--- a/src/HOL/SMT_Examples/SMT_Examples.thy Thu Mar 13 13:18:13 2014 +0100
+++ b/src/HOL/SMT_Examples/SMT_Examples.thy Thu Mar 13 13:18:13 2014 +0100
@@ -11,38 +11,40 @@
declare [[smt_certificates = "SMT_Examples.certs"]]
declare [[smt_read_only_certificates = true]]
+declare [[smt2_certificates = "SMT_Examples.certs2"]]
+declare [[smt2_read_only_certificates = true]]
section {* Propositional and first-order logic *}
-lemma "True" by smt
+lemma "True" by smt2
-lemma "p \<or> \<not>p" by smt
+lemma "p \<or> \<not>p" by smt2
-lemma "(p \<and> True) = p" by smt
+lemma "(p \<and> True) = p" by smt2
-lemma "(p \<or> q) \<and> \<not>p \<Longrightarrow> q" by smt
+lemma "(p \<or> q) \<and> \<not>p \<Longrightarrow> q" by smt2
lemma "(a \<and> b) \<or> (c \<and> d) \<Longrightarrow> (a \<and> b) \<or> (c \<and> d)"
- by smt
+ by smt2
-lemma "(p1 \<and> p2) \<or> p3 \<longrightarrow> (p1 \<longrightarrow> (p3 \<and> p2) \<or> (p1 \<and> p3)) \<or> p1" by smt
+lemma "(p1 \<and> p2) \<or> p3 \<longrightarrow> (p1 \<longrightarrow> (p3 \<and> p2) \<or> (p1 \<and> p3)) \<or> p1" by smt2
-lemma "P=P=P=P=P=P=P=P=P=P" by smt
+lemma "P = P = P = P = P = P = P = P = P = P" by smt2
lemma
- assumes "a | b | c | d"
- and "e | f | (a & d)"
- and "~(a | (c & ~c)) | b"
- and "~(b & (x | ~x)) | c"
- and "~(d | False) | c"
- and "~(c | (~p & (p | (q & ~q))))"
+ assumes "a \<or> b \<or> c \<or> d"
+ and "e \<or> f \<or> (a \<and> d)"
+ and "\<not> (a \<or> (c \<and> ~c)) \<or> b"
+ and "\<not> (b \<and> (x \<or> \<not> x)) \<or> c"
+ and "\<not> (d \<or> False) \<or> c"
+ and "\<not> (c \<or> (\<not> p \<and> (p \<or> (q \<and> \<not> q))))"
shows False
- using assms by smt
+ using assms by smt2
axiomatization symm_f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" where
symm_f: "symm_f x y = symm_f y x"
-lemma "a = a \<and> symm_f a b = symm_f b a" by (smt symm_f)
+lemma "a = a \<and> symm_f a b = symm_f b a" by (smt2 symm_f)
(*
Taken from ~~/src/HOL/ex/SAT_Examples.thy.
@@ -53,253 +55,253 @@
and "~x30"
and "~x29"
and "~x59"
- and "x1 | x31 | x0"
- and "x2 | x32 | x1"
- and "x3 | x33 | x2"
- and "x4 | x34 | x3"
- and "x35 | x4"
- and "x5 | x36 | x30"
- and "x6 | x37 | x5 | x31"
- and "x7 | x38 | x6 | x32"
- and "x8 | x39 | x7 | x33"
- and "x9 | x40 | x8 | x34"
- and "x41 | x9 | x35"
- and "x10 | x42 | x36"
- and "x11 | x43 | x10 | x37"
- and "x12 | x44 | x11 | x38"
- and "x13 | x45 | x12 | x39"
- and "x14 | x46 | x13 | x40"
- and "x47 | x14 | x41"
- and "x15 | x48 | x42"
- and "x16 | x49 | x15 | x43"
- and "x17 | x50 | x16 | x44"
- and "x18 | x51 | x17 | x45"
- and "x19 | x52 | x18 | x46"
- and "x53 | x19 | x47"
- and "x20 | x54 | x48"
- and "x21 | x55 | x20 | x49"
- and "x22 | x56 | x21 | x50"
- and "x23 | x57 | x22 | x51"
- and "x24 | x58 | x23 | x52"
- and "x59 | x24 | x53"
- and "x25 | x54"
- and "x26 | x25 | x55"
- and "x27 | x26 | x56"
- and "x28 | x27 | x57"
- and "x29 | x28 | x58"
- and "~x1 | ~x31"
- and "~x1 | ~x0"
- and "~x31 | ~x0"
- and "~x2 | ~x32"
- and "~x2 | ~x1"
- and "~x32 | ~x1"
- and "~x3 | ~x33"
- and "~x3 | ~x2"
- and "~x33 | ~x2"
- and "~x4 | ~x34"
- and "~x4 | ~x3"
- and "~x34 | ~x3"
- and "~x35 | ~x4"
- and "~x5 | ~x36"
- and "~x5 | ~x30"
- and "~x36 | ~x30"
- and "~x6 | ~x37"
- and "~x6 | ~x5"
- and "~x6 | ~x31"
- and "~x37 | ~x5"
- and "~x37 | ~x31"
- and "~x5 | ~x31"
- and "~x7 | ~x38"
- and "~x7 | ~x6"
- and "~x7 | ~x32"
- and "~x38 | ~x6"
- and "~x38 | ~x32"
- and "~x6 | ~x32"
- and "~x8 | ~x39"
- and "~x8 | ~x7"
- and "~x8 | ~x33"
- and "~x39 | ~x7"
- and "~x39 | ~x33"
- and "~x7 | ~x33"
- and "~x9 | ~x40"
- and "~x9 | ~x8"
- and "~x9 | ~x34"
- and "~x40 | ~x8"
- and "~x40 | ~x34"
- and "~x8 | ~x34"
- and "~x41 | ~x9"
- and "~x41 | ~x35"
- and "~x9 | ~x35"
- and "~x10 | ~x42"
- and "~x10 | ~x36"
- and "~x42 | ~x36"
- and "~x11 | ~x43"
- and "~x11 | ~x10"
- and "~x11 | ~x37"
- and "~x43 | ~x10"
- and "~x43 | ~x37"
- and "~x10 | ~x37"
- and "~x12 | ~x44"
- and "~x12 | ~x11"
- and "~x12 | ~x38"
- and "~x44 | ~x11"
- and "~x44 | ~x38"
- and "~x11 | ~x38"
- and "~x13 | ~x45"
- and "~x13 | ~x12"
- and "~x13 | ~x39"
- and "~x45 | ~x12"
- and "~x45 | ~x39"
- and "~x12 | ~x39"
- and "~x14 | ~x46"
- and "~x14 | ~x13"
- and "~x14 | ~x40"
- and "~x46 | ~x13"
- and "~x46 | ~x40"
- and "~x13 | ~x40"
- and "~x47 | ~x14"
- and "~x47 | ~x41"
- and "~x14 | ~x41"
- and "~x15 | ~x48"
- and "~x15 | ~x42"
- and "~x48 | ~x42"
- and "~x16 | ~x49"
- and "~x16 | ~x15"
- and "~x16 | ~x43"
- and "~x49 | ~x15"
- and "~x49 | ~x43"
- and "~x15 | ~x43"
- and "~x17 | ~x50"
- and "~x17 | ~x16"
- and "~x17 | ~x44"
- and "~x50 | ~x16"
- and "~x50 | ~x44"
- and "~x16 | ~x44"
- and "~x18 | ~x51"
- and "~x18 | ~x17"
- and "~x18 | ~x45"
- and "~x51 | ~x17"
- and "~x51 | ~x45"
- and "~x17 | ~x45"
- and "~x19 | ~x52"
- and "~x19 | ~x18"
- and "~x19 | ~x46"
- and "~x52 | ~x18"
- and "~x52 | ~x46"
- and "~x18 | ~x46"
- and "~x53 | ~x19"
- and "~x53 | ~x47"
- and "~x19 | ~x47"
- and "~x20 | ~x54"
- and "~x20 | ~x48"
- and "~x54 | ~x48"
- and "~x21 | ~x55"
- and "~x21 | ~x20"
- and "~x21 | ~x49"
- and "~x55 | ~x20"
- and "~x55 | ~x49"
- and "~x20 | ~x49"
- and "~x22 | ~x56"
- and "~x22 | ~x21"
- and "~x22 | ~x50"
- and "~x56 | ~x21"
- and "~x56 | ~x50"
- and "~x21 | ~x50"
- and "~x23 | ~x57"
- and "~x23 | ~x22"
- and "~x23 | ~x51"
- and "~x57 | ~x22"
- and "~x57 | ~x51"
- and "~x22 | ~x51"
- and "~x24 | ~x58"
- and "~x24 | ~x23"
- and "~x24 | ~x52"
- and "~x58 | ~x23"
- and "~x58 | ~x52"
- and "~x23 | ~x52"
- and "~x59 | ~x24"
- and "~x59 | ~x53"
- and "~x24 | ~x53"
- and "~x25 | ~x54"
- and "~x26 | ~x25"
- and "~x26 | ~x55"
- and "~x25 | ~x55"
- and "~x27 | ~x26"
- and "~x27 | ~x56"
- and "~x26 | ~x56"
- and "~x28 | ~x27"
- and "~x28 | ~x57"
- and "~x27 | ~x57"
- and "~x29 | ~x28"
- and "~x29 | ~x58"
- and "~x28 | ~x58"
+ and "x1 \<or> x31 \<or> x0"
+ and "x2 \<or> x32 \<or> x1"
+ and "x3 \<or> x33 \<or> x2"
+ and "x4 \<or> x34 \<or> x3"
+ and "x35 \<or> x4"
+ and "x5 \<or> x36 \<or> x30"
+ and "x6 \<or> x37 \<or> x5 \<or> x31"
+ and "x7 \<or> x38 \<or> x6 \<or> x32"
+ and "x8 \<or> x39 \<or> x7 \<or> x33"
+ and "x9 \<or> x40 \<or> x8 \<or> x34"
+ and "x41 \<or> x9 \<or> x35"
+ and "x10 \<or> x42 \<or> x36"
+ and "x11 \<or> x43 \<or> x10 \<or> x37"
+ and "x12 \<or> x44 \<or> x11 \<or> x38"
+ and "x13 \<or> x45 \<or> x12 \<or> x39"
+ and "x14 \<or> x46 \<or> x13 \<or> x40"
+ and "x47 \<or> x14 \<or> x41"
+ and "x15 \<or> x48 \<or> x42"
+ and "x16 \<or> x49 \<or> x15 \<or> x43"
+ and "x17 \<or> x50 \<or> x16 \<or> x44"
+ and "x18 \<or> x51 \<or> x17 \<or> x45"
+ and "x19 \<or> x52 \<or> x18 \<or> x46"
+ and "x53 \<or> x19 \<or> x47"
+ and "x20 \<or> x54 \<or> x48"
+ and "x21 \<or> x55 \<or> x20 \<or> x49"
+ and "x22 \<or> x56 \<or> x21 \<or> x50"
+ and "x23 \<or> x57 \<or> x22 \<or> x51"
+ and "x24 \<or> x58 \<or> x23 \<or> x52"
+ and "x59 \<or> x24 \<or> x53"
+ and "x25 \<or> x54"
+ and "x26 \<or> x25 \<or> x55"
+ and "x27 \<or> x26 \<or> x56"
+ and "x28 \<or> x27 \<or> x57"
+ and "x29 \<or> x28 \<or> x58"
+ and "~x1 \<or> ~x31"
+ and "~x1 \<or> ~x0"
+ and "~x31 \<or> ~x0"
+ and "~x2 \<or> ~x32"
+ and "~x2 \<or> ~x1"
+ and "~x32 \<or> ~x1"
+ and "~x3 \<or> ~x33"
+ and "~x3 \<or> ~x2"
+ and "~x33 \<or> ~x2"
+ and "~x4 \<or> ~x34"
+ and "~x4 \<or> ~x3"
+ and "~x34 \<or> ~x3"
+ and "~x35 \<or> ~x4"
+ and "~x5 \<or> ~x36"
+ and "~x5 \<or> ~x30"
+ and "~x36 \<or> ~x30"
+ and "~x6 \<or> ~x37"
+ and "~x6 \<or> ~x5"
+ and "~x6 \<or> ~x31"
+ and "~x37 \<or> ~x5"
+ and "~x37 \<or> ~x31"
+ and "~x5 \<or> ~x31"
+ and "~x7 \<or> ~x38"
+ and "~x7 \<or> ~x6"
+ and "~x7 \<or> ~x32"
+ and "~x38 \<or> ~x6"
+ and "~x38 \<or> ~x32"
+ and "~x6 \<or> ~x32"
+ and "~x8 \<or> ~x39"
+ and "~x8 \<or> ~x7"
+ and "~x8 \<or> ~x33"
+ and "~x39 \<or> ~x7"
+ and "~x39 \<or> ~x33"
+ and "~x7 \<or> ~x33"
+ and "~x9 \<or> ~x40"
+ and "~x9 \<or> ~x8"
+ and "~x9 \<or> ~x34"
+ and "~x40 \<or> ~x8"
+ and "~x40 \<or> ~x34"
+ and "~x8 \<or> ~x34"
+ and "~x41 \<or> ~x9"
+ and "~x41 \<or> ~x35"
+ and "~x9 \<or> ~x35"
+ and "~x10 \<or> ~x42"
+ and "~x10 \<or> ~x36"
+ and "~x42 \<or> ~x36"
+ and "~x11 \<or> ~x43"
+ and "~x11 \<or> ~x10"
+ and "~x11 \<or> ~x37"
+ and "~x43 \<or> ~x10"
+ and "~x43 \<or> ~x37"
+ and "~x10 \<or> ~x37"
+ and "~x12 \<or> ~x44"
+ and "~x12 \<or> ~x11"
+ and "~x12 \<or> ~x38"
+ and "~x44 \<or> ~x11"
+ and "~x44 \<or> ~x38"
+ and "~x11 \<or> ~x38"
+ and "~x13 \<or> ~x45"
+ and "~x13 \<or> ~x12"
+ and "~x13 \<or> ~x39"
+ and "~x45 \<or> ~x12"
+ and "~x45 \<or> ~x39"
+ and "~x12 \<or> ~x39"
+ and "~x14 \<or> ~x46"
+ and "~x14 \<or> ~x13"
+ and "~x14 \<or> ~x40"
+ and "~x46 \<or> ~x13"
+ and "~x46 \<or> ~x40"
+ and "~x13 \<or> ~x40"
+ and "~x47 \<or> ~x14"
+ and "~x47 \<or> ~x41"
+ and "~x14 \<or> ~x41"
+ and "~x15 \<or> ~x48"
+ and "~x15 \<or> ~x42"
+ and "~x48 \<or> ~x42"
+ and "~x16 \<or> ~x49"
+ and "~x16 \<or> ~x15"
+ and "~x16 \<or> ~x43"
+ and "~x49 \<or> ~x15"
+ and "~x49 \<or> ~x43"
+ and "~x15 \<or> ~x43"
+ and "~x17 \<or> ~x50"
+ and "~x17 \<or> ~x16"
+ and "~x17 \<or> ~x44"
+ and "~x50 \<or> ~x16"
+ and "~x50 \<or> ~x44"
+ and "~x16 \<or> ~x44"
+ and "~x18 \<or> ~x51"
+ and "~x18 \<or> ~x17"
+ and "~x18 \<or> ~x45"
+ and "~x51 \<or> ~x17"
+ and "~x51 \<or> ~x45"
+ and "~x17 \<or> ~x45"
+ and "~x19 \<or> ~x52"
+ and "~x19 \<or> ~x18"
+ and "~x19 \<or> ~x46"
+ and "~x52 \<or> ~x18"
+ and "~x52 \<or> ~x46"
+ and "~x18 \<or> ~x46"
+ and "~x53 \<or> ~x19"
+ and "~x53 \<or> ~x47"
+ and "~x19 \<or> ~x47"
+ and "~x20 \<or> ~x54"
+ and "~x20 \<or> ~x48"
+ and "~x54 \<or> ~x48"
+ and "~x21 \<or> ~x55"
+ and "~x21 \<or> ~x20"
+ and "~x21 \<or> ~x49"
+ and "~x55 \<or> ~x20"
+ and "~x55 \<or> ~x49"
+ and "~x20 \<or> ~x49"
+ and "~x22 \<or> ~x56"
+ and "~x22 \<or> ~x21"
+ and "~x22 \<or> ~x50"
+ and "~x56 \<or> ~x21"
+ and "~x56 \<or> ~x50"
+ and "~x21 \<or> ~x50"
+ and "~x23 \<or> ~x57"
+ and "~x23 \<or> ~x22"
+ and "~x23 \<or> ~x51"
+ and "~x57 \<or> ~x22"
+ and "~x57 \<or> ~x51"
+ and "~x22 \<or> ~x51"
+ and "~x24 \<or> ~x58"
+ and "~x24 \<or> ~x23"
+ and "~x24 \<or> ~x52"
+ and "~x58 \<or> ~x23"
+ and "~x58 \<or> ~x52"
+ and "~x23 \<or> ~x52"
+ and "~x59 \<or> ~x24"
+ and "~x59 \<or> ~x53"
+ and "~x24 \<or> ~x53"
+ and "~x25 \<or> ~x54"
+ and "~x26 \<or> ~x25"
+ and "~x26 \<or> ~x55"
+ and "~x25 \<or> ~x55"
+ and "~x27 \<or> ~x26"
+ and "~x27 \<or> ~x56"
+ and "~x26 \<or> ~x56"
+ and "~x28 \<or> ~x27"
+ and "~x28 \<or> ~x57"
+ and "~x27 \<or> ~x57"
+ and "~x29 \<or> ~x28"
+ and "~x29 \<or> ~x58"
+ and "~x28 \<or> ~x58"
shows False
- using assms by smt
+ using assms by smt (* smt2 FIXME: THM 0 *)
lemma "\<forall>x::int. P x \<longrightarrow> (\<forall>y::int. P x \<or> P y)"
- by smt
+ by smt2
lemma
assumes "(\<forall>x y. P x y = x)"
shows "(\<exists>y. P x y) = P x c"
- using assms by smt
+ using assms by smt (* smt2 FIXME: Option *)
lemma
assumes "(\<forall>x y. P x y = x)"
and "(\<forall>x. \<exists>y. P x y) = (\<forall>x. P x c)"
shows "(EX y. P x y) = P x c"
- using assms by smt
+ using assms by smt (* smt2 FIXME: Option *)
lemma
assumes "if P x then \<not>(\<exists>y. P y) else (\<forall>y. \<not>P y)"
shows "P x \<longrightarrow> P y"
- using assms by smt
+ using assms by smt2
section {* Arithmetic *}
subsection {* Linear arithmetic over integers and reals *}
-lemma "(3::int) = 3" by smt
+lemma "(3::int) = 3" by smt2
-lemma "(3::real) = 3" by smt
+lemma "(3::real) = 3" by smt2
-lemma "(3 :: int) + 1 = 4" by smt
+lemma "(3 :: int) + 1 = 4" by smt2
-lemma "x + (y + z) = y + (z + (x::int))" by smt
+lemma "x + (y + z) = y + (z + (x::int))" by smt2
-lemma "max (3::int) 8 > 5" by smt
+lemma "max (3::int) 8 > 5" by smt2
-lemma "abs (x :: real) + abs y \<ge> abs (x + y)" by smt
+lemma "abs (x :: real) + abs y \<ge> abs (x + y)" by smt2
-lemma "P ((2::int) < 3) = P True" by smt
+lemma "P ((2::int) < 3) = P True" by smt2
-lemma "x + 3 \<ge> 4 \<or> x < (1::int)" by smt
+lemma "x + 3 \<ge> 4 \<or> x < (1::int)" by smt2
lemma
assumes "x \<ge> (3::int)" and "y = x + 4"
shows "y - x > 0"
- using assms by smt
+ using assms by smt2
-lemma "let x = (2 :: int) in x + x \<noteq> 5" by smt
+lemma "let x = (2 :: int) in x + x \<noteq> 5" by smt2
lemma
fixes x :: real
assumes "3 * x + 7 * a < 4" and "3 < 2 * x"
shows "a < 0"
- using assms by smt
+ using assms by smt2
-lemma "(0 \<le> y + -1 * x \<or> \<not> 0 \<le> x \<or> 0 \<le> (x::int)) = (\<not> False)" by smt
+lemma "(0 \<le> y + -1 * x \<or> \<not> 0 \<le> x \<or> 0 \<le> (x::int)) = (\<not> False)" by smt2
lemma "
- (n < m & m < n') | (n < m & m = n') | (n < n' & n' < m) |
- (n = n' & n' < m) | (n = m & m < n') |
- (n' < m & m < n) | (n' < m & m = n) |
- (n' < n & n < m) | (n' = n & n < m) | (n' = m & m < n) |
- (m < n & n < n') | (m < n & n' = n) | (m < n' & n' < n) |
- (m = n & n < n') | (m = n' & n' < n) |
- (n' = m & m = (n::int))"
- by smt
+ (n < m \<and> m < n') \<or> (n < m \<and> m = n') \<or> (n < n' \<and> n' < m) \<or>
+ (n = n' \<and> n' < m) \<or> (n = m \<and> m < n') \<or>
+ (n' < m \<and> m < n) \<or> (n' < m \<and> m = n) \<or>
+ (n' < n \<and> n < m) \<or> (n' = n \<and> n < m) \<or> (n' = m \<and> m < n) \<or>
+ (m < n \<and> n < n') \<or> (m < n \<and> n' = n) \<or> (m < n' \<and> n' < n) \<or>
+ (m = n \<and> n < n') \<or> (m = n' \<and> n' < n) \<or>
+ (n' = m \<and> m = (n::int))"
+ by smt2
text{*
The following example was taken from HOL/ex/PresburgerEx.thy, where it says:
@@ -320,175 +322,173 @@
lemma "\<lbrakk> x3 = abs x2 - x1; x4 = abs x3 - x2; x5 = abs x4 - x3;
x6 = abs x5 - x4; x7 = abs x6 - x5; x8 = abs x7 - x6;
x9 = abs x8 - x7; x10 = abs x9 - x8; x11 = abs x10 - x9 \<rbrakk>
- \<Longrightarrow> x1 = x10 & x2 = (x11::int)"
- by smt
+ \<Longrightarrow> x1 = x10 \<and> x2 = (x11::int)"
+ by smt2
-lemma "let P = 2 * x + 1 > x + (x::real) in P \<or> False \<or> P" by smt
+lemma "let P = 2 * x + 1 > x + (x::real) in P \<or> False \<or> P" by smt2
lemma "x + (let y = x mod 2 in 2 * y + 1) \<ge> x + (1::int)"
- using [[z3_with_extensions]]
- by smt
+ using [[z3_new_extensions]]
+ by smt2
lemma "x + (let y = x mod 2 in y + y) < x + (3::int)"
- using [[z3_with_extensions]]
- by smt
+ using [[z3_new_extensions]]
+ by smt2
lemma
assumes "x \<noteq> (0::real)"
- shows "x + x \<noteq> (let P = (abs x > 1) in if P \<or> \<not>P then 4 else 2) * x"
- using assms by smt
+ shows "x + x \<noteq> (let P = (abs x > 1) in if P \<or> \<not> P then 4 else 2) * x"
+ using assms [[z3_new_extensions]] by smt2
lemma
assumes "(n + m) mod 2 = 0" and "n mod 4 = 3"
- shows "n mod 2 = 1 & m mod 2 = (1::int)"
- using assms [[z3_with_extensions]] by smt
-
+ shows "n mod 2 = 1 \<and> m mod 2 = (1::int)"
+ using assms [[z3_new_extensions]] by smt2
subsection {* Linear arithmetic with quantifiers *}
-lemma "~ (\<exists>x::int. False)" by smt
+lemma "~ (\<exists>x::int. False)" by smt2
-lemma "~ (\<exists>x::real. False)" by smt
+lemma "~ (\<exists>x::real. False)" by smt2
lemma "\<exists>x::int. 0 < x"
- using [[smt_oracle=true]] (* no Z3 proof *)
- by smt
+ using [[smt2_oracle=true]] (* no Z3 proof *)
+ by smt2
lemma "\<exists>x::real. 0 < x"
- using [[smt_oracle=true]] (* no Z3 proof *)
- by smt
+ using [[smt2_oracle=true]] (* no Z3 proof *)
+ by smt2
lemma "\<forall>x::int. \<exists>y. y > x"
- using [[smt_oracle=true]] (* no Z3 proof *)
- by smt
+ using [[smt2_oracle=true]] (* no Z3 proof *)
+ by smt2
-lemma "\<forall>x y::int. (x = 0 \<and> y = 1) \<longrightarrow> x \<noteq> y" by smt
+lemma "\<forall>x y::int. (x = 0 \<and> y = 1) \<longrightarrow> x \<noteq> y" by smt2
-lemma "\<exists>x::int. \<forall>y. x < y \<longrightarrow> y < 0 \<or> y >= 0" by smt
+lemma "\<exists>x::int. \<forall>y. x < y \<longrightarrow> y < 0 \<or> y >= 0" by smt2
-lemma "\<forall>x y::int. x < y \<longrightarrow> (2 * x + 1) < (2 * y)" by smt
+lemma "\<forall>x y::int. x < y \<longrightarrow> (2 * x + 1) < (2 * y)" by smt2
-lemma "\<forall>x y::int. (2 * x + 1) \<noteq> (2 * y)" by smt
+lemma "\<forall>x y::int. (2 * x + 1) \<noteq> (2 * y)" by smt2
-lemma "\<forall>x y::int. x + y > 2 \<or> x + y = 2 \<or> x + y < 2" by smt
+lemma "\<forall>x y::int. x + y > 2 \<or> x + y = 2 \<or> x + y < 2" by smt2
-lemma "\<forall>x::int. if x > 0 then x + 1 > 0 else 1 > x" by smt
+lemma "\<forall>x::int. if x > 0 then x + 1 > 0 else 1 > x" by smt2
-lemma "if (ALL x::int. x < 0 \<or> x > 0) then False else True" by smt
+lemma "if (ALL x::int. x < 0 \<or> x > 0) then False else True" by smt2
-lemma "(if (ALL x::int. x < 0 \<or> x > 0) then -1 else 3) > (0::int)" by smt
+lemma "(if (ALL x::int. x < 0 \<or> x > 0) then -1 else 3) > (0::int)" by smt2
-lemma "~ (\<exists>x y z::int. 4 * x + -6 * y = (1::int))" by smt
+lemma "~ (\<exists>x y z::int. 4 * x + -6 * y = (1::int))" by smt2
-lemma "\<exists>x::int. \<forall>x y. 0 < x \<and> 0 < y \<longrightarrow> (0::int) < x + y" by smt
+lemma "\<exists>x::int. \<forall>x y. 0 < x \<and> 0 < y \<longrightarrow> (0::int) < x + y" by smt2
-lemma "\<exists>u::int. \<forall>(x::int) y::real. 0 < x \<and> 0 < y \<longrightarrow> -1 < x" by smt
+lemma "\<exists>u::int. \<forall>(x::int) y::real. 0 < x \<and> 0 < y \<longrightarrow> -1 < x" by smt2
-lemma "\<exists>x::int. (\<forall>y. y \<ge> x \<longrightarrow> y > 0) \<longrightarrow> x > 0" by smt
+lemma "\<exists>x::int. (\<forall>y. y \<ge> x \<longrightarrow> y > 0) \<longrightarrow> x > 0" by smt2
-lemma "\<forall>x::int. SMT.trigger [[SMT.pat x]] (x < a \<longrightarrow> 2 * x < 2 * a)" by smt
+lemma "\<forall>x::int. SMT2.trigger [[SMT2.pat x]] (x < a \<longrightarrow> 2 * x < 2 * a)" by smt2
-lemma "\<forall>(a::int) b::int. 0 < b \<or> b < 1" by smt
+lemma "\<forall>(a::int) b::int. 0 < b \<or> b < 1" by smt2
subsection {* Non-linear arithmetic over integers and reals *}
lemma "a > (0::int) \<Longrightarrow> a*b > 0 \<Longrightarrow> b > 0"
- using [[smt_oracle, z3_with_extensions]]
- by smt
+ using [[smt2_oracle, z3_new_extensions]]
+ by smt2
lemma "(a::int) * (x + 1 + y) = a * x + a * (y + 1)"
- using [[z3_with_extensions]]
- by smt
+ using [[z3_new_extensions]]
+ by smt2
lemma "((x::real) * (1 + y) - x * (1 - y)) = (2 * x * y)"
- using [[z3_with_extensions]]
- by smt
+ using [[z3_new_extensions]]
+ by smt2
lemma
"(U::int) + (1 + p) * (b + e) + p * d =
U + (2 * (1 + p) * (b + e) + (1 + p) * d + d * p) - (1 + p) * (b + d + e)"
- using [[z3_with_extensions]]
- by smt
+ using [[z3_new_extensions]]
+ by smt2
-lemma [z3_rule]:
+lemma [z3_rule, z3_new_rule]:
fixes x :: "int"
assumes "x * y \<le> 0" and "\<not> y \<le> 0" and "\<not> x \<le> 0"
shows False
using assms by (metis mult_le_0_iff)
lemma "x * y \<le> (0 :: int) \<Longrightarrow> x \<le> 0 \<or> y \<le> 0"
- using [[z3_with_extensions]]
- by smt
-
+ using [[z3_with_extensions]] [[z3_new_extensions]]
+ by smt (* smt2 FIXME: "th-lemma" tactic fails *)
subsection {* Linear arithmetic for natural numbers *}
-lemma "2 * (x::nat) ~= 1" by smt
+lemma "2 * (x::nat) ~= 1" by smt2
-lemma "a < 3 \<Longrightarrow> (7::nat) > 2 * a" by smt
+lemma "a < 3 \<Longrightarrow> (7::nat) > 2 * a" by smt2
-lemma "let x = (1::nat) + y in x - y > 0 * x" by smt
+lemma "let x = (1::nat) + y in x - y > 0 * x" by smt2
lemma
"let x = (1::nat) + y in
let P = (if x > 0 then True else False) in
False \<or> P = (x - 1 = y) \<or> (\<not>P \<longrightarrow> False)"
- by smt
+ by smt2
-lemma "int (nat \<bar>x::int\<bar>) = \<bar>x\<bar>" by smt
+lemma "int (nat \<bar>x::int\<bar>) = \<bar>x\<bar>" by smt2
definition prime_nat :: "nat \<Rightarrow> bool" where
"prime_nat p = (1 < p \<and> (\<forall>m. m dvd p --> m = 1 \<or> m = p))"
-lemma "prime_nat (4*m + 1) \<Longrightarrow> m \<ge> (1::nat)" by (smt prime_nat_def)
+lemma "prime_nat (4*m + 1) \<Longrightarrow> m \<ge> (1::nat)" by (smt2 prime_nat_def)
section {* Pairs *}
lemma "fst (x, y) = a \<Longrightarrow> x = a"
using fst_conv
- by smt
+ by smt2
lemma "p1 = (x, y) \<and> p2 = (y, x) \<Longrightarrow> fst p1 = snd p2"
using fst_conv snd_conv
- by smt
+ by smt2
section {* Higher-order problems and recursion *}
lemma "i \<noteq> i1 \<and> i \<noteq> i2 \<Longrightarrow> (f (i1 := v1, i2 := v2)) i = f i"
using fun_upd_same fun_upd_apply
- by smt
+ by smt2
lemma "(f g (x::'a::type) = (g x \<and> True)) \<or> (f g x = True) \<or> (g x = True)"
- by smt
+ by smt2
-lemma "id x = x \<and> id True = True" by (smt id_def)
+lemma "id x = x \<and> id True = True" by (smt id_def) (* smt2 FIXME: Option *)
lemma "i \<noteq> i1 \<and> i \<noteq> i2 \<Longrightarrow> ((f (i1 := v1)) (i2 := v2)) i = f i"
using fun_upd_same fun_upd_apply
- by smt
+ by smt2
lemma
"f (\<exists>x. g x) \<Longrightarrow> True"
"f (\<forall>x. g x) \<Longrightarrow> True"
- by smt+
+ by smt2+
-lemma True using let_rsp by smt
+lemma True using let_rsp by smt2
-lemma "le = op \<le> \<Longrightarrow> le (3::int) 42" by smt
+lemma "le = op \<le> \<Longrightarrow> le (3::int) 42" by smt2
-lemma "map (\<lambda>i::nat. i + 1) [0, 1] = [1, 2]" by (smt list.map)
+lemma "map (\<lambda>i::nat. i + 1) [0, 1] = [1, 2]" by (smt2 list.map)
-lemma "(ALL x. P x) | ~ All P" by smt
+lemma "(ALL x. P x) \<or> ~ All P" by smt2
fun dec_10 :: "nat \<Rightarrow> nat" where
"dec_10 n = (if n < 10 then n else dec_10 (n - 10))"
-lemma "dec_10 (4 * dec_10 4) = 6" by (smt dec_10.simps)
+lemma "dec_10 (4 * dec_10 4) = 6" by (smt2 dec_10.simps)
axiomatization
@@ -505,35 +505,36 @@
(eval_dioph ks (map (\<lambda>x. x mod 2) xs) mod 2 = l mod 2 \<and>
eval_dioph ks (map (\<lambda>x. x div 2) xs) =
(l - eval_dioph ks (map (\<lambda>x. x mod 2) xs)) div 2)"
- using [[smt_oracle=true]] (*FIXME*)
- using [[z3_with_extensions]]
- by (smt eval_dioph_mod[where n=2] eval_dioph_div_mult[where n=2])
+ using [[smt2_oracle=true]] (*FIXME*)
+ using [[z3_new_extensions]]
+ by (smt2 eval_dioph_mod[where n=2] eval_dioph_div_mult[where n=2])
context complete_lattice
begin
lemma
- assumes "Sup { a | i::bool . True } \<le> Sup { b | i::bool . True }"
- and "Sup { b | i::bool . True } \<le> Sup { a | i::bool . True }"
- shows "Sup { a | i::bool . True } \<le> Sup { a | i::bool . True }"
- using assms by (smt order_trans)
+ assumes "Sup {a | i::bool. True} \<le> Sup {b | i::bool. True}"
+ and "Sup {b | i::bool. True} \<le> Sup {a | i::bool. True}"
+ shows "Sup {a | i::bool. True} \<le> Sup {a | i::bool. True}"
+ using assms by (smt2 order_trans)
end
-
section {* Monomorphization examples *}
definition Pred :: "'a \<Rightarrow> bool" where "Pred x = True"
-lemma poly_Pred: "Pred x \<and> (Pred [x] \<or> \<not>Pred[x])" by (simp add: Pred_def)
-lemma "Pred (1::int)" by (smt poly_Pred)
+
+lemma poly_Pred: "Pred x \<and> (Pred [x] \<or> \<not> Pred [x])" by (simp add: Pred_def)
+lemma "Pred (1::int)" by (smt2 poly_Pred)
axiomatization g :: "'a \<Rightarrow> nat"
axiomatization where
g1: "g (Some x) = g [x]" and
g2: "g None = g []" and
g3: "g xs = length xs"
-lemma "g (Some (3::int)) = g (Some True)" by (smt g1 g2 g3 list.size)
+
+lemma "g (Some (3::int)) = g (Some True)" by (smt g1 g2 g3 list.size) (* smt2 FIXME: Option *)
end