Add new theorems and proofs

EasyLean/Basic.lean

@@ -1,12 +1,2 @@
-import Mathlib.Data.Real.Basic
-import Mathlib.Tactic.Linarith
-
-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
+import EasyLean.Theorems.mathd_algebra_513
+import EasyLean.Theorems.eq_four

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 +1,23 @@
+import Mathlib
+
+theorem mathd_algebra_513
+    (a b : Real)
+    (h0 : 3 * a + 2 * b = 5)
+    (h1 : a + b = 2) :
+    a = 1 ∧ b = 1 := by
+  -- Subtract 2*(a+b) from h0 and then use h1 to simplify the right-hand side
+  have hsub : (3 * a + 2 * b) - 2 * (a + b) = (5 : Real) - 2 * 2 := by
+    have h := congrArg (fun x : Real => x - 2 * (a + b)) h0
+    simpa [h1] using h
+  -- Expand and normalize to get a = 1
+  have ha : a = 1 := by
+    have hsub' : (3 * a + 2 * b) - (2 * a + 2 * b) = (5 : Real) - 2 * 2 := by
+      simpa [mul_add] using hsub
+    have hnorm := hsub'
+    ring_nf at hnorm
+    exact hnorm
+  -- Get b from a + b = 2
+  have hb : b = 1 := by
+    have h1' : 1 + b = 2 := by simpa [ha] using h1
+    linarith [h1']
+  exact ⟨ha, hb⟩