# HG changeset patch
# User boehmes
# Date 1265130681 -3600
# Node ID faeee0e4ac5080f74edb0314a117a48a21c77f28
# Parent  e5cb3a0160942eea7ed62a0c9af58248ff02f099
updated SMT examples

diff -r e5cb3a016094 -r faeee0e4ac50 src/HOL/SMT/Examples/SMT_Examples.thy
--- a/src/HOL/SMT/Examples/SMT_Examples.thy	Tue Feb 02 18:10:41 2010 +0100
+++ b/src/HOL/SMT/Examples/SMT_Examples.thy	Tue Feb 02 18:11:21 2010 +0100
@@ -10,42 +10,34 @@
 
 declare [[smt_solver=z3, z3_proofs=true]]
 
+declare [[smt_certificates="$ISABELLE_SMT/Examples/SMT_Examples.certs"]]
+
 text {*
-To re-generate the certificates, replace the option 'smt_cert' with 'smt_keep'
-(while keeping the paths as they are) and let Isabelle process this theory.
+To avoid re-generation of certificates,
+the following option is set to "false":
 *}
 
+declare [[smt_record=false]] 
+
+
 
 section {* Propositional and first-order logic *}
 
-lemma "True"
-  using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_prop_01"]]
-  by smt
+lemma "True" by smt
 
-lemma "p \<or> \<not>p"
-  using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_prop_02"]]
-  by smt
+lemma "p \<or> \<not>p" by smt
 
-lemma "(p \<and> True) = p"
-  using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_prop_03"]]
-  by smt
+lemma "(p \<and> True) = p" by smt
 
-lemma "(p \<or> q) \<and> \<not>p \<Longrightarrow> q"
-  using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_prop_04"]]
-  by smt
+lemma "(p \<or> q) \<and> \<not>p \<Longrightarrow> q" by smt
 
 lemma "(a \<and> b) \<or> (c \<and> d) \<Longrightarrow> (a \<and> b) \<or> (c \<and> d)"
-  using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_prop_05"]]
   using [[z3_proofs=false]] (* no Z3 proof *)
   by smt
 
-lemma "(p1 \<and> p2) \<or> p3 \<longrightarrow> (p1 \<longrightarrow> (p3 \<and> p2) \<or> (p1 \<and> p3)) \<or> p1"
-  using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_prop_06"]]
-  by smt
+lemma "(p1 \<and> p2) \<or> p3 \<longrightarrow> (p1 \<longrightarrow> (p3 \<and> p2) \<or> (p1 \<and> p3)) \<or> p1" by smt
 
-lemma "P=P=P=P=P=P=P=P=P=P"
-  using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_prop_07"]]
-  by smt
+lemma "P=P=P=P=P=P=P=P=P=P" by smt
 
 lemma 
   assumes "a | b | c | d"
@@ -55,14 +47,11 @@
       and "~(d | False) | c"
       and "~(c | (~p & (p | (q & ~q))))"
   shows False
-  using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_prop_08"]]
   using assms by smt
 
 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"
-  using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_prop_09"]]
-  by (smt symm_f)
+lemma "a = a \<and> symm_f a b = symm_f b a" by (smt symm_f)
 
 (* 
 Taken from ~~/src/HOL/ex/SAT_Examples.thy.
@@ -254,106 +243,69 @@
   and "~x29 | ~x58"
   and "~x28 | ~x58"
   shows False
-  using assms
-  using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_prop_10"]]
-  by smt
+  using assms by smt
 
 lemma "\<forall>x::int. P x \<longrightarrow> (\<forall>y::int. P x \<or> P y)"
-  using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_fol_01"]]
   by smt
 
 lemma 
   assumes "(\<forall>x y. P x y = x)"
   shows "(\<exists>y. P x y) = P x c"
-  using assms 
-  using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_fol_02"]]
-  by smt
+  using assms by smt
 
 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
-  using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_fol_03"]]
-  by smt
+  using assms by smt
 
 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
-  using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_fol_04"]]
-  by smt
+  using assms by smt
 
 
 section {* Arithmetic *}
 
 subsection {* Linear arithmetic over integers and reals *}
 
-lemma "(3::int) = 3"
-  using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_linarith_01"]]
-  by smt
+lemma "(3::int) = 3" by smt
 
-lemma "(3::real) = 3"
-  using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_linarith_02"]]
-  by smt
+lemma "(3::real) = 3" by smt
 
-lemma "(3 :: int) + 1 = 4"
-  using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_linarith_03"]]
-  by smt
+lemma "(3 :: int) + 1 = 4" by smt
 
-lemma "x + (y + z) = y + (z + (x::int))"
-  using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_linarith_04"]]
-  by smt
+lemma "x + (y + z) = y + (z + (x::int))" by smt
 
-lemma "max (3::int) 8 > 5"
-  using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_linarith_05"]]
-  by smt
+lemma "max (3::int) 8 > 5" by smt
 
-lemma "abs (x :: real) + abs y \<ge> abs (x + y)"
-  using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_linarith_06"]]
-  by smt
+lemma "abs (x :: real) + abs y \<ge> abs (x + y)" by smt
 
-lemma "P ((2::int) < 3) = P True"
-  using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_linarith_07"]]
-  by smt
+lemma "P ((2::int) < 3) = P True" by smt
 
-lemma "x + 3 \<ge> 4 \<or> x < (1::int)"
-  using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_linarith_08"]]
-  by smt
+lemma "x + 3 \<ge> 4 \<or> x < (1::int)" by smt
 
 lemma
   assumes "x \<ge> (3::int)" and "y = x + 4"
   shows "y - x > 0" 
-  using assms
-  using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_linarith_09"]]
-  by smt
+  using assms by smt
 
-lemma "let x = (2 :: int) in x + x \<noteq> 5"
-  using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_linarith_10"]]
-  by smt
+lemma "let x = (2 :: int) in x + x \<noteq> 5" by smt
 
 lemma
   fixes x :: real
   assumes "3 * x + 7 * a < 4" and "3 < 2 * x"
   shows "a < 0"
-  using assms
-  using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_linarith_11"]]
-  by smt
+  using assms by smt
 
-lemma "(0 \<le> y + -1 * x \<or> \<not> 0 \<le> x \<or> 0 \<le> (x::int)) = (\<not> False)"
-  using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_linarith_12"]]
-  by smt
+lemma "(0 \<le> y + -1 * x \<or> \<not> 0 \<le> x \<or> 0 \<le> (x::int)) = (\<not> False)" by smt
 
-lemma "distinct [x < (3::int), 3 \<le> x]"
-  using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_linarith_13"]]
-  by smt
+lemma "distinct [x < (3::int), 3 \<le> x]" by smt
 
 lemma
   assumes "a > (0::int)"
   shows "distinct [a, a * 2, a - a]"
-  using assms
-  using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_linarith_14"]]
-  by smt
+  using assms by smt
 
 lemma "
   (n < m & m < n') | (n < m & m = n') | (n < n' & n' < m) |
@@ -363,7 +315,6 @@
   (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))"
-  using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_linarith_15"]]
   by smt
 
 text{* 
@@ -386,172 +337,109 @@
          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)"
-  using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_linarith_16"]]
   by smt
 
 
-lemma "let P = 2 * x + 1 > x + (x::real) in P \<or> False \<or> P"
-  using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_linarith_17"]]
-  by smt
+lemma "let P = 2 * x + 1 > x + (x::real) in P \<or> False \<or> P" by smt
+
+lemma "x + (let y = x mod 2 in 2 * y + 1) \<ge> x + (1::int)" by smt
 
-lemma "x + (let y = x mod 2 in 2 * y + 1) \<ge> x + (1::int)"
-  using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_linarith_18"]]
-  by smt
+lemma "x + (let y = x mod 2 in y + y) < x + (3::int)" by smt
 
-lemma "x + (let y = x mod 2 in y + y) < x + (3::int)"
-  using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_linarith_19"]]
-  by smt
-
-lemma 
+lemma
   assumes "x \<noteq> (0::real)"
   shows "x + x \<noteq> (let P = (abs x > 1) in if P \<or> \<not>P then 4 else 2) * x"
-  using assms
-  using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_linarith_20"]]
-  by smt
+  using assms by smt
 
 lemma                                                                         
   assumes "(n + m) mod 2 = 0" and "n mod 4 = 3"                               
   shows "n mod 2 = 1 & m mod 2 = (1::int)"      
-  using assms
-  using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_linarith_21"]]
-  by smt
+  using assms by smt
 
 
 subsection {* Linear arithmetic with quantifiers *}
 
-lemma "~ (\<exists>x::int. False)"
-  using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_arith_quant_01"]]
-  by smt
+lemma "~ (\<exists>x::int. False)" by smt
 
-lemma "~ (\<exists>x::real. False)"
-  using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_arith_quant_02"]]
-  by smt
+lemma "~ (\<exists>x::real. False)" by smt
 
 lemma "\<exists>x::int. 0 < x"
-  using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_arith_quant_03"]]
   using [[z3_proofs=false]] (* no Z3 proof *)
   by smt
 
 lemma "\<exists>x::real. 0 < x"
-  using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_arith_quant_04"]]
   using [[z3_proofs=false]] (* no Z3 proof *)
   by smt
 
 lemma "\<forall>x::int. \<exists>y. y > x"
-  using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_arith_quant_05"]]
   using [[z3_proofs=false]] (* no Z3 proof *)
   by smt
 
-lemma "\<forall>x y::int. (x = 0 \<and> y = 1) \<longrightarrow> x \<noteq> y"
-  using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_arith_quant_06"]]
-  by smt
+lemma "\<forall>x y::int. (x = 0 \<and> y = 1) \<longrightarrow> x \<noteq> y" by smt
 
-lemma "\<exists>x::int. \<forall>y. x < y \<longrightarrow> y < 0 \<or> y >= 0"
-  using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_arith_quant_07"]]
-  by smt
+lemma "\<exists>x::int. \<forall>y. x < y \<longrightarrow> y < 0 \<or> y >= 0" by smt
 
-lemma "\<forall>x y::int. x < y \<longrightarrow> (2 * x + 1) < (2 * y)"
-  using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_arith_quant_08"]]
-  by smt
+lemma "\<forall>x y::int. x < y \<longrightarrow> (2 * x + 1) < (2 * y)"  by smt
 
-lemma "\<forall>x y::int. (2 * x + 1) \<noteq> (2 * y)"
-  using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_arith_quant_09"]]
-  by smt
+lemma "\<forall>x y::int. (2 * x + 1) \<noteq> (2 * y)" by smt
 
-lemma "\<forall>x y::int. x + y > 2 \<or> x + y = 2 \<or> x + y < 2"
-  using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_arith_quant_10"]]
-  by smt
+lemma "\<forall>x y::int. x + y > 2 \<or> x + y = 2 \<or> x + y < 2" by smt
 
-lemma "\<forall>x::int. if x > 0 then x + 1 > 0 else 1 > x"
-  using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_arith_quant_11"]]
-  by smt
+lemma "\<forall>x::int. if x > 0 then x + 1 > 0 else 1 > x" by smt
 
-lemma "if (ALL x::int. x < 0 \<or> x > 0) then False else True"
-  using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_arith_quant_12"]]
-  by smt
+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 -1 else 3) > (0::int)"
-  using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_arith_quant_13"]]
-  by smt
+lemma "(if (ALL x::int. x < 0 \<or> x > 0) then -1 else 3) > (0::int)" by smt
 
-lemma "~ (\<exists>x y z::int. 4 * x + -6 * y = (1::int))"
-  using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_arith_quant_14"]]
-  by smt
+lemma "~ (\<exists>x y z::int. 4 * x + -6 * y = (1::int))" by smt
 
-lemma "\<exists>x::int. \<forall>x y. 0 < x \<and> 0 < y \<longrightarrow> (0::int) < x + y"
-  using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_arith_quant_15"]]
-  by smt
+lemma "\<exists>x::int. \<forall>x y. 0 < x \<and> 0 < y \<longrightarrow> (0::int) < x + y" by smt
 
-lemma "\<exists>u::int. \<forall>(x::int) y::real. 0 < x \<and> 0 < y \<longrightarrow> -1 < x"
-  using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_arith_quant_16"]]
-  by smt
+lemma "\<exists>u::int. \<forall>(x::int) y::real. 0 < x \<and> 0 < y \<longrightarrow> -1 < x" by smt
 
-lemma "\<exists>x::int. (\<forall>y. y \<ge> x \<longrightarrow> y > 0) \<longrightarrow> x > 0"
-  using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_arith_quant_17"]]
-  by smt
+lemma "\<exists>x::int. (\<forall>y. y \<ge> x \<longrightarrow> y > 0) \<longrightarrow> x > 0" by smt
 
-lemma "\<forall>x::int. trigger [pat x] (x < a \<longrightarrow> 2 * x < 2 * a)"
-  using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_arith_quant_18"]]
-  by smt
+lemma "\<forall>x::int. trigger [pat x] (x < a \<longrightarrow> 2 * x < 2 * a)" by smt
 
 
 subsection {* Non-linear arithmetic over integers and reals *}
 
 lemma "a > (0::int) \<Longrightarrow> a*b > 0 \<Longrightarrow> b > 0"
-  using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_nlarith_01"]]
   using [[z3_proofs=false]]  -- {* Isabelle's arithmetic decision procedures
     are too weak to automatically prove @{thm zero_less_mult_pos}. *}
   by smt
 
-lemma  "(a::int) * (x + 1 + y) = a * x + a * (y + 1)"
-  using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_nlarith_02"]]
-  by smt
+lemma  "(a::int) * (x + 1 + y) = a * x + a * (y + 1)" by smt
 
-lemma "((x::real) * (1 + y) - x * (1 - y)) = (2 * x * y)"
-  using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_nlarith_03"]]
-  by smt
+lemma "((x::real) * (1 + y) - x * (1 - y)) = (2 * x * y)" by smt
 
 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 [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_nlarith_04"]]
   by smt
 
 
 subsection {* Linear arithmetic for natural numbers *}
 
-lemma "2 * (x::nat) ~= 1"
-  using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_nat_arith_01"]]
-  by smt
+lemma "2 * (x::nat) ~= 1" by smt
 
-lemma "a < 3 \<Longrightarrow> (7::nat) > 2 * a"
-  using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_nat_arith_02"]]
-  by smt
+lemma "a < 3 \<Longrightarrow> (7::nat) > 2 * a" by smt
 
-lemma "let x = (1::nat) + y in x - y > 0 * x"
-  using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_nat_arith_03"]]
-  by smt
+lemma "let x = (1::nat) + y in x - y > 0 * x" by smt
 
 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)"
-  using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_nat_arith_04"]]
   by smt
 
-lemma "distinct [a + (1::nat), a * 2 + 3, a - a]"
-  using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_nat_arith_05"]]
-  by smt
+lemma "distinct [a + (1::nat), a * 2 + 3, a - a]" by smt
 
-lemma "int (nat \<bar>x::int\<bar>) = \<bar>x\<bar>"
-  using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_nat_arith_06"]]
-  by smt
+lemma "int (nat \<bar>x::int\<bar>) = \<bar>x\<bar>" by smt
 
 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)"
-  using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_nat_arith_07"]]
-  by (smt prime_nat_def)
+lemma "prime_nat (4*m + 1) \<Longrightarrow> m \<ge> (1::nat)" by (smt prime_nat_def)
 
 
 section {* Bitvectors *}
@@ -568,107 +456,59 @@
 
 subsection {* Bitvector arithmetic *}
 
-lemma "(27 :: 4 word) = -5" 
-  using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_bv_arith_01"]]
-  by smt
+lemma "(27 :: 4 word) = -5" by smt
 
-lemma "(27 :: 4 word) = 11"
-  using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_bv_arith_02"]]
-  by smt
+lemma "(27 :: 4 word) = 11" by smt
 
-lemma "23 < (27::8 word)" 
-  using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_bv_arith_03"]]
-  by smt
+lemma "23 < (27::8 word)" by smt
 
-lemma "27 + 11 = (6::5 word)" 
-  using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_bv_arith_04"]]
-  by smt
+lemma "27 + 11 = (6::5 word)" by smt
+
+lemma "7 * 3 = (21::8 word)" by smt
 
-lemma "7 * 3 = (21::8 word)" 
-  using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_bv_arith_05"]]
-  by smt
-lemma "11 - 27 = (-16::8 word)" 
-  using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_bv_arith_06"]]
-  by smt
+lemma "11 - 27 = (-16::8 word)" by smt
 
-lemma "- -11 = (11::5 word)" 
-  using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_bv_arith_07"]]
-  by smt
+lemma "- -11 = (11::5 word)" by smt
 
-lemma "-40 + 1 = (-39::7 word)" 
-  using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_bv_arith_08"]]
-  by smt
+lemma "-40 + 1 = (-39::7 word)" by smt
 
-lemma "a + 2 * b + c - b = (b + c) + (a :: 32 word)" 
-  using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_bv_arith_09"]]
-  by smt
+lemma "a + 2 * b + c - b = (b + c) + (a :: 32 word)" by smt
 
-lemma "x = (5 :: 4 word) \<Longrightarrow> 4 * x = 4" 
-  using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_bv_arith_10"]]
-  by smt
+lemma "x = (5 :: 4 word) \<Longrightarrow> 4 * x = 4" by smt
 
 
 subsection {* Bit-level logic *}
 
-lemma "0b110 AND 0b101 = (0b100 :: 32 word)"
-  using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_bv_bit_01"]]
-  by smt
+lemma "0b110 AND 0b101 = (0b100 :: 32 word)" by smt
 
-lemma "0b110 OR 0b011 = (0b111 :: 8 word)" 
-  using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_bv_bit_02"]]
-  by smt
+lemma "0b110 OR 0b011 = (0b111 :: 8 word)" by smt
 
-lemma "0xF0 XOR 0xFF = (0x0F :: 8 word)" 
-  using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_bv_bit_03"]]
-  by smt
+lemma "0xF0 XOR 0xFF = (0x0F :: 8 word)" by smt
 
-lemma "NOT (0xF0 :: 16 word) = 0xFF0F" 
-  using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_bv_bit_04"]]
-  by smt
+lemma "NOT (0xF0 :: 16 word) = 0xFF0F" by smt
 
-lemma "word_cat (27::4 word) (27::8 word) = (2843::12 word)" 
-  using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_bv_bit_05"]]
-  by smt
+lemma "word_cat (27::4 word) (27::8 word) = (2843::12 word)" by smt
 
 lemma "word_cat (0b0011::4 word) (0b1111::6word) = (0b0011001111 :: 10 word)"
-  using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_bv_bit_06"]]
-  by smt
-
-lemma "slice 1 (0b10110 :: 4 word) = (0b11 :: 2 word)" 
-  using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_bv_bit_07"]]
   by smt
 
-lemma "ucast (0b1010 :: 4 word) = (0b1010 :: 10 word)" 
-  using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_bv_bit_08"]]
-  by smt
+lemma "slice 1 (0b10110 :: 4 word) = (0b11 :: 2 word)" by smt
 
-lemma "scast (0b1010 :: 4 word) = (0b111010 :: 6 word)" 
-  using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_bv_bit_09"]]
-  by smt
+lemma "ucast (0b1010 :: 4 word) = (0b1010 :: 10 word)" by smt
 
-lemma "bv_lshr 0b10011 2 = (0b100::8 word)" 
-  using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_bv_bit_10"]]
-  by smt
+lemma "scast (0b1010 :: 4 word) = (0b111010 :: 6 word)" by smt
 
-lemma "bv_ashr 0b10011 2 = (0b100::8 word)" 
-  using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_bv_bit_11"]]
-  by smt
+lemma "bv_lshr 0b10011 2 = (0b100::8 word)" by smt
 
-lemma "word_rotr 2 0b0110 = (0b1001::4 word)" 
-  using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_bv_bit_12"]]
-  by smt
+lemma "bv_ashr 0b10011 2 = (0b100::8 word)" by smt
 
-lemma "word_rotl 1 0b1110 = (0b1101::4 word)" 
-  using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_bv_bit_13"]]
-  by smt
+lemma "word_rotr 2 0b0110 = (0b1001::4 word)" by smt
 
-lemma "(x AND 0xff00) OR (x AND 0x00ff) = (x::16 word)" 
-  using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_bv_bit_14"]]
-  by smt
+lemma "word_rotl 1 0b1110 = (0b1101::4 word)" by smt
 
-lemma "w < 256 \<Longrightarrow> (w :: 16 word) AND 0x00FF = w" 
-  using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_bv_bit_15"]]
-  by smt
+lemma "(x AND 0xff00) OR (x AND 0x00ff) = (x::16 word)" by smt
+
+lemma "w < 256 \<Longrightarrow> (w :: 16 word) AND 0x00FF = w" by smt
 
 end
 
@@ -681,57 +521,37 @@
   shows "\<forall>i::int. i < 0 \<longrightarrow> (\<forall>x::2 word. bv2int x > i)"
   using assms 
   using [[smt_solver=z3]]
-  using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_bv_01"]]
   by smt
 
 lemma "P (0 \<le> (a :: 4 word)) = P True"
   using [[smt_solver=z3, z3_proofs=false]]
-  using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_bv_02"]]
   by smt
 
 
 section {* Pairs *}
 
-lemma "fst (x, y) = a \<Longrightarrow> x = a"
-  using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_pair_01"]]
-  by smt
+lemma "fst (x, y) = a \<Longrightarrow> x = a" by smt
 
-lemma "p1 = (x, y) \<and> p2 = (y, x) \<Longrightarrow> fst p1 = snd p2"
-  using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_pair_02"]]
-  by smt
+lemma "p1 = (x, y) \<and> p2 = (y, x) \<Longrightarrow> fst p1 = snd p2" by smt
 
 
 section {* Higher-order problems and recursion *}
 
-lemma "i \<noteq> i1 \<and> i \<noteq> i2 \<Longrightarrow> (f (i1 := v1, i2 := v2)) i = f i"
-  using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_hol_01"]]
-  by smt
+lemma "i \<noteq> i1 \<and> i \<noteq> i2 \<Longrightarrow> (f (i1 := v1, i2 := v2)) i = f i" by smt
 
-lemma "(f g x = (g x \<and> True)) \<or> (f g x = True) \<or> (g x = True)"
-  using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_hol_02"]]
-  by smt
+lemma "(f g x = (g x \<and> True)) \<or> (f g x = True) \<or> (g x = True)" by smt
 
-lemma "id 3 = 3 \<and> id True = True"
-  using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_hol_03"]]
-  by (smt id_def)
+lemma "id 3 = 3 \<and> id True = True" by (smt id_def)
 
-lemma "i \<noteq> i1 \<and> i \<noteq> i2 \<Longrightarrow> ((f (i1 := v1)) (i2 := v2)) i = f i"
-  using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_hol_04"]]
-  by smt
+lemma "i \<noteq> i1 \<and> i \<noteq> i2 \<Longrightarrow> ((f (i1 := v1)) (i2 := v2)) i = f i" by smt
 
-lemma "map (\<lambda>i::nat. i + 1) [0, 1] = [1, 2]"
-  using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_hol_05"]]
-  by (smt map.simps)
+lemma "map (\<lambda>i::nat. i + 1) [0, 1] = [1, 2]" by (smt map.simps)
 
-lemma "(ALL x. P x) | ~ All P"
-  using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_hol_06"]]
-  by smt
+lemma "(ALL x. P x) | ~ All P" by smt
 
 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"
-  using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_hol_07"]]
-  by (smt dec_10.simps)
+lemma "dec_10 (4 * dec_10 4) = 6" by (smt dec_10.simps)
 
 axiomatization
   eval_dioph :: "int list \<Rightarrow> nat list \<Rightarrow> int"
@@ -747,7 +567,6 @@
    (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_cert="$ISABELLE_SMT/Examples/cert/z3_hol_08"]]
   by (smt eval_dioph_mod[where n=2] eval_dioph_div_mult[where n=2])
 
 
@@ -755,17 +574,13 @@
 
 definition P :: "'a \<Rightarrow> bool" where "P x = True"
 lemma poly_P: "P x \<and> (P [x] \<or> \<not>P[x])" by (simp add: P_def)
-lemma "P (1::int)"
-  using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_mono_01"]]
-  by (smt poly_P)
+lemma "P (1::int)" by (smt poly_P)
 
 consts g :: "'a \<Rightarrow> nat"
 axioms
   g1: "g (Some x) = g [x]"
   g2: "g None = g []"
   g3: "g xs = length xs"
-lemma "g (Some (3::int)) = g (Some True)"
-  using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_mono_02"]]
-  by (smt g1 g2 g3 list.size)
+lemma "g (Some (3::int)) = g (Some True)" by (smt g1 g2 g3 list.size)
 
 end