theory_file stringclasses 93
values | lemma_name stringlengths 6 83 | lemma_object stringlengths 7 12.2k | lemma_command stringlengths 20 1.57k ⌀ | template stringlengths 9 10.4k | greedy_predictions sequencelengths 1 1 | beam_search_top5_predictions sequencelengths 5 5 | min_greedy_hamming int64 0 9.02k | min_beam_search_top5_hamming int64 0 9.01k | min_hamming int64 0 9.01k | min_greedy_levenshtein int64 0 8.74k | min_beam_search_top5_levenshtein int64 0 8.81k | min_levenshtein int64 0 8.74k | max_greedy_jaccard float64 0.03 1 | max_beam_search_top5_jaccard float64 0.02 1 | max_jaccard float64 0.03 1 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cardinals/Ordinal_Arithmetic | Ordinal_Arithmetic.max_fun_diff_oprod | Well_order ?r \<Longrightarrow> Well_order ?s \<Longrightarrow> Well_order ?t \<Longrightarrow> Field ?r \<noteq> {} \<Longrightarrow> ?f \<noteq> ?g \<Longrightarrow> ?f \<in> FinFunc ?r (?s *o ?t) \<Longrightarrow> ?g \<in> FinFunc ?r (?s *o ?t) \<Longrightarrow> wo_rel.max_f... | lemma max_fun_diff_oprod:
assumes Field: "Field r \<noteq> {}" and "f \<noteq> g" "f \<in> FinFunc r (s *o t)" "g \<in> FinFunc r (s *o t)"
defines "m \<equiv> wo_rel.max_fun_diff t (rev_curr f) (rev_curr g)"
shows "wo_rel.max_fun_diff (s *o t) f g =
(wo_rel.max_fun_diff s (rev_curr f m) (rev_curr g m), m)" | \<lbrakk>?H1 x_1; ?H2 x_2; ?H3 x_3; ?H4 x_1 \<noteq> ?H5; x_4 \<noteq> x_5; x_4 \<in> ?H6 x_1 (?H7 x_2 x_3); x_5 \<in> ?H6 x_1 (?H7 x_2 x_3)\<rbrakk> \<Longrightarrow> ?H8 (?H7 x_2 x_3) x_4 x_5 = (?H9 x_2 (?H10 x_3 x_4 (?H11 x_3 (?H10 x_3 x_4) (?H10 x_3 x_5))) ... | [
"\\<lbrakk> ?H1 x_1; ?H2 x_2; ?H3 x_2 \\<noteq> ?H4\\<rbrakk> \\<Longrightarrow> ?H5 (?H6 x_1 x_2) (?H7 x_1 x_2) = ?H8 (?H5 x_1 x_2) (?H9 x_2)"
] | [
"\\<lbrakk> ?H1 x_1; ?H2 x_2; ?H3 x_2 \\<noteq> ?H4\\<rbrakk> \\<Longrightarrow> ?H5 (?H6 x_1 x_2) (?H7 x_1 x_2) = ?H8 (?H5 x_1 x_2) (?H9 x_2)",
"\\<lbrakk> ?H1 x_1; ?H2 x_2; ?H3 x_2 \\<noteq> ?H4\\<rbrakk> \\<Longrightarrow> ?H5 (?H6 x_1 x_2) (?H7 x_1 x_2) = ?H8 (?H5 x_1 x_2) (?H9 x_1 x_2)",
"\\<lbrakk> ?H1 x_... | 313 | 310 | 310 | 232 | 221 | 221 | 0.37883 | 0.413408 | 0.413408 |
Cardinals/Ordinal_Arithmetic | Ordinal_Arithmetic.max_fun_diff_eq_Inr(1) | Well_order ?r \<Longrightarrow> Well_order ?s \<Longrightarrow> Well_order ?t \<Longrightarrow> wo_rel.max_fun_diff (?s +o ?t) (case_sum ?f1.0 ?g1.0) (case_sum ?f2.0 ?g2.0) = Inr ?x \<Longrightarrow> case_sum ?f1.0 ?g1.0 \<noteq> case_sum ?f2.0 ?g2.0 \<Longrightarrow> case_sum... | lemma max_fun_diff_eq_Inr:
assumes "wo_rel.max_fun_diff (s +o t) (case_sum f1 g1) (case_sum f2 g2) = Inr x"
"case_sum f1 g1 \<noteq> case_sum f2 g2"
"case_sum f1 g1 \<in> FinFunc r (s +o t)" "case_sum f2 g2 \<in> FinFunc r (s +o t)"
shows "wo_rel.max_fun_diff t g1 g2 = x" (is ?P) "g1 \<noteq> g2" (is ?Q) | \<lbrakk>?H1 x_1; ?H2 x_2; ?H3 x_3; ?H4 (?H5 x_2 x_3) (?H6 x_4 x_5) (?H6 x_6 x_7) = ?H7 x_8; ?H6 x_4 x_5 \<noteq> ?H6 x_6 x_7; ?H6 x_4 x_5 \<in> ?H8 x_1 (?H5 x_2 x_3); ?H6 x_6 x_7 \<in> ?H8 x_1 (?H5 x_2 x_3)\<rbrakk> \<Longrightarrow> ?H9 x_3 x_5 x_7 = x_8 | [
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2) = ?H5 (\\<lambda>y_0. ?H6 (?H7 x_1 y_0) (?H8 x_2 (?H9 y_0))) (?H10 x_1)"
] | [
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2) = ?H5 (\\<lambda>y_0. ?H6 (\\<lambda>y_1. ?H7 (?H8 x_1 y_0 y_1))) (\\<lambda>y_2. ?H8 x_1 y_2 (?H9 x_2))",
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2) = ?H5 (\\<lambda>y_0. ?H6 (\\<lambda>y_1. ?H7 (?H8 x_1 ... | 224 | 224 | 224 | 173 | 168 | 168 | 0.450909 | 0.472603 | 0.472603 |
Cardinals/Ordinal_Arithmetic | Ordinal_Arithmetic.max_fun_diff_eq_Inl(1) | Well_order ?r \<Longrightarrow> Well_order ?s \<Longrightarrow> Well_order ?t \<Longrightarrow> wo_rel.max_fun_diff (?s +o ?t) (case_sum ?f1.0 ?g1.0) (case_sum ?f2.0 ?g2.0) = Inl ?x \<Longrightarrow> case_sum ?f1.0 ?g1.0 \<noteq> case_sum ?f2.0 ?g2.0 \<Longrightarrow> case_sum... | lemma max_fun_diff_eq_Inl:
assumes "wo_rel.max_fun_diff (s +o t) (case_sum f1 g1) (case_sum f2 g2) = Inl x"
"case_sum f1 g1 \<noteq> case_sum f2 g2"
"case_sum f1 g1 \<in> FinFunc r (s +o t)" "case_sum f2 g2 \<in> FinFunc r (s +o t)"
shows "wo_rel.max_fun_diff s f1 f2 = x" (is ?P) "g1 = g2" (is ?Q) | \<lbrakk>?H1 x_1; ?H2 x_2; ?H3 x_3; ?H4 (?H5 x_2 x_3) (?H6 x_4 x_5) (?H6 x_6 x_7) = ?H7 x_8; ?H6 x_4 x_5 \<noteq> ?H6 x_6 x_7; ?H6 x_4 x_5 \<in> ?H8 x_1 (?H5 x_2 x_3); ?H6 x_6 x_7 \<in> ?H8 x_1 (?H5 x_2 x_3)\<rbrakk> \<Longrightarrow> ?H9 x_2 x_4 x_6 = x_8 | [
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2) = ?H5 (\\<lambda>y_0. ?H6 (?H7 x_1 y_0) (?H8 x_2 (?H9 y_0))) (?H10 x_1) (?H11 x_2)"
] | [
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2) = ?H5 (\\<lambda>y_0. ?H6 (\\<lambda>y_1. ?H7 (y_0, y_1))) (\\<lambda>y_2. ?H8 (\\<lambda>y_3. ?H9 (y_2, y_3)))",
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2) = ?H5 (\\<lambda>y_0. ?H6 (\\<lambda>y_1. ?H7 (y... | 224 | 226 | 224 | 170 | 174 | 170 | 0.469534 | 0.429508 | 0.469534 |
Cardinals/Ordinal_Arithmetic | Ordinal_Arithmetic.max_fun_diff_eq_Inr(2) | Well_order ?r \<Longrightarrow> Well_order ?s \<Longrightarrow> Well_order ?t \<Longrightarrow> wo_rel.max_fun_diff (?s +o ?t) (case_sum ?f1.0 ?g1.0) (case_sum ?f2.0 ?g2.0) = Inr ?x \<Longrightarrow> case_sum ?f1.0 ?g1.0 \<noteq> case_sum ?f2.0 ?g2.0 \<Longrightarrow> case_sum... | lemma max_fun_diff_eq_Inr:
assumes "wo_rel.max_fun_diff (s +o t) (case_sum f1 g1) (case_sum f2 g2) = Inr x"
"case_sum f1 g1 \<noteq> case_sum f2 g2"
"case_sum f1 g1 \<in> FinFunc r (s +o t)" "case_sum f2 g2 \<in> FinFunc r (s +o t)"
shows "wo_rel.max_fun_diff t g1 g2 = x" (is ?P) "g1 \<noteq> g2" (is ?Q) | \<lbrakk>?H1 x_1; ?H2 x_2; ?H3 x_3; ?H4 (?H5 x_2 x_3) (?H6 x_4 x_5) (?H6 x_6 x_7) = ?H7 x_8; ?H6 x_4 x_5 \<noteq> ?H6 x_6 x_7; ?H6 x_4 x_5 \<in> ?H8 x_1 (?H5 x_2 x_3); ?H6 x_6 x_7 \<in> ?H8 x_1 (?H5 x_2 x_3)\<rbrakk> \<Longrightarrow> x_5 \<noteq> x_7 | [
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2) = ?H5 (\\<lambda>y_0. ?H6 (?H7 x_1 y_0) (?H8 x_2 (?H9 y_0))) (?H10 x_1)"
] | [
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2) = ?H5 (\\<lambda>y_0. ?H6 (\\<lambda>y_1. ?H7 (?H8 x_1 y_0 y_1))) (\\<lambda>y_2. ?H8 x_1 y_2 (?H9 x_2))",
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2) = ?H5 (\\<lambda>y_0. ?H6 (\\<lambda>y_1. ?H7 (?H8 x_1 ... | 219 | 219 | 219 | 170 | 169 | 169 | 0.453875 | 0.475694 | 0.475694 |
Cardinals/Ordinal_Arithmetic | Ordinal_Arithmetic.max_fun_diff_eq_Inl(2) | Well_order ?r \<Longrightarrow> Well_order ?s \<Longrightarrow> Well_order ?t \<Longrightarrow> wo_rel.max_fun_diff (?s +o ?t) (case_sum ?f1.0 ?g1.0) (case_sum ?f2.0 ?g2.0) = Inl ?x \<Longrightarrow> case_sum ?f1.0 ?g1.0 \<noteq> case_sum ?f2.0 ?g2.0 \<Longrightarrow> case_sum... | lemma max_fun_diff_eq_Inl:
assumes "wo_rel.max_fun_diff (s +o t) (case_sum f1 g1) (case_sum f2 g2) = Inl x"
"case_sum f1 g1 \<noteq> case_sum f2 g2"
"case_sum f1 g1 \<in> FinFunc r (s +o t)" "case_sum f2 g2 \<in> FinFunc r (s +o t)"
shows "wo_rel.max_fun_diff s f1 f2 = x" (is ?P) "g1 = g2" (is ?Q) | \<lbrakk>?H1 x_1; ?H2 x_2; ?H3 x_3; ?H4 (?H5 x_2 x_3) (?H6 x_4 x_5) (?H6 x_6 x_7) = ?H7 x_8; ?H6 x_4 x_5 \<noteq> ?H6 x_6 x_7; ?H6 x_4 x_5 \<in> ?H8 x_1 (?H5 x_2 x_3); ?H6 x_6 x_7 \<in> ?H8 x_1 (?H5 x_2 x_3)\<rbrakk> \<Longrightarrow> x_5 = x_7 | [
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2) = ?H5 (\\<lambda>y_0. ?H6 (?H7 x_1 y_0) (?H8 x_2 (?H9 y_0))) (?H10 x_1) (?H11 x_2)"
] | [
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2) = ?H5 (\\<lambda>y_0. ?H6 (\\<lambda>y_1. ?H7 (y_0, y_1))) (\\<lambda>y_2. ?H8 (\\<lambda>y_3. ?H9 (y_2, y_3)))",
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2) = ?H5 (\\<lambda>y_0. ?H6 (\\<lambda>y_1. ?H7 (y... | 212 | 214 | 212 | 159 | 169 | 159 | 0.485075 | 0.442177 | 0.485075 |
Cardinals/Ordinal_Arithmetic | Ordinal_Arithmetic.rev_curr_app_FinFunc | Well_order ?r \<Longrightarrow> Well_order ?s \<Longrightarrow> Well_order ?t \<Longrightarrow> ?f \<in> FinFunc ?r (?s *o ?t) \<Longrightarrow> ?z \<in> Field ?t \<Longrightarrow> rev_curr ?t ?f ?z \<in> FinFunc ?r ?s | lemma rev_curr_app_FinFunc[elim!]:
"\<lbrakk>f \<in> FinFunc r (s *o t); z \<in> Field t\<rbrakk> \<Longrightarrow> rev_curr f z \<in> FinFunc r s" | \<lbrakk>?H1 x_1; ?H2 x_2; ?H3 x_3; x_4 \<in> ?H4 x_1 (?H5 x_2 x_3); x_5 \<in> ?H6 x_3\<rbrakk> \<Longrightarrow> ?H7 x_3 x_4 x_5 \<in> ?H8 x_1 x_2 | [
"?H1 x_1 \\<Longrightarrow> ?H2 (?H3 x_1 (?H4 x_2 x_3)) (?H5 x_1 x_3) = ?H6 (?H7 (?H8 x_2 x_1) (?H9 x_3 x_1)) (?H10 x_3 x_1)"
] | [
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 (?H5 x_2)) (?H6 x_1 x_2) = ?H7 (?H3 x_1 x_2)",
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 (?H5 x_2)) (?H6 x_1 x_2) = ?H7 (?H8 (?H3 x_1 x_2)) (?H9 x_1)",
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H... | 141 | 118 | 118 | 97 | 87 | 87 | 0.578947 | 0.660494 | 0.660494 |
Cardinals/Ordinal_Arithmetic | Ordinal_Arithmetic.oexp_monoR | Well_order ?r \<Longrightarrow> Well_order ?s \<Longrightarrow> Well_order ?t \<Longrightarrow> oone <o ?r \<Longrightarrow> ?s <o ?t \<Longrightarrow> ?r ^o ?s <o ?r ^o ?t | lemma oexp_monoR:
assumes "oone <o r" "s <o t"
shows "r ^o s <o r ^o t" (is "?L <o ?R") | \<lbrakk>?H1 x_1; ?H2 x_2; ?H3 x_3; ?H4 ?H5 x_1; ?H6 x_2 x_3\<rbrakk> \<Longrightarrow> ?H7 (?H8 x_1 x_2) (?H9 x_1 x_3) | [
"?H1 x_1 \\<Longrightarrow> ?H2 x_1 ?H3 \\<and> ?H4 (?H5 x_1) ?H6"
] | [
"?H1 x_1 \\<Longrightarrow> ?H2 x_1 ?H3 \\<and> ?H4 (?H5 x_1) ?H6",
"\\<lbrakk> ?H1 x_1; ?H2 x_1 ?H3\\<rbrakk> \\<Longrightarrow> ?H4 (?H5 x_1)",
"?H1 x_1 \\<Longrightarrow> ?H2 x_1 ?H3 \\<and> ?H4 (?H5 x_1) ?H6",
"?H1 x_1 \\<Longrightarrow> ?H2 x_1 ?H3 \\<and> ?H4 (?H5 x_1) ?H6",
"?H1 x_1 \\<Longrightarrow... | 114 | 94 | 94 | 81 | 51 | 51 | 0.491803 | 0.591667 | 0.591667 |
Cardinals/Ordinal_Arithmetic | Ordinal_Arithmetic.oprod_monoR | Well_order ?r \<Longrightarrow> Well_order ?s \<Longrightarrow> Well_order ?t \<Longrightarrow> ozero <o ?r \<Longrightarrow> ?s <o ?t \<Longrightarrow> ?r *o ?s <o ?r *o ?t | lemma oprod_monoR:
assumes "ozero <o r" "s <o t"
shows "r *o s <o r *o t" (is "?L <o ?R") | \<lbrakk>?H1 x_1; ?H2 x_2; ?H3 x_3; ?H4 ?H5 x_1; ?H6 x_2 x_3\<rbrakk> \<Longrightarrow> ?H7 (?H8 x_1 x_2) (?H9 x_1 x_3) | [
"\\<lbrakk> ?H1 x_1; ?H2 x_2; ?H3 x_2 ?H4\\<rbrakk> \\<Longrightarrow> ?H5 (?H6 x_1 x_2)"
] | [
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2) ?H5",
"\\<lbrakk> ?H1 x_1; ?H2 x_2; ?H3 x_2 ?H4\\<rbrakk> \\<Longrightarrow> ?H5 (?H6 x_1 x_2)",
"\\<lbrakk> ?H1 x_1; ?H2 x_2; ?H3 x_1 ?H4; ?H5 x_2 ?H6\\<rbrakk> \\<Longrightarrow> ?H7 (?H8 x_1 x_2)",
"\\<lbrakk> ?H1 x_1; ?H2 x_2; ?H3... | 85 | 77 | 77 | 39 | 27 | 27 | 0.7 | 0.808333 | 0.808333 |
Cardinals/Ordinal_Arithmetic | Ordinal_Arithmetic.FinFunc_osum | Well_order ?r \<Longrightarrow> Well_order ?s \<Longrightarrow> Well_order ?t \<Longrightarrow> (?fg \<in> FinFunc ?r (?s +o ?t)) = (?fg \<circ> Inl \<in> FinFunc ?r ?s \<and> ?fg \<circ> Inr \<in> FinFunc ?r ?t) | lemma FinFunc_osum:
"fg \<in> FinFunc r (s +o t) = (fg o Inl \<in> FinFunc r s \<and> fg o Inr \<in> FinFunc r t)"
(is "?L = (?R1 \<and> ?R2)") | \<lbrakk>?H1 x_1; ?H2 x_2; ?H3 x_3\<rbrakk> \<Longrightarrow> (x_4 \<in> ?H4 x_1 (?H5 x_2 x_3)) = (?H6 x_4 ?H7 \<in> ?H8 x_1 x_2 \<and> ?H9 x_4 ?H10 \<in> ?H11 x_1 x_3) | [
"?H1 x_1 \\<Longrightarrow> ?H2 (?H3 x_1 x_2) = ?H4 (?H5 ?H6 ?H7) (?H8 ?H9 ?H10)"
] | [
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2) = ?H5 (?H6 ?H7 ?H8) (?H9 ?H10 ?H11)",
"?H1 x_1 \\<Longrightarrow> ?H2 (?H3 x_1 x_2) = ?H4 (?H5 ?H6 ?H7) (?H8 ?H9 ?H10)",
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2) = ?H5 (?H6 ?H7 ?H8) (?H9 ?H10 ?H11)",
... | 169 | 134 | 134 | 107 | 74 | 74 | 0.461538 | 0.633136 | 0.633136 |
Cardinals/Ordinal_Arithmetic | Ordinal_Arithmetic.fun_unequal_in_support | ?f \<noteq> ?g \<Longrightarrow> ?f \<in> Func ?A ?B \<Longrightarrow> ?g \<in> Func ?A ?C \<Longrightarrow> (support ?z ?A ?f \<union> support ?z ?A ?g) \<inter> {a. ?f a \<noteq> ?g a} \<noteq> {} | lemma fun_unequal_in_support:
assumes "f \<noteq> g" "f \<in> Func A B" "g \<in> Func A C"
shows "(support z A f \<union> support z A g) \<inter> {a. f a \<noteq> g a} \<noteq> {}" | \<lbrakk>x_1 \<noteq> x_2; x_1 \<in> ?H1 x_3 x_4; x_2 \<in> ?H1 x_3 x_5\<rbrakk> \<Longrightarrow> ?H2 (?H3 (?H4 x_6 x_3 x_1) (?H4 x_6 x_3 x_2)) (?H5 (\<lambda>y_0. x_1 y_0 \<noteq> x_2 y_0)) \<noteq> ?H6 | [
"\\<lbrakk>x_1 \\<in> ?H1 x_2 x_3; \\<And>y_0. \\<lbrakk>y_0 \\<in> x_2; x_1 y_0 \\<in> x_3\\<rbrakk> \\<Longrightarrow> x_4 y_0 \\<in> x_5; \\<And>y_1. \\<lbrakk>y_1 \\<in> x_2; x_1 y_1 \\<in> x_3\\<rbrakk> \\<Longrightarrow> x_4 y_1 \\<in> x_5; \\<And>y_2. \\<lbrakk>y_2 \\<in> x_2; x_1 y_2 \\<in> x_3\\<rbrakk> \\... | [
"x_1 \\<in> ?H1 x_2 x_3 \\<Longrightarrow> ?H2 (?H3 (?H4 x_1 x_2) (?H5 (\\<lambda>y_0. y_0 \\<in> x_2 \\<and> x_1 y_0 = x_4))) (?H6 (\\<lambda>y_1. y_1 \\<in> x_2 \\<and> x_1 y_1 \\<noteq> x_4)) \\<in> ?H1 x_2 x_3",
"x_1 \\<in> ?H1 x_2 x_3 \\<Longrightarrow> ?H2 (?H3 (?H4 x_1 x_2) (?H5 (\\<lambda>y_0. y_0 \\<in> ... | 829 | 193 | 193 | 698 | 141 | 141 | 0.195923 | 0.765217 | 0.765217 |
Cardinals/Ordinal_Arithmetic | Ordinal_Arithmetic.rev_curr_FinFunc | Well_order ?r \<Longrightarrow> Well_order ?s \<Longrightarrow> Well_order ?t \<Longrightarrow> Field ?r \<noteq> {} \<Longrightarrow> rev_curr ?t ` FinFunc ?r (?s *o ?t) = FinFunc (?r ^o ?s) ?t | lemma rev_curr_FinFunc:
assumes Field: "Field r \<noteq> {}"
shows "rev_curr ` (FinFunc r (s *o t)) = FinFunc (r ^o s) t" | \<lbrakk>?H1 x_1; ?H2 x_2; ?H3 x_3; ?H4 x_1 \<noteq> ?H5\<rbrakk> \<Longrightarrow> ?H6 (?H7 x_3) (?H8 x_1 (?H9 x_2 x_3)) = ?H10 (?H11 x_1 x_2) x_3 | [
"\\<lbrakk> ?H1 x_1; ?H2 x_2; ?H3 x_2 \\<noteq> ?H4\\<rbrakk> \\<Longrightarrow> ?H5 (?H6 (?H7 x_1 x_2)) (?H8 x_1 x_2) = ?H9 (?H10 x_1 x_2) (?H11 x_1 x_2)"
] | [
"\\<lbrakk> ?H1 x_1; ?H2 x_2; ?H3 x_2 \\<noteq> ?H4\\<rbrakk> \\<Longrightarrow> ?H5 (?H6 (?H7 x_1 x_2)) (?H8 x_1 x_2) = ?H9 (?H10 x_1 x_2) (?H11 x_1 x_2)",
"\\<lbrakk> ?H1 x_1; ?H2 x_2; ?H3 x_2 \\<noteq> ?H4\\<rbrakk> \\<Longrightarrow> ?H5 (?H6 (?H7 x_1 x_2)) (?H8 x_1 x_2) = ?H9 (?H10 x_1 x_2) (?H11 x_1 x_2)",
... | 104 | 104 | 104 | 43 | 43 | 43 | 0.928571 | 0.928571 | 0.928571 |
Cardinals/Ordinal_Arithmetic | Ordinal_Arithmetic.oprod_minus_Id | Total ?r \<Longrightarrow> \<not> ?r \<subseteq> Id \<Longrightarrow> Total ?r' \<Longrightarrow> \<not> ?r' \<subseteq> Id \<Longrightarrow> ?r *o ?r' - Id \<subseteq> (?r - Id) *o (?r' - Id) | lemma oprod_minus_Id:
assumes r: "Total r" "\<not> (r \<le> Id)" and r': "Total r'" "\<not> (r' \<le> Id)"
shows "(r *o r') - Id \<le> (r - Id) *o (r' - Id)" | \<lbrakk>?H1 x_1; \<not> ?H2 x_1 ?H3; ?H4 x_2; \<not> ?H5 x_2 ?H6\<rbrakk> \<Longrightarrow> ?H7 (?H8 (?H9 x_1 x_2) ?H10) (?H9 (?H11 x_1 ?H3) (?H12 x_2 ?H6)) | [
"\\<lbrakk> ?H1 x_1; ?H2 x_2 ?H3; ?H2 x_3 ?H3\\<rbrakk> \\<Longrightarrow> ?H2 (?H4 (?H5 x_1 x_2) (?H5 x_1 x_3)) (?H5 x_1 (?H6 x_2 x_3))"
] | [
"\\<lbrakk> ?H1 x_1; ?H1 x_2; ?H2 x_1 ?H3; ?H2 x_2 ?H3\\<rbrakk> \\<Longrightarrow> ?H1 (?H4 (?H5 x_1 x_2) (?H5 x_2 x_1))",
"\\<lbrakk> ?H1 x_1; ?H1 x_2; ?H2 x_1 ?H3; ?H2 x_2 ?H3\\<rbrakk> \\<Longrightarrow> ?H1 (?H4 (?H5 x_1 x_2) (?H5 x_2 x_1))",
"\\<lbrakk> ?H1 x_1; ?H1 x_2; ?H2 x_1 ?H3; ?H2 x_2 ?H3\\<rbrakk>... | 136 | 125 | 125 | 52 | 56 | 52 | 0.736527 | 0.72956 | 0.736527 |
Cardinals/Ordinal_Arithmetic | Ordinal_Arithmetic.osum_minus_Id | Total ?r \<Longrightarrow> \<not> ?r \<subseteq> Id \<Longrightarrow> Total ?r' \<Longrightarrow> \<not> ?r' \<subseteq> Id \<Longrightarrow> ?r +o ?r' - Id \<subseteq> (?r - Id) +o (?r' - Id) | lemma osum_minus_Id:
assumes r: "Total r" "\<not> (r \<le> Id)" and r': "Total r'" "\<not> (r' \<le> Id)"
shows "(r +o r') - Id \<le> (r - Id) +o (r' - Id)" | \<lbrakk>?H1 x_1; \<not> ?H2 x_1 ?H3; ?H4 x_2; \<not> ?H5 x_2 ?H6\<rbrakk> \<Longrightarrow> ?H7 (?H8 (?H9 x_1 x_2) ?H10) (?H9 (?H11 x_1 ?H3) (?H12 x_2 ?H6)) | [
"\\<lbrakk> ?H1 x_1; ?H1 x_2; ?H2 x_1 ?H3; ?H2 x_2 ?H3\\<rbrakk> \\<Longrightarrow> ?H1 (?H4 x_1 x_2)"
] | [
"\\<lbrakk> ?H1 x_1; ?H1 x_2; ?H2 x_1 ?H3; ?H2 x_2 ?H3\\<rbrakk> \\<Longrightarrow> ?H1 (?H4 (?H5 x_1 x_2) ?H3)",
"\\<lbrakk> ?H1 x_1; ?H1 x_2; ?H2 x_1 ?H3; ?H2 x_2 ?H3\\<rbrakk> \\<Longrightarrow> ?H1 (?H4 (?H5 x_1 x_2) ?H3)",
"\\<lbrakk> ?H1 x_1; ?H1 x_2; ?H2 x_1 ?H3; ?H2 x_2 ?H3\\<rbrakk> \\<Longrightarrow> ... | 138 | 135 | 135 | 72 | 64 | 64 | 0.613924 | 0.666667 | 0.666667 |
Cardinals/Ordinal_Arithmetic | Ordinal_Arithmetic.osum_monoR | Well_order ?r \<Longrightarrow> Well_order ?s \<Longrightarrow> Well_order ?t \<Longrightarrow> ?s <o ?t \<Longrightarrow> ?r +o ?s <o ?r +o ?t | lemma osum_monoR:
assumes "s <o t"
shows "r +o s <o r +o t" (is "?L <o ?R") | \<lbrakk>?H1 x_1; ?H2 x_2; ?H3 x_3; ?H4 x_2 x_3\<rbrakk> \<Longrightarrow> ?H5 (?H6 x_1 x_2) (?H7 x_1 x_3) | [
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2)"
] | [
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2)",
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2)",
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2)",
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2)"... | 80 | 80 | 80 | 38 | 38 | 38 | 0.663551 | 0.663551 | 0.663551 |
Cardinals/Ordinal_Arithmetic | Ordinal_Arithmetic.oprod_monoL | Well_order ?r \<Longrightarrow> Well_order ?s \<Longrightarrow> Well_order ?t \<Longrightarrow> ?r \<le>o ?s \<Longrightarrow> ?r *o ?t \<le>o ?s *o ?t | lemma oprod_monoL:
assumes "r \<le>o s"
shows "r *o t \<le>o s *o t" | \<lbrakk>?H1 x_1; ?H2 x_2; ?H3 x_3; ?H4 x_1 x_2\<rbrakk> \<Longrightarrow> ?H5 (?H6 x_1 x_3) (?H7 x_2 x_3) | [
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2)"
] | [
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2)",
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2)",
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2)",
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2)"... | 80 | 80 | 80 | 38 | 38 | 38 | 0.663551 | 0.663551 | 0.663551 |
Cardinals/Ordinal_Arithmetic | Ordinal_Arithmetic.osum_monoL | Well_order ?r \<Longrightarrow> Well_order ?s \<Longrightarrow> Well_order ?t \<Longrightarrow> ?r \<le>o ?s \<Longrightarrow> ?r +o ?t \<le>o ?s +o ?t | lemma osum_monoL:
assumes "r \<le>o s"
shows "r +o t \<le>o s +o t" | \<lbrakk>?H1 x_1; ?H2 x_2; ?H3 x_3; ?H4 x_1 x_2\<rbrakk> \<Longrightarrow> ?H5 (?H6 x_1 x_3) (?H7 x_2 x_3) | [
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2)"
] | [
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2)",
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2)",
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2)",
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2)"... | 80 | 80 | 80 | 38 | 38 | 38 | 0.663551 | 0.663551 | 0.663551 |
Cardinals/Ordinal_Arithmetic | Ordinal_Arithmetic.oexp_monoL | Well_order ?r \<Longrightarrow> Well_order ?s \<Longrightarrow> Well_order ?t \<Longrightarrow> ?r \<le>o ?s \<Longrightarrow> ?r ^o ?t \<le>o ?s ^o ?t | lemma oexp_monoL:
assumes "r \<le>o s"
shows "r ^o t \<le>o s ^o t" | \<lbrakk>?H1 x_1; ?H2 x_2; ?H3 x_3; ?H4 x_1 x_2\<rbrakk> \<Longrightarrow> ?H5 (?H6 x_1 x_3) (?H7 x_2 x_3) | [
"?H1 x_1 \\<Longrightarrow> ?H2 (?H3 x_1) (?H4 x_1)"
] | [
"?H1 x_1 \\<Longrightarrow> ?H2 (?H3 x_1) (?H4 x_1)",
"?H1 x_1 \\<Longrightarrow> ?H2 (?H3 x_1) (?H4 x_1)",
"?H1 x_1 \\<Longrightarrow> ?H2 (?H3 x_1) (?H4 x_1)",
"?H1 x_1 \\<Longrightarrow> ?H2 (?H3 x_1) (?H4 x_1)",
"?H1 x_1 \\<Longrightarrow> ?H2 (?H3 x_1) (?H4 x_1)"
] | 104 | 104 | 104 | 62 | 62 | 62 | 0.457944 | 0.457944 | 0.457944 |
Cardinals/Ordinal_Arithmetic | Ordinal_Arithmetic.oprod_wf_Id | Total ?r \<Longrightarrow> Total ?r' \<Longrightarrow> wf (?r - Id) \<Longrightarrow> wf (?r' - Id) \<Longrightarrow> wf (?r *o ?r' - Id) | lemma oprod_wf_Id:
assumes TOT: "Total r" and TOT': "Total r'" and WF: "wf(r - Id)" and WF': "wf(r' - Id)"
shows "wf ((r *o r') - Id)" | \<lbrakk>?H1 x_1; ?H2 x_2; ?H3 (?H4 x_1 ?H5); ?H6 (?H7 x_2 ?H8)\<rbrakk> \<Longrightarrow> ?H9 (?H10 (?H11 x_1 x_2) ?H12) | [
"\\<lbrakk> ?H1 x_1; ?H2 x_2; ?H3 (?H4 x_1 ?H5) \\<and> ?H3 (?H4 x_2 ?H5) \\<and> ?H6 x_1 x_2 = ?H7\\<rbrakk> \\<Longrightarrow> ?H8 (?H9 x_1 x_2)"
] | [
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 (?H5 x_1 x_2) ?H6)",
"\\<lbrakk> ?H1 x_1; ?H2 (?H3 x_1 ?H4); ?H1 x_2; ?H2 (?H3 x_2 ?H4)\\<rbrakk> \\<Longrightarrow> ?H1 (?H5 x_1 x_2)",
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 (?H5 x_1 x_2) ?H6)",
"\\<lbrakk> ?H1 x_... | 92 | 47 | 47 | 50 | 33 | 33 | 0.782313 | 0.862903 | 0.862903 |
Cardinals/Ordinal_Arithmetic | Ordinal_Arithmetic.osum_wf_Id | Total ?r \<Longrightarrow> Total ?r' \<Longrightarrow> wf (?r - Id) \<Longrightarrow> wf (?r' - Id) \<Longrightarrow> wf (?r +o ?r' - Id) | lemma osum_wf_Id:
assumes TOT: "Total r" and TOT': "Total r'" and WF: "wf(r - Id)" and WF': "wf(r' - Id)"
shows "wf ((r +o r') - Id)" | \<lbrakk>?H1 x_1; ?H2 x_2; ?H3 (?H4 x_1 ?H5); ?H6 (?H7 x_2 ?H8)\<rbrakk> \<Longrightarrow> ?H9 (?H10 (?H11 x_1 x_2) ?H12) | [
"\\<lbrakk> ?H1 x_1; ?H2 (?H3 x_1 ?H4)\\<rbrakk> \\<Longrightarrow> ?H1 (?H5 x_1 x_1)"
] | [
"\\<lbrakk> ?H1 x_1; ?H2 (?H3 x_1 ?H4); ?H1 x_2; ?H2 (?H3 x_2 ?H4)\\<rbrakk> \\<Longrightarrow> ?H1 (?H5 x_1 x_2)",
"\\<lbrakk> ?H1 x_1; ?H2 (?H3 x_1 ?H4); ?H1 x_2; ?H2 (?H3 x_2 ?H4)\\<rbrakk> \\<Longrightarrow> ?H1 (?H5 x_1 x_2)",
"\\<lbrakk> ?H1 x_1; ?H2 (?H3 x_1 ?H4); ?H1 x_2; ?H2 (?H3 x_2 ?H4)\\<rbrakk> \\<... | 97 | 47 | 47 | 44 | 33 | 33 | 0.663934 | 0.862903 | 0.862903 |
Cardinals/Ordinal_Arithmetic | Ordinal_Arithmetic.ozero_oexp | Well_order ?r \<Longrightarrow> Well_order ?s \<Longrightarrow> Well_order ?t \<Longrightarrow> (?s, ozero) \<notin> ordIso \<Longrightarrow> ozero ^o ?s =o ozero | lemma ozero_oexp: "\<not> (s =o ozero) \<Longrightarrow> ozero ^o s =o ozero" | \<lbrakk>?H1 x_1; ?H2 x_2; ?H3 x_3; ?H4 (x_2, ?H5) ?H6\<rbrakk> \<Longrightarrow> ?H7 (?H8 ?H9 x_2) ?H10 | [
"\\<lbrakk> ?H1 x_1; ?H1 x_2; ?H2 (?, ?H3) ?H4; (x_3, x_4) \\<in> ?H5\\<rbrakk> \\<Longrightarrow> ?H6 (?H7 x_1 x_3) (?H8 x_2 x_4)"
] | [
"\\<lbrakk> ?H1 x_1; ?H1 x_2; ?H2 (?H3, ?H4) ?H5\\<rbrakk> \\<Longrightarrow> ?H6 (?H7 x_1) (?H8 x_2)",
"\\<lbrakk> ?H1 x_1; ?H1 x_2; ?H2 (?H3, ?H4) ?H5\\<rbrakk> \\<Longrightarrow> ?H6 (?H7 x_1) (?H8 x_2)",
"\\<lbrakk> ?H1 x_1; ?H1 x_2; ?H2 (?H3, ?H4) ?H5\\<rbrakk> \\<Longrightarrow> ?H6 (?H7 x_1) (?H8 x_2)",
... | 92 | 74 | 74 | 38 | 29 | 29 | 0.782946 | 0.87037 | 0.87037 |
Cardinals/Ordinal_Arithmetic | Ordinal_Arithmetic.oprod_osum | Well_order ?r \<Longrightarrow> Well_order ?s \<Longrightarrow> Well_order ?t \<Longrightarrow> ?r *o (?s +o ?t) =o ?r *o ?s +o ?r *o ?t | lemma oprod_osum: "r *o (s +o t) =o r *o s +o r *o t" (is "?L =o ?R") | \<lbrakk>?H1 x_1; ?H2 x_2; ?H3 x_3\<rbrakk> \<Longrightarrow> ?H4 (?H5 x_1 (?H6 x_2 x_3)) (?H7 (?H8 x_1 x_2) (?H9 x_1 x_3)) | [
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 (?H5 x_1 x_2)) (?H6 x_1 x_2)"
] | [
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2) (?H5 x_1 x_2)",
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2) (?H5 x_1 x_2)",
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2) (?H5 x_1 x_2)",
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbr... | 94 | 91 | 91 | 39 | 42 | 39 | 0.733871 | 0.685484 | 0.733871 |
Cardinals/Ordinal_Arithmetic | Ordinal_Arithmetic.oexp_osum | Well_order ?r \<Longrightarrow> Well_order ?s \<Longrightarrow> Well_order ?t \<Longrightarrow> ?r ^o (?s +o ?t) =o ?r ^o ?s *o ?r ^o ?t | lemma oexp_osum: "r ^o (s +o t) =o (r ^o s) *o (r ^o t)" (is "?R =o ?L") | \<lbrakk>?H1 x_1; ?H2 x_2; ?H3 x_3\<rbrakk> \<Longrightarrow> ?H4 (?H5 x_1 (?H6 x_2 x_3)) (?H7 (?H8 x_1 x_2) (?H9 x_1 x_3)) | [
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 (?H5 x_1 x_2)) (?H6 (?H7 x_1 x_2))"
] | [
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 (?H5 x_1 x_2)) (?H6 (?H7 x_1 x_2))",
"\\<lbrakk> ?H1 x_1; ?H2 x_2; ?H3 x_3; ?H4 x_4\\<rbrakk> \\<Longrightarrow> ?H5 (?H6 (?H7 x_1 x_3) (?H8 x_2 x_4)) (?H9 (?H10 x_1 x_2) (?H11 x_3 x_4))",
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow... | 94 | 94 | 94 | 34 | 30 | 30 | 0.782258 | 0.855172 | 0.855172 |
Cardinals/Ordinal_Arithmetic | Ordinal_Arithmetic.ordLeq_oexp2 | Well_order ?r \<Longrightarrow> Well_order ?s \<Longrightarrow> Well_order ?t \<Longrightarrow> oone <o ?r \<Longrightarrow> ?s \<le>o ?r ^o ?s | lemma ordLeq_oexp2:
assumes "oone <o r"
shows "s \<le>o r ^o s" | \<lbrakk>?H1 x_1; ?H2 x_2; ?H3 x_3; ?H4 ?H5 x_1\<rbrakk> \<Longrightarrow> ?H6 x_2 (?H7 x_1 x_2) | [
"\\<lbrakk> ?H1 x_1; ?H2 x_1 x_2; ?H3 x_2 ?H4 \\<Longrightarrow> x_3\\<rbrakk> \\<Longrightarrow> ?H5 x_1 (?H6 x_2) x_3"
] | [
"\\<lbrakk> ?H1 x_1; ?H2 x_1 ?H3; ?H4 x_2\\<rbrakk> \\<Longrightarrow> ?H5 (?H6 x_1 x_2)",
"\\<lbrakk> ?H1 x_1; ?H2 x_1 ?H3; ?H4 x_2; ?H5 x_2 ?H6\\<rbrakk> \\<Longrightarrow> ?H7 (?H8 x_1 x_2)",
"\\<lbrakk> ?H1 x_1; ?H2 x_1 ?H3; ?H4 x_2; ?H5 x_2 ?H6\\<rbrakk> \\<Longrightarrow> ?H7 (?H8 x_1 x_2) (?H9 x_1 x_2)",... | 85 | 50 | 50 | 31 | 17 | 17 | 0.788136 | 0.94 | 0.94 |
Cardinals/Ordinal_Arithmetic | Ordinal_Arithmetic.oprod_assoc | Well_order ?r \<Longrightarrow> Well_order ?s \<Longrightarrow> Well_order ?t \<Longrightarrow> (?r *o ?s) *o ?t =o ?r *o ?s *o ?t | lemma oprod_assoc: "(r *o s) *o t =o r *o s *o t" (is "?L =o ?R") | \<lbrakk>?H1 x_1; ?H2 x_2; ?H3 x_3\<rbrakk> \<Longrightarrow> ?H4 (?H5 (?H6 x_1 x_2) x_3) (?H7 x_1 (?H8 x_2 x_3)) | [
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2)"
] | [
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2)",
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2)",
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2)",
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2)"... | 86 | 86 | 86 | 44 | 44 | 44 | 0.622807 | 0.622807 | 0.622807 |
Cardinals/Ordinal_Arithmetic | Ordinal_Arithmetic.osum_assoc | Well_order ?r \<Longrightarrow> Well_order ?s \<Longrightarrow> Well_order ?t \<Longrightarrow> (?r +o ?s) +o ?t =o ?r +o ?s +o ?t | lemma osum_assoc: "(r +o s) +o t =o r +o s +o t" (is "?L =o ?R") | \<lbrakk>?H1 x_1; ?H2 x_2; ?H3 x_3\<rbrakk> \<Longrightarrow> ?H4 (?H5 (?H6 x_1 x_2) x_3) (?H7 x_1 (?H8 x_2 x_3)) | [
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2)"
] | [
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2)",
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2)",
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2)",
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2)"... | 86 | 86 | 86 | 44 | 44 | 44 | 0.622807 | 0.622807 | 0.622807 |
Cardinals/Ordinal_Arithmetic | Ordinal_Arithmetic.oexp_oexp | Well_order ?r \<Longrightarrow> Well_order ?s \<Longrightarrow> Well_order ?t \<Longrightarrow> ?r ^o ?s ^o ?t =o ?r ^o (?s *o ?t) | lemma oexp_oexp: "(r ^o s) ^o t =o r ^o (s *o t)" (is "?R =o ?L") | \<lbrakk>?H1 x_1; ?H2 x_2; ?H3 x_3\<rbrakk> \<Longrightarrow> ?H4 (?H5 (?H6 x_1 x_2) x_3) (?H7 x_1 (?H8 x_2 x_3)) | [
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2) (?H5 x_1 x_2)"
] | [
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2) (?H5 x_1 x_2)",
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2) (?H5 x_1 x_2)",
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2) (?H5 x_1 x_2)",
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbr... | 73 | 73 | 73 | 32 | 32 | 32 | 0.745614 | 0.745614 | 0.745614 |
Cardinals/Ordinal_Arithmetic | Ordinal_Arithmetic.oprod_minus_Id2 | ?r' \<subseteq> Id \<Longrightarrow> ?r *o ?r' - Id \<subseteq> {((x1, y), x2, y). (x1, x2) \<in> ?r - Id \<and> y \<in> Field ?r'} | lemma oprod_minus_Id2:
"r' \<le> Id \<Longrightarrow> r *o r' - Id \<le> {((x1,y), (x2,y)). (x1, x2) \<in> (r - Id) \<and> y \<in> Field r'}" | ?H1 x_1 ?H2 \<Longrightarrow> ?H3 (?H4 (?H5 x_2 x_1) ?H6) (?H7 (?H8 (?H9 (\<lambda>y_0 y_1. ?H10 (\<lambda>y_2 y_3. (y_0, y_2) \<in> ?H11 x_2 ?H12 \<and> y_3 \<in> ?H13 x_1))))) | [
"?H1 x_1 ?H2 \\<Longrightarrow> ?H1 (?H3 (?H4 (?H5 x_2 x_1) (?H6 (?H7 x_2) (?H8 x_1))) (?H7 x_2)) (?H9 (?H10 (?H11 x_2) (?H12 (?H13 x_1) (?H14 (?H15 x_2) x_1))))"
] | [
"?H1 x_1 ?H2 \\<Longrightarrow> ?H1 (?H3 (?H4 (?H5 x_2 x_3) x_1) (?H6 (?H7 x_2) (?H8 x_3))) (?H6 (?H7 x_2) (?H8 x_3))",
"?H1 x_1 ?H2 \\<Longrightarrow> ?H1 (?H3 (?H4 (?H5 x_2 x_3) x_1) (?H6 (?H7 x_2) (?H8 x_3))) (?H6 (?H7 x_2) (?H8 x_3))",
"?H1 x_1 ?H2 \\<Longrightarrow> ?H1 (?H3 (?H4 (?H5 x_2 x_3) x_1) (?H6 (?... | 119 | 123 | 119 | 85 | 97 | 85 | 0.6 | 0.578378 | 0.6 |
Cardinals/Ordinal_Arithmetic | Ordinal_Arithmetic.oprod_minus_Id1 | ?r \<subseteq> Id \<Longrightarrow> ?r *o ?r' - Id \<subseteq> {((x, y1), x, y2). x \<in> Field ?r \<and> (y1, y2) \<in> ?r' - Id} | lemma oprod_minus_Id1:
"r \<le> Id \<Longrightarrow> r *o r' - Id \<le> {((x,y1), (x,y2)). x \<in> Field r \<and> (y1, y2) \<in> (r' - Id)}" | ?H1 x_1 ?H2 \<Longrightarrow> ?H3 (?H4 (?H5 x_1 x_2) ?H6) (?H7 (?H8 (?H9 (\<lambda>y_0 y_1. ?H10 (\<lambda>y_2 y_3. y_2 \<in> ?H11 x_1 \<and> (y_1, y_3) \<in> ?H12 x_2 ?H13))))) | [
"?H1 x_1 ?H2 \\<Longrightarrow> ?H1 (?H3 (?H4 (?H5 x_2 x_1) (?H6 (?H7 x_2) (?H8 x_1))) (?H7 x_2)) (?H9 (?H10 (?H11 x_2) (?H12 (?H13 x_1) (?H14 (?H15 x_2) x_1))))"
] | [
"?H1 x_1 ?H2 \\<Longrightarrow> ?H1 (?H3 (?H4 (?H5 x_2 x_3) x_1) (?H6 (?H7 x_2) (?H8 x_3))) (?H6 (?H7 x_2) (?H8 x_3))",
"?H1 x_1 ?H2 \\<Longrightarrow> ?H1 (?H3 (?H4 (?H5 x_2 x_3) x_1) (?H6 (?H7 x_2) (?H8 x_3))) (?H6 (?H7 x_2) (?H8 x_3))",
"?H1 x_1 ?H2 \\<Longrightarrow> ?H1 (?H3 (?H4 (?H5 x_2 x_3) x_1) (?H6 (?... | 122 | 123 | 122 | 85 | 97 | 85 | 0.607656 | 0.578378 | 0.607656 |
Cardinals/Ordinal_Arithmetic | Ordinal_Arithmetic.oprod_Preorder | Preorder ?r \<Longrightarrow> Preorder ?r' \<Longrightarrow> antisym ?r \<Longrightarrow> antisym ?r' \<Longrightarrow> Preorder (?r *o ?r') | lemma oprod_Preorder: "\<lbrakk>Preorder r; Preorder r'; antisym r; antisym r'\<rbrakk> \<Longrightarrow> Preorder (r *o r')" | \<lbrakk>?H1 x_1; ?H2 x_2; ?H3 x_1; ?H4 x_2\<rbrakk> \<Longrightarrow> ?H5 (?H6 x_1 x_2) | [
"\\<lbrakk> ?H1 x_1; ?H2 x_1\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_1)"
] | [
"\\<lbrakk> ?H1 x_1; ?H2 x_1\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_1)",
"\\<lbrakk> ?H1 x_1; ?H1 x_2; ?H2 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2)",
"\\<lbrakk> ?H1 x_1; ?H2 x_1\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_1)",
"\\<lbrakk> ?H1 x_1; ?H2 x_1\\<rbrakk> \\<Longrightarrow> ... | 61 | 5 | 5 | 21 | 5 | 5 | 0.777778 | 0.956044 | 0.956044 |
Cardinals/Ordinal_Arithmetic | Ordinal_Arithmetic.oprod_embed | Well_order ?r \<Longrightarrow> Well_order ?r' \<Longrightarrow> ?r' \<noteq> {} \<Longrightarrow> embed ?r (?r *o ?r') (\<lambda>x. (x, minim ?r' (Field ?r'))) | lemma oprod_embed:
assumes WELL: "Well_order r" and WELL': "Well_order r'" and "r' \<noteq> {}"
shows "embed r (r *o r') (\<lambda>x. (x, minim r' (Field r')))" (is "embed _ _ ?f") | \<lbrakk>?H1 x_1; ?H2 x_2; x_2 \<noteq> ?H3\<rbrakk> \<Longrightarrow> ?H4 x_1 (?H5 x_1 x_2) (\<lambda>y_0. (y_0, ?H6 x_2 (?H7 x_2))) | [
"\\<lbrakk> ?H1 x_1; ?H2 x_2; ?H3 x_2 x_1 x_3; ?H4 x_2 x_4 x_5; x_4 \\<noteq> ?H5\\<rbrakk> \\<Longrightarrow> ?H6 (?H7 (?H8 x_1 x_2) (?H9 x_4 x_5)) = ?H10 (?H11 x_1 x_4) (?H12 x_2 x_5)"
] | [
"\\<lbrakk> ?H1 x_1; ?H2 x_2; ?H3 x_1 x_2 x_3; ?H4 x_2 x_4 x_5; x_4 \\<noteq> ?H5\\<rbrakk> \\<Longrightarrow> ?H6 (?H7 x_1 (?H8 x_2)) (?H9 x_1 (?H10 x_4)) (\\<lambda>y_0. (x_3 y_0, x_5 y_0))",
"\\<lbrakk> ?H1 x_1; ?H2 x_2; ?H3 x_1 x_2 x_3; ?H4 x_2 x_4 x_5; x_4 \\<noteq> ?H5\\<rbrakk> \\<Longrightarrow> ?H6 (?H7 ... | 150 | 150 | 150 | 75 | 74 | 74 | 0.618557 | 0.730769 | 0.730769 |
Cardinals/Ordinal_Arithmetic | Ordinal_Arithmetic.osum_minus_Id2 | ?r' \<subseteq> Id \<Longrightarrow> ?r +o ?r' - Id \<subseteq> map_prod Inl Inl ` (?r - Id) \<union> Inl ` Field ?r \<times> Inr ` Field ?r' | lemma osum_minus_Id2:
"r' \<le> Id \<Longrightarrow> (r +o r') - Id \<le> (map_prod Inl Inl ` (r - Id)) \<union> (Inl ` Field r \<times> Inr ` Field r')" | ?H1 x_1 ?H2 \<Longrightarrow> ?H3 (?H4 (?H5 x_2 x_1) ?H6) (?H7 (?H8 (?H9 ?H10 ?H10) (?H11 x_2 ?H12)) (?H13 (?H14 ?H10 (?H15 x_2)) (?H16 ?H17 (?H18 x_1)))) | [
"?H1 ?H2 (?H3 (?H4 x_1 x_2) (?H5 x_1 x_2)) \\<Longrightarrow> ?H6 (?H7 (?H8 ?H9 ?H10) ?H11) (?H12 x_1 x_2) = ?H13 (?H14 x_1) (?H15 (?H16 x_2))"
] | [
"?H1 ?H2 (?H3 (?H4 x_1 x_2) (?H5 (?H6 x_1) (?H7 x_2))) \\<Longrightarrow> ?H8 (?H9 (?H10 x_1) (?H11 x_2)) (?H12 x_1) = ?H13 (?H14 x_2)",
"?H1 ?H2 (?H3 (?H4 x_1 x_2) (?H5 (?H6 x_1) (?H7 x_2))) \\<Longrightarrow> ?H8 (?H9 (?H10 x_1) (?H11 x_2)) (?H12 x_1) = ?H13 (?H14 x_2)",
"?H1 ?H2 (?H3 (?H4 x_1 x_2) (?H5 (?H6 ... | 135 | 147 | 135 | 83 | 86 | 83 | 0.8375 | 0.7875 | 0.8375 |
Cardinals/Ordinal_Arithmetic | Ordinal_Arithmetic.osum_minus_Id1 | ?r \<subseteq> Id \<Longrightarrow> ?r +o ?r' - Id \<subseteq> Inl ` Field ?r \<times> Inr ` Field ?r' \<union> map_prod Inr Inr ` (?r' - Id) | lemma osum_minus_Id1:
"r \<le> Id \<Longrightarrow> (r +o r') - Id \<le> (Inl ` Field r \<times> Inr ` Field r') \<union> (map_prod Inr Inr ` (r' - Id))" | ?H1 x_1 ?H2 \<Longrightarrow> ?H3 (?H4 (?H5 x_1 x_2) ?H6) (?H7 (?H8 (?H9 ?H10 (?H11 x_1)) (?H12 ?H13 (?H14 x_2))) (?H15 (?H16 ?H13 ?H13) (?H17 x_2 ?H18))) | [
"?H1 x_1 ?H2 \\<Longrightarrow> ?H3 (?H4 (?H5 x_2 x_3) (?H6 x_1 x_1)) (?H7 (?H8 ?H9 (?H10 x_1)) (?H11 (?H12 x_3) (?H13 (?H14 x_2) (?H15 x_3))))"
] | [
"?H1 ?H2 (?H3 (?H4 x_1 x_2) (?H5 (?H6 x_1) (?H7 x_2))) \\<Longrightarrow> ?H8 (?H9 x_1) (?H10 x_2) = ?H11 (?H12 ?H13 (?H14 x_1)) (?H15 ?H16 (?H7 x_2))",
"?H1 ?H2 (?H3 (?H4 x_1 x_2) (?H5 (?H6 x_1) (?H7 x_2))) \\<Longrightarrow> ?H8 (?H9 x_1) (?H10 x_2) = ?H11 (?H12 ?H13 (?H14 x_1)) (?H15 ?H16 (?H7 x_2))",
"?H1 ?... | 95 | 141 | 95 | 44 | 85 | 44 | 0.832298 | 0.8875 | 0.8875 |
Cardinals/Ordinal_Arithmetic | Ordinal_Arithmetic.fin_support_Field_osum | (?f \<in> fin_support ?z (Inl ` ?A \<union> Inr ` ?B)) = (?f \<circ> Inl \<in> fin_support ?z ?A \<and> ?f \<circ> Inr \<in> fin_support ?z ?B) | lemma fin_support_Field_osum:
"f \<in> fin_support z (Inl ` A \<union> Inr ` B) \<longleftrightarrow>
(f o Inl) \<in> fin_support z A \<and> (f o Inr) \<in> fin_support z B" (is "?L \<longleftrightarrow> ?R1 \<and> ?R2") | (x_1 \<in> ?H1 x_2 (?H2 (?H3 ?H4 x_3) (?H5 ?H6 x_4))) = (?H7 x_1 ?H4 \<in> ?H8 x_2 x_3 \<and> ?H9 x_1 ?H6 \<in> ?H10 x_2 x_4) | [
"?H1 x_1 x_2 = ?H2 (?H3 ?H4 (?H5 x_1)) (?H6 ?H7 x_2)"
] | [
"?H1 x_1 x_2 = ?H2 (?H3 ?H4 (?H5 x_1)) (?H6 ?H7 x_2)",
"?H1 x_1 x_2 = ?H2 (?H3 ?H4 (?H5 x_1)) (?H6 ?H7 x_2)",
"?H1 x_1 x_2 = ?H2 (?H3 ?H4 (?H5 x_1)) (?H6 ?H7 x_2)",
"?H1 x_1 x_2 = ?H2 (?H3 ?H4 (?H5 x_1)) (?H6 ?H7 x_2)",
"?H1 x_1 x_2 = ?H2 (?H3 ?H4 (?H5 x_1)) (?H6 ?H7 x_2)"
] | 113 | 113 | 113 | 83 | 83 | 83 | 0.408 | 0.408 | 0.408 |
Cardinals/Ordinal_Arithmetic | Ordinal_Arithmetic.oprod_trans | trans ?r \<Longrightarrow> trans ?r' \<Longrightarrow> antisym ?r \<Longrightarrow> antisym ?r' \<Longrightarrow> trans (?r *o ?r') | lemma oprod_trans:
assumes "trans r" "trans r'" "antisym r" "antisym r'"
shows "trans (r *o r')" | \<lbrakk>?H1 x_1; ?H2 x_2; ?H3 x_1; ?H4 x_2\<rbrakk> \<Longrightarrow> ?H5 (?H6 x_1 x_2) | [
"\\<lbrakk> ?H1 x_1; ?H2 x_1; ?H3 x_2; ?H4 x_2\\<rbrakk> \\<Longrightarrow> ?H5 (?H6 x_1 x_2)"
] | [
"\\<lbrakk> ?H1 x_1; ?H2 x_1\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_1)",
"\\<lbrakk> ?H1 x_1; ?H2 x_1; ?H3 x_2; ?H4 x_2\\<rbrakk> \\<Longrightarrow> ?H5 (?H6 x_1 x_2)",
"\\<lbrakk> ?H1 x_1; ?H2 x_1; ?H3 x_2\\<rbrakk> \\<Longrightarrow> ?H4 (?H5 x_1 x_2)",
"\\<lbrakk> ?H1 x_1; ?H2 x_1; ?H3 x_2; ?H4 x_2\\<... | 2 | 2 | 2 | 2 | 2 | 2 | 1 | 1 | 1 |
Cardinals/Ordinal_Arithmetic | Ordinal_Arithmetic.Func_upd | ?f \<in> Func ?A ?B \<Longrightarrow> ?x \<in> ?A \<Longrightarrow> ?y \<in> ?B \<Longrightarrow> ?f(?x := ?y) \<in> Func ?A ?B | lemma Func_upd: "\<lbrakk>f \<in> Func A B; x \<in> A; y \<in> B\<rbrakk> \<Longrightarrow> f(x := y) \<in> Func A B" | \<lbrakk>x_1 \<in> ?H1 x_2 x_3; x_4 \<in> x_2; x_5 \<in> x_3\<rbrakk> \<Longrightarrow> ?H2 x_1 x_4 x_5 \<in> ?H1 x_2 x_3 | [
"x_1 \\<in> ?H1 x_2 x_3 \\<Longrightarrow> ?H2 x_1 x_4 x_5 \\<in> ?H1 x_2 x_3"
] | [
"x_1 \\<in> ?H1 x_2 x_3 \\<Longrightarrow> ?H2 x_1 x_4 x_5 \\<in> ?H1 x_2 x_3",
"x_1 \\<in> ?H1 x_2 x_3 \\<Longrightarrow> ?H2 x_1 x_4 x_5 \\<in> ?H1 x_2 x_3",
"x_1 \\<in> ?H1 x_2 x_3 \\<Longrightarrow> ?H2 x_1 x_4 x_5 \\<in> ?H1 x_2 x_3",
"x_1 \\<in> ?H1 x_2 x_3 \\<Longrightarrow> ?H2 x_1 x_4 x_5 \\<in> ?H1 ... | 119 | 119 | 119 | 48 | 48 | 48 | 0.603306 | 0.603306 | 0.603306 |
Cardinals/Ordinal_Arithmetic | Ordinal_Arithmetic.oprod_ooneL | Well_order ?r \<Longrightarrow> Well_order ?s \<Longrightarrow> Well_order ?t \<Longrightarrow> oone *o ?r =o ?r | lemma oprod_ooneL: "oone *o r =o r" (is "?L =o ?R") | \<lbrakk>?H1 x_1; ?H2 x_2; ?H3 x_3\<rbrakk> \<Longrightarrow> ?H4 (?H5 ?H6 x_1) x_1 | [
"?H1 x_1 \\<Longrightarrow> ?H2 (?H3 x_1 ?H4) x_1"
] | [
"?H1 x_1 \\<Longrightarrow> ?H2 (?H3 x_1 ?H4) x_1",
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2) (?H5 ?H6 ?H7)",
"?H1 x_1 \\<Longrightarrow> ?H2 (?H3 x_1 ?H4) x_1",
"?H1 x_1 \\<Longrightarrow> ?H2 (?H3 x_1 ?H4) x_1",
"?H1 x_1 \\<Longrightarrow> ?H2 (?H3 x_1 ?H4) x_1"
] | 84 | 55 | 55 | 45 | 28 | 28 | 0.559524 | 0.877778 | 0.877778 |
Cardinals/Ordinal_Arithmetic | Ordinal_Arithmetic.oprod_ozeroL | Well_order ?r \<Longrightarrow> Well_order ?s \<Longrightarrow> Well_order ?t \<Longrightarrow> ozero *o ?r =o ozero | lemma oprod_ozeroL: "ozero *o r =o ozero" | \<lbrakk>?H1 x_1; ?H2 x_2; ?H3 x_3\<rbrakk> \<Longrightarrow> ?H4 (?H5 ?H6 x_1) ?H7 | [
"?H1 x_1 \\<Longrightarrow> ?H2 (?H3 x_1 ?H4) x_1"
] | [
"?H1 x_1 \\<Longrightarrow> ?H2 (?H3 x_1 ?H4) x_1",
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2) ?H5",
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2) (?H5 ?H6 ?H7)",
"?H1 x_1 \\<Longrightarrow> ?H2 (?H3 x_1 ?H4) x_1",
"?H1 x_1 \\<Longrightarrow> ?H... | 84 | 52 | 52 | 48 | 16 | 16 | 0.541176 | 0.942529 | 0.942529 |
Cardinals/Ordinal_Arithmetic | Ordinal_Arithmetic.oprod_ooneR | Well_order ?r \<Longrightarrow> Well_order ?s \<Longrightarrow> Well_order ?t \<Longrightarrow> ?r *o oone =o ?r | lemma oprod_ooneR: "r *o oone =o r" (is "?L =o ?R") | \<lbrakk>?H1 x_1; ?H2 x_2; ?H3 x_3\<rbrakk> \<Longrightarrow> ?H4 (?H5 x_1 ?H6) x_1 | [
"?H1 x_1 \\<Longrightarrow> ?H2 (?H3 x_1 ?H4) x_1"
] | [
"?H1 x_1 \\<Longrightarrow> ?H2 (?H3 x_1 ?H4) x_1",
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2) (?H5 ?H6 ?H7)",
"?H1 x_1 \\<Longrightarrow> ?H2 (?H3 x_1 ?H4) x_1",
"?H1 x_1 \\<Longrightarrow> ?H2 (?H3 x_1 ?H4) x_1",
"?H1 x_1 \\<Longrightarrow> ?H2 (?H3 x_1 ?H4) x_1"
] | 84 | 55 | 55 | 40 | 26 | 26 | 0.559524 | 0.877778 | 0.877778 |
Cardinals/Ordinal_Arithmetic | Ordinal_Arithmetic.oprod_ozeroR | Well_order ?r \<Longrightarrow> Well_order ?s \<Longrightarrow> Well_order ?t \<Longrightarrow> ?r *o ozero =o ozero | lemma oprod_ozeroR: "r *o ozero =o ozero" | \<lbrakk>?H1 x_1; ?H2 x_2; ?H3 x_3\<rbrakk> \<Longrightarrow> ?H4 (?H5 x_1 ?H6) ?H7 | [
"?H1 x_1 \\<Longrightarrow> ?H2 (?H3 x_1 ?H4) x_1"
] | [
"?H1 x_1 \\<Longrightarrow> ?H2 (?H3 x_1 ?H4) x_1",
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2) ?H5",
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2) (?H5 ?H6 ?H7)",
"?H1 x_1 \\<Longrightarrow> ?H2 (?H3 x_1 ?H4) x_1",
"?H1 x_1 \\<Longrightarrow> ?H... | 84 | 52 | 52 | 43 | 15 | 15 | 0.541176 | 0.942529 | 0.942529 |
Cardinals/Ordinal_Arithmetic | Ordinal_Arithmetic.osum_ozeroL | Well_order ?r \<Longrightarrow> Well_order ?s \<Longrightarrow> Well_order ?t \<Longrightarrow> ozero +o ?r =o ?r | lemma osum_ozeroL: "ozero +o r =o r" | \<lbrakk>?H1 x_1; ?H2 x_2; ?H3 x_3\<rbrakk> \<Longrightarrow> ?H4 (?H5 ?H6 x_1) x_1 | [
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2) ?H5 = (x_1 = ?H6 \\<or> x_2 = ?H7)"
] | [
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2) ?H5 = (x_1 = ?H6 \\<or> x_2 = ?H7)",
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2) ?H5 = (?H6 x_1 = ?H7 x_2)",
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2) ?H5 = (x_1 = ?H6 \\... | 73 | 65 | 65 | 44 | 34 | 34 | 0.766355 | 0.828283 | 0.828283 |
Cardinals/Ordinal_Arithmetic | Ordinal_Arithmetic.osum_ozeroR | Well_order ?r \<Longrightarrow> Well_order ?s \<Longrightarrow> Well_order ?t \<Longrightarrow> ?r +o ozero =o ?r | lemma osum_ozeroR: "r +o ozero =o r" | \<lbrakk>?H1 x_1; ?H2 x_2; ?H3 x_3\<rbrakk> \<Longrightarrow> ?H4 (?H5 x_1 ?H6) x_1 | [
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2) ?H5 = (x_1 = ?H6 \\<or> x_2 = ?H7)"
] | [
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2) ?H5 = (x_1 = ?H6 \\<or> x_2 = ?H7)",
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2) ?H5 = (?H6 x_1 = ?H7 x_2)",
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2) ?H5 = (x_1 = ?H6 \\... | 75 | 67 | 67 | 42 | 35 | 35 | 0.766355 | 0.828283 | 0.828283 |
Cardinals/Ordinal_Arithmetic | Ordinal_Arithmetic.oone_oexp | Well_order ?r \<Longrightarrow> Well_order ?s \<Longrightarrow> Well_order ?t \<Longrightarrow> oone ^o ?s =o oone | lemma oone_oexp: "oone ^o s =o oone" (is "?L =o ?R") | \<lbrakk>?H1 x_1; ?H2 x_2; ?H3 x_3\<rbrakk> \<Longrightarrow> ?H4 (?H5 ?H6 x_2) ?H6 | [
"?H1 x_1 \\<Longrightarrow> ?H2 (?H3 x_1) ?H4"
] | [
"?H1 x_1 \\<Longrightarrow> ?H2 (?H3 x_1) ?H4",
"?H1 x_1 \\<Longrightarrow> ?H2 (?H3 x_1 ?H4) x_1",
"?H1 x_1 \\<Longrightarrow> ?H2 (?H3 x_1) ?H4",
"?H1 x_1 \\<Longrightarrow> ?H2 (?H3 x_1) ?H4",
"?H1 x_1 \\<Longrightarrow> ?H2 (?H3 x_1) ?H4"
] | 84 | 84 | 84 | 45 | 45 | 45 | 0.494118 | 0.523256 | 0.523256 |
Cardinals/Ordinal_Arithmetic | Ordinal_Arithmetic.oprod_ordLeq | Well_order ?r \<Longrightarrow> Well_order ?r' \<Longrightarrow> ?r' \<noteq> {} \<Longrightarrow> ?r \<le>o ?r *o ?r' | null | \<lbrakk>?H1 x_1; ?H2 x_2; x_2 \<noteq> ?H3\<rbrakk> \<Longrightarrow> ?H4 x_1 (?H5 x_1 x_2) | [
"\\<lbrakk> ?H1 x_1; ?H2 x_2; x_1 \\<noteq> ?H3; x_2 \\<noteq> ?H4\\<rbrakk> \\<Longrightarrow> ?H5 (?H6 x_1 x_2) (?H7 x_1 x_2)"
] | [
"\\<lbrakk> ?H1 x_1; ?H2 x_2; x_2 \\<noteq> ?H3\\<rbrakk> \\<Longrightarrow> ?H4 (?H5 x_1 x_2)",
"\\<lbrakk> ?H1 x_1; ?H2 x_2; x_1 \\<noteq> ?H3; x_2 \\<noteq> ?H4\\<rbrakk> \\<Longrightarrow> ?H5 (?H6 x_1 x_2) (?H7 x_1 x_2)",
"\\<lbrakk> ?H1 x_1; ?H2 x_2; x_1 \\<noteq> ?H3; x_2 \\<noteq> ?H4\\<rbrakk> \\<Longr... | 75 | 14 | 14 | 31 | 4 | 4 | 0.768595 | 0.956989 | 0.956989 |
Cardinals/Ordinal_Arithmetic | Ordinal_Arithmetic.wf_extend_oprod1 | wf ?r \<Longrightarrow> wf {((x, y1), x, y2). x \<in> ?A \<and> (y1, y2) \<in> ?r} | lemma wf_extend_oprod1:
assumes "wf r"
shows "wf {((x,y1), (x,y2)) . x \<in> A \<and> (y1, y2) \<in> r}" | ?H1 x_1 \<Longrightarrow> ?H2 (?H3 (?H4 (?H5 (\<lambda>y_0 y_1. ?H6 (\<lambda>y_2 y_3. y_2 \<in> x_2 \<and> (y_1, y_3) \<in> x_1))))) | [
"?H1 (?H2 (?H3 (\\<lambda>y_0 y_1. x_1 y_0 \\<and> x_2 y_1))) \\<Longrightarrow> ?H4 (?H5 (?H6 (\\<lambda>y_2 y_3. (x_1 y_2, x_2 y_3))))"
] | [
"?H1 (?H2 (?H3 (\\<lambda>y_0 y_1. x_1 y_0 \\<and> x_2 y_1))) \\<Longrightarrow> ?H1 (?H2 (?H3 (\\<lambda>y_2 y_3. x_1 y_2 \\<and> x_3 y_3))) \\<and> ?H1 (?H2 (?H3 (\\<lambda>y_4 y_5. x_2 y_4 \\<and> x_3 y_5)))",
"?H1 (?H2 (?H3 (\\<lambda>y_0 y_1. x_1 y_0 \\<and> x_2 y_1))) \\<Longrightarrow> ?H1 (?H2 (?H3 (\\<la... | 124 | 187 | 124 | 100 | 118 | 100 | 0.87234 | 0.629268 | 0.87234 |
Cardinals/Ordinal_Arithmetic | Ordinal_Arithmetic.wf_extend_oprod2 | wf ?r \<Longrightarrow> wf {((x1, y), x2, y). (x1, x2) \<in> ?r \<and> y \<in> ?A} | lemma wf_extend_oprod2:
assumes "wf r"
shows "wf {((x1,y), (x2,y)) . (x1, x2) \<in> r \<and> y \<in> A}" | ?H1 x_1 \<Longrightarrow> ?H2 (?H3 (?H4 (?H5 (\<lambda>y_0 y_1. ?H6 (\<lambda>y_2 y_3. (y_0, y_2) \<in> x_1 \<and> y_3 \<in> x_2))))) | [
"?H1 (?H2 (?H3 (\\<lambda>y_0 y_1. x_1 y_0 \\<and> x_2 y_1))) \\<Longrightarrow> ?H4 (?H5 (?H6 (\\<lambda>y_2 y_3. (x_1 y_2, x_2 y_3))))"
] | [
"?H1 (?H2 (?H3 (\\<lambda>y_0 y_1. x_1 y_0 \\<and> x_2 y_1))) \\<Longrightarrow> ?H1 (?H2 (?H3 (\\<lambda>y_2 y_3. x_1 y_2 \\<and> x_3 y_3))) \\<and> ?H1 (?H2 (?H3 (\\<lambda>y_4 y_5. x_2 y_4 \\<and> x_3 y_5)))",
"?H1 (?H2 (?H3 (\\<lambda>y_0 y_1. x_1 y_0 \\<and> x_2 y_1))) \\<Longrightarrow> ?H1 (?H2 (?H3 (\\<la... | 122 | 191 | 122 | 100 | 122 | 100 | 0.87234 | 0.629268 | 0.87234 |
Cardinals/Ordinal_Arithmetic | Ordinal_Arithmetic.oprod_cong | ?t =o ?u \<Longrightarrow> ?r =o ?s \<Longrightarrow> ?t *o ?r =o ?u *o ?s | lemma oprod_cong:
assumes "t =o u" and "r =o s"
shows "t *o r =o u *o s" | \<lbrakk>?H1 x_1 x_2; ?H2 x_3 x_4\<rbrakk> \<Longrightarrow> ?H3 (?H4 x_1 x_3) (?H5 x_2 x_4) | [
"\\<lbrakk> ?H1 x_1 x_2; ?H2 x_3 x_4\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_3) (?H5 x_2 x_4)"
] | [
"\\<lbrakk> ?H1 x_1 x_2; ?H2 x_3 x_4\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_3) (?H5 x_2 x_4)",
"\\<lbrakk> ?H1 x_1 x_2; ?H2 x_3 x_4\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_3) (?H5 x_2 x_4)",
"\\<lbrakk> ?H1 x_1 x_2; ?H2 x_3 x_4\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_3) (?H5 x_2 x_4)",
"\\<lbrakk... | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 |
Cardinals/Ordinal_Arithmetic | Ordinal_Arithmetic.osum_cong | ?t =o ?u \<Longrightarrow> ?r =o ?s \<Longrightarrow> ?t +o ?r =o ?u +o ?s | lemma osum_cong:
assumes "t =o u" and "r =o s"
shows "t +o r =o u +o s" | \<lbrakk>?H1 x_1 x_2; ?H2 x_3 x_4\<rbrakk> \<Longrightarrow> ?H3 (?H4 x_1 x_3) (?H5 x_2 x_4) | [
"\\<lbrakk> ?H1 x_1 x_2; ?H2 x_3 x_4\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_3) (?H5 x_2 x_4)"
] | [
"\\<lbrakk> ?H1 x_1 x_2; ?H2 x_3 x_4\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_3) (?H5 x_2 x_4)",
"\\<lbrakk> ?H1 x_1 x_2; ?H2 x_3 x_4\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_3) (?H5 x_2 x_4)",
"\\<lbrakk> ?H1 x_1 x_2; ?H2 x_3 x_4\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_3) (?H5 x_2 x_4)",
"\\<lbrakk... | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 |
Cardinals/Ordinal_Arithmetic | Ordinal_Arithmetic.oprod_congR | ?r =o ?s \<Longrightarrow> Well_order ?t \<Longrightarrow> ?t *o ?r =o ?t *o ?s | lemma oprod_congR:
assumes "r =o s" and t: "Well_order t"
shows "t *o r =o t *o s" (is "?L =o ?R") | \<lbrakk>?H1 x_1 x_2; ?H2 x_3\<rbrakk> \<Longrightarrow> ?H3 (?H4 x_3 x_1) (?H5 x_3 x_2) | [
"\\<lbrakk> ?H1 x_1 x_2; ?H2 x_1; ?H3 x_3 x_4\\<rbrakk> \\<Longrightarrow> ?H4 (?H5 x_1 x_3) (?H6 x_2 x_4)"
] | [
"\\<lbrakk> ?H1 x_1 x_2; ?H2 x_1; ?H3 x_3 x_4\\<rbrakk> \\<Longrightarrow> ?H4 (?H5 x_1 x_3) (?H6 x_2 x_4)",
"\\<lbrakk> ?H1 x_1 x_2; ?H2 x_1; ?H3 x_3 x_4\\<rbrakk> \\<Longrightarrow> ?H4 (?H5 x_1 x_3) (?H6 x_2 x_4)",
"\\<lbrakk> ?H1 x_1 x_2; ?H2 x_1; ?H3 x_3 x_4\\<rbrakk> \\<Longrightarrow> ?H4 (?H5 x_1 x_3) (... | 71 | 71 | 71 | 20 | 20 | 20 | 0.854369 | 0.854369 | 0.854369 |
Cardinals/Ordinal_Arithmetic | Ordinal_Arithmetic.oprod_congL | ?r =o ?s \<Longrightarrow> Well_order ?t \<Longrightarrow> ?r *o ?t =o ?s *o ?t | lemma oprod_congL:
assumes "r =o s" and t: "Well_order t"
shows "r *o t =o s *o t" (is "?L =o ?R") | \<lbrakk>?H1 x_1 x_2; ?H2 x_3\<rbrakk> \<Longrightarrow> ?H3 (?H4 x_1 x_3) (?H5 x_2 x_3) | [
"\\<lbrakk> ?H1 x_1 x_2; ?H2 x_1; ?H3 x_3 x_4\\<rbrakk> \\<Longrightarrow> ?H4 (?H5 x_1 x_3) (?H6 x_2 x_4)"
] | [
"\\<lbrakk> ?H1 x_1 x_2; ?H2 x_1; ?H3 x_3 x_4\\<rbrakk> \\<Longrightarrow> ?H4 (?H5 x_1 x_3) (?H6 x_2 x_4)",
"\\<lbrakk> ?H1 x_1 x_2; ?H2 x_1; ?H3 x_3 x_4\\<rbrakk> \\<Longrightarrow> ?H4 (?H5 x_1 x_3) (?H6 x_2 x_4)",
"\\<lbrakk> ?H1 x_1 x_2; ?H2 x_1; ?H3 x_3 x_4\\<rbrakk> \\<Longrightarrow> ?H4 (?H5 x_1 x_3) (... | 71 | 71 | 71 | 17 | 17 | 17 | 0.854369 | 0.854369 | 0.854369 |
Cardinals/Ordinal_Arithmetic | Ordinal_Arithmetic.osum_congR | ?r =o ?s \<Longrightarrow> Well_order ?t \<Longrightarrow> ?t +o ?r =o ?t +o ?s | lemma osum_congR:
assumes "r =o s" and t: "Well_order t"
shows "t +o r =o t +o s" (is "?L =o ?R") | \<lbrakk>?H1 x_1 x_2; ?H2 x_3\<rbrakk> \<Longrightarrow> ?H3 (?H4 x_3 x_1) (?H5 x_3 x_2) | [
"\\<lbrakk> ?H1 x_1 x_2; ?H2 x_1; ?H3 x_3 x_2\\<rbrakk> \\<Longrightarrow> ?H4 (?H5 x_1 x_3) (?H6 x_2 x_3)"
] | [
"\\<lbrakk> ?H1 x_1 x_2; ?H2 x_1; ?H3 x_3 x_2\\<rbrakk> \\<Longrightarrow> ?H4 (?H5 x_1 x_3) (?H6 x_2 x_3)",
"\\<lbrakk> ?H1 x_1 x_2; ?H2 x_1; ?H3 x_3 x_2\\<rbrakk> \\<Longrightarrow> ?H4 (?H5 x_1 x_3) (?H6 x_2 x_3)",
"\\<lbrakk> ?H1 x_1 x_2; ?H2 x_1; ?H3 x_3 x_2\\<rbrakk> \\<Longrightarrow> ?H4 (?H5 x_1 x_3) (... | 71 | 71 | 71 | 20 | 20 | 20 | 0.872549 | 0.872549 | 0.872549 |
Cardinals/Ordinal_Arithmetic | Ordinal_Arithmetic.osum_congL | ?r =o ?s \<Longrightarrow> Well_order ?t \<Longrightarrow> ?r +o ?t =o ?s +o ?t | lemma osum_congL:
assumes "r =o s" and t: "Well_order t"
shows "r +o t =o s +o t" (is "?L =o ?R") | \<lbrakk>?H1 x_1 x_2; ?H2 x_3\<rbrakk> \<Longrightarrow> ?H3 (?H4 x_1 x_3) (?H5 x_2 x_3) | [
"\\<lbrakk> ?H1 x_1 x_2; ?H2 x_1; ?H3 x_3 x_2\\<rbrakk> \\<Longrightarrow> ?H4 (?H5 x_1 x_3) (?H6 x_2 x_3)"
] | [
"\\<lbrakk> ?H1 x_1 x_2; ?H2 x_1; ?H3 x_3 x_2\\<rbrakk> \\<Longrightarrow> ?H4 (?H5 x_1 x_3) (?H6 x_2 x_3)",
"\\<lbrakk> ?H1 x_1 x_2; ?H2 x_1; ?H3 x_3 x_2\\<rbrakk> \\<Longrightarrow> ?H4 (?H5 x_1 x_3) (?H6 x_2 x_3)",
"\\<lbrakk> ?H1 x_1 x_2; ?H2 x_1; ?H3 x_3 x_2\\<rbrakk> \\<Longrightarrow> ?H4 (?H5 x_1 x_3) (... | 71 | 71 | 71 | 16 | 16 | 16 | 0.872549 | 0.872549 | 0.872549 |
Cardinals/Ordinal_Arithmetic | Ordinal_Arithmetic.map_prod_ordIso | Well_order ?r \<Longrightarrow> inj_on ?f (Field ?r) \<Longrightarrow> map_prod ?f ?f ` ?r =o ?r | lemma map_prod_ordIso: "\<lbrakk>Well_order r; inj_on f (Field r)\<rbrakk> \<Longrightarrow> map_prod f f ` r =o r" | \<lbrakk>?H1 x_1; ?H2 x_2 (?H3 x_1)\<rbrakk> \<Longrightarrow> ?H4 (?H5 (?H6 x_2 x_2) x_1) x_1 | [
"\\<lbrakk> ?H1 x_1; ?H2 x_2 (?H3 x_1); ?H4 x_3 (?H5 x_2)\\<rbrakk> \\<Longrightarrow> ?H6 (?H7 (?H8 x_1 x_2) (?H9 x_3 x_2)) (?H10 x_1 x_3)"
] | [
"\\<lbrakk> ?H1 x_1; ?H2 x_2 (?H3 x_1); ?H4 x_3 (?H5 x_2)\\<rbrakk> \\<Longrightarrow> ?H6 (?H7 (?H8 x_1 x_2) (?H9 x_3 x_2)) (?H10 x_1 x_3)",
"\\<lbrakk> ?H1 x_1; ?H2 x_2 (?H3 x_1); ?H4 x_3 (?H5 x_2)\\<rbrakk> \\<Longrightarrow> ?H6 (?H7 (?H8 x_1 x_2) (?H9 x_3 x_2)) (?H10 x_1 x_3)",
"\\<lbrakk> ?H1 x_1; ?H2 x_2... | 95 | 95 | 95 | 45 | 45 | 45 | 0.703704 | 0.703704 | 0.703704 |
Cardinals/Ordinal_Arithmetic | Ordinal_Arithmetic.oone_ordIso_oexp | ?r =o oone \<Longrightarrow> Well_order ?s \<Longrightarrow> ?r ^o ?s =o oone | lemma oone_ordIso_oexp:
assumes "r =o oone" and s: "Well_order s"
shows "r ^o s =o oone" (is "?L =o ?R") | \<lbrakk>?H1 x_1 ?H2; ?H3 x_2\<rbrakk> \<Longrightarrow> ?H4 (?H5 x_1 x_2) ?H2 | [
"?H1 ?H2 x_1 \\<Longrightarrow> ?H3 (?H4 x_1)"
] | [
"?H1 ?H2 x_1 \\<Longrightarrow> ?H3 (?H4 x_1)",
"?H1 ?H2 x_1 \\<Longrightarrow> ?H3 (?H4 x_1)",
"?H1 ?H2 x_1 \\<Longrightarrow> ?H3 (?H4 x_1)",
"?H1 ?H2 x_1 \\<Longrightarrow> ?H3 (?H4 x_1)",
"?H1 ?H2 x_1 \\<Longrightarrow> ?H3 (?H4 x_1)"
] | 79 | 79 | 79 | 39 | 39 | 39 | 0.544304 | 0.544304 | 0.544304 |
Cardinals/Ordinal_Arithmetic | Ordinal_Arithmetic.oprod_Total | Total ?r \<Longrightarrow> Total ?r' \<Longrightarrow> Total (?r *o ?r') | lemma oprod_Total: "\<lbrakk>Total r; Total r'\<rbrakk> \<Longrightarrow> Total (r *o r')" | \<lbrakk>?H1 x_1; ?H2 x_2\<rbrakk> \<Longrightarrow> ?H3 (?H4 x_1 x_2) | [
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2)"
] | [
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2)",
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2)",
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2)",
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2)"... | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 |
Cardinals/Ordinal_Arithmetic | Ordinal_Arithmetic.osum_Total | Total ?r \<Longrightarrow> Total ?r' \<Longrightarrow> Total (?r +o ?r') | lemma osum_Total: "\<lbrakk>Total r; Total r'\<rbrakk> \<Longrightarrow> Total (r +o r')" | \<lbrakk>?H1 x_1; ?H2 x_2\<rbrakk> \<Longrightarrow> ?H3 (?H4 x_1 x_2) | [
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2)"
] | [
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2)",
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2)",
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2)",
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2)"... | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 |
Cardinals/Ordinal_Arithmetic | Ordinal_Arithmetic.oprod_Refl | Refl ?r \<Longrightarrow> Refl ?r' \<Longrightarrow> Refl (?r *o ?r') | lemma oprod_Refl:"\<lbrakk>Refl r; Refl r'\<rbrakk> \<Longrightarrow> Refl (r *o r')" | \<lbrakk>?H1 x_1; ?H2 x_2\<rbrakk> \<Longrightarrow> ?H3 (?H4 x_1 x_2) | [
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2)"
] | [
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2)",
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2)",
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2)",
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2)"... | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 |
Cardinals/Ordinal_Arithmetic | Ordinal_Arithmetic.osum_Refl | Refl ?r \<Longrightarrow> Refl ?r' \<Longrightarrow> Refl (?r +o ?r') | lemma osum_Refl:"\<lbrakk>Refl r; Refl r'\<rbrakk> \<Longrightarrow> Refl (r +o r')" | \<lbrakk>?H1 x_1; ?H2 x_2\<rbrakk> \<Longrightarrow> ?H3 (?H4 x_1 x_2) | [
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2)"
] | [
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2)",
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2)",
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2)",
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2)"... | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 |
Cardinals/Ordinal_Arithmetic | Ordinal_Arithmetic.oexp_Well_order | Well_order ?r \<Longrightarrow> Well_order ?s \<Longrightarrow> Well_order (?r ^o ?s) | lemma oexp_Well_order: "Well_order oexp" | \<lbrakk>?H1 x_1; ?H2 x_2\<rbrakk> \<Longrightarrow> ?H3 (?H4 x_1 x_2) | [
"?H1 x_1 \\<Longrightarrow> ?H2 (?H3 x_1)"
] | [
"?H1 x_1 \\<Longrightarrow> ?H2 (?H3 x_1)",
"?H1 x_1 \\<Longrightarrow> ?H2 (?H3 x_1)",
"?H1 x_1 \\<Longrightarrow> ?H2 (?H3 x_1)",
"?H1 x_1 \\<Longrightarrow> ?H2 (?H3 x_1)",
"?H1 x_1 \\<Longrightarrow> ?H2 (?H3 x_1)"
] | 71 | 71 | 71 | 34 | 34 | 34 | 0.549296 | 0.549296 | 0.549296 |
Cardinals/Ordinal_Arithmetic | Ordinal_Arithmetic.Field_oexp | Well_order ?r \<Longrightarrow> Well_order ?s \<Longrightarrow> Field (?r ^o ?s) = FinFunc ?r ?s | lemma Field_oexp: "Field oexp = FINFUNC" | \<lbrakk>?H1 x_1; ?H2 x_2\<rbrakk> \<Longrightarrow> ?H3 (?H4 x_1 x_2) = ?H5 x_1 x_2 | [
"\\<lbrakk> ?H1 x_1; ?H2 x_2; ?H3 x_1 = ?H4 x_2\\<rbrakk> \\<Longrightarrow> ?H5 (?H6 x_1 x_2)"
] | [
"?H1 x_1 \\<Longrightarrow> ?H2 (?H3 (?H4 x_1 x_2))",
"\\<lbrakk> ?H1 x_1; ?H2 x_2; ?H3 x_1 = ?H4 x_2\\<rbrakk> \\<Longrightarrow> ?H5 (?H6 x_1 x_2)",
"\\<lbrakk> ?H1 x_1; ?H2 x_2; ?H3 x_1 = ?H4 x_2\\<rbrakk> \\<Longrightarrow> ?H5 (?H6 x_1 x_2)",
"\\<lbrakk> ?H1 x_1; ?H2 x_2; ?H3 x_1 = ?H4 x_2\\<rbrakk> \\<L... | 59 | 59 | 59 | 35 | 35 | 35 | 0.944444 | 0.944444 | 0.944444 |
Cardinals/Ordinal_Arithmetic | Ordinal_Arithmetic.oprod_Well_order | Well_order ?r \<Longrightarrow> Well_order ?r' \<Longrightarrow> Well_order (?r *o ?r') | lemma oprod_Well_order:
assumes WELL: "Well_order r" and WELL': "Well_order r'"
shows "Well_order (r *o r')" | \<lbrakk>?H1 x_1; ?H2 x_2\<rbrakk> \<Longrightarrow> ?H3 (?H4 x_1 x_2) | [
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2)"
] | [
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2)",
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2)",
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2)",
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2)"... | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 |
Cardinals/Ordinal_Arithmetic | Ordinal_Arithmetic.osum_Well_order | Well_order ?r \<Longrightarrow> Well_order ?r' \<Longrightarrow> Well_order (?r +o ?r') | lemma osum_Well_order:
assumes WELL: "Well_order r" and WELL': "Well_order r'"
shows "Well_order (r +o r')" | \<lbrakk>?H1 x_1; ?H2 x_2\<rbrakk> \<Longrightarrow> ?H3 (?H4 x_1 x_2) | [
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2)"
] | [
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2)",
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2)",
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2)",
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2)"... | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 |
Cardinals/Ordinal_Arithmetic | Ordinal_Arithmetic.osum_Preorder | Preorder ?r \<Longrightarrow> Preorder ?r' \<Longrightarrow> Preorder (?r +o ?r') | lemma osum_Preorder: "\<lbrakk>Preorder r; Preorder r'\<rbrakk> \<Longrightarrow> Preorder (r +o r')" | \<lbrakk>?H1 x_1; ?H2 x_2\<rbrakk> \<Longrightarrow> ?H3 (?H4 x_1 x_2) | [
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2)"
] | [
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2)",
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2)",
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2)",
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2)"... | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 |
Cardinals/Ordinal_Arithmetic | Ordinal_Arithmetic.oprod_Partial_order | Partial_order ?r \<Longrightarrow> Partial_order ?r' \<Longrightarrow> Partial_order (?r *o ?r') | lemma oprod_Partial_order: "\<lbrakk>Partial_order r; Partial_order r'\<rbrakk> \<Longrightarrow> Partial_order (r *o r')" | \<lbrakk>?H1 x_1; ?H2 x_2\<rbrakk> \<Longrightarrow> ?H3 (?H4 x_1 x_2) | [
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2)"
] | [
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2)",
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2)",
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2)",
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2)"... | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 |
Cardinals/Ordinal_Arithmetic | Ordinal_Arithmetic.osum_Partial_order | Partial_order ?r \<Longrightarrow> Partial_order ?r' \<Longrightarrow> Partial_order (?r +o ?r') | lemma osum_Partial_order: "\<lbrakk>Partial_order r; Partial_order r'\<rbrakk> \<Longrightarrow> Partial_order (r +o r')" | \<lbrakk>?H1 x_1; ?H2 x_2\<rbrakk> \<Longrightarrow> ?H3 (?H4 x_1 x_2) | [
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2)"
] | [
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2)",
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2)",
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2)",
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2)"... | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 |
Cardinals/Ordinal_Arithmetic | Ordinal_Arithmetic.oprod_Linear_order | Linear_order ?r \<Longrightarrow> Linear_order ?r' \<Longrightarrow> Linear_order (?r *o ?r') | lemma oprod_Linear_order: "\<lbrakk>Linear_order r; Linear_order r'\<rbrakk> \<Longrightarrow> Linear_order (r *o r')" | \<lbrakk>?H1 x_1; ?H2 x_2\<rbrakk> \<Longrightarrow> ?H3 (?H4 x_1 x_2) | [
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2)"
] | [
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2)",
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2)",
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2)",
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2)"... | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 |
Cardinals/Ordinal_Arithmetic | Ordinal_Arithmetic.osum_Linear_order | Linear_order ?r \<Longrightarrow> Linear_order ?r' \<Longrightarrow> Linear_order (?r +o ?r') | lemma osum_Linear_order: "\<lbrakk>Linear_order r; Linear_order r'\<rbrakk> \<Longrightarrow> Linear_order (r +o r')" | \<lbrakk>?H1 x_1; ?H2 x_2\<rbrakk> \<Longrightarrow> ?H3 (?H4 x_1 x_2) | [
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2)"
] | [
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2)",
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2)",
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2)",
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2)"... | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 |
Cardinals/Ordinal_Arithmetic | Ordinal_Arithmetic.osum_ordLeqR | Well_order ?r \<Longrightarrow> Well_order ?s \<Longrightarrow> ?s \<le>o ?r +o ?s | lemma osum_ordLeqR: "Well_order r \<Longrightarrow> Well_order s \<Longrightarrow> s \<le>o r +o s" | \<lbrakk>?H1 x_1; ?H2 x_2\<rbrakk> \<Longrightarrow> ?H3 x_2 (?H4 x_1 x_2) | [
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2)"
] | [
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2)",
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2)",
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2)",
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2)"... | 14 | 14 | 14 | 4 | 4 | 4 | 0.946667 | 0.946667 | 0.946667 |
Cardinals/Ordinal_Arithmetic | Ordinal_Arithmetic.osum_ordLeqL | Well_order ?r \<Longrightarrow> Well_order ?r' \<Longrightarrow> ?r \<le>o ?r +o ?r' | null | \<lbrakk>?H1 x_1; ?H2 x_2\<rbrakk> \<Longrightarrow> ?H3 x_1 (?H4 x_1 x_2) | [
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2)"
] | [
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2)",
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2)",
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2)",
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2)"... | 14 | 14 | 14 | 4 | 4 | 4 | 0.946667 | 0.946667 | 0.946667 |
Cardinals/Ordinal_Arithmetic | Ordinal_Arithmetic.FinFuncD | ?f \<in> FinFunc ?r ?s \<Longrightarrow> ?x \<in> Field ?s \<Longrightarrow> ?f ?x \<in> Field ?r | lemma FinFuncD: "\<lbrakk>f \<in> FinFunc r s; x \<in> Field s\<rbrakk> \<Longrightarrow> f x \<in> Field r" | \<lbrakk>x_1 \<in> ?H1 x_2 x_3; x_4 \<in> ?H2 x_3\<rbrakk> \<Longrightarrow> x_1 x_4 \<in> ?H3 x_2 | [
"(?H1 x_1 x_2 = ?H1 x_3 x_4) = (x_2 = x_4 \\<and> x_1 = x_3 \\<or> (\\<exists>y_0. x_1 = ?H1 x_3 y_0 \\<and> x_3 \\<in> ?H2 x_1 \\<and> x_4 \\<in> ?H2 x_1))"
] | [
"(?H1 x_1 x_2 = ?H1 x_3 x_4) = (x_2 = x_4 \\<and> x_1 = x_3)",
"(?H1 x_1 x_2 = ?H1 x_3 x_2) = (x_2 \\<in> ?H2 x_1 \\<and> x_3 \\<in> ?H2 x_1 \\<and> ?H3 x_1 x_2 = ?H3 x_3 x_2)",
"(?H1 x_1 x_2 = ?H1 x_3 x_2) = (x_2 \\<in> ?H2 x_1 \\<and> x_3 \\<in> ?H2 x_1 \\<and> ?H3 x_1 x_2 = ?H3 x_3 x_2)",
"(?H1 x_1 x_2 = ?... | 140 | 91 | 91 | 96 | 61 | 61 | 0.464286 | 0.464286 | 0.464286 |
Cardinals/Ordinal_Arithmetic | Ordinal_Arithmetic.osum_embedL | Well_order ?r \<Longrightarrow> Well_order ?r' \<Longrightarrow> embed ?r (?r +o ?r') Inl | lemma osum_embedL:
assumes WELL: "Well_order r" and WELL': "Well_order r'"
shows "embed r (r +o r') Inl" | \<lbrakk>?H1 x_1; ?H2 x_2\<rbrakk> \<Longrightarrow> ?H3 x_1 (?H4 x_1 x_2) ?H5 | [
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2) (?H5 x_1 x_2) ?H6"
] | [
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2) (?H5 x_1 x_2) ?H6",
"\\<lbrakk> ?H1 x_1; ?H2 x_2; ?H3 x_1 x_2 x_3\\<rbrakk> \\<Longrightarrow> ?H4 (?H5 x_1 x_2) = ?H6 x_1 x_2",
"\\<lbrakk> ?H1 x_1; ?H2 x_2; ?H3 x_1 x_2 x_3\\<rbrakk> \\<Longrightarrow> ?H4 (?H5 x_1 x_2) = ?H6 x_1 x_2"... | 28 | 28 | 28 | 12 | 12 | 12 | 0.88764 | 0.88764 | 0.88764 |
Cardinals/Ordinal_Arithmetic | Ordinal_Arithmetic.FinFunc_singleton | FinFunc {(?z, ?z)} ?s = {\<lambda>x. if x \<in> Field ?s then ?z else undefined} | lemma FinFunc_singleton: "FinFunc {(z,z)} s = {\<lambda>x. if x \<in> Field s then z else undefined}" | ?H1 (?H2 (x_1, x_1) ?H3) x_2 = ?H4 (\<lambda>y_0. if y_0 \<in> ?H5 x_2 then x_1 else undefined) ?H6 | [
"?H1 x_1 (?H2 x_2 ?H3) = ?H1 x_1 ?H3"
] | [
"?H1 x_1 (?H2 x_2 ?H3) = ?H1 x_1 ?H4",
"?H1 x_1 (?H2 x_2 ?H3) = ?H1 x_1 ?H4",
"?H1 x_1 (?H2 x_2 ?H3) = ?H1 x_1 ?H4",
"?H1 x_1 (?H2 x_2 ?H3) = ?H1 x_1 ?H4",
"?H1 x_1 (?H2 x_2 ?H3) = ?H1 x_1 ?H4"
] | 95 | 95 | 95 | 71 | 71 | 71 | 0.34 | 0.353535 | 0.353535 |
Cardinals/Ordinal_Arithmetic | Ordinal_Arithmetic.oexp_empty2 | Well_order ?r \<Longrightarrow> ?r \<noteq> {} \<Longrightarrow> {} ^o ?r = {} | lemma oexp_empty2[simp]:
assumes "Well_order r" "r \<noteq> {}"
shows "{} ^o r = {}" | \<lbrakk>?H1 x_1; x_1 \<noteq> ?H2\<rbrakk> \<Longrightarrow> ?H3 ?H4 x_1 = ?H5 | [
"?H1 x_1 \\<Longrightarrow> ?H2 (?H3 x_1) = ?H4"
] | [
"?H1 x_1 \\<Longrightarrow> ?H2 (?H3 x_1) \\<noteq> ?H4",
"?H1 x_1 \\<Longrightarrow> ?H2 (?H3 x_1) \\<noteq> ?H4",
"?H1 x_1 \\<Longrightarrow> ?H2 (?H3 x_1) \\<noteq> ?H4",
"?H1 x_1 \\<Longrightarrow> ?H2 (?H3 x_1) \\<noteq> ?H4",
"?H1 x_1 \\<Longrightarrow> ?H2 (?H3 x_1) \\<noteq> ?H4"
] | 80 | 80 | 80 | 42 | 50 | 42 | 0.52439 | 0.609756 | 0.609756 |
Cardinals/Ordinal_Arithmetic | Ordinal_Arithmetic.oexp_empty | Well_order ?r \<Longrightarrow> ?r ^o {} = {(\<lambda>x. undefined, \<lambda>x. undefined)} | lemma oexp_empty[simp]:
assumes "Well_order r"
shows "r ^o {} = {(\<lambda>x. undefined, \<lambda>x. undefined)}" | ?H1 x_1 \<Longrightarrow> ?H2 x_1 ?H3 = ?H4 (\<lambda>y_0. undefined, \<lambda>y_1. undefined) ?H5 | [
"?H1 x_1 \\<Longrightarrow> ?H2 x_1 (?H3 x_2 ?H4) = ?H3 (?H5 x_2 x_3) ?H4"
] | [
"?H1 x_1 \\<Longrightarrow> ?H2 x_1 (?H3 x_2 ?H4) = ?H3 (?H5 x_2 x_3) ?H4",
"?H1 x_1 \\<Longrightarrow> ?H2 x_1 (?H3 x_2 ?H4) = ?H3 (?H5 x_2 x_3) ?H4",
"?H1 x_1 \\<Longrightarrow> ?H2 x_1 (?H3 x_2 ?H4) = ?H3 (?H5 x_2 x_3) ?H4",
"?H1 x_1 \\<Longrightarrow> ?H2 x_1 (?H3 x_2 ?H4) = ?H3 (?H5 x_2 x_3) ?H4",
"?H1... | 63 | 63 | 63 | 51 | 51 | 51 | 0.482456 | 0.482456 | 0.482456 |
Cardinals/Ordinal_Arithmetic | Ordinal_Arithmetic.support_upd_subset | support ?z ?A (?f(?x := ?y)) \<subseteq> support ?z ?A ?f \<union> {?x} | lemma support_upd_subset[simp]: "support z A (f(x := y)) \<subseteq> support z A f \<union> {x}" | ?H1 (?H2 x_1 x_2 (?H3 x_3 x_4 x_5)) (?H4 (?H2 x_1 x_2 x_3) (?H5 x_4 ?H6)) | [
"?H1 (?H2 x_1 x_2 (?H3 x_3 x_4 x_5)) (?H4 (?H2 x_1 x_2 x_3) (?H5 x_4 ?H6))"
] | [
"?H1 (?H2 x_1 x_2 (?H3 x_3 x_4 x_5)) (?H4 (?H2 x_1 x_2 x_3) (?H5 x_4 ?H6))",
"?H1 (?H2 x_1 x_2 (?H3 x_3 x_4 x_5)) (?H4 (?H2 x_1 x_2 x_3) (?H5 x_4 ?H6))",
"?H1 (?H2 x_1 x_2 (?H3 x_3 x_4 x_5)) (?H4 (?H2 x_1 x_2 x_3) (?H5 x_4 ?H6))",
"?H1 (?H2 x_1 x_2 (?H3 x_3 x_4 x_5)) (?H4 (?H2 x_1 x_2 x_3) (?H5 x_4 ?H6))",
... | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 |
Cardinals/Ordinal_Arithmetic | Ordinal_Arithmetic.support_upd | support ?z ?A (?f(?x := ?z)) = support ?z ?A ?f - {?x} | lemma support_upd[simp]: "support z A (f(x := z)) = support z A f - {x}" | ?H1 x_1 x_2 (?H2 x_3 x_4 x_1) = ?H3 (?H1 x_1 x_2 x_3) (?H4 x_4 ?H5) | [
"?H1 x_1 x_2 (?H2 x_3 x_1 x_4) = ?H3 (?H1 x_1 x_2 x_3) (?H4 x_4 ?H5)"
] | [
"?H1 x_1 x_2 (?H2 x_3 x_1 x_4) = ?H3 (?H1 x_1 x_2 x_3) (?H4 x_4 ?H5)",
"?H1 x_1 x_2 (?H2 x_3 x_1 x_4) = ?H3 (?H1 x_1 x_2 x_3) (?H4 x_4 ?H5)",
"?H1 x_1 x_2 (?H2 x_3 x_1 x_4) = ?H3 (?H1 x_1 x_2 x_3) (?H4 x_4 ?H5)",
"?H1 x_1 x_2 (?H2 x_3 x_1 x_4) = ?H3 (?H1 x_1 x_2 x_3) (?H4 x_4 ?H5)",
"?H1 x_1 x_2 (?H2 x_3 x_... | 2 | 2 | 2 | 2 | 2 | 2 | 1 | 1 | 1 |
Cardinals/Ordinal_Arithmetic | Ordinal_Arithmetic.support_Un | support ?z (?A \<union> ?B) ?f = support ?z ?A ?f \<union> support ?z ?B ?f | lemma support_Un[simp]: "support z (A \<union> B) f = support z A f \<union> support z B f" | ?H1 x_1 (?H2 x_2 x_3) x_4 = ?H2 (?H1 x_1 x_2 x_4) (?H1 x_1 x_3 x_4) | [
"?H1 x_1 (?H2 x_2 x_3) x_4 = ?H3 (?H1 x_1 x_2 x_4) (?H1 x_1 x_3 x_4)"
] | [
"?H1 x_1 (?H2 x_2 x_3) x_4 = ?H3 (?H1 x_1 x_2 x_4) (?H1 x_1 x_3 x_4)",
"?H1 x_1 (?H2 x_2 x_3) x_4 = ?H3 (?H1 x_1 x_2 x_4) (?H1 x_1 x_3 x_4)",
"?H1 x_1 (?H2 x_2 x_3) x_4 = ?H3 (?H1 x_1 x_2 x_4) (?H1 x_1 x_3 x_4)",
"?H1 x_1 (?H2 x_2 x_3) x_4 = ?H3 (?H1 x_1 x_2 x_4) (?H1 x_1 x_3 x_4)",
"?H1 x_1 (?H2 x_2 x_3) x... | 1 | 1 | 1 | 1 | 1 | 1 | 0.970588 | 0.970588 | 0.970588 |
Cardinals/Ordinal_Arithmetic | Ordinal_Arithmetic.oprod_wf | wf ?r \<Longrightarrow> wf ?r' \<Longrightarrow> wf (?r *o ?r') | lemma oprod_wf:
assumes WF: "wf r" and WF': "wf r'"
shows "wf (r *o r')" | \<lbrakk>?H1 x_1; ?H2 x_2\<rbrakk> \<Longrightarrow> ?H3 (?H4 x_1 x_2) | [
"?H1 (?H2 x_1 x_2) \\<Longrightarrow> ?H3 x_1"
] | [
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2)",
"?H1 (?H2 x_1 x_2) \\<Longrightarrow> ?H3 x_1",
"?H1 (?H2 x_1 x_2) \\<Longrightarrow> ?H3 x_1",
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2)",
"?H1 (?H2 x_1 x_2) \\<Longrightarrow> ?H3 x_1"
] | 68 | 0 | 0 | 35 | 0 | 0 | 0.605634 | 1 | 1 |
Cardinals/Ordinal_Arithmetic | Ordinal_Arithmetic.osum_wf | wf ?r \<Longrightarrow> wf ?r' \<Longrightarrow> wf (?r +o ?r') | lemma osum_wf:
assumes WF: "wf r" and WF': "wf r'"
shows "wf (r +o r')" | \<lbrakk>?H1 x_1; ?H2 x_2\<rbrakk> \<Longrightarrow> ?H3 (?H4 x_1 x_2) | [
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2)"
] | [
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2)",
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2)",
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2)",
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2)"... | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 |
Cardinals/Ordinal_Arithmetic | Ordinal_Arithmetic.osum_trans | trans ?r \<Longrightarrow> trans ?r' \<Longrightarrow> trans (?r +o ?r') | lemma osum_trans:
assumes TRANS: "trans r" and TRANS': "trans r'"
shows "trans (r +o r')" | \<lbrakk>?H1 x_1; ?H2 x_2\<rbrakk> \<Longrightarrow> ?H3 (?H4 x_1 x_2) | [
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2)"
] | [
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2)",
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2)",
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2)",
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2)"... | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 |
Cardinals/Ordinal_Arithmetic | Ordinal_Arithmetic.oprod_antisym | antisym ?r \<Longrightarrow> antisym ?r' \<Longrightarrow> antisym (?r *o ?r') | lemma oprod_antisym: "\<lbrakk>antisym r; antisym r'\<rbrakk> \<Longrightarrow> antisym (r *o r')" | \<lbrakk>?H1 x_1; ?H2 x_2\<rbrakk> \<Longrightarrow> ?H3 (?H4 x_1 x_2) | [
"?H1 (?H2 x_1 x_2)"
] | [
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2)",
"?H1 (?H2 x_1 x_2)",
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2)",
"?H1 (?H2 x_1 x_2)",
"?H1 (?H2 x_1 x_2)"
] | 71 | 0 | 0 | 55 | 0 | 0 | 0.239437 | 1 | 1 |
Cardinals/Ordinal_Arithmetic | Ordinal_Arithmetic.osum_antisym | antisym ?r \<Longrightarrow> antisym ?r' \<Longrightarrow> antisym (?r +o ?r') | lemma osum_antisym: "\<lbrakk>antisym r; antisym r'\<rbrakk> \<Longrightarrow> antisym (r +o r')" | \<lbrakk>?H1 x_1; ?H2 x_2\<rbrakk> \<Longrightarrow> ?H3 (?H4 x_1 x_2) | [
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2)"
] | [
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2)",
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2)",
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2)",
"\\<lbrakk> ?H1 x_1; ?H2 x_2\\<rbrakk> \\<Longrightarrow> ?H3 (?H4 x_1 x_2)"... | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 |
Cardinals/Ordinal_Arithmetic | Ordinal_Arithmetic.Field_osum | Field (?r +o ?r') = Inl ` Field ?r \<union> Inr ` Field ?r' | lemma Field_osum: "Field(r +o r') = Inl ` Field r \<union> Inr ` Field r'" | ?H1 (?H2 x_1 x_2) = ?H3 (?H4 ?H5 (?H6 x_1)) (?H7 ?H8 (?H9 x_2)) | [
"?H1 (?H2 x_1 x_2) = ?H3 (?H4 ?H5 (?H6 x_1)) (?H7 ?H8 (?H9 x_2))"
] | [
"?H1 (?H2 x_1 x_2) = ?H3 (?H4 ?H5 (?H6 x_1)) (?H7 ?H8 (?H9 x_2))",
"?H1 (?H2 x_1 x_2) = ?H3 (?H4 ?H5 (?H6 x_1)) (?H7 ?H8 (?H9 x_2))",
"?H1 (?H2 x_1 x_2) = ?H3 (?H4 ?H5 (?H6 x_1)) (?H7 ?H8 (?H9 x_2))",
"?H1 (?H2 x_1 x_2) = ?H3 (?H4 ?H5 (?H6 x_1)) (?H7 ?H8 (?H9 x_2))",
"?H1 (?H2 x_1 x_2) = ?H3 (?H4 ?H5 (?H6 x... | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 |
Cardinals/Ordinal_Arithmetic | Ordinal_Arithmetic.finite_support | ?f \<in> fin_support ?z ?A \<Longrightarrow> finite (support ?z ?A ?f) | lemma finite_support: "f \<in> fin_support z A \<Longrightarrow> finite (support z A f)" | x_1 \<in> ?H1 x_2 x_3 \<Longrightarrow> ?H2 (?H3 x_2 x_3 x_1) | [
"(x_1 \\<in> ?H1 x_2 x_3) = ?H2 (?H3 x_2 x_3 x_1)"
] | [
"x_1 \\<in> ?H1 x_2 x_3 \\<Longrightarrow> ?H2 (?H3 x_2 x_3 x_1)",
"(x_1 \\<in> ?H1 x_2 x_3) = ?H2 (?H3 x_2 x_3 x_1)",
"x_1 \\<in> ?H1 x_2 x_3 \\<Longrightarrow> ?H2 (?H3 x_2 x_3 x_1)",
"x_1 \\<in> ?H1 x_2 x_3 \\<Longrightarrow> ?H2 (?H3 x_2 x_3 x_1)",
"(x_1 \\<in> ?H1 x_2 x_3) = ?H2 (?H3 x_2 x_3 x_1)"
] | 61 | 0 | 0 | 19 | 0 | 0 | 0.6875 | 1 | 1 |
Cardinals/Ordinal_Arithmetic | Ordinal_Arithmetic.ozero_ordLeq | Well_order ?r \<Longrightarrow> ozero \<le>o ?r | lemma ozero_ordLeq:
assumes "Well_order r" shows "ozero \<le>o r" | ?H1 x_1 \<Longrightarrow> ?H2 ?H3 x_1 | [
"?H1 x_1 \\<Longrightarrow> ?H2 x_1 ?H3"
] | [
"?H1 x_1 \\<Longrightarrow> ?H2 x_1 ?H3",
"?H1 x_1 \\<Longrightarrow> ?H2 x_1 ?H3",
"?H1 x_1 \\<Longrightarrow> ?H2 x_1 ?H3",
"?H1 x_1 \\<Longrightarrow> ?H2 x_1 ?H3",
"?H1 x_1 \\<Longrightarrow> ?H2 x_1 ?H3"
] | 6 | 6 | 6 | 6 | 6 | 6 | 1 | 1 | 1 |
Cardinals/Ordinal_Arithmetic | Ordinal_Arithmetic.Field_oprod | Field (?r *o ?r') = Field ?r \<times> Field ?r' | lemma Field_oprod: "Field (r *o r') = Field r \<times> Field r'" | ?H1 (?H2 x_1 x_2) = ?H3 (?H4 x_1) (?H5 x_2) | [
"?H1 (?H2 x_1 x_2) = ?H3 (?H4 x_1) (?H5 x_2)"
] | [
"?H1 (?H2 x_1 x_2) = ?H3 (?H4 x_1) (?H5 x_2)",
"?H1 (?H2 x_1 x_2) = ?H3 (?H4 x_1) (?H5 x_2)",
"?H1 (?H2 x_1 x_2) = ?H3 (?H4 x_1) (?H5 x_2)",
"?H1 (?H2 x_1 x_2) = ?H3 (?H4 x_1) (?H5 x_2)",
"?H1 (?H2 x_1 x_2) = ?H3 (?H4 x_1) (?H5 x_2)"
] | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 |
Cardinals/Ordinal_Arithmetic | Ordinal_Arithmetic.Field_singleton | Field {(?z, ?z)} = {?z} | lemma Field_singleton[simp]: "Field {(z,z)} = {z}" | ?H1 (?H2 (x_1, x_1) ?H3) = ?H4 x_1 ?H5 | [
"?H1 (?H2 x_1 ?H3) = ?H4 x_1"
] | [
"?H1 (?H2 x_1 ?H3) = ?H4 x_1",
"?H1 x_1 = ?H2 x_2 ?H3",
"?H1 x_1 = ?H2 x_2 ?H3",
"?H1 x_1 = ?H2 x_2 ?H3",
"?H1 x_1 = ?H2 x_2 ?H3"
] | 26 | 26 | 26 | 11 | 11 | 11 | 0.710526 | 0.710526 | 0.710526 |
Cardinals/Ordinal_Arithmetic | Ordinal_Arithmetic.oone_ordIso | oone =o {(?x, ?x)} | lemma oone_ordIso: "oone =o {(x,x)}" | ?H1 ?H2 (?H3 (x_1, x_1) ?H4) | [
"?H1 ?H2 (?H3 (x_1, ?H4) ?H5)"
] | [
"?H1 ?H2 (?H3 (x_1, ?H4) ?H5)",
"?H1 ?H2 (?H3 (x_1, ?H4) ?H5)",
"?H1 ?H2 (?H3 (x_1, ?H4) ?H5)",
"?H1 ?H2 (?H3 (x_1, ?H4) ?H5)",
"?H1 ?H2 (?H3 (x_1, ?H4) ?H5)"
] | 4 | 4 | 4 | 4 | 4 | 4 | 0.806452 | 0.806452 | 0.806452 |
Cardinals/Ordinal_Arithmetic | Ordinal_Arithmetic.well_order_on_singleton | well_order_on {?x} {(?x, ?x)} | lemma well_order_on_singleton[simp]: "well_order_on {x} {(x, x)}" | ?H1 (?H2 x_1 ?H3) (?H4 (x_1, x_1) ?H5) | [
"?H1 x_1 (?H2 (x_2, x_3) ?H3)"
] | [
"?H1 x_1 (?H2 (x_2, x_3) ?H3)",
"?H1 x_1 (?H2 (x_2, x_3) ?H3)",
"?H1 x_1 (?H2 (x_2, x_3) ?H3)",
"?H1 x_1 (?H2 (x_2, x_3) ?H3)",
"?H1 x_1 (?H2 (x_2, x_3) ?H3)"
] | 33 | 33 | 33 | 14 | 14 | 14 | 0.65 | 0.65 | 0.65 |
Cardinals/Ordinal_Arithmetic | Ordinal_Arithmetic.iso_ozero_empty | (?r =o ozero) = (?r = {}) | lemma iso_ozero_empty[simp]: "r =o ozero = (r = {})" | ?H1 x_1 ?H2 = (x_1 = ?H3) | [
"?H1 ?H2 x_1 = (x_1 = ?H3)"
] | [
"?H1 ?H2 x_1 = (x_1 = ?H3)",
"?H1 ?H2 x_1 \\<Longrightarrow> x_1 = ?H3",
"?H1 ?H2 x_1 = (x_1 = ?H3)",
"?H1 ?H2 x_1 = (x_1 = ?H3)",
"?H1 ?H2 x_1 = (x_1 = ?H3)"
] | 6 | 6 | 6 | 6 | 6 | 6 | 1 | 1 | 1 |
Cardinals/Ordinal_Arithmetic | Ordinal_Arithmetic.dir_image_alt | dir_image ?r ?f = map_prod ?f ?f ` ?r | lemma dir_image_alt: "dir_image r f = map_prod f f ` r" | ?H1 x_1 x_2 = ?H2 (?H3 x_2 x_2) x_1 | [
"?H1 (?H2 x_1 x_2) x_3 = ?H3 (?H4 x_1 x_3) (\\<lambda>y_1. (x_1 y_1, x_2 y_1))"
] | [
"?H1 (?H2 x_1 x_2) x_3 = ?H3 (?H4 x_1 x_3) (\\<lambda>y_1. (x_1 y_1, x_2 y_1))",
"?H1 (?H2 x_1 x_2) x_3 = ?H3 (?H4 x_1 x_3) (\\<lambda>y_1. (x_1 y_1, x_2 y_1))",
"?H1 (?H2 x_1 x_2) x_3 = ?H3 (?H4 x_1 x_3) (\\<lambda>y_1. (x_1 y_1, x_2 y_1))",
"?H1 (?H2 x_1 x_2) x_3 = ?H3 (?H4 x_1 x_3) (\\<lambda>y_1. (x_1 y_1... | 71 | 71 | 71 | 45 | 45 | 45 | 0.441558 | 0.441558 | 0.441558 |
Cardinals/Ordinal_Arithmetic | Ordinal_Arithmetic.zero_singleton | zero {(?z, ?z)} = ?z | lemma zero_singleton[simp]: "zero {(z,z)} = z" | ?H1 (?H2 (x_1, x_1) ?H3) = x_1 | [
"?H1 \\<in> ?H2 x_1 ?H3"
] | [
"?H1 \\<in> ?H2 x_1 ?H3",
"?H1 \\<in> ?H2 x_1 ?H3",
"?H1 \\<in> ?H2 x_1 ?H3",
"?H1 \\<in> ?H2 x_1 ?H3",
"?H1 \\<in> ?H2 x_1 ?H3"
] | 26 | 26 | 26 | 18 | 18 | 18 | 0.457143 | 0.457143 | 0.457143 |
Cardinals/Ordinal_Arithmetic | Ordinal_Arithmetic.oprod_zero(2) | ?r *o {} = {} | lemma oprod_zero[simp]: "{} *o r = {}" "r *o {} = {}" | ?H1 x_1 ?H2 = ?H3 | [
"?H1 ?H2 x_1 = ?H3"
] | [
"?H1 ?H2 x_1 = ?H3",
"?H1 ?H2 x_1 = ?H3",
"?H1 ?H2 x_1 = ?H3",
"?H1 ?H2 x_1 = ?H3",
"?H1 ?H2 x_1 = ?H3"
] | 6 | 6 | 6 | 6 | 6 | 6 | 1 | 1 | 1 |
Cardinals/Ordinal_Arithmetic | Ordinal_Arithmetic.oprod_zero(1) | {} *o ?r = {} | lemma oprod_zero[simp]: "{} *o r = {}" "r *o {} = {}" | ?H1 ?H2 x_1 = ?H3 | [
"?H1 ?H2 x_1 = ?H3"
] | [
"?H1 ?H2 x_1 = ?H3",
"?H1 ?H2 x_1 = ?H3",
"?H1 ?H2 x_1 = ?H3",
"?H1 ?H2 x_1 = ?H3",
"?H1 ?H2 x_1 = ?H3"
] | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 |
Lattice/Bounds | Bounds.is_supI | ?x \<sqsubseteq> ?sup \<Longrightarrow> ?y \<sqsubseteq> ?sup \<Longrightarrow> (\<And>z. ?x \<sqsubseteq> z \<Longrightarrow> ?y \<sqsubseteq> z \<Longrightarrow> ?sup \<sqsubseteq> z) \<Longrightarrow> is_sup ?x ?y ?sup | lemma is_supI [intro?]: "x \<sqsubseteq> sup \<Longrightarrow> y \<sqsubseteq> sup \<Longrightarrow>
(\<And>z. x \<sqsubseteq> z \<Longrightarrow> y \<sqsubseteq> z \<Longrightarrow> sup \<sqsubseteq> z) \<Longrightarrow> is_sup x y sup" | \<lbrakk>?H1 x_1 x_2; ?H1 x_3 x_2; \<And>y_0. \<lbrakk>?H1 x_1 y_0; ?H1 x_3 y_0\<rbrakk> \<Longrightarrow> ?H1 x_2 y_0\<rbrakk> \<Longrightarrow> ?H2 x_1 x_3 x_2 | [
"\\<lbrakk> ?H1 x_1 x_2; ?H1 x_2 x_3; ?H2 x_1 x_3 x_4\\<rbrakk> \\<Longrightarrow> x_2 \\<le> x_4"
] | [
"\\<lbrakk> ?H1 x_1 x_2; ?H1 x_2 x_3; ?H2 x_1 x_3 x_4\\<rbrakk> \\<Longrightarrow> ?H2 x_1 x_2 x_4",
"\\<lbrakk> ?H1 x_1 x_2; ?H1 x_2 x_3; ?H2 x_1 x_3 x_4\\<rbrakk> \\<Longrightarrow> ?H2 x_1 x_2 x_4",
"\\<lbrakk> ?H1 x_1 x_2; ?H1 x_2 x_3; ?H2 x_1 x_3 x_4\\<rbrakk> \\<Longrightarrow> ?H2 x_1 x_2 x_4",
"\\<lbr... | 128 | 127 | 127 | 79 | 75 | 75 | 0.536145 | 0.557576 | 0.557576 |
Lattice/Bounds | Bounds.is_infI | ?inf \<sqsubseteq> ?x \<Longrightarrow> ?inf \<sqsubseteq> ?y \<Longrightarrow> (\<And>z. z \<sqsubseteq> ?x \<Longrightarrow> z \<sqsubseteq> ?y \<Longrightarrow> z \<sqsubseteq> ?inf) \<Longrightarrow> is_inf ?x ?y ?inf | lemma is_infI [intro?]: "inf \<sqsubseteq> x \<Longrightarrow> inf \<sqsubseteq> y \<Longrightarrow>
(\<And>z. z \<sqsubseteq> x \<Longrightarrow> z \<sqsubseteq> y \<Longrightarrow> z \<sqsubseteq> inf) \<Longrightarrow> is_inf x y inf" | \<lbrakk>?H1 x_1 x_2; ?H1 x_1 x_3; \<And>y_0. \<lbrakk>?H1 y_0 x_2; ?H1 y_0 x_3\<rbrakk> \<Longrightarrow> ?H1 y_0 x_1\<rbrakk> \<Longrightarrow> ?H2 x_2 x_3 x_1 | [
"\\<lbrakk> ?H1 x_1 x_2; ?H1 x_2 x_3; ?H2 x_1 x_3 x_4\\<rbrakk> \\<Longrightarrow> ?H2 x_1 x_2 x_4"
] | [
"\\<lbrakk> ?H1 x_1 x_2; ?H1 x_2 x_3; ?H2 x_1 x_3 x_4\\<rbrakk> \\<Longrightarrow> ?H2 x_1 x_2 x_4",
"\\<lbrakk> ?H1 x_1 x_2; ?H1 x_2 x_3; ?H2 x_1 x_3 x_4\\<rbrakk> \\<Longrightarrow> ?H2 x_1 x_2 x_4",
"\\<lbrakk> ?H1 x_1 x_2; ?H1 x_2 x_3; ?H2 x_1 x_3 x_4\\<rbrakk> \\<Longrightarrow> ?H2 x_1 x_2 x_4",
"\\<lbr... | 126 | 126 | 126 | 75 | 75 | 75 | 0.548193 | 0.548193 | 0.548193 |
Lattice/Bounds | Bounds.is_InfI | (\<And>x. x \<in> ?A \<Longrightarrow> ?inf \<sqsubseteq> x) \<Longrightarrow> (\<And>z. \<forall>x\<in>?A. z \<sqsubseteq> x \<Longrightarrow> z \<sqsubseteq> ?inf) \<Longrightarrow> is_Inf ?A ?inf | lemma is_InfI [intro?]: "(\<And>x. x \<in> A \<Longrightarrow> inf \<sqsubseteq> x) \<Longrightarrow>
(\<And>z. (\<forall>x \<in> A. z \<sqsubseteq> x) \<Longrightarrow> z \<sqsubseteq> inf) \<Longrightarrow> is_Inf A inf" | \<lbrakk>\<And>y_0. y_0 \<in> x_1 \<Longrightarrow> ?H1 x_2 y_0; \<And>y_1. \<forall>y_2\<in>x_1. ?H1 y_1 y_2 \<Longrightarrow> ?H1 y_1 x_2\<rbrakk> \<Longrightarrow> ?H2 x_1 x_2 | [
"\\<lbrakk> ?H1 x_1 x_2; ?H2 x_3 x_1\\<rbrakk> \\<Longrightarrow> ?H2 x_3 x_2"
] | [
"\\<lbrakk> ?H1 x_1 x_2; ?H2 x_3 x_1\\<rbrakk> \\<Longrightarrow> ?H2 x_3 x_2",
"\\<lbrakk> ?H1 x_1 x_2; ?H2 x_3 x_1\\<rbrakk> \\<Longrightarrow> ?H2 x_3 x_2",
"\\<lbrakk> ?H1 x_1 x_2; ?H2 x_3 x_1\\<rbrakk> \\<Longrightarrow> ?H2 x_3 x_2",
"\\<lbrakk> ?H1 x_1 x_2; ?H2 x_3 x_1\\<rbrakk> \\<Longrightarrow> ?H2 ... | 167 | 167 | 167 | 111 | 111 | 111 | 0.394444 | 0.394444 | 0.394444 |
Lattice/Bounds | Bounds.is_SupI | (\<And>x. x \<in> ?A \<Longrightarrow> x \<sqsubseteq> ?sup) \<Longrightarrow> (\<And>z. \<forall>x\<in>?A. x \<sqsubseteq> z \<Longrightarrow> ?sup \<sqsubseteq> z) \<Longrightarrow> is_Sup ?A ?sup | lemma is_SupI [intro?]: "(\<And>x. x \<in> A \<Longrightarrow> x \<sqsubseteq> sup) \<Longrightarrow>
(\<And>z. (\<forall>x \<in> A. x \<sqsubseteq> z) \<Longrightarrow> sup \<sqsubseteq> z) \<Longrightarrow> is_Sup A sup" | \<lbrakk>\<And>y_0. y_0 \<in> x_1 \<Longrightarrow> ?H1 y_0 x_2; \<And>y_1. \<forall>y_2\<in>x_1. ?H1 y_2 y_1 \<Longrightarrow> ?H1 x_2 y_1\<rbrakk> \<Longrightarrow> ?H2 x_1 x_2 | [
"\\<lbrakk> ?H1 x_1 x_2; ?H2 x_3 x_1\\<rbrakk> \\<Longrightarrow> ?H2 x_3 x_2"
] | [
"\\<lbrakk> ?H1 x_1 x_2; ?H2 x_3 x_1\\<rbrakk> \\<Longrightarrow> ?H2 x_3 x_2",
"\\<lbrakk> ?H1 x_1 x_2; ?H2 x_3 x_1\\<rbrakk> \\<Longrightarrow> ?H2 x_3 x_2",
"\\<lbrakk> ?H1 x_1 x_2; ?H2 x_3 x_1\\<rbrakk> \\<Longrightarrow> ?H2 x_3 x_2",
"\\<lbrakk> ?H1 x_1 x_2; ?H2 x_3 x_1\\<rbrakk> \\<Longrightarrow> ?H2 ... | 167 | 167 | 167 | 110 | 110 | 110 | 0.394444 | 0.394444 | 0.394444 |
Lattice/Bounds | Bounds.is_sup_upper | is_sup ?x ?y ?sup \<Longrightarrow> (?x \<sqsubseteq> ?sup \<Longrightarrow> ?y \<sqsubseteq> ?sup \<Longrightarrow> ?C) \<Longrightarrow> ?C | lemma is_sup_upper [elim?]:
"is_sup x y sup \<Longrightarrow> (x \<sqsubseteq> sup \<Longrightarrow> y \<sqsubseteq> sup \<Longrightarrow> C) \<Longrightarrow> C" | \<lbrakk>?H1 x_1 x_2 x_3; \<lbrakk>?H2 x_1 x_3; ?H2 x_2 x_3\<rbrakk> \<Longrightarrow> x_4\<rbrakk> \<Longrightarrow> x_4 | [
"?H1 x_1 x_2 x_3 \\<Longrightarrow> ?H2 x_1 x_3"
] | [
"?H1 x_1 x_2 x_3 \\<Longrightarrow> (\\<forall>y_0. ?H2 x_1 y_0 \\<longrightarrow> ?H2 x_2 y_0) \\<and> (\\<forall>y_1. ?H2 x_3 y_1 \\<longrightarrow> ?H2 x_2 y_1)",
"?H1 x_1 x_2 x_3 \\<Longrightarrow> (\\<forall>y_0. ?H2 x_1 y_0 \\<longrightarrow> ?H2 x_2 y_0) \\<and> (\\<forall>y_1. ?H2 x_3 y_1 \\<longrightarro... | 123 | 152 | 123 | 85 | 99 | 85 | 0.365854 | 0.576271 | 0.576271 |
Lattice/Bounds | Bounds.is_inf_lower | is_inf ?x ?y ?inf \<Longrightarrow> (?inf \<sqsubseteq> ?x \<Longrightarrow> ?inf \<sqsubseteq> ?y \<Longrightarrow> ?C) \<Longrightarrow> ?C | lemma is_inf_lower [elim?]:
"is_inf x y inf \<Longrightarrow> (inf \<sqsubseteq> x \<Longrightarrow> inf \<sqsubseteq> y \<Longrightarrow> C) \<Longrightarrow> C" | \<lbrakk>?H1 x_1 x_2 x_3; \<lbrakk>?H2 x_3 x_1; ?H2 x_3 x_2\<rbrakk> \<Longrightarrow> x_4\<rbrakk> \<Longrightarrow> x_4 | [
"?H1 x_1 x_2 x_3 \\<Longrightarrow> ?H2 x_1 x_2 \\<and> ?H2 x_1 x_3"
] | [
"?H1 x_1 x_2 x_3 \\<Longrightarrow> ?H2 x_1 x_2 \\<and> ?H2 x_1 x_3",
"?H1 x_1 x_2 x_3 \\<Longrightarrow> ?H2 x_1 x_2 \\<and> ?H2 x_1 x_3",
"?H1 x_1 x_2 x_3 \\<Longrightarrow> ?H2 x_1 x_2 \\<and> ?H2 x_1 x_3",
"?H1 x_1 x_2 x_3 \\<Longrightarrow> ?H2 x_1 x_2 \\<and> ?H2 x_1 x_3",
"?H1 x_1 x_2 x_3 \\<Longrigh... | 123 | 123 | 123 | 82 | 82 | 82 | 0.496 | 0.496 | 0.496 |
Lattice/Bounds | Bounds.is_Sup_least | is_Sup ?A ?sup \<Longrightarrow> (\<And>x. x \<in> ?A \<Longrightarrow> x \<sqsubseteq> ?z) \<Longrightarrow> ?sup \<sqsubseteq> ?z | lemma is_Sup_least [elim?]:
"is_Sup A sup \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> x \<sqsubseteq> z) \<Longrightarrow> sup \<sqsubseteq> z" | \<lbrakk>?H1 x_1 x_2; \<And>y_0. y_0 \<in> x_1 \<Longrightarrow> ?H2 y_0 x_3\<rbrakk> \<Longrightarrow> ?H2 x_2 x_3 | [
"?H1 x_1 x_2 = (\\<forall>y_0\\<in>x_1. ?H2 y_0 x_2)"
] | [
"?H1 x_1 x_2 = (\\<forall>y_0\\<in>x_1. ?H2 y_0 x_2)",
"?H1 x_1 x_2 = (\\<forall>y_0\\<in>x_1. ?H2 y_0 x_2)",
"?H1 x_1 x_2 = (\\<forall>y_0\\<in>x_1. ?H2 y_0 x_2)",
"?H1 x_1 x_2 = (\\<forall>y_0\\<in>x_1. ?H2 y_0 x_2)",
"?H1 x_1 x_2 = (\\<forall>y_0\\<in>x_1. ?H2 y_0 x_2)"
] | 115 | 115 | 115 | 81 | 81 | 81 | 0.363636 | 0.363636 | 0.363636 |
End of preview. Expand in Data Studio
README.md exists but content is empty.
- Downloads last month
- 12