hcju/isabelleps huggingface dataset:
simps_2: fixes M :: "'d"   and xvec :: "name list"   and N :: "'d" shows "object (M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle>) = Some N"
resolvent_def: fixes D :: "'a literal list"   and A :: "'a"   and CA :: "'a literal list"   and Ls :: "'a literal list" shows "resolvent D A CA Ls = map (\<lambda>M. M \<cdot>l the (mgu {insert A (atms_of (mset Ls))})) (remove_all CA Ls @ D)"
assms: shows "hn_refine \<Gamma> c \<Gamma>' R a"
inverse_arrowsI: fixes g :: "'s"   and f :: "'s" assumes "ide (g \<cdot> f)"   and "ide (f \<cdot> g)" shows "inverse_arrows f g"
derives_implies_leftderives: fixes \<NN> :: "symbol set"   and \<TT> :: "symbol set"   and \<RR> :: "(symbol \<times> symbol list) set"   and \<SS> :: "symbol"   and a :: "symbol list"   and b :: "symbol list" assumes "CFG \<NN> \<TT> \<RR> \<SS>"   and "CFG.derives \<NN> \<TT> \<RR> a b"   and "CFG.is_word \<TT> b" shows "CFG.leftderives \<NN> \<TT> \<RR> a b"
evalDdb_env_cong: fixes e :: "db"   and \<rho> :: "nat \<rightarrow> ValD"   and \<rho>' :: "nat \<rightarrow> ValD" assumes "\<forall>v. freedb e v \<longrightarrow> \<rho>\<cdot>v = \<rho>'\<cdot>v" shows "evalDdb e\<cdot>\<rho> = evalDdb e\<cdot>\<rho>'"
that: fixes g :: "'a \<Rightarrow> 'b" assumes "continuous_on U g"   and "g ` U \<subseteq> S"   and "\<And>x. x \<in> T \<Longrightarrow> g x = f x" shows "thesis"
z: shows "\<forall>y. O y z = (\<exists>x. F x \<and> O y x)"
eqelem_imp_iff: fixes x :: "'b"   and y :: "'b"   and A :: "'b set" assumes "x = y" shows "(x \<in> A) = (y \<in> A)"
subgroup_intersect: fixes H :: "'a set"   and H' :: "'a set" assumes "subgroup H (G\<lparr>carrier := H\<rparr>)"   and "subgroup H' (G\<lparr>carrier := H\<rparr>)" shows "subgroup (H \<inter> H') (G\<lparr>carrier := H\<rparr>)"
clock_commonKnowledge_def: fixes envObs :: "'b \<Rightarrow> 'c \<Rightarrow> 'd"   and cec :: "'c odlist" shows "clock_commonKnowledge envObs cec \<equiv> \<lambda>as s. let r = \<lambda>a. fromList (concat (map (\<lambda>s'. concat (map (\<lambda>s''. if envObs a s' = envObs a s'' then [(s', s'')] else []) (toList cec))) (toList cec))); R = toList (big_union r as) in fromList (memo_list_trancl R s)"
steps_refl: fixes P :: "'a"   and Cal :: "'a processCalculus" shows "P \<longmapsto>Cal* P"
Inf_pres_inf: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "Inf_pres f"   and "Inf_pres g" shows "Inf_pres (f \<sqinter> g)"
closed_substandard: fixes P :: "'b \<Rightarrow> bool" shows "closed {x. \<forall>i\<in>Basis. P i \<longrightarrow> x \<bullet> i = 0}"
bij_to_nat: shows "bij_betw mod_type_class.to_nat UNIV {0..<CARD('a)}"
bij_betw_inv_into_LEFT_RIGHT: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and A' :: "'c set"   and B :: "'b set"   and B' :: "'c set" assumes "bij_betw f A A'"   and "B \<subseteq> A"   and "f ` B = B'" shows "inv_into A f ` B' = B"
tyeta: shows "typ_of (Abs \<tau> (t $ Bv 0)) = Some (\<tau> \<rightarrow> \<tau>')"
infinite_rev_existence_ivl0_rewrites_1: fixes f :: "'b \<Rightarrow> 'b"   and X :: "'b set"   and x :: "'b" assumes "auto_ll_on_open f X" shows "({0..} \<subseteq> auto_ll_on_open.existence_ivl0 (- f) X x) = ({..0} \<subseteq> auto_ll_on_open.existence_ivl0 f X x)"
smult_r_null: fixes R :: "('e, 'f) ring_scheme"   and M :: "('e, 'g, 'h) module_scheme"   and a :: "'e" assumes "module R M"   and "a \<in> carrier R" shows "a \<odot>\<^bsub>M\<^esub> \<zero>\<^bsub>M\<^esub> = \<zero>\<^bsub>M\<^esub>"
inject_1: fixes x1 :: "'a set \<times> 'a set"   and y1 :: "'a set \<times> 'a set" shows "(\<N>\<^bsub>x1\<^esub> = \<N>\<^bsub>y1\<^esub>) = (x1 = y1)"
gh: fixes x :: "'b" assumes "x \<in> S \<inter> T" shows "g x = h x"
arr_cod_iff_arr: fixes f :: "'a \<times> 'a \<times> 'a" shows "CCC.arr (CCC.cod f) = CCC.arr f"
PP'_def: fixes t :: "'trm" shows "PP' t = subst P' t xx"
deduct_if_restricted_deduct: fixes M :: "('fun, 'var) Term.term set"   and P :: "('fun, 'var) Term.term \<Rightarrow> bool"   and m :: "('fun, 'var) Term.term" assumes "\<langle>M;P\<rangle> \<turnstile>\<^sub>r m" shows "M \<turnstile> m"
stable: shows "(\<box>(\<lambda>s. P s \<and> Q s \<longrightarrow> (\<circle>Q) s)) \<sigma>"
ordst_def: fixes X :: "char list"   and Y :: "char list" shows "ordst X Y \<equiv> length X \<le> length Y \<and> X \<noteq> [] \<and> Y \<noteq> [] \<and> of_char (last X) < of_char (last Y) \<or> length X < length Y"
rep_eq: fixes x :: "'b ttree set" shows "Rep_ttree (Either x) = insert [] (\<Union> (Rep_ttree ` x))"
bounded_inside: fixes S :: "'a set" assumes "bounded S" shows "bounded (inside S)"
uniform_limit_on_UNION: fixes S :: "'b set"   and f :: "'c \<Rightarrow> 'd \<Rightarrow> 'e"   and h :: "'b \<Rightarrow> 'd set"   and g :: "'d \<Rightarrow> 'e"   and F :: "'c filter" assumes "finite S"   and "\<And>s. s \<in> S \<Longrightarrow> uniform_limit (h s) f g F" shows "uniform_limit (\<Union> (h ` S)) f g F"
rotate1_hd_tl: fixes xs :: "'d list" assumes "xs \<noteq> []" shows "rotate1 xs = tl xs @ [hd xs]"
hlp1: shows "map wordinterval_to_set (build_ip_partition c rs) = map (\<lambda>x. \<Union> (set (map wordinterval_to_set x))) (groupWIs c rs)"
w_simps_3: shows "src \<tau>.chine = src (tab\<^sub>0 (local.dom \<tau>))"
demand_C_case: fixes f :: "C \<rightarrow> 'a" shows "demand (C_case\<cdot>f) = C\<cdot>(demand f)"
columns_rw: shows "columns (Gram_Schmidt_matrix A) = (\<lambda>x. column x (Gram_Schmidt_matrix A)) ` (UNIV - {ia})"
absorb_iff1: fixes b :: "'e"   and a :: "'e" shows "(b \<le> a) = (sup a b = a)"
hyps_3: fixes z :: "'a" assumes "rtrancl_path r z_ ys z" shows "rtrancl_path r y (xs @ ys) z"
iter_widen_pfp: fixes f :: "'a acom \<Rightarrow> 'a acom"   and c :: "'a acom"   and c' :: "'a acom" assumes "iter_widen f c = Some c'" shows "f c' \<sqsubseteq> c'"
red_lcl_incr: fixes e :: "char list exp"   and h\<^sub>0 :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and l\<^sub>0 :: "char list \<Rightarrow> val option"   and e' :: "char list exp"   and h\<^sub>1 :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and l\<^sub>1 :: "char list \<Rightarrow> val option"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list" assumes "P \<turnstile> \<langle>e,(h\<^sub>0, l\<^sub>0)\<rangle> \<rightarrow> \<langle>e',(h\<^sub>1, l\<^sub>1)\<rangle>" shows "dom l\<^sub>0 \<subseteq> dom l\<^sub>1"
refl: fixes A :: "('d \<times> ('e, 'f) cconstraint \<times> 'g \<times> 'e list \<times> 'd) set \<times> ('d \<Rightarrow> ('e, 'f) cconstraint)"   and l :: "'d"   and u :: "'e \<Rightarrow> 'f" shows "A \<turnstile> \<langle>l, u\<rangle> \<rightarrow>* \<langle>l, u\<rangle>"
lebesgue_on_UNIV_eq: shows "lebesgue_on UNIV = lebesgue"
simps_3: fixes s :: "Lambda_Z.term"   and t :: "Lambda_Z.term" shows "(App s t)\<^sup>\<bullet> = app_beta s\<^sup>\<bullet> t\<^sup>\<bullet>"
Always_eq_includes_reachable: fixes A :: "'b set" shows "Always A = {F. reachable F \<subseteq> A}"
flowD_KIR: fixes \<Delta> :: "('a, 'b) network_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal"   and x :: "'a" assumes "flow \<Delta> f"   and "x \<noteq> source \<Delta>"   and "x \<noteq> sink \<Delta>" shows "KIR f x"
sc_completion_shift: fixes s :: "('l, 'thread_id) locks \<times> (('thread_id \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('thread_id \<Rightarrow> 'w wait_set_status option) \<times> 'thread_id set"   and vs :: "'addr \<times> addr_loc \<Rightarrow> ('addr val \<times> bool) option"   and ttas :: "('thread_id \<times> ('l, 'thread_id, 'x, 'm, 'w, ('addr, 'thread_id) obs_event action) thread_action) list"   and s' :: "('l, 'thread_id) locks \<times> (('thread_id \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('thread_id \<Rightarrow> 'w wait_set_status option) \<times> 'thread_id set" assumes "sc_completion s vs"   and "s -\<triangleright>ttas\<rightarrow>* s'"   and "ta_seq_consist P vs (lconcat (lmap (\<lambda>(t, ta). llist_of \<lbrace>ta\<rbrace>\<^bsub>o\<^esub>) (llist_of ttas)))" shows "sc_completion s' (mrw_values P vs (concat (map (\<lambda>(t, ta). \<lbrace>ta\<rbrace>\<^bsub>o\<^esub>) ttas)))"
f14: shows "write_reg (div_comp_result instr (div_comp_temp_V instr (ucast (div_comp_temp_64bit instr (word_cat (cpu_reg_val Y (snd (fst (get_curr_win () s1)))) (user_reg_val (fst (fst (get_curr_win () s1))) rs1 (snd (fst (get_curr_win () s1))))) operand2 >> 32)) (ucast (div_comp_temp_64bit instr (word_cat (cpu_reg_val Y (snd (fst (get_curr_win () s1)))) (user_reg_val (fst (fst (get_curr_win () s1))) rs1 (snd (fst (get_curr_win () s1))))) operand2 >> 31))) (div_comp_temp_64bit instr (word_cat (cpu_reg_val Y (snd (fst (get_curr_win () s1)))) (user_reg_val (fst (fst (get_curr_win () s1))) rs1 (snd (fst (get_curr_win () s1))))) operand2)) (ucast (get_CWP (cpu_reg_val PSR s1))) rd (snd (fst (get_curr_win () s2))) = write_reg (div_comp_result instr (div_comp_temp_V instr (ucast (div_comp_temp_64bit instr (word_cat (cpu_reg_val Y (snd (fst (get_curr_win () s2)))) (user_reg_val (fst (fst (get_curr_win () s2))) rs1 (snd (fst (get_curr_win () s2))))) operand2 >> 32)) (ucast (div_comp_temp_64bit instr (word_cat (cpu_reg_val Y (snd (fst (get_curr_win () s2)))) (user_reg_val (fst (fst (get_curr_win () s2))) rs1 (snd (fst (get_curr_win () s2))))) operand2 >> 31))) (div_comp_temp_64bit instr (word_cat (cpu_reg_val Y (snd (fst (get_curr_win () s2)))) (user_reg_val (fst (fst (get_curr_win () s2))) rs1 (snd (fst (get_curr_win () s2))))) operand2)) (fst (fst (get_curr_win () s2))) rd (snd (fst (get_curr_win () s2)))"
inv_inv: fixes x :: "'a" shows "local.inv (local.inv x) = x"
DirProds_one_iso_sym: fixes G :: "'e"   and f :: "'e \<Rightarrow> ('d, 'f) monoid_scheme" shows "(\<lambda>x. \<lambda>_\<in>{G}. x) \<in> Group.iso (f G) (DirProds f {G})"
sq_norm_poly_pCons: fixes a :: "'b"   and p :: "'b poly" shows "\<parallel>pCons a p\<parallel>\<^sup>2 = sq_norm_conjugate a + \<parallel>p\<parallel>\<^sup>2"
finite_subset: fixes M :: "'a set"   and P :: "'a \<Rightarrow> bool" assumes "finite M" shows "finite {x \<in> M. P x}"
prime_prod_dvdI: 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 unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and A :: "'b set"   and f :: "'b \<Rightarrow> 'a"   and y :: "'a" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "finite A"   and "\<And>x. x \<in> A \<Longrightarrow> normalization_semidom.prime zero times one normalize (f x)"   and "\<And>x. x \<in> A \<Longrightarrow> dvd.dvd times (f x) y"   and "inj_on f A" shows "dvd.dvd times (comm_monoid_mult.prod times one f A) y"
complex_roots_complex_def: fixes p :: "complex poly" shows "complex_roots_complex p = (SOME as. Polynomial.smult (Polynomial.lead_coeff p) (\<Prod>a\<leftarrow>as. [:- a, 1:]) = p \<and> length as = degree p)"
simps_10: fixes rcvd' :: "('g \<Rightarrow> 'g \<Rightarrow> 'h option) \<Rightarrow> 'g \<Rightarrow> 'g \<Rightarrow> 'h option"   and round :: "'g \<Rightarrow> nat"   and state :: "'g \<Rightarrow> 'f"   and sent :: "'g \<Rightarrow> 'g set"   and rcvd :: "'g \<Rightarrow> 'g \<Rightarrow> 'h option"   and network :: "('g \<times> nat \<times> 'g \<times> 'h) set"   and more :: "'i" shows "rcvd_update rcvd' \<lparr>round = round, state = state, sent = sent, rcvd = rcvd, network = network, \<dots> = more\<rparr> = \<lparr>round = round, state = state, sent = sent, rcvd = rcvd' rcvd, network = network, \<dots> = more\<rparr>"
path_component_eq_empty: fixes S :: "'a set"   and x :: "'a" shows "(path_component_set S x = {}) = (x \<notin> S)"
Nil_in_star: fixes A :: "'b list set" shows "[] \<in> star A"
is_Hd_def: fixes tm :: "('a, 'b) tm" shows "is_Hd tm = (\<exists>x1. tm = Hd x1)"
fst_snd_echelon_form_of_column_k_le_nrows: fixes i :: "nat"   and A :: "(('c, 'd) vec, 'e) vec"   and bezout :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<times> 'c \<times> 'c \<times> 'c \<times> 'c"   and k :: "nat" assumes "i \<le> nrows A" shows "snd (echelon_form_of_column_k bezout (A, i) k) \<le> nrows A"
antisym: fixes b :: "'b"   and a :: "'b" assumes "order.greater_eq a b"   and "order.greater_eq b a" shows "a = b"
indspaceD_transform: fixes f :: "('f, 'g) aezfun \<Rightarrow> 'v"   and G :: "'g set"   and FH :: "('f, 'g) aezfun set"   and V :: "'v set"   and g :: "'g"   and x :: "('f, 'g) aezfun"   and a :: "'f" assumes "f \<in> BaseFGMod.indspace G FH V"   and "g \<in> G"   and "x \<in> FH" shows "f (a \<delta>\<delta> g * x) = a \<delta>\<delta> g \<cdot> f x"
elims_2: fixes x :: "'a list"   and xa :: "'a list" assumes "local.less_eq_list x xa" obtains ys :: "'a list" where "x = []" and "xa = ys__"   | xb :: "'a" and xs :: "'a list" and y :: "'a" and ys :: "'a list" where "x = xb # xs" and "xa = y # ys" and "xb < y \<or> xb = y \<and> local.less_eq_list xs ys"
good_item_simps_46: fixes X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and rhoa :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option"   and xsa :: "'varSort"   and xa :: "'var" assumes "good X"   and "goodEnv rhoa"   and "freshEnv xsa xa rhoa" shows "Abs xsa xa X $[rhoa] = Abs xsa xa X #[rhoa]"
3: shows "qAFresh ys y (X #[[x' \<and> x]]_ys)"
induct: fixes P :: "'b poly \<Rightarrow> 'b poly \<Rightarrow> bool"   and a0 :: "'b poly"   and a1 :: "'b poly" assumes "\<And>c c'. P (c)\<^sub>p (c')\<^sub>p"   and "\<And>c c' n' p'. P (c)\<^sub>p c' \<Longrightarrow> P (c)\<^sub>p (CN c' n' p')"   and "\<And>c n p c'. P c (c')\<^sub>p \<Longrightarrow> P (CN c n p) (c')\<^sub>p"   and "\<And>c n p c' n' p'. \<lbrakk>P c (CN c' n' p'); P (CN c n p) c'; P c c'; \<And>x. P p p'\<rbrakk> \<Longrightarrow> P (CN c n p) (CN c' n' p')"   and "\<And>v b. P (Bound v) b"   and "\<And>v va b. P (Add v va) b"   and "\<And>v va b. P (Sub v va) b"   and "\<And>v va b. P (Mul v va) b"   and "\<And>v b. P (Neg v) b"   and "\<And>v va b. P (Pw v va) b"   and "\<And>v va vb vc. P (CN v va vb) (Bound vc)"   and "\<And>v va vb vc vd. P (CN v va vb) (Add vc vd)"   and "\<And>v va vb vc vd. P (CN v va vb) (Sub vc vd)"   and "\<And>v va vb vc vd. P (CN v va vb) (Mul vc vd)"   and "\<And>v va vb vc. P (CN v va vb) (Neg vc)"   and "\<And>v va vb vc vd. P (CN v va vb) (Pw vc vd)"   and "\<And>a v. P a (Bound v)"   and "\<And>a v va. P a (Add v va)"   and "\<And>a v va. P a (Sub v va)"   and "\<And>a v va. P a (Mul v va)"   and "\<And>a v. P a (Neg v)"   and "\<And>a v va. P a (Pw v va)"   and "\<And>vc v va vb. P (Bound vc) (CN v va vb)"   and "\<And>vc vd v va vb. P (Add vc vd) (CN v va vb)"   and "\<And>vc vd v va vb. P (Sub vc vd) (CN v va vb)"   and "\<And>vc vd v va vb. P (Mul vc vd) (CN v va vb)"   and "\<And>vc v va vb. P (Neg vc) (CN v va vb)"   and "\<And>vc vd v va vb. P (Pw vc vd) (CN v va vb)" shows "P a0 a1"
wellformed1_sorted_aux: fixes x :: "('b, 'c) Combinators"   and p :: "('b, 'c) Combinators list"   and l :: "'b set list" assumes "wellformed_policy1 (x # p)" shows "wellformed_policy1 (FWNormalisationCore.insort x p l)"
TermEncode_axioms: shows "TermEncode var trm fmla Var FvarsT substT Fvars subst num Ops tenc"
subdivision_with_projI: fixes G :: "'b pair_pre_digraph"   and H :: "'b pair_pre_digraph" assumes "pair_sd G H" shows "subdivision_pair G H"
partitions_imp_finite_elements: fixes p :: "nat \<Rightarrow> nat"   and n :: "nat" assumes "p partitions n" shows "finite {i. 0 < p i}"
elementary_def: fixes s :: "'a set" shows "local.elementary s \<equiv> quorum s \<and> (\<forall>s'\<subset>s. \<not> quorum s')"
cobounded2: fixes a :: "'d"   and b :: "'d" shows "inf a b \<le> b"
prems_1: shows "ta_hb_consistent P E (LCons tob E')"
assms: shows "c \<in> {0..<length vs} \<rightarrow> carrier R"
group_axioms_def: fixes G :: "'c set"   and composition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit :: "'c" shows "Group_Theory.group_axioms G composition unit \<equiv> \<forall>u. u \<in> G \<longrightarrow> monoid.invertible G composition unit u"
not_before_in: fixes x :: "'a"   and xs :: "'a list"   and y :: "'a" assumes "x \<in> set xs"   and "y \<in> set xs" shows "(\<not> x < y in xs) = (y < x in xs \<or> x = y)"
emeasure_finite: fixes s :: "'a cfg"   and A :: "'a cfg stream set" shows "emeasure (MC.T s) A \<noteq> \<top>"
sq_norm_g2_i: shows "\<parallel>gs2.gso i\<parallel>\<^sup>2 = \<parallel>fs.gs.gso i\<parallel>\<^sup>2 * \<parallel>fs.gs.gso (i - 1)\<parallel>\<^sup>2 / \<parallel>gs2.gso (i - 1)\<parallel>\<^sup>2"
flowD_finite: fixes g :: "'v \<times> 'v \<Rightarrow> ennreal"   and e :: "'v \<times> 'v" assumes "flow \<Delta> g" shows "g e \<noteq> \<top>"
preserves_dom: fixes f :: "'a" assumes "C.arr f" shows "D.dom (G f) = G (C.dom f)"
assms_3: fixes i :: "V" assumes "i \<in>\<^sub>\<circ> I" shows "A i \<subseteq>\<^sub>\<circ> Vset \<alpha>"
absorb_iff2: fixes a :: "'c multiset"   and b :: "'c multiset" shows "(a \<subseteq># b) = (a \<union># b = b)"
positive_cblinfunI: fixes A :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'a" assumes "\<And>x. 0 \<le> x \<bullet>\<^sub>C (A *\<^sub>V x)" shows "0 \<le> A"
iso_imp_homomorphism: fixes h :: "'c \<Rightarrow> 'd"   and G :: "('c, 'e) monoid_scheme"   and H :: "('d, 'f) monoid_scheme" assumes "h \<in> Group.iso G H" shows "h \<in> hom G H"
IH: assumes "f \<in> F"   and "f \<noteq> 0"   and "v \<in> keys (tail p)"   and "lt f adds\<^sub>t v" shows "is_red F (tail p)"
cyclic_on_alldef: fixes f :: "'e \<Rightarrow> 'e"   and S :: "'e set" shows "cyclic_on f S = (S \<noteq> {} \<and> (\<forall>s\<in>S. S = orbit f s))"
set_object_value_consistent: fixes u :: "partition_id_t"   and s1 :: "state_t"   and s2 :: "state_t"   and x :: "obj_id_t"   and y :: "obj_t" assumes "vpeq u s1 s2" shows "vpeq u (set_object_value x y s1) (set_object_value x y s2)"
ih: fixes \<Gamma>3 :: "(name \<times> ty) list" assumes "valid \<Gamma>3"   and "(x, T1) # \<Gamma>1 \<subseteq> \<Gamma>3" shows "\<Gamma>3 \<turnstile> t : T2"
left_segment_congruence: fixes a :: "'p"   and b :: "'p"   and p :: "'p"   and q :: "'p"   and c :: "'p"   and d :: "'p" assumes "{a, b} = {p, q}"   and "p q \<equiv> c d" shows "a b \<equiv> c d"
soundness_wrt_ctx_equiv_aux: fixes e1 :: "exp"   and e2 :: "exp"   and C :: "ctx"   and ob :: "obs" assumes "E e1 = E e2"   and "FV (plug C e1) = {}"   and "FV (plug C e2) = {}"   and "plug C e1 \<Down> ob" shows "plug C e2 \<Down> ob"
last_snoc: fixes xs :: "'g list"   and x :: "'g" shows "last (xs @ [x]) = x"
list_all2_appendI: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'a list"   and b :: "'b list"   and c :: "'a list"   and d :: "'b list" assumes "list_all2 P a b"   and "list_all2 P c d" shows "list_all2 P (a @ c) (b @ d)"
simps_6: fixes w :: "nat \<Rightarrow> 'a set"   and \<phi> :: "'a ltl"   and \<psi> :: "'a ltl" shows "w \<Turnstile> \<phi> or \<psi> = (w \<Turnstile> \<phi> \<or> w \<Turnstile> \<psi>)"
map_add_Some_iff: fixes m :: "'b \<Rightarrow> 'a option"   and n :: "'b \<Rightarrow> 'a option"   and k :: "'b"   and x :: "'a" shows "((m ++ n) k = \<lfloor>x\<rfloor>) = (n k = \<lfloor>x\<rfloor> \<or> n k = None \<and> m k = \<lfloor>x\<rfloor>)"
up_ucast_inj: fixes x :: "'b word"   and y :: "'b word" assumes "UCAST('b \<rightarrow> 'a) x = UCAST('b \<rightarrow> 'a) y"   and "LENGTH('b) \<le> LENGTH('a)" shows "x = y"
tiebidsBridgingLemma': fixes x :: "'d \<times> 'e set"   and N :: "'d set"   and \<Omega> :: "'e set"   and a :: "('d \<times> 'e set) set" assumes "x \<in> N \<times> (Pow \<Omega> - {{}})" shows "tiebids' a N \<Omega> x = tiebidsAlg a N \<Omega> x"
put_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_ok: fixes h :: "(_) heap"   and ptr :: "(_) element_ptr"   and setter :: "('l \<Rightarrow> 'l) \<Rightarrow> ('c, 'a, 'b, 'e, 'g option) RElement_scheme \<Rightarrow> ('c, 'a, 'b, 'e, 'g option) RElement_scheme"   and v :: "'l" assumes "DocumentClass.type_wf h"   and "ptr |\<in>| element_ptr_kinds h" shows "h \<turnstile> ok put_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t ptr setter v"
RBT_inject: fixes x :: "('a, 'b) RBT_Impl.rbt"   and y :: "('a, 'b) RBT_Impl.rbt" assumes "x \<in> {t. is_rbt t}"   and "y \<in> {t. is_rbt t}" shows "(RBT x = RBT y) = (x = y)"
thunks_cong: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and \<Delta> :: "(Vars.var \<times> exp) list" assumes "map_of \<Gamma> = map_of \<Delta>" shows "thunks \<Gamma> = thunks \<Delta>"
env_restr_env_restr: fixes d1 :: "'c set"   and d2 :: "'c set"   and x :: "'c \<Rightarrow> 'd" shows "x f|` d2 f|` d1 = x f|` (d1 \<inter> d2)"
Gramian_determinant_1: fixes k :: "nat" assumes "k \<le> length fs" shows "rat_of_int (gs.Gramian_determinant fs k) = (\<Prod>j<k. \<parallel>fs.gs.gso j\<parallel>\<^sup>2)"
negligible_imp_sets: fixes S :: "'c set" assumes "negligible S" shows "S \<in> sets lebesgue"
explode_inject: fixes x :: "literal"   and y :: "literal" shows "(explode x = explode y) = (x = y)"
assms: shows "(f \<circ> vec has_integral y) {a $ 1..b $ 1}"
real_asymp_real_nat_transfer_1: fixes f :: "'a \<Rightarrow> nat"   and F :: "'a filter" assumes "LIM n F. real (f n) :> at_top" shows "filterlim f sequentially F"
helper_masked_ucast_reverse_generic: fixes m :: "nat"   and n :: "nat"   and b :: "16 word" assumes "m + 16 \<le> n"   and "n \<le> 128 - 16" shows "(UCAST(16 \<rightarrow> 128) b << n) && (mask 16 << m) = 0"
normalized_dst_ports_def2: fixes ms :: "'b common_primitive match_expr" shows "normalized_dst_ports ms = normalized_n_primitive (is_Dst_Ports, dst_ports_sel) (\<lambda>ps. case ps of L4Ports x pts \<Rightarrow> length pts \<le> 1) ms"
has_absolute_integral_change_of_variables: fixes S :: "(real, 'a) vec set"   and g :: "(real, 'a) vec \<Rightarrow> (real, 'a) vec"   and g' :: "(real, 'a) vec \<Rightarrow> (real, 'a) vec \<Rightarrow> (real, 'a) vec"   and f :: "(real, 'a) vec \<Rightarrow> (real, 'b) vec"   and b :: "(real, 'b) vec" assumes "S \<in> sets lebesgue"   and "\<And>x. x \<in> S \<Longrightarrow> (g has_derivative g' x) (at x within S)"   and "inj_on g S" shows "((\<lambda>x. \<bar>det (matrix (g' x))\<bar> *\<^sub>R f (g x)) absolutely_integrable_on S \<and> integral S (\<lambda>x. \<bar>det (matrix (g' x))\<bar> *\<^sub>R f (g x)) = b) = (f absolutely_integrable_on g ` S \<and> integral (g ` S) f = b)"
of_int_0_eq_iff: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and z :: "int" assumes "class.ring_char_0 one times plus zero minus uminus" shows "(zero = ring_1.of_int one plus zero minus z) = (z = 0)"
map_FGf_def: fixes l1 :: "'ac \<Rightarrow> 'am"   and l2 :: "'ad \<Rightarrow> 'an"   and co1 :: "'af \<Rightarrow> 'ao"   and co2 :: "'ag \<Rightarrow> 'ap"   and contra1 :: "'aq \<Rightarrow> 'ai"   and contra2 :: "'ar \<Rightarrow> 'aj" shows "map_FGf l1 l2 co1 co2 contra1 contra2 = map_F l1 l2 id co1 co2 id contra1 contra2 id"
indicator_def: fixes S :: "'d set"   and x :: "'d" shows "indicator S x = (if x \<in> S then 1::'b else (0::'b))"
below_env_deleteI: fixes f :: "'d \<Rightarrow> 'c"   and x :: "'d"   and g :: "'d \<Rightarrow> 'c" assumes "f x = \<bottom>"   and "f \<sqsubseteq> g" shows "f \<sqsubseteq> env_delete x g"
intros_10: fixes s :: "s"   and vs :: "v list"   and j :: "nat"   and i :: "nat" shows "\<lparr>s;vs;[$Get_global j]\<rparr> \<leadsto>_ i \<lparr>s;vs;[$C sglob_val s i j]\<rparr>"
carrier_open: shows "carrier\<^bsub>S\<^esub> open\<^bsub>S\<^esub>"
HNatInfinite_whn: shows "whn \<in> HNatInfinite"
quasi_inverse_hcomp: fixes f :: "'a"   and f' :: "'a" assumes "equivalence_map f"   and "equivalence_map f'"   and "equivalence_map (f \<star> f')" shows "(f \<star> f')\<^sup>~ \<cong> f'\<^sup>~ \<star> f\<^sup>~"
deriv: fixes \<xi> :: "real" assumes "x \<le> \<xi>"   and "\<xi> \<le> y" shows "\<exists>D. ((\<lambda>x. ln x - ((x - m)\<^sup>2 * f' / 2 + x / m)) has_real_derivative D) (at \<xi>) \<and> 0 \<le> D"