Pair Sequence System → Buchholz's ordinal notation Implementation

A matrix can be written in the following formats:

• (a,b)(c,d)⋯(y,z)
• ((a,b),(c,d),⋯,(y,z))

A term in Buchholz's ordinal notation can be written as follows:

• 0
• A principal term as: D_u a; D may be replaced with p, _ is optional, and the space is also optional if not immediately followed by a number.
• A sum as: (a₀,⋯,a_k)
• 1 as abbreviation of D_0 0, natural number n for (1,⋯,1) with n 1s, and w or ω for D_0 1
• Ω_n for positive integer n as abbreviation of D_n 0 and Ω with no number immediately following it for Ω_1; Ω may be replaced with W and _ is optional.
• ⊕ (or +) and × (or *) for the corresponding operations; × taking precedence over ⊕ and both operations break out of D_u a
• () to group a term, to write terms such as D_0 (D_1 0,D_1 0) as D_0(Ω×2)

Write i, j, k, m, n for integers; a=a₀;⋯;a_l for a sequence of integers; M for a matrix; Q=Q₀;⋯;Q_k for a sequence of matrices; t for a term in Buchholz's ordinal notation; s, c, b for strings.
Each line in the input is written as operation var1;var2;⋯, operation case insensitive. Following operations are implemented:

• Parent M;i;j - Outputs the integer k such that (i,k)<^Next_M(i,j)
• Parent M;i;j;k - Outputs whether (i,k)<^Next_M(i,j)
• Ancestor M;i;j - Outputs the integers k₀>k₁>⋯ such that (i,k_l)≤_M(i,j)
• Ancestor M;i;j;k - Outputs whether (i,k)≤_M(i,j)
• Pred M - Outputs Pred(M)
• Derp M - Outputs Derp(M)
• [] M;a - Outputs M[a₀];⋯;[a_l]
• IncrFirst M - Outputs IncrFirst(M)
• Classify M - Outputs M is in which of ZT_PS, PT_PS, or MT_PS
• P M - Outputs P(M)
• Unadmit M;j - Outputs whether M unadmits j
• Admit M;j - Outputs whether M admits j
• Admitted M - Outputs ℕ_M
• Adm M;j - Outputs Adm_M(j)
• Marked M;m - Outputs whether (M,m)∈T_PS^Marked
• MarkedReduced M;m - Outputs whether (M,m)∈RT_PS^Marked
• IdxSum Q - Outputs IdxSum(Q)
• TrMax M - Outputs TrMax(M)
• Br M - Outputs Br(M)
• FirstNodes M - Outputs FirstNodes(M)
• Joints M - Outputs Joints(M)
• Red M - Outputs Red(M)
• Reduced M - Outputs whether M∈RT_PS
• Standard M - Outputs whether M∈ST_PS
• Standard M;k - Outputs whether M∈S_kT_PS
• Descending Q - Outputs whether Q is descending
• Classify t - Outputs t is in which of {0}, PT_B, or MT_B
• Standard t - Outputs whether t∈OT_B
• [] t;a - Outputs t[a₀];⋯;[a_l]
• P t - Outputs P(t)
• RightNodes t - Outputs RightNodes(t)
• scb t;c - Outputs s and b such that (s,c,b) is a scb decomposition of t and if they are either type 0 scb decomposition, type 1 scb decomposition, or neither, if one exists
• scb t;s;c;b - Outputs whether (s,c,b) is the scb decomposition of t and if they are either type 0 scb decomposition, type 1 scb decomposition, or neither
• scb? t - Outputs t is either type 0 scb decomposable, type 1 scb decomposable, or neither
• scb! t - Outputs s, c, and b such that (s,c,b) is either the type 0 or the type 1 scb decomposition of t and which, if one exists
• Marked t;c - Outputs whether (t,c)∈T_B^Marked
• Trans M - Outputs Trans(M)
• Mark M;m - Outputs Mark(M,m)
• TransType M - Outputs M satisfies which of "M∈RT_PS and j₁=0", "M∈RT_PS and M∈PT_PS and j₁>0 and condition [(I)-(VI)]", "M∈RT_PS and M∈MT_PS and j₁>0", and "not M∈RT_PS"
• NextAdm M;i;j - Outputs the integer k such that (i,k)<^NextAdm_M(i,j)
• NextAdm M;i;j;k - Outputs whether (i,k)<^NextAdm_M(i,j)
• RightAnces M - Outputs RightAnces(M)
• StronglyPrincipal M - Outputs whether M∈DT_PS
• LastStep M - Outputs LastStep(M)
• Trans-1 t - Outputs Trans^-1(t)
• ? M - Outputs M
• ? t - Outputs t
• ! M - Outputs Trans(M)
• ! t - Outputs Trans^-1(t)