hcju/isabelleps huggingface dataset:
split: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "int \<Rightarrow> 'a"   and f2 :: "nat \<Rightarrow> 'a"   and item :: "item" shows "P (case item of VAL x \<Rightarrow> f1 x | HAN x \<Rightarrow> f2 x) = ((\<forall>x1. item = VAL x1 \<longrightarrow> P (f1 x1)) \<and> (\<forall>x2. item = HAN x2 \<longrightarrow> P (f2 x2)))"
twiddle_mkId: fixes s :: "Var \<Rightarrow> Val"   and h :: "(Location \<times> Class \<times> (Field \<times> Val) list) list" assumes "noLowDPs (s, h)" shows "(s, h) \<equiv>\<^sub>mkId h (s, h)"
LINEAR_UN: fixes b :: "'g"   and f :: "'j \<Rightarrow> ('f + 'g + 'h, 'i) Old_Datatype.node set" shows "LINEAR b (\<Union> (range f)) = (\<Union>x. LINEAR b (f x))"
step: fixes stk' :: "'addr val list"   and loc' :: "'addr val list"   and pc' :: "nat"   and xcp' :: "'addr option"   and stk'' :: "'addr val list"   and loc'' :: "'addr val list"   and pc'' :: "nat"   and xcp'' :: "'addr option" assumes "\<tau>Exec_movesr ci P t es h (stk, loc, pc, xcp) (stk', loc', pc', xcp')"   and "\<tau>exec_moves ci P t es h (stk', loc', pc', xcp') (stk'', loc'', pc'', xcp'')"   and "Q stk' loc' pc' xcp'" shows "Q stk'' loc'' pc'' xcp''"
residual_network_sel_3: fixes f :: "'v \<times> 'v \<Rightarrow> ennreal" shows "source (residual_network f) = source \<Delta>"
eq_inf_transp: fixes w\<^sub>1 :: "'action stream"   and w\<^sub>2 :: "'action stream"   and w\<^sub>3 :: "'action stream" assumes "w\<^sub>1 =\<^sub>I w\<^sub>2"   and "w\<^sub>2 =\<^sub>I w\<^sub>3" shows "w\<^sub>1 =\<^sub>I w\<^sub>3"
intros_2: fixes P :: "complex mat"   and U :: "complex mat" assumes "is_quantum_predicate P" shows "\<turnstile>\<^sub>p {adjoint U * P * U} Utrans U {P}"
filter_leI: fixes F' :: "'b filter"   and F :: "'b filter" assumes "\<And>P. eventually P F' \<Longrightarrow> eventually P F" shows "F \<le> F'"
memClosed: fixes p :: "(name \<times> name) list"   and x :: "'d"   and xs :: "'d list" shows "p \<bullet> x mem p \<bullet> xs = x mem xs"
CFGExit_wf_axioms: shows "CFGExit_wf sourcenode targetnode kind valid_edge (_Entry_) Def Use state_val (_Exit_)"
reach_is_node: fixes A :: "bool list bdd list \<times> bool list"   and n :: "nat"   and p :: "bool list"   and w :: "bool list list"   and q :: "bool list" assumes "aut_nfa A n"   and "nfa_reach A p w q"   and "nfa_is_node A p"   and "list_all (is_alph n) w" shows "nfa_is_node A q"
approx_f_e: fixes j :: "nat"   and x :: "'a" assumes "t j \<le> t0 + e1'" shows "dist (f (t j) x) (g (t j) x) \<le> sqrt (real DIM('a)) * 2 powr real_of_int (- e)"
surj_imp_inv_eq: fixes f :: "'e \<Rightarrow> 'd"   and g :: "'d \<Rightarrow> 'e" assumes "surj f"   and "\<And>x. g (f x) = x" shows "inv f = g"
assms_1: shows "t \<in> SMP (pair ` setops\<^sub>s\<^sub>s\<^sub>t S)"
IH: fixes n :: "nat" assumes "wf_formula n \<phi>" shows "lang n (rexp.Inter (rexp_of_alt n \<phi>) (ENC n (FOV \<phi>))) - {[]} = lang n (rexp.Inter (rexp_of_alt' n \<phi>) (ENC n (FOV \<phi>))) - {[]}"
filter_def: fixes P :: "'c \<Rightarrow> bool"   and A :: "'c set" shows "Set.filter P A = {a \<in> A. P a}"
linear: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "class.linorder less_eq less" shows "less_eq x y \<or> less_eq y x"
deg_eqI: fixes f :: "nat \<Rightarrow> 'a"   and n :: "nat" assumes "f \<in> carrier (UP R)"   and "deg R f \<le> n"   and "f n \<noteq> \<zero>" shows "deg R f = n"
fin: shows "finite {x. a < x \<and> x < b \<and> poly p x = 0}"
vimage_null_part_sets: fixes f :: "'a \<Rightarrow> 'c"   and N :: "'c measure"   and A :: "'c set" assumes "f \<in> M \<rightarrow>\<^sub>M N"   and "null_sets N \<subseteq> null_sets (distr M N f)"   and "A \<in> sets (completion N)" shows "f -` null_part N A \<inter> space M \<in> sets M"
wp_under_wlp_repeat: fixes a :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and n :: "nat" assumes "healthy (wp a)"   and "nearly_healthy (wlp a)"   and "wp_under_wlp a" shows "wp_under_wlp (repeat n a)"
homeomorphic_space: fixes X :: "'b topology"   and Y :: "'c topology" shows "(X homeomorphic_space Y) = (\<exists>f. homeomorphic_map X Y f)"
simps_1: fixes y :: "'a arctic_delta" shows "MinInfty_delta * y = MinInfty_delta"
bij_swap_compose_bij: fixes p :: "'a \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "bij p" shows "bij (a \<rightleftharpoons>\<^sub>F b \<circ> p)"
induct: fixes P :: "'c list list \<Rightarrow> bool"   and a0 :: "'c list list" assumes "P []"   and "\<And>xs. P [xs]"   and "\<And>v vb vc. P (merge_adj (v # vb # vc)) \<Longrightarrow> P (v # vb # vc)" shows "P a0"
hitting_paths_def: fixes G :: "('c, 'd) Graph_scheme"   and v0 :: "'c"   and v1 :: "'c"   and paths :: "'c list set" assumes "DisjointPaths G v0 v1 paths" shows "DisjointPaths.hitting_paths v0 v1 paths \<equiv> \<lambda>x. x \<noteq> v0 \<and> ((\<exists>xs\<in>paths. x \<in> set xs) \<or> x = v1)"
fps_deriv_fps_X: shows "fps_deriv fps_X = 1"
left_commute: fixes b :: "'c \<times> 'c \<Rightarrow> 'd"   and a :: "'c \<times> 'c \<Rightarrow> 'd"   and c :: "'c \<times> 'c \<Rightarrow> 'd" shows "b \<oplus> (a \<oplus> c) = a \<oplus> (b \<oplus> c)"
assms_3: shows "H \<turnstile> SyntaxN.All i (OrdP (Var i) IMP A IMP A(i::=SUCC (Var i)))"
ipd_pd_cd: fixes l :: "nat"   and \<pi> :: "nat \<Rightarrow> 'n"   and k :: "nat" assumes "l cd\<^bsup>\<pi>\<^esup>\<rightarrow> k" shows "ipd (\<pi> k) pd\<rightarrow> \<pi> l"
assms_4: shows "h \<turnstile> get_component ptr \<rightarrow>\<^sub>r c"
inf_sup_ord_2: fixes x :: "'b"   and y :: "'b" shows "inf x y \<le> y"
sdp_Apply: fixes f :: "'a \<Rightarrow> 'a" shows "sub_distrib_pconj (Apply f)"
\<R>\<^sub>G_comp: fixes \<alpha> :: "'d \<Rightarrow> 'd"   and \<alpha>' :: "'d \<Rightarrow> 'd"   and \<beta> :: "'e \<Rightarrow> 'e"   and \<beta>' :: "'e \<Rightarrow> 'e"   and s :: "'d \<Rightarrow> (('e \<Rightarrow> ('d, 'e, 'f) val option) \<times> ('e \<Rightarrow> ('d, 'e, 'f) val option) \<times> ('d, 'e, 'f) expr) option" assumes "bij \<alpha>"   and "bij \<alpha>'"   and "bij \<beta>"   and "bij \<beta>'" shows "\<R>\<^sub>G \<alpha>' \<beta>' (\<R>\<^sub>G \<alpha> \<beta> s) = \<R>\<^sub>G (\<alpha>' \<circ> \<alpha>) (\<beta>' \<circ> \<beta>) s"
length_Basis_list: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and sgn :: "'b \<Rightarrow> 'b"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and inner :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and Basis :: "'b set"   and abs :: "'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and Basis_list :: "'b list"   and eucl_down :: "int \<Rightarrow> 'b \<Rightarrow> 'b"   and eucl_truncate_down :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and eucl_truncate_up :: "nat \<Rightarrow> 'b \<Rightarrow> 'b" assumes "class.executable_euclidean_space Inf Sup minus dist norm plus zero uminus scaleR sgn uniformity open inner Basis abs inf sup less_eq less Basis_list eucl_down eucl_truncate_down eucl_truncate_up" shows "length Basis_list = card Basis"
put_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_disconnected_nodes_type_wf_preserved: fixes h :: "(_) heap"   and document_ptr :: "(_) document_ptr"   and v :: "(_) node_ptr list"   and h' :: "(_) heap" assumes "h \<turnstile> put_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t document_ptr disconnected_nodes_update v \<rightarrow>\<^sub>h h'" shows "DocumentClass.type_wf h = DocumentClass.type_wf h'"
lcos_m_assoc: fixes M :: "'a set"   and g :: "'a"   and h :: "'a" assumes "M \<subseteq> carrier G"   and "g \<in> carrier G"   and "h \<in> carrier G" shows "g <#\<^bsub>add_monoid G\<^esub> (h <#\<^bsub>add_monoid G\<^esub> M) = g \<oplus> h <#\<^bsub>add_monoid G\<^esub> M"
assms_2: shows "y \<in>\<^sub>i range_tm e t2"
empty_not_quorum: assumes "{} \<in> Quorum" shows "False"
simps: fixes xs :: "('a \<times> 'c) list"   and ko :: "'o" shows "tl_raw (xs, ko) = (tl xs, ko)"
r'': shows "Well_order r'' \<and> Field r'' = B - A"
invKey_def: shows "invKey \<equiv> SOME invKey. (\<forall>K. invKey (invKey K) = K) \<and> (all_symmetric \<longrightarrow> invKey = id)"
\<Omega>: shows "\<Omega> \<inter> space M \<in> sets M"
iface_conjunct_None: fixes i1 :: "iface"   and i2 :: "iface"   and p_i :: "char list" assumes "iface_conjunct i1 i2 = None" shows "\<not> (match_iface i1 p_i \<and> match_iface i2 p_i)"
l_cancel_one': fixes G :: "('c, 'd) monoid_scheme"   and x :: "'c"   and a :: "'c" assumes "Group.group G"   and "x \<in> carrier G"   and "a \<in> carrier G" shows "(x = x \<otimes>\<^bsub>G\<^esub> a) = (a = \<one>\<^bsub>G\<^esub>)"
introsAt: fixes index :: "nat" assumes "Suc 0 \<le> index"   and "index < length L'" shows "LeftDerivationIntrosAt (ladder_stepdown_\<alpha>_0 \<alpha> D L) (drop (ladder_stepdown_diff L) D) L' index"
simps_1: fixes x :: "var\<^sub>C"   and m :: "Mode"   and mds :: "Mode \<Rightarrow> var\<^sub>C set" shows "C.update_modes (x +=\<^sub>m m) mds = mds(m := insert x (mds m))"
norm_matrix_le_mult_op_norm: fixes A :: "(('b, 'd) vec, 'c) vec"   and x :: "('b, 'd) vec" shows "\<parallel>A *v x\<parallel> \<le> (\<parallel>A\<parallel>\<^sub>o\<^sub>p) * (\<parallel>x\<parallel>)"
bisim1ALength: fixes P :: "'a prog"   and h :: "'heap"   and a :: "'addr expr1"   and a' :: "'addr expr1"   and xs :: "'addr val list"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and xcp :: "'addr option" assumes "P,a,h \<turnstile> (a', xs) \<leftrightarrow> (stk, loc, pc, xcp)" shows "P,a\<bullet>length,h \<turnstile> (a'\<bullet>length, xs) \<leftrightarrow> (stk, loc, pc, xcp)"
tree_height_estimate_lower: fixes t :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree" assumes "SkewBinomialHeapStruc.tree_invar t" shows "2 ^ SkewBinomialHeapStruc.height_tree t \<le> size (SkewBinomialHeapStruc.tree_to_multiset t)"
assms_2: shows "\<S>\<bullet>\<C> \<turnstile> cs : [] _> [T_f64]"
Farkas'_Lemma_indexed: fixes c :: "nat \<Rightarrow> rat le_constraint"   and Is :: "nat set" assumes "lec_rel ` c ` Is \<subseteq> {Leq_Rel}"   and "finite Is" shows "(\<nexists>v. \<forall>i\<in>Is. v \<Turnstile>\<^sub>l\<^sub>e c i) = (\<exists>C const. (\<forall>i\<in>Is. 0 \<le> C i) \<and> (\<Sum>i\<in>Is. Le_Constraint Leq_Rel (C i *R lec_poly (c i)) (C i *R lec_const (c i))) = Le_Constraint Leq_Rel 0 const \<and> const < 0)"
process_mru_def: fixes vs :: "'d \<Rightarrow> 'c \<Rightarrow> 'e option" shows "process_mru vs \<equiv> \<lambda>a. mru_of_set vs {a}"
post_meas0_index_0_alice: fixes \<phi> :: "complex Matrix.mat"   and \<alpha> :: "complex"   and \<beta> :: "complex" assumes "state 1 \<phi>"   and "\<alpha> = \<phi> $$ (0, 0)"   and "\<beta> = \<phi> $$ (1, 0)" shows "post_meas0 3 (alice \<phi>) 0 = Tensor.mat_of_cols_list 8 [[\<alpha> / complex_of_real (sqrt 2), \<beta> / complex_of_real (sqrt 2), \<beta> / complex_of_real (sqrt 2), \<alpha> / complex_of_real (sqrt 2), 0, 0, 0, 0]]"
intros_16: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "'e \<Rightarrow> ('e, 'd, 'f) com"   and s :: "'e"   and t :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile> \<langle>c s,Normal s\<rangle> \<Rightarrow> t" shows "\<Gamma>\<turnstile> \<langle>DynCom c,Normal s\<rangle> \<Rightarrow> t"
inf_mono: fixes a :: "'b"   and c :: "'b"   and b :: "'b"   and d :: "'b" assumes "a \<le> c"   and "b \<le> d" shows "a \<sqinter> b \<le> c \<sqinter> d"
HNextPart_def: fixes s :: "state"   and s' :: "state" shows "HNextPart s s' = (chosen s' = (if chosen s \<noteq> NotAnInput \<or> (\<forall>p. outpt s' p = NotAnInput) then chosen s else outpt s' (SOME p. outpt s' p \<noteq> NotAnInput)) \<and> allInput s' = allInput s \<union> range (inpt s'))"
FiniteDetBroadcastEnvironment_axioms: shows "FiniteDetBroadcastEnvironment jkbp envInit envAction envTrans envVal agents envObsC"
subst_lsst_member: fixes x :: "'d strand_label \<times> ('e, 'f) stateful_strand_step"   and A :: "('d strand_label \<times> ('e, 'f) stateful_strand_step) list"   and \<delta> :: "'f \<Rightarrow> ('e, 'f) Term.term" assumes "x \<in> set A" shows "x \<cdot>\<^sub>l\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<delta> \<in> set (A \<cdot>\<^sub>l\<^sub>s\<^sub>s\<^sub>t \<delta>)"
P33A: shows "X \<noteq> y \<and> A' B' ReflectL X y"
RModuleIso_axioms_def: fixes M :: "'a set"   and T :: "'a \<Rightarrow> 'b"   and N :: "'b set" shows "RModuleIso_axioms M T N \<equiv> bij_betw T M N"
size_gen_2: shows "size_f Aodv_Basic.inv = 0"
simps_1: fixes ac :: "'a list" shows "quicksort_acc ac [] = ac"
module_vec_def: fixes ty :: "'b itself"   and n :: "nat" shows "module_vec ty n \<equiv> \<lparr>carrier = carrier_vec n, monoid.mult = undefined, one = undefined, zero = 0\<^sub>v n, add = (+), module.smult = (\<cdot>\<^sub>v)\<rparr>"
distinct_7: fixes x1 :: "'a"   and x5 :: "'a \<times> nat" shows "CFG_node x1 \<noteq> Actual_out x5"
compose_def: shows "(\<bullet>) \<equiv> \<lambda>uu uua. corec_dds (\<lambda>(S1, S2) a. (id (fst (run S2 (fst (run S1 a)))), Inr (snd (run S1 a), snd (run S2 (fst (run S1 a)))))) (uu, uua)"
distinct_256: fixes x14 :: "Syntax.term"   and x11 :: "Syntax.term" shows "Unroll x14 \<noteq> Prj1 x11"
pivot_fun_multrow: fixes A :: "'c mat"   and f :: "nat \<Rightarrow> nat"   and jj :: "nat"   and nr :: "nat"   and nc :: "nat"   and i0 :: "nat"   and a :: "'c" assumes "pivot_fun A f jj"   and "dim_row A = nr"   and "dim_col A = nc"   and "f i0 = jj"   and "jj \<le> nc" shows "pivot_fun (multrow i0 a A) f jj"
list_of_oalist_empty: fixes ko :: "'b nat_term_order" shows "list_of_oalist_ntm (OAlist_empty_ntm ko) = ([], ko)"
integral_mult_right_zero: fixes M :: "'c measure"   and c :: "'b"   and f :: "'c \<Rightarrow> 'b" shows "LINT x|M. c * f x = c * integral\<^sup>L M f"
simps_1: fixes \<mu> :: "'a" shows "Ide \<^bold>\<langle>\<mu>\<^bold>\<rangle>\<^sub>0 = obj \<mu>"
sum_abs: fixes f :: "'b \<Rightarrow> 'a"   and A :: "'b set" shows "\<bar>sum f A\<bar> \<le> (\<Sum>i\<in>A. \<bar>f i\<bar>)"
prems_1: shows "a \<in> insert x V \<rightarrow> carrier R"
simps_1: fixes c :: "'b list" shows "mk_cycles_path 0 c = []"
restrict_eq: fixes A :: "'d set" shows "restrict A = filter (\<lambda>(k, v). k \<in> A)"
Nonce_synth: fixes n :: "fresh_t"   and H :: "msg set" assumes "Nonce n \<in> synth H" obtains "Nonce n \<in> H"
simps_2: fixes subs :: "variable \<times> String.literal set \<Rightarrow> typ option" shows "raw_matches [] [] subs = Some subs"
intro: fixes subs :: "(('c \<times> nat) \<times> 'c \<times> nat) set"   and g :: "('c \<times> nat, 'd) rgraph_scheme" assumes "SubRel.vertices subs \<subseteq> Graph.vertices g" shows "sub_rel_of g subs"
prod_relI: fixes a :: "'a"   and a' :: "'b"   and R1 :: "('a \<times> 'b) set"   and b :: "'c"   and b' :: "'f"   and R2 :: "('c \<times> 'f) set" assumes "(a, a') \<in> R1"   and "(b, b') \<in> R2" shows "((a, b), a', b') \<in> R1 \<times>\<^sub>r R2"
edge_pred_succ: fixes a :: "'b" shows "edge_pred HM (edge_succ HM a) = a"
null_char: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" assumes "span_vertical_category C" shows "partial_magma.null (span_vertical_category.vcomp C) = \<lparr>Chn = partial_magma.null C, Dom = \<lparr>Leg0 = partial_magma.null C, Leg1 = partial_magma.null C\<rparr>, Cod = \<lparr>Leg0 = partial_magma.null C, Leg1 = partial_magma.null C\<rparr>\<rparr>"
set_map: fixes f :: "'f \<Rightarrow> 'e"   and v :: "'f list" shows "set (map f v) = f ` set v"
cpmi_eq: fixes D :: "int"   and P :: "int \<Rightarrow> bool"   and P1 :: "int \<Rightarrow> bool"   and B :: "int set" assumes "0 < D"   and "\<exists>z. \<forall>x<z. P x = P1 x"   and "\<forall>x. \<not> (\<exists>j\<in>{1..D}. \<exists>b\<in>B. P (b + j)) \<longrightarrow> P x \<longrightarrow> P (x - D)"   and "\<forall>x k. P1 x = P1 (x - k * D)" shows "(\<exists>x. P x) = ((\<exists>j\<in>{1..D}. P1 j) \<or> (\<exists>j\<in>{1..D}. \<exists>b\<in>B. P (b + j)))"
functor_axioms: shows "functor (\<cdot>\<^sub>D) (\<cdot>\<^sub>D) src\<^sub>D"
in_\<R>: shows "(\<langle>c\<^sub>A, mds\<^sub>A, mem\<^sub>A\<rangle>\<^sub>A, \<langle>c\<^sub>C, mds\<^sub>C, mem\<^sub>C\<rangle>\<^sub>C) \<in> \<R>"
call_return_node_edge: fixes a :: "'a"   and a' :: "'a" assumes "valid_edge a"   and "a' \<in> get_return_edges a" shows "\<exists>a''. valid_edge a'' \<and> sourcenode a'' = sourcenode a \<and> targetnode a'' = targetnode a' \<and> kind a'' = (\<lambda>cf. False)\<^sub>\<surd>"
pfun_minus_self: fixes f :: "('a, 'b) pfun" shows "f - f = {}\<^sub>p"
set_spmf_G_rel: fixes h :: "'pub_input"   and w :: "'witness" assumes "(h, w) \<in> set_spmf G" shows "(h, w) \<in> Rel"
inj: shows "\<forall>n. undiago (diago n) = n"
simps_1: fixes n :: "'a" shows "local.times n [] = []"
inner_Basis_minus: fixes b1 :: "'a"   and b2 :: "'a"   and v :: "'a" assumes "b1 \<in> S"   and "b2 \<in> S" shows "local.inner_Basis (b1 - b2) v = local.inner_Basis b1 v - local.inner_Basis b2 v"
set_trieP_insertP: fixes xs :: "bool list"   and t :: "trieP" shows "set_trieP (insertP xs t) = set_trieP t \<union> {xs}"
simps_3: fixes x2 :: "'c" shows "\<lfloor>x2\<rfloor> \<noteq> None"
map_of_set_pcs_notin: fixes C :: "'a"   and FDTs :: "(('b \<times> 'a) \<times> 'c) list"   and F :: "'b" assumes "C \<notin> (\<lambda>t. snd (fst t)) ` set FDTs" shows "map_of FDTs (F, C) = None"
non_atomic_Sign: fixes x :: "msg"   and y :: "msg" shows "composed (Sign x y)"
\<tau>move1_not_call1: fixes e :: "('a, 'b, 'addr) exp"   and a :: "'addr"   and M :: "String.literal"   and vs :: "'addr val list"   and P :: "'c prog"   and h :: "'heap" assumes "call1 e = \<lfloor>(a, M, vs)\<rfloor>" shows "\<tau>move1 P h e = (synthesized_call P h (a, M, vs) \<longrightarrow> \<tau>external' P h a M)"
simps_1: fixes \<Psi> :: "'b"   and \<Psi>' :: "'b" shows "insertAssertion (\<langle>\<epsilon>, \<Psi>\<rangle>) \<Psi>' = \<langle>\<epsilon>, \<Psi>' \<otimes> \<Psi>\<rangle>"
positions_in_row: fixes s :: "('c \<times> bool list) stream"   and i :: "nat" shows "positions_in_row s i = {p. snd (s !! p) ! i}"
dl_def: fixes r :: "('a \<times> 'a) set"   and \<sigma> :: "'a list" shows "r \<down>l \<sigma> = r \<down>s set \<sigma>"
simps_1: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and s :: "(('c \<Rightarrow> 'd option) \<times> 'e) list" assumes "CFG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main" shows "CFG.transfers procs [] s = s"
dim_def: fixes V :: "'b set" shows "local.dim V = (if \<exists>b\<subseteq>U\<^sub>V\<^sub>S. \<not> local.dependent b \<and> local.span b = local.span V then card (SOME b. b \<subseteq> U\<^sub>V\<^sub>S \<and> \<not> local.dependent b \<and> local.span b = local.span V) else 0)"
blinfun_compose_assoc: fixes a :: "'f \<Rightarrow>\<^sub>L 'd"   and b :: "'e \<Rightarrow>\<^sub>L 'f"   and c :: "'c \<Rightarrow>\<^sub>L 'e" shows "a o\<^sub>L b o\<^sub>L c = a o\<^sub>L (b o\<^sub>L c)"
quasi_inverses_some_quasi_inverse_1: fixes f :: "'d" assumes "D.equivalence_map f" shows "D.quasi_inverses f (D.some_quasi_inverse f)"
INF_nat_binary: fixes A :: "'c"   and B :: "'c" shows "inf A (INF x\<in>Collect ((<) 0). B) = inf A B"
span_0: fixes S :: "'b set" shows "(0::'b) \<in> span S"
assignNIlowlemma: fixes \<Gamma> :: "char list \<Rightarrow> secLevel option"   and s1 :: "char list \<Rightarrow> val option"   and s2 :: "char list \<Rightarrow> val option"   and V :: "char list"   and e :: "expr"   and s1' :: "char list \<Rightarrow> val option"   and s2' :: "char list \<Rightarrow> val option" assumes "\<Gamma> \<turnstile> s1 \<approx>\<^sub>L s2"   and "\<Gamma> V = Some Low"   and "\<Gamma> \<turnstile> e : Low"   and "s1' = s1(V := \<lbrakk>e\<rbrakk>s1)"   and "s2' = s2(V := \<lbrakk>e\<rbrakk>s2)" shows "\<Gamma> \<turnstile> s1' \<approx>\<^sub>L s2'"
asig_of_par: fixes A :: "('e set \<times> 'e set \<times> 'e set) \<times> 'f set \<times> ('f \<times> 'e \<times> 'f) set"   and B :: "('e set \<times> 'e set \<times> 'e set) \<times> 'g set \<times> ('g \<times> 'e \<times> 'g) set" shows "asig_of (A || B) = asig_comp (asig_of A) (asig_of B)"
cnt_0_D: fixes x :: "'c"   and xs :: "'c list" assumes "cnt x xs = 0" shows "x \<notin> set xs"
assms: fixes s :: "'a"   and a :: "'b"   and s' :: "'a" assumes "s \<in> reachable A I"   and "(s, a, s') \<in> automaton.trans A"   and "I a" shows "P (s, a, s')"
time_closest_pair_rec_conv_closest_pair_recurrence: fixes ps :: "(int \<times> int) list" shows "real (time (closest_pair_rec_tm ps)) \<le> closest_pair_recurrence (length ps)"
simps_4: fixes x21 :: "'b"   and x22 :: "'b" shows "DenyAll \<noteq> DenyAllFromTo x21 x22"
assms: fixes gs :: "'b list"   and bs :: "'b list"   and h :: "'c"   and ps :: "('d \<times> 'c \<times> 'b) list" shows "set (aps flag gs bs h ps) = set ps \<union> Pair flag ` ({h} \<times> (set gs \<union> set bs))"
MergeGuardsD_sound: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and F :: "'f set"   and P :: "'e set"   and c :: "('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set"   and n :: "nat" assumes "\<forall>n. \<Gamma>,\<Theta>\<Turnstile>n:\<^bsub>/F\<^esub> P merge_guards c Q,A" shows "\<Gamma>,\<Theta>\<Turnstile>n:\<^bsub>/F\<^esub> P c Q,A"
INF_lower: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and i :: "'b"   and A :: "'b set"   and f :: "'b \<Rightarrow> 'a" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "i \<in> A" shows "less_eq (Inf (f ` A)) (f i)"
cindependent_butterfly: fixes basisA :: "'c set"   and basisB :: "'d set" assumes "is_ortho_set basisA"   and "is_ortho_set basisB"   and "\<And>a. a \<in> basisA \<Longrightarrow> norm a = 1"   and "\<And>b. b \<in> basisB \<Longrightarrow> norm b = 1" shows "cindependent {butterfly a b |a b. a \<in> basisA \<and> b \<in> basisB}"
assms_5: shows "h \<turnstile> get_elements_by_tag_name ptr' name \<rightarrow>\<^sub>r results"
finite_basis_exists: assumes "W.fin_dim" shows "\<exists>\<beta>. finite \<beta> \<and> W.basis \<beta>"
f_expand_take_mult: fixes n :: "nat"   and k :: "nat"   and xs :: "'b message_af list" shows "xs \<odot> k \<down> (n * k) = xs \<down> n \<odot> k"
uniformity_vec_def: shows "uniformity = (INF e\<in>{0<..}. principal {(x, y). dist x y < e})"
equiv_Worder: fixes D :: "('c, 'd) Order_scheme"   and T :: "('e, 'f) Order_scheme" assumes "Worder D"   and "Order T"   and "ord_equiv D T" shows "Worder T"
gpv1_def: shows "gpv1 = Generative_Probabilistic_Value.Done x"
eq': shows "x\<^sup>2 - int D * y\<^sup>2 = 1"
gcd_mod: fixes a :: "'a"   and b :: "'a" assumes "a \<noteq> (0::'a)" shows "normalization_euclidean_semiring_class.gcd a (b mod a) = normalization_euclidean_semiring_class.gcd b a"
scale_ms_aux'_def: fixes c :: "'a"   and F :: "('a \<times> real) msllist" shows "scale_ms_aux' c F = scale_shift_ms_aux (c, 0) F"
skip: fixes m :: "'b \<Rightarrow> 'c"   and E :: "'a \<Rightarrow> ('b \<Rightarrow> 'c) \<Rightarrow> 'c"   and BMap :: "'c \<Rightarrow> bool" shows "((skip, m), [], m) \<in> MWLf_semantics.MWLfSteps_det E BMap"
dom_cod: fixes f :: "'s" assumes "S.arr f" shows "S.dom (S.cod f) = S.cod f"
hyps_3: shows "\<Gamma>\<turnstile>S<:T"