(* (c) Copyright 2006-2016 Microsoft Corporation and Inria.                  *)
(* Distributed under the terms of CeCILL-B.                                  *)
From mathcomp
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(* The seq type is the ssreflect type for sequences; it is an alias for the   *)
(* standard Coq list type. The ssreflect library equips it with many          *)
(* operations, as well as eqType and predType (and, later, choiceType)        *)
(* structures. The operations are geared towards reflection: they generally   *)
(* expect and provide boolean predicates, e.g., the membership predicate      *)
(* expects an eqType. To avoid any confusion we do not Import the Coq List    *)
(* module.                                                                    *)
(*   As there is no true subtyping in Coq, we don't use a type for non-empty  *)
(* sequences; rather, we pass explicitly the head and tail of the sequence.   *)
(*   The empty sequence is especially bothersome for subscripting, since it   *)
(* forces us to pass a default value. This default value can often be hidden  *)
(* by a notation.                                                             *)
(*   Here is the list of seq operations:                                      *)
(*  ** Constructors:                                                          *)
(*                        seq T == the type of sequences of items of type T.  *)
(*                       bitseq == seq bool.                                  *)
(*             [::], nil, Nil T == the empty sequence (of type T).            *)
(* x :: s, cons x s, Cons T x s == the sequence x followed by s (of type T).  *)
(*                       [:: x] == the singleton sequence.                    *)
(*           [:: x_0; ...; x_n] == the explicit sequence of the x_i.          *)
(*       [:: x_0, ..., x_n & s] == the sequence of the x_i, followed by s.    *)
(*                    rcons s x == the sequence s, followed by x.             *)
(*  All of the above, except rcons, can be used in patterns. We define a view *)
(* lastP and an induction principle last_ind that can be used to decompose    *)
(* or traverse a sequence in a right to left order. The view lemma lastP has  *)
(* a dependent family type, so the ssreflect tactic case/lastP: p => [|p' x]  *)
(* will generate two subgoals in which p has been replaced by [::] and by     *)
(* rcons p' x, respectively.                                                  *)
(*  ** Factories:                                                             *)
(*             nseq n x == a sequence of n x's.                               *)
(*          ncons n x s == a sequence of n x's, followed by s.                *)
(* seqn n x_0 ... x_n-1 == the sequence of the x_i; can be partially applied. *)
(*             iota m n == the sequence m, m + 1, ..., m + n - 1.             *)
(*            mkseq f n == the sequence f 0, f 1, ..., f (n - 1).             *)
(*  ** Sequential access:                                                     *)
(*      head x0 s == the head (zero'th item) of s if s is non-empty, else x0. *)
(*        ohead s == None if s is empty, else Some x when the head of s is x. *)
(*       behead s == s minus its head, i.e., s' if s = x :: s', else [::].    *)
(*       last x s == the last element of x :: s (which is non-empty).         *)
(*     belast x s == x :: s minus its last item.                              *)
(*  ** Dimensions:                                                            *)
(*         size s == the number of items (length) in s.                       *)
(*       shape ss == the sequence of sizes of the items of the sequence of    *)
(*                   sequences ss.                                            *)
(*  ** Random access:                                                         *)
(*         nth x0 s i == the item i of s (numbered from 0), or x0 if s does   *)
(*                       not have at least i+1 items (i.e., size x <= i)      *)
(*               s`_i == standard notation for nth x0 s i for a default x0,   *)
(*                       e.g., 0 for rings.                                   *)
(*   set_nth x0 s i y == s where item i has been changed to y; if s does not  *)
(*                       have an item i, it is first padded with copies of x0 *)
(*                       to size i+1.                                         *)
(*       incr_nth s i == the nat sequence s with item i incremented (s is     *)
(*                       first padded with 0's to size i+1, if needed).       *)
(*  ** Predicates:                                                            *)
(*          nilp s == s is [::].                                              *)
(*                 := (size s == 0).                                          *)
(*         x \in s == x appears in s (this requires an eqType for T).         *)
(*       index x s == the first index at which x appears in s, or size s if   *)
(*                    x \notin s.                                             *)
(*         has p s == the (applicative, boolean) predicate p holds for some   *)
(*                    item in s.                                              *)
(*         all p s == p holds for all items in s.                             *)
(*        find p s == the index of the first item in s for which p holds,     *)
(*                    or size s if no such item is found.                     *)
(*       count p s == the number of items of s for which p holds.             *)
(*   count_mem x s == the number of times x occurs in s := count (pred1 x) s. *)
(*      constant s == all items in s are identical (trivial if s = [::])      *)
(*          uniq s == all the items in s are pairwise different.              *)
(*    subseq s1 s2 == s1 is a subsequence of s2, i.e., s1 = mask m s2 for     *)
(*                    some m : bitseq (see below).                            *)
(*   perm_eq s1 s2 == s2 is a permutation of s1, i.e., s1 and s2 have the     *)
(*                    items (with the same repetitions), but possibly in a    *)
(*                    different order.                                        *)
(*  perm_eql s1 s2 <-> s1 and s2 behave identically on the left of perm_eq.   *)
(*  perm_eqr s1 s2 <-> s1 and s2 behave identically on the right of perm_eq.  *)
(*    These left/right transitive versions of perm_eq make it easier to chain *)
(* a sequence of equivalences.                                                *)
(*  ** Filtering:                                                             *)
(*           filter p s == the subsequence of s consisting of all the items   *)
(*                         for which the (boolean) predicate p holds.         *)
(*              rem x s == the subsequence of s, where the first occurrence   *)
(*                         of x has been removed (compare filter (predC1 x) s *)
(*                         where ALL occurrences of x are removed).           *)
(*              undup s == the subsequence of s containing only the first     *)
(*                         occurrence of each item in s, i.e., s with all     *)
(*                         duplicates removed.                                *)
(*             mask m s == the subsequence of s selected by m : bitseq, with  *)
(*                         item i of s selected by bit i in m (extra items or *)
(*                         bits are ignored.                                  *)
(*  ** Surgery:                                                               *)
(* s1 ++ s2, cat s1 s2 == the concatenation of s1 and s2.                     *)
(*            take n s == the sequence containing only the first n items of s *)
(*                        (or all of s if size s <= n).                       *)
(*            drop n s == s minus its first n items ([::] if size s <= n)     *)
(*             rot n s == s rotated left n times (or s if size s <= n).       *)
(*                     := drop n s ++ take n s                                *)
(*            rotr n s == s rotated right n times (or s if size s <= n).      *)
(*               rev s == the (linear time) reversal of s.                    *)
(*        catrev s1 s2 == the reversal of s1 followed by s2 (this is the      *)
(*                        recursive form of rev).                             *)
(*  ** Iterators: for s == [:: x_1, ..., x_n], t == [:: y_1, ..., y_m],       *)
(*        map f s == the sequence [:: f x_1, ..., f x_n].                     *)
(* allpairs f s t == the sequence of all the f x y, with x and y drawn from   *)
(*                  s and t, respectively, in row-major order.                *)
(*               := [:: f x_1 y_1; ...; f x_1 y_m; f x_2 y_1; ...; f x_n y_m] *)
(*      pmap pf s == the sequence [:: y_i1, ..., y_ik] where i1 < ... < ik,   *)
(*                   pf x_i = Some y_i, and pf x_j = None iff j is not in     *)
(*                   {i1, ..., ik}.                                           *)
(*   foldr f a s == the right fold of s by f (i.e., the natural iterator).    *)
(*               := f x_1 (f x_2 ... (f x_n a))                               *)
(*        sumn s == x_1 + (x_2 + ... + (x_n + 0)) (when s : seq nat).         *)
(*   foldl f a s == the left fold of s by f.                                  *)
(*               := f (f ... (f a x_1) ... x_n-1) x_n                         *)
(*   scanl f a s == the sequence of partial accumulators of foldl f a s.      *)
(*               := [:: f a x_1; ...; foldl f a s]                            *)
(* pairmap f a s == the sequence of f applied to consecutive items in a :: s. *)
(*               := [:: f a x_1; f x_1 x_2; ...; f x_n-1 x_n]                 *)
(*       zip s t == itemwise pairing of s and t (dropping any extra items).   *)
(*               := [:: (x_1, y_1); ...; (x_mn, y_mn)] with mn = minn n m.    *)
(*      unzip1 s == [:: (x_1).1; ...; (x_n).1] when s : seq (S * T).          *)
(*      unzip2 s == [:: (x_1).2; ...; (x_n).2] when s : seq (S * T).          *)
(*     flatten s == x_1 ++ ... ++ x_n ++ [::] when s : seq (seq T).           *)
(*   reshape r s == s reshaped into a sequence of sequences whose sizes are   *)
(*                  given by r (truncating if s is too long or too short).    *)
(*               := [:: [:: x_1; ...; x_r1];                                  *)
(*                      [:: x_(r1 + 1); ...; x_(r0 + r1)];                    *)
(*                      ...;                                                  *)
(*                      [:: x_(r1 + ... + r(k-1) + 1); ...; x_(r0 + ... rk)]] *)
(* flatten_index sh r c == the index, in flatten ss, of the item of indexes   *)
(*                  (r, c) in any sequence of sequences ss of shape sh        *)
(*               := sh_1 + sh_2 + ... + sh_r + c                              *)
(* reshape_index sh i == the index, in reshape sh s, of the sequence          *)
(*                  containing the i-th item of s.                            *)
(* reshape_offset sh i == the offset, in the (reshape_index sh i)-th          *)
(*                  sequence of reshape sh s of the i-th item of s            *)
(*  ** Notation for manifest comprehensions:                                  *)
(*         [seq x <- s | C] := filter (fun x => C) s.                         *)
(*         [seq E | x <- s] := map (fun x => E) s.                            *)
(* [seq E | x <- s, y <- t] := allpairs (fun x y => E) s t.                   *)
(*   [seq x <- s | C1 & C2] := [seq x <- s | C1 && C2].                       *)
(*     [seq E | x <- s & C] := [seq E | x <- [seq x | C]].                    *)
(*  --> The above allow optional type casts on the eigenvariables, as in      *)
(*  [seq x : T <- s | C] or [seq E | x : T <- s, y : U <- t]. The cast may be *)
(*  needed as type inference considers E or C before s.                       *)
(*   We are quite systematic in providing lemmas to rewrite any composition   *)
(* of two operations. "rev", whose simplifications are not natural, is        *)
(* protected with nosimpl.                                                    *)
(*  ** The following are equivalent:                                          *)
(*  [<-> P0; P1; ..; Pn] == P0, P1, ..., Pn are all equivalent                *)
(*                       := P0 -> P1 -> ... -> Pn -> P0                       *)
(*  if  T : [<-> P0; P1; ..; Pn]  is such an equivalence, and i, j are in nat *)
(*  then T i j is a proof of the equivalence Pi <-> Pj between Pi and Pj      *)
(*  when i (resp j) is out of bound, Pi (resp Pj) defaults to P0              *)
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Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Delimit Scope seq_scope with SEQ.
Open Scope seq_scope.
(* Inductive seq (T : Type) : Type := Nil | Cons of T & seq T. *)
Notation seq := list.
Prenex Implicits cons.
Notation Cons T := (@cons T) (only parsing).
Notation Nil T := (@nil T) (only parsing).
Bind Scope seq_scope with list.
Arguments cons _%type _ _%SEQ.
(* As :: and ++ are (improperly) declared in Init.datatypes, we only rebind   *)
(* them here.                                                                 *)
Infix "::" := cons : seq_scope.
(* GG - this triggers a camlp4 warning, as if this Notation had been Reserved *)
Notation "[ :: ]" := nil (at level 0, format "[ :: ]") : seq_scope.
Notation "[ :: x1 ]" := (x1 :: [::])
(at level 0, format "[ ::  x1 ]") : seq_scope.
Notation "[ :: x & s ]" := (x :: s) (at level 0, only parsing) : seq_scope.
Notation "[ :: x1 , x2 , .. , xn & s ]" := (x1 :: x2 :: .. (xn :: s) ..)
(at level 0, format
"'[hv' [ :: '['  x1 , '/'  x2 , '/'  .. , '/'  xn ']' '/ '  &  s ] ']'"
) : seq_scope.
Notation "[ :: x1 ; x2 ; .. ; xn ]" := (x1 :: x2 :: .. [:: xn] ..)
(at level 0, format "[ :: '['  x1 ; '/'  x2 ; '/'  .. ; '/'  xn ']' ]"
) : seq_scope.
Section Sequences.
Variable n0 : nat.  (* numerical parameter for take, drop et al *)
Variable T : Type.  (* must come before the implicit Type     *)
Variable x0 : T.    (* default for head/nth *)
Implicit Types x y z : T.
Implicit Types m n : nat.
Implicit Type s : seq T.
Fixpoint size s := if s is _ :: s' then (size s').+1 else 0.

Lemma size0nil s : size s = 0 -> s = [::]. Proof. by case: s. Qed.
Definition nilp s := size s == 0.