hcju/isabelleps huggingface dataset:
id: shows "mat_of_row (y1 @\<^sub>v y2) = four_block_mat (mat_of_row y1) (mat_of_row y2) (0\<^sub>m 0 nr1) (0\<^sub>m 0 nr2)"
Span_closed: fixes ms :: "'b list" assumes "set ms \<subseteq> V" shows "RSpan ms \<subseteq> V"
complement_part_not_part: fixes x :: "i"   and y :: "i" assumes "CEMC"   and "x \<noteq> u"   and "P y (\<midarrow> x)" shows "\<not> P y x"
Collect_case_prodD: fixes x :: "'d \<times> 'e"   and A :: "'d \<Rightarrow> 'e \<Rightarrow> bool" assumes "x \<in> {(x, y). A x y}" shows "A (fst x) (snd x)"
intro: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and get_shadow_root :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, (_) shadow_root_ptr option) prog"   and get_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and set_shadow_root :: "(_) element_ptr \<Rightarrow> (_) shadow_root_ptr option \<Rightarrow> ((_) heap, exception, unit) prog"   and set_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and 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 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 remove_shadow_root :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, unit) prog"   and remove_shadow_root_locs :: "(_) element_ptr \<Rightarrow> (_) shadow_root_ptr \<Rightarrow> ((_) heap, exception, unit) prog set" assumes "l_get_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf get_shadow_root get_shadow_root_locs"   and "l_set_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf set_shadow_root set_shadow_root_locs"   and "l_get_child_nodes type_wf known_ptr get_child_nodes get_child_nodes_locs"   and "l_get_disconnected_nodes type_wf get_disconnected_nodes get_disconnected_nodes_locs"   and "l_remove_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms get_child_nodes get_shadow_root set_shadow_root set_shadow_root_locs get_disconnected_nodes remove_shadow_root remove_shadow_root_locs" shows "l_remove_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_child_nodes get_child_nodes_locs get_shadow_root get_shadow_root_locs set_shadow_root set_shadow_root_locs get_disconnected_nodes get_disconnected_nodes_locs remove_shadow_root remove_shadow_root_locs type_wf known_ptr"
vbrelationE: fixes r :: "V" assumes "vbrelation r" obtains "vpairs r = r"
numeral_nat_2: fixes n :: "num" shows "numeral (num.Bit0 n) = Suc (numeral (Num.BitM n))"
basis_def: fixes A :: "'e set" shows "im.basis A = (im.lin_indpt A \<and> im.gen_set A \<and> A \<subseteq> carrier (W.vs imT))"
ell2_norm_L2_set: fixes x :: "'c \<Rightarrow> complex" assumes "has_ell2_norm x" shows "ell2_norm x = \<Squnion> (L2_set (cmod \<circ> x) ` {F. finite F})"
box_eucl_less: fixes a :: "'c"   and b :: "'c" shows "box a b = {x. eucl_less a x \<and> eucl_less x b}"
Proc_CFG_Exit_edge: fixes prog :: "cmd" obtains l :: "nat" and et :: "(char list, val, char list \<times> label, char list) edge_kind" where "prog \<turnstile> Label l -IEdge et\<rightarrow>\<^sub>p Exit" and "l \<le> #:prog"
hyps: fixes m :: "enat"   and n :: "enat" assumes "(m, n) \<in> Le_enat"   and "n < enat l" shows "m < enat l"
op_norm_def: fixes A :: "(('b, 'c) vec, 'd) vec" shows "\<parallel>A\<parallel>\<^sub>o\<^sub>p = Sup {\<parallel>A *v x\<parallel> |x. \<parallel>x\<parallel> = 1}"
interaction_bound_try_gpv: shows "interaction_bound consider (TRY gpv1 ELSE gpv2) \<le> interaction_bound consider gpv1 + interaction_bound consider gpv2"
components_are_equivalences: fixes a :: "'b" assumes "D.obj a" shows "D.equivalence_map (F.counit\<^sub>0 a)"
iFROM_cut_less: fixes n :: "nat"   and t :: "nat" shows "[n\<dots>] \<down>< t = (if t \<le> n then {} else [n\<dots>,t - Suc n])"
less_fset_def: fixes xs :: "'a fset"   and ys :: "'a fset" shows "xs < ys \<equiv> xs \<le> ys \<and> xs \<noteq> ys"
NormalizeD: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and F :: "'f set"   and P :: "'e set"   and c :: "('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set" assumes "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P Language.normalize c Q,A" shows "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P c Q,A"
E_def: shows "E = {x \<in> {0<..(int p * int q - 1) div 2}. x mod int p \<in> {0} \<and> x mod int q \<in> {(int q - 1) div 2<..int q - 1}}"
openin_clauses_2: fixes U :: "'a topology"   and S :: "'a set"   and T :: "'a set" assumes "openin U S"   and "openin U T" shows "openin U (S \<inter> T)"
is_ff_dghm_axioms: shows "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>f\<^sub>f\<^bsub>\<alpha>\<^esub> \<BB>"
insert_nth_remove_nth: fixes i :: "nat"   and xs :: "'b list" assumes "i < length xs" shows "insert_nth i (xs ! i) (remove_nth i xs) = xs"
Un_upper2: fixes B :: "'b set"   and A :: "'b set" shows "B \<subseteq> A \<union> B"
dvd_0_left: fixes times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and a :: "'c" assumes "class.comm_semiring_1 times one plus zero"   and "dvd.dvd times zero a" shows "a = zero"
hom: fixes r :: "real \<Rightarrow> 'c" assumes "path r"   and "path_image r \<subseteq> U"   and "pathstart r = z"   and "pathfinish r = z" shows "\<exists>b. homotopic_paths S (f \<circ> r) (linepath b b)"
set_7: fixes x7 :: "('a, 'b, 'c) expr" shows "RID\<^sub>E (Rfork x7) = RID\<^sub>E x7"
alphaAll_refl: fixes X :: "('b, 'c, 'varSort, 'var, 'd) qTerm"   and A :: "('b, 'c, 'varSort, 'var, 'd) qAbs" shows "(qGood X \<longrightarrow> X #= X) \<and> (qGoodAbs A \<longrightarrow> A $= A)"
set_merge_wrt: fixes rel :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and xs :: "'d list"   and ys :: "'d list" shows "set (merge_wrt rel xs ys) = set xs \<union> set ys"
fst_diff: fixes x :: "'d \<times> 'e"   and y :: "'d \<times> 'e" shows "fst (x - y) = fst x - fst y"
assms_2: fixes u :: "'t"   and v :: "'t" assumes "u \<in> keys p"   and "v \<in> keys r" shows "u \<prec>\<^sub>t v"
simps_8: fixes \<phi> :: "'a ltln"   and \<psi> :: "'a ltln" shows "rewrite_X_enat (\<phi> R\<^sub>n \<psi>) = combine mk_release (rewrite_X_enat \<phi>) (rewrite_X_enat \<psi>)"
compose: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and g :: "'b \<Rightarrow> 'c"   and h' :: "'d \<Rightarrow> 'b"   and G :: "'d filter" assumes "f \<in> \<omega>[F](g)"   and "filterlim h' F G" shows "(\<lambda>x. f (h' x)) \<in> \<omega>[G](\<lambda>x. g (h' x))"
eqvt_subs: fixes p :: "(name \<times> name) list"   and P :: "pi"   and a :: "name"   and b :: "name" shows "p \<bullet> P[a::=b] = (p \<bullet> P)[(p \<bullet> a)::=(p \<bullet> b)]"
merge_on_discrete: shows "merge_on UNIV id (=) merge_discrete"
replace_f: assumes "has_disc disc m" shows "\<forall>pts. \<not> has_disc disc2 (replace_f pts)"
sel_1: fixes x1 :: "'a atom formula list"   and x2 :: "'a atom formula list" shows "adds (Effect x1 x2) = x1"
list_update_append_lemma1: fixes i :: "nat"   and xs :: "'a list"   and v :: "'a"   and ys :: "'a list" assumes "i < length xs" shows "xs[i := v] @ ys = (xs @ ys)[i := v]"
bspec: fixes A :: "'d set"   and P :: "'d \<Rightarrow> bool"   and x :: "'d" assumes "\<forall>x\<in>A. P x"   and "x \<in> A" shows "P x"
return_sets_cong: fixes M :: "'c measure"   and N :: "'c measure" assumes "sets M = sets N" shows "return M = return N"
UNION_Cinfinite_bound: fixes I :: "'c set"   and r :: "('d \<times> 'd) set"   and A :: "'c \<Rightarrow> 'e set" assumes "|I| \<le>o r"   and "\<forall>i\<in>I. |A i| \<le>o r"   and "Cinfinite r" shows "|\<Union> (A ` I)| \<le>o r"
mayDiverge_Seq_R: fixes tval :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and c1 :: "('a, 'c) com"   and s :: "'b"   and s' :: "'b"   and c2 :: "('a, 'c) com" assumes "PL.MtransT tval aval (c1, s) s'"   and "PL.mayDiverge tval aval c2 s'" shows "PL.mayDiverge tval aval (c1 ;; c2) s"
smult_conv_mult: fixes s :: "'a"   and p :: "'a mpoly" shows "smult s p = monom 0 s * p"
simps_8: fixes uw :: "bdd" shows "Rmi_g 0 Trueif uw = False"
assms: shows "store_word_mem s addr data byte_mask asi = Some s' \<and> get_S (cpu_reg_val PSR s) = 0"
is_gcd: shows "is_gcd (\<lambda>a b. case bezout a b of (x, xa, u, v, gcd') \<Rightarrow> gcd')"
below_sctn_def: fixes X :: "'c set"   and sctn :: "'c sctn" shows "below_sctn X sctn = SPEC (\<lambda>b. b \<longrightarrow> X \<subseteq> below_halfspace sctn)"
FGModuleHomSet_def: fixes G :: "'c set"   and fgsmult :: "('d, 'c) aezfun \<Rightarrow> 'a \<Rightarrow> 'a"   and V :: "'a set"   and fgsmult' :: "('d, 'c) aezfun \<Rightarrow> 'b \<Rightarrow> 'b"   and W :: "'b set" shows "FGModuleHomSet G fgsmult V fgsmult' W \<equiv> {T. FGModuleHom G fgsmult V fgsmult' T} \<inter> {T. T ` V \<subseteq> W}"
maxsimpchain_Cons_reduce: fixes x :: "'a set"   and xs :: "'a set list" assumes "maxsimpchain (x # xs)" shows "maxsimpchain xs"
perm_bool_def: fixes pi :: "('a \<times> 'a) list"   and b :: "bool" shows "pi \<bullet> b = b"
add_in_gen_ideal: fixes R :: "'c set"   and addition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and multiplication :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and unit :: "'c"   and x :: "'c"   and a :: "'c"   and b :: "'c" assumes "comm_ring R addition multiplication zero unit"   and "x \<in> R"   and "a \<in> comm_ring.gen_ideal R multiplication x"   and "b \<in> comm_ring.gen_ideal R multiplication x" shows "addition a b \<in> comm_ring.gen_ideal R multiplication x"
sub_subsumed: fixes prb :: "('d, 'e, 'f) pre_RedBlack"   and sub :: "('d \<times> nat) \<times> 'd \<times> nat" assumes "RedBlack prb"   and "sub \<in> subs prb" shows "confs prb (subsumee sub) \<sqsubseteq> confs prb (subsumer sub)"
less_length_filter_update: fixes P :: "'a \<Rightarrow> bool"   and j :: "'a"   and list :: "'a list"   and i :: "nat" assumes "P j \<and> \<not> P (list ! i) \<and> i < length list" shows "length (filter P list) < length (filter P (list[i := j]))"
int_pow_closed: fixes G :: "('e, 'f) monoid_scheme"   and x :: "'e"   and i :: "int" assumes "Group.group G"   and "x \<in> carrier G" shows "x [^]\<^bsub>G\<^esub> i \<in> carrier G"
o_inv_o_cancel: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'e" assumes "inj f" shows "g \<circ> inv f \<circ> f = g"
update_eq: fixes xs :: "'h list"   and opt :: "('f, 'g, 'h) env option"   and env :: "('f, 'g, 'h) env" shows "update xs opt env = (case xs of [] \<Rightarrow> case opt of None \<Rightarrow> env | Some e \<Rightarrow> e | x # xs \<Rightarrow> case env of Val a \<Rightarrow> Val a | Env b es \<Rightarrow> case xs of [] \<Rightarrow> Env b (es(x := opt)) | y # ys \<Rightarrow> Env b (es(x := case es x of None \<Rightarrow> None | Some e \<Rightarrow> Some (update (y # ys) opt e))))"
thin0: fixes A :: "fmla"   and H :: "fmla set" assumes "{} \<turnstile> A" shows "H \<turnstile> A"
IH: fixes s :: "state" assumes "\<Gamma>,\<gamma>,p\<turnstile> \<langle>process_ret rs, s\<rangle> \<Rightarrow> Decision X" shows "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs, s\<rangle> \<Rightarrow> Decision X"
length_def: fixes as :: "'a iarray" shows "IArray.length as = List.length (list_of as)"
ennreal_diff_self: fixes a :: "ennreal" assumes "a \<noteq> top" shows "a - a = 0"
Says_S_message_form: fixes S :: "agent"   and A :: "agent"   and NA :: "nat"   and B :: "agent"   and K :: "nat"   and X :: "msg"   and evs :: "event list" assumes "Says S A (Crypt (shrK A) \<lbrace>Nonce NA, Agent B, Key K, X\<rbrace>) \<in> set evs"   and "evs \<in> ns_shared" shows "K \<notin> range shrK \<and> X = Crypt (shrK B) \<lbrace>Key K, Agent A\<rbrace> \<or> X \<in> analz (knows Spy evs)"
resNilLeft: fixes x :: "name"   and Rel :: "(ccs \<times> ccs) set" shows "\<lparr>\<nu>x\<rparr>\<zero> \<leadsto>[Rel] \<zero>"
init_def: shows "init \<equiv> $x = #0 \<and> $y = #0"
has_contour_integral_eqpath: fixes f :: "complex \<Rightarrow> complex"   and y :: "complex"   and p :: "real \<Rightarrow> complex"   and \<gamma> :: "real \<Rightarrow> complex" assumes "(f has_contour_integral y) p"   and "f contour_integrable_on \<gamma>"   and "contour_integral p f = contour_integral \<gamma> f" shows "(f has_contour_integral y) \<gamma>"
cr_pmf_def: shows "pmf_as_measure.cr_pmf \<equiv> \<lambda>x y. x = measure_pmf y"
one_reducible_to_via_def: shows "one_reducible_to_via = (\<lambda>A B f. total_recursive f \<and> inj f \<and> (\<forall>x. (x \<in> A) = (f x \<in> B)))"
inverse_unique: fixes u :: "'a"   and v' :: "'a"   and v :: "'a" assumes "u \<cdot> v' = \<one>"   and "v \<cdot> u = \<one>"   and "u \<in> M"   and "v \<in> M"   and "v' \<in> M" shows "v = v'"
elims: fixes x :: "linear_poly"   and xa :: "nat"   and y :: "nat \<times> linear_poly" assumes "linear_poly_to_eq x xa = y" obtains p :: "linear_poly" and v :: "nat" where "x = p" and "xa = v" and "y = (v, p)"
simps_1: fixes w :: "int \<Rightarrow> int \<Rightarrow> nat"   and i :: "int"   and j :: "int" shows "wpl.wpl w i j \<langle>\<rangle> = 0"
preimage_of_max_lideal: fixes \<ww>\<^sub>A :: "'a set"   and \<ww>\<^sub>B :: "'b set" assumes "max_lideal \<ww>\<^sub>A A addA multA zeroA oneA"   and "max_lideal \<ww>\<^sub>B B addB multB zeroB oneB" shows "f \<^sup>\<inverse> A \<ww>\<^sub>B = \<ww>\<^sub>A"
case_eq_if: fixes f :: "('d \<Rightarrow> ('e \<times> ('d, 'e) resource) spmf) \<Rightarrow> 'c"   and resource :: "('d, 'e) resource" shows "(case resource of Resource x \<Rightarrow> f x) = f (run_resource resource)"
M2_def: shows "M2 = M.summset {#lc ia \<cdot>\<^sub>v us[i := us ! i + c \<cdot>\<^sub>v us ! j] ! ia. ia \<in># mset_set ({0..<length us} - {i, j})#}"
mayDiverge_raw_coind: fixes phi :: "('test, 'atom) com \<Rightarrow> 'state \<Rightarrow> bool"   and c :: "('test, 'atom) com"   and s :: "'state" assumes "phi c s"   and "\<And>c s. phi c s \<Longrightarrow> \<exists>c' s'. (c, s) \<rightarrow>c (c', s') \<and> phi c' s'" shows "mayDiverge c s"
cinner_add_left: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "cinner (x + y) z = cinner x z + cinner y z"
assms_2: shows "strict_monoidal_extension_to_free_strict_monoidal_category C D T\<^sub>D \<alpha>\<^sub>D \<iota>\<^sub>D V F"
l_set_shadow_root_get_disconnected_nodes_def: fixes set_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_set_shadow_root_get_disconnected_nodes set_shadow_root_locs get_disconnected_nodes_locs \<equiv> \<forall>ptr h h' ptr'. \<forall>w\<in>set_shadow_root_locs ptr. h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> (\<forall>r\<in>get_disconnected_nodes_locs ptr'. r h h')"
fps_shift_fps_shift_reorder: fixes m :: "nat"   and n :: "nat"   and f :: "'a fps" shows "fps_shift m (fps_shift n f) = fps_shift n (fps_shift m f)"
eq_onp_to_eq: fixes P :: "'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "eq_onp P x y" shows "x = y"
quotient_of_square: fixes x :: "rat"   and a :: "int"   and b :: "int" assumes "quotient_of x = (a, b)" shows "quotient_of (x * x) = (a * a, b * b)"
mlex_iff: fixes x :: "'a"   and y :: "'a"   and f :: "'a \<Rightarrow> nat"   and R :: "('a \<times> 'a) set" shows "((x, y) \<in> f <*mlex*> R) = (f x < f y \<or> f x = f y \<and> (x, y) \<in> R)"
smc_1_components_2: fixes \<aa> :: "V"   and \<ff> :: "V" shows "smc_1 \<aa> \<ff>\<lparr>Arr\<rparr> = ZFC_in_HOL.set {\<ff>}"
mem_Ex2_rule: fixes f :: "nat \<Rightarrow> nat" shows "(f \<in> Ex2) = (\<exists>g. g \<in> CofGroups.Ex1 \<and> f = CONJ g)"
subset: shows "deadlocked s \<union> final_threads s \<subseteq> deadlocked s \<union> deadlocked s' \<union> final_threads s'"
DiffI: fixes c :: "'b"   and A :: "'b set"   and B :: "'b set" assumes "c \<in> A"   and "c \<notin> B" shows "c \<in> A - B"
bound_plus_of_nat: fixes a :: "'a"   and n :: "nat" assumes "(0::'a) \<le> a" shows "bound (a + of_nat n) = bound a + bound (of_nat n)"
fls_divide_def: fixes f :: "'a fls"   and g :: "'a fls" shows "f / g = fls_shift (fls_subdegree g - fls_subdegree f) (fps_to_fls (fls_base_factor_to_fps f / fls_base_factor_to_fps g))"
ds_p: shows "ds = [] \<or> ds = a # b \<and> (a \<longrightarrow> internal b T\<^sub>1) \<and> (\<not> a \<longrightarrow> internal b T\<^sub>2)"
restrict_map_on_disj': fixes h\<^sub>0 :: "'c \<Rightarrow> 'd option"   and h\<^sub>1 :: "'c \<Rightarrow> 'd option"   and S :: "'c set" assumes "h\<^sub>0 \<bottom> h\<^sub>1" shows "h\<^sub>0 \<bottom> h\<^sub>1 |` S"
run_one_step_basic_br_result: fixes d :: "nat"   and i :: "nat"   and s :: "s"   and vs :: "v list"   and ves :: "v list"   and x8 :: "nat"   and s' :: "s"   and vs' :: "v list"   and res :: "res_step" assumes "run_one_step d i (s, vs, ves, $Br x8) = (s', vs', res)" shows "\<exists>r vrs. res = RSBreak r vrs"
map_comp: fixes g1 :: "'g \<Rightarrow> 'e"   and g2 :: "'h \<Rightarrow> 'f"   and f1 :: "'i \<Rightarrow> 'g"   and f2 :: "'j \<Rightarrow> 'h"   and v :: "('i, 'j) T\<^sub>m" shows "map_T\<^sub>m g1 g2 (map_T\<^sub>m f1 f2 v) = map_T\<^sub>m (g1 \<circ> f1) (g2 \<circ> f2) v"
line_is_local_flow: fixes T :: "real set"   and c :: "'b" assumes "0 \<in> T"   and "is_interval T"   and "open T" shows "local_flow (\<lambda>s. c) T UNIV (\<lambda>t s. s + t *\<^sub>R c)"
var_subterm_ik\<^sub>s\<^sub>s\<^sub>t_is_fv\<^sub>s\<^sub>s\<^sub>t: fixes x :: "'d"   and A :: "('c, 'd) stateful_strand_step list" assumes "Var x \<sqsubseteq>\<^sub>s\<^sub>e\<^sub>t ik\<^sub>s\<^sub>s\<^sub>t A" shows "x \<in> fv\<^sub>s\<^sub>s\<^sub>t A"
dim_affine_hull: fixes S :: "'b set" shows "dim (affine hull S) = dim S"
rank_meld: fixes q1 :: "('c, 'd) SkewBinomialTree list"   and q2 :: "('c, 'd) SkewBinomialTree list" assumes "rank_skew_invar q1"   and "rank_skew_invar q2" shows "rank_skew_invar (meld q1 q2)"
smult_closed: fixes H :: "'e set"   and R :: "('f, 'g) ring_scheme"   and M :: "('f, 'e, 'h) module_scheme"   and a :: "'f"   and x :: "'e" assumes "Module.submodule H R M"   and "a \<in> carrier R"   and "x \<in> H" shows "a \<odot>\<^bsub>M\<^esub> x \<in> H"
poincare_between_nonstrict_1: fixes u :: "complex_homo"   and v :: "complex_homo" shows "poincare_between u u v"
notin_fset: fixes x :: "'d"   and S :: "'d fset" shows "(x |\<notin>| S) = (x \<notin> fset S)"
insertion_single: fixes f :: "nat \<Rightarrow> 'b"   and v :: "nat"   and n :: "nat"   and a :: "'b" shows "insertion f (MPoly_Type.monom (Poly_Mapping.single v n) a) = a * f v ^ n"
equivalent_objects_symmetric: fixes V :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and H :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<a> :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<i> :: "'c \<Rightarrow> 'c"   and src :: "'c \<Rightarrow> 'c"   and trg :: "'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "bicategory V H \<a> \<i> src trg"   and "bicategory.equivalent_objects V H \<a> \<i> src trg a b" shows "bicategory.equivalent_objects V H \<a> \<i> src trg b a"
range_tm_def: fixes e :: "nat \<Rightarrow> real"   and tm :: "taylor_model" shows "range_tm e tm = interval_of (insertion e (map_poly real_of_float (tm_poly tm))) + real_interval (tm_bound tm)"
filterlim_abs_real: shows "filterlim abs at_top at_top"
assms_3: shows "automaton.trans A = oseqp_sos \<Gamma> i"
terminates_implies_exec: 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 "\<exists>t. \<Gamma>\<turnstile> \<langle>c,s\<rangle> \<Rightarrow> t"
zero_le_divide_1_iff: fixes a :: "'a" shows "((0::'a) \<le> (1::'a) / a) = ((0::'a) \<le> a)"
RF_essential: fixes \<Gamma> :: "('a, 'b) web_scheme"   and S :: "'a set" shows "RF\<^bsub>\<Gamma>\<^esub> (\<E>\<^bsub>\<Gamma>\<^esub> S) = RF\<^bsub>\<Gamma>\<^esub> S"
norm_minus_cancel: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and sgn :: "'b \<Rightarrow> 'b"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and x :: "'b" assumes "class.real_normed_vector minus dist norm plus zero uminus scaleR sgn uniformity open" shows "norm (uminus x) = norm x"
ThrowThrow: fixes P :: "(char list list \<times> char list exp) prog"   and e :: "char list exp"   and s\<^sub>0 :: "heap \<times> sfields \<times> sheap"   and e' :: "char list exp"   and s\<^sub>1 :: "heap \<times> sfields \<times> sheap" assumes "P \<turnstile> \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw e',s\<^sub>1\<rangle>" shows "P \<turnstile> \<langle>throw e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw e',s\<^sub>1\<rangle>"
br_sv: fixes \<alpha> :: "'e \<Rightarrow> 'f"   and I :: "'e \<Rightarrow> bool" shows "single_valued (br \<alpha> I)"
induct: fixes x1 :: "com \<times> (char list \<Rightarrow> int)"   and x2 :: "nat"   and x3 :: "char list \<Rightarrow> int"   and P :: "com \<times> (char list \<Rightarrow> int) \<Rightarrow> nat \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool" assumes "x1 \<Rightarrow> x2 \<Down> x3"   and "\<And>s. P (SKIP, s) (Suc 0) s"   and "\<And>x a s. P (x ::= a, s) (Suc 0) (s[a/x])"   and "\<And>c1 s1 x s2 c2 y s3 z. \<lbrakk>(c1, s1) \<Rightarrow> x \<Down> s2; P (c1, s1) x s2; (c2, s2) \<Rightarrow> y \<Down> s3; P (c2, s2) y s3; z = x + y\<rbrakk> \<Longrightarrow> P (c1;; c2, s1) z s3"   and "\<And>b s c1 x t y c2. \<lbrakk>bval b s; (c1, s) \<Rightarrow> x \<Down> t; P (c1, s) x t; y = x + 1\<rbrakk> \<Longrightarrow> P (IF b THEN c1 ELSE c2, s) y t"   and "\<And>b s c2 x t y c1. \<lbrakk>\<not> bval b s; (c2, s) \<Rightarrow> x \<Down> t; P (c2, s) x t; y = x + 1\<rbrakk> \<Longrightarrow> P (IF b THEN c1 ELSE c2, s) y t"   and "\<And>b s c. \<not> bval b s \<Longrightarrow> P (WHILE b DO c, s) (Suc 0) s"   and "\<And>b s1 c x s2 y s3 z. \<lbrakk>bval b s1; (c, s1) \<Rightarrow> x \<Down> s2; P (c, s1) x s2; (WHILE b DO c, s2) \<Rightarrow> y \<Down> s3; P (WHILE b DO c, s2) y s3; 1 + x + y = z\<rbrakk> \<Longrightarrow> P (WHILE b DO c, s1) z s3" shows "P x1 x2 x3"
cfs_monom_mult_l': fixes f :: "nat \<Rightarrow> 'a"   and a :: "'a"   and n :: "nat"   and m :: "nat" assumes "f \<in> carrier P"   and "a \<in> carrier R"   and "n \<le> m" shows "(f \<otimes>\<^bsub>P\<^esub> monom P a n) m = a \<otimes> f (m - n)"
hom_mult: fixes x :: "'c"   and y :: "'c" shows "g (x * y) = g x * g y"
conv_galois_1: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(x ; y \<le> z) = (y \<le> - (x\<^sup>\<smile> ; - z))"
pr_dup_1: fixes a :: "'a" assumes "ide a" shows "\<pp>\<^sub>0[a, a] \<cdot> \<d>[a] = a"
Prop1A: fixes c :: "IMP"   and \<Phi> :: "(Var \<Rightarrow> Val) \<times> (Var \<Rightarrow> Val) \<Rightarrow> bool" assumes " \<Turnstile> c : Sec \<Phi> " shows "secure c"
case_eq_if: fixes f1 :: "'e \<Rightarrow> 'd"   and f2 :: "'f \<Rightarrow> 'd"   and sum :: "'e + 'f" shows "(case sum of Inl x \<Rightarrow> f1 x | Inr x \<Rightarrow> f2 x) = (if isl sum then f1 (projl sum) else f2 (projr sum))"
nonzero_mult_div_cancel_right: 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 b :: "'a"   and a :: "'a" assumes "class.semidom_divide divide plus minus zero times one"   and "b \<noteq> zero" shows "divide (times a b) b = a"
onorm_def: fixes f :: "'b \<Rightarrow> 'c" shows "onorm f = (SUP x. norm (f x) / norm x)"