Demostrar con Lean4 que si bc = ef, entonces ((ab)c)d = ((ae)f)d.<\/p>\n
Para ello, completar la siguiente teor\u00eda de Lean4:<\/p>\n
\r\nimport Mathlib.Data.Real.Basic\r\nimport Mathlib.Tactic\r\n\r\nexample\r\n (a b c d e f : \u211d)\r\n (h : b * c = e * f)\r\n : ((a * b) * c) * d = ((a * e) * f) * d :=\r\nby sorry\r\n<\/pre>\n<\/p>\n
Demostraci\u00f3n en lenguaje natural<\/b><\/p>\n
\nPor la siguiente cadena de igualdades
\n\\begin{align}
\n ((ab)c)d
\n &= (a(bc))d &&\\text{[por la asociativa]} \\\\
\n &= (a(ef))d &&\\text{[por la hip\u00f3tesis]} \\\\
\n &= ((ae)f)d &&\\text{[por la asociativa]}
\n\\end{align}<\/p>\n
Demostraciones con Lean<\/b><\/p>\n\r\nimport Mathlib.Data.Real.Basic\r\nimport Mathlib.Tactic\r\n\r\n-- 1\u00aa demostraci\u00f3n\r\nexample\r\n (a b c d e f : \u211d)\r\n (h : b * c = e * f)\r\n : ((a * b) * c) * d = ((a * e) * f) * d :=\r\ncalc\r\n ((a * b) * c) * d\r\n = (a * (b * c)) * d := by rw [mul_assoc a]\r\n _ = (a * (e * f)) * d := by rw [h]\r\n _ = ((a * e) * f) * d := by rw [\u2190mul_assoc a]\r\n\r\n-- 2\u00aa demostraci\u00f3n\r\nexample\r\n (a b c d e f : \u211d)\r\n (h : b * c = e * f)\r\n : ((a * b) * c) * d = ((a * e) * f) * d :=\r\nby\r\n rw [mul_assoc a]\r\n rw [h]\r\n rw [\u2190mul_assoc a]\r\n\r\n-- 3\u00aa demostraci\u00f3n\r\nexample\r\n (a b c d e f : \u211d)\r\n (h : b * c = e * f)\r\n : ((a * b) * c) * d = ((a * e) * f) * d :=\r\nby\r\n rw [mul_assoc a, h, \u2190mul_assoc a]\r\n<\/pre>\nDemostraciones interactivas<\/b><\/p>\n
Se puede interactuar con las demostraciones anteriores en Lean 4 Web<\/a>.<\/p>\nReferencias<\/b><\/p>\n