hcju/isabelleps huggingface dataset:
lemma3_5_2: fixes r :: "('b \<times> 'b) set"   and \<tau> :: "'b list"   and \<sigma> :: "'b list"   and \<sigma>' :: "'b list"   and \<tau>' :: "'b list"   and \<upsilon> :: "'b list"   and \<sigma>'' :: "'b list"   and \<upsilon>' :: "'b list" assumes "trans r"   and "irrefl r"   and "D r \<tau> \<sigma> \<sigma>' \<tau>'"   and "D r \<upsilon> \<sigma>' \<sigma>'' \<upsilon>'" shows "(r|\<sigma> @ \<tau>' @ \<upsilon>'|, r|\<sigma>| + r|\<tau> @ \<upsilon>|) \<in> mul_eq r"
thmD9: fixes k :: "nat"   and u :: "nat" assumes "items_le k (\<J> k u) = Gen (paths_le k (\<P> k u))"   and "\<T> k u = \<Z> k u"   and "k \<le> length Doc" shows "items_le k (\<J> k (Suc u)) \<subseteq> Gen (paths_le k (\<P> k (Suc u)))"
suffix_zero: fixes s :: "nat \<Rightarrow> 'a" shows "s |\<^sub>s 0 = s"
mapping_conv_bijective: fixes x :: "'a" shows "mapping x = bijective (x\<^sup>T)"
neq: fixes x :: "complex" assumes "Im x = 0" shows "\<i> + x \<noteq> 0"
mset_heap_ins_tree: fixes t :: "'b tree"   and ts :: "'b tree list" shows "mset_heap (ins_tree t ts) = mset_tree t + mset_heap ts"
db\<^sub>s\<^sub>s\<^sub>t_dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t: fixes T :: "('g strand_label \<times> ('e, 'f) stateful_strand_step) list"   and \<delta> :: "'f \<Rightarrow> ('e, 'f) Term.term"   and \<I> :: "'f \<Rightarrow> ('e, 'f) Term.term"   and D :: "(('e, 'f) Term.term \<times> ('e, 'f) Term.term) list" shows "db'\<^sub>l\<^sub>s\<^sub>s\<^sub>t (dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t (T \<cdot>\<^sub>l\<^sub>s\<^sub>s\<^sub>t \<delta>)) \<I> D = db'\<^sub>l\<^sub>s\<^sub>s\<^sub>t (T \<cdot>\<^sub>l\<^sub>s\<^sub>s\<^sub>t \<delta>) \<I> D"
simps_1: fixes r :: "'d set" shows "pre (AnnExpr r) = r"
size_abort_correct: fixes s :: "'s"   and m :: "nat" assumes "invar s" shows "g_size_abort m s = min m (card (dom (\<alpha> s)))"
size_inter: shows "int (size {#mset_set (bl \<inter> blv). bl \<in># filter_mset ((\<in>) p) (remove1_mset blv \<B>)#}) = \<Lambda> * (\<v> - 1) div (\<k> - 1) - 1"
witness_3: fixes b :: "'c" assumes "b \<in> T" shows "j (i b) = b"
h_4: fixes p :: "nat"   and X :: "'a topology"   and c :: "((nat \<Rightarrow> real) \<Rightarrow> 'a) \<Rightarrow>\<^sub>0 int" assumes "singular_chain p X c" shows "chain_boundary (Suc p) (h p c) + h (p - Suc 0) (chain_boundary p c) = singular_subdivision p c - c"
assms: shows "\<bar>Re (z0 - z)\<bar> < r"
span_linear_image: fixes f :: "'c \<Rightarrow> 'd"   and S :: "'c set" assumes "linear f" shows "span (f ` S) = f ` span S"
assms_5: shows "w * e\<^sup>T * top = bot"
induct: fixes P :: "pol \<Rightarrow> (mon \<times> pol) list \<Rightarrow> nat \<Rightarrow> bool"   and a0 :: "pol"   and a1 :: "(mon \<times> pol) list"   and a2 :: "nat" assumes "\<And>P1 n. P P1 [] n"   and "\<And>P1 M P2 mps n. (pnsubst P1 M P2 n = None \<Longrightarrow> P P1 mps n) \<Longrightarrow> P P1 ((M, P2) # mps) n" shows "P a0 a1 a2"
sim_actions_sym: fixes E :: "('a \<times> ('b, 'a) obs_event action) llist"   and E' :: "('a \<times> ('b, 'a) obs_event action) llist" assumes "E [\<approx>] E'" shows "E' [\<approx>] E"
finite_dimensional_real_vector_space_on_def: fixes S :: "'c set"   and basis :: "'c set" shows "finite_dimensional_real_vector_space_on S basis \<equiv> real_vector_space_on S \<and> finite_dimensional_real_vector_space_on_axioms S basis"
eq_divide_eq_numeral1_1: fixes a :: "'b"   and b :: "'b"   and w :: "num" shows "(a = b / numeral w) = (if numeral w \<noteq> (0::'b) then a * numeral w = b else a = (0::'b))"
IDFT_upper: fixes m :: "nat"   and i :: "nat"   and a :: "nat \<Rightarrow> complex" assumes "0 < m"   and "m \<le> i" shows "IDFT (2 * m) a i = IDFT m (\<lambda>i. a (2 * i)) (i - m) - (1 / FFT.root (2 * m)) ^ (i - m) * IDFT m (\<lambda>i. a (2 * i + 1)) (i - m)"
take_bit_signed_take_bit: 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 push_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and drop_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and take_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and "and" :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and or :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and xor :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and mask :: "nat \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and not :: "'b \<Rightarrow> 'b"   and m :: "nat"   and n :: "nat"   and a :: "'b" assumes "class.ring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask uminus not"   and "m \<le> Suc n" shows "take_bit m (ring_bit_operations.signed_take_bit zero times one bit take_bit or mask not n a) = take_bit m a"
sort_leq_refl: fixes cs :: "(String.literal \<times> String.literal) set"   and s :: "String.literal set" assumes "wf_subclass_loc cs"   and "sort_ex cs s" shows "sort_leq cs s s"
simps: fixes v :: "'a mod_ring poly"   and w :: "'a mod_ring poly"   and d :: "nat"   and res :: "(nat \<times> 'a mod_ring poly) list" shows "dist_degree_factorize_main v w d res = (if v = 1 then res else if degree v < d + d then (degree v, v) # res else let w = w ^ CARD('a) mod v; d = Suc d; gd = gcd (w - monom 1 1) v in if gd = 1 then dist_degree_factorize_main v w d res else let v' = v div gd in dist_degree_factorize_main v' (w mod v') d ((d, gd) # res))"
x1: fixes xa :: "'a"   and xb :: "'a"   and xc :: "'a" assumes "t \<notin> set ts"   and "disjoint (insert t (set ts))"   and "xa \<in> t"   and "xb \<in> s"   and "xb \<in> t"   and "xb \<notin> {}"   and "xc \<in> s"   and "xc \<notin> {}"   and "t \<inter> s \<in> set (partList4 (s - t) ts)" shows "\<not> t \<inter> s \<subseteq> \<Union> (set (partList4 (s - t) ts))"
simple_fw_join: fixes rs1 :: "'c simple_rule list"   and p :: "('c, 'd) simple_packet_scheme"   and rs2 :: "'c simple_rule list" shows "(simple_fw rs1 p = Decision FinalAllow \<and> simple_fw rs2 p = Decision FinalAllow) = (simple_fw (map (\<lambda>(u, a, b). SimpleRule u (if a = Accept \<and> b = Accept then Accept else Drop)) (generalized_fw_join (map simple_rule_dtor rs1) (map simple_rule_dtor rs2))) p = Decision FinalAllow)"
Transset_pred: fixes x :: "V" assumes "Transset x" shows "\<Squnion> (elts (ZFC_in_HOL.succ x)) = x"
valid_without_offending_flows: fixes F :: "('c \<times> 'c) set"   and sinvar :: "'c graph \<Rightarrow> ('c \<Rightarrow> 'd) \<Rightarrow> bool"   and G :: "'c graph"   and nP :: "'c \<Rightarrow> 'd" assumes "F \<in> SecurityInvariant_withOffendingFlows.set_offending_flows sinvar G nP" shows "sinvar (delete_edges G F) nP"
eq_onp_same_args: fixes P :: "'c \<Rightarrow> bool"   and x :: "'c" shows "eq_onp P x x = P x"
cycle_free_diag_dest: fixes M :: "nat \<Rightarrow> nat \<Rightarrow> 'c"   and n :: "nat" assumes "cycle_free M n" shows "\<forall>i xs. i \<le> n \<and> set xs \<subseteq> {0..n} \<longrightarrow> \<one> \<le> len M i i xs"
budan_fourier_interval: fixes a :: "real"   and b :: "real"   and p :: "real poly" assumes "a < b"   and "p \<noteq> 0" shows "int (proots_count p {x. a < x \<and> x \<le> b}) \<le> changes_itv_der a b p \<and> even (changes_itv_der a b p - int (proots_count p {x. a < x \<and> x \<le> b}))"
foldr_def: fixes f :: "'e \<times> 'f \<Rightarrow> 'd \<Rightarrow> 'd"   and t :: "('e, 'f) FingerTree"   and \<sigma> :: "'d" shows "local.foldr f t \<sigma> \<equiv> FingerTreeStruc.foldr f (Rep_FingerTree t) \<sigma>"
frag_cmul_cmul: fixes c :: "int"   and d :: "int"   and x :: "'c \<Rightarrow>\<^sub>0 int" shows "frag_cmul c (frag_cmul d x) = frag_cmul (c * d) x"
seqI: fixes g1 :: "('a2 \<Rightarrow> 'b, 'a2 \<Rightarrow> 'b) concrete_category.arr"   and f1 :: "('a2 \<Rightarrow> 'b, 'a2 \<Rightarrow> 'b) concrete_category.arr"   and g2 :: "'a2"   and f2 :: "'a2" assumes "A2_B.seq g1 f1 \<and> A2.seq g2 f2" shows "A2_BxA2.seq (g1, g2) (f1, f2)"
double_gauss_sum: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and n :: "nat" assumes "class.comm_semiring_1 times one plus zero" shows "times (numeral.numeral one plus (num.Bit0 num.One)) (comm_monoid_add.sum plus zero (semiring_1.of_nat one plus zero) {0..n}) = times (semiring_1.of_nat one plus zero n) (plus (semiring_1.of_nat one plus zero n) one)"
invar_update_with_trie: fixes t :: "('c, 'd) trie"   and ks :: "'c list"   and f :: "'d option \<Rightarrow> 'd" assumes "invar_trie t" shows "invar_trie (update_with_trie ks f t)"
l_adopt_node_wf_axioms_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and heap_is_wellformed :: "(_) heap \<Rightarrow> bool"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and known_ptrs :: "(_) heap \<Rightarrow> bool"   and adopt_node :: "(_) document_ptr \<Rightarrow> (_) node_ptr \<Rightarrow> ((_) heap, exception, unit) prog" shows "l_adopt_node_wf_axioms type_wf heap_is_wellformed get_child_nodes get_disconnected_nodes known_ptrs adopt_node \<equiv> ((\<forall>h document_ptr child h'. heap_is_wellformed h \<longrightarrow> h \<turnstile> adopt_node document_ptr child \<rightarrow>\<^sub>h h' \<longrightarrow> known_ptrs h \<longrightarrow> type_wf h \<longrightarrow> heap_is_wellformed h') \<and> (\<forall>h owner_document node_ptr h2 ptr children. heap_is_wellformed h \<longrightarrow> h \<turnstile> adopt_node owner_document node_ptr \<rightarrow>\<^sub>h h2 \<longrightarrow> h2 \<turnstile> get_child_nodes ptr \<rightarrow>\<^sub>r children \<longrightarrow> known_ptrs h \<longrightarrow> type_wf h \<longrightarrow> node_ptr \<notin> set children) \<and> (\<forall>h owner_document node_ptr h' disc_nodes. heap_is_wellformed h \<longrightarrow> h \<turnstile> adopt_node owner_document node_ptr \<rightarrow>\<^sub>h h' \<longrightarrow> h' \<turnstile> get_disconnected_nodes owner_document \<rightarrow>\<^sub>r disc_nodes \<longrightarrow> known_ptrs h \<longrightarrow> type_wf h \<longrightarrow> node_ptr \<in> set disc_nodes)) \<and> ((\<forall>h owner_document node h' ptr' children. heap_is_wellformed h \<longrightarrow> type_wf h \<longrightarrow> known_ptrs h \<longrightarrow> h \<turnstile> adopt_node owner_document node \<rightarrow>\<^sub>h h' \<longrightarrow> h \<turnstile> get_child_nodes ptr' \<rightarrow>\<^sub>r node # children \<longrightarrow> h' \<turnstile> get_child_nodes ptr' \<rightarrow>\<^sub>r children) \<and> (\<forall>h owner_document node. heap_is_wellformed h \<longrightarrow> known_ptrs h \<longrightarrow> type_wf h \<longrightarrow> h \<turnstile> ok adopt_node owner_document node \<longrightarrow> owner_document |\<in>| document_ptr_kinds h)) \<and> (\<forall>h document_ptr child h'. heap_is_wellformed h \<longrightarrow> h \<turnstile> adopt_node document_ptr child \<rightarrow>\<^sub>h h' \<longrightarrow> known_ptrs h \<longrightarrow> type_wf h \<longrightarrow> type_wf h') \<and> (\<forall>h document_ptr child h'. heap_is_wellformed h \<longrightarrow> h \<turnstile> adopt_node document_ptr child \<rightarrow>\<^sub>h h' \<longrightarrow> known_ptrs h \<longrightarrow> type_wf h \<longrightarrow> known_ptrs h')"
ae_filter_def: fixes M :: "'b measure" shows "ae_filter M = (INF N\<in>null_sets M. principal (space M - N))"
fps_of_poly_fps_X: shows "fps_of_poly [:0::'a, 1::'a:] = fps_X"
is_type_bool: fixes P :: "'a prog" shows "is_type P Boolean = True"
finite_Psegments_combine: fixes P :: "real \<Rightarrow> bool"   and a :: "real"   and b :: "real"   and c :: "real" assumes "finite_Psegments P a b"   and "finite_Psegments P b c"   and "b \<in> {a..c}"   and "closed ({x. P x} \<inter> {a..c})" shows "finite_Psegments P a c"
INT_E: fixes b :: "'e"   and B :: "'f \<Rightarrow> 'e set"   and A :: "'f set"   and a :: "'f" assumes "b \<in> \<Inter> (B ` A)" obtains "b \<in> B a"   | "a \<notin> A"
actionPar2Dest': fixes \<alpha> :: "'a action"   and P :: "('a, 'b, 'c) psi"   and \<beta> :: "'a action"   and Q :: "('a, 'b, 'c) psi"   and R :: "('a, 'b, 'c) psi" assumes "\<alpha> \<prec> P = \<beta> \<prec> Q \<parallel> R"   and "bn \<alpha> \<sharp>* Q"   and "bn \<beta> \<sharp>* Q" obtains T :: "('a, 'b, 'c) psi" where "P = Q \<parallel> T" and "\<alpha> \<prec> T = \<beta> \<prec> R"
Sup_upper2: fixes A :: "'al set"   and v :: "'al"   and u :: "'al" assumes "A \<subseteq> U"   and "v \<in> U"   and "u \<in> A"   and "v \<le>\<^sub>o\<^sub>w u" shows "v \<le>\<^sub>o\<^sub>w \<Squnion>\<^sub>o\<^sub>w A"
intros_1: fixes p :: "nat \<times> 'c"   and A :: "('d, 'c) nba"   and w :: "'d stream"   and V :: "(nat \<times> 'c) set" shows "p \<in> greachable A w V p"
wlp_loop_step_unitary: fixes body :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and P :: "'a \<Rightarrow> real"   and G :: "'a \<Rightarrow> bool" assumes "nearly_healthy (wlp body)"   and "\<And>P. unitary P \<Longrightarrow> unitary (t P)"   and "unitary P" shows "unitary (wlp (body ;; Embed t \<^bsub>\<guillemotleft> G \<guillemotright>\<^esub>\<oplus> Skip) P)"
is_lasso_def: fixes m :: "('Q, 'more) igb_graph_rec_scheme \<Rightarrow> 'b"   and L :: "('Q \<times> nat) lasso" shows "b_graph.is_lasso (degeneralize_ext m) L \<equiv> graph.is_lasso_pre (degeneralize_ext m) L \<and> set (lasso_cycle L) \<inter> bg_F (degeneralize_ext m) \<noteq> {}"
discI_3: fixes pr_operation' :: "pr_operation'"   and x3 :: "nat \<times> nat" assumes "pr_operation' = SAT_PUSH' x3" shows "is_SAT_PUSH' pr_operation'"
R01ia_def: shows "R01ia \<equiv> {(s, s'). a0n_state.signals s = l1_state.signals s'}"
dual_comp: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and dual :: "'c \<Rightarrow> 'c"   and dual_star :: "'c \<Rightarrow> 'c"   and omega :: "'c \<Rightarrow> 'c"   and star :: "'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and x :: "'c"   and y :: "'c" assumes "class.mbt_algebra one times inf less_eq less sup dual dual_star omega star bot top" shows "dual (times x y) = times (dual x) (dual y)"
iT_subset_same_7: fixes r :: "nat"   and m :: "nat"   and c :: "nat" shows "[ r, mod m, c ] \<subseteq> [ r, mod m ]"
prim_root_power: fixes x :: "'b list" assumes "x \<noteq> \<epsilon>" obtains i :: "nat" where "\<rho> x \<^sup>@ Suc i = x"
is_write_seen_def: fixes P :: "'a prog"   and E :: "('b \<times> ('c, 'b) obs_event action) llist"   and ws :: "nat \<Rightarrow> nat" shows "is_write_seen P E ws = (\<forall>a\<in>read_actions E. \<forall>ad al v. action_obs E a = NormalAction (ReadMem ad al v) \<longrightarrow> ws a \<in> write_actions E \<and> (ad, al) \<in> action_loc P E (ws a) \<and> value_written P E (ws a) (ad, al) = v \<and> \<not> P,E \<turnstile> a \<le>hb ws a \<and> (is_volatile P al \<longrightarrow> \<not> P,E \<turnstile> a \<le>so ws a) \<and> (\<forall>w'\<in>write_actions E. (ad, al) \<in> action_loc P E w' \<longrightarrow> P,E \<turnstile> ws a \<le>hb w' \<and> P,E \<turnstile> w' \<le>hb a \<or> is_volatile P al \<and> P,E \<turnstile> ws a \<le>so w' \<and> P,E \<turnstile> w' \<le>so a \<longrightarrow> w' = ws a))"
UTM_def: shows "UTM = (let (aprog, rs_pos, a_md) = rec_ci rec_F; abc_F = aprog [+] dummy_abc (Suc (Suc 0)) in t_wcode |+| tm_of abc_F @ shift (mopup (Suc (Suc 0))) (length (tm_of abc_F) div 2))"
spr_simAction_def: fixes kbp :: "('b, 'e, 'd) GC list"   and envVal :: "'c \<Rightarrow> 'e \<Rightarrow> bool" shows "SPRViewSingle.spr_simAction kbp envVal \<equiv> \<lambda>a X. concat (map (\<lambda>gc. if eval envVal X (guard gc) \<noteq> ODList.empty then [action gc] else []) kbp)"
assms_5: shows "u \<in> [M']\<^bsub>v,n\<^esub>"
perm: shows "\<forall>xs ys. xs <~~> ys \<longrightarrow> inDed xs = inDed ys"
induct: fixes P :: "fm \<Rightarrow> bool"   and a0 :: "fm" assumes "\<And>a. P (Lt a)"   and "\<And>a. P (Le a)"   and "\<And>a. P (Gt a)"   and "\<And>a. P (Ge a)"   and "\<And>a. P (Eq a)"   and "\<And>a. P (NEq a)"   and "\<And>i a. P (Dvd i a)"   and "\<And>i a. P (NDvd i a)"   and "\<And>p q. \<lbrakk>P p; P q\<rbrakk> \<Longrightarrow> P (And p q)"   and "\<And>p q. \<lbrakk>P p; P q\<rbrakk> \<Longrightarrow> P (Or p q)"   and "\<And>p q. \<lbrakk>P p; P q\<rbrakk> \<Longrightarrow> P (Imp p q)"   and "\<And>p q. \<lbrakk>P p; P q\<rbrakk> \<Longrightarrow> P (Iff p q)"   and "\<And>p. P p \<Longrightarrow> P (NOT p)"   and "P T"   and "P F"   and "\<And>v. P (E v)"   and "\<And>v. P (A v)" shows "P a0"
Ints_odd_less_0: fixes a :: "'a" assumes "a \<in> \<int>" shows "((1::'a) + a + a < (0::'a)) = (a < (0::'a))"
zero_less_double_add_iff_zero_less_single_add: fixes a :: "'b" shows "((0::'b) < a + a) = ((0::'b) < a)"
prod_Un: fixes A :: "'a set"   and B :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "finite A"   and "finite B"   and "\<forall>x\<in>A \<inter> B. f x \<noteq> (0::'b)" shows "prod f (A \<union> B) = prod f A * prod f B / prod f (A \<inter> B)"
polysub: fixes bs :: "'b list"   and p :: "'b poly"   and q :: "'b poly" shows "Ipoly bs (p -\<^sub>p q) = Ipoly bs p - Ipoly bs q"
a1: fixes x' :: "'x" assumes "pre x'"   and "(x', xa) \<in> V" shows "REC\<^sub>T body x' \<le> (if nofail (REC\<^sub>T body x') then M x' else FAIL)"
theFT_length: fixes n :: "nat"   and cl :: "('d, 'e, 'f) cmd list" assumes "n \<in> theFT cl" shows "n < length cl"
finprod_one_eqI: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "\<And>x. x \<in> A \<Longrightarrow> f x = (0::'c)" shows "sum f A = (0::'c)"
abs_inj: fixes m :: "nat"   and n :: "nat" assumes "abs_nat m = abs_nat n" shows "m = n"
is_Done_ABORT: shows "Resumption.resumption.is_Done ABORT"
pos_bounded: shows "\<exists>K>0. \<forall>x. norm (T x) \<le> norm x * K"
o_apply: fixes f :: "'n \<Rightarrow> 'm"   and g :: "'o \<Rightarrow> 'n"   and x :: "'o" shows "(f \<circ> g) x = f (g x)"
trancl_trans: fixes x :: "'k"   and y :: "'k"   and r :: "('k \<times> 'k) set"   and z :: "'k" assumes "(x, y) \<in> r\<^sup>+"   and "(y, z) \<in> r\<^sup>+" shows "(x, z) \<in> r\<^sup>+"
ivl_disj_un_one_8: fixes l :: "'a"   and u :: "'a" assumes "l \<le> u" shows "{l..<u} \<union> {u..} = {l..}"
simps_1: fixes x :: "'b equation" shows "negative_literal (Pos x) = False"
sign_changes_cong: fixes ps :: "real poly list"   and ps' :: "real poly list"   and x :: "real"   and y :: "real" assumes "length ps = length ps'"   and "\<forall>i<length ps. sgn (poly (ps ! i) x) = sgn (poly (ps' ! i) y)" shows "sign_changes ps x = sign_changes ps' y"
minus_nth_eucl: fixes x :: "'b"   and y :: "'b"   and i :: "nat" shows "(x - y) $\<^sub>e i = x $\<^sub>e i - y $\<^sub>e i"
card_seteq: fixes B :: "'d set"   and A :: "'d set" assumes "finite B"   and "A \<subseteq> B"   and "card B \<le> card A" shows "A = B"
mndet_distrib: fixes P :: "'a \<Rightarrow> 'a process" assumes "B - A \<noteq> {}"   and "A \<noteq> {}" shows "mndet (B - A \<union> A) P = (mndet (B - A) P |-| mndet A P)"
mod_poly_less: fixes x :: "'b poly"   and y :: "'b poly" assumes "degree x < degree y" shows "x mod y = x"
UN_iff: fixes b :: "'h"   and B :: "'i \<Rightarrow> 'h set"   and A :: "'i set" shows "(b \<in> \<Union> (B ` A)) = (\<exists>x\<in>A. b \<in> B x)"
IQ2: fixes l :: "char list \<Rightarrow> nat"   and s :: "char list \<Rightarrow> int" assumes "I l s"   and "bval b s" shows "pre C I l s \<and> 1 + preT C E s + time C s \<le> E s \<and> S s = S (postQ C s)"
sterms_psimps_1: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'c, 'a, 'd) seqp"   and p1 :: "('b, 'c, 'a, 'd) seqp"   and p2 :: "('b, 'c, 'a, 'd) seqp" assumes "wellformed \<Gamma>" shows "sterms \<Gamma> (p1 \<oplus> p2) = sterms \<Gamma> p1 \<union> sterms \<Gamma> p2"
heap_is_wellformed_one_parent: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and children :: "(_) node_ptr list"   and ptr' :: "(_) object_ptr"   and children' :: "(_) node_ptr list" assumes "heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M h"   and "h \<turnstile> Shadow_DOM.get_child_nodes ptr \<rightarrow>\<^sub>r children"   and "h \<turnstile> Shadow_DOM.get_child_nodes ptr' \<rightarrow>\<^sub>r children'"   and "set children \<inter> set children' \<noteq> {}" shows "ptr = ptr'"
collect_locksD: fixes l :: "'c"   and las :: "'c lock_actions" assumes "l \<in> collect_locks las" shows "Lock \<in> set (las $ l)"
ereal_minus_diff_eq: fixes x :: "ereal"   and y :: "ereal" assumes "x = \<infinity> \<longrightarrow> y \<noteq> \<infinity>"   and "x = - \<infinity> \<longrightarrow> y \<noteq> - \<infinity>" shows "- (x - y) = y - x"
block_design_axioms: shows "block_design \<V> \<B> \<k>"
wp1n_tl: fixes p :: "('a, 'b) Combinators list" assumes "wellformed_policy1_strong p" shows "p = DenyAll # tl p"
last_unchanged_or_empty_if_no_events: fixes t :: "('a, 'b, 'c) event list"   and i :: "nat"   and j :: "nat"   and cid :: "nat"   and p :: "'a"   and q :: "'a" assumes "trace init t final"   and "i \<le> j"   and "j \<le> length t"   and "msgs (S t i) cid \<noteq> []"   and "last (msgs (S t i) cid) = Marker"   and "ps (S t i) p \<noteq> None"   and "channel cid = Some (p, q)"   and "\<forall>k. i \<le> k \<and> k < j \<and> regular_event (t ! k) \<longrightarrow> occurs_on (t ! k) \<noteq> p" shows "msgs (S t j) cid = [] \<or> msgs (S t j) cid \<noteq> [] \<and> last (msgs (S t j) cid) = Marker"
Diff_Diff_Int: fixes A :: "'d set"   and B :: "'d set" shows "A - (A - B) = A \<inter> B"
hyps_3: shows "s2 \<in> rev ` A\<star>"
select_convs_1: fixes heap :: "nat \<Rightarrow> \<langle>'d\<rangle>\<^sub>\<bottom>"   and assocs :: "nat \<Rightarrow> \<langle>nat list list list\<rangle>\<^sub>\<bottom>"   and more :: "'e" shows "heap \<lparr>heap = heap, assocs = assocs, \<dots> = more\<rparr> = heap"
injD: fixes f :: "'a \<Rightarrow> 'b"   and x :: "'a"   and y :: "'a" assumes "inj f"   and "f x = f y" shows "x = y"
hom_inverse: fixes x :: "'a" shows "hom (inverse x) = inverse (hom x)"
conc_snd: fixes n :: "nat"   and w :: "'b list"   and x :: "nat \<Rightarrow> 'b" assumes "\<not> n < length w" shows "(w \<frown> x) n = x (n - length w)"
comp_ide_arr: fixes b :: "'c"   and f :: "'c" assumes "C.ide b"   and "C.seq b f" shows "b \<cdot>\<^sub>C f = f"
NoClassDefFoundC_def: shows "NoClassDefFoundC \<equiv> (NoClassDefFoundError, Object, [], [])"
set_integral_cont_up: fixes A :: "nat \<Rightarrow> 'b set"   and M :: "'b measure"   and f :: "'b \<Rightarrow> 'c" assumes "\<And>i. A i \<in> sets M"   and "incseq A"   and "set_integrable M (\<Union> (range A)) f" shows "(\<lambda>i. set_lebesgue_integral M (A i) f) \<longlonglongrightarrow> set_lebesgue_integral M (\<Union> (range A)) f"
simps_3: fixes a :: "'b set"   and b :: "'b set" shows "inf (RES a) (RES b) = RES (a \<inter> b)"
l_def_3: shows "\<S>\<bullet>\<C>' \<turnstile> LI : [] _> tls"
bounded_minus_image: fixes f :: "'c \<Rightarrow> 'b"   and s :: "'c set"   and g :: "'c \<Rightarrow> 'b" assumes "bounded (f ` s)"   and "bounded (g ` s)" shows "bounded ((\<lambda>x. f x - g x) ` s)"
equivalence: fixes S :: "'c set"   and P :: "'c set set" assumes "partition S P" shows "equivalence S (partition.Equivalence P)"
pwd_spy_sep_agr_rel: fixes n :: "nat" assumes "n \<in> bad_prikey \<inter> bad_id_shakey" shows "s\<^sub>0 \<Turnstile> fst (pwd_spy_sep_agr n)"
ntrunc_subsetD: fixes M :: "('a, 'b) node set"   and N :: "('a, 'b) node set" assumes "\<And>k. ntrunc k M \<subseteq> N" shows "M \<subseteq> N"
jj: shows "cs\<^bsup>\<pi>\<^esup> j = cs\<^bsup>\<pi>'\<^esup> j'"
power_transfer: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool" assumes "R (1::'a) (1::'b)"   and "rel_fun R (rel_fun R R) (*) (*)" shows "rel_fun R (rel_fun (=) R) (^) (^)"
model_1: shows "v \<Turnstile>\<^sub>t \<T> s"