"frag_start" is redundant, and every occurance can be replaced with cluster[0].dts - start_dts
The proof of no behaviour changes: (All line number below is based on commit bff7d662d728)
"frag_start" is read at 4 place (with all possible call stacks):
mov_write_packet
...
mov_flush_fragment
mov_write_moof_tag
mov_write_moof_tag_internal
mov_write_traf_tag
mov_write_tfxd_tag (#1)
mov_write_tfdt_tag (#2)
mov_add_tfra_entries (#3)
mov_write_sidx_tags
mov_write_sidx_tag (#4)
mov_write_trailer
mov_auto_flush_fragment
mov_flush_fragment
... (#1#2#3#4)
mov_write_sidx_tags
mov_write_sidx_tag (#4)
shift_data
compute_sidx_size
get_sidx_size
mov_write_sidx_tags
mov_write_sidx_tag (#4)
All read happens in "mov_write_trailer" and "mov_write_moof_tag". So we need to prove no behaviour change in these two
functions.
Condition 1: for every track that have "trk->entry == 0", trk->frag_start == trk->track_duration.
Condition 2: for every track that have "trk->entry > 0", trk->frag_start == trk->cluster[0].dts - trk->start_dts.
Definition 1: "Before flush" means just before the invocation of "mov_flush_fragment", except for the auto-flush case in
"mov_write_single_packet", which means before L5934.
Lemma 1: If Condition 1 & 2 is true before flush, Condition 1 & 2 is still true after "mov_flush_fragment" returns.
Proof:
No update to the tracks that have "trk->entry == 0" before flushing, so we only consider tracks that have "trk->entry > 0":
Case 1: !moov_written and moov will be written in this iteration
trk->entry = 0 L5366
trk->frag_start == trk->cluster[0].dts - trk->start_dts Lemma condition
trk->frag_start += trk->start_dts + trk->track_duration - trk->cluster[0].dts; L5363
So trk->entry == 0 && trk->frag_start == trk->track_duration
Case 2: !moov_written and moov will NOT be written in this iteration
nothing changed
Case 3: moov_written
trk->entry = 0 L5445
trk->frag_start == trk->cluster[0].dts - trk->start_dts Lemma condition
trk->frag_start += trk->start_dts + trk->track_duration - trk->cluster[0].dts; L5444
So trk->entry == 0 && trk->frag_start == trk->track_duration
Note that trk->track_duration may be updated for the tracks that have "trk->entry > 0" (mov_write_moov_tag will
update track_duration of "tmcd" track, but it must have 1 entry). But in all case, trk->frag_start is also updated
to consider the new value.
Lemma 2: If Condition 1 & 2 is true before "ff_mov_write_packet" invocation, Condition 1 & 2 is still true after it returns.
Proof:
Only the track corresponding to the pkt is updated, and no update to relevant variables if trk->entry > 0 before invocation.
So we only need to prove "trk->frag_start == trk->cluster[0].dts - trk->start_dts" after trk->entry increase from 0 to 1.
Case 1: trk->start_dts == AV_NOPTS_VALUE
Case 1.1: trk->frag_discont && use_editlist
trk->cluster[0].dts = pkt->dts at L5741
trk->frag_start = pkt->pts at L5785
trk->start_dts = pkt->dts - pkt->pts at L5786
So trk->frag_start == trk->cluster[0].dts - trk->start_dts
Case 1.2: trk->frag_discont && !use_editlist
trk->cluster[0].dts = pkt->dts at L5741
trk->frag_start = pkt->dts at L5790
trk->start_dts = 0 at L5791
So trk->frag_start == trk->cluster[0].dts - trk->start_dts
Case 1.3: !trk->frag_discont
trk->cluster[0].dts = pkt->dts at L5741
trk->frag_start = 0 init
trk->start_dts = pkt->dts at L5779
So trk->frag_start == trk->cluster[0].dts - trk->start_dts
Case 2: trk->start_dts != AV_NOPTS_VALUE
Case 2.1: trk->frag_discont
trk->cluster[0].dts = pkt->dts at L5741
trk->frag_start = pkt->dts - trk->start_dts at L5763
So trk->frag_start == trk->cluster[0].dts - trk->start_dts
Case 2.2: !trk->frag_discont
trk->cluster[0].dts = trk->start_dts + trk->track_duration at L5749
trk->track_duration == trk->frag_start Lemma condition
So trk->frag_start == trk->cluster[0].dts - trk->start_dts
Lemma 3: Condition 1 & 2 is true in all case before and after "ff_mov_write_packet" invocation, before flush and after
"mov_flush_fragment" returns.
Proof: All updates to relevant variable happen either in "ff_mov_write_packet", or during flush. And Condition 1 & 2
is true initially. So with lemma 1 & 2, we can prove this use induction.
Noticed that all read of "frag_start" only happen in "trk->entry > 0" branch. Now we need to prove Condition 2 is true
before each read.
Because no update to variables relevant to Condition 2 between "before flush" and "mov_write_moof_tag" invocation, we
can conclude Condition 2 is true before every invocation of "mov_write_moof_tag". No behaviour change in
"mov_write_moof_tag" is proved.
In "mov_write_trailer", No update to relevant variables after the last flush and before the invocation of
"mov_write_sidx_tag". So no behaviour change to "mov_write_trailer" is proved.
Q.E.D.
Signed-off-by: Hu Weiwen <sehuww@mail.scut.edu.cn>
Signed-off-by: Martin Storsjö <martin@martin.st>
4cdab8d022