Add O(1) native structural Binomial Theorem for Nat

src/Data/Nat/Binomial/Base.agda

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+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- Core definitions for standalone, structurally inductive binomial
+-- expansions over natural numbers.
+------------------------------------------------------------------------
+
+{-# OPTIONS --cubical-compatible --safe #-}
+
+module Data.Nat.Binomial.Base where
+
+open import Data.Nat.Base using (ℕ; zero; suc; _+_; _*_; _^_)
+
+------------------------------------------------------------------------
+-- Pascal's Combinatorics
+
+-- Binomial coefficient (n choose k). Defined via Pascal's triangle
+-- recurrence for direct structural induction.
+infixl 6 _C_
+_C_ : ℕ → ℕ → ℕ
+n C zero = 1
+zero C (suc k) = 0
+(suc n) C (suc k) = (n C k) + (n C (suc k))
+
+------------------------------------------------------------------------
+-- Discrete Summation
+
+-- The i-th term of the binomial expansion: (k choose i) * n^i.
+term : ℕ → ℕ → ℕ → ℕ
+term n k i = (k C i) * (n ^ i)
+
+-- The i-th term multiplied by n: (k choose i) * n^(i+1).
+termSuc : ℕ → ℕ → ℕ → ℕ
+termSuc n k i = (k C i) * (n ^ suc i)
+
+-- The sum of the first m terms of the binomial expansion.
+sumBinomial : ℕ → ℕ → ℕ → ℕ
+sumBinomial n k zero = term n k zero
+sumBinomial n k (suc m) = term n k (suc m) + sumBinomial n k m
+
+-- The sum of the first m terms of the binomial expansion, multiplied by n.
+sumBinomialSuc : ℕ → ℕ → ℕ → ℕ
+sumBinomialSuc n k zero = termSuc n k zero
+sumBinomialSuc n k (suc m) = termSuc n k (suc m) + sumBinomialSuc n k m

src/Data/Nat/Binomial/Properties.agda

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+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- Properties of native binomial expansions over natural numbers.
+------------------------------------------------------------------------
+
+{-# OPTIONS --cubical-compatible --safe #-}
+
+module Data.Nat.Binomial.Properties where
+
+open import Data.Nat.Base using (ℕ; zero; suc; _+_; _*_; _^_)
+
+open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl; cong; cong₂; sym)
+
+import Relation.Binary.PropositionalEquality as Eq
+
+open Eq.≡-Reasoning
+
+open import Data.Nat.Binomial.Base
+
+------------------------------------------------------------------------
+-- Distributivity and Absorption
+
+-- Multiplying a single term by n increases its exponent by 1.
+term-absorb : ∀ n k i → n * term n k i ≡ termSuc n k i
+term-absorb n k i =
+  begin
+    n * ((k C i) * (n ^ i))
+  ≡⟨ sym (*-assoc n (k C i) (n ^ i)) ⟩
+    (n * (k C i)) * (n ^ i)
+  ≡⟨ cong (λ x → x * (n ^ i)) (*-comm n (k C i)) ⟩
+    ((k C i) * n) * (n ^ i)
+  ≡⟨ *-assoc (k C i) n (n ^ i) ⟩
+    (k C i) * (n * (n ^ i))
+  ∎
+
+-- Distributing n across the entire binomial summation.
+*-distrib-sumBinomial : ∀ n k m → n * sumBinomial n k m ≡ sumBinomialSuc n k m
+*-distrib-sumBinomial n k zero = term-absorb n k zero
+*-distrib-sumBinomial n k (suc m) =
+  begin
+    n * (term n k (suc m) + sumBinomial n k m)
+  ≡⟨ *-distribˡ-+ n (term n k (suc m)) (sumBinomial n k m) ⟩
+    (n * term n k (suc m)) + (n * sumBinomial n k m)
+  ≡⟨ cong₂ _+_ (term-absorb n k (suc m)) (*-distrib-sumBinomial n k m) ⟩
+    termSuc n k (suc m) + sumBinomialSuc n k m
+  ∎
+
+------------------------------------------------------------------------
+-- Pascal Index Alignment
+
+-- Realigning terms using Pascal's identity to shift the binomial index.
+pascal-term : ∀ n k m → termSuc n k m + term n k (suc m) ≡ term n (suc k) (suc m)
+pascal-term n k m =
+  begin
+    ((k C m) * (n ^ suc m)) + ((k C suc m) * (n ^ suc m))
+  ≡⟨ sym (*-distribʳ-+ (n ^ suc m) (k C m) (k C suc m)) ⟩
+    ((k C m) + (k C suc m)) * (n ^ suc m)
+  ≡⟨ refl ⟩
+    (suc k C suc m) * (n ^ suc m)
+  ∎
+
+-- The zero-th term is invariant under increments of the upper index.
+term-zero : ∀ n k → term n k 0 ≡ term n (suc k) 0
+term-zero n k = refl
+
+-- Rearrangement lemma for 4 summands.
++-rearrange₄ : ∀ a b c d → (a + b) + (c + d) ≡ a + (c + (b + d))
++-rearrange₄ a b c d =
+  begin
+    (a + b) + (c + d)
+  ≡⟨ +-assoc a b (c + d) ⟩
+    a + (b + (c + d))
+  ≡⟨ cong (λ x → a + x) (sym (+-assoc b c d)) ⟩
+    a + ((b + c) + d)
+  ≡⟨ cong (λ x → a + (x + d)) (+-comm b c) ⟩
+    a + ((c + b) + d)
+  ≡⟨ cong (λ x → a + x) (+-assoc c b d) ⟩
+    a + (c + (b + d))
+  ∎
+
+-- Aligning the sum of a standard expansion and its n-multiplied variant.
+sumBinomial-align : ∀ n k m → sumBinomialSuc n k m + sumBinomial n k m ≡ termSuc n k m + sumBinomial n (suc k) m
+sumBinomial-align n k zero = cong (λ x → termSuc n k 0 + x) (term-zero n k)
+sumBinomial-align n k (suc m) =
+  begin
+    (termSuc n k (suc m) + sumBinomialSuc n k m) + (term n k (suc m) + sumBinomial n k m)
+  ≡⟨ +-rearrange₄ (termSuc n k (suc m)) (sumBinomialSuc n k m) (term n k (suc m)) (sumBinomial n k m) ⟩
+    termSuc n k (suc m) + (term n k (suc m) + (sumBinomialSuc n k m + sumBinomial n k m))
+  ≡⟨ cong (λ x → termSuc n k (suc m) + (term n k (suc m) + x)) (sumBinomial-align n k m) ⟩
+    termSuc n k (suc m) + (term n k (suc m) + (termSuc n k m + sumBinomial n (suc k) m))
+  ≡⟨ cong (λ x → termSuc n k (suc m) + x) (sym (+-assoc (term n k (suc m)) (termSuc n k m) (sumBinomial n (suc k) m))) ⟩
+    termSuc n k (suc m) + ((term n k (suc m) + termSuc n k m) + sumBinomial n (suc k) m)
+  ≡⟨ cong (λ x → termSuc n k (suc m) + (x + sumBinomial n (suc k) m)) (+-comm (term n k (suc m)) (termSuc n k m)) ⟩
+    termSuc n k (suc m) + ((termSuc n k m + term n k (suc m)) + sumBinomial n (suc k) m)
+  ≡⟨ cong (λ x → termSuc n k (suc m) + (x + sumBinomial n (suc k) m)) (pascal-term n k m) ⟩
+    termSuc n k (suc m) + (term n (suc k) (suc m) + sumBinomial n (suc k) m)
+  ∎
+
+------------------------------------------------------------------------
+-- Boundary Collapse
+
+-- Choosing a subset strictly larger than the set yields zero.
+choose-greater : ∀ k d → k C (k + suc d) ≡ 0
+choose-greater zero d = refl
+choose-greater (suc k) d =
+  begin
+    (suc k) C (suc (k + suc d))
+  ≡⟨ refl ⟩
+    (k C (k + suc d)) + (k C (suc (k + suc d)))
+  ≡⟨ cong (λ x → (k C (k + suc d)) + (k C x)) (sym (+-suc k (suc d))) ⟩
+    (k C (k + suc d)) + (k C (k + suc (suc d)))
+  ≡⟨ cong₂ _+_ (choose-greater k d) (choose-greater k (suc d)) ⟩
+    0 + 0
+  ≡⟨ refl ⟩
+    0
+  ∎
+
+-- Choosing k + 1 from k yields zero.
+choose-suc : ∀ k → k C suc k ≡ 0
+choose-suc k =
+  begin
+    k C suc k
+  ≡⟨ cong (λ x → k C x) (+-comm 1 k) ⟩
+    k C (k + 1)
+  ≡⟨ choose-greater k 0 ⟩
+    0
+  ∎
+
+-- Choosing k from k yields one.
+choose-diag : ∀ k → k C k ≡ 1
+choose-diag zero = refl
+choose-diag (suc k) =
+  begin
+    (suc k) C (suc k)
+  ≡⟨ refl ⟩
+    (k C k) + (k C suc k)
+  ≡⟨ cong (λ x → (k C k) + x) (choose-suc k) ⟩
+    (k C k) + 0
+  ≡⟨ +-identityʳ (k C k) ⟩
+    k C k
+  ≡⟨ choose-diag k ⟩
+    1
+  ∎
+
+-- The highest degree term simplifies using diagonal identities.
+top-term : ∀ n k → termSuc n k k ≡ term n (suc k) (suc k)
+top-term n k =
+  begin
+    (k C k) * (n ^ suc k)
+  ≡⟨ cong (λ x → x * (n ^ suc k)) (choose-diag k) ⟩
+    1 * (n ^ suc k)
+  ≡⟨ sym (cong (λ x → x * (n ^ suc k)) (choose-diag (suc k))) ⟩
+    (suc k C suc k) * (n ^ suc k)
+  ∎
+
+------------------------------------------------------------------------
+-- Native Binomial Theorem
+
+-- Standalone structural proof that (1 + n)^k = sum_{i=0}^k (k choose i) * n^i.
+binomial-theorem : ∀ n k → (suc n) ^ k ≡ sumBinomial n k k
+binomial-theorem n zero = refl
+binomial-theorem n (suc k) =
+  begin
+    (suc n) ^ (suc k)
+  ≡⟨ refl ⟩
+    (suc n) * ((suc n) ^ k)
+  ≡⟨ cong (λ x → (suc n) * x) (binomial-theorem n k) ⟩
+    (suc n) * sumBinomial n k k
+  ≡⟨ refl ⟩
+    sumBinomial n k k + n * sumBinomial n k k
+  ≡⟨ cong (λ x → sumBinomial n k k + x) (*-distrib-sumBinomial n k k) ⟩
+    sumBinomial n k k + sumBinomialSuc n k k
+  ≡⟨ +-comm (sumBinomial n k k) (sumBinomialSuc n k k) ⟩
+    sumBinomialSuc n k k + sumBinomial n k k
+  ≡⟨ sumBinomial-align n k k ⟩
+    termSuc n k k + sumBinomial n (suc k) k
+  ≡⟨ cong (λ x → x + sumBinomial n (suc k) k) (top-term n k) ⟩
+    term n (suc k) (suc k) + sumBinomial n (suc k) k
+  ≡⟨ refl ⟩
+    sumBinomial n (suc k) (suc k)
+  ∎