hcju/isabelleps huggingface dataset:
mat_assoc_test_9: fixes A :: "complex mat"   and n :: "nat"   and B :: "complex mat"   and C :: "complex mat"   and D :: "complex mat" assumes "A \<in> carrier_mat n n"   and "B \<in> carrier_mat n n"   and "C \<in> carrier_mat n n"   and "D \<in> carrier_mat n n" shows "A * (B - C) * D = A * B * D - A * C * D"
last_seg_impl_def: shows "last_seg_impl \<equiv> ASSERT (length B - 1 < length B) \<bind> (\<lambda>_. seg_set_impl (seg_start (length B - 1)) (seg_end (length B - 1)))"
rel_set_def: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool" shows "rel_set R = (\<lambda>A B. (\<forall>x\<in>A. \<exists>y\<in>B. R x y) \<and> (\<forall>y\<in>B. \<exists>x\<in>A. R x y))"
Rep_hyp2: fixes x :: "hyp2" shows "Rep_hyp2 x \<in> hyp2"
times_upcast_lower: fixes x :: "int"   and y :: "int" assumes "- 2147483648 \<le> x"   and "- 2147483648 \<le> y"   and "x \<le> 2147483648"   and "y \<le> 2147483648" shows "- 4611686018427387904 \<le> x * y"
squanderLowerBound_def: fixes g :: "graph" shows "squanderLowerBound g \<equiv> faceSquanderLowerBound g + ExcessNotAt g None"
to_ct_list_append_1: fixes ts :: "t list"   and as :: "ct list"   and bs :: "ct list" assumes "to_ct_list ts = as @ bs" shows "\<exists>as'. to_ct_list as' = as"
union_single_eq_member: fixes x :: "'b"   and M :: "'b multiset"   and N :: "'b multiset" assumes "x, M = N" shows "x \<in># N"
unbounded_complement_components: fixes C :: "complex set" assumes "C \<in> components (- S)"   and "connected S"   and "if bounded S then connected (frontier S) else \<forall>C\<in>components (frontier S). \<not> bounded C" shows "\<not> bounded C"
fres_canc1: fixes y :: "'a"   and x :: "'a" shows "(y \<leftarrow> x) \<cdot> x \<le> y"
pnormalize_Nil: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" assumes "class.semiring_0 plus zero times" shows "semiring_0.pnormalize zero [] = []"
directed_def: fixes X :: "'a set" shows "directed X = (\<forall>Y. finite Y \<and> Y \<subseteq> X \<longrightarrow> (\<exists>x\<in>X. \<forall>y\<in>Y. y \<le> x))"
mult_less_cancel_left1: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sgn :: "'c \<Rightarrow> 'c"   and abs :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and c :: "'c"   and b :: "'c" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less c (times c b) = ((less_eq zero c \<longrightarrow> less one b) \<and> (less_eq c zero \<longrightarrow> less b one))"
progress_ge_gen: assumes "Formula.future_bounded \<phi>" shows "\<exists>P j. dom P = {} \<and> range_mapping i j P \<and> i \<le> Monitor.progress \<sigma> P \<phi> j"
psubset_subset_trans: fixes A :: "'b set"   and B :: "'b set"   and C :: "'b set" assumes "A \<subset> B"   and "B \<subseteq> C" shows "A \<subset> C"
mtf2_forward_effect1: fixes q :: "'a"   and xs :: "'a list"   and i :: "nat"   and n :: "nat" assumes "q \<in> set xs"   and "distinct xs"   and "index xs q < i \<and> i < length xs" shows "index (mtf2 n q xs) (xs ! i) = index xs (xs ! i) \<and> index xs q < index (mtf2 n q xs) (xs ! i) \<and> index (mtf2 n q xs) (xs ! i) < length xs"
double_compl: fixes x :: "'a" shows "\<sim> (\<sim> x) = x"
f2: shows "(r |\<^bsub>rel\<^esub> s) \<otimes>\<^bsub>rec_rng_of_frac\<^esub> (r'' |\<^bsub>rel\<^esub> s'') = (r \<otimes> r'' |\<^bsub>rel\<^esub> s \<otimes> s'')"
Flow_axioms: shows "Flow c s t (augment (augmentingFlow p))"
result_def: fixes dc :: "nat option \<times> bool list" shows "result dc = foldl (\<noteq>) False (inversion dc)"
acyclic_subcls1: fixes wf_md :: "'b prog \<Rightarrow> char list \<Rightarrow> 'b mdecl \<Rightarrow> bool"   and P :: "'b prog" assumes "wf_prog wf_md P" shows "acyclic (subcls1 P)"
distinct_15: fixes x2 :: "nat"   and x51 :: "nat \<Rightarrow> nat \<Rightarrow> nat"   and x52 :: "exp"   and x53 :: "exp" shows "ENat x2 \<noteq> EPrim x51 x52 x53"
assms_2: shows "B \<equiv> snd (Gauss_Jordan_column_k (ia, A) k)"
subst_pre_skip: fixes \<sigma> :: "'b \<Rightarrow> 'b" shows "\<lceil>\<sigma>\<rceil>\<^sub>s \<dagger> II = \<langle>\<sigma>\<rangle>\<^sub>a"
set_less_def: fixes A :: "'a set"   and B :: "'a set" shows "(A \<sqsubset> B) = (A \<sqsubseteq> B \<and> \<not> B \<sqsubseteq> A)"
IH: assumes "(i_start, i_end) \<in> set (identify_blocks_main A i_b ((i_b, i_e) # list))"   and "\<And>i_s i_ea. (i_s, i_ea) \<in> set ((i_b, i_e) # list) \<Longrightarrow> i_s \<le> i_ea \<and> i_ea < k"   and "i_b \<le> k" shows "i_start \<le> i_end \<and> i_end < k"
closure_of_unique: fixes S :: "'c set"   and T :: "'c set"   and X :: "'c topology" assumes "S \<subseteq> T"   and "closedin X T"   and "\<And>T'. \<lbrakk>S \<subseteq> T'; closedin X T'\<rbrakk> \<Longrightarrow> T \<subseteq> T'" shows "X closure_of S = T"
cos_sqrt: shows "sqrt ((cos y)\<^sup>2) = cos y"
reachable_def: fixes m :: "'e"   and y :: "'h" shows "mut_m.reachable m y = (\<lambda>s. \<exists>x. x \<in> roots (s (mutator m)) \<union> ghost_honorary_root (s (mutator m)) \<union> mut_m.tso_store_refs m s \<and> (x reaches y) s)"
append_rows_mult_right: fixes A :: "'a mat"   and a :: "nat"   and n :: "nat"   and B :: "'a mat"   and b :: "nat"   and Q :: "'a mat" assumes "A \<in> carrier_mat a n"   and "B \<in> carrier_mat b n"   and "Q \<in> carrier_mat n n" shows "(A @\<^sub>r B) * Q = A * Q @\<^sub>r B * Q"
simps: fixes rf :: "recf" shows "rec_Minr rf = (let vl = arity rf in Let (rec_all (recf.id vl (vl - 1)) (Cn (Suc vl) rec_not [Cn (Suc vl) rf (get_fstn_args (Suc vl) (vl - 1) @ [recf.id (Suc vl) vl])])) rec_sigma)"
oInv_bound1: fixes a :: "ordinal" assumes "F 0 \<le> a" shows "F (oInv F a) \<le> a"
check_minimal_unsat_state_core: fixes s :: "('i, 'a) state" assumes "\<not> \<U> s"   and "\<Turnstile>\<^sub>n\<^sub>o\<^sub>l\<^sub>h\<^sub>s s"   and "\<diamond> s"   and "\<triangle> (\<T> s)"   and "\<nabla> s" shows "\<U> (check s) \<longrightarrow> minimal_unsat_state_core (check s)"
distinct_5: fixes x1 :: "'e" shows "prot_atom.Atom x1 \<noteq> AttackType"
ipurge_tr_aux_foldr_ref_aux: fixes U :: "'e set"   and V :: "'e set"   and I :: "('e \<times> 'e) set"   and D :: "'f \<Rightarrow> 'e"   and P :: "'f \<Rightarrow> bool"   and xs :: "'f list"   and X :: "'f set" assumes "U \<subseteq> V" shows "ipurge_ref_aux I D U (ipurge_tr_aux_foldr I D P V xs) X = ipurge_ref_aux I D U [] X"
antisym: fixes A :: "'a set"   and h :: "'a \<Rightarrow> 'b"   and bo :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and m :: "'a \<Rightarrow> 'a \<Rightarrow> 'a option"   and x :: "'a"   and y :: "'a" assumes "merge_on A h bo m"   and "bo x y"   and "bo y x" shows "x = y"
subset_eq: fixes A :: "'y set"   and B :: "'y set" shows "(A \<subseteq> B) = (\<forall>x\<in>A. x \<in> B)"
two_reduced_heads_imp_reduced_alt: fixes s :: "'w"   and t :: "'w"   and w :: "'w"   and as :: "'w list"   and bs :: "'w list" assumes "s \<noteq> t"   and "S_reduced_for w (s # as)"   and "S_reduced_for w (t # bs)" shows "\<exists>cs. S_reduced_for w (pair_relator_halflist s t @ cs)"
alpha_Tree_rel_relcompI: fixes x :: "('b, 'c, 'd, 'e) Tree"   and x' :: "('b, 'c, 'd, 'e) Tree"   and y :: "'f"   and R :: "(('b, 'c, 'd, 'e) Tree \<times> 'f) set" assumes "x =\<^sub>\<alpha> x'"   and "(x', y) \<in> R" shows "(x, y) \<in> FL_Validity.alpha_Tree_rel O R"
fv_strand_subst2: fixes S :: "('d, 'c) strand_step list"   and \<delta> :: "'c \<Rightarrow> ('d, 'c) Term.term" assumes "bvars\<^sub>s\<^sub>t S \<inter> (subst_domain \<delta> \<union> range_vars \<delta>) = {}" shows "fv\<^sub>s\<^sub>e\<^sub>t (\<delta> ` wfrestrictedvars\<^sub>s\<^sub>t S) = wfrestrictedvars\<^sub>s\<^sub>t (S \<cdot>\<^sub>s\<^sub>t \<delta>)"
d_in: shows "D \<in> P_of_state (lnth Sts i)"
cont_apply: fixes lubb :: "'e set \<Rightarrow> 'e"   and ordb :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'e \<Rightarrow> 'a"   and luba :: "'d set \<Rightarrow> 'd"   and orda :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and t :: "'d \<Rightarrow> 'e" assumes "\<And>x. cont lubb ordb Sup (\<le>) (f x)"   and "cont luba orda lubb ordb t"   and "\<And>y. cont luba orda Sup (\<le>) (\<lambda>x. f x y)"   and "monotone orda ordb t"   and "\<And>x. monotone ordb (\<le>) (f x)"   and "\<And>y. monotone orda (\<le>) (\<lambda>x. f x y)" shows "cont luba orda Sup (\<le>) (\<lambda>x. f x (t x))"
dim_row_of_iter_tensor: fixes n :: "nat"   and A :: "complex Matrix.mat" assumes "1 \<le> n" shows "dim_row (A \<otimes>\<^bsup>n\<^esup>) = dim_row A ^ n"
nat_term_compare_inverse: fixes x :: "'a nat_term_order" shows "Abs_nat_term_order (nat_term_compare x) = x"
NDVD_def: fixes i :: "int"   and c :: "int"   and t :: "MIR.num" shows "NDVD i c t = (if i = 0 then neq c t else if c = 0 then NDvd i t else if 0 < c then NDVDJ \<bar>i\<bar> c t else NDVDJ \<bar>i\<bar> (- c) (Neg t))"
reachable_trans: fixes u :: "'a"   and v :: "'a"   and w :: "'a" assumes "u \<rightarrow>\<^sup>*\<^bsub>G \<restriction> {v. u \<rightarrow>\<^sup>* v}\<^esub> v"   and "v \<rightarrow>\<^sup>*\<^bsub>G \<restriction> {v. u \<rightarrow>\<^sup>* v}\<^esub> w" shows "u \<rightarrow>\<^sup>*\<^bsub>G \<restriction> {v. u \<rightarrow>\<^sup>* v}\<^esub> w"
RECT_refine: fixes body :: "('c \<Rightarrow> 'd nres) \<Rightarrow> 'c \<Rightarrow> 'd nres"   and x :: "'c"   and x' :: "'e"   and R :: "('c \<times> 'e) set"   and S :: "('d \<times> 'f) set"   and body' :: "('e \<Rightarrow> 'f nres) \<Rightarrow> 'e \<Rightarrow> 'f nres" assumes "trimono body"   and "(x, x') \<in> R"   and "\<And>f f' x x'. \<lbrakk>\<And>x x'. (x, x') \<in> R \<Longrightarrow> f x \<le> \<Down> S (f' x'); (x, x') \<in> R\<rbrakk> \<Longrightarrow> body f x \<le> \<Down> S (body' f' x')" shows "REC\<^sub>T body x \<le> \<Down> S (REC\<^sub>T body' x')"
finite_iff_ordLess_natLeq: fixes A :: "'f set" shows "finite A = (|A| <o natLeq)"
ttl_tmap: fixes f :: "'c \<Rightarrow> 'a"   and g :: "'d \<Rightarrow> 'b"   and xs :: "('c, 'd) tllist" shows "ttl (tmap f g xs) = tmap f g (ttl xs)"
inf_left_dist_sup: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "- x \<sqinter> (- y \<squnion> - z) = - x \<sqinter> - y \<squnion> - x \<sqinter> - z"
adm_edges_def: fixes f :: "nat \<times> nat \<Rightarrow> 'capacity"   and l :: "nat \<Rightarrow> nat" shows "adm_edges f l \<equiv> {(u, v). (u, v) \<in> cfE_of f \<and> l u = l v + 1}"
linear: fixes x :: "'x \<Rightarrow>\<^sub>0 nat"   and y :: "'x \<Rightarrow>\<^sub>0 nat" shows "x \<preceq> y \<or> y \<preceq> x"
trm_prm_fvs: fixes \<pi> :: "'b prm"   and M :: "'b trm" shows "fvs (\<pi> \<cdot> M) = \<pi> {$} fvs M"
defined_args_valid: fixes \<tau> :: "'d state \<times> 'd state"   and x :: "'d state \<times> 'd state \<Rightarrow> Set('e)"   and y :: "'d state \<times> 'd state \<Rightarrow> 'e" shows "(\<tau> \<Turnstile> \<delta> x->excluding\<^sub>S\<^sub>e\<^sub>t(y)) = (\<tau> \<Turnstile> \<delta> x \<and> \<tau> \<Turnstile> \<upsilon> y)"
i_set_multp_i_set_mult_eq: fixes k :: "nat" shows "i_set_multp k = (\<lambda>x. x \<in> i_set_mult k)"
not_roofedE: fixes x :: "'a"   and G :: "('a, 'b) graph_scheme"   and B :: "'a set"   and S :: "'a set" assumes "x \<notin> roofed_gen G B S" obtains p :: "'a list" and y :: "'a" where "path G x p y" and "y \<in> B" and "\<And>z. z \<in> set (x # p) \<Longrightarrow> z \<notin> S"
lim_F: fixes x :: "real" assumes "isCont F x" shows "(\<lambda>n. f (r n) x) \<longlonglongrightarrow> F x"
dim_singleton: fixes x :: "'b" shows "dim {x} = (if x = (0::'b) then 0 else 1)"
simps: fixes G :: "('c, 'd) Graph_scheme"   and a :: "'c llist" assumes "Digraph G" shows "Digraph.maximal_path G a = (a = LNil \<or> (\<exists>v. a = LCons v LNil \<and> \<not> (\<exists>w\<in>V\<^bsub>G\<^esub>. v \<rightarrow>\<^bsub>G\<^esub> w)) \<or> (\<exists>Ps v. a = LCons v Ps \<and> \<not> lnull Ps \<and> Digraph.maximal_path G Ps))"
rank_greater_zero: fixes A :: "(('b, 'c) vec, 'd) vec" assumes "A \<noteq> 0" shows "0 < rank A"
arr_char: fixes f :: "'a \<times> 'a" shows "B.VV.arr f = (B.arr (fst f) \<and> B.arr (snd f) \<and> src (fst f) = trg (snd f))"
erH_inv: fixes P :: "'a set"   and R' :: "('a set, 'e) Ring_scheme"   and Y :: "'a set"   and g' :: "'a set" assumes "Idomain S"   and "ideal S P"   and "Ring R'"   and "PolynRg R' (S /\<^sub>r P) Y"   and "g' \<in> carrier R'" shows "\<exists>g\<in>carrier R. deg R S X g \<le> deg R' (S /\<^sub>r P) Y g' \<and> erH R S X R' (S /\<^sub>r P) Y (pj S P) g = g'"
isometryD_3: fixes f :: "'b \<Rightarrow> 'c" assumes "isometry f" shows "surj f"
assms_3: shows "(i, Expunge e mo i) \<in> set (node_deliver_messages xs)"
simps_1: fixes v :: "'a"   and t :: "'a ifex"   and e :: "'a ifex" shows "ifex_minimal (IF v t e) = (t \<noteq> e \<and> ifex_minimal t \<and> ifex_minimal e)"
att_or_snd_att: fixes x2 :: "'a attree list"   and x :: "'a set \<times> 'a set"   and a :: "'a attree" assumes "\<turnstile>x2 \<oplus>\<^sub>\<or>\<^bsup>x\<^esup>"   and "a \<in> set x2" shows "snd (attack a) \<subseteq> snd x"
h2: shows "minimal A (\<lambda>S. S \<subseteq> carrier V \<and> gen_set S)"
DERIV_arctan_suminf: fixes r :: "real"   and x :: "real" assumes "0 < r"   and "r < 1"   and "\<bar>x\<bar> < r" shows "((\<lambda>x. \<Sum>a. (- 1) ^ a * (1 / real (a * 2 + 1) * x ^ (a * 2 + 1))) has_real_derivative (\<Sum>n. (- 1) ^ n * x ^ (n * 2))) (at x)"
that: fixes A :: "fm" assumes "x = \<lbrakk>\<guillemotleft>A\<guillemotright>\<rbrakk>e" shows "thesis"
S6Retry: fixes p :: "PrIds"   and rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState" shows "\<turnstile> MClkRetry memCh crCh cst p \<and> HNext rmhist p \<and> $S6 rmhist p \<and> unchanged (e p, r p, m p) \<longrightarrow> S3 rmhist p$ \<and> unchanged (rmhist!p)"
preserves_arr: fixes f :: "'a" assumes "C.arr f" shows "C.arr (C.L f)"
init: shows "on_init param \<le>\<^sub>n SPEC (\<lambda>x. I (empty_state x))"
rename_gens_canceled: fixes g :: "'c \<Rightarrow> 'd"   and l :: "(bool \<times> 'c) list"   and f :: "bool \<Rightarrow> bool" assumes "inj_on g (snd ` set l)"   and "canceled l" shows "canceled (map (map_prod f g) l)"
Posix_elims_1: fixes s :: "'b list"   and v :: "'b val"   and P :: "bool" assumes "s \<in> Zero \<rightarrow> v" shows "P"
simps: fixes n :: "nat" shows "T'_mom_select n = (if n \<le> 20 then 463 else T'_mom_select (nat \<lceil>2 / 10 * real n\<rceil>) + T'_mom_select (nat \<lceil>7 / 10 * real n + 3\<rceil>) + 17 * n + 50)"
one_less_inverse_iff: fixes x :: "'a" shows "((1::'a) < inverse x) = ((0::'a) < x \<and> x < (1::'a))"
functor_category_def: fixes A :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" shows "functor_category A B \<equiv> category A \<and> category B"
denote_density_states: fixes \<rho> :: "complex mat"   and dims :: "nat list"   and S :: "com" assumes "\<rho> \<in> state_sig.density_states dims"   and "state_sig.well_com dims S" shows "state_sig.denote dims S \<rho> \<in> state_sig.density_states dims"
bind_def: fixes x :: "('d, 'f) state"   and f :: "'f \<Rightarrow> ('d, 'e) state" shows "local.bind x f = State (\<lambda>s. case run_state x s of (a, s') \<Rightarrow> run_state (f a) s')"
shadow_root_ptr_casts_commute3: fixes document_ptr :: "(_) document_ptr" assumes "is_shadow_root_ptr_kind document_ptr" shows "cast (the (cast document_ptr)) = document_ptr"
to_tree_order_ptrs_in_heap: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and to :: "(_) object_ptr list"   and ptr' :: "(_) object_ptr" assumes "Shadow_DOM.heap_is_wellformed h"   and "ShadowRootClass.type_wf h"   and "ShadowRootClass.known_ptrs h"   and "h \<turnstile> Shadow_DOM.to_tree_order ptr \<rightarrow>\<^sub>r to"   and "ptr' \<in> set to" shows "ptr' |\<in>| object_ptr_kinds h"
simps: fixes execute :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and enabled :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and a1 :: "'a stream"   and a2 :: "'b" shows "transition_system.run execute enabled a1 a2 = (\<exists>a p r. a1 = a ## r \<and> a2 = p \<and> enabled a p \<and> transition_system.run execute enabled r (execute a p))"
X_subseteq_XD_smallest: fixes ds :: "'b set"   and X :: "'a set" assumes "TruePref_tax.stable_on ds X" shows "X \<subseteq> TruePref_tax.XD_smallest ds X"
hom_sum: fixes f :: "'c \<Rightarrow> 'a poly"   and X :: "'c set" shows "map_poly hom (sum f X) = (\<Sum>x\<in>X. map_poly hom (f x))"
is_lb_def: fixes \<delta> :: "'a \<Rightarrow> 'b \<times> 'c \<Rightarrow> 'a"   and s :: "'a"   and s1 :: "'a"   and s2 :: "'a" shows "pre_RDR.is_lb \<delta> s s1 s2 \<equiv> pre_RDR.less_eq \<delta> s s2 \<and> pre_RDR.less_eq \<delta> s s1"
simps_2: fixes p :: "('psym, 'tp) epsym"   and Tl :: "('fsym, 'tp) efsym trm list" shows "GE.wtA (Pr p Tl) = (GE_wtPsym p \<and> list_all GE.wt Tl \<and> GE_parOf p = map GE.tpOf Tl)"
infnorm_neg: fixes x :: "'a" shows "infnorm (- x) = infnorm x"
in_set_impl_in_set_zip1: fixes xs :: "'a list"   and ys :: "'b list"   and x :: "'a" assumes "length xs = length ys"   and "x \<in> set xs" obtains y :: "'b" where "(x, y) \<in> set (zip xs ys)"
map_dual_nat_iso: fixes f :: "'b \<Rightarrow> 'c" shows "\<partial>\<^sub>F f \<circ> \<partial> = \<partial> \<circ> id f"
gv_2: shows "AE x in M. (\<lambda>xa. ereal (v xa x / real xa)) \<longlonglongrightarrow> gv x"
moebius_mu_not_squarefree: fixes n :: "nat" assumes "\<not> squarefree n" shows "moebius_mu n = (0::'a)"
forest_maximally_connected_incl_max1: fixes H :: "('v, 'w) graph" assumes "forest H"   and "subgraph H G"   and "\<forall>(a, w, b)\<in>E - edges H. \<not> forest (add_edge a w b H)" shows "maximally_connected H G"
\<tau>exec_1_d_def: fixes addr2thread_id :: "'a \<Rightarrow> 'b"   and thread_id2addr :: "'b \<Rightarrow> 'a"   and spurious_wakeups :: "bool"   and empty_heap :: "'c"   and allocate :: "'c \<Rightarrow> htype \<Rightarrow> ('c \<times> 'a) set"   and typeof_addr :: "'c \<Rightarrow> 'a \<Rightarrow> htype option"   and heap_read :: "'c \<Rightarrow> 'a \<Rightarrow> addr_loc \<Rightarrow> 'a val \<Rightarrow> bool"   and heap_write :: "'c \<Rightarrow> 'a \<Rightarrow> addr_loc \<Rightarrow> 'a val \<Rightarrow> 'c \<Rightarrow> bool"   and P :: "(nat \<times> nat \<times> 'a instr list \<times> (nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list) prog"   and t :: "'b"   and \<sigma> :: "'a option \<times> 'c \<times> 'a frame list"   and \<sigma>' :: "'a option \<times> 'c \<times> 'a frame list" shows "JVM_heap_base.\<tau>exec_1_d addr2thread_id thread_id2addr spurious_wakeups empty_heap allocate typeof_addr heap_read heap_write P t \<sigma> \<sigma>' = (JVM_heap_base.exec_1 addr2thread_id thread_id2addr spurious_wakeups empty_heap allocate typeof_addr heap_read heap_write P t \<sigma> \<lbrace>\<rbrace> \<sigma>' \<and> JVM_heap_base.\<tau>Move2 typeof_addr P \<sigma> \<and> JVM_heap_base.check typeof_addr P \<sigma>)"
bt_dfs_simp: fixes f :: "('d, 'e) bintree \<Rightarrow> 'c"   and t :: "('d, 'e) bintree" shows "bt_dfs f t = f t # bts_dfs f (children t)"
holds_of_OR: fixes \<Phi> :: "'b pltl set"   and \<sigma> :: "nat \<Rightarrow> 'b" assumes "finite \<Phi>" shows "\<sigma> \<Turnstile>\<^sub>p OR \<Phi> = (\<exists>\<phi>\<in>\<Phi>. \<sigma> \<Turnstile>\<^sub>p \<phi>)"
YFtorNTNatTrans: fixes C :: "('d, 'e, 'f) LSCategory_scheme"   and f :: "'e" assumes "LSCategory C"   and "f \<in> mor\<^bsub>C\<^esub>" shows "NatTrans (YFtorNT C f)"
Suc: shows "(\<Sum>k = 1..<2 ^ n. f (2 * 2 ^ Discrete.log k)) = (\<Sum>k<n. 2 ^ k * f (2 ^ Suc k))"
sign_r_pos_rec: fixes p :: "real poly"   and x :: "real" assumes "p \<noteq> 0" shows "sign_r_pos p x = (if poly p x = 0 then sign_r_pos (pderiv p) x else 0 < poly p x)"
leg0_composite: shows "trg_f.leg0 = f.leg0 \<cdot> trg_f.prj\<^sub>0"
invar_l_le_k: fixes k :: "'a"   and v :: "real"   and l :: "'a kdt"   and r :: "'a kdt" assumes "invar (Node k v l r)" shows "\<forall>p\<in>set_kdt l. p $ k \<le> v"
smallomegaI_filterlim_at_top_norm: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c"   and F :: "'b filter" assumes "LIM x F. norm (f x / g x) :> at_top" shows "f \<in> \<omega>[F](g)"
disc_4: fixes x21 :: "variable"   and x22 :: "String.literal set" shows "is_Tv (Tv x21 x22)"
Sec_wf: shows "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s Sec"
rel_gpv''_colossless_gpvD1: fixes A :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and C :: "'h \<Rightarrow> 'i \<Rightarrow> bool"   and R :: "'j \<Rightarrow> 'k \<Rightarrow> bool"   and gpv :: "('f, 'h, 'j) gpv"   and gpv' :: "('g, 'i, 'k) gpv"   and \<I> :: "('h, 'j) \<I>"   and \<I>' :: "('i, 'k) \<I>" assumes "rel_gpv'' A C R gpv gpv'"   and "colossless_gpv \<I> gpv"   and "rel_\<I> C R \<I> \<I>'" shows "colossless_gpv \<I>' gpv'"
ZObisT_trace2: fixes c :: "('test, 'atom) com"   and d :: "('test, 'atom) com"   and s :: "'state"   and t :: "'state"   and tr :: "(('test, 'atom) com \<times> 'state) list \<times> 'state" assumes "c \<approx>01T d"   and "s \<approx> t"   and "finTrace tr"   and "iconfig tr = (c, s)" shows "\<exists>tr'. finTrace tr' \<and> iconfig tr' = (d, t) \<and> fstate tr \<approx> fstate tr' \<and> lengthFT tr' \<le> lengthFT tr"
ResCong: fixes P :: "pi"   and Q :: "pi"   and x :: "name" assumes "(P, Q) \<in> bisim O bangRel weakBisim O bisim" shows "(<\<nu>x>P, <\<nu>x>Q) \<in> bisim O bangRel weakBisim O bisim"
con2: fixes env :: "v sem_env"   and cn :: "(char list, char list) id0 option"   and es :: "exp0 list"   and ck :: "bool"   and s :: "'b state" assumes "\<not> do_con_check (c env) cn (length es)" shows "evaluate ck env s (Con cn es) (s, Rerr (Rabort Rtype_error))"
tr_par_ik_eq: fixes B :: "('lbl strand_label \<times> ('fun, 'var) strand_step) list"   and A :: "('lbl strand_label \<times> ('fun, 'var) stateful_strand_step) list"   and D :: "('lbl strand_label \<times> ('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) list" assumes "B \<in> set (tr\<^sub>p\<^sub>c A D)" shows "ik\<^sub>s\<^sub>t (unlabel B) = ik\<^sub>l\<^sub>s\<^sub>s\<^sub>t A"
intro: 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"   and set_disconnected_nodes :: "(_) document_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and heap_is_wellformed :: "(_) heap \<Rightarrow> bool"   and parent_child_rel :: "(_) heap \<Rightarrow> ((_) object_ptr \<times> (_) object_ptr) set"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog" assumes "l_set_disconnected_nodes_get_disconnected_nodes type_wf get_disconnected_nodes get_disconnected_nodes_locs set_disconnected_nodes set_disconnected_nodes_locs"   and "l_heap_is_wellformed type_wf known_ptr heap_is_wellformed parent_child_rel get_child_nodes get_disconnected_nodes" shows "l_set_disconnected_nodes_get_disconnected_nodes_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr type_wf get_disconnected_nodes get_disconnected_nodes_locs set_disconnected_nodes set_disconnected_nodes_locs heap_is_wellformed parent_child_rel get_child_nodes"
simps_4: fixes \<A> :: "'a \<Rightarrow> bool"   and F :: "'a formula"   and G :: "'a formula" shows "\<A> \<Turnstile> F \<^bold>\<and> G = (\<A> \<Turnstile> F \<and> \<A> \<Turnstile> G)"
upper_floor_float_interval: fixes x :: "float interval" shows "upper (floor_float_interval x) = floor_fl (upper x)"
simps: fixes r :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and s :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and a1 :: "'d"   and a2 :: "'f" shows "(r OO s) a1 a2 = (\<exists>a b c. a1 = a \<and> a2 = c \<and> r a b \<and> s b c)"
CD_on_def: fixes ds :: "'b set"   and X :: "'a set" shows "CD_on ds X = (\<Union>d\<in>ds. Cd d X)"
sigma_algebra_sigma_sets: fixes a :: "'b set set"   and \<Omega> :: "'b set" assumes "a \<subseteq> Pow \<Omega>" shows "sigma_algebra \<Omega> (sigma_sets \<Omega> a)"