diff --git a/compiler/noirc_evaluator/src/ssa/acir_gen/acir_ir/generated_acir.rs b/compiler/noirc_evaluator/src/ssa/acir_gen/acir_ir/generated_acir.rs index a2a85498f16..bbba5bcac82 100644 --- a/compiler/noirc_evaluator/src/ssa/acir_gen/acir_ir/generated_acir.rs +++ b/compiler/noirc_evaluator/src/ssa/acir_gen/acir_ir/generated_acir.rs @@ -440,10 +440,9 @@ impl GeneratedAcir { // // When the predicate is 0, the equation always passes. // When the predicate is 1, the rhs must not be 0. - let rhs_is_zero = self.is_equal(&Expression::zero(), rhs); - let rhs_is_not_zero = &self.mul_with_witness(&rhs_is_zero.into(), predicate) - - &self.mul_with_witness(&Expression::zero(), predicate); - self.push_opcode(AcirOpcode::Arithmetic(rhs_is_not_zero)); + let rhs_is_zero = self.is_zero(rhs); + let rhs_is_not_zero = self.mul_with_witness(&rhs_is_zero.into(), predicate); + self.assert_is_zero(rhs_is_not_zero); // maximum bit size for q and for [r and rhs] let mut max_q_bits = max_bit_size; @@ -649,23 +648,26 @@ impl GeneratedAcir { /// Returns a `Witness` that is constrained to be: /// - `1` if `lhs == rhs` /// - `0` otherwise + pub(crate) fn is_equal(&mut self, lhs: &Expression, rhs: &Expression) -> Witness { + let t = lhs - rhs; + + self.is_zero(&t) + } + + /// Returns a `Witness` that is constrained to be: + /// - `1` if `t == 0` + /// - `0` otherwise /// - /// Intuition: the equality of two Expressions is linked to whether - /// their difference has an inverse; `a == b` implies that `a - b == 0` - /// which implies that a - b has no inverse. So if two variables are equal, - /// their difference will have no inverse. - /// - /// First, lets create a new variable that is equal to the difference - /// of the two expressions: `t = lhs - rhs` (constraint has been applied) + /// # Proof /// - /// Next lets create a new variable `y` which will be the Witness that we will ultimately - /// return indicating whether `lhs == rhs`. + /// First, let's create a new variable `y` which will be the Witness that we will ultimately + /// return indicating whether `t == 0`. /// Note: During this process we need to apply constraints that ensure that it is a boolean. /// But right now with no constraints applied to it, it is essentially a free variable. /// /// Next we apply the following constraint `y * t == 0`. /// This implies that either `y` or `t` or both is `0`. - /// - If `t == 0`, then this means that `lhs == rhs`. + /// - If `t == 0`, then by definition `t == 0`. /// - If `y == 0`, this does not mean anything at this point in time, due to it having no /// constraints. /// @@ -673,44 +675,46 @@ impl GeneratedAcir { /// This along with the previous `y * t == 0` constraint means that /// `y` or `t` needs to be zero, but they both cannot be zero. /// - /// This equation however falls short when lhs != rhs because then `t` + /// This equation however falls short when `t != 0` because then `t` /// may not be `1`. If `t` is non-zero, then `y` is also non-zero due to /// `y == 1 - t` and the equation `y * t == 0` fails. /// /// To fix, we introduce another free variable called `z` and apply the following /// constraint instead: `y == 1 - t * z`. /// - /// When `lhs == rhs`, `t` is `0` and so `y` is `1`. - /// When `lhs != rhs`, `t` is non-zero, however the prover can set `z = 1/t` - /// which will make `y = 1 - t * 1/t = 0`. + /// When `t == 0`, `y` is `1`. + /// When `t != 0`, the prover can set `z = 1/t` which will make `y = 1 - t * 1/t = 0`. /// - /// We now arrive at the conclusion that when `lhs == rhs`, `y` is `1` and when - /// `lhs != rhs`, then `y` is `0`. + /// We now arrive at the conclusion that when `t == 0`, `y` is `1` and when + /// `t != 0`, then `y` is `0`. /// - /// Bringing it all together, We introduce three variables `y`, `t` and `z`, + /// Bringing it all together, We introduce two variables `y` and `z`, /// With the following equations: - /// - `t == lhs - rhs` /// - `y == 1 - tz` (`z` is a value that is chosen to be the inverse of `t` by the prover) /// - `y * t == 0` /// /// Lets convince ourselves that the prover cannot prove an untrue statement. /// - /// Assume that `lhs == rhs`, can the prover return `y == 0`? + /// --- + /// Assume that `t == 0`, can the prover return `y == 0`? /// - /// When `lhs == rhs`, `t` is 0. There is no way to make `y` be zero - /// since `y = 1 - 0 * z = 1`. + /// When `t == 0`, there is no way to make `y` be zero since `y = 1 - 0 * z = 1`. /// - /// Assume that `lhs != rhs`, can the prover return `y == 1`? + /// --- + /// Assume that `t != 0`, can the prover return `y == 1`? /// - /// When `lhs != rhs`, then `t` is non-zero. /// By setting `z` to be `0`, we can make `y` equal to `1`. /// This is easily observed: `y = 1 - t * 0` /// Now since `y` is one, this means that `t` needs to be zero, or else `y * t == 0` will fail. - pub(crate) fn is_equal(&mut self, lhs: &Expression, rhs: &Expression) -> Witness { - let t = lhs - rhs; - // We avoid passing the expression to `self.brillig_inverse` directly because we need - // the `Witness` representation for constructing `y_is_boolean_constraint`. - let t_witness = self.get_or_create_witness(&t); + fn is_zero(&mut self, t_expr: &Expression) -> Witness { + // We're checking for equality with zero so we can negate the expression without changing the result. + // This is useful as it will sometimes allow us to simplify an expression down to a witness. + let t_witness = if let Some(witness) = t_expr.to_witness() { + witness + } else { + let negated_expr = t_expr * -FieldElement::one(); + self.get_or_create_witness(&negated_expr) + }; // Call the inversion directive, since we do not apply a constraint // the prover can choose anything here. diff --git a/tooling/nargo_cli/tests/acir_artifacts/2_div/target/acir.gz b/tooling/nargo_cli/tests/acir_artifacts/2_div/target/acir.gz index 6ddfd323e0f..5b64d6c475e 100644 Binary files a/tooling/nargo_cli/tests/acir_artifacts/2_div/target/acir.gz and b/tooling/nargo_cli/tests/acir_artifacts/2_div/target/acir.gz differ diff --git a/tooling/nargo_cli/tests/acir_artifacts/2_div/target/witness.gz b/tooling/nargo_cli/tests/acir_artifacts/2_div/target/witness.gz index a56f646c14d..fe5aad35bfc 100644 Binary files a/tooling/nargo_cli/tests/acir_artifacts/2_div/target/witness.gz and 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