hcju/isabelleps huggingface dataset:
continuous_on_closed_gen: fixes f :: "'c \<Rightarrow> 'b"   and S :: "'c set"   and T :: "'b set" assumes "f ` S \<subseteq> T" shows "continuous_on S f = (\<forall>U. closedin (top_of_set T) U \<longrightarrow> closedin (top_of_set S) (S \<inter> f -` U))"
simps_3: fixes n :: "nat" shows "count_derangements (Suc (Suc n)) = (n + 1) * (count_derangements (Suc n) + count_derangements n)"
Some_eq_bind_conv: fixes y :: "'c"   and x :: "'d option"   and f :: "'d \<Rightarrow> 'c option" shows "(Some y = x \<bind> f) = (\<exists>z. x = Some z \<and> f z = Some y)"
fun_rel_def: fixes A :: "('b \<times> 'd) set"   and B :: "('c \<times> 'e) set" shows "A \<rightarrow> B \<equiv> {(f, f'). \<forall>(a, a')\<in>A. (f a, f' a') \<in> B}"
divide_nonpos_nonneg: fixes x :: "'d"   and y :: "'d" assumes "x \<le> (0::'d)"   and "(0::'d) \<le> y" shows "x / y \<le> (0::'d)"
rell_G_eq_onp: fixes P1 :: "'n \<Rightarrow> bool"   and P2 :: "'o \<Rightarrow> bool" shows "rell_G (eq_onp P1) (eq_onp P2) = eq_onp (\<lambda>x. (\<forall>z\<in>set1_G x. P1 z) \<and> (\<forall>z\<in>set2_G x. P2 z))"
parity_assms: shows "even (degree v' G) \<and> v' \<noteq> v \<or> odd (degree v' G) \<and> v' = v"
WhileTrue: fixes tst :: "'test"   and s :: "'state"   and c :: "('test, 'atom) com" assumes "tval tst s" shows "(while tst do c, s) \<rightarrow>c (c ;; while tst do c, s)"
assms_4: fixes \<chi> :: "'a ltl" assumes "\<chi> \<in> dom \<pi>" shows "accepting_pair\<^sub>R (delta \<Sigma>) (initial \<phi>) (Acc \<Sigma> \<pi> \<chi>) w"
collision_d: fixes S :: "real set" shows "collision S = (\<exists>t\<in>S. d t = 0)"
subset_UNIV: fixes A :: "'b set" shows "order.greater_eq UNIV A"
si_1: shows "case si of None \<Rightarrow> True | Some ssi \<Rightarrow> p_sport p \<in> prefix_to_wordset ssi"
signed_real_of_word_sum_less_eq_numeral_iff: fixes v :: "'a Word.word"   and w :: "'b Word.word"   and n :: "num" shows "(signed_real_of_word v + signed_real_of_word w \<le> numeral n) = (sint v + sint w \<le> numeral n)"
elims_3: fixes blue :: "'a"   and brown :: "'a"   and green :: "'a"   and red :: "'a"   and guru :: "'b"   and x :: "nat"   and xa :: "'b"   and xb :: "'b \<Rightarrow> 'a" assumes "distinct [blue, brown, green, red]"   and "finite UNIV"   and "\<not> Blue_Eyes.leaves blue guru x xa xb" obtains n :: "nat" and p :: "'b" and w :: "'b \<Rightarrow> 'a" where "x = n" and "xa = p" and "xb = w" and "\<not> (\<forall>w'. Blue_Eyes.possible blue guru n p w w' \<longrightarrow> w' p = w p)"
rep_eq: fixes x :: "'a fls" shows "fls_regpart x = Abs_fps (\<lambda>n. fls_nth x (int n))"
knowledge_eval: fixes A :: "agent"   and evs :: "event list" shows "knows A evs = {Key (priEK A), Key (priSK A), Key (shrK A)} \<union> Key ` range pubK \<union> Key ` range pubSK \<union> {X. \<exists>S R. Says S R X \<in> set evs} \<union> {X. Notes A X \<in> set evs}"
linord_of_list_Linear_order: fixes xs :: "'b list"   and ys :: "'b set" assumes "distinct xs"   and "ys = set xs" shows "linear_order_on ys (linord_of_list xs)"
eventually_at_right_field: fixes P :: "'c \<Rightarrow> bool"   and x :: "'c" shows "eventually P (at_right x) = (\<exists>b>x. \<forall>y>x. y < b \<longrightarrow> P y)"
splitAt_not1: fixes v :: "'a"   and vs :: "'a list"   and ram :: "'a" assumes "v \<notin> set vs" shows "v \<notin> set (fst (splitAt ram vs))"
simple_restrict_none: fixes getIJ :: "(nat set \<times> 'b option list set) set \<Rightarrow> (nat set \<times> 'b option list set) set \<Rightarrow> nat set \<Rightarrow> nat set \<times> nat set"   and i :: "nat"   and X :: "'c option list"   and A :: "nat set" assumes "getIJ getIJ"   and "i < length X"   and "i \<notin> A" shows "restrict A X ! i = None"
hyps_1: fixes child_node :: "(_) node_ptr"   and parent :: "(_) object_ptr"   and ancestors :: "(_) object_ptr list" assumes "child = cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r child_node"   and "h \<turnstile> get_parent child_node \<rightarrow>\<^sub>r Some parent"   and "h \<turnstile> get_ancestors_si parent \<rightarrow>\<^sub>r ancestors"   and "ancestor \<in> set ancestors" shows "set ancestor_ancestors \<subseteq> set ancestors"
recvmsg_simps_3: fixes P :: "'e \<Rightarrow> bool"   and ip :: "nat"   and m :: "'e" shows "recvmsg P (unicast ip m) = True"
exhaust: fixes y :: "'a itrm" obtains x1 :: "'a" where "y = Opaque x1"   | x2 :: "dB" where "y = Pure x2"   | x31 :: "'a itrm" and x32 :: "'a itrm" where "y = x31 \<diamondop> x32"
alphaAll_preserves_qGoodAll1: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm"   and A :: "('d, 'e, 'varSort, 'var, 'f) qAbs" shows "(qGood X \<longrightarrow> (\<forall>X'. X #= X' \<longrightarrow> qGood X')) \<and> (qGoodAbs A \<longrightarrow> (\<forall>A'. A $= A' \<longrightarrow> qGoodAbs A'))"
IH_1: assumes "ssafe \<sigma>"   and "Oadmit \<sigma> ODE1 (BVO ODE1)" shows "ODE_sem I (Osubst ODE1 \<sigma>) (fst \<nu>) = ODE_sem (local.adjoint I \<sigma> \<nu>) ODE1 (fst \<nu>)"
proj_specific_def: fixes n :: "'b"   and t :: "('fun, 'var) Term.term"   and \<A> :: "('b strand_label \<times> ('fun, 'var) strand_step) list"   and Secrets :: "('fun, 'var) Term.term set" shows "proj_specific n t \<A> Secrets \<equiv> t \<in> GSMP (trms\<^sub>l\<^sub>s\<^sub>t (proj n \<A>)) - (Secrets \<union> {m. {} \<turnstile>\<^sub>c m})"
residue_closed: fixes p :: "int"   and b :: "nat \<Rightarrow> int"   and k :: "nat" assumes "padic_integers p"   and "b \<in> carrier (padic_int p)" shows "b k \<in> carrier (residue_ring (p ^ k))"
Suc_1: fixes A :: "'a mat" assumes "A \<in> carrier_mat n n"   and "I < n" shows "det (swap_col_to_front A I) = (- (1::'a)) ^ I * det A"
assms_2: fixes x :: "'a" assumes "x \<in> s" shows "(f has_derivative (\<lambda>h. 0::'b)) (at x within s)"
preserves_NewThread: fixes t :: "'t"   and x :: "'x"   and m :: "'m"   and ta :: "('l, 't, 'x, 'm, 'w, 'o) thread_action"   and x' :: "'x"   and m' :: "'m"   and t'' :: "'t"   and x'' :: "'x" assumes "t \<turnstile> \<langle>x, m\<rangle> -ta\<rightarrow> \<langle>x', m'\<rangle>"   and "P t x m"   and "NewThread t'' x'' m' \<in> set \<lbrace>ta\<rbrace>\<^bsub>t\<^esub>" shows "P t'' x'' m'"
pderiv_diff: fixes p :: "'b poly"   and q :: "'b poly" shows "pderiv (p - q) = pderiv p - pderiv q"
between_symmetry: fixes A :: "'p"   and B :: "'p"   and C :: "'p" assumes "Bet A B C" shows "Bet C B A"
spr_trans_cec: fixes t :: "('a, 'es, 'as) BEState Trace"   and ec :: "(('a, 'es, 'as) BEState \<times> ('a, 'es, 'as) BEState) odlist \<times> (('a, 'es, 'as) BEState \<times> ('a, 'es, 'as) BEState) odlist"   and a :: "'a" assumes "t \<in> SPR.jkbpC"   and "spr_simAbs ec = SPRdet.sim_equiv_class a t" shows "set (local.spr_trans (fst ec) (fst ec)) = rel_ext (\<lambda>uu_. \<exists>t' s. uu_ = (tFirst t', s) \<and> t' \<leadsto> s \<in> SPR.jkbpC \<and> tObsC t' = tObsC t)"
ibounded_byE: fixes \<A>1 :: "(('b \<times> 'b) \<times> 'c, 'b, 'd) gpv"   and \<A>2 :: "'d \<Rightarrow> 'c \<Rightarrow> (bool, 'b, 'd) gpv"   and q :: "enat" assumes "IND_CPA.ind_cpa.ibounded_by (\<A>1, \<A>2) q" obtains q1 :: "enat" and q2 :: "enat" where "q1 + q2 \<le> q" and "interaction_any_bounded_by \<A>1 q1" and "\<And>cipher \<sigma>. interaction_any_bounded_by (\<A>2 cipher \<sigma>) q2"
assms_2: shows "SAS_Plus_Semantics.is_parallel_solution_for_problem \<Psi> \<psi>"
\<beta>_fix: shows "LeftDerivationFix \<beta> (i + ix) D j \<gamma>"
rel_pmf_measureI: fixes p :: "'c pmf"   and q :: "'d pmf"   and R :: "'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "\<And>A. measure_pmf.prob p A \<le> measure_pmf.prob q {y. \<exists>x\<in>A. R x y}" shows "rel_pmf R p q"
a_get_child_nodes_def: fixes ptr :: "(_) object_ptr" shows "l_get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_child_nodes ptr = invoke a_get_child_nodes_tups ptr ()"
affected_swap: fixes a :: "'b"   and b :: "'b" assumes "a \<noteq> b" shows "affected (swap a b) = {a, b}"
minus_dvd_iff: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" assumes "class.comm_ring_1 times one plus zero minus uminus" shows "dvd.dvd times (uminus x) y = dvd.dvd times x y"
tail_ord_p: fixes p :: "'a \<Rightarrow>\<^sub>0 'c" assumes "p \<noteq> 0" shows "punit.ord_strict_p (punit.tail p) p"
child_eq: shows "sum F (lgrid (child b left d) {d} lm \<union> lgrid (child b right d) {d} lm) = sum F (lgrid (child b left d) {d} lm) + sum F (lgrid (child b right d) {d} lm)"
one_divide_eq_0_iff: fixes a :: "'a" shows "((1::'a) / a = (0::'a)) = (a = (0::'a))"
map_poly_hom_coeff_lift: fixes a :: "int" shows "of_int_poly [:a:] = [:of_int a:]"
set_cases_2: fixes e :: "'a"   and a :: "('b, 'a) ta_rule" assumes "e \<in> set2_ta_rule a" obtains z1 :: "'b" and z3 :: "'b list" where "a = z1 \<rightarrow> e z3"
i_closed: fixes a :: "'a" assumes "a \<in> carrier G" shows "\<rho> a \<in> carrier G"
univ_poly_units: fixes K :: "'a set" assumes "subfield K R" shows "Units (K [X]) = {[k] |k. k \<in> K - {\<zero>}}"
comp_left_one: fixes x :: "'c \<times> 'c \<Rightarrow> 'd" shows "mone \<odot> x = x"
is_extensional: fixes f :: "'c local.term set" assumes "\<not> \<F>C.arr f" shows "E.map f = D.null"
step: shows "\<Gamma>\<turnstile> (pcs, css, s) \<rightarrow> (cs', css'css, t)"
insertE: fixes a :: "'e"   and b :: "'e"   and A :: "'e set" assumes "a \<in> insert b A" obtains "a = b"   | "a \<in> A"
init_imp_prop_invs: fixes c :: "('b, 't, 'loc) Combined.configuration" assumes "InitConfig c" shows "\<forall>p. prop_invs (prop_config c p)"
basis_card_eq_dim: fixes B :: "'b set"   and V :: "'b set" assumes "B \<subseteq> V"   and "V \<subseteq> local.span B"   and "local.independent B" shows "card B = local.dim V"
Accessible_right_lang_eq: fixes q :: "hf" assumes "q \<in> accessible" shows "Accessible.right_lang q = right_lang q"
hyps: fixes i :: "nat" assumes "mem_idx x as = Some i" shows "\<forall>j<i. as ! j \<noteq> x"
card_H_eq: fixes G :: "('a, 'b) monoid_scheme"   and p :: "nat"   and a :: "nat"   and m :: "nat"   and M1 :: "'a set"   and M :: "'a set set" assumes "sylow_central G p a m M1 M" shows "card {g \<in> carrier G. M1 #>\<^bsub>G\<^esub> g = M1} = p ^ a"
not_Carmichael_numberD: fixes n :: "nat" assumes "\<not> Carmichael_number n"   and "\<not> prime n"   and "1 < n" shows "\<exists>a\<in>{2..<n}. fermat_witness a n \<and> coprime a n"
Quotient_refl2: fixes R :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and Abs :: "'d \<Rightarrow> 'e"   and Rep :: "'e \<Rightarrow> 'd"   and T :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and r :: "'d"   and s :: "'d" assumes "Quotient R Abs Rep T"   and "R r s" shows "R s s"
prime_gt_1_nat: fixes p :: "nat" assumes "normalization_semidom_class.prime p" shows "1 < p"
Sura_Bura_compact: fixes S :: "'b set"   and C :: "'b set" assumes "compact S"   and "C \<in> components S" shows "C = \<Inter> {T. C \<subseteq> T \<and> openin (top_of_set S) T \<and> closedin (top_of_set S) T}"
hoare_pre: fixes R :: "'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'b"   and Q :: "'b \<Rightarrow> bool"   and P :: "'b \<Rightarrow> bool" assumes "\<lbrace>R\<rbrace> f \<lbrace>Q\<rbrace>"   and "\<And>s. P s \<Longrightarrow> R s" shows "\<lbrace>P\<rbrace> f \<lbrace>Q\<rbrace>"
os2: fixes M1 :: "lam"   and M2 :: "lam"   and M3 :: "lam" assumes " M1 \<longrightarrow>\<^sub>1\<^sup>* M2"   and " M2 \<longrightarrow>\<^sub>1 M3" shows " M1 \<longrightarrow>\<^sub>1\<^sup>* M3"
BinOpThrow2: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and e\<^sub>1 :: "char list exp"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and v\<^sub>1 :: "val"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and e\<^sub>2 :: "char list exp"   and e :: "char list exp"   and s\<^sub>2 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and bop :: "bop" assumes "P \<turnstile> \<langle>e\<^sub>1,s\<^sub>0\<rangle> \<Rightarrow> \<langle>Val v\<^sub>1,s\<^sub>1\<rangle>"   and "P \<turnstile> \<langle>e\<^sub>2,s\<^sub>1\<rangle> \<Rightarrow> \<langle>throw e,s\<^sub>2\<rangle>" shows "P \<turnstile> \<langle>e\<^sub>1 \<guillemotleft>bop\<guillemotright> e\<^sub>2,s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw e,s\<^sub>2\<rangle>"
distrib_lincomb: fixes rs :: "'f list"   and ms :: "'v list" assumes "set rs \<subseteq> UNIV"   and "set ms \<subseteq> V" shows "T (rs \<bullet>\<cdot> ms) = rs \<bullet>\<star> map T ms"
conforms_raise_if: fixes c :: "bool"   and xn :: "xname"   and x :: "abopt"   and s :: "st"   and G :: "prog"   and L :: "lenv" shows "(raise_if c xn x, s)\<Colon>\<preceq>(G, L) = (x, s)\<Colon>\<preceq>(G, L)"
d: shows "wordinterval_to_set (prefix_to_wordinterval (PrefixMatch a LENGTH('a))) \<subseteq> wordinterval_to_set r"
snth_sset: fixes s :: "'a stream"   and n :: "nat" shows "s !! n \<in> i.R s"
agent_abs_tObsC_abs: fixes t :: "('a, 'es, 'as) BEState Trace"   and x :: "('a, 'es, 'as) BEState \<times> ('a, 'es, 'as) BEState"   and a :: "'a"   and y :: "('a, 'es, 'as) BEState \<times> ('a, 'es, 'as) BEState" assumes "t \<in> SPR.jkbpC"   and "x \<in> agent_abs a t"   and "y \<in> agent_abs a t" shows "(x, y) \<in> relations (spr_repMC (tObsC_abs t)) a"
absorb_iff1: fixes a :: "'a"   and b :: "'a" shows "(a \<sqsubseteq> b) = (ord.min (\<sqsubseteq>) a b = a)"
to_mregex_to_mregex_exec: shows "case to_mregex r of (mr, \<phi>s) \<Rightarrow> to_mregex_exec r [] = (Monitor.shift mr (length []), [] @ \<phi>s)"
o_apply: fixes f :: "'e \<Rightarrow> 'd"   and g :: "'f \<Rightarrow> 'e"   and x :: "'f" shows "(f \<circ> g) x = f (g x)"
2: shows "(X, X #[[x' \<and> x]]_xs) \<in> qSwapped"
ord_resolve_rename_empty_main_prem: fixes Cs :: "'a literal multiset list"   and AAs :: "'a multiset list"   and As :: "'a list"   and \<sigma> :: "'s"   and E :: "'a literal multiset" shows "\<not> ord_resolve_rename Cs {#} AAs As \<sigma> E"
arg_cong2: fixes a :: "'b"   and b :: "'b"   and c :: "'d"   and d :: "'d"   and f :: "'b \<Rightarrow> 'd \<Rightarrow> 'e" assumes "a = b"   and "c = d" shows "f a c = f b d"
map\<^sub>1_simps_4: fixes f :: "'c" assumes "C.ide f" shows "D.dom (\<rho>\<^sub>1 f) = H f \<star>\<^sub>D \<rho>\<^sub>0 (src\<^sub>C f)"
internal_iface_name_subset_def: fixes i1 :: "char list"   and i2 :: "char list" shows "internal_iface_name_subset i1 i2 = (case (iface_name_is_wildcard i1, iface_name_is_wildcard i2) of (True, True) \<Rightarrow> length i2 \<le> length i1 \<and> take (length i2 - 1) i1 = butlast i2 | (True, False) \<Rightarrow> False | (False, True) \<Rightarrow> take (length i2 - 1) i1 = butlast i2 | (False, False) \<Rightarrow> i1 = i2)"
read_only_share_acquired_all_shared: fixes \<O> :: "nat set"   and sb :: "'b memref list"   and \<S> :: "nat \<Rightarrow> bool option"   and a :: "nat" assumes "weak_sharing_consistent \<O> sb"   and "\<O> \<inter> read_only \<S> = {}"   and "a \<in> read_only (share sb \<S>)"   and "a \<in> \<O> \<union> all_acquired sb" shows "a \<in> all_shared sb"
intros: fixes n :: "'node"   and h :: "'outPort"   and p :: "'inPort" assumes "hyps n h = Some p" shows "h \<in> hyps_for' n p"
phase_rel_invD: fixes s :: "'a process_name \<Rightarrow> ('b, 'a, 'c, 'd) Model.local_state" assumes "phase_rel_inv s" shows "(\<forall>m. sys_ghost_hs_in_sync m s, sys_ghost_hs_phase s, gc_phase s, sys_phase s, filter is_mw_Phase (sys_mem_store_buffers gc s)) \<in> phase_rel"
eq: fixes x :: "nat" shows "(\<Sum>\<^sub>a(m, n)\<in>{(m, n). m * n = x}. f m * g n / nat_power x s) = dirichlet_prod f g x / nat_power x s"
document_ptr_kinds_eq2_h2: shows "|h2 \<turnstile> document_ptr_kinds_M|\<^sub>r = |h3 \<turnstile> document_ptr_kinds_M|\<^sub>r"
minf_8: fixes t :: "'d" shows "\<exists>z. \<forall>x<z. (t \<le> x) = False"
chop_assoc2: fixes r :: "real_int"   and r1 :: "real_int"   and r2 :: "real_int"   and r3 :: "real_int"   and r4 :: "real_int" shows "R_Chop(r,r1,r2) \<and> R_Chop(r1,r3,r4) \<longrightarrow> R_Chop(r,r3,Abs_real_int (left r4, right r2)) \<and> R_Chop(Abs_real_int (left r4, right r2),r4,r2)"
bounds_4: shows "\<B>\<^sub>u (l ! j) = \<B>\<^sub>u (l ! i)"
finfun_Abs_finfun_curry: fixes f :: "'a \<times> 'b \<Rightarrow> 'c" assumes "f \<in> finfun" shows "(\<lambda>a. Abs_finfun (curry f a)) \<in> finfun"
natural_transformation_axioms: shows "natural_transformation (\<cdot>\<^sub>A) (\<cdot>\<^sub>A) FG.map A.map \<epsilon>"
perp_in_per: fixes B :: "'p"   and A :: "'p"   and C :: "'p" assumes "B PerpAt A B B C " shows "Per A B C"
kth_state_def: fixes \<sigma> :: "'var \<Rightarrow> 'val"   and k :: "nat" shows "\<sigma>\<^bsup>k\<^esup> = snd ((local.step ^^ k) (local.entry, \<sigma>))"
underS_incl_iff: fixes r :: "('a \<times> 'a) set"   and a :: "'a"   and b :: "'a" assumes "Linear_order r"   and "a \<in> Field r"   and "b \<in> Field r" shows "(underS r a \<subseteq> underS r b) = ((a, b) \<in> r)"
MyhillNerode_dfa_def: fixes L :: "'b list set"   and R :: "('b list \<times> 'b list) set"   and A :: "'b list set"   and n :: "nat"   and h :: "'b list set \<Rightarrow> hf" shows "MyhillNerode_dfa L R A n h \<equiv> (equiv UNIV R \<and> right_invariant R) \<and> L = R `` A \<and> bij_betw h (UNIV // R) (hfset (ord_of n))"
the_inv_into_map_def: fixes V :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and x :: "'d" shows "the_inv_into_map V f x = (if x \<in> f ` V then Some (the_inv_into V f x) else None)"
intros: fixes P :: "pi"   and Q :: "pi" assumes "P \<leadsto><weakBisim> Q"   and "Q \<approx> P" shows "P \<approx> Q"
LeftDerivationIntro_examine_rule: fixes \<alpha> :: "symbol list"   and i :: "nat"   and r :: "symbol \<times> symbol list"   and ix :: "nat"   and D :: "(nat \<times> symbol \<times> symbol list) list"   and j :: "nat"   and \<gamma> :: "symbol list"   and \<alpha>1 :: "symbol list"   and M :: "symbol"   and \<alpha>2 :: "symbol list" assumes "LeftDerivationIntro \<alpha> i r ix D j \<gamma>"   and "splits_at \<alpha> i \<alpha>1 M \<alpha>2" shows "\<exists>\<eta>. M = fst r \<and> \<eta> = snd r \<and> (M, \<eta>) \<in> \<RR>"
zero_vec_def: shows "0 \<equiv> \<chi>i. (0::'d)"
emeasure_lim_sequence_emb_I0o: fixes x :: "'a"   and X :: "(nat \<Rightarrow> 'a) set"   and n :: "nat" assumes "x \<in> space M"   and "X \<in> sets (Pi\<^sub>M {0..<n} (\<lambda>i. M))" shows "emeasure (lim_sequence x) (PF.emb UNIV {0..<n} X) = emeasure (Ionescu_Tulcea.C (K' x) (\<lambda>_. M) 0 n (\<lambda>x. undefined)) X"
b_e_type_checker_complete: fixes \<C> :: "t_context"   and es :: "b_e list"   and tn :: "t list"   and tm :: "t list" assumes "\<C> \<turnstile> es : tn _> tm" shows "b_e_type_checker \<C> es (tn _> tm)"
IH: shows "[(n - 1) ^ m = n - 1] (mod n)"
OT_14_P1_sec: fixes R1_12_0T :: "nat \<Rightarrow> bool \<times> bool \<Rightarrow> bool \<Rightarrow> 'v_OT121 spmf"   and n :: "nat"   and m1 :: "bool \<times> bool \<times> bool \<times> bool"   and m2 :: "bool \<times> bool"   and D :: "(bool \<times> bool \<times> bool \<times> bool) \<times> (bool \<times> bool \<times> bool \<times> bool \<times> bool \<times> bool) \<times> 'v_OT121 \<times> 'v_OT121 \<times> 'v_OT121 \<Rightarrow> bool spmf" shows "OT_14.adv_P1 R1_12_0T n m1 m2 D \<le> 3 * adv_OT12 n"
1: shows "\<chi>.A.map \<circ> F = A'.map"
butlast_cs_not_cd: fixes \<pi> :: "nat \<Rightarrow> 'n"   and m :: "nat"   and n :: "nat" assumes "butlast (cs\<^bsup>\<pi>\<^esup> m) = butlast (cs\<^bsup>\<pi>\<^esup> n)" shows "\<not> m cd\<^bsup>\<pi>\<^esup>\<rightarrow> n"
Pbij_inverseD: fixes l1 :: "Location"   and l2 :: "Location"   and \<beta> :: "(Location \<times> Location) set" assumes "(l1, l2) \<in> Pbij_inverse \<beta>" shows "(l2, l1) \<in> \<beta>"
raw_ports_compress_dst_MatchExpr: fixes \<beta> :: "'b common_primitive \<Rightarrow> ('b, 'c) tagged_packet_scheme \<Rightarrow> ternaryvalue"   and pss :: "ipt_l4_ports list"   and m :: "ipt_l4_ports"   and \<alpha> :: "action \<Rightarrow> ('b, 'c) tagged_packet_scheme \<Rightarrow> bool"   and a :: "action"   and p :: "('b, 'c) tagged_packet_scheme" assumes "primitive_matcher_generic \<beta>"   and "l4_ports_compress pss = MatchExpr m" shows "matches (\<beta>, \<alpha>) (Match (Dst_Ports m)) a p = matches (\<beta>, \<alpha>) (alist_and (map (Pos \<circ> Dst_Ports) pss)) a p"
simps_4: fixes b :: "bexp"   and C\<^sub>1 :: "acom"   and C\<^sub>2 :: "acom"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> bool" shows "pre (IF b THEN C\<^sub>1 ELSE C\<^sub>2) Q = (\<lambda>s. if bval b s then pre C\<^sub>1 Q s else pre C\<^sub>2 Q s)"
fmlookup_fmmap_keys: fixes f :: "'d \<Rightarrow> 'e \<Rightarrow> 'c"   and m :: "('d, 'e) fmap"   and x :: "'d" shows "fmlookup (fmmap_keys f m) x = map_option (f x) (fmlookup m x)"
reds: shows "P \<turnstile> \<langle>es,s,b\<rangle> [\<rightarrow>]* \<langle>es'',s'',b''\<rangle>"
elims_2: fixes x :: "('f, 'g, 'e, 'd, 'h) qAbs" assumes "FixVars TYPE('d) TYPE('e)"   and "FixVars.qGoodAbs x" obtains xs :: "'e" and x :: "'d" and X :: "('f, 'g, 'e, 'd, 'h) qTerm" where "xa__ = qAbs xs x X" and "FixVars.qGood X"
WHILEIT_to_monadic: fixes I :: "'c \<Rightarrow> bool"   and b :: "'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'c nres"   and s :: "'c" shows "WHILE\<^sub>T\<^bsup>I\<^esup> b f s = monadic_WHILEIT I (\<lambda>s. RETURN (b s)) f s"
tmap_is_TNil: fixes xs :: "('a, 'b) tllist"   and f :: "'a \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'd" assumes "is_TNil xs" shows "tmap f g xs = TNil (g (terminal xs))"
\<mu>LTL_implies_valid_prefix: fixes \<phi> :: "'a ltln"   and w :: "nat \<Rightarrow> 'a set" assumes "\<phi> \<in> \<mu>LTL"   and "w \<Turnstile>\<^sub>n \<phi>" shows "\<exists>i. af \<phi> (prefix i w) \<sim> true\<^sub>n"
lemma_2_10_27: fixes a :: "'a"   and b :: "'a" shows "a \<le> (a r\<rightarrow> b) l\<rightarrow> a"
closure_subset: shows "closure C \<subseteq> inv_chart c ` cball (apply_chart c x) e"
\<tau>: shows "natural_transformation (\<cdot>\<^sup>o\<^sup>p) (\<cdot>\<^sub>S) (Y a) (Y a') \<tau>"
neg_equal_0_iff_equal: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and a :: "'c" assumes "class.group_add minus plus zero uminus" shows "(uminus a = zero) = (a = zero)"
memFreshChain_1: fixes p :: "'a"   and l :: "'a list"   and xvec :: "name list" assumes "p mem l"   and "xvec \<sharp>* l" shows "xvec \<sharp>* p"
h1: shows "\<forall>i<n. FlexRayController (nReturn i) recv (nC i) (nStore i) (nSend i) (nGet i)"
930: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "- (x \<sqinter> (y \<squnion> z)) \<sqinter> (x \<sqinter> y) \<squnion> - (x \<sqinter> (y \<squnion> z)) \<sqinter> (x \<sqinter> z) = bot"
eucl_truncate_down_real_def: fixes prec :: "nat"   and b :: "real" shows "eucl_truncate_down prec b = truncate_down prec b"
ANR_closed_Un: fixes S :: "'a set"   and T :: "'a set" assumes "closed S"   and "closed T"   and "ANR S"   and "ANR T"   and "ANR (S \<inter> T)" shows "ANR (S \<union> T)"
prj1_idem: fixes X :: "'M set"   and i :: "'M" assumes "X \<in> \<D>"   and "i \<in> Basis" shows "prj1 (vf X - uf X) = (vf X - uf X) \<bullet> i"
insert_commute: fixes x :: "'d"   and y :: "'d"   and A :: "'d set" shows "insert x (insert y A) = insert y (insert x A)"
simps_2: fixes n :: "nat" shows "seqseq (Suc n) = seqseq n \<circ> reduce (seqseq n) n"
semiconnex_irreflexive_def: fixes A :: "'c set"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool" shows "semiconnex_irreflexive A less \<equiv> semiconnex A less \<and> irreflexive A less"
is_tangle_make_vert_right: fixes xs :: "wall"   and n :: "nat" assumes "is_tangle_diagram xs" shows "is_tangle_diagram (xs \<otimes> basic (make_vert_block n))"