hcju/isabelleps huggingface dataset:
intros_1: fixes \<gamma> :: "'c set \<Rightarrow> 'd \<Rightarrow> bool"   and fe :: "('c, 'e) flow_entry_match"   and p :: "'d"   and ft1 :: "('c, 'e) flow_entry_match list"   and ft2 :: "('c, 'e) flow_entry_match list" assumes "\<gamma> (ofe_fields fe) p = True"   and "\<forall>fe'\<in>set (ft1 @ ft2). ofe_prio fe < ofe_prio fe' \<longrightarrow> \<gamma> (ofe_fields fe') p = False" shows "guha_table_semantics \<gamma> (ft1 @ fe # ft2) p (Some (ofe_action fe))"
BT_lemma_1_15: fixes F :: "'b poly"   and B :: "'b poly"   and G :: "'b poly"   and H :: "'b poly" assumes "F + B * G = H"   and "degree G \<le> degree F"   and "degree H < degree G \<or> H = 0 \<and> F \<noteq> 0 \<and> G \<noteq> 0"   and "degree H < degree G \<or> H \<noteq> 0" shows "subresultant (degree G - 1) F G = Polynomial.smult ((- (1::'b)) ^ (degree F - degree G + 1) * lead_coeff G ^ (degree F - degree G + 1)) H"
mkarr_in_hom: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and t :: "'b local.term" assumes "free_monoidal_category C"   and "monoidal_language.Arr C t" shows "partial_magma.in_hom (free_monoidal_category.comp C) (free_monoidal_category.mkarr C t) (free_monoidal_category.mkarr C (monoidal_language.Dom C t)) (free_monoidal_category.mkarr C (monoidal_language.Cod C t))"
flow: shows "flow \<Delta> (\<lambda>x. ennreal (f' x))"
subset_card: fixes x :: "'a set" assumes "x \<in># (\<Sum>b2\<in>#remove1_mset blv \<B>. mset_set {y. y \<subseteq> blv \<inter> b2 \<and> card y = 2})" shows "card x = 2"
prime_gt_1_int: fixes p :: "int" assumes "prime p" shows "1 < p"
hnorm_inverse: fixes a :: "'a star" shows "hnorm (inverse a) = inverse (hnorm a)"
hyps_2: assumes "\<Gamma>|\<^bsub>M\<^esub>\<turnstile> \<langle>c1,Normal s\<rangle> \<Rightarrow>\<notin>{Stuck}" shows "\<Gamma>\<turnstile>c1 \<down> Normal s"
subset_sublist_exists: fixes ys :: "'a set"   and xs :: "'a list" assumes "ys \<subseteq> set xs" shows "\<exists>zs. set zs = ys \<and> zs \<in> set (subseqs xs)"
subprob_algebra_cong: fixes M :: "'b measure"   and N :: "'b measure" assumes "sets M = sets N" shows "subprob_algebra M = subprob_algebra N"
quasicomm_1: fixes y :: "'a"   and x :: "'a" shows "(y \<sqinter> x \<le> x \<sqinter> top) = (top \<sqinter> x \<le> x \<sqinter> top)"
exec_until_stop_def: fixes joint_oracle :: "'f \<Rightarrow> 'g \<Rightarrow> 'd \<Rightarrow> (('e option \<times> 'f) option \<times> ('e option \<times> 'g) option) pmf"   and callee1 :: "'f \<Rightarrow> 'd \<Rightarrow> ('e option \<times> 'f) spmf" shows "exec_until_stop joint_oracle callee1 \<equiv> curry (curry (curry (spmf.fixp_fun (\<lambda>exec_until_stop (((gpv, s1), s2), b). if b then the_gpv gpv \<bind> (\<lambda>generat. case generat of Pure x \<Rightarrow> return_spmf (x, s1, s2) | IO out rpv \<Rightarrow> joint_oracle s1 s2 out \<bind> (\<lambda>(a, b). case a of None \<Rightarrow> return_pmf None | Some (r1, s1') \<Rightarrow> case b of Some (r2, s2') \<Rightarrow> case (r1, r2) of (None, None) \<Rightarrow> curry (curry (curry exec_until_stop)) (Generative_Probabilistic_Value.Done None) s1' s2' True | (None, _) \<Rightarrow> curry (curry (curry exec_until_stop)) (Generative_Probabilistic_Value.Done None) s1' s2' True | (Some r1', None) \<Rightarrow> curry (curry (curry exec_until_stop)) (rpv r1') s1' s2' False | (Some r1', Some r2') \<Rightarrow> curry (curry (curry exec_until_stop)) (rpv r1') s1' s2' True)) else the_gpv gpv \<bind> (\<lambda>generat. case generat of Pure x \<Rightarrow> return_spmf (None, s1, s2) | IO out rpv \<Rightarrow> callee1 s1 out \<bind> (\<lambda>(r1, s1'). case r1 of None \<Rightarrow> curry (curry (curry exec_until_stop)) (Generative_Probabilistic_Value.Done None) s1' s2 False | Some r1' \<Rightarrow> curry (curry (curry exec_until_stop)) (rpv r1') s1' s2 False))))))"
inject_1: fixes x11 :: "'c option"   and x12 :: "'d"   and y11 :: "'c option"   and y12 :: "'d" shows "(Insert x11 x12 = Insert y11 y12) = (x11 = y11 \<and> x12 = y12)"
simulation_silents1: fixes s1 :: "'s1"   and s2 :: "'s2"   and s1' :: "'s1" assumes "s1 \<approx> s2"   and "s1 -\<tau>1\<rightarrow>* s1'" shows "\<exists>s2'. s2 -\<tau>2\<rightarrow>* s2' \<and> s1' \<approx> s2'"
T_spaceM_stable_1: fixes x :: "'a" assumes "x \<in> space M" shows "T x \<in> space M"
lset_intros_2: fixes x :: "'c"   and xs :: "'c llist"   and x' :: "'c" assumes "x \<in> lset xs" shows "x \<in> lset (LCons x' xs)"
nn_integral_def: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> ennreal" shows "integral\<^sup>N M f = \<Squnion> (integral\<^sup>S M ` {g. simple_function M g \<and> g \<le> f})"
closed_subst_bv_no_change: fixes t :: "Core.term"   and u :: "Core.term" assumes "is_closed t" shows "subst_bv u t = t"
inf_prefix_le1: fixes xs :: "'b list"   and ys :: "'b list" shows "prefix (xs \<sqinter> ys) xs"
sminus_zero: fixes v :: "'a" shows "local.punit.sminus v (0::'a) = v"
comb_planar_impl_correct: fixes G :: "'c list \<times> ('c \<times> 'c) list" assumes "pair_graph (list_digraph G)"   and "distinct (fst G)"   and "distinct (snd G)" shows "comb_planar (with_proj (list_digraph G)) = comb_planar_impl G"
simps: fixes ap :: "abc_inst list" shows "tms_of ap = map (\<lambda>(x, y). ci (layout_of ap) x y) (tpairs_of ap)"
pair_commute_pmf: fixes x :: "'b pmf"   and y :: "'c pmf" shows "pair_pmf x y = map_pmf (\<lambda>(x, y). (y, x)) (pair_pmf y x)"
complementary_symm: fixes A :: "'a set"   and B :: "'a set" shows "H.complementary A B = H.complementary B A"
fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s_inI_5: fixes t :: "('a, 'b) Term.term"   and s :: "('c, 'b) Term.term"   and F :: "(('a, 'b) Term.term \<times> ('c, 'b) Term.term) list"   and x :: "'b" assumes "(t, s) \<in> set F"   and "x \<in> fv s" shows "x \<in> fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s F"
trans: fixes b :: "'loc \<times> 't"   and a :: "'loc \<times> 't"   and c :: "'loc \<times> 't" assumes "b\<le>\<^sub>pa"   and "c\<le>\<^sub>pb" shows "c\<le>\<^sub>pa"
univ_second_countable: obtains \<B> :: "'b set set" where "countable \<B>" and "\<And>C. C \<in> \<B> \<Longrightarrow> open C" and "\<And>S. open S \<Longrightarrow> \<exists>U\<subseteq>\<B>. S = \<Union> U"
g: shows "\<forall>z\<in>Z. S (g z) z \<and> (\<exists>x\<in>X. R x (g z))"
cyclic: fixes ccw :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and S :: "'b set"   and p :: "'b"   and q :: "'b"   and r :: "'b" assumes "ccw_system12 ccw S"   and "ccw p q r" shows "ccw q r p"
scalar_mult_def: fixes a :: "'b"   and m :: "'b matrix" shows "scalar_mult a m \<equiv> apply_matrix ((*) a) m"
disj_conj_distrib2: fixes y :: "'c"   and z :: "'c"   and x :: "'c" shows "y AND z OR x = (y OR x) AND (z OR x)"
proper_def: fixes S :: "'c set" shows "proper S \<equiv> \<forall>x r. compact (cball x r \<inter> S)"
sel_specD1: fixes sel :: "(('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list \<Rightarrow> (('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list \<Rightarrow> ((('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) \<times> ('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list \<Rightarrow> nat \<times> 'h \<Rightarrow> ((('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) \<times> ('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list"   and ps :: "((('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) \<times> ('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list"   and gs :: "(('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list"   and bs :: "(('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list"   and data :: "nat \<times> 'h" assumes "sel_spec sel"   and "ps \<noteq> []" shows "sel gs bs ps data \<noteq> []"
2: shows "rv \<in> fringe prb \<or> rv = tgt re"
TUA: shows "PUadmitFO \<sigma> \<alpha> U"
suspendable_prefix_invariant: fixes \<xi> :: "'a ltln"   and u :: "'a set list"   and w :: "nat \<Rightarrow> 'a set" assumes "suspendable \<xi>" shows "(u \<frown> w) \<Turnstile>\<^sub>n \<xi> = w \<Turnstile>\<^sub>n \<xi>"
set_spmf_sample_primes: shows "set_spmf sample_primes \<subseteq> {x. prime x \<and> 2 < x}"
same_length_Cons: fixes xs :: "'c list"   and ys :: "'d list"   and x :: "'c"   and y :: "'d" assumes "same_length xs ys" shows "same_length (x # xs) (y # ys)"
covering_space_lift_stronger: fixes C :: "'c set"   and p :: "'c \<Rightarrow> 'd"   and S :: "'d set"   and a :: "'c"   and z :: "'e"   and U :: "'e set"   and f :: "'e \<Rightarrow> 'd" assumes "covering_space C p S"   and "a \<in> C"   and "z \<in> U"   and "path_connected U"   and "locally path_connected U"   and "continuous_on U f"   and "f ` U \<subseteq> S"   and "f z = p a"   and "\<And>r. \<lbrakk>path r; path_image r \<subseteq> U; pathstart r = z; pathfinish r = z\<rbrakk> \<Longrightarrow> \<exists>b. homotopic_paths S (f \<circ> r) (linepath b b)" obtains g :: "'e \<Rightarrow> 'c" where "continuous_on U g" and "g ` U \<subseteq> C" and "g z = a" and "\<And>y. y \<in> U \<Longrightarrow> p (g y) = f y"
X: shows "(mk \<lparr>heap = \<sigma>'(oid \<mapsto> Object), assocs = A\<rparr> \<Turnstile> P (OclAllInstances_generic pre_post Type)) = (mk \<lparr>heap = \<sigma>', assocs = A\<rparr> \<Turnstile> \<lambda>_. P (\<lambda>_. OclAllInstances_generic pre_post Type (mk \<lparr>heap = \<sigma>'(oid \<mapsto> Object), assocs = A\<rparr>)) (mk \<lparr>heap = \<sigma>', assocs = A\<rparr>))"
wf: shows "wf {(X, Y). X \<subset> Y \<and> Y \<subseteq> alts}"
dim_null_space: fixes A :: "(('a, 'b) vec, 'c) vec" shows "vec.dim (null_space A) = finite_dimensional_vector_space.dimension cart_basis - rank A"
poslivenessI: fixes A :: "'b set"   and P :: "'b llist set" assumes "\<And>s. s \<in> A\<^sup>\<clubsuit> \<Longrightarrow> \<exists>t\<in>A\<^sup>\<infinity>. s @@ t \<in> P" shows "posliveness A P"
eq_is_equal: shows "(.=\<^bsub>\<P>\<^esub>) = (=)"
einterval_eq_Ici: fixes a :: "real" shows "einterval (ereal a) \<infinity> = {a<..}"
Pair_transfer: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and B :: "'c \<Rightarrow> 'd \<Rightarrow> bool" shows "(A ===> B ===> rel_prod A B) Pair Pair"
vTeT: fixes v :: "'a"   and e :: "'a" assumes "vector v"   and "e \<le> v * - v\<^sup>T" shows "v\<^sup>T * e\<^sup>T = bot"
deadlocked_elims: fixes t :: "'t"   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 "t \<in> deadlocked s" obtains x :: "'x" where "thr s t = \<lfloor>(x__, no_wait_locks)\<rfloor>" and "t \<turnstile> \<langle>x__, shr s\<rangle> \<wrong>" and "wset s t = None" and "\<And>LT. t \<turnstile> \<langle>x__, shr s\<rangle> LT \<wrong> \<Longrightarrow> \<exists>lt\<in>LT. must_wait s t lt (deadlocked s \<union> final_threads s)"   | x :: "'x" and ln :: "'l \<Rightarrow>f nat" where "thr s t = \<lfloor>(xa__, ln__)\<rfloor>" and "all_final_except s (deadlocked s)" and "waiting (wset s t)"   | x :: "'x" and ln :: "'l \<Rightarrow>f nat" and l :: "'l" and t' :: "'t" where "thr s t = \<lfloor>(x, ln)\<rfloor>" and "\<not> waiting (wset s t)" and "0 < ln $ l" and "has_lock (locks s $ l) t'" and "t \<noteq> t'" and "t' \<in> deadlocked s \<or> final_thread s t'"
T8_3: fixes w :: "i"   and \<psi> :: "\<Pi>\<^sub>3 \<Rightarrow> \<o>" shows "[\<^bold>\<forall>\<^sub>3x. \<psi> x in w] = (\<forall>x. [\<psi> x in w])"
pure_eventual_strong_release_simp: fixes \<phi> :: "'a ltln"   and \<psi> :: "'a ltln"   and w :: "nat \<Rightarrow> 'a set" assumes "pure_eventual \<phi>"   and "pure_eventual \<psi>" shows "w \<Turnstile>\<^sub>n \<phi> M\<^sub>n \<psi> = w \<Turnstile>\<^sub>n \<phi> and\<^sub>n \<psi>"
dgrad_set_le_trans: fixes d :: "'c \<Rightarrow> nat"   and S :: "'c set"   and T :: "'c set"   and U :: "'c set" assumes "dgrad_set_le d S T"   and "dgrad_set_le d T U" shows "dgrad_set_le d S U"
connected_closed: fixes s :: "'b set" shows "connected s = (\<nexists>A B. closed A \<and> closed B \<and> s \<subseteq> A \<union> B \<and> A \<inter> B \<inter> s = {} \<and> A \<inter> s \<noteq> {} \<and> B \<inter> s \<noteq> {})"
collapse_2: fixes sum :: "'h + 'i" assumes "\<not> isl sum" shows "Inr (projr sum) = sum"
simps_2: fixes m :: "'c match_expr"   and rs :: "'c rule list" shows "process_ret (Rule m Return # rs) = add_match (MatchNot m) (process_ret rs)"
list_all2_iff: fixes P :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and xs :: "'g list"   and ys :: "'h list" shows "list_all2 P xs ys = (length xs = length ys \<and> (\<forall>(x, y)\<in>set (zip xs ys). P x y))"
PREFER_I: fixes P :: "'g \<Rightarrow> bool"   and x :: "'g" assumes "P x" shows "PREFER P x"
semilattice_order_axioms: shows "semilattice_order max (\<lambda>x y. y \<le> x) (\<lambda>x y. y < x)"
neg_mono: fixes A :: "fmla"   and H :: "fmla set"   and B :: "fmla" assumes "insert A H \<turnstile> B" shows "insert (neg B) H \<turnstile> neg A"
Gromov_extension_isometry_inv_1: fixes f :: "'c \<Rightarrow> 'd" assumes "isometry f" shows "inv (Gromov_extension f) = Gromov_extension (inv f)"
tl_Nil: fixes xs :: "'g list" shows "(tl xs = []) = (xs = [] \<or> (\<exists>x. xs = [x]))"
has_integral_nonneg: fixes f :: "'c \<Rightarrow> real"   and i :: "real"   and S :: "'c set" assumes "(f has_integral i) S"   and "\<And>x. x \<in> S \<Longrightarrow> 0 \<le> f x" shows "0 \<le> i"
d2_var: fixes p :: "'a"   and x :: "'a" assumes "test p" shows "d (p \<sqinter> x) \<le> p"
cos_npi: fixes n :: "nat" shows "cos (real n * pi) = (- 1) ^ n"
Fvars_\<phi>G: shows "Fvars \<phi>G = {}"
reduced_GB_D4: fixes G :: "('t \<Rightarrow>\<^sub>0 'c) set"   and g :: "'t \<Rightarrow>\<^sub>0 'c" assumes "is_reduced_GB G"   and "g \<in> G" shows "g \<noteq> 0"
split_invar: fixes t :: "('b \<times> nat) tree" assumes "invar t" shows "split t = t"
topological_basis_open: fixes "open" :: "'d set \<Rightarrow> bool"   and B :: "'d set set"   and X :: "'d set" assumes "class.topological_space open"   and "topological_space.topological_basis open B"   and "X \<in> B" shows "open X"
stack_xlift_compxEs2: fixes n :: "nat"   and es :: "'a expr1 list"   and pc :: "nat"   and d :: "nat" shows "stack_xlift n (compxEs2 es pc d) = compxEs2 es pc (n + d)"
gallery_append_reduce1: fixes X :: "'b set set"   and xs :: "'b set list"   and ys :: "'b set list" assumes "ChamberComplex X"   and "SimplicialComplex.maxsimpchain X (xs @ ys)" shows "SimplicialComplex.maxsimpchain X xs"
op_DIM_def: shows "op_DIM TYPE('b) = DIM('b)"
thetaIfTT_RetrT: fixes indis :: "('a \<times> 'a) set"   and tval :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'a \<Rightarrow> 'a" assumes "PL_Indis indis" shows "PL_Indis.thetaIfTT tval aval indis \<subseteq> PL_Indis.RetrT tval aval indis (PL_Indis.thetaIfTT tval aval indis \<union> PL_Indis.BisT tval aval indis)"
Class_commutes_with_composition: fixes a :: "'a"   and b :: "'a" assumes "a \<in> R"   and "b \<in> R" shows "additive.Class a [\<cdot>] additive.Class b = additive.Class (a \<cdot> b)"
reachable_def: fixes A :: "('e, 'f, 'd, 'g) FSM_scheme" shows "reachable A \<equiv> transition_system.reachable (\<lambda>a p. snd a) (\<lambda>a p. snd a \<in> succ A (fst a) p)"
scale_left_diff_distrib: fixes a :: "real"   and b :: "real"   and x :: "'a" shows "(a - b) *\<^sub>R x = a *\<^sub>R x - b *\<^sub>R x"
indecomposable_\<omega>_power: fixes \<delta> :: "V" assumes "Ord \<delta>" shows "indecomposable (\<omega> \<up> \<delta>)"
that: fixes c :: "'a" assumes "homotopic_with_canon (\<lambda>z. True) (rel_frontier S) (rel_frontier T) f (\<lambda>x. c)" shows "thesis"
dvd_diff: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "class.comm_ring_1 times one plus zero minus uminus"   and "dvd.dvd times x y"   and "dvd.dvd times x z" shows "dvd.dvd times x (minus y z)"
ordered_insert_pairs_map: fixes x :: "'a"   and v :: "'b"   and ys :: "('a \<times> 'b) list" shows "map_of_alist (ordered_insert_pairs x v ys) =  map_of_alist ys { x \<rightarrow> v }"
in_homI: fixes f :: "'c local.term set"   and a :: "'c local.term set"   and b :: "'c local.term set" assumes "\<F>\<^sub>SC.arr f"   and "\<F>\<^sub>SC.dom f = a"   and "\<F>\<^sub>SC.cod f = b" shows "\<F>\<^sub>SC.in_hom f a b"
lifted_less_eq_def: fixes f :: "'a \<Rightarrow> 'a"   and g :: "'a \<Rightarrow> 'a" shows "f \<le>\<le> g \<equiv> \<forall>x. f x \<le> g x"
eval_single: fixes x :: "'a" shows "pred.eval (Predicate.single x) = (=) x"
arev_arev_raw: fixes G :: "('a, 'b) pre_digraph"   and arev :: "'b \<Rightarrow> 'b"   and a :: "'b" assumes "bidirected_digraph G arev"   and "a \<in> arcs G" shows "arev (arev a) = a"
chain_lprefix_ltl: fixes A :: "'c llist set" assumes "Complete_Partial_Order.chain (\<sqsubseteq>) A" shows "Complete_Partial_Order.chain (\<sqsubseteq>) (ltl ` (A \<inter> {xs. \<not> lnull xs}))"
satisifies_atom_restrict_to_Cons: fixes v :: "nat \<Rightarrow> 'a"   and as :: "('b \<times> 'a atom) list"   and I :: "'b set"   and i :: "'b"   and a :: "'a atom" assumes "v \<Turnstile>\<^sub>a\<^sub>s (snd ` (set as \<inter> I \<times> UNIV))"   and "i \<in> I \<Longrightarrow> v \<Turnstile>\<^sub>a a" shows "v \<Turnstile>\<^sub>a\<^sub>s (snd ` (set ((i, a) # as) \<inter> I \<times> UNIV))"
HFinite_inverse: fixes x :: "'a star" assumes "x \<in> HFinite"   and "x \<notin> Infinitesimal" shows "inverse x \<in> HFinite"
dd_lt_d: shows "\<forall>D'. D' \<in># DD \<longrightarrow> D' < D"
trivial_hom: fixes H :: "('b, 'c) monoid_scheme"   and G :: "('d, 'e) monoid_scheme" assumes "Group.group H" shows "(\<lambda>x. \<one>\<^bsub>H\<^esub>) \<in> hom G H"
M5exist_var2: fixes meets :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and \<I> :: "'a \<Rightarrow> bool"   and p :: "'a"   and q :: "'a" assumes "class.interval meets \<I>"   and "meets p q" shows "\<exists>r1 r2 r3 s t. meets r1 r2 \<and> meets r2 r3 \<and> meets r3 p \<and> meets p q \<and> meets q s \<and> meets r1 t \<and> meets t s"
outputPres: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and Rel :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set"   and M :: "'a"   and N :: "'a" assumes "(\<Psi>, P, Q) \<in> Rel" shows "\<Psi> \<rhd> M\<langle>N\<rangle>.P \<leadsto>[Rel] M\<langle>N\<rangle>.Q"
unit_closed: shows "\<zero>\<^bsub>V\<^esub> \<in> \<FF> V"
poly_divide_filterlim_at_bot: fixes p :: "real poly"   and q :: "real poly" assumes "p \<noteq> 0"   and "q \<noteq> 0" shows "LIM x at_bot. poly q x / poly p x :> if degree q < degree p then at 0 else if degree q = degree p then nhds (lead_coeff q / lead_coeff p) else if 0 < sgn_neg_inf q * sgn_neg_inf p then at_top else at_bot"
orderI: fixes a :: "'b"   and b :: "'b" assumes "a = a \<sqinter> b" shows "a \<le> b"
simps_1: fixes n :: "nat"   and k :: "nat"   and i :: "nat" shows "\<up>\<^sub>\<tau> n k (TVar i) = (if i < k then TVar i else TVar (i + n))"
size_simp1: fixes t :: "'c"   and ts :: "'c list" assumes "t \<in> set ts" shows "size t < Suc (size_list size ts)"
length_filter_le: fixes P :: "'a \<Rightarrow> bool"   and xs :: "'a list" shows "|filter P xs| \<le> |xs|"
ide_in_hom_2: fixes f :: "'a" assumes "B.ide f" shows "\<guillemotleft>f : f \<rightarrow>\<^sub>B f\<guillemotright>"
lappend_snocL1_conv_LCons2: fixes xs :: "'b llist"   and y :: "'b"   and ys :: "'b llist" shows "(xs @@ y ## LNil) @@ ys = xs @@ y ## ys"
rtrancl_on_trans: fixes a :: "'c"   and b :: "'c"   and F :: "'c set"   and r :: "('c \<times> 'c) set"   and c :: "'c" assumes "(a, b) \<in> rtrancl_on F r"   and "(b, c) \<in> rtrancl_on F r" shows "(a, c) \<in> rtrancl_on F r"
iprod_def: fixes xs :: "'b list"   and ys :: "'b list" shows "\<langle>xs,ys\<rangle> = sum_list (map2 (*) xs ys)"
hinsert_commute: fixes x :: "hf"   and y :: "hf"   and z :: "hf" shows "z \<triangleleft> y \<triangleleft> x = z \<triangleleft> x \<triangleleft> y"
3134: fixes x :: "'a"   and y :: "'a" shows "- (- x \<sqinter> - y) \<sqinter> y = y"
IH: assumes "\<forall>a\<in>set A \<union> B. case a of (j, p) \<Rightarrow> \<forall>a\<in>set A \<union> B. case a of (k, q) \<Rightarrow> p = q \<longrightarrow> j = k" shows "filter (\<lambda>d. d \<notin> snd ` B) (map snd A) = map snd (filter (\<lambda>d. d \<notin> B) A)"
tyenv_wellformed_preds_update: fixes P' :: "'BExp set"   and P'' :: "'BExp set"   and \<S> :: "'Var set \<times> 'Var set"   and mds :: "Mode \<Rightarrow> 'Var set"   and \<Gamma> :: "'Var \<Rightarrow> 'BExp set option"   and P :: "'BExp set" assumes "P' = P'' |` {v. stable \<S> v}"   and "tyenv_wellformed mds \<Gamma> \<S> P" shows "tyenv_wellformed mds \<Gamma> \<S> P'"
iT_defs_4: fixes r :: "nat"   and m :: "nat" shows "[ r, mod m ] \<equiv> {x. x mod m = r mod m \<and> r \<le> x}"
inflow_t_outflow_s: shows "sum max_flow (incoming t) = sum max_flow (outgoing s)"
subgraph_imp_subverts: fixes H :: "('e, 'f) pre_digraph"   and G :: "('e, 'f) pre_digraph" assumes "subgraph H G" shows "verts H \<subseteq> verts G"
arrI: fixes f :: "'a"   and a :: "'a"   and b :: "'a" assumes "\<guillemotleft>f : a \<rightarrow>\<^sub>C b\<guillemotright>" shows "C.arr f"
addition_correct': fixes m :: "nat"   and n :: "nat"   and p :: "nat"   and lm :: "nat list" assumes "m \<noteq> n"   and "max m n < p"   and "p < length lm"   and "lm ! p = 0" shows "\<exists>stp. case abc_steps_l (0, lm) (addition m n p) stp of (as, lm') \<Rightarrow> as = 7 \<and> addition_inv (as, lm') m n p lm"
ostep_invariantD: fixes A :: "('a \<times> 'b, 'c) automaton"   and S :: "'a \<Rightarrow> 'a \<Rightarrow> 'c \<Rightarrow> bool"   and U :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and P :: "('a \<times> 'b) \<times> 'c \<times> 'a \<times> 'b \<Rightarrow> bool"   and \<sigma> :: "'a"   and s :: "'b"   and a :: "'c"   and \<sigma>' :: "'a"   and s' :: "'b" assumes "A \<Turnstile>\<^sub>A (S, U \<rightarrow>) P"   and "(\<sigma>, s) \<in> oreachable A S U"   and "((\<sigma>, s), a, \<sigma>', s') \<in> automaton.trans A"   and "S \<sigma> \<sigma>' a" shows "P ((\<sigma>, s), a, \<sigma>', s')"
dist_nz: fixes dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.metric_space dist uniformity open" shows "(x \<noteq> y) = (0 < dist x y)"
partition_from_equivalence: shows "Congruence.partition (carrier S) classes"
trans: fixes P :: "('procS, 'procT) Proc"   and Q :: "('procS, 'procT) Proc"   and R :: "('procS, 'procT) Proc" assumes "P \<lesssim>\<lbrakk>\<cdot>\<rbrakk>R Q"   and "Q \<lesssim>\<lbrakk>\<cdot>\<rbrakk>R R" shows "P \<lesssim>\<lbrakk>\<cdot>\<rbrakk>R R"
group_decompose_adjoin_aux: fixes H :: "'a set" assumes "subgroup H G" shows "H = carrier G \<or> (\<exists>H' a. H \<subseteq> H' \<and> subgroup H' G \<and> a \<in> carrier G - H' \<and> carrier G = adjoin G H' a)"
find_min_p_min: fixes l :: "nat"   and xs :: "('c, 'd) bintree option list" assumes "is_binqueue l xs"   and "xs \<noteq> []"   and "normalized xs"   and "distinct (vals xs)"   and "distinct (prios xs)" shows "priority (the (find_min xs)) = the (PQ.priority (pqueue xs) (PQ.min (pqueue xs)))"
eval_embed_false: fixes P :: "'c \<Rightarrow> bool"   and s :: "'c" assumes "\<not> P s" shows "\<guillemotleft> P \<guillemotright> s = 0"
iso_lunit: fixes f :: "'a" assumes "C.ide f" shows "C.iso \<l>\<^sub>C[f]"
o_3: shows "xsrcp \<in> set (if fst (sports m) = 0 \<and> snd (sports m) = - 1 then [None] else if fst (sports m) \<le> snd (sports m) then map (Some \<circ> (\<lambda>pfx. (pfxm_prefix pfx, ~~ pfxm_mask pfx))) (wordinterval_CIDR_split_prefixmatch (WordInterval (fst (sports m)) (snd (sports m)))) else [])"
error_absE: fixes err :: "real"   and k :: "real" assumes "\<bar>err\<bar> \<le> k" obtains e :: "real" where "err = e * k" and "e \<in> {- 1..1}"
delete_Leaf: fixes T :: "bool list" shows "delete T Leaf = Leaf"
axioms_2: fixes N :: "'c set"   and M :: "'c set"   and composition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit :: "'c" assumes "submonoid N M composition unit" shows "submonoid_axioms N M composition unit"
fst_def: fixes prod :: "'g \<times> 'h" shows "fst prod = (case prod of (x1, x2) \<Rightarrow> x1)"
tail_add_arc: fixes G :: "('d, 'c) pre_digraph"   and a :: "'c" shows "tail (pre_digraph.add_arc G a) = tail G"