diff options
Diffstat (limited to 'theories/Numbers/Natural')
-rw-r--r-- | theories/Numbers/Natural/Abstract/NBase.v | 6 | ||||
-rw-r--r-- | theories/Numbers/Natural/Abstract/NOrder.v | 2 | ||||
-rw-r--r-- | theories/Numbers/Natural/Binary/NBinDefs.v | 4 |
3 files changed, 6 insertions, 6 deletions
diff --git a/theories/Numbers/Natural/Abstract/NBase.v b/theories/Numbers/Natural/Abstract/NBase.v index c7632d185..a0111a082 100644 --- a/theories/Numbers/Natural/Abstract/NBase.v +++ b/theories/Numbers/Natural/Abstract/NBase.v @@ -187,16 +187,16 @@ Qed. Theorem succ_pred : forall n : N, n ~= 0 -> S (P n) == n. Proof. cases n. -intro H; elimtype False; now apply H. +intro H; exfalso; now apply H. intros; now rewrite pred_succ. Qed. Theorem pred_inj : forall n m : N, n ~= 0 -> m ~= 0 -> P n == P m -> n == m. Proof. intros n m; cases n. -intros H; elimtype False; now apply H. +intros H; exfalso; now apply H. intros n _; cases m. -intros H; elimtype False; now apply H. +intros H; exfalso; now apply H. intros m H2 H3. do 2 rewrite pred_succ in H3. now rewrite H3. Qed. diff --git a/theories/Numbers/Natural/Abstract/NOrder.v b/theories/Numbers/Natural/Abstract/NOrder.v index f02baca2c..aee2cf8f7 100644 --- a/theories/Numbers/Natural/Abstract/NOrder.v +++ b/theories/Numbers/Natural/Abstract/NOrder.v @@ -455,7 +455,7 @@ Qed. Theorem lt_pred_l : forall n : N, n ~= 0 -> P n < n. Proof. cases n. -intro H; elimtype False; now apply H. +intro H; exfalso; now apply H. intros; rewrite pred_succ; apply lt_succ_diag_r. Qed. diff --git a/theories/Numbers/Natural/Binary/NBinDefs.v b/theories/Numbers/Natural/Binary/NBinDefs.v index e0f3fdf4b..c2c7767d5 100644 --- a/theories/Numbers/Natural/Binary/NBinDefs.v +++ b/theories/Numbers/Natural/Binary/NBinDefs.v @@ -160,11 +160,11 @@ Theorem NZlt_succ_r : forall n m : NZ, n < (NZsucc m) <-> n <= m. Proof. intros n m; unfold Nlt, Nle; destruct n as [| p]; destruct m as [| q]; simpl; split; intro H; try reflexivity; try discriminate. -destruct p; simpl; intros; discriminate. elimtype False; now apply H. +destruct p; simpl; intros; discriminate. exfalso; now apply H. apply -> Pcompare_p_Sq in H. destruct H as [H | H]. now rewrite H. now rewrite H, Pcompare_refl. apply <- Pcompare_p_Sq. case_eq ((p ?= q)%positive Eq); intro H1. -right; now apply Pcompare_Eq_eq. now left. elimtype False; now apply H. +right; now apply Pcompare_Eq_eq. now left. exfalso; now apply H. Qed. Theorem NZmin_l : forall n m : N, n <= m -> NZmin n m = n. |