hcju/isabelleps huggingface dataset:
mHom_compos: fixes M :: "('e, 'f, 'g) Module_scheme"   and R :: "('f, 'h) Ring_scheme"   and L :: "('i, 'f, 'j) Module_scheme"   and N :: "('k, 'f, 'l) Module_scheme"   and f :: "'i \<Rightarrow> 'e"   and g :: "'e \<Rightarrow> 'k" assumes "R module M"   and "R module L"   and "R module N"   and "f \<in> mHom R L M"   and "g \<in> mHom R M N" shows "compos L g f \<in> mHom R L N"
IH: fixes y :: "V"   and x :: "V" assumes "y \<in> elts y'"   and "x \<in> elts (TC y)" shows "(x, y) \<in> VWO"
lhr_fps_numerator_altdef: shows "lhr_fps_numerator (length fs + 1 - length cs) cs f = lhr_fps_numerator (length fs + 1 - length cs) cs ((!) fs)"
pw_eqI: fixes S :: "'b nres"   and S' :: "'b nres" assumes "nofail S = nofail S'"   and "\<And>x. inres S x = inres S' x" shows "S = S'"
topological_tendstoD: fixes "open" :: "'e set \<Rightarrow> bool"   and f :: "'f \<Rightarrow> 'e"   and l :: "'e"   and F :: "'f filter"   and S :: "'e set" assumes "class.topological_space open"   and "filterlim f (topological_space.nhds open l) F"   and "open S"   and "l \<in> S" shows "\<forall>\<^sub>F x in F. f x \<in> S"
eventually_defs: fixes F :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and w :: "nat \<Rightarrow> 'b" shows "(\<diamond>F) w = (\<exists>n. F (w |\<^sub>s n))"
lexordp_eq_antisym: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "'a list"   and ys :: "'a list" assumes "class.linorder less_eq less"   and "ord.lexordp_eq less xs ys"   and "ord.lexordp_eq less ys xs" shows "xs = ys"
isin: fixes t :: "('b \<times> color) tree"   and x :: "'b" assumes "rbt t \<and> Sorted_Less.sorted (Tree2.inorder t)" shows "isin t x = (x \<in> set (Tree2.inorder t))"
the_real_step_augment: fixes step :: "nat \<Rightarrow> atom list \<Rightarrow> atom fm list \<Rightarrow> atom fm"   and amount :: "nat"   and F :: "(atom list \<times> atom fm list \<times> nat) list"   and \<Gamma> :: "real list" assumes "\<And>xs var L F \<Gamma>. length xs = var \<Longrightarrow> (\<exists>x. eval (list_conj (map fm.Atom L @ F)) (xs @ x # \<Gamma>)) = (\<exists>x. eval (step var L F) (xs @ x # \<Gamma>))" shows "(\<exists>xs. length xs = amount \<and> eval (list_disj (map (\<lambda>(L, F, n). ExN n (list_conj (map fm.Atom L @ F))) F)) (xs @ \<Gamma>)) = eval (the_real_step_augment step amount F) \<Gamma>"
sip_None: assumes "ipcidr_conjunct sip1 sip2 = None" shows "\<not> simple_match_ip sip1 (p_src p) \<or> \<not> simple_match_ip sip2 (p_src p)"
ns\<^sub>2_1: shows "g \<turnstile> defNode g p'-ns\<^sub>2\<rightarrow>defNode g p"
tauLaw3StatImpLeft: fixes \<Psi> :: "'b"   and \<alpha> :: "'a prefix"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and Rel :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set" assumes "\<And>\<Psi>''. (\<Psi> \<otimes> \<Psi>'', \<alpha>\<cdot>(\<tau>.P), \<alpha>\<cdot>Q) \<in> Rel" shows "\<Psi> \<rhd> \<alpha>\<cdot>(\<tau>.P) \<lessapprox><Rel> \<alpha>\<cdot>Q"
lenv: shows "length v mod 2 = 0 \<and> last v = y \<and> v \<noteq> []"
upto_rec2: fixes i :: "int"   and j :: "int" assumes "i \<le> j" shows "[i..j] = [i..j - 1] @ [j]"
euclidean_eqI: fixes minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and dist :: "'d \<Rightarrow> 'd \<Rightarrow> real"   and norm :: "'d \<Rightarrow> real"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and uminus :: "'d \<Rightarrow> 'd"   and scaleR :: "real \<Rightarrow> 'd \<Rightarrow> 'd"   and sgn :: "'d \<Rightarrow> 'd"   and uniformity :: "('d \<times> 'd) filter"   and "open" :: "'d set \<Rightarrow> bool"   and inner :: "'d \<Rightarrow> 'd \<Rightarrow> real"   and Basis :: "'d set"   and x :: "'d"   and y :: "'d" assumes "class.euclidean_space minus dist norm plus zero uminus scaleR sgn uniformity open inner Basis"   and "\<And>b. b \<in> Basis \<Longrightarrow> inner x b = inner y b" shows "x = y"
unwindingrel_s1'_s1: shows "(s1', s1) \<in> ur"
sigma_sets_superset_generator: fixes A :: "'d set set"   and X :: "'d set" shows "A \<subseteq> sigma_sets X A"
assms_2: fixes x :: "real" assumes "b \<le> x" shows "f x \<le> u"
rcosets_non_empty: fixes H :: "'d set"   and G :: "('d, 'e) monoid_scheme"   and R :: "'d set" assumes "subgroup H G"   and "R \<in> rcosets\<^bsub>G\<^esub> H" shows "R \<noteq> {}"
h13: shows "(\<exists>(a', b', c')\<in>set c. a' \<noteq> 0 \<and> - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and> (\<forall>(d, e, f)\<in>set a. d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 + e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) + f = 0) \<and> (\<forall>(d, e, f)\<in>set b. d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 + e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) + f < 0) \<and> (\<forall>(d, e, f)\<in>set c. d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 + e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) + f \<le> 0) \<and> (\<forall>(d, e, f)\<in>set d. d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 + e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) + f \<noteq> 0)) \<longrightarrow> (\<exists>x. (\<forall>(a, b, c)\<in>set a. a * x\<^sup>2 + b * x + c = 0) \<and> (\<forall>(a, b, c)\<in>set b. a * x\<^sup>2 + b * x + c < 0) \<and> (\<forall>(a, b, c)\<in>set c. a * x\<^sup>2 + b * x + c \<le> 0) \<and> (\<forall>(a, b, c)\<in>set d. a * x\<^sup>2 + b * x + c \<noteq> 0))"
Cons_5: shows "lookup_pair (map_pair f xs) (fst (f (k, v))) = snd (f (k, lookup_pair xs k))"
cltn2_inverse_inv: fixes A :: "cltn2" shows "inv\<^bsub>\<lparr>carrier = UNIV, monoid.mult = cltn2_compose, one = cltn2_id\<rparr>\<^esub> A = cltn2_inverse A"
analz_conj_parts: fixes X :: "msg"   and H :: "msg set" shows "(X \<in> analz H \<and> X \<in> parts H) = (X \<in> analz H)"
Stable_def: fixes A :: "'c set" shows "Stable A \<equiv> A Co A"
linear_spanning_surjective_image: fixes f :: "'b \<Rightarrow> 'b"   and S :: "'b set" assumes "linear (*s) (*s) f"   and "UNIV \<subseteq> span S"   and "surj f" shows "UNIV \<subseteq> span (f ` S)"
AE_count_space: fixes P :: "'a \<Rightarrow> bool"   and A :: "'a set" shows "almost_everywhere (count_space A) P = (\<forall>x\<in>A. P x)"
linear_order_on_converse: fixes A :: "'a set"   and r :: "('a \<times> 'a) set" shows "linear_order_on A (r\<inverse>) = linear_order_on A r"
intros_2: fixes A :: "(C, BD) form" shows "([ \<LM> A  \<RM> \<Rightarrow>* \<Empt>],  \<LM> \<box> A  \<RM> \<Rightarrow>* \<Empt>) \<in> g3mod2"
concurrent_set_appendI: fixes a :: "'a"   and xs :: "'a list"   and ys :: "'a list" assumes "concurrent_set a xs"   and "concurrent_set a ys" shows "concurrent_set a (xs @ ys)"
wellformed_items_Predict: fixes I :: "item set"   and k :: "nat" assumes "wellformed_items I" shows "wellformed_items (Predict k I)"
elem_cc_restr: fixes x :: "Vars.var"   and y :: "Vars.var"   and S :: "Vars.var set"   and G :: "CoCalls" shows "x--y\<in>(G G|` S) = (x--y\<in>G \<and> x \<in> S \<and> y \<in> S)"
twoNetsDistinct_def: fixes a :: "'a set set"   and b :: "'b set set"   and c :: "'a set set"   and d :: "'b set set" shows "twoNetsDistinct a b c d = (netsDistinct a c \<or> netsDistinct b d)"
get_ancestors_si_parent_child_host_shadow_root_rel: fixes h :: "(_) heap"   and child :: "(_) object_ptr"   and ancestors :: "(_) object_ptr list"   and ptr :: "(_) object_ptr" assumes "heap_is_wellformed h"   and "known_ptrs h"   and "type_wf h"   and "h \<turnstile> get_ancestors_si child \<rightarrow>\<^sub>r ancestors"   and "(ptr, child) \<in> (parent_child_rel h \<union> local.a_host_shadow_root_rel h)\<^sup>*" shows "ptr \<in> set ancestors"
to_tree_order_def: shows "Shadow_DOM.to_tree_order \<equiv> l_to_tree_order\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_to_tree_order Shadow_DOM.get_child_nodes"
properties_def: fixes f :: "'b \<Rightarrow> 'b"   and x0 :: "'b"   and lambda :: "nat"   and mu :: "nat" shows "properties f x0 lambda mu \<equiv> fx0.properties f x0 lambda mu"
open_translation: fixes S :: "'c set"   and a :: "'c" assumes "open S" shows "open ((+) a ` S)"
calM_def: shows "calM \<equiv> {s. s \<subseteq> carrier G \<and> card s = q ^ 1}"
assms_1: shows "r \<in> A\<^sup>\<star>"
assms_3: shows "\<nexists>y. \<phi> i y \<down> \<and> e_length b \<le> y"
a_2: shows "real_of_float a \<le> f' (real_of_float u)"
sdiv_poly_smult: fixes c :: "'d"   and f :: "'d poly" assumes "c \<noteq> (0::'d)" shows "sdiv_poly (smult c f) c = f"
inj_map: fixes f :: "'c \<Rightarrow> 'd" assumes "inj f" shows "inj ((`\<^sub>c) f)"
sym_bis: fixes Retr :: "('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set" assumes "Bisim.mono Retr" shows "sym (bis Retr)"
Sup_in_diff: shows "Red_I (Sup_llist Ns) \<subseteq> Red_I (Sup_llist Ns - (Sup_llist Ns - Liminf_llist Ns))"
fold_eq: fixes z :: "'\<sigma>" shows "Finite_Set.fold f z (insert x S) = f x (Finite_Set.fold f z S)"
f: shows "f = steps0 (Suc 0, l, r) tinc_b"
P: fixes n :: "nat" assumes "P \<le> n" shows "E * F ^ n < fact (n - 1)"
init: fixes cfg1 :: "('p, 'v, 's) configuration" shows "reachable cfg1 cfg1"
submonoid_is_comm_monoid: fixes G :: "('e, 'f) monoid_scheme"   and H :: "'e set" assumes "Group.comm_monoid G"   and "submonoid H G" shows "Group.comm_monoid (G\<lparr>carrier := H\<rparr>)"
p_recurrence''': fixes k :: "nat" assumes "3 < k" shows "p k = p (k - 3) / 6 + p (k - 2) / 2 + p (k - 1)"
cases: fixes x :: "plan_action" obtains n :: "char list" and args :: "object list" where "x = PAction n args"
qFresh_imp_ex_qAFresh2: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm"   and xs :: "'varSort"   and x :: "'var"   and ys :: "'varSort"   and y :: "'var" assumes "qGood X"   and "qFresh xs x X"   and "qFresh ys y X" shows "\<exists>X'. qGood X' \<and> X #= X' \<and> qAFresh xs x X' \<and> qAFresh ys y X'"
assms_1: shows "(\<pi>\<^sub>2 r = 0) = (\<pi>\<^sub>3 r = unk)"
brnL_Int_lt: fixes n1 :: "nat"   and n2 :: "nat"   and cl :: "('test, 'atom, 'choice) cmd list" assumes "n1 < n2"   and "n2 < length cl" shows "{brnL cl n1..<brnL cl n1 + brn (cl ! n1)} \<inter> {brnL cl n2..<brnL cl n2 + brn (cl ! n2)} = {}"
of_char_Char: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and b0 :: "bool"   and b1 :: "bool"   and b2 :: "bool"   and b3 :: "bool"   and b4 :: "bool"   and b5 :: "bool"   and b6 :: "bool"   and b7 :: "bool" assumes "class.comm_semiring_1 times one plus zero" shows "comm_semiring_1.of_char times one plus zero (Char b0 b1 b2 b3 b4 b5 b6 b7) = comm_semiring_0.horner_sum plus zero times (zero_neq_one.of_bool one zero) (numeral.numeral one plus (num.Bit0 num.One)) [b0, b1, b2, b3, b4, b5, b6, b7]"
chine_in_hom: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<mu> :: "'b arrow_of_spans_data" assumes "arrow_of_spans C \<mu>" shows "partial_magma.in_hom C (Chn \<mu>) (span_in_category.apex C (Dom \<mu>)) (span_in_category.apex C (Cod \<mu>))"
outputCases: fixes \<Psi> :: "'b"   and M :: "'a"   and N :: "'a"   and P :: "('a, 'b, 'c) psi"   and \<alpha> :: "'a action"   and P' :: "('a, 'b, 'c) psi"   and Prop :: "'a action \<Rightarrow> ('a, 'b, 'c) psi \<Rightarrow> bool" assumes "\<Psi> \<rhd> M\<langle>N\<rangle>.P \<longmapsto> \<alpha> \<prec> P'"   and "\<And>K. \<Psi> \<turnstile> M \<leftrightarrow> K \<Longrightarrow> Prop (K\<langle>N\<rangle>) P" shows "Prop \<alpha> P'"
clear_locals_def: fixes \<xi> :: "state" shows "clear_locals \<xi> = \<xi>\<lparr>msg := SOME x. True, num := SOME x. True, sid := SOME x. True\<rparr>"
supp_h_diff_edges: fixes \<Delta> :: "('a, 'b) network_scheme"   and g :: "'a \<times> 'a \<Rightarrow> ennreal"   and i :: "nat" assumes "flow_network \<Delta> g" shows "support_flow (flow_network.h_diff \<Delta> g i) \<subseteq> \<^bold>E\<^bsub>\<Delta>\<^esub>"
tree_def: fixes G :: "('c, 'd) graph" shows "tree G \<equiv> forest G \<and> connected_graph G"
lex_total: fixes t :: "real \<times> real"   and q :: "real \<times> real" shows "Counterclockwise_2D_Arbitrary.lex t q \<and> t \<noteq> q \<or> Counterclockwise_2D_Arbitrary.lex q t \<and> t \<noteq> q \<or> t = q"
ts_c2': fixes x :: "nat"   and y :: "nat"   and qs :: "nat list"   and h :: "nat list"   and h0 :: "nat list" assumes "x \<noteq> y"   and "qs \<in> lang (seq [Atom x, Times (Atom y) (Atom x), Star (Times (Atom y) (Atom x)), Atom x])"   and "(\<exists>hs. h = [x, x] @ hs) \<or> h = []" shows "Partial_Cost_Model.T_on' (rTS h0) ([x, y], h) qs = length qs - 3 \<and> Partial_Cost_Model.config' (rTS h0) ([x, y], h) qs = ([x, y], rev qs @ h) \<and> (\<exists>hs. rev qs @ h = [x, x] @ hs)"
fres_galois: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(x \<sqinter> y \<le> z) = (x \<le> proto_near_quantale.fres Sup (\<le>) (\<sqinter>) z y)"
octo_of_real_sel_2: fixes x :: "real" shows "Im1 (octo_of_real x) = 0"
write_sops_append: fixes xs :: "'b memref list"   and ys :: "'b memref list" shows "write_sops (xs @ ys) = write_sops xs \<union> write_sops ys"
discrete_def: fixes X :: "'b set" shows "discrete X = (I = id)"
SUP_id_eq: fixes Sup :: "'a set \<Rightarrow> 'a"   and A :: "'a set" shows "Sup (id ` A) = Sup A"
poly_mapping_eqI: fixes f :: "'b \<Rightarrow>\<^sub>0 'c"   and g :: "'b \<Rightarrow>\<^sub>0 'c" assumes "\<And>k. lookup f k = lookup g k" shows "f = g"
ntr_unsplit: fixes w :: "('p, 'ba) label list list"   and wa :: "('p, 'ba) label list list"   and wb :: "('p, 'ba) label list list"   and ca :: "'n list multiset"   and ca' :: "'n list multiset"   and cb :: "'n list multiset"   and cb' :: "'n list multiset" assumes "w \<in> wa \<otimes>\<^bsub>\<alpha>n fg\<^esub> wb"   and "(ca, wa, ca') \<in> trcl (ntr fg)"   and "(cb, wb, cb') \<in> trcl (ntr fg)"   and "mon_c fg ca \<inter> (mon_c fg cb \<union> mon_ww fg wb) = {}"   and "mon_c fg cb \<inter> (mon_c fg ca \<union> mon_ww fg wa) = {}" shows "(ca + cb, w, ca' + cb') \<in> trcl (ntr fg)"
rel_lc_def: fixes lc :: "('c, 'ls) local_config"   and ld :: "('d, 'ls) local_config" shows "rel_lc lc ld \<equiv> rel_c (command lc) (command ld) \<and> local_config.state lc = local_config.state ld"
1: shows "meta_adjunction C D G.F G G.\<phi> G.\<psi>"
arctan_lower_31_def: shows "arctan_lower_31 \<equiv> \<lambda>x. - (pi / 2) - (33 + 170 * x\<^sup>2 + 165 * x ^ 4) * 7 * x / (5 * (5 + 105 * x\<^sup>2 + 315 * x ^ 4 + 231 * x ^ 6))"
rel_refl_strong: fixes x :: "('h, 'i, 'j) generat"   and R1a :: "'h \<Rightarrow> 'h \<Rightarrow> bool"   and R2a :: "'i \<Rightarrow> 'i \<Rightarrow> bool"   and R3a :: "'j \<Rightarrow> 'j \<Rightarrow> bool" assumes "\<And>z1. z1 \<in> generat_pures x \<Longrightarrow> R1a z1 z1"   and "\<And>z2. z2 \<in> generat_outs x \<Longrightarrow> R2a z2 z2"   and "\<And>z3. z3 \<in> generat_conts x \<Longrightarrow> R3a z3 z3" shows "rel_generat R1a R2a R3a x x"
lset_lfilter: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b llist" shows "lset (lfilter P xs) = {x \<in> lset xs. P x}"
lsbisE1: fixes xa :: "'d"   and x :: "'d"   and B1 :: "'d set"   and B2 :: "'e set"   and s1 :: "'d \<Rightarrow> ('f, 'd, 'e) F1"   and s2 :: "'e \<Rightarrow> ('g, 'd, 'e) F2" assumes "(xa, x) \<in> lsbis1 B1 B2 s1 s2" shows "\<exists>z\<in>F1in UNIV (lsbis1 B1 B2 s1 s2) (lsbis2 B1 B2 s1 s2). F1map id fst fst z = s1 xa \<and> F1map id snd snd z = s1 x"
npow_in_prime: fixes P :: "'a set"   and x :: "'a" assumes "prime_ideal R P"   and "x \<in> carrier R"   and "\<exists>n. x^\<^bsup>R n\<^esup> \<in> P" shows "x \<in> P"
Cons_1: fixes P :: "real \<times> real"   and Q :: "real \<times> real"   and Pc :: "real \<times> real" assumes "e \<in> UNIV \<rightarrow> {0<..<1}"   and "ccw'.sortedP 0 (P # Q # Rs)" shows "list_all (\<lambda>a. case a of (xi, xj) \<Rightarrow> ccw' xi xj (Pc + (\<Sum>P\<in>set (P # Q # Rs). e P *\<^sub>R P))) (polychain_of Pc (P # Q # Rs))"
P_def: shows "P = point_measure (Pow S_edges) (\<lambda>x. ennreal (p ^ card x * (1 - p) ^ card (S_edges - x)))"
db\<^sub>s\<^sub>s\<^sub>t_no_upd: fixes A :: "('c, 'd) stateful_strand_step list"   and I :: "'d \<Rightarrow> ('c, 'd) Term.term"   and D :: "(('c, 'd) Term.term \<times> ('c, 'd) Term.term) list" assumes "\<forall>a\<in>set A. \<not> is_Insert a \<and> \<not> is_Delete a" shows "db'\<^sub>s\<^sub>s\<^sub>t A I D = D"
not_disj_deMorgans: fixes P :: "'a upred"   and Q :: "'a upred" shows "(\<not> (P \<or> Q)) = (\<not> P \<and> \<not> Q)"
sepNetsID: fixes p :: "('a set set, 'b) Combinators list" assumes "noDenyAll1 p"   and "separated p" shows "separated (insertDenies p)"
simps_9: fixes x3 :: "'c"   and x2 :: "'b" shows "In3 x3 \<noteq> In2 x2"
split: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "'b \<Rightarrow> 'a"   and f2 :: "'a"   and f3 :: "'a"   and extended :: "'b extended" shows "P (case extended of Fin x \<Rightarrow> f1 x | \<infinity> \<Rightarrow> f2 | -\<infinity> \<Rightarrow> f3) = ((\<forall>x1. extended = Fin x1 \<longrightarrow> P (f1 x1)) \<and> (extended = \<infinity> \<longrightarrow> P f2) \<and> (extended = -\<infinity> \<longrightarrow> P f3))"
rational_root_test_2: fixes p :: "rat poly" assumes "rational_root_test p = None" shows "\<nexists>x. poly p x = 0"
flatten_focus: fixes X :: "'b set"   and p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" shows "flatten (focus X p) = p"
disjoint_family_subset: fixes A :: "'b \<Rightarrow> 'c set"   and B :: "'b \<Rightarrow> 'c set" assumes "disjoint_family A"   and "\<And>x. B x \<subseteq> A x" shows "disjoint_family B"
intros: fixes res' :: "'V \<Rightarrow> ('V \<times> 'W \<times> 'V) list option"   and res :: "'V \<Rightarrow> ('V \<times> 'W \<times> 'V) list option"   and v :: "'V"   and wl :: "'V set" assumes "\<forall>v'\<in>V. res' v' \<in> least_map path_weight' ({res v'} \<union> {Some (p @ [(v, w, v')]) |p w. res v = Some p \<and> (v, w, v') \<in> E})" shows "update_spec v (wl, res) (wl, res')"
assms_2: fixes P :: "'state"   and Q :: "'state" assumes "P \<in> S"   and "P \<sim>\<cdot> Q" shows "Q \<in> S"
domain: fixes A :: "'a set" shows "local.MaxR.MaxR_opt (A \<inter> Field r) = local.MaxR.MaxR_opt A"
Domainp_mod_ring_rel32: fixes p :: "int"   and pp :: "uint32" assumes "mod_ring_locale TYPE('b) p"   and "p = int_of_uint32 pp"   and "p \<le> 65535" shows "Domainp (mod_ring_locale.mod_ring_rel32 p) = (\<lambda>x. 0 \<le> x \<and> x < pp)"
foldl_append: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and a :: "'b"   and xs :: "'c list"   and ys :: "'c list" shows "foldl f a (xs @ ys) = foldl f (foldl f a xs) ys"
le_strict_1: fixes compare :: "'a \<rightarrow> 'a \<rightarrow> Ordering"   and x :: "'a" assumes "class.Ord_strict compare" shows "Ord.le compare\<cdot>\<bottom>\<cdot>x = \<bottom>"
fs_int_mu_d_Z_m_m: fixes n :: "nat"   and fs :: "int vec list"   and j :: "nat"   and ii :: "nat" assumes "fs_int_indpt n fs"   and "j < length fs"   and "ii < length fs" shows "rat_of_int (fs_int.d n fs (Suc j)) * gram_schmidt_fs.\<mu> n (map of_int_hom.vec_hom fs) ii j \<in> \<int>"
sep_conj_sep_impl: fixes P :: "'a \<Rightarrow> bool"   and h :: "'a"   and Q :: "'a \<Rightarrow> bool"   and R :: "'a \<Rightarrow> bool" assumes "P h"   and "\<And>h. (P \<and>* Q) h \<Longrightarrow> R h" shows "(Q \<longrightarrow>* R) h"
Q_def: fixes name :: "name"   and ts :: "value list"   and name' :: "name"   and us :: "value list" shows "Q (Vconstr name ts) (Vconstr name' us) = (name = name' \<and> list_all2 Q ts us)"
length_compl_induct: fixes P :: "'a list \<Rightarrow> bool"   and l :: "'a list" assumes "P []"   and "\<And>e l. (\<And>ll. length ll \<le> length l \<Longrightarrow> P ll) \<Longrightarrow> P (e # l)" shows "P l"
rotate1_map: fixes f :: "'c \<Rightarrow> 'b"   and xs :: "'c list" shows "rotate1 (map f xs) = map f (rotate1 xs)"
STL2_pr: fixes P :: "state \<Rightarrow> bool" shows "\<turnstile> \<box>P \<longrightarrow> Init P \<and> Init P$"
acdb': fixes i :: "'a" assumes "i \<in> Basis"   and "box c d \<subseteq> box a b"   and "\<And>i. i \<in> Basis \<Longrightarrow> c \<bullet> i < d \<bullet> i" shows "a \<bullet> i \<le> c \<bullet> i \<and> d \<bullet> i \<le> b \<bullet> i"
compact_continuous_image: fixes s :: "'e set"   and f :: "'e \<Rightarrow> 'f" assumes "continuous_on s f"   and "compact s" shows "compact (f ` s)"
a_elemrcos_carrier: fixes a :: "'a"   and a' :: "'a" assumes "a \<in> carrier G"   and "a' \<in> H +> a" shows "a' \<in> carrier G"
size_4: fixes x21 :: "'a"   and x22 :: "'a list" shows "|x21 # x22| = |x22| + Suc 0"
left: fixes \<sigma>' :: "'state" assumes "\<exists>\<sigma>1. \<sigma>1 \<in> small P \<sigma> \<and> \<sigma>' \<in> small_nstep P \<sigma>1 (Suc n)" shows "\<sigma>' \<in> small_nstep P \<sigma> (Suc (Suc n))"
empty_def: shows "RBT_Mapping2.empty \<equiv> Mapping_RBT rbt.Empty"
seq_inj: fixes f :: "'b \<Rightarrow> 'b"   and x :: "'b"   and i :: "nat"   and j :: "nat"   and p :: "nat"   and n :: "nat"   and q :: "nat" assumes "fx0.seq' f x i = fx0.seq' f x j"   and "p = i + n"   and "q = j + n" shows "fx0.seq' f x p = fx0.seq' f x q"
fimageI: fixes x :: "'x"   and A :: "'x fset"   and f :: "'x \<Rightarrow> 'y" assumes "x |\<in>| A" shows "f x |\<in>| f |`| A"
Neg: shows "\<forall>x\<in>P. - x \<in> P"
vTeT: fixes v :: "'a"   and e :: "'a" assumes "v \<sqinter> top = v"   and "e \<le> v \<sqinter> - id v" shows "id v \<sqinter> id e = bot"
WN_on_def: fixes r :: "('d \<times> 'd) set"   and A :: "'d set" shows "WN_on r A = (\<forall>a\<in>A. \<exists>b. (a, b) \<in> r\<^sup>!)"
max_prop_2: fixes XS :: "'a list set" assumes "XS \<noteq> {}"   and "finite XS" shows "\<forall>b'\<in>XS. length b' \<le> length (MAX XS)"
lepoll_trans2: fixes A :: "'a set"   and B :: "'b set"   and C :: "'c set" assumes "A \<lesssim> B"   and "B \<approx> C" shows "A \<lesssim> C"
eq1: shows "keys (monomial c t + p) = insert t (keys p)"
word_eq_iff_signed: fixes v :: "'c word"   and w :: "'c word" shows "(v = w) = (signed v = signed w)"
closed_Union: fixes "open" :: "'a set \<Rightarrow> bool"   and S :: "'a set set" assumes "class.topological_space open"   and "finite S"   and "\<forall>T\<in>S. topological_space.closed open T" shows "topological_space.closed open (\<Union> S)"
dgrad_p_set_le_trans: fixes d :: "'a \<Rightarrow> nat"   and F :: "('t \<Rightarrow>\<^sub>0 'f) set"   and G :: "('t \<Rightarrow>\<^sub>0 'f) set"   and H :: "('t \<Rightarrow>\<^sub>0 'f) set" assumes "dgrad_p_set_le d F G"   and "dgrad_p_set_le d G H" shows "dgrad_p_set_le d F H"
sdrop_fair: fixes rs :: "'rule stream"   and m :: "nat" assumes "fair rs" shows "fair (sdrop m rs)"
poly_deg_dehomogenize_le: fixes x :: "'b"   and p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" shows "poly_deg (dehomogenize x p) \<le> poly_deg p"
gcd_of_nat_gauss_int: fixes a :: "nat"   and b :: "nat" shows "gcd (of_nat a) (of_nat b) = of_nat (gcd a b)"
abs_eq: fixes xa :: "'a"   and x :: "'a poly" shows "Polynomial.smult xa x = Abs_poly (\<lambda>n. xa * poly.coeff x n)"
jordan_nf_order: fixes A :: "'b mat"   and n_as :: "(nat \<times> 'b) list"   and a :: "'b" assumes "jordan_nf A n_as" shows "order a (char_poly A) = sum_list (map fst (filter (\<lambda>na. snd na = a) n_as))"
asTerm_equal_imp_alpha: fixes qX :: "('a, 'b, 'varSort, 'var, 'c) qTerm"   and qY :: "('a, 'b, 'varSort, 'var, 'c) qTerm" assumes "qGood qX"   and "asTerm qX = asTerm qY" shows "qX #= qY"
pos_assign_poly: fixes \<alpha> :: "'c \<Rightarrow> 'd"   and p :: "('c monom \<times> 'd) list" assumes "pos_assign \<alpha>"   and "p \<ge>p zero_poly" shows "(0::'d) \<le> eval_poly \<alpha> p"
TESL_interp_unfold_stepwise_kills: fixes master :: "clock"   and slave :: "clock" shows "\<lbrakk> master kills slave \<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L = \<Inter> {Y. \<exists>n. Y = \<lbrakk> master kills slave \<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L\<^bsup>\<ge> n\<^esup>}"
minorM_row_def: fixes A :: "(('c, 'd) vec, 'e) vec"   and i :: "'e"   and j :: "'d"   and k :: "'e" shows "minorM_row A i j k = (\<chi>l. if k = i \<and> l = j then 1::'c else if k = i \<or> l = j then 0::'c else A $ k $ l)"
strict1: fixes y :: "'a state \<times> 'a state \<Rightarrow> 'b" shows "invalid->prepend\<^sub>S\<^sub>e\<^sub>q(y) = invalid"
V_alt: shows "V = fst ` E \<union> snd ` E"