hcju/isabelleps huggingface dataset:
card_of_singl_ordLeq: fixes A :: "'d set"   and b :: "'e" assumes "A \<noteq> {}" shows "|{b}| \<le>o |A|"
obsv_TS_def: shows "obsv_TS = \<lparr>init = obsv_init, trans = obsv_trans\<rparr>"
qtable_\<epsilon>_lax: fixes \<phi>s :: "Formula.formula list"   and mr :: "mregex"   and n :: "nat"   and R :: "event_data list set"   and \<sigma> :: "(String.literal \<times> event_data list) trace"   and V :: "String.literal \<Rightarrow> (nat \<Rightarrow> event_data list set) option"   and i :: "nat"   and rels :: "event_data option list set list"   and A :: "nat set"   and Q :: "event_data option list \<Rightarrow> bool"   and guard :: "event_data option list set" assumes "ok (length \<phi>s) mr"   and "list_all2 (\<lambda>\<phi>. qtable n (fv \<phi>) (mem_restr R) (\<lambda>v. Formula.sat \<sigma> V (map the v) i \<phi>)) \<phi>s rels"   and "fv_regex (from_mregex mr \<phi>s) \<subseteq> A"   and "qtable n A (mem_restr R) Q guard" shows "qtable n A (mem_restr R) (\<lambda>v. Regex.eps (Formula.sat \<sigma> V (map the v)) i (from_mregex mr \<phi>s) \<and> Q v) (\<epsilon>_lax guard rels mr)"
cases: fixes a1 :: "s_context"   and a2 :: "t list option"   and a3 :: "nat"   and a4 :: "v list"   and a5 :: "e list"   and a6 :: "t list" assumes "a1\<bullet>a2 \<tturnstile>_ a3 a4;a5 : a6" obtains i :: "nat" and \<S> :: "s_context" and tvs :: "t list" and vs :: "v list" and \<C> :: "t_context" and rs :: "t list option" and es :: "e list" and ts :: "t list" where "a1 = \<S>" and "a2 = rs" and "a3 = i" and "a4 = vs" and "a5 = es" and "a6 = ts" and "i < length (s_inst \<S>)" and "tvs = map typeof vs" and "\<C> = (s_inst \<S> ! i)\<lparr>local := local (s_inst \<S> ! i) @ tvs, return := rs\<rparr>" and "\<S>\<bullet>\<C> \<turnstile> es : [] _> ts" and "rs = Some ts \<or> rs = None"
rLarge: fixes i :: "nat" assumes "i < len" shows "hs i \<le> r"
keys_map_entry: fixes k :: "'b"   and f :: "'c \<Rightarrow> 'c"   and m :: "('b, 'c) mapping" shows "keys (map_entry k f m) = keys m"
fwi_canonical_extend: fixes n :: "nat"   and I :: "nat set"   and m :: "nat \<Rightarrow> nat \<Rightarrow> 'a"   and k :: "nat" assumes "canonical_subs n I m"   and "I \<subseteq> {0..n}"   and "(0::'a) \<le> m k k"   and "k \<le> n" shows "canonical_subs n (I \<union> {k}) (fwi m n k n n)"
1: shows "qGood (qOp delta (pickInp inp) (pickBinp binp))"
valid6: fixes X :: "'a state \<times> 'a state \<Rightarrow> 'b" shows "\<upsilon> \<delta> X = true"
ir_hoare_Skip_Skip: fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool" shows "ir_hoare P SKIP SKIP Q = (\<forall>s s'. Q s s' \<longrightarrow> P s s')"
jmpl_instr_result: fixes i :: "sparc_operation \<times> inst_operand list"   and s :: "'b sparc_state" assumes "fst i = ctrl_type JMPL" shows "snd (jmpl_instr i s) = False"
eq_iff: fixes x :: "'e list"   and y :: "'e list" shows "(x = y) = (prefix x y \<and> prefix y x)"
inv_eq_1_iff: fixes x :: "'c" assumes "x \<in> carrier M" shows "(\<ominus>\<^bsub>M\<^esub> x = \<zero>\<^bsub>M\<^esub>) = (x = \<zero>\<^bsub>M\<^esub>)"
r_neg: fixes x :: "'c" assumes "x \<in> carrier M" shows "x \<oplus>\<^bsub>M\<^esub> \<ominus>\<^bsub>M\<^esub> x = \<zero>\<^bsub>M\<^esub>"
L1_D_2: fixes f :: "'a \<Rightarrow> real"   and M :: "'a measure" assumes "f \<in> lspace M 1" shows "integrable M f"
t_n_closed: fixes x :: "'a" shows "!\<tt>\<tt> x = !x"
noicd: shows "\<nexists>j. m icd\<^bsup>\<pi>\<^esup>\<rightarrow> j"
g: shows "pdevs_val g (pdevs_of_list ys) = (\<Sum>i = length xs..<length xs + degree (pdevs_of_list ys). (if i < length xs then f i else g (i - length xs)) *\<^sub>R pdevs_apply (pdevs_of_list (xs @ ys)) i)"
ag_pOp_closed: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier M"   and "y \<in> carrier M" shows "x \<plusminus> y \<in> carrier M"
crel_vs_return_ext: fixes R :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "'d"   and y :: "'e" assumes "Transfer.Rel R x y" shows "Transfer.Rel (crel_vs R) x (State_Monad.return y)"
\<G>_af: fixes \<phi> :: "'b ltln"   and w :: "nat \<Rightarrow> 'b set"   and i :: "nat" shows "\<G> (af \<phi> (prefix i w)) (suffix i w) = \<G> \<phi> (suffix i w)"
ereal_of_enat_inf: fixes n :: "enat" shows "(ereal_of_enat n = \<infinity>) = (n = \<infinity>)"
complex_scaleC_def: fixes a :: "complex"   and x :: "complex" shows "a *\<^sub>C x = a * x"
ev_wait_one_weakly_step_consistent: fixes u :: "partition_id_t"   and s1 :: "state_t"   and s2 :: "state_t"   and tid :: "thread_id_t"   and ipt :: "int_point_t" assumes "vpeq u s1 s2"   and "vpeq (Step.partition tid) s1 s2"   and "atomic_step_invariant s1"   and "atomic_step_invariant s2"   and "atomic_step_precondition s1 (current s1) ipt"   and "atomic_step_precondition s1 (current s1) ipt" shows "vpeq u (atomic_step_ev_wait_one tid s1) (atomic_step_ev_wait_one tid s2)"
gamma_delta_beta_3: fixes F :: "nat \<Rightarrow> 'b fract poly"   and n :: "nat \<Rightarrow> nat"   and \<delta> :: "nat \<Rightarrow> nat"   and f :: "nat \<Rightarrow> 'b fract"   and k :: "nat"   and \<beta> :: "nat \<Rightarrow> 'b fract"   and G1 :: "'b poly"   and G2 :: "'b poly" assumes "subresultant_prs_locale F n \<delta> f k \<beta> G1 G2" shows "subresultant_prs_locale.\<gamma> n \<delta> f \<beta> 3 = (- 1) ^ (\<delta> 1 + 1) * \<beta> 3"
qSwap_preserves_qGood1: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm"   and zs :: "'varSort"   and x :: "'var"   and y :: "'var" assumes "qGood X" shows "qGood (X #[[x \<and> y]]_zs)"
independent_inj_image: fixes S :: "('b, 'c) vec set"   and A :: "(('b, 'c) vec, 'd) vec" assumes "vec.independent S"   and "inj ((*v) A)" shows "vec.independent ((*v) A ` S)"
diff_numeral_special_6: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and one :: "'b"   and m :: "num" assumes "class.neg_numeral minus plus zero uminus" shows "minus (numeral.numeral one plus m) (uminus one) = numeral.numeral one plus (Num.inc m)"
unx_some_lemmas_1: fixes u :: "'a"   and x :: "'a \<times> 'a" assumes "unapp u = Some x" shows "unconst u = None"
isPrefix_def: fixes p :: "'a list"   and t :: "'a list" shows "isPrefix p t = (\<exists>s. p @ s = t)"
vunion_comm: fixes A :: "V"   and B :: "V" shows "A \<union>\<^sub>\<circ> B = B \<union>\<^sub>\<circ> A"
is_confluent_on_def: fixes A :: "'a set" shows "is_confluent_on A = (\<forall>a\<in>A. \<forall>b1 b2. a \<rightarrow>\<^sup>* b1 \<and> a \<rightarrow>\<^sup>* b2 \<longrightarrow> b1 \<down>\<^sup>* b2)"
vec_of_list_list_of_vec: fixes a :: "'b vec" shows "vec_of_list (list_of_vec a) = a"
lossless_iff_set_pmf_None: fixes p :: "'d spmf" shows "lossless_spmf p = (None \<notin> set_pmf p)"
sccn: fixes n :: "nat" shows "singular_chain p (subtopology X S) ((singular_subdivision p ^^ n) c)"
distrib_right': fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "class.ab_near_semiring plus times" shows "times (plus x y) z = plus (times x z) (times y z)"
mndet_subset_FD: fixes A :: "'d set"   and B :: "'d set"   and P :: "'d process" assumes "A \<noteq> {}" shows "(\<sqinter>xa\<in>A \<union> B \<rightarrow>  P) \<le> (\<sqinter>xa\<in>A \<rightarrow>  P)"
eq_Fun_iff_incl_joinable: fixes f :: "'s"   and f' :: "'s" assumes "span f f'" shows "(Fun f = Fun f') = (\<exists>m m'. incl m \<and> incl m' \<and> seq m f \<and> seq m' f' \<and> m \<cdot> f = m' \<cdot> f')"
abs_eq: fixes x :: "nat" shows "mask x = natural_of_nat (mask x)"
assms_4: shows "angle b1 c1 a1 = angle b2 c2 a2"
map_of_aset_insert: fixes S :: "('b \<times> 'c) set"   and k :: "'b"   and v :: "'c" assumes "unique_keys_set (S \<union> {(k, v)})" shows "map_of_aset (S \<union> {(k, v)}) =  map_of_aset S { k \<rightarrow> v }"
word_div_lt_eq_0: fixes x :: "'b word"   and y :: "'b word" assumes "x < y" shows "x div y = 0"
Tstruct_def: fixes intT :: "'a \<Rightarrow> 'b \<Rightarrow> bool" shows "M.Tstruct intT \<equiv> \<forall>\<sigma>. \<exists>a. intT \<sigma> a"
comp_galcon_right_unit: fixes F :: "('a, 'b, unit, unit) galcon" shows "F \<circ>\<^sub>g I\<^sub>g \<X>\<^bsub>F\<^esub> = F"
Chernoff_onI: fixes A :: "'a set"   and f :: "'a set \<Rightarrow> 'a set" assumes "\<And>B C. \<lbrakk>B \<subseteq> A; C \<subseteq> B\<rbrakk> \<Longrightarrow> f B \<inter> C \<subseteq> f C" shows "Chernoff_on A f"
neg'_antitone: fixes x :: "'a"   and y :: "'a" assumes "x \<le> y" shows "-' y \<le> -' x"
ide_iso_in_hom: fixes a :: "'c" assumes "C.ide a" shows "\<guillemotleft>a : a \<cong> a\<guillemotright>"
dist_commute: fixes x :: "'b"   and y :: "'b" shows "dist x y = dist y x"
FOREACHoci_def: fixes R :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and \<Phi> :: "'d set \<Rightarrow> 'c \<Rightarrow> bool"   and S :: "'d set"   and c :: "'c \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'c \<Rightarrow> 'c nres"   and \<sigma>0 :: "'c" shows "FOREACH\<^sub>O\<^sub>C\<^bsup>R,\<Phi>\<^esup> S c f \<sigma>0 \<equiv> ASSERT (finite S) \<bind> (\<lambda>_. SPEC (\<lambda>xs. distinct xs \<and> S = set xs \<and> sorted_wrt R xs) \<bind> (\<lambda>xs. WHILE\<^sub>T\<^bsup>\<lambda>(it, \<sigma>). \<exists>xs'. xs = xs' @ it \<and> \<Phi> (set it) \<sigma>\<^esup> (FOREACH_cond c) (FOREACH_body f) (xs, \<sigma>0) \<bind> (\<lambda>(uu_, \<sigma>). RETURN \<sigma>)))"
relation_of_spec_t_f: fixes y :: "('c, 'd) action"   and x :: "('c, 'd) action"   and a :: "('c, 'd) alpha_rp_scheme"   and b :: "('c, 'd) alpha_rp_scheme" assumes "\<forall>a b. (action.relation_of y \<longrightarrow> action.relation_of x) (a, b)"   and "action.relation_of y\<^sup>t\<^sub>f (a\<lparr>tr := []\<rparr>, b)" shows "action.relation_of x\<^sup>t\<^sub>f (a\<lparr>tr := []\<rparr>, b)"
nonneg_lincomb_list_def: fixes c :: "nat \<Rightarrow> 'a"   and Vs :: "'a vec list"   and b :: "'a vec" shows "n.nonneg_lincomb_list c Vs b = (n.lincomb_list c Vs = b \<and> (\<forall>i<length Vs. (0::'a) \<le> c i))"
ang_vec_a_opposite2: fixes z1 :: "complex"   and z2 :: "complex" shows "\<angle>a z1 z2 = \<angle>a z1 (- z2)"
verify_def: fixes v_key :: "'a"   and m :: "nat"   and c :: "'a"   and d :: "nat" shows "verify v_key m c d = (c = \<^bold>g [^] d \<otimes> v_key [^] m)"
a: shows "a *\<^sub>R i \<noteq> (0::'b)"
D'_1: shows "D @ D' \<in> decomps\<^sub>e\<^sub>s\<^sub>t A N \<I>"
unless_apply_cong: fixes C :: "bool"   and C' :: "bool"   and s :: "'d"   and s' :: "'d"   and m :: "'d \<Rightarrow> (('e + unit) \<times> 'd) \<times> bool"   and m' :: "'d \<Rightarrow> (('e + unit) \<times> 'd) \<times> bool" assumes "C = C'"   and "s = s'"   and "\<not> C' \<Longrightarrow> m s' = m' s'" shows "unlessE C m s = unlessE C' m' s'"
Proc_CFG_Exit_no_sourcenode: fixes prog :: "cmd"   and et :: "p_edge_kind"   and n' :: "label" assumes "prog \<turnstile> Exit -et\<rightarrow>\<^sub>p n'" shows "False"
mono_generate: fixes K :: "'c set"   and H :: "'c set" assumes "K \<subseteq> H" shows "generate H K \<subseteq> generate H H"
that: fixes p :: "'a set set" assumes "p division_of cbox a b" shows "thesis"
\<alpha>r'_fresh: shows "\<alpha> r' \<notin> RID\<^sub>G (\<R>\<^sub>G \<alpha> \<beta> s)"
axioms_3: fixes known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptrs :: "(_) heap \<Rightarrow> bool"   and heap_is_wellformed :: "(_) heap \<Rightarrow> bool"   and parent_child_rel :: "(_) heap \<Rightarrow> ((_) object_ptr \<times> (_) object_ptr) set"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_parent :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr option) prog"   and get_parent_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_ancestors :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and get_ancestors_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_root_node :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr) prog"   and get_root_node_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" assumes "l_get_root_node_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr type_wf known_ptrs heap_is_wellformed parent_child_rel get_child_nodes get_child_nodes_locs get_disconnected_nodes get_parent get_parent_locs get_ancestors get_ancestors_locs get_root_node get_root_node_locs" shows "l_get_parent_wf type_wf known_ptr known_ptrs heap_is_wellformed parent_child_rel get_child_nodes get_parent"
port_open_def: shows "port_open = (\<lambda>L a p. not_before (is_ftp_close a) (is_ftp_port_request a p) L)"
helper_masked_ucast_equal_generic: fixes n :: "nat"   and b :: "16 word" assumes "n \<le> 128 - 16" shows "UCAST(128 \<rightarrow> 16) ((UCAST(16 \<rightarrow> 128) b << n) && (mask 16 << n) >> n) = b"
refl_on_def: fixes A :: "'c set"   and r :: "('c \<times> 'c) set" shows "refl_on A r = (r \<subseteq> A \<times> A \<and> (\<forall>x\<in>A. (x, x) \<in> r))"
abs_eq: fixes x :: "int extended \<times> int extended" shows "is_empty_ivl (abs_ivl x) = is_empty_rep x"
perm_boolI: fixes P :: "bool"   and pi :: "('a \<times> 'a) list" assumes "P" shows "pi \<bullet> P"
simps_2: fixes P :: "pi" shows "hnf (\<tau>.P) = True"
natural_isomorphisms_compose: fixes A :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and F :: "'a \<Rightarrow> 'b"   and G :: "'a \<Rightarrow> 'b"   and \<sigma> :: "'a \<Rightarrow> 'b"   and H :: "'a \<Rightarrow> 'b"   and \<tau> :: "'a \<Rightarrow> 'b" assumes "natural_isomorphism A B F G \<sigma>"   and "natural_isomorphism A B G H \<tau>" shows "natural_isomorphism A B F H (vertical_composite.map A B \<sigma> \<tau>)"
ds: shows "distinct [Xd d1h1, Xd d2h1, Xd d3h1]"
enum_all_finite_2_def: fixes P :: "finite_2 \<Rightarrow> bool" shows "enum_all P = (P a\<^sub>1 \<and> P a\<^sub>2)"
AE_upper_bound_inf_ennreal: fixes F :: "'b \<Rightarrow> ennreal"   and G :: "'b \<Rightarrow> ennreal"   and M :: "'b measure" assumes "\<And>e. 0 < e \<Longrightarrow> AE x in M. F x \<le> G x + ennreal e" shows "AE x in M. F x \<le> G x"
single_valued_confluent: fixes r :: "('a \<times> 'a) set"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "single_valued r"   and "(x, y) \<in> r\<^sup>*"   and "(x, z) \<in> r\<^sup>*" shows "(y, z) \<in> r\<^sup>* \<or> (z, y) \<in> r\<^sup>*"
SpecStuck: fixes s :: "'a"   and r :: "('a \<times> 'a) set"   and \<Gamma> :: "'b \<Rightarrow> ('a, 'b, 'c) com option" assumes "\<forall>t. (s, t) \<notin> r" shows "\<Gamma>\<turnstile> (Spec r, Normal s) \<rightarrow> (Skip, Stuck)"
filterlim_realpow_sequentially_at_top: fixes x :: "real" assumes "1 < x" shows "filterlim ((^) x) at_top sequentially"
lessb: fixes x :: "real"   and i :: "nat" assumes "x \<le> u i" shows "ereal x < b"
Throw: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and s :: "'b" shows "\<Gamma>\<turnstile> \<langle>Throw,Normal s\<rangle> \<Rightarrow> Abrupt s"
simps_6: fixes uu :: "Unboxed.type"   and x :: "'a" shows "norm_instr (ILoadUbx uu x) = Inca.instr.ILoad x"
assms_2: shows "J \<subseteq>\<^sub>\<circ> I"
fbd_zero_pres2: fixes f :: "'d \<Rightarrow> 'c set" shows "bot_pres (bd\<^sub>\<F> f)"
less_induct: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and P :: "'b \<Rightarrow> bool"   and a :: "'b" assumes "class.wellorder less_eq less"   and "\<And>x. (\<And>y. less y x \<Longrightarrow> P y) \<Longrightarrow> P x" shows "P a"
A_SET_INV_def: fixes G :: "('c, 'd) ring_scheme"   and H :: "'c set" shows "a_set_inv\<^bsub>G\<^esub> H = set_inv\<^bsub>add_monoid G\<^esub> H"
lproject_eq_limit_chain: fixes u :: "nat \<Rightarrow> 'a list"   and v :: "nat \<Rightarrow> 'a list"   and A :: "'a set" assumes "chain u"   and "chain v"   and "\<And>k. project A (u k) = project A (v k)" shows "lproject A (llist_of_stream (Word_Prefixes.limit u)) = lproject A (llist_of_stream (Word_Prefixes.limit v))"
degree_map_poly_hom: fixes p :: "int poly" shows "Polynomial.degree (of_int_poly p) = Polynomial.degree p"
nat_pow_pow: fixes x :: "'a"   and n :: "nat"   and m :: "nat" assumes "x \<in> carrier (G\<lparr>carrier := adjoin G H a\<rparr>)" shows "(x [^]\<^bsub>G\<lparr>carrier := adjoin G H a\<rparr>\<^esub> n) [^]\<^bsub>G\<lparr>carrier := adjoin G H a\<rparr>\<^esub> m = x [^]\<^bsub>G\<lparr>carrier := adjoin G H a\<rparr>\<^esub> (n * m)"
basis_gen_hull': fixes G :: "'b list set" shows "\<langle>\<BB> \<langle>G\<rangle>\<rangle> = \<langle>G\<rangle>"
carrier_mult_of: fixes R :: "('b, 'c) ring_scheme" shows "carrier (mult_of R) = carrier R - {\<zero>\<^bsub>R\<^esub>}"
vmatch_rel: fixes t\<^sub>1 :: "value"   and t\<^sub>2 :: "value"   and p :: "pat" assumes "Q t\<^sub>1 t\<^sub>2" shows "rel_option (fmrel Q) (vmatch p t\<^sub>1) (vmatch p t\<^sub>2)"
i_set0_i_set_conv: shows "i_set0 - {{}} = i_set"
nth_concat_splitE: fixes i :: "nat"   and ls :: "'a list list" assumes "i < length (concat ls)" obtains j :: "nat" and k :: "nat" where "j < length ls" and "k < length (ls ! j)" and "concat ls ! i = ls ! j ! k" and "i = length (concat (take j ls)) + k"
simps_1: fixes x :: "'c arctic_delta" shows "(MinInfty_delta \<le> x) = True"
preorder_def: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "class.preorder less_eq less \<equiv> (\<forall>x y. less x y = strict less_eq x y) \<and> (\<forall>x. less_eq x x) \<and> (\<forall>x y z. less_eq x y \<longrightarrow> less_eq y z \<longrightarrow> less_eq x z)"
offending_not_empty: fixes F :: "('v \<times> 'v) set"   and G :: "'v graph"   and nP :: "'v \<Rightarrow> 'a" assumes "F \<in> set_offending_flows G nP" shows "F \<noteq> {}"
of_int_0: fixes one :: "'q"   and times :: "'q \<Rightarrow> 'q \<Rightarrow> 'q"   and plus :: "'q \<Rightarrow> 'q \<Rightarrow> 'q"   and zero :: "'q"   and minus :: "'q \<Rightarrow> 'q \<Rightarrow> 'q"   and uminus :: "'q \<Rightarrow> 'q" assumes "class.ring_1 one times plus zero minus uminus" shows "ring_1.of_int one plus zero minus 0 = zero"
assms_6: shows "prv (imp \<phi>2 \<chi>2)"
ide_in_hom: fixes a :: "'d" shows "D.ide a = \<guillemotleft>a : a \<rightarrow>\<^sub>D a\<guillemotright>"
iTILL_I: fixes x :: "nat"   and n :: "nat" assumes "x \<le> n" shows "x \<in> [\<dots>n]"
Pd_shuffle_to_top_simps_3: fixes d :: "'b"   and ds' :: "'b set"   and Y :: "'a set" assumes "d \<notin> ds'" shows "Pd_shuffle_to_top ds' Y d = Pd d"
unique_session_keys: fixes B :: "agent"   and NA :: "msg"   and X :: "msg"   and NB :: "msg"   and K :: "msg"   and evs :: "event list"   and B' :: "agent"   and NA' :: "msg"   and X' :: "msg"   and NB' :: "msg" assumes "Says Server B \<lbrace>NA, X, Crypt (shrK B) \<lbrace>NB, K\<rbrace>\<rbrace> \<in> set evs"   and "Says Server B' \<lbrace>NA', X', Crypt (shrK B') \<lbrace>NB', K\<rbrace>\<rbrace> \<in> set evs"   and "evs \<in> otway" shows "X = X' \<and> B = B' \<and> NA = NA' \<and> NB = NB'"
in_set_conv_decomp_first: fixes x :: "'c"   and xs :: "'c list" shows "(x \<in> set xs) = (\<exists>ys zs. xs = ys \<cdot> x # zs \<and> x \<notin> set ys)"
IH2_1: fixes f :: "floatarith" assumes "f \<in> subterms fa2" shows "f \<in> Mapping.keys M2"
callee_invariant_run_resource: fixes \<I> :: "('c, 'd) \<I>" shows "callee_invariant_on run_resource (WT_resource \<I>) \<I>"
types_snocE: fixes e :: "nat \<Rightarrow> LambdaType.type"   and ts :: "dB list"   and t :: "dB"   and Ts :: "LambdaType.type list" assumes "e \<tturnstile> ts @ [t] : Ts" obtains Us :: "LambdaType.type list" and U :: "LambdaType.type" where "Ts = Us @ [U]" and "e \<tturnstile> ts : Us" and "e \<turnstile> t : U"
HAEvents_SAEvents_SAs: fixes HA :: "('b, 'a, 'c) hierauto" shows "\<Union> (SAEvents ` SAs HA) \<subseteq> HAEvents HA"
ucast_down_mult: fixes a :: "'b word"   and b :: "'b word" assumes "is_down UCAST('b \<rightarrow> 'c)" shows "UCAST('b \<rightarrow> 'c) (a * b) = UCAST('b \<rightarrow> 'c) a * UCAST('b \<rightarrow> 'c) b"
v: shows "vars_term_ms (SCF (Fun g ts)) \<subseteq># vars_term_ms (SCF (Fun f (bef @ s # aft)))"
inverse_unique: fixes a :: "int"   and b :: "int" assumes "a \<in> carrier R"   and "b \<in> carrier R"   and "a \<otimes> b = \<one>" shows "inv a = b"
swap_bij: fixes x :: "'d"   and x' :: "'d" shows "bij (id(x := x', x' := x))"
horiz: fixes i :: "nat"   and j :: "nat" shows "{(i, j), (i, j + 1)} \<in> domino"
INT_I: fixes A :: "'c set"   and b :: "'d"   and B :: "'c \<Rightarrow> 'd set" assumes "\<And>x. s2p A x \<Longrightarrow> s2p (B x) b" shows "s2p (\<Inter> (\<P> B A)) b"
cont: fixes x :: "'a"   and e :: "real" assumes "x \<in> S"   and "0 < e" shows "\<exists>d>0. \<forall>f\<in>\<F>. \<forall>x'\<in>S. dist x' x < d \<longrightarrow> dist (f x') (f x) < e"
cos_60: shows "cos (pi / 3) = 1 / 2"
iexp_eq1: fixes x :: "real"   and n :: "nat" shows "iexp x = (\<Sum>k\<le>n. (\<i> * complex_of_real x) ^ k / fact k) + \<i> ^ Suc n / fact n * (CLBINT s=0..ereal x. complex_of_real ((x - s) ^ n) * iexp s)"
preallocated_upd_obj: fixes h :: "heap"   and a :: "nat"   and C :: "char list"   and fs :: "Objects.fields"   and fs' :: "Objects.fields" assumes "preallocated h"   and "h a = \<lfloor>(C, fs)\<rfloor>" shows "preallocated (h(a \<mapsto> (C, fs')))"
payload_Enc: fixes X :: "msg"   and Y :: "msg" assumes "Enc X Y \<in> payload" obtains "X \<in> payload" and "Y \<in> payload"
igWlsAllDisj_def: fixes MOD :: "('c, 'd, 'varSort, 'sort, 'e, 'f, 'g, 'h, 'i) model_scheme" shows "igWlsAllDisj MOD \<equiv> igWlsDisj MOD \<and> igWlsAbsDisj MOD"
psi_inner_beta: shows "inner_prod \<psi> \<beta> = complex_of_real (sin (\<theta> / 2))"
top_in_upset: fixes x :: "'a" shows "top \<in> \<up>x"
vmatch_env: fixes pat :: "pat"   and v :: "value"   and env :: "(name, value) fmap" assumes "vmatch pat v = Some env"   and "vclosed v" shows "closed_venv env"
dmu: fixes j :: "nat"   and ii :: "nat" assumes "j < length fs"   and "ii < length fs" shows "rat_of_int (fs.d\<mu> ii j) = rat_of_int (fs'.d fs (Suc j)) * fs.gs.\<mu> ii j"
graeffe_one_step_def: fixes c :: "'a"   and f :: "'a poly" shows "graeffe_one_step c f = Polynomial.smult c (poly_square_subst (f * f \<circ>\<^sub>p [:0::'a, - (1::'a):]))"
mk_disjoint_cinsert: fixes a :: "'f"   and A :: "'f cset" assumes "cin a A" shows "\<exists>B. A = cinsert a B \<and> \<not> cin a B"
vsv_vimage_eqI: fixes a :: "V"   and Aa :: "V"   and f :: "V \<Rightarrow> V"   and b :: "V"   and A :: "V" assumes "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> (VLambda Aa f)"   and "VLambda Aa f\<lparr>a\<rparr> = b"   and "a \<in>\<^sub>\<circ> A" shows "b \<in>\<^sub>\<circ> VLambda Aa f `\<^sub>\<circ> A"
rel_cset_iff: fixes R :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and a :: "'e cset"   and b :: "'f cset" shows "rel_cset R a b = ((\<forall>t. cin t a \<longrightarrow> (\<exists>u. cin u b \<and> R t u)) \<and> (\<forall>t. cin t b \<longrightarrow> (\<exists>u. cin u a \<and> R u t)))"
swap_nth: fixes i :: "nat"   and l :: "'c list"   and j :: "nat"   and k :: "nat" assumes "i < length l"   and "j < length l"   and "k < length l" shows "swap l i j ! k = (if k = i then l ! j else if k = j then l ! i else l ! k)"
mk_all_list_def: fixes l :: "(variable \<times> typ) list"   and t :: "Core.term" shows "mk_all_list l t = fold (\<lambda>(n, T). mk_all n T) l t"
hom_mult: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier G"   and "y \<in> carrier G" shows "h (x \<otimes>\<^bsub>G\<^esub> y) = h x \<otimes>\<^bsub>H\<^esub> h y"