Hw3-SOLUTIONS - Homework 3 Solutions These valid and nicely...

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Homework 3 Solutions These valid, and nicely formatted, proofs are provided courtesy of Rebekah Leslie and Matt Griffin import Stdm cp = check_proof {- Problem 1 -} hwThm1 = Theorem [P, Q, R] (P `And` (Q `And` R)) proofThm1 = (Assume P, (Assume Q, Assume R) {--------------------} `AndI` (Q `And` R)) {---------------------------------------------------} `AndI` (P `And` (Q `And` R)) {- Problem 2 -} hwThm2 = Theorem [A `And` (Not A)] (FALSE) proofThm2 = (Assume(A `And` (Not A)) {----------------------------} `AndEL` A, Assume(A `And` (Not A)) {---------------------------------} `AndER` (Not A)) {---------------------------------------------------------------------------- -}`ImpE` (FALSE) {- Problem 3 -} hwThm3 = Theorem [A] ((A `Imp` FALSE) `Imp` FALSE) proofThm3 = (Assume A, Assume(A `Imp` FALSE)) {--------------------------------------------------------------------------- -------} `ImpE` FALSE {--------------------------------------------------------------------------- ------} `ImpI`
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((A `Imp` FALSE) `Imp` FALSE) {- Problem 4 -} hwThm4 = Theorem [A, A `Imp` B, B `Imp` C, C `Imp` D] (D) proofThm4 = (((Assume A, Assume(A `Imp` B)) {--------------------------} `ImpE` B, Assume(B `Imp` C)) {--------------------------------------------} `ImpE` C, Assume(C `Imp` D)) {--------------------------------------------------------------} `ImpE` D {- Problem 5 -} impChainThm = Theorem [A `Imp` B, B `Imp` C] (A `Imp` C) -- a theorem proved in lectures impChainRule = Use impChainThm -- theorem converted to inference rule hwThm5 = Theorem [A `Imp` B, Not B] (Not A) proofThm5 = [Assume(A `Imp` B), Assume(B `Imp` FALSE)] {--------------------------------------------} `impChainRule` (A `Imp` FALSE) {- Problem 6 -} hwThm6 = Theorem [(B `Or` (Not B)), A `Imp` B] (Not A `Or` B)
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