Commit c092ceb2 authored by Florian Rabe's avatar Florian Rabe

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parent b4e33143
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 ❙
......@@ -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❙
......
......@@ -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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