Write the code for the following proofs using LEAN Theorem Prover 3. Prove the following facts about indexed unions andintersections, using the theorems Inter.intro, Inter.elim,-- Union.intro, and Union.elim listed above.sectionvariables {IU : Type}variables (A : I + set U) (B : I + set U) (C : set U)--example : (n i, A i) n (n i, B i) s (n i, A in B i) :=sorryexample : C n (ui, A i) s ui, Cn A i :=sorryend-- 4. Prove the following fact about power sets. You can use the-- theorems subset.trans and subset.ref1sectionvariable {U : Type}variables A BC : set U-- For this exercise these two facts are usefulexample (h1 : AS B) (h2 : B s C) : A sC :=subset.trans h1 h2example : A SA :=subset.refl Aexample (h : AS B) : powerset A s powerset B :=sorryexample (h : powerset A s powerset B) : As B :=sorryend

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Answered: Write the code for the following proofs…

Engineering

Computer Science

Write the code for the following proofs using LEAN Theorem Prover

Database System Concepts

7th Edition

ISBN:9780078022159

Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Chapter1: Introduction

Section: Chapter Questions

Problem 1PE

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Question

Write the code for the following proofs using LEAN Theorem Prover

3. Prove the following facts about indexed unions and
intersections, using the theorems Inter.intro, Inter.elim,
-- Union.intro, and Union.elim listed above.
section
variables {IU : Type}
variables (A : I + set U) (B : I + set U) (C : set U)
--
example : (n i, A i) n (n i, B i) s (n i, A in B i) :=
sorry
example : C n (ui, A i) s ui, Cn A i :=
sorry
end
-- 4. Prove the following fact about power sets. You can use the
-- theorems subset.trans and subset.ref1
section
variable {U : Type}
variables A BC : set U
-- For this exercise these two facts are useful
example (h1 : AS B) (h2 : B s C) : A sC :=
subset.trans h1 h2
example : A SA :=
subset.refl A
example (h : AS B) : powerset A s powerset B :=
sorry
example (h : powerset A s powerset B) : As B :=
sorry
end

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