hcju/isabelleps huggingface dataset:
contI: fixes orda :: "'h \<Rightarrow> 'h \<Rightarrow> bool"   and f :: "'h \<Rightarrow> 'i"   and luba :: "'h set \<Rightarrow> 'h"   and lubb :: "'i set \<Rightarrow> 'i"   and ordb :: "'i \<Rightarrow> 'i \<Rightarrow> bool" assumes "\<And>Y. \<lbrakk>Complete_Partial_Order.chain orda Y; Y \<noteq> {}\<rbrakk> \<Longrightarrow> f (luba Y) = lubb (f ` Y)" shows "cont luba orda lubb ordb f"
range_varsE: fixes x :: "'a"   and \<sigma> :: "'a \<Rightarrow> ('b, 'a) Term.term" assumes "x \<in> range_vars \<sigma>" obtains t :: "('b, 'a) Term.term" where "x \<in> vars_term t" and "t \<in> subst_range \<sigma>"
Inv_Inv: fixes V :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and src :: "'a \<Rightarrow> 'a"   and trg :: "'a \<Rightarrow> 'a"   and t :: "'a local.term" assumes "bicategorical_language V src trg"   and "bicategorical_language.Can V src trg t" shows "bicategorical_language.Inv V (bicategorical_language.Inv V t) = t"
maxBalInp_def: fixes s :: "state"   and b :: "nat"   and v :: "InputsOrNi" shows "maxBalInp s b v = (\<forall>bk\<in>allBlocks s. b \<le> bal bk \<longrightarrow> inp bk = v)"
comp_tfr\<^sub>s\<^sub>s\<^sub>t_def: fixes arity :: "'a \<Rightarrow> nat"   and Ana :: "('a, 'b \<times> nat) Term.term \<Rightarrow> ('a, 'b \<times> nat) Term.term list \<times> 'c"   and \<Gamma> :: "('a, 'b \<times> nat) Term.term \<Rightarrow> ('d, 'e) Term.term"   and pair_fun :: "'a"   and M :: "('a, 'b \<times> nat) Term.term list"   and S :: "('a, 'b \<times> nat) stateful_strand_step list" shows "comp_tfr\<^sub>s\<^sub>s\<^sub>t arity Ana \<Gamma> pair_fun M S \<equiv> list_all (comp_tfr\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<Gamma> pair_fun) S \<and> list_all (wf\<^sub>t\<^sub>r\<^sub>m' arity) (trms_list\<^sub>s\<^sub>s\<^sub>t S) \<and> has_all_wt_instances_of \<Gamma> (trms\<^sub>s\<^sub>s\<^sub>t S \<union> pair' pair_fun ` setops\<^sub>s\<^sub>s\<^sub>t S) (set M) \<and> comp_tfr\<^sub>s\<^sub>e\<^sub>t arity Ana \<Gamma> M"
peirce: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(x ; y \<cdot> z\<^sup>T = 0) = (y ; z \<cdot> x\<^sup>T = 0)"
mod_mult_cong_right: 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 division_segment :: "'a \<Rightarrow> 'a"   and c :: "'a"   and a :: "'a"   and b :: "'a"   and d :: "'a" assumes "class.unique_euclidean_semiring divide plus minus zero times one modulo euclidean_size division_segment" shows "unique_euclidean_semiring.cong modulo (modulo c (times a b)) d a = unique_euclidean_semiring.cong modulo c d a"
monad_altc_optionT: fixes return :: "'a option \<Rightarrow> 'b"   and bind :: "'b \<Rightarrow> ('a option \<Rightarrow> 'b) \<Rightarrow> 'b"   and altc :: "'c cset \<Rightarrow> ('c \<Rightarrow> 'b) \<Rightarrow> 'b" assumes "monad_altc return bind altc" shows "monad_altc (return_option return) (bind_option return bind) (altc_option altc)"
simps_61: fixes x7 :: "jump"   and x31 :: "jump"   and x32 :: "stmt" shows "Jmp x7 \<noteq> x31\<bullet> x32"
center_on_geodesic_1: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "center x y z \<in> {x--y}"
summable_comparison_test: fixes g :: "nat \<Rightarrow> real" assumes "\<exists>N. \<forall>n\<ge>N. norm (f n) \<le> g n"   and "summable g" shows "summable f"
is_option_of_subset_is_Elem_of: shows "is_option_of \<subseteq> is_Elem_of\<^sup>+"
dom_eqI: fixes a :: "'b"   and f :: "'b" assumes "J0.ide a"   and "J0.seq f a" shows "J0.dom f = a"
emeasure_eq_measure: fixes i :: "'i"   and A :: "'b set" shows "emeasure (distr M (M' i) (X i)) A = ennreal (D'.prob i A)"
l3_inv10I: fixes implem :: "chan \<Rightarrow> msg"   and x :: "l3_state" assumes "sklvl3 implem"   and "secret x \<subseteq> payload" shows "x \<in> l3_inv10"
NoCycles_def: fixes F :: "('a, 'b, 'c) seqauto set"   and G :: "'a \<Rightarrow> ('a, 'b, 'c) seqauto set option" shows "NoCycles F G = (\<forall>S\<in>Pow (\<Union> (States ` F)). S \<noteq> {} \<longrightarrow> (\<exists>s\<in>S. S \<inter> \<Union> (States ` the (G s)) = {}))"
simps_10: fixes t :: "'a local.term"   and u :: "'a local.term"   and v :: "'a local.term" shows "Arr \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[t, u, v\<^bold>] = (Arr t \<and> Arr u \<and> Arr v \<and> Src t = Trg u \<and> Src u = Trg v)"
elims: fixes w :: "'a \<times> nat \<Rightarrow> nat"   and w0 :: "nat"   and scf :: "'a \<times> nat \<Rightarrow> nat \<Rightarrow> nat"   and least :: "'a \<Rightarrow> bool"   and pr_strict :: "'a \<times> nat \<Rightarrow> 'a \<times> nat \<Rightarrow> bool"   and pr_weak :: "'a \<times> nat \<Rightarrow> 'a \<times> nat \<Rightarrow> bool"   and x :: "('a, 'b) Term.term"   and xa :: "('a, 'b) Term.term"   and y :: "bool \<times> bool" assumes "kbo.kbo w w0 scf least pr_strict pr_weak x xa = y" obtains s :: "('a, 'b) Term.term" and t :: "('a, 'b) Term.term" where "x = s" and "xa = t" and "y = (if vars_term_ms (scf_term scf t) \<subseteq># vars_term_ms (scf_term scf s) \<and> weight_fun.weight w w0 scf t \<le> weight_fun.weight w w0 scf s then if weight_fun.weight w w0 scf t < weight_fun.weight w w0 scf s then (True, True) else case s of Var y \<Rightarrow> (False, case t of Var x \<Rightarrow> x = y | Fun g ts \<Rightarrow> ts = [] \<and> least g) | Fun f ss \<Rightarrow> case t of Var x \<Rightarrow> (True, True) | Fun g ts \<Rightarrow> if pr_strict (f, length ss) (g, length ts) then (True, True) else if pr_weak (f, length ss) (g, length ts) then lex_ext_unbounded (kbo.kbo w w0 scf least pr_strict pr_weak) ss ts else (False, False) else (False, False))"
Pbij_injective: fixes \<beta> :: "(Location \<times> Location) set" assumes "\<beta> \<in> Pbij" shows "\<forall>l l1 l2. (l1, l) \<in> \<beta> \<longrightarrow> (l2, l) \<in> \<beta> \<longrightarrow> l1 = l2"
cast_DEFL: fixes below :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and liftemb :: "'a\<^sub>\<bottom> \<rightarrow> udom\<^sub>\<bottom>"   and liftprj :: "udom\<^sub>\<bottom> \<rightarrow> 'a\<^sub>\<bottom>"   and liftdefl :: "'a itself \<Rightarrow> (udom\<^sub>\<bottom>) defl"   and emb :: "'a \<rightarrow> udom"   and prj :: "udom \<rightarrow> 'a"   and defl :: "'a itself \<Rightarrow> udom defl" assumes "class.domain below liftemb liftprj liftdefl emb prj defl" shows "cast\<cdot>(defl TYPE('a)) = emb oo prj"
h_le_f: fixes e :: "'v \<times> 'v" shows "h e \<le> f e"
rep_eq: fixes x :: "('a \<times> 'b) list" shows "fmlookup (fmap_of_list x) = map_of x"
comp_cod_arr: fixes f :: "'a"   and b :: "'a" assumes "L'.C.arr f"   and "L'.C.cod f = b" shows "local.Left a' b f = f"
Nmlize_Trg_2: fixes t :: "'a E.term" assumes "E.Arr t" shows "\<^bold>\<lfloor>E.Trg t\<^bold>\<rfloor> = E.Trg t"
sensors_ge: fixes e :: "cars"   and ts :: "traffic"   and c :: "cars" shows "0 < sensors e ts c"
reads_subset: fixes S :: "('l \<Rightarrow> 'l \<Rightarrow> bool) set"   and f :: "('l, 'm, 'n) prog"   and h :: "'l"   and h' :: "'l"   and S' :: "('l \<Rightarrow> 'l \<Rightarrow> bool) set" assumes "reads S f h h'"   and "\<forall>P\<in>S' - S. reflp P \<and> transp P"   and "S \<subseteq> S'" shows "reads S' f h h'"
absorb1: fixes b :: "'b"   and a :: "'b" assumes "b \<le> a" shows "sup a b = a"
emeasure_eq_measure: fixes A :: "'b set" shows "emeasure (distr M S X) A = ennreal (X.prob A)"
update_simps_2: fixes e :: "('f, 'g, 'h) env"   and env :: "('f, 'g, 'h) env" shows "update [] (Some e) env = e"
RPDi_Suc_alt: fixes i :: "nat"   and r :: "mregex" shows "RPDi (Suc i) r = \<Union> (RPD ` RPDi i r)"
Iyy: shows "Gromov_product_at py y' y = M"
assms_3: shows "\<I> \<turnstile>res res2 \<surd>"
weak_ranking_of_weak_ranking: fixes xs :: "'c set list" assumes "is_finite_weak_ranking xs" shows "weak_ranking (of_weak_ranking xs) = xs"
assms_2: shows "(g \<longlongrightarrow> 0) net"
convergent_of_real: fixes f :: "nat \<Rightarrow> real" assumes "convergent f" shows "convergent (\<lambda>n. of_real (f n))"
assms_3: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "f \<in> L F g" shows "(\<lambda>x. t x (f x)) \<in> L F (\<lambda>x. t x (g x))"
assms_1: shows "equivalent_on \<sigma> \<eta> (vars_of_eq e) I"
frp_cond_def: shows "frp_cond \<equiv> {(Q, W, \<delta>d). W \<noteq> []}"
fold_set_union: fixes A :: "'v set"   and B :: "'v set"   and X :: "'v \<Rightarrow> int pmf"   and x :: "'m" assumes "finite A"   and "finite B" shows "sample_vars X (A \<union> B) x = sample_vars X B (sample_vars X A x)"
proots_count_cong: fixes s :: "'b set"   and p :: "'b poly"   and q :: "'b poly" assumes "\<forall>x\<in>s. order x p = order x q"   and "p \<noteq> 0"   and "q \<noteq> 0" shows "proots_count p s = proots_count q s"
of_complex_mult: fixes x :: "complex"   and y :: "complex" shows "of_complex (x * y) = of_complex x * of_complex y"
antisym: fixes b :: "'b DBMEntry"   and a :: "'b DBMEntry" assumes "b \<preceq> a"   and "a \<preceq> b" shows "a = b"
list_rel_sv: fixes R :: "('a \<times> 'b) set" assumes "single_valued R" shows "single_valued (\<langle>R\<rangle>list_rel)"
responses_\<I>_empty_iff: fixes \<I> :: "('e, 'd) \<I>"   and out :: "'e" shows "(responses_\<I> \<I> out = {}) = (out \<notin> outs_\<I> \<I>)"
ev_alpha_norm_v: shows "map_vector cmod (\<alpha> *s v) = cmod \<alpha> *s map_vector cmod v"
pmat_extension_def: fixes m :: "'b mat" shows "ps2_P.pmat_extension m = tensor_P m (1\<^sub>m ps2_P.d2)"
OclInt0_def: shows "\<zero> = (\<lambda>_. \<lfloor>\<lfloor>0\<rfloor>\<rfloor>)"
Perp_perm: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Perp Bet Cong A B C D" shows "Tarski_neutral_dimensionless.Perp Bet Cong A B C D \<and> Tarski_neutral_dimensionless.Perp Bet Cong B A C D \<and> Tarski_neutral_dimensionless.Perp Bet Cong A B D C \<and> Tarski_neutral_dimensionless.Perp Bet Cong B A D C \<and> Tarski_neutral_dimensionless.Perp Bet Cong C D A B \<and> Tarski_neutral_dimensionless.Perp Bet Cong C D B A \<and> Tarski_neutral_dimensionless.Perp Bet Cong D C A B \<and> Tarski_neutral_dimensionless.Perp Bet Cong D C B A"
upper_density_all_times: fixes d :: "real" assumes "0 < d" shows "\<exists>c>0. ennreal (1 - d) < emeasure M {x \<in> space M. upper_asymptotic_density {n. \<exists>l\<in>{1..n}. u n x - u (n - l) x \<le> - c * real l} < d}"
SN_on_O_comm: fixes R :: "('b \<times> 'c) set"   and S :: "('c \<times> 'b) set"   and X :: "'c set" assumes "SN_on (R O S) (S `` X)" shows "SN_on (S O R) X"
intro_spec_refine_iff: fixes X :: "'b set"   and f :: "'b \<Rightarrow> 'a nres"   and R :: "('a \<times> 'c) set"   and M :: "'c nres" shows "(RES X \<bind> f \<le> \<Down> R M) = (\<forall>x\<in>X. f x \<le> \<Down> R M)"
natUnion_decompose: fixes p :: "'b"   and S :: "nat \<Rightarrow> 'b set"   and P :: "'b \<Rightarrow> bool" assumes "p \<in> natUnion S"   and "\<And>n p. p \<in> S n \<Longrightarrow> P p" shows "P p"
node_total_order_trans: fixes e1 :: "'evt"   and i :: "nat"   and e2 :: "'evt"   and e3 :: "'evt" assumes "e1 \<sqsubset>\<^sup>i e2"   and "e2 \<sqsubset>\<^sup>i e3" shows "e1 \<sqsubset>\<^sup>i e3"
Done_Done: fixes a :: "'d option"   and a' :: "'d option" assumes "option_ord a a'" shows "resumption_ord (Resumption.resumption.Done a) (Resumption.resumption.Done a')"
prems: shows "anybranch (Branching T\<^sub>1 T\<^sub>2) (\<lambda>p. red (ds @ p))"
natural_transformation_axioms: shows "natural_transformation (\<cdot>\<^sub>J) (\<cdot>) \<chi>'.A.map D (\<chi>.D.cones_map g \<chi>)"
bind_split_asm: fixes P :: "'b option \<Rightarrow> bool"   and m :: "'c option"   and f :: "'c \<Rightarrow> 'b option" shows "P (Option.bind m f) = (\<not> (m = None \<and> \<not> P None \<or> (\<exists>x. m = Some x \<and> \<not> P (f x))))"
exp_2dist_formula: fixes a :: "(real, 3) vec"   and b :: "(real, 3) vec" assumes "a \<noteq> 0"   and "b \<noteq> 0"   and "proj2_abs a \<in> hyp2"   and "proj2_abs b \<in> hyp2" shows "exp_2dist (proj2_abs a) (proj2_abs b) = (a \<bullet> (M *v b) + sqrt (quarter_discrim a b)) / (a \<bullet> (M *v b) - sqrt (quarter_discrim a b)) \<or> exp_2dist (proj2_abs a) (proj2_abs b) = (a \<bullet> (M *v b) - sqrt (quarter_discrim a b)) / (a \<bullet> (M *v b) + sqrt (quarter_discrim a b))"
sees_field_i_i_i_o_o_oE: fixes x :: "'a prog"   and xa :: "String.literal"   and xb :: "String.literal"   and xc :: "ty"   and xd :: "fmod"   and xe :: "String.literal" assumes "pred.eval (sees_field_i_i_i_o_o_o x xa xb) (xc, xd, xe)" obtains "x \<turnstile> xa sees xb:xc (xd) in xe"
cindex_poly_ubd_0: fixes p :: "real poly"   and q :: "real poly" assumes "p = 0 \<or> q = 0" shows "cindex_poly_ubd q p = 0"
round_num_denom_cost_def: fixes n :: "int"   and d :: "int" shows "round_num_denom_cost n d = ((2 * n + d) div (2 * d), 4)"
G12_def: shows "G12 \<equiv> {x \<in> U. g1 x - g2 x = (0::'c)}"
eval_simps_1: fixes c :: "real"   and x :: "real" shows "eval (const_expansion c) x = c"
simps_1: fixes x :: "name"   and a :: "coname"   and y :: "name"   and c :: "coname"   and P :: "trm" shows "Ax x a{y:=<c>.P} = (if x = y then Cut <c>.P y.Ax y a else Ax x a)"
vec_iff1: fixes R :: "('a \<times> 'a set) set" assumes "\<forall>a. (\<exists>A. (a, A) \<in> R) \<longrightarrow> (\<forall>A. (a, A) \<in> R)" shows "R \<cdot> 1\<^sub>\<pi> \<parallel> Multirelations.U = R"
lookup_tensor_vec: fixes "is" :: "nat list"   and ds :: "nat list"   and e :: "nat list \<Rightarrow> 'b" assumes "is \<lhd> ds" shows "lookup_base ds (tensor_vec_from_lookup ds e) is = e is"
le: fixes x :: "'a" assumes "x \<in> S - {least S}" shows "least S \<prec> x"
wt_int: fixes \<xi> :: "Preliminaries.var \<Rightarrow> 'univ"   and T :: "'fsym trm" assumes "wtE \<xi>"   and "wt T" shows "intT (tpOf T) (int \<xi> T)"
gf: shows "homotopic_with (\<lambda>h. h ` S \<subseteq> S) X X (g \<circ> f) id"
assms: shows "(\<lambda>x. g (f x)) \<in> M \<rightarrow>\<^sub>M N"
ex_def: shows "ex \<equiv> x - aform_val e (fst X)"
GEN_ALGO_tag_def: fixes P :: "'f" shows "GEN_ALGO_tag P \<equiv> P"
eq: fixes x :: "real" assumes "1 \<le> x" shows "\<psi> x - x = - sum_upto (dirichlet_prod \<mu> f) x - frac x - 2 * C"
reach_in: fixes G :: "('c, 'd) pre_digraph"   and S :: "'c set"   and u :: "'c"   and v :: "'c" assumes "max_reachable_set G S"   and "u \<in> S"   and "v \<in> S" shows "u \<rightarrow>\<^sup>*\<^bsub>G\<^esub> v"
HcompNml_Nml_Obj: fixes t :: "'a local.term"   and u :: "'a local.term" assumes "Nml t"   and "Obj u"   and "Src t = Trg u" shows "t \<^bold>\<lfloor>\<^bold>\<star>\<^bold>\<rfloor> u = t"
rep_eq: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and x :: "int" assumes "class.ring_1 one times plus zero minus uminus" shows "ring_1.of_int one plus zero minus x = (case Rep_Integ x of (i, j) \<Rightarrow> minus (semiring_1.of_nat one plus zero i) (semiring_1.of_nat one plus zero j))"
is_ground_cls_mono: fixes C :: "'a literal multiset"   and D :: "'a literal multiset" assumes "C \<subseteq># D"   and "is_ground_cls D" shows "is_ground_cls C"
subset_Collect_conv: fixes S :: "'b set"   and P :: "'b \<Rightarrow> bool" shows "(S \<subseteq> Collect P) = (\<forall>x\<in>S. P x)"
coeffs: shows "(1 + fps_X ^ p) ^ N $ k = (\<Sum>i = 0..k. int (N choose i) * fps_X ^ (p * i) $ k)"
add: fixes f :: "'f \<Rightarrow> 'g"   and x :: "'f"   and y :: "'f" assumes "Modules.additive f" shows "f (x + y) = f x + f y"
natfun_bigo_iff_Bseq: fixes f :: "nat \<Rightarrow> 'a" shows "(f \<in> O(\<lambda>_. 1::'a)) = Bseq f"
VVpstar: fixes v :: "'a"   and p :: "Player"   and W :: "'a set" assumes "v \<in> VV p**"   and "\<not> deadend v"   and "\<forall>w. v \<rightarrow> w \<longrightarrow> w \<in> attractor_inductive p W" shows "v \<in> attractor_inductive p W"
bin_rest_numeral_simps_2: shows "1 div 2 = 0"
Units_eq: shows "Units (homology_group p X) = carrier (homology_group p X)"
category_with_pullbacks_axioms: shows "category_with_pullbacks (\<cdot>)"
exec_strip_guards_to_exec: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and F :: "'f set"   and c :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate"   and t :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile> \<langle>strip_guards F c,s\<rangle> \<Rightarrow> t" shows "\<exists>t'. \<Gamma>\<turnstile> \<langle>c,s\<rangle> \<Rightarrow> t' \<and> (isFault t \<longrightarrow> isFault t') \<and> (t' \<in> Fault ` (- F) \<longrightarrow> t' = t) \<and> (\<not> isFault t' \<longrightarrow> t' = t)"
b: shows "\<chi> \<in> K \<^bold>* \<phi>"
stream_all_iff: fixes P :: "'d \<Rightarrow> bool"   and s :: "'d stream" shows "stream_all P s = Ball (sset s) P"
set_mult_mOp_closed: fixes A :: "'a set"   and B :: "'a set"   and x :: "'a" assumes "A \<subseteq> carrier R"   and "ideal R B"   and "x \<in> set_mult R A B" shows "-\<^sub>a x \<in> set_mult R A B"
power_diff: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and a :: "'b"   and n :: "nat"   and m :: "nat" assumes "class.semidom_divide divide plus minus zero times one"   and "a \<noteq> zero"   and "n \<le> m" shows "power.power one times a (m - n) = divide (power.power one times a m) (power.power one times a n)"
func_pre: fixes f :: "nat \<Rightarrow> 'h"   and n :: "nat"   and A :: "'h set" assumes "f \<in> {j. j \<le> Suc n} \<rightarrow> A" shows "f \<in> {j. j \<le> n} \<rightarrow> A"
refl: fixes x :: "'a" assumes "x \<in> A" shows "\<not> x \<bowtie> x"
schur: shows "schur_decomposition A3 es = (B, P', Q')"
f: shows "(0::'a) \<notin> f ` \<omega>_limit_set x"
trans: fixes b :: "('d, 'e) Term.term"   and a :: "('d, 'e) Term.term"   and c :: "('d, 'e) Term.term" assumes "b \<sqsubseteq> a"   and "c \<sqsubseteq> b" shows "c \<sqsubseteq> a"
ctr_2: fixes p :: "'a \<Rightarrow> bool"   and a :: "'a"   and g1 :: "'a \<Rightarrow> 'c"   and g21 :: "'a \<Rightarrow> 'b"   and g22 :: "'a \<Rightarrow> 'a" assumes "\<not> p a" shows "unfold_tllist p g1 g21 g22 a = TCons (g21 a) (unfold_tllist p g1 g21 g22 (g22 a))"
support_and: fixes P :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and Q :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool" shows "support (\<lambda>l s. P l s \<and> Q l s) \<subseteq> support P \<union> support Q"
control_state_ext_def: fixes break_status :: "bool"   and return_status :: "bool"   and more :: "'a" shows "\<lparr>break_status = break_status, return_status = return_status, \<dots> = more\<rparr> \<equiv> Record.iso_tuple_cons control_state_ext_Tuple_Iso (Record.iso_tuple_cons Record.tuple_iso_tuple break_status return_status) more"
assms_7: shows "host_apply s (t1s _> t2s) f vcs hs = Some (s', vcs')"
IH: fixes i :: "nat" assumes "infinite S"   and "i \<le> pick S n" shows "card {a \<in> S. a < i} \<le> n"
lock_ok_las'D: fixes ls :: "('c, 'd) locks"   and t :: "'d"   and las :: "'c lock_actions"   and l :: "'c" assumes "lock_ok_las' ls t las" shows "lock_actions_ok' (ls $ l) t (las $ l)"
factors_equal: fixes a :: "'c"   and b :: "'c"   and c :: "'c"   and d :: "'c" assumes "a = b"   and "c = d" shows "a + c = b + d"
assigns_if_const_b_simp: fixes e :: "expr"   and b :: "bool" assumes "constVal e = Some (Bool b)" shows "assigns_if b e = {}"
Always_reachable: fixes F :: "'b program" shows "F \<in> Always (reachable F)"
distinct_5: shows "cospan_shape.Null \<noteq> cospan_shape.TT"
simps: fixes cnf :: "lit uprod set"   and ls :: "lit set" assumes "construct_sa cnf" shows "construct_sa.extend cnf ls = (if vars_of_cnf cnf \<subseteq> var ` ls then ls else let x = SOME x. x \<in> vars_of_cnf cnf - var ` ls in if consistent (ls \<union> (imp_graph cnf)\<^sup>* `` {Pos x}) then construct_sa.extend cnf (ls \<union> (imp_graph cnf)\<^sup>* `` {Pos x}) else construct_sa.extend cnf (ls \<union> (imp_graph cnf)\<^sup>* `` {Neg x}))"
ts_invE: fixes P :: "'f \<Rightarrow> 'g \<Rightarrow> 'h \<Rightarrow> 'i \<Rightarrow> bool"   and I :: "'g \<Rightarrow> 'f option"   and ts :: "'g \<Rightarrow> ('h \<times> 'j \<Rightarrow>f nat) option"   and m :: "'i" assumes "ts_inv P I ts m" obtains "\<forall>t x ln. ts t = \<lfloor>(x, ln)\<rfloor> \<longrightarrow> (\<exists>i. I t = \<lfloor>i\<rfloor> \<and> P i t x m)"
strand_sem_split_3: fixes M :: "('fun, 'var) Term.term set"   and S :: "('fun, 'var) strand_step list"   and S' :: "('fun, 'var) strand_step list"   and \<theta> :: "'var \<Rightarrow> ('fun, 'var) Term.term" assumes "\<lbrakk>M; S @ S'\<rbrakk>\<^sub>d \<theta>" shows "\<lbrakk>M; S\<rbrakk>\<^sub>d \<theta>"
open_interval_def: fixes X :: "'a interval set" shows "open X = (\<forall>x\<in>X. \<exists>A B. open A \<and> open B \<and> lower x \<in> A \<and> upper x \<in> B \<and> Interval ` (A \<times> B) \<subseteq> X)"
PiE_insert_eq: fixes x :: "'c"   and S :: "'c set"   and T :: "'c \<Rightarrow> 'd set" shows "Pi\<^sub>E (insert x S) T = (\<lambda>(y, g). g(x := y)) ` (T x \<times> Pi\<^sub>E S T)"
tabulation_data_def: fixes V :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and H :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<a> :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<i> :: "'c \<Rightarrow> 'c"   and src :: "'c \<Rightarrow> 'c"   and trg :: "'c \<Rightarrow> 'c"   and r :: "'c"   and \<rho> :: "'c"   and f :: "'c"   and g :: "'c" shows "tabulation_data V H \<a> \<i> src trg r \<rho> f g \<equiv> bicategory V H \<a> \<i> src trg \<and> tabulation_data_axioms V H r \<rho> f g"
g_tendsto_1: shows "(g \<longlongrightarrow> 0) (at_left x)"
sg5a: shows "d + (t + (3 + k)) = t + 3 + k + d"
SUP_bot_conv_2: fixes Inf :: "'d set \<Rightarrow> 'd"   and Sup :: "'d set \<Rightarrow> 'd"   and inf :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sup :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and bot :: "'d"   and top :: "'d"   and B :: "'e \<Rightarrow> 'd"   and A :: "'e set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "(bot = Sup (B ` A)) = (\<forall>x\<in>A. B x = bot)"
op_type_alt_simps_9: fixes op :: "numeric_unop"   and \<tau> :: "'a OCL_Types.type"   and \<sigma> :: "'b OCL_Types.type" shows "numeric_unop_type op \<tau> \<sigma> = (op = UMinusOp \<and> \<sigma> = Real[1] \<and> \<tau> = Real[1] \<or> op = UMinusOp \<and> \<sigma> = Integer[1] \<and> \<tau> = UnlimitedNatural[1]\<midarrow>Integer[1] \<or> op = AbsOp \<and> \<sigma> = Real[1] \<and> \<tau> = Real[1] \<or> op = AbsOp \<and> \<sigma> = Integer[1] \<and> \<tau> = UnlimitedNatural[1]\<midarrow>Integer[1] \<or> op = FloorOp \<and> \<sigma> = Integer[1] \<and> \<tau> = UnlimitedNatural[1]\<midarrow>Real[1] \<or> op = RoundOp \<and> \<sigma> = Integer[1] \<and> \<tau> = UnlimitedNatural[1]\<midarrow>Real[1] \<or> op = numeric_unop.ToIntegerOp \<and> \<sigma> = Integer[1] \<and> \<tau> = UnlimitedNatural[1])"
while_option_unfold: fixes b :: "'c \<Rightarrow> bool"   and c :: "'c \<Rightarrow> 'c"   and s :: "'c" shows "while_option b c s = (if b s then while_option b c (c s) else Some s)"
cases: fixes a1 :: "(status \<times> 'x2) \<times> 'm2"   and a2 :: "('l, 't, status \<times> 'x2, 'm2, 'w, 'o action) thread_action"   and a3 :: "(status \<times> 'x2) \<times> 'm2" assumes "r2.init_fin_\<tau>move a1 a2 a3" obtains x :: "'x2" and m :: "'m2" and ta :: "('l, 't, 'x2, 'm2, 'w, 'o) thread_action" and x' :: "'x2" and m' :: "'m2" where "a1 = ((Running, x), m)" and "a2 = convert_TA_initial (convert_obs_initial ta)" and "a3 = ((Running, x'), m')" and "\<tau>move2 (x, m) ta (x', m')"
pairwise_orthogonal_insert: fixes S :: "'b set"   and x :: "'b" assumes "pairwise orthogonal S"   and "\<And>y. y \<in> S \<Longrightarrow> orthogonal x y" shows "pairwise orthogonal (insert x S)"