hcju/isabelleps huggingface dataset:
eventually_going_to_at_top_linorder: fixes P :: "'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b"   and A :: "'a set" shows "eventually P (f going_to at_top within A) = (\<exists>C. \<forall>x\<in>A. C \<le> f x \<longrightarrow> P x)"
last_message_Msg_eq_last: fixes xs :: "'d message_af list" assumes "xs \<noteq> []"   and "last xs \<noteq> NoMsg" shows "last_message xs = last xs"
simE_2: fixes P :: "pi"   and Rel :: "(pi \<times> pi) set"   and Q :: "pi"   and \<alpha> :: "freeRes"   and Q' :: "pi" assumes "P \<leadsto>[Rel] Q"   and "Q \<longmapsto> \<alpha> \<prec> Q'" shows "\<exists>P'. P \<longmapsto> \<alpha> \<prec> P' \<and> (P', Q') \<in> Rel"
f_Exec_Stream_Init_output_strictly_causal: fixes n :: "nat"   and xs :: "'d list"   and ys :: "'d list"   and output1 :: "'e list"   and output_fun :: "'f \<Rightarrow> 'e"   and trans_fun :: "'d \<Rightarrow> 'f \<Rightarrow> 'f"   and c :: "'f"   and output2 :: "'e list" assumes "xs \<down> n = ys \<down> n"   and "output1 = map output_fun (f_Exec_Comp_Stream_Init trans_fun xs c)"   and "output2 = map output_fun (f_Exec_Comp_Stream_Init trans_fun ys c)" shows "output1 \<down> Suc n = output2 \<down> Suc n"
refl_onD: fixes A :: "'a set"   and r :: "('a \<times> 'a) set"   and a :: "'a" assumes "refl_on A r"   and "a \<in> A" shows "(a, a) \<in> r"
case_two: fixes j :: "nat" assumes "\<exists>t. local.prefixes t j \<up>" shows "\<exists>t. (\<forall>t'\<le>t. local.prefixes t' j \<down>) \<and> (\<forall>t'>t. local.prefixes t' j \<up>)"
distinct_remdups: fixes xs :: "'d list" shows "distinct (remdups xs)"
bisimWeakCoinduct: fixes P :: "ccs"   and Q :: "ccs"   and X :: "(ccs \<times> ccs) set" assumes "(P, Q) \<in> X"   and "\<And>P Q. (P, Q) \<in> X \<Longrightarrow> P \<leadsto>[X] Q"   and "\<And>P Q. (P, Q) \<in> X \<Longrightarrow> (Q, P) \<in> X" shows "P \<sim> Q"
measurable_Pair: fixes f :: "'c \<Rightarrow> 'd"   and M :: "'c measure"   and M1 :: "'d measure"   and g :: "'c \<Rightarrow> 'e"   and M2 :: "'e measure" assumes "f \<in> M \<rightarrow>\<^sub>M M1"   and "g \<in> M \<rightarrow>\<^sub>M M2" shows "(\<lambda>x. (f x, g x)) \<in> M \<rightarrow>\<^sub>M M1 \<Otimes>\<^sub>M M2"
foreach_def: fixes u :: "'a"   and adjs :: "('v \<times> nat) list"   and Q\<pi> :: "('v \<Rightarrow> enat) \<times> ('v \<Rightarrow> 'a option)" shows "foreach u adjs Q\<pi> = foldr (foreach_body u) adjs Q\<pi>"
assms_2: shows "hn_ctxt B (snd x) (snd y) \<or>\<^sub>A hn_ctxt B' (snd x) (snd y) \<Longrightarrow>\<^sub>t hn_ctxt Bm (snd x) (snd y)"
deg1h: shows "MPoly_Type.degree (Var x) x = 1"
hseqE: fixes \<nu> :: "'a"   and \<mu> :: "'a" assumes "C.hseq \<nu> \<mu>" obtains "C.arr \<mu>" and "C.arr \<nu>" and "src\<^sub>C \<nu> = trg\<^sub>C \<mu>"
distinct_mset_union_mset: fixes D :: "'a multiset"   and C :: "'a multiset" shows "distinct_mset (D \<union># C) = (distinct_mset D \<and> distinct_mset C)"
sel_2: fixes x1 :: "(char list, char list, 'a) namespace"   and x2 :: "(char list, char list, nat \<times> tid_or_exn) namespace" shows "c (make_sem_env x1 x2) = x2"
PP_def: shows "PP = (SOME PP. similar_mat_wit (of_real_hom.mat_hom A) J (fst PP) (snd PP))"
TryCatchRedsFinal: fixes e\<^sub>1 :: "char list exp"   and s\<^sub>0 :: "heap \<times> sfields \<times> sheap"   and b\<^sub>0 :: "bool"   and a :: "nat"   and h\<^sub>1 :: "heap"   and l\<^sub>1 :: "sfields"   and sh\<^sub>1 :: "sheap"   and b\<^sub>1 :: "bool"   and P :: "(char list list \<times> char list exp) prog"   and D :: "char list"   and fs :: "Objects.fields"   and C :: "char list"   and e\<^sub>2 :: "char list exp"   and V :: "char list"   and e\<^sub>2' :: "char list exp"   and h\<^sub>2 :: "heap"   and l\<^sub>2 :: "sfields"   and sh\<^sub>2 :: "sheap"   and b\<^sub>2 :: "bool" assumes "P \<turnstile> \<langle>e\<^sub>1,s\<^sub>0,b\<^sub>0\<rangle> \<rightarrow>* \<langle>Throw a,(h\<^sub>1, l\<^sub>1, sh\<^sub>1),b\<^sub>1\<rangle>"   and "h\<^sub>1 a = \<lfloor>(D, fs)\<rfloor>"   and "P \<turnstile> D \<preceq>\<^sup>* C"   and "P \<turnstile> \<langle>e\<^sub>2,(h\<^sub>1, l\<^sub>1(V \<mapsto> Addr a), sh\<^sub>1),b\<^sub>1\<rangle> \<rightarrow>* \<langle>e\<^sub>2',(h\<^sub>2, l\<^sub>2, sh\<^sub>2),b\<^sub>2\<rangle>"   and "final e\<^sub>2'" shows "P \<turnstile> \<langle>try e\<^sub>1 catch(C V) e\<^sub>2,s\<^sub>0,b\<^sub>0\<rangle> \<rightarrow>* \<langle>e\<^sub>2',(h\<^sub>2, l\<^sub>2(V := l\<^sub>1 V), sh\<^sub>2),b\<^sub>2\<rangle>"
free_beta: fixes s :: "dB"   and t :: "dB"   and i :: "nat" assumes "s \<rightarrow>\<^sub>\<beta> t"   and "free t i" shows "free s i"
weakCongSimTransitive: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and Rel :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set"   and Rel' :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set"   and R :: "('a, 'b, 'c) psi"   and Rel'' :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set" assumes "(\<Psi>, P, Q) \<in> Rel"   and "\<Psi> \<rhd> P \<leadsto>\<guillemotleft>Rel\<guillemotright> Q"   and "\<Psi> \<rhd> Q \<leadsto>\<guillemotleft>Rel'\<guillemotright> R"   and "{(\<Psi>, P, R) |\<Psi> P R. \<exists>Q. (\<Psi>, P, Q) \<in> Rel \<and> (\<Psi>, Q, R) \<in> Rel'} \<subseteq> Rel''"   and "\<And>\<Psi> P Q. (\<Psi>, P, Q) \<in> Rel \<Longrightarrow> \<Psi> \<rhd> P \<leadsto><Rel> Q" shows "\<Psi> \<rhd> P \<leadsto>\<guillemotleft>Rel''\<guillemotright> R"
K_def: shows "K = 1 - (1 - norm z) / 2"
substn_subst_0': fixes s :: "Core.term"   and t :: "Core.term" shows "subst_bv s t = subst_bv2 t 0 s"
id_upt_k_def: fixes A :: "(('b, 'c) vec, 'c) vec"   and k :: "nat" shows "id_upt_k A k = (\<forall>i j. mod_type_class.to_nat i < k \<and> mod_type_class.to_nat j < k \<longrightarrow> (i = j \<longrightarrow> A $ i $ j = (1::'b)) \<and> (i \<noteq> j \<longrightarrow> A $ i $ j = (0::'b)))"
subterm_subst_unfold: fixes t :: "('a, 'b) Term.term"   and s :: "('a, 'c) Term.term"   and \<theta> :: "'c \<Rightarrow> ('a, 'b) Term.term" assumes "t \<sqsubseteq> s \<cdot> \<theta>" shows "(\<exists>s'. s' \<sqsubseteq> s \<and> t = s' \<cdot> \<theta>) \<or> (\<exists>x\<in>fv s. t \<sqsubset> \<theta> x)"
hyps: fixes \<AA> :: "'i"   and \<BB> :: "'i" assumes "local.wf (#\<^sub>V \<AA>) \<phi>"   and "#\<^sub>V \<AA> = #\<^sub>V \<BB>"   and "\<And>m k. LESS k m (#\<^sub>V \<AA>) \<Longrightarrow> m\<^bsup>\<AA>\<^esup>k = m\<^bsup>\<BB>\<^esup>k"   and "lformula \<phi>" shows "(\<AA> \<Turnstile> \<phi>) = (\<BB> \<Turnstile> \<phi>)"
reflects_isomorphic: fixes f :: "'a"   and f' :: "'a" assumes "ide f"   and "ide f'"   and "R f \<cong> R f'" shows "f \<cong> f'"
dual_ccpo_monoI: fixes f :: "'g \<Rightarrow> 'h" assumes "mono f" shows "monotone (\<lambda>x y. y \<le> x) (\<lambda>x y. y \<le> x) f"
x12: shows "\<forall>x\<in>f. p (fst x) \<le> Max (p ` fst ` f)"
basis_reduction_step_impl_1: fixes state :: "(int vec list \<times> int vec list) \<times> int iarray iarray \<times> int iarray"   and i :: "nat"   and fs :: "int vec list"   and upw :: "bool"   and upw' :: "bool"   and i' :: "nat"   and state' :: "(int vec list \<times> int vec list) \<times> int iarray iarray \<times> int iarray"   and fs' :: "int vec list" assumes "LLL_impl_inv state i fs"   and "LLL_invariant upw i fs"   and "LLL_Impl.basis_reduction_step \<alpha> n m upw i state = (upw', i', state')"   and "i < m"   and "fs' = fs_state state'" shows "LLL_impl_inv state' i' fs'"
strategy_conforming_path_exists: fixes v0 :: "'a"   and p :: "Player"   and \<sigma> :: "'a \<Rightarrow> 'a"   and \<sigma>' :: "'a \<Rightarrow> 'a" assumes "v0 \<in> V"   and "strategy p \<sigma>"   and "strategy p** \<sigma>'" obtains P :: "'a llist" where "vmc2_path G P v0 p \<sigma> \<sigma>'"
fpxs_shift_diff_left: fixes m :: "rat"   and n :: "rat"   and f :: "'b fpxs" shows "fpxs_shift (m - n) f = fpxs_shift m (fpxs_shift (- n) f)"
Conj\<^sub>\<alpha>_eq_iff: fixes tset1\<^sub>\<alpha> :: "('a, 'b, 'c) Formula.Tree\<^sub>\<alpha> set['a]"   and tset2\<^sub>\<alpha> :: "('a, 'b, 'c) Formula.Tree\<^sub>\<alpha> set['a]" shows "(Formula.Conj\<^sub>\<alpha> tset1\<^sub>\<alpha> = Formula.Conj\<^sub>\<alpha> tset2\<^sub>\<alpha>) = (tset1\<^sub>\<alpha> = tset2\<^sub>\<alpha>)"
finite_item: fixes x :: "('a, unit) Old_Datatype.node set" assumes "rep_set x" shows "finite_item x"
complex_i_not_numeral: fixes w :: "num" shows "\<i> \<noteq> numeral w"
surjective: fixes r :: "('m, 'n, 'o, 'p) RDocument_scheme" shows "r = \<lparr>RObject.nothing = RObject.nothing r, RDocument.nothing = RDocument.nothing r, doctype = doctype r, document_element = document_element r, disconnected_nodes = disconnected_nodes r, \<dots> = RDocument.more r\<rparr>"
union_single_eq_member: fixes x :: "'a"   and M :: "'a multiset"   and N :: "'a multiset" assumes "add_mset x M = N" shows "x \<in># N"
dag_setof_subsetD: fixes p :: "ref"   and low :: "ref \<Rightarrow> ref"   and high :: "ref \<Rightarrow> ref"   and t :: "dag"   and n :: "ref"   and nt :: "dag" assumes "Dag p low high t"   and "n \<in> set_of t"   and "Dag n low high nt" shows "set_of nt \<subseteq> set_of t"
clearjunk_delete: fixes x :: "'c"   and al :: "('c \<times> 'd) list" shows "clearjunk (delete x al) = delete x (clearjunk al)"
eval_wp_Abort: fixes P :: "'c \<Rightarrow> real" shows "wp Abort P = (\<lambda>s. 0)"
I: fixes P :: "'a \<Rightarrow> ('b \<times> 'b) set"   and P' :: "'a \<Rightarrow> ('b \<times> 'b) set"   and x :: "'b"   and y :: "'b" assumes "profile A Is P"   and "profile A Is P'"   and "x \<in> A"   and "y \<in> A"   and "\<And>i. \<lbrakk>i \<in> Is; x \<^bsub>(P i)\<^esub>\<prec> y\<rbrakk> \<Longrightarrow> x \<^bsub>(P' i)\<^esub>\<prec> y"   and "\<And>i. \<lbrakk>i \<in> Is; x \<^bsub>(P i)\<^esub>\<approx> y\<rbrakk> \<Longrightarrow> x \<^bsub>(P' i)\<^esub>\<preceq> y"   and "\<exists>k\<in>Is. x \<^bsub>(P k)\<^esub>\<approx> y \<and> x \<^bsub>(P' k)\<^esub>\<prec> y \<or> y \<^bsub>(P k)\<^esub>\<prec> x \<and> x \<^bsub>(P' k)\<^esub>\<preceq> y"   and "x \<^bsub>(scf P)\<^esub>\<preceq> y" shows "x \<^bsub>(scf P')\<^esub>\<prec> y"
Eats_cong1: fixes t :: "tm"   and t' :: "tm"   and u :: "tm"   and u' :: "tm" shows "{} \<turnstile> t EQ t' AND u EQ u' IMP Eats t u EQ Eats t' u'"
set_eq_fold: fixes x :: "'d"   and xs :: "'d list" shows "Max (set (x # xs)) = fold max xs x"
G_def: shows "G = ((- 1) ^ Suc n * bernoulli (Suc n) / fact (Suc n)) *\<^sub>R (C - f (real_of_int a))"
wbisim12: shows "delay_bisimulation_diverge_final trsys1 trsys2 bisim12 tlsim12 \<tau>move1 \<tau>move2 final1 final2"
bin_rest_set_bits: fixes f :: "nat \<Rightarrow> bool" assumes "wf_set_bits_int f" shows "set_bits f div 2 = set_bits (f \<circ> Suc)"
meval_elim: fixes cms :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list"   and mem :: "'Var \<Rightarrow> 'Val"   and k :: "nat"   and cms' :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list"   and mem' :: "'Var \<Rightarrow> 'Val" assumes "(cms, mem) \<leadsto>\<^bsub>k\<^esub> (cms', mem')" obtains cm' :: "'Com \<times> (Mode \<Rightarrow> 'Var set)" where "cms' = cms[k := cm']" and "(cms ! k, mem) \<leadsto> (cm', mem')" and "k < length cms"
new_path_follows_old_paths: fixes xs :: "'a list"   and w :: "'a"   and P :: "'a list" assumes "v0 \<leadsto>xs\<leadsto> w"   and "tl xs \<noteq> []"   and "v1 \<notin> set xs"   and "new_last \<notin> set xs"   and "P \<in> paths_with_new"   and "hd (tl xs) = hd (tl P)"   and "H.edges_of_walk xs \<subseteq> \<Union> (H.edges_of_walk ` paths_with_new)" shows "H.edges_of_walk xs \<subseteq> H.edges_of_walk P"
foundation1: fixes \<tau> :: "'d state \<times> 'd state" shows "\<tau> \<Turnstile> true"
simps_3: fixes \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and e :: "'c"   and p :: "'d" shows "Semantics.matches \<gamma> (Match e) p = \<gamma> e p"
icard_eq_enat_imp_card: fixes A :: "'a set"   and n :: "nat" assumes "icard A = enat n" shows "card A = n"
croots3: fixes p :: "complex poly" assumes "degree p = 3" shows "set (Cubic_Polynomials.croots3 p) = {x. poly p x = 0}"
truncate_f': shows "fds_truncate m f' = fds (\<lambda>n. if n = Suc 0 then 0::'a else fds_nth (fds_truncate m f) n)"
eval_ts: fixes G :: "prog"   and s :: "state"   and e :: "expr"   and v :: "val"   and s' :: "state"   and L :: "lenv"   and C :: "qtname"   and T :: "ty"   and A :: "assigned" assumes "G\<turnstile>s \<midarrow>e-\<succ>v\<rightarrow> s'"   and "wf_prog G"   and "s\<Colon>\<preceq>(G, L)"   and "\<lparr>prg = G, cls = C, lcl = L\<rparr>\<turnstile>e\<Colon>-T"   and "\<lparr>prg = G, cls = C, lcl = L\<rparr>\<turnstile> dom (locals (snd s)) \<guillemotright>In1l e\<guillemotright> A" shows "s'\<Colon>\<preceq>(G, L) \<and> (normal s' \<longrightarrow> G,snd s'\<turnstile>v\<Colon>\<preceq>T) \<and> error_free s = error_free s'"
fresh_fin_union: fixes X :: "'a set"   and Y :: "'a set"   and a :: "'b" assumes "pt TYPE('a) TYPE('b)"   and "at TYPE('b)"   and "fs TYPE('a) TYPE('b)"   and "finite X"   and "finite Y" shows "a \<sharp> (X \<union> Y) = (a \<sharp> X \<and> a \<sharp> Y)"
new_document_document_element: fixes h :: "(_) heap"   and h' :: "(_) heap"   and new_document_ptr :: "(_) document_ptr" assumes "h \<turnstile> new_document \<rightarrow>\<^sub>h h'"   and "h \<turnstile> new_document \<rightarrow>\<^sub>r new_document_ptr" shows "h' \<turnstile> get_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t new_document_ptr document_element \<rightarrow>\<^sub>r None"
WT2: shows "\<I> \<turnstile>res res2 \<surd>"
empty_def: shows "local.empty \<equiv> Abs_SkewBinomialHeap BsSkewBinomialHeapStruc.bs_empty"
\<sigma>t_def: fixes D :: "'c alpha_d_scheme \<times> 'c alpha_d_scheme \<Rightarrow> bool" shows "\<sigma>t D \<equiv> \<lambda>(A, A'). D (A, A'\<lparr>ok := True\<rparr>)"
cases: fixes a :: "'c"   and r :: "('c \<times> 'd) set" assumes "a \<in> Domain r" obtains a :: "'c" and b :: "'d" where "aa__ = a" and "(a, b) \<in> r"
f_Equiv_Exec_Stream_Acc_Output_set_eq: fixes k1 :: "nat"   and k2 :: "nat"   and equiv_states :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and localState1 :: "'f \<Rightarrow> 'd"   and c1 :: "'f"   and localState2 :: "'g \<Rightarrow> 'e"   and c2 :: "'g"   and A :: "'h set"   and input_fun1 :: "'h \<Rightarrow> 'i message_af"   and input_fun2 :: "'h \<Rightarrow> 'j message_af"   and output_fun1 :: "'f \<Rightarrow> 'k message_af"   and output_fun2 :: "'g \<Rightarrow> 'k message_af"   and trans_fun1 :: "'i message_af \<Rightarrow> 'f \<Rightarrow> 'f"   and trans_fun2 :: "'j message_af \<Rightarrow> 'g \<Rightarrow> 'g"   and input :: "'h list" assumes "0 < k1"   and "0 < k2"   and "equiv_states (localState1 c1) (localState2 c2)"   and "Equiv_Exec_stable_set A equiv_states localState1 localState2 input_fun1 input_fun2 output_fun1 output_fun2 trans_fun1 trans_fun2 k1 k2 c1 c2"   and "set input \<subseteq> A" shows "f_Exec_Comp_Stream_Acc_Output k1 output_fun1 trans_fun1 (map input_fun1 input) c1 = f_Exec_Comp_Stream_Acc_Output k2 output_fun2 trans_fun2 (map input_fun2 input) c2"
stable_ordIso2: fixes r :: "('a \<times> 'a) set"   and r' :: "('b \<times> 'b) set" assumes "stable r"   and "r =o r'" shows "stable r'"
IH: fixes p :: "'c spmf" shows "trace_callee run_resource (map_spmf (resource_of_oracle callee) p) xs x = trace_callee callee p xs x"
assms: shows "\<not> (w2 = NEG_INF \<and> w1 = NEG_INF \<or> w2 = POS_INF \<and> w1 = POS_INF) \<and> sint w2 \<le> sint w1"
tdev'_le: fixes x :: "'a"   and y :: "'a pdevs"   and p :: "nat" assumes "x \<le> tdev y" shows "x \<le> tdev' p y"
IH: fixes n :: "nat" assumes "n < 2 ^ k" shows "fds_nth (f ^ k) n = (0::'a)"
involved_ifp: shows "\<forall>s a. \<forall>d\<in>int.involved a. kprecondition s (the a) \<longrightarrow> ifp d (current s)"
one_chain_line_integral_eq_line_integral_on_sudivision_1: fixes one_chain1 :: "(int \<times> (real \<Rightarrow> real \<times> real)) set"   and subdiv :: "(int \<times> (real \<Rightarrow> real \<times> real)) set"   and F :: "real \<times> real \<Rightarrow> real \<times> real"   and basis :: "(real \<times> real) set" assumes "chain_subdiv_chain one_chain1 subdiv"   and "boundary_chain one_chain1"   and "boundary_chain subdiv"   and "\<forall>(k, \<gamma>)\<in>subdiv. line_integral_exists F basis \<gamma>"   and "\<forall>(k, \<gamma>)\<in>subdiv. valid_path \<gamma>"   and "finite one_chain1"   and "finite basis" shows "one_chain_line_integral F basis one_chain1 = one_chain_line_integral F basis subdiv"
cong_eq_0_not_coprime: fixes p :: "int"   and a :: "int" assumes "prime p"   and "[a = 0] (mod p)" shows "\<not> coprime a p"
i: shows "i < dim_row A + sum_list (map dim_row As)"
assms: shows "the (rt dip) \<sqsubseteq> r"
lookup_frag_of: fixes c :: "'b" shows "lookup (frag_of c) = (\<lambda>x. if x = c then 1 else 0)"
emeasure_Collect_distr: fixes X :: "'a \<Rightarrow> 'b"   and M :: "'a measure"   and N :: "'b measure"   and P :: "'b \<Rightarrow> bool" assumes "X \<in> M \<rightarrow>\<^sub>M N"   and "Measurable.pred N P" shows "emeasure (distr M N X) {x \<in> space N. P x} = emeasure M {x \<in> space M. P (X x)}"
IH_2: fixes lev :: "nat" assumes "\<And>x. lev < x \<Longrightarrow> \<not> loose_bvar1 t2 x"   and "\<not> occs (case v of (a, b) \<Rightarrow> Fv a b) t2" shows "bind_fv2 v lev (subst_bv1 t2 lev (case v of (a, b) \<Rightarrow> Fv a b)) = t2"
contractible_space_subtopology_singleton: fixes X :: "'a topology"   and a :: "'a" shows "contractible_space (subtopology X {a})"
paint_inv1: fixes t :: "('a, 'b) rbt" assumes "inv1l t" shows "inv1 (paint B t)"
remove_key_keys: fixes f :: "'b \<Rightarrow>\<^sub>0 'c"   and k :: "'b" shows "keys f - {k} = keys (remove_key k f)"
a_get_disconnected_document_locs_def: shows "get_disconnected_document_locs = \<Union> (range get_disconnected_nodes_locs) \<union> (\<Union>ptr. {preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t ptr RObject.nothing)})"
p_greaterI: fixes p' :: "real" assumes "1 < (\<Sum>i<k. as ! i * bs ! i powr p')" shows "p' < p"
splitAt_distinct_snd_aux: fixes vs :: "'a list"   and ram :: "'a" assumes "distinct vs" shows "distinct (snd (splitAt ram vs))"
z_2: shows "norm (z - k2) < norm (k1 - k2) / 2"
faces: shows "faces g'' = replace f' [f21] (faces g') @ [f12]"
type_global_elim: fixes cs :: "('Var, 'AExp, 'BExp) Stmt list" assumes "\<turnstile> cs" obtains "list_all (\<lambda>c. \<turnstile> \<Gamma>\<^sub>0 {c} \<Gamma>\<^sub>0) cs" and "\<forall>mem. list_all (\<lambda>cm. locally_sound_mode_use (cm, mem)) (add_initial_modes cs) \<and> globally_sound_mode_use (add_initial_modes cs, mem)"
IH2: assumes "simple_labels \<Gamma>"   and "wellformed \<Gamma>"   and "\<exists>pn. p2 \<in> subterms (\<Gamma> pn)"   and "l \<in> labels \<Gamma> p2" shows "\<forall>p'\<in>sterms \<Gamma> p2. l \<in> labels \<Gamma> p'"
single_comp_support: fixes asset :: "'b"   and qty :: "nat \<Rightarrow> 'c \<Rightarrow> real" shows "support_set (qty_single asset qty) \<subseteq> {asset}"
insertE: fixes a :: "'b"   and b :: "'b"   and A :: "'b set" assumes "(a::'b) \<in> (b::'b) \<triangleright> (A::'b set)" obtains "(a::'b) = (b::'b)"   | "(a::'b) \<in> (A::'b set)"
assms_7: shows "rk {A, B, A'} = 3"
ps_first_two: fixes ps :: "real poly list"   and p :: "real poly" assumes "sturm_seq ps p" obtains q :: "real poly" and ps' :: "real poly list" where "ps = p # q # ps'"
enat_of_option_le_conv: fixes m :: "nat option"   and n :: "nat option" shows "(Common.enat_of_option m \<le> Common.enat_of_option n) = (case (m, n) of (None, None) \<Rightarrow> True | (None, Some aa) \<Rightarrow> False | (Some aa, None) \<Rightarrow> True | (Some aa, Some b) \<Rightarrow> aa \<le> b)"
simps_8: fixes t :: "'c local.term" shows "Dom \<^bold>\<r>\<^sup>-\<^sup>1\<^bold>[t\<^bold>] = Dom t"
simps_2: fixes x2 :: "nat"   and y2 :: "nat" shows "(Intv x2 = Intv y2) = (x2 = y2)"
m_inv_mult_of: fixes x :: "'a" assumes "x \<in> carrier (mult_of R)" shows "inv\<^bsub>mult_of R\<^esub> x = inv x"
def_fold: fixes l :: "('a state \<times> 'a state \<Rightarrow> 'c) list" assumes "list_all (\<lambda>f. \<tau> \<Turnstile> \<upsilon> f) l" shows "\<tau> \<Turnstile> \<delta> foldl UML_Sequence.OclIncluding Sequence{} l"
pos_scaleC: fixes a :: "complex"   and X :: "'b \<Rightarrow>\<^sub>C\<^sub>L 'b" assumes "0 \<le> a"   and "pos X" shows "pos (a *\<^sub>C X)"
lm034: fixes P :: "('a \<times> 'b) set"   and Q :: "('a \<times> 'b) set" assumes "P +< Q = Q" shows "Domain P \<subseteq> Domain Q"
l3_dy_others_refines_skip: fixes M :: "msg" assumes "M \<notin> range LtK"   and "M \<notin> valid"   and "M \<notin> payload" shows "{R23s} Id, l3_dy M {> R23s}"
stabilizer_is_subgroup: fixes G :: "('c, 'd) monoid_scheme"   and \<phi> :: "'c \<Rightarrow> 'e \<Rightarrow> 'e"   and M :: "'e set"   and m :: "'e" assumes "group_action G \<phi> M"   and "m \<in> M" shows "subgroup (group_action.stabilizer G \<phi> m) G"
eval\<^sub>wp_eval\<^sub>w_eq: shows "eval\<^sub>wp = (\<lambda>x xa xb xc xd xe. \<langle>x, xa, xb\<rangle> \<leadsto> \<langle>xc, xd, xe\<rangle>)"
gt_imp_ge: fixes x :: "'a"   and y :: "'a" assumes "gt x y" shows "y \<le> x"
mismatchPres: fixes P :: "pi"   and Rel :: "(pi \<times> pi) set"   and Q :: "pi"   and Rel' :: "(pi \<times> pi) set"   and a :: "name"   and b :: "name" assumes "P \<leadsto><Rel> Q"   and "Rel \<subseteq> Rel'"   and "\<And>R S c d. \<lbrakk>(R, S) \<in> Rel; c \<noteq> d\<rbrakk> \<Longrightarrow> ([c\<noteq>d]R, S) \<in> Rel" shows "[a\<noteq>b]P \<leadsto><Rel'> [a\<noteq>b]Q"
AE_space: fixes M :: "'a measure" shows "AE x in M. x \<in> space M"
eq_zero_iff: fixes n :: "nat"   and \<chi> :: "nat \<Rightarrow> complex"   and x :: "nat" assumes "dcharacter n \<chi>" shows "(\<chi> x = 0) = (\<not> coprime x n)"
Rep_perm_uminus: fixes p :: "'a perm" shows "Rep_perm (- p) = inv (Rep_perm p)"
wf_no_infinite_down_chainE: fixes r :: "('b \<times> 'b) set"   and f :: "nat \<Rightarrow> 'b" assumes "wf r" obtains k :: "nat" where "(f (Suc k), f k) \<notin> r"
partial_deriv: shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P c Q,A"
homE_1: fixes A :: "'b"   and B :: "'b"   and f :: "'a" assumes "A \<in> Ob"   and "B \<in> Ob"   and "f \<in> Hom A B" shows "Dom f = A"
is_TNil_def: fixes tllist :: "('a, 'b) tllist" shows "is_TNil tllist = (\<exists>x1. tllist = TNil x1)"
intro: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) heap \<Rightarrow> bool"   and set_val :: "(_) character_data_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_val_locs :: "(_) character_data_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_tag_name :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, char list) prog"   and get_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" assumes "l_set_val\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M set_val set_val_locs"   and "l_get_tag_name\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_tag_name get_tag_name_locs" shows "l_set_val_get_tag_name\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M set_val set_val_locs get_tag_name get_tag_name_locs"
rb_aux_inv1_D2: fixes bs :: "('t \<Rightarrow>\<^sub>0 'b) list" assumes "rb_aux_inv1 bs" shows "0 \<notin> rep_list ` set bs"
continuous_on_product_coordinates: fixes i :: "'d" shows "continuous_on UNIV (\<lambda>x. x i)"
zmultiset_eqI: fixes A :: "'a zmultiset"   and B :: "'a zmultiset" assumes "\<And>x. zcount A x = zcount B x" shows "A = B"
inj_on_eq_iff: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and x :: "'a"   and y :: "'a" assumes "inj_on f A"   and "x \<in> A"   and "y \<in> A" shows "(f x = f y) = (x = y)"
eSuc_Sup: fixes A :: "enat set" assumes "A \<noteq> {}" shows "eSuc (\<Squnion> A) = \<Squnion> (eSuc ` A)"