hcju/isabelleps huggingface dataset:
corollary_1: fixes A :: "Points"   and B :: "Points"   and C :: "Points"   and A' :: "Points"   and B' :: "Points"   and C' :: "Points"   and M :: "Points"   and N :: "Points"   and P :: "Points"   and R :: "Points" assumes "desargues_config A B C A' B' C' M N P R"   and "is_pappus" shows "col M N P \<or> (incid A (line B' C') \<or> incid C' (line A B)) \<and> (incid C (line A' B') \<or> incid B' (line A C)) \<and> (incid B (line A' C') \<or> incid A' (line B C))"
NS_staticSecret_parts_agent_parts: fixes m :: "msg"   and C :: "agent"   and evs :: "event list"   and A :: "agent"   and D :: "agent" assumes "m \<in> parts (knows C evs)"   and "m \<in> staticSecret A"   and "A \<noteq> C"   and "evs \<in> ns_public" shows "m \<in> parts (knows D evs)"
1: shows "Collect (degen cs) \<inter> sset (w ||| k ## sscan (count cs) w k) \<noteq> {}"
ipset_from_cidr_alt2: fixes base :: "'b word"   and len :: "nat" shows "ipset_from_cidr base len = ipset_from_netmask base (~~ mask (LENGTH('b) - len))"
next_plane_subset: fixes g :: "graph"   and p :: "nat" assumes "\<forall>f\<in>\<F> g. vertices f \<noteq> []" shows "set (next_plane\<^bsub>p\<^esub> g) \<subseteq> set (next_plane0\<^bsub>p\<^esub> g)"
before_fst: fixes r2 :: "'a"   and vs :: "'a list"   and r1 :: "'a" assumes "r2 \<in> set vs"   and "r1 \<in> set (fst (splitAt r2 vs))" shows "before vs r1 r2"
downset_faithful: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.preorder less_eq less"   and "ord.downset less_eq x \<subseteq> ord.downset less_eq y" shows "less_eq x y"
Abs_castAbs: fixes xs :: "'varSort"   and s :: "'sort"   and A :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set" assumes "wlsAbs (xs, s) A" shows "Abs xs (fst (castAbs xs s A)) (snd (castAbs xs s A)) = A"
k: fixes v :: "'v"   and v' :: "'v" assumes "v \<in> V"   and "v' \<in> V"   and "nodes_connected G v v'" shows "nodes_connected H v v'"
iarray_update_of_fun: fixes f :: "nat \<Rightarrow> 'a"   and n :: "nat"   and i :: "nat"   and x :: "'a" shows "iarray_update (IArray.of_fun f n) i x = IArray.of_fun (f(i := x)) n"
lincomb_strip_while_0coeffs: fixes ms :: "'b list"   and as :: "'a list" assumes "set ms \<subseteq> V" shows "strip_while ((=) (0::'a)) as \<bullet>\<cdot> ms = as \<bullet>\<cdot> ms"
\<eta>_is_natural_transformation: shows "natural_transformation (\<cdot>\<^sub>C) (\<cdot>\<^sub>C) C.map F'G'.GF.map F'G'.\<eta>"
l_to_tree_order_wf_def: 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_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog" shows "l_to_tree_order_wf heap_is_wellformed parent_child_rel type_wf known_ptr known_ptrs to_tree_order get_parent get_child_nodes \<equiv> l_known_ptrs known_ptr known_ptrs \<and> l_to_tree_order_wf_axioms heap_is_wellformed parent_child_rel type_wf known_ptrs to_tree_order get_parent get_child_nodes"
inf_img_fin_domE': fixes f :: "'d \<Rightarrow> 'c"   and A :: "'d set" assumes "finite (f ` A)"   and "infinite A" obtains y :: "'c" where "y \<in> f ` A" and "infinite (f -` {y} \<inter> A)"
qstar_comm: fixes x :: "'a" shows "x \<cdot> qstar x = qstar x \<cdot> x"
power_Suc: fixes a :: "'a"   and n :: "nat" shows "a\<^bsup>Suc n\<^esup> = a \<cdot> a\<^bsup>n\<^esup>"
to_tree_order_parent: 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_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and h :: "(_) heap"   and ptr :: "(_) object_ptr"   and nodes :: "(_) object_ptr list"   and child :: "(_) node_ptr"   and parent :: "(_) object_ptr" assumes "l_to_tree_order_wf heap_is_wellformed parent_child_rel type_wf known_ptr known_ptrs to_tree_order get_parent get_child_nodes"   and "heap_is_wellformed h"   and "type_wf h"   and "known_ptrs h"   and "h \<turnstile> to_tree_order ptr \<rightarrow>\<^sub>r nodes"   and "h \<turnstile> get_parent child \<rightarrow>\<^sub>r Some parent"   and "parent \<in> set nodes" shows "cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r child \<in> set nodes"
s'_pos: fixes x :: "real" assumes "0 \<le> x"   and "x < delayed_safe_distance.other.t_stop"   and "0 \<le> other.s' \<delta>"   and "a\<^sub>o \<le> 0" shows "0 < delayed_safe_distance.other.s' x"
node_deliver_messages_def: fixes cs :: "('a \<times> ('a, 'b) operation) event list" shows "node_deliver_messages cs \<equiv> List.map_filter (\<lambda>e. case e of Broadcast x \<Rightarrow> Map.empty x | Deliver m \<Rightarrow> Some m) cs"
ds_ord_trans: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<sqsubseteq> y"   and "y \<sqsubseteq> z" shows "x \<sqsubseteq> z"
field: shows "field (R\<lparr>carrier := K\<rparr>)"
linear_sub_inv: fixes f :: "nat \<Rightarrow> 'a"   and d :: "'a"   and g :: "nat \<Rightarrow> 'a" assumes "f \<in> carrier (UP R)"   and "d \<in> Units R"   and "g = Cring_Poly.compose R f (monom (UP R) d 1)" shows "Cring_Poly.compose R g (monom (UP R) (inv d) 1) = f"
commute: fixes a :: "'a"   and b :: "'a" shows "normalization_euclidean_semiring_class.gcd a b = normalization_euclidean_semiring_class.gcd b a"
mult_numeral_1: fixes a :: "'c" shows "Numeral1 * a = a"
card_seteq: fixes B :: "'a set"   and A :: "'a set" assumes "finite B"   and "A \<subseteq> B"   and "card B \<le> card A" shows "A = B"
1: shows "lookup r = (\<lambda>k. q k when k \<in> A)"
assms_3: shows "x\<^sup>\<smile> ; z = 1"
assms: shows "eval the_singleton t t' n (\<diamond>\<^sub>b\<gamma>)"
rrs: shows "secure_refinement_simpler \<R>\<^sub>A \<R> P abs_steps"
init_reach: fixes T :: "('e, 'f) TS_scheme" shows "init T \<subseteq> reach T"
nth_alternate: fixes n :: "nat"   and xs :: "'b list" assumes "2 * n < length xs" shows "alternate xs ! n = xs ! (2 * n)"
Disj_IE1: fixes B :: "fm"   and H :: "fm set"   and C :: "fm"   and A :: "fm" assumes "insert B H \<turnstile> C" shows "insert (A OR B) H \<turnstile> A OR C"
nth_default_append_trailing: fixes dflt :: "'a"   and xs :: "'a list"   and n :: "nat" shows "nth_default dflt (xs @ replicate n dflt) = nth_default dflt xs"
simps_9: fixes f1 :: "'c"   and f2 :: "'d \<Rightarrow> 'c"   and f3 :: "'e \<Rightarrow> 'd \<Rightarrow> 'c" shows "(case Done of Done \<Rightarrow> f1 | Skip x \<Rightarrow> f2 x | Yield x xa \<Rightarrow> f3 x xa) = f1"
offending_notevalD: fixes F :: "('v \<times> 'v) set"   and G :: "'v graph"   and nP :: "'v \<Rightarrow> 'a" assumes "F \<in> set_offending_flows G nP" shows "\<not> sinvar G nP"
lift0: fixes i :: "nat"   and j :: "nat" shows "liftPoly i j 0 = 0"
res_neg_eq: fixes x :: "int" shows "\<ominus> x = - x mod m"
simps: fixes P :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a1 :: "'c list"   and a2 :: "'c list" shows "list_emb P a1 a2 = ((\<exists>ys. a1 = [] \<and> a2 = ys) \<or> (\<exists>xs ys y. a1 = xs \<and> a2 = y # ys \<and> list_emb P xs ys) \<or> (\<exists>x y xs ys. a1 = x # xs \<and> a2 = y # ys \<and> P x y \<and> list_emb P xs ys))"
finite1: fixes x :: "'v" shows "\<integral>\<^sup>+y\<in>X. (h_diff i (x, y) * indicator X x)\<partial>count_space UNIV \<noteq> \<top>"
tauChainStatEq: fixes P :: "('a, 'b, 'c) psi"   and P' :: "('a, 'b, 'c) psi"   and \<Psi> :: "'b"   and \<Psi>' :: "'b" assumes "\<Psi> \<rhd> P \<Longrightarrow>\<^sup>^\<^sub>\<tau> P'"   and "\<Psi> \<simeq> \<Psi>'" shows "\<Psi>' \<rhd> P \<Longrightarrow>\<^sup>^\<^sub>\<tau> P'"
Pi'_mem: fixes f :: "'d \<Rightarrow>\<^sub>F 'e"   and A :: "'d set"   and B :: "'d \<Rightarrow> 'e set"   and x :: "'d" assumes "f \<in> Pi' A B"   and "x \<in> A" shows "(f)\<^sub>F x \<in> B x"
arr_cod: fixes f :: "'c" assumes "C'.arr f" shows "C'.arr (C'.cod f)"
simps_1: fixes uu :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and uv :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and uw :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and ux :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> 'f) list \<times> 'g"   and uy :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f \<Rightarrow> 'g \<Rightarrow> 'g"   and uz :: "'g" shows "gb_schema_incr uu uv uw ux uy [] uz = []"
of_int_eq: fixes z :: "int" shows "of_int z = Abs_bit1 (z mod int CARD('b bit1))"
a: fixes d :: "real" assumes "0 < d"   and "sgn (poly p (x - d)) \<noteq> sgn (poly p (x + d))"   and "sgn (poly p (x - d)) \<noteq> 0"   and "0 \<noteq> sgn (poly p (x + d))"   and "\<forall>d'>0. d' \<le> d \<longrightarrow> sgn (poly p (x + d')) = sgn (poly p (x + d)) \<and> sgn (poly p (x - d')) = sgn (poly p (x - d))" shows "thesis"
prems_4: shows "\<forall>xs ys. as @ [a] = xs @ ys \<longrightarrow> \<not> same_level_path_rev_aux cs ys \<or> upd_rev_cs cs ys \<noteq> []"
All_mapping_def: fixes m :: "('b, 'c) mapping"   and P :: "'b \<Rightarrow> 'c \<Rightarrow> bool" shows "All_mapping m P = (\<forall>x. case lookup m x of None \<Rightarrow> True | Some y \<Rightarrow> P x y)"
elims: fixes x :: "('a \<Rightarrow> ('b, 'a, 'c) val option) \<times> ('a \<Rightarrow> ('b, 'a, 'c) val option) \<times> ('b, 'a, 'c) expr"   and y :: "'a set" assumes "doms x = y" obtains \<sigma> :: "'a \<Rightarrow> ('b, 'a, 'c) val option" and \<tau> :: "'a \<Rightarrow> ('b, 'a, 'c) val option" and e :: "('b, 'a, 'c) expr" where "x = (\<sigma>, \<tau>, e)" and "y = dom \<sigma> \<union> dom \<tau>"
admissible_def: fixes lub :: "'f set \<Rightarrow> 'f"   and ord :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and P :: "'f \<Rightarrow> bool" shows "ccpo.admissible lub ord P = (\<forall>A. Complete_Partial_Order.chain ord A \<longrightarrow> A \<noteq> {} \<longrightarrow> (\<forall>x\<in>A. P x) \<longrightarrow> P (lub A))"
iso_map\<^sub>1_ide: fixes f :: "'b" assumes "B.ide f" shows "C.iso (\<tau>\<^sub>1 f)"
yun_wrel_gcd: fixes F :: "int poly"   and c' :: "rat"   and f :: "rat poly"   and G :: "int poly"   and c :: "rat"   and g :: "rat poly"   and d :: "rat" assumes "yun_wrel F c' f"   and "yun_wrel G c g"   and "c' \<noteq> 0"   and "c \<noteq> 0"   and "d = rat_of_int (lead_coeff (gcd F G))"   and "d \<noteq> 0" shows "yun_wrel (gcd F G) d (gcd f g)"
mono_funpow: fixes Q :: "'c \<Rightarrow> 'c" assumes "mono Q" shows "incseq (\<lambda>i. (Q ^^ i) bot)"
finite_intersection_of_Int: fixes P :: "'c set \<Rightarrow> bool"   and S :: "'c set"   and T :: "'c set" assumes "(finite intersection_of P) S"   and "(finite intersection_of P) T" shows "(finite intersection_of P) (S \<inter> T)"
set_of_ivl_def: fixes x :: "'a \<times> 'a" shows "set_of_ivl x = {fst x..snd x}"
zero_not_one: fixes R :: "('c, 'd) ring_scheme" assumes "domain R" shows "\<zero>\<^bsub>R\<^esub> \<noteq> \<one>\<^bsub>R\<^esub>"
orthog_proj_exists: fixes M :: "'a set"   and h :: "'a" assumes "closed_csubspace M" shows "\<exists>k. h - k \<in> orthogonal_complement M \<and> k \<in> M"
smult_dvd: fixes p :: "'a poly"   and q :: "'a poly"   and a :: "'a" assumes "p dvd q"   and "a \<noteq> (0::'a)" shows "smult a p dvd q"
frsub: shows "frontier (connected_component_set (- closure A) b) \<subseteq> frontier A"
uint_add_ge0: fixes xa :: "'c word"   and x :: "'d word" shows "0 \<le> uint xa + uint x"
finite_measure_subalgebra_is_sigma_finite: fixes M :: "'b measure"   and F :: "'b measure" assumes "finite_measure_subalgebra M F" shows "sigma_finite_subalgebra M F"
heap_insert: fixes t :: "'b tree"   and x :: "'b" assumes "heap t" shows "heap (insert x t)"
project_stable_project_set: fixes G :: "'c program"   and C :: "'c set" assumes "G \<in> stable C" shows "project h C G \<in> stable (project_set h C)"
IH: fixes ts :: "'a \<Rightarrow> ('b \<times> 'c \<Rightarrow>f nat) option"   and I :: "'a \<Rightarrow> 'f option" assumes "thread_oks ts TAS"   and "I t = \<lfloor>i\<rfloor>"   and "ts t = \<lfloor>x\<rfloor>" shows "upd_invs I Q TAS t = \<lfloor>i\<rfloor>"
distinct_258: fixes x151 :: "floatarith"   and x152 :: "nat"   and x11a :: "floatarith" shows "x151 ^\<^sub>e x152 \<noteq> Sqrt x11a"
list_eq_iff: fixes xs :: "'a list"   and ys :: "'a list" shows "(xs = ys) = (length xs = length ys \<and> (\<forall>i<length xs. xs ! i = ys ! i))"
r_or_prim: shows "prim_recfn 2 r_or"
sgn_prod_def: fixes x :: "'d \<times> 'e" shows "sgn x = x /\<^sub>R norm x"
Q_min: fixes u :: "'v"   and d :: "nat" assumes "Q' Q \<pi> u u = enat d" shows "\<forall>v'\<in>S (A (Q' Q \<pi> u) (\<pi>' Q \<pi> u)). (u, v') \<in> edges rg \<longrightarrow> d \<le> w {u, v'}"
Sup: fixes lub :: "'b set \<Rightarrow> 'b"   and leq :: "('b \<times> 'b) set"   and f :: "'b \<Rightarrow> 'b"   and M :: "'b set"   and a :: "'b" assumes "bourbaki_witt_fixpoint lub leq f"   and "M \<in> Chains leq"   and "M \<noteq> {}"   and "\<And>x. x \<in> M \<Longrightarrow> x \<in> bourbaki_witt_fixpoint.iterates_above lub leq f a" shows "lub M \<in> bourbaki_witt_fixpoint.iterates_above lub leq f a"
lgridI: fixes p :: "(nat \<times> int) list"   and b :: "(nat \<times> int) list"   and ds :: "nat set"   and lm :: "nat" assumes "p \<in> grid b ds"   and "level p < lm" shows "p \<in> lgrid b ds lm"
mem_mod_w32_privilege: fixes s' :: "'b sparc_state"   and a1 :: "word8"   and a2 :: "word36"   and b :: "word4"   and d :: "word32"   and s :: "'b sparc_state" assumes "s' = mem_mod_w32 a1 a2 b d s \<and> get_S (cpu_reg_val PSR s) = 0" shows "get_S (cpu_reg_val PSR s') = 0"
pol_coeff_def: fixes S :: "('d, 'e) Ring_scheme"   and c :: "nat \<times> (nat \<Rightarrow> 'd)" shows "pol_coeff S c = (\<forall>j\<le>fst c. snd c j \<in> carrier S)"
convex_on_imp_above_tangent: fixes A :: "real set"   and f :: "real \<Rightarrow> real"   and c :: "real"   and x :: "real"   and f' :: "real" assumes "convex_on A f"   and "connected A"   and "c \<in> interior A"   and "x \<in> A"   and "(f has_real_derivative f') (at c within A)" shows "f' * (x - c) \<le> f x - f c"
is_terminal: fixes D :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and F :: "'a \<Rightarrow> 'b"   and x :: "'a"   and y :: "'b"   and e :: "'b"   and x' :: "'a"   and f :: "'b" assumes "terminal_arrow_from_functor D C F x y e"   and "arrow_from_functor D C F x' y f" shows "\<exists>!g. arrow_from_functor.is_coext D C F x e x' f g"
mult_nonneg_nonpos2: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.ordered_semiring_0 plus zero less_eq less times"   and "less_eq zero a"   and "less_eq b zero" shows "less_eq (times b a) zero"
add_cancel_right_right: fixes a :: "'ag"   and b :: "'ag" assumes "a \<in> U"   and "b \<in> U" shows "(a = a +\<^sub>o\<^sub>w b) = (b = 0\<^sub>o\<^sub>w)"
assms_1: shows "g \<equiv> \<lambda>r. poly pp \<circ> (\<lambda>x. complex_of_real (linepath (- r) (of_real r) x))"
dim_row_transfer_rule: fixes A :: "'b mat"   and A' :: "(('b, 'c) Finite_Cartesian_Product.vec, 'd) Finite_Cartesian_Product.vec" assumes "HMA_M A A'" shows "dim_row A = CARD('d)"
matrix_mul_diag_diag: fixes f :: "'e \<Rightarrow> 'd"   and g :: "'e \<Rightarrow> 'd" shows "diag_mat f ** diag_mat g = diag_mat (\<lambda>i. f i * g i)"
cancels_to_trans: fixes a :: "(bool \<times> 'c) list"   and b :: "(bool \<times> 'c) list"   and c :: "(bool \<times> 'c) list" assumes "cancels_to a b"   and "cancels_to b c" shows "cancels_to a c"
dd_def: shows "dd = (\<lambda>x. (1::'a) / x ^ (degree p - degree q))"
mcont_def: fixes luba :: "'c set \<Rightarrow> 'c"   and orda :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and lubb :: "'d set \<Rightarrow> 'd"   and ordb :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd" shows "mcont luba orda lubb ordb f = (monotone orda ordb f \<and> cont luba orda lubb ordb f)"
MN_refines: fixes L :: "'b list set"   and R :: "('b list \<times> 'b list) set"   and x :: "'b list"   and y :: "'b list" assumes "MyhillNerode L R"   and "(x, y) \<in> R" shows "(x \<in> L) = (y \<in> L)"
TrustCertVerify: fixes X :: "msg"   and evs :: "event list"   and KA :: "nat"   and nb :: "msg"   and B :: "agent"   and pms :: "msg"   and A :: "agent" assumes "X \<in> parts (knows Spy evs)"   and "X = Crypt (invKey KA) (Hash \<lbrace>nb, Agent B, pms\<rbrace>)"   and "certificate A KA \<in> parts (knows Spy evs)"   and "evs \<in> tls"   and "A \<notin> bad" shows "Says A B X \<in> set evs"
support_pre: fixes C :: "acom"   and Q :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool" shows "support (pre C Q) \<subseteq> support Q \<union> varacom C"
is_fmap_subset: fixes N :: "('a \<times> 'b) fset"   and M :: "('a \<times> 'b) fset" assumes "is_fmap N"   and "M |\<subseteq>| N" shows "is_fmap M"
lsc_at_mem: fixes x0 :: "'b"   and f :: "'b \<Rightarrow> 'c"   and x :: "nat \<Rightarrow> 'b"   and A :: "'c" assumes "lsc_at x0 f"   and "x \<longlonglongrightarrow> x0"   and "(f \<circ> x) \<longlonglongrightarrow> A" shows "f x0 \<le> A"
f_def: shows "f = (\<lambda>i. r * sin i + Im z - Im z0)"
disc_2: fixes x21 :: "'f"   and x22 :: "('f, 'g) tllist" shows "\<not> is_TNil (TCons x21 x22)"
Un_Int_distrib: fixes A :: "'b set"   and B :: "'b set"   and C :: "'b set" shows "A \<union> B \<inter> C = (A \<union> B) \<inter> (A \<union> C)"
if_mono: fixes orda :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and ordb :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and F :: "'b \<Rightarrow> 'c"   and G :: "'b \<Rightarrow> 'c"   and c :: "bool" assumes "monotone orda ordb F"   and "monotone orda ordb G" shows "monotone orda ordb (\<lambda>f. if c then F f else G f)"
Seq_red_nE: fixes c\<^sub>1 :: "com"   and c\<^sub>2 :: "com"   and s :: "char list \<Rightarrow> val option"   and n :: "nat"   and s' :: "char list \<Rightarrow> val option" assumes "\<langle>c\<^sub>1;; c\<^sub>2,s\<rangle> \<rightarrow>\<^bsup>n\<^esup> \<langle>Skip,s'\<rangle>" obtains i :: "nat" and j :: "nat" and s'' :: "char list \<Rightarrow> val option" where "\<langle>c\<^sub>1,s\<rangle> \<rightarrow>\<^bsup>i\<^esup> \<langle>Skip,s''\<rangle>" and "\<langle>c\<^sub>2,s''\<rangle> \<rightarrow>\<^bsup>j\<^esup> \<langle>Skip,s'\<rangle>" and "n = i + j + 1"
is_offending_flows_min_set_minimalize_offending_overapprox: fixes G :: "'v graph"   and ff :: "('v \<times> 'v) list"   and nP :: "'v \<Rightarrow> 'a" assumes "sinvar_mono"   and "wf_graph G"   and "is_offending_flows (set ff) G nP"   and "set ff \<subseteq> edges G"   and "distinct ff" shows "is_offending_flows_min_set (set (minimalize_offending_overapprox ff [] G nP)) G nP"
holdsIstate_def: fixes \<phi> :: "'state \<Rightarrow> bool" shows "holdsIstate \<phi> \<equiv> \<phi> istate"
WTrt_binop_simps_10: fixes P :: "'b prog"   and T1 :: "ty"   and T2 :: "ty"   and T :: "ty" shows "(P \<turnstile> T1\<guillemotleft>Div\<guillemotright>T2 : T) = (T1 = Integer \<and> T2 = Integer \<and> T = Integer)"
DERIV_inverse_fun: fixes f :: "'a \<Rightarrow> 'a"   and d :: "'a"   and x :: "'a"   and s :: "'a set" assumes "(f has_field_derivative d) (at x within s)"   and "f x \<noteq> (0::'a)" shows "((\<lambda>x. inverse (f x)) has_field_derivative - (d * inverse (f x ^ Suc (Suc 0)))) (at x within s)"
graeffe_poly_impl_def: fixes f :: "'a poly" shows "graeffe_poly_impl f = graeffe_poly_impl_main ((- (1::'a)) ^ degree f) f"
zero_less_diff: fixes n :: "nat"   and m :: "nat" shows "(zero_class.zero < n - m) = (m < n)"
0: shows "\<forall>a\<in>A. (a, b) \<in> r"
res_e0': shows "ord_resolve S CAs0' DA0' AAs0' As0' \<tau> E0'"
Loss_L5: fixes t :: "nat"   and a :: "nat \<Rightarrow> aType list"   and lose :: "nat \<Rightarrow> bool list"   and i2 :: "nat \<Rightarrow> ECall_Info list"   and y :: "nat \<Rightarrow> aType list"   and i :: "nat \<Rightarrow> ECall_Info list"   and t2 :: "nat" assumes "\<forall>t1\<le>t. a t1 = []"   and "Loss lose a i2 y i"   and "t2 \<le> t"   and "ts lose" shows "y t2 = []"
split_edges_sym: shows "\<exists>E. E \<inter> E\<inverse> = {} \<and> edges g = E \<union> E\<inverse>"
simps: fixes xs :: "'varSort"   and x :: "'var"   and X :: "('b, 'c, 'varSort, 'var, 'd) qTerm" shows "qGoodAbs (qAbs xs x X) = qGood X"
upper_triangular_imp_distinct: fixes A :: "'b mat"   and n :: "nat" assumes "A \<in> carrier_mat n n"   and "upper_triangular A"   and "(0::'b) \<notin> set (diag_mat A)" shows "distinct (rows A)"
prems: shows "t \<in> trms\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s (g # F \<cdot>\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s \<theta>)"
dom_iff: fixes m :: "'d \<Rightarrow> 'c option"   and x :: "'d" shows "(\<exists>y. m x = Some y) = (x \<in> dom m)"
eq_2: shows "typ_of f = Some (\<tau> \<rightarrow> \<tau>')"
fold_set_fold: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and y :: "'c"   and xs :: "'b list" assumes "comp_fun_idem f" shows "Finite_Set.fold f y (set xs) = fold f xs y"
Wbis_coind: fixes theta :: "(('test, 'atom) com \<times> ('test, 'atom) com) set" assumes "sym theta"   and "theta \<subseteq> Wretr (theta \<union> Wbis)" shows "theta \<subseteq> Wbis"
Lim_subseteq_BC: shows "LIM \<subseteq> BC"
deriv_G: fixes x :: "real" assumes "x \<in> {0<..<4}" shows "(G has_real_derivative sqrt ((4 - x) / x)) (at x)"
retraction: fixes S :: "'b set"   and T :: "'b set"   and r :: "'b \<Rightarrow> 'b" shows "retraction S T r = (T \<subseteq> S \<and> continuous_on S r \<and> r ` S = T \<and> (\<forall>x\<in>T. r x = x))"
consistent: fixes s :: "real" assumes "s \<in> {t0..t0 + e1'}" shows "consistent solution s (t0 + e1') (euler_C TYPE('a)) 1 (euler_increment f)"
cong_l_symmetric: fixes F :: "'a set"   and a :: "'a"   and b :: "'a" assumes "F \<in> filters" shows "cong_l F a b = cong_l F b a"