test(tests/lean/run/forest): define brec_on for forests

Almost everything explicit to get an idea of what needs to be generated automatically.

tests/lean/run/forest.lean

@@ -0,0 +1,108 @@
+import data.prod data.unit
+open prod
+
+inductive tree (A : Type) : Type :=
+node : A → forest A → tree A
+with forest : Type :=
+nil  : forest A,
+cons : tree A → forest A → forest A
+
+definition tree.below.{l₁ l₂}
+     (A : Type.{l₁})
+     (C₁ : tree A → Type.{l₂})
+     (C₂ : forest A → Type.{l₂})
+     (t : tree A) : Type.{max 1 l₂} :=
+  @tree.rec_on A
+    (λ t : tree A,   Type.{max 1 l₂})
+    (λ t : forest A, Type.{max 1 l₂})
+    t
+    (λ (a : A) (f : forest A) (r : Type.{max 1 l₂}), prod.{l₂ (max 1 l₂)} (C₂ f) r)
+    unit.{max 1 l₂}
+    (λ (t : tree A) (f : forest A) (rt : Type.{max 1 l₂}) (rf : Type.{max 1 l₂}),
+        prod.{(max 1 l₂) (max 1 l₂)} (prod.{l₂ (max 1 l₂)} (C₁ t) rt) (prod.{l₂ (max 1 l₂)} (C₂ f) rf))
+
+definition forest.below.{l₁ l₂}
+     (A : Type.{l₁})
+     (C₁ : tree A → Type.{l₂})
+     (C₂ : forest A → Type.{l₂})
+     (f : forest A) : Type.{max 1 l₂} :=
+  @forest.rec_on A
+    (λ t : tree A,   Type.{max 1 l₂})
+    (λ t : forest A, Type.{max 1 l₂})
+    f
+    (λ (a : A) (f : forest A) (r : Type.{max 1 l₂}), prod.{l₂ (max 1 l₂)} (C₂ f) r)
+    unit.{max 1 l₂}
+    (λ (t : tree A) (f : forest A) (rt : Type.{max 1 l₂}) (rf : Type.{max 1 l₂}),
+        prod.{(max 1 l₂) (max 1 l₂)} (prod.{l₂ (max 1 l₂)} (C₁ t) rt) (prod.{l₂ (max 1 l₂)} (C₂ f) rf))
+
+definition tree.brec_on.{l₁ l₂}
+     (A  : Type.{l₁})
+     (C₁ : tree A → Type.{l₂})
+     (C₂ : forest A → Type.{l₂})
+     (t  : tree A)
+     (F₁ : Π (t : tree A),   tree.below A C₁ C₂ t   → C₁ t)
+     (F₂ : Π (f : forest A), forest.below A C₁ C₂ f → C₂ f)
+  : C₁ t :=
+have general : prod.{l₂ (max 1 l₂)} (C₁ t) (tree.below A C₁ C₂ t), from
+  @tree.rec_on
+    A
+    (λ (t' : tree A),   prod.{l₂ (max 1 l₂)} (C₁ t') (tree.below A C₁ C₂ t'))
+    (λ (f' : forest A), prod.{l₂ (max 1 l₂)} (C₂ f') (forest.below A C₁ C₂ f'))
+    t
+    (λ (a : A) (f : forest A) (r : prod.{l₂ (max 1 l₂)} (C₂ f) (forest.below A C₁ C₂ f)),
+       have b : tree.below A C₁ C₂ (tree.node a f), from
+         r,
+       have c : C₁ (tree.node a f), from
+         F₁ (tree.node a f) b,
+       prod.mk.{l₂ (max 1 l₂)} c b)
+    (have b : forest.below A C₁ C₂ (forest.nil A), from
+      unit.star.{max 1 l₂},
+     have c : C₂ (forest.nil A), from
+      F₂ (forest.nil A) b,
+     prod.mk.{l₂ (max 1 l₂)} c b)
+    (λ (t : tree A)
+       (f : forest A)
+       (rt : prod.{l₂ (max 1 l₂)} (C₁ t) (tree.below A C₁ C₂ t))
+       (rf : prod.{l₂ (max 1 l₂)} (C₂ f) (forest.below A C₁ C₂ f)),
+     have b : forest.below A C₁ C₂ (forest.cons t f), from
+       prod.mk.{(max 1 l₂) (max 1 l₂)} rt rf,
+     have c : C₂ (forest.cons t f), from
+       F₂ (forest.cons t f) b,
+     prod.mk.{l₂ (max 1 l₂)} c b),
+pr₁ general
+
+definition forest.brec_on.{l₁ l₂}
+     (A  : Type.{l₁})
+     (C₁ : tree A → Type.{l₂})
+     (C₂ : forest A → Type.{l₂})
+     (f  : forest A)
+     (F₁ : Π (t : tree A),   tree.below A C₁ C₂ t   → C₁ t)
+     (F₂ : Π (f : forest A), forest.below A C₁ C₂ f → C₂ f)
+  : C₂ f :=
+have general : prod.{l₂ (max 1 l₂)} (C₂ f) (forest.below A C₁ C₂ f), from
+  @forest.rec_on
+    A
+    (λ (t' : tree A),   prod.{l₂ (max 1 l₂)} (C₁ t') (tree.below A C₁ C₂ t'))
+    (λ (f' : forest A), prod.{l₂ (max 1 l₂)} (C₂ f') (forest.below A C₁ C₂ f'))
+    f
+    (λ (a : A) (f : forest A) (r : prod.{l₂ (max 1 l₂)} (C₂ f) (forest.below A C₁ C₂ f)),
+       have b : tree.below A C₁ C₂ (tree.node a f), from
+         r,
+       have c : C₁ (tree.node a f), from
+         F₁ (tree.node a f) b,
+       prod.mk.{l₂ (max 1 l₂)} c b)
+    (have b : forest.below A C₁ C₂ (forest.nil A), from
+      unit.star.{max 1 l₂},
+     have c : C₂ (forest.nil A), from
+      F₂ (forest.nil A) b,
+     prod.mk.{l₂ (max 1 l₂)} c b)
+    (λ (t : tree A)
+       (f : forest A)
+       (rt : prod.{l₂ (max 1 l₂)} (C₁ t) (tree.below A C₁ C₂ t))
+       (rf : prod.{l₂ (max 1 l₂)} (C₂ f) (forest.below A C₁ C₂ f)),
+     have b : forest.below A C₁ C₂ (forest.cons t f), from
+       prod.mk.{(max 1 l₂) (max 1 l₂)} rt rf,
+     have c : C₂ (forest.cons t f), from
+       F₂ (forest.cons t f) b,
+     prod.mk.{l₂ (max 1 l₂)} c b),
+pr₁ general