Up to index of Isabelle/HOL
theory Groebner_Basis(* Title: HOL/Groebner_Basis.thy ID: $Id: Groebner_Basis.thy,v 1.12 2007/10/31 11:19:35 chaieb Exp $ Author: Amine Chaieb, TU Muenchen *) header {* Semiring normalization and Groebner Bases *} theory Groebner_Basis imports NatBin uses "Tools/Groebner_Basis/misc.ML" "Tools/Groebner_Basis/normalizer_data.ML" ("Tools/Groebner_Basis/normalizer.ML") ("Tools/Groebner_Basis/groebner.ML") begin subsection {* Semiring normalization *} setup NormalizerData.setup locale gb_semiring = fixes add mul pwr r0 r1 assumes add_a:"(add x (add y z) = add (add x y) z)" and add_c: "add x y = add y x" and add_0:"add r0 x = x" and mul_a:"mul x (mul y z) = mul (mul x y) z" and mul_c:"mul x y = mul y x" and mul_1:"mul r1 x = x" and mul_0:"mul r0 x = r0" and mul_d:"mul x (add y z) = add (mul x y) (mul x z)" and pwr_0:"pwr x 0 = r1" and pwr_Suc:"pwr x (Suc n) = mul x (pwr x n)" begin lemma mul_pwr:"mul (pwr x p) (pwr x q) = pwr x (p + q)" proof (induct p) case 0 then show ?case by (auto simp add: pwr_0 mul_1) next case Suc from this [symmetric] show ?case by (auto simp add: pwr_Suc mul_1 mul_a) qed lemma pwr_mul: "pwr (mul x y) q = mul (pwr x q) (pwr y q)" proof (induct q arbitrary: x y, auto simp add:pwr_0 pwr_Suc mul_1) fix q x y assume "!!x y. pwr (mul x y) q = mul (pwr x q) (pwr y q)" have "mul (mul x y) (mul (pwr x q) (pwr y q)) = mul x (mul y (mul (pwr x q) (pwr y q)))" by (simp add: mul_a) also have "… = (mul (mul y (mul (pwr y q) (pwr x q))) x)" by (simp add: mul_c) also have "… = (mul (mul y (pwr y q)) (mul (pwr x q) x))" by (simp add: mul_a) finally show "mul (mul x y) (mul (pwr x q) (pwr y q)) = mul (mul x (pwr x q)) (mul y (pwr y q))" by (simp add: mul_c) qed lemma pwr_pwr: "pwr (pwr x p) q = pwr x (p * q)" proof (induct p arbitrary: q) case 0 show ?case using pwr_Suc mul_1 pwr_0 by (induct q) auto next case Suc thus ?case by (auto simp add: mul_pwr [symmetric] pwr_mul pwr_Suc) qed subsubsection {* Declaring the abstract theory *} lemma semiring_ops: includes meta_term_syntax shows "TERM (add x y)" and "TERM (mul x y)" and "TERM (pwr x n)" and "TERM r0" and "TERM r1" by rule+ lemma semiring_rules: "add (mul a m) (mul b m) = mul (add a b) m" "add (mul a m) m = mul (add a r1) m" "add m (mul a m) = mul (add a r1) m" "add m m = mul (add r1 r1) m" "add r0 a = a" "add a r0 = a" "mul a b = mul b a" "mul (add a b) c = add (mul a c) (mul b c)" "mul r0 a = r0" "mul a r0 = r0" "mul r1 a = a" "mul a r1 = a" "mul (mul lx ly) (mul rx ry) = mul (mul lx rx) (mul ly ry)" "mul (mul lx ly) (mul rx ry) = mul lx (mul ly (mul rx ry))" "mul (mul lx ly) (mul rx ry) = mul rx (mul (mul lx ly) ry)" "mul (mul lx ly) rx = mul (mul lx rx) ly" "mul (mul lx ly) rx = mul lx (mul ly rx)" "mul lx (mul rx ry) = mul (mul lx rx) ry" "mul lx (mul rx ry) = mul rx (mul lx ry)" "add (add a b) (add c d) = add (add a c) (add b d)" "add (add a b) c = add a (add b c)" "add a (add c d) = add c (add a d)" "add (add a b) c = add (add a c) b" "add a c = add c a" "add a (add c d) = add (add a c) d" "mul (pwr x p) (pwr x q) = pwr x (p + q)" "mul x (pwr x q) = pwr x (Suc q)" "mul (pwr x q) x = pwr x (Suc q)" "mul x x = pwr x 2" "pwr (mul x y) q = mul (pwr x q) (pwr y q)" "pwr (pwr x p) q = pwr x (p * q)" "pwr x 0 = r1" "pwr x 1 = x" "mul x (add y z) = add (mul x y) (mul x z)" "pwr x (Suc q) = mul x (pwr x q)" "pwr x (2*n) = mul (pwr x n) (pwr x n)" "pwr x (Suc (2*n)) = mul x (mul (pwr x n) (pwr x n))" proof - show "add (mul a m) (mul b m) = mul (add a b) m" using mul_d mul_c by simp next show"add (mul a m) m = mul (add a r1) m" using mul_d mul_c mul_1 by simp next show "add m (mul a m) = mul (add a r1) m" using mul_c mul_d mul_1 add_c by simp next show "add m m = mul (add r1 r1) m" using mul_c mul_d mul_1 by simp next show "add r0 a = a" using add_0 by simp next show "add a r0 = a" using add_0 add_c by simp next show "mul a b = mul b a" using mul_c by simp next show "mul (add a b) c = add (mul a c) (mul b c)" using mul_c mul_d by simp next show "mul r0 a = r0" using mul_0 by simp next show "mul a r0 = r0" using mul_0 mul_c by simp next show "mul r1 a = a" using mul_1 by simp next show "mul a r1 = a" using mul_1 mul_c by simp next show "mul (mul lx ly) (mul rx ry) = mul (mul lx rx) (mul ly ry)" using mul_c mul_a by simp next show "mul (mul lx ly) (mul rx ry) = mul lx (mul ly (mul rx ry))" using mul_a by simp next have "mul (mul lx ly) (mul rx ry) = mul (mul rx ry) (mul lx ly)" by (rule mul_c) also have "… = mul rx (mul ry (mul lx ly))" using mul_a by simp finally show "mul (mul lx ly) (mul rx ry) = mul rx (mul (mul lx ly) ry)" using mul_c by simp next show "mul (mul lx ly) rx = mul (mul lx rx) ly" using mul_c mul_a by simp next show "mul (mul lx ly) rx = mul lx (mul ly rx)" by (simp add: mul_a) next show "mul lx (mul rx ry) = mul (mul lx rx) ry" by (simp add: mul_a ) next show "mul lx (mul rx ry) = mul rx (mul lx ry)" by (simp add: mul_a,simp add: mul_c) next show "add (add a b) (add c d) = add (add a c) (add b d)" using add_c add_a by simp next show "add (add a b) c = add a (add b c)" using add_a by simp next show "add a (add c d) = add c (add a d)" apply (simp add: add_a) by (simp only: add_c) next show "add (add a b) c = add (add a c) b" using add_a add_c by simp next show "add a c = add c a" by (rule add_c) next show "add a (add c d) = add (add a c) d" using add_a by simp next show "mul (pwr x p) (pwr x q) = pwr x (p + q)" by (rule mul_pwr) next show "mul x (pwr x q) = pwr x (Suc q)" using pwr_Suc by simp next show "mul (pwr x q) x = pwr x (Suc q)" using pwr_Suc mul_c by simp next show "mul x x = pwr x 2" by (simp add: nat_number pwr_Suc pwr_0 mul_1 mul_c) next show "pwr (mul x y) q = mul (pwr x q) (pwr y q)" by (rule pwr_mul) next show "pwr (pwr x p) q = pwr x (p * q)" by (rule pwr_pwr) next show "pwr x 0 = r1" using pwr_0 . next show "pwr x 1 = x" by (simp add: nat_number pwr_Suc pwr_0 mul_1 mul_c) next show "mul x (add y z) = add (mul x y) (mul x z)" using mul_d by simp next show "pwr x (Suc q) = mul x (pwr x q)" using pwr_Suc by simp next show "pwr x (2 * n) = mul (pwr x n) (pwr x n)" by (simp add: nat_number mul_pwr) next show "pwr x (Suc (2 * n)) = mul x (mul (pwr x n) (pwr x n))" by (simp add: nat_number pwr_Suc mul_pwr) qed lemma "axioms" [normalizer semiring ops: semiring_ops semiring rules: semiring_rules]: "gb_semiring add mul pwr r0 r1" by fact end interpretation class_semiring: gb_semiring ["op +" "op *" "op ^" "0::'a::{comm_semiring_1, recpower}" "1"] by unfold_locales (auto simp add: ring_simps power_Suc) lemmas nat_arith = add_nat_number_of diff_nat_number_of mult_nat_number_of eq_nat_number_of less_nat_number_of lemma not_iszero_Numeral1: "¬ iszero (Numeral1::'a::number_ring)" by (simp add: numeral_1_eq_1) lemmas comp_arith = Let_def arith_simps nat_arith rel_simps if_False if_True add_0 add_Suc add_number_of_left mult_number_of_left numeral_1_eq_1[symmetric] Suc_eq_add_numeral_1 numeral_0_eq_0[symmetric] numerals[symmetric] not_iszero_1 iszero_number_of_1 iszero_number_of_0 nonzero_number_of_Min iszero_number_of_Pls iszero_0 not_iszero_Numeral1 lemmas semiring_norm = comp_arith ML {* local open Conv; fun numeral_is_const ct = can HOLogic.dest_number (Thm.term_of ct); fun int_of_rat x = (case Rat.quotient_of_rat x of (i, 1) => i | _ => error "int_of_rat: bad int"); val numeral_conv = Simplifier.rewrite (HOL_basic_ss addsimps @{thms semiring_norm}) then_conv Simplifier.rewrite (HOL_basic_ss addsimps (@{thms numeral_1_eq_1} @ @{thms numeral_0_eq_0} @ @{thms numerals(1-2)})); in fun normalizer_funs key = NormalizerData.funs key {is_const = fn phi => numeral_is_const, dest_const = fn phi => fn ct => Rat.rat_of_int (snd (HOLogic.dest_number (Thm.term_of ct) handle TERM _ => error "ring_dest_const")), mk_const = fn phi => fn cT => fn x => Numeral.mk_cnumber cT (int_of_rat x), conv = fn phi => K numeral_conv} end *} declaration {* normalizer_funs @{thm class_semiring.axioms} *} locale gb_ring = gb_semiring + fixes sub :: "'a => 'a => 'a" and neg :: "'a => 'a" assumes neg_mul: "neg x = mul (neg r1) x" and sub_add: "sub x y = add x (neg y)" begin lemma ring_ops: includes meta_term_syntax shows "TERM (sub x y)" and "TERM (neg x)" . lemmas ring_rules = neg_mul sub_add lemma "axioms" [normalizer semiring ops: semiring_ops semiring rules: semiring_rules ring ops: ring_ops ring rules: ring_rules]: "gb_ring add mul pwr r0 r1 sub neg" by fact end interpretation class_ring: gb_ring ["op +" "op *" "op ^" "0::'a::{comm_semiring_1,recpower,number_ring}" 1 "op -" "uminus"] by unfold_locales simp_all declaration {* normalizer_funs @{thm class_ring.axioms} *} use "Tools/Groebner_Basis/normalizer.ML" method_setup sring_norm = {* Method.ctxt_args (fn ctxt => Method.SIMPLE_METHOD' (Normalizer.semiring_normalize_tac ctxt)) *} "semiring normalizer" locale gb_field = gb_ring + fixes divide :: "'a => 'a => 'a" and inverse:: "'a => 'a" assumes divide: "divide x y = mul x (inverse y)" and inverse: "inverse x = divide r1 x" begin lemma "axioms" [normalizer semiring ops: semiring_ops semiring rules: semiring_rules ring ops: ring_ops ring rules: ring_rules]: "gb_field add mul pwr r0 r1 sub neg divide inverse" by fact end subsection {* Groebner Bases *} locale semiringb = gb_semiring + assumes add_cancel: "add (x::'a) y = add x z <-> y = z" and add_mul_solve: "add (mul w y) (mul x z) = add (mul w z) (mul x y) <-> w = x ∨ y = z" begin lemma noteq_reduce: "a ≠ b ∧ c ≠ d <-> add (mul a c) (mul b d) ≠ add (mul a d) (mul b c)" proof- have "a ≠ b ∧ c ≠ d <-> ¬ (a = b ∨ c = d)" by simp also have "… <-> add (mul a c) (mul b d) ≠ add (mul a d) (mul b c)" using add_mul_solve by blast finally show "a ≠ b ∧ c ≠ d <-> add (mul a c) (mul b d) ≠ add (mul a d) (mul b c)" by simp qed lemma add_scale_eq_noteq: "[|r ≠ r0 ; (a = b) ∧ ~(c = d)|] ==> add a (mul r c) ≠ add b (mul r d)" proof(clarify) assume nz: "r≠ r0" and cnd: "c≠d" and eq: "add b (mul r c) = add b (mul r d)" hence "mul r c = mul r d" using cnd add_cancel by simp hence "add (mul r0 d) (mul r c) = add (mul r0 c) (mul r d)" using mul_0 add_cancel by simp thus "False" using add_mul_solve nz cnd by simp qed lemma add_r0_iff: " x = add x a <-> a = r0" proof- have "a = r0 <-> add x a = add x r0" by (simp add: add_cancel) thus "x = add x a <-> a = r0" by (auto simp add: add_c add_0) qed declare "axioms" [normalizer del] lemma "axioms" [normalizer semiring ops: semiring_ops semiring rules: semiring_rules idom rules: noteq_reduce add_scale_eq_noteq]: "semiringb add mul pwr r0 r1" by fact end locale ringb = semiringb + gb_ring + assumes subr0_iff: "sub x y = r0 <-> x = y" begin declare "axioms" [normalizer del] lemma "axioms" [normalizer semiring ops: semiring_ops semiring rules: semiring_rules ring ops: ring_ops ring rules: ring_rules idom rules: noteq_reduce add_scale_eq_noteq ideal rules: subr0_iff add_r0_iff]: "ringb add mul pwr r0 r1 sub neg" by fact end lemma no_zero_divirors_neq0: assumes az: "(a::'a::no_zero_divisors) ≠ 0" and ab: "a*b = 0" shows "b = 0" proof - { assume bz: "b ≠ 0" from no_zero_divisors [OF az bz] ab have False by blast } thus "b = 0" by blast qed interpretation class_ringb: ringb ["op +" "op *" "op ^" "0::'a::{idom,recpower,number_ring}" "1" "op -" "uminus"] proof(unfold_locales, simp add: ring_simps power_Suc, auto) fix w x y z ::"'a::{idom,recpower,number_ring}" assume p: "w * y + x * z = w * z + x * y" and ynz: "y ≠ z" hence ynz': "y - z ≠ 0" by simp from p have "w * y + x* z - w*z - x*y = 0" by simp hence "w* (y - z) - x * (y - z) = 0" by (simp add: ring_simps) hence "(y - z) * (w - x) = 0" by (simp add: ring_simps) with no_zero_divirors_neq0 [OF ynz'] have "w - x = 0" by blast thus "w = x" by simp qed declaration {* normalizer_funs @{thm class_ringb.axioms} *} interpretation natgb: semiringb ["op +" "op *" "op ^" "0::nat" "1"] proof (unfold_locales, simp add: ring_simps power_Suc) fix w x y z ::"nat" { assume p: "w * y + x * z = w * z + x * y" and ynz: "y ≠ z" hence "y < z ∨ y > z" by arith moreover { assume lt:"y <z" hence "∃k. z = y + k ∧ k > 0" by (rule_tac x="z - y" in exI, auto) then obtain k where kp: "k>0" and yz:"z = y + k" by blast from p have "(w * y + x *y) + x*k = (w * y + x*y) + w*k" by (simp add: yz ring_simps) hence "x*k = w*k" by simp hence "w = x" using kp by (simp add: mult_cancel2) } moreover { assume lt: "y >z" hence "∃k. y = z + k ∧ k>0" by (rule_tac x="y - z" in exI, auto) then obtain k where kp: "k>0" and yz:"y = z + k" by blast from p have "(w * z + x *z) + w*k = (w * z + x*z) + x*k" by (simp add: yz ring_simps) hence "w*k = x*k" by simp hence "w = x" using kp by (simp add: mult_cancel2)} ultimately have "w=x" by blast } thus "(w * y + x * z = w * z + x * y) = (w = x ∨ y = z)" by auto qed declaration {* normalizer_funs @{thm natgb.axioms} *} locale fieldgb = ringb + gb_field begin declare "axioms" [normalizer del] lemma "axioms" [normalizer semiring ops: semiring_ops semiring rules: semiring_rules ring ops: ring_ops ring rules: ring_rules idom rules: noteq_reduce add_scale_eq_noteq ideal rules: subr0_iff add_r0_iff]: "fieldgb add mul pwr r0 r1 sub neg divide inverse" by unfold_locales end lemmas bool_simps = simp_thms(1-34) lemma dnf: "(P & (Q | R)) = ((P&Q) | (P&R))" "((Q | R) & P) = ((Q&P) | (R&P))" "(P ∧ Q) = (Q ∧ P)" "(P ∨ Q) = (Q ∨ P)" by blast+ lemmas weak_dnf_simps = dnf bool_simps lemma nnf_simps: "(¬(P ∧ Q)) = (¬P ∨ ¬Q)" "(¬(P ∨ Q)) = (¬P ∧ ¬Q)" "(P --> Q) = (¬P ∨ Q)" "(P = Q) = ((P ∧ Q) ∨ (¬P ∧ ¬ Q))" "(¬ ¬(P)) = P" by blast+ lemma PFalse: "P ≡ False ==> ¬ P" "¬ P ==> (P ≡ False)" by auto use "Tools/Groebner_Basis/groebner.ML" method_setup algebra = {* let fun keyword k = Scan.lift (Args.$$$ k -- Args.colon) >> K () val addN = "add" val delN = "del" val any_keyword = keyword addN || keyword delN val thms = Scan.repeat (Scan.unless any_keyword Attrib.multi_thm) >> flat; in fn src => Method.syntax ((Scan.optional (keyword addN |-- thms) []) -- (Scan.optional (keyword delN |-- thms) [])) src #> (fn ((add_ths, del_ths), ctxt) => Method.SIMPLE_METHOD' (Groebner.algebra_tac add_ths del_ths ctxt)) end *} "solve polynomial equations over (semi)rings and ideal membership problems using Groebner bases" end
lemma mul_pwr:
mul (pwr x p) (pwr x q) = pwr x (p + q)
lemma pwr_mul:
pwr (mul x y) q = mul (pwr x q) (pwr y q)
lemma pwr_pwr:
pwr (pwr x p) q = pwr x (p * q)
lemma semiring_ops(1):
TERM add x y
and semiring_ops(2):
TERM mul x y
and semiring_ops(3):
TERM pwr x n
and semiring_ops(4):
TERM r0
and semiring_ops(5):
TERM r1
lemma semiring_rules:
add (mul a m) (mul b m) = mul (add a b) m
add (mul a m) m = mul (add a r1) m
add m (mul a m) = mul (add a r1) m
add m m = mul (add r1 r1) m
add r0 a = a
add a r0 = a
mul a b = mul b a
mul (add a b) c = add (mul a c) (mul b c)
mul r0 a = r0
mul a r0 = r0
mul r1 a = a
mul a r1 = a
mul (mul lx ly) (mul rx ry) = mul (mul lx rx) (mul ly ry)
mul (mul lx ly) (mul rx ry) = mul lx (mul ly (mul rx ry))
mul (mul lx ly) (mul rx ry) = mul rx (mul (mul lx ly) ry)
mul (mul lx ly) rx = mul (mul lx rx) ly
mul (mul lx ly) rx = mul lx (mul ly rx)
mul lx (mul rx ry) = mul (mul lx rx) ry
mul lx (mul rx ry) = mul rx (mul lx ry)
add (add a b) (add c d) = add (add a c) (add b d)
add (add a b) c = add a (add b c)
add a (add c d) = add c (add a d)
add (add a b) c = add (add a c) b
add a c = add c a
add a (add c d) = add (add a c) d
mul (pwr x p) (pwr x q) = pwr x (p + q)
mul x (pwr x q) = pwr x (Suc q)
mul (pwr x q) x = pwr x (Suc q)
mul x x = pwr x 2
pwr (mul x y) q = mul (pwr x q) (pwr y q)
pwr (pwr x p) q = pwr x (p * q)
pwr x 0 = r1
pwr x 1 = x
mul x (add y z) = add (mul x y) (mul x z)
pwr x (Suc q) = mul x (pwr x q)
pwr x (2 * n) = mul (pwr x n) (pwr x n)
pwr x (Suc (2 * n)) = mul x (mul (pwr x n) (pwr x n))
lemma axioms:
gb_semiring add mul pwr r0 r1
lemma nat_arith:
number_of v + number_of v' =
(if neg (number_of v) then number_of v'
else if neg (number_of v') then number_of v else number_of (v + v'))
number_of v - number_of v' =
(if neg (number_of v') then number_of v
else let d = number_of (v + - v') in if neg d then 0 else nat d)
number_of v * number_of v' =
(if neg (number_of v) then 0 else number_of (v * v'))
(number_of v = number_of v') =
(if neg (number_of v) then iszero (number_of v') ∨ neg (number_of v')
else if neg (number_of v') then iszero (number_of v)
else iszero (number_of (v + - v')))
(number_of v < number_of v') =
(if neg (number_of v) then neg (number_of (- v'))
else neg (number_of (v + - v')))
lemma not_iszero_Numeral1:
¬ iszero Numeral1
lemma comp_arith:
Let s f == f s
bit.B0 ≠ bit.B1
bit.B1 ≠ bit.B0
Numeral.Pls BIT bit.B0 = Numeral.Pls
Numeral.Min BIT bit.B1 = Numeral.Min
Numeral.pred Numeral.Pls = Numeral.Min
Numeral.pred Numeral.Min = Numeral.Min BIT bit.B0
Numeral.pred (k BIT bit.B1) = k BIT bit.B0
Numeral.pred (k BIT bit.B0) = Numeral.pred k BIT bit.B1
Numeral.succ Numeral.Pls = Numeral.Pls BIT bit.B1
Numeral.succ Numeral.Min = Numeral.Pls
Numeral.succ (k BIT bit.B1) = Numeral.succ k BIT bit.B0
Numeral.succ (k BIT bit.B0) = k BIT bit.B1
Numeral.Pls + k = k
Numeral.Min + k = Numeral.pred k
k BIT bit.B0 + l BIT b = (k + l) BIT b
k BIT bit.B1 + l BIT bit.B0 = (k + l) BIT bit.B1
k BIT bit.B1 + l BIT bit.B1 = (k + Numeral.succ l) BIT bit.B0
- Numeral.Pls = Numeral.Pls
- Numeral.Min = Numeral.Pls BIT bit.B1
- k BIT bit.B1 = Numeral.pred (- k) BIT bit.B1
- k BIT bit.B0 = (- k) BIT bit.B0
Numeral.Pls * w = Numeral.Pls
Numeral.Min * k = - k
k BIT bit.B1 * l = (k * l) BIT bit.B0 + l
k BIT bit.B0 * l = (k * l) BIT bit.B0
k + Numeral.Pls = k
k + Numeral.Min = Numeral.pred k
¦0::'a¦ = (0::'a)
¦1::'a¦ = (1::'a)
number_of v + number_of w = number_of (v + w)
- number_of w = number_of (- w)
- (1::'a) = (-1::'a)
number_of v * number_of w = number_of (v * w)
number_of v - number_of w = number_of (v + - w)
¦number_of x¦ = (if number_of x < (0::'a) then - number_of x else number_of x)
number_of v + number_of v' =
(if neg (number_of v) then number_of v'
else if neg (number_of v') then number_of v else number_of (v + v'))
number_of v - number_of v' =
(if neg (number_of v') then number_of v
else let d = number_of (v + - v') in if neg d then 0 else nat d)
number_of v * number_of v' =
(if neg (number_of v) then 0 else number_of (v * v'))
(number_of v = number_of v') =
(if neg (number_of v) then iszero (number_of v') ∨ neg (number_of v')
else if neg (number_of v') then iszero (number_of v)
else iszero (number_of (v + - v')))
(number_of v < number_of v') =
(if neg (number_of v) then neg (number_of (- v'))
else neg (number_of (v + - v')))
(number_of x = number_of y) = iszero (number_of (x + - y))
iszero (0::'a)
¬ iszero (-1::'a)
iszero (number_of (w BIT bit.B0)) = iszero (number_of w)
¬ iszero (number_of (w BIT bit.B1))
(number_of x < number_of y) = neg (number_of (x + - y))
¬ neg Numeral0
¬ neg (0::'a)
¬ neg (1::'a)
¬ iszero (1::'a)
neg (-1::'a)
neg (number_of (w BIT x)) = neg (number_of w)
(number_of x ≤ number_of y) = (¬ neg (number_of (y + - x)))
(if False then x else y) = y
(if True then x else y) = x
0 + n = n
Suc m + n = Suc (m + n)
number_of v + (number_of w + z) = number_of (v + w) + z
number_of v * (number_of w * z) = number_of (v * w) * z
(1::'a) = Numeral1
Suc n = n + 1
(0::'a) = Numeral0
0 = Numeral0
1 = Numeral1
Suc (Suc 0) = 2
¬ iszero (1::'a)
¬ iszero (number_of (w BIT bit.B1))
iszero (number_of (w BIT bit.B0)) = iszero (number_of w)
¬ iszero (-1::'a)
iszero Numeral0
iszero (0::'a)
¬ iszero Numeral1
lemma semiring_norm:
Let s f == f s
bit.B0 ≠ bit.B1
bit.B1 ≠ bit.B0
Numeral.Pls BIT bit.B0 = Numeral.Pls
Numeral.Min BIT bit.B1 = Numeral.Min
Numeral.pred Numeral.Pls = Numeral.Min
Numeral.pred Numeral.Min = Numeral.Min BIT bit.B0
Numeral.pred (k BIT bit.B1) = k BIT bit.B0
Numeral.pred (k BIT bit.B0) = Numeral.pred k BIT bit.B1
Numeral.succ Numeral.Pls = Numeral.Pls BIT bit.B1
Numeral.succ Numeral.Min = Numeral.Pls
Numeral.succ (k BIT bit.B1) = Numeral.succ k BIT bit.B0
Numeral.succ (k BIT bit.B0) = k BIT bit.B1
Numeral.Pls + k = k
Numeral.Min + k = Numeral.pred k
k BIT bit.B0 + l BIT b = (k + l) BIT b
k BIT bit.B1 + l BIT bit.B0 = (k + l) BIT bit.B1
k BIT bit.B1 + l BIT bit.B1 = (k + Numeral.succ l) BIT bit.B0
- Numeral.Pls = Numeral.Pls
- Numeral.Min = Numeral.Pls BIT bit.B1
- k BIT bit.B1 = Numeral.pred (- k) BIT bit.B1
- k BIT bit.B0 = (- k) BIT bit.B0
Numeral.Pls * w = Numeral.Pls
Numeral.Min * k = - k
k BIT bit.B1 * l = (k * l) BIT bit.B0 + l
k BIT bit.B0 * l = (k * l) BIT bit.B0
k + Numeral.Pls = k
k + Numeral.Min = Numeral.pred k
¦0::'a¦ = (0::'a)
¦1::'a¦ = (1::'a)
number_of v + number_of w = number_of (v + w)
- number_of w = number_of (- w)
- (1::'a) = (-1::'a)
number_of v * number_of w = number_of (v * w)
number_of v - number_of w = number_of (v + - w)
¦number_of x¦ = (if number_of x < (0::'a) then - number_of x else number_of x)
number_of v + number_of v' =
(if neg (number_of v) then number_of v'
else if neg (number_of v') then number_of v else number_of (v + v'))
number_of v - number_of v' =
(if neg (number_of v') then number_of v
else let d = number_of (v + - v') in if neg d then 0 else nat d)
number_of v * number_of v' =
(if neg (number_of v) then 0 else number_of (v * v'))
(number_of v = number_of v') =
(if neg (number_of v) then iszero (number_of v') ∨ neg (number_of v')
else if neg (number_of v') then iszero (number_of v)
else iszero (number_of (v + - v')))
(number_of v < number_of v') =
(if neg (number_of v) then neg (number_of (- v'))
else neg (number_of (v + - v')))
(number_of x = number_of y) = iszero (number_of (x + - y))
iszero (0::'a)
¬ iszero (-1::'a)
iszero (number_of (w BIT bit.B0)) = iszero (number_of w)
¬ iszero (number_of (w BIT bit.B1))
(number_of x < number_of y) = neg (number_of (x + - y))
¬ neg Numeral0
¬ neg (0::'a)
¬ neg (1::'a)
¬ iszero (1::'a)
neg (-1::'a)
neg (number_of (w BIT x)) = neg (number_of w)
(number_of x ≤ number_of y) = (¬ neg (number_of (y + - x)))
(if False then x else y) = y
(if True then x else y) = x
0 + n = n
Suc m + n = Suc (m + n)
number_of v + (number_of w + z) = number_of (v + w) + z
number_of v * (number_of w * z) = number_of (v * w) * z
(1::'a) = Numeral1
Suc n = n + 1
(0::'a) = Numeral0
0 = Numeral0
1 = Numeral1
Suc (Suc 0) = 2
¬ iszero (1::'a)
¬ iszero (number_of (w BIT bit.B1))
iszero (number_of (w BIT bit.B0)) = iszero (number_of w)
¬ iszero (-1::'a)
iszero Numeral0
iszero (0::'a)
¬ iszero Numeral1
lemma ring_ops(1):
TERM sub x y
and ring_ops(2):
TERM neg x
lemma ring_rules:
neg x = mul (neg r1) x
sub x y = add x (neg y)
lemma axioms:
gb_ring add mul pwr r0 r1 sub neg
lemma axioms:
gb_field add mul pwr r0 r1 sub neg divide inverse
lemma noteq_reduce:
(a ≠ b ∧ c ≠ d) = (add (mul a c) (mul b d) ≠ add (mul a d) (mul b c))
lemma add_scale_eq_noteq:
[| r ≠ r0; a = b ∧ c ≠ d |] ==> add a (mul r c) ≠ add b (mul r d)
lemma add_r0_iff:
(x = add x a) = (a = r0)
lemma axioms:
semiringb add mul pwr r0 r1
lemma axioms:
ringb add mul pwr r0 r1 sub neg
lemma no_zero_divirors_neq0:
[| a ≠ (0::'a); a * b = (0::'a) |] ==> b = (0::'a)
lemma axioms:
fieldgb add mul pwr r0 r1 sub neg divide inverse
lemma bool_simps:
(¬ ¬ P) = P
((¬ P) = (¬ Q)) = (P = Q)
(P ≠ Q) = (P = (¬ Q))
(P ∨ ¬ P) = True
(¬ P ∨ P) = True
(x = x) = True
(¬ True) = False
(¬ False) = True
(¬ P) ≠ P
P ≠ (¬ P)
(True = P) = P
(P = True) = P
(False = P) = (¬ P)
(P = False) = (¬ P)
(True --> P) = P
(False --> P) = True
(P --> True) = True
(P --> P) = True
(P --> False) = (¬ P)
(P --> ¬ P) = (¬ P)
(P ∧ True) = P
(True ∧ P) = P
(P ∧ False) = False
(False ∧ P) = False
(P ∧ P) = P
(P ∧ P ∧ Q) = (P ∧ Q)
(P ∧ ¬ P) = False
(¬ P ∧ P) = False
(P ∨ True) = True
(True ∨ P) = True
(P ∨ False) = P
(False ∨ P) = P
(P ∨ P) = P
(P ∨ P ∨ Q) = (P ∨ Q)
lemma dnf:
(P ∧ (Q ∨ R)) = (P ∧ Q ∨ P ∧ R)
((Q ∨ R) ∧ P) = (Q ∧ P ∨ R ∧ P)
(P ∧ Q) = (Q ∧ P)
(P ∨ Q) = (Q ∨ P)
lemma weak_dnf_simps:
(P ∧ (Q ∨ R)) = (P ∧ Q ∨ P ∧ R)
((Q ∨ R) ∧ P) = (Q ∧ P ∨ R ∧ P)
(P ∧ Q) = (Q ∧ P)
(P ∨ Q) = (Q ∨ P)
(¬ ¬ P) = P
((¬ P) = (¬ Q)) = (P = Q)
(P ≠ Q) = (P = (¬ Q))
(P ∨ ¬ P) = True
(¬ P ∨ P) = True
(x = x) = True
(¬ True) = False
(¬ False) = True
(¬ P) ≠ P
P ≠ (¬ P)
(True = P) = P
(P = True) = P
(False = P) = (¬ P)
(P = False) = (¬ P)
(True --> P) = P
(False --> P) = True
(P --> True) = True
(P --> P) = True
(P --> False) = (¬ P)
(P --> ¬ P) = (¬ P)
(P ∧ True) = P
(True ∧ P) = P
(P ∧ False) = False
(False ∧ P) = False
(P ∧ P) = P
(P ∧ P ∧ Q) = (P ∧ Q)
(P ∧ ¬ P) = False
(¬ P ∧ P) = False
(P ∨ True) = True
(True ∨ P) = True
(P ∨ False) = P
(False ∨ P) = P
(P ∨ P) = P
(P ∨ P ∨ Q) = (P ∨ Q)
lemma nnf_simps:
(¬ (P ∧ Q)) = (¬ P ∨ ¬ Q)
(¬ (P ∨ Q)) = (¬ P ∧ ¬ Q)
(P --> Q) = (¬ P ∨ Q)
(P = Q) = (P ∧ Q ∨ ¬ P ∧ ¬ Q)
(¬ ¬ P) = P
lemma PFalse:
P == False ==> ¬ P
¬ P ==> P == False