hcju/isabelleps huggingface dataset:
res_c1: fixes f :: "'a \<Rightarrow> 'a"   and x :: "'a" assumes "residuated f" shows "x \<le> residual f (f x)"
elims_2: fixes x :: "('c, 'd) BsSkewElem" assumes "elem_invar x" obtains e :: "'c" and a :: "'d" and q :: "('c, 'd) BsSkewBinomialTree list" where "x = Element e a q" and "(\<forall>x. x \<in># image_mset fst (SkewBinomialHeapStruc.queue_to_multiset (map bsmapt q)) \<longrightarrow> a \<le> eprio x \<and> elem_invar x) \<and> SkewBinomialHeapStruc.invar (map bsmapt q)"
ACI_norm_wf: fixes wf_atom :: "nat \<Rightarrow> 'c \<Rightarrow> bool"   and n :: "nat"   and r :: "'c rexp" shows "alphabet.wf wf_atom n \<guillemotleft>r\<guillemotright> = alphabet.wf wf_atom n r"
poly_mult_monom: fixes p :: "'a list"   and a :: "'a"   and n :: "nat" assumes "polynomial (carrier R) p"   and "a \<in> carrier R - {\<zero>}" shows "poly_mult (local.monom a n) p = (if p = [] then [] else poly_mult [a] p @ replicate n \<zero>)"
simps_2: fixes x :: "'h" shows "exception_of_option (Some x) = OK x"
fv_strand_subst': fixes S :: "('d, 'c) strand_step list"   and \<delta> :: "'c \<Rightarrow> ('d, 'c) Term.term" assumes "bvars\<^sub>s\<^sub>t S \<inter> (subst_domain \<delta> \<union> range_vars \<delta>) = {}" shows "fv\<^sub>s\<^sub>e\<^sub>t (\<delta> ` fv\<^sub>s\<^sub>t S) = fv\<^sub>s\<^sub>t (S \<cdot>\<^sub>s\<^sub>t \<delta>)"
even_and_iff: fixes 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 divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> '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 "and" :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and or :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and xor :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and mask :: "nat \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.semiring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask" shows "dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) (and a b) = (dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) a \<or> dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) b)"
elims_2: fixes x :: "('var, 'val, 'ret, 'pname) edge_kind list"   and xa :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" assumes "preds x xa" obtains s :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" where "x = []" and "xa = sa__"   | et :: "('var, 'val, 'ret, 'pname) edge_kind" and ets :: "('var, 'val, 'ret, 'pname) edge_kind list" and s :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" where "x = et # ets" and "xa = s" and "pred et s \<and> preds ets (transfer et s)"
bij_betw_inv_into_right: fixes f :: "'e \<Rightarrow> 'f"   and A :: "'e set"   and A' :: "'f set"   and a' :: "'f" assumes "bij_betw f A A'"   and "a' \<in> A'" shows "f (inv_into A f a') = a'"
hyps_3: shows "<\<nu>a>P \<Longrightarrow>\<^sub>\<tau> <\<nu>a>P''"
compatible_ordering_axioms: shows "compatible_ordering A (\<sqsubseteq>) (\<sqsubset>)"
std_then_implici_effs: fixes name :: "char list"   and pres :: "(nat \<times> nat) list"   and effs :: "((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list"   and layer :: "nat" assumes "is_standard_operator' (name, pres, effs, layer)" shows "implicit_pres effs = []"
inits_self_def: fixes xs :: "'c list" shows "inits_self xs = zip ([] # inits xs) xs"
line_integral_exists_on_rec_join: fixes l :: "(int \<times> (real \<Rightarrow> real \<times> real)) list"   and F :: "real \<times> real \<Rightarrow> real \<times> real"   and basis :: "(real \<times> real) set" assumes "boundary_chain (set l)"   and "valid_chain_list l"   and "\<And>k \<gamma>. (k, \<gamma>) \<in> set l \<Longrightarrow> valid_path \<gamma>"   and "\<forall>(k, \<gamma>)\<in>set l. line_integral_exists F basis \<gamma>" shows "line_integral_exists F basis (rec_join l)"
le_of_nat_term_order_alt: fixes to :: "'b nat_term_order" shows "le_of_nat_term_order to = (\<lambda>u v. ko.le (key_order_of_nat_term_order_inv to) v u)"
inject: fixes x1 :: "'a list"   and x2 :: "'a"   and y1 :: "'a list"   and y2 :: "'a" shows "(Infer x1 x2 = Infer y1 y2) = (x1 = y1 \<and> x2 = y2)"
cyclic_on_obtain_lists_succ: fixes f :: "'b \<Rightarrow> 'b"   and S :: "'b set"   and css :: "'b list list" assumes "f permutes S"   and "S = \<Union> (sset css)"   and "distincts css"   and "\<And>cs. cs \<in> set css \<Longrightarrow> cyclic_on f (set cs)" obtains xss :: "'b list list" where "f = lists_succ xss" and "distincts xss" and "map set xss = map set css" and "map hd xss = map hd css"
D_mono: fixes A :: "'a set option"   and A' :: "'a set option"   and e :: "'a exp" assumes "A \<sqsubseteq> A'"   and "\<D> e A" shows "\<D> e A'"
bit_numeral_Bit0_Suc_iff: fixes m :: "num"   and n :: "nat" shows "bit (numeral (num.Bit0 m)) (Suc n) = bit (numeral m) n"
simps_8: fixes P :: "pol"   and x :: "nat"   and R :: "pol"   and y :: "nat"   and Q :: "pol" shows "PX P x R \<langle>+\<rangle> Pinj y Q = (if y = 0 then PX P x R \<langle>+\<rangle> Q else if y = 1 then PX P x (R \<langle>+\<rangle> Q) else PX P x (R \<langle>+\<rangle> Pinj (y - 1) Q))"
Norm_def: fixes N :: "'b quasinorm"   and x :: "'b" shows "Norm N x = enn2real (eNorm N x)"
quotient_web_def: fixes \<Gamma> :: "('a, 'b) web_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal" shows "quotient_web \<Gamma> f = \<lparr>edge = \<lambda>x y. edge \<Gamma> x y \<and> x \<notin> RF\<^sup>\<circ>\<^bsub>\<Gamma>\<^esub> (TER\<^bsub>\<Gamma>\<^esub> f) \<and> y \<notin> RF\<^bsub>\<Gamma>\<^esub> (TER\<^bsub>\<Gamma>\<^esub> f), weight = \<lambda>x. if x \<in> RF\<^sup>\<circ>\<^bsub>\<Gamma>\<^esub> (TER\<^bsub>\<Gamma>\<^esub> f) \<or> x \<in> TER\<^bsub>\<Gamma>\<^esub> f \<inter> B \<Gamma> then 0 else weight \<Gamma> x, A = \<E>\<^bsub>\<Gamma>\<^esub> (TER\<^bsub>\<Gamma>\<^esub> f) - (B \<Gamma> - A \<Gamma>), B = B \<Gamma>, \<dots> = web.more \<Gamma>\<rparr>"
prems: shows "aeL f|` (- S) = aeR f|` (- S)"
simps_1: fixes pt :: "prim_ty" shows "default_val (PrimT pt) = defpval pt"
lambda_zero: shows "(\<lambda>h. 0::'e) = (*) (0::'e)"
ord_of_nat_in_Vset: fixes a :: "nat" shows "a\<^sub>\<nat> \<in>\<^sub>\<circ> Vset \<alpha>"
Rep_Abs_Mp'': fixes f :: "('b event list \<times> 'b event set) set"   and A :: "'b set"   and P :: "'b \<Rightarrow> 'b process"   and d :: "'b event list set" assumes "f = {(tr, ref). tr = [] \<and> ref \<inter> ev ` A = {}} \<union> {(tr, ref). tr \<noteq> [] \<and> hd tr \<in> ev ` A \<and> (\<exists>a. ev a = hd tr \<and> (tl tr, ref) \<in> F (P a))}"   and "d = {d. d \<noteq> [] \<and> hd d \<in> ev ` A \<and> (\<exists>a. ev a = hd d \<and> tl d \<in> D (P a))}" shows "Rep_process (Abs_process (f, d)) = (f, d)"
sigma_algebra_tail_events: fixes A :: "nat \<Rightarrow> 'a set set" assumes "\<And>i. sigma_algebra (space M) (A i)" shows "sigma_algebra (space M) (tail_events A)"
N0a: shows "n0 < length exec \<and> (\<forall>n'\<ge>n0. n' < length trace \<longrightarrow> execMsg n' \<noteq> msg) \<and> (n0 = 0 \<or> \<not> (\<forall>n'\<ge>n0 - 1. n' < length trace \<longrightarrow> execMsg n' \<noteq> msg))"
infinite_Icc: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" assumes "class.dense_linorder less_eq less"   and "less a b" shows "infinite (ord.atLeastAtMost less_eq a b)"
prefix_def: fixes xs :: "'d list"   and ys :: "'d list" shows "prefix xs ys = (\<exists>zs. ys = xs @ zs)"
partition_on_msetI: fixes P :: "'b multiset multiset"   and A :: "'b multiset" assumes "\<Sum>\<^sub># P = A"   and "{#} \<notin># P" shows "partition_on_mset A P"
msum_aform_Joints: fixes X :: "'b \<times> 'b pdevs"   and d :: "nat"   and XS :: "('b \<times> 'b pdevs) list"   and x :: "'b"   and xs :: "'b list"   and y :: "'b"   and Y :: "'b \<times> 'b pdevs" assumes "degree_aform X \<le> d"   and "\<And>X. X \<in> set XS \<Longrightarrow> degree_aform X \<le> d"   and "x # xs \<in> Joints (X # XS)"   and "y \<in> Affine Y" shows "(x + y) # x # xs \<in> Joints (msum_aform d X Y # X # XS)"
second: shows "(\<forall>(a, b, c)\<in>set les. \<exists>x. \<forall>y<x. a * y\<^sup>2 + b * y + c < 0) \<or> (\<exists>(a', b', c')\<in>set les. a' = 0 \<and> b' \<noteq> 0 \<and> (\<forall>(d, e, f)\<in>set les. \<exists>y'>- (c' / b'). \<forall>x\<in>{- (c' / b')<..y'}. d * x\<^sup>2 + e * x + f < 0) \<or> a' \<noteq> 0 \<and> 4 * a' * c' \<le> b'\<^sup>2 \<and> ((\<forall>(d, e, f)\<in>set les. \<exists>y'>(sqrt (b'\<^sup>2 - 4 * a' * c') - b') / (2 * a'). \<forall>x\<in>{(sqrt (b'\<^sup>2 - 4 * a' * c') - b') / (2 * a')<..y'}. d * x\<^sup>2 + e * x + f < 0) \<or> (\<forall>(d, e, f)\<in>set les. \<exists>y'>(- b' - sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'). \<forall>x\<in>{(- b' - sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}. d * x\<^sup>2 + e * x + f < 0))) \<longrightarrow> (\<exists>x. \<forall>(a, b, c)\<in>set les. a * x\<^sup>2 + b * x + c < 0)"
get_offending_flows_subseteq_edges: fixes M :: "'a SecurityInvariant_configured list"   and F :: "('a \<times> 'a) set"   and V :: "'a set"   and E :: "('a \<times> 'a) set" assumes "valid_reqs M"   and "F \<in> get_offending_flows M \<lparr>nodes = V, edges = E\<rparr>" shows "F \<subseteq> E"
coll: fixes x :: "real \<times> real" assumes "x \<in> set xs" shows "coll 0 z x"
monoidal_category_with_alternate_unit_def: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and T :: "'b \<times> 'b \<Rightarrow> 'b"   and \<alpha> :: "'b \<times> 'b \<times> 'b \<Rightarrow> 'b"   and \<iota> :: "'b"   and \<iota>\<^sub>1 :: "'b" shows "monoidal_category_with_alternate_unit C T \<alpha> \<iota> \<iota>\<^sub>1 \<equiv> monoidal_category C T \<alpha> \<iota> \<and> monoidal_category C T \<alpha> \<iota>\<^sub>1"
prob0_fst_qubits_of_jozsa_algo: shows "prob0_fst_qubits n jozsa_algo = (\<Sum>j\<in>{0, 1}. (cmod (jozsa_algo $$ (j, 0)))\<^sup>2)"
SUP_sup_distrib: fixes A :: "'g set"   and f :: "'g \<Rightarrow> 'h multiset"   and g :: "'g \<Rightarrow> 'h multiset" assumes "A \<noteq> {}"   and "subset_mset.bdd_above (f ` A)"   and "subset_mset.bdd_above (g ` A)" shows "Sup (f ` A) \<union># Sup (g ` A) = (SUP a\<in>A. f a \<union># g a)"
freshEnv_def2: fixes xs :: "'d"   and x :: "'e"   and rho :: "'d \<Rightarrow> 'e \<Rightarrow> ('f, 'g, 'd, 'e, 'h) qTerm set option" shows "freshEnv xs x rho = qFreshEnv xs x (pickE rho)"
Ball_set: fixes xs :: "'c list"   and P :: "'c \<Rightarrow> bool" shows "Ball (set xs) P = list_all P xs"
\<Y>_def: shows "\<Y> \<equiv> (\<lambda>U. U \<times> topspace Y) ` Collect (openin X)"
matrix_vector_mult_diff_rdistrib: fixes A :: "(('a, 'c) Finite_Cartesian_Product.vec, 'b) Finite_Cartesian_Product.vec"   and B :: "(('a, 'c) Finite_Cartesian_Product.vec, 'b) Finite_Cartesian_Product.vec"   and x :: "('a, 'c) Finite_Cartesian_Product.vec" shows "(A - B) *v x = A *v x - B *v x"
prems: shows "\<forall>i\<in>iset. i < n"
add_divide_eq_if_simps_4: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b"   and z :: "'b" assumes "class.division_ring divide inverse one times plus zero minus uminus" shows "minus a (divide b z) = (if z = zero then a else divide (minus (times a z) b) z)"
nn_cond_exp_F_meas: fixes f :: "'a \<Rightarrow> ennreal" assumes "f \<in> borel_measurable F" shows "AE x in M. f x = nn_cond_exp M F f x"
\<rho>2v2_subset_\<rho>v_inter_E2: shows "\<rho>2 \<V>2 \<subseteq> \<rho> \<V> \<inter> E\<^bsub>ES2\<^esub>"
lookup_map2_val_neutr: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and xs :: "('b, 'c) oalist_ntm"   and ys :: "('b, 'c) oalist_ntm"   and k :: "'b" assumes "\<And>k x. f k x (0::'c) = x"   and "\<And>k x. f k (0::'c) x = x" shows "OAlist_lookup_ntm (OAlist_map2_val_neutr_ntm f xs ys) k = f k (OAlist_lookup_ntm xs k) (OAlist_lookup_ntm ys k)"
r_kind: fixes e :: "nat" shows "eval r_kind [e] \<down>= encode_kind e"
h1: shows "set (characterize_consistent_signs_at_roots p [q]) \<subseteq> {[1], [0], [- 1]}"
suffix_plus_com: fixes s :: "nat \<Rightarrow> 'b"   and m :: "nat"   and n :: "nat" shows "s |\<^sub>s m |\<^sub>s n = s |\<^sub>s m + n"
rconjby_uminus: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a" assumes "class.group_add minus plus zero uminus" shows "plus (plus (uminus x) (uminus y)) x = uminus (plus (plus (uminus x) y) x)"
causal_broadcast: fixes m2 :: "'a \<times> ('a, 'b) operation"   and j :: "nat"   and m1 :: "'a \<times> ('a, 'b) operation"   and i :: "nat" assumes "Deliver m2 \<in> set (history j)"   and "Deliver m1 \<sqsubset>\<^sup>i Broadcast m2" shows "Deliver m1 \<sqsubset>\<^sup>j Deliver m2"
admissible_kbo_def: fixes w :: "'a \<times> nat \<Rightarrow> nat"   and w0 :: "nat"   and pr_strict :: "'a \<times> nat \<Rightarrow> 'a \<times> nat \<Rightarrow> bool"   and pr_weak :: "'a \<times> nat \<Rightarrow> 'a \<times> nat \<Rightarrow> bool"   and least :: "'a \<Rightarrow> bool"   and scf :: "'a \<times> nat \<Rightarrow> nat \<Rightarrow> nat" shows "admissible_kbo w w0 pr_strict pr_weak least scf \<equiv> (((\<forall>f. w0 \<le> w (f, 0)) \<and> 0 < w0) \<and> (\<forall>f g n. w (f, 1) = 0 \<longrightarrow> pr_weak (f, 1) (g, n)) \<and> (\<forall>f. least f = (w (f, 0) = w0 \<and> (\<forall>g. w (g, 0) = w0 \<longrightarrow> pr_weak (g, 0) (f, 0))))) \<and> ((\<forall>i n f. i < n \<longrightarrow> 0 < scf (f, n) i) \<and> (\<forall>fn. pr_weak fn fn)) \<and> (\<forall>fn gm hk. pr_weak fn gm \<longrightarrow> pr_weak gm hk \<longrightarrow> pr_weak fn hk) \<and> (\<forall>fn gm. pr_strict fn gm = (pr_weak fn gm \<and> \<not> pr_weak gm fn)) \<and> SN {(fn, gm). pr_strict fn gm}"
octo_of_real_sel_4: fixes x :: "real" shows "Im3 (octo_of_real x) = 0"
frontier_ci: fixes a :: "real"   and b :: "real" assumes "a < b" shows "frontier {a<..<b} = {a, b}"
rep_eq: fixes x :: "uint"   and xa :: "nat" shows "Rep_uint (x >>> xa) = signed_drop_bit xa (Rep_uint x)"
lfp_lowerbound: fixes Inf :: "'d set \<Rightarrow> 'd"   and Sup :: "'d set \<Rightarrow> 'd"   and inf :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sup :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and bot :: "'d"   and top :: "'d"   and f :: "'d \<Rightarrow> 'd"   and A :: "'d" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "less_eq (f A) A" shows "less_eq (complete_lattice.lfp Inf less_eq f) A"
sum_case_Inl: fixes f :: "hf \<Rightarrow> 'b"   and g :: "hf \<Rightarrow> 'b"   and x :: "hf" shows "sum_case f g (HF.Inl x) = f x"
propa_ss: shows "\<forall>(q, \<tau>)\<in>set (step (hd w) (ss ! hd w)). ssa ! q = \<tau> \<squnion>\<^bsub>f\<^esub> ss ! q"
iso_Field: fixes r :: "('d \<times> 'd) set"   and r' :: "('e \<times> 'e) set"   and f :: "'d \<Rightarrow> 'e" assumes "iso r r' f" shows "f ` Field r = Field r'"
assms_1: shows "t \<in>\<^sub>A frontier (c_imp c loc2)"
seqE: fixes g :: "'b"   and f :: "'b" assumes "D.seq g f" obtains "D.arr f" and "D.arr g" and "D.dom g = D.cod f"
ctr_simps_pred_prod_eq_cart: fixes A :: "'e set"   and B :: "'f set" shows "pred_prod (\<lambda>x. x \<in> A) (\<lambda>x. x \<in> B) = (\<lambda>x. x \<in> A \<times> B)"
hyps: assumes "xs \<noteq> []" shows "sum_list (map (vec_to_iarray \<circ> f) xs) = vec_to_iarray (sum_list (map f xs))"
Red_F_Bot: fixes B :: "'g"   and N :: "'g set" assumes "B \<in> Bot_G"   and "N \<Turnstile>G {B}" shows "N - Red_F_G N \<Turnstile>G {B}"
red_not_final_thread: fixes final :: "'a \<Rightarrow> bool"   and r :: "('d, 'b, 'a, 'c, 'e, 'f) semantics"   and convert_RA :: "'d \<Rightarrow>f nat \<Rightarrow> 'f list"   and s :: "('d, 'b) locks \<times> (('b \<Rightarrow> ('a \<times> 'd \<Rightarrow>f nat) option) \<times> 'c) \<times> ('b \<Rightarrow> 'e wait_set_status option) \<times> 'b set"   and t :: "'b"   and ta :: "('d, 'b, 'a, 'c, 'e, 'f) thread_action"   and s' :: "('d, 'b) locks \<times> (('b \<Rightarrow> ('a \<times> 'd \<Rightarrow>f nat) option) \<times> 'c) \<times> ('b \<Rightarrow> 'e wait_set_status option) \<times> 'b set" assumes "multithreaded final r"   and "multithreaded_base.redT final r convert_RA s (t, ta) s'" shows "final_thread.not_final_thread final s t"
intro: fixes assigned_nodes :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_tag_name :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, char list) prog"   and get_root_node :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr) prog"   and get_host :: "(_) shadow_root_ptr \<Rightarrow> ((_) heap, exception, (_) element_ptr) prog"   and find_slot :: "bool \<Rightarrow> (_) node_ptr \<Rightarrow> ((_) heap, exception, (_) element_ptr option) prog"   and flatten_dom :: "((_) heap, exception, unit) prog"   and remove :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, unit) prog"   and append_child :: "(_) object_ptr \<Rightarrow> (_) node_ptr \<Rightarrow> ((_) heap, exception, unit) prog"   and remove_shadow_root :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, unit) prog" assumes "assigned_nodes = l_assigned_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_assigned_nodes get_child_nodes get_tag_name get_root_node get_host find_slot"   and "flatten_dom = l_assigned_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_flatten_dom get_child_nodes get_tag_name get_root_node get_host find_slot remove append_child remove_shadow_root" shows "l_assigned_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms assigned_nodes flatten_dom get_child_nodes get_tag_name get_root_node get_host find_slot remove append_child remove_shadow_root"
wf_act_adds_dels_atoms: fixes P :: "ast_problem"   and effs :: "object ast_effect"   and ae :: "object atom formula" assumes "wf_ast_problem P"   and "ast_problem.wf_effect_inst P effs"   and "ae \<in> set (dels effs)" shows "is_predAtom ae"
simps: fixes pf :: "('d, 'c) lifted_pair dense"   and f :: "('d, 'c) phi" shows "get_dense pf f = (SOME z. Rep_lifted_pair (Rep_dense pf) f = (top, up_filter z))"
is_processT5_S2: fixes s :: "'b event list"   and X :: "'b event set"   and P :: "'b process"   and c :: "'b event" assumes "(s, X) \<in> F P"   and "(s @ [c], {}) \<notin> F P" shows "(s, X \<union> {c}) \<in> F P"
sequence_traces: fixes c\<^sub>1 :: "nat \<Rightarrow> ('a \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> unit \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list \<times> (nat \<Rightarrow> nat) \<times> (nat \<Rightarrow> bool option)"   and k :: "nat"   and c\<^sub>2 :: "nat \<Rightarrow> ('a \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> unit \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list \<times> (nat \<Rightarrow> nat) \<times> (nat \<Rightarrow> bool option)"   and m :: "nat"   and l :: "nat"   and c :: "nat \<Rightarrow> ('a \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> unit \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list \<times> (nat \<Rightarrow> nat) \<times> (nat \<Rightarrow> bool option)" assumes "trace c\<^sub>1 0 k"   and "trace c\<^sub>2 m l"   and "c\<^sub>2 m = c\<^sub>1 k"   and "c = (\<lambda>x. if x \<le> k then c\<^sub>1 x else c\<^sub>2 (m + x - k))" shows "trace c 0 (k + l)"
SNF_S': shows "Smith_normal_form_wrt S' \<Q>"
rotate1_rl: fixes l :: "'b list" shows "rotater1 (rotate1 l) = l"
qPsubst_qPsubstInp: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm option"   and delta :: "'f"   and qinp :: "'d \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm option"   and qbinp :: "'e \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qAbs option" shows "qOp delta qinp qbinp #[[rho]] = qOp delta (qinp %[[rho]]) (qbinp %%[[rho]])"
simps_27: fixes \<C> :: "t_context"   and i :: "nat"   and ts :: "checker_type" shows "check_single \<C> (Get_global i) ts = (if i < length (global \<C>) then type_update ts [] (Type [tg_t (global \<C> ! i)]) else Bot)"
ball_reg: fixes R :: "'d set"   and P :: "'d \<Rightarrow> bool"   and Q :: "'d \<Rightarrow> bool" assumes "\<forall>x. x \<in> R \<longrightarrow> P x \<longrightarrow> Q x"   and "Ball R P" shows "Ball R Q"
finite_field_factorization_def: fixes f :: "'b mod_ring poly" shows "finite_field_factorization f = (if degree f = 0 then (lead_coeff f, []) else let a = lead_coeff f; u = Polynomial.smult (inverse_class.inverse a) f; gs = if use_distinct_degree_factorization then distinct_degree_factorization u else [(1, u)]; (irr, hs) = List.partition (\<lambda>(i, f). degree f = i) gs in (a, map snd irr @ concat (map (\<lambda>(x, y). berlekamp_monic_factorization x y) hs)))"
arctan_def: fixes y :: "real" shows "arctan y = (THE x. - (pi / 2) < x \<and> x < pi / 2 \<and> tan x = y)"
less_le_not_le: fixes x :: "'b list"   and y :: "'b list" shows "strict_prefix x y = (prefix x y \<and> \<not> prefix y x)"
D: shows "emeasure (restrict_space lborel {0<..<1}) {\<omega> \<in> {0<..<1}. \<not> isCont M.I \<omega>} = 0"
CompDiag_Cod_Diag: fixes t :: "'a local.term" assumes "Diag t" shows "Cod t \<^bold>\<lfloor>\<^bold>\<cdot>\<^bold>\<rfloor> t = t"
Sigma_fm_imp_thm: fixes A :: "fm" assumes "Sigma_fm A"   and "ground_fm A"   and "eval_fm e0 A" shows "{} \<turnstile> A"
same_lab_conv_list_all: fixes ils :: "nat list"   and ls :: "t list list"   and ts :: "t list" assumes "same_lab ils ls = Some ts" shows "list_all (\<lambda>i. i < length ls \<and> ls ! i = ts) ils"
elims_2: fixes proper_interval :: "'a option \<Rightarrow> 'a option \<Rightarrow> bool"   and x :: "'a list" assumes "proper_intrvl.exhaustive proper_interval x" obtains xa :: "'a" and xs :: "'a list" where "x = xa # xs" and "\<not> proper_interval None (Some xa) \<and> proper_intrvl.exhaustive_above proper_interval xa xs"
set_partition_by_median: fixes l :: "(real, 'a) vec list"   and m :: "real"   and r :: "(real, 'a) vec list"   and k :: "'a"   and ps :: "(real, 'a) vec list" assumes "(l, m, r) = partition_by_median k ps" shows "set ps = set l \<union> set r"
is_rbt_def: fixes less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and t :: "('c, 'd) rbt" shows "ord.is_rbt less t = (inv1 t \<and> inv2 t \<and> color_of t = color.B \<and> ord.rbt_sorted less t)"
1: shows "(SOME a. a \<in> sources \<mu>) \<in> sources \<mu>"
run_asset_ii_rel: fixes s :: "(agent \<times> msg) set"   and n :: "nat" assumes "s\<^sub>0 \<Turnstile> s" shows "s \<Turnstile> fst (run_asset_ii n s)"
multi_psub_of_add_self: fixes A :: "'b multiset"   and x :: "'b" shows "A \<subset># add_mset x A"
finite_map_to_set: fixes m :: "'g \<Rightarrow> 'h option" shows "finite (map_to_set m) = finite (dom m)"
phi_j: shows "\<phi> j = j # a \<odot> \<up>\<^sup>\<infinity>"
us: shows "us \<equiv> getFrN (map snd txs) (map fst txs) [\<phi>] (length txs)"
dense_eq0_I: fixes x :: "'b" assumes "\<And>e. (0::'b) < e \<Longrightarrow> \<bar>x\<bar> \<le> e" shows "x = (0::'b)"
stateinterpol_FVF: fixes z :: "variable"   and e :: "fml"   and \<omega> :: "variable \<Rightarrow> real"   and \<omega>' :: "variable \<Rightarrow> real"   and S :: "variable set"   and I :: "interp" assumes "z \<notin> FVF e" shows "(stateinterpol \<omega> \<omega>' S \<in> fml_sem I e) = (stateinterpol \<omega> \<omega>' (insert z S) \<in> fml_sem I e)"
intro: fixes G :: "('a, 'b, 'c) sa_rec_scheme" assumes "graph G" shows "sa G"