hcju/isabelleps huggingface dataset:
simp_1: shows "karatsuba_main F n G m = (if n \<le> karatsuba_lower_bound \<or> m \<le> karatsuba_lower_bound then let ff = poly_of_list F in foldr (\<lambda>a p. smult a ff + pCons (0::'a) p) G 0 else let n2 = n div 2 in if n2 < m then case split_at n2 F of (f0, f1) \<Rightarrow> case split_at n2 G of (g0, g1) \<Rightarrow> let p1 = karatsuba_main f1 (n - n2) g1 (m - n2); p2 = karatsuba_main (coeffs_minus f1 f0) n2 (coeffs_minus g1 g0) n2; p3 = karatsuba_main f0 n2 g0 n2 in monom_mult (n2 + n2) p1 + (monom_mult n2 (p1 - p2 + p3) + p3) else case split_at n2 F of (f0, f1) \<Rightarrow> let p1 = karatsuba_main f1 (n - n2) G m in Let (karatsuba_main f0 n2 G m) ((+) (monom_mult n2 p1)))"
case_2: fixes f1 :: "'c"   and f2 :: "enat \<Rightarrow> 'c"   and x2 :: "enat" shows "(case eSuc x2 of 0 \<Rightarrow> f1 | eSuc x \<Rightarrow> f2 x) = f2 x2"
word_unat_eq_iff: fixes v :: "'b word"   and w :: "'b word" shows "(v = w) = (unat v = unat w)"
WT_gpv_iff_outs_gpv: fixes \<I> :: "('h, 'i) \<I>"   and gpv :: "('j, 'h, 'i) gpv" shows "\<I> \<turnstile>g gpv \<surd> = (outs_gpv \<I> gpv \<subseteq> outs_\<I> \<I>)"
bbind_def: shows "bbind \<equiv> \<Lambda> b g. fup\<cdot>g\<cdot>b"
cases: fixes x :: "char list \<times> fterm list" obtains p :: "char list" and ts :: "fterm list" where "x = (p, ts)"
continuous_within_Arcsin: fixes z :: "complex"   and s :: "complex set" assumes "Im z = 0 \<Longrightarrow> \<bar>Re z\<bar> < 1" shows "continuous (at z within s) Arcsin"
distinct_mset_count_less_1: fixes S :: "'b multiset" shows "distinct_mset S = (\<forall>a. count S a \<le> 1)"
edges_of_walkE: fixes G :: "('a, 'b) Graph_scheme"   and v :: "'a"   and w :: "'a"   and xs :: "'a list" assumes "Digraph G"   and "(v, w) \<in> Digraph.edges_of_walk xs" shows "\<exists>xs_pre xs_post. xs = xs_pre @ v # w # xs_post"
wtC_fax_unprot: fixes f :: "'fsym" assumes "wtFsym f"   and "\<not> unprot (resOf f)" shows "GE.wtC (fax f)"
counit_simps_4: shows "src \<psi>' = trg e"
lm057: fixes pair :: "'a \<times> 'b" shows "fst pair = snd (flip pair) \<and> snd pair = fst (flip pair)"
name_set_fresh_fresh: fixes pi :: "(name \<times> name) list"   and Xs :: "name set"   and Ys :: "name set"   and x :: "'a" assumes "set pi \<subseteq> Xs \<times> Ys"   and "Xs \<sharp>* x"   and "Ys \<sharp>* x" shows "pi \<bullet> x = x"
One_implies_Beta_star: fixes t :: "lam"   and s :: "lam" assumes " t \<longrightarrow>\<^sub>1 s" shows " t \<longrightarrow>\<^sub>\<beta>\<^sup>* s"
lower: fixes x :: "'c Gromov_completion"   and y :: "'c Gromov_completion" shows "extended_predist x y \<le> 2 * turn_into_distance extended_predist x y"
span_is_submodule: fixes S :: "'c set" assumes "S \<subseteq> carrier V" shows "LinearCombinations.submodule K (local.module.span S) V"
convergent_subsequence_is_convergent: fixes s :: "nat \<Rightarrow> nat \<Rightarrow> int"   and a :: "nat \<Rightarrow> int" assumes "s \<in> carrier Zp\<^bsup>\<omega>\<^esup>"   and "a = acc_point s" shows "Zp_converges_to (convergent_subseq s) (acc_point s)"
mod_exp_eq: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bit :: "'b \<Rightarrow> nat \<Rightarrow> bool"   and a :: "'b"   and m :: "nat"   and n :: "nat" assumes "class.semiring_bits plus minus zero times one divide modulo bit" shows "modulo (modulo a (power.power one times (numeral.numeral one plus (num.Bit0 num.One)) m)) (power.power one times (numeral.numeral one plus (num.Bit0 num.One)) n) = modulo a (power.power one times (numeral.numeral one plus (num.Bit0 num.One)) (min m n))"
BisT_transT: fixes c :: "('test, 'atom) com"   and d :: "('test, 'atom) com"   and s :: "'state"   and t :: "'state"   and s' :: "'state" assumes "c \<approx>T d"   and "mustT c s"   and "mustT d t"   and "s \<approx> t"   and "(c, s) \<rightarrow>t s'" obtains t' :: "'state" where "(d, t) \<rightarrow>*t t'" and "s' \<approx> t'"
of_nat_numeral: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and n :: "num" assumes "class.semiring_1 one times plus zero" shows "semiring_1.of_nat one plus zero (numeral n) = numeral.numeral one plus n"
True: fixes F :: "('k \<Rightarrow> bool) \<Rightarrow> bool" assumes "is_filter F" shows "F (\<lambda>x. True)"
subset: fixes H :: "'e set"   and G :: "('e, 'f) monoid_scheme" assumes "submonoid H G" shows "H \<subseteq> carrier G"
simps_13: fixes v :: "fm"   and va :: "fm" shows "not (Iff v va) = NOT (Iff v va)"
document_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)}) document_ptr_kinds_M h h'"
sigma_finite_countable: shows "\<exists>A. countable A \<and> A \<subseteq> sets lborel \<and> \<Union> A = space lborel \<and> (\<forall>a\<in>A. emeasure lborel a \<noteq> \<infinity>)"
fst_scaleR: fixes r :: "real"   and A :: "'d \<times> 'e" shows "fst (r *\<^sub>R A) = r *\<^sub>R fst A"
proto_Some: fixes proto :: "protocol" assumes "simple_proto_conjunct p1 p2 = Some proto" shows "match_proto proto (p_proto p) = (match_proto p1 (p_proto p) \<and> match_proto p2 (p_proto p))"
Ports_single_2: fixes \<beta> :: "'b common_primitive \<Rightarrow> ('b, 'c) tagged_packet_scheme \<Rightarrow> ternaryvalue"   and \<alpha> :: "action \<Rightarrow> ('b, 'c) tagged_packet_scheme \<Rightarrow> bool"   and proto :: "8 word"   and ps :: "(16 word \<times> 16 word) list"   and a :: "action"   and p :: "('b, 'c) tagged_packet_scheme" assumes "primitive_matcher_generic \<beta>" shows "matches (\<beta>, \<alpha>) (Match (Dst_Ports (L4Ports proto ps))) a p = (proto = p_proto p \<and> p_dport p \<in> ports_to_set ps)"
r2f_inj_iff: fixes f :: "'c \<Rightarrow> 'd set"   and g :: "'c \<Rightarrow> 'd set" shows "(\<R> f = \<R> g) = (f = g)"
hyps: shows "(filter P xs = xs) = (\<forall>x\<in>set xs. P x)"
fermat_test_not_Carmichael_number_def: fixes n :: "nat" shows "fermat_test_not_Carmichael_number n \<equiv> \<not> Carmichael_number n \<or> n < 3"
simps_2: fixes f :: "'a"   and ts :: "('a, 'b) Term.term list" shows "num_vars (Fun f ts) = sum_list (map num_vars ts)"
simps_2: fixes ik :: "msg set"   and secret :: "msg set"   and progress :: "fid_t \<Rightarrow> Runs.var set option"   and signals :: "signal \<Rightarrow> nat"   and more :: "'a" shows "l1_state.signals \<lparr>ik = ik, secret = secret, progress = progress, signals = signals, \<dots> = more\<rparr> = signals"
right_unique_vec_rel: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool" assumes "right_unique R" shows "right_unique (vec_rel R)"
inner_is_valid: fixes n :: "'node" assumes "inner_node n" shows "valid_node n"
wf_prog_acc_superD: fixes G :: "prog"   and C :: "qtname"   and c :: "class" assumes "wf_prog G"   and "class G C = Some c"   and "C \<noteq> Object" shows "is_acc_class G (pid C) (super c)"
inv: fixes gs :: "(nat, nat list list list) trie" assumes "gsopt = Some gs" shows "invar_trie gs"
Oops1: fixes evsO1 :: "event list"   and A :: "agent"   and authK :: "nat"   and Ta :: "nat"   and authTicket :: "msg" assumes "evsO1 \<in> kerbV"   and "A \<noteq> Spy"   and "Says Kas A \<lbrace>Crypt (shrK A) \<lbrace>Key authK, Agent Tgs, Number Ta\<rbrace>, authTicket\<rbrace> \<in> set evsO1"   and "expiredAK Ta evsO1" shows "Notes Spy \<lbrace>Agent A, Agent Tgs, Number Ta, Key authK\<rbrace> # evsO1 \<in> kerbV"
pwd_spy_id_prik_def: fixes n :: "nat" shows "pwd_spy_id_prik n \<equiv> case pwd_spy_dec n of (s, S, A, B, C, D) \<Rightarrow> (insert (Spy, \<langle>n, PriKey S\<rangle>) s, S, A, B, C, D)"
canonical_proj_ext_is_surj: fixes n :: "nat"   and I :: "nat \<Rightarrow> 'a set" assumes "\<And>i. i \<le> n \<Longrightarrow> ideal (I i) R"   and "\<And>i j. \<lbrakk>i \<le> n; j \<le> n; i \<noteq> j\<rbrakk> \<Longrightarrow> I i <+> I j = carrier R" shows "canonical_proj_ext I n ` carrier R = carrier (RDirProd_list (map (\<lambda>i. R Quot I i) [0..<Suc n]))"
finite_measure_subadditive: fixes A :: "'a set"   and B :: "'a set" assumes "A \<in> sets M"   and "B \<in> sets M" shows "Sigma_Algebra.measure M (A \<union> B) \<le> Sigma_Algebra.measure M A + Sigma_Algebra.measure M B"
cases: fixes a1 :: "'b"   and a2 :: "'b list" assumes "ListMem a1 a2" obtains x :: "'b" and xs :: "'b list" where "a1 = x__" and "a2 = x__ # xs__"   | x :: "'b" and xs :: "'b list" and y :: "'b" where "a1 = x" and "a2 = y # xs" and "ListMem x xs"
LD_def: fixes L :: "'a set"   and ars :: "('b \<times> 'b) set" shows "LD L ars = (\<exists>r lrs. ars = unlabel lrs \<and> trans r \<and> wf r \<and> (\<forall>P. local_peak lrs P \<longrightarrow> (\<exists>\<sigma>' \<tau>'. DD lrs r (fst P, snd P, \<sigma>', \<tau>'))))"
perp_distinct: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "A B Perp C D" shows "A \<noteq> B \<and> C \<noteq> D"
convert_atom_change: fixes xs' :: "real list"   and var :: "nat"   and At :: "atom"   and x :: "real"   and \<Gamma> :: "real list"   and x' :: "real" assumes "length xs' = var" shows "convert_atom var At (xs' @ x # \<Gamma>) = convert_atom var At (xs' @ x' # \<Gamma>)"
comm_add_vec: fixes v\<^sub>1 :: "'a vec"   and n :: "nat"   and v\<^sub>2 :: "'a vec" assumes "v\<^sub>1 \<in> carrier_vec n"   and "v\<^sub>2 \<in> carrier_vec n" shows "v\<^sub>1 + v\<^sub>2 = v\<^sub>2 + v\<^sub>1"
edc_pair: fixes x :: "nat"   and vs :: "nat list"   and y :: "nat"   and vs' :: "nat list"   and \<Gamma> :: "nat \<Rightarrow> pdf_type"   and \<delta> :: "cexpr" assumes "x \<in> set vs"   and "y \<in> set vs"   and "x \<noteq> y" shows "(vs, vs', \<Gamma>, \<delta>) \<turnstile>\<^sub>c <Var x ,  Var y> \<Rightarrow> marg_dens2_cexpr \<Gamma> vs x y \<delta>"
method_exit_def: fixes n :: "'node" shows "method_exit n \<equiv> n = (_Exit_) \<or> (\<exists>a Q p f. n = sourcenode a \<and> valid_edge a \<and> kind a = Q\<hookleftarrow>\<^bsub>p\<^esub>f)"
IH: fixes x :: "nat"   and xa :: "nat"   and xb :: "nat list"   and xc :: "nat"   and xd :: "nat list"   and xe :: "nat list" assumes "((x, xa, xb), n, i, is) \<in> measures [\<lambda>(n, i, is). n - i, \<lambda>(n, i, is). if is = [] then 1 else 0]"   and "2 \<le> xa"   and "2 \<le> x"   and "\<And>xb. \<lbrakk>2 \<le> xb; xb < xa\<rbrakk> \<Longrightarrow> \<not> xb dvd x"   and "\<And>x. \<lbrakk>xa \<le> x; x < xc; prime x\<rbrakk> \<Longrightarrow> x \<in> set xb"   and "xa \<le> xc"   and "sorted xb"   and "distinct xb"   and "candidate_invariant xc"   and "set xb \<subseteq> {xa..<xc}"   and "xd = prime_factorization_nat_main x xc xb xe" shows "\<exists>qs. xd = qs @ xe \<and> Ball (set qs) prime \<and> x = prod_list qs"
arr_cod_iff_arr: fixes f :: "'c" shows "C.arr (C.cod f) = C.arr f"
reduced_row_echelon_form_upt_k_intro: fixes k :: "nat"   and A :: "(('b, 'c) vec, 'a) vec" assumes "\<forall>i. is_zero_row_upt_k i k A \<longrightarrow> \<not> (\<exists>j>i. \<not> is_zero_row_upt_k j k A)"   and "\<forall>i. \<not> is_zero_row_upt_k i k A \<longrightarrow> A $ i $ (LEAST k. A $ i $ k \<noteq> (0::'b)) = (1::'b)"   and "\<forall>i. i < i + (1::'a) \<and> \<not> is_zero_row_upt_k i k A \<and> \<not> is_zero_row_upt_k (i + (1::'a)) k A \<longrightarrow> (LEAST n. A $ i $ n \<noteq> (0::'b)) < (LEAST n. A $ (i + (1::'a)) $ n \<noteq> (0::'b))"   and "\<forall>i. \<not> is_zero_row_upt_k i k A \<longrightarrow> (\<forall>j. i \<noteq> j \<longrightarrow> A $ j $ (LEAST n. A $ i $ n \<noteq> (0::'b)) = (0::'b))" shows "reduced_row_echelon_form_upt_k A k"
run_sample_env: fixes sample :: "'c pmf \<Rightarrow> ('c \<Rightarrow> 'a) \<Rightarrow> 'a"   and p :: "'c pmf"   and f :: "'c \<Rightarrow> ('b, 'a) envT"   and r :: "'b" shows "run_env (sample_env sample p f) r = sample p (\<lambda>x. run_env (f x) r)"
backward_finite_iff_msc: fixes x :: "'a" shows "backward_finite x = (many_strongly_connected x \<or> backward_terminating x)"
lex_ext_unbounded_iff: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> bool \<times> bool"   and xs :: "'b list"   and ys :: "'b list" shows "lex_ext_unbounded f xs ys = ((\<exists>i<length xs. i < length ys \<and> (\<forall>j<i. snd (f (xs ! j) (ys ! j))) \<and> fst (f (xs ! i) (ys ! i))) \<or> (\<forall>i<length ys. snd (f (xs ! i) (ys ! i))) \<and> length ys < length xs, (\<exists>i<length xs. i < length ys \<and> (\<forall>j<i. snd (f (xs ! j) (ys ! j))) \<and> fst (f (xs ! i) (ys ! i))) \<or> (\<forall>i<length ys. snd (f (xs ! i) (ys ! i))) \<and> length ys \<le> length xs)"
tree_map_of_distinct: fixes pres :: "('a \<times> 'b) list" shows "distinct (map fst (Tree2.inorder (M.tree_map_of' RBT_Set.empty pres)))"
Init: fixes s :: "'d"   and F :: "'d program" assumes "s \<in> Init F" shows "s \<in> reachable F"
valid_eval_match: fixes eval :: "v sem_env \<Rightarrow> exp0 \<Rightarrow> 'b state \<Rightarrow> 'b state \<times> (v, v) result"   and eval_list :: "v sem_env \<Rightarrow> exp0 list \<Rightarrow> 'b state \<Rightarrow> 'b state \<times> (v list, v) result"   and eval_match :: "v sem_env \<Rightarrow> v \<Rightarrow> (pat \<times> exp0) list \<Rightarrow> v \<Rightarrow> 'b state \<Rightarrow> 'b state \<times> (v, v) result"   and env :: "v sem_env"   and s :: "'b state"   and v :: "v"   and pes :: "(pat \<times> exp0) list"   and err_v :: "v" assumes "eval eval eval_list eval_match" shows "evaluate_match True env s v pes err_v (eval_match env v pes err_v s)"
simps_6: fixes x2 :: "Secrets"   and x1 :: "Keys" shows "sE x2 \<noteq> kE x1"
l_get_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   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" shows "l_get_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf get_disconnected_nodes get_disconnected_nodes_locs \<equiv> type_wf = DocumentClass.type_wf \<and> get_disconnected_nodes = l_get_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_disconnected_nodes \<and> get_disconnected_nodes_locs = l_get_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_disconnected_nodes_locs"
insert_minus_eq: fixes x :: "'a"   and y :: "'a"   and A :: "'a set" assumes "x \<noteq> y" shows "insert x A - {y} = insert x (A - {y})"
simps_20: fixes v :: "'b"   and va :: "bexp"   and vb :: "'b acom"   and vc :: "'b"   and vd :: "'b acom"   and ve :: "'b acom" shows "({v} WHILE va DO vb {vc} \<sqsubseteq> vd;; ve) = False"
nel_drop_emp: fixes u :: "'b list"   and S :: "'b list set" assumes "u \<noteq> \<epsilon>"   and "u \<in> S" shows "u \<in> S\<^sub>+"
connected_def: fixes g :: "'b ugraph" shows "Undirected_Graph.connected g \<equiv> nodes g \<times> nodes g \<subseteq> (edges g)\<^sup>*"
H_def: fixes p :: "'a"   and x :: "'a"   and q :: "'a" shows "kat.H (\<cdot>) (\<le>) ad p x q = \<lparr>ad (ad p)\<rparr>x\<lparr>x \<cdot> ad (ad q)\<rparr>"
c_tr_sinks: fixes I :: "('e \<times> 'e) set"   and D :: "'f \<Rightarrow> 'e"   and u :: "'e"   and step :: "'h \<Rightarrow> 'f \<Rightarrow> 'h"   and out :: "'h \<Rightarrow> 'f \<Rightarrow> 'g"   and s :: "'h"   and xs :: "'f list" shows "sinks I (c_dom D) u (c_tr step out s xs) = sinks I D u xs"
pp_eq_iff: fixes s :: "('b, 'c) pp"   and t :: "('b, 'c) pp" shows "(s = t) = (lookup_pp s = lookup_pp t)"
poly_of_pm_eq_zero_iff: fixes x :: "'d"   and p :: "('d \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" shows "(poly_of_pm x p = 0) = (keys p \<inter> .[{x}] = {})"
fBall_mono: fixes P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool"   and S :: "'a fset" assumes "P \<le> Q" shows "fBall S P \<le> fBall S Q"
simps_4: fixes p :: "real \<times> real \<times> real"   and L :: "atomUni list" shows "separateAtoms (LeqUni p # L) = (let (a, b, c, d) = separateAtoms L in (a, b, p # c, d))"
induct: fixes P :: "'a list \<Rightarrow> bool"   and a0 :: "'a list" assumes "P []"   and "\<And>x. P [x]"   and "\<And>x y xs. P ((x + y) # xs) \<Longrightarrow> P (x # y # xs)" shows "P a0"
adjoint_def: fixes f :: "'d \<Rightarrow> 'c" shows "adjoint f = (SOME f'. \<forall>x y. f x \<bullet> y = x \<bullet> f' y)"
convergent_imp_Bseq: fixes f :: "nat \<Rightarrow> 'e" assumes "convergent f" shows "Bseq f"
composites_are_arrows_1: shows "hseq \<nu> (\<pi> \<star> \<rho>)"
lformula_nFAND: fixes \<phi>s :: "('a, 'k) aformula list" shows "lformula (nFAND \<phi>s) = Ball (set \<phi>s) lformula"
Entry_cc_SDG_path_to_inner_node: fixes n :: "'node SDG_node" assumes "valid_SDG_node n"   and "parent_node n \<noteq> (_Exit_)" obtains ns :: "'node SDG_node list" where "CFG_node (_Entry_) cc-ns\<rightarrow>\<^sub>d* n"
dsW0k: fixes k :: "nat" shows "matrix_sum d (\<lambda>n. denote_while_n M0 M1 DS n \<rho>) k \<in> carrier_mat d d"
assms: shows "\<S>\<bullet>\<C> \<turnstile> [$Current_memory] : ts _> ts'"
3: shows "homeomorphism (cbox d b) (cbox v b) f3 g3"
host_shadow_root_rel_shadow_root: fixes h :: "(_) heap"   and host :: "(_) element_ptr"   and shadow_root_option :: "(_) shadow_root_ptr option"   and shadow_root :: "(_) shadow_root_ptr" assumes "h \<turnstile> get_shadow_root host \<rightarrow>\<^sub>r shadow_root_option" shows "(shadow_root_option = Some shadow_root) = ((cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r host, cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r shadow_root) \<in> host_shadow_root_rel h)"
finite_limit_intersection: fixes f :: "nat \<Rightarrow> 'c"   and A :: "nat set"   and B :: "'c set" assumes "finite (range f)"   and "\<And>x. (x \<in> A) = (f x \<in> B)" shows "finite A = (limit f \<inter> B = {})"
assms_3: shows "C = four_block_mat D (0\<^sub>m a b) (0\<^sub>m b a) (1\<^sub>m b)"
winding_number_constant: fixes \<gamma> :: "real \<Rightarrow> complex"   and S :: "complex set" assumes "path \<gamma>"   and "pathfinish \<gamma> = pathstart \<gamma>"   and "connected S"   and "S \<inter> path_image \<gamma> = {}" shows "winding_number \<gamma> constant_on S"
sys_block_sizes_uniform: shows "sys_block_sizes = {\<k>}"
uv_3: shows "int x = 2 * u * v \<and> int y = u\<^sup>2 - v\<^sup>2 \<or> int x = u\<^sup>2 - v\<^sup>2 \<and> int y = 2 * u * v"
finlsts_pred_def: shows "finlsts_pred \<equiv> {(r, s). r \<in> UNIV\<^sup>\<star> \<and> (\<exists>a. a ## r = s)}"
primes_M_mono: fixes x :: "real"   and y :: "real" assumes "x \<le> y" shows "\<MM> x \<le> \<MM> y"
cut_less_subset: fixes I :: "'a set"   and t :: "'a" shows "I \<down>< t \<subseteq> I"
isInj_def: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'c" shows "isInj h \<equiv> \<forall>s X Y. wls s X \<and> wls s Y \<longrightarrow> h X = h Y \<longrightarrow> X = Y"
cr_envT_stateTE: fixes s :: "'d"   and M :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and m1 :: "('d, 'e) envT"   and m2 :: "('d, 'f) stateT" assumes "cr_envT_stateT s M m1 m2" obtains f :: "'d \<Rightarrow> 'e" and g :: "'d \<Rightarrow> 'f" where "m1 = EnvT f" and "m2 = StateT g" and "(eq_onp ((=) s) ===> M) f g"
vsubstAbs_substAbs_compose1: fixes A :: "('a, 'b, 'varSort, 'var, 'c) qAbs set"   and Y :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and ys :: "'varSort"   and y1 :: "'var"   and y :: "'var" assumes "goodAbs A"   and "good Y" shows "A $[Y / y]_ys $[y1 // y]_ys = A $[Y #[y1 // y]_ys / y]_ys"
finfun_All_def: shows "finfun_All = finfun_All_except []"
Match_2: fixes P :: "pi"   and \<alpha> :: "freeRes"   and P' :: "pi"   and a :: "name" assumes "P \<Longrightarrow>\<^sub>l\<^sup>^\<alpha> \<prec> P'"   and "P \<noteq> P'" shows "[a\<frown>a]P \<Longrightarrow>\<^sub>l\<^sup>^\<alpha> \<prec> P'"
free_tv_W: fixes e :: "expr"   and A :: "type_scheme list"   and n :: "nat"   and S :: "nat \<Rightarrow> typ"   and t :: "typ"   and m :: "nat"   and v :: "nat" assumes "W e A n = Some (S, t, m)"   and "v \<in> free_tv S \<or> v \<in> free_tv t"   and "v < n" shows "v \<in> free_tv A"
lea_trans: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and A1 :: "'p"   and B1 :: "'p"   and C1 :: "'p"   and A2 :: "'p"   and B2 :: "'p"   and C2 :: "'p" assumes "A B C LeA A1 B1 C1"   and "A1 B1 C1 LeA A2 B2 C2" shows "A B C LeA A2 B2 C2"
iterate_bex_def: fixes it :: "(bool \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> bool \<Rightarrow> bool) \<Rightarrow> bool \<Rightarrow> bool"   and P :: "'b \<Rightarrow> bool" shows "iterate_bex it P = it Not (\<lambda>x \<sigma>. P x) False"
top_closed: shows "\<top>\<^bsub>L\<^esub> \<in> carrier L"
changes_scale: fixes n :: "nat"   and s :: "nat \<Rightarrow> 'a"   and xs :: "'a list" assumes "\<And>i. i < n \<Longrightarrow> (0::'a) < s i"   and "length xs \<le> n" shows "changes (map (\<lambda>i. s i * nth_default (0::'a) xs i) [0..<n]) = changes xs"
hpsafe_Assign_simps: fixes x :: "'c"   and e :: "('a, 'c) trm" shows "hpsafe (x := e) = dsafe e"
of_bl_Cons: fixes x :: "bool"   and xs :: "bool list" shows "of_bl (x # xs) = of_bool x * 2 ^ length xs + of_bl xs"
rBang: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and M :: "'a"   and xvec :: "name list"   and N :: "'a"   and P' :: "('a, 'b, 'c) psi"   and C :: "'d" assumes "\<Psi> \<rhd> P \<parallel> !P \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> P'"   and "guarded P"   and "\<And>C. Prop C \<Psi> (P \<parallel> !P) M xvec N P'" shows "Prop C \<Psi> (!P) M xvec N P'"
find_eigenvector: fixes A :: "'b mat"   and n :: "nat"   and e :: "'b" assumes "A \<in> carrier_mat n n"   and "eigenvalue A e" shows "eigenvector A (find_eigenvector A e) e"
col_bet2_cong1: fixes A :: "'p"   and B :: "'p"   and D :: "'p"   and C :: "'p" assumes "Col A B D"   and "Bet A C B"   and "Cong A B C D"   and "Bet C B D" shows "Cong A C D B"
vrat_vsubset_vreal: shows "\<rat>\<^sub>\<circ> \<subseteq>\<^sub>\<circ> \<real>\<^sub>\<circ>"
simps: fixes e :: "'a Point"   and x :: "'a Point"   and y :: "'a Point"   and z :: "'a Point" shows "coplanar e x y z = (\<exists>\<alpha> \<beta> \<gamma>. \<alpha> + \<beta> + \<gamma> = (1::'a) \<and> positionVector e = \<alpha> ** positionVector x \<oplus> \<beta> ** positionVector y \<oplus> \<gamma> ** positionVector z)"
Cauchy_real_Suc_diff: fixes f :: "nat \<Rightarrow> real"   and c :: "real"   and x :: "real" assumes "\<And>n. \<bar>f (Suc n) - f n\<bar> \<le> c / x ^ n"   and "1 < x" shows "Cauchy f"
dom_in_domains: fixes f :: "'a" assumes "C.domains f \<noteq> {}" shows "C.dom f \<in> C.domains f"
prob_eq_1: fixes A :: "'a set"   and M :: "'a pmf" assumes "A \<in> measure_pmf.events M" shows "(measure_pmf.prob M A = 1) = (AE x in measure_pmf M. x \<in> A)"
res: shows "finite_field_factorization_main p ff_ops f = (c, fs)"
conv_abscissa_truncate: fixes m :: "nat"   and f :: "'a fds" shows "conv_abscissa (fds_truncate m f) = - \<infinity>"
mod_diff_eq: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a"   and c :: "'a"   and b :: "'a" assumes "class.euclidean_ring_cancel divide plus minus zero times one modulo euclidean_size uminus" shows "modulo (minus (modulo a c) (modulo b c)) c = modulo (minus a b) c"
iWHILET_def: fixes bind :: "'c \<Rightarrow> ('b \<Rightarrow> 'd) \<Rightarrow> 'd"   and return :: "'b \<Rightarrow> 'd" shows "iWHILET bind return \<equiv> iWHILEIT bind return (\<lambda>_. True)"
assert_SE_def: fixes P :: "'b \<Rightarrow> bool" shows "assert\<^sub>S\<^sub>E P = (\<lambda>\<sigma>. if P \<sigma> then Some (True, \<sigma>) else None)"