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This is a repository copy of

Testing with guarantees and the failure of regression testing in

eXtreme Programming

.

White Rose Research Online URL for this paper:

http://eprints.whiterose.ac.uk/78799/

Proceedings Paper:

Simons, A.J.H. (2005) Testing with guarantees and the failure of regression testing in

eXtreme Programming. In: Baumeister, H., Marchesi, M. and Holcombe, M., (eds.)

Extreme Programming and Agile Processes in Software Engineering, Proceedings. 6th

International Conference, XP 2005, June 18-23, Sheffield, UK. Springer , 118 - 126. ISBN

978-3-540-26277-0

[email protected] https://eprints.whiterose.ac.uk/

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http://eprints.whiterose.ac.uk/

This is an author produced version of a paper published in Extreme

Programming and Agile Processes in Software Engineering, Proceedings.

White Rose Research Online URL for this paper:

http://eprints.whiterose.ac.uk/78799

Published paper

Simons, A.J.H. (2005) Testing with guarantees and the failure of regression

testing in eXtreme Programming. In: Baumeister, H., Marchesi, M. and Holcombe,

M., (eds.) Extreme Programming and Agile Processes in Software Engineering,

Proceedings. 6th International Conference, XP 2005, June 18-23, Sheffield, UK.

Springer , 118 - 126. ISBN 978-3-540-26277-0

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∧≈″≈→↵←•″≠ ⇑±≥°×≈←≈ ⊄″•← ⊂≈↑←⌠⊆≈←↑

⊆←↵←≈ ≥↵÷•″≈↑ ÷↵♠≈ ≡→≈↓↔≈″←×♣ ≈≈″ ↔↑≈… ↵↑ ≥±…≈×↑ ±≡ ±∂≈←↑ ±←÷ ←± ↵→←•↔×↵←≈ ←÷≈•→ …≈↑•≠″ ⊕ ℘ ↵″… ≥±→≈ ≡±→≥↵××♣ ↵↑ °↵→← ±≡ ↵ ←≈↑←•″≠ ≥≈←÷±… ⊕   ℘∫ ⋅″ ←÷≈ ≥±→≈ →•≠±→±↔↑ ↵°°→±↵÷≈↑ •← •↑ °±↑↑•×≈ ←± …≈♠≈×±° ↵ ″±←•±″ ±≡±≥°×≈←≈ ←≈↑← ±♠≈→↵≠≈ ↵↑≈… ±″ ←÷≈ ≈♦÷↵↔↑←•♠≈ ≈♦°×±→↵←•±″ ±≡ ←÷≈ ±∂≈←Ž↑ ↑←↵←≈↑ ↵″… ←→↵″↑•←•±″↑∫ ⊂÷≈ ≡±××±♥⌠ •″≠ •↑ ↵″ ↵…↵°←↵←•±″ ±≡ ←÷≈ ∪⌠↵÷•″≈ ←≈↑←•″≠ ≥≈←÷±… ⊕ ℘ ♥÷•÷ ±≡≡≈→↑ ↑←→±″≠≈→ ≠↔↵→↵″←≈≈↑ ←÷↵″ ±←÷≈→ ≥≈←÷±…↑ •″ ←÷↵← •←↑ ←≈↑←•″≠ ↵↑↑↔≥°←•±″↑ ↵→≈ ×≈↵→ ↵″… •← ←≈↑←↑ ″≈≠↵←•♠≈×♣≡±→ ←÷≈ ↵↑≈″≈ ±≡ ↵×× ↔″…≈↑•→≈… ≈÷↵♠•±↔→ ↵↑ ♥≈×× ↵↑°±↑•←•♠≈×♣≡±→ ←÷≈ °→≈↑⌠ ≈″≈ ±≡ ↵×× …≈↑•→≈… ≈÷↵♠•±↔→∫ ⋅← •↑ ♥≈×× ∝″±♥″ ←÷↵← °→±≠→↵≥≥≈→↑ ≡↵•× ←± ±″↑•…≈→ ←÷≈ ≡±→≥≈→ ↑•″≈ •← •↑ ÷↵→… ←± ←÷•″∝ ±≡ ↵×× ←÷≈ ↔″•″←≈″…≈… ♥↵♣↑ •″ ♥÷•÷ ↵ ↑♣↑←≈≥ ≥•≠÷← °±↑↑•×♣ ≈ ↵↔↑≈… ♥÷≈″ ←÷≈ ≡±↔↑ •↑ ±″ °±↑•←•♠≈ ±↔←±≥≈↑∫ ± ≥↵←←≈→ ÷±♥ ↑•≥°×≈ ↵″… …•→≈← ←÷≈ ÷↵″…⌠→↵≡←≈… ←≈↑←↑ ≥•≠÷← ↵°°≈↵→ •≡ ←÷≈♣ …±″Ž← ±♠≈→ ←÷≈ ±∂≈←Ž↑ ↑←↵←≈ ↑°↵≈ ←÷≈♣ ↵→≈•″±≥°×≈←≈ ↵″… ±″×♣ ±≡≡≈→ ↵ ≡↵×↑≈ ↑≈″↑≈ ±≡ ↑≈↔→•←♣∫

⇔↵→ו≈→ ±≥°↵″•±″ ♥±→∝ ↑÷±♥≈… ÷±♥ ≈↵↑♣ ↵″… °→↵←•↵× •← •↑ ←± ≠≈″≈→↵←≈ ↑•≥°×≈ ↑←↵←≈ ≥↵÷•″≈↑ …•→≈←×♣ ≡→±≥ ∪∠ ↑←±→♣ ↵→…↑ ⊕℘ ↵″… ⊄ ↔↑≈ ↵↑≈↑ ⊕℘∫ ⊂÷•↑ ↑↔°⌠ °±→←≈… ←÷≈ ↵↔←±≥↵←• ≠≈″≈→↵←•±″ ±≡ ±≥°×≈←≈↔↑≈→⌠↵≈°←↵″≈ ←≈↑←↑∫ ⋅″ ⊕ð℘ ←÷≈ ↵↔←±⌠ ≥↵←• ≠≈″≈→↵←•±″ ±≡ ±≥°×≈←≈↔″•← ←≈↑←↑≡→±≥ ↑←↵←≈ ≥↵÷•″≈↑ ♥↵↑ ↑÷±♥″ ←± ±↔←°≈→≡±→≥ ↵…⌠÷± ←≈↑← ↑→•°←•″≠∫ ⊂÷≈ ≡±↔↑ ±≡ ←÷≈ ↔→→≈″← °↵°≈→ •↑→≈≠→≈↑↑•±″ ←≈↑←•″≠↵″… ±≥⌠ °×≈←≈ ←≈↑←→≈≠≈″≈→↵←•±″•″ ←÷≈ ±″←≈♦← ±≡ ±∂≈← ≈♠±×↔←•±″∫

∫ ⊆←↵←≈⌠⇒↵↑≈… ⇐≈↑•≠″

⇓″ ±∂≈← ≈♦•↑←↑ •″ ↵ ↑≈→•≈↑ ±≡ ↑←↵←≈↑ ♥÷•÷ ↵→≈ ÷±↑≈″ ♣ ←÷≈ …≈↑•≠″≈→ ←± →≈≡×≈← ≥±…≈↑ •″ ♥÷•÷ •←↑ ≥≈←÷±…↑ →≈↵← …•≡≡≈→≈″←×♣ ←± ←÷≈ ↑↵≥≈ ≥≈↑↑↵≠≈↑∫ ⊂÷≈ ″↔≥≈→ ±≡ ↑←↵←≈↑ ↵″ ±∂≈← ↵″ ÷↵♠≈ …≈°≈″…↑ ±″ ←÷≈ •″…≈°≈″…≈″←×♣⌠♠↵→♣•″≠ ♠↵×↔≈↑ ±≡ ↵×× ±≡ •←↑ ↵←←→•↔←≈↑∫ ⊂÷≈ ←÷≈±→≈←•↵× ≥↵♦•≥↔≥ •↑ ←÷≈ ⇑↵→←≈↑•↵″ °→±…↔← ±≡ •←↑ ↵←←→•↔←≈ …±≥↵•″↑∫ ⊂♣°•↵××♣ ←÷•↑ ≡•″≈⌠≠→↵•″≈… ↵←←→•↔←≈ ↑°↵≈ •↑ °↵→←•←•±″≈… •″←± ±↵→↑≈→ ↑←↵←≈↑ •″ ♥÷•÷ ≈÷↵♠•±↔→ •↑ ↓↔↵ו←↵←•♠≈×♣ •←÷≈ ↑↵≥≈Œ •″ ≈↵÷ °↵→←•←•±″ ⊕ ℘∫ ⊃≈ …≈≡•″≈ ↑←↵←≈↑ ≥±→≈ ↵↑←→↵←×♣ ↵↑ °↵→←•←•±″↑ ±≡ ←÷≈ °→±…↔← ±≡ ←÷≈ →↵″≠≈↑ ±≡ ↵″ ±∂≈←Ž↑ ↵≈↑↑ ≥≈←÷±…↑ ⊕℘∫ ⊂÷•↑ °≈→≥•←↑ ←÷≈ …≈↑•≠″ ±≡ ↑←↵←≈ ≥↵÷•″≈↑ ≡±→ ≡↔××♣ ↵↑←→↵← ×↵↑↑≈↑ ↵″… •″←≈→≡↵≈↑ ♥÷•÷ ♣ …≈≡•″•←•±″ ÷↵♠≈ ″± ±″→≈←≈ ↵←←→•↔←≈↑∫

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↑←↵←≈∫ ⊃≈ ↵↑↑↔≥≈ ♣ ±″♠≈″←•±″ ←÷↵← ←÷≈ ↵≈↑↑ ≥≈←÷±…↑ ∞←±° ≈≥°←♣ ↑•ƒ≈≤ ♥÷•÷ ↵→≈ ″±← ↑÷±♥″ ↵×× ÷↵♠≈ •≥°×••← ←→↵″↑•←•±″↑ ×±±°•″≠ ≡→±≥ ≈↵÷ ↑←↵←≈ ↵∝ ←± •←↑≈×≡∫

⇔≥ °←♣

↑ •ƒ≈   ð

±→≥ ↵ ×

↑ •ƒ≈   ð °↔ ↑÷ ≈ 

° ↔ ↑÷ ≈ 

° ± °  ⊕↑ •ƒ≈    ℘ ° ± °  ⊕↑ •ƒ≈    ℘

°± °  ″ ≈ ♥

∨•≠∫ ∫ ⊆←↵←≈ ≥↵÷•″≈ ≡±→ ↵⊆←↵∝∫ ⊂÷≈ ←♥± ↑←↵←≈↑ ⇔≥°←♣ ±→≥↵× ↵→≈ …≈≡•″≈… ±″ ↵ °↵→←•←•±″ ±≡ ←÷≈ →↵″≠≈ ±≡ ←÷≈↑•ƒ≈↵≈↑↑ ≥≈←÷±…∫ ∇″×♣ ↑←↵←≈⌠≥±…•≡♣•″≠ ←→↵″↑•←•±″↑ ↵→≈ ↑÷±♥″ ≈♦°×••←×♣∫

⊂÷≈ ↑←↵←≈ ≥↵÷•″≈ •↑ …≈♠≈×±°≈… ♣ ±″↑•…≈→•″≠ ←÷≈ ≥±…≈↑ •″ ♥÷•÷ ≈→←↵•″ ≥≈←÷±…↑ ≈÷↵♠≈ …•≡≡≈→≈″←×♣∫ ¬≈→≈°±°↵″…←±°↵→≈ ×≈≠↵× •″ ←÷≈±→≥↵×↑←↵←≈ ↔← ↔″…≈≡•″≈… •″ ←÷≈⇔≥°←♣ ↑←↵←≈∫ ⊂÷≈→≈ •↑ ↵ ↑•″≠×≈ ←→↵″↑•←•±″ ←± ←÷≈ •″•←•↵× ↑←↵←≈ →≈°→≈↑≈″←•″≠ ±∂≈← ±″↑←→↔←•±″∫ ⊂→↵″↑•←•±″↑ ↵→≈ ←÷≈″ ↵……≈… ≡±→≈♠≈→♣≥≈←÷±… •″≈♠≈→♣↑←↵←≈∫ ⋅≡ ↵″ ±∂≈← •↑ ″± ×±″≠≈→ ↔↑≈↵×≈ •← ≥↵♣ ≈″←≈→ ↵ ≡•″↵× ↑←↵←≈ ↑↔÷ ↵↑ ←÷≈ ≈→→±→ ↑←↵←≈ ↵≡←≈→ ←÷≈ •××≈≠↵× °±°≡→±≥⇔≥°←♣∫ ⋅≡ ≥↔×←•°×≈ ←→↵″↑•←•±″↑ ±↔×… ≡•→≈ ≡±→ ←÷≈ ↑↵≥≈ ≥≈↑↑↵≠≈ →≈↓↔≈↑← ←÷≈ …≈↑•≠″≈→ ↑÷±↔×… →≈↑±×♠≈ ←÷•↑ ″±″…≈←≈→≥•″•↑≥ ♣ °×↵•″≠ ≠↔↵→…↑ ≥↔←↔↵××♣ ≈♦×↔↑•♠≈ ↵″… ≈♦÷↵↔↑←•♠≈ ±″…•←•±″↑ ±″ ←÷≈ ±″≡ו←•″≠ ←→↵″↑•←•±″↑ ↑↔÷ ↵↑ ±″ ←÷≈ ←♥±°±°←→↵″⌠ ↑•←•±″↑ ≡→±≥±→≥↵×∫ ⇑≈→←↵•″ …≈↑•≠″⌠≡±→⌠←≈↑← ±″…•←•±″↑ ≥↵♣ ↵°°×♣ ←± ≈″↑↔→≈ ←÷↵← ←÷≈ ≥↵÷•″≈ ↵″ ≈ …→•♠≈″ …≈←≈→≥•″•↑←•↵××♣ ←÷→±↔≠÷ ↵×× ±≡ •←↑ ↑←↵←≈↑ ↵″… ←→↵″↑•←•±″↑ ⊕℘∫ ∨±→ ≈♦↵≥°×≈ •″ ±→…≈→ ←± ∝″±♥ ♥÷≈″ ←÷≈ ≡•″↵×°±°←→↵″↑•←•±″ ≡→±≥±→≥↵×←±⇔≥°←♣•↑ →≈↵÷≈… ←÷≈ ↵≈↑↑±→ ↑•ƒ≈ •↑ →≈↓↔•→≈… ↵↑ ±″≈ ±≡ ⊆←↵∝Ž↑ ≥≈←÷±…↑ ↑± ←÷↵← ↑↔•←↵×≈ ≠↔↵→…↑ ≥↵♣ ≈ ±″↑←→↔←≈…∫ ⊂÷≈ ↵±♠≈ …≈↑•≠″ •↑ ×±≠•↵××♣±≥°×≈←≈ •″ ←÷≈ ↑≈″↑≈ ←÷↵← ←÷≈ ≥±…↵× ≈÷↵♠•±↔→ ±≡ ≈♠≈→♣ ≥≈←÷±… ≡→±≥ ≈♠≈→♣ ↑←↵←≈ •↑ ∝″±♥″∫

∫ ⇔×≈≥≈″←↑ ±≡ ←÷≈ ⊆←↵←≈ ±…≈×

⋅″ ≡±→≥↵× ↵°°→±↵÷≈↑ ←± ↑←↵←≈⌠↵↑≈… ←≈↑← ≠≈″≈→↵←•±″ ←≈↑←⌠↑≈←↑ ↵→≈ ±″↑←→↔←≈… ≡→±≥ ×↵″≠↔↵≠≈↑↑≈↓↔≈″≈↑ ±≡ ↑♣≥±×↑ …→↵♥″ ≡→±≥ ←÷≈ ←±←↵×↵×°÷↵≈←±≡ ←→↵″↑•←•±″ ×↵≈×↑∫ ⊂÷≈↵×°÷↵≈←•↑ ←÷≈ ↑≈← ±≡ ≥≈←÷±…↑ ≥Ρ ←÷↵← ↵″ ≈ ↵××≈… ±″ ←÷≈ •″←≈→≡↵≈ ±≡ ←÷≈ ±∂≈← •″×↔…•″≠ ↵″♣ •″÷≈→•←≈… ≥≈←÷±…↑∫ ∨±→ ←÷≈⊆←↵∝↑÷±♥″ •″ ≡•≠↔→≈  ←÷≈ ↵×°÷↵⌠ ≈←   ∞°↔↑÷°±°←±°≈≥°←♣↑•ƒ≈≤∫ ±←≈ ←÷↵←″≈♥•↑ ″±← ←≈÷″•↵××♣ •″ ←÷≈ ≥≈←÷±…⌠ •″←≈→≡↵≈ ±≡⊆←↵∝ ↔← →↵←÷≈→ ±″↑←→↔←↑ ←÷≈⊆←↵∝•″↑←↵″≈∫ ⊂÷≈ ±∂≈← →≈↑°±″…↑ ←± ↵×× ≥Ρ ↵″… ←± ″± ±←÷≈→ ≥≈←÷±…↑ ♥÷•÷ ↵→≈ →↔×≈… ±↔← ♣ ±≥°•×≈⌠←•≥≈ ↑♣″←↵♦ ÷≈∝⌠ •″≠∫ ⊂÷•↑ °↔←↑ ↵ ↔↑≈≡↔× ↔°°≈→ ±↔″… ±″ ←÷≈ ↑±°≈ ±≡ ″≈≠↵←•♠≈ ←≈↑←•″≠∫

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…≈←≈← ♥÷≈←÷≈→ ←÷≈ ∇⊄⊂ •↑ •″ ←÷↵← ↑←↵←≈∫ ∠→≈…•↵←≈↑ ≥↵♣ ≡→≈≈×♣ ↔↑≈ °↔ו ↵≈↑↑ ≥≈←÷⌠ ±…↑ •″←≈→″↵××♣ ≡±→ ≈♦↵≥°×≈•↑⇔≥°←♣↑ ↑∫↑•ƒ≈  ð∫ ∇↔→ ≈↵→ו≈→ …≈≡•″•←•±″ ±≡ ↑←↵←≈ ↑≈≈ ∫ ≈″↑↔→≈↑ ←÷↵← ↑↔÷ °→≈…•↵←≈↑ ≥↵♣ ↵×♥↵♣↑ ≈ …≈≡•″≈…∫

⊆≈↓↔≈″≈↑ ±≡ ≥≈←÷±…↑ …≈″±←≈… ≥ ≥ › ≥Ρ ≥↵♣ ≈ ±″↑←→↔←≈…∫ ↵″⌠

≠↔↵≠≈↑ð   › ↵→≈ ↑≈←↑ ±≡ ↑≈↓↔≈″≈↑ ±≡ ↑°≈•≡• ×≈″≠←÷↑∑ ←÷↵← •↑ ð•↑ ←÷≈ ↑≈← ±≡ ƒ≈→±⌠×≈″≠←÷ ↑≈↓↔≈″≈↑ ∞≤ ↵″… •↑ ←÷≈ ↑≈← ±≡ ↵×× ↔″•←⌠×≈″≠←÷ ↑≈↓↔≈″≈↑ ∞≥

≥ ›≤ ≈←∫ ⊂÷≈ •″≡•″•←≈ ×↵″≠↔↵≠≈  •↑ ←÷≈ ↔″•±″  ð

ΞΞΞ› ±″←↵•″⌠ •″≠ ↵×× ↵→•←→↵→♣⌠×≈″≠←÷ ↑≈↓↔≈″≈↑∫ ⇓ °→≈…•↵←≈ ×↵″≠↔↵≠≈ ∠  ∞° ° ›≤ •↑ ↵ ↑≈←

±≡ ↔″•←⌠×≈″≠←÷ °→≈…•↵←≈ ↑≈↓↔≈″≈↑ ←≈↑←•″≠ ≈♦÷↵↔↑←•♠≈×♣ ≡±→ ≈↵÷ ↑←↵←≈ ↑Ρ⊆∫

∫ ⇑±≥°×≈←≈ ⊄″•←⌠⊂≈↑← ∧≈″≈→↵←•±″

⊃÷≈″ ←≈↑←•″≠ ≡→±≥ ↵ ↑←↵←≈⌠↵↑≈… …≈↑•≠″ ←÷≈ ←≈↑←≈→ …→•♠≈↑ ←÷≈ ∇⊄⊂ •″←± ↵×× ±≡ •←↑ ↑←↵←≈↑ ↵″… ←÷≈″ ↵←←≈≥°←↑ ≈♠≈→♣ °±↑↑•×≈ ←→↵″↑•←•±″ ±←÷ ≈♦°≈←≈… ↵″… ↔″♥↵″←≈… ≡→±≥ ≈↵÷ ↑←↵←≈ ÷≈∝•″≠ ↵≡←≈→♥↵→…↑ ♥÷•÷ …≈↑←•″↵←•±″ ↑←↵←≈↑ ♥≈→≈ →≈↵÷≈…∫ ⊂÷≈ ∇⊄⊂ ↑÷±↔×… ≈♦÷••← •″…•↑←•″≠↔•↑÷↵×≈ ≈÷↵♠•±↔→ ≡→±≥ ←÷≈ …≈↑•≠″ ←± °↵↑↑ ←÷≈ ←≈↑←↑∫ ⋅← •↑ ↵↑↑↔≥≈… ←÷↵← ←÷≈ …≈↑•≠″ •↑ ↵≥•″•≥↵×↑←↵←≈ ≥↵÷•″≈ ♥•←÷ ″± …↔°×•↵←≈ ±→ →≈…↔″…↵″← ↑←↵←≈↑ ↔← ←÷≈ ←≈↑←≈… •≥°×≈≥≈″←↵←•±″ ≥↵♣ ≈ ″±″⌠≥•″•≥↵× ♥•←÷ ≥±→≈ ←÷↵″ ←÷≈ ≈♦°≈←≈… ↑←↵←≈↑∫ ⊂≈↑←•″≠ ≥↔↑← ←÷≈→≈≡±→≈ ←↵∝≈ ←÷•↑ •″←± ↵±↔″← ≈♦≈→•↑•″≠ ≥±→≈ ←÷↵″ ←÷≈ ≥•″•≥↵× ↑←↵←≈ ≥↵÷•″≈∫ ⊂÷≈↑≈ ″±←•±″↑ ↵→≈ ≡±→≥↵ו↑≈… ≈×±♥∫

⊂÷≈↑←↵←≈ ±♠≈→•↑ ↵ ↑≈← ⇑Σ ±″↑•↑←•″≠ ±≡ ←÷≈ ↑÷±→←≈↑← ↑≈↓↔≈″≈↑ ←÷↵← ♥•×× …→•♠≈ ←÷≈ ∇⊄⊂ •″←± ↵×× ±≡ •←↑ ↑←↵←≈↑∫ ⇑ •↑ ÷±↑≈″ ♣ •″↑°≈←•±″ ±→ ♣ ↵↔←±≥↵←• ≈♦°×±→↵←•±″ ±≡ ←÷≈ ≥±…≈×∫ ∨±→ ←÷≈⊆←↵∝↑÷±♥″ •″ ≡•≠↔→≈  ↵±♠≈ ⇑  ∞ °↔↑÷≤ •↑ ←÷≈ ↑≥↵××⌠ ≈↑← ↑←↵←≈ ±♠≈→ ♥÷•÷ ♥•×× ≈″←≈→ ←÷≈⇔≥°←♣↵″…±→≥↵×↑←↵←≈↑∫ ⇓″ •″•←•↵× ←≈↑←⌠↑≈← ⊂ð ↵•≥↑ ←± →≈↵÷ ↵″… ←÷≈″ ♠≈→•≡♣ ≈♠≈→♣ ↑←↵←≈∫ ⊇≈→•≡•↵←•±″ •↑ ↵±≥°×•↑÷≈… ♣ •″♠±∝•″≠ ≈↵÷ °→≈…•↵←≈ •″ ←÷≈ °→≈…•↵←≈ ×↵″≠↔↵≠≈ ∠ ↵≡←≈→ ≈♦°×±→•″≠ ≈↵÷ °↵←÷ •″ ←÷≈ ↑←↵←≈ ±♠≈→ ⇑ ↵ ←≈↑← ↑≈← …≈″±←≈… ♣ ⇑∴ ∠ ←÷≈ ±″↵←≈″↵←≈… °→±…↔←←÷↵← ↵°°≈″…↑ ≈♠≈→♣ ↑≈⌠ ↓↔≈″≈ •″ ∠ ←± ≈♠≈→♣ ↑≈↓↔≈″≈ •″ ⇑∫ ⇔♦↵←×♣ ±″≈ °→≈…•↵←≈ •″ ≈↵÷ ↑≈↓↔≈″≈ ↑÷±↔×… →≈←↔→″ ←→↔≈ ↵″… ↵×× ←÷≈ ±←÷≈→↑ ≡↵×↑≈ ↵↑ …≈←≈→≥•″≈… ≡→±≥ ←÷≈ ≥±…≈×∫

⊂ð ⇑∴∠ 

⇓ ≥±→≈ ↑±°÷•↑←•↵←≈… ←≈↑←⌠↑≈← ⊂ ↵•≥↑ ←± →≈↵÷ ≈♠≈→♣ ↑←↵←≈ ↵″… ↵×↑± ≈♦≈→•↑≈ ≈♠≈→♣ ↑•″≠×≈ ≥≈←÷±… •″ ≈♠≈→♣ ↑←↵←≈∫ ⊂÷•↑ •↑ ±″↑←→↔←≈… ≡→±≥ ←÷≈←→↵″↑•←•±″ ±♠≈→↵ ↑≈← ±≡ ↑≈↓↔≈″≈↑ ∏ ⇑Ξ⇑∴ ♥÷•÷ •″×↔…≈↑ ←÷≈ ↑←↵←≈ ±♠≈→ ⇑ ↵″… ←÷≈ ±″↵←≈″↵←≈… °→±…↔← ←≈→≥ ⇑∴ …≈″±←•″≠ ←÷≈ ↵←←≈≥°←≈… ≡•→•″≠ ±≡ ≈♠≈→♣ ↑•″≠×≈ ←→↵″↑•←•±″ ≡→±≥ ≈♠≈→♣ ↑←↵←≈∫ ⊂÷≈ ↑←↵←≈↑ →≈↵÷≈… ♣ ←÷≈ ←→↵″↑•←•±″ ±♠≈→ ↵→≈ ♠↵ו…↵←≈… ↵≠↵•″ ↔↑•″≠ ↵×× ↑•″≠×≈←±″ °→≈…•↵←≈ ↑≈↓↔≈″≈↑ °Ρ∠∫

⊂ ⇑Ξ⇑∴∴∠ 

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⊂ ⇑Ξ⇑∴Ξ⇑∴∴∠  ⋅″ ↵ ↑•≥•×↵→ ≡↵↑÷•±″ ≡↔→←÷≈→ ←≈↑←⌠↑≈←↑ ↵→≈ ±″↑←→↔←≈… ≡→±≥ ←÷≈ ↑←↵←≈ ±♠≈→ ⇑ ↵″… ×±♥⌠ ±→…≈→ ×↵″≠↔↵≠≈↑ ∝Σ∫ ⇔↵÷ ←≈↑←⌠↑≈← ↑↔↑↔≥≈↑ ←÷≈ ↑≥↵××≈→ ←≈↑←⌠↑≈←↑ ±≡ ×≈↑↑≈→ ↑±⌠ °÷•↑←•↵←•±″ •″ ←÷≈ ↑≈→•≈↑∫ ⋅″ ≠≈″≈→↵× ←÷≈ ↑≈→•≈↑ ↵″ ≈ ≡↵←±→•↑≈… ↵″… ≈♦°→≈↑↑≈… ≡±→ ←≈↑←⌠↑≈←↑ ±≡ ↵→•←→↵→♣ ↑±°÷•↑←•↵←•±″ ↵↑

⊂∝ ⇑∴ðΞΞ∫∫∫ ∝∴∠  ⊂÷≈ ≠≈″≈→↵× ≡±→≥↔×↵ …≈↑→•≈↑ ↵×× ←≈↑←⌠↑≈↓↔≈″≈↑ ↑←↵→←•″≠ ♥•←÷ ↵ ↑←↵←≈ ±♠≈→ ↵↔≠⌠ ≥≈″←≈… ♣ ≡•→•″≠ ↵×× ↑•″≠×≈ °↵•→↑ ←→•°×≈↑ ≈←∫ ±≡ ←→↵″↑•←•±″↑ ↵″… ←÷≈″ ♠≈→•≡♣•″≠ ←÷≈ →≈↵÷≈… ↑←↵←≈↑ ↔↑•″≠ ↑←↵←≈ °→≈…•↵←≈↑∫

⊂≈↑← ⇑±≥°×≈←≈″≈↑↑ ↵″… ∧↔↵→↵″←≈≈↑

⊂÷≈ ←≈↑←⌠↑≈←↑ °→±…↔≈… ♣ ←÷•↑ ↵×≠±→•←÷≥ ÷↵♠≈ •≥°±→←↵″← ±≥°×≈←≈″≈↑↑ °→±°≈→←•≈↑∫ ∨±→ ≈↵÷ ♠↵×↔≈ ±≡ ∝ ↑°≈•≡• ≠↔↵→↵″←≈≈↑ ↵→≈ ±←↵•″≈… ↵±↔← ←÷≈ •≥°×≈≥≈″←↵←•±″ ±″≈ ←≈↑←⌠ •″≠ •↑ ±♠≈→∫ ⇒≈×±♥ ♥≈ ↑÷±♥ ÷±♥ →≈≠→≈↑↑•±″ ←≈↑←•″≠ •″ ∪∠ …±≈↑ ″±← °→±♠•…≈ ←÷≈ ↑↵≥≈ ≠↔↵→↵″←≈≈↑ ↵≡←≈→ →≈←≈↑←•″≠∫ ¬±♥≈♠≈→ ≡±→ ↑•≥°×≈ ≈♦←≈″↑•±″↑ ←± ↵ ↑←↵←≈⌠↵↑≈… …≈↑•≠″ ←≈↑←↑ ≥↵♣ ≈ →≈⌠≠≈″≈→↵←≈… ♣ ←÷•↑ ↵×≠±→•←÷≥ ↵″… ↵×× ←÷≈ ↑↵≥≈ ≠↔↵→↵″←≈≈↑ ↔°÷≈×…∫

∫ ∧↔↵→↵″←≈≈↑ ↵≡←≈→ ⊂≈↑←•″≠

⊂÷≈ ↑≈← ⊂ð≠↔↵→↵″←≈≈↑ ←÷↵← ←÷≈ •≥°×≈≥≈″←↵←•±″ ÷↵↑↵← ×≈↵↑←↵×× ←÷≈ ↑←↵←≈↑ •″ ←÷≈ ↑°≈•≡•⌠ ↵←•±″∫ ⊂÷≈ ↑≈← ⊂≠↔↵→↵″←≈≈↑ ←÷•↑ ↵″… ←÷↵← ↵≥•″•≥↵× •≥°×≈≥≈″←↵←•±″ °→±♠•…≈↑ ≈♦⌠ ↵←×♣ ←÷≈ …≈↑•→≈… ↑←↵←≈⌠←→↵″↑•←•±″ ≈÷↵♠•±↔→∫ ⊂÷≈ →≈≥↵•″•″≠ ←≈↑←⌠↑≈←↑ ⊂∝°→±♠•…≈ ←÷≈ ↑↵≥≈ ≠↔↵→↵″←≈≈↑ ≡±→ ″±″⌠≥•″•≥↵× •≥°×≈≥≈″←↵←•±″↑ ↔″…≈→ ♥≈↵∝≈″•″≠ ↵↑↑↔≥°←•±″↑ ↵±↔← ←÷≈ ×≈♠≈× ±≡ …↔°×•↵←•±″ •″ ←÷≈ ↑←↵←≈↑ ↵″… ←→↵″↑•←•±″↑∫

⇓ ″±″⌠≥•″•≥↵× ±→→≈…↔″…↵″←•≥°×≈≥≈″←↵←•±″ •↑ ±″≈ ♥÷≈→≈ ↵ °→±≠→↵≥≥≈→ ÷↵↑ •″↵…⌠ ♠≈→←≈″←×♣ •″←→±…↔≈… ≈♦←→↵ •≠÷±↑←Œ ↑←↵←≈↑ ♥÷•÷ ≥↵♣ ±→ ≥↵♣ ″±← ≈ ≡↵•←÷≡↔× ±°•≈↑ ±≡ ↑←↵←≈↑ …≈↑•→≈… •″ ←÷≈ …≈↑•≠″∫ ⊂≈↑← ↑≈↓↔≈″≈↑ ≥↵♣ ×≈↵… •″←± ←÷≈↑≈ •≠÷±↑←Œ ↑←↵←≈↑ •≡ ←÷≈♣ ≈♦•↑← ↵″… ←÷≈ ∇⊄⊂ ≥↵♣ ←÷≈″ ≈÷↵♠≈ •″ ↑↔←×≈ ↔″≈♦°≈←≈… ♥↵♣↑ ≈♦÷••←•″≠ ≈♦←→↵ ±→ ≥•↑↑•″≠ ←→↵″↑•←•±″↑ ±→ →≈↵÷•″≠ ↔″≈♦°≈←≈… …≈↑←•″↵←•±″ ↑←↵←≈↑∫ ⇔↵÷ ←≈↑←⌠↑≈← ⊂∝°→±⌠ ♠•…≈↑ ±≥°×≈←≈ ±″≡•…≈″≈ ≡±→ ↑♣↑←≈≥↑ •″ ♥÷•÷ ÷↵•″↑ ±≡ …↔°×•↵←≈… ↑←↵←≈↑ …± ″±← ≈♦≈≈… ×≈″≠←÷ ∝⌠∫ ∨±→ ↑≥↵×× ♠↵×↔≈↑ ±≡ ∝ ↑↔÷ ↵↑ ∝ •← •↑ °±↑↑•×≈ ←± ÷↵♠≈ ↵ ♠≈→♣ ÷•≠÷ ×≈♠≈× ±≡ ±″≡•…≈″≈ •″ ←÷≈ ±→→≈← ↑←↵←≈⌠←→↵″↑•←•±″ ≈÷↵♠•±↔→ ±≡ ≈♠≈″ ↓↔•←≈ °≈→⌠ ♠≈→↑≈×♣⌠↑←→↔←↔→≈… •≥°×≈≥≈″←↵←•±″↑∫

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↑↵≥≈ ↑←↵←≈↑ ↵″… ←→↵″↑•←•±″↑ ±→ •≡ •← ÷↵↑ ≈♦←→↵ →≈…↔″…↵″← ↑←↵←≈↑ ↵″… ←→↵″↑•←•±″↑ ←÷≈″ ←÷≈↑≈ ↵→≈ ↑≈≥↵″←•↵××♣ •…≈″←•↵× …↔°×•↵←≈↑ ±≡ ←÷≈ •″←≈″…≈… ↑←↵←≈↑ •″ ←÷≈ ↑°≈•≡•↵←•±″∫

∫ ∈≈≡↵←±→•″≠ ⊆↔×↵↑↑•″≠ ↵″… ⊂≈↑← ⇑±♠≈→↵≠≈

∨•≠↔→≈  •××↔↑←→↵←≈↑ ←÷≈ ↑←↵←≈ ≥↵÷•″≈ ≡±→ ↵⇐♣″↵≥•⊆←↵∝ ↵″ ↵→→↵♣⌠↵↑≈… •≥°×≈≥≈″←↵⌠ ←•±″ ±≡ ↵⊆←↵∝∫ ⊃≈ ≥↵♣ ←÷•″∝ ±≡ ←÷•↑ ≈•←÷≈→ ↵↑ ↵ ↑↔×↵↑↑ …≈↑•≠″ ≡±→ ↵ ±″→≈←≈ ×↵↑↑ ←÷↵← •≥°×≈≥≈″←↑ ↵″ ↵↑←→↵←⊆←↵∝•″←≈→≡↵≈ ⊕℘ ±→ ↵↑ ↵ →≈≡↵←±→≈… …≈↑•≠″ ≡±→ ↵⊆←↵∝ ↵≡←≈→ ←÷≈ …≈•↑•±″ •↑ ←↵∝≈″ ←± ↑♥•←÷ ←± ↵″ ↵→→↵♣⌠↵↑≈… •≥°×≈≥≈″←↵←•±″∫ ⊂÷≈ ≥↵•″ …•≡⌠ ≡≈→≈″≈ ≈←♥≈≈″ ←÷•↑ ↵″… ←÷≈ ≈↵→ו≈→ ≥↵÷•″≈ •″ ≡•≠↔→≈  •↑ ←÷↵← ←÷≈ ±×…±→≥↵×↑←↵←≈ ″±♥ ±″×♣ ↑÷±♥″ ↵↑ ↵ …↵↑÷≈… →≈≠•±″ ÷↵↑ ≈≈″ °↵→←•←•±″≈… •″←± ←÷≈ ↑←↵←≈↑ ∞±↵…≈… ∨↔××≤ •″ ±→…≈→ ←± ↵°←↔→≈ ←÷≈ …•↑←•″← ≈÷↵♠•±↔→ ±≡°↔↑÷•″ ←÷≈∨↔××↑←↵←≈ ♥÷≈→≈ ←÷•↑ ←→•≠≠≈→↑ ↵ ≥≈≥±→♣ →≈↵××±↵←•±″∫ ⊂÷±↔≠÷ ←÷•↑ …≈↑•≠″ ↵°°≈↵→↑ ≥±→≈ ±≥°×≈♦ •← •↑ ≈↵↑•×♣ ≠≈″≈→↵←≈… ♣ ≡±××±♥•″≠ ←÷≈ ↑•≥°×≈ →↔×≈ ↑÷±♥ ←÷≈ ≈÷↵♠•±↔→ ±≡≈♠≈→♣≥≈←÷±… •″≈♠≈→♣ ↑←↵←≈ ↑≈×≡⌠←→↵″↑•←•±″↑ ≡±→ ←÷≈ ↵≈↑↑ ≥≈←÷±…↑ ↵→≈ •″≡≈→→≈… ↵↑ ↵±♠≈∫

⇔≥ °←♣

↑ •ƒ≈   ð

±↵ …≈ …

↑ •ƒ≈  ″ ° ↔ ↑÷ ≈ 

° ↔ ↑÷ ≈  ⊕↑ •ƒ≈   ″ ⌠℘

° ±°  ⊕↑ •ƒ≈    ℘ °± °  ⊕↑ •ƒ≈    ℘ °± ° 

″≈ ♥

∨↔××

↑ •ƒ≈   ″ ° ↔ ↑÷ ≈  ⊕↑•ƒ≈  ″ ⌠℘

° ± ° 

°↔ ↑÷ ≈  〉 →≈ ↑ •ƒ≈ 

∨•≠∫ ∫ ⊆←↵←≈ ≥↵÷•″≈ ≡±→ ↵⇐♣″↵≥•⊆←↵∝ ♥÷•÷ ±″≡±→≥↑ ←± ←÷≈ ≈÷↵♠•±↔→ ±≡ ←÷≈⊆←↵∝•″ ≡•≠∫ ∫ ⊂÷≈ ←♥± ↑←↵←≈↑ ±↵…≈… ∨↔×× °↵→←•←•±″ ←÷≈ ±×…±→≥↵×↑←↵←≈ •″ ≡•≠∫  →≈↑↔×←•″≠ •″ ←÷≈ →≈°×•↵←•±″ ±≡ •←↑ ←→↵″↑•←•±″↑∫ ⊂÷≈ ≈÷↵♠•±↔→ ±≡°↔↑÷•″ ←÷≈∨↔××↑←↵←≈ ≥↔↑← ≈ ←≈↑←≈…∫

⋅″→≈↵↑•″≠ ←÷≈ ↑←↵←≈⌠↑°↵≈ ÷↵↑ •≥°±→←↵″← •≥°×•↵←•±″↑ ≡±→ ←≈↑← ≠↔↵→↵″←≈≈↑∫ ⇑±″↑•…≈→ ←÷≈ ↑↔≡≡••≈″♣ ±≡ ←÷≈ ⊂←≈↑←⌠↑≈← ≠≈″≈→↵←≈… ≡→±≥ ←÷≈ ↵↑←→↵←⊆←↵∝↑°≈•≡•↵←•±″ •″ ≡•≠↔→≈ ∫ ⊂÷•↑ →±↔↑←×♣ ≠↔↵→↵″←≈≈↑ ←÷≈ ±→→≈← ≈÷↵♠•±↔→ ±≡ ↵ ↑•≥°×≈•″∝≈…⊆←↵∝•≥°×≈≥≈″⌠ ←↵←•±″ ♥•←÷ ⊆  ∞⇔≥°←♣ ±→≥↵×≤ ≈♠≈″ •″ ←÷≈ °→≈↑≈″≈ ±≡ •≠÷±↑←Œ ↑←↵←≈↑∫ ⊂♥•×× ≠≈″⌠ ≈→↵←≈ ±″≈ ↑≈↓↔≈″≈ °↔↑÷ °↔↑÷ °↔↑÷ •↑±→≥↵× ♥÷•÷ →±↔↑←×♣ ≈♦≈→•↑≈↑ °↔↑÷ °↔↑÷ ≡→±≥ ←÷≈±→≥↵×↑←↵←≈ ↵″… ♥•×× ≈♠≈″ …≈←≈← ↵ •≠÷±↑←Œ ±°♣ ±≡ ←÷≈±→≥↵×↑←↵←≈∫

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⋅″ ≠≈″≈→↵× ↑↔×↵↑↑≈↑ ÷↵♠≈ ↵ ×↵→≠≈→ ↑←↵←≈⌠↑°↵≈ ←÷↵″ ↑↔°≈→×↵↑↑≈↑ …↔≈ ←± ←÷≈•→ •″⌠ ←→±…↔←•±″ ±≡ ≥±→≈ ≥≈←÷±…↑ ↵≡≡≈←•″≠ ±∂≈← ↑←↵←≈↑∫ ⋅″ ⇑±±∝ ⇐↵″•≈×↑ ⊕℘ ↵″… ∧→≈≠±→Ž↑ ⊕ ℘ ↑←↵←≈ ≥±…≈×↑ ↑↔×↵↑↑•″≠ →≈↑↔×←↑ •″ ←÷≈ •″←→±…↔←•±″ ±≡ ″≈♥ ↑←↵←≈↑∫ ⋅″ ⊆•≥±″↑Ž •≥°→±♠≈… ≥±…≈× ±≡ ↑←↵←≈ →≈≡•″≈≥≈″← ⊕ ℘ ↑↔×↵↑↑•″≠ •↑ ↑÷±♥″ ↵×♥↵♣↑ ←± →≈↑↔×← •″ ←÷≈↑↔…•♠•↑•±″ ±≡ ≈♦•↑←•″≠ ↑←↵←≈↑↵↑ •″ ≡•≠↔→≈ ∫ ¬≈→≈ ±↔→ ″≈♥ ≡•″…•″≠ •↑ ←÷↵← ↑↵♠≈… ←≈↑←↑ ♥•×× ″±← ±″×♣ ≥•↑↑ ↑←↵←≈↑ ≡±→≥≈… •″ ←÷≈ ÷•×… ↔← ♥•×× ↵×↑± ≡↵•× ←± ←≈↑← ↵×× ±≡ ←÷≈ ←→↵″↑•←•±″↑ ←÷↵← ♥≈→≈ ←≈↑←≈… •″ ←÷≈ °↵→≈″← ↑•″≈ ←÷≈↑≈ ÷↵♠≈ ≈±≥≈ °↵→←•←•±″≈…∫ ⊂÷•↑ •↑ •≥°±→←↵″← ≡±→ →≈≠→≈↑↑•±″ ←≈↑←•″≠ ↑•″≈ •← ≥≈↵″↑ ←÷↵← ↑↵♠≈… ←≈↑←↑ ≈♦≈→•↑≈↑•≠″•≡•↵″←×♣ ×≈↑↑±≡ ←÷≈↑↵≥≈ ≈÷↵♠•±↔→•″ ←÷≈ ÷•×… ↵↑ ←÷≈♣ …•… •″ ←÷≈ °↵→≈″←∫

⊂± ↵÷•≈♠≈ ←÷≈ ↑↵≥≈ ×≈♠≈× ±≡ ±♠≈→↵≠≈ •← •↑ ″±← ↑↔≡≡••≈″← ←± ↑↔°°×♣ ≈♦←→↵ ←≈↑←↑ ≡±→ ←÷≈ ″≈♥ ≥≈←÷±…↑ ↔← →↵←÷≈→ •← •↑ ♠•←↵× ←± ←≈↑←↵×× ←÷≈ •″←≈→×≈↵♠•″≠↑±≡ ″≈♥ ≥≈←÷±…↑ ♥•←÷ ←÷≈ •″÷≈→•←≈… ≥≈←÷±…↑ ↑± ≈♦°×±→•″≠ ←÷≈ ↑←↵←≈⌠←→↵″↑•←•±″ …•↵≠→↵≥ ±≥°×≈←≈×♣∫ ⊂÷•↑ ↑•≥°×♣ ↵″″±← ≈ …±″≈ →≈ו↵×♣ ♣ ÷↔≥↵″ •″←↔•←•±″ ↵″… ≥↵″↔↵× ←≈↑←⌠↑→•°← →≈↵←•±″∫ ¬±♥≈♠≈→ •← ±↔×… ≈ …±″≈ ↵↔←±≥↵←•↵××♣ ↵″… →≈ו↵×♣ •≡ ←÷≈ ←≈↑←⌠↑≈←↑ ♥≈→≈→≈≠≈″≈→↵←≈… ≡±→ ←÷≈ ≥±…•≡•≈… ↑←↵←≈ ≥↵÷•″≈ …≈↑•≠″↑ ↔↑•″≠ ←÷≈ ↵×≠±→•←÷≥ ≡→±≥ ↑≈←•±″ ∫ ↵±♠≈∫

⇑±″×↔↑•±″↑

⊂÷≈ ↑←→≈″≠←÷ ±≡ ←÷≈ ≠↔↵→↵″←≈≈ ±←↵•″≈… ↵≡←≈→ →≈≠→≈↑↑•±″ ←≈↑←•″≠ •↑ ±♠≈→≈↑←•≥↵←≈… •″ ∪∠∫ ∈≈♣×≈… ←≈↑←⌠↑≈←↑ ↵×♥↵♣↑ ≈♦≈→•↑≈ ↑•≠″•≡•↵″←×♣ ×≈↑↑ ±≡ ←÷≈ →≈≡•″≈… ±∂≈← ←÷↵″ ←÷≈ ±→•≠•″↵× ↑↔÷ ←÷↵← →≈⌠←≈↑←≈… ±∂≈←↑ ≥↵♣ ≈ ±″↑•…≈→↵×♣ ×≈↑↑ ↑≈↔→≈ ≡±→ ←÷≈ ↑↵≥≈ ←≈↑←•″≠ ≈≡≡±→←∫ ⇓↑ ←÷≈ ↑←↵←≈⌠↑°↵≈ ±≡ ←÷≈ ≥±…•≡•≈… ±→ ≈♦←≈″…≈… ±∂≈← •″→≈↵↑≈↑ ←÷≈ ≠↔↵→↵″←≈≈ ±≡≡≈→≈… ♣ →≈←≈↑←•″≠ •↑ °→±≠→≈↑↑•♠≈×♣ ♥≈↵∝≈″≈…∫ ⊂÷•↑ ↔″…≈→≥•″≈↑ ←÷≈ ♠↵ו…⌠ •←♣ ±≡ °±°↔×↵→ →≈≠→≈↑↑•±″ ←≈↑←•″≠ ↵°°→±↵÷≈↑ ↑↔÷ ↵↑ °↵→↵××≈× …≈↑•≠″⌠↵″…⌠←≈↑← ←≈↑← ↑≈← •″÷≈→•←↵″≈ ↵″… →≈↔↑≈ ±≡ ↑↵♠≈… ←≈↑← ↑→•°←↑ •″ √⊄″•←∫

⇒♣ ±≥°↵→•↑±″ •≡ ±≥°×≈←≈ ←≈↑←⌠↑≈←↑ ↵→≈ ↵×♥↵♣↑ ≠≈″≈→↵←≈… ≡→±≥ ↑•≥°×≈ ↑←↵←≈ ≥↵⌠ ÷•″≈ …≈↑•≠″↑ •← •↑ °±↑↑•×≈ ←± °→±♠•…≈ ↑°≈•≡• ≠↔↵→↵″←≈≈↑ ≡±→ ≈♦↵≥°×≈ ↔° ←± ←÷≈ ∝ ±→ ∝ ±″≡•…≈″≈ ×≈♠≈× •″ ←÷≈ ∇⊄⊂∫ ⇓≡←≈→ →≈←≈↑←•″≠ ↵ ↑↔×↵↑↑≈… ±→ →≈≡↵←±→≈… ∇⊄⊂ ←÷≈ ↑↵≥≈ ≠↔↵→↵″←≈≈↑ ≥↵♣ ≈ ↔°÷≈×… ♣ ≠≈″≈→↵←•″≠ ≡→±≥ ←÷≈ →≈♠•↑≈… ↑←↵←≈ ≥↵÷•″≈ ←± ←÷≈ ↑↵≥≈ ±″≡•…≈″≈ ×≈♠≈×∫ ±← ±″×♣ ←÷•↑ ↔←↵××←÷≈ ±∂≈←↑ •″ ↵ ↑±≡←♥↵→≈ °→±∂≈← ≥↵♣ ≈ ↔″•←⌠←≈↑←≈… ←± ↵ ≠•♠≈″ ±″≡•…≈″≈ ×≈♠≈×∫ ⊂÷•↑ ″±←•±″ ±≡ ≠↔↵→↵″←≈≈… →≈°≈↵←↵×≈ ↓↔↵ו←♣•↑ ↵ ″≈♥ ↵″… •≥°±→←↵″← ±″≈°← •″ ±∂≈←⌠±→•≈″←≈… ←≈↑←•″≠∫

⊂÷•↑ →≈↑≈↵→÷ ♥↵↑ ↔″…≈→←↵∝≈″ ↵↑ °↵→← ±≡ ←÷≈ ∇⊂⋅⊇⇔ °→±∂≈← ↑↔°°±→←≈… ♣ ⊄∏ ⇔∠⊆∈⇑ ∧∈〉∫

∈≈≡≈→≈″≈↑

∫ ⊃≈××↑ ⇐∫ ⊂÷≈ ∈↔×≈↑ ↵″… ∠→↵←•≈↑ ±≡ ⇔♦←→≈≥≈ ∠→±≠→↵≥≥•″≠∫ ⊃≈↑•←≈ ↵″… ÷♣°≈→←≈♦← ↵→←•×≈

÷←←°〉〉♥♥♥∫≈♦←→≈≥≈°→±≠→↵≥≥•″≠∫±→≠〉→↔×≈↑∫÷←≥×

∫ ⇒≈∝ ∏∫ ⇔♦←→≈≥≈ ∠→±≠→↵≥≥•″≠ ⇔♦°×↵•″≈… ⇔≥→↵≈ ⇑÷↵″≠≈∫ ⇓……•↑±″⌠⊃≈↑×≈♣ ≈♥ ∩±→∝ ððð

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∫ ∏→↔÷←≈″ ∠∫ ⊂÷≈ ∈↵←•±″↵× ⊄″•≡•≈… ∠→±≈↑↑ ↵″ ⋅″←→±…↔←•±″∫ →…≈…″∫ ⇓……•↑±″⌠⊃≈↑×≈♣ ∈≈↵…•″≠ ðð

∫ ∇∂≈← ↵″↵≠≈≥≈″← ∧→±↔° ⊄ ∈≈↑±↔→≈ ∠↵≠≈∫ ⊃≈↑•←≈ ÷←←°〉〉♥♥♥∫±≥≠∫±→≠〉↔≥×〉

ðð

∫ ⊆←≈°÷≈″↑ ∫ ↵″… ∈±↑≈″≈→≠ ⇐∫ ⇔♦←→≈≥≈ ∠→±≠→↵≥≥•″≠ ∈≈≡↵←±→≈… ⊂÷≈ ⇑↵↑≈ ⇓≠↵•″↑← ∪∠∫ ⇓°→≈↑↑ ⇒≈→∝×≈♣ ðð

∫ ⇒≈∝ ∏∫ ∧↵≥≥↵ ⇔∫ ≈← ↵×∫ ⊂÷≈ √⊄″•← ∠→±∂≈←∫ ⊃≈↑•←≈÷←←°〉〉♥♥♥∫∂↔″•←∫±→≠〉ðð ∫ ⊆←±←←↑ ⇐∫ •″…↑≈♣ ∫ ↵″… ⇓″←×≈♣ ⇓∫ ⇓″ ⋅″≡±→≥↵× ≈←÷±… ≡±→ ⊆♣↑←≈≥↵←• √⊄″•← ⊂≈↑←

⇑↵↑≈ ≠≈″≈→↵←•±″∫ ≈←↔→≈ ±←≈↑ •″ ⇑±≥°↔←≈→ ⊆•≈″≈ ⊇±×∫ ∫ ⊆°→•″≠≈→ ⊇≈→×↵≠ ⇒≈→ו″ ¬≈•…≈×≈→≠ ≈♥ ∩±→∝ ðð ⌠

∫ ∧→≈≠±→ √∫ ⇐∫ ↵″… ∏±→↑±″ ⊂∫ ⋅″←≈≠→↵←•″≠ ∇∂≈←⌠∇→•≈″←≈… ⊂≈↑←•″≠ ↵″… ⇐≈♠≈×±°≥≈″← ∠→±≈↑↑≈↑∫ ⇑±≥≥↔″•↵←•±″↑ ±≡ ←÷≈ ⇓⇑ ⊇±×∫  ±∫   ⌠

ð∫ ∧→≈≠±→ √∫ ⇐∫ ↵″… ∏↵→≈ ⇓∫ ∠↵→↵××≈× ⇓→÷•←≈←↔→≈ ≡±→ ⇑±≥°±″≈″← ⊂≈↑←•″≠ ±≡ ∇∂≈←⌠ ±→•≈″←≈… ⊆±≡←♥↵→≈∫ ∠→±∫ ←÷ ⇓″″↔↵× ⊆±≡←♥↵→≈ ∉↔↵ו←♣ ⊃≈≈∝ ⊆±≡←♥↵→≈ ∈≈↑≈↵→÷ ⋅″∫ ⊆↵″ ∨→↵″•↑± ↵♣ 

∫ ⊃≈××↑ ⇐∫ ⊄″•← ⊂≈↑←↑ ≈↑↑±″↑ ≈↵→″≈… •″ ⊂÷≈ ∈↔×≈↑ ↵″… ∠→↵←•≈↑ ±≡ ⇔♦←→≈≥≈ ∠→±≠→↵≥⌠ ≥•″≠∫ ¬♣°≈→←≈♦← ↵→←•×≈÷←←°〉〉♥♥♥∫≈♦←→≈≥≈°→±≠→↵≥≥•″≠∫±→≠〉→↔×≈↑〉↔″•←←≈↑←↑∫÷←≥× ∫ ⊆÷↔≥↵″ ⊆∫ ⇓∫ ↵″… ∠•←← ⇐∫ ¬∫ ∇∂≈←⌠±→•≈″←≈… ⊆↔↑♣↑←≈≥ ⊆°≈•≡•↵←•±″∫ ⋅″ ∠→±≠→↵≥

⊆°≈•≡•↵←•±″ ↵″… ⊂→↵″↑≡±→≥↵←•±″∫ ⇔×↑≈♠•≈→ ⊆•≈″≈ ±→←÷ ¬±××↵″… 

∫ ⇑÷±♥ ⊂∫ ⊂≈↑←•″≠ ⊆±≡←♥↵→≈ ⇐≈↑•≠″ ±…≈×≈… ♣ ∨•″•←≈ ⊆←↵←≈ ↵÷•″≈↑∫ ⋅⇔⇔⇔ ⊂→↵″↑↵←•±″↑ ±″ ⊆±≡←♥↵→≈ ⇔″≠•″≈≈→•″≠ ⊇±×∫  ±∫   ⌠

∫ ⇒•″…≈→ ∈∫ ⊇∫ ⊂≈↑←•″≠ ∇∂≈←⌠∇→•≈″←≈… ⊆♣↑←≈≥↑ ↵ ⊆←↵←↔↑ ∈≈°±→←∫ →… ≈…″∫ ¬♣°≈→←≈♦← ↵→←•×≈

÷←←°〉〉♥♥♥∫→↑∫±≥〉°↵≠≈↑〉±±↑←↵←∫÷←≥×ðð

∫ ∧→≈≠±→ √∫ ⇐∫ ⇑±″↑←→↔←•″≠ ∨↔″←•±″↵× ⊂≈↑← ⇑↵↑≈↑ ⊄↑•″≠ ⋅″→≈≥≈″←↵××♣⌠⇐≈→•♠≈… ⊆←↵←≈ ↵÷•″≈↑∫ ∠→±∫ ←÷ ⋅″←≈→″↵←•±″↵× ⇑±″≡≈→≈″≈ ±″ ⊂≈↑←•″≠ ⇑±≥°↔←≈→ ⊆±≡←♥↵→≈∫ ⊄⊆∠⇐⋅ ⊃↵↑÷•″≠←±″ 

∫ ¬±×±≥≈ ⊃∫ ∫ ∫ ↵″… ⋅°↵←≈ ∨∫ ⇑±→→≈← ⊆♣↑←≈≥↑ ⇒↔•×…•″≠ ↵ ⇒↔↑•″≈↑↑ ∠→±≈↑↑ ⊆±×↔←•±″∫ ⇓°°×•≈… ⇑±≥°↔←•″≠ ⊆≈→•≈↑∫ ⊆°→•″≠≈→ ⊇≈→×↵≠ ⇒≈→ו″ ¬≈•…≈×≈→≠ ≈♥ ∩±→∝ 

∫ ⋅°↵←≈ ∨∫ ↵″… ¬±×±≥≈ ⊃∫ ∫ ∫ ⇓″ ⋅″←≈≠→↵←•±″ ⊂≈↑←•″≠ ≈←÷±… ←÷↵← •↑ ∠→±♠≈… ←± ∨•″… ⇓×× ∨↵↔×←↑∫ ⋅″←≈→″↵←•±″↵× √±↔→″↵× ±≡ ⇑±≥°↔←↵←•±″↵× ↵←÷≈≥↵←•↑ ⊇±×∫   ⌠ ∫ ¬±×±≥≈ ∫ ⇒±≠…↵″±♠ ∏∫ ↵″… ∧÷≈±→≠÷≈ ∫ ∨↔″←•±″↵× ⊂≈↑← ∧≈″≈→↵←•±″ ≡±→ ⇔♦←→≈≥≈

∠→±≠→↵≥≥•″≠∫ ∠→±∫ ″… ⋅″←≈→″↵←•±″↵× ⇑±″≡≈→≈″≈ ±″ ⇔♦←→≈≥≈ ∠→±≠→↵≥≥•″≠ ↵″… ∨×≈♦•×≈ ∠→±≈↑↑≈↑ •″ ⊆±≡←♥↵→≈ ⇔″≠•″≈≈→•″≠∫ ∪∠ ðð ⊆↵→…•″•↵ ⋅←↵×♣ ðð ð⌠

∫ ⇐→↵″•…•↑ ⇐∫ ⊂•≠∝↵ ∏∫ ↵″… ∏≈≡↵×↵↑ ∠∫ ∨±→≥↵× ±…≈×ו″≠ ±≡ ⊄↑≈ ⇑↵↑≈↑ ♥•←÷ ∪⌠≥↵÷•″≈↑∫ ∠→±∫ ↑←⊆±↔←÷⌠⇔↵↑← ⇔↔→±°≈↵″ ⊃±→∝↑÷±° ±″ ∨±→≥↵× ≈←÷±…↑∫ ⊆±↔←÷⌠⇔≈↵↑← ⇔↔→±°≈↵″ ∈≈⌠ ↑≈↵→÷ ⇑≈″←→≈ ⊂÷≈↑↑↵×±″•∝• ðð

ð∫ ¬±×±≥≈ ∫ ⊃÷≈→≈ ⇐± ⊄″•← ⊂≈↑←↑ ⇑±≥≈ ∨→±≥〈 ∠→±∫ ←÷∫ ⋅″←≈→″↵←•±″↵× ⇑±″≡≈→≈″≈ ±″ ⇔♦←→≈≥≈ ∠→±≠→↵≥≥•″≠ ↵″… ∨×≈♦•×≈ ∠→±≈↑↑≈↑ •″ ⊆±≡←♥↵→≈ ⇔″≠•″≈≈→•″≠∫ ∪∠ ðð ∧≈″±♠↵ ⋅←↵×♣ ≈←↔→≈ ±←≈↑ •″ ⇑±≥°↔←≈→ ⊆•≈″≈ ⊇±×∫ ∫⊆°→•″≠≈→ ⊇≈→×↵≠ ⇒≈→ו″ ¬≈•…≈×≈→≠ ≈♥ ∩±→∝ ðð ⌠

∫ ∧→≈≠±→ √∫ ⇐∫ ↵″… ⇐♣≈→ ⇐∫ ∫ ⇓ ±←≈ ±″ ⋅″÷≈→•←↵″≈ ↵″… ⊆←↵←≈ ↵÷•″≈↑∫ ⊆±≡←♥↵→≈ ⇔″≠•″≈≈→•″≠ ±←≈↑ ⊇±×∫  ±∫   ⌠

∫ ⊆•≥±″↑ ⇓∫ √∫ ¬∫ ⊆←↵″″≈←← ∫ ∠∫ ⇒±≠…↵″±♠ ∏∫ ⇔∫ ↵″… ¬±×±≥≈ ⊃∫ ∫ ∫ ∠×↔≠ ↵″… ∠×↵♣ ⊆↵≡≈×♣ ⇒≈÷↵♠•±↔→↵× ∈↔×≈↑ ≡±→ ⇑±≥°↵←••ו←♣∫ ∠→±∫ ←÷ ⋅⇓⊆⊂⇔⇐ ⋅″←≈→″↵←•±″↵× ⇑±″≡≈→≈″≈ ±″ ⊆±≡←♥↵→≈ ⇔″≠•″≈≈→•″≠ ↵″… ⇓°°×•↵←•±″↑∫ ⊆⇔⇓⌠ðð ⇑↵≥→•…≠≈ ðð ⌠ ∫ ⇒≈→″±← ⇒∫ ∧↵↔…≈× ∫⌠⇑∫ ↵″… ↵→→≈ ⇒∫ ⊆±≡←♥↵→≈ ⊂≈↑←•″≠ ⇒↵↑≈… ±″ ∨±→≥↵× ⊆°≈•≡•↵⌠

←•±″↑ ↵ ⊂÷≈±→♣ ↵″… ↵ ⊂±±×∫ ⊆±≡←♥↵→≈ ⇔″≠•″≈≈→•″≠ √±↔→″↵× ⊇±×∫  ±∫   ⌠ð ∫ ⇑±±∝ ⊆∫ ↵″… ⇐↵″•≈×↑ √∫ ⇐≈↑•≠″•″≠ ∇∂≈←⌠∇→•≈″←≈… ⊆♣↑←≈≥↑ ∇∂≈←⌠∇→•≈″←≈… ±…≈×ו″≠

♥•←÷ ⊆♣″←→±°♣∫ ∠→≈″←•≈ ¬↵×× ±″…±″ 

References

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