source: src/joint/blocks.ma @ 3129

include "joint/Joint.ma".
include "utilities/bindLists.ma".

(* inductive block_step (p : stmt_params) (globals : list ident) : Type[0] ≝
  | block_seq : joint_seq p globals → block_step p globals
  | block_skip : label → block_step p globals.

definition if_seq : ∀p,globals.∀A:Type[2].block_step p globals → A → A → A ≝
λp,g,A,s.match s with
[ block_seq _ ⇒ λx,y.x
| _ ⇒ λx,y.y
].

definition stmt_of_block_step : ∀p : stmt_params.∀globals.
  ∀s : block_step p globals.if_seq … s (succ p) unit → joint_statement p globals ≝
  λp,g,s.match s return λx.if_seq ??? x ?? → joint_statement ?? with
  [ block_seq s' ⇒ λnxt.sequential … s' nxt
  | block_skip l ⇒ λ_.GOTO … l
  ].

definition seq_to_block_step_list : ∀p : stmt_params.∀globals.
  list (joint_seq p globals) →
  list (block_step p globals) ≝ λp,globals.map ?? (block_seq ??).

coercion block_step_from_seq_list : ∀p : stmt_params.∀globals.
  ∀l:list (joint_seq p globals).
  list (block_step p globals) ≝
  seq_to_block_step_list
  on _l:list (joint_seq ??)
  to list (block_step ??).

definition is_inr : ∀A,B.A + B → bool ≝ λA,B,x.match x with
  [ inl _ ⇒ true
  | inr _ ⇒ false
  ].
definition is_inl : ∀A,B.A + B → bool ≝ λA,B,x.match x with
  [ inl _ ⇒ true
  | inr _ ⇒ false
  ].

definition skip_block ≝ λp,globals,A.
  (list (block_step p globals)) × A.*)

(* move *)
let rec bind_new_P R X (P : X → Prop) (b : bind_new R X) on b : Prop ≝
match b with
[ bret x ⇒ P x
| bnew f ⇒ ∀x.bind_new_P … P (f x)
].

definition BindNewP : ∀R.MonadPred (BindNew R) ≝
λR.mk_MonadPred (BindNew R) (bind_new_P R) ???.
[ #X #P #x #K @K
| #X #Y #Pin #Pout #m elim m -m
  [ #x #H #f #G @G @H
  | #g #IH #H #f #G #x @IH [ @H | @G ]
  ]
| #X #P #Q #H #m elim m -m
  [ #x #Px @H @Px
  | #g #IH #Pg #x @IH @Pg
  ]
]
qed.

definition not_empty : ∀X.list X → Prop ≝ λX,l.match l with [ nil ⇒ False | _ ⇒ True ].

definition split_on_last : ∀X.list X → option ((list X) × X) ≝
λX.foldr …
  (λel,acc.match acc with
    [ None ⇒ Some ? 〈[ ], el〉
    | Some pr ⇒ Some ? 〈el :: \fst pr, \snd pr〉
    ]) (None ?).

lemma split_on_last_inv : ∀X,l.
match split_on_last X l with
[ None ⇒ l = [ ]
| Some pr ⇒ l = \fst pr @ [ \snd pr]
].
#X #l elim l -l normalize nodelta [ // ]
#hd #tl
change with (match split_on_last ? tl in option  with [ _ ⇒ ?]) in
  match (split_on_last ? (hd :: tl)); cases (split_on_last ??)
normalize nodelta [2: * #pref #last ] #EQ >EQ normalize % qed.

lemma split_on_last_hit : ∀X,pref,last.
split_on_last X (pref @ [last]) = Some ? 〈pref, last〉.
#X #pref elim pref -pref normalize [//]
#hd #tl #IH #last >IH % qed.

(* the label input is to accomodate ERTLptr pass
   the postfix is for the case CALL ↦ instrs* ; CALL; instrs* *)
definition step_block ≝
  λp,globals.list (code_point p → joint_seq p globals) ×
             (code_point p → joint_step p globals) ×
             (list (joint_seq p globals)).

definition fin_block ≝ λp,globals.(list (joint_seq p globals))×(joint_fin_step p).

(*definition seq_to_skip_block :
  ∀p,g,A.seq_block p g A → skip_block p g A
 ≝ λp,g,A,b.〈\fst b, \snd b〉.

coercion skip_from_seq_block :
  ∀p,g,A.∀b : seq_block p g A.skip_block p g A ≝
  seq_to_skip_block on _b : seq_block ??? to skip_block ???.*)

definition bind_step_block ≝ λp.λglobals.
  bind_new register (step_block p globals).

unification hint 0 ≔ p, g;
P ≟ step_block p g
(*---------------------------------------*)⊢
bind_step_block p g ≡ bind_new register P.

definition bind_fin_block ≝ λp : stmt_params.λglobals.
  bind_new register (fin_block p globals).

unification hint 0 ≔ p : stmt_params, g;
P ≟ fin_block p g
(*---------------------------------------*)⊢
bind_fin_block p g ≡ bind_new register P.

definition bind_seq_list ≝ λp : stmt_params.λglobals.
  bind_new register (list (joint_seq p globals)).
unification hint 0 ≔ p : stmt_params, g;
S ≟ joint_seq p g,
L ≟ list S
(*---------------------------------------*)⊢
bind_seq_list p g ≡ bind_new register L.

definition add_dummy_variance : ∀X,Y : Type[0].list Y → list (X → Y) ≝ λX,Y.map … (λx.λ_.x).

definition ensure_step_block : ∀p : params.∀globals.
list (joint_seq p globals) → step_block p globals ≝
λp,g,l.
match l return λ_.step_block ?? with
[ nil ⇒ 〈[ ], λ_.NOOP ??, [ ]〉
| cons hd tl ⇒ 〈[ ], λ_.hd, tl〉
].

coercion step_block_from_seq_list nocomposites :
∀p : params.∀g.∀l : list (joint_seq p g).step_block p g ≝
ensure_step_block on _l : list (joint_seq ??) to step_block ??.

(*definition bind_skip_block ≝ λp : stmt_params.λglobals,A.
  bind_new (localsT p) (skip_block p globals A).
unification hint 0 ≔ p : stmt_params, g, A;
B ≟ skip_block p g A, R ≟ localsT p
(*---------------------------------------*)⊢
bind_skip_block p g A ≡ bind_new R B.

definition bind_seq_to_skip_block :
  ∀p,g,A.bind_seq_block p g A → bind_skip_block p g A ≝
  λp,g,A.m_map ? (seq_block p g A) (skip_block p g A)
    (λx.x).

coercion bind_skip_from_seq_block :
  ∀p,g,A.∀b:bind_seq_block p g A.bind_skip_block p g A ≝
  bind_seq_to_skip_block on _b : bind_seq_block ??? to bind_skip_block ???.*)
(*
definition block_classifier ≝
  λp,g.λb : other_block p g.
  seq_or_fin_step_classifier ?? (\snd b).
*)
(*definition seq_block_from_seq_list :
∀p : stmt_params.∀g.list (joint_seq p g) → seq_block p g (joint_step p g) ≝
λp,g,l.let 〈pre,post〉 ≝ split_on_last … (NOOP ??) l in 〈pre, (post : joint_step ??)〉.

definition bind_seq_block_from_bind_seq_list :
  ∀p : stmt_params.∀g.bind_new (localsT p) (list (joint_seq p g)) →
    bind_seq_block p g (joint_step p g) ≝ λp.λg.m_map … (seq_block_from_seq_list …).

definition bind_seq_block_step :
  ∀p,g.bind_seq_block p g (joint_step p g) →
    bind_seq_block p g (joint_step p g) + bind_seq_block p g (joint_fin_step p) ≝
  λp,g.inl ….
coercion bind_seq_block_from_step :
  ∀p,g.∀b:bind_seq_block p g (joint_step p g).
    bind_seq_block p g (joint_step p g) + bind_seq_block p g (joint_fin_step p) ≝
  bind_seq_block_step on _b : bind_seq_block ?? (joint_step ??) to
    bind_seq_block ?? (joint_step ??) + bind_seq_block ?? (joint_fin_step ?).

definition bind_seq_block_fin_step :
  ∀p,g.bind_seq_block p g (joint_fin_step p) →
    bind_seq_block p g (joint_step p g) + bind_seq_block p g (joint_fin_step p) ≝
  λp,g.inr ….
coercion bind_seq_block_from_fin_step :
  ∀p,g.∀b:bind_seq_block p g (joint_fin_step p).
    bind_seq_block p g (joint_step p g) + bind_seq_block p g (joint_fin_step p) ≝
  bind_seq_block_fin_step on _b : bind_seq_block ?? (joint_fin_step ?) to
    bind_seq_block ?? (joint_step ??) + bind_seq_block ?? (joint_fin_step ?).

definition seq_block_bind_seq_block :
  ∀p : stmt_params.∀g,A.seq_block p g A → bind_seq_block p g A ≝ λp,g,A.bret ….
coercion seq_block_to_bind_seq_block :
  ∀p : stmt_params.∀g,A.∀b:seq_block p g A.bind_seq_block p g A ≝
  seq_block_bind_seq_block
  on _b : seq_block ??? to bind_seq_block ???.

definition joint_step_seq_block : ∀p : stmt_params.∀g.joint_step p g → seq_block p g (joint_step p g) ≝
  λp,g,x.〈[ ], x〉.
coercion joint_step_to_seq_block : ∀p : stmt_params.∀g.∀b : joint_step p g.seq_block p g (joint_step p g) ≝
  joint_step_seq_block on _b : joint_step ?? to seq_block ?? (joint_step ??).

definition joint_fin_step_seq_block : ∀p : stmt_params.∀g.joint_fin_step p → seq_block p g (joint_fin_step p) ≝
  λp,g,x.〈[ ], x〉.
coercion joint_fin_step_to_seq_block : ∀p : stmt_params.∀g.∀b : joint_fin_step p.seq_block p g (joint_fin_step p) ≝
  joint_fin_step_seq_block on _b : joint_fin_step ? to seq_block ?? (joint_fin_step ?).

definition seq_list_seq_block :
  ∀p:stmt_params.∀g.list (joint_seq p g) → seq_block p g (joint_step p g) ≝
  λp,g,l.let pr ≝ split_on_last … (NOOP ??) l in 〈\fst pr, \snd pr〉.
coercion seq_list_to_seq_block :
  ∀p:stmt_params.∀g.∀l:list (joint_seq p g).seq_block p g (joint_step p g) ≝
  seq_list_seq_block on _l : list (joint_seq ??) to seq_block ?? (joint_step ??).

definition bind_seq_list_bind_seq_block :
  ∀p:stmt_params.∀g.bind_new (localsT p) (list (joint_seq p g)) → bind_seq_block p g (joint_step p g) ≝
  λp,g.m_map ??? (λx : list (joint_seq ??).(x : seq_block ???)).

coercion bind_seq_list_to_bind_seq_block :
  ∀p:stmt_params.∀g.∀l:bind_new (localsT p) (list (joint_seq p g)).bind_seq_block p g (joint_step p g) ≝
  bind_seq_list_bind_seq_block on _l : bind_new ? (list (joint_seq ??)) to bind_seq_block ?? (joint_step ??).
*)

notation > "x ~❨ B , l ❩~> y 'in' c" with precedence 56 for
  @{'block_in_code $c $x $B $l $y}.
  
notation < "hvbox(x ~❨ B , break l ❩~> y \nbsp 'in' \nbsp break c)" with precedence 56 for
  @{'block_in_code $c $x $B $l $y}.

notation > "x ~❨ B , l , r ❩~> y 'in' c" with precedence 56 for
  @{'bind_block_in_code $c $x $B $l $r $y}.
  
notation < "hvbox(x ~❨ B , break l , break r ❩~> y \nbsp 'in' \nbsp break c)" with precedence 56 for
  @{'bind_block_in_code $c $x $B $l $r $y}.

definition step_in_code ≝
  λp,globals.λc : codeT p globals.λsrc : code_point p.λs : joint_step p globals.
  λdst : code_point p.
  ∃nxt.stmt_at … c src = Some ? (sequential … s nxt) ∧
       point_of_succ … src nxt = dst.

definition fin_step_in_code ≝
  λp,globals.λc : codeT p globals.λsrc : code_point p.λs : joint_fin_step p.
  stmt_at … c src = Some ? (final … s).

let rec seq_list_in_code p globals (c : codeT p globals)
  (src : code_point p) (B : list (joint_seq p globals))
  (l : list (code_point p)) (dst : code_point p) on B : Prop ≝
  match B with
  [ nil ⇒ l = [ ] ∧ src = dst
  | cons hd tl ⇒
    ∃mid,rest.l = src :: rest ∧
    step_in_code …  c src hd mid ∧
    seq_list_in_code … c mid tl rest dst
  ].

(* leaving it out because it can be misleading
interpretation "step list in code" 'block_in_code c x B l y = (step_list_in_code ?? c x B l y).
*)

lemma seq_list_in_code_ne : ∀p,globals.∀c : codeT p globals.∀src,B,l,dst.
  not_empty ? B →
  seq_list_in_code … c src B l dst →
  ∃post.l = src :: post.
#p #globals #c #src * [ #l #dst * ] #hd #tl #l #dst #_ * #mid * #post ** #EQl #_ #_ %{EQl} qed.
(* #b lapply src -src elim b -b [ #src #l #dst * ]
#hd * [2: #hd' #tl ] #IH #src * [1,3: #dst #_ * ]
#mid * [ #dst #_ ** #EQ destruct #_ * |4: #mid' #tl #dst #_ ** #_ #_ * ]
[2: #dst #_ ** #EQ destruct normalize nodelta #_ #_ %{[ ]} % ]
#mid' #rest #dst #_ ** #EQ destruct #H1 #H2
cases (IH … I H2) #post normalize nodelta #EQ destruct %{(mid' :: post)} %
qed.
*)

lemma seq_list_in_code_append :
  ∀p,globals.∀c : codeT p globals.∀src,B1,l1,mid,B2,l2,dst.
  seq_list_in_code … c src B1 l1 mid →
  seq_list_in_code … c mid B2 l2 dst →
  seq_list_in_code … c src (B1@B2) (l1@l2) dst.
#p #globals #c #src #B1 lapply src -src elim B1 -B1
[ #src #l1 #mid #B2 #l2 #dst * #EQ1 #EQ2 destruct #H @H
| #hd #tl #IH #src #l1 #mid #B2 #l2 #dst * #mid' * #rest **
  #EQ destruct #H1 #H2 #H3 %{mid'} %{(rest@l2)} %{(IH … H2 H3)} %{H1} %
]
qed.

lemma seq_list_in_code_append_inv :
  ∀p,globals.∀c : codeT p globals.∀src,B1,B2,l,dst.
  seq_list_in_code … c src (B1@B2) l dst →
  ∃l1,mid,l2.l = l1 @ l2 ∧ seq_list_in_code … c src B1 l1 mid ∧
  seq_list_in_code … c mid B2 l2 dst.
#p #globals #c #src #B1 lapply src -src elim B1 -B1
[ #src #B2 #l #dst #H %{[ ]} %{src} %{l} %{H} % % %
| #hd #tl #IH #src #B2 #l2 #dst * #mid * #rest ** #EQ destruct
  #H1 #H2 elim (IH … H2) #l1 * #mid' * #l2 ** #EQ destruct #G1 #G2
  %{(src::l1)} %{mid'} %{l2} %{G2} %{(refl …)} % [| %[| %{G1} %{H1} % ]]
]
qed.

definition map_eval : ∀X,Y : Type[0].list (X → Y) → X → list Y ≝ λX,Y,l,x.map … (λf.f x) l.

definition step_block_in_code ≝
  λp,g.λc : codeT p g.λsrc.λb : step_block p g.λl,dst.
  let pref ≝ \fst (\fst b) in
  let stp ≝ \snd (\fst b) in
  let post ≝ \snd b in
  ∃l1,mid1,mid2,l2.
  l = l1 @ mid1 :: l2 ∧
  seq_list_in_code ?? c src (map_eval … pref mid1) l1 mid1 ∧
  step_in_code ?? c mid1 (stp mid1) mid2 ∧
  seq_list_in_code ?? c mid2 post l2 dst.

lemma map_compose : ∀X,Y,Z,f,g.∀l : list X.map Y Z f (map X Y g l) = map … (f∘g) l.
#X #Y #Z #f #g #l elim l -l normalize // qed.

lemma map_ext_eq : ∀X,Y,f,g.∀l : list X.(∀x.f x = g x) → map X Y f l = map X Y g l.
#X #Y #f #g #l #H elim l -l normalize // qed.

lemma map_id : ∀X.∀l : list X.map X X (λx.x) l = l.
#X #l elim l -l normalize // qed.

definition fin_block_in_code ≝
  λp,g.λc:codeT p g.λsrc.λB : fin_block p g.λl.λdst : unit.
  ∃pref,mid.l = pref @ [mid] ∧
  seq_list_in_code … c src (\fst B) pref mid ∧ fin_step_in_code … c mid (\snd B).

interpretation "step block in code" 'block_in_code c x B l y = (step_block_in_code ?? c x B l y).
interpretation "fin block in code" 'block_in_code c x B l y = (fin_block_in_code ?? c x B l y).

lemma coerced_step_list_in_code :
∀p : params.∀g,c,src.∀b : list (joint_seq p g).∀l,dst.
src ~❨b, l❩~> dst in c →
match b with
[ nil ⇒ step_in_code … c src (NOOP …) dst
| _ ⇒ seq_list_in_code … c src b l dst
].
#p #g #c #src * [2: #hd #tl]
#l #dst * #pref * #mid1 * #mid2 * #post *** #EQ * #EQ' #EQ''
#stp_in_code [ #tl_in | * #EQ''' #EQ''''] whd
destruct
[ % [| % [| %{tl_in} %{stp_in_code} % ]]
| @stp_in_code
]
qed.

let rec bind_new_instantiates B X
  (x : X) (b : bind_new B X) (l : list B) on b : Prop ≝
  match b with
  [ bret B ⇒
    match l with
    [ nil ⇒ x = B
    | _ ⇒ False
    ]
  | bnew f ⇒
    match l with
    [ nil ⇒ False
    | cons r l' ⇒
      bind_new_instantiates B X x (f r) l'
    ]
  ].

lemma bind_new_instantiates_bind_inv : ∀B,X,Y,y.
  ∀b : bind_new B X.∀f : X → bind_new B Y.∀l.
bind_new_instantiates B Y y (!y ← b; f y) l →
∃x,l1,l2.l = l1@l2 ∧ bind_new_instantiates B X x b l1 ∧
  bind_new_instantiates B Y y (f x) l2.
#B #X #Y #y #b elim b -b
[ #x #f #l >m_return_bind #H %{x} %{[ ]} %{l} %{H} %%
| #g #IH #f * [ * ] #hd #tl whd in ⊢ (%→?); #H
  cases (IH … H) #x * #l1 * #l2 ** #EQ #H1 #H2 %{x} %{(hd::l1)} %{l2}
  %{H2} %{H1} >EQ %
]
qed.

lemma bind_new_instantiates_bind : ∀B,X,Y,x,y.
  ∀b : bind_new B X.∀f : X → bind_new B Y.∀l1,l2.
bind_new_instantiates B X x b l1 →
bind_new_instantiates B Y y (f x) l2 →
bind_new_instantiates B Y y (!y ← b; f y) (l1@l2).
#B #X #Y #x #y #b elim b -b
[ #x' #f * [2: #hd #tl ] #l2 * #H @H
| #g #IH #f * [ #l2 * ] #hd #tl #l2 #H1 #H2 whd @IH assumption
]
qed.

definition bind_step_block_in_code ≝
  λp,g.λc:codeT p g.λsrc.λB : bind_step_block p g.λlbls,regs.λdst.
  ∃b.bind_new_instantiates … b B regs ∧ src ~❨b, lbls❩~> dst in c.

definition bind_fin_block_in_code ≝
  λp,g.λc:codeT p g.λsrc.λB : bind_fin_block p g.λlbls,regs.λdst.
  ∃b.bind_new_instantiates … b B regs ∧ src ~❨b, lbls❩~> dst in c.

interpretation "bound step block in code" 'bind_block_in_code c x B l r y = (bind_step_block_in_code ?? c x B l r y).
interpretation "bound fin block in code" 'bind_block_in_code c x B l r y = (bind_fin_block_in_code ?? c x B l r y).

(* generates ambiguity even if it shouldn't
interpretation "seq block step in code" 'block_in_code c x B l y = (seq_block_step_in_code ?? c x B l y).
interpretation "seq block fin step in code" 'block_in_code c x B l y = (seq_block_fin_step_in_code ?? c x B l y).
*)

(*

definition seq_block_append : 
  ∀p,g.
  ∀b1 : Σb.is_safe_block p g b.
  ∀b2 : seq_block p g.
  seq_block p g ≝ λp,g,b1,b2.
  〈match b1 with
  [ mk_Sig instr prf ⇒
    match \snd instr return λx.bool_to_Prop (is_inl … x) ∧ seq_or_fin_step_classifier … x = ? → ? with
    [ inl i ⇒ λprf.\fst b1 @ i :: \fst b2
    | inr _ ⇒ λprf.⊥
    ] prf
  ],\snd b2〉.
  cases prf #H1 #H2 assumption
  qed.

definition other_block_append : 
  ∀p,g.
  (Σb.block_classifier ?? b = cl_other) →
  other_block p g →
  other_block p g ≝ λp,g,b1,b2.
  〈\fst b1 @ «\snd b1, pi2 … b1» :: \fst b2, \snd b2〉.

definition seq_block_cons : ∀p : stmt_params.∀g.(Σs.step_classifier p g s = cl_other) →
  seq_block p g → seq_block p g ≝
  λp,g,x,b.〈x :: \fst b,\snd b〉.
definition other_block_cons : ∀p,g.
  (Σs.seq_or_fin_step_classifier p g s = cl_other) → other_block p g →
  other_block p g ≝
  λp,g,x,b.〈x :: \fst b,\snd b〉.

interpretation "seq block cons" 'cons x b = (seq_block_cons ?? x b).
interpretation "other block cons" 'vcons x b = (other_block_cons ?? x b).
interpretation "seq block append" 'append b1 b2 = (seq_block_append ?? b1 b2).
interpretation "other block append" 'vappend b1 b2 = (other_block_append ?? b1 b2).

definition step_to_block : ∀p,g.
  seq_or_fin_step p g → seq_block p g ≝ λp,g,s.〈[ ], s〉.

coercion block_from_step : ∀p,g.∀s : seq_or_fin_step p g.
  seq_block p g ≝ step_to_block on _s : seq_or_fin_step ?? to seq_block ??.

definition bind_seq_block_cons :
  ∀p : stmt_params.∀g,is_seq.
  (Σs.step_classifier p g s = cl_other) → bind_seq_block p g is_seq →
  bind_seq_block p g is_seq ≝
  λp,g,is_seq,x.m_map ??? (λb.〈x::\fst b,\snd b〉).

definition bind_other_block_cons :
  ∀p,g.(Σs.seq_or_fin_step_classifier p g s = cl_other) → bind_other_block p g → bind_other_block p g ≝
  λp,g,x.m_map … (other_block_cons … x).

let rec bind_pred_aux B X (P : X → Prop) (c : bind_new B X) on c : Prop ≝
  match c with
  [ bret x ⇒ P x
  | bnew f ⇒ ∀b.bind_pred_aux B X P (f b)
  ].

let rec bind_pred_inj_aux B X (P : X → Prop) (c : bind_new B X) on c :
  bind_pred_aux B X P c → bind_new B (Σx.P x) ≝
  match c return λx.bind_pred_aux B X P x → bind_new B (Σx.P x) with
  [ bret x ⇒ λprf.return «x, prf»
  | bnew f ⇒ λprf.bnew … (λx.bind_pred_inj_aux B X P (f x) (prf x))
  ].

definition bind_pred ≝ λB.
  mk_InjMonadPred (BindNew B)
    (mk_MonadPred ?
      (bind_pred_aux B)
      ???)
    (λX,P,c_sig.bind_pred_inj_aux B X P c_sig (pi2 … c_sig))
    ?.
[ #X #P #Q #H #y elim y [ #x @H | #f #IH #G #b @IH @G]
| #X #Y #Pin #Pout #m elim m [#x | #f #IH] #H #g #G [ @G @H | #b @(IH … G) @H]
| #X #P #x #Px @Px
| #X #P * #m elim m [#x | #f #IH] #H [ % | @bnew_proper #b @IH]
]
qed.

definition bind_seq_block_append :
  ∀p,g,is_seq.(Σb : bind_seq_block p g true.bind_pred ? (λb.step_classifier p g (\snd b) = cl_other) b) →
  bind_seq_block p g is_seq → bind_seq_block p g is_seq ≝
  λp,g,is_seq,b1,b2.
    !«p, prf» ← mp_inject … b1;
    !〈post, last〉 ← b2;
    return 〈\fst p @ «\snd p, prf» :: post, last〉.

definition bind_other_block_append :
  ∀p,g.(Σb : bind_other_block p g.bind_pred ?
    (λx.block_classifier ?? x = cl_other) b) →
  bind_other_block p g → bind_other_block p g ≝
  λp,g,b1.m_bin_op … (other_block_append ??) (mp_inject … b1).

interpretation "bind seq block cons" 'cons x b = (bind_seq_block_cons ??? x b).
interpretation "bind other block cons" 'vcons x b = (bind_other_block_cons ?? x b).
interpretation "bind seq block append" 'append b1 b2 = (bind_seq_block_append ??? b1 b2).
interpretation "bind other block append" 'vappend b1 b2 = (bind_other_block_append ?? b1 b2).

let rec instantiates_to B X
  (b : bind_new B X) (l : list B) (x : X) on b : Prop ≝
  match b with
  [ bret B ⇒
    match l with
    [ nil ⇒ x = B
    | _ ⇒ False
    ]
  | bnew f ⇒
    match l with
    [ nil ⇒ False
    | cons r l' ⇒
      instantiates_to B X (f r) l' x
    ]
  ].

lemma instantiates_to_bind_pred :
  ∀B,X,P,b,l,x.instantiates_to B X b l x → bind_pred B P b → P x.
#B #X #P #b elim b
[ #x * [ #y #EQ >EQ normalize // | #hd #tl #y *]
| #f #IH * [ #y * | #hd #tl normalize #b #H #G @(IH … H) @G ]
]
qed.

lemma seq_block_append_proof_irr :
  ∀p,g,b1,b1',b2.pi1 ?? b1 = pi1 ?? b1' →
    seq_block_append p g b1 b2 = seq_block_append p g b1' b2.
#p #g * #b1 #b1prf * #b1' #b1prf' #b2 #EQ destruct(EQ) %
qed.

lemma other_block_append_proof_irr :
  ∀p,g,b1,b1',b2.pi1 ?? b1 = pi1 ?? b1' →
  other_block_append p g b1 b2 = other_block_append p g b1' b2.
#p #g * #b1 #b1prf * #b1' #b1prf' #b2 #EQ destruct(EQ) %
qed.

(*
lemma is_seq_block_instance_append : ∀p,g,is_seq.
  ∀B1.
  ∀B2 : bind_seq_block p g is_seq.
  ∀l1,l2.
  ∀b1 : Σb.is_safe_block p g b.
  ∀b2 : seq_block p g.
  instantiates_to ? (seq_block p g) B1 l1 (pi1 … b1) →
  instantiates_to ? (seq_block p g) B2 l2 b2 →
  instantiates_to ? (seq_block p g) (B1 @ B2) (l1 @ l2) (b1 @ b2).
#p #g * #B1 elim B1 -B1
[ #B1 | #f1 #IH1 ]
#B1prf whd in B1prf;
#B2 * [2,4: #r1 #l1' ] #l2 #b1 #b2 [1,4: *]
whd in ⊢ (%→?);
[ @IH1
| #EQ destruct(EQ) lapply b2 -b2 lapply l2 -l2 elim B2 -B2
  [ #B2 | #f2 #IH2] * [2,4: #r2 #l2'] #b2 [1,4: *]
  whd in ⊢ (%→?);
  [ @IH2
  | #EQ' whd destruct @seq_block_append_proof_irr %
  ]
]
qed.

lemma is_other_block_instance_append : ∀p,g.
  ∀B1 : Σb.bind_pred ? (λx.block_classifier p g x = cl_other) b.
  ∀B2 : bind_other_block p g.
  ∀l1,l2.
  ∀b1 : Σb.block_classifier p g b = cl_other.
  ∀b2 : other_block p g.
  instantiates_to ? (other_block p g) B1 l1 (pi1 … b1) →
  instantiates_to ? (other_block p g) B2 l2 b2 →
  instantiates_to ? (other_block p g) (B1 @@ B2) (l1 @ l2) (b1 @@ b2).
#p #g * #B1 elim B1 -B1
[ #B1 | #f1 #IH1 ]
#B1prf whd in B1prf;
#B2 * [2,4: #r1 #l1' ] #l2 #b1 #b2 [1,4: *]
whd in ⊢ (%→?);
[ @IH1
| #EQ destruct(EQ) lapply b2 -b2 lapply l2 -l2 elim B2 -B2
  [ #B2 | #f2 #IH2] * [2,4: #r2 #l2'] #b2 [1,4: *]
  whd in ⊢ (%→?);
  [ @IH2
  | #EQ' whd destruct @other_block_append_proof_irr %
  ]
]
qed.

lemma other_fin_step_has_one_label :
  ∀p,g.∀s:(Σs.fin_step_classifier p g s = cl_other).
  match fin_step_labels ?? s with
  [ nil ⇒ False
  | cons _ tl ⇒
    match tl with
    [ nil ⇒ True
    | _ ⇒ False
    ]
  ].
#p #g ** [#lbl || #ext]
normalize
[3: cases (ext_fin_step_flows p ext)
  [* [2: #lbl' * [2: #lbl'' #tl']]] normalize nodelta ]
#EQ destruct %
qed.

definition label_of_other_fin_step : ∀p,g.
  (Σs.fin_step_classifier p g s = cl_other) → label ≝
λp,g,s.
match fin_step_labels p ? s return λx.match x with [ nil ⇒ ? | cons _ tl ⇒ ?] → ? with
[ nil ⇒ Ⓧ
| cons lbl tl ⇒ λ_.lbl
] (other_fin_step_has_one_label p g s).

(*
definition point_seq_transition : ∀p,g.codeT p g →
  code_point p → code_point p → Prop ≝
  λp,g,c,src,dst.
  match stmt_at … c src with
  [ Some stmt ⇒ match stmt with
    [ sequential sq nxt ⇒
      point_of_succ … src nxt = dst
    | final fn ⇒
      match fin_step_labels … fn with
      [ nil ⇒ False
      | cons lbl tl ⇒
        match tl with
        [ nil ⇒ point_of_label … c lbl = Some ? dst
        | _ ⇒ False
        ]
      ]
    ]
  | None ⇒ False
  ].

lemma point_seq_transition_label_of_other_fin_step :
  ∀p,c,src.∀s : (Σs.fin_step_classifier p s = cl_other).∀dst.
  stmt_at ?? c src = Some ? s →
  point_seq_transition p c src dst →
  point_of_label … c (label_of_other_fin_step p s) = Some ? dst.
#p #c #src ** [#lbl || #ext] #EQ1
#dst #EQ2
whd in match point_seq_transition; normalize nodelta
>EQ2 normalize nodelta whd in ⊢ (?→??(????%)?);
[#H @H | * ]
lapply (other_fin_step_has_one_label ? «ext,?»)
cases (fin_step_labels p ? ext) normalize nodelta [*]
#hd * normalize nodelta [2: #_ #_ *] *
#H @H
qed.

lemma point_seq_transition_succ :
  ∀p,c,src.∀s,nxt.∀dst.
  stmt_at ?? c src = Some ? (sequential ?? s nxt) →
  point_seq_transition p c src dst →
  point_of_succ … src nxt = dst.
#p #c #src #s #nxt #dst #EQ
whd in match point_seq_transition; normalize nodelta
>EQ normalize nodelta #H @H
qed.
*)

definition if_other : ∀p,g.∀A : Type[2].seq_or_fin_step p g → A → A → A ≝
  λp,g,A,c.match seq_or_fin_step_classifier p g c with
  [ cl_other ⇒ λx,y.x
  | _ ⇒ λx,y.y
  ].

definition other_step_in_code ≝
  λp,g.
  λc : codeT p g.
  λsrc : code_point p.
  λs : seq_or_fin_step p g.
  match s return λx.if_other p g ? x (code_point p) unit → Prop with
  [ inl s'' ⇒ λdst.∃n.stmt_at … c src = Some ? (sequential … s'' n) ∧ ?
  | inr s'' ⇒ λdst.stmt_at … c src = Some ? (final … s'') ∧ ?
  ].
  [ whd in dst; cases (seq_or_fin_step_classifier ???) in dst;
    normalize nodelta [1,2,3: #_ @True |*: #dst
      @(point_of_succ … src n = dst)]
  | whd in dst;
    lapply dst -dst
    lapply (refl … (seq_or_fin_step_classifier ?? (inr … s'')))
    cases (seq_or_fin_step_classifier ?? (inr … s'')) in ⊢ (???%→%);
    normalize nodelta
    [1,2,3: #_ #_ @True
    |*: #EQ #dst
      @(point_of_label … c (label_of_other_fin_step p g «s'', EQ») = Some ? dst)
    ]
  ]
qed.

definition if_other_sig :
  ∀p,g.∀B,C : Type[0].∀s : Σs.seq_or_fin_step_classifier p g s = cl_other.
  if_other p g ? s B C → B ≝
  λp,g,B,C.?.
  ** #s whd in match (if_other ??????);
  cases (seq_or_fin_step_classifier ???) normalize nodelta #EQ destruct(EQ)
  #x @x
qed.

definition if_other_block_sig :
  ∀p,g.∀B,C : Type[0].∀b : Σb.block_classifier p g b = cl_other.
  if_other p g ? (\snd b) B C → B ≝
  λp,g,B,C.?.
  ** #l #s
  #prf #x @(if_other_sig ???? «s, prf» x)
qed.

coercion other_sig_to_if nocomposites:
  ∀p,g.∀B,C : Type[0].∀s : Σs.seq_or_fin_step_classifier p g s = cl_other.
  ∀x : if_other p g ? s B C.B ≝ if_other_sig
  on _x : if_other ?? Type[0] ??? to ?.

coercion other_block_sig_to_if nocomposites:
  ∀p,g.∀B,C : Type[0].∀s : Σs.block_classifier p g s = cl_other.
  ∀x : if_other p g ? (\snd s) B C.B ≝ if_other_block_sig
  on _x : if_other ?? Type[0] (\snd ?) ?? to ?.

let rec other_list_in_code p g (c : codeT p g)
  src
  (b : list (Σs.seq_or_fin_step_classifier p g s = cl_other))
  dst on b : Prop ≝
  match b with
  [ nil ⇒ src = dst
  | cons hd tl ⇒ ∃mid.
    other_step_in_code p g c src hd mid ∧ other_list_in_code p g c mid tl dst
  ].

notation > "x ~❨ B ❩~> y 'in' c" with precedence 56 for
  @{'block_in_code $c $x $B $y}.
  
notation < "hvbox(x ~❨ B ❩~> y \nbsp 'in' \nbsp break c)" with precedence 56 for
  @{'block_in_code $c $x $B $y}.

interpretation "list in code" 'block_in_code c x B y =
  (other_list_in_code ?? c x B y).

definition other_block_in_code : ∀p,g.codeT p g →
  code_point p → ∀b : other_block p g.
    if_other … (\snd b) (code_point p) unit → Prop ≝
  λp,g,c,src,b,dst.
  ∃mid.src ~❨\fst b❩~> mid in c ∧
  other_step_in_code p g c mid (\snd b) dst.

interpretation "block in code" 'block_in_code c x B y =
  (other_block_in_code ?? c x B y).

lemma other_list_in_code_append : ∀p,g.∀c : codeT p g.
  ∀x.∀b1 : list ?.
  ∀y.∀b2 : list ?.∀z.
  x ~❨b1❩~> y in c→ y ~❨b2❩~> z in c → x ~❨b1@b2❩~> z in c.
#p#g#c#x#b1 lapply x -x
elim b1 [2: ** #hd #hd_prf #tl #IH] #x #y #b2 #z
[3: #EQ normalize in EQ; destruct #H @H]
* #mid * normalize nodelta [ *#n ] #H1 #H2 #H3
whd normalize nodelta %{mid}
%{(IH … H2 H3)}
[ %{n} ] @H1
qed.

lemma other_block_in_code_append : ∀p,g.∀c : codeT p g.∀x.
  ∀B1 : Σb.block_classifier p g b = cl_other.
  ∀y.
  ∀B2 : other_block p g.
  ∀z.
  x ~❨B1❩~> y in c → y ~❨B2❩~> z in c →
  x ~❨B1@@B2❩~> z in c.
#p#g#c #x ** #hd1 *#tl1 #tl1prf
#y * #hd2 #tl2 #z
* #mid1 * #H1 #H2
* #mid2 * #G1 #G2
%{mid2} %{G2}
whd in match (\fst ?);
@(other_list_in_code_append … H1)
%{y} %{H2 G1}
qed.

(*
definition instr_block_in_function :
  ∀p : evaluation_params.∀fn : joint_internal_function (globals p) p.
    code_point p →
    ∀b : bind_other_block p.
    ? → Prop ≝
 λp,fn,src,B,dst.
 ∃vars,B'.
  All ? (In ? (joint_if_locals … fn)) vars ∧
  instantiates_to … B vars B' ∧
  src ~❨B'❩~> dst in joint_if_code … fn.

interpretation "bind block in function" 'block_in_code fn x B y =
  (instr_block_in_function ? fn x B y).

lemma instr_block_in_function_trans :
  ∀p,fn,src.
  ∀B1 : ΣB.bind_pred ? (λb.block_classifier p b = cl_other) B.
  ∀mid.
  ∀B2 : bind_other_block p.
  ∀dst.
  src ~❨B1❩~> Some ? mid in fn →
  mid ~❨B2❩~> dst in fn →
  src ~❨B1@@B2❩~> dst in fn.
#p#fn#src*#B1#B1prf#mid#B2#dst
* #vars1 * #b1 ** #vars1_ok #b1B1 #b1_in
* #vars2 * #b2 ** #vars2_ok #b2B2 #b2_in
%{(vars1@vars2)} %{(«b1,instantiates_to_bind_pred … b1B1 B1prf» @@ b2)}
/4 by All_append, conj, is_other_block_instance_append, other_block_in_code_append/
qed.
*)
*)
*)