hcju/isabelleps huggingface dataset:
nat_power_diff: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and nat_power :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and n :: "nat"   and a :: "'a"   and b :: "'a" assumes "class.nat_power_field one times plus zero nat_power divide inverse minus uminus"   and "0 < n" shows "nat_power n (minus a b) = divide (nat_power n a) (nat_power n b)"
elim_singleton: fixes x :: "'a"   and s :: "'a"   and x' :: "'b"   and t :: "'b" assumes "x \<in> {s} \<and> x' \<in> {t}" obtains "x = s \<and> x' = t"
top_on_top: fixes X :: "char list set" shows "top_on\<^sub>o \<top> X"
sorted_list_of_set_2: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and A :: "'c set" assumes "class.linorder less_eq less" shows "linorder.sorted less_eq (linorder.sorted_list_of_set less_eq A)"
op_norm_diag_mat_eq: fixes f :: "'b \<Rightarrow> real" shows "\<parallel>diag_mat f\<parallel>\<^sub>o\<^sub>p = Max {\<bar>f i\<bar> |i. i \<in> UNIV}"
meissel_mertens_integral_le: fixes a :: "real" assumes "1 < a" shows "norm (integral {a<..} (\<lambda>t. (\<MM> t - ln t) / (t * (ln t)\<^sup>2))) \<le> 2 / ln a"
set_r_ar_cos_ker: fixes A :: "('c, 'd) Ring_scheme"   and R :: "('e, 'f) Ring_scheme"   and f :: "'c \<Rightarrow> 'e" assumes "Ring A"   and "Ring R"   and "f \<in> rHom A R" shows "set_rcs (b_ag A) (ker\<^bsub>A,R\<^esub> f) = set_ar_cos A (ker\<^bsub>A,R\<^esub> f)"
rule_sub_lem_1_d: fixes v :: "i"   and \<psi> :: "'c \<Rightarrow> \<o>"   and \<chi> :: "'c \<Rightarrow> \<o>" assumes "\<And>x. [\<^bold>\<box>(\<psi> x \<^bold>\<equiv> \<chi> x) in v]" shows "[(\<^bold>\<forall>\<alpha>. \<psi> \<alpha>) \<^bold>\<equiv> (\<^bold>\<forall>\<alpha>. \<chi> \<alpha>) in v]"
AlphaNegAnd: fixes A :: "('a, 'b) form"   and B :: "('a, 'b) form"   and G :: "('a, 'b) form list" assumes "\<turnstile> Neg A # Neg B # G" shows "\<turnstile> Neg (And A B) # G"
assms: shows "z \<in> set (fst (list_of_oalist xs))"
S3: shows "c_fst (c_hd (c_cons (c_pair x y) z)) = x"
set_less_aux_trans: fixes A :: "'a set"   and B :: "'a set"   and C :: "'a set" assumes "A \<sqsubset>' B"   and "B \<sqsubset>' C" shows "A \<sqsubset>' C"
restrict_assignment: fixes b :: "'b ifex"   and ass :: "'b \<Rightarrow> bool"   and var :: "'b"   and val :: "bool" shows "val_ifex b (ass(var := val)) = val_ifex (restrict b var val) ass"
b_pos: shows "\<forall>\<^sub>F x in at_top. 1 < b x"
nths_complement_ind_in_set: fixes A :: "nat set"   and B :: "nat set"   and xs :: "'b list" assumes "A \<inter> B = {}"   and "card (A \<union> B) = length xs" shows "nths xs (- ind_in_set (A \<union> B) ` A) = nths xs (ind_in_set (A \<union> B) ` B)"
compatibleI: fixes x :: "'a"   and z :: "'a"   and y :: "'a" assumes "x \<sqsubseteq> z"   and "y \<sqsubseteq> z"   and "\<And>a. \<lbrakk>x \<sqsubseteq> a; y \<sqsubseteq> a\<rbrakk> \<Longrightarrow> z \<sqsubseteq> a" shows "compatible x y"
dim_1: shows "dim_row (subresultant_mat J f g) = degree f - J + (degree g - J)"
induct: fixes P2 :: "'d tree list \<Rightarrow> bool"   and P1 :: "'d tree \<Rightarrow> bool"   and tree :: "'d tree"   and tree_list :: "'d tree list" assumes "\<And>x1 x2. P2 x2 \<Longrightarrow> P1 (NODE x1 x2)"   and "P2 []"   and "\<And>x1 x2. \<lbrakk>P1 x1; P2 x2\<rbrakk> \<Longrightarrow> P2 (x1 # x2)" shows "P1 tree \<and> P2 tree_list"
sum_7: fixes f :: "7 \<Rightarrow> 'a" shows "sum f UNIV = f 1 + f 2 + f 3 + f 4 + f 5 + f 6 + f 7"
to_tree_order_ptr_in_result: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and nodes :: "(_) object_ptr list" assumes "h \<turnstile> to_tree_order ptr \<rightarrow>\<^sub>r nodes" shows "ptr \<in> set nodes"
image_stretch_interval: fixes m :: "'b \<Rightarrow> real"   and a :: "'b"   and b :: "'b" shows "(\<lambda>x. \<Sum>k\<in>Basis. (m k * (x \<bullet> k)) *\<^sub>R k) ` cbox a b = (if cbox a b = {} then {} else cbox (\<Sum>k\<in>Basis. min (m k * (a \<bullet> k)) (m k * (b \<bullet> k)) *\<^sub>R k) (\<Sum>k\<in>Basis. max (m k * (a \<bullet> k)) (m k * (b \<bullet> k)) *\<^sub>R k))"
\<gamma>_rep_cases: fixes p :: "int extended \<times> int extended" shows "\<gamma>_rep p = (case p of (Fin i, Fin j) \<Rightarrow> {i..j} | (Fin i, \<infinity>) \<Rightarrow> {i..} | (Fin i, -\<infinity>) \<Rightarrow> {} | (\<infinity>, b) \<Rightarrow> {} | (-\<infinity>, Fin i) \<Rightarrow> {..i} | (-\<infinity>, \<infinity>) \<Rightarrow> UNIV | (-\<infinity>, -\<infinity>) \<Rightarrow> {})"
zero: assumes "vectorspace E" shows "(0::'a) \<in> H"
ide_in_hom: fixes a :: "'a" shows "A.ide a = \<guillemotleft>a : a \<rightarrow>\<^sub>A a\<guillemotright>"
dfa_of_pf_induct: fixes P :: "nat \<Rightarrow> pf \<Rightarrow> bool"   and a0 :: "nat"   and a1 :: "pf" assumes "\<And>n ks l. P n (Eq ks l)"   and "\<And>n ks l. P n (Le ks l)"   and "\<And>n p q. \<lbrakk>P n p; P n q\<rbrakk> \<Longrightarrow> P n (And p q)"   and "\<And>n p q. \<lbrakk>P n p; P n q\<rbrakk> \<Longrightarrow> P n (Or p q)"   and "\<And>n p q. \<lbrakk>P n p; P n q\<rbrakk> \<Longrightarrow> P n (Imp p q)"   and "\<And>n p. P (Suc n) p \<Longrightarrow> P n (Exist p)"   and "\<And>n p. P n (Neg (Exist (Neg p))) \<Longrightarrow> P n (Forall p)"   and "\<And>n p. P n p \<Longrightarrow> P n (Neg p)" shows "P a0 a1"
UN_box_eq_UNIV: shows "(\<Union>i. box (- (real i *\<^sub>R One)) (real i *\<^sub>R One)) = UNIV"
assms_1: fixes x :: "'a" shows "f x = \<langle>t, x\<rangle>"
drop_zip: fixes n :: "nat"   and xs :: "'a list"   and ys :: "'b list" shows "drop n (zip xs ys) = zip (drop n xs) (drop n ys)"
cong_power_int_code: fixes b :: "int"   and e :: "nat"   and x :: "int"   and m :: "int" shows "[b ^ e = x] (mod m) = (mod_exp b e m = x mod m)"
lappend_eq_lappend_conv: fixes xs :: "'b llist"   and us :: "'b llist"   and ys :: "'b llist"   and vs :: "'b llist" assumes "llength xs = llength us" shows "(xs $ ys = us $ vs) = (xs = us \<and> (lfinite xs \<longrightarrow> ys = vs))"
coeff_0_pol_0: fixes c :: "nat \<times> (nat \<Rightarrow> 'a)"   and k :: "nat" assumes "pol_coeff S c"   and "k \<le> fst c" shows "(\<forall>j\<le>k. snd c j = \<zero>\<^bsub>S\<^esub>) = (polyn_expr R X k c = \<zero>)"
abs_eq: shows "top = Linear_Aggregation_Algebras.unit.Abs_unit 1"
favorites_nonempty: fixes i :: "'agent" assumes "i \<in> agents" shows "favorites R i \<noteq> {}"
fresh_fact': fixes x :: "name"   and v :: "trm"   and t :: "trm" assumes "x \<sharp> v" shows "x \<sharp> t[x::=v]"
AT_fps_Suc: fixes f :: "nat \<Rightarrow> real"   and n :: "nat" shows "AT_fps f (Suc n) = (fps_X - 1) * fps_deriv (AT_fps f n)"
that: fixes V :: "'a set"   and E :: "('a \<times> 'a) set" assumes "g = graph V E"   and "finite V"   and "finite E"   and "E \<subseteq> V \<times> V"   and "sym E"   and "irrefl E" shows "thesis"
space_restrict_space: fixes M :: "'b measure"   and \<Omega> :: "'b set" shows "space (restrict_space M \<Omega>) = \<Omega> \<inter> space M"
valid_characteristic_formula: fixes P' :: "'state" shows "P' \<Turnstile> characteristic_weak_formula P = (\<forall>x\<in>{distinguishing_weak_formula P Q |Q. \<not> P \<equiv>\<cdot> Q}. P' \<Turnstile> x)"
fls_regpart_nth: fixes f :: "'a fls"   and n :: "nat" shows "fls_regpart f $ n = fls_nth f (int n)"
StepRecCond1: fixes prog :: "cmd"   and c :: "cmd"   and s :: "char list \<Rightarrow> val option"   and l :: "nat"   and c' :: "cmd"   and s' :: "char list \<Rightarrow> val option"   and l' :: "nat"   and b :: "expr"   and c\<^sub>2 :: "cmd" assumes "prog \<turnstile> \<langle>c,s,l\<rangle> \<leadsto> \<langle>c',s',l'\<rangle>" shows "if (b) prog else c\<^sub>2 \<turnstile> \<langle>c,s,l + 1\<rangle> \<leadsto> \<langle>c',s',l' + 1\<rangle>"
index_mat_multcol_5: fixes k :: "nat"   and a :: "'a"   and A :: "'a mat" shows "dim_col (multcol k a A) = dim_col A"
invertible_left_inverse: fixes u :: "'a" assumes "sub.invertible u"   and "u \<in> N" shows "sub.inverse u \<cdot> u = \<one>"
holomorphic_log: fixes f :: "complex \<Rightarrow> complex" assumes "connected S"   and "f holomorphic_on S"   and "\<And>z. z \<in> S \<Longrightarrow> f z \<noteq> 0"   and "\<And>f. f holomorphic_on S \<Longrightarrow> \<exists>h. \<forall>z\<in>S. (h has_field_derivative f z) (at z)" shows "\<exists>g. g holomorphic_on S \<and> (\<forall>z\<in>S. f z = exp (g z))"
subset_minus_empty: fixes A :: "'b set"   and B :: "'b set" assumes "A \<subseteq> B" shows "A - B = {}"
union_code: fixes xs :: "'c list"   and ys :: "'c list" shows "mset xs + mset ys = mset (xs @ ys)"
timpls_transformable_to_inv_1: fixes TI :: "('e set \<times> 'e set) list"   and f :: "('f, 'g, 'e) prot_fun"   and T :: "(('f, 'g, 'e) prot_fun, 'h) Term.term list"   and g :: "('f, 'g, 'e) prot_fun"   and S :: "(('f, 'g, 'e) prot_fun, 'h) Term.term list" assumes "timpls_transformable_to TI (Fun f T) (Fun g S)" shows "length T = length S"
trm_sound: fixes \<theta> :: "trm"   and \<nu>' :: "fin_string \<Rightarrow> real"   and r :: "real"   and \<nu> :: "fin_string + fin_string \<Rightarrow> 32 Word.word" assumes "[\<theta>]\<nu>' \<down> r"   and "\<nu> REP \<nu>'" shows "[\<theta>]<>\<nu> \<equiv>\<^sub>P r"
tendsto_add_ereal_PInf: fixes y :: "ereal"   and f :: "'b \<Rightarrow> ereal"   and F :: "'b filter"   and g :: "'b \<Rightarrow> ereal" assumes "y \<noteq> - \<infinity>"   and "(f \<longlongrightarrow> \<infinity>) F"   and "(g \<longlongrightarrow> y) F" shows "((\<lambda>x. f x + g x) \<longlongrightarrow> \<infinity>) F"
admits_diagonal_reduction_intro: fixes P :: "'b mat"   and A :: "'b mat"   and Q :: "'b mat" assumes "P \<in> carrier_mat (dim_row A) (dim_row A)"   and "Q \<in> carrier_mat (dim_col A) (dim_col A)"   and "invertible_mat P"   and "invertible_mat Q"   and "Smith_normal_form_mat (P * A * Q)" shows "admits_diagonal_reduction A"
d3_2: fixes p :: "('a \<times> 'a set) set" assumes "p tagged_division_of cbox a b"   and "d3 fine p" shows "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x *\<^sub>R g x) - P) < e"
real_of_float_min: fixes x :: "float"   and y :: "float" shows "real_of_float (min x y) = min (real_of_float x) (real_of_float y)"
CauchyD: fixes X :: "nat \<Rightarrow> 'a"   and e :: "real" assumes "Cauchy X"   and "0 < e" shows "\<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. norm (X m - X n) < e"
KS: fixes i :: "'b" assumes "i \<in> K" shows "x i \<in> S \<and> y i \<in> S"
dom_subset: shows "dom \<pi> \<subseteq> \<^bold>G \<phi>"
OF_same_priority_match2_def: fixes \<gamma> :: "'b set \<Rightarrow> 'c \<Rightarrow> bool"   and flow_entries :: "('b, 'a) flow_entry_match list"   and packet :: "'c" shows "OF_same_priority_match2 \<gamma> flow_entries packet \<equiv> let s = {ofe_action f |f. f \<in> set flow_entries \<and> \<gamma> (ofe_fields f) packet \<and> (\<forall>fo\<in>set flow_entries. ofe_prio f < ofe_prio fo \<longrightarrow> \<not> \<gamma> (ofe_fields fo) packet)} in case card s of 0 \<Rightarrow> NoAction | Suc 0 \<Rightarrow> Action (the_elem s) | Suc (Suc nata) \<Rightarrow> Undefined"
weak_lens_def: fixes x :: "'d \<Longrightarrow> 'e" shows "weak_lens x \<equiv> \<forall>\<sigma> v. get\<^bsub>x\<^esub> (put\<^bsub>x\<^esub> \<sigma> v) = v"
sign_simps_35: fixes am :: "'c"   and bm :: "'c"   and ck :: "'c" shows "(am + bm) * ck = am * ck + bm * ck"
simps_4: fixes uv :: "'e list"   and uw :: "(('d, 'e) Term.term \<times> ('d, 'e) Term.term) list"   and F' :: "(('d, 'e) Term.term \<times> ('d, 'e) Term.term) list" shows "setops\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<forall>uv\<langle>\<or>\<noteq>: uw \<or>\<notin>: F'\<rangle> = set F'"
rational_preference_def: fixes carrier :: "'a set"   and relation :: "('a \<times> 'a) set" shows "rational_preference carrier relation \<equiv> preference carrier relation \<and> rational_preference_axioms carrier relation"
AbstForm_imp_Ord: fixes v :: "hf"   and u :: "hf"   and x :: "hf"   and x' :: "hf" assumes "AbstForm v u x x'" shows "Ord v"
bound_child_parent_card_state_set_cons: fixes P :: "(('c, bool) fmap \<times> ('c, bool) fmap) set \<Rightarrow> bool"   and f :: "(('c, bool) fmap \<times> ('c, bool) fmap) set \<Rightarrow> nat" assumes "\<forall>PROB as s. P PROB \<and> as \<in> valid_plans PROB \<and> s \<in> valid_states PROB \<longrightarrow> (\<exists>as'. exec_plan s as = exec_plan s as' \<and> subseq as' as \<and> length as' < f PROB)" shows "\<forall>PROB s as. P PROB \<and> as \<in> valid_plans PROB \<and> s \<in> valid_states PROB \<longrightarrow> (\<exists>x. x \<in> PLS s as \<and> x < f PROB)"
vd1: fixes x :: "real" assumes "0 \<le> x"   and "x + a < 1"   and "x \<notin> (\<lambda>x. x - a) ` S" shows "vector_derivative (shiftpath a g) (at x) = vector_derivative g (at (x + a))"
no_ub: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a" assumes "class.dlo less_eq less" shows "\<exists>u. less x u"
set_mset_add_mset_insert: fixes a :: "'a"   and A :: "'a multiset" shows "set_mset (add_mset a A) = insert a (set_mset A)"
assms_2: shows "f \<approx>>\<^bsub>C\<^esub> g"
fun_app_lifted_def: fixes f\<^sub>T :: "('h \<Rightarrow> 'g Heap) Heap"   and x\<^sub>T :: "'h Heap" shows "Heap_Monad_Ext.fun_app_lifted f\<^sub>T x\<^sub>T \<equiv> f\<^sub>T \<bind> (\<bind>) x\<^sub>T"
left_inverse_linear: fixes scale :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and Basis :: "'e set"   and f :: "'e \<Rightarrow> 'e"   and g :: "'e \<Rightarrow> 'e" assumes "finite_dimensional_vector_space scale Basis"   and "linear scale scale f"   and "g \<circ> f = id" shows "linear scale scale g"
prop_prop2_3: fixes v :: "i"   and F :: "\<Pi>\<^sub>1" shows "[Propositional F \<^bold>\<rightarrow> \<^bold>\<box>Propositional F in v]"
assms: shows "P (\<lambda>n. f_inv (\<lambda>_. True) n + 1) = x"
assms_1: shows "x \<in>\<^sub>\<circ> vrrestriction D"
mult_neg_neg: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.linordered_ring_strict plus zero minus uminus less_eq less abs times"   and "less a zero"   and "less b zero" shows "less zero (times a b)"
phiNodes_of_base: fixes g :: "'g" shows "uninst_code.ssa.phiNodes_of (\<lambda>gb. p_g g s1_phis g) g = uninst_code.ssa.phiNodes_of (p_g g s1_phis) g"
comm_monoid_vec: fixes n :: "nat" shows "Group.comm_monoid (monoid_vec TYPE('a) n)"
is_env_model_ntou: fixes s :: "(nat \<times> form) list" shows "is_env (model s) ntou"
f_prime: fixes p :: "nat" assumes "prime p" shows "liouville_lambda p = - (1::'b)"
simps: fixes K :: "'c \<Rightarrow> 'c pmf"   and a1 :: "'c"   and a2 :: "'c stream" shows "MC_syntax.enabled K a1 a2 = (\<exists>\<omega> s. a1 = s \<and> a2 = \<omega> \<and> MC_syntax.enabled K (shd \<omega>) (stl \<omega>) \<and> shd \<omega> \<in> set_pmf (K s))"
ab_def_2: fixes n :: "nat" shows "A n = fst (AB n)"
aposs_tna_poss: fixes n :: "ant" assumes "n \<noteq> \<infinity>"   and "0 < n" shows "0 < tna n"
nsqn_sqn: fixes rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option"   and dip :: "nat" shows "nsqn rt dip \<le> sqn rt dip"
sat_dual_def: fixes A :: "'b mat"   and c :: "'b vec" shows "sat_dual A c = {y. [y\<^sub>v*A]=c \<and> (\<forall>i<dim_vec y. (0::'b) \<le> y $ i)}"
set_lfp_tail_rec_def: fixes f :: "'a set \<Rightarrow> 'a set"   and s :: "'a set" assumes "mono f" shows "set_lfp s f = (if f s \<subseteq> s then s else set_lfp (s \<union> f s) f)"
ctwo_ordLeq_Cinfinite: fixes r :: "('c \<times> 'c) set" assumes "Cinfinite r" shows "ctwo \<le>o r"
leq_antisym: fixes x :: "'b spmf"   and y :: "'b spmf" assumes "ord_spmf (=) x y"   and "ord_spmf (=) y x" shows "x = y"
card_2_iff: fixes S :: "'g set" shows "(card S = 2) = (\<exists>x y. S = {x, y} \<and> x \<noteq> y)"
emeasure_pmf_of_set: fixes S :: "'a set"   and A :: "'a set" assumes "S \<noteq> {}"   and "finite S" shows "emeasure (measure_pmf (pmf_of_set S)) A = ennreal (real (card (S \<inter> A)) / real (card S))"
RModuleEnd_def: fixes R :: "'a set"   and smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and M :: "'b set"   and T :: "'b \<Rightarrow> 'b" shows "RModuleEnd R smult M T \<equiv> RModuleHom R smult M smult T \<and> RModuleEnd_axioms M T"
suffixes_def: shows "suffixes \<equiv> rec_list [[]] (\<lambda>x xs xsa. xsa @ [x # xs])"
upper_dim: shows "\<forall>a b c p q. p \<noteq> q \<and> Cong a p a q \<and> Cong b p b q \<and> Cong c p c q \<longrightarrow> Bet a b c \<or> Bet b c a \<or> Bet c a b"
field_inverse: fixes a :: "'b" assumes "a \<noteq> (0::'b)" shows "inverse a * a = (1::'b)"
simps_4: fixes f1 :: "nat \<Rightarrow> 'a"   and f2 :: "'a"   and nat :: "nat" shows "(case enat nat of enat x \<Rightarrow> f1 x | \<infinity> \<Rightarrow> f2) = f1 nat"
Arg2pi_le_pi: fixes z :: "complex" shows "(Arg2pi z \<le> pi) = (0 \<le> Im z)"
max_prob_def: fixes s :: "'b"   and os :: "'a list" shows "max_prob s os = Max {emeasure (T' (I s)) {\<omega> \<in> space (stream_space (count_space UNIV)). \<exists>o \<omega>'. \<omega> = (s, o) ## zip as os @- \<omega>'} |as. length as = length os \<and> set as \<subseteq> \<S>}"
isNode2_def: fixes Kl :: "('b + 'c) list set"   and lab :: "('b + 'c) list \<Rightarrow> 'd + ('e, 'b, 'c) F2"   and kl :: "('b + 'c) list" shows "isNode2 Kl lab kl = (\<exists>x2. lab kl = Inr x2 \<and> F2set2 x2 = Succ1 Kl kl \<and> F2set3 x2 = Succ2 Kl kl)"
simps_1: fixes p1 :: "net_tree"   and p2 :: "net_tree" shows "wf_net_tree (p1 \<parallel> p2) = (net_tree_ips p1 \<inter> net_tree_ips p2 = {} \<and> wf_net_tree p1 \<and> wf_net_tree p2)"
if_doesnt_read: fixes x :: "'Var"   and e :: "'BExp"   and c\<^sub>1 :: "('Var, 'AExp, 'BExp) Stmt"   and c\<^sub>2 :: "('Var, 'AExp, 'BExp) Stmt"   and annos :: "'Var ModeUpd list" assumes "x \<notin> bexp_vars e" shows "doesnt_read (Stmt.If e c\<^sub>1 c\<^sub>2 \<otimes> annos) x"
locally_compact_Int: fixes S :: "'a set"   and t :: "'a set" assumes "locally compact S"   and "locally compact t" shows "locally compact (S \<inter> t)"
sel_14: fixes x1 :: "'b"   and x2 :: "'b"   and x3 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x4 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x5 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x6 :: "'b \<Rightarrow> 'b"   and x7 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x8 :: "'b \<Rightarrow> 'b"   and x9 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x10 :: "'b \<Rightarrow> 'b"   and x11 :: "'b \<Rightarrow> 'b"   and x12 :: "int \<Rightarrow> 'b"   and x13 :: "'b \<Rightarrow> int"   and x14 :: "'b \<Rightarrow> bool" shows "arith_ops_record.DP (Arith_Ops_Record x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14) = x14"
sets_measure_of_eq: fixes M :: "'b measure"   and \<mu> :: "'b set \<Rightarrow> ennreal" shows "sets (measure_of (space M) (sets M) \<mu>) = sets M"
us: shows "us \<equiv> getFrN (map snd txs) (tms r1 r2 # map fst txs) [] (length txs)"
R\<^sub>C_of_strong_low_bisim_mm: fixes \<R>\<^sub>A :: "((((Eg1.var, aexp, bexp) Stmt \<times> (Mode \<Rightarrow> Eg1.var set)) \<times> (Eg1.var \<Rightarrow> nat)) \<times> ((Eg1.var, aexp, bexp) Stmt \<times> (Mode \<Rightarrow> Eg1.var set)) \<times> (Eg1.var \<Rightarrow> nat)) set"   and \<R> :: "((((Eg1.var, aexp, bexp) Stmt \<times> (Mode \<Rightarrow> Eg1.var set)) \<times> (Eg1.var \<Rightarrow> nat)) \<times> ((var\<^sub>C, aexp\<^sub>C, bexp\<^sub>C) Stmt \<times> (Mode \<Rightarrow> var\<^sub>C set)) \<times> (var\<^sub>C \<Rightarrow> nat)) set"   and P :: "((((var\<^sub>C, aexp\<^sub>C, bexp\<^sub>C) Stmt \<times> (Mode \<Rightarrow> var\<^sub>C set)) \<times> (var\<^sub>C \<Rightarrow> nat)) \<times> ((var\<^sub>C, aexp\<^sub>C, bexp\<^sub>C) Stmt \<times> (Mode \<Rightarrow> var\<^sub>C set)) \<times> (var\<^sub>C \<Rightarrow> nat)) set" assumes "A.strong_low_bisim_mm \<R>\<^sub>A"   and "secure_refinement \<R>\<^sub>A \<R> P"   and "sym P" shows "conc.strong_low_bisim_mm (R\<^sub>C_of \<R>\<^sub>A \<R> P)"
Extend_subset: fixes S :: "(nat, 'b) form set"   and C :: "(nat, 'b) form set set"   and f :: "nat \<Rightarrow> (nat, 'b) form" shows "S \<subseteq> Extend S C f"
cond_nsg: fixes G :: "('e, 'f) Group_scheme"   and H :: "'e set" assumes "Group G"   and "G \<guillemotright> H "   and "\<forall>a\<in>carrier G. \<forall>h\<in>H. a \<cdot>\<^bsub>G\<^esub> h \<cdot>\<^bsub>G\<^esub> \<rho>\<^bsub>G\<^esub> a \<in> H" shows "G \<triangleright> H"
nvC_def: fixes c :: "('c, 'd) lit list" shows "nvC c \<equiv> \<Union> (set (map nvL c))"
kD_is_vD_and_iD: fixes rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option" shows "kD rt = vD rt \<union> iD rt"
l_type_wf\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_def: fixes type_wf :: "(_) heap \<Rightarrow> bool" shows "l_type_wf\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t type_wf \<equiv> \<forall>h. type_wf h \<longrightarrow> ShadowRootClass.type_wf h"
rep_eq: fixes x :: "\<Pi>\<^sub>2" shows "Semantics.d\<^sub>2 x = Some (eval\<Pi>\<^sub>2 x)"
simps_2: fixes f :: "'d \<Rightarrow> real \<Rightarrow> 'c"   and x1 :: "'d"   and x2 :: "real" shows "(case Sctn x1 x2 of Sctn x xa \<Rightarrow> f x xa) = f x1 x2"
ordered_resolvent_def: fixes P1 :: "'at Literal set"   and P2 :: "'at Literal set"   and C :: "'at Literal set" shows "ordered_resolvent P1 P2 C \<equiv> \<exists>A. C = P1 - {Pos A} \<union> (P2 - {Neg A}) \<and> strictly_maximal_literal P1 (Pos A) \<and> strictly_maximal_literal P2 (Neg A)"
outputPermFrame: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and M :: "'a"   and xvec :: "name list"   and N :: "'a"   and P' :: "('a, 'b, 'c) psi"   and p :: "(name \<times> name) list"   and yvec :: "name list"   and zvec :: "name list" assumes "\<Psi> \<rhd> P \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> P'"   and "set p \<subseteq> set yvec \<times> set zvec"   and "yvec \<sharp>* P"   and "zvec \<sharp>* P"   and "yvec \<sharp>* M"   and "zvec \<sharp>* M" shows "p \<bullet> \<Psi> \<rhd> P \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> P'"
distrib_right': fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(x + y) \<parallel> z = x \<parallel> z + y \<parallel> z"
elims: fixes x :: "call \<times> (nat \<Rightarrow> contour option) \<times> ((nat \<times> char list) \<times> contour \<Rightarrow> d option) \<times> contour"   and y :: "call \<times> (nat \<Rightarrow> 'a option) \<times> ((nat \<times> char list) \<times> 'a \<Rightarrow> 'a \<aproc> set) \<times> 'a" assumes "|x| = y" obtains c :: "call" and \<beta> :: "nat \<Rightarrow> contour option" and ve :: "(nat \<times> char list) \<times> contour \<Rightarrow> d option" and b :: "contour" where "x = (c, \<beta>, ve, b)" and "y = (c, |\<beta>|, |ve|, |b|)"
assms_4: shows "seg \<in> set (polychain_of Pc qs)"
sum_prod: fixes f :: "'c \<Rightarrow> 'a"   and g :: "'c \<Rightarrow> 'b"   and A :: "'c set" shows "(\<Sum>x\<in>A. (f x, g x)) = (sum f A, sum g A)"
inject_2: fixes x31 :: "'b ptrm"   and x32 :: "'b ptrm"   and y31 :: "'b ptrm"   and y32 :: "'b ptrm" shows "(PApp x31 x32 = PApp y31 y32) = (x31 = y31 \<and> x32 = y32)"
assms_1: shows "\<forall>(l, t, s)\<in>set D. (fv t \<union> fv s) \<inter> bvars\<^sub>l\<^sub>s\<^sub>s\<^sub>t A = {}"
forall_3': fixes P :: "3 \<Rightarrow> bool" shows "(\<forall>i. P i) = (P 0 \<and> P 1 \<and> P 2)"
intro: fixes G :: "('c, 'd) pre_digraph"   and M :: "'d pre_map"   and a :: "'d" assumes "bidel_arc G M a"   and "bidel_arc_tail_conn_axioms G M a" shows "bidel_arc_tail_conn G M a"
sum: shows "\<forall>w. O w z = (O w x \<or> O w y)"
simps_3: fixes c1 :: "('a, 'b) MWLsCom"   and c2 :: "('a, 'b) MWLsCom" shows "PPc (c1;c2) = PPc c1 @ PPc c2"