Commit 86748f17 authored by Sven Wille's avatar Sven Wille

finseq continued

parent b6140c48
......@@ -70,6 +70,9 @@ theory NaturalArithmetics : base:?Logic =
axiom_associativePlus : {x: ℕ,y,z} ⊦ x + (y + z) ≐ (x + y) + z ❘ # associative_plus_nat 1 2 3❙
axiom_leftUnitalPlus : {x : ℕ} ⊦ 0 + x ≐ x ❘ # leftunital_plus_nat ❙
lemma_rightUnitalPlus : {x : ℕ} ⊦ x + 0 ≐ x ❘ = axiom_plus1 ❘ # rightunital_plus_nat ❙
// added by sw ❙
lemma_lt_pred_lt : {x : ℕ , n : ℕ } ⊦ `?NaturalNumbers?succ x < n ⟶ ⊦ x < n ❘# lt_pred_lt 3❙
lemma_le_pred_le : {x : ℕ , n : ℕ } ⊦ (`?NaturalNumbers?succ : ℕ ⟶ ℕ) x ≤ n ⟶ ⊦ x ≤ n ❘# le_pred_le 3❙
......@@ -5,38 +5,30 @@ import base http://mathhub.info/MitM/Foundation ❚
import arith http://mathhub.info/MitM/smglom/arithmetics ❚
theory FinSequences : base:?Logic =
include arith:?RealArithmetics ❙
below : ℕ ⟶ type ❘ = [n] ⟨ x : ℕ | ⊦ x < n ⟩ ❙
// toBelow : {n : ℕ} below (Succ n) ❘# toB 1❙
// succBelow : {n : ℕ} below n ⟶ below (Succ n) ❘# succB 2❙
// belowIsSubtypeOfNat : {n : ℕ} (below n) <* ℕ ❙
// belowToNat : {n : ℕ} below (Succ n) ⟶ ℕ ❘# btn 1 2❙
// belowToNatBaseCase : {m : ℕ} ⊦ belowToNat m (toBelow m) ≐ m❙
// belowToNatRecCase : {m : ℕ, b : below (Succ m)} ⊦ belowToNat (Succ m) (succBelow (Succ m) b) ≐ belowToNat m b ❙
// belowToNatExample : ⊦ btn 3 (toB 3) ≐ 3❙
// belowToNatExample2 : ⊦ btn 4 (succB (toB 3)) ≐ 3 ❙
// belwoToNatExample2_2 : ⊦ belowToNat (Succ 3) (succBelow (Succ 3) (toBelow 3)) ≐ 3❙
// belowToNat2 : {n : ℕ} below (Succ n) ⟶ ℕ ❘= [n, b] n ❘# btn2 %I1 ❙
// belowToNat3 : {n : ℕ, m : ℕ} below n ⟶ ⊦ m < n ⟶ ℕ ❘= [n , m , b, p ] m ❘# btn3 %I1 ❙
// belowToNat4 : {n : ℕ} below n ⟶ ⟨ ℕ | [x] ⊦ x < n⟩ ❘# btn4 %I1❙
finSeq : ℕ ⟶ type ⟶ type ❘= [n , A] below n ⟶ A ❙
nonEmptyFinSeq_prop : {n : ℕ, A : type} finSeq n A ⟶ bool ❘= [n, A , f] 0 < n ❘# nefs %I1 %I2 ❙
disjointFinSeq_prop : {n : ℕ, A : type} finSeq n A ⟶ bool ❘= [n,A,fs] ∀[x] ∀[y] y ≐ x ∨ fs x ≠ fs y❙
// disjointFinSeq_prop_trivial : {A : type,f : finSeq 0 A} ⊦ disjointFinSeq_prop 0 A f ≐ true❙
// disjointFinSeq_prop_step // : {A : type , n : ℕ , f :
finSeq (succ n) A} ⊦ disjointFinSeq_prop (succ n) A f ≐ ∀ [m1 : below (succ n)] ∀ [m2 : below (succ n)] ⊦ m1 ≠ m2 ⟶ ⊦ true ❙
// belowShift : {n : ℕ , m : ℕ} below n ⟶ below (n + m)❙
// belowShift_base : {n, b} ⊦ belowShift n 0 b ≐ b❙
finSeqSum : {n : ℕ} finSeq n ℝ ⟶ ℝ❙
include arith:?NaturalArithmetics ❙
// below : ℕ ⟶ type ❘ = [n] ⟨ x : ℕ | ⊦ x < n⟩ ❙
finSeq : ℕ ⟶ type ⟶ type ❘= [n , A] {m : ℕ} ⊦ m < n ⟶ A ❙
disjointFinSeq_prop : {n : ℕ, A : type} finSeq n A ⟶ bool ❘= [n,A,fs] ∀[x] ∀[y] ∀ [h : ⊦ x < n] ∀ [h0 : ⊦ y < n] y ≐ x ∨ fs x h ≠ fs y h0❙
theory FinSeqSum : base:?Logic =
include ?FinSequences ❙
// this one does the actual recusive summation and is called by finSeqSum ❙
finSeqSumRec : {n : ℕ} {m : ℕ } ⊦ m < n ⟶ finSeq n ℝ ⟶ ℝ ❘# finSeqSumR 2 3 4❙
finSeqSumRecB : {n : ℕ} {sq : finSeq n ℝ} {h : ⊦ 0 < n} ⊦ finSeqSumR 0 h sq ≐ sq 0 h ❙
finSeqSumRecS : {n : ℕ} {m : ℕ} {sq : finSeq n ℝ} {h : ⊦ `arith:?NaturalNumbers?succ m < n}
⊦ finSeqSumR ( `arith:?NaturalNumbers?succ m) h sq ≐
sq (`arith:?NaturalNumbers?succ m) h + finSeqSumR m (lt_pred_lt h) sq❙
finSeqSum : {n : ℕ} finSeq n ℝ ⟶ ℝ❙
finSeqSumZero : {sq} ⊦ finSeqSum 0 sq ≐ 0 ❙
// finSeqSumFn : {n} {sq} ⊦ finSeqSum n sq ≐ finSeqSumRec n n sq❙
\ No newline at end of file
Markdown is supported
0%
or
You are about to add 0 people to the discussion. Proceed with caution.
Finish editing this message first!
Please register or to comment