hcju/isabelleps huggingface dataset:
exec_gpv_parametric: fixes S :: "'d \<Rightarrow> 'g \<Rightarrow> bool"   and CALL :: "'e \<Rightarrow> 'h \<Rightarrow> bool"   and A :: "'i \<Rightarrow> 'j \<Rightarrow> bool" shows "rel_fun (rel_fun S (rel_fun CALL (rel_spmf (rel_prod (=) S)))) (rel_fun (rel_gpv A CALL) (rel_fun S (rel_spmf (rel_prod A S)))) exec_gpv exec_gpv"
subseq_to_Suc: fixes i :: "nat"   and j :: "nat"   and w :: "nat \<Rightarrow> 'a" assumes "i \<le> j" shows "w [i \<rightarrow> Suc j] = w [i \<rightarrow> j] @ [w j]"
inf_sup_aci_5: fixes inf :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sup :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and x :: "'d"   and y :: "'d" assumes "class.lattice inf less_eq less sup" shows "sup x y = sup y x"
cblinfun_eqI: fixes x :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b"   and y :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b" assumes "\<And>i. x *\<^sub>V i = y *\<^sub>V i" shows "x = y"
incl_def: fixes f :: "'s" shows "S.incl f = (S.arr f \<and> S.set (S.dom f) \<subseteq> S.set (S.cod f) \<and> f = S.mkArr (S.set (S.dom f)) (S.set (S.cod f)) (\<lambda>x. x))"
clinear: fixes f :: "'b \<Rightarrow> 'c" assumes "bounded_clinear f" shows "clinear f"
downset_set_def: fixes X :: "'a set" shows "ord.downset_set (\<lambda>x y. y \<le> x) X = {y. \<exists>x\<in>X. x \<le> y}"
VSagree_refl: fixes \<nu> :: "(real, 'a) vec"   and A :: "'a set" shows "VSagree \<nu> \<nu> A"
induct: fixes P :: "('b \<Rightarrow> bool) \<Rightarrow> 'b \<Rightarrow> ('c, 'b) Node list \<Rightarrow> bool"   and a0 :: "'b \<Rightarrow> bool"   and a1 :: "'b"   and a2 :: "('c, 'b) Node list" assumes "\<And>p i a. P p i [a]"   and "\<And>p i a v va. (\<And>x. \<lbrakk>x = i + gmn a; \<not> p x\<rbrakk> \<Longrightarrow> P p x (v # va)) \<Longrightarrow> P p i (a # v # va)"   and "\<And>a b. P a b []" shows "P a0 a1 a2"
is_valid_problem_sas_plus_then_2: fixes \<Psi> :: "('a, 'b) sas_plus_problem" assumes "is_valid_problem_sas_plus \<Psi>" shows "\<forall>op\<in>set (\<Psi>\<^sub>\<O>\<^sub>+). is_valid_operator_sas_plus \<Psi> op"
simps: fixes xs :: "(nat \<times> 'b) list"   and m :: "nat option list" shows "index_of_pqueue (xs, m) = ((\<forall>i<length xs. fst (xs ! i) < length m \<and> m ! fst (xs ! i) = Some i) \<and> (\<forall>i k. k < length m \<longrightarrow> m ! k = Some i \<longrightarrow> i < length xs \<and> fst (xs ! i) = k))"
assms_10: shows "wf_tuple n (I \<union> J) z"
order_iff: fixes b :: "'b::semilattice_sup"   and a :: "'b::semilattice_sup" shows "((b::'b::semilattice_sup) \<le> (a::'b::semilattice_sup)) = (a = a \<squnion> b)"
sum_mset_sum_list: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and xs :: "'b list" assumes "class.comm_monoid_add plus zero" shows "comm_monoid_add.sum_mset plus zero (mset xs) = monoid_add.sum_list plus zero xs"
not_in_verts_G: shows "w \<notin> verts G"
fold_once_set_equiv: fixes xs :: "'b list"   and ys :: "'b list"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'c" assumes "xs \<approx> ys" shows "fold_once f xs = fold_once f ys"
r_coset_hom_Mod: shows "(#>) H \<in> hom G (G Mod H)"
WhileT: fixes s :: "'b"   and b :: "'b set"   and P :: "'b com" assumes "s \<in> b" shows "(Some (While b P), s) -c\<rightarrow> (Some (Seq P (While b P)), s)"
intro: fixes A :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and F :: "'a \<Rightarrow> 'b" assumes "\<And>f f'. \<lbrakk>partial_magma.arr A f \<and> partial_magma.arr A f' \<and> partial_magma.dom A f = partial_magma.dom A f' \<and> partial_magma.cod A f = partial_magma.cod A f'; F f = F f'\<rbrakk> \<Longrightarrow> f = f'" shows "faithful_functor_axioms A F"
inline1_in_sub_gpvs: fixes out :: "'call'"   and callee' :: "('ret \<times> 's, 'call', 'ret') rpv"   and rpv' :: "('b, 'call, 'ret) rpv"   and gpv :: "('b, 'call, 'ret) gpv"   and s :: "'s"   and x :: "'ret"   and s' :: "'s"   and input :: "'ret'" assumes "Inr (out, callee', rpv') \<in> set_spmf (inline1 callee gpv s)"   and "(x, s') \<in> results_gpv \<I>' (callee' input)"   and "input \<in> responses_\<I> \<I>' out"   and "\<I> \<turnstile>g gpv \<surd>"   and "I s" shows "rpv' x \<in> sub_gpvs \<I> gpv \<and> I s'"
splitAt_simpB': fixes a :: "'a"   and ram :: "'a"   and b :: "'a list" assumes "a \<noteq> ram" shows "fst (splitAt ram (a # b)) = a # fst (splitAt ram b)"
scale_zero_right: fixes a :: "real" shows "a *\<^sub>R 0 = 0"
indep_prob: shows "prob {x \<in> space M. a < min (X x) (Y x)} = exp (- a * (l + u))"
True_start_eps_starI: fixes A :: "bool list \<times> ('b option \<Rightarrow> bool list \<Rightarrow> bool list set) \<times> (bool list \<Rightarrow> bool)"   and p :: "bool list" assumes "fin A p" shows "(True # p, True # start A) \<in> eps (RegExp2NAe.star A)"
prSubstAll_def: fixes g :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> ('varSort \<Rightarrow> 'var \<Rightarrow> 'b) \<Rightarrow> 'b"   and gA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> ('varSort \<Rightarrow> 'var \<Rightarrow> 'b) \<Rightarrow> 'c"   and SEM :: "('d, 'e, 'f, 'sort, 'g, 'b, 'h) semDom_scheme" shows "prSubstAll g gA SEM \<equiv> prSubst g SEM \<and> prSubstAbs g gA SEM"
RDirProd_hom2_1: fixes f :: "'a \<Rightarrow> 'b"   and S :: "('a, 'c) ring_scheme"   and T :: "('b, 'd) ring_scheme"   and R :: "('e, 'f) ring_scheme" assumes "f \<in> ring_hom S T" shows "(\<lambda>(x, y). (x, f y)) \<in> ring_hom (RDirProd R S) (RDirProd R T)"
lim: shows "(\<lambda>w. f w * (w - z)) \<midarrow>z\<rightarrow> c"
strong_fermat_liar_imp_fermat_liar: fixes a :: "nat"   and n :: "nat" assumes "a \<in> {1..<n}"   and "1 < n"   and "strong_fermat_liar a n" shows "fermat_liar a n"
dom_restrict_def: fixes S :: "'a set"   and p :: "'a \<Rightarrow> 'b decision option" shows "S \<triangleleft> p \<equiv> \<lambda>x. if x \<in> S then p x else \<bottom>"
retraction_maps_def: fixes X :: "'b topology"   and Y :: "'c topology"   and f :: "'b \<Rightarrow> 'c"   and g :: "'c \<Rightarrow> 'b" shows "retraction_maps X Y f g \<equiv> continuous_map X Y f \<and> continuous_map Y X g \<and> (\<forall>x\<in>topspace Y. f (g x) = x)"
power2_eq_iff_nonneg: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and x :: "'b"   and y :: "'b" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "less_eq zero x"   and "less_eq zero y" shows "(power.power one times x 2 = power.power one times y 2) = (x = y)"
open_exclude_finite_2: fixes A :: "'b set"   and x :: "'b" assumes "finite A"   and "x \<in> A" shows "open_exclude_set x A \<inter> A = {x}"
conc_empty: fixes w :: "nat \<Rightarrow> 'a" shows "[] \<frown> w = w"
bij_is_surj: fixes f :: "'b \<Rightarrow> 'c" assumes "bij f" shows "surj f"
square_free_factor: fixes a :: "'b poly"   and p :: "'b poly" assumes "a dvd p"   and "square_free p" shows "square_free a"
exact_division: fixes x :: "int"   and y :: "int" assumes "of_int x / of_int y \<in> \<int>" shows "of_int (x div y) = of_int x / of_int y"
remove_paths_le_in_subset_Gen: fixes k :: "nat"   and I :: "item set"   and P :: "(symbol \<times> character list) list set" assumes "items_le k I = I"   and "I \<subseteq> Gen P" shows "I \<subseteq> Gen (paths_le k P)"
topology\<^sub>N_def: fixes N :: "'a quasinorm" shows "topology\<^sub>N N = topology (\<lambda>U. \<forall>x\<in>U. \<exists>e>0. \<forall>y. eNorm N (y - x) < e \<longrightarrow> y \<in> U)"
path_connected_space_path_component_set: fixes X :: "'a topology" shows "path_connected_space X = (\<forall>x\<in>topspace X. path_component_of_set X x = topspace X)"
rank_invar_meldUniq_strong: fixes bq1 :: "('c, 'd) SkewBinomialTree list"   and bq2 :: "('c, 'd) SkewBinomialTree list" assumes "rank_invar bq1"   and "rank_invar bq2" shows "rank_invar (meldUniq bq1 bq2) \<and> min (rank (hd bq1)) (rank (hd bq2)) \<le> rank (hd (meldUniq bq1 bq2))"
ac: fixes u :: "nat discr"   and v :: "nat discr"   and w :: "nat discr" shows "undiscr u * (undiscr v * undiscr w) = undiscr u * undiscr v * undiscr w"
list_all2_trans: fixes P1 :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and P2 :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and P3 :: "'c \<Rightarrow> 'e \<Rightarrow> bool"   and as :: "'c list"   and bs :: "'d list"   and cs :: "'e list" assumes "\<And>a b c. \<lbrakk>P1 a b; P2 b c\<rbrakk> \<Longrightarrow> P3 a c"   and "list_all2 P1 as bs"   and "list_all2 P2 bs cs" shows "list_all2 P3 as cs"
ext_inject: fixes name :: "nat"   and incoming :: "nat set"   and new :: "'b ltlr set"   and old :: "'b ltlr set"   and "next" :: "'b ltlr set"   and more :: "'c"   and name' :: "nat"   and incoming' :: "nat set"   and new' :: "'b ltlr set"   and old' :: "'b ltlr set"   and next' :: "'b ltlr set"   and more' :: "'c" shows "(\<lparr>name = name, incoming = incoming, new = new, old = old, next = next, \<dots> = more\<rparr> = \<lparr>name = name', incoming = incoming', new = new', old = old', next = next', \<dots> = more'\<rparr>) = (name = name' \<and> incoming = incoming' \<and> new = new' \<and> old = old' \<and> next = next' \<and> more = more')"
dom_empty_is_empty: fixes m :: "'c \<Rightarrow> 'd option" shows "(dom m = {}) = (m = Map.empty)"
distinct_indices_def: fixes as :: "('a \<times> 'b) list" shows "distinct_indices as = distinct (map fst as)"
sup_ccInf: fixes B :: "'a set"   and a :: "'a" assumes "countable B" shows "sup a (Inf B) = Inf (sup a ` B)"
finitesum_pOp_closed: fixes f :: "'f \<Rightarrow> 'a set"   and I :: "'f set"   and a :: "'a"   and b :: "'a" assumes "f \<in> I \<rightarrow> {X. submodule R M X}"   and "a \<in> finitesum M I f"   and "b \<in> finitesum M I f" shows "a \<plusminus> b \<in> finitesum M I f"
genideal_def: fixes R :: "('c, 'd) ring_scheme"   and S :: "'c set" shows "Idl\<^bsub>R\<^esub> S = \<Inter> {I. ideal I R \<and> S \<subseteq> I}"
trichotomy_nest: fixes a :: "'a set"   and c :: "'a set" assumes "a = c" shows "\<not> a \<lless> c \<and> \<not> c \<lless> a"
cases: fixes S :: "'o \<Rightarrow> 'p \<Rightarrow> bool"   and E :: "'q \<Rightarrow> 'r \<Rightarrow> bool"   and IA :: "'s \<Rightarrow> 't \<Rightarrow> bool"   and IU :: "'u \<Rightarrow> 'v \<Rightarrow> bool"   and OA :: "'w \<Rightarrow> 'x \<Rightarrow> bool"   and OU :: "'y \<Rightarrow> 'z \<Rightarrow> bool"   and M :: "'aa \<Rightarrow> 'ab \<Rightarrow> bool"   and a1 :: "('o, 'q, 's, 'u, 'w, 'y, 'aa) rest_scheme"   and a2 :: "('p, 'r, 't, 'v, 'x, 'z, 'ab) rest_scheme" assumes "rel_rest' S E IA IU OA OU M a1 a2" obtains rinit :: "'aa" and rfunc_adv :: "'o \<Rightarrow> 's \<Rightarrow> (('w \<times> 'q list) \<times> 'o) spmf" and rfunc_usr :: "'o \<Rightarrow> 'u \<Rightarrow> (('y \<times> 'q list) \<times> 'o) spmf" and rinit' :: "'ab" and rfunc_adv' :: "'p \<Rightarrow> 't \<Rightarrow> (('x \<times> 'r list) \<times> 'p) spmf" and rfunc_usr' :: "'p \<Rightarrow> 'v \<Rightarrow> (('z \<times> 'r list) \<times> 'p) spmf" where "a1 = Rest rinit rfunc_adv rfunc_usr" and "a2 = Rest rinit' rfunc_adv' rfunc_usr'" and "M rinit rinit'" and "(S ===> IA ===> rel_spmf (rel_prod (rel_prod OA (list_all2 E)) S)) rfunc_adv rfunc_adv'" and "(S ===> IU ===> rel_spmf (rel_prod (rel_prod OU (list_all2 E)) S)) rfunc_usr rfunc_usr'"
analz_insert: fixes X :: "msg"   and H :: "msg set" shows "insert X (analz H) \<subseteq> analz (insert X H)"
frac_less: fixes x :: "'a"   and y :: "'a"   and w :: "'a"   and z :: "'a" assumes "(0::'a) \<le> x"   and "x < y"   and "(0::'a) < w"   and "w \<le> z" shows "x / z < y / w"
uniPPV_uniPPc: fixes V :: "('a, 'b) MWLsCom list" assumes "unique_PPV V" shows "\<forall>i<length V. unique_PPc (V ! i)"
add_0_left: fixes a :: "'a" shows "(0::'a) + a = a"
bounded_closed_interval: fixes a :: "'a"   and b :: "'a" shows "bounded {a..b}"
not_in_iff: fixes x :: "'c"   and M :: "'c multiset" shows "(x \<notin># M) = (count M x = 0)"
assms: shows "update rt dip (dsn, dsk, flg, hops, sip) \<noteq> rt"
bind_ask2: fixes m :: "'m"   and f :: "'a \<Rightarrow> 'r \<Rightarrow> 'm" shows "bind m (\<lambda>x. ask (f x)) = ask (\<lambda>r. bind m (\<lambda>x. f x r))"
F: shows "\<forall>B\<in>P\<^sub>2 \<union> wrap B\<^sub>2. finite B"
abs_eq: fixes x :: "('b \<times> bool) list" assumes "eq_onp proper_signed_list x x" shows "- Abs_freeword x = Abs_freeword (rev (map flip_signed x))"
P1_ideal_ideal_eq: fixes x :: "'a \<times> 'a"   and y :: "bool"   and z :: "'f"   and \<A> :: "('a \<times> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'f \<Rightarrow> (('a \<times> 'a \<times> 'a) \<times> 'g) spmf) \<times> ('a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<Rightarrow> 'g \<Rightarrow> ((('a \<times> 'a) \<times> 'a \<times> 'a) \<times> 'g) spmf) \<times> ('g \<Rightarrow> 'e spmf)" shows "mal_def.ideal_view_1 x y z (P1_S1, P1_S2) \<A> = P1_ideal_model x y z \<A>"
d: shows "l \<cdot> r = subword w i (length l) \<cdot> subword w (i + length l) (j - length l)"
ba2_complement: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(x \<squnion> - y = z \<squnion> - z) = (x \<squnion> y = x)"
preserves_hseq: fixes \<mu> :: "'d"   and \<nu> :: "'d" assumes "D.arr \<mu> \<and> D.arr \<nu> \<and> src\<^sub>D \<mu> = trg\<^sub>D \<nu>" shows "C.arr (F \<mu>) \<and> C.arr (F \<nu>) \<and> src\<^sub>C (F \<mu>) = trg\<^sub>C (F \<nu>)"
wf_apply_effect: fixes e :: "object ast_effect"   and s :: "object atom formula set" assumes "wf_effect objT e"   and "wf_world_model s" shows "wf_world_model (apply_effect e s)"
execn_callFault: fixes \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and p :: "'e"   and bdy :: "('d, 'e, 'f) com"   and init :: "'d \<Rightarrow> 'd"   and s :: "'d"   and n :: "nat"   and f :: "'f"   and return :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and c :: "'d \<Rightarrow> 'd \<Rightarrow> ('d, 'e, 'f) com" assumes "\<Gamma> p = Some bdy"   and "\<Gamma>\<turnstile> \<langle>bdy,Normal (init s)\<rangle> =n\<Rightarrow> Fault f" shows "\<Gamma>\<turnstile> \<langle>call init p return c,Normal s\<rangle> =Suc n\<Rightarrow> Fault f"
inject: fixes x :: "'a list"   and ya :: "'a list" shows "(IArray x = IArray ya) = (x = ya)"
pcr_cr_eq: shows "pcr_generator (=) (=) = cr_generator"
fun_ordI: fixes ord :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'e" assumes "\<And>x. ord (f x) (g x)" shows "fun_ord ord f g"
cons1: shows "evaluate_top ck env s1 top0 (s2, Rval new_env) \<and> evaluate_prog ck (extend_dec_env new_env env) s2 tops (s3, r) \<and> (\<forall>extra cnt cnt' s' ra. clock s2 = extra + cnt \<and> clock s' = extra + cnt' \<and> (s3, r) = (s', ra) \<and> ck = True \<longrightarrow> evaluate_prog ck (extend_dec_env new_env env) (update_clock (\<lambda>_. cnt) s2) tops (update_clock (\<lambda>_. cnt') s', ra))"
\<theta>\<^sub>h': shows "\<guillemotleft>\<theta>\<^sub>h' : tab\<^sub>0 h \<star> w\<^sub>h' \<rightarrow> u\<^sub>h\<guillemotright>"
codomains_char: fixes f :: "'s" shows "S.codomains f = (if S.arr f then {S.cod f} else {})"
simps_9: fixes X :: "WS1S_Nameful.so"   and \<phi> :: "WS1S_Nameful.ws1s" shows "fos (All2 X \<phi>) = fos \<phi>"
12640: fixes x :: "'a"   and y :: "'a" shows "- x \<sqinter> - (- y \<sqinter> x) = - x"
add_to_coefficient_simps_If: fixes a :: "char list multiset"   and n :: "int"   and b :: "(char list multiset \<times> int) multiset" shows "add_to_coefficient (a, n) b = {#(a', uu_) \<in># b. a' \<noteq> a#} + (if n + \<Sum>\<^sub># (snd `# {#(a', uu_) \<in># b. a' = a#}) = 0 then {#} else {#(a, n + \<Sum>\<^sub># (snd `# {#(a', uu_) \<in># b. a' = a#}))#})"
fold_max_le_self: fixes y :: "'b"   and ys :: "'b list" shows "y \<le> fold max ys y"
all_inclG_2: shows "H1 \<subseteq> carrier G"
greaterThan_non_empty: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c" assumes "class.no_top less_eq less" shows "ord.greaterThan less x \<noteq> {}"
cons1: fixes ck :: "bool"   and env :: "v sem_env"   and s1 :: "'b state"   and e :: "exp0"   and s2 :: "'b state"   and v1 :: "v"   and es :: "exp0 list"   and s3 :: "'b state"   and vs :: "v list" assumes "evaluate ck env s1 e (s2, Rval v1) \<and> evaluate_list ck env s2 es (s3, Rval vs)" shows "evaluate_list ck env s1 (e # es) (s3, Rval (v1 # vs))"
strict_alt_def: fixes p :: "'a pmf"   and q :: "'a pmf" assumes "p \<in> \<P> outcomes"   and "q \<in> \<P> outcomes" shows "(p \<succeq> q \<and> (q, p) \<notin> relation) = ((\<Sum>z\<in>outcomes. pmf q z * u z) < (\<Sum>z\<in>outcomes. pmf p z * u z))"
WHILEI_body_def: fixes bind :: "'f \<Rightarrow> 'c \<Rightarrow> 'e"   and return :: "'d \<Rightarrow> 'e"   and I :: "'d \<Rightarrow> bool"   and b :: "'d \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'f" shows "WHILEI_body bind return I b f \<equiv> \<lambda>W s. if I s then if b s then bind (f s) W else return s else top"
SmartLet_def: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and e :: "exp" shows "SmartLet \<Gamma> e = (if \<Gamma> = [] then e else Let \<Gamma> e)"
Int_lower1: fixes A :: "'e set"   and B :: "'e set" shows "A \<inter> B \<subseteq> A"
ereal_uminus_zero_iff: fixes a :: "ereal" shows "(- a = 0) = (a = 0)"
Suc_m_minus_n: fixes n :: "nat"   and m :: "nat" shows "n \<le> m \<longrightarrow> Suc m - n = Suc (m - n)"
cycle_enter_leave_same: fixes G :: "('a, 'b) graph_scheme"   and p :: "'a list"   and x :: "'a" assumes "cycle G p" shows "card (set (filter (\<lambda>(x', y). x' = x) (cycle_edges p))) = card (set (filter (\<lambda>(x', y). y = x) (cycle_edges p)))"
pointed_conditionally_complete_iff_bounded_complete: fixes A :: "'a set" assumes "A \<noteq> {}" shows "(conditionally_complete A (\<sqsubseteq>) \<and> extremed A (\<lambda>x y. y \<sqsubseteq> x)) = bounded_complete A (\<sqsubseteq>)"
ex_raw_match_term_imp_tinst: fixes t2 :: "Core.term"   and t1 :: "Core.term" assumes "\<exists>subs. raw_match_term t2 t1 Map.empty = Some subs" shows "tinst t1 t2"
init_fin_delay_bisimulation_diverge: fixes t :: "'t" shows "delay_bisimulation_diverge (r1.init_fin t) (r2.init_fin t) (init_fin_bisim t) (ta_bisim init_fin_bisim) r1.init_fin_\<tau>move r2.init_fin_\<tau>move"
zero_real_alg_def: shows "0 \<equiv> of_rat_real_alg 0"
face_cycle_set_altdef: fixes a :: "'b" shows "GM.face_cycle_set a = {(GM.face_cycle_succ ^^ n) a |n. True}"
eq_fold: fixes A :: "'a set"   and x :: "'a" assumes "finite A" shows "\<Sqinter> insert x A = Finite_Set.fold (\<sqinter>) x A"
exists_unique_pirreducible_gen: fixes K :: "'a list set"   and h :: "'a list list \<Rightarrow> 'a list" assumes "subfield K (K [X])"   and "ring_hom_ring (K [X]\<^bsub>K [X]\<^esub>) (K [X]) h"   and "a_kernel (K [X]\<^bsub>K [X]\<^esub>) (K [X]) h \<noteq> {[]}"   and "a_kernel (K [X]\<^bsub>K [X]\<^esub>) (K [X]) h \<noteq> carrier (K [X]\<^bsub>K [X]\<^esub>)" shows "\<exists>!p. p \<in> carrier (K [X]\<^bsub>K [X]\<^esub>) \<and> pirreducible\<^bsub>K [X]\<^esub> K p \<and> lead_coeff p = \<one>\<^bsub>K [X]\<^esub> \<and> a_kernel (K [X]\<^bsub>K [X]\<^esub>) (K [X]) h = PIdl\<^bsub>K [X]\<^bsub>K [X]\<^esub>\<^esub> p"
mult_local_ring_at_def: shows "mult_local_ring_at \<equiv> local.mult_rel"
d'_subs: shows "set_mset D' \<subseteq> set_mset D"
lNact0: fixes c :: "'kid"   and t :: "nat \<Rightarrow> cnf" shows "\<langle>c \<Leftarrow> t\<rangle>\<^bsub>0\<^esub> = 0"
closure_finite_cspan: fixes T :: "'c set" assumes "finite T" shows "closure (cspan T) = cspan T"
substNext_code_alt_def: fixes p :: "'g \<Rightarrow> 'node \<times> 'val \<Rightarrow> 'val list option"   and g :: "'g"   and "next" :: "'node \<times> 'val" shows "uninst.substNext_code p g next = id(snd next := uninst.substitution_code p g next)"
starfun_exp_less_mono: fixes x :: "real star"   and y :: "real star" assumes "x < y" shows "(*f* exp) x < (*f* exp) y"
rtrancl_trancl_reflcl: fixes r :: "('a \<times> 'a) set" shows "r\<^sup>* = (r\<^sup>+)\<^sup>="
diff_diff: fixes C :: "'b set"   and A :: "'b set"   and B :: "'b set" shows "C - (A - B) = C - A \<union> C \<inter> B"
chainD: fixes M :: "('v \<times> 'v \<Rightarrow> ennreal) set" assumes "M \<in> Chains leq"   and "M \<noteq> {}" shows "\<Sqinter> M \<in> F"
sandwich: fixes y :: "'a" assumes "y \<in> s"   and "y \<noteq> x"   and "dist y x < e" shows "norm (f y - f x - f' (y - x)) / norm (y - x) \<le> H y"
uniform_discrete_insert: fixes a :: "'c"   and S :: "'c set" shows "uniform_discrete (insert a S) = uniform_discrete S"
prod_cases3: fixes y :: "'k \<times> 'l \<times> 'm" obtains a :: "'k" and b :: "'l" and c :: "'m" where "y = (a, b, c)"
h1: shows "mapping_of (a - b) = mapping_of a - mapping_of b"
sup_Un_eq2: fixes R :: "('b \<times> 'c) set"   and S :: "('b \<times> 'c) set" shows "sup (\<lambda>x y. (x, y) \<in> R) (\<lambda>x y. (x, y) \<in> S) = (\<lambda>x y. (x, y) \<in> R \<union> S)"
smulextp_mset_code: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> bool \<times> bool"   and xs :: "'a list"   and ys :: "'a list" shows "smulextp f (mset xs) (mset ys) = fst (mul_ext f xs ys)"
hyps_3: assumes "P \<turnstile> \<langle>e'',s''\<rangle> \<Rightarrow> \<langle>e',s'\<rangle>"   and "iconf (shp s2) e2"   and "P,shp s2 \<turnstile>\<^sub>b (e2,b2) \<surd>" shows "P \<turnstile> \<langle>e2,s2\<rangle> \<Rightarrow> \<langle>e',s'\<rangle>"
card_gr0_imp_finite: fixes A :: "'b set" assumes "0 < card A" shows "finite A"
ent_refl: fixes P :: "assn" shows "P \<Longrightarrow>\<^sub>A P"
poly_y_x_pCons: fixes a :: "'c poly"   and p :: "'c poly poly" shows "poly_y_x (pCons a p) = poly_lift a + map_poly (pCons (0::'c)) (poly_y_x p)"
b_e_type_call_1: fixes \<C> :: "t_context"   and e :: "b_e"   and ts :: "t list"   and ts' :: "t list"   and i :: "nat" assumes "\<C> \<turnstile> [e] : ts _> ts'"   and "e = Call i" shows "i < length (func_t \<C>)"
set_integrable_def: fixes M :: "'c measure"   and A :: "'c set"   and f :: "'c \<Rightarrow> 'd" shows "set_integrable M A f \<equiv> integrable M (\<lambda>x. indicat_real A x *\<^sub>R f x)"
return_only_one_intra_edge: fixes a :: "'a"   and Q' :: "('c \<Rightarrow> 'd option) \<times> 'e \<Rightarrow> bool"   and p :: "'f"   and f' :: "('c \<Rightarrow> 'd option) \<Rightarrow> ('c \<Rightarrow> 'd option) \<Rightarrow> 'c \<Rightarrow> 'd option" assumes "valid_edge a"   and "kind a = Q'\<hookleftarrow>\<^bsub>p\<^esub>f'" shows "\<exists>!a'. valid_edge a' \<and> targetnode a' = targetnode a \<and> intra_kind (kind a')"
cinsert_csubset: fixes x :: "'d"   and A :: "'d cset"   and B :: "'d cset" shows "csubset_eq (cinsert x A) B = (cin x B \<and> csubset_eq A B)"
invertible_def: fixes A :: "(('d, 'e) vec, 'f) vec" shows "invertible A = (\<exists>A'. A ** A' = mat (1::'d) \<and> A' ** A = mat (1::'d))"
cr_good_interp_def: shows "cr_good_interp \<equiv> \<lambda>x y. x = raw_interp y"
vc_sound': fixes C :: "acom"   and Q :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and P :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool" assumes "vc C Q"   and "finite (support Q)"   and "finite (varacom C)"   and "\<forall>l s. P l s \<longrightarrow> pre C Q l s" shows "\<turnstile>\<^sub>1 {P} strip C { time C \<Down> Q}"
dim_basis: fixes A :: "'a mod_ring poly set" assumes "finite A"   and "Berlekamp_subspace.basis A" shows "Berlekamp_subspace.dim = card A"
simps: fixes xs :: "('a \<times> 'e) list"   and ko :: "'o" shows "lookup_raw (xs, ko) = lookup_pair ko xs"
decided: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and rs\<^sub>1 :: "'c rule list"   and X :: "final_decision"   and rs\<^sub>2 :: "'c rule list" assumes "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs\<^sub>1, Undecided\<rangle> \<Rightarrow>\<^sub>s Decision X" shows "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs\<^sub>1 @ rs\<^sub>2, Undecided\<rangle> \<Rightarrow>\<^sub>s Decision X"