YES Confluence Proof

Confluence Proof

by csi

Input

The rewrite relation of the following TRS is considered.

c	→	c
h(a,h(f(b),c))	→	c

Proof

1 Redundant Rules Transformation

To prove that the TRS is (non-)confluent, we show (non-)confluence of the following modified system:

h(a,h(f(b),c))

→

All redundant rules that were added or removed can be simulated in 1 steps .

1.1 Redundant Rules Transformation

To prove that the TRS is (non-)confluent, we show (non-)confluence of the following modified system:

h(a,h(f(b),c))

→

All redundant rules that were added or removed can be simulated in 2 steps .

1.1.1 Critical Pair Closing System

Confluence is proven using the following terminating critical-pair-closing-system R:

There are no rules.

1.1.1.1 R is empty

There are no rules in the TRS. Hence, it is terminating.

Tool configuration

csi

version: csi 1.2.5 [hg: unknown]
strategy: (sorted -ms*; ( ((cr -kb;((( matrix -dim 1 -ib 3 -ob 5 | matrix -dim 2 -ib 2 -ob 3 | matrix -dim 3 -ib 1 -ob 1 | matrix -dim 3 -ib 1 -ob 3 | fail)[2]*);((dp;edg[0.5]?;(sccs | (sc || sct || {ur?;( (matrix -dp -ur -dim 1 -ib 3 -ob 5 | matrix -dp -ur -dim 2 -ib 2 -ob 3 | matrix -dp -ur -dim 3 -ib 1 -ob 1 | matrix -dp -ur -dim 3 -ib 1 -ob 3) || (kbo -ur -af | lpo -ur -af) || ( arctic -dp -ur -dim 2 -ib 2 -ob 2[2] | fail) || ( arctic -bz -dp -ur -dim 2 -ib 2 -ob 2[2] | fail) || fail) }restore || fail;(bounds -dp -rfc -qc || bounds -dp -all -rfc -qc || bounds -rfc -qc)[1] || fail ))*[6])! || (( kbo || (lpo | fail;(ref;lpo)) || fail;(bounds -rfc -qc) || fail)*[7])! || (rev;((dp;edg[0.5]?;(sccs | (sc || sct || {ur?;( (matrix -dp -ur -dim 1 -ib 3 -ob 5 | matrix -dp -ur -dim 2 -ib 2 -ob 3 | matrix -dp -ur -dim 3 -ib 1 -ob 1 | matrix -dp -ur -dim 3 -ib 1 -ob 3) || (kbo -ur -af | lpo -ur -af) || ( arctic -dp -ur -dim 2 -ib 2 -ob 2[2] | fail) || ( arctic -bz -dp -ur -dim 2 -ib 2 -ob 2[2] | fail) || fail) }restore || fail;(bounds -dp -rfc -qc || bounds -dp -all -rfc -qc || bounds -rfc -qc)[1] || fail ))*[6])! || (( kbo || (lpo | fail;(ref;lpo)) || fail;(bounds -rfc -qc) || fail)*[7])!)))))! || ((if linear then cr -closed -m -1;closed -strongly 7 else fail) || (if left-linear then cr -closed -m -1;(closed -development) else fail))! || (if linear then (cr -dup;(( lpo -quasi || (matrix -dim 1 -ib 3 -ob 4 | matrix -dim 2 -ib 2 -ob 2 | matrix -dim 3 -ib 1 -ob 2 | arctic -dim 2 -ib 2 -ob 2) || (if duplicating then fail else (bounds -rt || bounds -rt -qc))[1] || poly -ib 2 -ob 4 -nl2 -heuristic 1 || fail )[5]*);shift -lstar);(rule_labeling | rule_labeling -left)?;decreasing else fail)! || (if left-linear then (cr -dup;(( lpo -quasi || (matrix -dim 1 -ib 3 -ob 4 | matrix -dim 2 -ib 2 -ob 2 | matrix -dim 3 -ib 1 -ob 2 | arctic -dim 2 -ib 2 -ob 2) || (if duplicating then fail else (bounds -rt || bounds -rt -qc))[1] || poly -ib 2 -ob 4 -nl2 -heuristic 1 || fail )[5]*);shift -lstar);(rule_labeling | rule_labeling -left)?;decreasing else fail)! || (cr -cpcs2 -cpcscert; ((( matrix -dim 1 -ib 3 -ob 5 | matrix -dim 2 -ib 2 -ob 3 | matrix -dim 3 -ib 1 -ob 1 | matrix -dim 3 -ib 1 -ob 3 | fail)[2]*);((dp;edg[0.5]?;(sccs | (sc || sct || {ur?;( (matrix -dp -ur -dim 1 -ib 3 -ob 5 | matrix -dp -ur -dim 2 -ib 2 -ob 3 | matrix -dp -ur -dim 3 -ib 1 -ob 1 | matrix -dp -ur -dim 3 -ib 1 -ob 3) || (kbo -ur -af | lpo -ur -af) || ( arctic -dp -ur -dim 2 -ib 2 -ob 2[2] | fail) || ( arctic -bz -dp -ur -dim 2 -ib 2 -ob 2[2] | fail) || fail) }restore || fail;(bounds -dp -rfc -qc || bounds -dp -all -rfc -qc || bounds -rfc -qc)[1] || fail ))*[6])! || (( kbo || (lpo | fail;(ref;lpo)) || fail;(bounds -rfc -qc) || fail)*[7])! || (rev;((dp;edg[0.5]?;(sccs | (sc || sct || {ur?;( (matrix -dp -ur -dim 1 -ib 3 -ob 5 | matrix -dp -ur -dim 2 -ib 2 -ob 3 | matrix -dp -ur -dim 3 -ib 1 -ob 1 | matrix -dp -ur -dim 3 -ib 1 -ob 3) || (kbo -ur -af | lpo -ur -af) || ( arctic -dp -ur -dim 2 -ib 2 -ob 2[2] | fail) || ( arctic -bz -dp -ur -dim 2 -ib 2 -ob 2[2] | fail) || fail) }restore || fail;(bounds -dp -rfc -qc || bounds -dp -all -rfc -qc || bounds -rfc -qc)[1] || fail ))*[6])! || (( kbo || (lpo | fail;(ref;lpo)) || fail;(bounds -rfc -qc) || fail)*[7])!)))))!) || (( (nonconfluence -steps 0 -tcap -fun | nonconfluence -steps 2 -tcap -fun | nonconfluence -steps 25 -width 1 -tcap -fun) || (nonconfluence -steps 2 -tcap -var | nonconfluence -steps 25 -width 1 -tcap -var) || (nonconfluence -steps 0 -tree -cert -fun | nonconfluence -steps 0 -tree -cert -var | nonconfluence -steps 1 -tree -cert -fun | nonconfluence -steps 1 -tree -cert -var | nonconfluence -steps 2 -tree -cert -fun | nonconfluence -steps 2 -tree -cert -var | nonconfluence -steps 25 -tree -cert -fun | nonconfluence -steps 25 -tree -cert -var) )[6] | ((cr -m -1 -force);(redundant -narrowfwd -narrowbwd -size 7)))3*! || (((cr -m -1 -force);(redundant -remove 4)); ((cr -kb;((( matrix -dim 1 -ib 3 -ob 5 | matrix -dim 2 -ib 2 -ob 3 | matrix -dim 3 -ib 1 -ob 1 | matrix -dim 3 -ib 1 -ob 3 | fail)[2]*);((dp;edg[0.5]?;(sccs | (sc || sct || {ur?;( (matrix -dp -ur -dim 1 -ib 3 -ob 5 | matrix -dp -ur -dim 2 -ib 2 -ob 3 | matrix -dp -ur -dim 3 -ib 1 -ob 1 | matrix -dp -ur -dim 3 -ib 1 -ob 3) || (kbo -ur -af | lpo -ur -af) || ( arctic -dp -ur -dim 2 -ib 2 -ob 2[2] | fail) || ( arctic -bz -dp -ur -dim 2 -ib 2 -ob 2[2] | fail) || fail) }restore || fail;(bounds -dp -rfc -qc || bounds -dp -all -rfc -qc || bounds -rfc -qc)[1] || fail ))*[6])! || (( kbo || (lpo | fail;(ref;lpo)) || fail;(bounds -rfc -qc) || fail)*[7])! || (rev;((dp;edg[0.5]?;(sccs | (sc || sct || {ur?;( (matrix -dp -ur -dim 1 -ib 3 -ob 5 | matrix -dp -ur -dim 2 -ib 2 -ob 3 | matrix -dp -ur -dim 3 -ib 1 -ob 1 | matrix -dp -ur -dim 3 -ib 1 -ob 3) || (kbo -ur -af | lpo -ur -af) || ( arctic -dp -ur -dim 2 -ib 2 -ob 2[2] | fail) || ( arctic -bz -dp -ur -dim 2 -ib 2 -ob 2[2] | fail) || fail) }restore || fail;(bounds -dp -rfc -qc || bounds -dp -all -rfc -qc || bounds -rfc -qc)[1] || fail ))*[6])! || (( kbo || (lpo | fail;(ref;lpo)) || fail;(bounds -rfc -qc) || fail)*[7])!)))))! || ((if linear then cr -closed -m -1;closed -strongly 7 else fail) || (if left-linear then cr -closed -m -1;(closed -development) else fail))! || (if linear then (cr -dup;(( lpo -quasi || (matrix -dim 1 -ib 3 -ob 4 | matrix -dim 2 -ib 2 -ob 2 | matrix -dim 3 -ib 1 -ob 2 | arctic -dim 2 -ib 2 -ob 2) || (if duplicating then fail else (bounds -rt || bounds -rt -qc))[1] || poly -ib 2 -ob 4 -nl2 -heuristic 1 || fail )[5]*);shift -lstar);(rule_labeling | rule_labeling -left)?;decreasing else fail)! || (if left-linear then (cr -dup;(( lpo -quasi || (matrix -dim 1 -ib 3 -ob 4 | matrix -dim 2 -ib 2 -ob 2 | matrix -dim 3 -ib 1 -ob 2 | arctic -dim 2 -ib 2 -ob 2) || (if duplicating then fail else (bounds -rt || bounds -rt -qc))[1] || poly -ib 2 -ob 4 -nl2 -heuristic 1 || fail )[5]*);shift -lstar);(rule_labeling | rule_labeling -left)?;decreasing else fail)! || (cr -cpcs2 -cpcscert; ((( matrix -dim 1 -ib 3 -ob 5 | matrix -dim 2 -ib 2 -ob 3 | matrix -dim 3 -ib 1 -ob 1 | matrix -dim 3 -ib 1 -ob 3 | fail)[2]*);((dp;edg[0.5]?;(sccs | (sc || sct || {ur?;( (matrix -dp -ur -dim 1 -ib 3 -ob 5 | matrix -dp -ur -dim 2 -ib 2 -ob 3 | matrix -dp -ur -dim 3 -ib 1 -ob 1 | matrix -dp -ur -dim 3 -ib 1 -ob 3) || (kbo -ur -af | lpo -ur -af) || ( arctic -dp -ur -dim 2 -ib 2 -ob 2[2] | fail) || ( arctic -bz -dp -ur -dim 2 -ib 2 -ob 2[2] | fail) || fail) }restore || fail;(bounds -dp -rfc -qc || bounds -dp -all -rfc -qc || bounds -rfc -qc)[1] || fail ))*[6])! || (( kbo || (lpo | fail;(ref;lpo)) || fail;(bounds -rfc -qc) || fail)*[7])! || (rev;((dp;edg[0.5]?;(sccs | (sc || sct || {ur?;( (matrix -dp -ur -dim 1 -ib 3 -ob 5 | matrix -dp -ur -dim 2 -ib 2 -ob 3 | matrix -dp -ur -dim 3 -ib 1 -ob 1 | matrix -dp -ur -dim 3 -ib 1 -ob 3) || (kbo -ur -af | lpo -ur -af) || ( arctic -dp -ur -dim 2 -ib 2 -ob 2[2] | fail) || ( arctic -bz -dp -ur -dim 2 -ib 2 -ob 2[2] | fail) || fail) }restore || fail;(bounds -dp -rfc -qc || bounds -dp -all -rfc -qc || bounds -rfc -qc)[1] || fail ))*[6])! || (( kbo || (lpo | fail;(ref;lpo)) || fail;(bounds -rfc -qc) || fail)*[7])!)))))!))! || (((cr -force -redundant);(redundant)); ((cr -kb;((( matrix -dim 1 -ib 3 -ob 5 | matrix -dim 2 -ib 2 -ob 3 | matrix -dim 3 -ib 1 -ob 1 | matrix -dim 3 -ib 1 -ob 3 | fail)[2]*);((dp;edg[0.5]?;(sccs | (sc || sct || {ur?;( (matrix -dp -ur -dim 1 -ib 3 -ob 5 | matrix -dp -ur -dim 2 -ib 2 -ob 3 | matrix -dp -ur -dim 3 -ib 1 -ob 1 | matrix -dp -ur -dim 3 -ib 1 -ob 3) || (kbo -ur -af | lpo -ur -af) || ( arctic -dp -ur -dim 2 -ib 2 -ob 2[2] | fail) || ( arctic -bz -dp -ur -dim 2 -ib 2 -ob 2[2] | fail) || fail) }restore || fail;(bounds -dp -rfc -qc || bounds -dp -all -rfc -qc || bounds -rfc -qc)[1] || fail ))*[6])! || (( kbo || (lpo | fail;(ref;lpo)) || fail;(bounds -rfc -qc) || fail)*[7])! || (rev;((dp;edg[0.5]?;(sccs | (sc || sct || {ur?;( (matrix -dp -ur -dim 1 -ib 3 -ob 5 | matrix -dp -ur -dim 2 -ib 2 -ob 3 | matrix -dp -ur -dim 3 -ib 1 -ob 1 | matrix -dp -ur -dim 3 -ib 1 -ob 3) || (kbo -ur -af | lpo -ur -af) || ( arctic -dp -ur -dim 2 -ib 2 -ob 2[2] | fail) || ( arctic -bz -dp -ur -dim 2 -ib 2 -ob 2[2] | fail) || fail) }restore || fail;(bounds -dp -rfc -qc || bounds -dp -all -rfc -qc || bounds -rfc -qc)[1] || fail ))*[6])! || (( kbo || (lpo | fail;(ref;lpo)) || fail;(bounds -rfc -qc) || fail)*[7])!)))))! || ((if linear then cr -closed -m -1;closed -strongly 7 else fail) || (if left-linear then cr -closed -m -1;(closed -development) else fail))! || (if linear then (cr -dup;(( lpo -quasi || (matrix -dim 1 -ib 3 -ob 4 | matrix -dim 2 -ib 2 -ob 2 | matrix -dim 3 -ib 1 -ob 2 | arctic -dim 2 -ib 2 -ob 2) || (if duplicating then fail else (bounds -rt || bounds -rt -qc))[1] || poly -ib 2 -ob 4 -nl2 -heuristic 1 || fail )[5]*);shift -lstar);(rule_labeling | rule_labeling -left)?;decreasing else fail)! || (if left-linear then (cr -dup;(( lpo -quasi || (matrix -dim 1 -ib 3 -ob 4 | matrix -dim 2 -ib 2 -ob 2 | matrix -dim 3 -ib 1 -ob 2 | arctic -dim 2 -ib 2 -ob 2) || (if duplicating then fail else (bounds -rt || bounds -rt -qc))[1] || poly -ib 2 -ob 4 -nl2 -heuristic 1 || fail )[5]*);shift -lstar);(rule_labeling | rule_labeling -left)?;decreasing else fail)! || (cr -cpcs2 -cpcscert; ((( matrix -dim 1 -ib 3 -ob 5 | matrix -dim 2 -ib 2 -ob 3 | matrix -dim 3 -ib 1 -ob 1 | matrix -dim 3 -ib 1 -ob 3 | fail)[2]*);((dp;edg[0.5]?;(sccs | (sc || sct || {ur?;( (matrix -dp -ur -dim 1 -ib 3 -ob 5 | matrix -dp -ur -dim 2 -ib 2 -ob 3 | matrix -dp -ur -dim 3 -ib 1 -ob 1 | matrix -dp -ur -dim 3 -ib 1 -ob 3) || (kbo -ur -af | lpo -ur -af) || ( arctic -dp -ur -dim 2 -ib 2 -ob 2[2] | fail) || ( arctic -bz -dp -ur -dim 2 -ib 2 -ob 2[2] | fail) || fail) }restore || fail;(bounds -dp -rfc -qc || bounds -dp -all -rfc -qc || bounds -rfc -qc)[1] || fail ))*[6])! || (( kbo || (lpo | fail;(ref;lpo)) || fail;(bounds -rfc -qc) || fail)*[7])! || (rev;((dp;edg[0.5]?;(sccs | (sc || sct || {ur?;( (matrix -dp -ur -dim 1 -ib 3 -ob 5 | matrix -dp -ur -dim 2 -ib 2 -ob 3 | matrix -dp -ur -dim 3 -ib 1 -ob 1 | matrix -dp -ur -dim 3 -ib 1 -ob 3) || (kbo -ur -af | lpo -ur -af) || ( arctic -dp -ur -dim 2 -ib 2 -ob 2[2] | fail) || ( arctic -bz -dp -ur -dim 2 -ib 2 -ob 2[2] | fail) || fail) }restore || fail;(bounds -dp -rfc -qc || bounds -dp -all -rfc -qc || bounds -rfc -qc)[1] || fail ))*[6])! || (( kbo || (lpo | fail;(ref;lpo)) || fail;(bounds -rfc -qc) || fail)*[7])!)))))!)[15]?)3*! || (((cr -m -1 -force -redundant);(redundant -rhs)); ((cr -kb;((( matrix -dim 1 -ib 3 -ob 5 | matrix -dim 2 -ib 2 -ob 3 | matrix -dim 3 -ib 1 -ob 1 | matrix -dim 3 -ib 1 -ob 3 | fail)[2]*);((dp;edg[0.5]?;(sccs | (sc || sct || {ur?;( (matrix -dp -ur -dim 1 -ib 3 -ob 5 | matrix -dp -ur -dim 2 -ib 2 -ob 3 | matrix -dp -ur -dim 3 -ib 1 -ob 1 | matrix -dp -ur -dim 3 -ib 1 -ob 3) || (kbo -ur -af | lpo -ur -af) || ( arctic -dp -ur -dim 2 -ib 2 -ob 2[2] | fail) || ( arctic -bz -dp -ur -dim 2 -ib 2 -ob 2[2] | fail) || fail) }restore || fail;(bounds -dp -rfc -qc || bounds -dp -all -rfc -qc || bounds -rfc -qc)[1] || fail ))*[6])! || (( kbo || (lpo | fail;(ref;lpo)) || fail;(bounds -rfc -qc) || fail)*[7])! || (rev;((dp;edg[0.5]?;(sccs | (sc || sct || {ur?;( (matrix -dp -ur -dim 1 -ib 3 -ob 5 | matrix -dp -ur -dim 2 -ib 2 -ob 3 | matrix -dp -ur -dim 3 -ib 1 -ob 1 | matrix -dp -ur -dim 3 -ib 1 -ob 3) || (kbo -ur -af | lpo -ur -af) || ( arctic -dp -ur -dim 2 -ib 2 -ob 2[2] | fail) || ( arctic -bz -dp -ur -dim 2 -ib 2 -ob 2[2] | fail) || fail) }restore || fail;(bounds -dp -rfc -qc || bounds -dp -all -rfc -qc || bounds -rfc -qc)[1] || fail ))*[6])! || (( kbo || (lpo | fail;(ref;lpo)) || fail;(bounds -rfc -qc) || fail)*[7])!)))))! || ((if linear then cr -closed -m -1;closed -strongly 7 else fail) || (if left-linear then cr -closed -m -1;(closed -development) else fail))! || (if linear then (cr -dup;(( lpo -quasi || (matrix -dim 1 -ib 3 -ob 4 | matrix -dim 2 -ib 2 -ob 2 | matrix -dim 3 -ib 1 -ob 2 | arctic -dim 2 -ib 2 -ob 2) || (if duplicating then fail else (bounds -rt || bounds -rt -qc))[1] || poly -ib 2 -ob 4 -nl2 -heuristic 1 || fail )[5]*);shift -lstar);(rule_labeling | rule_labeling -left)?;decreasing else fail)! || (if left-linear then (cr -dup;(( lpo -quasi || (matrix -dim 1 -ib 3 -ob 4 | matrix -dim 2 -ib 2 -ob 2 | matrix -dim 3 -ib 1 -ob 2 | arctic -dim 2 -ib 2 -ob 2) || (if duplicating then fail else (bounds -rt || bounds -rt -qc))[1] || poly -ib 2 -ob 4 -nl2 -heuristic 1 || fail )[5]*);shift -lstar);(rule_labeling | rule_labeling -left)?;decreasing else fail)! || (cr -cpcs2 -cpcscert; ((( matrix -dim 1 -ib 3 -ob 5 | matrix -dim 2 -ib 2 -ob 3 | matrix -dim 3 -ib 1 -ob 1 | matrix -dim 3 -ib 1 -ob 3 | fail)[2]*);((dp;edg[0.5]?;(sccs | (sc || sct || {ur?;( (matrix -dp -ur -dim 1 -ib 3 -ob 5 | matrix -dp -ur -dim 2 -ib 2 -ob 3 | matrix -dp -ur -dim 3 -ib 1 -ob 1 | matrix -dp -ur -dim 3 -ib 1 -ob 3) || (kbo -ur -af | lpo -ur -af) || ( arctic -dp -ur -dim 2 -ib 2 -ob 2[2] | fail) || ( arctic -bz -dp -ur -dim 2 -ib 2 -ob 2[2] | fail) || fail) }restore || fail;(bounds -dp -rfc -qc || bounds -dp -all -rfc -qc || bounds -rfc -qc)[1] || fail ))*[6])! || (( kbo || (lpo | fail;(ref;lpo)) || fail;(bounds -rfc -qc) || fail)*[7])! || (rev;((dp;edg[0.5]?;(sccs | (sc || sct || {ur?;( (matrix -dp -ur -dim 1 -ib 3 -ob 5 | matrix -dp -ur -dim 2 -ib 2 -ob 3 | matrix -dp -ur -dim 3 -ib 1 -ob 1 | matrix -dp -ur -dim 3 -ib 1 -ob 3) || (kbo -ur -af | lpo -ur -af) || ( arctic -dp -ur -dim 2 -ib 2 -ob 2[2] | fail) || ( arctic -bz -dp -ur -dim 2 -ib 2 -ob 2[2] | fail) || fail) }restore || fail;(bounds -dp -rfc -qc || bounds -dp -all -rfc -qc || bounds -rfc -qc)[1] || fail ))*[6])! || (( kbo || (lpo | fail;(ref;lpo)) || fail;(bounds -rfc -qc) || fail)*[7])!)))))!)[15]?)3*! ))[54]