hcju/isabelleps huggingface dataset:
PScopeE: fixes \<pi> :: "char list \<Rightarrow> com option"   and \<pi>' :: "char list \<Rightarrow> com option"   and p :: "com"   and s :: "char list \<Rightarrow> int \<Rightarrow> int"   and t :: "char list \<Rightarrow> int \<Rightarrow> int" assumes "\<pi>: (PScope \<pi>' p, s) \<Rightarrow> t" obtains "\<pi>': (p, s) \<Rightarrow> t"
step_le_x: fixes as :: "real list"   and bs :: "real list"   and hs :: "(real \<Rightarrow> real) list"   and k :: "nat"   and x\<^sub>0 :: "real"   and x\<^sub>1 :: "real"   and hb :: "real"   and e :: "real"   and p :: "real"   and i :: "nat"   and x :: "real" assumes "akra_bazzi_real_recursion as bs hs k x\<^sub>0 x\<^sub>1 hb e p"   and "i < k"   and "x\<^sub>1 \<le> x" shows "bs ! i * x + (hs ! i) x \<le> x"
Sup: fixes Sup :: "'b set \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and M :: "'b set"   and f :: "'b \<Rightarrow> 'b" assumes "class.ccpo Sup less_eq less"   and "Complete_Partial_Order.chain less_eq M"   and "\<forall>x\<in>M. x \<in> ccpo.iterates Sup less_eq f" shows "Sup M \<in> ccpo.iterates Sup less_eq f"
add_cancel_left_right: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and a :: "'a"   and b :: "'a" assumes "class.cancel_comm_monoid_add plus minus zero" shows "(plus a b = a) = (b = zero)"
intro: assumes "u = (THE z. \<forall>w. P w z)"   and "\<exists>y. \<forall>x. P x y" shows "CMU_axioms"
ESem_fresh_cong_below: fixes e :: "'a"   and \<rho> :: "'b \<Rightarrow> 'c"   and \<rho>' :: "'b \<Rightarrow> 'c" assumes "\<rho> f|` fv e \<sqsubseteq> \<rho>' f|` fv e" shows "\<lbrakk> e \<rbrakk>\<^bsub>\<rho>\<^esub> \<sqsubseteq> \<lbrakk> e \<rbrakk>\<^bsub>\<rho>'\<^esub>"
card_before_le_index: fixes x :: "'b"   and xs :: "'b list" shows "card (before x xs) \<le> index xs x"
assms: shows "a loc' \<noteq> {#}\<^sub>z"
simps_4: fixes a\<^sub>1 :: "aexp"   and a\<^sub>2 :: "aexp" shows "vars (Times a\<^sub>1 a\<^sub>2) = vars a\<^sub>1 \<union> vars a\<^sub>2"
bex_Un: fixes A :: "'e set"   and B :: "'e set"   and P :: "'e \<Rightarrow> bool" shows "(\<exists>x\<in>A \<union> B. P x) = ((\<exists>x\<in>A. P x) \<or> (\<exists>x\<in>B. P x))"
emeasure_PiM_diagonal: fixes x :: "'b"   and A :: "'b set"   and y :: "'b"   and a :: "real"   and b :: "real" assumes "x \<in> A"   and "y \<in> A"   and "x \<noteq> y"   and "a < b"   and "finite A" shows "emeasure (Pi\<^sub>M A (\<lambda>_. uniform_measure lborel {a..b})) {h \<in> A \<rightarrow>\<^sub>E UNIV. h x = h y} = 0"
assms_5: shows "P \<turnstile> C sees M, NonStatic :  Ts\<rightarrow>T = (pns, body) in D"
CD_on_simps: fixes ds :: "'d set" shows "CD_on ds {} = {}"
tensor_mat_zero2: fixes m1 :: "'b mat" shows "tensor_mat m1 (0\<^sub>m d2 d2) = 0\<^sub>m d d"
hv_cmult: fixes a :: "real"   and x :: "(real, 7) vec" shows "Hv (a *\<^sub>R x) = a *\<^sub>R Hv x"
fresh_star_list: fixes xs :: "'c list"   and c :: "'d" shows "xs \<sharp>* c \<equiv> \<forall>x\<in>set xs. x \<sharp> c"
assms_2: shows "fv (t \<cdot> \<theta>) \<inter> X = {}"
kat_2_equiv: fixes y :: "'a"   and z :: "'a"   and x :: "'a" shows "(y \<cdot> ad z \<le> ad x \<cdot> y) = (d x \<cdot> y \<cdot> ad z = (zero_class.zero::'a))"
Smith_normal_form_upt_k_def: fixes A :: "(('a, 'b) vec, 'c) vec"   and k :: "nat" shows "Smith_normal_form_upt_k A k = ((\<forall>a b. to_nat a = to_nat b \<and> to_nat a + 1 < k \<and> to_nat b + 1 < k \<longrightarrow> A $ a $ b dvd A $ (a + (1::'c)) $ (b + (1::'b))) \<and> isDiagonal_upt_k A k)"
of_seq_code_3: fixes P :: "'b Predicate.pred"   and xq :: "'b Predicate.seq" shows "set_of_seq (seq.Join P xq) = set_of_pred P \<union> set_of_seq xq"
elims: fixes x :: "nat list"   and y :: "nat" assumes "Partial_Recursive.list_encode x = y" obtains "x = []" and "y = 0"   | xa :: "nat" and xs :: "nat list" where "x = xa # xs" and "y = Suc (prod_encode (length xs, tuple_encode (length xs) (xa # xs)))"
VSGeneral_def: shows "VSGeneral = opt \<circ> QE_dnf opt (\<lambda>x. gen_qe) \<circ> VSLuckiest \<circ> opt"
map_simp: fixes f :: "'c" assumes "C.arr f" shows "Cones.map f = S.mkArr (\<iota> ` D.cones (C.cod f)) (\<iota> ` D.cones (C.dom f)) (\<iota> \<circ> D.cones_map f \<circ> \<o>)"
problem_plan_bound_s_def: fixes PROB :: "(('c, 'd) fmap \<times> ('c, 'd) fmap) set"   and s :: "('c, 'd) fmap" shows "problem_plan_bound_s PROB s = Sup (MPLS_s PROB s)"
word_1_le_power: fixes n :: "nat" assumes "n < LENGTH('a)" shows "1 \<le> 2 ^ n"
tendsto_sum: fixes I :: "'a set"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and a :: "'a \<Rightarrow> 'c"   and F :: "'b filter" assumes "\<And>i. i \<in> I \<Longrightarrow> (f i \<longlongrightarrow> a i) F" shows "((\<lambda>x. \<Sum>i\<in>I. f i x) \<longlongrightarrow> sum a I) F"
assms_11: shows "R = (\<Union>(t, x)\<in>X. {merge xx t |xx. xx \<in> x})"
merge_apply_4: fixes J :: "'d set"   and I :: "'d set"   and i :: "'d"   and x :: "'d \<Rightarrow> 'e"   and y :: "'d \<Rightarrow> 'e" assumes "J \<inter> I = {}"   and "i \<in> J" shows "merge I J (x, y) i = y i"
sel_2: fixes x1 :: "'b word"   and x2 :: "nat" shows "pfxm_length (PrefixMatch x1 x2) = x2"
open_imp_locally_connected: fixes S :: "'b set" assumes "open S" shows "locally connected S"
transpose_mat_of_cols: fixes n :: "nat"   and vs :: "'a vec list" shows "(mat_of_cols n vs)\<^sup>T = mat_of_rows n vs"
1: shows "i + int (Max {k. k < nat (j - i + 1) \<and> (\<forall>n\<in>{i..j}. f ([i..j] ! k) \<le> f n)}) \<le> j"
is_span: shows "C.span src_\<mu>.cod.leg0 src_\<mu>.cod.leg1"
decseq_imp_monoseq: fixes X :: "nat \<Rightarrow> 'b" assumes "decseq X" shows "monoseq X"
apply_cltn2_Rep_hyp2: fixes J :: "cltn2"   and p :: "hyp2" assumes "is_K2_isometry J" shows "apply_cltn2 (Rep_hyp2 p) J \<in> hyp2"
spmf_of_pmf_pmf_of_set: fixes A :: "'a set" assumes "finite A"   and "A \<noteq> {}" shows "spmf_of_pmf (pmf_of_set A) = spmf_of_set A"
cases: fixes x :: "('d, 'c) Term.term" obtains xa :: "'c" where "x = Var xa"   | f :: "'d" and ts :: "('d, 'c) Term.term list" where "x = Fun f ts"
Times_cprod: fixes A :: "'b set"   and B :: "'c set" shows "|A \<times> B| =o |A| *c |B|"
SUP_cong: fixes A :: "'d set"   and B :: "'d set"   and C :: "'d \<Rightarrow> 'a"   and D :: "'d \<Rightarrow> 'a" assumes "A = B"   and "\<And>x. x \<in> B \<Longrightarrow> C x = D x" shows "Sup (C ` A) = Sup (D ` B)"
ipd_2: shows "\<forall>n\<in>{k..<m}. \<pi> n \<noteq> ipd (\<pi> k)"
assms_4: shows "kernel H K g \<subseteq> f ` carrier G"
a_set_child_nodes_locs_def: fixes ptr :: "(_) object_ptr" shows "a_set_child_nodes_locs ptr \<equiv> (if is_element_ptr_kind\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr then all_args (put_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t (the (cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr)) child_nodes_update) else {}) \<union> (if is_document_ptr_kind ptr then all_args (put_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t (the (cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr)) document_element_update) else {})"
ts_l_def: shows "\<S>\<bullet>\<C>\<lparr>label := [ts''] @ label \<C>\<rparr> \<turnstile> lfilledk : [] _> ts_l"
face_of_Int_Int: fixes F :: "'a set"   and T :: "'a set"   and F' :: "'a set"   and t' :: "'a set" assumes "F face_of T"   and "F' face_of t'" shows "F \<inter> F' face_of T \<inter> t'"
birkhoff_theorem_AE_nonergodic: fixes M :: "'b measure"   and T :: "'b \<Rightarrow> 'b"   and f :: "'b \<Rightarrow> real" assumes "fmpt M T"   and "integrable M f" shows "AE x in M. (\<lambda>n. qmpt.birkhoff_sum T f n x / real n) \<longlonglongrightarrow> real_cond_exp M (qmpt.Invariants M T) f x"
nat_le_iff: fixes x :: "int"   and n :: "nat" shows "(nat x \<le> n) = (x \<le> int n)"
locally_def: fixes P :: "'c set \<Rightarrow> bool"   and S :: "'c set" shows "locally P S \<equiv> \<forall>w x. openin (top_of_set S) w \<and> x \<in> w \<longrightarrow> (\<exists>u v. openin (top_of_set S) u \<and> P v \<and> x \<in> u \<and> u \<subseteq> v \<and> v \<subseteq> w)"
Q'_edges_def: shows "Q'_edges \<equiv> \<Union> (H.edges_of_walk ` Q') \<inter> B"
fcompare_def: fixes a :: "('a, 'b) IEEE.float"   and b :: "('a, 'b) IEEE.float" shows "fcompare a b = (if IEEE.is_nan a \<or> IEEE.is_nan b then Und else if is_infinity a \<and> sign a = 1 then if is_infinity b \<and> sign b = 1 then ccode.Eq else ccode.Lt else if is_infinity a \<and> sign a = 0 then if is_infinity b \<and> sign b = 0 then ccode.Eq else ccode.Gt else if is_infinity b \<and> sign b = 1 then ccode.Gt else if is_infinity b \<and> sign b = 0 then ccode.Lt else if valof a < valof b then ccode.Lt else if valof a = valof b then ccode.Eq else ccode.Gt)"
well_related_set_def: fixes A :: "'c set"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool" shows "well_related_set A less_eq \<equiv> \<forall>X\<subseteq>A. X \<noteq> {} \<longrightarrow> (\<exists>e. extreme X (\<lambda>x y. less_eq y x) e)"
P_2: shows "hd (tl xs) = hd (tl P)"
terminates_impl_not_inf: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile>c \<down> s" shows "\<not> AlternativeSmallStep.inf \<Gamma> [c] [] s"
not_False_in_image_Ball: fixes P :: "'c \<Rightarrow> bool"   and A :: "'c set" shows "(False \<notin> P ` A) = Ball A P"
lim_stream_eq: fixes x :: "'a" assumes "x \<in> space M" shows "lim_stream x = K x \<bind> (\<lambda>y. lim_stream y \<bind> (\<lambda>\<omega>. return (stream_space M) (y ## \<omega>)))"
cases: fixes a1 :: "('fun, 'var) Term.term set"   and a2 :: "'var \<Rightarrow> ('fun, 'var) Term.term"   and a3 :: "('fun, 'var) extstrand_step list" assumes "sem\<^sub>e\<^sub>s\<^sub>t_d a1 a2 a3" obtains M\<^sub>0 :: "('fun, 'var) Term.term set" and \<I> :: "'var \<Rightarrow> ('fun, 'var) Term.term" where "a1 = M\<^sub>0'__" and "a2 = \<I>'__" and "a3 = []"   | M\<^sub>0 :: "('fun, 'var) Term.term set" and \<I> :: "'var \<Rightarrow> ('fun, 'var) Term.term" and S :: "('fun, 'var) extstrand_step list" and t :: "('fun, 'var) Term.term" where "a1 = M\<^sub>0''__" and "a2 = \<I>''__" and "a3 = S__ @ [Step (send\<langle>t__\<rangle>\<^sub>s\<^sub>t)]" and "sem\<^sub>e\<^sub>s\<^sub>t_d M\<^sub>0''__ \<I>''__ S__" and "ik\<^sub>e\<^sub>s\<^sub>t S__ \<union> M\<^sub>0''__ \<cdot>\<^sub>s\<^sub>e\<^sub>t \<I>''__ \<turnstile> t__ \<cdot> \<I>''__"   | M\<^sub>0 :: "('fun, 'var) Term.term set" and \<I> :: "'var \<Rightarrow> ('fun, 'var) Term.term" and S :: "('fun, 'var) extstrand_step list" and t :: "('fun, 'var) Term.term" where "a1 = M\<^sub>0'''__" and "a2 = \<I>'''__" and "a3 = Sa__ @ [Step (receive\<langle>ta__\<rangle>\<^sub>s\<^sub>t)]" and "sem\<^sub>e\<^sub>s\<^sub>t_d M\<^sub>0'''__ \<I>'''__ Sa__"   | M\<^sub>0 :: "('fun, 'var) Term.term set" and \<I> :: "'var \<Rightarrow> ('fun, 'var) Term.term" and S :: "('fun, 'var) extstrand_step list" and t :: "('fun, 'var) Term.term" and t' :: "('fun, 'var) Term.term" and a :: "poscheckvariant" where "a1 = M\<^sub>0''''__" and "a2 = \<I>''''__" and "a3 = Sb__ @ [Step \<langle>a: tb__ \<doteq> t'\<rangle>\<^sub>s\<^sub>t]" and "sem\<^sub>e\<^sub>s\<^sub>t_d M\<^sub>0''''__ \<I>''''__ Sb__" and "Unifier \<I>''''__ tb__ t'"   | M\<^sub>0 :: "('fun, 'var) Term.term set" and \<I> :: "'var \<Rightarrow> ('fun, 'var) Term.term" and S :: "('fun, 'var) extstrand_step list" and X :: "'var list" and F :: "(('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) list" where "a1 = M\<^sub>0'''''__" and "a2 = \<I>'''''__" and "a3 = Sc__ @ [Step \<forall>X\<langle>\<or>\<noteq>: F\<rangle>\<^sub>s\<^sub>t]" and "sem\<^sub>e\<^sub>s\<^sub>t_d M\<^sub>0'''''__ \<I>'''''__ Sc__" and "ineq_model \<I>'''''__ X F"   | M\<^sub>0 :: "('fun, 'var) Term.term set" and \<I> :: "'var \<Rightarrow> ('fun, 'var) Term.term" and S :: "('fun, 'var) extstrand_step list" and t :: "('fun, 'var) Term.term" and K :: "('fun, 'var) Term.term list" and M :: "('fun, 'var) Term.term list" where "a1 = M\<^sub>0" and "a2 = \<I>" and "a3 = S @ [Decomp t]" and "sem\<^sub>e\<^sub>s\<^sub>t_d M\<^sub>0 \<I> S" and "ik\<^sub>e\<^sub>s\<^sub>t S \<union> M\<^sub>0 \<cdot>\<^sub>s\<^sub>e\<^sub>t \<I> \<turnstile> t \<cdot> \<I>" and "Ana t = (K, M)" and "\<And>k. k \<in> set K \<Longrightarrow> ik\<^sub>e\<^sub>s\<^sub>t S \<union> M\<^sub>0 \<cdot>\<^sub>s\<^sub>e\<^sub>t \<I> \<turnstile> k \<cdot> \<I>"
no_step_final: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate"   and c' :: "('e, 'd, 'f) com"   and s' :: "('e, 'f) xstate"   and P :: "bool" assumes "\<Gamma>\<turnstile> (c, s) \<rightarrow> (c', s')"   and "final (c, s)" shows "P"
strong_postdominate_def: fixes n' :: "'node"   and n :: "'node" shows "n' strongly-postdominates n \<equiv> n' postdominates n \<and> (\<exists>k\<ge>1. \<forall>as nx. n -as\<rightarrow>* nx \<and> k \<le> length as \<longrightarrow> n' \<in> set (sourcenodes as))"
base_in_hom_2: shows "\<guillemotleft>F r : F r \<Rightarrow>\<^sub>D F r\<guillemotright>"
cases: fixes \<Delta> :: "('a, 'b) network_scheme"   and S :: "'a set" assumes "cut \<Delta> S" obtains "source \<Delta> \<in> S" and "sink \<Delta> \<notin> S"
assms_2: shows "\<forall>m\<ge>0::'rows. A $ m $ mod_type_class.from_nat k = (0::'a)"
distinct_disc_19: fixes event :: "('a, 'b, 'c) event" assumes "isSnapshot event" shows "\<not> isRecvMarker event"
map_euler_incr_fas_nth: fixes X0 :: "floatarith list"   and d :: "nat"   and h :: "floatarith"   and CX :: "floatarith list" assumes "length X0 = d" shows "map (euler_incr_fas_nth X0 h CX) [0..<d] = euler_incr_fas X0 h CX"
invpst_upd: fixes t :: "(('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree"   and x :: "'c"   and y :: "'d" assumes "invpst t" shows "invpst (upd x y t)"
all_union_ts_def: fixes A :: "'a \<Rightarrow> 'b set"   and B :: "'b set" shows "(A \<Rightarrow>\<union>\<^sub>\<forall> B) = (\<lambda>k. A k \<union> B)"
symclpE: fixes r :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d"   and y :: "'d" assumes "symclp r x y" obtains "r x y"   | "r y x"
add_mult_distrib_mat: fixes A :: "'b mat"   and nr :: "nat"   and n :: "nat"   and B :: "'b mat"   and C :: "'b mat"   and nc :: "nat" assumes "A \<in> carrier_mat nr n"   and "B \<in> carrier_mat nr n"   and "C \<in> carrier_mat n nc" shows "(A + B) * C = A * C + B * C"
connected_inV_iff: fixes u :: "nat"   and v :: "nat" assumes "connected u v" shows "(v \<in> V) = (u \<in> V)"
field_le_mult_one_interval: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and 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 less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "\<And>z. \<lbrakk>less zero z; less z one\<rbrakk> \<Longrightarrow> less_eq (times z x) y" shows "less_eq x y"
in_homI: fixes f :: "'d"   and a :: "'d"   and b :: "'d" assumes "D.arr f"   and "D.dom f = a"   and "D.cod f = b" shows "\<guillemotleft>f : a \<rightarrow>\<^sub>D b\<guillemotright>"
vsp_W': shows "VectorSpace (\<sharp>\<cdot>) W'"
isSPath_nt_parallel_pf: fixes s :: "nat"   and p :: "(nat \<times> nat) list"   and t :: "nat" assumes "cf.isSimplePath s p t" shows "set p \<inter> (set p)\<inverse> = {}"
NSconvergent_NSCauchy: fixes X :: "nat \<Rightarrow> 'a" assumes "NSconvergent X" shows "NSCauchy X"
start_prog_start_shconf: fixes C :: "char list"   and M :: "char list"   and P :: "jvm_prog" shows "class_add P (start_class C M),start_heap P \<turnstile>\<^sub>s [Start \<mapsto> (Map.empty, Done)] \<surd>"
fmap_def: shows "fmap = COERCE((udom \<rightarrow> udom) \<rightarrow> udom\<cdot>'e \<rightarrow> udom\<cdot>'e, ('c \<rightarrow> 'd) \<rightarrow> 'c\<cdot>'e \<rightarrow> 'd\<cdot>'e)\<cdot>fmapU"
abs_o_rep: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and Abs :: "'a \<Rightarrow> 'b"   and Rep :: "'b \<Rightarrow> 'a" assumes "Quotient3 R Abs Rep" shows "Abs \<circ> Rep = id"
wt_elim_cases_6: fixes E :: "env"   and dt :: "loc \<Rightarrow> ty option"   and T' :: "ty"   and e :: "expr"   and T :: "tys" assumes "E,dt\<Turnstile>In1l (Cast T' e)\<Colon>T" obtains T :: "ty" where "Tb__ = Inl T'" and "E,dt\<Turnstile>e\<Colon>-T" and "is_acc_type (prg E) (pkg E) T'" and "prg E\<turnstile>T\<preceq>? T'"
funcset_image: fixes f :: "'e \<Rightarrow> 'f"   and A :: "'e set"   and B :: "'f set" assumes "f \<in> A \<rightarrow> B" shows "f ` A \<subseteq> B"
aform_err_to_aformE: fixes x :: "real"   and e :: "nat \<Rightarrow> real"   and X :: "(real \<times> real pdevs) \<times> real"   and n :: "nat" assumes "x \<in> aform_err e X"   and "degree_aform_err X \<le> n" obtains err :: "real" where "x = aform_val (e(n := err)) (aform_err_to_aform X n)" and "- 1 \<le> err" and "err \<le> 1"
simps_3: fixes h :: "'b tree" shows "U Del_min [h] = 3 * log 2 (real (size h + 1)) + 4"
l2_inv3_skr: fixes R :: "fid_t"   and K :: "msg" shows "{l2_inv3 \<inter> l2_inv2} l2_skr R K {> l2_inv3}"
assms_3: shows "i < dim_row (1\<^sub>m (dim_col G1 * dim_col G2))"
that: fixes y :: "'a" assumes "y \<in> set (local.filter_min xs)"   and "rel y x" shows "thesis"
assms_1: shows "x1 \<sqsubseteq> x2"
l_create_element\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def: fixes get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and set_disconnected_nodes :: "(_) document_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_tag_name :: "(_) element_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and type_wf :: "(_) heap \<Rightarrow> bool"   and create_element :: "(_) document_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, (_) element_ptr) prog"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool" shows "l_create_element\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_disconnected_nodes get_disconnected_nodes_locs set_disconnected_nodes set_tag_name set_tag_name_locs type_wf create_element known_ptr \<equiv> l_get_disconnected_nodes type_wf get_disconnected_nodes get_disconnected_nodes_locs \<and> l_set_tag_name type_wf set_tag_name set_tag_name_locs \<and> l_create_element\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms get_disconnected_nodes set_disconnected_nodes set_tag_name create_element known_ptr"
state_rank_None: fixes q :: "'b"   and n :: "nat" assumes "\<not> sink q" shows "(state_rank q n = None) = (oldest_token q n = None)"
operation_det: fixes classes :: "'a fset"   and associations :: "String.literal \<rightharpoonup>\<^sub>f String.literal \<rightharpoonup>\<^sub>f ('a \<times> nat \<times> enat \<times> bool \<times> bool)"   and operations :: "(String.literal \<times> 'b \<times> (String.literal \<times> 'b \<times> param_dir) list \<times> 'b \<times> bool \<times> 'c option) list"   and \<tau> :: "'b"   and name :: "String.literal"   and \<pi> :: "'b list"   and oper\<^sub>1 :: "String.literal \<times> 'b \<times> (String.literal \<times> 'b \<times> param_dir) list \<times> 'b \<times> bool \<times> 'c option"   and oper\<^sub>2 :: "String.literal \<times> 'b \<times> (String.literal \<times> 'b \<times> param_dir) list \<times> 'b \<times> bool \<times> 'c option" assumes "object_model classes associations"   and "unique_operation operations \<tau> name \<pi> oper\<^sub>1"   and "unique_operation operations \<tau> name \<pi> oper\<^sub>2" shows "oper\<^sub>1 = oper\<^sub>2"
map\<^sub>1_in_vhom: fixes f :: "'c" assumes "C.ide f" shows "\<guillemotleft>\<tau>\<^sub>1 f : G f \<star>\<^sub>D \<tau>\<^sub>0 (src\<^sub>C f) \<Rightarrow>\<^sub>D \<tau>\<^sub>0 (trg\<^sub>C f) \<star>\<^sub>D F f\<guillemotright>"
foldSetD_backwards: fixes A :: "'b set"   and z :: "'c"   and D :: "'c set"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and e :: "'c" assumes "A \<noteq> {}"   and "(A, z) \<in> foldSetD D f e" shows "\<exists>x y. x \<in> A \<and> (A\<setminus>{x}, y) \<in> foldSetD D f e \<and> z = f x y"
powr_upper_bound: fixes l :: "real"   and x :: "real"   and u :: "real"   and z :: "real" assumes "0 < l"   and "l \<le> x"   and "x \<le> u" shows "x powr z \<le> max (l powr z) (u powr z)"
invertible_lf_imp_invertible_matrix: fixes f :: "('a, 'b) vec \<Rightarrow> ('a, 'b) vec" assumes "invertible_lf (*s) (*s) f" shows "invertible (matrix f)"
del_list_idem: fixes x :: "'c"   and xs :: "('c \<times> 'd) list" assumes "x \<notin> set (map fst xs)" shows "del_list x xs = xs"
WTrtThrow: fixes P :: "(char list list \<times> char list exp) prog"   and h :: "heap"   and sh :: "sheap"   and E :: "char list \<Rightarrow> ty option"   and e :: "char list exp"   and T\<^sub>r :: "ty"   and T :: "ty" assumes "P,E,h,sh \<turnstile> e : T\<^sub>r"   and "is_refT T\<^sub>r" shows "P,E,h,sh \<turnstile> throw e : T"
mbufn_take_induct: fixes f :: "event_data option list set list \<Rightarrow> 'a \<Rightarrow> 'a"   and z :: "'a"   and buf :: "event_data option list set list list"   and z' :: "'a"   and buf' :: "event_data option list set list list"   and i :: "nat"   and js :: "nat list"   and Ps :: "(nat \<Rightarrow> event_data option list set \<Rightarrow> bool) list"   and U :: "nat \<Rightarrow> 'a \<Rightarrow> bool" assumes "mbufn_take f z buf = (z', buf')"   and "wf_mbufn i js Ps buf"   and "U i z"   and "\<And>k xs z. \<lbrakk>i \<le> k; Suc k \<le> Mini i js; list_all2 (\<lambda>P. P k) Ps xs; U k z\<rbrakk> \<Longrightarrow> U (Suc k) (f xs z)" shows "U (Mini i js) z'"
in_set_CollectI: fixes P :: "V \<Rightarrow> bool"   and x :: "V" assumes "P x"   and "small {x. P x}" shows "x \<in>\<^sub>\<circ> ZFC_in_HOL.set {x. P x}"
no_before_inI: fixes x :: "'c"   and y :: "'c"   and xs :: "'c list" assumes "x < y in xs" shows "(\<not> y < x in xs) = True"
ZN_mult: shows "(ZN ===> ZN ===> ZN) (*) (*)"
divisors_eq_dvd: fixes a :: "nat"   and n :: "nat" shows "(a \<in> divisors n) = (a dvd n)"
loc_reach\<^sub>A: shows "\<langle>c\<^sub>A', mds\<^sub>A_of mds\<^sub>C', mem\<^sub>A_of mem\<^sub>C'\<rangle>\<^sub>A \<in> abs.loc_reach \<langle>c\<^sub>A, mds\<^sub>A, mem\<^sub>A\<rangle>\<^sub>A"
enumerate_mono_iff: fixes S :: "'b set"   and m :: "nat"   and n :: "nat" assumes "infinite S" shows "(wellorder_class.enumerate S m < wellorder_class.enumerate S n) = (m < n)"
row_iarray_def: fixes k :: "nat"   and A :: "'b iarray iarray" shows "row_iarray k A = A !! k"
simps_2: fixes t :: "('a, 'b) BinomialHeapStruc.BinomialTree"   and ts :: "('a, 'b) BinomialHeapStruc.BinomialTree list" shows "BinomialHeapStruc.height_queue (t # ts) = max (Suc (BinomialHeapStruc.height_tree t)) (BinomialHeapStruc.height_queue ts)"
semiconfluentpD: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "semiconfluentp r"   and "r x y"   and "r\<^sup>*\<^sup>* x z" shows "\<exists>u. r\<^sup>*\<^sup>* y u \<and> r\<^sup>*\<^sup>* z u"
lift_Suc_mono_less_iff: fixes f :: "nat \<Rightarrow> 'x \<Rightarrow>\<^sub>0 nat"   and n :: "nat"   and m :: "nat" assumes "\<And>n. f n \<prec> f (Suc n)" shows "(f n \<prec> f m) = (n < m)"
H_parts_5: shows "arcs_ends H = HU \<times> HV \<union> HV \<times> HU"
S2: shows "PrimRecOp1 0 (\<lambda>x y. g_step y (c_fst x)) \<in> PrimRec1"