Reshuffle in preparation to break the dependency on N's implementation (#75)

Numbers/BinaryNaturals/Addition.agda

@@ -3,10 +3,13 @@
 open import LogicalFormulae
 open import Functions
 open import Lists.Lists
+open import Numbers.Naturals.Semiring
 open import Numbers.Naturals.Naturals
+open import Numbers.Naturals.Order
 open import Groups.Definition
 open import Numbers.BinaryNaturals.Definition
 open import Semirings.Definition
+open import Orders
 
 module Numbers.BinaryNaturals.Addition where
 
@@ -39,7 +42,7 @@ _+B_ : BinNat → BinNat → BinNat
     refine b pr with canonical b
     refine b pr | x :: bl = ::Inj pr
     t : NToBinNat (0 +N binNatToN (zero :: b)) ≡ zero :: b
-    t with orderIsTotal 0 (binNatToN b)
+    t with TotalOrder.totality ℕTotalOrder 0 (binNatToN b)
     t | inl (inl pos) = transitivity (doubleIsBitShift (binNatToN b) pos) (applyEquality (zero ::_) (transitivity (binToBin b) (equalityCommutative (refine b prB))))
     t | inl (inr ())
     ... | inr eq with binNatToNZero b (equalityCommutative eq)
@@ -61,9 +64,9 @@ _+B_ : BinNat → BinNat → BinNat
 
 +BIsInherited' : (a b : BinNat) → a +Binherit b ≡ canonical (a +B b)
 
-+BinheritLemma a b prA prB with orderIsTotal 0 (binNatToN a +N binNatToN b)
++BinheritLemma a b prA prB with TotalOrder.totality ℕTotalOrder 0 (binNatToN a +N binNatToN b)
 +BinheritLemma a b prA prB | inl (inl x) rewrite doubleIsBitShift (binNatToN a +N binNatToN b) x = applyEquality (one ::_) (+BIsInherited a b prA prB)
-+BinheritLemma a b prA prB | inr x with sumZeroImpliesOperandsZero (binNatToN a) (equalityCommutative x)
++BinheritLemma a b prA prB | inr x with sumZeroImpliesSummandsZero (equalityCommutative x)
 +BinheritLemma a b prA prB | inr x | fst ,, snd = ans2
   where
     bad : b ≡ []
@@ -75,10 +78,10 @@ _+B_ : BinNat → BinNat → BinNat
 
 +BIsInherited [] b _ prB = +BIsInherited[] b prB
 +BIsInherited (x :: a) [] prA _ = transitivity (applyEquality NToBinNat (Semiring.commutative ℕSemiring (binNatToN (x :: a)) 0)) (transitivity (binToBin (x :: a)) (equalityCommutative prA))
-+BIsInherited (zero :: as) (zero :: b) prA prB with orderIsTotal 0 (binNatToN as +N binNatToN b)
++BIsInherited (zero :: as) (zero :: b) prA prB with TotalOrder.totality ℕTotalOrder 0 (binNatToN as +N binNatToN b)
 ... | inl (inl 0<) rewrite Semiring.commutative ℕSemiring (binNatToN as) 0 | Semiring.commutative ℕSemiring (binNatToN b) 0 | Semiring.+Associative ℕSemiring (binNatToN as +N binNatToN as) (binNatToN b) (binNatToN b) | equalityCommutative (Semiring.+Associative ℕSemiring (binNatToN as) (binNatToN as) (binNatToN b)) | Semiring.commutative ℕSemiring (binNatToN as) (binNatToN b) | Semiring.+Associative ℕSemiring (binNatToN as) (binNatToN b) (binNatToN as) | equalityCommutative (Semiring.+Associative ℕSemiring (binNatToN as +N binNatToN b) (binNatToN as) (binNatToN b)) | Semiring.commutative ℕSemiring 0 ((binNatToN as +N binNatToN b) +N (binNatToN as +N binNatToN b)) | equalityCommutative (Semiring.+Associative ℕSemiring (binNatToN as +N binNatToN b) (binNatToN as +N binNatToN b) 0) = transitivity (doubleIsBitShift (binNatToN as +N binNatToN b) (identityOfIndiscernablesRight _<N_ 0< (Semiring.commutative ℕSemiring (binNatToN b) _))) (applyEquality (zero ::_) (+BIsInherited as b (canonicalDescends as prA) (canonicalDescends b prB)))
 +BIsInherited (zero :: as) (zero :: b) prA prB | inl (inr ())
-... | inr p with sumZeroImpliesOperandsZero (binNatToN as) (equalityCommutative p)
+... | inr p with sumZeroImpliesSummandsZero {binNatToN as} (equalityCommutative p)
 +BIsInherited (zero :: as) (zero :: b) prA prB | inr p | as=0 ,, b=0 rewrite as=0 | b=0 = exFalso ans
   where
     bad : (b : BinNat) → (pr : b ≡ canonical b) → (pr2 : binNatToN b ≡ 0) → b ≡ []
@@ -142,7 +145,7 @@ _+B_ : BinNat → BinNat → BinNat
   where
     ans : NToBinNat (2 *N (binNatToN as +N binNatToN bs)) ≡ canonical (zero :: (as +B bs))
     ans with inspect (binNatToN as +N binNatToN bs)
-    ans | zero with≡ x with sumZeroImpliesOperandsZero (binNatToN as) x
+    ans | zero with≡ x with sumZeroImpliesSummandsZero {binNatToN as} x
     ... | as=0 ,, bs=0 rewrite as=0 | bs=0 = foo
       where
         u : canonical (as +Binherit bs) ≡ []
@@ -162,7 +165,7 @@ _+B_ : BinNat → BinNat → BinNat
   where
     ans2 : incr (NToBinNat (2 *N (binNatToN as +N binNatToN bs))) ≡ one :: canonical (as +B bs)
     ans2 with inspect (binNatToN as +N binNatToN bs)
-    ans2 | zero with≡ x with sumZeroImpliesOperandsZero (binNatToN as) x
+    ans2 | zero with≡ x with sumZeroImpliesSummandsZero {binNatToN as} x
     ans2 | zero with≡ x | as=0 ,, bs=0 rewrite as=0 | bs=0 = applyEquality (one ::_) (transitivity t (+BIsInherited' as bs))
       where
         t : [] ≡ as +Binherit bs
@@ -172,7 +175,7 @@ _+B_ : BinNat → BinNat → BinNat
   where
     ans : incr (NToBinNat (2 *N (binNatToN as +N binNatToN bs))) ≡ one :: canonical (as +B bs)
     ans with inspect (binNatToN as +N binNatToN bs)
-    ans | zero with≡ x with sumZeroImpliesOperandsZero (binNatToN as) x
+    ans | zero with≡ x with sumZeroImpliesSummandsZero {binNatToN as} x
     ... | as=0 ,, bs=0 rewrite as=0 | bs=0 = applyEquality (one ::_) (transitivity t (+BIsInherited' as bs))
       where
         t : [] ≡ NToBinNat (binNatToN as +N binNatToN bs)
@@ -182,7 +185,7 @@ _+B_ : BinNat → BinNat → BinNat
   where
     ans : incr (incr (NToBinNat (2 *N (binNatToN as +N binNatToN bs)))) ≡ canonical (zero :: incr (as +B bs))
     ans with inspect (binNatToN as +N binNatToN bs)
-    ... | zero with≡ x with sumZeroImpliesOperandsZero (binNatToN as) x
+    ... | zero with≡ x with sumZeroImpliesSummandsZero {binNatToN as} x
     ans | zero with≡ x | as=0 ,, bs=0 rewrite as=0 | bs=0 = bar
       where
         u' : canonical (as +Binherit bs) ≡ []

Numbers/BinaryNaturals/Definition.agda

@@ -3,7 +3,8 @@
 open import LogicalFormulae
 open import Functions
 open import Lists.Lists
-open import Numbers.Naturals.Naturals
+open import Numbers.Naturals.Semiring
+open import Numbers.Naturals.Order
 open import Numbers.Naturals.Definition
 open import Groups.Definition
 open import Semirings.Definition
@@ -98,7 +99,7 @@ module Numbers.BinaryNaturals.Definition where
   canonicalAscends' {i} a pr = canonicalAscends {i} a (t a pr)
     where
       t : (a : BinNat) → (canonical a ≡ [] → False) → 0 <N binNatToN a
-      t a pr with orderIsTotal 0 (binNatToN a)
+      t a pr with TotalOrder.totality ℕTotalOrder 0 (binNatToN a)
       t a pr | inl (inl x) = x
       t a pr | inr x = exFalso (pr (binNatToNZero a (equalityCommutative x)))
 

Numbers/BinaryNaturals/Multiplication.agda

@@ -3,6 +3,7 @@
 open import LogicalFormulae
 open import Functions
 open import Lists.Lists
+open import Numbers.Naturals.Semiring
 open import Numbers.Naturals.Naturals
 open import Groups.Definition
 open import Numbers.BinaryNaturals.Definition
@@ -77,7 +78,7 @@ module Numbers.BinaryNaturals.Multiplication where
       t : binNatToN (as *B (zero :: bs)) +N binNatToN bs ≡ 0
       t = transitivity (equalityCommutative (nToN _)) (applyEquality binNatToN x)
       u : (binNatToN (as *B (zero :: bs)) ≡ 0) && (binNatToN bs ≡ 0)
-      u = sumZeroImpliesOperandsZero _ t
+      u = sumZeroImpliesSummandsZero t
       v : canonical bs ≡ []
       v with u
       ... | fst ,, snd = binNatToNZero bs snd

Numbers/BinaryNaturals/Order.agda

@@ -4,7 +4,9 @@ open import WellFoundedInduction
 open import LogicalFormulae
 open import Functions
 open import Lists.Lists
-open import Numbers.Naturals.Naturals
+open import Numbers.Naturals.Order
+open import Numbers.Naturals.Order.Lemmas
+open import Numbers.Naturals.Semiring
 open import Groups.Definition
 open import Numbers.BinaryNaturals.Definition
 open import Orders
@@ -27,7 +29,7 @@ module Numbers.BinaryNaturals.Order where
   badCompare'' ()
 
   _<BInherited_ : BinNat → BinNat → Compare
-  a <BInherited b with orderIsTotal (binNatToN a) (binNatToN b)
+  a <BInherited b with TotalOrder.totality ℕTotalOrder (binNatToN a) (binNatToN b)
   (a <BInherited b) | inl (inl x) = FirstLess
   (a <BInherited b) | inl (inr x) = FirstGreater
   (a <BInherited b) | inr x = Equal
@@ -283,24 +285,24 @@ module Numbers.BinaryNaturals.Order where
   chopFirstBit m n {one} FirstGreater = refl
 
   chopDouble : (a b : BinNat) (i : Bit) → (i :: a) <BInherited (i :: b) ≡ a <BInherited b
-  chopDouble a b i with orderIsTotal (binNatToN a) (binNatToN b)
-  chopDouble a b zero | inl (inl a<b) with orderIsTotal (2 *N binNatToN a) (2 *N binNatToN b)
+  chopDouble a b i with TotalOrder.totality ℕTotalOrder (binNatToN a) (binNatToN b)
+  chopDouble a b zero | inl (inl a<b) with TotalOrder.totality ℕTotalOrder (2 *N binNatToN a) (2 *N binNatToN b)
   chopDouble a b zero | inl (inl a<b) | inl (inl x) = refl
   chopDouble a b zero | inl (inl a<b) | inl (inr b<a) = exFalso (PartialOrder.irreflexive (TotalOrder.order ℕTotalOrder) (PartialOrder.<Transitive (TotalOrder.order ℕTotalOrder) b<a (lessRespectsMultiplicationLeft (binNatToN a) (binNatToN b) 2 a<b (le 1 refl))))
   chopDouble a b zero | inl (inl a<b) | inr a=b rewrite productCancelsLeft 2 (binNatToN a) (binNatToN b) (le 1 refl) a=b = exFalso (PartialOrder.irreflexive (TotalOrder.order ℕTotalOrder) a<b)
-  chopDouble a b one | inl (inl a<b) with orderIsTotal (2 *N binNatToN a) (2 *N binNatToN b)
+  chopDouble a b one | inl (inl a<b) with TotalOrder.totality ℕTotalOrder (2 *N binNatToN a) (2 *N binNatToN b)
   chopDouble a b one | inl (inl a<b) | inl (inl 2a<2b) = refl
   chopDouble a b one | inl (inl a<b) | inl (inr 2b<2a) = exFalso (PartialOrder.irreflexive (TotalOrder.order ℕTotalOrder) (PartialOrder.<Transitive (TotalOrder.order ℕTotalOrder) a<b (cancelInequalityLeft {2} 2b<2a)))
   chopDouble a b one | inl (inl a<b) | inr 2a=2b rewrite productCancelsLeft 2 (binNatToN a) (binNatToN b) (le 1 refl) 2a=2b = exFalso (PartialOrder.irreflexive (TotalOrder.order ℕTotalOrder) a<b)
-  chopDouble a b zero | inl (inr b<a) with orderIsTotal (2 *N binNatToN a) (2 *N binNatToN b)
+  chopDouble a b zero | inl (inr b<a) with TotalOrder.totality ℕTotalOrder (2 *N binNatToN a) (2 *N binNatToN b)
   chopDouble a b zero | inl (inr b<a) | inl (inl 2a<2b) = exFalso (PartialOrder.irreflexive (TotalOrder.order ℕTotalOrder) (PartialOrder.<Transitive (TotalOrder.order ℕTotalOrder) b<a (cancelInequalityLeft {2} {binNatToN a} {binNatToN b} 2a<2b)))
   chopDouble a b zero | inl (inr b<a) | inl (inr 2b<2a) = refl
   chopDouble a b zero | inl (inr b<a) | inr 2a=2b rewrite productCancelsLeft 2 (binNatToN a) (binNatToN b) (le 1 refl) 2a=2b = exFalso (PartialOrder.irreflexive (TotalOrder.order ℕTotalOrder) b<a)
-  chopDouble a b one | inl (inr b<a) with orderIsTotal (2 *N binNatToN a) (2 *N binNatToN b)
+  chopDouble a b one | inl (inr b<a) with TotalOrder.totality ℕTotalOrder (2 *N binNatToN a) (2 *N binNatToN b)
   chopDouble a b one | inl (inr b<a) | inl (inl 2a<2b) = exFalso (PartialOrder.irreflexive (TotalOrder.order ℕTotalOrder) (PartialOrder.<Transitive (TotalOrder.order ℕTotalOrder) b<a (cancelInequalityLeft {2} 2a<2b)))
   chopDouble a b one | inl (inr b<a) | inl (inr x) = refl
   chopDouble a b one | inl (inr b<a) | inr 2a=2b rewrite productCancelsLeft 2 (binNatToN a) (binNatToN b) (le 1 refl) 2a=2b = exFalso (PartialOrder.irreflexive (TotalOrder.order ℕTotalOrder) b<a)
-  chopDouble a b i | inr x with orderIsTotal (binNatToN (i :: a)) (binNatToN (i :: b))
+  chopDouble a b i | inr x with TotalOrder.totality ℕTotalOrder (binNatToN (i :: a)) (binNatToN (i :: b))
   chopDouble a b zero | inr a=b | inl (inl a<b) rewrite a=b = exFalso (PartialOrder.irreflexive (TotalOrder.order ℕTotalOrder) a<b)
   chopDouble a b one | inr a=b | inl (inl a<b) rewrite a=b = exFalso (PartialOrder.irreflexive (TotalOrder.order ℕTotalOrder) a<b)
   chopDouble a b zero | inr a=b | inl (inr b<a) rewrite a=b = exFalso (PartialOrder.irreflexive (TotalOrder.order ℕTotalOrder) b<a)
@@ -311,23 +313,23 @@ module Numbers.BinaryNaturals.Order where
   succNotLess {succ n} (le x proof) = succNotLess {n} (le x (succInjective (transitivity (applyEquality succ (transitivity (Semiring.commutative ℕSemiring (succ x) (succ n)) (transitivity (applyEquality succ (transitivity (Semiring.commutative ℕSemiring n (succ x)) (applyEquality succ (Semiring.commutative ℕSemiring x n)))) (Semiring.commutative ℕSemiring (succ (succ n)) x)))) proof)))
 
   <BIsInherited : (a b : BinNat) → a <BInherited b ≡ a <B b
-  <BIsInherited [] b with orderIsTotal 0 (binNatToN b)
+  <BIsInherited [] b with TotalOrder.totality ℕTotalOrder 0 (binNatToN b)
   <BIsInherited [] b | inl (inl x) with inspect (binNatToN b)
   <BIsInherited [] b | inl (inl x) | 0 with≡ pr rewrite binNatToNZero b pr | pr = exFalso (PartialOrder.irreflexive (TotalOrder.order ℕTotalOrder) x)
   <BIsInherited [] b | inl (inl x) | (succ bl) with≡ pr rewrite pr = equalityCommutative (zeroLess b λ p → zeroNotSucc bl b p pr)
   <BIsInherited [] b | inr 0=b rewrite canonicalSecond [] b Equal | binNatToNZero b (equalityCommutative 0=b) = refl
-  <BIsInherited (a :: as) [] with orderIsTotal (binNatToN (a :: as)) 0
+  <BIsInherited (a :: as) [] with TotalOrder.totality ℕTotalOrder (binNatToN (a :: as)) 0
   <BIsInherited (a :: as) [] | inl (inr x) with inspect (binNatToN (a :: as))
   <BIsInherited (a :: as) [] | inl (inr x) | zero with≡ pr rewrite binNatToNZero (a :: as) pr | pr = exFalso (PartialOrder.irreflexive (TotalOrder.order ℕTotalOrder) x)
   <BIsInherited (a :: as) [] | inl (inr x) | succ y with≡ pr rewrite pr = equalityCommutative (zeroLess' (a :: as) λ i → zeroNotSucc y (a :: as) i pr)
   <BIsInherited (a :: as) [] | inr x rewrite canonicalFirst (a :: as) [] Equal | binNatToNZero (a :: as) x = refl
   <BIsInherited (zero :: a) (zero :: b) = transitivity (chopDouble a b zero) (<BIsInherited a b)
-  <BIsInherited (zero :: a) (one :: b) with orderIsTotal (binNatToN (zero :: a)) (binNatToN (one :: b))
-  <BIsInherited (zero :: a) (one :: b) | inl (inl 2a<2b+1) with orderIsTotal (binNatToN a) (binNatToN b)
+  <BIsInherited (zero :: a) (one :: b) with TotalOrder.totality ℕTotalOrder (binNatToN (zero :: a)) (binNatToN (one :: b))
+  <BIsInherited (zero :: a) (one :: b) | inl (inl 2a<2b+1) with TotalOrder.totality ℕTotalOrder (binNatToN a) (binNatToN b)
   <BIsInherited (zero :: a) (one :: b) | inl (inl 2a<2b+1) | inl (inl a<b) = equalityCommutative (equalToFirstLess FirstLess a b (equalityCommutative indHyp))
      where
        t : a <BInherited b ≡ FirstLess
-       t with orderIsTotal (binNatToN a) (binNatToN b)
+       t with TotalOrder.totality ℕTotalOrder (binNatToN a) (binNatToN b)
        t | inl (inl x) = refl
        t | inl (inr x) = exFalso (PartialOrder.irreflexive (TotalOrder.order ℕTotalOrder) (PartialOrder.<Transitive (TotalOrder.order ℕTotalOrder) x a<b))
        t | inr x rewrite x = exFalso (PartialOrder.irreflexive (TotalOrder.order ℕTotalOrder) a<b)
@@ -335,12 +337,12 @@ module Numbers.BinaryNaturals.Order where
        indHyp = transitivity (equalityCommutative t) (<BIsInherited a b)
   <BIsInherited (zero :: a) (one :: b) | inl (inl 2a<2b+1) | inl (inr b<a) = exFalso (noIntegersBetweenXAndSuccX {2 *N binNatToN a} (2 *N binNatToN b) (lessRespectsMultiplicationLeft (binNatToN b) (binNatToN a) 2 b<a (le 1 refl)) 2a<2b+1)
   <BIsInherited (zero :: a) (one :: b) | inl (inl 2a<2b+1) | inr a=b rewrite a=b | canonicalFirst a b FirstLess | canonicalSecond (canonical a) b FirstLess | transitivity (equalityCommutative (binToBin a)) (transitivity (applyEquality NToBinNat a=b) (binToBin b)) = equalityCommutative (lemma1 (canonical b))
-  <BIsInherited (zero :: a) (one :: b) | inl (inr 2b+1<2a) with orderIsTotal (binNatToN a) (binNatToN b)
+  <BIsInherited (zero :: a) (one :: b) | inl (inr 2b+1<2a) with TotalOrder.totality ℕTotalOrder (binNatToN a) (binNatToN b)
   <BIsInherited (zero :: a) (one :: b) | inl (inr 2b+1<2a) | inl (inl a<b) = exFalso (PartialOrder.irreflexive (TotalOrder.order ℕTotalOrder) (PartialOrder.<Transitive (TotalOrder.order ℕTotalOrder) 2b+1<2a (PartialOrder.<Transitive (TotalOrder.order ℕTotalOrder) (lessRespectsMultiplicationLeft (binNatToN a) (binNatToN b) 2 a<b (le 1 refl)) (le zero refl))))
   <BIsInherited (zero :: a) (one :: b) | inl (inr 2b+1<2a) | inl (inr b<a) = equalityCommutative (equalToFirstGreater FirstLess a b (equalityCommutative indHyp))
     where
       t : a <BInherited b ≡ FirstGreater
-      t with orderIsTotal (binNatToN a) (binNatToN b)
+      t with TotalOrder.totality ℕTotalOrder (binNatToN a) (binNatToN b)
       t | inl (inl x) = exFalso (PartialOrder.irreflexive (TotalOrder.order ℕTotalOrder) (PartialOrder.<Transitive (TotalOrder.order ℕTotalOrder) x b<a))
       t | inl (inr x) = refl
       t | inr x rewrite x = exFalso (PartialOrder.irreflexive (TotalOrder.order ℕTotalOrder) b<a)
@@ -348,12 +350,12 @@ module Numbers.BinaryNaturals.Order where
       indHyp = transitivity (equalityCommutative t) (<BIsInherited a b)
   <BIsInherited (zero :: a) (one :: b) | inl (inr 2b+1<2a) | inr a=b rewrite a=b = exFalso (succNotLess 2b+1<2a)
   <BIsInherited (zero :: a) (one :: b) | inr 2a=2b+1 = exFalso (parity (binNatToN b) (binNatToN a) (equalityCommutative 2a=2b+1))
-  <BIsInherited (one :: a) (zero :: b) with orderIsTotal (binNatToN (one :: a)) (binNatToN (zero :: b))
-  <BIsInherited (one :: a) (zero :: b) | inl (inl 2a+1<2b) with orderIsTotal (binNatToN a) (binNatToN b)
+  <BIsInherited (one :: a) (zero :: b) with TotalOrder.totality ℕTotalOrder (binNatToN (one :: a)) (binNatToN (zero :: b))
+  <BIsInherited (one :: a) (zero :: b) | inl (inl 2a+1<2b) with TotalOrder.totality ℕTotalOrder (binNatToN a) (binNatToN b)
   <BIsInherited (one :: a) (zero :: b) | inl (inl 2a+1<2b) | inl (inl a<b) = equalityCommutative (equalToFirstLess FirstGreater a b (equalityCommutative indHyp))
     where
       t : a <BInherited b ≡ FirstLess
-      t with orderIsTotal (binNatToN a) (binNatToN b)
+      t with TotalOrder.totality ℕTotalOrder (binNatToN a) (binNatToN b)
       t | inl (inr x) = exFalso (PartialOrder.irreflexive (TotalOrder.order ℕTotalOrder) (PartialOrder.<Transitive (TotalOrder.order ℕTotalOrder) x a<b))
       t | inl (inl x) = refl
       t | inr x rewrite x = exFalso (PartialOrder.irreflexive (TotalOrder.order ℕTotalOrder) a<b)
@@ -361,12 +363,12 @@ module Numbers.BinaryNaturals.Order where
       indHyp = transitivity (equalityCommutative t) (<BIsInherited a b)
   <BIsInherited (one :: a) (zero :: b) | inl (inl 2a+1<2b) | inl (inr b<a) = exFalso (PartialOrder.irreflexive (TotalOrder.order ℕTotalOrder) (PartialOrder.<Transitive (TotalOrder.order ℕTotalOrder) 2a+1<2b (PartialOrder.<Transitive (TotalOrder.order ℕTotalOrder) (lessRespectsMultiplicationLeft (binNatToN b) (binNatToN a) 2 b<a (le 1 refl)) (le zero refl))))
   <BIsInherited (one :: a) (zero :: b) | inl (inl 2a+1<2b) | inr a=b rewrite a=b = exFalso (succNotLess 2a+1<2b)
-  <BIsInherited (one :: a) (zero :: b) | inl (inr 2b<2a+1) with orderIsTotal (binNatToN a) (binNatToN b)
+  <BIsInherited (one :: a) (zero :: b) | inl (inr 2b<2a+1) with TotalOrder.totality ℕTotalOrder (binNatToN a) (binNatToN b)
   <BIsInherited (one :: a) (zero :: b) | inl (inr 2b<2a+1) | inl (inl a<b) = exFalso (noIntegersBetweenXAndSuccX {2 *N binNatToN b} (2 *N binNatToN a) (lessRespectsMultiplicationLeft (binNatToN a) (binNatToN b) 2 a<b (le 1 refl)) 2b<2a+1)
   <BIsInherited (one :: a) (zero :: b) | inl (inr 2b<2a+1) | inl (inr b<a) = equalityCommutative (equalToFirstGreater FirstGreater a b (equalityCommutative indHyp))
     where
       t : a <BInherited b ≡ FirstGreater
-      t with orderIsTotal (binNatToN a) (binNatToN b)
+      t with TotalOrder.totality ℕTotalOrder (binNatToN a) (binNatToN b)
       t | inl (inl x) = exFalso (PartialOrder.irreflexive (TotalOrder.order ℕTotalOrder) (PartialOrder.<Transitive (TotalOrder.order ℕTotalOrder) x b<a))
       t | inl (inr x) = refl
       t | inr x rewrite x = exFalso (PartialOrder.irreflexive (TotalOrder.order ℕTotalOrder) b<a)

Numbers/BinaryNaturals/Subtraction.agda

@@ -2,6 +2,7 @@
 
 open import LogicalFormulae
 open import Lists.Lists
+open import Numbers.Naturals.Semiring
 open import Numbers.Naturals.Naturals
 open import Numbers.Naturals.Order
 open import Numbers.BinaryNaturals.Definition