Structure ratRingTheory
signature ratRingTheory =
sig
type thm = Thm.thm
(* Definitions *)
val rat_interp_p_def : thm
val rat_polynom_normalize_def : thm
val rat_polynom_simplify_def : thm
val rat_r_canonical_sum_merge_def : thm
val rat_r_canonical_sum_prod_def : thm
val rat_r_canonical_sum_scalar2_def : thm
val rat_r_canonical_sum_scalar3_def : thm
val rat_r_canonical_sum_scalar_def : thm
val rat_r_canonical_sum_simplify_def : thm
val rat_r_ics_aux_def : thm
val rat_r_interp_cs_def : thm
val rat_r_interp_m_def : thm
val rat_r_interp_sp_def : thm
val rat_r_interp_vl_def : thm
val rat_r_ivl_aux_def : thm
val rat_r_monom_insert_def : thm
val rat_r_spolynom_normalize_def : thm
val rat_r_spolynom_simplify_def : thm
val rat_r_varlist_insert_def : thm
(* Theorems *)
val RAT_IS_RING : thm
val rat_ring_thms : thm
val ratRing_grammars : type_grammar.grammar * term_grammar.grammar
(*
[rat] Parent theory of "ratRing"
[rat_interp_p_def] Definition
⊢ rat_interp_p = interp_p (ring 0 1 $+ $* numeric_negate)
[rat_polynom_normalize_def] Definition
⊢ rat_polynom_normalize =
polynom_normalize (ring 0 1 $+ $* numeric_negate)
[rat_polynom_simplify_def] Definition
⊢ rat_polynom_simplify =
polynom_simplify (ring 0 1 $+ $* numeric_negate)
[rat_r_canonical_sum_merge_def] Definition
⊢ rat_r_canonical_sum_merge =
r_canonical_sum_merge (ring 0 1 $+ $* numeric_negate)
[rat_r_canonical_sum_prod_def] Definition
⊢ rat_r_canonical_sum_prod =
r_canonical_sum_prod (ring 0 1 $+ $* numeric_negate)
[rat_r_canonical_sum_scalar2_def] Definition
⊢ rat_r_canonical_sum_scalar2 =
r_canonical_sum_scalar2 (ring 0 1 $+ $* numeric_negate)
[rat_r_canonical_sum_scalar3_def] Definition
⊢ rat_r_canonical_sum_scalar3 =
r_canonical_sum_scalar3 (ring 0 1 $+ $* numeric_negate)
[rat_r_canonical_sum_scalar_def] Definition
⊢ rat_r_canonical_sum_scalar =
r_canonical_sum_scalar (ring 0 1 $+ $* numeric_negate)
[rat_r_canonical_sum_simplify_def] Definition
⊢ rat_r_canonical_sum_simplify =
r_canonical_sum_simplify (ring 0 1 $+ $* numeric_negate)
[rat_r_ics_aux_def] Definition
⊢ rat_r_ics_aux = r_ics_aux (ring 0 1 $+ $* numeric_negate)
[rat_r_interp_cs_def] Definition
⊢ rat_r_interp_cs = r_interp_cs (ring 0 1 $+ $* numeric_negate)
[rat_r_interp_m_def] Definition
⊢ rat_r_interp_m = r_interp_m (ring 0 1 $+ $* numeric_negate)
[rat_r_interp_sp_def] Definition
⊢ rat_r_interp_sp = r_interp_sp (ring 0 1 $+ $* numeric_negate)
[rat_r_interp_vl_def] Definition
⊢ rat_r_interp_vl = r_interp_vl (ring 0 1 $+ $* numeric_negate)
[rat_r_ivl_aux_def] Definition
⊢ rat_r_ivl_aux = r_ivl_aux (ring 0 1 $+ $* numeric_negate)
[rat_r_monom_insert_def] Definition
⊢ rat_r_monom_insert = r_monom_insert (ring 0 1 $+ $* numeric_negate)
[rat_r_spolynom_normalize_def] Definition
⊢ rat_r_spolynom_normalize =
r_spolynom_normalize (ring 0 1 $+ $* numeric_negate)
[rat_r_spolynom_simplify_def] Definition
⊢ rat_r_spolynom_simplify =
r_spolynom_simplify (ring 0 1 $+ $* numeric_negate)
[rat_r_varlist_insert_def] Definition
⊢ rat_r_varlist_insert =
r_varlist_insert (ring 0 1 $+ $* numeric_negate)
[RAT_IS_RING] Theorem
⊢ is_ring (ring 0 1 $+ $* numeric_negate)
[rat_ring_thms] Theorem
⊢ is_ring (ring 0 1 $+ $* numeric_negate) ∧
(∀vm p.
rat_interp_p vm p =
rat_r_interp_cs vm (rat_polynom_simplify p)) ∧
(((∀vm c. rat_interp_p vm (Pconst c) = c) ∧
(∀vm i. rat_interp_p vm (Pvar i) = varmap_find i vm) ∧
(∀vm p1 p2.
rat_interp_p vm (Pplus p1 p2) =
rat_interp_p vm p1 + rat_interp_p vm p2) ∧
(∀vm p1 p2.
rat_interp_p vm (Pmult p1 p2) =
rat_interp_p vm p1 * rat_interp_p vm p2) ∧
∀vm p1. rat_interp_p vm (Popp p1) = -rat_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.
rat_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
(rat_r_canonical_sum_merge t1 (Cons_monom c2 l2 t2))
| EQUAL =>
Cons_monom (c1 + c2) l1 (rat_r_canonical_sum_merge t1 t2)
| GREATER =>
Cons_monom c2 l2
(rat_r_canonical_sum_merge (Cons_monom c1 l1 t1) t2)) ∧
(∀t2 t1 l2 l1 c1.
rat_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
(rat_r_canonical_sum_merge t1 (Cons_varlist l2 t2))
| EQUAL =>
Cons_monom (c1 + 1) l1 (rat_r_canonical_sum_merge t1 t2)
| GREATER =>
Cons_varlist l2
(rat_r_canonical_sum_merge (Cons_monom c1 l1 t1) t2)) ∧
(∀t2 t1 l2 l1 c2.
rat_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
(rat_r_canonical_sum_merge t1 (Cons_monom c2 l2 t2))
| EQUAL =>
Cons_monom (1 + c2) l1 (rat_r_canonical_sum_merge t1 t2)
| GREATER =>
Cons_monom c2 l2
(rat_r_canonical_sum_merge (Cons_varlist l1 t1) t2)) ∧
(∀t2 t1 l2 l1.
rat_r_canonical_sum_merge (Cons_varlist l1 t1)
(Cons_varlist l2 t2) =
case list_compare index_compare l1 l2 of
LESS =>
Cons_varlist l1
(rat_r_canonical_sum_merge t1 (Cons_varlist l2 t2))
| EQUAL =>
Cons_monom (1 + 1) l1 (rat_r_canonical_sum_merge t1 t2)
| GREATER =>
Cons_varlist l2
(rat_r_canonical_sum_merge (Cons_varlist l1 t1) t2)) ∧
(∀s1. rat_r_canonical_sum_merge s1 Nil_monom = s1) ∧
(∀v6 v5 v4.
rat_r_canonical_sum_merge Nil_monom (Cons_monom v4 v5 v6) =
Cons_monom v4 v5 v6) ∧
∀v8 v7.
rat_r_canonical_sum_merge Nil_monom (Cons_varlist v7 v8) =
Cons_varlist v7 v8) ∧
((∀t2 l2 l1 c2 c1.
rat_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 (rat_r_monom_insert c1 l1 t2)) ∧
(∀t2 l2 l1 c1.
rat_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 + 1) l1 t2
| GREATER => Cons_varlist l2 (rat_r_monom_insert c1 l1 t2)) ∧
∀l1 c1.
rat_r_monom_insert c1 l1 Nil_monom =
Cons_monom c1 l1 Nil_monom) ∧
((∀t2 l2 l1 c2.
rat_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 (1 + c2) l1 t2
| GREATER => Cons_monom c2 l2 (rat_r_varlist_insert l1 t2)) ∧
(∀t2 l2 l1.
rat_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 (1 + 1) l1 t2
| GREATER => Cons_varlist l2 (rat_r_varlist_insert l1 t2)) ∧
∀l1. rat_r_varlist_insert l1 Nil_monom = Cons_varlist l1 Nil_monom) ∧
((∀c0 c l t.
rat_r_canonical_sum_scalar c0 (Cons_monom c l t) =
Cons_monom (c0 * c) l (rat_r_canonical_sum_scalar c0 t)) ∧
(∀c0 l t.
rat_r_canonical_sum_scalar c0 (Cons_varlist l t) =
Cons_monom c0 l (rat_r_canonical_sum_scalar c0 t)) ∧
∀c0. rat_r_canonical_sum_scalar c0 Nil_monom = Nil_monom) ∧
((∀l0 c l t.
rat_r_canonical_sum_scalar2 l0 (Cons_monom c l t) =
rat_r_monom_insert c (list_merge index_lt l0 l)
(rat_r_canonical_sum_scalar2 l0 t)) ∧
(∀l0 l t.
rat_r_canonical_sum_scalar2 l0 (Cons_varlist l t) =
rat_r_varlist_insert (list_merge index_lt l0 l)
(rat_r_canonical_sum_scalar2 l0 t)) ∧
∀l0. rat_r_canonical_sum_scalar2 l0 Nil_monom = Nil_monom) ∧
((∀c0 l0 c l t.
rat_r_canonical_sum_scalar3 c0 l0 (Cons_monom c l t) =
rat_r_monom_insert (c0 * c) (list_merge index_lt l0 l)
(rat_r_canonical_sum_scalar3 c0 l0 t)) ∧
(∀c0 l0 l t.
rat_r_canonical_sum_scalar3 c0 l0 (Cons_varlist l t) =
rat_r_monom_insert c0 (list_merge index_lt l0 l)
(rat_r_canonical_sum_scalar3 c0 l0 t)) ∧
∀c0 l0. rat_r_canonical_sum_scalar3 c0 l0 Nil_monom = Nil_monom) ∧
((∀c1 l1 t1 s2.
rat_r_canonical_sum_prod (Cons_monom c1 l1 t1) s2 =
rat_r_canonical_sum_merge
(rat_r_canonical_sum_scalar3 c1 l1 s2)
(rat_r_canonical_sum_prod t1 s2)) ∧
(∀l1 t1 s2.
rat_r_canonical_sum_prod (Cons_varlist l1 t1) s2 =
rat_r_canonical_sum_merge (rat_r_canonical_sum_scalar2 l1 s2)
(rat_r_canonical_sum_prod t1 s2)) ∧
∀s2. rat_r_canonical_sum_prod Nil_monom s2 = Nil_monom) ∧
((∀c l t.
rat_r_canonical_sum_simplify (Cons_monom c l t) =
if c = 0 then rat_r_canonical_sum_simplify t
else if c = 1 then
Cons_varlist l (rat_r_canonical_sum_simplify t)
else Cons_monom c l (rat_r_canonical_sum_simplify t)) ∧
(∀l t.
rat_r_canonical_sum_simplify (Cons_varlist l t) =
Cons_varlist l (rat_r_canonical_sum_simplify t)) ∧
rat_r_canonical_sum_simplify Nil_monom = Nil_monom) ∧
((∀vm x. rat_r_ivl_aux vm x [] = varmap_find x vm) ∧
∀vm x x' t'.
rat_r_ivl_aux vm x (x'::t') =
varmap_find x vm * rat_r_ivl_aux vm x' t') ∧
((∀vm. rat_r_interp_vl vm [] = 1) ∧
∀vm x t. rat_r_interp_vl vm (x::t) = rat_r_ivl_aux vm x t) ∧
((∀vm c. rat_r_interp_m vm c [] = c) ∧
∀vm c x t. rat_r_interp_m vm c (x::t) = c * rat_r_ivl_aux vm x t) ∧
((∀vm a. rat_r_ics_aux vm a Nil_monom = a) ∧
(∀vm a l t.
rat_r_ics_aux vm a (Cons_varlist l t) =
a + rat_r_ics_aux vm (rat_r_interp_vl vm l) t) ∧
∀vm a c l t.
rat_r_ics_aux vm a (Cons_monom c l t) =
a + rat_r_ics_aux vm (rat_r_interp_m vm c l) t) ∧
((∀vm. rat_r_interp_cs vm Nil_monom = 0) ∧
(∀vm l t.
rat_r_interp_cs vm (Cons_varlist l t) =
rat_r_ics_aux vm (rat_r_interp_vl vm l) t) ∧
∀vm c l t.
rat_r_interp_cs vm (Cons_monom c l t) =
rat_r_ics_aux vm (rat_r_interp_m vm c l) t) ∧
((∀i. rat_polynom_normalize (Pvar i) = Cons_varlist [i] Nil_monom) ∧
(∀c. rat_polynom_normalize (Pconst c) = Cons_monom c [] Nil_monom) ∧
(∀pl pr.
rat_polynom_normalize (Pplus pl pr) =
rat_r_canonical_sum_merge (rat_polynom_normalize pl)
(rat_polynom_normalize pr)) ∧
(∀pl pr.
rat_polynom_normalize (Pmult pl pr) =
rat_r_canonical_sum_prod (rat_polynom_normalize pl)
(rat_polynom_normalize pr)) ∧
∀p.
rat_polynom_normalize (Popp p) =
rat_r_canonical_sum_scalar3 (-1) [] (rat_polynom_normalize p)) ∧
∀x.
rat_polynom_simplify x =
rat_r_canonical_sum_simplify (rat_polynom_normalize x)
*)
end
HOL 4, Kananaskis-13