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MMT
examples
Commits
c092ceb2
Commit
c092ceb2
authored
Oct 16, 2018
by
Florian Rabe
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-15
bands.mmt
source/bands.mmt
+42
-11
fol.mmt
source/logic/fol.mmt
+2
-2
magmas.mmt
source/magmas.mmt
+2
-2
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source/bands.mmt
View file @
c092ceb2
namespace http://cds.omdoc.org/examples❚
/T regular bands❚
/T regular bands and views between them
mostly following the wikipedia page on Bands
Notes:
- some names that were not given on Wikipedia are made up
- most views do not actually use (all of) the Band properties
- LeftX and RightX are stronger than X, not weaker
❚
theory Regular : ?FOLEQNatDed =
include ?Band ❙
...
...
@@ -31,22 +38,22 @@ theory RightRegular : ?FOLEQNatDed =
implicit view Regular2LeftRegular1 : ?Regular -> ?LeftRegular1 =
include ?Band = ?Band❙
regular = [x,y,z] congT
([u] u∘z) leftregular1
❙
regular = [x,y,z] congT
leftregular1 [u]u∘z
❙
❚
implicit view LeftRegular12LeftRegular : ?LeftRegular1 -> ?LeftRegular =
include ?Band = ?Band❙
leftregular1 = [x,y,z] congT
([u] u∘y) leftregular
❙
leftregular1 = [x,y,z] congT
leftregular [u]u∘y
❙
❚
implicit view Regular2RightRegular1 : ?Regular -> ?RightRegular1 =
include ?Band = ?Band❙
regular = [x,y,z] trans3 assoc5 (congT (
[u] z∘u) trans3 assoc4R rightregular1 assoc
) assoc4R❙
regular = [x,y,z] trans3 assoc5 (congT (
trans3 assoc4R rightregular1 assoc) [u]z∘u
) assoc4R❙
❚
implicit view RightRegular12RightRegular : ?RightRegular1 -> ?RightRegular =
include ?Band = ?Band❙
rightregular1 = [x,y,z] trans3 assoc4 (congT (
[u] x∘u) trans assocR rightregular
) assocR❙
rightregular1 = [x,y,z] trans3 assoc4 (congT (
trans assocR rightregular) [u] x∘u
) assocR❙
❚
/T normal bands ❚
...
...
@@ -54,6 +61,9 @@ implicit view RightRegular12RightRegular : ?RightRegular1 -> ?RightRegular =
theory Normal : ?FOLEQNatDed =
include ?Band ❙
normal: {x,y,z} ⊦ z∘(x∘y)∘z ≐ z∘(y∘x)∘z ❙
lr: {x,y,z} ⊦ z∘x∘z∘y ≐ z∘(z∘x)∘y ❘= [x,y,z] trans (congT assoc [u]u∘y) normal❙
❚
theory LeftNormal : ?FOLEQNatDed =
...
...
@@ -70,34 +80,35 @@ theory RightNormal : ?FOLEQNatDed =
implicit view Normal2LeftNormal : ?Normal -> ?LeftNormal =
include ?Band = ?Band❙
normal = [x,y,z] congT
([u] u∘z) leftnormal
❙
normal = [x,y,z] congT
leftnormal [u]u∘z
❙
❚
implicit view Normal2RightNormal : ?Normal -> ?RightNormal =
include ?Band = ?Band❙
normal = [x,y,z] trans3 assoc (congT
([u] z∘u) rightnormal
) assocR❙
normal = [x,y,z] trans3 assoc (congT
rightnormal [u]z∘u
) assocR❙
❚
implicit view LeftNormal2Semilattice : ?LeftNormal -> ?Semilattice =
include ?Band = ?Band❙
leftnormal = [x,y,z] congT
([u] z∘u) comm
❙
leftnormal = [x,y,z] congT
comm [u]z∘u
❙
❚
implicit view RightNormal2Semilattice : ?RightNormal -> ?Semilattice =
include ?Band = ?Band❙
rightnormal = [x,y,z] congT
([u] u∘z) comm
❙
rightnormal = [x,y,z] congT
comm [u]u∘z
❙
❚
/T views from regular to normal band theories ❚
implicit view LeftRegular12Normal : ?LeftRegular1 -> ?Normal =
include ?Band = ?Band❙
// zxzy = zxzyzxzy = zx yz zxzy = zxy zx zy = zxy xz zy = zxy zxy = zxy❙
leftregular1 = [x,y,z] _❙
❚
implicit view LeftRegular2LeftNormal : ?LeftRegular -> ?LeftNormal =
include ?Band = ?Band❙
leftregular = [x,
y,z] _
❙
leftregular = [x,
z] trans4 assoc leftnormal assocR (congT idempotent [u]u∘x)
❙
❚
implicit view RightRegular12Normal : ?RightRegular1 -> ?Normal =
...
...
@@ -107,5 +118,25 @@ implicit view RightRegular12Normal : ?RightRegular1 -> ?Normal =
implicit view RightRegular2RightNormal : ?RightRegular -> ?RightNormal =
include ?Band = ?Band❙
rightregular = [x,y,z] _❙
rightregular = [y,z] trans3 rightnormal assoc (congT idempotent [u]y∘u)❙
❚
/T rectangular bands (the axioms here already imply idempotency)❚
theory Rectangular : ?FOLEQNatDed =
include ?Band ❙
rectangular: {x,y} ⊦ x∘y∘x ≐ x ❙
// provable ❙
rectangularAny: {x,y,z} ⊦ x∘y∘z ≐ x∘z❙
❚
theory LeftSingular : ?FOLEQNatDed =
include ?Band ❙
leftsingular: {x,y} ⊦ x∘y ≐ x ❙
❚
theory RightSingular : ?FOLEQNatDed =
include ?Band ❙
rightsingular: {x,y} ⊦ x∘y ≐ y ❙
❚
source/logic/fol.mmt
View file @
c092ceb2
...
...
@@ -28,8 +28,8 @@ theory FOLEQNatDed : http://cds.omdoc.org/urtheories?LF =
include ?FOLEQ❙
include ?FOLNatDed ❙
refl : {x} ⊦ x ≐ x❙
congT : {x,y}
{T: term ⟶ term} ⊦ x ≐ y ⟶
⊦ (T x) ≐ (T y) ❘# congT 3 4❙
congF : {x,y}
{F: term ⟶ prop} ⊦ x ≐ y ⟶
⊦ (F x) ⟶ ⊦ (F y)❘# congF 3 4 5❙
congT : {x,y}
⊦ x ≐ y ⟶ {T: term ⟶ term}
⊦ (T x) ≐ (T y) ❘# congT 3 4❙
congF : {x,y}
⊦ x ≐ y ⟶ {F: term ⟶ prop}
⊦ (F x) ⟶ ⊦ (F y)❘# congF 3 4 5❙
sym : {x,y} ⊦ x ≐ y ⟶ ⊦ y ≐ x❙
trans : {x,y,z} ⊦ x ≐ y ⟶ ⊦ y ≐ z ⟶ ⊦ x ≐ z❙
...
...
source/magmas.mmt
View file @
c092ceb2
...
...
@@ -12,11 +12,11 @@ theory Semigroup : ?FOLEQNatDed =
= [x,y,z] sym assoc❙
assoc4: {w,x,y,z} ⊦ ((w∘x)∘y)∘z ≐ w∘(x∘(y∘z))❘
= [w,x,y,z] trans3 (congT
([u]u∘z) assoc) assoc (congT ([u]w∘u) assoc
)❙
= [w,x,y,z] trans3 (congT
assoc [u]u∘z) assoc (congT assoc [u]w∘u
)❙
assoc4R: {w,x,y,z} ⊦ w∘(x∘(y∘z)) ≐ ((w∘x)∘y)∘z❘
= [w,x,y,z] sym assoc4❙
assoc5: {v,w,x,y,z} ⊦ (((v∘w)∘x)∘y)∘z ≐ v∘(w∘(x∘(y∘z)))❘
= [v,w,x,y,z] trans3 (congT
([u]u∘z) assoc4) assoc (congT ([u]v∘u) assoc4
)❙
= [v,w,x,y,z] trans3 (congT
assoc4 [u]u∘z) assoc (congT assoc4 [u]v∘u
)❙
❚
theory Commutative : ?FOLEQNatDed =
...
...
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