Learning.Hist.map_comp
Lemma
No docstring.
Types
-
๐ : Type u_1 -
๐' : Type u_2 -
๐'' : Type u_3 -
๐ : Type u_4 -
๐' : Type u_5 -
๐'' : Type u_6 -
๐จ : Type u_7 -
๐จ' : Type u_8 -
๐จ'' : Type u_9
Given
-
fo : ๐ โ ๐' -
fa : ๐ โ ๐' -
fy : ๐จ โ ๐จ' -
go : ๐' โ ๐'' -
ga : ๐' โ ๐'' -
gy : ๐จ' โ ๐จ'' -
n : โ -
h : Hist ๐ ๐ ๐จ nHistory ofncomplete rounds;n = 0is the empty history.
Then
map go ga gy (map fo fa fy h) = map (go โ fo) (ga โ fa) (gy โ fy) hNat : TypeThe natural numbers, starting at zero. This type is special-cased by both the kernel and the compiler, and overridden with an efficient implementation. Both use a fast arbitrary-precision arithmetic library (usually [GMP](https://gmplib.org/)); at runtime, `Nat` values that are sufficiently small are unboxed.
Learning.Hist : Type u_5 โ Type u_6 โ Type u_7 โ โ โ Type (max (max u_7 u_6) u_5)History of `n` complete rounds; `n = 0` is the empty history.Go to its page
Eq : {ฮฑ : Sort u_1} โ ฮฑ โ ฮฑ โ PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (ฮฑ : Type) (a b c d : ฮฑ)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (ฮฑ : Type) (a b : ฮฑ) (p : ฮฑ โ Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (ฮฑ : Type) (a b : ฮฑ) (p : ฮฑ โ Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 โธ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
Conventions for notations in identifiers:
* The recommended spelling of `=` in identifiers is `eq`.Learning.Hist.map : {๐ : Type u_1} โ
{๐' : Type u_2} โ
{๐ : Type u_4} โ
{๐' : Type u_5} โ
{๐จ : Type u_7} โ
{๐จ' : Type u_8} โ (๐ โ ๐') โ (๐ โ ๐') โ (๐จ โ ๐จ') โ {n : โ} โ Learning.Hist ๐ ๐ ๐จ n โ Learning.Hist ๐' ๐' ๐จ' nTransport a history round-wise.Go to its page
Function.comp : {ฮฑ : Sort u} โ {ฮฒ : Sort v} โ {ฮด : Sort w} โ (ฮฒ โ ฮด) โ (ฮฑ โ ฮฒ) โ ฮฑ โ ฮดFunction composition, usually written with the infix operator `โ`. A new function is created from two existing functions, where one function's output is used as input to the other. Examples: * `Function.comp List.reverse (List.drop 2) [3, 2, 4, 1] = [1, 4]` * `(List.reverse โ List.drop 2) [3, 2, 4, 1] = [1, 4]` Conventions for notations in identifiers: * The recommended spelling of `โ` in identifiers is `comp`.
Code
lemma Hist.map_comp {n : โ} (h : Hist ๐ ๐ ๐จ n) :
Hist.map go ga gy (Hist.map fo fa fy h) = Hist.map (go โ fo) (ga โ fa) (gy โ fy) hProof
rfl
New in v4.34.0-rc2-74-ge05e4f3 (2026-09-10), and its meaning has not changed since.
Self-contained, with its dependencies inlined and proofs replaced by sorry: download the raw file ยท open it in the Lean web editor.
Dependency graph
Audit surface: 7 project declarations, 8 external constants
โ Proved: no sorry anywhere in its closure
This is the tool's own reading of one build's recorded axioms, and it is not robust against an author who wants it to pass. Checking meant to be relied on should go through Comparator, which replays the proof through the kernel from an export against an explicit list of permitted axioms.