# Theory Parametric_Ferrante_Rackoff

```(*  Title:      HOL/Decision_Procs/Parametric_Ferrante_Rackoff.thy
Author:     Amine Chaieb
*)

section ‹A formalization of Ferrante and Rackoff's procedure with polynomial parameters, see Paper in CALCULEMUS 2008›

theory Parametric_Ferrante_Rackoff
imports
Reflected_Multivariate_Polynomial
Dense_Linear_Order
DP_Library
"HOL-Library.Code_Target_Numeral"
begin

subsection ‹Terms›

datatype (plugins del: size) tm = CP poly | Bound nat | Add tm tm | Mul poly tm
| Neg tm | Sub tm tm | CNP nat poly tm

instantiation tm :: size
begin

primrec size_tm :: "tm ⇒ nat"
where
"size_tm (CP c) = polysize c"
| "size_tm (Bound n) = 1"
| "size_tm (Neg a) = 1 + size_tm a"
| "size_tm (Add a b) = 1 + size_tm a + size_tm b"
| "size_tm (Sub a b) = 3 + size_tm a + size_tm b"
| "size_tm (Mul c a) = 1 + polysize c + size_tm a"
| "size_tm (CNP n c a) = 3 + polysize c + size_tm a "

instance ..

end

text ‹Semantics of terms tm.›
primrec Itm :: "'a::field_char_0 list ⇒ 'a list ⇒ tm ⇒ 'a"
where
"Itm vs bs (CP c) = (Ipoly vs c)"
| "Itm vs bs (Bound n) = bs!n"
| "Itm vs bs (Neg a) = -(Itm vs bs a)"
| "Itm vs bs (Add a b) = Itm vs bs a + Itm vs bs b"
| "Itm vs bs (Sub a b) = Itm vs bs a - Itm vs bs b"
| "Itm vs bs (Mul c a) = (Ipoly vs c) * Itm vs bs a"
| "Itm vs bs (CNP n c t) = (Ipoly vs c)*(bs!n) + Itm vs bs t"

fun allpolys :: "(poly ⇒ bool) ⇒ tm ⇒ bool"
where
"allpolys P (CP c) = P c"
| "allpolys P (CNP n c p) = (P c ∧ allpolys P p)"
| "allpolys P (Mul c p) = (P c ∧ allpolys P p)"
| "allpolys P (Neg p) = allpolys P p"
| "allpolys P (Add p q) = (allpolys P p ∧ allpolys P q)"
| "allpolys P (Sub p q) = (allpolys P p ∧ allpolys P q)"
| "allpolys P p = True"

primrec tmboundslt :: "nat ⇒ tm ⇒ bool"
where
"tmboundslt n (CP c) = True"
| "tmboundslt n (Bound m) = (m < n)"
| "tmboundslt n (CNP m c a) = (m < n ∧ tmboundslt n a)"
| "tmboundslt n (Neg a) = tmboundslt n a"
| "tmboundslt n (Add a b) = (tmboundslt n a ∧ tmboundslt n b)"
| "tmboundslt n (Sub a b) = (tmboundslt n a ∧ tmboundslt n b)"
| "tmboundslt n (Mul i a) = tmboundslt n a"

primrec tmbound0 :: "tm ⇒ bool"  ― ‹a ‹tm› is ∗‹independent› of Bound 0›
where
"tmbound0 (CP c) = True"
| "tmbound0 (Bound n) = (n>0)"
| "tmbound0 (CNP n c a) = (n≠0 ∧ tmbound0 a)"
| "tmbound0 (Neg a) = tmbound0 a"
| "tmbound0 (Add a b) = (tmbound0 a ∧ tmbound0 b)"
| "tmbound0 (Sub a b) = (tmbound0 a ∧ tmbound0 b)"
| "tmbound0 (Mul i a) = tmbound0 a"

lemma tmbound0_I:
assumes "tmbound0 a"
shows "Itm vs (b#bs) a = Itm vs (b'#bs) a"
using assms by (induct a rule: tm.induct) auto

primrec tmbound :: "nat ⇒ tm ⇒ bool"  ― ‹a ‹tm› is ∗‹independent› of Bound n›
where
"tmbound n (CP c) = True"
| "tmbound n (Bound m) = (n ≠ m)"
| "tmbound n (CNP m c a) = (n≠m ∧ tmbound n a)"
| "tmbound n (Neg a) = tmbound n a"
| "tmbound n (Add a b) = (tmbound n a ∧ tmbound n b)"
| "tmbound n (Sub a b) = (tmbound n a ∧ tmbound n b)"
| "tmbound n (Mul i a) = tmbound n a"

lemma tmbound0_tmbound_iff: "tmbound 0 t = tmbound0 t"
by (induct t) auto

lemma tmbound_I:
assumes bnd: "tmboundslt (length bs) t"
and nb: "tmbound n t"
and le: "n ≤ length bs"
shows "Itm vs (bs[n:=x]) t = Itm vs bs t"
using nb le bnd
by (induct t rule: tm.induct) auto

fun decrtm0 :: "tm ⇒ tm"
where
"decrtm0 (Bound n) = Bound (n - 1)"
| "decrtm0 (Neg a) = Neg (decrtm0 a)"
| "decrtm0 (Sub a b) = Sub (decrtm0 a) (decrtm0 b)"
| "decrtm0 (Mul c a) = Mul c (decrtm0 a)"
| "decrtm0 (CNP n c a) = CNP (n - 1) c (decrtm0 a)"
| "decrtm0 a = a"

fun incrtm0 :: "tm ⇒ tm"
where
"incrtm0 (Bound n) = Bound (n + 1)"
| "incrtm0 (Neg a) = Neg (incrtm0 a)"
| "incrtm0 (Sub a b) = Sub (incrtm0 a) (incrtm0 b)"
| "incrtm0 (Mul c a) = Mul c (incrtm0 a)"
| "incrtm0 (CNP n c a) = CNP (n + 1) c (incrtm0 a)"
| "incrtm0 a = a"

lemma decrtm0:
assumes nb: "tmbound0 t"
shows "Itm vs (x # bs) t = Itm vs bs (decrtm0 t)"
using nb by (induct t rule: decrtm0.induct) simp_all

lemma incrtm0: "Itm vs (x#bs) (incrtm0 t) = Itm vs bs t"
by (induct t rule: decrtm0.induct) simp_all

primrec decrtm :: "nat ⇒ tm ⇒ tm"
where
"decrtm m (CP c) = (CP c)"
| "decrtm m (Bound n) = (if n < m then Bound n else Bound (n - 1))"
| "decrtm m (Neg a) = Neg (decrtm m a)"
| "decrtm m (Add a b) = Add (decrtm m a) (decrtm m b)"
| "decrtm m (Sub a b) = Sub (decrtm m a) (decrtm m b)"
| "decrtm m (Mul c a) = Mul c (decrtm m a)"
| "decrtm m (CNP n c a) = (if n < m then CNP n c (decrtm m a) else CNP (n - 1) c (decrtm m a))"

primrec removen :: "nat ⇒ 'a list ⇒ 'a list"
where
"removen n [] = []"
| "removen n (x#xs) = (if n=0 then xs else (x#(removen (n - 1) xs)))"

lemma removen_same: "n ≥ length xs ⟹ removen n xs = xs"
by (induct xs arbitrary: n) auto

lemma nth_length_exceeds: "n ≥ length xs ⟹ xs!n = []!(n - length xs)"
by (induct xs arbitrary: n) auto

lemma removen_length: "length (removen n xs) = (if n ≥ length xs then length xs else length xs - 1)"
by (induct xs arbitrary: n) auto

lemma removen_nth:
"(removen n xs)!m =
(if n ≥ length xs then xs!m
else if m < n then xs!m
else if m ≤ length xs then xs!(Suc m)
else []!(m - (length xs - 1)))"
proof (induct xs arbitrary: n m)
case Nil
then show ?case by simp
next
case (Cons x xs)
let ?l = "length (x # xs)"
consider "n ≥ ?l" | "n < ?l" by arith
then show ?case
proof cases
case 1
with removen_same[OF this] show ?thesis by simp
next
case nl: 2
consider "m < n" | "m ≥ n" by arith
then show ?thesis
proof cases
case 1
then show ?thesis
using Cons by (cases m) auto
next
case 2
consider "m ≤ ?l" | "m > ?l" by arith
then show ?thesis
proof cases
case 1
then show ?thesis
using Cons by (cases m) auto
next
case ml: 2
have th: "length (removen n (x # xs)) = length xs"
using removen_length[where n = n and xs= "x # xs"] nl by simp
with ml have "m ≥ length (removen n (x # xs))"
by auto
from th nth_length_exceeds[OF this] have "(removen n (x # xs))!m = [] ! (m - length xs)"
by auto
then have "(removen n (x # xs))!m = [] ! (m - (length (x # xs) - 1))"
by auto
then show ?thesis
using ml nl by auto
qed
qed
qed
qed

lemma decrtm:
assumes bnd: "tmboundslt (length bs) t"
and nb: "tmbound m t"
and nle: "m ≤ length bs"
shows "Itm vs (removen m bs) (decrtm m t) = Itm vs bs t"
using bnd nb nle by (induct t rule: tm.induct) (auto simp: removen_nth)

primrec tmsubst0:: "tm ⇒ tm ⇒ tm"
where
"tmsubst0 t (CP c) = CP c"
| "tmsubst0 t (Bound n) = (if n=0 then t else Bound n)"
| "tmsubst0 t (CNP n c a) = (if n=0 then Add (Mul c t) (tmsubst0 t a) else CNP n c (tmsubst0 t a))"
| "tmsubst0 t (Neg a) = Neg (tmsubst0 t a)"
| "tmsubst0 t (Add a b) = Add (tmsubst0 t a) (tmsubst0 t b)"
| "tmsubst0 t (Sub a b) = Sub (tmsubst0 t a) (tmsubst0 t b)"
| "tmsubst0 t (Mul i a) = Mul i (tmsubst0 t a)"

lemma tmsubst0: "Itm vs (x # bs) (tmsubst0 t a) = Itm vs (Itm vs (x # bs) t # bs) a"
by (induct a rule: tm.induct) auto

lemma tmsubst0_nb: "tmbound0 t ⟹ tmbound0 (tmsubst0 t a)"
by (induct a rule: tm.induct) auto

primrec tmsubst:: "nat ⇒ tm ⇒ tm ⇒ tm"
where
"tmsubst n t (CP c) = CP c"
| "tmsubst n t (Bound m) = (if n=m then t else Bound m)"
| "tmsubst n t (CNP m c a) =
(if n = m then Add (Mul c t) (tmsubst n t a) else CNP m c (tmsubst n t a))"
| "tmsubst n t (Neg a) = Neg (tmsubst n t a)"
| "tmsubst n t (Add a b) = Add (tmsubst n t a) (tmsubst n t b)"
| "tmsubst n t (Sub a b) = Sub (tmsubst n t a) (tmsubst n t b)"
| "tmsubst n t (Mul i a) = Mul i (tmsubst n t a)"

lemma tmsubst:
assumes nb: "tmboundslt (length bs) a"
and nlt: "n ≤ length bs"
shows "Itm vs bs (tmsubst n t a) = Itm vs (bs[n:= Itm vs bs t]) a"
using nb nlt
by (induct a rule: tm.induct) auto

lemma tmsubst_nb0:
assumes tnb: "tmbound0 t"
shows "tmbound0 (tmsubst 0 t a)"
using tnb
by (induct a rule: tm.induct) auto

lemma tmsubst_nb:
assumes tnb: "tmbound m t"
shows "tmbound m (tmsubst m t a)"
using tnb
by (induct a rule: tm.induct) auto

lemma incrtm0_tmbound: "tmbound n t ⟹ tmbound (Suc n) (incrtm0 t)"
by (induct t) auto

text ‹Simplification.›

fun tmadd:: "tm ⇒ tm ⇒ tm"
where
"tmadd (CNP n1 c1 r1) (CNP n2 c2 r2) =
(if n1 = n2 then
let c = c1 +⇩p c2
in if c = 0⇩p then tmadd r1 r2 else CNP n1 c (tmadd r1 r2)
else if n1 ≤ n2 then (CNP n1 c1 (tmadd r1 (CNP n2 c2 r2)))
else (CNP n2 c2 (tmadd (CNP n1 c1 r1) r2)))"
| "tmadd (CNP n1 c1 r1) t = CNP n1 c1 (tmadd r1 t)"
| "tmadd t (CNP n2 c2 r2) = CNP n2 c2 (tmadd t r2)"
| "tmadd (CP b1) (CP b2) = CP (b1 +⇩p b2)"

proof (induct t s rule: tmadd.induct)
case (1 n1 c1 r1 n2 c2 r2)
show ?case
proof (cases "c1 +⇩p c2 = 0⇩p")
case 0: True
show ?thesis
proof (cases "n1 ≤ n2")
case True
with 0 show ?thesis
by (metis INum_int(2) Ipoly.simps(1) comm_semiring_class.distrib mult_eq_0_iff polyadd)
qed (use 0 1 in auto)
next
case False
then show ?thesis
using 1 comm_semiring_class.distrib by auto
qed
qed auto

lemma tmadd_nb0[simp]: "tmbound0 t ⟹ tmbound0 s ⟹ tmbound0 (tmadd t s)"
by (induct t s rule: tmadd.induct) (auto simp: Let_def)

lemma tmadd_nb[simp]: "tmbound n t ⟹ tmbound n s ⟹ tmbound n (tmadd t s)"
by (induct t s rule: tmadd.induct) (auto simp: Let_def)

lemma tmadd_blt[simp]: "tmboundslt n t ⟹ tmboundslt n s ⟹ tmboundslt n (tmadd t s)"
by (induct t s rule: tmadd.induct) (auto simp: Let_def)

"allpolys isnpoly t ⟹ allpolys isnpoly s ⟹ allpolys isnpoly (tmadd t s)"

fun tmmul:: "tm ⇒ poly ⇒ tm"
where
"tmmul (CP j) = (λi. CP (i *⇩p j))"
| "tmmul (CNP n c a) = (λi. CNP n (i *⇩p c) (tmmul a i))"
| "tmmul t = (λi. Mul i t)"

lemma tmmul[simp]: "Itm vs bs (tmmul t i) = Itm vs bs (Mul i t)"
by (induct t arbitrary: i rule: tmmul.induct) (simp_all add: field_simps)

lemma tmmul_nb0[simp]: "tmbound0 t ⟹ tmbound0 (tmmul t i)"
by (induct t arbitrary: i rule: tmmul.induct) auto

lemma tmmul_nb[simp]: "tmbound n t ⟹ tmbound n (tmmul t i)"
by (induct t arbitrary: n rule: tmmul.induct) auto

lemma tmmul_blt[simp]: "tmboundslt n t ⟹ tmboundslt n (tmmul t i)"
by (induct t arbitrary: i rule: tmmul.induct) (auto simp: Let_def)

lemma tmmul_allpolys_npoly[simp]:
assumes "SORT_CONSTRAINT('a::field_char_0)"
shows "allpolys isnpoly t ⟹ isnpoly c ⟹ allpolys isnpoly (tmmul t c)"
by (induct t rule: tmmul.induct) (simp_all add: Let_def polymul_norm)

definition tmneg :: "tm ⇒ tm"
where "tmneg t ≡ tmmul t (C (- 1,1))"

definition tmsub :: "tm ⇒ tm ⇒ tm"
where "tmsub s t ≡ (if s = t then CP 0⇩p else tmadd s (tmneg t))"

lemma tmneg[simp]: "Itm vs bs (tmneg t) = Itm vs bs (Neg t)"
using tmneg_def[of t] by simp

lemma tmneg_nb0[simp]: "tmbound0 t ⟹ tmbound0 (tmneg t)"
using tmneg_def by simp

lemma tmneg_nb[simp]: "tmbound n t ⟹ tmbound n (tmneg t)"
using tmneg_def by simp

lemma tmneg_blt[simp]: "tmboundslt n t ⟹ tmboundslt n (tmneg t)"
using tmneg_def by simp

lemma [simp]: "isnpoly (C (-1, 1))"

lemma tmneg_allpolys_npoly[simp]:
assumes "SORT_CONSTRAINT('a::field_char_0)"
shows "allpolys isnpoly t ⟹ allpolys isnpoly (tmneg t)"
by (auto simp: tmneg_def)

lemma tmsub[simp]: "Itm vs bs (tmsub a b) = Itm vs bs (Sub a b)"
using tmsub_def by simp

lemma tmsub_nb0[simp]: "tmbound0 t ⟹ tmbound0 s ⟹ tmbound0 (tmsub t s)"
using tmsub_def by simp

lemma tmsub_nb[simp]: "tmbound n t ⟹ tmbound n s ⟹ tmbound n (tmsub t s)"
using tmsub_def by simp

lemma tmsub_blt[simp]: "tmboundslt n t ⟹ tmboundslt n s ⟹ tmboundslt n (tmsub t s)"
using tmsub_def by simp

lemma tmsub_allpolys_npoly[simp]:
assumes "SORT_CONSTRAINT('a::field_char_0)"
shows "allpolys isnpoly t ⟹ allpolys isnpoly s ⟹ allpolys isnpoly (tmsub t s)"

fun simptm :: "tm ⇒ tm"
where
"simptm (CP j) = CP (polynate j)"
| "simptm (Bound n) = CNP n (1)⇩p (CP 0⇩p)"
| "simptm (Neg t) = tmneg (simptm t)"
| "simptm (Sub t s) = tmsub (simptm t) (simptm s)"
| "simptm (Mul i t) =
(let i' = polynate i in if i' = 0⇩p then CP 0⇩p else tmmul (simptm t) i')"
| "simptm (CNP n c t) =
(let c' = polynate c in if c' = 0⇩p then simptm t else tmadd (CNP n c' (CP 0⇩p)) (simptm t))"

lemma polynate_stupid:
assumes "SORT_CONSTRAINT('a::field_char_0)"
shows "polynate t = 0⇩p ⟹ Ipoly bs t = (0::'a)"
by (metis INum_int(2) Ipoly.simps(1) polynate)

lemma simptm_ci[simp]: "Itm vs bs (simptm t) = Itm vs bs t"
by (induct t rule: simptm.induct) (auto simp: Let_def polynate_stupid)

lemma simptm_tmbound0[simp]: "tmbound0 t ⟹ tmbound0 (simptm t)"
by (induct t rule: simptm.induct) (auto simp: Let_def)

lemma simptm_nb[simp]: "tmbound n t ⟹ tmbound n (simptm t)"
by (induct t rule: simptm.induct) (auto simp: Let_def)

lemma simptm_nlt[simp]: "tmboundslt n t ⟹ tmboundslt n (simptm t)"
by (induct t rule: simptm.induct) (auto simp: Let_def)

lemma [simp]: "isnpoly 0⇩p"
and [simp]: "isnpoly (C (1, 1))"

lemma simptm_allpolys_npoly[simp]:
assumes "SORT_CONSTRAINT('a::field_char_0)"
shows "allpolys isnpoly (simptm p)"
by (induct p rule: simptm.induct) (auto simp: Let_def)

declare let_cong[fundef_cong del]

fun split0 :: "tm ⇒ poly × tm"
where
"split0 (Bound 0) = ((1)⇩p, CP 0⇩p)"
| "split0 (CNP 0 c t) = (let (c', t') = split0 t in (c +⇩p c', t'))"
| "split0 (Neg t) = (let (c, t') = split0 t in (~⇩p c, Neg t'))"
| "split0 (CNP n c t) = (let (c', t') = split0 t in (c', CNP n c t'))"
| "split0 (Add s t) = (let (c1, s') = split0 s; (c2, t') = split0 t in (c1 +⇩p c2, Add s' t'))"
| "split0 (Sub s t) = (let (c1, s') = split0 s; (c2, t') = split0 t in (c1 -⇩p c2, Sub s' t'))"
| "split0 (Mul c t) = (let (c', t') = split0 t in (c *⇩p c', Mul c t'))"
| "split0 t = (0⇩p, t)"

declare let_cong[fundef_cong]

lemma split0_stupid[simp]: "∃x y. (x, y) = split0 p"
using prod.collapse by blast

lemma split0:
"tmbound 0 (snd (split0 t)) ∧ Itm vs bs (CNP 0 (fst (split0 t)) (snd (split0 t))) = Itm vs bs t"
proof (induct t rule: split0.induct)
case (7 c t)
then show ?case
by (simp add: Let_def split_def mult.assoc flip: distrib_left)
qed (auto simp: Let_def split_def field_simps)

lemma split0_ci: "split0 t = (c',t') ⟹ Itm vs bs t = Itm vs bs (CNP 0 c' t')"
proof -
fix c' t'
assume "split0 t = (c', t')"
then have "c' = fst (split0 t)" "t' = snd (split0 t)"
by auto
with split0[where t="t" and bs="bs"] show "Itm vs bs t = Itm vs bs (CNP 0 c' t')"
by simp
qed

lemma split0_nb0:
assumes "SORT_CONSTRAINT('a::field_char_0)"
shows "split0 t = (c',t') ⟹  tmbound 0 t'"
proof -
fix c' t'
assume "split0 t = (c', t')"
then have "c' = fst (split0 t)" "t' = snd (split0 t)"
by auto
with conjunct1[OF split0[where t="t"]] show "tmbound 0 t'"
by simp
qed

lemma split0_nb0'[simp]:
assumes "SORT_CONSTRAINT('a::field_char_0)"
shows "tmbound0 (snd (split0 t))"
using split0_nb0[of t "fst (split0 t)" "snd (split0 t)"]

lemma split0_nb:
assumes nb: "tmbound n t"
shows "tmbound n (snd (split0 t))"
using nb by (induct t rule: split0.induct) (auto simp: Let_def split_def)

lemma split0_blt:
assumes nb: "tmboundslt n t"
shows "tmboundslt n (snd (split0 t))"
using nb by (induct t rule: split0.induct) (auto simp: Let_def split_def)

lemma tmbound_split0: "tmbound 0 t ⟹ Ipoly vs (fst (split0 t)) = 0"
by (induct t rule: split0.induct) (auto simp: Let_def split_def)

lemma tmboundslt_split0: "tmboundslt n t ⟹ Ipoly vs (fst (split0 t)) = 0 ∨ n > 0"
by (induct t rule: split0.induct) (auto simp: Let_def split_def)

lemma tmboundslt0_split0: "tmboundslt 0 t ⟹ Ipoly vs (fst (split0 t)) = 0"
by (induct t rule: split0.induct) (auto simp: Let_def split_def)

lemma allpolys_split0: "allpolys isnpoly p ⟹ allpolys isnpoly (snd (split0 p))"
by (induct p rule: split0.induct) (auto simp  add: isnpoly_def Let_def split_def)

lemma isnpoly_fst_split0:
assumes "SORT_CONSTRAINT('a::field_char_0)"
shows "allpolys isnpoly p ⟹ isnpoly (fst (split0 p))"
by (induct p rule: split0.induct)

subsection ‹Formulae›

datatype (plugins del: size) fm = T | F | Le tm | Lt tm | Eq tm | NEq tm |
Not fm | And fm fm | Or fm fm | Imp fm fm | Iff fm fm | E fm | A fm

instantiation fm :: size
begin

primrec size_fm :: "fm ⇒ nat"
where
"size_fm (Not p) = 1 + size_fm p"
| "size_fm (And p q) = 1 + size_fm p + size_fm q"
| "size_fm (Or p q) = 1 + size_fm p + size_fm q"
| "size_fm (Imp p q) = 3 + size_fm p + size_fm q"
| "size_fm (Iff p q) = 3 + 2 * (size_fm p + size_fm q)"
| "size_fm (E p) = 1 + size_fm p"
| "size_fm (A p) = 4 + size_fm p"
| "size_fm T = 1"
| "size_fm F = 1"
| "size_fm (Le _) = 1"
| "size_fm (Lt _) = 1"
| "size_fm (Eq _) = 1"
| "size_fm (NEq _) = 1"

instance ..

end

lemma fmsize_pos [simp]: "size p > 0" for p :: fm
by (induct p) simp_all

text ‹Semantics of formulae (fm).›
primrec Ifm ::"'a::linordered_field list ⇒ 'a list ⇒ fm ⇒ bool"
where
"Ifm vs bs T = True"
| "Ifm vs bs F = False"
| "Ifm vs bs (Lt a) = (Itm vs bs a < 0)"
| "Ifm vs bs (Le a) = (Itm vs bs a ≤ 0)"
| "Ifm vs bs (Eq a) = (Itm vs bs a = 0)"
| "Ifm vs bs (NEq a) = (Itm vs bs a ≠ 0)"
| "Ifm vs bs (Not p) = (¬ (Ifm vs bs p))"
| "Ifm vs bs (And p q) = (Ifm vs bs p ∧ Ifm vs bs q)"
| "Ifm vs bs (Or p q) = (Ifm vs bs p ∨ Ifm vs bs q)"
| "Ifm vs bs (Imp p q) = ((Ifm vs bs p) ⟶ (Ifm vs bs q))"
| "Ifm vs bs (Iff p q) = (Ifm vs bs p = Ifm vs bs q)"
| "Ifm vs bs (E p) = (∃x. Ifm vs (x#bs) p)"
| "Ifm vs bs (A p) = (∀x. Ifm vs (x#bs) p)"

fun not:: "fm ⇒ fm"
where
"not (Not (Not p)) = not p"
| "not (Not p) = p"
| "not T = F"
| "not F = T"
| "not (Lt t) = Le (tmneg t)"
| "not (Le t) = Lt (tmneg t)"
| "not (Eq t) = NEq t"
| "not (NEq t) = Eq t"
| "not p = Not p"

lemma not[simp]: "Ifm vs bs (not p) = Ifm vs bs (Not p)"
by (induct p rule: not.induct) auto

definition conj :: "fm ⇒ fm ⇒ fm"
where "conj p q ≡
(if p = F ∨ q = F then F
else if p = T then q
else if q = T then p
else if p = q then p
else And p q)"

lemma conj[simp]: "Ifm vs bs (conj p q) = Ifm vs bs (And p q)"
by (cases "p=F ∨ q=F", simp_all add: conj_def) (cases p, simp_all)

definition disj :: "fm ⇒ fm ⇒ fm"
where "disj p q ≡
(if (p = T ∨ q = T) then T
else if p = F then q
else if q = F then p
else if p = q then p
else Or p q)"

lemma disj[simp]: "Ifm vs bs (disj p q) = Ifm vs bs (Or p q)"
by (cases "p = T ∨ q = T", simp_all add: disj_def) (cases p, simp_all)

definition imp :: "fm ⇒ fm ⇒ fm"
where "imp p q ≡
(if p = F ∨ q = T ∨ p = q then T
else if p = T then q
else if q = F then not p
else Imp p q)"

lemma imp[simp]: "Ifm vs bs (imp p q) = Ifm vs bs (Imp p q)"
by (cases "p = F ∨ q = T") (simp_all add: imp_def)

definition iff :: "fm ⇒ fm ⇒ fm"
where "iff p q ≡
(if p = q then T
else if p = Not q ∨ Not p = q then F
else if p = F then not q
else if q = F then not p
else if p = T then q
else if q = T then p
else Iff p q)"

lemma iff[simp]: "Ifm vs bs (iff p q) = Ifm vs bs (Iff p q)"
by (unfold iff_def, cases "p = q", simp, cases "p = Not q", simp) (cases "Not p= q", auto)

text ‹Quantifier freeness.›
fun qfree:: "fm ⇒ bool"
where
"qfree (E p) = False"
| "qfree (A p) = False"
| "qfree (Not p) = qfree p"
| "qfree (And p q) = (qfree p ∧ qfree q)"
| "qfree (Or  p q) = (qfree p ∧ qfree q)"
| "qfree (Imp p q) = (qfree p ∧ qfree q)"
| "qfree (Iff p q) = (qfree p ∧ qfree q)"
| "qfree p = True"

text ‹Boundedness and substitution.›
primrec boundslt :: "nat ⇒ fm ⇒ bool"
where
"boundslt n T = True"
| "boundslt n F = True"
| "boundslt n (Lt t) = tmboundslt n t"
| "boundslt n (Le t) = tmboundslt n t"
| "boundslt n (Eq t) = tmboundslt n t"
| "boundslt n (NEq t) = tmboundslt n t"
| "boundslt n (Not p) = boundslt n p"
| "boundslt n (And p q) = (boundslt n p ∧ boundslt n q)"
| "boundslt n (Or p q) = (boundslt n p ∧ boundslt n q)"
| "boundslt n (Imp p q) = ((boundslt n p) ∧ (boundslt n q))"
| "boundslt n (Iff p q) = (boundslt n p ∧ boundslt n q)"
| "boundslt n (E p) = boundslt (Suc n) p"
| "boundslt n (A p) = boundslt (Suc n) p"

fun bound0:: "fm ⇒ bool"  ― ‹a formula is independent of Bound 0›
where
"bound0 T = True"
| "bound0 F = True"
| "bound0 (Lt a) = tmbound0 a"
| "bound0 (Le a) = tmbound0 a"
| "bound0 (Eq a) = tmbound0 a"
| "bound0 (NEq a) = tmbound0 a"
| "bound0 (Not p) = bound0 p"
| "bound0 (And p q) = (bound0 p ∧ bound0 q)"
| "bound0 (Or p q) = (bound0 p ∧ bound0 q)"
| "bound0 (Imp p q) = ((bound0 p) ∧ (bound0 q))"
| "bound0 (Iff p q) = (bound0 p ∧ bound0 q)"
| "bound0 p = False"

lemma bound0_I:
assumes bp: "bound0 p"
shows "Ifm vs (b#bs) p = Ifm vs (b'#bs) p"
using bp tmbound0_I[where b="b" and bs="bs" and b'="b'"]
by (induct p rule: bound0.induct) auto

primrec bound:: "nat ⇒ fm ⇒ bool"  ― ‹a formula is independent of Bound n›
where
"bound m T = True"
| "bound m F = True"
| "bound m (Lt t) = tmbound m t"
| "bound m (Le t) = tmbound m t"
| "bound m (Eq t) = tmbound m t"
| "bound m (NEq t) = tmbound m t"
| "bound m (Not p) = bound m p"
| "bound m (And p q) = (bound m p ∧ bound m q)"
| "bound m (Or p q) = (bound m p ∧ bound m q)"
| "bound m (Imp p q) = ((bound m p) ∧ (bound m q))"
| "bound m (Iff p q) = (bound m p ∧ bound m q)"
| "bound m (E p) = bound (Suc m) p"
| "bound m (A p) = bound (Suc m) p"

lemma bound_I:
assumes bnd: "boundslt (length bs) p"
and nb: "bound n p"
and le: "n ≤ length bs"
shows "Ifm vs (bs[n:=x]) p = Ifm vs bs p"
using bnd nb le tmbound_I[where bs=bs and vs = vs]
proof (induct p arbitrary: bs n rule: fm.induct)
case (E p bs n)
have "Ifm vs ((y#bs)[Suc n:=x]) p = Ifm vs (y#bs) p" for y
proof -
from E have bnd: "boundslt (length (y#bs)) p"
and nb: "bound (Suc n) p" and le: "Suc n ≤ length (y#bs)" by simp+
from E.hyps[OF bnd nb le tmbound_I] show ?thesis .
qed
then show ?case by simp
next
case (A p bs n)
have "Ifm vs ((y#bs)[Suc n:=x]) p = Ifm vs (y#bs) p" for y
proof -
from A have bnd: "boundslt (length (y#bs)) p"
and nb: "bound (Suc n) p"
and le: "Suc n ≤ length (y#bs)"
by simp_all
from A.hyps[OF bnd nb le tmbound_I] show ?thesis .
qed
then show ?case by simp
qed auto

fun decr0 :: "fm ⇒ fm"
where
"decr0 (Lt a) = Lt (decrtm0 a)"
| "decr0 (Le a) = Le (decrtm0 a)"
| "decr0 (Eq a) = Eq (decrtm0 a)"
| "decr0 (NEq a) = NEq (decrtm0 a)"
| "decr0 (Not p) = Not (decr0 p)"
| "decr0 (And p q) = conj (decr0 p) (decr0 q)"
| "decr0 (Or p q) = disj (decr0 p) (decr0 q)"
| "decr0 (Imp p q) = imp (decr0 p) (decr0 q)"
| "decr0 (Iff p q) = iff (decr0 p) (decr0 q)"
| "decr0 p = p"

lemma decr0:
assumes "bound0 p"
shows "Ifm vs (x#bs) p = Ifm vs bs (decr0 p)"
using assms by (induct p rule: decr0.induct) (simp_all add: decrtm0)

primrec decr :: "nat ⇒ fm ⇒ fm"
where
"decr m T = T"
| "decr m F = F"
| "decr m (Lt t) = (Lt (decrtm m t))"
| "decr m (Le t) = (Le (decrtm m t))"
| "decr m (Eq t) = (Eq (decrtm m t))"
| "decr m (NEq t) = (NEq (decrtm m t))"
| "decr m (Not p) = Not (decr m p)"
| "decr m (And p q) = conj (decr m p) (decr m q)"
| "decr m (Or p q) = disj (decr m p) (decr m q)"
| "decr m (Imp p q) = imp (decr m p) (decr m q)"
| "decr m (Iff p q) = iff (decr m p) (decr m q)"
| "decr m (E p) = E (decr (Suc m) p)"
| "decr m (A p) = A (decr (Suc m) p)"

lemma decr:
assumes bnd: "boundslt (length bs) p"
and nb: "bound m p"
and nle: "m < length bs"
shows "Ifm vs (removen m bs) (decr m p) = Ifm vs bs p"
using bnd nb nle
proof (induct p arbitrary: bs m rule: fm.induct)
case (E p bs m)
have "Ifm vs (removen (Suc m) (x#bs)) (decr (Suc m) p) = Ifm vs (x#bs) p" for x
proof -
from E
have bnd: "boundslt (length (x#bs)) p"
and nb: "bound (Suc m) p"
and nle: "Suc m < length (x#bs)"
by auto
from E(1)[OF bnd nb nle] show ?thesis .
qed
then show ?case by auto
next
case (A p bs m)
have "Ifm vs (removen (Suc m) (x#bs)) (decr (Suc m) p) = Ifm vs (x#bs) p" for x
proof -
from A
have bnd: "boundslt (length (x#bs)) p"
and nb: "bound (Suc m) p"
and nle: "Suc m < length (x#bs)"
by auto
from A(1)[OF bnd nb nle] show ?thesis .
qed
then show ?case by auto
qed (auto simp: decrtm removen_nth)

primrec subst0 :: "tm ⇒ fm ⇒ fm"
where
"subst0 t T = T"
| "subst0 t F = F"
| "subst0 t (Lt a) = Lt (tmsubst0 t a)"
| "subst0 t (Le a) = Le (tmsubst0 t a)"
| "subst0 t (Eq a) = Eq (tmsubst0 t a)"
| "subst0 t (NEq a) = NEq (tmsubst0 t a)"
| "subst0 t (Not p) = Not (subst0 t p)"
| "subst0 t (And p q) = And (subst0 t p) (subst0 t q)"
| "subst0 t (Or p q) = Or (subst0 t p) (subst0 t q)"
| "subst0 t (Imp p q) = Imp (subst0 t p)  (subst0 t q)"
| "subst0 t (Iff p q) = Iff (subst0 t p) (subst0 t q)"
| "subst0 t (E p) = E p"
| "subst0 t (A p) = A p"

lemma subst0:
assumes qf: "qfree p"
shows "Ifm vs (x # bs) (subst0 t p) = Ifm vs ((Itm vs (x # bs) t) # bs) p"
using qf tmsubst0[where x="x" and bs="bs" and t="t"]
by (induct p rule: fm.induct) auto

lemma subst0_nb:
assumes bp: "tmbound0 t"
and qf: "qfree p"
shows "bound0 (subst0 t p)"
using qf tmsubst0_nb[OF bp] bp by (induct p rule: fm.induct) auto

primrec subst:: "nat ⇒ tm ⇒ fm ⇒ fm"
where
"subst n t T = T"
| "subst n t F = F"
| "subst n t (Lt a) = Lt (tmsubst n t a)"
| "subst n t (Le a) = Le (tmsubst n t a)"
| "subst n t (Eq a) = Eq (tmsubst n t a)"
| "subst n t (NEq a) = NEq (tmsubst n t a)"
| "subst n t (Not p) = Not (subst n t p)"
| "subst n t (And p q) = And (subst n t p) (subst n t q)"
| "subst n t (Or p q) = Or (subst n t p) (subst n t q)"
| "subst n t (Imp p q) = Imp (subst n t p)  (subst n t q)"
| "subst n t (Iff p q) = Iff (subst n t p) (subst n t q)"
| "subst n t (E p) = E (subst (Suc n) (incrtm0 t) p)"
| "subst n t (A p) = A (subst (Suc n) (incrtm0 t) p)"

lemma subst:
assumes nb: "boundslt (length bs) p"
and nlm: "n ≤ length bs"
shows "Ifm vs bs (subst n t p) = Ifm vs (bs[n:= Itm vs bs t]) p"
using nb nlm
proof (induct p arbitrary: bs n t rule: fm.induct)
case (E p bs n)
have "Ifm vs (x#bs) (subst (Suc n) (incrtm0 t) p) =
Ifm vs (x#bs[n:= Itm vs bs t]) p" for x
proof -
from E have bn: "boundslt (length (x#bs)) p"
by simp
from E have nlm: "Suc n ≤ length (x#bs)"
by simp
from E(1)[OF bn nlm]
have "Ifm vs (x#bs) (subst (Suc n) (incrtm0 t) p) =
Ifm vs ((x#bs)[Suc n:= Itm vs (x#bs) (incrtm0 t)]) p"
by simp
then show ?thesis
by (simp add: incrtm0[where x="x" and bs="bs" and t="t"])
qed
then show ?case by simp
next
case (A p bs n)
have "Ifm vs (x#bs) (subst (Suc n) (incrtm0 t) p) =
Ifm vs (x#bs[n:= Itm vs bs t]) p" for x
proof -
from A have bn: "boundslt (length (x#bs)) p"
by simp
from A have nlm: "Suc n ≤ length (x#bs)"
by simp
from A(1)[OF bn nlm]
have "Ifm vs (x#bs) (subst (Suc n) (incrtm0 t) p) =
Ifm vs ((x#bs)[Suc n:= Itm vs (x#bs) (incrtm0 t)]) p"
by simp
then show ?thesis
by (simp add: incrtm0[where x="x" and bs="bs" and t="t"])
qed
then show ?case by simp
qed (auto simp: tmsubst)

lemma subst_nb:
assumes "tmbound m t"
shows "bound m (subst m t p)"
using assms tmsubst_nb incrtm0_tmbound by (induct p arbitrary: m t rule: fm.induct) auto

lemma not_qf[simp]: "qfree p ⟹ qfree (not p)"
by (induct p rule: not.induct) auto
lemma not_bn0[simp]: "bound0 p ⟹ bound0 (not p)"
by (induct p rule: not.induct) auto
lemma not_nb[simp]: "bound n p ⟹ bound n (not p)"
by (induct p rule: not.induct) auto
lemma not_blt[simp]: "boundslt n p ⟹ boundslt n (not p)"
by (induct p rule: not.induct) auto

lemma conj_qf[simp]: "qfree p ⟹ qfree q ⟹ qfree (conj p q)"
using conj_def by auto
lemma conj_nb0[simp]: "bound0 p ⟹ bound0 q ⟹ bound0 (conj p q)"
using conj_def by auto
lemma conj_nb[simp]: "bound n p ⟹ bound n q ⟹ bound n (conj p q)"
using conj_def by auto
lemma conj_blt[simp]: "boundslt n p ⟹ boundslt n q ⟹ boundslt n (conj p q)"
using conj_def by auto

lemma disj_qf[simp]: "qfree p ⟹ qfree q ⟹ qfree (disj p q)"
using disj_def by auto
lemma disj_nb0[simp]: "bound0 p ⟹ bound0 q ⟹ bound0 (disj p q)"
using disj_def by auto
lemma disj_nb[simp]: "bound n p ⟹ bound n q ⟹ bound n (disj p q)"
using disj_def by auto
lemma disj_blt[simp]: "boundslt n p ⟹ boundslt n q ⟹ boundslt n (disj p q)"
using disj_def by auto

lemma imp_qf[simp]: "qfree p ⟹ qfree q ⟹ qfree (imp p q)"
using imp_def by (cases "p = F ∨ q = T") (simp_all add: imp_def)
lemma imp_nb0[simp]: "bound0 p ⟹ bound0 q ⟹ bound0 (imp p q)"
using imp_def by (cases "p = F ∨ q = T ∨ p = q") (simp_all add: imp_def)
lemma imp_nb[simp]: "bound n p ⟹ bound n q ⟹ bound n (imp p q)"
using imp_def by (cases "p = F ∨ q = T ∨ p = q") (simp_all add: imp_def)
lemma imp_blt[simp]: "boundslt n p ⟹ boundslt n q ⟹ boundslt n (imp p q)"
using imp_def by auto

lemma iff_qf[simp]: "qfree p ⟹ qfree q ⟹ qfree (iff p q)"
unfolding iff_def by (cases "p = q") auto
lemma iff_nb0[simp]: "bound0 p ⟹ bound0 q ⟹ bound0 (iff p q)"
using iff_def unfolding iff_def by (cases "p = q") auto
lemma iff_nb[simp]: "bound n p ⟹ bound n q ⟹ bound n (iff p q)"
using iff_def unfolding iff_def by (cases "p = q") auto
lemma iff_blt[simp]: "boundslt n p ⟹ boundslt n q ⟹ boundslt n (iff p q)"
using iff_def by auto
lemma decr0_qf: "bound0 p ⟹ qfree (decr0 p)"
by (induct p) simp_all

fun isatom :: "fm ⇒ bool"  ― ‹test for atomicity›
where
"isatom T = True"
| "isatom F = True"
| "isatom (Lt a) = True"
| "isatom (Le a) = True"
| "isatom (Eq a) = True"
| "isatom (NEq a) = True"
| "isatom p = False"

lemma bound0_qf: "bound0 p ⟹ qfree p"
by (induct p) simp_all

definition djf :: "('a ⇒ fm) ⇒ 'a ⇒ fm ⇒ fm"
where "djf f p q ≡
(if q = T then T
else if q = F then f p
else (let fp = f p in case fp of T ⇒ T | F ⇒ q | _ ⇒ Or (f p) q))"

definition evaldjf :: "('a ⇒ fm) ⇒ 'a list ⇒ fm"
where "evaldjf f ps ≡ foldr (djf f) ps F"

lemma djf_Or: "Ifm vs bs (djf f p q) = Ifm vs bs (Or (f p) q)"
by (cases "f p") (simp_all add: Let_def djf_def)

lemma evaldjf_ex: "Ifm vs bs (evaldjf f ps) ⟷ (∃p ∈ set ps. Ifm vs bs (f p))"
by (induct ps) (simp_all add: evaldjf_def djf_Or)

lemma evaldjf_bound0:
assumes "∀x ∈ set xs. bound0 (f x)"
shows "bound0 (evaldjf f xs)"
using assms
proof (induct xs)
case Nil
then show ?case
next
case (Cons a xs)
then show ?case
by (cases "f a") (simp_all add: evaldjf_def djf_def Let_def)
qed

lemma evaldjf_qf:
assumes "∀x∈ set xs. qfree (f x)"
shows "qfree (evaldjf f xs)"
using assms
proof (induct xs)
case Nil
then show ?case