chore: improve doc string

This PR improves the theorems used to justify the steps performed by
the inequality offset module. See new test for examples of how they are
going to be used.

src/Init/Grind/Offset.lean

@@ -7,7 +7,7 @@ prelude
 import Init.Core
 import Init.Omega
 
-namespace Lean.Grind
+namespace Lean.Grind.Offset
 
 abbrev Var := Nat
 abbrev Context := Lean.RArray Nat
@@ -22,7 +22,7 @@ structure Cnstr where
   y : Var
   k : Nat  := 0
   l : Bool := true
-  deriving Repr, BEq, Inhabited
+  deriving Repr, DecidableEq, Inhabited
 
 def Cnstr.denote (c : Cnstr) (ctx : Context) : Prop :=
   if c.l then
@@ -82,51 +82,84 @@ def Cnstr.isFalse (c : Cnstr) : Bool := c.x == c.y && c.k != 0 && c.l == true
 theorem Cnstr.of_isFalse (ctx : Context) {c : Cnstr} : c.isFalse = true → ¬c.denote ctx := by
   cases c; simp [isFalse]; intros; simp [*, denote]; omega
 
-def Certificate := List Cnstr
+def Cnstrs := List Cnstr
 
-def Certificate.denote' (ctx : Context) (c₁ : Cnstr) (c₂ : Certificate) : Prop :=
+def Cnstrs.denoteAnd' (ctx : Context) (c₁ : Cnstr) (c₂ : Cnstrs) : Prop :=
   match c₂ with
   | [] => c₁.denote ctx
-  | c::cs => c₁.denote ctx ∧ Certificate.denote' ctx c cs
+  | c::cs => c₁.denote ctx ∧ Cnstrs.denoteAnd' ctx c cs
 
-theorem Certificate.denote'_trans (ctx : Context) (c₁ c : Cnstr) (cs : Certificate) : c₁.denote ctx → denote' ctx c cs → denote' ctx (c₁.trans c) cs := by
+theorem Cnstrs.denote'_trans (ctx : Context) (c₁ c : Cnstr) (cs : Cnstrs) : c₁.denote ctx → denoteAnd' ctx c cs → denoteAnd' ctx (c₁.trans c) cs := by
   induction cs
-  next => simp [denote', *]; apply Cnstr.denote_trans
-  next c cs ih => simp [denote']; intros; simp [*]
+  next => simp [denoteAnd', *]; apply Cnstr.denote_trans
+  next c cs ih => simp [denoteAnd']; intros; simp [*]
 
-def Certificate.trans' (c₁ : Cnstr) (c₂ : Certificate) : Cnstr :=
+def Cnstrs.trans' (c₁ : Cnstr) (c₂ : Cnstrs) : Cnstr :=
   match c₂ with
   | [] => c₁
   | c::c₂ => trans' (c₁.trans c) c₂
 
-@[simp] theorem Certificate.denote'_trans' (ctx : Context) (c₁ : Cnstr) (c₂ : Certificate) : denote' ctx c₁ c₂ → (trans' c₁ c₂).denote ctx := by
+@[simp] theorem Cnstrs.denote'_trans' (ctx : Context) (c₁ : Cnstr) (c₂ : Cnstrs) : denoteAnd' ctx c₁ c₂ → (trans' c₁ c₂).denote ctx := by
   induction c₂ generalizing c₁
-  next => intros; simp_all [trans', denote']
-  next c cs ih => simp [denote']; intros; simp [trans']; apply ih; apply denote'_trans <;> assumption
+  next => intros; simp_all [trans', denoteAnd']
+  next c cs ih => simp [denoteAnd']; intros; simp [trans']; apply ih; apply denote'_trans <;> assumption
 
-def Certificate.denote (ctx : Context) (c : Certificate) : Prop :=
+def Cnstrs.denoteAnd (ctx : Context) (c : Cnstrs) : Prop :=
   match c with
   | [] => True
-  | c::cs => denote' ctx c cs
+  | c::cs => denoteAnd' ctx c cs
 
-def Certificate.trans (c : Certificate) : Cnstr :=
+def Cnstrs.trans (c : Cnstrs) : Cnstr :=
   match c with
   | []    => trivialCnstr
   | c::cs => trans' c cs
 
-theorem Certificate.denote_trans {ctx : Context} {c : Certificate} : c.denote ctx → c.trans.denote ctx := by
-  cases c <;> simp [*, trans, Certificate.denote] <;> intros <;> simp [*]
+theorem Cnstrs.of_denoteAnd_trans {ctx : Context} {c : Cnstrs} : c.denoteAnd ctx → c.trans.denote ctx := by
+  cases c <;> simp [*, trans, denoteAnd] <;> intros <;> simp [*]
 
-def Certificate.isFalse (c : Certificate) : Bool :=
+def Cnstrs.isFalse (c : Cnstrs) : Bool :=
   c.trans.isFalse
 
-theorem Certificate.unsat (ctx : Context) (c : Certificate) : c.isFalse = true → ¬ c.denote ctx := by
+theorem Cnstrs.unsat' (ctx : Context) (c : Cnstrs) : c.isFalse = true → ¬ c.denoteAnd ctx := by
   simp [isFalse]; intro h₁ h₂
-  have := Certificate.denote_trans h₂
+  have := of_denoteAnd_trans h₂
   have := Cnstr.of_isFalse ctx h₁
   contradiction
 
-theorem Certificate.imp (ctx : Context) (c : Certificate) : c.denote ctx → c.trans.denote ctx := by
-  apply denote_trans
+/-- `denote ctx [c_1, ..., c_n] C` is `c_1.denote ctx → ... → c_n.denote ctx → C` -/
+def Cnstrs.denote (ctx : Context) (cs : Cnstrs) (C : Prop) : Prop :=
+  match cs with
+  | [] => C
+  | c::cs => c.denote ctx → denote ctx cs C
 
-end Lean.Grind
+theorem Cnstrs.not_denoteAnd'_eq (ctx : Context) (c : Cnstr) (cs : Cnstrs) (C : Prop) : (denoteAnd' ctx c cs → C) = denote ctx (c::cs) C := by
+  simp [denote]
+  induction cs generalizing c
+  next => simp [denoteAnd', denote]
+  next c' cs ih =>
+    simp [denoteAnd', denote, *]
+
+theorem Cnstrs.not_denoteAnd_eq (ctx : Context) (cs : Cnstrs) (C : Prop) : (denoteAnd ctx cs → C) = denote ctx cs C := by
+  cases cs
+  next => simp [denoteAnd, denote]
+  next c cs => apply not_denoteAnd'_eq
+
+def Cnstr.isImpliedBy (cs : Cnstrs) (c : Cnstr) : Bool :=
+  cs.trans == c
+
+/-! Main theorems used by `grind`. -/
+
+/-- Auxiliary theorem used by `grind` to prove that a system of offset inequalities is unsatisfiable. -/
+theorem Cnstrs.unsat (ctx : Context) (cs : Cnstrs) : cs.isFalse = true → cs.denote ctx False := by
+  intro h
+  rw [← not_denoteAnd_eq]
+  apply unsat'
+  assumption
+
+/-- Auxiliary theorem used by `grind` to prove an implied offset inequality. -/
+theorem Cnstrs.imp (ctx : Context) (cs : Cnstrs) (c : Cnstr) (h : c.isImpliedBy cs = true)  : cs.denote ctx (c.denote ctx) := by
+  rw [← eq_of_beq h]
+  rw [← not_denoteAnd_eq]
+  apply of_denoteAnd_trans
+
+end Lean.Grind.Offset

tests/lean/run/grind_offset_ineq_thms.lean

@@ -0,0 +1,31 @@
+import Lean
+
+elab tk:"#R[" ts:term,* "]" : term => do
+  let ts : Array Lean.Syntax := ts
+  let es ← ts.mapM fun stx => Lean.Elab.Term.elabTerm stx none
+  if h : 0 < es.size then
+    return (Lean.RArray.toExpr (← Lean.Meta.inferType es[0]!) id (Lean.RArray.ofArray es h))
+  else
+    throwErrorAt tk "RArray cannot be empty"
+
+open Lean.Grind.Offset
+
+macro "C[" "#" x:term:max " ≤ " "#" y:term:max "]" : term => `({ x := $x, y := $y : Cnstr })
+macro "C[" "#" x:term:max " + " k:term:max " ≤ " "#" y:term:max "]" : term => `({ x := $x, y := $y, k := $k : Cnstr })
+macro "C[" "#" x:term:max " ≤ " "#"y:term:max " + " k:term:max "]" : term => `({ x := $x, y := $y, k := $k, l := false : Cnstr })
+
+example (x y z : Nat) : x + 2 ≤ y → y ≤ z → z + 1 ≤ x → False :=
+  Cnstrs.unsat #R[x, y, z] [
+    C[ #0 + 2 ≤ #1 ],
+    C[ #1 ≤ #2 ],
+    C[ #2 + 1 ≤ #0 ]
+  ] rfl
+
+example (x y z w : Nat) : x + 2 ≤ y → y ≤ z → z ≤ w + 7 → x ≤ w + 5 :=
+  Cnstrs.imp #R[x, y, z, w] [
+    C[ #0 + 2 ≤ #1 ],
+    C[ #1 ≤ #2 ],
+    C[ #2 ≤ #3 + 7]
+  ]
+  C[ #0 ≤ #3 + 5 ]
+  rfl