hcju/isabelleps huggingface dataset:
verticesFrom_splitAt_v_fst: fixes f :: "face"   and v :: "nat" assumes "distinct (verticesFrom f v)" shows "fst (splitAt v (verticesFrom f v)) = []"
IH: assumes "distinct (map fst ps)"   and "k \<notin> set (map fst ps)" shows "fmap_of_list (insort_key fst (k, v) ps) = fmap_of_list ps(k $$:= v)"
bounded_by_nneg: fixes b :: "real"   and P :: "'a \<Rightarrow> real" assumes "bounded_by b P"   and "nneg P" shows "0 \<le> b"
Suc_le_length_iff: fixes n :: "nat"   and xs :: "'e list" shows "(Suc n \<le> length xs) = (\<exists>x ys. xs = x # ys \<and> n \<le> length ys)"
T_length_eq: fixes xs :: "'a list" shows "T_length xs = length xs + 1"
E_def: shows "E = lambda * (4 * deltaG TYPE('a)) + C + D"
elims: fixes x :: "'a \<Rightarrow> bool"   and xa :: "'a list"   and y :: "'a list option" assumes "delete_first x xa = y" obtains P :: "'a \<Rightarrow> bool" where "x = P__" and "xa = []" and "y = None"   | P :: "'a \<Rightarrow> bool" and xb :: "'a" and xs :: "'a list" where "x = P" and "xa = xb # xs" and "y = (if P xb then Some xs else map_option ((#) xb) (delete_first P xs))"
ring_simprules_20: fixes R :: "('b, 'c) ring_scheme"   and x :: "'b" assumes "ring R"   and "x \<in> carrier R" shows "\<ominus>\<^bsub>R\<^esub> (\<ominus>\<^bsub>R\<^esub> x) = x"
Unlock1SynchronizedFail: fixes uf :: "bool"   and xs :: "'addr val list"   and V :: "nat"   and a' :: "'addr"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and a :: "'addr"   and v :: "'addr val"   and h :: "'heap" assumes "uf"   and "xs ! V = Addr a'"   and "V < length xs" shows "uf,P,t \<turnstile>1 \<langle>insync\<^bsub>V\<^esub> (a) Val v,(h, xs)\<rangle> -\<lbrace>(UnlockFail, a')\<rbrace>\<rightarrow> \<langle>Throw (addr_of_sys_xcpt IllegalMonitorState),(h, xs)\<rangle>"
simps_2: fixes c :: "'b poly"   and n :: "nat"   and p :: "'b poly" shows "isnpolyh (CN c n p) = (\<lambda>k. k \<le> n \<and> isnpolyh c (Suc n) \<and> isnpolyh p n \<and> p \<noteq> 0\<^sub>p)"
nonInterference_path_to_Low: fixes s :: "'state"   and s' :: "'state"   and S :: "'node set"   and as :: "'edge list"   and as' :: "'edge list" assumes "s \<approx>\<^sub>L s'"   and "(_High_) \<notin> backward_slice S"   and "(_Low_) \<in> S"   and "(_Entry_) -as\<rightarrow>* (_Low_)"   and "preds (kinds as) s"   and "(_Entry_) -as'\<rightarrow>* (_Low_)"   and "preds (kinds as') s'" shows "transfers (kinds as) s \<approx>\<^sub>L transfers (kinds as') s'"
assms_2: shows "stack c\<^sub>2 \<lesssim> stack c\<^sub>3"
rep_eq: fixes x :: "('c, 'd) lifted_boolean_algebra"   and xa :: "('c, 'd) lifted_boolean_algebra" shows "(x < xa) = ((\<forall>f. Rep_lifted_boolean_algebra x f \<le> Rep_lifted_boolean_algebra xa f) \<and> \<not> (\<forall>f. Rep_lifted_boolean_algebra xa f \<le> Rep_lifted_boolean_algebra x f))"
zs_1: shows "len M j j zs \<le> len M j j ys"
shift_snth_ge: fixes xs :: "'d list"   and p :: "nat"   and s :: "'d stream" assumes "length xs \<le> p" shows "(xs @- s) !! p = s !! (p - length xs)"
hn_refine_cons_pre: fixes P :: "assn"   and P' :: "assn"   and c :: "'a Heap"   and Q :: "assn"   and R :: "'b \<Rightarrow> 'a \<Rightarrow> assn"   and m :: "'b nres" assumes "P \<Longrightarrow>\<^sub>t P'"   and "hn_refine P' c Q R m" shows "hn_refine P c Q R m"
inj_linear_imp_inv_linear: fixes f :: "'a \<Rightarrow> 'a" assumes "linear f"   and "inj f" shows "linear (inv f)"
rep_eq: fixes x :: "nat"   and xa :: "nat"   and xb :: "nat \<Rightarrow>\<^sub>0 'b" shows "lookup (lowerPowers x xa xb) = (\<lambda>xc. if xc \<in> {x..<x + xa} then 0::'b else lookup xb (dec_above x xa xc))"
top_unique: fixes x :: "'a"   and y :: "'a" shows "x \<squnion> - x = y \<squnion> - y"
case_2: fixes f1 :: "bool \<Rightarrow> 'a"   and f2 :: "nat \<Rightarrow> 'a"   and x2 :: "nat" shows "(case Indet x2 of Det x \<Rightarrow> f1 x | Indet x \<Rightarrow> f2 x) = f2 x2"
simps_25: fixes vc :: "'a"   and vd :: "bexp"   and ve :: "'a acom"   and vf :: "'a"   and v :: "char list"   and va :: "aexp"   and vb :: "'a" shows "({vc} WHILE vd DO ve {vf} \<sqsubseteq> v ::= va {vb}) = False"
not_alg_dvd_sum: fixes A :: "'a set"   and x' :: "'a"   and y :: "'b"   and f :: "'a \<Rightarrow> 'b" assumes "\<And>x. x \<in> A - {x'} \<Longrightarrow> y alg_dvd f x"   and "\<not> y alg_dvd f x'"   and "x' \<in> A"   and "finite A" shows "\<not> y alg_dvd sum f A"
dom_in_domains: fixes f :: "'a" assumes "H.domains f \<noteq> {}" shows "H.dom f \<in> H.domains f"
sylow_eq: fixes G :: "('a, 'b) monoid_scheme"   and p :: "nat"   and a :: "nat"   and m :: "nat" shows "sylow G p a m = (Group.group G \<and> sylow_axioms G p a m)"
finprod_0: fixes G :: "('c, 'd) monoid_scheme"   and f :: "nat \<Rightarrow> 'c" assumes "Group.comm_monoid G"   and "f \<in> {0} \<rightarrow> carrier G" shows "finprod G f {..0} = f 0"
f_inv_f: fixes x :: "'b" shows "hom (inv hom x) = x"
fimage_fconst_on_vsubset_const: fixes A :: "V"   and c :: "V"   and B :: "V" shows "fconst_on A c `\<^sub>\<bullet> B \<subseteq>\<^sub>\<circ> set {c}"
nolb_def: fixes f :: "atom fm"   and xs :: "real list"   and l :: "real"   and x :: "real" shows "nolb f xs l x = (\<forall>y\<in>{l<..<x}. y \<notin> {(r - \<langle>cs,xs\<rangle>) / c |r c cs. Less r (c # cs) \<in> set (atoms\<^sub>0 f) \<and> 0 < c})"
IH_1: fixes stk :: "int list" shows "ccomp c \<turnstile> (0, s1, stk) \<rightarrow>* (size (ccomp c), s2, stk)"
ring_ow_axioms: shows "ring_ow U (+\<^sub>o\<^sub>w) 0\<^sub>o\<^sub>w (-\<^sub>o\<^sub>w) uminus (*\<^sub>o\<^sub>w)"
igFreshIGAbs2STR_def: fixes MOD :: "('h, 'i, 'j, 'k, 'l, 'm, 'n, 'o, 'p) model_scheme" shows "igFreshIGAbs2STR MOD \<equiv> \<forall>ys y xs x X. igFresh MOD ys y X \<longrightarrow> igFreshAbs MOD ys y (igAbs MOD xs x X)"
asm: shows "good_context s \<and> s'' = delayed_pool_write s"
assms_2: shows "(f \<longlongrightarrow> l) (at x within T)"
an_Suc: fixes n :: "nat" shows "an (Suc n) = an n + 1"
NIS_short: shows "\<forall>t1 t2. NIA t1 t2 \<longrightarrow> NIC t1 t2"
simps_3: fixes cts :: "ct list"   and ts :: "t list" shows "checker_type_suffix (TopType cts) (Type ts) = ct_suffix cts (to_ct_list ts)"
runiq_alt: fixes R :: "('c \<times> 'd) set" shows "runiq R = (\<forall>x. trivial (R `` {x}))"
pruned_def: fixes \<alpha>n :: "'b \<Rightarrow> 'c list"   and defs :: "'b \<Rightarrow> 'c \<Rightarrow> 'd set"   and "uses" :: "'b \<Rightarrow> 'c \<Rightarrow> 'd set"   and phis :: "'b \<Rightarrow> 'c \<times> 'd \<Rightarrow> 'd list option"   and g :: "'b" shows "CFG_SSA_wf_base.pruned \<alpha>n defs uses phis g = (\<forall>n\<in>set (\<alpha>n g). \<forall>val. val \<in> CFG_SSA_base.phiDefs phis g n \<longrightarrow> CFG_SSA_wf_base.liveVal \<alpha>n defs uses phis g val)"
add_order_add_eq1: fixes s :: "'a"   and t :: "'a" assumes "add_order (s + t) = 1" shows "t = - s"
to_nat_from_nat: fixes j :: "'d"   and k :: "nat" assumes "mod_type_class.to_nat j = k" shows "mod_type_class.from_nat k = j"
is_none_bind: fixes f :: "'d option"   and g :: "'d \<Rightarrow> 'c option" shows "Option.is_none (f \<bind> g) = (Option.is_none f \<or> Option.is_none (g (the f)))"
distinct_G1: shows "valid_unMultigraph.is_trail G1 v ps v'"
simps_4: fixes E_ES :: "'a set"   and I_ES :: "'a set"   and O_ES :: "'a set"   and Tr_ES :: "'a list set"   and more :: "'b" shows "Tr_ES \<lparr>E_ES = E_ES, I_ES = I_ES, O_ES = O_ES, Tr_ES = Tr_ES, \<dots> = more\<rparr> = Tr_ES"
image_image_mset: fixes f :: "'b \<Rightarrow> 'a"   and g :: "'c \<Rightarrow> 'b"   and A :: "'c multiset" shows "image_mset f (image_mset g A) = {#f (g x). x \<in># A#}"
kernel_from_subgroup_generated: fixes S :: "'b set"   and G :: "('b, 'c) monoid_scheme"   and H :: "('d, 'e) monoid_scheme"   and f :: "'b \<Rightarrow> 'd" assumes "subgroup S G" shows "kernel (subgroup_generated G S) H f = kernel G H f \<inter> S"
permutes_empty: fixes p :: "'b \<Rightarrow> 'b" shows "(p permutes {}) = (p = id)"
f_def: shows "f \<equiv> \<lambda>k. [k] \<odot> 0\<^sup>\<infinity>"
EL\<^sub>N_def: fixes n :: "nat" shows "EL\<^sub>N n = measure_pmf.expectation (R\<^sub>N n 1 1) real"
succ_chain_rvar_valuation: fixes l :: "('i, 'a) state list"   and i :: "nat"   and j :: "nat" assumes "rel_chain l {(s, s'). \<triangle> (\<T> s) \<and> \<diamond> s \<and> \<Turnstile>\<^sub>n\<^sub>o\<^sub>l\<^sub>h\<^sub>s s \<and> \<nabla> s \<and> s \<succ>\<^sub>x s' \<and> \<B>\<^sub>i s' = \<B>\<^sub>i s \<and> \<U>\<^sub>c s' = \<U>\<^sub>c s}"   and "i \<le> j"   and "j < length l" shows "\<forall>x\<in>rvars (\<T> (l ! j)). \<langle>\<V> (l ! j)\<rangle> x = \<langle>\<V> (l ! i)\<rangle> x \<or> \<langle>\<V> (l ! j)\<rangle> x = the (\<B>\<^sub>l (l ! i) x) \<or> \<langle>\<V> (l ! j)\<rangle> x = the (\<B>\<^sub>u (l ! i) x)"
arc_univalent: fixes x :: "'a" assumes "arc x" shows "univalent x"
disjI: fixes P :: "bool"   and Q :: "bool" assumes "\<not> P \<Longrightarrow> Q" shows "P \<or> Q"
V'_def: shows "V' = V - attractor p (deadends p**) - attractor p** (deadends p****)"
lang\<^sub>M\<^sub>2\<^sub>L_rexp_of_rexp_of': fixes n :: "nat"   and \<phi> :: "'a formula" assumes "wf_formula n \<phi>" shows "lang n (rexp_of n \<phi>) - {[]} = lang n (rexp_of' n \<phi>) - {[]}"
get\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def: fixes element_ptr :: "(_) element_ptr"   and h :: "(_) heap" shows "get element_ptr h = Option.bind (get (cast element_ptr) h) cast"
finite_imageI: fixes F :: "'e set"   and h :: "'e \<Rightarrow> 'f" assumes "finite F" shows "finite (h ` F)"
that: fixes A :: "fm" assumes "x = \<lbrakk>\<guillemotleft>A\<guillemotright>\<rbrakk>e"   and "x' = \<lbrakk>\<guillemotleft>A(decode_Var v::=u)\<guillemotright>\<rbrakk>e" shows "thesis"
runit_simps_4: fixes f :: "'b" assumes "D.ide f" shows "D.dom \<r>\<^sub>D[f] = f \<star>\<^sub>D src\<^sub>D f"
pOp_assocTr43: fixes a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" assumes "a \<in> carrier M"   and "b \<in> carrier M"   and "c \<in> carrier M"   and "d \<in> carrier M" shows "a \<plusminus> b \<plusminus> (c \<plusminus> d) = a \<plusminus> (b \<plusminus> c) \<plusminus> d"
wfE_weaken: fixes \<Delta> :: "binding list"   and \<Gamma> :: "binding list"   and B :: "binding" assumes "\<Delta> @ \<Gamma> \<turnstile>\<^sub>w\<^sub>f"   and "\<Gamma> \<turnstile>\<^sub>w\<^sub>f\<^sub>B B" shows "\<up>\<^sub>e (Suc 0) 0 \<Delta> @ B \<Colon> \<Gamma> \<turnstile>\<^sub>w\<^sub>f"
low_mds_eq_sym: fixes mds :: "Mode \<Rightarrow> 'Var\<^sub>C set"   and mem\<^sub>1 :: "'Var\<^sub>C \<Rightarrow> 'Val"   and mem\<^sub>2 :: "'Var\<^sub>C \<Rightarrow> 'Val" assumes "conc.low_mds_eq mds mem\<^sub>1 mem\<^sub>2" shows "conc.low_mds_eq mds mem\<^sub>2 mem\<^sub>1"
M: shows "n = 0 \<or> n \<in> {0<..length (xs @ [y])}"
surjective: fixes r :: "'a prog_scheme" shows "r = \<lparr>para = para r, body = body r, \<dots> = more r\<rparr>"
splitAt_dist_ram21: fixes vs :: "'c list"   and a :: "'c list"   and ram1 :: "'c"   and b :: "'c list"   and ram2 :: "'c"   and c :: "'c list" assumes "distinct vs"   and "vs = a @ ram1 # b @ ram2 # c" shows "(a, b) = splitAt ram1 (fst (splitAt ram2 vs))"
remove_child_heap_is_wellformed_preserved_3: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and child :: "(_) node_ptr"   and h' :: "(_) heap" assumes "heap_is_wellformed h"   and "h \<turnstile> remove_child ptr child \<rightarrow>\<^sub>h h'"   and "known_ptrs h"   and "type_wf h" shows "heap_is_wellformed h'"
pos: fixes t :: "real" assumes "t \<in> T" shows "0 < t"
sources_sinks: fixes I :: "('c \<times> 'c) set"   and D :: "'d \<Rightarrow> 'c"   and u :: "'c"   and xs :: "'d list" shows "sources I D u xs = sinks (I\<inverse>) D u (rev xs)"
that: shows "\<bar>x - aform_val e (fst Xe)\<bar> \<le> snd Xe"
assms: shows "\<exists>p i. j = make_full_observation e None r p i"
in_degree_app_iso_eq: fixes hom :: "('a, 'b, 'e, 'f) digraph_isomorphism"   and u :: "'a" assumes "digraph_isomorphism hom"   and "u \<in> verts G" shows "in_degree (app_iso hom G) (iso_verts hom u) = in_degree G u"
of_nat_monomial: fixes p :: "nat" shows "of_nat p = monomial p (0::'a)"
eqfs: shows "\<forall>w. disc_rfr_proc r (Suc 0) w = geom_proc (Suc 0) w"
emeasure_eq_1_AE: fixes S :: "'a set"   and M :: "'a pmf" assumes "S \<in> measure_pmf.events M"   and "AE x in measure_pmf M. x \<in> S" shows "emeasure (measure_pmf M) S = 1"
isDERIV_FDERIV_floatarith: fixes x :: "nat"   and fa :: "floatarith"   and vs :: "real list"   and ds :: "floatarith list"   and xs :: "nat list" assumes "isDERIV x fa vs"   and "\<And>i. i < length ds \<Longrightarrow> isDERIV x (ds ! i) vs"   and "length xs = length ds" shows "isDERIV x (FDERIV_floatarith fa xs ds) vs"
colossless: assumes "x = None"   and "\<not> colossless_gpv \<I>_full gpv" shows "thesis"
test2: fixes s :: "real set"   and g :: "real \<Rightarrow> 'a"   and u :: "real"   and v :: "real"   and x :: "real" assumes "\<And>x. x \<in> {0..1} - s \<Longrightarrow> g differentiable at x"   and "finite s"   and "u \<in> {0..1}"   and "v \<in> {0..1}"   and "u \<le> v"   and "x \<in> {0..1}"   and "x \<notin> (\<lambda>t. (v - u) *\<^sub>R t + u) -` s" shows "vector_derivative (\<lambda>x. g ((v - u) * x + u)) (at x within {0..1}) = (v - u) *\<^sub>R vector_derivative g (at ((v - u) * x + u) within {0..1})"
sdrop_wait: fixes \<phi> :: "'a stream \<Rightarrow> bool"   and xs :: "'a stream" assumes "ev \<phi> xs" shows "\<phi> (sdrop (wait \<phi> xs) xs)"
mk_type_def: fixes ty :: "typ" shows "mk_type ty = Ct STR ''Pure.type'' (itselfT ty)"
simps_2: fixes I :: "('a \<times> 'a) set"   and D :: "'b \<Rightarrow> 'a"   and u :: "'a"   and xs :: "'b list"   and x :: "'b" shows "sinks I D u (xs @ [x]) = (if (u, D x) \<in> I \<or> (\<exists>v\<in>sinks I D u xs. (v, D x) \<in> I) then insert (D x) (sinks I D u xs) else sinks I D u xs)"
distinct_map_filter: fixes f :: "'e \<Rightarrow> 'd"   and xs :: "'e list"   and P :: "'e \<Rightarrow> bool" assumes "distinct (map f xs)" shows "distinct (map f (filter P xs))"
trisuma_perm_231: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p"   and E :: "'p"   and F :: "'p" assumes "A B C TriSumA D E F" shows "B C A TriSumA D E F"
mult_subgroups: fixes H :: "'a set"   and K :: "'a set" assumes "subgroup H (add_monoid G)"   and "subgroup K (add_monoid G)" shows "subgroup (H <#>\<^bsub>add_monoid G\<^esub> K) (add_monoid G)"
listset_empty_iff: fixes xs :: "'b set list" shows "(listset xs = {}) = ({} \<in> set xs)"
domains_subsume_globally_def: fixes subst :: "('a, 'b, 'c) expr \<Rightarrow> 'c \<Rightarrow> ('a, 'b, 'c) expr \<Rightarrow> ('a, 'b, 'c) expr"   and s :: "'a \<Rightarrow> (('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('a, 'b, 'c) expr) option" assumes "substitution subst" shows "substitution.domains_subsume_globally s = (\<forall>r ls. s r = Some ls \<longrightarrow> substitution.domains_subsume ls)"
Sum_def: shows "Sum = [:0::'a, 1::'a:] \<cdot>\<^sub>m 1\<^sub>m n + map_mat (\<lambda>a. [:- a:]) A"
floor_Fract: fixes a :: "int"   and b :: "int" shows "\<lfloor>Fract a b\<rfloor> = a div b"
IntE: fixes c :: "'b"   and A :: "'b set"   and B :: "'b set" assumes "c \<in> A \<inter> B" obtains "c \<in> A" and "c \<in> B"
intros_2: fixes step :: "'e \<Rightarrow> 'f \<Rightarrow> 'e"   and out :: "'e \<Rightarrow> 'f \<Rightarrow> 'g"   and s\<^sub>0 :: "'e"   and xps :: "('f \<times> 'g) list"   and uu :: "('f \<times> 'g) set"   and s :: "'e"   and x :: "'f" assumes "c_failuresp step out s\<^sub>0 xps uu"   and "s = foldl step s\<^sub>0 (map fst xps)" shows "c_failuresp step out s\<^sub>0 (xps @ [(x, out s x)]) {(y, p). p \<noteq> out (step s x) y}"
translation_exist: fixes M :: "'c set"   and composition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit :: "'c"   and f :: "'c \<Rightarrow> 'c" assumes "left_translations_of_monoid M composition unit"   and "f \<in> left_translations_of_monoid.translation M composition ` M" shows "\<exists>a\<in>M. f = left_translations_of_monoid.translation M composition a"
ro_unowned: shows "read_only_unowned \<S> (t # ts)"
length_init: shows "\<forall>x\<in>set signs. length x = length qs"
psubstBinp_empBinp: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set option" shows "empBinp %%[rho] = empBinp"
schem_valid: shows "valid ($\<phi> p (singleton \<theta>) \<leftrightarrow> $\<phi> p (singleton \<theta>'))"
int_emp: fixes i :: "'a" assumes "(0::'a) < i" shows "{..<i} \<noteq> {}"
eq_fin_le_inf_concat_end: fixes ind :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and w\<^sub>1 :: "'a list"   and w\<^sub>2 :: "'a list"   and w :: "'a stream" assumes "traces ind"   and "(traces.eq_swap ind)\<^sup>*\<^sup>* w\<^sub>1 w\<^sub>2" shows "traces.le_inf ind (w\<^sub>1 @- w) (w\<^sub>2 @- w)"
action_tid_def: fixes E :: "('a \<times> ('b, 'a) obs_event action) llist"   and a :: "nat" shows "action_tid E a = fst (lnth E a)"
chamber_map: fixes C :: "'a set" assumes "folding_g.chamber C" shows "folding_g.chamber (g ` C)"
v0_v1_Digraph_axioms: shows "v0_v1_Digraph G v0 v1"
ei_def: shows "ei = (\<lambda>j. local.xij_f i (i + j) * local.pprod k i (i + j))"
shift_bounds_nat_ivl: fixes g :: "nat \<Rightarrow> 'b"   and m :: "nat"   and k :: "nat"   and n :: "nat" shows "sum g {m + k..<n + k} = (\<Sum>i = m..<n. g (i + k))"
Inf_pres_map_dual_var: fixes f :: "'c \<Rightarrow> 'd" shows "Inf_pres f = Sup_pres (\<partial>\<^sub>F f)"
size_7: fixes x1 :: "lit" shows "size (Litv x1) = 0"
leq_antisym: fixes x :: "'a llist"   and y :: "'a llist" assumes "lprefix x y"   and "lprefix y x" shows "x = y"
right_iff_left: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<sigma> :: "'b"   and \<tau> :: "'b" assumes "weak_composition V H" shows "weak_composition.right V H \<sigma> \<tau> = weak_composition.left V H \<tau> \<sigma>"
fully_abstract_wrt_preorders_impl_source_target_relation_is_trans: fixes SRel :: "('procS \<times> 'procS) set"   and TRel :: "('procT \<times> 'procT) set" assumes "\<forall>S1 S2. ((S1, S2) \<in> SRel) = ((\<lbrakk>S1\<rbrakk>, \<lbrakk>S2\<rbrakk>) \<in> TRel)" shows "\<exists>Rel. (\<forall>S. (SourceTerm S, TargetTerm (\<lbrakk>S\<rbrakk>)) \<in> Rel) \<and> SRel = _Collect (S1, S2) ((SourceTerm S1, SourceTerm S2) \<in> Rel) \<and> TRel = _Collect (T1, T2) ((TargetTerm T1, TargetTerm T2) \<in> Rel) \<and> (refl SRel \<and> trans TRel) = trans (Rel \<union> _Collect (P, Q) (\<exists>S. \<lbrakk>S\<rbrakk> \<in>T P \<and> S \<in>S Q))"
assms: shows "wadjust_loop_right_move m rs (c, Oc # list)"
sstar_rec_isotone: fixes y :: "'a" shows "isotone (Sf y)"
Abs_hf_cases: fixes x :: "hf" obtains y :: "nat" where "x = Abs_hf y" and "y \<in> UNIV"
HC_intros_9: fixes F :: "'b formula"   and G :: "'b formula" shows "AX10 \<turnstile>\<^sub>H ((F::'b formula) \<^bold>\<and> (G::'b formula)) \<^bold>\<rightarrow> G"
subsetD: fixes A :: "'h set"   and B :: "'h set"   and c :: "'h" assumes "A \<subseteq> B"   and "c \<in> A" shows "c \<in> B"
GSubspace_eigenspace: fixes G :: "'a set"   and smult :: "('b, 'a) aezfun \<Rightarrow> 'c \<Rightarrow> 'c"   and V :: "'c set"   and T :: "'c \<Rightarrow> 'c"   and e :: "'b" assumes "FGModuleHom G smult V smult T" shows "RModule (aezfun_setspan G) smult {v \<in> V. T v = aezfun_scalar_mult.fsmult smult e v} \<and> {v \<in> V. T v = aezfun_scalar_mult.fsmult smult e v} \<subseteq> V"
nth_rotater: assumes "n < length xs" shows "rotater 1 xs ! n = xs ! ((n + (length xs - 1 mod length xs)) mod length xs)"
ex_less_Suc: fixes j :: "nat"   and P :: "nat \<Rightarrow> bool" shows "(\<exists>i<Suc j. P i) = (P 0 \<or> (\<exists>i<j. P (Suc i)))"
succ_ex_1: fixes R :: "('c \<Rightarrow> real) set" assumes "R \<in> \<R>" shows "succ \<R> R \<in> \<R>"
assms: shows "Controller_L s (fin_inf_append [Zero] l) l z"
full_functor_axioms: shows "full_functor (\<cdot>\<^sub>B) (\<cdot>) UP"
id: shows "adjoint M0 * M0 + adjoint M1 * M1 = 1\<^sub>m d"
wf_fun_pair_ineqs_map: fixes X :: "'a set"   and A :: "('fun, 'a) strand_step list"   and Y :: "'a list"   and t :: "('fun, 'a) Term.term"   and s :: "('fun, 'a) Term.term"   and D :: "(('fun, 'a) Term.term \<times> ('fun, 'a) Term.term) list" assumes "wf\<^sub>s\<^sub>t X A" shows "wf\<^sub>s\<^sub>t X (map (\<lambda>d. \<forall>Y\<langle>\<or>\<noteq>: [(pair (t, s), pair d)]\<rangle>\<^sub>s\<^sub>t) D @ A)"
bigo_const: fixes c :: "'c"   and F :: "'b filter" shows "(\<lambda>_. c) \<in> O[F](\<lambda>_. 1::'c)"
loop_iterate_equiv: fixes \<alpha> :: "game" shows "game_equiv \<alpha>** (Skip \<union>\<union> (\<alpha> ;; \<alpha>**))"
cross_ratio_not_inf: fixes z :: "complex_homo"   and w :: "complex_homo"   and u :: "complex_homo"   and v :: "complex_homo" assumes "z \<noteq> w"   and "u \<noteq> v" shows "cross_ratio z u v w \<noteq> \<infinity>\<^sub>h"
WT_ideal: shows "\<I>_ideal \<oplus>\<^sub>\<I> (\<I>_common_core \<oplus>\<^sub>\<I> \<I>_common_rest) \<turnstile>res ideal_resource \<surd>"
Eq\<kappa>_prop: fixes v :: "i"   and x :: "\<kappa>"   and y :: "\<kappa>"   and \<phi> :: "\<kappa> \<Rightarrow> \<o>" assumes "[x \<^bold>=\<^sub>\<kappa> y in v]" shows "[\<phi> x in v] = [\<phi> y in v]"
coprimeI: fixes p :: "'a"   and q :: "'a" assumes "\<And>r. \<lbrakk>r dvd p; r dvd q\<rbrakk> \<Longrightarrow> r dvd (1::'a)" shows "comm_monoid_mult_class.coprime p q"
empty_plan_is_valid: fixes PROB :: "'a set" shows "[] \<in> valid_plans PROB"
return: shows "spmf (funct m1 m2 \<bind> (\<lambda>(o1, o2). S2 m2 o2 \<bind> D)) True = spmf (funct m1 m2 \<bind> (\<lambda>(o1, o2). S2 m2 o2 \<bind> (\<lambda>view. D view \<bind> return_spmf))) True"
eventually_lbound_finite: fixes A :: "'a set set"   and P :: "'a set \<Rightarrow> bool" assumes "finite A"   and "A \<noteq> {}"   and "A \<subseteq> I" shows "eventually P (lbound A) = (\<exists>w0. w0 \<in> I \<and> (\<forall>a\<in>A. w0 \<subseteq> a) \<and> (\<forall>w. w \<subseteq> w0 \<and> w \<in> I \<longrightarrow> P w))"
outgoing_def: fixes v :: "nat" shows "h.cf.outgoing v \<equiv> {(v, u) |u. (v, u) \<in> h.cf.E}"
assms: fixes v :: "i" assumes "[\<chi> in v]" shows "[\<phi> \<^bold>\<rightarrow> \<^bold>\<box>\<psi> in v]"
assms_3: shows "r \<in> dgrad_p_set d m"
union: fixes r :: "('a \<times> 'a) set"   and A :: "'a set"   and B :: "'a set" assumes "MaxR r" shows "MaxR.MaxR_opt r (A \<union> B) = (case MaxR.MaxR_opt r A of None \<Rightarrow> MaxR.MaxR_opt r B | Some mA \<Rightarrow> Some (case MaxR.MaxR_opt r B of None \<Rightarrow> mA | Some mB \<Rightarrow> MaxR.maxR r mA mB))"
c: fixes a :: "'a extended \<Rightarrow> 'a sort_key extended" assumes "a_conditions (insert_messages (M \<union> {Delete m})) a" shows "sorted_wrt (<) (map a (ext_ids t))"
step: fixes x :: "'a node \<times> 'a node set" assumes "((\<mu> R\<^sub>r \<eta> \<in> old (fst x) \<longrightarrow> {\<mu>, \<eta>} \<subseteq> old (fst x) \<union> new (fst x) \<or> \<eta> \<in> old (fst x) \<union> new (fst x) \<and> \<mu> R\<^sub>r \<eta> \<in> next (fst x)) \<and> (\<forall>nd\<in>snd x. \<mu> R\<^sub>r \<eta> \<in> old nd \<and> name nd \<in> incoming (fst x) \<longrightarrow> {\<mu>, \<eta>} \<subseteq> old nd \<or> \<eta> \<in> old nd \<and> \<mu> R\<^sub>r \<eta> \<in> old (fst x) \<union> new (fst x))) \<and> (\<forall>nd\<in>snd x. \<forall>nd'\<in>snd x. \<mu> R\<^sub>r \<eta> \<in> old nd \<and> name nd \<in> incoming nd' \<longrightarrow> {\<mu>, \<eta>} \<subseteq> old nd \<or> \<eta> \<in> old nd \<and> \<mu> R\<^sub>r \<eta> \<in> old nd') \<and> ((\<forall>nm\<in>incoming (fst x). nm < name (fst x)) \<and> 0 < name (fst x) \<and> (\<forall>q\<in>snd x. name q < name (fst x) \<and> (\<forall>nm\<in>incoming q. nm < name (fst x)))) \<and> expand_name_ident (snd x)" shows "f x \<le> SPEC (\<lambda>r. \<forall>nd\<in>snd r. \<forall>nd'\<in>snd r. \<mu> R\<^sub>r \<eta> \<in> old nd \<and> name nd \<in> incoming nd' \<longrightarrow> {\<mu>, \<eta>} \<subseteq> old nd \<or> \<eta> \<in> old nd \<and> \<mu> R\<^sub>r \<eta> \<in> old nd')"
assms_5: shows "InvariantNoDecisionsWhenConflict (getF state) (getM state) (currentLevel (getM state))"
gt_sub_fun: fixes s :: "('s, 'v) tm"   and t :: "('s, 'v) tm" shows "App s t >\<^sub>t s"
OUT: fixes x :: "'v" shows "d_OUT f x \<le> weight \<Gamma> x"
K_def: shows "K = (\<lambda>(x, r). ball x r) ` (\<Union> (f ` (\<rat> \<inter> {0<..})) \<times> \<rat>)"
assms: shows "(s, a, s') \<in> automaton.trans (pnet np \<langle>i; R\<rangle>)"