hcju/isabelleps huggingface dataset:
get_min_min: fixes h :: "'b heap"   and x :: "'b" assumes "h \<noteq> Empty"   and "pheap h"   and "x \<in># mset_heap h" shows "get_min h \<le> x"
primeI: fixes P :: "state \<Rightarrow> bool" shows "\<turnstile> \<box>P \<longrightarrow> Init P$"
simulation1: fixes trsys1 :: "'a \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> bool"   and trsys2 :: "'c \<Rightarrow> 'd \<Rightarrow> 'c \<Rightarrow> bool"   and bisim :: "'a \<Rightarrow> 'c \<Rightarrow> bool"   and tlsim :: "'b \<Rightarrow> 'd \<Rightarrow> bool"   and s1 :: "'a"   and s2 :: "'c"   and tl1 :: "'b"   and s1' :: "'a" assumes "bisimulation trsys1 trsys2 bisim tlsim"   and "bisim s1 s2"   and "trsys1 s1 tl1 s1'" shows "\<exists>s2' tl2. trsys2 s2 tl2 s2' \<and> bisim s1' s2' \<and> tlsim tl1 tl2"
is_imag_arg2: fixes z :: "complex" assumes "is_imag z"   and "z \<noteq> 0" shows "arg z = pi / 2 \<or> arg z = - pi / 2"
cong: fixes oalist_of_list :: "('e \<times> 'f) list \<times> 'c \<Rightarrow> 'd" shows "oalist_abstract.empty oalist_of_list \<equiv> oalist_abstract.empty oalist_of_list"
valid_typing: fixes \<Gamma> :: "(name \<times> ty) list"   and t :: "trm"   and T :: "ty" assumes "\<Gamma> \<turnstile> t : T" shows "valid \<Gamma>"
antipar_1: shows "trg (trg f) = src (trg f)"
mset_sublist_eq: fixes xs' :: "'c list"   and xs :: "'c list"   and lo :: "nat"   and hi :: "nat" assumes "mset xs' = mset xs"   and "\<And>i. i < lo \<Longrightarrow> xs' ! i = xs ! i"   and "\<And>j. \<lbrakk>hi < j; j < length xs\<rbrakk> \<Longrightarrow> xs' ! j = xs ! j"   and "lo \<le> hi"   and "hi < length xs" shows "set (WB_Sort.sublist xs' lo hi) = set (WB_Sort.sublist xs lo hi)"
no_loops: fixes v :: "'c" shows "(v, v) \<notin> E\<^bsub>T\<^esub>"
axioms_2: fixes return :: "'a \<Rightarrow> 'b"   and bind :: "'b \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b"   and get :: "('c \<Rightarrow> 'b) \<Rightarrow> 'b"   and put :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and sample :: "'d pmf \<Rightarrow> ('d \<Rightarrow> 'b) \<Rightarrow> 'b" assumes "monad_state_prob return bind get put sample" shows "monad_prob return bind sample"
prob_space_bind: fixes N :: "'a \<Rightarrow> 'c measure"   and S :: "'c measure" assumes "AE x in f. prob_space (N x)"   and "f.random_variable (subprob_algebra S) N" shows "prob_space (f \<bind> N)"
simps: fixes weights :: "'a \<Rightarrow> 'a \<Rightarrow> 'b antichain"   and a1 :: "'a"   and a2 :: "'a"   and a3 :: "('a \<times> 'b \<times> 'a) list" assumes "graph weights" shows "graph.path weights a1 a2 a3 = ((\<exists>l1 l2. a1 = l1 \<and> a2 = l2 \<and> a3 = [] \<and> l1 = l2) \<or> (\<exists>l1 l2 xs lbl l3. a1 = l1 \<and> a2 = l3 \<and> a3 = xs @ [(l2, lbl, l3)] \<and> graph.path weights l1 l2 xs \<and> lbl \<in>\<^sub>A weights l2 l3))"
Cauchy_iff: fixes X :: "nat \<Rightarrow> 'b" shows "Cauchy X = (\<forall>e>0. \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. norm (X m - X n) < e)"
kingman_theorem_nonergodic_3: fixes u :: "nat \<Rightarrow> 'a \<Rightarrow> real" assumes "subcocycle u"   and "- \<infinity> < subcocycle_avg_ereal u" shows "(\<lambda>n. \<integral>\<^sup>+ x. ennreal \<bar>u n x / real n - subcocycle_lim u x\<bar> \<partial>M) \<longlonglongrightarrow> 0"
simps_4: fixes \<A> :: "'a ltl set"   and \<phi> :: "'a ltl"   and \<psi> :: "'a ltl" shows "\<A> \<Turnstile>\<^sub>P \<phi> or \<psi> = (\<A> \<Turnstile>\<^sub>P \<phi> \<or> \<A> \<Turnstile>\<^sub>P \<psi>)"
intro: fixes R :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "wfP R" shows "well_founded R"
sub: fixes i :: "name"   and t :: "trm"   and u :: "trm"   and v :: "trm"   and x :: "trm" assumes "atom i \<sharp> tms" shows "(P t u v)(i::=x) = P (subst i x t) (subst i x u) (subst i x v)"
f_shrink_nth: fixes n :: "nat"   and xs :: "'d message_af list"   and k :: "nat" assumes "n < length xs div k" shows "xs \<div> k ! n = last_message (xs \<up> (n * k) \<down> k)"
partially_ordered_set_axioms: shows "partially_ordered_set A (\<lambda>x y. y \<sqsubseteq> x)"
WHILE_bounded_aux1: fixes cond :: "'c \<Rightarrow> bool"   and n :: "nat"   and step :: "'c \<Rightarrow> 'c nres"   and m :: "'c nres" shows "filter_nb cond (bounded_WHILE n cond step m) \<le> m \<bind> WHILE cond step"
asymptotics6: shows "\<forall>\<^sub>F x in at_top. hb / ln x powr (1 + e) < b / 2"
complement_1: fixes x :: "'a"   and y :: "'a" shows "- - x = - (- x \<squnion> - y) \<squnion> - (- x \<squnion> - - y)"
SC_siso_imp_SC_ZObis: fixes c :: "('test, 'atom) com" assumes "SC_siso c" shows "SC_ZObis c"
nonneg_bounded: shows "\<exists>K\<ge>0. \<forall>a b. norm (cblinfun_apply (f a) b) \<le> norm a * norm b * K"
simps_2: fixes uu :: "int" shows "(Num_arc uu \<le> MinInfty) = False"
sum_wt_ParT_sub_notWtFT_finished: fixes cl :: "('test, 'atom, 'choice) cmd list"   and n :: "nat"   and I :: "nat set"   and s :: "'state" assumes "properL cl"   and "WtFT cl \<noteq> 1"   and "n < length cl"   and "finished (cl ! n)"   and "I \<subseteq> {..<brn (cl ! n)}" shows "sum (wt (ParT cl) s) ((+) (brnL cl n) ` I) = 0"
l2_inv8_def: shows "l2_inv8 \<equiv> {s. secret s \<subseteq> {Exp (Exp Gen (NonceF (R $ N))) (NonceF (R' $ N')) |N N' R R'. R = test \<and> R' \<in> partners}}"
Map_comp: fixes f :: "('a, 'a set) Maps.arr"   and g :: "('a, 'a set) Maps.arr" assumes "Maps.arr f"   and "Maps.arr g"   and "Maps.Dom g = Maps.Cod f" shows "Maps.Map (g \<odot> f) = Maps.Comp (Maps.Map g) (Maps.Map f)"
arrow: fixes D :: "'ag \<Rightarrow> 'ag \<Rightarrow> 'ag"   and C :: "'ah \<Rightarrow> 'ah \<Rightarrow> 'ah"   and F :: "'ag \<Rightarrow> 'ah"   and x :: "'ag"   and y :: "'ah"   and e :: "'ah" assumes "arrow_from_functor D C F x y e" shows "partial_magma.ide D x \<and> partial_magma.in_hom C e (F x) y"
H: fixes n :: "int" assumes "\<And>i. \<bar>i\<bar> < \<bar>n\<bar> \<Longrightarrow> P i" shows "P n"
\<alpha>'_not_neg: shows "\<not> \<alpha>' < 0"
is_lasso_pre_def: fixes L :: "'Q lasso" shows "is_lasso_pre L \<equiv> lasso_v0 L \<in> V0 \<and> path E (lasso_v0 L) (lasso_reach L) (lasso_va L) \<and> path E (lasso_va L) (lasso_cycle L) (lasso_va L)"
lift_poly_fun_2_def: fixes f :: "('a \<Rightarrow>\<^sub>0 'c) \<Rightarrow> ('a \<Rightarrow>\<^sub>0 'c) \<Rightarrow> 'a \<Rightarrow>\<^sub>0 'c"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" shows "lift_poly_fun_2 f p q = atomize_poly (mapp_2 (\<lambda>_. f) (vectorize_poly p) (vectorize_poly q))"
cons: fixes a :: "'c \<times> 'd"   and A :: "('c, 'd) nba"   and p :: "'d"   and r :: "('c \<times> 'd) list" assumes "fst a \<in> alphabet A \<and> snd a \<in> transition A (fst a) p"   and "path A r (snd a)" shows "path A (a # r) p"
on_finish_refine: fixes v :: "'a"   and s0 :: "('a, 'c) state_scheme"   and s :: "('a, 'c) state_scheme"   and vi :: "'a"   and si :: "('a, 'd) simple_state_scheme"   and si' :: "('a, 'd) simple_state_scheme"   and s' :: "('a, 'c) state_scheme" assumes "a.a.gen_dfs.pre_finish v s0 s"   and "(vi, v) \<in> Id"   and "(si, s) \<in> \<langle>ES\<rangle>simple_state_rel"   and "(si', s') \<in> \<langle>ES\<rangle>simple_state_rel"   and "nf_inres (gbs_finish a.a.gbs v s) s'" shows "on_finish parami vi si' \<le> \<Down> ES (on_finish param v s')"
ex_star_eq: fixes P :: "'c star \<Rightarrow> bool" shows "(\<exists>x. P x) = (\<exists>X. P (star_n X))"
plane_rel_def: fixes A :: "('c \<times> 'd) set" shows "\<langle>A\<rangle>plane_rel = \<langle>A\<rangle>sctn_rel O br plane_of top"
bres_galois_imp: fixes x :: "'b \<times> 'c"   and y :: "'b \<times> 'c"   and z :: "'b \<times> 'c" assumes "less_eq_prod (times_prod' x y) z" shows "less_eq_prod y (prod.bres x z)"
list_all2_append: fixes xs :: "'a list"   and ys :: "'b list"   and P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and us :: "'a list"   and vs :: "'b list" assumes "length xs = length ys" shows "list_all2 P (xs @ us) (ys @ vs) = (list_all2 P xs ys \<and> list_all2 P us vs)"
N\<V>_inter_E1_is_N\<V>1: shows "N\<^bsub>\<V>\<^esub> \<inter> E\<^bsub>ES1\<^esub> = N\<^bsub>\<V>1\<^esub>"
SubstFormP_Neg: fixes v :: "tm"   and i :: "tm"   and x :: "tm"   and x' :: "tm" shows "{SubstFormP v i x x'} \<turnstile> SubstFormP v i (Q_Neg x) (Q_Neg x')"
manifold_eucl_carrier: shows "manifold_eucl.dest.carrier = UNIV"
shows_list_gen_assoc: fixes xs :: "'c list"   and elt :: "'c \<Rightarrow> char list \<Rightarrow> char list"   and e :: "char list"   and l :: "char list"   and sep :: "char list"   and r :: "char list"   and s :: "char list"   and t :: "char list" assumes "\<And>r s. \<forall>x\<in>set xs. elt x r @ s = elt x (r @ s)" shows "shows_list_gen elt e l sep r xs s @ t = shows_list_gen elt e l sep r xs (s @ t)"
and_and_not: fixes a :: "'b word"   and b :: "'b word" shows "(a && b) && ~~ b = 0"
INT_I: fixes A :: "'a set"   and b :: "'b"   and B :: "'a \<Rightarrow> 'b set" assumes "\<And>x. x \<in> A \<Longrightarrow> b \<in> B x" shows "b \<in> \<Inter> (B ` A)"
that: fixes p' :: "int poly" assumes "p = of_int_poly p'" shows "thesis"
exec_gpv_def: fixes callee :: "'d \<Rightarrow> 'b \<Rightarrow> ('c \<times> 'd) spmf" shows "exec_gpv callee \<equiv> curry (spmf.fixp_fun (\<lambda>exec_gpv (c, s). the_gpv c \<bind> case_generat (\<lambda>x. return_spmf (x, s)) (\<lambda>out c. callee s out \<bind> (\<lambda>(x, y). curry exec_gpv (c x) y))))"
identify_block_le: fixes A :: "'b mat"   and i :: "nat" shows "??.Jordan_Normal_Form_Existence.identify_block A i \<le> i"
simps_2: fixes x :: "'c"   and xs :: "'c list" shows "distinct (x # xs) = (x \<notin> set xs \<and> distinct xs)"
d1_2: shows "\<exists>j<e_length (f \<triangleright> n\<^sub>1). 0 < the (eval r_result1 [e_length (f \<triangleright> n\<^sub>1), i, e_take (Suc j) (f \<triangleright> n\<^sub>1)])"
add_decreasing2: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and c :: "'b"   and a :: "'b"   and b :: "'b" assumes "class.ordered_comm_monoid_add plus zero less_eq less"   and "less_eq c zero"   and "less_eq a b" shows "less_eq (plus a c) b"
transpose_carrier_mat: fixes A :: "'b mat"   and nc :: "nat"   and nr :: "nat" shows "(A\<^sup>T \<in> carrier_mat nc nr) = (A \<in> carrier_mat nr nc)"
invariantI: fixes F :: "'b program"   and A :: "'b set" assumes "Init F \<subseteq> A"   and "F \<in> stable A" shows "F \<in> invariant A"
COMP_def: fixes g :: "(bool, 'arr list) concrete_category.arr"   and f :: "(bool, 'arr list) concrete_category.arr" shows "g \<cdot> f \<equiv> if (f \<noteq> Null \<and> Dom f \<in> {False, True} \<and> Cod f \<in> {False, True} \<and> Map f \<in> Collect (G.path (Dom f) (Cod f))) \<and> (g \<noteq> Null \<and> Dom g \<in> {False, True} \<and> Cod g \<in> {False, True} \<and> Map g \<in> Collect (G.path (Dom g) (Cod g))) \<and> Dom g = Cod f then MkArr (Dom f) (Cod g) (Map f @ Map g) else Null"
assms: shows "(\<forall>va. virt_to_phys va (mmu s2) (mem s1) = virt_to_phys va (mmu s2) (mem s2)) \<and> (\<forall>pa. (\<exists>va b. virt_to_phys va (mmu s2) (mem s2) = Some (pa, b) \<and> mmu_readable (get_acc_flag b) 10) \<longrightarrow> mem_equal s1 s2 pa) \<and> mmu s1 = mmu s2 \<and> virt_to_phys va (mmu s2) ((mem s1)(10 := mem s1 10(addr \<mapsto> val), 11 := (mem s1 11)(addr := None))) = Some (pa, b) \<and> mmu_readable (get_acc_flag b) 10"
stabiliser_subgroup_corollary: fixes G :: "('d, 'e) monoid_scheme"   and action :: "'d \<Rightarrow> 'f \<Rightarrow> 'f"   and g :: "'d"   and h :: "'d"   and x :: "'f" assumes "orbit_stabiliser G action"   and "g \<in> carrier G"   and "h \<in> carrier G" shows "(action g x = action h x) = (inv\<^bsub>G\<^esub> g \<otimes>\<^bsub>G\<^esub> h \<in> orbit_stabiliser.stabiliser G action x)"
head_isnpolyh: fixes p :: "poly"   and n0 :: "nat" assumes "isnpolyh p n0" shows "isnpolyh (head p) n0"
i_cut_Int_conv_1: fixes I :: "'a set"   and t :: "'a" shows "I \<down>\<le> t = I \<inter> {..t}"
span_some_basis: fixes V :: "'b set" assumes "local.subspace V" shows "local.span (some_basis V) = V"
simps_9: fixes t :: "'c local.term"   and u :: "'c local.term"   and v :: "'c local.term" shows "Arr \<^bold>\<a>\<^bold>[t, u, v\<^bold>] = (Arr t \<and> Arr u \<and> Arr v)"
natural_extension_map_def: shows "natural_extension_map = int_left_shift"
updates_disjoint: fixes X :: "'a set"   and x :: "'a" assumes "finite X"   and "x \<notin> X" shows "fmupd x True ` {s. fmdom' s = X} \<inter> fmupd x False ` {s. fmdom' s = X} = {}"
local_quality_increases: fixes i :: "nat" shows "paodv i \<TTurnstile>\<^sub>A (recvmsg rreq_rrep_sn \<rightarrow>) onll \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V (\<lambda>((\<xi>, uu_), uu_, \<xi>', uu_). quality_increases \<xi> \<xi>')"
Approximation1_5Tr4: fixes n :: "nat"   and vv :: "nat \<Rightarrow> 'a \<Rightarrow> ant"   and aa :: "'a"   and x :: "'a"   and j :: "nat"   and m :: "nat" assumes "vals_nonequiv K (Suc n) vv"   and "aa \<in> carrier K"   and "x \<in> carrier K"   and "Ostrowski_elem K (Suc n) vv x"   and "j \<le> Suc n" shows "vv j (aa \<cdot>\<^sub>r x^\<^bsup>K m\<^esup>) = vv j aa + int m *\<^sub>a vv j x"
acc_lengths_concat_injective: fixes as' :: "'a list list"   and as :: "'a list list"   and n :: "nat" assumes "concat as' = concat as"   and "acc_lengths n as' = acc_lengths n as" shows "as' = as"
ltl_llist_of: fixes xs :: "'c list" shows "ltl (llist_of xs) = llist_of (tl xs)"
atm_of_lit_in_atms_of: fixes L :: "'a literal"   and C :: "'a literal multiset" assumes "L \<in># C" shows "atm_of L \<in> atms_of C"
subgroup_generated_group_carrier: fixes G :: "('a, 'b) monoid_scheme" assumes "Group.group G" shows "subgroup_generated G (carrier G) = G"
run_pause_state: fixes pause :: "'c \<Rightarrow> ('d \<Rightarrow> 'a) \<Rightarrow> 'a"   and out :: "'c"   and c :: "'d \<Rightarrow> ('b, 'a) stateT"   and s :: "'b" shows "run_state (pause_state pause out c) s = pause out (\<lambda>i. run_state (c i) s)"
pred_inject_3: fixes P :: "'a \<Rightarrow> bool"   and a :: "'a"   and aa :: "'a ifex"   and ab :: "'a ifex" shows "pred_ifex P (IF a aa ab) = (P a \<and> pred_ifex P aa \<and> pred_ifex P ab)"
nn_integral_has_integral_lebesgue': fixes \<Omega> :: "'a set"   and f :: "'a \<Rightarrow> real"   and I :: "real" assumes "\<And>x. x \<in> \<Omega> \<Longrightarrow> 0 \<le> f x"   and "(f has_integral I) \<Omega>" shows "\<integral>\<^sup>+x\<in>\<Omega>. ennreal (f x)\<partial>lborel = ennreal I"
atLeastLessThan_def: fixes less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and l :: "'g"   and u :: "'g" shows "ord.atLeastLessThan less_eq less l u \<equiv> ord.atLeast less_eq l \<inter> ord.lessThan less u"
sat_precond_as_proj_3: fixes vs :: "'d set"   and a :: "('d, 'e) fmap \<times> ('d, 'e) fmap"   and s :: "('d, 'e) fmap" assumes "fmdom' (fmrestrict_set vs (snd a)) = {}" shows "fmrestrict_set vs (state_succ s a) = fmrestrict_set vs s"
LocalSecretsComposition_ine1_s: fixes s :: "Secrets"   and PQ :: "specID"   and P :: "specID"   and Q :: "specID" assumes "sKS s \<in> LocalSecrets PQ"   and "subcomponents PQ = {P, Q}"   and "correctCompositionLoc PQ"   and "\<not> ine Q (sE s)"   and "sKS s \<notin> LocalSecrets P"   and "sKS s \<notin> LocalSecrets Q" shows "ine P (sE s)"
iTILL_minus: fixes x :: "nat"   and n :: "nat"   and k :: "nat" assumes "x \<in> [\<dots>n]" shows "x - k \<in> [\<dots>n]"
CFG_axioms: fixes prog :: "cmd" shows "CFG sourcenode targetnode (valid_edge prog) (_Entry_)"
f_trivial_C0: fixes v :: "'a" assumes "v \<in> C0" shows "f v = v"
simps_2: fixes x :: "'c list"   and xs :: "'c list list" shows "concat (x # xs) = x @ concat xs"
7: shows "sset (gtrace r v) \<subseteq> clean A w (graph A w k)"
frequently_at_topE: fixes P :: "nat \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<forall>n. frequently (P n) at_top" obtains s :: "nat \<Rightarrow> 'b" where "\<And>i. P i (s i)" and "strict_mono s"
bound_typ_instance_trans: fixes t :: "typ"   and sch :: "type_scheme"   and sch' :: "type_scheme" assumes "t <| sch"   and "sch \<le> sch'" shows "t <| sch'"
simps_2: fixes h :: "nat" shows "Um (Suc h) = N1 ` T h"
mult_1s_2: fixes a :: "'b" shows "a * Numeral1 = a"
i_cut_defs_4: fixes I :: "'a set"   and t :: "'a" shows "I \<down>> t \<equiv> {x \<in> I. t < x}"
Ftor: shows "Ftor (YFtor C @@ Y) : Op C \<longrightarrow> SET"
nonempty_standard_simplex: fixes p :: "nat" shows "standard_simplex p \<noteq> {}"
relcompI: fixes a :: "'a"   and b :: "'b"   and r :: "('a \<times> 'b) set"   and c :: "'c"   and s :: "('b \<times> 'c) set" assumes "(a, b) \<in> r"   and "(b, c) \<in> s" shows "(a, c) \<in> r O s"
Subring_zero_ring_zero: fixes S :: "('a, 'g) Ring_scheme" assumes "Subring R S" shows "\<zero>\<^bsub>S\<^esub> = \<zero>"
cut: fixes H :: "fmla set"   and A :: "fmla"   and H' :: "fmla set"   and B :: "fmla" assumes "H \<turnstile> A"   and "insert A H' \<turnstile> B" shows "H \<union> H' \<turnstile> B"
homeomorphism_imp_open_map: fixes S :: "'b set"   and T :: "'c set"   and f :: "'b \<Rightarrow> 'c"   and g :: "'c \<Rightarrow> 'b"   and U :: "'b set" assumes "homeomorphism S T f g"   and "openin (top_of_set S) U" shows "openin (top_of_set T) (f ` U)"
unstar_def: fixes b :: "bool star" shows "unstar b = (b = star_of True)"
rank_of_mono: fixes carrier :: "'b set"   and indep :: "'b set \<Rightarrow> bool"   and X :: "'b set"   and Y :: "'b set" assumes "matroid carrier indep"   and "X \<subseteq> Y"   and "Y \<subseteq> carrier" shows "indep_system.lower_rank_of indep X \<le> indep_system.lower_rank_of indep Y"
assms: shows "P \<rightarrow>\<beta>\<^sup>* P'"
g_3: fixes x :: "('a \<Rightarrow> real) \<Rightarrow> real" assumes "x \<in> sub.tangent_space p" shows "g (inclusion.push_forward x) = x"
proots_within_times: fixes p :: "'a poly"   and q :: "'a poly"   and s :: "'a set" shows "proots_within (p * q) s = proots_within p s \<union> proots_within q s"
hom_induced_empty_hom: fixes p :: "int"   and X :: "'a topology"   and X' :: "'b topology"   and f :: "'a \<Rightarrow> 'b" shows "hom_induced p X {} X' {} f \<in> hom (homology_group p X) (homology_group p X')"
circuit_in_subset_carrier: fixes \<E> :: "'a set"   and X :: "'a set" assumes "\<E> \<subseteq> \<E>"   and "\<E>.circuit_in \<E> X" shows "X \<subseteq> \<E>"
list_all2_mono: fixes P :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'b list"   and ys :: "'c list"   and Q :: "'b \<Rightarrow> 'c \<Rightarrow> bool" assumes "list_all2 P xs ys"   and "\<And>xs ys. P xs ys \<Longrightarrow> Q xs ys" shows "list_all2 Q xs ys"
confc\<^sub>0: shows "conf_clinit P sh ((stk, loc, C, M, pc, Calling C' Cs) # frs)"
assms: shows "secure_refinement \<R>\<^sub>A \<R> P"
of_rat_of_int: shows "of_rat \<circ> rat_of_int = of_int"
parts_analz: fixes H :: "msg set" shows "parts (analz H) = parts H"
Substitution: fixes t :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term"   and M :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term set"   and \<delta> :: "('fun, 'atom) Term.term \<times> nat \<Rightarrow> ('fun, ('fun, 'atom) Term.term \<times> nat) Term.term" assumes "t \<in> SMP M"   and "wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<delta>"   and "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<delta>)" shows "t \<cdot> \<delta> \<in> SMP M"
anonymity_prefs_from_table: fixes agents :: "'a set"   and alts :: "'b set"   and scf :: "('a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'b set"   and xs :: "('a \<times> 'b set list) list"   and ys :: "('a \<times> 'b set list) list" assumes "anonymous_scf agents alts scf"   and "prefs_from_table_wf agents alts xs"   and "prefs_from_table_wf agents alts ys"   and "mset (map snd xs) = mset (map snd ys)" shows "scf (prefs_from_table xs) = scf (prefs_from_table ys)"
that: fixes p' :: "('a, 'b, 'c, 'd) seqp" assumes "p = {l}broadcast(fmsg) . p'"   and "l' \<in> labels \<Gamma> qq" shows "thesis"
hdw_in_nodes: shows "hd w \<in> set (g_V G)"
indets_extend_indets: fixes p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" shows "indets (extend_indets p) = Some ` indets p"
InitBlockReds_aux: fixes wf_md :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list \<Rightarrow> char list \<Rightarrow> char list \<times> ty list \<times> ty \<times> char list list \<times> expr \<Rightarrow> bool"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and e :: "expr"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and e' :: "expr"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and E :: "char list \<Rightarrow> ty option"   and V :: "char list"   and T :: "ty" assumes "wf_prog wf_md P"   and "P,E(V \<mapsto> T) \<turnstile> \<langle>e,s\<rangle> \<rightarrow>* \<langle>e',s'\<rangle>" shows "\<forall>h l h' l' v v'. s = (h, l(V \<mapsto> v')) \<longrightarrow> P \<turnstile> T casts v to v'  \<longrightarrow> s' = (h', l') \<longrightarrow> (\<exists>v'' w. P,E \<turnstile> \<langle>{V:T; V:=Val v;; e},(h, l)\<rangle> \<rightarrow>* \<langle>{V:T; V:=Val v'';; e'},(h', l'(V := l V))\<rangle> \<and> P \<turnstile> T casts v'' to w )"
wf13: shows "mat nr ncc (mat_mult nr m1 m3)"
lift_subst_lt: fixes i :: "nat"   and j :: "nat"   and t :: "db"   and s :: "db" assumes "i < j + 1" shows "lift (t<s/j>) i = lift t i<lift s i/j + 1>"
seq_offset_neg: fixes f :: "nat \<Rightarrow> 'b"   and l :: "'b"   and k :: "nat" assumes "f \<longlonglongrightarrow> l" shows "(\<lambda>i. f (i - k)) \<longlonglongrightarrow> l"
bin_lcp_fst_mismatch_pref: shows "\<alpha> \<cdot> [c\<^sub>0] \<le>p u\<^sub>0 \<cdot> \<alpha>"
simps_66: fixes x8 :: "'a OCL_Types.type"   and x5 :: "'a OCL_Types.type" shows "Sequence x8 \<noteq> Set x5"
ThrowNull: fixes P :: "(char list list \<times> char list exp) prog"   and e :: "char list exp"   and s\<^sub>0 :: "heap \<times> sfields \<times> sheap"   and s\<^sub>1 :: "heap \<times> sfields \<times> sheap" assumes "P \<turnstile> \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>null,s\<^sub>1\<rangle>" shows "P \<turnstile> \<langle>throw e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>THROW NullPointer,s\<^sub>1\<rangle>"
attach_compose: fixes conv1 :: "('a, 'b, 'e, 'f) converter"   and conv2 :: "('e, 'f, 'c, 'd) converter"   and res :: "('c, 'd) resource" shows "attach (comp_converter conv1 conv2) res = attach conv1 (attach conv2 res)"
inducts: fixes \<Gamma> :: "('a, 'b) web_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal" assumes "web_flow \<Gamma> f" obtains "current \<Gamma> f" and "\<And>x. \<lbrakk>x \<in> \<^bold>V\<^bsub>\<Gamma>\<^esub>; x \<notin> A \<Gamma>; x \<notin> B \<Gamma>\<rbrakk> \<Longrightarrow> KIR f x"
spmf_embed_spmf: fixes f :: "'c \<Rightarrow> real"   and x :: "'c" assumes "\<integral>\<^sup>+ x. ennreal (f x) \<partial>count_space UNIV \<le> 1" shows "spmf (embed_spmf f) x = max 0 (f x)"
REFL: fixes r :: "('b \<times> 'b) set" assumes "wo_rel r" shows "Refl r"
transform2: fixes P :: "'a state \<times> 'a state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>"   and Q :: "'a state \<times> 'a state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>"   and \<tau> :: "'a state \<times> 'a state" assumes "P = Q" shows "(\<tau> \<Turnstile> P) = (\<tau> \<Turnstile> Q)"
ln_upper_7_neg: fixes x :: "real" assumes "0 < x"   and "x \<le> 1" shows "ln x \<le> ln_upper_7 x"
vimage2p_def: fixes f :: "'b \<Rightarrow> 'f"   and g :: "'d \<Rightarrow> 'g"   and R :: "'f \<Rightarrow> 'g \<Rightarrow> 'e" shows "vimage2p f g R = (\<lambda>x y. R (f x) (g y))"
secureEQUIV: fixes p :: "L"   and G :: "Var \<Rightarrow> L"   and c :: "IMP"   and H :: "Var \<Rightarrow> L" shows "secure p G c H = (\<forall>q. secure1 p G c H \<and> secure2 q G c H)"
assms_1: shows "F = [f, f', f'']\<^sub>\<circ>"
prems_9: shows "is_tangent p\<^sub>1 (local.add a p\<^sub>2 p\<^sub>3)"