# 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 \ \p" - using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_prop_02"]] - by smt +lemma "p \ \p" by smt -lemma "(p \ True) = p" - using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_prop_03"]] - by smt +lemma "(p \ True) = p" by smt -lemma "(p \ q) \ \p \ q" - using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_prop_04"]] - by smt +lemma "(p \ q) \ \p \ q" by smt lemma "(a \ b) \ (c \ d) \ (a \ b) \ (c \ d)" - using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_prop_05"]] using [[z3_proofs=false]] (* no Z3 proof *) by smt -lemma "(p1 \ p2) \ p3 \ (p1 \ (p3 \ p2) \ (p1 \ p3)) \ p1" - using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_prop_06"]] - by smt +lemma "(p1 \ p2) \ p3 \ (p1 \ (p3 \ p2) \ (p1 \ p3)) \ p1" by smt -lemma "P=P=P=P=P=P=P=P=P=P" - using [[smt_cert="$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="$ISABELLE_SMT/Examples/cert/z3_prop_08"]] using assms by smt axiomatization symm_f :: "'a \ 'a \ 'a" where symm_f: "symm_f x y = symm_f y x" -lemma "a = a \ symm_f a b = symm_f b a" - using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_prop_09"]] - by (smt symm_f) +lemma "a = a \ 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="$ISABELLE_SMT/Examples/cert/z3_prop_10"]] - by smt + using assms by smt lemma "\x::int. P x \ (\y::int. P x \ P y)" - using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_fol_01"]] by smt lemma assumes "(\x y. P x y = x)" shows "(\y. P x y) = P x c" - using assms - using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_fol_02"]] - by smt + using assms by smt lemma assumes "(\x y. P x y = x)" and "(\x. \y. P x y) = (\x. P x c)" shows "(EX y. P x y) = P x c" - using assms - using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_fol_03"]] - by smt + using assms by smt lemma assumes "if P x then \(\y. P y) else (\y. \P y)" shows "P x \ 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 \ abs (x + y)" - using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_linarith_06"]] - by smt +lemma "abs (x :: real) + abs y \ abs (x + y)" by smt -lemma "P ((2::int) < 3) = P True" - using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_linarith_07"]] - by smt +lemma "P ((2::int) < 3) = P True" by smt -lemma "x + 3 \ 4 \ x < (1::int)" - using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_linarith_08"]] - by smt +lemma "x + 3 \ 4 \ x < (1::int)" by smt lemma assumes "x \ (3::int)" and "y = x + 4" shows "y - x > 0" - using assms - using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_linarith_09"]] - by smt + using assms by smt -lemma "let x = (2 :: int) in x + x \ 5" - using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_linarith_10"]] - by smt +lemma "let x = (2 :: int) in x + x \ 5" by smt lemma fixes x :: real assumes "3 * x + 7 * a < 4" and "3 < 2 * x" shows "a < 0" - using assms - using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_linarith_11"]] - by smt + using assms by smt -lemma "(0 \ y + -1 * x \ \ 0 \ x \ 0 \ (x::int)) = (\ False)" - using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_linarith_12"]] - by smt +lemma "(0 \ y + -1 * x \ \ 0 \ x \ 0 \ (x::int)) = (\ False)" by smt -lemma "distinct [x < (3::int), 3 \ x]" - using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_linarith_13"]] - by smt +lemma "distinct [x < (3::int), 3 \ 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 \ \ x1 = x10 & x2 = (x11::int)" - using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_linarith_16"]] by smt -lemma "let P = 2 * x + 1 > x + (x::real) in P \ False \ P" - using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_linarith_17"]] - by smt +lemma "let P = 2 * x + 1 > x + (x::real) in P \ False \ P" by smt + +lemma "x + (let y = x mod 2 in 2 * y + 1) \ x + (1::int)" by smt -lemma "x + (let y = x mod 2 in 2 * y + 1) \ 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 \ (0::real)" shows "x + x \ (let P = (abs x > 1) in if P \ \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 "~ (\x::int. False)" - using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_arith_quant_01"]] - by smt +lemma "~ (\x::int. False)" by smt -lemma "~ (\x::real. False)" - using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_arith_quant_02"]] - by smt +lemma "~ (\x::real. False)" by smt lemma "\x::int. 0 < x" - using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_arith_quant_03"]] using [[z3_proofs=false]] (* no Z3 proof *) by smt lemma "\x::real. 0 < x" - using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_arith_quant_04"]] using [[z3_proofs=false]] (* no Z3 proof *) by smt lemma "\x::int. \y. y > x" - using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_arith_quant_05"]] using [[z3_proofs=false]] (* no Z3 proof *) by smt -lemma "\x y::int. (x = 0 \ y = 1) \ x \ y" - using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_arith_quant_06"]] - by smt +lemma "\x y::int. (x = 0 \ y = 1) \ x \ y" by smt -lemma "\x::int. \y. x < y \ y < 0 \ y >= 0" - using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_arith_quant_07"]] - by smt +lemma "\x::int. \y. x < y \ y < 0 \ y >= 0" by smt -lemma "\x y::int. x < y \ (2 * x + 1) < (2 * y)" - using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_arith_quant_08"]] - by smt +lemma "\x y::int. x < y \ (2 * x + 1) < (2 * y)" by smt -lemma "\x y::int. (2 * x + 1) \ (2 * y)" - using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_arith_quant_09"]] - by smt +lemma "\x y::int. (2 * x + 1) \ (2 * y)" by smt -lemma "\x y::int. x + y > 2 \ x + y = 2 \ x + y < 2" - using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_arith_quant_10"]] - by smt +lemma "\x y::int. x + y > 2 \ x + y = 2 \ x + y < 2" by smt -lemma "\x::int. if x > 0 then x + 1 > 0 else 1 > x" - using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_arith_quant_11"]] - by smt +lemma "\x::int. if x > 0 then x + 1 > 0 else 1 > x" by smt -lemma "if (ALL x::int. x < 0 \ x > 0) then False else True" - using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_arith_quant_12"]] - by smt +lemma "if (ALL x::int. x < 0 \ x > 0) then False else True" by smt -lemma "(if (ALL x::int. x < 0 \ x > 0) then -1 else 3) > (0::int)" - using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_arith_quant_13"]] - by smt +lemma "(if (ALL x::int. x < 0 \ x > 0) then -1 else 3) > (0::int)" by smt -lemma "~ (\x y z::int. 4 * x + -6 * y = (1::int))" - using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_arith_quant_14"]] - by smt +lemma "~ (\x y z::int. 4 * x + -6 * y = (1::int))" by smt -lemma "\x::int. \x y. 0 < x \ 0 < y \ (0::int) < x + y" - using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_arith_quant_15"]] - by smt +lemma "\x::int. \x y. 0 < x \ 0 < y \ (0::int) < x + y" by smt -lemma "\u::int. \(x::int) y::real. 0 < x \ 0 < y \ -1 < x" - using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_arith_quant_16"]] - by smt +lemma "\u::int. \(x::int) y::real. 0 < x \ 0 < y \ -1 < x" by smt -lemma "\x::int. (\y. y \ x \ y > 0) \ x > 0" - using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_arith_quant_17"]] - by smt +lemma "\x::int. (\y. y \ x \ y > 0) \ x > 0" by smt -lemma "\x::int. trigger [pat x] (x < a \ 2 * x < 2 * a)" - using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_arith_quant_18"]] - by smt +lemma "\x::int. trigger [pat x] (x < a \ 2 * x < 2 * a)" by smt subsection {* Non-linear arithmetic over integers and reals *} lemma "a > (0::int) \ a*b > 0 \ 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 \ (7::nat) > 2 * a" - using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_nat_arith_02"]] - by smt +lemma "a < 3 \ (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 \ P = (x - 1 = y) \ (\P \ 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 \x::int\) = \x\" - using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_nat_arith_06"]] - by smt +lemma "int (nat \x::int\) = \x\" by smt definition prime_nat :: "nat \ bool" where "prime_nat p = (1 < p \ (\m. m dvd p --> m = 1 \ m = p))" -lemma "prime_nat (4*m + 1) \ m \ (1::nat)" - using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_nat_arith_07"]] - by (smt prime_nat_def) +lemma "prime_nat (4*m + 1) \ m \ (1::nat)" by (smt prime_nat_def) section {* Bitvectors *} @@ -568,107 +456,59 @@ subsection {* Bitvector arithmetic *} -lemma "(27 :: 4 word) = -5" - using [[smt_cert="$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) \ 4 * x = 4" - using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_bv_arith_10"]] - by smt +lemma "x = (5 :: 4 word) \ 4 * x = 4" by smt subsection {* Bit-level logic *} -lemma "0b110 AND 0b101 = (0b100 :: 32 word)" - using [[smt_cert="$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="$ISABELLE_SMT/Examples/cert/z3_bv_bit_14"]] - by smt +lemma "word_rotl 1 0b1110 = (0b1101::4 word)" by smt -lemma "w < 256 \ (w :: 16 word) AND 0x00FF = w" - using [[smt_cert="$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 \ (w :: 16 word) AND 0x00FF = w" by smt end @@ -681,57 +521,37 @@ shows "\i::int. i < 0 \ (\x::2 word. bv2int x > i)" using assms using [[smt_solver=z3]] - using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_bv_01"]] by smt lemma "P (0 \ (a :: 4 word)) = P True" using [[smt_solver=z3, z3_proofs=false]] - using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_bv_02"]] by smt section {* Pairs *} -lemma "fst (x, y) = a \ x = a" - using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_pair_01"]] - by smt +lemma "fst (x, y) = a \ x = a" by smt -lemma "p1 = (x, y) \ p2 = (y, x) \ fst p1 = snd p2" - using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_pair_02"]] - by smt +lemma "p1 = (x, y) \ p2 = (y, x) \ fst p1 = snd p2" by smt section {* Higher-order problems and recursion *} -lemma "i \ i1 \ i \ i2 \ (f (i1 := v1, i2 := v2)) i = f i" - using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_hol_01"]] - by smt +lemma "i \ i1 \ i \ i2 \ (f (i1 := v1, i2 := v2)) i = f i" by smt -lemma "(f g x = (g x \ True)) \ (f g x = True) \ (g x = True)" - using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_hol_02"]] - by smt +lemma "(f g x = (g x \ True)) \ (f g x = True) \ (g x = True)" by smt -lemma "id 3 = 3 \ id True = True" - using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_hol_03"]] - by (smt id_def) +lemma "id 3 = 3 \ id True = True" by (smt id_def) -lemma "i \ i1 \ i \ i2 \ ((f (i1 := v1)) (i2 := v2)) i = f i" - using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_hol_04"]] - by smt +lemma "i \ i1 \ i \ i2 \ ((f (i1 := v1)) (i2 := v2)) i = f i" by smt -lemma "map (\i::nat. i + 1) [0, 1] = [1, 2]" - using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_hol_05"]] - by (smt map.simps) +lemma "map (\i::nat. i + 1) [0, 1] = [1, 2]" by (smt map.simps) -lemma "(ALL x. P x) | ~ All P" - using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_hol_06"]] - by smt +lemma "(ALL x. P x) | ~ All P" by smt fun dec_10 :: "nat \ 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="$ISABELLE_SMT/Examples/cert/z3_hol_07"]] - by (smt dec_10.simps) +lemma "dec_10 (4 * dec_10 4) = 6" by (smt dec_10.simps) axiomatization eval_dioph :: "int list \ nat list \ int" @@ -747,7 +567,6 @@ (eval_dioph ks (map (\x. x mod 2) xs) mod 2 = l mod 2 \ eval_dioph ks (map (\x. x div 2) xs) = (l - eval_dioph ks (map (\x. x mod 2) xs)) div 2)" - using [[smt_cert="$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 \ bool" where "P x = True" lemma poly_P: "P x \ (P [x] \ \P[x])" by (simp add: P_def) -lemma "P (1::int)" - using [[smt_cert="$ISABELLE_SMT/Examples/cert/z3_mono_01"]] - by (smt poly_P) +lemma "P (1::int)" by (smt poly_P) consts g :: "'a \ nat" axioms g1: "g (Some x) = g [x]" g2: "g None = g []" g3: "g xs = length xs" -lemma "g (Some (3::int)) = g (Some True)" - using [[smt_cert="$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