hcju/isabelleps huggingface dataset:
cases: fixes x :: "real_alg_2 list" obtains xa :: "real_alg_2" and xs :: "real_alg_2 list" where "x = xa # xs"   | "x = []"
simps_1: fixes \<alpha> :: "game" shows "Loopo (Agame \<alpha>) = Agame \<alpha>**"
nn_integral_eq_integral: fixes M :: "'a measure"   and f :: "'a \<Rightarrow> real" assumes "integrable M f"   and "AE x in M. 0 \<le> f x" shows "\<integral>\<^sup>+ x. ennreal (f x) \<partial>M = ennreal (integral\<^sup>L M f)"
pow_int_add: fixes x :: "'b"   and a :: "int"   and b :: "int" assumes "x \<noteq> (0::'b)" shows "pow_int x (a + b) = pow_int x a * pow_int x b"
is_gcdE: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and x :: "'a"   and a :: "'a"   and b :: "'a" assumes "dvd.is_gcd times x a b" obtains "dvd.dvd times x a" and "dvd.dvd times x b" and "\<And>y. \<lbrakk>dvd.dvd times y a; dvd.dvd times y b\<rbrakk> \<Longrightarrow> dvd.dvd times y x"
star_sim1: fixes z :: "'a"   and x :: "'a"   and y :: "'a" assumes "z \<cdot> x \<le> y \<cdot> z" shows "z \<cdot> x\<^sup>\<star> \<le> y\<^sup>\<star> \<cdot> z"
chamber_system_simplices: fixes X :: "'b set set" assumes "ChamberComplex X" shows "order.greater_eq X (ChamberComplex.chamber_system X)"
closure_sum: fixes S :: "'d set"   and T :: "'d set" shows "closure S + closure T \<subseteq> closure (S + T)"
empty: fixes \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and m :: "'c match_expr"   and p :: "'d"   and \<Gamma> :: "char list \<Rightarrow> 'c rule list option" assumes "matches \<gamma> m p" shows "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>[Rule m Empty], Undecided\<rangle> \<Rightarrow> Undecided"
exclude_one_is_pseudominimal: fixes w :: "'w"   and t :: "'w" assumes "w \<in> W"   and "t \<in> S" shows "dual_order.pseudominimal_in \<P> (w +o \<langle>S - {t}\<rangle>)"
arith_special_3: shows "(1::'a) + (1::'a) = (2::'a)"
routing_ipassmt_wi_distinct: shows "distinct (map fst (routing_ipassmt_wi rtbl))"
revcast_eq_of_bl: fixes w :: "'c word" shows "revcast w = of_bl (takefill False LENGTH('b) (to_bl w))"
elims_1: fixes x :: "ereal"   and xa :: "ereal"   and y :: "bool" assumes "(x < xa) = y" obtains xb :: "real" and ya :: "real" where "x = ereal xb__" and "xa = ereal ya" and "y = (xb__ < ya)"   | a :: "ereal" where "x = \<infinity>" and "xa = a__" and "\<not> y"   | a :: "ereal" where "x = a" and "xa = - \<infinity>" and "\<not> y"   | xb :: "real" where "x = ereal xb" and "xa = \<infinity>" and "y"   | r :: "real" where "x = - \<infinity>" and "xa = ereal r" and "y"   | "x = - \<infinity>" and "xa = \<infinity>" and "y"
lift_def: fixes f :: "'d \<Rightarrow> 'a"   and w :: "(bool \<times> 'd) list" shows "lift f w = m_concat (map (lift_gi f) w)"
accs_reduce_b_acc: fixes \<delta> :: "('a, 'b) ta_rule set"   and t :: "'b tree"   and q :: "'a" assumes "accs \<delta> t q" shows "accs (reduce_rules \<delta> (b_accessible \<delta>)) t q"
pmf_prob_code_unfold: fixes p :: "'b pmf" shows "pmf_prob p = measure_pmf.prob p"
transpose_row_def: fixes A :: "(('d, 'f) vec, 'e) vec"   and i :: "'f" shows "transpose_row A i = (\<chi>j. A $ j $ i)"
Vfrom_succ_Ord: fixes i :: "V"   and a :: "V" assumes "Ord i" shows "Vfrom a (ZFC_in_HOL.succ i) = a \<squnion> VPow (Vfrom a i)"
assms_4: shows "\<forall>x\<in>U\<^sub>3. g x \<in> U"
pmf_scale_spmf_None: fixes k :: "real"   and p :: "'c spmf" shows "pmf (scale_spmf k p) None = 1 - min 1 (max 0 k * (1 - pmf p None))"
dns5: fixes x :: "'a"   and y :: "'a" shows "ad (ad x) \<cdot> (ad (ad x) + ad (ad y)) = ad (ad x)"
fps_deriv_def: fixes f :: "'b fps" shows "fps_deriv f = Abs_fps (\<lambda>n. of_nat (n + 1) * f $ (n + 1))"
rrep_dip_in_vD: fixes i :: "nat" shows "paodv i \<TTurnstile> onl \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V (\<lambda>(\<xi>, l). l \<in> {PRrep-:4} \<longrightarrow> dip \<xi> \<in> vD (rt \<xi>))"
of_bool_odd_eq_mod_2: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c" assumes "class.semiring_parity plus minus zero times one divide modulo" shows "zero_neq_one.of_bool one zero (\<not> dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) a) = modulo a (numeral.numeral one plus (num.Bit0 num.One))"
p\<^sub>0_simps: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and h :: "'b"   and k :: "'b" assumes "cospan_of_maps_in_bicategory_of_spans V H \<a> \<i> src trg h k" shows "trg (bicategory_of_spans.tab\<^sub>0 V H \<a> \<i> src trg (H (chosen_right_adjoints.some_right_adjoint V H \<a> \<i> src trg k) h)) = src h"
singular_relboundary_diff: fixes p :: "nat"   and X :: "'c topology"   and S :: "'c set"   and a :: "((nat \<Rightarrow> real) \<Rightarrow> 'c) \<Rightarrow>\<^sub>0 int"   and b :: "((nat \<Rightarrow> real) \<Rightarrow> 'c) \<Rightarrow>\<^sub>0 int" assumes "singular_relboundary p X S a"   and "singular_relboundary p X S b" shows "singular_relboundary p X S (a - b)"
select_convs_1: fixes statename :: "nat option"   and datastate :: "nat \<Rightarrow>f value option"   and action :: "String.literal \<times> value list"   and "output" :: "value option list"   and more :: "'a" shows "statename \<lparr>statename = statename, datastate = datastate, action = action, output = output, \<dots> = more\<rparr> = statename"
rep_eq: fixes x :: "'b mat_impl" shows "Rep_mat (mat_impl x) = (case Rep_mat_impl x of (nr, nc, m) \<Rightarrow> (nr, nc, mk_mat nr nc (\<lambda>(i, j). m !! i !! j)))"
double_fminus: fixes A :: "'c fset"   and B :: "'c fset"   and C :: "'c fset" assumes "A |\<subseteq>| B"   and "B |\<subseteq>| C" shows "B |-| (C |-| A) = A"
COD_comp: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b local.term set"   and g :: "'b local.term set" assumes "free_monoidal_category C"   and "free_monoidal_category.ARR C f"   and "free_monoidal_category.ARR C g"   and "monoidal_language.Dom C (free_monoidal_category.rep C f) = monoidal_language.Cod C (free_monoidal_category.rep C g)" shows "monoidal_language.Cod C (free_monoidal_category.rep C (free_monoidal_category.comp C f g)) = monoidal_language.Cod C (free_monoidal_category.rep C f)"
field_frac_mul: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier K"   and "y \<in> carrier K"   and "y \<noteq> \<zero>" shows "x = x \<cdot>\<^sub>r y\<^bsup>\<hyphen> K\<^esup> \<cdot>\<^sub>r y"
J: shows "card {i. i < CARD('nc) \<and> i \<in> J} = CARD('nc2)"
return_def: fixes a :: "'e" shows "return a \<equiv> State (Pair a)"
bvars\<^sub>s\<^sub>s\<^sub>t_def: fixes S :: "('f, 'e) stateful_strand_step list" shows "bvars\<^sub>s\<^sub>s\<^sub>t S \<equiv> \<Union> (set (map (set \<circ> bvars\<^sub>s\<^sub>s\<^sub>t\<^sub>p) S))"
vanishes_minus: fixes X :: "nat \<Rightarrow> rat" assumes "Real.vanishes X" shows "Real.vanishes (\<lambda>n. - X n)"
set_of_ivl_def: fixes x :: "'c \<times> 'c" shows "set_of_ivl x = {fst x..snd x}"
less_cSup_iff: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and X :: "'b set"   and y :: "'b" assumes "class.conditionally_complete_linorder Inf Sup inf less_eq less sup"   and "X \<noteq> {}"   and "preorder.bdd_above less_eq X" shows "less y (Sup X) = (\<exists>x\<in>X. less y x)"
less_gexp_aux_antisym: fixes x :: "'a gexp"   and y :: "'a gexp" shows "local.less_gexp_aux x y = (\<not> local.less_gexp_aux y x \<and> x \<noteq> y)"
elims_2: fixes init_s :: "(nat \<times> linear_poly) list \<Rightarrow> 'a"   and assert_s :: "'b \<times> 'c atom \<Rightarrow> 'a \<Rightarrow> 'b list + 'a"   and check_s :: "'a \<Rightarrow> 'b list + 'a"   and solution_s :: "'a \<Rightarrow> (nat, 'c) mapping"   and checkpoint_s :: "'a \<Rightarrow> 'd"   and backtrack_s :: "'d \<Rightarrow> 'a \<Rightarrow> 'a"   and invariant_s :: "(nat \<times> linear_poly) list \<Rightarrow> ('b \<times> 'c atom) set \<Rightarrow> 'a \<Rightarrow> bool"   and checked_s :: "(nat \<times> linear_poly) list \<Rightarrow> ('b \<times> 'c atom) set \<Rightarrow> 'a \<Rightarrow> bool"   and preprocess :: "('b \<times> 'c ns_constraint) list \<Rightarrow> (nat \<times> linear_poly) list \<times> ('b \<times> 'c atom) list \<times> ((nat, 'c) mapping \<Rightarrow> (nat, 'c) mapping) \<times> 'b list"   and x :: "('b \<times> 'c ns_constraint) list"   and xa :: "'b set"   and xb :: "(('b, ('b \<times> 'c atom) list) mapping \<times> ((nat, 'c) mapping \<Rightarrow> (nat, 'c) mapping) \<times> 'b list) \<times> 'a" assumes "Incremental_Atom_Ops_For_NS_Constraint_Ops init_s assert_s check_s solution_s checkpoint_s backtrack_s invariant_s checked_s preprocess"   and "Incremental_Atom_Ops_For_NS_Constraint_Ops.checked_nsc checked_s preprocess x xa xb" obtains nsc :: "('b \<times> 'c ns_constraint) list" and J :: "'b set" and asi :: "('b, ('b \<times> 'c atom) list) mapping" and tv :: "(nat, 'c) mapping \<Rightarrow> (nat, 'c) mapping" and ui :: "'b list" and s :: "'a" where "x = nsc" and "xa = J" and "xb = ((asi, tv, ui), s)" and "case preprocess nsc of (t, as, tv', ui') \<Rightarrow> Incremental_Atom_Ops_For_NS_Constraint_Ops_Default.invariant_as_asi (set as) asi tv tv' ui ui' \<and> checked_s t (set as \<inter> J \<times> UNIV) s \<and> J \<inter> set ui = {}"
poly_no_roots_leq_less: fixes a :: "real"   and b :: "real"   and p :: "real poly" shows "(\<forall>x. a \<le> x \<and> x < b \<longrightarrow> poly p x \<noteq> 0) = (b \<le> a \<or> p \<noteq> 0 \<and> poly p a \<noteq> 0 \<and> count_roots_between p a b = (if a < b \<and> poly p b = 0 then 1 else 0))"
Tree\<^sub>\<alpha>_free_6: fixes f1 :: "'d"   and \<phi> :: "'b"   and f2 :: "'d"   and \<alpha> :: "'c"   and t\<^sub>\<alpha> :: "('a, 'b, 'c, 'd) Tree\<^sub>\<alpha>" shows "Pred\<^sub>\<alpha> f1 \<phi> \<noteq> Act\<^sub>\<alpha> f2 \<alpha> t\<^sub>\<alpha>"
card_of_Un_ordLess_infinite: fixes C :: "'a set"   and A :: "'b set"   and B :: "'b set" assumes "infinite C"   and "|A| <o |C|"   and "|B| <o |C|" shows "|A \<union> B| <o |C|"
lm153: fixes X :: "'a set"   and f :: "'a \<Rightarrow> 'b" shows "graph X f = Graph f || X"
select_index_inter: fixes n :: "nat"   and i :: "nat" shows "{k |k. select_index n i k} \<inter> {k |k. k < 2 ^ n \<and> \<not> select_index n i k} = {}"
osum_def: fixes r :: "('d \<times> 'd) set"   and r' :: "('e \<times> 'e) set" shows "r +o r' = map_prod Inl Inl ` r \<union> map_prod Inr Inr ` r' \<union> {(Inl a, Inr a') |a a'. a \<in> Field r \<and> a' \<in> Field r'}"
induct: fixes P :: "nat \<Rightarrow> bool"   and a0 :: "nat" assumes "P 0"   and "P 1"   and "\<And>n. \<lbrakk>2 \<le> n; P (nat \<lceil>real n / 2\<rceil>)\<rbrakk> \<Longrightarrow> P n" shows "P a0"
cSUP_mono: fixes A :: "'c set"   and g :: "'e \<Rightarrow> 'd multiset"   and B :: "'e set"   and f :: "'c \<Rightarrow> 'd multiset" assumes "A \<noteq> {}"   and "subset_mset.bdd_above (g ` B)"   and "\<And>n. n \<in> A \<Longrightarrow> \<exists>m\<in>B. f n \<subseteq># g m" shows "Sup (f ` A) \<subseteq># Sup (g ` B)"
formulasI: fixes S :: "'a set"   and x :: "'a formula" assumes "\<And>A B. \<forall>i\<in>S. (i \<in> A) = (i \<in> B) \<Longrightarrow> (A \<in> Rep_formula x) = (B \<in> Rep_formula x)" shows "x \<in> formulas S"
minus_zero: shows "\<ominus>\<^bsub>K [X]\<^esub> \<zero>\<^bsub>K [X]\<^esub> = \<zero>\<^bsub>K [X]\<^esub>"
relator_freewords: shows "Abs_freelist ` R = Abs_freeword ` map (charpair S) ` R"
gcd_k: shows "gcd m n * (m div gcd m n) = m"
fmrel_on_fsubset: fixes S :: "'a fset"   and R :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and m :: "('a, 'b) fmap"   and n :: "('a, 'c) fmap"   and T :: "'a fset" assumes "fmrel_on_fset S R m n"   and "T |\<subseteq>| S" shows "fmrel_on_fset T R m n"
simps_5: fixes x31 :: "'a"   and x32 :: "'a ifex"   and x33 :: "'a ifex" shows "IF x31 x32 x33 \<noteq> Trueif"
insert_inter_insert: fixes a :: "'b"   and A :: "'b set"   and B :: "'b set" shows "insert a A \<inter> insert a B = insert a (A \<inter> B)"
convex_imp_path_connected: fixes S :: "'a set" assumes "convex S" shows "path_connected S"
map_of_concat''': fixes k :: "'a"   and l :: "('a \<times> 'b) list list" assumes "\<exists>i. k \<in> dom (map_of (l ! i)) \<and> i < length l" shows "k \<in> dom (map_of (concat l))"
star_denest: fixes x :: "'a"   and y :: "'a" shows "(x + y)\<^sup>\<star> = (x\<^sup>\<star> ; y\<^sup>\<star>)\<^sup>\<star>"
loc_unconc: fixes a :: "'a el_step list"   and b :: "'a el_step list" shows "loc (a @ b) = loc a @ loc b"
rel_Grp: fixes A1 :: "'q set"   and f1 :: "'q \<Rightarrow> 's"   and A2 :: "'r set"   and f2 :: "'r \<Rightarrow> 't" shows "rel_sum (BNF_Def.Grp A1 f1) (BNF_Def.Grp A2 f2) = BNF_Def.Grp {x. Basic_BNFs.setl x \<subseteq> A1 \<and> Basic_BNFs.setr x \<subseteq> A2} (map_sum f1 f2)"
dbm_entry_dbm_min': fixes u :: "'c \<Rightarrow> 'd"   and c1 :: "'c"   and c2 :: "'c"   and a :: "'d DBMEntry"   and b :: "'d DBMEntry" assumes "dbm_entry_val u (Some c1) (Some c2) (min a b)" shows "dbm_entry_val u (Some c1) (Some c2) a"
assms_12: shows "InvariantVarsM (getM state) F0 Vbl"
assms_2: assumes "C1.arr (fst f) \<and> C2.arr (snd f)" shows "T"
val_ifex: fixes b :: "'b ltln"   and s :: "'b ltln \<Rightarrow> bool" shows "val_ifex (ifex_of_ltl b) s = {x. s x} \<Turnstile>\<^sub>P b"
set_tag_name_get_child_nodes: fixes ptr :: "(_) element_ptr"   and h :: "(_) heap"   and h' :: "(_) heap"   and ptr' :: "(_) object_ptr" shows "\<forall>w\<in>set_tag_name_locs ptr. h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> (\<forall>r\<in>get_child_nodes_locs\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M ptr'. r h h')"
\<a>_def: fixes ps :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list" shows "\<a> ps = (LEAST k. standard_decomp k ps)"
primepow_even_altdef: fixes n :: "nat" shows "primepow_even n = (primepow n \<and> even (multiplicity (aprimedivisor n) n))"
scale_right_distrib: fixes a :: "'a"   and x :: "'b"   and y :: "'b" shows "a *s (x + y) = a *s x + a *s y"
var_Monotone: fixes b :: "BExpr" shows "Monotone (\<lambda>\<Phi> (s, t). (b, \<Phi>, s, t) \<in> var)"
local_lipschitz_def: fixes T :: "'g set"   and X :: "'h set"   and f :: "'g \<Rightarrow> 'h \<Rightarrow> 'i" shows "local_lipschitz T X f \<equiv> \<forall>x\<in>X. \<forall>t\<in>T. \<exists>u>0. \<exists>L. \<forall>t\<in>cball t u \<inter> T. L-lipschitz_on (cball x u \<inter> X) (f t)"
mod_add_left_eq: fixes divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and modulo :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and euclidean_size :: "'d \<Rightarrow> nat"   and a :: "'d"   and c :: "'d"   and b :: "'d" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size" shows "modulo (plus (modulo a c) b) c = modulo (plus a b) c"
field_le_epsilon: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "\<And>e. less zero e \<Longrightarrow> less_eq x (plus y e)" shows "less_eq x y"
is_parallel_solution_for_problem_operator_set: fixes \<Pi> :: "'c strips_problem"   and \<pi> :: "'c strips_operator list list"   and ops :: "'c strips_operator list"   and op :: "'c strips_operator" assumes "STRIPS_Semantics.is_parallel_solution_for_problem \<Pi> \<pi>"   and "ops \<in> set \<pi>"   and "op \<in> set ops" shows "op \<in> set (\<Pi>\<^sub>\<O>)"
gf: shows "homotopic_with (\<lambda>h. True) X X (g \<circ> f) id"
in_deg_r: fixes v :: "'a" assumes "v \<in> verts G" shows "int (in_degree G v) = \<r>"
lift_dnfeq_qe_def: shows "lift_dnfeq_qe = lift_dnf_qe \<circ> lift_eq_qe"
interior_interior: fixes S :: "'a set" shows "interior (interior S) = interior S"
cobounded_unique_unbounded_components: fixes S :: "'a set"   and c :: "'a set"   and c' :: "'a set" assumes "bounded (- S)"   and "c \<in> components S"   and "\<not> bounded c"   and "c' \<in> components S"   and "\<not> bounded c'"   and "2 \<le> DIM('a)" shows "c' = c"
msgs_unchanged_if_snapshotted_RecvMarker_for_other_is: fixes c :: "('a, 'b, 'c) configuration"   and ev :: "('a, 'b, 'c) event"   and c' :: "('a, 'b, 'c) configuration"   and i :: "nat"   and p :: "'a"   and q :: "'a"   and i' :: "nat" assumes "c \<turnstile> ev \<mapsto> c'"   and "ev = RecvMarker i p q"   and "ps c p \<noteq> None"   and "i' \<noteq> i" shows "msgs c i' = msgs c' i'"
return_None_bind_spmf: fixes f :: "'b \<Rightarrow> 'a spmf" shows "return_pmf None \<bind> f = return_pmf None"
simps_3: fixes cx :: "int"   and cy :: "int"   and more :: "'b" shows "point.more \<lparr>cx = cx, cy = cy, \<dots> = more\<rparr> = more"
countable_Un: fixes A :: "'k set"   and B :: "'k set" assumes "countable A"   and "countable B" shows "countable (A \<union> B)"
simps_2: fixes u :: "'b"   and e :: "'b \<times> 'b"   and es :: "('b \<times> 'b) list" shows "pawalk_verts u (e # es) = fst e # pawalk_verts (snd e) es"
rtrancl_converseI: fixes y :: "'b"   and x :: "'b"   and r :: "('b \<times> 'b) set" assumes "(y, x) \<in> r\<^sup>*" shows "(x, y) \<in> (r\<inverse>)\<^sup>*"
lossless_gpv_induct: fixes \<I> :: "('d, 'e) \<I>"   and gpv :: "('f, 'd, 'e) gpv"   and P :: "('f, 'd, 'e) gpv \<Rightarrow> bool" assumes "lossless_gpv \<I> gpv"   and "\<And>p. \<lbrakk>lossless_spmf p; \<And>out c input. \<lbrakk>IO out c \<in> set_spmf p; input \<in> responses_\<I> \<I> out\<rbrakk> \<Longrightarrow> lossless_gpv \<I> (c input); \<And>out c input. \<lbrakk>IO out c \<in> set_spmf p; input \<in> responses_\<I> \<I> out\<rbrakk> \<Longrightarrow> P (c input)\<rbrakk> \<Longrightarrow> P (GPV p)" shows "P gpv"
M_def: shows "M = (MAX (s, t)\<in>SIGMA x:acc_on (- H) `` {s}. set_pmf (K x). \<rho> s + \<iota> s t)"
Inf_closed': fixes U :: "'a set"   and Inf :: "'a set \<Rightarrow> 'a" assumes "Inf_ow U Inf" shows "\<forall>x\<subseteq>U. Inf x \<in> U"
simps: fixes s :: "String.literal"   and S :: "String.literal set" shows "variant_name s S = (let s' = fresh_name' S in (s', insert s' S))"
parts_crypt_1: fixes K :: "key"   and X :: "msg" shows "parts {Crypt K X} \<subseteq> insert (Crypt K X) (parts {X})"
eq_refl: fixes x :: "'b"   and y :: "'b" assumes "x = y" shows "x \<sqsubseteq> y"
get_shadow_root_locs_impl: shows "get_shadow_root_locs = get_shadow_root_locs"
Image_def: fixes r :: "('c \<times> 'b) set"   and s :: "'c set" shows "r `` s = rel_ext (\<lambda>y. \<exists>x\<in>s. (x, y) \<in> r)"
upper_sup: fixes A :: "'a interval"   and B :: "'a interval" shows "upper (sup A B) = sup (upper A) (upper B)"
clause_semantics_def: fixes \<A> :: "'a \<Rightarrow> bool"   and C :: "'a literal set" shows "clause_semantics \<A> C \<equiv> \<exists>L\<in>C. lit_semantics \<A> L"
dom_def: fixes m :: "'a \<Rightarrow> 'b option" shows "Map.dom m = {a. m a \<noteq> None}"
edges_ins_edge': fixes u :: "'b"   and v :: "'b"   and g :: "'b ugraph" assumes "u \<noteq> v" shows "edges (ins_edge (u, v) g) = {(u, v), (v, u)} \<union> edges g"
prime_factorization: fixes p :: "nat" assumes "noninert_gauss_int_prime p" obtains q1 :: "gauss_int" and q2 :: "gauss_int" where "prime q1" and "prime q2" and "prime_factorization (of_nat p) = {#q1, q2#}" and "gauss_int_norm q1 = p" and "gauss_int_norm q2 = p" and "q2 = \<i>\<^sub>\<int> * gauss_cnj q1" and "0 < ReZ q1" and "0 < ImZ q1" and "0 < ReZ q1" and "0 < ImZ q2"
prob_grw_axioms_def: fixes geom_proc :: "nat \<Rightarrow> bool stream \<Rightarrow> real"   and u :: "real"   and d :: "real"   and init :: "real" shows "prob_grw_axioms geom_proc u d init \<equiv> geom_proc = geom_rand_walk u d init"
induced_map_measure_preserving: fixes A :: "'a set"   and W :: "'a set" assumes "A \<in> sets M"   and "W \<in> sets M" shows "emeasure M (induced_map A --` W) = emeasure M W"
ltl_simps_2: fixes x21 :: "'b"   and x22 :: "'b llist" shows "ltl (x21 % x22) = x22"
foundation7: fixes \<tau> :: "'d state \<times> 'd state"   and x :: "'d state \<times> 'd state \<Rightarrow> 'e" shows "(\<tau> \<Turnstile> not (\<delta> x)) = (\<tau> |\<noteq> \<delta> x)"
left_commute: fixes b :: "'a"   and a :: "'a"   and c :: "'a" shows "b \<cdot> (a \<cdot> c) = a \<cdot> (b \<cdot> c)"
MM12: fixes P :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and v :: "'b \<Rightarrow> 'c" shows "\<turnstile> \<box>[\<box>[P]_v]_v = \<box>[P]_v"
jge': shows "\<forall>i'<k'. k' + n' cd\<^bsup>\<pi>'\<^esup>\<rightarrow> i' \<longrightarrow> i' \<le> j'"
exponential_distributedD_gt: fixes X :: "real \<Rightarrow> real"   and l :: "real"   and a :: "real" assumes "distributed (exponential l) lborel X (\<lambda>x. ennreal (exponential_density l x))"   and "0 \<le> a"   and "0 < l" shows "prob {x \<in> space (exponential l). a < X x} = exp (- a * l)"
ltake_lappend_llength: fixes r :: "'b llist"   and A :: "'b set"   and s :: "'b llist" assumes "r \<in> A\<^sup>\<star>" shows "(r @@ s) \<down> LList2.llength r = r"
listrel_Nil1: fixes xs :: "'h list"   and r :: "('g \<times> 'h) set" assumes "([], xs) \<in> listrel r" obtains "xs = []"
msb_nth: fixes w :: "'a word" shows "msb w = w !! (LENGTH('a) - 1)"
HFun_Sigma_graph_aux_def: shows "HFun_Sigma_graph_aux \<equiv> lfp (\<lambda>p x1 x2. \<exists>HFun_Sigma_aux z r z' x y x' y'. x1 = r \<and> x2 = All2 z r (All2 z' r (SyntaxN.Ex x (SyntaxN.Ex y (SyntaxN.Ex x' (SyntaxN.Ex y' (Var z EQ HPair (Var x) (Var y) AND Var z' EQ HPair (Var x') (Var y') AND OrdP (Var x) AND OrdP (Var x') AND (Var x EQ Var x' IMP Var y EQ Var y'))))))) \<and> atom z \<sharp> (r, z', x, y, x', y') \<and> atom z' \<sharp> (r, x, y, x', y') \<and> atom x \<sharp> (r, y, x', y') \<and> atom y \<sharp> (r, x', y') \<and> atom x' \<sharp> (r, y') \<and> atom y' \<sharp> r)"
entry_valid: fixes p :: "'b" shows "proc_of fg (entry fg p) = p"
iter_step_mono: fixes seq :: "'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice"   and nil :: "'b::refinement_lattice"   and c :: "'b::refinement_lattice" assumes "finite_or_infinite_iteration (seq::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) (nil::'b::refinement_lattice)" shows "mono (\<lambda>x::'b::refinement_lattice. (nil::'b::refinement_lattice) \<sqinter> (seq::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) (c::'b::refinement_lattice) x)"
wti_mono: fixes s\<^sub>2 :: "'a"   and s\<^sub>1 :: "'a"   and pc :: "nat" assumes "s\<^sub>2 \<sqsubseteq>\<^bsub>r\<^esub> s\<^sub>1"   and "pc < length \<tau>s"   and "s\<^sub>1 \<in> A"   and "s\<^sub>2 \<in> A" shows "wti c pc s\<^sub>2 \<sqsubseteq>\<^bsub>r\<^esub> wti c pc s\<^sub>1"
17: shows "nat (padic_val p f) + nat (padic_val p g) < Suc (nat (padic_val p f + padic_val p g))"