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27 Aug 2008: The derivation! B Promotion! B! B;! B; C Contraction!B; C Cut!; Cis reduced to3 The exceptions are the cases where (L) is the (second) rule above the cut. ... B! B;! B; C Cut!;! B; C Cut!;! ; C Contraction!; Cor to the symmetric one where we cut against

19 Aug 2008: L(A, B) L′(A′, B′) - L(A B′, B B′) L′(A A′, A B′). ... L(A A′, B A′) L′(B A′, B B′)? comp- L(A A′, B B′).

21 Aug 2008: This easily reduces to(x)(x(b):b(d):chd i j! x(a00):a00h 2i). 1. 2. 3. ... c. d. e. f. (((, ), )),Figure 5: Trace of G.which is to(x)(x(b):b(d):ch di j x(a00):a00h 2i j!

8 Mar 2008: Our first step is to obtain the missing part (iii)b of Lemma 9.3.

21 Aug 2008: p = cod: B B).The first case is of interest in the context of Classical Domain Theory, while the second one istypical of SDT. ... Example 1.12 Given a quasi-topos B, let p = cod: B B (and CI = B/I), then:. •

18 Aug 2008: A; W-L ; ;B W-R ; A;A; A; C-L ; ;B;B ;B C-R :Naturalities implicit in proof nets in tandem with our reduction principle for logicalcuts suggest a ... A B A A B BA;B A BA _ A;B;B A B;A BA _ A;B A B (3).

21 Aug 2008: p = cod: B B).The first case is of interest in the context of Classical Domain Theory, while the second one istypical of SDT. ... Example 1.12 Given a quasi-topos B, let p = cod: B B (and CI = B/I), then:. •

22 Jan 2008: fixBgf = g.fixCf(Ag). • (Diagonal)For f : AB B B. fixB(fixBf) = fixBB(B.f). ... When f : A B B we show the variables by writing f(a,b).

28 Aug 2008: RA¥Tr¥T@U $µU U$¡$¢$ª R$TT¡b3rU [ U U$¡$7U$7¥T9¥qµ¢ZN$9µ9¡bTT¡b¡b=[Ð<;Ò7¥T$ª«Z¥T&¡b=Ñ5¥N$ &¡Ä$ r$N¡ @7$9] 7 @R7 rHU7. ... j7 ;U'(jPj$Ê5ª Ê 1F] &Å)¡$$Uª2¡$7$@? ¡bµ$¡b ¡b ¡$@T! $j¡$'U¥ÂT ¡b.ª #))º.

28 Aug 2008: RA¥Tr¥T@U $µU U$¡$¢$ª R$TT¡b3rU [ U U$¡$7U$7¥T9¥qµ¢ZN$9µ9¡bTT¡b¡b=[Ð<;Ò7¥T$ª«Z¥T&¡b=Ñ5¥N$ &¡Ä$ r$N¡ @7$9] 7 @R7 rHU7. ... j7 ;U'(jPj$Ê5ª Ê 1F] &Å)¡$$Uª2¡$7$@? ¡bµ$¡b ¡b ¡$@T! $j¡$'U¥ÂT ¡b.ª #))º.

19 Aug 2008: A B. S''OO. OOOOOO. OOOO eAB // P(A B). S]. PA PB. ... def. (〈a〉, 〈〉. ), S. (ι2(b). )=def. (〈〉, 〈b〉. )for a A and b B.

18 Aug 2008: Theorem 4.1 Every symmetric monoidal adjunction from A to B lifts to an ad-. ... Theorem 4.2 Given symmetric monoidal categories A and B and given an ordi-.

27 Aug 2008: rDwhich commutes to! B C. rD WeakeningD Commutation of Contraction! B [!B][!B]C ContractionC. ... rDwhich commutes to 29! B [!B][!B]C. rD ContractionDAgain, rather than presenting the above deductions with terms attached, we give (all) theterm

27 Aug 2008: rDwhich commutes to! B C. rD WeakeningD Commutation of Contraction! B [!B][!B]C ContractionC. ... rDwhich commutes to 29! B [!B][!B]C. rD ContractionDAgain, rather than presenting the above deductions with terms attached, we give (all) theterm

13 Aug 2008: We write objects as A = (A R), B = (B R) and so on. ... We write objects of RDill as A = (A R), B = (B R) and soon.

29 Sep 2008: 6. unit- [B,AB]. [e, 1]. 6. k- [[B,B], [B,AB]]. [[B,e], 1]. ... If T is pseudo-commutative this map hasa section. Proposition 12. Given a T-algebra B = (B,b) and a small category X, thecomposite.

8 Jan 2008: a‖ = max1jn. |aj|. defines a norm on Fn.(iii) Show that, if a, b Fn, then. ... f (a) = 1 when a A. f (b) = 0 when b B.

1 Feb 2008: Curve is a continuous map from a closed interval γ : [a,b] C. ... γf dz. • reparameterisation: if φ: [a′,b′] [a,b] is C1 and φ(a′) = a, φ(b′) = b thenif δ = γ φ: [a′,b′] U, have.

13 May 2008: Commutativity A B = B AA B = B A. Complement A A = UA A = {}. Idempotency A A = AA A = A. Identity A {} = AA U = A. ... N-tuples are defined in terms of the Cartesianproduct of sets. • Cartesian product:A B = {〈a,b〉 | a A,b B}.

7 Aug 2008: Combining effects: sum and tensor. Martin Hyland,1 Gordon Plotkin2 and John Power2? 1 Dept. of Mathematics, University of Cambridge, Cambridge CB3 0WB, England.email: M.Hyland@dpmms.cam.ac.uk. 2 Laboratory for the Foundations of Computer Science,