hcju/isabelleps huggingface dataset:
mod_add_self1: fixes divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and modulo :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and euclidean_size :: "'d \<Rightarrow> nat"   and b :: "'d"   and a :: "'d" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size" shows "modulo (plus b a) b = modulo a b"
exhaust_sel: fixes insert_message :: "('a, 'b) insert_message" obtains "insert_message = InsertMessage (P insert_message) (I insert_message) (S insert_message) (\<Sigma> insert_message)"
reduced_IF_depends: fixes t :: "'c ifex"   and X :: "'c set" assumes "reduced t X"   and "t \<noteq> Trueif"   and "t \<noteq> Falseif" shows "\<exists>s1 s2. val_ifex t s1 \<noteq> val_ifex t s2"
prime_elem_iff_irreducible: fixes x :: "'a" shows "prime_elem x = irreducible x"
R_assignl: fixes P :: "(char list \<Rightarrow> 'a) \<Rightarrow> bool"   and P' :: "(char list \<Rightarrow> 'a) \<Rightarrow> bool"   and v :: "char list"   and e :: "(char list \<Rightarrow> 'a) \<Rightarrow> 'a"   and Q :: "(char list \<Rightarrow> 'a) \<Rightarrow> bool" assumes "\<forall>s. P s \<longrightarrow> P' (s(v := e s))" shows "(v ::= e) ; rel_R \<lceil>P'\<rceil> \<lceil>Q\<rceil> \<subseteq> rel_R \<lceil>P\<rceil> \<lceil>Q\<rceil>"
axioms_1: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f0 :: "'b"   and f1 :: "'b"   and p0 :: "'b"   and p1 :: "'b" assumes "binary_product_cone C f0 f1 p0 p1" shows "category C"
coeffs_map_poly_hom: fixes p :: "'b mod_ring poly" shows "coeffs (to_int_poly p) = map to_int_mod_ring (coeffs p)"
cont: shows "continuous_on dom (\<lambda>x. flow0 x t)"
e_type_comp_conc2: fixes \<S> :: "s_context"   and \<C> :: "t_context"   and es :: "e list"   and es' :: "e list"   and es'' :: "e list"   and t1s :: "t list"   and t2s :: "t list" assumes "\<S>\<bullet>\<C> \<turnstile> es @ es' @ es'' : t1s _> t2s" shows "\<exists>ts' ts''. \<S>\<bullet>\<C> \<turnstile> es : t1s _> ts' \<and> \<S>\<bullet>\<C> \<turnstile> es' : ts' _> ts'' \<and> \<S>\<bullet>\<C> \<turnstile> es'' : ts'' _> t2s"
injI: fixes f :: "'b \<Rightarrow> 'c" assumes "\<And>x y. f x = f y \<Longrightarrow> x = y" shows "inj f"
word_sless_msb_less: fixes x :: "'a word"   and y :: "'a word" shows "(x <s y) = ((msb y \<longrightarrow> msb x) \<and> (msb x \<and> \<not> msb y \<or> x < y))"
assms_6: shows "ptr \<noteq> cast\<^sub>c\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>_\<^sub>d\<^sub>a\<^sub>t\<^sub>a\<^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 |h \<turnstile> create_character_data document_ptr text|\<^sub>r"
leg0_is_map: shows "is_left_adjoint s\<^sub>0"
lim1: shows "(((\<lambda>x. - 1 / ln x) \<circ> real_of_ereal) \<longlongrightarrow> - (1 / ln a)) (at_right (ereal a))"
cases: fixes P :: "real \<Rightarrow> bool"   and a1 :: "real"   and a2 :: "real" assumes "finite_Psegments P a1 a2" obtains a :: "real" and b :: "real" where "a1 = a__" and "a2 = b__" and "b__ \<le> a__"   | s :: "real" and a :: "real" and b :: "real" where "a1 = aa__" and "a2 = ba__" and "sa__ \<in> {aa__..<ba__}" and "sa__ = aa__ \<or> P sa__" and "\<forall>t\<in>{sa__<..<ba__}. P t" and "finite_Psegments P aa__ sa__"   | s :: "real" and a :: "real" and b :: "real" where "a1 = a" and "a2 = b" and "s \<in> {a..<b}" and "s = a \<or> P s" and "\<forall>t\<in>{s<..<b}. \<not> P t" and "finite_Psegments P a s"
p\<^sub>1: shows "on_curvep a b p\<^sub>1"
dist_les_7: fixes cfun' :: "ValD \<rightarrow> ValD" shows "ValFF \<notsqsubseteq> ValF\<cdot>cfun'"
sum_mono2: 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 B :: "'c set"   and A :: "'c set"   and f :: "'c \<Rightarrow> 'b" assumes "class.ordered_comm_monoid_add plus zero less_eq less"   and "finite B"   and "A \<subseteq> B"   and "\<And>b. b \<in> B - A \<Longrightarrow> less_eq zero (f b)" shows "less_eq (comm_monoid_add.sum plus zero f A) (comm_monoid_add.sum plus zero f B)"
id_take_nth_drop: fixes i :: "nat"   and xs :: "'b list" assumes "i < length xs" shows "xs = xs \<down> i @ xs ! i # xs \<up> Suc i"
NotInDom_graph_aux_def: shows "NotInDom_graph_aux \<equiv> lfp (\<lambda>p x1 x2. \<exists>NotInDom_aux z t r. x1 = (t, r) \<and> x2 = SyntaxN.All z (Neg (HPair t (Var z) IN r)) \<and> atom z \<sharp> (t, r))"
assoc: fixes f :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and a :: "'g"   and b :: "'g"   and c :: "'g" assumes "semigroup f" shows "f (f a b) c = f a (f b c)"
RepFun_0: fixes f :: "hf \<Rightarrow> hf" shows "RepFun 0 f = 0"
snd_apfst: fixes f :: "'d \<Rightarrow> 'c"   and x :: "'d \<times> 'b" shows "snd (apfst f x) = snd x"
oseq_receiveTE: fixes \<sigma> :: "nat \<Rightarrow> 'e"   and l :: "'h"   and u\<^sub>m\<^sub>s\<^sub>g :: "'f \<Rightarrow> 'e \<Rightarrow> 'e"   and p :: "('e, 'f, 'g, 'h) seqp"   and a :: "'f seq_action"   and \<sigma>' :: "nat \<Rightarrow> 'e"   and q :: "('e, 'f, 'g, 'h) seqp"   and \<Gamma> :: "'g \<Rightarrow> ('e, 'f, 'g, 'h) seqp"   and i :: "nat" assumes "((\<sigma>, {l}receive(u\<^sub>m\<^sub>s\<^sub>g) . p), a, \<sigma>', q) \<in> oseqp_sos \<Gamma> i" obtains msg :: "'f" where "a = receive msg" and "\<sigma>' i = u\<^sub>m\<^sub>s\<^sub>g msg (\<sigma> i)" and "q = p"
acyclic_4a_def: fixes x :: "'a" shows "acyclic_4a x \<equiv> \<forall>y\<le>x. x \<sqinter> y\<^sup>\<star> \<le> y"
map_comp0: fixes g1 :: "'g \<Rightarrow> 'e"   and f1 :: "'c \<Rightarrow> 'g"   and g2 :: "'h \<Rightarrow> 'f"   and f2 :: "'d \<Rightarrow> 'h" shows "map_F0 (g1 \<circ> f1) (g2 \<circ> f2) = map_F0 g1 g2 \<circ> map_F0 f1 f2"
deg_pm_single: fixes x :: "'d"   and k :: "'c" shows "deg_pm (monomial k x) = k"
numeral_One: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c" assumes "class.numeral plus" shows "numeral.numeral one plus num.One = one"
weakCongI: fixes substTerm :: "'d \<Rightarrow> name list \<Rightarrow> 'd list \<Rightarrow> 'd"   and substAssert :: "'e \<Rightarrow> name list \<Rightarrow> 'd list \<Rightarrow> 'e"   and substCond :: "'f \<Rightarrow> name list \<Rightarrow> 'd list \<Rightarrow> 'f"   and SCompose' :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and SImp' :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and SBottom' :: "'e"   and SChanEq' :: "'d \<Rightarrow> 'd \<Rightarrow> 'f"   and P :: "('d, 'e, 'f) psi"   and Q :: "('d, 'e, 'f) psi" assumes "env substTerm substAssert substCond SCompose' SImp' SBottom' SChanEq'"   and "\<And>\<Psi> \<sigma>. wellFormedSubst \<sigma> \<Longrightarrow> env.weakPsiCongruence substTerm substAssert substCond SCompose' SImp' SBottom' SChanEq' \<Psi> (substPsi.seqSubs substTerm substAssert substCond P \<sigma>) (substPsi.seqSubs substTerm substAssert substCond Q \<sigma>)" shows "env.weakCongruence substTerm substAssert substCond SCompose' SImp' SBottom' SChanEq' P Q"
bla: fixes s :: "char list \<Rightarrow> int" assumes "P s" shows "\<exists>s' m. (c, s) \<Rightarrow> m \<Down> s' \<and> m \<le> k * n \<and> Q s'"
con_rews_2: fixes a :: "'a" shows "Ok\<cdot>a \<noteq> \<bottom>"
of_fun_cases: fixes x :: "('d, 'c) sq_matrix" obtains y :: "'c \<Rightarrow> 'c \<Rightarrow> 'd" where "x = of_fun y" and "y \<in> UNIV"
Units_l_inv_ex: fixes x :: "'a set" assumes "x \<in> Units (ideals_set R)" shows "\<exists>y\<in>carrier (ideals_set R). y \<otimes>\<^bsub>ideals_set R\<^esub> x = \<one>\<^bsub>ideals_set R\<^esub>"
indicator_eq_1_iff: fixes A :: "'b set"   and x :: "'b" shows "(indicator A x = (1::'a)) = (x \<in> A)"
HM: shows "(a, b) \<Turnstile> is_hashtable l ht"
5: shows "\<forall>a\<in>sset (smap f (gtrace (sdrop n r) (gtarget (stake n r) v))). odd a"
Callee: fixes G :: "prog"   and A :: "'b triple set"   and P :: "'b assn"   and l :: "lname \<Rightarrow> val option"   and e :: "expr"   and Q :: "'b assn" shows "G,A\<turnstile>{Normal P} Callee l e-\<succ> {Q}"
run_reachable: fixes x\<^sub>0 :: "'a"   and xs :: "'a stream" assumes "run (x\<^sub>0 ## xs)" shows "pred_stream (reaches x\<^sub>0) xs"
group_closure_scalar_mult_right: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and s :: "'a"   and S :: "'a set"   and n :: "nat" assumes "class.comm_ring_1 times one plus zero minus uminus"   and "s \<in> ab_group_add.group_closure zero minus S" shows "times s (semiring_1.of_nat one plus zero n) \<in> ab_group_add.group_closure zero minus S"
union_ring_is_abelian_group: assumes "C \<noteq> {}" shows "cring (union_ring C)"
And_Not: fixes \<phi> :: "formula"   and \<psi> :: "formula" assumes "safe_formula \<phi>"   and "\<not> safe_assignment (fv \<phi>) (Neg \<psi>)"   and "\<not> safe_formula (Neg \<psi>)"   and "fv (Neg \<psi>) \<subseteq> fv \<phi>"   and "\<not> is_constraint (Neg \<psi>)"   and "safe_formula \<psi>"   and "P \<phi>"   and "P \<psi>" shows "P (And \<phi> (Neg \<psi>))"
inf_sup_aci_1: fixes x :: "'b \<times> 'c"   and y :: "'b \<times> 'c" shows "inf_prod x y = inf_prod y x"
coprime_approx_main_def: fixes p :: "int"   and ff_ops :: "'a arith_ops_record"   and f :: "int poly"   and g :: "int poly" shows "coprime_approx_main p ff_ops f g = (gcd_poly_i ff_ops (of_int_poly_i ff_ops (poly_mod.Mp p f)) (of_int_poly_i ff_ops (poly_mod.Mp p g)) = one_poly_i ff_ops)"
intro: fixes n :: "nat"   and \<chi> :: "nat \<Rightarrow> complex" assumes "dcharacter n \<chi>"   and "nonprimitive_dchar_axioms n \<chi>" shows "nonprimitive_dchar n \<chi>"
gtr2gtrp_s: fixes s :: "'a list"   and c :: "'a list multiset"   and e :: "('b, 'c) label list"   and c' :: "'a list multiset"   and fg :: "('a, 'b, 'c, 'd, 'e) flowgraph_rec_scheme" assumes "(add_mset s c, e, c') \<in> ntr fg" obtains s' :: "'a list" and ce' :: "'a list multiset" and ee :: "('b, 'c) label list el_step" where "c' = add_mset s' ce'" and "e = le_rem_s ee" and "((s, c), ee, s', ce') \<in> ntrp fg"
const_res_range: fixes s :: "nat \<Rightarrow> nat \<Rightarrow> int"   and k :: "nat" assumes "s \<in> carrier Zp\<^bsup>\<omega>\<^esup>"   and "0 < k" shows "const_res k s \<in> carrier (residue_ring (p ^ k))"
getRel_map_fn: fixes a2 :: "'b"   and G :: "('c, 'b) labeled_graph"   and b2 :: "'b"   and l :: "'c"   and f :: "'b \<Rightarrow> 'd"   and a :: "'d"   and b :: "'d" assumes "a2 \<in> vertices G"   and "b2 \<in> vertices G"   and "(a2, b2) \<in> getRel l G"   and "f a2 = a"   and "f b2 = b" shows "(a, b) \<in> getRel l (map_graph_fn G f)"
assms_5: shows "check_AND (x0, x1) (a0, a1) s (z0, z1)"
map_ds_def: fixes f :: "'d \<Rightarrow> 'e" shows "map_ds f = ds \<circ> f \<circ> SSup"
hindered_by: shows "hindered_by (\<Gamma>\<lparr>weight := weight \<Gamma> - u\<rparr>) \<epsilon>"
bind_option_simps_2: fixes fail :: "'c"   and x :: "'d"   and f :: "'d \<Rightarrow> 'c" shows "monad.bind_option fail (Some x) f = f x"
Hwit22_def: fixes b :: "'q"   and c :: "'o" shows "Hwit22 b c = wit2_G (wit2_F2 c) (wit_F3 b c)"
simps: fixes m :: "nat"   and lm :: "nat list"   and l :: "cell list"   and r :: "cell list" shows "wprepare_loop_start_in_middle m lm (l, r) = (\<exists>rn mr lm1. rev l @ r = Oc \<up> Suc m @ Bk # Bk # <lm> @ Bk \<up> rn \<and> l \<noteq> [] \<and> r = Oc \<up> mr @ Bk # <lm1> @ Bk \<up> rn \<and> lm1 \<noteq> [])"
C_def: shows "C = max D (2 * c * (real n / real (n - 1)) / sin \<alpha> ^ (n - 1))"
exhaust_sel: fixes exetheory :: "exetheory" obtains "exetheory = ExeTheory (exesig exetheory) (exeaxioms_of exetheory)"
simps_1: fixes x :: "('b, 'c) Combinators"   and xs :: "('b, 'c) Combinators list" shows "firstList (x # xs) = first_bothNet x"
A'_in_carrier: shows "A' \<subseteq> carrier_vec n"
simps_1: fixes P :: "'d \<Rightarrow> 'e \<Rightarrow> bool" shows "list_all_zip P [] [] = True"
floor_le_ceiling: fixes x :: "'d" shows "\<lfloor>x\<rfloor> \<le> \<lceil>x\<rceil>"
ereal_eq_0_2: fixes r :: "real" shows "(0 = ereal r) = (r = 0)"
mult_subgroup_generated: fixes G :: "('f, 'g) monoid_scheme"   and S :: "'f set" shows "(\<otimes>\<^bsub>subgroup_generated G S\<^esub>) = (\<otimes>\<^bsub>G\<^esub>)"
left_connected: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<nu> :: "'b"   and \<nu>' :: "'b"   and \<mu> :: "'b" assumes "weak_composition V H"   and "partial_magma.arr V (V \<nu> \<nu>')" shows "(H \<nu> \<mu> \<noteq> partial_magma.null V) = (H \<nu>' \<mu> \<noteq> partial_magma.null V)"
row_to_poly_vec_times: fixes ts :: "'a list"   and A :: "'c mat"   and v :: "'c vec" assumes "distinct ts"   and "length ts = dim_col A" shows "row_to_poly ts (v \<^sub>v* A) = (\<Sum>i = 0..<dim_row A. v $ i \<cdot> row_to_poly ts (row A i))"
MPair_analz: fixes X :: "msg"   and Y :: "msg"   and H :: "msg set" assumes "\<lbrace>X, Y\<rbrace> \<in> analz H" obtains "X \<in> analz H" and "Y \<in> analz H"
stutter_sampler_def: fixes f :: "nat \<Rightarrow> nat"   and \<sigma> :: "nat \<Rightarrow> 'b" shows "stutter_sampler f \<sigma> \<equiv> f 0 = 0 \<and> strict_mono f \<and> (\<forall>k n. f k < n \<and> n < f (Suc k) \<longrightarrow> \<sigma> n = \<sigma> (f k))"
elims: fixes x :: "'a list"   and xa :: "'a \<times> 'a option"   and y :: "'a list" assumes "insert_spec x xa = y" obtains xs :: "'a list" and oid :: "'a" where "x = xs__" and "xa = (oid__, None)" and "y = oid__ # xs__"   | oid :: "'a" and v :: "'a" where "x = []" and "xa = (oida__, Some v)" and "y = []"   | xb :: "'a" and xs :: "'a list" and oid :: "'a" and ref :: "'a" where "x = xb # xs" and "xa = (oid, Some ref)" and "y = (if xb = ref then xb # oid # xs else xb # insert_spec xs (oid, Some ref))"
divides_ff_trans: fixes x :: "'b fract"   and y :: "'b fract"   and z :: "'b fract" assumes "divides_ff x y"   and "divides_ff y z" shows "divides_ff x z"
fun_rel_def: fixes A :: "('d \<times> 'f) set"   and B :: "('e \<times> 'g) set" shows "A \<rightarrow> B \<equiv> {(f, f'). \<forall>(a, a')\<in>A. (f a, f' a') \<in> B}"
rep_eq: fixes x :: "uint64" shows "bit x = bit (Rep_uint64 x)"
gt_trans: fixes y :: "int"   and x :: "int"   and z :: "int" assumes "y < x"   and "z < y" shows "z < x"
less_le_trans: fixes less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "'e"   and y :: "'e"   and z :: "'e" assumes "class.preorder less_eq less"   and "less x y"   and "less_eq y z" shows "less x z"
norm_minus_commute: 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 a :: "'c"   and b :: "'c" assumes "class.real_normed_vector minus dist norm plus zero uminus scaleR sgn uniformity open" shows "norm (minus a b) = norm (minus b a)"
assms: shows "fst a \<subseteq>\<^sub>f s"
that: fixes c :: "real" assumes "((\<lambda>x. \<MM> x - ln x) \<longlongrightarrow> c) at_top" shows "thesis"
bisim1ThrowNull: fixes P :: "'a prog"   and h :: "'heap"   and e :: "'addr expr1"   and xs :: "'addr val list" shows "P,throw e,h \<turnstile> (Throw (addr_of_sys_xcpt NullPointer), xs) \<leftrightarrow> ([Null], xs, length (compE2 e), \<lfloor>addr_of_sys_xcpt NullPointer\<rfloor>)"
add_le_same_cancel1: fixes b :: "'a multiset"   and a :: "'a multiset" shows "(b + a \<subseteq># b) = (a \<subseteq># {#})"
all_acquired_dropWhile_subset: fixes P :: "'b memref \<Rightarrow> bool"   and sb :: "'b memref list" shows "all_acquired (dropWhile P sb) \<subseteq> all_acquired sb"
multisets_supp_finite: fixes M :: "'a multiset" shows "finite (supp M)"
hyp2_extend_segment_unique: fixes a :: "hyp2"   and b :: "hyp2"   and c :: "hyp2"   and d :: "hyp2" assumes "a \<noteq> b"   and "B\<^sub>K a b c"   and "B\<^sub>K a b d"   and "b c \<congruent>\<^sub>K b d" shows "c = d"
inject_2: fixes x2 :: "'a \<times> nat"   and y2 :: "'a \<times> nat" shows "(Formal_in x2 = Formal_in y2) = (x2 = y2)"
simps: fixes b :: "'b" shows "the_In2 (In2 b) = b"
flatten_times: fixes p :: "'a \<Rightarrow>\<^sub>0 'a \<Rightarrow>\<^sub>0 'b"   and q :: "'a \<Rightarrow>\<^sub>0 'a \<Rightarrow>\<^sub>0 'b" shows "flatten (p * q) = flatten p * flatten q"
return_pmf_None_eq_bind: fixes p :: "'c spmf"   and f :: "'c \<Rightarrow> 'b spmf" shows "(return_pmf None = p \<bind> f) = (\<forall>x\<in>set_spmf p. f x = return_pmf None)"
hom_add: fixes c :: "'c"   and x :: "'c"   and y :: "'c" shows "c * (x + y) = c * x + c * y"
borel_predict_stoch_proc_borel_measurable: fixes M :: "'c measure"   and F :: "nat \<Rightarrow> 'c measure"   and X :: "nat \<Rightarrow> 'c \<Rightarrow> 'd" assumes "disc_filtr_prob_space M F"   and "borel_predict_stoch_proc F X" shows "\<forall>n. X n \<in> borel_measurable M"
proj2_not_self_incident: fixes p :: "proj2" shows "\<not> proj2_incident p (P2L p)"
simps_2: fixes n :: "nat"   and S :: "(nat \<times> nat) set"   and x :: "nat"   and xs :: "nat list" shows "is_path n S (x # xs) = (if xs = [] then x < n else ((x, hd xs) \<in> S \<or> (hd xs, x) \<in> S) \<and> is_path n S xs)"
map_pair_id: fixes x :: "'o"   and xs :: "('a \<times> 'c) list" assumes "oalist_inv_raw x xs" shows "map_pair id xs = xs"
healthy_intros_9: fixes f :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real"   and g :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real" assumes "healthy (wp f)"   and "healthy (wp g)" shows "healthy (wp (f \<Sqinter> g))"
cases: fixes a1 :: "'node SDG_node set"   and a2 :: "'edge \<Rightarrow> ('var, 'val, 'ret, 'pname) edge_kind"   and a3 :: "'node list"   and a4 :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and a5 :: "'edge list"   and a6 :: "'node list"   and a7 :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" assumes "a1,a2 \<turnstile> (a3,a4) =a5\<Rightarrow>\<^sub>\<tau> (a6,a7)" obtains ms :: "'node list" and s :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" and S :: "'node SDG_node set" and f :: "'edge \<Rightarrow> ('var, 'val, 'ret, 'pname) edge_kind" where "a1 = Sa__" and "a2 = f__" and "a3 = msa__" and "a4 = sa__" and "a5 = []" and "a6 = msa__" and "a7 = sa__" and "length msa__ = length sa__"   | S :: "'node SDG_node set" and f :: "'edge \<Rightarrow> ('var, 'val, 'ret, 'pname) edge_kind" and ms :: "'node list" and s :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" and a :: "'edge" and ms' :: "'node list" and s' :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" and as :: "'edge list" and ms'' :: "'node list" and s'' :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" where "a1 = S" and "a2 = f" and "a3 = ms" and "a4 = s" and "a5 = a # as" and "a6 = ms''" and "a7 = s''" and "S,f \<turnstile> (ms,s) -a\<rightarrow>\<^sub>\<tau> (ms',s')" and "S,f \<turnstile> (ms',s') =as\<Rightarrow>\<^sub>\<tau> (ms'',s'')"
as_empty_refine: fixes R :: "('c \<times> 'd) set" shows "(as_empty (), []) \<in> \<langle>R\<rangle>as_rel"
indx: shows "del_point_blocks x index {y, z} = \<B> index {y, z}"
simps_7: fixes v :: "char list"   and va :: "aexp"   and vb :: "'b"   and vc :: "'b acom"   and vd :: "'b acom" shows "(v ::= va {vb} \<sqsubseteq> vc;; vd) = False"
ok2_empty_mrexp2: fixes r :: "'b rexp" shows "ok2 (empty_mrexp2 r)"
set_eq: shows "range_permutation A B `` {f} = (\<lambda>p x. if x \<in> A then p (f x) else undefined) ` {p. p permutes B}"
keys1: shows "keys (monomial (lc p) (lt p)) = {lt p}"
limseq_even_odd: fixes f :: "nat \<Rightarrow> 'a"   and l :: "'a" assumes "(\<lambda>n. f (2 * n)) \<longlonglongrightarrow> l"   and "(\<lambda>n. f (2 * n + 1)) \<longlonglongrightarrow> l" shows "f \<longlonglongrightarrow> l"
subst_comp_set_image: fixes \<delta> :: "'c \<Rightarrow> ('a, 'd) Term.term"   and \<theta> :: "'d \<Rightarrow> ('a, 'b) Term.term"   and X :: "'c set" shows "(\<delta> \<circ>\<^sub>s \<theta>) ` X = \<delta> ` X \<cdot>\<^sub>s\<^sub>e\<^sub>t \<theta>"
tick_count_suc: fixes r :: "'b run"   and c :: "clock"   and n :: "nat" shows "tick_count r c (Suc n) = (if hamlet (Rep_run r (Suc n) c) then Suc (tick_count r c n) else tick_count r c n)"
coeffs_map_poly_hom: fixes p :: "'a poly" shows "coeffs (map_poly hom p) = map hom (coeffs p)"
extend_def: fixes uu :: "unit"   and b :: "bool"   and bs :: "bool list" shows "extend uu b bs \<equiv> b # bs"
simps_2: fixes v :: "'a"   and v' :: "'a aexp" shows "aexp_constrains (V v) v' = (V v = v')"
balance_cl1: fixes l :: "('a, 'b) rbt"   and r :: "('a, 'b) rbt"   and k :: "'a"   and v :: "'b" assumes "cl_inv' l"   and "cl_inv r" shows "cl_inv (balance l k v r)"
sem_def: fixes c :: "'b com" shows "sem c \<equiv> \<lambda>s. {t. \<exists>Ts. (c, s) -P*\<rightarrow> (Parallel Ts, t) \<and> All_None Ts}"
assms_2: shows "y * z\<^sup>+ * y = bot"
subset_imageE: fixes B :: "'f set"   and f :: "'g \<Rightarrow> 'f"   and A :: "'g set" assumes "B \<subseteq> f ` A" obtains C :: "'g set" where "C \<subseteq> A" and "B = f ` C"
representation_zero: fixes scale :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and basis :: "'c set" assumes "module scale" shows "module.representation scale basis (0::'c) = (\<lambda>b. 0::'b)"
list_of_oalist_OAlist_tc_filter: fixes P :: "'a \<times> 'b \<Rightarrow> bool"   and xs :: "('a, 'b) oalist_tc" shows "list_of_oalist_tc (OAlist_tc_filter P xs) = filter P (list_of_oalist_tc xs)"
set_integral_diff_2: fixes M :: "'b measure"   and A :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "set_integrable M A f"   and "set_integrable M A g" shows "\<integral>x\<in>A. (f x - g x)\<partial>M = set_lebesgue_integral M A f - set_lebesgue_integral M A g"
hyps_2: shows "sh' = sh(C \<mapsto> (sfs, Error))"
countable_trees: fixes A :: "'d set" assumes "countable A" shows "countable (trees A)"
ode_d_na_def: shows "ode_d_na = (\<lambda>tx. ode_d1 (snd tx) o\<^sub>L snd_blinfun)"
rep_eq: fixes x :: "('c, 'd) lifted_pair"   and xa :: "('c, 'd) lifted_pair" shows "(x \<le> xa) = (\<forall>f. triple.pairs_less_eq (Rep_lifted_pair x f) (Rep_lifted_pair xa f))"
square_integrable_approximate_continuous: fixes f :: "real \<Rightarrow> real"   and S :: "real set"   and e :: "real" assumes "f square_integrable S"   and "S \<in> lmeasurable"   and "0 < e" obtains g :: "real \<Rightarrow> real" where "continuous_on UNIV g" and "g square_integrable S" and "l2norm S (\<lambda>x. f x - g x) < e"
i_shrink_1: fixes f :: "nat \<Rightarrow> 'd message_af" shows "f \<div> Suc 0 = f"
Collect_mono_iff: fixes P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool" shows "(p2s P \<subseteq> p2s Q) = (\<forall>x. P x \<longrightarrow> Q x)"
frestriction_flrestriction: fixes r :: "V"   and A :: "V" shows "(r \<restriction>\<^sub>\<bullet> A) \<restriction>\<^sup>l\<^sub>\<bullet> A = r \<restriction>\<^sub>\<bullet> A"
to_gbg_alt1_3: fixes m :: "'a" shows "gbg_F (to_gbg_ext m) = F"
empty_eq_union: fixes M :: "'a multiset"   and N :: "'a multiset" shows "(\<Empt> = M + N) = (M = \<Empt> \<and> N = \<Empt>)"
inverse_arrows_sym: fixes f :: "'a \<times> 'a"   and g :: "'a \<times> 'a" shows "CC.inverse_arrows f g = CC.inverse_arrows g f"
progress_LN2: fixes j :: "nat"   and lholed :: "Lholed"   and e1s :: "e list"   and lfilled :: "e list"   and e2s :: "e list" assumes "Lfilled j lholed e1s lfilled" shows "\<exists>lfilled'. Lfilled j lholed e2s lfilled'"
iuntilI: fixes Q :: "nat \<Rightarrow> bool"   and t :: "nat"   and I :: "nat set"   and P :: "nat \<Rightarrow> bool" assumes "Q t"   and "\<And>t'. t' \<in> I \<down>< t \<Longrightarrow> P t'"   and "t \<in> I" shows "P t'. t' \<U> t I. Q t"
cong: fixes \<Delta> :: "('b, 'c) network_scheme" shows "flow_network.enum_cycle \<Delta> \<equiv> flow_network.enum_cycle \<Delta>"