hcju/isabelleps huggingface dataset:
shift_keeps_length: fixes r :: "real_int"   and x :: "real" shows "\<parallel>r\<parallel> = \<parallel> shift r x\<parallel>"
emb_step_under_args_head: fixes p :: "dir list"   and d :: "dir"   and t :: "('a, 'b) tm" assumes "\<not> list_all (\<lambda>x. x = dir.Left) p" shows "head (emb_step_at p d t) = head t"
cases: fixes ci :: "('addr, 'heap) check_instr"   and P :: "(nat \<times> nat \<times> 'addr instr list \<times> (nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list) prog"   and ins :: "'addr instr list"   and xt :: "(nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list"   and t :: "'thread_id"   and a1 :: "'heap"   and a2 :: "'addr val list \<times> 'addr val list \<times> nat \<times> 'addr option"   and a3 :: "('addr, 'thread_id, 'heap) jvm_thread_action"   and a4 :: "'heap"   and a5 :: "'addr val list \<times> 'addr val list \<times> nat \<times> 'addr option" assumes "exec_meth ci P ins xt t a1 a2 a3 a4 a5" obtains ta :: "('addr, 'thread_id, 'heap) jvm_thread_action" and xcp :: "'addr option" and h' :: "'heap" and stk' :: "'addr val list" and loc' :: "'addr val list" and pc' :: "nat" and pc :: "nat" and h :: "'heap" and stk :: "'addr val list" and loc :: "'addr val list" where "a1 = ha__" and "a2 = (stka__, loca__, pca__, None)" and "a3 = ta" and "a4 = h'" and "a5 = (stk', loc', pc'__, xcpa__)" and "(ta, xcpa__, h', [(stk', loc', undefined, undefined, pc'__)]) \<in> exec_instr (ins ! pca__) P t ha__ stka__ loca__ undefined undefined pca__ []" and "pca__ < length ins" and "ci_app ci (ins ! pca__) P ha__ stka__ loca__ undefined undefined pca__ []"   | h :: "'heap" and xcp :: "'addr" and pc :: "nat" and pc' :: "nat" and d :: "nat" and stk :: "'addr val list" and loc :: "'addr val list" where "a1 = h" and "a2 = (stk, loc, pc, \<lfloor>xcp\<rfloor>)" and "a3 = \<lbrace>\<rbrace>" and "a4 = h" and "a5 = (Addr xcp # drop (length stk - d) stk, loc, pc', None)" and "match_ex_table P (cname_of h xcp) pc xt = \<lfloor>(pc', d)\<rfloor>" and "d \<le> length stk"
all_croots: fixes n :: "nat"   and x :: "complex" assumes "n \<noteq> 0" shows "set (all_croots n x) = {y. y ^ n = x}"
intros_2: fixes vs :: "e list"   and lholed :: "Lholed"   and n :: "nat"   and es' :: "e list"   and l :: "Lholed"   and es'' :: "e list"   and k :: "nat"   and es :: "e list"   and lfilledk :: "e list" assumes "const_list vs"   and "lholed = LRec vs n es' l es''"   and "Lfilled_exact k l es lfilledk" shows "Lfilled_exact (k + 1) lholed es (vs @ [Label n es' lfilledk] @ es'')"
distinct_43: fixes x31 :: "('a, 'b, 'c, 'd) seqp"   and x32 :: "('a, 'b, 'c, 'd) seqp"   and x81 :: "'d"   and x82 :: "'a \<Rightarrow> nat"   and x83 :: "('a, 'b, 'c, 'd) seqp" shows "x31 \<oplus> x32 \<noteq> {x81}deliver(x82) . x83"
term_pp_rel_trans: fixes r :: "'t \<Rightarrow> 't \<Rightarrow> bool"   and a :: "'t \<times> 'a"   and b :: "'t \<times> 'a"   and c :: "'t \<times> 'a" assumes "ord_term_lin.is_le_rel r"   and "term_pp_rel r a b"   and "term_pp_rel r b c" shows "term_pp_rel r a c"
sr_imp_eigen_vector_main: shows "sr *s u = A *v u"
not_loose_bvar_imp_not_loose_bvar1_all_greater: fixes t :: "Core.term"   and lev :: "nat"   and x :: "nat" assumes "\<not> loose_bvar t lev"   and "lev < x" shows "\<not> loose_bvar1 t x"
group_add_axioms_def: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b" shows "class.group_add_axioms minus plus zero uminus \<equiv> (\<forall>a. plus (uminus a) a = zero) \<and> (\<forall>a b. plus a (uminus b) = minus a b)"
append_butlast_last_id: fixes xs :: "'e list" assumes "xs \<noteq> []" shows "butlast xs @ [last xs] = xs"
upper_triangular_def: fixes A :: "'b mat" shows "Matrix.upper_triangular A \<equiv> \<forall>i<dim_row A. \<forall>j<i. A $$ (i, j) = (0::'b)"
RG_aezdeltafun_closed: fixes g :: "'g"   and r :: "'a" assumes "g \<in> H" shows "r \<delta>\<delta> g \<in> Supgroup.group_ring"
multiset_filter_subset: fixes f :: "'a \<Rightarrow> bool"   and M :: "'a multiset" shows "filter_mset f M \<subseteq># M"
map_lists_mono: fixes f :: "'b \<Rightarrow> 'a"   and A :: "'b set"   and B :: "'a set" assumes "f ` A \<subseteq> B" shows "map f ` lists A \<subseteq> lists B"
fresh_fun_eqvt: fixes f :: "'a"   and a :: "atom" assumes "eqvt f" shows "a \<sharp> f"
mat_rel_def: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'a mat"   and B :: "'b mat" shows "mat_rel R A B \<equiv> dim_row A = dim_row B \<and> dim_col A = dim_col B \<and> (\<forall>i j. i < dim_row B \<longrightarrow> j < dim_col B \<longrightarrow> R (A $$ (i, j)) (B $$ (i, j)))"
direct_decomp_singleton: fixes A :: "'b set" shows "direct_decomp A [A]"
mult_right_mono_neg: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and b :: "'c"   and a :: "'c"   and c :: "'c" assumes "class.ordered_ring plus minus zero less_eq less times uminus"   and "less_eq b a"   and "less_eq c zero" shows "less_eq (times a c) (times b c)"
set_simps2: fixes x :: "'a"   and xs :: "'a list" shows "set (x # xs) = {x} \<union> set xs"
BinOpRedsThrow2: fixes e\<^sub>1 :: "char list exp"   and s\<^sub>0 :: "heap \<times> sfields \<times> sheap"   and b\<^sub>0 :: "bool"   and v\<^sub>1 :: "val"   and s\<^sub>1 :: "heap \<times> sfields \<times> sheap"   and b\<^sub>1 :: "bool"   and P :: "(char list list \<times> char list exp) prog"   and e\<^sub>2 :: "char list exp"   and e :: "char list exp"   and s\<^sub>2 :: "heap \<times> sfields \<times> sheap"   and b\<^sub>2 :: "bool"   and bop :: "bop" assumes "P \<turnstile> \<langle>e\<^sub>1,s\<^sub>0,b\<^sub>0\<rangle> \<rightarrow>* \<langle>Val v\<^sub>1,s\<^sub>1,b\<^sub>1\<rangle>"   and "P \<turnstile> \<langle>e\<^sub>2,s\<^sub>1,b\<^sub>1\<rangle> \<rightarrow>* \<langle>throw e,s\<^sub>2,b\<^sub>2\<rangle>" shows "P \<turnstile> \<langle>e\<^sub>1 \<guillemotleft>bop\<guillemotright> e\<^sub>2,s\<^sub>0,b\<^sub>0\<rangle> \<rightarrow>* \<langle>throw e,s\<^sub>2,b\<^sub>2\<rangle>"
set_vzero_eq_ord_of_nat_vone: shows "set {[]\<^sub>\<circ>} = 1\<^sub>\<nat>"
star_inductl_eq: fixes z :: "'a"   and x :: "'a"   and y :: "'a" assumes "z + x \<cdot> y = y" shows "x\<^sup>\<star> \<cdot> z \<le> y"
inv\<^sub>2_def: fixes P\<^sub>1 :: "'a set set"   and P\<^sub>2 :: "'a set set"   and B\<^sub>1 :: "'a set"   and B\<^sub>2 :: "'a set"   and V :: "'a set" shows "inv\<^sub>2 P\<^sub>1 P\<^sub>2 B\<^sub>1 B\<^sub>2 V = (inv\<^sub>1 P\<^sub>1 P\<^sub>2 B\<^sub>1 B\<^sub>2 V \<and> (V \<inter> L \<noteq> {} \<longrightarrow> (\<forall>B\<in>P\<^sub>1 \<union> wrap B\<^sub>1. B \<inter> L \<noteq> {})) \<and> bij_exists P\<^sub>1 (\<Union> (P\<^sub>2 \<union> wrap B\<^sub>2)) \<and> 2 * card P\<^sub>2 \<le> card (\<Union> P\<^sub>2))"
map_FGl_def: fixes l1 :: "'p \<Rightarrow> 'ac"   and l2 :: "'q \<Rightarrow> 'ad"   and l3 :: "'w \<Rightarrow> 'ai"   and co1 :: "'r \<Rightarrow> 'ae"   and co2 :: "'s \<Rightarrow> 'af"   and co3 :: "'x \<Rightarrow> 'aj"   and co4 :: "'y \<Rightarrow> 'ak"   and contra1 :: "'ag \<Rightarrow> 't"   and contra2 :: "'ah \<Rightarrow> 'u"   and contra3 :: "'al \<Rightarrow> 'z"   and contra4 :: "'am \<Rightarrow> 'aa" shows "map_FGl l1 l2 l3 co1 co2 co3 co4 contra1 contra2 contra3 contra4 = map_F (map_G l1 l2 co1 co2 contra1 contra2) l1 l3 co1 co3 co4 contra1 contra3 contra4"
Right_Coset_def: fixes composition :: "'d \<Rightarrow> 'e \<Rightarrow> 'c"   and H :: "'d set"   and x :: "'e" shows "coset_notation.Right_Coset composition H x = {composition h x |h. h \<in> H}"
adds_plus_right: fixes a :: "'a"   and c :: "'a"   and b :: "'a" assumes "a adds c" shows "a adds b + c"
simps_2: fixes plus :: "'h \<Rightarrow> 'h \<Rightarrow> 'h"   and one :: "'h"   and n :: "num" assumes "class.numeral plus" shows "numeral.numeral one plus (num.Bit0 n) = plus (numeral.numeral one plus n) (numeral.numeral one plus n)"
CallObj: fixes E :: "char list \<Rightarrow> ty option"   and e :: "expr"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and e' :: "expr"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and Copt :: "char list option"   and M :: "char list"   and es :: "expr list" assumes "P,E \<turnstile> \<langle>e,s\<rangle> \<rightarrow> \<langle>e',s'\<rangle>" shows "P,E \<turnstile> \<langle>Call e Copt M es,s\<rangle> \<rightarrow> \<langle>Call e' Copt M es,s'\<rangle>"
sequent_intuitonistic_semantics: fixes \<Gamma> :: "'b formula multiset"   and \<delta> :: "'b formula" shows "\<Turnstile> \<Gamma> \<Rightarrow> \<delta>, {#} = set_mset \<Gamma> \<TTurnstile> \<delta>"
sty_objI: fixes P :: "cld list"   and ty :: "ty"   and ctxclds :: "(ctx \<times> cld) list" assumes "find_path_ty P ty (Some ctxclds)" shows "sty_one P ty ty_top"
a1: shows "\<forall>p\<in>set Ps. \<exists>\<Phi>' \<Psi>'. p = ( \<Phi>' \<oplus> F \<nabla> [x].A \<Rightarrow>* \<Psi>')"
iso_is_arr: fixes f :: "'a \<times> 'a" assumes "CC.iso f" shows "CC.arr f"
Domainp_set: fixes T :: "'c \<Rightarrow> 'd \<Rightarrow> bool" shows "Domainp (rel_set T) = (\<lambda>A. Ball A (Domainp T))"
eq1: shows "lpp ` (hom_deg_set z A - {0}) = lpp ` B"
insert_def: fixes x :: "'a"   and xs :: "'a list" shows "List.insert x xs = (if x \<in> set xs then xs else x \<Colon> xs)"
essentially_equalI: fixes fs1 :: "'a list"   and fs1' :: "'a list"   and fs2 :: "'a list" assumes "fs1 <~~> fs1'"   and "fs1' [\<sim>] fs2" shows "essentially_equal G fs1 fs2"
list_of_oalist_inject: fixes x :: "('a, 'b) oalist"   and y :: "('a, 'b) oalist" shows "(list_of_oalist x = list_of_oalist y) = (x = y)"
iD: fixes i :: "nat" assumes "i < DIM('a)" shows "isDERIV i fa (list_of_eucl x @ params)"
trunc_pdevs_raw_def: fixes p :: "nat"   and x :: "nat \<Rightarrow> 'a"   and i :: "nat" shows "trunc_pdevs_raw p x i = eucl_truncate_down p (x i)"
trans: fixes b :: "'a::order"   and a :: "'a::order"   and c :: "'a::order" assumes "(b::'a::order) \<le> (a::'a::order)"   and "(c::'a::order) \<le> (b::'a::order)" shows "(c::'a::order) \<le> (a::'a::order)"
PfP_implies_SubstForm_PfP: fixes H :: "fm set"   and y :: "tm"   and x :: "tm"   and t :: "tm"   and z :: "tm" assumes "H \<turnstile> PfP y"   and "H \<turnstile> SubstFormP x t y z" shows "H \<turnstile> PfP z"
group_axioms_def: fixes G :: "('b, 'c) monoid_scheme" shows "group_axioms G \<equiv> carrier G \<subseteq> Units G"
right_idem: fixes a :: "'b"   and b :: "'b" shows "a \<squnion> b \<squnion> b = a \<squnion> b"
in_env_smaller: fixes x :: "char list"   and e :: "'b environment"   and a :: "char list"   and X :: "'b" assumes "x \<in> env_dom (e\<lparr>a:X\<rparr>)"   and "x \<noteq> a" shows "x \<in> env_dom e"
g: shows "\<forall>\<^sub>F n in sequentially. g n = - sum (residue (\<lambda>z. f z / z ^ Suc n)) S"
linear_irreducible_field: fixes p :: "'a poly" assumes "degree p = 1" shows "irreducible p"
assms_3: shows "ind_in_set A a = ind_in_set A b"
untime_nth_eq_filter: fixes n :: "nat"   and s :: "'b message_af list" assumes "n < untime_length s" shows "Msg (untime s ! n) = filter (\<lambda>x. x \<noteq> NoMsg) s ! n"
chain_map_ident: fixes p :: "nat"   and X :: "'b topology"   and c :: "((nat \<Rightarrow> real) \<Rightarrow> 'b) \<Rightarrow>\<^sub>0 int" assumes "singular_chain p X c" shows "chain_map p id c = c"
weak_GB_is_strong_GB_dgrad_p_set: fixes d :: "'a \<Rightarrow> nat"   and G :: "('t \<Rightarrow>\<^sub>0 'c) set"   and m :: "nat" assumes "dickson_grading d"   and "G \<subseteq> dgrad_p_set d m"   and "\<And>f. \<lbrakk>f \<in> pmdl G; f \<in> dgrad_p_set d m\<rbrakk> \<Longrightarrow> (red G)\<^sup>*\<^sup>* f 0" shows "is_Groebner_basis G"
heap_sort_mset_step: fixes l :: "int"   and m :: "int"   and r :: "int"   and arr' :: "int \<Rightarrow> int"   and arr :: "int \<Rightarrow> int"   and mod_arr :: "int \<Rightarrow> int" assumes "l < m"   and "m \<le> r"   and "arr_mset arr' l (m - 1) = arr_mset arr l m - {#Min_mset (arr_mset arr l m)#}"   and "arr = arr' on - {l..<m}"   and "mod_arr = arr'(m - 1 := Min_mset (arr_mset arr l m))" shows "arr_mset arr l r = arr_mset mod_arr l r"
neg1: fixes A :: "'a"   and k :: "'a"   and a :: "'a" assumes "A \<equiv> k + a" shows "- A \<equiv> - k + - a"
finite_imageD: fixes f :: "'f \<Rightarrow> 'e"   and A :: "'f set" assumes "finite (f ` A)"   and "inj_on f A" shows "finite A"
complete_cmono: fixes CC :: "'b set set"   and DD :: "'b set set" assumes "CC \<subseteq> DD" shows "DD-complete \<le> CC-complete"
cobounded1: fixes a :: "'b word"   and b :: "'b word" shows "a \<le>s signed.max a b"
sets_lebesgue_continuous_image: fixes T :: "'c set"   and S :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "T \<in> sets lebesgue"   and "continuous_on S f"   and "\<And>T. \<lbrakk>negligible T; T \<subseteq> S\<rbrakk> \<Longrightarrow> negligible (f ` T)"   and "T \<subseteq> S" shows "f ` T \<in> sets lebesgue"
cases: fixes x :: "nat list" obtains "x = []"   | n :: "nat" and ns :: "nat list" where "x = Suc (Suc (Suc (Suc n))) # nsa__"   | ns :: "nat list" where "x = 0 # nsb__"   | ns :: "nat list" where "x = Suc 0 # nsc__"   | ns :: "nat list" where "x = Suc (Suc 0) # nsd__"   | ns :: "nat list" where "x = Suc (Suc (Suc 0)) # ns"
ComplD: fixes c :: "'a"   and A :: "'a set" assumes "c \<in> - A" shows "c \<notin> A"
zero_below_greatest: shows "\<forall>m\<ge>(GREATEST n. \<not> is_zero_row_upt_k n k A) + (1::'rows). A $ m $ mod_type_class.from_nat k = (0::'a)"
dg_prod_is_arrE: fixes a :: "V"   and b :: "V"   and f :: "V" assumes "f : a \<mapsto>\<^bsub>dg_prod I \<AA>\<^esub> b" obtains "vsv f" and "\<D>\<^sub>\<circ> f = I" and "vsv a" and "\<D>\<^sub>\<circ> a = I" and "vsv b" and "\<D>\<^sub>\<circ> b = I" and "\<And>i. i \<in>\<^sub>\<circ> I \<Longrightarrow> f\<lparr>i\<rparr> : a\<lparr>i\<rparr> \<mapsto>\<^bsub>\<AA> i\<^esub> b\<lparr>i\<rparr>"
LeadsTo_weaken_L: fixes F :: "'c program"   and A :: "'c set"   and A' :: "'c set"   and B :: "'c set" assumes "F \<in> A \<longmapsto>w A'"   and "B \<subseteq> A" shows "F \<in> B \<longmapsto>w A'"
flow_has_derivative: fixes t :: "real"   and x0a :: "'c"   and t0 :: "real"   and x0 :: "'c \<Rightarrow>\<^sub>L 'c" assumes "t \<in> mvar.existence_ivl x0a t0 x0" shows "(mvar.flow x0a t0 x0 has_derivative (\<lambda>i. i *\<^sub>R (vareq x0a t o\<^sub>L mvar.flow x0a t0 x0 t))) (at t)"
set_get_min_rest: fixes ts :: "'b tree list"   and t' :: "'b tree"   and ts' :: "'b tree list" assumes "get_min_rest ts = (t', ts')"   and "ts \<noteq> []" shows "set ts = Set.insert t' (set ts')"
Ints_minus: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a" assumes "class.ring_1 one times plus zero minus uminus"   and "a \<in> ring_1.Ints one plus zero minus" shows "uminus a \<in> ring_1.Ints one plus zero minus"
reduce_system_matrix_signs_helper: fixes S :: "nat list"   and signs :: "rat list list"   and p :: "real poly"   and subsets :: "nat list list" assumes "\<forall>x. List.member S x \<longrightarrow> x < length signs"   and "p \<noteq> 0" shows "M_mat (take_indices signs S) subsets = take_cols_from_matrix (M_mat signs subsets) S"
hyperpow_hypnat_of_nat: fixes x :: "'a star"   and n :: "nat" shows "x pow hypnat_of_nat n = x ^ n"
cfun_below_iff: fixes f :: "'b \<rightarrow> 'c"   and g :: "'b \<rightarrow> 'c" shows "(f \<sqsubseteq> g) = (\<forall>x. f\<cdot>x \<sqsubseteq> g\<cdot>x)"
Dickson_termE: fixes d :: "'a \<Rightarrow> nat"   and f :: "nat \<Rightarrow> 't"   and m :: "nat" assumes "dickson_grading d"   and "finite (component_of_term ` range f)"   and "pp_of_term ` range f \<subseteq> dgrad_set d m" obtains i :: "nat" and j :: "nat" where "i < j" and "f i adds\<^sub>t f j"
IH: fixes \<sigma> :: "nat \<Rightarrow> 'a"   and s :: "'b net_state"   and t :: "'b net_state" assumes "st = (\<sigma>, SubnetS s t)" shows "P \<sigma> s t"
vector_add_component: fixes x :: "('b, 'c) Finite_Cartesian_Product.vec"   and y :: "('b, 'c) Finite_Cartesian_Product.vec"   and i :: "'c" shows "(x + y) $h i = x $h i + y $h i"
assms_1: shows "(j, i) \<notin> r \<or> j = i"
rel_reflp: fixes R :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "reflp R" shows "reflp (rel_option R)"
find_def: shows "find \<equiv> \<lambda>uu uua. rec_list (\<lambda>_. None) (\<lambda>x xs xsa P. if P x then Some x else xsa P) uua uu"
insert_Diff: fixes a :: "'b"   and A :: "'b set" assumes "a \<in> A" shows "a \<triangleright> A - {a} = A"
Im_power2: fixes x :: "complex" shows "Im (x\<^sup>2) = 2 * Re x * Im x"
cond_unit_F: fixes P :: "'b \<times> 'b \<Rightarrow> bool"   and Q :: "'b \<times> 'b \<Rightarrow> bool" shows "(P \<triangleleft> false \<triangleright>  Q) = Q"
abs_eq: fixes xa :: "int"   and x :: "int" assumes "eq_onp (\<lambda>x. x \<in> {0..<int CARD('b)}) xa xa"   and "eq_onp (\<lambda>x. x \<in> {0..<int CARD('b)}) x x" shows "Abs_mod_ring xa + Abs_mod_ring x = Abs_mod_ring ((xa + x) mod int CARD('b))"
hom_relboundary_empty: fixes p :: "int"   and X :: "'b topology"   and S :: "'b set" shows "hom_relboundary p X S {} = hom_boundary p X S"
cauchy_def: fixes S :: "nat \<Rightarrow> 'b" shows "Cauchy S = (\<forall>e>0. \<exists>N. \<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (S m) (S n) < e)"
abelian_groupI: fixes R :: "('e, 'f) ring_scheme" assumes "\<And>x y. \<lbrakk>x \<in> carrier R; y \<in> carrier R\<rbrakk> \<Longrightarrow> x \<oplus>\<^bsub>R\<^esub> y \<in> carrier R"   and "\<zero>\<^bsub>R\<^esub> \<in> carrier R"   and "\<And>x y z. \<lbrakk>x \<in> carrier R; y \<in> carrier R; z \<in> carrier R\<rbrakk> \<Longrightarrow> x \<oplus>\<^bsub>R\<^esub> y \<oplus>\<^bsub>R\<^esub> z = x \<oplus>\<^bsub>R\<^esub> (y \<oplus>\<^bsub>R\<^esub> z)"   and "\<And>x y. \<lbrakk>x \<in> carrier R; y \<in> carrier R\<rbrakk> \<Longrightarrow> x \<oplus>\<^bsub>R\<^esub> y = y \<oplus>\<^bsub>R\<^esub> x"   and "\<And>x. x \<in> carrier R \<Longrightarrow> \<zero>\<^bsub>R\<^esub> \<oplus>\<^bsub>R\<^esub> x = x"   and "\<And>x. x \<in> carrier R \<Longrightarrow> \<exists>y\<in>carrier R. y \<oplus>\<^bsub>R\<^esub> x = \<zero>\<^bsub>R\<^esub>" shows "abelian_group R"
r'_def: shows "r' = (\<lambda>x. sum_upto (\<lambda>x. ln (real x)) x - x * ln x)"
interface_wiring: fixes cnv_advr :: "('a, 'd, 'g, 'l) converter"   and cnv_send :: "('b, 'e, 'h + 'i, 'm + 'n) converter"   and cnv_recv :: "('c, 'f, 'j + 'k, 'o + 'p) converter"   and res2_send :: "'q \<Rightarrow> 'h \<Rightarrow> ('m \<times> 'q) spmf"   and res2_recv :: "'q \<Rightarrow> 'j \<Rightarrow> ('o \<times> 'q) spmf"   and res2_s :: "'q"   and res1_advr :: "'r \<Rightarrow> 'g \<Rightarrow> ('l \<times> 'r) spmf"   and res1_send :: "'r \<Rightarrow> 'i \<Rightarrow> ('n \<times> 'r) spmf"   and res1_recv :: "'r \<Rightarrow> 'k \<Rightarrow> ('p \<times> 'r) spmf"   and res1_s :: "'r" shows "cnv_advr |\<^sub>= cnv_send |\<^sub>= cnv_recv \<rhd> 1\<^sub>C |\<^sub>= parallel_wiring \<rhd> parallel_resource1_wiring \<rhd> RES (res2_send \<oplus>\<^sub>O res2_recv) res2_s \<parallel> RES (res1_advr \<oplus>\<^sub>O res1_send \<oplus>\<^sub>O res1_recv) res1_s = cnv_advr |\<^sub>= cnv_send |\<^sub>= cnv_recv \<rhd> RES (\<dagger>res1_advr \<oplus>\<^sub>O (res2_send\<dagger> \<oplus>\<^sub>O \<dagger>res1_send) \<oplus>\<^sub>O res2_recv\<dagger> \<oplus>\<^sub>O \<dagger>res1_recv) (res2_s, res1_s)"
mod_ring_uminus64: shows "(mod_ring_rel64 ===> mod_ring_rel64) (uminus_p64 pp) uminus"
abs_trieP_isinP: fixes t :: "trieP"   and ks :: "bool list" shows "isinP t ks = isin (abs_trieP t) ks"
countable_wtPsym: shows "countable {p. wtPsym p}"
assms: shows "F \<hookrightarrow>\<^sub>F G"
vertex_conv_special_coset: fixes S :: "'b set"   and X :: "'b set" assumes "CoxeterComplex S"   and "X \<in> \<Union> (CoxeterComplex.TheComplex S)" shows "\<exists>w s. w \<in> \<langle>S\<rangle> \<and> s \<in> S \<and> X = w +o \<langle>S - {s}\<rangle>"
fact_2: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b" assumes "class.semiring_char_0 one times plus zero" shows "semiring_char_0.fact one plus zero 2 = numeral.numeral one plus (num.Bit0 num.One)"
top_closed: shows "(\<cdot>) \<in> carrier G \<rightarrow> carrier G \<rightarrow> carrier G"
floor_diff_one: fixes x :: "'b" shows "\<lfloor>x - (1::'b)\<rfloor> = \<lfloor>x\<rfloor> - 1"
Fun_comp: fixes g :: "'s"   and f :: "'s" assumes "seq g f" shows "Fun (g \<cdot> f) = restrict (Fun g \<circ> Fun f) (local.set (local.dom f))"
succ_rvar_valuation_id: fixes eq_idx_for_lvar :: "(nat \<times> linear_poly) list \<Rightarrow> nat \<Rightarrow> nat"   and min_lvar_not_in_bounds :: "('a, 'b) state \<Rightarrow> nat option"   and min_rvar_incdec_eq :: "('a, 'b) Direction \<Rightarrow> ('a, 'b) state \<Rightarrow> nat \<times> linear_poly \<Rightarrow> 'a list + nat"   and pivot_and_update :: "nat \<Rightarrow> nat \<Rightarrow> 'b \<Rightarrow> ('a, 'b) state \<Rightarrow> ('a, 'b) state"   and s :: "('a, 'b) state"   and s' :: "('a, 'b) state"   and x :: "nat" assumes "PivotUpdateMinVars eq_idx_for_lvar min_lvar_not_in_bounds min_rvar_incdec_eq pivot_and_update"   and "\<triangle> (\<T> s) \<and> \<diamond> s \<and> \<Turnstile>\<^sub>n\<^sub>o\<^sub>l\<^sub>h\<^sub>s s \<and> \<nabla> s \<and> PivotUpdateMinVars.gt_state eq_idx_for_lvar min_lvar_not_in_bounds min_rvar_incdec_eq pivot_and_update s s' \<and> \<B>\<^sub>i s' = \<B>\<^sub>i s \<and> \<U>\<^sub>c s' = \<U>\<^sub>c s"   and "x \<in> rvars (\<T> s)"   and "x \<in> rvars (\<T> s')" shows "\<langle>\<V> s\<rangle> x = \<langle>\<V> s'\<rangle> x"
assms_2: shows "\<forall>i. num_unique_vars (f i) < num_unique_vars (f (Suc i))"
Cons_1: fixes i :: "nat"   and j :: "nat"   and k :: "nat" assumes "i < j"   and "j < length bs" shows "ord_term_lin.max (term_of_pair (punit.lt (bs ! i), k + j)) (term_of_pair (punit.lt (bs ! j), k + i)) \<in> set (Koszul_syz_sigs_aux bs k)"
bounded_clinear_axioms_def: fixes f :: "'c \<Rightarrow> 'd" shows "bounded_clinear_axioms f \<equiv> \<exists>K. \<forall>x. norm (f x) \<le> norm x * K"
mkIde_as_mkArr: fixes S :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and A :: "'a set" assumes "replete_set_category S"   and "A \<subseteq> Collect (category.terminal S)" shows "set_category_given_img.mkArr S (set_category.some_img S (cardSuc (cmax |UNIV| natLeq))) (cardSuc (cmax |UNIV| natLeq)) A A (\<lambda>x. x) = set_category_given_img.mkIde S (set_category.some_img S (cardSuc (cmax |UNIV| natLeq))) (cardSuc (cmax |UNIV| natLeq)) A"
x_axis: fixes x :: "complex" assumes "is_real x"   and "0 < Re x" shows "P (of_complex x)"
invertible_Units: fixes A :: "'b mat"   and b :: "'c" assumes "invertible_mat A" shows "A \<in> Units (ring_mat TYPE('b) (dim_row A) b)"
elims: fixes x :: "('b, 'c) Combinators"   and y :: "'b set" assumes "first_bothNet x = y" obtains xa :: "('b, 'c) Combinators" and ya :: "('b, 'c) Combinators" where "x = xa \<oplus> ya" and "y = first_bothNet xa"   | "x = DenyAll" and "y = bothNet DenyAll"   | v :: "'b" and va :: "'b" where "x = DenyAllFromTo v__ va__" and "y = bothNet (DenyAllFromTo v__ va__)"   | v :: "'b" and va :: "'b" and vb :: "'c" where "x = AllowPortFromTo v va vb" and "y = bothNet (AllowPortFromTo v va vb)"
simps_3: fixes \<phi> :: "'aprop dfmla"   and env :: "pvar \<Rightarrow> nat" shows "sem (Neg \<phi>) env = neg (sem \<phi> env)"
vimage_vsingleton_in: fixes a :: "V"   and A :: "V"   and b :: "V" assumes "a \<in>\<^sub>\<circ> A" shows "set {\<langle>a, b\<rangle>} `\<^sub>\<circ> A = set {b}"
field_hom_def: fixes hom :: "'a \<Rightarrow> 'b" shows "field_hom hom \<equiv> idom_hom hom"