hcju/isabelleps huggingface dataset:
shadow_root_ptr_kinds_simp: fixes shadow_root_ptr :: "(_) shadow_root_ptr"   and shadow_root :: "(_) Object"   and h :: "(_) heap" shows "shadow_root_ptr_kinds (Heap (fmupd (cast shadow_root_ptr) shadow_root (the_heap h))) = {|shadow_root_ptr|} |\<union>| shadow_root_ptr_kinds h"
ring_of_sets_iff: fixes \<Omega> :: "'c set"   and M :: "'c set set" shows "ring_of_sets \<Omega> M = (M \<subseteq> Pow \<Omega> \<and> {} \<in> M \<and> (\<forall>a\<in>M. \<forall>b\<in>M. a \<union> b \<in> M) \<and> (\<forall>a\<in>M. \<forall>b\<in>M. a - b \<in> M))"
S_b_eq: fixes a :: "'a" assumes "a \<in> set as" shows "S_b a = set (bs a)"
make_fmap_union: fixes xs :: "'a fset"   and v :: "'b"   and ys :: "'a fset" shows "[ xs |=> v ] ++\<^sub>f [ ys |=> v ] = [ xs |\<union>| ys |=> v ]"
intros: fixes a :: "'a"   and b :: "'c"   and r :: "('a \<times> 'c) set"   and c :: "'d"   and s :: "('c \<times> 'd) set" assumes "(a, b) \<in> r"   and "(b, c) \<in> s" shows "(a, c) \<in> r O s"
case_4: fixes f1 :: "'a"   and f2 :: "('b \<Rightarrow> 'b) \<Rightarrow> 'a"   and f3 :: "('b \<times> 'b) set \<Rightarrow> 'a"   and f4 :: "('b, 'c, 'd) com \<Rightarrow> ('b, 'c, 'd) com \<Rightarrow> 'a"   and f5 :: "'b set \<Rightarrow> ('b, 'c, 'd) com \<Rightarrow> ('b, 'c, 'd) com \<Rightarrow> 'a"   and f6 :: "'b set \<Rightarrow> ('b, 'c, 'd) com \<Rightarrow> 'a"   and f7 :: "'c \<Rightarrow> 'a"   and f8 :: "('b \<Rightarrow> ('b, 'c, 'd) com) \<Rightarrow> 'a"   and f9 :: "'d \<Rightarrow> 'b set \<Rightarrow> ('b, 'c, 'd) com \<Rightarrow> 'a"   and f10 :: "'a"   and f11 :: "('b, 'c, 'd) com \<Rightarrow> ('b, 'c, 'd) com \<Rightarrow> 'a"   and x41 :: "('b, 'c, 'd) com"   and x42 :: "('b, 'c, 'd) com" shows "(case Seq x41 x42 of Skip \<Rightarrow> f1 | Basic x \<Rightarrow> f2 x | Spec x \<Rightarrow> f3 x | Seq x xa \<Rightarrow> f4 x xa | Cond x xa xb \<Rightarrow> f5 x xa xb | While x xa \<Rightarrow> f6 x xa | Call x \<Rightarrow> f7 x | DynCom x \<Rightarrow> f8 x | Guard x xa xb \<Rightarrow> f9 x xa xb | Throw \<Rightarrow> f10 | Catch x xa \<Rightarrow> f11 x xa) = f4 x41 x42"
sep_conj_commute: fixes P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" shows "(P \<and>* Q) = (Q \<and>* P)"
simps_6: fixes a :: "name"   and b :: "name"   and P :: "pi" shows "summands ([a\<noteq>b]P) = {}"
martingaleAE: fixes M :: "'c measure"   and F :: "'d \<Rightarrow> 'c measure"   and X :: "'d \<Rightarrow> 'c \<Rightarrow> real"   and t :: "'d"   and s :: "'d" assumes "martingale M F X"   and "t \<le> s" shows "AEeq M (real_cond_exp M (F t) (X s)) (X t)"
start_heap_sys_xcpts: fixes C :: "char list"   and P :: "'a prog" assumes "C \<in> sys_xcpts" shows "start_heap P (addr_of_sys_xcpt C) = \<lfloor>blank P C\<rfloor>"
domain_eq: shows "Domainp pcr_unit = (\<lambda>x. x \<in> {0..1})"
PC_def: fixes a :: "bool \<Rightarrow> ('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real"   and P :: "'b \<Rightarrow> real"   and b :: "bool \<Rightarrow> ('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real" shows "a \<^bsub>P\<^esub>\<oplus> b \<equiv> \<lambda>ab Q s. P s * a ab Q s + (1 - P s) * b ab Q s"
line_integral_on_path: shows "one_chain_line_integral F {i} \<gamma> = one_chain_line_integral F {i} \<V> + one_chain_line_integral F {i} (two_chain_horizontal_boundary two_chain)"
open_segment_translation: fixes c :: "'b"   and a :: "'b"   and b :: "'b" shows "open_segment (c + a) (c + b) = (+) c ` open_segment a b"
perm_refl: fixes l :: "'a list" shows "l <~~> l"
Inf_fun_def: fixes A :: "('b \<Rightarrow> 'c) set" shows "\<Sqinter>A = (\<lambda>x. \<Sqinter>f\<in>A. f x)"
child_invariant: fixes d' :: "nat"   and p :: "(nat \<times> int) list"   and dir :: "dir"   and d :: "nat" assumes "d' < length p" shows "(child p dir d ! d' = p ! d') = (d \<noteq> d')"
birkhoff_sum_cocycle: fixes M :: "'b measure"   and T :: "'b \<Rightarrow> 'b"   and f :: "'b \<Rightarrow> 'c"   and n :: "nat"   and m :: "nat"   and x :: "'b" assumes "qmpt M T" shows "qmpt.birkhoff_sum T f (n + m) x = qmpt.birkhoff_sum T f n x + qmpt.birkhoff_sum T f m ((T ^^ n) x)"
Transaction_Transaction': shows "Transaction = (\<lambda>cm pm views. Transaction' ((cm, pm), views))"
vec_of_list_index: fixes xs :: "'b list"   and j :: "nat" shows "vec_of_list xs $ j = xs ! j"
pivot_tableau_def: fixes x\<^sub>i :: "nat"   and x\<^sub>j :: "nat"   and t :: "(nat \<times> linear_poly) list" shows "pivot_tableau_code x\<^sub>i x\<^sub>j t \<equiv> let eqa = eq_for_lvar_code t x\<^sub>i; eq' = pivot_eq eqa x\<^sub>j in map (\<lambda>e. if lhs e = lhs eqa then eq' else subst_var_eq_code x\<^sub>j (rhs eq') e) t"
rep_eq: fixes x :: "linear_poly" shows "vars x = fun_vars (Rep_linear_poly x)"
filter_join_False_insert: fixes a :: "'a"   and A :: "'a set"   and m :: "('a, 'b) mapping" shows "filter_join False (insert a A) m = filter_join False A (Mapping.delete a m)"
csum_def: fixes r1 :: "('b \<times> 'b) set"   and r2 :: "('c \<times> 'c) set" shows "r1 +c r2 \<equiv> |Field r1 <+> Field r2|"
mc_ClockAlg_def: shows "mc_ClockAlg \<equiv> mkClockAuto agents jkbp envInit envAction envTrans envVal envObs"
int_o: fixes \<xi> :: "Preliminaries.var \<Rightarrow> 'univ"   and \<rho> :: "Preliminaries.var \<Rightarrow> 'fsym trm"   and T :: "'fsym trm" shows "int (int \<xi> \<circ> \<rho>) T = int \<xi> (subst \<rho> T)"
asymp_equiv_divide: fixes f1 :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g1 :: "'a \<Rightarrow> 'b"   and f2 :: "'a \<Rightarrow> 'b"   and g2 :: "'a \<Rightarrow> 'b" assumes "f1 \<sim>[F] g1"   and "f2 \<sim>[F] g2" shows "(\<lambda>x. f1 x / f2 x) \<sim>[F] (\<lambda>x. g1 x / g2 x)"
sign_simps_9: fixes ab :: "'b"   and bb :: "'b"   and cb :: "'b" shows "(ab - bb = cb) = (ab = cb + bb)"
infinite_card_of_insert: fixes A :: "'b set"   and a :: "'b" assumes "infinite A" shows "|insert a A| =o |A|"
point_ext_def: fixes cx :: "int"   and cy :: "int"   and more :: "'a" shows "\<lparr>cx = cx, cy = cy, \<dots> = more\<rparr> \<equiv> Record.iso_tuple_cons point_ext_Tuple_Iso (Record.iso_tuple_cons Record.tuple_iso_tuple cx cy) more"
real_roots_of_rat_poly_1: fixes p :: "rat poly" assumes "p \<noteq> 0" shows "set (real_roots_of_rat_poly p) = {x. rpoly p x = 0}"
vimage_restrict_preseve_mono: fixes J :: "'b set"   and I :: "'b set"   and A :: "('b \<Rightarrow> 'c) set"   and S :: "'b \<Rightarrow> 'c set"   and B :: "('b \<Rightarrow> 'c) set" assumes "J \<subseteq> I"   and "A \<subseteq> Pi\<^sub>E J S"   and "B \<subseteq> Pi\<^sub>E J S"   and "Pi\<^sub>E I S \<noteq> {}"   and "(\<lambda>x. restrict x J) -` A \<inter> Pi\<^sub>E I S \<subseteq> (\<lambda>x. restrict x J) -` B \<inter> Pi\<^sub>E I S" shows "A \<subseteq> B"
that_1: fixes n :: "nat" assumes "r = C\<^bsup>n\<^esup>" shows "thesis"
analz_parts: fixes H :: "msg set" shows "analz (parts H) = parts H"
lm02: fixes X :: "'c set" assumes "finite X" shows "trivial X = (card X \<le> 1)"
simps: fixes a1 :: "('f \<times> 'l) set"   and a2 :: "('f \<times> 'l) set" shows "(a1 \<leadsto>GC a2) = ((\<exists>N1 N M N2 M'. a1 = N1 \<and> a2 = N2 \<and> N1 = N \<union> M \<and> N2 = N \<union> M' \<and> M \<subseteq> Red_F_\<G> (N \<union> M') \<and> active_subset M' = {}) \<or> (\<exists>N1 N C L N2 M. a1 = N1 \<and> a2 = N2 \<and> N1 = N \<union> {(C, L)} \<and> N2 = N \<union> {(C, active)} \<union> M \<and> L \<noteq> active \<and> active_subset M = {} \<and> no_labels.Inf_between (fst ` active_subset N) {C} \<subseteq> no_labels.Red_I_\<G> (fst ` (N \<union> {(C, active)} \<union> M))))"
floor_log_nat_eq_if: fixes b :: "nat"   and n :: "nat"   and k :: "nat" assumes "b ^ n \<le> k"   and "k < b ^ (n + 1)"   and "2 \<le> b" shows "\<lfloor>log (real b) (real k)\<rfloor> = int n"
rem_conless_sat: fixes s :: "'a \<Rightarrow> 'b option"   and ops :: "('a, 'b) sas_plus_operator list" shows "sat_preconds_as s (rem_condless_ops s ops)"
fst_delete_max_eq_ritem: fixes t :: "'b tree" assumes "t \<noteq> ET" shows "fst (delete_max t) = rightmost_item t"
NatTransMapsTo: fixes \<eta> :: "('d, 'e, 'f, 'g, 'h, 'i) NatTrans"   and F :: "('d, 'e, 'f, 'g, 'h, 'i) Functor"   and G :: "('d, 'e, 'f, 'g, 'h, 'i) Functor"   and X :: "'d" assumes "NT \<eta> : F \<Longrightarrow> G"   and "X \<in> obj\<^bsub>CatDom F\<^esub>" shows "\<eta> $$ X maps\<^bsub>CatCod G\<^esub> F @@ X to G @@ X"
oalist_inv_list_of_oalist_ntm: fixes xs :: "('b, 'c) oalist_ntm" shows "ko_ntm.oalist_inv (list_of_oalist_ntm xs)"
coeff_pcompose_x_pow_n: fixes n :: "nat"   and f :: "'a poly"   and i :: "nat" assumes "n \<noteq> 0" shows "poly.coeff (f \<circ>\<^sub>p Polynomial.monom (1::'a) n) (n * i) = poly.coeff f i"
map_comp: fixes g1 :: "'e \<Rightarrow> 'c"   and g2 :: "'f \<Rightarrow> 'd"   and f1 :: "'g \<Rightarrow> 'e"   and f2 :: "'h \<Rightarrow> 'f"   and v :: "'g \<times> 'h" shows "map_prod g1 g2 (map_prod f1 f2 v) = map_prod (g1 \<circ> f1) (g2 \<circ> f2) v"
norm_iff: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and g :: "'b \<Rightarrow> 'c" shows "((\<lambda>x. norm (f x)) \<in> O[F](\<lambda>x. norm (g x))) = (f \<in> O[F](g))"
prod_induct3: fixes P :: "'b \<times> 'c \<times> 'd \<Rightarrow> bool"   and x :: "'b \<times> 'c \<times> 'd" assumes "\<And>a b c. P (a, b, c)" shows "P x"
assms: shows "ct_suffix (xs @ to_ct_list ys) (to_ct_list zs)"
assms_1: fixes a :: "('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" assumes "a \<in> A" shows "- a \<in> A"
galleryD_adj: fixes X :: "'b set set"   and xs :: "'b set list" assumes "ChamberComplex X"   and "SimplicialComplex.maxsimpchain X xs" shows "adjacentchain xs"
is_hashmap'_distinct: fixes l :: "('a \<times> 'b) list list"   and ht :: "('a, 'b) hashtable" shows "is_hashtable l ht \<Longrightarrow>\<^sub>A is_hashtable l ht * \<up> (distinct (map fst (concat l)))"
red_unionI2: fixes G :: "('a \<Rightarrow>\<^sub>0 'd) set"   and p :: "'a \<Rightarrow>\<^sub>0 'd"   and q :: "'a \<Rightarrow>\<^sub>0 'd"   and F :: "('a \<Rightarrow>\<^sub>0 'd) set" assumes "punit.red G p q" shows "punit.red (F \<union> G) p q"
eventually_const_iff: fixes P :: "bool"   and F :: "'b filter" shows "(\<forall>\<^sub>F x in F. P) = (P \<or> F = bot)"
\<E>_\<E>: fixes S :: "'c set"   and \<Gamma> :: "('c, 'd) web_scheme" shows "\<E>\<^bsub>\<Gamma>\<^esub> (\<E>\<^bsub>\<Gamma>\<^esub> S) = \<E>\<^bsub>\<Gamma>\<^esub> S"
half_largest_elements_mono: fixes ds :: "nat list"   and d :: "nat" shows "sum_list (drop (length ds div 2) (sort ds)) \<le> sum_list (drop (Suc (length ds) div 2) (insort d (sort ds)))"
fits_Null: fixes G :: "prog"   and s :: "st"   and T :: "ty" shows "G,s\<turnstile>Null fits T"
same_state_def: fixes c :: "('b com option list \<times> 'b) list"   and clist :: "('b com option \<times> 'b) list list" shows "same_state c clist \<equiv> \<forall>i<length clist. \<forall>j<length c. snd (c ! j) = snd (clist ! i ! j)"
reachable_invar: fixes s :: "'s" assumes "gen_rwof' s" shows "set (stack s) \<subseteq> reachable \<and> pending s \<subseteq> E \<and> set (stack s) \<subseteq> gen_discovered s \<and> distinct (stack s) \<and> pending s \<subseteq> set (stack s) \<times> UNIV"
ole_def: fixes D :: "('c, 'd) Order_scheme"   and a :: "'c"   and b :: "'c" shows "a \<preceq>\<^bsub>D\<^esub> b = ((a, b) \<in> rel D)"
upto_enum_set_conv: fixes a :: "'b word"   and b :: "'b word" shows "set [a .e. b] = {x. a \<le> x \<and> x \<le> b}"
ax: shows "ax = \<lbrakk>quot_dbfm (abst_dbfm (decode_Var v) 0 (trans_fm [] A))\<rbrakk>e"
cmp_simps_2: fixes f :: "'a"   and g :: "'a" assumes "C.ide f"   and "C.ide g"   and "src\<^sub>C f = trg\<^sub>C g" shows "src\<^sub>D (\<Phi>\<^sub>G (f, g)) = src\<^sub>D (G g)"
igSwapInp_def: fixes MOD :: "('m, 'n, 'o, 'p, 'q, 'r, 'l, 's, 't) model_scheme"   and zs :: "'o"   and z1 :: "'r"   and z2 :: "'r"   and inp :: "'k \<Rightarrow> 'l option" shows "igSwapInp MOD zs z1 z2 inp \<equiv> lift (igSwap MOD zs z1 z2) inp"
zero_ext_g: fixes y :: "'a" assumes "y \<in> FG - supp g" shows "g y * h (- y + x) = (0::'b)"
conjugate_unit_circle_set: shows "conjugate ` circline_set unit_circle = circline_set unit_circle"
eff_S: fixes s :: "'state"   and r :: "'rule"   and sl :: "'state fset"   and s' :: "'state" assumes "s \<in> S"   and "r \<in> R"   and "eff r s sl"   and "s' |\<in>| sl" shows "s' \<in> S"
square_base: fixes b :: "proper_base" shows "get_base (square_base b) = get_base b * get_base b"
L8_Controller: fixes y :: "nat \<Rightarrow> nat list"   and l :: "nat \<Rightarrow> bit"   and z :: "nat \<Rightarrow> bit list"   and t :: "nat" assumes "Controller_L y (fin_inf_append [Zero] l) l z" shows "z t = [] \<or> z t = [Zero] \<or> z t = [One]"
mem_existence_ivl_iv_defined_1: fixes t :: "real"   and t0 :: "real"   and x0 :: "'a" assumes "t \<in> existence_ivl t0 x0" shows "t0 \<in> T"
Prec_FL_def: fixes Cl1 :: "'a literal multiset \<times> label"   and Cl2 :: "'a literal multiset \<times> label" shows "(Cl1 \<sqsubset> Cl2) = (fst Cl1 \<prec>\<cdot> fst Cl2 \<or> fst Cl1 \<doteq> fst Cl2 \<and> snd Cl1 \<sqsubset>l snd Cl2)"
wlp_whileI': fixes I :: "(char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool"   and s\<^sub>0 :: "char list \<Rightarrow> int \<Rightarrow> int"   and b :: "bexp"   and \<pi> :: "char list \<Rightarrow> com option"   and c :: "com"   and Q :: "(char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool" assumes "I s\<^sub>0"   and "\<And>s. I s \<Longrightarrow> if bval b s then wlp \<pi> c I s else Q s" shows "wlp \<pi> (WHILE b DO c) Q s\<^sub>0"
is_chain_def: fixes f :: "nat \<Rightarrow> 'b set" shows "is_chain f \<equiv> \<forall>n. f n \<subseteq> f (Suc n)"
hvzk_h_sub_rewrite: fixes h :: "'a"   and w :: "nat"   and z :: "nat"   and c :: "nat" assumes "h = \<^bold>g [^] w"   and "z < order \<G>" shows "\<^bold>g [^] (z + order \<G> * w * c - w * c) = \<^bold>g [^] z \<otimes> inv (h [^] c)"
iNext_induct_rule: fixes P :: "nat \<Rightarrow> bool"   and I :: "nat set"   and t :: "nat" assumes "P (iMin I)"   and "\<box> t I. P t \<longrightarrow> iNext t I P"   and "t \<in> I" shows "P t"
cardSuc_ordLess_ordLeq: fixes r :: "('a \<times> 'a) set"   and r' :: "('b \<times> 'b) set" assumes "Card_order r"   and "Card_order r'" shows "(r <o r') = (cardSuc r \<le>o r')"
abs_conv_abscissa_def: fixes f :: "'c fds" shows "abs_conv_abscissa f = (INF s\<in>{s. fds_abs_converges f s}. ereal (s \<bullet> (1::'c)))"
loopfree_digraph_axioms: shows "loopfree_digraph G"
Min_insert: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a set"   and x :: "'a" assumes "class.linorder less_eq less"   and "finite A"   and "A \<noteq> {}" shows "linorder.Min less_eq (insert x A) = ord.min less_eq x (linorder.Min less_eq A)"
cases: fixes x :: "tm" obtains y :: "name" and z :: "name" and w :: "name" and xa :: "tm" where "atom y \<sharp> (z, w, xa)" and "atom z \<sharp> (w, xa)" and "atom w \<sharp> xa" and "x = xa"
ik\<^sub>s\<^sub>s\<^sub>t_subst: fixes A :: "('d, 'e) stateful_strand_step list"   and \<delta> :: "'e \<Rightarrow> ('d, 'e) Term.term" shows "ik\<^sub>s\<^sub>s\<^sub>t (A \<cdot>\<^sub>s\<^sub>s\<^sub>t \<delta>) = ik\<^sub>s\<^sub>s\<^sub>t A \<cdot>\<^sub>s\<^sub>e\<^sub>t \<delta>"
antisym_conv3: fixes less_eq :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and less :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and y :: "'f"   and x :: "'f" assumes "class.linorder less_eq less"   and "\<not> less y x" shows "(\<not> less x y) = (x = y)"
sinvar_mono_imp_sinvar_mono: fixes sinvar :: "'c graph \<Rightarrow> ('c \<Rightarrow> 'd) \<Rightarrow> bool"   and N :: "'c set"   and E :: "('c \<times> 'c) set"   and E' :: "('c \<times> 'c) set"   and nP :: "'c \<Rightarrow> 'd" assumes "SecurityInvariant_withOffendingFlows.sinvar_mono sinvar"   and "wf_graph \<lparr>nodes = N, edges = E\<rparr>"   and "E' \<subseteq> E"   and "sinvar \<lparr>nodes = N, edges = E\<rparr> nP" shows "sinvar \<lparr>nodes = N, edges = E'\<rparr> nP"
bind_return: fixes f :: "'b Heap" shows "f \<bind> return = f"
rep_eq: fixes x :: "'a coreflexive"   and xa :: "'a coreflexive" shows "Rep_coreflexive (x * xa) = Rep_coreflexive x * Rep_coreflexive xa"
seq_extract_empty: fixes xs :: "'b list" shows "{} \<upharpoonleft>\<^sub>l xs = []"
val_normif: fixes s :: "'d \<Rightarrow> bool"   and env :: "('d, bool) mapping"   and t :: "'d ifex"   and t1 :: "'d ifex"   and t2 :: "'d ifex" assumes "agree s env" shows "val_ifex (normif env t t1 t2) s = val_ifex (if val_ifex t s then t1 else t2) s"
5: shows "ms \<langle>g\<rangle> [l] \<odot> [l] \<langle>star \<circ> f\<rangle> [l] \<preceq> ms \<langle>g\<rangle> [l]"
assms_1: shows "dom \<pi>\<^sub>\<A> \<subseteq> \<^bold>G \<phi>"
weakInputFreshChainDerivative: fixes \<Psi> :: "'b"   and Q :: "('a, 'b, 'c) psi"   and P :: "('a, 'b, 'c) psi"   and M :: "'a"   and N :: "'a"   and P' :: "('a, 'b, 'c) psi"   and xvec :: "name list" assumes "\<Psi> : Q \<rhd> P \<Longrightarrow>M\<lparr>N\<rparr> \<prec> P'"   and "xvec \<sharp>* P"   and "xvec \<sharp>* N" shows "xvec \<sharp>* P'"
nest: fixes m :: "'a"   and n :: "'a" assumes "m \<le> n" shows "S n \<subseteq> S m"
le_f_make_mono: fixes f :: "nat \<Rightarrow> ordinal"   and m :: "nat"   and n :: "nat" assumes "\<forall>n. f n < oLimit f"   and "m \<le> make_mono f n" shows "f m \<le> f (make_mono f n)"
antisymD: fixes r :: "('c \<times> 'c) set"   and a :: "'c"   and b :: "'c" assumes "antisym r"   and "(a, b) \<in> r"   and "(b, a) \<in> r" shows "a = b"
type_bexpr_elim: fixes \<Gamma> :: "'Var \<Rightarrow> Sec option"   and e :: "'BExp"   and t :: "Sec" assumes "\<Gamma> \<turnstile>\<^sub>b e \<in> t " obtains "t = max_dom (to_total \<Gamma> ` bexp_vars e)"
simps_2: fixes \<Gamma> :: "('a, 'b) Term.term \<Rightarrow> ('c, 'd) Term.term"   and X :: "'b list"   and F :: "(('a, 'b) Term.term \<times> ('a, 'b) Term.term) list" shows "comp_tfr\<^sub>s\<^sub>t\<^sub>p \<Gamma> \<forall>X\<langle>\<or>\<noteq>: F\<rangle>\<^sub>s\<^sub>t = ((\<forall>x\<in>fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s F - set X. is_Var (\<Gamma> (Var x))) \<or> (\<forall>u\<in>subterms\<^sub>s\<^sub>e\<^sub>t (trms\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s F). is_Fun u \<longrightarrow> args u = [] \<or> (\<exists>s\<in>set (args u). s \<notin> Var ` set X)))"
simps_9: fixes \<nu> :: "'a ltln" shows "pure_universal (X\<^sub>n \<nu>) = pure_universal \<nu>"
euclidean_isCont: fixes x :: "'d"   and f :: "'d \<Rightarrow> 'c" assumes "\<And>b. b \<in> Basis \<Longrightarrow> isCont (\<lambda>x. (f x \<bullet> b) *\<^sub>R b) x" shows "isCont f x"
nop_onto_axioms: shows "nop_onto A (2\<^sub>\<nat>) f"
lfp_eq_fixp: fixes f :: "'d \<Rightarrow> 'd" assumes "mono f" shows "\<nu> f = ccpo_class.fixp f"
2: shows "dom\<^bsub>C\<^esub> (f ;;\<^bsub>C\<^esub> g) = dom\<^bsub>C\<^esub> f"
simps_1: fixes x :: "QDelta ns_constraint list \<times> (('a, ('a \<times> QDelta atom) list) mapping \<times> ((nat, QDelta) mapping \<Rightarrow> (nat, QDelta) mapping) \<times> 'a list) \<times> ('a, QDelta) state"   and ya :: "QDelta ns_constraint list \<times> (('a, ('a \<times> QDelta atom) list) mapping \<times> ((nat, QDelta) mapping \<Rightarrow> (nat, QDelta) mapping) \<times> 'a list) \<times> ('a, QDelta) state" shows "(Simplex_State x = Simplex_State ya) = (x = ya)"
chain_Seq2: fixes Y :: "nat \<Rightarrow> 'c process"   and S :: "'c process" assumes "chain Y" shows "chain (\<lambda>i. S `;` Y i)"
span_of_maps_axioms: shows "span_of_maps (\<cdot>) (\<star>) \<a> \<i> src trg s\<^sub>0 s\<^sub>1"
Axiom_of_Infinity: shows "\<omega> \<in>\<^sub>\<circ> Vset \<alpha>"
not_Neg_cases: fixes \<phi> :: "Formula.formula"   and x :: "'a"   and f :: "Formula.formula \<Rightarrow> 'a" assumes "\<forall>\<psi>. \<phi> \<noteq> formula.Neg \<psi>" shows "(case \<phi> of formula.Neg \<psi> \<Rightarrow> f \<psi> | _ \<Rightarrow> x) = x"
labeling_consistent: fixes a :: "'b"   and b :: "'f" assumes "(a, b) \<in> R2" shows "s1.b.L a = s2.b.L b"
Done_bind_gpv: fixes a :: "'g"   and f :: "'g \<Rightarrow> ('d, 'e, 'f) gpv" shows "Generative_Probabilistic_Value.Done a \<bind> f = f a"
ih_10: shows "InvariantConflictFlagCharacterization (getConflictFlag state') (getF state') (getM state')"
r_4: shows "\<nexists>fs. set fs \<subseteq> carrier (Ring_Divisibility.mult_of R) \<and> wfactors (Ring_Divisibility.mult_of R) fs r"
extend_set_mono: fixes A :: "'d set"   and B :: "'d set"   and h :: "'d \<times> 'f \<Rightarrow> 'e" assumes "A \<subseteq> B" shows "extend_set h A \<subseteq> extend_set h B"
ncloses_def: fixes \<theta>_n :: "(name \<times> coname \<times> trm) list"   and \<Gamma> :: "(name \<times> ty) list" shows "\<theta>_n ncloses \<Gamma> \<equiv> \<forall>x B. (x, B) \<in> set \<Gamma> \<longrightarrow> (\<exists>c P. \<theta>_n nmaps x to Some (c, P) \<and> <c>:P \<in> \<parallel><B>\<parallel>)"
rel_reflp: fixes R :: "'d \<Rightarrow> 'd \<Rightarrow> bool" assumes "reflp R" shows "reflp (rel_option R)"
rel_FGco_pos_distr_imp: fixes Co1 :: "'p \<Rightarrow> 'w \<Rightarrow> bool"   and Co2 :: "'q \<Rightarrow> 'x \<Rightarrow> bool"   and Co3 :: "'r \<Rightarrow> 'y \<Rightarrow> bool"   and Co4 :: "'s \<Rightarrow> 'z \<Rightarrow> bool"   and Contra1 :: "'t \<Rightarrow> 'aa \<Rightarrow> bool"   and Contra2 :: "'u \<Rightarrow> 'ab \<Rightarrow> bool"   and Co1' :: "'w \<Rightarrow> 'ac \<Rightarrow> bool"   and Co2' :: "'x \<Rightarrow> 'ad \<Rightarrow> bool"   and Co3' :: "'y \<Rightarrow> 'ae \<Rightarrow> bool"   and Co4' :: "'z \<Rightarrow> 'af \<Rightarrow> bool"   and Contra1' :: "'aa \<Rightarrow> 'ag \<Rightarrow> bool"   and Contra2' :: "'ab \<Rightarrow> 'ah \<Rightarrow> bool"   and Co6 :: "'ai \<Rightarrow> 'aj \<Rightarrow> bool"   and Co6' :: "'aj \<Rightarrow> 'ak \<Rightarrow> bool"   and Contra3 :: "'al \<Rightarrow> 'am \<Rightarrow> bool"   and Contra3' :: "'am \<Rightarrow> 'an \<Rightarrow> bool"   and Contra4 :: "'ao \<Rightarrow> 'ap \<Rightarrow> bool"   and Contra4' :: "'ap \<Rightarrow> 'aq \<Rightarrow> bool"   and tytok_F :: "('ar \<times> 'as \<times> 'at \<times> 'p \<times> 'w \<times> 'ac \<times> 'au \<times> 'av \<times> 'aw \<times> 'ax) itself"   and tytok_G :: "('p \<times> 'w \<times> 'ac \<times> 'q \<times> 'x \<times> 'ad \<times> 'v) itself"   and Co5 :: "'au \<Rightarrow> 'av \<Rightarrow> bool"   and Co5' :: "'av \<Rightarrow> 'aw \<Rightarrow> bool"   and tytok_FGco :: "('ar \<times> 'as \<times> 'at \<times> 'v \<times> 'ax) itself" assumes "rel_F_pos_distr_cond (rel_G Co1 Co2 Co3 Co4 Contra1 Contra2) (rel_G Co1' Co2' Co3' Co4' Contra1' Contra2') Co3 Co3' Co6 Co6' Contra1 Contra1' Contra3 Contra3' Contra4 Contra4' tytok_F"   and "rel_G_pos_distr_cond Co3 Co3' Co4 Co4' Contra1 Contra1' Contra2 Contra2' tytok_G" shows "rel_FGco_pos_distr_cond Co1 Co1' Co2 Co2' Co3 Co3' Co4 Co4' Co5 Co5' Co6 Co6' Contra1 Contra1' Contra2 Contra2' Contra3 Contra3' Contra4 Contra4' tytok_FGco"
simps: fixes xs :: "'b list"   and fs :: "'b list"   and i :: "nat"   and n :: "nat" shows "b xs fs i n = (if n \<le> i then [:c xs fs n:] else b xs fs (Suc i) n * X xs i + [:c xs fs i:])"
IH_real: fixes p :: "real poly"   and x :: "real" assumes "P p" shows "P (p * [:- x, 1:])"
size_neq: fixes x :: "'b formula" shows "size x \<noteq> 0"
coarser_partitions_with_list_distinct: fixes ps :: "'b set list"   and x :: "'b"   and Q :: "'b set list" assumes "ps \<in> set (coarser_partitions_with_list x Q)"   and "distinct Q"   and "is_non_overlapping (set Q)"   and "{x} \<notin> set Q" shows "distinct ps"
right_idem: fixes a :: "'a"   and b :: "'a" shows "min (min a b) b = min a b"
l_null: fixes R :: "('c, 'd) ring_scheme"   and x :: "'c" assumes "semiring R"   and "x \<in> carrier R" shows "\<zero>\<^bsub>R\<^esub> \<otimes>\<^bsub>R\<^esub> x = \<zero>\<^bsub>R\<^esub>"
scalar_matrix_vector_assoc: fixes k :: "'b"   and A :: "(('b, 'd) vec, 'c) vec"   and v :: "('b, 'd) vec" shows "k *s (A *v v) = k *k A *v v"
RSubmodule_eigenspace: fixes R :: "'a set"   and smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and M :: "'b set"   and T :: "'b \<Rightarrow> 'b"   and r :: "'a" assumes "RModuleHom R smult M smult T"   and "r \<in> R"   and "\<And>s m. \<lbrakk>s \<in> R; m \<in> M\<rbrakk> \<Longrightarrow> smult s (smult r m) = smult r (smult s m)" shows "RModule R smult {m \<in> M. T m = smult r m} \<and> {m \<in> M. T m = smult r m} \<subseteq> M"