hcju/isabelleps huggingface dataset:
append_cols_split: fixes A :: "'a mat" assumes "0 < dim_col A" shows "A = mat_of_cols (dim_row A) [col A 0] @\<^sub>c mat_of_cols (dim_row A) (map (col A) [1..<dim_col A])"
add_diff_cancel_right_vec: fixes a :: "'b vec"   and n :: "nat"   and b :: "'b vec" assumes "a \<in> carrier_vec n"   and "b \<in> carrier_vec n" shows "a + b - b = a"
axioms_1: fixes smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and M :: "'b set"   and T :: "'b \<Rightarrow> 'b" assumes "ModuleEnd smult M T" shows "ModuleHom smult M smult T"
ex_mk_circline: fixes H :: "circline" shows "\<exists>A B C D. H = mk_circline A B C D \<and> hermitean (A, B, C, D) \<and> (A, B, C, D) \<noteq> mat_zero"
lookup_not_edom: fixes x :: "'d"   and m :: "'d \<Rightarrow> 'e" assumes "x \<notin> edom m" shows "m x = \<bottom>"
xy_Ur: shows "set {x, y} \<in>\<^sub>\<circ> \<Union>\<^sub>\<circ> r"
15: shows "padic_mult p f g (Suc (nat (padic_val p f + padic_val p g))) mod p ^ Suc (nat (padic_val p f + padic_val p g)) = i * j * p ^ (nat (padic_val p f) + nat (padic_val p g)) mod p ^ Suc (nat (padic_val p f + padic_val p g))"
intros_1: fixes r :: "Vertex \<Rightarrow> Vertex" assumes "r \<in> simple_rotations" shows "complex_rotationsp r"
outcomes_sum: fixes f :: "nat \<Rightarrow> real"   and n :: "nat"   and i :: "nat" shows "sum f {k |k. select_index n i k} + sum f {k |k. k < 2 ^ n \<and> \<not> select_index n i k} = sum f {0..<2 ^ n}"
solution_set_def: fixes A :: "(('d, 'e) vec, 'f) vec"   and b :: "('d, 'f) vec" shows "solution_set A b = {x. is_solution x A b}"
f_foldl_abs_alt_def: fixes \<phi> :: "'b ltl"   and w :: "'b set list" shows "af_G_abs_opt.f_foldl_abs (Abs \<phi>) w = foldl af_G_abs_opt.f_abs (Abs \<phi>) w"
assms_8: shows "(Var x' - p)\<^sup>2 - (Var x' - p) \<in> More_Modules.ideal polynomial_bool"
simps_6: fixes x3 :: "nat \<times> nat" shows "NONSAT_PUSH' \<noteq> SAT_PUSH' x3"
Inf_aform'_le_Sup_aform': fixes p :: "nat"   and X :: "'a \<times> 'a pdevs" shows "Inf_aform' p X \<le> Sup_aform' p X"
sublist_equal_part: fixes xs :: "'b list"   and ys :: "'b list" assumes "prefix xs ys" shows "take (length xs) ys = xs"
less_imp_neq: fixes x :: "'b list"   and y :: "'b list" assumes "strict_prefix x y" shows "x \<noteq> y"
merge0_def: fixes I' :: "('a \<times> 'b) list"   and I :: "('a \<times> 'b) list" shows "merge0 I' I \<equiv> filter (\<lambda>xy. fst xy \<notin> fst ` set I) I' @ I"
zero_is_irreducible_mult: shows "irreducible (Ring_Divisibility.mult_of R) \<zero>"
simps_2: fixes \<delta> :: "'e \<Rightarrow> 'f \<times> 'g \<Rightarrow> 'e"   and s :: "'e"   and r :: "'f \<times> 'g"   and rs :: "('f \<times> 'g) list" shows "pre_RDR.exec \<delta> s (Cons r rs) = \<delta> (pre_RDR.exec \<delta> s rs) r"
a: shows "h x \<oplus>\<^bsub>S\<^esub> \<ominus>\<^bsub>S\<^esub> h a = \<zero>\<^bsub>S\<^esub>"
hyps: fixes \<S> :: "nat \<Rightarrow> bool option"   and \<O> :: "nat set"   and L :: "nat set"   and R :: "nat set" assumes "sharing_consistent \<S> \<O> sb"   and "all_shared sb \<inter> L = {}"   and "all_shared sb \<inter> A = {}"   and "all_acquired sb \<inter> R = {}"   and "all_unshared sb \<inter> R = {}"   and "all_shared sb \<inter> R = {}" shows "share sb (\<S> \<oplus>\<^bsub>W\<^esub> R \<ominus>\<^bsub>A\<^esub> L) = share sb \<S> \<oplus>\<^bsub>W\<^esub> R \<ominus>\<^bsub>A\<^esub> L"
poly_higher_pderiv_aux1: fixes k :: "nat"   and n :: "nat"   and c :: "'a"   and p :: "'a poly" assumes "k < n" shows "poly ((pderiv ^^ k) ([:- c, 1::'a:] ^ n * p)) c = (0::'a)"
INum_int_2: shows "INum 0\<^sub>N = (0::'a)"
cmp_simps'_3: fixes \<mu> :: "'a"   and \<nu> :: "'a" assumes "C.arr \<mu>"   and "C.arr \<nu>"   and "src\<^sub>C \<mu> = trg\<^sub>C \<nu>" shows "trg\<^sub>D (\<Phi> (\<mu>, \<nu>)) = map\<^sub>0 (trg\<^sub>C \<mu>)"
inf_comp: fixes one :: "'b"   and times :: "'b \<Rightarrow> '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 dual :: "'b \<Rightarrow> 'b"   and dual_star :: "'b \<Rightarrow> 'b"   and omega :: "'b \<Rightarrow> 'b"   and star :: "'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "class.mbt_algebra one times inf less_eq less sup dual dual_star omega star bot top" shows "times (inf x y) z = inf (times x z) (times y z)"
1: shows "\<forall>m<length t + length u. \<forall>x xa. m = length x + length xa \<longrightarrow> (a \<in> set xa \<and> a \<notin> set x \<or> a \<in> set x \<and> a \<notin> set xa) \<and> a \<in> A \<longrightarrow> setinterleaving (xa, A, x) = {}"
subst_offending_flows: fixes G :: "'v graph"   and nP :: "'v \<Rightarrow> 'b" shows "SecurityInvariant_withOffendingFlows.set_offending_flows (\<lambda>G nP. c_sinvar m G) G nP = c_offending_flows m G"
P_mod_n_1: fixes t :: "'a"   and g :: "'a"   and n :: "nat" assumes "Idomain S"   and "t \<in> carrier S"   and "g \<in> carrier R"   and "P_mod R S X (S \<diamondsuit>\<^sub>p (t^\<^bsup>S Suc n\<^esup>)) g" shows "P_mod R S X (S \<diamondsuit>\<^sub>p t) g"
real_ivl_eqs_5: fixes r :: "real" assumes "0 < r" shows "ball 0 r = {- r<--<r}"
fpxs_of_fls_eq_0_iff: fixes f :: "'b fls" shows "(fpxs_of_fls f = 0) = (f = 0)"
smult_prod: fixes a :: "'a"   and b :: "'a poly" shows "Polynomial.smult a b = monom a 0 * b"
simps_2: fixes oper :: "op0"   and es :: "exp0 list" shows "is_cupcake_exp (App oper es) = (oper = Opapp \<and> list_all is_cupcake_exp es)"
get_M_pure: fixes ptr :: "(_) element_ptr"   and getter :: "('n, 'l, 'm, 'o, 'r option) RElement_scheme \<Rightarrow> 'u"   and h :: "(_) heap" shows "pure (get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t ptr getter) h"
inf_aci_1: fixes inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.semilattice_inf inf less_eq less" shows "inf x y = inf y x"
False: shows "\<not> length l * load_factor \<le> the_size ht * 100"
measure_preserving_distr': fixes f :: "'b \<Rightarrow> 'c"   and M :: "'b measure"   and N :: "'c measure" assumes "f \<in> M \<rightarrow>\<^sub>M N" shows "f \<in> measure_preserving M (distr M N f)"
coreflexive_one_closed: shows "matrix_idempotent_semiring.coreflexive mone"
mono_mult: fixes r :: "('a \<times> 'a) set"   and r' :: "('a \<times> 'a) set" assumes "r \<subseteq> r'" shows "mult r \<subseteq> mult r'"
add_nonneg_nonneg: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'d"   and b :: "'d" assumes "class.ordered_comm_monoid_add plus zero less_eq less"   and "less_eq zero a"   and "less_eq zero b" shows "less_eq zero (plus a b)"
rift_in_space_time: fixes I :: "('a, 'b, 'c) interp"   and \<theta> :: "('a, 'c) trm"   and ODE :: "('a, 'c) ODE"   and sol :: "real \<Rightarrow> (real, 'c) vec"   and t :: "real"   and b :: "(real, 'c) vec"   and \<psi> :: "('a, 'b, 'c) formula"   and s :: "real" assumes "is_interp I"   and "dfree \<theta>"   and "osafe ODE"   and "(sol solves_ode (\<lambda>_. ODE_sem I ODE)) {0..t} {x. mk_v I ODE (sol 0, b) x \<in> fml_sem I \<psi>}"   and "FVT \<theta> \<subseteq> semBV I ODE"   and "s \<in> {0..t}" shows "((\<lambda>t. sterm_sem I \<theta> (fst (mk_v I ODE (sol 0, b) (sol t)))) has_derivative (\<lambda>t'. t' * frechet I \<theta> (fst (mk_v I ODE (sol 0, b) (sol s))) (snd (mk_v I ODE (sol 0, b) (sol s))))) (at s within {0..t})"
nth_minus1: fixes n :: "nat" shows "(- 1) !! n = (n < LENGTH('b))"
Bag_mset: fixes xs :: "('a, nat) alist" shows "Bag xs = mset (ass_list_to_single_list (alist.impl_of xs))"
SReal_HFinite_diff_Infinitesimal: fixes x :: "real star" assumes "x \<in> \<real>"   and "x \<noteq> 0" shows "x \<in> HFinite - Infinitesimal"
WTrt_binop_simps_6: fixes P :: "'b prog"   and T1 :: "ty"   and T2 :: "ty"   and T :: "ty" shows "(P \<turnstile> T1\<guillemotleft>GreaterOrEqual\<guillemotright>T2 : T) = (T1 = Integer \<and> T2 = Integer \<and> T = Boolean)"
f1: shows "rk {a, b, \<gamma>} = 2"
in_hhomI: fixes \<mu> :: "'d"   and a :: "'d"   and b :: "'d" assumes "D.arr \<mu>"   and "src\<^sub>D \<mu> = a"   and "trg\<^sub>D \<mu> = b" shows "\<guillemotleft>\<mu> : a \<rightarrow>\<^sub>D b\<guillemotright>"
iter_def: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and step :: "nat \<Rightarrow> 'a \<Rightarrow> (nat \<times> 'a) list"   and \<tau>s :: "'a list"   and w :: "nat list" shows "iter f step \<tau>s w = while (\<lambda>(\<tau>s, w). w \<noteq> []) (\<lambda>(\<tau>s, w). let p = hd w in propa f (step p (\<tau>s ! p)) \<tau>s (tl w)) (\<tau>s, w)"
continuous_on_dist: fixes s :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "continuous_on s f"   and "continuous_on s g" shows "continuous_on s (\<lambda>x. dist (f x) (g x))"
content_smult_int: fixes a :: "int"   and p :: "int poly" shows "content (smult a p) = \<bar>a\<bar> * content p"
elims_2: fixes x :: "cmd"   and xa :: "w_node LDCFG_node"   and xb :: "cmd" assumes "labels_LDCFG_nodes x xa xb" obtains prog :: "cmd" and n :: "w_node" and c :: "cmd" where "x = prog" and "xa = LDCFG_node.Node n" and "xb = c" and "labels_nodes prog n c"
SystemCESubst_def: shows "SystemCESubst = \<lparr>SFunctions = Map.empty, SPredicates = Map.empty, SContexts = \<lambda>C. if C = pid1 then Some ((Geq (Const 0) (Const 0) && Geq ($f fid1 local.empty) (Const 0)) \<rightarrow> [[DiffAssign vid1 ($f fid1 local.empty)]]Pc (Inr ())) else None, SPrograms = Map.empty, SODEs = Map.empty\<rparr>"
component_of_term_sminus: fixes v :: "'t"   and t :: "'a" shows "component_of_term (v \<ominus> t) = component_of_term v"
filter_min_minimal: fixes rel :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and xs :: "'c list"   and y :: "'c" assumes "transp rel"   and "x \<in> set (filter_min rel xs)"   and "y \<in> set (filter_min rel xs)"   and "rel x y" shows "x = y"
l_set_tag_name_get_child_nodes_axioms_def: fixes set_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_set_tag_name_get_child_nodes_axioms set_tag_name_locs get_child_nodes_locs \<equiv> \<forall>ptr h h' ptr'. \<forall>w\<in>set_tag_name_locs ptr. h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> (\<forall>r\<in>get_child_nodes_locs ptr'. r h h')"
of_nat_code_if: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and n :: "nat" assumes "class.semiring_1 one times plus zero" shows "semiring_1.of_nat one plus zero n = (if n = 0 then zero else let (m, q) = divmod_nat n 2; m' = times (numeral.numeral one plus (num.Bit0 num.One)) (semiring_1.of_nat one plus zero m) in if q = 0 then m' else plus m' one)"
fun_upd_eqD: fixes f :: "'f \<Rightarrow> 'g"   and x :: "'f"   and y :: "'g"   and g :: "'f \<Rightarrow> 'g"   and z :: "'g" assumes "f(x := y) = g(x := z)" shows "y = z"
is_singleton_def: fixes A :: "'d set" shows "is_singleton A = (\<exists>x. A = {x})"
fmlookup_dom'_iff: fixes x :: "'c"   and m :: "('c, 'd) fmap" shows "(x \<in> fmdom' m) = (\<exists>a. fmlookup m x = Some a)"
subset_segment: fixes D :: "('c, 'd) Order_scheme"   and T :: "'c set"   and a :: "'c" assumes "Worder D"   and "T \<subseteq> carrier D"   and "\<forall>b\<in>T. \<forall>x. x \<prec>\<^bsub>D\<^esub> b \<and> x \<in> carrier D \<longrightarrow> x \<in> T"   and "minimum_elem D (carrier D - T) a" shows "T = segment D a"
sim_semantic_equivalence: fixes M :: "('b, 'c, 'd) KripkeStructure"   and M' :: "('b, 'c, 'e) KripkeStructure"   and f :: "'d \<Rightarrow> 'e"   and u :: "'d"   and \<phi> :: "('b, 'c) Kform" assumes "kripke M"   and "sim M M' f"   and "u \<in> worlds M" shows "M, u \<Turnstile> \<phi> = M', f u \<Turnstile> \<phi>"
map_of_rev_zip_upto_length_eq_None: fixes i :: "nat"   and B :: "'d list"   and d :: "nat" assumes "\<not> i < length B"   and "d = length B" shows "map_of (rev (zip [0..<d] B)) i = None"
length_take: fixes n :: "nat"   and xs :: "'f list" shows "length (take n xs) = min (length xs) n"
mHom_l_inv: fixes N :: "('g, 'b, 'h) Module_scheme"   and f :: "'a \<Rightarrow> 'g" assumes "R module N"   and "f \<in> mHom R M N" shows "tOp_mHom R M N (iOp_mHom R M N f) f = mzeromap M N"
fo_elig: shows "eligible S_G \<sigma> As (D + negs (mset As))"
append_Cons_nth_middle: fixes i :: "nat"   and xs :: "'a list"   and y :: "'a"   and zs :: "'a list" assumes "i = length xs" shows "(xs @ y # zs) ! i = y"
cycle_free_alt_in_nodes: fixes g :: "'b ugraph" shows "cycle_free g \<equiv> \<nexists>p u. p \<noteq> [] \<and> u \<in> nodes g \<and> simple p \<and> path g u p u"
dghm_ObjMap_app_in_HomCod_Obj: fixes a :: "V" assumes "a \<in>\<^sub>\<circ> op_dg \<AA>\<lparr>Obj\<rparr>" shows "\<FF>\<lparr>ObjMap\<rparr>\<lparr>a\<rparr> \<in>\<^sub>\<circ> \<BB>\<lparr>Obj\<rparr>"
prj1_simps_1: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and prj0 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and prj1 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and f :: "'a"   and g :: "'a" assumes "elementary_category_with_pullbacks C prj0 prj1"   and "partial_magma.arr C f \<and> partial_magma.arr C g \<and> partial_magma.cod C f = partial_magma.cod C g" shows "partial_magma.arr C (prj1 f g)"
linord_of_listP_append: fixes x :: "'a"   and y :: "'a"   and xs :: "'a list"   and ys :: "'a list" shows "linord_of_listP x y (xs @ ys) = (linord_of_listP x y xs \<or> linord_of_listP x y ys \<or> y \<in> set xs \<and> x \<in> set ys)"
intros_3: fixes \<Gamma> :: "'b formula multiset"   and F :: "'b formula"   and \<Delta> :: "'b formula multiset" assumes "\<Gamma> \<Rightarrow> F, \<Delta>" shows "\<^bold>\<not> F, \<Gamma> \<Rightarrow> \<Delta>"
path_append_elim: fixes r :: "'action list"   and s :: "'action list"   and p :: "'state" assumes "R.path (r @ s) p" obtains "R.path r p" and "R.path s (target r p)"
extremum_uniqueI: fixes a :: "'a upred" assumes "false \<sqsubseteq> a" shows "a = false"
transfer_get: shows "local.rel_state (=) State_Monad.get heap_get"
set_plus_closed: fixes G :: "'a set"   and A :: "'a set"   and B :: "'a set" assumes "Group G"   and "A \<subseteq> G"   and "B \<subseteq> G" shows "A + B \<subseteq> G"
agsum_add2Tr: fixes a :: "'a"   and n :: "nat" assumes "a \<in> carrier A" shows "\<zero> = aSum A n a \<plusminus> aSum A n (-\<^sub>a a)"
y2: shows "y\<^sub>2 [^] 2 = x\<^sub>2 [^] 3 \<oplus> a \<otimes> x\<^sub>2 \<oplus> b"
mat1I_def: fixes ze :: "'c"   and on :: "'c"   and n :: "nat" shows "mat1I ze on n \<equiv> map (vec1I ze on n) [0..<n]"
inf_commute: fixes inf :: "'ap \<Rightarrow> 'ap \<Rightarrow> 'ap"   and less_eq :: "'ap \<Rightarrow> 'ap \<Rightarrow> bool"   and less :: "'ap \<Rightarrow> 'ap \<Rightarrow> bool"   and x :: "'ap"   and y :: "'ap" assumes "class.semilattice_inf inf less_eq less" shows "inf x y = inf y x"
convex_hull_singleton: fixes a :: "'c" shows "convex hull {a} = {a}"
notifyWatchesLoopPreservedVariables: fixes state :: "State"   and Wl :: "nat list"   and literal :: "Literal"   and newWl :: "nat list" assumes "InvariantWatchesEl (getF state) (getWatch1 state) (getWatch2 state)"   and "\<forall>c. c \<in> set Wl \<longrightarrow> 0 \<le> c \<and> c < length (getF state)" shows "let state' = notifyWatches_loop literal Wl newWl state in getM state' = getM state \<and> getF state' = getF state \<and> getSATFlag state' = getSATFlag state \<and> isPrefix (getQ state) (getQ state')"
F_preserves_cod: fixes AA :: "('d, 'e, 'f) category_scheme"   and BB :: "('g, 'h, 'i) category_scheme"   and F :: "('d, 'e, 'g, 'h) functor" assumes "Functor F : AA \<longrightarrow> BB" shows "\<forall>f\<in>Ar\<^bsub>AA\<^esub>. F \<^bsub>\<o>\<^esub> (Cod\<^bsub>AA\<^esub> f) = Cod\<^bsub>BB\<^esub> F \<^bsub>\<a>\<^esub> f"
assms_1: shows "ipresIGWlsAbs hA MOD MOD'"
fps_divide_diff: fixes f :: "'a fps"   and g :: "'a fps"   and h :: "'a fps" shows "(f - g) / h = f / h - g / h"
isZeroK_def: shows "isZeroK \<equiv> \<Lambda> nK. bindK\<cdot>nK\<cdot>(\<Lambda> (ValKN\<cdot>n). unitK\<cdot>(if n = 0 then ValKTT else ValKFF))"
dvd_lcm_I2_nat: fixes k :: "nat"   and n :: "nat"   and m :: "nat" assumes "k dvd n" shows "k dvd lcm m n"
equal_fun_def: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" shows "equal_class.equal f g = (\<forall>x\<in>set enum. f x = g x)"
constant_def: fixes T :: "Unary_PCF.type" shows "constant T = \<Lambda>\<langle>\<B>\<rangle> close0_Var (''bool'', \<B>) (constant0 T)"
openin_subtopology_Int: fixes X :: "'c topology"   and S :: "'c set"   and T :: "'c set" assumes "openin X S" shows "openin (subtopology X T) (S \<inter> T)"
map_index: fixes f :: "nat \<Rightarrow> 'e \<Rightarrow> 'd"   and xs :: "'e list" shows "map_index f xs = map2 f [0..<length xs] xs"
unit_SBE_def: fixes e :: "'c" shows "(returning e) = (\<lambda>\<sigma>. Some {(e, \<sigma>)})"
IH_2: shows "vars_of (t2 \<lhd> \<sigma>) = \<Union> {V. \<exists>x. x \<in> vars_of t2 \<and> V = vars_of (Var x \<lhd> \<sigma>)}"
iteratei_def: fixes S :: "'s" shows "iteratei S \<equiv> it_to_it (list_it S)"
lnth_lappend: fixes xs :: "'a llist"   and ys :: "'a llist"   and n :: "nat" shows "lnth (lappend xs ys) n = (if enat n < llength xs then lnth xs n else lnth ys (n - the_enat (llength xs)))"
attach_shadow_root_def: fixes element_ptr :: "(_) element_ptr"   and shadow_root_mode :: "shadow_root_mode" shows "attach_shadow_root element_ptr shadow_root_mode = Heap_Error_Monad.bind (get_tag_name element_ptr) (\<lambda>tag. Heap_Error_Monad.bind (if tag \<notin> safe_shadow_root_element_types then error NotSupportedError else return ()) (\<lambda>_. Heap_Error_Monad.bind (get_shadow_root element_ptr) (\<lambda>prev_shadow_root. Heap_Error_Monad.bind (if prev_shadow_root \<noteq> None then error NotSupportedError else return ()) (\<lambda>_. Heap_Error_Monad.bind new\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M (\<lambda>new_shadow_root_ptr. Heap_Error_Monad.bind (set_mode new_shadow_root_ptr shadow_root_mode) (\<lambda>_. Heap_Error_Monad.bind (set_shadow_root element_ptr (Some new_shadow_root_ptr)) (\<lambda>_. return new_shadow_root_ptr)))))))"
ind_from_1: fixes n :: "nat"   and P :: "nat \<Rightarrow> bool" assumes "0 < n"   and "P 1"   and "\<And>n. \<lbrakk>0 < n; P n\<rbrakk> \<Longrightarrow> P (Suc n)" shows "P n"
NonceF_synth: fixes n :: "fresh_t"   and H :: "msg set" assumes "NonceF n \<in> synth H" obtains "NonceF n \<in> H"
Max_ge_image: fixes S :: "'c set"   and x :: "'c"   and f :: "'c \<Rightarrow> 'd" assumes "finite S"   and "x \<in> S" shows "f x \<le> Max (f ` S)"
edges: fixes w :: "'a set \<Rightarrow> nat"   and g :: "'a ugraph"   and r :: "'a"   and Q :: "'a \<Rightarrow> enat"   and \<pi> :: "'a \<Rightarrow> 'a option"   and u :: "'a"   and v :: "'a" assumes "Prim_Invar2_ctd_loc w g r Q \<pi>"   and "\<pi> u = Some v" shows "(u, v) \<in> edges (Prim.rg g r)"
leD: fixes y :: "'a word"   and x :: "'a word" assumes "y \<le>s x" shows "\<not> x <s y"
COPY'_rec: shows "COPY' = (left`?`x \<rightarrow> right`!`x \<rightarrow> COPY')"
Bell_def: fixes n :: "nat" shows "Bell n = card {P. partition_on {0..<n} P}"
mset_subset_eq_count: fixes A :: "'b multiset"   and B :: "'b multiset"   and a :: "'b" assumes "A \<subseteq># B" shows "count A a \<le> count B a"
fin: shows "\<forall>i\<in>{0..<Suc l}. finite {p. p permutes {0..<Suc l} \<and> p i = j}"
env_restr_override_onI: fixes S2 :: "'d set"   and S :: "'d set"   and \<rho>1 :: "'d \<Rightarrow> 'e"   and \<rho>2 :: "'d \<Rightarrow> 'e" assumes "- S2 \<subseteq> S" shows "\<rho>1 f|` S ++\<^bsub>S2\<^esub> \<rho>2 = \<rho>1 ++\<^bsub>S2\<^esub> \<rho>2"
typ_of_Abs_bind_fv: fixes A :: "Core.term"   and ty :: "typ"   and bT :: "typ"   and v :: "variable" assumes "typ_of A = Some ty" shows "typ_of (Abs_fv v bT A) = Some (bT \<rightarrow> ty)"
assms_2: fixes R :: "ccs"   and S :: "ccs" assumes "(R, S) \<in> X" shows "R \<leadsto>\<^sup>^<(X \<union> weakBisimulation)> S"
assms_4: shows "\<forall>m\<in>set ms. S m = T m"
PTrans: shows "P \<Longrightarrow>\<^sub>l\<^sup>^a<\<nu>x> \<prec> P'"
NI_unrelated_total_def: shows "int.NI_unrelated_total \<equiv> \<forall>execs a n. int.realistic_executions execs \<longrightarrow> (let s_f = int.run_total n s0 execs in output_f s_f a = output_f (int.run_total n s0 (int.purge execs (current s_f))) a \<and> current s_f = current (int.run_total n s0 (int.purge execs (current s_f))))"
star: fixes x :: "'a" assumes "x \<in> S" shows "open_segment (0::'a) x \<subseteq> rel_interior S"
n_eq_length: shows "n = IArray.length (matrix_to_iarray A)"
valid_W_inv_mo_co_unlock: fixes s :: "'d process_name \<Rightarrow> ('e, 'd, 'f, 'g) Model.local_state"   and p :: "'d process_name" assumes "sys_mem_lock s = Some p"   and "sys_mem_store_buffers p s = []"   and "\<And>r. r \<in> ghost_honorary_grey (s p) \<Longrightarrow> marked r s"   and "valid_W_inv s" shows "valid_W_inv (s(sys := mem_lock_update Map.empty (s sys)))"
altc_nondet_def: fixes mergec :: "'f cset \<Rightarrow> ('f \<Rightarrow> 'e) \<Rightarrow> 'e"   and A :: "'f cset"   and f :: "'f \<Rightarrow> ('d, 'e) nondetT" shows "cset_nondetM_base.altc_nondet mergec A f = NondetT (mergec A (run_nondet \<circ> f))"
sinks_aux_single_event: fixes I :: "('b \<times> 'b) set"   and D :: "'c \<Rightarrow> 'b"   and U :: "'b set"   and x :: "'c" shows "sinks_aux I D U [x] = (if \<exists>v\<in>U. (v, D x) \<in> I then insert (D x) U else U)"
strict_boundedE: fixes a :: "'c"   and b :: "'c"   and c :: "'c" assumes "a < min b c" obtains "a < b" and "a < c"
assms: fixes x :: "'a set" assumes "x \<in> U" shows "f x = g x"
ennreal_Ex_less_of_nat: fixes x :: "ennreal" assumes "x < top" shows "\<exists>n. x < of_nat n"
lazy_sequence_eqI: fixes xq :: "'a lazy_sequence"   and yq :: "'a lazy_sequence" assumes "list_of_lazy_sequence xq = list_of_lazy_sequence yq" shows "xq = yq"
discr_discr0: fixes indis :: "('a \<times> 'a) set"   and tval :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'a \<Rightarrow> 'a"   and c :: "('b, 'c) com" assumes "PL_Indis indis"   and "PL_Indis.discr tval aval indis c" shows "PL_Indis.discr0 tval aval indis c"
Schur_Im: fixes G :: "'a set"   and smult :: "('b, 'a) aezfun \<Rightarrow> 'c \<Rightarrow> 'c"   and V :: "'c set"   and smult' :: "('b, 'a) aezfun \<Rightarrow> 'd \<Rightarrow> 'd"   and W :: "'d set"   and T :: "'c \<Rightarrow> 'd" assumes "FinGroupRepresentation G smult V"   and "IrrFinGroupRepresentation G smult' W"   and "FGModuleHom G smult V smult' T"   and "T ` V \<subseteq> W"   and "T ` V \<noteq> 0" shows "T ` V = W"
domain_1'': fixes x :: "'a"   and y :: "'a" shows "ad (ad (x \<cdot> y)) \<le> ad (ad x)"
assms_6: shows "(iface::iface) \<in> dom (ipassmt::iface \<Rightarrow> ('i word \<times> nat) list option)"
drop_Suc_Cons: fixes n :: "nat"   and x :: "'k"   and xs :: "'k list" shows "drop (Suc n) (x # xs) = drop n xs"
simps_8: fixes \<sigma> :: "(char list \<times> 'c list) trace"   and v :: "'c list"   and i :: "nat"   and \<phi> :: "'c MFOTL.formula"   and I :: "\<I>"   and \<psi> :: "'c MFOTL.formula" shows "MFOTL.sat \<sigma> v i (formula.Since \<phi> I \<psi>) = (\<exists>j\<le>i. mem (\<tau> \<sigma> i - \<tau> \<sigma> j) I \<and> MFOTL.sat \<sigma> v j \<psi> \<and> (\<forall>k\<in>{j<..i}. MFOTL.sat \<sigma> v k \<phi>))"
M: fixes x :: "'a"   and m :: "nat"   and n :: "nat" assumes "x \<in> X"   and "M \<le> m"   and "m < n" shows "dist (f m x) (f n x) < e"
newton_interpolation_poly: fixes xs_ys :: "('a \<times> 'a) list"   and p :: "'a poly"   and x :: "'a"   and y :: "'a" assumes "distinct (map fst xs_ys)"   and "p = newton_interpolation_poly xs_ys"   and "(x, y) \<in> set xs_ys" shows "poly p x = y"
failingSubProps: fixes subs :: "'b \<Rightarrow> 'b set"   and P :: "(nat \<times> 'b) set \<Rightarrow> bool"   and gamma :: "'b" assumes "inherited subs P"   and "\<not> P (tree subs gamma)"   and "\<not> terminal subs gamma"   and "fans subs" shows "failingSub subs P gamma \<in> subs gamma \<and> \<not> P (tree subs (failingSub subs P gamma))"