(** * Memory injections *)

(** A memory state [m1] injects into another memory state [m2] via the
  memory injection [f] if the following conditions hold:
- each access in [m2] that corresponds to a valid access in [m1]
  is itself valid;
- the memory value associated in [m1] to an accessible address
  must inject into [m2]'s memory value at the corersponding address;
- unallocated blocks in [m1] must be mapped to [None] by [f];
- if [f b = Some(b', delta)], [b'] must be valid in [m2];
- distinct blocks in [m1] are mapped to non-overlapping sub-blocks in [m2];
- the sizes of [m2]'s blocks are representable with unsigned machine integers;
- pointers that could be represented using unsigned machine integers remain
  representable after the injection.
*)

Record inject' (f: meminj) (m1 m2: mem) : Prop :=
  mk_inject {
    mi_inj:
      mem_inj f m1 m2;
    mi_freeblocks:
      forall b, ~(valid_block m1 b) -> f b = None;
    mi_mappedblocks:
      forall b b' delta, f b = Some(b', delta) -> valid_block m2 b';
    mi_no_overlap:
      meminj_no_overlap f m1;
    mi_representable:
      forall b b' delta ofs,
      f b = Some(b', delta) ->
      perm m1 b (Ptrofs.unsigned ofs) Max Nonempty \/ perm m1 b (Ptrofs.unsigned ofs - 1) Max Nonempty ->
      delta >= 0 /\ 0 <= Ptrofs.unsigned ofs + delta <= Ptrofs.max_unsigned;
    mi_perm_inv:
      forall b1 ofs b2 delta k p,
      f b1 = Some(b2, delta) ->
      perm m2 b2 (ofs + delta) k p ->
      perm m1 b1 ofs k p \/ ~perm m1 b1 ofs Max Nonempty
  }.(** * Memory injections *)

(** A memory state [m1] injects into another memory state [m2] via the
  memory injection [f] if the following conditions hold:
- each access in [m2] that corresponds to a valid access in [m1]
  is itself valid;
- the memory value associated in [m1] to an accessible address
  must inject into [m2]'s memory value at the corersponding address;
- unallocated blocks in [m1] must be mapped to [None] by [f];
- if [f b = Some(b', delta)], [b'] must be valid in [m2];
- distinct blocks in [m1] are mapped to non-overlapping sub-blocks in [m2];
- the sizes of [m2]'s blocks are representable with unsigned machine integers;
- pointers that could be represented using unsigned machine integers remain
  representable after the injection.
*)

Record inject' (f: meminj) (m1 m2: mem) : Prop :=
  mk_inject {
    mi_inj:
      mem_inj f m1 m2;
    mi_freeblocks:
      forall b, ~(valid_block m1 b) -> f b = None;
    mi_mappedblocks:
      forall b b' delta, f b = Some(b', delta) -> valid_block m2 b';
    mi_no_overlap:
      meminj_no_overlap f m1;
    mi_representable:
      forall b b' delta ofs,
      f b = Some(b', delta) ->
      perm m1 b (Ptrofs.unsigned ofs) Max Nonempty \/ perm m1 b (Ptrofs.unsigned ofs - 1) Max Nonempty ->
      delta >= 0 /\ 0 <= Ptrofs.unsigned ofs + delta <= Ptrofs.max_unsigned;
    mi_perm_inv:
      forall b1 ofs b2 delta k p,
      f b1 = Some(b2, delta) ->
      perm m2 b2 (ofs + delta) k p ->
      perm m1 b1 ofs k p \/ ~perm m1 b1 ofs Max Nonempty
  }.(** * Memory injections *)

(** A memory state [m1] injects into another memory state [m2] via the
  memory injection [f] if the following conditions hold:
- each access in [m2] that corresponds to a valid access in [m1]
  is itself valid;
- the memory value associated in [m1] to an accessible address
  must inject into [m2]'s memory value at the corersponding address;
- unallocated blocks in [m1] must be mapped to [None] by [f];
- if [f b = Some(b', delta)], [b'] must be valid in [m2];
- distinct blocks in [m1] are mapped to non-overlapping sub-blocks in [m2];
- the sizes of [m2]'s blocks are representable with unsigned machine integers;
- pointers that could be represented using unsigned machine integers remain
  representable after the injection.
*)

Record inject' (f: meminj) (m1 m2: mem) : Prop :=
  mk_inject {
    mi_inj:
      mem_inj f m1 m2;
    mi_freeblocks:
      forall b, ~(valid_block m1 b) -> f b = None;
    mi_mappedblocks:
      forall b b' delta, f b = Some(b', delta) -> valid_block m2 b';
    mi_no_overlap:
      meminj_no_overlap f m1;
    mi_representable:
      forall b b' delta ofs,
      f b = Some(b', delta) ->
      perm m1 b (Ptrofs.unsigned ofs) Max Nonempty \/ perm m1 b (Ptrofs.unsigned ofs - 1) Max Nonempty ->
      delta >= 0 /\ 0 <= Ptrofs.unsigned ofs + delta <= Ptrofs.max_unsigned;
    mi_perm_inv:
      forall b1 ofs b2 delta k p,
      f b1 = Some(b2, delta) ->
      perm m2 b2 (ofs + delta) k p ->
      perm m1 b1 ofs k p \/ ~perm m1 b1 ofs Max Nonempty
  }.(** * Memory injections *)

(** A memory state [m1] injects into another memory state [m2] via the
  memory injection [f] if the following conditions hold:
- each access in [m2] that corresponds to a valid access in [m1]
  is itself valid;
- the memory value associated in [m1] to an accessible address
  must inject into [m2]'s memory value at the corersponding address;
- unallocated blocks in [m1] must be mapped to [None] by [f];
- if [f b = Some(b', delta)], [b'] must be valid in [m2];
- distinct blocks in [m1] are mapped to non-overlapping sub-blocks in [m2];
- the sizes of [m2]'s blocks are representable with unsigned machine integers;
- pointers that could be represented using unsigned machine integers remain
  representable after the injection.
*)

Record inject' (f: meminj) (m1 m2: mem) : Prop :=
  mk_inject {
    mi_inj:
      mem_inj f m1 m2;
    mi_freeblocks:
      forall b, ~(valid_block m1 b) -> f b = None;
    mi_mappedblocks:
      forall b b' delta, f b = Some(b', delta) -> valid_block m2 b';
    mi_no_overlap:
      meminj_no_overlap f m1;
    mi_representable:
      forall b b' delta ofs,
      f b = Some(b', delta) ->
      perm m1 b (Ptrofs.unsigned ofs) Max Nonempty \/ perm m1 b (Ptrofs.unsigned ofs - 1) Max Nonempty ->
      delta >= 0 /\ 0 <= Ptrofs.unsigned ofs + delta <= Ptrofs.max_unsigned;
    mi_perm_inv:
      forall b1 ofs b2 delta k p,
      f b1 = Some(b2, delta) ->
      perm m2 b2 (ofs + delta) k p ->
      perm m1 b1 ofs k p \/ ~perm m1 b1 ofs Max Nonempty
  }.(** * Memory injections *)

(** A memory state [m1] injects into another memory state [m2] via the
  memory injection [f] if the following conditions hold:
- each access in [m2] that corresponds to a valid access in [m1]
  is itself valid;
- the memory value associated in [m1] to an accessible address
  must inject into [m2]'s memory value at the corersponding address;
- unallocated blocks in [m1] must be mapped to [None] by [f];
- if [f b = Some(b', delta)], [b'] must be valid in [m2];
- distinct blocks in [m1] are mapped to non-overlapping sub-blocks in [m2];
- the sizes of [m2]'s blocks are representable with unsigned machine integers;
- pointers that could be represented using unsigned machine integers remain
  representable after the injection.
*)

Record inject' (f: meminj) (m1 m2: mem) : Prop :=
  mk_inject {
    mi_inj:
      mem_inj f m1 m2;
    mi_freeblocks:
      forall b, ~(valid_block m1 b) -> f b = None;
    mi_mappedblocks:
      forall b b' delta, f b = Some(b', delta) -> valid_block m2 b';
    mi_no_overlap:
      meminj_no_overlap f m1;
    mi_representable:
      forall b b' delta ofs,
      f b = Some(b', delta) ->
      perm m1 b (Ptrofs.unsigned ofs) Max Nonempty \/ perm m1 b (Ptrofs.unsigned ofs - 1) Max Nonempty ->
      delta >= 0 /\ 0 <= Ptrofs.unsigned ofs + delta <= Ptrofs.max_unsigned;
    mi_perm_inv:
      forall b1 ofs b2 delta k p,
      f b1 = Some(b2, delta) ->
      perm m2 b2 (ofs + delta) k p ->
      perm m1 b1 ofs k p \/ ~perm m1 b1 ofs Max Nonempty
  }.(** * Memory injections *)

(** A memory state [m1] injects into another memory state [m2] via the
  memory injection [f] if the following conditions hold:
- each access in [m2] that corresponds to a valid access in [m1]
  is itself valid;
- the memory value associated in [m1] to an accessible address
  must inject into [m2]'s memory value at the corersponding address;
- unallocated blocks in [m1] must be mapped to [None] by [f];
- if [f b = Some(b', delta)], [b'] must be valid in [m2];
- distinct blocks in [m1] are mapped to non-overlapping sub-blocks in [m2];
- the sizes of [m2]'s blocks are representable with unsigned machine integers;
- pointers that could be represented using unsigned machine integers remain
  representable after the injection.
*)

Record inject' (f: meminj) (m1 m2: mem) : Prop :=
  mk_inject {
    mi_inj:
      mem_inj f m1 m2;
    mi_freeblocks:
      forall b, ~(valid_block m1 b) -> f b = None;
    mi_mappedblocks:
      forall b b' delta, f b = Some(b', delta) -> valid_block m2 b';
    mi_no_overlap:
      meminj_no_overlap f m1;
    mi_representable:
      forall b b' delta ofs,
      f b = Some(b', delta) ->
      perm m1 b (Ptrofs.unsigned ofs) Max Nonempty \/ perm m1 b (Ptrofs.unsigned ofs - 1) Max Nonempty ->
      delta >= 0 /\ 0 <= Ptrofs.unsigned ofs + delta <= Ptrofs.max_unsigned;
    mi_perm_inv:
      forall b1 ofs b2 delta k p,
      f b1 = Some(b2, delta) ->
      perm m2 b2 (ofs + delta) k p ->
      perm m1 b1 ofs k p \/ ~perm m1 b1 ofs Max Nonempty
  }.(** * Memory injections *)

(** A memory state [m1] injects into another memory state [m2] via the
  memory injection [f] if the following conditions hold:
- each access in [m2] that corresponds to a valid access in [m1]
  is itself valid;
- the memory value associated in [m1] to an accessible address
  must inject into [m2]'s memory value at the corersponding address;
- unallocated blocks in [m1] must be mapped to [None] by [f];
- if [f b = Some(b', delta)], [b'] must be valid in [m2];
- distinct blocks in [m1] are mapped to non-overlapping sub-blocks in [m2];
- the sizes of [m2]'s blocks are representable with unsigned machine integers;
- pointers that could be represented using unsigned machine integers remain
  representable after the injection.
*)

Record inject' (f: meminj) (m1 m2: mem) : Prop :=
  mk_inject {
    mi_inj:
      mem_inj f m1 m2;
    mi_freeblocks:
      forall b, ~(valid_block m1 b) -> f b = None;
    mi_mappedblocks:
      forall b b' delta, f b = Some(b', delta) -> valid_block m2 b';
    mi_no_overlap:
      meminj_no_overlap f m1;
    mi_representable:
      forall b b' delta ofs,
      f b = Some(b', delta) ->
      perm m1 b (Ptrofs.unsigned ofs) Max Nonempty \/ perm m1 b (Ptrofs.unsigned ofs - 1) Max Nonempty ->
      delta >= 0 /\ 0 <= Ptrofs.unsigned ofs + delta <= Ptrofs.max_unsigned;
    mi_perm_inv:
      forall b1 ofs b2 delta k p,
      f b1 = Some(b2, delta) ->
      perm m2 b2 (ofs + delta) k p ->
      perm m1 b1 ofs k p \/ ~perm m1 b1 ofs Max Nonempty
  }.(** * Memory injections *)

(** A memory state [m1] injects into another memory state [m2] via the
  memory injection [f] if the following conditions hold:
- each access in [m2] that corresponds to a valid access in [m1]
  is itself valid;
- the memory value associated in [m1] to an accessible address
  must inject into [m2]'s memory value at the corersponding address;
- unallocated blocks in [m1] must be mapped to [None] by [f];
- if [f b = Some(b', delta)], [b'] must be valid in [m2];
- distinct blocks in [m1] are mapped to non-overlapping sub-blocks in [m2];
- the sizes of [m2]'s blocks are representable with unsigned machine integers;
- pointers that could be represented using unsigned machine integers remain
  representable after the injection.
*)

Record inject' (f: meminj) (m1 m2: mem) : Prop :=
  mk_inject {
    mi_inj:
      mem_inj f m1 m2;
    mi_freeblocks:
      forall b, ~(valid_block m1 b) -> f b = None;
    mi_mappedblocks:
      forall b b' delta, f b = Some(b', delta) -> valid_block m2 b';
    mi_no_overlap:
      meminj_no_overlap f m1;
    mi_representable:
      forall b b' delta ofs,
      f b = Some(b', delta) ->
      perm m1 b (Ptrofs.unsigned ofs) Max Nonempty \/ perm m1 b (Ptrofs.unsigned ofs - 1) Max Nonempty ->
      delta >= 0 /\ 0 <= Ptrofs.unsigned ofs + delta <= Ptrofs.max_unsigned;
    mi_perm_inv:
      forall b1 ofs b2 delta k p,
      f b1 = Some(b2, delta) ->
      perm m2 b2 (ofs + delta) k p ->
      perm m1 b1 ofs k p \/ ~perm m1 b1 ofs Max Nonempty
  }.Definition inject := inject'.

Theorem alloc_parallel_inject: forall f m1 m2 lo1 hi1 m1' b1 lo2 hi2, inject f m1 m2 -> alloc m1 lo1 hi1 = (m1', b1) -> lo2 <= lo1 -> hi1 <= hi2 -> exists f', exists m2', exists b2, alloc m2 lo2 hi2 = (m2', b2) /\ inject f' m1' m2' /\ inject_incr f f' /\ f' b1 = Some(b2, 0) /\ (forall b, b <> b1 -> f' b = f b).