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Theorem List for Quantum Logic Explorer - 1201-1215   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremxxdp53 1201 Part of proof (5)=>(3) in Day/Pickering 1982.
p2 =< (a2 v b2)   &   c0 = ((a1 v a2) ^ (b1 v b2))   &   c1 = ((a0 v a2) ^ (b0 v b2))   &   c2 = ((a0 v a1) ^ (b0 v b1))   &   d = (a2 v (a0 ^ (a1 v b1)))   &   e = (b0 ^ (a0 v p0))   &   p = (((a0 v b0) ^ (a1 v b1)) ^ (a2 v b2))   &   p0 = ((a1 v b1) ^ (a2 v b2))   &   p2 = ((a0 v b0) ^ (a1 v b1))   =>   p =< (a0 v (b0 ^ (b1 v (c2 ^ (c0 v c1)))))
 
Theoremxdp45lem 1202 Part of proof (4)=>(5) in Day/Pickering 1982.
p2 =< (a2 v b2)   &   c0 = ((a1 v a2) ^ (b1 v b2))   &   c1 = ((a0 v a2) ^ (b0 v b2))   &   c2 = ((a0 v a1) ^ (b0 v b1))   &   d = (a2 v (a0 ^ (a1 v b1)))   &   e = (b0 ^ (a0 v p0))   &   p = (((a0 v b0) ^ (a1 v b1)) ^ (a2 v b2))   &   p0 = ((a1 v b1) ^ (a2 v b2))   &   p2 = ((a0 v b0) ^ (a1 v b1))   =>   ((a0 v a1) ^ ((b0 ^ (a0 v p0)) v b1)) =< ((c0 v c1) v (b1 ^ (a0 v a1)))
 
Theoremxdp43lem 1203 Part of proof (4)=>(3) in Day/Pickering 1982.
p2 =< (a2 v b2)   &   c0 = ((a1 v a2) ^ (b1 v b2))   &   c1 = ((a0 v a2) ^ (b0 v b2))   &   c2 = ((a0 v a1) ^ (b0 v b1))   &   d = (a2 v (a0 ^ (a1 v b1)))   &   e = (b0 ^ (a0 v p0))   &   p = (((a0 v b0) ^ (a1 v b1)) ^ (a2 v b2))   &   p0 = ((a1 v b1) ^ (a2 v b2))   &   p2 = ((a0 v b0) ^ (a1 v b1))   =>   p =< (a0 v (b0 ^ (b1 v (c2 ^ (c0 v c1)))))
 
Theoremxdp45 1204 Part of proof (4)=>(5) in Day/Pickering 1982.
c0 = ((a1 v a2) ^ (b1 v b2))   &   c1 = ((a0 v a2) ^ (b0 v b2))   &   c2 = ((a0 v a1) ^ (b0 v b1))   &   d = (a2 v (a0 ^ (a1 v b1)))   &   e = (b0 ^ (a0 v p0))   &   p = (((a0 v b0) ^ (a1 v b1)) ^ (a2 v b2))   &   p0 = ((a1 v b1) ^ (a2 v b2))   &   p2 = ((a0 v b0) ^ (a1 v b1))   =>   ((a0 v a1) ^ ((b0 ^ (a0 v p0)) v b1)) =< ((c0 v c1) v (b1 ^ (a0 v a1)))
 
Theoremxdp43 1205 Part of proof (4)=>(3) in Day/Pickering 1982.
c0 = ((a1 v a2) ^ (b1 v b2))   &   c1 = ((a0 v a2) ^ (b0 v b2))   &   c2 = ((a0 v a1) ^ (b0 v b1))   &   d = (a2 v (a0 ^ (a1 v b1)))   &   e = (b0 ^ (a0 v p0))   &   p = (((a0 v b0) ^ (a1 v b1)) ^ (a2 v b2))   &   p0 = ((a1 v b1) ^ (a2 v b2))   &   p2 = ((a0 v b0) ^ (a1 v b1))   =>   p =< (a0 v (b0 ^ (b1 v (c2 ^ (c0 v c1)))))
 
Theorem3dp43 1206 "3OA" version of xdp43 1205. Changed a2 to a1 and b2 to b1.
c0 = ((a1 v a1) ^ (b1 v b1))   &   c1 = ((a0 v a1) ^ (b0 v b1))   &   c2 = ((a0 v a1) ^ (b0 v b1))   &   d = (a1 v (a0 ^ (a1 v b1)))   &   e = (b0 ^ (a0 v p0))   &   p = (((a0 v b0) ^ (a1 v b1)) ^ (a1 v b1))   &   p0 = ((a1 v b1) ^ (a1 v b1))   &   p2 = ((a0 v b0) ^ (a1 v b1))   =>   p =< (a0 v (b0 ^ (b1 v (c2 ^ (c0 v c1)))))
 
Theoremoadp35lemg 1207 Part of proof (3)=>(5) in Day/Pickering 1982.
c0 = ((a1 v a2) ^ (b1 v b2))   &   c1 = ((a0 v a2) ^ (b0 v b2))   &   c2 = ((a0 v a1) ^ (b0 v b1))   &   p0 = ((a1 v b1) ^ (a2 v b2))   &   p = (((a0 v b0) ^ (a1 v b1)) ^ (a2 v b2))   =>   p =< (a0 v (b0 ^ (b1 v (c2 ^ (c0 v c1)))))
 
Theoremoadp35lemf 1208 Part of proof (3)=>(5) in Day/Pickering 1982.
c0 = ((a1 v a2) ^ (b1 v b2))   &   c1 = ((a0 v a2) ^ (b0 v b2))   &   c2 = ((a0 v a1) ^ (b0 v b1))   &   p0 = ((a1 v b1) ^ (a2 v b2))   &   p = (((a0 v b0) ^ (a1 v b1)) ^ (a2 v b2))   =>   (a0 v p) =< (a0 v (b0 ^ (b1 v (c2 ^ (c0 v c1)))))
 
Theoremoadp35lemc 1209 Part of proof (3)=>(5) in Day/Pickering 1982.
c0 = ((a1 v a2) ^ (b1 v b2))   &   c1 = ((a0 v a2) ^ (b0 v b2))   &   c2 = ((a0 v a1) ^ (b0 v b1))   &   p0 = ((a1 v b1) ^ (a2 v b2))   &   p = (((a0 v b0) ^ (a1 v b1)) ^ (a2 v b2))   =>   (b0 ^ (((a0 ^ b0) v b1) v (c2 ^ (c0 v c1)))) = (b0 ^ (b1 v (c2 ^ (c0 v c1))))
 
Theoremoadp35 1210 Part of theorem from Alan Day and Doug Pickering, "A note on the Arguesian lattice identity," Studia Sci. Math. Hungar. 19:303-305 (1982). (3)=>(5)
c0 = ((a1 v a2) ^ (b1 v b2))   &   c1 = ((a0 v a2) ^ (b0 v b2))   &   p0 = ((a1 v b1) ^ (a2 v b2))   =>   ((a0 v a1) ^ ((b0 ^ (a0 v p0)) v b1)) =< ((c0 v c1) v (b1 ^ (a0 v a1)))
 
Theoremtestmod 1211 A modular law experiment.
(((c v a) v ((b v c) ^ (d v a))) ^ (a v (b ^ (d v ((a v c) ^ (b v d)))))) = ((b ^ ((((a v c) v ((b v c) ^ (d v a))) ^ d) v ((a v c) ^ (b v d)))) v a)
 
Theoremtestmod1 1212 A modular law experiment.
(((c v a) v ((b v c) ^ (d v a))) ^ (a v (b ^ (d v ((a v c) ^ (b v d)))))) = (a v (b ^ (((a v c) ^ (b v d)) v (d ^ ((a v c) v ((b v c) ^ (d v a)))))))
 
Theoremtestmod2 1213 A modular law experiment.
((a v b) ^ (a v (c v d))) = (a v (b ^ (((a v c) ^ (b v d)) v (d ^ ((a v c) v ((b v c) ^ (d v a)))))))
 
Theoremtestmod2expanded 1214 A modular law experiment.
((a v b) ^ (a v (c v d))) = (a v (b ^ (((a v c) ^ (b v d)) v (d ^ ((a v c) v ((b v c) ^ (d v a)))))))
 
Theoremtestmod3 1215 A modular law experiment.
(((c v a) v ((b v c) ^ (d v a))) ^ (a v (b ^ (d v ((a v c) ^ (b v d)))))) = (a v (((c v a) v ((b v c) ^ (d v a))) ^ (b ^ (d v ((a v c) ^ (b v d))))))
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