hcju/isabelleps huggingface dataset:
R1_equiv_entailment: fixes c :: "('Var, 'AExp, 'BExp) Stmt"   and mds :: "Mode \<Rightarrow> 'Var set"   and mem :: "'Var \<Rightarrow> 'Val"   and c' :: "('Var, 'AExp, 'BExp) Stmt"   and mds' :: "Mode \<Rightarrow> 'Var set"   and mem' :: "'Var \<Rightarrow> 'Val"   and \<Gamma>' :: "'Var \<Rightarrow> 'BExp set option"   and \<S>' :: "'Var set \<times> 'Var set"   and P' :: "'BExp set"   and \<Gamma>'' :: "'Var \<Rightarrow> 'BExp set option"   and P'' :: "'BExp set" assumes "\<langle>c, mds, mem\<rangle> \<R>\<^sup>1\<^bsub>\<Gamma>',\<S>',P'\<^esub> \<langle>c', mds', mem'\<rangle>"   and "\<Gamma>' =:\<^sub>P' \<Gamma>''"   and "P' \<turnstile> P''"   and "\<forall>mds. tyenv_wellformed mds \<Gamma>' \<S>' P' \<longrightarrow> tyenv_wellformed mds \<Gamma>'' \<S>' P''" shows "\<langle>c, mds, mem\<rangle> \<R>\<^sup>1\<^bsub>\<Gamma>'',\<S>',P''\<^esub> \<langle>c', mds', mem'\<rangle>"
hyps: fixes f :: "'a"   and n :: "nat" shows "nth_default (0::'a) (pderiv_coeffs_code f xs) n = (f + of_nat n) * nth_default (0::'a) xs n"
ex_toFull: fixes wtFsym :: "'a \<Rightarrow> bool"   and wtPsym :: "'b \<Rightarrow> bool"   and arOf :: "'a \<Rightarrow> 'c list"   and resOf :: "'a \<Rightarrow> 'c"   and parOf :: "'b \<Rightarrow> 'c list"   and \<Phi> :: "('a, 'b) lit list set"   and intT :: "'c \<Rightarrow> univ \<Rightarrow> bool"   and intF :: "'a \<Rightarrow> univ list \<Rightarrow> univ"   and intP :: "'b \<Rightarrow> univ list \<Rightarrow> bool"   and \<sigma> :: "'c" assumes "InfModel wtFsym wtPsym arOf resOf parOf \<Phi> intT intF intP" shows "\<exists>F. bij_betw F {a. intT \<sigma> a} UNIV"
plus_ivl_def: shows "(+) \<equiv> map_fun rep_ivl (map_fun rep_ivl abs_ivl) plus_rep"
assoc: fixes a :: "'b::semilattice_inf"   and b :: "'b::semilattice_inf"   and c :: "'b::semilattice_inf" shows "(a::'b::semilattice_inf) \<sqinter> (b::'b::semilattice_inf) \<sqinter> (c::'b::semilattice_inf) = a \<sqinter> (b \<sqinter> c)"
tendsto_PInfty: fixes f :: "'b \<Rightarrow> ereal"   and F :: "'b filter" shows "(f \<longlongrightarrow> \<infinity>) F = (\<forall>r. \<forall>\<^sub>F x in F. ereal r < f x)"
collapse_2: fixes hd :: "('c, 'd) hd" assumes "is_Sym hd" shows "Sym (hd.sym hd) = hd"
prems_3: shows "abs.strong_low_bisim_mm \<R>\<^sub>A"
complement_path_components_of_Union: fixes C :: "'c set"   and X :: "'c topology" assumes "C \<in> path_components_of X" shows "topspace X - C = \<Union> (path_components_of X - {C})"
double_compl: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and x :: "'c" assumes "class.boolean_algebra minus uminus inf less_eq less sup bot top" shows "uminus (uminus x) = x"
poly_cpoly_of_real_iff: fixes pR :: "real poly"   and pI :: "real poly"   and t :: "real" shows "(poly (cpoly_of pR pI) (complex_of_real t) = 0) = (poly pR t = 0 \<and> poly pI t = 0)"
r_ce: shows "ce_rel_to_set r \<in> ce_sets"
top_higher: fixes x :: "'a" assumes "x \<in> carrier L" shows "x \<sqsubseteq>\<^bsub>L\<^esub> \<top>\<^bsub>L\<^esub>"
var_is_subterm: fixes x :: "'c"   and t :: "('d, 'c) Term.term" assumes "x \<in> fv t" shows "Var x \<sqsubseteq> t"
mprod_Suc: fixes n :: "nat"   and e :: "nat \<Rightarrow> int"   and f :: "nat \<Rightarrow> 'a" assumes "\<forall>j\<le>Suc n. e j \<in> Zset"   and "\<forall>j\<le>Suc n. f j \<in> carrier K - {\<zero>}" shows "mprod_exp K e f (Suc n) = mprod_exp K e f n \<cdot>\<^sub>r f (Suc n)\<^bsub>K\<^esub>\<^bsup>e (Suc n)\<^esup>"
seqE: fixes g :: "'a1"   and f :: "'a1" assumes "A1.seq g f" obtains "A1.arr f" and "A1.arr g" and "A1.dom g = A1.cod f"
induct: fixes P :: "fm \<Rightarrow> bool"   and a0 :: "fm" assumes "\<And>t. P (Lt t)"   and "\<And>t. P (Le t)"   and "\<And>t. P (Eq t)"   and "\<And>t. P (NEq t)"   and "\<And>p q. \<lbrakk>P p; P q\<rbrakk> \<Longrightarrow> P (And p q)"   and "\<And>p q. \<lbrakk>P p; P q\<rbrakk> \<Longrightarrow> P (Or p q)"   and "\<And>p q. \<lbrakk>P (NOT p); P q\<rbrakk> \<Longrightarrow> P (Imp p q)"   and "\<And>p q. \<lbrakk>P p; P q; P (NOT p); P (NOT q)\<rbrakk> \<Longrightarrow> P (Iff p q)"   and "\<And>p q. \<lbrakk>P (NOT p); P (NOT q)\<rbrakk> \<Longrightarrow> P (NOT (And p q))"   and "\<And>p q. \<lbrakk>P (NOT p); P (NOT q)\<rbrakk> \<Longrightarrow> P (NOT (Or p q))"   and "\<And>p q. \<lbrakk>P p; P (NOT q)\<rbrakk> \<Longrightarrow> P (NOT (Imp p q))"   and "\<And>p q. \<lbrakk>P p; P (NOT q); P (NOT p); P q\<rbrakk> \<Longrightarrow> P (NOT (Iff p q))"   and "\<And>t. P (NOT (Eq t))"   and "\<And>t. P (NOT (NEq t))"   and "\<And>t. P (NOT (Le t))"   and "\<And>t. P (NOT (Lt t))"   and "\<And>p. P p \<Longrightarrow> P (NOT (NOT p))"   and "P (NOT T)"   and "P (NOT F)"   and "P T"   and "P F"   and "\<And>va. P (NOT (E va))"   and "\<And>va. P (NOT (A va))"   and "\<And>v. P (E v)"   and "\<And>v. P (A v)" shows "P a0"
IH: shows "fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s (G \<cdot>\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s \<theta>) = fv\<^sub>s\<^sub>e\<^sub>t (\<theta> ` fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s G)"
extend_restrict_indets: fixes p :: "('c option \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd" assumes "None \<notin> indets p" shows "extend_indets (restrict_indets p) = p"
keys_add: fixes f :: "'c \<Rightarrow>\<^sub>0 'd"   and g :: "'c \<Rightarrow>\<^sub>0 'd" shows "keys (f + g) \<subseteq> keys f \<union> keys g"
CONSTRAINT_SLOT_def: fixes x :: "prop" shows "CONSTRAINT_SLOT PROP x \<equiv> PROP x"
happ_hlam_3: fixes X :: "hf"   and A :: "hf"   and B :: "hf"   and C :: "hf"   and F :: "hf" assumes "X \<^bold>\<in> A"   and "hfun (A * B) C F" shows "happ (hlam A B C F) X = \<lbrace>yz \<^bold>\<in> B * C. \<langle>\<langle>X, hfst yz\<rangle>, hsnd yz\<rangle> \<^bold>\<in> F\<rbrace>"
Infinitesimal_less_SReal: fixes x :: "real star"   and y :: "real star" assumes "x \<in> \<real>"   and "y \<in> Infinitesimal"   and "0 < x" shows "y < x"
comp_fun_commute_axioms: shows "comp_fun_commute (\<otimes>\<^sub>m)"
L_transform_Act: fixes f :: "'d"   and \<alpha> :: "'c"   and x :: "('a, 'b, 'c, 'd) FL_Formula.formula" shows "L_transform (FL_Formula.Act f \<alpha> x) = Formula.Act (Eff f) (Formula.Act (L_action.Act \<alpha>) (L_transform x))"
conth: shows "continuous_map (prod_topology (top_of_set {0..1}) S) U h"
simps_163: fixes x121 :: "expr"   and x122 :: "expr"   and x71 :: "char list"   and x72 :: "expr" shows "x121;; x122 \<noteq> x71:=x72"
exec_meth_stk_splits: fixes P :: "'addr J1_prog"   and h :: "'heap"   and Es :: "'addr expr1 list"   and es :: "'addr expr1 list"   and xs :: "'addr val list"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and xcp :: "'addr option"   and STK :: "'addr val list"   and t :: "'thread_id"   and ta :: "('addr, 'thread_id, 'heap) jvm_thread_action"   and h' :: "'heap"   and stk' :: "'addr val list"   and loc' :: "'addr val list"   and pc' :: "nat"   and xcp' :: "'addr option" assumes "P,Es,h \<turnstile> (es, xs) [\<leftrightarrow>] (stk, loc, pc, xcp)"   and "exec_meth_d (compP2 P) (compEs2 Es) (stack_xlift (length STK) (compxEs2 Es 0 0)) t h (stk @ STK, loc, pc, xcp) ta h' (stk', loc', pc', xcp')" shows "\<exists>stk''. stk' = stk'' @ STK \<and> exec_meth_d (compP2 P) (compEs2 Es) (compxEs2 Es 0 0) t h (stk, loc, pc, xcp) ta h' (stk'', loc', pc', xcp')"
Span_var_pow_base: fixes K :: "'a set"   and n :: "nat" assumes "subfield K R" shows "ring.Span (K [X]) (poly_of_const ` K) (ring.exp_base (K [X]) X n) = {q \<in> carrier (K [X]). length q \<le> n}"
assms_1: shows "x \<in> fv (\<theta> y)"
inject_1: fixes x3 :: "bool"   and y3 :: "bool" shows "(Bool x3 = Bool y3) = (x3 = y3)"
mat_of_row_dim_row_1: fixes A :: "'b mat" shows "(dim_row A = 1) = (A = mat_of_row (row A 0))"
exp3Min: fixes a :: "nat" shows "0 < exp 3 a"
simps_2: fixes fj :: "'a"   and xj :: "'a"   and xis :: "'a list" shows "local.divided_differences_impl [] fj xj xis = [fj]"
rsapss_sign_def: fixes m :: "bit list"   and e :: "nat"   and n :: "nat" shows "rsapss_sign m e n = (if emsapss_encode m (length (nat_to_bv n) - 1) = [] then [] else rsapss_sign_help1 (bv_to_nat (emsapss_encode m (length (nat_to_bv n) - 1))) e n)"
indspaceD_add: fixes f :: "('r, 'g) aezfun \<Rightarrow> 'v"   and G :: "'g set"   and FH :: "('r, 'g) aezfun set"   and V :: "'v set"   and x :: "('r, 'g) aezfun"   and y :: "('r, 'g) aezfun" assumes "f \<in> indspace G FH V"   and "x \<in> FH"   and "y \<in> FH" shows "f (x + y) = f x + f y"
preimage_def: fixes f :: "'c \<Rightarrow> 'd option"   and B :: "'d set" shows "preimage f B = {x \<in> dom f. the (f x) \<in> B}"
inj_on_insert: fixes f :: "'d \<Rightarrow> 'e"   and a :: "'d"   and A :: "'d set" shows "inj_on f (insert a A) = (inj_on f A \<and> f a \<notin> f ` (A - {a}))"
induct: fixes P :: "'a poly \<Rightarrow> 'a poly \<Rightarrow> bool"   and a0 :: "'a poly"   and a1 :: "'a poly" assumes "\<And>c c'. P (c)\<^sub>p (c')\<^sub>p"   and "\<And>c c' n' p'. P (c)\<^sub>p c' \<Longrightarrow> P (c)\<^sub>p (CN c' n' p')"   and "\<And>c n p c'. P c (c')\<^sub>p \<Longrightarrow> P (CN c n p) (c')\<^sub>p"   and "\<And>c n p c' n' p'. \<lbrakk>P c (CN c' n' p'); P (CN c n p) c'; P c c'; \<And>x. P p p'\<rbrakk> \<Longrightarrow> P (CN c n p) (CN c' n' p')"   and "\<And>v b. P (poly.Bound v) b"   and "\<And>v va b. P (Add v va) b"   and "\<And>v va b. P (Sub v va) b"   and "\<And>v va b. P (Mul v va) b"   and "\<And>v b. P (Neg v) b"   and "\<And>v va b. P (Pw v va) b"   and "\<And>v va vb vc. P (CN v va vb) (poly.Bound vc)"   and "\<And>v va vb vc vd. P (CN v va vb) (Add vc vd)"   and "\<And>v va vb vc vd. P (CN v va vb) (Sub vc vd)"   and "\<And>v va vb vc vd. P (CN v va vb) (Mul vc vd)"   and "\<And>v va vb vc. P (CN v va vb) (Neg vc)"   and "\<And>v va vb vc vd. P (CN v va vb) (Pw vc vd)"   and "\<And>a v. P a (poly.Bound v)"   and "\<And>a v va. P a (Add v va)"   and "\<And>a v va. P a (Sub v va)"   and "\<And>a v va. P a (Mul v va)"   and "\<And>a v. P a (Neg v)"   and "\<And>a v va. P a (Pw v va)"   and "\<And>vc v va vb. P (poly.Bound vc) (CN v va vb)"   and "\<And>vc vd v va vb. P (Add vc vd) (CN v va vb)"   and "\<And>vc vd v va vb. P (Sub vc vd) (CN v va vb)"   and "\<And>vc vd v va vb. P (Mul vc vd) (CN v va vb)"   and "\<And>vc v va vb. P (Neg vc) (CN v va vb)"   and "\<And>vc vd v va vb. P (Pw vc vd) (CN v va vb)" shows "P a0 a1"
fully_faithful_functor_def: fixes A :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and F :: "'a \<Rightarrow> 'b" shows "fully_faithful_functor A B F \<equiv> faithful_functor A B F \<and> full_functor A B F"
B_invariant: fixes A :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and B :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sim :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and PA :: "'c \<Rightarrow> bool"   and QA :: "'c \<Rightarrow> bool"   and PB :: "'d \<Rightarrow> bool"   and QB :: "'d \<Rightarrow> bool"   and a :: "'d"   and b :: "'d" assumes "Bisimulation_Invariants A B sim PA QA PB QB"   and "PB a"   and "B a b" shows "QB b"
lookup6: fixes a :: "coname"   and \<Delta> :: "(coname \<times> ty) list"   and y :: "name"   and x :: "name"   and c :: "coname"   and P :: "trm" assumes "\<not> a \<sharp> idc \<Delta> y" shows "lookupb x a (idc \<Delta> y) c P = Cut <c>.P y.Ax y a"
subdegree_uminus: fixes f :: "'a fps" shows "subdegree (- f) = subdegree f"
vec_eq_0_iff: fixes v :: "'b vec"   and n :: "nat" shows "(v = 0\<^sub>v n) = (n = dim_vec v \<and> (n = 0 \<or> set (list_of_vec v) = {0::'b}))"
transient_def: fixes A :: "'b set" shows "transient A \<equiv> {F. \<exists>act\<in>Acts F. act `` A \<subseteq> - A}"
reachable_in_vertsE: fixes G :: "('e, 'f) pre_digraph"   and u :: "'e"   and v :: "'e" assumes "u \<rightarrow>\<^sup>*\<^bsub>G\<^esub> v" obtains "u \<in> verts G" and "v \<in> verts G"
AAcc_\<tau>red1t_xt1: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and h :: "'heap"   and a :: "'addr expr1"   and xs :: "'addr val list"   and a' :: "'addr expr1"   and xs' :: "'addr val list"   and i :: "'addr expr1" assumes "\<tau>red1gt uf P t h (a, xs) (a', xs')" shows "\<tau>red1gt uf P t h (a\<lfloor>i\<rceil>, xs) (a'\<lfloor>i\<rceil>, xs')"
mvar_existence_ivl_eq_existence_ivl: fixes t :: "real"   and x0 :: "'c" assumes "t \<in> existence_ivl0 x0" shows "mvar.existence_ivl x0 t = (\<lambda>_. existence_ivl0 x0)"
ideal_generated_dvd_subset: 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 S :: "'b set"   and d :: "'b" assumes "class.comm_ring_1 times one plus zero minus uminus"   and "\<forall>x\<in>S. dvd.dvd times d x"   and "finite S" shows "ring.ideal_generated plus zero uminus times S \<subseteq> ring.ideal_generated plus zero uminus times {d}"
digitToList_empty: fixes d :: "('a, 'b) Digit" shows "FingerTreeStruc.digitToList d \<noteq> []"
assms_2: fixes i :: "nat"   and j :: "nat" assumes "i \<le> j" shows "g i \<subseteq> g j"
PartDom_def: fixes d :: "'b dataspace"   and n :: "nat" shows "d !D! n = Rep_dataspace d ! n"
fmpred_empty: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool" shows "fmpred P fmempty"
bind_map_gpv: fixes f :: "'e \<Rightarrow> 'd"   and gpv :: "('e, 'b, 'c) gpv"   and g :: "'d \<Rightarrow> ('a, 'b, 'c) gpv" shows "bind_gpv (map_gpv f id gpv) g = bind_gpv gpv (g \<circ> f)"
eq_on_trans: fixes r :: "'a \<Rightarrow> 'b"   and s :: "'a \<Rightarrow> 'b"   and X :: "'a set"   and t :: "'a \<Rightarrow> 'b" assumes "r = s on X"   and "s = t on X" shows "r = t on X"
norm1_scaleR: fixes r :: "real"   and v :: "('b, 'c) Finite_Cartesian_Product.vec" shows "norm1 (r *\<^sub>R v) = \<bar>r\<bar> * norm1 v"
sum_plus_right_isotone: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<noteq> bot \<and> sum_class.sum x \<le> sum_class.sum y \<longrightarrow> sum_class.sum z + sum_class.sum x \<le> sum_class.sum z + sum_class.sum y"
assms_2: shows "h \<turnstile> f \<rightarrow>\<^sub>h h'"
to_fun_plus: fixes g :: "nat \<Rightarrow> 'a"   and f :: "nat \<Rightarrow> 'a"   and x :: "'a" assumes "g \<in> carrier P"   and "f \<in> carrier P"   and "x \<in> carrier R" shows "to_fun (f \<oplus>\<^bsub>P\<^esub> g) x = to_fun f x \<oplus> to_fun g x"
bicategory_axioms: shows "bicategory (\<cdot>) Cop.H Cop.\<a> Cop.\<i> Cop.src Cop.trg"
uminus_closed: fixes G :: "'a set"   and g :: "'a" assumes "Group G"   and "g \<in> G" shows "- g \<in> G"
confs_widens: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'a) list) list"   and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and vs :: "val list"   and Ts :: "ty list"   and Ts' :: "ty list" assumes "P,h \<turnstile> vs [:\<le>] Ts"   and "P \<turnstile> Ts [\<le>] Ts'" shows "P,h \<turnstile> vs [:\<le>] Ts'"
comm_monoid_mult_hom_axioms: shows "comm_monoid_mult_hom hom"
UN_Un_distrib: fixes A :: "'d \<Rightarrow> 'c set"   and B :: "'d \<Rightarrow> 'c set"   and I :: "'d set" shows "(\<Union>i\<in>I. A i \<union> B i) = \<Union> (A ` I) \<union> \<Union> (B ` I)"
distinct_3: fixes x31 :: "char list"   and x32 :: "char list" shows "SKIP \<noteq> x31[] ::= x32"
traverse_impl_def: fixes E :: "('b \<times> 'b) list" shows "traverse_impl E = dds_of (\<lambda>visited A. (map snd (filter (\<lambda>(x, y). x \<in> set A \<and> y |\<notin>| visited) E), visited |\<union>| fset_of_list A)) {||}"
mcp_unique: fixes z :: "'e list"   and W :: "'e list set"   and p :: "'e list"   and p' :: "'e list" assumes "mcp z W p"   and "mcp z W p'" shows "p = p'"
a: shows "ip \<in> get_all_matching_src_ips iface m"
bind_const': fixes M :: "'d measure"   and N :: "'e measure" assumes "prob_space M"   and "subprob_space N" shows "M \<bind> (\<lambda>x. N) = N"
vec_list: fixes v :: "'b vec" shows "vec_of_list (list_of_vec v) = v"
get_child_nodes_locs_def: shows "get_child_nodes_locs \<equiv> l_get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_child_nodes_locs"
less_eq_quadnat_def: fixes A :: "quadnat"   and B :: "quadnat" shows "(A \<le> B) = (case A of Quad (a1, b1, c1, d1) \<Rightarrow> case B of Quad (a2, b2, c2, d2) \<Rightarrow> if a1 = a2 then if b1 = b2 then if c1 = c2 then d1 \<le> d2 else c1 < c2 else b1 < b2 else a1 < a2)"
endo_inj_surj: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'a" assumes "finite A"   and "f ` A \<subseteq> A"   and "inj_on f A" shows "f ` A = A"
elims: fixes x :: "'b \<Rightarrow> 'a"   and xa :: "'b \<times> 'b \<times> 'b \<times> 'b"   and y :: "'a \<times> 'a \<times> 'a \<times> 'a" assumes "mat_map x xa = y" obtains f :: "'b \<Rightarrow> 'a" and a :: "'b" and b :: "'b" and c :: "'b" and d :: "'b" where "x = f" and "xa = (a, b, c, d)" and "y = (f a, f b, f c, f d)"
permute_lift_def: fixes p :: "perm"   and x :: "'c lift" shows "p \<bullet> x = (case x of \<bottom> \<Rightarrow> \<bottom> | Def x \<Rightarrow> Def (p \<bullet> x))"
irfb_inj: fixes \<phi> :: "'c set \<Rightarrow> 'd set"   and \<psi> :: "'c set \<Rightarrow> 'd set" assumes "Inf_pres \<phi>"   and "Inf_pres \<psi>"   and "fb\<^sup>-\<^sub>\<R> \<phi> = fb\<^sup>-\<^sub>\<R> \<psi>" shows "\<phi> = \<psi>"
assms_3: shows "(f \<circ> x) \<longlonglongrightarrow> A"
MGF_implies_complete: fixes c :: "stmt"   and P :: "assn"   and Q :: "assn" assumes "\<forall>Z. {} |\<turnstile> {MGT c Z}"   and "\<Turnstile> {P} c {Q}" shows "{} \<turnstile> {P} c {Q}"
refine_to_conj: fixes c :: "'b::refinement_lattice"   and d\<^sub>0 :: "'b::refinement_lattice"   and d\<^sub>1 :: "'b::refinement_lattice" assumes "(c::'b::refinement_lattice) \<sqsubseteq> (d\<^sub>0::'b::refinement_lattice)"   and "(c::'b::refinement_lattice) \<sqsubseteq> (d\<^sub>1::'b::refinement_lattice)" shows "(c::'b::refinement_lattice) \<sqsubseteq> (d\<^sub>0::'b::refinement_lattice) \<iinter> (d\<^sub>1::'b::refinement_lattice)"
assms: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<cdot> (d y \<cdot> z) = x \<cdot> d y \<cdot> z"
hyps: assumes "m \<le> n" shows "x\<^bsub>0\<^esub>\<^bsup>m\<^esup> \<cdot> (x\<^bsup>n\<^esup>)\<^sup>\<star> = (x\<^bsup>n\<^esup>)\<^sup>\<star> \<cdot> x\<^bsub>0\<^esub>\<^bsup>m\<^esup>"
connected_closedin_eq: fixes S :: "'b set" shows "connected S = (\<nexists>E1 E2. closedin (top_of_set S) E1 \<and> closedin (top_of_set S) E2 \<and> E1 \<union> E2 = S \<and> E1 \<inter> E2 = {} \<and> E1 \<noteq> {} \<and> E2 \<noteq> {})"
elims_3: fixes x :: "'a \<times> 'b \<times> 'c \<times> 'd" assumes "\<not> mat_diagonal x" obtains A :: "'a" and B :: "'b" and C :: "'c" and D :: "'d" where "x = (A, B, C, D)" and "\<not> (B = (0::'b) \<and> C = (0::'c))"
apply_times: fixes f :: "'b perm"   and g :: "'b perm" shows "apply (f * g) = apply f \<circ> apply g"
priorities_set: fixes q :: "('d, 'c) pq" shows "set \<parallel>q\<parallel> = snd ` set (alist_of q)"
prefixE: fixes xs :: "'c list"   and ys :: "'c list" assumes "prefix xs ys" obtains zs :: "'c list" where "ys = xs @ zs"
push_bit_nat_def: fixes n :: "nat"   and m :: "nat" shows "push_bit n m = m * 2 ^ n"
bigtheta_def: fixes F :: "'a filter"   and g :: "'a \<Rightarrow> 'b" shows "\<Theta>[F](g) = O[F](g) \<inter> \<Omega>[F](g)"
reachable_state_def: fixes sys :: "('g, 'h, 'i, 'j, 'k, 'l) pre_system_scheme"   and s :: "('g, 'h, 'i, 'j, 'k) system_state" shows "reachable_state sys s = (\<exists>\<sigma> i. prerun sys \<sigma> \<and> \<sigma> i = s)"
restrict_indets_pp_plus: fixes s :: "'c option \<Rightarrow>\<^sub>0 nat"   and t :: "'c option \<Rightarrow>\<^sub>0 nat" shows "restrict_indets_pp (s + t) = restrict_indets_pp s + restrict_indets_pp t"
sound_wrt_small_step: fixes e :: "exp"   and v :: "bval" assumes "[] \<turnstile> e \<Down> v"   and "FV e = {}" shows "\<exists>v' ob. e \<longrightarrow>* v' \<and> isval v' \<and> observe v' ob \<and> bs_observe v ob"
card_suc: fixes b :: "nat" assumes "b < 2" shows "card (evnodd (a \<union> t) b) = Suc (card (evnodd t b))"
scalar_prod_smult_right: fixes w :: "'c vec"   and v :: "'c vec"   and k :: "'c" assumes "dim_vec w = dim_vec v" shows "w \<bullet> (k \<cdot>\<^sub>v v) = k * (w \<bullet> v)"
hequiv_refl: fixes i :: "'c"   and H :: "(('d, 'c) fm list \<times> 'c) set" assumes "i \<in> names H" shows "hequiv H i i"
bij_lens_def: fixes x :: "'b \<Longrightarrow> 'c" shows "bij_lens x \<equiv> weak_lens x \<and> bij_lens_axioms x"
remove1_idem: fixes x :: "'c"   and xs :: "'c list" assumes "x \<notin> set xs" shows "remove1 x xs = xs"
primefact_variant: fixes ps :: "'a list"   and n :: "'a" shows "primefact ps n = (foldr (*) ps (1::'a) = n \<and> list_all prime ps)"
l9_8_2: fixes P :: "'p"   and Q :: "'p"   and A :: "'p"   and C :: "'p"   and B :: "'p" assumes "P Q TS A C"   and "P Q OS A B" shows "P Q TS B C"
terminal_tllist_of_llist_lfinite: fixes xs :: "'a llist"   and b :: "'b" assumes "lfinite xs" shows "terminal (tllist_of_llist b xs) = b"
dvd_mod_imp_dvd: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and c :: "'a"   and a :: "'a"   and b :: "'a" assumes "class.semidom_modulo divide plus minus zero times one modulo"   and "dvd.dvd times c (modulo a b)"   and "dvd.dvd times c b" shows "dvd.dvd times c a"
PUpdate_def: fixes L :: "(('e, 'f) expr \<times> ('d data \<Rightarrow> bool) \<times> 'f set \<times> 'd pupdate) set" shows "Expr.PUpdate L = Expr.pupdate ` L"
rel_cases: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'b equation"   and b :: "'c equation" assumes "rel_equation R a b" obtains x1a :: "'b trm" and x2a :: "'b trm" and y1a :: "'c trm" and y2a :: "'c trm" where "a = Eq x1a x2a" and "b = Eq y1a y2a" and "rel_trm R x1a y1a" and "rel_trm R x2a y2a"
diff_add_eq_ereal: fixes a :: "ereal"   and b :: "ereal"   and c :: "ereal" shows "a - b + c = a + c - b"
sorted_wrt1: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b" shows "sorted_wrt P [x] = True"
regular_event_preserves_process_snapshots: fixes c :: "('a, 'b, 'c) configuration"   and ev :: "('a, 'b, 'c) event"   and c' :: "('a, 'b, 'c) configuration"   and r :: "'a" assumes "c \<turnstile> ev \<mapsto> c'"   and "regular_event ev" shows "ps c r = ps c' r"
simps_3: fixes e :: "('b, 'c, 'd) exp" shows "extRet2J e RetStaySame = e"
wf_cdecl_all_standard_classes: shows "Ball (set standard_classes) (wf_cdecl tprg)"
next_state_def: fixes current :: "'a \<Rightarrow> 'b"   and control :: "'a \<Rightarrow> 'b \<Rightarrow> 'c list list \<Rightarrow> 'c option \<times> 'c list list \<times> 'a"   and s :: "'a"   and execs :: "'b \<Rightarrow> 'c list list" shows "Kernel.next_state current control s execs = snd (snd (control s (current s) (execs (current s))))"
add_mono1: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and 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 a :: "'a"   and b :: "'a" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero"   and "less a b" shows "less (plus a one) (plus b one)"
assms_13: shows "InvariantCEntailed (getConflictFlag state) F0' (getC state)"
assms_1: shows "n'' -as''\<rightarrow>\<^sub>c\<^sub>d n'"
disc_asian_option_lgeom: fixes T :: "nat"   and w :: "bool stream" shows "discounted_value r (\<lambda>m. asian_option T) T w = det_discount r T * (\<Sum>i = 1..T. lgeom_proc u d init (take i (stake T w))) / real T"
xs_a: shows "set_mset (mset xs) \<subseteq> A"
exhaust: fixes y :: "'e\<cdot>('f, 'g) errorT" obtains "y = \<bottom>"   | app :: "'e\<cdot>'g error\<cdot>'f" where "y = ErrorT\<cdot>app" and "app \<noteq> \<bottom>"
g_bigtheta: shows "g \<in> \<Theta>(\<lambda>x. real x powr p * ln (real x) powr (p' - 1))"