hcju/isabelleps huggingface dataset:
F: fixes x :: "real" assumes "min a b \<le> x"   and "x \<le> max a b" shows "(F has_vector_derivative f x) (at x within {min a b..max a b})"
fin_substn_nrename: fixes M :: "trm"   and x :: "name"   and y :: "name"   and P :: "trm"   and c :: "coname" assumes "fin M x"   and "x \<noteq> y"   and "x \<sharp> P" shows "M[x\<turnstile>n>y]{y:=<c>.P} = Cut <c>.P x.M{y:=<c>.P}"
bfs2_op_def: fixes am :: "nat \<Rightarrow> nat list"   and cf :: "nat \<times> nat \<Rightarrow> int" shows "bfs2_op am cf \<equiv> Graph.bfs2 cf (rg_succ2 am cf) s t"
lcoeff_closed: fixes p :: "nat \<Rightarrow> 'a" assumes "p \<in> carrier P" shows "lcoeff p \<in> carrier R"
convex_explicit: fixes S :: "'c set" shows "convex S = (\<forall>t u. finite t \<and> t \<subseteq> S \<and> (\<forall>x\<in>t. 0 \<le> u x) \<and> sum u t = 1 \<longrightarrow> (\<Sum>x\<in>t. u x *\<^sub>R x) \<in> S)"
assms_5: fixes i :: "'a" assumes "i \<in> A" shows "\<exists>x\<in>B. g x \<le>\<^sub>o\<^sub>w f i"
injectivity: fixes x :: "'a"   and y :: "'a" assumes "[:x:] = [:y:]" shows "x = y"
image_mmap_subset_hd_F: fixes n :: "nat" shows "mmap ` {n..} \<subseteq> hd (F (Suc n))"
Sim: shows "P \<leadsto>\<^sup>^<Rel> Q"
assms_1: shows "h \<turnstile> get_elements_by_tag_name ptr name \<rightarrow>\<^sub>r results"
simps_6: fixes K\<^sub>1 :: "clock"   and K\<^sub>2 :: "clock" shows "\<lbrakk> K\<^sub>1 weakly precedes K\<^sub>2 \<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L = {\<rho>. \<forall>n. #\<^sub>\<le> \<rho> K\<^sub>2 n \<le> #\<^sub>\<le> \<rho> K\<^sub>1 n}"
add_nonpos_neg: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" assumes "class.ordered_comm_monoid_add plus zero less_eq less"   and "less_eq a zero"   and "less b zero" shows "less (plus a b) zero"
ltakeWhile_K_False: fixes xs :: "'c llist" shows "ltakeWhile (\<lambda>_. False) xs = LNil"
intro: fixes X :: "'c set"   and is_open :: "'c set \<Rightarrow> bool"   and S :: "'c set" assumes "topological_space X is_open"   and "ind_topology_axioms X S" shows "ind_topology X is_open S"
control_vars_are_A_vars: shows "\<C>\<^sub>C = Eg2_var\<^sub>C_of_Eg1 ` \<C>"
vector_transpose_matrix: fixes x :: "('b, 'd) Finite_Cartesian_Product.vec"   and A :: "(('b, 'd) Finite_Cartesian_Product.vec, 'c) Finite_Cartesian_Product.vec" shows "x v* Finite_Cartesian_Product.transpose A = A *v x"
assms: shows "\<Gamma>,\<gamma>,p\<turnstile> rs1 \<Rightarrow>\<^sub>r Decision X"
rep_eq: fixes x :: "'b ratfps" shows "fps_of_ratfps x = fps_of_poly (numerator (Rep_ratfps x)) / fps_of_poly (denominator (Rep_ratfps x))"
terminates_block: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and bdy :: "('e, 'd, 'f) com"   and init :: "'e \<Rightarrow> 'e"   and s :: "'e"   and c :: "'e \<Rightarrow> 'e \<Rightarrow> ('e, 'd, 'f) com"   and return :: "'e \<Rightarrow> 'e \<Rightarrow> 'e" assumes "\<Gamma>\<turnstile>bdy \<down> Normal (init s)"   and "\<forall>t. \<Gamma>\<turnstile> \<langle>bdy,Normal (init s)\<rangle> \<Rightarrow> Normal t \<longrightarrow> \<Gamma>\<turnstile>c s t \<down> Normal (return s t)" shows "\<Gamma>\<turnstile>block init bdy return c \<down> Normal s"
create_closure: fixes x :: "'c \<Longrightarrow> 'd"   and v :: "'c" assumes "weak_lens x" shows "create\<^bsub>x\<^esub> v \<in> \<S>\<^bsub>x\<^esub>"
clear: shows "set (take_indices (cols (M_mat_R signs subsets)) (find_nonzeros_from_input_vec (solve_for_lhs_R p qs subsets (M_mat_R signs subsets)))) \<subseteq> set (cols (M_mat_R signs subsets))"
monom_degree: fixes x :: "'c"   and m :: "'d monom" shows "eval_monom (\<lambda>_. x) m = x ^ monom_degree m"
tameEnumFilter_subseteq_TameEnum: fixes p :: "nat"   and t :: "(nat, nat list list list) trie" assumes "tameEnumFilter p = Some t" shows "set_tries t \<subseteq> fgraph ` TameEnum\<^bsub>p\<^esub>"
assms_22: shows "InvariantVarsF (getF state) F0 Vbl"
complex_eq_iff: fixes x :: "complex"   and y :: "complex" shows "(x = y) = (Re x = Re y \<and> Im x = Im y)"
exists_maximal_superset: fixes C :: "'b fm set set"   and V :: "'b fm set" assumes "consistency C"   and "finite_char C"   and "V \<in> C" obtains W :: "'b fm set" where "V \<subseteq> W" and "W \<in> C" and "maximal W C"
cod_is_obj: fixes x :: "'arr" assumes "x \<in> Arr" shows "C x \<in> Obj"
ll_def: shows "ll \<equiv> if degree q < degree p then at 0 else if degree q = degree p then nhds (lead_coeff q / lead_coeff p) else if 0 < sgn_neg_inf q * sgn_neg_inf p then at_top else at_bot"
subset_UnE: fixes C :: "'a set"   and A :: "'a set"   and B :: "'a set" assumes "C \<subseteq> A \<union> B" obtains A' :: "'a set" and B' :: "'a set" where "A' \<subseteq> A" and "B' \<subseteq> B" and "C = A' \<union> B'"
rev_nth_Cons: fixes xs :: "'b list"   and v :: "nat"   and y :: "'b"   and x :: "'b" assumes "xs !. v = Some y" shows "(x # xs) !. v = Some y"
dist_commute_lessI: fixes dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and y :: "'a"   and x :: "'a"   and e :: "real" assumes "class.metric_space dist uniformity open"   and "dist y x < e" shows "dist x y < e"
rel_fundefs_Some1: fixes f :: "'a \<Rightarrow> ('b, 'c, 'd, 'op) Std.instr fundef option"   and g :: "'a \<Rightarrow> ('b, 'c, 'd, 'op, 'opinl) Inca.instr fundef option"   and x :: "'a"   and y :: "('b, 'c, 'd, 'op) Std.instr fundef" assumes "rel_fundefs f g"   and "f x = Some y" shows "\<exists>z. g x = Some z \<and> rel_fundef (\<lambda>x y. x = norm_instr y) y z"
create_character_data_is_strongly_dom_component_safe_step: fixes h :: "(_) heap"   and document_ptr :: "(_) document_ptr"   and "text" :: "char list"   and h' :: "(_) heap"   and ptr :: "(_) object_ptr"   and getter :: "(_) RObject_ext \<Rightarrow> 'l" assumes "heap_is_wellformed h"   and "type_wf h"   and "known_ptrs h"   and "h \<turnstile> create_character_data document_ptr text \<rightarrow>\<^sub>h h'"   and "ptr \<notin> set |h \<turnstile> get_scdom_component (cast\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r document_ptr)|\<^sub>r"   and "ptr \<noteq> cast\<^sub>c\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>_\<^sub>d\<^sub>a\<^sub>t\<^sub>a\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r |h \<turnstile> create_character_data document_ptr text|\<^sub>r" shows "preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t ptr getter) h h'"
that_2: fixes x :: "'a" assumes "x \<in> T" shows "f x = g x"
p': shows "is_path_undir (add_edge a w b G) x p' v'"
induct: fixes P :: "MIR.num \<Rightarrow> bool"   and a0 :: "MIR.num" assumes "\<And>c. P (C c)"   and "\<And>n. P (Bound n)"   and "\<And>n c a. P (Add (Mul c (Bound n)) a) \<Longrightarrow> P (CN n c a)"   and "\<And>c a b. P (Add (Mul c (Floor a)) b) \<Longrightarrow> P (CF c a b)"   and "\<And>a. P a \<Longrightarrow> P (Neg a)"   and "\<And>a b. \<lbrakk>P a; \<And>x xa y. \<lbrakk>x = zsplit0 a; (xa, y) = x\<rbrakk> \<Longrightarrow> P b\<rbrakk> \<Longrightarrow> P (Add a b)"   and "\<And>a b. \<lbrakk>P a; \<And>x xa y. \<lbrakk>x = zsplit0 a; (xa, y) = x\<rbrakk> \<Longrightarrow> P b\<rbrakk> \<Longrightarrow> P (Sub a b)"   and "\<And>i a. P a \<Longrightarrow> P (Mul i a)"   and "\<And>a. P a \<Longrightarrow> P (Floor a)" shows "P a0"
lmap_mono: fixes f :: "'c \<Rightarrow> 'd"   and xs :: "'c llist" shows "mono_llist (\<lambda>lmap. case xs of LNil \<Rightarrow> LNil | LCons x xs \<Rightarrow> LCons (f x) (lmap xs))"
a: shows "((0::'a) \<le> a \<or> a \<le> (0::'a)) \<and> ((0::'a) \<le> b \<or> b \<le> (0::'a))"
at_within_open_subset: fixes "open" :: "'b set \<Rightarrow> bool"   and a :: "'b"   and S :: "'b set"   and T :: "'b set" assumes "class.topological_space open"   and "a \<in> S"   and "open S"   and "S \<subseteq> T" shows "topological_space.at_within open a T = topological_space.at_within open a UNIV"
less_eq_def: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.plus_ord plus less_eq less" shows "less_eq x y = (plus x y = y)"
preserves_dom: fixes f :: "'c" assumes "Cop.arr f" shows "S.dom (\<Psi>.map f) = Cones.map (Cop.dom f)"
cont_intersect2: fixes y :: "'b ttree" shows "cont ((\<inter>\<inter>) y)"
nv2C_def: fixes c :: "('b, 'psym) lit list" shows "nv2C c \<equiv> \<Union> (set (map nv2L c))"
length_compl_induct: fixes P :: "'d list \<Rightarrow> bool"   and l :: "'d list" assumes "P []"   and "\<And>e l. (\<And>ll. length ll \<le> length l \<Longrightarrow> P ll) \<Longrightarrow> P (e # l)" shows "P l"
elims: fixes x :: "('b, 'c) RBT_Impl.rbt"   and y :: "'b" assumes "rbt_min_opt x = y" obtains c :: "color" and k :: "'b" and v :: "'c" and rt :: "('b, 'c) RBT_Impl.rbt" where "x = Branch c__ rbt.Empty k__ v__ rt__" and "y = k__"   | c :: "color" and lc :: "color" and llc :: "('b, 'c) RBT_Impl.rbt" and lk :: "'b" and lv :: "'c" and lrt :: "('b, 'c) RBT_Impl.rbt" and k :: "'b" and v :: "'c" and rt :: "('b, 'c) RBT_Impl.rbt" where "x = Branch c (Branch lc llc lk lv lrt) k v rt" and "y = rbt_min_opt (Branch lc llc lk lv lrt)"   | "x = rbt.Empty" and "y = undefined"
AE_upper_bound_inf: fixes F :: "'b \<Rightarrow> real"   and G :: "'b \<Rightarrow> real"   and M :: "'b measure" assumes "\<And>e. 0 < e \<Longrightarrow> AE x in M. F x \<le> G x + e" shows "AE x in M. F x \<le> G x"
Rats_abs_nat_div_natE: fixes x :: "real" assumes "x \<in> \<rat>" obtains m :: "nat" and n :: "nat" where "n \<noteq> 0" and "\<bar>x\<bar> = real m / real n" and "coprime m n"
fmdom'_notD: fixes x :: "'d"   and m :: "('d, 'e) fmap" assumes "x \<notin> fmdom' m" shows "fmlookup m x = None"
zero_blinfun_def: shows "0 \<equiv> Blinfun (\<lambda>x. 0::'b)"
g4: shows "nextsuffix (t |\<^sub>s m) = t |\<^sub>s m |\<^sub>s Suc (nextnat (t |\<^sub>s m) - 1)"
arc_set_balanced_def: fixes u :: "'a"   and A :: "'b set"   and v :: "'a" shows "arc_set_balanced u A v \<equiv> if u = v then \<forall>w\<in>verts G. arc_set_balance w A = 0 else (\<forall>w\<in>verts G. w \<noteq> u \<and> w \<noteq> v \<longrightarrow> arc_set_balance w A = 0) \<and> arc_set_balance u A = - 1 \<and> arc_set_balance v A = 1"
le_iff_inf: fixes x :: "'a"   and y :: "'a" shows "(x \<le> y) = (x \<sqinter> y = x)"
assms_2: shows "queue_\<alpha> queue \<noteq> []"
image_Suc_atLeastAtMost: fixes i :: "nat"   and j :: "nat" shows "Suc ` {i..j} = {Suc i..Suc j}"
clauseEntailsLiteral_def: fixes clause :: "Literal list"   and literal :: "Literal" shows "clauseEntailsLiteral clause literal \<equiv> \<forall>valuation. model valuation clause \<longrightarrow> literalTrue literal valuation"
simps_3: fixes \<phi> :: "'a ltln"   and \<psi> :: "'a ltln" shows "rewrite_syn_imp (\<phi> U\<^sub>n \<psi>) = (if \<phi> \<turnstile>\<^sub>s \<psi> then rewrite_syn_imp \<psi> else if not\<^sub>n \<phi> \<turnstile>\<^sub>s \<psi> then mk_finally (rewrite_syn_imp \<psi>) else mk_until (rewrite_syn_imp \<phi>) (rewrite_syn_imp \<psi>))"
mod_mult_left_eq: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and euclidean_size :: "'b \<Rightarrow> nat"   and a :: "'b"   and c :: "'b"   and b :: "'b" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size" shows "modulo (times (modulo a c) b) c = modulo (times a b) c"
map_drop_set_def: fixes A :: "'c set" shows "map_drop_set A = map_filter (\<lambda>a. a \<notin> A)"
convex_lincomb_list_def: fixes c :: "nat \<Rightarrow> 'a"   and Vs :: "'a vec list"   and b :: "'a vec" shows "n.convex_lincomb_list c Vs b = (n.nonneg_lincomb_list c Vs b \<and> sum c {0..<length Vs} = (1::'a))"
assignment_rhs_append: fixes S :: "('a, 'b) strand_step list"   and S' :: "('a, 'b) strand_step list" shows "assignment_rhs\<^sub>s\<^sub>t (S @ S') = assignment_rhs\<^sub>s\<^sub>t S \<union> assignment_rhs\<^sub>s\<^sub>t S'"
pow_un2: shows "\<mu> \<circ> \<eta> = id"
zcount_pointstamps_cm_all': fixes c :: "('b, 't, 'c) Propagate.configuration_scheme"   and \<Delta> :: "('b \<times> 't) zmultiset"   and loc :: "'b"   and x :: "'t" shows "zcount (c_pts (cm_all' c \<Delta>) loc) x = zcount (c_pts c loc) x + zcount \<Delta> (loc, x)"
EmptyMap_the: fixes x :: "'d"   and S :: "'d set" assumes "x \<in> S" shows "the (EmptyMap S x) = {}"
imp_simps_5: fixes P :: "fm" shows "imp P P = T"
fromEnum_unat: fixes x :: "'b word" shows "fromEnum x = unat x"
add_canc1: fixes x :: "'c"   and y :: "'c" assumes "D x y" shows "rquot (x \<oplus> y) x = y"
semiring_norm_170: fixes v :: "num"   and w :: "num"   and y :: "'b" shows "- numeral v * (numeral w * y) = - numeral (v * w) * y"
strict_implies_order: fixes a :: "'a"   and b :: "'a" assumes "lt a b" shows "le a b"
nonempty_fine_imp_less: fixes \<delta> :: "real \<Rightarrow> real"   and a :: "real"   and b :: "real"   and D :: "(real \<times> real \<times> real) list" assumes "fine \<delta> (a, b) D"   and "D \<noteq> []" shows "a < b"
poly_pos_between_leq_less: fixes a :: "real"   and b :: "real"   and p :: "real poly" shows "(\<forall>x. a \<le> x \<and> x < b \<longrightarrow> 0 < poly p x) = (b \<le> a \<or> p \<noteq> 0 \<and> 0 < poly p a \<and> count_roots_between p a b = (if a < b \<and> poly p b = 0 then 1 else 0))"
R: shows "dfa_reach (binop_dfa f A B) 0 bs m"
vo_start_geq_ve: assumes "\<delta> \<le> other.t_stop"   and "v\<^sub>e \<le> other.s' \<delta>" shows "u \<delta> < other.s \<delta>"
ts''_def_1: shows "\<S>\<bullet>\<C> \<turnstile> ves : ts _> ts''"
preserves_unity: shows "D.isomorphic D'.unity (F C'.unity)"
gen_model_tc_rels: fixes M :: "('e, 'f, 'g) KripkeStructure"   and w' :: "'g"   and w'' :: "'g"   and w :: "'g"   and as :: "'e set" assumes "kripke M"   and "(w', w'') \<in> (\<Union> (relations (gen_model M w) ` as))\<^sup>+" shows "(w', w'') \<in> (\<Union> (relations M ` as))\<^sup>+"
exec_1I: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> nat \<times> nat \<times> instr list \<times> (nat \<times> nat \<times> char list \<times> nat \<times> nat) list) list) list"   and \<sigma> :: "nat option \<times> (nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (val list \<times> val list \<times> char list \<times> char list \<times> nat) list"   and \<sigma>' :: "nat option \<times> (nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (val list \<times> val list \<times> char list \<times> char list \<times> nat) list" assumes "exec (P, \<sigma>) = \<lfloor>\<sigma>'\<rfloor>" shows "exec_1p P \<sigma> \<sigma>'"
proj_arcs_G_simps_1: shows "proj_arcs_G uv = uw"
l1_trans_def: shows "l1_trans \<equiv> \<Union>m Ra Rb A B K KE. (l1_step1 Ra A B \<union> l1_step2 Rb A B KE \<union> l1_step3 Ra A B K \<union> l1_learn m)\<^sup>="
nrename_ax2: fixes N :: "trm"   and x :: "name"   and y :: "name"   and z :: "name"   and c :: "coname" assumes "N[x\<turnstile>n>y] = Ax z c" shows "\<exists>z. N = Ax z c"
simps: fixes s1 :: "'e"   and s2 :: "('a, 'f, 'g, 'h) SLin_state_scheme" shows "P12 (s1, s2) = (initialized s2 \<longrightarrow> \<Sqinter> initVals s2 \<preceq> dstate s2)"
restr_rel_def: fixes R :: "'c set" shows "restr_rel R \<equiv> {(s, s'). (ss_stack s, ss_stack s') \<in> \<langle>Id \<times>\<^sub>r {(U, U'). U - R = U'}\<rangle>list_rel \<and> on_stack s = on_stack s' \<and> visited s = visited s' \<union> R \<and> visited s' \<inter> R = {} \<and> simple_state.more s = simple_state.more s'}"
consistent_add_witness: fixes S :: "('a, 'b) fm set"   and p :: "('a, 'b) fm"   and a :: "'a" assumes "consistent S"   and "(\<^bold>\<not> \<^bold>\<forall> p) \<in> S"   and "a \<notin> params S" shows "consistent ({\<^bold>\<not> p\<langle>\<^bold>\<dagger> a []/0\<rangle>} \<union> S)"
surj_on_mult': fixes x :: "nat"   and q :: "nat" assumes "coprime x q"   and "inj_on (\<lambda>b. x * b mod q) ({..<q} - {0})" shows "(\<lambda>b. x * b mod q) ` ({..<q} - {0}) = {..<q} - {0}"
fair_def: fixes Ns :: "('f \<times> 'l) set llist" shows "empty_ord.fair Ns = (Inf_from (Liminf_llist Ns) \<subseteq> Sup_llist (lmap Red_I_\<G> Ns))"
initState_into_used: fixes X :: "msg"   and B :: "agent"   and evs :: "event list" assumes "X \<in> parts (initState B)" shows "X \<in> used evs"
left_comp: fixes T :: "'f \<Rightarrow> 'e"   and U :: "'d \<Rightarrow> 'f"   and T' :: "'g \<Rightarrow> 'e"   and U' :: "'d \<Rightarrow> 'g"   and S :: "'e \<Rightarrow> 'h" assumes "T \<circ> U = T' \<circ> U'" shows "S \<circ> T \<circ> U = S \<circ> T' \<circ> U'"
satisfiable_prefix_implies_\<nu>LTL: fixes \<phi> :: "'a ltln"   and w :: "nat \<Rightarrow> 'a set" assumes "\<phi> \<in> \<nu>LTL"   and "\<nexists>i. af \<phi> (prefix i w) \<sim>\<^sub>C false\<^sub>n" shows "w \<Turnstile>\<^sub>n \<phi>"
assms_2: fixes v :: "'a" assumes "v \<in> verts G" shows "in_degree G v = \<r>"
focus_monomial: fixes X :: "'a set"   and t :: "'a \<Rightarrow>\<^sub>0 nat"   and c :: "'b" shows "focus X (monomial c t) = monomial (monomial c (except t X)) (except t (- X))"
P1: shows "C A B CongA F D G0 \<and> F D OS G0 E"
ntrunc_subsetD: fixes M :: "('f, 'g) Old_Datatype.node set"   and N :: "('f, 'g) Old_Datatype.node set" assumes "\<And>k. ntrunc k M \<subseteq> N" shows "M \<subseteq> N"
frag_extend_add: fixes f :: "'d \<Rightarrow> 'c \<Rightarrow>\<^sub>0 int"   and a :: "'d \<Rightarrow>\<^sub>0 int"   and b :: "'d \<Rightarrow>\<^sub>0 int" shows "frag_extend f (a + b) = frag_extend f a + frag_extend f b"
wp_If: fixes b :: "bexp"   and c\<^sub>1 :: "com"   and c\<^sub>2 :: "com"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> bool" shows "wp (IF b THEN c\<^sub>1 ELSE c\<^sub>2) Q = (\<lambda>s. if bval b s then wp c\<^sub>1 Q s else wp c\<^sub>2 Q s)"
set_intros_2: fixes x22 :: "'a"   and x21 :: "'a tree"   and x23 :: "'a tree" shows "x22 \<in> set_tree \<langle>x21, x22, x23\<rangle>"
independent_explicit_finite_subsets: fixes A :: "'b set" shows "cindependent A = (\<forall>S\<subseteq>A. finite S \<longrightarrow> (\<forall>u. (\<Sum>v\<in>S. u v *\<^sub>C v) = (0::'b) \<longrightarrow> (\<forall>v\<in>S. u v = 0)))"
Pi_def: fixes A :: "'f set"   and B :: "'f \<Rightarrow> 'g set" shows "Pi A B = {f. \<forall>x. x \<in> A \<longrightarrow> f x \<in> B x}"
nth_digit_general_def: fixes num :: "nat"   and i :: "nat"   and base :: "nat" shows "nth_digit_general num i base = num div base ^ i mod base"
elims_3: fixes x :: "('b, 'c) Combinators list"   and xa :: "'b set list" assumes "\<not> FWNormalisationCore.sorted x xa" obtains xb :: "('b, 'c) Combinators" and y :: "('b, 'c) Combinators" and zs :: "('b, 'c) Combinators list" and l :: "'b set list" where "x = xb # y # zs" and "xa = l" and "\<not> (smaller xb y l \<and> FWNormalisationCore.sorted (y # zs) l)"
dbm_regions': fixes S :: "('a \<Rightarrow> real) set"   and M :: "nat \<Rightarrow> nat \<Rightarrow> real DBMEntry" assumes "S = [M]\<^bsub>v,n\<^esub> \<and> dbm_int M n"   and "normalized M"   and "S \<subseteq> V" shows "\<exists>U\<subseteq>\<R>. S = \<Union> U"
csol_2: fixes T :: "real set"   and f :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and X :: "'b set"   and t :: "real"   and t0 :: "real"   and x0 :: "'b" assumes "ll_on_open_it T f X"   and "t \<in> ll_on_open.existence_ivl T f X t0 x0" shows "{t0--t} \<subseteq> T"
isin_random_bst: fixes A :: "'a set"   and t :: "'a tree"   and x :: "'a" assumes "finite A"   and "t \<in> set_pmf (random_bst A)" shows "isin t x = (x \<in> A)"
divergence_refine_def: fixes P :: "'a process"   and Q :: "'a process" shows "P \<sqsubseteq>\<^sub>D Q \<equiv> D Q \<subseteq> D P"
qSwap_preserves_alpha: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm"   and X' :: "('d, 'e, 'varSort, 'var, 'f) qTerm"   and zs :: "'varSort"   and z1 :: "'var"   and z2 :: "'var" assumes "qGood X \<or> qGood X'"   and "X #= X'" shows "X #[[z1 \<and> z2]]_zs #= X' #[[z1 \<and> z2]]_zs"
heyting_stone: fixes x :: "'a" shows "- x \<squnion> - - x = top"
replicate_pred_Cons_length: fixes n :: "nat"   and x :: "'d"   and a :: "'d" assumes "0 < n" shows "length (x # a\<^bsup>n - Suc 0\<^esup>) = n"
get_child_nodes_locs_def: shows "Shadow_DOM.get_child_nodes_locs \<equiv> l_get_child_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_child_nodes_locs"
main_2: assumes "y \<noteq> 0" shows "b ^ (y - 1) < x"
simps_3: fixes x :: "'a" shows "merge_option (None, Some x) = Some x"
simps_3: fixes \<delta> :: "int"   and c\<^sub>0 :: "int \<times> int"   and c\<^sub>1 :: "int \<times> int"   and p\<^sub>0 :: "int \<times> int"   and v :: "int \<times> int"   and va :: "(int \<times> int) list" shows "find_closest_pair_code (\<delta>, c\<^sub>0, c\<^sub>1) (p\<^sub>0 # v # va) = (let (\<delta>', p\<^sub>1) = find_closest_code p\<^sub>0 \<delta> (v # va) in if \<delta> \<le> \<delta>' then find_closest_pair_code (\<delta>, c\<^sub>0, c\<^sub>1) (v # va) else find_closest_pair_code (\<delta>', p\<^sub>0, p\<^sub>1) (v # va))"
tagrel_e: fixes \<Gamma> :: "'c constr list"   and n :: "nat"   and K\<^sub>1 :: "clock"   and K\<^sub>2 :: "clock"   and R :: "'c tag_const \<times> 'c tag_const \<Rightarrow> bool"   and \<Psi> :: "'c TESL_atomic list"   and \<Phi> :: "'c TESL_atomic list" shows "\<Gamma>, n \<turnstile> (time-relation \<lfloor>K\<^sub>1, K\<^sub>2\<rfloor> \<in> R) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^sub>e \<lfloor>\<tau>\<^sub>v\<^sub>a\<^sub>r (K\<^sub>1, n), \<tau>\<^sub>v\<^sub>a\<^sub>r (K\<^sub>2, n)\<rfloor> \<in> R # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (time-relation \<lfloor>K\<^sub>1, K\<^sub>2\<rfloor> \<in> R) # \<Phi>"
cancel1_append_same2: fixes xs :: "'d list"   and ys :: "'d list"   and zs :: "'d list" assumes "cancel1 xs ys" shows "cancel1 (xs @ zs) (ys @ zs)"
preserves_hseq: fixes \<mu> :: "'a"   and \<nu> :: "'a" assumes "C.arr \<mu> \<and> C.arr \<nu> \<and> src\<^sub>C \<mu> = trg\<^sub>C \<nu>" shows "D.arr (G \<mu>) \<and> D.arr (G \<nu>) \<and> src\<^sub>D (G \<mu>) = trg\<^sub>D (G \<nu>)"
prime: fixes x :: "'a" assumes "x \<in># A" shows "prime x"
empty_antichain_def: shows "{}\<^sub>A \<equiv> antichain {}"
simps_4: fixes G :: "prog"   and T :: "ty"   and dynC :: "qtname"   and static_membr :: "bool" shows "G,ArrayT T \<turnstile> dynC valid_lookup_cls_for static_membr = (dynC = Object)"
size_add_mset: fixes a :: "'a"   and A :: "'a multiset" shows "size (add_mset a A) = Suc (size A)"
assms_2: shows "(v \<longlongrightarrow> m) F"
nth_take: fixes i :: "nat"   and n :: "nat"   and xs :: "'c list" assumes "i < n" shows "take n xs ! i = xs ! i"
simplex_3: fixes cs :: "constraint list"   and v :: "(nat, rat) mapping" assumes "simplex cs = Inr v" shows "\<langle>v\<rangle> \<Turnstile>\<^sub>c\<^sub>s set cs"
max_mapping_def: fixes P :: "'d \<Rightarrow> 'c option"   and P' :: "'d \<Rightarrow> 'c option"   and x :: "'d" shows "max_mapping P P' x = (case (P x, P' x) of (None, None) \<Rightarrow> None | (None, Some x) \<Rightarrow> Map.empty x | (Some x, None) \<Rightarrow> None | (Some x, Some y) \<Rightarrow> Some (max x y))"
map_def: fixes A :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and b :: "'b"   and f :: "'a" assumes "constant_functor A B b" shows "constant_functor.map A B b f = (if partial_magma.arr A f then b else partial_magma.null B)"
ennreal_minus_le_iff: fixes a :: "ennreal"   and b :: "ennreal"   and c :: "ennreal" shows "(a - b \<le> c) = (a \<le> b + c \<and> (a = \<top> \<and> b = \<top> \<longrightarrow> c = \<top>))"