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 + B) P inv[A + B] (A + B) == A + B P inv[A + B] (A + B) P inv[A + B] == P inv[A + B] (A^{T} + B^{T}) (P inv[A + B])^{T} == P inv[A + B] (A + B) (P inv[A + B])^{T} (A^{T} + B^{T}) == (A + B) P inv[A + B] (P inv[A + B])^{T} == P inv[A^{T} + B^{T}] (A^{T} + B^{T}) (A + B) P inv[A + B] == A^{T} + B^{T} (A + B) P inv[A + B] P inv[A^{T} + B^{T}] == P inv[A^{T} + B^{T}] File Name = PinvAPlusB 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 + B] < P inv[A^{T} + B^{T}] < (P inv[A + B])^{T}

YOUR SESSION HAS DIGESTED THE FOLLOWING RELATIONS

THE FOLLOWING VARIABLES HAVE BEEN SOLVED FOR: {(P inv[A])^{T} , (P inv[B])^{T} , (P inv[A + B])^{T} } The corresponding rules are the following: (P inv[A])^{T} → P inv[A^{T}]

(P inv[B])^{T} → P inv[B^{T}]

(P inv[A + B])^{T} → P inv[A^{T} + B^{T}] The expressions with unknown variables {} and knowns {A, B, P inv[A], P inv[B], P inv[A+B], P inv[A^{T}], P inv[B^{T}], P inv[A^{T} +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 + B] B → −1 P inv[A + B] A + A^{T} P inv[A^{T} + B^{T}] + B^{T} P inv[A^{T} + B^{T}] P inv[A^{T}] A^{T} → A P inv[A] P inv[B^{T}] B^{T} → B P inv[B] P inv[A^{T} + B^{T}] B^{T} → A P inv[A + B] + B P inv[A + B] − P inv[A^{T} + B^{T}] A^{T} 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 P inv[A + B] P inv[A^{T} + B^{T}] → −1 A P inv[A + B] P inv[A^{T} + B^{T}] + P inv[A^{T} + B^{T}] B B^{T} P inv[B^{T}] → B B B^{T} P inv[A^{T} +B^{T}] → A+B −A A^{T} P inv[A^{T} +B^{T}]−A B^{T} P inv[A^{T} +B^{T}]−B A^{T} P inv[A^{T} +B^{T}] 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} B P inv[A + B] → −1 A^{T} A P inv[A + B] − A^{T} B P inv[A + B] − B^{T} A P inv[A + B] + A^{T} + B^{T} B^{T} P inv[B^{T}] P inv[B] → P inv[B] B^{T} P inv[A^{T} + B^{T}] P inv[A + B] → −1 A^{T} P inv[A^{T} + B^{T}] P inv[A + B] + P inv[A + B]

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

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

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

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

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

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

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

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

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 + B], P inv[A^{T}], P inv[B^{T}], P inv[A^{T} + 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 0:00:01 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 0:00:02 The time for SmallBasis was 0:00:00 The time for CreateCategories was 0:00:02 The time for NCCV was 0:00:00 The time for RegularOutput was 0:00:00 The time for everything so far was 0:00:05