hcju/isabelleps huggingface dataset:
is_character_data_ptr\<^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_def: fixes ptr :: "(_) character_data_ptr" shows "is_character_data_ptr ptr = (case ptr of character_data_ptr.Ref x \<Rightarrow> True | character_data_ptr.Ext character_data_ptr \<Rightarrow> False)"
Parallel: fixes i :: "nat"   and cs :: "('d, 'e, 'f) com list"   and \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and s :: "('d, 'f) xstate"   and c' :: "('d, 'e, 'f) com"   and s' :: "('d, 'f) xstate" assumes "i < length cs"   and "\<Gamma>\<turnstile> (cs ! i, s) \<rightarrow> (c', s')" shows "\<Gamma>\<turnstile> (Parallel cs, s) \<rightarrow> (Parallel (cs[i := c']), s')"
additive_right: fixes prod :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and a :: "'a" assumes "bounded_bilinear prod" shows "additive (prod a)"
exact_seq_cons_iff: fixes G :: "'a monoid"   and K :: "'a monoid"   and H :: "'a monoid"   and l :: "'a monoid list"   and g :: "'a \<Rightarrow> 'a"   and h :: "'a \<Rightarrow> 'a"   and q :: "('a \<Rightarrow> 'a) list" shows "exact_seq (G # K # H # l, g # h # q) = (exact_seq (K # H # l, h # q) \<and> Group.group G \<and> g \<in> hom K G \<and> kernel K G g = h ` carrier H)"
select_min_def: fixes xs :: "(nat \<times> 'a) list" shows "local.select_min xs = (case xs of (i, p) # ys \<Rightarrow> local.select_min_main (measure p) i ys)"
add_assoc: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<oplus> y \<oplus> z = x \<oplus> (y \<oplus> z)"
get_owner_document_ptr_in_heap: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr" assumes "h \<turnstile> ok get_owner_document ptr" shows "ptr |\<in>| object_ptr_kinds h"
R19_1: shows "pmf (sds R19) a = 1 / 2"
prems_1: assumes "remove1 c (u @ [a] @ [b] @ v) = remove1 c (u @ [b] @ [a] @ v)" shows "thesis"
mult_assoc: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "a \<cdot> b \<cdot> c = a \<cdot> (b \<cdot> c)"
map_splitE: fixes f :: "'c \<Rightarrow> 'b"   and xs :: "'c list"   and ys :: "'b list"   and zs :: "'b list" assumes "map f xs = ys @ zs" obtains us :: "'c list" and vs :: "'c list" where "xs = us @ vs" and "map f us = ys" and "map f vs = zs"
Sigma_empty2: fixes A :: "'a set" shows "A \<times> {} = {}"
sublists_fin: fixes x :: "'b list" shows "finite {t. \<exists>t1 t2. x = t1 @ t @ t2}"
inject_1: fixes x1 :: "'c"   and y1 :: "'c" shows "(Content x1 = Content y1) = (x1 = y1)"
assms_2: shows "\<turnstile> imply ps (p \<^bold>\<longrightarrow> q)"
pimage: shows "\<forall>p\<in>path_image \<gamma>. Im p \<le> Im z"
empty_def: shows "local.empty \<equiv> Abs_FingerTree Empty"
cong: shows "\<forall>i\<in>A. [y mod prod m A = u i] (mod m i)"
int_pow_eq: fixes x :: "'a"   and m :: "int"   and n :: "int" assumes "x \<in> carrier \<G>" shows "(x [^] m = x [^] n) = (int (ord x) dvd n - m)"
le_ccINF_iff: fixes A :: "'b set"   and u :: "'c"   and f :: "'b \<Rightarrow> 'c" assumes "countable A" shows "(u \<le> Inf (f ` A)) = (\<forall>i\<in>A. u \<le> f i)"
simps_15: fixes x1 :: "'b" shows "set_literal (x1\<^sup>+) = {x1}"
fin_2: shows "\<integral>\<^sup>+ x. ennreal (- f x) \<partial>M \<noteq> \<infinity>"
treewidth_upper_bound_ex: fixes T :: "nat Graph"   and bag :: "nat \<Rightarrow> nat set"   and n :: "nat" assumes "TreeDecomposition G T bag"   and "TreeDecomposition.width T bag \<le> n" shows "treewidth \<le> n"
iMOD_plus_divisor_mult: fixes x :: "nat"   and r :: "nat"   and m :: "nat"   and k :: "nat" assumes "x \<in> [ r, mod m ]" shows "x + k * m \<in> [ r, mod m ]"
elims_1: fixes x :: "'b btree"   and y :: "bool" assumes "nonempty_lasttreebal x = y" obtains "x = Leaf" and "y"   | ts :: "('b btree \<times> 'b) list" and t :: "'b btree" where "x = Node ts t" and "y = ((\<exists>ls tsub tsep. ts = ls @ [(tsub, tsep)] \<and> height tsub = height t) \<and> nonempty_lasttreebal t)"
infTp_I_intT: fixes \<sigma> :: "'tp"   and a :: "univ" assumes "infTp \<sigma>" shows "I_intT \<sigma> a = intT \<sigma> a"
prems_1: shows "strict_sorted (concat (as1 @ as2))"
has_lock_has_lock_t_eq: fixes l :: "('b \<times> nat) option"   and t :: "'b"   and t' :: "'b" assumes "has_lock l t"   and "has_lock l t'" shows "t = t'"
map_add_def: fixes m1 :: "'e \<Rightarrow> 'f option"   and m2 :: "'e \<Rightarrow> 'f option" shows "m2 \<Oplus> m1 = (\<lambda>x. case m2 x of \<bottom> \<Rightarrow> m1 x | \<lfloor>y\<rfloor> \<Rightarrow> \<lfloor>y\<rfloor>)"
nonempty_ex: fixes A :: "'c set" assumes "A \<noteq> {}" shows "\<exists>x. x \<in> A"
indefinite_integral_continuous: fixes f :: "'a \<Rightarrow> 'b"   and a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a"   and \<epsilon> :: "real" assumes "f integrable_on cbox a b"   and "c \<in> cbox a b"   and "d \<in> cbox a b"   and "0 < \<epsilon>" obtains \<delta> :: "real" where "0 < \<delta>" and "\<And>c' d'. \<lbrakk>c' \<in> cbox a b; d' \<in> cbox a b; norm (c' - c) \<le> \<delta>; norm (d' - d) \<le> \<delta>\<rbrakk> \<Longrightarrow> norm (integral (cbox c' d') f - integral (cbox c d) f) < \<epsilon>"
inv_is_inverse: fixes f :: "'a \<times> 'a" assumes "B.VV.iso f" shows "B.VV.inverse_arrows f (B.VV.inv f)"
many_strongly_connected_iff_2: fixes x :: "'a" shows "many_strongly_connected x = (x\<^sup>T \<le> x\<^sup>+)"
h_def: shows "h \<equiv> \<lambda>z. if z \<in> X then ZFC_Cardinals.Inl (f z) else ZFC_Cardinals.Inr (g z)"
lstrict_prefix_code_2: fixes y :: "'b"   and ys :: "'b llist" shows "lstrict_prefix LNil (LCons y ys) = True"
at_map_atom_pres2: fixes x :: "'a" assumes "atom x" shows "atom (at_map x)"
ipresIGSubstAll_def: fixes h :: "'c \<Rightarrow> 'd"   and hA :: "'e \<Rightarrow> 'f"   and MOD :: "('g, 'h, 'varSort, 'sort, 'i, 'j, 'c, 'e, 'k) model_scheme"   and MOD' :: "('l, 'm, 'varSort, 'n, 'o, 'j, 'd, 'f, 'p) model_scheme" shows "ipresIGSubstAll h hA MOD MOD' \<equiv> ipresIGSubst h MOD MOD' \<and> ipresIGSubstAbs h hA MOD MOD'"
setmonotone_comp: fixes f :: "'a set \<Rightarrow> 'a set"   and g :: "'a set \<Rightarrow> 'a set" assumes "setmonotone f"   and "setmonotone g" shows "setmonotone (g \<circ> f)"
axioms_5: fixes get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and set_child_nodes :: "(_) object_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_parent :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr option) prog"   and get_parent_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   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 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_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and remove_child :: "(_) object_ptr \<Rightarrow> (_) node_ptr \<Rightarrow> ((_) heap, exception, unit) prog"   and remove_child_locs :: "(_) object_ptr \<Rightarrow> (_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and remove :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, unit) prog"   and type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and known_ptrs :: "(_) heap \<Rightarrow> bool" assumes "l_remove_child\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_child_nodes get_child_nodes_locs set_child_nodes set_child_nodes_locs get_parent get_parent_locs get_owner_document get_disconnected_nodes get_disconnected_nodes_locs set_disconnected_nodes set_disconnected_nodes_locs remove_child remove_child_locs remove type_wf known_ptr known_ptrs" shows "l_set_disconnected_nodes_get_disconnected_nodes type_wf get_disconnected_nodes get_disconnected_nodes_locs set_disconnected_nodes set_disconnected_nodes_locs"
prj0_def: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and f :: "'a"   and g :: "'a" assumes "category_with_pullbacks C" shows "category_with_pullbacks.prj0 C f g \<equiv> if partial_magma.arr C f \<and> partial_magma.arr C g \<and> partial_magma.cod C f = partial_magma.cod C g then snd (SOME x. limit_cone cospan_shape.comp C (cospan_diagram.map C f g) (partial_magma.dom C (fst x)) (cospan_diagram.mkCone C f g (fst x) (snd x))) else partial_magma.null C"
set_bd_1: fixes x :: "('d, 'e, 'f) F2" shows "|F2set1 x| \<le>o bd_F2"
continuous_logarithm_on_simply_connected: fixes S :: "'a set"   and f :: "'a \<Rightarrow> complex" assumes "continuous_on S f"   and "simply_connected S"   and "locally path_connected S"   and "\<And>z. z \<in> S \<Longrightarrow> f z \<noteq> 0" obtains g :: "'a \<Rightarrow> complex" where "continuous_on S g" and "\<And>x. x \<in> S \<Longrightarrow> f x = exp (g x)"
ennreal_of_enat_inj: fixes i :: "enat"   and j :: "enat" shows "(ennreal_of_enat i = ennreal_of_enat j) = (i = j)"
option_rel_simp_3: fixes x :: "'a"   and y :: "'b"   and R :: "('a \<times> 'b) set" shows "((Some x, Some y) \<in> \<langle>R\<rangle>option_rel) = ((x, y) \<in> R)"
integral_def: fixes f :: "'b \<Rightarrow> real"   and M :: "'b set set \<times> ('b set \<Rightarrow> real)" assumes "integrable f M" shows "\<integral> f \<partial>M = (THE i. i \<in> nnfis (pp f) M) - (THE j. j \<in> nnfis (np f) M)"
ctrm_smult: fixes f :: "nat \<Rightarrow> 'a"   and a :: "'a" assumes "f \<in> carrier P"   and "a \<in> carrier R" shows "monom P ((a \<odot>\<^bsub>P\<^esub> f) 0) 0 = a \<odot>\<^bsub>P\<^esub> monom P (f 0) 0"
hyps: fixes s' :: "nat"   and l' :: "cell list"   and r' :: "cell list" assumes "steps0 (1, l, r) B stp = (s', l', r')" shows "steps0 (Suc (length A div 2), l, r) (A |+| B) stp = (if s' = 0 then 0 else s' + length A div 2, l', r')"
AnalBinds_lookup: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and ae :: "Vars.var \<Rightarrow> 'a\<^sub>\<bottom>"   and x :: "Vars.var" shows "(AnalBinds \<Gamma>\<cdot>ae) x = (case map_of \<Gamma> x of None \<Rightarrow> \<bottom> | Some e \<Rightarrow> fup\<cdot>(exp e)\<cdot>(ae x))"
finite_subset: fixes A :: "'k set"   and B :: "'k set" assumes "A \<subseteq> B"   and "finite B" shows "finite A"
ag_eq_sol2: fixes a :: "'a"   and x :: "'a"   and b :: "'a" assumes "a \<in> carrier A"   and "x \<in> carrier A"   and "b \<in> carrier A"   and "x \<plusminus> a = b" shows "x = b \<plusminus> -\<^sub>a a"
inducts: fixes x :: "'b list"   and C :: "'b list set"   and b :: "'b list"   and P :: "'b list \<Rightarrow> bool" assumes "x \<in> no_head_gen C b"   and "\<And>u. \<lbrakk>u \<in> C; u \<noteq> b\<rbrakk> \<Longrightarrow> P u"   and "\<And>u. \<lbrakk>u \<in> no_head_gen C b; P u\<rbrakk> \<Longrightarrow> P (u \<cdot> b)" shows "P x"
put_writer_def: fixes put :: "'g \<Rightarrow> 'f \<Rightarrow> 'f"   and s :: "'g"   and m :: "('d, 'e, 'f) writerT" shows "put_writer put s m = WriterT (put s (run_writer m))"
65616: fixes x :: "'a"   and y :: "'a" shows "- (- x \<squnion> y) = - y \<sqinter> - ((top \<squnion> y) \<sqinter> - x)"
is_int_t_exists: fixes t :: "t" assumes "is_int_t t" shows "t = T_i32 \<or> t = T_i64"
simps_6: fixes f1 :: "\<omega> \<Rightarrow> 'a"   and f2 :: "\<sigma> \<Rightarrow> 'a"   and x2 :: "\<sigma>" shows "(case \<sigma>\<upsilon> x2 of \<omega>\<upsilon> x \<Rightarrow> f1 x | \<sigma>\<upsilon> x \<Rightarrow> f2 x) = f2 x2"
snoc: shows "nths (nths xs A) (ind_in_set A ` B) = nths xs B"
less: fixes xs :: "'a list" assumes "length xs < length xsa__" shows "P xs"
prod_algebraE: fixes A :: "('b \<Rightarrow> 'c) set"   and I :: "'b set"   and M :: "'b \<Rightarrow> 'c measure" assumes "A \<in> prod_algebra I M" obtains J :: "'b set" and E :: "'b \<Rightarrow> 'c set" where "A = prod_emb I M J (Pi\<^sub>E J E)" and "finite J" and "J \<noteq> {} \<or> I = {}" and "J \<subseteq> I" and "\<And>i. i \<in> J \<Longrightarrow> E i \<in> sets (M i)"
that: fixes m :: "nat" assumes "n = m\<^sub>\<nat>" shows "thesis"
mult_compose: fixes x :: "'e \<Rightarrow> 'd"   and y :: "'e \<Rightarrow> 'd"   and f :: "'c \<Rightarrow> 'e" shows "x * y \<circ> f = (x \<circ> f) * (y \<circ> f)"
False: shows "mem_context_val_w32 9 (ucast (ptp OR ucast (ucast (va >> 12)))) (mem s1) \<noteq> None"
assoc: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "inf (inf a b) c = inf a (inf b c)"
next_inI: fixes u :: "'a"   and v :: "'a" assumes "u \<in> {a..<b}"   and "v \<in> A"   and "u < v"   and "{u<..<v} \<inter> A = {}" shows "next_in A u = v"
sat_def: fixes \<AA> :: "'i"   and \<phi> :: "('a, 'k) aformula" shows "sat \<AA> \<phi> \<equiv> sat_vars_gen (\<lambda>k P n. True) UNIV \<AA> \<phi>"
lcosets_carrier: fixes H :: "'d set"   and G :: "('d, 'e) monoid_scheme"   and X :: "'d set" assumes "subgroup H G"   and "Group.group G"   and "X \<in> lcosets\<^bsub>G\<^esub> H" shows "X \<subseteq> carrier G"
Ex: shows "\<exists>n. drop n (map \<pi> (stake min s)) = replicate (length (map \<pi> (stake min s)) - n) (\<pi> x)"
degree_div: fixes a :: "'b poly"   and b :: "'b poly" assumes "a dvd b" shows "degree (b div a) = degree b - degree a"
DSourcesS13: shows "DSources level2 sS13 = {sS12}"
subsumes_def: fixes C :: "'a Literal set"   and D :: "'a Literal set" shows "subsumes C D \<equiv> C \<subseteq> D"
countable_as_injective_image: fixes A :: "'b set" assumes "countable A"   and "infinite A" obtains f :: "nat \<Rightarrow> 'b" where "A = range f" and "inj f"
arrI: fixes f1 :: "'a"   and f2 :: "'a \<times> 'a" assumes "arr f1"   and "CC.arr f2" shows "CCC.arr (f1, f2)"
det_nz_iff_inj: fixes f :: "(real, 'a) vec \<Rightarrow> (real, 'a) vec" assumes "linear f" shows "(det (matrix f) \<noteq> 0) = inj f"
simps_10: fixes v :: "'a acom"   and va :: "'a acom"   and vb :: "'a" shows "(v;; va \<sqsubseteq> SKIP {vb}) = False"
id_comp: fixes g :: "'n \<Rightarrow> 'o" shows "id \<circ> g = g"
explode_cases: fixes y :: "char list" assumes "y \<in> {cs. \<forall>c\<in>set cs. \<not> digit7 c}" obtains x :: "literal" where "y = explode x"
crsp_step_dec_b_suc_pre: fixes ly :: "nat list"   and ap :: "abc_inst list"   and as :: "nat"   and lm :: "nat list"   and s :: "nat"   and l :: "cell list"   and r :: "cell list"   and ires :: "cell list"   and n :: "nat"   and la :: "cell list"   and ra :: "cell list"   and e :: "nat"   and f :: "nat \<Rightarrow> (nat \<times> cell list \<times> cell list) \<times> nat \<times> nat"   and P :: "(nat \<times> cell list \<times> cell list) \<times> nat \<times> nat \<Rightarrow> bool"   and Q :: "(nat \<times> cell list \<times> cell list) \<times> nat \<times> nat \<Rightarrow> bool" assumes "ly = layout_of ap"   and "crsp ly (as, lm) (s, l, r) ires"   and "inv_locate_a (as, lm) (n, la, ra) ires"   and "abc_fetch as ap = Some (Dec n e)"   and "0 < abc_lm_v lm n"   and "f = (\<lambda>stp. (steps (start_of ly as + 2 * n, la, ra) (ci ly (start_of ly as) (Dec n e), start_of ly as - Suc 0) stp, start_of ly as, n))"   and "P = (\<lambda>((s, l, r), ss, x). s = start_of ly as + 2 * n + 16)"   and "Q = (\<lambda>((s, l, r), ss, x). dec_inv_2 ly x e (as, lm) (s, l, r) ires)" shows "\<exists>stp. P (f stp) \<and> Q (f stp)"
induct: fixes P :: "'b negation_type list \<Rightarrow> bool"   and a0 :: "'b negation_type list" assumes "P []"   and "\<And>x xs. P xs \<Longrightarrow> P (Pos x # xs)"   and "\<And>v xs. P xs \<Longrightarrow> P (Neg v # xs)" shows "P a0"
list_all2_refl: fixes P :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and xs :: "'e list" assumes "\<And>x. P x x" shows "list_all2 P xs xs"
shift_factor0: fixes f :: "nat \<Rightarrow> 'a"   and k :: "nat" assumes "f \<in> carrier P"   and "Suc k \<le> deg R f" shows "deg R (f \<ominus>\<^bsub>P\<^esub> shift (Suc k) f \<otimes>\<^bsub>P\<^esub> X_poly R [^]\<^bsub>P\<^esub> Suc k) < Suc k"
OAlist_tc_min_key_val_def: fixes xs :: "('c, 'd) oalist_tc" shows "OAlist_tc_min_key_val xs = hd (list_of_oalist_tc xs)"
One_fract_def: shows "1 = Fraction_Field.Fract (1::'a) (1::'a)"
last_block: fixes i :: "nat"   and t :: "nat" shows "e_nth (block i t) (gap i t) = 0"
isFDERIV_approx_def: fixes p :: "nat"   and n :: "nat"   and xs :: "nat list"   and fas :: "floatarith list"   and vs :: "float interval option list" shows "isFDERIV_approx p n xs fas vs = ((\<forall>i<n. \<forall>j<n. isDERIV_approx p (xs ! i) (fas ! j) vs) \<and> length fas = n \<and> length xs = n)"
strict_implies_not_eq: fixes b :: "'a"   and a :: "'a" assumes "lt b a" shows "a \<noteq> b"
dghm_tdghm_comp_NTMap_app: fixes x :: "V"   and \<HH> :: "V" assumes "x \<in>\<^sub>\<circ> \<BB>\<lparr>Obj\<rparr>" shows "(\<HH> \<circ>\<^sub>D\<^sub>G\<^sub>H\<^sub>M\<^sub>-\<^sub>T\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<NN>)\<lparr>NTMap\<rparr>\<lparr>x\<rparr> = \<HH>\<lparr>ArrMap\<rparr>\<lparr>\<NN>\<lparr>NTMap\<rparr>\<lparr>x\<rparr>\<rparr>"
round_num_denom_d\<mu>_d: fixes n :: "nat"   and fs :: "int vec list"   and j :: "nat"   and i :: "nat" assumes "fs_int_indpt n fs"   and "j \<le> i"   and "i < length fs" shows "round_num_denom (fs_int_indpt.d\<mu> n fs i j) (fs_int.d n fs (Suc j)) = round (gram_schmidt_fs.\<mu> n (map of_int_hom.vec_hom fs) i j)"
state_sim_trans: fixes s :: "'b"   and A :: "('b \<Rightarrow> bool) set"   and t :: "'b"   and u :: "'b" assumes "s ~A~ t"   and "t ~A~ u" shows "s ~A~ u"
hyps_3: shows "\<Gamma>\<turnstile> \<langle>c',s'\<rangle> \<Rightarrow> t"
rep_eq: fixes x :: "('b, 'a) finfun" shows "finfun_default x = finfun_default_aux (finfun_apply x)"
padic_zero_def: fixes p :: "int" shows "padic_zero p \<equiv> \<lambda>n. 0"
lexordp_iff: fixes xs :: "'a list"   and ys :: "'a list" shows "lexordp xs ys = ((\<exists>x vs. ys = xs @ x # vs) \<or> (\<exists>us a b vs ws. a < b \<and> xs = us @ a # vs \<and> ys = us @ b # ws))"
tendsto_dist: fixes f :: "'b \<Rightarrow> 'c"   and l :: "'c"   and F :: "'b filter"   and g :: "'b \<Rightarrow> 'c"   and m :: "'c" assumes "(f \<longlongrightarrow> l) F"   and "(g \<longlongrightarrow> m) F" shows "((\<lambda>x. dist (f x) (g x)) \<longlongrightarrow> dist l m) F"
hnr_bind: fixes \<Gamma> :: "assn"   and m' :: "'c Heap"   and \<Gamma>1 :: "assn"   and Rh :: "'d \<Rightarrow> 'c \<Rightarrow> assn"   and m :: "'d nres"   and f' :: "'c \<Rightarrow> 'e Heap"   and \<Gamma>2 :: "'d \<Rightarrow> 'c \<Rightarrow> assn"   and R :: "'f \<Rightarrow> 'e \<Rightarrow> assn"   and f :: "'d \<Rightarrow> 'f nres"   and \<Gamma>' :: "assn"   and Rx :: "'d \<Rightarrow> 'c \<Rightarrow> assn" assumes "hn_refine \<Gamma> m' \<Gamma>1 Rh m"   and "\<And>x x'. RETURN x \<le> m \<Longrightarrow> hn_refine (\<Gamma>1 * hn_ctxt Rh x x') (f' x') (\<Gamma>2 x x') R (f x)"   and "\<And>x x'. \<Gamma>2 x x' \<Longrightarrow>\<^sub>t \<Gamma>' * hn_ctxt Rx x x'" shows "hn_refine \<Gamma> (m' \<bind> f') \<Gamma>' R (m \<bind> f)"
merge_property': fixes t1 :: "'e list"   and t2 :: "'e list" assumes "set t1 \<subseteq> E\<^bsub>ES1\<^esub>"   and "set t2 \<subseteq> E\<^bsub>ES2\<^esub>"   and "t1 \<upharpoonleft> E\<^bsub>ES2\<^esub> = t2 \<upharpoonleft> E\<^bsub>ES1\<^esub>"   and "t1 \<upharpoonleft> V\<^bsub>\<V>\<^esub> = []"   and "t2 \<upharpoonleft> V\<^bsub>\<V>\<^esub> = []"   and "t1 \<upharpoonleft> C\<^bsub>\<V>\<^esub> = []"   and "t2 \<upharpoonleft> C\<^bsub>\<V>\<^esub> = []" shows "\<exists>t. t \<upharpoonleft> E\<^bsub>ES1\<^esub> = t1 \<and> t \<upharpoonleft> E\<^bsub>ES2\<^esub> = t2 \<and> t \<upharpoonleft> V\<^bsub>\<V>\<^esub> = [] \<and> t \<upharpoonleft> C\<^bsub>\<V>\<^esub> = [] \<and> set t \<subseteq> E\<^bsub>ES1\<^esub> \<union> E\<^bsub>ES2\<^esub>"
no_overlaps_ne1: fixes \<gamma> :: "'a set \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "('a, 'c) flow_entry_match list"   and a :: "('a, 'c) flow_entry_match"   and y :: "('a, 'c) flow_entry_match list"   and b :: "('a, 'c) flow_entry_match"   and z :: "('a, 'c) flow_entry_match list" assumes "no_overlaps \<gamma> (x @ a # y @ b # z)"   and "(\<exists>p. \<gamma> (ofe_fields a) p) \<or> (\<exists>p. \<gamma> (ofe_fields b) p)" shows "a \<noteq> b"
bitor_zero: fixes y :: "bit" shows "(\<zero> \<or>\<^sub>b y) = y"
rw: shows "Hermite_of_row_i ass res A i $ j $ (LEAST n. A $ i $ n \<noteq> (0::'b)) = row_add M j i ((res (M $ i $ (LEAST n. A $ i $ n \<noteq> (0::'b))) (M $ j $ (LEAST n. A $ i $ n \<noteq> (0::'b))) - M $ j $ (LEAST n. A $ i $ n \<noteq> (0::'b))) div M $ i $ (LEAST n. A $ i $ n \<noteq> (0::'b))) $ j $ (LEAST n. A $ i $ n \<noteq> (0::'b))"
new_def: fixes n :: "nat"   and x :: "'b" shows "Array.new n x = Heap_Monad.heap (Array.alloc (replicate n x))"
sdrop_smap: fixes n :: "nat"   and f :: "'d \<Rightarrow> 'c"   and s :: "'d stream" shows "sdrop n (smap f s) = smap f (sdrop n s)"
no_qf: shows "qfree (list_conj (snd (partition bound0 (conjuncts p))))"
equiv_subst_var_eq: fixes subst_var :: "nat \<Rightarrow> linear_poly \<Rightarrow> linear_poly \<Rightarrow> linear_poly"   and v :: "nat \<Rightarrow> 'a"   and x\<^sub>j :: "nat"   and lp' :: "linear_poly"   and eq :: "nat \<times> linear_poly" assumes "SubstVar TYPE('a) subst_var"   and "v \<Turnstile>\<^sub>e (x\<^sub>j, lp')" shows "v \<Turnstile>\<^sub>e eq = v \<Turnstile>\<^sub>e SubstVar.subst_var_eq subst_var x\<^sub>j lp' eq"
disjnt_Un1: fixes A :: "'b set"   and B :: "'b set"   and C :: "'b set" shows "disjnt (A \<union> B) C = (disjnt A C \<and> disjnt B C)"
simps_1: fixes jkbp :: "'b \<Rightarrow> ('b, 'c, 'd) GC list"   and jview :: "'b \<Rightarrow> 'e Trace \<Rightarrow> 'f"   and envInit :: "'e list"   and envAction :: "'e \<Rightarrow> 'g list"   and envTrans :: "'g \<Rightarrow> ('b \<Rightarrow> 'd) \<Rightarrow> 'e \<Rightarrow> 'e"   and envVal :: "'e \<Rightarrow> 'c \<Rightarrow> bool" assumes "PreEnvironmentJView jkbp jview" shows "PreEnvironmentJView.jkbpCn jkbp envInit envAction envTrans envVal jview 0 = {tInit s |s. s \<in> set envInit}"
ideD_3: fixes a :: "('a S.E.term, 'a) S.arr" assumes "S.ide a" shows "S.cod a = a"
o_apply: fixes f :: "'g \<Rightarrow> 'f"   and g :: "'h \<Rightarrow> 'g"   and x :: "'h" shows "(f \<circ> g) x = f (g x)"
attach_converter_of_resource_conv_parallel_resource2: fixes res :: "('b, 'd) resource"   and res' :: "('a, 'c) resource" shows "1\<^sub>C |\<^sub>\<propto> converter_of_resource res \<rhd> res' = res' \<parallel> res"
prems_1: shows "Root_Balanced_Tree.split_min \<langle>l, y, r\<rangle> = (x, t')"
fset_eqI: fixes A :: "'b fset"   and B :: "'b fset" assumes "\<And>x. (x |\<in>| A) = (x |\<in>| B)" shows "A = B"
ld_le_2ld: fixes x :: "real"   and y :: "real" assumes "0 \<le> x"   and "0 \<le> y" shows "log 2 (1 + x + y) \<le> 1 + log 2 (1 + x) + log 2 (1 + y)"
limitin_eventually: fixes l :: "'b"   and X :: "'b topology"   and f :: "'c \<Rightarrow> 'b"   and F :: "'c filter" assumes "l \<in> topspace X"   and "\<forall>\<^sub>F x in F. f x = l" shows "limitin X f l F"
cInf_less_iff: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and X :: "'b set"   and y :: "'b" assumes "class.conditionally_complete_linorder Inf Sup inf less_eq less sup"   and "X \<noteq> {}"   and "preorder.bdd_below less_eq X" shows "less (Inf X) y = (\<exists>x\<in>X. less x y)"
\<gamma>1_def: fixes x :: "real" shows "\<gamma>1 x = \<pi> (2 * \<rho> * \<beta> + 2 * \<Lambda>) (2 * \<Lambda> + x + 2 * \<rho> * (rmax + \<beta>))"
img: fixes bl :: "'a set" assumes "bl \<in># \<B>" shows "(inv_into \<V> \<pi> \<circ> \<pi>) ` bl = bl"
False: shows "\<not> (n = 0 \<or> rai_ub x = 0)"