hcju/isabelleps huggingface dataset:
simps_22: fixes e1 :: "'a aexp"   and e2 :: "'a aexp"   and v :: "value" shows "local.less_aexp_aux (Times e1 e2) (L v) = False"
lt_max_keys: fixes v :: "'a"   and p :: "'a \<Rightarrow>\<^sub>0 'c" assumes "v \<in> keys p" shows "v \<preceq> punit.lt p"
that: fixes U :: "(nat \<times> (nat, nat) Op) set" assumes "finite U"   and "deterministic U"   and "0 \<notin> fst ` U"   and "1 \<notin> fst ` U"   and "rec_inseparable {uu_. \<exists>x vs'. uu_ = x \<and> ((2, \<lambda>n. if n = 0 then x else 0), 0, vs') \<in> (step U)\<^sup>*} {uu_. \<exists>x vs'. uu_ = x \<and> ((2, \<lambda>n. if n = 0 then x else 0), 1, vs') \<in> (step U)\<^sup>*}" shows "thesis"
is_Read\<^sub>s\<^sub>b_simps_2: fixes volatile :: "bool"   and a :: "nat"   and sop :: "nat set \<times> ((nat \<Rightarrow> nat option) \<Rightarrow> nat)"   and v :: "nat"   and A :: "nat set"   and L :: "nat set"   and R :: "nat set"   and W :: "nat set" shows "is_Read\<^sub>s\<^sub>b (Write\<^sub>s\<^sub>b volatile a sop v A L R W) = False"
Disjoint_cong: fixes H :: "fm set"   and t :: "tm"   and t' :: "tm"   and u :: "tm"   and u' :: "tm" assumes "H \<turnstile> t EQ t'"   and "H \<turnstile> u EQ u'" shows "H \<turnstile> Disjoint t u IFF Disjoint t' u'"
minus_poly_rev_list: fixes q :: "'b list"   and p :: "'b list" assumes "length q \<le> length p" shows "Poly (rev (minus_poly_rev_list (rev p) (rev q))) = Poly p - monom (1::'b) (length p - length q) * Poly q"
lift_def: fixes x :: "V"   and y :: "V" shows "lift x y \<equiv> ZFC_in_HOL.set ((+) x ` elts y)"
compute_flow_val_aux_def: fixes am :: "nat \<Rightarrow> nat list"   and cf :: "nat \<times> nat \<Rightarrow> int" shows "compute_flow_val_aux am cf \<equiv> am_get am s \<bind> (\<lambda>succs. sum_impl (\<lambda>v. cf_get cf (v, s)) (set succs))"
t0_2: shows "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<theta>)"
hyps_2: fixes t :: "taylor_model" assumes "approx_tm prec ord I a r env = Some t" shows "interpret_floatarith r xs \<in>\<^sub>i range_tm e t"
qSwapBinp_def: fixes xs :: "'h"   and x :: "'i"   and y :: "'i"   and qbinp :: "'e \<Rightarrow> ('f, 'g, 'h, 'i, 'j) qAbs option" shows "qSwapBinp xs x y qbinp \<equiv> lift (qSwapAbs xs x y) qbinp"
finite_atLeastAtMost_integer: fixes l :: "integer"   and u :: "integer" shows "finite {l..u}"
inner_dirsum_double_left0: fixes A :: "'a set" shows "0 \<oplus> A = A"
neg_diag_empty_spec: fixes i :: "nat"   and M :: "nat \<Rightarrow> nat \<Rightarrow> 'c DBMEntry" assumes "i \<le> n"   and "M i i < \<one>" shows "[M]\<^bsub>v,n\<^esub> = {}"
assms_2: shows "R = Ax \<union> g3up \<union> g3mod1 \<union> g3mod2"
SASP_to_DIMACS_def: fixes h :: "nat"   and prob :: "(char list \<times> nat option \<times> char list list) list \<times> nat list \<times> (nat \<times> nat) list \<times> (char list \<times> (nat \<times> nat) list \<times> ((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list \<times> nat) list" shows "SASP_to_DIMACS h prob \<equiv> cnf_to_dimacs (map_formula (cnf_to_dimacs.var_to_dimacs (Suc h) (Suc (length (ast_problem.ast\<delta> prob)))) (\<Phi>\<^sub>\<forall> \<phi> prob_with_noop (ast_problem.abs_prob prob)  h))"
assms_2: shows "f $ subdegree f = g $ subdegree g"
strict_order_iff: fixes a :: "'b word"   and b :: "'b word" shows "(a <s b) = (a = signed.min a b \<and> a \<noteq> b)"
prems_9: shows "leftderives \<alpha> (terminals (drop r q))"
lless_cong_r: fixes x :: "'a"   and y :: "'a"   and y' :: "'a" assumes "x \<sqsubset>\<^bsub>division_rel G\<^esub> y"   and "y .=\<^bsub>division_rel G\<^esub> y'"   and "x \<in> carrier (division_rel G)"   and "y \<in> carrier (division_rel G)"   and "y' \<in> carrier (division_rel G)" shows "x \<sqsubset>\<^bsub>division_rel G\<^esub> y'"
supp: fixes G :: "'a set"   and smult :: "('b, 'a) aezfun \<Rightarrow> 'c \<Rightarrow> 'c"   and V :: "'c set"   and smult' :: "('b, 'a) aezfun \<Rightarrow> 'd \<Rightarrow> 'd"   and T :: "'c \<Rightarrow> 'd" assumes "FGModuleHom G smult V smult' T" shows "supp T \<subseteq> V"
foo'_ax1: fixes x :: "'d"   and y :: "bool"   and z :: "bool" assumes "foo' x y z" shows "z \<and> y"
sp_Cons: fixes g :: "('c, 'd) rgraph_scheme"   and v1 :: "'c"   and e :: "'c edge"   and es :: "'c edge list"   and v2 :: "'c" shows "Graph.subpath g v1 (e # es) v2 = (src e = v1 \<and> e \<in> edges g \<and> Graph.subpath g (tgt e) es v2)"
set_eqI: fixes A :: "'c set"   and B :: "'c set" assumes "\<And>x. (x \<in> A) = (x \<in> B)" shows "A = B"
cong: fixes f :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and z :: "'d" shows "abel_semigroup_set.F f z \<equiv> abel_semigroup_set.F f z"
subst_atm_comp_subst: fixes A :: "'a"   and \<sigma> :: "'s"   and \<tau> :: "'s" shows "A \<cdot>a (\<sigma> \<odot> \<tau>) = A \<cdot>a \<sigma> \<cdot>a \<tau>"
take_drop: fixes n :: "nat"   and m :: "nat"   and xs :: "'a list" shows "take n (drop m xs) = drop m (take (n + m) xs)"
prems: shows "Oadmit \<sigma> (OProd ODE1 ODE2) U"
minus_lt: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleRat :: "rat \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.linordered_rational_vector less_eq less plus zero minus uminus scaleRat" shows "less a b = less (minus a b) zero"
nz2: shows "inverse (1 + \<i> * z) \<noteq> 0"
omega_of_real_transfer: fixes a :: "nat" shows "cr_omega (a\<^sub>\<nat>) a"
is_nth_power_nat_def: shows "is_nth_power_nat = is_nth_power"
test_de_morgan: fixes x :: "'a"   and y :: "'a" shows "n (n n x \<cdot> n n y) = n x + n y"
sup_least: fixes y :: "'d multiset"   and x :: "'d multiset"   and z :: "'d multiset" assumes "y \<subseteq># x"   and "z \<subseteq># x" shows "y \<union># z \<subseteq># x"
assms: shows "(u, v) \<in> R\<^sup>*"
one_add_one: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d" assumes "class.numeral plus" shows "plus one one = numeral.numeral one plus (num.Bit0 num.One)"
map2_val_compat_def: fixes comp :: "'a \<Rightarrow> 'a \<Rightarrow> order"   and f :: "('a \<times> 'b) list \<Rightarrow> ('a \<times> 'c) list" assumes "comparator comp" shows "comparator.map2_val_compat comp f = (\<forall>zs. comparator.oalist_inv_raw comp zs \<longrightarrow> comparator.oalist_inv_raw comp (f zs) \<and> fst ` set (f zs) \<subseteq> fst ` set zs)"
Match: fixes P :: "pi"   and Rs :: "Late_Semantics.residual"   and b :: "name" assumes "P \<longmapsto>\<^sub>l Rs" shows "[b\<frown>b]P \<longmapsto>\<^sub>l Rs"
ran_override: fixes A :: "'d \<Rightarrow> 'e option"   and B :: "'d \<Rightarrow> 'e option" assumes "dom A \<inter> dom B = {}" shows "ran (A ++ B) = ran A \<union> ran B"
sig_red_zeroI: fixes sing_reg :: "'t \<Rightarrow> 't \<Rightarrow> bool"   and F :: "('t \<Rightarrow>\<^sub>0 'b) set"   and r :: "'t \<Rightarrow>\<^sub>0 'b"   and s :: "'t \<Rightarrow>\<^sub>0 'b" assumes "(sig_red sing_reg (\<preceq>) F)\<^sup>*\<^sup>* r s"   and "rep_list s = 0" shows "sig_red_zero sing_reg F r"
cone_alt_def: fixes Vs :: "'a vec set" assumes "Vs \<subseteq> carrier_vec n" shows "cone Vs = {x. \<exists>Ws. set Ws \<subseteq> Vs \<and> x \<in> cone_list Ws}"
\<Delta>\<^sub>d_def: shows "\<Delta>\<^sub>d = (\<Sum>i\<in>I. \<Sum>j | j \<in> I \<and> i \<noteq> j \<and> \<not> indep (A i) (A j). prob (A i \<inter> A j))"
assms_3: fixes i :: "nat" assumes "i < length as'"   and "i < length bs'" shows "f (as' ! i) (bs' ! i) = g (as' ! i) (bs' ! i)"
of_complex_numeral: fixes w :: "num" shows "of_complex (numeral w) = numeral w"
bij_betw_image_mset_set: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and B :: "'b set" assumes "bij_betw f A B" shows "image_mset f (mset_set A) = mset_set B"
lookup_default_mapping_of_pmf: fixes p :: "'b pmf"   and x :: "'b" shows "Mapping.lookup_default 0 (mapping_of_pmf p) x = pmf p x"
set_intros_3: fixes x12 :: "'f"   and x11 :: "'g"   and x13 :: "'h" shows "x12 \<in> set2_new_thread_action (NewThread x11 x12 x13)"
assms_2: fixes c :: "((nat \<Rightarrow> real) \<Rightarrow> 'a) \<Rightarrow>\<^sub>0 int" assumes "singular_chain (p - 1) (subtopology X S) c" shows "singular_chain p (subtopology X' T) (h' c)"
inject_2: fixes x2 :: "trm"   and x1 :: "trm"   and y2 :: "trm"   and y1 :: "trm" shows "(App x2 x1 = App y2 y1) = (x2 = y2 \<and> x1 = y1)"
xor_list_def: fixes xs :: "'b list"   and ys :: "'b list" shows "xs [\<oplus>] ys = map2 (\<oplus>) xs ys"
pick_def: shows "pick = (\<lambda>c. SOME k. enat k \<le> llength t \<and> j \<le> k \<and> Marker \<notin> set (msgs (coS t k) c))"
drop_update_until: shows "drop (length done) (update_until args rel1 rel2 nt auxlist) = map (\<lambda>(t, a1, a2). (t, if pos then join a1 True rel1 else a1 \<union> rel1, if left I \<le> nt - t then a2 \<union> join rel2 pos a1 else a2)) (drop (length done) auxlist) @ [(nt, rel1, if left I = 0 then rel2 else empty_table)]"
inf_th_OT12_P2: fixes funct_OT12 :: "bool \<times> bool \<Rightarrow> bool \<Rightarrow> (unit \<times> bool) spmf"   and m0 :: "bool"   and m1 :: "bool"   and \<sigma> :: "bool" shows "sim_det_def.perfect_sec_P2 R2_OT12 S2_OT12 funct_OT12 (m0, m1) \<sigma>"
unitary_simps_1: fixes A :: "'a mat"   and n :: "nat" assumes "A \<in> carrier_mat n n"   and "unitary A" shows "adjoint A * A = 1\<^sub>m n"
Nil_is_rev_conv: fixes xs :: "'f list" shows "([] = rev xs) = (xs = [])"
trnl\<^sub>\<epsilon>_comp: fixes u :: "'a"   and \<mu> :: "'a"   and \<nu> :: "'a" assumes "ide u"   and "seq \<mu> \<nu>"   and "src k = trg \<mu>" shows "k.trnl\<^sub>\<epsilon> u (\<mu> \<cdot> \<nu>) = k.trnl\<^sub>\<epsilon> u \<mu> \<cdot> (k \<star> \<nu>)"
obtain_comp_block_orig: fixes bl1 :: "'a set" assumes "bl1 \<in># fin.complement_blocks" obtains bl2 :: "'a set" where "bl2 \<in># \<B>\<^sup>C" and "bl1 = bl2\<^sup>c"
assn_times_assoc: fixes P :: "assn"   and Q :: "assn"   and R :: "assn" shows "P * Q * R = P * (Q * R)"
consistent\<^sub>F_insortTree: fixes t :: "'a tree"   and ts :: "'a tree list" shows "consistent\<^sub>F (insortTree t ts) = consistent\<^sub>F (t # ts)"
real_root_lt_0_iff: fixes n :: "nat"   and x :: "real" assumes "0 < n" shows "(root n x < 0) = (x < 0)"
abs_eq: shows "1\<^sub>L = Blinfun (\<lambda>x. x)"
nnf: shows "\<forall>m'\<in>set (normalize_primitive_extract (disc, sel) C f m). normalized_nnf_match m'"
k_decr: fixes k :: "int" shows "f (of_int (k - 1)) - g (of_int (k - 1)) = f (of_int k) - g (of_int k)"
diff_in_polynomial_bool_pac_idealI: fixes p :: "int mpoly"   and A :: "int mpoly set"   and p' :: "int mpoly" assumes "p \<in> pac_ideal A"   and "p - p' \<in> More_Modules.ideal polynomial_bool" shows "p' \<in> pac_ideal A"
that: fixes A :: "fm" assumes "x = \<lbrakk>\<guillemotleft>A\<guillemotright>\<rbrakk>e"   and "{} \<turnstile> A" shows "thesis"
assms_3: shows "\<forall>x\<in>s. g (f x) = x"
component_def: fixes F :: "'b program"   and H :: "'b program" shows "(F \<le> H) = (\<exists>G. F \<squnion> G = H)"
dg_GRPH_Dom_vrange: fixes \<alpha> :: "V" shows "\<R>\<^sub>\<circ> (dg_GRPH \<alpha>\<lparr>Dom\<rparr>) \<subseteq>\<^sub>\<circ> dg_GRPH \<alpha>\<lparr>Obj\<rparr>"
rel_interior_eq: fixes s :: "'a set" shows "(rel_interior s = s) = openin (top_of_set (affine hull s)) s"
Lim: fixes f :: "'b \<Rightarrow> 'c"   and l :: "'c"   and net :: "'b filter" shows "(f \<longlongrightarrow> l) net = (net = bot \<or> (\<forall>e>0. \<forall>\<^sub>F x in net. dist (f x) l < e))"
integrable_linear: fixes f :: "'a \<Rightarrow> 'b"   and S :: "'a set"   and h :: "'b \<Rightarrow> 'c" assumes "f integrable_on S"   and "bounded_linear h" shows "h \<circ> f integrable_on S"
assms_1: shows "c \<equiv> min (\<Sqinter> (f ` results_gpv \<I> gpv)) fail"
strict_uv_prop_def: fixes X :: "'b program set" shows "strict_uv_prop X \<equiv> \<bottom> \<in> X \<and> (\<forall>F G. F ok G \<longrightarrow> (F \<in> X \<and> G \<in> X) = (F \<squnion> G \<in> X))"
assms_3: shows "distinct_of_odd p i \<noteq> 0"
card_older_seniors_monotonic: fixes x :: "nat"   and n :: "nat"   and m :: "nat" assumes "x \<le> n" shows "card (older_seniors x (n + m)) \<le> card (older_seniors x n)"
linordered_field_no_lb: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sgn :: "'c \<Rightarrow> 'c"   and abs :: "'c \<Rightarrow> 'c" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "\<forall>x. \<exists>y. less y x"
scale_sum_left: fixes f :: "'d \<Rightarrow> 'a"   and A :: "'d set"   and x :: "'b" shows "sum f A *a x = (\<Sum>a\<in>A. f a *a x)"
restrict_index_out: fixes getIJ :: "(nat set \<times> 'b option list set) set \<Rightarrow> (nat set \<times> 'b option list set) set \<Rightarrow> nat set \<Rightarrow> nat set \<times> nat set"   and i :: "nat"   and X :: "'c option list"   and I :: "nat set" assumes "getIJ getIJ"   and "i < length X"   and "i \<notin> I" shows "restrict I X ! i = None"
case_2: fixes f1 :: "'a"   and f2 :: "'a" shows "(case SO of FO \<Rightarrow> f1 | SO \<Rightarrow> f2) = f2"
disjointI: fixes A :: "'a set"   and B :: "'a set" assumes "\<And>x. \<lbrakk>x \<in> A; x \<in> B\<rbrakk> \<Longrightarrow> False" shows "A \<inter> B = {}"
n_kat_2_opp: fixes y :: "'a"   and z :: "'a"   and x :: "'a" shows "(y \<cdot> !z = !x \<cdot> y \<cdot> !z) = (y \<cdot> !z \<le> !x \<cdot> y)"
matrix_inv_right: fixes M :: "(('d, 'e) Finite_Cartesian_Product.vec, 'f) Finite_Cartesian_Product.vec" assumes "invertible M" shows "M ** matrix_inv M = mat (1::'d)"
assms_8: shows "InvariantCEntailed (getConflictFlag state) F0 (getC state)"
dg_op_dg_op_dg: fixes \<alpha> :: "V"   and \<CC> :: "V" assumes "digraph \<alpha> \<CC>" shows "op_dg (op_dg \<CC>) = \<CC>"
simps_1: fixes v :: "nat" shows "used_tmps (Const v) = 0"
update_simps_1: fixes ip :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option"   and r :: "nat \<times> k \<times> f \<times> nat \<times> nat" assumes "ip \<notin> kD rt" shows "update rt ip r = rt(ip \<mapsto> r)"
euclidean_inner: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and norm :: "'c \<Rightarrow> real"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and scaleR :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and sgn :: "'c \<Rightarrow> 'c"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and inner :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and Basis :: "'c set"   and x :: "'c"   and y :: "'c" assumes "class.euclidean_space minus dist norm plus zero uminus scaleR sgn uniformity open inner Basis" shows "inner x y = (\<Sum>b\<in>Basis. inner x b * inner y b)"
is_nextElem_rotate_eq: fixes m :: "nat"   and ls :: "'a list"   and x :: "'a"   and y :: "'a" shows "is_nextElem (rotate m ls) x y = is_nextElem ls x y"
rel_interior_openin: fixes s :: "'b set" assumes "openin (top_of_set (affine hull s)) s" shows "rel_interior s = s"
PEI_3: shows "(\<lambda>i. ValFF) \<in> unlr PE.delta"
pos_oriented: fixes H :: "ocircline" shows "pos_oriented H \<or> pos_oriented (opposite_ocircline H)"
intro: fixes D :: "'a set"   and cmp :: "'a \<Rightarrow> 'a \<Rightarrow> comp_res" assumes "\<And>x y. \<lbrakk>x \<in> D; y \<in> D\<rbrakk> \<Longrightarrow> (cmp x y = LESS) = (cmp y x = GREATER)"   and "\<And>x. x \<in> D \<Longrightarrow> cmp x x = EQUAL"   and "\<And>x y z. \<lbrakk>x \<in> D; y \<in> D; z \<in> D; cmp x y = LESS; cmp y z = LESS\<rbrakk> \<Longrightarrow> cmp x z = LESS"   and "\<And>x y z. \<lbrakk>x \<in> D; y \<in> D; z \<in> D; cmp x y = LESS; cmp y z = EQUAL\<rbrakk> \<Longrightarrow> cmp x z = LESS"   and "\<And>x y z. \<lbrakk>x \<in> D; y \<in> D; z \<in> D; cmp x y = EQUAL; cmp y z = LESS\<rbrakk> \<Longrightarrow> cmp x z = LESS"   and "\<And>x y z. \<lbrakk>x \<in> D; y \<in> D; z \<in> D; cmp x y = EQUAL; cmp y z = EQUAL\<rbrakk> \<Longrightarrow> cmp x z = EQUAL" shows "linorder_on D cmp"
sort_key_eq_sort_key: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "'b list"   and ys :: "'b list"   and f :: "'b \<Rightarrow> 'a" assumes "class.linorder less_eq less"   and "mset xs = mset ys"   and "inj_on f (set xs)" shows "linorder.sort_key less_eq f xs = linorder.sort_key less_eq f ys"
reducedAllocation: fixes a :: "('c \<times> 'd set) set"   and b :: "('c \<times> 'd set) set" assumes "a \<in> allocationsUniverse" shows "a - b \<in> allocationsUniverse"
shift_pow: fixes u :: "'b list"   and v :: "'b list"   and k :: "nat" shows "(u \<cdot> v) \<^sup>@ k \<cdot> u = u \<cdot> (v \<cdot> u) \<^sup>@ k"
elims_2: fixes x :: "'a extC"   and xa :: "'a extC" assumes "x < xa" obtains x :: "'a extC" and y :: "'a extC" where "xa__ = x" and "xa = y" and "x \<le> y \<and> \<not> y \<le> x"
elims_1: fixes x :: "('i \<times> constraint) list"   and xa :: "'i set"   and xb :: "'a ns_constraint list \<times> 's"   and y :: "bool" assumes "checked_cs x xa xb = y" obtains cs :: "('i \<times> constraint) list" and J :: "'i set" and ds :: "'a ns_constraint list" and s :: "'s" where "x = cs" and "xa = J" and "xb = (ds, s)" and "y = (ds = map snd (to_ns cs) \<and> checked_nsc (to_ns cs) J s)"
analz_Un_implSecureSet_aux_aux: fixes G :: "msg set"   and H :: "msg set"   and Ag :: "(agent \<times> agent) set" assumes "Enc_keys_clean (G \<union> H)" shows "analz (implSecureSet_aux Ag G \<union> H) \<subseteq> implSecureSet_aux Ag G \<union> AencSet (PairSet Tags (PairSet AgentSet G)) (pubK ` Ag `` UNIV) \<union> PairSet Tags (PairSet AgentSet G) \<union> Tags \<union> PairSet AgentSet G \<union> synth (analz (G \<union> H))"
adopt_node_removes_child: fixes h :: "(_) heap"   and owner_document :: "(_) document_ptr"   and node_ptr :: "(_) node_ptr"   and h2 :: "(_) heap"   and ptr :: "(_) object_ptr"   and children :: "(_) node_ptr list" assumes "heap_is_wellformed h"   and "h \<turnstile> adopt_node owner_document node_ptr \<rightarrow>\<^sub>h h2"   and "h2 \<turnstile> get_child_nodes ptr \<rightarrow>\<^sub>r children"   and "known_ptrs h"   and "type_wf h" shows "node_ptr \<notin> set children"
fg_1: shows "OpposedThinChamberComplexFoldings X f g A"
isUIPExistsBackjumpLevel: fixes c :: "Literal list"   and M :: "(Literal \<times> bool) list"   and l :: "Literal" assumes "clauseFalse c (elements M)"   and "isUIP l c M"   and "uniq (elements M)"   and "0 < elementLevel (opposite l) M" shows "\<exists>level. isBackjumpLevel level l c M"
polyhedral_cone_def: fixes A :: "'a mat" shows "nr.polyhedral_cone A = {x \<in> carrier_vec nr. A *\<^sub>v x \<le> 0\<^sub>v (dim_row A)}"
arg_min_natI: fixes P :: "'a \<Rightarrow> bool"   and k :: "'a"   and m :: "'a \<Rightarrow> nat" assumes "P k" shows "P (arg_min m P)"
flift2_Def: fixes f :: "'c \<Rightarrow> 'b"   and x :: "'c" shows "flift2 f\<cdot>(Def x) = Def (f x)"
build_rel_def: fixes \<alpha> :: "'h \<Rightarrow> 'i"   and I :: "'h \<Rightarrow> bool" shows "br \<alpha> I \<equiv> {(c, a). a = \<alpha> c \<and> I c}"
ring_hom_one: fixes h :: "'a \<Rightarrow> 'b"   and R :: "('a, 'c) ring_scheme"   and S :: "('b, 'd) ring_scheme" assumes "h \<in> ring_hom R S" shows "h \<one>\<^bsub>R\<^esub> = \<one>\<^bsub>S\<^esub>"
fps_inv_deriv: fixes a :: "'a fps" assumes "fps_nth a 0 = (0::'a)"   and "fps_nth a 1 \<noteq> (0::'a)" shows "fps_deriv (fps_inv a) = inverse (fps_deriv a oo fps_inv a)"
depth_wrap_Id: fixes f :: "recf"   and a :: "nat"   and n :: "nat" assumes "recfn a f" shows "depth (wrap_Id f n) = depth f + n"
IH: assumes "\<not> G'.reaches1 a y" shows "\<exists>c d. a \<rightarrow>* c \<and> c \<rightarrow> d \<and> \<not> E' c d \<and> d \<rightarrow>* y"
lookup_zero: fixes k :: "'b" shows "lookup 0 k = (0::'a)"
polymul_properties_1: fixes p :: "'b poly"   and n0 :: "nat"   and q :: "'b poly"   and n1 :: "nat"   and m :: "nat" assumes "isnpolyh p n0"   and "isnpolyh q n1"   and "m \<le> min n0 n1" shows "isnpolyh (p *\<^sub>p q) (min n0 n1)"
shift_inj: fixes cval :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and indis :: "('b \<times> 'b) set"   and cl :: "('c, 'd, 'a) cmd list"   and n :: "nat"   and I1 :: "nat set"   and I2 :: "nat set" assumes "PL_Indis cval indis" shows "(PL_Indis.shift cl n I1 = PL_Indis.shift cl n I2) = (I1 = I2)"
trace'_eqI_sim: fixes S :: "'a spmf \<Rightarrow> 'b spmf \<Rightarrow> bool"   and p :: "'a spmf"   and q :: "'b spmf"   and A :: "'c set"   and callee1 :: "'a \<Rightarrow> 'c \<Rightarrow> ('d \<times> 'a) spmf"   and callee2 :: "'b \<Rightarrow> 'c \<Rightarrow> ('d \<times> 'b) spmf" assumes "S p q"   and "\<And>p q a. \<lbrakk>S p q; a \<in> A\<rbrakk> \<Longrightarrow> p \<bind> (\<lambda>s. map_spmf fst (callee1 s a)) = q \<bind> (\<lambda>s. map_spmf fst (callee2 s a))"   and "\<And>p q a res b s'. \<lbrakk>S p q; a \<in> A; res \<in> set_spmf q; (b, s') \<in> set_spmf (callee2 res a)\<rbrakk> \<Longrightarrow> S (cond_spmf_fst (p \<bind> (\<lambda>s. callee1 s a)) b) (cond_spmf_fst (q \<bind> (\<lambda>s. callee2 s a)) b)" shows "trace_callee_eq callee1 callee2 A p q"
dg_Dom_vrange: shows "\<R>\<^sub>\<circ> (\<CC>\<lparr>Dom\<rparr>) \<subseteq>\<^sub>\<circ> \<CC>\<lparr>Obj\<rparr>"
assms: shows "\<forall>p\<in>set ps. sgn (poly p x) = sgn (poly p y)"
integral_restrict_UNIV: fixes S :: "'b set"   and f :: "'b \<Rightarrow> 'a" shows "integral UNIV (\<lambda>x. if x \<in> S then f x else (0::'a)) = integral S f"
exhaust: fixes y :: "yxml_of_term" obtains "y = YXML"
solves_odeD_1: fixes y :: "real \<Rightarrow> 'a"   and f :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and T :: "real set"   and X :: "'a set" assumes "(y solves_ode f) T X" shows "(y has_vderiv_on (\<lambda>t. f t (y t))) T"
lookup_map_scale: fixes c :: "'d"   and p :: "'c \<Rightarrow>\<^sub>0 'd" shows "lookup (c \<cdot> p) = (\<lambda>x. c * lookup p x)"
float_distinct_9: fixes a :: "('a, 'b) IEEE.float" shows "\<not> (is_denormal a \<and> is_zero a)"
DSourcesA92_L1: shows "DSources level1 sA92 = {sA81}"
tnm_def: fixes R :: "('g, 'h) Ring_scheme"   and P :: "('e \<times> 'f, 'g, 'i) Module_scheme"   and M :: "('e, 'g, 'i) Module_scheme"   and N :: "('f, 'g, 'i) Module_scheme" shows "tnm R P M N = compose (M \<times>\<^sub>c N) (mpj P (TR\<^bsub>R\<^esub> M N P)) (tau R M N P)"
compress_bottom_inv: fixes w1 :: "wall"   and w2 :: "wall" assumes "compress_bottom w1 w2" shows "kauff_mat w1 = kauff_mat w2"
s: shows "\<forall>i. simple_bochner_integrable M (s i)"
omit_branch_id: fixes branch :: "(('b, 'c) fm list \<times> 'c) list" shows "omit_branch {} branch = branch"
assms: shows "VPow A \<in>\<^sub>\<circ> Vset i"
sterms_not_empty: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'c, 'a, 'd) seqp"   and p :: "('b, 'c, 'a, 'd) seqp" assumes "wellformed \<Gamma>" shows "sterms \<Gamma> p \<noteq> {}"
IF2set: shows "IF2set \<circ> ctor2 = IF2col \<circ> F2map id IF1set IF2set"
cone_axioms: shows "GE.cone (G a) (G \<circ> \<chi>)"
cc_subs: shows "CC \<subseteq> N - Red_F N"
is_operator_applicable_in_def: fixes s :: "'a \<Rightarrow> bool option"   and op :: "'a strips_operator" shows "STRIPS_Representation.is_operator_applicable_in s op \<equiv> Let (strips_operator.precondition_of op) (list_all (\<lambda>v. s v = Some True))"
streams_shd: fixes s :: "'b stream"   and A :: "'b set" assumes "s \<in> streams A" shows "shd s \<in> A"
q2_def: shows "q2 = except q (- {u. deg_pm u < poly_deg q})"
lang\<^sub>W\<^sub>S\<^sub>1\<^sub>S_rexp_of: fixes n :: "nat"   and \<phi> :: "'a formula" assumes "wf_formula n \<phi>" shows "lang\<^sub>W\<^sub>S\<^sub>1\<^sub>S n \<phi> = lang n (rexp_of n \<phi>)"