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Diffstat (limited to 'kvx/FPDivision64.v')
-rw-r--r-- | kvx/FPDivision64.v | 2670 |
1 files changed, 2670 insertions, 0 deletions
diff --git a/kvx/FPDivision64.v b/kvx/FPDivision64.v new file mode 100644 index 00000000..298831a0 --- /dev/null +++ b/kvx/FPDivision64.v @@ -0,0 +1,2670 @@ +(* +This needs a special gappa script + +#!/bin/sh +/home/monniaux/.opam/4.12.0+flambda/bin/gappa -Eprecision=100 "$@" + +in PATH before the normal gappa + *) + +From Flocq Require Import Core Digits Operations Round Bracket Sterbenz + Binary Round_odd Bits. +Require Archi. +Require Import Coqlib. +Require Import Compopts. +Require Import AST. +Require Import Integers. +Require Import Floats. +Require Import Op. +Require Import CminorSel. +Require Import OpHelpers. +Require Import ExtFloats. +Require Import DecBoolOps. +Require Import Chunks. +Require Import Builtins. +Require Import Values Globalenvs. +Require Compopts. +Require Import Psatz. +Require Import IEEE754_extra. +Require Import Memory. + +From Gappa Require Import Gappa_tactic. + +Definition approx_inv_longu b := + let invb_s := ExtValues.invfs (Val.singleoffloat (Val.maketotal (Val.floatoflongu b))) in + let invb_d := Val.floatofsingle invb_s in + let b_d := Val.maketotal (Val.floatoflongu b) in + let one := Vfloat (ExtFloat.one) in + let alpha := ExtValues.fmsubf one invb_d b_d in + ExtValues.fmaddf invb_d alpha invb_d. + +Lemma Rabs_relax: + forall b b' (INEQ : (b < b')%R) x, + (-b <= x <= b)%R -> (Rabs x < b')%R. +Proof. + intros. + apply Rabs_lt. + lra. +Qed. + +Local Notation "'rd'" := (round radix2 (FLT_exp (-1074) 53) ZnearestE). +Local Notation "'rs'" := (round radix2 (FLT_exp (-149) 24) ZnearestE). + +Definition approx_inv_rel_thresh := (1049/72057594037927936)%R. +Theorem approx_inv_longu_correct_rel : + forall b, + (0 < Int64.unsigned b)%Z -> + exists (f : float), + (approx_inv_longu (Vlong b)) = Vfloat f /\ + is_finite _ _ f = true /\ (Rabs(IZR (Int64.unsigned b) * (B2R _ _ f) - 1) <= approx_inv_rel_thresh)%R. +Proof. + intros b NONZ. + unfold approx_inv_longu. + cbn. + econstructor. + split. + reflexivity. + Local Transparent Float.neg Float.of_single Float32.of_longu Float32.div Float.of_longu Float32.of_int Float.of_int Float.to_single. + unfold Float.fma, Float.neg, Float.of_single, Float32.of_longu, ExtFloat32.inv, Float32.div, Float.of_longu, ExtFloat32.one, Float32.of_int, ExtFloat.one, Float.of_int, Float.to_single. + set (re := (@eq_refl Datatypes.comparison Lt)). + change (Int.signed (Int.repr 1)) with 1%Z. + set (b' := Int64.unsigned b) in *. + pose proof (Int64.unsigned_range b) as RANGE. + change Int64.modulus with 18446744073709551616%Z in RANGE. + assert(1 <= IZR b' <= 18446744073709551616)%R as RANGE'. + { split; apply IZR_le; lia. + } + + assert (-16777216 <= 1 <= 16777216)%Z as SILLY by lia. + destruct (BofZ_exact 24 128 re re 1 SILLY) as (C0R & C0F & _). + clear SILLY. + set (one_s := (BofZ 24 128 re re 1)) in *. + + pose proof (BofZ_correct 53 1024 re re b') as C5. + rewrite Rlt_bool_true in C5; cycle 1. + { clear C5. + eapply (Rabs_relax (bpow radix2 64)). + { apply bpow_lt. lia. } + cbn. + gappa. + } + cbn in C5. + destruct C5 as (C5R & C5F & C5S). + set (b_d := (BofZ 53 1024 re re b')) in *. + + pose proof (Bconv_correct 53 1024 24 128 re re Float.to_single_nan mode_NE b_d C5F) as C1. + rewrite Rlt_bool_true in C1; cycle 1. + { clear C1. + apply (Rabs_relax (bpow radix2 64)). + { apply bpow_lt. lia. } + rewrite C5R. + cbn. + gappa. + } + cbn in C1. + destruct C1 as (C1R & C1F & C1S). + set (b_s := (Bconv 53 1024 24 128 re re Float.to_single_nan mode_NE b_d)) in *. + assert(1 <= B2R 24 128 b_s <= 18446744073709551616)%R as b_s_RANGE. + { rewrite C1R. + rewrite C5R. + cbn. + gappa. + } + assert(B2R 24 128 b_s <> 0)%R as b_s_NONZ by lra. + + pose proof (Bdiv_correct 24 128 re re Float32.binop_nan mode_NE one_s b_s b_s_NONZ) as C2. + rewrite Rlt_bool_true in C2; cycle 1. + { clear C2. + apply Rabs_relax with (b := 1%R). + { cbn; lra. } + rewrite C0R. + set (r_b_s := B2R 24 128 b_s) in *. + cbn. + gappa. + } + + destruct C2 as (C2R & C2F & _). + set (invb_s := (Bdiv 24 128 re re Float32.binop_nan mode_NE one_s b_s)) in *. + rewrite C0F in C2F. + + assert ((1/18446744073709551616 <= B2R 24 128 invb_s <= 1)%R) as invb_s_RANGE. + { rewrite C2R. + set (r_b_s := B2R 24 128 b_s) in *. + rewrite C0R. + cbn. + gappa. + } + + pose proof (Bconv_correct 24 128 53 1024 re re Float.of_single_nan mode_NE invb_s C2F) as C3. + rewrite Rlt_bool_true in C3; cycle 1. + { clear C3. + set (r_invb_s := (B2R 24 128 invb_s)) in *. + apply Rabs_relax with (b := 1%R). + { replace 1%R with (bpow radix2 0)%R by reflexivity. + apply bpow_lt. + lia. + } + cbn. + gappa. + } + + destruct C3 as (C3R & C3F & _). + set (invb_d := (Bconv 24 128 53 1024 re re Float.of_single_nan mode_NE invb_s)) in *. + assert ((1/18446744073709551616 <= B2R 53 1024 invb_d <= 1)%R) as invb_d_RANGE. + { + rewrite C3R. + set (r_invb_s := B2R 24 128 invb_s) in *. + cbn. + gappa. + } + + pose proof (is_finite_Bopp 53 1024 Float.neg_nan invb_d) as opp_finite. + rewrite C3F in opp_finite. + + pose proof (BofZ_correct 53 1024 re re 1) as C4. + rewrite Rlt_bool_true in C4; cycle 1. + { clear C4. + cbn. + eapply (Rabs_relax (IZR 18446744073709551616)). + lra. + set (b'' := IZR b') in *. + gappa. + } + destruct C4 as (C4R & C4F & _). + + assert(1 <= B2R 53 1024 b_d <= 18446744073709551616)%R as b_d_RANGE. + { rewrite C5R. + gappa. + } + + pose proof (Bfma_correct 53 1024 re re Float.fma_nan mode_NE + (Bopp 53 1024 Float.neg_nan invb_d) (BofZ 53 1024 re re b') + (BofZ 53 1024 re re 1) opp_finite C5F C4F) as C6. + rewrite Rlt_bool_true in C6; cycle 1. + { clear C6. + rewrite C4R. + rewrite B2R_Bopp. + cbn. + eapply (Rabs_relax (IZR 18446744073709551616)). + { lra. } + fold invb_d. + fold b_d. + set (r_invb_d := B2R 53 1024 invb_d) in *. + set (r_b_d := B2R 53 1024 b_d) in *. + gappa. + } + fold b_d in C6. + destruct C6 as (C6R & C6F & _). + + pose proof (Bfma_correct 53 1024 re re Float.fma_nan mode_NE + (Bfma 53 1024 re re Float.fma_nan mode_NE + (Bopp 53 1024 Float.neg_nan invb_d) b_d (BofZ 53 1024 re re 1)) + invb_d invb_d C6F C3F C3F) as C7. + rewrite Rlt_bool_true in C7; cycle 1. + { clear C7. + rewrite C6R. + rewrite B2R_Bopp. + eapply (Rabs_relax (bpow radix2 64)). + { apply bpow_lt. lia. } + rewrite C4R. + cbn. + set (r_invb_d := B2R 53 1024 invb_d) in *. + set (r_b_d := B2R 53 1024 b_d) in *. + gappa. + } + destruct C7 as (C7R & C7F & _). + + split. assumption. + rewrite C7R. + rewrite C6R. + rewrite C5R. + rewrite C4R. + rewrite B2R_Bopp. + rewrite C3R. + rewrite C2R. + rewrite C1R. + rewrite C5R. + rewrite C0R. + cbn. + set(b1 := IZR b') in *. + + replace (rd 1) with 1%R by gappa. + replace (rd (rs (1 / rs (rd b1)))) with + ((((rd (rs (1 / rs (rd b1))) - (/b1))/(/b1))+1)*(/ b1))%R ; cycle 1. + { field. lra. } + set (er0 := ((rd (rs (1 / rs (rd b1))) - (/b1))/(/b1))%R). + replace (rd b1) with ((((rd b1) - b1)/b1 + 1) * b1)%R; cycle 1. + { field. lra. } + set (er1 := (((rd b1) - b1)/b1)%R). + replace (- ((er0 + 1) * / b1) * ((er1 + 1) * b1) + 1)%R + with (1 - (er0 + 1)*(er1 + 1))%R ; cycle 1. + { field. lra. } + set (z0 := (1 - (er0 + 1) * (er1 + 1))%R). + assert (Rabs er0 <= 257/2147483648)%R as er0_ABS. + { unfold er0. + gappa. + } + assert (Rabs er1 <= 1/9007199254740992)%R as er1_ABS. + { unfold er1. + gappa. + } + replace (rd z0) with ((rd(z0)-z0)+z0)%R by ring. + set (ea0 := (rd(z0)-z0)%R). + assert (Rabs ea0 <= 1/75557863725914323419136)%R as ea0_ABS. + { unfold ea0. unfold z0. + gappa. + } + set (z1 := ((ea0 + z0) * ((er0 + 1) * / b1) + (er0 + 1) * / b1)%R). + replace (rd z1) with ((((rd z1)-z1)/z1+1)*z1)%R; cycle 1. + { field. + unfold z1. + unfold z0. + gappa. + } + set (er2 := ((rd z1 - z1) / z1)%R). + assert (Rabs er2 <= 1/9007199254740992)%R as er2_ABS. + { unfold er2. + unfold z1, z0. + gappa. + } + unfold z1, z0. + replace (b1 * + ((er2 + 1) * + ((ea0 + (1 - (er0 + 1) * (er1 + 1))) * ((er0 + 1) * / b1) + + (er0 + 1) * / b1)) - 1)%R + with (-er0*er0*er1*er2 - er0*er0*er1 + ea0*er0*er2 - er0*er0*er2 - 2*er0*er1*er2 + ea0*er0 - er0*er0 - 2*er0*er1 + ea0*er2 - er1*er2 + ea0 - er1 + er2)%R; cycle 1. + { field. lra. } + unfold approx_inv_rel_thresh. + gappa. +Qed. + +Definition step1_real_inv_longu b := + let invb_s := ExtValues.invfs (Val.singleoffloat (Val.maketotal (Val.floatoflongu b))) in + Val.floatofsingle invb_s. + +Definition step1_real_inv_thresh := (3/33554432)%R. +(* 8.94069671630859e-8 *) + +Theorem step1_real_inv_longu_correct : + forall b, + (0 < Int64.unsigned b)%Z -> + exists (f : float), + (step1_real_inv_longu (Vlong b)) = Vfloat f /\ + (B2R _ _ f) = (rd (rs (1 / rs (rd (IZR (Int64.unsigned b)))))) /\ + is_finite _ _ f = true /\ + Bsign _ _ f = false. +Proof. + intros b NONZ. + unfold step1_real_inv_longu. + cbn. + econstructor. + split. + reflexivity. + Local Transparent Float.neg Float.of_single Float32.of_longu Float32.div Float.of_longu Float32.of_int Float.of_int Float.to_single. + unfold Float.fma, Float.neg, Float.of_single, Float32.of_longu, ExtFloat32.inv, Float32.div, Float.of_longu, ExtFloat32.one, Float32.of_int, ExtFloat.one, Float.of_int, Float.to_single. + set (re := (@eq_refl Datatypes.comparison Lt)). + change (Int.signed (Int.repr 1)) with 1%Z. + set (b' := Int64.unsigned b) in *. + pose proof (Int64.unsigned_range b) as RANGE. + change Int64.modulus with 18446744073709551616%Z in RANGE. + assert(1 <= IZR b' <= 18446744073709551616)%R as RANGE'. + { split; apply IZR_le; lia. + } + + assert (-16777216 <= 1 <= 16777216)%Z as SILLY by lia. + destruct (BofZ_exact 24 128 re re 1 SILLY) as (C0R & C0F & C0S). + clear SILLY. + set (one_s := (BofZ 24 128 re re 1)) in *. + + pose proof (BofZ_correct 53 1024 re re b') as C0'. + rewrite Rlt_bool_true in C0'; cycle 1. + { apply (Rabs_relax (bpow radix2 64)). + { apply bpow_lt. lia. } + cbn. + gappa. + } + cbn in C0'. + destruct C0' as (C0'R & C0'F & C0'S). + set (b_d := (BofZ 53 1024 re re b')) in *. + + pose proof (Bconv_correct 53 1024 24 128 re re Float.to_single_nan mode_NE b_d C0'F) as C1. + rewrite C0'R in C1. + rewrite C0'S in C1. + rewrite Rlt_bool_true in C1; cycle 1. + { clear C1. + eapply (Rabs_relax (bpow radix2 64)). + { apply bpow_lt. lia. } + cbn. + gappa. + } + destruct C1 as (C1R & C1F & C1S). + set (b_s := (Bconv 53 1024 24 128 re re Float.to_single_nan mode_NE b_d)) in *. + + assert(1 <= B2R 24 128 b_s <= 18446744073709551616)%R as b_s_RANGE. + { rewrite C1R. + cbn. + gappa. + } + assert(B2R 24 128 b_s <> 0)%R as b_s_NONZ by lra. + + pose proof (Bdiv_correct 24 128 re re Float32.binop_nan mode_NE one_s b_s b_s_NONZ) as C2. + rewrite Rlt_bool_true in C2; cycle 1. + { clear C2. + apply Rabs_relax with (b := 1%R). + { cbn; lra. } + rewrite C0R. + set (r_b_s := B2R 24 128 b_s) in *. + cbn. + gappa. + } + rewrite C1R in C2. + rewrite C1S in C2. + rewrite C0S in C2. + destruct C2 as (C2R & C2F & C2Sz). + change (1 <? 0)%Z with false in C2Sz. + replace (b' <? 0)%Z with false in C2Sz by lia. + change (xorb false false) with false in C2Sz. + set (invb_s := (Bdiv 24 128 re re Float32.binop_nan mode_NE one_s b_s)) in *. + rewrite C0F in C2F. + assert (is_nan 24 128 invb_s = false) as NAN. + { apply is_finite_not_is_nan. + assumption. + } + pose proof (C2Sz NAN) as C2S. + clear C2Sz. + + assert ((1/18446744073709551616 <= B2R 24 128 invb_s <= 1)%R) as invb_s_RANGE. + { rewrite C2R. + set (r_b_s := B2R 24 128 b_s) in *. + rewrite C0R. + cbn. + gappa. + } + + pose proof (Bconv_correct 24 128 53 1024 re re Float.of_single_nan mode_NE invb_s C2F) as C3. + rewrite Rlt_bool_true in C3; cycle 1. + { clear C3. + set (r_invb_s := (B2R 24 128 invb_s)) in *. + apply Rabs_relax with (b := 1%R). + { replace 1%R with (bpow radix2 0)%R by reflexivity. + apply bpow_lt. + lia. + } + cbn. + gappa. + } + destruct C3 as (C3R & C3F & C3S). + set (invb_d := (Bconv 24 128 53 1024 re re Float.of_single_nan mode_NE invb_s)) in *. + assert ((1/18446744073709551616 <= B2R 53 1024 invb_d <= 1)%R) as invb_d_RANGE. + { + rewrite C3R. + set (r_invb_s := B2R 24 128 invb_s) in *. + cbn. + gappa. + } + rewrite C2S in C3S. + rewrite C2R in C3R. + rewrite C0R in C3R. + + auto. +Qed. + +Theorem step1_real_inv_longu_correct1 : + forall b, + (Int64.unsigned b = 1%Z) -> + exists f, + (step1_real_inv_longu (Vlong b)) = Vfloat f /\ + (B2R _ _ f) = 1%R /\ + is_finite _ _ f = true /\ + Bsign _ _ f = false. +Proof. + intros b EQ1. + assert (0 < Int64.unsigned b)%Z as b_RANGE by lia. + destruct (step1_real_inv_longu_correct b b_RANGE) as (f & C1E & C1R & C1F & C1S). + rewrite EQ1 in C1R. + exists f. + repeat split; try assumption. + rewrite C1R. + gappa. +Qed. + +Lemma Bsign_false_nonneg: + forall prec emax f, + (Bsign prec emax f) = false -> (0 <= (B2R prec emax f))%R. +Proof. + intros until f. intro SIGN. + destruct f. + 1, 2, 3: cbn; lra. + cbn. + apply F2R_ge_0. + cbn. + cbn in SIGN. + rewrite SIGN. + cbn. + lia. +Qed. + +Lemma Znearest_IZR_le : + forall rnd n x, (IZR n <= x)%R -> (n <= Znearest rnd x)%Z. +Proof. + intros until x. intro ORDER. + pose proof (Znearest_ge_floor rnd x). + pose proof (Zfloor_le _ _ ORDER) as KK. + rewrite Zfloor_IZR in KK. + lia. +Qed. + +Lemma Znearest_le_IZR : + forall rnd n x, (x <= IZR n)%R -> (Znearest rnd x <= n)%Z. +Proof. + intros until x. intro ORDER. + pose proof (Znearest_le_ceil rnd x). + pose proof (Zceil_le _ _ ORDER) as KK. + rewrite Zceil_IZR in KK. + lia. +Qed. + +Definition step1_real_div_longu a b := + Val.mulf (Val.maketotal (Val.floatoflongu a)) (step1_real_inv_longu b). + +Definition step1_div_longu a b := + Val.maketotal (Val.longuoffloat_ne (step1_real_div_longu a b)). + +Definition step1_real_quotient (a b : R) := + rd ((rd (a)) * (rd (rs (1 / rs (rd (b)))))). + +Theorem step1_real_div_longu_correct: + forall a b, + (1 < Int64.unsigned b)%Z -> + exists (q : float), + (step1_real_div_longu (Vlong a) (Vlong b)) = Vfloat q /\ + (B2R _ _ q) = step1_real_quotient (IZR (Int64.unsigned a)) + (IZR (Int64.unsigned b)) /\ + is_finite _ _ q = true /\ + Bsign _ _ q = false. +Proof. + intros a b b_NON01. + assert (0 < Int64.unsigned b)%Z as b_NON0 by lia. + destruct (step1_real_inv_longu_correct b b_NON0) as (f & C1E & C1R & C1F & C1S). + unfold step1_real_div_longu. + rewrite C1E. + cbn. + set (b' := Int64.unsigned b) in *. + Local Transparent Float.mul. + unfold Float.mul, Float.of_longu. + econstructor. + split. reflexivity. + set (a' := Int64.unsigned a) in *. + set (re := (@eq_refl Datatypes.comparison Lt)). + + pose proof (Int64.unsigned_range a) as a_RANGE. + change Int64.modulus with 18446744073709551616%Z in a_RANGE. + assert (0 <= IZR a' <= 18446744073709551615)%R as IZR_a_RANGE. + { split; apply IZR_le; lia. } + pose proof (Int64.unsigned_range b) as b_RANGE. + change Int64.modulus with 18446744073709551616%Z in b_RANGE. + assert (2 <= IZR b' <= 18446744073709551615)%R as IZR_b_RANGE. + { split; apply IZR_le; lia. } + + pose proof (BofZ_correct 53 1024 re re a') as C2. + rewrite Rlt_bool_true in C2; cycle 1. + { clear C2. + apply Rabs_relax with (b := bpow radix2 64). + { apply bpow_lt. lia. } + cbn. + gappa. + } + destruct C2 as (C2R & C2F & C2S). + rewrite Zlt_bool_false in C2S by lia. + + pose proof (Bmult_correct 53 1024 re re Float.binop_nan mode_NE (BofZ 53 1024 re re a') f) as C3. + rewrite C1S in C3. + rewrite C2S in C3. + rewrite C1F in C3. + rewrite C2F in C3. + rewrite C1R in C3. + rewrite C2R in C3. + rewrite Rlt_bool_true in C3; cycle 1. + { apply Rabs_relax with (b := bpow radix2 64). + { apply bpow_lt ; lia. } + cbn. + gappa. + } + cbn in C3. + destruct C3 as (C3R & C3F & C3Sz). + assert (is_nan 53 1024 + (Bmult 53 1024 re re Float.binop_nan mode_NE + (BofZ 53 1024 re re a') f) = false) as NAN. + { apply is_finite_not_is_nan. + assumption. } + pose proof (C3Sz NAN) as C3S. + clear NAN C3Sz. + + auto. +Qed. + +Definition smallb_thresh := 4398046511104%Z. + +Definition smallb_approx_real_range := 2200000000000%R. +Lemma step1_smallb_real : + forall a b + (a_RANGE : (1 <= a <= 18446744073709551615)%R) + (b_RANGE : (1 <= b <= IZR smallb_thresh)%R), + (Rabs((step1_real_quotient a b) * b - a) <= smallb_approx_real_range)%R. +Proof. + intros. + unfold smallb_thresh in b_RANGE. + unfold smallb_approx_real_range. + unfold step1_real_quotient. + set (q := ((rd (a)) * (rd (rs (1 / rs (rd b)))))%R) in *. + replace ((rd q) *b - a)%R with + ((rd(q)-q)/q * rd(a) * (1 + (rd (rs (1 / rs (rd b))) - 1/b)/(1/b)) + + (rd (a)) * ((rd (rs (1 / rs (rd b))) - 1 / b) / (1/b)) + + (rd(a) - a))%R; cycle 1. + { unfold q. + field. + split. lra. + split. gappa. + gappa. + } + unfold q. + gappa. +Qed. + +Lemma step1_div_longu_a0 : + forall b, + (0 < Int64.unsigned b)%Z -> + (step1_div_longu (Vlong Int64.zero) (Vlong b)) = Vlong Int64.zero. +Proof. + intros b b_NOT0. + unfold step1_div_longu. + unfold step1_real_div_longu. + destruct (step1_real_inv_longu_correct b b_NOT0) as + (f & C1E & C1R & C1F & C1S). + rewrite C1E. + cbn. + unfold Float.to_longu_ne, Float.of_longu, Float.mul. + rewrite Int64.unsigned_zero. + set (re := (@eq_refl Datatypes.comparison Lt)). + assert (- 2 ^ 53 <= 0 <= 2 ^ 53)%Z as SILLY by lia. + destruct (BofZ_exact 53 1024 re re 0 SILLY) as (C2R & C2F & C2S). + + pose proof (Bmult_correct 53 1024 re re Float.binop_nan mode_NE + (BofZ 53 1024 re re 0) f) as C3. + rewrite C1F in C3. + rewrite C2F in C3. + rewrite C1S in C3. + rewrite C2S in C3. + rewrite Z.ltb_irrefl in C3. + rewrite Rlt_bool_true in C3; cycle 1. + { clear C3. + apply Rabs_relax with (b := bpow radix2 64). + { apply bpow_lt. lia. } + cbn. + rewrite Rmult_0_l. + gappa. + } + rewrite C2R in C3. + rewrite Rmult_0_l in C3. + destruct C3 as (C3R & C3F & C3Sz). + change (true && true) with true in C3F. + change (xorb false false) with false in C3Sz. + assert (is_nan 53 1024 + (Bmult 53 1024 re re Float.binop_nan mode_NE + (BofZ 53 1024 re re 0) f) = false) as NAN. + { apply is_finite_not_is_nan. + assumption. + } + pose proof (C3Sz NAN) as C3S. + clear NAN C3Sz. + pose proof ((ZofB_ne_range_correct 53 1024 + (Bmult 53 1024 re re Float.binop_nan mode_NE + (BofZ 53 1024 re re 0) f) 0 Int64.max_unsigned)) as C4. + rewrite C3R in C4. + replace (round radix2 (FLT_exp (3 - 1024 - 53) 53) (round_mode mode_NE) 0) + with 0%R in C4 by (cbn ; gappa). + rewrite Znearest_IZR in C4. + cbn zeta in C4. + rewrite Z.leb_refl in C4. + change (0 <=? Int64.max_unsigned)%Z with true in C4. + rewrite andb_true_r in C4. + rewrite andb_true_r in C4. + rewrite C3F in C4. + rewrite C4. + reflexivity. +Qed. + +Lemma step1_div_longu_correct_anyb : + forall a b + (b_NOT01 : (1 < Int64.unsigned b)%Z), + exists (q : int64), + (step1_div_longu (Vlong a) (Vlong b)) = Vlong q. +Proof. + intros. + + pose proof (Int64.unsigned_range a) as a_RANGE. + pose proof (Int64.unsigned_range b) as b_RANGE. + change Int64.modulus with 18446744073709551616%Z in *. + set (a' := Int64.unsigned a) in *. + set (b' := Int64.unsigned b) in *. + assert (0 <= IZR a' <= 18446744073709551615)%R as a_RANGE'. + { split; apply IZR_le; lia. } + assert (2 <= IZR b' <= 18446744073709551615)%R as b_RANGE'. + { split; apply IZR_le; lia. } + + assert (0 < b')%Z as b_NOT0 by lia. + + destruct (Z_le_gt_dec a' 0). + { assert (a' = 0%Z) as ZERO by lia. + replace a with Int64.zero; cycle 1. + { + unfold a' in ZERO. + unfold Int64.zero. + rewrite <- ZERO. + apply Int64.repr_unsigned. + } + exists Int64.zero. + apply step1_div_longu_a0. + exact b_NOT0. + } + + unfold step1_div_longu. + unfold step1_real_div_longu. + destruct (step1_real_inv_longu_correct b b_NOT0) as (f & C1E & C1R & C1F & C1S). + rewrite C1E. + cbn. + unfold Float.of_longu, Float.mul, Float.to_longu_ne. + set (re := (@eq_refl Datatypes.comparison Lt)). + fold a'. + fold b' in C1R. + pose proof (BofZ_correct 53 1024 re re a') as C2. + rewrite Rlt_bool_true in C2; cycle 1. + { clear C2. + apply Rabs_relax with (b := bpow radix2 64). + { apply bpow_lt. lia. } + cbn. + gappa. + } + cbn in C2. + destruct C2 as (C2R & C2F & C2S). + pose proof (Bmult_correct 53 1024 re re Float.binop_nan mode_NE + (BofZ 53 1024 re re a') f) as C3. + rewrite C2R in C3. + rewrite C2F in C3. + rewrite C2S in C3. + rewrite C1R in C3. + rewrite C1F in C3. + rewrite C1S in C3. + rewrite Rlt_bool_true in C3; cycle 1. + { clear C3. + apply Rabs_relax with (b := bpow radix2 64). + { apply bpow_lt. lia. } + cbn. + gappa. + } + cbn in C3. + destruct C3 as (C3R & C3F & _). + pose proof (ZofB_ne_range_correct 53 1024 + (Bmult 53 1024 re re Float.binop_nan mode_NE + (BofZ 53 1024 re re a') f) 0 Int64.max_unsigned) as C4. + rewrite C3R in C4. + rewrite C3F in C4. + cbn zeta in C4. + rewrite Zle_bool_true in C4 ; cycle 1. + { clear C4. + apply Znearest_lub. + gappa. + } + rewrite Zle_bool_true in C4 ; cycle 1. + { clear C4. + apply Znearest_glb. + cbn. + gappa. + } + rewrite C4. + cbn. + eauto. +Qed. + +Definition smallb_approx_range := 4400000000000%Z. +Lemma step1_div_longu_correct : + forall a b, + (1 < Int64.unsigned b <= smallb_thresh)%Z -> + exists (q : int64), + (step1_div_longu (Vlong a) (Vlong b)) = Vlong q /\ + (Z.abs (Int64.unsigned a - Int64.unsigned b*Int64.unsigned q) <= smallb_approx_range)%Z. +Proof. + intros a b b_RANGE. + + pose proof (Int64.unsigned_range a) as a_RANGE. + change Int64.modulus with 18446744073709551616%Z in a_RANGE. + set (a' := Int64.unsigned a) in *. + set (b' := Int64.unsigned b) in *. + + destruct (Z_le_gt_dec a' 0). + { assert (a' = 0%Z) as ZERO by lia. + exists Int64.zero. + rewrite ZERO. + rewrite Int64.unsigned_zero. + replace (Z.abs (0 - b' * 0))%Z with 0%Z by lia. + replace a with Int64.zero; cycle 1. + { + unfold a' in ZERO. + unfold Int64.zero. + rewrite <- ZERO. + apply Int64.repr_unsigned. + } + split. + { apply step1_div_longu_a0. + lia. + } + unfold smallb_approx_range. + lia. + } + + unfold step1_div_longu. + assert (1 < b')%Z as b_NOT01 by lia. + destruct (step1_real_div_longu_correct a b b_NOT01) as (q & C1E & C1R & C1F & C1S). + rewrite C1E. cbn. + unfold Float.to_longu_ne. + pose proof (ZofB_ne_range_correct 53 1024 q 0 Int64.max_unsigned) as C2. + rewrite C1F in C2. + + + assert (1 <= IZR a' <= 18446744073709551615)%R as a_RANGE'. + { split; apply IZR_le; lia. } + assert (2 <= IZR b' <= IZR smallb_thresh)%R as b_RANGE'. + { split; apply IZR_le; lia. } + assert (1 <= IZR b' <= IZR smallb_thresh)%R as b_RANGE'' by lra. + pose proof (step1_smallb_real (IZR a') (IZR b') a_RANGE' b_RANGE'') as DELTA. + fold a' in C1R. + fold b' in C1R. + rewrite <- C1R in DELTA. + + assert (0 <= B2R _ _ q)%R as q_NONNEG. + { apply Bsign_false_nonneg. assumption. } + cbn in C2. + rewrite Zle_bool_true in C2; cycle 1. + { apply Znearest_IZR_le. assumption. } + assert (B2R _ _ q <= 9223376000000000000)%R as q_SMALL. + { replace (B2R _ _ q) with + ((IZR a' / IZR b') + (B2R _ _ q * IZR b' - IZR a') / IZR b')%R; cycle 1. + { field. lra. } + unfold smallb_approx_real_range in DELTA. + unfold smallb_thresh in b_RANGE'. + set (y := (B2R 53 1024 q * IZR b' - IZR a')%R) in *. + gappa. + } + rewrite Zle_bool_true in C2; cycle 1. + { apply Znearest_le_IZR. lra. } + cbn in C2. + + change Int64.max_unsigned with 18446744073709551615%Z. + rewrite C2. + cbn. + + econstructor. split. reflexivity. + rewrite Int64.unsigned_repr; cycle 1. + { split. + - apply Znearest_IZR_le. lra. + - apply Znearest_le_IZR. + change Int64.max_unsigned with 18446744073709551615%Z. + lra. + } + apply le_IZR. + rewrite abs_IZR. + unfold smallb_approx_real_range, smallb_approx_range, smallb_thresh in *. + rewrite minus_IZR. + rewrite mult_IZR. + set (q_r := B2R 53 1024 q) in *. + assert (Rabs (IZR (ZnearestE q_r) - q_r) <= / 2)%R as NEAR + by apply Znearest_imp2. + set (q_i := IZR (ZnearestE q_r)) in *. + replace (IZR a' - IZR b' * q_i)%R with + (-(IZR b' * (q_i - q_r)) - (q_r * IZR b' - IZR a'))%R by ring. + set (delta1 := (q_i - q_r)%R) in *. + set (delta2 := (q_r * IZR b' - IZR a')%R) in *. + gappa. +Qed. + +Lemma le_IZR3 : + forall n m p : Z, (IZR n <= IZR m <= IZR p)%R -> (n <= m <= p)%Z. +Proof. + intros ; split ; apply le_IZR ; lra. +Qed. + +Definition mostb_thresh := 18446740000000000000%Z. +Lemma step1_div_longu_correct_mostb : + forall a b, + (1 < Int64.unsigned b <= mostb_thresh)%Z -> + exists (q : int64), + (step1_div_longu (Vlong a) (Vlong b)) = Vlong q /\ + (Int64.min_signed <= (Int64.unsigned a - Int64.unsigned b*Int64.unsigned q) <= Int64.max_signed)%Z. +Proof. + intros a b b_RANGE. + + pose proof (Int64.unsigned_range a) as a_RANGE. + change Int64.modulus with 18446744073709551616%Z in a_RANGE. + set (a' := Int64.unsigned a) in *. + set (b' := Int64.unsigned b) in *. + + destruct (Z_le_gt_dec a' 0). + { assert (a' = 0%Z) as ZERO by lia. + exists Int64.zero. + rewrite ZERO. + rewrite Int64.unsigned_zero. + replace (Z.abs (0 - b' * 0))%Z with 0%Z by lia. + replace a with Int64.zero; cycle 1. + { + unfold a' in ZERO. + unfold Int64.zero. + rewrite <- ZERO. + apply Int64.repr_unsigned. + } + split. + { apply step1_div_longu_a0. + lia. + } + change Int64.min_signed with (-9223372036854775808)%Z. + change Int64.max_signed with ( 9223372036854775807)%Z. + lia. + } + + unfold step1_div_longu. + assert (1 < b')%Z as b_NOT01 by lia. + destruct (step1_real_div_longu_correct a b b_NOT01) as (q & C1E & C1R & C1F & C1S). + rewrite C1E. cbn. + unfold Float.to_longu_ne. + pose proof (ZofB_ne_range_correct 53 1024 q 0 Int64.max_unsigned) as C2. + rewrite C1F in C2. + + + assert (1 <= IZR a' <= 18446744073709551615)%R as a_RANGE'. + { split; apply IZR_le; lia. } + assert (2 <= IZR b' <= IZR mostb_thresh)%R as b_RANGE'. + { split; apply IZR_le; lia. } + assert (1 <= IZR b' <= IZR mostb_thresh)%R as b_RANGE'' by lra. + cbn zeta in C2. + rewrite C2. + cbn. + rewrite C1R. + unfold step1_real_quotient. + fold a' b'. + unfold mostb_thresh in *. + + rewrite Zle_bool_true ; cycle 1. + { apply Znearest_IZR_le. + gappa. + } + rewrite Zle_bool_true ; cycle 1. + { apply Znearest_le_IZR. + gappa. + } + cbn. + econstructor; split. reflexivity. + set (q_r := (rd (rd (IZR a') * rd (rs (1 / rs ( rd (IZR b'))))))%R). + assert (Rabs (IZR (ZnearestE q_r) - q_r) <= /2)%R as NEAR by apply Znearest_imp2. + set (delta1 := (IZR (ZnearestE q_r) - q_r)%R) in NEAR. + apply le_IZR3. + rewrite minus_IZR. + rewrite mult_IZR. + rewrite Int64.unsigned_repr ; cycle 1. + { split. + - apply Znearest_IZR_le. unfold q_r. + gappa. + - apply Znearest_le_IZR. unfold q_r. + change Int64.max_unsigned with 18446744073709551615%Z. + gappa. + } + replace (IZR (ZnearestE q_r)) with ((IZR (ZnearestE q_r) - q_r) + q_r)%R by ring. + fold delta1. + unfold q_r. + set (a1 := IZR a') in *. + set (b1 := IZR b') in *. + replace (rd (rd a1 * rd (rs (1 / rs (rd b1)))))%R with + ((((rd (rd a1 * rd (rs (1 / rs (rd b1))))-(a1 * (1 / b1))) / (a1 * (1 / b1)))+1) * (a1 / b1))%R; + cycle 1. + { field. lra. } + set (delta2 := ((rd (rd a1 * rd (rs (1 / rs (rd b1))))-(a1 * (1 / b1))) / (a1 * (1 / b1)))%R) in *. + (* assert (Rabs (delta2) <= 1/4194304)%R. + { unfold delta2. gappa. } *) + replace (a1 - b1 * (delta1 + (delta2 + 1) * (a1 / b1)))%R with + (-b1*delta1 - a1*delta2)%R; cycle 1. + { field. lra. } + unfold delta2. + gappa. +Qed. + +Lemma find_quotient: + forall (a b : Z) + (b_POS : (0 < b)%Z) + (qr : R) + (GAP : (Rabs (IZR a / IZR b - qr) < /2)%R), + (a / b)%Z = + let q := ZnearestE qr in + if (b*q >? a)%Z + then (q-1)%Z + else q. +Proof. + intros. + set (q := ZnearestE qr). + cbn zeta. + set (b' := IZR b) in *. + set (a' := IZR a) in *. + assert (1 <= b')%R as b_POS'. + { apply IZR_le. + lia. + } + + pose proof (Znearest_imp2 (fun x : Z => negb (Z.even x)) qr) as ROUND. + fold q in ROUND. + set (q' := IZR q) in *. + + pose proof (Rabs_triang (a' / b' - qr) + (qr - q'))%R as TRIANGLE. + replace ((a' / b' - qr) + (qr - q'))%R with + (a' / b' - q')%R in TRIANGLE by ring. + rewrite <- Rabs_Ropp in ROUND. + replace (- (q' - qr))%R with (qr - q')%R in ROUND by ring. + assert (Z.abs (a - b*q) < b)%Z as DELTA. + { apply lt_IZR. + rewrite <- Rabs_Zabs. + rewrite minus_IZR. + rewrite mult_IZR. + fold a' q' b'. + apply Rmult_lt_reg_r with (r := (/b')%R). + { apply Rinv_0_lt_compat. lra. } + rewrite Rinv_r by lra. + replace (/ b')%R with (/ Rabs(b'))%R ; cycle 1. + { f_equal. + apply Rabs_pos_eq. lra. } + rewrite <- Rabs_Rinv by lra. + rewrite <- Rabs_mult. + replace ((a' - b' * q') * / b')%R with (a'/b' - q')%R by (field ; lra). + lra. + } + + pose proof (Zgt_cases (b * q) a)%Z as CASE. + destruct (_ >? _)%Z. + { apply Zdiv_unique with (b := (a - (q-1)*b)%Z). + ring. + split; lia. + } + + apply Zdiv_unique with (b := (a - q*b)%Z). + ring. + split; lia. +Qed. + +Definition step2_real_div_long a b := + Val.mulf (Val.maketotal (Val.floatoflong a)) (approx_inv_longu b). + +Definition smalla_thresh := 34184372088832%Z. + +Lemma step2_real_div_long_smalla_correct : + forall (a b : int64) + (a_SMALL : (Z.abs (Int64.signed a) <= smalla_thresh)%Z) + (b_NOT0 : (0 < Int64.unsigned b)%Z), + exists (q : float), + (step2_real_div_long (Vlong a) (Vlong b)) = Vfloat q /\ + (Rabs ((B2R _ _ q) - (IZR (Int64.signed a)) / (IZR (Int64.unsigned b))) <= (32767/65536))%R /\ + is_finite _ _ q = true. +Proof. + intros. + unfold step2_real_div_long. + destruct (approx_inv_longu_correct_rel b b_NOT0) as (f & C0E & C0F & C0R). + rewrite C0E. + econstructor. + split. reflexivity. + Local Transparent Float.of_long. + unfold Float.mul, Float.of_long. + set (re := (@eq_refl Datatypes.comparison Lt)) in *. + pose proof (Int64.unsigned_range b) as b_RANGE. + change Int64.modulus with 18446744073709551616%Z in b_RANGE. + set (a' := Int64.signed a) in *. + set (b' := Int64.unsigned b) in *. + assert (1 <= IZR b' <= 18446744073709551615)%R as b_RANGE'. + { split; apply IZR_le; lia. + } + assert(Rabs (IZR a') <= IZR smalla_thresh)%R as a_RANGE'. + { rewrite Rabs_Zabs. + apply IZR_le. + assumption. + } + assert (- 2 ^ 53 <= a' <= 2 ^ 53)%Z as SILLY. + { unfold smalla_thresh in a_SMALL. + apply Z.abs_le. + lia. + } + destruct (BofZ_exact 53 1024 re re (Int64.signed a) SILLY) as (C1R & C1F & C1S). + fold a' in C1R, C1F, C1S. + pose proof (Bmult_correct 53 1024 re re Float.binop_nan mode_NE (BofZ 53 1024 re re a') f) as C2. + rewrite Rlt_bool_true in C2 ; cycle 1. + { clear C2. + apply Rabs_relax with (b := bpow radix2 53). + { apply bpow_lt. lia. } + cbn. + rewrite C1R. + unfold approx_inv_rel_thresh in C0R. + replace (B2R 53 1024 f) with + ((1/IZR b') * ((IZR b' * B2R 53 1024 f - 1) + 1))%R ; cycle 1. + { field. lra. } + unfold smalla_thresh in *. + gappa. + } + rewrite C0F in C2. + rewrite C1R in C2. + rewrite C1F in C2. + rewrite C1S in C2. + cbn in C2. + destruct C2 as (C2R & C2F & _). + split. + 2: exact C2F. + rewrite C2R. + replace (IZR a' * (B2R 53 1024 f))%R with + ((IZR a'/IZR b') * ((IZR b' * B2R 53 1024 f - 1) + 1))%R ; cycle 1. + { field. lra. } + set (delta1 := (IZR b' * B2R 53 1024 f - 1)%R) in *. + set (q1 := (IZR a' / IZR b' * (delta1 + 1))%R). + replace (rd q1) with (((rd q1) - q1) + q1)%R by ring. + set (delta2 := ((rd q1) - q1)%R). + unfold q1. + replace (delta2 + IZR a' / IZR b' * (delta1 + 1) - IZR a' / IZR b')%R with + (delta2 + (IZR a' / IZR b') * delta1)%R by ring. + unfold delta2. + unfold q1. + unfold approx_inv_rel_thresh in *. + unfold smalla_thresh in *. + gappa. +Qed. + +Definition step2_div_long' a b := + Val.maketotal (Val.longoffloat_ne (step2_real_div_long a b)). + +Definition step2_div_long a b := + let q := step2_div_long' a b in + Val.select (Val.cmpl_bool Clt (Val.subl a (Val.mull q b)) (Vlong Int64.zero)) + (Val.addl q (Vlong Int64.mone)) q Tlong. + +Lemma function_ite : + forall {A B : Type} (fn : A->B) (b : bool) (x y: A), + fn (if b then x else y) = (if b then fn x else fn y). +Proof. + intros. + destruct b; reflexivity. +Qed. + +Lemma normalize_ite : + forall ty (b : bool) x y, + Val.normalize (if b then x else y) ty = + (if b then Val.normalize x ty else Val.normalize y ty). +Proof. + intros. + destruct b; reflexivity. +Qed. + + +Lemma int64_mul_signed_unsigned: + forall x y : int64, + Int64.mul x y = Int64.repr (Int64.signed x * Int64.unsigned y). +Proof. + intros. + unfold Int64.mul. + apply Int64.eqm_samerepr. + apply Int64.eqm_mult. + - apply Int64.eqm_sym. + apply Int64.eqm_signed_unsigned. + - apply Int64.eqm_refl. +Qed. + +Lemma int64_eqm_signed_repr: + forall z : Z, Int64.eqm z (Int64.signed (Int64.repr z)). +Proof. + intros. + apply Int64.eqm_trans with (y := Int64.unsigned (Int64.repr z)). + - apply Int64.eqm_unsigned_repr. + - apply Int64.eqm_sym. + apply Int64.eqm_signed_unsigned. +Qed. + +Lemma signed_repr_sub: + forall x y, + Int64.repr (Int64.signed (Int64.repr x) - y) = + Int64.repr (x - y). +Proof. + intros. + apply Int64.eqm_samerepr. + apply Int64.eqm_sub. + - apply Int64.eqm_sym. + apply int64_eqm_signed_repr. + - apply Int64.eqm_refl. +Qed. + +Lemma signed_repr_sub2: + forall x y, + Int64.repr (x - Int64.signed (Int64.repr y)) = + Int64.repr (x - y). +Proof. + intros. + apply Int64.eqm_samerepr. + apply Int64.eqm_sub. + - apply Int64.eqm_refl. + - apply Int64.eqm_sym. + apply int64_eqm_signed_repr. +Qed. + +Lemma step2_div_long_smalla_correct : + forall a b + (a_SMALL : (Z.abs (Int64.signed a) <= smalla_thresh)%Z) + (b_NOT0 : (0 < Int64.unsigned b)%Z) + (b_NOT_VERY_BIG : (Int64.unsigned b <= Int64.max_signed)%Z), + step2_div_long (Vlong a) (Vlong b) = Vlong (Int64.repr (Int64.signed a / Int64.unsigned b))%Z. +Proof. + intros. + pose proof (Int64.unsigned_range b) as b_RANGE. + change Int64.modulus with 18446744073709551616%Z in b_RANGE. + set (a' := (Int64.signed a)) in *. + set (b' := (Int64.unsigned b)) in *. + assert (Rabs (IZR a') <= IZR smalla_thresh)%R as a_RANGE'. + { rewrite Rabs_Zabs. + apply IZR_le. + assumption. + } + assert (1 <= IZR b' <= 18446744073709551615)%R as b_RANGE'. + { split; apply IZR_le; lia. + } + destruct (step2_real_div_long_smalla_correct a b a_SMALL b_NOT0) as (q & C1R & C1E & C1F). + fold a' b' in C1E. + assert ((Int64.min_signed <=? ZnearestE (B2R 53 1024 q))=true)%Z as q_LOW. + { apply Zle_imp_le_bool. + change Int64.min_signed with (-9223372036854775808)%Z. + apply Znearest_lub. + set (q' := B2R 53 1024 q) in *. + replace q' with (IZR a' / IZR b' + (q' - IZR a' / IZR b'))%R by ring. + unfold smalla_thresh in a_RANGE'. + gappa. + } + assert ((ZnearestE (B2R 53 1024 q) <=? Int64.max_signed)=true)%Z as q_HIGH. + { apply Zle_imp_le_bool. + change Int64.max_signed with (9223372036854775807)%Z. + apply Znearest_glb. + set (q' := B2R 53 1024 q) in *. + replace q' with (IZR a' / IZR b' + (q' - IZR a' / IZR b'))%R by ring. + unfold smalla_thresh in a_RANGE'. + gappa. + } + unfold step2_div_long, step2_div_long'. + rewrite C1R. + cbn. + unfold Float.to_long_ne. + rewrite (ZofB_ne_range_correct _ _ q Int64.min_signed Int64.max_signed). + rewrite C1F. + rewrite q_LOW. + rewrite q_HIGH. + cbn. + rewrite normalize_ite. + cbn. + rewrite <- (function_ite Vlong). + f_equal. + unfold Int64.lt. + set (q' := B2R 53 1024 q) in *. + fold a'. + assert (Int64.signed (Int64.repr (ZnearestE q')) = ZnearestE q') as q_SIGNED. + { apply Int64.signed_repr. + split; lia. + } + rewrite Int64.add_signed. + rewrite q_SIGNED. + rewrite Int64.signed_mone. + rewrite Int64.signed_zero. + rewrite <- (function_ite Int64.repr). + f_equal. + replace (ZnearestE q' + -1)%Z with (ZnearestE q' - 1)%Z by ring. + + set (q'' := (ZnearestE q')) in *. + fold a'. + rewrite int64_mul_signed_unsigned. + rewrite q_SIGNED. + fold b'. + + rewrite Int64.sub_signed. + fold a'. + rewrite signed_repr_sub2. + + assert ((Rabs (IZR a' / IZR b' - q') < / 2)%R) as HALF. + { replace (IZR a' / IZR b' - q')%R with + (-(q' - IZR a' / IZR b'))%R by ring. + rewrite Rabs_Ropp. + lra. + } + pose proof (find_quotient a' b' b_NOT0 q' HALF) as QUOTIENT. + fold q'' in QUOTIENT. + cbn zeta in QUOTIENT. + + assert (b' <> 0)%Z as NONZ by lia. + pose proof (Zmod_eq_full a' b' NONZ) as MOD. + assert (b' > 0)%Z as b_GT0 by lia. + pose proof (Z_mod_lt a' b' b_GT0) as MOD_LT. + destruct (Z_lt_dec a' (b' * q'')) as [LT | GE]. + { replace (b' * q'' >? a')%Z with true in QUOTIENT by lia. + replace q'' with (1 + (a' / b'))%Z by lia. + replace (a' - (1 + a' / b') * b')%Z + with ((a' - a' / b' * b')-b')%Z by ring. + rewrite <- MOD. + rewrite Int64.signed_repr; cycle 1. + { change Int64.min_signed with (-9223372036854775808)%Z in *. + change Int64.max_signed with (9223372036854775807)%Z in *. + lia. + } + rewrite zlt_true by lia. + ring. + } + replace (b' * q'' >? a')%Z with false in QUOTIENT by lia. + rewrite <- QUOTIENT. + replace (a' / b' * b' - a')%Z with (-(a' - a' / b' * b'))%Z by ring. + rewrite <- MOD. + rewrite Int64.signed_repr ; cycle 1. + { change Int64.min_signed with (-9223372036854775808)%Z in *. + change Int64.max_signed with (9223372036854775807)%Z in *. + lia. + } + rewrite zlt_false by lia. + reflexivity. +Qed. + +Definition twostep_div_longu a b := + let q1 := step1_div_longu a b in + let q2 := step2_div_long (Val.subl a (Val.mull b q1)) b in + Val.addl q1 q2. + +Lemma unsigned_repr_sub : + forall a b, + Int64.repr (a - b) = Int64.repr (a - Int64.unsigned (Int64.repr b)). +Proof. + intros. + apply Int64.eqm_samerepr. + apply Int64.eqm_sub. + - apply Int64.eqm_refl. + - apply Int64.eqm_unsigned_repr. +Qed. + +Lemma unsigned_repr_add : + forall a b, + Int64.repr (a + b) = Int64.repr (a + Int64.unsigned (Int64.repr b)). +Proof. + intros. + apply Int64.eqm_samerepr. + apply Int64.eqm_add. + - apply Int64.eqm_refl. + - apply Int64.eqm_unsigned_repr. +Qed. + +Lemma twostep_div_longu_smallb_correct : + forall a b + (b_RANGE : (1 < Int64.unsigned b <= smallb_thresh)%Z), + (twostep_div_longu (Vlong a) (Vlong b)) = + (Val.maketotal (Val.divlu (Vlong a) (Vlong b))). +Proof. + intros. + unfold twostep_div_longu. + destruct (step1_div_longu_correct a b b_RANGE) as (q1 & C1R & C1E). + rewrite C1R. + set (q1' := Int64.unsigned q1) in *. + set (b' := Int64.unsigned b) in *. + set (a' := Int64.unsigned a) in *. + assert ( Z.abs (Int64.signed (Int64.sub a (Int64.mul b q1))) <= smalla_thresh)%Z as r1_SMALL. + { unfold smalla_thresh, smallb_approx_range in *. + unfold Int64.sub, Int64.mul. + fold q1' b' a'. + rewrite <- unsigned_repr_sub. + rewrite Int64.signed_repr ; cycle 1. + { change Int64.min_signed with (-9223372036854775808)%Z. + change Int64.max_signed with (9223372036854775807)%Z. + lia. + } + lia. + } + assert (0 < b')%Z as b_NOT0 by lia. + assert (b' <= Int64.max_signed)%Z as b_NOTBIG. + { change Int64.max_signed with (9223372036854775807)%Z. + unfold smallb_thresh in b_RANGE. + lia. + } + cbn. + rewrite (step2_div_long_smalla_correct (Int64.sub a (Int64.mul b q1)) b r1_SMALL b_NOT0 b_NOTBIG). + unfold Int64.add, Int64.sub, Int64.mul, Int64.divu. + fold q1' b' a'. + rewrite <- unsigned_repr_sub. + rewrite <- unsigned_repr_add. + rewrite Int64.signed_repr ; cycle 1. + { + change Int64.min_signed with (-9223372036854775808)%Z. + change Int64.max_signed with (9223372036854775807)%Z. + unfold smallb_approx_range in *. + lia. + } + rewrite Z.add_comm. + rewrite <- Z.div_add by lia. + replace (a' - b' * q1' + q1' * b')%Z with a' by ring. + rewrite Int64.eq_false ; cycle 1. + { intro Z. unfold b' in b_NOT0. rewrite Z in b_NOT0. + rewrite Int64.unsigned_zero in b_NOT0. + lia. + } + reflexivity. +Qed. + + +Lemma step2_real_div_long_bigb_correct : + forall (a b : int64) + (b_BIG : ((Int64.unsigned b) > smallb_thresh)%Z), + exists (q : float), + (step2_real_div_long (Vlong a) (Vlong b)) = Vfloat q /\ + (Rabs ((B2R _ _ q) - (IZR (Int64.signed a)) / (IZR (Int64.unsigned b))) <= (32767/65536))%R /\ + is_finite _ _ q = true. +Proof. + intros. + unfold step2_real_div_long. + assert (0 < Int64.unsigned b)%Z as b_NOT0 by (unfold smallb_thresh in *; lia). + destruct (approx_inv_longu_correct_rel b b_NOT0) as (f & C0E & C0F & C0R). + rewrite C0E. + econstructor. + split. reflexivity. + Local Transparent Float.of_long. + unfold Float.mul, Float.of_long. + set (re := (@eq_refl Datatypes.comparison Lt)) in *. + pose proof (Int64.unsigned_range b) as b_RANGE. + change Int64.modulus with 18446744073709551616%Z in b_RANGE. + pose proof (Int64.signed_range a) as a_RANGE. + set (a' := Int64.signed a) in *. + set (b' := Int64.unsigned b) in *. + assert (IZR (1 + smallb_thresh) <= IZR b' <= 18446744073709551615)%R as b_RANGE'. + { split; apply IZR_le; lia. + } + assert(IZR Int64.min_signed <= IZR a' <= IZR Int64.max_signed)%R as a_RANGE'. + { split; apply IZR_le; lia. + } + change Int64.min_signed with (-9223372036854775808)%Z in a_RANGE'. + change Int64.max_signed with (9223372036854775807)%Z in a_RANGE'. + pose proof (BofZ_correct 53 1024 re re a') as C1. + rewrite Rlt_bool_true in C1 ; cycle 1. + { clear C1. + apply Rabs_relax with (b := bpow radix2 64). + { apply bpow_lt; lia. } + cbn. + gappa. + } + cbn in C1. + destruct C1 as (C1R & C1F & C1S). + + unfold smallb_thresh in b_RANGE'; cbn in b_RANGE'. + + pose proof (Bmult_correct 53 1024 re re Float.binop_nan mode_NE (BofZ 53 1024 re re a') f) as C2. + rewrite Rlt_bool_true in C2 ; cycle 1. + { clear C2. + apply Rabs_relax with (b := bpow radix2 53). + { apply bpow_lt. lia. } + cbn. + rewrite C1R. + unfold approx_inv_rel_thresh in C0R. + replace (B2R 53 1024 f) with + ((1/IZR b') * ((IZR b' * B2R 53 1024 f - 1) + 1))%R ; cycle 1. + { field. lra. } + gappa. + } + rewrite C0F in C2. + rewrite C1R in C2. + rewrite C1F in C2. + rewrite C1S in C2. + cbn in C2. + destruct C2 as (C2R & C2F & _). + split. + 2: exact C2F. + rewrite C2R. + set (f' := (B2R 53 1024 f)) in *. + replace (rd(rd (IZR a') * f') - IZR a' / IZR b')%R with + ((rd(rd (IZR a') * f') - IZR a' * f') + IZR a' / IZR b' * (IZR b' * f' - 1))%R ; cycle 1. + { field. lra. } + unfold approx_inv_rel_thresh in *. + gappa. +Qed. + +Lemma step2_div_long_bigb_correct : + forall a b + (b_BIG : ((Int64.unsigned b) > smallb_thresh)%Z) + (b_NOT_TOO_BIG : ((Int64.unsigned b) <= Int64.max_signed)%Z), + step2_div_long (Vlong a) (Vlong b) = Vlong (Int64.repr (Int64.signed a / Int64.unsigned b))%Z. +Proof. + intros. + pose proof (Int64.unsigned_range b) as b_RANGE. + change Int64.modulus with 18446744073709551616%Z in b_RANGE. + pose proof (Int64.signed_range a) as a_RANGE. + set (a' := (Int64.signed a)) in *. + set (b' := (Int64.unsigned b)) in *. + assert (IZR (1 + smallb_thresh) <= IZR b' <= 18446744073709551615)%R as b_RANGE'. + { split; apply IZR_le; lia. + } + assert(IZR Int64.min_signed <= IZR a' <= IZR Int64.max_signed)%R as a_RANGE'. + { split; apply IZR_le; lia. + } + unfold smallb_thresh in *; cbn in b_RANGE'. + change Int64.min_signed with (-9223372036854775808)%Z in *. + change Int64.max_signed with (9223372036854775807)%Z in *. + assert (0 < b')%Z as b_NOT0 by lia. + + destruct (step2_real_div_long_bigb_correct a b b_BIG) as (q & C1R & C1E & C1F). + fold a' b' in C1E. + assert ((Int64.min_signed <=? ZnearestE (B2R 53 1024 q))=true)%Z as q_LOW. + { apply Zle_imp_le_bool. + change Int64.min_signed with (-9223372036854775808)%Z. + apply Znearest_lub. + set (q' := B2R 53 1024 q) in *. + replace q' with (IZR a' / IZR b' + (q' - IZR a' / IZR b'))%R by ring. + gappa. + } + assert ((ZnearestE (B2R 53 1024 q) <=? Int64.max_signed)=true)%Z as q_HIGH. + { apply Zle_imp_le_bool. + change Int64.max_signed with (9223372036854775807)%Z. + apply Znearest_glb. + set (q' := B2R 53 1024 q) in *. + replace q' with (IZR a' / IZR b' + (q' - IZR a' / IZR b'))%R by ring. + gappa. + } + unfold step2_div_long, step2_div_long'. + rewrite C1R. + cbn. + unfold Float.to_long_ne. + rewrite (ZofB_ne_range_correct _ _ q Int64.min_signed Int64.max_signed). + rewrite C1F. + rewrite q_LOW. + rewrite q_HIGH. + cbn. + rewrite normalize_ite. + cbn. + rewrite <- (function_ite Vlong). + f_equal. + unfold Int64.lt. + set (q' := B2R 53 1024 q) in *. + fold a'. + assert (Int64.signed (Int64.repr (ZnearestE q')) = ZnearestE q') as q_SIGNED. + { apply Int64.signed_repr. + split; lia. + } + rewrite Int64.add_signed. + rewrite q_SIGNED. + rewrite Int64.signed_mone. + rewrite Int64.signed_zero. + rewrite <- (function_ite Int64.repr). + f_equal. + replace (ZnearestE q' + -1)%Z with (ZnearestE q' - 1)%Z by ring. + + set (q'' := (ZnearestE q')) in *. + fold a'. + rewrite int64_mul_signed_unsigned. + rewrite q_SIGNED. + fold b'. + + rewrite Int64.sub_signed. + fold a'. + rewrite signed_repr_sub2. + + assert ((Rabs (IZR a' / IZR b' - q') < / 2)%R) as HALF. + { replace (IZR a' / IZR b' - q')%R with + (-(q' - IZR a' / IZR b'))%R by ring. + rewrite Rabs_Ropp. + lra. + } + pose proof (find_quotient a' b' b_NOT0 q' HALF) as QUOTIENT. + fold q'' in QUOTIENT. + cbn zeta in QUOTIENT. + + assert (b' <> 0)%Z as NONZ by lia. + pose proof (Zmod_eq_full a' b' NONZ) as MOD. + assert (b' > 0)%Z as b_GT0 by lia. + pose proof (Z_mod_lt a' b' b_GT0) as MOD_LT. + destruct (Z_lt_dec a' (b' * q'')) as [LT | GE]. + { replace (b' * q'' >? a')%Z with true in QUOTIENT by lia. + replace q'' with (1 + (a' / b'))%Z by lia. + replace (a' - (1 + a' / b') * b')%Z + with ((a' - a' / b' * b')-b')%Z by ring. + rewrite <- MOD. + rewrite Int64.signed_repr; cycle 1. + { change Int64.min_signed with (-9223372036854775808)%Z in *. + change Int64.max_signed with (9223372036854775807)%Z in *. + lia. + } + rewrite zlt_true by lia. + ring. + } + replace (b' * q'' >? a')%Z with false in QUOTIENT by lia. + rewrite <- QUOTIENT. + replace (a' / b' * b' - a')%Z with (-(a' - a' / b' * b'))%Z by ring. + rewrite <- MOD. + rewrite Int64.signed_repr ; cycle 1. + { change Int64.min_signed with (-9223372036854775808)%Z in *. + change Int64.max_signed with (9223372036854775807)%Z in *. + lia. + } + rewrite zlt_false by lia. + reflexivity. +Qed. + +Definition step2_real_div_longu a b := + Val.mulf (Val.maketotal (Val.floatoflongu a)) (approx_inv_longu b). + +Definition step2_div_longu' a b := + Val.maketotal (Val.longuoffloat_ne (step2_real_div_longu a b)). + +Definition step2_div_longu a b := + let q := step2_div_longu' a b in + Val.select (Val.cmpl_bool Cgt (Val.subl (Val.mull q b) a) (Vlong Int64.zero)) + (Val.addl q (Vlong Int64.mone)) q Tlong. + +Lemma step2_real_div_longu_bigb_correct : + forall (a b : int64) + (b_BIG : ((Int64.unsigned b) > smallb_thresh)%Z), + exists (q : float), + (step2_real_div_longu (Vlong a) (Vlong b)) = Vfloat q /\ + (Rabs ((B2R _ _ q) - (IZR (Int64.unsigned a)) / (IZR (Int64.unsigned b))) <= (32767/65536))%R /\ + is_finite _ _ q = true. +Proof. + intros. + unfold step2_real_div_longu. + assert (0 < Int64.unsigned b)%Z as b_NOT0 by (unfold smallb_thresh in *; lia). + destruct (approx_inv_longu_correct_rel b b_NOT0) as (f & C0E & C0F & C0R). + rewrite C0E. + econstructor. + split. reflexivity. + Local Transparent Float.of_longu. + unfold Float.mul, Float.of_longu. + set (re := (@eq_refl Datatypes.comparison Lt)) in *. + pose proof (Int64.unsigned_range b) as b_RANGE. + pose proof (Int64.unsigned_range a) as a_RANGE. + change Int64.modulus with 18446744073709551616%Z in *. + set (a' := Int64.unsigned a) in *. + set (b' := Int64.unsigned b) in *. + assert (IZR (1 + smallb_thresh) <= IZR b' <= 18446744073709551615)%R as b_RANGE'. + { split; apply IZR_le; lia. + } + assert(0 <= IZR a' <= 18446744073709551615)%R as a_RANGE'. + { split; apply IZR_le; lia. + } + pose proof (BofZ_correct 53 1024 re re a') as C1. + rewrite Rlt_bool_true in C1 ; cycle 1. + { clear C1. + apply Rabs_relax with (b := bpow radix2 64). + { apply bpow_lt; lia. } + cbn. + gappa. + } + cbn in C1. + destruct C1 as (C1R & C1F & C1S). + + unfold smallb_thresh in b_RANGE'; cbn in b_RANGE'. + + pose proof (Bmult_correct 53 1024 re re Float.binop_nan mode_NE (BofZ 53 1024 re re a') f) as C2. + rewrite Rlt_bool_true in C2 ; cycle 1. + { clear C2. + apply Rabs_relax with (b := bpow radix2 53). + { apply bpow_lt. lia. } + cbn. + rewrite C1R. + unfold approx_inv_rel_thresh in C0R. + replace (B2R 53 1024 f) with + ((1/IZR b') * ((IZR b' * B2R 53 1024 f - 1) + 1))%R ; cycle 1. + { field. lra. } + gappa. + } + rewrite C0F in C2. + rewrite C1R in C2. + rewrite C1F in C2. + rewrite C1S in C2. + cbn in C2. + destruct C2 as (C2R & C2F & _). + split. + 2: exact C2F. + rewrite C2R. + set (f' := (B2R 53 1024 f)) in *. + replace (rd(rd (IZR a') * f') - IZR a' / IZR b')%R with + ((rd(rd (IZR a') * f') - IZR a' * f') + IZR a' / IZR b' * (IZR b' * f' - 1))%R ; cycle 1. + { field. lra. } + unfold approx_inv_rel_thresh in *. + gappa. +Qed. + +Lemma repr_unsigned_mul: + forall a b, + (Int64.repr (Int64.unsigned (Int64.repr a) * b)) = Int64.repr (a * b). +Proof. + intros. + apply Int64.eqm_samerepr. + apply Int64.eqm_mult. + - apply Int64.eqm_sym. apply Int64.eqm_unsigned_repr. + - apply Int64.eqm_refl. +Qed. + +Lemma repr_unsigned_sub: + forall a b, + (Int64.repr (Int64.unsigned (Int64.repr a) - b)) = Int64.repr (a - b). +Proof. + intros. + apply Int64.eqm_samerepr. + apply Int64.eqm_sub. + - apply Int64.eqm_sym. apply Int64.eqm_unsigned_repr. + - apply Int64.eqm_refl. +Qed. + +Lemma repr_unsigned_add: + forall a b, + (Int64.repr (Int64.unsigned (Int64.repr a) + b)) = Int64.repr (a + b). +Proof. + intros. + apply Int64.eqm_samerepr. + apply Int64.eqm_add. + - apply Int64.eqm_sym. apply Int64.eqm_unsigned_repr. + - apply Int64.eqm_refl. +Qed. + +Lemma step2_div_longu_bigb_correct : + forall a b + (b_BIG : ((Int64.unsigned b) > smallb_thresh)%Z) + (b_NOT_TOO_BIG : ((Int64.unsigned b) <= Int64.max_signed)%Z), + step2_div_longu (Vlong a) (Vlong b) = Vlong (Int64.repr (Int64.unsigned a / Int64.unsigned b))%Z. +Proof. + intros. + pose proof (Int64.unsigned_range b) as b_RANGE. + pose proof (Int64.unsigned_range a) as a_RANGE. + change Int64.modulus with 18446744073709551616%Z in *. + set (a' := (Int64.unsigned a)) in *. + set (b' := (Int64.unsigned b)) in *. + assert (IZR (1 + smallb_thresh) <= IZR b' <= 18446744073709551615)%R as b_RANGE'. + { split; apply IZR_le; lia. + } + assert(0 <= IZR a' <= 18446744073709551615)%R as a_RANGE'. + { split; apply IZR_le; lia. + } + unfold smallb_thresh in *; cbn in b_RANGE'. + assert (0 < b')%Z as b_NOT0 by lia. + + destruct (step2_real_div_longu_bigb_correct a b b_BIG) as (q & C1R & C1E & C1F). + fold a' b' in C1E. + + assert ((0 <=? ZnearestE (B2R 53 1024 q))=true)%Z as q_LOW. + { apply Zle_imp_le_bool. + set (q' := B2R 53 1024 q) in *. + assert (-32767 / 65536 <= q')%R as LOWROUND. + { replace q' with (IZR a' / IZR b' + (q' - IZR a' / IZR b'))%R by ring. + gappa. + } + destruct (Rcase_abs q'). + { replace (ZnearestE q') with 0%Z. lia. + symmetry. + apply Znearest_imp. + apply Rabs_lt. + split; lra. + } + apply Znearest_lub. + lra. + } + assert ((ZnearestE (B2R 53 1024 q) <=? Int64.max_unsigned)=true)%Z as q_HIGH. + { apply Zle_imp_le_bool. + change Int64.max_unsigned with (18446744073709551615)%Z. + apply Znearest_glb. + set (q' := B2R 53 1024 q) in *. + replace q' with (IZR a' / IZR b' + (q' - IZR a' / IZR b'))%R by ring. + gappa. + } + + unfold step2_div_longu, step2_div_longu'. + rewrite C1R. + cbn. + unfold Float.to_longu_ne. + rewrite (ZofB_ne_range_correct _ _ q _ _). + rewrite C1F. + rewrite q_LOW. + rewrite q_HIGH. + cbn. + rewrite normalize_ite. + cbn. + rewrite <- (function_ite Vlong). + f_equal. + unfold Int64.lt. + set (q' := B2R 53 1024 q) in *. + fold a'. + rewrite Int64.signed_zero. + set (q'' := (ZnearestE q')) in *. + assert ((Rabs (IZR a' / IZR b' - q') < / 2)%R) as HALF. + { replace (IZR a' / IZR b' - q')%R with + (-(q' - IZR a' / IZR b'))%R by ring. + rewrite Rabs_Ropp. + lra. + } + pose proof (find_quotient a' b' b_NOT0 q' HALF) as QUOTIENT. + fold q'' in QUOTIENT. + cbn zeta in QUOTIENT. + + assert (b' <> 0)%Z as NONZ by lia. + pose proof (Zmod_eq_full a' b' NONZ) as MOD. + assert (b' > 0)%Z as b_GT0 by lia. + pose proof (Z_mod_lt a' b' b_GT0) as MOD_LT. + destruct (Z_lt_dec a' (b' * q'')) as [LT | GE]. + { replace (b' * q'' >? a')%Z with true in QUOTIENT by lia. + unfold Int64.sub, Int64.mul. + fold a' b'. + replace q'' with (1 + a'/b')%Z by lia. + rewrite repr_unsigned_mul. + rewrite repr_unsigned_sub. + + replace ((1 + a' / b') * b' - a')%Z with (b' - (a' - a' / b' * b'))%Z by ring. + rewrite <- MOD. + rewrite Int64.signed_repr ; cycle 1. + { change Int64.max_signed with 9223372036854775807%Z in *. + change Int64.min_signed with (-9223372036854775808)%Z in *. + lia. + } + rewrite zlt_true by lia. + replace q'' with (1 + (a' / b'))%Z by lia. + apply Int64.eqm_samerepr. + apply Int64.eqm_trans with (y := ((1 + a' / b') + (-1))%Z). + { apply Int64.eqm_add. + apply Int64.eqm_sym. + apply Int64.eqm_unsigned_repr. + rewrite Int64.unsigned_mone. + replace (-1)%Z with (0 - 1)%Z by ring. + apply Int64.eqm_add. + exists 1%Z. + lia. + apply Int64.eqm_refl. + } + replace (1 + a' / b' + -1)%Z with (a'/b')%Z by ring. + apply Int64.eqm_refl. + } + replace (b' * q'' >? a')%Z with false in QUOTIENT by lia. + rewrite <- QUOTIENT. + unfold Int64.sub, Int64.mul. + fold a' b'. + rewrite repr_unsigned_mul. + rewrite repr_unsigned_sub. + replace (a' / b' * b' - a')%Z with (- (a' mod b'))%Z by lia. + rewrite Int64.signed_repr ; cycle 1. + { change Int64.max_signed with 9223372036854775807%Z in *. + change Int64.min_signed with (-9223372036854775808)%Z in *. + lia. + } + rewrite zlt_false by lia. + reflexivity. +Qed. + +Lemma one_bigb_div : + forall a b + (b_RANGE : (9223372036854775808 <= b < 18446744073709551616)%Z) + (ORDER : (b <= a < 18446744073709551616)%Z), + (a / b = 1)%Z. +Proof. + intros. + assert (((a - b) / b) = 0)%Z as ZERO. + { apply Zdiv_small. lia. + } + replace a with (1 * b + (a - b))%Z by ring. + rewrite Z.div_add_l by lia. + rewrite ZERO. + ring. +Qed. + +Lemma repr_unsigned_sub2: + forall a b, + (Int64.repr (a - Int64.unsigned (Int64.repr b))) = Int64.repr (a - b). +Proof. + intros. + apply Int64.eqm_samerepr. + apply Int64.eqm_sub. + - apply Int64.eqm_refl. + - apply Int64.eqm_sym. apply Int64.eqm_unsigned_repr. +Qed. + +Lemma repr_unsigned_add2: + forall a b, + (Int64.repr (a + Int64.unsigned (Int64.repr b))) = Int64.repr (a + b). +Proof. + intros. + apply Int64.eqm_samerepr. + apply Int64.eqm_add. + - apply Int64.eqm_refl. + - apply Int64.eqm_sym. apply Int64.eqm_unsigned_repr. +Qed. + +Lemma twostep_div_longu_mostb_correct : + forall a b + (b_RANGE : (1 < Int64.unsigned b <= Int64.max_signed)%Z), + (twostep_div_longu (Vlong a) (Vlong b)) = + (Val.maketotal (Val.divlu (Vlong a) (Vlong b))). +Proof. + intros. + destruct (Z_le_gt_dec (Int64.unsigned b) smallb_thresh). + { apply twostep_div_longu_smallb_correct. lia. } + set (a' := Int64.unsigned a). + set (b' := Int64.unsigned b). + assert (0 < b')%Z as b_NOT0 by lia. + cbn. + rewrite Int64.eq_false ; cycle 1. + { intro Z. unfold b' in b_NOT0. rewrite Z in b_NOT0. + rewrite Int64.unsigned_zero in b_NOT0. + lia. + } + cbn. + + unfold twostep_div_longu. + assert (1 < Int64.unsigned b <= mostb_thresh)%Z as MOST_B. + { unfold mostb_thresh. + change Int64.max_signed with 9223372036854775807%Z in b_RANGE. + lia. + } + destruct (step1_div_longu_correct_mostb a b MOST_B) as + (q & step1_REW & step1_RANGE). + rewrite step1_REW. + cbn. + rewrite step2_div_long_bigb_correct; cycle 1. + 1, 2: lia. + f_equal. + + unfold Int64.sub, Int64.mul. + rewrite repr_unsigned_sub2. + rewrite Int64.signed_repr by lia. + unfold Int64.add, Int64.divu. + fold a' b'. + set (q' := Int64.unsigned q) in *. + rewrite repr_unsigned_add2. + rewrite <- Z.div_add_l by lia. + f_equal. f_equal. + ring. +Qed. + +Definition full2_div_longu a b m := + let is_big := Val.cmpl_bool Clt b (Vlong Int64.zero) in + let is_one := Val.cmplu_bool (Mem.valid_pointer m) Cle b (Vlong Int64.one) in + let is_special := Val.or (Val.of_optbool is_big) (Val.of_optbool is_one) in + let if_big := Val.longofintu (Val.of_optbool (Val.cmplu_bool (Mem.valid_pointer m) Cge a b)) in + let if_special := Val.select is_big if_big a Tlong in + Val.select (Val.cmp_bool Cne is_special (Vint Int.zero)) + if_special + (twostep_div_longu a b) Tlong. + +Lemma full2_div_longu_correct : + forall a b m + (b_NOT0 : (0 < Int64.unsigned b)%Z), + full2_div_longu (Vlong a) (Vlong b) m = Vlong (Int64.repr (Int64.unsigned a / Int64.unsigned b))%Z. +Proof. + + Local Opaque twostep_div_longu. + intros. + unfold full2_div_longu. + cbn. + unfold Int64.lt, Int64.ltu. + pose proof (Int64.unsigned_range a). + pose proof (Int64.unsigned_range b). + set (a' := Int64.unsigned a) in *. + set (b' := Int64.unsigned b) in *. + rewrite Int64.signed_zero. + rewrite Int64.unsigned_one. + destruct zlt. + { replace (Val.cmp_bool Cne + (Val.or Vtrue + (if negb (if zlt 1 b' then true else false) then Vtrue else Vfalse)) + (Vint Int.zero)) with (Some true) ; cycle 1. + { destruct zlt; reflexivity. + } + cbn. + destruct zlt; cbn. + { rewrite Zdiv_small by lia. + reflexivity. + } + rewrite one_bigb_div. + reflexivity. + { + change Int64.modulus with 18446744073709551616%Z in *. + split. 2: lia. + unfold b'. + rewrite Int64.unsigned_signed. + unfold Int64.lt. + rewrite Int64.signed_zero. + rewrite zlt_true by lia. + pose proof (Int64.signed_range b). + change Int64.min_signed with (-9223372036854775808)%Z in *. + change Int64.max_signed with (9223372036854775807)%Z in *. + change Int64.modulus with 18446744073709551616%Z in *. + lia. + } + change Int64.modulus with 18446744073709551616%Z in *. + lia. + } + destruct zlt; cbn. + { change (negb (Int.eq (Int.or Int.zero Int.zero) Int.zero)) with false. + cbn. + rewrite twostep_div_longu_mostb_correct. + { + cbn. + unfold Int64.eq. + fold b'. + rewrite Int64.unsigned_zero. + rewrite zeq_false by lia. + reflexivity. + } + + change Int64.modulus with 18446744073709551616%Z in *. + split. lia. + rewrite Int64.unsigned_signed. + unfold Int64.lt. + rewrite Int64.signed_zero. + rewrite zlt_false by lia. + pose proof (Int64.signed_range b). + change Int64.min_signed with (-9223372036854775808)%Z in *. + change Int64.max_signed with (9223372036854775807)%Z in *. + change Int64.modulus with 18446744073709551616%Z in *. + lia. + } + change (negb (Int.eq (Int.or Int.zero Int.one) Int.zero)) with true. + cbn. + replace b' with 1%Z by lia. + rewrite Z.div_1_r. + unfold a'. + rewrite Int64.repr_unsigned. + reflexivity. +Qed. + +Definition step2_mod_long a b := + let q := step2_div_long' a b in + let r := Val.subl a (Val.mull q b) in + Val.select (Val.cmpl_bool Clt r (Vlong Int64.zero)) + (Val.addl r b) r Tlong. + +Definition twostep_mod_longu a b := + let q1 := step1_div_longu a b in + step2_mod_long (Val.subl a (Val.mull b q1)) b. + +Lemma vlong_eq: forall a b, (Vlong a) = (Vlong b) -> a = b. +Proof. + intros a b EQ. + congruence. +Qed. + +Lemma move_repr_in_mod : + forall a b c, + Int64.repr (a - b * c)%Z = + Int64.repr (a - b * Int64.unsigned (Int64.repr c))%Z. +Proof. + intros. + apply Int64.eqm_samerepr. + auto 10 with ints. +Qed. + +Lemma twostep_mod_longu_mostb_correct : + forall a b + (b_RANGE : (1 < Int64.unsigned b <= Int64.max_signed)%Z), + (twostep_mod_longu (Vlong a) (Vlong b)) = + (Val.maketotal (Val.modlu (Vlong a) (Vlong b))). +Proof. + intros. + Local Transparent twostep_div_longu. + pose proof (twostep_div_longu_mostb_correct a b b_RANGE) as div_correct. + unfold twostep_div_longu, twostep_mod_longu, step2_div_long, step2_mod_long in *. + set (q1 := (step1_div_longu (Vlong a) (Vlong b))) in *. + set (q2 := (step2_div_long' (Val.subl (Vlong a) (Val.mull (Vlong b) q1)) (Vlong b))) in *. + cbn in div_correct. + cbn. + unfold Int64.eq in *. + change (Int64.unsigned Int64.zero) with 0%Z in *. + rewrite zeq_false by lia. + rewrite zeq_false in div_correct by lia. + cbn in div_correct. + cbn. + destruct q1 as [ | | q1l | | | ] ; cbn in *; try discriminate. + destruct q2 as [ | | q2l | | | ] ; cbn in *; try discriminate. + rewrite <- (function_ite Vlong). + rewrite <- (function_ite Vlong) in div_correct. + cbn. cbn in div_correct. + unfold Int64.lt, Int64.sub, Int64.mul, Int64.add, Int64.divu, Int64.modu in *. + set (a' := Int64.unsigned a) in *. + set (b' := Int64.unsigned b) in *. + set (q1' := Int64.unsigned q1l) in *. + set (q2' := Int64.unsigned q2l) in *. + change (Int64.signed Int64.zero) with 0%Z in *. + replace (Int64.repr + (Int64.unsigned + (Int64.repr (a' - Int64.unsigned (Int64.repr (b' * q1')))) - + Int64.unsigned (Int64.repr (q2' * b')))) + with (Int64.repr (a' - (b' * q1') - (q2' * b')))%Z in * ; cycle 1. + { + apply Int64.eqm_samerepr. + auto 10 with ints. + } + replace (a' - b' * q1' - q2' * b')%Z with (a' - b' * (q1' + q2'))%Z in * by ring. + f_equal. + apply vlong_eq in div_correct. + rewrite Z.mod_eq by lia. + rewrite (move_repr_in_mod a' b' (a' / b'))%Z. + rewrite <- div_correct. + clear div_correct. + rewrite <- (move_repr_in_mod a' b')%Z. + + destruct zlt as [NEG | POS]. + 2: reflexivity. + rewrite repr_unsigned_add. + replace (a' - b' * (q1' + q2') + b')%Z with (a' - b' * (q1' + q2' - 1))%Z by ring. + apply Int64.eqm_samerepr. + assert (Int64.eqm (Int64.unsigned (Int64.repr (q2' + Int64.unsigned Int64.mone))) + (q2' -1))%Z as EQM. + { apply Int64.eqm_trans with (y := (q2' + Int64.unsigned Int64.mone)%Z). + apply Int64.eqm_sym. + apply Int64.eqm_unsigned_repr. + apply Int64.eqm_add. + apply Int64.eqm_refl. + exists (1)%Z. + reflexivity. + } + replace (q1' + q2' - 1)%Z with (q1' + (q2' - 1))%Z by ring. + auto with ints. +Qed. + +Definition full2_mod_longu a b m := + let is_big := Val.cmpl_bool Clt b (Vlong Int64.zero) in + let is_one := Val.cmplu_bool (Mem.valid_pointer m) Cle b (Vlong Int64.one) in + let is_special := Val.or (Val.of_optbool is_big) (Val.of_optbool is_one) in + let if_big := Val.select (Val.cmplu_bool (Mem.valid_pointer m) Cge a b) (Val.subl a b) a Tlong in + let if_special := Val.select is_big if_big (Vlong Int64.zero) Tlong in + Val.select (Val.cmp_bool Cne is_special (Vint Int.zero)) + if_special + (twostep_mod_longu a b) Tlong. + +Lemma full2_mod_longu_correct : + forall a b m + (b_NOT0 : (0 < Int64.unsigned b)%Z), + full2_mod_longu (Vlong a) (Vlong b) m = Vlong (Int64.repr ((Int64.unsigned a) mod (Int64.unsigned b)))%Z. +Proof. + + Local Opaque twostep_mod_longu. + intros. + unfold full2_mod_longu. + cbn. + unfold Int64.lt, Int64.ltu. + pose proof (Int64.unsigned_range a). + pose proof (Int64.unsigned_range b). + set (a' := Int64.unsigned a) in *. + set (b' := Int64.unsigned b) in *. + rewrite Int64.signed_zero. + rewrite Int64.unsigned_one. + destruct zlt. + { replace (Val.cmp_bool Cne + (Val.or Vtrue + (if negb (if zlt 1 b' then true else false) then Vtrue else Vfalse)) + (Vint Int.zero)) with (Some true) ; cycle 1. + { destruct zlt; reflexivity. + } + cbn. + rewrite Z.mod_eq by lia. + + destruct zlt; cbn. + { rewrite Zdiv_small by lia. + replace (a' - b' * 0)%Z with a' by ring. + unfold a'. + rewrite Int64.repr_unsigned. + reflexivity. + } + rewrite one_bigb_div. + { unfold Int64.sub. + fold a' b'. + repeat f_equal. ring. + } + { + change Int64.modulus with 18446744073709551616%Z in *. + split. 2: lia. + unfold b'. + rewrite Int64.unsigned_signed. + unfold Int64.lt. + rewrite Int64.signed_zero. + rewrite zlt_true by lia. + pose proof (Int64.signed_range b). + change Int64.min_signed with (-9223372036854775808)%Z in *. + change Int64.max_signed with (9223372036854775807)%Z in *. + change Int64.modulus with 18446744073709551616%Z in *. + lia. + } + change Int64.modulus with 18446744073709551616%Z in *. + lia. + } + destruct zlt; cbn. + { change (negb (Int.eq (Int.or Int.zero Int.zero) Int.zero)) with false. + cbn. + rewrite twostep_mod_longu_mostb_correct. + { + cbn. + unfold Int64.eq. + fold b'. + rewrite Int64.unsigned_zero. + rewrite zeq_false by lia. + reflexivity. + } + + change Int64.modulus with 18446744073709551616%Z in *. + split. lia. + rewrite Int64.unsigned_signed. + unfold Int64.lt. + rewrite Int64.signed_zero. + rewrite zlt_false by lia. + pose proof (Int64.signed_range b). + change Int64.min_signed with (-9223372036854775808)%Z in *. + change Int64.max_signed with (9223372036854775807)%Z in *. + change Int64.modulus with 18446744073709551616%Z in *. + lia. + } + change (negb (Int.eq (Int.or Int.zero Int.one) Int.zero)) with true. + cbn. + replace b' with 1%Z by lia. + rewrite Z.mod_1_r. + reflexivity. +Qed. + +Open Scope cminorsel_scope. +Definition e_invfs a := Eop Oinvfs (a ::: Enil). +Definition e_float_of_longu a := Eop Ofloatoflongu (a ::: Enil). +Definition e_float_of_long a := Eop Ofloatoflong (a ::: Enil). +Definition e_float_of_single a := Eop Ofloatofsingle (a ::: Enil). +Definition e_single_of_float a := Eop Osingleoffloat (a ::: Enil). +Definition e_long_of_float_ne a := Eop Olongoffloat_ne (a ::: Enil). +Definition e_longu_of_float_ne a := Eop Olonguoffloat_ne (a ::: Enil). +Definition e_mulf a b := Eop Omulf (a ::: b ::: Enil). +Definition e_float_const c := Eop (Ofloatconst c) Enil. +Definition e_fmaddf a b c := Eop Ofmaddf (a ::: b ::: c ::: Enil). +Definition e_fmsubf a b c := Eop Ofmsubf (a ::: b ::: c ::: Enil). +Definition e_addlimm a b := Eop (Oaddlimm b) (a ::: Enil). +Definition e_msubl a b c := Eop Omsubl (a ::: b ::: c ::: Enil). +Definition e_ite ty c vc v1 v2 := Eop (Osel c ty) (v1 ::: v2 ::: vc ::: Enil). +Definition e_cmplimm c v n := Eop (Ocmp (Ccomplimm c n)) (v ::: Enil). +Definition e_cmpluimm c v n := Eop (Ocmp (Ccompluimm c n)) (v ::: Enil). +Definition e_addl a b := Eop Oaddl (a ::: b ::: Enil). +Definition e_or a b := Eop Oor (a ::: b ::: Enil). +Definition e_cast32unsigned a := Eop Ocast32unsigned (a ::: Enil). +Definition e_cmplu c a b := Eop (Ocmp (Ccomplu c)) (a ::: b ::: Enil). + +Definition a_var1 := Eletvar (4%nat). +Definition a_d_var1 := Eletvar (3%nat). +Definition b_var1 := Eletvar (2%nat). +Definition b_d_var1 := Eletvar (1%nat). +Definition binv_d_var1 := Eletvar (0%nat). + +Definition e_setup1 a b rest := + Elet a (Elet (e_float_of_longu (Eletvar 0%nat)) + (Elet (lift (lift b)) (Elet (e_float_of_longu (Eletvar 0%nat)) + (Elet (e_float_of_single (e_invfs (e_single_of_float (Eletvar 0%nat)))) + rest)))). +Definition e_step1 := e_longu_of_float_ne (e_mulf a_d_var1 binv_d_var1). + +Lemma e_step1_correct : + forall (ge : genv) (sp: val) cmenv memenv (le : letenv) + (expr_a : expr) (a : int64) (EVAL_a : eval_expr ge sp cmenv memenv le expr_a (Vlong a)) + (expr_b : expr) (b : int64) (EVAL_b : eval_expr ge sp cmenv memenv le expr_b (Vlong b)), + (eval_expr ge sp cmenv memenv le (e_setup1 expr_a expr_b (e_step1)) + (step1_div_longu (Vlong a) (Vlong b))). +Proof. + intros. + unfold e_setup1, step1_div_longu. + repeat econstructor. + { eassumption. } + { cbn. apply eval_lift. apply eval_lift. eassumption. } +Qed. + +Definition e_setup2 a b rest := (e_setup1 a b (Elet e_step1 rest)). + +Definition a_var2 := Eletvar (5%nat). +Definition a_d_var2 := Eletvar (4%nat). +Definition b_var2 := Eletvar (3%nat). +Definition b_d_var2 := Eletvar (2%nat). +Definition binv_d_var2 := Eletvar (1%nat). +Definition step1_var2 := Eletvar (0%nat). + +Definition e_step2 := e_msubl a_var2 b_var2 step1_var2. + +Definition e_setup3 a b rest := (e_setup2 a b (Elet e_step2 rest)). + +Definition a_var3 := Eletvar (6%nat). +Definition a_d_var3 := Eletvar (5%nat). +Definition b_var3 := Eletvar (4%nat). +Definition b_d_var3 := Eletvar (3%nat). +Definition binv_d_var3 := Eletvar (2%nat). +Definition step1_var3 := Eletvar (1%nat). +Definition step2_var3 := Eletvar (0%nat). + +Definition e_step3 := + e_long_of_float_ne + (e_mulf (e_float_of_long step2_var3) + (e_fmaddf + binv_d_var3 + (e_fmsubf (e_float_const ExtFloat.one) + binv_d_var3 + b_d_var3 ) + binv_d_var3)). + +Lemma e_step3_correct : + forall (ge : genv) (sp: val) cmenv memenv (le : letenv) + (expr_a : expr) (a : int64) (EVAL_a : eval_expr ge sp cmenv memenv le expr_a (Vlong a)) + (expr_b : expr) (b : int64) (EVAL_b : eval_expr ge sp cmenv memenv le expr_b (Vlong b)), + (eval_expr ge sp cmenv memenv le (e_setup3 expr_a expr_b (e_step3)) + (step2_div_long' (Val.subl (Vlong a) (Val.mull (Vlong b) (step1_div_longu (Vlong a) (Vlong b)))) (Vlong b))). +Proof. +intros. +unfold e_setup2, e_setup1, e_step2, step2_div_long', step2_real_div_long, approx_inv_longu. +repeat (econstructor + apply eval_lift + eassumption). +Qed. + +Definition e_setup4 a b rest := (e_setup3 a b (Elet e_step3 rest)). + +Definition a_var4 := Eletvar (7%nat). +Definition a_d_var4 := Eletvar (6%nat). +Definition b_var4 := Eletvar (5%nat). +Definition b_d_var4 := Eletvar (4%nat). +Definition binv_d_var4 := Eletvar (3%nat). +Definition step1_var4 := Eletvar (2%nat). +Definition step2_var4 := Eletvar (1%nat). +Definition step3_var4 := Eletvar (0%nat). + +Definition e_step4 := + e_ite Tlong (Ccompl0 Clt) (e_msubl step2_var4 step3_var4 b_var4) + (e_addlimm step3_var4 Int64.mone) step3_var4. + +Lemma e_step4_correct : + forall (ge : genv) (sp: val) cmenv memenv (le : letenv) + (expr_a : expr) (a : int64) (EVAL_a : eval_expr ge sp cmenv memenv le expr_a (Vlong a)) + (expr_b : expr) (b : int64) (EVAL_b : eval_expr ge sp cmenv memenv le expr_b (Vlong b)), + (eval_expr ge sp cmenv memenv le (e_setup4 expr_a expr_b (e_step4)) + (step2_div_long (Val.subl (Vlong a) (Val.mull (Vlong b) (step1_div_longu (Vlong a) (Vlong b)))) (Vlong b))). +Proof. +intros. +unfold e_setup2, e_setup1, e_step2, step2_div_long, step2_div_long', step2_real_div_long, approx_inv_longu, step1_div_longu. +repeat (econstructor + apply eval_lift + eassumption). +Qed. + +Definition e_setup5 a b rest := (e_setup4 a b (Elet e_step4 rest)). + +Definition a_var5 := Eletvar (8%nat). +Definition a_d_var5 := Eletvar (7%nat). +Definition b_var5 := Eletvar (6%nat). +Definition b_d_var5 := Eletvar (5%nat). +Definition binv_d_var5 := Eletvar (4%nat). +Definition step1_var5 := Eletvar (3%nat). +Definition step2_var5 := Eletvar (2%nat). +Definition step3_var5 := Eletvar (1%nat). +Definition step4_var5 := Eletvar (0%nat). + +Definition e_step5 := e_addl step1_var5 step4_var5. + +Lemma e_step5_correct : + forall (ge : genv) (sp: val) cmenv memenv (le : letenv) + (expr_a : expr) (a : int64) (EVAL_a : eval_expr ge sp cmenv memenv le expr_a (Vlong a)) + (expr_b : expr) (b : int64) (EVAL_b : eval_expr ge sp cmenv memenv le expr_b (Vlong b)), + (eval_expr ge sp cmenv memenv le (e_setup5 expr_a expr_b (e_step5)) + (twostep_div_longu (Vlong a) (Vlong b))). +Proof. + intros. + Local Transparent twostep_div_longu. + repeat unfold e_setup2, e_setup1, e_step2, step2_div_long, step2_div_long', step2_real_div_long, approx_inv_longu, step1_div_longu, twostep_div_longu. +repeat (econstructor + apply eval_lift + eassumption). +Qed. + +Definition e_setup6 a b rest := (e_setup5 a b (Elet e_step5 rest)). + +Definition a_var6 := Eletvar (9%nat). +Definition a_d_var6 := Eletvar (8%nat). +Definition b_var6 := Eletvar (7%nat). +Definition b_d_var6 := Eletvar (6%nat). +Definition binv_d_var6 := Eletvar (5%nat). +Definition step1_var6 := Eletvar (4%nat). +Definition step2_var6 := Eletvar (3%nat). +Definition step3_var6 := Eletvar (2%nat). +Definition step4_var6 := Eletvar (1%nat). +Definition twostep_var6 := Eletvar (0%nat). + +Definition e_step6 := e_cmplimm Clt b_var6 Int64.zero. + +Definition e_setup7 a b rest := e_setup6 a b (Elet e_step6 rest). + +Definition a_var7 := Eletvar (10%nat). +Definition a_d_var7 := Eletvar (9%nat). +Definition b_var7 := Eletvar (8%nat). +Definition b_d_var7 := Eletvar (7%nat). +Definition binv_d_var7 := Eletvar (6%nat). +Definition step1_var7 := Eletvar (5%nat). +Definition step2_var7 := Eletvar (5%nat). +Definition step3_var7 := Eletvar (3%nat). +Definition step4_var7 := Eletvar (2%nat). +Definition twostep_var7 := Eletvar (1%nat). +Definition is_big_var7 := Eletvar (0%nat). + +Definition e_is_one := e_cmpluimm Cle b_var7 Int64.one. +Definition e_is_special := e_or is_big_var7 e_is_one. +Definition e_if_big := e_cast32unsigned (e_cmplu Cge a_var7 b_var7). +Definition e_if_special := e_ite Tlong (Ccompu0 Cne) is_big_var7 e_if_big a_var7. +Definition e_step7 := e_ite Tlong (Ccompu0 Cne) e_is_special e_if_special twostep_var7. + +Lemma e_step7_correct : + forall (ge : genv) (sp: val) cmenv memenv (le : letenv) + (expr_a : expr) (a : int64) (EVAL_a : eval_expr ge sp cmenv memenv le expr_a (Vlong a)) + (expr_b : expr) (b : int64) (EVAL_b : eval_expr ge sp cmenv memenv le expr_b (Vlong b)), + (eval_expr ge sp cmenv memenv le (e_setup7 expr_a expr_b (e_step7)) + (full2_div_longu (Vlong a) (Vlong b) memenv)). +Proof. + intros. + Local Transparent full2_div_longu. + repeat unfold e_setup2, e_setup1, e_step2, step2_div_long, step2_div_long', step2_real_div_long, approx_inv_longu, step1_div_longu, twostep_div_longu, full2_div_longu. + repeat (econstructor + apply eval_lift + eassumption). + cbn. + repeat f_equal. + destruct (Int64.lt b Int64.zero); cbn; change (Int.eq Int.one Int.zero) with false; change (Int.eq Int.zero Int.zero) with true; cbn; reflexivity. +Qed. + +Definition fp_divu64 a b := e_setup7 a b e_step7. + +Theorem fp_divu64_correct : + forall (ge : genv) (sp: val) cmenv memenv + (le : letenv) (expr_a expr_b : expr) (a b : int64) + (EVAL_a : eval_expr ge sp cmenv memenv le expr_a (Vlong a)) + (EVAL_b : eval_expr ge sp cmenv memenv le expr_b (Vlong b)) + (b_nz : (Int64.unsigned b > 0)%Z), + eval_expr ge sp cmenv memenv le (fp_divu64 expr_a expr_b) + (Vlong (Int64.divu a b)). +Proof. + intros. + unfold Int64.divu. + rewrite <- full2_div_longu_correct with (m := memenv) by lia. + apply e_step7_correct; assumption. +Qed. + +Definition fp_modu64 a b := Elet a (Elet (lift b) (e_msubl (Eletvar 1%nat) (Eletvar 0%nat) + (fp_divu64 (Eletvar 1%nat) (Eletvar 0%nat)))). + +Theorem fp_modu64_correct : + forall (ge : genv) (sp: val) cmenv memenv + (le : letenv) (expr_a expr_b : expr) (a b : int64) + (EVAL_a : eval_expr ge sp cmenv memenv le expr_a (Vlong a)) + (EVAL_b : eval_expr ge sp cmenv memenv le expr_b (Vlong b)) + (b_nz : (Int64.unsigned b > 0)%Z), + eval_expr ge sp cmenv memenv le (fp_modu64 expr_a expr_b) + (Vlong (Int64.modu a b)). +Proof. + intros. + rewrite Int64.modu_divu; cycle 1. + { intro Z. + subst. + rewrite Int64.unsigned_zero in b_nz. + lia. + } + unfold fp_modu64. + Local Opaque fp_divu64. + repeat (econstructor + apply eval_lift + eassumption). + { apply fp_divu64_correct; + repeat (econstructor + apply eval_lift + eassumption). + } + cbn. + rewrite Int64.mul_commut. + reflexivity. +Qed. + +Definition e_is_negl a := Eop (Ocmp (Ccomplimm Clt Int64.zero)) (a ::: Enil). +Definition e_xorw a b := Eop Oxor (a ::: b ::: Enil). +Definition e_negl a := Eop Onegl (a ::: Enil). +Definition e_absl a := Eop (Oabsdifflimm Int64.zero) (a ::: Enil). + +Definition fp_divs64 a b := + Elet a (Elet (lift b) + (Elet (fp_divu64 (e_absl (Eletvar (1%nat))) (e_absl (Eletvar (0%nat)))) + (e_ite Tlong (Ccompu0 Cne) (e_xorw (e_is_negl (Eletvar 2%nat)) + (e_is_negl (Eletvar 1%nat))) + (e_negl (Eletvar 0%nat)) (Eletvar 0%nat)))). + +Lemma nonneg_signed_unsigned: + forall x (x_NONNEG : (Int64.signed x >= 0)%Z), + (Int64.signed x) = (Int64.unsigned x). +Proof. + intros. + pose proof (Int64.unsigned_range x). + unfold Int64.signed in *. + destruct zlt. reflexivity. + change Int64.modulus with 18446744073709551616%Z in *. + change Int64.half_modulus with 9223372036854775808%Z in *. + lia. +Qed. + +Lemma long_min_signed_unsigned : + (- Int64.min_signed < Int64.max_unsigned)%Z. +Proof. + reflexivity. +Qed. + +Lemma long_divs_divu : + forall a b + (b_NOT0 : (Int64.signed b <> 0)%Z), + Int64.divs a b = if xorb (Int64.lt a Int64.zero) + (Int64.lt b Int64.zero) + then Int64.neg (Int64.divu (ExtValues.long_abs a) + (ExtValues.long_abs b)) + else Int64.divu (ExtValues.long_abs a) (ExtValues.long_abs b). +Proof. + intros. + unfold Int64.divs, Int64.divu, Int64.lt, ExtValues.long_abs. + pose proof (Int64.signed_range a) as a_RANGE. + pose proof (Int64.signed_range b) as b_RANGE. + change (Int64.signed Int64.zero) with 0%Z. + destruct zlt. + - cbn. rewrite (Z.abs_neq (Int64.signed a)) by lia. + rewrite (Int64.unsigned_repr (- Int64.signed a)); cycle 1. + { pose proof long_min_signed_unsigned. lia. } + + destruct zlt. + + rewrite (Z.abs_neq (Int64.signed b)) by lia. + rewrite Int64.unsigned_repr ; cycle 1. + { pose proof long_min_signed_unsigned. lia. } + rewrite <- (Z.opp_involutive (Int64.signed b)) at 1. + rewrite Z.quot_opp_r by lia. + rewrite <- (Z.opp_involutive (Int64.signed a)) at 1. + rewrite Z.quot_opp_l by lia. + rewrite Z.quot_div_nonneg by lia. + rewrite Z.opp_involutive. + reflexivity. + + + rewrite (Z.abs_eq (Int64.signed b)) by lia. + rewrite Int64.unsigned_repr ; cycle 1. + { pose proof Int64.max_signed_unsigned. lia. } + rewrite <- (Z.opp_involutive (Int64.signed a)) at 1. + rewrite Z.quot_opp_l by lia. + rewrite Z.quot_div_nonneg by lia. + rewrite Int64.neg_repr. + reflexivity. + + - cbn. rewrite (Z.abs_eq (Int64.signed a)) by lia. + rewrite (Int64.unsigned_repr (Int64.signed a)); cycle 1. + { pose proof Int64.max_signed_unsigned. lia. } + destruct zlt. + + rewrite (Z.abs_neq (Int64.signed b)) by lia. + rewrite Int64.unsigned_repr ; cycle 1. + { pose proof long_min_signed_unsigned. lia. } + rewrite Int64.neg_repr. + rewrite <- (Z.opp_involutive (Int64.signed b)) at 1. + rewrite Z.quot_opp_r by lia. + rewrite Z.quot_div_nonneg by lia. + reflexivity. + + + rewrite (Z.abs_eq (Int64.signed b)) by lia. + rewrite Int64.unsigned_repr ; cycle 1. + { pose proof Int64.max_signed_unsigned. lia. } + rewrite Z.quot_div_nonneg by lia. + reflexivity. +Qed. + +Lemma nonzero_unsigned_signed : + forall b, (Int64.unsigned b > 0 -> Int64.signed b <> 0)%Z. +Proof. + intros b GT EQ. + rewrite Int64.unsigned_signed in GT. + unfold Int64.lt in GT. + rewrite Int64.signed_zero in GT. + destruct zlt in GT; lia. +Qed. + +Theorem fp_divs64_correct : + forall (ge : genv) (sp: val) cmenv memenv + (le : letenv) (expr_a expr_b : expr) (a b : int64) + (EVAL_a : eval_expr ge sp cmenv memenv le expr_a (Vlong a)) + (EVAL_b : eval_expr ge sp cmenv memenv le expr_b (Vlong b)) + (b_nz : (Int64.unsigned b > 0)%Z), + eval_expr ge sp cmenv memenv le (fp_divs64 expr_a expr_b) + (Vlong (Int64.divs a b)). +Proof. + intros. + unfold fp_divs64. + Local Opaque fp_divu64. + repeat (econstructor + apply eval_lift + eassumption). + apply fp_divu64_correct. + all: repeat (econstructor + apply eval_lift + eassumption). + { unfold ExtValues.long_absdiff, ExtValues.Z_abs_diff. + rewrite Int64.signed_zero. rewrite Z.sub_0_r. + rewrite Int64.unsigned_repr. + { pose proof (nonzero_unsigned_signed b b_nz). + lia. + } + pose proof Int64.max_signed_unsigned. + pose proof long_min_signed_unsigned. + pose proof (Int64.signed_range b). + lia. + } + cbn. + rewrite long_divs_divu ; cycle 1. + { apply nonzero_unsigned_signed. assumption. } + unfold Int64.lt, ExtValues.long_abs, ExtValues.long_absdiff, ExtValues.Z_abs_diff. + change (Int64.signed Int64.zero) with 0%Z. + repeat rewrite Z.sub_0_r. + destruct zlt; destruct zlt; reflexivity. +Qed. + +Lemma long_mods_modu : + forall a b + (b_NOT0 : (Int64.signed b <> 0)%Z), + Int64.mods a b = if Int64.lt a Int64.zero + then Int64.neg (Int64.modu (ExtValues.long_abs a) + (ExtValues.long_abs b)) + else Int64.modu (ExtValues.long_abs a) (ExtValues.long_abs b). +Proof. + intros. + unfold Int64.mods, Int64.modu, Int64.lt, ExtValues.long_abs. + pose proof (Int64.signed_range a) as a_RANGE. + pose proof (Int64.signed_range b) as b_RANGE. + change (Int64.signed Int64.zero) with 0%Z. + destruct zlt. + - cbn. rewrite (Z.abs_neq (Int64.signed a)) by lia. + rewrite (Int64.unsigned_repr (- Int64.signed a)); cycle 1. + { pose proof long_min_signed_unsigned. lia. } + + destruct (zlt (Int64.signed b) 0%Z). + + rewrite (Z.abs_neq (Int64.signed b)) by lia. + rewrite Int64.unsigned_repr ; cycle 1. + { pose proof long_min_signed_unsigned. lia. } + rewrite <- (Z.opp_involutive (Int64.signed b)) at 1. + rewrite Z.rem_opp_r by lia. + rewrite <- (Z.opp_involutive (Int64.signed a)) at 1. + rewrite Z.rem_opp_l by lia. + rewrite Z.rem_mod_nonneg by lia. + rewrite Int64.neg_repr. + reflexivity. + + + rewrite (Z.abs_eq (Int64.signed b)) by lia. + rewrite Int64.unsigned_repr ; cycle 1. + { pose proof Int64.max_signed_unsigned. lia. } + rewrite <- (Z.opp_involutive (Int64.signed a)) at 1. + rewrite Z.rem_opp_l by lia. + rewrite Z.rem_mod_nonneg by lia. + rewrite Int64.neg_repr. + reflexivity. + + - cbn. rewrite (Z.abs_eq (Int64.signed a)) by lia. + rewrite (Int64.unsigned_repr (Int64.signed a)); cycle 1. + { pose proof Int64.max_signed_unsigned. lia. } + destruct (zlt (Int64.signed b) 0%Z). + + rewrite (Z.abs_neq (Int64.signed b)) by lia. + rewrite Int64.unsigned_repr ; cycle 1. + { pose proof long_min_signed_unsigned. lia. } + rewrite <- (Z.opp_involutive (Int64.signed b)) at 1. + rewrite Z.rem_opp_r by lia. + rewrite Z.rem_mod_nonneg by lia. + reflexivity. + + + rewrite (Z.abs_eq (Int64.signed b)) by lia. + rewrite Int64.unsigned_repr ; cycle 1. + { pose proof Int64.max_signed_unsigned. lia. } + rewrite Z.rem_mod_nonneg by lia. + reflexivity. +Qed. + +Definition fp_mods64z a b := + Elet a (Elet (lift b) + (Elet (fp_modu64 (e_absl (Eletvar (1%nat))) (e_absl (Eletvar (0%nat)))) + (e_ite Tlong (Ccompl0 Clt) (Eletvar 2%nat) + (e_negl (Eletvar 0%nat)) (Eletvar 0%nat)))). + +Theorem fp_mods64z_correct : + forall (ge : genv) (sp: val) cmenv memenv + (le : letenv) (expr_a expr_b : expr) (a b : int64) + (EVAL_a : eval_expr ge sp cmenv memenv le expr_a (Vlong a)) + (EVAL_b : eval_expr ge sp cmenv memenv le expr_b (Vlong b)) + (b_nz : (Int64.unsigned b > 0)%Z), + eval_expr ge sp cmenv memenv le (fp_mods64z expr_a expr_b) + (Vlong (Int64.mods a b)). +Proof. + intros. + unfold fp_mods64z. + Local Opaque fp_modu64. + repeat (econstructor + apply eval_lift + eassumption). + apply fp_modu64_correct. + all: repeat (econstructor + apply eval_lift + eassumption). + { unfold ExtValues.long_absdiff, ExtValues.Z_abs_diff. + rewrite Int64.signed_zero. rewrite Z.sub_0_r. + rewrite Int64.unsigned_repr. + { pose proof (nonzero_unsigned_signed b b_nz). + lia. + } + pose proof Int64.max_signed_unsigned. + pose proof long_min_signed_unsigned. + pose proof (Int64.signed_range b). + lia. + } + cbn. + rewrite long_mods_modu ; cycle 1. + { apply nonzero_unsigned_signed. assumption. } + unfold Int64.lt, ExtValues.long_abs, ExtValues.long_absdiff, ExtValues.Z_abs_diff. + change (Int64.signed Int64.zero) with 0%Z. + repeat rewrite Z.sub_0_r. + destruct zlt; reflexivity. +Qed. + +Definition fp_mods64 a b := + Elet a (Elet (lift b) + (Elet (fp_divs64 (Eletvar (1%nat)) (Eletvar (0%nat))) + (e_msubl (Eletvar 2%nat) (Eletvar 1%nat) (Eletvar 0%nat)))). + +Theorem fp_mods64_correct : + forall (ge : genv) (sp: val) cmenv memenv + (le : letenv) (expr_a expr_b : expr) (a b : int64) + (EVAL_a : eval_expr ge sp cmenv memenv le expr_a (Vlong a)) + (EVAL_b : eval_expr ge sp cmenv memenv le expr_b (Vlong b)) + (b_nz : (Int64.unsigned b > 0)%Z), + eval_expr ge sp cmenv memenv le (fp_mods64 expr_a expr_b) + (Vlong (Int64.mods a b)). +Proof. + intros. + rewrite Int64.mods_divs. + unfold fp_mods64. + Local Opaque fp_divs64. + repeat (econstructor + apply eval_lift + eassumption). + { apply fp_divs64_correct; + repeat (econstructor + apply eval_lift + eassumption). + } + cbn. + rewrite Int64.mul_commut. + reflexivity. +Qed. |