hcju/isabelleps huggingface dataset:
constructive_security_aux_obsf_def: fixes real_resource :: "('y + 'z, 'aa + 'ab) resource"   and ideal_resource :: "('ac + 'z, 'ad + 'ab) resource"   and sim :: "('y, 'aa, 'ac, 'ad) converter"   and \<I>_real :: "('y, 'aa) \<I>"   and \<I>_ideal :: "('ac, 'ad) \<I>"   and \<I>_common :: "('z, 'ab) \<I>"   and adv :: "real" shows "constructive_security_aux_obsf real_resource ideal_resource sim \<I>_real \<I>_ideal \<I>_common adv \<equiv> (\<I>_real \<oplus>\<^sub>\<I> \<I>_common \<turnstile>res real_resource \<surd> \<and> \<I>_ideal \<oplus>\<^sub>\<I> \<I>_common \<turnstile>res ideal_resource \<surd>) \<and> \<I>_real, \<I>_ideal \<turnstile>\<^sub>C sim \<surd> \<and> pfinite_converter \<I>_real \<I>_ideal sim \<and> 0 \<le> adv"
SUP_ereal_add_left: fixes I :: "'a set"   and c :: "ereal"   and f :: "'a \<Rightarrow> ereal" assumes "I \<noteq> {}"   and "c \<noteq> - \<infinity>" shows "(SUP i\<in>I. f i + c) = Sup (f ` I) + c"
h_props_1: shows "\<H> \<subseteq> two_chain_horizontal_boundary two_chain"
lunit_tensor: fixes a :: "'d"   and b :: "'d" assumes "ide a"   and "ide b" shows "\<l>[a \<otimes> b] = (\<l>[a] \<otimes> b) \<cdot> \<a>\<^sup>-\<^sup>1[\<I>, a, b]"
sub_addD: fixes t :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real"   and P :: "'b \<Rightarrow> real"   and Q :: "'b \<Rightarrow> real"   and s :: "'b" assumes "sub_add t"   and "sound P"   and "sound Q" shows "t P s + t Q s \<le> t (\<lambda>s'. P s' + Q s') s"
simps_5: fixes v :: "Ferrack.num" shows "not (Le v) = NOT (Le v)"
seqI': fixes f :: "(('c + ('j \<Rightarrow> 'c)) set, 'c + ('j \<Rightarrow> 'c) \<Rightarrow> 'c + ('j \<Rightarrow> 'c)) concrete_category.arr"   and a :: "(('c + ('j \<Rightarrow> 'c)) set, 'c + ('j \<Rightarrow> 'c) \<Rightarrow> 'c + ('j \<Rightarrow> 'c)) concrete_category.arr"   and b :: "(('c + ('j \<Rightarrow> 'c)) set, 'c + ('j \<Rightarrow> 'c) \<Rightarrow> 'c + ('j \<Rightarrow> 'c)) concrete_category.arr"   and g :: "(('c + ('j \<Rightarrow> 'c)) set, 'c + ('j \<Rightarrow> 'c) \<Rightarrow> 'c + ('j \<Rightarrow> 'c)) concrete_category.arr"   and c :: "(('c + ('j \<Rightarrow> 'c)) set, 'c + ('j \<Rightarrow> 'c) \<Rightarrow> 'c + ('j \<Rightarrow> 'c)) concrete_category.arr" assumes "\<guillemotleft>f : a \<rightarrow>\<^sub>S b\<guillemotright>"   and "\<guillemotleft>g : b \<rightarrow>\<^sub>S c\<guillemotright>" shows "S.seq g f"
is_invar_def: fixes I :: "'s \<Rightarrow> bool" shows "is_invar I \<equiv> I s\<^sub>0 \<and> (\<forall>s s'. reachable s \<and> I s \<and> step s s' \<longrightarrow> I s')"
D_def: shows "D = (\<lambda>n. return_time_function A -` {n + 1})"
aux_lemma2: fixes Ia :: "'a set"   and i :: "'a"   and j :: "'a" assumes "ideal Ia R"   and "i \<in> Ia"   and "j \<in> carrier R"   and "i \<oplus> j = \<one>" shows "Ia +> j = \<one>\<^bsub>R Quot Ia\<^esub>"
map_comp_def: fixes f :: "'f \<Rightarrow> 'e option"   and g :: "'d \<Rightarrow> 'f option" shows "f \<circ>\<^sub>m g = (\<lambda>k. case g k of \<bottom> \<Rightarrow> \<bottom> | \<lfloor>v\<rfloor> \<Rightarrow> f v)"
1: shows "g \<ominus>\<^bsub>UP R\<^esub> h = to_polynomial R (g 0)"
trail_Nil_iff: fixes u :: "'a"   and v :: "'a" shows "trail u [] v = (u = v \<and> u \<in> verts G)"
rel_witness_gpv1: fixes A :: "'j \<Rightarrow> 'k \<Rightarrow> bool"   and C :: "'l \<Rightarrow> 'm \<Rightarrow> bool"   and R :: "'n \<Rightarrow> 'p \<Rightarrow> bool"   and R' :: "'p \<Rightarrow> 'o \<Rightarrow> bool"   and gpv :: "('j, 'l, 'n) gpv"   and gpv' :: "('k, 'm, 'o) gpv" assumes "rel_gpv'' A C (R OO R') gpv gpv'"   and "left_unique R"   and "right_total R"   and "right_unique R'"   and "left_total R'" shows "rel_gpv'' (\<lambda>a (a', b). a = a' \<and> A a' b) (\<lambda>c (c', d). c = c' \<and> C c' d) R gpv (rel_witness_gpv A C R R' (gpv, gpv'))"
avar_upd_no_jump: fixes upd :: "val \<Rightarrow> state \<Rightarrow> state"   and G :: "prog"   and i :: "val"   and a :: "val"   and s' :: "state"   and s :: "state"   and j :: "jump"   and val :: "val" assumes "upd = snd (fst (avar G i a s'))"   and "fst s \<noteq> Some (Jump j)" shows "fst (upd val s) \<noteq> Some (Jump j)"
le_\<I>_def: fixes \<I> :: "('h, 'i) \<I>"   and \<I>' :: "('h, 'i) \<I>" shows "(\<I> \<le> \<I>') = (outs_\<I> \<I> \<subseteq> outs_\<I> \<I>' \<and> (\<forall>x\<in>outs_\<I> \<I>. responses_\<I> \<I>' x \<subseteq> responses_\<I> \<I> x))"
R12s_def: shows "R12s \<equiv> {(s, s'). s = med12s s'}"
inj_on_arc_to_ends: shows "inj_on (arc_to_ends H) (arcs H)"
F_def: shows "F \<equiv> \<lambda>results r. case r of Resumption.resumption.Done x \<Rightarrow> set_option x | Resumption.resumption.Pause out c \<Rightarrow> \<Union>input. results (c input)"
D_eq_2: fixes r :: "('a \<times> 'a) set"   and \<tau> :: "'a list"   and \<sigma> :: "'a list"   and \<sigma>' :: "'a list"   and \<tau>' :: "'a list" assumes "trans r"   and "irrefl r"   and "Decreasing_Diagrams.D r \<tau> \<sigma> \<sigma>' \<tau>'" shows "(r|\<sigma>'| -s r \<down>l \<tau>, r|\<sigma>|) \<in> mul_eq r"
inj_on_def: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set" shows "inj_on f A = (\<forall>x\<in>A. \<forall>y\<in>A. f x = f y \<longrightarrow> x = y)"
Gauss_Jordan_in_ij_0': fixes A :: "(('b, 'c) vec, 'a) vec"   and j :: "'c"   and i :: "'a" assumes "\<exists>n. A $ n $ j \<noteq> (0::'b) \<and> i \<le> n" shows "\<forall>a. a \<noteq> i \<longrightarrow> Gauss_Jordan_in_ij A i j $ a $ j = (0::'b)"
field_cont_on_top_edge_image: shows "continuous_on {p. \<exists>x. x \<in> cbox a b \<and> p = (g1 x, x)} F"
st1_def: shows "st1 = [Frame f pc \<Sigma>1]"
diff_right: fixes prod :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and a :: "'a"   and b :: "'b"   and b' :: "'b" assumes "bounded_bilinear prod" shows "prod a (b - b') = prod a b - prod a b'"
cadd_le_mono: fixes \<kappa>' :: "V"   and \<kappa> :: "V"   and \<mu>' :: "V"   and \<mu> :: "V" assumes "\<kappa>' \<le> \<kappa>"   and "\<mu>' \<le> \<mu>" shows "\<kappa>' \<oplus> \<mu>' \<le> \<kappa> \<oplus> \<mu>"
numeral_6_eq_6: shows "6 = Suc (Suc (Suc (Suc (Suc (Suc 0)))))"
agree_FUNION: fixes fm :: "('d, 'e) fmap"   and fm1 :: "('d, 'e) fmap"   and fm2 :: "('d, 'e) fmap" assumes "agree fm fm1"   and "agree fm fm2" shows "agree fm (fm1 ++ fm2)"
max_reg_Plus: fixes a1 :: "vname aexp"   and a2 :: "vname aexp" shows "max_reg (Plus a1 a2) = max (max_reg a1) (max_reg a2)"
minimize_def: shows "minimize = minimize_wrt (\<lambda>(x, y) (u, v). \<not> x @ y <\<^sub>v u @ v)"
select_convs_3: fixes states :: "hf"   and init :: "hf"   and final :: "hf"   and "next" :: "hf \<Rightarrow> 'b \<Rightarrow> hf \<Rightarrow> bool"   and more :: "'c" shows "final \<lparr>states = states, init = init, final = final, next = next, \<dots> = more\<rparr> = final"
wfPUNIVI: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<And>P x. \<forall>x. (\<forall>y. r y x \<longrightarrow> P y) \<longrightarrow> P x \<Longrightarrow> P x" shows "wfP r"
eq_Tuple2_def: shows "eq = (\<Lambda> \<langle>x1, y1\<rangle> \<langle>x2, y2\<rangle>. eq\<cdot>x1\<cdot>x2 andalso eq\<cdot>y1\<cdot>y2)"
subdg_cod_simp: fixes f :: "V" assumes "f \<in>\<^sub>\<circ> \<AA>\<lparr>Arr\<rparr>" shows "\<AA>\<lparr>Cod\<rparr>\<lparr>f\<rparr> = \<BB>\<lparr>Cod\<rparr>\<lparr>f\<rparr>"
assms_4: shows "(\<delta>', c\<^sub>0', c\<^sub>1') = combine_code (\<delta>\<^sub>L', p\<^sub>0\<^sub>L, p\<^sub>1\<^sub>L) (\<delta>\<^sub>R', p\<^sub>0\<^sub>R, p\<^sub>1\<^sub>R) l ps"
CanonicalInterpretation_def: fixes T :: "(ZF, ZF) Axioms" shows "CI T \<equiv> \<lparr>ISignature = aS\<^bsub>T\<^esub>, ICategory = CanonicalCat T, ITypes = \<lambda>A. A, IFunctions = \<lambda>f. [sDom\<^bsub>aS\<^bsub>T\<^esub>\<^esub> f,f E@ Vx,sCod\<^bsub>aS\<^bsub>T\<^esub>\<^esub> f]\<^bsub>T\<^esub>\<rparr>"
vec_eq_iff: fixes x :: "('a, 'b) vec"   and y :: "('a, 'b) vec" shows "(x = y) = (\<forall>i. x $ i = y $ i)"
abs_eq: fixes xa :: "'c ltl"   and x :: "'c ltl" shows "Abs xa \<up>\<longrightarrow>\<^sub>P Abs x = xa \<longrightarrow>\<^sub>P x"
Union_image_empty: fixes A :: "'c set"   and f :: "'d \<Rightarrow> 'c set" shows "A \<union> \<Union> (f ` {}) = A"
row_space_eq: fixes A :: "'a mat"   and nr :: "nat" assumes "A \<in> carrier_mat nr n" shows "row_space A = {w \<in> carrier_vec (dim_col A). \<exists>y\<in>carrier_vec (dim_row A). A\<^sup>T *\<^sub>v y = w}"
axxdiv2: fixes x :: "int" shows "(1 + x + x) div 2 = x \<and> (0 + x + x) div 2 = x"
Equals_def: fixes \<theta> :: "('a, 'c) trm"   and \<theta>' :: "('a, 'c) trm" shows "Equals \<theta> \<theta>' = (Geq \<theta> \<theta>' && Geq \<theta>' \<theta>)"
cobounded_has_bounded_component: fixes S :: "'a set" assumes "bounded (- S)"   and "\<not> connected S"   and "2 \<le> DIM('a)" obtains C :: "'a set" where "C \<in> components S" and "bounded C"
BC: shows "\<nu>.dom.leg0 \<cdot> doms.Prj\<^sub>0\<^sub>1 = \<pi>.dom.leg1 \<cdot> doms.Prj\<^sub>0"
IK_nf_real_card: shows "card ((\<lambda>f. f RRR) ` {f. IK_nf f}) = 7"
pathfinish_circlepath: fixes z :: "complex"   and r :: "real" shows "pathfinish (circlepath z r) = z + complex_of_real r"
set_child_nodes_get_disconnected_nodes: fixes ptr :: "(_) object_ptr"   and h :: "(_) heap"   and h' :: "(_) heap"   and ptr' :: "(_) document_ptr" shows "\<forall>w\<in>set_child_nodes_locs ptr. h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> (\<forall>r\<in>get_disconnected_nodes_locs ptr'. r h h')"
altc_sample_state_def: fixes altc_sample :: "'d \<Rightarrow> ('e \<Rightarrow> 'm) \<Rightarrow> 'm"   and p :: "'d"   and f :: "'e \<Rightarrow> ('s, 'm) stateT" shows "local.altc_sample_state altc_sample p f = StateT (\<lambda>s. altc_sample p (\<lambda>x. run_state (f x) s))"
push_def: fixes f :: "nat \<times> nat \<Rightarrow> 'capacity"   and l :: "nat \<Rightarrow> nat" shows "push f l \<equiv> \<lambda>(u, v). assert (push_precond f l (u, v)) \<bind> (\<lambda>_. assert (Labeling c s t f l) \<bind> (\<lambda>_. return (push_effect f (u, v))))"
lowner_le_trace: fixes A :: "complex mat"   and n :: "nat"   and B :: "complex mat" assumes "A \<in> carrier_mat n n"   and "B \<in> carrier_mat n n" shows "(A \<le>\<^sub>L B) = (\<forall>\<rho>\<in>carrier_mat n n. partial_density_operator \<rho> \<longrightarrow> trace (A * \<rho>) \<le> trace (B * \<rho>))"
concat_map_distinct: fixes f :: "'b \<Rightarrow> 'a list"   and xs :: "'b list"   and y :: "'a" assumes "distinct (concat (map f xs))"   and "y \<in> set (concat (map f xs))" shows "\<exists>!x. x \<in> set xs \<and> y \<in> set (f x)"
wgraph_specs_4: shows "\<alpha>w empty = (\<lambda>_. 0)"
retract_of_locally_compact: fixes S :: "'a set"   and T :: "'a set" assumes "locally compact S"   and "T retract_of S" shows "locally compact T"
takeWhile_eq_Nil_conv: fixes P :: "'a \<Rightarrow> bool"   and xs :: "'a list" shows "(takeWhile P xs = []) = (xs = [] \<or> \<not> P (hd xs))"
dot_cross_self_2: fixes x :: "(real, 3) vec"   and y :: "(real, 3) vec" shows "x \<bullet> y \<times> x = 0"
vfsequence_vremove_vcons_vfsequence: fixes xs' :: "V"   and x :: "V" assumes "xs #\<^sub>\<circ> x = xs' #\<^sub>\<circ> x" shows "vfsequence xs'"
H_def: shows "H \<equiv> complex_of_real (1 / sqrt 2) \<cdot>\<^sub>m Matrix.mat 2 2 (\<lambda>(i, j). if i \<noteq> j then 1 else if i = 0 then 1 else - 1)"
intro: fixes num' :: "int \<Rightarrow> 'b"   and plus' :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<gamma> :: "'b \<Rightarrow> int set" assumes "Gamma num' plus' \<gamma>" shows "Abs_Int num' plus' \<gamma>"
juncts_eq_set_juncts_list_2: fixes \<phi> :: "('c, 'd) aformula" shows "conjuncts \<phi> = set (conjuncts_list \<phi>)"
c: shows "\<Psi> (l', u') i' = \<Psi> (max l l', min u u') i'"
sWlsInp_def: fixes SEM :: "('h, 'i, 'j, 'sort, 'k, 'l, 'm) semDom_scheme"   and delta :: "'opSym"   and sinp :: "'index \<Rightarrow> 'l option" shows "sWlsInp SEM delta sinp \<equiv> wlsOpS delta \<and> sameDom (arOf delta) sinp \<and> liftAll2 (sWls SEM) (arOf delta) sinp"
pseudonatural_transformation_axioms: shows "pseudonatural_transformation (\<cdot>\<^sub>B) (\<star>\<^sub>B) \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B (\<cdot>\<^sub>C) (\<star>\<^sub>C) \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C F \<Phi>\<^sub>F G \<Phi>\<^sub>G \<tau>\<^sub>0 \<tau>\<^sub>1"
anonymous_sds_axioms_def: fixes agents :: "'a set"   and alts :: "'b set"   and sds :: "('a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'b pmf" shows "anonymous_sds_axioms agents alts sds \<equiv> \<forall>\<pi> R. \<pi> permutes agents \<longrightarrow> pref_profile_wf agents alts R \<longrightarrow> sds (R \<circ> \<pi>) = sds R"
gt_imp_vars: fixes ground_heads_var :: "'a \<Rightarrow> 'b set"   and gt_sym :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and \<epsilon> :: "nat"   and \<delta> :: "nat"   and extf :: "'b \<Rightarrow> (('b, 'a) tm \<Rightarrow> ('b, 'a) tm \<Rightarrow> bool) \<Rightarrow> ('b, 'a) tm list \<Rightarrow> ('b, 'a) tm list \<Rightarrow> bool"   and arity_sym :: "'b \<Rightarrow> enat"   and arity_var :: "'a \<Rightarrow> enat"   and wt_sym :: "'b \<Rightarrow> hmultiset"   and coef_sym :: "'b \<Rightarrow> nat \<Rightarrow> hmultiset"   and t :: "('b, 'a) tm"   and s :: "('b, 'a) tm" assumes "tkbo_coefs ground_heads_var gt_sym \<epsilon> \<delta> extf arity_sym arity_var wt_sym coef_sym"   and "tkbo_coefs.gt ground_heads_var gt_sym \<delta> extf arity_sym wt_sym coef_sym t s" shows "vars s \<subseteq> vars t"
pair_list_sig_of_pair: fixes bs :: "('t \<Rightarrow>\<^sub>0 'b) list"   and ss :: "'t list"   and ps :: "(('t \<Rightarrow>\<^sub>0 'b) \<times> ('t \<Rightarrow>\<^sub>0 'b) + nat) list"   and p :: "('t \<Rightarrow>\<^sub>0 'b) \<times> ('t \<Rightarrow>\<^sub>0 'b) + nat" assumes "rb_aux_inv (bs, ss, ps)"   and "p \<in> set ps" shows "sig_of_pair p = lt (poly_of_pair p)"
real_arch_simple: fixes x :: "'a" shows "\<exists>n. x \<le> of_nat n"
1: shows "prod_list (length As # ds) = length (concat (map vec As))"
termFSbMorph_def: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'c"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'd"   and MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd, 'e) Recursion.model_scheme" shows "termFSbMorph h hA MOD \<equiv> presWlsAll h hA MOD \<and> presCons h hA MOD \<and> presFreshAll h hA MOD \<and> presSubstAll h hA MOD"
NTMap_def: shows "NTMap = []\<^sub>\<circ>"
zero_le_even_power: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and n :: "nat"   and a :: "'a" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one"   and "even n" shows "less_eq zero (power.power one times a n)"
app_Some: fixes i :: "instr"   and P :: "'b prog"   and mxs :: "nat"   and T\<^sub>r :: "ty"   and pc :: "nat"   and mpc :: "nat"   and xt :: "ex_table"   and \<tau> :: "ty list \<times> ty err list" shows "app i P mxs T\<^sub>r pc mpc xt \<lfloor>\<tau>\<rfloor> = (app\<^sub>i (i, P, pc, mxs, T\<^sub>r, \<tau>) \<and> xcpt_app i P pc mxs xt \<tau> \<and> (\<forall>(pc', s')\<in>set (eff i P pc xt \<lfloor>\<tau>\<rfloor>). pc' < mpc))"
disc_29: fixes x51 :: "'b rexp"   and x52 :: "'b rexp" shows "is_Times (Times x51 x52)"
rotate_5: shows "round_robin_invar (t' # queue, n0) (dom (thr s))"
summandsZero': fixes P :: "pi" assumes "summands P = {}"   and "uhnf P" shows "P = \<zero>"
exec_Normal_elim_cases_8: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c1 :: "('e, 'd, 'f) com"   and c2 :: "('e, 'd, 'f) com"   and s :: "'e"   and t :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile> \<langle>Seq c1 c2,Normal s\<rangle> \<Rightarrow> t" obtains s' :: "('e, 'f) xstate" where "\<Gamma>\<turnstile> \<langle>c1,Normal s\<rangle> \<Rightarrow> s'" and "\<Gamma>\<turnstile> \<langle>c2,s'\<rangle> \<Rightarrow> t"
freeword_funlift_uminus_Abs_freeletter: fixes f :: "'b \<Rightarrow> 'a"   and s :: "'b" shows "freeword_funlift f (Abs_freeword [pairfalse s]) = - f s"
normalize_prod_mset: fixes A :: "'a multiset" shows "normalize (\<Prod>\<^sub># A) = \<Prod>\<^sub># (image_mset normalize A)"
suminf_cong: fixes f :: "nat \<Rightarrow> 'b"   and g :: "nat \<Rightarrow> 'b" assumes "\<And>n. f n = g n" shows "suminf f = suminf g"
Alim: shows "(\<lambda>n. u n / real n) \<longlonglongrightarrow> A"
adhCE: fixes x :: "'a"   and A :: "'a set"   and U :: "'a set" assumes "x adh A" obtains "U \<notin> nhds x"   | "U \<inter> A \<noteq> {}"
vec_lambda_beta: fixes g :: "'d \<Rightarrow> 'c"   and i :: "'d" shows "vec_lambda g $ i = g i"
llcp_def: fixes xs :: "'c llist"   and ys :: "'c llist" shows "llcp xs ys = enat_unfold (\<lambda>(xs, ys). lnull xs \<or> lnull ys \<or> lhd xs \<noteq> lhd ys) (map_prod ltl ltl) (xs, ys)"
no_tvsT_imp_tsubsT_unchanged: fixes T :: "typ"   and \<rho> :: "variable \<Rightarrow> String.literal set \<Rightarrow> typ" assumes "tvsT T = {}" shows "tsubstT T \<rho> = T"
mono_assert_fun: fixes p :: "'b" shows "mono (assert_fun p)"
pinf_6: fixes t :: "'c" shows "\<exists>z. \<forall>x>z. (x \<le> t) = False"
CR_iff_meet_subset_join: fixes r :: "('b \<times> 'b) set" shows "CR r = (r\<^sup>\<up> \<subseteq> r\<^sup>\<down>)"
cases: fixes a1 :: "'d set"   and a2 :: "'d set"   and a3 :: "('d, 'e) ta_rule \<Rightarrow> nat option"   and a4 :: "'d set"   and a5 :: "'d set"   and a6 :: "('d, 'e) ta_rule \<Rightarrow> nat option"   and \<delta> :: "('d, 'e) ta_rule set" assumes "((a1, a2, a3), a4, a5, a6) \<in> br'_step \<delta>" obtains q :: "'d" and W :: "'d set" and Q' :: "'d set" and Q :: "'d set" and rcm :: "('d, 'e) ta_rule \<Rightarrow> nat option" and W' :: "'d set" and rcm' :: "('d, 'e) ta_rule \<Rightarrow> nat option" where "a1 = Q" and "a2 = W" and "a3 = rcm" and "a4 = Q'" and "a5 = W'" and "a6 = rcm'" and "q \<in> W" and "Q' = Q \<union> {lhs r |r. r \<in> \<delta> \<and> q \<in> set (rhsq r) \<and> the (rcm r) \<le> 1}" and "W' = W - {q} \<union> ({lhs r |r. r \<in> \<delta> \<and> q \<in> set (rhsq r) \<and> the (rcm r) \<le> 1} - Q)" and "\<And>r. r \<in> \<delta> \<Longrightarrow> rcm' r = (if q \<in> set (rhsq r) then Some (the (rcm r) - 1) else rcm r)"
maxBlk_allInput: fixes s :: "state"   and p :: "Proc" assumes "Inv2a s"   and "maxBlk s p \<in> nonInitBlks s p" shows "inp (maxBlk s p) \<in> allInput s"
LeftDerives1_implies_leftderives1: fixes u :: "symbol list"   and i :: "nat"   and r :: "symbol \<times> symbol list"   and v :: "symbol list" assumes "LeftDerives1 u i r v" shows "leftderives1 u v"
pair_in_listE: fixes ps :: "(bool \<times> (('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) \<times> ('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list"   and i :: "nat"   and j :: "nat" assumes "pair_in_list ps i j" obtains p :: "'e \<Rightarrow>\<^sub>0 'f" and q :: "'e \<Rightarrow>\<^sub>0 'f" and a :: "'g" and b :: "'g" where "((p, i, a), q, j, b) \<in>\<^sub>p snd ` set ps"
tl_rev_take: fixes ok :: "nat"   and f :: "'b list"   and x :: "'b" assumes "0 < ok"   and "ok \<le> length f" shows "rev (take ok (x # f)) = tl (rev (take ok f)) @ [x]"
push_bit_eq_mult: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bit :: "'c \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and drop_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and take_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and n :: "nat"   and a :: "'c" assumes "class.semiring_bit_shifts plus minus zero times one divide modulo bit push_bit drop_bit take_bit" shows "push_bit n a = times a (power.power one times (numeral.numeral one plus (num.Bit0 num.One)) n)"
c_pair_mono2: fixes y1 :: "nat"   and y2 :: "nat"   and x :: "nat" assumes "y1 \<le> y2" shows "c_pair x y1 \<le> c_pair x y2"
SUP_const: fixes Inf :: "'g set \<Rightarrow> 'g"   and Sup :: "'g set \<Rightarrow> 'g"   and inf :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and sup :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and bot :: "'g"   and top :: "'g"   and A :: "'h set"   and f :: "'g" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "A \<noteq> {}" shows "Sup ((\<lambda>i. f) ` A) = f"
gap_effect_def: fixes l :: "'b \<Rightarrow> nat"   and k :: "nat" shows "gap_effect l k \<equiv> \<lambda>v. if k < l v \<and> l v < card V then card V + 1 else l v"
classes_changed_cons_eq: fixes t :: "'b cdecl"   and P :: "'b prog"   and P' :: "'b prog" shows "classes_changed (t # P) P' = classes_changed P P' - {fst t} \<union> (if class_changed [t] P' (fst t) then {fst t} else {})"
simps_3: fixes u :: "'c local.term"   and v :: "'c local.term"   and w :: "'c local.term"   and x :: "'c local.term" shows "(u \<^bold>\<star> v) \<^bold>\<lfloor>\<^bold>\<cdot>\<^bold>\<rfloor> (w \<^bold>\<star> x) = u \<^bold>\<lfloor>\<^bold>\<cdot>\<^bold>\<rfloor> w \<^bold>\<star> v \<^bold>\<lfloor>\<^bold>\<cdot>\<^bold>\<rfloor> x"
parabolic_isometryD_1: fixes f :: "'b \<Rightarrow> 'b" assumes "parabolic_isometry f" shows "isometry f"
Sup_le_iff: fixes A :: "('d \<times> 'e) set"   and b :: "'d \<times> 'e" shows "less_eq_prod (Sup_prod A) b = (\<forall>a\<in>A. less_eq_prod a b)"
indspace_def: fixes G :: "'g set"   and FH :: "('f, 'g) aezfun set"   and V :: "'v set" shows "BaseFGMod.indspace G FH V = addfunset FH V \<inter> BaseFGMod.smultfunset G FH"
LSCategory_axioms_def: fixes C :: "('d, 'e, 'f) LSCategory_scheme" shows "LSCategory_axioms C \<equiv> (\<forall>x y. x \<in> mor\<^bsub>C\<^esub> \<longrightarrow> y \<in> mor\<^bsub>C\<^esub> \<longrightarrow> m2z\<^bsub>C\<^esub>x = m2z\<^bsub>C\<^esub>y \<longrightarrow> x = y) \<and> (\<forall>X Y. X \<in> obj\<^bsub>C\<^esub> \<longrightarrow> Y \<in> obj\<^bsub>C\<^esub> \<longrightarrow> HOMCollection C X Y \<in> range explode) \<and> mor2ZF C \<in> extensional (mor\<^bsub>C\<^esub>)"
scale_zero_left: fixes x :: "'a" shows "0 *\<^sub>R x = (0::'a)"
update_by_pair_less: fixes v :: "'c"   and xs :: "('a \<times> 'c) list"   and k :: "'a" assumes "v \<noteq> (0::'c)"   and "xs = [] \<or> comp k (fst (hd xs)) = Lt" shows "update_by_pair (k, v) xs = (k, v) # xs"
elims_2: fixes x :: "'a ifex"   and xa :: "'a \<Rightarrow> bool" assumes "val_ifex x xa" obtains s :: "'a \<Rightarrow> bool" where "x = Trueif" and "xa = s__"   | n :: "'a" and t1 :: "'a ifex" and t2 :: "'a ifex" and s :: "'a \<Rightarrow> bool" where "x = IF n t1 t2" and "xa = s" and "if s n then val_ifex t1 s else val_ifex t2 s"
lower_character: fixes \<chi> :: "'a \<Rightarrow> complex" assumes "character (G\<lparr>carrier := adjoin G H a\<rparr>) \<chi>" shows "character (G\<lparr>carrier := H\<rparr>) (\<lambda>x. if x \<in> H then \<chi> x else 0)"
interval_union_commute: fixes A :: "'a interval"   and B :: "'a interval" shows "sup A B = sup B A"
M'_1: shows "\<forall>u. u \<turnstile>\<^bsub>v,n\<^esub> \<lambda>i' j'. if i' = 0 \<and> j' = v c2 then Le r else if i' = v c2 \<and> j' = 0 then Le (- r) else M i' j' \<longrightarrow> u \<turnstile>\<^bsub>v,n\<^esub> M"
normalize_dst_ips_def: shows "normalize_dst_ips = normalize_primitive_extract (is_Dst, dst_sel) Dst ipt_iprange_compress"
arev_neq: fixes G :: "('c, 'd) pre_digraph"   and arev :: "'d \<Rightarrow> 'd"   and a :: "'d" assumes "bidirected_digraph G arev"   and "a \<in> arcs G" shows "arev a \<noteq> a"
one_dim_iso_idem: fixes x :: "'c" shows "one_dim_iso (one_dim_iso x) = one_dim_iso x"
cod_src: fixes \<mu> :: "'c" assumes "C.arr \<mu>" shows "C.cod (src\<^sub>C \<mu>) = src\<^sub>C \<mu>"
assms: shows "homotopic_with_canon (\<lambda>h. True) S (sphere 0 1) f (\<lambda>t. a)"
fds_converges_mult_const_left: fixes f :: "'b fds"   and s :: "'b"   and c :: "'b" assumes "fds_converges f s" shows "fds_converges (fds_const c * f) s"