hcju/isabelleps huggingface dataset:
rec: fixes f11 :: "'b \<Rightarrow> 'c"   and f12 :: "'b bexp \<Rightarrow> 'b aexp \<Rightarrow> 'b aexp \<Rightarrow> 'a \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and f2 :: "'b aexp \<Rightarrow> 'b aexp \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'a"   and x21 :: "'b aexp"   and x22 :: "'b aexp" shows "rec_bexp f11 f12 f2 (Equal x21 x22) = f2 x21 x22 (rec_aexp f11 f12 f2 x21) (rec_aexp f11 f12 f2 x22)"
steps_add: fixes P :: "'b"   and Cal :: "'b processCalculus"   and Q :: "'b"   and R :: "'b" assumes "P \<longmapsto>Cal* Q"   and "Q \<longmapsto>Cal* R" shows "P \<longmapsto>Cal* R"
SigmaE2: fixes a :: "'c"   and b :: "'d"   and A :: "'c set"   and B :: "'c \<Rightarrow> 'd set" assumes "(a, b) \<in> Sigma A B" obtains "a \<in> A" and "b \<in> B a"
simps_1: fixes p :: "'t \<Rightarrow>\<^sub>0 'b"   and q :: "'t \<Rightarrow>\<^sub>0 'b" shows "poly_of_pair (Inl (p, q)) = spair p q"
reflection: shows "rel_weakly_reflects_barbs (indRelLSTPO SRel TRel) (STCalWB SWB TWB)"
that: fixes T :: "'a set" assumes "open T"   and "S \<subseteq> T"   and "T - S \<in> lmeasurable"   and "emeasure lebesgue (T - S) < ennreal e" shows "thesis"
x_le: shows "2 powr real_of_int (- p) \<le> x"
ps_not_Nil: fixes ps :: "real poly list" assumes "quasi_sturm_seq ps" shows "ps \<noteq> []"
Pi_cong: fixes A :: "'h set"   and f :: "'h \<Rightarrow> 'i"   and g :: "'h \<Rightarrow> 'i"   and B :: "'h \<Rightarrow> 'i set" assumes "\<And>w. w \<in> A \<Longrightarrow> f w = g w" shows "(f \<in> Pi A B) = (g \<in> Pi A B)"
card_update: fixes X :: "'a set"   and v :: "'a"   and b :: "bool" assumes "finite X"   and "v \<notin> X" shows "card (fmupd v b ` {s. fmdom' s = X}) = card {s. fmdom' s = X}"
assms_3: shows "Q * X \<Longrightarrow>\<^sub>A R"
smultfunset_def: fixes G :: "'g set"   and FH :: "('r, 'g) aezfun set" shows "smultfunset G FH \<equiv> {f. \<forall>a. \<forall>g\<in>G. \<forall>x\<in>FH. f (a \<delta>\<delta> g * x) = (a \<delta>\<delta> g) \<cdot> f x}"
bilinear_cross7: shows "bilinear (\<times>\<^sub>7)"
is_basis_def: fixes S :: "('a, 'b) vec set" shows "is_basis S \<equiv> vec.independent S \<and> vec.span S = UNIV"
assms: shows "xa \<in> Joints (msum_aforms' x y)"
iso_class_eqI: fixes f :: "'a"   and g :: "'a" assumes "f \<cong> g" shows "\<lbrakk>f\<rbrakk> = \<lbrakk>g\<rbrakk>"
l_get_scdom_component\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def: fixes get_scdom_component :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and is_strongly_scdom_component_safe :: "(_) object_ptr set \<Rightarrow> (_) object_ptr set \<Rightarrow> (_) heap \<Rightarrow> (_) heap \<Rightarrow> bool"   and is_weakly_scdom_component_safe :: "(_) object_ptr set \<Rightarrow> (_) object_ptr set \<Rightarrow> (_) heap \<Rightarrow> (_) heap \<Rightarrow> bool"   and get_owner_document :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) document_ptr) prog"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and to_tree_order :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog" shows "l_get_scdom_component\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_scdom_component is_strongly_scdom_component_safe is_weakly_scdom_component_safe get_owner_document get_disconnected_nodes to_tree_order \<equiv> get_scdom_component = l_get_scdom_component\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_scdom_component get_owner_document get_disconnected_nodes to_tree_order \<and> is_strongly_scdom_component_safe = l_get_scdom_component\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_is_strongly_scdom_component_safe get_owner_document get_disconnected_nodes to_tree_order \<and> is_weakly_scdom_component_safe = l_get_scdom_component\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_is_weakly_scdom_component_safe get_owner_document get_disconnected_nodes to_tree_order"
pick: fixes x :: "'a"   and y :: "'a" assumes "R\<^sup>+\<^sup>+ x y" shows "chain R (llist_of (x # local.pick x y @ [y]))"
monic_dvdm_constant: fixes u :: "int poly"   and k :: "int" assumes "u dvdm [:k:]"   and "monic u"   and "0 < degree u" shows "k mod m = 0"
after_is_final: fixes c :: "nat \<times> cell list \<times> cell list"   and p :: "(action \<times> nat) list \<times> nat"   and n :: "nat" assumes "is_final c" shows "is_final (steps c p n)"
in_S: fixes x :: "'a stream" shows "x \<in> space (stream_space (count_space UNIV))"
edge'_simps_2: fixes y :: "'v vertex"   and x :: "'v vertex ??.MFMC_Flow_Attainability.flow_attainability.node" shows "??.local.\<Psi>.edge' (??.MFMC_Flow_Attainability.flow_attainability.node.Inner y) x = (\<exists>ya. x = ??.MFMC_Flow_Attainability.flow_attainability.node.Inner ya \<and> edge \<Psi> y ya)"
word_no_log_defs_3: fixes a :: "num"   and b :: "num" shows "numeral a AND numeral b = word_of_int (numeral a AND numeral b)"
arccos_def: fixes y :: "real" shows "arccos y = (THE x. 0 \<le> x \<and> x \<le> pi \<and> cos x = y)"
Rep_MonoTran: fixes x :: "'b MonoTran" shows "Rep_MonoTran x \<in> {f. mono f}"
assms_2: shows "term_ok \<Theta> (mk_eq s t)"
founded_def: fixes subs :: "'a \<Rightarrow> 'a set"   and Pred :: "'a \<Rightarrow> bool" shows "founded subs Pred = (\<lambda>A. \<forall>(n, delta)\<in>A. terminal subs delta \<longrightarrow> Pred delta)"
Ind_3: shows "ra setinterleaves ((trace_hide t A, trace_hide u A), C)"
word_length_def: fixes A :: "'a set"   and a :: "'a" shows "word_length A a \<equiv> length (arg_min length (word_for A a))"
Inter_skip1: fixes P :: "'a process" shows "(P|||SKIP) = P"
UN_singleton: fixes A :: "'d set" shows "(\<Union>x\<in>A. {x}) = A"
szip_sconst_smap_fst: fixes a :: "'a"   and xs :: "'b stream" shows "sconst a ||| xs = smap (Pair a) xs"
nonzero_minus_divide_divide: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and b :: "'b"   and a :: "'b" assumes "class.division_ring divide inverse one times plus zero minus uminus"   and "b \<noteq> zero" shows "divide (uminus a) (uminus b) = divide a b"
finite_measure_Union: fixes A :: "('s \<times> 's option) stream set"   and s :: "'s \<times> 's option"   and B :: "('s \<times> 's option) stream set" assumes "A \<in> KN.T.events s"   and "B \<in> KN.T.events s"   and "A \<inter> B = {}" shows "KN.T.prob s (A \<union> B) = KN.T.prob s A + KN.T.prob s B"
A_modifies_sec: shows "\<forall>x. dma mem\<^sub>1 [\<parallel>\<^sub>1 A] x = Low \<and> (x \<in> mds AsmNoReadOrWrite \<longrightarrow> x \<in> \<C>) \<longrightarrow> mem\<^sub>1 [\<parallel>\<^sub>1 A] x = mem\<^sub>2 [\<parallel>\<^sub>2 A] x"
continuous_sqrt_imp_simply_connected: assumes "connected S"   and "\<And>f. \<lbrakk>continuous_on S f; \<forall>z\<in>S. f z \<noteq> 0\<rbrakk> \<Longrightarrow> \<exists>g. continuous_on S g \<and> (\<forall>z\<in>S. f z = (g z)\<^sup>2)" shows "simply_connected S"
ctor_prism_def: fixes ctor :: "'a \<Rightarrow> 'b"   and disc :: "'b \<Rightarrow> bool"   and sel :: "'b \<Rightarrow> 'a" shows "ctor_prism ctor disc sel = \<lparr>prism_match = \<lambda>d. if disc d then Some (sel d) else None, prism_build = ctor\<rparr>"
case_tllist_Lazy_tllist: fixes n :: "'b \<Rightarrow> 'a"   and c :: "'c \<Rightarrow> ('c, 'b) tllist \<Rightarrow> 'a"   and xs :: "unit \<Rightarrow> 'c \<times> ('c, 'b) tllist + 'b" shows "(case Lazy_tllist xs of TNil x \<Rightarrow> n x | TCons x xa \<Rightarrow> c x xa) = (case xs () of Inl (x, ys) \<Rightarrow> c x ys | Inr b \<Rightarrow> n b)"
curryI: fixes f :: "'k \<times> 'l \<Rightarrow> bool"   and a :: "'k"   and b :: "'l" assumes "f (a, b)" shows "curry f a b"
Partition_parts1: fixes m :: "nat" shows "Partition (Suc m) (Suc 0) = 1"
not_ord_pI: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and v :: "'t"   and q :: "'t \<Rightarrow>\<^sub>0 'c" assumes "lookup p v \<noteq> lookup q v"   and "lookup p v \<noteq> (0::'c)"   and "\<And>u. v \<prec>\<^sub>t u \<Longrightarrow> lookup p u = lookup q u" shows "\<not> p \<preceq>\<^sub>p q"
comp_2_def: fixes p2 :: "'f \<Rightarrow> 'e decision option"   and p1 :: "'d \<Rightarrow> 'f decision option" shows "p2 \<circ>\<^sub>2 p1 \<equiv> map_option flat_2 \<circ> (lift p2 \<circ>\<^sub>m p1)"
pre_cases: fixes l :: "('b \<times> nat) tree"   and x :: "'b"   and lv :: "nat"   and r :: "('b \<times> nat) tree" assumes "pre_adjust \<langle>l, (x, lv), r\<rangle>" obtains "invar l \<and> invar r \<and> lv = Suc (lvl r) \<and> lvl l = lvl r"   | "invar l \<and> invar r \<and> lv = lvl r \<and> Suc (lvl l) = lvl r \<and> sngl r"   | "invar l \<and> invar r \<and> lv = Suc (Suc (lvl r)) \<and> lv = Suc (lvl l)"   | "invar l \<and> invar r \<and> lv = Suc (lvl r) \<and> lv = Suc (Suc (lvl l))"   | "invar l \<and> invar r \<and> lv = lvl r \<and> lv = Suc (Suc (lvl l)) \<and> sngl r"
enum_prot_atom_def: shows "enum_class.enum \<equiv> map prot_atom.Atom enum_class.enum @ [Value, SetType, AttackType, Bottom, OccursSecType]"
Iagree_Contexts: fixes I :: "('d, 'e, 'f) interp"   and J :: "('d, 'e, 'f) interp"   and V :: "('d + 'e + 'f) set"   and C :: "'e" assumes "Iagree I J V"   and "Inr (Inl C) \<in> V" shows "Contexts I C = Contexts J C"
LIMSEQ_divide_realpow_zero: fixes x :: "real"   and a :: "real" assumes "1 < x" shows "(\<lambda>n. a / x ^ n) \<longlonglongrightarrow> 0"
F_def: shows "F = [f, f', f'']\<^sub>\<circ>"
nfoldli_filter_deforestation: fixes P :: "'d \<Rightarrow> bool"   and xs :: "'d list"   and c :: "'c \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'c \<Rightarrow> 'c nres"   and s :: "'c" shows "nfoldli (filter P xs) c f s = nfoldli xs c (\<lambda>x s. if P x then f x s else RETURN s) s"
shows_list_poly_def: fixes ps :: "'a poly list" shows "shows_list ps = showsp_list shows_prec 0 ps"
FInter_def: fixes xs :: "'b fset fset" shows "\<Inter>\<^sub>fxs = Abs_fset (\<Inter> (fset ` \<langle>xs\<rangle>\<^sub>f))"
inv\<^sub>2E_3: fixes m :: "nat"   and t :: "nat \<Rightarrow> nat"   and n :: "nat"   and T :: "nat \<Rightarrow> nat"   and A :: "nat \<Rightarrow> nat set"   and j :: "nat"   and T' :: "nat \<Rightarrow> nat"   and A' :: "nat \<Rightarrow> nat set" assumes "LoadBalancing m"   and "LoadBalancing.inv\<^sub>2 t m n T A j"   and "LoadBalancing.lb t m T' A' j" shows "real (f_Max\<^sub>0 T m) \<le> 3 / 2 * real (f_Max\<^sub>0 T' m)"
map_id: fixes t :: "('g, 'h) tllist" shows "tmap id id t = t"
gf_U'_pos: fixes z :: "real"   and x :: "'s"   and y :: "'s" assumes "0 < z"   and "z < 1"   and "U x y \<noteq> 0" shows "0 < gf_U' x y z"
successor_is_node: fixes n :: "'node"   and g :: "'g"   and n' :: "'node" assumes "n \<in> set (predecessors g n')"   and "n \<in> set (\<alpha>n g)"   and "invar g" shows "n' \<in> set (\<alpha>n g)"
PAC_checker_specification_spec_trans: fixes spec :: "int mpoly"   and A :: "nat set \<times> int mpoly multiset"   and st :: "status"   and x2 :: "nat set \<times> int mpoly multiset"   and st' :: "status"   and x1a :: "nat set \<times> int mpoly multiset" assumes "PAC_checker_specification_spec spec A (st, x2)"   and "PAC_checker_specification_step_spec A spec x2 (st', x1a)" shows "PAC_checker_specification_spec spec A (st', x1a)"
in_carrier_4: fixes m :: "'c multiset" shows "Q' m \<in> carrier R"
extremum_unique: fixes a :: "'b upred" shows "(a \<sqsubseteq> true) = (a = true)"
negate_embed: fixes P :: "'a \<Rightarrow> bool"   and s :: "'a" shows "\<guillemotleft> \<N> P \<guillemotright> s = 1 - \<guillemotleft> P \<guillemotright> s"
div_0: fixes a :: "'a" shows "(0::'a) div a = (0::'a)"
convex_hull_singleton: fixes a :: "'b" shows "convex hull {a} = {a}"
dvd_power_same: fixes x :: "'a"   and y :: "'a"   and n :: "nat" assumes "x dvd y" shows "x ^ n dvd y ^ n"
p_eq: shows "p = q * [:- c, 1::'a:] + [:d:]"
special_cosets_lcoset_shift: fixes w :: "'w" assumes "w \<in> W" shows "(+o) w ` \<P> = \<P>"
fold_invar: fixes M :: "'a set"   and P :: "'b \<Rightarrow> bool"   and z :: "'b"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b" assumes "finite M"   and "P z"   and "\<forall>z. \<forall>x\<in>M. P z \<longrightarrow> P (f x z)"   and "comp_fun_commute f" shows "P (Finite_Set.fold f z M)"
lm098: fixes aa :: "('c \<times> 'd set) set"   and N :: "'c set"   and G :: "'d set"   and bids :: "'c \<times> 'd set \<Rightarrow> nat" assumes "aa \<subseteq> N \<times> (Pow G - {{}})" shows "sum (summedBidVector (real \<circ> bids) N G) aa = real (sum (summedBidVector bids N G) aa)"
changes_drop_dup: fixes xs :: "'a list"   and ys :: "'a list" assumes "xs \<noteq> []"   and "ys \<noteq> [] \<longrightarrow> last xs = hd ys" shows "changes (xs @ ys) = changes (xs @ tl ys)"
Limsup_bounded: fixes X :: "'b \<Rightarrow> 'c"   and C :: "'c"   and F :: "'b filter" assumes "\<forall>\<^sub>F n in F. X n \<le> C" shows "Limsup F X \<le> C"
oreturn_wp: fixes P :: "'d \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'d" shows "ovalid (P x) (oreturn x) P"
hyp2: shows "fst (foldl Gauss_Jordan_column_k_iarrays (0, matrix_to_iarray A) [0..<Suc k]) = fst (foldl Gauss_Jordan_column_k (0, A) [0..<Suc k])"
prems_5: shows "literalFalse literal (elements (getM state))"
unsat_stateful_def: fixes \<A> :: "('a \<times> ('fun, 'var) stateful_strand_step) list" shows "unsat_stateful \<A> \<equiv> \<forall>\<I>. subst_domain \<I> = \<V> \<and> ground (subst_range \<I>) \<longrightarrow> \<not> \<lbrakk>{}; {}; unlabel \<A>\<rbrakk>\<^sub>s \<I>"
transD: fixes r :: "('g \<times> 'g) set"   and x :: "'g"   and y :: "'g"   and z :: "'g" assumes "trans r"   and "(x, y) \<in> r"   and "(y, z) \<in> r" shows "(x, z) \<in> r"
assms_3: shows "(I, J) = getIJ Q Qn V"
list_all2_fusion_def: fixes P :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and g1 :: "('b, 'd) generator"   and g2 :: "('c, 'e) generator"   and s1 :: "'d"   and s2 :: "'e" shows "list_all2_fusion P g1 g2 s1 s2 = list_all2 P (list.unfoldr g1 s1) (list.unfoldr g2 s2)"
F_to_idx_correct: fixes F :: "'b set set" shows "F_to_idx F \<le> SPEC (\<lambda>(num_acc, acc). F = {{q. i \<in> acc q} |i. i < num_acc} \<and> \<Union> (range acc) \<subseteq> {0..<num_acc})"
simps_3: fixes opt :: "atom fm \<Rightarrow> atom fm"   and step :: "nat \<Rightarrow> nat \<Rightarrow> atom list \<Rightarrow> atom fm list \<Rightarrow> atom fm"   and \<phi> :: "atom fm" shows "QE_dnf opt step (Neg \<phi>) = neg (QE_dnf opt step \<phi>)"
in_homE: fixes f :: "'b"   and a :: "'b"   and b :: "'b" assumes "\<guillemotleft>f : a \<rightarrow>\<^sub>B b\<guillemotright>" obtains "B.arr f" and "B.dom f = a" and "B.cod f = b"
hvzk_xr'_rewrite: fixes r :: "nat"   and w :: "nat"   and c :: "nat" assumes "r < order \<G>" shows "((w * c + r) mod order \<G> mod order \<G> + order \<G> * w * c - w * c) mod order \<G> = r"
power2_eq_square: fixes a :: "'b" shows "a\<^sup>2 = a * a"
WCFG_Entry_Exit: fixes prog :: "cmd" shows "prog \<turnstile> (_Entry_) -(\<lambda>s. False)\<^sub>\<surd>\<rightarrow> (_Exit_)"
inv_all_eq_inv_part: fixes p :: "'a mat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bool"   and A :: "'a mat" shows "local.inv_all p A = local.inv_part p A n n"
ln_Gamma_series_def: fixes z :: "'a"   and n :: "nat" shows "ln_Gamma_series z n = z * ln (of_nat n) - ln z - (\<Sum>k = 1..n. ln (z / of_nat k + (1::'a)))"
D4_2: fixes F :: "\<Pi>\<^sub>2" shows "d\<^sub>2 (\<^bold>\<lambda>\<^sup>2 (\<lambda>x y. \<lparr>F,x\<^sup>P,y\<^sup>P\<rparr>)) = d\<^sub>2 F"
surj_pair: fixes p :: "'f \<times> 'g" shows "\<exists>x y. p = (x, y)"
mono_seq_T_right: fixes S :: "'a process"   and S' :: "'a process"   and P :: "'a process" assumes "S \<sqsubseteq>\<^sub>T S'" shows "(P `;` S) \<sqsubseteq>\<^sub>T (P `;` S')"
sqrt_nat_floor_code: fixes x :: "nat" shows "sqrt_nat_floor x = sqrt_int_floor_pos (int x)"
r_rev_def: shows "r_rev \<equiv> Cn 1 r_ifz [recf.Id 1 0, Z, Cn 1 S [Cn 1 r_prod_encode [Cn 1 r_dec [r_length], Cn 1 r_rev_aux [Cn 1 r_dec [r_length], recf.Id 1 0]]]]"
matrix_mul_rid: fixes A :: "(('a, 'b) Finite_Cartesian_Product.vec, 'c) Finite_Cartesian_Product.vec" shows "A ** Finite_Cartesian_Product.mat (1::'a) = A"
set_map: fixes f :: "'c \<Rightarrow> 'b"   and xs :: "'c list" shows "set (map f xs) = f ` set xs"
invalidVertexList_def: fixes g :: "graph"   and f :: "face"   and vs :: "nat option list" shows "invalidVertexList g f vs \<equiv> \<exists>i<|vs| - 1. case vs ! i of None \<Rightarrow> False | Some a \<Rightarrow> case vs ! (i + 1) of None \<Rightarrow> False | Some b \<Rightarrow> is_duplicateEdge g f a b"
cbelow_on_second_below: fixes A :: "'c set"   and ord :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and z :: "'c"   and rel :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c" assumes "cbelow_on A ord z rel a b" shows "ord b z"
eq_snd_iff: fixes b :: "'d"   and p :: "'e \<times> 'd" shows "(b = snd p) = (\<exists>a. p = (a, b))"
in_replicate_mset: fixes x :: "'b"   and n :: "nat"   and y :: "'b" shows "(x \<in># replicate_mset n y) = (0 < n \<and> x = y)"
record_simps_5: fixes f :: "nat \<Rightarrow> nat"   and x :: "'b state" shows "defined_mods (update_clock f x) = defined_mods x"
lunit_simps_4: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and f :: "'b" assumes "bicategory V H \<a> \<i> src trg"   and "partial_magma.ide V f" shows "partial_magma.dom V (bicategory.lunit V H \<a> \<i> src trg f) = H (trg f) f"
assms_2: shows "n \<in>\<^sub>\<circ> \<omega>"
geodesic_segment_topology_2: fixes G :: "'b set" assumes "geodesic_segment G" shows "connected G"
Not_def: shows "Formula.Not \<equiv> (Formula.formula.Rep_formula ---> Formula.formula.Abs_formula) Formula.Not\<^sub>\<alpha>"
induct: fixes P :: "'b poly \<Rightarrow> 'b poly \<Rightarrow> bool"   and a0 :: "'b poly"   and a1 :: "'b poly" assumes "\<And>c c'. P (c)\<^sub>p (c')\<^sub>p"   and "\<And>c c' n' p'. \<lbrakk>P (c)\<^sub>p c'; P (c)\<^sub>p p'\<rbrakk> \<Longrightarrow> P (c)\<^sub>p (CN c' n' p')"   and "\<And>c n p c'. \<lbrakk>P c (c')\<^sub>p; P p (c')\<^sub>p\<rbrakk> \<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 p (CN c' n' p'); P (CN c n p) c'; P (CN c n p) p'; P (CN c n p) c'; P (CN c n p) p'\<rbrakk> \<Longrightarrow> P (CN c n p) (CN c' n' p')"   and "\<And>v b. P (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) (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 (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 (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"
D_invar_impl_def: fixes Q :: "'b"   and V :: "'a" shows "D_invar_impl Q V \<equiv> Q_invar Q \<and> M_invar V \<and> D_invar' (Q_lookup Q) (M_lookup V)"
that_2: fixes u :: "nat"   and p' :: "(nat \<times> nat) list" assumes "p = (s, u) # p'"   and "(s, u) \<in> E"   and "isPath u p' t" shows "thesis"
Taylor_up: fixes n :: "nat"   and diff :: "nat \<Rightarrow> real \<Rightarrow> real"   and f :: "real \<Rightarrow> real"   and a :: "real"   and b :: "real"   and c :: "real" assumes "0 < n"   and "diff 0 = f"   and "\<forall>m t. m < n \<and> a \<le> t \<and> t \<le> b \<longrightarrow> (diff m has_real_derivative diff (Suc m) t) (at t)"   and "a \<le> c"   and "c < b" shows "\<exists>t>c. t < b \<and> f b = (\<Sum>m<n. diff m c / fact m * (b - c) ^ m) + diff n t / fact n * (b - c) ^ n"
l_to_tree_order_wf_axioms: shows "l_to_tree_order_wf Shadow_DOM.heap_is_wellformed Shadow_DOM.parent_child_rel ShadowRootClass.type_wf ShadowRootClass.known_ptr ShadowRootClass.known_ptrs Shadow_DOM.to_tree_order Shadow_DOM.get_parent Shadow_DOM.get_child_nodes"
Setcompr_eq_image: fixes f :: "'f \<Rightarrow> 'e"   and A :: "'f set" shows "{f x |x. x \<in> A} = f ` A"
leftderives_start: shows "leftderives [\<SS>] (terminals (take r q) @ [N] @ \<gamma>)"
sorted_on_insort_key_on: fixes f :: "'b \<Rightarrow> 'a"   and x :: "'b"   and xs :: "'b list" assumes "f ` set (x # xs) \<subseteq> Field r" shows "sorted_on r (map f (insort_key_on r f x xs)) = sorted_on r (map f xs)"
upper_bound_def: fixes D :: "('c, 'd) Order_scheme"   and S :: "'c set"   and b :: "'c" shows "ub\<^bsub>D\<^esub> S b = (b \<in> carrier D \<and> (\<forall>s\<in>S. s \<preceq>\<^bsub>D\<^esub> b))"
word_ao_dist2: fixes x :: "'a word"   and y :: "'a word"   and z :: "'a word" shows "x AND (y OR z) = x AND y OR x AND z"
eq\<rho>_mono: fixes Aexp :: "exp \<Rightarrow> Arity \<rightarrow> Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and Aheap :: "(Vars.var \<times> exp) list \<Rightarrow> exp \<Rightarrow> Arity \<rightarrow> Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and ae' :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and \<rho>1 :: "Vars.var \<Rightarrow> Value"   and \<rho>2 :: "Vars.var \<Rightarrow> Value" assumes "ArityAnalysisLetSafe Aexp Aheap"   and "ae \<sqsubseteq> ae'"   and "eq\<rho> ae' \<rho>1 \<rho>2" shows "eq\<rho> ae \<rho>1 \<rho>2"
awalkE': fixes u :: "'a"   and p :: "'b list"   and v :: "'a" assumes "awalk u p v" obtains "set (awalk_verts u p) \<subseteq> verts G" and "set p \<subseteq> arcs G" and "cas u p v" and "awhd u p = u" and "awlast u p = v" and "u \<in> verts G" and "v \<in> verts G"
iso_class_eqI: fixes f :: "'a"   and g :: "'a" assumes "f \<cong>\<^sub>B g" shows "\<lbrakk>f\<rbrakk>\<^sub>B = \<lbrakk>g\<rbrakk>\<^sub>B"
pseudominimal_inD1: fixes P :: "'b set"   and x :: "'b" assumes "dual_order.pseudominimal_in P x" shows "x \<in> P"
imp_to_meta: fixes P :: "bool"   and Q :: "bool" shows "P \<longrightarrow> Q \<equiv> (P \<Longrightarrow> Q)"
App_def: fixes x2 :: "trm"   and x1 :: "trm" shows "App x2 x1 \<equiv> Abs_trm (App_Rep (Rep_trm x2) (Rep_trm x1))"
interval_lemma1: fixes S :: "'a set"   and a :: "'a"   and b :: "'a"   and x :: "'a" assumes "S \<subseteq> interval r a b"   and "x \<in> S" shows "(a, x) \<in> r"
R2'_a_def: shows "R2'_a \<equiv> {: p, t, lnk, lbl, mrk \<leadsto> stk, mrk' . p = head stk \<and> t = head (tail stk) \<and> stack lnk lbl t (tail stk) \<and> link0 = link_0 lnk lbl p (tail stk) \<and> label0 = label_0 lbl (tail stk) \<and> nil \<notin> set stk \<and> mrk' = mrk :}"
dist_eqs_2: fixes a :: "'b"   and list :: "['b]" shows "a : list \<noteq> []"
assms_1: shows "\<S>\<bullet>None \<tturnstile>_ i vs;es : ts"
convergent_prod_ignore_initial_segment: fixes f :: "nat \<Rightarrow> 'e"   and m :: "nat" assumes "convergent_prod f" shows "convergent_prod (\<lambda>n. f (n + m))"
rcis_divide: fixes r1 :: "real"   and a :: "real"   and r2 :: "real"   and b :: "real" shows "rcis r1 a / rcis r2 b = rcis (r1 / r2) (a - b)"
sfilter_shd_LEAST: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b stream" assumes "ev (holds P) xs" shows "shd (sfilter P xs) = xs !! (LEAST n. P (xs !! n))"
g_nonneg: fixes t :: "real" assumes "t \<in> {a..b}" shows "0 \<le> g t"
N5_lattice_def: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "N5_lattice a b c = (a \<sqinter> c = b \<sqinter> c \<and> a < b \<and> a \<squnion> c = b \<squnion> c)"
has_field_derivative_def: fixes f :: "'b \<Rightarrow> 'b"   and D :: "'b"   and F :: "'b filter" shows "(f has_field_derivative D) F = (f has_derivative (*) D) F"