(* Title: HOL/Decision_Procs/Reflective_Field.thy
Author: Stefan Berghofer
Reducing equalities in fields to equalities in rings.
*)
theory Reflective_Field
imports Commutative_Ring
begin
datatype fexpr =
FCnst int
| FVar nat
| FAdd fexpr fexpr
| FSub fexpr fexpr
| FMul fexpr fexpr
| FNeg fexpr
| FDiv fexpr fexpr
| FPow fexpr nat
fun (in field) nth_el :: "'a list \<Rightarrow> nat \<Rightarrow> 'a" where
"nth_el [] n = \<zero>"
| "nth_el (x # xs) 0 = x"
| "nth_el (x # xs) (Suc n) = nth_el xs n"
lemma (in field) nth_el_Cons:
"nth_el (x # xs) n = (if n = 0 then x else nth_el xs (n - 1))"
by (cases n) simp_all
lemma (in field) nth_el_closed [simp]:
"in_carrier xs \<Longrightarrow> nth_el xs n \<in> carrier R"
by (induct xs n rule: nth_el.induct) (simp_all add: in_carrier_def)
primrec (in field) feval :: "'a list \<Rightarrow> fexpr \<Rightarrow> 'a"
where
"feval xs (FCnst c) = \<guillemotleft>c\<guillemotright>"
| "feval xs (FVar n) = nth_el xs n"
| "feval xs (FAdd a b) = feval xs a \<oplus> feval xs b"
| "feval xs (FSub a b) = feval xs a \<ominus> feval xs b"
| "feval xs (FMul a b) = feval xs a \<otimes> feval xs b"
| "feval xs (FNeg a) = \<ominus> feval xs a"
| "feval xs (FDiv a b) = feval xs a \<oslash> feval xs b"
| "feval xs (FPow a n) = feval xs a (^) n"
lemma (in field) feval_Cnst:
"feval xs (FCnst 0) = \<zero>"
"feval xs (FCnst 1) = \<one>"
"feval xs (FCnst (numeral n)) = \<guillemotleft>numeral n\<guillemotright>"
by simp_all
datatype pexpr =
PExpr1 pexpr1
| PExpr2 pexpr2
and pexpr1 =
PCnst int
| PVar nat
| PAdd pexpr pexpr
| PSub pexpr pexpr
| PNeg pexpr
and pexpr2 =
PMul pexpr pexpr
| PPow pexpr nat
lemma pexpr_cases [case_names PCnst PVar PAdd PSub PNeg PMul PPow]:
assumes
"\<And>c. e = PExpr1 (PCnst c) \<Longrightarrow> P"
"\<And>n. e = PExpr1 (PVar n) \<Longrightarrow> P"
"\<And>e1 e2. e = PExpr1 (PAdd e1 e2) \<Longrightarrow> P"
"\<And>e1 e2. e = PExpr1 (PSub e1 e2) \<Longrightarrow> P"
"\<And>e'. e = PExpr1 (PNeg e') \<Longrightarrow> P"
"\<And>e1 e2. e = PExpr2 (PMul e1 e2) \<Longrightarrow> P"
"\<And>e' n. e = PExpr2 (PPow e' n) \<Longrightarrow> P"
shows P
proof (cases e)
case (PExpr1 e')
then show ?thesis
apply (cases e')
apply simp_all
apply (erule assms)+
done
next
case (PExpr2 e')
then show ?thesis
apply (cases e')
apply simp_all
apply (erule assms)+
done
qed
lemmas pexpr_cases2 = pexpr_cases [case_product pexpr_cases]
fun (in field) peval :: "'a list \<Rightarrow> pexpr \<Rightarrow> 'a"
where
"peval xs (PExpr1 (PCnst c)) = \<guillemotleft>c\<guillemotright>"
| "peval xs (PExpr1 (PVar n)) = nth_el xs n"
| "peval xs (PExpr1 (PAdd a b)) = peval xs a \<oplus> peval xs b"
| "peval xs (PExpr1 (PSub a b)) = peval xs a \<ominus> peval xs b"
| "peval xs (PExpr1 (PNeg a)) = \<ominus> peval xs a"
| "peval xs (PExpr2 (PMul a b)) = peval xs a \<otimes> peval xs b"
| "peval xs (PExpr2 (PPow a n)) = peval xs a (^) n"
lemma (in field) peval_Cnst:
"peval xs (PExpr1 (PCnst 0)) = \<zero>"
"peval xs (PExpr1 (PCnst 1)) = \<one>"
"peval xs (PExpr1 (PCnst (numeral n))) = \<guillemotleft>numeral n\<guillemotright>"
"peval xs (PExpr1 (PCnst (- numeral n))) = \<ominus> \<guillemotleft>numeral n\<guillemotright>"
by simp_all
lemma (in field) peval_closed [simp]:
"in_carrier xs \<Longrightarrow> peval xs e \<in> carrier R"
"in_carrier xs \<Longrightarrow> peval xs (PExpr1 e1) \<in> carrier R"
"in_carrier xs \<Longrightarrow> peval xs (PExpr2 e2) \<in> carrier R"
by (induct e and e1 and e2) simp_all
definition npepow :: "pexpr \<Rightarrow> nat \<Rightarrow> pexpr"
where
"npepow e n =
(if n = 0 then PExpr1 (PCnst 1)
else if n = 1 then e
else (case e of
PExpr1 (PCnst c) \<Rightarrow> PExpr1 (PCnst (c ^ n))
| _ \<Rightarrow> PExpr2 (PPow e n)))"
lemma (in field) npepow_correct:
"in_carrier xs \<Longrightarrow> peval xs (npepow e n) = peval xs (PExpr2 (PPow e n))"
by (cases e rule: pexpr_cases)
(simp_all add: npepow_def)
fun npemul :: "pexpr \<Rightarrow> pexpr \<Rightarrow> pexpr"
where
"npemul x y = (case x of
PExpr1 (PCnst c) \<Rightarrow>
if c = 0 then x
else if c = 1 then y else
(case y of
PExpr1 (PCnst d) \<Rightarrow> PExpr1 (PCnst (c * d))
| _ \<Rightarrow> PExpr2 (PMul x y))
| PExpr2 (PPow e1 n) \<Rightarrow>
(case y of
PExpr2 (PPow e2 m) \<Rightarrow>
if n = m then npepow (npemul e1 e2) n
else PExpr2 (PMul x y)
| PExpr1 (PCnst d) \<Rightarrow>
if d = 0 then y
else if d = 1 then x
else PExpr2 (PMul x y)
| _ \<Rightarrow> PExpr2 (PMul x y))
| _ \<Rightarrow> (case y of
PExpr1 (PCnst d) \<Rightarrow>
if d = 0 then y
else if d = 1 then x
else PExpr2 (PMul x y)
| _ \<Rightarrow> PExpr2 (PMul x y)))"
lemma (in field) npemul_correct:
"in_carrier xs \<Longrightarrow> peval xs (npemul e1 e2) = peval xs (PExpr2 (PMul e1 e2))"
proof (induct e1 e2 rule: npemul.induct)
case (1 x y)
then show ?case
proof (cases x y rule: pexpr_cases2)
case (PPow_PPow e n e' m)
then show ?thesis
by (simp add: 1 npepow_correct nat_pow_distr
npemul.simps [of "PExpr2 (PPow e n)" "PExpr2 (PPow e' m)"]
del: npemul.simps)
qed simp_all
qed
declare npemul.simps [simp del]
definition npeadd :: "pexpr \<Rightarrow> pexpr \<Rightarrow> pexpr"
where
"npeadd x y = (case x of
PExpr1 (PCnst c) \<Rightarrow>
if c = 0 then y else
(case y of
PExpr1 (PCnst d) \<Rightarrow> PExpr1 (PCnst (c + d))
| _ \<Rightarrow> PExpr1 (PAdd x y))
| _ \<Rightarrow> (case y of
PExpr1 (PCnst d) \<Rightarrow>
if d = 0 then x
else PExpr1 (PAdd x y)
| _ \<Rightarrow> PExpr1 (PAdd x y)))"
lemma (in field) npeadd_correct:
"in_carrier xs \<Longrightarrow> peval xs (npeadd e1 e2) = peval xs (PExpr1 (PAdd e1 e2))"
by (cases e1 e2 rule: pexpr_cases2) (simp_all add: npeadd_def)
definition npesub :: "pexpr \<Rightarrow> pexpr \<Rightarrow> pexpr"
where
"npesub x y = (case y of
PExpr1 (PCnst d) \<Rightarrow>
if d = 0 then x else
(case x of
PExpr1 (PCnst c) \<Rightarrow> PExpr1 (PCnst (c - d))
| _ \<Rightarrow> PExpr1 (PSub x y))
| _ \<Rightarrow> (case x of
PExpr1 (PCnst c) \<Rightarrow>
if c = 0 then PExpr1 (PNeg y)
else PExpr1 (PSub x y)
| _ \<Rightarrow> PExpr1 (PSub x y)))"
lemma (in field) npesub_correct:
"in_carrier xs \<Longrightarrow> peval xs (npesub e1 e2) = peval xs (PExpr1 (PSub e1 e2))"
by (cases e1 e2 rule: pexpr_cases2) (simp_all add: npesub_def)
definition npeneg :: "pexpr \<Rightarrow> pexpr"
where
"npeneg e = (case e of
PExpr1 (PCnst c) \<Rightarrow> PExpr1 (PCnst (- c))
| _ \<Rightarrow> PExpr1 (PNeg e))"
lemma (in field) npeneg_correct:
"peval xs (npeneg e) = peval xs (PExpr1 (PNeg e))"
by (cases e rule: pexpr_cases) (simp_all add: npeneg_def)
lemma option_pair_cases [case_names None Some]:
assumes
"x = None \<Longrightarrow> P"
"\<And>p q. x = Some (p, q) \<Longrightarrow> P"
shows P
proof (cases x)
case None
then show ?thesis by (rule assms)
next
case (Some r)
then show ?thesis
apply (cases r)
apply simp
by (rule assms)
qed
fun isin :: "pexpr \<Rightarrow> nat \<Rightarrow> pexpr \<Rightarrow> nat \<Rightarrow> (nat * pexpr) option"
where
"isin e n (PExpr2 (PMul e1 e2)) m =
(case isin e n e1 m of
Some (k, e3) \<Rightarrow>
if k = 0 then Some (0, npemul e3 (npepow e2 m))
else (case isin e k e2 m of
Some (l, e4) \<Rightarrow> Some (l, npemul e3 e4)
| None \<Rightarrow> Some (k, npemul e3 (npepow e2 m)))
| None \<Rightarrow> (case isin e n e2 m of
Some (k, e3) \<Rightarrow> Some (k, npemul (npepow e1 m) e3)
| None \<Rightarrow> None))"
| "isin e n (PExpr2 (PPow e' k)) m =
(if k = 0 then None else isin e n e' (k * m))"
| "isin (PExpr1 e) n (PExpr1 e') m =
(if e = e' then
if n >= m then Some (n - m, PExpr1 (PCnst 1))
else Some (0, npepow (PExpr1 e) (m - n))
else None)"
| "isin (PExpr2 e) n (PExpr1 e') m = None"
lemma (in field) isin_correct:
assumes "in_carrier xs"
and "isin e n e' m = Some (p, e'')"
shows
"peval xs (PExpr2 (PPow e' m)) =
peval xs (PExpr2 (PMul (PExpr2 (PPow e (n - p))) e''))"
"p \<le> n"
using assms
by (induct e n e' m arbitrary: p e'' rule: isin.induct)
(force
simp add:
nat_pow_distr nat_pow_pow nat_pow_mult m_ac
npemul_correct npepow_correct
split: option.split_asm prod.split_asm if_split_asm)+
lemma (in field) isin_correct':
"in_carrier xs \<Longrightarrow> isin e n e' 1 = Some (p, e'') \<Longrightarrow>
peval xs e' = peval xs e (^) (n - p) \<otimes> peval xs e''"
"in_carrier xs \<Longrightarrow> isin e n e' 1 = Some (p, e'') \<Longrightarrow> p \<le> n"
using isin_correct [where m=1]
by simp_all
fun split_aux :: "pexpr \<Rightarrow> nat \<Rightarrow> pexpr \<Rightarrow> pexpr \<times> pexpr \<times> pexpr"
where
"split_aux (PExpr2 (PMul e1 e2)) n e =
(let
(left1, common1, right1) = split_aux e1 n e;
(left2, common2, right2) = split_aux e2 n right1
in (npemul left1 left2, npemul common1 common2, right2))"
| "split_aux (PExpr2 (PPow e' m)) n e =
(if m = 0 then (PExpr1 (PCnst 1), PExpr1 (PCnst 1), e)
else split_aux e' (m * n) e)"
| "split_aux (PExpr1 e') n e =
(case isin (PExpr1 e') n e 1 of
Some (m, e'') \<Rightarrow>
(if m = 0 then (PExpr1 (PCnst 1), npepow (PExpr1 e') n, e'')
else (npepow (PExpr1 e') m, npepow (PExpr1 e') (n - m), e''))
| None \<Rightarrow> (npepow (PExpr1 e') n, PExpr1 (PCnst 1), e))"
hide_const Left Right
abbreviation Left :: "pexpr \<Rightarrow> pexpr \<Rightarrow> pexpr" where
"Left e1 e2 \<equiv> fst (split_aux e1 (Suc 0) e2)"
abbreviation Common :: "pexpr \<Rightarrow> pexpr \<Rightarrow> pexpr" where
"Common e1 e2 \<equiv> fst (snd (split_aux e1 (Suc 0) e2))"
abbreviation Right :: "pexpr \<Rightarrow> pexpr \<Rightarrow> pexpr" where
"Right e1 e2 \<equiv> snd (snd (split_aux e1 (Suc 0) e2))"
lemma split_aux_induct [case_names 1 2 3]:
assumes I1: "\<And>e1 e2 n e. P e1 n e \<Longrightarrow> P e2 n (snd (snd (split_aux e1 n e))) \<Longrightarrow>
P (PExpr2 (PMul e1 e2)) n e"
and I2: "\<And>e' m n e. (m \<noteq> 0 \<Longrightarrow> P e' (m * n) e) \<Longrightarrow> P (PExpr2 (PPow e' m)) n e"
and I3: "\<And>e' n e. P (PExpr1 e') n e"
shows "P x y z"
proof (induct x y z rule: split_aux.induct)
case 1
from 1(1) 1(2) [OF refl prod.collapse prod.collapse]
show ?case by (rule I1)
next
case 2
then show ?case by (rule I2)
next
case 3
then show ?case by (rule I3)
qed
lemma (in field) split_aux_correct:
"in_carrier xs \<Longrightarrow>
peval xs (PExpr2 (PPow e\<^sub>1 n)) =
peval xs (PExpr2 (PMul (fst (split_aux e\<^sub>1 n e\<^sub>2)) (fst (snd (split_aux e\<^sub>1 n e\<^sub>2)))))"
"in_carrier xs \<Longrightarrow>
peval xs e\<^sub>2 =
peval xs (PExpr2 (PMul (snd (snd (split_aux e\<^sub>1 n e\<^sub>2))) (fst (snd (split_aux e\<^sub>1 n e\<^sub>2)))))"
by (induct e\<^sub>1 n e\<^sub>2 rule: split_aux_induct)
(auto simp add: split_beta
nat_pow_distr nat_pow_pow nat_pow_mult m_ac
npemul_correct npepow_correct isin_correct'
split: option.split)
lemma (in field) split_aux_correct':
"in_carrier xs \<Longrightarrow>
peval xs e\<^sub>1 = peval xs (Left e\<^sub>1 e\<^sub>2) \<otimes> peval xs (Common e\<^sub>1 e\<^sub>2)"
"in_carrier xs \<Longrightarrow>
peval xs e\<^sub>2 = peval xs (Right e\<^sub>1 e\<^sub>2) \<otimes> peval xs (Common e\<^sub>1 e\<^sub>2)"
using split_aux_correct [where n=1]
by simp_all
fun fnorm :: "fexpr \<Rightarrow> pexpr \<times> pexpr \<times> pexpr list"
where
"fnorm (FCnst c) = (PExpr1 (PCnst c), PExpr1 (PCnst 1), [])"
| "fnorm (FVar n) = (PExpr1 (PVar n), PExpr1 (PCnst 1), [])"
| "fnorm (FAdd e1 e2) =
(let
(xn, xd, xc) = fnorm e1;
(yn, yd, yc) = fnorm e2;
(left, common, right) = split_aux xd 1 yd
in
(npeadd (npemul xn right) (npemul yn left),
npemul left (npemul right common),
List.union xc yc))"
| "fnorm (FSub e1 e2) =
(let
(xn, xd, xc) = fnorm e1;
(yn, yd, yc) = fnorm e2;
(left, common, right) = split_aux xd 1 yd
in
(npesub (npemul xn right) (npemul yn left),
npemul left (npemul right common),
List.union xc yc))"
| "fnorm (FMul e1 e2) =
(let
(xn, xd, xc) = fnorm e1;
(yn, yd, yc) = fnorm e2;
(left1, common1, right1) = split_aux xn 1 yd;
(left2, common2, right2) = split_aux yn 1 xd
in
(npemul left1 left2,
npemul right2 right1,
List.union xc yc))"
| "fnorm (FNeg e) =
(let (n, d, c) = fnorm e
in (npeneg n, d, c))"
| "fnorm (FDiv e1 e2) =
(let
(xn, xd, xc) = fnorm e1;
(yn, yd, yc) = fnorm e2;
(left1, common1, right1) = split_aux xn 1 yn;
(left2, common2, right2) = split_aux xd 1 yd
in
(npemul left1 right2,
npemul left2 right1,
List.insert yn (List.union xc yc)))"
| "fnorm (FPow e m) =
(let (n, d, c) = fnorm e
in (npepow n m, npepow d m, c))"
abbreviation Num :: "fexpr \<Rightarrow> pexpr" where
"Num e \<equiv> fst (fnorm e)"
abbreviation Denom :: "fexpr \<Rightarrow> pexpr" where
"Denom e \<equiv> fst (snd (fnorm e))"
abbreviation Cond :: "fexpr \<Rightarrow> pexpr list" where
"Cond e \<equiv> snd (snd (fnorm e))"
primrec (in field) nonzero :: "'a list \<Rightarrow> pexpr list \<Rightarrow> bool"
where
"nonzero xs [] = True"
| "nonzero xs (p # ps) = (peval xs p \<noteq> \<zero> \<and> nonzero xs ps)"
lemma (in field) nonzero_singleton:
"nonzero xs [p] = (peval xs p \<noteq> \<zero>)"
by simp
lemma (in field) nonzero_append:
"nonzero xs (ps @ qs) = (nonzero xs ps \<and> nonzero xs qs)"
by (induct ps) simp_all
lemma (in field) nonzero_idempotent:
"p \<in> set ps \<Longrightarrow> (peval xs p \<noteq> \<zero> \<and> nonzero xs ps) = nonzero xs ps"
by (induct ps) auto
lemma (in field) nonzero_insert:
"nonzero xs (List.insert p ps) = (peval xs p \<noteq> \<zero> \<and> nonzero xs ps)"
by (simp add: List.insert_def nonzero_idempotent)
lemma (in field) nonzero_union:
"nonzero xs (List.union ps qs) = (nonzero xs ps \<and> nonzero xs qs)"
by (induct ps rule: rev_induct)
(auto simp add: List.union_def nonzero_insert nonzero_append)
lemma (in field) fnorm_correct:
assumes "in_carrier xs"
and "nonzero xs (Cond e)"
shows "feval xs e = peval xs (Num e) \<oslash> peval xs (Denom e)"
and "peval xs (Denom e) \<noteq> \<zero>"
using assms
proof (induct e)
case (FCnst c) {
case 1
show ?case by simp
next
case 2
show ?case by simp
}
next
case (FVar n) {
case 1
then show ?case by simp
next
case 2
show ?case by simp
}
next
case (FAdd e1 e2)
note split = split_aux_correct' [where xs=xs and
e\<^sub>1="Denom e1" and e\<^sub>2="Denom e2"]
{
case 1
let ?left = "peval xs (Left (Denom e1) (Denom e2))"
let ?common = "peval xs (Common (Denom e1) (Denom e2))"
let ?right = "peval xs (Right (Denom e1) (Denom e2))"
from 1 FAdd
have "feval xs (FAdd e1 e2) =
(?common \<otimes> (peval xs (Num e1) \<otimes> ?right \<oplus> peval xs (Num e2) \<otimes> ?left)) \<oslash>
(?common \<otimes> (?left \<otimes> (?right \<otimes> ?common)))"
by (simp add: split_beta split nonzero_union
add_frac_eq r_distr m_ac)
also from 1 FAdd have "\<dots> =
peval xs (Num (FAdd e1 e2)) \<oslash> peval xs (Denom (FAdd e1 e2))"
by (simp add: split_beta split nonzero_union npeadd_correct npemul_correct integral_iff)
finally show ?case .
next
case 2
with FAdd show ?case
by (simp add: split_beta split nonzero_union npemul_correct integral_iff)
}
next
case (FSub e1 e2)
note split = split_aux_correct' [where xs=xs and
e\<^sub>1="Denom e1" and e\<^sub>2="Denom e2"]
{
case 1
let ?left = "peval xs (Left (Denom e1) (Denom e2))"
let ?common = "peval xs (Common (Denom e1) (Denom e2))"
let ?right = "peval xs (Right (Denom e1) (Denom e2))"
from 1 FSub
have "feval xs (FSub e1 e2) =
(?common \<otimes> (peval xs (Num e1) \<otimes> ?right \<ominus> peval xs (Num e2) \<otimes> ?left)) \<oslash>
(?common \<otimes> (?left \<otimes> (?right \<otimes> ?common)))"
by (simp add: split_beta split nonzero_union
diff_frac_eq r_diff_distr m_ac)
also from 1 FSub have "\<dots> =
peval xs (Num (FSub e1 e2)) \<oslash> peval xs (Denom (FSub e1 e2))"
by (simp add: split_beta split nonzero_union npesub_correct npemul_correct integral_iff)
finally show ?case .
next
case 2
with FSub show ?case
by (simp add: split_beta split nonzero_union npemul_correct integral_iff)
}
next
case (FMul e1 e2)
note split =
split_aux_correct' [where xs=xs and
e\<^sub>1="Num e1" and e\<^sub>2="Denom e2"]
split_aux_correct' [where xs=xs and
e\<^sub>1="Num e2" and e\<^sub>2="Denom e1"]
{
case 1
let ?left\<^sub>1 = "peval xs (Left (Num e1) (Denom e2))"
let ?common\<^sub>1 = "peval xs (Common (Num e1) (Denom e2))"
let ?right\<^sub>1 = "peval xs (Right (Num e1) (Denom e2))"
let ?left\<^sub>2 = "peval xs (Left (Num e2) (Denom e1))"
let ?common\<^sub>2 = "peval xs (Common (Num e2) (Denom e1))"
let ?right\<^sub>2 = "peval xs (Right (Num e2) (Denom e1))"
from 1 FMul
have "feval xs (FMul e1 e2) =
((?common\<^sub>1 \<otimes> ?common\<^sub>2) \<otimes> (?left\<^sub>1 \<otimes> ?left\<^sub>2)) \<oslash>
((?common\<^sub>1 \<otimes> ?common\<^sub>2) \<otimes> (?right\<^sub>2 \<otimes> ?right\<^sub>1))"
by (simp add: split_beta split nonzero_union
nonzero_divide_divide_eq_left m_ac)
also from 1 FMul have "\<dots> =
peval xs (Num (FMul e1 e2)) \<oslash> peval xs (Denom (FMul e1 e2))"
by (simp add: split_beta split nonzero_union npemul_correct integral_iff)
finally show ?case .
next
case 2
with FMul show ?case
by (simp add: split_beta split nonzero_union npemul_correct integral_iff)
}
next
case (FNeg e)
{
case 1
with FNeg show ?case
by (simp add: split_beta npeneg_correct)
next
case 2
with FNeg show ?case
by (simp add: split_beta)
}
next
case (FDiv e1 e2)
note split =
split_aux_correct' [where xs=xs and
e\<^sub>1="Num e1" and e\<^sub>2="Num e2"]
split_aux_correct' [where xs=xs and
e\<^sub>1="Denom e1" and e\<^sub>2="Denom e2"]
{
case 1
let ?left\<^sub>1 = "peval xs (Left (Num e1) (Num e2))"
let ?common\<^sub>1 = "peval xs (Common (Num e1) (Num e2))"
let ?right\<^sub>1 = "peval xs (Right (Num e1) (Num e2))"
let ?left\<^sub>2 = "peval xs (Left (Denom e1) (Denom e2))"
let ?common\<^sub>2 = "peval xs (Common (Denom e1) (Denom e2))"
let ?right\<^sub>2 = "peval xs (Right (Denom e1) (Denom e2))"
from 1 FDiv
have "feval xs (FDiv e1 e2) =
((?common\<^sub>1 \<otimes> ?common\<^sub>2) \<otimes> (?left\<^sub>1 \<otimes> ?right\<^sub>2)) \<oslash>
((?common\<^sub>1 \<otimes> ?common\<^sub>2) \<otimes> (?left\<^sub>2 \<otimes> ?right\<^sub>1))"
by (simp add: split_beta split nonzero_union nonzero_insert
nonzero_divide_divide_eq m_ac)
also from 1 FDiv have "\<dots> =
peval xs (Num (FDiv e1 e2)) \<oslash> peval xs (Denom (FDiv e1 e2))"
by (simp add: split_beta split nonzero_union nonzero_insert npemul_correct integral_iff)
finally show ?case .
next
case 2
with FDiv show ?case
by (simp add: split_beta split nonzero_union nonzero_insert npemul_correct integral_iff)
}
next
case (FPow e n)
{
case 1
with FPow show ?case
by (simp add: split_beta nonzero_power_divide npepow_correct)
next
case 2
with FPow show ?case
by (simp add: split_beta npepow_correct)
}
qed
lemma (in field) feval_eq0:
assumes "in_carrier xs"
and "fnorm e = (n, d, c)"
and "nonzero xs c"
and "peval xs n = \<zero>"
shows "feval xs e = \<zero>"
using assms fnorm_correct [of xs e]
by simp
lemma (in field) fexpr_in_carrier:
assumes "in_carrier xs"
and "nonzero xs (Cond e)"
shows "feval xs e \<in> carrier R"
using assms
proof (induct e)
case (FDiv e1 e2)
then have "feval xs e1 \<in> carrier R" "feval xs e2 \<in> carrier R"
"peval xs (Num e2) \<noteq> \<zero>" "nonzero xs (Cond e2)"
by (simp_all add: nonzero_union nonzero_insert split: prod.split_asm)
from \<open>in_carrier xs\<close> \<open>nonzero xs (Cond e2)\<close>
have "feval xs e2 = peval xs (Num e2) \<oslash> peval xs (Denom e2)"
by (rule fnorm_correct)
moreover from \<open>in_carrier xs\<close> \<open>nonzero xs (Cond e2)\<close>
have "peval xs (Denom e2) \<noteq> \<zero>" by (rule fnorm_correct)
ultimately have "feval xs e2 \<noteq> \<zero>" using \<open>peval xs (Num e2) \<noteq> \<zero>\<close> \<open>in_carrier xs\<close>
by (simp add: divide_eq_0_iff)
with \<open>feval xs e1 \<in> carrier R\<close> \<open>feval xs e2 \<in> carrier R\<close>
show ?case by simp
qed (simp_all add: nonzero_union split: prod.split_asm)
lemma (in field) feval_eq:
assumes "in_carrier xs"
and "fnorm (FSub e e') = (n, d, c)"
and "nonzero xs c"
shows "(feval xs e = feval xs e') = (peval xs n = \<zero>)"
proof -
from assms have "nonzero xs (Cond e)" "nonzero xs (Cond e')"
by (auto simp add: nonzero_union split: prod.split_asm)
with assms fnorm_correct [of xs "FSub e e'"]
have "feval xs e \<ominus> feval xs e' = peval xs n \<oslash> peval xs d"
"peval xs d \<noteq> \<zero>"
by simp_all
show ?thesis
proof
assume "feval xs e = feval xs e'"
with \<open>feval xs e \<ominus> feval xs e' = peval xs n \<oslash> peval xs d\<close>
\<open>in_carrier xs\<close> \<open>nonzero xs (Cond e')\<close>
have "peval xs n \<oslash> peval xs d = \<zero>"
by (simp add: fexpr_in_carrier minus_eq r_neg)
with \<open>peval xs d \<noteq> \<zero>\<close> \<open>in_carrier xs\<close>
show "peval xs n = \<zero>"
by (simp add: divide_eq_0_iff)
next
assume "peval xs n = \<zero>"
with \<open>feval xs e \<ominus> feval xs e' = peval xs n \<oslash> peval xs d\<close> \<open>peval xs d \<noteq> \<zero>\<close>
\<open>nonzero xs (Cond e)\<close> \<open>nonzero xs (Cond e')\<close> \<open>in_carrier xs\<close>
show "feval xs e = feval xs e'"
by (simp add: eq_diff0 fexpr_in_carrier)
qed
qed
code_reflect Field_Code
datatypes fexpr = FCnst | FVar | FAdd | FSub | FMul | FNeg | FDiv | FPow
and pexpr = PExpr1 | PExpr2
and pexpr1 = PCnst | PVar | PAdd | PSub | PNeg
and pexpr2 = PMul | PPow
functions fnorm
term_of_fexpr_inst.term_of_fexpr
term_of_pexpr_inst.term_of_pexpr
equal_pexpr_inst.equal_pexpr
definition field_codegen_aux :: "(pexpr \<times> pexpr list) itself" where
"field_codegen_aux = (Code_Evaluation.TERM_OF_EQUAL::(pexpr \<times> pexpr list) itself)"
ML \<open>
signature FIELD_TAC =
sig
structure Field_Simps:
sig
type T
val get: Context.generic -> T
val put: T -> Context.generic -> Context.generic
val map: (T -> T) -> Context.generic -> Context.generic
end
val eq_field_simps:
(term * (thm list * thm list * thm list * thm * thm)) *
(term * (thm list * thm list * thm list * thm * thm)) -> bool
val field_tac: bool -> Proof.context -> int -> tactic
end
structure Field_Tac : FIELD_TAC =
struct
open Ring_Tac;
fun field_struct (Const (@{const_name Ring.ring.add}, _) $ R $ _ $ _) = SOME R
| field_struct (Const (@{const_name Ring.a_minus}, _) $ R $ _ $ _) = SOME R
| field_struct (Const (@{const_name Group.monoid.mult}, _) $ R $ _ $ _) = SOME R
| field_struct (Const (@{const_name Ring.a_inv}, _) $ R $ _) = SOME R
| field_struct (Const (@{const_name Group.pow}, _) $ R $ _ $ _) = SOME R
| field_struct (Const (@{const_name Algebra_Aux.m_div}, _) $ R $ _ $ _) = SOME R
| field_struct (Const (@{const_name Ring.ring.zero}, _) $ R) = SOME R
| field_struct (Const (@{const_name Group.monoid.one}, _) $ R) = SOME R
| field_struct (Const (@{const_name Algebra_Aux.of_integer}, _) $ R $ _) = SOME R
| field_struct _ = NONE;
fun reif_fexpr vs (Const (@{const_name Ring.ring.add}, _) $ _ $ a $ b) =
@{const FAdd} $ reif_fexpr vs a $ reif_fexpr vs b
| reif_fexpr vs (Const (@{const_name Ring.a_minus}, _) $ _ $ a $ b) =
@{const FSub} $ reif_fexpr vs a $ reif_fexpr vs b
| reif_fexpr vs (Const (@{const_name Group.monoid.mult}, _) $ _ $ a $ b) =
@{const FMul} $ reif_fexpr vs a $ reif_fexpr vs b
| reif_fexpr vs (Const (@{const_name Ring.a_inv}, _) $ _ $ a) =
@{const FNeg} $ reif_fexpr vs a
| reif_fexpr vs (Const (@{const_name Group.pow}, _) $ _ $ a $ n) =
@{const FPow} $ reif_fexpr vs a $ n
| reif_fexpr vs (Const (@{const_name Algebra_Aux.m_div}, _) $ _ $ a $ b) =
@{const FDiv} $ reif_fexpr vs a $ reif_fexpr vs b
| reif_fexpr vs (Free x) =
@{const FVar} $ HOLogic.mk_number HOLogic.natT (find_index (equal x) vs)
| reif_fexpr vs (Const (@{const_name Ring.ring.zero}, _) $ _) =
@{term "FCnst 0"}
| reif_fexpr vs (Const (@{const_name Group.monoid.one}, _) $ _) =
@{term "FCnst 1"}
| reif_fexpr vs (Const (@{const_name Algebra_Aux.of_integer}, _) $ _ $ n) =
@{const FCnst} $ n
| reif_fexpr _ _ = error "reif_fexpr: bad expression";
fun reif_fexpr' vs (Const (@{const_name Groups.plus}, _) $ a $ b) =
@{const FAdd} $ reif_fexpr' vs a $ reif_fexpr' vs b
| reif_fexpr' vs (Const (@{const_name Groups.minus}, _) $ a $ b) =
@{const FSub} $ reif_fexpr' vs a $ reif_fexpr' vs b
| reif_fexpr' vs (Const (@{const_name Groups.times}, _) $ a $ b) =
@{const FMul} $ reif_fexpr' vs a $ reif_fexpr' vs b
| reif_fexpr' vs (Const (@{const_name Groups.uminus}, _) $ a) =
@{const FNeg} $ reif_fexpr' vs a
| reif_fexpr' vs (Const (@{const_name Power.power}, _) $ a $ n) =
@{const FPow} $ reif_fexpr' vs a $ n
| reif_fexpr' vs (Const (@{const_name divide}, _) $ a $ b) =
@{const FDiv} $ reif_fexpr' vs a $ reif_fexpr' vs b
| reif_fexpr' vs (Free x) =
@{const FVar} $ HOLogic.mk_number HOLogic.natT (find_index (equal x) vs)
| reif_fexpr' vs (Const (@{const_name zero_class.zero}, _)) =
@{term "FCnst 0"}
| reif_fexpr' vs (Const (@{const_name one_class.one}, _)) =
@{term "FCnst 1"}
| reif_fexpr' vs (Const (@{const_name numeral}, _) $ b) =
@{const FCnst} $ (@{const numeral (int)} $ b)
| reif_fexpr' _ _ = error "reif_fexpr: bad expression";
fun eq_field_simps
((t, (ths1, ths2, ths3, th4, th)),
(t', (ths1', ths2', ths3', th4', th'))) =
t aconv t' andalso
eq_list Thm.eq_thm (ths1, ths1') andalso
eq_list Thm.eq_thm (ths2, ths2') andalso
eq_list Thm.eq_thm (ths3, ths3') andalso
Thm.eq_thm (th4, th4') andalso
Thm.eq_thm (th, th');
structure Field_Simps = Generic_Data
(struct
type T = (term * (thm list * thm list * thm list * thm * thm)) Net.net
val empty = Net.empty
val extend = I
val merge = Net.merge eq_field_simps
end);
fun get_field_simps ctxt optcT t =
(case get_matching_rules ctxt (Field_Simps.get (Context.Proof ctxt)) t of
SOME (ths1, ths2, ths3, th4, th) =>
let val tr =
Thm.transfer (Proof_Context.theory_of ctxt) #>
(case optcT of NONE => I | SOME cT => inst [cT] [] #> norm)
in (map tr ths1, map tr ths2, map tr ths3, tr th4, tr th) end
| NONE => error "get_field_simps: lookup failed");
fun nth_el_conv (_, _, _, nth_el_Cons, _) =
let
val a = type_of_eqn nth_el_Cons;
val If_conv_a = If_conv a;
fun conv ys n = (case strip_app ys of
(@{const_name Cons}, [x, xs]) =>
transitive'
(inst [] [x, xs, n] nth_el_Cons)
(If_conv_a (args2 nat_eq_conv)
Thm.reflexive
(cong2' conv Thm.reflexive (args2 nat_minus_conv))))
in conv end;
fun feval_conv (rls as
([feval_simps_1, feval_simps_2, feval_simps_3,
feval_simps_4, feval_simps_5, feval_simps_6,
feval_simps_7, feval_simps_8, feval_simps_9,
feval_simps_10, feval_simps_11],
_, _, _, _)) =
let
val nth_el_conv' = nth_el_conv rls;
fun conv xs x = (case strip_app x of
(@{const_name FCnst}, [c]) => (case strip_app c of
(@{const_name zero_class.zero}, _) => inst [] [xs] feval_simps_9
| (@{const_name one_class.one}, _) => inst [] [xs] feval_simps_10
| (@{const_name numeral}, [n]) => inst [] [xs, n] feval_simps_11
| _ => inst [] [xs, c] feval_simps_1)
| (@{const_name FVar}, [n]) =>
transitive' (inst [] [xs, n] feval_simps_2) (args2 nth_el_conv')
| (@{const_name FAdd}, [a, b]) =>
transitive' (inst [] [xs, a, b] feval_simps_3)
(cong2 (args2 conv) (args2 conv))
| (@{const_name FSub}, [a, b]) =>
transitive' (inst [] [xs, a, b] feval_simps_4)
(cong2 (args2 conv) (args2 conv))
| (@{const_name FMul}, [a, b]) =>
transitive' (inst [] [xs, a, b] feval_simps_5)
(cong2 (args2 conv) (args2 conv))
| (@{const_name FNeg}, [a]) =>
transitive' (inst [] [xs, a] feval_simps_6)
(cong1 (args2 conv))
| (@{const_name FDiv}, [a, b]) =>
transitive' (inst [] [xs, a, b] feval_simps_7)
(cong2 (args2 conv) (args2 conv))
| (@{const_name FPow}, [a, n]) =>
transitive' (inst [] [xs, a, n] feval_simps_8)
(cong2 (args2 conv) Thm.reflexive))
in conv end;
fun peval_conv (rls as
(_,
[peval_simps_1, peval_simps_2, peval_simps_3,
peval_simps_4, peval_simps_5, peval_simps_6,
peval_simps_7, peval_simps_8, peval_simps_9,
peval_simps_10, peval_simps_11],
_, _, _)) =
let
val nth_el_conv' = nth_el_conv rls;
fun conv xs x = (case strip_app x of
(@{const_name PExpr1}, [e]) => (case strip_app e of
(@{const_name PCnst}, [c]) => (case strip_numeral c of
(@{const_name zero_class.zero}, _) => inst [] [xs] peval_simps_8
| (@{const_name one_class.one}, _) => inst [] [xs] peval_simps_9
| (@{const_name numeral}, [n]) => inst [] [xs, n] peval_simps_10
| (@{const_name uminus}, [n]) => inst [] [xs, n] peval_simps_11
| _ => inst [] [xs, c] peval_simps_1)
| (@{const_name PVar}, [n]) =>
transitive' (inst [] [xs, n] peval_simps_2) (args2 nth_el_conv')
| (@{const_name PAdd}, [a, b]) =>
transitive' (inst [] [xs, a, b] peval_simps_3)
(cong2 (args2 conv) (args2 conv))
| (@{const_name PSub}, [a, b]) =>
transitive' (inst [] [xs, a, b] peval_simps_4)
(cong2 (args2 conv) (args2 conv))
| (@{const_name PNeg}, [a]) =>
transitive' (inst [] [xs, a] peval_simps_5)
(cong1 (args2 conv)))
| (@{const_name PExpr2}, [e]) => (case strip_app e of
(@{const_name PMul}, [a, b]) =>
transitive' (inst [] [xs, a, b] peval_simps_6)
(cong2 (args2 conv) (args2 conv))
| (@{const_name PPow}, [a, n]) =>
transitive' (inst [] [xs, a, n] peval_simps_7)
(cong2 (args2 conv) Thm.reflexive)))
in conv end;
fun nonzero_conv (rls as
(_, _,
[nonzero_Nil, nonzero_Cons, nonzero_singleton],
_, _)) =
let
val peval_conv' = peval_conv rls;
fun conv xs qs = (case strip_app qs of
(@{const_name Nil}, []) => inst [] [xs] nonzero_Nil
| (@{const_name Cons}, [p, ps]) => (case Thm.term_of ps of
Const (@{const_name Nil}, _) =>
transitive' (inst [] [xs, p] nonzero_singleton)
(cong1 (cong2 (args2 peval_conv') Thm.reflexive))
| _ => transitive' (inst [] [xs, p, ps] nonzero_Cons)
(cong2 (cong1 (cong2 (args2 peval_conv') Thm.reflexive)) (args2 conv))))
in conv end;
val cv = Code_Evaluation.static_conv
{ctxt = @{context},
consts =
[@{const_name nat_of_integer},
@{const_name fnorm}, @{const_name field_codegen_aux}]};
fun field_tac in_prem ctxt =
SUBGOAL (fn (g, i) =>
let
val (prems, concl) = Logic.strip_horn g;
fun find_eq s = (case s of
(_ $ (Const (@{const_name HOL.eq}, Type (_, [T, _])) $ t $ u)) =>
(case (field_struct t, field_struct u) of
(SOME R, _) => SOME ((t, u), R, T, NONE, mk_in_carrier ctxt R [], reif_fexpr)
| (_, SOME R) => SOME ((t, u), R, T, NONE, mk_in_carrier ctxt R [], reif_fexpr)
| _ =>
if Sign.of_sort (Proof_Context.theory_of ctxt) (T, @{sort field})
then SOME ((t, u), mk_ring T, T, SOME T, K @{thm in_carrier_trivial}, reif_fexpr')
else NONE)
| _ => NONE);
val ((t, u), R, T, optT, mkic, reif) =
(case get_first find_eq
(if in_prem then prems else [concl]) of
SOME q => q
| NONE => error "cannot determine field");
val rls as (_, _, _, _, feval_eq) =
get_field_simps ctxt (Option.map (Thm.ctyp_of ctxt) optT) R;
val xs = [] |> Term.add_frees t |> Term.add_frees u |> filter (equal T o snd);
val cxs = Thm.cterm_of ctxt (HOLogic.mk_list T (map Free xs));
val ce = Thm.cterm_of ctxt (reif xs t);
val ce' = Thm.cterm_of ctxt (reif xs u);
val fnorm = cv ctxt
(Thm.apply @{cterm fnorm} (Thm.apply (Thm.apply @{cterm FSub} ce) ce'));
val (_, [n, dc]) = strip_app (Thm.rhs_of fnorm);
val (_, [_, c]) = strip_app dc;
val th =
Conv.fconv_rule (Conv.concl_conv 1 (Conv.arg_conv
(binop_conv
(binop_conv
(K (feval_conv rls cxs ce)) (K (feval_conv rls cxs ce')))
(Conv.arg1_conv (K (peval_conv rls cxs n))))))
([mkic xs,
mk_obj_eq fnorm,
mk_obj_eq (nonzero_conv rls cxs c) RS @{thm iffD2}] MRS
feval_eq);
val th' = Drule.rotate_prems 1
(th RS (if in_prem then @{thm iffD1} else @{thm iffD2}));
in
if in_prem then
dresolve_tac ctxt [th'] 1 THEN defer_tac 1
else
resolve_tac ctxt [th'] 1
end);
end
\<close>
context field begin
local_setup \<open>
Local_Theory.declaration {syntax = false, pervasive = false}
(fn phi => Field_Tac.Field_Simps.map (Ring_Tac.insert_rules Field_Tac.eq_field_simps
(Morphism.term phi @{term R},
(Morphism.fact phi @{thms feval.simps [meta] feval_Cnst [meta]},
Morphism.fact phi @{thms peval.simps [meta] peval_Cnst [meta]},
Morphism.fact phi @{thms nonzero.simps [meta] nonzero_singleton [meta]},
singleton (Morphism.fact phi) @{thm nth_el_Cons [meta]},
singleton (Morphism.fact phi) @{thm feval_eq}))))
\<close>
end
method_setup field = \<open>
Scan.lift (Args.mode "prems") -- Attrib.thms >> (fn (in_prem, thms) => fn ctxt =>
SIMPLE_METHOD' (Field_Tac.field_tac in_prem ctxt THEN' Ring_Tac.ring_tac in_prem thms ctxt))
\<close> "reduce equations over fields to equations over rings"
end