From 3f1ef225cec35c31fd6279020cf20dd6ac938c69 Mon Sep 17 00:00:00 2001
From: arshiamoeini <arshiamoeini82@gmail.com>
Date: Fri, 2 Feb 2024 13:33:42 +0330
Subject: [PATCH] add pigeonhole_principle

---
 library/Automata.v                 | 2 ++
 library/EnumerativeCombinatorics.v | 4 ++++
 library/List.v                     | 1 +
 3 files changed, 7 insertions(+)

diff --git a/library/Automata.v b/library/Automata.v
index 810e3b4..3476578 100644
--- a/library/Automata.v
+++ b/library/Automata.v
@@ -106,6 +106,7 @@ Axiom run_dfa_nil: ∀ sigma, ∀ A: DFA sigma, ∀ u, run_dfa A u "" = u;
 Axiom run_dfa_cons: ∀ sigma, ∀ A: DFA sigma, ∀ u, ∀ s, ∀ c, ∀ f: char → ℤ, f = nth (λ a: char, - 1) (edges_of_DFA A) u → ∀ v, v = f c → run_dfa A u (c :: s) = run_dfa A v s;
 
 Todo run_dfa_append: ∀ sigma: list char, ∀ A: DFA sigma, ∀ u, ∀ a b, run_dfa A u (a + b) = run_dfa A (run_dfa A u a) b;
+Axiom run_dfa_range: ∀ sigma: list char, ∀ A: DFA sigma, ∀ u, ∀ s, IsWord sigma s -> run_dfa A u s ∈ {i | 0 ≤ i ∧ i < |A| };
 
 Axiom #1 Ldfa: ∀ sigma, DFA sigma → set (list char);
 Axiom Ldfa_unfold: ∀ sigma, ∀ A: DFA sigma, ∀ s, s ∈ Ldfa A -> IsWord sigma s ∧ run_dfa A (start_of_DFA A) s ∈ accept_nodes_of_DFA A;
@@ -197,6 +198,7 @@ Suggest hyp default apply is_decidable_unfold in $n with label Destruct;
 
 Axiom #2 computable: ∀ X Y: U, (X → Y) → U;
 Axiom computable_concat: computable (plus (list char));
+Axiom computable_neg: computable (neg ℤ);
 Axiom computable_const: ∀ X Y: U, ∀ a: Y, computable (λ x: X, a);
 Suggest goal default apply computable_const with label Trivial;
 
diff --git a/library/EnumerativeCombinatorics.v b/library/EnumerativeCombinatorics.v
index 956f111..280e48b 100644
--- a/library/EnumerativeCombinatorics.v
+++ b/library/EnumerativeCombinatorics.v
@@ -1175,6 +1175,10 @@ Proof;
 Qed;
 Todo valid_paren_cnt_right: ∀ a, valid_paren a -> ∀ k, 2 * k = |a| -> cnt ')' a = k;
 
+Todo pigeonhole_principle: ∀ pigeons pigeonholes: U, ∀ A: set pigeons, ∀ B: set pigeonholes, 
+    |B| < |A| → ∀ f: pigeons → pigeonholes, projection A f ⊆ B 
+    → ∃ a b: pigeons, a ∈ A ∧ b ∈ A ∧ ~ a = b ∧ f a = f b;
+
 Theorem count_of_lists: ∀ T: U, ∀ S: set T, ∀ m, 0 < m -> |S| = m -> ∀ n, 0 ≤ n -> |{ l: list T | member_set l ⊆ S ∧ |l| = n }| = m ^ n;
 Proof;
     intros;
diff --git a/library/List.v b/library/List.v
index 8115f19..e0f3bd1 100644
--- a/library/List.v
+++ b/library/List.v
@@ -241,6 +241,7 @@ Todo skipn_append_l: ∀ T: U, ∀ a b: list T, ∀ i: ℤ, |a| ≥ i -> skipn (
 Todo skipn_append_r: ∀ T: U, ∀ a b: list T, ∀ i: ℤ, i ≥ 0 -> skipn (a + b) (|a| + i) = skipn b i;
 Todo skipn_append_l_len: ∀ T: U, ∀ a b: list T, skipn (a + b) (|a|) = b;
 Todo skipn_len: ∀ T: U, ∀ l: list T, skipn l (|l|) = [];
+Todo len_skipn: ∀ T: U, ∀ l: list T, ∀ i, 0 ≤ i -> i ≤ |l| -> |skipn l i| = |l| - i;
 
 Todo firstn_skipn: ∀ T: U, ∀ l: list T, ∀ i: ℤ, l = firstn l i + skipn l i;