hcju/isabelleps huggingface dataset:
long_power': fixes r :: "'b list"   and x :: "'c list" assumes "r \<noteq> \<epsilon>" shows "\<^bold>|x\<^bold>| < \<^bold>|r \<^sup>@ Suc \<^bold>|x\<^bold>|\<^bold>|"
to_bool_from_bool: fixes r :: "bool" shows "to_bool (from_bool r) = r"
c_nth_def: shows "c_nth = (\<lambda>x n. c_hd (c_drop n x))"
simps_2: fixes x :: "'a"   and xs :: "'a list" shows "butlast (x # xs) = (if xs = [] then [] else x # butlast xs)"
ds_eq: shows "{d \<in> {0..<dm} \<union> ds. d < length (start dm)} = {0..<dm}"
solve_factored_ratfps': fixes fctrs :: "'a \<times> ('a \<times> nat) list"   and q :: "'a poly"   and p :: "'a poly" assumes "is_alt_factorization_of fctrs q"   and "q \<noteq> 0" shows "Abs_fps (interp_ratfps_solution (solve_factored_ratfps' p fctrs)) = fps_of_poly p / fps_of_poly q"
emb_step_hsize: fixes t :: "('a, 'b) tm"   and s :: "('a, 'b) tm" assumes "t \<rightarrow>\<^sub>e\<^sub>m\<^sub>b s" shows "hsize s < hsize t"
rb_aux_inv1_D4: fixes bs :: "('t \<Rightarrow>\<^sub>0 'b) list"   and i :: "nat" assumes "rb_aux_inv1 bs"   and "i < length bs" shows "\<not> is_sig_red (\<prec>\<^sub>t) (\<preceq>) (set (drop (Suc i) bs)) (bs ! i)"
fiter_qiter1: fixes f :: "'b iso"   and g :: "'b iso"   and h :: "'b iso" shows "Abs_iso (fiter_fun (Rep_iso f) (Rep_iso g) (Rep_iso h)) = qiter_fun f g h"
c_manifolds_axioms: shows "c_manifolds k (charts_submanifold S) charts"
assms_1: shows "\<Union> P1 \<inter> \<Union> P2 = {}"
winding_number_cindex_pathE_aux: fixes g :: "real \<Rightarrow> complex"   and z :: "complex" assumes "finite_ReZ_segments g z"   and "valid_path g"   and "z \<notin> path_image g"   and "Re (g 1) = Re z"   and "Re (g 0) = Re z" shows "2 * Re (winding_number g z) = - cindex_pathE g z"
sub_const: fixes g :: "nat \<Rightarrow> 'a"   and f :: "nat \<Rightarrow> 'a" assumes "g \<in> carrier P"   and "f \<in> carrier P"   and "deg R f = 0" shows "Cring_Poly.compose R f g = f"
aux6_var: fixes x :: "'a"   and y :: "'a" shows "(- x + y) \<cdot> x = x \<cdot> y"
rtrancl_Un_rtrancl: fixes R :: "('a \<times> 'a) set"   and S :: "('a \<times> 'a) set" shows "(R\<^sup>* \<union> S\<^sup>*)\<^sup>* = (R \<union> S)\<^sup>*"
\<pi>_le_self: fixes x :: "real" assumes "0 \<le> x" shows "\<pi> x \<le> x"
upset_set_def: fixes X :: "'a set" shows "ord.upset_set (\<lambda>x y. y \<le> x) X = {y. \<exists>x\<in>X. y \<le> x}"
assigns_var_simp: fixes v :: "Term.var" shows "assigns (In2 v) = assignsV v"
natural_transformation_axioms: shows "natural_transformation (\<cdot>\<^sub>D) (\<cdot>\<^sub>D) ((\<otimes>\<^sub>D) \<I>\<^sub>D) ((\<otimes>\<^sub>D) \<I>\<^sub>D) ((\<otimes>\<^sub>D) \<I>\<^sub>D)"
130: fixes x :: "'a" shows "- x \<squnion> - - - x = - x"
epigraph_def: fixes S :: "'c set"   and f :: "'c \<Rightarrow> real" shows "epigraph S f = {xy. fst xy \<in> S \<and> f (fst xy) \<le> snd xy}"
inv_partition_preserv: fixes S :: "'a set"   and X :: "'a set"   and P :: "'a set set"   and x :: "'a" assumes "inv_partition S X P"   and "x \<in> V - X" shows "inv_partition (insert x S) (X \<union> neighbors x \<union> {x}) (insert ({x} \<union> neighbors x) P)"
Bseq_isUb: fixes X :: "nat \<Rightarrow> real" assumes "Bseq X" shows "\<exists>U. isUb UNIV {x. \<exists>n. X n = x} U"
function_to_finmap_axioms: shows "function_to_finmap J f f'"
factorize_rat_poly_monic_irreducible_monic: fixes f :: "rat poly"   and c :: "rat"   and fs :: "(rat poly \<times> nat) list"   and fi :: "rat poly"   and i :: "nat" assumes "factorize_rat_poly_monic f = (c, fs)"   and "(fi, i) \<in> set fs" shows "irreducible fi \<and> monic fi"
TID_nonneg: fixes x :: "'a"   and y :: "'a" shows "0 \<le> turn_into_distance f x y"
compact_entailment: fixes \<Gamma> :: "'b formula set"   and F :: "'b formula" assumes "\<Gamma> \<TTurnstile> F" shows "\<exists>\<Gamma>'. finite \<Gamma>' \<and> \<Gamma>' \<subseteq> \<Gamma> \<and> \<Gamma>' \<TTurnstile> F"
atLeastatMost_empty_iff2: fixes a :: "'d DBMEntry"   and b :: "'d DBMEntry" shows "({} = linordered_monoid.atLeastAtMost a b) = (\<not> a \<preceq> b)"
alphaAbs_ex_equal_or_qFresh_def: fixes xs :: "'f"   and x :: "'g"   and X :: "('h, 'i, 'f, 'g, 'j) qTerm"   and xs' :: "'f"   and x' :: "'g"   and X' :: "('h, 'i, 'f, 'g, 'j) qTerm" shows "alphaAbs_ex_equal_or_qFresh xs x X xs' x' X' \<equiv> xs = xs' \<and> (\<exists>y. (y = x \<or> qFresh xs y X) \<and> (y = x' \<or> qFresh xs y X') \<and> X #[[y \<and> x]]_xs #= X' #[[y \<and> x']]_xs)"
2: shows "\<Union> (Range a) \<subseteq> \<Union> (Range (a - (X \<union> {i}) \<times> Range a)) \<union> \<Union> (a `` (X \<union> {i})) \<and> \<Union> (Range (a - (X \<union> {i}) \<times> Range a)) \<union> \<Union> (Range ({(i, \<Union> (a `` (X \<union> {i})))} - {(i, {})})) \<subseteq> \<Union> (Range a)"
min_elt_mem: fixes B :: "'a set" assumes "B \<subseteq> A"   and "B \<noteq> {}" shows "min_elt B \<in> B"
feasible_path_inclusion: assumes "RedBlack prb" shows "{as. \<exists>l. path (black prb) as l \<and> feasible (confs prb (root (red prb))) (trace as (labelling (black prb)))} \<subseteq> RedBlack_paths prb"
alpha_Tree_transp: shows "transp (=\<^sub>\<alpha>)"
ide_cod: fixes f :: "('a, 'a set) Maps.arr" assumes "Maps.arr f" shows "Maps.ide (Maps.cod f)"
max_mult_distrib_right: fixes x :: "'a"   and y :: "'a"   and p :: "'a" shows "max x y * p = (if (0::'a) \<le> p then max (x * p) (y * p) else min (x * p) (y * p))"
assms: shows "Term \<lbrakk>quot_dbtm u\<rbrakk>e"
disc_1: fixes n :: "enat"   and xs :: "'c llist" assumes "n = 0 \<or> lnull xs" shows "lnull (ltake n xs)"
connected_def: fixes "open" :: "'c set \<Rightarrow> bool"   and S :: "'c set" assumes "class.topological_space open" shows "topological_space.connected open S = (\<nexists>A B. open A \<and> open B \<and> S \<subseteq> A \<union> B \<and> A \<inter> B \<inter> S = {} \<and> A \<inter> S \<noteq> {} \<and> B \<inter> S \<noteq> {})"
open_greaterThan: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and "open" :: "'a set \<Rightarrow> bool"   and a :: "'a" assumes "class.order_topology less_eq less open" shows "open (ord.greaterThan less a)"
effect_deterministic_2: fixes f :: "'b Heap"   and h :: "heap"   and h' :: "heap"   and a :: "'b"   and h'' :: "heap"   and b :: "'b" assumes "effect f h h' a"   and "effect f h h'' b" shows "h' = h''"
subset_divisors_adds: fixes a :: "'a"   and b :: "'a" shows "({c. c adds a} \<subseteq> {c. c adds b}) = (a adds b)"
maxc: fixes A :: "'a set" assumes "P A" shows "A \<subseteq> S"
HNextPart_Inv1: fixes s :: "state"   and s' :: "state" assumes "HInv1 s"   and "HNextPart s s'"   and "Inv1 s'" shows "HInv1 s'"
productive_imp_produces_Max_literal: fixes C :: "'a literal multiset" assumes "production C \<noteq> {}" shows "production C = {atm_of (Max_mset C)}"
maxsimp_vertices: fixes x :: "'a set" assumes "chamber x" shows "order.greater_eq (\<Union> X) x"
vimage_Suc_insert_0: fixes A :: "nat set" shows "Suc -` insert 0 A = Suc -` A"
intro: fixes A :: "V"   and f :: "V" assumes "nop_bij A (1\<^sub>\<nat>) f" shows "unop_bij A f"
linear_continuous_on: fixes f :: "'d \<Rightarrow> 'e"   and s :: "'d set" assumes "bounded_linear f" shows "continuous_on s f"
set_intros_1: fixes x2 :: "'b" shows "x2 \<in> set_ptrm (PVar x2)"
bottomD_2: fixes P :: "'a set"   and x :: "'a" assumes "domain.has_bottom P"   and "x \<in> P" shows "x \<succeq> domain.bottom P"
m_lcancel: fixes R :: "('b, 'c) ring_scheme"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "domain R"   and "a \<noteq> \<zero>\<^bsub>R\<^esub>"   and "a \<in> carrier R"   and "b \<in> carrier R"   and "c \<in> carrier R" shows "(a \<otimes>\<^bsub>R\<^esub> b = a \<otimes>\<^bsub>R\<^esub> c) = (b = c)"
norm_eq_on_real_2_vec: fixes x :: "(real, 2) vec" shows "norm x = sqrt ((x $ 1)\<^sup>2 + (x $ 2)\<^sup>2)"
fps_of_ratfps_quot_to_ratfps: fixes y :: "'b poly"   and x :: "'b poly" assumes "constant_term y \<noteq> (0::'b)" shows "fps_of_ratfps (quot_to_ratfps (x, y)) = fps_of_poly x / fps_of_poly y"
row_space_def: fixes A :: "(('a, 'b) vec, 'c) vec" shows "row_space A = vec.span (rows A)"
L\<^sub>1_def: fixes \<phi> :: "'b ltln"   and X :: "'b ltln set" shows "L\<^sub>1 \<phi> X = {w. \<exists>i. suffix i w \<Turnstile>\<^sub>n af \<phi> (prefix i w)[X]\<^sub>\<nu>}"
pac_ideal_add: fixes p :: "int mpoly"   and A :: "int mpoly multiset"   and q :: "int mpoly" assumes "p \<in># A"   and "q \<in># A" shows "p + q \<in> pac_ideal (set_mset A)"
map_iterator_rev_linord_rule_P: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and it :: "('d \<Rightarrow> bool) \<Rightarrow> ('c \<times> 'e \<Rightarrow> 'd \<Rightarrow> 'd) \<Rightarrow> 'd \<Rightarrow> 'd"   and m :: "'c \<Rightarrow> 'e option"   and I :: "'c set \<Rightarrow> 'd \<Rightarrow> bool"   and \<sigma>0 :: "'d"   and c :: "'d \<Rightarrow> bool"   and f :: "'c \<times> 'e \<Rightarrow> 'd \<Rightarrow> 'd"   and P :: "'d \<Rightarrow> bool" assumes "class.linorder less_eq less"   and "linorder.set_iterator_map_rev_linord less_eq it (map_to_set m)"   and "I (dom m) \<sigma>0"   and "\<And>k v it \<sigma>. \<lbrakk>c \<sigma>; k \<in> it; m k = Some v; it \<subseteq> dom m; I it \<sigma>; \<And>k'. k' \<in> it \<Longrightarrow> less_eq k' k; \<And>k'. k' \<in> dom m - it \<Longrightarrow> less_eq k k'\<rbrakk> \<Longrightarrow> I (it - {k}) (f (k, v) \<sigma>)"   and "\<And>\<sigma>. I {} \<sigma> \<Longrightarrow> P \<sigma>"   and "\<And>\<sigma> it. \<lbrakk>it \<subseteq> dom m; it \<noteq> {}; \<not> c \<sigma>; I it \<sigma>; \<And>k k'. \<lbrakk>k \<in> dom m - it; k' \<in> it\<rbrakk> \<Longrightarrow> less_eq k' k\<rbrakk> \<Longrightarrow> P \<sigma>" shows "P (it c f \<sigma>0)"
p'm': shows "(case d' of L \<Rightarrow> m \<otimes> LL | R \<Rightarrow> m \<otimes> UR) \<otimes> p'm' = p'm"
Dom_preserves_Diag: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and t :: "'a local.term" assumes "monoidal_language C"   and "monoidal_language.Diag C t" shows "monoidal_language.Diag C (monoidal_language.Dom C t)"
fresh_star_set: fixes xs :: "'d set"   and c :: "'e" shows "xs \<sharp>* c \<equiv> \<forall>x\<in>xs. x \<sharp> c"
line_integral_on_reverse_path_2: fixes \<gamma> :: "real \<Rightarrow> 'a"   and F :: "'a \<Rightarrow> 'a"   and basis :: "'a set" assumes "valid_path \<gamma>"   and "line_integral_exists F basis \<gamma>" shows "line_integral_exists F basis (reversepath \<gamma>)"
implements_represents: fixes jp :: "'a \<Rightarrow> ('obs, 'aAct, 'b) Protocol" assumes "SPR.implements jp" shows "SPR.represents (SPR.jpTraces jp)"
eventually_abs_ge_0: fixes f :: "real \<Rightarrow> real" shows "\<forall>\<^sub>F x in at_top. 0 \<le> \<bar>f x\<bar>"
exec_gpv_pauses: fixes callee :: "'a \<Rightarrow> 'b \<Rightarrow> ('c \<times> 'a) spmf"   and xs :: "'b list"   and s :: "'a" shows "exec_gpv callee (pauses xs) s = map_spmf (Pair ()) (foldl_spmf (map_fun id (map_fun id (map_spmf snd)) callee) (return_spmf s) xs)"
segmentD_orbit: fixes x :: "'c"   and f :: "'c \<Rightarrow> 'c"   and y :: "'c"   and z :: "'c" assumes "x \<in> segment f y z" shows "x \<in> orbit f y"
S1: shows "c_unfold (Suc k) u = c_fst u # c_unfold k (c_snd u)"
sup_aci_1: fixes x :: "'b"   and y :: "'b" shows "x + y = y + x"
ring_hom_def: fixes R :: "('d, 'f) ring_scheme"   and S :: "('e, 'g) ring_scheme" shows "ring_hom R S = {h \<in> carrier R \<rightarrow> carrier S. (\<forall>x y. x \<in> carrier R \<and> y \<in> carrier R \<longrightarrow> h (x \<otimes>\<^bsub>R\<^esub> y) = h x \<otimes>\<^bsub>S\<^esub> h y \<and> h (x \<oplus>\<^bsub>R\<^esub> y) = h x \<oplus>\<^bsub>S\<^esub> h y) \<and> h \<one>\<^bsub>R\<^esub> = \<one>\<^bsub>S\<^esub>}"
hyps_2: fixes w :: "letter list" assumes "is_prefix (w # ws) f" shows "\<exists>i j. emb (f i) (f j) \<and> i < j"
h_def: shows "h = (\<lambda>x. max (- f x) 0)"
B_2: fixes x :: "'a" shows "norm (h x) \<le> norm x * B"
infTp_I_intT: fixes \<sigma> :: "'tp"   and a :: "univ" assumes "infTp \<sigma>" shows "I_intT \<sigma> a = Ik_intT \<sigma> a"
dom_one: shows "ad (ad (1::'b)) = (1::'b)"
norm_cvec_def: fixes z :: "complex \<times> complex" shows "\<langle>z\<rangle> = sqrt (Re \<langle>z,z\<rangle>)"
add_2_eq_Suc: fixes n :: "nat" shows "2 + n = Suc (Suc n)"
unit_unique_upto_unique_iso: shows "\<exists>!f. \<guillemotleft>f : \<I>\<^sub>D \<rightarrow>\<^sub>D \<I>\<^sub>1\<guillemotright> \<and> D.iso f \<and> f \<cdot>\<^sub>D \<iota>\<^sub>D = \<iota>\<^sub>1 \<cdot>\<^sub>D (f \<otimes>\<^sub>D f)"
exec_gpv_fixp_induct: fixes P :: "(('a, 'b, 'c) gpv \<Rightarrow> 'd \<Rightarrow> ('a \<times> 'd) spmf) \<Rightarrow> bool"   and callee :: "'d \<Rightarrow> 'b \<Rightarrow> ('c \<times> 'd) spmf" assumes "spmf.admissible (\<lambda>f. P (\<lambda>c s. f (c, s)))"   and "P (\<lambda>_ _. return_pmf None)"   and "\<And>exec_gpv. P exec_gpv \<Longrightarrow> P (\<lambda>c s. bind_spmf (the_gpv c) (case_generat (\<lambda>x. return_spmf (x, s)) (\<lambda>out c. bind_spmf (callee s out) (\<lambda>(x, y). exec_gpv (c x) y))))" shows "P (exec_gpv callee)"
reduce_system_matrix_subsets_helper_R: fixes p :: "real poly"   and signs :: "rat list list"   and subsets :: "(nat list \<times> nat list) list"   and S :: "nat list" assumes "p \<noteq> 0"   and "length signs \<le> length subsets"   and "\<forall>x. List.member S x \<longrightarrow> x < length subsets" shows "M_mat_R signs (take_indices subsets S) = take_rows_from_matrix (M_mat_R signs subsets) S"
D_imp_front_tickFree: fixes s :: "'a event list"   and P :: "'a process" assumes "s \<in> D P" shows "front_tickFree s"
span_neg: fixes K :: "('b, 'c) ring_scheme"   and V :: "('b, 'd, 'e) module_scheme"   and U :: "'d set"   and v :: "'d" assumes "vectorspace K V"   and "U \<subseteq> carrier V"   and "v \<in> LinearCombinations.module.span K V U" shows "\<ominus>\<^bsub>V\<^esub> v \<in> LinearCombinations.module.span K V U"
self_in_orbit_trans: fixes s :: "'b"   and f :: "'b \<Rightarrow> 'b"   and t :: "'b" assumes "s \<in> orbit f s"   and "t \<in> orbit f s" shows "t \<in> orbit f t"
equal_poly_mapping_def: fixes p :: "'c \<Rightarrow>\<^sub>0 'd"   and q :: "'c \<Rightarrow>\<^sub>0 'd" shows "equal_class.equal p q \<equiv> \<forall>t. lookup p t = lookup q t"
analz_Un_implInsecSet_aux: fixes G :: "msg set"   and H :: "msg set" assumes "Enc_keys_clean (G \<union> H)" shows "analz (implInsecSet_aux G \<union> H) \<subseteq> synth (analz (G \<union> H)) \<union> - payload"
simps_3: fixes x :: "nat"   and v :: "nat"   and va :: "nat list" shows "c_fold (x # v # va) = c_pair x (c_fold (v # va))"
y_sub_ys: shows "Y \<subseteq># mset ys"
preprm_inv_compose: fixes \<pi> :: "('b \<times> 'b) list" shows "preprm_compose (preprm_inv \<pi>) \<pi> =p preprm_id"
has_derivative_cinner: fixes f :: "'a \<Rightarrow> 'b"   and f' :: "'a \<Rightarrow> 'b"   and x :: "'a"   and s :: "'a set"   and g :: "'a \<Rightarrow> 'b"   and g' :: "'a \<Rightarrow> 'b" assumes "(f has_derivative f') (at x within s)"   and "(g has_derivative g') (at x within s)" shows "((\<lambda>x. cinner (f x) (g x)) has_derivative (\<lambda>h. cinner (f x) (g' h) + cinner (f' h) (g x))) (at x within s)"
SC_discr_imp_SC_WbisT: fixes indis :: "('a \<times> 'a) set"   and c :: "('b, 'c) com"   and aval :: "'c \<Rightarrow> 'a \<Rightarrow> 'a"   and tval :: "'b \<Rightarrow> 'a \<Rightarrow> bool" assumes "PL_Indis indis"   and "noWhile c"   and "PL_Indis.SC_discr aval indis c" shows "PL_Indis.SC_WbisT tval aval indis c"
groupF_concat_set: fixes f :: "'c \<Rightarrow> 'd"   and xs :: "'c list" shows "set (concat (groupF f xs)) = set xs"
equivE1_PL: fixes A :: "'a set"   and p :: "'a"   and q :: "'a" assumes "A \<turnstile> p .\<longleftrightarrow>. q" shows "A \<turnstile> p .\<longrightarrow>. q"
continuous_ge_on_Ioo: fixes c :: "real"   and d :: "real"   and g :: "real \<Rightarrow> real"   and a :: "real"   and x :: "real" assumes "continuous_on {c..d} g"   and "\<And>x. x \<in> {c<..<d} \<Longrightarrow> a \<le> g x"   and "c < d"   and "x \<in> {c..d}" shows "a \<le> g x"
comp_arr_ide: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and f :: "'b" assumes "partial_magma C"   and "partial_magma.ide C a"   and "partial_magma.arr C (C f a)" shows "C f a = f"
hull_def: fixes S :: "'c set \<Rightarrow> bool"   and s :: "'c set" shows "S hull s = \<Inter> {t. S t \<and> s \<subseteq> t}"
append_rows_nth_1: fixes A :: "'b mat"   and nr1 :: "nat"   and nc :: "nat"   and B :: "'b mat"   and nr2 :: "nat"   and i :: "nat" assumes "A \<in> carrier_mat nr1 nc"   and "B \<in> carrier_mat nr2 nc"   and "i < nr1" shows "row (A @\<^sub>r B) i = row A i"
Field_Union: fixes R :: "('b \<times> 'b) set set" shows "Field (\<Union> R) = \<Union> (Field ` R)"
is_empty_empty: fixes q :: "('c, 'd) pq" shows "PQ.is_empty q = (q = PQ.empty)"
nprv_addImpLemmaI: fixes \<phi>1 :: "'fmla"   and \<phi>2 :: "'fmla"   and F :: "'fmla set" assumes "prv (imp \<phi>1 \<phi>2)"   and "F \<subseteq> fmla"   and "finite F"   and "\<phi>1 \<in> fmla"   and "\<phi>2 \<in> fmla"   and "nprv F \<phi>1" shows "nprv F \<phi>2"
induced_subgraphI': fixes H :: "('c, 'd) pre_digraph"   and G :: "('c, 'd) pre_digraph" assumes "subgraph H G"   and "\<And>H'. subgraph H' G \<Longrightarrow> verts H' \<noteq> verts H \<or> arcs H' \<subseteq> arcs H" shows "induced_subgraph H G"
ind_topology_is_open_self: shows "ind_topology S is_open S"
Inf_empty: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "Inf {} = top"
1: shows "[\<lparr>A!,x\<^sup>P\<rparr> \<^bold>& (\<^bold>\<forall>F. \<lbrace>x\<^sup>P,F\<rbrace>) in v]"
T_on'_append: fixes step :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'a"   and t :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> nat"   and A :: "('a \<Rightarrow> 'd) \<times> ('a \<times> 'd \<Rightarrow> 'b \<Rightarrow> 'c \<times> 'd)"   and s :: "'a \<times> 'd"   and xs :: "'b list"   and ys :: "'b list" shows "On_Off.T_on' step t A s (xs @ ys) = On_Off.T_on' step t A s xs + On_Off.T_on' step t A (On_Off.config' step A s xs) ys"
distinct_103: fixes x8 :: "'a \<Rightarrow> ('a, 'b, 'c) com"   and x111 :: "('a, 'b, 'c) com"   and x112 :: "('a, 'b, 'c) com" shows "DynCom x8 \<noteq> Catch x111 x112"
inline1_right_gpv: fixes callee :: "'b \<Rightarrow> 'h \<Rightarrow> ('g \<times> 'b, 'd, 'f) gpv"   and gpv :: "('a, 'h, 'g) gpv"   and s :: "'b" shows "inline1 (\<lambda>s q. right_gpv (callee s q)) gpv s = map_spmf (map_sum id (map_prod Inr (map_prod right_rpv id))) (inline1 callee gpv s)"
local_unique_solution: fixes x0 :: "'a" assumes "0 \<in> UNIV"   and "x0 \<in> X" obtains et :: "real" and ex :: "real" and B :: "real" and L :: "real" where "0 < et" and "0 < ex" and "cball 0 et \<subseteq> UNIV" and "cball x0 ex \<subseteq> X" and "unique_on_cylinder 0 (cball 0 et) x0 ex (\<lambda>_. f) B L"
count_pre_disj: fixes B :: "'c set"   and i :: "nat"   and j :: "nat"   and Y :: "'d \<Rightarrow> 'c" assumes "countable B"   and "i \<noteq> j" shows "countable_preimages B Y i \<inter> countable_preimages B Y j = {}"
Assign4: fixes x :: "char list"   and a :: "aexp"   and Q :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool" shows "\<turnstile>\<^sub>3\<^sub>a {(\<lambda>(ps, t). x \<in> dom ps \<and> vars a \<subseteq> dom ps \<and> Q (ps(x \<mapsto> paval a ps), t)) \<and>* $ 1} x ::= a { Q}"
rep_eq: fixes x :: "'a"   and xa :: "'a" shows "bounds_of_interval (Ivl x xa) = (min x xa, xa)"
strict_trans: fixes b :: "'b list"   and a :: "'b list"   and c :: "'b list" assumes "b <s a"   and "c <s b" shows "c <s a"
trg_dom: fixes \<mu> :: "'b" assumes "B.arr \<mu>" shows "trg\<^sub>B (B.dom \<mu>) = trg\<^sub>B \<mu>"
tendsto_within_subset: fixes f :: "'b \<Rightarrow> 'c"   and l :: "'c filter"   and x :: "'b"   and S :: "'b set"   and T :: "'b set" assumes "filterlim f l (at x within S)"   and "T \<subseteq> S" shows "filterlim f l (at x within T)"
arcs_G: fixes G :: "('a, 'b) pre_digraph"   and rev_G :: "'b \<Rightarrow> 'b"   and H :: "('a, 'b) pre_digraph"   and rev_H :: "'b \<Rightarrow> 'b"   and u :: "'a"   and v :: "'a"   and w :: "'a"   and uv :: "'b"   and uw :: "'b"   and vw :: "'b" assumes "subdiv_step G rev_G H rev_H u v w uv uw vw" shows "arcs G = arcs H \<union> {uv, rev_G uv} - {uw, rev_H uw, vw, rev_H vw}"
transp_less: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "class.preorder less_eq less" shows "transp less"
entails_ex_post: fixes P :: "assn"   and Q :: "'a \<Rightarrow> assn" assumes "\<not> (P \<Longrightarrow>\<^sub>A \<exists>\<^sub>Ax. Q x)" shows "\<forall>x. \<not> (P \<Longrightarrow>\<^sub>A Q x)"
ProductSubset: fixes u :: "ZF" shows "HOLZF.subset (Product u) (Power (CartProd u (HOLZF.Sum u)))"
evalO_addO: fixes b :: "nat"   and n :: "Ord"   and m :: "Ord" shows "evalO b (addO n m) = evalO b n + evalO b m"
sum_mset_conv_sum: fixes f :: "'d \<Rightarrow> 'c"   and A :: "'d multiset" shows "\<Sum>\<^sub># (image_mset f A) = (\<Sum>x\<in>set_mset A. of_nat (count A x) * f x)"
select_convs_2: fixes inpt :: "Proc \<Rightarrow> InputsOrNi"   and outpt :: "Proc \<Rightarrow> InputsOrNi"   and disk :: "Disk \<Rightarrow> Proc \<Rightarrow> DiskBlock"   and dblock :: "Proc \<Rightarrow> DiskBlock"   and phase :: "Proc \<Rightarrow> nat"   and disksWritten :: "Proc \<Rightarrow> Disk set"   and blocksRead :: "Proc \<Rightarrow> Disk \<Rightarrow> BlockProc set"   and allInput :: "InputsOrNi set"   and chosen :: "InputsOrNi"   and more :: "'a" shows "outpt \<lparr>inpt = inpt, outpt = outpt, disk = disk, dblock = dblock, phase = phase, disksWritten = disksWritten, blocksRead = blocksRead, allInput = allInput, chosen = chosen, \<dots> = more\<rparr> = outpt"
inner_ge_zero: fixes x :: "'b" shows "0 \<le> real_of (inner x x)"
primeness_condition_monoid_def: fixes G :: "('c, 'd) monoid_scheme" shows "primeness_condition_monoid G \<equiv> comm_monoid_cancel G \<and> primeness_condition_monoid_axioms G"
rep_eq: fixes x :: "'b ttree"   and xa :: "'b ttree" shows "Rep_ttree (x \<inter>\<inter> xa) = Rep_ttree x \<inter> Rep_ttree xa"
in_set_butlast_appendI: fixes x :: "'d"   and xs :: "'d list"   and ys :: "'d list" assumes "x \<in> set (butlast xs) \<or> x \<in> set (butlast ys)" shows "x \<in> set (butlast (xs @ ys))"
SPN_r_SPN_s_apex_eq: shows "SPN_r_SPN_s.apex = Maps.MkIde (src \<rho>\<sigma>.tab)"
simps_7: fixes n :: "nat"   and r :: "'b rexp" shows "local.wf n (Star r) = local.wf n r"
leg1_uniquely_isomorphic_2: shows "\<exists>!\<nu>. \<guillemotleft>\<nu> : r\<^sub>1 \<Rightarrow> s\<^sub>1 \<star> \<mu>.chine\<guillemotright>"
horiz_vector_helper_zer_ind_R2: fixes p :: "real poly"   and x :: "real"   and sign :: "rat list"   and qs :: "real poly list"   and I :: "nat list" assumes "p \<noteq> 0"   and "x \<in> {x. poly p x = 0}"   and "sign = consistent_sign_vec qs x"   and "list_constr I (length qs)" shows "(prod_list (map ((!) sign) I) = 0) = (poly (prod_list (retrieve_polys qs I)) x = 0)"
locked_preserved: fixes s :: "v_state"   and r :: "nat"   and s' :: "v_state"   and v_f :: "process \<Rightarrow> val option" assumes "s \<in> Vinv1"   and "r = next_round s"   and "votes s' = (votes s)(r := v_f)" shows "locked s \<subseteq> locked s'"
mtf2_q_passes: fixes q :: "'a"   and xs :: "'a list"   and n :: "nat"   and x :: "'a" assumes "q \<in> set xs"   and "distinct xs"   and "index xs q - n \<le> index xs x"   and "index xs x < index xs q" shows "q < x in mtf2 n q xs"
sets_pre_sigma: fixes T :: "'a \<Rightarrow> 't" assumes "stopping_time F T" shows "sets (pre_sigma T) = {A. \<forall>t. {\<omega> \<in> A. T \<omega> \<le> t} \<in> sets (F t)}"
PSP_Stable: fixes F :: "'b program"   and A :: "'b set"   and A' :: "'b set"   and B :: "'b set" assumes "F \<in> A \<longmapsto>w A'"   and "F \<in> Stable B" shows "F \<in> A \<inter> B \<longmapsto>w A' \<inter> B"