hcju/isabelleps huggingface dataset:
indaut_selfcomp_halftriv_f: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'b"   and C0 :: "'b set" assumes "OpposedThinChamberComplexFoldings X f g C0" shows "fixespointwise (OpposedThinChamberComplexFoldings.induced_automorphism X f g \<circ> OpposedThinChamberComplexFoldings.induced_automorphism X f g) (\<Union> (f \<turnstile> X))"
while_nfoldli: fixes c :: "'c \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'c \<Rightarrow> 'c nres"   and l :: "'d list"   and \<sigma> :: "'c" shows "WHILE\<^sub>T (FOREACH_cond c) (\<lambda>x. ASSERT (FOREACH_cond c x) \<bind> (\<lambda>_. FOREACH_body f x)) (l, \<sigma>) \<bind> (\<lambda>(uu_, \<sigma>). RETURN \<sigma>) \<le> nfoldli l c f \<sigma>"
strict_trans: fixes carrier :: "'a set"   and relation :: "('a \<times> 'a) set"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "rational_preference carrier relation"   and "x \<succeq>[relation] y \<and> (y, x) \<notin> relation"   and "y \<succeq>[relation] z \<and> (z, y) \<notin> relation" shows "x \<succeq>[relation] z \<and> (z, x) \<notin> relation"
pdevs_inner2s_def: fixes xs :: "'b pdevs list"   and a :: "real list"   and b :: "real list" shows "pdevs_inner2s xs a b = prod_of_pdevs (inner_pdevs xs a) (inner_pdevs xs b)"
nabla_omega_U: fixes x :: "'a" assumes "\<And>x y z. x \<cdot> (d y \<cdot> z) = x \<cdot> d y \<cdot> z" shows "\<nu> x\<^sup>\<omega> = \<nabla> (\<nu> x) \<cdot> U"
that: fixes q :: "'b"   and qs :: "'b list" assumes "q \<rightarrow> f qs \<in> \<delta>" shows "thesis"
atLeastLessThan0: fixes m :: "nat" shows "{m..<0} = {}"
nat_term_compD2: fixes cmp :: "'a \<Rightarrow> 'a \<Rightarrow> order"   and u :: "'a"   and v :: "'a" assumes "nat_term_comp cmp"   and "fst (rep_nat_term u) = fst (rep_nat_term v)"   and "snd (rep_nat_term u) < snd (rep_nat_term v)" shows "cmp u v = Lt"
intros_2: fixes A :: "'e set"   and a :: "'e" assumes "finite A" shows "finite (insert a A)"
fresh_star_list_cons_1: fixes xvec :: "name list"   and x :: "'a"   and xs :: "'a list" shows "xvec \<sharp>* (x # xs) = (xvec \<sharp>* x \<and> xvec \<sharp>* xs)"
subgraph_restrict: fixes G :: "('d, 'e) labeled_graph" shows "subgraph G (restrict G) = graph G"
simps: fixes h :: "heap"   and as :: "nat set" shows "in_range_assn (pHeap h as) = (\<forall>a\<in>as. a < lim h)"
mult_numeral_1_right: fixes a :: "'a" shows "a \<cdot> Numeral1 = a"
par_comm: fixes P :: "'a process"   and Q :: "'a process" shows "(P||Q) = (Q||P)"
GSMP_is_homogeneous: fixes \<A> :: "('a strand_label \<times> ('fun, 'var) strand_step) list"   and Sec :: "('fun, 'var) Term.term set"   and t :: "('fun, 'var) Term.term" assumes "\<forall>l l'. l \<noteq> l' \<longrightarrow> GSMP_disjoint (trms\<^sub>l\<^sub>s\<^sub>t (proj l \<A>)) (trms\<^sub>l\<^sub>s\<^sub>t (proj l' \<A>)) Sec"   and "t \<in> GSMP (trms\<^sub>l\<^sub>s\<^sub>t \<A>)"   and "t \<notin> Sec" shows "homogeneous\<^sub>l\<^sub>s\<^sub>t t \<A> Sec"
assms_3: shows "(i, Append i e) \<in> set (node_deliver_messages xs)"
inner_pos_move_base: fixes a :: "'b"   and b :: "'b"   and n :: "'b"   and x :: "'b"   and y :: "'b" assumes "(a - b) \<bullet> n = 0"   and "0 < (x - y) \<bullet> n"   and "y \<in> {a--b}" shows "0 < (x - a) \<bullet> n"
x'_def: shows "x' \<equiv> fst (normalize_quot (x, y))"
root_poly_Im_def: fixes p :: "int poly" shows "root_poly_Im p = (let fs = factors_of_int_poly (poly_add p (poly_uminus p)) in remdups (if \<exists>f\<in>set fs. coeff f 0 = 0 then [[:0, 1:]] else []) @ concat (map (\<lambda>f. if coeff f 0 \<noteq> 0 then [cf_pos_poly (poly_div f poly_2i)] else []) fs))"
that: fixes g :: "complex \<Rightarrow> complex" assumes "g holomorphic_on f ` S"   and "\<And>z. z \<in> S \<Longrightarrow> deriv f z * deriv g (f z) = 1"   and "\<And>z. z \<in> S \<Longrightarrow> g (f z) = z" shows "thesis"
noread_exists_change_respecting: fixes X :: "'Var set"   and c :: "'Com"   and mds :: "Mode \<Rightarrow> 'Var set"   and mem :: "'Var \<Rightarrow> 'Val"   and c' :: "'Com"   and mds' :: "Mode \<Rightarrow> 'Var set"   and mem' :: "'Var \<Rightarrow> 'Val" assumes "finite X"   and "\<langle>c, mds, mem\<rangle> \<leadsto> \<langle>c', mds', mem'\<rangle>"   and "\<forall>x\<in>X. doesnt_read c x" shows "\<exists>g. change_respecting \<langle>c, mds, mem\<rangle> \<langle>c', mds', mem'\<rangle> X g"
path_connected_openin_diff_countable: fixes S :: "'b set"   and T :: "'b set" assumes "connected S"   and "openin (top_of_set (affine hull S)) S"   and "\<not> collinear S"   and "countable T" shows "path_connected (S - T)"
span_mono: fixes scale :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and A :: "'c set"   and B :: "'c set" assumes "Modules.module scale"   and "A \<subseteq> B" shows "module.span scale A \<subseteq> module.span scale B"
continuous_closed_imp_quotient_map: fixes X :: "'c topology"   and X' :: "'d topology"   and f :: "'c \<Rightarrow> 'd" assumes "continuous_map X X' f"   and "closed_map X X' f"   and "f ` topspace X = topspace X'" shows "quotient_map X X' f"
CfgIsReachable: shows "last exec' \<turnstile> msg \<mapsto> cMsg"
assm_1: fixes i :: "'a set" assumes "i \<in> I" shows "finite (pfn i)"
fun_add_eval_car: fixes R :: "('a, 'b) ring_scheme"   and a :: "'c"   and S :: "'c set"   and f :: "'c \<Rightarrow> 'a"   and g :: "'c \<Rightarrow> 'a" assumes "ring_functions R"   and "a \<in> S"   and "f \<in> carrier (function_ring S R)"   and "g \<in> carrier (function_ring S R)" shows "(f \<oplus>\<^bsub>function_ring S R\<^esub> g) a = f a \<oplus>\<^bsub>R\<^esub> g a"
one_q: shows "q.eq_m (D * S + H * T) 1"
flow0_defined: fixes xa :: "real"   and x0 :: "'c" assumes "xa \<in> existence_ivl0 x0" shows "flow0 x0 xa \<in> Csafe"
induct: fixes P2 :: "'a hterm list \<Rightarrow> bool"   and P1 :: "'a hterm \<Rightarrow> bool"   and hterm :: "'a hterm" assumes "\<And>x1 x2. P2 x2 \<Longrightarrow> P1 (HApp x1 x2)"   and "P2 []"   and "\<And>x1 x2. \<lbrakk>P1 x1; P2 x2\<rbrakk> \<Longrightarrow> P2 (x1 # x2)" shows "P1 hterm"
chamber_morphism: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b" assumes "ChamberComplexAutomorphism X f" shows "ChamberComplexMorphism X X f"
nonneg: shows "\<forall>x\<in>A. 0 \<le> f x"
distinct_3: fixes x11 :: "'a"   and x12 :: "'b"   and x31 :: "('a, 'b, 'c) Stmt"   and x32 :: "'a ModeUpd" shows "x11 \<leftarrow> x12 \<noteq> x31@[x32]"
lcos_self: fixes x :: "Vertex \<Rightarrow> Vertex"   and H :: "(Vertex \<Rightarrow> Vertex) set" assumes "x \<in> carrier tetrahedral_group"   and "subgroup H tetrahedral_group" shows "x \<in> x <#\<^bsub>tetrahedral_group\<^esub> H"
update_by_fun_def: fixes k :: "'a"   and f :: "'b \<Rightarrow> 'b"   and xs :: "'x" shows "update_by_fun k f xs = oalist_of_list (update_by_fun_raw k f (list_of_oalist xs))"
surj_def: fixes f :: "'i \<Rightarrow> 'h" shows "surj f = (\<forall>y. \<exists>x. y = f x)"
LeftDerivationFix_grow_prefix: fixes b1 :: "symbol list"   and X :: "symbol"   and b2 :: "symbol list"   and D :: "(nat \<times> symbol \<times> symbol list) list"   and j :: "nat"   and c :: "symbol list"   and prefix :: "symbol list"   and e :: "nat"   and r :: "symbol \<times> symbol list" assumes "LeftDerivationFix (b1 @ [X] @ b2) (length b1) D j c"   and "LeftDerives1 prefix e r b1" shows "LeftDerivationFix (prefix @ [X] @ b2) (length prefix) ((e, r) # D) j c"
UN_Un: fixes M :: "'d \<Rightarrow> 'c set"   and A :: "'d set"   and B :: "'d set" shows "\<mu> (\<P> M (A \<union> B)) = \<mu> (\<P> M A) \<union> \<mu> (\<P> M B)"
inf_sup_aci_1: fixes x :: "'a"   and y :: "'a" shows "x \<sqinter> y = y \<sqinter> x"
interval_dual: fixes x :: "'a"   and y :: "'a" assumes "x \<in> A"   and "y \<in> A" shows "interval r x y = interval (order (dual cl)) y x"
zero_hom_0_axioms: shows "zero_hom_0 poly_lift"
abs_eq: fixes xa :: "'a"   and x :: "'a" shows "\<langle>xa \<leftrightarrow> x\<rangle> = Perm (Fun.swap xa x id)"
SuppA_def: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p"   and E :: "'p"   and F :: "'p" shows "A B C SuppA D E F  \<equiv> A \<noteq> B \<and> (\<exists>A'. Bet A B A' \<and> D E F CongA C B A')"
p'_def: shows "p' = map_poly (fst \<circ> quot_of_fract) (smult (to_fract (Lcm (snd ` quot_of_fract ` set (coeffs p)))) p)"
simps_5: fixes h :: "heap"   and a :: "nat" shows "typeof\<^bsub>h\<^esub> (Addr a) = (case h a of None \<Rightarrow> None | \<lfloor>(C, fs)\<rfloor> \<Rightarrow> \<lfloor>Class C\<rfloor>)"
hermitean_def: fixes A :: "complex \<times> complex \<times> complex \<times> complex" shows "hermitean A = (mat_adj A = A)"
combine_per_simp: fixes P :: "'b \<Rightarrow> bool"   and R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" shows "combine_per P R x y = (P x \<and> P y \<and> R x y)"
sum_mod_2_cases_2: shows "(f 0 + f 1) mod 2 = 1 \<longrightarrow> is_balanced"
unitary_operator_keep_trace: fixes U :: "complex mat"   and n :: "nat"   and A :: "complex mat" assumes "U \<in> carrier_mat n n"   and "A \<in> carrier_mat n n"   and "unitary U" shows "trace A = trace (adjoint U * A * U)"
dghm_is_tiny_dghm_if_ge_Limit: fixes \<beta> :: "V" assumes "\<Z> \<beta>"   and "\<alpha> \<in>\<^sub>\<circ> \<beta>" shows "\<GG> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>t\<^sub>i\<^sub>n\<^sub>y\<^bsub>\<beta>\<^esub> \<BB>"
induct: fixes P :: "'a rose_tree \<Rightarrow> bool"   and a0 :: "'a rose_tree" assumes "\<And>data subtrees. (\<And>x. x \<in> set subtrees \<Longrightarrow> P x) \<Longrightarrow> P (Tree (data, subtrees))" shows "P a0"
encR: fixes S :: "'procS"   and SRel :: "('procS \<times> 'procS) set"   and TRel :: "('procT \<times> 'procT) set" shows "SourceTerm S \<R>\<lbrakk>\<cdot>\<rbrakk>R<SRel,TRel> TargetTerm (\<lbrakk>S\<rbrakk>)"
eigenspace_def: fixes a :: "complex"   and A :: "'b \<Rightarrow>\<^sub>C\<^sub>L 'b" shows "eigenspace a A = kernel (A - a *\<^sub>C id_cblinfun)"
inv_list_succ_eq: fixes xs :: "'b list" assumes "distinct xs" shows "inv (list_succ xs) = list_succ (rev xs)"
refine: fixes E :: "'a \<Rightarrow> 'a list"   and onstack :: "('a, unit) ArrayHashMap.hashmap"   and V :: "('a, unit) ArrayHashMap.hashmap"   and u :: "'a" shows "RETURN (code_red_dfs_ahs E onstack V u) \<le> impl_red_dfs_ahs E onstack V u"
Collect_minI_ex_cut: fixes I :: "'a set"   and P :: "'a \<Rightarrow> bool" assumes "\<exists>k\<in>I. P k" shows "\<exists>x\<in>I. P x \<and> (\<forall>y\<in>I \<down>< x. \<not> P y)"
expectation_gpv_const_le: fixes \<I> :: "('h, 'i) \<I>"   and gpv :: "('j, 'h, 'i) gpv"   and fail :: "ennreal"   and c :: "ennreal" assumes "\<I> \<turnstile>g gpv \<surd>" shows "expectation_gpv fail \<I> (\<lambda>_. c) gpv \<le> max c fail"
lemma2b': fixes p :: "bool list"   and q :: "bool list"   and R :: "bool list \<times> ('b option \<Rightarrow> bool list \<Rightarrow> bool list set) \<times> (bool list \<Rightarrow> bool)"   and L :: "bool list \<times> ('b option \<Rightarrow> bool list \<Rightarrow> bool list set) \<times> (bool list \<Rightarrow> bool)" assumes "(p, q) \<in> (eps R)\<^sup>*" shows "(False # p, False # q) \<in> (eps (RegExp2NAe.conc L R))\<^sup>*"
unitary_monom_eq_var_pow: fixes K :: "'a list set"   and n :: "nat" assumes "subring K (K [X])" shows "UP.monom \<one>\<^bsub>K [X]\<^esub> n = X\<^bsub>K [X]\<^esub> [^]\<^bsub>K [X]\<^bsub>K [X]\<^esub>\<^esub> n"
ih1: fixes \<Gamma> :: "(name \<times> ty) list"   and s :: "trm"   and t :: "trm"   and u :: "trm" assumes "\<Gamma> \<turnstile> s is t : T\<^sub>1"   and "\<Gamma> \<turnstile> t is u : T\<^sub>1" shows "\<Gamma> \<turnstile> s is u : T\<^sub>1"
hyps: shows "ffact n (x + y) = (\<Sum>k = 0..n. (n choose k) * ffact k x * ffact (n - k) y)"
cases: fixes a :: "nat set" assumes "a \<in> i_set_cont" obtains n :: "nat" where "a = [n__\<dots>]"   | n :: "nat" where "a = [\<dots>na__]"   | n :: "nat" and d :: "nat" where "a = [n\<dots>,d]"
sba3_inf_complement_bot: fixes x :: "'a" shows "- x \<sqinter> x = bot"
pinf_7: fixes t :: "'b" shows "\<exists>z. \<forall>x>z. (t < x) = True"
subspace_hyperplane: fixes a :: "'b" shows "subspace {x. a \<bullet> x = 0}"
simps_3: fixes Vs :: "char list list"   and v :: "val" shows "compE\<^sub>1 Vs (Val v) = Val v"
gridgen_lgrid_eq: fixes p :: "(nat \<times> int) list"   and ds :: "nat set"   and l :: "nat" shows "set (gridgen p ds l) = lgrid p ds (level p + l)"
zero_le_imp_eq_int: fixes k :: "Quotient_Int.int" assumes "0 < k" shows "\<exists>n>0. k = of_nat n"
bij_betw_Pow: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and B :: "'b set" assumes "bij_betw f A B" shows "bij_betw ((`) f) (Pow A) (Pow B)"
ipresWls_def: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'i"   and MOD :: "('j, 'k, 'l, 'sort, 'm, 'n, 'i, 'o, 'p) model_scheme" shows "ipresWls h MOD \<equiv> \<forall>s X. wls s X \<longrightarrow> igWls MOD s (h X)"
b: fixes i :: "nat"   and l :: "nat" shows "(i < Suc l) = (i \<le> l)"
ladder_n_monotone: fixes D :: "(nat \<times> symbol \<times> symbol list) list"   and L :: "(nat \<times> nat \<times> nat) list"   and u :: "nat"   and v :: "nat" assumes "is_ladder D L"   and "u \<le> v"   and "v < length L" shows "ladder_n L u \<le> ladder_n L v"
finst_def: fixes body :: "form"   and w :: "nat" shows "finst body w = subst (\<lambda>v. case v of 0 \<Rightarrow> w | Suc n \<Rightarrow> n) body"
locally_sound_mode_use_def: fixes lc :: "(('Var, 'AExp, 'BExp) Stmt \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)" shows "locally_sound_mode_use lc = (\<forall>c' mds' mem'. \<langle>c', mds', mem'\<rangle> \<in> loc_reach lc \<longrightarrow> (\<forall>x. (x \<in> mds' GuarNoRead \<longrightarrow> doesnt_read c' x) \<and> (x \<in> mds' GuarNoWrite \<longrightarrow> doesnt_modify c' x)))"
is_gcd_def: fixes x :: "'a"   and a :: "'a"   and b :: "'a" shows "is_gcd x a b \<equiv> x dvd a \<and> x dvd b \<and> (\<forall>y. y dvd a \<longrightarrow> y dvd b \<longrightarrow> y dvd x)"
dim_col: fixes A :: "'b mat"   and i :: "nat" shows "dim_vec (col A i) = dim_row A"
max_list_Max: fixes xs :: "'a list" assumes "xs \<noteq> []" shows "max_list xs = Max (set xs)"
eq_onpI: fixes P :: "'c \<Rightarrow> bool"   and x :: "'c" assumes "P x" shows "eq_onp P x x"
mset_concat_merge_pairs: fixes key :: "'a \<Rightarrow> 'b"   and xs :: "'a list list" shows "mset (concat (merge_pairs key xs)) = mset (concat xs)"
inf_sup_absorb: fixes inf :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and sup :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and x :: "'e"   and y :: "'e" assumes "class.lattice inf less_eq less sup" shows "inf x (sup x y) = x"
is_extensional: fixes f :: "'j" assumes "\<not> J.arr f" shows "\<chi> f = S.null"
rec_add_def: shows "rec_add = PR (recf.Id 1 0) (CN S [recf.Id 3 1])"
congruent_triangleD_6: fixes a1 :: "'b"   and b1 :: "'b"   and c1 :: "'b"   and a2 :: "'c"   and b2 :: "'c"   and c2 :: "'c" assumes "congruent_triangle a1 b1 c1 a2 b2 c2" shows "dist c1 b1 = dist b2 c2"
image_mset_single: fixes f :: "'b \<Rightarrow> 'a"   and x :: "'b" shows "image_mset f (x, {#}) = f x, {#}"
lookup_filter: fixes P :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and m :: "('b, 'a) mapping"   and k :: "'b" shows "Mapping.lookup (Mapping.filter P m) k = (case Mapping.lookup m k of None \<Rightarrow> None | Some v \<Rightarrow> if P k v then Some v else None)"
intros_3: fixes m1 :: "(nat \<times> nat) convnet"   and m2 :: "(nat \<times> nat) convnet" assumes "output_size m1 = output_size m2"   and "valid_net m1"   and "valid_net m2" shows "valid_net (Pool m1 m2)"
tJ: fixes \<psi> :: "'fmla" assumes "\<psi> \<in> fmla"   and "Fvars \<psi> = {xx}" shows "tJ \<psi> \<in> trm"
conway_monoid_one: fixes x :: "('b, 'c) boffa_mon" assumes "conway_assm2 x" shows "x\<^bsub>1::'b\<^esub> = (1::'c) + x\<^bsub>1::'b\<^esub>"
notin_segment_midpoint: fixes a :: "'c"   and b :: "'c" assumes "a \<noteq> b" shows "a \<notin> closed_segment (midpoint a b) b"
first_countable_topology_ow_def: fixes U :: "'c set"   and \<tau> :: "'c set \<Rightarrow> bool" shows "first_countable_topology_ow U \<tau> \<equiv> topological_space_ow U \<tau> \<and> first_countable_topology_ow_axioms U \<tau>"
normalizer_imp_subgroup: fixes H :: "'a set" assumes "H \<subseteq> carrier G" shows "subgroup (normalizer G H) G"
merge_apply_2: fixes I :: "'b set"   and J :: "'b set"   and i :: "'b"   and x :: "'b \<Rightarrow> 'c"   and y :: "'b \<Rightarrow> 'c" assumes "I \<inter> J = {}"   and "i \<in> J" shows "merge I J (x, y) i = y i"
ins0_neq_Unbal: fixes bal_i :: "nat \<Rightarrow> nat \<Rightarrow> bool"   and t :: "'b tree"   and n :: "nat"   and a :: "'b"   and t' :: "'b tree" assumes "RBTi1 TYPE('b) bal_i"   and "size t \<le> n" shows "RBTi1.ins bal_i n 0 a t \<noteq> Unbal t'"
lb_cos_minus: fixes x :: "float"   and prec :: "nat" assumes "- pi \<le> real_of_float x"   and "real_of_float x \<le> 0" shows "cos (real_of_float (- x)) \<in> {real_of_float (lb_cos prec (- x))..real_of_float (ub_cos prec (- x))}"
swaprows_carrier: fixes k :: "nat"   and l :: "nat"   and A :: "'a mat"   and n :: "nat"   and nc :: "nat" shows "(swaprows k l A \<in> carrier_mat n nc) = (A \<in> carrier_mat n nc)"
DirProd_def: fixes G :: "('a, 'c) monoid_scheme"   and H :: "('b, 'd) monoid_scheme" shows "G \<times>\<times> H = \<lparr>carrier = carrier G \<times> carrier H, mult = \<lambda>(g, h) (g', h'). (g \<otimes>\<^bsub>G\<^esub> g', h \<otimes>\<^bsub>H\<^esub> h'), one = (\<one>\<^bsub>G\<^esub>, \<one>\<^bsub>H\<^esub>)\<rparr>"
finite_Lim_measure_decseq: fixes A :: "nat \<Rightarrow> ('s \<times> 's option) stream set"   and s :: "'s \<times> 's option" assumes "range A \<subseteq> KN.T.events s"   and "decseq A" shows "(\<lambda>n. KN.T.prob s (A n)) \<longlonglongrightarrow> KN.T.prob s (\<Inter> (range A))"
hyps_3: shows "(\<forall>q\<in>F. q \<noteq> 0) \<longrightarrow> \<Prod>F \<noteq> 0"
assms_3: fixes M :: "nat \<Rightarrow> 'a" assumes "\<And>i. P (M i)" shows "P (Sup (range M))"
zero_neq_one_def: fixes one :: "'b"   and zero :: "'b" shows "class.zero_neq_one one zero \<equiv> zero \<noteq> one"
unit_div_1_unit: fixes a :: "'a" assumes "is_unit a" shows "is_unit ((1::'a) div a)"
col_upd_col: fixes M :: "('c, 'd) sq_matrix"   and i :: "'d"   and x :: "('c, 'd) vec" shows "col (upd_col M i x) i = x"
vdef: shows "add_point x = \<V> \<union> {x}"
Cons: fixes x :: "'a"   and y :: "'a"   and xs :: "'a list" assumes "E x y"   and "steps (y # xs)" shows "steps (x # y # xs)"
tensorprod_subtensor_base: fixes f :: "'c \<Rightarrow> 'b list"   and xss :: "'c list list" shows "concat (map f (concat xss)) = concat (map (\<lambda>xs. concat (map f xs)) xss)"
assms_2: shows "d \<turnstile> ev' \<mapsto> e"
unit_factor_pCons: fixes a :: "'a"   and p :: "'a poly" shows "unit_factor (pCons a p) = (if p = 0 then [:unit_factor a:] else unit_factor p)"
ParCompose: fixes P :: "pi"   and Q :: "pi"   and R :: "pi"   and T :: "pi" assumes "P \<approx> Q"   and "(R, T) \<in> Strong_Early_Bisim.bisim O bangRel weakBisim O Strong_Early_Bisim.bisim" shows "(P \<parallel> R, Q \<parallel> T) \<in> Strong_Early_Bisim.bisim O bangRel weakBisim O Strong_Early_Bisim.bisim"
useful_tautologies_1: fixes v :: "i"   and \<phi> :: "\<o>" shows "[\<^bold>\<not>\<^bold>\<not>\<phi> \<^bold>\<rightarrow> \<phi> in v]"
pid_system_automaton'_simps_3: shows "g_V0 pid_system_automaton' = {pid_init}"
B: shows "vid_on (\<BB>\<lparr>Obj\<rparr>)\<lparr>B\<rparr> = B"
u_def: shows "u = (if gcd_nat.greater_eq m 2 then s else t)"
Par1B_2: fixes P :: "pi"   and u :: "name"   and P'' :: "pi"   and a :: "name"   and x :: "name"   and P' :: "pi"   and Q :: "pi" assumes "P \<Longrightarrow>\<^sub>lu in P''\<rightarrow>a<x> \<prec> P'"   and "x \<sharp> Q" shows "P \<parallel> Q \<Longrightarrow>\<^sub>lu in P'' \<parallel> Q\<rightarrow>a<x> \<prec> P' \<parallel> Q"
mk_linear_orders_nth: fixes r :: "'a list"   and C :: "'a set"   and B :: "'a set"   and i :: "nat" assumes "r \<in> local.mk_linear_orders C B"   and "B \<subseteq> A"   and "i < length r" shows "r ! i \<in> f (B - set (take i r))"
choiceSet_def: fixes A :: "'c set"   and r :: "('c \<times> 'c) set" shows "choiceSet A r \<equiv> {x \<in> A. \<forall>y\<in>A. x \<^bsub>r\<^esub>\<preceq> y}"
qc_wf_relto_iff: fixes R :: "('b \<times> 'b) set"   and S :: "('b \<times> 'b) set" assumes "R O S \<subseteq> (R \<union> S)\<^sup>* O R" shows "wf (S\<^sup>* O R O S\<^sup>*) = wf R"
ring_isom_def: fixes hom :: "'a \<Rightarrow> 'b" shows "ring_isom hom \<equiv> inj_ring_hom hom \<and> surjective hom"
that: fixes l :: "nat"   and j :: "nat" assumes "is_kth_obs \<pi> l j"   and "\<nexists>j'. cs\<^bsup>\<pi>\<^esup> j = cs\<^bsup>\<pi>'\<^esup> j'" shows "thesis"
homotopic_with_canon_on_empty: fixes t :: "'c set"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" shows "homotopic_with_canon (\<lambda>x. True) {} t f g"
hyps_3: shows "\<exists>P. (P, F) \<in> sumComposeSet"
simps_1: fixes d :: "nat"   and p :: "(nat \<times> int) list"   and \<alpha> :: "(nat \<times> int) list \<Rightarrow> real" shows "up' d 0 p \<alpha> = ((0, 0), \<alpha>)"
winning_region_def: fixes G :: "('c, 'd) ParityGame_scheme"   and p :: "Player" assumes "ParityGame G" shows "ParityGame.winning_region G p \<equiv> {v \<in> V\<^bsub>G\<^esub>. \<exists>\<sigma>. ParityGame.strategy G p \<sigma> \<and> ParityGame.winning_strategy G p \<sigma> v}"
subseteq_qle_def: fixes qle :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and M :: "'b set"   and N :: "'b set" assumes "quasi_order qle" shows "quasi_order.subseteq_qle qle M N \<equiv> \<forall>x\<in>M. quasi_order.in_qle qle x N"
simps_1: fixes access :: "acc_modi"   and static :: "bool"   and more :: "'a" shows "static \<lparr>access = access, static = static, \<dots> = more\<rparr> = static"
src_vcomp: fixes \<nu> :: "'a"   and \<mu> :: "'a" assumes "B.seq \<nu> \<mu>" shows "src\<^sub>B (\<nu> \<cdot>\<^sub>B \<mu>) = src\<^sub>B \<nu>"
\<phi>_F_char: fixes g :: "'b"   and y' :: "'b"   and y :: "'b" assumes "\<guillemotleft>g : y' \<rightarrow>\<^sub>B y\<guillemotright>" shows "G (F g) = \<eta>o y \<cdot>\<^sub>B g"
fresh_at_def': fixes t :: "('form, 'rule, 'c, 'd) itnode rose_tree"   and "is" :: "nat list" shows "fresh_at t is = (if is = [] then {} else freshenLC (iAnnot (tree_at t (butlast is))) ` a_fresh (inPorts' (iNodeOf (tree_at t (butlast is))) ! last is))"
iffs: fixes black :: "bool"   and more :: "'a"   and black' :: "bool"   and more' :: "'a" shows "(\<lparr>black = black, \<dots> = more\<rparr> = \<lparr>black = black', \<dots> = more'\<rparr>) = (black = black' \<and> more = more')"
walk_takeWhile_edge: fixes xs :: "'a list"   and v :: "'a" assumes "walk (xs @ [v])"   and "xs \<noteq> []"   and "hd xs \<noteq> v" shows "last (takeWhile (\<lambda>x. x \<noteq> v) xs) \<rightarrow> v"
distinct_15: fixes x31 :: "('a, 'b, 'c) Stmt"   and x32 :: "'a ModeUpd" shows "Skip \<noteq> x31@[x32]"
prime_elem_mono: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and p :: "'b"   and q :: "'b" assumes "class.algebraic_semidom divide plus minus zero times one"   and "comm_semiring_1.prime_elem times one zero p"   and "\<not> dvd.dvd times q one"   and "dvd.dvd times q p" shows "comm_semiring_1.prime_elem times one zero q"
dist: shows "distinct (map of_int_hom.vec_hom fs')"
intensional_simps_20: fixes y :: "'b \<Rightarrow> bool" shows "y \<and> y \<equiv> y"
cinner_simps_6: fixes x :: "'a"   and r :: "complex"   and y :: "'a" shows "x \<bullet>\<^sub>C r *\<^sub>C y = r * (x \<bullet>\<^sub>C y)"
props_5: shows "\<forall>(k, \<gamma>)\<in>ps1. point_path \<gamma>"
equiv_norm_Arr: fixes t :: "'c local.term" assumes "Arr t" shows "local.equiv \<^bold>\<parallel>t\<^bold>\<parallel> t"
sees_field_fun: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list"   and C :: "char list"   and F :: "char list"   and T :: "ty"   and D :: "char list"   and T' :: "ty"   and D' :: "char list" assumes "P \<turnstile> C sees F:T in D"   and "P \<turnstile> C sees F:T' in D'" shows "T' = T \<and> D' = D"