1	# GENERATED by tools/generators/rw.py from ../../../tools/generators/secp256k1_hand/sel.src; edit the .src
2	import Base
3	import ../../../spec/lib/common.bend as C
4	import ../../../spec/crypto/secp256k1/field.bend as FS
5	import ../../../spec/crypto/secp256k1/curve.bend as CV
6	import ../../../src/crypto/secp256k1/limbs.bend as L
7	import ../../../src/crypto/secp256k1/point.bend as P
8	import ../../lib/logic.bend as Lg
9	import ../../lib/lemmas/proofs/nat_algebra.bend as NA
10	import ./fieldops.bend as FO
11	import ./bitsv.bend as BV
12	import ./bitsrel.bend as BR
13	import ./point.bend as PT
14	
15	# Selection (b ? p : q, b in {0, 1}) and small facts for the point proofs.
16	def bit01(n: Nat) -> {Nat.is_lt(C.bit(n), 2n) == True{} : Bool}:
17	  match n:
18	    case 0n:
19	      {==}
20	    case 1n:
21	      {==}
22	    case 2n+m:
23	      bit01(m)
24	
25	def sb01(i: Nat, +off: Nat, +k: Nat) -> {BV.all01(BR.sb(i, off, k)) == True{} : Bool}:
26	  match i:
27	    case 0n:
28	      {==}
29	    case 1n+ +j:
30	      Lg.and_intro(Nat.is_lt(CV.bit(Nat.add(off, j), k), 2n), BV.all01(BR.sb(j, off, k)), bit01(C.high(Nat.add(off, j), k)), sb01(j, off, k))
31	
32	def sel0(+one: Nat, n: Nat, xs: List<&2, Nat>, ys: List<&2, Nat>, +hx: {FS.limbs16(one, n, xs) == True{} : Bool}, +hy: {FS.limbs16(one, n, ys) == True{} : Bool}) -> {L.select(0n, xs, ys) == ys : List<&2, Nat>}:
33	  match n xs:
34	    case 0n Nil{}:
35	      match ys:
36	        case Nil{}:
37	          {==}
38	        case y <> t:
39	          Empty.absurd({L.select(0n, [], y <> t) == y <> t : List<&2, Nat>}, Lg.false_true(hy))
40	    case 0n x <> t:
41	      Empty.absurd({L.select(0n, x <> t, ys) == ys : List<&2, Nat>}, Lg.false_true(hx))
42	    case 1n+k Nil{}:
43	      Empty.absurd({L.select(0n, [], ys) == ys : List<&2, Nat>}, Lg.false_true(hx))
44	    case 1n+ +k +x <> +xt:
45	      match ys:
46	        case Nil{}:
47	          Empty.absurd({L.select(0n, x <> xt, []) == [] : List<&2, Nat>}, Lg.false_true(hy))
48	        case +y <> +yt:
49	          +hxt = Lg.and_right(Nat.is_lt(x, C.shift(16n, one)), FS.limbs16(one, k, xt), hx)
50	          +hyt = Lg.and_right(Nat.is_lt(y, C.shift(16n, one)), FS.limbs16(one, k, yt), hy)
51	          %Equal.sym(List<&2, Nat>, L.select(0n, xt, yt), yt, sel0(one, k, xt, yt, hxt, hyt)) : {Nat.add(Nat.mul(x, 0n), Nat.mul(y, 1n)) <> _ == y <> yt : List<&2, Nat>}
52	          %Equal.sym(Nat, Nat.mul(x, 0n), 0n, NA.mul_zero(x)) : {Nat.add(_, Nat.mul(y, 1n)) <> yt == y <> yt : List<&2, Nat>}
53	          %Equal.sym(Nat, Nat.mul(y, 1n), y, NA.mul_one(y)) : {Nat.add(0n, _) <> yt == y <> yt : List<&2, Nat>}
54	          {==}
55	
56	def sel1(+one: Nat, n: Nat, xs: List<&2, Nat>, ys: List<&2, Nat>, +hx: {FS.limbs16(one, n, xs) == True{} : Bool}, +hy: {FS.limbs16(one, n, ys) == True{} : Bool}) -> {L.select(1n, xs, ys) == xs : List<&2, Nat>}:
57	  match n xs:
58	    case 0n Nil{}:
59	      match ys:
60	        case Nil{}:
61	          {==}
62	        case y <> t:
63	          Empty.absurd({L.select(1n, [], y <> t) == [] : List<&2, Nat>}, Lg.false_true(hy))
64	    case 0n x <> t:
65	      Empty.absurd({L.select(1n, x <> t, ys) == x <> t : List<&2, Nat>}, Lg.false_true(hx))
66	    case 1n+k Nil{}:
67	      Empty.absurd({L.select(1n, [], ys) == [] : List<&2, Nat>}, Lg.false_true(hx))
68	    case 1n+ +k +x <> +xt:
69	      match ys:
70	        case Nil{}:
71	          Empty.absurd({L.select(1n, x <> xt, []) == x <> xt : List<&2, Nat>}, Lg.false_true(hy))
72	        case +y <> +yt:
73	          +hxt = Lg.and_right(Nat.is_lt(x, C.shift(16n, one)), FS.limbs16(one, k, xt), hx)
74	          +hyt = Lg.and_right(Nat.is_lt(y, C.shift(16n, one)), FS.limbs16(one, k, yt), hy)
75	          %Equal.sym(List<&2, Nat>, L.select(1n, xt, yt), xt, sel1(one, k, xt, yt, hxt, hyt)) : {Nat.add(Nat.mul(x, 1n), Nat.mul(y, 0n)) <> _ == x <> xt : List<&2, Nat>}
76	          %Equal.sym(Nat, Nat.mul(y, 0n), 0n, NA.mul_zero(y)) : {Nat.add(Nat.mul(x, 1n), _) <> xt == x <> xt : List<&2, Nat>}
77	          %Equal.sym(Nat, Nat.mul(x, 1n), x, NA.mul_one(x)) : {Nat.add(_, 0n) <> xt == x <> xt : List<&2, Nat>}
78	          %Equal.sym(Nat, Nat.add(x, 0n), x, NA.add_zero(x)) : {_ <> xt == x <> xt : List<&2, Nat>}
79	          {==}
80	
81	def l16(+one: Nat, +x: List<&2, Nat>, +h: {PT.red(one, x) == True{} : Bool}) -> {FS.limbs16(one, 16n, x) == True{} : Bool}:
82	  Lg.and_left(FS.limbs16(one, 16n, x), Nat.is_lt(FS.value(one, x), FS.prime(one)), h)
83	
84	def rx(+one: Nat, +x: List<&2, Nat>, +y: List<&2, Nat>, +z: List<&2, Nat>, +h: {PT.pred(one, P.Point{x, y, z}) == True{} : Bool}) -> {FS.limbs16(one, 16n, x) == True{} : Bool}:
85	  l16(one, x, Lg.and_left(PT.red(one, x), Bool.and(PT.red(one, y), PT.red(one, z)), h))
86	
87	def ry(+one: Nat, +x: List<&2, Nat>, +y: List<&2, Nat>, +z: List<&2, Nat>, +h: {PT.pred(one, P.Point{x, y, z}) == True{} : Bool}) -> {FS.limbs16(one, 16n, y) == True{} : Bool}:
88	  l16(one, y, Lg.and_left(PT.red(one, y), PT.red(one, z), Lg.and_right(PT.red(one, x), Bool.and(PT.red(one, y), PT.red(one, z)), h)))
89	
90	def rz(+one: Nat, +x: List<&2, Nat>, +y: List<&2, Nat>, +z: List<&2, Nat>, +h: {PT.pred(one, P.Point{x, y, z}) == True{} : Bool}) -> {FS.limbs16(one, 16n, z) == True{} : Bool}:
91	  l16(one, z, Lg.and_right(PT.red(one, y), PT.red(one, z), Lg.and_right(PT.red(one, x), Bool.and(PT.red(one, y), PT.red(one, z)), h)))
92	
93	# b = 0 picks the second point, b = 1 the first
94	def psel0(+one: Nat, a: P.Point, b: P.Point, +ha: {PT.pred(one, a) == True{} : Bool}, +hb: {PT.pred(one, b) == True{} : Bool}) -> {P.select(0n, a, b) == b : P.Point}:
95	  match a b:
96	    case P.Point{+x1, +y1, +z1} P.Point{+x2, +y2, +z2}:
97	      %Equal.sym(List<&2, Nat>, L.select(0n, x1, x2), x2, sel0(one, 16n, x1, x2, rx(one, x1, y1, z1, ha), rx(one, x2, y2, z2, hb))) : {P.Point{_, L.select(0n, y1, y2), L.select(0n, z1, z2)} == P.Point{x2, y2, z2} : P.Point}
98	      %Equal.sym(List<&2, Nat>, L.select(0n, y1, y2), y2, sel0(one, 16n, y1, y2, ry(one, x1, y1, z1, ha), ry(one, x2, y2, z2, hb))) : {P.Point{x2, _, L.select(0n, z1, z2)} == P.Point{x2, y2, z2} : P.Point}
99	      %Equal.sym(List<&2, Nat>, L.select(0n, z1, z2), z2, sel0(one, 16n, z1, z2, rz(one, x1, y1, z1, ha), rz(one, x2, y2, z2, hb))) : {P.Point{x2, y2, _} == P.Point{x2, y2, z2} : P.Point}
100	      {==}
101	
102	def psel1(+one: Nat, a: P.Point, b: P.Point, +ha: {PT.pred(one, a) == True{} : Bool}, +hb: {PT.pred(one, b) == True{} : Bool}) -> {P.select(1n, a, b) == a : P.Point}:
103	  match a b:
104	    case P.Point{+x1, +y1, +z1} P.Point{+x2, +y2, +z2}:
105	      %Equal.sym(List<&2, Nat>, L.select(1n, x1, x2), x1, sel1(one, 16n, x1, x2, rx(one, x1, y1, z1, ha), rx(one, x2, y2, z2, hb))) : {P.Point{_, L.select(1n, y1, y2), L.select(1n, z1, z2)} == P.Point{x1, y1, z1} : P.Point}
106	      %Equal.sym(List<&2, Nat>, L.select(1n, y1, y2), y1, sel1(one, 16n, y1, y2, ry(one, x1, y1, z1, ha), ry(one, x2, y2, z2, hb))) : {P.Point{x1, _, L.select(1n, z1, z2)} == P.Point{x1, y1, z1} : P.Point}
107	      %Equal.sym(List<&2, Nat>, L.select(1n, z1, z2), z1, sel1(one, 16n, z1, z2, rz(one, x1, y1, z1, ha), rz(one, x2, y2, z2, hb))) : {P.Point{x1, y1, _} == P.Point{x1, y1, z1} : P.Point}
108	      {==}
109	
110	def one_lt(+one: Nat, +h1: {one == 1n : Nat}) -> {Nat.is_lt(1n, FS.prime(one)) == True{} : Bool}:
111	  FO.small_lt_p(one, h1, 1n, {==})
112	
113	def lt2(c: Nat) -> {Nat.is_lt(2n+c, 2n) == False{} : Bool}:
114	  match c:
115	    case 0n:
116	      {==}
117	    case 1n+d:
118	      {==}
119	
120	def not01(+c: Nat, +h: {Nat.is_lt(2n+c, 2n) == True{} : Bool}) -> Empty:
121	  Lg.false_true(Equal.trans(Bool, False{}, Nat.is_lt(2n+c, 2n), True{}, Equal.sym(Bool, Nat.is_lt(2n+c, 2n), False{}, lt2(c)), h))
122	
123	def red_of(+one: Nat, +x: List<&2, Nat>, +hl: {FS.limbs16(one, 16n, x) == True{} : Bool}, +hb: {Nat.is_lt(FS.value(one, x), FS.prime(one)) == True{} : Bool}) -> {PT.red(one, x) == True{} : Bool}:
124	  Lg.and_intro(FS.limbs16(one, 16n, x), Nat.is_lt(FS.value(one, x), FS.prime(one)), hl, hb)
125	
126	def sl16(+one: Nat, +x: List<&2, Nat>, +h: {FS.reduced(one, FS.order(one), x) == True{} : Bool}) -> {FS.limbs16(one, 16n, x) == True{} : Bool}:
127	  Lg.and_left(FS.limbs16(one, 16n, x), Nat.is_lt(FS.value(one, x), FS.order(one)), h)