diff options
-rw-r--r-- | _CoqProject | 3 | ||||
-rw-r--r-- | src/Demo.v | 206 | ||||
-rw-r--r-- | src/Util/Decidable/Bool2Prop.v | 61 | ||||
-rw-r--r-- | src/Util/ListUtil.v | 98 | ||||
-rw-r--r-- | src/Util/QUtil.v | 19 |
5 files changed, 347 insertions, 40 deletions
diff --git a/_CoqProject b/_CoqProject index 7819439ff..c4f414b6c 100644 --- a/_CoqProject +++ b/_CoqProject @@ -3,6 +3,7 @@ -arg "-compat 8.6" Bedrock/Nomega.v Bedrock/Word.v +src/Demo.v src/Algebra/Field.v src/Algebra/Field_test.v src/Algebra/Group.v @@ -276,6 +277,7 @@ src/Util/Option.v src/Util/PartiallyReifiedProp.v src/Util/PointedProp.v src/Util/Prod.v +src/Util/QUtil.v src/Util/Relations.v src/Util/Sigma.v src/Util/Sum.v @@ -287,6 +289,7 @@ src/Util/Unit.v src/Util/WordUtil.v src/Util/ZRange.v src/Util/ZUtil.v +src/Util/Decidable/Bool2Prop.v src/Util/ForLoop/Instances.v src/Util/ForLoop/InvariantFramework.v src/Util/ForLoop/Tests.v diff --git a/src/Demo.v b/src/Demo.v new file mode 100644 index 000000000..fc048d051 --- /dev/null +++ b/src/Demo.v @@ -0,0 +1,206 @@ +(* Following http://adam.chlipala.net/theses/andreser.pdf chapter 3 *) +Require Import Coq.ZArith.ZArith Coq.micromega.Lia Crypto.Algebra.Nsatz. +Require Import Crypto.Util.Tactics Crypto.Util.Decidable. +Require Import Crypto.Util.Tuple Crypto.Util.Prod Crypto.Util.LetIn. +Require Import Crypto.Util.ListUtil Coq.Lists.List Crypto.Util.NatUtil. +Require Import QArith.QArith_base QArith.Qround Crypto.Util.QUtil. +Require Import Crypto.Algebra.Ring Crypto.Util.Decidable.Bool2Prop. +Import ListNotations. Local Open Scope Z_scope. + +Definition runtime_mul := Z.mul. +Definition runtime_add := Z.add. +Delimit Scope runtime_scope with RT. +Infix "*" := runtime_mul : runtime_scope. +Infix "+" := runtime_add : runtime_scope. + +Module Associational. + Definition eval (p:list (Z*Z)) : Z := + fold_right Z.add 0%Z (map (fun t => fst t * snd t) p). + + Lemma eval_nil : eval nil = 0. + Proof. trivial. Qed. + Lemma eval_cons p q : eval (p::q) = fst p * snd p + eval q. + Proof. trivial. Qed. + Lemma eval_app p q: eval (p++q) = eval p + eval q. + Proof. induction p; rewrite <-?List.app_comm_cons; + rewrite ?eval_nil, ?eval_cons; nsatz. Qed. + + Hint Rewrite eval_nil eval_cons eval_app : push_eval. + Local Ltac push := autorewrite with + push_eval push_map push_partition push_flat_map + push_fold_right push_nth_default cancel_pair. + + Lemma eval_map_mul (a x:Z) (p:list (Z*Z)) + : eval (List.map (fun t => (a*fst t, x*snd t)) p) = a*x*eval p. + Proof. induction p; push; nsatz. Qed. + Hint Rewrite eval_map_mul : push_eval. + + Definition mul (p q:list (Z*Z)) : list (Z*Z) := + flat_map (fun t => + map (fun t' => + (fst t * fst t', (snd t * snd t')%RT)) + q) p. + Lemma eval_mul p q : eval (mul p q) = eval p * eval q. + Proof. induction p; cbv [mul]; push; nsatz. Qed. + Hint Rewrite eval_mul : push_eval. + + Example base10_2digit_mul (a0:Z) (a1:Z) (b0:Z) (b1:Z) : + {ab| eval ab = eval [(10,a1);(1,a0)] * eval [(10,b1);(1,b0)]}. + eexists ?[ab]. + (* Goal: eval ?ab = eval [(10,a1);(1,a0)] * eval [(10,b1);(1,b0)] *) + rewrite <-eval_mul. + (* Goal: eval ?ab = eval (mul [(10,a1);(1,a0)] [(10,b1);(1,b0)]) *) + cbv -[runtime_mul eval]. + (* Goal: eval ?ab = eval [(100,(a1*b1));(10,a1*b0);(10,a0*b1);(1,a0*b0)]%RT *) + trivial. Defined. + + Definition split (s:Z) (p:list (Z*Z)) : list (Z*Z) * list (Z*Z) + := let hi_lo := partition (fun t => fst t mod s =? 0) p in + (snd hi_lo, map (fun t => (fst t / s, snd t)) (fst hi_lo)). + Lemma eval_split s p (s_nz:s<>0) : + eval (fst (split s p)) + s * eval (snd (split s p)) = eval p. + Proof. cbv [split]; induction p; + repeat match goal with + | |- context[?a/?b] => + unique pose proof (Z_div_exact_full_2 a b ltac:(trivial) ltac:(trivial)) + | _ => progress push + | _ => progress break_match + | _ => progress nsatz end. Qed. + + Lemma reduction_rule a b s c (modulus_nz:s-c<>0) : + (a + s * b) mod (s - c) = (a + c * b) mod (s - c). + Proof. replace (a + s * b) with ((a + c*b) + b*(s-c)) by nsatz. + rewrite Z.add_mod,Z_mod_mult,Z.add_0_r,Z.mod_mod;trivial. Qed. + + Definition reduce (s:Z) (c:list _) (p:list _) : list (Z*Z) := + let lo_hi := split s p in fst lo_hi ++ mul c (snd lo_hi). + + Lemma eval_reduce s c p (s_nz:s<>0) (modulus_nz:s-eval c<>0) : + eval (reduce s c p) mod (s - eval c) = eval p mod (s - eval c). + Proof. cbv [reduce]; push. + rewrite <-reduction_rule, eval_split; trivial. Qed. + Hint Rewrite eval_reduce : push_eval. +End Associational. + +Module Positional. Section Positional. + Context (weight : nat -> Z) + (weight_0 : weight 0%nat = 1) + (weight_nz : forall i, weight i <> 0). + + Definition to_associational {n:nat} (xs:tuple Z n) : list (Z*Z) + := combine (map weight (List.seq 0 n)) (Tuple.to_list n xs). + Definition eval {n} x := Associational.eval (@to_associational n x). + Lemma eval_to_associational {n} x : + Associational.eval (@to_associational n x) = eval x. + Proof. trivial. Qed. + + (* SKIP over this: zeros, add_to_nth *) + Local Ltac push := autorewrite with push_eval push_map distr_length + push_flat_map push_fold_right push_nth_default cancel_pair natsimplify. + Program Definition zeros n : tuple Z n + := Tuple.from_list n (List.map (fun _ => 0) (List.seq 0 n)) _. + Next Obligation. push; reflexivity. Qed. + Lemma eval_zeros n : eval (zeros n) = 0. + Proof. + cbv [eval Associational.eval to_associational zeros]; + rewrite Tuple.to_list_from_list. + generalize dependent (List.seq 0 n); intro xs. + induction xs; simpl; nsatz. Qed. + Program Definition add_to_nth {n} i x : tuple Z n -> tuple Z n + := Tuple.on_tuple (ListUtil.update_nth i (runtime_add x)) _. + Next Obligation. apply ListUtil.length_update_nth. Defined. + Lemma eval_add_to_nth {n} (i:nat) (H:(i<n)%nat) (x:Z) (xs:tuple Z n) : + eval (add_to_nth i x xs) = weight i * x + eval xs. + Proof. + cbv [eval to_associational add_to_nth Tuple.on_tuple runtime_add]. + rewrite !Tuple.to_list_from_list. + rewrite ListUtil.combine_update_nth_r at 1. + rewrite <-(update_nth_id i (List.combine _ _)) at 2. + rewrite <-!(ListUtil.splice_nth_equiv_update_nth_update _ _ + (weight 0, 0)) by (push; lia); cbv [ListUtil.splice_nth id]. + repeat match goal with + | _ => progress push + | _ => progress break_match + | _ => progress (apply Zminus_eq; ring_simplify) + | _ => rewrite <-ListUtil.map_nth_default_always + end; lia. Qed. + Hint Rewrite @eval_add_to_nth eval_zeros : push_eval. + + Fixpoint place (t:Z*Z) (i:nat) : nat * Z := + if dec (fst t mod weight i = 0) + then (i, let c := fst t / weight i in (c * snd t)%RT) + else match i with S i' => place t i' | O => (O, fst t * snd t)%RT end. + Lemma place_in_range (t:Z*Z) (n:nat) : (fst (place t n) < S n)%nat. + Proof. induction n; cbv [place] in *; break_match; autorewrite with cancel_pair; try omega. Qed. + Lemma weight_place t i : weight (fst (place t i)) * snd (place t i) = fst t * snd t. + Proof. induction i; cbv [place] in *; break_match; push; + repeat match goal with |- context[?a/?b] => + unique pose proof (Z_div_exact_full_2 a b ltac:(auto) ltac:(auto)) + end; nsatz. Qed. + Hint Rewrite weight_place : push_eval. + + Definition from_associational n (p:list (Z*Z)) := + List.fold_right (fun t => + let p := place t (pred n) in + add_to_nth (fst p) (snd p) ) (zeros n) p. + Lemma eval_from_associational {n} p (n_nz:n<>O) : + eval (from_associational n p) = Associational.eval p. + Proof. induction p; cbv [from_associational] in *; push; try + pose proof place_in_range a (pred n); try omega; try nsatz. Qed. + Hint Rewrite @eval_from_associational : push_eval. + + Section mulmod. + Context (m:Z) (m_nz:m <> 0) (s:Z) (s_nz:s <> 0) + (c:list (Z*Z)) (Hm:m = s - Associational.eval c). + Definition mulmod {n} (a b:tuple Z n) : tuple Z n + := let a_a := to_associational a in + let b_a := to_associational b in + let ab_a := Associational.mul a_a b_a in + let abm_a := Associational.reduce s c ab_a in + from_associational n abm_a. + Lemma eval_mulmod {n} (H:(n<>0)%nat) (f g:tuple Z n) : + eval (mulmod f g) mod m = (eval f * eval g) mod m. + Proof. cbv [mulmod]; rewrite Hm in *; push; trivial. Qed. + End mulmod. +End Positional. End Positional. + +Import Associational Positional. +Local Coercion Z.of_nat : nat >-> Z. +Local Coercion QArith_base.inject_Z : Z >-> Q. + +Definition w (i:nat) : Z := 2^Qceiling((25+1/2)*i). + +Example base_25_5_mul (f g:tuple Z 10) : + { fg : tuple Z 10 | (eval w fg) mod (2^255-19) + = (eval w f * eval w g) mod (2^255-19) }. + (* manually assign names to limbs for pretty-printing *) + destruct f as [[[[[[[[[f9 f8]f7]f6]f5]f4]f3]f2]f1]f0]. + destruct g as [[[[[[[[[g9 g8]g7]g6]g5]g4]g3]g2]g1]g0]. + eexists ?[fg]. + erewrite <-eval_mulmod with (s:=2^255) (c:=[(1,19)]) + by (try eapply pow_ceil_mul_nat_nonzero; vm_decide). +(* eval w ?fg mod (2 ^ 255 - 19) = *) +(* eval w *) +(* (mulmod w (2^255) [(1, 19)] (f9,f8,f7,f6,f5,f4,f3,f2,f1,f0) *) +(* (g9,g8,g7,g6,g5,g4,g3,g2,g1,g0)) mod (2^255 - 19) *) + eapply f_equal2; [|trivial]. eapply f_equal. +(* ?fg = *) +(* mulmod w (2 ^ 255) [(1, 19)] (f9, f8, f7, f6, f5, f4, f3, f2, f1, f0) *) +(* (g9, g8, g7, g6, g5, g4, g3, g2, g1, g0) *) + cbv -[runtime_mul runtime_add]; cbv [runtime_mul runtime_add]. + ring_simplify_subterms. +(* ?fg = + (f0*g9+ f1*g8+ f2*g7+ f3*g6+ f4*g5+ f5*g4+ f6*g3+ f7*g2+ f8*g1+ f9*g0, + f0*g8+ 2*f1*g7+ f2*g6+ 2*f3*g5+ f4*g4+ 2*f5*g3+ f6*g2+ 2*f7*g1+ f8*g0+ 38*f9*g9, + f0*g7+ f1*g6+ f2*g5+ f3*g4+ f4*g3+ f5*g2+ f6*g1+ f7*g0+ 19*f8*g9+ 19*f9*g8, + f0*g6+ 2*f1*g5+ f2*g4+ 2*f3*g3+ f4*g2+ 2*f5*g1+ f6*g0+ 38*f7*g9+ 19*f8*g8+ 38*f9*g7, + f0*g5+ f1*g4+ f2*g3+ f3*g2+ f4*g1+ f5*g0+ 19*f6*g9+ 19*f7*g8+ 19*f8*g7+ 19*f9*g6, + f0*g4+ 2*f1*g3+ f2*g2+ 2*f3*g1+ f4*g0+ 38*f5*g9+ 19*f6*g8+ 38*f7*g7+ 19*f8*g6+ 38*f9*g5, + f0*g3+ f1*g2+ f2*g1+ f3*g0+ 19*f4*g9+ 19*f5*g8+ 19*f6*g7+ 19*f7*g6+ 19*f8*g5+ 19*f9*g4, + f0*g2+ 2*f1*g1+ f2*g0+ 38*f3*g9+ 19*f4*g8+ 38*f5*g7+ 19*f6*g6+ 38*f7*g5+ 19*f8*g4+ 38*f9*g3, + f0*g1+ f1*g0+ 19*f2*g9+ 19*f3*g8+ 19*f4*g7+ 19*f5*g6+ 19*f6*g5+ 19*f7*g4+ 19*f8*g3+ 19*f9*g2, + f0*g0+ 38*f1*g9+ 19*f2*g8+ 38*f3*g7+ 19*f4*g6+ 38*f5*g5+ 19*f6*g4+ 38*f7*g3+ 19*f8*g2+ 38*f9*g1) *) + trivial. +Defined. + +(* Eval cbv on this one would produce an ugly term due to the use of [destruct] *) diff --git a/src/Util/Decidable/Bool2Prop.v b/src/Util/Decidable/Bool2Prop.v new file mode 100644 index 000000000..072d5c568 --- /dev/null +++ b/src/Util/Decidable/Bool2Prop.v @@ -0,0 +1,61 @@ +Require Coq.ZArith.ZArith. + +Lemma unit_eq (x y:unit) : x = y. destruct x, y; reflexivity. Qed. +Hint Resolve unit_eq. + +Hint Extern 0 (_ = _ :> bool) => ( + match goal with + | [H:Bool.eqb ?a ?b = true |- ?a = ?b ] => apply (proj1 (Bool.eqb_true_iff _ _) H) + | [H:Bool.eqb ?b ?a = true |- ?a = ?b ] => symmetry; apply (proj1 (Bool.eqb_true_iff _ _) H) + end). + +Hint Extern 0 (_ = _ :> nat) => ( + match goal with + | [H:Nat.eqb ?a ?b = true |- ?a = ?b ] => apply (proj1 (PeanoNat.Nat.eqb_eq _ _) H) + | [H:Nat.eqb ?b ?a = true |- ?a = ?b ] => symmetry; apply (proj1 (PeanoNat.Nat.eqb_eq _ _) H) + end). + +Hint Extern 0 (_ <= _) => ( + match goal with + | [H:Nat.ltb ?a ?b = true |- ?a < ?b ] => apply (proj1 (PeanoNat.Nat.ltb_lt _ _) H) + end). + +Hint Extern 0 (_ <= _) => ( + match goal with + | [H:Nat.leb ?a ?b = true |- ?a <= ?b ] => apply (proj1 (PeanoNat.Nat.leb_le _ _) H) + end). + +Hint Extern 0 (_ = _ :> BinNums.N) => ( + match goal with + | [H:BinNat.N.eqb ?a ?b = true |- ?a = ?b ] => apply (proj1 (BinNat.N.eqb_eq _ _) H) + | [H:BinNat.N.eqb ?b ?a = true |- ?a = ?b ] => symmetry; apply (proj1 (BinNat.N.eqb_eq _ _) H) + end). +Hint Extern 0 (_ = _ :> BinInt.Z) => ( + match goal with + | [H:BinInt.Z.eqb ?a ?b = true |- ?a = ?b ] => apply (proj1 (BinInt.Z.eqb_eq _ _) H) + | [H:BinInt.Z.eqb ?a ?b = true |- ?b = ?a ] => symmetry; apply (proj1 (BinInt.Z.eqb_eq _ _) H) + end). +Hint Extern 0 (BinInt.Z.lt _ _) => ( + match goal with + | [H:BinInt.Z.ltb ?a ?b = true |- BinInt.Z.lt ?a ?b ] => apply (proj1 (BinInt.Z.ltb_lt _ _) H) + | [H:BinInt.Z.gtb ?b ?a = true |- BinInt.Z.lt ?a ?b ] => apply (proj1 (BinInt.Z.gtb_lt _ _) H) + end). +Hint Extern 0 (BinInt.Z.le _ _) => ( + match goal with + | [H:BinInt.Z.leb ?a ?b = true |- BinInt.Z.le ?a ?b ] => apply (proj1 (BinInt.Z.leb_le _ _) H) + | [H:BinInt.Z.geb ?b ?a = true |- BinInt.Z.le ?a ?b ] => apply (proj1 (BinInt.Z.geb_le _ _) H) + end). + +Hint Extern 0 (_ = _ :> BinPos.positive) => ( + match goal with + | [H:BinPos.Pos.eqb ?a ?b = true |- ?a = ?b ] => apply (proj1 (BinPos.Pos.eqb_eq _ _) H) + | [H:BinPos.Pos.eqb ?a ?b = true |- ?b = ?a ] => symmetry; apply (proj1 (BinPos.Pos.eqb_eq _ _) H) + end). +Hint Extern 0 (BinPos.Pos.lt _ _) => ( + match goal with + | [H:BinPos.Pos.ltb ?a ?b = true |- BinPos.Pos.lt ?a ?b ] => apply (proj1 (BinPos.Pos.ltb_lt _ _) H) + end). +Hint Extern 0 (BinPos.Pos.le _ _) => ( + match goal with + | [H:BinPos.Pos.leb ?a ?b = true |- BinPos.Pos.le ?a ?b ] => apply (proj1 (BinPos.Pos.leb_le _ _) H) + end).
\ No newline at end of file diff --git a/src/Util/ListUtil.v b/src/Util/ListUtil.v index 6c6f96761..ee4fb0b9b 100644 --- a/src/Util/ListUtil.v +++ b/src/Util/ListUtil.v @@ -19,6 +19,10 @@ Create HintDb simpl_firstn discriminated. Create HintDb simpl_skipn discriminated. Create HintDb simpl_fold_right discriminated. Create HintDb simpl_sum_firstn discriminated. +Create HintDb push_map discriminated. +Create HintDb push_flat_map discriminated. +Create HintDb push_fold_right discriminated. +Create HintDb push_partition discriminated. Create HintDb pull_nth_error discriminated. Create HintDb push_nth_error discriminated. Create HintDb pull_nth_default discriminated. @@ -70,11 +74,45 @@ Module Export List. Variables (A : Type) (B : Type). Variable f : A -> B. + Lemma map_nil : forall A B (f : A -> B), map f nil = nil. + Proof. reflexivity. Qed. Lemma map_cons (x:A)(l:list A) : map f (x::l) = (f x) :: (map f l). Proof using Type. reflexivity. Qed. End Map. + Hint Rewrite @map_cons @map_nil : push_map. + + Section FlatMap. + Lemma flat_map_nil {A B} (f:A->list B) : List.flat_map f (@nil A) = nil. + Proof. reflexivity. Qed. + Lemma flat_map_cons {A B} (f:A->list B) x xs : + (List.flat_map f (x::xs) = (f x++List.flat_map f xs))%list. + Proof. reflexivity. Qed. + End FlatMap. + Hint Rewrite @flat_map_cons @flat_map_nil : push_flat_map. + + Section FoldRight. + Context {A B} (f:B->A->A). + Lemma fold_right_nil : forall {A B} (f:B->A->A) a, + List.fold_right f a nil = a. + Proof. reflexivity. Qed. + Lemma fold_right_cons : forall a b bs, + fold_right f a (b::bs) = f b (fold_right f a bs). + Proof. reflexivity. Qed. + End FoldRight. + Hint Rewrite @fold_right_nil @fold_right_cons : simpl_fold_right push_fold_right. + + Section Partition. + Lemma partition_nil {A} (f:A->_) : partition f nil = (nil, nil). + Proof. reflexivity. Qed. + Lemma partition_cons {A} (f:A->_) x xs : partition f (x::xs) = + if f x + then (x :: (fst (partition f xs)), (snd (partition f xs))) + else ((fst (partition f xs)), x :: (snd (partition f xs))). + Proof. cbv [partition]; break_match; reflexivity. Qed. + End Partition. + Hint Rewrite @partition_nil @partition_cons : push_partition. Lemma in_seq len start n : In n (seq start len) <-> start <= n < start+len. @@ -357,22 +395,6 @@ Proof. omega. Qed. -Lemma map_nil : forall A B (f : A -> B), map f nil = nil. -Proof. reflexivity. Qed. - -(* Note: this is a duplicate of a lemma that exists in 8.5, included for - 8.4 support *) -Lemma In_nth : forall {A} (x : A) d xs, In x xs -> - exists i, i < length xs /\ nth i xs d = x. -Proof. - induction xs as [|?? IHxs]; intros; - match goal with H : In _ _ |- _ => simpl in H; destruct H end. - + subst. exists 0. simpl; split; auto || omega. - + destruct IHxs as [i [ ]]; auto. - exists (S i). - split; auto; simpl; try omega. -Qed. - Hint Rewrite @map_nth_default using omega : push_nth_default. Ltac nth_tac := @@ -481,21 +503,6 @@ Qed. Hint Rewrite @length_update_nth : distr_length. -(** TODO: this is in the stdlib in 8.5; remove this when we move to 8.5-only *) -Lemma nth_error_None : forall (A : Type) (l : list A) (n : nat), nth_error l n = None <-> length l <= n. -Proof. - intros A l n. - destruct (le_lt_dec (length l) n) as [H|H]; - split; intro H'; - try omega; - try (apply nth_error_length_error in H; tauto); - try (apply nth_error_error_length in H'; omega). -Qed. - -(** TODO: this is in the stdlib in 8.5; remove this when we move to 8.5-only *) -Lemma nth_error_Some : forall (A : Type) (l : list A) (n : nat), nth_error l n <> None <-> n < length l. -Proof. intros; rewrite nth_error_None; split; omega. Qed. - Lemma nth_set_nth : forall m {T} (xs:list T) (n:nat) x, nth_error (set_nth m x xs) n = if eq_nat_dec n m @@ -742,6 +749,17 @@ Proof. induction xs, xs', ys, ys'; boring; omega. Qed. +Lemma map_fst_combine {A B} (xs:list A) (ys:list B) : List.map fst (List.combine xs ys) = List.firstn (length ys) xs. +Proof. + revert xs; induction ys; destruct xs; simpl; solve [ trivial | congruence ]. +Qed. + +Lemma map_snd_combine {A B} (xs:list A) (ys:list B) : List.map snd (List.combine xs ys) = List.firstn (length xs) ys. +Proof. + revert xs; induction ys; destruct xs; simpl; solve [ trivial | congruence ]. +Qed. +Hint Rewrite @map_fst_combine @map_snd_combine : push_map. + Lemma skipn_nil : forall {A} n, skipn n nil = @nil A. Proof. destruct n; auto. Qed. @@ -873,14 +891,6 @@ Qed. Hint Rewrite @skipn_length : distr_length. -Lemma fold_right_cons : forall {A B} (f:B->A->A) a b bs, - fold_right f a (b::bs) = f b (fold_right f a bs). -Proof. - reflexivity. -Qed. - -Hint Rewrite @fold_right_cons : simpl_fold_right. - Lemma length_cons : forall {T} (x:T) xs, length (x::xs) = S (length xs). reflexivity. Qed. @@ -1456,6 +1466,14 @@ induction m; intros. omega. Qed. +Lemma nth_default_seq_inbounds d s n i (H:(i < n)%nat) : + List.nth_default d (List.seq s n) i = (s+i)%nat. +Proof. + progress cbv [List.nth_default]. + rewrite nth_error_seq. + break_innermost_match; solve [ trivial | omega ]. +Qed. +Hint Rewrite @nth_default_seq_inbounds using lia : push_nth_default. Lemma sum_firstn_prefix_le' : forall l n m, (forall x, In x l -> (0 <= x)%Z) -> (sum_firstn l n <= sum_firstn l (n + m))%Z. @@ -1756,4 +1774,4 @@ Lemma map2_map {A B C A' B'} (f:A -> B -> C) (g:A' -> A) (h:B' -> B) (xs:list A' Proof. revert ys; induction xs as [|a xs IHxs]; destruct ys; intros; try reflexivity; []. simpl. rewrite IHxs. reflexivity. -Qed. +Qed.
\ No newline at end of file diff --git a/src/Util/QUtil.v b/src/Util/QUtil.v new file mode 100644 index 000000000..453cad308 --- /dev/null +++ b/src/Util/QUtil.v @@ -0,0 +1,19 @@ +Require Import Coq.ZArith.ZArith Coq.QArith.QArith QArith.Qround. + +Local Open Scope Z_scope. + +Lemma pow_ceil_mul_nat_nonzero b f + (b_nz:b <> 0) + (f_pos:(0 <= f)%Q) + : forall i, b^Qceiling (f * inject_Z (Z.of_nat i)) <> 0. +Proof. + intros. + eapply Z.pow_nonzero; try omega. + rewrite Zle_Qle. + eapply Qle_trans; [|eapply Qle_ceiling]. + eapply Qle_trans; [|eapply (Qmult_le_compat_r 0)]; trivial. + cbv; discriminate. + apply (Qle_trans _ (inject_Z 0)); [eapply Qle_refl|]. + rewrite <-Zle_Qle. + eapply Zle_0_nat. +Qed.
\ No newline at end of file |