hcju/isabelleps huggingface dataset:
HCP_adv_def: fixes A :: "'index \<Rightarrow> bool \<Rightarrow> bool \<Rightarrow> (bool \<times> 'index \<times> bool \<times> bool \<Rightarrow> bool spmf) \<Rightarrow> 'range \<Rightarrow> bool spmf"   and \<sigma> :: "bool"   and b\<^sub>\<sigma> :: "bool"   and D :: "bool \<times> 'index \<times> bool \<times> bool \<Rightarrow> bool spmf" shows "local.etp.HCP_adv A \<sigma> b\<^sub>\<sigma> D = \<bar>spmf (local.etp.HCP_game A \<sigma> b\<^sub>\<sigma> D) True - 1 / 2\<bar>"
S_O_NS_1: fixes S :: "('a \<times> 'a) set"   and NS :: "('a \<times> 'a) set" assumes "order_pair S NS" shows "S O NS = S"
Gamma_minus_one_half_real: shows "Gamma (- (1 / 2)) = - 2 * sqrt pi"
lift_def: fixes P :: "'b set set"   and F :: "'b set \<Rightarrow> 'a set"   and II :: "'b set" shows "lift P F II \<equiv> \<Union> {F I |I. I \<in> P \<and> I \<subseteq> II}"
plus_float_def: fixes a :: "('a, 'b) IEEE.float"   and b :: "('a, 'b) IEEE.float" shows "a + b = fadd To_nearest a b"
prime_power_canonical: fixes p :: "nat"   and m :: "nat" assumes "prime p"   and "0 < m" shows "\<exists>k n. \<not> p dvd n \<and> m = n * p ^ k"
valid_eval_step_mmuaux': fixes args :: "Monitor.args"   and cur :: "nat"   and dt :: "nat"   and aux :: "nat \<times> nat queue \<times> nat \<times> bool list \<times> bool list \<times> ('a option list, nat) mapping \<times> (nat, ('a option list, nat + nat) mapping) mapping \<times> 'a option list set list \<times> nat"   and auxlist :: "(nat \<times> 'a option list set \<times> 'a option list set) list"   and ts :: "nat"   and tss'' :: "nat list" assumes "valid_mmuaux' args cur dt aux auxlist"   and "lin_ts_mmuaux aux = ts # tss''"   and "enat ts + right (args_ivl args) < enat dt" shows "valid_mmuaux' args cur dt (eval_step_mmuaux aux) auxlist \<and> lin_ts_mmuaux (eval_step_mmuaux aux) = tss''"
A1: shows "P \<R>\<mapsto><Rel,Cal> Q"
infinite_set_less_eq_Complement: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set"   and B :: "'b set" assumes "class.linorder less_eq less"   and "finite A"   and "finite B"   and "infinite UNIV" shows "ord.set_less_eq less_eq A (- B)"
ucast_ucast_mask_shift: fixes a :: "nat"   and b :: "nat"   and p :: "'c word" assumes "a \<le> LENGTH('b) + b" shows "UCAST('b \<rightarrow> 'c) (UCAST('c \<rightarrow> 'b) (p AND mask a >> b)) = p AND mask a >> b"
Rep_uint32_numeral: fixes n :: "num" shows "Rep_uint32 (numeral n) = numeral n"
descendants_oob_head: fixes k :: "'b"   and i :: "'b"   and ps :: "('c, 'b) fm list"   and a :: "'b"   and branch :: "(('c, 'b) fm list \<times> 'b) list"   and xs :: "(nat \<times> nat) set" assumes "descendants k i ((ps, a) # branch) xs" shows "(length branch, length ps) \<notin> xs"
Y_def: fixes M :: "'a measure"   and I :: "'b set"   and A :: "'b \<Rightarrow> 'a set"   and x :: "'a" assumes "prob_space_with_indicators M I A" shows "prob_space_with_indicators.Y I A x = (\<Sum>i\<in>I. prob_space_with_indicators.X A i x)"
p0: shows "p \<lbrace> v \<rbrace> = (0::'a)"
subgame_\<omega>: fixes V' :: "'a set" shows "\<omega>\<^bsub>subgame V'\<^esub> = \<omega>"
inst_cfun_pcpo: shows "\<bottom> = (\<Lambda> x. \<bottom>)"
G4: shows "dom (\<Psi>\<^sub>I\<^sub>+) = set (\<Psi>\<^sub>\<V>\<^sub>+)"
elemI: fixes a' :: "'d"   and A :: "'d set"   and R :: "('d, 'e) eq_object_scheme"   and a :: "'d" assumes "a' \<in> A"   and "a .=\<^bsub>R\<^esub> a'" shows "a .\<in>\<^bsub>R\<^esub> A"
iface_name_is_wildcard_alt: fixes eth :: "char list" shows "iface_name_is_wildcard eth = (eth \<noteq> [] \<and> last eth = CHR ''+'')"
E_valid_2: shows "snd ` snd ` E \<subseteq> V"
SemWhileT: fixes b :: "BExpr"   and s :: "(Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list"   and c :: "OBJ"   and n :: "nat"   and r :: "(Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list"   and m :: "nat"   and t :: "(Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list"   and k :: "nat" assumes "evalB b (fst s)"   and " s , c \<rightarrow>\<^sub>n  r "   and " r , While b c \<rightarrow>\<^sub>m  t "   and "k = max n m + 1" shows " s , While b c \<rightarrow>\<^sub>k  t "
mult_def: fixes r :: "('k \<times> 'k) set" shows "mult r = (mult1 r)\<^sup>+"
c_def: shows "c = b - 2 * pi * real_of_int \<lceil>(b - pi) / (2 * pi)\<rceil>"
indeps_7: shows "mrg_left \<bowtie> mrg_right"
diff_subset_eq_self: fixes M :: "'a multiset"   and N :: "'a multiset" shows "M - N \<subseteq># M"
simps_2: fixes l :: "'b tree"   and a :: "'b"   and r :: "'b tree" shows "mset_tree \<langle>l, a, r\<rangle> = {#a#} + mset_tree l + mset_tree r"
add_left_imp_eq: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and a :: "'d"   and b :: "'d"   and c :: "'d" assumes "class.cancel_semigroup_add plus"   and "plus a b = plus a c" shows "b = c"
between_simp2: fixes vs :: "'a list"   and ram1 :: "'a"   and ram2 :: "'a" assumes "before vs ram1 ram2"   and "pre_between vs ram1 ram2" shows "between vs ram2 ram1 = snd (splitAt ram2 vs) @ fst (splitAt ram1 vs)"
simps_11: fixes va :: "pol"   and vb :: "nat"   and vd :: "pol"   and ve :: "nat"   and vf :: "pol"   and v :: "nat"   and Q :: "pol" shows "isnorm (PX (PX va vb (PX vd ve vf)) (Suc v) Q) = (isnorm (PX va vb (PX vd ve vf)) \<and> isnorm Q)"
cod_comp: fixes g :: "'a"   and f :: "'a" assumes "A.seq g f" shows "A.cod (A g f) = A.cod g"
lift_keys_def: fixes i :: "nat"   and xs :: "('t, 'c) oalist_ntm" shows "lift_keys i xs = oalist_of_list_ntm (map_raw (\<lambda>kv. (map_component ((+) i) (fst kv), snd kv)) (list_of_oalist_ntm xs))"
uncurry_rec_nat: fixes fn :: "'f \<Rightarrow> 'g \<Rightarrow> 'e"   and fs :: "nat \<Rightarrow> ('f \<Rightarrow> 'g \<Rightarrow> 'e) \<Rightarrow> 'f \<Rightarrow> 'g \<Rightarrow> 'e"   and n :: "nat"   and a :: "'f"   and b :: "'g" shows "rec_nat fn fs n a b = rec_nat (\<lambda>(a, b). fn a b) (\<lambda>n rr (a, b). fs n (\<lambda>a b. rr (a, b)) a b) n (a, b)"
process_prop_2_def: fixes P :: "('c list \<times> 'c set) set \<times> 'c list set" shows "process_prop_2 P \<equiv> \<forall>xs x X. (xs @ [x], X) \<in> fst P \<longrightarrow> (xs, {}) \<in> fst P"
lta_right_comm: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p"   and E :: "'p"   and F :: "'p" assumes "A B C LtA D E F" shows "A B C LtA F E D"
chart_image_eq_vimage: fixes X :: "'c set"   and c :: "('c, 'd) chart" assumes "X \<subseteq> domain c" shows "apply_chart c ` X = inv_chart c -` X \<inter> codomain c"
IfThenElse: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and scrut :: "exp"   and L :: "Vars.var list"   and \<Delta> :: "(Vars.var \<times> exp) list"   and b :: "bool"   and e\<^sub>1 :: "exp"   and e\<^sub>2 :: "exp"   and \<Theta> :: "(Vars.var \<times> exp) list"   and z :: "exp" assumes "\<Gamma> : scrut \<Down>\<^bsub>L\<^esub> \<Delta> : Bool b"   and "\<Delta> : (if b then e\<^sub>1 else e\<^sub>2) \<Down>\<^bsub>L\<^esub> \<Theta> : z" shows "\<Gamma> : (scrut ? e\<^sub>1 : e\<^sub>2) \<Down>\<^bsub>L\<^esub> \<Theta> : z"
compact_interval: fixes a :: "'e"   and b :: "'e" shows "compact {a..b}"
assms: shows "f \<in> F t \<rightarrow>\<^sub>M N"
existence_ivl_reverse: fixes t :: "real"   and t0 :: "real"   and x0 :: "real" assumes "t \<in> existence_ivl t0 x0" shows "t0 \<in> existence_ivl t (flow t0 x0 t)"
to_conjugate_space_cases: fixes x :: "'d conjugate_space" obtains y :: "'d" where "x = to_conjugate_space y" and "y \<in> UNIV"
congruence: fixes e :: "exp"   and e' :: "exp"   and C :: "ctx" assumes "E e = E e'" shows "E (plug C e) = E (plug C e')"
eq2: fixes a :: "'b"   and b :: "'e"   and ys :: "('b \<times> 'e) list"   and c :: "'b" shows "lookup_dflt ((a, b) # ys) c = (if c = a then b else lookup_dflt ys c)"
Init_simps2_3: fixes P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" shows "Init (P \<longrightarrow> Q) \<equiv> Init P \<longrightarrow> Init Q"
lcm_unique: fixes one :: "'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and a :: "'a"   and d :: "'a"   and b :: "'a" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize" shows "(dvd.dvd times a d \<and> dvd.dvd times b d \<and> normalize d = d \<and> (\<forall>e. dvd.dvd times a e \<and> dvd.dvd times b e \<longrightarrow> dvd.dvd times d e)) = (d = lcm a b)"
domains_simp: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b" assumes "category C"   and "partial_magma.arr C f" shows "partial_magma.domains C f = {partial_magma.dom C f}"
simps_2: fixes edge_rev :: "'c \<Rightarrow> 'c"   and edge_succ :: "'c \<Rightarrow> 'c"   and more :: "'d" shows "edge_succ \<lparr>edge_rev = edge_rev, edge_succ = edge_succ, \<dots> = more\<rparr> = edge_succ"
all_post: fixes self :: "'\<AA> state \<times> '\<AA> state \<Rightarrow> '\<alpha>0"   and a1 :: "'\<AA> state \<times> '\<AA> state \<Rightarrow> '\<alpha>1"   and a2 :: "'\<AA> state \<times> '\<AA> state \<Rightarrow> '\<alpha>2" shows "\<forall>\<sigma> \<sigma>' \<sigma>''. ((\<sigma>, \<sigma>') \<Turnstile> PRE self a1 a2) = ((\<sigma>, \<sigma>'') \<Turnstile> PRE self a1 a2)"
gbinomial_absorption: fixes k :: "nat"   and a :: "'a" shows "of_nat (Suc k) * (a gchoose Suc k) = a * (a - (1::'a) gchoose k)"
assertWeaken: fixes \<Psi> :: "'b"   and \<Psi>' :: "'b" shows "\<Psi> \<hookrightarrow> \<Psi> \<otimes> \<Psi>'"
swap: fixes g :: "'b \<Rightarrow> 'c \<Rightarrow> 'a"   and B :: "'c set"   and A :: "'b set" shows "(\<Sum>i\<in>A. sum (g i) B) = (\<Sum>j\<in>B. \<Sum>i\<in>A. g i j)"
simps_10: fixes b :: "nat"   and \<phi> :: "formula" shows "fvi b (Exists \<phi>) = fvi (Suc b) \<phi>"
replace1: fixes f :: "'a"   and f' :: "'a"   and fs :: "'a list"   and ls :: "'a list" assumes "f \<in> set (replace f' fs ls)"   and "f \<notin> set ls" shows "f \<in> set fs"
base_empty: fixes p :: "(nat \<times> int) list"   and dm :: "nat" assumes "p \<in> sparsegrid' dm" shows "base {} p = p"
group_comm_groupI: fixes G :: "('b, 'c) monoid_scheme" assumes "Group.group G"   and "\<And>x y. \<lbrakk>x \<in> carrier G; y \<in> carrier G\<rbrakk> \<Longrightarrow> x \<otimes>\<^bsub>G\<^esub> y = y \<otimes>\<^bsub>G\<^esub> x" shows "comm_group G"
sublens_pres_indep': fixes X :: "'e \<Longrightarrow> 'f"   and Y :: "'g \<Longrightarrow> 'f"   and Z :: "'h \<Longrightarrow> 'f" assumes "X \<subseteq>\<^sub>L Y"   and "Z \<bowtie> Y" shows "Z \<bowtie> X"
evalRR_monom: fixes a :: "'a"   and x :: "'a"   and d :: "nat" assumes "a \<in> carrier R"   and "x \<in> carrier R" shows "eval R R id x (monom (UP R) a d) = a \<otimes> x [^] d"
ArityAnalysisSafe_axioms_def: fixes Aexp :: "exp \<Rightarrow> Arity \<rightarrow> Vars.var \<Rightarrow> Arity\<^sub>\<bottom>" shows "ArityAnalysisSafe_axioms Aexp \<equiv> ((\<forall>n x. up\<cdot>n \<sqsubseteq> (Aexp (Var x)\<cdot>n) x) \<and> (\<forall>e n x. Aexp e\<cdot>(inc\<cdot>n) \<squnion> esing x\<cdot>(up\<cdot>0) \<sqsubseteq> Aexp (App e x)\<cdot>n)) \<and> (\<forall>y e n. env_delete y (Aexp e\<cdot>(pred\<cdot>n)) \<sqsubseteq> Aexp (Lam [y]. e)\<cdot>n) \<and> (\<forall>scrut e1 a e2. Aexp scrut\<cdot>0 \<squnion> Aexp e1\<cdot>a \<squnion> Aexp e2\<cdot>a \<sqsubseteq> Aexp (scrut ? e1 : e2)\<cdot>a)"
smult_assoc: fixes r :: "'f"   and s :: "'f"   and m :: "'v" assumes "r \<in> UNIV"   and "s \<in> UNIV"   and "m \<in> V" shows "(r \<cdot>\<cdot> (\<cdot>) s) m = (r * s) \<cdot> m"
sel_2: fixes x2 :: "bool" shows "extract_bool (BoolVal x2) = x2"
seq_compactE: fixes S :: "'b set"   and f :: "nat \<Rightarrow> 'b" assumes "seq_compact S"   and "\<forall>n. f n \<in> S" obtains l :: "'b" and r :: "nat \<Rightarrow> nat" where "l \<in> S" and "strict_mono r" and "(f \<circ> r) \<longlonglongrightarrow> l"
outgoing'_def: fixes k :: "nat set" shows "outgoing' k \<equiv> {(v, u) |u v. u \<notin> k \<and> v \<in> k \<and> (v, u) \<in> E}"
mmupd_in_upd: fixes k :: "'a"   and K :: "'a set"   and m :: "'a \<Rightarrow> 'b option"   and v :: "'b" assumes "k \<in> K" shows "map_mmupd m K v k = Some v"
atm_of_uminus: fixes L :: "'a literal" shows "atm_of (- L) = atm_of L"
inorder_balR: fixes ba :: "bal"   and r :: "('b \<times> bal) tree"   and l :: "('b \<times> bal) tree"   and a :: "'b" assumes "ba = Rh \<longrightarrow> r \<noteq> \<langle>\<rangle>"   and "avl r" shows "Tree2.inorder (balR l a ba r) = Tree2.inorder l @ a # Tree2.inorder r"
partial_magma_axioms: shows "partial_magma (\<cdot>\<^sub>[\<^sub>J\<^sub>,\<^sub>C\<^sub>])"
distr_pair_swap: shows "M2 \<Otimes>\<^sub>M M1 = distr (M1 \<Otimes>\<^sub>M M2) (M2 \<Otimes>\<^sub>M M1) (\<lambda>(x, y). (y, x))"
simps_10: fixes t :: "'c local.term"   and u :: "'c local.term"   and v :: "'c local.term" shows "Arr \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[t, u, v\<^bold>] = (Arr t \<and> Arr u \<and> Arr v \<and> Src t = Trg u \<and> Src u = Trg v)"
pterm_to_sterm_shadows: fixes t :: "pterm" assumes "pre_strong_term_class.wellformed t" shows "shadows_consts t = shadows_consts (pterm_to_sterm t)"
Na: fixes n :: "nat" assumes "Na \<le> n" shows "ereal (M + deltaG TYPE('a)) \<le> extended_Gromov_product_at basepoint (to_Gromov_completion ((f ^^ n) basepoint)) (attracting_fixed_point f)"
vector_one: fixes x :: "('a, 1) vec" shows "x = (\<chi>i. x $ 1)"
encL: fixes Enc :: "'a \<Rightarrow> 'b"   and S :: "'a" shows "(TargetTerm (Enc S), SourceTerm S) \<in> encoding.indRelEQ Enc"
final_def: fixes cfg :: "('a, 'b, 'c) com \<times> ('a, 'c) xstate" shows "final cfg = (fst cfg = Skip \<or> fst cfg = Throw \<and> (\<exists>s. snd cfg = Normal s))"
sees_wwf_nsub_RI: fixes P :: "(char list list \<times> char list exp) prog"   and C :: "char list"   and M :: "char list"   and b :: "staticb"   and Ts :: "ty list"   and T :: "ty"   and pns :: "char list list"   and body :: "char list exp"   and D :: "char list" assumes "wwf_J_prog P"   and "P \<turnstile> C sees M, b :  Ts\<rightarrow>T = (pns, body) in D" shows "\<not> sub_RI body"
one_chain_line_integral_def: fixes F :: "real \<times> real \<Rightarrow> real \<times> real"   and b :: "(real \<times> real) set"   and C :: "(int \<times> (real \<Rightarrow> real \<times> real)) set" shows "one_chain_line_integral F b C \<equiv> \<Sum>(k, g)\<in>C. real_of_int k * line_integral F b g"
Cons_1: fixes v1 :: "'a" assumes "subpath g v1 es v2"   and "e \<in> set es" shows "\<exists>es'. es = es' @ [e] \<and> e \<notin> set es'"
iwlsFSbSwTR_iterSTR_ipresAbs: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model" assumes "iwlsFSbSwTR MOD" shows "ipresAbs (iterSTR MOD) (iterAbsSTR MOD) MOD"
assms_1: shows "bij_betw enum {0..<card A} A"
type_less_eq_x_OclSuper_intro: fixes \<tau> :: "'a OCL_Types.type" shows "\<tau> \<le> OclSuper"
bcontfunE: fixes f :: "'c \<Rightarrow> 'd" assumes "f \<in> bcontfun" obtains g :: "'c \<Rightarrow>\<^sub>C 'd" where "f = apply_bcontfun g"
A_B_step: fixes a :: "'a"   and b :: "'a"   and a' :: "'b" assumes "A a b"   and "PA a"   and "PB a'"   and "sim1 a a'" shows "\<exists>b'. B a' b' \<and> sim1 b b'"
proper_it_unfold: fixes it :: "('a list \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'a list \<Rightarrow> 'a list) \<Rightarrow> 'a list \<Rightarrow> 'a list"   and it' :: "('b \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'b" assumes "proper_it it it'" shows "it' = foldli (it (\<lambda>_. True) (\<lambda>x l. l @ [x]) [])"
simps_2: fixes s :: "'a wordinterval"   and ss :: "'a wordinterval list"   and ts :: "'a wordinterval list" shows "partitioningIps (s # ss) ts = partIps s (partitioningIps ss ts)"
compatible_def: fixes f :: "('s \<Rightarrow> 's \<Rightarrow> bool) \<Rightarrow> 's \<Rightarrow> 's \<Rightarrow> bool" shows "compatible f = (mono f \<and> (\<forall>R S. R \<rightarrow> S \<longrightarrow> f R \<rightarrow> f S))"
sort_key_def: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'b"   and xs :: "'c list" assumes "class.linorder less_eq less" shows "linorder.sort_key less_eq f xs = foldr (linorder.insort_key less_eq f) xs []"
SumA_def: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a"   and E :: "'a"   and F :: "'a"   and G :: "'a"   and H :: "'a"   and I :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong" shows "Tarski_neutral_dimensionless.SumA Bet Cong A B C D E F G H I \<equiv> \<exists>J. Tarski_neutral_dimensionless.CongA Bet Cong C B J D E F \<and> \<not> Tarski_neutral_dimensionless.OS Bet B C A J \<and> Tarski_neutral_dimensionless.Coplanar Bet A B C J \<and> Tarski_neutral_dimensionless.CongA Bet Cong A B J G H I"
strand_substI_1: fixes \<theta> :: "'a \<Rightarrow> ('b, 'a) Term.term"   and S :: "('b, 'a) strand_step list" assumes "subst_domain \<theta> \<inter> fv\<^sub>s\<^sub>t S = {}" shows "S \<cdot>\<^sub>s\<^sub>t \<theta> = S"
count_pre_meas: fixes M :: "'c measure"   and N :: "'d measure"   and Y :: "'c \<Rightarrow> 'd"   and B :: "'d set" assumes "point_measurable M (space N) Y"   and "B \<subseteq> space N"   and "countable B" shows "\<forall>i. countable_preimages B Y i \<inter> space M \<in> sets M"
epath_rev: fixes E :: "'b uprod set"   and y :: "'b"   and p :: "'b uprod list"   and x :: "'b" shows "epath E y (rev p) x = epath E x p y"
assms_3: shows "SubRel.subpath (add_edge g e) v es (tgt e) subs"
elims: fixes x :: "nat"   and xa :: "'a list"   and y :: "nat" assumes "T_chop x xa = y" obtains uu_ :: "'a list" where "x = 0" and "xa = uu_" and "y = 1"   | v :: "nat" where "x = Suc v__" and "xa = []" and "y = 1"   | v :: "nat" and va :: "'a" and vb :: "'a list" where "x = Suc v" and "xa = va # vb" and "y = T_take (Suc v) (va # vb) + T_drop (Suc v) (va # vb) + T_chop (Suc v) (drop (Suc v) (va # vb))"
execute_serial_plan_sas_plus_is_execute_parallel_plan_sas_plus_iv: fixes \<psi> :: "('c, 'd) sas_plus_operator list"   and \<Psi> :: "('c, 'd) sas_plus_problem" assumes "\<forall>op\<in>set \<psi>. op \<in> set (\<Psi>\<^sub>\<O>\<^sub>+)" shows "\<forall>ops\<in>set (List_Supplement.embed \<psi>). \<forall>op\<in>set ops. op \<in> set (\<Psi>\<^sub>\<O>\<^sub>+)"
simps_2: fixes e :: "expr"   and es :: "expr list"   and A :: "char list set option" shows "\<D>s (e # es) A = (\<D> e A \<and> \<D>s es (A \<squnion> \<A> e))"
CL_imp_PO: fixes c :: "'b potype" assumes "c \<in> CompleteLattice" shows "c \<in> PartialOrder"
app_simps_3: fixes size :: "'b \<Rightarrow> nat"   and frees :: "'b \<Rightarrow> name fset"   and subst :: "'b \<Rightarrow> (name, 'b) fmap \<Rightarrow> 'b"   and "consts" :: "'b \<Rightarrow> name fset"   and app :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unapp :: "'b \<Rightarrow> ('b \<times> 'b) option"   and const :: "name \<Rightarrow> 'b"   and unconst :: "'b \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'b"   and unfree :: "'b \<Rightarrow> name option"   and t :: "'b"   and u :: "'b" assumes "class.pre_term size frees subst consts app unapp const unconst free unfree" shows "pre_term.is_app unapp (app t u)"
power_strict_mono: fixes a :: "'a"   and b :: "'a"   and n :: "nat" assumes "a < b"   and "(0::'a) \<le> a"   and "0 < n" shows "a ^ n < b ^ n"
F_seq: fixes P :: "'c process"   and Q :: "'c process" shows "F (P `;` Q) = {(t, X). (t, X \<union> {tick}) \<in> F P \<and> tickFree t} \<union> {(t, X). \<exists>t1 t2. t = t1 @ t2 \<and> t1 @ [tick] \<in> T P \<and> (t2, X) \<in> F Q} \<union> {(t, X). \<exists>t1 t2. t = t1 @ t2 \<and> t1 \<in> D P \<and> tickFree t1 \<and> front_tickFree t2} \<union> {(t, X). \<exists>t1 t2. t = t1 @ t2 \<and> t1 @ [tick] \<in> T P \<and> t2 \<in> D Q}"
distinct_160: fixes x11 :: "Syntax.term"   and x6 :: "Syntax.term" shows "Prj1 x11 \<noteq> Inj2 x6"
finite_char_def: fixes C :: "'b set set" shows "finite_char C \<equiv> \<forall>S. (S \<in> C) = (\<forall>S'. finite S' \<longrightarrow> S' \<subseteq> S \<longrightarrow> S' \<in> C)"
compactin_Un: fixes X :: "'a topology"   and S :: "'a set"   and T :: "'a set" assumes "compactin X S"   and "compactin X T" shows "compactin X (S \<union> T)"
opnet_sync_action_subnet_oreachable: fixes \<sigma> :: "nat \<Rightarrow> 'd"   and s :: "'e net_state"   and t :: "'e net_state"   and onp :: "nat \<Rightarrow> ((nat \<Rightarrow> 'd) \<times> 'e, 'f seq_action) automaton"   and p\<^sub>1 :: "net_tree"   and p\<^sub>2 :: "net_tree"   and I :: "(nat \<Rightarrow> 'd) \<Rightarrow> 'f \<Rightarrow> bool"   and U :: "'d \<Rightarrow> 'd \<Rightarrow> bool" assumes "(\<sigma>, SubnetS s t) \<in> oreachable (opnet onp (p\<^sub>1 \<parallel> p\<^sub>2)) (\<lambda>\<sigma> _. oarrivemsg I \<sigma>) (other U (net_tree_ips (p\<^sub>1 \<parallel> p\<^sub>2)))"   and "\<And>\<xi>. U \<xi> \<xi>"   and "opnet onp p\<^sub>1 \<Turnstile>\<^sub>A (\<lambda>\<sigma> _. oarrivemsg I \<sigma>, other U (net_tree_ips p\<^sub>1) \<rightarrow>) globala (\<lambda>(\<sigma>, a, \<sigma>'). castmsg (I \<sigma>) a \<and> (a = \<tau> \<or> (\<exists>i d. a = i:deliver(d)) \<longrightarrow> (\<forall>i\<in>net_tree_ips p\<^sub>1. U (\<sigma> i) (\<sigma>' i)) \<and> (\<forall>i. i \<notin> net_tree_ips p\<^sub>1 \<longrightarrow> \<sigma>' i = \<sigma> i)))"   and "opnet onp p\<^sub>2 \<Turnstile>\<^sub>A (\<lambda>\<sigma> _. oarrivemsg I \<sigma>, other U (net_tree_ips p\<^sub>2) \<rightarrow>) globala (\<lambda>(\<sigma>, a, \<sigma>'). castmsg (I \<sigma>) a \<and> (a = \<tau> \<or> (\<exists>i d. a = i:deliver(d)) \<longrightarrow> (\<forall>i\<in>net_tree_ips p\<^sub>2. U (\<sigma> i) (\<sigma>' i)) \<and> (\<forall>i. i \<notin> net_tree_ips p\<^sub>2 \<longrightarrow> \<sigma>' i = \<sigma> i)))" shows "(\<sigma>, s) \<in> oreachable (opnet onp p\<^sub>1) (\<lambda>\<sigma> _. oarrivemsg I \<sigma>) (other U (net_tree_ips p\<^sub>1)) \<and> (\<sigma>, t) \<in> oreachable (opnet onp p\<^sub>2) (\<lambda>\<sigma> _. oarrivemsg I \<sigma>) (other U (net_tree_ips p\<^sub>2)) \<and> net_tree_ips p\<^sub>1 \<inter> net_tree_ips p\<^sub>2 = {}"
L: shows "A \<subseteq> carrier \<lparr>carrier = {H. subgroup H G}, eq = (=), le = (\<subseteq>)\<rparr>"
set_rev_mp: fixes c :: "'b"   and A :: "'b set"   and B :: "'b set" assumes "c \<in> A"   and "A \<subseteq> B" shows "c \<in> B"
mem_set_ar_cos: fixes I :: "'a set"   and a :: "'a" assumes "ideal R I"   and "a \<in> carrier R" shows "a \<uplus>\<^bsub>R\<^esub> I \<in> set_ar_cos R I"
unique_factorization_mI: fixes f :: "int poly"   and c :: "int"   and fs :: "int poly multiset" assumes "p.factorization_m f (c, fs)"   and "\<And>d gs. p.factorization_m f (d, gs) \<Longrightarrow> p.Mf (d, gs) = p.Mf (c, fs)" shows "p.unique_factorization_m f (c, fs)"
take_eq_Nil: fixes n :: "nat"   and xs :: "'a list" shows "(take n xs = []) = (n = 0 \<or> xs = [])"
n: shows "nsphere n = subtopology (subtopology (nsphere (Suc n)) {x. x (Suc n) \<le> 0}) {x. x (Suc n) = 0}"
inv_step: fixes C :: "nat set"   and R :: "'a set" assumes "local.inv C R"   and "R \<noteq> {}" shows "local.inv (C \<union> {min_arg R m}) (R - S (min_arg R m))"
lang\<^sub>M\<^sub>2\<^sub>L_rexp_of'_rexp_of'': fixes n :: "nat"   and \<phi> :: "'a formula" assumes "wf_formula n \<phi>" shows "lang n (rexp_of' n \<phi>) = lang n (rexp_of'' n \<phi>)"