clarification of covering

README.md

@@ -219,32 +219,48 @@ then any function in `intsec` maps `a_I_intsec` into `d`. If `a_I_intsec`
 happens to intersect with `c`, it "covers" a part of `c`. I.e. the set `I` can
 be understood as promising to map a part of `c` into `d`.
 
-Defining this formally, we say that `I_cand ⊆ {1..n}` is a (d-)*candidate* if,
+Defining this formally, we say that `I_cand ⊆ {1..n}` is a (`d`-)*candidate* if,
 ```
 ∩[i ∈ I_cand] b_i ⊆ d
 ```
-Now, letting `IC = {I_cand_1..I_cand_k}` be the set of candidates, it is fairly
+We can thus see a candidate `I_cand` as a combination, or more specifically,
-simple to see that
+intersection of the function types `{a_i -> b_i | i ∈ I_cand}`, that according
+to the reasoning above promises to map a part of the domain into `d`, namely
 ```
-intsec <: c -> d
+∩[i ∈ I_cand] a_i
 ```
-iff
+A candidate thus "covers" part of the domain or formally speaking, `I`
+(`d`-)covers a domain `s` if `I` is a (`d`-)candidate and
+```
+s ⊆ ∩[i ∈ I] a_i
+```
+If `IS ⊆ P({1..n})` then we say that `IS` (`d`-)covers `s` if all of `s` is
+covered by some `I ∈ IS`, that is
+```
+s ⊆ ∪[I ∈ IS | I is d-candidate] ∩[i ∈ I] a_i
+```
+
+Given the function types `{a_i -> b_i | i ∈ {1..n}}` and its set of candidates
+`IC = {I_cand_1..I_cand_k}`, it is fairly easy to see that
+```
+&[i ∈ {1..n}] (a_i -> b_i)  <:  c -> d
+```
+iff `IC` `d`-covers `c`:
 ```
 c   ⊆   ( ∪[I_cand ∈ IC]  ∩[i ∈ I_cand] a_i )
 ```
-Using some pro gamer moves, we can see that this is equivalent to
+Using some *pro gamer moves*, we can see that this is equivalent to
 ```
 c   ⊆   ( ∩[I_cand ∈ IC]  ∪[i ∉ I_cand] a_i )
 ```
 This gives us the final form for our subtyping rule
-
 ```
 c, γ*    ⊢    a_i | ... | a_n, δ*
 forall I ⊆ {1..n}.
 	&[i ∈ I] b_i, γ*    ⊢    d, δ*
 	  or
 	c, γ*    ⊢    |[i ∉ I] a_i, δ*
-----
+---- [loads-o-fun]
 a_1 -> b_1, ..., a_n -> b_n, γ ⊢ c -> d, δ
 ```