hcju/isabelleps huggingface dataset:
split_asm: fixes P :: "'k \<Rightarrow> bool"   and f1 :: "'k"   and f2 :: "nat \<Rightarrow> 'k"   and nat :: "nat" shows "P (case nat of 0 \<Rightarrow> f1 | Suc x \<Rightarrow> f2 x) = (\<not> (nat = 0 \<and> \<not> P f1 \<or> (\<exists>x2. nat = Suc x2 \<and> \<not> P (f2 x2))))"
vimage_const: fixes c :: "'d"   and A :: "'d set" shows "(\<lambda>x. c) -` A = (if c \<in> A then UNIV else {})"
divide: fixes e :: "'b \<times> 'b"   and H :: "'b pair_pre_digraph"   and w :: "'b"   and G :: "'b pair_pre_digraph" assumes "e \<in> parcs H"   and "w \<notin> pverts H"   and "pair_sd G H" shows "pair_sd G (subdivide H e w)"
retraction_if_Img_eq_Cod_1: fixes S :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<AA> :: "('c \<times> 'c) set"   and g :: "'b" assumes "set_category S \<AA>"   and "partial_magma.arr S g"   and "set_category_given_img.Fun S (set_category.some_img S \<AA>) g ` set_category_data.set S (set_category.some_img S \<AA>) (partial_magma.dom S g) = set_category_data.set S (set_category.some_img S \<AA>) (partial_magma.cod S g)" shows "category.retraction S g"
meval_sched_sound_mode_use: fixes gc :: "('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) list \<times> ('Var\<^sub>A \<Rightarrow> 'Val)"   and sched :: "nat list"   and gc' :: "('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) list \<times> ('Var\<^sub>A \<Rightarrow> 'Val)" assumes "abs.sound_mode_use gc"   and "abs.meval_sched_abv gc sched gc'" shows "abs.sound_mode_use gc'"
acyclic_1_def: fixes x :: "'a" shows "acyclic_1 x \<equiv> \<forall>y. orientation x y \<longrightarrow> pd_kleene_allegory_class.acyclic y"
partition_on_notemptyI: fixes A :: "'b set"   and P :: "'b set set" assumes "partition_on A P"   and "A \<noteq> {}" shows "P \<noteq> {}"
abel_semigroup_axioms: shows "abel_semigroup (\<^bold>\<sqinter>)"
agree1': shows "Vagree \<nu> \<nu>' (FVT t1)"
that: fixes p :: "(name \<times> name) list"   and T :: "('a, 'b, 'c) psi" assumes "set p \<subseteq> set xvec \<times> set yvec"   and "P = T \<parallel> (p \<bullet> R)"   and "\<lparr>\<nu>*xvec\<rparr>M \<prec>' T = \<lparr>\<nu>*yvec\<rparr>N \<prec>' Q" shows "thesis"
isleq: shows "\<forall>(r, a)\<in>set C. a \<in> as \<and> is_leq_ns (r *R nsc_of_atom a) \<and> r \<noteq> 0"
weight_le_less_iff: fixes s :: "('f, 'b) Term.term"   and t :: "('f, 'b) Term.term" assumes "weight_le s t" shows "weight_less s t = (weight s < weight t)"
vars1'_def: shows "c.vars1' = ind_in_set c.vars0 ` vars2"
split_max_induct: fixes P :: "('b \<times> nat) tree \<Rightarrow> bool"   and a0 :: "('b \<times> nat) tree" assumes "\<And>l a uu_ r. (r \<noteq> \<langle>\<rangle> \<Longrightarrow> P r) \<Longrightarrow> P \<langle>l, (a, uu_), r\<rangle>"   and "P \<langle>\<rangle>" shows "P a0"
that: fixes ns :: "'node SDG_node list" assumes "CFG_node n cd-ns\<rightarrow>\<^sub>d* CFG_node n'"   and "ns \<noteq> []"   and "\<forall>n''\<in>set ns. parent_node n'' \<in> set (sourcenodes as)" shows "thesis"
subformulas_in_subformulas_2: fixes G :: "'a formula"   and H :: "'a formula"   and F :: "'a formula" assumes "G \<^bold>\<or> H \<in> set (subformulae F)" shows "G \<in> set (subformulae F) \<and> H \<in> set (subformulae F)"
distinct_remove1: fixes equal :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'c list"   and x :: "'c" assumes "equal equal"   and "equal_base.list_distinct equal xs" shows "equal_base.list_distinct equal (equal_base.list_remove1 equal x xs)"
jvtt': shows "spr_jview a t = spr_jview a t'"
distinct_33: fixes x41 :: "('a, 'b, 'c) Stmt"   and x42 :: "('a, 'b, 'c) Stmt"   and x61 :: "'c"   and x62 :: "('a, 'b, 'c) Stmt" shows "x41 ;; x42 \<noteq> While x61 x62"
rep_eq: fixes x :: "double"   and xa :: "double" shows "fcompare_double x xa = fcompare (rep_double x) (rep_double xa)"
While_aux: fixes b :: "'a set"   and i :: "'a set"   and v :: "'a \<Rightarrow> nat"   and c :: "'a com"   and s :: "'a option"   and s' :: "'a option"   and I :: "'a set" assumes "Sem (While b i v c) s s'"   and "\<forall>s s'. Sem c s s' \<longrightarrow> s \<in> Some ` (I \<inter> b) \<longrightarrow> s' \<in> Some ` I"   and "s \<in> Some ` I" shows "s' \<in> Some ` (I \<inter> - b)"
subset_Compl_singleton: fixes A :: "'b set"   and b :: "'b" shows "(A \<subseteq> - {b}) = (b \<notin> A)"
PPs_closed_adds: fixes s :: "'c \<Rightarrow>\<^sub>0 nat"   and X :: "'c set"   and t :: "'c \<Rightarrow>\<^sub>0 nat" assumes "s \<in> .[X]"   and "t adds s" shows "t \<in> .[X]"
sas_plus_problem_has_serial_solution_iff_i': fixes \<Psi> :: "('a, 'b) sas_plus_problem"   and \<A> :: "sat_plan_variable \<Rightarrow> bool"   and t :: "nat" assumes "is_valid_problem_sas_plus \<Psi>"   and "\<A> \<Turnstile> \<Phi>\<^sub>\<forall> \<phi> prob_with_noop \<Psi>  t" shows "SAS_Plus_Semantics.is_serial_solution_for_problem \<Psi> (rem_noops (map (strips_op_to_sasp (prob_with_noop \<Psi>)) (concat (\<Phi>\<inverse> \<phi> prob_with_noop \<Psi>  \<A> t))))"
det_identical_cols: fixes A :: "'b mat"   and n :: "nat"   and i :: "nat"   and j :: "nat" assumes "A \<in> carrier_mat n n"   and "i \<noteq> j"   and "i < n"   and "j < n"   and "col A i = col A j" shows "det A = (0::'b)"
M2: fixes F :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and G :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and v :: "'a \<Rightarrow> 'b" assumes "|~ F \<longrightarrow> G" shows "\<turnstile> \<box>[F]_v \<longrightarrow> \<box>[G]_v"
simple_integral_cmult_indicator: fixes A :: "'b set"   and M :: "'b measure"   and c :: "ennreal" assumes "A \<in> sets M" shows "\<integral>\<^sup>S x. c * indicator A x \<partial>M = c * emeasure M A"
con_comp_sinks_filter: fixes I :: "('e \<times> 'e) set"   and D :: "'g \<Rightarrow> 'e"   and E :: "'h \<Rightarrow> 'e"   and p :: "'g \<Rightarrow> 'f"   and q :: "'h \<Rightarrow> 'f"   and u :: "'e option"   and xs :: "'f list" shows "sinks (con_comp_pol I) (con_comp_map D E p q) u (filter (\<lambda>x. x \<in> range p \<union> range q) xs) = sinks (con_comp_pol I) (con_comp_map D E p q) u xs \<inter> range Some"
fls_base_factor_nonzero: fixes f :: "'a fls" assumes "f \<noteq> 0" shows "fls_base_factor f \<noteq> 0"
Cons_1: fixes L' :: "atomUni list" assumes "convert_atom_list var L (xs' @ x # xs) = Some L'"   and "length xs' = var" shows "(\<forall>f\<in>set L. aEval f (xs' @ x # xs)) = (\<forall>f\<in>set L'. aEvalUni f x)"
TryThrow: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and e\<^sub>1 :: "char list exp"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and a :: "nat"   and h\<^sub>1 :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and l\<^sub>1 :: "char list \<Rightarrow> val option"   and D :: "char list"   and fs :: "char list \<times> char list \<Rightarrow> val option"   and C :: "char list"   and V :: "char list"   and e\<^sub>2 :: "char list exp" assumes "P \<turnstile> \<langle>e\<^sub>1,s\<^sub>0\<rangle> \<Rightarrow> \<langle>Throw a,(h\<^sub>1, l\<^sub>1)\<rangle>"   and "h\<^sub>1 a = \<lfloor>(D, fs)\<rfloor>"   and "(D, C) \<notin> (subcls1 P)\<^sup>*" shows "P \<turnstile> \<langle>try e\<^sub>1 catch(C V) e\<^sub>2,s\<^sub>0\<rangle> \<Rightarrow> \<langle>Throw a,(h\<^sub>1, l\<^sub>1)\<rangle>"
type_to_set_2: fixes p :: "nat" assumes "\<exists>Rep Abs. type_definition Rep Abs {0..<int p}"   and "1 < p" shows "p = CARD('b)"
polyfun_det_deep_model: shows "polyfun {..<weight_space_dim} (\<lambda>f. det (witness_submatrix f))"
cases: fixes a :: "'a"   and outPorts :: "'b \<Rightarrow> 'c fset"   and vertices :: "'a fset"   and nodeOf :: "'a \<Rightarrow> 'b"   and edges :: "(('a \<times> 'c) \<times> 'a \<times> 'd) set"   and ps :: "'a \<times> 'd" assumes "a \<in> Pre_Port_Graph.scope outPorts vertices nodeOf edges ps" obtains v :: "'a" where "a = v" and "v |\<in>| vertices" and "\<And>pth v'. \<lbrakk>Pre_Port_Graph.path edges v v' pth; Vertex_Graph.terminal_vertex outPorts vertices nodeOf v'\<rbrakk> \<Longrightarrow> ps \<in> snd ` set pth"
g: fixes i :: "nat" shows "g i = eval_poly (\<lambda>x. 0::'a) (f i)"
ln_lower_13_def: shows "ln_lower_13 \<equiv> \<lambda>x. - ln_upper_13 (inverse x)"
wf_subset: fixes r :: "('e \<times> 'e) set"   and p :: "('e \<times> 'e) set" assumes "wf r"   and "p \<subseteq> r" shows "wf p"
numeric_unop_type_det: fixes op :: "numeric_unop"   and \<tau> :: "'a OCL_Types.type"   and \<sigma>\<^sub>1 :: "'b OCL_Types.type"   and \<sigma>\<^sub>2 :: "'b OCL_Types.type" assumes "numeric_unop_type op \<tau> \<sigma>\<^sub>1"   and "numeric_unop_type op \<tau> \<sigma>\<^sub>2" shows "\<sigma>\<^sub>1 = \<sigma>\<^sub>2"
Subset_graph_aux_def: shows "Subset_graph_aux \<equiv> lfp (\<lambda>p x1 x2. \<exists>Subset_aux z t u. x1 = (t, u) \<and> x2 = All2 z t (Var z IN u) \<and> atom z \<sharp> (t, u))"
prems_1: shows "t \<in> ik\<^sub>e\<^sub>s\<^sub>t (A @ [Decomp t']) \<cdot>\<^sub>s\<^sub>e\<^sub>t \<I>"
cl6: fixes x :: "'a"   and z :: "'a" shows "x \<cdot> (0::'a) \<cdot> z = x \<cdot> (0::'a)"
isometry_def: fixes f :: "'b \<Rightarrow> 'c" shows "isometry f \<equiv> isometry_on UNIV f \<and> surj f"
inj_on_index: fixes xs :: "'b list" shows "inj_on (index xs) (set xs)"
subst_def: shows "subst \<equiv> \<lambda>x y s. lam_rec (\<lambda>x. if x = y then s else Var x) (\<lambda>t\<^sub>1 t\<^sub>2. (\<degree>)) (\<lambda>x t. Lam x) x"
add_diff_cancel_right': fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "cancel_ab_semigroup_add plus minus" shows "minus (plus a b) b = a"
fs_int_mu_d_Z_m_m: fixes j :: "nat"   and ii :: "nat" assumes "j < length fs"   and "ii < length fs" shows "rat_of_int (fs.d fs (Suc j)) * fs.gs.\<mu> ii j \<in> \<int>"
AF_lfp: fixes p :: "'b set" shows "\<^bold>A\<^bold>F p = lfp (\<lambda>s. p \<union> \<^bold>A\<^bold>X s)"
coeff_monom: fixes R :: "('c, 'd) ring_scheme"   and a :: "'c"   and m :: "nat"   and n :: "nat" assumes "UP_ring R"   and "a \<in> carrier R" shows "coeff (UP R) (monom (UP R) a m) n = (if m = n then a else \<zero>\<^bsub>R\<^esub>)"
rep_eq: fixes x :: "'c \<Rightarrow> 'a upred"   and xa :: "'c \<Rightarrow> ('b, 'a) uexpr" shows "\<lbrakk>USUP x xa\<rbrakk>\<^sub>e = (\<lambda>b. \<Squnion>{uu_. \<exists>xb. uu_ = \<lbrakk>xa xb\<rbrakk>\<^sub>e b \<and> \<lbrakk>x xb\<rbrakk>\<^sub>e b})"
sfirst_0: fixes P :: "'c stream \<Rightarrow> bool"   and \<omega> :: "'c stream" assumes "P \<omega>" shows "sfirst P \<omega> = 0"
assms_1: shows "b_e_type_checker \<C> es (t1s _> t2s)"
asIMOD_igOpIPresIGWls: fixes SEM :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'b) semDom" assumes "sOpPrSWls SEM"   and "sWlsNE SEM" shows "igOpIPresIGWls (asIMOD SEM)"
case_sum_o_map_sum: fixes f :: "'h \<Rightarrow> 'g"   and g :: "'i \<Rightarrow> 'g"   and h1 :: "'e \<Rightarrow> 'h"   and h2 :: "'f \<Rightarrow> 'i" shows "case_sum f g \<circ> map_sum h1 h2 = case_sum (f \<circ> h1) (g \<circ> h2)"
in_borel_measurable: fixes f :: "'a \<Rightarrow> 'b"   and M :: "'a measure" shows "(f \<in> borel_measurable M) = (\<forall>S\<in>sigma_sets UNIV {S. open S}. f -` S \<inter> space M \<in> sets M)"
summable_geometric_iff: fixes c :: "'b" shows "summable ((^) c) = (norm c < 1)"
simps_2: fixes h :: "'b \<times> bool"   and t :: "('b \<times> bool) list"   and l :: "nat"   and cl :: "nat" shows "prefixToLevel_aux (h # t) l cl = (if marked h then if l \<le> cl then [] else h # prefixToLevel_aux t l (cl + 1) else h # prefixToLevel_aux t l cl)"
iOp_mHom_closed: fixes N :: "('g, 'b, 'h) Module_scheme"   and f :: "'a \<Rightarrow> 'g" assumes "R module N"   and "f \<in> mHom R M N" shows "iOp_mHom R M N f \<in> mHom R M N"
length_derivation_shift: fixes D :: "(nat \<times> symbol \<times> symbol list) list"   and left :: "nat"   and right :: "nat" shows "length (derivation_shift D left right) = length D"
l_heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def: fixes known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and type_wf :: "(_) heap \<Rightarrow> bool"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and heap_is_wellformed :: "(_) heap \<Rightarrow> bool"   and parent_child_rel :: "(_) heap \<Rightarrow> ((_) object_ptr \<times> (_) object_ptr) set" shows "l_heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr type_wf get_child_nodes get_child_nodes_locs get_disconnected_nodes get_disconnected_nodes_locs heap_is_wellformed parent_child_rel \<equiv> l_get_child_nodes type_wf known_ptr get_child_nodes get_child_nodes_locs \<and> l_get_disconnected_nodes type_wf get_disconnected_nodes get_disconnected_nodes_locs \<and> l_heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms get_child_nodes get_disconnected_nodes heap_is_wellformed parent_child_rel"
rank_mul_le_right: fixes A :: "((real, 'c) vec, 'b) vec"   and B :: "((real, 'a) vec, 'c) vec" shows "rank (A ** B) \<le> rank B"
subst_true: fixes \<sigma> :: "'a \<Rightarrow> 'b" shows "\<sigma> \<dagger> true = true"
less_eq_quasinorm_def: fixes N1 :: "'c quasinorm"   and N2 :: "'c quasinorm" shows "(N1 \<subseteq>\<^sub>N N2) = (\<exists>C\<ge>0. \<forall>f. eNorm N2 f \<le> ennreal C * eNorm N1 f)"
cont: fixes x :: "real" assumes "a < x" shows "isCont (\<lambda>t. 1 / (t * (ln t)\<^sup>2)) x"
sets_vimage_Sup_eq: fixes M :: "'c measure set"   and f :: "'d \<Rightarrow> 'c"   and X :: "'d set"   and Y :: "'c set" assumes "M \<noteq> {}"   and "f \<in> X \<rightarrow> Y"   and "\<And>m. m \<in> M \<Longrightarrow> space m = Y" shows "sets (vimage_algebra X f (Sup M)) = sets (Sup (vimage_algebra X f ` M))"
ideal_generated_subset_generator: fixes A :: "'a set"   and I :: "'a set" assumes "ideal_generated A = I" shows "A \<subseteq> I"
eq_add_iff2: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and e :: "'a"   and c :: "'a"   and b :: "'a"   and d :: "'a" assumes "class.ring plus zero minus uminus times" shows "(plus (times a e) c = plus (times b e) d) = (c = plus (times (minus b a) e) d)"
discI: fixes vertex :: "'a vertex"   and x3 :: "'a" assumes "vertex = \<langle>x3\<rangle>" shows "is_Inner vertex"
bind_runit: fixes x :: "'a option" shows "x \<bind> Some = x"
has_sgnx_imp_sgnx: fixes f :: "real \<Rightarrow> real"   and c :: "real"   and F :: "real filter" assumes "(f has_sgnx c) F"   and "F \<noteq> bot" shows "sgnx f F = c"
CHAR_def: shows "CHAR \<equiv> semiring_char TYPE('a)"
sup_exists: fixes A :: "'a hrel set" assumes "A \<subseteq> carrier (fpl \<P> \<H>)" shows "\<exists>s. is_lub (fpl \<P> \<H>) s A"
Keys_not_empty: fixes a :: "'e \<Rightarrow>\<^sub>0 'f"   and A :: "('e \<Rightarrow>\<^sub>0 'f) set" assumes "a \<in> A"   and "a \<noteq> 0" shows "Keys A \<noteq> {}"
rel_inject: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'a list"   and y :: "'b list" shows "rel_iarray R (IArray x) (IArray y) = list_all2 R x y"
sum_upto_divisor_sum2: fixes f :: "nat \<Rightarrow> real"   and x :: "real" shows "sum_upto (\<lambda>n. \<Sum>d | d dvd n. f d) x = sum_upto (\<lambda>x. sum_upto f (x / real x)) x"
measurable_marg_dens2': fixes x :: "nat"   and y :: "nat" assumes "x \<in> V"   and "y \<in> V" shows "(\<lambda>(xa, ya). marg_dens2 \<Y> x y xa ya) \<in> borel_measurable (state_measure V' \<Gamma> \<Otimes>\<^sub>M stock_measure (PRODUCT (\<Gamma> x) (\<Gamma> y)))"
ioa_implements_def: fixes A :: "('e, 'f) ioa"   and B :: "('g, 'f) ioa" shows "A =<| B \<equiv> inputs (ioa.asig A) = inputs (ioa.asig B) \<and> outputs (ioa.asig A) = outputs (ioa.asig B) \<and> local.traces A \<subseteq> local.traces B"
sorted_sublist_wrt_def: fixes R :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'c list"   and lo :: "nat"   and hi :: "nat" shows "sorted_sublist_wrt R xs lo hi = sorted_wrt R (WB_Sort.sublist xs lo hi)"
iso_char: fixes \<phi> :: "'a" assumes "Right.arr \<phi>" shows "Right.iso \<phi> = local.iso \<phi>"
P1: fixes i :: "nat" assumes "i \<le> k" shows "i \<in> set js"
KeyWithNonce_def: fixes K :: "nat"   and NB :: "nat"   and evs :: "event list" shows "KeyWithNonce K NB evs \<equiv> \<exists>A B na X. Says Server A \<lbrace>Crypt (shrK A) \<lbrace>Agent B, Key K, na, Nonce NB\<rbrace>, X\<rbrace> \<in> set evs"
monotone_onI: fixes X :: "'c set"   and r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and s :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd" assumes "\<And>x y. \<lbrakk>x \<in> X; y \<in> X; r x y\<rbrakk> \<Longrightarrow> s (f x) (f y)" shows "monotone_on X r s f"
assms_2: shows "offs_num (length ns) xs index key mi ma i = 0"
simps_1: fixes comp :: "'a \<Rightarrow> 'a \<Rightarrow> order"   and x :: "'a" assumes "comparator comp" shows "comparator.lookup_pair comp [] x = (0::'b)"
ccw'_sortedP_translateI: fixes x0 :: "real \<times> real"   and x :: "real \<times> real"   and g :: "'a \<Rightarrow> real \<times> real"   and xs :: "'a list" assumes "ccw'.sortedP (x0 - x) (map g xs)" shows "ccw'.sortedP x0 (map ((+) x \<circ> g) xs)"
per_init_def: fixes D :: "'b set" shows "per_init D \<equiv> {(i, i) |i. i \<in> D}"
sum_right_ideal: fixes X :: "'a set"   and U :: "'a set"   and f :: "'a \<Rightarrow> 'a" assumes "right_ideal X"   and "U \<subseteq> X"   and "finite U" shows "(\<Sum>i\<in>U. i * f i) \<in> X"
cmp_simps_5: fixes f :: "'c"   and g :: "'c" assumes "C.ide f"   and "C.ide g"   and "src\<^sub>C f = trg\<^sub>C g" shows "D.cod (\<Phi>\<^sub>F (f, g)) = F (f \<star>\<^sub>C g)"
intro: fixes wtFsym :: "'a \<Rightarrow> bool"   and wtPsym :: "'b \<Rightarrow> bool"   and arOf :: "'a \<Rightarrow> 'c list"   and resOf :: "'a \<Rightarrow> 'c"   and intT :: "'c \<Rightarrow> univ \<Rightarrow> bool"   and intF :: "'a \<Rightarrow> univ list \<Rightarrow> univ"   and intP :: "'b \<Rightarrow> univ list \<Rightarrow> bool" assumes "FullStruct wtFsym wtPsym arOf resOf intT intF intP" shows "M_FullStruct wtFsym wtPsym arOf resOf intT intF intP"
times_enat_simps_4: fixes m :: "nat" shows "enat m * \<infinity> = (if m = 0 then 0 else \<infinity>)"
equality: fixes r :: "('b, 'c) Line_scheme"   and r' :: "('b, 'c) Line_scheme" assumes "basepoint r = basepoint r'"   and "direction r = direction r'"   and "Line.more r = Line.more r'" shows "r = r'"
mset_bubblesort: fixes xs :: "'a list" shows "mset (bubblesort xs) = mset xs"
dis_rews_7: shows "is_LT\<cdot>LT = TT"
Algorithm_axioms: shows "Algorithm jkbp envInit envAction envTrans envVal jview envObs jviewInit jviewIncr simf simRels simVal simAbs simObs simInit KBP.k_succs simAction aOps tOps"
assms_3: fixes A :: "'c set" assumes "A \<in> sets M" shows "set_lebesgue_integral M A f = set_lebesgue_integral M A f'"
pid_run_conv: fixes r :: "nat \<Rightarrow> ('c, 'ls, 'gs) global_config" shows "sa.is_run r = (\<exists>r'. r = pidgc_\<alpha> \<circ> r' \<and> pid.is_run r')"
lc: shows "insertion (nth_default 0 (xs' @ x # xs)) (liftPoly 0 0 c) = C"
that: fixes e' :: "nat \<Rightarrow> real" assumes "pdevs_val e (pdevs_of_list (filter p xs)) = pdevs_val e' (pdevs_of_list xs)"   and "e' \<in> UNIV \<rightarrow> I" shows "thesis"
S_eq_1: fixes x :: "'a" assumes "\<forall>\<omega>\<in>streams (space M). T (x ## \<omega>) = 0" shows "S x = lim_stream x"
has_unique_favorites_altdef: shows "has_unique_favorites R = (\<forall>i\<in>agents. is_singleton (favorites R i))"
\<iota>FL_def: shows "\<iota>FL = Infer (zip (prems_of \<iota>F) Ll) (concl_of \<iota>F, L0)"
not_le_imp_less: fixes y :: "'c word"   and x :: "'c word" assumes "\<not> y \<le>s x" shows "x <s y"
two_component_val_process: fixes arb_pf :: "'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> real"   and x :: "'c"   and a :: "nat \<Rightarrow> 'd \<Rightarrow> real"   and y :: "'c"   and b :: "nat \<Rightarrow> 'd \<Rightarrow> real"   and Mkt :: "('d, 'c) discrete_market"   and n :: "nat"   and w :: "'d" assumes "arb_pf = (\<lambda>x n w. 0)(x := a, y := b)"   and "portfolio arb_pf"   and "x \<noteq> y"   and "\<exists>n w. a n w \<noteq> 0"   and "\<exists>n w. b n w \<noteq> 0" shows "val_process Mkt arb_pf n w = prices Mkt y n w * b (Suc n) w + prices Mkt x n w * a (Suc n) w"
attractor_set_VVpstar: fixes G :: "('c, 'd) ParityGame_scheme"   and v :: "'c"   and p :: "Player"   and W :: "'c set" assumes "ParityGame G"   and "\<not> \<not> (\<exists>w\<in>V\<^bsub>G\<^esub>. v \<rightarrow>\<^bsub>G\<^esub> w)"   and "\<And>w. v \<rightarrow>\<^bsub>G\<^esub> w \<Longrightarrow> w \<in> ParityGame.attractor G p W" shows "v \<in> ParityGame.attractor G p W"
powr_one_gt_zero_iff: fixes x :: "real" shows "(x powr 1 = x) = (0 \<le> x)"
external_to_internal_invariant_vimage: fixes S :: "('f, 'e) spec"   and Q :: "'e set"   and P :: "'f set" assumes "oreach S \<subseteq> Q"   and "obs S -` Q \<subseteq> P" shows "reach S \<subseteq> P"
deg_minus_eq: fixes p :: "nat \<Rightarrow> 'a"   and q :: "nat \<Rightarrow> 'a" assumes "p \<in> carrier P"   and "q \<in> carrier P"   and "deg R q \<noteq> deg R p" shows "deg R (p \<ominus>\<^bsub>P\<^esub> q) = max (deg R p) (deg R q)"
strategyproofness: shows "pmf (sds R40) b + pmf (sds R40) a \<le> pmf (sds R9) b + pmf (sds R9) a"
object_ptr_kinds_M_reads: fixes h :: "(_) heap"   and h' :: "(_) heap" shows "reads (\<Union>object_ptr. {preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t object_ptr RObject.nothing)}) object_ptr_kinds_M h h'"
ub: fixes x :: "'c"   and y :: "'c" shows "\<exists>z\<le>x. z \<le> y"
invar_cons_down: fixes t :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree"   and bq :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree list" assumes "SkewBinomialHeapStruc.invar (t # bq)" shows "SkewBinomialHeapStruc.invar bq"
rule_charn2: fixes p :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list"   and c :: "(int \<times> int \<times> protocol) set set"   and d :: "(int \<times> int \<times> protocol) set set"   and po :: "protocol \<times> int"   and x :: "int \<times> (int \<times> int \<times> protocol) \<times> (int \<times> int \<times> protocol) \<times> DummyContent" assumes "allNetsDistinct p"   and "wellformed_policy1 p"   and "singleCombinators p"   and "wellformed_policy3Pr p"   and "AllowPortFromTo c d po \<in> set p"   and "x \<in> dom (Cp (AllowPortFromTo c d po))" shows "applied_rule_rev Cp x p = \<lfloor>AllowPortFromTo c d po\<rfloor>"
lm127: shows "\<forall>n. set (perm2 [] n) = set []"
rel_gpv_bindI: fixes A :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and C :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and gpv :: "('d, 'f, 'h) gpv"   and gpv' :: "('e, 'g, 'h) gpv"   and B :: "'i \<Rightarrow> 'j \<Rightarrow> bool"   and f :: "'d \<Rightarrow> ('i, 'f, 'h) gpv"   and g :: "'e \<Rightarrow> ('j, 'g, 'h) gpv" assumes "rel_gpv A C gpv gpv'"   and "\<And>x y. A x y \<Longrightarrow> rel_gpv B C (f x) (g y)" shows "rel_gpv B C (gpv \<bind> f) (gpv' \<bind> g)"
dfs_eq_rtrancl: fixes y :: "nat \<times> nat"   and x :: "nat \<times> nat" assumes "prod_is_node A B y"   and "prod_is_node A B x" shows "prod_memb y (dfs (prod_empt A B) [x]) = ((x, y) \<in> (succsr (prod_succs A B))\<^sup>*)"
some_snapshot_def: fixes t :: "('a, 'b, 'c) event llist"   and p :: "'a" shows "some_snapshot t p = (SOME i. ps (coS t i) p \<noteq> None \<and> enat i \<le> llength t)"