hcju/isabelleps huggingface dataset:
sinks_aux_cons: fixes I :: "('e \<times> 'e) set"   and D :: "'f \<Rightarrow> 'e"   and U :: "'e set"   and x :: "'f"   and xs :: "'f list" shows "sinks_aux I D U (x # xs) = (if \<exists>v\<in>U. (v, D x) \<in> I then sinks_aux I D (insert (D x) U) xs else sinks_aux I D U xs)"
compact_space_topspace_empty: fixes X :: "'c topology" assumes "topspace X = {}" shows "compact_space X"
is_ring_from_is_homomorphism: fixes U :: "'a set" assumes "is_open U" shows "ring (\<FF> U) +\<^bsub>U\<^esub> \<cdot>\<^bsub>U\<^esub> \<zero>\<^bsub>U\<^esub> \<one>\<^bsub>U\<^esub>"
has_least_wf_mdecl: 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 C :: "char list"   and M :: "char list"   and m :: "ty list \<times> ty \<times> char list list \<times> expr"   and Cs :: "char list list" assumes "wf_prog wf_md P"   and "P \<turnstile> C has least M = m via Cs" shows "wf_mdecl wf_md P (last Cs) (M, m)"
inter_guards_Skip: fixes c2 :: "('d, 'e, 'f) com"   and c :: "('d, 'e, 'f) com" shows "((Skip \<inter>\<^sub>g c2) = Some c) = (c2 = Skip \<and> c = Skip)"
max_0_1_1: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "ord.max less_eq zero one = one"
Un_insert_right: fixes A :: "'b set"   and a :: "'b"   and B :: "'b set" shows "A \<union> Set.insert a B = Set.insert a (A \<union> B)"
z: shows "(\<forall>w. w = x \<or> w = y \<longrightarrow> P w z) \<and> (\<forall>w. P w z \<longrightarrow> (\<exists>v. (v = x \<or> v = y) \<and> O w v))"
int_pow_one: fixes z :: "int" shows "[z] \<cdot>\<^bsub>P\<^esub> \<zero>\<^bsub>P\<^esub> = \<zero>\<^bsub>P\<^esub>"
rv_edge_slice_kinds: fixes a :: "'edge"   and S :: "'node SDG_node set"   and s :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and s' :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and as :: "'edge list"   and asx :: "'edge list" assumes "valid_edge a"   and "intra_kind (kind a)"   and "\<forall>V\<in>rv S (CFG_node (sourcenode a)). state_val s V = state_val s' V"   and "preds (slice_kinds S (a # as)) s"   and "preds (slice_kinds S (a # asx)) s'" shows "\<forall>V\<in>rv S (CFG_node (targetnode a)). state_val (transfer (slice_kind S a) s) V = state_val (transfer (slice_kind S a) s') V"
transitive_bot_closed: shows "bot \<sqinter> bot \<le> bot"
induct: fixes P :: "('b, 'c) Combinators list \<Rightarrow> bool"   and a0 :: "('b, 'c) Combinators list" assumes "\<And>x. P x \<Longrightarrow> P (DenyAll # x)"   and "\<And>v va y z. \<lbrakk>first_bothNet (DenyAllFromTo v va) = first_bothNet y \<Longrightarrow> P (DenyAllFromTo v va \<oplus> y # z); first_bothNet (DenyAllFromTo v va) \<noteq> first_bothNet y \<Longrightarrow> P (y # z)\<rbrakk> \<Longrightarrow> P (DenyAllFromTo v va # y # z)"   and "\<And>v va vb y z. \<lbrakk>first_bothNet (AllowPortFromTo v va vb) = first_bothNet y \<Longrightarrow> P (AllowPortFromTo v va vb \<oplus> y # z); first_bothNet (AllowPortFromTo v va vb) \<noteq> first_bothNet y \<Longrightarrow> P (y # z)\<rbrakk> \<Longrightarrow> P (AllowPortFromTo v va vb # y # z)"   and "\<And>v va y z. \<lbrakk>first_bothNet (v \<oplus> va) = first_bothNet y \<Longrightarrow> P ((v \<oplus> va) \<oplus> y # z); first_bothNet (v \<oplus> va) \<noteq> first_bothNet y \<Longrightarrow> P (y # z)\<rbrakk> \<Longrightarrow> P (v \<oplus> va # y # z)"   and "P []"   and "\<And>vb vc. P [DenyAllFromTo vb vc]"   and "\<And>vb vc vd. P [AllowPortFromTo vb vc vd]"   and "\<And>vb vc. P [vb \<oplus> vc]" shows "P a0"
id: shows "D n = (\<Sum>(a, i)\<in>as - UNIV \<times> {0..<Suc n}. (\<Prod>(b, j)\<in>as - UNIV \<times> {0..<n} - {(a, i)}. b) * smult (of_nat (i - n)) (pderiv a))"
less_1: fixes p :: "'a list" assumes "degree p < degree pa__"   and "p \<in> carrier (poly_ring R)" shows "size (roots p) \<le> degree p"
h5: shows "invertible_mat (M_mat_R (get_signs_R (snd (combine_systems_R p (qs1, calculate_data_R p qs1) (qs2, calculate_data_R p qs2)))) (get_subsets_R (snd (combine_systems_R p (qs1, calculate_data_R p qs1) (qs2, calculate_data_R p qs2)))))"
iftrue: fixes b :: "'exp"   and m :: "'id \<Rightarrow> 'val"   and \<iota> :: "nat"   and c1 :: "('exp, 'id) MWLsCom"   and c2 :: "('exp, 'id) MWLsCom" assumes "BMap (E b m) = True" shows "SRabbr (if\<^bsub>\<iota>\<^esub> b then c1 else c2 fi) m [] (Some c1) m"
less_pair_def: fixes a :: "'a \<times> 'b"   and b :: "'a \<times> 'b" shows "(a \<sqsubset> b) = (fst a \<sqsubset> fst b \<or> fst a = fst b \<and> snd a \<sqsubset> snd b)"
scalar_prod_add_distrib: fixes v\<^sub>1 :: "'b vec"   and n :: "nat"   and v\<^sub>2 :: "'b vec"   and v\<^sub>3 :: "'b vec" assumes "v\<^sub>1 \<in> carrier_vec n"   and "v\<^sub>2 \<in> carrier_vec n"   and "v\<^sub>3 \<in> carrier_vec n" shows "v\<^sub>1 \<bullet> (v\<^sub>2 + v\<^sub>3) = v\<^sub>1 \<bullet> v\<^sub>2 + v\<^sub>1 \<bullet> v\<^sub>3"
sturm_ext_interval: fixes a :: "real"   and b :: "real"   and p :: "real poly" assumes "a < b"   and "poly p a \<noteq> 0"   and "poly p b \<noteq> 0" shows "int (proots_count p {x. a < x \<and> x < b}) = changes_itv_smods_ext a b p (pderiv p)"
bottom_least: fixes L :: "('l, 'm) gorder_scheme" assumes "weak_partial_order_bottom L" shows "least L \<bottom>\<^bsub>L\<^esub> (carrier L)"
sup_of_two_exists: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier (inv_gorder L)"   and "y \<in> carrier (inv_gorder L)" shows "\<exists>s. is_lub (inv_gorder L) s {x, y}"
pCons: shows "presCons (rec MOD) (recAbs MOD) MOD"
sup_def: fixes L :: "('a, 'b) gorder_scheme"   and A :: "'a set" shows "\<Squnion>\<^bsub>L\<^esub>A = (SOME x. is_lub L x A)"
evalHeap_reorder: fixes \<Gamma> :: "('d \<times> 'e) list"   and \<Delta> :: "('d \<times> 'e) list"   and h :: "'e \<Rightarrow> 'f" assumes "map_of \<Gamma> = map_of \<Delta>" shows "evalHeap \<Gamma> h = evalHeap \<Delta> h"
\<sigma>: shows "\<sigma> \<in> ptrm (Suc 0)"
cases: fixes x :: "('d, 'c) trie" obtains v :: "'c option" and m :: "('d \<times> ('d, 'c) trie) list" where "x = Trie v m"
convex_on_continuous: fixes s :: "'c set"   and f :: "'c \<Rightarrow> real" assumes "open s"   and "convex_on s f" shows "continuous_on s f"
simps_22: fixes v :: "'a"   and va :: "'a rexp"   and vb :: "'a rexp" shows "nTimes (Atom v) (Times va vb) = Times (Atom v) (Times va vb)"
lunit_char_eqn: fixes a :: "'a" assumes "ide a" shows "local.prod \<one> (lunit a) = local.prod \<iota> a \<cdot> assoc' \<one> \<one> a"
in_Basis_index_Basis_list: fixes i :: "'a" assumes "i \<in> Basis" shows "i = Basis_list ! index Basis_list i"
scaleC_scaleC: fixes a :: "complex"   and b :: "complex"   and x :: "'b" shows "a *\<^sub>C b *\<^sub>C x = (a * b) *\<^sub>C x"
elims_2: fixes x :: "Monitor.args"   and xa :: "nat"   and xb :: "nat \<times> nat \<times> bool list \<times> bool list \<times> (nat \<times> 'a option list set) queue \<times> (nat \<times> 'a option list set) queue \<times> ('a option list, nat) mapping \<times> ('a option list, nat) mapping"   and xc :: "(nat \<times> 'a option list set) list" assumes "valid_mmsaux x xa xb xc" obtains args :: "Monitor.args" and cur :: "nat" and nt :: "nat" and gc :: "nat" and maskL :: "bool list" and maskR :: "bool list" and data_prev :: "(nat \<times> 'a option list set) queue" and data_in :: "(nat \<times> 'a option list set) queue" and tuple_in :: "('a option list, nat) mapping" and tuple_since :: "('a option list, nat) mapping" and ys :: "(nat \<times> 'a option list set) list" where "x = args" and "xa = cur" and "xb = (nt, gc, maskL, maskR, data_prev, data_in, tuple_in, tuple_since)" and "xc = ys" and "args_L args \<subseteq> args_R args \<and> maskL = join_mask (args_n args) (args_L args) \<and> maskR = join_mask (args_n args) (args_R args) \<and> (\<forall>(t, y)\<in>set ys. table (args_n args) (args_R args) y) \<and> table (args_n args) (args_R args) (Mapping.keys tuple_in) \<and> table (args_n args) (args_R args) (Mapping.keys tuple_since) \<and> Ball (\<Union> (snd ` set (linearize data_prev))) (wf_tuple (args_n args) (args_R args)) \<and> cur = nt \<and> ts_tuple_rel (set ys) = {tas \<in> ts_tuple_rel (set (linearize data_prev) \<union> set (linearize data_in)). valid_tuple tuple_since tas} \<and> sorted (map fst (linearize data_prev)) \<and> (\<forall>t\<in>fst ` set (linearize data_prev). t \<le> nt \<and> nt - t < left (args_ivl args)) \<and> sorted (map fst (linearize data_in)) \<and> (\<forall>t\<in>fst ` set (linearize data_in). t \<le> nt \<and> left (args_ivl args) \<le> nt - t) \<and> (\<forall>as. Mapping.lookup tuple_in as = safe_max (fst ` {tas \<in> ts_tuple_rel (set (linearize data_in)). valid_tuple tuple_since tas \<and> as = snd tas})) \<and> (\<forall>as\<in>Mapping.keys tuple_since. case Mapping.lookup tuple_since as of Some t \<Rightarrow> t \<le> nt)"
Tinv_qmpt: assumes "invertible_qmpt" shows "qmpt M Tinv"
assms_4: fixes \<xi> :: "'a"   and \<xi>' :: "'a" assumes "S \<xi> \<xi>'" shows "U \<xi> \<xi>'"
lipschitz_on_def: fixes C :: "real"   and U :: "'c set"   and f :: "'c \<Rightarrow> 'd" shows "C-lipschitz_on U f = (0 \<le> C \<and> (\<forall>x\<in>U. \<forall>y\<in>U. dist (f x) (f y) \<le> C * dist x y))"
sigma_sets_into_sp: fixes A :: "'c set set"   and sp :: "'c set"   and x :: "'c set" assumes "A \<subseteq> Pow sp"   and "x \<in> sigma_sets sp A" shows "x \<subseteq> sp"
assms_2: shows "\<forall>X\<langle>\<or>\<noteq>: F\<rangle>\<^sub>s\<^sub>t # S \<in> \<S>"
proj_list_3_same: fixes xs :: "'b list"   and ys :: "'c list"   and z :: "'c"   and zs :: "'c list"   and ys' :: "'b list"   and z' :: "'b"   and zs' :: "'b list" assumes "proj_list_3 xs (ys, z, zs) = (ys', z', zs')"   and "length xs = length ys + 1 + length zs" shows "xs = ys' @ z' # zs'"
homeomorphic_contractible_eq: fixes S :: "'b set"   and T :: "'c set" assumes "S homeomorphic T" shows "contractible S = contractible T"
open_greaterThan: fixes a :: "'a" shows "open {a<..}"
card_quotient_disjoint: fixes A :: "'a set"   and r :: "('a \<times> 'a) set" assumes "finite A"   and "inj_on (\<lambda>x. {x} // r) A" shows "card (A // r) = card A"
vt_2: shows "t = (\<theta>1 \<circ>\<^sub>s \<theta>2) v"
mod_less: fixes m :: "nat"   and n :: "nat" assumes "m < n" shows "m mod n = m"
simps_3: fixes C\<^sub>1 :: "acom"   and C\<^sub>2 :: "acom" shows "strip (C\<^sub>1;; C\<^sub>2) = strip C\<^sub>1;; strip C\<^sub>2"
indspaceI: fixes f :: "('r, 'g) aezfun \<Rightarrow> 'v"   and FH :: "('r, 'g) aezfun set"   and V :: "'v set"   and G :: "'g set" assumes "f \<in> addfunset FH V"   and "f \<in> smultfunset G FH" shows "f \<in> indspace G FH V"
least_map_subset: fixes f :: "'b \<Rightarrow> 'c"   and S :: "'b set" shows "least_map f S \<subseteq> S"
adjoint_pair_antipar_4: fixes f :: "'a"   and g :: "'a" assumes "C.adjoint_pair f g" shows "src\<^sub>C g = trg\<^sub>C f"
vconn_implies_hpar_2: fixes V :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and src :: "'c \<Rightarrow> 'c"   and trg :: "'c \<Rightarrow> 'c"   and \<mu> :: "'c"   and f :: "'c"   and g :: "'c" assumes "horizontal_homs V src trg"   and "partial_magma.in_hom V \<mu> f g" shows "trg \<mu> = trg f"
function_ring_defs_5: shows "\<zero>\<^bsub>F\<^esub> = function_zero S R"
pick_insert2: fixes a :: "nat"   and I :: "nat set"   and i :: "nat"   and a' :: "nat" assumes "a \<notin> I"   and "i < card I"   and "pick (insert a I) a' = a"   and "a' \<le> i"   and "a' < card I + 1" shows "pick (insert a I) i < pick I i"
removing_attractor_induces_no_deadends: fixes G :: "('c, 'd) ParityGame_scheme"   and v :: "'c"   and S :: "'c set"   and p :: "Player"   and W :: "'c set"   and w :: "'c" assumes "ParityGame G"   and "v \<in> S - ParityGame.attractor G p W"   and "v \<rightarrow>\<^bsub>G\<^esub> w"   and "w \<in> S"   and "\<And>w. \<lbrakk>v \<in> (if p** = Even then V0\<^bsub>G\<^esub> else V\<^bsub>G\<^esub> - V0\<^bsub>G\<^esub>); v \<rightarrow>\<^bsub>G\<^esub> w\<rbrakk> \<Longrightarrow> w \<in> S" shows "\<exists>w\<in>S - ParityGame.attractor G p W. v \<rightarrow>\<^bsub>G\<^esub> w"
assms_1: shows "\<forall>w\<in>space M. f w = g w"
lv_rel_le: shows "(list_all2 (\<le>), (\<le>)) \<in> lv_rel \<rightarrow> lv_rel \<rightarrow> bool_rel"
supp_fst: fixes x :: "name"   and \<theta> :: "(name \<times> trm) list" assumes "x \<in> supp (map fst \<theta>)" shows "x \<in> supp \<theta>"
cmod_less: fixes a :: "int"   and n :: "int" assumes "a < n"   and "- n < a" shows "cmod (of_int a) < real_of_int n"
emeasure_interval_measure_Ioc_eq: fixes F :: "real \<Rightarrow> real"   and a :: "real"   and b :: "real" assumes "\<And>x y. x \<le> y \<Longrightarrow> F x \<le> F y"   and "\<And>a. continuous (at_right a) F" shows "emeasure (interval_measure F) {a<..b} = ennreal (if a \<le> b then F b - F a else 0)"
bounded: shows "\<exists>K. \<forall>x. norm (g x) \<le> norm x * K"
Nset_nonempty: fixes n :: "nat" shows "{i. i \<le> n} \<noteq> {}"
set_shadow_root_get_shadow_root: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and set_shadow_root :: "(_) element_ptr \<Rightarrow> (_) shadow_root_ptr option \<Rightarrow> ((_) heap, exception, unit) prog"   and set_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_shadow_root :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, (_) shadow_root_ptr option) prog"   and get_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and h :: "(_) heap"   and ptr :: "(_) element_ptr"   and shadow_root_ptr_opt :: "(_) shadow_root_ptr option"   and h' :: "(_) heap" assumes "l_set_shadow_root_get_shadow_root type_wf set_shadow_root set_shadow_root_locs get_shadow_root get_shadow_root_locs"   and "type_wf h"   and "h \<turnstile> set_shadow_root ptr shadow_root_ptr_opt \<rightarrow>\<^sub>h h'" shows "h' \<turnstile> get_shadow_root ptr \<rightarrow>\<^sub>r shadow_root_ptr_opt"
reduction_4: assumes "\<alpha> = 4 / 3" shows "reduction = 1"
\<sigma>_def: fixes P :: "'a list" shows "\<sigma> P = map indexed_const P"
fun_relD1: fixes f :: "'a \<Rightarrow> 'b"   and f' :: "'c \<Rightarrow> 'f"   and Ra :: "('a \<times> 'c) set"   and Rr :: "('b \<times> 'f) set"   and x :: "'a"   and r :: "'b" assumes "(f, f') \<in> Ra \<rightarrow> Rr"   and "f x = r" shows "\<forall>x'. (x, x') \<in> Ra \<longrightarrow> (r, f' x') \<in> Rr"
solvable_iff_trivial_derived_seq: fixes G :: "('a, 'b) monoid_scheme" assumes "Group.group G" shows "solvable G = (\<exists>n. (derived G ^^ n) (carrier G) = {\<one>\<^bsub>G\<^esub>})"
iT_Minus_finite: fixes k :: "nat"   and I :: "nat set" shows "finite (k \<ominus> I)"
is_gcd_def: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and x :: "'a"   and a :: "'a"   and b :: "'a" shows "dvd.is_gcd times x a b \<equiv> dvd.dvd times x a \<and> dvd.dvd times x b \<and> (\<forall>y. dvd.dvd times y a \<longrightarrow> dvd.dvd times y b \<longrightarrow> dvd.dvd times y x)"
oless_def: fixes D :: "('c, 'd) Order_scheme"   and a :: "'c"   and b :: "'c" shows "a \<prec>\<^bsub>D\<^esub> b \<equiv> a \<preceq>\<^bsub>D\<^esub> b \<and> a \<noteq> b"
assms_4: fixes i :: "nat" assumes "i < n" shows "w $ i \<le> v $ i"
z'_min: fixes y :: "('t \<Rightarrow>\<^sub>0 'b) list" assumes "(y, z') \<in> rb_aux_term1" shows "y \<notin> fst ` Q"
F_sync: fixes P :: "'c process"   and A :: "'c set"   and Q :: "'c process" shows "F (P \<lbrakk>A\<rbrakk> Q) = {(s, R). \<exists>t u X Y. (t, X) \<in> F P \<and> (u, Y) \<in> F Q \<and> s setinterleaves ((t, u), ev ` A \<union> {tick}) \<and> R = (X \<union> Y) \<inter> (ev ` A \<union> {tick}) \<union> X \<inter> Y} \<union> {(s, R). \<exists>t u r v. front_tickFree v \<and> (tickFree r \<or> v = []) \<and> s = r @ v \<and> r setinterleaves ((t, u), ev ` A \<union> {tick}) \<and> (t \<in> D P \<and> u \<in> T Q \<or> t \<in> D Q \<and> u \<in> T P)}"
terminates_to_terminates_sequence_flatten: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile>c \<down> s" shows "\<Gamma>\<turnstile>sequence Seq (flatten c) \<down> s"
L3_5ii_lemma3: fixes c1 :: "'a com"   and c2 :: "'a com"   and s :: "'a"   and Ts :: "('a ann_com option \<times> 'a set) list"   and t :: "'a" assumes "(Seq c1 c2, s) -P*\<rightarrow> (Parallel Ts, t)"   and "All_None Ts" shows "\<exists>y Rs. (c1, s) -P*\<rightarrow> (Parallel Rs, y) \<and> All_None Rs \<and> (c2, y) -P*\<rightarrow> (Parallel Ts, t)"
cmp_simps'_1: fixes \<mu> :: "'b"   and \<nu> :: "'b" assumes "B.arr \<mu>"   and "B.arr \<nu>"   and "src\<^sub>B \<mu> = trg\<^sub>B \<nu>" shows "C.arr (\<Phi>\<^sub>F (\<mu>, \<nu>))"
list_gcd_greatest: fixes xs :: "'b list"   and y :: "'b" assumes "\<And>x. x \<in> set xs \<Longrightarrow> y dvd x" shows "y dvd gcd_list xs"
pw_bind_leI: fixes S :: "'b nres"   and M :: "'c nres"   and f :: "'c \<Rightarrow> 'b nres" assumes "nofail S \<Longrightarrow> nofail M"   and "\<And>x. \<lbrakk>nofail M; inres M x\<rbrakk> \<Longrightarrow> f x \<le> S" shows "M \<bind> f \<le> S"
sheaf_spec_morphisms_def: fixes U :: "nat set set"   and V :: "nat set set" shows "cring0.sheaf_spec_morphisms U V \<equiv> \<lambda>s\<in>cring0.sheaf_spec U. restrict s V"
int_simps_20: fixes P :: "'b \<Rightarrow> bool" shows "\<turnstile> (P \<and> P) = P"
has_derivative_at_withinI: fixes f :: "'e \<Rightarrow> 'f"   and f' :: "'e \<Rightarrow> 'f"   and x :: "'e"   and s :: "'e set" assumes "(f has_derivative f') (at x)" shows "(f has_derivative f') (at x within s)"
cases: fixes edges :: "(('a \<times> 'b) \<times> 'a \<times> 'c) set"   and a1 :: "'a"   and a2 :: "'a"   and a3 :: "(('a \<times> 'b) \<times> 'a \<times> 'c) list" assumes "Pre_Port_Graph.path edges a1 a2 a3" obtains v :: "'a" where "a1 = v" and "a2 = v" and "a3 = []"   | e :: "('a \<times> 'b) \<times> 'a \<times> 'c" and v' :: "'a" and pth :: "(('a \<times> 'b) \<times> 'a \<times> 'c) list" where "a1 = Pre_Port_Graph.edge_begin e" and "a2 = v'" and "a3 = e # pth" and "e \<in> edges" and "Pre_Port_Graph.path edges (Pre_Port_Graph.edge_end e) v' pth"
omega_least: fixes x :: "'a"   and z :: "'a"   and y :: "'a" assumes "x * z \<sqinter> y \<le> z" shows "x ^ \<omega> * y \<le> z"
b: shows "(\<lambda>n. f (v n)) \<longlonglongrightarrow> f y"
assms_2: shows "l' \<equiv> relabel_effect f l u"
\<beta>_def: shows "\<beta> = SyntaxN.Ex j (KRP \<guillemotleft>Var i\<guillemotright> (Var i) (Var j) AND \<alpha>(i::=Var j))"
assign_Norm_Some: fixes f :: "'a \<Rightarrow> state \<Rightarrow> state"   and v :: "'a"   and s :: "st"   and y :: "abrupt" assumes "fst (f v (Norm s)) = Some y" shows "assign f v (Norm s) = (Some y, s)"
non_Cauchy_unbounded: fixes a :: "nat \<Rightarrow> real"   and e :: "real" assumes "\<And>n. 0 \<le> a n"   and "0 < e"   and "\<forall>M. \<exists>m n. M \<le> m \<and> M \<le> n \<and> n < m \<and> e \<le> sum a {Suc n..m}" shows "(\<lambda>x. ereal (sum a {0..x})) \<longlonglongrightarrow> \<infinity>"
in_homI: fixes f :: "(bool, 'arr list) concrete_category.arr"   and a :: "(bool, 'arr list) concrete_category.arr"   and b :: "(bool, 'arr list) concrete_category.arr" assumes "arr f"   and "local.dom f = a"   and "cod f = b" shows "\<guillemotleft>f : a \<rightarrow> b\<guillemotright>"
weave_def: fixes A :: "nat set"   and xs :: "'b list"   and ys :: "'b list" shows "weave A xs ys = map (\<lambda>i. if i \<in> A then xs ! card {a \<in> A. a < i} else ys ! card {a \<in> - A. a < i}) [0..<length xs + length ys]"
rewrite_modal_atoms: fixes \<phi> :: "'a ltln" shows "atoms_ltln (rewrite_modal \<phi>) \<subseteq> atoms_ltln \<phi>"
compl_top_eq: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c" assumes "class.boolean_algebra minus uminus inf less_eq less sup bot top" shows "uminus top = bot"
eq_\<I>_converterD_WT1: fixes \<I> :: "('a, 'b) \<I>"   and \<I>' :: "('c, 'd) \<I>"   and conv1 :: "('a, 'b, 'c, 'd) converter"   and conv2 :: "('a, 'b, 'c, 'd) converter" assumes "\<I>,\<I>' \<turnstile>\<^sub>C conv1 \<sim> conv2"   and "\<I>, \<I>' \<turnstile>\<^sub>C conv1 \<surd>" shows "\<I>, \<I>' \<turnstile>\<^sub>C conv2 \<surd>"
base: fixes x :: "nat"   and y :: "nat"   and rel :: "(nat \<times> nat) set" assumes "(x, y) \<in> rel" shows "Steps rel 0 (x, y)"
se_field: fixes e :: "exp"   and fi :: "nat" shows "e \<in> isubexprs(FieldProj e fi)"
outsp_gpv_outs_gpv_eq: fixes \<I> :: "('e, 'f) \<I>"   and x :: "'e" shows "outsp_gpv \<I> x = (\<lambda>gpv. x \<in> outs_gpv \<I> gpv)"
lsl_def: fixes x :: "'b" shows "\<nu> x = \<top> \<cdot> x"
compliant_stateful_ACS_only_state_violations': fixes \<T> :: "'b stateful_policy"   and G :: "'b graph"   and M :: "'b SecurityInvariant_configured list" assumes "stateful_policy_compliance \<T> G M" shows "\<forall>F\<in>get_offending_flows M (stateful_policy_to_network_graph \<T>). F \<subseteq> backflows (flows_state \<T>)"
map_graph_fn_eqI: fixes G :: "('b, 'a) labeled_graph"   and f :: "'a \<Rightarrow> 'c"   and g :: "'a \<Rightarrow> 'c" assumes "\<And>x. x \<in> vertices G \<Longrightarrow> f x = g x" shows "map_graph_fn G f = map_graph_fn G g"
intros_7: shows "Integer \<sqsubset>\<^sub>B Real"
beta_cases_3: fixes T :: "typ"   and r :: "Core.term"   and s :: "Core.term" assumes "Abs T r \<rightarrow>\<^sub>\<beta> s" obtains t :: "Core.term" where "s = Abs T t" and "r \<rightarrow>\<^sub>\<beta> t"
prems: shows "safe_reach_upto (Suc n) safe_free_flowing c\<^sub>0"
szip_sset_elim: fixes a :: "'c"   and b :: "'d"   and u :: "'c stream"   and v :: "'d stream" assumes "(a, b) \<in> sset (u ||| v)" obtains "a \<in> sset u" and "b \<in> sset v"
x0_ge_1: shows "1 \<le> x\<^sub>0"
permutation_of_list_permutes: fixes xs :: "('b \<times> 'b) list"   and A :: "'b set" assumes "list_permutes xs A" shows "permutation_of_list xs permutes A"
left_space: fixes ts :: "traffic"   and v :: "view"   and c :: "cars" shows "left (space ts v c) = pos ts c"
c'_def: shows "c' = (\<lambda>i. fls_X_intpow ((int N - int i) * a) * c i)"
bi_total_def: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool" shows "bi_total R = ((\<forall>x. \<exists>y. R x y) \<and> (\<forall>y. \<exists>x. R x y))"
zip_eq_Nil_iff: fixes xs :: "'k list"   and ys :: "'l list" shows "(zip xs ys = []) = (xs = [] \<or> ys = [])"
map_of_is_SomeI: fixes xys :: "('c \<times> 'd) list"   and x :: "'c"   and y :: "'d" assumes "distinct (list.map fst xys)"   and "(x, y) \<in> set xys" shows "map_of xys x = Some y"
crdt_ops_ref_less_last: fixes xs :: "('c \<times> 'd) list"   and oid :: "'c"   and oper :: "'d"   and deps :: "'d \<Rightarrow> 'c set"   and r :: "'c" assumes "crdt_ops (xs @ [(oid, oper)]) deps"   and "r \<in> deps oper" shows "r < oid"
mprefix_Par_distr: fixes A :: "'a set"   and C :: "'a set"   and B :: "'a set"   and P :: "'a \<Rightarrow> 'a process"   and Q :: "'a \<Rightarrow> 'a process" assumes "A \<subseteq> C"   and "B \<subseteq> C" shows "(Mprefix A P \<lbrakk>C\<rbrakk> Mprefix B Q) = Mprefix (A \<inter> B) (\<lambda>x. P x \<lbrakk>C\<rbrakk> Q x)"
bf_xor_def: fixes a :: "'b \<Rightarrow> bool"   and b :: "'b \<Rightarrow> bool" shows "bf_xor a b = bf_not (bf_biimp a b)"
uniformity_Gromov_completion_def: shows "uniformity = (INF e\<in>{0<..}. principal {(x, y). dist x y < e})"
M1_add_M0: shows "M1 + M0 = 1\<^sub>m K"
lexordp_eq_simps_3: fixes less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and xs :: "'a list" shows "ord.lexordp_eq less (x # xs) [] = False"