diff options
Diffstat (limited to 'theories/Numbers/Integer/Abstract/ZOrder.v')
| -rw-r--r-- | theories/Numbers/Integer/Abstract/ZOrder.v | 18 |
1 files changed, 9 insertions, 9 deletions
diff --git a/theories/Numbers/Integer/Abstract/ZOrder.v b/theories/Numbers/Integer/Abstract/ZOrder.v index e0ef2f15d9..7eed9a8eeb 100644 --- a/theories/Numbers/Integer/Abstract/ZOrder.v +++ b/theories/Numbers/Integer/Abstract/ZOrder.v @@ -2,7 +2,7 @@ Require Export ZTimes. Module ZOrderPropFunct (Import ZAxiomsMod : ZAxiomsSig). Module Export ZTimesPropMod := ZTimesPropFunct ZAxiomsMod. -Open Local Scope NatIntScope. +Open Local Scope IntScope. (* Axioms *) @@ -140,21 +140,21 @@ Proof NZneq_succ_iter_l. in the induction step *) Theorem Zright_induction : - forall A : Z -> Prop, predicate_wd E A -> + forall A : Z -> Prop, predicate_wd Zeq A -> forall z : Z, A z -> (forall n : Z, z <= n -> A n -> A (S n)) -> forall n : Z, z <= n -> A n. Proof NZright_induction. Theorem Zleft_induction : - forall A : Z -> Prop, predicate_wd E A -> + forall A : Z -> Prop, predicate_wd Zeq A -> forall z : Z, A z -> (forall n : Z, n < z -> A (S n) -> A n) -> forall n : Z, n <= z -> A n. Proof NZleft_induction. Theorem Zorder_induction : - forall A : Z -> Prop, predicate_wd E A -> + forall A : Z -> Prop, predicate_wd Zeq A -> forall z : Z, A z -> (forall n : Z, z <= n -> A n -> A (S n)) -> (forall n : Z, n < z -> A (S n) -> A n) -> @@ -162,7 +162,7 @@ Theorem Zorder_induction : Proof NZorder_induction. Theorem Zorder_induction' : - forall A : Z -> Prop, predicate_wd E A -> + forall A : Z -> Prop, predicate_wd Zeq A -> forall z : Z, A z -> (forall n : Z, z <= n -> A n -> A (S n)) -> (forall n : Z, n <= z -> A n -> A (P n)) -> @@ -175,7 +175,7 @@ unfold predicate_wd, fun_wd in A_wd; apply -> (A_wd (P (S m)) m); Qed. Theorem Zright_induction' : - forall A : Z -> Prop, predicate_wd E A -> + forall A : Z -> Prop, predicate_wd Zeq A -> forall z : Z, (forall n : Z, n <= z -> A n) -> (forall n : Z, z <= n -> A n -> A (S n)) -> @@ -183,7 +183,7 @@ Theorem Zright_induction' : Proof NZright_induction'. Theorem Zstrong_right_induction' : - forall A : Z -> Prop, predicate_wd E A -> + forall A : Z -> Prop, predicate_wd Zeq A -> forall z : Z, (forall n : Z, n <= z -> A n) -> (forall n : Z, z <= n -> (forall m : Z, z <= m -> m < n -> A m) -> A n) -> @@ -193,7 +193,7 @@ Proof NZstrong_right_induction'. (** Elimintation principle for < *) Theorem Zlt_ind : - forall A : Z -> Prop, predicate_wd E A -> + forall A : Z -> Prop, predicate_wd Zeq A -> forall n : Z, A (S n) -> (forall m : Z, n < m -> A m -> A (S m)) -> @@ -203,7 +203,7 @@ Proof NZlt_ind. (** Elimintation principle for <= *) Theorem Zle_ind : - forall A : Z -> Prop, predicate_wd E A -> + forall A : Z -> Prop, predicate_wd Zeq A -> forall n : Z, A n -> (forall m : Z, n <= m -> A m -> A (S m)) -> |