perm.bend source
perm.bend on the hub · documented module
# bend-mathlib/perm.bend: step-list permutations over bendlib-kernel-list, plus sort value defs and proofs.# Resolves the unpublished kernel by its real content hash (devlib); see packages/bendlib-kernel-list/README.md.import Baseimport 0xb5c8145e53a6a127d611f45f602666ec/list.bend as Kdef internal_shift(+s: List<&2, Nat>) -> List<&2, Nat>: match s: case Nil{}: Nil{} case +i <> rest: (1n+i) <> internal_shift(rest)def internal_apply_app(-A: Data, +s1: List<&2, Nat>, +s2: List<&2, Nat>, +xs: List<&2, A>) -> {K.apply(A, List.append(&2, Nat, s1, s2), xs) == K.apply(A, s2, K.apply(A, s1, xs)) : List<&2, A>}: match s1: case Nil{}: {==} case +i <> rest: internal_apply_app(A, rest, s2, K.swap_at(A, i, xs))def internal_apply_shift(-A: Data, +s: List<&2, Nat>, +x: A, +xs: List<&2, A>) -> {K.apply(A, internal_shift(s), x <> xs) == x <> K.apply(A, s, xs) : List<&2, A>}: match s: case Nil{}: {==} case +i <> rest: internal_apply_shift(A, rest, x, K.swap_at(A, i, xs))# Permutation is reflexive.def perm_refl(-A: Data, -xs: List<&2, A>) -> K.perm(A, xs, xs): (Nil{}, {==})# The empty list is a permutation of itself.def perm_nil(-A: Data) -> K.perm(A, Nil{}, Nil{}): (Nil{}, {==})# Swapping the two leading elements is a permutation.def perm_swap(-A: Data, -x: A, -y: A, -t: List<&2, A>) -> K.perm(A, x <> y <> t, y <> x <> t): (0n <> Nil{}, {==})def internal_trans_eq(-A: Data, +s1: List<&2, Nat>, +s2: List<&2, Nat>, +xs: List<&2, A>, -ys: List<&2, A>, -zs: List<&2, A>, e1: K.perm_steps(A, s1, xs, ys), e2: K.perm_steps(A, s2, ys, zs)) -> K.perm_steps(A, List.append(&2, Nat, s1, s2), xs, zs): %Equal.sym(List<&2, A>, K.apply(A, List.append(&2, Nat, s1, s2), xs), K.apply(A, s2, K.apply(A, s1, xs)), internal_apply_app(A, s1, s2, xs)) : {_ == zs : List<&2, A>} %Equal.sym(List<&2, A>, K.apply(A, s1, xs), ys, e1) : {K.apply(A, s2, _) == zs : List<&2, A>} e2# Permutation is transitive.def perm_trans(-A: Data, +xs: List<&2, A>, -ys: List<&2, A>, -zs: List<&2, A>, h1: K.perm(A, xs, ys), h2: K.perm(A, ys, zs)) -> K.perm(A, xs, zs): (s1, e1) = h1 (s2, e2) = h2 +s1 = s1 +s2 = s2 (List.append(&2, Nat, s1, s2), internal_trans_eq(A, s1, s2, xs, ys, zs, e1, e2))def internal_cons_eq(-A: Data, +s: List<&2, Nat>, +x: A, +xs: List<&2, A>, -ys: List<&2, A>, e: K.perm_steps(A, s, xs, ys)) -> K.perm_steps(A, internal_shift(s), x <> xs, x <> ys): %Equal.sym(List<&2, A>, K.apply(A, internal_shift(s), x <> xs), x <> K.apply(A, s, xs), internal_apply_shift(A, s, x, xs)) : {_ == x <> ys : List<&2, A>} %Equal.sym(List<&2, A>, K.apply(A, s, xs), ys, e) : {x <> _ == x <> ys : List<&2, A>} {==}# Prepending the same element preserves permutation.def perm_cons(-A: Data, +x: A, +xs: List<&2, A>, -ys: List<&2, A>, h: K.perm(A, xs, ys)) -> K.perm(A, x <> xs, x <> ys): (s, e) = h +s = s (internal_shift(s), internal_cons_eq(A, s, x, xs, ys, e))# A Perm hypothesis in existential form is reusable.def perm_dup(-A: Data, -xs: List<&2, A>, -ys: List<&2, A>, +h: K.perm(A, xs, ys)) -> K.perm(A, xs, ys) & K.perm(A, xs, ys): (h, h)# Insertion into a list, generic comparator (structural, Base-only branching via Bool.pick).def insert_by(~A: Data, ~le: A -> A -> Bool, +x: A, xs: List<&2, A>) -> List<&2, A>: match xs: case Nil{}: x <> Nil{} case +y <> +t: Bool.pick(List<&2, A>, le(x, y), x <> y <> t, y <> insert_by(~A, ~le, x, t))def isort_by(~A: Data, ~le: A -> A -> Bool, xs: List<&2, A>) -> List<&2, A>: match xs: case Nil{}: Nil{} case +x <> t: insert_by(~A, ~le, x, isort_by(~A, ~le, t))def internal_ins_pick(-A: Data, b: Bool, +x: A, +y: A, +t: List<&2, A>, -r: List<&2, A>, ih: K.perm(A, x <> t, r)) -> K.perm(A, x <> y <> t, Bool.pick(List<&2, A>, b, x <> y <> t, y <> r)): match b: case True{}: perm_refl(A, x <> y <> t) case False{}: perm_trans(A, x <> y <> t, y <> x <> t, y <> r, perm_swap(A, x, y, t), perm_cons(A, y, x <> t, r, ih))# Inserting x into xs permutes x <> xs.def insert_by_perm(~A: Data, ~le: A -> A -> Bool, +x: A, +xs: List<&2, A>) -> K.perm(A, x <> xs, insert_by(~A, ~le, x, xs)): match xs: case Nil{}: perm_refl(A, x <> Nil{}) case +y <> t: internal_ins_pick(A, le(x, y), x, y, t, insert_by(~A, ~le, x, t), insert_by_perm(~A, ~le, x, t))# Insertion sort permutes its input.def isort_by_perm(~A: Data, ~le: A -> A -> Bool, +xs: List<&2, A>) -> K.perm(A, xs, isort_by(~A, ~le, xs)): match xs: case Nil{}: perm_refl(A, Nil{}) case +x <> t: perm_trans(A, x <> t, x <> isort_by(~A, ~le, t), isort_by(~A, ~le, x <> t), perm_cons(A, x, t, isort_by(~A, ~le, t), isort_by_perm(~A, ~le, t)), insert_by_perm(~A, ~le, x, isort_by(~A, ~le, t)))# Closed instance: instantiate at Nat and at String.def isort_by_perm_nat(+xs: List<&2, Nat>) -> K.perm(Nat, xs, isort_by(~Nat, ~Nat.is_le, xs)): isort_by_perm(~Nat, ~Nat.is_le, xs)def internal_append_nil(-A: Data, +ys: List<&2, A>) -> {List.append(&2, A, ys, Nil{}) == ys : List<&2, A>}: match ys: case Nil{}: {==} case +y <> t: %Equal.sym(List<&2, A>, List.append(&2, A, t, Nil{}), t, internal_append_nil(A, t)) : {y <> _ == y <> t : List<&2, A>} {==}# Appending Nil is a permutation (steps: none).def perm_append_nil(-A: Data, +xs: List<&2, A>) -> K.perm(A, List.append(&2, A, xs, Nil{}), xs): (Nil{}, internal_append_nil(A, xs))# Moving an element from the middle to the front is a permutation.def perm_move(-A: Data, +xs: List<&2, A>, +y: A, +t: List<&2, A>) -> K.perm(A, List.append(&2, A, xs, y <> t), y <> List.append(&2, A, xs, t)): match xs: case Nil{}: perm_refl(A, y <> t) case +x <> r: perm_trans(A, x <> List.append(&2, A, r, y <> t), x <> y <> List.append(&2, A, r, t), y <> x <> List.append(&2, A, r, t), perm_cons(A, x, List.append(&2, A, r, y <> t), y <> List.append(&2, A, r, t), perm_move(A, r, y, t)), perm_swap(A, x, y, List.append(&2, A, r, t)))def merge_by(~A: Data, ~le: A -> A -> Bool, xs: List<&2, A>, ys: List<&2, A>) -> List<&2, A>: match xs ys: case Nil{} _: ys case +x <> +xt Nil{}: x <> xt case +x <> +xt +y <> +yt: Bool.pick(List<&2, A>, le(x, y), x <> merge_by(~A, ~le, xt, y <> yt), y <> merge_by(~A, ~le, x <> xt, yt))def internal_merge_pick(-A: Data, b: Bool, +x: A, +xt: List<&2, A>, +y: A, +yt: List<&2, A>, -m1: List<&2, A>, -m2: List<&2, A>, ih1: K.perm(A, List.append(&2, A, xt, y <> yt), m1), ih2: K.perm(A, x <> List.append(&2, A, xt, yt), m2)) -> K.perm(A, x <> List.append(&2, A, xt, y <> yt), Bool.pick(List<&2, A>, b, x <> m1, y <> m2)): match b: case True{}: perm_cons(A, x, List.append(&2, A, xt, y <> yt), m1, ih1) case False{}: perm_trans(A, x <> List.append(&2, A, xt, y <> yt), y <> x <> List.append(&2, A, xt, yt), y <> m2, perm_move(A, x <> xt, y, yt), perm_cons(A, y, x <> List.append(&2, A, xt, yt), m2, ih2))# Merging permutes the concatenation.def merge_by_perm(~A: Data, ~le: A -> A -> Bool, +xs: List<&2, A>, +ys: List<&2, A>) -> K.perm(A, List.append(&2, A, xs, ys), merge_by(~A, ~le, xs, ys)): match xs ys: case Nil{} _: perm_refl(A, ys) case +x <> +xt Nil{}: perm_cons(A, x, List.append(&2, A, xt, Nil{}), xt, perm_append_nil(A, xt)) case +x <> +xt +y <> +yt: internal_merge_pick(A, le(x, y), x, xt, y, yt, merge_by(~A, ~le, xt, y <> yt), merge_by(~A, ~le, x <> xt, yt), merge_by_perm(~A, ~le, xt, y <> yt), merge_by_perm(~A, ~le, x <> xt, yt))def merge_by_perm_nat(+xs: List<&2, Nat>, +ys: List<&2, Nat>) -> K.perm(Nat, List.append(&2, Nat, xs, ys), merge_by(~Nat, ~Nat.is_le, xs, ys)): merge_by_perm(~Nat, ~Nat.is_le, xs, ys)# Moving the front element into the middle is a permutation.def perm_move_rev(-A: Data, +xs: List<&2, A>, +y: A, +t: List<&2, A>) -> K.perm(A, y <> List.append(&2, A, xs, t), List.append(&2, A, xs, y <> t)): match xs: case Nil{}: perm_refl(A, y <> t) case +x <> r: perm_trans(A, y <> x <> List.append(&2, A, r, t), x <> y <> List.append(&2, A, r, t), x <> List.append(&2, A, r, y <> t), perm_swap(A, y, x, List.append(&2, A, r, t)), perm_cons(A, x, y <> List.append(&2, A, r, t), List.append(&2, A, r, y <> t), perm_move_rev(A, r, y, t)))# Appending commutes up to permutation (derived without symmetry).def perm_append_comm(-A: Data, +xs: List<&2, A>, +ys: List<&2, A>) -> K.perm(A, List.append(&2, A, xs, ys), List.append(&2, A, ys, xs)): match xs: case Nil{}: (Nil{}, Equal.sym(List<&2, A>, List.append(&2, A, ys, Nil{}), ys, internal_append_nil(A, ys))) case +x <> r: perm_trans(A, x <> List.append(&2, A, r, ys), x <> List.append(&2, A, ys, r), List.append(&2, A, ys, x <> r), perm_cons(A, x, List.append(&2, A, r, ys), List.append(&2, A, ys, r), perm_append_comm(A, r, ys)), perm_move_rev(A, ys, x, r))# Permuting the right operand of an append.def perm_append_right(-A: Data, +xs: List<&2, A>, +ys: List<&2, A>, -zs: List<&2, A>, h: K.perm(A, ys, zs)) -> K.perm(A, List.append(&2, A, xs, ys), List.append(&2, A, xs, zs)): match xs: case Nil{}: h case +x <> r: perm_cons(A, x, List.append(&2, A, r, ys), List.append(&2, A, r, zs), perm_append_right(A, r, ys, zs, h))# Permuting both operands of an append.def perm_append(-A: Data, +xs: List<&2, A>, +xs2: List<&2, A>, +ys: List<&2, A>, -ys2: List<&2, A>, p: K.perm(A, xs, xs2), q: K.perm(A, ys, ys2)) -> K.perm(A, List.append(&2, A, xs, ys), List.append(&2, A, xs2, ys2)): perm_trans(A, List.append(&2, A, xs, ys), List.append(&2, A, ys, xs), List.append(&2, A, xs2, ys2), perm_append_comm(A, xs, ys), perm_trans(A, List.append(&2, A, ys, xs), List.append(&2, A, ys, xs2), List.append(&2, A, xs2, ys2), perm_append_right(A, ys, xs, xs2, p), perm_trans(A, List.append(&2, A, ys, xs2), List.append(&2, A, xs2, ys), List.append(&2, A, xs2, ys2), perm_append_comm(A, ys, xs2), perm_append_right(A, xs2, ys, ys2, q))))def evens(-A: Data, xs: List<&2, A>) -> List<&2, A>: match xs: case Nil{}: Nil{} case +x <> r: match r: case Nil{}: x <> Nil{} case +y <> t: x <> evens(A, t)def odds(-A: Data, xs: List<&2, A>) -> List<&2, A>: match xs: case Nil{}: Nil{} case +x <> r: match r: case Nil{}: Nil{} case +y <> t: y <> odds(A, t)# Splitting into evens and odds is a permutation.def split_perm(-A: Data, +xs: List<&2, A>) -> K.perm(A, xs, List.append(&2, A, evens(A, xs), odds(A, xs))): match xs: case Nil{}: perm_refl(A, Nil{}) case +x <> r: match r: case Nil{}: perm_refl(A, x <> Nil{}) case +y <> t: perm_cons(A, x, y <> t, List.append(&2, A, evens(A, t), y <> odds(A, t)), perm_trans(A, y <> t, y <> List.append(&2, A, evens(A, t), odds(A, t)), List.append(&2, A, evens(A, t), y <> odds(A, t)), perm_cons(A, y, t, List.append(&2, A, evens(A, t), odds(A, t)), split_perm(A, t)), perm_move_rev(A, evens(A, t), y, odds(A, t))))# Merge sort with fuel (the fuel only bounds recursion; it never affects the permutation).def msort_by(~A: Data, ~le: A -> A -> Bool, fuel: Nat, +xs: List<&2, A>) -> List<&2, A>: match fuel: case 0n: xs case 1n++f: merge_by(~A, ~le, msort_by(~A, ~le, f, evens(A, xs)), msort_by(~A, ~le, f, odds(A, xs)))# Merge sort permutes its input, for any fuel.def msort_by_perm(~A: Data, ~le: A -> A -> Bool, fuel: Nat, +xs: List<&2, A>) -> K.perm(A, xs, msort_by(~A, ~le, fuel, xs)): match fuel: case 0n: perm_refl(A, xs) case 1n++f: perm_trans(A, xs, List.append(&2, A, evens(A, xs), odds(A, xs)), msort_by(~A, ~le, 1n+f, xs), split_perm(A, xs), perm_trans(A, List.append(&2, A, evens(A, xs), odds(A, xs)), List.append(&2, A, msort_by(~A, ~le, f, evens(A, xs)), msort_by(~A, ~le, f, odds(A, xs))), msort_by(~A, ~le, 1n+f, xs), perm_append(A, evens(A, xs), msort_by(~A, ~le, f, evens(A, xs)), odds(A, xs), msort_by(~A, ~le, f, odds(A, xs)), msort_by_perm(~A, ~le, f, evens(A, xs)), msort_by_perm(~A, ~le, f, odds(A, xs))), merge_by_perm(~A, ~le, msort_by(~A, ~le, f, evens(A, xs)), msort_by(~A, ~le, f, odds(A, xs)))))def sort_by(~A: Data, ~le: A -> A -> Bool, +xs: List<&2, A>) -> List<&2, A>: msort_by(~A, ~le, List.length(&2, A, xs), xs)# The sort permutes its input.def sort_by_perm(~A: Data, ~le: A -> A -> Bool, +xs: List<&2, A>) -> K.perm(A, xs, sort_by(~A, ~le, xs)): msort_by_perm(~A, ~le, List.length(&2, A, xs), xs)def sort_by_perm_nat(+xs: List<&2, Nat>) -> K.perm(Nat, xs, sort_by(~Nat, ~Nat.is_le, xs)): sort_by_perm(~Nat, ~Nat.is_le, xs)def internal_sort_nat_test() -> {sort_by(~Nat, ~Nat.is_le, [5n, 1n, 4n, 2n, 3n, 1n]) == [1n, 1n, 2n, 3n, 4n, 5n] : List<&2, Nat>}: {==}def internal_swap_head_invol(-A: Data, xs: List<&2, A>) -> {K.swap_head(A, K.swap_head(A, xs)) == xs : List<&2, A>}: match xs: case Nil{}: {==} case +x <> t: match t: case Nil{}: {==} case +y <> u: {==}def internal_swap_at_invol(-A: Data, +i: Nat, +xs: List<&2, A>) -> {K.swap_at(A, i, K.swap_at(A, i, xs)) == xs : List<&2, A>}: match i: case 0n: internal_swap_head_invol(A, xs) case 1n+p: match xs: case Nil{}: {==} case +x <> t: %Equal.sym(List<&2, A>, K.swap_at(A, p, K.swap_at(A, p, t)), t, internal_swap_at_invol(A, p, t)) : {x <> _ == x <> t : List<&2, A>} {==}def internal_invert(+s: List<&2, Nat>) -> List<&2, Nat>: match s: case Nil{}: Nil{} case +i <> rest: List.append(&2, Nat, internal_invert(rest), i <> Nil{})def internal_apply_invert(-A: Data, +s: List<&2, Nat>, +xs: List<&2, A>) -> {K.apply(A, internal_invert(s), K.apply(A, s, xs)) == xs : List<&2, A>}: match s: case Nil{}: {==} case +i <> rest: %Equal.sym(List<&2, A>, K.apply(A, List.append(&2, Nat, internal_invert(rest), i <> Nil{}), K.apply(A, rest, K.swap_at(A, i, xs))), K.apply(A, i <> Nil{}, K.apply(A, internal_invert(rest), K.apply(A, rest, K.swap_at(A, i, xs)))), internal_apply_app(A, internal_invert(rest), i <> Nil{}, K.apply(A, rest, K.swap_at(A, i, xs)))) : {_ == xs : List<&2, A>} %Equal.sym(List<&2, A>, K.apply(A, internal_invert(rest), K.apply(A, rest, K.swap_at(A, i, xs))), K.swap_at(A, i, xs), internal_apply_invert(A, rest, K.swap_at(A, i, xs))) : {K.apply(A, i <> Nil{}, _) == xs : List<&2, A>} internal_swap_at_invol(A, i, xs)def internal_sym_eq(-A: Data, +s: List<&2, Nat>, +xs: List<&2, A>, -ys: List<&2, A>, e: K.perm_steps(A, s, xs, ys)) -> K.perm_steps(A, internal_invert(s), ys, xs): %e : {K.apply(A, internal_invert(s), _) == xs : List<&2, A>} internal_apply_invert(A, s, xs)# Permutation is symmetric.def perm_sym(-A: Data, +xs: List<&2, A>, -ys: List<&2, A>, h: K.perm(A, xs, ys)) -> K.perm(A, ys, xs): (s, e) = h +s = s (internal_invert(s), internal_sym_eq(A, s, xs, ys, e))# Permutation preserves length.def internal_swap_head_length(-A: Data, xs: List<&2, A>) -> {List.length(&2, A, K.swap_head(A, xs)) == List.length(&2, A, xs) : Nat}: match xs: case Nil{}: {==} case +x <> t: match t: case Nil{}: {==} case +y <> u: {==}def internal_swap_at_length(-A: Data, +i: Nat, +xs: List<&2, A>) -> {List.length(&2, A, K.swap_at(A, i, xs)) == List.length(&2, A, xs) : Nat}: match i: case 0n: internal_swap_head_length(A, xs) case 1n+p: match xs: case Nil{}: {==} case +x <> t: %Equal.sym(Nat, List.length(&2, A, K.swap_at(A, p, t)), List.length(&2, A, t), internal_swap_at_length(A, p, t)) : {1n+_ == 1n+List.length(&2, A, t) : Nat} {==}def internal_apply_length(-A: Data, +s: List<&2, Nat>, +xs: List<&2, A>) -> {List.length(&2, A, K.apply(A, s, xs)) == List.length(&2, A, xs) : Nat}: match s: case Nil{}: {==} case +i <> rest: Equal.trans(Nat, List.length(&2, A, K.apply(A, rest, K.swap_at(A, i, xs))), List.length(&2, A, K.swap_at(A, i, xs)), List.length(&2, A, xs), internal_apply_length(A, rest, K.swap_at(A, i, xs)), internal_swap_at_length(A, i, xs))def internal_length_eq(-A: Data, +s: List<&2, Nat>, +xs: List<&2, A>, -ys: List<&2, A>, e: K.perm_steps(A, s, xs, ys)) -> {List.length(&2, A, ys) == List.length(&2, A, xs) : Nat}: %e : {List.length(&2, A, _) == List.length(&2, A, xs) : Nat} internal_apply_length(A, s, xs)def perm_length(-A: Data, +xs: List<&2, A>, -ys: List<&2, A>, h: K.perm(A, xs, ys)) -> {List.length(&2, A, ys) == List.length(&2, A, xs) : Nat}: (s, e) = h internal_length_eq(A, s, xs, ys, e)def internal_perm_nil_nat() -> K.perm(Nat, Nil{}, Nil{}): perm_nil(Nat)def internal_perm_dup_nat(+xs: List<&2, Nat>, -ys: List<&2, Nat>, h: K.perm(Nat, xs, ys)) -> K.perm(Nat, xs, ys) & K.perm(Nat, xs, ys): perm_dup(Nat, xs, ys, h)def internal_perm_sym_nat(+xs: List<&2, Nat>, -ys: List<&2, Nat>, h: K.perm(Nat, xs, ys)) -> K.perm(Nat, ys, xs): perm_sym(Nat, xs, ys, h)def internal_perm_length_nat(+xs: List<&2, Nat>, -ys: List<&2, Nat>, h: K.perm(Nat, xs, ys)) -> {List.length(&2, Nat, ys) == List.length(&2, Nat, xs) : Nat}: perm_length(Nat, xs, ys, h)