hcju/isabelleps huggingface dataset:
splus_mono_strict_left: fixes s :: "'a"   and t :: "'a"   and v :: "'t" assumes "s \<prec> t" shows "s \<oplus> v \<prec>\<^sub>t t \<oplus> v"
\<Phi>o_in_hom: fixes J :: "'ag \<Rightarrow> 'ag \<Rightarrow> 'ag"   and C :: "'ah \<Rightarrow> 'ah \<Rightarrow> 'ah"   and D :: "'ag \<Rightarrow> 'ah"   and a :: "'ah"   and \<chi> :: "'ag \<Rightarrow> 'ah"   and a' :: "'ah" assumes "limit_cone J C D a \<chi>"   and "partial_magma.ide C a'" shows "partial_magma.in_hom replete_setcat.comp (limit_cone.\<Phi>o J C D a \<chi> a') (set_category_given_img.mkIde replete_setcat.comp (set_category.some_img replete_setcat.comp (cardSuc (cmax |UNIV| natLeq))) (cardSuc (cmax |UNIV| natLeq)) (hom_functor.set C (\<lambda>_. replete_setcat.UP \<circ> Inl) (a', a))) (set_category_given_img.mkIde replete_setcat.comp (set_category.some_img replete_setcat.comp (cardSuc (cmax |UNIV| natLeq))) (cardSuc (cmax |UNIV| natLeq)) ((replete_setcat.UP \<circ> Inr) ` {\<chi>. Limit.cone J C D a' \<chi>}))"
exp_pow_int: fixes x :: "'b"   and n :: "nat" shows "x ^ n = pow_int x (int n)"
natural_isomorphism_axioms: shows "natural_isomorphism VVV.comp (\<cdot>) HoHV HoVH (\<lambda>\<mu>\<nu>\<tau>. \<a>[fst \<mu>\<nu>\<tau>, fst (snd \<mu>\<nu>\<tau>), snd (snd \<mu>\<nu>\<tau>)])"
simps_1: fixes prnt :: "(nat \<times> nat \<times> char list) list"   and p :: "nat"   and slp :: "floatarith list"   and X :: "(real \<times> real pdevs) list list" shows "prove_nonneg prnt 0 p slp X = (let _ = if prnt \<noteq> [] then print STR ''# depth limit exceeded\<newline>'' else () in False)"
subset_basis_in_imp_indep_in: fixes X :: "'a set"   and Y :: "'a set" assumes "basis_in \<E> X"   and "Y \<subseteq> X" shows "indep_in \<E> Y"
less_interval_def: fixes x :: "'b interval"   and y :: "'b interval" shows "(x < y) = (x \<le> y \<and> \<not> y \<le> x)"
exhaust: fixes y :: "state" obtains "y = Undecided"   | x2 :: "final_decision" where "y = Decision x2"
beukers_nn_integral1_different: assumes "s \<le> r"   and "s < r" shows "beukers_nn_integral1 r s = ennreal ((\<Sum>k\<in>{s<..r}. 1 / real (k\<^sup>2)) / real (r - s))"
Cons_1: fixes state :: "State"   and newWl :: "nat list" assumes "InvariantWatchesEl (getF state) (getWatch1 state) (getWatch2 state)"   and "InvariantWatchesDiffer (getF state) (getWatch1 state) (getWatch2 state)"   and "InvariantConsistent (getM state)"   and "InvariantUniq (getM state)"   and "InvariantWatchCharacterization (getF state) (getWatch1 state) (getWatch2 state) M"   and "\<forall>c. c \<in> set Wl' \<longrightarrow> 0 \<le> c \<and> c < length (getF state)"   and "getM state = M @ [(opposite literal, decision)]"   and "uniq Wl'"   and "\<forall>c. c \<in> set Wl' \<longrightarrow> Some literal = getWatch1 state c \<or> Some literal = getWatch2 state c" shows "let state' = notifyWatches_loop literal Wl' newWl state in \<forall>c. c \<in> set Wl' \<longrightarrow> (\<forall>w1 w2. Some w1 = getWatch1 state' c \<and> Some w2 = getWatch2 state' c \<longrightarrow> watchCharacterizationCondition w1 w2 (getM state') (getF state' ! c) \<and> watchCharacterizationCondition w2 w1 (getM state') (getF state' ! c))"
fail_option_def: fixes return :: "'d option \<Rightarrow> 'e" shows "fail_option return = OptionT (return None)"
lagrange_finite: fixes G :: "('a, 'b) monoid_scheme"   and H :: "'a set" assumes "Group.group G"   and "finite (carrier G)"   and "subgroup H G" shows "card (rcosets\<^bsub>G\<^esub> H) * card H = Coset.order G"
him: shows "h ` ({0..1} \<times> T) \<subseteq> U"
rule: fixes wf_md :: "'a prog \<Rightarrow> char list \<Rightarrow> 'a mdecl \<Rightarrow> bool"   and Ca :: "char list"   and M :: "char list"   and b :: "staticb"   and Ts :: "ty list"   and T :: "ty"   and m :: "'a"   and bd :: "'a mdecl" assumes "wf_prog wf_md P"   and "P \<turnstile> Ca sees M, b :  Ts\<rightarrow>T = m in Ca"   and "set Ts \<subseteq> types P"   and "bd = (M, b, Ts, T, m)"   and "A P Ca bd" shows "B P Ca bd"
is_SDG_path_intra_CFG_path: fixes n :: "'node SDG_node"   and ns :: "'node SDG_node list"   and n' :: "'node SDG_node" assumes "n is-ns\<rightarrow>\<^sub>d* n'" obtains as :: "'edge list" where "parent_node n -as\<rightarrow>\<^sub>\<iota>* parent_node n'"
l_one: fixes G :: "('f, 'g) monoid_scheme"   and x :: "'f" assumes "Group.monoid G"   and "x \<in> carrier G" shows "\<one>\<^bsub>G\<^esub> \<otimes>\<^bsub>G\<^esub> x = x"
split_at_eq_val_split_at_tm: fixes n :: "nat"   and xs :: "'a list" shows "Time_Monad.val (split_at_tm n xs) = split_at n xs"
inext_all_le_fix: fixes I :: "nat set"   and n :: "nat" assumes "\<forall>x\<in>I. x \<le> n" shows "inext n I = n"
i: shows "(PROP P) \<Longrightarrow> (PROP Q)"
GrpE: fixes A :: "'d set"   and f :: "'d \<Rightarrow> 'e"   and x :: "'d"   and y :: "'e" assumes "BNF_Def.Grp A f x y" obtains "f x = y" and "x \<in> A"
partial_function_image: fixes ord :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and Lub :: "'a set \<Rightarrow> 'a"   and f :: "'b \<Rightarrow> 'a"   and g :: "'a \<Rightarrow> 'b" assumes "partial_function_definitions ord Lub"   and "\<And>x y. f x = f y \<Longrightarrow> x = y"   and "\<And>x. f (g x) = x" shows "partial_function_definitions (img_ord f ord) (img_lub f g Lub)"
reachable_weaken_TT: fixes s :: "'e"   and A :: "('e, 'f) automaton"   and I :: "'f \<Rightarrow> bool" assumes "s \<in> reachable A I" shows "s \<in> reachable A TT"
L1_Boiler: fixes x :: "nat \<Rightarrow> bit list"   and s :: "nat \<Rightarrow> nat list"   and y :: "nat \<Rightarrow> nat list" assumes "SteamBoiler x s y"   and "ts x" shows "ts s"
convergence_norm_G: fixes x :: "'s"   and y :: "'s"   and z :: "'a" assumes "convergence_G x y z" shows "summable (\<lambda>n. p x y n * norm z ^ n)"
tendsto_mult_filterlim_at_infinity: fixes f :: "'c \<Rightarrow> 'd"   and c :: "'d"   and F :: "'c filter"   and g :: "'c \<Rightarrow> 'd" assumes "(f \<longlongrightarrow> c) F"   and "c \<noteq> (0::'d)"   and "filterlim g at_infinity F" shows "LIM x F. f x * g x :> at_infinity"
sel_3: fixes x21 :: "'a"   and x22 :: "'a list" shows "tl (x21 # x22) = x22"
prefix_eq_prefix_take: fixes xs :: "'b list"   and ys :: "'b list" shows "prefix xs ys = (ys \<down> length xs = xs)"
merge_updates: fixes qs :: "('c \<times> 'd) list"   and ps :: "('c \<times> 'd) list" shows "merge qs ps = updates (rev (map fst ps)) (rev (map snd ps)) qs"
ss_wt_ParT_UNlift01: fixes cl :: "('test, 'atom, 'choice) cmd list"   and dl :: "('test, 'atom, 'choice) cmd list"   and II :: "nat set"   and P :: "nat \<Rightarrow> nat set set"   and F :: "nat \<Rightarrow> nat set \<Rightarrow> nat set"   and s :: "'state"   and t :: "'state" assumes "length cl = length dl"   and "properL cl"   and "properL dl"   and "II \<in> UNpart01 cl dl P - {BrnFT cl dl}"   and "\<And>n. n < length cl \<Longrightarrow> part {..<brn (cl ! n)} (P n) \<and> {} \<notin> P n"   and "\<And>n. n < length dl \<Longrightarrow> part {..<brn (dl ! n)} (F n ` P n)"   and "\<And>n I. \<lbrakk>n < length cl; I \<in> P n\<rbrakk> \<Longrightarrow> sum (wt (cl ! n) s) I = sum (wt (dl ! n) t) (F n I)"   and "s \<approx> t"   and "sum (wt (ParT cl) s) (BrnFT cl dl) < 1"   and "sum (wt (ParT dl) t) (BrnFT dl cl) < 1" shows "sum (wt (ParT cl) s) II / (1 - sum (wt (ParT cl) s) (BrnFT cl dl)) = sum (wt (ParT dl) t) (UNlift01 cl dl P F II) / (1 - sum (wt (ParT dl) t) (BrnFT dl cl))"
implSecure_synth_analz: fixes H :: "msg set"   and A :: "agent"   and B :: "agent"   and M :: "msg" assumes "H \<subseteq> payload \<union> sym.valid \<union> range LtK \<union> Tags"   and "sym.implSecure A B M \<in> synth (analz H)" shows "sym.implSecure A B M \<in> H \<or> M \<in> synth (analz H) \<and> (A, B) \<in> broken H"
subset_insert: fixes x :: "'b"   and A :: "'b set"   and B :: "'b set" assumes "x \<notin> A" shows "order.greater_eq (insert x B) A = order.greater_eq B A"
\<chi>_def: shows "\<chi> \<equiv> substT r t x"
list2FWpolicys_eq_sym: fixes Filter :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list" assumes "Filter \<noteq> []" shows "Cp (list2policyR (rev Filter)) = Cp (list2FWpolicy Filter)"
predict_stoch_proc_def: fixes F :: "nat \<Rightarrow> 'c measure"   and X :: "nat \<Rightarrow> 'c \<Rightarrow> 'd"   and N :: "'d measure" shows "predict_stoch_proc F X N = (X 0 \<in> F 0 \<rightarrow>\<^sub>M N \<and> (\<forall>n. X (Suc n) \<in> F n \<rightarrow>\<^sub>M N))"
focus_zero: fixes X :: "'b set" shows "focus X 0 = 0"
sum_A: shows "(\<lambda>x. \<Sum>\<^sub>ay\<in>B x. norm (f (x, y))) abs_summable_on A"
fixp_above_unfold: fixes a :: "'v \<times> 'v \<Rightarrow> ennreal" assumes "a \<in> Field leq" shows "fixp_above a = trim1 (fixp_above a)"
ereal_all_split: fixes P :: "ereal \<Rightarrow> bool" shows "(\<forall>x. P x) = (P \<infinity> \<and> (\<forall>x. P (ereal x)) \<and> P (- \<infinity>))"
gen_dfs_post_aux: fixes u0 :: "'a"   and x :: "'a"   and E :: "('a \<times> 'a) set"   and y :: "'a"   and S :: "'a set"   and V :: "'a set" assumes "(u0, x) \<in> E"   and "(x, y) \<in> (E - UNIV \<times> insert u0 S)\<^sup>*"   and "S \<subseteq> V"   and "x \<notin> V" shows "(u0, y) \<in> (E - UNIV \<times> S)\<^sup>*"
node_props_eq_node_props_formaldef: shows "BLPtrusted.node_props_formaldef = BLPtrusted.node_props"
ChamberComplexApartmentSystemTriangle_axioms: shows "ChamberComplexApartmentSystemTriangle X \<A> A B B' C D E z"
split_asm: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "'a"   and f2 :: "'b \<Rightarrow> 'b list \<Rightarrow> 'a"   and list :: "'b list" shows "P (case list of [] \<Rightarrow> f1 | x # xa \<Rightarrow> f2 x xa) = (\<not> (list = [] \<and> \<not> P f1 \<or> (\<exists>x21 x22. list = x21 # x22 \<and> \<not> P (f2 x21 x22))))"
HBT_def: fixes m :: "nat" shows "HBT m \<equiv> 0 < m"
ComposeR: fixes T :: "('fun, 'var) Term.term list"   and f :: "'fun"   and M :: "('fun, 'var) Term.term set"   and Q :: "('fun, 'var) Term.term \<Rightarrow> bool" assumes "length T = arity f"   and "public f"   and "\<And>t. t \<in> set T \<Longrightarrow> \<langle>M;Q\<rangle> \<turnstile>\<^sub>r t"   and "Q (Fun f T)" shows "\<langle>M;Q\<rangle> \<turnstile>\<^sub>r Fun f T"
singular_system: fixes a11 :: "'a"   and a12 :: "'a"   and a21 :: "'a"   and a22 :: "'a"   and x0 :: "'a \<times> 'a"   and b1 :: "'a"   and b2 :: "'a"   and x :: "'a \<times> 'a" assumes "det2 a11 a12 a21 a22 = (0::'a)"   and "a11 \<noteq> (0::'a) \<or> a12 \<noteq> (0::'a)"   and "a11 * fst x0 + a12 * snd x0 = b1"   and "a21 * fst x0 + a22 * snd x0 = b2"   and "a11 * fst x + a12 * snd x = b1" shows "a21 * fst x + a22 * snd x = b2"
conjugate_iff_def: fixes f :: "'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a" assumes "residuated f" shows "(f x \<sqinter> y = \<bottom>) = (x \<sqinter> conjugate f y = \<bottom>)"
set_sift_down: fixes l :: "'b tree"   and a :: "'b"   and r :: "'b tree" assumes "braun \<langle>l, a, r\<rangle>" shows "set_tree (sift_down l a r) = {a} \<union> (set_tree l \<union> set_tree r)"
swapFourSyms_def: fixes t :: "'a tree"   and a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" shows "swapFourSyms t a b c d = (if a = d then swapSyms t b c else if b = c then swapSyms t a d else swapSyms (swapSyms t a c) b d)"
assms_3: shows "Y \<in> M \<rightarrow>\<^sub>M N"
disc_12: fixes x2 :: "'b" shows "is_Prim \<^bold>\<langle>x2\<^bold>\<rangle>"
reachable_mono: fixes H :: "('a, 'b) pre_digraph"   and u :: "'a"   and v :: "'a" assumes "u \<rightarrow>\<^sup>*\<^bsub>H\<^esub> v"   and "subgraph H (G \<restriction> S)" shows "u \<rightarrow>\<^sup>*\<^bsub>G \<restriction> S\<^esub> v"
tau_steps_iff_tau_transition: fixes P :: "'state"   and x :: "('idx, 'pred, 'act) formula" shows "(\<exists>n. P \<Turnstile> tau_steps x n) = (\<exists>P'. P \<Rightarrow> P' \<and> P' \<Turnstile> x)"
wset_actions_ok_def: fixes ws :: "'b \<Rightarrow> 'c wait_set_status option"   and t :: "'b"   and was :: "('b, 'c) wait_set_action list" shows "wset_actions_ok ws t was = (if Notified \<in> set was then ws t = \<lfloor>PostWS WSNotified\<rfloor> else if WokenUp \<in> set was then ws t = \<lfloor>PostWS WSWokenUp\<rfloor> else ws t = None)"
gen_new_keys_not_used: fixes K :: "nat"   and evs :: "event list"   and KK :: "nat set" assumes "Key K \<notin> used evs"   and "K \<in> symKeys"   and "evs \<in> set_pur" shows "Key K \<notin> used evs \<longrightarrow> K \<in> symKeys \<longrightarrow> K \<notin> keysFor (parts (Key ` KK \<union> knows Spy evs))"
prems_2: fixes f :: "(nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real"   and x :: "nat \<Rightarrow> real"   and y :: "nat \<Rightarrow> real" assumes "f \<in> Poly_Mapping.keys c"   and "x \<in> standard_simplex (Suc p)"   and "y \<in> standard_simplex (Suc p)" shows "\<bar>f x k - f y k\<bar> \<le> d"
index_unit_vec_1: fixes i :: "nat"   and n :: "nat"   and j :: "nat" assumes "i < n"   and "j < n" shows "unit_vec n i $ j = (if j = i then 1::'a else (0::'a))"
surjective: fixes r :: "('f, 'g, 'h, 'i) CHOAlgorithm_scheme" shows "r = \<lparr>CinitState = CinitState r, sendMsg = sendMsg r, CnextState = CnextState r, \<dots> = CHOAlgorithm.more r\<rparr>"
div_add_self1: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and 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 modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and euclidean_size :: "'b \<Rightarrow> nat"   and b :: "'b"   and a :: "'b" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size"   and "b \<noteq> zero" shows "divide (plus b a) b = plus (divide a b) one"
assms: shows "(\<xi>, p) \<in> \<sigma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V i"
VectorSpaceHomSet_is_fmaps_in_GroupHomSet: fixes smult' :: "'f \<Rightarrow> 'a \<Rightarrow> 'a"   and W :: "'a set" shows "VectorSpaceHomSet (\<cdot>) V smult' W = GroupHomSet V W \<inter> {T. \<forall>a. \<forall>v\<in>V. T (a \<cdot> v) = smult' a (T v)}"
tseytin_tran1: fixes fresh :: "'a set \<Rightarrow> 'a"   and G :: "'a formula"   and F :: "'a formula"   and H :: "'a formula"   and S :: "'a formula \<Rightarrow> 'a formula"   and \<A> :: "'a \<Rightarrow> bool" assumes "freshstuff_sema fresh"   and "G \<in> set (subformulae F)"   and "H \<in> set (freshnats.tseytin_tran1 S G)"   and "\<forall>J\<in>set (subformulae F). freshstuff_sema.tseytin_update fresh \<A> F \<Turnstile> J = freshstuff_sema.tseytin_update fresh \<A> F \<Turnstile> S J" shows "freshstuff_sema.tseytin_update fresh \<A> F \<Turnstile> H"
intros_1: fixes xs :: "'c option list"   and P :: "'c option process"   and X :: "'c option set"   and Q :: "'c option process" assumes "xs \<notin> sentences P"   and "(xs, X) \<in> failures P"   and "None \<notin> set xs" shows "(xs, X) \<in> seq_comp_failures P Q"
median_le_5_in_set: fixes xs :: "'b list" assumes "xs \<noteq> []"   and "length xs \<le> 5" shows "median_le_5 xs \<in> set xs"
mkChannels_def: fixes g :: "'a gState_scheme"   and p :: "pState"   and cs :: "channel list" shows "mkChannels g p cs = (if cs = [] then (g, p) else let l = length (gState.channels g) in if max_channels < l + length cs then abort STR ''Too much channels'' (\<lambda>_. (g, p)) else let cs\<^sub>p = map integer_of_nat [l..<l + length cs]; g' = g\<lparr>gState.channels := gState.channels g @ cs\<rparr> in Let (p\<lparr>pState.channels := pState.channels p @ cs\<^sub>p\<rparr>) (Pair g'))"
sublinear_wp_DC: fixes x :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and y :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" assumes "sublinear (wp x)"   and "sublinear (wp y)" shows "sublinear (wp (x \<Sqinter> y))"
fundchamber_W_image_chamber: fixes w :: "'a permutation" assumes "w \<in> \<langle>\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g}\<rangle>" shows "chamber (w `\<rightarrow> C0)"
extendID: fixes S :: "'b sequent"   and i :: "nat"   and \<Gamma> :: "'b form multiset"   and \<Delta> :: "'b form multiset" assumes "extend S ( \<LM> At i  \<RM> \<Rightarrow>* \<LM> At i  \<RM>) = ( \<Gamma> \<Rightarrow>* \<Delta>)" shows "At i \<in># \<Gamma> \<and> At i \<in># \<Delta>"
iarray_length_of_fun: fixes f :: "nat \<Rightarrow> 'a"   and n :: "nat" shows "IArray.length (IArray.of_fun f n) = n"
opp_powsum_def: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and m :: "nat"   and n :: "nat" assumes "class.dioid_one_zero plus times one zero less_eq less" shows "dioid_one_zero.powsum plus (times.opp_mult times) one zero x m n = dioid_one_zero.powsum plus times one zero x m n"
vcgaIsAllocationAllocatingGoodsOnly: fixes \<Omega> :: "'c list"   and N :: "integer set"   and b :: "integer \<times> 'c set \<Rightarrow> 'd"   and r :: "integer" assumes "distinct \<Omega>"   and "set \<Omega> \<noteq> {}"   and "finite N" shows "Outside' {seller} (the_elem (argmax (sum (summedBidVector (pseudoAllocation (randomEl (takeAll (\<lambda>x. winningAllocationRel (N \<union> {seller}) (set \<Omega>) ((\<in>) x) b) (allAllocationsAlg (N \<union> {seller}) \<Omega>)) r) <| ((N \<union> {seller}) \<times> finestpart (set \<Omega>))) (N \<union> {seller}) (set \<Omega>))) (maximalStrictAllocations N (set \<Omega>) b))) \<in> allocationsUniverse \<and> \<Union> (Range (Outside' {seller} (the_elem (argmax (sum (summedBidVector (pseudoAllocation (randomEl (takeAll (\<lambda>x. winningAllocationRel (N \<union> {seller}) (set \<Omega>) ((\<in>) x) b) (allAllocationsAlg (N \<union> {seller}) \<Omega>)) r) <| ((N \<union> {seller}) \<times> finestpart (set \<Omega>))) (N \<union> {seller}) (set \<Omega>))) (maximalStrictAllocations N (set \<Omega>) b))))) \<subseteq> set \<Omega>"
set_update_distinct: fixes xs :: "'a list"   and n :: "nat"   and x :: "'a" assumes "distinct xs"   and "n < length xs" shows "set (xs[n := x]) = insert x (set xs - {xs ! n})"
hyps_2: fixes V :: "char list" assumes "(l(V__ \<mapsto> w')) V = \<lfloor>v\<rfloor>"   and "(E(V__ \<mapsto> T')) V = \<lfloor>T\<rfloor>"   and "P \<turnstile> T casts v'' to v "   and "l' V = \<lfloor>v'\<rfloor>" shows "P \<turnstile> T casts v' to v' "
path_2: fixes G :: "('c, 'd) Graph_scheme"   and v :: "'c"   and w :: "'c" assumes "Graph G"   and "v \<rightarrow>\<^bsub>G\<^esub> w"   and "v \<noteq> w" shows "Graph.walk G [v, w] \<and> distinct [v, w]"
count_mset_set_1: fixes A :: "'a set"   and x :: "'a" assumes "finite A"   and "x \<in> A" shows "count (mset_set A) x = 1"
invariance_of_dimension_Euclidean_space: fixes m :: "nat"   and n :: "nat" shows "(Euclidean_space m homeomorphic_space Euclidean_space n) = (m = n)"
fair_implies_Liminf_saturated: fixes Ns :: "('a literal multiset \<times> label) set llist" assumes "chain (\<rhd>L) Ns"   and "FL.empty_ord.fair Ns" shows "FL.empty_ord.saturated (Liminf_llist Ns)"
new_keys_not_analzd: fixes K :: "nat"   and evs :: "event list" assumes "K \<in> symKeys"   and "evs \<in> yahalom"   and "Key K \<notin> used evs" shows "K \<notin> keysFor (analz (knows Spy evs))"
rel_F_pos_distr_cond_eq: fixes tytok :: "('v \<times> 'w \<times> 'x \<times> 'y \<times> 'z \<times> 'aa \<times> 'ab \<times> 'ac \<times> 'ad \<times> 'ae) itself" shows "rel_F_pos_distr_cond (=) (=) (=) (=) (=) (=) (=) (=) (=) (=) (=) (=) tytok"
bind_map_pmf: fixes f :: "'e \<Rightarrow> 'd"   and M :: "'e pmf"   and g :: "'d \<Rightarrow> 'b pmf" shows "map_pmf f M \<bind> g = M \<bind> (\<lambda>x. g (f x))"
assms_1: shows "\<gamma> C1_differentiable_on {0..1}"
simps_1: fixes carrier :: "'c set"   and more :: "'d" shows "carrier \<lparr>carrier = carrier, \<dots> = more\<rparr> = carrier"
borel_measurable_times: fixes f :: "'b \<Rightarrow> 'c"   and M :: "'b measure"   and g :: "'b \<Rightarrow> 'c" assumes "f \<in> borel_measurable M"   and "g \<in> borel_measurable M" shows "(\<lambda>x. f x * g x) \<in> borel_measurable M"
enc_wf_interp: fixes I :: "(nat + nat set) list"   and \<phi> :: "'a formula"   and w :: "'a list" assumes "wf_formula (length I) \<phi>"   and "wf_interp_for_formula (w, I) \<phi>" shows "wf_interp_for_formula (dec_word (enc (w, I)), dec_interp (length I) (FOV \<phi>) (enc (w, I))) \<phi>"
shadow_root_ptr_kinds_eq_h: shows "shadow_root_ptr_kinds h2 = shadow_root_ptr_kinds h"
IH: fixes c :: "nat \<Rightarrow> 'a" assumes "set Xs \<subseteq> carrier_vec n" shows "vec_first (n.lincomb_list c Xs) m = m.lincomb_list c (map (\<lambda>v. vec_first v m) Xs)"
assms_3: fixes x :: "nat \<Rightarrow> 'a" shows "A x \<in> carrier_mat m n"
extensional0_0: fixes X :: "'c set" shows "extensional0 X 0"
iadd_iless_mono: fixes i :: "enat"   and j :: "enat"   and k :: "enat"   and l :: "enat" assumes "i < j"   and "k < l" shows "i + k < j + l"
lift_prime_elem_poly: fixes c :: "'b" assumes "prime_elem c" shows "prime_elem [:c:]"
isomorphicE: fixes a :: "'a"   and a' :: "'a" assumes "Left_a.isomorphic a a'" obtains f :: "'a" where "Left_a.in_hom f a a' \<and> Left_a.iso f"
landau_function_family_chain_def: fixes F :: "'a filter"   and gs :: "'b list"   and get_fun :: "'b \<Rightarrow> 'a \<Rightarrow> real" shows "landau_function_family_chain F gs get_fun \<equiv> F \<noteq> bot \<and> (\<forall>g. g \<in> set (map get_fun gs) \<longrightarrow> filterlim g at_top F) \<and> landau_dominating_chain F (map get_fun gs)"
C: shows "(\<lambda>k. of_nat k powr z * fact k / pochhammer z (k + 1)) \<longlonglongrightarrow> Gamma z"
in_set_polychain_of_imp_sum_list: fixes z :: "'a \<times> 'a"   and Pc :: "'a"   and Ps :: "'a list" assumes "z \<in> set (polychain_of Pc Ps)" obtains d :: "nat" where "z = (Pc + sum_list (take d Ps), Pc + sum_list (take (Suc d) Ps))"
lleq_head: fixes L2 :: "char list"   and L1 :: "char list"   and x :: "char"   and w :: "char list" assumes "\<And>z. lleq_charlist L2 z \<Longrightarrow> lleq_charlist L1 z"   and "lleq_charlist L1 L2"   and "lleq_charlist (x # L2) w" shows "lleq_charlist (x # L1) w"
sup_compl_top_left1: fixes x :: "'a"   and y :: "'a" shows "- x \<squnion> (x \<squnion> y) = \<top>"
assms_4: shows "x \<in> circline_set H \<inter> circline_set x_axis"
Gromov_completion_rel_const: fixes x :: "'a" shows "Gromov_completion_rel (\<lambda>n. x) (\<lambda>n. x)"
count_repeat_mset: fixes i :: "nat"   and A :: "'k multiset"   and a :: "'k" shows "count (repeat_mset i A) a = i * count A a"
normalize_match_preserves_normalized_n_primitive: fixes disc_sel :: "('b \<Rightarrow> bool) \<times> ('b \<Rightarrow> 'c)"   and f :: "'c \<Rightarrow> bool"   and rst :: "'b match_expr" assumes "normalized_n_primitive disc_sel f rst" shows "\<forall>m\<in>set (normalize_match rst). normalized_n_primitive disc_sel f m"
assms: shows "(\<Union>x\<in>set S. case x of receive\<langle>x\<rangle> \<Rightarrow> fv x | \<langle>assign: x \<doteq> t'\<rangle> \<Rightarrow> fv t' | \<langle>check: x \<doteq> t'\<rangle> \<Rightarrow> {} | insert\<langle>t,t'\<rangle> \<Rightarrow> fv t \<union> fv t' | _ \<Rightarrow> {}) \<subseteq> V"
llast_llast_body: shows "llast = fix\<cdot>llast_body"
term_cases: fixes t :: "'c" obtains name :: "name" where "t = free namea__"   | name :: "name" where "t = const name"   | u\<^sub>1 :: "'c" and u\<^sub>2 :: "'c" where "t = app u\<^sub>1 u\<^sub>2"   | "unfree t = None" and "unapp t = None" and "unconst t = None"
mono_mndet_FD_set: fixes A :: "'a set"   and B :: "'a set"   and P :: "'a \<Rightarrow> 'a process" assumes "A \<noteq> {}"   and "A \<subseteq> B" shows "mndet B P \<sqsubseteq>\<^sub>F\<^sub>D mndet A P"
ipurge_tr_rev_ipurge_tr_sources_aux_1: fixes D :: "'d \<Rightarrow> 'c"   and ys :: "'d list"   and I :: "('c \<times> 'c) set"   and U :: "'c set"   and zs :: "'d list"   and xs :: "'d list" assumes "\<not> (\<exists>v\<in>D ` set ys. \<exists>u\<in>sources_aux I D U zs. (v, u) \<in> I)" shows "ipurge_tr_rev_aux I D U (xs @ ys @ zs) = ipurge_tr_rev_aux I D U (xs @ ipurge_tr_aux I D (D ` set ys) zs)"
gc_invar_def: fixes Ns :: "('f \<times> 'l) set llist"   and i :: "enat" shows "gc_invar Ns i = (Inf_from (active_subset (Liminf_upto_llist Ns i)) \<subseteq> Sup_upto_llist (lmap Red_I_\<G> Ns) i)"
inv_image_def: fixes r :: "('b \<times> 'b) set"   and f :: "'a \<Rightarrow> 'b" shows "inv_image r f = {(x, y). (f x, f y) \<in> r}"