Commit 39f3a83b authored by Sven Wille's avatar Sven Wille

subtypes in naturals

parent fa67ab94
......@@ -72,7 +72,19 @@ theory NaturalArithmetics : base:?Logic =
lemma_rightUnitalPlus : {x : ℕ} ⊦ x + 0 ≐ x ❘ = axiom_plus1 ❘ # rightunital_plus_nat ❙
// added by sw ❙
/T usefull lemmata for simple arithmetics ❙
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❙
// added by sw ❙
// this one is needed to create a possibly empty subset of the natural numbers (this way we can create empty finite sequences)❙
/T finite subset of the natural numbers ❙
between : ℕ ⟶ ℕ ⟶ type ❘= [f,t] ⟨ x : ℕ | ⊦ f ≤ x ∧ x ≤ t⟩ ❙
// convenience function for between ❙
betweenOneAnd : ℕ ⟶ type ❘= [t] ⟨ x : ℕ | ⊦ 1 ≤ x ∧ x ≤ t⟩ ❙
/T convert explicitely a natural number into an "between" ❙
//this function is needed in context wehere a natural number "n" is needed as both natural number and between ❙
// added by sw ❙
natToBetween : {from : ℕ}{to : ℕ} {v : ℕ } ⊦ from ≤ v ∧ v ≤ to ⟶ between from to ❘= [v,h] v❙
......@@ -8,7 +8,8 @@ theory FinSequences : base:?Logic =
include arith:?NaturalArithmetics ❙
// below : ℕ ⟶ type ❘ = [n] ⟨ x : ℕ | ⊦ x < n⟩ ❙
finSeq : ℕ ⟶ type ⟶ type ❘= [n , A] {m : ℕ} ⊦ m < n ⟶ A ❙
/T finite sequences ❙
finiteSequence : ℕ ⟶ type ⟶ type ❘= [n , A] between 1 n ⟶ A ❘# finSeq 1 2 ❙
// 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❙
......@@ -22,15 +23,20 @@ theory FinSeqSum : base:?Logic =
include arith:?NaturalArithmetics ❙
// this one does the actual recusive summation and is called by finSeqSum ❙
finSeqSumRec : {n : ℕ} {m : ℕ } ⊦ `arith:?NaturalArithmetics?lessthan m n ⟶ finSeq n ℝ ⟶ ℝ ❘# finSeqSumR 2 3 4❙
finSeqSumRecB : {n : ℕ} {sq : finSeq n ℝ} {h : ⊦ `arith:?NaturalArithmetics?lessthan 0 n} ⊦ finSeqSumRec n 0 h sq ≐ sq 0 h
// finSeqSumRec : {n : ℕ} {m : ℕ } ⊦ `arith:?NaturalArithmetics?lessthan m n ⟶ finSeq n ℝ ⟶ ℝ ❘# finSeqSumR 2 3 4❙
// finSeqSumRecB : {n : ℕ} {sq : finSeq n ℝ} {h : ⊦ `arith:?NaturalArithmetics?lessthan 0 n} ⊦ finSeqSumRec n 0 h sq ≐ 0
// 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❙
finSeqSumRec : {n : ℕ} betweenOneAnd n ⟶ finSeq n ℝ ⟶ ℝ ❘# finSeqSumR 2 3 4❙
finSeqSumRecB : {n : ℕ} {sq : finSeq n ℝ} ⊦ finSeqSumRec n 1 sq ≐ sq 1 ❙
// 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❙
finSeqSumFn : {n : ℕ} {sq : finSeq n ℝ} ⊦ 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