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Is it possible to define a mutual inductive type and coinductive type in Coq? I can do this in Agda by putting the declarations at the top but Coq only seems to have with

re: G proof 

It's like a little logic puzzle and I got sniped

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re: G proof 

If you interpret r, s, t, u as sets (might as well), then ⊳ is some sort of pointwise order (every element of r is greater than every element of s), and ⊏ (not <, oops) is the subset relation
Axiom 4 says that if r is pointwise greater than s and u is a subset of s, then r is pointwise greater than u, which makes sense
Axiom 5 says that there is a set r s.t. if there is a set pointwise smaller than z, then z is a subset of r
Personally I think ax5 isn't independent of ax1 and ax4
I bet Axiom 5 could be proven from the other axioms
I bet I can find a shorter proof

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G proof 

Looks like there's two relations, _ ⊳ _ and _ < _
The first is meant to be some sort of order and the second one is some sort of subtyping relation


  • S r ≡ r ⊳ r
  • C r ≡ ∃s. r ⊳ s
  • O r ≡ ∀s. r ⊳ s


  • ax1: ∀r. ∃v. v ⊳ r
  • ax2: ∀r. ∀s. ∀t. (r ⊳ s) → (s ⊳ t) → (r ⊳ t)
  • ax3: ∀r. ∀s. (r ⊳ s) → (s ⊳ r) → r = s
  • ax4: ∀r. ∀s. (r ⊳ s) → (u < s) → r ⊳ u
  • ax5: ∃r. ∀z. C z → z < r, i.e. ∃r. ∀z. ∃s. z ⊳ s → z < r

Let's see... ax1 says that there always exists a greater element, ax2 is transitivity, ax3 is antisymmetry, and ax4 and ax5... well I can't tell what < is meant to be
In any case the G theorem just says that there's a unique maximal element that's reflexive under ⊳

Anyway, and is and, or is or, nor is nor, so is implication, for is inverse implication, and yet is like the inverse of but

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They're all coordinating conjunctions but it doesn't seem like for or so could be used to coordinate anything but independent clauses, i.e. not noun phrases

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In school we learned about FANBOYS:

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"and" is a conjunction. "or" is a conjunction. what about "but"

In contrast, a well-founded data type like [Integer] lives entirely in the moment.

McBride writes the funniest sentences

genitals, Coq 

Okay people like this

But maybe you should ask yourself why you see what-you-don't-want-to-see in something that has never been related to what-you-don't-want-to-see...
(If the shape of the Coq logo makes you think to what-you-don't-want-to-see, maybe it's time to go to the doctor...?)

People like this annoy me so much

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"It type checks but I have no idea what it means" is my slogan right now

genitals, Coq 

They're doing the "yeah I agree" reply-all email chain on the Coq mailing list about Coq sounding like "cock" and the logo looking like a beige cock

(inspecting gender) Wow..... this is a shit type system...

There's no good way of pronouncing Coq anyway
Say it in English and it sounds like cock
Say it in French and it sounds like cuck

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