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This 11 page Class Notes was uploaded by Kathleen Cartwright on Monday September 28, 2015. The Class Notes belongs to CIS500 at University of Pennsylvania taught by B.Pierce in Fall. Since its upload, it has received 29 views. For similar materials see /class/215371/cis500-university-of-pennsylvania in Computer Information Technology at University of Pennsylvania.
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Date Created: 09/28/15
900 Jaqoloo 98 009 SID 9008 1191 009 SID K 9005 19q0100 95 N suowepuno 1 01910qu J 900 Jaqowo 98 009 SID snmopeQ wepqumq paXmXIdUIIS Last time Last time we talked about proving type soundness for the simply typed lambda calculus with booleans This involved showing 6 Canonical Forms 6 Inversion Lemmas 9 Progress Theorem 9 Preservation Theorem C18 500 26 October7 2005 3 Preservation Theorem Ifl t T and t gt t39 then I t39 T We got to the point Where we needed a substitution lemma to show the preservation theorem IfXStTandI sSthen XI gtstT C18 500 26 October 2005 Typing rules T I true Bool T TRUE F I false Bool T FALSE FI t11BOOl FI tziT FI t3iT T IF F I if t1 then t2 else t3 T XT E F T VAR F IX T F XT1 I tz ITZ T ABS F I AXT1t2 1T1 gtT2 FI t1IT11 gtT12 FI tz 1T1 T APP F PM 62 T12 C18 500 26 October 2005 5 Strengthened induction hypothesis Lemma Types are preserved under substitition IfT XStTandFI sSthenF XI gtstT C18 500 26 October 2005 6 Lemmas about the context Lemma Permutation If F I t T and A is a permutation of F then A I t T Lemma Weakening If F I t T and X 93 domU then T X S I t T k C18 500 26 October 2005 7 Important points about type soundness for ST LC 9 Typing rules need a context to deal with variables 6 Progress lemma must have an empty context to be true 6 Preservation lemma need not see TAPL 6 Application case of preservation lemma needs substitution lemma 6 Substitution lemma must be strenthened to non empty contexts 6 Substitution lemma cannot be proved by induction on typing derivation C18 500 26 October7 2005 8 900 Jaqowo 98 009 SID 39SQEBBHEBUBI Buymmm md 991 09 UO unit unit Unit N ew typing rules The Unit type F I unit Unit terms constant unit values constant um t types unit type C18 500 26 October 2005
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