hcju/isabelleps huggingface dataset:
path_Entry_target_2: fixes prog :: "cmd"   and n :: "w_node"   and as :: "(w_node \<times> (char list \<Rightarrow> val option) edge_kind \<times> w_node) list" assumes "While_CFG.path prog n as (_Entry_)" shows "as = []"
Df_Cons: fixes a :: "'a"   and i :: "nat" assumes "a \<in> closed_segment X (X + H)"   and "i < n" shows "((\<lambda>a. Df a i H) has_derivative Df a (Suc i)) (at a within G)"
acomplete_sym: fixes l :: "'b tree"   and x :: "'b"   and r :: "'b tree"   and y :: "'b" assumes "acomplete \<langle>l, x, r\<rangle>" shows "acomplete \<langle>r, y, l\<rangle>"
sign_changes_Cons_Cons_0: fixes x :: "'a"   and xs :: "'a list" shows "sign_changes (x # (0::'a) # xs) = sign_changes (x # xs)"
valid_Entry_path_ascending_path: fixes as :: "'edge list"   and n :: "'node" assumes "(_Entry_) -as\<rightarrow>\<^sub>\<surd>* n" obtains as' :: "'edge list" where "(_Entry_) -as'\<rightarrow>\<^sub>\<surd>* n" and "set (sourcenodes as') \<subseteq> set (sourcenodes as)" and "\<forall>a'\<in>set as'. intra_kind (kind a') \<or> (\<exists>Q r p fs. kind a' = Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs)"
crit_pair_zero: fixes q :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'a" shows "fst (punit.crit_pair 0 q) = 0"
action_proj_inter_i: fixes V :: "'d set"   and W :: "'d set"   and f :: "('d, 'e) fmap" shows "fmrestrict_set V (fmrestrict_set W f) = fmrestrict_set (V \<inter> W) f"
subspace_sum: fixes A :: "('b \<times> 'c) set"   and B :: "'d set"   and f :: "'d \<Rightarrow> 'b \<times> 'c" assumes "p.subspace A"   and "\<And>x. x \<in> B \<Longrightarrow> f x \<in> A" shows "sum f B \<in> A"
simps_1: fixes F :: "'f set" shows "strip_guards F Skip = Skip"
fun_apps_Nil: fixes s :: "('a, 'b) tm" shows "fun (apps s []) = fun s"
igWlsBinp_defs_1: fixes MOD :: "('c, 'd, 'varSort, 'sort, 'e, 'f, 'g, 'h, 'i) model_scheme"   and delta :: "'opSym"   and binp :: "'bindex \<Rightarrow> 'h option" shows "igWlsBinp MOD delta binp \<equiv> wlsOpS delta \<and> sameDom (barOf delta) binp \<and> liftAll2 (igWlsAbs MOD) (barOf delta) binp"
evalS_def: fixes M :: "model"   and phi :: "vbl \<Rightarrow> object"   and fs :: "formula list" shows "evalS M phi fs = (\<exists>f\<in>set fs. evalF M phi f = True)"
sorted_upt: fixes m :: "nat"   and n :: "nat" shows "sorted [m..<n]"
marked_deletions_def: shows "marked_deletions = (\<lambda>s. \<forall>x. x \<in> set (sys_mem_store_buffers (mutator m) s) \<longrightarrow> marked_deletion x s)"
that: fixes d :: "real" assumes "0 < d"   and "\<And>x'. \<lbrakk>x' \<in> s; dist x' x \<le> d\<rbrakk> \<Longrightarrow> dist (f x') (f x) < e" shows "thesis"
naturality: fixes \<mu> :: "'a" assumes "C.arr \<mu>" shows "\<tau>\<^sub>1 (C.cod \<mu>) \<cdot>\<^sub>D (G \<mu> \<star>\<^sub>D \<tau>\<^sub>0 (src\<^sub>C \<mu>)) = (\<tau>\<^sub>0 (trg\<^sub>C \<mu>) \<star>\<^sub>D F \<mu>) \<cdot>\<^sub>D \<tau>\<^sub>1 (C.dom \<mu>)"
zip_eq_Nil_iff: fixes xs :: "'f list"   and ys :: "'g list" shows "(zip xs ys = []) = (xs = [] \<or> ys = [])"
UP_pre_univ_prop_axioms: shows "UP_pre_univ_prop R S h"
divide_le_eq: fixes divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and inverse :: "'d \<Rightarrow> 'd"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sgn :: "'d \<Rightarrow> 'd"   and abs :: "'d \<Rightarrow> 'd"   and b :: "'d"   and c :: "'d"   and a :: "'d" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "less_eq (divide b c) a = (if less zero c then less_eq b (times a c) else if less c zero then less_eq (times a c) b else less_eq zero a)"
lincomb_index: fixes i :: "nat"   and X :: "'a vec set"   and a :: "'a vec \<Rightarrow> 'a" assumes "i < n + 1"   and "X \<subseteq> carrier_vec (n + 1)" shows "next_dim.lincomb a X $ i = (\<Sum>x\<in>X. a x * x $ i)"
is_functor: shows "functor J (\<cdot>) \<chi>f.A.map"
tr_tfr: fixes A' :: "('fun, 'var) strand_step list"   and A :: "('fun, 'var) stateful_strand_step list" assumes "A' \<in> set (tr A [])"   and "tfr\<^sub>s\<^sub>s\<^sub>t A"   and "fv\<^sub>s\<^sub>s\<^sub>t A \<inter> bvars\<^sub>s\<^sub>s\<^sub>t A = {}" shows "tfr\<^sub>s\<^sub>t A'"
weakly_fair_unless_until: fixes enabled :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and taken :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" shows "(\<lambda>s. weakly_fair enabled taken s \<and> (\<box>(\<lambda>s. enabled s \<longrightarrow> (enabled \<W> taken) s)) s) = \<box>(\<lambda>s. enabled s \<longrightarrow> (enabled \<U> taken) s)"
runit'_simps_5: fixes f :: "'a" assumes "ide f" shows "cod \<r>\<^sup>-\<^sup>1[f] = f \<star> src f"
simps_6: fixes P :: "jvm_prog"   and h :: "heap"   and stk :: "val list"   and loc :: "val list"   and C :: "char list"   and M :: "char list"   and pc :: "nat"   and frs :: "frame list"   and sh :: "sheap" shows "exec_step P h stk loc C M pc (Called []) frs sh = exec_instr (instrs_of P C M ! pc) P h stk loc C M pc (Called []) frs sh"
elims: fixes x :: "'c arctic_delta"   and xa :: "'c arctic_delta"   and y :: "'c arctic_delta" assumes "x * xa = y" obtains ya :: "'c arctic_delta" where "x = MinInfty_delta" and "xa = ya__" and "y = MinInfty_delta"   | v :: "'c" where "x = Num_arc_delta v" and "xa = MinInfty_delta" and "y = MinInfty_delta"   | xb :: "'c" and ya :: "'c" where "x = Num_arc_delta xb" and "xa = Num_arc_delta ya" and "y = Num_arc_delta (xb + ya)"
chop_reduce: fixes n :: "nat"   and xs :: "'a list" assumes "0 < n"   and "xs \<noteq> []" shows "chop n xs = take n xs # chop n (drop n xs)"
has_derivative_transform_eventually: fixes f :: "'c \<Rightarrow> 'd"   and f' :: "'c \<Rightarrow> 'd"   and x :: "'c"   and s :: "'c set"   and g :: "'c \<Rightarrow> 'd" assumes "(f has_derivative f') (at x within s)"   and "\<forall>\<^sub>F x' in at x within s. f x' = g x'"   and "f x = g x"   and "x \<in> s" shows "(g has_derivative f') (at x within s)"
E_wfD_1: fixes v :: "'v"   and v' :: "'v" assumes "(v, v') \<in> E" shows "v \<in> V"
last_ConsL: fixes xs :: "'a list"   and x :: "'a" assumes "xs = []" shows "last (x # xs) = x"
bounded_iff: fixes b :: "'e multiset"   and c :: "'e multiset"   and a :: "'e multiset" shows "(b \<union># c \<subseteq># a) = (b \<subseteq># a \<and> c \<subseteq># a)"
simps_2: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" assumes "monoidal_language C" shows "monoidal_language.Dom C \<^bold>\<I> = \<^bold>\<I>"
CFG_call_determ: fixes a :: "'edge"   and Q :: "('var \<Rightarrow> 'val option) \<times> 'ret \<Rightarrow> bool"   and r :: "'ret"   and p :: "'pname"   and fs :: "(('var \<Rightarrow> 'val option) \<Rightarrow> 'val option) list"   and a' :: "'edge"   and Q' :: "('var \<Rightarrow> 'val option) \<times> 'ret \<Rightarrow> bool"   and r' :: "'ret"   and p' :: "'pname"   and fs' :: "(('var \<Rightarrow> 'val option) \<Rightarrow> 'val option) list"   and s :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" assumes "valid_edge a"   and "kind a = Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs"   and "valid_edge a'"   and "kind a' = Q':r'\<hookrightarrow>\<^bsub>p'\<^esub>fs'"   and "sourcenode a = sourcenode a'"   and "pred (kind a) s"   and "pred (kind a') s" shows "a = a'"
conflictClauseIsDecreasedByExplain_1: fixes stateA :: "State"   and stateB :: "State" assumes "appliedExplain stateA stateB" shows "getM stateA = getM stateB"
not_fresh_floatariths_map: fixes f :: "'a \<Rightarrow> floatarith"   and xs :: "'a list"   and i :: "nat" shows "(\<not> fresh_floatariths (map f xs) i) = (\<exists>x\<in>set xs. \<not> fresh_floatarith (f x) i)"
assms: shows "b \<in> arcs H \<inter> G.face_cycle_set a'"
a_star_AndL2: fixes M :: "trm"   and M' :: "trm"   and x :: "name"   and y :: "name" assumes "M \<longrightarrow>\<^sub>a* M'" shows "AndL2 x.M y \<longrightarrow>\<^sub>a* AndL2 x.M' y"
assms_3: shows "(c, s) \<rightarrow>c (c', s')"
prv_inj_suc: fixes t :: "'trm"   and t' :: "'trm" assumes "t \<in> atrm"   and "t' \<in> atrm" shows "prv (imp (eql (suc t) (suc t')) (eql t t'))"
image_Collect_subsetI: fixes P :: "'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c"   and B :: "'c set" assumes "\<And>x. P x \<Longrightarrow> f x \<in> B" shows "\<P> f {x. P x} \<subseteq> B"
assms_3: shows "matcher_agree_on_exact_matches \<gamma> common_matcher"
zero_le_ceiling: fixes x :: "'b" shows "(0 \<le> \<lceil>x\<rceil>) = (- (1::'b) < x)"
SourceExistsDSource_level: fixes S :: "CSet"   and i :: "AbstrLevelsID"   and C :: "CSet" assumes "S \<in> Sources i C" shows "\<exists>Z\<in>AbstrLevel i. S \<in> DSources i Z"
X: shows "distributed M MX X (\<lambda>x. ennreal (Px x))"
min_wpl_simps_1: fixes j :: "int"   and i :: "int"   and w :: "int \<Rightarrow> int \<Rightarrow> nat" assumes "j < i" shows "Optimal_BST.min_wpl w i j = 0"
sum_set_upt_conv_sum_list_nat: fixes f :: "nat \<Rightarrow> 'a"   and m :: "nat"   and n :: "nat" shows "sum f (set [m..<n]) = sum_list (map f [m..<n])"
it_to_it_fold: fixes it :: "('c list \<Rightarrow> bool) \<Rightarrow> ('c \<Rightarrow> 'c list \<Rightarrow> 'c list) \<Rightarrow> 'c list \<Rightarrow> 'c list"   and it' :: "('d \<Rightarrow> bool) \<Rightarrow> ('c \<Rightarrow> 'd \<Rightarrow> 'd) \<Rightarrow> 'd \<Rightarrow> 'd" assumes "proper_it it it'" shows "it_to_it it = it'"
separated_ExcessNotAtRecList: fixes g :: "graph"   and E :: "nat \<Rightarrow> nat"   and ps :: "(nat \<times> nat) list" assumes "minGraphProps g"   and "final g"   and "isTable E (vertices g) ps" shows "separated g (set (ExcessNotAtRecList ps g))"
UN_extend_simps_9: fixes C :: "'e \<Rightarrow> 'c set"   and B :: "'d \<Rightarrow> 'e set"   and A :: "'d set" shows "(\<Union>x\<in>A. \<Union> (C ` B x)) = \<Union> (C ` \<Union> (B ` A))"
Sup_nres_def: fixes X :: "'a nres set" shows "Sup X \<equiv> if FAILi \<in> X then FAILi else RES (\<Union> {x. RES x \<in> X})"
float_cases: shows "is_nan a \<or> is_infinity a \<or> is_normal a \<or> is_denormal a \<or> is_zero a"
le_oLimit: fixes f :: "nat \<Rightarrow> ordinal"   and n :: "nat" shows "f n \<le> oLimit f"
monom_inv_Cons: fixes x :: "'a"   and p :: "nat"   and m :: "('a \<times> nat) list"   and y :: "'a" assumes "monom_inv ((x, p) # m)"   and "y \<le> x" shows "y \<notin> fst ` set m"
induct: fixes P :: "nat \<Rightarrow> bool"   and nat :: "nat" assumes "P 0"   and "\<And>nat. P nat \<Longrightarrow> P (Suc nat)" shows "P nat"
alw_shift: fixes \<phi> :: "'b stream \<Rightarrow> bool"   and xl :: "'b list"   and xs :: "'b stream" assumes "alw \<phi> (xl @- xs)" shows "alw \<phi> xs"
Count_def: shows "Count \<equiv> \<lbrace>\<acute>Proper \<and> Roots \<subseteq> Blacks \<acute>M \<and> \<acute>obc \<subseteq> Blacks \<acute>Ma \<and> Blacks \<acute>Ma \<subseteq> Blacks \<acute>M \<and> \<acute>bc \<subseteq> Blacks \<acute>M \<and> length \<acute>Ma = length \<acute>M \<and> (\<acute>obc \<subset> Blacks \<acute>Ma \<or> \<acute>Safe) \<and> \<acute>bc = {}\<rbrace> \<acute>ind := 0;; \<lbrace>\<acute>Proper \<and> Roots \<subseteq> Blacks \<acute>M \<and> \<acute>obc \<subseteq> Blacks \<acute>Ma \<and> Blacks \<acute>Ma \<subseteq> Blacks \<acute>M \<and> \<acute>bc \<subseteq> Blacks \<acute>M \<and> length \<acute>Ma = length \<acute>M \<and> (\<acute>obc \<subset> Blacks \<acute>Ma \<or> \<acute>Safe) \<and> \<acute>bc = {} \<and> \<acute>ind = 0\<rbrace>  WHILE \<acute>ind < length \<acute>M INV \<lbrace>\<acute>Proper \<and> Roots \<subseteq> Blacks \<acute>M \<and> \<acute>obc \<subseteq> Blacks \<acute>Ma \<and> Blacks \<acute>Ma \<subseteq> Blacks \<acute>M \<and> \<acute>bc \<subseteq> Blacks \<acute>M \<and> length \<acute>Ma = length \<acute>M \<and> \<acute>CountInv \<acute>ind \<and> (\<acute>obc \<subset> Blacks \<acute>Ma \<or> \<acute>Safe) \<and> \<acute>ind \<le> length \<acute>M\<rbrace>  DO \<lbrace>\<acute>Proper \<and> Roots \<subseteq> Blacks \<acute>M \<and> \<acute>obc \<subseteq> Blacks \<acute>Ma \<and> Blacks \<acute>Ma \<subseteq> Blacks \<acute>M \<and> \<acute>bc \<subseteq> Blacks \<acute>M \<and> length \<acute>Ma = length \<acute>M \<and> \<acute>CountInv \<acute>ind \<and> (\<acute>obc \<subset> Blacks \<acute>Ma \<or> \<acute>Safe) \<and> \<acute>ind < length \<acute>M\<rbrace>  IF \<acute>M ! \<acute>ind = Black THEN \<lbrace>\<acute>Proper \<and> Roots \<subseteq> Blacks \<acute>M \<and> \<acute>obc \<subseteq> Blacks \<acute>Ma \<and> Blacks \<acute>Ma \<subseteq> Blacks \<acute>M \<and> \<acute>bc \<subseteq> Blacks \<acute>M \<and> length \<acute>Ma = length \<acute>M \<and> \<acute>CountInv \<acute>ind \<and> (\<acute>obc \<subset> Blacks \<acute>Ma \<or> \<acute>Safe) \<and> \<acute>ind < length \<acute>M \<and> \<acute>M ! \<acute>ind = Black\<rbrace> \<acute>bc := insert \<acute>ind \<acute>bc FI;; \<lbrace>\<acute>Proper \<and> Roots \<subseteq> Blacks \<acute>M \<and> \<acute>obc \<subseteq> Blacks \<acute>Ma \<and> Blacks \<acute>Ma \<subseteq> Blacks \<acute>M \<and> \<acute>bc \<subseteq> Blacks \<acute>M \<and> length \<acute>Ma = length \<acute>M \<and> \<acute>CountInv (\<acute>ind + 1) \<and> (\<acute>obc \<subset> Blacks \<acute>Ma \<or> \<acute>Safe) \<and> \<acute>ind < length \<acute>M\<rbrace> \<acute>ind := \<acute>ind + 1 OD"
integrable_inner_left: fixes c :: "'c"   and M :: "'d measure"   and f :: "'d \<Rightarrow> 'c" assumes "c \<noteq> (0::'c) \<Longrightarrow> integrable M f" shows "integrable M (\<lambda>x. f x \<bullet> c)"
or_foldl_sem: fixes \<Gamma> :: "('a, 'b, 'c) formula list"   and \<phi> :: "('a, 'b, 'c) formula"   and \<nu> :: "(real, 'c) vec \<times> (real, 'c) vec"   and I :: "('a, 'b, 'c) interp" assumes "List.member \<Gamma> \<phi>"   and "\<nu> \<in> fml_sem I \<phi>" shows "\<nu> \<in> fml_sem I (foldr (||) \<Gamma> FF)"
less_2: fixes xs :: "'a list"   and x :: "'a"   and ys :: "'a list"   and x' :: "'a"   and ys' :: "'a list" assumes "ls = xs @ [x] @ ys"   and "ls' = xs @ [x'] @ ys'"   and "x \<noteq> x'" shows "thesis"
wf_prog_compPI: fixes P :: "'a prog"   and wf\<^sub>1 :: "'a prog \<Rightarrow> char list \<Rightarrow> 'a mdecl \<Rightarrow> bool"   and wf\<^sub>2 :: "'b prog \<Rightarrow> char list \<Rightarrow> 'b mdecl \<Rightarrow> bool"   and f :: "staticb \<Rightarrow> 'a \<Rightarrow> 'b" assumes "\<And>C M b Ts T m. \<lbrakk>P \<turnstile> C sees M, b :  Ts\<rightarrow>T = m in C; wf_mdecl wf\<^sub>1 P C (M, b, Ts, T, m)\<rbrakk> \<Longrightarrow> wf_mdecl wf\<^sub>2 (compP f P) C (M, b, Ts, T, f b m)"   and "wf_prog wf\<^sub>1 P" shows "wf_prog wf\<^sub>2 (compP f P)"
Compl_partition: fixes A :: "'a set" shows "A \<union> - A = UNIV"
update_error_return: fixes m :: "'c + 'b"   and c :: "'c \<Rightarrow> 'a"   and x :: "'b" shows "((try m catch (\<lambda>x. Inl (c x))) = Inr x) = (m = Inr x)"
simps: fixes p :: "'a Point"   and l :: "'a Line" shows "inLine p l = collinear p (basepoint l) (basepoint l \<leadsto> direction l)"
Rep_inverse: fixes x :: "'b word" shows "of_bl (to_bl x) = x"
axioms_1: fixes x :: "'d \<Longrightarrow> 'e" assumes "wb_lens x" shows "weak_lens x"
continuous_attains_sup: fixes s :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "compact s"   and "s \<noteq> {}"   and "continuous_on s f" shows "\<exists>x\<in>s. \<forall>y\<in>s. f y \<le> f x"
security_P2: fixes m0 :: "'grp"   and \<eta> :: "nat"   and m1 :: "'grp"   and \<sigma> :: "bool" assumes "m0 \<in> carrier (\<G> \<eta>)"   and "m1 \<in> carrier (\<G> \<eta>)" shows "sim_def.perfect_sec_P2 \<eta> (m0, m1) \<sigma>"
monom_mult_unfold_1: fixes n :: "nat"   and f :: "'b poly" shows "monom (1::'b) n * f = monom_mult n f"
lim_0: fixes R :: "'d \<Rightarrow> 'd \<Rightarrow> real"   and t :: "real"   and s :: "'d" assumes "transition_rates R" shows "discrete_Markov_process.lim_stream (borel \<Otimes>\<^sub>M count_space UNIV) (transition_rates.K R) (t, s) = distr (discrete_Markov_process.lim_stream (borel \<Otimes>\<^sub>M count_space UNIV) (transition_rates.K R) (0, s)) (stream_space (borel \<Otimes>\<^sub>M count_space UNIV)) (smap (\<lambda>(t', s). (t' + t, s)))"
jkbpC_def: shows "Clock.jkbpC \<equiv> \<Union> (range Clock.jkbpCn)"
reaches: fixes E :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and E' :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c" assumes "Subgraph E E'"   and "E'\<^sup>*\<^sup>* a b" shows "E\<^sup>*\<^sup>* a b"
assms_2: shows "a t \<noteq> [vc_com]"
code: fixes B :: "'b \<Rightarrow> 'c set"   and x :: "'a"   and gpv :: "('a, 'b, 'c) gpv" shows "mk_lossless_gpv B x gpv = GPV (map_spmf (\<lambda>generat. case generat of Pure x \<Rightarrow> Pure x | IO out c \<Rightarrow> IO out (\<lambda>input. if input \<in> B out then mk_lossless_gpv B x (c input) else Generative_Probabilistic_Value.Done x)) (the_gpv gpv))"
winning_path_drop_add: fixes P :: "'a llist"   and p :: "Player"   and n :: "nat" assumes "G'.valid_path P"   and "G'.winning_path p (ldropn n P)"   and "enat n < llength P" shows "G'.winning_path p P"
maskedDB_zero_def: fixes maskedDB :: "bit list"   and emBits :: "nat" shows "maskedDB_zero maskedDB emBits = bv_prepend (roundup emBits 8 * 8 - emBits) \<zero> (drop (roundup emBits 8 * 8 - emBits) maskedDB)"
disc_1: fixes x1 :: "'b \<Rightarrow> ('b, 'c, 'd) sp\<^sub>\<mu>" shows "is_Get (Get x1)"
F2set2_bd: fixes x :: "('e, 'd, 'f) F2" shows "|F2set2 x| \<le>o IFbd"
response_def: fixes r :: "nat \<times> nat"   and w :: "nat \<times> nat"   and e :: "nat" shows "response r w e = (let (r1, r2) = r; (x1, x2) = w; z1 = (e * x1 + r1) mod order \<G>; z2 = (e * x2 + r2) mod order \<G> in return_spmf (z1, z2))"
poly_ident_mult: fixes t :: "'a list" shows "(1::'a) %* t = t"
neighbors_in_E: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and s :: "nat"   and t :: "nat"   and u :: "nat"   and fo :: "nat \<times> nat \<Rightarrow> 'a"   and lo :: "nat \<Rightarrow> nat"   and no :: "nat \<Rightarrow> nat set"   and f :: "nat \<times> nat \<Rightarrow> 'a"   and l :: "nat \<Rightarrow> nat"   and n :: "nat \<Rightarrow> nat set"   and v :: "nat" assumes "discharge_invar c s t u fo lo no f l n"   and "v \<in> n u" shows "(u, v) \<in> Graph.E c \<union> (Graph.E c)\<inverse>"
lspace_approximate_bounded: fixes f :: "'a \<Rightarrow> real"   and p :: "real"   and S :: "'a set"   and e :: "real" assumes "f \<in> lspace (lebesgue_on S) (ennreal p)"   and "S \<in> lmeasurable"   and "0 < p"   and "0 < e" obtains g :: "'a \<Rightarrow> real" where "g \<in> lspace (lebesgue_on S) (ennreal p)" and "bounded (g ` S)" and "lnorm (lebesgue_on S) p (f - g) < e"
wp_eval_3: fixes P :: "'c \<Rightarrow> real" shows "wp Skip P = P"
igVarIPresIGWls_def: fixes MOD :: "('b, 'c, 'varSort, 'sort, 'd, 'e, 'f, 'g, 'h) model_scheme" shows "igVarIPresIGWls MOD \<equiv> \<forall>xs x. igWls MOD (asSort xs) (igVar MOD xs x)"
read_buffer_def: shows "A.read_buffer \<equiv> (Skip@[control_var +=\<^sub>m AsmNoWrite]) ;; (Skip@[temp +=\<^sub>m AsmNoReadOrWrite]) ;; (temp \<leftarrow> aexp.Load buffer) ;; Stmt.If (bexp.Eq control_var 0) (low_var \<leftarrow> aexp.Load temp) (high_var \<leftarrow> aexp.Load temp) ;; (temp \<leftarrow> aexp.Const 0) ;; Skip@[temp -=\<^sub>m AsmNoReadOrWrite]"
a: shows "\<forall>a b. [(a, b)] \<bullet> y = y"
alw_smap: fixes P :: "'c stream \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'c"   and s :: "'d stream" shows "alw P (smap f s) = alw (\<lambda>x. P (smap f x)) s"
conservative_axioms: shows "conservative (restrict_space M A) (induced_map A)"
map_ran_delete: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and x :: "'b"   and \<Gamma> :: "('b \<times> 'c) list" shows "map_ran f (delete x \<Gamma>) = delete x (map_ran f \<Gamma>)"
redundant_infer_def: fixes N :: "'b literal multiset set"   and \<gamma> :: "'b inference" shows "standard_redundancy_criterion.redundant_infer N \<gamma> = (\<exists>DD. set_mset DD \<subseteq> N \<and> (\<forall>I. I \<Turnstile>m DD + side_prems_of \<gamma> \<longrightarrow> I \<Turnstile> concl_of \<gamma>) \<and> (\<forall>D. D \<in># DD \<longrightarrow> D < main_prem_of \<gamma>))"
f_shrink_defs_1: fixes xs :: "'d message_af list"   and k :: "nat" shows "xs \<div> k \<equiv> f_aggregate xs k last_message"
mreq_endNone: fixes C :: "'a \<Rightarrow> 'b \<Rightarrow> 'c option"   and x :: "'b"   and b :: "'a list"   and c :: "'a list"   and a :: "'a" assumes "applied_rule_rev C x b = \<bottom>"   and "applied_rule_rev C x c = \<bottom>" shows "applied_rule_rev C x (a # b) = applied_rule_rev C x (a # c)"
ereal_pos_distrib: fixes c :: "ereal"   and a :: "ereal"   and b :: "ereal" assumes "0 \<le> c"   and "c \<noteq> \<infinity>" shows "c * (a + b) = c * a + c * b"
prems_3: shows "finite (varacom ({A} WHILE b DO C))"
linear_order_imp_linorder: fixes leq :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "linear_order {(A, B). leq A B}" shows "class.linorder leq (\<lambda>x y. leq x y \<and> \<not> leq y x)"
healthy_boundedD: fixes t :: "('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real"   and P :: "'c \<Rightarrow> real" assumes "healthy t"   and "sound P" shows "Expectations.bounded (t P)"
bij_rename: fixes h :: "'c \<Rightarrow> 'd" assumes "bij h" shows "bij (rename h)"
confTs_widen: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'a) list) list"   and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and loc :: "val list"   and LT :: "ty err list"   and LT' :: "ty err list" assumes "P,h \<turnstile> loc [:\<le>\<^sub>\<top>] LT"   and "P \<turnstile> LT [\<le>\<^sub>\<top>] LT'" shows "P,h \<turnstile> loc [:\<le>\<^sub>\<top>] LT'"
FOLD_bin_ne_def: fixes b :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and X :: "'a set" shows "FOLD_bin_ne b X = ASSERT (X \<noteq> {}) \<bind> (\<lambda>_. SPEC (\<lambda>x. x \<in> X) \<bind> (\<lambda>x. let X = X - {x} in FOREACH X (\<lambda>x r. RETURN (b x r)) x))"
not_in_smaller: fixes e :: "'b environment"   and a :: "char list"   and X :: "'b" assumes "ok (e\<lparr>a:X\<rparr>)" shows "a \<notin> env_dom e"
size_list_estimation: fixes x :: "'g"   and xs :: "'g list"   and y :: "nat"   and f :: "'g \<Rightarrow> nat" assumes "x \<in> set xs"   and "y < f x" shows "y < size_list f xs"
sum_list_distinct_conv_sum_set: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and xs :: "'d list"   and f :: "'d \<Rightarrow> 'e" assumes "class.monoid_add plus zero"   and "distinct xs" shows "sum_list (map f xs) = sum f (set xs)"
pair: fixes A :: "'a trm"   and x :: "'a"   and M :: "'a trm"   and A' :: "'a trm"   and B :: "'a trm"   and B' :: "'a trm" assumes "substitutes A x M A'"   and "substitutes B x M B'" shows "substitutes (SimplyTyped.Pair A B) x M (SimplyTyped.Pair A' B')"
n: shows "univ_list ! n = univ_list ! i \<and> n < length univ_list"
fMin_fold: fixes h :: "'a"   and t :: "'a list" shows "fMin (fset_of_list (h # t)) = fold min t h"
gt_wf: shows "wfP (\<lambda>s t. local.kbo_std.gtw t s)"
functor_category_axioms: shows "functor_category (\<cdot>\<^sub>A\<^sub>2) (\<cdot>\<^sub>B)"
Span_same_set: fixes K :: "'a set"   and Us :: "'a list"   and Vs :: "'a list" assumes "subfield K R"   and "set Us \<subseteq> carrier R"   and "set Us = set Vs" shows "R.Span K Us = R.Span K Vs"
perfect_hiding_ind_cpa_def: fixes key_gen :: "('b \<times> 'c) spmf"   and commit :: "'b \<Rightarrow> 'd \<Rightarrow> ('e \<times> 'f) spmf"   and valid_msg :: "'d \<Rightarrow> bool"   and \<A> :: "('c \<Rightarrow> (('d \<times> 'd) \<times> 'g) spmf) \<times> ('e \<Rightarrow> 'g \<Rightarrow> bool spmf)" shows "abstract_commitment.perfect_hiding_ind_cpa key_gen commit valid_msg \<A> \<equiv> abstract_commitment.hiding_advantage_ind_cpa key_gen commit valid_msg \<A> = 0"
exhaust: fixes y :: "'b kdt" obtains x1 :: "(real, 'b) vec" where "y = Leaf x1"   | x21 :: "'b" and x22 :: "real" and x23 :: "'b kdt" and x24 :: "'b kdt" where "y = Node x21 x22 x23 x24"
not_disj_deMorgans: fixes P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool" shows "(\<not> (P \<or> Q)) = (\<not> P \<and> \<not> Q)"
hd_rev: fixes xs :: "'a list" assumes "xs \<noteq> []" shows "hd (rev xs) = last xs"