Require Import Coq.ZArith.ZArith Coq.ZArith.BinIntDef. Require Import Coq.Lists.List. Import ListNotations. Require Import Crypto.NewBaseSystem. Import B. Require Import Crypto.ModularArithmetic.PrimeFieldTheorems. Require Import Crypto.Util.Tactics Crypto.Util.Decidable. Require Import Crypto.Util.LetIn Crypto.Util.ZUtil. Require Crypto.Util.Tuple. Local Notation tuple := Tuple.tuple. Local Open Scope list_scope. Local Open Scope Z_scope. Local Coercion Z.of_nat : nat >-> Z. (*** Modulus : 2^255-19 Base: 25.5 ***) Section Ops25p5. Local Infix "^" := tuple : type_scope. (* These definitions will need to be passed as Ltac arguments (or cleverly inferred) when things are eventually automated *) Definition wt := fun i : nat => 2^(25 * (i / 2) + 26 * ((i + 1) / 2)). Definition sz := 10%nat. Definition s : Z := 2^255. Definition c : list B.limb := [(1, 19)]. Definition coef_div_modulus := 2. (* add 2*modulus before subtracting *) Definition carry_chain1 := Eval vm_compute in (seq 0 (pred sz)). Definition carry_chain2 := ([0;1])%nat. (* These definitions are inferred from those above *) Definition m := Eval vm_compute in Z.to_pos (s - Associational.eval c). (* modulus *) Definition sz2 := Eval vm_compute in ((sz * 2) - 1)%nat. Definition coef := Eval vm_compute in (@Positional.encode wt modulo div sz (coef_div_modulus * (s-Associational.eval c))). (* subtraction coefficient *) Definition coef_mod : mod_eq m (Positional.eval (n:=sz) wt coef) 0 := eq_refl. Lemma sz_nonzero : sz <> 0%nat. Proof. vm_decide. Qed. Lemma wt_nonzero i : wt i <> 0. Proof. apply Z.pow_nonzero; zero_bounds. Qed. Lemma wt_divides_chain1 i (H:In i carry_chain1) : wt (S i) / wt i <> 0. Proof. cbv [In carry_chain1] in H. repeat match goal with H : _ \/ _ |- _ => destruct H end; try (exfalso; assumption); subst; try vm_decide. Qed. Lemma wt_divides_chain2 i (H:In i carry_chain2) : wt (S i) / wt i <> 0. Proof. cbv [In carry_chain2] in H. repeat match goal with H : _ \/ _ |- _ => destruct H end; try (exfalso; assumption); subst; try vm_decide. Qed. Lemma wt_divides_full i : wt (S i) / wt i <> 0. Proof. cbv [wt]. match goal with |- _ ^ ?x / _ ^ ?y <> _ => assert (0 <= y <= x) end. { rewrite Nat2Z.inj_succ. split; [zero_bounds|]. apply Z.add_le_mono; (apply Z.mul_le_mono_nonneg_l; [zero_bounds|]); apply Z.div_le_mono; omega. } rewrite <-Z.pow_sub_r by omega. apply Z.pow_nonzero; omega. Qed. Definition zero_sig : { zero : Z^sz | Positional.Fdecode (m:=m) wt zero = 0%F}. Proof. let t := eval vm_compute in (Positional.encode (n:=sz) (modulo:=modulo) (div:=div) wt 0) in exists t; vm_decide. Defined. Definition one_sig : { one : Z^sz | Positional.Fdecode (m:=m) wt one = 1%F}. Proof. let t := eval vm_compute in (Positional.encode (n:=sz) (modulo:=modulo) (div:=div) wt 1) in exists t; vm_decide. Defined. Definition add_sig : { add : (Z^sz -> Z^sz -> Z^sz)%type | forall a b : Z^sz, let eval := Positional.Fdecode (m:=m) wt in eval (add a b) = (eval a + eval b)%F }. Proof. eexists; cbv beta zeta; intros. pose proof wt_nonzero. let x := constr:( Positional.add_cps (n := sz) wt a b id) in solve_op_F wt x. reflexivity. Defined. Definition sub_sig : {sub : (Z^sz -> Z^sz -> Z^sz)%type | forall a b : Z^sz, let eval := Positional.Fdecode (m:=m) wt in eval (sub a b) = (eval a - eval b)%F}. Proof. eexists; cbv beta zeta; intros. pose proof wt_nonzero. let x := constr:( Positional.sub_cps (n:=sz) (coef := coef) wt a b id) in solve_op_F wt x. reflexivity. Defined. Definition opp_sig : {opp : (Z^sz -> Z^sz)%type | forall a : Z^sz, let eval := Positional.Fdecode (m := m) wt in eval (opp a) = F.opp (eval a)}. Proof. eexists; cbv beta zeta; intros. pose proof wt_nonzero. let x := constr:( Positional.opp_cps (n:=sz) (coef := coef) wt a id) in solve_op_F wt x. reflexivity. Defined. Definition mul_sig : {mul : (Z^sz -> Z^sz -> Z^sz)%type | forall a b : Z^sz, let eval := Positional.Fdecode (m := m) wt in eval (mul a b) = (eval a * eval b)%F}. Proof. eexists; cbv beta zeta; intros. pose proof wt_nonzero. let x := constr:( Positional.mul_cps (n:=sz) (m:=sz2) wt a b (fun ab => Positional.reduce_cps (n:=sz) (m:=sz2) wt s c ab id)) in solve_op_F wt x. reflexivity. (* rough breakdown of synthesis time *) (* 1.2s for side conditions -- should improve significantly when [chained_carries] gets a correctness lemma *) (* basesystem_partial_evaluation_RHS (primarily vm_compute): 1.8s, which gets re-computed during defined *) (* doing [cbv -[Let_In runtime_add runtime_mul]] took 37s *) Defined. (* 3s *) (* Performs a full carry loop (as specified by carry_chain) *) Definition carry_sig : {carry : (Z^sz -> Z^sz)%type | forall a : Z^sz, let eval := Positional.Fdecode (m := m) wt in eval (carry a) = eval a}. Proof. eexists; cbv beta zeta; intros. pose proof wt_nonzero. pose proof wt_divides_chain1. pose proof div_mod. pose proof wt_divides_chain2. let x := constr:( Positional.chained_carries_cps (n:=sz) (div:=div)(modulo:=modulo) wt a carry_chain1 (fun r => Positional.carry_reduce_cps (n:=sz) (div:=div) (modulo:=modulo) wt s c r (fun rrr => Positional.chained_carries_cps (n:=sz) (div:=div) (modulo:=modulo) wt rrr carry_chain2 id ))) in solve_op_F wt x. reflexivity. Defined. Definition ring_25p5 := (Ring.ring_by_isomorphism (F := F m) (H := Z^sz) (phi := Positional.Fencode wt) (phi' := Positional.Fdecode wt) (zero := proj1_sig zero_sig) (one := proj1_sig one_sig) (opp := proj1_sig opp_sig) (add := proj1_sig add_sig) (sub := proj1_sig sub_sig) (mul := proj1_sig mul_sig) (phi'_zero := proj2_sig zero_sig) (phi'_one := proj2_sig one_sig) (phi'_opp := proj2_sig opp_sig) (Positional.Fdecode_Fencode_id (sz_nonzero := sz_nonzero) (div_mod := div_mod) wt eq_refl wt_nonzero wt_divides_full) (Positional.eq_Feq_iff wt) (proj2_sig add_sig) (proj2_sig sub_sig) (proj2_sig mul_sig) ). (* Eval cbv [proj1_sig add_sig] in (proj1_sig add_sig). Eval cbv [proj1_sig sub_sig] in (proj1_sig sub_sig). Eval cbv [proj1_sig opp_sig] in (proj1_sig opp_sig). Eval cbv [proj1_sig mul_sig] in (proj1_sig mul_sig). Eval cbv [proj1_sig carry_sig] in (proj1_sig carry_sig). *) End Ops25p5.