ENH: Simplify solving logic

This is a combination of multiple WIP commits:

- Semantic group names - G, H
- Simplify fan<>()
- Template int deduction
- Remove _recontext_gens
- Move g_gens logic into caller
- Move subgroup logic into caller
- Move recontext logic into coller
- Move _generators, _recontext_gens to anonymous namespace
- Simplify hull, tile

tc/include/tc/core.hpp

@@ -92,6 +92,7 @@ namespace tc {
     private:
         size_t _rank;
         std::vector<size_t> _mults;
+        std::vector<size_t> _gens;
 
     public:
         Group(Group const &) = default;
@@ -108,7 +109,11 @@ namespace tc {
 
         [[nodiscard]] size_t rank() const;
 
-        [[nodiscard]] Group sub(std::vector<size_t> const &idxs) const;
+        [[nodiscard]] std::vector<size_t> gens() const;
+
+        [[nodiscard]] Group sub(std::vector<size_t> const &gens) const;
+
+        [[nodiscard]] std::vector<Group> subs(size_t rank) const;
 
         [[nodiscard]] Cosets solve(std::vector<size_t> const &idxs = {}, size_t bound = SIZE_MAX) const;
     };

tc/src/group.cpp

@@ -1,9 +1,12 @@
 #include <tc/core.hpp>
 
 #include <cassert>
+#include <numeric>
 
 namespace tc {
-    Group::Group(size_t rank) : _rank(rank), _mults(_rank * _rank, 2) {
+    Group::Group(size_t rank) : _rank(rank), _mults(_rank * _rank, 2), _gens(_rank) {
+        std::iota(_gens.begin(), _gens.end(), 0);
+
         for (int idx = 0; idx < rank; ++idx) {
             set(idx, idx, 1);
         }
@@ -28,15 +31,42 @@ namespace tc {
         return _rank;
     }
 
-    [[nodiscard]] Group Group::sub(std::vector<size_t> const &idxs) const {
-        Group res(idxs.size());
+    [[nodiscard]] std::vector<size_t> Group::gens() const {
+        return _gens;
+    }
 
-        for (int i = 0; i < idxs.size(); ++i) {
-            for (int j = i; j < idxs.size(); ++j) {
-                res.set(i, j, get(idxs[i], idxs[j]));
+    [[nodiscard]] Group Group::sub(std::vector<size_t> const &gens) const {
+        Group res(gens.size());
+        res._gens = gens;
+
+        for (int i = 0; i < gens.size(); ++i) {
+            for (int j = i; j < gens.size(); ++j) {
+                res.set(i, j, get(gens[i], gens[j]));
             }
         }
 
         return res;
     }
+
+    [[nodiscard]] std::vector<Group> Group::subs(size_t rank) const {
+        std::vector<bool> mask(_rank, false);
+        std::fill(mask.begin(), mask.begin() + rank, true);
+
+        std::vector<size_t> sub_gens(rank);
+
+        std::vector<Group> res;
+
+        do {
+            for (int i = 0, j = 0; i < _rank; ++i) {
+                if (mask[i]) sub_gens[j++] = i;
+            }
+            res.push_back(sub(sub_gens));
+        } while (std::next_permutation(
+            mask.begin(),
+            mask.end(),
+            std::greater<>()
+        ));
+
+        return res;
+    }
 }