hcju/isabelleps huggingface dataset:
hcomplex_numeral_hcmod: fixes v :: "num" shows "hcmod (numeral v) = numeral v"
fin_gr_fgsts: shows "lfinite (lmap (\<lambda>St. grounding_of_clss (wrp.N_of_wstate St \<union> wrp.P_of_wstate St \<union> wrp.Q_of_wstate St)) sswSts)"
assms: shows "S \<le> SPEC (\<lambda>x. \<exists>x'. (x, x') \<in> R \<and> \<Phi> x')"
iv: shows "\<phi> 0 \<le> \<phi> 1"
intro: fixes f :: "'b \<Rightarrow> 'c" assumes "linear f"   and "bounded_linear_axioms f" shows "bounded_linear f"
inf_continuous_gfp: fixes F :: "'c \<Rightarrow> 'c" assumes "inf_continuous F" shows "gfp F = (INF i. (F ^^ i) top)"
sum: shows "take (p + q) (take p w \<cdot> w) = take p w \<cdot> take q w"
pgwt_arity: fixes t :: "('fun, 'var) Term.term"   and f :: "'fun"   and T :: "('fun, 'var) Term.term list" assumes "public_ground_wf_term t"   and "Fun f T \<sqsubseteq> t" shows "arity f = length T"
f_Exec_Stream_expand_aggregate_map_take: fixes n :: "nat"   and f :: "'e \<Rightarrow> 'd"   and trans_fun :: "'f message_af \<Rightarrow> 'e \<Rightarrow> 'e"   and xs :: "'f message_af list"   and k :: "nat"   and c :: "'e"   and ag :: "'d list \<Rightarrow> 'd" shows "f_aggregate (map f (f_Exec_Comp_Stream trans_fun (xs \<odot> k) c)) k ag \<down> n = f_aggregate (map f (f_Exec_Comp_Stream trans_fun (xs \<down> n \<odot> k) c)) k ag"
ns_mul_ext_mono: fixes ns :: "('a \<times> 'a) set"   and ns' :: "('a \<times> 'a) set"   and s :: "('a \<times> 'a) set"   and s' :: "('a \<times> 'a) set" assumes "ns \<subseteq> ns'"   and "s \<subseteq> s'" shows "ns_mul_ext ns s \<subseteq> ns_mul_ext ns' s'"
P3: fixes i :: "nat" assumes "k < i" shows "i \<notin> set js"
le_add_same_cancel1: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'d"   and b :: "'d" assumes "class.ordered_ab_semigroup_monoid_add_imp_le plus zero minus less_eq less" shows "less_eq a (plus a b) = less_eq zero b"
take_add: fixes i :: "nat"   and j :: "nat"   and xs :: "'c list" shows "take (i + j) xs = take i xs @ take j (drop i xs)"
F_sup_def: fixes f :: "'s \<Rightarrow> ennreal" shows "F_sup f = (\<lambda>s\<in>S. if s \<in> S2 then 1 else if s \<in> S1 then \<Squnion>D\<in>K s. integral\<^sup>N (measure_pmf D) f else 0)"
Lp_AE_limit: fixes g :: "'b \<Rightarrow> real"   and M :: "'b measure"   and f :: "nat \<Rightarrow> 'b \<Rightarrow> real"   and p :: "ennreal" assumes "g \<in> borel_measurable M"   and "AE x in M. (\<lambda>n. f n x) \<longlonglongrightarrow> g x" shows "eNorm (\<LL> p M) g \<le> liminf (\<lambda>n. eNorm (\<LL> p M) (f n))"
assms: fixes i :: "nat" assumes "i \<in> s" shows "enat i < llength w"
s_pre: fixes p :: "nat"   and n :: "nat"   and base :: "nat \<Rightarrow> nat"   and upd :: "nat \<Rightarrow> nat"   and s :: "(nat \<Rightarrow> nat) set" assumes "kuhn_simplex p n base upd s" shows "s = (\<lambda>i j. if j \<in> upd ` {..<i} then Suc (base j) else base j) ` {..n}"
g_def: shows "g \<equiv> to_fract G * Fraction_Field.Fract (1::'a) r"
axioms_3: fixes step1 :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and step2 :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and final1 :: "'d \<Rightarrow> bool"   and final2 :: "'e \<Rightarrow> bool"   and load1 :: "'f \<Rightarrow> 'd \<Rightarrow> bool"   and load2 :: "'g \<Rightarrow> 'e \<Rightarrow> bool"   and order :: "'h \<Rightarrow> 'h \<Rightarrow> bool"   and match :: "'h \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> bool"   and compile :: "'f \<Rightarrow> 'g option" assumes "compiler step1 step2 final1 final2 load1 load2 order match compile" shows "backward_simulation step1 step2 final1 final2 order match"
f: shows "f = (\<lambda>stp. (steps (Suc (start_of ly as) + 2 * n, la, ra) (ci ly (start_of ly as) (Dec n e), start_of ly as - Suc 0) stp, start_of ly as, n))"
geodesic_quasi_isometric_image: fixes lambda :: "real"   and C :: "real"   and f :: "'c \<Rightarrow> 'd"   and G :: "'c set"   and x :: "'c"   and y :: "'c" assumes "lambda C -quasi_isometry f"   and "geodesic_segment_between G x y" shows "hausdorff_distance (f ` G) {f x--f y} \<le> 92 * lambda\<^sup>2 * (C + deltaG TYPE('d))"
induct: fixes x1 :: "trm"   and x2 :: "trm"   and P :: "trm \<Rightarrow> trm \<Rightarrow> bool" assumes "x1 \<longlonglongrightarrow>\<^sup>* x2"   and "\<And>s. P s s"   and "\<And>s t u. \<lbrakk>s \<longlonglongrightarrow> t; t \<longlonglongrightarrow>\<^sup>* u; P t u\<rbrakk> \<Longrightarrow> P s u" shows "P x1 x2"
set_mr_cos_mem: fixes H :: "'a set"   and m :: "'a" assumes "submodule R M H"   and "m \<in> carrier M" shows "m \<uplus>\<^bsub>M\<^esub> H \<in> set_mr_cos M H"
asset_seskey_shakey: fixes s :: "(agent \<times> msg) set"   and n :: "nat"   and SK :: "nat option \<times> nat set \<times> nat set" assumes "s\<^sub>0 \<Turnstile> s"   and "n \<notin> bad_shakey"   and "(Asset n, SesKey SK) \<in> s" shows "SesKey SK \<notin> spied s"
C: shows "C = - \<i> / 2 \<cdot>\<^sub>m (A - adjoint A)"
ts_ts_os: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "A B TS C D"   and "C D TS A B" shows "A C OS B D"
bind_sample1: fixes p :: "'p pmf"   and f :: "'p \<Rightarrow> 'm"   and g :: "'a \<times> 'w list \<Rightarrow> 'm" shows "bind (sample p f) g = sample p (\<lambda>x. bind (f x) g)"
clock_set_cong: fixes r :: "'a list"   and u :: "'a \<Rightarrow> 'b"   and d :: "'b" assumes "\<forall>c\<in>set r. u c = d" shows "[r\<rightarrow>d]u = u"
is_moebius_similarity': fixes a :: "complex_homo"   and b :: "complex_homo" assumes "a \<noteq> 0\<^sub>h"   and "a \<noteq> \<infinity>\<^sub>h"   and "b \<noteq> \<infinity>\<^sub>h" shows "(\<lambda>z. a *\<^sub>h z +\<^sub>h b) = moebius_pt (moebius_similarity (to_complex a) (to_complex b))"
cases: fixes \<Gamma> :: "('a, 'b) web_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal"   and g :: "'a \<times> 'a \<Rightarrow> ennreal" assumes "trimming \<Gamma> f g" obtains "current \<Gamma> g" and "wave \<Gamma> g" and "g \<le> f" and "\<And>x. \<lbrakk>x \<in> RF\<^sup>\<circ>\<^bsub>\<Gamma>\<^esub> (TER\<^bsub>\<Gamma>\<^esub> f); x \<notin> A \<Gamma>\<rbrakk> \<Longrightarrow> KIR g x" and "\<E>\<^bsub>\<Gamma>\<^esub> (TER\<^bsub>\<Gamma>\<^esub> g) - A \<Gamma> = \<E>\<^bsub>\<Gamma>\<^esub> (TER\<^bsub>\<Gamma>\<^esub> f) - A \<Gamma>"
SA_Rcd: fixes \<Gamma> :: "binding list"   and fs :: "(char list \<times> POPLmarkRecord.type) list"   and fs' :: "(char list \<times> POPLmarkRecord.type) list" assumes "\<Gamma> \<turnstile>\<^sub>w\<^sub>f"   and "\<Gamma> \<turnstile>\<^sub>w\<^sub>f RcdT fs"   and "unique fs'"   and "\<forall>(l, T)\<in>set fs'. \<exists>S. (l, S) \<in> set fs \<and> \<Gamma> \<turnstile> S <: T" shows "\<Gamma> \<turnstile> RcdT fs <: RcdT fs'"
new_notin: fixes A :: "ref set" assumes "finite A" shows "new A \<notin> A"
bst_baliR: fixes l :: "('b \<times> color) tree"   and r :: "('b \<times> color) tree"   and a :: "'b" assumes "Tree2.bst l"   and "Tree2.bst r"   and "\<forall>x\<in>Tree2.set_tree l. x < a"   and "\<forall>x\<in>Tree2.set_tree r. a < x" shows "Tree2.bst (baliR l a r)"
zer_left_mult: fixes id :: "'b"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zer :: "'b"   and g :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inver :: "'b \<Rightarrow> 'b"   and x :: "'b" assumes "plus_mult id f zer g inver" shows "f zer x = zer"
homotopic_with_imp_subset1: fixes P :: "('b \<Rightarrow> 'c) \<Rightarrow> bool"   and X :: "'b set"   and Y :: "'c set"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "homotopic_with_canon P X Y f g" shows "f ` X \<subseteq> Y"
le_minus_one_simps_2: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less_eq (uminus one) one"
has_derivative_inverse_basic: fixes f :: "'a \<Rightarrow> 'b"   and f' :: "'a \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'a"   and y :: "'b"   and g' :: "'b \<Rightarrow> 'a"   and T :: "'b set" assumes "(f has_derivative f') (at (g y))"   and "bounded_linear g'"   and "g' \<circ> f' = id"   and "isCont g y"   and "open T"   and "y \<in> T"   and "\<And>z. z \<in> T \<Longrightarrow> f (g z) = z" shows "(g has_derivative g') (at y)"
mig_real_def: fixes a :: "'a"   and b :: "'a" shows "mig_real a b = (if a \<le> (0::'a) \<and> (0::'a) \<le> b \<or> b \<le> (0::'a) \<and> (0::'a) \<le> a then 0::'a else min \<bar>a\<bar> \<bar>b\<bar>)"
pfp_bot_least: fixes c :: "com"   and f :: "'a option acom \<Rightarrow> 'a option acom"   and C' :: "'a option acom"   and C :: "'a option acom" assumes "\<forall>x\<in>{C. strip C = c}. \<forall>y\<in>{C. strip C = c}. x \<le> y \<longrightarrow> f x \<le> f y"   and "\<forall>C. C \<in> {C. strip C = c} \<longrightarrow> f C \<in> {C. strip C = c}"   and "f C' \<le> C'"   and "strip C' = c"   and "pfp f (bot c) = Some C" shows "C \<le> C'"
bound_main_lemma_s: fixes PROB :: "(('c, bool) fmap \<times> ('c, bool) fmap) set"   and s :: "('c, bool) fmap" assumes "finite PROB"   and "s \<in> valid_states PROB" shows "problem_plan_bound_s PROB s \<le> 2 ^ card (prob_dom PROB) - 1"
emeasure_lebesgue_affine: fixes m :: "real"   and \<delta> :: "'a"   and S :: "'a set" shows "emeasure lebesgue ((\<lambda>x. m *\<^sub>R x + \<delta>) ` S) = ennreal (\<bar>m\<bar> ^ DIM('a)) * emeasure lebesgue S"
Rep_bit1_inverse: fixes x :: "'b bit1" shows "Abs_bit1 (Rep_bit1 x) = x"
simps: fixes acc :: "'a list"   and A :: "'a set" shows "permutations_of_set_aux acc A = (if infinite A then {} else if A = {} then {acc} else \<Union>x\<in>A. permutations_of_set_aux (x # acc) (A - {x}))"
index_mult_mat_3: fixes A :: "'a mat"   and B :: "'a mat" shows "dim_col (A * B) = dim_col B"
entt_disjD2: fixes A :: "assn"   and B :: "assn"   and C :: "assn" assumes "A \<or>\<^sub>A B \<Longrightarrow>\<^sub>t C" shows "B \<Longrightarrow>\<^sub>t C"
sol_dim_in_min_lp: fixes x :: "'a vec"   and c :: "'a vec"   and A :: "'a mat" assumes "x \<in> optimal_set (\<lambda>x y. x \<bullet> c \<le> y \<bullet> c) (sat_dual A c)" shows "dim_vec x = dim_row A"
xfer_lrun_aux2: fixes cl :: "nat \<Rightarrow> 'l"   and cs :: "nat \<Rightarrow> 'c" assumes "C.is_lrun cl cs" obtains as :: "nat \<Rightarrow> 'a" where "A.is_lrun cl as" and "\<forall>i. R (as i) (cs i)"
interior_open: fixes S :: "'a set" assumes "open S" shows "interior S = S"
Q: shows "Q = Mod_Type_Connect.from_hma\<^sub>m Q_HMA"
WhileRedsThrow: fixes b :: "char list exp"   and s :: "heap \<times> sfields \<times> sheap"   and b\<^sub>0 :: "bool"   and e :: "char list exp"   and s' :: "heap \<times> sfields \<times> sheap"   and b' :: "bool"   and P :: "(char list list \<times> char list exp) prog"   and c :: "char list exp" assumes "P \<turnstile> \<langle>b,s,b\<^sub>0\<rangle> \<rightarrow>* \<langle>throw e,s',b'\<rangle>" shows "P \<turnstile> \<langle>while (b) c,s,b\<^sub>0\<rangle> \<rightarrow>* \<langle>throw e,s',b'\<rangle>"
distinct_35: fixes x4 :: "nat"   and x71 :: "nat"   and x72 :: "msg" shows "Key x4 \<noteq> Crypt x71 x72"
split: fixes P :: "'d \<Rightarrow> bool"   and f :: "'e \<Rightarrow> 'e stream \<Rightarrow> 'd"   and stream :: "'e stream" shows "P (case stream of x ## xa \<Rightarrow> f x xa) = (\<forall>x1 x2. stream = x1 ## x2 \<longrightarrow> P (f x1 x2))"
non_empty_newQuery: fixes Q1 :: "(nat set \<times> 'b option list set) set"   and J :: "nat set"   and Q0 :: "(nat set \<times> 'b option list set) set"   and Q2 :: "(nat set \<times> 'b option list set) set"   and I :: "nat set"   and t :: "'b option list"   and n :: "nat" assumes "Q1 = New_max.filterQuery J Q0"   and "Q2 = New_max.newQuery J Q1 (I, t)"   and "\<forall>X\<in>Q0. wf_atable n X" shows "non_empty_query Q2"
OUTfromVCorrect1SYSTEM_def: shows "OUTfromVCorrect1SYSTEM \<equiv> \<forall>x. OUTfromVCorrect1 x"
ipresSubst_def: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'b"   and MOD :: "('c, 'd, 'varSort, 'e, 'f, 'var, 'b, 'g, 'h) model_scheme" shows "ipresSubst h MOD \<equiv> \<forall>ys Y y s X. wls (asSort ys) Y \<and> wls s X \<longrightarrow> h (X #[Y / y]_ys) = igSubst MOD ys (h Y) y (h X)"
map_id: fixes t :: "'g \<Rightarrow> 'h" shows "id \<circ> t = t"
h2: shows "\<forall>i<n. FlexRayController (nReturn i) recv (nC i) (nStore i) (nSend i) (nGet i)"
exp_eq_zero_iff: fixes n :: "nat" shows "(2 ^ n = 0) = (LENGTH('k) \<le> n)"
gf: fixes n :: "nat" shows "norm (g n) \<le> f n"
antisym_conv3: fixes less_eq :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and less :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and y :: "'k"   and x :: "'k" assumes "class.linorder less_eq less"   and "\<not> less y x" shows "(\<not> less x y) = (x = y)"
ie: shows "te q (i + 1) \<le> te p (i + 1)"
rank_eq_state_rank: fixes \<Sigma> :: "'c set"   and \<delta> :: "'d \<Rightarrow> 'c \<Rightarrow> 'd"   and q\<^sub>0 :: "'d"   and w :: "nat \<Rightarrow> 'c"   and x :: "nat"   and n :: "nat" assumes "semi_mojmir \<Sigma> \<delta> q\<^sub>0 w"   and "x \<le> n" shows "semi_mojmir_def.rank \<Sigma> \<delta> q\<^sub>0 w x n = semi_mojmir_def.state_rank \<Sigma> \<delta> q\<^sub>0 w (semi_mojmir_def.token_run \<delta> q\<^sub>0 w x n) n"
l_remove_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms_def: fixes get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_shadow_root :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, (_) shadow_root_ptr option) prog"   and set_shadow_root :: "(_) element_ptr \<Rightarrow> (_) shadow_root_ptr option \<Rightarrow> ((_) heap, exception, unit) prog"   and set_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and remove_shadow_root :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, unit) prog"   and remove_shadow_root_locs :: "(_) element_ptr \<Rightarrow> (_) shadow_root_ptr \<Rightarrow> ((_) heap, exception, unit) prog set" shows "l_remove_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms get_child_nodes get_shadow_root set_shadow_root set_shadow_root_locs remove_shadow_root remove_shadow_root_locs \<equiv> remove_shadow_root = l_remove_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_remove_shadow_root get_child_nodes get_shadow_root set_shadow_root \<and> remove_shadow_root_locs = l_remove_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_remove_shadow_root_locs set_shadow_root_locs"
tarjan_disc_def: fixes v :: "'v"   and s :: "'v tarjan_state" shows "tarjan_disc v s = RETURN \<lparr>sccs = sccs s, lowlink = lowlink s(v \<mapsto> the_discovered s v), tj_stack = v # tj_stack s\<rparr>"
hyps: fixes \<R>' :: "nat \<Rightarrow> bool option"   and \<R> :: "nat \<Rightarrow> bool option"   and \<S> :: "nat set" assumes "dom \<R>' = dom \<R>" shows "dom (release sb \<S> \<R>') = dom (release sb \<S> \<R>)"
rel_gpv''_conversep: fixes A :: "'k \<Rightarrow> 'h \<Rightarrow> bool"   and C :: "'l \<Rightarrow> 'i \<Rightarrow> bool"   and R :: "'m \<Rightarrow> 'j \<Rightarrow> bool" shows "rel_gpv'' A\<inverse>\<inverse> C\<inverse>\<inverse> R\<inverse>\<inverse> = (rel_gpv'' A C R)\<inverse>\<inverse>"
simps_2: fixes program_step :: "(nat \<Rightarrow> nat option) \<Rightarrow> 'b \<Rightarrow> 'b \<times> instr list \<Rightarrow> bool"   and \<theta> :: "nat \<Rightarrow> nat option"   and p :: "'b"   and r :: "'b memref"   and rs :: "'b memref list" shows "program.history_consistent program_step \<theta> p (r # rs) = (case r of Write\<^sub>s\<^sub>b vol a (D, f) v x xa xb xc \<Rightarrow> D \<subseteq> dom \<theta> \<and> f \<theta> = v \<and> D \<inter> read_tmps rs = {} \<and> program.history_consistent program_step \<theta> p rs | Read\<^sub>s\<^sub>b vol a t v \<Rightarrow> case \<theta> t of None \<Rightarrow> False | Some v' \<Rightarrow> v = v' \<and> program.history_consistent program_step \<theta> p rs | Prog\<^sub>s\<^sub>b p\<^sub>1 p\<^sub>2 is \<Rightarrow> p\<^sub>1 = p \<and> program_step (\<theta> |` (dom \<theta> - read_tmps rs)) p\<^sub>1 (p\<^sub>2, is) \<and> program.history_consistent program_step \<theta> p\<^sub>2 rs | Ghost\<^sub>s\<^sub>b set1 set2 set3 set4 \<Rightarrow> program.history_consistent program_step \<theta> p rs)"
pp_prod_iff: fixes w :: "'a ppath"   and X :: "'a ppath set"   and Y :: "'a ppath set" shows "(w \<in> pp_prod X Y) = (\<exists>u v. w = pp_fusion u v \<and> u \<in> X \<and> v \<in> Y \<and> pp_last u = pp_first v)"
dual_leq: fixes x :: "'b"   and y :: "'b" shows "(dual x \<sqsubseteq> dual y) = (y \<sqsubseteq> x)"
equals_sem: fixes \<nu> :: "(real, 'a) vec \<times> (real, 'a) vec"   and I :: "('b, 'c, 'a) interp"   and \<theta> :: "('b, 'a) trm"   and \<theta>' :: "('b, 'a) trm" shows "(\<nu> \<in> fml_sem I (Equals \<theta> \<theta>')) = (dterm_sem I \<theta> \<nu> = dterm_sem I \<theta>' \<nu>)"
max_flow_min_cut''_bounded: assumes "\<And>y. \<not> edge \<Delta>'' (sink \<Delta>'') y"   and "\<And>x. \<not> edge \<Delta>'' x (source \<Delta>'')"   and "\<And>x. \<not> edge \<Delta>'' x x"   and "\<And>e. e \<in> \<^bold>E\<^bsub>\<Delta>''\<^esub> \<Longrightarrow> 0 < capacity \<Delta>'' e" shows "\<exists>f S. flow \<Delta>'' f \<and> MFMC_Network.cut \<Delta>'' S \<and> orthogonal \<Delta>'' f S"
wt_elim_cases_12: fixes E :: "env"   and dt :: "loc \<Rightarrow> ty option"   and va :: "Term.var"   and T :: "tys" assumes "E,dt\<Turnstile>In1l (Acc va)\<Colon>T" obtains Ta :: "ty" where "T = Inl Ta" and "E,dt\<Turnstile>va\<Colon>=Ta"
Push_Node_def: shows "Push_Node \<equiv> \<lambda>n x. Abs_Node (apfst (Old_Datatype.Push n) (Rep_Node x))"
Rep_Abs_D: fixes P :: "'c process"   and Q :: "'c process" shows "Rep_process (Abs_process ({(s, X). s = [] \<and> (s, X) \<in> F P \<inter> F Q} \<union> {(s, X). s \<noteq> [] \<and> (s, X) \<in> F P \<union> F Q} \<union> {(s, X). s = [] \<and> s \<in> D P \<union> D Q} \<union> {(s, X). s = [] \<and> tick \<notin> X \<and> [tick] \<in> T P \<union> T Q}, D P \<union> D Q)) = ({(s, X). s = [] \<and> (s, X) \<in> F P \<inter> F Q} \<union> {(s, X). s \<noteq> [] \<and> (s, X) \<in> F P \<union> F Q} \<union> {(s, X). s = [] \<and> s \<in> D P \<union> D Q} \<union> {(s, X). s = [] \<and> tick \<notin> X \<and> [tick] \<in> T P \<union> T Q}, D P \<union> D Q)"
eventually_filtercomap_at_top_linorder: fixes P :: "'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b" shows "eventually P (filtercomap f at_top) = (\<exists>N. \<forall>x. N \<le> f x \<longrightarrow> P x)"
simps_4: fixes \<xi> :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltlc" shows "\<xi> \<Turnstile>\<^sub>c not\<^sub>c \<phi> = (\<not> \<xi> \<Turnstile>\<^sub>c \<phi>)"
s_mono: fixes u :: "real"   and t :: "real" assumes "u \<le> t"   and "0 \<le> u" shows "delayed_safe_distance.other.s u \<le> delayed_safe_distance.other.s t"
dnf_modified_eval: fixes \<phi> :: "atom fm"   and xs :: "real list" shows "(\<exists>(al, fl, n)\<in>set (dnf_modified \<phi>). \<exists>L. length L = n \<and> (\<forall>a\<in>set al. aEval a (L @ xs)) \<and> (\<forall>f\<in>set fl. eval f (L @ xs))) = eval \<phi> xs"
scon_def: fixes \<phi>s :: "(('state \<times> 'aprop set) stream list \<Rightarrow> bool) set"   and \<pi>l :: "('state \<times> 'aprop set) stream list" shows "scon \<phi>s \<pi>l \<equiv> \<forall>\<phi>\<in>\<phi>s. \<phi> \<pi>l"
double_zero: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c" assumes "class.linordered_ab_group_add plus zero minus uminus less_eq less" shows "(plus a a = zero) = (a = zero)"
eval_fds_one: shows "eval_fds 1 = (\<lambda>_. 1::'a)"
T_Var: fixes \<Gamma> :: "binding list"   and i :: "nat"   and U :: "POPLmark.type"   and T :: "POPLmark.type" assumes "\<Gamma> \<turnstile>\<^sub>w\<^sub>f"   and "\<Gamma>\<langle>i\<rangle> = \<lfloor>VarB U\<rfloor>"   and "T = \<up>\<^sub>\<tau> (Suc i) 0 U" shows "\<Gamma> \<turnstile> Var i : T"
sum_mono2: fixes B :: "'b set"   and A :: "'b set"   and f :: "'b \<Rightarrow> 'c multiset" assumes "finite B"   and "A \<subseteq> B"   and "\<And>b. b \<in> B - A \<Longrightarrow> {#} \<subseteq># f b" shows "comm_monoid_add.sum (+) {#} f A \<subseteq># comm_monoid_add.sum (+) {#} f B"
i_set_min_i_UN_set_min_eq: shows "i_set_min = i_UN_set_min"
ceiling_le_numeral: fixes x :: "'b"   and v :: "num" shows "(\<lceil>x\<rceil> \<le> numeral v) = (x \<le> numeral v)"
inc_def: shows "inc = (\<lambda>(n, fs). (Suc n, fs))"
all_cong1: fixes P :: "'a \<Rightarrow> bool"   and P' :: "'a \<Rightarrow> bool" assumes "\<And>x. P x = P' x" shows "(\<forall>x. P x) = (\<forall>x. P' x)"
SNaI: fixes M :: "trm" assumes "\<And>M'. M \<longrightarrow>\<^sub>a M' \<Longrightarrow> SNa M'" shows "SNa M"
NE_intT: fixes intT :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and \<sigma> :: "'a" assumes "M.Tstruct intT" shows "\<exists>a. intT \<sigma> a"
vimage2pI: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'a"   and x :: "'c"   and g :: "'d \<Rightarrow> 'b"   and y :: "'d" assumes "R (f x) (g y)" shows "vimage2p f g R x y"
simps_3: fixes v :: "'b"   and va :: "'b"   and y :: "('b, 'c) Combinators list" shows "removeShadowRules2 (DenyAllFromTo v va # y) = DenyAllFromTo v va # removeShadowRules2 y"
equiv_funE: fixes f :: "nat \<Rightarrow> nat"   and g :: "nat \<Rightarrow> nat" assumes "f \<cong> g" obtains n :: "nat" and c\<^sub>1 :: "nat" and c\<^sub>2 :: "nat" where "0 < c\<^sub>1" and "0 < c\<^sub>2" and "\<And>m. n < m \<Longrightarrow> f m \<le> c\<^sub>1 * g m \<and> g m \<le> c\<^sub>2 * f m"
expl_cond_expect_const: fixes Y :: "'d \<Rightarrow> 'c"   and w :: "'d"   and y :: "'d"   and M :: "'d measure"   and X :: "'d \<Rightarrow> real" assumes "Y w = Y y" shows "expl_cond_expect M Y X w = expl_cond_expect M Y X y"
rel_sel: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'c option"   and b :: "'d option" shows "rel_option R a b = ((a = None) = (b = None) \<and> (a \<noteq> None \<longrightarrow> b \<noteq> None \<longrightarrow> R (the a) (the b)))"
prod_mset_singleton: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and x :: "'b" assumes "class.comm_monoid_mult times one" shows "comm_monoid_mult.prod_mset times one {#x#} = x"
simps: fixes f :: "('d, 'c) trm" shows "FVDiff f = \<Union> (primify ` FVT f)"
abs_eq: fixes xa :: "'g"   and x :: "('e, 'f, 'g) Tree" shows "Act\<^sub>\<alpha> xa (abs_Tree\<^sub>\<alpha> x) = abs_Tree\<^sub>\<alpha> (tAct xa x)"
jw_Unroll_inv: fixes \<Gamma> :: "(Syntax.var, ty) fmap"   and e :: "Syntax.term"   and \<tau> :: "ty"   and \<alpha> :: "Syntax.tvar" assumes "\<Gamma> \<turnstile>\<^sub>W Unroll e : \<tau>"   and "atom \<alpha> \<sharp> (\<Gamma>, \<tau>)" obtains \<tau>'' :: "ty" where "\<tau> = subst_type \<tau>'' (Mu \<alpha> \<tau>'') \<alpha>" and "\<Gamma> \<turnstile>\<^sub>W e : Mu \<alpha> \<tau>''"
double_SN: fixes a :: "trm"   and b :: "trm"   and P :: "trm \<Rightarrow> trm \<Rightarrow> bool" assumes "SN a"   and "SN b"   and "\<And>x z. \<lbrakk>\<And>y. x \<mapsto> y \<Longrightarrow> P y z; \<And>u. z \<mapsto> u \<Longrightarrow> P x u\<rbrakk> \<Longrightarrow> P x z" shows "P a b"
wy_def: shows "wy = geodesic_segment_param {e--y} e t"
Arg2pi_eq_0_pi: fixes z :: "complex" shows "(Arg2pi z = 0 \<or> Arg2pi z = pi) = (z \<in> \<real>)"
simps_3: fixes r :: "'k \<Rightarrow> 'v \<Rightarrow> nat \<Rightarrow> bool"   and \<AA> :: "'i"   and \<phi> :: "('a, 'k) aformula" shows "satisfies_gen r \<AA> (FNot \<phi>) = (\<not> satisfies_gen r \<AA> \<phi>)"