hcju/isabelleps huggingface dataset:
simps_2: fixes insts :: "((variable \<times> String.literal set) \<times> typ) list"   and idn :: "variable"   and S :: "String.literal set" shows "subst_typ insts (Tv idn S) = the_default (Tv idn S) (lookup (\<lambda>x. x = (idn, S)) insts)"
lexext_Nil: fixes gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and y :: "'b"   and ys :: "'b list" shows "lexext gt (y # ys) []"
div_dvd_div: 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 a :: "'a"   and b :: "'a"   and c :: "'a" assumes "class.algebraic_semidom divide plus minus zero times one"   and "dvd.dvd times a b"   and "dvd.dvd times a c" shows "dvd.dvd times (divide b a) (divide c a) = dvd.dvd times b c"
is_strongly_scdom_component_safe_def: fixes S\<^sub>a\<^sub>r\<^sub>g :: "(_) object_ptr set"   and S\<^sub>r\<^sub>e\<^sub>s\<^sub>u\<^sub>l\<^sub>t :: "(_) object_ptr set"   and h :: "(_) heap"   and h' :: "(_) heap" shows "is_strongly_scdom_component_safe S\<^sub>a\<^sub>r\<^sub>g S\<^sub>r\<^sub>e\<^sub>s\<^sub>u\<^sub>l\<^sub>t h h' = (let removed_pointers = fset (object_ptr_kinds h) - fset (object_ptr_kinds h'); added_pointers = fset (object_ptr_kinds h') - fset (object_ptr_kinds h); arg_components = \<Union>ptr\<in>(\<Union>ptr\<in>S\<^sub>a\<^sub>r\<^sub>g. set |h \<turnstile> local.a_get_scdom_component ptr|\<^sub>r) \<inter> fset (object_ptr_kinds h). set |h \<turnstile> local.a_get_scdom_component ptr|\<^sub>r; arg_components' = \<Union>ptr\<in>(\<Union>ptr\<in>S\<^sub>a\<^sub>r\<^sub>g. set |h \<turnstile> local.a_get_scdom_component ptr|\<^sub>r) \<inter> fset (object_ptr_kinds h'). set |h' \<turnstile> local.a_get_scdom_component ptr|\<^sub>r in removed_pointers \<subseteq> arg_components \<and> added_pointers \<subseteq> arg_components' \<and> S\<^sub>r\<^sub>e\<^sub>s\<^sub>u\<^sub>l\<^sub>t \<subseteq> arg_components' \<and> (\<forall>outside_ptr\<in>fset (object_ptr_kinds h) \<inter> fset (object_ptr_kinds h') - (\<Union>ptr\<in>S\<^sub>a\<^sub>r\<^sub>g. set |h \<turnstile> local.a_get_scdom_component ptr|\<^sub>r). preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t outside_ptr id) h h'))"
instInp: fixes \<tau> :: "'fmla"   and t :: "'trm"   and n :: "nat" assumes "\<tau> \<in> ptrm (Suc 0)"   and "t \<in> trm"   and "FvarsT t = Variable ` {Suc 0..n}" shows "instInp \<tau> t \<in> ptrm n"
no_matching_Goto_append: fixes \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and rs1 :: "'c rule list"   and rs2 :: "'c rule list" shows "no_matching_Goto \<gamma> p (rs1 @ rs2) = (no_matching_Goto \<gamma> p rs1 \<and> no_matching_Goto \<gamma> p rs2)"
with_proj_simps_3: fixes G :: "'c pair_pre_digraph" shows "arcs_ends (with_proj G) = parcs G"
intro: fixes invar :: "'c \<Rightarrow> bool"   and iterateoi :: "'c \<Rightarrow> ('d \<Rightarrow> bool) \<Rightarrow> ('e \<times> 'f \<Rightarrow> 'd \<Rightarrow> 'd) \<Rightarrow> 'd \<Rightarrow> 'd"   and \<alpha> :: "'c \<Rightarrow> 'e \<Rightarrow> 'f option" assumes "\<And>m. invar m \<Longrightarrow> map_iterator_linord (iterateoi m) (\<alpha> m)" shows "map_iterateoi_axioms \<alpha> invar iterateoi"
a_lcos_mult_one: fixes M :: "'a set" assumes "M \<subseteq> carrier R" shows "\<zero> <+ M = M"
DisjS: fixes v :: "i"   and \<phi> :: "\<o>"   and \<psi> :: "\<o>" shows "[\<phi> \<^bold>\<or> \<psi> in v] = ([\<phi> in v] \<or> [\<psi> in v])"
rotation_pi_2_y_axis: shows "moebius_circline (moebius_rotation (pi / 2)) y_axis = x_axis"
mono_thms_4: fixes x :: "'a" shows "x \<le> x"
constant_rep_design_axioms: shows "constant_rep_design \<V> \<B> \<r>"
dghm_const_components_4: fixes \<CC> :: "V"   and \<DD> :: "V"   and a :: "V"   and f :: "V" shows "dghm_const \<CC> \<DD> a f\<lparr>HomCod\<rparr> = \<DD>"
arctan_half: fixes x :: "real" shows "arctan x = 2 * arctan (x / (1 + sqrt (1 + x\<^sup>2)))"
intro: fixes A :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and B :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and F :: "'b \<Rightarrow> 'c"   and G :: "'b \<Rightarrow> 'c"   and t :: "'b \<Rightarrow> 'c" assumes "category A"   and "category B"   and "functor A B F"   and "functor A B G"   and "transformation_by_components_axioms A B F G t" shows "transformation_by_components A B F G t"
surj_imp_surj_swap: fixes f :: "'b \<Rightarrow> 'a"   and a :: "'b"   and b :: "'b" assumes "surj f" shows "surj (Fun.swap a b f)"
uint8_sshiftr_def: fixes x :: "uint8"   and n :: "integer" shows "uint8_sshiftr x n = (if n < 0 \<or> 8 \<le> n then undefined (>>>) x n else x >>> nat_of_integer n)"
rep_eq: fixes x :: "'b" shows "($) (K$ x) = (\<lambda>xa. x)"
hyps: fixes a :: "'a"   and as' :: "'a list" assumes "wfactors G as a"   and "wfactors G as' a"   and "a \<in> carrier G"   and "set as \<subseteq> carrier G"   and "set as' \<subseteq> carrier G" shows "essentially_equal G as as'"
node2_def: fixes nd1 :: "('c, 'd) Node"   and nd2 :: "('c, 'd) Node" shows "node2 nd1 nd2 = Node2 (gmn nd1 + gmn nd2) nd1 nd2"
rep_eq: fixes x :: "'c \<rightharpoonup>\<^sub>f 'd" shows "fmdom' x = dom (fmlookup x)"
subseq_from_cnf_less_eq: fixes ks :: "ordinal list"   and ls :: "ordinal list" assumes "subseq ks ls" shows "from_cnf ks \<le> from_cnf ls"
brouwer_order_def: shows "brouwer_order = brouwer_pred\<^sup>+"
finestPart: fixes X :: "'b set" shows "finestpart X = {{x} |x. x \<in> X}"
prod_2_def: fixes p1 :: "'d \<Rightarrow> 'f decision option"   and p2 :: "'e \<Rightarrow> 'g decision option" shows "p1 \<Otimes>\<^sub>2 p2 \<equiv> \<lambda>(x, y). case p1 x of \<bottom> \<Rightarrow> \<bottom> | \<lfloor>allow d1\<rfloor> \<Rightarrow> case p2 y of \<bottom> \<Rightarrow> \<bottom> | \<lfloor>allow d2\<rfloor> \<Rightarrow> \<lfloor>allow (d1, d2)\<rfloor> | \<lfloor>deny d2\<rfloor> \<Rightarrow> \<lfloor>deny (d1, d2)\<rfloor> | \<lfloor>deny d1\<rfloor> \<Rightarrow> case p2 y of \<bottom> \<Rightarrow> \<bottom> | \<lfloor>allow d2\<rfloor> \<Rightarrow> \<lfloor>allow (d1, d2)\<rfloor> | \<lfloor>deny d2\<rfloor> \<Rightarrow> \<lfloor>deny (d1, d2)\<rfloor>"
undual_dual: fixes x :: "'a" shows "undual (dual x) = x"
msubstneq_def: fixes c :: "poly"   and t :: "tm"   and d :: "poly"   and s :: "tm"   and a :: "poly"   and r :: "tm" shows "msubstneq c t d s a r = evaldjf (\<lambda>(x, y). Parametric_Ferrante_Rackoff.conj x y) [let cd = c *\<^sub>p d in (NEq (CP cd), NEq (tm.Add (tm.Mul (~\<^sub>p a) (tm.Add (tm.Mul d t) (tm.Mul c s))) (tm.Mul ((2)\<^sub>p *\<^sub>p cd) r))), (Parametric_Ferrante_Rackoff.conj (Eq (CP c)) (NEq (CP d)), NEq (tm.Add (tm.Mul (~\<^sub>p a) s) (tm.Mul ((2)\<^sub>p *\<^sub>p d) r))), (Parametric_Ferrante_Rackoff.conj (NEq (CP c)) (Eq (CP d)), NEq (tm.Add (tm.Mul (~\<^sub>p a) t) (tm.Mul ((2)\<^sub>p *\<^sub>p c) r))), (Parametric_Ferrante_Rackoff.conj (Eq (CP c)) (Eq (CP d)), NEq r)]"
cases: fixes a :: "enat" assumes "a \<in> enat_set" obtains "a = 0"   | n :: "enat" where "a = eSuc n" and "n \<in> enat_set"
closure_of_Un: fixes X :: "'c topology"   and S :: "'c set"   and T :: "'c set" shows "X closure_of (S \<union> T) = X closure_of S \<union> X closure_of T"
oalist_inv_sort_oalist_raw: fixes xs :: "('a \<times> 'b) list \<times> 'a nat_term_order" shows "ko_ntm.oalist_inv (sort_oalist_ko_ntm xs)"
f_Exec_State_Idle_append_replicate_NoMsg_gr_output: fixes localState :: "'d \<Rightarrow> 'e"   and output_fun :: "'d \<Rightarrow> 'f message_af"   and trans_fun :: "'g message_af \<Rightarrow> 'd \<Rightarrow> 'd"   and xs :: "'g message_af list"   and m :: "nat"   and c :: "'d"   and n :: "nat" assumes "State_Idle localState output_fun trans_fun (localState (f_Exec_Comp trans_fun (xs @ NoMsg\<^bsup>m\<^esup>) c))"   and "m < n" shows "output_fun (f_Exec_Comp trans_fun (xs @ NoMsg\<^bsup>n\<^esup>) c) = NoMsg"
\<Delta>'_countable_network: assumes "\<alpha> \<noteq> \<top>" shows "countable_network \<Delta>'"
tObsC_abs_jview_eq: fixes a :: "'a"   and t' :: "('a, 'es, 'as) BEState Trace"   and t :: "('a, 'es, 'as) BEState Trace" assumes "spr_jview a t' = spr_jview a t" shows "tObsC_abs t = tObsC_abs t'"
mset_lt_single_iff: fixes x :: "'b"   and y :: "'b" shows "({#x#} < {#y#}) = (x < y)"
hyps_3: shows "p\<^sub>3 = Point x\<^sub>3 y\<^sub>3"
invertible_right_inverse2: fixes u :: "'a"   and v :: "'a" assumes "invertible u"   and "u \<in> M"   and "v \<in> M" shows "u \<cdot> (inverse u \<cdot> v) = v"
hyps: assumes "n < length l" shows "<is_hashtable l ht> hm_it_adjust n ht <\<lambda>j. is_hashtable l ht * \<up> (j \<le> n \<and> concat (take (Suc n) l) = concat (take (Suc j) l) \<and> (j = 0 \<or> l ! j \<noteq> []))>"
simps_2: fixes i :: "name"   and x :: "tm"   and k :: "name" shows "subst i x (Var k) = (if i = k then x else Var k)"
ring_of_sets_axioms: shows "ring_of_sets \<Omega> (lambda_system \<Omega> M f)"
assoc'_simps_1: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "ide a"   and "ide b"   and "ide c" shows "arr (assoc' a b c)"
S1: shows "x1 = x2 \<and> index_of_c_pair_n y1 = index_of_c_pair_n y2"
freeword: fixes x :: "'b freeword" shows "freeword x \<in> {as. proper_signed_list as}"
divides_mult_rI: fixes G :: "('c, 'd) monoid_scheme"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "Group.comm_monoid G"   and "a divides\<^bsub>G\<^esub> b"   and "a \<in> carrier G"   and "b \<in> carrier G"   and "c \<in> carrier G" shows "a \<otimes>\<^bsub>G\<^esub> c divides\<^bsub>G\<^esub> b \<otimes>\<^bsub>G\<^esub> c"
length_matrix: fixes nr :: "nat"   and nc :: "nat"   and y :: "'a list"   and ys :: "'a list list"   and v :: "'a list"   and k :: "nat"   and x :: "'a list"   and xs :: "'a list list" assumes "mat nr nc (y # ys)"   and "length v = k"   and "vec_mat_Tensor v (y # ys) = x # xs" shows "vec (nr * k) x"
var_not_in_eval: fixes \<phi> :: "atom fm" shows "\<forall>var L. freeIn var \<phi> \<longrightarrow> (\<exists>x. eval \<phi> (L[var := x])) = (\<forall>x. eval \<phi> (L[var := x]))"
src_dom: fixes \<mu> :: "'a" assumes "B.arr \<mu>" shows "src\<^sub>B (B.dom \<mu>) = src\<^sub>B \<mu>"
nr_def: shows "nr \<equiv> addpre r (\<pi>\<^sub>7 s)"
prop_atoms_notin_4: fixes \<phi>\<^sub>1 :: "'b ltln"   and \<phi>\<^sub>2 :: "'b ltln"   and \<phi> :: "'b ltln" shows "\<phi>\<^sub>1 or\<^sub>n \<phi>\<^sub>2 \<notin> prop_atoms \<phi>"
fls_base_factor_mult_base_factor_1: fixes f :: "'a fls"   and g :: "'a fls" shows "fls_base_factor (f * fls_base_factor g) = fls_base_factor (f * g)"
intros_2: fixes m :: "real mat convnet"   and A :: "real mat" assumes "shared_weight_net m" shows "shared_weight_net (Conv A m)"
isNotDSource_EmptyDAcc: fixes i :: "AbstrLevelsID"   and S :: "CSet" assumes "isNotDSource i S" shows "DAcc i S = {}"
gbm_from_list_def: fixes m1_ops :: "('d, 'g, 'f, 'h) map_ops_scheme"   and m2_ops :: "('d, 'i, 'g, 'j) map_ops_scheme"   and s3_ops :: "('e, 'i, 'k) set_ops_scheme" shows "GraphByMapDefs.gbm_from_list m1_ops m2_ops s3_ops \<equiv> gga_from_list (GraphByMapDefs.gbm_empty m1_ops) (GraphByMapDefs.gbm_add_node m1_ops m2_ops) (GraphByMapDefs.gbm_add_edge m1_ops m2_ops s3_ops)"
lgrid_finite: fixes b :: "(nat \<times> int) list"   and ds :: "nat set"   and lm :: "nat" shows "finite (lgrid b ds lm)"
protCl_fsym: fixes resPf :: "'fsym \<Rightarrow> 'tp"   and f :: "'fsym" assumes "protCl (resPf f)" shows "list_all protCl (arOf f)"
sparsegrid'_length: fixes b :: "(nat \<times> int) list"   and dm :: "nat" assumes "b \<in> sparsegrid' dm" shows "length b = dm"
rc': shows "root_cond (p, l', r') (real_of_1 (p, l, r))"
simps_3: fixes uu :: "('e set \<times> 'e set) list"   and v :: "('f, 'g, 'e) prot_fun"   and va :: "(('f, 'g, 'e) prot_fun, 'h) Term.term list"   and vb :: "'h" shows "timpls_transformable_to uu (Fun v va) (Var vb) = False"
ends: fixes i :: "'b"   and j :: "'b" assumes "\<I> i"   and "\<I> j" shows "((i, j) \<in> f \<union> f\<inverse> \<union> e) = (END i = END j)"
set_band_filter_aux: fixes p\<^sub>0 :: "int \<times> int"   and ps\<^sub>L :: "(int \<times> int) set"   and p\<^sub>1 :: "int \<times> int"   and ps\<^sub>R :: "(int \<times> int) set"   and \<delta> :: "real"   and ps :: "(int \<times> int) list"   and l :: "int"   and ps' :: "(int \<times> int) list" assumes "p\<^sub>0 \<in> ps\<^sub>L"   and "p\<^sub>1 \<in> ps\<^sub>R"   and "p\<^sub>0 \<noteq> p\<^sub>1"   and "dist (case p\<^sub>0 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case p\<^sub>1 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) < \<delta>"   and "set ps = ps\<^sub>L \<union> ps\<^sub>R"   and "\<forall>p\<in>ps\<^sub>L. fst p \<le> l"   and "\<forall>p\<in>ps\<^sub>R. l \<le> fst p"   and "ps' = filter (\<lambda>p. real_of_int l - \<delta> < real_of_int (fst p) \<and> real_of_int (fst p) < real_of_int l + \<delta>) ps" shows "p\<^sub>0 \<in> set ps' \<and> p\<^sub>1 \<in> set ps'"
S: fixes p :: "'s1 spmf"   and q :: "'s2 spmf"   and a :: "'a"   and s :: "'s2"   and b :: "'b"   and s' :: "'s2" assumes "S p q"   and "a \<in> A"   and "s \<in> set_spmf q"   and "(b, s') \<in> set_spmf (callee2 s a)" shows "S (cond_spmf_fst (p \<bind> (\<lambda>s. callee1 s a)) b) (cond_spmf_fst (q \<bind> (\<lambda>s. callee2 s a)) b)"
adm_subst: fixes t :: "'a \<Rightarrow> 'b"   and P :: "'b \<Rightarrow> bool" assumes "cont t"   and "adm P" shows "adm (\<lambda>x. P (t x))"
hyps: assumes "is_tangle_diagram xs" shows "is_tangle_diagram (basic (make_vert_block n) \<otimes> xs)"
qtable_r\<epsilon>_strict: fixes mr :: "mregex"   and \<phi>s :: "Formula.formula list"   and A :: "nat set"   and n :: "nat"   and R :: "event_data list set"   and \<sigma> :: "(String.literal \<times> event_data list) trace"   and V :: "String.literal \<Rightarrow> (nat \<Rightarrow> event_data list set) option"   and i :: "nat"   and rels :: "event_data option list set list" assumes "safe_regex Past Strict (from_mregex mr \<phi>s)"   and "ok (length \<phi>s) mr"   and "A = fv_regex (from_mregex mr \<phi>s)"   and "list_all2 (\<lambda>\<phi>. qtable n (fv \<phi>) (mem_restr R) (\<lambda>v. Formula.sat \<sigma> V (map the v) i \<phi>)) \<phi>s rels" shows "qtable n A (mem_restr R) (\<lambda>v. Regex.eps (Formula.sat \<sigma> V (map the v)) i (from_mregex mr \<phi>s)) (r\<epsilon>_strict n rels mr)"
order_iff: fixes a :: "'b DBMEntry"   and b :: "'b DBMEntry" shows "(a \<preceq> b) = (a = linordered_monoid.min a b)"
sublist_trans: fixes l1 :: "'b list"   and l2 :: "'b list"   and l3 :: "'b list" assumes "subseq l1 l2"   and "subseq l2 l3" shows "subseq l1 l3"
offending_partition_subset_empty: fixes V :: "'a set"   and E :: "('a \<times> 'a) set"   and X :: "('a \<times> 'a) set"   and nP :: "'a \<Rightarrow> 'b" assumes "\<forall>F\<in>set_offending_flows \<lparr>nodes = V, edges = E \<union> X\<rparr> nP. F \<subseteq> X"   and "wf_graph \<lparr>nodes = V, edges = E \<union> X\<rparr>"   and "E \<inter> X = {}" shows "set_offending_flows \<lparr>nodes = V, edges = E\<rparr> nP = {}"
drop_zero_ms: fixes C :: "'a"   and e :: "real"   and xs :: "('a \<times> real) msllist"   and f :: "real \<Rightarrow> real"   and basis :: "(real \<Rightarrow> real) list" assumes "\<forall>\<^sub>F x in at_top. eval C x = 0"   and "is_expansion (MS (MSLCons (C, e) xs) f) basis" shows "is_expansion (MS xs f) basis"
assms_2: shows "(m, lam) \<in> set n_as"
measure_notin_sets: fixes A :: "'d set"   and M :: "'d measure" assumes "A \<notin> sets M" shows "Sigma_Algebra.measure M A = 0"
Beta_def: fixes a :: "'a"   and b :: "'a" shows "Beta a b = Gamma a * Gamma b / Gamma (a + b)"
make_map_def: fixes ks :: "'a fset"   and v :: "'b" shows "make_map ks v \<equiv> \<lambda>k. if k \<in> fset ks then Some v else None"
fact1: shows "good_context s \<and> fetch_instruction (delayed_pool_write s) = Inr v1 \<and> (\<exists>e. decode_instruction v1 = Inl e)"
z_1: shows "z \<in> (interval_bij (a, b) (- 1, 1) \<circ> f) ` {0..1}"
igFreshBinp_def: fixes MOD :: "('a, 'b, 'c, 'd, 'e, 'f, 'g, 'h, 'i) Iteration.model_scheme"   and ys :: "'c"   and y :: "'f"   and binp :: "'j \<Rightarrow> 'h option" shows "igFreshBinp MOD ys y binp \<equiv> liftAll (igFreshAbs MOD ys y) binp"
cases: fixes x :: "'b btree" obtains "x = Leaf"   | ts :: "('b btree \<times> 'b) list" and t :: "'b btree" where "x = Node ts t"
simps_1: fixes uu :: "('b \<times> 'b) set"   and uv :: "'a \<Rightarrow> 'b"   and uw :: "'b" shows "ipurge_tr_rev uu uv uw [] = []"
lmeasurable_compact: fixes S :: "'b set" assumes "compact S" shows "S \<in> lmeasurable"
not_ffs_1: fixes x :: "f" shows "(x \<noteq> val) = (x = Aodv_Basic.inv)"
map_update_dj_axioms: shows "map_update_dj m2.\<alpha> m2.invar m2.update_dj"
distrib: fixes a :: "'e"   and b :: "'e"   and c :: "'e" shows "(a + b) \<cdot> c = a \<cdot> c + b \<cdot> c"
assms_3: shows "(x, y) = split_i_j_x (dim_col A) l X"
path_connected_path_image: fixes p :: "real \<Rightarrow> 'a" assumes "path p" shows "path_connected (path_image p)"
qstar_inductr: fixes z :: "'b \<times> 'c"   and y :: "'b \<times> 'c"   and x :: "'b \<times> 'c" assumes "less_eq_prod (sup_prod z (times_prod' y x)) y" shows "less_eq_prod (times_prod' z (prod.qstar x)) y"
basis_enum_of_vec_cblinfun_apply: fixes M :: "complex mat"   and x :: "complex vec" assumes "M \<in> carrier_mat (length canonical_basis) (length canonical_basis)"   and "dim_vec x = length canonical_basis" shows "basis_enum_of_vec (M *\<^sub>v x) = cblinfun_of_mat M *\<^sub>V basis_enum_of_vec x"
ss: fixes n :: "'b" assumes "n \<in> N" shows "sum f1 (B1 n) = sum f2 (B2 n)"
f_x1: fixes x :: "'a" shows "f *\<^sub>v r *\<^sub>R x = r *\<^sub>R (f *\<^sub>v x)"
coeffs_eq_iff: fixes p :: "'b poly"   and q :: "'b poly" shows "(p = q) = (coeffs p = coeffs q)"
less_irrefl: fixes x :: "'a" shows "\<not> cmp2.lt x x"
assms_1: shows "P \<turnstile> \<langle>e\<^sub>0,s\<^sub>0,b\<^sub>0\<rangle> \<rightarrow>* \<langle>throw a,s\<^sub>1,b\<^sub>1\<rangle>"
success_guardI: fixes P :: "heap \<Rightarrow> bool"   and h :: "heap"   and f :: "heap \<Rightarrow> 'a \<times> heap" assumes "P h" shows "success (Heap_Monad.guard P f) h"
evalmat_def: fixes P :: "'c poly"   and A :: "(('c, 'd) vec, 'd) vec" shows "evalmat P A = (\<Sum>i\<in>{n. n \<le> degree P}. coeff P i *k matpow A i)"
hyps_1: fixes j :: "int"   and i :: "int" assumes "n = nat (j - i)"   and "i \<le> j" shows "[i..j] = [i..j - 1] @ [j]"
class_Object: fixes G :: "prog" assumes "wf_prog G" shows "class G Object = Some \<lparr>access = Public, cfields = [], methods = Object_mdecls, init = Skip, super = undefined, superIfs = []\<rparr>"
is_projection_on_kernel_is_orthogonal_complement: fixes \<pi> :: "'b \<Rightarrow> 'b"   and M :: "'b set" assumes "is_projection_on \<pi> M"   and "closed_csubspace M" shows "\<pi> -` {0::'b} = orthogonal_complement M"
that: fixes f :: "'a \<Rightarrow> 'a"   and g :: "'a \<Rightarrow> 'a" assumes "homeomorphism (cball a r \<inter> T) (cball a r \<inter> T) f g"   and "f a = u"   and "\<And>x. x \<in> sphere a r \<Longrightarrow> f x = x" shows "thesis"
c_dom_def: fixes D :: "'b \<Rightarrow> 'a"   and xp :: "'b \<times> 'c" shows "c_dom D xp \<equiv> D (fst xp)"
Quotient3_refl2: fixes R :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and Abs :: "'e \<Rightarrow> 'f"   and Rep :: "'f \<Rightarrow> 'e"   and r :: "'e"   and s :: "'e" assumes "Quotient3 R Abs Rep"   and "R r s" shows "R s s"
fun_belowD: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b"   and x :: "'a" assumes "f \<sqsubseteq> g" shows "f x \<sqsubseteq> g x"
prefix_contains_msg: fixes es :: "'msg event list"   and i :: "nat"   and m :: "'msg" assumes "es prefix of i"   and "m \<in> set (node_deliver_messages es)" shows "Deliver m \<in> set es"
radj_adj: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'a" assumes "f \<stileturn> g" shows "g = radj f"
subset_eq: fixes A :: "'d set"   and B :: "'d set" shows "(A \<subseteq> B) = (\<forall>x\<in>A. x \<in> B)"
p: shows "valid_path p \<and> z \<notin> path_image p \<and> pathfinish p = pathstart p \<and> (\<forall>t\<in>{0..1}. cmod (\<gamma> t - p t) < e) \<and> contour_integral p (\<lambda>w. 1 / (w - z)) = complex_of_real (2 * pi) * \<i> * n"
bijective_open_imp_homeomorphic_map: fixes X :: "'c topology"   and Y :: "'d topology"   and f :: "'c \<Rightarrow> 'd" assumes "continuous_map X Y f"   and "open_map X Y f"   and "f ` topspace X = topspace Y"   and "inj_on f (topspace X)" shows "homeomorphic_map X Y f"
eventually_nhds_within_iff_sequentially: fixes P :: "'b \<Rightarrow> bool"   and a :: "'b"   and s :: "'b set" shows "eventually P (inf (nhds a) (principal s)) = (\<forall>f. (\<forall>n. f n \<in> s) \<and> f \<longlonglongrightarrow> a \<longrightarrow> (\<forall>\<^sub>F n in sequentially. P (f n)))"
finite_measure_eq_sum_singleton: fixes S :: "nat set set set" assumes "finite S"   and "\<And>x. x \<in> S \<Longrightarrow> {x} \<in> events" shows "prob S = (\<Sum>x\<in>S. prob {x})"
geqb: fixes x :: "'a" assumes "x \<in> S" shows "b \<le> f x"
poly_cond2_no_rat_root: fixes p :: "int poly"   and x :: "rat" assumes "poly_cond2 p" shows "poly (real_of_int_poly p) (real_of_rat x) \<noteq> 0"
intersect_segment_xline_le: fixes prec :: "nat"   and y :: "(real \<times> real) \<times> real \<times> real"   and xl :: "real"   and m0 :: "real"   and M0 :: "real" assumes "intersect_segment_xline prec y xl = Some (m0, M0)" shows "m0 \<le> M0"
dom_char: fixes f :: "(bool, bool list) concrete_category.arr" shows "local.dom f = (if f = j0 \<or> f = j1 then Zero else if arr f then f else null)"
drop_Suc: shows "drop k (replicate (Suc k) (2::'a)) = [2::'a]"
star_of_def: fixes x :: "'c" shows "star_of x \<equiv> star_n (\<lambda>n. x)"
map_carrier_mat: fixes f :: "'d \<Rightarrow> 'c"   and A :: "'d mat"   and nr :: "nat"   and nc :: "nat" shows "(map_mat f A \<in> carrier_mat nr nc) = (A \<in> carrier_mat nr nc)"
inj_on_id2: fixes A :: "'b set" shows "inj_on (\<lambda>x. x) A"
lr_PE_arg_rel: shows "PE_arg_rel \<in> unlr PE.delta"
step_Normal_elim_cases_17: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c1 :: "('e, 'd, 'f) com"   and c2 :: "('e, 'd, 'f) com"   and cs :: "('e, 'd, 'f) com list"   and css :: "(('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list"   and s :: "'e"   and u :: "('e, 'd, 'f) com list \<times> (('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list \<times> ('e, 'f) xstate" assumes "\<Gamma>\<turnstile> (Catch c1 c2 # cs, css, Normal s) \<rightarrow> u" obtains "u = ([c1], (cs, c2 # cs) # css, Normal s)"
minus_id_iff: fixes t :: "'b \<Rightarrow>\<^sub>0 nat"   and s :: "'b \<Rightarrow>\<^sub>0 nat" shows "(t - s = t) = (keys t \<inter> keys s = {})"
alpha2: shows "(qinp', qinp'') \<in> alphaInp"
fmdrop_lookup: fixes a :: "'f"   and m :: "'f \<rightharpoonup>\<^sub>f 'e" shows "fmlookup (fmdrop a m) a = None"
division_ofD_6: fixes s :: "'a set set"   and i :: "'a set" assumes "s division_of i" shows "\<Union> s = i"
Sup_le_iff: fixes A :: "'b set"   and b :: "'b" shows "(\<Squnion> A \<le> b) = (\<forall>a\<in>A. a \<le> b)"
cp_def: fixes P :: "('a state \<times> 'a state \<Rightarrow> 'b) \<Rightarrow> 'a state \<times> 'a state \<Rightarrow> 'c" shows "cp P \<equiv> \<exists>f. \<forall>X \<tau>. P X \<tau> = f (X \<tau>) \<tau>"
AXIOMS_in_CANDs_2: fixes B :: "ty" shows "AXIOMSc B \<subseteq> \<parallel><B>\<parallel>"
intro: fixes hom :: "'a \<Rightarrow> 'b" assumes "comm_monoid_add_hom hom"   and "monoid_mult_hom hom" shows "semiring_hom hom"
fields_rec': fixes C :: "qtname"   and c :: "class" assumes "class tprg C = Some c" shows "DeclConcepts.fields tprg C = map (\<lambda>(fn, ft). ((fn, C), ft)) (cfields c) @ (if C = Object then [] else DeclConcepts.fields tprg (super c))"
isomorphic_transitive: fixes f :: "'b"   and g :: "'b"   and h :: "'b" assumes "f \<cong>\<^sub>D g"   and "g \<cong>\<^sub>D h" shows "f \<cong>\<^sub>D h"
currentD_outside: fixes \<Gamma> :: "('k, 'l) web_scheme"   and f :: "'k \<times> 'k \<Rightarrow> ennreal"   and x :: "'k"   and y :: "'k" assumes "current \<Gamma> f"   and "\<not> edge \<Gamma> x y" shows "f (x, y) = 0"
e1: shows "items_le k (\<I> k) = natUnion (\<lambda>u. items_le k (\<J> k u))"
convex_halfspace_gt: fixes b :: "real"   and a :: "'a" shows "convex {x. b < a \<bullet> x}"
card_of_Times_infinite_simps_1: fixes A :: "'a set"   and B :: "'b set" assumes "infinite A"   and "B \<noteq> {}"   and "|B| \<le>o |A|" shows "|A \<times> B| =o |A|"