Input = A P inv[A] A == A P inv[A] A P inv[A] == P inv[A] A^{T} (P inv[A])^{T} == P inv[A] A (P inv[A])^{T} A^{T} == A P inv[A] (P inv[A])^{T} == P inv[A^{T}] A^{T} A P inv[A] == A^{T} A P inv[A] P inv[A^{T}] == P inv[A^{T}] B P inv[B] B == B P inv[B] B P inv[B] == P inv[B] B^{T} (P inv[B])^{T} == P inv[B] B (P inv[B])^{T} B^{T} == B P inv[B] (P inv[B])^{T} == P inv[B^{T}] B^{T} B P inv[B] == B^{T} B P inv[B] P inv[B^{T}] == P inv[B^{T}] (A P inv[A] + P inv[B] B) P inv[A P inv[A] + P inv[B] B] (A P inv[A] + P inv[B] B) == A P inv[A] + P inv[B] B P inv[A P inv[A]+P inv[B] B] (A P inv[A]+P inv[B] B) P inv[A P inv[A]+P inv[B] B] == P inv[A P inv[A]+ P inv[B] B] (B^{T} (P inv[B])^{T} +(P inv[A])^{T} A^{T}) (P inv[A P inv[A] + P inv[B] B])^{T} == P inv[A P inv[A]+P inv[B] B] (A P inv[A]+ P inv[B] B) (P inv[A P inv[A] + P inv[B] B])^{T} (B^{T} (P inv[B])^{T} +(P inv[A])^{T} A^{T}) == (A P inv[A]+P inv[B] B) P inv[A P inv[A]+ P inv[B] B] (P inv[A P inv[A] + P inv[B] B])^{T} == P inv[B^{T} (P inv[B])^{T} + (P inv[A])^{T} A^{T}] (B^{T} (P inv[B])^{T} +(P inv[A])^{T} A^{T}) (A P inv[A]+P inv[B] B) P inv[A P inv[A]+P inv[B] B] == B^{T} (P inv[B])^{T} + (P inv[A])^{T} A^{T} (A P inv[A]+P inv[B] B) P inv[A P inv[A]+P inv[B] B] P inv[B^{T} (P inv[B])^{T} +(P inv[A])^{T} A^{T}] == P inv[B^{T} (P inv[B])^{T} + (P inv[A])^{T} A^{T}] File Name = APinvAPlusPinvBB NCMakeGB Iterations = 6 NCMakeGB on Digested Iterations = 0 SmallBasis Iterations = 0 SmallBasis on Knowns Iterations = 0 Deselect→ {} UserSelect→ {} UserUnknowns→ {} NCShortFormulas→-1 RR→False RRByCat→False SB→False SBByCat→False DegreeCap→-1 DegreeSumCap→-1 DegreeCapSB→-1 DegreeSumCapSB→-1 NCCV→False THE ORDER IS NOW THE FOLLOWING: A < B < A^{T} < B^{T} < P inv[A] < P inv[B] < P inv[A^{T}] < P inv[B^{T}] < (P inv[A])^{T} < (P inv[B])^{T} < P inv[A P inv[A]+P inv[B] B] < P inv[B^{T} (P inv[B])^{T} +(P inv[A])^{T} A^{T}] < (P inv[A P inv[A] + P inv[B] B])^{T}

YOUR SESSION HAS DIGESTED THE FOLLOWING RELATIONS

THE FOLLOWING VARIABLES HAVE BEEN SOLVED FOR:

{P inv[B^{T} (P inv[B])^{T} +(P inv[A])^{T} A^{T}], (P inv[A])^{T} , (P inv[B])^{T} , (P inv[A P inv[A] + P inv[B] B])^{T} }

The corresponding rules are the following: P inv[B^{T} (P inv[B])^{T} + (P inv[A])^{T} A^{T}] → P inv[A P inv[A] + P inv[B] B] (P inv[A])^{T} → P inv[A^{T}] (P inv[B])^{T} → P inv[B^{T}] (P inv[A P inv[A] + P inv[B] B])^{T} → P inv[A P inv[A] + P inv[B] B]

The expressions with unknown variables {} and knowns {A, B, P inv[A], P inv[B], P inv[A P inv[A]+P inv[B] B], P inv[A^{T}], P inv[B^{T}], A^{T} , B^{T} } P inv[A] A → A^{T} P inv[A^{T}] P inv[B] B → B^{T} P inv[B^{T}] P inv[A^{T}] A^{T} → A P inv[A] P inv[B^{T}] B^{T} → B P inv[B] A P inv[A] P inv[A^{T}] → P inv[A^{T}] A A^{T} P inv[A^{T}] → A B P inv[B] P inv[B^{T}] → P inv[B^{T}] B B^{T} P inv[B^{T}] → B

{P inv[A P inv[A] + P inv[B] B] B^{T} P inv[B^{T}] → A P inv[A] P inv[A P inv[A] + P inv[B] B] − \atop P inv[A P inv[A] + P inv[B] B] A P inv[A] + B^{T} P inv[B^{T}] P inv[A P inv[A] + P inv[B] B]}

A^{T} A P inv[A] → A^{T} A^{T} P inv[A^{T}] P inv[A] → P inv[A] B^{T} B P inv[B] → B^{T} B^{T} P inv[B^{T}] P inv[B] → P inv[B] P inv[A P inv[A] + P inv[B] B] A P inv[A] P inv[B] → −1 P inv[A P inv[A] + P inv[B] B] P inv[B] + A P inv[A] P inv[A P inv[A] + P inv[B] B] P inv[B] + B^{T} P inv[B^{T}] P inv[A P inv[A] + P inv[B] B] P inv[B]

{P inv[A P inv[A] + P inv[B] B] A P inv[A] B^{T} → −1 P inv[A P inv[A] + P inv[B] B] B^{T} + \atop A P inv[A] P inv[A P inv[A] + P inv[B] B] B^{T} + B^{T} P inv[B^{T}] P inv[A P inv[A] + P inv[B] B] B^{T}}

B^{T} P inv[B^{T}] P inv[A P inv[A] + P inv[B] B] P inv[A P inv[A] + P inv[B] B] → −1 A P inv[A] P inv[A P inv[A] + P inv[B] B] P inv[A P inv[A] + P inv[B] B] + P inv[A P inv[A] + P inv[B] B]

{B A P inv[A] P inv[A P inv[A] + P inv[B] B] P inv[A P inv[A] + P inv[B] B] → B P inv[A P inv[A] + \atop P inv[B] B] − B P inv[A P inv[A] + P inv[B] B] P inv[A P inv[A] + P inv[B] B]}

P inv[A P inv[A] + P inv[B] B] A P inv[A] P inv[A P inv[A] + P inv[B] B] P inv[A P inv[A] + P inv[B] B] → \frac{1}{8} A P inv[A] + \frac{1}{2} P inv[A P inv[A] + P inv[B] B] P inv[A P inv[A] + P inv[B] B] − \frac{1}{8} A P inv[A] P inv[A P inv[A] + P inv[B] B] − \frac{1}{4} P inv[A P inv[A] + P inv[B] B] A P inv[A] + \frac{1}{4} B^{T} P inv[B^{T}] P inv[A P inv[A] + P inv[B] B] + \frac{1}{2} A P inv[A] P inv[A P inv[A] + P inv[B] B] P inv[A P inv[A] + P inv[B] B] + P inv[A P inv[A] + P inv[B] B] A P inv[A] P inv[A P inv[A] + P inv[B] B] − \frac{1}{8} A P inv[A] B^{T} P inv[B^{T}] P inv[A P inv[A] + P inv[B] B] − \frac{3}{4} P inv[A P inv[A] + P inv[B] B] P inv[A P inv[A] + P inv[B] B] P inv[A P inv[A] + P inv[B] B] A P inv[A] P inv[A P inv[A] + P inv[B] B] → −\frac{1}{8} A P inv[A] + \frac{1}{2} P inv[A P inv[A] + P inv[B] B] P inv[A P inv[A] + P inv[B] B] + \frac{1}{8} A P inv[A] P inv[A P inv[A] + P inv[B] B] + \frac{1}{2} P inv[A P inv[A] + P inv[B] B] A P inv[A] P inv[A P inv[A] + P inv[B] B] + \frac{1}{2} P inv[A P inv[A] + P inv[B] B] P inv[A P inv[A] + P inv[B] B] A P inv[A] + \frac{1}{8} A P inv[A] B^{T} P inv[B^{T}] P inv[A P inv[A] + P inv[B] B] − \frac{1}{2} P inv[A P inv[A] + P inv[B] B]

P inv[B^{T}] A P inv[A] P inv[A P inv[A] + P inv[B] B] P inv[A P inv[A] + P inv[B] B] → P inv[B^{T}] P inv[A P inv[A]+P inv[B] B]−P inv[B^{T}] P inv[A P inv[A]+P inv[B] B] P inv[A P inv[A]+ P inv[B] B]

B^{T} P inv[B^{T}] A P inv[A] P inv[A P inv[A] + P inv[B] B] → A P inv[A] + B^{T} P inv[B^{T}] − A P inv[A] P inv[A P inv[A] + P inv[B] B] − B^{T} P inv[B^{T}] P inv[A P inv[A] + P inv[B] B] − A P inv[A] B^{T} P inv[B^{T}] P inv[A P inv[A] + P inv[B] B]

B^{T} P inv[B^{T}] P inv[A P inv[A]+P inv[B] B] A P inv[A] → 2 A P inv[A]−A P inv[A] P inv[A P inv[A]+ P inv[B] B] − 2 P inv[A P inv[A] + P inv[B] B] A P inv[A] + 2 B^{T} P inv[B^{T}] P inv[A P inv[A] + P inv[B] B] + 4 P inv[A P inv[A] + P inv[B] B] A P inv[A] P inv[A P inv[A] + P inv[B] B] − A P inv[A] P inv[A P inv[A] + P inv[B] B] A P inv[A] − A P inv[A] B^{T} P inv[B^{T}] P inv[A P inv[A] + P inv[B] B] − 2 P inv[A P inv[A] + P inv[B] B]

A P inv[A] B^{T} P inv[B^{T}] P inv[A P inv[A]+P inv[B] B] A → 2 A−4 P inv[A P inv[A]+P inv[B] B] A− 2 A P inv[A] P inv[A P inv[A] + P inv[B] B] A + B^{T} P inv[B^{T}] P inv[A P inv[A] + P inv[B] B] A + 4 P inv[A P inv[A] + P inv[B] B] A P inv[A] P inv[A P inv[A] + P inv[B] B] A

A P inv[A] B^{T} P inv[B^{T}] P inv[A P inv[A] + P inv[B] B] P inv[B] → A P inv[A] P inv[B] − 2 A P inv[A] P inv[A P inv[A] + P inv[B] B] P inv[B] + B^{T} P inv[B^{T}] P inv[A P inv[A] + P inv[B] B] P inv[B] + 4 P inv[A P inv[A] + P inv[B] B] A P inv[A] P inv[A P inv[A] + P inv[B] B] P inv[B] − P inv[B]

A P inv[A] B^{T} P inv[B^{T}] P inv[A P inv[A] + P inv[B] B] P inv[A^{T}] → −4 P inv[A P inv[A] +

P inv[B] B] P inv[A^{T}]  −  2 A P inv[A] P inv[A P inv[A]  +  P inv[B] B] P inv[A^{T}]  +
B^{T} P inv[B^{T}] P inv[A P inv[A]  +  P inv[B] B] P inv[A^{T}]  +  4 P inv[A P inv[A]  +

P inv[B] B] A P inv[A] P inv[A P inv[A] + P inv[B] B] P inv[A^{T}] + 2 P inv[A^{T}]

A P inv[A] B^{T} P inv[B^{T}] P inv[A P inv[A] + P inv[B] B] B^{T}  →  A P inv[A] B^{T} −

2 A P inv[A] P inv[A P inv[A] + P inv[B] B] B^{T} + B^{T} P inv[B^{T}] P inv[A P inv[A] + P inv[B] B] B^{T} + 4 P inv[A P inv[A] + P inv[B] B] A P inv[A] P inv[A P inv[A] + P inv[B] B] B^{T} − B^{T}

B A P inv[A] P inv[A P inv[A] + P inv[B] B] A P inv[A] → −1 B + B P inv[A P inv[A] + P inv[B] B] + B A P inv[A] + B A P inv[A] P inv[A P inv[A] + P inv[B] B] − 3 B P inv[A P inv[A] + P inv[B] B] A P inv[A] + 4 B P inv[A P inv[A] + P inv[B] B] A P inv[A] P inv[A P inv[A] + P inv[B] B]

{B A P inv[A] B^{T} P inv[B^{T}] P inv[A P inv[A] + P inv[B] B] → B − B P inv[A P inv[A] + P inv[B] B] + \atop B A P inv[A] − 2 B A P inv[A] P inv[A P inv[A] + P inv[B] B]}

{B P inv[A P inv[A]+P inv[B] B] A P inv[A] P inv[A P inv[A]+P inv[B] B] A → \frac{1}{2} B P inv[A P inv[A]+ \atop P inv[B] B] A}

{B P inv[A P inv[A] + P inv[B] B] A P inv[A] P inv[A P inv[A] + P inv[B] B] P inv[B] → \frac{1}{2} B P inv[B] − \atop \frac{1}{2} B P inv[A P inv[A] + P inv[B] B] P inv[B]}

{B P inv[A P inv[A] + P inv[B] B] A P inv[A] P inv[A P inv[A] + P inv[B] B] P inv[A^{T}] → \atop \frac{1}{2} B P inv[A P inv[A] + P inv[B] B] P inv[A^{T}]}

{B P inv[A P inv[A] + P inv[B] B] A P inv[A] P inv[A P inv[A] + P inv[B] B] B^{T} → \frac{1}{2} B B^{T} − \atop \frac{1}{2} B P inv[A P inv[A] + P inv[B] B] B^{T}}

{P inv[A] P inv[A P inv[A] + P inv[B] B] A P inv[A] P inv[A P inv[A] + P inv[B] B] A → \atop −\frac{1}{2} A^{T} P inv[A^{T}] + \frac{3}{2} P inv[A] P inv[A P inv[A] + P inv[B] B] A}

{P inv[A] P inv[A P inv[A] + P inv[B] B] A P inv[A] P inv[A P inv[A] + P inv[B] B] P inv[B] → \atop \frac{1}{2} P inv[A] P inv[A P inv[A] + P inv[B] B] P inv[B]}

{P inv[A] P inv[A P inv[A] + P inv[B] B] A P inv[A] P inv[A P inv[A] + P inv[B] B] P inv[A^{T}] → \atop −\frac{1}{2} P inv[A] P inv[A^{T}] + \frac{3}{2} P inv[A] P inv[A P inv[A] + P inv[B] B] P inv[A^{T}]}

{P inv[A] P inv[A P inv[A] + P inv[B] B] A P inv[A] P inv[A P inv[A] + P inv[B] B] B^{T} → \atop \frac{1}{2} P inv[A] P inv[A P inv[A] + P inv[B] B] B^{T}}

P inv[A P inv[A] + P inv[B] B] A P inv[A] P inv[A P inv[A] + P inv[B] B] A P inv[A] → −\frac{1}{2} A P inv[A] − \frac{1}{2} A P inv[A] P inv[A P inv[A] + P inv[B] B] + \frac{3}{2} P inv[A P inv[A] + P inv[B] B] A P inv[A] − \frac{1}{2} B^{T} P inv[B^{T}] P inv[A P inv[A] + P inv[B] B] − P inv[A P inv[A] + P inv[B] B] A P inv[A] P inv[A P inv[A] + P inv[B] B] + \frac{1}{2} A P inv[A] B^{T} P inv[B^{T}] P inv[A P inv[A] + P inv[B] B] + A P inv[A] P inv[A P inv[A] + P inv[B] B] A P inv[A] P inv[A P inv[A] + P inv[B] B] + \frac{1}{2} P inv[A P inv[A] + P inv[B] B]

P inv[B^{T}] A P inv[A] P inv[A P inv[A] + P inv[B] B] A P inv[A] → P inv[B^{T}] P inv[A P inv[A] + P inv[B] B] + P inv[B^{T}] A P inv[A] + P inv[B^{T}] A P inv[A] P inv[A P inv[A] + P inv[B] B] − 3 P inv[B^{T}] P inv[A P inv[A] + P inv[B] B] A P inv[A] + 4 P inv[B^{T}] P inv[A P inv[A] + P inv[B] B] A P inv[A] P inv[A P inv[A] + P inv[B] B] − P inv[B^{T}]

{P inv[B^{T}] A P inv[A] B^{T} P inv[B^{T}] P inv[A P inv[A] + P inv[B] B] → −1 P inv[B^{T}] P inv[A P inv[A] + \atop P inv[B] B]+P inv[B^{T}] A P inv[A]−2 P inv[B^{T}] A P inv[A] P inv[A P inv[A]+P inv[B] B]+P inv[B^{T}]}

{P inv[B^{T}] P inv[A P inv[A] + P inv[B] B] A P inv[A] P inv[A P inv[A] + P inv[B] B] A → \atop \frac{1}{2} P inv[B^{T}] P inv[A P inv[A] + P inv[B] B] A}

{P inv[B^{T}] P inv[A P inv[A] + P inv[B] B] A P inv[A] P inv[A P inv[A] + P inv[B] B] P inv[B] → \atop \frac{1}{2} P inv[B^{T}] P inv[B] − \frac{1}{2} P inv[B^{T}] P inv[A P inv[A] + P inv[B] B] P inv[B]}

{P inv[B^{T}] P inv[A P inv[A] + P inv[B] B] A P inv[A] P inv[A P inv[A] + P inv[B] B] P inv[A^{T}] → \atop \frac{1}{2} P inv[B^{T}] P inv[A P inv[A] + P inv[B] B] P inv[A^{T}]}

{P inv[B^{T}] P inv[A P inv[A] + P inv[B] B] A P inv[A] P inv[A P inv[A] + P inv[B] B] B^{T} → \atop \frac{1}{2} B P inv[B] − \frac{1}{2} P inv[B^{T}] P inv[A P inv[A] + P inv[B] B] B^{T}}

{A^{T} P inv[A P inv[A] + P inv[B] B] A P inv[A] P inv[A P inv[A] + P inv[B] B] A → −\frac{1}{2} A^{T} A + \atop \frac{3}{2} A^{T} P inv[A P inv[A] + P inv[B] B] A}

{A^{T} P inv[A P inv[A] + P inv[B] B] A P inv[A] P inv[A P inv[A] + P inv[B] B] P inv[B] → \atop \frac{1}{2} A^{T} P inv[A P inv[A] + P inv[B] B] P inv[B]}

{A^{T} P inv[A P inv[A] + P inv[B] B] A P inv[A] P inv[A P inv[A] + P inv[B] B] P inv[A^{T}] → \atop −\frac{1}{2} A^{T} P inv[A^{T}] + \frac{3}{2} A^{T} P inv[A P inv[A] + P inv[B] B] P inv[A^{T}]}

{A^{T} P inv[A P inv[A] + P inv[B] B] A P inv[A] P inv[A P inv[A] + P inv[B] B] B^{T} → \atop \frac{1}{2} A^{T} P inv[A P inv[A] + P inv[B] B] B^{T}}

USER CREATIONS APPEAR BELOW

SOME RELATIONS WHICH APPEAR BELOW MAY BE UNDIGESTED

THE FOLLOWING VARIABLES HAVE NOT BEEN SOLVED FOR: {A, B, P inv[A], P inv[B], P inv[A P inv[A] + P inv[B] B], P inv[A^{T}], P inv[B^{T}], A^{T} , B^{T} }

The time for preliminaries was 0:00:00 The time for NCMakeGB 1 was 0:00:00 The time for Remove Redundant 1 was 0:00:00 The time for the main NCMakeGB was 0:00:08 The time for Remove Redundant 2 was 0:00:00 The time for reducing unknowns was 0:00:00 The time for clean up basis was 0:00:00 The time for SmallBasis was 0:00:01 The time for CreateCategories was 0:00:07 The time for NCCV was 0:00:00 The time for RegularOutput was 0:00:01 The time for everything so far was 0:00:17