Lots of without-K (#110)

Fields/Orders/Lemmas.agda

@@ -120,16 +120,7 @@ abstract
   IntegralDomain.intDom orderedFieldIsIntDom = decidedIntDom R orderedFieldIntDom
   IntegralDomain.nontrivial orderedFieldIsIntDom pr = Field.nontrivial F (Equivalence.symmetric (Setoid.eq S) pr)
 
-  fromNIncreasing : (n : ℕ) → (fromN n) < (fromN (succ n))
-  fromNIncreasing zero = <WellDefined reflexive (symmetric identRight) (0<1 nontrivial)
-  fromNIncreasing (succ n) = <WellDefined groupIsAbelian groupIsAbelian (orderRespectsAddition (fromNIncreasing n) 1R)
-
-  fromNPreservesOrder : {a b : ℕ} → (a <N b) → (fromN a) < (fromN b)
-  fromNPreservesOrder {zero} {succ zero} a<b = fromNIncreasing 0
-  fromNPreservesOrder {zero} {succ (succ b)} a<b = <Transitive (fromNPreservesOrder (succIsPositive b)) (fromNIncreasing (succ b))
-  fromNPreservesOrder {succ a} {succ b} a<b = <WellDefined groupIsAbelian groupIsAbelian (orderRespectsAddition (fromNPreservesOrder (canRemoveSuccFrom<N a<b)) 1R)
-
   charZero : (n : ℕ) → (0R ∼ (fromN (succ n))) → False
-  charZero n 0=sn = irreflexive (<WellDefined 0=sn reflexive (fromNPreservesOrder (succIsPositive n)))
+  charZero n 0=sn = irreflexive (<WellDefined 0=sn reflexive (fromNPreservesOrder nontrivial (succIsPositive n)))
   charZero' : (n : ℕ) → ((fromN (succ n)) ∼ 0R) → False
   charZero' n pr = charZero n (symmetric pr)

Fields/Orders/Limits/Definition.agda

@@ -54,6 +54,19 @@ abstract
   halfHalves : {a : A} → (1/2 * (a + a)) ∼ a
   halfHalves {a} = transitive (*WellDefined reflexive (+WellDefined (symmetric identIsIdent) (symmetric identIsIdent))) (transitive (*WellDefined reflexive (symmetric *DistributesOver+')) (transitive *Associative (transitive (*WellDefined 1/2is1/2 reflexive) identIsIdent)))
 
+abstract
+  bothNearImpliesNear : {t r s : A} (e : A) .(0<e : 0R < e) → (abs (r + inverse t) < e) → (abs (s + inverse t) < e) → abs (r + inverse s) < (e + e)
+  bothNearImpliesNear {t} {r} {s} e 0<e rNearT sNearT = u
+    where
+      pr : ((abs (r + inverse t)) + (abs (s + inverse t))) < (e + e)
+      pr = ringAddInequalities rNearT sNearT
+      t' : (abs (r + inverse t) + abs (t + inverse s)) < (e + e)
+      t' = <WellDefined (+WellDefined reflexive (transitive (absNegation _) (absWellDefined _ _ (transitive invContravariant (+WellDefined (invTwice _) reflexive))))) reflexive pr
+      u : abs (r + inverse s) < (e + e)
+      u with triangleInequality (r + inverse t) (t + inverse s)
+      ... | inl t'' = <Transitive (<WellDefined (absWellDefined _ _ (transitive +Associative (+WellDefined (transitive (symmetric +Associative) (transitive (+WellDefined reflexive invLeft) identRight)) reflexive))) reflexive t'') t'
+      ... | inr eq = <WellDefined (transitive (symmetric eq) (absWellDefined _ _ (transitive +Associative (+WellDefined (transitive (symmetric +Associative) (transitive (+WellDefined reflexive invLeft) identRight)) reflexive)))) reflexive t'
+
 private
   limitsUniqueLemma : (x : Sequence A) {r s : A} (xr : x ~> r) (xs : x ~> s) (r<s : r < s) → False
   limitsUniqueLemma x {r} {s} xr xs r<s = irreflexive (<WellDefined reflexive (symmetric prE) u')

Fields/Orders/Limits/Lemmas.agda

@@ -0,0 +1,64 @@
+{-# OPTIONS --safe --warning=error --without-K --guardedness #-}
+
+open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
+
+open import LogicalFormulae
+open import Setoids.Subset
+open import Setoids.Setoids
+open import Setoids.Orders
+open import Sets.EquivalenceRelations
+open import Rings.Orders.Total.Definition
+open import Rings.Orders.Partial.Definition
+open import Rings.Definition
+open import Fields.Fields
+open import Groups.Definition
+open import Sequences
+open import Numbers.Naturals.Semiring
+open import Numbers.Naturals.Order
+open import Semirings.Definition
+open import Functions
+open import Fields.Orders.Total.Definition
+open import Numbers.Primes.PrimeNumbers
+
+module Fields.Orders.Limits.Lemmas {a b c : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} {_*_ : A → A → A} {_<_ : Rel {_} {c} A} {R : Ring S _+_ _*_} {pOrder : SetoidPartialOrder S _<_} {F : Field R} {pRing : PartiallyOrderedRing R pOrder} (oF : TotallyOrderedField F pRing) where
+
+open Ring R
+open TotallyOrderedField oF
+open TotallyOrderedRing oRing
+open PartiallyOrderedRing pRing
+open import Rings.Orders.Total.Lemmas oRing
+open import Rings.Orders.Partial.Lemmas pRing
+open SetoidTotalOrder total
+open SetoidPartialOrder pOrder
+open Group additiveGroup
+open import Groups.Lemmas additiveGroup
+open Setoid S
+open Equivalence eq
+open Field F
+open import Fields.CauchyCompletion.Definition (TotallyOrderedField.oRing oF) F
+open import Fields.Orders.Limits.Definition oF
+open import Fields.Lemmas F
+open import Fields.Orders.Total.Lemmas oF
+open import Rings.Characteristic R
+open import Rings.InitialRing R
+
+private
+  2!=3 : 2 ≡ 3 → False
+  2!=3 ()
+
+convergentSequenceCauchy : (nontrivial : 0R ∼ 1R → False) → {a : Sequence A} → {r : A} → a ~> r → (decidedCharacteristic : ((1R + 1R) ∼ 0R) || (((1R + 1R) ∼ 0R) → False)) → cauchy a
+convergentSequenceCauchy nontrivial {a} {r} a->r (inl 2=0) ε x with 1/nPositive 2 λ t → 2!=3 (characteristicWellDefined nontrivial {2} {3} twoIsPrime threeIsPrime (transitive (transitive +Associative identRight) 2=0) t)
+... | 0<1/3 with allInvertible (fromN 3) λ t → 2!=3 (characteristicWellDefined nontrivial {2} {3} twoIsPrime threeIsPrime (transitive (transitive +Associative identRight) 2=0) t)
+... | 1/3 , pr1/3 with a->r (1/3 * ε) (orderRespectsMultiplication 0<1/3 x)
+... | N , pr = N , λ {m} {n} → ans m n
+  where
+    2/3 : (1/3 + 1/3) < 1R
+    2/3 = <WellDefined reflexive (transitive (transitive (+WellDefined (transitive (symmetric identIsIdent) *Commutative) (transitive (+WellDefined (transitive (symmetric identIsIdent) *Commutative) (transitive (+WellDefined (symmetric (transitive *Commutative identIsIdent)) (symmetric timesZero)) (symmetric *DistributesOver+))) (symmetric *DistributesOver+))) (symmetric *DistributesOver+)) pr1/3) (<WellDefined reflexive (transitive (+WellDefined reflexive (symmetric identRight)) (symmetric +Associative)) (orderRespectsAddition (<WellDefined identLeft reflexive (orderRespectsAddition 0<1/3 1/3)) 1/3))
+    ans : (m n : ℕ) → N <N m → N <N n → abs (index a m + inverse (index a n)) < ε
+    ans m n N<m N<n = <Transitive (bothNearImpliesNear {r} (1/3 * ε) (orderRespectsMultiplication 0<1/3 x) (pr m N<m) (pr n N<n)) (<WellDefined *DistributesOver+' identIsIdent (ringCanMultiplyByPositive x 2/3))
+convergentSequenceCauchy _ {a} {r} a->r (inr 2!=0) e 0<e with halve 2!=0 e
+... | e/2 , prE/2 with a->r e/2 (halvePositive' prE/2 0<e)
+... | N , pr = N , λ {m} {n} → ans m n
+  where
+    ans : (m n : ℕ) → N <N m → N <N n → abs (index a m + inverse (index a n)) < e
+    ans m n N<m N<n = <WellDefined reflexive prE/2 (bothNearImpliesNear {r} e/2 (halvePositive' prE/2 0<e) (pr m N<m) (pr n N<n))

Fields/Orders/Partial/Lemmas.agda

@@ -0,0 +1,28 @@
+{-# OPTIONS --safe --warning=error --without-K #-}
+
+open import Rings.Definition
+open import Rings.Orders.Partial.Definition
+open import Setoids.Setoids
+open import Setoids.Orders
+open import Functions
+open import Fields.Fields
+open import Fields.Orders.Partial.Definition
+open import Numbers.Naturals.Semiring
+open import Sets.EquivalenceRelations
+open import LogicalFormulae
+open import Groups.Definition
+
+open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
+
+module Fields.Orders.Partial.Lemmas {m n : _} {A : Set m} {S : Setoid {m} {n} A} {_+_ : A → A → A} {_*_ : A → A → A} {R : Ring S _+_ _*_} {F : Field R} {p : _} {_<_ : Rel {_} {p} A} {pOrder : SetoidPartialOrder S _<_} (oF : PartiallyOrderedField F pOrder) where
+
+open Ring R
+open Group additiveGroup
+open Setoid S
+open Equivalence eq
+open Field F
+open PartiallyOrderedField oF
+open SetoidPartialOrder pOrder
+open import Rings.Orders.Partial.Lemmas oRing
+open import Rings.InitialRing R
+

Fields/Orders/Total/Lemmas.agda

@@ -0,0 +1,44 @@
+{-# OPTIONS --safe --warning=error --without-K #-}
+
+open import Rings.Definition
+open import Rings.Orders.Partial.Definition
+open import Rings.Orders.Total.Definition
+open import Setoids.Setoids
+open import Setoids.Orders
+open import Functions
+open import Fields.Fields
+open import Fields.Orders.Total.Definition
+open import Numbers.Naturals.Semiring
+open import Numbers.Naturals.Order
+open import Sets.EquivalenceRelations
+open import LogicalFormulae
+open import Groups.Definition
+
+open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
+
+module Fields.Orders.Total.Lemmas {m n : _} {A : Set m} {S : Setoid {m} {n} A} {_+_ : A → A → A} {_*_ : A → A → A} {R : Ring S _+_ _*_} {F : Field R} {p : _} {_<_ : Rel {_} {p} A} {pOrder : SetoidPartialOrder S _<_} {oR : PartiallyOrderedRing R pOrder} (oF : TotallyOrderedField F oR) where
+
+open Ring R
+open Group additiveGroup
+open Setoid S
+open Equivalence eq
+open Field F
+open TotallyOrderedField oF
+open TotallyOrderedRing oRing
+open PartiallyOrderedRing oR
+open SetoidTotalOrder total
+open SetoidPartialOrder pOrder
+open import Rings.InitialRing R
+open import Rings.Orders.Total.Lemmas oRing
+open import Rings.Orders.Partial.Lemmas oR
+open import Rings.Lemmas R
+open import Groups.Lemmas additiveGroup
+
+1/nPositive : (n : ℕ) → (charNotN : (fromN (succ n) ∼ 0R) → False) → 0R < underlying (allInvertible _ charNotN)
+1/nPositive 0 nNot0 with allInvertible (fromN 1) nNot0
+... | 1/1 , pr1 = <WellDefined reflexive (transitive (symmetric pr1) (transitive (transitive (*WellDefined reflexive identRight) *Commutative) identIsIdent)) (0<1 λ i → nNot0 (transitive identRight (symmetric i)))
+1/nPositive (succ n) nNot0 with allInvertible (fromN (succ (succ n))) nNot0
+... | 1/n , pr1/n with totality 0R 1/n
+... | inr x = exFalso (nNot0 (oneZeroImpliesAllZero (transitive (symmetric (transitive (*WellDefined (symmetric x) reflexive) timesZero')) pr1/n)))
+... | inl (inl x) = x
+... | inl (inr x) = exFalso (1<0False (<WellDefined pr1/n timesZero' (ringCanMultiplyByPositive (fromNPreservesOrder (anyComparisonImpliesNontrivial x) (succIsPositive (succ n))) x)))