### (0) Obligation:

Runtime Complexity TRS:
The TRS R consists of the following rules:

f(x, a(b(y))) → f(a(b(x)), y)
f(x, b(c(y))) → f(b(c(x)), y)
f(x, c(a(y))) → f(c(a(x)), y)
f(a(x), y) → f(x, a(y))
f(b(x), y) → f(x, b(y))
f(c(x), y) → f(x, c(y))

Rewrite Strategy: FULL

### (1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n1):
The rewrite sequence
f(x, a(b(y))) →+ f(a(b(x)), y)
gives rise to a decreasing loop by considering the right hand sides subterm at position [].
The pumping substitution is [y / a(b(y))].
The result substitution is [x / a(b(x))].

### (3) RenamingProof (EQUIVALENT transformation)

Renamed function symbols to avoid clashes with predefined symbol.

### (4) Obligation:

Runtime Complexity Relative TRS:
The TRS R consists of the following rules:

f(x, a(b(y))) → f(a(b(x)), y)
f(x, b(c(y))) → f(b(c(x)), y)
f(x, c(a(y))) → f(c(a(x)), y)
f(a(x), y) → f(x, a(y))
f(b(x), y) → f(x, b(y))
f(c(x), y) → f(x, c(y))

S is empty.
Rewrite Strategy: FULL

Infered types.

### (6) Obligation:

TRS:
Rules:
f(x, a(b(y))) → f(a(b(x)), y)
f(x, b(c(y))) → f(b(c(x)), y)
f(x, c(a(y))) → f(c(a(x)), y)
f(a(x), y) → f(x, a(y))
f(b(x), y) → f(x, b(y))
f(c(x), y) → f(x, c(y))

Types:
f :: b:a:c → b:a:c → f
a :: b:a:c → b:a:c
b :: b:a:c → b:a:c
c :: b:a:c → b:a:c
hole_f1_0 :: f
hole_b:a:c2_0 :: b:a:c
gen_b:a:c3_0 :: Nat → b:a:c

### (7) OrderProof (LOWER BOUND(ID) transformation)

Heuristically decided to analyse the following defined symbols:
f

### (8) Obligation:

TRS:
Rules:
f(x, a(b(y))) → f(a(b(x)), y)
f(x, b(c(y))) → f(b(c(x)), y)
f(x, c(a(y))) → f(c(a(x)), y)
f(a(x), y) → f(x, a(y))
f(b(x), y) → f(x, b(y))
f(c(x), y) → f(x, c(y))

Types:
f :: b:a:c → b:a:c → f
a :: b:a:c → b:a:c
b :: b:a:c → b:a:c
c :: b:a:c → b:a:c
hole_f1_0 :: f
hole_b:a:c2_0 :: b:a:c
gen_b:a:c3_0 :: Nat → b:a:c

Generator Equations:
gen_b:a:c3_0(0) ⇔ hole_b:a:c2_0
gen_b:a:c3_0(+(x, 1)) ⇔ a(gen_b:a:c3_0(x))

The following defined symbols remain to be analysed:
f

### (9) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
f(gen_b:a:c3_0(+(1, n5_0)), gen_b:a:c3_0(b)) → *4_0, rt ∈ Ω(n50)

Induction Base:
f(gen_b:a:c3_0(+(1, 0)), gen_b:a:c3_0(b))

Induction Step:
f(gen_b:a:c3_0(+(1, +(n5_0, 1))), gen_b:a:c3_0(b)) →RΩ(1)
f(gen_b:a:c3_0(+(1, n5_0)), a(gen_b:a:c3_0(b))) →IH
*4_0

We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).

### (11) Obligation:

TRS:
Rules:
f(x, a(b(y))) → f(a(b(x)), y)
f(x, b(c(y))) → f(b(c(x)), y)
f(x, c(a(y))) → f(c(a(x)), y)
f(a(x), y) → f(x, a(y))
f(b(x), y) → f(x, b(y))
f(c(x), y) → f(x, c(y))

Types:
f :: b:a:c → b:a:c → f
a :: b:a:c → b:a:c
b :: b:a:c → b:a:c
c :: b:a:c → b:a:c
hole_f1_0 :: f
hole_b:a:c2_0 :: b:a:c
gen_b:a:c3_0 :: Nat → b:a:c

Lemmas:
f(gen_b:a:c3_0(+(1, n5_0)), gen_b:a:c3_0(b)) → *4_0, rt ∈ Ω(n50)

Generator Equations:
gen_b:a:c3_0(0) ⇔ hole_b:a:c2_0
gen_b:a:c3_0(+(x, 1)) ⇔ a(gen_b:a:c3_0(x))

No more defined symbols left to analyse.

### (12) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
f(gen_b:a:c3_0(+(1, n5_0)), gen_b:a:c3_0(b)) → *4_0, rt ∈ Ω(n50)

### (14) Obligation:

TRS:
Rules:
f(x, a(b(y))) → f(a(b(x)), y)
f(x, b(c(y))) → f(b(c(x)), y)
f(x, c(a(y))) → f(c(a(x)), y)
f(a(x), y) → f(x, a(y))
f(b(x), y) → f(x, b(y))
f(c(x), y) → f(x, c(y))

Types:
f :: b:a:c → b:a:c → f
a :: b:a:c → b:a:c
b :: b:a:c → b:a:c
c :: b:a:c → b:a:c
hole_f1_0 :: f
hole_b:a:c2_0 :: b:a:c
gen_b:a:c3_0 :: Nat → b:a:c

Lemmas:
f(gen_b:a:c3_0(+(1, n5_0)), gen_b:a:c3_0(b)) → *4_0, rt ∈ Ω(n50)

Generator Equations:
gen_b:a:c3_0(0) ⇔ hole_b:a:c2_0
gen_b:a:c3_0(+(x, 1)) ⇔ a(gen_b:a:c3_0(x))

No more defined symbols left to analyse.

### (15) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
f(gen_b:a:c3_0(+(1, n5_0)), gen_b:a:c3_0(b)) → *4_0, rt ∈ Ω(n50)
{"ft_lang_label":"__label__en","ft_lang_prob":0.50339717,"math_prob":0.99991333,"size":4085,"snap":"2021-31-2021-39","text_gpt3_token_len":1830,"char_repetition_ratio":0.20166625,"word_repetition_ratio":0.6820084,"special_character_ratio":0.49204406,"punctuation_ratio":0.28068805,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":1.0000063,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-07-27T01:25:16Z\",\"WARC-Record-ID\":\"<urn:uuid:e650fc00-bf30-44c8-a6d4-365a868f5924>\",\"Content-Length\":\"23913\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:0abfd612-5cf0-404c-a93f-1e1e78783634>\",\"WARC-Concurrent-To\":\"<urn:uuid:c39499f3-9b3b-42d0-8d1a-2430ab34104b>\",\"WARC-IP-Address\":\"137.226.37.212\",\"WARC-Target-URI\":\"https://aprove.informatik.rwth-aachen.de/eval/FullComplexity/newtpdb/Runtime_Complexity_Full_Rewriting/Zantema_05/z24.xml.AProVE%20Lower.html\",\"WARC-Payload-Digest\":\"sha1:3VWWNAOVLWF6GIRK4BJ6M4BGSWYPJMJY\",\"WARC-Block-Digest\":\"sha1:A7MLCKH6SVHKWAIQHCN7LQNQFRXCI6NM\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-31/CC-MAIN-2021-31_segments_1627046152168.38_warc_CC-MAIN-20210727010203-20210727040203-00341.warc.gz\"}"}