ii, 6) that "lust affects to be called surfeit and abundance." But surfeit regards meat and drink, while abundance refers to riches. Objection 1: It would seem that the matter of lust is not only venereal desires and pleasures. Ergo luxuria non est proprie circa concupiscentias et voluptates venereas. Sed satietas pertinet ad cibos et potus, abundantia autem ad divitias. Dicit enim Augustinus, in libro Confess., quod luxuria ad satietatem atque abundantiam se cupit vocari. Videtur quod materia luxuriae non sit solum concupiscentiae et delectationes venereae. Whether the matter of lust is only venereal desires and pleasures?Īd primum sic proceditur. Quarto, utrum luxuria sit vitium capitale. Tertio, utrum luxuria sit peccatum mortale. Secundo, utrum omnis concubitus sit illicitus. Under the first head there are five points of inquiry: We must next consider the vice of lust which is opposed to chastity: It also uses the standard O_S lemma (of type forall n : nat, 0 S n), so (O_S _ contra) has type type False.Fathers of the English Dominican Provinceĭeinde considerandum est de vitio luxuriae, quod opponitur castitati. This solution uses the so-called "convoy pattern" which is described in several Stackoverflow answers and in the CPDT book by Adam Chlipala. | vnil _ => fun contra => False_rect _ (O_S _ contra) Here is a stub for this function: Fixpoint head (v : vec A (S n)) : A := But for vectors it is especially easy to ask for non-empty input: we put this restriction in the type of the input v : vec A (S n), where S n ensures that the length of the input vector is not zero. There is many approaches to deal with the case of the empty list as the input of head for lists. Now, we might want to provide the analogue of the head function on regular (non-indexed) lists.
Inductive vec (A : Type) : nat -> Type := It is defined in the standard library, but let me give it here for context. Let us consider the hello-world of dependently typed programming, namely length-indexed lists (a.k.a. Would giving up on 'ex falso' change anything w.r.t.
AB FALSO QUOD LIBET DEFINE HOW TO
It's hard for me to see what the computational content of such an axiom is, especially when it says "Given that you have constructed the unconstructable, you now get (for free!) a construction of anything you choose to imagine)." I don't know how to write a program to do such a thing, and Coq's program extraction doesn't seem to have a consistent perspective on the matter either. I'm asking this question because the connection between proofs and programs (Propositions-as-Types / Curry-Howard Isomorphism) seems well-understood for the logical connectives generally, but PEP is an axiom in Heyting and Kolmogorov's formalizations of intuitionistic logic ( Kolmogorov and Brouwer on constructive implication and
_ = Prelude.error "Logical or arity value used"
This principal seems to be necessary for proving things like disjunctive syllogism: Lemma disj_syll : forall (A B: Prop), A \/ B -> (not B) -> A. The induction principle generated from a data type definition with no constructors is PEP: False_ind =įun (P : Prop) (f : False) => match f return P with
Coq seems to assume ex falso quodlibet / the principle of explosion (PEP).
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