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Diffstat (limited to 'theories/Num/LeProps.v')
-rw-r--r-- | theories/Num/LeProps.v | 67 |
1 files changed, 67 insertions, 0 deletions
diff --git a/theories/Num/LeProps.v b/theories/Num/LeProps.v new file mode 100644 index 000000000..0b46c3576 --- /dev/null +++ b/theories/Num/LeProps.v @@ -0,0 +1,67 @@ +Require Export LtProps. +Require Export LeAxioms. + +(*s Properties of the relation [<=] *) + +Lemma lt_le : (x,y:N)(x<y)->(x<=y). +Auto with num. +Save. + +Lemma eq_le : (x,y:N)(x=y)->(x<=y). +Auto with num. +Save. + +Lemma le_eq_compat : (x1,x2,y1,y2:N)(x1=y1)->(x2=y2)->(x1<=x2)->(y1<=y2). +Intros x1 x2 y1 y2 eq1 eq2 le1; Case le_lt_or_eq with 1:=le1; Intro. +EAuto with num. +Apply eq_le; Apply eq_trans with x1; EAuto with num. +Save. +Hints Resolve le_eq_compat : num. + +Lemma le_trans : (x,y,z:N)(x<=y)->(y<=z)->(x<=z). +Intros x y z le1 le2. +Case le_lt_or_eq with 1:=le1; Intro. +Case le_lt_or_eq with 1:=le2; EAuto with num. +EAuto with num. +Save. +Hints Resolve le_trans : num. + +(*s compatibility with equality, addition and successor *) +Lemma le_add_compat_l : (x,y,z:N)(x<=y)->((x+z)<=(y+z)). +Intros x y z le1. +Case le_lt_or_eq with 1:=le1; EAuto with num. +Save. +Hints Resolve le_add_compat_l : num. + +Lemma le_add_compat_r : (x,y,z:N)(x<=y)->((z+x)<=(z+y)). +Intros x y z H; Apply le_eq_compat with (x+z) (y+z); Auto with num. +Save. +Hints Resolve le_add_compat_r : num. + +Lemma le_S_compat : (x,y:N)(x<=y)->((S x)<=(S y)). +Intros x y le1. +Case le_lt_or_eq with 1:=le1; EAuto with num. +Save. +Hints Resolve le_S_compat : num. + +Lemma le_lt_x_Sy : (x,y:N)(x<=y)->(x<(S y)). +Intros x y le1. +Case le_lt_or_eq with 1:=le1; Auto with num. +Save. +Hints Immediate le_lt_x_Sy : num. + +Lemma le_Sx_y_lt : (x,y:N)((S x)<=y)->(x<y). +Intros x y le1. +Case le_lt_or_eq with 1:=le1; EAuto with num. +Save. +Hints Immediate le_Sx_y_lt : num. + +Lemma lt_le_trans : (x,y,z:N)(x<y)->(y<=z)->(x<z). +Intros x y z lt1 le1; Case le_lt_or_eq with 1:= le1; EAuto with num. +Save. + +Lemma le_lt_trans : (x,y,z:N)(x<=y)->(y<z)->(x<=z). +Intros x y z le1 lt1; Case le_lt_or_eq with 1:= le1; EAuto with num. +Save. +Hints Immediate lt_le_trans le_lt_trans : num. + |