Isabelle 如何在归纳法中引用变量

我有一个与列表匹配的函数：

fun merge where
  ‹merge [] [] = []› |
  ‹merge (v#vs) [] = (v#vs)› |
  ‹merge [] (v#vs) = (v#vs)› |
  ‹merge (x#xs) (y#ys) =
    (if x < y then x # merge xs (y#ys) else y # merge (x#xs) ys)›

问题是，我可以证明

sorted（merge xs[]）

的这种情况，但这并不满足

sorted（merge（v#vs）[]）

这是证明的目标

因此，我如何将

xs

固定为

v#vs

，或者为该特定案例参考案例分析

xs

因此，如何将xs固定为

v#vs

，或者以其他方式引用案例 针对该特定案例分析了

xs

Isabelle2020参考手册（Isar ref）第6.5节对此进行了解释：

。。。通过使用显式形式

case（cy1…ym）

。。。证据 作者可以选择适合当前环境的本地名称 上下文

因此，在您的情况下，您可以使用类似于

fun merge where
  ‹merge [] [] = []› 
| ‹merge (v#vs) [] = (v#vs)› 
| ‹merge [] (v#vs) = (v#vs)› 
| ‹merge (x#xs) (y#ys) = 
    (if x < y then x # merge xs (y#ys) else y # merge (x#xs) ys)›

lemma sorted_merge:
  assumes s_xs: "sorted(xs)" and s_ys: "sorted(ys)"
  shows ‹sorted(merge xs ys)›  
  using assms
proof(induction xs ys rule: merge.induct[case_names base xs ys both])
  case base then show ?case by auto
next
  case (xs v vs) then show ?case by auto
next
  case (ys v vs) then show ?case by auto
next
  case (both x xs y ys)
  show ?case 
  proof(cases ‹x < y›)
    case True with both(1,3,4) show ?thesis by (induction xs) force+
  next
    case False with both(2,3,4) show ?thesis by (induction ys) force+
  qed      
qed

fun合并在哪里
媫合并[]]=[]›
|è合并（v#vs）[]=（v#vs）›
|è合并[]（v#vs）=（v#vs）›
|èmerge（x#xs）（y#ys）=
（如果x

---

伊莎贝尔版本：伊莎贝尔2020

fun merge where
  ‹merge [] [] = []› 
| ‹merge (v#vs) [] = (v#vs)› 
| ‹merge [] (v#vs) = (v#vs)› 
| ‹merge (x#xs) (y#ys) = 
    (if x < y then x # merge xs (y#ys) else y # merge (x#xs) ys)›

lemma sorted_merge:
  assumes s_xs: "sorted(xs)" and s_ys: "sorted(ys)"
  shows ‹sorted(merge xs ys)›  
  using assms
proof(induction xs ys rule: merge.induct[case_names base xs ys both])
  case base then show ?case by auto
next
  case (xs v vs) then show ?case by auto
next
  case (ys v vs) then show ?case by auto
next
  case (both x xs y ys)
  show ?case 
  proof(cases ‹x < y›)
    case True with both(1,3,4) show ?thesis by (induction xs) force+
  next
    case False with both(2,3,4) show ?thesis by (induction ys) force+
  qed      
qed