PCM20210728_SICP_1.1.1_Expressions.jl

### A Pluto.jl notebook ###
# v0.20.4

using Markdown
using InteractiveUtils

# ╔═╡ a5837b6b-a4d0-45bb-976c-cecda8e6781b
begin 
	using Pluto
	using Plots	
	using GraphRecipes
	#----------------------------------------------------------------
	println("pkgversion(Pluto)         = ", pkgversion(Pluto))
	println("pkgversion(Plots)         = ", pkgversion(Plots))
	println("pkgversion(GraphRecipes)  = ", pkgversion(GraphRecipes))
end # begin

# ╔═╡ 0b714da0-b003-11ef-36b9-c1998185b8b1
md"
====================================================================================
#### SICP: [1.1.1_Expressions](https://mitp-content-server.mit.edu/books/content/sectbyfn/books_pres_0/6515/sicp.zip/full-text/book/book-Z-H-10.html#%_sec_1.1.1)
##### file: PCM20210728\_SICP\_1.1.1\_Expressions.jl
##### Julia/Pluto.jl: 1.11.2/0.20.4 by PCM *** 2025/01/11 ***

====================================================================================
"


# ╔═╡ 53196c5e-41a1-4e37-9405-5e2ee4d8c58e
md"
##### 0. Introduction
*One easy way to get started at programming is to examine some typical interactions with* ... a Julia/Pluto compiler/notebook system. *Imagine that you are sitting at a computer terminal. You type an expression, and the* Julia/Pluto system *responds by displaying the result of its evaluating that expression.*(SICP, 1996, with some modifications by us)

First we try to be code Julia as close as possible to the *functional* style of SICP in [*MIT-Scheme*](https://en.wikipedia.org/wiki/MIT/GNU_Scheme) and especially [*Scheme*](https://en.wikipedia.org/wiki/Scheme_(programming_language)), then we introduce *idiomatic* Julia elements to document its the elegance and expressiveness.

In case of any doubt regarding *syntax* and *semantics* of Julia's constructs you can consult the [Julia doc](https://docs.julialang.org/en/v1/).


"

# ╔═╡ 991bea2b-cb9b-4f95-8085-58eb16f7f775
md"
---
##### 1. Topics

*Topics* dealt are:

- [*expression*](https://en.wikipedia.org/wiki/Expression_(computer_science)), *evaluation*, *value*
- [*numeral*](- https://en.wikipedia.org/wiki/Numeral_system), *Number*
- [*literal*](https://en.wikipedia.org/wiki/Literal_(computer_programming))
- [*types*](https://en.wikipedia.org/wiki/Data_type): $Int64, Float64, Real$
- *dynamic*, *static* types
- *combination*, *compound expression*
- *function call*, [*operator*](https://en.wikipedia.org/wiki/Operator_(computer_programming)): $+, - *, /, div, ÷, ==, ===$
- *prefix*, *infix* and *postfix* notation
- [*Polish notation*](https://en.wikipedia.org/wiki/Polish_notation)
- [*infix* notation](https://en.wikipedia.org/wiki/Infix_notation)
- [*reverse Polish notation*](https://en.wikipedia.org/wiki/Reverse_Polish_notation)
- *Kantorovic* trees, [*algebraic expression* tree](https://en.wikipedia.org/wiki/Binary_expression_tree)
- type *conversion*
- tests of *identity*: $==$, $===$
- [*operator associativity*](https://en.wikipedia.org/wiki/Operator_associativity), *left*-associativity, *right*-associativity 
- *supertype, subtypes, type hierarchy*
- *built-in* functions: $typeof(.), typejoin(.,.), typemin(.), typemax(.)$

"

# ╔═╡ 30996adf-ecc2-4132-ae86-3c02ce8ddff4
md"
---
##### 2. Libraries
"

# ╔═╡ 3ecd4f01-805a-4c67-8897-fb57016b50ae
md"
---
##### 3.  SICP-Scheme-like *functional* Julia: 
###### 3.1 *Simple* Numeric *Expressions*

*Expressions* are evaluated to a *value* (like e.g. a *number*):

$evaluation: <expression> \mapsto <value> \square$

When *numeric* expressions are *simple* they are *numerals* (a special kind of [*literals*](https://en.wikipedia.org/wiki/Literal_(computer_programming))) and evaluated to *numbers*:

$<expression> ::= <numeric\ expression> | <nonnumeric\ expression>.$

$<numeric\ expression> ::=$ 
$<simple\ numeric\ expression> | <numeric\ combination>.$

$<simple\ numeric\ expression> ::= <numeral>.$

$evaluation: <simple\ numeric\ expression> \mapsto <number> \square$

$evaluation: <numeral> \mapsto <number> \square$

Concepts enclosed in $<...>$ are *nonterminals* symbols; they are replaced by *terminals* of a *grammatic*, like:

$<value>\ ::=\ 12\ |\ 13.0\ |\ 23E-5\ |\ 23.f0\ |\ ...$

When they are *numeric combinations* they are evaluated in *several* steps to a *number*:

$evaluation: <numeric\ combination> \mapsto <number> \square$

"

# ╔═╡ 09690a94-3733-4e75-b1c9-e72826a571bc
md"
---
###### 3.2 Simple Numeric Expressions: *Numeral* $\mapsto$ *Number*

Numerals in Julia are *dynamic* typed. [Types](https://en.wikipedia.org/wiki/Type_system) are sets of elements of a kind *declared* by the programmer. 

*Dynamic* types are *not* declared *in the code* but appear *in the evaluation process* (e.g. *with* or *without* decimal point when a numeric type). In contrast to that *static* types are explicitly declared in the code *before* the computation process starts. 

Languages with *dynamic* typing are e.g. Scheme, Javascript and Smalltalk, wheras *static* types appear in Java, C++, etc. *Julia* code can be written in *both* styles.
"

# ╔═╡ 198ce2f2-988c-4387-a73d-60f5d785c3bc
486                                                     # dynamic typed ==> 486 

# ╔═╡ dc5f6985-d4ae-4336-9f66-185bfed79a05
typeof(486)                                             # dynamic typed ==> Int64 

# ╔═╡ 9fdcb71c-da9f-4b20-b606-a1ce7e3d8456
486.                                                    #  dynamic typed ==> 486.0

# ╔═╡ 17468ebe-4b5c-4ff1-a067-fae14a5e332f
typeof(486.)                                            #  dynamic typed ==> Float64

# ╔═╡ 6ddd5a28-6ed9-4115-98ef-b0047bc2dfe5
486f0                                                   #  dynamic typed ==> 486.0f0

# ╔═╡ 373d489f-c853-4ba1-9ed5-7923a7ca4b16
typeof(486f0)                                           #  dynamic typed ==> Float32

# ╔═╡ 0d95e7f8-a75f-4a09-9134-f4833f7ea561
486E-2                                                  #  dynamic typed ==> 4.86

# ╔═╡ a947f3c3-8e9b-402f-84fc-91c0fbd5df22
typeof(486E0)                                           #  dynamic typed ==> Float64

# ╔═╡ 579560f8-5e67-4ba1-bccd-8c9aa05b2530
md"
###### 3.3 *Compound Prefix* Expressions (= *Combinations*) are *Function Calls*
*Flat (= non-nested)* and *nested* expressions have several names. They are called *compound prefix* expressions or *combinations*. They have the *syntax* and *semantics* of *function calls*. 

These can be written in *linear* form with parentheses in different ways, such as in *prefix*, *infix* and *postfix* notation. While *SICP* (= *MIT Scheme*) accepts only *prefix* expressions, *postfix* notation is used in some pocket calculators. 

Julia also understands the *infix* notation learnt in school when dealing with arithmetic expressions. This makes mathematical expressions and formulae coded in Julia easy to understand for non-CS domain experts. Parentheses are used to control the evaluation process.

When they are *numeric combinations* they are evaluated in *several* steps to a *number*:

$evaluation: <numeric\ combination> \mapsto <number> \square$

*Numeric combinations* are *numeric function calls* with at least *one* operand. This constraint can be tested by yourself (see below):

$<numeric\ combination> ::= <numeric\ function\ call>$

###### 3.3.1 *Flat* Numeric Expressions (= *Flat Numeric Combinations*)

The *numeric function call* consists of the *numeric function symbol* or *numeric function name* (= $<operator>$) and one or more *arguments* (= $<operands>$):

$<numeric\ function\ call> ::= <numeric\ operator> \left(<operand>...\right).$

$evaluation: <numeric\ operator>\left(<operand>\right) \mapsto <number> \square$

$evaluation: <numeric\ operator>\left(<operand>,...\right)\; \mapsto \; <number> \square$

"

# ╔═╡ 866f4a84-5479-44aa-b915-4c3e10a16d48
md"
###### 3.3.2 *Nested* Numeric Expressions (= *Nested Numeric Combinations*)

*Nested* numeric combinations contain a *numeric function call* in the place of an *operand*.

$evaluation:$
$<numeric\ operator>\left(..., <operator>\left(<operand>\right),...\right)\; \mapsto \; <number>\square$

$evaluation:$
$<numeric\ operator>\left(..., <operator>\left(<operand>,...\right),...\right)\;\mapsto\; <number>\square$

$<numeric\ operator> ::= +, -, ..., sin, ...$

$<operand> ::= e.g.: -3, 1, -2.0, 3.141, ...$

$<number> ::= e.g.: -1, 4.0, -6.2, 8.2, ...$

"

# ╔═╡ 015dbbd5-a3b0-407a-9ff9-b5a0bee3713a
md"
---
###### 3.4 *Flat Numeric* Expressions
"

# ╔═╡ e7125587-8e19-4201-9faf-e5fcc6d9f0bd
+(486)                                                    # ==> 486

# ╔═╡ 9cc6ba2b-b25d-495f-b4b1-c71e0b897d41
-(486.)                                                   # -486.0

# ╔═╡ 2741242e-fc52-4918-a884-af95a40626ed
-(0.)                                                     # -0.0

# ╔═╡ f46897fa-abdc-4a21-846e-0195b0172fa0
# Math: 137 + 349                                         # ==> 486
# SICP-Scheme: (+ 137 349)                                # ==> 486
# numeric prefix expression Julia:
+(137, 349)                                               # ==> 486 --> :)

# ╔═╡ 3794e29c-696b-4e18-8cc2-2a36489ccd5c
typeof( +(137, 349))                                      # ==> Int64 

# ╔═╡ 86072e36-d2f7-4dd7-8461-103ad10090b7
# Math: 2.7 + 10                                          # ==> 12.7
# SICP: (+ 2.7 10)                                        # ==> 12.7
# numeric prefix expression Julia:
+(2.7, 10)                                                # ==> 12.7 --> :)

# ╔═╡ 575db3c8-3ccc-4d89-9bc8-f25a39cf87d3
# Math: 21 + 35 + 12 + 7                                  # ==> 75
# SICP: (+ 21 35 12 7)                                    # ==> 75
# numeric prefix expression Julia:
+(21, 35, 12, 7)                                          # ==> 75 --> :)

# ╔═╡ d7e0156f-5b8c-48c9-8ce1-bbb93f6ed6d8
# Math: 1000 - 334                                        # ==> 666
# SICP-Scheme: (- 1000 334)                               # ==> 666
# numeric prefix expression Julia:
-(1000, 334)                                              # ==> 666 --> :)

# ╔═╡ 79e083b5-e2a0-426f-968a-1f6c59eac390
typeof( -(1000, 334))                                     # ==> Int64 

# ╔═╡ c45887c0-ddbf-4d09-b89b-af83216e51cf
# Math: 5 * 99                                            # ==> 495
# SICP-Scheme: (* 5 99)                                   # ==> 495
# numeric prefix expression Julia:
*(5, 99)                                                  # ==> 495 --> :)

# ╔═╡ 873a3284-043b-4499-b1ce-8a87605e5a09
typeof( *(5, 99))                                         # ==> Int64 --> :)

# ╔═╡ e806ccc9-77e6-4a8a-9eec-0b1a643a5b9a
# Math: 25 * 4 * 12                                       # ==> 1200
# SICP: (* 25 4 12)                                       # ==> 1200
# numeric prefix expression Julia:
*(25, 4, 12)                                              # ==> 1200 --> :)      

# ╔═╡ 6568b2a5-bc32-4d22-9c88-c8b1f1fee5af
# Math: 10 / 5                                            # ==> 2
# SICP-Scheme: (/ 10 5)                                   # ==> 2
# numeric prefix expression Julia:
/(10, 5)                                                  # ==> 2.0 --> !!

# ╔═╡ 6dfc1c6c-9b77-4df8-9133-63c988696161
typeof( /(10, 5))                                         # ==> Float64 --> !!

# ╔═╡ f7a38676-8f9e-43ab-8cf4-af58e6e2c89c
# Math: 10 / 5                                            # ==> 2
# SICP: (/ 10 5)                                          # ==> 2
# numeric prefix expression Julia:
div(10, 5)                                                # ==> 2 --> :)

# ╔═╡ 0f5bb248-9230-41a6-ad18-f68636100328
# Math: 10 / 5                                            # ==> 2
# SICP: (/ 10 5)                                          # ==> 2
# numeric prefix expression Julia:
÷(10, 5)                                                  # ==> 2 --> :)

# ╔═╡ db568d2c-ac58-425d-acbe-553a072d941d
md"
###### *Boolean* Expressions  
$evaluation: (Number, Number, ==) \rightarrow Boolean$
$evaluation: (Number, Number, ===) \rightarrow Boolean$
$Boolean = \{true, false\}$

"

# ╔═╡ 20850ab6-70a9-44a3-a6f0-de55c071309e
==(2, 2.0)                                                # test of equality

# ╔═╡ 7bdaae8b-21bf-4e2d-ba03-10b9385062db
===(2, 2.0)                                               # test of equality

# ╔═╡ 0a5dc1b5-c8c5-424b-8712-1d7394a4d07a
==(÷, div)                                                # ==> true 

# ╔═╡ 8c64d38b-98e2-4b45-bdc7-e1772efdc965
===(÷, div)                                               # ==> true 

# ╔═╡ d955da7c-2570-4808-8e6b-cb8564c2f0cd
md"
###### Type conversion
$evaluation: (Float64, Int64, /) \rightarrow Float64$
"

# ╔═╡ 9f83cded-8b07-459b-847e-87e3aca59da5
# Math: 10.0 / 5                                          # ==> 2.0
# SICP: (/ 10.0 5)                                        # ==> 2.0
# numeric prefix expression Julia:
/(10.0, 5)                                                # ==> 2.0 --> :)

# ╔═╡ b3af6b38-3b06-4966-8e88-3da2fe720e3e
md"
---
###### 3.5 *Nested  Numeric* Expressions (*Compound* Combinations)

$<numeric\ combination> ::=$ 
$<numeric\ operator>\left(<operand>,..., <operator>\left(<operand>\right),...\right)\;|$

$<numeric\ operator>\left(<operand>,..., <operator>\left(<operand>,...\right),...\right).$

"

# ╔═╡ 62a20957-b08b-48a9-968b-e65038cc520a
md"
###### [*Operator associativity*](https://en.wikipedia.org/wiki/Operator_associativity): *left*-associativity of operator '-'
"

# ╔═╡ 2507313b-a2c7-452e-9288-a4da8bfb0ff9
# *left*-associativity of operator '-'
# Math: 1000 - 300 - 34                                   # ==> 666
# SICP: (- 1000 300 34)                                   # ==> 666
# prefix Julia:
# -(1000, 300, 34) ==> MethodError: no method matching -(::Int64, ::Int64, ::Int64)  
-(1000, -(300, 34))                                       # ==> 734 --> :(

# ╔═╡ aaa8d90b-b0d9-4e8b-aff6-a286910afc65
# left-associativity of operator '-'
# Math: 1000 - 300 - 34                                   # ==> 666
# SICP: (- 1000 300 34)                                   # ==> 666
# prefix Julia:
-(-(1000, 300), 34)                                       # ==> 666 --> :)

# ╔═╡ 3c945bff-e78a-433d-8a4d-118b6915ffc4
# Math: 3 * 5 + (10 - 6)                                  # ==> 19
# SICP: (+ (* 3 5) (- 10 6))                              # ==> 19
# prefix Julia:
+( *(3, 5), -(10, 6))                                     # ==> 19 =/= 666 --> :(

# ╔═╡ 63571818-e93c-4122-a998-59c040df93da
# Math: (3 * ((2 * 4) + (3 + 5))) + ((10 - 7) + 6) =
#       (3 * (   8    +    8   )) + (    3    + 6) =
#       (3 *         16         ) + (       9    ) =
#       (       48              ) + (       9    ) =
#               48                +         9      =
#                                57
#
# SICP: (+ (* 3 (+ (* 2 4) (+ 3 5))) (+ (- 10 7) 6))    ==> 57
#
#       (+ 
#	       (* 
#		      3
#		        (+ 
#			       (* 2 4)
#			       (+ 3 5)))
#	       (+ 
#		       (- 10 7)
#	           6))                                      ==> 57
#
# prefix Julia:
+( *(3, +( *(2, 4), +(3, 5))), +( -(10, 7), 6))       # ==> 57

# ╔═╡ 07d566e1-77a2-45b1-be22-5521d1f65d2e
md"
###### 3.6 *Nested* Combination with *Prefix* Operators as a *nonlinear* Data Flow (*Kantorovic*) tree
(Bauer & Wössner, 1981, p.21)
As a tree the *Kantorovic* tree posses *two* types of *nodes*: *leafs* and *roots*. *Leafs* are *<operands>* (like: 2, 4, ...) and *roots* are *<operators>* (like: $*, +, ...$). Edges are between *leafs* and *roots* (like: 5 -- +) or between *roots* (like: + -- +).
"

# ╔═╡ 9fe7665f-faf4-4633-be95-62126d7d6c90
# SICP: (+ (* 3 (+ (* 2 4) (+ 3 5))) (+ (- 10 7) 6))            # ==> 57
# prefix Julia:
# +( *(3, +( *(2, 4), +(3, 5))), +( -(10, 7), 6))
  +(
	*(
		3, 
		+(
			*(2, 4), 
			+(3, 5))), 
	+(
		-(10, 7), 
		6))                                                     # ==> 57 --> :)

# ╔═╡ 9cfedf46-4325-4972-9c35-d8e604003eaa
begin
	function plotBinaryTree!(markOfLeftLeaf, markOfRightLeaf, markOfRoot, coordinateXOfRootMark, coordinateYOfRootMark; widthOfTree=1.5, heightOfTree=2, fontSize=12) 
		annotate!(
			(coordinateXOfRootMark - widthOfTree/2, 
			 coordinateYOfRootMark + heightOfTree, 
			 text(markOfLeftLeaf, fontSize, :blue))) # mark of left vertical arm 'x'
		plot!([
			(coordinateXOfRootMark -widthOfTree/2,   
			coordinateYOfRootMark + heightOfTree - heightOfTree/6),  
			(coordinateXOfRootMark - widthOfTree/2, coordinateYOfRootMark + heightOfTree/2)], 
			lw=1, linecolor=:black) #  left vertical arm '|'
		annotate!(
			(coordinateXOfRootMark + widthOfTree/2, 
			coordinateYOfRootMark + heightOfTree, 
			text(markOfRightLeaf, fontSize, :blue))) # mark of right vertical arm 'x'
		plot!([
			(coordinateXOfRootMark + widthOfTree/2, 
			coordinateYOfRootMark + heightOfTree - heightOfTree/6), 
			(coordinateXOfRootMark + widthOfTree/2, coordinateYOfRootMark + heightOfTree/2)], 
			lw=1, linecolor=:black) #  right vertical arm '|'
		plot!([
			(coordinateXOfRootMark - widthOfTree/2, coordinateYOfRootMark + heightOfTree/2), 
			(coordinateXOfRootMark + widthOfTree/2, coordinateYOfRootMark + heightOfTree/2)], 
			lw=1, linecolor=:black) #  horizontal bar '-'
		plot!([
			(coordinateXOfRootMark, coordinateYOfRootMark + heightOfTree/2), 
			(coordinateXOfRootMark, coordinateYOfRootMark + heightOfTree/6)], 
			lw=1.5, linecolor=:black) #  middle vertical arm '|'
		annotate!(
			(coordinateXOfRootMark, coordinateYOfRootMark, text(markOfRoot, fontSize, :blue))) # mark of root 'x'
	end # function plotBinaryTree!
	#-----------------------------------------------------------------------------
	plot(size=(800, 400), xlim=(-1, 11), ylim=(0, 12), legend=:false, ticks=:none, title="Kantorovic Tree of expression +( *(3, +( *(2, 4), +(3, 5))), +( -(10, 7), 6))", titlefontsize=12)
	plotBinaryTree!( "2", "4", "*",  1.5,  8.0)
	plotBinaryTree!( "3", "5", "+",  4.0,  8.0)
	plotBinaryTree!( "*", "+", "+",  2.75, 6.0; widthOfTree=2.5)
	plotBinaryTree!("10", "7", "--", 7.0,  6.0; widthOfTree=2.5)
	plotBinaryTree!( "3", "+", "*",  1.5,  4.0; widthOfTree=2.5)
	plotBinaryTree!( "-", "6", "+",  8.25, 4.0; widthOfTree=2.5)
	plotBinaryTree!( "*", "+", "+",  4.86, 2.0; widthOfTree=6.75)
end # begin

# ╔═╡ 56f8b0f1-04ce-4111-84e5-e265c5c0209b
md"
###### Fig. 1.1.1.1: *Kantorovic* tree (Bauer & Wössner, 1981, p.21)

The abstract representation of an expression in *linear* form is the *nonlinear* graphic *Kantorovic* tree. By various *search* strategies in the *expression tree* we can generate the *linear surface* representation of the expression.

- [*prefix*-expression]((https://en.wikipedia.org/wiki/Polish_notation)) (from root to leaves): 
$+( *(3, +( *( 2, 4), +(3, 5))), +( -(10, 7), 6))$

- [*infix*-expression](https://en.wikipedia.org/wiki/Infix_notation) (from leaf to root to leaf):
$(((2*4)+(3+5))*3)+((10-7)+6)$

- [*postfix*-expression](https://en.wikipedia.org/wiki/Reverse_Polish_notation) (from leaves to root):
$((3,((2,4)*,(3, 5)+)+)*,((10,7)-,6)+)+$

"

# ╔═╡ 36ceefbb-a164-49cf-a499-19c6c336be23
md"
---
##### 4. *Idiomatic* Julia
###### 4.1 *Numeric* type *hierarchy*, *builtin* functions $typejoin, subtypes,...$
"

# ╔═╡ 751b2fc4-298b-479f-9974-67578d4c5f24
typejoin(Int64, Float64)                                  # ==> Real

# ╔═╡ 0f0ce1d6-c474-4cc4-8673-5117fb05039e
Int64 <: Real                                             # ==> true

# ╔═╡ 16a261cb-b368-4296-9452-490a0247b593
Float64 <: Real                                           # ==> true

# ╔═╡ 838f513c-fbaa-4233-a766-79f592df74d8
subtypes(Real)               # ==> there are many more types than we used here !

# ╔═╡ 901c4920-b1d8-48dc-a100-d7ae25fa51cb
function makeMyTypeTreeReal()
	g = [
		0 0 0;
		1 0 0;
		1 0 0]
	names = [" Real ", " Int64 ", "Float64"]
	edgelabels = Dict{Tuple, String}((2, 1) => "supertype", (3, 1) => "supertype")
	graphplot(g, x=[1.0, -0.0, +2.0], y = [0.7, 0, 0], edgelabel=edgelabels, nodeshape=:rect, nodecolor=:aqua, names=names, lw=2, nodesize=0.15, fontsize=8, title="Fig. 1.1.1.2 Excerpt of Type Tree: Real", titlefontsize=12, size=(800, 400))
end # function makeMyTypeTreeReal()

# ╔═╡ f3d35d58-5889-44b8-be7c-4ca3c838bf86
makeMyTypeTreeReal()

# ╔═╡ 34265915-b300-4291-a327-4b2f512950fc
isprimitivetype(Int64)

# ╔═╡ 7e379802-13ac-4f26-97ad-644aa42b0b82
isprimitivetype(Float64)

# ╔═╡ ba6e6b36-3e29-47cc-b89d-81044f1e21be
typemin(Int64), typemax(Int64)

# ╔═╡ 69ed7655-db45-4075-8041-1f86b8c0ce67
typemin(Float64), typemax(Float64)

# ╔═╡ 1b51d453-6197-46ea-8aaa-545bbbcbf332
md"
---
###### 4.2 *Type* tree of *functions*
"

# ╔═╡ 45269af2-373c-4178-a527-b0d0064f5a06
isprimitivetype(Function)

# ╔═╡ 049a95bf-3f4d-4e07-9ae2-97718475c3b3
typeof(+)

# ╔═╡ c680cf4f-15ec-4f40-aa9b-68ca83447efb
md"""
"A *singleton type* in Julia is a type that has exactly *one* instance. Julia assigns each function its own unique singleton type. This type represents the function itself, rather its arguments or behavior" (ChatGPT, 2024/12/06)
"""

# ╔═╡ c97ebd78-ff4f-46f1-b2d4-bb2a87f268f9
typeof(==)

# ╔═╡ 74bb6da5-143f-450a-a4c0-23f6535bf91a
typeof(===)

# ╔═╡ b53099c7-267f-4f3e-aac3-b01f00ba814e
typeof(+) <: Function

# ╔═╡ 3a215113-f3da-4675-80cd-78e3c0742225
function makeMyTypeTreeFunction()
	g = [
		0 0 0 0;
		1 0 0 0;
		1 0 0 0;
	    1 0 0 0]
	names = ["Function", "  +  ", "  -  ", "==="]
	edgelabels = Dict{Tuple, String}((2, 1) => "supertype", (3, 1) => "supertype", (4, 1) => "supertype")
	graphplot(g, x=[1.5, -0.0, +1.5, 3.0], y = [0.7, 0, 0, 0], edgelabel=edgelabels, nodeshape=:rect, nodecolor=:aqua, names=names, lw=2, nodesize=0.15, fontsize=8, title="Fig. 1.1.1.3 Excerpt of Type Tree: Function", titlefontsize=12, size=(800, 300))
end # function makeMyTypeTreeFunction

# ╔═╡ 7adbad5d-6f70-4b63-93dc-977e71bff457
makeMyTypeTreeFunction()

# ╔═╡ a270e918-1d26-4d66-92bd-54faeb0b4005
md"
---
###### 4.3 *Infix* operators and expressions
"

# ╔═╡ 9e498769-9f09-4a0b-9cb9-e01b03706f20
# Math: 137 + 349                                         # ==> 486
# SICP: (+ 137 349)                                       # ==> 486
# prefix Julia: +(137, 349)                               # ==> 486 --> :)
# infix Julia:
137 + 349                                                 # ==> 486 --> :)

# ╔═╡ fe4ea16b-98a9-4ca6-84d9-071e3c16fb9b
# Math: 1000 - 334                                        # ==> 666
# SICP: (- 1000 334)                                      # ==> 666
# prefix Julia: -(1000, 334)                              # ==> 666 --> :)
# infix Julia:
1000 - 334                                                # ==> 666 --> :)

# ╔═╡ ae1d20ae-8738-412b-bfc9-3ee4a670360f
# left-associativity of operator '-'
# Math: 1000 - 300 - 34                                   # ==> 666
# SICP: (- 1000 300 34)                                   # ==> 666
# prefix Julia: -(-(1000, 300), 34)                       # ==> 666 --> :)
# infix Julia:
(1000 - 300) - 34 == 1000 - 300 - 34                      # ==> 666 --> :)

# ╔═╡ b8dcf0cd-7183-4da2-aab1-154f8c31e510
(1000 - 300) - 34 === 1000 - 300 - 34                     # ==> 666 --> :)

# ╔═╡ a3b868e5-ff5d-44a0-a5e6-1191867fb2e4
# Math: 5 * 99                                            # ==> 495
# SICP: (* 5 99)                                          # ==> 495
# prefix Julia: *(5, 99)                                  # ==> 495 --> :)
# infix Julia: 
5 * 99                                                    # ==> 495 --> :)

# ╔═╡ 5f9f3c23-735f-4f24-814c-f332a0308df4
# Math: 10 / 5                                            # ==> 2
# SICP: (/ 10 5)                                          # ==> 2
# prefix Julia: /(10, 5)                                  # ==> 2.0 --> !!           
# infix Julia:
10 / 5                                                    # ==> 2.0 --> !!

# ╔═╡ af131abe-cad2-48e6-a3b4-1143dc22bea0
# Math: 10 / 5                                            # ==> 2
# SICP: (/ 10 5)                                          # ==> 2
# prefix Julia: ÷(10, 5)                                  # ==> 2 --> :)
# infix Julia:
10 ÷ 5                                                    # ==> 2 --> :)

# ╔═╡ a220eb4c-5f3e-4cc8-8bd1-5f5d245dfde9
# Math: 10.0 / 5                                          # ==> 2.0
# SICP: (/ 10.0 5)                                        # ==> 2.0
# prefix Julia: /(10.0, 5)                                # ==> 2.0 --> :)
# infix Julia:
10.0 / 5                                                  # ==> 2.0 --> !!

# ╔═╡ 0f7e5745-a2ee-4969-bbb0-11affbd01fef
# Math: 2.7 + 10                                          # ==> 12.7
# SICP: (+ 2.7 10)                                        # ==> 12.7
# prefix Julia: +(2.7, 10)                                # ==> 12.7 --> :)
# infix Julia: 
2.7 + 10.0                                                # ==> 12.7 --> :)

# ╔═╡ 97028ddc-11a1-4e83-86e2-cacdddbb9370
# Math: 21 + 35 + 12 + 7                                  # ==> 75
# SICP: (+ 21 35 12 7)                                    # ==> 75
# prefix Julia: +(21, 35, 12, 7)                          # ==> 75 --> :)
# infix Julia: 
21 + 35 + 12 + 7                                          # ==> 75 --> :)

# ╔═╡ ccb70ed9-2a4d-4aae-a4ab-54477a9cd1e9
# Math: 25 * 4 * 12                                       # ==> 1200
# SICP: (* 25 4 12)                                       # ==> 1200
# prefix Julia: *(25, 4, 12)                              # ==> 1200 --> :)  
# infix Julia:
25 * 4 * 12                                               # ==> 1200 --> :) 

# ╔═╡ 0d07dacf-9004-4f0d-a39c-4fc2a5a430a0
# Math: 3 * 5 + (10 - 6)                                  # ==> 19
# SICP: (+ (* 3 5) (- 10 6))                              # ==> 19
# prefix Julia: +(*(3, 5), -(10, 6))                      # ==> 19 --> :)
# infix Julia:
((3 * 5) + (10 - 6)) == 3 * 5 + (10 - 6)                  # ==> 19 --> :)

# ╔═╡ 9d009aba-7015-401b-aaf2-0243b81f7d9a
# SICP: (+ (* 3 (+ (* 2 4) (+ 3 5))) (+ (- 10 7) 6))      # ==> 57
# prefix Julia:
# +(*(3, +(*(2, 4), +(3, 5))), +(-(10, 7), 6))
# +(
#	*(
#		3, 
#		+(
#			*(2, 4), 
#			+(3, 5))), 
#	+(
#		-(10, 7), 
#		6))                                                # ==> 57 --> :)
# infix Julia:
((3 * ((2 * 4) + (3 + 5))) + ((10 - 7) + 6)) ==
3 * (2 * 4 + (3 + 5)) + 10 - 7 + 6                         # ==> 57 --> :)

# ╔═╡ 41414ac2-6102-4472-9854-4c046da59cd6
((3 * ((2 * 4) + (3 + 5))) + ((10 - 7) + 6))

# ╔═╡ 2f6afa41-9c1a-403a-9a91-eb697e5f48bd
3 * (2 * 4 + (3 + 5)) + 10 - 7 + 6 

# ╔═╡ 6cf5f760-e2fc-498a-b73e-aad0aae95059
3(2 * 4 + (3 + 5)) + 10 - 7 + 6 

# ╔═╡ 477f6294-c4d2-49ee-8b9f-45b24cd696e9
3(2 * 4 + 3 + 5) + 10 - 7 + 6                              # mostly condensed

# ╔═╡ d2dd2233-7afb-4222-96bb-86ae61ba5e72
md"
---
##### 5. Summary

We have evaluated *flat (= non-nested)* and *nested* numeric expressions by interpreting expressions as *function calls*. These can be written in *linear* form with parentheses in different ways, such as in *prefix*, *infix* and *postfix* notation. While *SICP* (= *MIT schema*) only knows *prefix* expressions, Julia also understands the *infix* notation already learnt in school. Parentheses are used to control the evaluation process.

As an alternative to the *linear* form, expressions can also be documented in *non-linear* graphical, parenthesis-free form with *Kantorovic* (= *algebraic* expression) trees. The three linear forms can be read from these trees using various *search* strategies. The evaluation process is here controlled by the edges of the graph.

As a further topic, we discussed a small excerpt from Julia's type hierarchy. In our calculations, we came across $Int64, Float64$, and $Function$. The first and second are subtypes of the supertype $Real$.

"

# ╔═╡ 9e0c051d-5502-4ac2-98ba-aa4fee74df33
md"
---
##### 6. References

- **Abelson, H., Sussman, G.J. & Sussman, J.**; [*Structure and Interpretation of Computer Programs*](https://mitp-content-server.mit.edu/books/content/sectbyfn/books_pres_0/6515/sicp.zip/full-text/book/book.html), Cambridge, Mass.: MIT Press, (2/e), 2016; last visit 2025/01/09), Cambridge, Mass.: MIT Press, (2/e), 1996; last visit 2025/01/09

- **Abelson, H., Sussman, G.J. & Sussman, J.**; [*Structure and Interpretation of Computer Programs*](https://web.mit.edu/6.001/6.037/sicp.pdf), Cambridge, Mass.: MIT Press, (2/e), 2016; last visit 2025/01/09

- **Bauer, F.L. & Wössner, H.**; *Algorithmic Language and Program Development*, Heidelberg: Springer, 1981

- **JuliHub**; [*Julia doc*](https://docs.julialang.org/en/v1/); last visit 2025/01/09

- **Wikipedia**; [*Literal*](https://en.wikipedia.org/wiki/Literal_(computer_programming)); last visit 2025/01/11

- **Wikipedia**; [*MIT/GNU-Scheme*](https://en.wikipedia.org/wiki/MIT/GNU_Scheme); last visit 2024/12/04

- **Wikipedia**; [*Scheme*](https://en.wikipedia.org/wiki/Scheme_(programming_language)); last visit 2025/01/09

- **Wikipedia**; [*Type System*](https://en.wikipedia.org/wiki/Type_system); last visit 2025/01/09

"

# ╔═╡ f6448ce3-ee6c-4835-984a-20d0e08e425d
md"
====================================================================================

This is a **draft** under the Attribution-NonCommercial-ShareAlike 4.0 International **(CC BY-NC-SA 4.0)** license. Comments, suggestions for improvement and bug reports are welcome: **claus.moebus(@)uol.de**

====================================================================================
"

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git-tree-sha1 = "ecda72ccaf6a67c190c9adf27034ee569bccbc3a"
uuid = "ffd25f8a-64ca-5728-b0f7-c24cf3aae800"
version = "5.6.3+1"

[[deps.Xorg_libICE_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "326b4fea307b0b39892b3e85fa451692eda8d46c"
uuid = "f67eecfb-183a-506d-b269-f58e52b52d7c"
version = "1.1.1+0"

[[deps.Xorg_libSM_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libICE_jll"]
git-tree-sha1 = "3796722887072218eabafb494a13c963209754ce"
uuid = "c834827a-8449-5923-a945-d239c165b7dd"
version = "1.2.4+0"

[[deps.Xorg_libX11_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libxcb_jll", "Xorg_xtrans_jll"]
git-tree-sha1 = "ff1fdd02e71717c7418deb1c42f487529d0b9574"
uuid = "4f6342f7-b3d2-589e-9d20-edeb45f2b2bc"
version = "1.8.6+2"

[[deps.Xorg_libXau_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "7966eb654d74306e553ce28b9aea17969fc1966c"
uuid = "0c0b7dd1-d40b-584c-a123-a41640f87eec"
version = "1.0.11+2"

[[deps.Xorg_libXcursor_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libXfixes_jll", "Xorg_libXrender_jll"]
git-tree-sha1 = "807c226eaf3651e7b2c468f687ac788291f9a89b"
uuid = "935fb764-8cf2-53bf-bb30-45bb1f8bf724"
version = "1.2.3+0"

[[deps.Xorg_libXdmcp_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "6a0d3b4248b01faa44509c5ea363881d3ad3f5eb"
uuid = "a3789734-cfe1-5b06-b2d0-1dd0d9d62d05"
version = "1.1.4+2"

[[deps.Xorg_libXext_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libX11_jll"]
git-tree-sha1 = "fb3f116a4efb81aecf8c415e9423869c6ee0f21f"
uuid = "1082639a-0dae-5f34-9b06-72781eeb8cb3"
version = "1.3.6+2"

[[deps.Xorg_libXfixes_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libX11_jll"]
git-tree-sha1 = "6fcc21d5aea1a0b7cce6cab3e62246abd1949b86"
uuid = "d091e8ba-531a-589c-9de9-94069b037ed8"
version = "6.0.0+0"

[[deps.Xorg_libXi_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libXext_jll", "Xorg_libXfixes_jll"]
git-tree-sha1 = "984b313b049c89739075b8e2a94407076de17449"
uuid = "a51aa0fd-4e3c-5386-b890-e753decda492"
version = "1.8.2+0"

[[deps.Xorg_libXinerama_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libXext_jll"]
git-tree-sha1 = "a1a7eaf6c3b5b05cb903e35e8372049b107ac729"
uuid = "d1454406-59df-5ea1-beac-c340f2130bc3"
version = "1.1.5+0"

[[deps.Xorg_libXrandr_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libXext_jll", "Xorg_libXrender_jll"]
git-tree-sha1 = "b6f664b7b2f6a39689d822a6300b14df4668f0f4"
uuid = "ec84b674-ba8e-5d96-8ba1-2a689ba10484"
version = "1.5.4+0"

[[deps.Xorg_libXrender_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libX11_jll"]
git-tree-sha1 = "a490c6212a0e90d2d55111ac956f7c4fa9c277a6"
uuid = "ea2f1a96-1ddc-540d-b46f-429655e07cfa"
version = "0.9.11+1"

[[deps.Xorg_libpthread_stubs_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "9c7539767c23ed0db32fd50916d8f5807ee11af8"
uuid = "14d82f49-176c-5ed1-bb49-ad3f5cbd8c74"
version = "0.1.1+2"

[[deps.Xorg_libxcb_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "XSLT_jll", "Xorg_libXau_jll", "Xorg_libXdmcp_jll", "Xorg_libpthread_stubs_jll"]
git-tree-sha1 = "b4678b3c5ee394ae6422c415b231b8015c85542f"
uuid = "c7cfdc94-dc32-55de-ac96-5a1b8d977c5b"
version = "1.17.0+2"

[[deps.Xorg_libxkbfile_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libX11_jll"]
git-tree-sha1 = "dbc53e4cf7701c6c7047c51e17d6e64df55dca94"
uuid = "cc61e674-0454-545c-8b26-ed2c68acab7a"
version = "1.1.2+1"

[[deps.Xorg_xcb_util_cursor_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_xcb_util_image_jll", "Xorg_xcb_util_jll", "Xorg_xcb_util_renderutil_jll"]
git-tree-sha1 = "04341cb870f29dcd5e39055f895c39d016e18ccd"
uuid = "e920d4aa-a673-5f3a-b3d7-f755a4d47c43"
version = "0.1.4+0"

[[deps.Xorg_xcb_util_image_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "0fab0a40349ba1cba2c1da699243396ff8e94b97"
uuid = "12413925-8142-5f55-bb0e-6d7ca50bb09b"
version = "0.4.0+1"

[[deps.Xorg_xcb_util_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libxcb_jll"]
git-tree-sha1 = "e7fd7b2881fa2eaa72717420894d3938177862d1"
uuid = "2def613f-5ad1-5310-b15b-b15d46f528f5"
version = "0.4.0+1"

[[deps.Xorg_xcb_util_keysyms_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "d1151e2c45a544f32441a567d1690e701ec89b00"
uuid = "975044d2-76e6-5fbe-bf08-97ce7c6574c7"
version = "0.4.0+1"

[[deps.Xorg_xcb_util_renderutil_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "dfd7a8f38d4613b6a575253b3174dd991ca6183e"
uuid = "0d47668e-0667-5a69-a72c-f761630bfb7e"
version = "0.3.9+1"

[[deps.Xorg_xcb_util_wm_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "e78d10aab01a4a154142c5006ed44fd9e8e31b67"
uuid = "c22f9ab0-d5fe-5066-847c-f4bb1cd4e361"
version = "0.4.1+1"

[[deps.Xorg_xkbcomp_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libxkbfile_jll"]
git-tree-sha1 = "ab2221d309eda71020cdda67a973aa582aa85d69"
uuid = "35661453-b289-5fab-8a00-3d9160c6a3a4"
version = "1.4.6+1"

[[deps.Xorg_xkeyboard_config_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_xkbcomp_jll"]
git-tree-sha1 = "691634e5453ad362044e2ad653e79f3ee3bb98c3"
uuid = "33bec58e-1273-512f-9401-5d533626f822"
version = "2.39.0+0"

[[deps.Xorg_xtrans_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "26ded386f85de26df35524639e525c2018f68ddd"
uuid = "c5fb5394-a638-5e4d-96e5-b29de1b5cf10"
version = "1.5.0+2"

[[deps.Zlib_jll]]
deps = ["Libdl"]
uuid = "83775a58-1f1d-513f-b197-d71354ab007a"
version = "1.2.13+1"

[[deps.Zstd_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "7dc5adc3f9bfb9b091b7952f4f6048b7e37acafc"
uuid = "3161d3a3-bdf6-5164-811a-617609db77b4"
version = "1.5.6+2"

[[deps.eudev_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "gperf_jll"]
git-tree-sha1 = "431b678a28ebb559d224c0b6b6d01afce87c51ba"
uuid = "35ca27e7-8b34-5b7f-bca9-bdc33f59eb06"
version = "3.2.9+0"

[[deps.fzf_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "6e50f145003024df4f5cb96c7fce79466741d601"
uuid = "214eeab7-80f7-51ab-84ad-2988db7cef09"
version = "0.56.3+0"

[[deps.gperf_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "0ba42241cb6809f1a278d0bcb976e0483c3f1f2d"
uuid = "1a1c6b14-54f6-533d-8383-74cd7377aa70"
version = "3.1.1+1"

[[deps.libaom_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "1827acba325fdcdf1d2647fc8d5301dd9ba43a9d"
uuid = "a4ae2306-e953-59d6-aa16-d00cac43593b"
version = "3.9.0+0"

[[deps.libass_jll]]
deps = ["Artifacts", "Bzip2_jll", "FreeType2_jll", "FriBidi_jll", "HarfBuzz_jll", "JLLWrappers", "Libdl", "Zlib_jll"]
git-tree-sha1 = "e17c115d55c5fbb7e52ebedb427a0dca79d4484e"
uuid = "0ac62f75-1d6f-5e53-bd7c-93b484bb37c0"
version = "0.15.2+0"

[[deps.libblastrampoline_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "8e850b90-86db-534c-a0d3-1478176c7d93"
version = "5.11.0+0"

[[deps.libdecor_jll]]
deps = ["Artifacts", "Dbus_jll", "JLLWrappers", "Libdl", "Libglvnd_jll", "Pango_jll", "Wayland_jll", "xkbcommon_jll"]
git-tree-sha1 = "9bf7903af251d2050b467f76bdbe57ce541f7f4f"
uuid = "1183f4f0-6f2a-5f1a-908b-139f9cdfea6f"
version = "0.2.2+0"

[[deps.libevdev_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "141fe65dc3efabb0b1d5ba74e91f6ad26f84cc22"
uuid = "2db6ffa8-e38f-5e21-84af-90c45d0032cc"
version = "1.11.0+0"

[[deps.libfdk_aac_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "8a22cf860a7d27e4f3498a0fe0811a7957badb38"
uuid = "f638f0a6-7fb0-5443-88ba-1cc74229b280"
version = "2.0.3+0"

[[deps.libinput_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "eudev_jll", "libevdev_jll", "mtdev_jll"]
git-tree-sha1 = "ad50e5b90f222cfe78aa3d5183a20a12de1322ce"
uuid = "36db933b-70db-51c0-b978-0f229ee0e533"
version = "1.18.0+0"

[[deps.libpng_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Zlib_jll"]
git-tree-sha1 = "9c42636e3205e555e5785e902387be0061e7efc1"
uuid = "b53b4c65-9356-5827-b1ea-8c7a1a84506f"
version = "1.6.44+1"

[[deps.libvorbis_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Ogg_jll", "Pkg"]
git-tree-sha1 = "490376214c4721cdaca654041f635213c6165cb3"
uuid = "f27f6e37-5d2b-51aa-960f-b287f2bc3b7a"
version = "1.3.7+2"

[[deps.mtdev_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "814e154bdb7be91d78b6802843f76b6ece642f11"
uuid = "009596ad-96f7-51b1-9f1b-5ce2d5e8a71e"
version = "1.1.6+0"

[[deps.nghttp2_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "8e850ede-7688-5339-a07c-302acd2aaf8d"
version = "1.59.0+0"

[[deps.p7zip_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "3f19e933-33d8-53b3-aaab-bd5110c3b7a0"
version = "17.4.0+2"

[[deps.x264_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "4fea590b89e6ec504593146bf8b988b2c00922b2"
uuid = "1270edf5-f2f9-52d2-97e9-ab00b5d0237a"
version = "2021.5.5+0"

[[deps.x265_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "ee567a171cce03570d77ad3a43e90218e38937a9"
uuid = "dfaa095f-4041-5dcd-9319-2fabd8486b76"
version = "3.5.0+0"

[[deps.xkbcommon_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Wayland_jll", "Wayland_protocols_jll", "Xorg_libxcb_jll", "Xorg_xkeyboard_config_jll"]
git-tree-sha1 = "63406453ed9b33a0df95d570816d5366c92b7809"
uuid = "d8fb68d0-12a3-5cfd-a85a-d49703b185fd"
version = "1.4.1+2"
"""

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