Add new theorems and proofs

EasyLean/Basic.lean

@@ -1,12 +1,5 @@
-import Mathlib.Data.Real.Basic
-import Mathlib.Tactic.Linarith
+import Mathlib
+import EasyLean.Theorems.mathd_algebra_513
+import EasyLean.Theorems.eq_four
 
-theorem mathd_algebra_513
-  (a b : ℝ)
-  (h₀ : 3 * a + 2 * b = 5)
-  (h₁ : a + b = 2) :
-  a = 1 ∧ b = 1 := by
-  sorry
-
-theorem eq_four : ∀ a b c d : Nat, a = b → a = d → a = c → c = b := by
-  sorry
+-- Additional project-wide declarations can be added here if needed.

EasyLean/Theorems/eq_four.lean

@@ -0,0 +1,5 @@
+import Mathlib
+
+theorem eq_four : ∀ a b c d : Nat, a = b → a = d → a = c → c = b := by
+  intro a b c d hab had hac
+  exact hac.symm.trans hab

EasyLean/Theorems/mathd_algebra_513.lean

@@ -0,0 +1,22 @@
+import Mathlib
+
+open scoped Real
+
+theorem mathd_algebra_513 (a b : Real) (h0 : 3 * a + 2 * b = 5) (h1 : a + b = 2) : a = 1 ∧ b = 1 := by
+  have hb : b = 2 - a := by
+    have ht : (a + b) - a = 2 - a := by
+      simpa using congrArg (fun t => t - a) h1
+    have hx : (a + b) - a = b := by
+      simpa using add_sub_cancel a b
+    simpa [hx] using ht
+  have h2 : 3 * a + 2 * (2 - a) = 5 := by
+    simpa [hb] using h0
+  have h3 : 4 + a = 5 := by
+    have h2' := h2
+    ring_nf at h2'
+    simpa using h2'
+  have ha : a = 1 := by
+    linarith [h3]
+  have hb1 : b = 1 := by
+    linarith [h1, ha]
+  exact ⟨ha, hb1⟩