hcju/isabelleps huggingface dataset:
rel_F_mono': fixes L1 :: "'n \<Rightarrow> 'o \<Rightarrow> bool"   and L2 :: "'p \<Rightarrow> 'q \<Rightarrow> bool"   and L3 :: "'r \<Rightarrow> 's \<Rightarrow> bool"   and Co1 :: "'t \<Rightarrow> 'u \<Rightarrow> bool"   and Co2 :: "'v \<Rightarrow> 'w \<Rightarrow> bool"   and Co3 :: "'x \<Rightarrow> 'y \<Rightarrow> bool"   and Contra1 :: "'z \<Rightarrow> 'aa \<Rightarrow> bool"   and Contra2 :: "'ab \<Rightarrow> 'ac \<Rightarrow> bool"   and Contra3 :: "'ad \<Rightarrow> 'ae \<Rightarrow> bool"   and x :: "('n, 'p, 'r, 't, 'v, 'x, 'z, 'ab, 'ad, 'af) F"   and y :: "('o, 'q, 's, 'u, 'w, 'y, 'aa, 'ac, 'ae, 'af) F"   and L1' :: "'n \<Rightarrow> 'o \<Rightarrow> bool"   and L2' :: "'p \<Rightarrow> 'q \<Rightarrow> bool"   and L3' :: "'r \<Rightarrow> 's \<Rightarrow> bool"   and Co1' :: "'t \<Rightarrow> 'u \<Rightarrow> bool"   and Co2' :: "'v \<Rightarrow> 'w \<Rightarrow> bool"   and Co3' :: "'x \<Rightarrow> 'y \<Rightarrow> bool"   and Contra1' :: "'z \<Rightarrow> 'aa \<Rightarrow> bool"   and Contra2' :: "'ab \<Rightarrow> 'ac \<Rightarrow> bool"   and Contra3' :: "'ad \<Rightarrow> 'ae \<Rightarrow> bool" assumes "rel_F L1 L2 L3 Co1 Co2 Co3 Contra1 Contra2 Contra3 x y"   and "L1 \<le> L1'"   and "L2 \<le> L2'"   and "L3 \<le> L3'"   and "Co1 \<le> Co1'"   and "Co2 \<le> Co2'"   and "Co3 \<le> Co3'"   and "Contra1' \<le> Contra1"   and "Contra2' \<le> Contra2"   and "Contra3' \<le> Contra3" shows "rel_F L1' L2' L3' Co1' Co2' Co3' Contra1' Contra2' Contra3' x y"
assms_2: shows "(\<sigma>, s) \<in> oreachable A (\<lambda>s s'. I) U"
hd_m: shows "head_\<omega> m1 < head_\<omega> m2"
weakBisimParPresSym: fixes substTerm :: "'d \<Rightarrow> name list \<Rightarrow> 'd list \<Rightarrow> 'd"   and substAssert :: "'e \<Rightarrow> name list \<Rightarrow> 'd list \<Rightarrow> 'e"   and substCond :: "'f \<Rightarrow> name list \<Rightarrow> 'd list \<Rightarrow> 'f"   and SCompose' :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and SImp' :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and SBottom' :: "'e"   and SChanEq' :: "'d \<Rightarrow> 'd \<Rightarrow> 'f"   and \<Psi> :: "'e"   and P :: "('d, 'e, 'f) psi"   and Q :: "('d, 'e, 'f) psi"   and R :: "('d, 'e, 'f) psi" assumes "env substTerm substAssert substCond SCompose' SImp' SBottom' SChanEq'"   and "(\<Psi>, P, Q) \<in> env.weakBisim substTerm substAssert substCond SCompose' SImp' SBottom' SChanEq'" shows "(\<Psi>, R \<parallel> P, R \<parallel> Q) \<in> env.weakBisim substTerm substAssert substCond SCompose' SImp' SBottom' SChanEq'"
outs_plus_\<I>: fixes \<I>l :: "('y, 'aa) \<I>"   and \<I>r :: "('z, 'ab) \<I>" shows "outs_\<I> (\<I>l \<oplus>\<^sub>\<I> \<I>r) = outs_\<I> \<I>l <+> outs_\<I> \<I>r"
scale_zero_left: fixes x :: "'a" shows "(0::'a) * x = (0::'a)"
to_hma_from_hma\<^sub>m: fixes v :: "(('a, 'b) Finite_Cartesian_Product.vec, 'c) Finite_Cartesian_Product.vec" shows "to_hma\<^sub>m (from_hma\<^sub>m v) = v"
simps_3: fixes C :: "char list"   and e :: "expr"   and A :: "char list set option" shows "\<D> (\<lparr>C\<rparr>e) A = \<D> e A"
\<R>\<^sub>G_id: fixes s :: "'a \<Rightarrow> (('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('a, 'b, 'c) expr) option" shows "\<R>\<^sub>G id id s = s"
f_def: fixes x :: "'a" shows "f x = (THE y. R x y)"
T_nn_integral_preserving: fixes f :: "'a \<Rightarrow> ennreal" assumes "f \<in> borel_measurable M" shows "\<integral>\<^sup>+ x. f (T x) \<partial>M = integral\<^sup>N M f"
unit_div_mult_self: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and a :: "'c"   and b :: "'c" assumes "class.algebraic_semidom divide plus minus zero times one"   and "dvd.dvd times a one" shows "times (divide b a) a = b"
Int_def: fixes A :: "'b set"   and B :: "'b set" shows "A \<inter> B = {x \<in> A. x \<in> B}"
Aexp_App: fixes e :: "exp"   and n :: "Arity"   and x :: "Vars.var" shows "\<A>\<^bsub>inc\<cdot>n\<^esub> e \<squnion> esing x\<cdot>(up\<cdot>0) \<sqsubseteq> \<A>\<^bsub>n\<^esub> (App e x)"
compl_inf: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and x :: "'b"   and y :: "'b" assumes "class.boolean_algebra minus uminus inf less_eq less sup bot top" shows "uminus (inf x y) = sup (uminus x) (uminus y)"
IVT': fixes f :: "'c \<Rightarrow> 'b"   and a :: "'c"   and y :: "'b"   and b :: "'c" assumes "f a \<le> y"   and "y \<le> f b"   and "a \<le> b"   and "continuous_on {a..b} f" shows "\<exists>x\<ge>a. x \<le> b \<and> f x = y"
power_minus: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> '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 n :: "nat" assumes "class.ring_1 one times plus zero minus uminus" shows "power.power one times (uminus a) n = times (power.power one times (uminus one) n) (power.power one times a n)"
vseq_implies_hpar_1: fixes \<nu> :: "'c"   and \<mu> :: "'c" assumes "C.seq \<nu> \<mu>" shows "src\<^sub>C \<nu> = src\<^sub>C \<mu>"
bot_ivl_def: shows "\<bottom> = Abs_Int2_ivl.empty"
assms_1: shows "D \<in> N_of_state St \<union> P_of_state St \<union> Q_of_state St"
2: shows "{x. (x = f p \<or> x \<in> P) \<and> x \<sqsubset> f p} = P"
FinFunc_def: fixes r :: "('e \<times> 'e) set"   and s :: "('d \<times> 'd) set" shows "FinFunc r s = Func (Field s) (Field r) \<inter> fin_support (zero r) (Field s)"
wlp_measure_expand_m: fixes m :: "nat"   and n :: "nat"   and M :: "nat \<Rightarrow> complex mat"   and S :: "com list"   and P :: "complex mat" assumes "m \<le> n"   and "well_com (Measure n M S)" shows "wlp (Measure m M S) P = matrix_sum d (\<lambda>k. adjoint (M k) * wlp (S ! k) P * M k) m"
positive_one: fixes n :: "nat" shows "positive (1\<^sub>m n)"
sublist_length: fixes l :: "'a list"   and l' :: "'a list" assumes "subseq l l'" shows "length l \<le> length l'"
weight_def: fixes LV :: "nat set"   and f :: "nat \<Rightarrow> nat" shows "weight LV f = f 1 + (\<Sum>i\<in>LV. f i div 2)"
NAMING_HINT_def: fixes s :: "char list \<Rightarrow> int \<Rightarrow> int"   and x :: "char list"   and n :: "char list" shows "NAMING_HINT s x n \<equiv> True"
PrefixListMonotonicity: fixes l1 :: "'b list"   and l2 :: "'b list" assumes "prefixList l1 l2" shows "length l1 < length l2"
set_child_nodes_get_disconnected_nodes_is_l_set_child_nodes_get_disconnected_nodes: shows "l_set_child_nodes_get_disconnected_nodes ShadowRootClass.type_wf Shadow_DOM.set_child_nodes Shadow_DOM.set_child_nodes_locs get_disconnected_nodes get_disconnected_nodes_locs"
g_1: fixes x :: "'a" assumes "x \<in> S" shows "(\<lambda>n. f n x) sums g x"
is_tm_tdghm_op: shows "op_tdghm \<NN> : op_dghm \<GG> \<mapsto>\<^sub>D\<^sub>G\<^sub>H\<^sub>M\<^sub>.\<^sub>t\<^sub>m op_dghm \<FF> : op_dg \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>t\<^sub>m\<^bsub>\<alpha>\<^esub> op_dg \<BB>"
pre_set_pick_def: shows "pre_set_pick \<equiv> \<lambda>s. s \<noteq> {}"
simps_5: fixes M :: "('b, 'c) model"   and g :: "'d \<Rightarrow> 'b"   and w :: "'b"   and p :: "('c, 'd) fm" shows "(M, g, w \<Turnstile> \<^bold>\<diamond> p) = (\<exists>v\<in>R M w. M, g, v \<Turnstile> p)"
inf_demorgan_2: fixes uminus :: "'b \<Rightarrow> 'b"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and top :: "'b"   and x :: "'b"   and y :: "'b" assumes "class.subset_boolean_algebra_5_extended uminus sup minus inf bot less_eq less top" shows "uminus (inf x y) = sup (uminus x) (uminus y)"
induct: fixes P :: "(nat \<Rightarrow> 'n) \<Rightarrow> nat \<Rightarrow> bool"   and a0 :: "nat \<Rightarrow> 'n"   and a1 :: "nat" assumes "\<And>\<pi> n. (Ex (is_icdi n \<pi>) \<Longrightarrow> P \<pi> (The (is_icdi n \<pi>))) \<Longrightarrow> P \<pi> n" shows "P a0 a1"
mut_m_not_idle_no_fM_writeD: fixes s :: "'h process_name \<Rightarrow> ('e, 'h, 'f, 'g) Model.local_state"   and p :: "'h process_name"   and fl :: "bool"   and ws :: "('e, 'f, 'g) mem_store_action list"   and m :: "'h" assumes "sys_mem_store_buffers p s = mw_fM fl # ws"   and "ghost_hs_phase (s (mutator m)) \<noteq> hp_Idle"   and "fM_rel_inv s"   and "handshake_phase_inv s"   and "tso_store_inv s"   and "p \<noteq> sys" shows "False"
upred_lattice_le: fixes P :: "'a upred"   and Q :: "'a upred" shows "le \<P> P Q = (P \<sqsubseteq> Q)"
v11_vconverse_app_app: fixes a :: "V" assumes "a \<in>\<^sub>\<circ> \<AA>\<lparr>Obj\<rparr>" shows "(\<FF>\<lparr>ObjMap\<rparr>)\<inverse>\<^sub>\<circ>\<lparr>\<FF>\<lparr>ObjMap\<rparr>\<lparr>a\<rparr>\<rparr> = a"
simps: fixes p :: "nat"   and r :: "nat"   and a :: "nat" shows "newrght p r a = (if a = 0 then newrgt0 [p, r] else if a = 1 then newrgt1 [p, r] else if a = 2 then newrgt2 [p, r] else if a = 3 then newrgt3 [p, r] else r)"
ipdk_pdl: shows "ipd (\<pi> k) pd\<rightarrow> \<pi> l"
disc_1: fixes x11 :: "char list"   and x12 :: "'a list" shows "sign (Pos x11 x12)"
intro: fixes Rep :: "'b \<Rightarrow> 'c"   and A :: "'c set"   and Abs :: "'c \<Rightarrow> 'b" assumes "\<And>x. Rep x \<in> A"   and "\<And>x. Abs (Rep x) = x"   and "\<And>y. y \<in> A \<Longrightarrow> Rep (Abs y) = y" shows "type_definition Rep Abs A"
degree_eq_zeroE: fixes p :: "'b poly" assumes "degree p = 0" obtains a :: "'b" where "p = [:a:]"
orthogonal_sum: fixes S :: "(real, 'a) vec set"   and v :: "(real, 'a) vec"   and c :: "(real, 'a) vec \<Rightarrow> real" assumes "\<And>w. w \<in> S \<Longrightarrow> orthogonal v w" shows "orthogonal v (\<Sum>w\<in>S. c w *s w)"
normalized_find_min_exists: fixes xs :: "('c, 'd) bintree option list" assumes "normalized xs"   and "xs \<noteq> []" shows "\<exists>t. find_min xs = Some t"
SUCC_eqvt: fixes p :: "perm"   and a :: "tm" shows "p \<bullet> SUCC a = SUCC (p \<bullet> a)"
prj_sym_1: fixes a0 :: "'a"   and a1 :: "'a" assumes "ide a0"   and "ide a1" shows "pr0 a1 a0 \<cdot> local.sym a0 a1 = pr1 a0 a1"
assms_2: fixes i :: "'a" assumes "i \<in> I" shows "Group.group (Gs i)"
assms: fixes a :: "'a poly" assumes "a \<in> set as" shows "monic a"
Dijkstra_Impl_Adts_axioms: shows "Dijkstra_Impl_Adts G.\<alpha> G.invar G_succ G_empty G_add_edge RBT_Set.empty RBT_Map.update RBT_Map.delete lookup M.invar PST_RBT.empty PST_RBT.update PST_RBT.delete PM.invar lookup rbt_is_empty pst_getmin"
powhr_minus: fixes x :: "real star"   and a :: "real star" shows "x powhr - a = inverse (x powhr a)"
sep_conj_sep_true: fixes P :: "'a \<Rightarrow> bool"   and s :: "'a" assumes "P s" shows "(P ** sep_true) s"
simps_4: fixes x4 :: "nat"   and y4 :: "nat" shows "(Key x4 = Key y4) = (x4 = y4)"
length_negHorbit_list: fixes gs :: "'g list"   and T :: "'a \<Rightarrow> ('f, 'g) aezfun \<Rightarrow> 'v"   and as :: "'a list" shows "length (negHorbit_list gs T as) = length gs"
simps: fixes var :: "nat"   and p :: "real mpoly" shows "split_p var p = And (fm.Atom (Eq (isolate_variable_sparse p var 2))) (And (fm.Atom (Eq (isolate_variable_sparse p var 1))) (fm.Atom (Eq (isolate_variable_sparse p var 0))))"
\<delta>\<^sub>L_def: fixes \<Sigma> :: "'b list"   and \<delta> :: "'a \<Rightarrow> 'b \<Rightarrow> 'a"   and q\<^sub>0 :: "'a" shows "\<delta>\<^sub>L \<Sigma> \<delta> q\<^sub>0 = set (let start = map (\<lambda>\<nu>. (q\<^sub>0, \<nu>, \<delta> q\<^sub>0 \<nu>)) \<Sigma>; succ = \<lambda>(uu_, uu_, q). map (\<lambda>\<nu>. (q, \<nu>, \<delta> q \<nu>)) \<Sigma> in list_dfs succ [] start)"
non_constructive_selector: fixes Rs :: "(('b, 'c) labeled_graph \<times> ('b, 'c) labeled_graph) set" assumes "set_of_graph_rules Rs" shows "valid_selector Rs (non_constructive_selector Rs)"
diff_divide_distrib: fixes divide :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and inverse :: "'e \<Rightarrow> 'e"   and one :: "'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and uminus :: "'e \<Rightarrow> 'e"   and a :: "'e"   and b :: "'e"   and c :: "'e" assumes "class.division_ring divide inverse one times plus zero minus uminus" shows "divide (minus a b) c = minus (divide a c) (divide b c)"
symmetry_rule_def: fixes t :: "'c" shows "symmetry_rule t = transl_rule (A_Cnv (A_Lbl t) \<sqsubseteq> A_Lbl t)"
exhaust: fixes y :: "state" obtains "y = s0"   | "y = s1"
subterms_of_an_atomic_term: fixes t :: "'b trm" assumes "is_a_variable t \<or> is_a_constant t" shows "subterms_of t = {t}"
case_2: fixes f1 :: "'f"   and f2 :: "'g \<Rightarrow> 'g list \<Rightarrow> 'f"   and x21 :: "'g"   and x22 :: "'g list" shows "(case x21 # x22 of [] \<Rightarrow> f1 | x # xa \<Rightarrow> f2 x xa) = f2 x21 x22"
intros: fixes \<I> :: "('a, 'b) \<I>"   and \<I>' :: "('c, 'd) \<I>"   and conv :: "('a, 'b, 'c, 'd) converter" assumes "\<And>q. q \<in> outs_\<I> \<I> \<Longrightarrow> \<I>' \<turnstile>g run_converter conv q \<surd>"   and "\<And>q r conv'. \<lbrakk>q \<in> outs_\<I> \<I>; (r, conv') \<in> results_gpv \<I>' (run_converter conv q)\<rbrakk> \<Longrightarrow> r \<in> responses_\<I> \<I> q \<and> \<I>, \<I>' \<turnstile>\<^sub>C conv' \<surd>" shows "\<I>, \<I>' \<turnstile>\<^sub>C conv \<surd>"
domain_eq: shows "Domainp (pcr_fs_set (=)) = (\<lambda>x. finite (supp x))"
simps_1: fixes st :: "'a" shows "local.getGMState (Idle st) = st"
lr_POR_arg2_rel: shows "POR_arg2_rel \<in> unlr POR.delta"
kzkki: fixes i :: "nat" assumes "i < st.d" shows "ket_plus_k (Suc k) $ i = 1 / csqrt (2 ^ Suc (Suc k))"
out_bet__out: fixes P :: "'p"   and Q :: "'p"   and A :: "'p"   and B :: "'p" assumes "Bet P Q A"   and "Q Out A B" shows "P Out A B"
denote_while_add: fixes \<rho>\<^sub>1 :: "complex mat"   and \<rho>\<^sub>2 :: "complex mat"   and x1 :: "nat \<Rightarrow> complex mat"   and x2 :: "com" assumes "\<rho>\<^sub>1 \<in> carrier_mat d d"   and "\<rho>\<^sub>2 \<in> carrier_mat d d"   and "x1 0 \<in> carrier_mat d d"   and "x1 1 \<in> carrier_mat d d"   and "partial_density_operator \<rho>\<^sub>1"   and "partial_density_operator \<rho>\<^sub>2"   and "trace (\<rho>\<^sub>1 + \<rho>\<^sub>2) \<le> 1"   and "well_com x2"   and "measurement d 2 x1"   and "\<And>\<rho>\<^sub>1 \<rho>\<^sub>2. \<lbrakk>\<rho>\<^sub>1 \<in> carrier_mat d d; \<rho>\<^sub>2 \<in> carrier_mat d d; partial_density_operator \<rho>\<^sub>1; partial_density_operator \<rho>\<^sub>2; trace (\<rho>\<^sub>1 + \<rho>\<^sub>2) \<le> 1\<rbrakk> \<Longrightarrow> denote x2 (\<rho>\<^sub>1 + \<rho>\<^sub>2) = denote x2 \<rho>\<^sub>1 + denote x2 \<rho>\<^sub>2" shows "denote_while (x1 0) (x1 1) (denote x2) (\<rho>\<^sub>1 + \<rho>\<^sub>2) = denote_while (x1 0) (x1 1) (denote x2) \<rho>\<^sub>1 + denote_while (x1 0) (x1 1) (denote x2) \<rho>\<^sub>2"
hom_compose: fixes f :: "'g \<Rightarrow> 'h"   and G :: "('g, 'i) monoid_scheme"   and H :: "('h, 'j) monoid_scheme"   and g :: "'h \<Rightarrow> 'k"   and I :: "('k, 'l) monoid_scheme" assumes "f \<in> hom G H"   and "g \<in> hom H I" shows "g \<circ> f \<in> hom G I"
id_FSwSbImorph: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'c, 'a, 'b, 'd) model_scheme" shows "FSwSbImorph id id MOD MOD"
cos_monotone_0_pi: fixes y :: "real"   and x :: "real" assumes "0 \<le> y"   and "y < x"   and "x \<le> pi" shows "cos x < cos y"
sub_BitM_One_eq: fixes n :: "num" shows "neg_numeral_class.sub (Num.BitM n) num.One = 2 * neg_numeral_class.sub n num.One"
notInCC_bot: fixes x :: "Vars.var"   and y :: "Vars.var" shows "x--y\<in>\<bottom> = False"
replicateE: fixes lm1 :: "nat list"   and am :: "nat list"   and m :: "nat"   and lm2 :: "nat list"   and RR :: "bool" assumes "\<not> length lm1 < length am"   and "am @ 0 \<up> (length lm1 - length am) @ [0] = lm1 @ m # lm2"   and "0 < m" shows "RR"
satPB_def: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ"   and \<Phi> :: "(('fsym, 'tp) efsym, 'psym) lit list set" shows "TE.satPB \<xi> \<Phi> \<equiv> \<forall>c\<in>\<Phi>. TE.satC \<xi> c"
fast_select_correct_aux: fixes k :: "nat"   and xs :: "'b list" assumes "fast_select_dom (k, xs)"   and "k < length xs" shows "fast_select k xs = select k xs"
matrix_change_of_basis_works: fixes X :: "(('a, 'b) vec, 'b) vec"   and Y :: "(('a, 'b) vec, 'b) vec"   and v :: "('a, 'b) vec" assumes "is_basis (set_of_vector X)"   and "is_basis (set_of_vector Y)" shows "matrix_change_of_basis X Y *v coord X v = coord Y v"
assms_2: shows "n\<^sub>1 \<le> n\<^sub>2"
unit_vec_bracket: fixes A :: "complex mat"   and n :: "nat"   and i :: "nat" assumes "A \<in> carrier_mat n n"   and "i < n" shows "inner_prod (unit_vec n i) (A *\<^sub>v unit_vec n i) = A $$ (i, i)"
exec_Normal_elim_cases_5: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and f :: "'f"   and g :: "'e set"   and c :: "('e, 'd, 'f) com"   and s :: "'e"   and t :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile> \<langle>Guard f g c,Normal s\<rangle> \<Rightarrow> t" obtains "s \<in> g" and "\<Gamma>\<turnstile> \<langle>c,Normal s\<rangle> \<Rightarrow> t"   | "t = Fault f" and "s \<notin> g"
fun_below_iff: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" shows "(f \<sqsubseteq> g) = (\<forall>x. f x \<sqsubseteq> g x)"
fv_Pair: fixes x :: "'c"   and y :: "'d" shows "fv (x, y) = fv x \<union> fv y"
Borsukian_circle: fixes S :: "'a set" shows "Borsukian S = (\<forall>f. continuous_on S f \<and> f ` S \<subseteq> sphere 0 1 \<longrightarrow> (\<exists>a. homotopic_with_canon (\<lambda>h. True) S (sphere 0 1) f (\<lambda>x. a)))"
cod_src_eq_dom_src: shows "assoc'_fgh.cod.src = assoc'_fgh.dsrc"
UN_iff: fixes b :: "'d"   and B :: "'e \<Rightarrow> 'd set"   and A :: "'e set" shows "(b \<in> \<Union> (B ` A)) = (\<exists>x\<in>A. b \<in> B x)"
simps: fixes m :: "nat"   and xs :: "nat list" shows "inpt m xs = trpl 0 1 (strt xs)"
assms: fixes e :: "real" assumes "0 < e" shows "AE x in M. F x \<le> G x + ereal e"
Q: fixes f :: "'a \<Rightarrow> 'c" assumes "continuous_on s f"   and "f ` s \<subseteq> U"   and "P f" shows "Q (f \<circ> k)"
usolves_odeI: fixes y :: "real \<Rightarrow> 'b"   and f :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and T :: "real set"   and X :: "'b set"   and t0 :: "real" assumes "(y solves_ode f) T X"   and "t0 \<in> T"   and "is_interval T"   and "\<And>z t. \<lbrakk>{t0--t} \<subseteq> T; (z solves_ode f) {t0--t} X; z t0 = y t0\<rbrakk> \<Longrightarrow> z t = y t" shows "(y usolves_ode f from t0) T X"
ffun_upd_ext: fixes x :: "'a"   and f :: "('a, 'b) ffun" assumes "x \<in> fdom f" shows "f(x \<mapsto> f(x)\<^sub>f)\<^sub>f = f"
prems: shows "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (\<Union> (trms\<^sub>e\<^sub>s\<^sub>t\<^sub>p ` set (x # A)))"
Bool: fixes A :: "fm"   and H :: "fm set" assumes "A \<in> boolean_axioms" shows "H \<turnstile> A"
proj2_intersection_def: fixes l :: "proj2_line"   and m :: "proj2_line" shows "proj2_intersection l m \<equiv> L2P (proj2_line_through (L2P l) (L2P m))"
cSUP_least: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'd multiset"   and M :: "'d multiset" assumes "A \<noteq> {}"   and "\<And>x. x \<in> A \<Longrightarrow> f x \<subseteq># M" shows "Sup (f ` A) \<subseteq># M"
rtrancl_impl: fixes rel :: "('b \<times> 'b) list"   and as :: "'b list" shows "set (RTLI.rtrancl_i rel as) = {b. \<exists>a\<in>set as. (a, b) \<in> (set rel)\<^sup>*}"
fact2: assumes "(\<nexists>e. fetch_instruction s'' = Inl e) \<and> (\<nexists>v1. fetch_instruction s'' = Inr v1)" shows "False"
that_2: fixes f :: "Value \<rightarrow> Value" assumes "x = Fn\<cdot>f"   and "y = \<bottom>" shows "thesis"
mult_eq_1_iff: fixes m :: "nat"   and n :: "nat" shows "(m * n = Suc 0) = (m = Suc 0 \<and> n = Suc 0)"
B: shows "next_dom_events P D u (ipurge_tr_rev I D u xs) = next_dom_events P D u xs \<and> ref_dom_events P D u (ipurge_tr_rev I D u xs) = ref_dom_events P D u xs"
convex: fixes S :: "'a set" shows "convex S = (\<forall>k u x. (\<forall>i. 1 \<le> i \<and> i \<le> k \<longrightarrow> 0 \<le> u i \<and> x i \<in> S) \<and> sum u {1..k} = 1 \<longrightarrow> (\<Sum>i = 1..k. u i *\<^sub>R x i) \<in> S)"
wf_syscls_class_Thread: fixes P :: "'b prog" assumes "wf_syscls P" shows "\<exists>C fs ms. class P Thread = \<lfloor>(C, fs, ms)\<rfloor>"
measure_induct_rule: fixes f :: "'c \<Rightarrow> 'd"   and P :: "'c \<Rightarrow> bool"   and a :: "'c" assumes "\<And>x. (\<And>y. f y < f x \<Longrightarrow> P y) \<Longrightarrow> P x" shows "P a"
complex_cnj: fixes a :: "real"   and b :: "real" shows "cnj (Complex a b) = Complex a (- b)"
meet_bot1: fixes y :: "'a" shows "\<bottom> \<sqinter> y = \<bottom>"
cSUP_least: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'a"   and M :: "'a" assumes "A \<noteq> {}"   and "\<And>x. x \<in> A \<Longrightarrow> f x \<le> M" shows "Sup (f ` A) \<le> M"
address_of_def: fixes r :: "'b memref" shows "address_of r = (case r of Write\<^sub>s\<^sub>b volatile a x v xa xb xc xd \<Rightarrow> {a} | Read\<^sub>s\<^sub>b volatile a t v \<Rightarrow> {a} | _ \<Rightarrow> {})"
target_alt_def: fixes execute :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and r :: "'e list"   and p :: "'d" shows "fold execute r p = last (p # scan execute r p)"
solution_nat_abs_nat_abs: fixes D :: "nat"   and x :: "int"   and y :: "int" assumes "pell D" shows "pell.solution D (nat \<bar>x\<bar>, nat \<bar>y\<bar>) = pell.solution D (x, y)"
decl_eval\<^sub>w': fixes mem' :: "var\<^sub>C \<Rightarrow> nat"   and mem :: "var\<^sub>C \<Rightarrow> nat"   and mds' :: "Mode \<Rightarrow> var\<^sub>C set"   and upd :: "var\<^sub>C ModeUpd"   and mds :: "Mode \<Rightarrow> var\<^sub>C set" assumes "mem' = mem"   and "mds' = C.update_modes upd mds" shows "eval_abv\<^sub>C \<langle>Skip@[upd], mds, mem\<rangle>\<^sub>C \<langle>Stop, mds', mem'\<rangle>\<^sub>C"
seq_char': fixes g :: "'c local.term set"   and f :: "'c local.term set" shows "\<F>C.seq g f = (\<F>C.ARR f \<and> \<F>C.ARR g \<and> \<F>C.DOM g = \<F>C.COD f)"
top_greatest: fixes a :: "'b \<times> 'c" shows "less_eq_prod a top_prod"
imonad1: fixes u :: "unit"   and f :: "unit \<Rightarrow> 'e nres" shows "RETURN u \<bind> f = f u"
n_add_op_def: fixes x :: "'a"   and y :: "'a" shows "x \<oplus> y \<equiv> n (n x \<cdot> n y)"
strict_mono_less: fixes f :: "'a \<Rightarrow> 'b"   and x :: "'a"   and y :: "'a" assumes "strict_mono f" shows "(f x < f y) = (x < y)"
bij_betw_combine: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and B :: "'b set"   and C :: "'a set"   and D :: "'b set" assumes "bij_betw f A B"   and "bij_betw f C D"   and "B \<inter> D = {}" shows "bij_betw f (A \<union> C) (B \<union> D)"
B: shows "p\<^sub>1 = p\<^sub>1'"
subterms_inclusion_refl: fixes E :: "'a trm set" shows "subterms_inclusion E E"
set2_G_map: fixes l1 :: "'n \<Rightarrow> 'v"   and l2 :: "'o \<Rightarrow> 'u" shows "set2_G \<circ> mapl_G l1 l2 = (`) l2 \<circ> set2_G"
matricize_def: fixes rmodes :: "nat set"   and T :: "'m tensor" shows "matricize rmodes T = mat (prod_list (nths (dims T) rmodes)) (prod_list (nths (dims T) (- rmodes))) (\<lambda>(r, c). lookup T (weave rmodes (digit_encode (nths (dims T) rmodes) r) (digit_encode (nths (dims T) (- rmodes)) c)))"