theory_file stringclasses 2
values | lemma_name stringlengths 16 52 | lemma_command stringlengths 0 1.3k | lemma_object stringlengths 11 391 | template stringlengths 17 688 | symbols listlengths 1 21 | types listlengths 1 21 | defs listlengths 1 16 | output_key stringclasses 1
value | input stringlengths 162 3.2k | output stringlengths 28 408 |
|---|---|---|---|---|---|---|---|---|---|---|
Octonions/Cross_Product_7 | Cross_Product_7.not_equal_vector7 | lemma not_equal_vector7 :
fixes x::"real^7" and y::"real^7"
assumes "x = vector[x1,x2,x3,x4,x5,x6,x7] " and "y= vector [y1,y2,y3,y4,y5,y6,y7]"
and "x$1 \<noteq> y$1 \<or> x$2 \<noteq> y$2 \<or> x$3 \<noteq> y$3 \<or> x$4 \<noteq> y$4 \<or> x$5 \<noteq> y$5 \<or> x$6 \<noteq> y$6 \<or> x$7 \<noteq> y$7 "
sh... | ?x = vector [ ?x1.0, ?x2.0, ?x3.0, ?x4.0, ?x5.0, ?x6.0, ?x7.0] \<Longrightarrow> ?y = vector [ ?y1.0, ?y2.0, ?y3.0, ?y4.0, ?y5.0, ?y6.0, ?y7.0] \<Longrightarrow> ?x $ 1 \<noteq> ?y $ 1 \<or> ?x $ 2 \<noteq> ?y $ 2 \<or> ?x $ 3 \<noteq> ?y $ 3 \<or> ?x $ 4 \<noteq> ?y $ 4 \<or> ?x $ 5 \<noteq> ?y $ 5 \<or> ?x $ 6 \<note... | \<lbrakk>x_1 = ?H1 (?H2 x_2 (?H2 x_3 (?H2 x_4 (?H2 x_5 (?H2 x_6 (?H2 x_7 (?H2 x_8 ?H3))))))); x_9 = ?H1 (?H2 x_10 (?H2 x_11 (?H2 x_12 (?H2 x_13 (?H2 x_14 (?H2 x_15 (?H2 x_16 ?H3))))))); ?H4 x_1 ?H5 \<noteq> ?H4 x_9 ?H5 \<or> ?H4 x_1 (?H6 (?H7 ?H8)) \<noteq> ?H4 x_9 (?H6 (?H7 ?H8)) \<or> ?H4 x_1 (?H6 (?H9 ?H8)) \<noteq>... | [
"Num.num.Bit1",
"Num.num.One",
"Num.num.Bit0",
"Num.numeral_class.numeral",
"Groups.one_class.one",
"Finite_Cartesian_Product.vec.vec_nth",
"List.list.Nil",
"List.list.Cons",
"Cartesian_Space.vector"
] | [
"num \\<Rightarrow> num",
"num",
"num \\<Rightarrow> num",
"num \\<Rightarrow> 'a",
"'a",
"('a, 'b) vec \\<Rightarrow> 'b \\<Rightarrow> 'a",
"'a list",
"'a \\<Rightarrow> 'a list \\<Rightarrow> 'a list",
"'a list \\<Rightarrow> ('a, 'b) vec"
] | [
"datatype num = One | Bit0 num | Bit1 num",
"primrec numeral :: \\<open>num \\<Rightarrow> 'a\\<close>\n where\n numeral_One: \\<open>numeral One = 1\\<close>\n | numeral_Bit0: \\<open>numeral (Bit0 n) = numeral n + numeral n\\<close>\n | numeral_Bit1: \\<open>numeral (Bit1 n) = numeral n + numeral n + 1\\<... | lemma_object | ###symbols
Num.num.Bit1 :::: num \<Rightarrow> num
Num.num.One :::: num
Num.num.Bit0 :::: num \<Rightarrow> num
Num.numeral_class.numeral :::: num \<Rightarrow> 'a
Groups.one_class.one :::: 'a
Finite_Cartesian_Product.vec.vec_nth :::: ('a, 'b) vec \<Rightarrow> 'b \<Rightarrow> 'a
List.list.Nil :::: 'a list
List... |
###output
?x = vector [ ?x1.0, ?x2.0, ?x3.0, ?x4.0, ?x5.0, ?x6.0, ?x7.0] \<Longrightarrow> ?y = vector [ ?y1.0, ?y2.0, ?y3.0, ?y4.0, ?y5.0, ?y6.0, ?y7.0] \<Longrightarrow> ?x $ 1 \<noteq> ?y $ 1 \<or> ?x $ 2 \<noteq> ?y $ 2 \<or> ?x $ 3 \<noteq> ?y $ 3 \<or> ?x $ 4 \<noteq> ?y $ 4 \<or> ?x $ 5 \<noteq> ?y $ 5 \<or> ?x... |
Octonions/Cross_Product_7 | Cross_Product_7.equal_vector7 | lemma equal_vector7:
fixes x::"real^7" and y::"real^7"
assumes "x = vector[x1,x2,x3,x4,x5,x6,x7] " and "y= vector [y1,y2,y3,y4,y5,y6,y7]"
and "x = y"
shows "x$1 = y$1 \<and> x$2 = y$2 \<and> x$3 = y$3 \<and> x$4 = y$4 \<and> x$5 = y$5 \<and> x$6 = y$6 \<and> x$7 = y$7 " | ?x = vector [ ?x1.0, ?x2.0, ?x3.0, ?x4.0, ?x5.0, ?x6.0, ?x7.0] \<Longrightarrow> ?y = vector [ ?y1.0, ?y2.0, ?y3.0, ?y4.0, ?y5.0, ?y6.0, ?y7.0] \<Longrightarrow> ?x = ?y \<Longrightarrow> ?x $ 1 = ?y $ 1 \<and> ?x $ 2 = ?y $ 2 \<and> ?x $ 3 = ?y $ 3 \<and> ?x $ 4 = ?y $ 4 \<and> ?x $ 5 = ?y $ 5 \<and> ?x $ 6 = ?y $ 6 \... | \<lbrakk>x_1 = ?H1 (?H2 x_2 (?H2 x_3 (?H2 x_4 (?H2 x_5 (?H2 x_6 (?H2 x_7 (?H2 x_8 ?H3))))))); x_9 = ?H1 (?H2 x_10 (?H2 x_11 (?H2 x_12 (?H2 x_13 (?H2 x_14 (?H2 x_15 (?H2 x_16 ?H3))))))); x_1 = x_9\<rbrakk> \<Longrightarrow> ?H4 x_1 ?H5 = ?H4 x_9 ?H5 \<and> ?H4 x_1 (?H6 (?H7 ?H8)) = ?H4 x_9 (?H6 (?H7 ?H8)) \<and> ?H4 x_1... | [
"Num.num.Bit1",
"Num.num.One",
"Num.num.Bit0",
"Num.numeral_class.numeral",
"Groups.one_class.one",
"Finite_Cartesian_Product.vec.vec_nth",
"List.list.Nil",
"List.list.Cons",
"Cartesian_Space.vector"
] | [
"num \\<Rightarrow> num",
"num",
"num \\<Rightarrow> num",
"num \\<Rightarrow> 'a",
"'a",
"('a, 'b) vec \\<Rightarrow> 'b \\<Rightarrow> 'a",
"'a list",
"'a \\<Rightarrow> 'a list \\<Rightarrow> 'a list",
"'a list \\<Rightarrow> ('a, 'b) vec"
] | [
"datatype num = One | Bit0 num | Bit1 num",
"primrec numeral :: \\<open>num \\<Rightarrow> 'a\\<close>\n where\n numeral_One: \\<open>numeral One = 1\\<close>\n | numeral_Bit0: \\<open>numeral (Bit0 n) = numeral n + numeral n\\<close>\n | numeral_Bit1: \\<open>numeral (Bit1 n) = numeral n + numeral n + 1\\<... | lemma_object | ###symbols
Num.num.Bit1 :::: num \<Rightarrow> num
Num.num.One :::: num
Num.num.Bit0 :::: num \<Rightarrow> num
Num.numeral_class.numeral :::: num \<Rightarrow> 'a
Groups.one_class.one :::: 'a
Finite_Cartesian_Product.vec.vec_nth :::: ('a, 'b) vec \<Rightarrow> 'b \<Rightarrow> 'a
List.list.Nil :::: 'a list
List... |
###output
?x = vector [ ?x1.0, ?x2.0, ?x3.0, ?x4.0, ?x5.0, ?x6.0, ?x7.0] \<Longrightarrow> ?y = vector [ ?y1.0, ?y2.0, ?y3.0, ?y4.0, ?y5.0, ?y6.0, ?y7.0] \<Longrightarrow> ?x = ?y \<Longrightarrow> ?x $ 1 = ?y $ 1 \<and> ?x $ 2 = ?y $ 2 \<and> ?x $ 3 = ?y $ 3 \<and> ?x $ 4 = ?y $ 4 \<and> ?x $ 5 = ?y $ 5 \<and> ?x $ 6... |
Octonions/Cross_Product_7 | Cross_Product_7.cross7_components(1) | lemma cross7_components:
"(x \<times>\<^sub>7 y)$1 = x$2 * y$4 - x$4 * y$2 + x$3 * y$7 - x$7 * y$3 + x$5 * y$6 - x$6 * y$5 "
"(x \<times>\<^sub>7 y)$2 = x$4 * y$1 - x$1 * y$4 + x$3 * y$5 - x$5 * y$3 + x$6 * y$7 - x$7 * y$6 "
"(x \<times>\<^sub>7 y)$3 = x$5 * y$2 - x$2 * y$5 + x$4 * y$6 - x$6 * y$4 + x$7... | (?x \<times>\<^sub>7 ?y) $ 1 = ?x $ 2 * ?y $ 4 - ?x $ 4 * ?y $ 2 + ?x $ 3 * ?y $ 7 - ?x $ 7 * ?y $ 3 + ?x $ 5 * ?y $ 6 - ?x $ 6 * ?y $ 5 | ?H1 (?H2 x_1 x_2) ?H3 = ?H4 (?H5 (?H4 (?H5 (?H4 (?H6 (?H1 x_1 (?H7 (?H8 ?H9))) (?H1 x_2 (?H7 (?H8 (?H8 ?H9))))) (?H6 (?H1 x_1 (?H7 (?H8 (?H8 ?H9)))) (?H1 x_2 (?H7 (?H8 ?H9))))) (?H6 (?H1 x_1 (?H7 (?H10 ?H9))) (?H1 x_2 (?H7 (?H10 (?H10 ?H9)))))) (?H6 (?H1 x_1 (?H7 (?H10 (?H10 ?H9)))) (?H1 x_2 (?H7 (?H10 ?H9))))) (?H6 (?... | [
"Num.num.Bit1",
"Num.num.One",
"Num.num.Bit0",
"Num.numeral_class.numeral",
"Groups.times_class.times",
"Groups.plus_class.plus",
"Groups.minus_class.minus",
"Groups.one_class.one",
"Cross_Product_7.cross7",
"Finite_Cartesian_Product.vec.vec_nth"
] | [
"num \\<Rightarrow> num",
"num",
"num \\<Rightarrow> num",
"num \\<Rightarrow> 'a",
"'a \\<Rightarrow> 'a \\<Rightarrow> 'a",
"'a \\<Rightarrow> 'a \\<Rightarrow> 'a",
"'a \\<Rightarrow> 'a \\<Rightarrow> 'a",
"'a",
"(real, 7) vec \\<Rightarrow> (real, 7) vec \\<Rightarrow> (real, 7) vec",
"('a, '... | [
"datatype num = One | Bit0 num | Bit1 num",
"primrec numeral :: \\<open>num \\<Rightarrow> 'a\\<close>\n where\n numeral_One: \\<open>numeral One = 1\\<close>\n | numeral_Bit0: \\<open>numeral (Bit0 n) = numeral n + numeral n\\<close>\n | numeral_Bit1: \\<open>numeral (Bit1 n) = numeral n + numeral n + 1\\<... | lemma_object | ###symbols
Num.num.Bit1 :::: num \<Rightarrow> num
Num.num.One :::: num
Num.num.Bit0 :::: num \<Rightarrow> num
Num.numeral_class.numeral :::: num \<Rightarrow> 'a
Groups.times_class.times :::: 'a \<Rightarrow> 'a \<Rightarrow> 'a
Groups.plus_class.plus :::: 'a \<Rightarrow> 'a \<Rightarrow> 'a
Groups.minus_class... |
###output
(?x \<times>\<^sub>7 ?y) $ 1 = ?x $ 2 * ?y $ 4 - ?x $ 4 * ?y $ 2 + ?x $ 3 * ?y $ 7 - ?x $ 7 * ?y $ 3 + ?x $ 5 * ?y $ 6 - ?x $ 6 * ?y $ 5###end |
Octonions/Cross_Product_7 | Cross_Product_7.cross7_basis_nonzero | lemma cross7_basis_nonzero:
"\<not> (u \<times>\<^sub>7 axis 1 1 = 0) \<or> \<not> (u \<times>\<^sub>7 axis 2 1 = 0) \<or> \<not> (u \<times>\<^sub>7 axis 3 1 = 0)
\<or> \<not> (u \<times>\<^sub>7 axis 4 1 = 0) \<or> \<not> (u \<times>\<^sub>7 axis 5 1 = 0 ) \<or> \<not> (u \<times>\<^sub>7 axis 6 1 = 0 )
\<or>... | ?u \<times>\<^sub>7 axis 1 1 \<noteq> 0 \<or> ?u \<times>\<^sub>7 axis 2 1 \<noteq> 0 \<or> ?u \<times>\<^sub>7 axis 3 1 \<noteq> 0 \<or> ?u \<times>\<^sub>7 axis 4 1 \<noteq> 0 \<or> ?u \<times>\<^sub>7 axis 5 1 \<noteq> 0 \<or> ?u \<times>\<^sub>7 axis 6 1 \<noteq> 0 \<or> ?u \<times>\<^sub>7 axis 7 1 \<noteq> 0 \<Lo... | ?H1 x_1 (?H2 ?H3 ?H3) \<noteq> ?H4 \<or> ?H1 x_1 (?H2 (?H5 (?H6 ?H7)) ?H3) \<noteq> ?H4 \<or> ?H1 x_1 (?H2 (?H5 (?H8 ?H7)) ?H3) \<noteq> ?H4 \<or> ?H1 x_1 (?H2 (?H5 (?H6 (?H6 ?H7))) ?H3) \<noteq> ?H4 \<or> ?H1 x_1 (?H2 (?H5 (?H8 (?H6 ?H7))) ?H3) \<noteq> ?H4 \<or> ?H1 x_1 (?H2 (?H5 (?H6 (?H8 ?H7))) ?H3) \<noteq> ?H4 \<... | [
"Num.num.Bit1",
"Num.num.One",
"Num.num.Bit0",
"Num.numeral_class.numeral",
"Groups.zero_class.zero",
"Groups.one_class.one",
"Finite_Cartesian_Product.axis",
"Cross_Product_7.cross7"
] | [
"num \\<Rightarrow> num",
"num",
"num \\<Rightarrow> num",
"num \\<Rightarrow> 'a",
"'a",
"'a",
"'a \\<Rightarrow> 'b \\<Rightarrow> ('b, 'a) vec",
"(real, 7) vec \\<Rightarrow> (real, 7) vec \\<Rightarrow> (real, 7) vec"
] | [
"datatype num = One | Bit0 num | Bit1 num",
"primrec numeral :: \\<open>num \\<Rightarrow> 'a\\<close>\n where\n numeral_One: \\<open>numeral One = 1\\<close>\n | numeral_Bit0: \\<open>numeral (Bit0 n) = numeral n + numeral n\\<close>\n | numeral_Bit1: \\<open>numeral (Bit1 n) = numeral n + numeral n + 1\\<... | lemma_object | ###symbols
Num.num.Bit1 :::: num \<Rightarrow> num
Num.num.One :::: num
Num.num.Bit0 :::: num \<Rightarrow> num
Num.numeral_class.numeral :::: num \<Rightarrow> 'a
Groups.zero_class.zero :::: 'a
Groups.one_class.one :::: 'a
Finite_Cartesian_Product.axis :::: 'a \<Rightarrow> 'b \<Rightarrow> ('b, 'a) vec
Cross_P... |
###output
?u \<times>\<^sub>7 axis 1 1 \<noteq> 0 \<or> ?u \<times>\<^sub>7 axis 2 1 \<noteq> 0 \<or> ?u \<times>\<^sub>7 axis 3 1 \<noteq> 0 \<or> ?u \<times>\<^sub>7 axis 4 1 \<noteq> 0 \<or> ?u \<times>\<^sub>7 axis 5 1 \<noteq> 0 \<or> ?u \<times>\<^sub>7 axis 6 1 \<noteq> 0 \<or> ?u \<times>\<^sub>7 axis 7 1 \<no... |
Octonions/Cross_Product_7 | Cross_Product_7.cross7_components(3) | lemma cross7_components:
"(x \<times>\<^sub>7 y)$1 = x$2 * y$4 - x$4 * y$2 + x$3 * y$7 - x$7 * y$3 + x$5 * y$6 - x$6 * y$5 "
"(x \<times>\<^sub>7 y)$2 = x$4 * y$1 - x$1 * y$4 + x$3 * y$5 - x$5 * y$3 + x$6 * y$7 - x$7 * y$6 "
"(x \<times>\<^sub>7 y)$3 = x$5 * y$2 - x$2 * y$5 + x$4 * y$6 - x$6 * y$4 + x$7... | (?x \<times>\<^sub>7 ?y) $ 3 = ?x $ 5 * ?y $ 2 - ?x $ 2 * ?y $ 5 + ?x $ 4 * ?y $ 6 - ?x $ 6 * ?y $ 4 + ?x $ 7 * ?y $ 1 - ?x $ 1 * ?y $ 7 | ?H1 (?H2 x_1 x_2) (?H3 (?H4 ?H5)) = ?H6 (?H7 (?H6 (?H7 (?H6 (?H8 (?H1 x_1 (?H3 (?H4 (?H9 ?H5)))) (?H1 x_2 (?H3 (?H9 ?H5)))) (?H8 (?H1 x_1 (?H3 (?H9 ?H5))) (?H1 x_2 (?H3 (?H4 (?H9 ?H5)))))) (?H8 (?H1 x_1 (?H3 (?H9 (?H9 ?H5)))) (?H1 x_2 (?H3 (?H9 (?H4 ?H5)))))) (?H8 (?H1 x_1 (?H3 (?H9 (?H4 ?H5)))) (?H1 x_2 (?H3 (?H9 (?H9... | [
"Groups.one_class.one",
"Num.num.Bit0",
"Groups.times_class.times",
"Groups.plus_class.plus",
"Groups.minus_class.minus",
"Num.num.One",
"Num.num.Bit1",
"Num.numeral_class.numeral",
"Cross_Product_7.cross7",
"Finite_Cartesian_Product.vec.vec_nth"
] | [
"'a",
"num \\<Rightarrow> num",
"'a \\<Rightarrow> 'a \\<Rightarrow> 'a",
"'a \\<Rightarrow> 'a \\<Rightarrow> 'a",
"'a \\<Rightarrow> 'a \\<Rightarrow> 'a",
"num",
"num \\<Rightarrow> num",
"num \\<Rightarrow> 'a",
"(real, 7) vec \\<Rightarrow> (real, 7) vec \\<Rightarrow> (real, 7) vec",
"('a, '... | [
"class one =\n fixes one :: 'a (\\<open>1\\<close>)",
"datatype num = One | Bit0 num | Bit1 num",
"class times =\n fixes times :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\" (infixl \\<open>*\\<close> 70)",
"class plus =\n fixes plus :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\" (infixl \\<open>+\\<clos... | lemma_object | ###symbols
Groups.one_class.one :::: 'a
Num.num.Bit0 :::: num \<Rightarrow> num
Groups.times_class.times :::: 'a \<Rightarrow> 'a \<Rightarrow> 'a
Groups.plus_class.plus :::: 'a \<Rightarrow> 'a \<Rightarrow> 'a
Groups.minus_class.minus :::: 'a \<Rightarrow> 'a \<Rightarrow> 'a
Num.num.One :::: num
Num.num.Bit1 :... |
###output
(?x \<times>\<^sub>7 ?y) $ 3 = ?x $ 5 * ?y $ 2 - ?x $ 2 * ?y $ 5 + ?x $ 4 * ?y $ 6 - ?x $ 6 * ?y $ 4 + ?x $ 7 * ?y $ 1 - ?x $ 1 * ?y $ 7###end |
Octonions/Cross_Product_7 | Cross_Product_7.cross7_components(2) | lemma cross7_components:
"(x \<times>\<^sub>7 y)$1 = x$2 * y$4 - x$4 * y$2 + x$3 * y$7 - x$7 * y$3 + x$5 * y$6 - x$6 * y$5 "
"(x \<times>\<^sub>7 y)$2 = x$4 * y$1 - x$1 * y$4 + x$3 * y$5 - x$5 * y$3 + x$6 * y$7 - x$7 * y$6 "
"(x \<times>\<^sub>7 y)$3 = x$5 * y$2 - x$2 * y$5 + x$4 * y$6 - x$6 * y$4 + x$7... | (?x \<times>\<^sub>7 ?y) $ 2 = ?x $ 4 * ?y $ 1 - ?x $ 1 * ?y $ 4 + ?x $ 3 * ?y $ 5 - ?x $ 5 * ?y $ 3 + ?x $ 6 * ?y $ 7 - ?x $ 7 * ?y $ 6 | ?H1 (?H2 x_1 x_2) (?H3 (?H4 ?H5)) = ?H6 (?H7 (?H6 (?H7 (?H6 (?H8 (?H1 x_1 (?H3 (?H4 (?H4 ?H5)))) (?H1 x_2 ?H9)) (?H8 (?H1 x_1 ?H9) (?H1 x_2 (?H3 (?H4 (?H4 ?H5)))))) (?H8 (?H1 x_1 (?H3 (?H10 ?H5))) (?H1 x_2 (?H3 (?H10 (?H4 ?H5)))))) (?H8 (?H1 x_1 (?H3 (?H10 (?H4 ?H5)))) (?H1 x_2 (?H3 (?H10 ?H5))))) (?H8 (?H1 x_1 (?H3 (?... | [
"Num.num.Bit1",
"Groups.one_class.one",
"Groups.times_class.times",
"Groups.plus_class.plus",
"Groups.minus_class.minus",
"Num.num.One",
"Num.num.Bit0",
"Num.numeral_class.numeral",
"Cross_Product_7.cross7",
"Finite_Cartesian_Product.vec.vec_nth"
] | [
"num \\<Rightarrow> num",
"'a",
"'a \\<Rightarrow> 'a \\<Rightarrow> 'a",
"'a \\<Rightarrow> 'a \\<Rightarrow> 'a",
"'a \\<Rightarrow> 'a \\<Rightarrow> 'a",
"num",
"num \\<Rightarrow> num",
"num \\<Rightarrow> 'a",
"(real, 7) vec \\<Rightarrow> (real, 7) vec \\<Rightarrow> (real, 7) vec",
"('a, '... | [
"datatype num = One | Bit0 num | Bit1 num",
"class one =\n fixes one :: 'a (\\<open>1\\<close>)",
"class times =\n fixes times :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\" (infixl \\<open>*\\<close> 70)",
"class plus =\n fixes plus :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\" (infixl \\<open>+\\<clos... | lemma_object | ###symbols
Num.num.Bit1 :::: num \<Rightarrow> num
Groups.one_class.one :::: 'a
Groups.times_class.times :::: 'a \<Rightarrow> 'a \<Rightarrow> 'a
Groups.plus_class.plus :::: 'a \<Rightarrow> 'a \<Rightarrow> 'a
Groups.minus_class.minus :::: 'a \<Rightarrow> 'a \<Rightarrow> 'a
Num.num.One :::: num
Num.num.Bit0 :... |
###output
(?x \<times>\<^sub>7 ?y) $ 2 = ?x $ 4 * ?y $ 1 - ?x $ 1 * ?y $ 4 + ?x $ 3 * ?y $ 5 - ?x $ 5 * ?y $ 3 + ?x $ 6 * ?y $ 7 - ?x $ 7 * ?y $ 6###end |
Octonions/Cross_Product_7 | Cross_Product_7.cross7_components(7) | lemma cross7_components:
"(x \<times>\<^sub>7 y)$1 = x$2 * y$4 - x$4 * y$2 + x$3 * y$7 - x$7 * y$3 + x$5 * y$6 - x$6 * y$5 "
"(x \<times>\<^sub>7 y)$2 = x$4 * y$1 - x$1 * y$4 + x$3 * y$5 - x$5 * y$3 + x$6 * y$7 - x$7 * y$6 "
"(x \<times>\<^sub>7 y)$3 = x$5 * y$2 - x$2 * y$5 + x$4 * y$6 - x$6 * y$4 + x$7... | (?x \<times>\<^sub>7 ?y) $ 7 = ?x $ 1 * ?y $ 3 - ?x $ 3 * ?y $ 1 + ?x $ 4 * ?y $ 5 - ?x $ 5 * ?y $ 4 + ?x $ 2 * ?y $ 6 - ?x $ 6 * ?y $ 2 | ?H1 (?H2 x_1 x_2) (?H3 (?H4 (?H4 ?H5))) = ?H6 (?H7 (?H6 (?H7 (?H6 (?H8 (?H1 x_1 ?H9) (?H1 x_2 (?H3 (?H4 ?H5)))) (?H8 (?H1 x_1 (?H3 (?H4 ?H5))) (?H1 x_2 ?H9))) (?H8 (?H1 x_1 (?H3 (?H10 (?H10 ?H5)))) (?H1 x_2 (?H3 (?H4 (?H10 ?H5)))))) (?H8 (?H1 x_1 (?H3 (?H4 (?H10 ?H5)))) (?H1 x_2 (?H3 (?H10 (?H10 ?H5)))))) (?H8 (?H1 x_1... | [
"Num.num.Bit0",
"Groups.one_class.one",
"Groups.times_class.times",
"Groups.plus_class.plus",
"Groups.minus_class.minus",
"Num.num.One",
"Num.num.Bit1",
"Num.numeral_class.numeral",
"Cross_Product_7.cross7",
"Finite_Cartesian_Product.vec.vec_nth"
] | [
"num \\<Rightarrow> num",
"'a",
"'a \\<Rightarrow> 'a \\<Rightarrow> 'a",
"'a \\<Rightarrow> 'a \\<Rightarrow> 'a",
"'a \\<Rightarrow> 'a \\<Rightarrow> 'a",
"num",
"num \\<Rightarrow> num",
"num \\<Rightarrow> 'a",
"(real, 7) vec \\<Rightarrow> (real, 7) vec \\<Rightarrow> (real, 7) vec",
"('a, '... | [
"datatype num = One | Bit0 num | Bit1 num",
"class one =\n fixes one :: 'a (\\<open>1\\<close>)",
"class times =\n fixes times :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\" (infixl \\<open>*\\<close> 70)",
"class plus =\n fixes plus :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\" (infixl \\<open>+\\<clos... | lemma_object | ###symbols
Num.num.Bit0 :::: num \<Rightarrow> num
Groups.one_class.one :::: 'a
Groups.times_class.times :::: 'a \<Rightarrow> 'a \<Rightarrow> 'a
Groups.plus_class.plus :::: 'a \<Rightarrow> 'a \<Rightarrow> 'a
Groups.minus_class.minus :::: 'a \<Rightarrow> 'a \<Rightarrow> 'a
Num.num.One :::: num
Num.num.Bit1 :... |
###output
(?x \<times>\<^sub>7 ?y) $ 7 = ?x $ 1 * ?y $ 3 - ?x $ 3 * ?y $ 1 + ?x $ 4 * ?y $ 5 - ?x $ 5 * ?y $ 4 + ?x $ 2 * ?y $ 6 - ?x $ 6 * ?y $ 2###end |
Octonions/Cross_Product_7 | Cross_Product_7.cross7_components(5) | lemma cross7_components:
"(x \<times>\<^sub>7 y)$1 = x$2 * y$4 - x$4 * y$2 + x$3 * y$7 - x$7 * y$3 + x$5 * y$6 - x$6 * y$5 "
"(x \<times>\<^sub>7 y)$2 = x$4 * y$1 - x$1 * y$4 + x$3 * y$5 - x$5 * y$3 + x$6 * y$7 - x$7 * y$6 "
"(x \<times>\<^sub>7 y)$3 = x$5 * y$2 - x$2 * y$5 + x$4 * y$6 - x$6 * y$4 + x$7... | (?x \<times>\<^sub>7 ?y) $ 5 = ?x $ 6 * ?y $ 1 - ?x $ 1 * ?y $ 6 + ?x $ 2 * ?y $ 3 - ?x $ 3 * ?y $ 2 + ?x $ 7 * ?y $ 4 - ?x $ 4 * ?y $ 7 | ?H1 (?H2 x_1 x_2) (?H3 (?H4 (?H5 ?H6))) = ?H7 (?H8 (?H7 (?H8 (?H7 (?H9 (?H1 x_1 (?H3 (?H5 (?H4 ?H6)))) (?H1 x_2 ?H10)) (?H9 (?H1 x_1 ?H10) (?H1 x_2 (?H3 (?H5 (?H4 ?H6)))))) (?H9 (?H1 x_1 (?H3 (?H5 ?H6))) (?H1 x_2 (?H3 (?H4 ?H6))))) (?H9 (?H1 x_1 (?H3 (?H4 ?H6))) (?H1 x_2 (?H3 (?H5 ?H6))))) (?H9 (?H1 x_1 (?H3 (?H4 (?H4 ... | [
"Groups.one_class.one",
"Groups.times_class.times",
"Groups.plus_class.plus",
"Groups.minus_class.minus",
"Num.num.One",
"Num.num.Bit0",
"Num.num.Bit1",
"Num.numeral_class.numeral",
"Cross_Product_7.cross7",
"Finite_Cartesian_Product.vec.vec_nth"
] | [
"'a",
"'a \\<Rightarrow> 'a \\<Rightarrow> 'a",
"'a \\<Rightarrow> 'a \\<Rightarrow> 'a",
"'a \\<Rightarrow> 'a \\<Rightarrow> 'a",
"num",
"num \\<Rightarrow> num",
"num \\<Rightarrow> num",
"num \\<Rightarrow> 'a",
"(real, 7) vec \\<Rightarrow> (real, 7) vec \\<Rightarrow> (real, 7) vec",
"('a, '... | [
"class one =\n fixes one :: 'a (\\<open>1\\<close>)",
"class times =\n fixes times :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\" (infixl \\<open>*\\<close> 70)",
"class plus =\n fixes plus :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\" (infixl \\<open>+\\<close> 65)",
"class minus =\n fixes minus :: \"... | lemma_object | ###symbols
Groups.one_class.one :::: 'a
Groups.times_class.times :::: 'a \<Rightarrow> 'a \<Rightarrow> 'a
Groups.plus_class.plus :::: 'a \<Rightarrow> 'a \<Rightarrow> 'a
Groups.minus_class.minus :::: 'a \<Rightarrow> 'a \<Rightarrow> 'a
Num.num.One :::: num
Num.num.Bit0 :::: num \<Rightarrow> num
Num.num.Bit1 :... |
###output
(?x \<times>\<^sub>7 ?y) $ 5 = ?x $ 6 * ?y $ 1 - ?x $ 1 * ?y $ 6 + ?x $ 2 * ?y $ 3 - ?x $ 3 * ?y $ 2 + ?x $ 7 * ?y $ 4 - ?x $ 4 * ?y $ 7###end |
Octonions/Cross_Product_7 | Cross_Product_7.cross7_components(6) | lemma cross7_components:
"(x \<times>\<^sub>7 y)$1 = x$2 * y$4 - x$4 * y$2 + x$3 * y$7 - x$7 * y$3 + x$5 * y$6 - x$6 * y$5 "
"(x \<times>\<^sub>7 y)$2 = x$4 * y$1 - x$1 * y$4 + x$3 * y$5 - x$5 * y$3 + x$6 * y$7 - x$7 * y$6 "
"(x \<times>\<^sub>7 y)$3 = x$5 * y$2 - x$2 * y$5 + x$4 * y$6 - x$6 * y$4 + x$7... | (?x \<times>\<^sub>7 ?y) $ 6 = ?x $ 1 * ?y $ 5 - ?x $ 5 * ?y $ 1 + ?x $ 7 * ?y $ 2 - ?x $ 2 * ?y $ 7 + ?x $ 3 * ?y $ 4 - ?x $ 4 * ?y $ 3 | ?H1 (?H2 x_1 x_2) (?H3 (?H4 (?H5 ?H6))) = ?H7 (?H8 (?H7 (?H8 (?H7 (?H9 (?H1 x_1 ?H10) (?H1 x_2 (?H3 (?H5 (?H4 ?H6))))) (?H9 (?H1 x_1 (?H3 (?H5 (?H4 ?H6)))) (?H1 x_2 ?H10))) (?H9 (?H1 x_1 (?H3 (?H5 (?H5 ?H6)))) (?H1 x_2 (?H3 (?H4 ?H6))))) (?H9 (?H1 x_1 (?H3 (?H4 ?H6))) (?H1 x_2 (?H3 (?H5 (?H5 ?H6)))))) (?H9 (?H1 x_1 (?H... | [
"Groups.one_class.one",
"Groups.times_class.times",
"Groups.plus_class.plus",
"Groups.minus_class.minus",
"Num.num.One",
"Num.num.Bit1",
"Num.num.Bit0",
"Num.numeral_class.numeral",
"Cross_Product_7.cross7",
"Finite_Cartesian_Product.vec.vec_nth"
] | [
"'a",
"'a \\<Rightarrow> 'a \\<Rightarrow> 'a",
"'a \\<Rightarrow> 'a \\<Rightarrow> 'a",
"'a \\<Rightarrow> 'a \\<Rightarrow> 'a",
"num",
"num \\<Rightarrow> num",
"num \\<Rightarrow> num",
"num \\<Rightarrow> 'a",
"(real, 7) vec \\<Rightarrow> (real, 7) vec \\<Rightarrow> (real, 7) vec",
"('a, '... | [
"class one =\n fixes one :: 'a (\\<open>1\\<close>)",
"class times =\n fixes times :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\" (infixl \\<open>*\\<close> 70)",
"class plus =\n fixes plus :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\" (infixl \\<open>+\\<close> 65)",
"class minus =\n fixes minus :: \"... | lemma_object | ###symbols
Groups.one_class.one :::: 'a
Groups.times_class.times :::: 'a \<Rightarrow> 'a \<Rightarrow> 'a
Groups.plus_class.plus :::: 'a \<Rightarrow> 'a \<Rightarrow> 'a
Groups.minus_class.minus :::: 'a \<Rightarrow> 'a \<Rightarrow> 'a
Num.num.One :::: num
Num.num.Bit1 :::: num \<Rightarrow> num
Num.num.Bit0 :... |
###output
(?x \<times>\<^sub>7 ?y) $ 6 = ?x $ 1 * ?y $ 5 - ?x $ 5 * ?y $ 1 + ?x $ 7 * ?y $ 2 - ?x $ 2 * ?y $ 7 + ?x $ 3 * ?y $ 4 - ?x $ 4 * ?y $ 3###end |
Octonions/Cross_Product_7 | Cross_Product_7.cross7_components(4) | lemma cross7_components:
"(x \<times>\<^sub>7 y)$1 = x$2 * y$4 - x$4 * y$2 + x$3 * y$7 - x$7 * y$3 + x$5 * y$6 - x$6 * y$5 "
"(x \<times>\<^sub>7 y)$2 = x$4 * y$1 - x$1 * y$4 + x$3 * y$5 - x$5 * y$3 + x$6 * y$7 - x$7 * y$6 "
"(x \<times>\<^sub>7 y)$3 = x$5 * y$2 - x$2 * y$5 + x$4 * y$6 - x$6 * y$4 + x$7... | (?x \<times>\<^sub>7 ?y) $ 4 = ?x $ 1 * ?y $ 2 - ?x $ 2 * ?y $ 1 + ?x $ 6 * ?y $ 3 - ?x $ 3 * ?y $ 6 + ?x $ 5 * ?y $ 7 - ?x $ 7 * ?y $ 5 | ?H1 (?H2 x_1 x_2) (?H3 (?H4 (?H4 ?H5))) = ?H6 (?H7 (?H6 (?H7 (?H6 (?H8 (?H1 x_1 ?H9) (?H1 x_2 (?H3 (?H4 ?H5)))) (?H8 (?H1 x_1 (?H3 (?H4 ?H5))) (?H1 x_2 ?H9))) (?H8 (?H1 x_1 (?H3 (?H4 (?H10 ?H5)))) (?H1 x_2 (?H3 (?H10 ?H5))))) (?H8 (?H1 x_1 (?H3 (?H10 ?H5))) (?H1 x_2 (?H3 (?H4 (?H10 ?H5)))))) (?H8 (?H1 x_1 (?H3 (?H10 (?... | [
"Num.num.Bit1",
"Groups.one_class.one",
"Groups.times_class.times",
"Groups.plus_class.plus",
"Groups.minus_class.minus",
"Num.num.One",
"Num.num.Bit0",
"Num.numeral_class.numeral",
"Cross_Product_7.cross7",
"Finite_Cartesian_Product.vec.vec_nth"
] | [
"num \\<Rightarrow> num",
"'a",
"'a \\<Rightarrow> 'a \\<Rightarrow> 'a",
"'a \\<Rightarrow> 'a \\<Rightarrow> 'a",
"'a \\<Rightarrow> 'a \\<Rightarrow> 'a",
"num",
"num \\<Rightarrow> num",
"num \\<Rightarrow> 'a",
"(real, 7) vec \\<Rightarrow> (real, 7) vec \\<Rightarrow> (real, 7) vec",
"('a, '... | [
"datatype num = One | Bit0 num | Bit1 num",
"class one =\n fixes one :: 'a (\\<open>1\\<close>)",
"class times =\n fixes times :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\" (infixl \\<open>*\\<close> 70)",
"class plus =\n fixes plus :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\" (infixl \\<open>+\\<clos... | lemma_object | ###symbols
Num.num.Bit1 :::: num \<Rightarrow> num
Groups.one_class.one :::: 'a
Groups.times_class.times :::: 'a \<Rightarrow> 'a \<Rightarrow> 'a
Groups.plus_class.plus :::: 'a \<Rightarrow> 'a \<Rightarrow> 'a
Groups.minus_class.minus :::: 'a \<Rightarrow> 'a \<Rightarrow> 'a
Num.num.One :::: num
Num.num.Bit0 :... |
###output
(?x \<times>\<^sub>7 ?y) $ 4 = ?x $ 1 * ?y $ 2 - ?x $ 2 * ?y $ 1 + ?x $ 6 * ?y $ 3 - ?x $ 3 * ?y $ 6 + ?x $ 5 * ?y $ 7 - ?x $ 7 * ?y $ 5###end |
Octonions/Cross_Product_7 | Cross_Product_7.cross7_basis_zero | lemma cross7_basis_zero:
" u=0 \<Longrightarrow> (u \<times>\<^sub>7 axis 1 1 = 0) \<and> (u \<times>\<^sub>7 axis 2 1 = 0) \<and> (u \<times>\<^sub>7 axis 3 1 = 0)
\<and> (u \<times>\<^sub>7 axis 4 1 = 0) \<and> (u \<times>\<^sub>7 axis 5 1 = 0 ) \<and> (u \<times>\<^sub>7 axis 6 1 = 0 )
\<and> (u \<times>\<^sub>... | ?u = 0 \<Longrightarrow> ?u \<times>\<^sub>7 axis 1 1 = 0 \<and> ?u \<times>\<^sub>7 axis 2 1 = 0 \<and> ?u \<times>\<^sub>7 axis 3 1 = 0 \<and> ?u \<times>\<^sub>7 axis 4 1 = 0 \<and> ?u \<times>\<^sub>7 axis 5 1 = 0 \<and> ?u \<times>\<^sub>7 axis 6 1 = 0 \<and> ?u \<times>\<^sub>7 axis 7 1 = 0 | x_1 = ?H1 \<Longrightarrow> ?H2 x_1 (?H3 ?H4 ?H4) = ?H1 \<and> ?H2 x_1 (?H3 (?H5 (?H6 ?H7)) ?H4) = ?H1 \<and> ?H2 x_1 (?H3 (?H5 (?H8 ?H7)) ?H4) = ?H1 \<and> ?H2 x_1 (?H3 (?H5 (?H6 (?H6 ?H7))) ?H4) = ?H1 \<and> ?H2 x_1 (?H3 (?H5 (?H8 (?H6 ?H7))) ?H4) = ?H1 \<and> ?H2 x_1 (?H3 (?H5 (?H6 (?H8 ?H7))) ?H4) = ?H1 \<and> ?H2 ... | [
"Num.num.Bit1",
"Num.num.One",
"Num.num.Bit0",
"Num.numeral_class.numeral",
"Groups.one_class.one",
"Finite_Cartesian_Product.axis",
"Cross_Product_7.cross7",
"Groups.zero_class.zero"
] | [
"num \\<Rightarrow> num",
"num",
"num \\<Rightarrow> num",
"num \\<Rightarrow> 'a",
"'a",
"'a \\<Rightarrow> 'b \\<Rightarrow> ('b, 'a) vec",
"(real, 7) vec \\<Rightarrow> (real, 7) vec \\<Rightarrow> (real, 7) vec",
"'a"
] | [
"datatype num = One | Bit0 num | Bit1 num",
"primrec numeral :: \\<open>num \\<Rightarrow> 'a\\<close>\n where\n numeral_One: \\<open>numeral One = 1\\<close>\n | numeral_Bit0: \\<open>numeral (Bit0 n) = numeral n + numeral n\\<close>\n | numeral_Bit1: \\<open>numeral (Bit1 n) = numeral n + numeral n + 1\\<... | lemma_object | ###symbols
Num.num.Bit1 :::: num \<Rightarrow> num
Num.num.One :::: num
Num.num.Bit0 :::: num \<Rightarrow> num
Num.numeral_class.numeral :::: num \<Rightarrow> 'a
Groups.one_class.one :::: 'a
Finite_Cartesian_Product.axis :::: 'a \<Rightarrow> 'b \<Rightarrow> ('b, 'a) vec
Cross_Product_7.cross7 :::: (real, 7) v... |
###output
?u = 0 \<Longrightarrow> ?u \<times>\<^sub>7 axis 1 1 = 0 \<and> ?u \<times>\<^sub>7 axis 2 1 = 0 \<and> ?u \<times>\<^sub>7 axis 3 1 = 0 \<and> ?u \<times>\<^sub>7 axis 4 1 = 0 \<and> ?u \<times>\<^sub>7 axis 5 1 = 0 \<and> ?u \<times>\<^sub>7 axis 6 1 = 0 \<and> ?u \<times>\<^sub>7 axis 7 1 = 0###end |
Octonions/Cross_Product_7 | Cross_Product_7.exhaust_7 | lemma exhaust_7:
fixes x :: 7
shows "x = 1 \<or> x = 2 \<or> x = 3 \<or> x = 4 \<or> x = 5 \<or> x = 6 \<or> x = 7 " | ?x = 1 \<or> ?x = 2 \<or> ?x = 3 \<or> ?x = 4 \<or> ?x = 5 \<or> ?x = 6 \<or> ?x = 7 | x_1 = ?H1 \<or> x_1 = ?H2 (?H3 ?H4) \<or> x_1 = ?H2 (?H5 ?H4) \<or> x_1 = ?H2 (?H3 (?H3 ?H4)) \<or> x_1 = ?H2 (?H5 (?H3 ?H4)) \<or> x_1 = ?H2 (?H3 (?H5 ?H4)) \<or> x_1 = ?H2 (?H5 (?H5 ?H4)) | [
"Num.num.Bit1",
"Num.num.One",
"Num.num.Bit0",
"Num.numeral_class.numeral",
"Groups.one_class.one"
] | [
"num \\<Rightarrow> num",
"num",
"num \\<Rightarrow> num",
"num \\<Rightarrow> 'a",
"'a"
] | [
"datatype num = One | Bit0 num | Bit1 num",
"primrec numeral :: \\<open>num \\<Rightarrow> 'a\\<close>\n where\n numeral_One: \\<open>numeral One = 1\\<close>\n | numeral_Bit0: \\<open>numeral (Bit0 n) = numeral n + numeral n\\<close>\n | numeral_Bit1: \\<open>numeral (Bit1 n) = numeral n + numeral n + 1\\<... | lemma_object | ###symbols
Num.num.Bit1 :::: num \<Rightarrow> num
Num.num.One :::: num
Num.num.Bit0 :::: num \<Rightarrow> num
Num.numeral_class.numeral :::: num \<Rightarrow> 'a
Groups.one_class.one :::: 'a
###defs
datatype num = One | Bit0 num | Bit1 num
primrec numeral :: \<open>num \<Rightarrow> 'a\<close>
where
numera... |
###output
?x = 1 \<or> ?x = 2 \<or> ?x = 3 \<or> ?x = 4 \<or> ?x = 5 \<or> ?x = 6 \<or> ?x = 7###end |
Octonions/Cross_Product_7 | Cross_Product_7.forall_7 | lemma forall_7: "(\<forall>i::7. P i) \<longleftrightarrow> P 1 \<and> P 2 \<and> P 3\<and> P 4 \<and> P 5 \<and> P 6\<and> P 7 " | (\<forall>i. ?P i) = (?P 1 \<and> ?P 2 \<and> ?P 3 \<and> ?P 4 \<and> ?P 5 \<and> ?P 6 \<and> ?P 7) | (\<forall>y_0. x_1 y_0) = (x_1 ?H1 \<and> x_1 (?H2 (?H3 ?H4)) \<and> x_1 (?H2 (?H5 ?H4)) \<and> x_1 (?H2 (?H3 (?H3 ?H4))) \<and> x_1 (?H2 (?H5 (?H3 ?H4))) \<and> x_1 (?H2 (?H3 (?H5 ?H4))) \<and> x_1 (?H2 (?H5 (?H5 ?H4)))) | [
"Num.num.Bit1",
"Num.num.One",
"Num.num.Bit0",
"Num.numeral_class.numeral",
"Groups.one_class.one"
] | [
"num \\<Rightarrow> num",
"num",
"num \\<Rightarrow> num",
"num \\<Rightarrow> 'a",
"'a"
] | [
"datatype num = One | Bit0 num | Bit1 num",
"primrec numeral :: \\<open>num \\<Rightarrow> 'a\\<close>\n where\n numeral_One: \\<open>numeral One = 1\\<close>\n | numeral_Bit0: \\<open>numeral (Bit0 n) = numeral n + numeral n\\<close>\n | numeral_Bit1: \\<open>numeral (Bit1 n) = numeral n + numeral n + 1\\<... | lemma_object | ###symbols
Num.num.Bit1 :::: num \<Rightarrow> num
Num.num.One :::: num
Num.num.Bit0 :::: num \<Rightarrow> num
Num.numeral_class.numeral :::: num \<Rightarrow> 'a
Groups.one_class.one :::: 'a
###defs
datatype num = One | Bit0 num | Bit1 num
primrec numeral :: \<open>num \<Rightarrow> 'a\<close>
where
numera... |
###output
(\<forall>i. ?P i) = (?P 1 \<and> ?P 2 \<and> ?P 3 \<and> ?P 4 \<and> ?P 5 \<and> ?P 6 \<and> ?P 7)###end |
Octonions/Cross_Product_7 | Cross_Product_7.sum_7 | lemma sum_7: "sum f (UNIV::7 set) = f 1 + f 2 + f 3 + f 4 + f 5 + f 6 + f 7" | sum ?f UNIV = ?f 1 + ?f 2 + ?f 3 + ?f 4 + ?f 5 + ?f 6 + ?f 7 | ?H1 x_1 ?H2 = ?H3 (?H3 (?H3 (?H3 (?H3 (?H3 (x_1 ?H4) (x_1 (?H5 (?H6 ?H7)))) (x_1 (?H5 (?H8 ?H7)))) (x_1 (?H5 (?H6 (?H6 ?H7))))) (x_1 (?H5 (?H8 (?H6 ?H7))))) (x_1 (?H5 (?H6 (?H8 ?H7))))) (x_1 (?H5 (?H8 (?H8 ?H7)))) | [
"Num.num.Bit1",
"Num.num.One",
"Num.num.Bit0",
"Num.numeral_class.numeral",
"Groups.one_class.one",
"Groups.plus_class.plus",
"Set.UNIV",
"Groups_Big.comm_monoid_add_class.sum"
] | [
"num \\<Rightarrow> num",
"num",
"num \\<Rightarrow> num",
"num \\<Rightarrow> 'a",
"'a",
"'a \\<Rightarrow> 'a \\<Rightarrow> 'a",
"'a set",
"('a \\<Rightarrow> 'b) \\<Rightarrow> 'a set \\<Rightarrow> 'b"
] | [
"datatype num = One | Bit0 num | Bit1 num",
"primrec numeral :: \\<open>num \\<Rightarrow> 'a\\<close>\n where\n numeral_One: \\<open>numeral One = 1\\<close>\n | numeral_Bit0: \\<open>numeral (Bit0 n) = numeral n + numeral n\\<close>\n | numeral_Bit1: \\<open>numeral (Bit1 n) = numeral n + numeral n + 1\\<... | lemma_object | ###symbols
Num.num.Bit1 :::: num \<Rightarrow> num
Num.num.One :::: num
Num.num.Bit0 :::: num \<Rightarrow> num
Num.numeral_class.numeral :::: num \<Rightarrow> 'a
Groups.one_class.one :::: 'a
Groups.plus_class.plus :::: 'a \<Rightarrow> 'a \<Rightarrow> 'a
Set.UNIV :::: 'a set
Groups_Big.comm_monoid_add_class.s... |
###output
sum ?f UNIV = ?f 1 + ?f 2 + ?f 3 + ?f 4 + ?f 5 + ?f 6 + ?f 7###end |
Octonions/Cross_Product_7 | Cross_Product_7.forall_vector_7 | lemma forall_vector_7:
"(\<forall>v::'a::zero^7. P v) \<longleftrightarrow> (\<forall>x1 x2 x3 x4 x5 x6 x7. P(vector[x1, x2, x3, x4, x5, x6, x7]))" | (\<forall>v. ?P v) = (\<forall>x1 x2 x3 x4 x5 x6 x7. ?P (vector [x1, x2, x3, x4, x5, x6, x7])) | (\<forall>y_0. x_1 y_0) = (\<forall>y_1 y_2 y_3 y_4 y_5 y_6 y_7. x_1 (?H1 (?H2 y_1 (?H2 y_2 (?H2 y_3 (?H2 y_4 (?H2 y_5 (?H2 y_6 (?H2 y_7 ?H3))))))))) | [
"List.list.Nil",
"List.list.Cons",
"Cartesian_Space.vector"
] | [
"'a list",
"'a \\<Rightarrow> 'a list \\<Rightarrow> 'a list",
"'a list \\<Rightarrow> ('a, 'b) vec"
] | [
"definition \"vector l = (\\<chi> i. foldr (\\<lambda>x f n. fun_upd (f (n+1)) n x) l (\\<lambda>n x. 0) 1 i)\""
] | lemma_object | ###symbols
List.list.Nil :::: 'a list
List.list.Cons :::: 'a \<Rightarrow> 'a list \<Rightarrow> 'a list
Cartesian_Space.vector :::: 'a list \<Rightarrow> ('a, 'b) vec
###defs
definition "vector l = (\<chi> i. foldr (\<lambda>x f n. fun_upd (f (n+1)) n x) l (\<lambda>n x. 0) 1 i)" |
###output
(\<forall>v. ?P v) = (\<forall>x1 x2 x3 x4 x5 x6 x7. ?P (vector [x1, x2, x3, x4, x5, x6, x7]))###end |
Octonions/Cross_Product_7 | Cross_Product_7.norm_cross7_dot_magnitude | lemma norm_cross7_dot_magnitude: "(norm (x \<times>\<^sub>7 y))\<^sup>2 = (norm x)\<^sup>2 * (norm y)\<^sup>2 - (x \<bullet> y)\<^sup>2" | (norm (?x \<times>\<^sub>7 ?y))\<^sup>2 = (norm ?x)\<^sup>2 * (norm ?y)\<^sup>2 - (?x \<bullet> ?y)\<^sup>2 | ?H1 (?H2 (?H3 x_1 x_2)) = ?H4 (?H5 (?H1 (?H2 x_1)) (?H1 (?H2 x_2))) (?H1 (?H6 x_1 x_2)) | [
"Inner_Product.real_inner_class.inner",
"Groups.times_class.times",
"Groups.minus_class.minus",
"Cross_Product_7.cross7",
"Real_Vector_Spaces.norm_class.norm",
"Power.power_class.power2"
] | [
"'a \\<Rightarrow> 'a \\<Rightarrow> real",
"'a \\<Rightarrow> 'a \\<Rightarrow> 'a",
"'a \\<Rightarrow> 'a \\<Rightarrow> 'a",
"(real, 7) vec \\<Rightarrow> (real, 7) vec \\<Rightarrow> (real, 7) vec",
"'a \\<Rightarrow> real",
"'a \\<Rightarrow> 'a"
] | [
"class real_inner = real_vector + sgn_div_norm + dist_norm + uniformity_dist + open_uniformity +\n fixes inner :: \"'a \\<Rightarrow> 'a \\<Rightarrow> real\"\n assumes inner_commute: \"inner x y = inner y x\"\n and inner_add_left: \"inner (x + y) z = inner x z + inner y z\"\n and inner_scaleR_left [simp]: \"in... | lemma_object | ###symbols
Inner_Product.real_inner_class.inner :::: 'a \<Rightarrow> 'a \<Rightarrow> real
Groups.times_class.times :::: 'a \<Rightarrow> 'a \<Rightarrow> 'a
Groups.minus_class.minus :::: 'a \<Rightarrow> 'a \<Rightarrow> 'a
Cross_Product_7.cross7 :::: (real, 7) vec \<Rightarrow> (real, 7) vec \<Rightarrow> (real, ... |
###output
(norm (?x \<times>\<^sub>7 ?y))\<^sup>2 = (norm ?x)\<^sup>2 * (norm ?y)\<^sup>2 - (?x \<bullet> ?y)\<^sup>2###end |
Octonions/Cross_Product_7 | Cross_Product_7.vector_7(7) | lemma vector_7 [simp]:
"(vector [x1,x2,x3,x4,x5,x6,x7] ::('a::zero)^7)$1 = x1"
"(vector [x1,x2,x3,x4,x5,x6,x7] ::('a::zero)^7)$2 = x2"
"(vector [x1,x2,x3,x4,x5,x6,x7] ::('a::zero)^7)$3 = x3"
"(vector [x1,x2,x3,x4,x5,x6,x7] ::('a::zero)^7)$4 = x4"
"(vector [x1,x2,x3,x4,x5,x6,x7] ::('a::zero)^7)$5 = x5"
"(vector [x... | vector [ ?x1.0, ?x2.0, ?x3.0, ?x4.0, ?x5.0, ?x6.0, ?x7.0] $ 7 = ?x7.0 | ?H1 (?H2 (?H3 x_1 (?H3 x_2 (?H3 x_3 (?H3 x_4 (?H3 x_5 (?H3 x_6 (?H3 x_7 ?H4)))))))) (?H5 (?H6 (?H6 ?H7))) = x_7 | [
"Num.num.One",
"Num.num.Bit1",
"Num.numeral_class.numeral",
"List.list.Nil",
"List.list.Cons",
"Cartesian_Space.vector",
"Finite_Cartesian_Product.vec.vec_nth"
] | [
"num",
"num \\<Rightarrow> num",
"num \\<Rightarrow> 'a",
"'a list",
"'a \\<Rightarrow> 'a list \\<Rightarrow> 'a list",
"'a list \\<Rightarrow> ('a, 'b) vec",
"('a, 'b) vec \\<Rightarrow> 'b \\<Rightarrow> 'a"
] | [
"datatype num = One | Bit0 num | Bit1 num",
"primrec numeral :: \\<open>num \\<Rightarrow> 'a\\<close>\n where\n numeral_One: \\<open>numeral One = 1\\<close>\n | numeral_Bit0: \\<open>numeral (Bit0 n) = numeral n + numeral n\\<close>\n | numeral_Bit1: \\<open>numeral (Bit1 n) = numeral n + numeral n + 1\\<... | lemma_object | ###symbols
Num.num.One :::: num
Num.num.Bit1 :::: num \<Rightarrow> num
Num.numeral_class.numeral :::: num \<Rightarrow> 'a
List.list.Nil :::: 'a list
List.list.Cons :::: 'a \<Rightarrow> 'a list \<Rightarrow> 'a list
Cartesian_Space.vector :::: 'a list \<Rightarrow> ('a, 'b) vec
Finite_Cartesian_Product.vec.vec_... |
###output
vector [ ?x1.0, ?x2.0, ?x3.0, ?x4.0, ?x5.0, ?x6.0, ?x7.0] $ 7 = ?x7.0###end |
Octonions/Cross_Product_7 | Cross_Product_7.vector_7(5) | lemma vector_7 [simp]:
"(vector [x1,x2,x3,x4,x5,x6,x7] ::('a::zero)^7)$1 = x1"
"(vector [x1,x2,x3,x4,x5,x6,x7] ::('a::zero)^7)$2 = x2"
"(vector [x1,x2,x3,x4,x5,x6,x7] ::('a::zero)^7)$3 = x3"
"(vector [x1,x2,x3,x4,x5,x6,x7] ::('a::zero)^7)$4 = x4"
"(vector [x1,x2,x3,x4,x5,x6,x7] ::('a::zero)^7)$5 = x5"
"(vector [x... | vector [ ?x1.0, ?x2.0, ?x3.0, ?x4.0, ?x5.0, ?x6.0, ?x7.0] $ 5 = ?x5.0 | ?H1 (?H2 (?H3 x_1 (?H3 x_2 (?H3 x_3 (?H3 x_4 (?H3 x_5 (?H3 x_6 (?H3 x_7 ?H4)))))))) (?H5 (?H6 (?H7 ?H8))) = x_5 | [
"Num.num.One",
"Num.num.Bit0",
"Num.num.Bit1",
"Num.numeral_class.numeral",
"List.list.Nil",
"List.list.Cons",
"Cartesian_Space.vector",
"Finite_Cartesian_Product.vec.vec_nth"
] | [
"num",
"num \\<Rightarrow> num",
"num \\<Rightarrow> num",
"num \\<Rightarrow> 'a",
"'a list",
"'a \\<Rightarrow> 'a list \\<Rightarrow> 'a list",
"'a list \\<Rightarrow> ('a, 'b) vec",
"('a, 'b) vec \\<Rightarrow> 'b \\<Rightarrow> 'a"
] | [
"datatype num = One | Bit0 num | Bit1 num",
"primrec numeral :: \\<open>num \\<Rightarrow> 'a\\<close>\n where\n numeral_One: \\<open>numeral One = 1\\<close>\n | numeral_Bit0: \\<open>numeral (Bit0 n) = numeral n + numeral n\\<close>\n | numeral_Bit1: \\<open>numeral (Bit1 n) = numeral n + numeral n + 1\\<... | lemma_object | ###symbols
Num.num.One :::: num
Num.num.Bit0 :::: num \<Rightarrow> num
Num.num.Bit1 :::: num \<Rightarrow> num
Num.numeral_class.numeral :::: num \<Rightarrow> 'a
List.list.Nil :::: 'a list
List.list.Cons :::: 'a \<Rightarrow> 'a list \<Rightarrow> 'a list
Cartesian_Space.vector :::: 'a list \<Rightarrow> ('a, '... |
###output
vector [ ?x1.0, ?x2.0, ?x3.0, ?x4.0, ?x5.0, ?x6.0, ?x7.0] $ 5 = ?x5.0###end |
Octonions/Cross_Product_7 | Cross_Product_7.vector_7(6) | lemma vector_7 [simp]:
"(vector [x1,x2,x3,x4,x5,x6,x7] ::('a::zero)^7)$1 = x1"
"(vector [x1,x2,x3,x4,x5,x6,x7] ::('a::zero)^7)$2 = x2"
"(vector [x1,x2,x3,x4,x5,x6,x7] ::('a::zero)^7)$3 = x3"
"(vector [x1,x2,x3,x4,x5,x6,x7] ::('a::zero)^7)$4 = x4"
"(vector [x1,x2,x3,x4,x5,x6,x7] ::('a::zero)^7)$5 = x5"
"(vector [x... | vector [ ?x1.0, ?x2.0, ?x3.0, ?x4.0, ?x5.0, ?x6.0, ?x7.0] $ 6 = ?x6.0 | ?H1 (?H2 (?H3 x_1 (?H3 x_2 (?H3 x_3 (?H3 x_4 (?H3 x_5 (?H3 x_6 (?H3 x_7 ?H4)))))))) (?H5 (?H6 (?H7 ?H8))) = x_6 | [
"Num.num.One",
"Num.num.Bit1",
"Num.num.Bit0",
"Num.numeral_class.numeral",
"List.list.Nil",
"List.list.Cons",
"Cartesian_Space.vector",
"Finite_Cartesian_Product.vec.vec_nth"
] | [
"num",
"num \\<Rightarrow> num",
"num \\<Rightarrow> num",
"num \\<Rightarrow> 'a",
"'a list",
"'a \\<Rightarrow> 'a list \\<Rightarrow> 'a list",
"'a list \\<Rightarrow> ('a, 'b) vec",
"('a, 'b) vec \\<Rightarrow> 'b \\<Rightarrow> 'a"
] | [
"datatype num = One | Bit0 num | Bit1 num",
"primrec numeral :: \\<open>num \\<Rightarrow> 'a\\<close>\n where\n numeral_One: \\<open>numeral One = 1\\<close>\n | numeral_Bit0: \\<open>numeral (Bit0 n) = numeral n + numeral n\\<close>\n | numeral_Bit1: \\<open>numeral (Bit1 n) = numeral n + numeral n + 1\\<... | lemma_object | ###symbols
Num.num.One :::: num
Num.num.Bit1 :::: num \<Rightarrow> num
Num.num.Bit0 :::: num \<Rightarrow> num
Num.numeral_class.numeral :::: num \<Rightarrow> 'a
List.list.Nil :::: 'a list
List.list.Cons :::: 'a \<Rightarrow> 'a list \<Rightarrow> 'a list
Cartesian_Space.vector :::: 'a list \<Rightarrow> ('a, '... |
###output
vector [ ?x1.0, ?x2.0, ?x3.0, ?x4.0, ?x5.0, ?x6.0, ?x7.0] $ 6 = ?x6.0###end |
Octonions/Cross_Product_7 | Cross_Product_7.vector_7(4) | lemma vector_7 [simp]:
"(vector [x1,x2,x3,x4,x5,x6,x7] ::('a::zero)^7)$1 = x1"
"(vector [x1,x2,x3,x4,x5,x6,x7] ::('a::zero)^7)$2 = x2"
"(vector [x1,x2,x3,x4,x5,x6,x7] ::('a::zero)^7)$3 = x3"
"(vector [x1,x2,x3,x4,x5,x6,x7] ::('a::zero)^7)$4 = x4"
"(vector [x1,x2,x3,x4,x5,x6,x7] ::('a::zero)^7)$5 = x5"
"(vector [x... | vector [ ?x1.0, ?x2.0, ?x3.0, ?x4.0, ?x5.0, ?x6.0, ?x7.0] $ 4 = ?x4.0 | ?H1 (?H2 (?H3 x_1 (?H3 x_2 (?H3 x_3 (?H3 x_4 (?H3 x_5 (?H3 x_6 (?H3 x_7 ?H4)))))))) (?H5 (?H6 (?H6 ?H7))) = x_4 | [
"Num.num.One",
"Num.num.Bit0",
"Num.numeral_class.numeral",
"List.list.Nil",
"List.list.Cons",
"Cartesian_Space.vector",
"Finite_Cartesian_Product.vec.vec_nth"
] | [
"num",
"num \\<Rightarrow> num",
"num \\<Rightarrow> 'a",
"'a list",
"'a \\<Rightarrow> 'a list \\<Rightarrow> 'a list",
"'a list \\<Rightarrow> ('a, 'b) vec",
"('a, 'b) vec \\<Rightarrow> 'b \\<Rightarrow> 'a"
] | [
"datatype num = One | Bit0 num | Bit1 num",
"primrec numeral :: \\<open>num \\<Rightarrow> 'a\\<close>\n where\n numeral_One: \\<open>numeral One = 1\\<close>\n | numeral_Bit0: \\<open>numeral (Bit0 n) = numeral n + numeral n\\<close>\n | numeral_Bit1: \\<open>numeral (Bit1 n) = numeral n + numeral n + 1\\<... | lemma_object | ###symbols
Num.num.One :::: num
Num.num.Bit0 :::: num \<Rightarrow> num
Num.numeral_class.numeral :::: num \<Rightarrow> 'a
List.list.Nil :::: 'a list
List.list.Cons :::: 'a \<Rightarrow> 'a list \<Rightarrow> 'a list
Cartesian_Space.vector :::: 'a list \<Rightarrow> ('a, 'b) vec
Finite_Cartesian_Product.vec.vec_... |
###output
vector [ ?x1.0, ?x2.0, ?x3.0, ?x4.0, ?x5.0, ?x6.0, ?x7.0] $ 4 = ?x4.0###end |
Octonions/Cross_Product_7 | Cross_Product_7.vector_7(3) | lemma vector_7 [simp]:
"(vector [x1,x2,x3,x4,x5,x6,x7] ::('a::zero)^7)$1 = x1"
"(vector [x1,x2,x3,x4,x5,x6,x7] ::('a::zero)^7)$2 = x2"
"(vector [x1,x2,x3,x4,x5,x6,x7] ::('a::zero)^7)$3 = x3"
"(vector [x1,x2,x3,x4,x5,x6,x7] ::('a::zero)^7)$4 = x4"
"(vector [x1,x2,x3,x4,x5,x6,x7] ::('a::zero)^7)$5 = x5"
"(vector [x... | vector [ ?x1.0, ?x2.0, ?x3.0, ?x4.0, ?x5.0, ?x6.0, ?x7.0] $ 3 = ?x3.0 | ?H1 (?H2 (?H3 x_1 (?H3 x_2 (?H3 x_3 (?H3 x_4 (?H3 x_5 (?H3 x_6 (?H3 x_7 ?H4)))))))) (?H5 (?H6 ?H7)) = x_3 | [
"Num.num.One",
"Num.num.Bit1",
"Num.numeral_class.numeral",
"List.list.Nil",
"List.list.Cons",
"Cartesian_Space.vector",
"Finite_Cartesian_Product.vec.vec_nth"
] | [
"num",
"num \\<Rightarrow> num",
"num \\<Rightarrow> 'a",
"'a list",
"'a \\<Rightarrow> 'a list \\<Rightarrow> 'a list",
"'a list \\<Rightarrow> ('a, 'b) vec",
"('a, 'b) vec \\<Rightarrow> 'b \\<Rightarrow> 'a"
] | [
"datatype num = One | Bit0 num | Bit1 num",
"primrec numeral :: \\<open>num \\<Rightarrow> 'a\\<close>\n where\n numeral_One: \\<open>numeral One = 1\\<close>\n | numeral_Bit0: \\<open>numeral (Bit0 n) = numeral n + numeral n\\<close>\n | numeral_Bit1: \\<open>numeral (Bit1 n) = numeral n + numeral n + 1\\<... | lemma_object | ###symbols
Num.num.One :::: num
Num.num.Bit1 :::: num \<Rightarrow> num
Num.numeral_class.numeral :::: num \<Rightarrow> 'a
List.list.Nil :::: 'a list
List.list.Cons :::: 'a \<Rightarrow> 'a list \<Rightarrow> 'a list
Cartesian_Space.vector :::: 'a list \<Rightarrow> ('a, 'b) vec
Finite_Cartesian_Product.vec.vec_... |
###output
vector [ ?x1.0, ?x2.0, ?x3.0, ?x4.0, ?x5.0, ?x6.0, ?x7.0] $ 3 = ?x3.0###end |
Octonions/Cross_Product_7 | Cross_Product_7.vector_7(2) | lemma vector_7 [simp]:
"(vector [x1,x2,x3,x4,x5,x6,x7] ::('a::zero)^7)$1 = x1"
"(vector [x1,x2,x3,x4,x5,x6,x7] ::('a::zero)^7)$2 = x2"
"(vector [x1,x2,x3,x4,x5,x6,x7] ::('a::zero)^7)$3 = x3"
"(vector [x1,x2,x3,x4,x5,x6,x7] ::('a::zero)^7)$4 = x4"
"(vector [x1,x2,x3,x4,x5,x6,x7] ::('a::zero)^7)$5 = x5"
"(vector [x... | vector [ ?x1.0, ?x2.0, ?x3.0, ?x4.0, ?x5.0, ?x6.0, ?x7.0] $ 2 = ?x2.0 | ?H1 (?H2 (?H3 x_1 (?H3 x_2 (?H3 x_3 (?H3 x_4 (?H3 x_5 (?H3 x_6 (?H3 x_7 ?H4)))))))) (?H5 (?H6 ?H7)) = x_2 | [
"Num.num.One",
"Num.num.Bit0",
"Num.numeral_class.numeral",
"List.list.Nil",
"List.list.Cons",
"Cartesian_Space.vector",
"Finite_Cartesian_Product.vec.vec_nth"
] | [
"num",
"num \\<Rightarrow> num",
"num \\<Rightarrow> 'a",
"'a list",
"'a \\<Rightarrow> 'a list \\<Rightarrow> 'a list",
"'a list \\<Rightarrow> ('a, 'b) vec",
"('a, 'b) vec \\<Rightarrow> 'b \\<Rightarrow> 'a"
] | [
"datatype num = One | Bit0 num | Bit1 num",
"primrec numeral :: \\<open>num \\<Rightarrow> 'a\\<close>\n where\n numeral_One: \\<open>numeral One = 1\\<close>\n | numeral_Bit0: \\<open>numeral (Bit0 n) = numeral n + numeral n\\<close>\n | numeral_Bit1: \\<open>numeral (Bit1 n) = numeral n + numeral n + 1\\<... | lemma_object | ###symbols
Num.num.One :::: num
Num.num.Bit0 :::: num \<Rightarrow> num
Num.numeral_class.numeral :::: num \<Rightarrow> 'a
List.list.Nil :::: 'a list
List.list.Cons :::: 'a \<Rightarrow> 'a list \<Rightarrow> 'a list
Cartesian_Space.vector :::: 'a list \<Rightarrow> ('a, 'b) vec
Finite_Cartesian_Product.vec.vec_... |
###output
vector [ ?x1.0, ?x2.0, ?x3.0, ?x4.0, ?x5.0, ?x6.0, ?x7.0] $ 2 = ?x2.0###end |
Octonions/Cross_Product_7 | Cross_Product_7.vector_7(1) | lemma vector_7 [simp]:
"(vector [x1,x2,x3,x4,x5,x6,x7] ::('a::zero)^7)$1 = x1"
"(vector [x1,x2,x3,x4,x5,x6,x7] ::('a::zero)^7)$2 = x2"
"(vector [x1,x2,x3,x4,x5,x6,x7] ::('a::zero)^7)$3 = x3"
"(vector [x1,x2,x3,x4,x5,x6,x7] ::('a::zero)^7)$4 = x4"
"(vector [x1,x2,x3,x4,x5,x6,x7] ::('a::zero)^7)$5 = x5"
"(vector [x... | vector [ ?x1.0, ?x2.0, ?x3.0, ?x4.0, ?x5.0, ?x6.0, ?x7.0] $ 1 = ?x1.0 | ?H1 (?H2 (?H3 x_1 (?H3 x_2 (?H3 x_3 (?H3 x_4 (?H3 x_5 (?H3 x_6 (?H3 x_7 ?H4)))))))) ?H5 = x_1 | [
"Groups.one_class.one",
"List.list.Nil",
"List.list.Cons",
"Cartesian_Space.vector",
"Finite_Cartesian_Product.vec.vec_nth"
] | [
"'a",
"'a list",
"'a \\<Rightarrow> 'a list \\<Rightarrow> 'a list",
"'a list \\<Rightarrow> ('a, 'b) vec",
"('a, 'b) vec \\<Rightarrow> 'b \\<Rightarrow> 'a"
] | [
"class one =\n fixes one :: 'a (\\<open>1\\<close>)",
"definition \"vector l = (\\<chi> i. foldr (\\<lambda>x f n. fun_upd (f (n+1)) n x) l (\\<lambda>n x. 0) 1 i)\"",
"definition \"vec x = (\\<chi> i. x)\""
] | lemma_object | ###symbols
Groups.one_class.one :::: 'a
List.list.Nil :::: 'a list
List.list.Cons :::: 'a \<Rightarrow> 'a list \<Rightarrow> 'a list
Cartesian_Space.vector :::: 'a list \<Rightarrow> ('a, 'b) vec
Finite_Cartesian_Product.vec.vec_nth :::: ('a, 'b) vec \<Rightarrow> 'b \<Rightarrow> 'a
###defs
class one =
fixes on... |
###output
vector [ ?x1.0, ?x2.0, ?x3.0, ?x4.0, ?x5.0, ?x6.0, ?x7.0] $ 1 = ?x1.0###end |
Octonions/Cross_Product_7 | Cross_Product_7.norm_and_cross7_eq_0 | lemma norm_and_cross7_eq_0:
"x \<bullet> y = 0 \<and> x \<times>\<^sub>7 y = 0 \<longleftrightarrow> x = 0 \<or> y = 0" (is "?lhs = ?rhs") | (?x \<bullet> ?y = 0 \<and> ?x \<times>\<^sub>7 ?y = 0) = (?x = 0 \<or> ?y = 0) | (?H1 x_1 x_2 = ?H2 \<and> ?H3 x_1 x_2 = ?H2) = (x_1 = ?H2 \<or> x_2 = ?H2) | [
"Cross_Product_7.cross7",
"Groups.zero_class.zero",
"Inner_Product.real_inner_class.inner"
] | [
"(real, 7) vec \\<Rightarrow> (real, 7) vec \\<Rightarrow> (real, 7) vec",
"'a",
"'a \\<Rightarrow> 'a \\<Rightarrow> real"
] | [
"definition cross7 :: \"[real^7, real^7] \\<Rightarrow> real^7\" (infixr \\<open>\\<times>\\<^sub>7\\<close> 80)\n where \"a \\<times>\\<^sub>7 b \\<equiv>\n vector [a$2 * b$4 - a$4 * b$2 + a$3 * b$7 - a$7 * b$3 + a$5 * b$6 - a$6 * b$5 ,\n a$3 * b$5 - a$5 * b$3 + a$4 * b$1 - a$1 * b$4 + a$6 * ... | lemma_object | ###symbols
Cross_Product_7.cross7 :::: (real, 7) vec \<Rightarrow> (real, 7) vec \<Rightarrow> (real, 7) vec
Groups.zero_class.zero :::: 'a
Inner_Product.real_inner_class.inner :::: 'a \<Rightarrow> 'a \<Rightarrow> real
###defs
definition cross7 :: "[real^7, real^7] \<Rightarrow> real^7" (infixr \<open>\<times>\<^s... |
###output
(?x \<bullet> ?y = 0 \<and> ?x \<times>\<^sub>7 ?y = 0) = (?x = 0 \<or> ?y = 0)###end |
Octonions/Cross_Product_7 | Cross_Product_7.continuous_on_cross | lemma continuous_on_cross:
fixes f :: "'a::t2_space \<Rightarrow> real^7"
shows "\<lbrakk>continuous_on S f; continuous_on S g\<rbrakk> \<Longrightarrow> continuous_on S (\<lambda>x. f x \<times>\<^sub>7 g x)" | continuous_on ?S ?f \<Longrightarrow> continuous_on ?S ?g \<Longrightarrow> continuous_on ?S (\<lambda>x. ?f x \<times>\<^sub>7 ?g x) | \<lbrakk> ?H1 x_1 x_2; ?H1 x_1 x_3\<rbrakk> \<Longrightarrow> ?H1 x_1 (\<lambda>y_0. ?H2 (x_2 y_0) (x_3 y_0)) | [
"Cross_Product_7.cross7",
"Topological_Spaces.continuous_on"
] | [
"(real, 7) vec \\<Rightarrow> (real, 7) vec \\<Rightarrow> (real, 7) vec",
"'a set \\<Rightarrow> ('a \\<Rightarrow> 'b) \\<Rightarrow> bool"
] | [
"definition cross7 :: \"[real^7, real^7] \\<Rightarrow> real^7\" (infixr \\<open>\\<times>\\<^sub>7\\<close> 80)\n where \"a \\<times>\\<^sub>7 b \\<equiv>\n vector [a$2 * b$4 - a$4 * b$2 + a$3 * b$7 - a$7 * b$3 + a$5 * b$6 - a$6 * b$5 ,\n a$3 * b$5 - a$5 * b$3 + a$4 * b$1 - a$1 * b$4 + a$6 * ... | lemma_object | ###symbols
Cross_Product_7.cross7 :::: (real, 7) vec \<Rightarrow> (real, 7) vec \<Rightarrow> (real, 7) vec
Topological_Spaces.continuous_on :::: 'a set \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> bool
###defs
definition cross7 :: "[real^7, real^7] \<Rightarrow> real^7" (infixr \<open>\<times>\<^sub>7\<close> ... |
###output
continuous_on ?S ?f \<Longrightarrow> continuous_on ?S ?g \<Longrightarrow> continuous_on ?S (\<lambda>x. ?f x \<times>\<^sub>7 ?g x)###end |
Octonions/Cross_Product_7 | Cross_Product_7.continuous_cross7 | lemma continuous_cross7: "\<lbrakk>continuous F f; continuous F g\<rbrakk> \<Longrightarrow> continuous F (\<lambda>x. f x \<times>\<^sub>7 g x)" | continuous ?F ?f \<Longrightarrow> continuous ?F ?g \<Longrightarrow> continuous ?F (\<lambda>x. ?f x \<times>\<^sub>7 ?g x) | \<lbrakk> ?H1 x_1 x_2; ?H1 x_1 x_3\<rbrakk> \<Longrightarrow> ?H1 x_1 (\<lambda>y_0. ?H2 (x_2 y_0) (x_3 y_0)) | [
"Cross_Product_7.cross7",
"Topological_Spaces.continuous"
] | [
"(real, 7) vec \\<Rightarrow> (real, 7) vec \\<Rightarrow> (real, 7) vec",
"'a filter \\<Rightarrow> ('a \\<Rightarrow> 'b) \\<Rightarrow> bool"
] | [
"definition cross7 :: \"[real^7, real^7] \\<Rightarrow> real^7\" (infixr \\<open>\\<times>\\<^sub>7\\<close> 80)\n where \"a \\<times>\\<^sub>7 b \\<equiv>\n vector [a$2 * b$4 - a$4 * b$2 + a$3 * b$7 - a$7 * b$3 + a$5 * b$6 - a$6 * b$5 ,\n a$3 * b$5 - a$5 * b$3 + a$4 * b$1 - a$1 * b$4 + a$6 * ... | lemma_object | ###symbols
Cross_Product_7.cross7 :::: (real, 7) vec \<Rightarrow> (real, 7) vec \<Rightarrow> (real, 7) vec
Topological_Spaces.continuous :::: 'a filter \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> bool
###defs
definition cross7 :: "[real^7, real^7] \<Rightarrow> real^7" (infixr \<open>\<times>\<^sub>7\<close> ... |
###output
continuous ?F ?f \<Longrightarrow> continuous ?F ?g \<Longrightarrow> continuous ?F (\<lambda>x. ?f x \<times>\<^sub>7 ?g x)###end |
Octonions/Cross_Product_7 | Cross_Product_7.norm_square_vec_eq | lemma norm_square_vec_eq: "norm x ^ 2 = (\<Sum>i\<in>UNIV. x $ i ^ 2)" | (norm ?x)\<^sup>2 = (\<Sum>i\<in>UNIV. (?x $ i)\<^sup>2) | ?H1 (?H2 x_1) = ?H3 (\<lambda>y_0. ?H1 (?H4 x_1 y_0)) ?H5 | [
"Set.UNIV",
"Finite_Cartesian_Product.vec.vec_nth",
"Groups_Big.comm_monoid_add_class.sum",
"Real_Vector_Spaces.norm_class.norm",
"Power.power_class.power2"
] | [
"'a set",
"('a, 'b) vec \\<Rightarrow> 'b \\<Rightarrow> 'a",
"('a \\<Rightarrow> 'b) \\<Rightarrow> 'a set \\<Rightarrow> 'b",
"'a \\<Rightarrow> real",
"'a \\<Rightarrow> 'a"
] | [
"abbreviation UNIV :: \"'a set\"\n where \"UNIV \\<equiv> top\"",
"definition \"vec x = (\\<chi> i. x)\"",
"class norm =\n fixes norm :: \"'a \\<Rightarrow> real\"",
"primrec power :: \"'a \\<Rightarrow> nat \\<Rightarrow> 'a\" (infixr \\<open>^\\<close> 80)\n where\n power_0: \"a ^ 0 = 1\"\n | power_... | lemma_object | ###symbols
Set.UNIV :::: 'a set
Finite_Cartesian_Product.vec.vec_nth :::: ('a, 'b) vec \<Rightarrow> 'b \<Rightarrow> 'a
Groups_Big.comm_monoid_add_class.sum :::: ('a \<Rightarrow> 'b) \<Rightarrow> 'a set \<Rightarrow> 'b
Real_Vector_Spaces.norm_class.norm :::: 'a \<Rightarrow> real
Power.power_class.power2 :::: '... |
###output
(norm ?x)\<^sup>2 = (\<Sum>i\<in>UNIV. (?x $ i)\<^sup>2)###end |
Octonions/Cross_Product_7 | Cross_Product_7.cross7_eq_0 | lemma cross7_eq_0: "x \<times>\<^sub>7 y = 0 \<longleftrightarrow> collinear {0, x, y}" | (?x \<times>\<^sub>7 ?y = 0) = collinear {0, ?x, ?y} | (?H1 x_1 x_2 = ?H2) = ?H3 (?H4 ?H2 (?H4 x_1 (?H4 x_2 ?H5))) | [
"Set.empty",
"Set.insert",
"Linear_Algebra.collinear",
"Groups.zero_class.zero",
"Cross_Product_7.cross7"
] | [
"'a set",
"'a \\<Rightarrow> 'a set \\<Rightarrow> 'a set",
"'a set \\<Rightarrow> bool",
"'a",
"(real, 7) vec \\<Rightarrow> (real, 7) vec \\<Rightarrow> (real, 7) vec"
] | [
"abbreviation empty :: \"'a set\" (\\<open>{}\\<close>)\n where \"{} \\<equiv> bot\"",
"definition insert :: \"'a \\<Rightarrow> 'a set \\<Rightarrow> 'a set\"\n where insert_compr: \"insert a B = {x. x = a \\<or> x \\<in> B}\"",
"class zero =\n fixes zero :: 'a (\\<open>0\\<close>)",
"definition cross7 :... | lemma_object | ###symbols
Set.empty :::: 'a set
Set.insert :::: 'a \<Rightarrow> 'a set \<Rightarrow> 'a set
Linear_Algebra.collinear :::: 'a set \<Rightarrow> bool
Groups.zero_class.zero :::: 'a
Cross_Product_7.cross7 :::: (real, 7) vec \<Rightarrow> (real, 7) vec \<Rightarrow> (real, 7) vec
###defs
abbreviation empty :: "'a set... |
###output
(?x \<times>\<^sub>7 ?y = 0) = collinear {0, ?x, ?y}###end |
Octonions/Cross_Product_7 | Cross_Product_7.axis_nth_neq | lemma axis_nth_neq [simp]: "i \<noteq> j \<Longrightarrow> axis i x $ j = 0" | ?i \<noteq> ?j \<Longrightarrow> axis ?i ?x $ ?j = (0:: ?'b) | x_1 \<noteq> x_2 \<Longrightarrow> ?H1 (?H2 x_1 x_3) x_2 = ?H3 | [
"Groups.zero_class.zero",
"Finite_Cartesian_Product.axis",
"Finite_Cartesian_Product.vec.vec_nth"
] | [
"'a",
"'a \\<Rightarrow> 'b \\<Rightarrow> ('b, 'a) vec",
"('a, 'b) vec \\<Rightarrow> 'b \\<Rightarrow> 'a"
] | [
"class zero =\n fixes zero :: 'a (\\<open>0\\<close>)",
"definition \"vec x = (\\<chi> i. x)\""
] | lemma_object | ###symbols
Groups.zero_class.zero :::: 'a
Finite_Cartesian_Product.axis :::: 'a \<Rightarrow> 'b \<Rightarrow> ('b, 'a) vec
Finite_Cartesian_Product.vec.vec_nth :::: ('a, 'b) vec \<Rightarrow> 'b \<Rightarrow> 'a
###defs
class zero =
fixes zero :: 'a (\<open>0\<close>)
definition "vec x = (\<chi> i. x)" |
###output
?i \<noteq> ?j \<Longrightarrow> axis ?i ?x $ ?j = (0:: ?'b)###end |
Octonions/Cross_Product_7 | Cross_Product_7.cross7_add_left | lemma cross7_add_left: "(x + y) \<times>\<^sub>7 z = (x \<times>\<^sub>7 z) + (y \<times>\<^sub>7 z)"
and cross7_add_right: "x \<times>\<^sub>7 (y + z) = (x \<times>\<^sub>7 y) + (x \<times>\<^sub>7 z)" | (?x + ?y) \<times>\<^sub>7 ?z = ?x \<times>\<^sub>7 ?z + ?y \<times>\<^sub>7 ?z | ?H1 (?H2 x_1 x_2) x_3 = ?H2 (?H1 x_1 x_3) (?H1 x_2 x_3) | [
"Groups.plus_class.plus",
"Cross_Product_7.cross7"
] | [
"'a \\<Rightarrow> 'a \\<Rightarrow> 'a",
"(real, 7) vec \\<Rightarrow> (real, 7) vec \\<Rightarrow> (real, 7) vec"
] | [
"class plus =\n fixes plus :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\" (infixl \\<open>+\\<close> 65)",
"definition cross7 :: \"[real^7, real^7] \\<Rightarrow> real^7\" (infixr \\<open>\\<times>\\<^sub>7\\<close> 80)\n where \"a \\<times>\\<^sub>7 b \\<equiv>\n vector [a$2 * b$4 - a$4 * b$2 + a$3 * b$7 ... | lemma_object | ###symbols
Groups.plus_class.plus :::: 'a \<Rightarrow> 'a \<Rightarrow> 'a
Cross_Product_7.cross7 :::: (real, 7) vec \<Rightarrow> (real, 7) vec \<Rightarrow> (real, 7) vec
###defs
class plus =
fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl \<open>+\<close> 65)
definition cross7 :: "[real^7, real^7]... |
###output
(?x + ?y) \<times>\<^sub>7 ?z = ?x \<times>\<^sub>7 ?z + ?y \<times>\<^sub>7 ?z###end |
Octonions/Cross_Product_7 | Cross_Product_7.left_diff_distrib | lemma left_diff_distrib: "(x - y) \<times>\<^sub>7 z = x \<times>\<^sub>7 z - y \<times>\<^sub>7 z"
and right_diff_distrib: "x \<times>\<^sub>7 (y - z) = x \<times>\<^sub>7 y - x \<times>\<^sub>7 z" | (?x - ?y) \<times>\<^sub>7 ?z = ?x \<times>\<^sub>7 ?z - ?y \<times>\<^sub>7 ?z | ?H1 (?H2 x_1 x_2) x_3 = ?H2 (?H1 x_1 x_3) (?H1 x_2 x_3) | [
"Groups.minus_class.minus",
"Cross_Product_7.cross7"
] | [
"'a \\<Rightarrow> 'a \\<Rightarrow> 'a",
"(real, 7) vec \\<Rightarrow> (real, 7) vec \\<Rightarrow> (real, 7) vec"
] | [
"class minus =\n fixes minus :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\" (infixl \\<open>-\\<close> 65)",
"definition cross7 :: \"[real^7, real^7] \\<Rightarrow> real^7\" (infixr \\<open>\\<times>\\<^sub>7\\<close> 80)\n where \"a \\<times>\\<^sub>7 b \\<equiv>\n vector [a$2 * b$4 - a$4 * b$2 + a$3 * b$... | lemma_object | ###symbols
Groups.minus_class.minus :::: 'a \<Rightarrow> 'a \<Rightarrow> 'a
Cross_Product_7.cross7 :::: (real, 7) vec \<Rightarrow> (real, 7) vec \<Rightarrow> (real, 7) vec
###defs
class minus =
fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl \<open>-\<close> 65)
definition cross7 :: "[real^7, rea... |
###output
(?x - ?y) \<times>\<^sub>7 ?z = ?x \<times>\<^sub>7 ?z - ?y \<times>\<^sub>7 ?z###end |
Octonions/Cross_Product_7 | Cross_Product_7.cross7_add_right | lemma cross7_add_left: "(x + y) \<times>\<^sub>7 z = (x \<times>\<^sub>7 z) + (y \<times>\<^sub>7 z)"
and cross7_add_right: "x \<times>\<^sub>7 (y + z) = (x \<times>\<^sub>7 y) + (x \<times>\<^sub>7 z)" | ?x \<times>\<^sub>7 (?y + ?z) = ?x \<times>\<^sub>7 ?y + ?x \<times>\<^sub>7 ?z | ?H1 x_1 (?H2 x_2 x_3) = ?H2 (?H1 x_1 x_2) (?H1 x_1 x_3) | [
"Groups.plus_class.plus",
"Cross_Product_7.cross7"
] | [
"'a \\<Rightarrow> 'a \\<Rightarrow> 'a",
"(real, 7) vec \\<Rightarrow> (real, 7) vec \\<Rightarrow> (real, 7) vec"
] | [
"class plus =\n fixes plus :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\" (infixl \\<open>+\\<close> 65)",
"definition cross7 :: \"[real^7, real^7] \\<Rightarrow> real^7\" (infixr \\<open>\\<times>\\<^sub>7\\<close> 80)\n where \"a \\<times>\\<^sub>7 b \\<equiv>\n vector [a$2 * b$4 - a$4 * b$2 + a$3 * b$7 ... | lemma_object | ###symbols
Groups.plus_class.plus :::: 'a \<Rightarrow> 'a \<Rightarrow> 'a
Cross_Product_7.cross7 :::: (real, 7) vec \<Rightarrow> (real, 7) vec \<Rightarrow> (real, 7) vec
###defs
class plus =
fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl \<open>+\<close> 65)
definition cross7 :: "[real^7, real^7]... |
###output
?x \<times>\<^sub>7 (?y + ?z) = ?x \<times>\<^sub>7 ?y + ?x \<times>\<^sub>7 ?z###end |
Octonions/Cross_Product_7 | Cross_Product_7.right_diff_distrib | lemma left_diff_distrib: "(x - y) \<times>\<^sub>7 z = x \<times>\<^sub>7 z - y \<times>\<^sub>7 z"
and right_diff_distrib: "x \<times>\<^sub>7 (y - z) = x \<times>\<^sub>7 y - x \<times>\<^sub>7 z" | ?x \<times>\<^sub>7 (?y - ?z) = ?x \<times>\<^sub>7 ?y - ?x \<times>\<^sub>7 ?z | ?H1 x_1 (?H2 x_2 x_3) = ?H2 (?H1 x_1 x_2) (?H1 x_1 x_3) | [
"Groups.minus_class.minus",
"Cross_Product_7.cross7"
] | [
"'a \\<Rightarrow> 'a \\<Rightarrow> 'a",
"(real, 7) vec \\<Rightarrow> (real, 7) vec \\<Rightarrow> (real, 7) vec"
] | [
"class minus =\n fixes minus :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\" (infixl \\<open>-\\<close> 65)",
"definition cross7 :: \"[real^7, real^7] \\<Rightarrow> real^7\" (infixr \\<open>\\<times>\\<^sub>7\\<close> 80)\n where \"a \\<times>\\<^sub>7 b \\<equiv>\n vector [a$2 * b$4 - a$4 * b$2 + a$3 * b$... | lemma_object | ###symbols
Groups.minus_class.minus :::: 'a \<Rightarrow> 'a \<Rightarrow> 'a
Cross_Product_7.cross7 :::: (real, 7) vec \<Rightarrow> (real, 7) vec \<Rightarrow> (real, 7) vec
###defs
class minus =
fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl \<open>-\<close> 65)
definition cross7 :: "[real^7, rea... |
###output
?x \<times>\<^sub>7 (?y - ?z) = ?x \<times>\<^sub>7 ?y - ?x \<times>\<^sub>7 ?z###end |
Octonions/Cross_Product_7 | Cross_Product_7.cross7_mult_left | lemma cross7_mult_left: "(c *\<^sub>R x) \<times>\<^sub>7 y = c *\<^sub>R (x \<times>\<^sub>7 y)"
and cross7_mult_right: "x \<times>\<^sub>7 (c *\<^sub>R y) = c *\<^sub>R (x \<times>\<^sub>7 y)" | (?c *\<^sub>R ?x) \<times>\<^sub>7 ?y = ?c *\<^sub>R ?x \<times>\<^sub>7 ?y | ?H1 (?H2 x_1 x_2) x_3 = ?H2 x_1 (?H1 x_2 x_3) | [
"Real_Vector_Spaces.scaleR_class.scaleR",
"Cross_Product_7.cross7"
] | [
"real \\<Rightarrow> 'a \\<Rightarrow> 'a",
"(real, 7) vec \\<Rightarrow> (real, 7) vec \\<Rightarrow> (real, 7) vec"
] | [
"class scaleR =\n fixes scaleR :: \"real \\<Rightarrow> 'a \\<Rightarrow> 'a\" (infixr \\<open>*\\<^sub>R\\<close> 75)\nbegin",
"definition cross7 :: \"[real^7, real^7] \\<Rightarrow> real^7\" (infixr \\<open>\\<times>\\<^sub>7\\<close> 80)\n where \"a \\<times>\\<^sub>7 b \\<equiv>\n vector [a$2 * b$4 - a$... | lemma_object | ###symbols
Real_Vector_Spaces.scaleR_class.scaleR :::: real \<Rightarrow> 'a \<Rightarrow> 'a
Cross_Product_7.cross7 :::: (real, 7) vec \<Rightarrow> (real, 7) vec \<Rightarrow> (real, 7) vec
###defs
class scaleR =
fixes scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a" (infixr \<open>*\<^sub>R\<close> 75)
begin
d... |
###output
(?c *\<^sub>R ?x) \<times>\<^sub>7 ?y = ?c *\<^sub>R ?x \<times>\<^sub>7 ?y###end |
Octonions/Cross_Product_7 | Cross_Product_7.cross7_mult_right | lemma cross7_mult_left: "(c *\<^sub>R x) \<times>\<^sub>7 y = c *\<^sub>R (x \<times>\<^sub>7 y)"
and cross7_mult_right: "x \<times>\<^sub>7 (c *\<^sub>R y) = c *\<^sub>R (x \<times>\<^sub>7 y)" | ?x \<times>\<^sub>7 (?c *\<^sub>R ?y) = ?c *\<^sub>R ?x \<times>\<^sub>7 ?y | ?H1 x_1 (?H2 x_2 x_3) = ?H2 x_2 (?H1 x_1 x_3) | [
"Real_Vector_Spaces.scaleR_class.scaleR",
"Cross_Product_7.cross7"
] | [
"real \\<Rightarrow> 'a \\<Rightarrow> 'a",
"(real, 7) vec \\<Rightarrow> (real, 7) vec \\<Rightarrow> (real, 7) vec"
] | [
"class scaleR =\n fixes scaleR :: \"real \\<Rightarrow> 'a \\<Rightarrow> 'a\" (infixr \\<open>*\\<^sub>R\\<close> 75)\nbegin",
"definition cross7 :: \"[real^7, real^7] \\<Rightarrow> real^7\" (infixr \\<open>\\<times>\\<^sub>7\\<close> 80)\n where \"a \\<times>\\<^sub>7 b \\<equiv>\n vector [a$2 * b$4 - a$... | lemma_object | ###symbols
Real_Vector_Spaces.scaleR_class.scaleR :::: real \<Rightarrow> 'a \<Rightarrow> 'a
Cross_Product_7.cross7 :::: (real, 7) vec \<Rightarrow> (real, 7) vec \<Rightarrow> (real, 7) vec
###defs
class scaleR =
fixes scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a" (infixr \<open>*\<^sub>R\<close> 75)
begin
d... |
###output
?x \<times>\<^sub>7 (?c *\<^sub>R ?y) = ?c *\<^sub>R ?x \<times>\<^sub>7 ?y###end |
Octonions/Cross_Product_7 | Cross_Product_7.cross7_triple1 | lemma cross7_triple1: "(x \<times>\<^sub>7 y) \<bullet> z = (y \<times>\<^sub>7 z) \<bullet> x"
and cross7_triple2: "(x \<times>\<^sub>7 y) \<bullet> z = x \<bullet> (y \<times>\<^sub>7 z) " | ?x \<times>\<^sub>7 ?y \<bullet> ?z = ?y \<times>\<^sub>7 ?z \<bullet> ?x | ?H1 (?H2 x_1 x_2) x_3 = ?H1 (?H2 x_2 x_3) x_1 | [
"Cross_Product_7.cross7",
"Inner_Product.real_inner_class.inner"
] | [
"(real, 7) vec \\<Rightarrow> (real, 7) vec \\<Rightarrow> (real, 7) vec",
"'a \\<Rightarrow> 'a \\<Rightarrow> real"
] | [
"definition cross7 :: \"[real^7, real^7] \\<Rightarrow> real^7\" (infixr \\<open>\\<times>\\<^sub>7\\<close> 80)\n where \"a \\<times>\\<^sub>7 b \\<equiv>\n vector [a$2 * b$4 - a$4 * b$2 + a$3 * b$7 - a$7 * b$3 + a$5 * b$6 - a$6 * b$5 ,\n a$3 * b$5 - a$5 * b$3 + a$4 * b$1 - a$1 * b$4 + a$6 * ... | lemma_object | ###symbols
Cross_Product_7.cross7 :::: (real, 7) vec \<Rightarrow> (real, 7) vec \<Rightarrow> (real, 7) vec
Inner_Product.real_inner_class.inner :::: 'a \<Rightarrow> 'a \<Rightarrow> real
###defs
definition cross7 :: "[real^7, real^7] \<Rightarrow> real^7" (infixr \<open>\<times>\<^sub>7\<close> 80)
where "a \<ti... |
###output
?x \<times>\<^sub>7 ?y \<bullet> ?z = ?y \<times>\<^sub>7 ?z \<bullet> ?x###end |
Octonions/Cross_Product_7 | Cross_Product_7.cross7_triple2 | lemma cross7_triple1: "(x \<times>\<^sub>7 y) \<bullet> z = (y \<times>\<^sub>7 z) \<bullet> x"
and cross7_triple2: "(x \<times>\<^sub>7 y) \<bullet> z = x \<bullet> (y \<times>\<^sub>7 z) " | ?x \<times>\<^sub>7 ?y \<bullet> ?z = ?x \<bullet> ?y \<times>\<^sub>7 ?z | ?H1 (?H2 x_1 x_2) x_3 = ?H1 x_1 (?H2 x_2 x_3) | [
"Cross_Product_7.cross7",
"Inner_Product.real_inner_class.inner"
] | [
"(real, 7) vec \\<Rightarrow> (real, 7) vec \\<Rightarrow> (real, 7) vec",
"'a \\<Rightarrow> 'a \\<Rightarrow> real"
] | [
"definition cross7 :: \"[real^7, real^7] \\<Rightarrow> real^7\" (infixr \\<open>\\<times>\\<^sub>7\\<close> 80)\n where \"a \\<times>\\<^sub>7 b \\<equiv>\n vector [a$2 * b$4 - a$4 * b$2 + a$3 * b$7 - a$7 * b$3 + a$5 * b$6 - a$6 * b$5 ,\n a$3 * b$5 - a$5 * b$3 + a$4 * b$1 - a$1 * b$4 + a$6 * ... | lemma_object | ###symbols
Cross_Product_7.cross7 :::: (real, 7) vec \<Rightarrow> (real, 7) vec \<Rightarrow> (real, 7) vec
Inner_Product.real_inner_class.inner :::: 'a \<Rightarrow> 'a \<Rightarrow> real
###defs
definition cross7 :: "[real^7, real^7] \<Rightarrow> real^7" (infixr \<open>\<times>\<^sub>7\<close> 80)
where "a \<ti... |
###output
?x \<times>\<^sub>7 ?y \<bullet> ?z = ?x \<bullet> ?y \<times>\<^sub>7 ?z###end |
Octonions/Cross_Product_7 | Cross_Product_7.scalar7_triple2 | lemma scalar7_triple1: "x \<bullet> (y \<times>\<^sub>7 z) = y \<bullet> (z \<times>\<^sub>7 x)"
and scalar7_triple2: "x \<bullet> (y \<times>\<^sub>7 z) = z \<bullet> (x \<times>\<^sub>7 y ) " | ?x \<bullet> ?y \<times>\<^sub>7 ?z = ?z \<bullet> ?x \<times>\<^sub>7 ?y | ?H1 x_1 (?H2 x_2 x_3) = ?H1 x_3 (?H2 x_1 x_2) | [
"Cross_Product_7.cross7",
"Inner_Product.real_inner_class.inner"
] | [
"(real, 7) vec \\<Rightarrow> (real, 7) vec \\<Rightarrow> (real, 7) vec",
"'a \\<Rightarrow> 'a \\<Rightarrow> real"
] | [
"definition cross7 :: \"[real^7, real^7] \\<Rightarrow> real^7\" (infixr \\<open>\\<times>\\<^sub>7\\<close> 80)\n where \"a \\<times>\\<^sub>7 b \\<equiv>\n vector [a$2 * b$4 - a$4 * b$2 + a$3 * b$7 - a$7 * b$3 + a$5 * b$6 - a$6 * b$5 ,\n a$3 * b$5 - a$5 * b$3 + a$4 * b$1 - a$1 * b$4 + a$6 * ... | lemma_object | ###symbols
Cross_Product_7.cross7 :::: (real, 7) vec \<Rightarrow> (real, 7) vec \<Rightarrow> (real, 7) vec
Inner_Product.real_inner_class.inner :::: 'a \<Rightarrow> 'a \<Rightarrow> real
###defs
definition cross7 :: "[real^7, real^7] \<Rightarrow> real^7" (infixr \<open>\<times>\<^sub>7\<close> 80)
where "a \<ti... |
###output
?x \<bullet> ?y \<times>\<^sub>7 ?z = ?z \<bullet> ?x \<times>\<^sub>7 ?y###end |
Octonions/Cross_Product_7 | Cross_Product_7.scalar7_triple1 | lemma scalar7_triple1: "x \<bullet> (y \<times>\<^sub>7 z) = y \<bullet> (z \<times>\<^sub>7 x)"
and scalar7_triple2: "x \<bullet> (y \<times>\<^sub>7 z) = z \<bullet> (x \<times>\<^sub>7 y ) " | ?x \<bullet> ?y \<times>\<^sub>7 ?z = ?y \<bullet> ?z \<times>\<^sub>7 ?x | ?H1 x_1 (?H2 x_2 x_3) = ?H1 x_2 (?H2 x_3 x_1) | [
"Cross_Product_7.cross7",
"Inner_Product.real_inner_class.inner"
] | [
"(real, 7) vec \\<Rightarrow> (real, 7) vec \\<Rightarrow> (real, 7) vec",
"'a \\<Rightarrow> 'a \\<Rightarrow> real"
] | [
"definition cross7 :: \"[real^7, real^7] \\<Rightarrow> real^7\" (infixr \\<open>\\<times>\\<^sub>7\\<close> 80)\n where \"a \\<times>\\<^sub>7 b \\<equiv>\n vector [a$2 * b$4 - a$4 * b$2 + a$3 * b$7 - a$7 * b$3 + a$5 * b$6 - a$6 * b$5 ,\n a$3 * b$5 - a$5 * b$3 + a$4 * b$1 - a$1 * b$4 + a$6 * ... | lemma_object | ###symbols
Cross_Product_7.cross7 :::: (real, 7) vec \<Rightarrow> (real, 7) vec \<Rightarrow> (real, 7) vec
Inner_Product.real_inner_class.inner :::: 'a \<Rightarrow> 'a \<Rightarrow> real
###defs
definition cross7 :: "[real^7, real^7] \<Rightarrow> real^7" (infixr \<open>\<times>\<^sub>7\<close> 80)
where "a \<ti... |
###output
?x \<bullet> ?y \<times>\<^sub>7 ?z = ?y \<bullet> ?z \<times>\<^sub>7 ?x###end |
Octonions/Cross_Product_7 | Cross_Product_7.cross7_minus_left | lemma cross7_minus_left [simp]: "(-x) \<times>\<^sub>7 y = - (x \<times>\<^sub>7 y)"
and cross7_minus_right [simp]: "x \<times>\<^sub>7 -y = - (x \<times>\<^sub>7 y)" | (- ?x) \<times>\<^sub>7 ?y = - (?x \<times>\<^sub>7 ?y) | ?H1 (?H2 x_1) x_2 = ?H2 (?H1 x_1 x_2) | [
"Groups.uminus_class.uminus",
"Cross_Product_7.cross7"
] | [
"'a \\<Rightarrow> 'a",
"(real, 7) vec \\<Rightarrow> (real, 7) vec \\<Rightarrow> (real, 7) vec"
] | [
"class uminus =\n fixes uminus :: \"'a \\<Rightarrow> 'a\" (\\<open>(\\<open>open_block notation=\\<open>prefix -\\<close>\\<close>- _)\\<close> [81] 80)",
"definition cross7 :: \"[real^7, real^7] \\<Rightarrow> real^7\" (infixr \\<open>\\<times>\\<^sub>7\\<close> 80)\n where \"a \\<times>\\<^sub>7 b \\<equiv... | lemma_object | ###symbols
Groups.uminus_class.uminus :::: 'a \<Rightarrow> 'a
Cross_Product_7.cross7 :::: (real, 7) vec \<Rightarrow> (real, 7) vec \<Rightarrow> (real, 7) vec
###defs
class uminus =
fixes uminus :: "'a \<Rightarrow> 'a" (\<open>(\<open>open_block notation=\<open>prefix -\<close>\<close>- _)\<close> [81] 80)
defi... |
###output
(- ?x) \<times>\<^sub>7 ?y = - (?x \<times>\<^sub>7 ?y)###end |
Octonions/Cross_Product_7 | Cross_Product_7.cross7_minus_right | lemma cross7_minus_left [simp]: "(-x) \<times>\<^sub>7 y = - (x \<times>\<^sub>7 y)"
and cross7_minus_right [simp]: "x \<times>\<^sub>7 -y = - (x \<times>\<^sub>7 y)" | ?x \<times>\<^sub>7 - ?y = - (?x \<times>\<^sub>7 ?y) | ?H1 x_1 (?H2 x_2) = ?H2 (?H1 x_1 x_2) | [
"Groups.uminus_class.uminus",
"Cross_Product_7.cross7"
] | [
"'a \\<Rightarrow> 'a",
"(real, 7) vec \\<Rightarrow> (real, 7) vec \\<Rightarrow> (real, 7) vec"
] | [
"class uminus =\n fixes uminus :: \"'a \\<Rightarrow> 'a\" (\\<open>(\\<open>open_block notation=\\<open>prefix -\\<close>\\<close>- _)\\<close> [81] 80)",
"definition cross7 :: \"[real^7, real^7] \\<Rightarrow> real^7\" (infixr \\<open>\\<times>\\<^sub>7\\<close> 80)\n where \"a \\<times>\\<^sub>7 b \\<equiv... | lemma_object | ###symbols
Groups.uminus_class.uminus :::: 'a \<Rightarrow> 'a
Cross_Product_7.cross7 :::: (real, 7) vec \<Rightarrow> (real, 7) vec \<Rightarrow> (real, 7) vec
###defs
class uminus =
fixes uminus :: "'a \<Rightarrow> 'a" (\<open>(\<open>open_block notation=\<open>prefix -\<close>\<close>- _)\<close> [81] 80)
defi... |
###output
?x \<times>\<^sub>7 - ?y = - (?x \<times>\<^sub>7 ?y)###end |
Octonions/Cross_Product_7 | Cross_Product_7.cross7_eq_self(2) | lemma cross7_eq_self: "x \<times>\<^sub>7 y = x \<longleftrightarrow> x = 0" "x \<times>\<^sub>7 y = y \<longleftrightarrow> y = 0" | (?x \<times>\<^sub>7 ?y = ?y) = (?y = 0) | (?H1 x_1 x_2 = x_2) = (x_2 = ?H2) | [
"Groups.zero_class.zero",
"Cross_Product_7.cross7"
] | [
"'a",
"(real, 7) vec \\<Rightarrow> (real, 7) vec \\<Rightarrow> (real, 7) vec"
] | [
"class zero =\n fixes zero :: 'a (\\<open>0\\<close>)",
"definition cross7 :: \"[real^7, real^7] \\<Rightarrow> real^7\" (infixr \\<open>\\<times>\\<^sub>7\\<close> 80)\n where \"a \\<times>\\<^sub>7 b \\<equiv>\n vector [a$2 * b$4 - a$4 * b$2 + a$3 * b$7 - a$7 * b$3 + a$5 * b$6 - a$6 * b$5 ,\n ... | lemma_object | ###symbols
Groups.zero_class.zero :::: 'a
Cross_Product_7.cross7 :::: (real, 7) vec \<Rightarrow> (real, 7) vec \<Rightarrow> (real, 7) vec
###defs
class zero =
fixes zero :: 'a (\<open>0\<close>)
definition cross7 :: "[real^7, real^7] \<Rightarrow> real^7" (infixr \<open>\<times>\<^sub>7\<close> 80)
where "a \... |
###output
(?x \<times>\<^sub>7 ?y = ?y) = (?y = 0)###end |
Octonions/Cross_Product_7 | Cross_Product_7.cross7_eq_self(1) | lemma cross7_eq_self: "x \<times>\<^sub>7 y = x \<longleftrightarrow> x = 0" "x \<times>\<^sub>7 y = y \<longleftrightarrow> y = 0" | (?x \<times>\<^sub>7 ?y = ?x) = (?x = 0) | (?H1 x_1 x_2 = x_1) = (x_1 = ?H2) | [
"Groups.zero_class.zero",
"Cross_Product_7.cross7"
] | [
"'a",
"(real, 7) vec \\<Rightarrow> (real, 7) vec \\<Rightarrow> (real, 7) vec"
] | [
"class zero =\n fixes zero :: 'a (\\<open>0\\<close>)",
"definition cross7 :: \"[real^7, real^7] \\<Rightarrow> real^7\" (infixr \\<open>\\<times>\\<^sub>7\\<close> 80)\n where \"a \\<times>\\<^sub>7 b \\<equiv>\n vector [a$2 * b$4 - a$4 * b$2 + a$3 * b$7 - a$7 * b$3 + a$5 * b$6 - a$6 * b$5 ,\n ... | lemma_object | ###symbols
Groups.zero_class.zero :::: 'a
Cross_Product_7.cross7 :::: (real, 7) vec \<Rightarrow> (real, 7) vec \<Rightarrow> (real, 7) vec
###defs
class zero =
fixes zero :: 'a (\<open>0\<close>)
definition cross7 :: "[real^7, real^7] \<Rightarrow> real^7" (infixr \<open>\<times>\<^sub>7\<close> 80)
where "a \... |
###output
(?x \<times>\<^sub>7 ?y = ?x) = (?x = 0)###end |
Octonions/Cross_Product_7 | Cross_Product_7.cross7_skew | lemma cross7_skew: "(x \<times>\<^sub>7 y) = -(y \<times>\<^sub>7 x)" | ?x \<times>\<^sub>7 ?y = - (?y \<times>\<^sub>7 ?x) | ?H1 x_1 x_2 = ?H2 (?H1 x_2 x_1) | [
"Groups.uminus_class.uminus",
"Cross_Product_7.cross7"
] | [
"'a \\<Rightarrow> 'a",
"(real, 7) vec \\<Rightarrow> (real, 7) vec \\<Rightarrow> (real, 7) vec"
] | [
"class uminus =\n fixes uminus :: \"'a \\<Rightarrow> 'a\" (\\<open>(\\<open>open_block notation=\\<open>prefix -\\<close>\\<close>- _)\\<close> [81] 80)",
"definition cross7 :: \"[real^7, real^7] \\<Rightarrow> real^7\" (infixr \\<open>\\<times>\\<^sub>7\\<close> 80)\n where \"a \\<times>\\<^sub>7 b \\<equiv... | lemma_object | ###symbols
Groups.uminus_class.uminus :::: 'a \<Rightarrow> 'a
Cross_Product_7.cross7 :::: (real, 7) vec \<Rightarrow> (real, 7) vec \<Rightarrow> (real, 7) vec
###defs
class uminus =
fixes uminus :: "'a \<Rightarrow> 'a" (\<open>(\<open>open_block notation=\<open>prefix -\<close>\<close>- _)\<close> [81] 80)
defi... |
###output
?x \<times>\<^sub>7 ?y = - (?y \<times>\<^sub>7 ?x)###end |
Octonions/Cross_Product_7 | Cross_Product_7.dot_cross7_self(4) | lemma dot_cross7_self: "x \<bullet> (x \<times>\<^sub>7 y) = 0" "x \<bullet> (y \<times>\<^sub>7 x) = 0" "(x \<times>\<^sub>7 y) \<bullet> y = 0" "(y \<times>\<^sub>7 x) \<bullet> y = 0" | ?y \<times>\<^sub>7 ?x \<bullet> ?y = 0 | ?H1 (?H2 x_1 x_2) x_1 = ?H3 | [
"Groups.zero_class.zero",
"Cross_Product_7.cross7",
"Inner_Product.real_inner_class.inner"
] | [
"'a",
"(real, 7) vec \\<Rightarrow> (real, 7) vec \\<Rightarrow> (real, 7) vec",
"'a \\<Rightarrow> 'a \\<Rightarrow> real"
] | [
"class zero =\n fixes zero :: 'a (\\<open>0\\<close>)",
"definition cross7 :: \"[real^7, real^7] \\<Rightarrow> real^7\" (infixr \\<open>\\<times>\\<^sub>7\\<close> 80)\n where \"a \\<times>\\<^sub>7 b \\<equiv>\n vector [a$2 * b$4 - a$4 * b$2 + a$3 * b$7 - a$7 * b$3 + a$5 * b$6 - a$6 * b$5 ,\n ... | lemma_object | ###symbols
Groups.zero_class.zero :::: 'a
Cross_Product_7.cross7 :::: (real, 7) vec \<Rightarrow> (real, 7) vec \<Rightarrow> (real, 7) vec
Inner_Product.real_inner_class.inner :::: 'a \<Rightarrow> 'a \<Rightarrow> real
###defs
class zero =
fixes zero :: 'a (\<open>0\<close>)
definition cross7 :: "[real^7, real^... |
###output
?y \<times>\<^sub>7 ?x \<bullet> ?y = 0###end |
Octonions/Cross_Product_7 | Cross_Product_7.dot_cross7_self(3) | lemma dot_cross7_self: "x \<bullet> (x \<times>\<^sub>7 y) = 0" "x \<bullet> (y \<times>\<^sub>7 x) = 0" "(x \<times>\<^sub>7 y) \<bullet> y = 0" "(y \<times>\<^sub>7 x) \<bullet> y = 0" | ?x \<times>\<^sub>7 ?y \<bullet> ?y = 0 | ?H1 (?H2 x_1 x_2) x_2 = ?H3 | [
"Groups.zero_class.zero",
"Cross_Product_7.cross7",
"Inner_Product.real_inner_class.inner"
] | [
"'a",
"(real, 7) vec \\<Rightarrow> (real, 7) vec \\<Rightarrow> (real, 7) vec",
"'a \\<Rightarrow> 'a \\<Rightarrow> real"
] | [
"class zero =\n fixes zero :: 'a (\\<open>0\\<close>)",
"definition cross7 :: \"[real^7, real^7] \\<Rightarrow> real^7\" (infixr \\<open>\\<times>\\<^sub>7\\<close> 80)\n where \"a \\<times>\\<^sub>7 b \\<equiv>\n vector [a$2 * b$4 - a$4 * b$2 + a$3 * b$7 - a$7 * b$3 + a$5 * b$6 - a$6 * b$5 ,\n ... | lemma_object | ###symbols
Groups.zero_class.zero :::: 'a
Cross_Product_7.cross7 :::: (real, 7) vec \<Rightarrow> (real, 7) vec \<Rightarrow> (real, 7) vec
Inner_Product.real_inner_class.inner :::: 'a \<Rightarrow> 'a \<Rightarrow> real
###defs
class zero =
fixes zero :: 'a (\<open>0\<close>)
definition cross7 :: "[real^7, real^... |
###output
?x \<times>\<^sub>7 ?y \<bullet> ?y = 0###end |
Octonions/Cross_Product_7 | Cross_Product_7.dot_cross7_self(2) | lemma dot_cross7_self: "x \<bullet> (x \<times>\<^sub>7 y) = 0" "x \<bullet> (y \<times>\<^sub>7 x) = 0" "(x \<times>\<^sub>7 y) \<bullet> y = 0" "(y \<times>\<^sub>7 x) \<bullet> y = 0" | ?x \<bullet> ?y \<times>\<^sub>7 ?x = 0 | ?H1 x_1 (?H2 x_2 x_1) = ?H3 | [
"Groups.zero_class.zero",
"Cross_Product_7.cross7",
"Inner_Product.real_inner_class.inner"
] | [
"'a",
"(real, 7) vec \\<Rightarrow> (real, 7) vec \\<Rightarrow> (real, 7) vec",
"'a \\<Rightarrow> 'a \\<Rightarrow> real"
] | [
"class zero =\n fixes zero :: 'a (\\<open>0\\<close>)",
"definition cross7 :: \"[real^7, real^7] \\<Rightarrow> real^7\" (infixr \\<open>\\<times>\\<^sub>7\\<close> 80)\n where \"a \\<times>\\<^sub>7 b \\<equiv>\n vector [a$2 * b$4 - a$4 * b$2 + a$3 * b$7 - a$7 * b$3 + a$5 * b$6 - a$6 * b$5 ,\n ... | lemma_object | ###symbols
Groups.zero_class.zero :::: 'a
Cross_Product_7.cross7 :::: (real, 7) vec \<Rightarrow> (real, 7) vec \<Rightarrow> (real, 7) vec
Inner_Product.real_inner_class.inner :::: 'a \<Rightarrow> 'a \<Rightarrow> real
###defs
class zero =
fixes zero :: 'a (\<open>0\<close>)
definition cross7 :: "[real^7, real^... |
###output
?x \<bullet> ?y \<times>\<^sub>7 ?x = 0###end |
Octonions/Cross_Product_7 | Cross_Product_7.dot_cross7_self(1) | lemma dot_cross7_self: "x \<bullet> (x \<times>\<^sub>7 y) = 0" "x \<bullet> (y \<times>\<^sub>7 x) = 0" "(x \<times>\<^sub>7 y) \<bullet> y = 0" "(y \<times>\<^sub>7 x) \<bullet> y = 0" | ?x \<bullet> ?x \<times>\<^sub>7 ?y = 0 | ?H1 x_1 (?H2 x_1 x_2) = ?H3 | [
"Groups.zero_class.zero",
"Cross_Product_7.cross7",
"Inner_Product.real_inner_class.inner"
] | [
"'a",
"(real, 7) vec \\<Rightarrow> (real, 7) vec \\<Rightarrow> (real, 7) vec",
"'a \\<Rightarrow> 'a \\<Rightarrow> real"
] | [
"class zero =\n fixes zero :: 'a (\\<open>0\\<close>)",
"definition cross7 :: \"[real^7, real^7] \\<Rightarrow> real^7\" (infixr \\<open>\\<times>\\<^sub>7\\<close> 80)\n where \"a \\<times>\\<^sub>7 b \\<equiv>\n vector [a$2 * b$4 - a$4 * b$2 + a$3 * b$7 - a$7 * b$3 + a$5 * b$6 - a$6 * b$5 ,\n ... | lemma_object | ###symbols
Groups.zero_class.zero :::: 'a
Cross_Product_7.cross7 :::: (real, 7) vec \<Rightarrow> (real, 7) vec \<Rightarrow> (real, 7) vec
Inner_Product.real_inner_class.inner :::: 'a \<Rightarrow> 'a \<Rightarrow> real
###defs
class zero =
fixes zero :: 'a (\<open>0\<close>)
definition cross7 :: "[real^7, real^... |
###output
?x \<bullet> ?x \<times>\<^sub>7 ?y = 0###end |
Octonions/Cross_Product_7 | Cross_Product_7.orthogonal_cross7(2) | lemma orthogonal_cross7: "orthogonal (x \<times>\<^sub>7 y) x" "orthogonal (x \<times>\<^sub>7 y) y"
"orthogonal y (x\<times>\<^sub>7 y)" "orthogonal (x \<times>\<^sub>7 y) x" | orthogonal (?x \<times>\<^sub>7 ?y) ?y | ?H1 (?H2 x_1 x_2) x_2 | [
"Cross_Product_7.cross7",
"Linear_Algebra.real_inner_class.orthogonal"
] | [
"(real, 7) vec \\<Rightarrow> (real, 7) vec \\<Rightarrow> (real, 7) vec",
"'a \\<Rightarrow> 'a \\<Rightarrow> bool"
] | [
"definition cross7 :: \"[real^7, real^7] \\<Rightarrow> real^7\" (infixr \\<open>\\<times>\\<^sub>7\\<close> 80)\n where \"a \\<times>\\<^sub>7 b \\<equiv>\n vector [a$2 * b$4 - a$4 * b$2 + a$3 * b$7 - a$7 * b$3 + a$5 * b$6 - a$6 * b$5 ,\n a$3 * b$5 - a$5 * b$3 + a$4 * b$1 - a$1 * b$4 + a$6 * ... | lemma_object | ###symbols
Cross_Product_7.cross7 :::: (real, 7) vec \<Rightarrow> (real, 7) vec \<Rightarrow> (real, 7) vec
Linear_Algebra.real_inner_class.orthogonal :::: 'a \<Rightarrow> 'a \<Rightarrow> bool
###defs
definition cross7 :: "[real^7, real^7] \<Rightarrow> real^7" (infixr \<open>\<times>\<^sub>7\<close> 80)
where "... |
###output
orthogonal (?x \<times>\<^sub>7 ?y) ?y###end |
Octonions/Cross_Product_7 | Cross_Product_7.orthogonal_cross7(1) | lemma orthogonal_cross7: "orthogonal (x \<times>\<^sub>7 y) x" "orthogonal (x \<times>\<^sub>7 y) y"
"orthogonal y (x\<times>\<^sub>7 y)" "orthogonal (x \<times>\<^sub>7 y) x" | orthogonal (?x \<times>\<^sub>7 ?y) ?x | ?H1 (?H2 x_1 x_2) x_1 | [
"Cross_Product_7.cross7",
"Linear_Algebra.real_inner_class.orthogonal"
] | [
"(real, 7) vec \\<Rightarrow> (real, 7) vec \\<Rightarrow> (real, 7) vec",
"'a \\<Rightarrow> 'a \\<Rightarrow> bool"
] | [
"definition cross7 :: \"[real^7, real^7] \\<Rightarrow> real^7\" (infixr \\<open>\\<times>\\<^sub>7\\<close> 80)\n where \"a \\<times>\\<^sub>7 b \\<equiv>\n vector [a$2 * b$4 - a$4 * b$2 + a$3 * b$7 - a$7 * b$3 + a$5 * b$6 - a$6 * b$5 ,\n a$3 * b$5 - a$5 * b$3 + a$4 * b$1 - a$1 * b$4 + a$6 * ... | lemma_object | ###symbols
Cross_Product_7.cross7 :::: (real, 7) vec \<Rightarrow> (real, 7) vec \<Rightarrow> (real, 7) vec
Linear_Algebra.real_inner_class.orthogonal :::: 'a \<Rightarrow> 'a \<Rightarrow> bool
###defs
definition cross7 :: "[real^7, real^7] \<Rightarrow> real^7" (infixr \<open>\<times>\<^sub>7\<close> 80)
where "... |
###output
orthogonal (?x \<times>\<^sub>7 ?y) ?x###end |
Octonions/Cross_Product_7 | Cross_Product_7.orthogonal_cross7(3) | lemma orthogonal_cross7: "orthogonal (x \<times>\<^sub>7 y) x" "orthogonal (x \<times>\<^sub>7 y) y"
"orthogonal y (x\<times>\<^sub>7 y)" "orthogonal (x \<times>\<^sub>7 y) x" | orthogonal ?y (?x \<times>\<^sub>7 ?y) | ?H1 x_1 (?H2 x_2 x_1) | [
"Cross_Product_7.cross7",
"Linear_Algebra.real_inner_class.orthogonal"
] | [
"(real, 7) vec \\<Rightarrow> (real, 7) vec \\<Rightarrow> (real, 7) vec",
"'a \\<Rightarrow> 'a \\<Rightarrow> bool"
] | [
"definition cross7 :: \"[real^7, real^7] \\<Rightarrow> real^7\" (infixr \\<open>\\<times>\\<^sub>7\\<close> 80)\n where \"a \\<times>\\<^sub>7 b \\<equiv>\n vector [a$2 * b$4 - a$4 * b$2 + a$3 * b$7 - a$7 * b$3 + a$5 * b$6 - a$6 * b$5 ,\n a$3 * b$5 - a$5 * b$3 + a$4 * b$1 - a$1 * b$4 + a$6 * ... | lemma_object | ###symbols
Cross_Product_7.cross7 :::: (real, 7) vec \<Rightarrow> (real, 7) vec \<Rightarrow> (real, 7) vec
Linear_Algebra.real_inner_class.orthogonal :::: 'a \<Rightarrow> 'a \<Rightarrow> bool
###defs
definition cross7 :: "[real^7, real^7] \<Rightarrow> real^7" (infixr \<open>\<times>\<^sub>7\<close> 80)
where "... |
###output
orthogonal ?y (?x \<times>\<^sub>7 ?y)###end |
Octonions/Cross_Product_7 | Cross_Product_7.cross7_refl | lemma cross7_refl [simp]: "x \<times>\<^sub>7 x = 0" | ?x \<times>\<^sub>7 ?x = 0 | ?H1 x_1 x_1 = ?H2 | [
"Groups.zero_class.zero",
"Cross_Product_7.cross7"
] | [
"'a",
"(real, 7) vec \\<Rightarrow> (real, 7) vec \\<Rightarrow> (real, 7) vec"
] | [
"class zero =\n fixes zero :: 'a (\\<open>0\\<close>)",
"definition cross7 :: \"[real^7, real^7] \\<Rightarrow> real^7\" (infixr \\<open>\\<times>\\<^sub>7\\<close> 80)\n where \"a \\<times>\\<^sub>7 b \\<equiv>\n vector [a$2 * b$4 - a$4 * b$2 + a$3 * b$7 - a$7 * b$3 + a$5 * b$6 - a$6 * b$5 ,\n ... | lemma_object | ###symbols
Groups.zero_class.zero :::: 'a
Cross_Product_7.cross7 :::: (real, 7) vec \<Rightarrow> (real, 7) vec \<Rightarrow> (real, 7) vec
###defs
class zero =
fixes zero :: 'a (\<open>0\<close>)
definition cross7 :: "[real^7, real^7] \<Rightarrow> real^7" (infixr \<open>\<times>\<^sub>7\<close> 80)
where "a \... |
###output
?x \<times>\<^sub>7 ?x = 0###end |
Octonions/Cross_Product_7 | Cross_Product_7.cross7_zero_right | lemma cross7_zero_left [simp]: "0 \<times>\<^sub>7 x = 0"
and cross7_zero_right [simp]: "x \<times>\<^sub>7 0 = 0" | ?x \<times>\<^sub>7 0 = 0 | ?H1 x_1 ?H2 = ?H2 | [
"Groups.zero_class.zero",
"Cross_Product_7.cross7"
] | [
"'a",
"(real, 7) vec \\<Rightarrow> (real, 7) vec \\<Rightarrow> (real, 7) vec"
] | [
"class zero =\n fixes zero :: 'a (\\<open>0\\<close>)",
"definition cross7 :: \"[real^7, real^7] \\<Rightarrow> real^7\" (infixr \\<open>\\<times>\\<^sub>7\\<close> 80)\n where \"a \\<times>\\<^sub>7 b \\<equiv>\n vector [a$2 * b$4 - a$4 * b$2 + a$3 * b$7 - a$7 * b$3 + a$5 * b$6 - a$6 * b$5 ,\n ... | lemma_object | ###symbols
Groups.zero_class.zero :::: 'a
Cross_Product_7.cross7 :::: (real, 7) vec \<Rightarrow> (real, 7) vec \<Rightarrow> (real, 7) vec
###defs
class zero =
fixes zero :: 'a (\<open>0\<close>)
definition cross7 :: "[real^7, real^7] \<Rightarrow> real^7" (infixr \<open>\<times>\<^sub>7\<close> 80)
where "a \... |
###output
?x \<times>\<^sub>7 0 = 0###end |
Octonions/Cross_Product_7 | Cross_Product_7.cross7_zero_left | lemma cross7_zero_left [simp]: "0 \<times>\<^sub>7 x = 0"
and cross7_zero_right [simp]: "x \<times>\<^sub>7 0 = 0" | 0 \<times>\<^sub>7 ?x = 0 | ?H1 ?H2 x_1 = ?H2 | [
"Groups.zero_class.zero",
"Cross_Product_7.cross7"
] | [
"'a",
"(real, 7) vec \\<Rightarrow> (real, 7) vec \\<Rightarrow> (real, 7) vec"
] | [
"class zero =\n fixes zero :: 'a (\\<open>0\\<close>)",
"definition cross7 :: \"[real^7, real^7] \\<Rightarrow> real^7\" (infixr \\<open>\\<times>\\<^sub>7\\<close> 80)\n where \"a \\<times>\\<^sub>7 b \\<equiv>\n vector [a$2 * b$4 - a$4 * b$2 + a$3 * b$7 - a$7 * b$3 + a$5 * b$6 - a$6 * b$5 ,\n ... | lemma_object | ###symbols
Groups.zero_class.zero :::: 'a
Cross_Product_7.cross7 :::: (real, 7) vec \<Rightarrow> (real, 7) vec \<Rightarrow> (real, 7) vec
###defs
class zero =
fixes zero :: 'a (\<open>0\<close>)
definition cross7 :: "[real^7, real^7] \<Rightarrow> real^7" (infixr \<open>\<times>\<^sub>7\<close> 80)
where "a \... |
###output
0 \<times>\<^sub>7 ?x = 0###end |
Octonions/Octonions | Octonions.norm_octo_squared | lemma norm_octo_squared:
"norm x ^ 2 = Ree x ^ 2 + Im1 x ^ 2 + Im2 x ^ 2 + Im3 x ^ 2 +
Im4 x ^ 2 + Im5 x ^ 2 + Im6 x ^ 2 + Im7 x ^ 2" | (norm ?x)\<^sup>2 = (Ree ?x)\<^sup>2 + (Im1 ?x)\<^sup>2 + (Im2 ?x)\<^sup>2 + (Im3 ?x)\<^sup>2 + (Im4 ?x)\<^sup>2 + (Im5 ?x)\<^sup>2 + (Im6 ?x)\<^sup>2 + (Im7 ?x)\<^sup>2 | ?H1 (?H2 x_1) = ?H3 (?H3 (?H3 (?H3 (?H3 (?H3 (?H3 (?H1 (?H4 x_1)) (?H1 (?H5 x_1))) (?H1 (?H6 x_1))) (?H1 (?H7 x_1))) (?H1 (?H8 x_1))) (?H1 (?H9 x_1))) (?H1 (?H10 x_1))) (?H1 (?H11 x_1)) | [
"Octonions.octo.Im7",
"Octonions.octo.Im6",
"Octonions.octo.Im5",
"Octonions.octo.Im4",
"Octonions.octo.Im3",
"Octonions.octo.Im2",
"Octonions.octo.Im1",
"Octonions.octo.Ree",
"Groups.plus_class.plus",
"Real_Vector_Spaces.norm_class.norm",
"Power.power_class.power2"
] | [
"octo \\<Rightarrow> real",
"octo \\<Rightarrow> real",
"octo \\<Rightarrow> real",
"octo \\<Rightarrow> real",
"octo \\<Rightarrow> real",
"octo \\<Rightarrow> real",
"octo \\<Rightarrow> real",
"octo \\<Rightarrow> real",
"'a \\<Rightarrow> 'a \\<Rightarrow> 'a",
"'a \\<Rightarrow> real",
"'a ... | [
"codatatype octo =\n Octo (Ree: real) (Im1: real) (Im2: real) (Im3: real) (Im4: real) \n (Im5: real) (Im6: real) (Im7: real)",
"class plus =\n fixes plus :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\" (infixl \\<open>+\\<close> 65)",
"class norm =\n fixes norm :: \"'a \\<Rightarrow> real\"",
"primrec... | lemma_object | ###symbols
Octonions.octo.Im7 :::: octo \<Rightarrow> real
Octonions.octo.Im6 :::: octo \<Rightarrow> real
Octonions.octo.Im5 :::: octo \<Rightarrow> real
Octonions.octo.Im4 :::: octo \<Rightarrow> real
Octonions.octo.Im3 :::: octo \<Rightarrow> real
Octonions.octo.Im2 :::: octo \<Rightarrow> real
Octonions.octo.... |
###output
(norm ?x)\<^sup>2 = (Ree ?x)\<^sup>2 + (Im1 ?x)\<^sup>2 + (Im2 ?x)\<^sup>2 + (Im3 ?x)\<^sup>2 + (Im4 ?x)\<^sup>2 + (Im5 ?x)\<^sup>2 + (Im6 ?x)\<^sup>2 + (Im7 ?x)\<^sup>2###end |
Octonions/Octonions | Octonions.octo_eq_0_iff | lemma octo_eq_0_iff:
"x = 0 \<longleftrightarrow> Ree x ^ 2 + Im1 x ^ 2 + Im2 x ^ 2 + Im3 x ^ 2 +
Im4 x ^ 2 + Im5 x ^ 2 + Im6 x ^ 2 + Im7 x ^ 2 = 0" | (?x = 0) = ((Ree ?x)\<^sup>2 + (Im1 ?x)\<^sup>2 + (Im2 ?x)\<^sup>2 + (Im3 ?x)\<^sup>2 + (Im4 ?x)\<^sup>2 + (Im5 ?x)\<^sup>2 + (Im6 ?x)\<^sup>2 + (Im7 ?x)\<^sup>2 = 0) | (x_1 = ?H1) = (?H2 (?H2 (?H2 (?H2 (?H2 (?H2 (?H2 (?H3 (?H4 x_1)) (?H3 (?H5 x_1))) (?H3 (?H6 x_1))) (?H3 (?H7 x_1))) (?H3 (?H8 x_1))) (?H3 (?H9 x_1))) (?H3 (?H10 x_1))) (?H3 (?H11 x_1)) = ?H1) | [
"Octonions.octo.Im7",
"Octonions.octo.Im6",
"Octonions.octo.Im5",
"Octonions.octo.Im4",
"Octonions.octo.Im3",
"Octonions.octo.Im2",
"Octonions.octo.Im1",
"Octonions.octo.Ree",
"Power.power_class.power2",
"Groups.plus_class.plus",
"Groups.zero_class.zero"
] | [
"octo \\<Rightarrow> real",
"octo \\<Rightarrow> real",
"octo \\<Rightarrow> real",
"octo \\<Rightarrow> real",
"octo \\<Rightarrow> real",
"octo \\<Rightarrow> real",
"octo \\<Rightarrow> real",
"octo \\<Rightarrow> real",
"'a \\<Rightarrow> 'a",
"'a \\<Rightarrow> 'a \\<Rightarrow> 'a",
"'a"
] | [
"codatatype octo =\n Octo (Ree: real) (Im1: real) (Im2: real) (Im3: real) (Im4: real) \n (Im5: real) (Im6: real) (Im7: real)",
"primrec power :: \"'a \\<Rightarrow> nat \\<Rightarrow> 'a\" (infixr \\<open>^\\<close> 80)\n where\n power_0: \"a ^ 0 = 1\"\n | power_Suc: \"a ^ Suc n = a * a ^ n\"",
"cl... | lemma_object | ###symbols
Octonions.octo.Im7 :::: octo \<Rightarrow> real
Octonions.octo.Im6 :::: octo \<Rightarrow> real
Octonions.octo.Im5 :::: octo \<Rightarrow> real
Octonions.octo.Im4 :::: octo \<Rightarrow> real
Octonions.octo.Im3 :::: octo \<Rightarrow> real
Octonions.octo.Im2 :::: octo \<Rightarrow> real
Octonions.octo.... |
###output
(?x = 0) = ((Ree ?x)\<^sup>2 + (Im1 ?x)\<^sup>2 + (Im2 ?x)\<^sup>2 + (Im3 ?x)\<^sup>2 + (Im4 ?x)\<^sup>2 + (Im5 ?x)\<^sup>2 + (Im6 ?x)\<^sup>2 + (Im7 ?x)\<^sup>2 = 0)###end |
Octonions/Octonions | Octonions.octo_eqI | lemma octo_eqI [intro?]:
"\<lbrakk>Ree x = Ree y; Im1 x = Im1 y; Im2 x = Im2 y; Im3 x = Im3 y;
Im4 x = Im4 y;Im5 x = Im5 y; Im6 x = Im6 y; Im7 x = Im7 y\<rbrakk> \<Longrightarrow> x = y" | Ree ?x = Ree ?y \<Longrightarrow> Im1 ?x = Im1 ?y \<Longrightarrow> Im2 ?x = Im2 ?y \<Longrightarrow> Im3 ?x = Im3 ?y \<Longrightarrow> Im4 ?x = Im4 ?y \<Longrightarrow> Im5 ?x = Im5 ?y \<Longrightarrow> Im6 ?x = Im6 ?y \<Longrightarrow> Im7 ?x = Im7 ?y \<Longrightarrow> ?x = ?y | \<lbrakk> ?H1 x_1 = ?H1 x_2; ?H2 x_1 = ?H2 x_2; ?H3 x_1 = ?H3 x_2; ?H4 x_1 = ?H4 x_2; ?H5 x_1 = ?H5 x_2; ?H6 x_1 = ?H6 x_2; ?H7 x_1 = ?H7 x_2; ?H8 x_1 = ?H8 x_2\<rbrakk> \<Longrightarrow> x_1 = x_2 | [
"Octonions.octo.Im7",
"Octonions.octo.Im6",
"Octonions.octo.Im5",
"Octonions.octo.Im4",
"Octonions.octo.Im3",
"Octonions.octo.Im2",
"Octonions.octo.Im1",
"Octonions.octo.Ree"
] | [
"octo \\<Rightarrow> real",
"octo \\<Rightarrow> real",
"octo \\<Rightarrow> real",
"octo \\<Rightarrow> real",
"octo \\<Rightarrow> real",
"octo \\<Rightarrow> real",
"octo \\<Rightarrow> real",
"octo \\<Rightarrow> real"
] | [
"codatatype octo =\n Octo (Ree: real) (Im1: real) (Im2: real) (Im3: real) (Im4: real) \n (Im5: real) (Im6: real) (Im7: real)"
] | lemma_object | ###symbols
Octonions.octo.Im7 :::: octo \<Rightarrow> real
Octonions.octo.Im6 :::: octo \<Rightarrow> real
Octonions.octo.Im5 :::: octo \<Rightarrow> real
Octonions.octo.Im4 :::: octo \<Rightarrow> real
Octonions.octo.Im3 :::: octo \<Rightarrow> real
Octonions.octo.Im2 :::: octo \<Rightarrow> real
Octonions.octo.... |
###output
Ree ?x = Ree ?y \<Longrightarrow> Im1 ?x = Im1 ?y \<Longrightarrow> Im2 ?x = Im2 ?y \<Longrightarrow> Im3 ?x = Im3 ?y \<Longrightarrow> Im4 ?x = Im4 ?y \<Longrightarrow> Im5 ?x = Im5 ?y \<Longrightarrow> Im6 ?x = Im6 ?y \<Longrightarrow> Im7 ?x = Im7 ?y \<Longrightarrow> ?x = ?y###end |
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