hcju/isabelleps huggingface dataset:
FIN_wrt_def: fixes \<psi> :: "nat \<Rightarrow> nat \<Rightarrow> nat option" shows "FIN_wrt \<psi> \<equiv> {U. \<exists>s. learn_fin \<psi> U s}"
sets_le_imp_space_le: fixes A :: "'b measure"   and B :: "'b measure" assumes "sets A \<subseteq> sets B" shows "space A \<subseteq> space B"
rep_eq: fixes x :: "'b Filters.filter"   and xa :: "'b Filters.filter" shows "(x \<le> xa) = (Filters.filter.Rep_filter x \<subseteq> Filters.filter.Rep_filter xa)"
matched_is_SDG_path: fixes n :: "'node SDG_node"   and ns :: "'node SDG_node list"   and n' :: "'node SDG_node" assumes "matched n ns n'" obtains ns' :: "'node SDG_node list" where "n is-ns'\<rightarrow>\<^sub>d* n'"
chain_group_def: fixes p :: "nat"   and X :: "'b topology" shows "chain_group p X \<equiv> free_Abelian_group (singular_simplex_set p X)"
split_strip_while_append: fixes P :: "'f \<Rightarrow> bool"   and xs :: "'f list" obtains ys :: "'f list" and zs :: "'f list" where "strip_while P xs = ys" and "\<forall>x\<in>set zs. P x" and "xs = ys @ zs"
Eq\<^sub>2I: fixes F :: "\<Pi>\<^sub>2"   and G :: "\<Pi>\<^sub>2"   and v :: "i" assumes "F = G" shows "[F \<^bold>=\<^sub>2 G in v]"
coprime_nat_abs_right_iff: fixes n :: "nat"   and k :: "int" shows "coprime n (nat \<bar>k\<bar>) = coprime (int n) k"
weakPsiCongOutputPres: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and M :: "'a"   and N :: "'a" assumes "\<Psi> \<rhd> P \<doteq> Q" shows "\<Psi> \<rhd> M\<langle>N\<rangle>.P \<doteq> M\<langle>N\<rangle>.Q"
insertE: fixes a :: "'k"   and b :: "'k"   and A :: "'k set" assumes "a \<in> insert b A" obtains "a = b"   | "a \<in> A"
intro: fixes \<eta> :: "('g, 'i, 'j, 'h, 'k, 'l) NatTrans" assumes "($$) \<eta> \<in> extensional (obj\<^bsub>NTCatDom \<eta>\<^esub>)" shows "NatTransExt \<eta>"
hseqE: fixes \<nu> :: "'b"   and \<mu> :: "'b" assumes "B.hseq \<nu> \<mu>" obtains "B.arr \<mu>" and "B.arr \<nu>" and "src\<^sub>B \<nu> = trg\<^sub>B \<mu>"
refl_symm_closure_is_symm_refl_closure: fixes Rel :: "('a \<times> 'a) set" shows "symcl (Rel\<^sup>=) = (symcl Rel)\<^sup>="
squash_idem: fixes x :: "'a" shows "squash (squash x) = squash x"
dioph_ineq_succs_def: fixes n :: "nat"   and ks :: "int list"   and m :: "int" shows "dioph_ineq_succs n ks m = map (\<lambda>xs. (m - eval_dioph ks xs) div 2) (mk_nat_vecs n)"
EQ: shows "GEN_OP eq (=) (Rv \<rightarrow> Rv \<rightarrow> bool_rel)"
path_image_nonempty: fixes g :: "real \<Rightarrow> 'b" shows "path_image g \<noteq> {}"
fls_subdegree_leI: fixes f :: "'a fls"   and n :: "int" assumes "fls_nth f n \<noteq> (0::'a)" shows "fls_subdegree f \<le> n"
is_ipd_def: fixes y :: "'n"   and x :: "'n" shows "(y ipd\<rightarrow> x) = (x \<noteq> y \<and> y pd\<rightarrow> x \<and> (\<forall>z. z \<noteq> x \<and> z pd\<rightarrow> x \<longrightarrow> z pd\<rightarrow> y))"
card_dep_pair_set: fixes A :: "'a set"   and f :: "'a set \<Rightarrow> 'b set"   and n :: "nat"   and g :: "'a set \<Rightarrow> nat" assumes "finite A"   and "\<And>a. a \<subseteq> A \<Longrightarrow> finite (f a)" shows "card {(a, b). a \<subseteq> A \<and> card a = n \<and> b \<subseteq> f a \<and> card b = g a} = (\<Sum>a | a \<subseteq> A \<and> card a = n. card (f a) choose g a)"
successI: fixes f :: "'a Heap"   and h :: "heap" assumes "execute f h \<noteq> None" shows "success f h"
Bcard: fixes k :: "nat"   and n :: "nat" assumes "N k \<le> n"   and "n < N (Suc k)" shows "real (card (B \<inter> {..<n})) \<le> (l + (1 / 2) ^ k) * real n"
of_real_eq_0_iff: fixes x :: "real" shows "(of_real x = (0::'c)) = (x = 0)"
diamond_gen_eq_rot_diamond: fixes x :: "real"   and y :: "real" assumes "0 \<le> x"   and "x \<le> 1"   and "0 \<le> y"   and "y \<le> 1" shows "diamond_cube_gen (x, y) = rot_diamond_cube ((x - 1 / 2) / (2 * diamond_y_gen (x_coord (rot_x x y))) + 1 / 2, rot_x x y)"
rtranclp_rtrancl_eq: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" shows "r\<^sup>*\<^sup>* x y = ((x, y) \<in> {(x, y). r x y}\<^sup>*)"
assms: shows "\<forall>i j k. f (i, j) \<noteq> bot \<and> f (i, k) \<noteq> bot \<longrightarrow> j = k"
dmu_ij_state: fixes state :: "(int vec list \<times> int vec list) \<times> int iarray iarray \<times> int iarray"   and i :: "nat"   and fs :: "int vec list"   and upw :: "bool"   and f :: "int vec list \<times> int vec list"   and mu :: "int iarray iarray"   and ds :: "int iarray"   and j :: "nat"   and ii :: "nat" assumes "LLL_impl_inv state i fs"   and "LLL_invariant upw i fs"   and "state = (f, mu, ds)"   and "j < ii"   and "ii < m" shows "dmu_ij_state state ii j = d\<mu> fs ii j"
continuous_on_iff: fixes s :: "'a set"   and f :: "'a \<Rightarrow> 'b" shows "continuous_on s f = (\<forall>x\<in>s. \<forall>e>0. \<exists>d>0. \<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist (f x') (f x) < e)"
eq_app_right_eq: fixes L :: "'b list set" shows "eq_app_right L `` L = L"
control_dependence_path: fixes n :: "'node"   and n' :: "'node" assumes "n controls n'" obtains as :: "'edge list" where "n -as\<rightarrow>\<^sub>\<iota>* n'" and "as \<noteq> []"
DS: fixes \<rho>\<^sub>1 :: "complex mat"   and \<rho>\<^sub>2 :: "complex mat" assumes "\<rho>\<^sub>1 \<in> carrier_mat d d"   and "\<rho>\<^sub>2 \<in> carrier_mat d d"   and "partial_density_operator \<rho>\<^sub>1"   and "partial_density_operator \<rho>\<^sub>2"   and "trace (\<rho>\<^sub>1 + \<rho>\<^sub>2) \<le> 1" shows "denote x2 (\<rho>\<^sub>1 + \<rho>\<^sub>2) = denote x2 \<rho>\<^sub>1 + denote x2 \<rho>\<^sub>2"
select_convs_1: fixes cx :: "int"   and cy :: "int"   and black :: "bool"   and more :: "'a" shows "black \<lparr>cx = cx, cy = cy, black = black, \<dots> = more\<rparr> = black"
1: shows "consistent \<sigma>' \<sigma>\<^sub>s\<^sub>y\<^sub>m (store c')"
comm_groupE_4: fixes G :: "('a, 'b) monoid_scheme"   and x :: "'a"   and y :: "'a" assumes "comm_group G"   and "x \<in> carrier G"   and "y \<in> carrier G" shows "x \<otimes>\<^bsub>G\<^esub> y = y \<otimes>\<^bsub>G\<^esub> x"
Mapping_lookup_filter_Some: fixes P :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and k :: "'b"   and m :: "('b, 'a) mapping" assumes "\<And>v. P k v" shows "Mapping.lookup (Mapping.filter P m) k = Mapping.lookup m k"
simps: fixes a :: "'a"   and H :: "'a set" shows "(a \<in> generate H) = (a = \<one> \<or> (\<exists>a. a = a \<and> a \<in> H) \<or> (\<exists>a. a = local.inverse a \<and> a \<in> H) \<or> (\<exists>a b. a = a \<cdot> b \<and> a \<in> generate H \<and> b \<in> generate H))"
relH_dist_union: fixes as :: "nat set"   and as' :: "nat set"   and h :: "heap"   and h' :: "heap" assumes "relH (as \<union> as') h h'" shows "relH as h h' \<and> relH as' h h'"
id: fixes i :: "nat" assumes "i < nc" shows "col m1 i = col m2 i"
semantics_pltl_sugar_4: fixes w :: "nat \<Rightarrow> 'b"   and \<phi> :: "'b pltl"   and \<psi> :: "'b pltl" shows "w \<Turnstile>\<^sub>p \<phi> and\<^sub>p \<psi> = (w \<Turnstile>\<^sub>p \<phi> \<and> w \<Turnstile>\<^sub>p \<psi>)"
set_mset_union: fixes M :: "'d multiset"   and N :: "'d multiset" shows "set_mset (M + N) = set_mset M \<union> set_mset N"
intros_2: fixes c :: "(('fsym, 'tp) efsym, ('psym, 'tp) epsym) lit list"   and \<sigma> :: "'tp" assumes "\<And>l x. \<lbrakk>l \<in> set c; x \<in> nv2L l; GE.tpOfV x = \<sigma>\<rbrakk> \<Longrightarrow> isGuard x (grdOf c l x)" shows "mcalc2 \<sigma> c"
diff_ant_def: fixes z :: "ant"   and w :: "ant" shows "z - w \<equiv> z + - w"
Lp_Norm_2: fixes p :: "real"   and f :: "'b \<Rightarrow> real"   and M :: "'b measure" assumes "0 < p"   and "f \<in> borel_measurable M" shows "Norm (\<LL> (ennreal p) M) f powr p = LINT x|M. \<bar>f x\<bar> powr p"
monad_stateT: fixes return :: "'a \<times> 'b \<Rightarrow> 'c"   and bind :: "'c \<Rightarrow> ('a \<times> 'b \<Rightarrow> 'c) \<Rightarrow> 'c" assumes "monad return bind" shows "monad (return_state return) (bind_state bind)"
log_of_power_eq: fixes m :: "nat"   and b :: "real"   and n :: "nat" assumes "real m = b ^ n"   and "1 < b" shows "real n = log b (real m)"
upset_set_def: fixes X :: "'a set" shows "\<Up> X = {y. \<exists>x\<in>X. x \<le> y}"
uncomputable_def: fixes code :: "(action \<times> nat) list \<Rightarrow> nat"   and H :: "(action \<times> nat) list" shows "uncomputable code H \<equiv> tm_wf0 H \<and> (\<forall>M ns. halts M ns \<longrightarrow> {\<lambda>tp. tp = ([Bk], <(code M, ns)>)} H {\<lambda>tp. \<exists>k. tp = (Bk \<up> k, <0>)}) \<and> (\<forall>M ns. \<not> halts M ns \<longrightarrow> {\<lambda>tp. tp = ([Bk], <(code M, ns)>)} H {\<lambda>tp. \<exists>k. tp = (Bk \<up> k, <1>)})"
is_sig_GB_in_mono: fixes d :: "'a \<Rightarrow> nat"   and G :: "('t \<Rightarrow>\<^sub>0 'b) set"   and u :: "'t"   and G' :: "('t \<Rightarrow>\<^sub>0 'b) set" assumes "is_sig_GB_in d G u"   and "G \<subseteq> G'" shows "is_sig_GB_in d G' u"
indep_set_def: fixes M :: "'c measure"   and A :: "'c set set"   and B :: "'c set set" assumes "prob_space M" shows "prob_space.indep_set M A B = prob_space.indep_sets M (case_bool A B) UNIV"
freshs_floatarith_max_Var_floatarithI: fixes xs :: "nat list"   and f :: "floatarith" assumes "\<And>x. x \<in> set xs \<Longrightarrow> max_Var_floatarith f \<le> x" shows "freshs_floatarith f xs"
scaleR_sum_list: fixes a :: "real"   and xs :: "'a list" shows "a *\<^sub>R sum_list xs = sum_list (map ((*\<^sub>R) a) xs)"
return_def: shows "return = COERCE(udom \<rightarrow> udom\<cdot>'b, 'a \<rightarrow> 'a\<cdot>'b)\<cdot>returnU"
neq: fixes n :: "nat" shows "s + of_nat n \<noteq> 0"
DSourcesA9_L0: shows "DSources level0 sA9 = {sA8}"
psubsetI: fixes A :: "'a set"   and B :: "'a set" assumes "A \<subseteq> B"   and "A \<noteq> B" shows "A \<subset> B"
minor: fixes \<beta> :: "V"   and \<gamma> :: "V" assumes "\<beta> < \<alpha>"   and "\<gamma> < \<alpha>"   and "Ord \<gamma>" shows "\<beta> + \<gamma> < \<alpha>"
psubst_def: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('e, 'f, 'varSort, 'var, 'g) qTerm set option"   and X :: "('e, 'f, 'varSort, 'var, 'g) qTerm set" shows "X #[rho] \<equiv> asTerm pick X #[[pickE rho]]"
group_Int_image_ker: fixes f :: "'c \<Rightarrow> 'd"   and G :: "('c, 'e) monoid_scheme"   and H :: "('d, 'f) monoid_scheme"   and g :: "'d \<Rightarrow> 'g"   and K :: "('g, 'h) monoid_scheme" assumes "f \<in> hom G H"   and "g \<in> hom H K"   and "inj_on (g \<circ> f) (carrier G)"   and "Group.group G"   and "Group.group H"   and "Group.group K" shows "f ` carrier G \<inter> kernel H K g = {\<one>\<^bsub>H\<^esub>}"
dghm_up_ArrMap_vrange: fixes \<CC> :: "V"   and \<phi> :: "V \<Rightarrow> V" assumes "\<And>i. i \<in>\<^sub>\<circ> I \<Longrightarrow> \<phi> i : \<CC> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<AA> i" shows "\<R>\<^sub>\<circ> (dghm_up I \<AA> \<CC> \<phi>\<lparr>ArrMap\<rparr>) \<subseteq>\<^sub>\<circ> dg_prod I \<AA>\<lparr>Arr\<rparr>"
periodic_set_cos_linear: fixes a :: "real"   and c :: "real"   and b :: "real" assumes "a \<noteq> 0"   and "c \<noteq> 0" shows "periodic_set (roots (\<lambda>x. a * cos (x / c) + b)) (2 * c * pi)"
strict_trans: fixes b :: "'b"   and a :: "'b"   and c :: "'b" assumes "b < a"   and "c < b" shows "c < a"
alpha_imp_asTerm_equal: fixes qX :: "('d, 'e, 'varSort, 'var, 'f) qTerm"   and qY :: "('d, 'e, 'varSort, 'var, 'f) qTerm" assumes "qGood qX"   and "qX #= qY" shows "asTerm qX = asTerm qY"
divide_eq_1_iff: fixes divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and inverse :: "'d \<Rightarrow> 'd"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and a :: "'d"   and b :: "'d" assumes "class.field divide inverse times one plus zero minus uminus" shows "(divide a b = one) = (b \<noteq> zero \<and> a = b)"
ltl_lang_equiv_def: fixes \<phi> :: "'b ltln"   and \<psi> :: "'b ltln" shows "\<phi> \<sim>\<^sub>L \<psi> \<equiv> \<forall>w. w \<Turnstile>\<^sub>n \<phi> = w \<Turnstile>\<^sub>n \<psi>"
foldr_Cons: fixes f :: "'b \<Rightarrow> 'a \<Rightarrow> 'a"   and x :: "'b"   and xs :: "'b list" shows "foldr f (x # xs) = f x \<circ> foldr f xs"
cod_lunit: fixes a :: "'c local.term set" assumes "ide a" shows "cod \<l>[a] = a"
Subset_D: fixes H :: "fm set"   and t :: "tm"   and u :: "tm"   and a :: "tm" assumes "H \<turnstile> t SUBS u"   and "H \<turnstile> a IN t" shows "H \<turnstile> a IN u"
Pbij_inverse_Inverse: fixes \<gamma> :: "(Location \<times> Location) set"   and \<beta> :: "(Location \<times> Location) set"   and l :: "Location"   and ll :: "Location" assumes "\<gamma> = Pbij_inverse \<beta>" shows "((l, ll) \<in> \<beta>) = ((ll, l) \<in> \<gamma>)"
atLeastLessThan_iff: fixes i :: "'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and l :: "'c"   and u :: "'c" shows "(i \<in> ord.atLeastLessThan less_eq less l u) = (less_eq l i \<and> less i u)"
coclop_coextensive_var: fixes f :: "'b \<Rightarrow> 'b"   and x :: "'b" assumes "coclop f" shows "f x \<le> x"
accept\<^sub>M_def: fixes \<phi> :: "'b ltl"   and \<G> :: "'b ltl set"   and w :: "nat \<Rightarrow> 'b set" shows "accept\<^sub>M \<phi> \<G> w \<equiv> \<forall>\<^sub>\<infinity>j. \<forall>S. (\<forall>\<psi>. G \<psi> \<in> \<G> \<longrightarrow> S \<Turnstile>\<^sub>P G \<psi> \<and> S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (\<F> \<psi> w \<G> j)) \<longrightarrow> S \<Turnstile>\<^sub>P af \<phi> (prefix j w)"
Skolem_holds_def: fixes \<phi> :: "('sf, 'sc, 'sz) formula"   and var :: "'sz" shows "Skolem_holds \<phi> var \<equiv> valid \<phi> \<longrightarrow> valid (Forall var \<phi>)"
exhaust: fixes y :: "('b, 'c) sAbs" obtains x1 :: "'b" and x2 :: "'c \<Rightarrow> 'c" where "y = sAbs x1 x2"
pos_prod_lt: fixes c :: "'b"   and x :: "'b" assumes "(0::'b) < c" shows "c * x < (0::'b) \<equiv> x < (0::'b)"
bind_is_OK_I2: fixes h :: "'a"   and f :: "('a, 'b, 'c) prog"   and g :: "'c \<Rightarrow> ('a, 'b, 'd) prog" assumes "h \<turnstile> ok f"   and "\<And>x h'. \<lbrakk>h \<turnstile> f \<rightarrow>\<^sub>r x; h \<turnstile> f \<rightarrow>\<^sub>h h'\<rbrakk> \<Longrightarrow> h' \<turnstile> ok g x" shows "h \<turnstile> ok (f \<bind> g)"
exhaust: fixes y :: "Integer" obtains "y = \<bottom>"   | int :: "int" where "y = MkI\<cdot>int"
While_1: fixes P :: "complex mat"   and Q :: "complex mat" assumes "well_com S"   and "is_quantum_predicate P"   and "is_quantum_predicate Q"   and "P \<le>\<^sub>L Q" shows "is_quantum_predicate (wlp S P) \<and> wlp S P \<le>\<^sub>L wlp S Q"
pre_oldest_configuration_range: fixes \<Sigma> :: "'c set"   and \<delta> :: "'d \<Rightarrow> 'c \<Rightarrow> 'd"   and q\<^sub>0 :: "'d"   and w :: "nat \<Rightarrow> 'c"   and q :: "'d"   and n :: "nat" assumes "semi_mojmir \<Sigma> \<delta> q\<^sub>0 w" shows "semi_mojmir.pre_oldest_tokens \<delta> q\<^sub>0 w q n \<subseteq> {0..Suc n}"
simps_1: fixes \<C> :: "'a"   and \<sigma> :: "'a basic_type" shows "\<langle>\<C>\<rangle>\<^sub>\<T> \<squnion> \<sigma> = (case \<sigma> of OclVoid \<Rightarrow> \<langle>\<C>\<rangle>\<^sub>\<T> | \<langle>\<D>\<rangle>\<^sub>\<T> \<Rightarrow> \<langle>\<C> \<squnion> \<D>\<rangle>\<^sub>\<T> | _ \<Rightarrow> OclAny)"
couSim: shows "weak_barbed_coupled_simulation (indRelLTPO TRel) (STCalWB SWB TWB)"
constrains_imp_subset: fixes F :: "'c program"   and A :: "'c set"   and A' :: "'c set" assumes "F \<in> A co A'" shows "A \<subseteq> A'"
AllowedActs_project: fixes h :: "'c \<times> 'd \<Rightarrow> 'e"   and C :: "'e set"   and F :: "'e program" assumes "Extend h" shows "AllowedActs (project h C F) = {act. Restrict (project_set h C) act \<in> project_act h ` Restrict C ` AllowedActs F}"
focus_times: fixes X :: "'a set"   and p :: "('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and q :: "('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" shows "focus X (p * q) = focus X p * focus X q"
prems_2: shows "(m # ms) \<langle>g\<rangle> [l] \<odot> [l] \<langle>f\<rangle> [l] \<preceq> (m # ms) \<langle>g\<rangle> [l]"
list_singleton_induct: fixes P :: "'a list \<Rightarrow> bool"   and xs :: "'a list" assumes "P []"   and "\<And>x. P [x]"   and "\<And>x y xs. P (y # xs) \<Longrightarrow> P (x # y # xs)" shows "P xs"
Trg_Trg: fixes t :: "'a local.term" assumes "Arr t" shows "Trg (Trg t) = Trg t"
assms_2: shows "\<guillemotleft>f : a \<rightarrow> b\<guillemotright>"
bst_joinL: fixes l :: "('b \<times> color) tree"   and a :: "'b"   and n :: "color"   and r :: "('b \<times> color) tree" assumes "Tree2.bst \<langle>l, (a, n), r\<rangle>"   and "bheight l \<le> bheight r" shows "Tree2.bst (joinL l a r)"
f: fixes t :: "nat" shows "eval f [t, i, x] \<down>= (if x < e_length (block i t) \<and> x \<noteq> gap i t then 0 else 1)"
comm_monoid_axioms: shows "comm_monoid (+) (0::'a)"
nn_integral_restrict_space: fixes \<Omega> :: "'c set"   and M :: "'c measure"   and f :: "'c \<Rightarrow> ennreal" assumes "\<Omega> \<inter> space M \<in> sets M" shows "integral\<^sup>N (restrict_space M \<Omega>) f = \<integral>\<^sup>+ x. f x * indicator \<Omega> x \<partial>M"
conv: shows "(\<lambda>n. trace (f n)) \<longlonglongrightarrow> trace lowner_lub"
insert_Diff_single: fixes a :: "'a"   and A :: "'a set" shows "insert a (A - {a}) = insert a A"
Bcontfun_inject: fixes x :: "'c \<Rightarrow> 'd"   and y :: "'c \<Rightarrow> 'd" assumes "x \<in> bcontfun"   and "y \<in> bcontfun" shows "(Bcontfun x = Bcontfun y) = (x = y)"
exists_nat_var: shows "\<exists>f. inj f \<and> range f \<subseteq> var"
step: fixes gpv :: "('a, 'b, 'c) gpv" assumes "X gpv" shows "lossless_spmf (the_gpv gpv) \<and> (\<forall>out c input. IO out c \<in> set_spmf (the_gpv gpv) \<longrightarrow> input \<in> responses_\<I> \<I> out \<longrightarrow> X (c input) \<or> colossless_gpv \<I> (c input))"
subst_atm_list_single: fixes A :: "'a"   and \<sigma> :: "'s" shows "[A] \<cdot>al \<sigma> = [A \<cdot>a \<sigma>]"
concat_in: fixes i :: "nat"   and xs :: "'a list list" assumes "i < length (concat xs)" shows "\<exists>k<length xs. concat xs ! i \<in> set (xs ! k)"
induct: fixes x1 :: "string_ternop"   and x2 :: "'a OCL_Types.type"   and x3 :: "'b OCL_Types.type"   and x4 :: "'c OCL_Types.type"   and x5 :: "'d OCL_Types.type"   and P :: "string_ternop \<Rightarrow> 'a OCL_Types.type \<Rightarrow> 'b OCL_Types.type \<Rightarrow> 'c OCL_Types.type \<Rightarrow> 'd OCL_Types.type \<Rightarrow> bool" assumes "string_ternop_type x1 x2 x3 x4 x5"   and "\<And>\<sigma> \<rho>. \<lbrakk>\<sigma> = UnlimitedNatural[1]\<midarrow>Integer[1]; \<rho> = UnlimitedNatural[1]\<midarrow>Integer[1]\<rbrakk> \<Longrightarrow> P SubstringOp String[1] \<sigma> \<rho> String[1]" shows "P x1 x2 x3 x4 x5"
cauchy_not_vanishes: fixes X :: "nat \<Rightarrow> rat" assumes "Real.cauchy X"   and "\<not> Real.vanishes X" shows "\<exists>b>0. \<exists>k. \<forall>n\<ge>k. b < \<bar>X n\<bar>"
thetaIf_sym: fixes indis :: "('a \<times> 'a) set"   and tval :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'a \<Rightarrow> 'a" assumes "PL_Indis indis" shows "sym (PL_Indis.thetaIf tval aval indis)"
lt_in_keys: fixes p :: "('x option \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'a" assumes "p \<noteq> 0" shows "pp_pm.extended_ord.lpp p \<in> keys p"
fresh_not_fic: fixes a :: "coname"   and M :: "trm" assumes "a \<sharp> M" shows "\<not> fic M a"
l_get_dom_component_get_scdom_component\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms_def: fixes known_ptr :: "(_) object_ptr \<Rightarrow> bool" shows "l_get_dom_component_get_scdom_component\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms known_ptr \<equiv> known_ptr = ShadowRootClass.known_ptr"
append_init_def: fixes s1 :: "'a"   and s2 :: "'b" shows "append_init s1 s2 = Inl (s1, s2)"
gallery_double_cross_not_minimal_Cons1: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b"   and B :: "'b set"   and C :: "'b set"   and D :: "'b set"   and Cs :: "'b set list" assumes "ThinChamberComplexFolding X f"   and "B \<in> f \<turnstile> ChamberComplex.chamber_system X"   and "C \<in> ChamberComplex.chamber_system X - f \<turnstile> ChamberComplex.chamber_system X"   and "D \<in> f \<turnstile> ChamberComplex.chamber_system X"   and "SimplicialComplex.maxsimpchain X (B # C # Cs @ [D])" shows "\<not> SimplicialComplex.min_maxsimpchain X (B # C # Cs @ [D])"
simps: fixes b :: "proper_base"   and x :: "int" shows "log_main b x = (if x < get_base b then (0, 1) else case log_main (square_base b) x of (z, bz) \<Rightarrow> let l = 2 * z; bz1 = bz * get_base b in if x < bz1 then (l, bz) else (Suc l, bz1))"
insertion_code: fixes i :: "nat \<Rightarrow> 'a"   and mp :: "'a mpoly" shows "insertion i mp = (\<Sum>m\<in>monomials mp. coeff mp m * (\<Prod>k\<in>keys m. i k ^ lookup m k))"
INT_extend_simps_8: fixes B :: "'b \<Rightarrow> 'a set"   and A :: "'b set set" shows "(\<Inter>y\<in>A. \<Inter> (B ` y)) = \<Inter> (B ` \<Union> A)"
quicksort_term2: fixes r :: "nat"   and l :: "nat"   and xs :: "'b list"   and x :: "nat \<times> 'b list"   and p :: "nat"   and xs2 :: "'b list" assumes "\<not> r \<le> l"   and "\<not> length xs \<le> r"   and "x = Quicksort.partition xs l r"   and "(p, xs2) = x" shows "r - Suc p < r - l"
Fvars_fm_simps_7: fixes A :: "fm" shows "Fvars (Neg A) = Fvars A"
arr_cod: fixes f :: "'c" assumes "Cop.arr f" shows "Cop.arr (Cop.cod f)"
irreducible_dvd_degree: fixes f :: "'a poly"   and g :: "'a poly" assumes "f dvd g"   and "irreducible g"   and "0 < degree f" shows "degree f = degree g"
mono_mprefix_FD: fixes A :: "'b set"   and P :: "'b \<Rightarrow> 'b process"   and P' :: "'b \<Rightarrow> 'b process" assumes "\<forall>x\<in>A. P x \<le> P' x" shows "Mprefix A P \<le> Mprefix A P'"
l1_iagreement_Init_def: shows "l2_iagreement_Init \<equiv> {s. \<forall>A B N. signalsInit s (Commit A B N) \<le> signalsInit s (Running A B N)}"
chine_assoc_props_2: shows "fgh.Prj\<^sub>1 \<cdot> fgh.chine_assoc = fgh.Prj\<^sub>1\<^sub>1"
UNION_in_sets: fixes A :: "nat \<Rightarrow> 'a set"   and n :: "nat" assumes "range A \<subseteq> M" shows "\<Union> (A ` {0..<n}) \<in> M"
AE_all_countable: fixes P :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and M :: "'a measure" shows "(AE x in M. \<forall>i. P i x) = (\<forall>i. almost_everywhere M (P i))"
heap_upds_ok_Nil: fixes \<Gamma> :: "(Vars.var \<times> 'a) list" shows "heap_upds_ok (\<Gamma>, [])"
all_def: fixes p :: "'b \<Rightarrow> bool"   and as :: "'b iarray" shows "IArray.all p as = (\<forall>a\<in>set (IArray.list_of as). p a)"
induct: fixes P :: "\<I> \<Rightarrow> 'a option list set list \<Rightarrow> nat list \<Rightarrow> bool"   and a0 :: "\<I>"   and a1 :: "'a option list set list"   and a2 :: "nat list" assumes "\<And>I ts. P I [] ts"   and "\<And>I v va. P I (v # va) []"   and "\<And>I v va t. P I (v # va) [t]"   and "\<And>I x xs t t' ts. P I xs (t' # ts) \<Longrightarrow> P I (x # xs) (t # t' # ts)" shows "P a0 a1 a2"
ipresWls_def: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'c"   and MOD :: "('d, 'e, 'f, 'sort, 'g, 'h, 'c, 'i, 'j) Iteration.model_scheme" shows "ipresWls h MOD \<equiv> \<forall>s X. wls s X \<longrightarrow> igWls MOD s (h X)"
0: shows "rawpsubstT (rawpsubstT (substT r (Var u) x) (zip (map Var us) xs)) (zip ts us) = rawpsubstT (rawpsubstT (substT r (Var u) x) (zip (map Var vs) xs)) (zip ts vs)"
inner_sum_left: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and inner :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and f :: "'b \<Rightarrow> 'a"   and A :: "'b set"   and y :: "'a" assumes "class.real_inner minus dist norm plus zero uminus scaleR sgn uniformity open inner" shows "inner (comm_monoid_add.sum plus zero f A) y = (\<Sum>x\<in>A. inner (f x) y)"
less_matrix_def: fixes A :: "('c, 'd, 'e) matrix"   and B :: "('c, 'd, 'e) matrix" shows "A < B \<equiv> A \<le> B \<and> A \<noteq> B"