hcju/isabelleps huggingface dataset:
Diff_triv: fixes A :: "'e set"   and B :: "'e set" assumes "A \<inter> B = {}" shows "A - B = A"
inv_exec': fixes ss :: "'t list"   and f :: "'op" assumes "\<forall>s\<in>set ss. inv' s"   and "length ss = arity f" shows "inv' (exec' f ss)"
Dom_comp: fixes Obj :: "'a set"   and Hom :: "'a \<Rightarrow> 'a \<Rightarrow> 'b set"   and Id :: "'a \<Rightarrow> 'b"   and Comp :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "('a, 'b) concrete_category.arr"   and g :: "('a, 'b) concrete_category.arr" assumes "concrete_category Obj Hom Id Comp"   and "partial_magma.arr (concrete_category.COMP Obj Hom Comp) f"   and "partial_magma.arr (concrete_category.COMP Obj Hom Comp) g"   and "concrete_category.Dom g = concrete_category.Cod f" shows "concrete_category.Dom (concrete_category.COMP Obj Hom Comp g f) = concrete_category.Dom f"
type_wellformed_def: fixes t :: "'BExp set" shows "type_wellformed t \<equiv> vars_of_type t \<subseteq> \<C>"
pop_closed: shows "(\<plusminus>) \<in> carrier A \<rightarrow> carrier A \<rightarrow> carrier A"
round_robin_reschedule_Some_SomeD: fixes I :: "(('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set) set"   and t0 :: "'t"   and queue :: "'t list"   and n0 :: "nat"   and s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and t :: "'t"   and ta :: "('l, 't, 'x, 'm, 'w, 'o) thread_action"   and x' :: "'x"   and m' :: "'m"   and \<sigma>' :: "'t list \<times> nat" assumes "deterministic I"   and "round_robin_reschedule t0 queue n0 s = \<lfloor>(t, \<lfloor>(ta, x', m')\<rfloor>, \<sigma>')\<rfloor>"   and "t0 \<in> set queue"   and "s \<in> I" shows "\<exists>x. thr s t = \<lfloor>(x, no_wait_locks)\<rfloor> \<and> t \<turnstile> \<langle>x, shr s\<rangle> -ta\<rightarrow> \<langle>x', m'\<rangle> \<and> actions_ok s t ta"
simp2: shows "\<forall>y. a * (- b / a) * y + a * (- c / a) = - b * y + a * (- c / a)"
lzip_conv_lzip_ltake_min_llength: fixes xs :: "'c llist"   and ys :: "'d llist" shows "lzip xs ys = lzip (ltake (min (llength xs) (llength ys)) xs) (ltake (min (llength xs) (llength ys)) ys)"
inject_3: fixes x31 :: "'a itrm"   and x32 :: "'a itrm"   and y31 :: "'a itrm"   and y32 :: "'a itrm" shows "(x31 \<diamondop> x32 = y31 \<diamondop> y32) = (x31 = y31 \<and> x32 = y32)"
A: shows "\<not> (\<forall>i xs. i \<le> n \<and> set xs \<subseteq> {0..n} \<longrightarrow> (0::'a) \<le> len m i i xs)"
rga_ops_rem_last: fixes xs :: "('a \<times> 'a option) list"   and x :: "'a \<times> 'a option" assumes "rga_ops (xs @ [x])" shows "rga_ops xs"
comp_inv_arr: fixes f :: "'a"   and g :: "'a" assumes "B.inverse_arrows f g" shows "g \<cdot>\<^sub>B f = B.dom f"
solution_set_homogeneous_zero: shows "solution_set A 0 = {0}"
r_gtz: shows "0 < poly [:c, b, a:] r"
oarrivemsg_simps_2: fixes P :: "(nat \<Rightarrow> 'a) \<Rightarrow> 'b \<Rightarrow> bool"   and \<sigma> :: "nat \<Rightarrow> 'a"   and d :: "nat"   and m :: "nat" shows "oarrivemsg P \<sigma> (d:deliver(m)) = True"
integrable_indicator: fixes A :: "'b set"   and M :: "'b measure"   and c :: "'c" assumes "A \<in> sets M"   and "emeasure M A < \<infinity>" shows "integrable M (\<lambda>x. indicat_real A x *\<^sub>R c)"
GIrrRep_repset_def: shows "ActingGroup.GIrrRep_repset \<equiv> 0 \<union> set (ActingGroup.remisodups ActingGroup.FG_constituents)"
vequalityD2: fixes A :: "V"   and B :: "V" assumes "A = B" shows "B \<subseteq>\<^sub>\<circ> A"
induct: fixes P :: "'b poly \<Rightarrow> nat \<Rightarrow> bool"   and a0 :: "'b poly"   and a1 :: "nat" assumes "\<And>p. P p 0"   and "\<And>p v. (\<And>x. x \<in> {1..Suc v} \<Longrightarrow> P p (Suc v - x)) \<Longrightarrow> P p (Suc v)" shows "P a0 a1"
matches_alist_and: fixes \<gamma> :: "('a \<Rightarrow> 'b \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'b \<Rightarrow> bool)"   and l :: "'a negation_type list"   and a :: "action"   and p :: "'b" shows "matches \<gamma> (alist_and l) a p = ((\<forall>m\<in>set (getPos l). matches \<gamma> (Match m) a p) \<and> (\<forall>m\<in>set (getNeg l). matches \<gamma> (MatchNot (Match m)) a p))"
sol1: shows "(sol solves_ode (\<lambda>_. ODE_sem I (OVar vid1))) {0..t} {x. mk_v I (OVar vid1) (sol 0, b) x \<in> fml_sem I ($\<phi> vid1 local.empty)}"
proots_sphere_pos_interval: fixes p :: "real poly"   and a :: "real"   and b :: "real" assumes "p \<noteq> 0"   and "a < b" shows "proots_count p {x. a < x \<and> x < b} = proots_count (fcompose p [:a, b:] [:1, 1:]) {x. 0 < x}"
sums_finite: fixes N :: "nat set"   and f :: "nat \<Rightarrow> 'a" assumes "finite N"   and "\<And>n. n \<notin> N \<Longrightarrow> f n = (0::'a)" shows "f sums sum f N"
if_semi2: fixes c' :: "com"   and s :: "loc \<Rightarrow> nat"   and t :: "loc \<Rightarrow> nat"   and e :: "exp"   and c1 :: "com"   and c :: "com"   and c2 :: "com" assumes "(c', s) -[eval]-> t"   and "c' = IF e THEN c1;;c ELSE c2;;c" shows "(IF e THEN c1 ELSE c2;;c, s) -[eval]-> t"
semantics_ltlc_sugar_2: fixes \<xi> :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltlc" shows "\<xi> \<Turnstile>\<^sub>c F\<^sub>c \<phi> = \<xi> \<Turnstile>\<^sub>c true\<^sub>c U\<^sub>c \<phi>"
unit_simps_1: shows "D.seq (D.inv (\<Phi> (g, f))) (F \<eta> \<cdot>\<^sub>D unit (src\<^sub>C f))"
incomp_def: fixes x :: "'a"   and y :: "'a" shows "incomp x y = (\<not> x \<le> y \<and> \<not> y \<le> x)"
coeff_p': fixes i :: "nat" shows "coeff p' i = (if i = n then 1 else c ^ (n - i - 1) * coeff p i)"
bit_horner_sum_bit_iff: 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 division_segment :: "'a \<Rightarrow> 'a"   and bit :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and drop_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and take_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and bs :: "bool list"   and n :: "nat" assumes "class.unique_euclidean_semiring_with_bit_shifts divide plus minus zero times one modulo euclidean_size division_segment bit push_bit drop_bit take_bit" shows "bit (comm_semiring_0.horner_sum plus zero times (zero_neq_one.of_bool one zero) (numeral.numeral one plus (num.Bit0 num.One)) bs) n = (n < length bs \<and> bs ! n)"
type_less_eq_x_Set_intro: fixes \<tau> :: "'a OCL_Types.type"   and \<rho> :: "'a OCL_Types.type"   and \<sigma> :: "'a OCL_Types.type" assumes "\<tau> = Set \<rho>"   and "\<rho> \<le> \<sigma>" shows "\<tau> \<le> Set \<sigma>"
monad_def: fixes return :: "'d \<Rightarrow> 'e"   and bind :: "'e \<Rightarrow> ('d \<Rightarrow> 'e) \<Rightarrow> 'e" shows "monad return bind \<equiv> (\<forall>x f g. bind (bind x f) g = bind x (\<lambda>y. bind (f y) g)) \<and> (\<forall>x f. bind (return x) f = f x) \<and> (\<forall>x. bind x return = x)"
rel_optionT'_phantom: fixes A :: "'g \<Rightarrow> 'h \<Rightarrow> bool" shows "rel_optionT' A = rel_optionT' \<top>"
sym_rtrancl: fixes r :: "('a \<times> 'a) set" assumes "sym r" shows "sym (r\<^sup>*)"
simps_2: fixes v :: "nat" shows "nlen (Suc v) = 1 + nlen (Suc v div 10)"
subint1_acyclic: fixes G :: "prog" assumes "ws_prog G" shows "acyclic (subint1 G)"
assms_1: shows "(if \<sigma> then b0 else b1) = True"
comp_in_homI': fixes f :: "'c"   and g :: "'c"   and a :: "'c"   and c :: "'c" assumes "C.arr f"   and "C.arr g"   and "C.dom f = a"   and "C.cod g = c"   and "C.dom g = C.cod f" shows "\<guillemotleft>g \<cdot> f : a \<rightarrow> c\<guillemotright>"
Cons_4: shows "indexed_eval_aux (P # Ps) i m \<noteq> \<zero>"
to_gbg_alt1_1: fixes m :: "'b" shows "g_E (to_gbg_ext m) = E"
hom_eq: fixes G :: "('b, 'c) monoid_scheme"   and f :: "'b \<Rightarrow> 'd"   and H :: "('d, 'e) monoid_scheme"   and f' :: "'b \<Rightarrow> 'd" assumes "Group.group G"   and "f \<in> hom G H"   and "\<And>x. x \<in> carrier G \<Longrightarrow> f' x = f x" shows "f' \<in> hom G H"
add_right: fixes a :: "'c \<Rightarrow>\<^sub>L 'b"   and b :: "'c"   and b' :: "'c" shows "blinfun_apply a (b + b') = blinfun_apply a b + blinfun_apply a b'"
evals\<^sub>1_preserves_elen: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> nat exp) list) list"   and es :: "nat exp list"   and s :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> val list"   and es' :: "nat exp list"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> val list" assumes "P \<turnstile>\<^sub>1 \<langle>es,s\<rangle> [\<Rightarrow>] \<langle>es',s'\<rangle>" shows "length es = length es'"
assms: shows "\<exists>r. Some r = d\<^sub>0 p \<and> ex0 r v"
onode_comp_def: fixes i :: "nat"   and onp :: "((nat \<Rightarrow> 'd) \<times> 'e, 'f seq_action) automaton"   and R\<^sub>i :: "nat set" shows "\<langle>i : onp : R\<^sub>i\<rangle>\<^sub>o \<equiv> \<lparr>init = {(\<sigma>, NodeS i s R\<^sub>i) |\<sigma> s. (\<sigma>, s) \<in> init onp}, trans = onode_sos (automaton.trans onp)\<rparr>"
is_natural_2: fixes f :: "'a" assumes "A.arr f" shows "A.map (A.cod f) \<cdot>\<^sub>A A.map f = A.map f"
while_simps_1: fixes s :: "'a" assumes "guard s" shows "local.while s = body s \<bind> local.while"
eval_var_monom: fixes \<alpha> :: "'c \<Rightarrow> 'b"   and x :: "'c" shows "eval_monom \<alpha> (var_monom x) = \<alpha> x"
sets_uniform_count_measure_eq_UNIV_2: shows "(UNIV = sets (uniform_count_measure UNIV)) = True"
Left_equals_Right_coset_implies_normality: assumes "\<And>g. g \<in> G \<Longrightarrow> g \<cdot>| Normal = Normal |\<cdot> g" shows "normal_subgroup Normal G (\<cdot>) \<one>"
term_ok_eta_red_step: fixes t :: "Core.term"   and \<Theta> :: "((String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)) \<times> Core.term set"   and T :: "typ" assumes "\<not> is_dependent t"   and "term_ok \<Theta> (Abs T (t $ Bv 0))" shows "term_ok \<Theta> (decr 0 t)"
index_mat_swapcols_2: fixes k :: "nat"   and l :: "nat"   and A :: "'b mat" shows "dim_row (swapcols k l A) = dim_row A"
less_imp_of_nat_less: fixes m :: "nat"   and n :: "nat" assumes "m < n" shows "of_nat m < of_nat n"
simps: fixes x :: "nat" shows "Np x = Min {y. y \<le> Suc (x!) \<and> x < y \<and> Prime y}"
time_closest_pair_rec_tm_simps_1: fixes xs :: "(int \<times> int) list" assumes "length xs \<le> 3" shows "time (closest_pair_rec_tm xs) = 1 + time (length_tm xs) + time (mergesort_tm snd xs) + time (closest_pair_bf_tm xs)"
Ifun_def: fixes f :: "('b \<Rightarrow> 'c) star" shows "Ifun f \<equiv> \<lambda>x. Abs_star (\<Union>F\<in>Rep_star f. \<Union>X\<in>Rep_star x. starrel `` {\<lambda>n. F n (X n)})"
replete_set_category_def: fixes S :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" shows "replete_set_category S \<equiv> set_category S (cardSuc (cmax |UNIV| natLeq))"
proof_sound: fixes Pf :: "(('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list) \<times> (nat \<times> ('sf, 'sc, 'sz) step) list" assumes "proof_ok Pf" shows "sound (proof_result Pf)"
hyps_1: shows "P \<turnstile> \<langle>es1,s1,b1\<rangle> [\<rightarrow>] \<langle>es2,s2,b2\<rangle>"
Gcsrp_def: fixes G :: "('e, 'f) Group_scheme"   and N :: "'e set" shows "Gcsrp G N = \<lparr>carrier = gcsrp_map G N ` set_rcs G N, Group.top = gcsrp_top G N, iop = gcsrp_iop G N, one = gcsrp_one G N\<rparr>"
init_state: shows "active_subset (snd (lhd Ns)) = {}"
C3_a: fixes x :: "'a" shows "x l\<rightarrow> (0::'a) = -l x"
funcset_id: fixes A :: "'c set" shows "(\<lambda>x. x) \<in> A \<rightarrow> A"
simps_8: fixes \<Sigma> :: "'b list"   and w :: "'b list"   and I :: "(nat + nat set) list"   and \<phi> :: "'b formula" assumes "formula \<Sigma>" shows "formula.satisfies \<Sigma> (w, I) (FEXISTS \<phi>) = (\<exists>P. finite P \<and> formula.satisfies \<Sigma> (w, Inr P # I) \<phi>)"
exec_0: fixes P :: "instr list"   and c :: "int \<times> (char list \<Rightarrow> int) \<times> int list" shows "P \<turnstile> c \<rightarrow>^0 c"
eval_neg: fixes f :: "atom fm"   and L :: "real list" shows "(\<not> eval (neg f) L) = eval f L"
leg1_simps_2: shows "arr r\<^sub>0s\<^sub>1.p\<^sub>1"
inf1I: fixes A :: "'d \<Rightarrow> bool"   and x :: "'d"   and B :: "'d \<Rightarrow> bool" assumes "A x"   and "B x" shows "(A \<sqinter> B) x"
ltl_prop_equiv_false_implied_by_false: fixes y :: "'b ltln"   and x :: "'b ltln" assumes "y \<sim>\<^sub>P false\<^sub>n"   and "x \<longrightarrow>\<^sub>P y" shows "x \<sim>\<^sub>P false\<^sub>n"
equivE: fixes A :: "'c set"   and r :: "('c \<times> 'c) set" assumes "equiv A r" obtains "refl_on A r" and "sym r" and "trans r"
fork_state_update_def: fixes i :: "nat"   and s :: "nat"   and e :: "dining_event" shows "Up\<^sub>f i s e = (if e = picks i i then 1 else if e = picks ((i + 1) mod N) i then 2 else 0)"
iptables_goto_bigstep_deterministic: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and rs :: "'c rule list"   and s :: "state"   and t :: "state"   and t' :: "state" assumes "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>rs, s\<rangle> \<Rightarrow> t"   and "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>rs, s\<rangle> \<Rightarrow> t'" shows "t = t'"
beta_bound2: fixes p :: "nat"   and i :: "nat"   and q :: "nat" assumes "correct p (te p (i + 1))"   and "correct q (te q i)" shows "te p (i + 1) - te q i \<le> rmax + \<beta>"
swapcols_carrier: fixes l :: "nat"   and k :: "nat"   and A :: "'a mat"   and n :: "nat"   and m :: "nat" shows "(swapcols l k A \<in> carrier_mat n m) = (A \<in> carrier_mat n m)"
assms: shows "split_block (two_block_non_interfering A B) (dim_row A) (dim_col A) = (Q1, Q2, Q3, Q4)"
mor_UNIV: fixes s1 :: "('f, 'd, 'e) F1 \<Rightarrow> 'd"   and s2 :: "('f, 'd, 'e) F2 \<Rightarrow> 'e"   and s1' :: "('f, 'g, 'h) F1 \<Rightarrow> 'g"   and s2' :: "('f, 'g, 'h) F2 \<Rightarrow> 'h"   and f :: "'d \<Rightarrow> 'g"   and g :: "'e \<Rightarrow> 'h" shows "mor UNIV UNIV s1 s2 UNIV UNIV s1' s2' f g = (f \<circ> s1 = s1' \<circ> F1map id f g \<and> g \<circ> s2 = s2' \<circ> F2map id f g)"
of_bl_to_bl_shift_mask: shows "(of_bl (to_bl b) << n) && mask (16 + m) = 0"
permutes_sing: fixes p :: "'b \<Rightarrow> 'b"   and a :: "'b" shows "(p permutes {a}) = (p = id)"
LstSeqP_single: fixes H :: "fm set"   and x :: "tm" shows "H \<turnstile> LstSeqP (Eats Zero (HPair Zero x)) Zero x"
delete_edges_wf: fixes G :: "'b graph"   and E :: "('b \<times> 'b) set" assumes "wf_graph G" shows "wf_graph (FiniteGraph.delete_edges G E)"
dvdm_add: fixes u :: "int poly"   and a :: "int poly"   and b :: "int poly" assumes "u dvdm a"   and "u dvdm b" shows "u dvdm a + b"
aqgrp_def: fixes A :: "('c, 'd) aGroup_scheme"   and H :: "'c set" shows "aqgrp A H = \<lparr>carrier = set_rcs (b_ag A) H, pop = c_top (b_ag A) H, mop = c_iop (b_ag A) H, zero = H\<rparr>"
surj_imp_inv_eq: fixes f :: "'d \<Rightarrow> 'c"   and g :: "'c \<Rightarrow> 'd" assumes "surj f"   and "\<And>x. g (f x) = x" shows "inv f = g"
inject_14: fixes x151 :: "floatarith"   and x152 :: "nat"   and y151 :: "floatarith"   and y152 :: "nat" shows "(Power x151 x152 = Power y151 y152) = (x151 = y151 \<and> x152 = y152)"
bilinear_continuous_compose: fixes F :: "'b filter"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'd"   and h :: "'c \<Rightarrow> 'd \<Rightarrow> 'e" assumes "continuous F f"   and "continuous F g"   and "bilinear h" shows "continuous F (\<lambda>x. h (f x) (g x))"
subringE_6: fixes H :: "'c set"   and R :: "('c, 'd) ring_scheme"   and h1 :: "'c"   and h2 :: "'c" assumes "subring H R"   and "h1 \<in> H"   and "h2 \<in> H" shows "h1 \<otimes>\<^bsub>R\<^esub> h2 \<in> H"
simps_5: fixes t :: "('c, 'd) Term.term"   and t' :: "('c, 'd) Term.term" shows "trms\<^sub>s\<^sub>s\<^sub>t\<^sub>p (delete\<langle>t,t'\<rangle>) = {t, t'}"
continuous_on_open_Un: fixes s :: "'a set"   and t :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "open s"   and "open t"   and "continuous_on s f"   and "continuous_on t f" shows "continuous_on (s \<union> t) f"
inverse_eq_divide: fixes a :: "'a" shows "inverse a = (1::'a) / a"
externals_of_par_extra: fixes a :: "'e"   and A1 :: "('e set \<times> 'e set \<times> 'e set) \<times> 'f set \<times> ('f \<times> 'e \<times> 'f) set"   and A2 :: "('e set \<times> 'e set \<times> 'e set) \<times> 'g set \<times> ('g \<times> 'e \<times> 'g) set" shows "(a \<in> externals (asig_of (A1 || A2))) = (a \<in> externals (asig_of A1) \<and> a \<in> externals (asig_of A2) \<or> a \<in> externals (asig_of A1) \<and> a \<notin> externals (asig_of A2) \<or> a \<notin> externals (asig_of A1) \<and> a \<in> externals (asig_of A2))"
cong_ex: fixes f1 :: "'b \<Rightarrow> 'c"   and f2 :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and g1 :: "'b \<Rightarrow> 'c"   and g2 :: "'b \<Rightarrow> 'c" assumes "\<forall>\<^sub>F x in F. f1 x = f2 x"   and "\<forall>\<^sub>F x in F. g1 x = g2 x" shows "(f1 \<in> o[F](g1)) = (f2 \<in> o[F](g2))"
word_msb_def: fixes a :: "'b word" shows "msb a = (bin_sign (sint a) = - 1)"
NS: fixes i :: "nat" assumes "i < length ss"   and "i < length ts"   and "snd (r (ss ! i) (ts ! i))" shows "snd (r (map f ss ! i) (map f ts ! i))"
qq_cong: fixes a :: "nat" shows "[a = q * q + a] (mod q)"
init_fin_simulation_Wakeup1: fixes t :: "'t"   and sx1 :: "status \<times> 'x1"   and m1 :: "'m1"   and sx2 :: "status \<times> 'x2"   and m2 :: "'m2"   and ta1 :: "('l, 't, status \<times> 'x1, 'm1, 'w, 'o action) thread_action"   and sx1' :: "status \<times> 'x1"   and m1' :: "'m1" assumes "t \<turnstile> (sx1, m1) \<approx>i (sx2, m2)"   and "sx1 \<approx>iw sx2"   and "r1.init_fin t (sx1, m1) ta1 (sx1', m1')"   and "Notified \<in> set \<lbrace>ta1\<rbrace>\<^bsub>w\<^esub> \<or> WokenUp \<in> set \<lbrace>ta1\<rbrace>\<^bsub>w\<^esub>" shows "\<exists>ta2 sx2' m2'. r2.init_fin t (sx2, m2) ta2 (sx2', m2') \<and> t \<turnstile> (sx1', m1') \<approx>i (sx2', m2') \<and> ta_bisim init_fin_bisim ta1 ta2"
univalent_union: fixes f_a :: "('e \<times> 'f) set"   and f_b :: "('e \<times> 'f) set" assumes "univalent f_a"   and "univalent f_b"   and "Domain f_a \<inter> Domain f_b = Domain (f_a \<inter> f_b)" shows "univalent (f_a \<union> f_b)"
VWF_non_refl: fixes x :: "V" shows "(x, x) \<notin> VWF"
SIod_isom_Iod: fixes D :: "'c Order"   and T :: "'c set" assumes "Order D"   and "T \<subseteq> carrier D" shows "ord_isom (SIod D T) (Iod D T) (\<lambda>x\<in>T. x)"
GE_Guard_all: fixes f :: "'fsym"   and al :: "univ list" assumes "wtFsym f"   and "list_all2 eintT (arOf f) al" shows "(\<not> unprot (resOf f) \<and> \<not> protCl (resOf f) \<longrightarrow> eintP (Guard (resOf f)) [eintF (Oldf f) al]) \<and> (protCl (resOf f) \<longrightarrow> list_all2 (eintP \<circ> Guard) (arOf f) (map singl al) \<longrightarrow> eintP (Guard (resOf f)) [eintF (Oldf f) al])"
carr_3: shows "v \<noteq> 0\<^sub>v n"
not_inf_Fault': fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and f :: "nat \<Rightarrow> ('e, 'd, 'f) com list \<times> (('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list \<times> ('e, 'f) xstate"   and k :: "nat"   and cs :: "('e, 'd, 'f) com list"   and css :: "(('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list"   and m :: "'f" assumes "\<forall>i. \<Gamma>\<turnstile> f i \<rightarrow> f (Suc i)"   and "f k = (cs, css, Fault m)" shows "False"
simps_2: fixes f :: "'d \<Rightarrow> 'c \<Rightarrow> 'c"   and xs :: "'d stream"   and a :: "'c" shows "stl (sscan f xs a) = sscan f (stl xs) (f (shd xs) a)"
star_zero_def: shows "0 \<equiv> star_of (0::'d)"
fbox_mult: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "antidomain_semiringl.fbox a (\<cdot>) (x \<cdot> y) z = antidomain_semiringl.fbox a (\<cdot>) x (antidomain_semiringl.fbox a (\<cdot>) y z)"
assms_2: shows "prv (eql (Num m) (Num n))"
bit_not_iff: fixes 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 divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bit :: "'c \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and drop_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and take_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and "and" :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and or :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and xor :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and mask :: "nat \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and not :: "'c \<Rightarrow> 'c"   and a :: "'c"   and n :: "nat" assumes "class.ring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask uminus not" shows "bit (not a) n = (power.power one times (numeral.numeral one plus (num.Bit0 num.One)) n \<noteq> zero \<and> \<not> bit a n)"
add_mset_eq_add_mset: fixes a :: "'c"   and M :: "'c multiset"   and b :: "'c"   and M' :: "'c multiset" shows "(add_mset a M = add_mset b M') = (a = b \<and> M = M' \<or> a \<noteq> b \<and> b \<in># M \<and> add_mset a (remove1_mset b M) = M')"
parAssoc: fixes P :: "pi"   and Q :: "pi"   and R :: "pi" shows "(P \<parallel> Q) \<parallel> R \<simeq> P \<parallel> Q \<parallel> R"
elims: fixes x :: "'b interrupt_action"   and xa :: "'b set"   and y :: "'b set" assumes "collect_interrupt x xa = y" obtains t :: "'b" and Ts :: "'b set" where "x = IsInterrupted ta__ True" and "xa = Ts__" and "y = insert ta__ Ts__"   | t :: "'b" and Ts :: "'b set" where "x = Interrupt t" and "xa = Tsa__" and "y = Tsa__ - {t}"   | v :: "'b" and Ts :: "'b set" where "x = IsInterrupted v__ False" and "xa = Tsb__" and "y = Tsb__"   | v :: "'b" and Ts :: "'b set" where "x = ClearInterrupt v" and "xa = Ts" and "y = Ts"
record_simps_26: fixes f :: "tid_or_exn set \<Rightarrow> tid_or_exn set"   and x :: "'b state" shows "clock (update_defined_types f x) = clock x"
o: fixes \<Gamma> :: "'a formula multiset" assumes "is_nnf_mset \<Gamma>" shows "set R' = R \<and> set (map set S') = S \<and> (disj_of_clause R', image_mset disj_of_clause (mset S') + \<Gamma> \<Rightarrow>\<^sub>n \<longrightarrow> image_mset disj_of_clause (mset S') + \<Gamma> \<Rightarrow>\<^sub>n)"
induct: fixes P :: "('a \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> (char list \<times> 'a) list \<Rightarrow> (char list \<times> 'a) list \<Rightarrow> bool"   and a0 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a1 :: "(char list \<times> 'a) list"   and a2 :: "(char list \<times> 'a) list" assumes "\<And>f ps2. P f [] ps2"   and "\<And>f x y ps1 ps2. (\<And>xa. xa = fun_rep ps2 x \<Longrightarrow> P f ps1 ps2) \<Longrightarrow> P f ((x, y) # ps1) ps2" shows "P a0 a1 a2"
Sup_insert: fixes a :: "'b"   and A :: "'b set" shows "Sup (insert a A) = sup a (Sup A)"
distinct_244: fixes x151 :: "floatarith"   and x152 :: "nat" shows "Power x151 x152 \<noteq> Pi"
r_max_alt_def: fixes t :: "('b, unit) RBT.rbt" shows "r_max t = fold1_keys max t"
WP1ConcEnd: fixes xs :: "('b, 'c) Combinators list"   and xa :: "('b, 'c) Combinators" assumes "wellformed_policy1 (xs @ [xa])" shows "wellformed_policy1 xs"
right_inv_of_unitary_transpose: fixes U :: "complex mat" assumes "unitary U" shows "U\<^sup>t * U\<^sup>t\<^sup>\<dagger> = 1\<^sub>m (dim_col U)"
subset_refl: fixes A :: "'g set" shows "A \<subseteq> A"
disc_2: fixes x21 :: "nat"   and x22 :: "'a"   and x23 :: "'a"   and x24 :: "'b"   and x25 :: "'b"   and x26 :: "'c" shows "\<not> isTrans (Send x21 x22 x23 x24 x25 x26)"
finite_subsets_at_top_def: fixes A :: "'c set" shows "finite_subsets_at_top A = (INF X\<in>{X. finite X \<and> X \<subseteq> A}. principal {Y. finite Y \<and> X \<subseteq> Y \<and> Y \<subseteq> A})"