hcju/isabelleps huggingface dataset:
codomains_comp: fixes g :: "'a"   and f :: "'a" assumes "C.seq g f" shows "C.codomains (g \<cdot>\<^sub>C f) = C.codomains g"
constant_rep_designI: fixes \<r> :: "int" assumes "\<And>x. x \<in> \<V>\<^sup>+ \<Longrightarrow> \<B>\<^sup>+ rep x = \<r>" shows "constant_rep_design \<V>\<^sup>+ \<B>\<^sup>+ \<r>"
intros: fixes X :: "'b set"   and u :: "'b \<Rightarrow> real"   and I :: "'b \<Rightarrow> intv"   and X\<^sub>0 :: "'b set"   and r :: "('b \<times> 'b) set"   and J :: "'b \<Rightarrow> 'b \<Rightarrow> intv'" assumes "\<forall>x\<in>X. 0 \<le> u x"   and "\<forall>x\<in>X. intv_elem x u (I x)"   and "X\<^sub>0 = {x \<in> X. \<exists>d. I x = Intv d}"   and "\<forall>x\<in>X\<^sub>0. \<forall>y\<in>X\<^sub>0. ((x, y) \<in> r) = (frac (u x) \<le> frac (u y))"   and "\<forall>x\<in>X. \<forall>y\<in>X. isGreater (I x) \<or> isGreater (I y) \<longrightarrow> intv'_elem x y u (J x y)" shows "u \<in> region X I J r"
AAA: shows "rotate_A \<noteq> rotate_A \<circ> rotate_A"
len_def: fixes sensors :: "cars \<Rightarrow> traffic \<Rightarrow> cars \<Rightarrow> real"   and v :: "view"   and ts :: "traffic"   and c :: "cars" assumes "sensors sensors" shows "sensors.len sensors v ts c \<equiv> if right (ext v) < left (sensors.space sensors ts v c) then Abs_real_int (right (ext v), right (ext v)) else if right (sensors.space sensors ts v c) < left (ext v) then Abs_real_int (left (ext v), left (ext v)) else Abs_real_int (max (left (ext v)) (left (sensors.space sensors ts v c)), min (right (ext v)) (right (sensors.space sensors ts v c)))"
bit_a'': fixes a :: "'a"   and b :: "'a"   and x :: "'a"   and y :: "'a"   and s :: "('a list \<times> bool list \<times> 'a list) pmf"   and qs :: "'a list" assumes "a \<noteq> b"   and "{a, b} = {x, y}"   and "BIT_inv s a [x, y]"   and "set qs \<subseteq> {a, b}"   and "qs \<in> lang (seq [question (Atom a), Atom b, Atom b])" shows "BIT_inv (Partial_Cost_Model.config'_rand BIT s qs) (last qs) [x, y] \<and> T\<^sub>p_on_rand' BIT s qs = 15 / 10"
bet_col1: fixes A :: "'p"   and B :: "'p"   and D :: "'p"   and C :: "'p" assumes "Bet A B D"   and "Bet A C D" shows "Col A B C"
w_2: shows "z = dual_order.poset_simplex_map (PreCoxeterSystem.special_cosets (\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g})) {w}"
UN_simps_5: fixes A :: "'a set"   and B :: "'b \<Rightarrow> 'a set"   and C :: "'b set" shows "(\<Union>\<^bsub>x\<in>C\<^esub> A \<inter> B x) = A \<inter> \<Union> (B ` C)"
map_poly_eq_0_iff: fixes f :: "'c \<Rightarrow> 'b"   and p :: "'c poly" assumes "f (0::'c) = (0::'b)"   and "\<And>x. \<lbrakk>x \<in> set (coeffs p); x \<noteq> (0::'c)\<rbrakk> \<Longrightarrow> f x \<noteq> (0::'b)" shows "(map_poly f p = 0) = (p = 0)"
\<iota>_ri: shows "\<iota> \<in> Red_I N"
takeWhile_cong: fixes l :: "'c list"   and k :: "'c list"   and P :: "'c \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> bool" assumes "l = k"   and "\<And>x. x \<in> set l \<Longrightarrow> P x = Q x" shows "takeWhile P l = takeWhile Q k"
polya_vinogradov_inequality_explicit: fixes x :: "nat" assumes "\<chi> \<noteq> principal_dchar n" shows "cmod (sum \<chi> {1..x}) < sqrt (real conductor) * ln (real conductor) * real (divisor_count (n div conductor))"
assms_2: shows "has_density M (count_space UNIV) f"
assn_assoc_2: fixes x :: "assn"   and y :: "assn"   and z :: "assn" shows "(x \<and>\<^sub>A y) \<and>\<^sub>A z = x \<and>\<^sub>A y \<and>\<^sub>A z"
v11_eq_iff: fixes r :: "V"   and a :: "V"   and b :: "V" assumes "v11 r"   and "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> r"   and "b \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> r" shows "(r\<lparr>a\<rparr> = r\<lparr>b\<rparr>) = (a = b)"
group_closure_add: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and s :: "'a"   and S :: "'a set"   and t :: "'a" assumes "class.ab_group_add plus zero minus uminus"   and "s \<in> ab_group_add.group_closure zero minus S"   and "t \<in> ab_group_add.group_closure zero minus S" shows "plus s t \<in> ab_group_add.group_closure zero minus S"
Imp3: fixes p :: "('a, 'b) fm"   and q :: "('a, 'b) fm"   and r :: "('a, 'b) fm"   and s :: "('a, 'b) fm" shows "\<turnstile> (p \<^bold>\<longrightarrow> q \<^bold>\<longrightarrow> r) \<^bold>\<longrightarrow> (s \<^bold>\<longrightarrow> p) \<^bold>\<longrightarrow> (s \<^bold>\<longrightarrow> q) \<^bold>\<longrightarrow> s \<^bold>\<longrightarrow> r"
ln_upper_3_pos: fixes x :: "real" assumes "1 \<le> x" shows "ln x \<le> ln_upper_3 x"
sum_SDG_edge_valid_SDG_node_1: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and Exit :: "'b"   and Def :: "'b \<Rightarrow> 'c set"   and Use :: "'b \<Rightarrow> 'c set"   and ParamDefs :: "'b \<Rightarrow> 'c list"   and ParamUses :: "'b \<Rightarrow> 'c set list"   and n :: "'b SDG_node"   and Vopt :: "'c option"   and popt :: "('f \<times> bool) option"   and b :: "bool"   and n' :: "'b SDG_node" assumes "SDG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit Def Use ParamDefs ParamUses"   and "SDG.sum_SDG_edge sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses n Vopt popt b n'" shows "SDG.valid_SDG_node sourcenode targetnode kind valid_edge procs n"
dvd_div_eq_cancel: 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 a :: "'b"   and c :: "'b"   and b :: "'b" assumes "class.semidom_divide divide plus minus zero times one"   and "divide a c = divide b c"   and "dvd.dvd times c a"   and "dvd.dvd times c b" shows "a = b"
floor_Fract: fixes a :: "int"   and b :: "int" shows "\<lfloor>Rat.Fract a b\<rfloor> = a div b"
both_contains_arg2: fixes t' :: "'b ttree"   and t :: "'b ttree" shows "paths t' \<subseteq> paths (t \<otimes>\<otimes> t')"
thetaSeqWD_Wretr: fixes indis :: "('a \<times> 'a) set"   and tval :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'a \<Rightarrow> 'a" assumes "PL_Indis indis" shows "PL_Indis.thetaSeqWD tval aval indis \<subseteq> PL_Indis.Wretr tval aval indis (PL_Indis.thetaSeqWD tval aval indis \<union> PL_Indis.Wbis tval aval indis)"
ordered_sel_resolvent_upon_is_resolvent: fixes atom_ordering :: "('a \<times> 'a) set"   and Sel :: "'a set"   and P1 :: "'a Literal set"   and P2 :: "'a Literal set"   and C :: "'a Literal set"   and A :: "'a" assumes "propositional_atoms atom_ordering"   and "propositional_atoms.ordered_sel_resolvent_upon atom_ordering Sel P1 P2 C A" shows "propositional_atoms.ordered_sel_resolvent atom_ordering Sel P1 P2 C"
evalG: shows "sinvar \<lparr>nodes = V, edges = E\<rparr> nP"
proots_card_usphere_eq: fixes r :: "real"   and p :: "complex poly"   and z0 :: "complex" assumes "0 < r" shows "card (proots_within p (sphere z0 r)) = card (proots_within (p \<circ>\<^sub>p [:z0, complex_of_real r:]) (sphere 0 1))"
minus_diff_minus: fixes a :: "'a"   and b :: "'a" shows "- a - - b = - (a - b)"
vector_matrix_mult_iarray_def: fixes x :: "'b iarray"   and A :: "'b iarray iarray" shows "x v*i A = IArray.of_fun (\<lambda>j. \<Sum>i = 0..<IArray.length x. A !! i !! j * x !! i) (ncols_iarray A)"
wf_imp_SN: fixes A :: "('c \<times> 'c) set" assumes "wf (A\<inverse>)" shows "SN A"
is_parser_parse_attribute_value: shows "is_parser parse_attribute_value"
in_listsI: fixes xs :: "'c list"   and A :: "'c set" assumes "\<forall>x\<in>set xs. x \<in> A" shows "xs \<in> lists A"
Cons_replicate_eq: fixes x :: "'g"   and xs :: "'g list"   and n :: "nat"   and y :: "'g" shows "(x # xs = replicate n y) = (x = y \<and> 0 < n \<and> xs = replicate (n - 1) x)"
inv_from_upto_at_all_ev_block: fixes A :: "'a mat"   and k :: "nat"   and p :: "'a mat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bool" assumes "local.inv_upto local.jb A k"   and "local.inv_from local.uppert A k"   and "local.inv_at p A k"   and "local.ev_block n A"   and "\<And>i. \<lbrakk>i < n; p A i k\<rbrakk> \<Longrightarrow> local.uppert A i k"   and "k < n" shows "local.inv_all local.uppert A"
Length_SNOCs: fixes i :: "nat"   and x :: "'x"   and \<AA> :: "'i" shows "Length ((SNOC x ^^ i) \<AA>) = Length \<AA> + i"
lappend_lnull2: fixes ys :: "'c llist"   and xs :: "'c llist" assumes "lnull ys" shows "lappend xs ys = xs"
matched_intra_CFG_path: fixes n :: "'node SDG_node"   and ns :: "'node SDG_node list"   and n' :: "'node SDG_node" assumes "matched n ns n'" obtains as :: "'edge list" where "parent_node n -as\<rightarrow>\<^sub>\<iota>* parent_node n'"
min_dnf_prop_atoms: fixes \<Phi> :: "'b ltln fset"   and \<phi> :: "'b ltln" assumes "\<Phi> \<in> min_dnf \<phi>" shows "fset \<Phi> \<subseteq> prop_atoms \<phi>"
vcomp_atE1: fixes s :: "V"   and r :: "V"   and a :: "V"   and c :: "V" assumes "(s \<circ>\<^sub>\<circ> r)\<lparr>a\<rparr> = c"   and "vsv r"   and "vsv s"   and "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> r"   and "r\<lparr>a\<rparr> \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> s" obtains "\<exists>b. s\<lparr>b\<rparr> = c \<and> r\<lparr>a\<rparr> = b"
path_snoc: fixes v :: "'vertex"   and v' :: "'vertex"   and pth1 :: "(('vertex \<times> ('form, 'var) out_port) \<times> 'vertex \<times> ('form, 'var) antecedent) list"   and e :: "('vertex \<times> ('form, 'var) out_port) \<times> 'vertex \<times> ('form, 'var) antecedent" shows "path v v' (pth1 @ [e]) = (e \<in> edges \<and> path v (edge_begin e) pth1 \<and> edge_end e = v')"
mappend_assoc: fixes below :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and liftemb :: "'e\<^sub>\<bottom> \<rightarrow> udom\<^sub>\<bottom>"   and liftprj :: "udom\<^sub>\<bottom> \<rightarrow> 'e\<^sub>\<bottom>"   and liftdefl :: "'e itself \<Rightarrow> (udom\<^sub>\<bottom>) defl"   and emb :: "'e \<rightarrow> udom"   and prj :: "udom \<rightarrow> 'e"   and defl :: "'e itself \<Rightarrow> udom defl"   and mempty :: "'e"   and mappend :: "'e \<rightarrow> 'e \<rightarrow> 'e"   and xs :: "'e"   and ys :: "'e"   and zs :: "'e" assumes "class.monoid below liftemb liftprj liftdefl emb prj defl mempty mappend" shows "mappend\<cdot>(mappend\<cdot>xs\<cdot>ys)\<cdot>zs = mappend\<cdot>xs\<cdot>(mappend\<cdot>ys\<cdot>zs)"
data_dependency_consistent_instrs: fixes i :: "nat"   and p :: "'a"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "'a memref list"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option" assumes "i < length ts\<^sub>s\<^sub>b"   and "ts\<^sub>s\<^sub>b ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>)" shows "data_dependency_consistent_instrs (dom \<theta>) is"
lex_transI: fixes r :: "('g \<times> 'g) set" assumes "trans r" shows "trans (lex r)"
preserves_dom: fixes f :: "'b" assumes "arr f" shows "local.dom (\<ll>'.map f) = local.map (local.dom f)"
wf_lex_prod: fixes ra :: "('d \<times> 'd) set"   and rb :: "('e \<times> 'e) set" assumes "wf ra"   and "wf rb" shows "wf (ra <*lex*> rb)"
valid_expr_NormalI: fixes A :: "'b triple set"   and G :: "prog"   and t :: "expr"   and P :: "'b assn"   and Q :: "'b assn" assumes "\<And>n s0 L accC T C v s1 Y Z. \<lbrakk>\<forall>t\<in>A. G\<Turnstile>n\<Colon>t; s0\<Colon>\<preceq>(G, L); normal s0; \<lparr>prg = G, cls = accC, lcl = L\<rparr>\<turnstile>t\<Colon>-T; \<lparr>prg = G, cls = accC, lcl = L\<rparr>\<turnstile> dom (locals (snd s0)) \<guillemotright>In1l t\<guillemotright> C; G\<turnstile>s0 \<midarrow>t-\<succ>v\<midarrow>n\<rightarrow> s1; Normal P Y s0 Z\<rbrakk> \<Longrightarrow> Q (In1 v) s1 Z \<and> s1\<Colon>\<preceq>(G, L)" shows "G,A|\<Turnstile>\<Colon>{{Normal P} t-\<succ> {Q}}"
base: fixes f :: "'b \<Rightarrow> 'b"   and x :: "'b" shows "f x \<in> orbit f x"
list_decode_init: fixes f :: "nat \<Rightarrow> nat option"   and n :: "nat" shows "Partial_Recursive.list_decode (f \<triangleright> n) = prefix f n"
nAct_le_proj: fixes c :: "'id"   and n :: "enat"   and t :: "cnf llist" shows "\<langle>c #\<^bsub>n\<^esub>t\<rangle> \<le> llength (\<pi>\<^bsub>c\<^esub>t)"
case_eq_if: fixes f1 :: "'b"   and f2 :: "'b"   and f3 :: "bool \<Rightarrow> 'c \<Rightarrow> 'b"   and f4 :: "'c mrexp3 \<Rightarrow> 'c mrexp3 \<Rightarrow> 'c set \<Rightarrow> bool \<Rightarrow> 'b"   and f5 :: "'c mrexp3 \<Rightarrow> 'c mrexp3 \<Rightarrow> 'c set \<Rightarrow> bool \<Rightarrow> 'b"   and f6 :: "'c mrexp3 \<Rightarrow> 'c set \<Rightarrow> 'b"   and mrexp3 :: "'c mrexp3" shows "(case mrexp3 of Zero3 \<Rightarrow> f1 | One3 \<Rightarrow> f2 | Atom3 x xa \<Rightarrow> f3 x xa | Plus3 x xa xb xc \<Rightarrow> f4 x xa xb xc | Times3 x xa xb xc \<Rightarrow> f5 x xa xb xc | Star3 x xa \<Rightarrow> f6 x xa) = (if mrexp3 = Zero3 then f1 else if mrexp3 = One3 then f2 else if is_Atom3 mrexp3 then f3 (un_Atom31 mrexp3) (un_Atom32 mrexp3) else if is_Plus3 mrexp3 then f4 (un_Plus31 mrexp3) (un_Plus32 mrexp3) (fin1 mrexp3) (nul mrexp3) else if is_Times3 mrexp3 then f5 (un_Times31 mrexp3) (un_Times32 mrexp3) (fin1 mrexp3) (nul mrexp3) else f6 (un_Star31 mrexp3) (fin1 mrexp3))"
games_gfp_imp_lfp: fixes g :: "ZF" shows "g \<in> games_gfp \<longrightarrow> g \<in> games_lfp"
hyps_1: shows "oalist_inv_raw ((k', v') # xs)"
sel: fixes x :: "'d \<Rightarrow> 'e" shows "run_env (EnvT x) = x"
simps: fixes a1 :: "ccs"   and a2 :: "ccs" shows "bangPred a1 a2 = ((\<exists>P. a1 = P \<and> a2 = !P) \<or> (\<exists>P. a1 = P \<and> a2 = P \<parallel> !P))"
finite_sorted_distinct_unique: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a set" assumes "class.linorder less_eq less"   and "finite A" shows "\<exists>!xs. set xs = A \<and> linorder.sorted less_eq xs \<and> distinct xs"
I: fixes x :: "('a, 'b, 'c) sp\<^sub>\<mu>" assumes "\<forall>y. (\<exists>a f. y = f a \<and> x = Get f) \<longrightarrow> P y" shows "P x"
of_nat_simps_3: fixes m :: "nat" shows "of_nat (Suc m) = (1::'b) + of_nat m"
case_1: fixes f1 :: "'b"   and f2 :: "'c \<Rightarrow> 'c \<Rightarrow> 'b" shows "(case Infinity of Infinity \<Rightarrow> f1 | Point x xa \<Rightarrow> f2 x xa) = f1"
closed_csubspace_inter: fixes A :: "'b set"   and B :: "'b set" assumes "closed_csubspace A"   and "closed_csubspace B" shows "closed_csubspace (A \<inter> B)"
vec_lambda_inverse: fixes y :: "'a \<Rightarrow> 'b" assumes "y \<in> UNIV" shows "($) (vec_lambda y) = y"
leq_dual_def: fixes x' :: "'b dual"   and y' :: "'b dual" shows "x' \<sqsubseteq> y' \<equiv> undual y' \<sqsubseteq> undual x'"
right_neutral: fixes a :: "'b upred" shows "(a \<or> false) = a"
finite_same_card_bij: fixes A :: "'b set"   and B :: "'c set" assumes "finite A"   and "finite B"   and "card A = card B" shows "\<exists>h. bij_betw h A B"
assms_3: shows "set (vals xs) \<inter> set (vals [x]) = {}"
Ints_numeral: 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 n :: "num" assumes "class.ring_1 one times plus zero minus uminus" shows "numeral.numeral one plus n \<in> ring_1.Ints one plus zero minus"
has_limits_if_has_products: fixes J :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" assumes "category J"   and "has_products (Collect (partial_magma.ide J))"   and "has_products (Collect (partial_magma.arr J))" shows "has_limits_of_shape J"
zip_append_left: fixes xs :: "'a list"   and ys :: "'a list"   and zs :: "'b list" shows "zip (xs @ ys) zs = zip xs zs @ zip ys (drop (length xs) zs)"
simps_20: fixes vb :: "'a"   and vc :: "value list"   and v :: "'a aexp"   and va :: "'a aexp" shows "local.less_gexp_aux (In vb vc) (Gt v va) = False"
expand: fixes option :: "\<langle>'a\<rangle>\<^sub>\<bottom>"   and option' :: "\<langle>'a\<rangle>\<^sub>\<bottom>" assumes "(option = \<bottom>) = (option' = \<bottom>)"   and "\<lbrakk>option \<noteq> \<bottom>; option' \<noteq> \<bottom>\<rbrakk> \<Longrightarrow> the option = the option'" shows "option = option'"
implies_order: fixes x :: "'a"   and y :: "'a" shows "(x \<le> y) = (x \<leadsto> y = top)"
sgn_of_float: fixes x :: "real" assumes "x \<in> float" shows "sgn x \<in> float"
assms_2: shows "4 \<le> rk {A, B, C, A', B', C', P, Q}"
assms_1: shows "fds_abs_converges (fds_norm f) s"
sum_list_abs: fixes abs :: "'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b list" assumes "class.ordered_ab_group_add_abs abs plus zero minus uminus less_eq less" shows "less_eq (abs (monoid_add.sum_list plus zero xs)) (monoid_add.sum_list plus zero (map abs xs))"
resNilRight: fixes Rel :: "(ccs \<times> ccs) set"   and x :: "name" shows "\<zero> \<leadsto>[Rel] \<lparr>\<nu>x\<rparr>\<zero>"
Sigma_lepoll_mono: fixes A :: "'b set"   and C :: "'b set"   and B :: "'b \<Rightarrow> 'c set"   and D :: "'b \<Rightarrow> 'd set" assumes "A \<subseteq> C"   and "\<And>x. x \<in> A \<Longrightarrow> B x \<lesssim> D x" shows "Sigma A B \<lesssim> Sigma C D"
div_mult_self4: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and b :: "'a"   and c :: "'a"   and a :: "'a" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size"   and "b \<noteq> zero" shows "divide (plus (times b c) a) b = plus c (divide a b)"
fv_supp_heap: fixes \<Gamma> :: "(Vars.var \<times> exp) list" shows "supp \<Gamma> = atom ` fv \<Gamma>"
Proc_CFG_targetlabel_less_num_nodes: fixes prog :: "cmd"   and n :: "label"   and et :: "p_edge_kind"   and l :: "nat" assumes "prog \<turnstile> n -et\<rightarrow>\<^sub>p Label l" shows "l < #:prog"
fds_abs_converges_exp: fixes f :: "'a fds"   and s :: "'a" assumes "fds_abs_converges f s" shows "fds_abs_converges (fds_exp f) s"
equivalence_top_closed: shows "(top \<le> top \<and> top \<sqinter> top \<le> top) \<and> local.comp_inf.symmetric top"
le_Sup_iff: 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 x :: "'b"   and A :: "'b set" assumes "class.complete_linorder Inf Sup inf less_eq less sup bot top" shows "less_eq x (Sup A) = (\<forall>y. less y x \<longrightarrow> (\<exists>a\<in>A. less y a))"
check_precond_r_def: fixes s\<^sub>e :: "real"   and v\<^sub>e :: "real"   and a\<^sub>e :: "real"   and s\<^sub>o :: "real"   and v\<^sub>o :: "real"   and a\<^sub>o :: "real"   and \<delta> :: "real" shows "check_precond_r s\<^sub>e v\<^sub>e a\<^sub>e s\<^sub>o v\<^sub>o a\<^sub>o \<delta> = (s\<^sub>e < s\<^sub>o \<and> 0 \<le> v\<^sub>e \<and> 0 \<le> v\<^sub>o \<and> a\<^sub>e < 0 \<and> a\<^sub>o < 0 \<and> 0 < \<delta>)"
finite_is_methd: fixes G :: "prog" assumes "ws_prog G" shows "finite {(x, y). is_methd G x y}"
p_id_st: fixes R :: "('a \<times> 'a set) set" shows "R \<cdot> 1\<^sub>\<pi> = {(a, {}) |a. \<exists>B. (a, B) \<in> R}"
mset_set_eq_iff: fixes A :: "'a set"   and B :: "'a set" assumes "finite A"   and "finite B" shows "(mset_set A = mset_set B) = (A = B)"
mat_mult_row_def: fixes m :: "(('d, 'f) vec, 'g) vec"   and m' :: "(('d, 'e) vec, 'f) vec"   and f :: "'g" shows "mat_mult_row m m' f = (\<chi>c. \<Sum>k\<in>UNIV. m $ f $ k * m' $ k $ c)"
wf_snd_append: fixes V :: "'c set"   and S :: "('d, 'c) strand_step list"   and t :: "('d, 'c) Term.term" shows "wf\<^sub>s\<^sub>t V (S @ [send\<langle>t\<rangle>\<^sub>s\<^sub>t]) = wf\<^sub>s\<^sub>t V S"
E18: fixes F :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool" shows "\<turnstile> \<box>\<diamond>\<box>F \<longrightarrow> \<diamond>\<box>F"
indRelT_symm: fixes TRel :: "('procT \<times> 'procT) set" assumes "sym TRel" shows "sym (indRelT TRel)"
rev_curr_FinFunc: assumes "Field r \<noteq> {}" shows "local.rev_curr ` r_st.FINFUNC = rst.FINFUNC"
support: shows "pmf (sds R28) b = 0 \<or> pmf (sds R28) c = 0"
HStartBallot_HInv4c_p: fixes s :: "state"   and s' :: "state"   and p :: "Proc" assumes "HStartBallot s s' p"   and "HInv4c s p" shows "HInv4c s' p"
emeasure_bind_pmf: fixes M :: "'d pmf"   and N :: "'d \<Rightarrow> 'c pmf"   and X :: "'c set" shows "emeasure (measure_pmf (M \<bind> N)) X = \<integral>\<^sup>+ x. emeasure (measure_pmf (N x)) X \<partial>measure_pmf M"
semivalid_implem_axioms: shows "semivalid_implem implem"
map_cong0: fixes x :: "'c set['d]"   and f :: "'c \<Rightarrow> 'e"   and g :: "'c \<Rightarrow> 'e" assumes "\<And>z. z \<in> set_bset x \<Longrightarrow> f z = g z" shows "map_bset f x = map_bset g x"
assms_2: shows "fv\<^sub>s\<^sub>t \<A> \<inter> bvars\<^sub>s\<^sub>t \<A> = {}"
intensional_rews_3: fixes f :: "'d \<Rightarrow> 'e \<Rightarrow> 'c"   and x :: "'f \<Rightarrow> 'd"   and y :: "'f \<Rightarrow> 'e"   and w :: "'f" shows "(f<x, y>) w \<equiv> f (x w) (y w)"
prefix_scons_snil: fixes x :: "'a"   and xs :: "[:'a:]" shows "prefix\<cdot>(x :# xs)\<cdot>[::] \<noteq> TT"
sets_lebesgue_on_refl: fixes S :: "'c set" shows "S \<in> sets (lebesgue_on S)"
cop_F_HM_def: fixes ds :: "'d set" shows "cop_F_HM ds = (\<lambda>(XD, XH). (- Rf CH XH, XH \<union> CD_on ds (- Rf CH XH)))"
noStupid: fixes x :: "nat"   and l :: "nat" assumes "x < length Strat"   and "l < length (snd (Strat ! x))" shows "Suc (snd (Strat ! x) ! l) < length init"
power_aform_err_def: fixes p :: "nat"   and X :: "(real \<times> real pdevs) \<times> real"   and n :: "nat" shows "power_aform_err p X n = (if n = 0 then ((1, zero_pdevs), 0) else if n = 1 then X else let x0 = float_of (fst (fst X)); xs = snd (fst X); xe = float_of (snd X); C = the (approx p (floatarith.Num x0 ^\<^sub>e n) []); (c, ce) = mid_err C; NX = the (approx p (floatarith.Num (of_nat n) * floatarith.Num x0 ^\<^sub>e (n - 1)) []); (nx, nxe) = mid_err NX; Y = scaleR_pdevs nx xs; (Y', Y_err) = trunc_bound_pdevs p Y; t = tdev' p xs; Ye = truncate_up p (nxe * t); err = the (approx p (floatarith.Num (of_nat n) * floatarith.Num xe * floatarith.Abs (floatarith.Num x0) ^\<^sub>e (n - 1) + Sum\<^sub>e (\<lambda>k. floatarith.Num (of_nat (n choose k)) * floatarith.Abs (floatarith.Num x0) ^\<^sub>e (n - k) * (floatarith.Num xe + floatarith.Num (float_of t)) ^\<^sub>e k) [2..<Suc n]) []); ERR = upper err in ((c, Y'), sum_list' p [ce, Y_err, Ye, real_of_float ERR]))"
igFreshIGAbs1_def: fixes MOD :: "('b, 'c, 'varSort, 'sort, 'd, 'e, 'f, 'g, 'h) model_scheme" shows "igFreshIGAbs1 MOD \<equiv> \<forall>ys y s X. isInBar (ys, s) \<and> igWls MOD s X \<longrightarrow> igFreshAbs MOD ys y (igAbs MOD ys y X)"
per_root_trans: fixes w :: "'b list"   and u :: "'b list"   and t :: "'b list" assumes "w \<le>p u\<^sup>\<omega>"   and "u \<in> t*" shows "w \<le>p t\<^sup>\<omega>"
\<G>_properties_2: fixes \<G> :: "'b ltl set"   and \<phi> :: "'b ltl" assumes "\<G> \<subseteq> \<^bold>G \<phi>" shows "Only_G \<G>"
asymptotics5: shows "\<forall>\<^sub>F x in at_top. ln (b * x - hb * x * ln x powr - (1 + e)) powr (- e / 2) < 1"
pre_sigma_def: fixes T :: "'a \<Rightarrow> 't" shows "pre_sigma T = sigma \<Omega> {A. \<forall>t. {\<omega> \<in> A. T \<omega> \<le> t} \<in> sets (F t)}"
pfixGe_refl: fixes x :: "nat list" shows "x pfixGe x"