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User:Jon Awbrey/SCRATCHPAD (view source)

Revision as of 19:16, 10 February 2009

597 bytes removed ,  19:16, 10 February 2009
→‎Fact 2.1
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|- style="height:50px; text-align:center"
 
|- style="height:50px; text-align:center"
 
| style="width:80%" |  
 
| style="width:80%" |  
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| style="width:20%; border-left:1px solid black" | <math>\operatorname{Fact~1}</math>
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| style="width:20%; border-left:1px solid black" | <math>\operatorname{Fact~2.1}</math>
 
|}
 
|}
 
|-
 
|-
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{| align="center" cellpadding="0" cellspacing="0" width="100%"
 
{| align="center" cellpadding="0" cellspacing="0" width="100%"
 
|- style="height:50px"
 
|- style="height:50px"
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| style="border-top:1px solid black; width:2%" | &nbsp;
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| width="2%"  style="border-top:1px solid black" | &nbsp;
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| style="border-top:1px solid black; width:18%" | <math>\text{If}\!</math>
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| width="10%" style="border-top:1px solid black" | <math>\text{If}\!</math>
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| style="border-top:1px solid black; width:60%" | <math>P, Q ~\subseteq~ X</math>
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| width="68%" style="border-top:1px solid black" | <math>L ~\subseteq~ O \times S \times I</math>
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| style="border-top:1px solid black; border-left:1px solid black" | &nbsp;
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| width="20%" style="border-top:1px solid black; border-left:1px solid black" | &nbsp;
 
|- style="height:50px"
 
|- style="height:50px"
 
| &nbsp;
 
| &nbsp;
 
| <math>\text{then}\!</math>
 
| <math>\text{then}\!</math>
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| <math>\text{the following are equivalent:}\!</math>
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| <math>\text{the following are identical subsets of}~ S \times I :</math>
 
| style="border-left:1px solid black" | &nbsp;
 
| style="border-left:1px solid black" | &nbsp;
 
|}
 
|}
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{| align="center" cellpadding="0" cellspacing="0" width="100%"
 
{| align="center" cellpadding="0" cellspacing="0" width="100%"
 
|- style="height:60px"
 
|- style="height:60px"
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| style="width:2%; border-top:1px solid black" | &nbsp;
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| width="2%" style="border-top:1px solid black" | &nbsp;
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| style="width:14%; border-top:1px solid black" | <math>\operatorname{F1a.}</math>
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| width="10%" style="border-top:1px solid black" | <math>\operatorname{F2.1a.}</math>
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| style="width:64%; border-top:1px solid black" | <math>s \quad \Leftrightarrow \quad (P ~=~ Q)</math>
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| width="68%" style="border-top:1px solid black" | <math>\operatorname{Der}^L</math>
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| style="width:20%; border-top:1px solid black; border-left:1px solid black; text-align:center" |
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| width="20%" style="border-top:1px solid black; border-left:1px solid black; text-align:center" | <math>\operatorname{F2.1a~:~D13a}</math>
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<math>\operatorname{F1a~:~R9a}</math>
   
|- style="height:20px"
 
|- style="height:20px"
 
| &nbsp;
 
| &nbsp;
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|- style="height:40px"
 
|- style="height:40px"
 
| &nbsp;
 
| &nbsp;
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| <math>\operatorname{F1b.}</math>
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| <math>\operatorname{F2.1b.}</math>
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| <math>s \quad \Leftrightarrow \quad (\upharpoonleft P \upharpoonright ~=~ \upharpoonleft Q \upharpoonright)</math>
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| <math>\operatorname{Der}(L)</math>
 
| style="border-left:1px solid black; text-align:center" |
 
| style="border-left:1px solid black; text-align:center" |
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<math>\operatorname{F1b~:~R9b}</math>
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<math>\operatorname{F2.1b~:~D13b}</math>
 
|- style="height:20px"
 
|- style="height:20px"
 
| &nbsp;
 
| &nbsp;
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|- style="height:60px"
 
|- style="height:60px"
 
| &nbsp;
 
| &nbsp;
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| <math>\operatorname{F1c.}</math>
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| <math>\operatorname{F2.1c.}</math>
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| <math>s \quad \Leftrightarrow \quad \overset{X}{\underset{x}{\forall}}~ (\upharpoonleft P \upharpoonright (x) ~=~ \upharpoonleft Q \upharpoonright (x))</math>
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| <math>\{ (x, y) \in S \times I ~:~ \operatorname{Den}(L, x) ~=~ \operatorname{Den}(L, y) \}</math>
 
| style="border-left:1px solid black; text-align:center" |
 
| style="border-left:1px solid black; text-align:center" |
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<math>\operatorname{F1c~:~R9c}</math>
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<p><math>\operatorname{F2.1c~:~D13c}</math></p>
 +
<p><math>\operatorname{F2.1c~:~R9a}</math></p>
 
|- style="height:20px"
 
|- style="height:20px"
 
| &nbsp;
 
| &nbsp;
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| &nbsp;
 
| &nbsp;
 
| style="border-left:1px solid black; text-align:center" | <math>::\!</math>
 
| style="border-left:1px solid black; text-align:center" | <math>::\!</math>
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|- style="height:60px"
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|- style="height:40px"
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| &nbsp;
 +
| <math>\operatorname{F2.1d.}</math>
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| <math>\{ (x, y) \in S \times I ~:~ \upharpoonleft \operatorname{Den}(L, x) \upharpoonright ~=~ \upharpoonleft \operatorname{Den}(L, y) \upharpoonright \}</math>
 +
| style="border-left:1px solid black; text-align:center" |
 +
<math>\operatorname{F2.1d~:~R9b}</math>
 +
|- style="height:20px"
 +
| &nbsp;
 +
| &nbsp;
 +
| &nbsp;
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| style="border-left:1px solid black; text-align:center" | <math>::\!</math>
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|- style="height:40px"
 
| &nbsp;
 
| &nbsp;
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| <math>\operatorname{F1d.}</math>
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| <math>\operatorname{F2.1e.}</math>
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| <math>s \quad \Leftrightarrow \quad \operatorname{Conj_x^X}~ (\upharpoonleft P \upharpoonright (x) ~=~ \upharpoonleft Q \upharpoonright (x))</math>
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| <math>\ldots</math>
 
| style="border-left:1px solid black; text-align:center" |
 
| style="border-left:1px solid black; text-align:center" |
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<p><math>\operatorname{F1d~:~R9d}</math></p>
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<math>\operatorname{F2.1e~:~R9c}</math>
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<p><math>\operatorname{F1d~:~R8a}</math></p>
   
|- style="height:20px"
 
|- style="height:20px"
 
| &nbsp;
 
| &nbsp;
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| &nbsp;
 
| &nbsp;
 
| style="border-left:1px solid black; text-align:center" | <math>::\!</math>
 
| style="border-left:1px solid black; text-align:center" | <math>::\!</math>
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|- style="height:60px"
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|- style="height:40px"
 
| &nbsp;
 
| &nbsp;
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| <math>\operatorname{F1e.}</math>
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| <math>\operatorname{F2.1f.}</math>
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| <math>\downharpoonleft s \downharpoonright \quad = \quad \downharpoonleft \operatorname{Conj_x^X}~ (\upharpoonleft P \upharpoonright (x) ~=~ \upharpoonleft Q \upharpoonright (x)) \downharpoonright</math>
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| <math>\ldots</math>
 
| style="border-left:1px solid black; text-align:center" |
 
| style="border-left:1px solid black; text-align:center" |
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<p><math>\operatorname{F1e~:~R8b}</math></p>
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<math>\operatorname{F2.1f~:~R9d}</math>
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<p><math>\operatorname{F1e~:~\_\_?\_\_}</math></p>
   
|- style="height:20px"
 
|- style="height:20px"
 
| &nbsp;
 
| &nbsp;
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| &nbsp;
 
| &nbsp;
 
| style="border-left:1px solid black; text-align:center" | <math>::\!</math>
 
| style="border-left:1px solid black; text-align:center" | <math>::\!</math>
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|- style="height:60px"
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|- style="height:40px"
 
| &nbsp;
 
| &nbsp;
−
| <math>\operatorname{F1f.}</math>
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| <math>\operatorname{F2.1g.}</math>
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| <math>\downharpoonleft s \downharpoonright \quad = \quad \operatorname{Conj_x^X}~ \downharpoonleft (\upharpoonleft P \upharpoonright (x) ~=~ \upharpoonleft Q \upharpoonright (x)) \downharpoonright</math>
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| <math>\ldots</math>
 
| style="border-left:1px solid black; text-align:center" |
 
| style="border-left:1px solid black; text-align:center" |
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<p><math>\operatorname{F1f~:~\_\_?\_\_}</math></p>
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<math>\operatorname{F2.1g~:~R9e}</math>
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<p><math>\operatorname{F1f~:~\_\_?\_\_}</math></p>
   
|- style="height:20px"
 
|- style="height:20px"
 
| &nbsp;
 
| &nbsp;
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|- style="height:60px"
 
|- style="height:60px"
 
| &nbsp;
 
| &nbsp;
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| <math>\operatorname{F1g.}</math>
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| <math>\operatorname{F2.1h.}</math>
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| <math>\downharpoonleft s \downharpoonright \quad = \quad \operatorname{Conj_x^X}~ \underline{((}~ \upharpoonleft P \upharpoonright (x) ~,~ \upharpoonleft Q \upharpoonright (x) ~\underline{))}</math>
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| <math>\ldots</math>
 
| style="border-left:1px solid black; text-align:center" |
 
| style="border-left:1px solid black; text-align:center" |
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<p><math>\operatorname{F1g~:~\_\_?\_\_}</math></p>
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<p><math>\operatorname{F2.1h~:~R9f}</math></p>
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<p><math>\operatorname{F1g~:~\_\_?\_\_}</math></p>
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<p><math>\operatorname{F2.1h~:~D12e}</math></p>
 
|- style="height:20px"
 
|- style="height:20px"
 
| &nbsp;
 
| &nbsp;
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|- style="height:40px"
 
|- style="height:40px"
 
| &nbsp;
 
| &nbsp;
−
| <math>\operatorname{F1h.}</math>
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| <math>\operatorname{F2.1i.}</math>
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| <math>\downharpoonleft s \downharpoonright \quad = \quad \operatorname{Conj_x^X}~ \underline{((}~ \upharpoonleft P \upharpoonright ~,~ \upharpoonleft Q \upharpoonright ~\underline{))}^\$ (x)</math>
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| <math>\ldots</math>
 
| style="border-left:1px solid black; text-align:center" |
 
| style="border-left:1px solid black; text-align:center" |
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<math>\operatorname{F1h~:~~\_\_?\_\_}</math>
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<math>\operatorname{F2.1i~:~D12a}</math>
 
|}
 
|}
 
|}
 
|}
Jon Awbrey
12,080

edits

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