diff --git a/algebra/CMonoids.v b/algebra/CMonoids.v index d2f89e4e..85abb211 100644 --- a/algebra/CMonoids.v +++ b/algebra/CMonoids.v @@ -253,12 +253,12 @@ Proof. astepl ((Inv f iso_imp_bij) (f a' [+] f b')). astepl ((Inv f iso_imp_bij) ( f ( a'[+] b'))). set (H3:= (inv2 M1 M2 f iso_imp_bij (a'[+]b'))). - astepl (a'[+]b'). astepr (a'[+] b'). intuition. + astepl (a'[+]b'). astepr (a'[+] b'). intuition; auto with *. set (H4:=(inv2 M1 M2 f iso_imp_bij a')). apply csbf_wd. - astepr (Inv f iso_imp_bij (f a')); intuition. - astepr (Inv f iso_imp_bij (f b')). set (H5:= (inv2 M1 M2 f iso_imp_bij b')); intuition. - intuition. + astepr (Inv f iso_imp_bij (f a')); intuition; auto with *. + astepr (Inv f iso_imp_bij (f b')). set (H5:= (inv2 M1 M2 f iso_imp_bij b')); intuition; auto with *. + intuition; auto with *. apply Inv_bij. Qed. @@ -335,11 +335,11 @@ Proof. induction s. simpl. replace (k+0) with k. - intuition. + intuition; auto with *. intuition. simpl. replace (k+((l-k)+s*(l-k))) with (l + s*(l-k)). - 2:intuition. + 2:intuition; auto with *. set (H1:= (power_plus M u l (s*(l-k)))). astepr (csbf_fun (csg_crr (cm_crr M)) (csg_crr (cm_crr M)) (csg_crr (cm_crr M)) (csg_op (c:=cm_crr M)) (power_CMonoid u l) (power_CMonoid u (s * (l - k)))). @@ -374,7 +374,7 @@ Proof. intros H4. set (H6:=(Z_div_mod_eq_full (n-k)(l-k))). cut (((n - k) mod (l - k))= (n-k)%Z -((l - k) * ((n - k) / (l - k))))%Z. - 2:intuition. + 2:intuition; auto with *. set (H7:=(mod_nat_correct (n-k) (l-k) H2)). intro H8. cut {s:nat | (mod_nat (n-k)(l-k) H2)=(n-k)-s*(l-k) and s*(l-k)<=(n-k)}. @@ -391,7 +391,7 @@ Proof. rewrite -> (power_plus M u (k+(s*(l-k))) ((n-k)-s*(l-k))). rewrite -> (power_plus M u k (n-k-s*(l-k))). setoid_replace (power_CMonoid u (k + s * (l - k))) with (power_CMonoid u k). now reflexivity. - unfold canonical_names.equiv. now intuition. + unfold canonical_names.equiv. now intuition; auto with *. cut (n=k+(n-k)). intro H10. cut (n=((k+(n-k))+(s*(l-k)-s*(l-k)))). @@ -408,18 +408,18 @@ Proof. intro H11. rewrite H11. now intuition. - now intuition. + now intuition; auto with *. cut (n=n+(k-k)). intro H10. cut (n+(k-k)=k+(n-k)). intro H11. now rewrite<- H10 in H11. apply minus3. - split; now intuition. + split; now intuition; auto with *. cut ((k-k)=0). intro H10. now rewrite H10. - now intuition. + now intuition; auto with *. simpl. cut (l-k>0). intro H9. @@ -435,17 +435,17 @@ Proof. elim H10'. clear H10'. intros H10' H10''. - 3:intuition. + 3:intuition; auto with *. cut ((n-k)= q*(l-k)+ (mod_nat (n-k)(l-k) H2)). intro H11. - intuition. + intuition; auto with *. cut (r= (mod_nat (n-k)(l-k)H2)). intro H11. now rewrite<- H11. simpl. cut ((Z_of_nat r)=(mod_nat (n - k) (l - k) H2)). intro H11. - intuition. + intuition; auto with *. rewrite<- H7. apply nat_Z_div with (n-k) q (l-k) ((Z_of_nat n - Z_of_nat k) / (Z_of_nat l - Z_of_nat k))%Z. exact H10'. @@ -459,17 +459,17 @@ Proof. set (H16:=(inj_minus1 n k H14)). rewrite H16. exact H6. - intuition. - intuition. + intuition; auto with *. + intuition; auto with *. set (H17:=(Z_mod_lt (Z_of_nat (n-k)) (Z_of_nat (l-k)))). - intuition. + intuition; auto with *. elim H10. clear H10. intros r H10. elim H10. clear H10. intros H10 H10'. - intuition. + intuition; auto with *. Qed. Lemma cyc_imp_comm: forall (M:CMonoid)(H:(cyclic M)), (commutes (@csg_op M)). @@ -491,7 +491,7 @@ Proof. replace (nx+ny) with (ny+nx). rewrite -> (power_plus M c0 ny nx). now apply eq_reflexive. - intuition. + intuition; auto with *. Qed. Lemma weakly_inj1: @@ -544,12 +544,12 @@ Proof. split. intuition. right. - intuition. - intuition. + intuition; auto with *. + intuition; auto with *. clear orex. intro orex. - intuition. - intuition. + intuition; auto with *. + intuition auto with *. clear H5. intro H5. cut False. @@ -587,13 +587,13 @@ Proof. split. intuition. right. - intuition. - intuition. + intuition auto with *. + intuition auto with *. clear orex. intro orex. - intuition. - intuition. - intuition. + intuition; auto with *. + intuition auto with *. + intuition; auto with *. Qed. @@ -716,14 +716,14 @@ Proof. simpl. split. split. - intuition. + intuition; auto with *. intros a b. case a. intros a0 a1. case b. intros b0 b1. simpl. - intuition. + intuition; auto with *. unfold bijective. split. unfold injective. @@ -741,7 +741,7 @@ Proof. intros a0 a1. exists (pairT a1 a0). simpl. - intuition. + intuition; auto with *. Qed. End p71E2b2. @@ -768,7 +768,7 @@ Proof. unfold eq_fun in |- *. unfold id_un_op in |- *. simpl in |- *. - intuition. + intuition; auto with *. Qed. Lemma id_is_lft_unit : @@ -781,7 +781,7 @@ Proof. unfold eq_fun in |- *. unfold id_un_op in |- *. simpl in |- *. - intuition. + intuition; auto with *. Qed. Definition FS_is_CMonoid (A : CSetoid) := @@ -813,7 +813,7 @@ Proof. induction a. apply eq_fm_reflexive. simpl. - intuition. + intuition; auto with *. Qed. Section Th12. @@ -835,7 +835,7 @@ Proof. simpl. intuition. simpl. - intuition. + intuition; auto with *. Qed. Lemma nil_is_lft_unit: (is_lft_unit (app_as_csb_fun A) (empty_word A)). @@ -847,7 +847,7 @@ Proof. simpl. intuition. simpl. - intuition. + intuition; auto with *. Qed. Definition free_monoid_is_CMonoid: @@ -943,7 +943,7 @@ Proof. unfold Dbrack. exists (@nil M). simpl. - intuition. + intuition; auto with *. Qed. @@ -958,7 +958,7 @@ Proof. simpl. astepr (a [+] ( (cm_Sum k)[+](cm_Sum l))). apply csbf_wd_unfolded. - intuition. + intuition; auto with *. exact IHk. Qed. @@ -986,7 +986,7 @@ Qed. Lemma cm_Sum_units (a: list M): (forall x, In x a -> x [=] [0]) -> cm_Sum a [=] [0]. Proof with intuition. clear D. - induction a. intuition. + induction a. intuition; auto with *. intros E. simpl. rewrite IHa... diff --git a/algebra/COrdFields2.v b/algebra/COrdFields2.v index 26b6a87a..243a7039 100644 --- a/algebra/COrdFields2.v +++ b/algebra/COrdFields2.v @@ -739,7 +739,7 @@ Proof. astepr ((Two:R)[^]S n). astepl (([1]:R)[^]S n). apply nexp_resp_less. - intuition. + intuition; auto with *. apply less_leEq. apply pos_one. apply one_less_two. diff --git a/algebra/CSemiGroups.v b/algebra/CSemiGroups.v index 0b83eb1f..a765aef0 100644 --- a/algebra/CSemiGroups.v +++ b/algebra/CSemiGroups.v @@ -378,7 +378,7 @@ Proof. intros x y. apply eq_fm_reflexive. simpl. - intuition. + intuition; auto with *. Qed. Definition Astar_as_CSemiGroup:= diff --git a/algebra/CSetoidFun.v b/algebra/CSetoidFun.v index 41c23795..43fd5464 100644 --- a/algebra/CSetoidFun.v +++ b/algebra/CSetoidFun.v @@ -231,7 +231,7 @@ Proof. simpl in |- *. unfold eq_fun in |- *. simpl in |- *. - intuition. + intuition; auto with *. Qed. Section unary_and_binary_function_composition. @@ -579,7 +579,7 @@ Proof. simpl. intuition. simpl. - intuition. + intuition; auto with *. Qed. End p66E2b4. diff --git a/liouville/Liouville.v b/liouville/Liouville.v index a50e7d91..e046cbd0 100644 --- a/liouville/Liouville.v +++ b/liouville/Liouville.v @@ -239,8 +239,8 @@ Proof. unfold Z.abs. destruct p. destruct Hap; reflexivity. - simpl; unfold Qle; simpl; intuition. - simpl; unfold Qle; simpl; intuition. + simpl; unfold Qle; simpl; intuition; auto with *. + simpl; unfold Qle; simpl; intuition; auto with *. Qed. Lemma Liouville_lemma5 : forall (p : Z_as_CRing) (q : positive), (zx2qx P) ! (p#q)%Q [#] [0] -> diff --git a/logic/CLogic.v b/logic/CLogic.v index 168472a6..55c2df1c 100644 --- a/logic/CLogic.v +++ b/logic/CLogic.v @@ -1610,7 +1610,7 @@ Lemma CForall_indexed {A} (P: A -> Type) (l: list A): CForall P l -> Proof. intros X i. revert l X. - induction i; destruct l; simpl in *; intuition; exfalso; inversion H. + induction i; destruct l; simpl in *; intuition; auto with *; exfalso; inversion H. Qed. Lemma CForall_map {A B} (P: B -> Type) (f: A -> B) (l: list A): @@ -1639,24 +1639,24 @@ Qed. Lemma CNoDup_indexed {T} (R: T -> T -> Type) (Rsym: Csymmetric _ R) (l: list T) (d: T): CNoDup R l -> forall i j, (i < length l)%nat -> (j < length l)%nat -> i <> j -> R (nth i l d) (nth j l d). -Proof with intuition. - induction l; simpl... - exfalso... +Proof. + induction l; simpl; intuition. + exfalso; auto with *. destruct i. - destruct j... - apply (CForall_indexed (R a) l)... - destruct j... + destruct j; intuition. + apply (CForall_indexed (R a) l); auto with *. + destruct j; auto with *. apply Rsym. - apply (CForall_indexed (R a) l)... + apply (CForall_indexed (R a) l); auto with *. Qed. Lemma CNoDup_map {A B: Type} (R: B -> B -> Type) (f: A -> B): forall l, CNoDup (fun x y => R (f x) (f y)) l IFF CNoDup R (map f l). -Proof with auto; intuition. - induction l; simpl... +Proof. + induction l; simpl; auto; intuition; auto with *. split; intro; split. apply IHl, X. - apply CForall_map... + apply CForall_map; auto; intuition. apply IHl, X. - apply CForall_map... + apply CForall_map; auto; intuition. Qed. diff --git a/logic/CornBasics.v b/logic/CornBasics.v index 75bfc556..dd044746 100644 --- a/logic/CornBasics.v +++ b/logic/CornBasics.v @@ -161,7 +161,7 @@ Lemma minus3:forall (a b c:nat),(c<=b<=a)-> a+(b-c)=b+(a-c). Proof. intros a b d H. cut ((Z_of_nat a) + ((Z_of_nat b) - (Z_of_nat d)) = (Z_of_nat b) + ((Z_of_nat a) - (Z_of_nat d)))%Z. - 2:intuition. + 2:intuition; auto with *. intro H1. elim H. intros H2 H3. @@ -169,10 +169,10 @@ Proof. rewrite<- H4 in H1. cut (d <=a). intro H5. - 2:intuition. + 2:intuition; auto with *. set (H6:=(inj_minus1 a d H5)). rewrite<- H6 in H1. - intuition. + intuition; auto with *. Qed. Lemma minus4:forall (a b c d:nat), (d<=c<=b)-> @@ -182,17 +182,17 @@ Proof. cut (((Z_of_nat a)+(Z_of_nat b))+((Z_of_nat c0)-(Z_of_nat d))= ((Z_of_nat a)+(Z_of_nat c0))+((Z_of_nat b)-(Z_of_nat d)))%Z. intro H0. - 2:intuition. + 2:intuition; auto with *. elim H. intros H1 H2. set (H3:=(inj_minus1 c0 d H1)). rewrite<- H3 in H0. cut (d<=b). - 2:intuition. + 2:intuition; auto with *. intro H4. set (H5:=(inj_minus1 b d H4)). rewrite<- H5 in H0. - intuition. + intuition; auto with *. Qed. (** The power function does not exist in the standard library *) @@ -367,21 +367,21 @@ Lemma surj_eq:forall (n m:nat), ((Z_of_nat n)=(Z_of_nat m))%Z -> n=m. Proof. intros n m. - intuition. + intuition; auto with *. Qed. Lemma surj_le:forall (n m:nat), ((Z_of_nat n)<=(Z_of_nat m))%Z -> n<=m. Proof. intros n m. - intuition. + intuition; auto with *. Qed. Lemma surj_lt:forall (n m:nat), ((Z_of_nat n)<(Z_of_nat m))%Z -> n p p+(l-q) (=) ==> (=) ==> iff) genball. - Proof with auto; intuition. + Proof with auto; intuition; auto with *. unfold genball. intros u e' E. destruct u, e'. @@ -226,7 +226,7 @@ Section genball. exfalso. revert q2. rewrite <- E, q1. apply Qlt_irrefl. intros ?? A ?? B. rewrite A, B... repeat intro... - intuition. + intuition; auto with *. repeat intro. reflexivity. Qed. @@ -250,7 +250,7 @@ Proof. intros; now apply genball_Proper. Qed. repeat intro. unfold genball. destruct q... - destruct Qdec_sign as [[|]|]; intuition... + destruct Qdec_sign as [[|]|]; intuition; auto with *. apply (Qlt_not_le q 0)... Qed. diff --git a/metric2/list_separates.v b/metric2/list_separates.v index 4f498df8..0ca7a6ec 100644 --- a/metric2/list_separates.v +++ b/metric2/list_separates.v @@ -36,7 +36,7 @@ Qed. #[global] Instance separates_Proper {A}: Proper (@Permutation _ ==> SetoidPermutation (pair_rel eq (@Permutation _))) (@separates A). -Proof with simpl; auto; intuition. +Proof with simpl; auto; intuition; auto with *. intros ?? P. induction P... 3:eauto. diff --git a/metrics/CMetricSpaces.v b/metrics/CMetricSpaces.v index 12d64f5f..a20ff07e 100644 --- a/metrics/CMetricSpaces.v +++ b/metrics/CMetricSpaces.v @@ -578,7 +578,7 @@ Lemma nz : forall n m : nat, n <= Nat.max n m. Proof. intro n. intro m. - intuition. + intuition; auto with *. Qed. (* end hide *) diff --git a/metrics/CPMSTheory.v b/metrics/CPMSTheory.v index 38a3dd8d..a920b840 100644 --- a/metrics/CPMSTheory.v +++ b/metrics/CPMSTheory.v @@ -132,7 +132,7 @@ Proof. intros. right. rewrite b. - intuition. + intuition; auto with *. Qed. (** @@ -660,8 +660,8 @@ Proof. apply leEq_transitive with (xj[-d]from_SubPsMetricSpace X P x0). apply eq_imp_leEq. apply csbf_wd_unfolded. - intuition. - intuition. + intuition; auto with *. + intuition; auto with *. exact q. apply plus_resp_leEq. apply leEq_transitive with (from_SubPsMetricSpace X P x[-d]xj). diff --git a/metrics/CPseudoMSpaces.v b/metrics/CPseudoMSpaces.v index daf8ac66..870fc473 100644 --- a/metrics/CPseudoMSpaces.v +++ b/metrics/CPseudoMSpaces.v @@ -290,7 +290,7 @@ Proof. unfold Zero_fun in |- *. simpl in |- *. unfold zero_fun in |- *. - intuition. + intuition; auto with *. Qed. Lemma zero_fun_nneg : forall X : CSetoid, nneg (Zero_fun X). @@ -302,7 +302,7 @@ Proof. simpl in |- *. unfold zero_fun in |- *. apply eq_imp_leEq. - intuition. + intuition; auto with *. Qed. Lemma zero_fun_pos_imp_ap : forall X : CSetoid, pos_imp_ap (Zero_fun X). diff --git a/metrics/ContFunctions.v b/metrics/ContFunctions.v index f452ab33..85280603 100644 --- a/metrics/ContFunctions.v +++ b/metrics/ContFunctions.v @@ -183,7 +183,7 @@ Proof. apply recip_resp_less. unfold Snring in |- *. apply nring_pos. - intuition. + intuition; auto with *. apply nat_less_bin_nexp. unfold one_div_succ in |- *. unfold Snring in |- *. diff --git a/metrics/IR_CPMSpace.v b/metrics/IR_CPMSpace.v index a31b0869..b778e4c9 100644 --- a/metrics/IR_CPMSpace.v +++ b/metrics/IR_CPMSpace.v @@ -277,14 +277,14 @@ Proof. apply great_nexp_resp_le. apply less_leEq. apply one_less_two. - intuition. + intuition; auto with *. apply ax_d_nneg. apply CPsMetricSpace_is_CPsMetricSpace. apply mult_resp_leEq_rht. apply great_nexp_resp_le. apply less_leEq. apply one_less_two. - intuition. + intuition; auto with *. apply ax_d_nneg. apply CPsMetricSpace_is_CPsMetricSpace. apply eq_imp_leEq. @@ -350,7 +350,7 @@ Proof. astepr (Two:IR). apply less_leEq. apply one_less_two. - intuition. + intuition; auto with *. apply leEq_transitive with ((OneR[/] Two:IR[//]H1)[^]S n[+]([1][/] Two:IR[//]H1)[^]S n). apply plus_resp_leEq_lft. apply less_leEq. @@ -374,7 +374,7 @@ Proof. astepr (Two:IR). apply less_leEq. apply one_less_two. - intuition. + intuition; auto with *. apply eq_imp_leEq. astepl ((Two:IR)[*]([1][/] Two:IR[//]H1)[^]S n). astepl ((Two:IR)[*]([1][^]S n[/] (Two:IR)[^]S n[//]nexp_resp_ap_zero (S n) H1)). @@ -442,7 +442,7 @@ Proof. apply less_leEq. apply one_less_two. rational. - intuition. + intuition; auto with *. apply leEq_transitive with ((OneR[/] [0][+][1][+][1][//]H1)[^]S n[+] ([1][/] [0][+][1][+][1][//]H1)[^]S n). apply plus_resp_leEq_lft. @@ -467,7 +467,7 @@ Proof. apply less_leEq. apply one_less_two. rational. - intuition. + intuition; auto with *. apply eq_imp_leEq. astepl ((Two:IR)[*]([1][/] [0][+][1][+][1][//]H1)[^]S n). astepr ((OneR[/] [0][+][1][+][1][//]H1)[^]n). @@ -494,4 +494,5 @@ Proof. rational. Qed. + End Addition. diff --git a/metrics/Prod_Sub.v b/metrics/Prod_Sub.v index 47c5e087..cbed5c0e 100644 --- a/metrics/Prod_Sub.v +++ b/metrics/Prod_Sub.v @@ -398,14 +398,14 @@ Proof. apply rev_tri_ineq. apply csf_wd. apply cg_minus_wd. - intuition. + intuition; auto with *. apply ax_d_com. apply CPsMetricSpace_is_CPsMetricSpace. apply csf_wd. apply cg_minus_wd. apply ax_d_com. apply CPsMetricSpace_is_CPsMetricSpace. - intuition. + intuition; auto with *. Qed. diff --git a/model/Zmod/Cmod.v b/model/Zmod/Cmod.v index 88801695..59b901a2 100644 --- a/model/Zmod/Cmod.v +++ b/model/Zmod/Cmod.v @@ -63,9 +63,9 @@ Proof. cut False. intuition. cut (Zneg p < 0)%Z. - intuition. + intuition; auto with *. unfold Z.lt. - intuition. + intuition; auto with *. Qed. @@ -98,7 +98,7 @@ Proof. simpl. intro x. set (H1:= (inject_nat_convert x)). - intuition. + intuition; auto with *. Qed. Lemma nat_Z_div:forall (a b c r:nat)(b' r':Z), @@ -107,7 +107,7 @@ Proof. intros a b c0 r b' r' H H1 H2 H3. cut (c0>0)%Z. intro H5. - 2:intuition. + 2:intuition; auto with *. set (H4:=(Z_div_mod_eq_full (Z_of_nat a) (Z_of_nat c0))). cut ((Z_of_nat a mod (Z_of_nat c0))%Z = r'). intro H6. @@ -124,12 +124,12 @@ Proof. rewrite H9. rewrite H8. apply Zmod_small. - intuition. + intuition; auto with *. intuition. rewrite H2. replace (Z_of_nat c0 * b' + r')%Z with ( b'*Z_of_nat c0 + r')%Z. - 2:intuition. + 2:intuition; auto with *. set (H8:= (Zmod_cancel_multiple c0 r' b' H5)). rewrite H8. - apply Zmod_small; intuition. + apply Zmod_small; intuition; auto with *. Qed. diff --git a/model/Zmod/IrrCrit.v b/model/Zmod/IrrCrit.v index e9fec357..6ffc696b 100644 --- a/model/Zmod/IrrCrit.v +++ b/model/Zmod/IrrCrit.v @@ -155,7 +155,7 @@ Hint Resolve mult_zero : algebra. Lemma fp_resp_zero : zxfpx(cpoly_zero Z_as_CRing)[=](cpoly_zero fp). Proof. - intuition. + intuition; auto with *. Qed. Lemma fpx_resp_mult_cr : forall (c:Z_as_CRing)(f:zx), @@ -163,13 +163,13 @@ Lemma fpx_resp_mult_cr : forall (c:Z_as_CRing)(f:zx), (zxfpx (cpoly_mult_cr_cs _ f c)). Proof. induction f as [|c0 f]. - intuition. + intuition; auto with *. astepr (zxfpx ((c[*]c0)[+X*](cpoly_mult_cr_cs _ f c))). astepr ((zfp (c[*]c0))[+X*](zxfpx (cpoly_mult_cr_cs _ f c))). astepr (((zfp c)[*](zfp c0))[+X*](zxfpx (cpoly_mult_cr_cs _ f c))). astepr (((zfp c)[*](zfp c0))[+X*](cpoly_mult_cr_cs fp (zxfpx f) (zfp c))). astepr (cpoly_mult_cr_cs fp ((zfp c0)[+X*](zxfpx f)) (zfp c)). - intuition. + intuition; auto with *. Qed. #[global] @@ -180,9 +180,9 @@ Lemma fpx_resp_plus : forall f g:zx, (zxfpx (cpoly_plus_op _ f g)). Proof. induction f as [|c f]. - intuition. + intuition; auto with *. induction g as [|c0 g]. - intuition. + intuition; auto with *. astepl (cpoly_plus fp (zxfpx (c[+X*]f)) (zxfpx (c0[+X*]g))). astepr (zxfpx (cpoly_plus_op _ (c[+X*]f) (c0[+X*]g))). astepr (zxfpx ((c[+]c0)[+X*](cpoly_plus_op _ f g))). @@ -203,7 +203,7 @@ Proof. astepl (cpoly_mult_op fp (cpoly_zero fp)(zxfpx g)). astepl (cpoly_zero fp). astepr (zxfpx (cpoly_zero Z_as_CRing)). - astepr (cpoly_zero fp); intuition. + astepr (cpoly_zero fp); intuition; auto with *. induction g as [|c0 g]. astepl (cpoly_mult_op fp (zxfpx (c[+X*]f)) (cpoly_zero fp)). astepl (cpoly_zero fp). @@ -231,7 +231,7 @@ Proof. (([0]:fp)[+X*](zxfpx (cpoly_mult_op _ f (c0[+X*]g))))). astepl ((zfp (c[*]c0)[+]([0]:fp))[+X*](cpoly_plus_op fp (zxfpx (cpoly_mult_cr_cs _ g c)) (zxfpx (cpoly_mult_op _ f (c0[+X*]g))))). - intuition. + intuition; auto with *. Qed. #[global] @@ -241,9 +241,9 @@ Lemma fpx_resp_coef : forall (f:zx)(n:nat), (zfp (nth_coeff n f)) [=] (nth_coeff n (zxfpx f)). Proof. induction f. - intuition. + intuition; auto with *. induction n. - intuition. + intuition; auto with *. astepl (zfp (nth_coeff n f)). astepr (nth_coeff n (zxfpx f)). apply (IHf n). @@ -281,17 +281,17 @@ Proof. unfold monic. split. astepl (zfp (nth_coeff m f)). - assert ([1][=]nth_coeff m f); intuition. + assert ([1][=]nth_coeff m f); intuition; auto with *. simpl in H. rewrite <- H. - intuition. + intuition; auto with *. red. intros. astepl (zfp (nth_coeff m0 f)). - assert ([0][=]nth_coeff m0 f); intuition. + assert ([0][=]nth_coeff m0 f); intuition; auto with *. simpl in H0. rewrite <- H0. - intuition. + intuition; auto with *. Qed. #[global] @@ -309,11 +309,11 @@ Proof. intros Hfok Hfred2; elim Hfred2. intros g Hgok Hfred3; elim Hfred3. intros h Hhok Hfgh; unfold reducible. - intuition. + intuition; auto with *. exists (zxfpx g). - intuition. + intuition; auto with *. exists (zxfpx h). - intuition. + intuition; auto with *. astepr (zxfpx (cpoly_mult_op _ g h)). apply fpxeq_wd. exact Hfgh. diff --git a/model/Zmod/ZBasics.v b/model/Zmod/ZBasics.v index 9cbdf08b..08c76cbb 100644 --- a/model/Zmod/ZBasics.v +++ b/model/Zmod/ZBasics.v @@ -257,7 +257,7 @@ Section zineq. Lemma Zgt_Zge: forall (n m:Z), (n>m)%Z -> (n>=m)%Z. Proof. intros n m. - intuition. + intuition; auto with *. Qed. Lemma Zle_antisymm : forall a b : Z, (a >= b)%Z -> (b >= a)%Z -> a = b. @@ -519,7 +519,7 @@ Qed. Lemma Zcompat_lt_plus: forall (n m p:Z),(n < m)%Z-> (p+n < p+m)%Z. Proof. intros n m p. - intuition. + intuition; auto with *. Qed. Transparent Zplus. @@ -531,7 +531,7 @@ Proof. simpl. set (H:=(succ_nat m)). rewrite H. - intuition. + intuition; auto with *. Qed. Opaque Zplus. diff --git a/model/monoids/Zmonoid.v b/model/monoids/Zmonoid.v index a9d0a6ff..2e1886a9 100644 --- a/model/monoids/Zmonoid.v +++ b/model/monoids/Zmonoid.v @@ -68,7 +68,7 @@ Proof. simpl. split. reflexivity. - intuition. + intuition; auto with *. Qed. Definition Z_is_CMonoid := Build_is_CMonoid diff --git a/model/setoids/Nfinsetoid.v b/model/setoids/Nfinsetoid.v index dc864f2d..c1c7ed61 100644 --- a/model/setoids/Nfinsetoid.v +++ b/model/setoids/Nfinsetoid.v @@ -121,7 +121,7 @@ Proof. red in |- *. unfold not in |- *. unfold Not in |- *. - intuition. + intuition; auto with *. Qed. Definition less (n : nat) := diff --git a/model/setoids/Zfinsetoid.v b/model/setoids/Zfinsetoid.v index 6620e0ad..47e6a0b3 100644 --- a/model/setoids/Zfinsetoid.v +++ b/model/setoids/Zfinsetoid.v @@ -122,7 +122,7 @@ Proof. red in |- *. unfold not in |- *. unfold Not in |- *. - intuition. + intuition; auto with *. Qed. Definition Zless (n : Z) := diff --git a/model/setoids/decsetoid.v b/model/setoids/decsetoid.v index b2e6812d..2188fa79 100644 --- a/model/setoids/decsetoid.v +++ b/model/setoids/decsetoid.v @@ -40,8 +40,8 @@ Section contents. Instance ap_apart: Apartness ap. Proof with auto. apply Build_Apartness. - do 2 intro. intuition. - do 4 intro. intuition. + do 2 intro. intuition; auto with *. + do 4 intro. intuition; auto with *. intros x y H z. destruct (eq_dec x z)... destruct (eq_dec z y)... diff --git a/model/structures/Nsec.v b/model/structures/Nsec.v index 24994f1e..bf485b15 100644 --- a/model/structures/Nsec.v +++ b/model/structures/Nsec.v @@ -237,7 +237,7 @@ Proof. clear H1. intuition. intuition. - intuition. + intuition; auto with *. Qed. Lemma lexi_dec:(forall (k i l:nat), @@ -262,12 +262,12 @@ Proof. right. unfold Not. intro H2. - intuition. + intuition; auto with *. intro H1. left. right. intuition. - intuition. + intuition; auto with *. Qed. diff --git a/model/structures/Qinf.v b/model/structures/Qinf.v index 5761eefa..fde1055f 100644 --- a/model/structures/Qinf.v +++ b/model/structures/Qinf.v @@ -26,8 +26,8 @@ Instance setoid: Setoid T. Proof with intuition. constructor. intros []... - intros [] [] ?... - intros [x|] [y|] [z|] ??... + intros [] [] ?; intuition; auto with *. + intros [x|] [y|] [z|] ??; intuition; auto with *. change (x = z). transitivity y... Qed. @@ -42,7 +42,7 @@ Definition le (x y: T): Prop := #[global] Instance: Proper (=) le. Proof. - intros [|] [|] E [|] [|] F; intuition; simpl; try reflexivity. + intros [|] [|] E [|] [|] F; intuition; auto with *; simpl; try reflexivity. unfold equiv in * |-. simpl in *. now rewrite E, F. Qed. diff --git a/model/structures/QnnInf.v b/model/structures/QnnInf.v index 1699ece1..abd1a09a 100644 --- a/model/structures/QnnInf.v +++ b/model/structures/QnnInf.v @@ -17,8 +17,8 @@ Global Instance: Equivalence eq. Proof with intuition. unfold eq. split; repeat intro. - destruct x... - destruct x, y... + destruct x; intuition; auto with *. + destruct x, y; intuition; auto with *. destruct x, y, z... transitivity q0... Qed. @@ -40,7 +40,7 @@ Section liftM2. Definition liftM2 (x y: T): T := bind x (fun x' => bind y (fun y' => Finite (f x' y'))). Global Instance liftM2_Proper: Proper (eq ==> eq ==> eq) liftM2. - Proof with intuition. intros [] [] ? [] [] ?... simpl. apply p... Qed. + Proof with intuition. intros [] [] ? [] [] ?; intuition; auto with *. simpl. apply p... Qed. Lemma assoc: (forall x y z, f x (f y z) == f (f x y) z)%Qnn -> diff --git a/model/structures/Qsec.v b/model/structures/Qsec.v index 149af4cd..13b580f8 100644 --- a/model/structures/Qsec.v +++ b/model/structures/Qsec.v @@ -257,7 +257,7 @@ Qed. Lemma Qlt_gives_apartness : forall x y : Q, Iff (x/=y) ((x eq n m). - { intros i j H. intuition. } + { intros i j H. intuition; auto with *. } apply H. clear H. apply Zmult_reg_l with (Z.of_nat (nat_of_P b)); [rewrite positive_nat_Z; auto with *|]. diff --git a/stdlib_omissions/List.v b/stdlib_omissions/List.v index f86401a6..1278f624 100644 --- a/stdlib_omissions/List.v +++ b/stdlib_omissions/List.v @@ -70,9 +70,9 @@ Proof with auto. apply NoDup_cons. intros [?|?]... subst y. - intuition. + intuition; auto with *. apply NoDup_cons... - intuition. + intuition; auto with *. Qed. #[global] diff --git a/stdlib_omissions/Q.v b/stdlib_omissions/Q.v index 4c731c04..469b5b74 100644 --- a/stdlib_omissions/Q.v +++ b/stdlib_omissions/Q.v @@ -118,8 +118,8 @@ Proof with intuition. split; intro E. setoid_replace x with (x + z - z)%Q by (simpl; ring). setoid_replace y with (y + z - z)%Q by (simpl; ring). - rewrite E... - rewrite E... + rewrite E; intuition; auto with *. + rewrite E; auto with *. Qed. Lemma Qminus_eq (x y: Q): (x - y == 0 <-> x == y)%Q. @@ -460,7 +460,7 @@ Proof with intuition. intros. apply Qlt_le_trans with (inject_Z (Z_of_nat n) + 1). do 2 rewrite (Qplus_comm (inject_Z (Z_of_nat n))). - apply Qplus_lt_l... + apply Qplus_lt_l; auto with *. pose proof (inj_le _ _ H). rewrite Zle_Qle in H0. rewrite <- S_Qplus... diff --git a/util/Qsums.v b/util/Qsums.v index d50f7ce6..99f3a04f 100644 --- a/util/Qsums.v +++ b/util/Qsums.v @@ -153,7 +153,7 @@ Proof with intuition. with (e * (1#n)). apply Qball_Qabs... unfold canonical_names.equiv, stdlib_rationals.Q_eq. - rewrite <- Zpos_eq_Z_of_nat_o_nat_of_P... + rewrite <- Zpos_eq_Z_of_nat_o_nat_of_P; intuition; auto with *. Qed. Lemma Qmult_Σ (f: nat -> Q) n (k: nat): diff --git a/util/SetoidPermutation.v b/util/SetoidPermutation.v index aade0a2e..3e90498b 100644 --- a/util/SetoidPermutation.v +++ b/util/SetoidPermutation.v @@ -20,9 +20,9 @@ Section def. Proof with eauto; intuition. constructor... intro l. - induction l... + induction l; eauto; intuition; auto with *. intros x y H. - induction H... + induction H; eauto; intuition; auto with *. Qed. Global Instance: Proper (list_eq e ==> list_eq e ==> iff) SetoidPermutation. @@ -69,7 +69,7 @@ Lemma SetoidPermutation_meaning {A} (R: relation A) `{!Equivalence R} (x y: list Proof with auto. split. intro H. induction H. - exists nil. intuition. + exists nil. intuition; auto with *. destruct IHSetoidPermutation as [?[??]]. exists (y :: x0). repeat split... @@ -82,7 +82,7 @@ Proof with auto. exists x1. split. transitivity x; intuition. - transitivity x0; intuition. + transitivity x0; intuition; auto with *. intros [?[E?]]. rewrite E. symmetry. apply SetoidPermutation_from_Permutation... apply _. @@ -99,11 +99,11 @@ Proof with simpl; auto; try reflexivity. apply s_perm_trans with (x y0 :: x x0 :: map y l). apply s_perm_skip... apply s_perm_skip... - induction l... intuition. + induction l... intuition; auto with *. apply s_perm_trans with (y y0 :: y x0 :: map y l)... unfold respectful in *. - apply s_perm_skip. intuition. - apply s_perm_skip... intuition. + apply s_perm_skip. intuition; auto with *. + apply s_perm_skip... intuition; auto with *. apply s_perm_trans with (map y l')... apply s_perm_trans with (map x l')... clear IHX1 IHX2 X1 X2.