hcju/isabelleps huggingface dataset:
inner_Basis: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and sgn :: "'b \<Rightarrow> 'b"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and inner :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and Basis :: "'b set"   and u :: "'b"   and v :: "'b" assumes "class.euclidean_space minus dist norm plus zero uminus scaleR sgn uniformity open inner Basis"   and "u \<in> Basis"   and "v \<in> Basis" shows "inner u v = (if u = v then 1 else 0)"
intro: fixes \<alpha> :: "'a \<Rightarrow> 'b \<times> 'b \<Rightarrow> enat"   and invar :: "'a \<Rightarrow> bool"   and succ :: "'a \<Rightarrow> 'b \<Rightarrow> (nat \<times> 'b) list"   and empty_graph :: "'a"   and add_edge :: "'b \<times> 'b \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a" assumes "adt_wgraph \<alpha> invar succ empty_graph add_edge" shows "wgraph_from_list_algo \<alpha> invar succ empty_graph add_edge"
list_conj: fixes bs :: "real list"   and ps :: "fm list" shows "Ifm bs (list_conj ps) = (\<forall>p\<in>set ps. Ifm bs p)"
assms: shows "sources \<nu> \<inter> targets \<mu> \<noteq> {}"
Y_FG: shows "\<forall>\<psi>\<in>Y. w \<Turnstile>\<^sub>n F\<^sub>n (G\<^sub>n \<psi>[X]\<^sub>\<nu>)"
ide_cod: fixes f :: "'d" assumes "D.arr f" shows "D.ide (D.cod f)"
add_decreasing2: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and c :: "'d"   and a :: "'d"   and b :: "'d" assumes "class.ordered_comm_monoid_add plus zero less_eq less"   and "less_eq c zero"   and "less_eq a b" shows "less_eq (plus a c) b"
nat_le_linear: fixes m :: "nat"   and n :: "nat" shows "m \<le> n \<or> n \<le> m"
assms: fixes \<sigma>' :: "'a \<Rightarrow> ('b, 'c) Term.term option" shows "((P, \<sigma>), {#}, \<sigma>') \<notin> matchrel\<^sup>*"
senior_states_psubset_card_mono: fixes \<Sigma> :: "'c set"   and \<delta> :: "'d \<Rightarrow> 'c \<Rightarrow> 'd"   and q\<^sub>0 :: "'d"   and w :: "nat \<Rightarrow> 'c"   and A :: "'d set"   and q :: "'d"   and n :: "nat" assumes "semi_mojmir \<Sigma> \<delta> q\<^sub>0 w"   and "A \<subset> semi_mojmir_def.senior_states \<Sigma> \<delta> q\<^sub>0 w q n" shows "card A < card (semi_mojmir_def.senior_states \<Sigma> \<delta> q\<^sub>0 w q n)"
before_dist_eq: fixes vs :: "'a list"   and ram1 :: "'a"   and ram2 :: "'a" assumes "distinct vs"   and "ram1 \<in> set vs"   and "ram2 \<in> set vs"   and "ram1 \<noteq> ram2" shows "(\<not> before vs ram1 ram2) = before vs ram2 ram1"
rbt_eqI: fixes t1 :: "('a, 'b) RBT.rbt"   and t2 :: "('a, 'b) RBT.rbt" assumes "impl_of t1 = impl_of t2" shows "t1 = t2"
degree_pcompose_le: fixes p :: "'b poly"   and q :: "'b poly" shows "degree (p \<circ>\<^sub>p q) \<le> degree p * degree q"
identity_atomic: fixes x :: "i" shows "atomic ((\<^bold>=) x)"
sym: fixes x :: "'a"   and y :: "'a" shows "f x y = f y x"
lasso_v0_def: fixes L :: "('d, 'e) lasso_scheme" shows "lasso_v0 L \<equiv> case lasso_reach L of [] \<Rightarrow> lasso_va L | v0 # x \<Rightarrow> v0"
supp_at_base: fixes a :: "'b" shows "supp a = {atom a}"
term_brackets_Nil: fixes term_bracket :: "nat \<Rightarrow> dB \<Rightarrow> dB option"   and t :: "dB" assumes "bracket_abstraction term_bracket" shows "bracket_abstraction.term_brackets term_bracket [] t = Some t"
assms_1: shows "X \<in> M \<rightarrow>\<^sub>M N"
evaln_ExprI: fixes G :: "prog"   and s :: "state"   and e :: "expr"   and n :: "nat"   and v :: "val"   and s' :: "state" assumes "G\<turnstile>s \<midarrow>e-\<succ>v\<midarrow>n\<rightarrow> s'" shows "G\<turnstile>s \<midarrow>Expr e\<midarrow>n\<rightarrow> s'"
F'rel_def: fixes R1 :: "'s \<Rightarrow> 't \<Rightarrow> bool"   and R2 :: "'u \<Rightarrow> 'v \<Rightarrow> bool"   and R3 :: "'o \<Rightarrow> 'q \<Rightarrow> bool"   and R4 :: "'p \<Rightarrow> 'r \<Rightarrow> bool" shows "F'rel R1 R2 R3 R4 = (BNF_Def.Grp {x. {} \<subseteq> {(x, y). R1 x y} \<and> {} \<subseteq> {(x, y). R2 x y} \<and> F'set3 x \<subseteq> {(x, y). R3 x y} \<and> F'set4 x \<subseteq> {(x, y). R4 x y}} (Fmap fst fst))\<inverse>\<inverse> OO BNF_Def.Grp {x. {} \<subseteq> {(x, y). R1 x y} \<and> {} \<subseteq> {(x, y). R2 x y} \<and> F'set3 x \<subseteq> {(x, y). R3 x y} \<and> F'set4 x \<subseteq> {(x, y). R4 x y}} (Fmap snd snd)"
neutral_const: fixes A :: "'d set" shows "(\<Sum>_\<in>A. (0::'c)) = (0::'c)"
assms_6: shows "\<forall>(k2, \<gamma>2)\<in>one_chain2. \<forall>b\<in>basis. continuous_on (path_image \<gamma>2) (\<lambda>x. F x \<bullet> b)"
chain_lnth_rel: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "'a llist"   and j :: "nat" assumes "chain R xs"   and "enat (Suc j) < llength xs" shows "R (lnth xs j) (lnth xs (Suc j))"
assms: shows "list_all (\<lambda>x. x = y) xs"
lift_is_hom: fixes G :: "('c, 'd) monoid_scheme"   and f :: "'e \<Rightarrow> 'c"   and gens :: "'e set" assumes "Group.group G"   and "f \<in> gens \<rightarrow> carrier G" shows "group.lift G f \<in> hom \<F>\<^bsub>gens\<^esub> G"
env_restr_mono: fixes m1 :: "'a \<Rightarrow> 'b"   and m2 :: "'a \<Rightarrow> 'b"   and S :: "'a set" assumes "m1 \<sqsubseteq> m2" shows "m1 f|` S \<sqsubseteq> m2 f|` S"
EM_def: shows "EM \<equiv> M \<and> EM_axioms"
span_diff: fixes x :: "'a"   and S :: "'a set"   and y :: "'a" assumes "x \<in> More_Modules.ideal S"   and "y \<in> More_Modules.ideal S" shows "x - y \<in> More_Modules.ideal S"
axioms_2: fixes V\<^sub>B :: "'r \<Rightarrow> 'r \<Rightarrow> 'r"   and H\<^sub>B :: "'r \<Rightarrow> 'r \<Rightarrow> 'r"   and \<a>\<^sub>B :: "'r \<Rightarrow> 'r \<Rightarrow> 'r \<Rightarrow> 'r"   and \<i>\<^sub>B :: "'r \<Rightarrow> 'r"   and src\<^sub>B :: "'r \<Rightarrow> 'r"   and trg\<^sub>B :: "'r \<Rightarrow> 'r"   and V\<^sub>C :: "'s \<Rightarrow> 's \<Rightarrow> 's"   and H\<^sub>C :: "'s \<Rightarrow> 's \<Rightarrow> 's"   and \<a>\<^sub>C :: "'s \<Rightarrow> 's \<Rightarrow> 's \<Rightarrow> 's"   and \<i>\<^sub>C :: "'s \<Rightarrow> 's"   and src\<^sub>C :: "'s \<Rightarrow> 's"   and trg\<^sub>C :: "'s \<Rightarrow> 's"   and V\<^sub>D :: "'t \<Rightarrow> 't \<Rightarrow> 't"   and H\<^sub>D :: "'t \<Rightarrow> 't \<Rightarrow> 't"   and \<a>\<^sub>D :: "'t \<Rightarrow> 't \<Rightarrow> 't \<Rightarrow> 't"   and \<i>\<^sub>D :: "'t \<Rightarrow> 't"   and src\<^sub>D :: "'t \<Rightarrow> 't"   and trg\<^sub>D :: "'t \<Rightarrow> 't"   and F :: "'r \<Rightarrow> 's"   and \<Phi>\<^sub>F :: "'r \<times> 'r \<Rightarrow> 's"   and G :: "'s \<Rightarrow> 't"   and \<Phi>\<^sub>G :: "'s \<times> 's \<Rightarrow> 't" assumes "composite_pseudofunctor V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>\<^sub>F G \<Phi>\<^sub>G" shows "bicategory V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C"
res_seq_subseq': fixes s :: "nat \<Rightarrow> nat \<Rightarrow> int"   and k :: "nat" assumes "s \<in> carrier Zp\<^bsup>\<omega>\<^esup>" shows "is_subseq_of s (res_seq s k)"
dvd_imp_mult_div_cancel_left: fixes a :: "'b"   and b :: "'b" assumes "a dvd b" shows "a * (b div a) = b"
"3_7": shows "(0::'a) < last (x # y # xs)"
Pxyz: shows "simple_distributed M (\<lambda>x. (X x, Y x, Z x)) Pxyz"
axes: fixes x :: "complex"   and y :: "complex" assumes "is_real x"   and "0 \<le> Re x"   and "Re x < 1"   and "is_imag y"   and "0 \<le> Im y"   and "Im y < 1" shows "P 0\<^sub>h (of_complex x) (of_complex y)"
simps_1: fixes uu :: "'b"   and uv :: "'b"   and F :: "'a" shows "Map (MkArr uu uv F) = F"
lt_comm: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Lt Bet Cong A B C D" shows "Tarski_neutral_dimensionless.Lt Bet Cong B A D C"
psi_ubound_log: fixes n :: "nat" shows "psi n \<le> 551 / 256 * ln 2 * real n"
execn_elim_cases_5: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c1 :: "('e, 'd, 'f) com"   and c2 :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate"   and n :: "nat"   and t :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile> \<langle>Seq c1 c2,s\<rangle> =n\<Rightarrow> t" obtains f :: "'f" where "sb__ = Fault f" and "t = Fault f"   | s :: "'e" and s' :: "('e, 'f) xstate" where "sb__ = Normal sc__" and "\<Gamma>\<turnstile> \<langle>c1,Normal sc__\<rangle> =n\<Rightarrow> s'" and "\<Gamma>\<turnstile> \<langle>c2,s'\<rangle> =n\<Rightarrow> t"   | "sb__ = Stuck" and "t = Stuck"   | s :: "'e" where "sb__ = Abrupt s" and "t = Abrupt s"
smult_carrier_vec: fixes a :: "'a"   and v :: "'a vec"   and n :: "nat" shows "(a \<cdot>\<^sub>v v \<in> carrier_vec n) = (v \<in> carrier_vec n)"
csl: shows "cs\<^bsup>\<pi>\<^esup> l = [\<pi> l]"
pseudonatural_transformation_def: fixes V\<^sub>C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and H\<^sub>C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<a>\<^sub>C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<i>\<^sub>C :: "'c \<Rightarrow> 'c"   and src\<^sub>C :: "'c \<Rightarrow> 'c"   and trg\<^sub>C :: "'c \<Rightarrow> 'c"   and V\<^sub>D :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and H\<^sub>D :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and \<a>\<^sub>D :: "'d \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and \<i>\<^sub>D :: "'d \<Rightarrow> 'd"   and src\<^sub>D :: "'d \<Rightarrow> 'd"   and trg\<^sub>D :: "'d \<Rightarrow> 'd"   and F :: "'c \<Rightarrow> 'd"   and \<Phi>\<^sub>F :: "'c \<times> 'c \<Rightarrow> 'd"   and G :: "'c \<Rightarrow> 'd"   and \<Phi>\<^sub>G :: "'c \<times> 'c \<Rightarrow> 'd"   and \<tau>\<^sub>0 :: "'c \<Rightarrow> 'd"   and \<tau>\<^sub>1 :: "'c \<Rightarrow> 'd" shows "pseudonatural_transformation V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>\<^sub>F G \<Phi>\<^sub>G \<tau>\<^sub>0 \<tau>\<^sub>1 \<equiv> (bicategory V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C \<and> bicategory V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D) \<and> pseudofunctor V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>\<^sub>F \<and> pseudofunctor V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D G \<Phi>\<^sub>G \<and> pseudonatural_transformation_axioms V\<^sub>C H\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>\<^sub>F G \<Phi>\<^sub>G \<tau>\<^sub>0 \<tau>\<^sub>1"
rel_interior_closure_convex_shrink: fixes S :: "'b set"   and c :: "'b"   and x :: "'b"   and e :: "real" assumes "convex S"   and "c \<in> rel_interior S"   and "x \<in> closure S"   and "0 < e"   and "e \<le> 1" shows "x - e *\<^sub>R (x - c) \<in> rel_interior S"
B2_def: fixes m1 :: "'a option"   and m2 :: "'a option"   and \<alpha> :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "B2 m1 m2 \<alpha> \<equiv> m1 \<rightharpoonup> (\<lambda>a. m2 \<rightharpoonup> \<alpha> a)"
ClassI: fixes a :: "'a"   and b :: "'a" assumes "(a, b) \<in> Ring_Congruence" shows "a \<in> additive.Class b"
stable_imp_take_bit_eq: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bit :: "'c \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and drop_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and take_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c"   and n :: "nat" assumes "class.semiring_bit_shifts plus minus zero times one divide modulo bit push_bit drop_bit take_bit"   and "divide a (numeral.numeral one plus (num.Bit0 num.One)) = a" shows "take_bit n a = (if dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) a then zero else minus (power.power one times (numeral.numeral one plus (num.Bit0 num.One)) n) one)"
extensional_funcset_fun_upd_inj_onI: fixes f :: "'c \<Rightarrow> 'd"   and S :: "'c set"   and T :: "'d set"   and a :: "'d"   and x :: "'c" assumes "f \<in> S \<rightarrow>\<^sub>E T - {a}"   and "inj_on f S" shows "inj_on (f(x := a)) S"
analz_Un_AencSet2: fixes H :: "msg set"   and Ag :: "agent set"   and G :: "msg set" assumes "Enc_keys_clean H"   and "priK ` Ag \<inter> synth (analz H) = {}" shows "analz (AencSet G (pubK ` Ag) \<union> H) \<subseteq> AencSet G (pubK ` Ag) \<union> analz H"
nat_to_string___is_nat_string: fixes n :: "nat" shows "is_nat_string (nat_to_string n)"
invariant_NewThread: fixes t :: "'t"   and x :: "'x"   and m :: "'m"   and ta :: "('l, 't, 'x, 'm, 'w, 'o) thread_action"   and x' :: "'x"   and m' :: "'m"   and i :: "'i"   and t'' :: "'t"   and x'' :: "'x" assumes "t \<turnstile> \<langle>x, m\<rangle> -ta\<rightarrow> \<langle>x', m'\<rangle>"   and "P i t x m"   and "NewThread t'' x'' m' \<in> set \<lbrace>ta\<rbrace>\<^bsub>t\<^esub>" shows "\<exists>i''. P i'' t'' x'' m'"
dim_span: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and S :: "'b set" assumes "vector_space scale" shows "vector_space.dim scale (module.span scale S) = vector_space.dim scale S"
StrongEq_sym: fixes X :: "'d state \<times> 'd state \<Rightarrow> 'e"   and Y :: "'d state \<times> 'd state \<Rightarrow> 'e" shows "(X \<triangleq> Y) = (Y \<triangleq> X)"
section_def: fixes f :: "'b" shows "B.section f \<equiv> \<exists>g. B.ide (g \<cdot>\<^sub>B f)"
filterlim_tan_at_left: shows "filterlim tan at_top (at_left (pi / 2))"
inj_Inr: fixes A :: "'b set" shows "inj_on Inr A"
del_arc_def: fixes G :: "('c, 'd) pre_digraph"   and a :: "'d" shows "pre_digraph.del_arc G a = \<lparr>verts = verts G, arcs = arcs G - {a}, tail = tail G, head = head G\<rparr>"
analz_Un: fixes G :: "msg set"   and H :: "msg set" shows "analz G \<union> analz H \<subseteq> analz (G \<union> H)"
input: shows "input \<in> responses_\<I> \<I> out"
abelian_FactGroup: fixes N :: "'a set" assumes "subgroup N G" shows "comm_group (G Mod N)"
halloc_gext: fixes G :: "prog"   and s1 :: "state"   and oi :: "obj_tag"   and a :: "loc"   and s2 :: "state" assumes "G\<turnstile>s1 \<midarrow>halloc oi\<succ>a\<rightarrow> s2" shows "snd s1\<le>|snd s2"
upE: fixes p :: "'b\<^sub>\<bottom>" obtains "p = \<bottom>"   | x :: "'b" where "p = up\<cdot>x"
bidirected_digraph: shows "bidirected_digraph H (edge_rev HM)"
cyclic_group_assoc: fixes G :: "('a, 'b) cyclic_group_scheme"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "Cyclic_Group.cyclic_group G"   and "a \<in> carrier G"   and "b \<in> carrier G"   and "c \<in> carrier G" shows "a \<otimes>\<^bsub>G\<^esub> b \<otimes>\<^bsub>G\<^esub> c = a \<otimes>\<^bsub>G\<^esub> (b \<otimes>\<^bsub>G\<^esub> c)"
padic_int_is_domain: fixes p :: "int" assumes "prime p" shows "domain (padic_int p)"
is_le_relE: fixes rel :: "'t \<Rightarrow> 't \<Rightarrow> bool" assumes "ord_term_lin.is_le_rel rel" obtains "rel = (=)"   | "rel = (\<preceq>\<^sub>t)"   | "rel = (\<prec>\<^sub>t)"
row_length_constant: fixes id :: "'a"   and f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zer :: "'a"   and g :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inver :: "'a \<Rightarrow> 'a"   and nr :: "nat"   and nc :: "nat"   and A :: "'a list list"   and j :: "nat" assumes "plus_mult id f zer g inver"   and "Matrix_Legacy.mat nr nc A"   and "j < length A" shows "length (A ! j) = mult.row_length A"
assms_2: shows "foldr (\<otimes>) fs \<one> = a"
HSem_cong: fixes heap2 :: "('b \<times> 'a) list"   and ESem1 :: "'a \<Rightarrow> ('b \<Rightarrow> 'c) \<rightarrow> 'c"   and ESem2 :: "'a \<Rightarrow> ('b \<Rightarrow> 'c) \<rightarrow> 'c"   and heap1 :: "('b \<times> 'a) list" assumes "\<And>e. e \<in> snd ` set heap2 \<Longrightarrow> ESem1 e = ESem2 e"   and "heap1 = heap2" shows "has_ESem.HSem ESem1 heap1 = has_ESem.HSem ESem2 heap2"
Set_filter_union: fixes P :: "'b \<Rightarrow> bool"   and A :: "'b set"   and B :: "'b set" shows "Set.filter P (A \<union> B) = Set.filter P A \<union> Set.filter P B"
list_rel_def: fixes R :: "('b \<times> 'c) set" shows "\<langle>R\<rangle>list_rel \<equiv> {(l, l'). list_all2 (\<lambda>x x'. (x, x') \<in> R) l l'}"
list_combE_1: fixes f :: "'a"   and xs :: "'a list" assumes "\<not> shadows_consts (list_comb f xs)" shows "\<not> shadows_consts f"
nodes_def: fixes A :: "('e, 'f, 'd, 'g) FSM_scheme" shows "nodes A \<equiv> transition_system_initial.nodes (\<lambda>a p. snd a) (\<lambda>a p. snd a \<in> succ A (fst a) p) (\<lambda>p. p = initial A)"
edg_simps_4: fixes v :: "'v vertex"   and x :: "'v"   and y :: "'v" shows "edg v (Edge x y) = (v = Vertex x \<and> edge \<Delta> x y)"
bangPres: fixes P :: "ccs"   and Q :: "ccs"   and Rel :: "(ccs \<times> ccs) set" assumes "(P, Q) \<in> Rel"   and "\<And>R S. (R, S) \<in> Rel \<Longrightarrow> R \<leadsto>[Rel] S" shows "!P \<leadsto>[bangRel Rel] !Q"
l12_l_OK: fixes l :: "'a list" assumes "l \<in> set (l1 # l2 # ls)" shows "distinct l \<and> sorted l"
2: shows "sum t_mtf {Suc n..<length rs} = int (T (s_mtf (Suc n)) (drop (Suc n) rs) (drop (Suc n) (off MTF s0 rs)))"
assms_1: shows "p \<in> dgrad_p_set d m"
extend_restrict_indets: fixes p :: "('b option \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" assumes "None \<notin> indets p" shows "extend_indets (restrict_indets p) = p"
inf_commute: fixes x :: "'a multiset"   and y :: "'a multiset" shows "x \<inter># y = y \<inter># x"
\<eta>_in_terms_of_\<phi>: fixes y :: "'d" assumes "D.ide y" shows "\<eta> y = \<phi> y (F y)"
merge_length_le: fixes as :: "nat list list"   and bs :: "nat list list"   and us :: "nat list list"   and vs :: "nat list list" assumes "merge as bs us vs" shows "length vs \<le> length us"
C_eq_RS2_mr: fixes x :: "int \<times> (int \<times> int \<times> protocol) \<times> (int \<times> int \<times> protocol) \<times> DummyContent"   and p :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list" shows "applied_rule_rev Cp x (removeShadowRules2 p) = applied_rule_rev Cp x p"
cmult_2: fixes n :: "V" assumes "Card n" shows "ord_of_nat 2 \<otimes> n = n \<oplus> n"
orbit_char: fixes G :: "('c, 'd) monoid_scheme"   and \<phi> :: "'c \<Rightarrow> 'e \<Rightarrow> 'e"   and M :: "'e set"   and m :: "'e" assumes "group_action G \<phi> M"   and "m \<in> M" shows "group_action.orbit G \<phi> M m = {n. \<exists>g. g \<in> carrier G \<and> \<phi> g m = n}"
path_append_eq_up_to: fixes \<pi> :: "nat \<Rightarrow> 'n"   and k :: "nat"   and \<pi>' :: "nat \<Rightarrow> 'n" shows "(\<pi> @\<^bsup>k\<^esup> \<pi>') =\<^bsub>k\<^esub> \<pi>"
hyps_3: shows "is_unit (\<Prod>i\<in>S. U $ i $ i) = (\<forall>i\<in>S. is_unit (U $ i $ i))"
comp_fun_commute_axioms: shows "comp_fun_commute (\<lambda>a b. b * Var (\<phi> a))"
pred_alt_def_1: fixes P :: "(name, value) fmap \<Rightarrow> (Term_Class.term \<times> sterm) list \<Rightarrow> bool"   and Q :: "name \<Rightarrow> bool"   and R :: "name fset \<Rightarrow> bool"   and name :: "name"   and vs :: "value list" shows "value_pred.pred P Q R (Vconstr name vs) = (Q name \<and> list_all (value_pred.pred P Q R) vs)"
yoneda_functor_axioms: shows "yoneda_functor (\<cdot>) (\<cdot>\<^sub>S) \<phi>"
rep_eq: fixes x :: "'a \<Rightarrow> 'c \<Rightarrow> 'b"   and xa :: "('a, 'c) fmap" shows "fmlookup (fmmap_keys x xa) = (\<lambda>a. map_option (x a) (fmlookup xa a))"
enum_set_get_one_enum_Ex_Some: fixes S :: "'b set" assumes "S \<noteq> {}" shows "\<exists>x. enum_set_get_one enum_class.enum S = Some x"
lex_pp_trans: fixes s :: "('a, 'b) pp"   and t :: "('a, 'b) pp"   and u :: "('a, 'b) pp" assumes "lex_pp s t"   and "lex_pp t u" shows "lex_pp s u"
lub: fixes A :: "'b set"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'b"   and z :: "'b"   and y :: "'b" assumes "Semilat A r f"   and "x \<sqsubseteq>\<^bsub>r\<^esub> z"   and "y \<sqsubseteq>\<^bsub>r\<^esub> z"   and "x \<in> A"   and "y \<in> A"   and "z \<in> A" shows "x \<squnion>\<^bsub>f\<^esub> y \<sqsubseteq>\<^bsub>r\<^esub> z"
Sup_upper2: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and u :: "'b"   and A :: "'b set"   and v :: "'b" assumes "class.complete_lattice (Inf::'b set \<Rightarrow> 'b) (Sup::'b set \<Rightarrow> 'b) (inf::'b \<Rightarrow> 'b \<Rightarrow> 'b) (less_eq::'b \<Rightarrow> 'b \<Rightarrow> bool) (less::'b \<Rightarrow> 'b \<Rightarrow> bool) (sup::'b \<Rightarrow> 'b \<Rightarrow> 'b) (bot::'b) (top::'b)"   and "(u::'b) \<in> (A::'b set)"   and "(less_eq::'b \<Rightarrow> 'b \<Rightarrow> bool) (v::'b) (u::'b)" shows "(less_eq::'b \<Rightarrow> 'b \<Rightarrow> bool) (v::'b) ((Sup::'b set \<Rightarrow> 'b) (A::'b set))"
toplevel_summands_lang: fixes r :: "'b rexp"   and s :: "'b rexp" assumes "r \<in> toplevel_summands s" shows "lang r \<subseteq> lang s"
denote_trace: fixes dims :: "nat list"   and S :: "com"   and \<rho> :: "complex mat" assumes "state_sig.well_com dims S"   and "\<rho> \<in> carrier_mat (state_sig.d dims) (state_sig.d dims)"   and "partial_density_operator \<rho>" shows "trace (state_sig.denote dims S \<rho>) \<le> trace \<rho>"
hyp1: shows "matrix_to_iarray (echelon_form_of_upt_k A k bezout) = echelon_form_of_upt_k_iarrays (matrix_to_iarray A) k bezout"
fst_map_prod: fixes f :: "'g \<Rightarrow> 'e"   and g :: "'h \<Rightarrow> 'f"   and x :: "'g \<times> 'h" shows "fst (map_prod f g x) = f (fst x)"
gt_param: fixes q :: "real" assumes "q = (1 + r - d) / (u - d)"   and "viable_market Mkt" shows "0 < q"
redT_updT_nta_bisim_inv: fixes ta1 :: "('t, 'x1, 'm1) new_thread_action"   and ta2 :: "('t, 'x2, 'm2) new_thread_action"   and ts1 :: "'t \<Rightarrow> ('x1 \<times> 'c \<Rightarrow>f nat) option"   and T :: "'t"   and ts2 :: "'t \<Rightarrow> ('x2 \<times> 'd \<Rightarrow>f nat) option" assumes "nta_bisim bisim ta1 ta2"   and "(ts1 T = None) = (ts2 T = None)" shows "(redT_updT ts1 ta1 T = None) = (redT_updT ts2 ta2 T = None)"
inject_2: fixes x2 :: "'e"   and y2 :: "'e" shows "(Inr x2 = Inr y2) = (x2 = y2)"
neville_aitken_interpolation_impl: fixes x_fs :: "('b \<times> 'b) list" assumes "x_fs \<noteq> []" shows "neville_aitken_interpolation_poly x_fs = neville_aitken (\<lambda>i. fst (x_fs ! i)) (\<lambda>i. snd (x_fs ! i)) (length x_fs - 1)"
pr_key_analz: fixes evs :: "event list"   and S :: "agent \<times> nat \<times> nat \<Rightarrow> session"   and A :: "msg set"   and U :: "msg set"   and K :: "nat" assumes "(evs, S, A, U) \<in> protocol" shows "(Key K \<in> analz (A \<union> spies evs)) = (Key K \<in> A)"
rec_valu_def: shows "rec_valu = Cn 1 rec_minus [Cn 1 rec_lg [s, constn 2], constn 1]"
linear_scale: fixes k :: "enat"   and f :: "('c \<Rightarrow> real) \<Rightarrow> real"   and b :: "'c \<Rightarrow> real"   and r :: "real" assumes "\<forall>x. x \<in> manifold_eucl.diff_fun_space k \<longrightarrow> f x \<in> UNIV"   and "b \<in> manifold_eucl.diff_fun_space k"   and "linear_on (manifold_eucl.diff_fun_space k) UNIV (*\<^sub>R) (*\<^sub>R) f" shows "f (r *\<^sub>R b) = r *\<^sub>R f b"
norm_box_stack: fixes \<Sigma> :: "('dyn, 'ubx1, 'ubx2) unboxed list" shows "norm_stack (map Subx.box_operand \<Sigma>) = norm_stack \<Sigma>"
trace_adjoint_eq_u: fixes A :: "complex mat" shows "trace (A * adjoint A) = complex_of_real (\<Sum>i = 0..<dim_row A. \<Sum>j = 0..<dim_col A. (cmod (A $$ (i, j)))\<^sup>2)"