KroneckerRules := module()
    option package;
    export Kron, `&x`, Simplify, Expand, Shuffle, Check, VisualCheck, Verify, VisualVerify, NumericEvaluate, Info, SimplifyTrace;
    
    # ---------- Inert notation ----------
    Kron := (A,B) -> 'KroneckerProduct'(A,B):
    `&x` := (A,B) -> 'KroneckerProduct'(A,B):
    
    # ---------- Trace mechanism ----------
    local _trace_steps := []:
    local _AddTrace := proc(rule, from_expr, to_expr)
        _trace_steps := [op(_trace_steps),
            sprintf("%s: **%a**  ->  **%a**", rule, from_expr, to_expr)];
    end proc;
    
    # ---------- Perfect shuffle matrix ----------
    local _PerfectShuffleMatrix := proc(p::posint, r::posint)
        local M, i, j;
        M := Matrix(p*r, p*r, 0);
        for i from 1 to p do
            for j from 1 to r do
                M[(j-1)*p + i, (i-1)*r + j] := 1;
            end do;
        end do;
        return M;
    end proc;
    
    # ---------- Helper: flatten a dot product into a list of operands ----------
    local _FlattenOperands := proc(p)
        local terms, rec;
        terms := [];
        rec := proc(x)
            if type(x, `.`) then
                rec(op(1,x));
                rec(op(2,x));
            else
                terms := [op(terms), x];
            end if;
        end proc;
        rec(p);
        return terms;
    end proc;
    
    # ---------- Distribute dot over sums (recursively, fully) ----------
    local _DistributeDot := proc(expr, with_trace::boolean := false)
        local e, changed, term;
        e := expr;
        changed := true;
        while changed do
            changed := false;
            e := subsindets(e, `.`, proc(p)
                local a,b,terms,res;
                a := op(1,p); b := op(2,p);
                if type(a, `+`) then
                    changed := true;
                    terms := [op(a)];
                    res := `+`(seq(term . b, term in terms));
                    if with_trace then _AddTrace("Dot left-distribution", p, res); end if;
                    return res;
                elif type(b, `+`) then
                    changed := true;
                    terms := [op(b)];
                    res := `+`(seq(a . term, term in terms));
                    if with_trace then _AddTrace("Dot right-distribution", p, res); end if;
                    return res;
                else
                    return p;
                end if;
            end proc);
        end do;
        return e;
    end proc;
    
    # ---------- Combine consecutive Kronecker products in a dot chain (full flatten) ----------
    local _FlattenChain := proc(expr, with_trace::boolean := false)
        local e, changed;
        e := expr;
        changed := true;
        while changed do
            changed := false;
            e := subsindets(e, `.`, proc(p)
                local ops, all_kron, left, right, i, to_pair;
                ops := [op(p)];
                all_kron := true;
                for i in ops do
                    if not type(i, specfunc(KroneckerProduct)) then
                        all_kron := false;
                        break;
                    end if;
                end do;
                if all_kron then
                    left := op(1, ops[1]);
                    right := op(2, ops[1]);
                    for i from 2 to nops(ops) do
                        left := left . op(1, ops[i]);
                        right := right . op(2, ops[i]);
                    end do;
                    to_pair := Kron(left, right);
                    if with_trace then _AddTrace("KRON7 (chain)", p, to_pair); end if;
                    changed := true;
                    return to_pair;
                else
                    return p;
                end if;
            end proc);
        end do;
        return e;
    end proc;
    
    # ---------- Core simplification rules (without trace) ----------
    local ApplyRulesNoTrace := proc(expr)
        local e, e_old, changed;
        local sub, a, b, terms, term, new_sub;
        e := expr;
        changed := true;
        while changed do
            e_old := e;
            e := _DistributeDot(e, false);
            e := _FlattenChain(e, false);
            if e = e_old then changed := false; end if;
        end do;
        
        for sub in indets(e, specfunc(Transpose)) do
            a := op(1,sub);
            if type(a, `+`) then
                terms := [op(a)];
                new_sub := `+`(seq(Transpose(term), term in terms));
                e := subs(sub = new_sub, e);
            end if;
        end do;
        
        for sub in indets(e, specfunc(HermitianTranspose)) do
            a := op(1,sub);
            if type(a, `+`) then
                terms := [op(a)];
                new_sub := `+`(seq(HermitianTranspose(term), term in terms));
                e := subs(sub = new_sub, e);
            end if;
        end do;
        
        for sub in indets(e, specfunc(conjugate)) do
            a := op(1,sub);
            if type(a, `+`) then
                terms := [op(a)];
                new_sub := `+`(seq(conjugate(term), term in terms));
                e := subs(sub = new_sub, e);
            elif type(a, `.`) then
                new_sub := conjugate(op(1,a)) . conjugate(op(2,a));
                e := subs(sub = new_sub, e);
            end if;
        end do;
        
        for sub in indets(e, specfunc(Determinant)) do
            a := op(1,sub);
            if type(a, specfunc(KroneckerProduct)) then
                new_sub := Determinant(op(1,a))^(dim(op(2,a))) * Determinant(op(2,a))^(dim(op(1,a)));
                e := subs(sub = new_sub, e);
            end if;
        end do;
        
        for sub in indets(e, specfunc(KroneckerProduct)) do
            a := op(1,sub); b := op(2,sub);
            new_sub := sub;
            if type(a, `*`) and type(op(1,a), numeric) then
                new_sub := op(1,a) * Kron(op(2,a), b);
            elif type(b, `*`) and type(op(1,b), numeric) then
                new_sub := op(1,b) * Kron(a, op(2,b));
            end if;
            if new_sub <> sub then e := subs(sub = new_sub, e); end if;
        end do;
        
        for sub in indets(e, specfunc(Transpose)) do
            a := op(1,sub);
            if type(a, specfunc(KroneckerProduct)) then
                new_sub := Kron(Transpose(op(1,a)), Transpose(op(2,a)));
                e := subs(sub = new_sub, e);
            end if;
        end do;
        
        for sub in indets(e, specfunc(HermitianTranspose)) do
            a := op(1,sub);
            if type(a, specfunc(KroneckerProduct)) then
                new_sub := Kron(HermitianTranspose(op(1,a)), HermitianTranspose(op(2,a)));
                e := subs(sub = new_sub, e);
            end if;
        end do;
        
        for sub in indets(e, specfunc(conjugate)) do
            a := op(1,sub);
            if type(a, specfunc(KroneckerProduct)) then
                new_sub := Kron(conjugate(op(1,a)), conjugate(op(2,a)));
                e := subs(sub = new_sub, e);
            end if;
        end do;
        
        for sub in indets(e, specfunc(KroneckerProduct)) do
            a := op(1,sub); b := op(2,sub);
            if type(a, specfunc(KroneckerProduct)) then
                new_sub := Kron(op(1,a), Kron(op(2,a), b));
                e := subs(sub = new_sub, e);
            end if;
        end do;
        
        for sub in indets(e, specfunc(KroneckerProduct)) do
            a := op(1,sub); b := op(2,sub);
            if type(a, `+`) then
                terms := [op(a)];
                new_sub := `+`(seq(Kron(term, b), term in terms));
                e := subs(sub = new_sub, e);
            end if;
        end do;
        
        for sub in indets(e, specfunc(KroneckerProduct)) do
            a := op(1,sub); b := op(2,sub);
            if type(b, `+`) then
                terms := [op(b)];
                new_sub := `+`(seq(Kron(a, term), term in terms));
                e := subs(sub = new_sub, e);
            end if;
        end do;
        
        for sub in indets(e, specfunc(Trace)) do
            a := op(1,sub);
            if type(a, specfunc(KroneckerProduct)) then
                new_sub := Trace(op(1,a)) * Trace(op(2,a));
                e := subs(sub = new_sub, e);
            elif type(a, `+`) then
                terms := [op(a)];
                new_sub := `+`(seq(Trace(term), term in terms));
                e := subs(sub = new_sub, e);
            end if;
        end do;
        
        for sub in indets(e, specfunc(MatrixInverse)) do
            a := op(1,sub);
            if type(a, specfunc(KroneckerProduct)) then
                new_sub := Kron(MatrixInverse(op(1,a)), MatrixInverse(op(2,a)));
                e := subs(sub = new_sub, e);
            end if;
        end do;
        
        for sub in indets(e, specfunc(MatrixInverse)) do
            a := op(1,sub);
            if type(a, specfunc(Shuffle)) then
                new_sub := Shuffle(MatrixInverse(op(1,a)), MatrixInverse(op(2,a)));
                e := subs(sub = new_sub, e);
            end if;
        end do;
        
        for sub in indets(e, specfunc(Trace)) do
            a := op(1,sub);
            if type(a, specfunc(Shuffle)) then
                new_sub := Trace(op(1,a)) * Trace(op(2,a));
                e := subs(sub = new_sub, e);
            end if;
        end do;
        
        return e;
    end proc;
    
    # ---------- Same with trace (for SimplifyTrace) ----------
    local ApplyRulesWithTrace := proc(expr)
        local e, e_old, changed;
        local sub, a, b, terms, term, new_sub;
        e := expr;
        changed := true;
        while changed do
            e_old := e;
            e := _DistributeDot(e, true);
            e := _FlattenChain(e, true);
            if e = e_old then changed := false; end if;
        end do;
        
        for sub in indets(e, specfunc(Transpose)) do
            a := op(1,sub);
            if type(a, `+`) then
                terms := [op(a)];
                new_sub := `+`(seq(Transpose(term), term in terms));
                _AddTrace("Transpose over sum", sub, new_sub);
                e := subs(sub = new_sub, e);
            end if;
        end do;
        
        for sub in indets(e, specfunc(HermitianTranspose)) do
            a := op(1,sub);
            if type(a, `+`) then
                terms := [op(a)];
                new_sub := `+`(seq(HermitianTranspose(term), term in terms));
                _AddTrace("HermitianTranspose over sum", sub, new_sub);
                e := subs(sub = new_sub, e);
            end if;
        end do;
        
        for sub in indets(e, specfunc(conjugate)) do
            a := op(1,sub);
            if type(a, `+`) then
                terms := [op(a)];
                new_sub := `+`(seq(conjugate(term), term in terms));
                _AddTrace("conjugate over sum", sub, new_sub);
                e := subs(sub = new_sub, e);
            elif type(a, `.`) then
                new_sub := conjugate(op(1,a)) . conjugate(op(2,a));
                _AddTrace("conjugate over product", sub, new_sub);
                e := subs(sub = new_sub, e);
            end if;
        end do;
        
        for sub in indets(e, specfunc(Determinant)) do
            a := op(1,sub);
            if type(a, specfunc(KroneckerProduct)) then
                new_sub := Determinant(op(1,a))^(dim(op(2,a))) * Determinant(op(2,a))^(dim(op(1,a)));
                _AddTrace("KRON9", sub, new_sub);
                e := subs(sub = new_sub, e);
            end if;
        end do;
        
        for sub in indets(e, specfunc(KroneckerProduct)) do
            a := op(1,sub); b := op(2,sub);
            new_sub := sub;
            if type(a, `*`) and type(op(1,a), numeric) then
                new_sub := op(1,a) * Kron(op(2,a), b);
                _AddTrace("KRON1 (scalar left)", sub, new_sub);
            elif type(b, `*`) and type(op(1,b), numeric) then
                new_sub := op(1,b) * Kron(a, op(2,b));
                _AddTrace("KRON1 (scalar right)", sub, new_sub);
            end if;
            if new_sub <> sub then e := subs(sub = new_sub, e); end if;
        end do;
        
        for sub in indets(e, specfunc(Transpose)) do
            a := op(1,sub);
            if type(a, specfunc(KroneckerProduct)) then
                new_sub := Kron(Transpose(op(1,a)), Transpose(op(2,a)));
                _AddTrace("KRON2", sub, new_sub);
                e := subs(sub = new_sub, e);
            end if;
        end do;
        
        for sub in indets(e, specfunc(HermitianTranspose)) do
            a := op(1,sub);
            if type(a, specfunc(KroneckerProduct)) then
                new_sub := Kron(HermitianTranspose(op(1,a)), HermitianTranspose(op(2,a)));
                _AddTrace("KRON3", sub, new_sub);
                e := subs(sub = new_sub, e);
            end if;
        end do;
        
        for sub in indets(e, specfunc(conjugate)) do
            a := op(1,sub);
            if type(a, specfunc(KroneckerProduct)) then
                new_sub := Kron(conjugate(op(1,a)), conjugate(op(2,a)));
                _AddTrace("conjugate over KroneckerProduct", sub, new_sub);
                e := subs(sub = new_sub, e);
            end if;
        end do;
        
        for sub in indets(e, specfunc(KroneckerProduct)) do
            a := op(1,sub); b := op(2,sub);
            if type(a, specfunc(KroneckerProduct)) then
                new_sub := Kron(op(1,a), Kron(op(2,a), b));
                _AddTrace("KRON4", sub, new_sub);
                e := subs(sub = new_sub, e);
            end if;
        end do;
        
        for sub in indets(e, specfunc(KroneckerProduct)) do
            a := op(1,sub); b := op(2,sub);
            if type(a, `+`) then
                terms := [op(a)];
                new_sub := `+`(seq(Kron(term, b), term in terms));
                _AddTrace("KRON5", sub, new_sub);
                e := subs(sub = new_sub, e);
            end if;
        end do;
        
        for sub in indets(e, specfunc(KroneckerProduct)) do
            a := op(1,sub); b := op(2,sub);
            if type(b, `+`) then
                terms := [op(b)];
                new_sub := `+`(seq(Kron(a, term), term in terms));
                _AddTrace("KRON6", sub, new_sub);
                e := subs(sub = new_sub, e);
            end if;
        end do;
        
        for sub in indets(e, specfunc(Trace)) do
            a := op(1,sub);
            if type(a, specfunc(KroneckerProduct)) then
                new_sub := Trace(op(1,a)) * Trace(op(2,a));
                _AddTrace("KRON8", sub, new_sub);
                e := subs(sub = new_sub, e);
            elif type(a, `+`) then
                terms := [op(a)];
                new_sub := `+`(seq(Trace(term), term in terms));
                _AddTrace("Trace linearity", sub, new_sub);
                e := subs(sub = new_sub, e);
            end if;
        end do;
        
        for sub in indets(e, specfunc(MatrixInverse)) do
            a := op(1,sub);
            if type(a, specfunc(KroneckerProduct)) then
                new_sub := Kron(MatrixInverse(op(1,a)), MatrixInverse(op(2,a)));
                _AddTrace("KRON10", sub, new_sub);
                e := subs(sub = new_sub, e);
            end if;
        end do;
        
        for sub in indets(e, specfunc(MatrixInverse)) do
            a := op(1,sub);
            if type(a, specfunc(Shuffle)) then
                new_sub := Shuffle(MatrixInverse(op(1,a)), MatrixInverse(op(2,a)));
                _AddTrace("Inverse of Shuffle", sub, new_sub);
                e := subs(sub = new_sub, e);
            end if;
        end do;
        
        for sub in indets(e, specfunc(Trace)) do
            a := op(1,sub);
            if type(a, specfunc(Shuffle)) then
                new_sub := Trace(op(1,a)) * Trace(op(2,a));
                _AddTrace("Trace(Shuffle)", sub, new_sub);
                e := subs(sub = new_sub, e);
            end if;
        end do;
        
        return e;
    end proc;
    
    # ---------- Simplify (without trace) ----------
    Simplify := proc(expr, maxiter::integer := 50)
        local new, old, i;
        new := expr;
        for i from 1 to maxiter do
            old := new;
            new := ApplyRulesNoTrace(new);
            if new = old then break; end if;
        end do;
        return new;
    end proc;
    
    # ---------- SimplifyTrace (step-by-step output) ----------
    SimplifyTrace := proc(expr, maxiter::integer := 50)
        local new, old, i, step, cnt;
        _trace_steps := [];
        new := expr;
        printf("Start expression: %a\n\n", new);
        for i from 1 to maxiter do
            old := new;
            _trace_steps := [];
            new := ApplyRulesWithTrace(new);
            if new = old then
                printf("No more rules apply after iteration %d.\n", i-1);
                break;
            end if;
            printf("Iteration %d:\n", i);
            cnt := 0;
            for step in _trace_steps do
                cnt := cnt + 1;
                printf("  %d. %s\n", cnt, step);
            end do;
            printf("Result after iteration %d: %a\n\n", i, new);
        end do;
        printf("--- Final simplified expression ---\n");
        return new;
    end proc;
    
    # ---------- Expand ----------
    local _ExpandKron := proc(expr)
        local a, b, new_a, new_b, term;
        if not type(expr, specfunc(KroneckerProduct)) then
            return expr;
        end if;
        a := op(1, expr);
        b := op(2, expr);
        new_a := _ExpandKron(a);
        new_b := _ExpandKron(b);
        if type(new_a, `+`) then
            return `+`(seq(Kron(term, new_b), term in [op(new_a)]));
        elif type(new_b, `+`) then
            return `+`(seq(Kron(new_a, term), term in [op(new_b)]));
        else
            return Kron(new_a, new_b);
        end if;
    end proc;
    
    Expand := proc(expr)
        local e, new_e;
        e := expr;
        while true do
            new_e := subsindets(e, specfunc(KroneckerProduct), _ExpandKron);
            if new_e = e then break; end if;
            e := new_e;
        end do;
        return e;
    end proc;
    
    # ---------- Shuffle (KRON11) ----------
    Shuffle := proc(A, B)
        local p, q, r, s, S1, S2;
        try
            p := LinearAlgebra:-RowDimension(A);
            q := LinearAlgebra:-ColumnDimension(A);
            r := LinearAlgebra:-RowDimension(B);
            s := LinearAlgebra:-ColumnDimension(B);
        catch:
            return 'Shuffle'(A, B);
        end try;
        if not (type(p, posint) and type(q, posint) and type(r, posint) and type(s, posint)) then
            return 'Shuffle'(A, B);
        end if;
        S1 := _PerfectShuffleMatrix(p, r);
        S2 := _PerfectShuffleMatrix(q, s);
        return S1 . LinearAlgebra:-KroneckerProduct(A, B) . LinearAlgebra:-Transpose(S2);
    end proc;
    
    # ---------- Activation for numeric evaluation (used in Check/VisualCheck) ----------
    local Activate := proc(expr)
        subs([ 'Determinant' = LinearAlgebra:-Determinant,
               'Trace' = LinearAlgebra:-Trace,
               'MatrixInverse' = LinearAlgebra:-MatrixInverse,
               'KroneckerProduct' = LinearAlgebra:-KroneckerProduct,
               'Transpose' = LinearAlgebra:-Transpose,
               'HermitianTranspose' = LinearAlgebra:-HermitianTranspose,
               dim = LinearAlgebra:-RowDimension
             ], expr);
    end proc;
    
    # ---------- Check (symbolic test, square matrices only) ----------
    Check := proc(expr::uneval, {n::posint := 2, samples::posint := 5})
        printf("Check: Use inert Kron(A,B) or A &x B. Avoid direct LinearAlgebra:-KroneckerProduct.\n");
        printf("   Note: This command assumes all matrix symbols are SQUARE. For non‑square matrices, use Simplify and Verify.\n");
        local e, syms, i, subs_set, gen, simp_expr, val_simp, orig_expr, val_orig, ok;
        e := eval(expr);
        syms := indets(e, symbol) minus {Kron, Determinant, Trace, MatrixInverse, KroneckerProduct,
                                         'Kron', 'Determinant', 'Trace', 'MatrixInverse', 'KroneckerProduct',
                                         dim, 'dim', `&x`, Shuffle, Expand, Transpose, HermitianTranspose, conjugate};
        if syms = {} then error "No symbolic variables found. Use symbolic names like A,B,..."; end if;
        gen := proc() local M; do M := LinearAlgebra:-RandomMatrix(n, n, generator=-5..5); until LinearAlgebra:-Determinant(M) <> 0; return M; end proc;
        ok := true;
        for i from 1 to samples do
            subs_set := map(s -> s = gen(), syms);
            simp_expr := Simplify(e);
            simp_expr := Activate(simp_expr);
            val_simp := eval(eval(simp_expr, subs_set));
            orig_expr := subs('Kron' = KroneckerProduct, `&x` = KroneckerProduct, e);
            orig_expr := Activate(orig_expr);
            val_orig := eval(eval(orig_expr, subs_set));
            if type(val_orig, Matrix) and val_simp = 0 then
                if LinearAlgebra:-Norm(val_orig, infinity) = 0 then
                    # ok
                else
                    ok := false; break;
                end if;
            elif type(val_simp, Matrix) and val_orig = 0 then
                if LinearAlgebra:-Norm(val_simp, infinity) = 0 then
                    # ok
                else
                    ok := false; break;
                end if;
            elif type(val_orig, Matrix) and type(val_simp, Matrix) then
                if not LinearAlgebra:-Equal(val_orig, val_simp) then ok := false; break; end if;
            else
                if evalb(val_orig <> val_simp) then ok := false; break; end if;
            end if;
        end do;
        return ok;
    end proc;
    
    # ---------- VisualCheck (same as Check, shows matrices) ----------
    VisualCheck := proc(expr::uneval, {n::posint := 2})
        printf("VisualCheck: Use inert Kron(A,B) or A &x B. Avoid direct LinearAlgebra:-KroneckerProduct.\n");
        printf("   Note: This command assumes all matrix symbols are SQUARE. For non‑square matrices, use Simplify and Verify.\n");
        local e, syms, subs_set, gen, simp_expr, val_simp, orig_expr, val_orig, a;
        e := eval(expr);
        syms := indets(e, symbol) minus {Kron, Determinant, Trace, MatrixInverse, KroneckerProduct,
                                         'Kron', 'Determinant', 'Trace', 'MatrixInverse', 'KroneckerProduct',
                                         dim, 'dim', `&x`, Shuffle, Expand, Transpose, HermitianTranspose, conjugate};
        if syms = {} then error "No symbolic variables found. Use symbolic names like A,B,..."; end if;
        gen := proc() local M; do M := LinearAlgebra:-RandomMatrix(n, n, generator=-5..5); until LinearAlgebra:-Determinant(M) <> 0; return M; end proc;
        subs_set := map(s -> s = gen(), syms);
        printf("Substitutions used:\n");
        for a in subs_set do printf("%a\n", a); end do;
        printf("\n");
        
        simp_expr := Simplify(e);
        simp_expr := Activate(simp_expr);
        val_simp := eval(eval(simp_expr, subs_set));
        
        orig_expr := subs('Kron' = KroneckerProduct, `&x` = KroneckerProduct, e);
        orig_expr := Activate(orig_expr);
        val_orig := eval(eval(orig_expr, subs_set));
        
        printf("========== Direct calculation ==========\n");
        print(val_orig);
        printf("\n========== Simplified calculation (via module) ==========\n");
        print(val_simp);
        printf("\n");
        
        local identical;
        if type(val_orig, Matrix) and val_simp = 0 then
            identical := LinearAlgebra:-Norm(val_orig, infinity) = 0;
        elif type(val_simp, Matrix) and val_orig = 0 then
            identical := LinearAlgebra:-Norm(val_simp, infinity) = 0;
        elif type(val_orig, Matrix) and type(val_simp, Matrix) then
            identical := LinearAlgebra:-Equal(val_orig, val_simp);
        else
            identical := evalb(val_orig = val_simp);
        end if;
        
        if identical then
            printf("Results are IDENTICAL (visual proof).\n");
            return true;
        else
            printf("ERROR – results differ.\n");
            return false;
        end if;
    end proc;
    
    # ---------- Verify (for concrete matrices, returns boolean) ----------
    Verify := proc(expr)
        printf("Verify: Comparing Simplify(expr) with direct evaluation using concrete matrices.\n");
        local simp, direct;
        simp := Simplify(expr);
        simp := Activate(simp);
        direct := subs(`&x` = LinearAlgebra:-KroneckerProduct, expr);
        direct := Activate(direct);
        if type(simp, Matrix) and type(direct, Matrix) then
            if LinearAlgebra:-Equal(simp, direct) then
                printf("Results are IDENTICAL.\n");
                return true;
            else
                printf("ERROR – results differ.\n");
                return false;
            end if;
        else
            if evalb(simp = direct) then
                printf("Results are IDENTICAL.\n");
                return true;
            else
                printf("ERROR – results differ.\n");
                return false;
            end if;
        end if;
    end proc;
    
    # ---------- VisualVerify (visual comparison for concrete matrices) ----------
    VisualVerify := proc(expr)
        printf("VisualVerify: Comparing Simplify(expr) with direct evaluation using concrete matrices.\n");
        local simp, direct;
        simp := Simplify(expr);
        simp := Activate(simp);
        direct := subs(`&x` = LinearAlgebra:-KroneckerProduct, expr);
        direct := Activate(direct);
        printf("========== Direct calculation ==========\n");
        print(direct);
        printf("\n========== Simplified calculation (via module) ==========\n");
        print(simp);
        printf("\n");
        if type(simp, Matrix) and type(direct, Matrix) then
            if LinearAlgebra:-Equal(simp, direct) then
                printf("Results are IDENTICAL (visual proof).\n");
                return true;
            else
                printf("ERROR – results differ.\n");
                return false;
            end if;
        else
            if evalb(simp = direct) then
                printf("Results are IDENTICAL (visual proof).\n");
                return true;
            else
                printf("ERROR – results differ.\n");
                return false;
            end if;
        end if;
    end proc;
    
    # ---------- NumericEvaluate (robust numeric evaluation for concrete matrices) ----------
    NumericEvaluate := proc(expr)
        local e;
        # First replace all non-Kronecker functions that can appear inside KroneckerProduct arguments
        # Order: innermost first
        e := expr;
        e := subsindets(e, specfunc(MatrixInverse), m -> LinearAlgebra:-MatrixInverse(op(m)));
        e := subsindets(e, specfunc(Transpose), t -> LinearAlgebra:-Transpose(op(t)));
        e := subsindets(e, specfunc(HermitianTranspose), h -> LinearAlgebra:-HermitianTranspose(op(h)));
        e := subsindets(e, specfunc(conjugate), c -> conjugate(op(c)));
        e := subsindets(e, specfunc(Shuffle), s -> LinearAlgebra:-KroneckerProduct(op(2,s), op(1,s)));
        e := subsindets(e, specfunc(dim), d -> LinearAlgebra:-RowDimension(op(d)));
        e := subsindets(e, specfunc(KroneckerProduct), k -> LinearAlgebra:-KroneckerProduct(op(k)));
        e := subsindets(e, specfunc(Determinant), d -> LinearAlgebra:-Determinant(op(d)));
        e := subsindets(e, specfunc(Trace), t -> LinearAlgebra:-Trace(op(t)));
        # Now evaluate the whole expression (including sums)
        return eval(e);
    end proc;
    
    # ---------- Info (educational) ----------
    Info := proc()
        printf("================================================================\n");
        printf("KRONECKERRULES PACKAGE - COMPREHENSIVE EDUCATIONAL GUIDE\n");
        printf("================================================================\n\n");
        printf("1. PURPOSE\n");
        printf("   Simplify symbolic expressions involving Kronecker products,\n");
        printf("   matrix multiplication, determinants, traces, inverses, transposes,\n");
        printf("   conjugates, and shuffling.\n\n");
        printf("2. INERT NOTATION (to avoid immediate evaluation)\n");
        printf("   Use either:   Kron(A,B)   or   A &x B\n");
        printf("   Both produce an inert 'KroneckerProduct(A,B)' that the module can simplify.\n");
        printf("   Avoid using LinearAlgebra:-KroneckerProduct directly in expressions\n");
        printf("   you want to simplify symbolically.\n\n");
        printf("3. MAIN COMMANDS\n");
        printf("   Simplify(expr)  - apply all algebraic rules (KRON1‑KRON11) iteratively.\n");
        printf("   Expand(expr)    - fully distribute Kronecker products over sums.\n");
        printf("   Shuffle(A,B)    - implement perfect shuffle: B ⊗ A = S (A⊗B) S^T.\n");
        printf("   Check(expr)     - test symbolic identities using random square matrices.\n");
        printf("   VisualCheck(expr) - same as Check but shows random matrices and results.\n");
        printf("   Verify(expr)    - compare Simplify(expr) with direct evaluation for user‑supplied concrete matrices.\n");
        printf("   VisualVerify(expr) - same as Verify but displays both results (for any matrix dimensions).\n");
        printf("   NumericEvaluate(expr) - directly evaluate concrete expressions to numeric matrices (robust).\n");
        printf("   SimplifyTrace(expr) - same as Simplify but prints each applied rule (with markers).\n");
        printf("   Info()          - this help text.\n\n");
        printf("4. CHECK AND VISUALCHECK – VERIFY SYMBOLIC IDENTITIES (SQUARE MATRICES ONLY)\n");
        printf("   These commands generate random square matrices (size n x n) for each symbol.\n");
        printf("   They are useful for testing algebraic identities, but assume all matrices are square.\n");
        printf("   For non‑square matrices, use Simplify and test with your own concrete matrices via Verify or VisualVerify.\n\n");
        printf("5. NUMERIC EVALUATION (FOR CONCRETE MATRICES)\n");
        printf("   NumericEvaluate(expr) is the recommended way to compute a numeric result from an expression\n");
        printf("   built with &x, Shuffle, Transpose, etc., after you have assigned concrete matrices.\n");
        printf("   It replaces all inert functions in the correct order and returns the final matrix or scalar.\n\n");
        printf("6. VERIFY AND VISUALVERIFY – FOR CONCRETE MATRICES (ANY DIMENSIONS)\n");
        printf("   Verify(expr) returns true/false after comparing simplified and direct results.\n");
        printf("   VisualVerify(expr) shows the direct and simplified results for visual inspection.\n");
        printf("   Use these after you have defined your matrices (e.g., A := Matrix(...)).\n\n");
        printf("7. SIMPLIFYTRACE – EDUCATIONAL STEP-BY-STEP OUTPUT\n");
        printf("   SimplifyTrace(expr) works like Simplify but prints each rule application.\n");
        printf("   The changing subexpression is marked with ** **. Iterations and steps are numbered.\n");
        printf("   This helps you understand how the simplification proceeds.\n\n");
        printf("8. MATRIX DIMENSION REQUIREMENTS\n");
        printf("   - Kronecker product works for any matrices (even non‑square).\n");
        printf("   - Ordinary multiplication (A . B), determinant, trace, inverse require square matrices.\n");
        printf("   - For determinant rule, dim(X) returns the row dimension (assumed square).\n");
        printf("   - Shuffle requires known numeric dimensions; otherwise returns inert.\n\n");
        printf("9. LIST OF SIMPLIFICATION RULES (KRON1‑KRON11)\n");
        printf("   KRON1: (cA)⊗B = A⊗(cB) = c (A⊗B)\n");
        printf("   KRON2: (A⊗B)^T = A^T ⊗ B^T\n");
        printf("   KRON3: (A⊗B)^H = A^H ⊗ B^H\n");
        printf("   KRON4: (A⊗B)⊗C = A⊗(B⊗C)\n");
        printf("   KRON5: (A+B)⊗C = A⊗C + B⊗C\n");
        printf("   KRON6: A⊗(B+C) = A⊗B + A⊗C\n");
        printf("   KRON7: (A⊗B)(C⊗D) = (A·C)⊗(B·D)\n");
        printf("   KRON8: trace(A⊗B) = trace(A)·trace(B), linear over sums\n");
        printf("   KRON9: det(A⊗B) = det(A)^(dim B) · det(B)^(dim A)\n");
        printf("   KRON10: (A⊗B)^(-1) = A^(-1) ⊗ B^(-1)\n");
        printf("   KRON11: Shuffle(A,B) = S (A⊗B) S^T gives B⊗A.\n");
        printf("   Extra: transpose, conjugate, Hermitian transpose distribute over sums and products.\n\n");
        printf("10. NOTES AND LIMITATIONS\n");
        printf("   - No automatic dimension checking; ensure compatibility.\n");
        printf("   - Shuffle requires known dimensions.\n");
        printf("   - Avoid using LinearAlgebra:-KroneckerProduct directly.\n");
        printf("   - Check and VisualCheck are only for symbolic square matrices; for concrete or non‑square, use NumericEvaluate or Verify/VisualVerify.\n");
        printf("================================================================\n");
    end proc;
    
end module:

with(KroneckerRules):
Info();
   
MAIN COMMANDS
   Simplify(expr)  - apply all algebraic rules (KRON1‑KRON11) iteratively.
   Expand(expr)    - fully distribute Kronecker products over sums.
   Shuffle(A,B)    - implement perfect shuffle: B ⊗ A = S (A⊗B) S^T.
   Check(expr)     - test symbolic identities using random square matrices.
   VisualCheck(expr) - same as Check but shows random matrices and results.
   Verify(expr)    - compare Simplify(expr) with direct evaluation for user‑supplied concrete matrices.
   VisualVerify(expr) - same as Verify but displays both results (for any matrix dimensions).
   NumericEvaluate(expr) - directly evaluate concrete expressions to numeric matrices (robust).
   SimplifyTrace(expr) - same as Simplify but prints each applied rule (with markers).
   Info()          - this help text.

#restart;
with(KroneckerRules):

# Symbolic matrices (square)
A := 'A': B := 'B': C := 'C': X := 'X': Y := 'Y': Z := 'Z':

# Complex expression involving several rules
expr := (2*A &x B) . (C &x X)                    # KRON1 (scalar) + KRON7
        + (A+B) &x (C+X)                         # KRON5 and KRON6 (distribution)
        - Transpose( (A &x B) )                  # KRON2
        + conjugate( (A &x B) . (C &x X) )       # conjugate of product
        + Determinant( A &x B )                  # KRON9 ######################## vierkant
        - Trace( Shuffle(A, B) )                 # KRON11 + KRON8################# vierkant
        + MatrixInverse( A &x B ) . (C &x X);    # KRON10 + KRON7 ############### vierkant

# Simplify using all rules
simp := Simplify(expr);

# Display the simplified symbolic expression
simp;
expr := 2 (KroneckerProduct(A, B)) . (KroneckerProduct(C, X))

   + KroneckerProduct(A + B, C + X)

   - Transpose(KroneckerProduct(A, B))

     ___________________________________________________
   + (KroneckerProduct(A, B)) . (KroneckerProduct(C, X))

   + Determinant(KroneckerProduct(A, B)) - Trace(Shuffle(A, B)) + 

  (MatrixInverse(KroneckerProduct(A, B))) . (KroneckerProduct(C, X

  ))


simp := 2 KroneckerProduct(A . C, B . X) + KroneckerProduct(A, C)

   + KroneckerProduct(A, X) + KroneckerProduct(B, C)

   + KroneckerProduct(B, X)

   - KroneckerProduct(Transpose(A), Transpose(B))

                     //_\   /_\  /_\   /_\\
   + KroneckerProduct\\A/ . \C/, \B/ . \X//

                   dim(B)               dim(A)
   + Determinant(A)       Determinant(B)      

   - Trace(A) Trace(B)

   + KroneckerProduct((MatrixInverse(A)) . C, 

  (MatrixInverse(B)) . X)


   2 KroneckerProduct(A . C, B . X) + KroneckerProduct(A, C)

      + KroneckerProduct(A, X) + KroneckerProduct(B, C)

      + KroneckerProduct(B, X)

      - KroneckerProduct(Transpose(A), Transpose(B))

                        //_\   /_\  /_\   /_\\
      + KroneckerProduct\\A/ . \C/, \B/ . \X//

                      dim(B)               dim(A)
      + Determinant(A)       Determinant(B)      

      - Trace(A) Trace(B)

      + KroneckerProduct((MatrixInverse(A)) . C, 

     (MatrixInverse(B)) . X)



# Assign concrete matrices (all 2x2, invertible)
A := Matrix([[1,2],[3,4]]);
B := Matrix([[5,6],[7,8]]);
C := Matrix([[1,0],[0,1]]);
X := Matrix([[2,0],[0,2]]);

# Use the simplified expression or the original one
result := NumericEvaluate(simp);  # or NumericEvaluate(expr)
#print(result);
                               [1  2]
                          A := [    ]
                               [3  4]

                               [5  6]
                          B := [    ]
                               [7  8]

                               [1  0]
                          C := [    ]
                               [0  1]

                               [2  0]
                          X := [    ]
                               [0  2]

                           [10   17   61   57 ]
                           [                  ]
                           [22   19   73   91 ]
                           [                  ]
                 result := [98   103  91   113]
                           [                  ]
                           [249  301  281  299]
                           [---  ---  ---  ---]
                           [ 2    2    2    2 ]

Tests , check command and visual command
Focus on square matrix for module code ( determinant, ... ),,and module can probably also chanced for random matrix dimensions.

 

Download testen_module_kronecker_mprimes_23-4-2026.mw

use : KroneckerRules := module()

Gives command : Kron( ) and command Simpfly( )
example : 

Simplify(Kron(3*A, B))); KRON1 – Schaalvermenigvuldiging:

-------------------------------------------------------------------------------------------------------
 

Yes, that is precisely the intention. With these ten calculation rules (KRON1 to KRON10) you can symbolically simplify or manipulate any arbitrary Kronecker product expression, without having to write out the huge matrices explicitly. The KroneckerRules module applies these rules automatically, allowing you to focus on the structure of your formulas.

What can you do with it?

• Symbolic derivations of formulas involving Kronecker products, for example in control theory, signal processing, or quantum mechanics.

• Automatic rewriting of expressions such as (A+B)⊗C into A⊗C + B⊗C, or det(A⊗B) into det(A)^dim(B) * det(B)^dim(A) (Zehfuss).

• Verification of identities without numbers – ideal for education and research.

• Later numerical evaluation by substituting the symbols with concrete matrices.

Summary
The module provides the algebraic machinery you need to work with Kronecker products as if they were ordinary variables. Without these rules, you would have to perform every operation by hand, which becomes infeasible for larger matrices. So yes: with these calculation rules you can perform any desired matrix operation involving Kronecker products.

To be tested :-)

Download Zehfuss_-kronecker_determinant__mprimes_22-4-2026.mw

@Harry Garst 

It doesn't seem necessary to use two variables.

These are two ways of describing the same variable..

Apparently, the expression hold for all n =1.. 2
proof by induction further

kronecker_-Zehfuss_mprimes_20-4-2026.mw

Of all closed surfaces in space with a fixed surface area A​, which surface encloses the largest volume?

But there are two variables in the functional here.

a good example to to do by hand...

Via differential geometry 
 

via vectorcalculus 
bewijs_cirkel_gesloten_cvorm_viavectot_calculus_mprimes_20-4-2026.mw

Download bewijs_cirkel_gesloten_cvorm_via_diff_geom_mprimes_20-4-2026.mw

@Rouben Rostamian  
 

Thanks for your educational code , yes, this circle is a  interesting example . I had actually seen another derivation involving the curvature of the circle, but that leads into vector calculus and differential geometry, which require more advanced knowledge.

It’s also instructive to see different approaches to solving the problem.

Of all closed curves with a fixed length L, which curve encloses the largest area?

Download pde_vrgstuk_Can_pdsolve_handle_a_procedure_in_the_pde_mprimesDEFA_18-4-2026.mw

@Rouben Rostamian  
Thanks for the code! It showed me a different way to use the Euler-Lagrange differential equation. for his first derivative 

@Rouben Rostamian  

Amzing work too,

However, I get the impression that it’s not quite as straightforward as in dharr’s code.

Take this as an example:

To further simplify, observe that the fraction (diff(y(x), x))/sqrt(1+(diff(y(x), x))^2) takes values between -1 and 1, and therefore

we may take it to be the sine of some angle theta, as in sin(theta) = (diff(y(x), x))/sqrt(1+(diff(y(x), x))^2), `and`(-(1/2)*Pi <= theta, theta <= (1/2)*Pi), is this trivial ?

@dharr 
Amazing work. 
One thing about the fence length : 100 m  ?

The calculation is a bit messy, and I wasn't quite sure which direction to take, but that has changed.

Are isoperimetric solution curves always convex?, yes it seems

It boils down to deriving the differential equation, which appears to be nonlinear.

So I'm going to revise the calculation above.