feat(hott/init/prod): show lex is well-founded in HoTT

hott/init/prod.hlean

@@ -0,0 +1,83 @@
+/-
+Copyright (c) 2014 Microsoft Corporation. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+
+Module: data.prod.decl
+Author: Leonardo de Moura, Jeremy Avigad
+-/
+prelude
+import init.wf
+
+definition pair := @prod.mk
+
+namespace prod
+  notation A * B := prod A B
+  notation A × B := prod A B
+  namespace low_precedence_times
+    reserve infixr `*`:30  -- conflicts with notation for multiplication
+    infixr `*` := prod
+  end low_precedence_times
+
+  notation `pr₁` := pr1
+  notation `pr₂` := pr2
+
+  -- notation for n-ary tuples
+  notation `(` h `,` t:(foldl `,` (e r, prod.mk r e) h) `)` := t
+
+  open well_founded
+
+  section
+  variables {A B : Type}
+  variable  (Ra  : A → A → Type)
+  variable  (Rb  : B → B → Type)
+
+  -- Lexicographical order based on Ra and Rb
+  inductive lex : A × B → A × B → Type :=
+  left  : ∀{a₁ b₁} a₂ b₂, Ra a₁ a₂ → lex (a₁, b₁) (a₂, b₂),
+  right : ∀a {b₁ b₂},     Rb b₁ b₂ → lex (a, b₁)  (a, b₂)
+
+  -- Relational product based on Ra and Rb
+  inductive rprod : A × B → A × B → Type :=
+  intro : ∀{a₁ b₁ a₂ b₂}, Ra a₁ a₂ → Rb b₁ b₂ → rprod (a₁, b₁) (a₂, b₂)
+  end
+
+  context
+  parameters {A B : Type}
+  parameters {Ra  : A → A → Type} {Rb  : B → B → Type}
+  infix `≺`:50 := lex Ra Rb
+
+  definition lex.accessible {a} (aca : acc Ra a) (acb : ∀b, acc Rb b): ∀b, acc (lex Ra Rb) (a, b) :=
+  acc.rec_on aca
+    (λxa aca (iHa : ∀y, Ra y xa → ∀b, acc (lex Ra Rb) (y, b)),
+      λb, acc.rec_on (acb b)
+        (λxb acb
+          (iHb : ∀y, Rb y xb → acc (lex Ra Rb) (xa, y)),
+          acc.intro (xa, xb) (λp (lt : p ≺ (xa, xb)),
+            have aux : xa = xa → xb = xb → acc (lex Ra Rb) p, from
+              @lex.rec_on A B Ra Rb (λp₁ p₂ h, pr₁ p₂ = xa → pr₂ p₂ = xb → acc (lex Ra Rb) p₁)
+                         p (xa, xb) lt
+                (λa₁ b₁ a₂ b₂ (H : Ra a₁ a₂) (eq₂ : a₂ = xa) (eq₃ : b₂ = xb),
+                  show acc (lex Ra Rb) (a₁, b₁), from
+                  have Ra₁  : Ra a₁ xa, from eq.rec_on eq₂ H,
+                  iHa a₁ Ra₁ b₁)
+                (λa b₁ b₂ (H : Rb b₁ b₂) (eq₂ : a = xa) (eq₃ : b₂ = xb),
+                  show acc (lex Ra Rb) (a, b₁), from
+                  have Rb₁  : Rb b₁ xb, from eq.rec_on eq₃ H,
+                  have eq₂' : xa = a, from eq.rec_on eq₂ rfl,
+                  eq.rec_on eq₂' (iHb b₁ Rb₁)),
+            aux rfl rfl)))
+
+  -- The lexicographical order of well founded relations is well-founded
+  definition lex.wf (Ha : well_founded Ra) (Hb : well_founded Rb) : well_founded (lex Ra Rb) :=
+  well_founded.intro (λp, destruct p (λa b, lex.accessible (Ha a) (well_founded.apply Hb) b))
+
+  -- Relational product is a subrelation of the lex
+  definition rprod.sub_lex : ∀ a b, rprod Ra Rb a b → lex Ra Rb a b :=
+  λa b H, rprod.rec_on H (λ a₁ b₁ a₂ b₂ H₁ H₂, lex.left Rb a₂ b₂ H₁)
+
+  -- The relational product of well founded relations is well-founded
+  definition rprod.wf (Ha : well_founded Ra) (Hb : well_founded Rb) : well_founded (rprod Ra Rb) :=
+  subrelation.wf (rprod.sub_lex) (lex.wf Ha Hb)
+
+  end
+end prod