hcju/isabelleps huggingface dataset:
comp_ide_arr: fixes b :: "'a"   and f :: "'a" assumes "H.ide b"   and "H.seq b f" shows "b \<star> f = f"
simps_13: fixes p :: "nat"   and a :: "floatarith"   and vs :: "((real \<times> real pdevs) \<times> real) list" shows "approx_floatarith p (Sqrt a) vs = approx_floatarith p a vs \<bind> sqrt_aform_err p"
substitutes_varE': fixes x :: "'a"   and y :: "'a"   and M :: "'a trm"   and X :: "'a trm" assumes "substitutes (Var x) y M X" obtains "Var xa__ = Unit" and "X = Unit"   | "Var xa__ = Var ya__" and "X = M"   | x :: "'a" where "Var xa__ = Var x" and "X = Var x" and "x \<noteq> ya__"   | A :: "'a trm" and A' :: "'a trm" and B :: "'a trm" and B' :: "'a trm" where "Var xa__ = App A__ B__" and "X = App A'__ B'__" and "substitutes A__ ya__ M A'__" and "substitutes B__ ya__ M B'__"   | y :: "'a" and A :: "'a trm" and A' :: "'a trm" and T :: "PreSimplyTyped.type" where "Var xa__ = Fn y T Aa__" and "X = Fn y T A'a__" and "ya__ \<noteq> y" and "y \<notin> fvs M" and "substitutes Aa__ ya__ M A'a__"   | A :: "'a trm" and A' :: "'a trm" and B :: "'a trm" and B' :: "'a trm" where "Var xa__ = SimplyTyped.Pair A B" and "X = SimplyTyped.Pair A' B'" and "substitutes A ya__ M A'" and "substitutes B ya__ M B'"   | P :: "'a trm" and P' :: "'a trm" where "Var xa__ = Fst P__" and "X = Fst P'__" and "substitutes P__ ya__ M P'__"   | P :: "'a trm" and P' :: "'a trm" where "Var xa__ = Snd P" and "X = Snd P'" and "substitutes P ya__ M P'"
llength_LNil: shows "llength []\<^sub>l = 0"
real_root_ge_zero: fixes x :: "real"   and n :: "nat" assumes "0 \<le> x" shows "0 \<le> root n x"
simps_2: fixes x :: "'b equation" shows "positive_literal (Neg x) = False"
assms_3: shows "(a, b) \<in> set (ordered_map S)"
retract_of_closed: fixes T :: "'a set"   and S :: "'a set" assumes "closed T"   and "S retract_of T" shows "closed S"
matrix_to_iarray_fst_Gauss_Jordan_column_k_det_P: fixes i :: "nat"   and A :: "(('b, 'c) vec, 'd) vec"   and k :: "nat"   and n :: "'b" assumes "i \<le> nrows A"   and "k < ncols A" shows "fst (Gauss_Jordan_column_k_det_P (n, i, A) k) = fst (Gauss_Jordan_column_k_det_P_iarrays (n, i, matrix_to_iarray A) k)"
sxalloc_gext: fixes G :: "prog"   and s1 :: "state"   and s2 :: "state" assumes "G\<turnstile>s1 \<midarrow>sxalloc\<rightarrow> s2" shows "snd s1\<le>|snd s2"
isometry_on_cong: fixes X :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "isometry_on X f"   and "\<And>x. x \<in> X \<Longrightarrow> g x = f x" shows "isometry_on X g"
var_imp_symvar_var: fixes v :: "'c"   and e :: "('c \<Rightarrow> 'd) \<Rightarrow> 'd"   and s :: "'c \<Rightarrow> nat" assumes "v \<in> Aexp.vars e" shows "symvar v s \<in> Aexp.vars (adapt_aexp e s)"
subclseq_C_m: shows "G\<turnstile>C\<preceq>\<^sub>C declclass m"
mann4: fixes y :: "'a"   and z :: "'a"   and x :: "'a" shows "- (y \<squnion> z) = - (- (- (- (- x \<squnion> y) \<squnion> x \<squnion> y \<squnion> y) \<squnion> - (- x \<squnion> y) \<squnion> - (y \<squnion> z) \<squnion> z) \<squnion> z)"
restrict_nonempty_left: fixes k :: "'c"   and ks :: "'c list"   and f :: "'c \<times> 'c \<Rightarrow> 'd"   and ls :: "'c list" shows "(k # ks) \<langle>f\<rangle> ls = [k] \<langle>f\<rangle> ls \<oplus> ks \<langle>f\<rangle> ls"
acyclic_irreflexive_star_antisymmetric: fixes x :: "'a" shows "pd_kleene_allegory_class.acyclic x = (irreflexive x \<and> antisymmetric (x\<^sup>\<star>))"
Mp_unique_factorization_m: fixes f :: "int poly"   and cfs :: "int \<times> int poly multiset" shows "q.unique_factorization_m (q.Mp f) cfs = q.unique_factorization_m f cfs"
cases: fixes b :: "nat"   and a :: "C" assumes "hbase_extp b a" obtains n :: "C" where "a = na__" and "hbasep b na__"   | n :: "C" and m :: "C list" where "a = C (n # m)" and "hbase_extp b n" and "hbasep b (C m)" and "\<And>m'. m' \<in> set m \<Longrightarrow> evalC b n \<le> evalC b m'"
add_mono_ennreal: fixes x :: "ennreal"   and y :: "real"   and x' :: "ennreal"   and y' :: "real" assumes "x < ennreal y"   and "x' < ennreal y'" shows "x + x' < ennreal (y + y')"
Cd_domain: fixes d :: "'b"   and X :: "'a set" shows "Cd d (X \<inter> Field (Pd d)) = Cd d X"
Max_in: fixes A :: "'a set" assumes "finite A"   and "A \<noteq> {}" shows "Max A \<in> A"
loc_imps_fw_M_in_implications: fixes c :: "('loc, 't) configuration"   and loc1 :: "'loc"   and loc2 :: "'loc"   and M :: "'t zmultiset"   and xs :: "('loc \<times> 'sum \<times> 'loc) list"   and t :: "'t" assumes "loc_imps_fw c loc1 loc2 M xs"   and "inv_imps_work_sum c"   and "inv_implications_nonneg c"   and "\<And>loc. c_work c loc = {#}\<^sub>z"   and "0 < zcount M t" shows "\<exists>s\<le>t. s \<in>\<^sub>A frontier (c_imp c loc2)"
eq: shows "mset_set {insert x g |g. g \<in> \<G>} = image_mset (insert x) (mset_set \<G>)"
sdom_union: fixes m :: "'c \<Rightarrow> 'd set"   and n :: "'c \<Rightarrow> 'd set" shows "sdom m \<union>. n = sdom m \<union> sdom n"
1: fixes f :: "('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" assumes "f \<notin> F'" shows "q f = 0"
before_final: fixes tp :: "cell list \<times> cell list"   and A :: "(action \<times> nat) list"   and n :: "nat"   and tp' :: "cell list \<times> cell list" assumes "steps0 (1, tp) A n = (0, tp')" shows "\<exists>n'. \<not> is_final (steps0 (1, tp) A n') \<and> steps0 (1, tp) A (Suc n') = (0, tp')"
transfer_rule_of_int: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool" assumes "R (0::'b) (0::'c)"   and "R (1::'b) (1::'c)"   and "(R ===> R ===> R) (+) (+)"   and "(R ===> R) uminus uminus" shows "((=) ===> R) of_int of_int"
simps_1: fixes signals :: "signal \<Rightarrow> nat"   and more :: "'a" shows "signals \<lparr>signals = signals, \<dots> = more\<rparr> = signals"
t_def: shows "t \<equiv> [:- a, 1::'a:]"
eval_times: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and is_expansion :: "'a \<Rightarrow> (real \<Rightarrow> real) list \<Rightarrow> bool"   and expansion_level :: "'a itself \<Rightarrow> nat"   and eval :: "'a \<Rightarrow> real \<Rightarrow> real"   and zero_expansion :: "'a"   and const_expansion :: "real \<Rightarrow> 'a"   and powr_expansion :: "bool \<Rightarrow> 'a \<Rightarrow> real \<Rightarrow> 'a"   and power_expansion :: "bool \<Rightarrow> 'a \<Rightarrow> nat \<Rightarrow> 'a"   and trimmed :: "'a \<Rightarrow> bool"   and dominant_term :: "'a \<Rightarrow> real \<times> real list"   and F :: "'a"   and G :: "'a" assumes "class.multiseries divide inverse minus plus times uminus is_expansion expansion_level eval zero_expansion const_expansion powr_expansion power_expansion trimmed dominant_term" shows "eval (times F G) = (\<lambda>x. eval F x * eval G x)"
c: assumes "x \<noteq> y"   and "\<not> rel x y" shows "filter P (merge_wrt rel (x # xs) ys) = filter P ys"
SIod_self_le: fixes D :: "'c Order"   and E :: "'c Order" assumes "Order D"   and "Order E"   and "carrier E \<subseteq> carrier D"   and "\<forall>a\<in>carrier E. \<forall>b\<in>carrier E. a \<preceq>\<^bsub>E\<^esub> b = a \<preceq>\<^bsub>D\<^esub> b" shows "E = SIod D (carrier E)"
If_cases: fixes A :: "'b set"   and P :: "'b \<Rightarrow> bool"   and h :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "finite A" shows "(\<Sum>x\<in>A. if P x then h x else g x) = sum h (A \<inter> {x. P x}) + sum g (A \<inter> - {x. P x})"
sum_B: shows "\<forall>x\<in>A. (\<lambda>y. f (x, y)) abs_summable_on B x"
f'': fixes tx :: "real \<times> 'a" assumes "tx \<in> T \<times> X" shows "(f' has_derivative ($) (f'' tx)) (at tx)"
inv_inv: fixes x :: "'b" assumes "x \<in> carrier (G i)" shows "inv\<^bsub>G i\<^esub> (inv\<^bsub>G i\<^esub> x) = x"
strategyproofness_1: shows "(pmf (sds R28) b + pmf (sds R28) a < pmf (sds R39) c + pmf (sds R39) a \<or> pmf (sds R28) b + (pmf (sds R28) a + pmf (sds R28) d) < pmf (sds R39) c + (pmf (sds R39) a + pmf (sds R39) d)) \<or> pmf (sds R28) b + pmf (sds R28) a = pmf (sds R39) c + pmf (sds R39) a \<and> pmf (sds R28) b + (pmf (sds R28) a + pmf (sds R28) d) = pmf (sds R39) c + (pmf (sds R39) a + pmf (sds R39) d)"
induct: fixes P :: "Arity list \<Rightarrow> stack_elem list \<Rightarrow> bool"   and a0 :: "Arity list"   and a1 :: "stack_elem list" assumes "\<And>uu_. P uu_ []"   and "\<And>a as e1 e2 S. P as S \<Longrightarrow> P (a # as) (Alts e1 e2 # S)"   and "\<And>as x S. P as S \<Longrightarrow> P as (Upd x # S)"   and "\<And>as x S. P as S \<Longrightarrow> P as (Arg x # S)"   and "\<And>v va S. P [] S \<Longrightarrow> P [] (Alts v va # S)"   and "\<And>v S. P [] S \<Longrightarrow> P [] (Dummy v # S)"   and "\<And>as v S. P as S \<Longrightarrow> P as (Dummy v # S)" shows "P a0 a1"
ideI: fixes a :: "'c local.term set" assumes "\<F>C.arr a \<and> (\<F>C.ide (\<F>C.dom a) \<and> Diag (\<F>C.DOM (\<F>C.dom a))) \<and> \<F>C.ide (\<F>C.cod a) \<and> Diag (\<F>C.DOM (\<F>C.cod a))"   and "\<F>C.ide a" shows "ide a"
leg1_simps_4: shows "trg rs.p\<^sub>1 = trg ((tab\<^sub>0 r)\<^sup>* \<star> tab\<^sub>1 s)"
rcosets_carrier: fixes X :: "'a set" assumes "Group.group G"   and "X \<in> rcosets H" shows "X \<subseteq> carrier G"
one_side_not_col124: fixes A :: "'p"   and B :: "'p"   and X :: "'p"   and Y :: "'p" assumes "A B OS X Y" shows "\<not> Col A B Y"
seq_char: fixes g :: "'c"   and f :: "'c" shows "Cop.seq g f = C.seq f g"
dec_unclocked_ignore: fixes ck :: "bool"   and mn :: "char list list"   and env :: "v sem_env"   and s :: "'b state"   and d :: "dec"   and res :: "'b state \<times> (v sem_env, v) result" assumes "evaluate_dec ck mn env s d res" shows "\<forall>s' r count. res = (s', r) \<and> r \<noteq> Rerr (Rabort Rtimeout_error) \<longrightarrow> evaluate_dec False mn env (update_clock (\<lambda>_. count) s) d (update_clock (\<lambda>_. count) s', r)"
carrier_direct_lim_def: shows "carrier_direct_lim \<equiv> rel.Partition"
brc_rcm_init_def: fixes \<delta> :: "('a, 'b) ta_rule dlist" shows "brc_rcm_init \<delta> \<equiv> ls.iterate \<delta> (\<lambda>r. hm.update r (length (remdups (rhsq r)))) (hm.empty ())"
induct: fixes P1 :: "trm \<Rightarrow> bool"   and P3 :: "(char list \<times> trm) list \<Rightarrow> bool"   and P2 :: "char list \<times> trm \<Rightarrow> bool"   and trm :: "trm" assumes "\<And>x. P1 (Var x)"   and "\<And>x1 x2. P1 x2 \<Longrightarrow> P1 (\<lambda>:x1. x2)"   and "\<And>x1 x2. P1 x2 \<Longrightarrow> P1 (\<lambda><:x1. x2)"   and "\<And>x1 x2. \<lbrakk>P1 x1; P1 x2\<rbrakk> \<Longrightarrow> P1 (x1 \<bullet> x2)"   and "\<And>x1 x2. P1 x1 \<Longrightarrow> P1 (x1 \<bullet>\<^sub>\<tau> x2)"   and "\<And>x. P3 x \<Longrightarrow> P1 (Rcd x)"   and "\<And>x1 x2. P1 x1 \<Longrightarrow> P1 (x1..x2)"   and "\<And>x1 x2 x3. \<lbrakk>P1 x2; P1 x3\<rbrakk> \<Longrightarrow> P1 (LET x1 = x2 IN x3)"   and "\<And>x1 x2. P1 x2 \<Longrightarrow> P2 (x1, x2)"   and "P3 []"   and "\<And>x1 x2. \<lbrakk>P2 x1; P3 x2\<rbrakk> \<Longrightarrow> P3 (x1 \<Colon> x2)" shows "P1 trm"
subcls1I: fixes P :: "'b prog"   and C :: "char list"   and D :: "char list"   and rest :: "fdecl list \<times> 'b mdecl list" assumes "class P C = \<lfloor>(D, rest)\<rfloor>"   and "C \<noteq> Object" shows "subcls1p P C D"
union: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and z :: "'a"   and M :: "'a multiset"   and N :: "'a multiset" assumes "comm_monoid_mset f z" shows "comm_monoid_mset.F f z (M + N) = f (comm_monoid_mset.F f z M) (comm_monoid_mset.F f z N)"
type_definition_def: fixes Rep :: "'d \<Rightarrow> 'e"   and Abs :: "'e \<Rightarrow> 'd"   and A :: "'e set" shows "type_definition Rep Abs A \<equiv> (\<forall>x. Rep x \<in> A) \<and> (\<forall>x. Abs (Rep x) = x) \<and> (\<forall>y. y \<in> A \<longrightarrow> Rep (Abs y) = y)"
boffa_var: fixes x :: "'a" assumes "x ; x \<le> x" shows "x\<^sup>\<star> = 1' + x"
lfinite_inf_llist: fixes f :: "nat \<Rightarrow> 'c" shows "\<not> lfinite (inf_llist f)"
homeomorphic_space_refl: fixes X :: "'b topology" shows "X homeomorphic_space X"
B_2: fixes n :: "nat" shows "\<exists>N. C n \<inter> {N..} \<subseteq> B"
add_eq_0_iff_both_eq_0: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.canonically_ordered_monoid_add plus zero less_eq less" shows "(plus x y = zero) = (x = zero \<and> y = zero)"
unity_def: shows "\<I>\<^sub>1 \<equiv> cod \<iota>\<^sub>1"
assms_2: shows "\<forall>x\<in>#M. \<exists>y\<in>#N. (x, y) \<in> r"
hm_it_init_rule: fixes m :: "'a \<Rightarrow> 'b option"   and ht :: "('a, 'b) hashtable" shows "<is_hashmap m ht> hm_it_init ht <hm_is_it m ht m>\<^sub>t"
ringE_1: fixes R :: "('e, 'f) ring_scheme" assumes "ring R" shows "abelian_group R"
is_rewrite_ordD3: fixes pair_of_term :: "'c \<Rightarrow> 'd \<times> nat"   and term_of_pair :: "'d \<times> nat \<Rightarrow> 'c"   and ord :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_strict :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_term :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and ord_term_strict :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and fs :: "('d \<Rightarrow>\<^sub>0 'e) list"   and rword :: "'c \<times> ('d \<Rightarrow>\<^sub>0 'e) \<Rightarrow> 'c \<times> ('d \<Rightarrow>\<^sub>0 'e) \<Rightarrow> bool"   and a :: "'c \<times> ('d \<Rightarrow>\<^sub>0 'e)"   and b :: "'c \<times> ('d \<Rightarrow>\<^sub>0 'e)" assumes "qpm_inf_term pair_of_term term_of_pair ord ord_strict ord_term ord_term_strict"   and "qpm_inf_term.is_rewrite_ord pair_of_term term_of_pair ord ord_strict ord_term ord_term_strict fs rword" obtains "rword a b"   | "\<not> rword a b" and "rword b a"
dom_iso: fixes x :: "'a"   and y :: "'a" assumes "x \<le> y" shows "d x \<le> d y"
adt_finite_wgraph_axioms: shows "adt_finite_wgraph G.invar G_succ G_empty G_add_edge G.\<alpha>"
qTermLess_ingoreFirst2_wf: shows "wf (inv_image qTermLess aux_qFresh_ignoreFirst2)"
shiftr_integer_conv_div_pow2: fixes x :: "integer"   and n :: "nat" shows "x >> n = x div 2 ^ n"
MakeForm_def: fixes y :: "hf"   and u :: "hf"   and w :: "hf" shows "MakeForm y u w \<equiv> y = q_Disj u w \<or> y = q_Neg u \<or> (\<exists>v u'. AbstForm v 0 u u' \<and> y = q_Ex u')"
hyp2_incident_perp_foot_same_point: fixes a :: "proj2"   and l :: "proj2_line" assumes "a \<in> hyp2"   and "proj2_incident a l" shows "perp_foot a l = a"
ifSomeE: fixes c :: "bool"   and x :: "'c"   and y :: "'c" assumes "(if c then Some x else None) = Some y" obtains "c" and "x = y"
rel_gpv_colossless_gpvD2: fixes A :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and C :: "'h \<Rightarrow> 'i \<Rightarrow> bool"   and gpv :: "('f, 'h, 'j) gpv"   and gpv' :: "('g, 'i, 'j) gpv"   and \<I>' :: "('i, 'j) \<I>"   and \<I> :: "('h, 'j) \<I>" assumes "rel_gpv A C gpv gpv'"   and "colossless_gpv \<I>' gpv'"   and "rel_\<I> C (=) \<I> \<I>'" shows "colossless_gpv \<I> gpv"
dirichlet_prod_mangoldt1_floor_bigo: shows "(\<lambda>x. dirichlet_prod' (\<lambda>n. ind prime n * ln (real n)) (\<lambda>x. real_of_int \<lfloor>x\<rfloor>) x - x * ln x) \<in> O(\<lambda>x. x)"
\<mu>: shows "arr \<mu> \<and> \<guillemotleft>\<mu> : local.dom \<mu> \<Rightarrow> cod \<mu>\<guillemotright> \<and> local.iso \<mu> \<and> \<guillemotleft>local.inv \<mu> : cod \<mu> \<Rightarrow> local.dom \<mu>\<guillemotright>"
pprod_def: fixes xs :: "'b list"   and k :: "nat"   and i :: "nat"   and j :: "nat" shows "pprod xs k i j = prod_list (map (xd xs k) [i..<j])"
hom_nat_pow: fixes x :: "nat \<Rightarrow> 'a"   and n :: "nat" assumes "x \<in> carrier P" shows "Eval (x [^]\<^bsub>P\<^esub> n) = Eval x [^]\<^bsub>S\<^esub> n"
representation_basis: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and basis :: "'b set"   and b :: "'b" assumes "Modules.module scale"   and "\<not> module.dependent scale basis"   and "b \<in> basis" shows "module.representation scale basis b = (\<lambda>v. if v = b then 1::'a else (0::'a))"
wf_norm_ACI: fixes idx :: "'n"   and \<phi> :: "('a, 'k) aformula" shows "local.wf idx \<langle>\<phi>\<rangle> = local.wf idx \<phi>"
sum_rel_id: fixes Rl :: "('c \<times> 'c) set"   and Rr :: "('d \<times> 'd) set" assumes "Rl = Id"   and "Rr = Id" shows "\<langle>Rl, Rr\<rangle>sum_rel = Id"
image_diff_atLeastAtMost: fixes d :: "'b"   and a :: "'b"   and b :: "'b" shows "(-) d ` {a..b} = {d - b..d - a}"
simps_2: fixes x :: "'g"   and xs :: "'g list" shows "remdups (x # xs) = (if x \<in> set xs then remdups xs else x # remdups xs)"
prf'_reduction_def: fixes \<A> :: "(bool, '\<beta>, '\<gamma>) gpv" shows "prf'_reduction \<A> = lift_spmf seed_gen \<bind> (\<lambda>seed. inline (reduction_oracle seed) \<A> () \<bind> (\<lambda>(b, \<sigma>). Generative_Probabilistic_Value.Done b))"
TryThrow: fixes P :: "(char list list \<times> char list exp) prog"   and e\<^sub>1 :: "char list exp"   and s\<^sub>0 :: "heap \<times> sfields \<times> sheap"   and a :: "nat"   and h\<^sub>1 :: "heap"   and l\<^sub>1 :: "sfields"   and sh\<^sub>1 :: "sheap"   and D :: "char list"   and fs :: "Objects.fields"   and C :: "char list"   and V :: "char list"   and e\<^sub>2 :: "char list exp" assumes "P \<turnstile> \<langle>e\<^sub>1,s\<^sub>0\<rangle> \<Rightarrow> \<langle>Throw a,(h\<^sub>1, l\<^sub>1, sh\<^sub>1)\<rangle>"   and "h\<^sub>1 a = \<lfloor>(D, fs)\<rfloor>"   and "(D, C) \<notin> (subcls1 P)\<^sup>*" shows "P \<turnstile> \<langle>try e\<^sub>1 catch(C V) e\<^sub>2,s\<^sub>0\<rangle> \<Rightarrow> \<langle>Throw a,(h\<^sub>1, l\<^sub>1, sh\<^sub>1)\<rangle>"
reflp_def: fixes r :: "'e \<Rightarrow> 'e \<Rightarrow> bool" shows "reflp r = (\<forall>x. r x x)"
flatten_dom_def: shows "flatten_dom = Heap_Error_Monad.bind (Heap_Error_Monad.bind element_ptr_kinds_M (map_filter_M2 (\<lambda>element_ptr. Heap_Error_Monad.bind (get_tag_name element_ptr) (\<lambda>tag. Heap_Error_Monad.bind (assigned_nodes element_ptr) (\<lambda>assigned_nodes. if tag = ''slot'' \<and> assigned_nodes \<noteq> [] then return (Some (element_ptr, assigned_nodes)) else return None))))) (\<lambda>tups. Heap_Error_Monad.bind (forall_M (\<lambda>(slot, assigned_nodes). Heap_Error_Monad.bind (Heap_Error_Monad.bind (get_child_nodes (cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r slot)) (forall_M remove)) (\<lambda>_. forall_M (append_child (cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r slot)) assigned_nodes)) tups) (\<lambda>_. Heap_Error_Monad.bind (Heap_Error_Monad.bind shadow_root_ptr_kinds_M (forall_M (\<lambda>shadow_root_ptr. Heap_Error_Monad.bind (get_host shadow_root_ptr) (\<lambda>host. Heap_Error_Monad.bind (Heap_Error_Monad.bind (get_child_nodes (cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r host)) (forall_M remove)) (\<lambda>_. Heap_Error_Monad.bind (Heap_Error_Monad.bind (get_child_nodes (cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r shadow_root_ptr)) (forall_M (append_child (cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r host)))) (\<lambda>_. remove_shadow_root host)))))) (\<lambda>_. return ())))"
l\<^sub>1inc\<^sub>1: shows "l\<^sub>1 \<in> C\<^sub>1"
face_of_affine_eq: fixes S :: "'b set"   and T :: "'b set" assumes "affine S" shows "(T face_of S) = (T = {} \<or> T = S)"
singular_relboundary_add: fixes p :: "nat"   and X :: "'b topology"   and S :: "'b set"   and a :: "((nat \<Rightarrow> real) \<Rightarrow> 'b) \<Rightarrow>\<^sub>0 int"   and b :: "((nat \<Rightarrow> real) \<Rightarrow> 'b) \<Rightarrow>\<^sub>0 int" assumes "singular_relboundary p X S a"   and "singular_relboundary p X S b" shows "singular_relboundary p X S (a + b)"
V_def: shows "V \<equiv> {u. \<exists>v. (u, v) \<in> E \<or> (v, u) \<in> E}"
mtype_f_def: fixes P :: "cld list"   and ty :: "ty"   and m :: "char list" shows "mtype_f P ty m = (case find_meth_def_f P ty m of None \<Rightarrow> None | Some (ctx, meth_def_def (meth_sig_def cl m' vds) mb) \<Rightarrow> case find_type_f P ctx cl of None \<Rightarrow> None | Some ty' \<Rightarrow> case lift_opts (map (\<lambda>vd. case vd of vd_def clk vark \<Rightarrow> find_type_f P ctx clk) vds) of None \<Rightarrow> None | Some tys \<Rightarrow> Some (mty_def tys ty'))"
tPB_def: shows "tPB = tC ` \<Phi> \<union> Fax \<union> Wax"
measure_lub_spmf: fixes A :: "'a set" assumes "Y \<noteq> {}" shows "Sigma_Algebra.measure (measure_spmf local.lub_spmf) A = (\<Squnion>y\<in>Y. Sigma_Algebra.measure (measure_spmf y) A)"
satisfies_tableau_Cons: fixes v :: "nat \<Rightarrow> 'a"   and t :: "(nat \<times> linear_poly) list"   and e :: "nat \<times> linear_poly" assumes "v \<Turnstile>\<^sub>t t"   and "v \<Turnstile>\<^sub>e e" shows "v \<Turnstile>\<^sub>t (e # t)"
I2d: fixes s :: "state"   and s' :: "state" assumes "HNext s s'"   and "HInv1 s \<and> HInv2 s \<and> HInv2 s' \<and> HInv4 s" shows "HInv4 s'"
Dube_is_hom_GB_bound: fixes X :: "'a option set"   and F :: "(('a option \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c) set" assumes "finite X"   and "finite F"   and "F \<subseteq> P[X]" shows "extended_ord.is_hom_GB_bound F (Dube (card X) (maxdeg F))"
induct: fixes L1 :: "'n \<Rightarrow> 'o \<Rightarrow> bool"   and Co1 :: "'p \<Rightarrow> 'q \<Rightarrow> bool"   and Co2 :: "'r \<Rightarrow> 's \<Rightarrow> bool"   and Contra1 :: "'t \<Rightarrow> 'u \<Rightarrow> bool"   and Contra2 :: "'v \<Rightarrow> 'w \<Rightarrow> bool"   and x1 :: "('n, 'p, 'r, 't, 'v, 'x) T"   and x2 :: "('o, 'q, 's, 'u, 'w, 'x) T"   and P :: "('n, 'p, 'r, 't, 'v, 'x) T \<Rightarrow> ('o, 'q, 's, 'u, 'w, 'x) T \<Rightarrow> bool" assumes "rel_T L1 Co1 Co2 Contra1 Contra2 x1 x2"   and "\<And>x y. rel_G (\<lambda>x1 x2. rel_T L1 Co1 Co2 Contra1 Contra2 x1 x2 \<and> P x1 x2) L1 Co1 Co2 Contra1 Contra2 x y \<Longrightarrow> P (C_T x) (C_T y)" shows "P x1 x2"
big_def: fixes small :: "'a \<Rightarrow> 'b \<Rightarrow> 'b set"   and endset :: "'b set"   and P :: "'a"   and \<sigma> :: "'b" assumes "Semantics small endset" shows "Semantics.big small endset P \<sigma> \<equiv> {\<sigma>'. \<exists>n. \<sigma>' \<in> Semantics.small_nstep small P \<sigma> n \<and> \<sigma>' \<in> endset}"
inject_1: fixes x1 :: "'b"   and y1 :: "'b" shows "(x1\<^sup>+ = y1\<^sup>+) = (x1 = y1)"
corec_gpv_parametric': fixes S :: "'m \<Rightarrow> 'q \<Rightarrow> bool"   and A :: "'n \<Rightarrow> 'r \<Rightarrow> bool"   and C :: "'o \<Rightarrow> 's \<Rightarrow> bool"   and R :: "'p \<Rightarrow> 't \<Rightarrow> bool" shows "((S ===> rel_spmf (rel_generat A C (R ===> rel_sum (rel_gpv'' A C R) S))) ===> S ===> rel_gpv'' A C R) corec_gpv corec_gpv"
plus_minus_no_overflow_ab: fixes x :: "'b word"   and ab :: "'b word"   and c :: "'b word" assumes "x \<le> ab - c"   and "c \<le> ab" shows "x \<le> x + c"
homeomorphic_space: fixes X :: "'c topology"   and Y :: "'d topology" shows "(X homeomorphic_space Y) = (\<exists>f. homeomorphic_map X Y f)"
I0: fixes s :: "'d"   and z :: "'e nres" assumes "s \<in> S"   and "SPEC (I {s}) \<le> z" shows "f s \<le> z"
nlists_eq_empty_iff: fixes A :: "'b set"   and n :: "nat" shows "(nlists A n = {}) = (0 < n \<and> A = {})"
card_deg_sect_Suc: fixes X :: "'b set"   and d :: "nat" assumes "finite X" shows "card (deg_sect X (Suc d)) = d + card X choose Suc d"
lex_lexs_trans: fixes x :: "real \<times> real"   and y :: "real \<times> real"   and z :: "real \<times> real" assumes "Counterclockwise_2D_Arbitrary.lex x y"   and "lexs y z" shows "lexs x z"