hcju/isabelleps huggingface dataset:
cod_assoc: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "ide a"   and "ide b"   and "ide c" shows "cod \<a>[a, b, c] = a \<otimes> b \<otimes> c"
landau_symbol_axioms: shows "landau_symbol smallomega smallomega smallomega"
assms: fixes \<pi> :: "('state \<times> 'aprop set) stream"   and \<pi>' :: "('state \<times> 'aprop set) stream" shows "wfp AP' \<pi> \<and> wfp AP' \<pi>' \<longrightarrow> \<phi> [] [\<pi>, \<pi>'] = f (stateOf \<pi>) (stateOf \<pi>') \<and> \<psi> [] [\<pi>, \<pi>'] = g (stateOf \<pi>) (stateOf \<pi>')"
Mem_SUCC_I1: fixes H :: "fm set"   and u :: "tm"   and t :: "tm" assumes "H \<turnstile> u IN t" shows "H \<turnstile> u IN SUCC t"
rel_fun_def: fixes A :: "'b \<Rightarrow> 'd \<Rightarrow> bool"   and B :: "'c \<Rightarrow> 'e \<Rightarrow> bool" shows "rel_fun A B = (\<lambda>f g. \<forall>x y. A x y \<longrightarrow> B (f x) (g y))"
sets_lim_sequence: fixes x :: "'a" assumes "x \<in> space M" shows "sets (lim_sequence x) = sets (Pi\<^sub>M UNIV (\<lambda>i. M))"
bezout_matrix_def: fixes 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 A :: "(('b, 'd) Finite_Cartesian_Product.vec, 'c) Finite_Cartesian_Product.vec"   and a :: "'c"   and b :: "'c"   and j :: "'d"   and bezout :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<times> 'b \<times> 'b \<times> 'b \<times> 'b" assumes "class.bezout_ring times one plus zero minus uminus" shows "bezout_ring.bezout_matrix one zero A a b j bezout = (\<chi>x y. let (p, q, u, v, d) = bezout (A $h a $h j) (A $h b $h j) in if x = a \<and> y = a then p else if x = a \<and> y = b then q else if x = b \<and> y = a then u else if x = b \<and> y = b then v else if x = y then one else zero)"
ring_r_one: fixes x :: "'a" assumes "x \<in> carrier R" shows "x \<cdot>\<^sub>r 1\<^sub>r = x"
elims_3: fixes x :: "real_min"   and xa :: "real_min" assumes "\<not> x \<le> xa" obtains x :: "real" and y :: "real" where "xa__ = Rmin x" and "xa = Rmin y" and "\<not> x \<le> y"
simps_9: fixes n :: "nat"   and r :: "'a atom rexp"   and s :: "'a atom rexp" shows "local.wf n (rexp.Inter r s) = (local.wf n r \<and> local.wf n s)"
assms: fixes I :: "'a set" assumes "ideal I R" shows "\<exists>A\<subseteq>carrier R. finite A \<and> I = Idl A"
numeral_Integer_neq_bottom: fixes k :: "num" shows "numeral k \<noteq> \<bottom>"
monoidI: fixes G :: "('a, 'b) monoid_scheme" assumes "\<And>x y. \<lbrakk>x \<in> carrier G; y \<in> carrier G\<rbrakk> \<Longrightarrow> x \<otimes>\<^bsub>G\<^esub> y \<in> carrier G"   and "\<one>\<^bsub>G\<^esub> \<in> carrier G"   and "\<And>x y z. \<lbrakk>x \<in> carrier G; y \<in> carrier G; z \<in> carrier G\<rbrakk> \<Longrightarrow> x \<otimes>\<^bsub>G\<^esub> y \<otimes>\<^bsub>G\<^esub> z = x \<otimes>\<^bsub>G\<^esub> (y \<otimes>\<^bsub>G\<^esub> z)"   and "\<And>x. x \<in> carrier G \<Longrightarrow> \<one>\<^bsub>G\<^esub> \<otimes>\<^bsub>G\<^esub> x = x"   and "\<And>x. x \<in> carrier G \<Longrightarrow> x \<otimes>\<^bsub>G\<^esub> \<one>\<^bsub>G\<^esub> = x" shows "Group.monoid G"
hyp: shows "\<forall>m. Suc m \<le> n \<longrightarrow> (\<forall>t. G,A\<turnstile>{=:m} t\<succ> {G\<rightarrow>})"
assms_2: fixes x :: "'a" assumes "x \<in> X" shows "Max Y < x"
simps: fixes s :: "'b"   and f :: "'d \<rightarrow> 'c"   and h :: "'b \<rightarrow> ('d, 'b) Step" assumes "s \<noteq> \<bottom>" shows "mapS\<cdot>f\<cdot>(Stream\<cdot>h\<cdot>s) = Stream\<cdot>(mapStep\<cdot>f\<cdot>h)\<cdot>s"
if_cong: fixes b :: "bool"   and c :: "bool"   and x :: "'a"   and u :: "'a"   and y :: "'a"   and v :: "'a" assumes "b = c"   and "c \<Longrightarrow> x = u"   and "\<not> c \<Longrightarrow> y = v" shows "(if b then x else y) = (if c then u else v)"
prems_1: shows "ws'' t = \<lfloor>w'\<rfloor>"
scalar_prod_smult_right: fixes w :: "'b vec"   and v :: "'b vec"   and k :: "'b" assumes "dim_vec w = dim_vec v" shows "w \<bullet> (k \<cdot>\<^sub>v v) = k * (w \<bullet> v)"
assms: shows "\<nexists>\<theta>. Unifier \<theta> t t'"
Nil_eq_concat_conv: fixes xss :: "'a list list" shows "([] = concat xss) = (\<forall>xs\<in>list.set xss. xs = [])"
cons_exec_def: fixes e :: "'i \<times> 'j list"   and p :: "'j" shows "cons_exec e p \<equiv> (fst e, Cons p (snd e))"
list_bind_cong: fixes xs :: "'c list"   and ys :: "'c list"   and f :: "'c \<Rightarrow> 'd list"   and g :: "'c \<Rightarrow> 'd list" assumes "xs = ys"   and "\<And>x. x \<in> set xs \<Longrightarrow> f x = g x" shows "List.bind xs f = List.bind ys g"
Lim_transform2: fixes f :: "'c \<Rightarrow> 'd"   and a :: "'d"   and F :: "'c filter"   and g :: "'c \<Rightarrow> 'd" assumes "(f \<longlongrightarrow> a) F"   and "((\<lambda>x. f x - g x) \<longlongrightarrow> (0::'d)) F" shows "(g \<longlongrightarrow> a) F"
suminf_pos_iff: assumes "summable f"   and "\<And>n. (0::'a) \<le> f n" shows "((0::'a) < suminf f) = (\<exists>i. (0::'a) < f i)"
M1: fixes meets :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and \<I> :: "'a \<Rightarrow> bool"   and p :: "'a"   and q :: "'a"   and s :: "'a"   and r :: "'a" assumes "class.interval meets \<I>"   and "meets p q"   and "meets p s"   and "meets r q" shows "meets r s"
changes_diff_def: shows "changes_diff \<equiv> \<lambda>x. changes_poly_at (p # q # - (p mod q) # ps) x - changes_poly_at (q # - (p mod q) # ps) x"
ivl_closure_diff_del: fixes s :: "real set"   and a :: "real"   and b :: "real"   and x :: "real" assumes "finite s"   and "a < b"   and "x \<in> {a..b} - s" shows "x \<in> closure ({a..b} - s - {x})"
ssubst_empty: fixes t :: "tm"   and F :: "name \<Rightarrow> tm" shows "ssubst t {} F = t"
frange_def: fixes r :: "V" shows "\<R>\<^sub>\<bullet> r = set {b. \<exists>a. [a, b]\<^sub>\<circ> \<in>\<^sub>\<circ> r}"
absorb2: fixes b :: "'e"   and a :: "'e" assumes "b \<le> a" shows "min a b = b"
assms_4: fixes x :: "nat" assumes "x \<in> B" shows "prime x"
vec_iff2: fixes R :: "('a \<times> 'a set) set" assumes "R \<cdot> 1\<^sub>\<pi> \<parallel> Multirelations.U = R" shows "\<forall>a. (\<exists>A. (a, A) \<in> R) \<longrightarrow> (\<forall>A. (a, A) \<in> R)"
eventually_at_right_to_0: fixes P :: "real \<Rightarrow> bool"   and a :: "real" shows "eventually P (at_right a) = (\<forall>\<^sub>F x in at_right 0. P (x + a))"
Indets_pp_def: fixes ps :: "(('x, nat) pp \<Rightarrow>\<^sub>0 'b) list" shows "Indets_pp ps = remdups (concat (map indets_pp ps))"
lattice_of_altdef: fixes vs :: "'a vec list" assumes "set vs \<subseteq> carrier_vec n" shows "lattice_of vs = range (\<lambda>c. \<Oplus>\<^bsub>V\<^esub>v\<in>set vs. of_int (c v) \<cdot>\<^sub>v v)"
pdouble_correct: fixes a :: "int"   and p :: "int \<times> int \<times> int" assumes "a \<in> carrier R"   and "in_carrierp p" shows "make_affine (pdouble a p) = local.add a (make_affine p) (make_affine p)"
ts_inv_okE2: fixes ts :: "'f \<Rightarrow> ('g \<times> 'h \<Rightarrow>f nat) option"   and I :: "'f \<Rightarrow> 'i option" assumes "ts_inv_ok ts I" obtains "\<forall>t. (\<exists>v. ts t = \<lfloor>v\<rfloor>) = (\<exists>v. I t = \<lfloor>v\<rfloor>)"
assms_3: shows "\<^bold>\<lfloor>u\<^bold>\<rfloor> \<noteq> \<^bold>\<I>"
Rep_def: shows "Rep \<equiv> quot_type.rep Rep_ltl_prop_equiv_quotient"
Abs_linear_poly_inverse: fixes y :: "nat \<Rightarrow> rat" assumes "y \<in> {c. Abstract_Linear_Poly.inv c}" shows "Rep_linear_poly (Abs_linear_poly y) = y"
assms: shows "insert_seq start (xs @ [(xi, xr), (yi, yr)])"
upclosed_set_def: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and X :: "'b set" shows "ord.upclosed_set less_eq X = (X \<in> ord.upsets less_eq)"
DomainI: fixes a :: "'d"   and b :: "'e"   and r :: "('d \<times> 'e) set" assumes "(a, b) \<in> r" shows "a \<in> Domain r"
neq_Nil_rev_conv: fixes l :: "'a list" shows "(l \<noteq> []) = (\<exists>xs x. l = xs @ [x])"
finite_extension_incl: fixes K :: "'a set"   and xs :: "'a list" assumes "K \<subseteq> carrier (L\<lparr>carrier := A\<rparr>)"   and "set xs \<subseteq> carrier (L\<lparr>carrier := A\<rparr>)" shows "K \<subseteq> Hom.finite_extension K xs"
value_no_step: fixes \<pi>\<^sub>1 :: "Syntax.term list"   and v :: "Syntax.term"   and m :: "mode"   and \<pi>\<^sub>2 :: "Syntax.term list"   and t :: "Syntax.term" assumes "\<lless>\<pi>\<^sub>1, v\<ggreater> m\<rightarrow> \<lless>\<pi>\<^sub>2, t\<ggreater>"   and "value v" shows "False"
fold_congs_3: fixes r :: "('b, 'c) routing_rule_scheme"   and r' :: "('b, 'c) routing_rule_scheme"   and v' :: "'b routing_action"   and f :: "'b routing_action \<Rightarrow> 'b routing_action"   and f' :: "'b routing_action \<Rightarrow> 'b routing_action" assumes "r = r'"   and "routing_action r' = v'"   and "\<And>v. v' = v \<Longrightarrow> f v = f' v" shows "routing_action_update f r = routing_action_update f' r'"
card_compose_bij: fixes f :: "'c \<Rightarrow> 'c"   and A :: "'c set"   and P :: "'c \<Rightarrow> bool" assumes "bij_betw f A A" shows "card {a \<in> A. P (f a)} = card {a \<in> A. P a}"
fast_dice_roll_def: fixes n :: "nat" shows "Fast_Dice_Roll.fast_dice_roll n \<equiv> curry (spmf.fixp_fun (\<lambda>fast_dice_roll (v, c). if n \<le> v then if c < n then return_spmf c else curry fast_dice_roll (v - n) (c - n) else coin_spmf \<bind> (\<lambda>b. curry fast_dice_roll (2 * v) (2 * c + (if b then 1 else 0)))))"
elims: fixes x :: "'e \<Rightarrow> 'd"   and xa :: "'e list"   and xb :: "'d list"   and y :: "'d list" assumes "map_tailrec_rev x xa xb = y" obtains f :: "'e \<Rightarrow> 'd" and bs :: "'d list" where "x = f__" and "xa = []" and "xb = bs__" and "y = bs__"   | f :: "'e \<Rightarrow> 'd" and a :: "'e" and as :: "'e list" and bs :: "'d list" where "x = f" and "xa = a # as" and "xb = bs" and "y = map_tailrec_rev f as (f a # bs)"
Sup_finite_3_def: fixes A :: "finite_3 set" shows "\<Squnion> A = (if a\<^sub>3 \<in> A then a\<^sub>3 else if a\<^sub>2 \<in> A then a\<^sub>2 else a\<^sub>1)"
job_lower_bound_machine: fixes m :: "nat"   and t :: "nat \<Rightarrow> nat"   and T :: "nat \<Rightarrow> nat"   and A :: "nat \<Rightarrow> nat set"   and j :: "nat"   and x :: "nat" assumes "LoadBalancing m"   and "LoadBalancing.lb t m T A j"   and "x \<in> {1..j}" shows "\<exists>y\<in>{1..m}. t x \<le> T y"
finite_imageD2: fixes f :: "'b \<Rightarrow> 'a"   and A :: "'b set"   and B :: "'b set" assumes "finite (f ` A)"   and "inj_on f (A - B)"   and "finite B" shows "finite A"
non_pivot_base_def: fixes uminus :: "'c \<Rightarrow> 'c"   and zero :: "'c"   and one :: "'c"   and A :: "'c mat"   and pivots :: "(nat \<times> nat) list" shows "non_pivot_base_gen uminus zero one A pivots \<equiv> let nr = dim_row A; nc = dim_col A; invers = map_of (map prod.swap pivots) in (\<lambda>qj. vec nc (\<lambda>i. if i = qj then one else case invers i of None \<Rightarrow> zero | Some j \<Rightarrow> uminus (A $$ (j, qj))))"
ext_inject: fixes pid :: "pname"   and tid :: "tname"   and more :: "'a"   and pid' :: "pname"   and tid' :: "tname"   and more' :: "'a" shows "(\<lparr>pid = pid, tid = tid, \<dots> = more\<rparr> = \<lparr>pid = pid', tid = tid', \<dots> = more'\<rparr>) = (pid = pid' \<and> tid = tid' \<and> more = more')"
index_eqI: fixes i :: "nat"   and xs :: "'a list"   and x :: "'a" assumes "i \<le> length xs"   and "\<forall>j<i. xs ! j \<noteq> x"   and "i < length xs \<Longrightarrow> xs ! i = x" shows "index xs x = i"
Id_on_def: fixes A :: "'b set" shows "s2r A = (\<Union>x\<in>A. {(x, x)})"
mop_list_tl_def: shows "mop_list_tl \<equiv> mop pre_list_tl (RETURN \<circ> op_list_tl)"
has_lock_may_lock: fixes l :: "('b \<times> nat) option"   and t :: "'b" assumes "has_lock l t" shows "may_lock l t"
fImage_mono: fixes r' :: "('b \<times> 'c) fset"   and r :: "('b \<times> 'c) fset"   and A' :: "'b fset"   and A :: "'b fset" assumes "r' |\<subseteq>| r"   and "A' |\<subseteq>| A" shows "fImage r' A' |\<subseteq>| fImage r A"
assms: shows "P \<subseteq> 1\<^sub>\<sigma>"
last_cs: fixes \<pi> :: "nat \<Rightarrow> 'n"   and i :: "nat" shows "last (cs\<^bsup>\<pi>\<^esup> i) = \<pi> i"
root_rat_floor: fixes p :: "nat"   and x :: "rat" shows "root_rat_floor p x = \<lfloor>root p (real_of_rat x)\<rfloor>"
var: fixes i :: "'a" shows "Collect ((\<in>) i) \<in> fba"
X_in_eqs: shows "(X, rhs) \<in> Init (UNIV // \<approx>A)"
revg_fun: fixes a :: "'a list"   and b :: "'a list" shows "revg a b = rev a @ b"
qs: fixes b :: "real" assumes "4 * b \<le> 3"   and "\<not> b * 2 \<le> 1" shows "q (4 * b - 2) \<in> s"
card_disjoint_shuffles: fixes xs :: "'a list"   and ys :: "'a list" assumes "set xs \<inter> set ys = {}" shows "card (shuffles xs ys) = length xs + length ys choose length xs"
sel_1: fixes x1 :: "'b \<Rightarrow> char list \<Rightarrow> bool"   and x2 :: "'c \<Rightarrow> 'b \<Rightarrow> 'b set" shows "\<pi> (Kripke x1 x2) = x1"
MGFn_Loop: fixes G :: "prog"   and A :: "state triple set"   and n :: "nat"   and e :: "expr"   and c :: "stmt"   and l :: "label" assumes "G,A\<turnstile>{=:n} In1l e\<succ> {G\<rightarrow>}"   and "G,A\<turnstile>{=:n} In1r c\<succ> {G\<rightarrow>}"   and "wf_prog G" shows "G,A\<turnstile>{=:n} In1r (l\<bullet> While(e) c)\<succ> {G\<rightarrow>}"
infdist_eq_setdist: fixes x :: "'a"   and A :: "'a set" shows "infdist x A = setdist {x} A"
rep_eq: fixes x :: "nat"   and xa :: "integer" shows "int_of_integer (take_bit x xa) = take_bit x (int_of_integer xa)"
1: shows "(r ; x \<cdot> 1') ; 1 \<le> x\<^sup>T ; 1"
bit_yxyx: fixes x :: "'a"   and y :: "'a"   and init :: "'a list"   and qs :: "'a list"   and r :: "'a list" assumes "x \<noteq> y"   and "init \<in> {[x, y], [y, x]}"   and "qs \<in> lang (seq [Times (Atom y) (Atom x), Star (Times (Atom y) (Atom x))])" shows "T\<^sub>p_on_rand' BIT (type0 init x y) (qs @ r) = 75 / 10\<^sup>2 * real (length qs) + T\<^sub>p_on_rand' BIT (type1 init x y) r \<and> Partial_Cost_Model.config'_rand BIT (type0 init x y) qs = type1 init x y"
mssnth_ln_series_stream_aux: fixes b :: "bool"   and x :: "real"   and n :: "nat" shows "mssnth (ln_series_stream_aux b x) n = (if b then - ((- 1) ^ n) * inverse (x + real n) else (- 1) ^ n * inverse (x + real n))"
residue_minus_car: fixes y :: "int"   and k :: "nat"   and x :: "int" assumes "y \<in> carrier (residue_ring (p ^ k))" shows "x \<ominus>\<^bsub>residue_ring (p ^ k)\<^esub> y = (x - y) mod p ^ k"
paste_def: fixes P :: "('c \<times> 'd) set"   and Q :: "('c \<times> 'd) set" shows "P +< Q = P outside Domain Q \<union> Q"
smallest_accepting_rank\<^sub>\<R>_properties: fixes \<Sigma> :: "'b set"   and \<delta> :: "'c \<Rightarrow> 'b \<Rightarrow> 'c"   and q\<^sub>0 :: "'c"   and w :: "nat \<Rightarrow> 'b"   and F :: "'c set"   and i :: "nat" assumes "mojmir_to_rabin \<Sigma> \<delta> q\<^sub>0 w F"   and "mojmir_to_rabin.smallest_accepting_rank\<^sub>\<R> \<Sigma> \<delta> q\<^sub>0 w F = Some i" shows "accepting_pair\<^sub>R (semi_mojmir_def.step \<Sigma> \<delta> q\<^sub>0) (semi_mojmir_def.initial q\<^sub>0) (mojmir_to_rabin_def.fail\<^sub>R \<Sigma> \<delta> q\<^sub>0 F \<union> mojmir_to_rabin_def.merge\<^sub>R \<delta> q\<^sub>0 F i, mojmir_to_rabin_def.succeed\<^sub>R \<delta> q\<^sub>0 F i) w"
Conseq: fixes C :: "(('a \<Rightarrow> state \<Rightarrow> bool) \<times> com \<times> ('a \<Rightarrow> state \<Rightarrow> bool)) set"   and P' :: "'a \<Rightarrow> state \<Rightarrow> bool"   and c :: "com"   and Q' :: "'a \<Rightarrow> state \<Rightarrow> bool"   and P :: "'a \<Rightarrow> state \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> state \<Rightarrow> bool" assumes "C \<turnstile> {P'} c {Q'}"   and "\<forall>s t. (\<forall>z. P' z s \<longrightarrow> Q' z t) \<longrightarrow> (\<forall>z. P z s \<longrightarrow> Q z t)" shows "C \<turnstile> {P} c {Q}"
supp_eqvt_at: fixes f :: "'b \<Rightarrow> 'c"   and x :: "'b" assumes "eqvt_at f x"   and "finite (supp x)" shows "supp (f x) \<subseteq> supp x"
strongBisimWeakBisim: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi" assumes "\<Psi> \<rhd> P \<sim> Q" shows "\<Psi> \<rhd> P \<approx> Q"
complete_tcsigs_imp_exe_complete_tcsigs: fixes arss :: "(String.literal \<times> (String.literal \<times> String.literal set list) list) list"   and cs :: "(String.literal \<times> String.literal) list" assumes "exe_ars_conds arss"   and "complete_tcsigs (set cs) (translate_ars arss)" shows "exe_complete_tcsigs cs arss"
assms: fixes x :: "V" assumes "x \<in>\<^sub>\<circ> A" shows "x \<subseteq>\<^sub>\<circ> B"
isomorphic_unit_right: fixes f :: "'a" assumes "C.ide f" shows "f \<star>\<^sub>C src\<^sub>C f \<cong>\<^sub>C f"
prems: shows "isin \<langle>x1_, (a, b_), x3_\<rangle> x"
HInv_def: fixes s :: "state" shows "HInv s = (HInv1 s \<and> HInv2 s \<and> HInv3 s \<and> HInv4 s \<and> HInv5 s \<and> HInv6 s)"
compress_normalize_input_interfaces_def: fixes m :: "'c common_primitive match_expr" shows "compress_normalize_input_interfaces m \<equiv> compress_normalize_primitive (is_Iiface, iiface_sel) IIface compress_interfaces m"
tree_path_impl_parenthesis: fixes v :: "'v"   and w :: "'v" assumes "(v, w) \<in> (tree_edges s)\<^sup>+"   and "v \<in> dom (finished s)" shows "w \<in> dom (finished s) \<and> the_discovered s v < the_discovered s w \<and> the_finished s w < the_finished s v"
inf_aci_1: fixes inf :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and less_eq :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and less :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and x :: "'b::type"   and y :: "'b::type" assumes "class.semilattice_inf (inf::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (less::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool)" shows "(inf::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (x::'b::type) (y::'b::type) = inf y x"
NonUniformExecutionBase: fixes cfg :: "('p, 'v, 's) configuration" assumes "initial cfg"   and "nonUniform cfg" shows "execution trans sends start [cfg] [] \<and> nonUniform (last [cfg]) \<and> (\<exists>cfgList' msgList'. nonUniform (last cfgList') \<and> prefixList [cfg] cfgList' \<and> prefixList [] msgList' \<and> execution trans sends start cfgList' msgList' \<and> (\<exists>msg'. execution.minimalEnabled [cfg] [] msg' \<and> msg' \<in> set msgList'))"
hyps_2: assumes "f a \<noteq> x" shows "inv f x \<in> segment f a x"
tendsto_add_const_iff: fixes c :: "'c"   and f :: "'b \<Rightarrow> 'c"   and d :: "'c"   and F :: "'b filter" shows "((\<lambda>x. c + f x) \<longlongrightarrow> c + d) F = (f \<longlongrightarrow> d) F"
fmapU_error_simps_1: fixes f :: "udom \<rightarrow> udom" shows "fmapU\<cdot>f\<cdot>\<bottom> = \<bottom>"
ChamberComplexMorphism_axioms_def: fixes X :: "'b set set"   and Y :: "'c set set"   and f :: "'b \<Rightarrow> 'c" shows "ChamberComplexMorphism_axioms X Y f \<equiv> (\<forall>C. SimplicialComplex.maxsimp X C \<longrightarrow> SimplicialComplex.maxsimp Y (f ` C)) \<and> (\<forall>C. SimplicialComplex.maxsimp X C \<longrightarrow> card (f ` C) = card C)"
simps_4: fixes as :: "Arity list"   and x :: "Vars.var"   and S :: "stack_elem list" shows "AEstack as (Arg x # S) = esing x\<cdot>(up\<cdot>0) \<squnion> AEstack as S"
simps: fixes r :: "nat"   and x\<^sub>1 :: "'b"   and ts\<^sub>1 :: "'b tree list"   and r' :: "nat"   and x\<^sub>2 :: "'b"   and ts\<^sub>2 :: "'b tree list" shows "link (Node r x\<^sub>1 ts\<^sub>1) (Node r' x\<^sub>2 ts\<^sub>2) = (let t\<^sub>2 = Node r' x\<^sub>2 ts\<^sub>2; t\<^sub>1 = Node r x\<^sub>1 ts\<^sub>1 in if x\<^sub>1 \<le> x\<^sub>2 then Node (r + 1) x\<^sub>1 (t\<^sub>2 # ts\<^sub>1) else Node (r + 1) x\<^sub>2 (t\<^sub>1 # ts\<^sub>2))"
F_2: fixes n :: "nat" shows "f = f1 n + f2 n"
compute_inferences: fixes C :: "'a literal list"   and i :: "nat"   and P :: "('a literal list \<times> nat) list"   and Q :: "('a literal list \<times> nat) list"   and n :: "nat" assumes "(C, i) \<in> set P"   and "\<forall>(D, j)\<in>#mset (map (apfst mset) P). weight (mset C, i) \<le> weight (D, j)" shows "wstate_of_dstate ([], P, Q, n) \<leadsto>\<^sub>w wstate_of_dstate (map (\<lambda>D. (D, n)) (remdups_gen mset (resolve_rename C C @ concat (map (resolve_rename_either_way C \<circ> fst) Q))), filter (\<lambda>(D, j). mset D \<noteq> mset C) P, (C, i) # Q, Suc n)"
interleavings_Cons: fixes x :: "'a"   and xs :: "'a list"   and y :: "'a"   and ys :: "'a list" shows "{zs. interleavings (x # xs) (y # ys) zs} = {x # zs |zs. interleavings xs (y # ys) zs} \<union> {y # zs |zs. interleavings (x # xs) ys zs}"
hom_induced_chain_map_gen: fixes X :: "'b topology"   and Y :: "'c topology"   and f :: "'b \<Rightarrow> 'c"   and S :: "'b set"   and T :: "'c set"   and p :: "nat"   and c :: "((nat \<Rightarrow> real) \<Rightarrow> 'b) \<Rightarrow>\<^sub>0 int" assumes "continuous_map X Y f"   and "f ` (topspace X \<inter> S) \<subseteq> T"   and "singular_relcycle p X S c" shows "hom_induced (int p) X S Y T f (homologous_rel_set p X S c) = homologous_rel_set p Y T (chain_map p f c)"
assms_4: shows "rk {C, D, M} \<le> 2"
antisymmetric_axioms: shows "antisymmetric B (\<sqsubseteq>)"
integrable_monotone_convergence: fixes M :: "'b measure"   and f :: "nat \<Rightarrow> 'b \<Rightarrow> real"   and u :: "'b \<Rightarrow> real"   and x :: "real" assumes "\<And>i. integrable M (f i)"   and "AE x in M. incseq (\<lambda>n. f n x)"   and "AE x in M. (\<lambda>i. f i x) \<longlonglongrightarrow> u x"   and "(\<lambda>i. integral\<^sup>L M (f i)) \<longlonglongrightarrow> x"   and "u \<in> borel_measurable M" shows "integrable M u"
image_mset_single: fixes f :: "'d \<Rightarrow> 'c"   and x :: "'d" shows "image_mset f {#x#} = {#f x#}"
d1: shows "fst (crit_pair g1 g2) = 0 \<or> lt (fst (crit_pair g1 g2)) \<prec>\<^sub>t term_of_pair (lcs (lp g1) (lp g2), component_of_term (lt g1))"
Ln_unique: fixes z :: "complex"   and w :: "complex" assumes "exp z = w"   and "- pi < Im z"   and "Im z \<le> pi" shows "Ln w = z"
set_not_fresh: fixes x :: "'a"   and L :: "'a list" assumes "x \<in> set L" shows "\<not> atom x \<sharp> L"
residual_order: shows "int (card (bl\<^sup>c)) = \<v> - \<k>"
is_partial_heap_def: fixes cmp :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and heap :: "int \<Rightarrow> 'b"   and l :: "int"   and m :: "int"   and r :: "int" shows "is_partial_heap cmp heap l m r = (\<forall>x. bounded m r x \<longrightarrow> bounded m r (parent l x) \<longrightarrow> cmp (heap (parent l x)) (heap x))"
assms_2: shows "w \<le> y \<squnion> z"
disj_conj_distrib: fixes x :: "'c"   and y :: "'c"   and z :: "'c" shows "x OR y AND z = (x OR y) AND (x OR z)"
while_red_nE: fixes b :: "expr"   and cx :: "com"   and s :: "char list \<Rightarrow> val option"   and n :: "nat"   and s' :: "char list \<Rightarrow> val option" assumes "\<langle>while (b) cx,s\<rangle> \<rightarrow>\<^bsup>n\<^esup> \<langle>Skip,s'\<rangle>" shows "\<lbrakk>b\<rbrakk>s = Some false \<and> s = s' \<and> n = 1 \<or> (\<exists>i j s''. \<lbrakk>b\<rbrakk>s = Some true \<and> \<langle>cx,s\<rangle> \<rightarrow>\<^bsup>i\<^esup> \<langle>Skip,s''\<rangle> \<and> \<langle>while (b) cx,s''\<rangle> \<rightarrow>\<^bsup>j\<^esup> \<langle>Skip,s'\<rangle> \<and> n = i + j + 2)"
mult_poly_p_elim: fixes q :: "(char list multiset \<times> int) multiset"   and A :: "(char list multiset \<times> int) multiset"   and r :: "(char list multiset \<times> int) multiset"   and B :: "(char list multiset \<times> int) multiset"   and r' :: "(char list multiset \<times> int) multiset" assumes "mult_poly_p q (A, r) (B, r')" obtains xs :: "char list multiset" and n :: "int" where "A = add_mset (xs, n) B" and "r' = {#case x of (ys, m) \<Rightarrow> (remdups_mset (xs + ys), n * m). x \<in># q#} + r"
edge_kind_cases: fixes et :: "('g, 'h, 'i, 'j) edge_kind" obtains "intra_kind et"   | Q :: "('g \<Rightarrow> 'h option) \<times> 'i \<Rightarrow> bool" and r :: "'i" and p :: "'j" and fs :: "(('g \<Rightarrow> 'h option) \<Rightarrow> 'h option) list" where "et = Q__:r\<hookrightarrow>\<^bsub>p__\<^esub>fs"   | Q :: "('g \<Rightarrow> 'h option) \<times> 'i \<Rightarrow> bool" and p :: "'j" and f :: "('g \<Rightarrow> 'h option) \<Rightarrow> ('g \<Rightarrow> 'h option) \<Rightarrow> 'g \<Rightarrow> 'h option" where "et = Q\<hookleftarrow>\<^bsub>p\<^esub>f"