hcju/isabelleps huggingface dataset:
linear_order_on_converse: fixes A :: "'d set"   and r :: "('d \<times> 'd) set" shows "linear_order_on A (r\<inverse>) = linear_order_on A r"
add_poly_p_add_to_result: assumes "add_poly_p\<^sup>*\<^sup>* (A, Aa, {#}) ({#}, {#}, r)" shows "add_poly_p\<^sup>*\<^sup>* (A, Aa, {#(ys, m)#} + {#}) ({#}, {#}, {#(ys, m)#} + r)"
eval_fds_deriv_zeta: fixes s :: "complex" assumes "1 < Re s" shows "eval_fds (fds_deriv fds_zeta) s = deriv zeta s"
e: shows "on_triple f `` {e} \<subseteq> edges (S (max i j))"
evaln_elim_cases_5: fixes G :: "prog"   and s :: "st"   and n :: "nat"   and v :: "vals"   and s' :: "state" assumes "G\<turnstile>Norm s \<midarrow>In3 []\<succ>\<midarrow>n\<rightarrow> (v, s')" obtains "v = In3 []" and "s' = Norm s"
poly_eq_poly_eq_iff: fixes p :: "'a poly"   and q :: "'a poly" shows "(poly p = poly q) = (p = q)"
f4_red_aux_spoly_reducible: fixes ps :: "((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and bs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and p :: "('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f"   and q :: "('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f" assumes "set ps \<subseteq> set bs \<times> set bs"   and "(p, q) \<in> set ps" shows "(red (fst ` set bs \<union> set (f4_red_aux bs ps)))\<^sup>*\<^sup>* (spoly (fst p) (fst q)) 0"
R: fixes U :: "'a set set" assumes "U \<in> Filters"   and "F \<subseteq> U"   and "ultra U" shows "R"
circle_cube_is_only_horizontal_div_of_rot: shows "only_horizontal_division (boundary circle_cube) {rot_circle_cube}"
point_abscissa_add: fixes A :: "point"   and B :: "point" shows "abscissa (A + B) = abscissa A + abscissa B"
simps: fixes uu :: "'b ext0" shows "uu\<^sup>\<star> = ext0.Top"
dghm_HomCod: shows "\<GG>\<lparr>HomCod\<rparr> = \<CC>"
surj_iff: fixes f :: "'d \<Rightarrow> 'c" shows "surj f = (f \<circ> inv f = id)"
Suc_1: fixes s :: "nat set" assumes "enat i < llength (lselect s w)" shows "lselect s w ?! i = w ?! nth_least s i"
isVar_or_isOp: fixes s :: "'sort"   and X :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set" assumes "wls s X" shows "isVar s X \<or> isOp s X"
set_image_mset: fixes f :: "'d \<Rightarrow> 'c"   and M :: "'d multiset" shows "set_mset (image_mset f M) = f ` set_mset M"
Rep_uint_numeral: fixes n :: "num" shows "Rep_uint (numeral n) = numeral n"
children_eq2: fixes ptr' :: "(_) object_ptr" assumes "ptr' \<noteq> cast\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r new_document_ptr" shows "|h \<turnstile> get_child_nodes ptr'|\<^sub>r = |h' \<turnstile> get_child_nodes ptr'|\<^sub>r"
octo_inner_1_right: fixes x :: "octo" shows "x \<bullet> 1 = Ree x"
leg_simps_4: shows "C.dom \<nu>.dom.leg1 = \<nu>.dom.apex"
vars_term_subst: fixes t :: "('b, 'c) Term.term"   and \<sigma> :: "'c \<Rightarrow> ('b, 'a) Term.term" shows "vars_term (t \<cdot> \<sigma>) = \<Union> (vars_term ` \<sigma> ` vars_term t)"
presheaf_of_rings_axioms: shows "presheaf_of_rings X is_open \<FF> \<rho> b add_str mult_str zero_str one_str"
simps: fixes a1 :: "state"   and a2 :: "com list"   and a3 :: "state" shows "(a1 =a2\<Rightarrow> a3) = ((\<exists>s. a1 = s \<and> a2 = [] \<and> a3 = s) \<or> (\<exists>s c t cs u. a1 = s \<and> a2 = c # cs \<and> a3 = u \<and> s -c\<rightarrow> t \<and> t =cs\<Rightarrow> u))"
A: fixes U :: "'a set" shows "U \<subseteq> unaffected_domains I D (D ` set ys) zs \<longrightarrow> ipurge_tr_rev_aux I D U (xs @ ys @ zs) = ipurge_tr_rev_aux I D U (xs @ ipurge_tr_aux I D (D ` set ys) zs)"
four_composable_identity_arrows_of_spans_axioms: shows "four_composable_identity_arrows_of_spans (\<cdot>) prj0 prj1 \<mu> \<nu> \<pi> \<rho>"
mu_Gramian_beta_def: fixes j :: "nat"   and i :: "nat" assumes "j < i"   and "i < length fs" shows "\<mu> i j = (fs ! i \<bullet> fs ! j - (\<Sum>k = 0..<j. \<mu> j k * \<mu> i k * \<beta> fs k)) / \<beta> fs j"
Suc: shows "a ^ n $ n = (a $ 1) ^ n"
bij_betw_imp_surj_on: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and B :: "'c set" assumes "bij_betw f A B" shows "f ` A = B"
surjective: fixes r :: "('a, 'b) rgraph_scheme" shows "r = \<lparr>root = root r, edges = edges r, \<dots> = rgraph.more r\<rparr>"
UN_atLeast_UNIV: shows "\<Union> (range atLeast) = UNIV"
inj_f: fixes f :: "'b \<Rightarrow> 'c" assumes "injective f" shows "inj f"
perm_count_conv: fixes A :: "'a list"   and B :: "'a list" shows "(A <~~> B) = (\<forall>x. PermutationLemmas.count x A = PermutationLemmas.count x B)"
conj_def: fixes p :: "fm"   and q :: "fm" shows "Parametric_Ferrante_Rackoff.conj p q \<equiv> if p = F \<or> q = F then F else if p = T then q else if q = T then p else if p = q then p else And p q"
reduced_ifex: fixes b :: "'a bexp" shows "reduced (Boolean_Expression_Example.ifex_of b) {}"
l_remove_shadow_root_get_tag_name\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) heap \<Rightarrow> bool"   and get_tag_name :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, char list) prog"   and get_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   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_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 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"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool" shows "l_remove_shadow_root_get_tag_name\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_tag_name get_tag_name_locs 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 known_ptr \<equiv> l_get_tag_name\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_tag_name get_tag_name_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 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"
push_bit_and: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bit :: "'b \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and drop_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and take_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and "and" :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and or :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and xor :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and mask :: "nat \<Rightarrow> 'b"   and n :: "nat"   and a :: "'b"   and b :: "'b" assumes "class.semiring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask" shows "push_bit n (and a b) = and (push_bit n a) (push_bit n b)"
map_zip_replicate_False_xor: fixes n :: "nat"   and xs :: "bool list" assumes "n = length xs" shows "map2 (\<lambda>x y. x = (\<not> y)) xs (replicate n False) = xs"
under_UnderS_trans: fixes r :: "('a \<times> 'a) set"   and a :: "'a"   and b :: "'a"   and C :: "'a set" assumes "trans r"   and "antisym r"   and "a \<in> under r b"   and "b \<in> UnderS r C" shows "a \<in> UnderS r C"
healthy_intros_24: fixes a :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and n :: "nat" assumes "nearly_healthy (wlp a)" shows "nearly_healthy (wlp (repeat n a))"
prev_in_def: fixes A :: "'c set"   and u :: "'c" shows "prev_in A u = Max (A \<inter> {..<u})"
gSwapGAbs_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd, 'e) Recursion.model_scheme" shows "gSwapGAbs MOD \<equiv> \<forall>zs z1 z2 xs x s X' X. isInBar (xs, s) \<and> wls s X' \<and> gWls MOD s X \<longrightarrow> gSwapAbs MOD zs z1 z2 (Abs xs x X') (gAbs MOD xs x X' X) = gAbs MOD xs (x @xs[z1 \<and> z2]_zs) (X' #[z1 \<and> z2]_zs) (gSwap MOD zs z1 z2 X' X)"
map_add_le_mapI: fixes f :: "'a \<Rightarrow> 'b option"   and h :: "'a \<Rightarrow> 'b option"   and g :: "'a \<Rightarrow> 'b option" assumes "f \<subseteq>\<^sub>m h"   and "g \<subseteq>\<^sub>m h" shows "f ++ g \<subseteq>\<^sub>m h"
subst4Chain: fixes xvec :: "name list"   and Tvec :: "'a list"   and T :: "'b" assumes "length xvec = length Tvec"   and "distinct xvec"   and "xvec \<sharp>* Tvec" shows "xvec \<sharp>* substAssert T xvec Tvec"
beukers_nn_integral3_pos: shows "0 < local.beukers_nn_integral3"
substT_subtype: fixes \<Delta> :: "binding list"   and Q :: "POPLmarkRecord.type"   and \<Gamma> :: "binding list"   and S :: "POPLmarkRecord.type"   and T :: "POPLmarkRecord.type"   and P :: "POPLmarkRecord.type" assumes "\<Delta> @ TVarB Q \<Colon> \<Gamma> \<turnstile> S <: T"   and "\<Gamma> \<turnstile> P <: Q" shows "\<Delta>[0 \<mapsto>\<^sub>\<tau> P]\<^sub>e @ \<Gamma> \<turnstile> S[\<parallel>\<Delta>\<parallel> \<mapsto>\<^sub>\<tau> P]\<^sub>\<tau> <: T[\<parallel>\<Delta>\<parallel> \<mapsto>\<^sub>\<tau> P]\<^sub>\<tau>"
c1: shows "(fst \<delta>1''', map (Pair True) (snd \<delta>1''')) \<in> conv ars"
2: fixes xa :: "real" assumes "signs_at fss xa ! i = 0"   and "i \<le> length (removeAt i fss)" shows "signs_at fss xa \<in> insertAt i 0 ` signs_at (removeAt i fss) ` {x. poly (fss ! i) x = 0}"
distinct_5: fixes x2 :: "'a" shows "\<lbrakk>x2\<rbrakk> \<noteq> \<stileturn>"
ord_min: fixes m :: "nat"   and x :: "'a" assumes "1 \<le> m"   and "x \<in> carrier G"   and "x [^] m = \<one>" shows "ord x \<le> m"
is_lub_def: fixes r :: "('b \<times> 'b) set"   and x :: "'b"   and y :: "'b"   and u :: "'b" shows "is_lub r x y u = (is_ub r x y u \<and> (\<forall>z. is_ub r x y z \<longrightarrow> (u, z) \<in> r))"
p_recurrence: fixes k :: "nat" assumes "0 < k" shows "p k = - of_nat k * (- 1) ^ k * e k - (\<Sum>i = 1..<k. (- 1) ^ i * e i * p (k - i))"
fa_intros_6: fixes p :: "'b \<Rightarrow> 'a \<Rightarrow> real"   and a :: "'a \<Rightarrow> bool \<Rightarrow> ('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real" assumes "\<And>x s. x \<in> supp (p s) \<Longrightarrow> Transformers.additive (wp (a x))"   and "\<And>s. finite (supp (p s))" shows "Transformers.additive (wp (SetPC a p))"
parent_child_rel_node_ptr: fixes parent :: "(_) object_ptr"   and child :: "(_) object_ptr"   and h :: "(_) heap" assumes "(parent, child) \<in> parent_child_rel h" shows "is_node_ptr_kind child"
adds_term_refl: fixes v :: "'t" shows "v adds\<^sub>t v"
image_add_cball: fixes b :: "'c"   and a :: "'c"   and r :: "real" shows "(+) b ` cball a r = cball (a + b) r"
imps: shows "\<forall>pr. price_structure der matur \<pi> pr \<longrightarrow> (\<exists>x Mkt2 p. x \<notin> stocks Mkt \<and> coincides_on Mkt Mkt2 (stocks Mkt) \<and> prices Mkt2 x = pr \<and> portfolio p \<and> support_set p \<subseteq> stocks Mkt \<union> {x} \<and> arbitrage_process Mkt2 p)"
is_icdi_def: fixes prob :: "('a, 'b, 'c, 'd) ifc_problem"   and n :: "nat"   and \<pi> :: "nat \<Rightarrow> 'a"   and n' :: "nat" shows "IFC_def.is_icdi prob n \<pi> n' = (IFC_def.is_path prob \<pi> \<and> IFC_def.is_cdi prob n \<pi> n' \<and> (\<forall>m\<in>{n'<..<n}. \<not> IFC_def.is_cdi prob n \<pi> m))"
aff_dim_sing: fixes a :: "'a" shows "aff_dim {a} = 0"
Method_def: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list"   and C :: "char list"   and M :: "char list"   and Ts :: "ty list"   and T :: "ty"   and m :: "'b"   and D :: "char list" shows "P \<turnstile> C sees M: Ts\<rightarrow>T = m in D \<equiv> \<exists>Mm. P \<turnstile> C sees_methods Mm \<and> Mm M = \<lfloor>((Ts, T, m), D)\<rfloor>"
IMPLEFT_eqvt_name: fixes pi :: "(name \<times> name) list"   and A :: "ty"   and B :: "ty"   and X :: "ctrm set"   and Y :: "ntrm set" shows "pi \<bullet> IMPLEFT (A IMP B) X Y = IMPLEFT (A IMP B) (pi \<bullet> X) (pi \<bullet> Y)"
octo_of_real_sel_6: fixes x :: "real" shows "Im5 (octo_of_real x) = 0"
fds_integral_fds_deriv: fixes f :: "'a fds" shows "fds_integral (fds_nth f 1) (fds_deriv f) = f"
cases: fixes x :: "nat" obtains "x = 0"   | "x = Suc 0"   | v :: "nat" where "x = Suc (Suc v)"
R: shows "zimage Abs_game (zexplode r) = right_options g"
indRelRSTPO_impl_SRel_and_TRel_weakly_reflect_barbs_2: fixes SRel :: "('procS \<times> 'procS) set"   and TRel :: "('procT \<times> 'procT) set" assumes "rel_weakly_reflects_barbs (indRelRSTPO SRel TRel) (STCalWB SWB TWB)" shows "rel_weakly_reflects_barbs TRel TWB"
is1_replicate: shows "is1 \<lhd> replicate (2 ^ (length (tl rs) - 2)) (last rs)"
nth_take: fixes i :: "nat"   and n :: "nat"   and xs :: "'d list" assumes "i < n" shows "take n xs ! i = xs ! i"
induct_list012: fixes P :: "'c list \<Rightarrow> bool"   and xs :: "'c list" assumes "P []"   and "\<And>x. P [x]"   and "\<And>x y zs. \<lbrakk>P zs; P (y # zs)\<rbrakk> \<Longrightarrow> P (x # y # zs)" shows "P xs"
conc_subset_lists: fixes A :: "'c list set"   and S :: "'c set"   and B :: "'c list set" assumes "A \<subseteq> lists S"   and "B \<subseteq> lists S" shows "A @@ B \<subseteq> lists S"
assms_4: shows "(_Entry_) -as\<rightarrow>* (_Exit_)"
wf_inst_effect: fixes Q :: "variable \<Rightarrow> PDDL_STRIPS_Semantics.type option"   and f :: "variable \<Rightarrow> object"   and \<phi> :: "PDDL_STRIPS_Semantics.term ast_effect" assumes "\<And>x T. is_of_type Q x T \<Longrightarrow> is_of_type objT (f x) T"   and "wf_effect (ty_term Q constT) \<phi>" shows "wf_effect objT ((map_ast_effect \<circ> subst_term) f \<phi>)"
assms_2: shows "other nos_inc {i} \<sigma> \<sigma>'"
fv\<^sub>s\<^sub>e\<^sub>t_union: fixes M :: "('d, 'c) Term.term set"   and N :: "('d, 'c) Term.term set" shows "fv\<^sub>s\<^sub>e\<^sub>t (M \<union> N) = fv\<^sub>s\<^sub>e\<^sub>t M \<union> fv\<^sub>s\<^sub>e\<^sub>t N"
fun_eq_iff: fixes f :: "'g \<Rightarrow> 'h"   and g :: "'g \<Rightarrow> 'h" shows "(f = g) = (\<forall>x. f x = g x)"
four_distinct_props_2: fixes v :: "i"   and p :: "\<o>" shows "[Contingent p \<^bold>\<rightarrow> \<^bold>\<not>(\<^bold>\<exists>q. NonContingent q \<^bold>& q \<^bold>= p) in v]"
extf_args_single: shows "\<forall>f\<in>local.ground_heads (head s). extf f (>\<^sub>t) (args s @ [t']) (args s @ [t])"
sumlist_as_summset: fixes xs :: "'a vec list" assumes "set xs \<subseteq> carrier_vec n" shows "M.sumlist xs = M.summset (mset xs)"
sb_dec_lem: fixes k :: "nat"   and a :: "int" assumes "0 \<le> - (2 ^ k) + a" shows "(a + 2 ^ k) mod (2 * 2 ^ k) \<le> - (2 ^ k) + a"
heap_lupd: fixes x :: "vname"   and y :: "val"   and s :: "state" shows "heap (lupd(x\<mapsto>y) s) = heap s"
hoare_trans_Un: fixes P :: "'a set"   and R1 :: "('a \<times> 'a) set"   and R2 :: "('a \<times> 'a) set"   and Q :: "'a set" shows "{P} R1 \<union> R2 {> Q} = ({P} R1 {> Q} \<and> {P} R2 {> Q})"
cong_symmetry: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "Cong A B C D" shows "Cong C D A B"
singleton_lepoll: fixes x :: "'a"   and y :: "'b"   and A :: "'b set" shows "{x} \<lesssim> insert y A"
rep_eq: fixes x :: "'a dual"   and xa :: "'a dual" shows "undual (sup x xa) = inf (undual x) (undual xa)"
simps_4: fixes x :: "'b trm"   and y :: "'b trm"   and "next" :: "indices list"   and u :: "'b trm"   and S :: "'b trm" shows "replace_subterm (x \<cdot> y) (indices.Left # next) u S = (\<exists>S1. replace_subterm x next u S1 \<and> S = S1 \<cdot> y)"
pred_set: fixes P :: "'c \<Rightarrow> bool" shows "list_all P = (\<lambda>x. Ball (set x) P)"
Cod_comp: fixes f :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr"   and g :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr" assumes "arr f"   and "arr g"   and "Dom g = Cod f" shows "Cod (g \<cdot> f) = Cod g"
ins2_ins_3: fixes n :: "nat"   and d :: "nat"   and x :: "'a"   and t :: "'a tree"   and t' :: "'a tree" assumes "ins n d x t = Unbal t'" shows "ins2 n d x t = Unbal2 t' (size t') (height t')"
xd_def: fixes i :: "nat"   and j :: "nat" shows "local.xd i j = local.x i - local.x j"
fresh: shows "atom ` domA \<Gamma> \<sharp>* \<Delta>"
l13: fixes x :: "'a"   and y :: "'a" shows "- - (- - x \<sqinter> y) \<le> - - x"
AT_fps_def: fixes f :: "nat \<Rightarrow> real"   and n :: "nat" shows "AT_fps f n = (fps_X - 1) * Abs_fps (gen_akiyama_tanigawa f n)"
span_listI: fixes scale :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and qs :: "'c list"   and bs :: "'d list" assumes "Modules.module scale" shows "sum_list (map2 scale qs bs) \<in> module.span scale (set bs)"
DomInterAllowsMT: fixes a :: "(int \<times> int \<times> protocol) set set"   and b :: "(int \<times> int \<times> protocol) set set"   and c :: "(int \<times> int \<times> protocol) set set"   and d :: "(int \<times> int \<times> protocol) set set"   and p :: "protocol \<times> int"   and po :: "protocol \<times> int" assumes "twoNetsDistinct a b c d" shows "dom (Cp (AllowPortFromTo a b p)) \<inter> dom (Cp (AllowPortFromTo c d po)) = {}"
select_convs_2: fixes tdir :: "'b"   and xdir :: "'b"   and ydir :: "'b"   and zdir :: "'b"   and more :: "'c" shows "xdir \<lparr>tdir = tdir, xdir = xdir, ydir = ydir, zdir = zdir, \<dots> = more\<rparr> = xdir"
simps_6: fixes f1 :: "'b \<Rightarrow> 'a"   and f2 :: "abort \<Rightarrow> 'a"   and x2 :: "abort" shows "(case Rabort x2 of Rraise x \<Rightarrow> f1 x | Rabort x \<Rightarrow> f2 x) = f2 x2"
intro: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and s :: "nat"   and t :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'a" assumes "Preflow c s t f"   and "Flow_axioms c s t f" shows "Flow c s t f"
cblinfun_image_def: shows "(*\<^sub>S) \<equiv> map_fun (*\<^sub>V) (map_fun space_as_set Abs_clinear_space) (\<lambda>A S. closure (A ` S))"
g: shows "steps (s, l, r) (ci ly (start_of ly as) ins, start_of ly as - 1) stp = (s1, l1, r1)"
assms_2: shows "\<forall>(v', a')\<in>set (effect_of op). v' \<noteq> v"
closed_mono: fixes B :: "'b set"   and B' :: "'b set"   and T :: "'b set"   and act :: "('b \<times> 'b) set"   and M :: "'b set"   and C :: "'b set set" assumes "B \<subseteq> B'"   and "T \<inter> (B \<union> wp act M) \<in> C"   and "B' \<in> C"   and "T \<in> C"   and "lattice C" shows "T \<inter> (B' \<union> wp act M) \<in> C"
Class_equivalence: fixes a :: "'a \<times> 'a"   and b :: "'a \<times> 'a" assumes "a \<in> R \<times> S"   and "b \<in> R \<times> S" shows "(rel.Class a = rel.Class b) = ((a, b) \<in> {(x, y). (x, y) \<in> (R \<times> S) \<times> R \<times> S \<and> x \<sim> y})"
pp_first_pp_fusion: fixes x :: "'a ppath"   and y :: "'a ppath" assumes "pp_last x = pp_first y" shows "pp_first (pp_fusion x y) = pp_first x"
rep_eq: shows "poly.coeff 0 = (\<lambda>_. 0::'b)"
some_betw4a: fixes P :: "'a set"   and a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" assumes "P \<in> \<P>"   and "a \<in> P"   and "b \<in> P"   and "c \<in> P"   and "d \<in> P"   and "a \<noteq> b \<and> a \<noteq> c \<and> a \<noteq> d \<and> b \<noteq> c \<and> b \<noteq> d \<and> c \<noteq> d"   and "\<not> ([[a b c d]] \<or> [[a b d c]] \<or> [[a c b d]] \<or> [[a c d b]] \<or> [[a d b c]] \<or> [[a d c b]])" shows "[[b a c d]] \<or> [[b a d c]] \<or> [[b c a d]] \<or> [[b d a c]] \<or> [[c a b d]] \<or> [[c b a d]]"
step: fixes f :: "'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" assumes "orbitp f x y" shows "orbitp f x (f y)"
subalgebra_sigma_finite: fixes M :: "'c measure"   and N :: "'c measure" assumes "prob_space M"   and "subalgebra M N" shows "sigma_finite_subalgebra M N"
inj_int_decode: fixes A :: "nat set" shows "inj_on int_decode A"
finite_zcount_neg: fixes M :: "'a zmultiset" shows "finite {t. zcount M t < 0}"
arr_MkArr: fixes F :: "'a \<Rightarrow> 'b"   and G :: "'a \<Rightarrow> 'b"   and \<tau> :: "'a \<Rightarrow> 'b" shows "A_B.arr (A_B.MkArr F G \<tau>) = natural_transformation (\<cdot>\<^sub>A) (\<cdot>\<^sub>B) F G \<tau>"
prodag_def: fixes I :: "'d set"   and A :: "'d \<Rightarrow> ('e, 'f) aGroup_scheme" shows "a\<Pi>\<^bsub>I\<^esub> A = \<lparr>carrier = carr_prodag I A, pop = prod_pOp I A, mop = prod_mOp I A, zero = prod_zero I A\<rparr>"
mem_diff: shows "\<forall>x. x \<in> mds AsmNoWrite \<longrightarrow> mem\<^sub>1 x = mem\<^sub>2 x"
fold_insert_remove: fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and A :: "'a set"   and z :: "'b"   and x :: "'a" assumes "comp_fun_commute f"   and "finite A" shows "Finite_Set.fold f z (insert x A) = f x (Finite_Set.fold f z (A - {x}))"
SourcesLevelX: fixes i :: "AbstrLevelsID"   and X :: "CSet" shows "Sources i X \<subseteq> AbstrLevel i"
interior_maximal: fixes T :: "'a set"   and S :: "'a set" assumes "T \<subseteq> S"   and "open T" shows "T \<subseteq> interior S"
wf_cdecl_supD: fixes wf_md :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list \<Rightarrow> char list \<Rightarrow> char list \<times> ty list \<times> ty \<times> char list list \<times> expr \<Rightarrow> bool"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and C :: "char list"   and Bs :: "base list"   and r :: "(char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list"   and D :: "char list" assumes "wf_cdecl wf_md P (C, Bs, r)"   and "D \<in> baseClasses Bs" shows "is_class P D"
Bseq_def: fixes X :: "nat \<Rightarrow> 'b" shows "Bseq X = (\<exists>K>0. \<forall>n. norm (X n) \<le> K)"
pairwise_image: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'c"   and s :: "'d set" shows "pairwise r (f ` s) = pairwise (\<lambda>x y. f x \<noteq> f y \<longrightarrow> r (f x) (f y)) s"
index_mat_addrow_5: fixes ad :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and mul :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and k :: "nat"   and l :: "nat"   and A :: "'a mat" shows "dim_col (mat_addrow_gen ad mul a k l A) = dim_col A"
eq: shows "inner_prod a (outer_prod b c *\<^sub>v d) = (\<Sum>i = 0..<n. \<Sum>j = 0..<m. conjugate (a $ i) * b $ i * conjugate (c $ j) * d $ j)"
assms_1: shows "\<forall>l l'. l \<noteq> l' \<longrightarrow> GSMP_disjoint (trms\<^sub>l\<^sub>s\<^sub>t (proj l A)) (trms\<^sub>l\<^sub>s\<^sub>t (proj l' A)) Sec"
inj_on_imp_Ker0: assumes "inj_on T G" shows "Ker = 0"
map_sorted_distinct_set_unique: fixes f :: "'b \<Rightarrow> 'a"   and xs :: "'b list"   and ys :: "'b list" assumes "inj_on f (set xs \<union> set ys)"   and "sorted (map f xs)"   and "distinct (map f xs)"   and "sorted (map f ys)"   and "distinct (map f ys)"   and "set xs = set ys" shows "xs = ys"