (! u1 (th_holds (iff f1 f2)) + (th_holds (impl f2 f1)))))) + +(declare not_iff_elim_1 + (! f1 formula + (! f2 formula + (! u1 (th_holds (not f1)) + (! u2 (th_holds (not (iff f1 f2))) + (th_holds f2)))))) + +(declare not_iff_elim_2 + (! f1 formula + (! f2 formula + (! u1 (th_holds f1) + (! u2 (th_holds (not (iff f1 f2))) + (th_holds (not f2))))))) + +;; ite elimination + +(declare ite_elim_1 + (! a formula + (! b formula + (! c formula + (! u1 (th_holds a) + (! u2 (th_holds (ifte a b c)) + (th_holds b))))))) + +(declare ite_elim_2 + (! a formula + (! b formula + (! c formula + (! u1 (th_holds (not a)) + (! u2 (th_holds (ifte a b c)) + (th_holds c))))))) + +(declare ite_elim_3 + (! a formula + (! b formula + (! c formula + (! u1 (th_holds (not b)) + (! u2 (th_holds (ifte a b c)) + (th_holds c))))))) + +(declare ite_elim_2n + (! a formula + (! b formula + (! c formula + (! u1 (th_holds a) + (! u2 (th_holds (ifte (not a) b c)) + (th_holds c))))))) + +(declare not_ite_elim_1 + (! a formula + (! b formula + (! c formula + (! u1 (th_holds a) + (! u2 (th_holds (not (ifte a b c))) + (th_holds (not b)))))))) + +(declare not_ite_elim_2 + (! a formula + (! b formula + (! c formula + (! u1 (th_holds (not a)) + (! u2 (th_holds (not (ifte a b c))) + (th_holds (not c)))))))) + +(declare not_ite_elim_3 + (! a formula + (! b formula + (! c formula + (! u1 (th_holds b) + (! u2 (th_holds (not (ifte a b c))) + (th_holds (not c)))))))) + +(declare not_ite_elim_2n + (! a formula + (! b formula + (! c formula + (! u1 (th_holds a) + (! u2 (th_holds (not (ifte (not a) b c))) + (th_holds (not c)))))))) + + + +;; How to do CNF for disjunctions of theory literals. +;; +;; Given theory literals F1....Fn, and an input formula A of the form (th_holds (or F1 (or F2 .... (or F{n-1} Fn))))). +;; +;; We introduce atoms a1...an for literals F1...Fn, mapping them to boolean literals v1...vn. +;; Do this at the beginning of the proof: +;; +;; (decl_atom F1 (\ v1 (\ a1 +;; (decl_atom F2 (\ v2 (\ a2 +;; .... +;; (decl_atom Fn (\ vn (\ an +;; +;; Our input A is clausified by the following proof: +;; +;;(satlem _ _ +;;(asf _ _ _ a1 (\ l1 +;;(asf _ _ _ a2 (\ l2 +;;... +;;(asf _ _ _ an (\ ln +;;(clausify_false +;; +;; (contra _ +;; (or_elim_1 _ _ l{n-1} +;; ... +;; (or_elim_1 _ _ l2 +;; (or_elim_1 _ _ l1 A))))) ln) +;; +;;))))))) (\ C +;; +;; We now have the free variable C, which should be the clause (v1 V ... V vn). +;; +;; We also need to consider the polarity of literals, say we have A of the form (th_holds (or (not F1) (or F2 (not F3)))). +;; Where necessary, we use "ast" instead of "asf", introduce negations by "not_not_intro" for pattern matching, and flip +;; the arguments of contra: +;; +;;(satlem _ _ +;;(ast _ _ _ a1 (\ l1 +;;(asf _ _ _ a2 (\ l2 +;;(ast _ _ _ a3 (\ l3 +;;(clausify_false +;; +;; (contra _ l3 +;; (or_elim_1 _ _ l2 +;; (or_elim_1 _ _ (not_not_intro l1) A)))) +;; +;;))))))) (\ C +;; +;; C should be the clause (~v1 V v2 V ~v3 ) \ No newline at end of file diff --git a/proofs/signatures/th_base.plf b/proofs/signatures/th_base.plf new file mode 100755 index 000000000..e66990de4 --- /dev/null +++ b/proofs/signatures/th_base.plf @@ -0,0 +1,107 @@ +;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; +; +; Atomization / Clausification +; +;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; + +; binding between an LF var and an (atomic) formula +(declare atom (! v var (! p formula type))) + +(declare decl_atom + (! f formula + (! u (! v var + (! a (atom v f) + (holds cln))) + (holds cln)))) + +; direct clausify +(declare clausify_form + (! f formula + (! v var + (! a (atom v f) + (! u (th_holds f) + (holds (clc (pos v) cln))))))) + +(declare clausify_form_not + (! f formula + (! v var + (! a (atom v f) + (! u (th_holds (not f)) + (holds (clc (neg v) cln))))))) + +(declare clausify_false + (! u (th_holds false) + (holds cln))) + + +(declare th_let_pf + (! f formula + (! u (th_holds f) + (! u2 (! v (th_holds f) (holds cln)) + (holds cln))))) + +;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; +; +; Theory reasoning +; - make a series of assumptions and then derive a contradiction (or false) +; - then the assumptions yield a formula like "v1 -> v2 -> ... -> vn -> false" +; - In CNF, it becomes a clause: "~v1, ~v2, ..., ~vn" +; +;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; + +(declare ast + (! v var + (! f formula + (! C clause + (! r (atom v f) ;this is specified + (! u (! o (th_holds f) + (holds C)) + (holds (clc (neg v) C)))))))) + +(declare asf + (! v var + (! f formula + (! C clause + (! r (atom v f) + (! u (! o (th_holds (not f)) + (holds C)) + (holds (clc (pos v) C)))))))) + +;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; +; +; Theory of Equality and Congruence Closure +; +;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; + +; temporary : +(declare trust (th_holds false)) + +(declare refl + (! s sort + (! t (term s) + (th_holds (= s t t))))) + +(declare symm (! s sort + (! x (term s) + (! y (term s) + (! u (th_holds (= _ x y)) + (th_holds (= _ y x))))))) + +(declare trans (! s sort + (! x (term s) + (! y (term s) + (! z (term s) + (! u (th_holds (= _ x y)) + (! u (th_holds (= _ y z)) + (th_holds (= _ x z))))))))) + +(declare cong (! s1 sort + (! s2 sort + (! a1 (term (arrow s1 s2)) + (! b1 (term (arrow s1 s2)) + (! a2 (term s1) + (! b2 (term s1) + (! u1 (th_holds (= _ a1 b1)) + (! u2 (th_holds (= _ a2 b2)) + (th_holds (= _ (apply _ _ a1 a2) (apply _ _ b1 b2)))))))))))) + -- 2.30.2