hcju/isabelleps huggingface dataset:
w_simps_6: shows "cod TTfgh_TfTgh.chine = TTfgh_TfTgh.chine"
lunit_char_2: fixes f :: "'c" assumes "C.ide f" shows "C.L \<l>\<^sub>C[f] = (\<i>\<^sub>C[trg\<^sub>C f] \<star>\<^sub>C f) \<cdot>\<^sub>C \<a>\<^sub>C\<^sup>-\<^sup>1[trg\<^sub>C f, trg\<^sub>C f, f]"
functor_axioms: shows "functor (\<cdot>) (\<cdot>) A.map"
cast_document_not_node_2: fixes node :: "(\<lparr>RElement.nothing :: unit, tag_name :: char list, child_nodes :: (_) node_ptr list, attrs :: (char list, char list) fmap, shadow_root_opt :: (_) shadow_root_ptr option, \<dots> :: 'q option\<rparr> + (_) RCharacterData_ext + 's) RNode_scheme"   and document :: "('m, 'n, 'o, 't option) RDocument_scheme" shows "cast node \<noteq> cast document"
assms_1: fixes z :: "'a" assumes "norm z < K" shows "summable (\<lambda>n. c n * z ^ n)"
generate_valid_stateful_policy_IFSACS_2_select_simps_2: fixes G :: "'b graph"   and M :: "'b SecurityInvariant_configured list"   and edgesList :: "('b \<times> 'b) list" shows "flows_fix (generate_valid_stateful_policy_IFSACS_2 G M edgesList) = edges G"
seqsim_notstutstep: fixes s :: "nat \<Rightarrow> 'a"   and n :: "nat" assumes "\<not> stutstep s n" shows "s |\<^sub>s Suc n \<approx> nextsuffix (s |\<^sub>s n)"
label_incr_ge: fixes l :: "nat"   and n :: "w_node"   and i :: "nat" assumes "(_ l _) = n \<oplus> i" shows "i \<le> l"
intros_1: fixes execute :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and enabled :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and p :: "'b" shows "transition_system.path execute enabled [] p"
assms: fixes x :: "V" assumes "x \<in> elts A" shows "b x \<in> elts (B x)"
Nats_subset_Ints: 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" assumes "class.ring_1 one times plus zero minus uminus" shows "semiring_1.Nats one plus zero \<subseteq> ring_1.Ints one plus zero minus"
assms_4: shows "almost_everywhere (T s) (ev (HLD H))"
has_integral_add: fixes f :: "'a \<Rightarrow> 'b"   and k :: "'b"   and S :: "'a set"   and g :: "'a \<Rightarrow> 'b"   and l :: "'b" assumes "(f has_integral k) S"   and "(g has_integral l) S" shows "((\<lambda>x. f x + g x) has_integral k + l) S"
chine_assoc_props_4: shows "cods.Prj\<^sub>0\<^sub>0 \<cdot> cods.chine_assoc = cods.Prj\<^sub>0"
card_of_ordIso: fixes A :: "'a set"   and B :: "'b set" shows "(\<exists>f. bij_betw f A B) = ordIso2 (card_of A) (card_of B)"
ASSUME_refine_right: fixes \<Phi> :: "bool"   and S :: "'b nres"   and R :: "('b \<times> 'c) set"   and S' :: "'c nres" assumes "\<Phi>"   and "\<Phi> \<Longrightarrow> S \<le> \<Down> R S'" shows "S \<le> \<Down> R (ASSUME \<Phi> \<bind> (\<lambda>_. S'))"
differentiable_of_real: fixes x :: "real"   and A :: "real set" shows "of_real differentiable at x within A"
rotate_minus1: fixes n :: "nat"   and m :: "nat"   and ls :: "'c list"   and ms :: "'c list" assumes "0 < n"   and "0 < m"   and "rotate n ls = rotate m ms" shows "rotate (n - 1) ls = rotate (m - 1) ms"
R12_d: shows "pmf (sds R12) d = 1 - pmf (sds R12) a"
finite_roots_unity: fixes n :: "nat" assumes "1 \<le> n" shows "finite {z. z ^ n = (1::'c)}"
prems_2: shows "ifex_sat (IF v t e) = Some ass"
simps_15: fixes x21 :: "'a"   and x22 :: "'a list" shows "set (x21 # x22) = {x21} \<union> set x22"
floor_minus_divide_eq_div_numeral: fixes a :: "num"   and b :: "num" shows "\<lfloor>- (numeral a / numeral b)\<rfloor> = - numeral a div numeral b"
Always_imp_Stable: fixes F :: "'b program"   and A :: "'b set" assumes "F \<in> Always A" shows "F \<in> Stable A"
outPortsRule_def: fixes r :: "'rule" shows "outPortsRule r = ffUnion ((\<lambda>a. (\<lambda>h. Hyp h a) |`| a_hyps a) |`| f_antecedent r) |\<union>| Reg |`| f_consequent r"
mult_imp_div_pos_le: fixes y :: "'a"   and x :: "'a"   and z :: "'a" assumes "(0::'a) < y"   and "x \<le> z * y" shows "x / y \<le> z"
has_integral_bound: fixes B :: "real"   and f :: "'b \<Rightarrow> 'c"   and i :: "'c"   and a :: "'b"   and b :: "'b" assumes "0 \<le> B"   and "(f has_integral i) (cbox a b)"   and "\<And>x. x \<in> cbox a b \<Longrightarrow> norm (f x) \<le> B" shows "norm i \<le> B * content (cbox a b)"
Cons_nth_drop_Suc: fixes i :: "nat"   and xs :: "'b list" assumes "i < \<^bold>|xs\<^bold>|" shows "xs ! i # drop (Suc i) xs = drop i xs"
in_Macaulay_listE: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and ps :: "('t \<Rightarrow>\<^sub>0 'c) list"   and f :: "nat \<Rightarrow> nat" assumes "p \<in> set (Macaulay_list ps)"   and "pivot_fun (row_echelon (Macaulay_mat ps)) f (dim_col (row_echelon (Macaulay_mat ps)))" obtains i :: "nat" where "i < dim_row (row_echelon (Macaulay_mat ps))" and "p = mat_to_polys (Keys_to_list ps) (row_echelon (Macaulay_mat ps)) ! i" and "f i < dim_col (row_echelon (Macaulay_mat ps))"
Lam_def: fixes x1 :: "name"   and x2 :: "lam" shows "Lam [x1].x2 \<equiv> Abs_lam (Lam_Rep ([x1].Rep_lam x2))"
coboundedI: fixes A :: "complex set"   and a :: "complex" assumes "finite A"   and "a \<in> A" shows "a \<le>\<^sub>\<complex> complex_lex.Max A"
h_eq: fixes k :: "nat"   and m :: "nat" assumes "0 < k"   and "k < 2 ^ m" shows "h (real k / 2 ^ m) = f (c (real k / 2 ^ m))"
abs_eq: fixes xa :: "'c"   and x :: "'d \<Rightarrow> 'd \<Rightarrow> 'c" shows "xa *\<^sub>S of_fun x = of_fun (\<lambda>i j. xa * x i j)"
induct_step: fixes st :: "'a" shows "(Pre \<and> \<lceil>e\<rceil>\<^sub>< =\<^sub>u \<guillemotleft>st\<guillemotright> \<Rightarrow> Post) \<sqsubseteq> B (Pre \<and> (\<lceil>e\<rceil>\<^sub><, \<guillemotleft>st\<guillemotright>)\<^sub>u \<in>\<^sub>u \<guillemotleft>R\<guillemotright> \<Rightarrow> Post)"
R: fixes i :: "'a"   and x :: "'c \<times> 'd \<Rightarrow>\<^sub>L 'c" assumes "i \<in> I"   and "x \<in> X1 i" shows "x \<in> R"
assms: shows "st \<in> reachable (closed (pnet np \<langle>ii; R\<^sub>i\<rangle>)) TT"
tendsto_matpower: fixes f :: "'a \<Rightarrow> (('b, 'c) Finite_Cartesian_Product.vec, 'c) Finite_Cartesian_Product.vec"   and a :: "(('b, 'c) Finite_Cartesian_Product.vec, 'c) Finite_Cartesian_Product.vec"   and F :: "'a filter"   and n :: "nat" assumes "(f \<longlongrightarrow> a) F" shows "((\<lambda>x. matpow (f x) n) \<longlongrightarrow> matpow a n) F"
one_root_a_gt0: fixes b :: "real"   and a :: "real"   and r :: "real"   and c :: "real"   and y' :: "real" assumes "b = 2 * a * r"   and "\<not> a < 0"   and "a * r\<^sup>2 - 2 * a * r * r + c = 0"   and "- r < y'" shows "\<exists>x\<in>{- r<..y'}. \<not> a * x\<^sup>2 + 2 * a * r * x + c < 0"
path_component_def: fixes S :: "'c set"   and x :: "'c"   and y :: "'c" shows "path_component S x y \<equiv> \<exists>g. path g \<and> path_image g \<subseteq> S \<and> pathstart g = x \<and> pathfinish g = y"
retraction_map_def: fixes X :: "'a topology"   and Y :: "'b topology"   and f :: "'a \<Rightarrow> 'b" shows "retraction_map X Y f \<equiv> \<exists>g. retraction_maps X Y f g"
UM_Model_def: fixes wtFsym :: "'a \<Rightarrow> bool"   and wtPsym :: "'b \<Rightarrow> bool"   and arOf :: "'a \<Rightarrow> 'c list"   and resOf :: "'a \<Rightarrow> 'c"   and parOf :: "'b \<Rightarrow> 'c list"   and \<Phi> :: "('a, 'b) lit list set"   and D :: "univ \<Rightarrow> bool"   and intF :: "'a \<Rightarrow> univ list \<Rightarrow> univ"   and intP :: "'b \<Rightarrow> univ list \<Rightarrow> bool" shows "UM_Model wtFsym wtPsym arOf resOf parOf \<Phi> D intF intP \<equiv> M_Problem wtFsym wtPsym arOf resOf parOf \<Phi> \<and> UM_Struct wtFsym wtPsym arOf parOf D intF intP \<and> CU.Model wtFsym wtPsym (length \<circ> arOf) (length \<circ> parOf) \<Phi> D intF intP"
Collect_finite_eq_lists: shows "Collect finite = set ` lists UNIV"
ih: fixes y :: "('s, 'v) tm" assumes "size y < size s"   and "y >\<^sub>t y" shows "False"
simps_2: fixes h :: "'b \<Rightarrow> 'b ref"   and x :: "'b ref"   and a :: "'b"   and as :: "'b list"   and y :: "'b ref" shows "Path h x (a # as) y = (x = Ref a \<and> Path h (h a) as y)"
inj2: shows "inj_on (\<lambda>s. (s, inv p s)) S"
suffix_take: fixes xs :: "'a list"   and ys :: "'a list" assumes "suffix xs ys" shows "ys = take (length ys - length xs) ys @ xs"
abs_inner: fixes Inf :: "'e set \<Rightarrow> 'e"   and Sup :: "'e set \<Rightarrow> 'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and dist :: "'e \<Rightarrow> 'e \<Rightarrow> real"   and norm :: "'e \<Rightarrow> real"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and uminus :: "'e \<Rightarrow> 'e"   and scaleR :: "real \<Rightarrow> 'e \<Rightarrow> 'e"   and sgn :: "'e \<Rightarrow> 'e"   and uniformity :: "('e \<times> 'e) filter"   and "open" :: "'e set \<Rightarrow> bool"   and inner :: "'e \<Rightarrow> 'e \<Rightarrow> real"   and Basis :: "'e set"   and abs :: "'e \<Rightarrow> 'e"   and inf :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and sup :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and i :: "'e"   and x :: "'e" assumes "class.ordered_euclidean_space Inf Sup minus dist norm plus zero uminus scaleR sgn uniformity open inner Basis abs inf sup less_eq less"   and "i \<in> Basis" shows "inner (abs x) i = \<bar>inner x i\<bar>"
contg: fixes n :: "nat" shows "continuous_on UNIV (g n)"
eval\<kappa>_inverse: fixes x :: "\<kappa>" shows "make\<kappa> (eval\<kappa> x) = x"
index_mat_multrow_4: fixes mul :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and k :: "nat"   and a :: "'b"   and A :: "'b mat" shows "dim_row (mat_multrow_gen mul k a A) = dim_row A"
update_def: fixes a :: "'c"   and b :: "'d"   and t :: "('c \<times> 'd) tree23" shows "update a b t = treeI (upd a b t)"
group_add_ow_axioms_def: fixes U :: "'b set"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b" shows "group_add_ow_axioms U minus plus zero uminus \<equiv> (\<forall>a. a \<in> U \<longrightarrow> plus (uminus a) a = zero) \<and> (\<forall>a b. a \<in> U \<longrightarrow> b \<in> U \<longrightarrow> plus a (uminus b) = minus a b)"
OclANY_valid_args_valid'': fixes X :: "'a state \<times> 'a state \<Rightarrow> Set('b)" shows "\<upsilon> X->any\<^sub>S\<^sub>e\<^sub>t() = \<upsilon> X"
l3_inv9I: fixes x :: "l3_state" assumes "\<And>A B M. implSecure A B M \<in> synth (analz (ik x)) \<and> M \<in> payload \<Longrightarrow> implSecure A B M \<in> ik x \<or> M \<in> synth (analz (ik x)) \<and> (A \<in> l3_state.bad x \<or> B \<in> l3_state.bad x)" shows "x \<in> l3_inv9"
xIy: assumes "x \<^bsub>r\<^esub>\<approx> y" shows "thesis"
tendsto_unique: fixes "open" :: "'a set \<Rightarrow> bool"   and F :: "'b filter"   and f :: "'b \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.t2_space open"   and "F \<noteq> bot"   and "filterlim f (topological_space.nhds open a) F"   and "filterlim f (topological_space.nhds open b) F" shows "a = b"
fds_abs_converges_ln: fixes f :: "'a fds"   and s :: "'a"   and l :: "'a" assumes "fds_abs_converges (fds_deriv f / f) s" shows "fds_abs_converges (fds_ln l f) s"
timpls_transformable_to_pred_inv_4: fixes A :: "'e set set"   and f :: "('f, 'g, 'e) prot_fun"   and T :: "(('f, 'g, 'e) prot_fun, 'h) Term.term list"   and t :: "(('f, 'g, 'e) prot_fun, 'h) Term.term" assumes "timpls_transformable_to_pred A (Fun f T) t"   and "\<not> is_Abs f" shows "f = the_Fun t"
uint_cong: fixes x :: "'a word"   and y :: "'a word" assumes "x = y" shows "uint x = uint y"
size1_imbal2: fixes t :: "'b tree" assumes "\<not> bal_log t"   and "bal_log (hchild t)"   and "t \<noteq> \<langle>\<rangle>" shows "real (size1 t) < 2 powr (1 / c) / (2 - 2 powr (1 / c)) * real (imbal t + 1)"
wf_hull_rel_relcomp: fixes R :: "('e \<times> 'e) set" assumes "wf R"   and "eqvt R" shows "wf (hull_rel O R)"
first_edge_first: fixes G :: "('c, 'd) Graph_scheme"   and v :: "'c"   and xs :: "'c list"   and w :: "'c"   and w' :: "'c" assumes "Digraph G"   and "v \<leadsto>xs\<leadsto>\<^bsub>G\<^esub> w"   and "(v, w') \<in> Digraph.edges_of_walk xs" shows "H.first_edge_of_walk xs = (v, w')"
dvdnumcoeff_aux2: fixes t :: "MIR.num" assumes "1 < numgcd t" shows "dvdnumcoeff t (numgcd t) \<and> 0 < numgcd t"
g_def: shows "g = (\<lambda>(i, n). (i + 1, n + 1 + i))"
ind_all: shows "\<forall>i\<le>Suc x. \<forall>xc. {\<lambda>nl. nl = xs @ i # 0 \<up> (fft - far) @ xc} fap {\<lambda>nl. nl = xs @ i # rec_exec f (xs @ [i]) # 0 \<up> (fft - Suc far) @ xc}"
queue_correct_deq: fixes q :: "'b queue" assumes "invar q" shows "list (deq q) = tl (list q)"
diff_add_eq: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "class.ab_group_add plus zero minus uminus" shows "plus (minus a b) c = minus (plus a c) b"
mk_base_unit_def: fixes t :: "'e itself"   and s :: "'f itself" shows "mk_base_unit t s = 1"
finite_equiv_class: fixes A :: "'c set"   and r :: "('c \<times> 'c) set"   and X :: "'c set" assumes "finite A"   and "r \<subseteq> A \<times> A"   and "X \<in> A // r" shows "finite X"
Vset_succ: fixes i :: "V" assumes "Ord i" shows "Vset (ZFC_in_HOL.succ i) = VPow (Vset i)"
inner_axis': fixes i :: "'b"   and y :: "'a"   and x :: "('a, 'b) vec" shows "inner (axis i y) x = inner y (x $ i)"
split_indicator: fixes P :: "'k \<Rightarrow> bool"   and S :: "'l set"   and x :: "'l" shows "P (indicator S x) = ((x \<in> S \<longrightarrow> P (1::'k)) \<and> (x \<notin> S \<longrightarrow> P (0::'k)))"
arev_in_arcs: fixes G :: "('b, 'c) pre_digraph"   and arev :: "'c \<Rightarrow> 'c"   and a :: "'c" assumes "bidirected_digraph G arev"   and "a \<in> arcs G" shows "arev a \<in> arcs G"
pseudo_proj_element_prob_pref: fixes w :: "bool stream"   and n :: "nat"   and c :: "bool" assumes "w \<in> range (pseudo_proj_True n)" shows "emeasure M {y \<in> space M. \<exists>x\<in>pseudo_proj_True n -` {w}. y = c ## x} = ennreal (prob_component p (c ## w) 0) * emeasure M (pseudo_proj_True n -` {w} \<inter> space M)"
Merge_insert: fixes x :: "'a\<^sub>m"   and A :: "'a\<^sub>m set" shows "Merge (insert x A) = (if A = {} then Some x else Merge A \<bind> m x)"
vmatch_vmatch'_eq: fixes p :: "pat"   and v :: "value" shows "vmatch p v = vmatch' p v"
keys_frag_extend: fixes f :: "'c \<Rightarrow> 'b \<Rightarrow>\<^sub>0 int"   and c :: "'c \<Rightarrow>\<^sub>0 int" shows "keys (frag_extend f c) \<subseteq> (\<Union>x\<in>keys c. keys (f x))"
Range_Id_on: fixes A :: "'a set" shows "Range (Id_on A) = A"
continuous_within_tan: fixes z :: "'a"   and s :: "'a set" assumes "cos z \<noteq> (0::'a)" shows "continuous (at z within s) tan"
weak_Nullstellensatz_aux_1: fixes I :: "'c set"   and g :: "'c \<Rightarrow> 'd"   and B :: "'d set"   and f :: "'c \<Rightarrow> 'd"   and m :: "'c \<Rightarrow> nat" assumes "\<And>i. i \<in> I \<Longrightarrow> g i \<in> ideal B" obtains c :: "'d" where "c \<in> ideal B" and "(\<Prod>i\<in>I. (f i + g i) ^ m i) = (\<Prod>i\<in>I. f i ^ m i) + c"
higher_differentiable_on_norm: fixes S :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and n :: "nat" assumes "higher_differentiable_on S f n"   and "(0::'c) \<notin> f ` S"   and "open S" shows "higher_differentiable_on S (\<lambda>x. norm (f x)) n"
\<psi>_eqI: fixes f :: "'d" assumes "\<guillemotleft>f : \<I>\<^sub>D \<rightarrow>\<^sub>D F \<I>\<^sub>C\<guillemotright>"   and "D.iso f"   and "f \<cdot>\<^sub>D \<iota>\<^sub>D = \<iota>\<^sub>1 \<cdot>\<^sub>D (f \<otimes>\<^sub>D f)" shows "f = \<psi>"
restrict_relation_UNIV: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "restrict_relation UNIV R = R"
measurable_measure_of: fixes N :: "'c set set"   and \<Omega> :: "'c set"   and f :: "'d \<Rightarrow> 'c"   and M :: "'d measure"   and \<mu> :: "'c set \<Rightarrow> ennreal" assumes "N \<subseteq> Pow \<Omega>"   and "f \<in> space M \<rightarrow> \<Omega>"   and "\<And>y. y \<in> N \<Longrightarrow> f -` y \<inter> space M \<in> sets M" shows "f \<in> M \<rightarrow>\<^sub>M measure_of \<Omega> N \<mu>"
cases: fixes a1 :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and a2 :: "'t \<times> ('l, 't, 'x, 'm, 'w, 'o) thread_action"   and a3 :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" assumes "m\<tau>move a1 a2 a3" obtains s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" and t :: "'t" and x :: "'x" and s' :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" and x' :: "'x" and ln' :: "'l \<Rightarrow>f nat" and ta :: "('l, 't, 'x, 'm, 'w, 'o) thread_action" where "a1 = s" and "a2 = (t, ta)" and "a3 = s'" and "thr s t = \<lfloor>(x, no_wait_locks)\<rfloor>" and "thr s' t = \<lfloor>(x', ln')\<rfloor>" and "\<tau>move (x, shr s) ta (x', shr s')"
NS_staticSecret_parts_agent: fixes m :: "msg"   and C :: "agent"   and evs :: "event list"   and A :: "agent" assumes "m \<in> parts (knows C evs)"   and "m \<in> staticSecret A"   and "C \<noteq> A"   and "evs \<in> ns_public" shows "\<exists>B X. Says A B X \<in> set evs \<and> m \<in> parts {X}"
Keys_Un: fixes A :: "('a \<Rightarrow>\<^sub>0 'b) set"   and B :: "('a \<Rightarrow>\<^sub>0 'b) set" shows "Keys (A \<union> B) = Keys A \<union> Keys B"
sync_rules_4: fixes c :: "'b \<Rightarrow> 'c"   and C :: "'c set"   and a :: "'b"   and P :: "'c process"   and Q :: "'b \<Rightarrow> 'c process" assumes "\<And>y. c y \<in> C"   and "inj c" shows "(c`!`a \<rightarrow> P \<lbrakk>C\<rbrakk> read c UNIV Q) = (c`!`a \<rightarrow> (P \<lbrakk>C\<rbrakk> Q a))"
intro: fixes prognosis :: "(Vars.var \<Rightarrow> Arity\<^sub>\<bottom>) \<Rightarrow> Arity list \<Rightarrow> Arity \<Rightarrow> (Vars.var \<times> exp) list \<times> exp \<times> stack_elem list \<Rightarrow> Vars.var \<Rightarrow> one\<^sub>\<bottom>" assumes "\<And>ae as a \<Gamma> e S. edom (prognosis ae as a (\<Gamma>, e, S)) \<subseteq> fv \<Gamma> \<union> fv e \<union> fv S" shows "CardinalityPrognosisEdom prognosis"
continuous_ge_on_closure: fixes s :: "'c set"   and f :: "'c \<Rightarrow> real"   and x :: "'c"   and a :: "real" assumes "continuous_on (closure s) f"   and "x \<in> closure s"   and "\<And>x. x \<in> s \<Longrightarrow> a \<le> f x" shows "a \<le> f x"
gt_subst: fixes ground_heads_var :: "'a \<Rightarrow> 'b set"   and gt_sym :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and \<epsilon> :: "nat"   and \<delta> :: "nat"   and extf :: "'b \<Rightarrow> (('b, 'a) tm \<Rightarrow> ('b, 'a) tm \<Rightarrow> bool) \<Rightarrow> ('b, 'a) tm list \<Rightarrow> ('b, 'a) tm list \<Rightarrow> bool"   and arity_sym :: "'b \<Rightarrow> enat"   and arity_var :: "'a \<Rightarrow> enat"   and wt_sym :: "'b \<Rightarrow> nat"   and \<rho> :: "'a \<Rightarrow> ('b, 'a) tm"   and t :: "('b, 'a) tm"   and s :: "('b, 'a) tm" assumes "kbo_std ground_heads_var gt_sym \<epsilon> \<delta> extf arity_sym arity_var wt_sym"   and "ground_heads.wary_subst arity_sym arity_var ground_heads_var \<rho>"   and "arity.wary arity_sym arity_var t"   and "arity.wary arity_sym arity_var s"   and "Lambda_Free_KBO_Std.kbo_std.gt ground_heads_var gt_sym \<delta> extf arity_sym wt_sym t s" shows "Lambda_Free_KBO_Std.kbo_std.gt ground_heads_var gt_sym \<delta> extf arity_sym wt_sym (subst \<rho> t) (subst \<rho> s)"
M3: fixes meets :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and \<I> :: "'b \<Rightarrow> bool"   and p :: "'b" assumes "class.interval meets \<I>"   and "\<I> p" shows "\<exists>q r. meets q p \<and> meets p r"
coinduct_strong: fixes R :: "('h, 'i, 'j) gpv \<Rightarrow> ('h, 'i, 'j) gpv \<Rightarrow> bool"   and gpv :: "('h, 'i, 'j) gpv"   and gpv' :: "('h, 'i, 'j) gpv" assumes "R gpv gpv'"   and "\<And>gpv gpv'. R gpv gpv' \<Longrightarrow> rel_spmf (rel_generat (=) (=) (rel_fun (=) (\<lambda>gpv gpv'. R gpv gpv' \<or> gpv = gpv'))) (the_gpv gpv) (the_gpv gpv')" shows "gpv = gpv'"
minus_mult_cancel: fixes x :: "'a"   and a :: "real" assumes "x \<in> V" shows "- a \<cdot> - x = a \<cdot> x"
sec_prik_tok_prik: shows "range Sec_PriK \<inter> range Tok_PriK = {}"
simps_2: fixes x :: "'a" shows "primify (Inr x) = {Inl x, Inr x}"
rotation_bij_corollary: fixes r :: "Vertex \<Rightarrow> Vertex"   and x :: "Vertex"   and y :: "Vertex" assumes "r \<in> complex_rotations" shows "(r x \<noteq> r y) = (x \<noteq> y)"
lowner_is_lub_dim: fixes M :: "complex mat" assumes "local.lowner_is_lub M" shows "M \<in> carrier_mat dim dim"
dictatesx_def: fixes i :: "indi"   and c :: "alt" shows "i dictates_except c \<equiv> \<forall>a b. c \<notin> {a, b} \<longrightarrow> i dictates a < b"
inorder_inv_delete: fixes t :: "'t"   and x :: "'a" assumes "inv t \<and> sorted (inorder t)" shows "inv (delete x t)"
abs_conv_abscissa_leI_weak: fixes d :: "ereal"   and f :: "'a fds" assumes "\<And>x. d < ereal x \<Longrightarrow> fds_abs_converges f (of_real x)" shows "abs_conv_abscissa f \<le> d"
t1_space_alt: fixes X :: "'b topology" shows "t1_space X = (\<forall>x\<in>topspace X. \<forall>y\<in>topspace X. x \<noteq> y \<longrightarrow> (\<exists>U. closedin X U \<and> x \<in> U \<and> y \<notin> U))"
f_chain: shows "\<forall>i\<in>{m..<n}. (f i, f (Suc i)) \<in> R"
monofun_mHomTr: fixes h :: "'a"   and H :: "'a set"   and a :: "'b"   and r :: "'b" assumes "h \<in> H"   and "free_generator R M H"   and "a \<in> carrier R"   and "r \<in> carrier R"   and "a \<cdot>\<^sub>s h = r \<cdot>\<^sub>s h" shows "a = r"
distinct_21: fixes x21 :: "floatarith"   and x22 :: "floatarith"   and x23 :: "form"   and x71 :: "form"   and x72 :: "form" shows "Assign x21 x22 x23 \<noteq> Disj x71 x72"
prod_algebra_def: fixes I :: "'d set"   and M :: "'d \<Rightarrow> 'e measure" shows "prod_algebra I M = (\<lambda>(J, X). prod_emb I M J (Pi\<^sub>E J X)) ` {(J, X). (J \<noteq> {} \<or> I = {}) \<and> finite J \<and> J \<subseteq> I \<and> X \<in> (\<Pi> j\<in>J. sets (M j))}"
assms: shows "\<Omega> \<inter> space M \<in> sets M"
foldli_transform: fixes iti :: "('d \<Rightarrow> bool) \<Rightarrow> ('e \<Rightarrow> 'd \<Rightarrow> 'd) \<Rightarrow> 'd \<Rightarrow> 'd"   and S0 :: "'e set"   and R :: "'e \<Rightarrow> 'e \<Rightarrow> bool" assumes "set_iterator_genord iti S0 R" shows "\<exists>l0. distinct l0 \<and> S0 = set l0 \<and> sorted_wrt R l0 \<and> iti = foldli l0"
abs_diff_less_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 (abs (minus x a)) r = (less (minus a r) x \<and> less x (plus a r))"
monom_mult_i_def: fixes ops :: "'b arith_ops_record"   and n :: "nat"   and xs :: "'b list" shows "monom_mult_i ops n xs = (if xs = [] then xs else replicate n (arith_ops_record.zero ops) @ xs)"
cdi_prefix: fixes prob :: "('a, 'b, 'c, 'd) ifc_problem"   and n :: "nat"   and \<pi> :: "nat \<Rightarrow> 'a"   and m :: "nat"   and n' :: "nat" assumes "IFC prob"   and "IFC_def.is_cdi prob n \<pi> m"   and "m < n'"   and "n' \<le> n" shows "IFC_def.is_cdi prob n' \<pi> m"
substitute'_Nil: fixes f :: "'b \<Rightarrow> 'b ttree"   and T :: "'b set"   and t :: "'b ttree" shows "substitute' f T t [] = True"
rep_eq: fixes x :: "'a list" shows "Rep_vec_impl (vec_of_list_impl x) = (length x, IArray x)"
eval: shows "P \<turnstile> \<langle>body,(h, [pns [\<mapsto>] vs], sh)\<rangle> \<Rightarrow> \<langle>e',(h', l', sh')\<rangle>"