hcju/isabelleps huggingface dataset:
is_sig_red_addsE: fixes sing_reg :: "'t \<Rightarrow> 't \<Rightarrow> bool"   and top_tail :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and F :: "('t \<Rightarrow>\<^sub>0 'b) set"   and p :: "'t \<Rightarrow>\<^sub>0 'b" assumes "is_sig_red sing_reg top_tail F p" obtains f :: "'t \<Rightarrow>\<^sub>0 'b" and t :: "'a" where "f \<in> F" and "t \<in> keys (rep_list p)" and "rep_list f \<noteq> 0" and "punit.lt (rep_list f) adds t" and "sing_reg (t \<oplus> lt f) (punit.lt (rep_list f) \<oplus> lt p)" and "top_tail t (punit.lt (rep_list p))"
LS_def: shows "LS = {y. \<exists>x\<in>S. - a + x = y}"
fold_mset_add_mset: fixes f :: "'k \<Rightarrow> 'l \<Rightarrow> 'l"   and s :: "'l"   and x :: "'k"   and M :: "'k multiset" assumes "comp_fun_commute f" shows "fold_mset f s (add_mset x M) = f x (fold_mset f s M)"
rel_map_on_set_def: fixes S :: "'c set"   and P :: "'d \<Rightarrow> 'e \<Rightarrow> bool" shows "rel_map_on_set S P = rel_fun (eq_onp (\<lambda>x. x \<in> S)) (rel_option P)"
preserves_cod: fixes f :: "'c local.term set" assumes "\<F>C.arr f" shows "D.cod (E.map f) = E.map (\<F>C.cod f)"
fps_nth_Lambert_W: fixes n :: "nat" shows "fps_Lambert_W $ n = (if n = 0 then 0 else (- real n) ^ (n - 1) / fact n)"
min_height_size: fixes kdt :: "'b kdt" shows "2 ^ min_height kdt \<le> KD_Tree.size_kdt kdt"
gcs_comm: fixes s :: "'a"   and t :: "'a" shows "gcs s t = gcs t s"
type_conf_Integer: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and e :: "expr" shows "(P,E,h \<turnstile> e :\<^bsub>NT\<^esub> Integer) = (P,E,h \<turnstile> e : Integer)"
\<tau>moves2_stk_append: fixes P :: "'a prog"   and h :: "'heap"   and stk :: "'addr val list"   and es :: "'addr expr1 list"   and pc :: "nat"   and xcp :: "'addr option"   and vs :: "'addr val list" assumes "\<tau>moves2 P h stk es pc xcp" shows "\<tau>moves2 P h (stk @ vs) es pc xcp"
cone_list_def: fixes Vs :: "'a vec list" shows "m.cone_list Vs = {b. \<exists>c. m.nonneg_lincomb_list c Vs b}"
action_select: fixes a :: "('f, 'd) expr"   and b :: "'e data \<Rightarrow> bool"   and c :: "'d set \<times> 'e pupdate" shows "action (a, b, c) = c"
D5_2: fixes \<phi> :: "\<kappa> \<Rightarrow> \<kappa> \<Rightarrow> \<o>"   and r :: "\<upsilon> \<Rightarrow> \<upsilon> \<Rightarrow> j \<Rightarrow> i \<Rightarrow> bool"   and o\<^sub>1 :: "\<nu>"   and x :: "\<kappa>"   and o\<^sub>2 :: "\<nu>"   and y :: "\<kappa>"   and w :: "i" assumes "IsProperInXY \<phi>" shows "Some r = d\<^sub>2 (\<^bold>\<lambda>\<^sup>2 (\<lambda>x y. \<phi> (x\<^sup>P) (y\<^sup>P))) \<and> Some o\<^sub>1 = d\<^sub>\<kappa> x \<and> Some o\<^sub>2 = d\<^sub>\<kappa> y \<longrightarrow> ((o\<^sub>1, o\<^sub>2) \<in> ex2 r w) = [\<phi> x y in w]"
vfsequence_vdomain: shows "\<D>\<^sub>\<circ> [a, b]\<^sub>\<circ> = vcard [a, b]\<^sub>\<circ>"
xcpt_eff: fixes i :: "instr"   and P :: "'a prog"   and pc :: "nat"   and \<tau> :: "ty list \<times> ty err list"   and et :: "ex_table" shows "xcpt_eff i P pc \<tau> et = (let (ST, LT) = \<tau> in map (\<lambda>(f, t, C, h, d). (h, \<lfloor>(Class C # drop (length ST - d) ST, LT)\<rfloor>)) (relevant_entries P i pc et))"
rk_triple_le: fixes A :: "Points"   and B :: "Points"   and C :: "Points" shows "rk {A, B, C} \<le> 3"
exec_step_ind_NewOOM_Called: fixes h :: "heap"   and C :: "char list"   and P :: "jvm_prog"   and stk :: "val list"   and loc :: "val list"   and C\<^sub>0 :: "char list"   and M\<^sub>0 :: "char list"   and pc :: "nat"   and Cs :: "char list list"   and frs :: "frame list"   and sh :: "sheap" assumes "new_Addr h = None" shows "exec_step_ind (StepI (New C)) P h stk loc C\<^sub>0 M\<^sub>0 pc (Called Cs) frs sh (\<lfloor>addr_of_sys_xcpt OutOfMemory\<rfloor>, h, (stk, loc, C\<^sub>0, M\<^sub>0, pc, No_ics) # frs, sh)"
extremally_disconnected_def: fixes X :: "'a set" shows "extremally_disconnected X = (K \<circ> I \<circ> K = I \<circ> K)"
fps_normalize_def: fixes f :: "'a fps" shows "normalize f = (if f = 0 then 0 else fps_X ^ subdegree f)"
ex_large_of_nat: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and y :: "'b" assumes "class.large_ordered_semiring_1 times one plus zero less_eq" shows "\<exists>x. less_eq y (semiring_1.of_nat one plus zero x)"
ev_eq_lfp: fixes P :: "'a stream \<Rightarrow> bool" shows "ev P = lfp (\<lambda>F \<omega>. P \<omega> \<or> \<not> P \<omega> \<and> F (stl \<omega>))"
approx_tse_form'_le: fixes prec :: "nat"   and t :: "nat"   and a :: "floatarith"   and b :: "floatarith"   and s :: "nat"   and ivl :: "float interval"   and x :: "real" assumes "approx_tse_form' prec t (Add a (Minus b)) s ivl (\<lambda>ivl. 0 \<le> lower ivl)"   and "x \<in>\<^sub>r ivl" shows "interpret_floatarith b [x] \<le> interpret_floatarith a [x]"
21: shows "{0..<x} \<times> {0..<y} \<subseteq> (\<lambda>ia. (ia div y, ia mod y)) ` {0..<x * y}"
wffp_purgeIdle: fixes do :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'a"   and sl :: "('a, 'b, 'c) state list" assumes "GM_sec_model.wffp do sl"   and "GM_sec_model.purgeIdle sl \<noteq> []" shows "GM_sec_model.wffp do (GM_sec_model.purgeIdle sl)"
vertexI2: fixes \<Gamma> :: "('k, 'l) graph_scheme"   and x :: "'k"   and y :: "'k" assumes "edge \<Gamma> x y" shows "vertex \<Gamma> y"
1: shows "((case_option {None} ((`) Some) \<circ> f)(x := {None})) x \<noteq> {}"
lossless: fixes s :: "'a" assumes "guard s" shows "lossless_spmf (body s)"
finite_Lim_measure_incseq: fixes A :: "nat \<Rightarrow> 'a set" assumes "range A \<subseteq> sets M"   and "incseq A" shows "(\<lambda>i. Sigma_Algebra.measure M (A i)) \<longlonglongrightarrow> Sigma_Algebra.measure M (\<Union> (range A))"
distinct_19: fixes x21 :: "'c"   and x22 :: "'c \<Rightarrow> ('a, 'c) trm"   and x61 :: "('a, 'b, 'c) hp"   and x62 :: "('a, 'b, 'c) formula" shows "$\<phi> x21 x22 \<noteq> (\<langle> x61 \<rangle> x62)"
\<eta>_in_terms_of_\<phi>: fixes y :: "'b" assumes "Adj.D.ide y" shows "Adj'.\<eta> y = \<phi>' y (F y)"
path_vector_deriv_line_integrals: shows "\<forall>x\<in>{0..1}. ((\<lambda>x. \<gamma> x \<bullet> i) has_vector_derivative vector_derivative (\<lambda>x. \<gamma> x \<bullet> i) (at x)) (at x)"
append_eq_Cons_conv: fixes ys :: "'c list"   and zs :: "'c list"   and x :: "'c"   and xs :: "'c list" shows "(ys \<cdot> zs = x # xs) = (ys = [] \<and> zs = x # xs \<or> (\<exists>ys'. ys = x # ys' \<and> ys' \<cdot> zs = xs))"
simps: fixes a :: "'a list" shows "array_length (Array a) = length a"
hd_sort_remdups: fixes l :: "'b list" shows "hd (sort (remdups l)) = hd (sort l)"
triangle_num_ge: fixes n :: "nat" shows "n \<le> triangle_num n"
simps_7: fixes t :: "'c local.term" shows "Inv \<^bold>\<r>\<^bold>[t\<^bold>] = \<^bold>\<r>\<^sup>-\<^sup>1\<^bold>[Inv t\<^bold>]"
insert_3: shows "eval_poly hom (sum p A) x = (\<Sum>k\<in>A. eval_poly hom (p k) x)"
intro: fixes S1 :: "'c set"   and basis :: "'c set"   and S2 :: "'d set" assumes "finite_dimensional_real_vector_space_on S1 basis"   and "real_vector_space_on S2" shows "finite_dimensional_real_vector_space_pair_1_on S1 S2 basis"
dist_of_int: fixes m :: "int"   and n :: "int" shows "dist (of_int m) (of_int n) = real_of_int \<bar>m - n\<bar>"
inject_2: fixes x2 :: "Vars.var"   and y2 :: "Vars.var" shows "(Arg x2 = Arg y2) = (x2 = y2)"
convex_hull_carrier: fixes Vs :: "'b vec set"   and n :: "nat" assumes "Vs \<subseteq> carrier_vec n" shows "gram_schmidt.convex_hull n Vs \<subseteq> carrier_vec n"
rvars_pivot_eq: fixes x\<^sub>j :: "nat"   and eq :: "nat \<times> linear_poly" assumes "x\<^sub>j \<in> rvars_eq eq"   and "lhs eq \<notin> rvars_eq eq" shows "rvars_eq (pivot_eq eq x\<^sub>j) = {lhs eq} \<union> (rvars_eq eq - {x\<^sub>j})"
wf_dual_rexp_dual_of: fixes \<Sigma> :: "'a list"   and n :: "nat"   and r :: "'a atom rexp" assumes "WS1S_Equivalence_Checking.wf \<Sigma> n r" shows "wf_dual \<Sigma> n (rexp_dual_of r)"
inf_top_left: fixes inf :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and less_eq :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and less :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and top :: "'b::type"   and x :: "'b::type" assumes "class.bounded_semilattice_inf_top (inf::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (less::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (top::'b::type)" shows "(inf::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (top::'b::type) (x::'b::type) = x"
really_stupid: fixes s :: "tm" shows "\<not> (\<forall>c1 s'. (c1, s') \<noteq> split0 s)"
Class_closed2: fixes a :: "'a \<times> 'a" assumes "a \<in> R \<times> S" shows "rel.Class a \<subseteq> R \<times> S"
composable: shows "src \<nu> = trg (\<pi> \<star> \<rho>)"
\<G>\<F>_semantics: fixes \<phi> :: "'b ltln"   and w :: "nat \<Rightarrow> 'b set" shows "\<G>\<F> \<phi> w = {\<psi> \<in> subformulas\<^sub>\<mu> \<phi>. w \<Turnstile>\<^sub>n G\<^sub>n (F\<^sub>n \<psi>)}"
open_domain: fixes c :: "('a, 'b) chart" shows "open (domain c)"
assms_4: fixes h :: "'a"   and h' :: "'a" shows "\<forall>w\<in>SW. h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> (\<forall>r\<in>SR. r h h')"
ih1: fixes a :: "coname"   and M :: "trm" assumes "<a>:M \<in> \<parallel><A>\<parallel>" shows "SNa M"
simps_2: fixes k :: "'c"   and k' :: "'c"   and v :: "'d"   and xs :: "('c \<times> 'd) list" shows "delete_aux k ((k', v) # xs) = (if k = k' then xs else (k', v) # delete_aux k xs)"
V_2: fixes l :: "nat"   and i :: "nat" assumes "l \<le> i" shows "V i \<bullet> (T ^^ l) (0::'a) \<le> (- A + eps i) * real l"
dim_gen_eigenspace_max_jordan_block: fixes A :: "'a mat"   and n_as :: "(nat \<times> 'a) list"   and l :: "'a"   and d :: "nat" assumes "jordan_nf A n_as" shows "(dim_gen_eigenspace A l d = order l (char_poly A)) = (\<forall>n. (n, l) \<in> set n_as \<longrightarrow> n \<le> d)"
abs_eq: shows "o_circline_point_0 = abs_ocircline circline_point_0_clmat"
ceuclidean_representation_sum_fun: fixes scaleR :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and scaleC :: "complex \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and norm :: "'c \<Rightarrow> real"   and sgn :: "'c \<Rightarrow> 'c"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and cinner :: "'c \<Rightarrow> 'c \<Rightarrow> complex"   and CBasis :: "'c set"   and f :: "'d \<Rightarrow> 'c" assumes "class.ceuclidean_space scaleR scaleC plus zero minus uminus dist norm sgn uniformity open cinner CBasis" shows "(\<lambda>x. comm_monoid_add.sum plus zero (\<lambda>b. scaleC (cinner b (f x)) b) CBasis) = f"
ign_res_eq_def: fixes P :: "'b assn"   and w :: "vals" shows "P\<down>=w \<equiv> \<lambda>Y:. P\<down> \<and>. (\<lambda>s. Y = w)"
rupture_def: fixes R :: "('c, 'd) ring_scheme"   and K :: "'c set"   and p :: "'c list" shows "Rupt\<^bsub>R\<^esub> K p = K [X]\<^bsub>R\<^esub> Quot PIdl\<^bsub>K [X]\<^bsub>R\<^esub>\<^esub> p"
simps_2: fixes b1 :: "bool"   and v :: "'b aexp"   and va :: "'b aexp" shows "GExp_Lexorder.less_gexp_aux (Bc b1) (Eq v va) = True"
corthogonal_mat_def: fixes A :: "'b mat" shows "corthogonal_mat A \<equiv> let B = adjoint A * A in diagonal_mat B \<and> (\<forall>i<dim_col A. B $$ (i, i) \<noteq> (0::'b))"
a_sum_le_U_sum: fixes ot :: "'op rose_tree" assumes "local.wf ot" shows "acost_sum ot \<le> U_sum ot"
H_1: shows "cauchy_in\<^sub>N bcontfun\<^sub>N u"
dg_prod_is_arrD_2: fixes a :: "V"   and b :: "V"   and f :: "V" assumes "f : a \<mapsto>\<^bsub>dg_prod I \<AA>\<^esub> b" shows "\<D>\<^sub>\<circ> f = I"
nested_submodules: fixes W :: "'c set"   and X :: "'c set" assumes "LinearCombinations.submodule K W V"   and "LinearCombinations.submodule K X V"   and "X \<subseteq> W" shows "LinearCombinations.submodule K X (vs W)"
simps_1: fixes f1 :: "'f \<Rightarrow> 'd"   and f2 :: "'g \<Rightarrow> 'e"   and a :: "'f" shows "map_sum f1 f2 (Inl a) = Inl (f1 a)"
E: shows "E \<in> (\<Pi> i\<in>J. sets (M i))"
vsubst_commute: fixes X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and xs :: "'varSort"   and xs' :: "'varSort"   and x :: "'var"   and y :: "'var"   and x' :: "'var"   and y' :: "'var" assumes "good X"   and "xs \<noteq> xs' \<or> {x, y} \<inter> {x', y'} = {}"   and "fresh xs x X"   and "fresh xs' x' X" shows "X #[x // y]_xs #[x' // y']_xs' = X #[x' // y']_xs' #[x // y]_xs"
not_in_arcs_G: shows "{uw, rev_H uw, vw, rev_H vw} \<inter> arcs G = {}"
intros_1: fixes nid :: "'nid"   and t :: "nat \<Rightarrow> cnf"   and n :: "nat" assumes "\<parallel>nid\<parallel>\<^bsub>t n\<^esub>" shows "(n, nid) \<in> his t n nid"
wfactors_exist: fixes G :: "('c, 'd) monoid_scheme"   and a :: "'c" assumes "factorial_monoid G"   and "a \<in> carrier G" shows "\<exists>fs. set fs \<subseteq> carrier G \<and> wfactors G fs a"
neighbourhood_base_at_with_subset: fixes X :: "'b topology"   and U :: "'b set"   and x :: "'b"   and P :: "'b set \<Rightarrow> bool" assumes "openin X U"   and "x \<in> U" shows "neighbourhood_base_at x P X = neighbourhood_base_at x (\<lambda>T. T \<subseteq> U \<and> P T) X"
cases: fixes a :: "'a list" assumes "steps a" obtains x :: "'a" where "a = [xa__]"   | x :: "'a" and y :: "'a" and xs :: "'a list" where "a = x # y # xs" and "x \<rightarrow> y" and "steps (y # xs)"
concat_tl: fixes x :: "'b list"   and xs :: "'b list list"   and G :: "'b list set" assumes "x # xs \<in> lists G" shows "concat xs \<in> \<langle>G\<rangle>"
j_def: shows "j = (LEAST j. i \<le> j \<and> n \<in> acc (fst (r j)))"
BV_correct_initial: fixes \<Phi> :: "char list \<Rightarrow> char list \<Rightarrow> (ty list \<times> ty err list) option list"   and P :: "jvm_prog"   and C :: "char list"   and M :: "char list"   and m :: "jvm_method"   and \<Phi>' :: "char list \<Rightarrow> char list \<Rightarrow> (ty list \<times> ty err list) option list" assumes "wf_jvm_prog\<^bsub>\<Phi>\<^esub> P"   and "\<not> is_class P Start"   and "P \<turnstile> C sees M, Static :  []\<rightarrow>Void = m in C"   and "M \<noteq> clinit"   and "\<Phi>' Start start_m = start_\<phi>\<^sub>m" shows "class_add P (start_class C M),\<Phi>' |- start_state P [ok]"
ListReds2: fixes es :: "char list exp list"   and s :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and es' :: "char list exp list"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and 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 v :: "val" assumes "P \<turnstile> \<langle>es,s\<rangle> [\<rightarrow>]* \<langle>es',s'\<rangle>" shows "P \<turnstile> \<langle>Val v # es,s\<rangle> [\<rightarrow>]* \<langle>Val v # es',s'\<rangle>"
InitPhase_HInv3_p: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and q :: "Proc"   and d :: "Disk" assumes "InitializePhase s s' p"   and "HInv3_L s' p q d" shows "HInv3_R s' p q d"
skip_elim: fixes mds :: "Mode \<Rightarrow> 'Var set"   and mem :: "'Var \<Rightarrow> 'Val"   and c' :: "('Var, 'AExp, 'BExp) Stmt"   and mds' :: "Mode \<Rightarrow> 'Var set"   and mem' :: "'Var \<Rightarrow> 'Val" assumes "\<langle>Skip, mds, mem\<rangle>\<^sub>w \<leadsto>\<^sub>w \<langle>c', mds', mem'\<rangle>\<^sub>w" shows "c' = Stop \<and> mds = mds' \<and> mem = mem'"
var_def: fixes g :: "'g" shows "var g \<equiv> fst"
rank_le_1_product_entries: fixes A :: "'b mat"   and n :: "nat"   and nc :: "nat"   and f :: "nat \<Rightarrow> 'b"   and g :: "nat \<Rightarrow> 'b" assumes "A \<in> carrier_mat n nc"   and "\<And>r c. \<lbrakk>r < dim_row A; c < dim_col A\<rbrakk> \<Longrightarrow> A $$ (r, c) = f r * g c" shows "vec_space.rank n A \<le> 1"
prikey_spied: fixes s :: "(agent \<times> msg) set"   and K :: "nat" assumes "s\<^sub>0 \<Turnstile> s"   and "PriKey K \<in> parts (used s)" shows "PriKey K \<in> spied s"
convergent_ignore_initial_segment: fixes f :: "nat \<Rightarrow> 'c"   and m :: "nat" shows "convergent (\<lambda>n. f (n + m)) = convergent f"
div_dvd_iff_mult: fixes b :: "'a"   and a :: "'a"   and c :: "'a" assumes "b \<noteq> (0::'a)"   and "b dvd a" shows "(a div b dvd c) = (a dvd c * b)"
option_neq_cases: fixes x :: "'a option"   and y :: "'a option" assumes "x \<noteq> y" obtains a :: "'a" where "x = None" and "y = Some a__"   | a :: "'a" where "x = Some aa__" and "y = None"   | a :: "'a" and b :: "'a" where "x = Some a" and "y = Some b" and "a \<noteq> b"
translation_invertible: fixes G :: "'c set"   and composition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit :: "'c"   and a :: "'c" assumes "right_translations_of_group G composition unit"   and "a \<in> G" shows "monoid.invertible (right_translations_of_group.translation G composition ` G) (compose G) (identity G) (right_translations_of_group.translation G composition a)"
induct: fixes x1 :: "call_kind"   and x2 :: "typeop"   and x3 :: "'a OCL_Types.type"   and x4 :: "'a OCL_Types.type"   and x5 :: "'a OCL_Types.type"   and P :: "call_kind \<Rightarrow> typeop \<Rightarrow> 'a OCL_Types.type \<Rightarrow> 'a OCL_Types.type \<Rightarrow> 'a OCL_Types.type \<Rightarrow> bool" assumes "typeop_type x1 x2 x3 x4 x5"   and "\<And>\<sigma> \<tau>. \<sigma> < \<tau> \<or> \<tau> < \<sigma> \<Longrightarrow> P DotCall OclAsTypeOp \<tau> \<sigma> \<sigma>"   and "\<And>\<sigma> \<tau>. \<sigma> < \<tau> \<Longrightarrow> P DotCall OclIsTypeOfOp \<tau> \<sigma> Boolean[1]"   and "\<And>\<sigma> \<tau>. \<sigma> < \<tau> \<Longrightarrow> P DotCall OclIsKindOfOp \<tau> \<sigma> Boolean[1]"   and "\<And>\<tau> \<rho> \<sigma> \<upsilon>. \<lbrakk>element_type \<tau> \<rho>; \<sigma> < \<rho>; update_element_type \<tau> \<sigma> \<upsilon>\<rbrakk> \<Longrightarrow> P ArrowCall SelectByKindOp \<tau> \<sigma> \<upsilon>"   and "\<And>\<tau> \<rho> \<sigma> \<upsilon>. \<lbrakk>element_type \<tau> \<rho>; \<sigma> < \<rho>; update_element_type \<tau> \<sigma> \<upsilon>\<rbrakk> \<Longrightarrow> P ArrowCall SelectByTypeOp \<tau> \<sigma> \<upsilon>" shows "P x1 x2 x3 x4 x5"
sin_angle_zero_iff: fixes a :: "'b"   and b :: "'b"   and c :: "'b" shows "(sin (angle a b c) = 0) = (angle a b c \<in> {0, pi})"
not_reach_out: fixes G :: "('c, 'd) pre_digraph"   and S :: "'c set"   and u :: "'c"   and v :: "'c" assumes "max_reachable_set G S"   and "u \<in> S"   and "v \<notin> S" shows "\<not> u \<rightarrow>\<^sup>*\<^bsub>G\<^esub> v \<or> \<not> v \<rightarrow>\<^sup>*\<^bsub>G\<^esub> u"
info_rel_def: fixes I :: "('a \<times> 'e) set"   and S :: "('c \<times> 'd) set" shows "\<langle>I, S\<rangle>info_rel = (I \<times>\<^sub>r S) O br snd top"
it_to_list_def: fixes it :: "'c \<Rightarrow> ('d \<Rightarrow> bool) \<Rightarrow> ('e \<Rightarrow> 'e list \<Rightarrow> 'e list) \<Rightarrow> 'f list \<Rightarrow> 'b"   and s :: "'c" shows "it_to_list it s \<equiv> it s (\<lambda>_. True) (\<lambda>x l. l @ [x]) []"
convergence_G_def: fixes s :: "'s"   and t :: "'s"   and z :: "'a" shows "convergence_G s t z = summable (\<lambda>n. p s t n * norm z ^ n)"
cw_len_length: fixes w :: "'b list" shows "cw_len_concat w = length (enc w)"
dynamically_complete: fixes B :: "'a"   and Ns :: "'a set llist" assumes "B \<in> Bot"   and "chain (\<rhd>) Ns"   and "fair Ns"   and "lhd Ns \<Turnstile> {B}" shows "\<exists>i\<in>{i. enat i < llength Ns}. \<exists>B'\<in>Bot. B' \<in> lnth Ns i"
simple_ports_conjunct_correct: fixes p1 :: "16 word \<times> 16 word"   and pkt :: "16 word"   and p2 :: "16 word \<times> 16 word" shows "(simple_match_port p1 pkt \<and> simple_match_port p2 pkt) = simple_match_port (simpl_ports_conjunct p1 p2) pkt"
proj_def: fixes n :: "'d"   and S :: "('d strand_label \<times> 'e) list" shows "proj n S \<equiv> filter (\<lambda>s. is_LabelN n s \<or> is_LabelS s) S"
trans1: fixes l :: "nat"   and a1 :: "nat"   and a2 :: "nat"   and a3 :: "nat"   and a4 :: "nat"   and r :: "nat" assumes "l = a1 + a2 + (a3 + a4)"   and "a1 + a3 = r" shows "l = r + a2 + a4"
compat_NS_S_point: fixes S :: "('a \<times> 'a) set"   and NS :: "('a \<times> 'a) set"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "compat_pair S NS"   and "(x, y) \<in> NS"   and "(y, z) \<in> S" shows "(x, z) \<in> S"
arrow_of_spans_of_maps_axioms: shows "arrow_of_spans_of_maps (\<cdot>) (\<star>) \<a> \<i> src trg (tab\<^sub>0 f) (tab\<^sub>1 f) (tab\<^sub>0 f) (tab\<^sub>1 f) f.chine"
E_TAbs: fixes T\<^sub>1\<^sub>1 :: "POPLmarkRecord.type"   and t\<^sub>1\<^sub>2 :: "trm"   and T\<^sub>2 :: "POPLmarkRecord.type" shows "(\<lambda><:T\<^sub>1\<^sub>1. t\<^sub>1\<^sub>2) \<bullet>\<^sub>\<tau> T\<^sub>2 \<longmapsto>\<^sub>c t\<^sub>1\<^sub>2[0 \<mapsto>\<^sub>\<tau> T\<^sub>2]"
prems_7: shows "sorted_wrt (\<lambda>x y. y < x) \<alpha>s"
assoc: fixes id :: "'b"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "Matrix_Tensor.mult id f" shows "f (f a b) c = f a (f b c)"
arc_to_block_is_inj_in_arcs: fixes v :: "'a" shows "inj_on arc_to_block (in_arcs G v)"
Rep_inverse: fixes Rep :: "'e \<Rightarrow> 'f"   and Abs :: "'f \<Rightarrow> 'e"   and A :: "'f set"   and x :: "'e" assumes "type_definition Rep Abs A" shows "Abs (Rep x) = x"
simps_2: fixes n :: "nat"   and rai :: "int poly \<times> rat \<times> rat" shows "to_rat_2 (Irrational n rai) = None"
lunit'_naturality: fixes f :: "'d" assumes "arr f" shows "(\<I> \<otimes> f) \<cdot> \<l>\<^sup>-\<^sup>1[local.dom f] = \<l>\<^sup>-\<^sup>1[cod f] \<cdot> f"
record_simps_5: fixes f :: "'h \<Rightarrow> 'f"   and g :: "'i \<Rightarrow> 'h"   and x :: "('i, 'g) test1" shows "update_field_t11 f (update_field_t11 g x) = update_field_t11 (\<lambda>a. f (g a)) x"
arr_char: fixes f :: "'a" shows "hom.arr f = \<guillemotleft>f : src (src \<mu>) \<rightarrow> src e\<guillemotright>"
inj_idom_hom_def: fixes hom :: "'a \<Rightarrow> 'b" shows "inj_idom_hom hom \<equiv> idom_hom hom \<and> zero_hom_0 hom"
Mp_coeff: fixes f :: "int poly"   and i :: "nat" shows "coeff (pll.Mp f) i = pll.M (coeff f i)"
complements_do_not_falsify: fixes l\<^sub>1 :: "fterm literal"   and C\<^sub>1' :: "fterm literal set"   and l\<^sub>2 :: "fterm literal"   and G :: "bool list" assumes "l\<^sub>1 \<in> C\<^sub>1'"   and "l\<^sub>2 \<in> C\<^sub>1'"   and "falsifies\<^sub>g G C\<^sub>1'" shows "l\<^sub>1 \<noteq> l\<^sub>2\<^sup>c"
collapse_1: fixes "term" :: "'a local.term" assumes "is_Prim term" shows "\<^bold>\<langle>un_Prim term\<^bold>\<rangle> = term"
nth_equalityI: fixes xs :: "'b list"   and ys :: "'b list" assumes "\<^bold>|xs\<^bold>| = \<^bold>|ys\<^bold>|"   and "\<And>i. i < \<^bold>|xs\<^bold>| \<Longrightarrow> xs ! i = ys ! i" shows "xs = ys"
map_upds_apply_eq_Some: fixes m :: "'d \<Rightarrow> 'c option"   and xs :: "'d list"   and ys :: "'c list"   and x :: "'d"   and y :: "'c" shows "((m(xs [\<mapsto>] ys)) x = Some y) = (let xs' = take (length ys) xs in if x \<in> set xs' then ys ! index xs' x = y else m x = Some y)"
arr_dom: fixes f :: "'a \<times> 'a" assumes "H.VV.arr f" shows "H.VV.arr (H.VV.dom f)"
hyps_2: shows "\<exists>x\<in>grid b ds. p \<in> grid x ds'"
uniform_limit_iff: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd"   and S :: "'c set"   and l :: "'c \<Rightarrow> 'd"   and F :: "'b filter" shows "uniform_limit S f l F = (\<forall>e>0. \<forall>\<^sub>F n in F. \<forall>x\<in>S. dist (f n x) (l x) < e)"
asym: fixes a :: "'b DBMEntry"   and b :: "'b DBMEntry" assumes "a \<prec> b"   and "b \<prec> a" shows "False"
32: shows "bprv (imp (SS \<langle>\<chi>\<rangle> \<langle>\<psi>\<rangle>) (all yy' (imp (SS \<langle>\<chi>\<rangle> (Var yy')) (eql \<langle>\<psi>\<rangle> (Var yy')))))"