hcju/isabelleps huggingface dataset:
init_strong_tricolour_inv: fixes s :: "'h process_name \<Rightarrow> ('i, 'j, 'k, ('e, 'h, 'f, 'g) Model.local_state) com list \<times> 'j option \<times> ('e, 'h, 'f, 'g) Model.local_state" assumes "obj_mark ` ran (sys_heap \<lparr>GST = s, HST = []\<rparr>\<down>) \<subseteq> {gc_fM \<lparr>GST = s, HST = []\<rparr>\<down>}"   and "sys_fM \<lparr>GST = s, HST = []\<rparr>\<down> = gc_fM \<lparr>GST = s, HST = []\<rparr>\<down>" shows "strong_tricolour_inv \<lparr>GST = s, HST = []\<rparr>\<down>"
l_to_tree_order_axioms: shows "l_to_tree_order to_tree_order"
merge_mono: fixes A :: "assn"   and A' :: "assn"   and B :: "assn"   and B' :: "assn"   and C :: "assn" assumes "A \<Longrightarrow>\<^sub>t A'"   and "B \<Longrightarrow>\<^sub>t B'"   and "A' \<or>\<^sub>A B' \<Longrightarrow>\<^sub>t C" shows "A \<or>\<^sub>A B \<Longrightarrow>\<^sub>t C"
maxsimp_def: fixes x :: "'a set" shows "maxsimp x \<equiv> x \<in> X \<and> (\<forall>z\<in>X. order.greater_eq z x \<longrightarrow> z = x)"
leg_simps_4: shows "C.dom \<nu>\<pi>.leg1 = \<nu>\<pi>.apex"
type_substitution_aux: fixes \<Delta> :: "(name \<times> ty) list"   and x :: "name"   and T' :: "ty"   and \<Gamma> :: "(name \<times> ty) list"   and e :: "lam"   and T :: "ty"   and e' :: "lam" assumes "\<Delta> @ [(x, T')] @ \<Gamma> \<turnstile> e : T"   and "\<Gamma> \<turnstile> e' : T'" shows "\<Delta> @ \<Gamma> \<turnstile> e[x::=e'] : T"
Sup_upper: 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 x :: "'a"   and A :: "'a set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "s2p A x" shows "less_eq x (Sup A)"
assms: shows "Act\<^sub>\<alpha> \<alpha> t\<^sub>\<alpha> = Act\<^sub>\<alpha> \<alpha>' t\<^sub>\<alpha>'"
verit_la_generic: fixes a :: "int"   and x :: "int" shows "a \<le> x \<or> a = x \<or> x \<le> a"
that: fixes q :: "gauss_int" assumes "prime q"   and "prime_elem (gauss_cnj q)"   and "0 < ReZ q"   and "0 < ImZ q"   and "of_nat p = q * gauss_cnj q"   and "gauss_int_norm q = p" shows "thesis"
cSUP_least: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b multiset"   and M :: "'b multiset" assumes "A \<noteq> {}"   and "\<And>x. x \<in> A \<Longrightarrow> f x \<subseteq># M" shows "\<Squnion> (f ` A) \<subseteq># M"
bij_1: fixes x :: "'b" assumes "P x" shows "f (g x) = x"
comp: fixes f :: "'d \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'd" shows "map f \<circ> map g = map (f \<circ> g)"
path_weight_refl: fixes loc :: "'vtx" shows "(0::'lbl) \<in>\<^sub>A path_weight loc loc"
obs_eq: shows "obs_intra m \<lfloor>HRB_slice S\<rfloor>\<^bsub>CFG\<^esub> = obs_intra m' \<lfloor>HRB_slice S\<rfloor>\<^bsub>CFG\<^esub>"
Zfun_def: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter" shows "Zfun f F = (\<forall>r>0. \<forall>\<^sub>F x in F. norm (f x) < r)"
convex_real_interval_5: fixes a :: "real"   and b :: "real" shows "convex {a..b}"
isLam_obtain_fresh: fixes z :: "exp"   and c :: "'b" assumes "isLam z" obtains y :: "Vars.var" and e' :: "exp" where "z = Lam [y]. e'" and "atom y \<sharp> c"
HOLCF_trans_rules_2: fixes x :: "'c"   and y :: "'c" assumes "x \<sqsubseteq> y"   and "y \<sqsubseteq> x" shows "x = y"
chi_not_principal: shows "\<chi> \<noteq> principal_dchar n"
valid_init_mmsaux: fixes L :: "nat set"   and R :: "nat set"   and I :: "\<I>"   and n :: "nat"   and b :: "bool" assumes "L \<subseteq> R" shows "valid_mmsaux (init_args I n L R b) 0 (init_mmsaux (init_args I n L R b)) []"
IH: shows "TS M2 M1 \<Omega> V m i = \<Union> (C M2 M1 \<Omega> V m ` set [0..<Suc i])"
fd: shows "dom\<^bsub>CanonicalCat' T\<^esub> f = X"
cSUP_upper2: fixes f :: "'e \<Rightarrow> 'd multiset"   and A :: "'e set"   and x :: "'e"   and u :: "'d multiset" assumes "subset_mset.bdd_above (f ` A)"   and "x \<in> A"   and "u \<subseteq># f x" shows "u \<subseteq># \<Squnion> (f ` A)"
change_on_1_hyp_change: fixes i :: "nat"   and t :: "nat" assumes "change_on_1 (block i t) (gap i t)" shows "\<sigma> i (\<tau> i \<triangleright> (t + 1)) \<noteq> \<sigma> i (\<tau> i \<triangleright> (gap i t - 1))"
zeta_conv_hurwitz_zeta_multiplication: assumes "0 < 2"   and "s \<noteq> 1" shows "of_nat 2 powr s * zeta s = (\<Sum>n = 1..2. hurwitz_zeta (real n / real 2) s)"
transpI: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<And>x y z. \<lbrakk>r x y; r y z\<rbrakk> \<Longrightarrow> r x z" shows "transp r"
openin_open_Int: fixes S :: "'c set"   and U :: "'c set" assumes "open S" shows "openin (top_of_set U) (U \<inter> S)"
ivl_disj_int_two_5: fixes l :: "'b"   and m :: "'b"   and u :: "'b" shows "{l<..<m} \<inter> {m..u} = {}"
next_primes_def: fixes n :: "nat" shows "next_primes n = (if n = 0 then next_candidates 0 else let (m, ps) = next_candidates n in (m, filter prime ps))"
C_def: shows "C = max 0 (SUP s\<in>closure U. cmod (pochhammer s (Suc (2 * n))))"
allUses_in_allVars: fixes \<alpha>e :: "'a \<Rightarrow> ('b \<times> 'c \<times> 'b) set"   and \<alpha>n :: "'a \<Rightarrow> 'b list"   and invar :: "'a \<Rightarrow> bool"   and inEdges' :: "'a \<Rightarrow> 'b \<Rightarrow> ('b \<times> 'c) list"   and Entry :: "'a \<Rightarrow> 'b"   and defs :: "'a \<Rightarrow> 'b \<Rightarrow> 'd set"   and "uses" :: "'a \<Rightarrow> 'b \<Rightarrow> 'd set"   and phis :: "'a \<Rightarrow> 'b \<times> 'd \<Rightarrow> 'd list option"   and v :: "'d"   and g :: "'a"   and n :: "'b" assumes "CFG_SSA \<alpha>e \<alpha>n invar inEdges' Entry defs uses phis"   and "v \<in> CFG_SSA_base.allUses \<alpha>n inEdges' uses phis g n"   and "n \<in> set (\<alpha>n g)" shows "v \<in> CFG_SSA_base.allVars \<alpha>n inEdges' defs uses phis g"
ev_fork_idx: fixes e :: "dining_event"   and i :: "nat"   and s :: "nat" assumes "e \<in> Tr\<^sub>f i s" shows "fork e = i"
no_spoofing: shows "no_spoofing TYPE('pkt_ext) ipassmt rs"
bin_to_bl_drop_bit: fixes k :: "nat"   and m :: "nat"   and n :: "nat"   and c :: "int" assumes "k = m + n" shows "bin_to_bl m (drop_bit n c) = take m (bin_to_bl k c)"
init_tableau_valuated: fixes init :: "(nat \<times> linear_poly) list \<Rightarrow> ('a, 'b) state"   and t :: "(nat \<times> linear_poly) list" assumes "Init init" shows "\<nabla> (init t)"
add_left: fixes a :: "'a"   and a' :: "'a"   and b :: "'b" shows "(a + a') ** b = a ** b + a' ** b"
filterlim_atI: fixes "open" :: "'b set \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'b"   and c :: "'b"   and F :: "'c filter" assumes "class.topological_space open"   and "filterlim f (topological_space.nhds open c) F"   and "\<forall>\<^sub>F x in F. f x \<noteq> c" shows "filterlim f (topological_space.at_within open c UNIV) F"
assms_1: shows "orient_lit_inst L t s polarity \<theta>"
\<alpha>_props_1: shows "distinct (map (\<lambda>(k, v). (k, \<alpha> v)) lc)"
inter_gamma_subset_gamma_inf: fixes a1 :: "'av"   and a2 :: "'av" shows "\<gamma> a1 \<inter> \<gamma> a2 \<subseteq> \<gamma> (a1 \<sqinter> a2)"
quadratic_linear_at_most_2_intersections: fixes a11 :: "complex"   and a12 :: "complex"   and a22 :: "complex"   and k1 :: "complex"   and k2 :: "complex"   and qf :: "complex \<Rightarrow> complex \<Rightarrow> complex"   and b1 :: "complex"   and b2 :: "complex"   and c :: "complex"   and lf :: "complex \<Rightarrow> complex \<Rightarrow> complex"   and n :: "complex"   and x1 :: "complex"   and y1 :: "complex"   and x2 :: "complex"   and y2 :: "complex"   and x :: "complex"   and y :: "complex" assumes "(a11, a12, a22) \<noteq> (0, 0, 0)"   and "(k1, k2) \<noteq> (0, 0)"   and "a11 * k2\<^sup>2 - 2 * a12 * k1 * k2 + a22 * k1\<^sup>2 \<noteq> 0"   and "qf = (\<lambda>x y. a11 * x\<^sup>2 + 2 * a12 * x * y + a22 * y\<^sup>2 + b1 * x + b2 * y + c)"   and "lf = (\<lambda>x y. k1 * x + k2 * y + n)"   and "qf x1 y1 = 0"   and "lf x1 y1 = 0"   and "qf x2 y2 = 0"   and "lf x2 y2 = 0"   and "(x1, y1) \<noteq> (x2, y2)"   and "qf x y = 0"   and "lf x y = 0" shows "(x, y) = (x1, y1) \<or> (x, y) = (x2, y2)"
lb_mem: shows "length bef \<in> set (\<sigma> (f, Suc (length bef + length aft)))"
closed_Inter: fixes "open" :: "'b set \<Rightarrow> bool"   and K :: "'b set set" assumes "class.topological_space open"   and "\<forall>S\<in>K. topological_space.closed open S" shows "topological_space.closed open (\<Inter> K)"
Spy_not_see_NB: fixes B :: "agent"   and A :: "agent"   and NA :: "nat"   and NB :: "nat"   and evs :: "event list" assumes "Says B A (Crypt (pubK A) \<lbrace>Nonce NA, Nonce NB\<rbrace>) \<in> set evs"   and "\<forall>C. Says A C (Crypt (pubK C) (Nonce NB)) \<notin> set evs"   and "A \<notin> bad"   and "B \<notin> bad"   and "evs \<in> ns_public" shows "Nonce NB \<notin> analz (knows Spy evs)"
assms_2: fixes r :: "rat" assumes "r < r'" shows "fpxs_nth f r = (0::'a)"
sorted_poly_list_rel_ConsD: fixes ys :: "char list list"   and n :: "int"   and p :: "(char list list \<times> int) list"   and a :: "(char list multiset \<times> int) multiset"   and S :: "char list list \<Rightarrow> char list list \<Rightarrow> bool" assumes "((ys, n) # p, a) \<in> sorted_poly_list_rel S" shows "(p, remove1_mset (mset ys, n) a) \<in> sorted_poly_list_rel S \<and> (mset ys, n) \<in># a \<and> (\<forall>x\<in>set p. S ys (fst x)) \<and> sorted_wrt var_order ys \<and> distinct ys \<and> ys \<notin> set (map fst p) \<and> n \<noteq> 0 \<and> nonzero_coeffs a"
zmod_minus1: fixes b :: "int" assumes "0 < b" shows "- 1 mod b = b - 1"
pred_mapping_def: fixes Q :: "'d \<Rightarrow> bool" shows "pred_mapping Q = pred_fun (\<lambda>_. True) (pred_option Q)"
simps: fixes x :: "nat"   and l :: "cell list"   and r :: "cell list" shows "inv_loop5_exit x (l, r) = (\<exists>i j. i + j = Suc x \<and> 0 < i \<and> 0 < j \<and> (l, r) = (Bk \<up> (j - 1) @ Oc \<up> i, Bk # Oc \<up> j))"
vec_rel_def: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and A :: "'b vec"   and B :: "'c vec" shows "vec_rel R A B \<equiv> dim_vec A = dim_vec B \<and> (\<forall>i<dim_vec B. R (A $ i) (B $ i))"
hbase_tl': fixes n :: "C"   and ns :: "C list"   and b :: "nat" assumes "C (n # ns) \<in> hbase b" shows "C ns \<in> hbase b"
crb_def: fixes p :: "real poly" shows "crb p = \<lceil>2 + max_list_non_empty (map (\<lambda>i. norm (coeff p i)) [0..<degree p]) / norm (lead_coeff p)\<rceil>"
assms_2: fixes U :: "'a set" assumes "U \<subseteq> T" shows "openin (top_of_set S) (S \<inter> f -` U) = openin (top_of_set T) U"
le_square: fixes m :: "nat" shows "m \<le> m * m"
LIM_NSLIM_iff: fixes f :: "'c \<Rightarrow> 'd"   and L :: "'d"   and x :: "'c" shows "f \<midarrow>x\<rightarrow> L = f \<midarrow>x\<rightarrow>\<^sub>N\<^sub>S L"
ord_option_simps_4: fixes ord :: "'h \<Rightarrow> 'i \<Rightarrow> bool"   and x :: "'h" shows "ord_option ord (Some x) None = False"
prems_2: shows "(l # ls) ! i = (p, Read volatile a t # is, \<theta>, sb, \<D>, \<O>)"
e_def: shows "e = (case ms_aux_hd_exp xs of None \<Rightarrow> e' - 1 | Some e \<Rightarrow> (e' + e) / 2)"
poly_rel_pdivmod_monic: fixes Y :: "'a poly"   and x :: "'i list"   and X :: "'a poly"   and y :: "'i list" assumes "monic Y"   and "poly_rel x X"   and "poly_rel y Y" shows "rel_prod poly_rel poly_rel (pdivmod_monic_i ops x y) (pdivmod_monic X Y)"
assms: fixes x :: "'var" assumes "x \<in> domA h" shows "\<lbrakk> the (map_of h x) \<rbrakk>\<^bsub>r\<^esub> \<sqsubseteq> r x"
simps_1: fixes G :: "prog"   and S :: "qtname" shows "mheads G S NullT = (\<lambda>sig. {})"
effect_deterministic_2: fixes f :: "'a Heap"   and h :: "heap"   and h' :: "heap"   and a :: "'a"   and h'' :: "heap"   and b :: "'a" assumes "effect f h h' a"   and "effect f h h'' b" shows "h' = h''"
hi: shows "hom_induced (int p) (nsphere p) {} (nsphere p) {} f a = a [^]\<^bsub>reduced_homology_group (int p) (nsphere p)\<^esub> d"
finsum_insert: fixes F :: "'c set"   and a :: "'c"   and f :: "'c \<Rightarrow> 'a vec" assumes "finite F"   and "a \<notin> F"   and "f \<in> F \<rightarrow> carrier_vec nc"   and "f a \<in> carrier_vec nc" shows "finsum NC.V f (insert a F) = f a + finsum NC.V f F"
parts_auth_data: fixes x :: "rat"   and y :: "rat"   and H :: "msg set" shows "parts (insert (Auth_Data x y) H) = {Pri_AgrK x, Pri_AgrK y, Auth_Data x y} \<union> parts H"
lset_llist_of: fixes xs :: "'c list" shows "lset (llist_of xs) = set xs"
b: fixes x :: "'a \<times> 'b \<times> 'c" assumes "\<And>y. ((fst y, fst (snd y), snd (snd y)), fst x, fst (snd x), snd (snd x)) \<in> r1 <*lex*> r2 <*lex*> r3 \<longrightarrow> P y" shows "P x"
virt_to_phys_unchanged: fixes s2 :: "('b, 'c) sparc_state_scheme"   and s1 :: "('d, 'e) sparc_state_scheme"   and addr :: "word36"   and val :: "word8" assumes "\<forall>va. virt_to_phys va (mmu s2) (mem s1) = virt_to_phys va (mmu s2) (mem s2)" shows "\<forall>va. virt_to_phys va (mmu s2) ((mem s1)(10 := mem s1 10(addr \<mapsto> val), 11 := (mem s1 11)(addr := None))) = virt_to_phys va (mmu s2) ((mem s2)(10 := mem s2 10(addr \<mapsto> val), 11 := (mem s2 11)(addr := None)))"
Class_self: fixes a :: "'a" assumes "a \<in> G" shows "a \<in> Class a"
atS_simps_3: fixes l :: "'g"   and s :: "('h, 'g, 'i, 'j, 'k) system_state"   and p :: "'i"   and ls :: "'g set" assumes "at p l s"   and "l \<in> ls" shows "atS p ls s"
hfunction_shift_iff: fixes f :: "hf"   and delta :: "hf" shows "hfunction (shift f delta) = hfunction f"
unif_def: shows "unif = {(x, y). \<exists>\<sigma>. UNIF1 \<sigma> x y}"
nat_pow_0: fixes x :: "nat \<Rightarrow> int" shows "x [^] 0 = \<one>"
u': shows "adjoint U * 1\<^sub>m d * U = 1\<^sub>m d"
height_del: fixes t :: "'b tree23"   and x :: "'b" assumes "complete t" shows "hD (del x t) = height t"
subgame_\<omega>_subset: fixes V' :: "'a set" shows "\<omega>\<^bsub>subgame V'\<^esub> ` V\<^bsub>subgame V'\<^esub> \<subseteq> \<omega> ` V"
bounded_antilinear_axioms_def: fixes f :: "'d \<Rightarrow> 'e" shows "bounded_antilinear_axioms f \<equiv> \<exists>K. \<forall>x. norm (f x) \<le> norm x * K"
elims_2: fixes x :: "('a, 'e, 'f, 'g) SLin_state_scheme \<times> ('a, 'h, 'i, 'j) SLin_state_scheme" assumes "P9 x" obtains s1 :: "('a, 'e, 'f, 'g) SLin_state_scheme" and s2 :: "('a, 'h, 'i, 'j) SLin_state_scheme" where "x = (s1, s2)" and "initialized s2 \<longrightarrow> dstate s1 \<preceq> dstate s2"
simps_8: fixes p :: "fm"   and q :: "fm" shows "bound0 (And p q) = (bound0 p \<and> bound0 q)"
e_1: shows "\<Gamma> \<turnstile>\<^sub>c e : REAL"
d: fixes n :: "nat"   and fs :: "'a vec list" shows "vec_module.lattice_of n fs = range (\<lambda>c. abelian_monoid.sumlist (module_vec TYPE('a) n) (map (\<lambda>i. of_int (c i) \<cdot>\<^sub>v fs ! i) [0..<length fs]))"
scale: fixes r :: "'a"   and b :: "('a, 'n) vec" shows "f (r *s b) = r *s f b"
partial_object_ext_def: fixes carrier :: "'c set"   and more :: "'d" shows "\<lparr>carrier = carrier, \<dots> = more\<rparr> \<equiv> Record.iso_tuple_cons partial_object_ext_Tuple_Iso carrier more"
epath_to_depath': fixes E :: "'b uprod set"   and u :: "'b"   and p :: "'b uprod list"   and v :: "'b" assumes "epath E u p v" shows "\<exists>dp. depath E u dp v"
\<omega>_limit_set_in_compact_compact: shows "compact (\<omega>_limit_set x)"
apply_adaptation\<^sub>A_1: fixes mem\<^sub>1\<^sub>C :: "'Var\<^sub>C \<Rightarrow> 'Val"   and A\<^sub>C :: "'Var\<^sub>C \<Rightarrow> ('Val \<times> 'Val) option" shows "mem\<^sub>A_of mem\<^sub>1\<^sub>C [\<parallel>\<^sub>1 A\<^sub>A_of A\<^sub>C] = mem\<^sub>A_of mem\<^sub>1\<^sub>C [\<parallel>\<^sub>1 A\<^sub>C]"
y2_prop: shows "r < y2 \<and> (\<forall>(d, e, f)\<in>set c. \<forall>x\<in>{r<..y2}. d * x\<^sup>2 + e * x + f \<le> 0)"
polyn_minus: fixes c :: "nat \<times> (nat \<Rightarrow> 'a)"   and k :: "nat" assumes "pol_coeff S c"   and "k \<le> fst c" shows "-\<^sub>a polyn_expr R X k c = polyn_expr R X k (fst c, \<lambda>j. -\<^sub>a\<^bsub>S\<^esub> snd c j)"
strengthen_pre: fixes P' :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and P :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and c :: "com"   and e :: "(char list \<Rightarrow> int) \<Rightarrow> nat"   and Q :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool" assumes "\<forall>l s. P' l s \<longrightarrow> P l s"   and "\<turnstile>\<^sub>1 {P} c { e \<Down> Q}" shows "\<turnstile>\<^sub>1 {P'} c { e \<Down> Q}"
row_length_def: fixes id :: "'b"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and xs :: "'b list list" assumes "Matrix_Tensor.mult id f" shows "mult.row_length xs \<equiv> if xs = [] then 0 else length (hd xs)"
\<Delta>''_flow_attainability: assumes "flow_attainability_axioms \<Delta>" shows "flow_attainability \<Delta>''"
K_right_mp: fixes ps :: "'b fm list"   and p :: "'b fm"   and q :: "'b fm" assumes "\<turnstile> imply ps p"   and "\<turnstile> imply ps (p \<^bold>\<longrightarrow> q)" shows "\<turnstile> imply ps q"
s0_\<beta>\<alpha>_s': shows "s0\<^bsub>SES\<^esub> (\<beta> @ \<alpha>)\<Longrightarrow>\<^bsub>SES\<^esub> s'"
hyps: fixes "is" :: "nat list" shows "map fst (filter (\<lambda>p. P (Suc (snd p))) (zip xs is)) = map fst (filter (\<lambda>p. P (snd p)) (zip xs (map Suc is)))"
redT_updT'_init_fin_descend_thr_eq_None: fixes ts :: "'e \<Rightarrow> (('h \<times> 'c) \<times> 'd \<Rightarrow>f nat) option"   and nt :: "('e, 'f, 'g) new_thread_action"   and t :: "'e" shows "(redT_updT' (init_fin_descend_thr ts) nt t = None) = (redT_updT' ts nt t = None)"
VSagree_refl: fixes \<nu> :: "(real, 'd) vec"   and A :: "'d set" shows "VSagree \<nu> \<nu> A"
Expr: fixes E :: "env"   and dt :: "loc \<Rightarrow> ty option"   and e :: "expr"   and T :: "ty" assumes "E,dt\<Turnstile>e\<Colon>-T" shows "E,dt\<Turnstile>Expr e\<Colon>\<surd>"
klt: assumes "k < z" shows "0 < a * z\<^sup>2 + b * z + c"
apd3: fixes x :: "'a" shows "ad (ad x) + ad x = (1::'a)"
assms_3: shows "f ` \<Omega> = \<Omega>"
Nil: fixes \<M> :: "('fun, 'var) Term.term set"   and \<N> :: "('fun, 'var) Term.term set"   and \<I> :: "'var \<Rightarrow> ('fun, 'var) Term.term" shows "[] \<in> decomps\<^sub>e\<^sub>s\<^sub>t \<M> \<N> \<I>"
in_RSpan_obtain_same_length_coeffs: fixes n :: "'m"   and ms :: "'m list" assumes "n \<in> RSpan ms" shows "\<exists>rs. set rs \<subseteq> R \<and> length rs = length ms \<and> n = rs \<bullet>\<cdot> ms"
compact_nest: fixes F :: "'a \<Rightarrow> 'b set" assumes "\<And>n. compact (F n)"   and "\<And>n. F n \<noteq> {}"   and "\<And>m n. m \<le> n \<Longrightarrow> F n \<subseteq> F m" shows "\<Inter> (range F) \<noteq> {}"
bi_unique_rel_12_21: shows "bi_unique rel_12_21"
assms_1: shows "Finite_And_Cyclic_Groups.cyclic_group H h"
spmf_pred_mono_strong: fixes P :: "'c \<Rightarrow> bool"   and p :: "'c spmf"   and P' :: "'c \<Rightarrow> bool" assumes "pred_spmf P p"   and "\<And>a. \<lbrakk>a \<in> set_spmf p; P a\<rbrakk> \<Longrightarrow> P' a" shows "pred_spmf P' p"
interleave_order: fixes s :: "'a event list"   and t1 :: "'a event list"   and t2 :: "'a event list"   and C :: "'a event set"   and u :: "'a event list" assumes "s setinterleaves ((t1 @ t2, u), C)" shows "set t2 \<subseteq> set (drop (length t1) s)"
set_tree_delete: fixes t :: "('a \<times> 'b) tree"   and x :: "'a" assumes "Tree2.bst t" shows "Tree2.set_tree (delete x t) = Tree2.set_tree t - {x}"
ProcModifyReturnSameFaults_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 init :: "'e \<Rightarrow> 'e"   and p :: "'d"   and return' :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and c :: "'e \<Rightarrow> 'e \<Rightarrow> ('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set"   and Modif :: "'e \<Rightarrow> 'e set"   and ModifAbr :: "'e \<Rightarrow> 'e set"   and return :: "'e \<Rightarrow> 'e \<Rightarrow> 'e" assumes "\<Gamma>,\<Theta>\<Turnstile>\<^sub>t\<^bsub>/F\<^esub> P call init p return' c Q,A"   and "\<forall>\<sigma>. \<Gamma>,\<Theta>\<Turnstile>\<^bsub>/F\<^esub> {\<sigma>} Call p (Modif \<sigma>),(ModifAbr \<sigma>)"   and "\<forall>s t. t \<in> Modif (init s) \<longrightarrow> return' s t = return s t"   and "\<forall>s t. t \<in> ModifAbr (init s) \<longrightarrow> return' s t = return s t" shows "\<Gamma>,\<Theta>\<Turnstile>\<^sub>t\<^bsub>/F\<^esub> P call init p return c Q,A"
reduction_chunk: fixes q :: "'state"   and a :: "'action"   and v :: "'action stream" assumes "q \<in> nodes"   and "run ([a] @- v) q" obtains b :: "'action list" and b\<^sub>1 :: "'action list" and b\<^sub>2 :: "'action list" and u :: "'action stream" where "R.path (b @ [a]) q" and "Ind {a} (set b)" and "set b \<subseteq> invisible" and "b =\<^sub>F b\<^sub>1 @ b\<^sub>2" and "b\<^sub>1 @- u =\<^sub>I v" and "Ind (set b\<^sub>2) (sset u)"
SXcpt_def: shows "SXcpt = (\<lambda>x. \<lparr>pid = java_lang, tid = SXcpt' x\<rparr>)"
overflow_imp_lsb: fixes x :: "'b word" assumes "x + 1 = 0" shows "x !! 0"
inj_trancl_image: fixes f :: "'a \<Rightarrow> 'b"   and i :: "'a"   and j :: "'a"   and R :: "('a \<times> 'a) set" assumes "inj f" shows "((f i, f j) \<in> (map_prod f f ` R)\<^sup>+) = ((i, j) \<in> R\<^sup>+)"
mult_right_cancel: fixes c :: "'a"   and a :: "'a"   and b :: "'a" assumes "c \<noteq> (0::'a)" shows "(a * c = b * c) = (a = b)"
OC_iff_weak_reduction_bisimulation: fixes TRel :: "('procT \<times> 'procT) set" shows "(operational_corresponding (TRel\<^sup>*) \<and> weak_reduction_bisimulation (TRel\<^sup>+) Target) = (\<exists>Rel. (\<forall>S. (SourceTerm S, TargetTerm (\<lbrakk>S\<rbrakk>)) \<in> Rel) \<and> (\<forall>T1 T2. (T1, T2) \<in> TRel \<longrightarrow> (TargetTerm T1, TargetTerm T2) \<in> Rel) \<and> (\<forall>T1 T2. (TargetTerm T1, TargetTerm T2) \<in> Rel \<longrightarrow> (T1, T2) \<in> TRel\<^sup>+) \<and> (\<forall>S T. (SourceTerm S, TargetTerm T) \<in> Rel \<longrightarrow> (\<lbrakk>S\<rbrakk>, T) \<in> TRel\<^sup>*) \<and> weak_reduction_bisimulation Rel (STCal Source Target))"
in_keys_iff: fixes x :: "'c"   and s :: "'c \<Rightarrow>\<^sub>0 'd" shows "(x \<in> Poly_Mapping.keys s) = (poly_mapping.lookup s x \<noteq> (0::'d))"
elims_3: fixes x :: "'b heap" assumes "\<not> pheap x" obtains xa :: "'b" and hs :: "'b heap list" where "x = Hp xa hs" and "\<not> (\<forall>h\<in>set hs. Multiset.Ball (mset_heap h) ((\<le>) xa) \<and> pheap h)"
sin_le_one: fixes x :: "real" shows "sin x \<le> 1"
inf_le2: fixes x :: "'a"   and y :: "'a" shows "x \<cdot> y \<le> y"
symmetric_conv: fixes G :: "('c, 'd) pre_digraph" shows "symmetric G = (\<forall>e1\<in>arcs G. \<exists>e2\<in>arcs G. tail G e1 = head G e2 \<and> head G e1 = tail G e2)"
strict_implies_order: fixes b :: "'b"   and a :: "'b" assumes "(b::'b) < (a::'b)" shows "(b::'b) \<le> (a::'b)"
mem_eq: shows "\<forall>a\<in>A \<union> W. m' a = m a"
poincare_distance_formula'_ge_1: fixes u :: "complex_homo"   and v :: "complex_homo" assumes "u \<in> unit_disc"   and "v \<in> unit_disc" shows "1 \<le> poincare_distance_formula' (to_complex u) (to_complex v)"
mopup_mod2: fixes k :: "nat" shows "is_even (length (mopup k))"
foldl_cong: fixes a :: "'e"   and b :: "'e"   and l :: "'f list"   and k :: "'f list"   and f :: "'e \<Rightarrow> 'f \<Rightarrow> 'e"   and g :: "'e \<Rightarrow> 'f \<Rightarrow> 'e" assumes "a = b"   and "l = k"   and "\<And>a x. x \<in> set l \<Longrightarrow> f a x = g a x" shows "foldl f a l = foldl g b k"
OUTfromVCorrect1_data19: shows "OUTfromVCorrect1 data19"
f_carrier: fixes i :: "nat" assumes "i < length (map of_int_hom.vec_hom fs)" shows "map of_int_hom.vec_hom fs ! i \<in> carrier_vec n"
preserves_adjunction: fixes f :: "'c"   and g :: "'c"   and \<eta> :: "'c"   and \<epsilon> :: "'c" assumes "adjunction_in_bicategory (\<cdot>\<^sub>C) (\<star>\<^sub>C) \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C f g \<eta> \<epsilon>" shows "adjunction_in_bicategory (\<cdot>\<^sub>D) (\<star>\<^sub>D) \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D (F f) (F g) (D.inv (\<Phi> (g, f)) \<cdot>\<^sub>D F \<eta> \<cdot>\<^sub>D unit (src\<^sub>C f)) (D.inv (unit (trg\<^sub>C f)) \<cdot>\<^sub>D F \<epsilon> \<cdot>\<^sub>D \<Phi> (f, g))"
assms_2: shows "C \<notin> Liminf_llist Ns"
poly_lift_def: shows "poly_lift = map_poly (\<lambda>a. [:a:])"