@@ -9,83 +9,90 @@ import Interfaces.Verified
99||| A prof that -0 = 0 in any verified group.
1010inverseNeutralIsNeutral : VerifiedGroup t => inverse (the t A.neutral) = A.neutral
1111inverseNeutralIsNeutral {t} =
12− let zero = the t neutral in
13− let step1 = cong {f = (<+> inverse zero)} (the (zero = zero) Refl) in
14− let step2 = replace {P = \x => zero <+> inverse zero = x} (groupInverseIsInverseL zero) step1 in
12+ let zero = the t neutral
13+ step1 = cong {f = (<+> inverse zero)} (the (zero = zero) Refl)
14+ step2 = replace {P = \x => zero <+> inverse zero = x} (groupInverseIsInverseL zero) step1
15+ in
1516 replace {P = \x => x = neutral} (monoidNeutralIsNeutralR $ inverse zero) step2
1617
1718||| A proof that -(-x) = x in any verified group.
1819inverseSquaredIsIdentity : VerifiedGroup t => (x : t) -> inverse (inverse x) = x
1920inverseSquaredIsIdentity {t} x =
20− let zero = the t neutral in
21− let step1 = cong {f = (x <+>)} (groupInverseIsInverseL (inverse x)) in
22− let step2 = replace {P = \r => r = x <+> neutral} (semigroupOpIsAssociative x (inverse x) (inverse $ inverse x)) step1 in
23− let step3 = replace {P = \r => r <+> inverse (inverse x) = x <+> neutral} (groupInverseIsInverseL x) step2 in
24− let step4 = replace {P = \r => r = x <+> neutral} (monoidNeutralIsNeutralR (inverse $ inverse x)) step3 in
21+ let zero = the t neutral
22+ step1 = cong {f = (x <+>)} (groupInverseIsInverseL (inverse x))
23+ step2 = replace {P = \r => r = x <+> neutral} (semigroupOpIsAssociative x (inverse x) (inverse $ inverse x)) step1
24+ step3 = replace {P = \r => r <+> inverse (inverse x) = x <+> neutral} (groupInverseIsInverseL x) step2
25+ step4 = replace {P = \r => r = x <+> neutral} (monoidNeutralIsNeutralR (inverse $ inverse x)) step3
26+ in
2527 replace {P = \r => inverse (inverse x) = r} (monoidNeutralIsNeutralL x) step4
2628
2729||| A proof that -(x + y) = -x - y in any verified abelian group.
2830inverseDistributesOverGroupOp : VerifiedAbelianGroup t => (l, r : t) -> inverse (l <+> r) = inverse l <+> inverse r
2931inverseDistributesOverGroupOp {t} l r =
30− let step1 = replace {P = \x => x = neutral} (sym $ groupInverseIsInverseL (l <+> r)) $ the (the t neutral = neutral) Refl in
31− let step2 = cong {f = ((inverse r) <+>) . ((inverse l) <+>)} step1 in
32− let step3 = replace {P = \x => inverse r <+> (inverse l <+> (l <+> r <+> inverse (l <+> r))) = inverse r <+> x} (monoidNeutralIsNeutralL (inverse l)) step2 in
33− let step4 = replace {P = \x => x = inverse r <+> inverse l} (semigroupOpIsAssociative (inverse r) (inverse l) (l <+> r <+> inverse (l <+> r))) step3 in
34− let step5 = replace {P = \x => x = inverse r <+> inverse l} (semigroupOpIsAssociative (inverse r <+> inverse l) (l <+> r) (inverse (l <+> r))) step4 in
35− let step6 = replace {P = \x => x <+> inverse (l <+> r) = inverse r <+> inverse l} (semigroupOpIsAssociative (inverse r <+> inverse l) l r) step5 in
36− let step7 = replace {P = \x => x <+> r <+> inverse (l <+> r) = inverse r <+> inverse l} (sym $ semigroupOpIsAssociative (inverse r) (inverse l) l) step6 in
37− let step8 = replace {P = \x => inverse r <+> x <+> r <+> inverse (l <+> r) = inverse r <+> inverse l} (groupInverseIsInverseR l) step7 in
38− let step9 = replace {P = \x => x <+> r <+> inverse (l <+> r) = inverse r <+> inverse l} (monoidNeutralIsNeutralL (inverse r)) step8 in
39− let step10 = replace {P = \x => x <+> inverse (l <+> r) = inverse r <+> inverse l} (groupInverseIsInverseR r) step9 in
40− let step11 = replace {P = \x => x = inverse r <+> inverse l} (monoidNeutralIsNeutralR (inverse (l <+> r))) step10 in
32+ let step1 = replace {P = \x => x = neutral} (sym $ groupInverseIsInverseL (l <+> r)) $ the (the t neutral = neutral) Refl
33+ step2 = cong {f = ((inverse r) <+>) . ((inverse l) <+>)} step1
34+ step3 = replace {P = \x => inverse r <+> (inverse l <+> (l <+> r <+> inverse (l <+> r))) = inverse r <+> x} (monoidNeutralIsNeutralL (inverse l)) step2
35+ step4 = replace {P = \x => x = inverse r <+> inverse l} (semigroupOpIsAssociative (inverse r) (inverse l) (l <+> r <+> inverse (l <+> r))) step3
36+ step5 = replace {P = \x => x = inverse r <+> inverse l} (semigroupOpIsAssociative (inverse r <+> inverse l) (l <+> r) (inverse (l <+> r))) step4
37+ step6 = replace {P = \x => x <+> inverse (l <+> r) = inverse r <+> inverse l} (semigroupOpIsAssociative (inverse r <+> inverse l) l r) step5
38+ step7 = replace {P = \x => x <+> r <+> inverse (l <+> r) = inverse r <+> inverse l} (sym $ semigroupOpIsAssociative (inverse r) (inverse l) l) step6
39+ step8 = replace {P = \x => inverse r <+> x <+> r <+> inverse (l <+> r) = inverse r <+> inverse l} (groupInverseIsInverseR l) step7
40+ step9 = replace {P = \x => x <+> r <+> inverse (l <+> r) = inverse r <+> inverse l} (monoidNeutralIsNeutralL (inverse r)) step8
41+ step10 = replace {P = \x => x <+> inverse (l <+> r) = inverse r <+> inverse l} (groupInverseIsInverseR r) step9
42+ step11 = replace {P = \x => x = inverse r <+> inverse l} (monoidNeutralIsNeutralR (inverse (l <+> r))) step10
43+ in
4144 replace {P = \x => inverse (l <+> r) = x} (abelianGroupOpIsCommutative (inverse r) (inverse l)) step11
4245
4346||| A proof that anything multiplied by zero yields zero back.
4447multNeutralAbsorbingL : VerifiedRingWithUnity t => (r : t) -> A.neutral <.> r = A.neutral
4548multNeutralAbsorbingL r =
46− let step1 = the (unity <.> r = r) (ringWithUnityIsUnityR r) in
47− let step2 = replace {P = \x => x <.> r = r} (sym $ monoidNeutralIsNeutralR unity) step1 in
48− let step3 = replace {P = \x => x = r} (ringOpIsDistributiveR neutral unity r) step2 in
49− let step4 = replace {P = \x => neutral <.> r <+> x = r} (ringWithUnityIsUnityR r) step3 in
50− let step5 = cong {f = \x => x <+> inverse r} step4 in
51− let step6 = replace {P = \x => x = r <+> inverse r} (sym $ semigroupOpIsAssociative (neutral <.> r) r (inverse r)) step5 in
52− let step7 = replace {P = \x => neutral <.> r <+> x = x} (groupInverseIsInverseL r) step6 in
49+ let step1 = the (unity <.> r = r) (ringWithUnityIsUnityR r)
50+ step2 = replace {P = \x => x <.> r = r} (sym $ monoidNeutralIsNeutralR unity) step1
51+ step3 = replace {P = \x => x = r} (ringOpIsDistributiveR neutral unity r) step2
52+ step4 = replace {P = \x => neutral <.> r <+> x = r} (ringWithUnityIsUnityR r) step3
53+ step5 = cong {f = \x => x <+> inverse r} step4
54+ step6 = replace {P = \x => x = r <+> inverse r} (sym $ semigroupOpIsAssociative (neutral <.> r) r (inverse r)) step5
55+ step7 = replace {P = \x => neutral <.> r <+> x = x} (groupInverseIsInverseL r) step6
56+ in
5357 replace {P = \x => x = neutral} (monoidNeutralIsNeutralL (neutral <.> r)) step7
5458
5559||| A proof that anything multiplied by zero yields zero back.
5660multNeutralAbsorbingR : VerifiedRingWithUnity t => (l : t) -> l <.> A.neutral = A.neutral
5761multNeutralAbsorbingR l =
58− let step1 = the (l <.> unity = l) (ringWithUnityIsUnityL l) in
59− let step2 = replace {P = \x => l <.> x = l} (sym $ monoidNeutralIsNeutralR unity) step1 in
60− let step3 = replace {P = \x => x = l} (ringOpIsDistributiveL l neutral unity) step2 in
61− let step4 = replace {P = \x => l <.> neutral <+> x = l} (ringWithUnityIsUnityL l) step3 in
62− let step5 = cong {f = \x => x <+> inverse l} step4 in
63− let step6 = replace {P = \x => x = l <+> inverse l} (sym $ semigroupOpIsAssociative (l <.> neutral) l (inverse l)) step5 in
64− let step7 = replace {P = \x => l <.> neutral <+> x = x} (groupInverseIsInverseL l) step6 in
62+ let step1 = the (l <.> unity = l) (ringWithUnityIsUnityL l)
63+ step2 = replace {P = \x => l <.> x = l} (sym $ monoidNeutralIsNeutralR unity) step1
64+ step3 = replace {P = \x => x = l} (ringOpIsDistributiveL l neutral unity) step2
65+ step4 = replace {P = \x => l <.> neutral <+> x = l} (ringWithUnityIsUnityL l) step3
66+ step5 = cong {f = \x => x <+> inverse l} step4
67+ step6 = replace {P = \x => x = l <+> inverse l} (sym $ semigroupOpIsAssociative (l <.> neutral) l (inverse l)) step5
68+ step7 = replace {P = \x => l <.> neutral <+> x = x} (groupInverseIsInverseL l) step6
69+ in
6570 replace {P = \x => x = neutral} (monoidNeutralIsNeutralL (l <.> neutral)) step7
6671
6772||| A proof that inverse operator can be extracted before multiplication
6873||| (-x)y = -(xy)
6974multInverseInversesL : VerifiedRingWithUnity t => (l, r : t) -> inverse l <.> r = inverse (l <.> r)
7075multInverseInversesL l r =
71− let step1 = replace {P = \x => x <.> r = neutral} (sym $ groupInverseIsInverseL l) (multNeutralAbsorbingL r) in
72− let step2 = replace {P = \x => x = neutral} (ringOpIsDistributiveR l (inverse l) r) step1 in
73− let step3 = cong {f = ((inverse (l <.> r)) <+>)} step2 in
74− let step4 = replace {P = \x => inverse (l <.> r) <+> (l <.> r <+> inverse l <.> r) = x} (monoidNeutralIsNeutralL (inverse (l <.> r))) step3 in
75− let step5 = replace {P = \x => x = inverse (l <.> r)} (semigroupOpIsAssociative (inverse (l <.> r)) (l <.> r) (inverse l <.> r)) step4 in
76− let step6 = replace {P = \x => x <+> inverse l <.> r = inverse (l <.> r)} (groupInverseIsInverseR (l <.> r)) step5 in
76+ let step1 = replace {P = \x => x <.> r = neutral} (sym $ groupInverseIsInverseL l) (multNeutralAbsorbingL r)
77+ step2 = replace {P = \x => x = neutral} (ringOpIsDistributiveR l (inverse l) r) step1
78+ step3 = cong {f = ((inverse (l <.> r)) <+>)} step2
79+ step4 = replace {P = \x => inverse (l <.> r) <+> (l <.> r <+> inverse l <.> r) = x} (monoidNeutralIsNeutralL (inverse (l <.> r))) step3
80+ step5 = replace {P = \x => x = inverse (l <.> r)} (semigroupOpIsAssociative (inverse (l <.> r)) (l <.> r) (inverse l <.> r)) step4
81+ step6 = replace {P = \x => x <+> inverse l <.> r = inverse (l <.> r)} (groupInverseIsInverseR (l <.> r)) step5
82+ in
7783 replace {P = \x => x = inverse (l <.> r)} (monoidNeutralIsNeutralR (inverse l <.> r)) step6
7884
7985||| A proof that inverse operator can be extracted before multiplication
8086||| x(-y) = -(xy)
8187multInverseInversesR : VerifiedRingWithUnity t => (l, r : t) -> l <.> inverse r = inverse (l <.> r)
8288multInverseInversesR l r =
83− let step1 = replace {P = \x => l <.> x = neutral} (sym $ groupInverseIsInverseL r) (multNeutralAbsorbingR l) in
84− let step2 = replace {P = \x => x = neutral} (ringOpIsDistributiveL l r (inverse r)) step1 in
85− let step3 = cong {f = ((inverse (l <.> r)) <+>)} step2 in
86− let step4 = replace {P = \x => inverse (l <.> r) <+> (l <.> r <+> l <.> inverse r) = x} (monoidNeutralIsNeutralL (inverse (l <.> r))) step3 in
87− let step5 = replace {P = \x => x = inverse (l <.> r)} (semigroupOpIsAssociative (inverse (l <.> r)) (l <.> r) (l <.> inverse r)) step4 in
88− let step6 = replace {P = \x => x <+> l <.> inverse r = inverse (l <.> r)} (groupInverseIsInverseR (l <.> r)) step5 in
89+ let step1 = replace {P = \x => l <.> x = neutral} (sym $ groupInverseIsInverseL r) (multNeutralAbsorbingR l)
90+ step2 = replace {P = \x => x = neutral} (ringOpIsDistributiveL l r (inverse r)) step1
91+ step3 = cong {f = ((inverse (l <.> r)) <+>)} step2
92+ step4 = replace {P = \x => inverse (l <.> r) <+> (l <.> r <+> l <.> inverse r) = x} (monoidNeutralIsNeutralL (inverse (l <.> r))) step3
93+ step5 = replace {P = \x => x = inverse (l <.> r)} (semigroupOpIsAssociative (inverse (l <.> r)) (l <.> r) (l <.> inverse r)) step4
94+ step6 = replace {P = \x => x <+> l <.> inverse r = inverse (l <.> r)} (groupInverseIsInverseR (l <.> r)) step5
95+ in
8996 replace {P = \x => x = inverse (l <.> r)} (monoidNeutralIsNeutralR (l <.> inverse r)) step6
9097
9198||| A proof that multiplication of inverses is the same as multiplication of original