hcju/isabelleps huggingface dataset:
simps_5: fixes n :: "nat"   and f :: "recf"   and g :: "recf" shows "encode (Pr n f g) = quad_encode 4 n (encode f) (encode g)"
le_imp_less_or_eq: fixes a :: "'a"   and b :: "'a" assumes "a \<in> carrier D"   and "b \<in> carrier D" shows "a \<preceq> b = (a \<prec> b \<or> a = b)"
inv_f_eq_iff: fixes x :: "'a"   and y :: "'b" shows "(inv (inv hom) x = y) = (x = inv hom y)"
set_remove1_subset: fixes x :: "'a"   and xs :: "'a list" shows "set (remove1 x xs) \<subseteq> set xs"
LIM_unique: fixes f :: "'b \<Rightarrow> 'c"   and L :: "'c"   and a :: "'b"   and M :: "'c" assumes "f \<midarrow>a\<rightarrow> L"   and "f \<midarrow>a\<rightarrow> M" shows "L = M"
b: shows "(x * e1 + (q - s1)) mod q = (x * y + x * q - x * b + (q * q + q * q + q - (x * y + x * (q - b) + (q - r)) mod q)) mod q"
abs_summable_on_subset: fixes f :: "'b \<Rightarrow> 'c"   and B :: "'b set"   and A :: "'b set" assumes "f abs_summable_on B"   and "A \<subseteq> B" shows "f abs_summable_on A"
xy: fixes y :: "real" assumes "x \<le> y * y"   and "y \<noteq> 0" shows "x * (2 * (y * y)) \<le> x * x + y * (y * (y * y))"
has_derivative_inverse_strong_x: fixes S :: "'a set"   and g :: "'a \<Rightarrow> 'a"   and y :: "'a"   and f :: "'a \<Rightarrow> 'a"   and f' :: "'a \<Rightarrow> 'a"   and g' :: "'a \<Rightarrow> 'a" assumes "open S"   and "g y \<in> S"   and "continuous_on S f"   and "\<And>x. x \<in> S \<Longrightarrow> g (f x) = x"   and "(f has_derivative f') (at (g y))"   and "f' \<circ> g' = id"   and "f (g y) = y" shows "(g has_derivative g') (at y)"
vconst_on_vempty: fixes c :: "V" shows "vconst_on 0 c = 0"
deadlock_no_red: fixes s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and t :: "'t"   and ta :: "('l, 't, 'x, 'm, 'w, 'o) thread_action"   and s' :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" assumes "s -t\<triangleright>ta\<rightarrow> s'"   and "deadlock s" shows "False"
ordIso_ordLeq_trans: fixes r :: "('c \<times> 'c) set"   and r' :: "('d \<times> 'd) set"   and r'' :: "('e \<times> 'e) set" assumes "r =o r'"   and "r' \<le>o r''" shows "r \<le>o r''"
line_integral_exists_on_chain1: shows "\<forall>(k, \<gamma>)\<in>one_chain1. line_integral_exists F basis \<gamma>"
assms_1: shows "m \<in>\<^sub>\<circ> \<omega>"
h_diff_le_h_plus: fixes \<Delta> :: "('a, 'b) network_scheme"   and g :: "'a \<times> 'a \<Rightarrow> ennreal"   and i :: "nat"   and e :: "'a \<times> 'a" assumes "flow_network \<Delta> g" shows "flow_network.h_diff \<Delta> g i e \<le> flow_network.h_plus \<Delta> g i e"
lead_coeff_power: fixes p :: "'b poly"   and n :: "nat" shows "lead_coeff (p ^ n) = lead_coeff p ^ n"
adjoint_transpose_left_2: fixes u :: "'a"   and v :: "'a" assumes "ide u"   and "ide v"   and "src k = trg v"   and "src k\<^sup>* = trg u" shows "k.trnl\<^sub>\<epsilon> u \<in> hom v (k\<^sup>* \<star> u) \<rightarrow> hom (k \<star> v) u"
set_disconnected_nodes_ok: fixes h :: "(_) heap"   and document_ptr :: "(_) document_ptr"   and node_ptrs :: "(_) node_ptr list" assumes "DocumentClass.type_wf h"   and "document_ptr |\<in>| document_ptr_kinds h" shows "h \<turnstile> ok set_disconnected_nodes document_ptr node_ptrs"
cong_add_trans_4: fixes x :: "'a"   and b :: "'a"   and a :: "'a"   and n :: "'a"   and y :: "'a" assumes "[x + b = a] (mod n)"   and "[x = y] (mod n)" shows "[y + b = a] (mod n)"
rref_rank: fixes A :: "(('a, 'b) vec, 'c) vec" assumes "reduced_row_echelon_form A" shows "rank A = card {row i A |i. row i A \<noteq> 0}"
r_ifle: fixes a :: "nat"   and b :: "nat"   and v\<^sub>0 :: "nat"   and v\<^sub>1 :: "nat" shows "eval r_ifle [a, b, v\<^sub>0, v\<^sub>1] \<down>= (if a \<le> b then v\<^sub>0 else v\<^sub>1)"
of_complex_of_real_eq: fixes n :: "real" shows "of_complex (complex_of_real n) = of_real n"
comp_inv: fixes f :: "int \<Rightarrow> int" assumes "f \<in> CofGroups.Ex1" shows "inv upOne \<circ> f \<in> CofGroups.Ex1"
foldD_closed: fixes B :: "'e set"   and D :: "'f set"   and f :: "'e \<Rightarrow> 'f \<Rightarrow> 'f"   and A :: "'e set"   and e :: "'f" assumes "LCD B D f"   and "finite A"   and "e \<in> D"   and "A \<subseteq> B" shows "foldD D f e A \<in> D"
assms_2: shows "w \<in> \<langle>G\<rangle>"
hyps_2: shows "Fun f T \<sqsubseteq>\<^sub>s\<^sub>e\<^sub>t M \<union> N"
WT_auth: fixes \<eta> :: "nat"   and s :: "bool list cstate" assumes "pred_cstate (\<lambda>x. length x = \<eta>) s" shows "\<I>_real \<eta> \<turnstile>c CIPHER.AUTH.auth_oracle s \<surd>"
\<i>_in_hom_2: fixes a :: "'a" assumes "B.obj a" shows "\<guillemotleft>UP \<i>\<^sub>B[a] \<cdot> cmp\<^sub>U\<^sub>P (a, a) : UP a \<star> UP a \<Rightarrow> UP a\<guillemotright>"
sat_test_def: fixes ifex_of :: "'d \<Rightarrow> 'e ifex"   and b :: "'d" shows "sat_test ifex_of b = (ifex_of b \<noteq> Falseif)"
bconf_Cast: fixes P :: "'a prog"   and sh :: "sheap"   and C :: "char list"   and e :: "'b exp"   and b :: "bool" shows "(P,sh \<turnstile>\<^sub>b (Cast C e,b) \<surd>) = (P,sh \<turnstile>\<^sub>b (e,b) \<surd>)"
less_eSuc_iff: fixes a :: "enat"   and b :: "enat" shows "(a < eSuc b) = (a \<le> b \<and> a \<noteq> \<infinity>)"
hyps_3: shows "d' \<in> {d} \<union> ds"
map_of_is_SomeI: fixes xys :: "('c \<times> 'd) list"   and x :: "'c"   and y :: "'d" assumes "distinct (map fst xys)"   and "(x, y) \<in> set xys" shows "Map.map_of xys x = Some y"
adjoint_pair_antipar_2: fixes f :: "('a S.E.term, 'a) S.arr"   and g :: "('a S.E.term, 'a) S.arr" assumes "S.adjoint_pair f g" shows "S.ide g"
mult_strict_right_mono: fixes plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and a :: "'e"   and b :: "'e"   and c :: "'e" assumes "class.linordered_semiring_strict plus zero minus less_eq less times"   and "less a b"   and "less zero c" shows "less (times a c) (times b c)"
latin_rect_iff: fixes m :: "nat"   and n :: "nat"   and s :: "(latin_type \<Rightarrow> nat) set" shows "(m \<le> n \<and> partial_latin_square s n \<and> card s = n * m \<and> (\<forall>e\<in>s. e Row < m)) = latin_rect s m n"
plane_rel_def: fixes A :: "('a \<times> 'c) set" shows "\<langle>A\<rangle>plane_rel = \<langle>A\<rangle>sctn_rel O br plane_of top"
head_prefixes: fixes list :: "'b list" shows "prefixes list ! 0 = []"
assms_2: fixes k :: "'a" assumes "k \<in> A"   and "\<And>l. l \<in> A \<Longrightarrow> k \<le> l" shows "P k"
projection_empty_implies_absence_of_events: fixes l :: "'b list"   and X :: "'b set" assumes "l \<upharpoonleft> X = []" shows "X \<inter> set l = {}"
conv_concat_labels: fixes ss1 :: "'a \<times> (bool \<times> 'b \<times> 'a) list"   and ars :: "('a \<times> 'b \<times> 'a) set"   and ss2 :: "'a \<times> (bool \<times> 'b \<times> 'a) list"   and S :: "'b set"   and T :: "'b set" assumes "ss1 \<in> conv ars"   and "ss2 \<in> conv ars"   and "set (labels_conv ss1) \<subseteq> S"   and "set (labels_conv ss2) \<subseteq> T" shows "set (labels_conv (fst ss1, snd ss1 @ snd ss2)) \<subseteq> S \<union> T"
assms_3: shows "eval_mmuaux args nt aux = (res, aux')"
cp_def: shows "cp = (\<lambda>r. cindex_pathE (poly p \<circ> (\<lambda>x. complex_of_real (linepath (- r) (of_real r) x))) 0)"
tapp: fixes \<Gamma> :: "'a \<Rightarrow> PreSimplyTyped.type option"   and f :: "'a trm"   and \<tau> :: "PreSimplyTyped.type"   and \<sigma> :: "PreSimplyTyped.type"   and x :: "'a trm" assumes "\<Gamma> \<turnstile> f : TArr \<tau> \<sigma>"   and "\<Gamma> \<turnstile> x : \<tau>" shows "\<Gamma> \<turnstile> App f x : \<sigma>"
WT_gpv_outs_gpvI: fixes \<I> :: "('d, 'e) \<I>"   and gpv :: "('f, 'd, 'e) gpv" assumes "outs_gpv \<I> gpv \<subseteq> outs_\<I> \<I>" shows "\<I> \<turnstile>g gpv \<surd>"
col_two_sides_bet: fixes A :: "'p"   and X :: "'p"   and Y :: "'p"   and B :: "'p" assumes "Col A X Y"   and "A B TS X Y" shows "Bet X A Y"
Dup_all_is_branch: fixes p :: "('b, 'c) fm"   and i :: "'c"   and branch :: "(('b, 'c) fm list \<times> 'c) list"   and xs :: "(nat \<times> nat) set" assumes "Dup p i branch xs" shows "all_is_branch p i branch xs"
independent_explicit_module: fixes s :: "'b set" shows "independent s = (\<forall>t u v. finite t \<longrightarrow> t \<subseteq> s \<longrightarrow> (\<Sum>v\<in>t. u v *s v) = (0::'b) \<longrightarrow> v \<in> t \<longrightarrow> u v = (0::'a))"
induct: fixes P :: "pat \<Rightarrow> bool"   and pat :: "pat" assumes "\<And>x1 x2. P (PVar x1 x2)"   and "\<And>x1 x2. \<lbrakk>P x1; P x2\<rbrakk> \<Longrightarrow> P \<langle>\<langle>x1, x2\<rangle>\<rangle>" shows "P pat"
fst_diag_fst: shows "fst \<circ> ((\<lambda>x. (x, x)) \<circ> fst) = fst"
assms_1: shows "subst_domain \<theta> \<inter> range_vars \<theta> = {}"
simpl_times_elements: fixes id :: "'b"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and i :: "nat"   and xs :: "'c list"   and v :: "'b list"   and a :: "'b" assumes "Matrix_Tensor.mult id f"   and "i < length xs" shows "i < length v \<longrightarrow> mult.times f a v ! i = f a (v ! i)"
SUP_apply: fixes f :: "'d \<Rightarrow> 'c \<Rightarrow> 'b"   and A :: "'d set"   and x :: "'c" shows "(\<Squnion> (f ` A)) x = (\<Squnion>y\<in>A. f y x)"
allAllocationsEquivalenceExtended: shows "soldAllocations = soldAllocations' \<and> soldAllocations' = soldAllocations'' \<and> soldAllocations'' = soldAllocations'''"
induct: fixes x1 :: "('a, 'b) ta_rule set"   and x2 :: "'b tree"   and x3 :: "'a"   and P :: "('a, 'b) ta_rule set \<Rightarrow> 'b tree \<Rightarrow> 'a \<Rightarrow> bool" assumes "accs x1 x2 x3"   and "\<And>q f qs \<delta> ts. \<lbrakk>q \<rightarrow> f qs \<in> \<delta>; length ts = length qs; \<And>i. i < length qs \<Longrightarrow> accs \<delta> (ts ! i) (qs ! i); \<And>i. i < length qs \<Longrightarrow> P \<delta> (ts ! i) (qs ! i)\<rbrakk> \<Longrightarrow> P \<delta> (NODE f ts) q" shows "P x1 x2 x3"
is_derivation_add: fixes x :: "('a \<Rightarrow> real) \<Rightarrow> real"   and p :: "'a"   and y :: "('a \<Rightarrow> real) \<Rightarrow> real" assumes "is_derivation x p"   and "is_derivation y p" shows "is_derivation (x + y) p"
plus_mwb_lens: fixes x :: "'a \<Longrightarrow> 'b"   and y :: "'c \<Longrightarrow> 'b" assumes "mwb_lens x"   and "mwb_lens y"   and "x \<bowtie> y" shows "mwb_lens (x +\<^sub>L y)"
tendsto_min: fixes a :: "nat \<Rightarrow> real"   and x :: "real"   and b :: "nat \<Rightarrow> real"   and y :: "real" assumes "a \<longlonglongrightarrow> x"   and "b \<longlonglongrightarrow> y" shows "(\<lambda>i. min (a i) (b i)) \<longlonglongrightarrow> min x y"
I': shows "I = (\<lambda>(ps, n). vars b \<subseteq> dom ps \<and> I' (ps, n))"
smooth_compat_refl: fixes k :: "enat"   and x :: "('c, 'd) chart" shows "k-smooth_compat x x"
finite_dimensionI: fixes n :: "nat"   and K :: "'a list set set"   and E :: "'a list set set" assumes "Rupt.dimension n K E" shows "Rupt.finite_dimension K E"
t2_space_quotient: fixes p :: "'c \<Rightarrow> 'b" assumes "open = openin (map_topology p euclidean)"   and "surj p"   and "\<And>X. open X \<Longrightarrow> open (p ` X)"   and "closed {(x, y). p x = p y}" shows "class.t2_space open"
x_less_exp: fixes y :: "nat"   and x :: "nat" assumes "Suc 0 < y" shows "x < y ^ x"
splitFace_holds_faceListAt_len: fixes g' :: "graph"   and v :: "nat"   and a :: "nat"   and f' :: "face"   and vs :: "nat list" assumes "pre_splitFace g' v a f' vs"   and "minGraphProps g'" shows "faceListAt_len (snd (snd (splitFace g' v a f' vs)))"
generate_eqI: fixes A :: "'a set"   and B :: "'a set" assumes "A \<subseteq> carrier G"   and "B \<subseteq> carrier G"   and "A \<subseteq> generate G B"   and "B \<subseteq> generate G A" shows "generate G A = generate G B"
xs'_def: shows "xs' \<equiv> filter (\<lambda>z. snd z \<noteq> 0) xs"
simps: fixes g :: "('c, 'd) generator"   and s :: "'d" shows "list.unfoldr g s = (if list.has_next g s then let (a, s') = list.next g s in a # list.unfoldr g s' else [])"
lex_comp': fixes x :: "'a"   and y :: "'a" shows "lex_comp' x y = comp_of_ord lex_pp (rep_nat_pp x) (rep_nat_pp y)"
the_elem_image_unique: fixes A :: "'d set"   and f :: "'d \<Rightarrow> 'e"   and x :: "'d" assumes "A \<noteq> {}"   and "\<And>y. y \<in> A \<Longrightarrow> f y = f x" shows "the_elem (f ` A) = f x"
map_sq_matrix_mult_eq_smult: fixes a :: "'b"   and M :: "('b, 'c) sq_matrix" shows "map_sq_matrix ((*) a) M = a *\<^sub>S M"
dualr_iff: fixes cl :: "'b potype"   and x :: "'b"   and y :: "'b" assumes "PO cl" shows "((x, y) \<in> order (dual cl)) = ((y, x) \<in> order cl)"
final_state_correct2: fixes ts' :: "'a list"   and ms :: "nat \<Rightarrow> 'a option"   and r :: "'b" assumes "reachable (ts', ms, r)"   and "final (ts', ms, r)" shows "r = fold_mset f start (mset ts)"
delta_snoc: fixes A :: "'c \<times> ('d \<Rightarrow> 'c \<Rightarrow> 'c) \<times> ('c \<Rightarrow> bool)"   and xs :: "'d list"   and y :: "'d"   and q :: "'c" shows "delta A (xs @ [y]) q = next A y (delta A xs q)"
VUnion_vempty_conv_right: fixes A :: "V" shows "(\<Union>\<^sub>\<circ> A = 0) = (\<forall>x\<in>elts A. x = 0)"
28423: fixes x :: "'a"   and y :: "'a" shows "- (- x \<sqinter> - y) = - (- y \<sqinter> - x)"
ncf3: fixes y :: "real" assumes "x \<le> y"   and "y \<le> 0" shows "0 < numer_cf3 y"
ptc: shows "pt TYPE('x \<Rightarrow> 'a) TYPE('x)"
cases: fixes a :: "nat set" assumes "a \<in> i_set_cont_min" obtains n :: "nat" where "a = [n__\<dots>]"   | n :: "nat" and d :: "nat" where "a = [n\<dots>,d]"
t1_space_def: fixes X :: "'c topology" shows "t1_space X \<equiv> \<forall>x\<in>topspace X. \<forall>y\<in>topspace X. x \<noteq> y \<longrightarrow> (\<exists>U. openin X U \<and> x \<in> U \<and> y \<notin> U)"
yun_wrel_div: fixes F :: "int poly"   and c :: "rat"   and f :: "rat poly"   and G :: "int poly"   and d :: "rat"   and g :: "rat poly" assumes "yun_wrel F c f"   and "yun_wrel G d g"   and "G dvd F"   and "g dvd f"   and "G \<noteq> 0" shows "yun_wrel (F div G) (c / d) (f div g)"
fminus_idemp: fixes A :: "'b fset"   and B :: "'b fset" shows "A |-| B |-| B = A |-| B"
IH: fixes gpv :: "('c, 'a, 'b) gpv"   and n :: "enat"   and s :: "'s" assumes "interaction_bounded_by consider gpv n"   and "I s"   and "\<I> \<turnstile>g gpv \<surd>" shows "set_spmf (exec_gpv' gpv s) \<subseteq> {a. case a of (x, s') \<Rightarrow> count s' \<le> n + count s}"
pre_pure_rule: fixes P :: "assn"   and b :: "bool"   and f :: "'c Heap"   and Q :: "'c \<Rightarrow> assn" shows "(\<not> <P * \<up> b> f <Q>) = (\<not> <P> f <Q> \<and> b)"
lc_monic: fixes p :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" assumes "p \<noteq> 0" shows "lcf (punit.monic p) = (1::'b)"
1: shows "C.cospan f.dom.leg0 \<mu>.dom.leg1"
unwinding_implies_isecure_total: shows "int.isecure_total"
assms_2: fixes i :: "nat" assumes "i < lo" shows "xs' ! i = xs ! i"
exFresh_s_p_cof: fixes L :: "char list set" assumes "finite L" shows "\<exists>s p. s \<notin> L \<and> p \<notin> L \<and> s \<noteq> p"
simps_2: fixes \<alpha> :: "real"   and l :: "'b tree"   and uu :: "'b"   and r :: "'b tree" assumes "WBT0 \<alpha>" shows "WBT0.wbt \<alpha> \<langle>l, uu, r\<rangle> = (WBT0.wbt \<alpha> l \<and> WBT0.wbt \<alpha> r \<and> (let ratio = real (size1 l) / real (size1 l + size1 r) in \<alpha> \<le> ratio \<and> ratio \<le> 1 - \<alpha>))"
all_events_refusal: fixes s :: "'b event list"   and P :: "'b process" assumes "(s, {tick} \<union> ev ` events_of P) \<in> F P" shows "(s, UNIV) \<in> F P"
A: shows "-l x l\<rightarrow> x l\<rightarrow> (0::'a) = (1::'a)"
left_neutral: fixes a :: "'c multiset" shows "{#} \<union># a = a"
prems_2: shows "c \<in> {0..<length (v # vs)} \<rightarrow> carrier R"
hyps_5: fixes \<rho> :: "Vars.var \<Rightarrow> Value" assumes "fv (as @ \<Gamma>, body) \<subseteq> set L \<union> domA (as @ \<Gamma>)" shows "(\<lbrace>as @ \<Gamma>\<rbrace>\<rho>) f|` domA (as @ \<Gamma>) = (\<lbrace>\<Delta>\<rbrace>\<rho>) f|` domA (as @ \<Gamma>)"
down_def: shows "down = lift (\<lambda>d l p. down' d l p 0 0)"
strict_trans: fixes a :: "nat list"   and b :: "nat list"   and c :: "nat list" assumes "a <\<^sub>v b"   and "b <\<^sub>v c" shows "a <\<^sub>v c"
aexec: shows "steps0 (s, l, r) A stpa = (Suc (length A div 2), la, ra)"
cs_not_ipd: fixes k :: "nat"   and \<pi> :: "nat \<Rightarrow> 'n"   and j :: "nat" assumes "k cd\<^bsup>\<pi>\<^esup>\<rightarrow> j \<and> ipd (\<pi> j) \<noteq> ipd (\<pi> k)" shows "cs\<^bsup>\<pi>\<^esup> (GREATEST j. k cd\<^bsup>\<pi>\<^esup>\<rightarrow> j \<and> ipd (\<pi> j) \<noteq> ipd (\<pi> k)) = filter (\<lambda>n. ipd n \<noteq> ipd (\<pi> k)) (cs\<^bsup>\<pi>\<^esup> k)"
\<sigma>'_def: shows "\<sigma>' = (\<lambda>v. Suc (v + Max (vars_clause_list (subst_cls_lists Cs (renamings_apart Cs)) \<union> {0})))"
loc_imps_fw_implication_in_M: fixes summary :: "'a \<Rightarrow> 'a \<Rightarrow> 'b antichain"   and results_in :: "'c \<Rightarrow> 'b \<Rightarrow> 'c"   and c :: "('a, 'c) configuration"   and loc1 :: "'a"   and loc2 :: "'a"   and M :: "'c zmultiset"   and xs :: "('a \<times> 'b \<times> 'a) list"   and t :: "'c" assumes "dataflow_topology summary results_in"   and "dataflow_topology.inv_imps_work_sum summary results_in c"   and "inv_implications_nonneg c"   and "dataflow_topology.loc_imps_fw summary results_in c loc1 loc2 M xs"   and "0 < zcount (c_imp c loc1) t" shows "0 < zcount M (results_in t (foldr (+) (map (\<lambda>(s, l, t). l) xs) (0::'b)))"
conv_self_conjugate: fixes x :: "'a"   and y :: "'a" shows "(x\<^sup>T \<cdot> y = 0) = (x \<cdot> y\<^sup>T = 0)"
A_def: shows "A \<equiv> (rev \<circ> sorted_list_of_set) ` Pow (set nodes)"
point_has_unique_group: fixes x :: "'a" assumes "x \<in> \<V>" shows "\<exists>!G. x \<in> G \<and> G \<in> \<G>"
bind_returns_heap_E: fixes h :: "'aq"   and f :: "('aq, 'ar, 'at) prog"   and g :: "'at \<Rightarrow> ('aq, 'ar, 'as) prog"   and h'' :: "'aq" assumes "h \<turnstile> Heap_Error_Monad.bind f g \<rightarrow>\<^sub>h h''" obtains x :: "'at" and h' :: "'aq" where "h \<turnstile> f \<rightarrow>\<^sub>r x" and "h \<turnstile> f \<rightarrow>\<^sub>h h'" and "h' \<turnstile> g x \<rightarrow>\<^sub>h h''"
only_horizontal_division_1: shows "only_horizontal_division one_chain_typeII two_chain_typeII"
fg: fixes x :: "'a"   and y :: "'b" assumes "R x y" shows "rel_pmf P (f x) (g y)"
unop_pdevs_raw_def: fixes f :: "'c \<Rightarrow> 'b"   and x :: "nat \<Rightarrow> 'c"   and i :: "nat" shows "unop_pdevs_raw f x i = (if x i = (0::'c) then 0::'b else f (x i))"
par_beta_lift: fixes t :: "dB"   and t' :: "dB"   and n :: "nat" assumes "t => t'" shows "lift t n => lift t' n"
h: shows "h \<in> Group.iso (sum_group (path_components_of X) (\<lambda>S. homology_group 0 (subtopology X S))) (homology_group 0 X)"
a1: shows "(user_seq_exe n s1 \<and> user_seq_exe n s2 \<longrightarrow> (\<exists>t1. Some t1 = SEQ n s1 \<and> (\<exists>t2. Some t2 = SEQ n s2 \<and> get_S (cpu_reg_val PSR t1) = 0 \<and> get_S (cpu_reg_val PSR t2) = 0 \<and> low_equal t1 t2))) \<and> user_seq_exe (Suc n) s1 \<and> user_seq_exe (Suc n) s2"
hyps: fixes u :: "symbol list"   and v :: "symbol list"   and w :: "symbol list" assumes "Derivation (u @ v) D w"   and "derivation_ge D (length u)" shows "Derivation v (derivation_shift D (length u) 0) (drop (length u) w)"
abs_eq: fixes y :: "32 word" shows "Abs_uint32' y = Abs_uint32 y"
y: shows "e_length b \<le> y \<and> \<phi> i y \<down>"
gd_nat_term_def: fixes pair_of_term :: "'b \<Rightarrow> 'c \<times> 'd"   and term_of_pair :: "'c \<times> 'd \<Rightarrow> 'b"   and cmp_term :: "'b nat_term_order" shows "gd_nat_term pair_of_term term_of_pair cmp_term \<equiv> gd_term pair_of_term term_of_pair (\<lambda>s t. le_of_nat_term_order cmp_term (term_of_pair (s, the_min)) (term_of_pair (t, the_min))) (\<lambda>s t. lt_of_nat_term_order cmp_term (term_of_pair (s, the_min)) (term_of_pair (t, the_min))) (le_of_nat_term_order cmp_term) (lt_of_nat_term_order cmp_term) \<and> gd_nat_term_axioms pair_of_term term_of_pair"
assms_1: shows "\<^bold>g [^] d' \<otimes> (\<^bold>g [^] y) [^] m' = \<^bold>g [^] d \<otimes> (\<^bold>g [^] y) [^] m"
list_update_same_conv: fixes i :: "nat"   and xs :: "'a list"   and x :: "'a" assumes "i \<sqsubset> length xs" shows "(xs[i := x] = xs) = (xs ! i = x)"
if_case_mvup: fixes b :: "bool"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'a"   and p :: "'b \<times> 'c"   and e :: "'a" shows "(if b then case p of (x1, x2) \<Rightarrow> f x1 x2 else e) = (case p of (x1, x2) \<Rightarrow> if b then f x1 x2 else e)"
simps_2: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> assn"   and a :: "'a"   and as :: "'a list"   and c :: "'b"   and cs :: "'b list" shows "list_assn P (a # as) (c # cs) = P a c * list_assn P as cs"
deg_fss: shows "\<forall>f\<in>set fss. 0 < degree f"
covering_lemma: fixes S :: "'a set"   and a :: "'a"   and b :: "'a"   and g :: "'a \<Rightarrow> 'a set" assumes "S \<subseteq> cbox a b"   and "box a b \<noteq> {}"   and "gauge g" obtains \<D> :: "'a set set" where "countable \<D>" and "\<Union> \<D> \<subseteq> cbox a b" and "\<And>K. K \<in> \<D> \<Longrightarrow> interior K \<noteq> {} \<and> (\<exists>c d. K = cbox c d)" and "pairwise (\<lambda>A B. interior A \<inter> interior B = {}) \<D>" and "\<And>K. K \<in> \<D> \<Longrightarrow> \<exists>x\<in>S \<inter> K. K \<subseteq> g x" and "\<And>u v. cbox u v \<in> \<D> \<Longrightarrow> \<exists>n. \<forall>i\<in>Basis. v \<bullet> i - u \<bullet> i = (b \<bullet> i - a \<bullet> i) / 2 ^ n" and "S \<subseteq> \<Union> \<D>"
is_zero_row_def': fixes i :: "'b"   and A :: "(('c, 'd) vec, 'b) vec" shows "is_zero_row i A = (\<forall>j. A $ i $ j = (0::'c))"
Zfun_sum: fixes s :: "'d set"   and f :: "'d \<Rightarrow> 'e \<Rightarrow> 'f"   and F :: "'e filter" assumes "finite s"   and "\<And>i. i \<in> s \<Longrightarrow> Zfun (f i) F" shows "Zfun (\<lambda>x. \<Sum>i\<in>s. f i x) F"
comp1: fixes prod :: "'d \<Rightarrow> 'e \<Rightarrow> 'f"   and g :: "'g \<Rightarrow> 'd" assumes "bounded_bilinear prod"   and "bounded_linear g" shows "bounded_bilinear (\<lambda>x. prod (g x))"
InitState_States: fixes A :: "('d, 'e, 'f) seqauto" shows "InitState A \<in> States A"
eval_fds_truncate: fixes n :: "nat"   and f :: "'b fds"   and s :: "'b" shows "eval_fds (fds_truncate n f) s = (\<Sum>k = 1..n. fds_nth f k / nat_power k s)"
\<Phi>_in_hom: fixes f :: "'s" assumes "S.arr f" shows "\<Phi> f \<in> S'.hom (\<Phi>o (S.dom f)) (\<Phi>o (S.cod f))"
add_eq_0_iff: fixes a :: "'c"   and b :: "'c" shows "(a + b = (0::'c)) = (b = - a)"
underS_def: fixes r :: "('d \<times> 'd) set"   and a :: "'d" shows "underS r a \<equiv> {b. b \<noteq> a \<and> (b, a) \<in> r}"
step: fixes x :: "'a"   and f :: "'a \<Rightarrow> 'a" assumes "x \<in> ccpo_class.iterates f" shows "f x \<in> ccpo_class.iterates f"
pivot_def: fixes x\<^sub>i :: "nat"   and x\<^sub>j :: "nat"   and s :: "('c, 'a) state" shows "pivot x\<^sub>i x\<^sub>j s \<equiv> \<T>_update (pivot_tableau x\<^sub>i x\<^sub>j (\<T> s)) s"
pair_bidirected_digraph_axioms: shows "pair_bidirected_digraph G"
bounded_connected_Compl_real: fixes S :: "real set" assumes "bounded S"   and "connected (- S)" shows "S = {}"
induct: fixes P :: "'a list \<Rightarrow> bool"   and a0 :: "'a list" assumes "\<And>i j cs. P (j # cs) \<Longrightarrow> P (i # j # cs)"   and "P []"   and "\<And>v. P [v]" shows "P a0"
wt_args_def: fixes i :: "nat"   and A :: "'v pvar \<Rightarrow> zhmultiset"   and \<zeta> :: "('s, 'v) hd"   and ss :: "('s, 'v) tm list" shows "wt_args i A \<zeta> ss = sum_list (map ((\<lambda>p. eval_tpoly A (map_tpoly (\<lambda>x. x) zhmset_of p)) \<circ> (\<lambda>(s, i). PMult [coef_hd \<zeta> i, wt s])) (zip ss [i..<i + length ss]))"
b: shows "\<forall>xa. b xa \<in> \<rat> \<and> x \<bullet> xa < b xa \<and> b xa - x \<bullet> xa < e'"
SetDC_def: fixes f :: "'d \<Rightarrow> bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real"   and S :: "'c \<Rightarrow> 'd set" shows "SetDC f S \<equiv> \<lambda>ab P s. INF a\<in>S s. f a ab P s"
add_order_def: fixes a :: "'a" shows "add_order a \<equiv> if \<exists>n. order.greater n 0 \<and> a +^ n = (0::'a) then LEAST n. order.greater n 0 \<and> a +^ n = (0::'a) else 0"