hcju/isabelleps huggingface dataset:
invimage_iff: fixes a :: "V"   and r :: "V"   and A :: "V" shows "(a \<in>\<^sub>\<circ> r -`\<^sub>\<circ> A) = (\<exists>b\<in>elts A. \<langle>a, b\<rangle> \<in>\<^sub>\<circ> r)"
subset_fst_snd: fixes A :: "('b \<times> 'c) set" shows "A \<subseteq> fst ` A \<times> final_b ` A"
map_equal: fixes m :: "'c \<Rightarrow> 'd option"   and m' :: "'c \<Rightarrow> 'd option" assumes "dom m = dom m'"   and "\<And>x. x \<in> dom m \<Longrightarrow> m x = m' x" shows "m = m'"
norm_assertion_simps_16: fixes Q :: "'a \<Rightarrow> assn"   and R :: "assn" shows "(\<exists>\<^sub>Ax. Q x) * R = (\<exists>\<^sub>Ax. Q x * R)"
prefix_Nil: fixes xs :: "'b list" shows "(xs \<le>p \<epsilon>) = (xs = \<epsilon>)"
finite_merge_t: fixes i :: "nat" shows "finite (local.merge i) = finite (merge_t i)"
lex_pm_alt: fixes s :: "'c \<Rightarrow>\<^sub>0 'd"   and t :: "'c \<Rightarrow>\<^sub>0 'd" shows "lex_pm s t = (s = t \<or> (\<exists>x. lookup s x < lookup t x \<and> (\<forall>y<x. lookup s y = lookup t y)))"
power_mono: fixes L1 :: "'a list set"   and L2 :: "'a list set"   and n :: "nat" assumes "L1 \<subseteq> L2" shows "L1 ^^ n \<subseteq> L2 ^^ n"
comp_ide_self: fixes a :: "'a" assumes "V.ide a" shows "V a a = a"
permutation_conv_induced_automorph: fixes f :: "'a \<Rightarrow> 'a"   and g :: "'a \<Rightarrow> 'a" assumes "(f, g) \<in> fundfoldpairs" shows "(\<rightarrow>) (Abs_induced_automorph f g) = induced_automorph f g"
intros_2: fixes s :: "Core.term"   and t :: "Core.term"   and u :: "Core.term" assumes "s \<rightarrow>\<^sub>\<beta> t" shows "s $ u \<rightarrow>\<^sub>\<beta> t $ u"
summable_iff_convergent': fixes f :: "nat \<Rightarrow> 'b" shows "summable f = convergent (\<lambda>n. sum f {..n})"
set_sel_3: fixes a :: "('g, 'h, 'i) generat" assumes "\<not> is_Pure a" shows "continuation a \<in> generat_conts a"
tosfw_1: shows "_Collect p ((common_matcher, in_doubt_allow),p\<turnstile> \<langle>rs, Undecided\<rangle> \<Rightarrow>\<^sub>\<alpha> Decision FinalAllow \<and> match_tcp_flags ipt_tcp_syn (p_tcp_flags p) \<and> p_tag_ctstate p = CT_New) \<subseteq> _Collect p (simple_fw (to_simple_firewall_without_interfaces ipassmt rtblo rs) p = Decision FinalAllow \<and> match_tcp_flags ipt_tcp_syn (p_tcp_flags p) \<and> p_tag_ctstate p = CT_New)"
simps_5: fixes f1 :: "vname \<Rightarrow> 'a"   and f2 :: "'a" shows "(case Res of VNam x \<Rightarrow> f1 x | Res \<Rightarrow> f2) = f2"
prolongation_def: fixes a :: "'a"   and b :: "'a" shows "prolongation a b \<equiv> {x. \<exists>ab. [[a b x]] \<and> x \<in> ab \<and> path ab a b}"
add_ac_3: fixes b :: "('a \<times> 'a set) set"   and a :: "('a \<times> 'a set) set"   and c :: "('a \<times> 'a set) set" shows "b \<union> (a \<union> c) = a \<union> (b \<union> c)"
rtrancl_trans: fixes x :: "'d"   and y :: "'d"   and r :: "('d \<times> 'd) set"   and z :: "'d" assumes "(x, y) \<in> r\<^sup>*"   and "(y, z) \<in> r\<^sup>*" shows "(x, z) \<in> r\<^sup>*"
match_1: fixes h :: "'a \<times> 'a"   and g :: "'a \<times> 'a"   and f :: "'a \<times> 'a" assumes "VoV.seq h g"   and "VoV.seq (VoV.comp h g) f" shows "VoV.seq g f"
single_iteration_inv': fixes i' :: "nat"   and i :: "nat"   and j' :: "nat"   and n :: "nat"   and m :: "nat \<Rightarrow> nat \<Rightarrow> 'a"   and k :: "nat"   and j :: "nat" assumes "i' < i"   and "j' \<le> n" shows "fwi m n k i j i' j' = fwi m n k i' j' i' j'"
get_child_nodes_is_l_get_child_nodes: shows "l_get_child_nodes ShadowRootClass.type_wf ShadowRootClass.known_ptr Shadow_DOM.get_child_nodes Shadow_DOM.get_child_nodes_locs"
ide_mkIde: fixes f0 :: "'a"   and f1 :: "'a" assumes "C.span f0 f1" shows "V.ide (mkIde f0 f1)"
simps_15: fixes v :: "Cooper.num" shows "qfree (NEq v) = True"
safe_distance_axioms: shows "safe_distance a\<^sub>e v\<^sub>e (ego.q \<delta>) a\<^sub>o (other.s' \<delta>) (other.s \<delta>)"
oghoare_intros_24: fixes \<Theta> :: "'b \<Rightarrow> ('a, 'b, 'c) ann list option"   and p :: "'b"   and as :: "('a, 'b, 'c) ann list"   and n :: "nat"   and P'' :: "('a, 'b, 'c) ann"   and \<Gamma> :: "'b \<Rightarrow> ('a, 'b, 'c) com option"   and b :: "('a, 'b, 'c) com"   and F :: "'c set"   and P :: "'a set"   and Q :: "'a set"   and A :: "'a set"   and r :: "'a set" assumes "\<Theta> p = Some as"   and "as ! n = P''"   and "\<Gamma> p = Some b"   and "n < length as"   and "\<Gamma>, \<Theta> \<tturnstile>\<^bsub>/F\<^esub> P P'' b Q, A" shows "\<Gamma>, \<Theta> \<tturnstile>\<^bsub>/F\<^esub> P (AnnCall r n) (Call p) Q, A"
Sup_le_iff: fixes A :: "'a set"   and b :: "'a" shows "(\<Squnion>A \<le> b) = (\<forall>a\<in>A. a \<le> b)"
sq_mtx_zero_ith: fixes i :: "'a" shows "0 $$ i = 0"
FG_fdd0_closed: fixes a :: "'f" shows "a \<delta>\<delta> (0::'g) \<in> FG"
oalist_inv_raw_update_by_pair: fixes xs :: "('a \<times> 'c) list"   and kv :: "'a \<times> 'c" assumes "oalist_inv_raw xs" shows "oalist_inv_raw (update_by_pair kv xs)"
tree_map_of_works: fixes empty :: "'a"   and update :: "'b \<Rightarrow> 'c \<Rightarrow> 'a \<Rightarrow> 'a"   and delete :: "'b \<Rightarrow> 'a \<Rightarrow> 'a"   and lookup :: "'a \<Rightarrow> 'b \<Rightarrow> 'c option"   and invar :: "'a \<Rightarrow> bool"   and pres :: "('b \<times> 'c) list"   and x :: "'b" assumes "Map empty update delete lookup invar" shows "lookup (tree_map_of update empty pres) x = AList_Upd_Del.map_of pres x"
lub_upper: fixes A :: "('f, 'g, 'h) Resumption.resumption set"   and x :: "('f, 'g, 'h) Resumption.resumption" assumes "Complete_Partial_Order.chain resumption_ord A"   and "x \<in> A" shows "resumption_ord x (resumption_lub A)"
double'_def: fixes dict :: "'b dict_plus"   and x :: "'b" shows "double' dict x = param_plus dict x x"
equivalence_in_bicategory_axioms: shows "internal_equivalence (h \<star> f) (g \<star> k) \<eta> \<epsilon>"
g1: shows "\<^bold>g [^] Suc n = \<one>"
mono: shows "mono (\<lambda>p x1 x2. (\<exists>y ys. x1 = [] \<and> x2 = y # ys) \<or> (\<exists>x y xs ys. x1 = x # xs \<and> x2 = y # ys \<and> mk_less ordb x y) \<or> (\<exists>x y xs ys. x1 = x # xs \<and> x2 = y # ys \<and> \<not> mk_less ordb x y \<and> \<not> mk_less ordb y x \<and> p xs ys))"
ltl_lfilter: fixes P :: "'c \<Rightarrow> bool"   and xs :: "'c llist" shows "ltl (lfilter P xs) = lfilter P (ltl (ldropWhile (Not \<circ> P) xs))"
pendulum_inv: fixes r :: "real"   and G :: "(real, 2) vec \<Rightarrow> bool" shows "\<lceil>\<lambda>s. r\<^sup>2 = (s $ 1)\<^sup>2 + (s $ 2)\<^sup>2\<rceil> \<subseteq> wp x\<acute>=\<lambda>t s. fball (- s $ 1) s & G on \<lambda>s. {t. 0 \<le> t} UNIV @ 0 \<lceil>\<lambda>s. r\<^sup>2 = (s $ 1)\<^sup>2 + (s $ 2)\<^sup>2\<rceil>"
rc_new_arg: fixes CT :: "nat \<Rightarrow> classDef option"   and ei :: "exp"   and ei' :: "exp"   and C :: "nat"   and el :: "exp list"   and er :: "exp list" assumes "CT \<turnstile> ei \<rightarrow> ei'" shows "CT \<turnstile> New C (el @ ei # er) \<rightarrow> New C (el @ ei' # er)"
simps_5: fixes opubx :: "'opubx" shows "generalize_instr (IOpUbx opubx) = IOpInl (\<BB>\<oo>\<xx> opubx)"
cases: fixes a :: "['b]" assumes "finite_list a" obtains "a = []"   | x :: "'b" and xs :: "['b]" where "a = x : xs" and "finite_list xs"
alpha_Tree_symp: shows "symp (=\<^sub>\<alpha>)"
cast': shows "e' = \<lparr>C\<rparr>e\<^sub>1"
symKeys_invKey_eq: fixes K :: "nat" assumes "K \<in> symKeys" shows "invKey K = K"
numeral_eq_eint: fixes k :: "num" shows "eint (numeral k) = eint (numeral k)"
U_def: shows "U = {x \<in> space N. \<not> P x}"
presheaf_of_rings_axioms: shows "presheaf_of_rings S is_open local.im_sheaf im_sheaf_morphisms b add_im_sheaf mult_im_sheaf zero_im_sheaf one_im_sheaf"
some_trg_dom: fixes \<mu> :: "'a" assumes "arr \<mu>" shows "some_trg (local.dom \<mu>) = some_trg \<mu>"
1: fixes \<Gamma> :: "'a formula multiset" assumes "\<^bold>\<And>map disj_of_clause (cnf_lists F), \<Gamma> \<Rightarrow>\<^sub>n"   and "\<^bold>\<And>map disj_of_clause (cnf_lists G), \<Gamma> \<Rightarrow>\<^sub>n"   and "is_nnf_mset \<Gamma>" shows "\<^bold>\<And>map disj_of_clause (concat (map (\<lambda>f. map ((@) f) (cnf_lists G)) (cnf_lists F))), \<Gamma> \<Rightarrow>\<^sub>n"
inv_upto_mono: fixes n :: "nat"   and k :: "nat"   and p :: "'b mat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bool"   and A :: "'b mat"   and q :: "'b mat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bool" assumes "\<And>i j. \<lbrakk>i < n; j < k; p A i j\<rbrakk> \<Longrightarrow> q A i j"   and "inv_upto n p A k" shows "inv_upto n q A k"
cone_decompI: fixes T :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set"   and ps :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list" assumes "direct_decomp T (map cone ps)" shows "cone_decomp T ps"
rep_eq: fixes x :: "linear_poly" shows "Abstract_Linear_Poly.coeff x = fun_coeff (Rep_linear_poly x)"
mu_rec_total_utp_rule: fixes R :: "('a \<times> 'a) set"   and B :: "('b, 'c) urel \<Rightarrow> ('b, 'c) urel"   and Pre :: "('b, 'c) urel"   and e :: "('a, 'b) uexpr"   and Post :: "('b, 'c) urel" assumes "wf R"   and "Monotonic B"   and "\<And>st. (Pre \<and> \<lceil>e\<rceil>\<^sub>< =\<^sub>u \<guillemotleft>st\<guillemotright> \<Rightarrow> Post) \<sqsubseteq> B (Pre \<and> (\<lceil>e\<rceil>\<^sub><, \<guillemotleft>st\<guillemotright>)\<^sub>u \<in>\<^sub>u \<guillemotleft>R\<guillemotright> \<Rightarrow> Post)" shows "(Pre \<Rightarrow> Post) \<sqsubseteq> \<mu> B"
has_integral': fixes f :: "'a \<Rightarrow> 'b"   and i :: "'b"   and S :: "'a set" shows "(f has_integral i) S = (\<forall>e>0. \<exists>B>0. \<forall>a b. ball (0::'a) B \<subseteq> cbox a b \<longrightarrow> (\<exists>z. ((\<lambda>x. if x \<in> S then f x else (0::'b)) has_integral z) (cbox a b) \<and> norm (z - i) < e))"
isometry_classical_operator: fixes \<pi> :: "'c \<Rightarrow> 'd" assumes "inj \<pi>" shows "isometry (classical_operator (Some \<circ> \<pi>))"
assms_2: shows "strictly_maximal_literal P2 (Neg A)"
image_map_upd: fixes x :: "'a"   and A :: "'a set"   and m :: "'a \<Rightarrow> 'b option"   and y :: "'b" assumes "x \<notin> A" shows "m(x \<mapsto> y) ` A = m ` A"
distinct_fst_append: fixes x :: "'a"   and xs :: "('a \<times> 'b) list"   and ys :: "('a \<times> 'b) list" assumes "x \<in> set (map fst xs)"   and "distinct (map fst (xs @ ys))" shows "x \<notin> set (map fst ys)"
exhaust: fixes y :: "('c, 'd) label" obtains "y = Skip"   | x2 :: "('c \<Rightarrow> 'd) \<Rightarrow> bool" where "y = Assume x2"   | x31 :: "'c" and x32 :: "('c \<Rightarrow> 'd) \<Rightarrow> 'd" where "y = Assign x31 x32"
content_cball: fixes r :: "real"   and c :: "'c" assumes "0 \<le> r" shows "content (cball c r) = unit_ball_vol (real DIM('c)) * r ^ DIM('c)"
inf_vector_comp: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<sqinter> (y \<sqinter> top) \<sqinter> z = y \<sqinter> top \<sqinter> (x \<sqinter> z)"
log_Suc_zero: shows "Discrete.log (Suc 0) = 0"
sfilter_SCons_decomp: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b stream"   and x :: "'b"   and zs :: "'b stream" assumes "sfilter P xs = x ## zs"   and "ev (holds P) xs" shows "\<exists>ys' zs'. xs = ys' @- x ## zs' \<and> list_all (Not \<circ> P) ys' \<and> P x \<and> sfilter P zs' = zs"
facetrelI_card: fixes y :: "'b set"   and x :: "'b set" assumes "order.greater_eq x y"   and "card (x - y) = 1" shows "y \<lhd> x"
finite_h': fixes i :: "nat"   and e :: "'v \<times> 'v" shows "h_plus (Suc i) e - h_minus i e \<noteq> \<top>"
simps_3: fixes x1 :: "nat" shows "(_Entry_) \<noteq> (_ x1 _)"
axioms_4: fixes known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and type_wf :: "(_) heap \<Rightarrow> bool"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and heap_is_wellformed :: "(_) heap \<Rightarrow> bool"   and parent_child_rel :: "(_) heap \<Rightarrow> ((_) object_ptr \<times> (_) object_ptr) set"   and set_val :: "(_) character_data_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_val_locs :: "(_) character_data_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and set_disconnected_nodes :: "(_) document_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and create_character_data :: "(_) document_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, (_) character_data_ptr) prog"   and known_ptrs :: "(_) heap \<Rightarrow> bool" assumes "l_create_character_data_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr type_wf get_child_nodes get_child_nodes_locs get_disconnected_nodes get_disconnected_nodes_locs heap_is_wellformed parent_child_rel set_val set_val_locs set_disconnected_nodes set_disconnected_nodes_locs create_character_data known_ptrs" shows "l_create_character_data\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_disconnected_nodes get_disconnected_nodes_locs set_disconnected_nodes set_val set_val_locs type_wf create_character_data known_ptr"
CatchSteps: fixes \<Gamma> :: "'b \<Rightarrow> ('a, 'b, 'c) com option"   and cfg\<^sub>1 :: "('a, 'b, 'c) com \<times> ('a, 'c) xstate"   and cfg\<^sub>2 :: "('a, 'b, 'c) com \<times> ('a, 'c) xstate"   and c\<^sub>1 :: "('a, 'b, 'c) com"   and s :: "('a, 'c) xstate"   and c\<^sub>1' :: "('a, 'b, 'c) com"   and s' :: "('a, 'c) xstate"   and c\<^sub>2 :: "('a, 'b, 'c) com" assumes "\<Gamma>\<turnstile> cfg\<^sub>1 \<rightarrow>\<^sup>* cfg\<^sub>2"   and "cfg\<^sub>1 = (c\<^sub>1, s)"   and "cfg\<^sub>2 = (c\<^sub>1', s')" shows "\<Gamma>\<turnstile> (Catch c\<^sub>1 c\<^sub>2, s) \<rightarrow>\<^sup>* (Catch c\<^sub>1' c\<^sub>2, s')"
eq_Union: shows "{dc \<in> dc_crypto. inversion dc = xs} = \<Union> {{dc \<in> dc_crypto. payer dc = Some i \<and> inversion dc = xs} |i. i < n}"
Dag_unique_ex_conjI: fixes p :: "ref"   and l :: "ref \<Rightarrow> ref"   and r :: "ref \<Rightarrow> ref"   and t :: "dag"   and P :: "dag \<Rightarrow> bool" assumes "Dag p l r t"   and "P t" shows "\<exists>t. Dag p l r t \<and> P t"
intensional_simps_11: fixes P :: "'a \<Rightarrow> bool" shows "#False \<longrightarrow> P \<equiv> #True"
P2_security_rsa: fixes D :: "bool \<times> (nat \<times> nat) \<times> bool \<times> bool \<Rightarrow> bool spmf"   and n :: "nat"   and m2 :: "bool"   and HCP_ad :: "real"   and m1 :: "bool \<times> bool" assumes "\<forall>a. lossless_spmf (D a)"   and "\<And>b\<^sub>\<sigma>. etp_rsa.HCP_adv n etp.\<A> m2 b\<^sub>\<sigma> D \<le> HCP_ad" shows "etp.OT_12.adv_P2 n m1 m2 D \<le> 2 * HCP_ad"
F_inj_def: shows "local.F_inj = {f \<in> {0..<n} \<rightarrow> {0..<m}. (\<forall>i. i \<notin> {0..<n} \<longrightarrow> f i = i) \<and> inj_on f {0..<n}}"
atms_of_plus: fixes C :: "'b literal multiset"   and D :: "'b literal multiset" shows "atms_of (C + D) = atms_of C \<union> atms_of D"
def: fixes upw :: "bool"   and i :: "nat"   and fs :: "int vec list" shows "basis_reduction_step upw i fs = (if i = 0 then (True, Suc i, fs) else let fs' = basis_reduction_add_rows upw i fs in if \<parallel>gram_schmidt_fs.gso n (map of_int_hom.vec_hom fs') (i - 1)\<parallel>\<^sup>2 \<le> \<alpha> * \<parallel>gram_schmidt_fs.gso n (map of_int_hom.vec_hom fs') i\<parallel>\<^sup>2 then (True, Suc i, fs') else basis_reduction_swap i fs')"
SA_arrow: fixes \<Gamma> :: "binding list"   and T\<^sub>1 :: "POPLmark.type"   and S\<^sub>1 :: "POPLmark.type"   and S\<^sub>2 :: "POPLmark.type"   and T\<^sub>2 :: "POPLmark.type" assumes "\<Gamma> \<turnstile> T\<^sub>1 <: S\<^sub>1"   and "\<Gamma> \<turnstile> S\<^sub>2 <: T\<^sub>2" shows "\<Gamma> \<turnstile> S\<^sub>1 \<rightarrow> S\<^sub>2 <: T\<^sub>1 \<rightarrow> T\<^sub>2"
Padmit_Test_simps: fixes \<sigma> :: "('a, 'b, 'c) subst"   and p :: "('a, 'b, 'c) formula" shows "Padmit \<sigma> (? p) = Fadmit \<sigma> p"
two_composable_arrows_of_spans_axioms: shows "two_composable_arrows_of_spans (\<cdot>) prj0 prj1 g h"
GPicard6: fixes M :: "complex set"   and z :: "complex"   and a :: "complex"   and f :: "complex \<Rightarrow> complex" assumes "open M"   and "z \<in> M"   and "a \<noteq> 0"   and "f holomorphic_on M - {z}"   and "\<And>w. w \<in> M - {z} \<Longrightarrow> f w \<noteq> 0 \<and> f w \<noteq> a" obtains r :: "real" where "0 < r" and "ball z r \<subseteq> M" and "bounded (f ` (ball z r - {z})) \<or> bounded ((inverse \<circ> f) ` (ball z r - {z}))"
ldropWhile_cong: fixes xs :: "'c llist"   and ys :: "'c llist"   and P :: "'c \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> bool" assumes "xs = ys"   and "\<And>x. x \<in> lset ys \<Longrightarrow> P x = Q x" shows "ldropWhile P xs = ldropWhile Q ys"
isState_purgeIdle: fixes sll :: "('a, 'b, 'c) state list" shows "(\<exists>sl. GM_sec_model.purgeIdle sl = sll) = list_all isState sll"
homeomorphic_convex_compact_cball: fixes S :: "'c set"   and e :: "real"   and b :: "'c" assumes "convex S"   and "compact S"   and "interior S \<noteq> {}"   and "0 < e" shows "S homeomorphic cball b e"
dep3_event_permute_2: fixes Q :: "'a set"   and R :: "'a set"   and S :: "'a set"   and x :: "'a" assumes "dep3_event Q R S x" shows "dep3_event R Q S x"
val_Zp_ultrametric_eq': fixes x :: "nat \<Rightarrow> int"   and y :: "nat \<Rightarrow> int" assumes "x \<in> carrier Zp"   and "y \<in> carrier Zp"   and "val_Zp y < val_Zp x" shows "val_Zp (x \<ominus> y) = val_Zp y"
t'_1: shows "t' \<in> Fault ` (- F') \<longrightarrow> t' = t"
assms_3: shows "((\<sigma>, p), a, \<sigma>', q) \<in> oseqp_sos \<Gamma> i"
Gamma_complex_of_real: fixes x :: "real" shows "Gamma (complex_of_real x) = complex_of_real (Gamma x)"
pred_inject_1: fixes P1 :: "'a \<Rightarrow> bool"   and P2 :: "'b \<Rightarrow> bool"   and a :: "'a" shows "pred_efsym P1 P2 (Oldf a) = P1 a"
ereal_plus_1_4: shows "- \<infinity> + 1 = - \<infinity>"
not_dbm_lt_lt_impl: fixes a :: "'d"   and b :: "'d" assumes "\<not> Lt a \<prec> Lt b" shows "b \<le> a"
set_mode_def: shows "set_mode \<equiv> l_set_mode\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_set_mode"
piecewise_C1_differentiable_on_pair: fixes f :: "real \<Rightarrow> 'a"   and S :: "real set"   and g :: "real \<Rightarrow> 'b" assumes "f piecewise_C1_differentiable_on S"   and "g piecewise_C1_differentiable_on S" shows "(\<lambda>x. (f x, g x)) piecewise_C1_differentiable_on S"
not_iff_asym: fixes x :: "'a"   and y :: "'a" assumes "x \<in> X"   and "y \<in> X" shows "(\<not> x \<sqsubseteq> y) = (y \<sqsubset> x)"
SYNTH_def: fixes f :: "'h \<Rightarrow> 'i nres"   and R :: "(('j \<Rightarrow> 'k Heap) \<times> ('h \<Rightarrow> 'i nres)) set" shows "SYNTH f R \<equiv> True"
order_on_defs_1: fixes A :: "'b set"   and r :: "('b \<times> 'b) set" shows "preorder_on A r \<equiv> refl_on A r \<and> trans r"
multiple_block_multiplicity: fixes n :: "nat"   and bl :: "'a set" shows "count (multiple_blocks n) bl = multiplicity bl * n"
tg_def: shows "e' 0 \<in>\<^sub>i range_tm e tg"
Rep_formula_eqvt: fixes x :: "perm"   and xa :: "('a, 'b, 'c, 'd) formula" shows "x \<bullet> Rep_formula xa = Rep_formula (x \<bullet> xa)"
cauchy_not_vanishes_cases: fixes X :: "nat \<Rightarrow> rat" assumes "Real.cauchy X"   and "\<not> Real.vanishes X" shows "\<exists>b>0. \<exists>k. (\<forall>n\<ge>k. b < - X n) \<or> (\<forall>n\<ge>k. b < X n)"
subring_is_ring: fixes H :: "'a set" assumes "subring H R" shows "ring (R\<lparr>carrier := H\<rparr>)"
weak_transition_def: fixes P :: "'state"   and \<alpha> :: "('act, 'pred) S_action"   and P' :: "'state" shows "P \<Rightarrow>\<langle>\<alpha>\<rangle>\<^sub>S P' \<equiv> if \<alpha> = S_action.Act \<tau> then P \<Rightarrow>\<^sub>S P' else P \<Rightarrow>{\<alpha>}\<^sub>S P'"
leq: fixes i :: "nat" assumes "i < Suc n" shows "min i n \<le> k"
aux_lemma: fixes S :: "'a set" assumes "finite S"   and "S \<subseteq> \<Union> C" shows "\<exists>I. I \<in> C \<and> S \<subseteq> I"
a: fixes n :: "nat" shows "ennreal (1 - 1 / real (n + 2)) \<le> emeasure M V"
a2: shows "snd a \<otimes> snd h [^] e' = g' [^] z'"
is_sup_least: fixes le :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a"   and s :: "'a"   and z :: "'a" assumes "dlat le"   and "dpo.is_sup le x y s"   and "le x z"   and "le y z" shows "le s z"
bex_triv_one_point1: fixes A :: "'b set"   and a :: "'b" shows "(\<exists>x\<in>A. x = a) = (a \<in> A)"
thetaZOSeqI_ZOretr: shows "thetaZOSeqI \<subseteq> ZOretr (thetaZOSeqI \<union> ZObis)"
continuous_on_compose: fixes s :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'c" assumes "continuous_on s f"   and "continuous_on (f ` s) g" shows "continuous_on s (g \<circ> f)"
div_mult_self2: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and b :: "'a"   and a :: "'a"   and c :: "'a" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size"   and "b \<noteq> zero" shows "divide (plus a (times b c)) b = plus c (divide a b)"
assms_4: shows "h \<turnstile> create_document \<rightarrow>\<^sub>h h'"
l2product_def: fixes S :: "'a set"   and f :: "'a \<Rightarrow> real"   and g :: "'a \<Rightarrow> real" shows "l2product S f g \<equiv> LINT x|lebesgue_on S. f x * g x"
map_replicate_False: fixes n :: "nat"   and xs :: "bool list" assumes "n = length xs" shows "map2 (\<and>) xs (replicate n False) = replicate n False"
acc_run_sim2: fixes r :: "nat \<Rightarrow> 'v" assumes "G'.is_acc_run r" shows "is_acc_run (fi \<circ> r)"
T_def: shows "T = {x. \<exists>a\<in>S. \<exists>d>0. (\<forall>y\<in>U. dist y a < d \<longrightarrow> y \<in> S) \<and> dist x a < d}"
t_dvd_iff: fixes u :: "'a poly" shows "(t dvd u) = (poly u a = (0::'a))"
step_correct: fixes f :: "recf"   and xs :: "nat list"   and rest :: "(recf \<times> nat list \<times> nat list) list" assumes "valid ((f, xs, []) # rest)" shows "correct ((f, xs, []) # rest, None)"
wellformed1_sortedQ: fixes p :: "('b, 'c) Combinators list"   and l :: "'b set list" assumes "wellformed_policy1 p" shows "wellformed_policy1 (qsort p l)"
weight_mono: fixes g :: "'a ugraph"   and g' :: "'a ugraph"   and w :: "'a set \<Rightarrow> nat" assumes "edges g \<subseteq> edges g'" shows "weight w g \<le> weight w g'"
fun_sum_apply: fixes u :: "'c \<Rightarrow> 'd \<Rightarrow> 'b"   and I :: "'c set"   and x :: "'d" shows "sum u I x = (\<Sum>i\<in>I. u i x)"
pcompose_smult: fixes a :: "'b"   and p :: "'b poly"   and r :: "'b poly" shows "smult a p \<circ>\<^sub>p r = smult a (p \<circ>\<^sub>p r)"
existence_ivl_trans': fixes t :: "real"   and s :: "real"   and x0 :: "'a" assumes "t + s \<in> existence_ivl0 x0"   and "t \<in> existence_ivl0 x0" shows "s \<in> existence_ivl0 (flow0 x0 t)"
prod_emb_Int: fixes M :: "'b set"   and L :: "'b \<Rightarrow> 'c measure"   and K :: "'b set"   and A :: "('b \<Rightarrow> 'c) set"   and B :: "('b \<Rightarrow> 'c) set" shows "prod_emb M L K (A \<inter> B) = prod_emb M L K A \<inter> prod_emb M L K B"
rel_mono_strong: fixes R1 :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and R2 :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "('a, 'c) tllist"   and y :: "('b, 'd) tllist"   and R1a :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and R2a :: "'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "tllist_all2 R1 R2 x y"   and "\<And>z1 y1. \<lbrakk>z1 \<in> tset x; y1 \<in> tset y; R1 z1 y1\<rbrakk> \<Longrightarrow> R1a z1 y1"   and "\<And>z2 y2. \<lbrakk>z2 \<in> set2_tllist x; y2 \<in> set2_tllist y; R2 z2 y2\<rbrakk> \<Longrightarrow> R2a z2 y2" shows "tllist_all2 R1a R2a x y"
inf_set_def: fixes A :: "'b set"   and B :: "'b set" shows "A \<inter> B = p2s ((\<lambda>x. x \<in> A) \<sqinter> (\<lambda>x. x \<in> B))"