movenc: Get rid of frag_start

"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>

2 files changed, 4 insertions, 21 deletions

diff --git a/libavformat/movenc.c b/libavformat/movenc.c
index a460cd9ada..18410c70fa 100644
--- a/libavformat/movenc.c
+++ b/libavformat/movenc.c
@@ -4483,8 +4483,7 @@ static int mov_write_tfxd_tag(AVIOContext *pb, MOVTrack *track)
     avio_write(pb, uuid, sizeof(uuid));
     avio_w8(pb, 1);
     avio_wb24(pb, 0);
-    avio_wb64(pb, track->start_dts + track->frag_start +
-                  track->cluster[0].cts);
+    avio_wb64(pb, track->cluster[0].dts + track->cluster[0].cts);
     avio_wb64(pb, track->end_pts -
                   (track->cluster[0].dts + track->cluster[0].cts));
 
@@ -4563,8 +4562,7 @@ static int mov_add_tfra_entries(AVIOContext *pb, MOVMuxContext *mov, int tracks,
         info->size     = size;
         // Try to recreate the original pts for the first packet
         // from the fields we have stored
-        info->time     = track->start_dts + track->frag_start +
-                         track->cluster[0].cts;
+        info->time     = track->cluster[0].dts + track->cluster[0].cts;
         info->duration = track->end_pts -
                          (track->cluster[0].dts + track->cluster[0].cts);
         // If the pts is less than zero, we will have trimmed
@@ -4602,7 +4600,7 @@ static int mov_write_tfdt_tag(AVIOContext *pb, MOVTrack *track)
     ffio_wfourcc(pb, "tfdt");
     avio_w8(pb, 1); /* version */
     avio_wb24(pb, 0);
-    avio_wb64(pb, track->frag_start);
+    avio_wb64(pb, track->cluster[0].dts - track->start_dts);
     return update_size(pb, pos);
 }
 
@@ -4679,8 +4677,7 @@ static int mov_write_sidx_tag(AVIOContext *pb,
 
     if (track->entry) {
         entries = 1;
-        presentation_time = track->start_dts + track->frag_start +
-                            track->cluster[0].cts;
+        presentation_time = track->cluster[0].dts + track->cluster[0].cts;
         duration = track->end_pts -
                    (track->cluster[0].dts + track->cluster[0].cts);
         starts_with_SAP = track->cluster[0].flags & MOV_SYNC_SAMPLE;
@@ -5359,10 +5356,6 @@ static int mov_flush_fragment(AVFormatContext *s, int force)
         mov->moov_written = 1;
         mov->mdat_size = 0;
         for (i = 0; i < mov->nb_streams; i++) {
-            if (mov->tracks[i].entry)
-                mov->tracks[i].frag_start += mov->tracks[i].start_dts +
-                                             mov->tracks[i].track_duration -
-                                             mov->tracks[i].cluster[0].dts;
             mov->tracks[i].entry = 0;
             mov->tracks[i].end_reliable = 0;
         }
@@ -5416,11 +5409,7 @@ static int mov_flush_fragment(AVFormatContext *s, int force)
         MOVTrack *track = &mov->tracks[i];
         int buf_size, write_moof = 1, moof_tracks = -1;
         uint8_t *buf;
-        int64_t duration = 0;
 
-        if (track->entry)
-            duration = track->start_dts + track->track_duration -
-                       track->cluster[0].dts;
         if (mov->flags & FF_MOV_FLAG_SEPARATE_MOOF) {
             if (!track->entry)
                 continue;
@@ -5440,8 +5429,6 @@ static int mov_flush_fragment(AVFormatContext *s, int force)
             ffio_wfourcc(s->pb, "mdat");
         }
 
-        if (track->entry)
-            track->frag_start += duration;
         track->entry = 0;
         track->entries_flushed = 0;
         track->end_reliable = 0;
@@ -5760,7 +5747,6 @@ int ff_mov_write_packet(AVFormatContext *s, AVPacket *pkt)
             /* New fragment, but discontinuous from previous fragments.
              * Pretend the duration sum of the earlier fragments is
              * pkt->dts - trk->start_dts. */
-            trk->frag_start = pkt->dts - trk->start_dts;
             trk->end_pts = AV_NOPTS_VALUE;
             trk->frag_discont = 0;
         }
@@ -5782,12 +5768,10 @@ int ff_mov_write_packet(AVFormatContext *s, AVPacket *pkt)
                 /* Pretend the whole stream started at pts=0, with earlier fragments
                  * already written. If the stream started at pts=0, the duration sum
                  * of earlier fragments would have been pkt->pts. */
-                trk->frag_start = pkt->pts;
                 trk->start_dts  = pkt->dts - pkt->pts;
             } else {
                 /* Pretend the whole stream started at dts=0, with earlier fragments
                  * already written, with a duration summing up to pkt->dts. */
-                trk->frag_start = pkt->dts;
                 trk->start_dts  = 0;
             }
             trk->frag_discont = 0;
diff --git a/libavformat/movenc.h b/libavformat/movenc.h
index af1ea0bce6..daeaad1cc6 100644
--- a/libavformat/movenc.h
+++ b/libavformat/movenc.h
@@ -138,7 +138,6 @@ typedef struct MOVTrack {
 
     AVIOContext *mdat_buf;
     int64_t     data_offset;
-    int64_t     frag_start;
     int         frag_discont;
     int         entries_flushed;