PCM20210729_SICP_1.1.6_Conditional_Expressions_and_Predicates.jl

### A Pluto.jl notebook ###
# v0.19.12

using Markdown
using InteractiveUtils

# ╔═╡ 4ba04c25-a29b-4bf0-be10-be3f20279f28
md"
===================================================================================
 ### SICP: [1.1.6 Conditional Expressions and Predicates](https://sarabander.github.io/sicp/html/1_002e1.xhtml#g_t1_002e1_002e6)

 ###### file: PCM20210729\_SICP\_1.1.6\_Conditional\_Expressions\_and_Predicates.jl

 ###### Julia/Pluto.jl-code (1.8.0/19.11) by PCM *** 2022/11/08 ***
===================================================================================
"

# ╔═╡ e8d5627a-1e6a-48a4-ad50-6013870b8890
md"
###### Implication *Lucky Luke* and *me*
People whisper: *Lucky Luke is the man who shoots faster than his shadow*. 

After some deep thinking I come up with this implication: *If Lucky Luke shoots faster than his shadow, I also can do the very same.* 

This implication is true from two different viewpoints. The *first* is the view of a *psychologist*. (S)he knows that human skill to differentiate two time-sequential events is imperfect. So *both* events in the implication are *false*. Thus the whole implication is *true*.

The *second* viewpoint is that of a *physicist*. From that view *both* events are *true*. The action is always a tiny little moment earlier than the generated shadow on a reflecting wall or other canvas. This is explicated in more detail by [*Nándor Bokor*](https://iopscience.iop.org/article/10.1088/0031-9120/50/6/733)(2015). So this implication is also *true*.

---

"

# ╔═╡ 33f31484-b07f-4550-a829-37f1d4f7feba
md"
#### 1.1.6.1 SICP-Scheme-like *functional* Julia
"

# ╔═╡ 20a147db-1759-4445-a60b-f75e9374864f
md"
---

$$\begin{array}{ccc}
\hline
(cond & (<p_1> & <e_1>) \\
      & (<p_1> & <e_1>) \\
      & ...    & ...    \\
      & (<p_n> & <e_n>)) \\
\hline
\end{array}$$

**Fig. 1.1.6.1** SICP (Scheme/Lisp) conditional expression with *non-strict* evaluation

---

"

# ╔═╡ b649e901-d3bf-4bf0-9ba8-69cd4e6bf550
md"
###### *case analysis* for absolute value function $$|x|$$
"

# ╔═╡ 8287786a-0c31-4d58-83fe-ff4d75513ebe
md"
$$|x| = \cases{\;\;\; x \text{ if } x \gt 0 \cr \;\;\; 0  \text{ if } x = 0 \cr -x \text{ if } x < 0}$$
" 

# ╔═╡ 638667fc-0381-47a5-9288-e10b973e10b7
abs1(x) =  
	x  > 0 ?  x : 
	x == 0 ?  0 :
	x  < 0 ? -x : 
	"unidentified case"

# ╔═╡ c0df9843-2a0c-4c00-a3ee-78a153bde39f
abs1(+4.9)

# ╔═╡ 93ac3435-1b97-4a36-8868-69a40e6bfe67
abs1(0.0)

# ╔═╡ 5eaff842-ff87-445a-9749-052ef1ec8cb9
abs1(-4.9)

# ╔═╡ 50969205-37b7-4d35-8fa9-fa151b11632d
abs2(x) = 
	if x > 0 
		x 
	elseif x == 0 
		0 
	else 
		-x
	end # if

# ╔═╡ e1ef77b1-03a2-478b-8fbd-17556c5e51bd
abs2(4.9)

# ╔═╡ 49ac622a-3846-4096-ae67-7380363d395d
abs2(0)

# ╔═╡ 126c883d-243c-4529-bc5a-d23f2337f67f
abs2(-4.9)

# ╔═╡ 17ffb5ef-1b1a-49f6-8f1d-cc14580781af
abs3(x) = 
	if x > 0
		x
	else -x
	end # if

# ╔═╡ 461a492a-6a79-4df5-ac46-a8b915e2afc5
abs3(4.9)

# ╔═╡ 0339e437-491f-4a04-b3ff-4bf6aa8aceaa
abs3(0)

# ╔═╡ b0d5154a-e71d-4212-bf2c-58a13fadf1e7
abs3(-4.9)

# ╔═╡ 752bc49a-35f2-48ea-8ea5-a05783996b3f
abs4(x) = x > 0 ? x : -x

# ╔═╡ 3e388e69-2c1a-4d37-906f-4c67797b27b3
abs4(4.9)

# ╔═╡ 2ab60d27-1cd4-4098-9ebb-08bdc3eeaae9
abs4(0)

# ╔═╡ bd28e726-cc8f-4aa9-bee1-d7f969a60aa4
abs4(-4.9)

# ╔═╡ 7efcf92c-963e-428e-9393-547bd58a7eea
md"
##### *Binary* Boolean operators with *nonstrict* ('short circuit') evaluation
"

# ╔═╡ f379ad14-d42d-469e-a341-d8082b0f0b83
md"
###### 1st *binary* method with *nonstrict* evaluation of function 'and'
"

# ╔═╡ 6ea1c6c7-56f4-404d-825b-4ddd18e08915
and(x, y) = x && y  # definition of *binary* function 'and' 

# ╔═╡ 848fd3c8-d2df-4e9c-985e-8093f2cfced9
md"
###### 1st *binary* method with *nonstrict* evaluation of function 'or'
"

# ╔═╡ b3ba0af4-ad8f-4d10-b8ca-336b8df614e3
or(x, y) = x || y

# ╔═╡ 69f36630-a57f-4c63-8e01-f38b0a1649a4
not(x) = !x

# ╔═╡ a7e1951f-bf97-4966-b93b-104c7b165f67
not(true)

# ╔═╡ d41e6ce1-1192-495a-8bfd-74936101efa3
not(false)

# ╔═╡ 4188358b-3a15-4bae-addc-54da9bfc87b8
md"
###### compound predicate $$x \ge y := (x > y) \lor (x = y)$$
"

# ╔═╡ fad8b34d-30e1-4ff8-a180-59efde3925fa
geEq2(x, y) = x > y || x == y

# ╔═╡ ad41d505-3803-420f-a077-5f147e32b83e
geEq2(3, 5)

# ╔═╡ 2a1fbc65-f991-46e3-8057-35f8085fc394
geEq2(5, 5)

# ╔═╡ d2a400fa-a179-4ae5-94af-8e1af14ae116
geEq2(7, 5)

# ╔═╡ e40e61d9-692e-4838-a84d-11c51911e3d4
md"
###### compound predicate $$x \ge y := (x \not \lt y)$$
"

# ╔═╡ 7c2f185f-e363-4a8e-8b52-80fef2622aa7
geEq3(x, y) = not(x < y)

# ╔═╡ a96202d4-6f52-40cc-a421-e81bb98c363b
geEq3(5, 3)

# ╔═╡ 5907313d-43b1-4c67-84dd-ea0f5c1f3da8
geEq3(3, 3)

# ╔═╡ b153aba7-acee-4b0e-84e7-1e8aa6f5eea9
geEq3(3, 5)

# ╔═╡ adb0c3c3-97b9-45f1-ae42-e05539dbc427
geEq4(x, y) = !(x < y)

# ╔═╡ e668030b-8d70-4bd2-887c-1bafb6fe5928
geEq4(5, 3)

# ╔═╡ 5decbbfe-402e-472a-9e22-402f7178d3d0
geEq4(3, 3)

# ╔═╡ 6f5a6658-4128-4829-be58-4831b1496d0b
geEq4(3, 5)

# ╔═╡ 18c20c8c-842a-4f03-b3a7-0d5442ae905e
geEq5(x, y) = !<(x, y)

# ╔═╡ d91629d0-79e1-4ddd-9a28-3bc1a9948ff5
geEq5(5, 3)

# ╔═╡ c77e9bd1-5964-411f-831c-9b76b05ff640
geEq5(3, 3)

# ╔═╡ 93fb8d81-c942-420d-ab91-8371a4af3062
geEq5(3, 5)

# ╔═╡ 8208a2ee-8698-4a8c-81e2-3624abc4f7d3
md"
###### n-ary Boolean operators with *strict* evaluation
###### function application forms of *bitwise* '&' and '|'
"

# ╔═╡ c40f2fe6-c8ac-4ace-bb54-148506bdb7a1
(&)(true, true, true)

# ╔═╡ 5f74d11d-f384-4453-b268-5c81ee77cf65
(&)(true, true, false)

# ╔═╡ cf1ae6ed-d21a-4ca3-bfee-cbf0e1549667
(|)(true, true, true)

# ╔═╡ f3cc88a0-e5e0-4e75-8953-bd361937140b
(|)(true, true, false)

# ╔═╡ 22f11800-1d96-46d4-9ac0-4ae817f1b4f1
(|)(false, false, false)

# ╔═╡ 7ad09c29-c9cf-49c9-988f-83b7e4b05393
md"
###### 2nd *nary* method with *strict* evaluation of function 'and'
"

# ╔═╡ f3e3edfa-c5c5-4c1a-805d-c46370298c62
and(x, y, rest...) = (&)(x, y, rest...)  # 2nd (!) method with strict evaluation

# ╔═╡ 70cb087f-17d8-4457-90f1-7cf1d16d9e63
and(true, true)

# ╔═╡ 995d50b4-4e53-4787-b730-df40db2029b1
and(true, false)

# ╔═╡ 9e0cac6c-1121-4e0a-8750-7ad5736974c9
and(true, true, false)

# ╔═╡ 5414db5c-5cc7-4bd5-8c19-aa12e43dfee2
and(true, true, true)

# ╔═╡ 01e00375-4405-4ecb-abf2-8fcb51d6caa4
md"
###### 2nd *nary* method with *strict* evaluation of function 'or'
"

# ╔═╡ d7ff540f-4b3c-4f4e-a7ad-eb5487fd98d5
or(x, y, rest...) = (|)(x, y, rest...)  # 2nd (!) method with strict evaluation

# ╔═╡ 42c5e112-db6f-4874-a227-5a06d8054102
or(false, true)

# ╔═╡ 63bd6526-ea6b-4b48-9456-4c5f05045ef1
or(false, false)

# ╔═╡ ce79a92e-9bb9-48cd-834c-34eb61c67d18
geEq1(x, y) = or((x > y), (x == y))

# ╔═╡ 91eeb0b4-c921-4672-8bee-1d18b2c3c13c
geEq1(5, 3)

# ╔═╡ 234fcd0f-cebc-4adb-8730-3cd2fad15ba0
geEq1(3, 3)

# ╔═╡ 081df546-68eb-4609-8671-e1a6395d9bb9
geEq1(2, 3)

# ╔═╡ 18ee8e44-abd2-4b6e-a66f-5be2dcda2978
or(true, true, false)

# ╔═╡ 6877d30c-7b37-4ec5-9fc7-d6b5fbe382bd
or(false, false, false)

# ╔═╡ 9326271a-ce21-4254-ab26-a42f87d18bc5
md"
---
#### 1.1.6.2 *idiomatic* Julia ...
##### ... with *infix* operators and *chaining* comparisons
$$5 < 2 < 10 := (5 < 2) \land (2 < 10)$$
"

# ╔═╡ 96004b6e-d0f7-4afa-8ee2-f61aa84af474
(5 < 2) && (2 < 10)  

# ╔═╡ b00ca9fe-70fa-4d24-808f-581965d76df6
5 < 2 < 10  

# ╔═╡ cedbf13f-be78-43d7-879a-63971f02c8f5
((5 < 2) && (2 < 10)) == (5 < 2 < 10)

# ╔═╡ f6ee94f3-a049-44a1-91ba-bb6daf3d0140
((5 < 2) && (2 < 10)) === (5 < 2 < 10)

# ╔═╡ be1bb1e0-52f5-4158-9625-aa9f900b3d02
5 < 6 < 10 

# ╔═╡ faeff098-044b-49bc-85a3-c0c655de80a6
(5 < 6) && (6 < 10)

# ╔═╡ f7b6234a-1951-4111-9095-418e6a2a4d49
(5 < 6 < 10) == ((5 < 6) && (6 < 10))

# ╔═╡ c1d0f438-b1d0-44d3-8836-858fcb229935
md"
---
###### Example: Implication

$$\forall x,y: positive:  (x^2 \lt y^2) \Longrightarrow (x \lt y)$$

This implication is proven for *positive* $$x,y$$ in Arens et al. (2018, p.29) in a tutorial style in three alternative ways (*direct, indirect*, and *contradiction*).
This statement is even true for *nonpositive* $$x, y$$

$$\forall x,y: x < y:  (x^2 \lt y^2) \Longrightarrow (x \lt y)$$
"

# ╔═╡ 91d1967b-5fc1-42db-83f2-65858a04f748
let
	x = 0; y = 1
	#--------------
	if x^2 < y^2
		#----------
		if x < y
			true
		else
			false 
		end # if
		#----------
	else
		true
	end # if
	#--------------
end # let

# ╔═╡ f6bb9860-73c8-4d45-9eb9-76bbf9e4674e
let
	x = 0; y = 1
	#------------------------------------------
	x^2 < y^2 ? (x < y ? true : false) : true
	#------------------------------------------
end # let

# ╔═╡ dbc6f577-b2d1-4396-af18-e5066af805d1
let # is 'false' because x^2 < y^2 and not(x < y)
	x = 0; y = -1
	#------------------------------------------
	x^2 < y^2 ? (x < y ? true : false) : true
	#------------------------------------------
end # let

# ╔═╡ 5a755ee8-4dde-42d0-af88-1d8ae231a887
let # is 'true' because not(x^2 < y^2)
	x = -1; y = 0
	#------------------------------------------
	x^2 < y^2 ? (x < y ? true : false) : true
	#------------------------------------------
end # let

# ╔═╡ 18a5a84c-c07f-487b-a671-bb7daf4f0ce3
let # 'false' because x^2 < y^2 and not(x < y)
	x = -2; y = -3
	#------------------------------------------
	x^2 < y^2 ? (x < y ? true : false) : true
	#------------------------------------------
end # let

# ╔═╡ 49272f94-62e9-4e91-8397-bc328b63f63d
let # is 'true' because not(x^2 < y^2)
	x = 3; y = 3
	#--------------
	if x^2 < y^2
		#----------
		if x < y
			true
		else
			false 
		end # if
		#----------
	else
		true
	end # if
	#--------------
end # let

# ╔═╡ 49c8f3c1-a043-44c3-9c01-b26c9a1e7f24
let # is 'true' because not(x^2 < y^2)
	x = 3; y = 3
	#------------------------------------------
	x^2 < y^2 ? (x < y ? true : false) : true
	#------------------------------------------
end # let

# ╔═╡ 4d30686d-0dc1-4d62-aa8a-1409587f2654
let # is 'true' because not(x^2 < y^2)
	x = 3; y = 2
	#--------------
	if x^2 < y^2
		#----------
		if x < y
			true
		else
			false 
		end # if
		#----------
	else
		true
	end # if
	#--------------
end # let

# ╔═╡ 706a89f9-2c14-4b54-819e-2d009e777507
let # is 'true' because not(x^2 < y^2)
	x = 3; y = 2
	#------------------------------------------
	x^2 < y^2 ? (x < y ? true : false) : true
	#------------------------------------------
end # let

# ╔═╡ 9a5d6d10-27d7-4cac-a950-6e5668955067
md"
---
##### References

- **Abelson, H., Sussman, G.J. & Sussman, J.**, Structure and Interpretation of Computer Programs, Cambridge, Mass.: MIT Press, (2/e), 1996, [https://sarabander.github.io/sicp/](https://sarabander.github.io/sicp/), last visit 2022/08/23 

- **Arens, T., Hettlich, F., Karpfinger, Chr., Kockelkorn, U., Lichtenegger, K., & Stachel, H.,** Mathematik, 4/e, Heidelberg, Springer Spektrum, [https://doi.org/10.1007/978-3-662-56741-8](https://doi.org/10.1007/978-3-662-56741-8)

- **Nándor Bokor**, 2015 Phys. Educ. 50 733, online [*here*](https://iopscience.iop.org/article/10.1088/0031-9120/50/6/733); last visit 2022/11/08
"

# ╔═╡ e89faf5f-04a8-44c3-a941-381d41782d40
md"
---
##### end of ch. 1.1.6
===================================================================================

This is a **draft** under the [Attribution-NonCommercial-ShareAlike 4.0 International (CC BY-NC-SA 4.0)](https://creativecommons.org/licenses/by-nc-sa/4.0/) license. Comments, suggestions for improvement and bug reports are welcome: **claus.moebus(@)uol.de**

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

# ╔═╡ 00000000-0000-0000-0000-000000000001
PLUTO_PROJECT_TOML_CONTENTS = """
[deps]
"""

# ╔═╡ 00000000-0000-0000-0000-000000000002
PLUTO_MANIFEST_TOML_CONTENTS = """
# This file is machine-generated - editing it directly is not advised

julia_version = "1.8.2"
manifest_format = "2.0"
project_hash = "da39a3ee5e6b4b0d3255bfef95601890afd80709"

[deps]
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# ╔═╡ Cell order:
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