hcju/isabelleps huggingface dataset:
not_lock_ok_las'_conv: fixes ls :: "('c, 'd) locks"   and t :: "'d"   and las :: "'c lock_actions" shows "(\<not> lock_ok_las' ls t las) = (\<exists>l. \<not> lock_actions_ok' (ls $ l) t (las $ l))"
bra_vec_def: fixes v :: "complex vec" shows "bra_vec v \<equiv> row_vec v\<^sup>\<star>"
ZObis_coind: fixes theta :: "(('test, 'atom) com \<times> ('test, 'atom) com) set" assumes "sym theta"   and "theta \<subseteq> ZOretr (theta \<union> ZObis)" shows "theta \<subseteq> ZObis"
transfer_ex: fixes p :: "'b star \<Rightarrow> bool"   and P :: "nat \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<And>X. p (star_n X) \<equiv> \<forall>\<^sub>F n in \<U>. P n (X n)" shows "\<exists>x. p x \<equiv> \<forall>\<^sub>F n in \<U>. \<exists>x. P n x"
C1r: fixes y :: "'a"   and x :: "'a"   and z :: "'a" assumes "y \<cdot> x \<le> z \<cdot> y" shows "y \<cdot> x\<^sup>\<star> \<le> z\<^sup>\<star> \<cdot> y"
SHNat_eq: shows "\<nat> = {n. \<exists>N. n = hypnat_of_nat N}"
reg\<^sub>C_load_helper: fixes x\<^sub>C :: "var\<^sub>C"   and x :: "Eg1.var"   and mds :: "Mode \<Rightarrow> var\<^sub>C set"   and mem :: "var\<^sub>C \<Rightarrow> nat"   and v :: "nat" assumes "x\<^sub>C \<noteq> Eg2_var\<^sub>C_of_Eg1 x" shows "(\<langle>reg\<^sub>C \<leftarrow> aexp\<^sub>C.Load (Eg2_var\<^sub>C_of_Eg1 x), mds, mem(x\<^sub>C := v)\<rangle>\<^sub>C, \<langle>Stop, mds, mem(x\<^sub>C := v, reg\<^sub>C := mem (Eg2_var\<^sub>C_of_Eg1 x))\<rangle>\<^sub>C) = (\<langle>reg\<^sub>C \<leftarrow> aexp\<^sub>C.Load (Eg2_var\<^sub>C_of_Eg1 x), mds, mem(x\<^sub>C := v)\<rangle>\<^sub>C, \<langle>Stop, mds, mem(x\<^sub>C := v, reg\<^sub>C := (mem(x\<^sub>C := v)) (Eg2_var\<^sub>C_of_Eg1 x))\<rangle>\<^sub>C)"
bij_1: shows "bij_betw f {..<length xs} {..<length ys}"
M_MR: shows "(\<rightarrow>\<^sub>M) (M \<Otimes>\<^sub>M R) = (\<rightarrow>\<^sub>M) (N \<Otimes>\<^sub>M R)"
va1: shows "(\<lambda>x. f (g x) * vector_derivative g (at x)) integrable_on {a..1}"
sel_1: fixes x1 :: "real"   and x2 :: "real"   and x3 :: "real"   and x4 :: "real" shows "quat.Re (Quat x1 x2 x3 x4) = x1"
pairwise_insert: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and s :: "'b set" shows "pairwise r (insert x s) = ((\<forall>y. y \<in> s \<and> y \<noteq> x \<longrightarrow> r x y \<and> r y x) \<and> pairwise r s)"
poly_y_x_const: fixes a :: "'a" shows "poly_y_x [:[:a:]:] = [:[:a:]:]"
app_invimageD: fixes a :: "V"   and r :: "V"   and A :: "V" assumes "a \<in>\<^sub>\<circ> r -`\<^sub>\<circ> A" shows "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> (r \<restriction>\<^sup>r\<^sub>\<circ> A)"
ccompare_option_def: shows "ccompare \<equiv> case ID ccompare of None \<Rightarrow> None | Some comp_'a \<Rightarrow> Some (comparator_option comp_'a)"
is_strongly_dom_component_safe_def: fixes S\<^sub>a\<^sub>r\<^sub>g :: "(_) object_ptr set"   and S\<^sub>r\<^sub>e\<^sub>s\<^sub>u\<^sub>l\<^sub>t :: "(_) object_ptr set"   and h :: "(_) heap"   and h' :: "(_) heap" shows "is_strongly_dom_component_safe S\<^sub>a\<^sub>r\<^sub>g S\<^sub>r\<^sub>e\<^sub>s\<^sub>u\<^sub>l\<^sub>t h h' = (let removed_pointers = fset (object_ptr_kinds h) - fset (object_ptr_kinds h'); added_pointers = fset (object_ptr_kinds h') - fset (object_ptr_kinds h); arg_components = \<Union>ptr\<in>(\<Union>ptr\<in>S\<^sub>a\<^sub>r\<^sub>g. set |h \<turnstile> local.a_get_dom_component ptr|\<^sub>r) \<inter> fset (object_ptr_kinds h). set |h \<turnstile> local.a_get_dom_component ptr|\<^sub>r; arg_components' = \<Union>ptr\<in>(\<Union>ptr\<in>S\<^sub>a\<^sub>r\<^sub>g. set |h \<turnstile> local.a_get_dom_component ptr|\<^sub>r) \<inter> fset (object_ptr_kinds h'). set |h' \<turnstile> local.a_get_dom_component ptr|\<^sub>r in removed_pointers \<subseteq> arg_components \<and> added_pointers \<subseteq> arg_components' \<and> S\<^sub>r\<^sub>e\<^sub>s\<^sub>u\<^sub>l\<^sub>t \<subseteq> arg_components' \<and> (\<forall>outside_ptr\<in>fset (object_ptr_kinds h) \<inter> fset (object_ptr_kinds h') - (\<Union>ptr\<in>S\<^sub>a\<^sub>r\<^sub>g. set |h \<turnstile> local.a_get_dom_component ptr|\<^sub>r). preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t outside_ptr id) h h'))"
valid_data_dependency_axioms: shows "valid_data_dependency ts\<^sub>s\<^sub>b"
amor_mtf_ub: fixes x :: "'b"   and ys :: "'b list"   and xs :: "'b list" assumes "x \<in> set ys"   and "set xs = set ys" shows "int (card (before x xs \<inter> before x ys)) - int (card (after x xs \<inter> before x ys)) \<le> 2 * int (index xs x) - int (card (before x ys))"
F_fix: shows "F = (\<mu> F. \<Lambda> x. \<^ps> g\<cdot>x \<union> F\<cdot>(\<^ps> R\<cdot>x))"
fsafe_Context_simps: fixes C :: "'b"   and p :: "('a, 'b, 'c) formula" shows "fsafe (InContext C p) = fsafe p"
S_2: fixes x :: "real" assumes "x \<in> {0..1} - S" shows "g differentiable at x within {0..1}"
leg0_simps_3: shows "trg (s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) = src (r \<star> s)"
consPerm: fixes x :: "name"   and y :: "name"   and p :: "(name \<times> name) list"   and C :: "'a" shows "((x, y) # p) \<bullet> C = [(x, y)] \<bullet> p \<bullet> C"
len_take: shows "length (take (Suc j) \<beta>) = Suc j"
base_eq: shows "(Abs ^^ n) (unlift_vars n x) \<leftrightarrow> (Abs ^^ n) (unlift_vars n y)"
vdash_properties_6: fixes v :: "i"   and \<phi> :: "\<o>"   and \<psi> :: "\<o>" assumes "[\<phi> in v]"   and "[\<phi> \<^bold>\<rightarrow> \<psi> in v]" shows "[\<psi> in v]"
edge_preserving: fixes R :: "('b \<times> 'c) set"   and E :: "('d \<times> 'b \<times> 'b) set"   and G :: "('d \<times> 'c \<times> 'c) set" shows "edge_preserving R E G = (on_triple R `` E \<subseteq> G)"
reachable_V: shows "G'.reachable \<subseteq> G'.V"
real_hyp2_C_def: fixes p :: "hyp2"   and q :: "hyp2"   and r :: "hyp2"   and s :: "hyp2" shows "p q \<congruent>\<^sub>K r s \<equiv> \<exists>A. is_K2_isometry A \<and> hyp2_cltn2 p A = r \<and> hyp2_cltn2 q A = s"
sigma_finite_incseq: fixes i :: "'i" obtains A :: "nat \<Rightarrow> 'a set" where "range A \<subseteq> sets (M i)" and "\<Union> (range A) = space (M i)" and "\<And>ia. emeasure (M i) (A ia) \<noteq> \<infinity>" and "incseq A"
irreducible_mult_unit_right: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a"   and p :: "'a" assumes "class.idom times one plus zero minus uminus"   and "dvd.dvd times a one" shows "comm_semiring_1.irreducible times one zero (times p a) = comm_semiring_1.irreducible times one zero p"
bdd_above_uminus: fixes X :: "'c set" shows "bdd_above (uminus ` X) = bdd_below X"
Int_Un_distrib: fixes A :: "'c set"   and B :: "'c set"   and C :: "'c set" shows "A \<inter> (B \<union> C) = A \<inter> B \<union> A \<inter> C"
Borsukian_componentwise_eq: fixes S :: "'a set" assumes "locally connected S \<or> compact S" shows "Borsukian S = (\<forall>C\<in>components S. Borsukian C)"
square_free_int_imp_separable_mod_ring: fixes f :: "int poly" assumes "square_free f"   and "separable_bound f < int CARD('a)" shows "separable (of_int_poly f)"
s': shows "s' = envTrans eact' aact' (tLast t')"
distinct_set_conv_list: fixes xs :: "'a list"   and g :: "'a \<Rightarrow> 'b" assumes "distinct xs" shows "sum g (set xs) = sum_list (map g xs)"
pred_inject_2: fixes P :: "'e \<Rightarrow> bool"   and a :: "'e" shows "pred_option P (Some a) = P a"
enc_atom_bool_take_drop_False: fixes r :: "nat"   and I :: "('c + 'c set) list"   and p :: "'c" assumes "r < length I"   and "case I ! r of Inl p' \<Rightarrow> p \<noteq> p' | Inr P \<Rightarrow> p \<notin> P" shows "enc_atom_bool I p = take r (enc_atom_bool I p) @ False # drop (Suc r) (enc_atom_bool I p)"
mult_nonneg_nonneg: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "class.ordered_semiring_0 plus zero less_eq less times"   and "less_eq zero a"   and "less_eq zero b" shows "less_eq zero (times a b)"
unique_rootD_1: fixes plr :: "int poly \<times> rat \<times> rat" assumes "\<exists>!x. root_cond plr x" shows "real_of_rat (rai_lb plr) \<le> real_of_1 plr"
strict_mono_onD: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and r :: "'a"   and s :: "'a" assumes "strict_mono_on f A"   and "r \<in> A"   and "s \<in> A"   and "r < s" shows "f r < f s"
not_less: fixes less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and x :: "'g"   and y :: "'g" assumes "class.linorder less_eq less" shows "(\<not> less x y) = less_eq y x"
simps_4: fixes a :: "'a poly"   and b :: "'a poly" shows "num_params (Sub a b) = max (num_params a) (num_params b)"
ineq_3: shows "real_of_rat l' \<le> real_of_rat r'"
mset_replicate: fixes n :: "nat"   and x :: "'c" shows "mset (replicate n x) = replicate_mset n x"
atLeastAtMost_singleton: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b" assumes "class.order less_eq less" shows "ord.atLeastAtMost less_eq a a = {a}"
simps_8: fixes a :: "Ferrack.num" shows "bound0 (NEq a) = numbound0 a"
c_1: shows "mat_of_cols nc ss *\<^sub>v v = A\<^sup>H *\<^sub>v c"
region_def: shows "region \<equiv> \<lambda>X I r. {x. regionp (\<lambda>x. x \<in> X) I (\<lambda>x xa. (x, xa) \<in> r) x}"
pState_inv_def: fixes prog :: "program"   and p :: "pState" shows "pState_inv prog p = (pid p \<le> max_procs \<and> pState.idx p < IArray.length (states prog) \<and> IArray.length (states prog) = IArray.length (processes prog) \<and> pc p < IArray.length (states prog !! pState.idx p) \<and> set (pState.channels p) \<subseteq> {- 1..<integer_of_nat max_channels} \<and> length (pState.channels p) \<le> max_channels \<and> vardict_inv (states prog !! pState.idx p) (processes prog !! pState.idx p) (pState.vars p))"
ide_composite: shows "ide (\<mu> \<star> \<nu>)"
uv': shows "u \<cdot> r = r \<cdot> v"
weakly_bisimilar_reflp: shows "reflp (\<approx>\<cdot>)"
hyps_1: shows "atom ` domA as \<sharp>* \<Gamma>"
measurable_C: fixes n :: "nat"   and m :: "nat" shows "C n m \<in> Pi\<^sub>M {0..<n} M \<rightarrow>\<^sub>M subprob_algebra (Pi\<^sub>M {0..<n + m} M)"
le_top: fixes x :: "'b \<times> 'c" shows "less_eq_prod x (times_prod' top_prod x)"
simps_1: fixes x :: "'b" shows "weight (Var x) = w0"
traces_refinement: fixes sched\<^sub>C :: "nat list"   and gc\<^sub>C :: "('Com\<^sub>C \<times> (Mode \<Rightarrow> 'Var\<^sub>C set)) list \<times> ('Var\<^sub>C \<Rightarrow> 'Val)"   and gc\<^sub>C' :: "('Com\<^sub>C \<times> (Mode \<Rightarrow> 'Var\<^sub>C set)) list \<times> ('Var\<^sub>C \<Rightarrow> 'Val)"   and gc\<^sub>A :: "('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) list \<times> ('Var\<^sub>A \<Rightarrow> 'Val)" assumes "conc.meval_sched_abv gc\<^sub>C sched\<^sub>C gc\<^sub>C'"   and "length (fst gc\<^sub>A) = length cms"   and "length (fst gc\<^sub>C) = length cms"   and "\<And>i. i < length cms \<Longrightarrow> ((fst gc\<^sub>A ! i, snd gc\<^sub>A), fst gc\<^sub>C ! i, snd gc\<^sub>C) \<in> \<R>_rel (cms ! i)"   and "abs.sound_mode_use gc\<^sub>A"   and "modes_respect_priv (map snd (fst gc\<^sub>C))" shows "\<exists>sched\<^sub>A gc\<^sub>A'. abs.meval_sched_abv gc\<^sub>A sched\<^sub>A gc\<^sub>A' \<and> (\<forall>i<length cms. ((fst gc\<^sub>A' ! i, snd gc\<^sub>A'), fst gc\<^sub>C' ! i, snd gc\<^sub>C') \<in> \<R>_rel (cms ! i)) \<and> abs.sound_mode_use gc\<^sub>A'"
antisym: fixes b :: "'a list"   and a :: "'a list" assumes "Sublist.prefix b a"   and "Sublist.prefix a b" shows "a = b"
simps_2: fixes s :: "'b stream"   and n :: "nat" shows "s !! Suc n = stl s !! n"
a: shows "limsup (\<lambda>n. u n + v n + - u n) \<le> limsup (\<lambda>n. u n + v n) + limsup (\<lambda>n. - u n)"
cases: fixes r :: "('b, 'c, 'd) spr_simWorld" obtains sprFst :: "('b, 'c, 'd) BEState" and sprLst :: "('b, 'c, 'd) BEState" and sprCRel :: "(('b, 'c, 'd) BEState \<times> ('b, 'c, 'd) BEState) set" where "r = \<lparr>sprFst = sprFst, sprLst = sprLst, sprCRel = sprCRel\<rparr>"
pseudonatural_transformation_axioms: shows "pseudonatural_transformation (\<cdot>\<^sub>C) (\<star>\<^sub>C) \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C (\<cdot>\<^sub>D) (\<star>\<^sub>D) \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D G \<Phi>\<^sub>G H \<Phi>\<^sub>H \<rho>\<^sub>0 \<rho>\<^sub>1"
m_def: shows "m \<equiv> integral S (\<lambda>x. \<bar>det (matrix (f' x))\<bar>)"
UN_extend_simps_6: fixes A :: "'c \<Rightarrow> 'b set"   and C :: "'c set"   and B :: "'b set" shows "\<Union> (A ` C) - B = (\<Union>x\<in>C. A x - B)"
SN: shows "SN_on (S\<^sup>* O R O S\<^sup>*) T"
wfTransitionRelation: fixes decisionVars :: "nat set"   and state0 :: "State"   and F0 :: "Literal list list" assumes "finite decisionVars"   and "isInitialState state0 F0"   and "\<exists>learnL. wf learnL \<and> (\<forall>stateA stateB. appliedLearn stateA stateB \<longrightarrow> (getF stateB, getF stateA) \<in> learnL)" shows "wf {(stateB, stateA). (state0, stateA) \<in> transitionRelation decisionVars \<and> transition stateA stateB decisionVars}"
halfspace_Int_eq_2: fixes b :: "real"   and a :: "'b" shows "{x. b \<le> a \<bullet> x} \<inter> {x. a \<bullet> x \<le> b} = {x. a \<bullet> x = b}"
keys_is_none_rep: fixes k :: "'b"   and m :: "('b, 'c) mapping" shows "(k \<in> keys m) = (\<not> Option.is_none (lookup m k))"
mult_add_distrib_mat: fixes A :: "'b mat"   and nr :: "nat"   and n :: "nat"   and B :: "'b mat"   and nc :: "nat"   and C :: "'b mat" assumes "A \<in> carrier_mat nr n"   and "B \<in> carrier_mat n nc"   and "C \<in> carrier_mat n nc" shows "A * (B + C) = A * B + A * C"
simps_1: fixes \<alpha> :: "'d \<Rightarrow> 'c" shows "eval_poly \<alpha> [] = (0::'c)"
hom_sup: fixes f :: "'b \<Rightarrow> 'a"   and x :: "'b formula"   and y :: "'b formula" shows "hom f (x \<squnion> y) = hom f x \<squnion> hom f y"
is_interval_neg_translationI: fixes X :: "'b set"   and x :: "'b" assumes "is_interval X" shows "is_interval ((-) x ` X)"
isEmpty_def: fixes t :: "('a, 'b) FingerTreeStruc" shows "FingerTreeStruc.isEmpty t = (t = Empty)"
d_OUT_0: fixes x :: "'k" shows "d_OUT (\<lambda>_. 0) x = 0"
split_completion: fixes A :: "'b set"   and M :: "'b measure" assumes "A \<in> sets (completion M)" shows "split_completion M A (main_part M A, null_part M A)"
set_map: fixes f :: "'e \<Rightarrow> 'd"   and v :: "'e stream" shows "sset (smap f v) = f ` sset v"
fcomp_rel_fid_on: fixes r :: "V"   and A :: "V" shows "r \<circ>\<^sub>\<bullet> fid_on A = r \<restriction>\<^sup>l\<^sub>\<bullet> A"
finals_def: fixes es :: "'a exp list" shows "finals es \<equiv> (\<exists>vs. es = map Val vs) \<or> (\<exists>vs a es'. es = map Val vs @ Throw a # es')"
cInf_atLeast: fixes x :: "'b multiset" shows "\<Sqinter> subset_mset.atLeast x = x"
assms_2: fixes v :: "'t" assumes "v \<in> keys p" shows "d (pp_of_term v) \<le> n"
elims_1: fixes x :: "arctic"   and xa :: "arctic"   and y :: "bool" assumes "(x < xa) = y" obtains xb :: "arctic" where "x = MinInfty" and "xa = xb__" and "y"   | uu_ :: "int" where "x = Num_arc uu_" and "xa = MinInfty" and "\<not> y"   | ya :: "int" and xb :: "int" where "x = Num_arc ya" and "xa = Num_arc xb" and "y = (ya < xb)"
sappend_strict2: fixes xs :: "[:'r:]" shows "xs :@ \<bottom> = \<bottom>"
induct: fixes P1 :: "Term.var \<Rightarrow> bool"   and P2 :: "expr \<Rightarrow> bool"   and P3 :: "stmt \<Rightarrow> bool"   and var :: "Term.var" assumes "\<And>x. P1 (LVar x)"   and "\<And>x1 x2 x3 x4 x5. P2 x4 \<Longrightarrow> P1 ({x1,x2,x3}x4..x5)"   and "\<And>x1 x2. \<lbrakk>P2 x1; P2 x2\<rbrakk> \<Longrightarrow> P1 (x1.[x2])"   and "\<And>x1 x2. \<lbrakk>P3 x1; P1 x2\<rbrakk> \<Longrightarrow> P1 (InsInitV x1 x2)"   and "\<And>x. P2 (NewC x)"   and "\<And>x1 x2. P2 x2 \<Longrightarrow> P2 (New x1[x2])"   and "\<And>x1 x2. P2 x2 \<Longrightarrow> P2 (Cast x1 x2)"   and "\<And>x1 x2. P2 x1 \<Longrightarrow> P2 (x1 InstOf x2)"   and "\<And>x. P2 (Lit x)"   and "\<And>x1 x2. P2 x2 \<Longrightarrow> P2 (UnOp x1 x2)"   and "\<And>x1 x2 x3. \<lbrakk>P2 x2; P2 x3\<rbrakk> \<Longrightarrow> P2 (BinOp x1 x2 x3)"   and "P2 Super"   and "\<And>x. P1 x \<Longrightarrow> P2 (Acc x)"   and "\<And>x1 x2. \<lbrakk>P1 x1; P2 x2\<rbrakk> \<Longrightarrow> P2 (x1:=x2)"   and "\<And>x1 x2 x3. \<lbrakk>P2 x1; P2 x2; P2 x3\<rbrakk> \<Longrightarrow> P2 (x1 ? x2 : x3)"   and "\<And>x1 x2 x3 x4 x5 x6 x7. \<lbrakk>P2 x4; \<And>x7a. x7a \<in> set x7 \<Longrightarrow> P2 x7a\<rbrakk> \<Longrightarrow> P2 ({x1,x2,x3}x4\<cdot>x5( {x6}x7))"   and "\<And>x1 x2. P2 (Methd x1 x2)"   and "\<And>x1 x2. P3 x2 \<Longrightarrow> P2 (Body x1 x2)"   and "\<And>x1 x2. \<lbrakk>P3 x1; P2 x2\<rbrakk> \<Longrightarrow> P2 (InsInitE x1 x2)"   and "\<And>x1 x2. P2 x2 \<Longrightarrow> P2 (Callee x1 x2)"   and "P3 Skip"   and "\<And>x. P2 x \<Longrightarrow> P3 (Expr x)"   and "\<And>x1 x2. P3 x2 \<Longrightarrow> P3 (x1\<bullet> x2)"   and "\<And>x1 x2. \<lbrakk>P3 x1; P3 x2\<rbrakk> \<Longrightarrow> P3 (x1;; x2)"   and "\<And>x1 x2 x3. \<lbrakk>P2 x1; P3 x2; P3 x3\<rbrakk> \<Longrightarrow> P3 (If(x1) x2 Else x3)"   and "\<And>x1 x2 x3. \<lbrakk>P2 x2; P3 x3\<rbrakk> \<Longrightarrow> P3 (x1\<bullet> While(x2) x3)"   and "\<And>x. P3 (Jmp x)"   and "\<And>x. P2 x \<Longrightarrow> P3 (Throw x)"   and "\<And>x1 x2 x3 x4. \<lbrakk>P3 x1; P3 x4\<rbrakk> \<Longrightarrow> P3 (Try x1 Catch(x2 x3) x4)"   and "\<And>x1 x2. \<lbrakk>P3 x1; P3 x2\<rbrakk> \<Longrightarrow> P3 (x1 Finally x2)"   and "\<And>x1 x2. P3 x2 \<Longrightarrow> P3 (FinA x1 x2)"   and "\<And>x. P3 (Init x)" shows "P1 var"
\<K>_finite: fixes K :: "nat set"   and j0 :: "nat"   and j :: "nat" assumes "K \<in> \<K> j0 j" shows "finite K"
eval_wfe: fixes \<Gamma> :: "char list set"   and e :: "expr"   and \<E> :: "char list \<Rightarrow> nat option" assumes "wfe \<Gamma> e"   and "match \<E> \<Gamma>" shows "eval \<E> e \<noteq> None"
W: shows "W \<equiv> {v. [v ^ CARD('a) = v] (mod u)}"
not_digit7_ascii_of: fixes c :: "char" shows "\<not> digit7 (ascii_of c)"
DF_def: fixes A :: "'a set" shows "DF A \<equiv> \<mu> x. \<sqinter>xa\<in>A \<rightarrow>  x"
coherent_Vcomp: fixes t :: "'c local.term"   and u :: "'c local.term" assumes "Arr t"   and "Arr u"   and "Dom t = Cod u"   and "coherent t"   and "coherent u" shows "coherent (t \<^bold>\<cdot> u)"
set_del_list: fixes xs :: "'b list"   and x :: "'b" assumes "Sorted_Less.sorted xs" shows "set (del_list x xs) = set xs - {x}"
cong_diff_iff_cong_0: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and uminus :: "'a \<Rightarrow> 'a"   and division_segment :: "'a \<Rightarrow> 'a"   and b :: "'a"   and c :: "'a"   and a :: "'a" assumes "class.unique_euclidean_ring divide plus minus zero times one modulo euclidean_size uminus division_segment" shows "unique_euclidean_semiring.cong modulo (minus b c) zero a = unique_euclidean_semiring.cong modulo b c a"
pos_mean: fixes xs :: "real list" assumes "pos xs" shows "0 < mean xs"
simps: fixes x :: "real"   and y :: "real" shows "abscissa (Point x y) = x"
map_le_trans: fixes m1 :: "'g \<Rightarrow> 'h option"   and m2 :: "'g \<Rightarrow> 'h option"   and m3 :: "'g \<Rightarrow> 'h option" assumes "m1 \<subseteq>\<^sub>m m2"   and "m2 \<subseteq>\<^sub>m m3" shows "m1 \<subseteq>\<^sub>m m3"
g: shows "(\<Sum>x\<in>UNIV. g x *s Y $ x) = f v"
closure_mono: fixes S :: "'c set"   and T :: "'c set" assumes "S \<subseteq> T" shows "closure S \<subseteq> closure T"
line_through_hyp2_pole_not_in_hyp2: fixes a :: "proj2"   and l :: "proj2_line" assumes "a \<in> hyp2"   and "proj2_incident a l" shows "pole l \<notin> hyp2"
that: fixes h' :: "heap \<times> nat set" assumes "h' \<Turnstile> P" shows "thesis"
norm_power: fixes x :: "'b"   and n :: "nat" shows "norm (x ^ n) = norm x ^ n"
last_message_f_last_message_hold: fixes xs :: "'b message_af list"   and k :: "nat" shows "last_message (xs \<longmapsto> k) = last_message xs"
that: fixes P' :: "ccs" assumes "P \<Longrightarrow>\<^sub>\<tau> P'"   and "(P', Q') \<in> Rel" shows "thesis"
fresh_Pair: fixes a :: "atom"   and x :: "'d"   and y :: "'e" shows "a \<sharp> (x, y) = (a \<sharp> x \<and> a \<sharp> y)"
less_next: fixes A :: "ordinal set"   and x :: "ordinal" assumes "normal_set A" shows "x < (LEAST z. z \<in> A \<and> x < z)"
Standard_abs: fixes x :: "'a star" assumes "x \<in> Standard" shows "\<bar>x\<bar> \<in> Standard"
le_sqrt_iff: fixes x :: "nat"   and y :: "nat" shows "(x \<le> Discrete.sqrt y) = (x\<^sup>2 \<le> y)"
insert_before_is_strongly_dom_component_safe_step: fixes h :: "(_) heap"   and ptr' :: "(_) object_ptr"   and child :: "(_) node_ptr"   and ref :: "(_) node_ptr option"   and h' :: "(_) heap"   and ptr :: "(_) object_ptr"   and getter :: "(_) RObject_ext \<Rightarrow> 'm" assumes "heap_is_wellformed h"   and "type_wf h"   and "known_ptrs h"   and "h \<turnstile> insert_before ptr' child ref \<rightarrow>\<^sub>h h'"   and "ptr \<notin> set |h \<turnstile> get_scdom_component ptr'|\<^sub>r"   and "ptr \<notin> set |h \<turnstile> get_scdom_component (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)|\<^sub>r" shows "preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t ptr getter) h h'"
minus_divide_left: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "class.division_ring divide inverse one times plus zero minus uminus" shows "uminus (divide a b) = divide (uminus a) b"
C_eq_until_separatedQ: fixes p :: "((int \<times> int) set set, int) Combinators"   and l :: "(int \<times> int) set set set list" assumes "DenyAll \<in> set (policy2list p)"   and "all_in_list (policy2list p) l"   and "allNetsDistinct (policy2list p)" shows "C (list2FWpolicy (separate (qsort (removeShadowRules2 (remdups (rm_MT_rules C (insertDeny (removeShadowRules1 (policy2list p)))))) l))) = C p"
bit_0: fixes a :: "'a" shows "bit a 0 = odd a"
assms_2: fixes T :: "'a set" assumes "P T" shows "Q T"
dg_Set_is_arrD_3: fixes \<alpha> :: "V"   and A :: "V"   and B :: "V"   and S :: "V" assumes "S : A \<mapsto>\<^bsub>dg_Set \<alpha>\<^esub> B" shows "S\<lparr>ArrCod\<rparr> = B"
msubstltpos_def: fixes c :: "poly"   and t :: "tm"   and a :: "poly"   and b :: "tm" shows "msubstltpos c t a b = Lt (tm.Add (tm.Mul a t) (tm.Mul c b))"
Aexp_e: fixes a :: "Arity" shows "Aexp_syn' a e = esing x\<cdot>(up\<cdot>a) \<squnion> esing y\<cdot>(up\<cdot>a)"
rel_sum_simps_3: fixes R1 :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and R2 :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and a2 :: "'e"   and b1 :: "'d" shows "rel_sum R1 R2 (Inr a2) (Inl b1) = False"
dom_comp: fixes g :: "('obj, 'obj) concrete_category.arr"   and f :: "('obj, 'obj) concrete_category.arr" assumes "seq g f" shows "local.dom (g \<cdot> f) = local.dom f"
p_Mp_pl_Mp: fixes k :: "int poly" shows "p.eq_m (pl.Mp k) k"
foreach_body_def: fixes u :: "'a" shows "foreach_body u \<equiv> \<lambda>(v, d) (Q, \<pi>). if v = r then (Q, \<pi>) else case (Q v, \<pi> v) of (enat d', x) \<Rightarrow> if d < d' then (Q(v := enat d), \<pi>(v \<mapsto> u)) else (Q, \<pi>) | (\<infinity>, None) \<Rightarrow> (Q(v := enat d), \<pi>(v \<mapsto> u)) | (\<infinity>, Some xa) \<Rightarrow> (Q, \<pi>)"
c1_on_open_euclidean_axioms: shows "c1_on_open_euclidean f f' Y"
troeger_3: fixes x :: "'a"   and y :: "'a" shows "(x \<squnion> y * bot)\<^sup>\<circ> = x\<^sup>\<circ> * Rf y bot"
ground_fm_aux_def: fixes A :: "fm"   and S :: "atom set" shows "ground_fm_aux A S \<equiv> supp A \<subseteq> S"
P_def: shows "P \<equiv> \<lambda>b. f b = TRUE"