hcju/isabelleps huggingface dataset:
abs_diff_le_iff: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and x :: "'a"   and a :: "'a"   and r :: "'a" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less_eq (abs (minus x a)) r = (less_eq (minus a r) x \<and> less_eq x (plus a r))"
all_val: fixes p :: "linear_poly" assumes "\<forall>v. \<exists>v'. (\<forall>x\<in>vars p. v' x = v x) \<and> p \<lbrace> v' \<rbrace> = (0::'b)" shows "p = 0"
atLeastAtMost_subseteq_atLeastLessThan_iff: fixes a :: "'c multiset"   and b :: "'c multiset"   and c :: "'c multiset"   and d :: "'c multiset" shows "(subset_mset.atLeastAtMost a b \<subseteq> subset_mset.atLeastLessThan c d) = (a \<subseteq># b \<longrightarrow> c \<subseteq># a \<and> b \<subset># d)"
SbisL_def: fixes cl :: "('test, 'atom, 'choice) cmd list"   and dl :: "('test, 'atom, 'choice) cmd list" shows "SbisL cl dl \<equiv> length cl = length dl \<and> (\<forall>n<length cl. cl ! n \<approx>s dl ! n)"
prems_1: shows "s1 \<subseteq>\<^sub>f s2"
is_processT5_S2a: fixes s :: "'b event list"   and X :: "'b event set"   and P :: "'b process"   and c :: "'b event" assumes "(s, X) \<in> F P"   and "(s, X \<union> {c}) \<notin> F P" shows "(s @ [c], {}) \<in> F P"
is_wt_instance_of_condD'': fixes s :: "('fun, 'var) Term.term"   and t :: "('fun, 'var) Term.term"   and x :: "'var" assumes "wf\<^sub>t\<^sub>r\<^sub>m s"   and "is_wt_instance_of_cond \<Gamma> t s"   and "t = Var x" shows "\<exists>y. s = Var y \<and> \<Gamma> (Var y) = \<Gamma> (Var x)"
Call\<^sub>1: fixes P :: "nat exp prog"   and e :: "nat exp"   and s\<^sub>0 :: "heap \<times> val list \<times> sheap"   and a :: "nat"   and s\<^sub>1 :: "heap \<times> val list \<times> sheap"   and es :: "nat exp list"   and vs :: "val list"   and h\<^sub>2 :: "heap"   and ls\<^sub>2 :: "val list"   and sh\<^sub>2 :: "sheap"   and C :: "char list"   and fs :: "Objects.fields"   and M :: "char list"   and Ts :: "ty list"   and T :: "ty"   and body :: "nat exp"   and D :: "char list"   and ls\<^sub>2' :: "val list"   and e' :: "nat exp"   and h\<^sub>3 :: "heap"   and ls\<^sub>3 :: "val list"   and sh\<^sub>3 :: "sheap" assumes "P \<turnstile>\<^sub>1 \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>addr a,s\<^sub>1\<rangle>"   and "P \<turnstile>\<^sub>1 \<langle>es,s\<^sub>1\<rangle> [\<Rightarrow>] \<langle>map Val vs,(h\<^sub>2, ls\<^sub>2, sh\<^sub>2)\<rangle>"   and "h\<^sub>2 a = \<lfloor>(C, fs)\<rfloor>"   and "P \<turnstile> C sees M, NonStatic :  Ts\<rightarrow>T = body in D"   and "length vs = length Ts"   and "ls\<^sub>2' = Addr a # vs @ replicate (max_vars body) undefined"   and "P \<turnstile>\<^sub>1 \<langle>body,(h\<^sub>2, ls\<^sub>2', sh\<^sub>2)\<rangle> \<Rightarrow> \<langle>e',(h\<^sub>3, ls\<^sub>3, sh\<^sub>3)\<rangle>" shows "P \<turnstile>\<^sub>1 \<langle>e\<bullet>M(es),s\<^sub>0\<rangle> \<Rightarrow> \<langle>e',(h\<^sub>3, ls\<^sub>2, sh\<^sub>3)\<rangle>"
strict_iff_order: fixes b :: "nat list"   and a :: "nat list" shows "(b <\<^sub>v a) = (b \<le>\<^sub>v a \<and> a \<noteq> b)"
tendsto_scaleR: fixes f :: "'b \<Rightarrow> real"   and a :: "real"   and F :: "'b filter"   and g :: "'b \<Rightarrow> 'c"   and b :: "'c" assumes "(f \<longlongrightarrow> a) F"   and "(g \<longlongrightarrow> b) F" shows "((\<lambda>x. f x *\<^sub>R g x) \<longlongrightarrow> a *\<^sub>R b) F"
surjec_ind_bijec: fixes A :: "('c, 'd) Ring_scheme"   and R :: "('e, 'f) Ring_scheme"   and f :: "'c \<Rightarrow> 'e" assumes "Ring A"   and "Ring R"   and "f \<in> rHom A R"   and "surjec\<^bsub>A,R\<^esub> f" shows "bijec\<^bsub>A /\<^sub>r (ker\<^bsub>A,R\<^esub> f),R\<^esub> (f\<degree>\<^bsub>A,R\<^esub>)"
gr0_conv_Suc: fixes n :: "nat" shows "order.greater n 0 = (\<exists>m. n = Suc m)"
vconn_implies_hpar_2: fixes \<mu> :: "'a"   and f :: "'a"   and g :: "'a" assumes "\<guillemotleft>\<mu> : f \<Rightarrow> g\<guillemotright>" shows "trg \<mu> = trg f"
simps: fixes a :: "A" shows "body'\<cdot>(B\<cdot>a) = B\<cdot>A"
less_cSupE: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and y :: "'a"   and X :: "'a set" assumes "class.conditionally_complete_linorder Inf Sup inf less_eq less sup"   and "less y (Sup X)"   and "X \<noteq> {}" obtains x :: "'a" where "x \<in> X" and "less y x"
\<tau>Execst_induct: fixes ci :: "('addr, 'heap) check_instr"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and es :: "'addr expr1 list"   and h :: "'heap"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and xcp :: "'addr option"   and stk'' :: "'addr val list"   and loc'' :: "'addr val list"   and pc'' :: "nat"   and xcp'' :: "'addr option"   and Q :: "'addr val list \<Rightarrow> 'addr val list \<Rightarrow> nat \<Rightarrow> 'addr option \<Rightarrow> bool" assumes "\<tau>Exec_movest ci P t es h (stk, loc, pc, xcp) (stk'', loc'', pc'', xcp'')"   and "\<And>stk' loc' pc' xcp'. \<tau>exec_moves ci P t es h (stk, loc, pc, xcp) (stk', loc', pc', xcp') \<Longrightarrow> Q stk' loc' pc' xcp'"   and "\<And>stk' loc' pc' xcp' stk'' loc'' pc'' xcp''. \<lbrakk>\<tau>Exec_movest ci P t es h (stk, loc, pc, xcp) (stk', loc', pc', xcp'); \<tau>exec_moves ci P t es h (stk', loc', pc', xcp') (stk'', loc'', pc'', xcp''); Q stk' loc' pc' xcp'\<rbrakk> \<Longrightarrow> Q stk'' loc'' pc'' xcp''" shows "Q stk'' loc'' pc'' xcp''"
id0: shows "0\<^sub>v (nc + (nr + nr)) = 0\<^sub>v nc @\<^sub>v 0\<^sub>v nr @\<^sub>v 0\<^sub>v nr"
stable_onI: fixes ds :: "'b set"   and X :: "'a set" assumes "individually_rational_on ds X"   and "stable_no_blocking_on ds X" shows "stable_on ds X"
init_fin_delay_bisimulation_final_base: fixes t :: "'t" shows "delay_bisimulation_final_base (r1.init_fin t) (r2.init_fin t) (init_fin_bisim t) r1.init_fin_\<tau>move r2.init_fin_\<tau>move (\<lambda>(x1, m). r1.init_fin_final x1) (\<lambda>(x2, m). r2.init_fin_final x2)"
orbit_stabiliser_bij: fixes x :: "'b" shows "bij_betw (\<lambda>H. rep H \<odot> x) (carrier (G LMod stabiliser x)) (orbit x)"
Cauchy_uniform: fixes X :: "nat \<Rightarrow> 'a" shows "Cauchy X = cauchy_filter (filtermap X sequentially)"
map_comp: fixes g :: "'e \<Rightarrow> 'd"   and f :: "'f \<Rightarrow> 'e"   and v :: "'f stream" shows "smap g (smap f v) = smap (g \<circ> f) v"
le_append: fixes xs :: "nat list"   and vs :: "nat list"   and ys :: "nat list"   and ws :: "nat list" assumes "length xs = length vs" shows "(xs @ ys \<le>\<^sub>v vs @ ws) = (xs \<le>\<^sub>v vs \<and> ys \<le>\<^sub>v ws)"
strictI: fixes f :: "'b \<rightarrow> 'a"   and x :: "'b" assumes "f\<cdot>x = \<bottom>" shows "f\<cdot>\<bottom> = \<bottom>"
strict_subseq_def: fixes xs :: "'c list"   and ys :: "'c list" shows "strict_subseq xs ys = (xs \<noteq> ys \<and> subseq xs ys)"
smult_subtensor: fixes A :: "'b tensor"   and i :: "nat"   and \<alpha> :: "'b" assumes "dims A \<noteq> []"   and "i < hd (dims A)" shows "\<alpha> \<cdot> subtensor A i = subtensor (\<alpha> \<cdot> A) i"
less_2_cases: fixes n :: "nat" assumes "n < 2" shows "n = 0 \<or> n = Suc 0"
simps: fixes k :: "complex"   and x :: "complex"   and y :: "complex" shows "k *\<^sub>s\<^sub>v (x, y) = (k * x, k * y)"
ag_eq_diffzero: fixes A :: "('d, 'e) aGroup_scheme"   and a :: "'d"   and b :: "'d" assumes "aGroup A"   and "a \<in> carrier A"   and "b \<in> carrier A" shows "(a = b) = (a \<plusminus>\<^bsub>A\<^esub> -\<^sub>a\<^bsub>A\<^esub> b = \<zero>\<^bsub>A\<^esub>)"
assms_2: shows "i \<in> proj (hequiv_rel A H) a"
sl8: shows "- 2 * e * b * f \<in> radical_sqrt"
word_arith_nat_div: fixes a :: "'a word"   and b :: "'a word" shows "a div b = word_of_nat (unat a div unat b)"
nprv_dsjE: fixes F :: "'fmla set"   and \<phi> :: "'fmla"   and \<psi> :: "'fmla"   and \<chi> :: "'fmla" assumes "nprv F (dsj \<phi> \<psi>)"   and "nprv (insert \<phi> F) \<chi>"   and "nprv (insert \<psi> F) \<chi>"   and "F \<subseteq> fmla"   and "finite F"   and "\<phi> \<in> fmla"   and "\<psi> \<in> fmla"   and "\<chi> \<in> fmla" shows "nprv F \<chi>"
weak_coinduct: fixes P :: "pi"   and Q :: "pi"   and X :: "(pi \<times> pi) set" assumes "(P, Q) \<in> X"   and "\<And>P Q. (P, Q) \<in> X \<Longrightarrow> P \<leadsto>\<^sub>e[X] Q \<and> (Q, P) \<in> X" shows "P \<sim>\<^sub>e Q"
x_def: shows "x = \<langle>a, b\<rangle>"
cInf_lower: fixes x :: "'a multiset"   and X :: "'a multiset set" assumes "x \<in> X"   and "subset_mset.bdd_below X" shows "Inf X \<subseteq># x"
mat_arc_posI_def: fixes ap :: "'b \<Rightarrow> bool"   and A :: "'b mat" shows "mat_arc_posI ap A \<equiv> ap (A $$ (0, 0))"
ord_resolve_sound: fixes CAs :: "'a literal multiset list"   and DA :: "'a literal multiset"   and AAs :: "'a multiset list"   and As :: "'a list"   and E :: "'a literal multiset"   and I :: "'a set" assumes "gr.ord_resolve CAs DA AAs As E"   and "I \<Turnstile>m mset CAs"   and "I \<Turnstile> DA" shows "I \<Turnstile> E"
P_conforms: shows "path_conforms_with_strategy p P \<sigma>"
not_add_mem_right: fixes x :: "V"   and y :: "V" shows "x + y \<notin> elts x"
sup_option_def: fixes x :: "'a option"   and y :: "'a option" shows "x \<squnion> y = (case x of None \<Rightarrow> y | Some x' \<Rightarrow> case y of None \<Rightarrow> x | Some y \<Rightarrow> Some (x' \<squnion> y))"
map_def: fixes f :: "'b \<times> 'b" shows "B.VV.map f = (if B.VV.arr f then f else B.VV.null)"
inorder_upd: fixes t :: "(('c \<times> 'd) \<times> color) tree"   and x :: "'c"   and y :: "'d" assumes "sorted1 (Tree2.inorder t)" shows "Tree2.inorder (upd x y t) = upd_list x y (Tree2.inorder t)"
nths_minus_upt_eq_drop: fixes l :: "'a list"   and n :: "nat" shows "nths l (- {..<n}) = drop n l"
shortest_path_least: fixes w :: "'a \<times> 'a \<Rightarrow> enat"   and s :: "'a"   and p :: "enat list"   and u :: "'a" assumes "WGraph.path w s p u" shows "WGraph.\<delta> w s u \<le> sum_list p"
A_1: shows "A \<subseteq> C \<and> A \<subseteq> f A"
lnth_0_conv_lhd: fixes xs :: "'b llist" assumes "\<not> lnull xs" shows "lnth xs 0 = lhd xs"
ipc_weakly_step_consistent: fixes u :: "partition_id_t"   and s1 :: "state_t"   and s2 :: "state_t"   and tid :: "thread_id_t"   and ipt :: "int_point_t"   and dir :: "ipc_direction_t"   and stage :: "ipc_stage_t"   and partner :: "thread_id_t"   and page :: "page_t" assumes "vpeq u s1 s2"   and "vpeq (Step.partition tid) s1 s2"   and "atomic_step_invariant s1"   and "atomic_step_invariant s2"   and "atomic_step_precondition s1 tid ipt"   and "atomic_step_precondition s1 tid ipt"   and "ipt = SK_IPC dir stage partner page" shows "vpeq u (atomic_step_ipc tid dir stage partner page s1) (atomic_step_ipc tid dir stage partner page s2)"
assms: shows "\<forall>i\<in>frees t. i < k \<or> k + n \<le> i"
assms_2: shows "\<S>\<bullet>\<C> \<turnstile> [$C v1, $C v2, $e] : ts _> ts'"
scc_of_is_scc: fixes E :: "('a \<times> 'a) set"   and v :: "'a" shows "is_scc E (scc_of E v)"
qtable_nonempty_empty_table: fixes n :: "nat"   and R :: "'a option list \<Rightarrow> bool"   and P :: "'a option list \<Rightarrow> bool"   and X :: "'a option list set"   and x :: "'a option list" assumes "qtable n {} R P X"   and "x \<in> X" shows "qtable n {} R (\<lambda>v. \<not> P v) empty_table"
lang_flatten_PLUS: fixes X :: "'b rexp set" assumes "finite X" shows "lang (flatten PLUS X) = \<Union> (lang ` X)"
preserves_dom: fixes f :: "'a \<times> 'a" assumes "B.VV.arr f" shows "local.dom (cmp\<^sub>U\<^sub>P f) = HoUP_UP.map (B.VV.dom f)"
dsafe: shows "dsafe ($f fid1 (\<lambda>i. Const 0))"
L_ss: fixes q :: "'Q"   and l :: "'L" assumes "L q l" shows "q \<in> V"
le_list_trans: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b list"   and ys :: "'b list"   and zs :: "'b list" assumes "order r"   and "xs [\<sqsubseteq>\<^bsub>r\<^esub>] ys"   and "ys [\<sqsubseteq>\<^bsub>r\<^esub>] zs" shows "xs [\<sqsubseteq>\<^bsub>r\<^esub>] zs"
simps_2: fixes n :: "nat" shows "ns (Suc n) = newton_step (ns n)"
inf_commute: fixes inf :: "'a::type \<Rightarrow> 'a::type \<Rightarrow> 'a::type"   and less_eq :: "'a::type \<Rightarrow> 'a::type \<Rightarrow> bool"   and less :: "'a::type \<Rightarrow> 'a::type \<Rightarrow> bool"   and x :: "'a::type"   and y :: "'a::type" assumes "class.semilattice_inf (inf::'a::type \<Rightarrow> 'a::type \<Rightarrow> 'a::type) (less_eq::'a::type \<Rightarrow> 'a::type \<Rightarrow> bool) (less::'a::type \<Rightarrow> 'a::type \<Rightarrow> bool)" shows "(inf::'a::type \<Rightarrow> 'a::type \<Rightarrow> 'a::type) (x::'a::type) (y::'a::type) = inf y x"
prime_nprod_exc: fixes P :: "'a set"   and n :: "nat"   and f :: "nat \<Rightarrow> 'a" assumes "prime_ideal R P"   and "\<forall>i\<le>n. f i \<in> carrier R"   and "\<forall>l\<le>n. f l \<notin> P" shows "e\<Pi>\<^bsub>R,n\<^esub> f \<notin> P"
intro: fixes M :: "('a, 'b) message set"   and m :: "'a delete_message" assumes "consistent (M \<union> {Delete m})" shows "integrate_insert_commute_delete_axioms M m"
is_borel_def: fixes f :: "'b \<Rightarrow> 'c"   and M :: "'b measure" shows "Borel_Space.is_borel f M = (f \<in> borel_measurable M)"
omega_sum_lex_less_iff_cases: fixes ms :: "nat list"   and m :: "nat"   and ns :: "nat list"   and n :: "nat" shows "(((length ms, omega_sum (m # ms)), length ns, omega_sum (n # ns)) \<in> less_than <*lex*> VWF) = (length ms < length ns \<or> length ms = length ns \<and> m < n \<or> m = n \<and> ((length ms, omega_sum ms), length ns, omega_sum ns) \<in> less_than <*lex*> VWF)"
simps_8: fixes f1 :: "'c"   and f2 :: "'c"   and f3 :: "'c" shows "(case TernaryFalse of TernaryTrue \<Rightarrow> f1 | TernaryFalse \<Rightarrow> f2 | TernaryUnknown \<Rightarrow> f3) = f2"
perp_right_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.Perp Bet Cong A B C D" shows "Tarski_neutral_dimensionless.Perp Bet Cong A B D C"
OUadmitFO_def: fixes \<sigma> :: "'b \<Rightarrow> ('c, 'd) trm"   and \<theta> :: "('c + 'b, 'd) ODE"   and U :: "('d + 'd) set" shows "OUadmitFO \<sigma> \<theta> U = ((\<Union>i\<in>{i. Inl (Inr i) \<in> SIGO \<theta>}. FVT (\<sigma> i)) \<inter> U = {})"
no_output_none: fixes e :: "((nat \<times> nat) \<times> transition) fset"   and r :: "nat \<Rightarrow>f value option"   and p :: "value option list"   and t :: "(String.literal \<times> value list) stream" shows "nxt (alw (output_eq [])) (make_full_observation e None r p t)"
prems: shows "set (a # p1) \<subseteq> carrier R"
greaterThanLessThan_empty: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and l :: "'c"   and k :: "'c" assumes "class.order less_eq less"   and "less_eq l k" shows "ord.greaterThanLessThan less k l = {}"
cut_ge_Int_left: fixes A :: "'a set"   and B :: "'a set"   and t :: "'a" shows "(A \<inter> B) \<down>\<ge> t = A \<down>\<ge> t \<inter> B"
simps_1: shows "ext0.Bot + ext0.Bot = ext0.Val (0::'a)"
reaching_path_without_repetition: fixes M :: "('d, 'e, 'f) FSM"   and q2 :: "'f"   and q1 :: "'f" assumes "well_formed M"   and "q2 \<in> reachable M q1"   and "q1 \<in> nodes M" shows "\<exists>p. path M p q1 \<and> target p q1 = q2 \<and> distinct (q1 # states p q1)"
arrI: fixes f :: "'a" assumes "left \<omega> f" shows "S.arr f"
eucl_rel_int_iff: fixes k :: "int"   and l :: "int"   and q :: "int"   and r :: "int" shows "eucl_rel_int k l (q, r) = (k = l * q + r \<and> (if 0 < l then 0 \<le> r \<and> r < l else if l < 0 then l < r \<and> r \<le> 0 else q = 0))"
ccw_half_segments_of_aform_1: fixes xi :: "real \<times> real"   and xj :: "real \<times> real"   and X :: "(real \<times> real) \<times> (real \<times> real) pdevs"   and c :: "real \<times> real"   and d :: "real \<times> real" assumes "(xi, xj) \<in> set (half_segments_of_aform X)"   and "(c, d) \<in> set (half_segments_of_aform X)" shows "ccw xi xj c"
l_get_scdom_component_remove_child\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_get_scdom_component_remove_child\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_owner_document heap_is_wellformed parent_child_rel DocumentClass.type_wf DocumentClass.known_ptr DocumentClass.known_ptrs get_scdom_component is_strongly_scdom_component_safe is_weakly_scdom_component_safe get_dom_component is_strongly_dom_component_safe is_weakly_dom_component_safe to_tree_order get_parent get_parent_locs get_child_nodes get_child_nodes_locs get_root_node get_root_node_locs get_ancestors get_ancestors_locs get_disconnected_nodes get_disconnected_nodes_locs get_element_by_id get_elements_by_tag_name set_child_nodes set_child_nodes_locs set_disconnected_nodes set_disconnected_nodes_locs remove_child remove_child_locs remove"
sets_lebesgue_on_continuous_image: fixes S :: "'c set"   and X :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "S \<in> sets lebesgue"   and "X \<in> sets (lebesgue_on S)"   and "continuous_on S f"   and "\<And>T. \<lbrakk>negligible T; T \<subseteq> S\<rbrakk> \<Longrightarrow> negligible (f ` T)" shows "f ` X \<in> sets (lebesgue_on (f ` S))"
f_Exec_Comp_Stream_Acc_LocalState_Init_def: fixes k :: "nat"   and localState :: "'e \<Rightarrow> 'd"   and trans_fun :: "'f message_af \<Rightarrow> 'e \<Rightarrow> 'e"   and xs :: "'f message_af list"   and c :: "'e" shows "f_Exec_Comp_Stream_Acc_LocalState_Init k localState trans_fun xs c \<equiv> localState c # f_Exec_Comp_Stream_Acc_LocalState k localState trans_fun xs c"
IH': fixes I :: "('a, 'b, 'c) interp"   and J :: "('a, 'b, 'c) interp"   and \<nu> :: "(real, 'c) vec \<times> (real, 'c) vec"   and \<nu>' :: "(real, 'c) vec \<times> (real, 'c) vec" assumes "Iagree I J (SIGF \<phi>)"   and "Vagree \<nu> \<nu>' (FVF \<phi>)" shows "(\<nu> \<in> fml_sem I \<phi>) = (\<nu>' \<in> fml_sem J \<phi>)"
ideal_points_def: fixes H :: "circline" shows "ideal_points H = circline_intersection H unit_circle"
preserves_section_retraction: fixes e :: "'a"   and m :: "'a" assumes "S.ide (e \<cdot>\<^sub>S m)" shows "S.ide (R e \<cdot>\<^sub>S R m)"
assms: fixes f :: "nat \<Rightarrow> 'a" assumes "\<forall>n. f n \<in> S" shows "\<exists>l\<in>S. \<exists>r. strict_mono r \<and> (f \<circ> r) \<longlonglongrightarrow> l"
ipurge_tr_ref_aux_failures_general: fixes P :: "'e process"   and I :: "('f \<times> 'f) set"   and D :: "'e \<Rightarrow> 'f"   and xs :: "'e list"   and ys :: "'e list"   and Y :: "'e set"   and U :: "'f set" assumes "secure P I D"   and "(xs @ ys, Y) \<in> failures P" shows "(xs @ ipurge_tr_aux I D U ys, ipurge_ref_aux I D U ys Y) \<in> failures P"
elims: fixes x :: "int poly \<times> rat \<times> rat"   and y :: "real_alg_2" assumes "local.root_1 x = y" obtains p :: "int poly" and l :: "rat" and r :: "rat" where "x = (p, l, r)" and "y = (if n = 0 \<or> r = 0 then Rational 0 else if 0 < r then local.root_pos_1 (p, l, r) else uminus_2 (local.root_pos_1 (uminus_1 (p, l, r))))"
z_3: shows "cosh y - cosh x = (y - x) * sinh z"
3: shows "asymmetric t \<and> t \<le> - - y \<and> t\<^sup>T\<^sup>\<star> * (x * top) = (- - y)\<^sup>T\<^sup>\<star> * (x * top)"
case_2: fixes f1 :: "'d \<Rightarrow> 'd \<Rightarrow> 'c"   and f2 :: "'e \<Rightarrow> 'c"   and x2 :: "'e" shows "(case Leaf x2 of Branch x xa \<Rightarrow> f1 x xa | Leaf x \<Rightarrow> f2 x) = f2 x2"
fits_Addr_RefT: fixes G :: "prog"   and s :: "st"   and a :: "loc"   and t :: "ref_ty" shows "G,s\<turnstile>Addr a fits RefT t = G\<turnstile>obj_ty (the (heap s a))\<preceq>RefT t"
intro: fixes M :: "'b measure" assumes "emeasure M (space M) \<noteq> top" shows "finite_measure_axioms M"
shift_def: fixes cl :: "('test, 'atom, 'choice) cmd list"   and n :: "nat" shows "local.shift cl n \<equiv> (`) ((+) (brnL cl n))"
supplement1_Jacobi': fixes n :: "int" assumes "odd n"   and "1 < n" shows "Jacobi (- 1) n = (if n mod 4 = 1 then 1 else - 1)"
N1_enabled_at_g: shows "\<turnstile> pc1 = #g \<longrightarrow> Enabled (<N1>_(x, y, sem, pc1, pc2))"
higher_zero: fixes t :: "'t" shows "higher 0 t = 0"
weakBisimWeakCoinductAux: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and X :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set" assumes "(\<Psi>, P, Q) \<in> X"   and "\<And>\<Psi> P Q. (\<Psi>, P, Q) \<in> X \<Longrightarrow> \<Psi> \<rhd> P \<lessapprox><X> Q \<and> \<Psi> \<rhd> P \<leadsto><X> Q \<and> (\<forall>\<Psi>'. (\<Psi> \<otimes> \<Psi>', P, Q) \<in> X) \<and> (\<Psi>, Q, P) \<in> X" shows "\<Psi> \<rhd> P \<approx> Q"
integrable_count_space_nat_iff: fixes f :: "nat \<Rightarrow> 'd" shows "integrable (count_space UNIV) f = summable (\<lambda>x. norm (f x))"
app_def: fixes s :: "('c, 'd) FingerTree"   and t :: "('c, 'd) FingerTree" shows "app s t = Abs_FingerTree (FingerTreeStruc.app (Rep_FingerTree s) (Rep_FingerTree t))"
PiM_empty: fixes M :: "'d \<Rightarrow> 'e measure" shows "Pi\<^sub>M {} M = count_space {\<lambda>_. undefined}"
generate_in_carrier: fixes H :: "'a set"   and h :: "'a" assumes "H \<subseteq> carrier G"   and "h \<in> generate G H" shows "h \<in> carrier G"
regular_linorder_matrix: fixes f :: "'c \<times> 'c \<Rightarrow> 'd" shows "matrix_p_algebra.regular f = (\<forall>e. f e = bot \<or> f e = top)"
relChain_def: fixes r :: "('a \<times> 'a) set"   and As :: "'a \<Rightarrow> 'b" shows "relChain r As \<equiv> \<forall>i j. (i, j) \<in> r \<longrightarrow> As i \<le> As j"
last_in_set: fixes as :: "'a list" assumes "as \<noteq> []" shows "last as \<in> set as"
assms_5: shows "\<forall>b\<in>basis. line_integral_exists F {b} \<gamma>2"
lhs: shows "((x + a) mod q * b + q - (a * b + q - r) mod q) mod q = ((x + a) mod q * b + (q - (a * b + (q - r)) mod q)) mod q"
dsg2: fixes x :: "'a"   and y :: "'a" shows "ad (ad (x \<cdot> ad (ad y))) = ad (ad (x \<cdot> y))"
assms_3: fixes y :: "('a, 'b) pre_digraph" assumes "x \<noteq> y"   and "subgraph x y"   and "subgraph y G" shows "\<not> P y"
Box_desc_encode_1: fixes v :: "i"   and \<phi> :: "\<Pi>\<^sub>1 \<Rightarrow> \<o>"   and G :: "\<Pi>\<^sub>1" shows "[\<^bold>\<box>\<phi> G \<^bold>\<rightarrow> \<lbrace>\<^bold>\<iota>x. \<lparr>A!,x\<^sup>P\<rparr> \<^bold>& (\<^bold>\<forall>F. \<lbrace>x\<^sup>P,F\<rbrace> \<^bold>\<equiv> \<phi> F),G\<rbrace> in v]"
lang_UP: fixes r :: "'b rexp" shows "lang (UP r) = SUPSEQ (lang r)"
are_endpoints_in_S_swap_34: fixes p :: "proj2"   and q :: "proj2"   and a :: "proj2"   and b :: "proj2" assumes "are_endpoints_in_S p q a b" shows "are_endpoints_in_S p q b a"
S2_le_S1: fixes n :: "nat" shows "{(i, j). i + j < n} \<subseteq> {..<n} \<times> {..<n}"
eval_fds_logderiv_zeta_2: fixes s :: "complex" assumes "1 < Re s" shows "deriv zeta s / zeta s = - (\<Sum>\<^sub>ap | prime p. complex_of_real (ln (real p)) / (of_nat p powr s - 1))"
apartments_have_many_foldings3: fixes A :: "'a set set"   and C :: "'a set"   and D :: "'a set" assumes "A \<in> \<A>"   and "chamber C"   and "chamber D"   and "C \<sim> D"   and "C \<noteq> D"   and "C \<in> A"   and "D \<in> A" shows "\<exists>f g. OpposedThinChamberComplexFoldings A f g C \<and> D = g ` C"
show_law_rat: fixes r :: "rat" shows "show_law showsp_rat r"