hcju/isabelleps huggingface dataset:
DSQR_ALT: shows "dsqr = {r \<in> \<delta>. qw \<in> set (rhsq r) \<and> set (rhsq r) \<subseteq> dom Q - (W - {qw})}"
skippable: shows "(rg::'a::refinement_lattice) \<sqsubseteq> skip"
strategy_conforming_path_exists_single: fixes v0 :: "'a"   and p :: "Player"   and \<sigma> :: "'a \<Rightarrow> 'a" assumes "v0 \<in> V"   and "strategy p \<sigma>" obtains P :: "'a llist" where "vmc_path G P v0 p \<sigma>"
\<pi>_def: shows "\<pi> \<equiv> into_WW 0 \<circ> Cantor_\<omega>\<omega>"
take_rews_1: fixes n :: "nat"   and cfun :: "Value \<rightarrow> Value" shows "\<psi>\<^sup>D\<^bsub>Suc n\<^esub>\<cdot>(Fn\<cdot>cfun) = Fn\<cdot>(cfun_map\<cdot>\<psi>\<^sup>D\<^bsub>n\<^esub>\<cdot>\<psi>\<^sup>D\<^bsub>n\<^esub>\<cdot>cfun)"
Suc_pred: fixes n :: "nat" assumes "0 \<sqsubset> n" shows "Suc (n - Suc 0) = n"
S5RPC_enabled: fixes rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState"   and p :: "PrIds" shows "\<turnstile> $S5 rmhist p \<and> ImpNext p \<and> [HNext rmhist p]_(c p, r p, m p, rmhist!p) \<longrightarrow> $Enabled (<RPCNext crCh rmCh rst p>_(r p))"
Sup_le_iff: fixes A :: "('b \<times> 'c) set"   and b :: "'b \<times> 'c" shows "less_eq_prod (Sup_prod A) b = (\<forall>a\<in>A. less_eq_prod a b)"
CallObj: fixes e :: "char list exp"   and s :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and e' :: "char list exp"   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 M :: "char list"   and es :: "char list exp list" assumes "P \<turnstile> \<langle>e,s\<rangle> \<rightarrow> \<langle>e',s'\<rangle>" shows "P \<turnstile> \<langle>e\<bullet>M(es),s\<rangle> \<rightarrow> \<langle>e'\<bullet>M(es),s'\<rangle>"
arr_char: fixes f :: "'a \<times> 'a \<times> 'a" shows "CCC.arr f = (C.arr (fst f) \<and> CC.arr (snd f))"
subdivFace_def: fixes g :: "graph"   and f :: "face"   and vos :: "nat option list" shows "subdivFace g f vos \<equiv> subdivFace' g f (the (hd vos)) 0 (tl vos)"
wens_set_imp_subset: fixes A :: "'b set"   and F :: "'b program"   and B :: "'b set" assumes "A \<in> wens_set F B" shows "B \<subseteq> A"
ArrowCallNE: fixes \<Gamma> :: "String.literal \<rightharpoonup>\<^sub>f 'a OCL_Types.type"   and src :: "'a expr"   and call :: "'a call_expr"   and b :: "'a expr" assumes "\<Gamma> \<turnstile> Call src ArrowCall call \<Rrightarrow> b" obtains src\<^sub>2 :: "'a expr" and \<tau> :: "'a OCL_Types.type" and \<sigma> :: "'a OCL_Types.type" and call\<^sub>2 :: "'a call_expr" where "b = Call (OperationCall src\<^sub>2__ DotCall (Inl (Inl OclAsSetOp)) []) ArrowCall call\<^sub>2__" and "\<Gamma> \<turnstile> src \<Rrightarrow> src\<^sub>2__" and "\<Gamma> \<turnstile>\<^sub>E src\<^sub>2__ : \<tau>__" and "\<tau>__ \<le> OclAny[?] \<or> \<tau>__ \<le> Tuple fmempty" and "\<Gamma> \<turnstile>\<^sub>E OperationCall src\<^sub>2__ DotCall (Inl (Inl OclAsSetOp)) [] : \<sigma>" and "(\<Gamma>, \<sigma>) \<turnstile>\<^sub>C call \<Rrightarrow> call\<^sub>2__"   | src\<^sub>2 :: "'a expr" and \<tau> :: "'a OCL_Types.type" and uu_ :: "'a OCL_Types.type" and call\<^sub>2 :: "'a call_expr" where "b = Call src\<^sub>2 ArrowCall call\<^sub>2" and "\<Gamma> \<turnstile> src \<Rrightarrow> src\<^sub>2" and "\<Gamma> \<turnstile>\<^sub>E src\<^sub>2 : \<tau>" and "element_type \<tau> uu_" and "(\<Gamma>, \<tau>) \<turnstile>\<^sub>C call \<Rrightarrow> call\<^sub>2"
bij_nat_of_b: shows "\<forall>n. nat_of_b (inv nat_of_b n) = n"
3: shows "of_complex (\<i> / 2) \<in> unit_disc"
continuous_map_span_sum: fixes I :: "'a set"   and b :: "'a \<Rightarrow> 'b"   and B :: "'b set" assumes "\<And>i. i \<in> I \<Longrightarrow> b i \<in> B" shows "continuous_map euclidean (top_of_set (span B)) (\<lambda>x. \<Sum>i\<in>I. x i *\<^sub>R b i)"
insert_body_commutes: fixes e1 :: "'c \<times> 'd \<times> bool"   and e2 :: "'c \<times> 'd \<times> bool"   and xs :: "('c \<times> 'd \<times> bool) list" assumes "fst e1 \<noteq> fst e2" shows "insert_body (insert_body xs e1) e2 = insert_body (insert_body xs e2) e1"
weak_get_via_put: fixes s :: "'b" assumes "s \<in> \<S>" shows "get s = (THE v. put s v = s)"
r_comp_rtrancl_eq: fixes r :: "('c \<times> 'c) set" shows "r O r\<^sup>* = r\<^sup>* O r"
of_int_poly_eq_iff: fixes p :: "int poly"   and q :: "int poly" shows "(of_int_poly p = of_int_poly q) = (p = q)"
monoseq_le: fixes a :: "nat \<Rightarrow> 'a"   and x :: "'a" assumes "monoseq a"   and "a \<longlonglongrightarrow> x" shows "(\<forall>n. a n \<le> x) \<and> (\<forall>m n. m \<le> n \<longrightarrow> a m \<le> a n) \<or> (\<forall>n. x \<le> a n) \<and> (\<forall>m n. m \<le> n \<longrightarrow> a n \<le> a m)"
norm_ket_k: fixes k :: "nat" shows "inner_prod (ket_k k) (ket_k k) \<le> 1"
final_PLUS: fixes xs :: "'c rexp list" shows "final (PLUS xs) = (\<exists>r\<in>set xs. final r)"
that: fixes y :: "'b" assumes "y \<in> T"   and "g y = y" shows "thesis"
swap_args2_def: fixes f :: "'d \<Rightarrow> 'e \<Rightarrow> 'b"   and x :: "'e"   and y :: "'d" shows "swap_args2 f x y \<equiv> f y x"
take_bit_take_bit: fixes 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 divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bit :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and drop_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and take_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and m :: "nat"   and n :: "nat"   and a :: "'a" assumes "class.semiring_bit_shifts plus minus zero times one divide modulo bit push_bit drop_bit take_bit" shows "take_bit m (take_bit n a) = take_bit (min m n) a"
Nil2: fixes xs :: "'b list" assumes "as = xs"   and "bs = []" shows "P"
rdBy_def: fixes s :: "state"   and p :: "Proc"   and q :: "Proc"   and d :: "Disk" shows "rdBy s p q d = {br \<in> blocksRead s q d. proc br = p}"
assms: shows "x \<in> \<int>\<^sub>\<le>\<^sub>0"
axioms_1: fixes A :: "'ag \<Rightarrow> 'ag \<Rightarrow> 'ag"   and B :: "'ah \<Rightarrow> 'ah \<Rightarrow> 'ah"   and g :: "'ah" assumes "constant_transformation A B g" shows "category A"
Ifun_star_n: fixes F :: "nat \<Rightarrow> 'd \<Rightarrow> 'c"   and X :: "nat \<Rightarrow> 'd" shows "star_n F \<star> star_n X = star_n (\<lambda>n. F n (X n))"
zero_in_Polys: fixes X :: "'c set" shows "0 \<in> P[X]"
assms_1: shows "\<S>\<bullet>None \<tturnstile>_ i vs;cs_es : ts'"
closure_closure: fixes S :: "'c set" shows "closure (closure S) = closure S"
simps_1: fixes k :: "'a" shows "to_mini_formula (Atom k) = Atom k"
assms_6: shows "r' \<equiv> r \<union> {(x, y). x \<in> Z \<and> y \<in> X \<and> intv_const (I x) < k x \<and> isIntv (I' y)}"
Stirling_Suc_n_n: fixes n :: "nat" shows "Stirling (Suc n) n = Suc n choose 2"
safe_combined_implies_safe_combined2: fixes c :: "('b, 't, 'loc) Combined.configuration" assumes "cri.InvCapsNonneg (exchange_config c)"   and "safe_combined c" shows "safe_combined2 c"
assms_1: shows "g \<cdot> f \<noteq> null"
Mp_1: fixes m :: "int" assumes "poly_mod_2 m" shows "poly_mod.Mp m 1 = 1"
comp_Abs_inverse: fixes f :: "nat \<Rightarrow> 'b"   and g :: "int \<Rightarrow> 'b" assumes "f \<circ> nat = g" shows "g \<circ> int = f"
permutes_sing: fixes p :: "'a \<Rightarrow> 'a"   and a :: "'a" shows "(p permutes {a}) = (p = id)"
flip_fresh_fresh: fixes a :: "'c"   and x :: "'b"   and b :: "'c" assumes "atom a \<sharp> x"   and "atom b \<sharp> x" shows "(a \<leftrightarrow> b) \<bullet> x = x"
simps_2: fixes k :: "nat" shows "ORD_OF (Suc k) = SUCC (ORD_OF k)"
inf_sup_aci_5: fixes inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a" assumes "class.lattice inf less_eq less sup" shows "sup x y = sup y x"
le_inf_iff: fixes x :: "'b"   and y :: "'b"   and z :: "'b" shows "(x \<le> y \<sqinter> z) = (x \<le> y \<and> x \<le> z)"
hyp: assumes "finals es" shows "P,E \<turnstile> \<langle>es,s\<rangle> [\<Rightarrow>] \<langle>es,s\<rangle>"
HSem_restr: fixes h :: "('b \<times> 'a) list"   and \<rho> :: "'b \<Rightarrow> 'c" shows "\<lbrace> h \<rbrace>\<rho> f|` (- domA h) = \<lbrace> h \<rbrace>\<rho>"
simps_1: fixes f :: "'a \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and k :: "'a"   and v :: "'c"   and as :: "('a \<times> 'c) list"   and k' :: "'a"   and v' :: "'c"   and bs :: "('a \<times> 'c) list" shows "sinter_with f ((k, v) # as) ((k', v') # bs) = (if k' < k then sinter_with f ((k, v) # as) bs else if k < k' then sinter_with f as ((k', v') # bs) else (k, f k v v') # sinter_with f as bs)"
\<theta>\<^sub>\<tau>: shows "\<guillemotleft>\<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] : t\<^sub>0 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> u \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>"
mod_ring_mod: shows "(mod_ring_rel ===> mod_ring_rel ===> mod_ring_rel) (\<lambda>x y. if y = 0 then x else 0) (mod)"
assms: fixes x1 :: "'a"   and x2 :: "'b" assumes "\<And>y1 y2. meas y1 y2 < meas x1 x2 \<Longrightarrow> S y1 y2" shows "S x1 x2"
arr_eqI': fixes f :: "'s"   and f' :: "'s" assumes "par f f'"   and "\<And>x. \<guillemotleft>x : unity \<rightarrow> local.dom f\<guillemotright> \<Longrightarrow> f \<cdot> x = f' \<cdot> x" shows "f = f'"
assms_2: fixes i :: "nat" assumes "i \<in> t" shows "enat i < llength (lselect s w)"
1: shows "x * (1::'a)\<^sup>\<circ> \<le> L \<squnion> x"
length_restrict: fixes A :: "nat set"   and v :: "'b option list" shows "length (restrict A v) = length v"
sum_closed: fixes A :: "'a set"   and X :: "'b set"   and f :: "'b \<Rightarrow> 'a" assumes "ring_closed A"   and "\<And>x. x \<in> X \<Longrightarrow> f x \<in> A" shows "sum f X \<in> A"
bal_split_right_2: fixes ls :: "('b btree \<times> 'b) list"   and rs :: "('b btree \<times> 'b) list"   and t :: "'b btree" assumes "bal (Node (ls @ rs) t)" shows "height (Node rs t) = height (Node (ls @ rs) t)"
unique_paths_implies_no_cycles: fixes G :: "('a, 'b) Graph_scheme"   and xs :: "'a list" assumes "Graph G"   and "\<And>v w. Graph.connected G v w \<Longrightarrow> \<exists>!xs. v \<leadsto>xs\<leadsto>\<^bsub>G\<^esub> w" shows "\<not> Graph.cycle G xs"
CD: shows "C \<inter> D \<in> null_sets lborel"
empty_rs_stateD: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and s :: "state"   and t :: "state" assumes "\<Gamma>,\<gamma>,p\<turnstile> \<langle>[], s\<rangle> \<Rightarrow>\<^sub>s t" shows "t = s"
indices_state_def: fixes s :: "('b, 'c) state" shows "indices_state s = {i. \<exists>x b. Mapping.lookup (\<B>\<^sub>i\<^sub>l s) x = Some (i, b) \<or> Mapping.lookup (\<B>\<^sub>i\<^sub>u s) x = Some (i, b)}"
from_nat_mono: fixes i :: "nat"   and j :: "nat" assumes "i < j"   and "j < CARD('a)" shows "mod_type_class.from_nat i < mod_type_class.from_nat j"
Fun: fixes x2a :: "('f, 'a) Term.term" assumes "x2a \<in> set ts" shows "NS x2a (Fun f [])"
rcm_C: fixes a :: "'a"   and b :: "'a" shows "(a \<le> b) = (a r\<rightarrow> b = (1::'a))"
prems: fixes i :: "nat" assumes "i \<le> Suc (Suc n)" shows "f i \<le> Suc n"
fun_eq_on_set_and_comp_imp_eq: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c"   and A :: "'b set" assumes "fun_eq_on f g A"   and "fun_eq_on f g (- A)" shows "f = g"
PRED_closed: fixes v :: "nat"   and u :: "nat" assumes "PRED v = Some u" shows "u \<in> dom PRED"
Max_ge: fixes A :: "'a word set"   and x :: "'a word" assumes "finite A"   and "x \<in> A" shows "x \<le>s signed.Max A"
eq: shows "f \<equiv> C (fixp_fun (\<lambda>f. U (F (C f))))"
ext1_ext2_is_not_act1: fixes A2 :: "('e set \<times> 'e set \<times> 'e set) \<times> 'f set \<times> ('f \<times> 'e \<times> 'f) set"   and A1 :: "('e set \<times> 'e set \<times> 'e set) \<times> 'g set \<times> ('g \<times> 'e \<times> 'g) set"   and a :: "'e" assumes "compat_ioas A2 A1"   and "a \<in> externals (asig_of A1)"   and "a \<notin> externals (asig_of A2)" shows "a \<notin> actions (asig_of A2)"
rule_nonterminal_type: fixes \<NN> :: "symbol set"   and \<TT> :: "symbol set"   and \<RR> :: "(symbol \<times> symbol list) set"   and \<SS> :: "symbol"   and N :: "symbol"   and \<alpha> :: "symbol list" assumes "CFG \<NN> \<TT> \<RR> \<SS>"   and "(N, \<alpha>) \<in> \<RR>" shows "CFG.is_nonterminal \<NN> N"
eq: shows "b = ((1::'a) / k) *s a - ((1::'a) / k) *s (a - k *s b)"
ap_pmf_homo: fixes f :: "'e \<Rightarrow> 'd"   and x :: "'e" shows "return_pmf f \<diamondop> return_pmf x = return_pmf (f x)"
wp1n_RS2: fixes p :: "('b, 'c) Combinators list" assumes "wellformed_policy1_strong p" shows "wellformed_policy1_strong (removeShadowRules2 p)"
obj_simps_3: fixes a :: "'d" assumes "D.obj a" shows "trg\<^sub>D a = a"
matchTrans: fixes a :: "name"   and b :: "name"   and P :: "pi"   and Rs :: "residual" assumes "[a\<frown>b]P \<longmapsto> Rs"   and "a \<noteq> b" shows "False"
measure_Union_le: fixes F :: "'c set set"   and M :: "'c measure" assumes "finite F"   and "\<And>S. S \<in> F \<Longrightarrow> S \<in> sets M" shows "Sigma_Algebra.measure M (\<Union> F) \<le> sum (Sigma_Algebra.measure M) F"
simps: fixes S :: "'b literal multiset \<Rightarrow> 'b literal multiset"   and subst_atm :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and id_subst :: "'c"   and comp_subst :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and renamings_apart :: "'b literal multiset list \<Rightarrow> 'c list"   and atm_of_atms :: "'b list \<Rightarrow> 'b"   and mgu :: "'b set set \<Rightarrow> 'c option"   and less_atm :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and timestamp_factor :: "nat"   and size_atm :: "'b \<Rightarrow> nat"   and size_factor :: "nat"   and C :: "'b literal multiset"   and i :: "nat" assumes "weighted_FO_resolution_prover_with_size_timestamp_factors S subst_atm id_subst comp_subst renamings_apart atm_of_atms mgu less_atm timestamp_factor" shows "weighted_FO_resolution_prover_with_size_timestamp_factors.weight size_atm size_factor timestamp_factor (C, i) = size_factor * size_multiset (size_literal size_atm) C + timestamp_factor * i"
igSubstAbsIPresIGWlsAbsSTR_imp_igSubstAbsIPresIGWlsAbs: fixes Delta :: "('b, 'c, 'd, 'e, 'f) signature"   and MOD :: "('g, 'h, 'd, 'e, 'i, 'j, 'k, 'l, 'm) model_scheme" assumes "FixSyn TYPE('a) Delta"   and "FixSyn.igSubstAbsIPresIGWlsAbsSTR Delta MOD" shows "FixSyn.igSubstAbsIPresIGWlsAbs Delta MOD"
right_neutral: fixes a :: "'b" shows "a + (zero_class.zero::'b) = a"
Rep_ccs: fixes x :: "ccs" shows "Rep_ccs x \<in> Collect ccs_Rep_set"
get_dom_component_no_overlap: fixes heap_is_wellformed :: "(_) heap \<Rightarrow> bool"   and parent_child_rel :: "(_) heap \<Rightarrow> ((_) object_ptr \<times> (_) object_ptr) set"   and type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and known_ptrs :: "(_) heap \<Rightarrow> bool"   and to_tree_order :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and get_parent :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr option) prog"   and get_parent_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_dom_component :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and is_strongly_dom_component_safe :: "(_) object_ptr set \<Rightarrow> (_) object_ptr set \<Rightarrow> (_) heap \<Rightarrow> (_) heap \<Rightarrow> bool"   and is_weakly_dom_component_safe :: "(_) object_ptr set \<Rightarrow> (_) object_ptr set \<Rightarrow> (_) heap \<Rightarrow> (_) heap \<Rightarrow> bool"   and get_root_node :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr) prog"   and get_root_node_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_ancestors :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and get_ancestors_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_element_by_id :: "(_) object_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, (_) element_ptr option) prog"   and get_elements_by_tag_name :: "(_) object_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, (_) element_ptr list) prog"   and h :: "(_) heap"   and ptr :: "(_) object_ptr"   and c :: "(_) object_ptr list"   and ptr' :: "(_) object_ptr"   and c' :: "(_) object_ptr list" assumes "l_get_dom_component\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M heap_is_wellformed parent_child_rel type_wf known_ptr known_ptrs to_tree_order get_parent get_parent_locs get_child_nodes get_child_nodes_locs get_dom_component is_strongly_dom_component_safe is_weakly_dom_component_safe get_root_node get_root_node_locs get_ancestors get_ancestors_locs get_disconnected_nodes get_element_by_id get_elements_by_tag_name"   and "heap_is_wellformed h"   and "type_wf h"   and "known_ptrs h"   and "h \<turnstile> get_dom_component ptr \<rightarrow>\<^sub>r c"   and "h \<turnstile> get_dom_component ptr' \<rightarrow>\<^sub>r c'" shows "set c \<inter> set c' = {} \<or> c = c'"
R: fixes A :: "'a stream measure"   and B :: "'a stream measure" assumes "R A B" shows "\<exists>K\<in>space (prob_algebra M). \<exists>A'\<in>M \<rightarrow>\<^sub>M prob_algebra (stream_space M). \<exists>B'\<in>M \<rightarrow>\<^sub>M prob_algebra (stream_space M). (AE y in K. R (A' y) (B' y) \<or> A' y = B' y) \<and> A = K \<bind> (\<lambda>y. A' y \<bind> (\<lambda>\<omega>. return (stream_space M) (y ## \<omega>))) \<and> B = K \<bind> (\<lambda>y. B' y \<bind> (\<lambda>\<omega>. return (stream_space M) (y ## \<omega>)))"
chg_map_def: fixes f :: "'d \<Rightarrow> 'd"   and a :: "'c"   and m :: "'c \<Rightarrow> 'd option" shows "chg_map f a m = (case m a of None \<Rightarrow> m | Some b \<Rightarrow> m(a \<mapsto> f b))"
ts_okE: fixes P :: "'f \<Rightarrow> 'g \<Rightarrow> 'h \<Rightarrow> bool"   and ts :: "'f \<Rightarrow> ('g \<times> 'i \<Rightarrow>f nat) option"   and m :: "'h" assumes "ts_ok P ts m" obtains "\<And>t x ln. ts t = \<lfloor>(x, ln)\<rfloor> \<Longrightarrow> P t x m"
swapVal_def: fixes zs :: "'varSort"   and z1 :: "'var"   and z2 :: "'var"   and val :: "'varSort \<Rightarrow> 'var \<Rightarrow> 'g" shows "val ^[z1 \<and> z2]_zs \<equiv> \<lambda>xs x. val xs (x @xs[z1 \<and> z2]_zs)"
plus_uint64_def: shows "(+) \<equiv> map_fun Rep_uint64 (map_fun Rep_uint64 Abs_uint64) (+)"
induct: fixes P :: "clam \<Rightarrow> bool"   and clam :: "clam" assumes "P \<box>"   and "\<And>clam lam. P clam \<Longrightarrow> P (CAppL clam lam)"   and "\<And>lam clam. P clam \<Longrightarrow> P (CAppR lam clam)"   and "\<And>name clam. P clam \<Longrightarrow> P (CLam [name].clam)" shows "P clam"
assms_2: shows "VectorSpaceHom (\<star>) W smult'' S"
K_wf: fixes s :: "'s" shows "K s \<noteq> {}"
anno_annotate: fixes p :: "nat"   and c :: "com"   and f :: "nat \<Rightarrow> 'b" assumes "p < asize c" shows "anno (annotate f c) p = f p"
f_Exec_Stream_Nil: fixes trans_fun :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and c :: "'d" shows "f_Exec_Comp_Stream trans_fun [] c = []"
incrIndexList_help81: fixes l :: "nat"   and m :: "nat"   and nmax :: "nat" shows "\<not> incrIndexList [l] m nmax"
failure_divergence_refine_def: fixes P :: "'b process"   and Q :: "'b process" shows "P \<sqsubseteq>\<^sub>F\<^sub>D Q \<equiv> P \<le> Q"
simps_4: fixes n :: "nat"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and y :: "'b"   and x :: "'b" assumes "n \<noteq> 1"   and "odd n" shows "efficient_funpow f y x n = efficient_funpow f (f x y) (f x x) (n div 2)"
plus_trilean_assoc: fixes a :: "trilean"   and b :: "trilean"   and c :: "trilean" shows "a \<or>? b \<or>? c = a \<or>? (b \<or>? c)"
le_of_nat_term_order: fixes to :: "'b nat_term_order"   and x :: "'b"   and y :: "'b" shows "le_of_nat_term_order to x y = (nat_term_compare to x y \<noteq> Gt)"
prim_subgraph_inv: fixes e :: "'a"   and v :: "'a"   and g :: "'a"   and t :: "'a" assumes "e \<le> v \<sqinter> - id v \<sqinter> g"   and "t \<le> v \<sqinter> id v \<sqinter> g" shows "t \<squnion> e \<le> (v \<squnion> id e \<sqinter> top) \<sqinter> id (v \<squnion> id e \<sqinter> top) \<sqinter> g"
pred_map: fixes Q1 :: "'d \<Rightarrow> bool"   and Q2 :: "'e \<Rightarrow> bool"   and f1 :: "'g \<Rightarrow> 'd"   and f2 :: "'h \<Rightarrow> 'e"   and x :: "('g, 'h, 'f) gpv" shows "pred_gpv Q1 Q2 (map_gpv f1 f2 x) = pred_gpv (Q1 \<circ> f1) (Q2 \<circ> f2) x"
route2match_def: fixes r :: "('c, 'd) routing_rule_scheme" shows "route2match r = \<lparr>iiface = ifaceAny, oiface = ifaceAny, src = (0, 0), dst = (pfxm_prefix (routing_match r), routing_prefix r), proto = ProtoAny, sports = (0, - 1), dports = (0, - 1)\<rparr>"
subtopology_UNIV: fixes U :: "'a topology" shows "subtopology U UNIV = U"
Dup_all_is_branch: fixes p :: "('c, 'd) fm"   and i :: "'d"   and branch :: "(('c, 'd) fm list \<times> 'd) list"   and xs :: "(nat \<times> nat) set" assumes "Dup p i branch xs" shows "all_is_branch p i branch xs"
cnf_form_semantics: fixes F :: "'b formula"   and \<A> :: "'b \<Rightarrow> bool" assumes "is_nnf F" shows "\<A> \<Turnstile> cnf_form_of F = \<A> \<Turnstile> F"
propositions_lemma_2: fixes v :: "i"   and \<phi> :: "\<o>" shows "[\<^bold>\<lambda>\<^sup>0 \<phi> \<^bold>\<equiv> \<phi> in v]"
handshake_obj_fields_markedD: fixes s :: "(('f, 'g, 'h) response, char list, 'e process_name, 'e process_name \<times> ('f, 'e, 'g, 'h) request_op, ('f, 'e, 'g, 'h) Model.local_state) system_state" assumes "atS gc obj_fields_marked_locs s"   and "handshake_invL s" shows "sys_ghost_hs_phase s\<down> = hp_IdleMarkSweep \<and> All (ghost_hs_in_sync (s\<down> sys))"
simps_5: fixes f1 :: "'c"   and f2 :: "'d \<Rightarrow> 'd llist \<Rightarrow> 'c"   and x21 :: "'d"   and x22 :: "'d llist" shows "(case LCons x21 x22 of LNil \<Rightarrow> f1 | LCons x xa \<Rightarrow> f2 x xa) = f2 x21 x22"
cil_cons2: fixes w :: "'a list"   and wa :: "'a list"   and \<alpha> :: "'a \<Rightarrow> 'b set \<times> 'b set"   and wb :: "'a list"   and e :: "'a" assumes "w \<in> wa \<otimes>\<^bsub>\<alpha>\<^esub> wb"   and "fst (\<alpha> e) \<inter> mon_pl (map \<alpha> wa) = {}" shows "e # w \<in> wa \<otimes>\<^bsub>\<alpha>\<^esub> e # wb"
2: shows "rawpsubstT (rawpsubstT r2 (zip (map Var us) (map snd txs))) (zip (map fst txs) us) = rawpsubstT (rawpsubstT r2 (zip (map Var vs2) (map snd txs))) (zip (map fst txs) vs2)"
ch: shows "\<forall>H\<in>set (C # Fs @ [F]). SimplicialComplex.maxsimp A H"
seq'_def: fixes f :: "'b \<Rightarrow> 'b"   and x :: "'b"   and i :: "nat" shows "fx0.seq' f x i \<equiv> (f ^^ i) x"
simps_2: fixes n :: "nat"   and i :: "nat" shows "tuple_decode (Suc n) i = pdec1 i # tuple_decode n (pdec2 i)"
substs_preserve_ground_clause: fixes C :: "'b literal set"   and \<sigma> :: "('b \<times> 'b trm) list" assumes "ground_clause C" shows "subst_cl C \<sigma> = C"
subspace_sum: fixes A :: "'b set"   and B :: "'c set"   and f :: "'c \<Rightarrow> 'b" assumes "subspace A"   and "\<And>x. x \<in> B \<Longrightarrow> f x \<in> A" shows "sum f B \<in> A"