hcju/isabelleps huggingface dataset:
Qg_unit_closed: fixes N :: "'a set" assumes "G \<triangleright> N" shows "N \<in> set_rcs G N"
assms_3: shows "column k (Gram_Schmidt_column_k A (mod_type_class.to_nat k)) = column i (Gram_Schmidt_column_k A (mod_type_class.to_nat k))"
mset_exists_imply: fixes x :: "'b"   and f :: "'c \<Rightarrow> 'b"   and A :: "'c multiset" assumes "x \<in># image_mset f A" shows "\<exists>y\<in>#A. x = f y"
is_ground_subst_atm_list_member: fixes As :: "'a list"   and i :: "nat"   and \<sigma> :: "'s" assumes "is_ground_atm_list As"   and "i < length As" shows "As ! i \<cdot>a \<sigma> = As ! i"
plus_finite_2_def: fixes x :: "finite_2"   and y :: "finite_2" shows "x + y = (case (x, y) of (a\<^sub>1, a\<^sub>1) \<Rightarrow> a\<^sub>1 | (a\<^sub>1, a\<^sub>2) \<Rightarrow> a\<^sub>2 | (a\<^sub>2, a\<^sub>1) \<Rightarrow> a\<^sub>2 | (a\<^sub>2, a\<^sub>2) \<Rightarrow> a\<^sub>1)"
mem_UnitsD: fixes u :: "'a" assumes "u \<in> units.sub.Units" shows "units.sub.invertible u \<and> u \<in> Units"
asym: fixes b :: "'t \<Rightarrow>\<^sub>0 'c"   and a :: "'t \<Rightarrow>\<^sub>0 'c" assumes "ord_strict_p b a"   and "ord_strict_p a b" shows "False"
suffix_plus_com: fixes s :: "nat \<Rightarrow> 'a"   and m :: "nat"   and n :: "nat" shows "s |\<^sub>s m |\<^sub>s n = s |\<^sub>s m + n"
bigo_abs3: fixes f :: "'a \<Rightarrow> 'b" shows "O(f) = O(\<lambda>x. \<bar>f x\<bar>)"
injective_iff: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and x :: "'c"   and y :: "'c" assumes "inj_on f A"   and "x \<in> A"   and "y \<in> A" shows "(x = y) = (f x = f y)"
Cons_1: fixes ys :: "'a list"   and zs :: "'a list" assumes "list_emb P (xs @ ys) zs" shows "\<exists>us vs. zs = us @ vs \<and> list_emb P xs us \<and> list_emb P ys vs"
content_primitive_part: fixes p :: "'a poly" assumes "p \<noteq> 0" shows "content (primitive_part p) = (1::'a)"
d_IN_0: fixes x :: "'a" shows "d_IN (\<lambda>_. 0) x = 0"
not_le_imp_less: fixes y :: "'t"   and x :: "'t" assumes "\<not> y \<preceq>\<^sub>t x" shows "x \<prec>\<^sub>t y"
WCFG_CondTrue: fixes b :: "expr"   and c\<^sub>1 :: "cmd"   and c\<^sub>2 :: "cmd" shows "if (b) c\<^sub>1 else c\<^sub>2 \<turnstile> (_ 0 _) -(\<lambda>s. interpret b s = Some true)\<^sub>\<surd>\<rightarrow> (_ 0 _) \<oplus> 1"
lm12: fixes x :: "'a"   and y :: "'b"   and P :: "('a \<times> 'b) set" shows "((x, y) \<in> P) = (y \<in> P `` {x})"
solve_depressed_cubic_real_def: fixes e :: "real"   and f :: "real" shows "solve_depressed_cubic_real e f = (case solve_depressed_cubic_Cardano_real e f of None \<Rightarrow> map Re (solve_depressed_cubic_complex (cor e) (cor f)) | Some y \<Rightarrow> [y])"
distinct_indices_atoms_def: fixes as :: "('a \<times> 'b atom) set" shows "distinct_indices_atoms as = (\<forall>i a b. (i, a) \<in> as \<longrightarrow> (i, b) \<in> as \<longrightarrow> atom_var a = atom_var b \<and> atom_const a = atom_const b)"
coarser_partitions_with_list_def: fixes new_el :: "'b"   and P :: "'b set list" shows "coarser_partitions_with_list new_el P = ({new_el} # P) # map (insert_into_member_list new_el P) P"
sorted_spvec_minus_spvec: fixes v :: "(nat \<times> 'a) list" assumes "sorted_spvec v" shows "sorted_spvec (minus_spvec v)"
T_join_all: fixes ts :: "'a tree23s" shows "T_join_all ts \<le> 2 * len ts"
l5_3: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and D :: "'a"   and C :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Bet A B D"   and "Bet A C D" shows "Bet A B C \<or> Bet A C B"
lindemann_weierstrass_integral_bound: fixes C :: "real"   and u :: "complex"   and f :: "complex poly" assumes "0 \<le> C"   and "\<And>t. t \<in> closed_segment 0 u \<Longrightarrow> cmod (poly f t) \<le> C" shows "cmod (lindemann_weierstrass_aux.I f u) \<le> cmod u * exp (cmod u) * C"
find_prime_def: fixes f :: "nat \<Rightarrow> bool" shows "find_prime f = find_prime_main f 0 []"
loop_has_cd: fixes prob :: "('a, 'b, 'c, 'd) ifc_problem"   and \<pi> :: "nat \<Rightarrow> 'a"   and i :: "nat" assumes "IFC prob"   and "IFC_def.is_path prob \<pi>"   and "0 < i"   and "\<pi> i = \<pi> 0"   and "\<pi> 0 \<noteq> IFC_def.return prob" shows "\<exists>k<i. IFC_def.is_cdi prob i \<pi> k"
le_Suc_eq: fixes m :: "nat"   and n :: "nat" shows "order.greater_eq (Suc n) m = (order.greater_eq n m \<or> m = Suc n)"
remove_child_removes_first_child: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and node_ptr :: "(_) node_ptr"   and children :: "(_) node_ptr list"   and h' :: "(_) heap" assumes "heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M h"   and "type_wf h"   and "known_ptrs h"   and "h \<turnstile> get_child_nodes ptr \<rightarrow>\<^sub>r node_ptr # children"   and "h \<turnstile> remove_child ptr node_ptr \<rightarrow>\<^sub>h h'" shows "h' \<turnstile> get_child_nodes ptr \<rightarrow>\<^sub>r children"
set_child_nodes_locs_impl: shows "set_child_nodes_locs = a_set_child_nodes_locs"
mem: shows "\<bar>coeff p 0\<bar> \<in> set (map (\<lambda>i. \<bar>coeff p i\<bar>) [0..<n])"
binomial_n_0: fixes n :: "nat" shows "n choose 0 = 1"
LNil_suff: fixes A :: "'b set"   and s :: "'b llist" shows "(LNil \<in> suff A s) = (s = LNil)"
kcomp_def: fixes f :: "'c \<Rightarrow> 'e set"   and g :: "'e \<Rightarrow> 'd set" shows "f \<circ>\<^sub>K g = \<mu> \<circ> \<P> g \<circ> f"
valid2_last_Cons: fixes step :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<times> 'a"   and tr :: "('a, 'b, 'c) trans list"   and tran :: "('a, 'b, 'c) trans" assumes "IO_Automaton.valid2 step tr" shows "last (tran # tr) = last tr"
open_Times: fixes S :: "'e set"   and T :: "'f set" assumes "open S"   and "open T" shows "open (S \<times> T)"
norm: fixes k :: "nat" shows "cmod (\<Phi> k) = (if coprime k d then 1 else 0)"
prim_pyth_triple_aux: fixes v :: "nat"   and u :: "nat" assumes "v \<le> u" shows "(2 * u * v)\<^sup>2 + (u\<^sup>2 - v\<^sup>2)\<^sup>2 = (u\<^sup>2 + v\<^sup>2)\<^sup>2"
m1_inv2i_servD: fixes x :: "'a m1r_state_scheme"   and Ra :: "fid_t"   and A :: "agent"   and B :: "agent"   and Kab :: "key"   and nla :: "atom list"   and Na :: "fresh_t" assumes "runs x Ra = Some (Init, [A, B], aKey Kab # nla)"   and "Na = Ra $ na"   and "x \<in> m1_inv2i_serv"   and "A \<notin> bad" shows "\<exists>Rs. Kab = sesK (Rs $ sk) \<and> runs x Rs = Some (Serv, [A, B], [aNon Na])"
P_rec: shows "P = (\<lambda>s. Mprefix (Tr\<^sub>1 s) (\<lambda>e. P (Up\<^sub>1 s e)))"
DBM_zone_repr_def: fixes M :: "nat \<Rightarrow> nat \<Rightarrow> 'd DBMEntry"   and v :: "'c \<Rightarrow> nat"   and n :: "nat" shows "[M]\<^bsub>v,n\<^esub> = {u. u \<turnstile>\<^bsub>v,n\<^esub> M}"
ide_left: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and f :: "'b"   and g :: "'b"   and \<eta> :: "'b"   and \<epsilon> :: "'b" assumes "adjunction_data_in_bicategory V H \<a> \<i> src trg f g \<eta> \<epsilon>" shows "partial_magma.ide V f"
rd_is_CSP: fixes r :: "('c, 'd) alpha_rp_scheme \<times> ('c, 'd) alpha_rp_scheme \<Rightarrow> bool"   and p :: "('c, 'd) alpha_rp_scheme \<times> ('c, 'd) alpha_rp_scheme \<Rightarrow> bool" assumes "\<forall>a b. r (a, b\<lparr>ok := True\<rparr>) \<longrightarrow> r (a, b\<lparr>ok := False\<rparr>)" shows "is_CSP_process (R (r \<turnstile> p))"
compatible_head: fixes G :: "('a, 'b) pre_digraph"   and H :: "('a, 'b) pre_digraph" assumes "compatible G H" shows "head G = head H"
mono_det_T: fixes P :: "'a process"   and P' :: "'a process"   and S :: "'a process"   and S' :: "'a process" assumes "P \<sqsubseteq>\<^sub>T P'"   and "S \<sqsubseteq>\<^sub>T S'" shows "(P [+] S) \<sqsubseteq>\<^sub>T (P' [+] S')"
sine_law_triangle: fixes a :: "'b"   and b :: "'b"   and c :: "'b" shows "sin (angle a b c) * dist b c = sin (angle b a c) * dist a c"
not_in_net_ips_fst_init_missing: fixes i :: "nat"   and \<sigma> :: "((state \<times> 'a) \<times> 'b) net_state" assumes "i \<notin> net_ips \<sigma>" shows "fst (initmissing (netgmap fst \<sigma>)) i = toy_init i"
f5: shows "rk {A, B, C, A', B', C', P, Q} \<le> rk {P, Q, R, A', A, B, C, B', C'}"
in_subterms_eq: fixes x :: "('a, 'b) Term.term"   and y :: "('a, 'b) Term.term" assumes "x \<sqsubseteq> y"   and "y \<sqsubseteq> x" shows "subterms x = subterms y"
update_unsat_core_id: fixes s :: "('i, 'a) state"   and x :: "nat"   and c :: "'a" assumes "\<triangle> (\<T> s)"   and "\<nabla> s"   and "x \<notin> lvars (\<T> s)" shows "\<U>\<^sub>c (update x c s) = \<U>\<^sub>c s"
from_hma\<^sub>m_def: fixes a :: "(('d, 'e) Finite_Cartesian_Product.vec, 'f) Finite_Cartesian_Product.vec" shows "HMA_Connect.from_hma\<^sub>m a = Matrix.mat CARD('f) CARD('e) (\<lambda>(i, j). a $h Bij_Nat.from_nat i $h Bij_Nat.from_nat j)"
chamber_retraction2: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b"   and C :: "'b set" assumes "ChamberComplexFolding X f"   and "C \<in> f \<turnstile> ChamberComplex.chamber_system X" shows "f ` C = C"
mod_ring_inverse64: shows "(mod_ring_rel64 ===> mod_ring_rel64) (inverse_p64 pp) inverse"
filter_map: fixes P :: "'k \<Rightarrow> bool"   and f :: "'l \<Rightarrow> 'k"   and xs :: "'l list" shows "filter P (map f xs) = map f (filter (P \<circ> f) xs)"
dropWhile_eq_self_iff: fixes P :: "'c \<Rightarrow> bool"   and xs :: "'c list" shows "(dropWhile P xs = xs) = (xs = [] \<or> \<not> P (hd xs))"
cos_eq_periodic_intro: fixes x :: "real"   and y :: "real"   and k :: "int" assumes "x - y = 2 * real_of_int k * pi \<or> x + y = 2 * real_of_int k * pi" shows "cos x = cos y"
in_mset_sum_list2: fixes L :: "'b"   and Ci :: "'b multiset list" assumes "L \<in># sum_list Ci" obtains LL :: "'b multiset" where "LL \<in> set Ci" and "L \<in># LL"
single_valuedI: fixes r :: "('c \<times> 'd) set" assumes "\<And>x y z. \<lbrakk>(x, y) \<in> r; (x, z) \<in> r\<rbrakk> \<Longrightarrow> y = z" shows "single_valued r"
linear_dependent_inj_imageD: fixes f :: "('c, 'd) vec \<Rightarrow> ('c, 'e) vec"   and s :: "('c, 'd) vec set" assumes "Vector_Spaces.linear (*s) (*s) f"   and "vec.dependent (f ` s)"   and "inj_on f (vec.span s)" shows "vec.dependent s"
diff: fixes A :: "(('b, 'd) vec, 'c) vec"   and x :: "('b, 'd) vec"   and y :: "('b, 'd) vec" shows "A *v (x - y) = A *v x - A *v y"
quat_inner_j_left: fixes x :: "quat" shows "\<j> \<bullet> x = Im2 x"
A: fixes Y :: "'a set" assumes "Y \<subseteq> X"   and "Y \<in> sets A" shows "emeasure A Y \<le> emeasure C Y"
map_is_Nil_conv: fixes f :: "'d \<Rightarrow> 'c"   and xs :: "'d list" shows "(map f xs = []) = (xs = [])"
reducible\<^sub>dE: fixes p :: "'b poly" assumes "\<not> irreducible\<^sub>d p" obtains "degree p = 0"   | q :: "'b poly" and r :: "'b poly" where "0 < degree q" and "degree q < degree p" and "0 < degree r" and "degree r < degree p" and "p = q * r"
swap0_eq: fixes b :: "nat"   and i :: "nat"   and p :: "nat \<Rightarrow>\<^sub>0 'b"   and x :: "nat" shows "lookup (swap0 b i p) x = lookup p (swap b i x)"
path_cons: fixes xs :: "'a list"   and v :: "'a" assumes "path xs"   and "xs \<noteq> []"   and "v \<rightarrow> hd xs"   and "v \<notin> set xs" shows "path (v # xs)"
simps_2: fixes f :: "'a \<Rightarrow> 'a"   and x :: "'a" shows "stl (siterate f x) = siterate f (f x)"
hyps: fixes qs :: "real poly list" assumes "length qs < length qsa__" shows "p \<noteq> 0 \<and> 0 < length qs \<longrightarrow> satisfies_properties_R p qs (get_subsets_R (calculate_data_R p qs)) (get_signs_R (calculate_data_R p qs)) (get_matrix_R (calculate_data_R p qs))"
field_accessible_fromD: fixes G :: "prog"   and accC :: "qtname"   and membr :: "qtname \<times> memberdecl"   and C :: "qtname" assumes "G \<turnstile> membr of C accessible_from accC"   and "is_field membr" shows "G \<turnstile> membr member_of C \<and> G \<turnstile> Class C accessible_in pid accC \<and> G \<turnstile> membr in C permits_acc_from accC"
\<F>_eq_\<S>: fixes \<Sigma> :: "'b set set"   and w :: "nat \<Rightarrow> 'b set"   and \<G> :: "'b ltl set"   and \<psi> :: "'b ltl" assumes "finite \<Sigma>"   and "range w \<subseteq> \<Sigma>"   and "closed \<G> w"   and "G \<psi> \<in> \<G>" shows "\<forall>\<^sub>\<infinity>j. \<forall>S. (S \<Turnstile>\<^sub>P \<F> \<psi> w \<G> j \<and> \<G> \<subseteq> S) = (\<forall>q. q \<in> mojmir_def.\<S> \<Sigma> \<up>af\<^sub>G (Abs \<psi>) w {q. \<G> \<Turnstile>\<^sub>P Rep q} j \<longrightarrow> S \<Turnstile>\<^sub>P Rep q)"
apex_def: shows "trg_\<mu>.cod.apex \<equiv> C.dom trg_\<mu>.cod.leg0"
isNewArr_simps_1: fixes C :: "CTypeId" shows "isNewArr (new_instance C) = False"
other_quality_increases: fixes I :: "'a set"   and \<sigma> :: "'a \<Rightarrow> state"   and \<sigma>' :: "'a \<Rightarrow> state" assumes "other quality_increases I \<sigma> \<sigma>'" shows "\<forall>j. quality_increases (\<sigma> j) (\<sigma>' j)"
elims: fixes x :: "'edge list"   and xa :: "'edge list"   and y :: "'edge list" assumes "upd_cs x xa = y" obtains cs :: "'edge list" where "x = cs__" and "xa = []" and "y = cs__"   | cs :: "'edge list" and a :: "'edge" and as :: "'edge list" where "x = cs" and "xa = a # as" and "y = (case kind a of Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs \<Rightarrow> upd_cs (a # cs) as | Q\<hookleftarrow>\<^bsub>p\<^esub>f \<Rightarrow> case cs of [] \<Rightarrow> upd_cs cs as | c' # cs' \<Rightarrow> upd_cs cs' as | _ \<Rightarrow> upd_cs cs as)"
fbox_frame: fixes p :: "'a"   and x :: "'a"   and q :: "'a"   and r :: "'a" assumes "d p \<cdot> x \<le> x \<cdot> d p"   and "d q \<le>  |x] r" shows "d p \<cdot> d q \<le>  |x] (d p \<cdot> d r)"
final_setFinal: fixes f :: "face" shows "final (setFinal f)"
monotone_on_cmono: fixes A :: "'b set"   and B :: "'b set" assumes "A \<subseteq> B" shows "monotone_on B \<le> monotone_on A"
subgroupE_1: fixes H :: "(nat \<Rightarrow> nat) set" assumes "subgroup H (alt_group n)" shows "H \<subseteq> carrier (alt_group n)"
max_flow_min_cut'_bounded: assumes "\<And>x. \<not> edge \<Delta>'' x (source \<Delta>'')"   and "\<And>y. \<not> edge \<Delta>'' (sink \<Delta>'') y"   and "\<And>x y. edge \<Delta>'' x y \<Longrightarrow> (\<exists>z. edge \<Delta>'' y z) \<or> (\<exists>z. edge \<Delta>'' z x)"   and "\<not> edge \<Delta>'' (source \<Delta>'') (sink \<Delta>'')"   and "\<And>x. \<not> edge \<Delta>'' x x"   and "\<And>e. e \<in> \<^bold>E\<^bsub>\<Delta>''\<^esub> \<Longrightarrow> 0 < capacity \<Delta>'' e" shows "\<exists>f S. flow \<Delta>'' f \<and> MFMC_Network.cut \<Delta>'' S \<and> orthogonal \<Delta>'' f S"
contour_integrable_diff: fixes f1 :: "complex \<Rightarrow> complex"   and g :: "real \<Rightarrow> complex"   and f2 :: "complex \<Rightarrow> complex" assumes "f1 contour_integrable_on g"   and "f2 contour_integrable_on g" shows "(\<lambda>x. f1 x - f2 x) contour_integrable_on g"
monom_one: shows "monom (UP R) \<one> 0 = \<one>\<^bsub>UP R\<^esub>"
evalK_body_def: shows "evalK_body \<equiv> \<Lambda> r e. case undiscr e of const n \<Rightarrow> returnK\<cdot>(Discr n) | add e1 e2 \<Rightarrow> addK\<cdot>(r\<cdot>(Discr e1))\<cdot>(r\<cdot>(Discr e2)) | expr.disj e1 e2 \<Rightarrow> disjK\<cdot>(r\<cdot>(Discr e1))\<cdot>(r\<cdot>(Discr e2)) | fail \<Rightarrow> failK"
wf_measure: fixes f :: "'a \<Rightarrow> nat" shows "wf (Wellfounded.measure f)"
trim_alt: fixes eff :: "'a \<Rightarrow> 'b \<Rightarrow> 'b fset \<Rightarrow> bool"   and rules :: "'a stream"   and S :: "'b set"   and s :: "'b"   and rs :: "'a stream" assumes "RuleSystem eff rules S"   and "s \<in> S"   and "i.fair rules rs" shows "RuleSystem_Defs.trim eff rs s = sdrop (RuleSystem.minWait eff rs s) rs"
lossless_bind_gpvI: fixes \<I> :: "('d, 'e) \<I>"   and gpv :: "('f, 'd, 'e) gpv"   and f :: "'f \<Rightarrow> ('g, 'd, 'e) gpv" assumes "lossless_gpv \<I> gpv"   and "\<And>x. x \<in> results_gpv \<I> gpv \<Longrightarrow> lossless_gpv \<I> (f x)" shows "lossless_gpv \<I> (gpv \<bind> f)"
evenperm_unique: fixes n :: "nat"   and p :: "'b \<Rightarrow> 'b"   and b :: "bool" assumes "swapidseq n p"   and "even n = b" shows "evenperm p = b"
upt: shows "{0..<Suc (length N)} = insert 0 {1..<Suc (length N)}"
H_genus: shows "pre_digraph_map.euler_genus H HM = 0"
simps: fixes a1 :: "'effect fs_set"   and a2 :: "('b, 'pred, 'act, 'effect) formula" shows "(a2 \<in> \<A>[a1]) = ((\<exists>xset F. a1 = F \<and> a2 = Conj xset \<and> finite (supp xset) \<and> (\<forall>x. x \<in> set_bset xset \<longrightarrow> x \<in> \<A>[F])) \<or> (\<exists>F x. a1 = F \<and> a2 = FL_Formula.Not x \<and> x \<in> \<A>[F]) \<or> (\<exists>f F \<phi>. a1 = F \<and> a2 = Pred f \<phi> \<and> f \<in>\<^sub>f\<^sub>s F) \<or> (\<exists>f F \<alpha> x. a1 = F \<and> a2 = Act f \<alpha> x \<and> f \<in>\<^sub>f\<^sub>s F \<and> bn \<alpha> \<sharp>* (F, f) \<and> x \<in> \<A>[L (\<alpha>, F, f)]))"
inject_2: fixes x2 :: "'c"   and y2 :: "'c" shows "(Lt x2 = Lt y2) = (x2 = y2)"
IH_2: fixes Y :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm" assumes "qSkel Y = qSkel X" shows "\<forall>X'. Y #= X' \<longrightarrow> X' #= Y"
properfactor_mult_rI: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "properfactor G a b"   and "a \<in> carrier G"   and "c \<in> carrier G" shows "properfactor G (a \<otimes> c) (b \<otimes> c)"
semiring_simprules_9: fixes R :: "('e, 'f) ring_scheme"   and x :: "'e" assumes "semiring R"   and "x \<in> carrier R" shows "\<one>\<^bsub>R\<^esub> \<otimes>\<^bsub>R\<^esub> x = x"
freshen_closed: fixes f :: "'form"   and a :: "nat" assumes "lconsts f = {}" shows "freshen a f = f"
exi_def: fixes F :: "('state \<times> 'aprop set) stream \<Rightarrow> ('state \<times> 'aprop set) stream list \<Rightarrow> bool"   and \<pi>l :: "('state \<times> 'aprop set) stream list" shows "exi F \<pi>l \<equiv> \<exists>\<pi>. wfp AP' \<pi> \<and> stateOf \<pi> = (if \<pi>l \<noteq> [] then stateOf (last \<pi>l) else s0) \<and> F \<pi> \<pi>l"
inst: fixes \<phi> :: "'fmla"   and t :: "'trm" assumes "\<phi> \<in> fmla"   and "t \<in> trm" shows "inst \<phi> t \<in> fmla"
zero_neq_numeral: fixes one :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and n :: "num" assumes "class.semiring_char_0 one times plus zero" shows "zero \<noteq> numeral.numeral one plus n"
1: shows "\<forall>f K. finite {(c, K d) |c d. 0 < v c \<and> v c \<le> n \<and> f M (v c) = K d}"
Abs_dnelist_inject: fixes x :: "'a list"   and y :: "'a list" assumes "x \<in> {xs. distinct xs \<and> xs \<noteq> []}"   and "y \<in> {xs. distinct xs \<and> xs \<noteq> []}" shows "(Abs_dnelist x = Abs_dnelist y) = (x = y)"
Abs: fixes r :: "lam"   and r' :: "lam"   and x :: "name" assumes "r \<rightarrow>\<^sub>l r'" shows "(Lam [x].r) \<rightarrow>\<^sub>l (Lam [x].r')"
dom_def: fixes f :: "'a2" shows "C2.dom f = (if C2.domains f \<noteq> {} then SOME a. a \<in> C2.domains f else C2.null)"
wfactors_cong_r: fixes fs :: "'a list"   and a :: "'a"   and a' :: "'a" assumes "wfactors G fs a"   and "a \<sim> a'"   and "a \<in> carrier G"   and "a' \<in> carrier G"   and "set fs \<subseteq> carrier G" shows "wfactors G fs a'"
extr_mono_chan: fixes G :: "chan set"   and H :: "chan set"   and bad :: "agent set"   and IK :: "msg set" assumes "G \<subseteq> H" shows "extr bad IK G \<subseteq> extr bad IK H"
cfun_eq_iff: fixes f :: "'c \<rightarrow> 'd"   and g :: "'c \<rightarrow> 'd" shows "(f = g) = (\<forall>x. f\<cdot>x = g\<cdot>x)"
approx_closed_set_with_set_zero_measure_boundary: fixes M :: "'a measure"   and S :: "'a set"   and epsilon :: "real" assumes "finite_borel_measure' M"   and "closed S"   and "0 < epsilon"   and "S \<noteq> {}" shows "\<exists>r<epsilon. 0 < r \<and> Sigma_Algebra.measure M {x. infdist x S = r} = 0 \<and> Sigma_Algebra.measure M {x. infdist x S \<le> r} < Sigma_Algebra.measure M S + epsilon"
LFP_fixed_point: fixes L :: "('a, 'b) gorder_scheme"   and f :: "'a \<Rightarrow> 'a" assumes "weak_complete_lattice L"   and "Mono\<^bsub>L\<^esub> f"   and "f \<in> carrier L \<rightarrow> carrier L" shows "LFP\<^bsub>L\<^esub> f \<in> fps L f"
lta__lea: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p"   and E :: "'p"   and F :: "'p" assumes "A B C LtA D E F" shows "A B C LeA D E F"
otherassms_6: shows "(\<xi>, pp) \<in> reachable A I"
simps_10: fixes t :: "'c local.term"   and u :: "'c local.term"   and v :: "'c local.term" shows "\<^bold>\<lfloor>\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[t, u, v\<^bold>]\<^bold>\<rfloor> = \<^bold>\<lfloor>t\<^bold>\<rfloor> \<^bold>\<lfloor>\<^bold>\<otimes>\<^bold>\<rfloor> \<^bold>\<lfloor>u\<^bold>\<rfloor> \<^bold>\<lfloor>\<^bold>\<otimes>\<^bold>\<rfloor> \<^bold>\<lfloor>v\<^bold>\<rfloor>"
assms_3: shows "g \<in> B \<rightarrow> A"
order_less_le_subst2: fixes a :: "'c"   and b :: "'c"   and f :: "'c \<Rightarrow> 'd"   and c :: "'d" assumes "a < b"   and "f b \<le> c"   and "\<And>x y. x < y \<Longrightarrow> f x < f y" shows "f a < c"
continuous_compose_quotient_map_eq: fixes X :: "'d topology"   and X' :: "'e topology"   and f :: "'d \<Rightarrow> 'e"   and X'' :: "'f topology"   and g :: "'e \<Rightarrow> 'f" assumes "quotient_map X X' f" shows "continuous_map X X'' (g \<circ> f) = continuous_map X' X'' g"
dividesE: fixes G :: "('c, 'd) monoid_scheme"   and a :: "'c"   and b :: "'c" assumes "a divides\<^bsub>G\<^esub> b" obtains c :: "'c" where "b = a \<otimes>\<^bsub>G\<^esub> c" and "c \<in> carrier G"
term_templateInduct_fresh: fixes vars :: "'varSort \<Rightarrow> 'var set"   and terms :: "('a, 'b, 'varSort, 'var, 'c) qTerm set set"   and abs :: "('a, 'b, 'varSort, 'var, 'c) qAbs set set"   and envs :: "('varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option) set"   and rel :: "(('a, 'b, 'varSort, 'var, 'c) qTerm set \<times> ('a, 'b, 'varSort, 'var, 'c) qTerm set) set"   and phi :: "('a, 'b, 'varSort, 'var, 'c) qTerm set \<Rightarrow> bool"   and phiAbs :: "('a, 'b, 'varSort, 'var, 'c) qAbs set \<Rightarrow> bool"   and X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and A :: "('a, 'b, 'varSort, 'var, 'c) qAbs set" assumes "\<And>xs. (|vars xs| <o |UNIV| \<or> finite (vars xs)) \<and> (|terms| <o |UNIV| \<or> finite terms) \<and> Ball terms good \<and> (|abs| <o |UNIV| \<or> finite abs) \<and> Ball abs goodAbs \<and> (|envs| <o |UNIV| \<or> finite envs) \<and> Ball envs goodEnv"   and "\<And>X Y. \<lbrakk>good X; (X, Y) \<in> rel\<rbrakk> \<Longrightarrow> good Y \<and> skel Y = skel X"   and "\<And>xs x. phi (Var xs x)"   and "\<And>delta inp binp. \<lbrakk>goodInp inp; goodBinp binp; liftAll phi inp; liftAll phiAbs binp\<rbrakk> \<Longrightarrow> phi (Op delta inp binp)"   and "\<And>xs x X. \<lbrakk>good X; x \<notin> vars xs; \<And>Y. Y \<in> terms \<Longrightarrow> fresh xs x Y; \<And>A. A \<in> abs \<Longrightarrow> freshAbs xs x A; \<And>rho. rho \<in> envs \<Longrightarrow> freshEnv xs x rho; \<And>Y. (X, Y) \<in> rel \<Longrightarrow> phi Y\<rbrakk> \<Longrightarrow> phiAbs (Abs xs x X)" shows "(good X \<longrightarrow> phi X) \<and> (goodAbs A \<longrightarrow> phiAbs A)"
simps: fixes n :: "nat"   and l :: "cell list"   and r :: "cell list" shows "inv_loop5 n (l, r) = (inv_loop5_loop n (l, r) \<or> inv_loop5_exit n (l, r))"
simp: shows "Abs \<tau> (mk_eq' \<tau>' (f $ Bv 0) (g $ Bv 0)) \<bullet> Fv x \<tau> = mk_eq (f $ Fv x \<tau>) (g $ Fv x \<tau>)"
f'_def: shows "f' \<equiv> \<lambda>y. \<integral>\<^sup>+ x. f <|x, y|> \<partial>stock_measure t1"
exhaust: fixes y :: "('a, 'b) FingerTreeStruc" obtains "y = Empty"   | x2 :: "('a, 'b) Node" where "y = Single x2"   | x31 :: "'b" and x32 :: "('a, 'b) Digit" and x33 :: "('a, 'b) FingerTreeStruc" and x34 :: "('a, 'b) Digit" where "y = Deep x31 x32 x33 x34"
P0: shows "Cring_Poly.compose R f g = b \<odot>\<^bsub>P\<^esub> g [^]\<^bsub>P\<^esub> n"
cases: fixes r :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and a1 :: "'k"   and a2 :: "'k" assumes "r\<^sup>*\<^sup>* a1 a2" obtains a :: "'k" where "a1 = aa__" and "a2 = aa__"   | a :: "'k" and b :: "'k" and c :: "'k" where "a1 = a" and "a2 = c" and "r\<^sup>*\<^sup>* a b" and "r b c"