(* (c) Copyright 2006-2016 Microsoft Corporation and Inria.                  *)
(* Distributed under the terms of CeCILL-B.                                  *)
From mathcomp
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(*    The basic theory of paths over an eqType; this file is essentially a    *)
(* complement to seq.v. Paths are non-empty sequences that obey a progression *)
(* relation. They are passed around in three parts: the head and tail of the  *)
(* sequence, and a proof of (boolean) predicate asserting the progression.    *)
(* This "exploded" view is rarely embarrassing, as the first two parameters   *)
(* are usually inferred from the type of the third; on the contrary, it saves *)
(* the hassle of constantly constructing and destructing a dependent record.  *)
(*    We define similarly cycles, for which we allow the empty sequence,      *)
(* which represents a non-rooted empty cycle; by contrast, the "empty" path   *)
(* from a point x is the one-item sequence containing only x.                 *)
(*   We allow duplicates; uniqueness, if desired (as is the case for several  *)
(* geometric constructions), must be asserted separately. We do provide       *)
(* shorthand, but only for cycles, because the equational properties of       *)
(* "path" and "uniq" are unfortunately  incompatible (esp. wrt "cat").        *)
(*    We define notations for the common cases of function paths, where the   *)
(* progress relation is actually a function. In detail:                       *)
(*   path e x p == x :: p is an e-path [:: x_0; x_1; ... ; x_n], i.e., we     *)
(*                 e x_i x_{i+1} for all i < n. The path x :: p starts at x   *)
(*                 and ends at last x p.                                      *)
(*  fpath f x p == x :: p is an f-path, where f is a function, i.e., p is of  *)
(*                 the form [:: f x; f (f x); ...]. This is just a notation   *)
(*                 for path (frel f) x p.                                     *)
(*   sorted e s == s is an e-sorted sequence: either s = [::], or s = x :: p  *)
(*                 is an e-path (this is oten used with e = leq or ltn).      *)
(*    cycle e c == c is an e-cycle: either c = [::], or c = x :: p with       *)
(*                 x :: (rcons p x) an e-path.                                *)
(*   fcycle f c == c is an f-cycle, for a function f.                         *)
(* traject f x n == the f-path of size n starting at x                        *)
(*              := [:: x; f x; ...; iter n.-1 f x]                            *)
(* looping f x n == the f-paths of size greater than n starting at x loop     *)
(*                 back, or, equivalently, traject f x n contains all         *)
(*                 iterates of f at x.                                        *)
(* merge e s1 s2 == the e-sorted merge of sequences s1 and s2: this is always *)
(*                 a permutation of s1 ++ s2, and is e-sorted when s1 and s2  *)
(*                 are and e is total.                                        *)
(*     sort e s == a permutation of the sequence s, that is e-sorted when e   *)
(*                 is total (computed by a merge sort with the merge function *)
(*                 above).                                                    *)
(*   mem2 s x y == x, then y occur in the sequence (path) s; this is          *)
(*                 non-strict: mem2 s x x = (x \in s).                        *)
(*     next c x == the successor of the first occurrence of x in the sequence *)
(*                 c (viewed as a cycle), or x if x \notin c.                 *)
(*     prev c x == the predecessor of the first occurrence of x in the        *)
(*                 sequence c (viewed as a cycle), or x if x \notin c.        *)
(*    arc c x y == the sub-arc of the sequece c (viewed as a cycle) starting  *)
(*                 at the first occurrence of x in c, and ending just before  *)
(*                 the next ocurrence of y (in cycle order); arc c x y        *)
(*                 returns an unspecified sub-arc of c if x and y do not both *)
(*                 occur in c.                                                *)
(*  ucycle e c <-> ucycleb e c (ucycle e c is a Coercion target of type Prop) *)
(* ufcycle f c <-> c is a simple f-cycle, for a function f.                   *)
(*  shorten x p == the tail a duplicate-free subpath of x :: p with the same  *)
(*                 endpoints (x and last x p), obtained by removing all loops *)
(*                 from x :: p.                                               *)
(* rel_base e e' h b <-> the function h is a functor from relation e to       *)
(*                 relation e', EXCEPT at points whose image under h satisfy  *)
(*                 the "base" predicate b:                                    *)
(*                    e' (h x) (h y) = e x y UNLESS b (h x) holds             *)
(*                 This is the statement of the side condition of the path    *)
(*                 functorial mapping lemma map_path.                         *)
(* fun_base f f' h b <-> the function h is a functor from function f to f',   *)
(*                 except at the preimage of predicate b under h.             *)
(* We also provide three segmenting dependently-typed lemmas (splitP, splitPl *)
(* and splitPr) whose elimination split a path x0 :: p at an internal point x *)
(* as follows:                                                                *)
(*  - splitP applies when x \in p; it replaces p with (rcons p1 x ++ p2), so  *)
(*    that x appears explicitly at the end of the left part. The elimination  *)
(*    of splitP will also simultaneously replace take (index x p) with p1 and *)
(*    drop (index x p).+1 p with p2.                                          *)
(*  - splitPl applies when x \in x0 :: p; it replaces p with p1 ++ p2 and     *)
(*    simulaneously generates an equation x = last x0 p.                      *)
(*  - splitPr applies when x \in p; it replaces p with (p1 ++ x :: p2), so x  *)
(*    appears explicitly at the start of the right part.                      *)
(* The parts p1 and p2 are computed using index/take/drop in all cases, but   *)
(* only splitP attemps to subsitute the explicit values. The substitution of  *)
(* p can be deferred using the dependent equation generation feature of       *)
(* ssreflect, e.g.: case/splitPr def_p: {1}p / x_in_p => [p1 p2] generates    *)
(* the equation p = p1 ++ p2 instead of performing the substitution outright. *)
(*   Similarly, eliminating the loop removal lemma shortenP simultaneously    *)
(* replaces shorten e x p with a fresh constant p', and last x p with         *)
(* last x p'.                                                                 *)
(*   Note that although all "path" functions actually operate on the          *)
(* underlying sequence, we provide a series of lemmas that define their       *)
(* interaction with thepath and cycle predicates, e.g., the cat_path equation *)
(* can be used to split the path predicate after splitting the underlying     *)
(* sequence.                                                                  *)
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Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Section Paths.
Variables (n0 : nat) (T : Type).
Section Path.
Variables (x0_cycle : T) (e : rel T).
Fixpoint path x (p : seq T) :=
if p is y :: p' then e x y && path y p' else true.

Lemma cycle_path p : cycle p = path (last x0_cycle p) p.