hcju/isabelleps huggingface dataset:
can_signal_def: fixes s :: "'a l1_state_scheme"   and A :: "agent"   and B :: "agent" shows "can_signal s A B \<equiv> (A = test_owner \<and> B = test_partner \<or> B = test_owner \<and> A = test_partner) \<and> \<not> test_ended s"
coboundedI1: fixes a :: "'b"   and c :: "'b"   and b :: "'b" assumes "order.greater_eq c a" shows "order.greater_eq c (inf a b)"
nan_not_infinity: fixes x :: "('a, 'b) IEEE.float" assumes "IEEE.is_nan x" shows "\<not> is_infinity x"
discI_2: fixes literal :: "'a literal"   and x2 :: "'a" assumes "literal = Neg x2" shows "is_neg literal"
ms4: fixes e1 :: "lam"   and Es1 :: "ctx list"   and e2 :: "lam"   and Es2 :: "ctx list" assumes "<e1,Es1> \<mapsto> <e2,Es2>" shows "<e1,Es1> \<mapsto>* <e2,Es2>"
M: shows "(map (inv q) (filter (\<lambda>x. x \<in> range q) (xs @ y # ys)), inv q ` S) \<in> failures Q"
eq_eqvt: fixes p :: "perm"   and x :: "'a"   and y :: "'a" shows "p \<bullet> (x = y) = (p \<bullet> x = p \<bullet> y)"
conj_UINF_ind_dist: fixes P :: "'a upred"   and F :: "'b \<Rightarrow> 'a upred" shows "(P \<and> (\<Sqinter> Q \<bullet> F Q)) = (\<Sqinter> Q \<bullet> P \<and> F Q)"
extend_1: fixes K :: "'a set" assumes "K \<in> \<D>" shows "extend K \<noteq> {}"
simps_4: fixes less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and proper_interval :: "'a option \<Rightarrow> 'a option \<Rightarrow> bool"   and ao :: "'a option"   and x :: "'a"   and xs :: "'a list"   and y :: "'a"   and ys :: "'a list" shows "proper_intrvl.set_less_aux_Compl less proper_interval ao (x # xs) (y # ys) = (if less x y then proper_interval ao (Some x) \<or> proper_intrvl.set_less_aux_Compl less proper_interval (Some x) xs (y # ys) else if less y x then proper_interval ao (Some y) \<or> proper_intrvl.set_less_aux_Compl less proper_interval (Some y) (x # xs) ys else proper_interval ao (Some y))"
arcs_ends_app_iso_eq: fixes G :: "('c, 'd) pre_digraph"   and hom :: "('c, 'd, 'e, 'f) digraph_isomorphism" assumes "wf_digraph G"   and "pre_digraph.digraph_isomorphism G hom" shows "arcs_ends (app_iso hom G) = (\<lambda>(u, v). (iso_verts hom u, iso_verts hom v)) ` arcs_ends G"
hinsert_1: assumes "x \<noteq> 0"   and "\<And>i. i \<^bold>\<in> x \<Longrightarrow> Ord i" shows "\<Squnion>x \<^bold>\<in> x"
insert_eq_iff: fixes a :: "'d"   and A :: "'d set"   and b :: "'d"   and B :: "'d set" assumes "a \<notin> A"   and "b \<notin> B" shows "(insert a A = insert b B) = (if a = b then A = B else \<exists>C. A = insert b C \<and> b \<notin> C \<and> B = insert a C \<and> a \<notin> C)"
prems_3: fixes lholed :: "Lholed"   and i :: "nat" shows "\<not> Lfilled 0 lholed [$Br i] (cs @ ($* es))"
mds\<^sub>A_def: shows "mds\<^sub>A = mds\<^sub>A_of mds\<^sub>C"
lower_def: fixes p :: "'a \<Rightarrow>\<^sub>0 'c"   and t :: "'a" shows "punit.lower p t = except p {s. t \<preceq> s}"
leg1_composite: shows "trg_trg.leg1 = trg.leg1 \<cdot> trg_trg.prj\<^sub>1"
sum_strict_mono: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b multiset"   and g :: "'a \<Rightarrow> 'b multiset" assumes "finite A"   and "A \<noteq> {}"   and "\<And>x. x \<in> A \<Longrightarrow> f x \<subset># g x" shows "comm_monoid_add.sum (+) {#} f A \<subset># comm_monoid_add.sum (+) {#} g A"
foo_sig_undeclared_in_Object: shows "tprg\<turnstile> mid foo_sig undeclared_in Object"
residue_mult_group_kernel_partition_6: fixes m :: "nat"   and n :: "nat" assumes "1 < m"   and "0 < n"   and "m dvd n" shows "bij_betw (\<lambda>b. the_elem ((\<lambda>k. k mod m) ` b)) (rcosets\<^bsub>residue_mult_group n\<^esub> kernel (residue_mult_group n) (residue_mult_group m) (\<lambda>k. k mod m)) (carrier (residue_mult_group m))"
GroupHom_axioms_def: fixes G :: "'a set"   and T :: "'a \<Rightarrow> 'b" shows "GroupHom_axioms G T \<equiv> (\<forall>g g'. g \<in> G \<longrightarrow> g' \<in> G \<longrightarrow> T (g + g') = T g + T g') \<and> supp T \<subseteq> G"
not_outside_1: fixes p :: "'a pmf"   and q :: "'a pmf" assumes "p \<succeq> q" shows "p \<in> \<P> outcomes"
remap_polys_polynomial_bool_remap_polys_change_all: fixes spec :: "int mpoly"   and \<V> :: "nat set"   and A :: "(nat, int mpoly) fmap" shows "remap_polys_polynomial_bool spec \<V> A \<le> remap_polys_change_all spec \<V> A"
ostep_invariant_def: fixes A :: "('e \<times> 'f, 'g) automaton"   and S :: "'e \<Rightarrow> 'e \<Rightarrow> 'g \<Rightarrow> bool"   and U :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and P :: "('e \<times> 'f) \<times> 'g \<times> 'e \<times> 'f \<Rightarrow> bool" shows "(A \<Turnstile>\<^sub>A (S, U \<rightarrow>) P) = (\<forall>s\<in>oreachable A S U. \<forall>a s'. (s, a, s') \<in> automaton.trans A \<and> S (fst s) (fst s') a \<longrightarrow> P (s, a, s'))"
adm_below_monic_exists: fixes P :: "'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b" assumes "adm P"   and "below_monic f"   and "cont f" shows "adm (\<lambda>x. \<exists>y. x = f y \<and> P y)"
gen_lset_code_1: fixes A :: "'a set" shows "gen_lset A LNil = A"
simps_6: fixes uv :: "checker_type" shows "produce Bot uv = Bot"
omega_sum_\<omega>\<omega>: fixes \<gamma> :: "V" assumes "\<gamma> \<in> elts (\<omega> \<up> \<omega>)" obtains ns :: "nat list" where "\<gamma> = omega_sum ns" and "normal ns"
expected_utility_weak_ranking: fixes carrier :: "'b set"   and le :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and u :: "'b \<Rightarrow> real"   and p :: "'b pmf" assumes "vnm_utility carrier le u"   and "p \<in> lotteries_on carrier" shows "measure_pmf.expectation p u = (\<Sum>A\<leftarrow>weak_ranking le. u (SOME x. x \<in> A) * measure_pmf.prob p A)"
axioms_1: fixes p :: "real"   and M :: "bool stream measure"   and F :: "nat \<Rightarrow> bool stream measure" assumes "infinite_cts_filtration p M F" shows "infinite_coin_toss_space p M"
inner_axis: fixes x :: "('a, 'b) vec"   and i :: "'b"   and y :: "'a" shows "x \<bullet> axis i y = x $ i \<bullet> y"
sublist_subset: fixes l1 :: "'a list"   and l2 :: "'a list" assumes "subseq l1 l2" shows "set l1 \<subseteq> set l2"
tendsto_unique': fixes F :: "'b filter"   and f :: "'b \<Rightarrow> 'a" assumes "F \<noteq> bot" shows "\<exists>\<^sub>\<le>\<^sub>1 l. (f \<longlongrightarrow> l) F"
bconf_def: fixes P :: "'a prog"   and sh :: "sheap"   and e :: "'b exp"   and b :: "bool" shows "P,sh \<turnstile>\<^sub>b (e,b) \<surd> \<equiv> b \<longrightarrow> (\<exists>C. icheck P C (ss_exp e) \<and> initPD sh C)"
der: fixes x :: "real" assumes "a \<le> x"   and "x \<le> b" shows "\<exists>y. (f has_real_derivative y) (at x) \<and> y < 0"
shrinking: fixes U :: "('a \<times> 'a) set set set"   and \<pp> :: "('a \<times> 'a) set set"   and s :: "('a \<times> 'a) set set \<Rightarrow> (('a \<times> 'a) set \<times> ('a \<times> 'a) set) set"   and t :: "('a \<times> 'a) set set \<Rightarrow> (('a \<times> 'a) set \<times> ('a \<times> 'a) set) set" assumes "local.local.is_zariski_open U"   and "\<pp> \<in> U"   and "s \<in> local.local.sheaf_spec U"   and "t \<in> local.local.sheaf_spec U" obtains V :: "('a \<times> 'a) set set set" and a :: "('a \<times> 'a) set" and f :: "('a \<times> 'a) set" and b :: "('a \<times> 'a) set" and g :: "('a \<times> 'a) set" where "local.local.is_zariski_open V" and "V \<subseteq> U" and "\<pp> \<in> V" and "a \<in> local.carrier_quotient_ring" and "f \<in> local.carrier_quotient_ring" and "b \<in> local.carrier_quotient_ring" and "g \<in> local.carrier_quotient_ring" and "f \<notin> \<pp>" and "g \<notin> \<pp>" and "\<And>\<qq>. \<qq> \<in> V \<Longrightarrow> f \<notin> \<qq> \<and> s \<qq> = quotient_ring.frac (local.carrier_quotient_ring\<setminus>\<qq>) local.carrier_quotient_ring local.add_rel local.mult_rel local.zero_rel a f" and "\<And>\<qq>. \<qq> \<in> V \<Longrightarrow> g \<notin> \<qq> \<and> t \<qq> = quotient_ring.frac (local.carrier_quotient_ring\<setminus>\<qq>) local.carrier_quotient_ring local.add_rel local.mult_rel local.zero_rel b g"
simps_6: fixes v :: "nat"   and uw :: "bdd" shows "Rmi_g (Suc v) Falseif uw = False"
R29_ac_2: shows "pmf (sds R29) d = 1 / 2"
simps_2: fixes s1 :: "'b stream"   and s2 :: "'c stream" shows "stl (s1 ||| s2) = stl s1 ||| stl s2"
skl1_step1_def: fixes Ra :: "fid_t"   and A :: "agent"   and B :: "agent" shows "skl1_step1 Ra A B \<equiv> {(s, s'). Ra \<notin> dom (progress s) \<and> guessed_runs Ra = \<lparr>role = Init, owner = A, partner = B\<rparr> \<and> s' = s\<lparr>progress := progress s(Ra \<mapsto> {xnx, xni, xgnx})\<rparr>}"
dup_def: fixes k :: "int" shows "Int.dup k = k + k"
preserves_iso: fixes f :: "'a" assumes "S.iso f" shows "S.iso (L f)"
ball_min_Int: fixes a :: "'b"   and r :: "real"   and s :: "real" shows "ball a (min r s) = ball a r \<inter> ball a s"
rScope: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and M :: "'a"   and xvec :: "name list"   and N :: "'a"   and P' :: "('a, 'b, 'c) psi"   and x :: "name"   and C :: "'d" assumes "\<Psi> \<rhd> P \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> P'"   and "\<And>C. Prop C \<Psi> P M (\<lparr>\<nu>*xvec\<rparr>N \<prec>' P')"   and "x \<sharp> \<Psi>"   and "x \<sharp> M"   and "x \<sharp> xvec"   and "x \<sharp> N"   and "xvec \<sharp>* \<Psi>"   and "xvec \<sharp>* P"   and "xvec \<sharp>* M"   and "x \<sharp> C"   and "xvec \<sharp>* C" shows "Prop C \<Psi> (\<lparr>\<nu>x\<rparr>P) M (\<lparr>\<nu>*xvec\<rparr>N \<prec>' \<lparr>\<nu>x\<rparr>P')"
simps: fixes s :: "'s" shows "local.lunstream s = (case g s of Done \<Rightarrow> LNil | Skip x \<Rightarrow> local.lunstream x | Yield x s' \<Rightarrow> LCons x (local.lunstream s'))"
cross7_mult_right: fixes x :: "(real, 7) vec"   and c :: "real"   and y :: "(real, 7) vec" shows "x \<times>\<^sub>7 (c *\<^sub>R y) = c *\<^sub>R x \<times>\<^sub>7 y"
list_all3_cons: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'a list"   and ys :: "'b list"   and zs :: "'c list"   and x :: "'a"   and y :: "'b"   and z :: "'c" assumes "list_all3 P xs ys zs"   and "P x y z" shows "list_all3 P (x # xs) (y # ys) (z # zs)"
eval_fm_inject: fixes e :: "(name, hf) finfun"   and A :: "fm"   and B :: "fm" shows "(\<lbrakk>\<guillemotleft>A\<guillemotright>\<rbrakk>e = \<lbrakk>\<guillemotleft>B\<guillemotright>\<rbrakk>e) = (A = B)"
comp_assoc: fixes h :: "'d"   and g :: "'d"   and f :: "'d" shows "D'.comp (D'.comp h g) f = D'.comp h (D'.comp g f)"
assms_2: shows "a \<in> hyp2 \<union> S"
insert_spec_set: fixes ref :: "'a"   and xs :: "'a list"   and oid :: "'a" assumes "ref \<in> set xs" shows "set (insert_spec xs (oid, Some ref)) = set xs \<union> {oid}"
leg1_is_map: shows "is_left_adjoint (tab\<^sub>1 s)"
induct: fixes P :: "nat \<Rightarrow> 'a list list \<Rightarrow> bool"   and a0 :: "nat"   and a1 :: "'a list list" assumes "\<And>acc. P acc []"   and "\<And>acc l ls. P (acc + length l) ls \<Longrightarrow> P acc (l # ls)" shows "P a0 a1"
inv_embed: fixes x :: "'b" shows "inv_embed (to_fract x) = x"
bin_ops_same_2: fixes x :: "int" shows "x OR x = x"
flatten_pi_2_length: fixes \<phi> :: "'b ltln"   and N :: "'b ltln set" shows "size \<phi>[N]\<^sub>\<Pi>\<^sub>2 \<le> 2 ^ (size \<phi> + 1)"
ring_hom_closed: fixes h :: "'c \<Rightarrow> 'd"   and R :: "('c, 'e) ring_scheme"   and S :: "('d, 'f) ring_scheme"   and x :: "'c" assumes "h \<in> ring_hom R S"   and "x \<in> carrier R" shows "h x \<in> carrier S"
ah_il_commute: fixes h1 :: "'a \<Rightarrow> 'a set"   and h2 :: "'a \<Rightarrow> 'a set" assumes "h1 [*] h2" shows "h2 [*] h1"
image_func1: fixes f :: "'e \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'e" shows "\<P> (f \<circ> g) = \<P> f \<circ> \<P> g"
rhom_to_rimg: fixes A :: "('c, 'd) Ring_scheme"   and R :: "('e, 'f) Ring_scheme"   and f :: "'c \<Rightarrow> 'e" assumes "Ring A"   and "Ring R"   and "f \<in> rHom A R" shows "f \<in> rHom A (rimg A R f)"
Qpres: fixes P :: "'prog"   and \<sigma> :: "'state"   and \<sigma>' :: "'state" assumes "Q P \<sigma>"   and "\<sigma>' \<in> small P \<sigma>" shows "Q P \<sigma>'"
monom_one: shows "monom 0 (1::'a) = 1"
succ_0_plus_eq: fixes \<alpha> :: "V" assumes "\<alpha> \<in>\<^sub>\<circ> \<omega>" shows "ZFC_in_HOL.succ 0 + \<alpha> = ZFC_in_HOL.succ \<alpha>"
cmp_def: shows "Tfg.cmp = Tfg_Hfg.chine"
simps_12: fixes e\<^sub>1 :: "expr"   and e\<^sub>2 :: "expr" shows "fv (e\<^sub>1;; e\<^sub>2) = fv e\<^sub>1 \<union> fv e\<^sub>2"
B_def: shows "B = (\<lambda>n. if n = 0 then space M - \<Union> (range A) else A (n - 1))"
nres_of_simps_3: fixes x :: "'c" shows "nres_of (dRETURN x) = RETURN x"
Shallow_axioms: shows "Shallow S s0 \<delta> AP L AP'"
enum_UNIV: fixes enum :: "'a list"   and enum_all :: "('a \<Rightarrow> bool) \<Rightarrow> bool"   and enum_ex :: "('a \<Rightarrow> bool) \<Rightarrow> bool" assumes "class.enum enum enum_all enum_ex" shows "set enum = UNIV"
cnj_mix_real: fixes z1 :: "complex"   and z2 :: "complex" shows "is_real (cnj_mix z1 z2)"
in_nres_rel_iff: fixes a :: "'g nres"   and b :: "'h nres"   and R :: "('g \<times> 'h) set" shows "((a, b) \<in> \<langle>R\<rangle>nres_rel) = (a \<le> \<Down> R b)"
append_eq_Cons_conv: fixes ys :: "'g list"   and zs :: "'g list"   and x :: "'g"   and xs :: "'g list" shows "(ys @ zs = x # xs) = (ys = [] \<and> zs = x # xs \<or> (\<exists>ys'. ys = x # ys' \<and> ys' @ zs = xs))"
natToVertexList_hd: fixes es :: "nat list"   and f :: "face"   and v :: "nat" assumes "incrIndexList es |es| |vertices f|" shows "hd (natToVertexList v f es) = Some v"
assms_3: shows "\<forall>y\<in>t. (0::'a) \<le> g y"
xorci_def: fixes a :: "nat"   and b :: "nat"   and s :: "bddi" shows "xorci a b s \<equiv> notci b s \<bind> (\<lambda>(nb, s). iteci_lu a nb b s)"
mult_numeral_left_semiring_numeral: fixes v :: "num"   and w :: "num"   and z :: "'a" shows "numeral v * (numeral w * z) = numeral (v * w) * z"
cd_impl_icd_cd: fixes i :: "nat"   and \<pi> :: "nat \<Rightarrow> 'n"   and l :: "nat"   and k :: "nat" assumes "i cd\<^bsup>\<pi>\<^esup>\<rightarrow> l"   and "i icd\<^bsup>\<pi>\<^esup>\<rightarrow> k"   and "\<not> i icd\<^bsup>\<pi>\<^esup>\<rightarrow> l" shows "k cd\<^bsup>\<pi>\<^esup>\<rightarrow> l"
upper_cos_float_interval: fixes p :: "nat"   and x :: "float interval" shows "upper (cos_float_interval p x) = snd (bnds_cos p (lower x) (upper x))"
not_pair_prm: fixes P :: "'b trm"   and \<pi> :: "'b prm"   and A :: "'b trm"   and B :: "'b trm" assumes "\<And>A B. P \<noteq> SimplyTyped.Pair A B" shows "\<pi> \<cdot> P \<noteq> SimplyTyped.Pair A B"
"5_1": assumes "\<And>l. l \<in> set ls \<Longrightarrow> sorted l"   and "\<And>l. l \<in> set (merge l1 l2 # acc2) \<Longrightarrow> sorted l" shows "sorted (merge_list (merge l1 l2 # acc2) ls) \<and> mset (merge_list (merge l1 l2 # acc2) ls) = mset (concat ((merge l1 l2 # acc2) @ ls)) \<and> set (merge_list (merge l1 l2 # acc2) ls) = set (concat ((merge l1 l2 # acc2) @ ls))"
ii: shows "\<forall>x\<in>E - E'. \<not> forest \<lparr>nodes = fst ` E \<union> (snd \<circ> snd) ` E, edges = insert x E'\<rparr>"
mopup_inv_start: fixes n :: "nat"   and am :: "nat list"   and ires :: "cell list"   and k :: "nat" assumes "n < length am" shows "mopup_inv (Suc 0, Bk # Bk # ires, <am> @ Bk \<up> k) am n ires"
of_real_arccos: fixes x :: "real" assumes "\<bar>x\<bar> \<le> 1" shows "complex_of_real (arccos x) = Arccos (complex_of_real x)"
asym: fixes x :: "'a"   and y :: "'a" assumes "x \<sqsubset> y"   and "y \<sqsubset> x"   and "x \<in> A"   and "y \<in> A" shows "False"
A5: shows "\<lfloor>(\<lambda>w. \<P> (\<lambda>x. \<forall>xa. xa x \<and> (\<forall>xb. xb x \<longrightarrow> (\<^bold>\<box>(\<lambda>v. \<forall>x. existsAt x v \<longrightarrow> xa x \<longrightarrow> xb x)) w) \<longrightarrow> (\<^bold>\<box>(\<lambda>v. \<exists>x. existsAt x v \<and> xa x)) w) w)\<rfloor>"
rel_conversep: fixes R1 :: "'g \<Rightarrow> 'd \<Rightarrow> bool"   and R2 :: "'h \<Rightarrow> 'e \<Rightarrow> bool"   and R3 :: "'i \<Rightarrow> 'f \<Rightarrow> bool" shows "F1rel R1\<inverse>\<inverse> R2\<inverse>\<inverse> R3\<inverse>\<inverse> = (F1rel R1 R2 R3)\<inverse>\<inverse>"
shift_def: fixes f :: "hf"   and delta :: "hf" shows "shift f delta = \<lbrace>v . u \<^bold>\<in> f, \<exists>n y. u = \<langle>n, y\<rangle> \<and> v = \<langle>delta @+ n, y\<rangle>\<rbrace>"
diff_add_assoc2: fixes k :: "nat"   and j :: "nat"   and i :: "nat" assumes "k \<le> j" shows "j + i - k = j - k + i"
rev_eq_flow: fixes y :: "'a"   and t :: "real" shows "rev.flow0 y t = flow0 y (- t)"
cases: fixes a :: "'a llist" assumes "maximal_path a" obtains "a = LNil"   | v :: "'a" where "a = LCons v__ LNil" and "deadend v__"   | Ps :: "'a llist" and v :: "'a" where "a = LCons v Ps" and "\<not> lnull Ps" and "maximal_path Ps"
Catch_decomp: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and p\<^sub>1 :: "('e, 'd, 'f) com"   and p\<^sub>2 :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate"   and p' :: "('e, 'd, 'f) com"   and s' :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile> (Catch p\<^sub>1 p\<^sub>2, s) \<rightarrow> (p', s')"   and "p\<^sub>1 \<noteq> Skip"   and "p\<^sub>1 \<noteq> Throw" shows "\<exists>p\<^sub>1'. \<Gamma>\<turnstile> (p\<^sub>1, s) \<rightarrow> (p\<^sub>1', s') \<and> p' = Catch p\<^sub>1' p\<^sub>2"
m: fixes x :: "'i list"   and x' :: "'a mod_ring poly" assumes "poly_rel x x'" shows "poly_rel (m x) (x' mod m')"
lunit_coherence: fixes a :: "'c" assumes "C.ide a" shows "D'.lunit (F a) = F (C'.lunit a) \<cdot>\<^sub>D \<phi> (snd (C'.unity, a), fst (C'.unity, a)) \<cdot>\<^sub>D D'.T (F'.\<psi>, F a)"
comp_ide_self: fixes a :: "'b" assumes "J1.ide a" shows "J1 a a = a"
int_msb_or: fixes x :: "int"   and y :: "int" shows "msb (x OR y) = (msb x \<or> msb y)"
is_vector_def: fixes x :: "'a" shows "is_vector x \<equiv> x = x ; 1"
incseq_SucD: fixes A :: "nat \<Rightarrow> 'a"   and i :: "nat" assumes "incseq A" shows "A i \<le> A (Suc i)"
thetaSCh_Sretr: fixes ch :: "'choice"   and c1 :: "('test, 'atom, 'choice) cmd"   and d1 :: "('test, 'atom, 'choice) cmd"   and c2 :: "('test, 'atom, 'choice) cmd"   and d2 :: "('test, 'atom, 'choice) cmd" assumes "compatCh ch"   and "c1 \<approx>s d1"   and "c2 \<approx>s d2" shows "thetaSCh ch c1 c2 d1 d2 \<subseteq> Sretr (thetaSCh ch c1 c2 d1 d2 \<union> Sbis)"
assms_2: shows "y \<sqsubseteq> \<rho>' x"
Compl_Diff_eq: fixes A :: "'a set"   and B :: "'a set" shows "- (A - B) = - A \<union> B"
intro: fixes return :: "'a \<Rightarrow> 'b"   and bind :: "'b \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b"   and get :: "('c \<Rightarrow> 'b) \<Rightarrow> 'b"   and put :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and sample :: "'d pmf \<Rightarrow> ('d \<Rightarrow> 'b) \<Rightarrow> 'b" assumes "monad_state return bind get put"   and "monad_prob return bind sample"   and "monad_state_prob_axioms get sample" shows "monad_state_prob return bind get put sample"
compose_Id: fixes g :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and B :: "'c set" assumes "g \<in> A \<rightarrow> B"   and "g \<in> extensional A" shows "compose A g (\<lambda>x\<in>A. x) = g"
prems: shows "b \<le> a \<or> p \<noteq> 0 \<and> count_roots_between p a b = (if poly p b = 0 then 1 else 0)"
ccINF_greatest: fixes A :: "'d set"   and u :: "'a"   and f :: "'d \<Rightarrow> 'a" assumes "countable A"   and "\<And>i. i \<in> A \<Longrightarrow> u \<le> f i" shows "u \<le> Inf (f ` A)"
thetaParZOL2_ZOretr: shows "thetaParZOL2 \<subseteq> ZOretr (thetaParZOL2 \<union> ZObis)"
select_convs_2: fixes cx :: "int"   and cy :: "int"   and more :: "'a" shows "cy \<lparr>cx = cx, cy = cy, \<dots> = more\<rparr> = cy"
Ex_finite_integrable_function: fixes M :: "'b measure" assumes "sigma_finite_measure M" shows "\<exists>h\<in>borel_measurable M. integral\<^sup>N M h \<noteq> \<infinity> \<and> (\<forall>x\<in>space M. 0 < h x \<and> h x < \<infinity>)"
iprev_mono2_if_neq_iMin: fixes n :: "nat"   and I :: "nat set" assumes "n \<in> I"   and "iMin I \<noteq> n" shows "iprev n I < n"
distinct_60: fixes x151 :: "floatarith"   and x152 :: "nat"   and x2 :: "floatarith" shows "x151 ^\<^sub>e x152 \<noteq> Minus x2"
aa_def: shows "aa = (\<Prod>j = 1..n. real_of_int (a j))"
mon_def: fixes G :: "('b, 'd) monoid_scheme"   and H :: "('c, 'e) monoid_scheme" shows "mon G H = {f \<in> hom G H. inj_on f (carrier G)}"
type_update_select_top_exists: fixes xs :: "checker_type"   and tm :: "ct list" assumes "type_update_select xs = TopType tm" shows "\<exists>tn. xs = TopType tn"
simps_2: fixes p :: "nat" shows "simplicial_subdivision (Suc p) = frag_extend (\<lambda>f. simplicial_cone p (\<lambda>i. (\<Sum>j\<le>Suc p. simplicial_vertex j f i) / (real p + 2)) (simplicial_subdivision p (chain_boundary (Suc p) (frag_of f))))"
strict_trans: fixes b :: "char"   and a :: "char"   and c :: "char" assumes "less_char b a"   and "less_char c b" shows "less_char c a"
self_in_downset: fixes x :: "'c" shows "x \<in> \<up>x"
B2_def: shows "B2 = B \<union> {UNIV}"
livePhi': fixes g :: "'g"   and v :: "'val"   and vs :: "'val list"   and v' :: "'val" assumes "liveVal' g (v # vs)"   and "phiArg g v v'" shows "liveVal' g (v' # v # vs)"
Im_Ln_eq: fixes z :: "complex" assumes "z \<noteq> 0" shows "Im (Ln z) = (if Re z \<noteq> 0 then if 0 < Re z then arctan (Im z / Re z) else if 0 \<le> Im z then arctan (Im z / Re z) + pi else arctan (Im z / Re z) - pi else if 0 < Im z then pi / 2 else - pi / 2)"
P_def: shows "P \<equiv> \<lambda>l u x. f x \<in> {l1 x..u1 x} \<longrightarrow> g x \<in> {l2 x..u2 x} \<longrightarrow> f x * g x \<in> {l..u}"
hyps_2: shows "P \<turnstile> D \<preceq>\<^sup>* C"
upd_incoming__elem: fixes nd :: "('b, 'c) node_scheme"   and n :: "('b, 'd) node_scheme"   and ns :: "('b, 'c) node_scheme set" assumes "nd \<in> upd_incoming n ns" shows "nd \<in> ns \<or> (\<exists>nd'\<in>ns. nd = nd'\<lparr>incoming := incoming n \<union> incoming nd'\<rparr> \<and> old nd' = old n \<and> next nd' = next n)"
f: shows "(\<Sum>i\<in>U. f i * i) = d"
fab: fixes w :: "complex" assumes "w \<in> M - {z}" shows "f w \<noteq> a \<and> f w \<noteq> b"
ln_Gamma_series_complex_converges'': fixes z :: "complex" assumes "z \<notin> \<int>\<^sub>\<le>\<^sub>0" shows "convergent (ln_Gamma_series z)"
lfinite_llength_enat: fixes xs :: "'b llist" assumes "lfinite xs" shows "\<exists>n. llength xs = enat n"
simps_22: fixes x1 :: "'b"   and x8 :: "'b local.term" shows "\<^bold>\<langle>x1\<^bold>\<rangle> \<noteq> \<^bold>\<r>\<^sup>-\<^sup>1\<^bold>[x8\<^bold>]"
simple_match_valid_def: fixes m :: "('a, 'b) simple_match_scheme" shows "simple_match_valid m \<equiv> ({p. simple_match_port (sports m) p} \<noteq> UNIV \<or> {p. simple_match_port (dports m) p} \<noteq> UNIV \<longrightarrow> proto m \<in> Proto ` {TCP, UDP, L4_Protocol.SCTP}) \<and> valid_prefix_fw (src m) \<and> valid_prefix_fw (dst m)"
AE_lim_stream: fixes x :: "real \<times> 'a" shows "AE \<omega> in K.lim_stream x. \<forall>i. snd ((x ## \<omega>) !! i) \<in> (SIGMA x:UNIV. set_pmf (J x))\<^sup>* `` {snd x} \<and> snd (\<omega> !! i) \<in> set_pmf (J (snd ((x ## \<omega>) !! i))) \<and> fst ((x ## \<omega>) !! i) < fst (\<omega> !! i)"
mpoly_coeff_uminus: fixes p :: "'b mpoly"   and m :: "nat \<Rightarrow>\<^sub>0 nat" shows "MPoly_Type.coeff (- p) m = - MPoly_Type.coeff p m"
lSup_insert_LNil: fixes Y :: "'b llist set" shows "lSup (insert <> Y) = lSup Y"
exchange_acyclic: fixes v :: "'a"   and e :: "'a"   and w :: "'a" assumes "v \<sqinter> top = v"   and "e \<le> v \<sqinter> - id v"   and "w \<sqinter> v \<le> v"   and "w \<sqinter> top \<le> - top" shows "(w \<sqinter> - (w \<sqinter> - id v \<sqinter> (top \<sqinter> e \<sqinter> top)) \<squnion> id (w \<sqinter> (- v \<sqinter> - id v) \<sqinter> (top \<sqinter> e \<sqinter> top)) \<squnion> e) \<sqinter> top \<le> - top"
assoc'_is_natural_2: fixes V :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and H :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<a> :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<i> :: "'c \<Rightarrow> 'c"   and src :: "'c \<Rightarrow> 'c"   and trg :: "'c \<Rightarrow> 'c"   and \<mu> :: "'c"   and \<nu> :: "'c"   and \<tau> :: "'c" assumes "bicategory V H \<a> \<i> src trg"   and "partial_magma.arr V \<mu>"   and "partial_magma.arr V \<nu>"   and "partial_magma.arr V \<tau>"   and "src \<mu> = trg \<nu>"   and "src \<nu> = trg \<tau>" shows "bicategory.\<a>' V H \<a> src trg \<mu> \<nu> \<tau> = V (bicategory.\<a>' V H \<a> src trg (partial_magma.cod V \<mu>) (partial_magma.cod V \<nu>) (partial_magma.cod V \<tau>)) (H \<mu> (H \<nu> \<tau>))"