Lots of without-K (#110)

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))