Structure integerRingTheory
signature integerRingTheory =
sig
type thm = Thm.thm
(* Definitions *)
val int_interp_p_def : thm
val int_polynom_normalize_def : thm
val int_polynom_simplify_def : thm
val int_r_canonical_sum_merge_def : thm
val int_r_canonical_sum_prod_def : thm
val int_r_canonical_sum_scalar2_def : thm
val int_r_canonical_sum_scalar3_def : thm
val int_r_canonical_sum_scalar_def : thm
val int_r_canonical_sum_simplify_def : thm
val int_r_ics_aux_def : thm
val int_r_interp_cs_def : thm
val int_r_interp_m_def : thm
val int_r_interp_sp_def : thm
val int_r_interp_vl_def : thm
val int_r_ivl_aux_def : thm
val int_r_monom_insert_def : thm
val int_r_spolynom_normalize_def : thm
val int_r_spolynom_simplify_def : thm
val int_r_varlist_insert_def : thm
(* Theorems *)
val int_calculate : thm
val int_is_ring : thm
val int_rewrites : thm
val int_ring_thms : thm
val integerRing_grammars : type_grammar.grammar * term_grammar.grammar
(*
[integer] Parent theory of "integerRing"
[ringNorm] Parent theory of "integerRing"
[int_interp_p_def] Definition
⊢ int_interp_p = interp_p (ring int_0 int_1 $+ $* numeric_negate)
[int_polynom_normalize_def] Definition
⊢ int_polynom_normalize =
polynom_normalize (ring int_0 int_1 $+ $* numeric_negate)
[int_polynom_simplify_def] Definition
⊢ int_polynom_simplify =
polynom_simplify (ring int_0 int_1 $+ $* numeric_negate)
[int_r_canonical_sum_merge_def] Definition
⊢ int_r_canonical_sum_merge =
r_canonical_sum_merge (ring int_0 int_1 $+ $* numeric_negate)
[int_r_canonical_sum_prod_def] Definition
⊢ int_r_canonical_sum_prod =
r_canonical_sum_prod (ring int_0 int_1 $+ $* numeric_negate)
[int_r_canonical_sum_scalar2_def] Definition
⊢ int_r_canonical_sum_scalar2 =
r_canonical_sum_scalar2 (ring int_0 int_1 $+ $* numeric_negate)
[int_r_canonical_sum_scalar3_def] Definition
⊢ int_r_canonical_sum_scalar3 =
r_canonical_sum_scalar3 (ring int_0 int_1 $+ $* numeric_negate)
[int_r_canonical_sum_scalar_def] Definition
⊢ int_r_canonical_sum_scalar =
r_canonical_sum_scalar (ring int_0 int_1 $+ $* numeric_negate)
[int_r_canonical_sum_simplify_def] Definition
⊢ int_r_canonical_sum_simplify =
r_canonical_sum_simplify (ring int_0 int_1 $+ $* numeric_negate)
[int_r_ics_aux_def] Definition
⊢ int_r_ics_aux = r_ics_aux (ring int_0 int_1 $+ $* numeric_negate)
[int_r_interp_cs_def] Definition
⊢ int_r_interp_cs =
r_interp_cs (ring int_0 int_1 $+ $* numeric_negate)
[int_r_interp_m_def] Definition
⊢ int_r_interp_m = r_interp_m (ring int_0 int_1 $+ $* numeric_negate)
[int_r_interp_sp_def] Definition
⊢ int_r_interp_sp =
r_interp_sp (ring int_0 int_1 $+ $* numeric_negate)
[int_r_interp_vl_def] Definition
⊢ int_r_interp_vl =
r_interp_vl (ring int_0 int_1 $+ $* numeric_negate)
[int_r_ivl_aux_def] Definition
⊢ int_r_ivl_aux = r_ivl_aux (ring int_0 int_1 $+ $* numeric_negate)
[int_r_monom_insert_def] Definition
⊢ int_r_monom_insert =
r_monom_insert (ring int_0 int_1 $+ $* numeric_negate)
[int_r_spolynom_normalize_def] Definition
⊢ int_r_spolynom_normalize =
r_spolynom_normalize (ring int_0 int_1 $+ $* numeric_negate)
[int_r_spolynom_simplify_def] Definition
⊢ int_r_spolynom_simplify =
r_spolynom_simplify (ring int_0 int_1 $+ $* numeric_negate)
[int_r_varlist_insert_def] Definition
⊢ int_r_varlist_insert =
r_varlist_insert (ring int_0 int_1 $+ $* numeric_negate)
[int_calculate] Theorem
⊢ (&n + &m = &(n + m)) ∧
(-&n + &m = if n ≤ m then &(m − n) else -&(n − m)) ∧
(&n + -&m = if m ≤ n then &(n − m) else -&(m − n)) ∧
(-&n + -&m = -&(n + m)) ∧ (&n * &m = &(n * m)) ∧
(-&n * &m = -&(n * m)) ∧ (&n * -&m = -&(n * m)) ∧
(-&n * -&m = &(n * m)) ∧ ((&n = &m) ⇔ (n = m)) ∧
((&n = -&m) ⇔ (n = 0) ∧ (m = 0)) ∧
((-&n = &m) ⇔ (n = 0) ∧ (m = 0)) ∧ ((-&n = -&m) ⇔ (n = m)) ∧
(--x = x) ∧ (-0 = 0)
[int_is_ring] Theorem
⊢ is_ring (ring int_0 int_1 $+ $* numeric_negate)
[int_rewrites] Theorem
⊢ ((&n + &m = &(n + m)) ∧
(-&n + &m = if n ≤ m then &(m − n) else -&(n − m)) ∧
(&n + -&m = if m ≤ n then &(n − m) else -&(m − n)) ∧
(-&n + -&m = -&(n + m)) ∧ (&n * &m = &(n * m)) ∧
(-&n * &m = -&(n * m)) ∧ (&n * -&m = -&(n * m)) ∧
(-&n * -&m = &(n * m)) ∧ ((&n = &m) ⇔ (n = m)) ∧
((&n = -&m) ⇔ (n = 0) ∧ (m = 0)) ∧
((-&n = &m) ⇔ (n = 0) ∧ (m = 0)) ∧ ((-&n = -&m) ⇔ (n = m)) ∧
(--x = x) ∧ (-0 = 0)) ∧ (int_0 = 0) ∧ (int_1 = 1) ∧
(∀n m.
(ZERO < BIT1 n ⇔ T) ∧ (ZERO < BIT2 n ⇔ T) ∧ (n < ZERO ⇔ F) ∧
(BIT1 n < BIT1 m ⇔ n < m) ∧ (BIT2 n < BIT2 m ⇔ n < m) ∧
(BIT1 n < BIT2 m ⇔ ¬(m < n)) ∧ (BIT2 n < BIT1 m ⇔ n < m)) ∧
(∀n m.
(ZERO ≤ n ⇔ T) ∧ (BIT1 n ≤ ZERO ⇔ F) ∧ (BIT2 n ≤ ZERO ⇔ F) ∧
(BIT1 n ≤ BIT1 m ⇔ n ≤ m) ∧ (BIT1 n ≤ BIT2 m ⇔ n ≤ m) ∧
(BIT2 n ≤ BIT1 m ⇔ ¬(m ≤ n)) ∧ (BIT2 n ≤ BIT2 m ⇔ n ≤ m)) ∧
(∀n m.
NUMERAL (n − m) =
if m < n then NUMERAL (numeral$iSUB T n m) else 0) ∧
(∀b n m.
(numeral$iSUB b ZERO x = ZERO) ∧ (numeral$iSUB T n ZERO = n) ∧
(numeral$iSUB F (BIT1 n) ZERO = numeral$iDUB n) ∧
(numeral$iSUB T (BIT1 n) (BIT1 m) =
numeral$iDUB (numeral$iSUB T n m)) ∧
(numeral$iSUB F (BIT1 n) (BIT1 m) = BIT1 (numeral$iSUB F n m)) ∧
(numeral$iSUB T (BIT1 n) (BIT2 m) = BIT1 (numeral$iSUB F n m)) ∧
(numeral$iSUB F (BIT1 n) (BIT2 m) =
numeral$iDUB (numeral$iSUB F n m)) ∧
(numeral$iSUB F (BIT2 n) ZERO = BIT1 n) ∧
(numeral$iSUB T (BIT2 n) (BIT1 m) = BIT1 (numeral$iSUB T n m)) ∧
(numeral$iSUB F (BIT2 n) (BIT1 m) =
numeral$iDUB (numeral$iSUB T n m)) ∧
(numeral$iSUB T (BIT2 n) (BIT2 m) =
numeral$iDUB (numeral$iSUB T n m)) ∧
(numeral$iSUB F (BIT2 n) (BIT2 m) = BIT1 (numeral$iSUB F n m))) ∧
∀t. (T ∧ t ⇔ t) ∧ (t ∧ T ⇔ t) ∧ (F ∧ t ⇔ F) ∧ (t ∧ F ⇔ F) ∧
(t ∧ t ⇔ t)
[int_ring_thms] Theorem
⊢ is_ring (ring int_0 int_1 $+ $* numeric_negate) ∧
(∀vm p.
int_interp_p vm p = int_r_interp_cs vm (int_polynom_simplify p)) ∧
(((∀vm c. int_interp_p vm (Pconst c) = c) ∧
(∀vm i. int_interp_p vm (Pvar i) = varmap_find i vm) ∧
(∀vm p1 p2.
int_interp_p vm (Pplus p1 p2) =
int_interp_p vm p1 + int_interp_p vm p2) ∧
(∀vm p1 p2.
int_interp_p vm (Pmult p1 p2) =
int_interp_p vm p1 * int_interp_p vm p2) ∧
∀vm p1. int_interp_p vm (Popp p1) = -int_interp_p vm p1) ∧
(∀x v2 v1. varmap_find End_idx (Node_vm x v1 v2) = x) ∧
(∀x v2 v1 i1.
varmap_find (Right_idx i1) (Node_vm x v1 v2) =
varmap_find i1 v2) ∧
(∀x v2 v1 i1.
varmap_find (Left_idx i1) (Node_vm x v1 v2) = varmap_find i1 v1) ∧
∀i. varmap_find i Empty_vm = @x. T) ∧
((∀t2 t1 l2 l1 c2 c1.
int_r_canonical_sum_merge (Cons_monom c1 l1 t1)
(Cons_monom c2 l2 t2) =
case list_compare index_compare l1 l2 of
LESS =>
Cons_monom c1 l1
(int_r_canonical_sum_merge t1 (Cons_monom c2 l2 t2))
| EQUAL =>
Cons_monom (c1 + c2) l1 (int_r_canonical_sum_merge t1 t2)
| GREATER =>
Cons_monom c2 l2
(int_r_canonical_sum_merge (Cons_monom c1 l1 t1) t2)) ∧
(∀t2 t1 l2 l1 c1.
int_r_canonical_sum_merge (Cons_monom c1 l1 t1)
(Cons_varlist l2 t2) =
case list_compare index_compare l1 l2 of
LESS =>
Cons_monom c1 l1
(int_r_canonical_sum_merge t1 (Cons_varlist l2 t2))
| EQUAL =>
Cons_monom (c1 + int_1) l1 (int_r_canonical_sum_merge t1 t2)
| GREATER =>
Cons_varlist l2
(int_r_canonical_sum_merge (Cons_monom c1 l1 t1) t2)) ∧
(∀t2 t1 l2 l1 c2.
int_r_canonical_sum_merge (Cons_varlist l1 t1)
(Cons_monom c2 l2 t2) =
case list_compare index_compare l1 l2 of
LESS =>
Cons_varlist l1
(int_r_canonical_sum_merge t1 (Cons_monom c2 l2 t2))
| EQUAL =>
Cons_monom (int_1 + c2) l1 (int_r_canonical_sum_merge t1 t2)
| GREATER =>
Cons_monom c2 l2
(int_r_canonical_sum_merge (Cons_varlist l1 t1) t2)) ∧
(∀t2 t1 l2 l1.
int_r_canonical_sum_merge (Cons_varlist l1 t1)
(Cons_varlist l2 t2) =
case list_compare index_compare l1 l2 of
LESS =>
Cons_varlist l1
(int_r_canonical_sum_merge t1 (Cons_varlist l2 t2))
| EQUAL =>
Cons_monom (int_1 + int_1) l1
(int_r_canonical_sum_merge t1 t2)
| GREATER =>
Cons_varlist l2
(int_r_canonical_sum_merge (Cons_varlist l1 t1) t2)) ∧
(∀s1. int_r_canonical_sum_merge s1 Nil_monom = s1) ∧
(∀v6 v5 v4.
int_r_canonical_sum_merge Nil_monom (Cons_monom v4 v5 v6) =
Cons_monom v4 v5 v6) ∧
∀v8 v7.
int_r_canonical_sum_merge Nil_monom (Cons_varlist v7 v8) =
Cons_varlist v7 v8) ∧
((∀t2 l2 l1 c2 c1.
int_r_monom_insert c1 l1 (Cons_monom c2 l2 t2) =
case list_compare index_compare l1 l2 of
LESS => Cons_monom c1 l1 (Cons_monom c2 l2 t2)
| EQUAL => Cons_monom (c1 + c2) l1 t2
| GREATER => Cons_monom c2 l2 (int_r_monom_insert c1 l1 t2)) ∧
(∀t2 l2 l1 c1.
int_r_monom_insert c1 l1 (Cons_varlist l2 t2) =
case list_compare index_compare l1 l2 of
LESS => Cons_monom c1 l1 (Cons_varlist l2 t2)
| EQUAL => Cons_monom (c1 + int_1) l1 t2
| GREATER => Cons_varlist l2 (int_r_monom_insert c1 l1 t2)) ∧
∀l1 c1.
int_r_monom_insert c1 l1 Nil_monom = Cons_monom c1 l1 Nil_monom) ∧
((∀t2 l2 l1 c2.
int_r_varlist_insert l1 (Cons_monom c2 l2 t2) =
case list_compare index_compare l1 l2 of
LESS => Cons_varlist l1 (Cons_monom c2 l2 t2)
| EQUAL => Cons_monom (int_1 + c2) l1 t2
| GREATER => Cons_monom c2 l2 (int_r_varlist_insert l1 t2)) ∧
(∀t2 l2 l1.
int_r_varlist_insert l1 (Cons_varlist l2 t2) =
case list_compare index_compare l1 l2 of
LESS => Cons_varlist l1 (Cons_varlist l2 t2)
| EQUAL => Cons_monom (int_1 + int_1) l1 t2
| GREATER => Cons_varlist l2 (int_r_varlist_insert l1 t2)) ∧
∀l1. int_r_varlist_insert l1 Nil_monom = Cons_varlist l1 Nil_monom) ∧
((∀c0 c l t.
int_r_canonical_sum_scalar c0 (Cons_monom c l t) =
Cons_monom (c0 * c) l (int_r_canonical_sum_scalar c0 t)) ∧
(∀c0 l t.
int_r_canonical_sum_scalar c0 (Cons_varlist l t) =
Cons_monom c0 l (int_r_canonical_sum_scalar c0 t)) ∧
∀c0. int_r_canonical_sum_scalar c0 Nil_monom = Nil_monom) ∧
((∀l0 c l t.
int_r_canonical_sum_scalar2 l0 (Cons_monom c l t) =
int_r_monom_insert c (list_merge index_lt l0 l)
(int_r_canonical_sum_scalar2 l0 t)) ∧
(∀l0 l t.
int_r_canonical_sum_scalar2 l0 (Cons_varlist l t) =
int_r_varlist_insert (list_merge index_lt l0 l)
(int_r_canonical_sum_scalar2 l0 t)) ∧
∀l0. int_r_canonical_sum_scalar2 l0 Nil_monom = Nil_monom) ∧
((∀c0 l0 c l t.
int_r_canonical_sum_scalar3 c0 l0 (Cons_monom c l t) =
int_r_monom_insert (c0 * c) (list_merge index_lt l0 l)
(int_r_canonical_sum_scalar3 c0 l0 t)) ∧
(∀c0 l0 l t.
int_r_canonical_sum_scalar3 c0 l0 (Cons_varlist l t) =
int_r_monom_insert c0 (list_merge index_lt l0 l)
(int_r_canonical_sum_scalar3 c0 l0 t)) ∧
∀c0 l0. int_r_canonical_sum_scalar3 c0 l0 Nil_monom = Nil_monom) ∧
((∀c1 l1 t1 s2.
int_r_canonical_sum_prod (Cons_monom c1 l1 t1) s2 =
int_r_canonical_sum_merge
(int_r_canonical_sum_scalar3 c1 l1 s2)
(int_r_canonical_sum_prod t1 s2)) ∧
(∀l1 t1 s2.
int_r_canonical_sum_prod (Cons_varlist l1 t1) s2 =
int_r_canonical_sum_merge (int_r_canonical_sum_scalar2 l1 s2)
(int_r_canonical_sum_prod t1 s2)) ∧
∀s2. int_r_canonical_sum_prod Nil_monom s2 = Nil_monom) ∧
((∀c l t.
int_r_canonical_sum_simplify (Cons_monom c l t) =
if c = int_0 then int_r_canonical_sum_simplify t
else if c = int_1 then
Cons_varlist l (int_r_canonical_sum_simplify t)
else Cons_monom c l (int_r_canonical_sum_simplify t)) ∧
(∀l t.
int_r_canonical_sum_simplify (Cons_varlist l t) =
Cons_varlist l (int_r_canonical_sum_simplify t)) ∧
(int_r_canonical_sum_simplify Nil_monom = Nil_monom)) ∧
((∀vm x. int_r_ivl_aux vm x [] = varmap_find x vm) ∧
∀vm x x' t'.
int_r_ivl_aux vm x (x'::t') =
varmap_find x vm * int_r_ivl_aux vm x' t') ∧
((∀vm. int_r_interp_vl vm [] = int_1) ∧
∀vm x t. int_r_interp_vl vm (x::t) = int_r_ivl_aux vm x t) ∧
((∀vm c. int_r_interp_m vm c [] = c) ∧
∀vm c x t. int_r_interp_m vm c (x::t) = c * int_r_ivl_aux vm x t) ∧
((∀vm a. int_r_ics_aux vm a Nil_monom = a) ∧
(∀vm a l t.
int_r_ics_aux vm a (Cons_varlist l t) =
a + int_r_ics_aux vm (int_r_interp_vl vm l) t) ∧
∀vm a c l t.
int_r_ics_aux vm a (Cons_monom c l t) =
a + int_r_ics_aux vm (int_r_interp_m vm c l) t) ∧
((∀vm. int_r_interp_cs vm Nil_monom = int_0) ∧
(∀vm l t.
int_r_interp_cs vm (Cons_varlist l t) =
int_r_ics_aux vm (int_r_interp_vl vm l) t) ∧
∀vm c l t.
int_r_interp_cs vm (Cons_monom c l t) =
int_r_ics_aux vm (int_r_interp_m vm c l) t) ∧
((∀i. int_polynom_normalize (Pvar i) = Cons_varlist [i] Nil_monom) ∧
(∀c. int_polynom_normalize (Pconst c) = Cons_monom c [] Nil_monom) ∧
(∀pl pr.
int_polynom_normalize (Pplus pl pr) =
int_r_canonical_sum_merge (int_polynom_normalize pl)
(int_polynom_normalize pr)) ∧
(∀pl pr.
int_polynom_normalize (Pmult pl pr) =
int_r_canonical_sum_prod (int_polynom_normalize pl)
(int_polynom_normalize pr)) ∧
∀p. int_polynom_normalize (Popp p) =
int_r_canonical_sum_scalar3 (-int_1) []
(int_polynom_normalize p)) ∧
∀x. int_polynom_simplify x =
int_r_canonical_sum_simplify (int_polynom_normalize x)
*)
end
HOL 4, Kananaskis-14