/- ────────────────────────────────────────────────────────────────────────────── Proofs/ToBytesSpec.lean — total-correctness spec of `FieldElement51::to_bytes` (phase 2 of the signature apex: the "to_bytes canonicity" brick). THE THEOREM (`to_bytes_spec`): for any field element a, to_bytes a = ok s with bytesVal s = feVal a mod p — the emitted 32 bytes are the CANONICAL little-endian encoding of the represented residue. Canonicity (bytesVal s < p, top bit clear) and injectivity ("equal residues ⇒ equal bytes, equal bytes ⇒ equal residues") are corollaries (`to_bytes_lt`, at the end), because bytesVal s is pinned to the residue itself. STRUCTURE: one straight-line symbolic execution (~150 machine ops — the longest walk in the repo, but loop-free), consuming the pure ℕ lemmas of Proofs/ToBytesMath.lean at exactly four joints: q_telescope — the 5-rung carry chain that computes q = (h+19)/2²⁵⁵; q_facts — q ∈ {0,1} (also feeds the no-overflow side conditions); carry_pack + q_mod_p — the final carry pass assembles (h+19q) mod 2²⁵⁵ = h mod p; bytes_pack — the 32 shift/mask extractions reassemble the value. The four boundary bytes (6, 12, 19, 25) OR the high bits of one limb with the low bits of the next; disjointness turns each OR into + (Nat.two_pow_add_eq_or_of_lt, the ScalarBytesSpec idiom). Rust: `FieldElement51::to_bytes`, curve25519-dalek/src/backend/serial/u64/ field.rs:368-450 (incl. its trailing debug-assert that the top bit is clear — discharged, not assumed). Gen: gen/CurveField/Funs.lean `backend.serial.u64.field.FieldElement51.as_bytes`. ────────────────────────────────────────────────────────────────────────────── -/ import Proofs.ReduceSpec import Proofs.ToBytesMath open Aeneas Aeneas.Std Result open curve25519_dalek set_option maxHeartbeats 8000000 set_option linter.unusedSimpArgs false set_option maxRecDepth 8000 namespace CurveFieldProofs open Aeneas.Std.WP abbrev fe_to_bytes := backend.serial.u64.field.FieldElement51.as_bytes /-- Little-endian value of a 32-byte array (match-style, like `feVal`). -/ def bytesVal (s : Std.Array Std.U8 32#usize) : ℕ := match (↑s : List Std.U8) with | [b0, b1, b2, b3, b4, b5, b6, b7, b8, b9, b10, b11, b12, b13, b14, b15, b16, b17, b18, b19, b20, b21, b22, b23, b24, b25, b26, b27, b28, b29, b30, b31] => b0.val + b1.val * 2^8 + b2.val * 2^16 + b3.val * 2^24 + b4.val * 2^32 + b5.val * 2^40 + b6.val * 2^48 + b7.val * 2^56 + b8.val * 2^64 + b9.val * 2^72 + b10.val * 2^80 + b11.val * 2^88 + b12.val * 2^96 + b13.val * 2^104 + b14.val * 2^112 + b15.val * 2^120 + b16.val * 2^128 + b17.val * 2^136 + b18.val * 2^144 + b19.val * 2^152 + b20.val * 2^160 + b21.val * 2^168 + b22.val * 2^176 + b23.val * 2^184 + b24.val * 2^192 + b25.val * 2^200 + b26.val * 2^208 + b27.val * 2^216 + b28.val * 2^224 + b29.val * 2^232 + b30.val * 2^240 + b31.val * 2^248 | _ => 0 /-- **`to_bytes` is the canonical encoder**: it always succeeds and its output bytes denote exactly the represented residue mod p. -/ theorem to_bytes_spec (a : Fe) (l0 l1 l2 l3 l4 : U64) (hl : (↑a : List U64) = [l0, l1, l2, l3, l4]) : fe_to_bytes a ⦃ s => bytesVal s = feVal a % P ⦄ := by unfold fe_to_bytes backend.serial.u64.field.FieldElement51.as_bytes -- weak reduction: limbs < 2⁵¹ + 19·2¹³, value preserved mod p (exactly) step with (reduce_spec a l0 l1 l2 l3 l4 hl) as ⟨fe, hbnd, hval⟩ obtain ⟨m0, m1, m2, m3, m4, hm⟩ := Fe.exists_limbs fe rw [Bnd_eq fe m0 m1 m2 m3 m4 _ hm] at hbnd obtain ⟨hbm0, hbm1, hbm2, hbm3, hbm4⟩ := hbnd rw [feVal_eq fe m0 m1 m2 m3 m4 hm] at hval -- h := the weakly-reduced value; h < 2p have hh2p : limbsVal m0 m1 m2 m3 m4 < 2 * P := by unfold limbsVal P omega -- ── the q pass: q = (h+19)/2²⁵⁵ ───────────────────────────────────────── step as ⟨i, hi⟩ simp [hm] at hi step as ⟨i1, hi1⟩ rw [hi] at hi1 step as ⟨q, hq⟩ step as ⟨i2, hi2⟩ simp [hm] at hi2 step as ⟨i3, hi3⟩ rw [hi2] at hi3 step as ⟨q1, hq1⟩ step as ⟨i4, hi4⟩ simp [hm] at hi4 step as ⟨i5, hi5⟩ rw [hi4] at hi5 step as ⟨q2, hq2⟩ step as ⟨i6, hi6⟩ simp [hm] at hi6 step as ⟨i7, hi7⟩ rw [hi6] at hi7 step as ⟨q3, hq3⟩ step as ⟨i8, hi8⟩ simp [hm] at hi8 step as ⟨i9, hi9⟩ rw [hi8] at hi9 step as ⟨q4, hq4⟩ -- q4 = (h + 19) / 2²⁵⁵, and q4 ≤ 1 have hq4v : q4.val = (limbsVal m0 m1 m2 m3 m4 + 19) / 2^255 := by rw [hq4, nat_shr, hi9, hq3, nat_shr, hi7, hq2, nat_shr, hi5, hq1, nat_shr, hi3, hq, nat_shr, hi1] have := q_telescope (m0.val + 19) m1.val m2.val m3.val m4.val unfold limbsVal omega have hq4le : q4.val ≤ 1 := by rw [hq4v]; exact (q_facts _ hh2p).1 -- fold 19q into limb 0 step as ⟨i10, hi10⟩ step as ⟨i11, hi11⟩ rw [hi] at hi11 step as ⟨limbs, hlimbs⟩ have hll : (↑limbs : List U64) = [i11, m1, m2, m3, m4] := by simp only [hlimbs, Array.set_val_eq, hm] rfl have hi11v : i11.val = m0.val + 19 * q4.val := by rw [hi11, hi10] -- the local mask constant 2⁵¹ − 1 step as ⟨i12, hi12⟩ step as ⟨mask, hmask⟩ have hmaskv : mask.val = 2251799813685247 := by rw [hmask, hi12] simp [Nat.shiftLeft_eq] scalar_tac have hmask_mod : ∀ n : ℕ, n &&& mask.val = n % 2^51 := by intro n rw [hmaskv, nat_and_mask] norm_num -- ── the carry pass: normalize to radix 2⁵¹, dropping the 2²⁵⁵ carry ──── -- round 0→1 step as ⟨i13, hi13⟩ simp [hll] at hi13 step as ⟨i14, hi14⟩ rw [hi13] at hi14 step as ⟨i15, hi15⟩ simp [hll] at hi15 step as ⟨i16, hi16⟩ rw [hi15] at hi16 step as ⟨limbs1, hlimbs1⟩ have hll1 : (↑limbs1 : List U64) = [i11, i16, m2, m3, m4] := by simp only [hlimbs1, Array.set_val_eq, hll]; rfl step as ⟨i17, hi17⟩ simp [hll1] at hi17 step as ⟨i18, hi18⟩ rw [hi17] at hi18 step as ⟨limbs2, hlimbs2⟩ have hll2 : (↑limbs2 : List U64) = [i18, i16, m2, m3, m4] := by simp only [hlimbs2, Array.set_val_eq, hll1]; rfl -- round 1→2 step as ⟨i19, hi19⟩ simp [hll2] at hi19 step as ⟨i20, hi20⟩ rw [hi19] at hi20 step as ⟨i21, hi21⟩ simp [hll2] at hi21 step as ⟨i22, hi22⟩ rw [hi21] at hi22 step as ⟨limbs3, hlimbs3⟩ have hll3 : (↑limbs3 : List U64) = [i18, i16, i22, m3, m4] := by simp only [hlimbs3, Array.set_val_eq, hll2]; rfl step as ⟨i23, hi23⟩ simp [hll3] at hi23 step as ⟨i24, hi24⟩ rw [hi23] at hi24 step as ⟨limbs4, hlimbs4⟩ have hll4 : (↑limbs4 : List U64) = [i18, i24, i22, m3, m4] := by simp only [hlimbs4, Array.set_val_eq, hll3]; rfl -- round 2→3 step as ⟨i25, hi25⟩ simp [hll4] at hi25 step as ⟨i26, hi26⟩ rw [hi25] at hi26 step as ⟨i27, hi27⟩ simp [hll4] at hi27 step as ⟨i28, hi28⟩ rw [hi27] at hi28 step as ⟨limbs5, hlimbs5⟩ have hll5 : (↑limbs5 : List U64) = [i18, i24, i22, i28, m4] := by simp only [hlimbs5, Array.set_val_eq, hll4]; rfl step as ⟨i29, hi29⟩ simp [hll5] at hi29 step as ⟨i30, hi30⟩ rw [hi29] at hi30 step as ⟨limbs6, hlimbs6⟩ have hll6 : (↑limbs6 : List U64) = [i18, i24, i30, i28, m4] := by simp only [hlimbs6, Array.set_val_eq, hll5]; rfl -- round 3→4 step as ⟨i31, hi31⟩ simp [hll6] at hi31 step as ⟨i32, hi32⟩ rw [hi31] at hi32 step as ⟨i33, hi33⟩ simp [hll6] at hi33 step as ⟨i34, hi34⟩ rw [hi33] at hi34 step as ⟨limbs7, hlimbs7⟩ have hll7 : (↑limbs7 : List U64) = [i18, i24, i30, i28, i34] := by simp only [hlimbs7, Array.set_val_eq, hll6]; rfl step as ⟨i35, hi35⟩ simp [hll7] at hi35 step as ⟨i36, hi36⟩ rw [hi35] at hi36 step as ⟨limbs8, hlimbs8⟩ have hll8 : (↑limbs8 : List U64) = [i18, i24, i30, i36, i34] := by simp only [hlimbs8, Array.set_val_eq, hll7]; rfl -- top slot: final mask step as ⟨i37, hi37⟩ simp [hll8] at hi37 step as ⟨i38, hi38⟩ rw [hi37] at hi38 step as ⟨limbs9, hlimbs9⟩ have hll9 : (↑limbs9 : List U64) = [i18, i24, i30, i36, i38] := by simp only [hlimbs9, Array.set_val_eq, hll8]; rfl -- ── final-limb values in the exact nested q/r forms carry_pack expects ── have hf0v : i18.val = i11.val % 2^51 := by rw [hi18, UScalar.val_and, hmask_mod] have hi14v : i14.val = i11.val / 2^51 := by rw [hi14, nat_shr] have hi16v : i16.val = m1.val + i11.val / 2^51 := by rw [hi16, hi14v] have hf1v : i24.val = (m1.val + i11.val / 2^51) % 2^51 := by rw [hi24, UScalar.val_and, hmask_mod, hi16v] have hi20v : i20.val = (m1.val + i11.val / 2^51) / 2^51 := by rw [hi20, nat_shr, hi16v] have hi22v : i22.val = m2.val + (m1.val + i11.val / 2^51) / 2^51 := by rw [hi22, hi20v] have hf2v : i30.val = (m2.val + (m1.val + i11.val / 2^51) / 2^51) % 2^51 := by rw [hi30, UScalar.val_and, hmask_mod, hi22v] have hi26v : i26.val = (m2.val + (m1.val + i11.val / 2^51) / 2^51) / 2^51 := by rw [hi26, nat_shr, hi22v] have hi28v : i28.val = m3.val + (m2.val + (m1.val + i11.val / 2^51) / 2^51) / 2^51 := by rw [hi28, hi26v] have hf3v : i36.val = (m3.val + (m2.val + (m1.val + i11.val / 2^51) / 2^51) / 2^51) % 2^51 := by rw [hi36, UScalar.val_and, hmask_mod, hi28v] have hi32v : i32.val = (m3.val + (m2.val + (m1.val + i11.val / 2^51) / 2^51) / 2^51) / 2^51 := by rw [hi32, nat_shr, hi28v] have hi34v : i34.val = m4.val + (m3.val + (m2.val + (m1.val + i11.val / 2^51) / 2^51) / 2^51) / 2^51 := by rw [hi34, hi32v] have hf4v : i38.val = (m4.val + (m3.val + (m2.val + (m1.val + i11.val / 2^51) / 2^51) / 2^51) / 2^51) % 2^51 := by rw [hi38, UScalar.val_and, hmask_mod, hi34v] have hf0lt : i18.val < 2^51 := by rw [hf0v]; exact Nat.mod_lt _ (by norm_num) have hf1lt : i24.val < 2^51 := by rw [hf1v]; exact Nat.mod_lt _ (by norm_num) have hf2lt : i30.val < 2^51 := by rw [hf2v]; exact Nat.mod_lt _ (by norm_num) have hf3lt : i36.val < 2^51 := by rw [hf3v]; exact Nat.mod_lt _ (by norm_num) have hf4lt : i38.val < 2^51 := by rw [hf4v]; exact Nat.mod_lt _ (by norm_num) clear hi14 hi16 hi20 hi22 hi26 hi28 hi32 hi34 hi18 hi24 hi30 hi36 hi38 clear hi14v hi16v hi20v hi22v hi26v hi28v hi32v hi34v clear hlimbs hlimbs1 hlimbs2 hlimbs3 hlimbs4 hlimbs5 hlimbs6 hlimbs7 hlimbs8 hlimbs9 clear hll hll1 hll2 hll3 hll4 hll5 hll6 hll7 hll8 -- ── the 32 byte extractions ────────────────────────────────────────────── -- byte 0 (limb 0 read + low byte) step as ⟨v0, hv0⟩ simp [hll9] at hv0 step as ⟨b0, hb0⟩ rw [hv0] at hb0 have hb0v : b0.val = i18.val % 2^8 := by rw [hb0, UScalar.cast_val_eq] norm_num [UScalarTy.numBits] step as ⟨s1, hs1⟩ have hsl0 : (↑s1 : List Std.U8) = [b0, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8] := by simp only [hs1, Array.set_val_eq, Array.repeat_val] rfl clear hb0 hs1 -- byte 1 (limb 0 >> 8) step as ⟨x1, hx1⟩ rw [hv0] at hx1 step as ⟨b1, hb1⟩ have hb1v : b1.val = i18.val / 2^8 % 2^8 := by rw [hb1, UScalar.cast_val_eq, hx1, nat_shr] norm_num [UScalarTy.numBits] step as ⟨s2, hs2⟩ have hsl1 : (↑s2 : List Std.U8) = [b0, b1, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8] := by simp only [hs2, Array.set_val_eq, hsl0] rfl clear hx1 hb1 hs2 hsl0 -- byte 2 (limb 0 >> 16) step as ⟨x2, hx2⟩ rw [hv0] at hx2 step as ⟨b2, hb2⟩ have hb2v : b2.val = i18.val / 2^16 % 2^8 := by rw [hb2, UScalar.cast_val_eq, hx2, nat_shr] norm_num [UScalarTy.numBits] step as ⟨s3, hs3⟩ have hsl2 : (↑s3 : List Std.U8) = [b0, b1, b2, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8] := by simp only [hs3, Array.set_val_eq, hsl1] rfl clear hx2 hb2 hs3 hsl1 -- byte 3 (limb 0 >> 24) step as ⟨x3, hx3⟩ rw [hv0] at hx3 step as ⟨b3, hb3⟩ have hb3v : b3.val = i18.val / 2^24 % 2^8 := by rw [hb3, UScalar.cast_val_eq, hx3, nat_shr] norm_num [UScalarTy.numBits] step as ⟨s4, hs4⟩ have hsl3 : (↑s4 : List Std.U8) = [b0, b1, b2, b3, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8] := by simp only [hs4, Array.set_val_eq, hsl2] rfl clear hx3 hb3 hs4 hsl2 -- byte 4 (limb 0 >> 32) step as ⟨x4, hx4⟩ rw [hv0] at hx4 step as ⟨b4, hb4⟩ have hb4v : b4.val = i18.val / 2^32 % 2^8 := by rw [hb4, UScalar.cast_val_eq, hx4, nat_shr] norm_num [UScalarTy.numBits] step as ⟨s5, hs5⟩ have hsl4 : (↑s5 : List Std.U8) = [b0, b1, b2, b3, b4, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8] := by simp only [hs5, Array.set_val_eq, hsl3] rfl clear hx4 hb4 hs5 hsl3 -- byte 5 (limb 0 >> 40) step as ⟨x5, hx5⟩ rw [hv0] at hx5 step as ⟨b5, hb5⟩ have hb5v : b5.val = i18.val / 2^40 % 2^8 := by rw [hb5, UScalar.cast_val_eq, hx5, nat_shr] norm_num [UScalarTy.numBits] step as ⟨s6, hs6⟩ have hsl5 : (↑s6 : List Std.U8) = [b0, b1, b2, b3, b4, b5, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8] := by simp only [hs6, Array.set_val_eq, hsl4] rfl clear hx5 hb5 hs6 hsl4 -- byte 6 (boundary: limb 0 >> 48 | limb 1 << 3) step as ⟨x6, hx6⟩ rw [hv0] at hx6 step as ⟨v1, hv1⟩ simp [hll9] at hv1 step as ⟨y6, hy6⟩ rw [hv1] at hy6 have hy6v : y6.val = i24.val * 2^3 := by rw [hy6] simp only [Nat.shiftLeft_eq] rw [Nat.mod_eq_of_lt (show i24.val * 2^3 < U64.size by scalar_tac)] step as ⟨z6, hz6⟩ step as ⟨b6, hb6⟩ have hb6v : b6.val = i18.val / 2^48 + (i24.val % 2^5) * 2^3 := by rw [hb6, UScalar.cast_val_eq] norm_num [UScalarTy.numBits] rw [hz6, UScalar.val_or, hx6, nat_shr, hy6v] have hxlt : i18.val / 2^48 < 2^3 := by omega have hor := Nat.two_pow_add_eq_or_of_lt (b := i18.val / 2^48) (i := 3) hxlt (i24.val) have hadd : i18.val / 2^48 ||| i24.val * 2^3 = i18.val / 2^48 + i24.val * 2^3 := by calc i18.val / 2^48 ||| i24.val * 2^3 = i18.val / 2^48 ||| 2^3 * i24.val := by rw [Nat.mul_comm] _ = 2^3 * i24.val ||| i18.val / 2^48 := Nat.lor_comm _ _ _ = 2^3 * i24.val + i18.val / 2^48 := hor.symm _ = i18.val / 2^48 + i24.val * 2^3 := by ring rw [hadd] omega step as ⟨s7, hs7⟩ have hsl6 : (↑s7 : List Std.U8) = [b0, b1, b2, b3, b4, b5, b6, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8] := by simp only [hs7, Array.set_val_eq, hsl5] rfl clear hx6 hy6 hy6v hz6 hb6 hs7 hsl5 -- byte 7 (limb 1 >> 5) step as ⟨x7, hx7⟩ rw [hv1] at hx7 step as ⟨b7, hb7⟩ have hb7v : b7.val = i24.val / 2^5 % 2^8 := by rw [hb7, UScalar.cast_val_eq, hx7, nat_shr] norm_num [UScalarTy.numBits] step as ⟨s8, hs8⟩ have hsl7 : (↑s8 : List Std.U8) = [b0, b1, b2, b3, b4, b5, b6, b7, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8] := by simp only [hs8, Array.set_val_eq, hsl6] rfl clear hx7 hb7 hs8 hsl6 -- byte 8 (limb 1 >> 13) step as ⟨x8, hx8⟩ rw [hv1] at hx8 step as ⟨b8, hb8⟩ have hb8v : b8.val = i24.val / 2^13 % 2^8 := by rw [hb8, UScalar.cast_val_eq, hx8, nat_shr] norm_num [UScalarTy.numBits] step as ⟨s9, hs9⟩ have hsl8 : (↑s9 : List Std.U8) = [b0, b1, b2, b3, b4, b5, b6, b7, b8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8] := by simp only [hs9, Array.set_val_eq, hsl7] rfl clear hx8 hb8 hs9 hsl7 -- byte 9 (limb 1 >> 21) step as ⟨x9, hx9⟩ rw [hv1] at hx9 step as ⟨b9, hb9⟩ have hb9v : b9.val = i24.val / 2^21 % 2^8 := by rw [hb9, UScalar.cast_val_eq, hx9, nat_shr] norm_num [UScalarTy.numBits] step as ⟨s10, hs10⟩ have hsl9 : (↑s10 : List Std.U8) = [b0, b1, b2, b3, b4, b5, b6, b7, b8, b9, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8] := by simp only [hs10, Array.set_val_eq, hsl8] rfl clear hx9 hb9 hs10 hsl8 -- byte 10 (limb 1 >> 29) step as ⟨x10, hx10⟩ rw [hv1] at hx10 step as ⟨b10, hb10⟩ have hb10v : b10.val = i24.val / 2^29 % 2^8 := by rw [hb10, UScalar.cast_val_eq, hx10, nat_shr] norm_num [UScalarTy.numBits] step as ⟨s11, hs11⟩ have hsl10 : (↑s11 : List Std.U8) = [b0, b1, b2, b3, b4, b5, b6, b7, b8, b9, b10, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8] := by simp only [hs11, Array.set_val_eq, hsl9] rfl clear hx10 hb10 hs11 hsl9 -- byte 11 (limb 1 >> 37) step as ⟨x11, hx11⟩ rw [hv1] at hx11 step as ⟨b11, hb11⟩ have hb11v : b11.val = i24.val / 2^37 % 2^8 := by rw [hb11, UScalar.cast_val_eq, hx11, nat_shr] norm_num [UScalarTy.numBits] step as ⟨s12, hs12⟩ have hsl11 : (↑s12 : List Std.U8) = [b0, b1, b2, b3, b4, b5, b6, b7, b8, b9, b10, b11, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8] := by simp only [hs12, Array.set_val_eq, hsl10] rfl clear hx11 hb11 hs12 hsl10 -- byte 12 (boundary: limb 1 >> 45 | limb 2 << 6) step as ⟨x12, hx12⟩ rw [hv1] at hx12 step as ⟨v2, hv2⟩ simp [hll9] at hv2 step as ⟨y12, hy12⟩ rw [hv2] at hy12 have hy12v : y12.val = i30.val * 2^6 := by rw [hy12] simp only [Nat.shiftLeft_eq] rw [Nat.mod_eq_of_lt (show i30.val * 2^6 < U64.size by scalar_tac)] step as ⟨z12, hz12⟩ step as ⟨b12, hb12⟩ have hb12v : b12.val = i24.val / 2^45 + (i30.val % 2^2) * 2^6 := by rw [hb12, UScalar.cast_val_eq] norm_num [UScalarTy.numBits] rw [hz12, UScalar.val_or, hx12, nat_shr, hy12v] have hxlt : i24.val / 2^45 < 2^6 := by omega have hor := Nat.two_pow_add_eq_or_of_lt (b := i24.val / 2^45) (i := 6) hxlt (i30.val) have hadd : i24.val / 2^45 ||| i30.val * 2^6 = i24.val / 2^45 + i30.val * 2^6 := by calc i24.val / 2^45 ||| i30.val * 2^6 = i24.val / 2^45 ||| 2^6 * i30.val := by rw [Nat.mul_comm] _ = 2^6 * i30.val ||| i24.val / 2^45 := Nat.lor_comm _ _ _ = 2^6 * i30.val + i24.val / 2^45 := hor.symm _ = i24.val / 2^45 + i30.val * 2^6 := by ring rw [hadd] omega step as ⟨s13, hs13⟩ have hsl12 : (↑s13 : List Std.U8) = [b0, b1, b2, b3, b4, b5, b6, b7, b8, b9, b10, b11, b12, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8] := by simp only [hs13, Array.set_val_eq, hsl11] rfl clear hx12 hy12 hy12v hz12 hb12 hs13 hsl11 -- byte 13 (limb 2 >> 2) step as ⟨x13, hx13⟩ rw [hv2] at hx13 step as ⟨b13, hb13⟩ have hb13v : b13.val = i30.val / 2^2 % 2^8 := by rw [hb13, UScalar.cast_val_eq, hx13, nat_shr] norm_num [UScalarTy.numBits] step as ⟨s14, hs14⟩ have hsl13 : (↑s14 : List Std.U8) = [b0, b1, b2, b3, b4, b5, b6, b7, b8, b9, b10, b11, b12, b13, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8] := by simp only [hs14, Array.set_val_eq, hsl12] rfl clear hx13 hb13 hs14 hsl12 -- byte 14 (limb 2 >> 10) step as ⟨x14, hx14⟩ rw [hv2] at hx14 step as ⟨b14, hb14⟩ have hb14v : b14.val = i30.val / 2^10 % 2^8 := by rw [hb14, UScalar.cast_val_eq, hx14, nat_shr] norm_num [UScalarTy.numBits] step as ⟨s15, hs15⟩ have hsl14 : (↑s15 : List Std.U8) = [b0, b1, b2, b3, b4, b5, b6, b7, b8, b9, b10, b11, b12, b13, b14, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8] := by simp only [hs15, Array.set_val_eq, hsl13] rfl clear hx14 hb14 hs15 hsl13 -- byte 15 (limb 2 >> 18) step as ⟨x15, hx15⟩ rw [hv2] at hx15 step as ⟨b15, hb15⟩ have hb15v : b15.val = i30.val / 2^18 % 2^8 := by rw [hb15, UScalar.cast_val_eq, hx15, nat_shr] norm_num [UScalarTy.numBits] step as ⟨s16, hs16⟩ have hsl15 : (↑s16 : List Std.U8) = [b0, b1, b2, b3, b4, b5, b6, b7, b8, b9, b10, b11, b12, b13, b14, b15, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8] := by simp only [hs16, Array.set_val_eq, hsl14] rfl clear hx15 hb15 hs16 hsl14 -- byte 16 (limb 2 >> 26) step as ⟨x16, hx16⟩ rw [hv2] at hx16 step as ⟨b16, hb16⟩ have hb16v : b16.val = i30.val / 2^26 % 2^8 := by rw [hb16, UScalar.cast_val_eq, hx16, nat_shr] norm_num [UScalarTy.numBits] step as ⟨s17, hs17⟩ have hsl16 : (↑s17 : List Std.U8) = [b0, b1, b2, b3, b4, b5, b6, b7, b8, b9, b10, b11, b12, b13, b14, b15, b16, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8] := by simp only [hs17, Array.set_val_eq, hsl15] rfl clear hx16 hb16 hs17 hsl15 -- byte 17 (limb 2 >> 34) step as ⟨x17, hx17⟩ rw [hv2] at hx17 step as ⟨b17, hb17⟩ have hb17v : b17.val = i30.val / 2^34 % 2^8 := by rw [hb17, UScalar.cast_val_eq, hx17, nat_shr] norm_num [UScalarTy.numBits] step as ⟨s18, hs18⟩ have hsl17 : (↑s18 : List Std.U8) = [b0, b1, b2, b3, b4, b5, b6, b7, b8, b9, b10, b11, b12, b13, b14, b15, b16, b17, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8] := by simp only [hs18, Array.set_val_eq, hsl16] rfl clear hx17 hb17 hs18 hsl16 -- byte 18 (limb 2 >> 42) step as ⟨x18, hx18⟩ rw [hv2] at hx18 step as ⟨b18, hb18⟩ have hb18v : b18.val = i30.val / 2^42 % 2^8 := by rw [hb18, UScalar.cast_val_eq, hx18, nat_shr] norm_num [UScalarTy.numBits] step as ⟨s19, hs19⟩ have hsl18 : (↑s19 : List Std.U8) = [b0, b1, b2, b3, b4, b5, b6, b7, b8, b9, b10, b11, b12, b13, b14, b15, b16, b17, b18, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8] := by simp only [hs19, Array.set_val_eq, hsl17] rfl clear hx18 hb18 hs19 hsl17 -- byte 19 (boundary: limb 2 >> 50 | limb 3 << 1) step as ⟨x19, hx19⟩ rw [hv2] at hx19 step as ⟨v3, hv3⟩ simp [hll9] at hv3 step as ⟨y19, hy19⟩ rw [hv3] at hy19 have hy19v : y19.val = i36.val * 2^1 := by rw [hy19] simp only [Nat.shiftLeft_eq] rw [Nat.mod_eq_of_lt (show i36.val * 2^1 < U64.size by scalar_tac)] step as ⟨z19, hz19⟩ step as ⟨b19, hb19⟩ have hb19v : b19.val = i30.val / 2^50 + (i36.val % 2^7) * 2^1 := by rw [hb19, UScalar.cast_val_eq] norm_num [UScalarTy.numBits] rw [hz19, UScalar.val_or, hx19, nat_shr, hy19v] have hxlt : i30.val / 2^50 < 2^1 := by omega have hor := Nat.two_pow_add_eq_or_of_lt (b := i30.val / 2^50) (i := 1) hxlt (i36.val) have hadd : i30.val / 2^50 ||| i36.val * 2^1 = i30.val / 2^50 + i36.val * 2^1 := by calc i30.val / 2^50 ||| i36.val * 2^1 = i30.val / 2^50 ||| 2^1 * i36.val := by rw [Nat.mul_comm] _ = 2^1 * i36.val ||| i30.val / 2^50 := Nat.lor_comm _ _ _ = 2^1 * i36.val + i30.val / 2^50 := hor.symm _ = i30.val / 2^50 + i36.val * 2^1 := by ring rw [hadd] omega step as ⟨s20, hs20⟩ have hsl19 : (↑s20 : List Std.U8) = [b0, b1, b2, b3, b4, b5, b6, b7, b8, b9, b10, b11, b12, b13, b14, b15, b16, b17, b18, b19, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8] := by simp only [hs20, Array.set_val_eq, hsl18] rfl clear hx19 hy19 hy19v hz19 hb19 hs20 hsl18 -- byte 20 (limb 3 >> 7) step as ⟨x20, hx20⟩ rw [hv3] at hx20 step as ⟨b20, hb20⟩ have hb20v : b20.val = i36.val / 2^7 % 2^8 := by rw [hb20, UScalar.cast_val_eq, hx20, nat_shr] norm_num [UScalarTy.numBits] step as ⟨s21, hs21⟩ have hsl20 : (↑s21 : List Std.U8) = [b0, b1, b2, b3, b4, b5, b6, b7, b8, b9, b10, b11, b12, b13, b14, b15, b16, b17, b18, b19, b20, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8] := by simp only [hs21, Array.set_val_eq, hsl19] rfl clear hx20 hb20 hs21 hsl19 -- byte 21 (limb 3 >> 15) step as ⟨x21, hx21⟩ rw [hv3] at hx21 step as ⟨b21, hb21⟩ have hb21v : b21.val = i36.val / 2^15 % 2^8 := by rw [hb21, UScalar.cast_val_eq, hx21, nat_shr] norm_num [UScalarTy.numBits] step as ⟨s22, hs22⟩ have hsl21 : (↑s22 : List Std.U8) = [b0, b1, b2, b3, b4, b5, b6, b7, b8, b9, b10, b11, b12, b13, b14, b15, b16, b17, b18, b19, b20, b21, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8] := by simp only [hs22, Array.set_val_eq, hsl20] rfl clear hx21 hb21 hs22 hsl20 -- byte 22 (limb 3 >> 23) step as ⟨x22, hx22⟩ rw [hv3] at hx22 step as ⟨b22, hb22⟩ have hb22v : b22.val = i36.val / 2^23 % 2^8 := by rw [hb22, UScalar.cast_val_eq, hx22, nat_shr] norm_num [UScalarTy.numBits] step as ⟨s23, hs23⟩ have hsl22 : (↑s23 : List Std.U8) = [b0, b1, b2, b3, b4, b5, b6, b7, b8, b9, b10, b11, b12, b13, b14, b15, b16, b17, b18, b19, b20, b21, b22, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8] := by simp only [hs23, Array.set_val_eq, hsl21] rfl clear hx22 hb22 hs23 hsl21 -- byte 23 (limb 3 >> 31) step as ⟨x23, hx23⟩ rw [hv3] at hx23 step as ⟨b23, hb23⟩ have hb23v : b23.val = i36.val / 2^31 % 2^8 := by rw [hb23, UScalar.cast_val_eq, hx23, nat_shr] norm_num [UScalarTy.numBits] step as ⟨s24, hs24⟩ have hsl23 : (↑s24 : List Std.U8) = [b0, b1, b2, b3, b4, b5, b6, b7, b8, b9, b10, b11, b12, b13, b14, b15, b16, b17, b18, b19, b20, b21, b22, b23, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8] := by simp only [hs24, Array.set_val_eq, hsl22] rfl clear hx23 hb23 hs24 hsl22 -- byte 24 (limb 3 >> 39) step as ⟨x24, hx24⟩ rw [hv3] at hx24 step as ⟨b24, hb24⟩ have hb24v : b24.val = i36.val / 2^39 % 2^8 := by rw [hb24, UScalar.cast_val_eq, hx24, nat_shr] norm_num [UScalarTy.numBits] step as ⟨s25, hs25⟩ have hsl24 : (↑s25 : List Std.U8) = [b0, b1, b2, b3, b4, b5, b6, b7, b8, b9, b10, b11, b12, b13, b14, b15, b16, b17, b18, b19, b20, b21, b22, b23, b24, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8] := by simp only [hs25, Array.set_val_eq, hsl23] rfl clear hx24 hb24 hs25 hsl23 -- byte 25 (boundary: limb 3 >> 47 | limb 4 << 4) step as ⟨x25, hx25⟩ rw [hv3] at hx25 step as ⟨v4, hv4⟩ simp [hll9] at hv4 step as ⟨y25, hy25⟩ rw [hv4] at hy25 have hy25v : y25.val = i38.val * 2^4 := by rw [hy25] simp only [Nat.shiftLeft_eq] rw [Nat.mod_eq_of_lt (show i38.val * 2^4 < U64.size by scalar_tac)] step as ⟨z25, hz25⟩ step as ⟨b25, hb25⟩ have hb25v : b25.val = i36.val / 2^47 + (i38.val % 2^4) * 2^4 := by rw [hb25, UScalar.cast_val_eq] norm_num [UScalarTy.numBits] rw [hz25, UScalar.val_or, hx25, nat_shr, hy25v] have hxlt : i36.val / 2^47 < 2^4 := by omega have hor := Nat.two_pow_add_eq_or_of_lt (b := i36.val / 2^47) (i := 4) hxlt (i38.val) have hadd : i36.val / 2^47 ||| i38.val * 2^4 = i36.val / 2^47 + i38.val * 2^4 := by calc i36.val / 2^47 ||| i38.val * 2^4 = i36.val / 2^47 ||| 2^4 * i38.val := by rw [Nat.mul_comm] _ = 2^4 * i38.val ||| i36.val / 2^47 := Nat.lor_comm _ _ _ = 2^4 * i38.val + i36.val / 2^47 := hor.symm _ = i36.val / 2^47 + i38.val * 2^4 := by ring rw [hadd] omega step as ⟨s26, hs26⟩ have hsl25 : (↑s26 : List Std.U8) = [b0, b1, b2, b3, b4, b5, b6, b7, b8, b9, b10, b11, b12, b13, b14, b15, b16, b17, b18, b19, b20, b21, b22, b23, b24, b25, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8] := by simp only [hs26, Array.set_val_eq, hsl24] rfl clear hx25 hy25 hy25v hz25 hb25 hs26 hsl24 -- byte 26 (limb 4 >> 4) step as ⟨x26, hx26⟩ rw [hv4] at hx26 step as ⟨b26, hb26⟩ have hb26v : b26.val = i38.val / 2^4 % 2^8 := by rw [hb26, UScalar.cast_val_eq, hx26, nat_shr] norm_num [UScalarTy.numBits] step as ⟨s27, hs27⟩ have hsl26 : (↑s27 : List Std.U8) = [b0, b1, b2, b3, b4, b5, b6, b7, b8, b9, b10, b11, b12, b13, b14, b15, b16, b17, b18, b19, b20, b21, b22, b23, b24, b25, b26, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8] := by simp only [hs27, Array.set_val_eq, hsl25] rfl clear hx26 hb26 hs27 hsl25 -- byte 27 (limb 4 >> 12) step as ⟨x27, hx27⟩ rw [hv4] at hx27 step as ⟨b27, hb27⟩ have hb27v : b27.val = i38.val / 2^12 % 2^8 := by rw [hb27, UScalar.cast_val_eq, hx27, nat_shr] norm_num [UScalarTy.numBits] step as ⟨s28, hs28⟩ have hsl27 : (↑s28 : List Std.U8) = [b0, b1, b2, b3, b4, b5, b6, b7, b8, b9, b10, b11, b12, b13, b14, b15, b16, b17, b18, b19, b20, b21, b22, b23, b24, b25, b26, b27, 0#u8, 0#u8, 0#u8, 0#u8] := by simp only [hs28, Array.set_val_eq, hsl26] rfl clear hx27 hb27 hs28 hsl26 -- byte 28 (limb 4 >> 20) step as ⟨x28, hx28⟩ rw [hv4] at hx28 step as ⟨b28, hb28⟩ have hb28v : b28.val = i38.val / 2^20 % 2^8 := by rw [hb28, UScalar.cast_val_eq, hx28, nat_shr] norm_num [UScalarTy.numBits] step as ⟨s29, hs29⟩ have hsl28 : (↑s29 : List Std.U8) = [b0, b1, b2, b3, b4, b5, b6, b7, b8, b9, b10, b11, b12, b13, b14, b15, b16, b17, b18, b19, b20, b21, b22, b23, b24, b25, b26, b27, b28, 0#u8, 0#u8, 0#u8] := by simp only [hs29, Array.set_val_eq, hsl27] rfl clear hx28 hb28 hs29 hsl27 -- byte 29 (limb 4 >> 28) step as ⟨x29, hx29⟩ rw [hv4] at hx29 step as ⟨b29, hb29⟩ have hb29v : b29.val = i38.val / 2^28 % 2^8 := by rw [hb29, UScalar.cast_val_eq, hx29, nat_shr] norm_num [UScalarTy.numBits] step as ⟨s30, hs30⟩ have hsl29 : (↑s30 : List Std.U8) = [b0, b1, b2, b3, b4, b5, b6, b7, b8, b9, b10, b11, b12, b13, b14, b15, b16, b17, b18, b19, b20, b21, b22, b23, b24, b25, b26, b27, b28, b29, 0#u8, 0#u8] := by simp only [hs30, Array.set_val_eq, hsl28] rfl clear hx29 hb29 hs30 hsl28 -- byte 30 (limb 4 >> 36) step as ⟨x30, hx30⟩ rw [hv4] at hx30 step as ⟨b30, hb30⟩ have hb30v : b30.val = i38.val / 2^36 % 2^8 := by rw [hb30, UScalar.cast_val_eq, hx30, nat_shr] norm_num [UScalarTy.numBits] step as ⟨s31, hs31⟩ have hsl30 : (↑s31 : List Std.U8) = [b0, b1, b2, b3, b4, b5, b6, b7, b8, b9, b10, b11, b12, b13, b14, b15, b16, b17, b18, b19, b20, b21, b22, b23, b24, b25, b26, b27, b28, b29, b30, 0#u8] := by simp only [hs31, Array.set_val_eq, hsl29] rfl clear hx30 hb30 hs31 hsl29 -- byte 31 (limb 4 >> 44) step as ⟨x31, hx31⟩ rw [hv4] at hx31 step as ⟨b31, hb31⟩ have hb31v : b31.val = i38.val / 2^44 := by rw [hb31, UScalar.cast_val_eq, hx31, nat_shr] norm_num [UScalarTy.numBits] omega step as ⟨s32, hs32⟩ have hsl31 : (↑s32 : List Std.U8) = [b0, b1, b2, b3, b4, b5, b6, b7, b8, b9, b10, b11, b12, b13, b14, b15, b16, b17, b18, b19, b20, b21, b22, b23, b24, b25, b26, b27, b28, b29, b30, b31] := by simp only [hs32, Array.set_val_eq, hsl30] rfl clear hx31 hb31 hs32 hsl30 -- ── the trailing debug-assert: the top bit of byte 31 is clear ─────────── step as ⟨t115, ht115⟩ simp [hsl31] at ht115 step as ⟨t116, ht116⟩ rw [ht115] at ht116 have hb31lt : b31.val < 2^7 := by rw [hb31v]; omega have ht116z : t116 = 0#u8 := by have hval : t116.val = 0 := by rw [ht116, UScalar.val_and] have h128 : (128#u8).val = 2^7 := by norm_num rw [h128, Nat.and_two_pow] have htb : b31.val.testBit 7 = false := Nat.testBit_lt_two_pow hb31lt rw [htb] simp scalar_tac rw [ht116z] step try simp only [spec_ok] -- ── assembly: bytes → limbs → mod-2²⁵⁵ → mod-p ────────────────────────── have hsum : bytesVal s32 = i18.val + i24.val * 2^51 + i30.val * 2^102 + i36.val * 2^153 + i38.val * 2^204 := by simp only [bytesVal, hsl31] rw [hb0v, hb1v, hb2v, hb3v, hb4v, hb5v, hb6v, hb7v, hb8v, hb9v, hb10v, hb11v, hb12v, hb13v, hb14v, hb15v, hb16v, hb17v, hb18v, hb19v, hb20v, hb21v, hb22v, hb23v, hb24v, hb25v, hb26v, hb27v, hb28v, hb29v, hb30v, hb31v] exact bytes_pack i18.val i24.val i30.val i36.val i38.val hf0lt hf1lt hf2lt hf3lt hf4lt have hmod255 : i18.val + i24.val * 2^51 + i30.val * 2^102 + i36.val * 2^153 + i38.val * 2^204 = (i11.val + m1.val * 2^51 + m2.val * 2^102 + m3.val * 2^153 + m4.val * 2^204) % 2^255 := by rw [hf0v, hf1v, hf2v, hf3v, hf4v] exact carry_pack i11.val m1.val m2.val m3.val m4.val have hmodp : (i11.val + m1.val * 2^51 + m2.val * 2^102 + m3.val * 2^153 + m4.val * 2^204) % 2^255 = limbsVal m0 m1 m2 m3 m4 % P := by have hre : i11.val + m1.val * 2^51 + m2.val * 2^102 + m3.val * 2^153 + m4.val * 2^204 = limbsVal m0 m1 m2 m3 m4 + 19 * ((limbsVal m0 m1 m2 m3 m4 + 19) / 2^255) := by rw [hi11v, hq4v] unfold limbsVal ring rw [hre] exact q_mod_p _ hh2p have hfe_a : feVal a % P = limbsVal m0 m1 m2 m3 m4 % P := by rw [← hval] rw [Nat.add_mul_mod_self_left] rw [hsum, hmod255, hmodp, hfe_a] end CurveFieldProofs