J. Cederquist, Th. Coquand, and S. Negri. Hahn-banach in type theory. to appear in 25 years in Constructive Type Theory, 1998. Home/Search Document Not in Database Summary Related Articles Check |
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J. Cederquist, Th. Coquand, and S. Negri. Hahn-banach in type theory. to appear in 25 years in Constructive Type Theory, 1998.
A Direct Proof of the Localic Hahn-Banach Theorem - Coquand (1999) Self-citation (Coquand) (Correct)
....a 2 (r 0 ; s 0 ) a 2 (r; s) if r r 0 s 0 s follows from r 0 x r x if r r 0 : The following theorem for instance follows directly from our interpretation. This proposition is proved by a classical reasoning in [MP91] and as an application of the localic HahnBanach in [CCN98, Ver86]. Theorem 5.1 If a 2 ( Gamma1; 1) then a 2 N(1) Proof. By our interpretation, we have Gamma1 a where is the entailment relation defined above. Hence there exists s 0 and r 0 and x such that x 2 N(1) and rx = sa and Gammar = Gammas. It follows that a = x and so a 2 N(1) So far, ....
....: F 8) that x 2 (p; q) and x 2 (p; q 0 ) are equivalent if x 2 N(q) and x 2 N(q 0 ) Theorem 5.2 The translations of the axioms (A1) A2) A3) are provable in the theory (F 1) F 8) 4 The theorem of Hahn Banach is also a direct application of our interpretation. We recall [MP91, CCN98] that we have to define when a basic open p i x i is covered by a family of basic open W j k q j k y j k . This is the case iff for any choice p 0 i p i we can find a finite number of indices j 1 ; j n and q 0j k q j k such that for any choice of k 1 ; k n we ....
J. Cederquist, Th. Coquand, and S. Negri. Hahn-banach in type theory. to appear in 25 years in Constructive Type Theory, 1998. 5
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