-- THIS FILE WAS AUTOMATICALLY GENERATED BY AENEAS
-- [pasta_curves]: external functions.
-- This is a template file: rename it to "FunsExternal.lean" and fill the holes.
import Aeneas
import PallasFp.Types
open Aeneas Aeneas.Std Result ControlFlow Error
set_option linter.dupNamespace false
set_option linter.hashCommand false
set_option linter.unusedVariables false

/- You can set the `maxHeartbeats` value with the `-max-heartbeats` CLI option -/
set_option maxHeartbeats 1000000

/- You can set the `maxRecDepth` value with the `-max-recdepth` CLI option -/
set_option maxRecDepth 2048
open pasta_curves

/-- [core::array::{impl core::convert::AsRef<[T]> for [T; N]}::as_ref]:
    Source: '/rustc/library/core/src/array/mod.rs', lines 208:4-208:28
    Name pattern: [core::array::{core::convert::AsRef<[@T; @N], [@T]>}::as_ref]
    Visibility: public -/
@[rust_fun "core::array::{core::convert::AsRef<[@T; @N], [@T]>}::as_ref"]
axiom Array.Insts.CoreConvertAsRefSlice.as_ref
  {T : Type} {N : Std.Usize} : Array T N → Result (Slice T)

/-- [core::array::{impl core::convert::AsMut<[T]> for [T; N]}::as_mut]:
    Source: '/rustc/library/core/src/array/mod.rs', lines 217:4-217:36
    Name pattern: [core::array::{core::convert::AsMut<[@T; @N], [@T]>}::as_mut]
    Visibility: public -/
@[rust_fun "core::array::{core::convert::AsMut<[@T; @N], [@T]>}::as_mut"]
axiom Array.Insts.CoreConvertAsMutSlice.as_mut
  {T : Type} {N : Std.Usize} :
  Array T N → Result ((Slice T) × (Slice T → Array T N))

/-- [core::borrow::{impl core::borrow::Borrow<T> for T}::borrow]:
    Source: '/rustc/library/core/src/borrow.rs', lines 214:4-214:26
    Name pattern: [core::borrow::{core::borrow::Borrow<@T, @T>}::borrow]
    Visibility: public -/
@[rust_fun "core::borrow::{core::borrow::Borrow<@T, @T>}::borrow"]
axiom core.borrow.Borrow.Blanket.borrow {T : Type} : T → Result T

/-- [core::borrow::{impl core::borrow::Borrow<T> for &'_0 T}::borrow]:
    Source: '/rustc/library/core/src/borrow.rs', lines 230:4-230:26
    Name pattern: [core::borrow::{core::borrow::Borrow<&'0 @T, @T>}::borrow]
    Visibility: public -/
@[rust_fun "core::borrow::{core::borrow::Borrow<&'0 @T, @T>}::borrow"]
axiom Shared0T.Insts.CoreBorrowBorrow.borrow {T : Type} : T → Result T

/-- [core::convert::{impl core::convert::AsRef<U> for &'_0 T}::as_ref]:
    Source: '/rustc/library/core/src/convert/mod.rs', lines 717:4-717:26
    Name pattern: [core::convert::{core::convert::AsRef<&'0 @T, @U>}::as_ref]
    Visibility: public -/
@[rust_fun "core::convert::{core::convert::AsRef<&'0 @T, @U>}::as_ref"]
axiom Shared0T.Insts.CoreConvertAsRef.as_ref
  {T : Type} {U : Type} (AsRefInst : core.convert.AsRef T U) : T → Result U

/-- [ff::Field::sqrt]:
    Source: '/cargo/registry/src/index.crates.io-1949cf8c6b5b557f/ff-0.13.1/src/lib.rs', lines 144:4-144:36
    Name pattern: [ff::Field::sqrt]
    Visibility: public -/
@[rust_fun "ff::Field::sqrt"]
axiom ff.Field.sqrt.default
  {Self : Type} (FieldInst : ff.Field Self) :
  Self → Result (subtle.CtOption Self)

/-- [ff::Field::pow_vartime]:
    Source: '/cargo/registry/src/index.crates.io-1949cf8c6b5b557f/ff-0.13.1/src/lib.rs', lines 178:4-178:58
    Name pattern: [ff::Field::pow_vartime]
    Visibility: public -/
@[rust_fun "ff::Field::pow_vartime"]
axiom ff.Field.pow_vartime.default
  {Self : Type} {S : Type} (FieldInst : ff.Field Self)
  (coreconvertAsRefSSliceU64Inst : core.convert.AsRef S (Slice Std.U64)) :
  Self → S → Result Self

/-- [ff::PrimeField::from_u128]:
    Source: '/cargo/registry/src/index.crates.io-1949cf8c6b5b557f/ff-0.13.1/src/lib.rs', lines 254:4-254:33
    Name pattern: [ff::PrimeField::from_u128]
    Visibility: public -/
@[rust_fun "ff::PrimeField::from_u128"]
axiom ff.PrimeField.from_u128.default
  {Self : Type} {Clause0_Repr : Type} (PrimeFieldInst : ff.PrimeField Self
  Clause0_Repr) :
  Std.U128 → Result Self

/-- [subtle::{subtle::Choice}::unwrap_u8]:
    Source: '/cargo/registry/src/index.crates.io-1949cf8c6b5b557f/subtle-2.6.1/src/lib.rs', lines 133:4-133:33
    Name pattern: [subtle::{subtle::Choice}::unwrap_u8]
    Visibility: public -/
@[rust_fun "subtle::{subtle::Choice}::unwrap_u8"]
axiom subtle.Choice.unwrap_u8 : subtle.Choice → Result Std.U8

/-- [subtle::{impl core::ops::bit::BitAnd<subtle::Choice, subtle::Choice> for subtle::Choice}::bitand]:
    Source: '/cargo/registry/src/index.crates.io-1949cf8c6b5b557f/subtle-2.6.1/src/lib.rs', lines 162:4-162:42
    Name pattern: [subtle::{core::ops::bit::BitAnd<subtle::Choice, subtle::Choice, subtle::Choice>}::bitand]
    Visibility: public -/
@[rust_fun
  "subtle::{core::ops::bit::BitAnd<subtle::Choice, subtle::Choice, subtle::Choice>}::bitand"]
axiom subtle.Choice.Insts.CoreOpsBitBitAndChoiceChoice.bitand
  : subtle.Choice → subtle.Choice → Result subtle.Choice

/-- [subtle::{impl core::ops::bit::Not<subtle::Choice> for subtle::Choice}::not]:
    Source: '/cargo/registry/src/index.crates.io-1949cf8c6b5b557f/subtle-2.6.1/src/lib.rs', lines 207:4-207:26
    Name pattern: [subtle::{core::ops::bit::Not<subtle::Choice, subtle::Choice>}::not]
    Visibility: public -/
@[rust_fun
  "subtle::{core::ops::bit::Not<subtle::Choice, subtle::Choice>}::not"]
axiom subtle.Choice.Insts.CoreOpsBitNotChoice.not
  : subtle.Choice → Result subtle.Choice

/-- [subtle::{impl core::convert::From<u8> for subtle::Choice}::from]:
    Source: '/cargo/registry/src/index.crates.io-1949cf8c6b5b557f/subtle-2.6.1/src/lib.rs', lines 238:4-238:32
    Name pattern: [subtle::{core::convert::From<subtle::Choice, u8>}::from]
    Visibility: public -/
@[rust_fun "subtle::{core::convert::From<subtle::Choice, u8>}::from"]
axiom subtle.Choice.Insts.CoreConvertFromU8.from
  : Std.U8 → Result subtle.Choice

/-- [subtle::{impl subtle::ConstantTimeEq for u64}::ct_eq]:
    Source: '/cargo/registry/src/index.crates.io-1949cf8c6b5b557f/subtle-2.6.1/src/lib.rs', lines 348:12-348:51
    Name pattern: [subtle::{subtle::ConstantTimeEq<u64>}::ct_eq]
    Visibility: public -/
@[rust_fun "subtle::{subtle::ConstantTimeEq<u64>}::ct_eq"]
axiom U64.Insts.SubtleConstantTimeEq.ct_eq
  : Std.U64 → Std.U64 → Result subtle.Choice

/-- [subtle::{impl subtle::ConditionallySelectable for u64}::conditional_select]:
    Source: '/cargo/registry/src/index.crates.io-1949cf8c6b5b557f/subtle-2.6.1/src/lib.rs', lines 513:12-513:77
    Name pattern: [subtle::{subtle::ConditionallySelectable<u64>}::conditional_select]
    Visibility: public -/
@[rust_fun
  "subtle::{subtle::ConditionallySelectable<u64>}::conditional_select"]
axiom U64.Insts.SubtleConditionallySelectable.conditional_select
  : Std.U64 → Std.U64 → subtle.Choice → Result Std.U64

/-- [subtle::{subtle::CtOption<T>}::new]:
    Source: '/cargo/registry/src/index.crates.io-1949cf8c6b5b557f/subtle-2.6.1/src/lib.rs', lines 678:4-678:56
    Name pattern: [subtle::{subtle::CtOption<@T>}::new]
    Visibility: public -/
@[rust_fun "subtle::{subtle::CtOption<@T>}::new"]
axiom subtle.CtOption.new
  {T : Type} : T → subtle.Choice → Result (subtle.CtOption T)

/-- [pasta_curves::fields::fp::{impl core::fmt::Debug for pasta_curves::fields::fp::Fp}::fmt]:
    Source: 'src/fields/fp.rs', lines 34:4-41:5
    Visibility: public -/
axiom fields.fp.Fp.Insts.CoreFmtDebug.fmt
  :
  fields.fp.Fp → core.fmt.Formatter → Result ((core.result.Result Unit
    core.fmt.Error) × core.fmt.Formatter)

/-- [pasta_curves::fields::fp::{impl core::cmp::PartialOrd<pasta_curves::fields::fp::Fp> for pasta_curves::fields::fp::Fp}::partial_cmp]:
    Source: 'src/fields/fp.rs', lines 92:4-94:5
    Visibility: public -/
axiom fields.fp.Fp.Insts.CoreCmpPartialOrdFp.partial_cmp
  : fields.fp.Fp → fields.fp.Fp → Result (Option Ordering)

/-- [pasta_curves::fields::fp::{impl core::cmp::Ord for pasta_curves::fields::fp::Fp}::cmp]:
    Source: 'src/fields/fp.rs', lines 77:4-88:5
    Visibility: public -/
axiom fields.fp.Fp.Insts.CoreCmpOrd.cmp
  : fields.fp.Fp → fields.fp.Fp → Result Ordering

/-- [pasta_curves::fields::fp::{impl core::iter::traits::accum::Sum<T> for pasta_curves::fields::fp::Fp}::sum]:
    Source: 'src/fields/fp.rs', lines 179:4-181:5
    Visibility: public -/
axiom fields.fp.Fp.Insts.CoreIterTraitsAccumSum.sum
  {T : Type} {I : Type} (coreborrowBorrowTFpInst : core.borrow.Borrow T
  fields.fp.Fp) (coreitertraitsiteratorIteratorInst :
  core.iter.traits.iterator.Iterator I T) :
  I → Result fields.fp.Fp

/-- [pasta_curves::fields::fp::{impl core::iter::traits::accum::Product<T> for pasta_curves::fields::fp::Fp}::product]:
    Source: 'src/fields/fp.rs', lines 185:4-187:5
    Visibility: public -/
axiom fields.fp.Fp.Insts.CoreIterTraitsAccumProduct.product
  {T : Type} {I : Type} (coreborrowBorrowTFpInst : core.borrow.Borrow T
  fields.fp.Fp) (coreitertraitsiteratorIteratorInst :
  core.iter.traits.iterator.Iterator I T) :
  I → Result fields.fp.Fp

/-- [pasta_curves::fields::fp::{impl ff::Field for pasta_curves::fields::fp::Fp}::sqrt]:
    Source: 'src/fields/fp.rs', lines 566:4-575:5
    Visibility: public -/
axiom fields.fp.Fp.Insts.FfField.sqrt
  : fields.fp.Fp → Result (subtle.CtOption fields.fp.Fp)

/-- [pasta_curves::fields::fp::{impl ff::Field for pasta_curves::fields::fp::Fp}::sqrt_ratio]:
    Source: 'src/fields/fp.rs', lines 550:4-558:5
    Visibility: public -/
axiom fields.fp.Fp.Insts.FfField.sqrt_ratio
  : fields.fp.Fp → fields.fp.Fp → Result (subtle.Choice × fields.fp.Fp)

/-- [pasta_curves::fields::fp::{impl ff::Field for pasta_curves::fields::fp::Fp}::random]:
    Source: 'src/fields/fp.rs', lines 528:4-539:5
    Visibility: public -/
axiom fields.fp.Fp.Insts.FfField.random
  {T0 : Type} (rand_coreRngCoreInst : rand_core.RngCore T0) :
  T0 → Result fields.fp.Fp

/-- [pasta_curves::fields::fp::{impl pasta_curves::arithmetic::fields::SqrtTableHelpers<[u8; 32usize]> for pasta_curves::fields::fp::Fp}::get_lower_32]:
    Source: 'src/fields/fp.rs', lines 780:4-785:5 -/
axiom
  fields.fp.Fp.Insts.Pasta_curvesArithmeticFieldsSqrtTableHelpersArrayU832.get_lower_32
  : fields.fp.Fp → Result Std.U32

/-- [pasta_curves::fields::fp::{impl pasta_curves::arithmetic::fields::SqrtTableHelpers<[u8; 32usize]> for pasta_curves::fields::fp::Fp}::pow_by_t_minus1_over2]:
    Source: 'src/fields/fp.rs', lines 749:4-778:5 -/
axiom
  fields.fp.Fp.Insts.Pasta_curvesArithmeticFieldsSqrtTableHelpersArrayU832.pow_by_t_minus1_over2
  : fields.fp.Fp → Result fields.fp.Fp

/-- [pasta_curves::fields::fp::{impl ff::WithSmallOrderMulGroup<[u8; 32usize], 3u8> for pasta_curves::fields::fp::Fp}::ZETA]
    Source: 'src/fields/fp.rs', lines 789:4-794:7
    Visibility: public -/
axiom fields.fp.Fp.Insts.FfWithSmallOrderMulGroupArrayU8323.ZETA
  : Result fields.fp.Fp

/-- [pasta_curves::fields::fp::{impl ff::FromUniformBytes<[u8; 32usize], 64usize> for pasta_curves::fields::fp::Fp}::from_uniform_bytes]:
    Source: 'src/fields/fp.rs', lines 800:4-811:5
    Visibility: public -/
axiom fields.fp.Fp.Insts.FfFromUniformBytesArrayU83264.from_uniform_bytes
  : Array Std.U8 64#usize → Result fields.fp.Fp