hcju/isabelleps huggingface dataset:
inline1_gpv_stop_obsf_converter: fixes gpv :: "('a, 'b, 'c) gpv"   and conv :: "('b, 'c, 'd, 'e) converter" shows "bind_spmf (inline1 run_converter (map_gpv' id id option_of_exception (gpv_stop gpv)) (obsf_converter conv)) (\<lambda>xy. case xy of Inl (None, conv') \<Rightarrow> return_pmf None | Inl (Some x, conv') \<Rightarrow> return_spmf (Inl (x, conv')) | Inr y \<Rightarrow> return_spmf (Inr y)) = map_spmf (map_sum (apsnd obsf_converter) (apsnd (map_prod (\<lambda>rpv input. case input of Fault \<Rightarrow> Done (Fault, const_converter Fault) | OK input' \<Rightarrow> map_gpv' (\<lambda>res. case res of None \<Rightarrow> (Fault, const_converter Fault) | Some (x, conv') \<Rightarrow> (OK x, obsf_converter conv')) id option_of_exception (TRY gpv_stop (rpv input') ELSE Done None)) (\<lambda>rpv input. case input of Fault \<Rightarrow> Done None | OK input' \<Rightarrow> map_gpv' id id option_of_exception (gpv_stop (rpv input')))))) (inline1 run_converter gpv conv)"
Ycomb_def: shows "Ycomb \<equiv> AbsN 0 (App (App Y_delta (Var 0)) (App Y_delta (Var 0)))"
eq_linorder_on_axioms_def: fixes D :: "'a set"   and cmp :: "'a \<Rightarrow> 'a \<Rightarrow> comp_res" shows "eq_linorder_on_axioms D cmp \<equiv> \<forall>x y. x \<in> D \<longrightarrow> y \<in> D \<longrightarrow> cmp x y = EQUAL \<longrightarrow> x = y"
point_singleton: fixes p :: "'a"   and v :: "'a" assumes "is_point p"   and "is_vector v"   and "v \<noteq> 0"   and "v \<le> p" shows "v = p"
lsteps_height: fixes xs :: "'a list"   and l :: "nat"   and f :: "'a \<Rightarrow> nat"   and up :: "nat" assumes "\<exists>x\<in>set xs. l \<le> f x" shows "lsteps xs f l up 0 + up * l = up * Max (f ` set xs \<union> {0})"
Pderiv_def: fixes a :: "'b"   and R :: "'b rexp set" shows "Pderiv a R \<equiv> pderiv_set a R"
diamond_cube_boundary_valid: fixes k :: "int"   and \<gamma> :: "real \<Rightarrow> real \<times> real" assumes "(k, \<gamma>) \<in> boundary diamond_cube" shows "valid_path \<gamma>"
finite_same_card_bij: fixes A :: "'c set"   and B :: "'d set" assumes "finite A"   and "finite B"   and "card A = card B" shows "\<exists>h. bij_betw h A B"
of_int_mult: fixes w :: "int"   and z :: "int" shows "of_int (w * z) = of_int w * of_int z"
fls_nth_inject: fixes x :: "'b fls"   and y :: "'b fls" shows "(fls_nth x = fls_nth y) = (x = y)"
congruent_triangle_asa_12: fixes a1 :: "'b"   and b1 :: "'b"   and c1 :: "'b"   and a2 :: "'c"   and b2 :: "'c"   and c2 :: "'c" assumes "angle a1 b1 c1 = angle a2 b2 c2"   and "dist a1 b1 = dist a2 b2"   and "angle b1 a1 c1 = angle b2 a2 c2"   and "\<not> collinear {a1, b1, c1}" shows "dist c1 b1 = dist c2 b2"
less_imp_not_less: fixes x :: "'a word"   and y :: "'a word" assumes "x <s y" shows "(\<not> y <s x) = True"
push_down_state_rank_tokens: fixes q :: "'b"   and n :: "nat"   and i :: "nat" assumes "state_rank q n = Some i" shows "configuration q n \<noteq> {}"
length_partition_by_median_6: fixes ps :: "(real, 'a) vec list"   and l :: "(real, 'a) vec list"   and m :: "real"   and r :: "(real, 'a) vec list"   and k :: "'a" assumes "0 < length ps"   and "\<forall>k. distinct (map (\<lambda>p. p $ k) ps)"   and "(l, m, r) = partition_by_median k ps" shows "length r < length ps"
CCs_empty: shows "CCs {} = {{v} |v. v \<in> V}"
mangoldt_primepow: fixes p :: "nat"   and k :: "nat" assumes "prime p" shows "mangoldt (p ^ k) = (if 0 < k then of_real (ln (real p)) else (0::'b))"
Max_gr_iff: fixes A :: "'a word set"   and x :: "'a word" assumes "finite A"   and "A \<noteq> {}" shows "(x <s signed.Max A) = (\<exists>a\<in>A. x <s a)"
Z_def: shows "Z = (\<Union>p\<in>P. {z. poly p z = (0::'a)})"
NSLIMSEQ_realpow_zero: fixes x :: "real" assumes "0 \<le> x"   and "x < 1" shows "(^) x \<longlonglongrightarrow>\<^sub>N\<^sub>S 0"
intro: fixes type_wf :: "(_) heap \<Rightarrow> bool"   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" assumes "type_wf = ShadowRootClass.type_wf"   and "set_shadow_root = set_shadow_root"   and "set_shadow_root_locs = set_shadow_root_locs" shows "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"
comm: fixes a :: "'d"   and b :: "'d" shows "a + b = b + a"
in_support_on: fixes x :: "'a"   and S :: "'a set"   and f :: "'a \<Rightarrow> 'b" shows "(x \<in> support_on S f) = (x \<in> S \<and> f x \<noteq> (0::'b))"
r_constant_alt: shows "\<forall>x\<in>\<V>. \<B> rep x = \<Lambda> * (\<v> - 1) div (\<k> - 1)"
func_zero: shows "0 = (\<lambda>x. 0::'d)"
ToCT_def: fixes f :: "'a \<times> 'a \<times> 'a" shows "T.ToCT f \<equiv> if CCC.arr f then fst f \<otimes> fst (snd f) \<otimes> snd (snd f) else null"
gr_zeroI_hmset: fixes M :: "hmultiset" assumes "M = 0 \<Longrightarrow> False" shows "0 < M"
rexp_of'_def: fixes \<Sigma> :: "'b list"   and n :: "nat"   and \<phi> :: "'b formula" assumes "formula \<Sigma>" shows "formula.rexp_of' \<Sigma> n \<phi> = rexp.Inter (formula.rexp_of_alt \<Sigma> n \<phi>) (formula.ENC n (FOV \<phi>))"
fname_simp: fixes n :: "vname"   and c :: "'a" shows "fname (n, c) = n"
intro: fixes G :: "'a set" assumes "G \<noteq> {}"   and "\<And>g h. \<lbrakk>g \<in> G; h \<in> G\<rbrakk> \<Longrightarrow> g - h \<in> G" shows "Group G"
nz_prod_eq: fixes c :: "'a"   and x :: "'a" assumes "c \<noteq> (0::'a)" shows "c * x = (0::'a) \<equiv> x = (0::'a)"
is_category_with_binary_products: shows "category_with_binary_products (\<cdot>)"
take_all_iff: fixes n :: "nat"   and xs :: "'d list" shows "(xs \<down> n = xs) = (length xs \<le> n)"
map_ran_def: fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b" shows "map_ran f = map (\<lambda>(k, v). (k, f k v))"
rtrancl_tab_imp_rtrancl_path: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and ys :: "'a list"   and x :: "'a"   and y :: "'a" assumes "rtrancl_tab r ys x y" obtains xs :: "'a list" where "rtrancl_path r x xs y"
weighted_matroid_def: fixes carrier :: "'c set"   and indep :: "'c set \<Rightarrow> bool" shows "weighted_matroid carrier indep \<equiv> matroid carrier indep"
le_supI1: fixes x :: "'a"   and a :: "'a"   and b :: "'a" assumes "x \<le> a" shows "x \<le> a \<squnion> b"
G: fixes i :: "'a" assumes "i \<in> I" shows "Group.group (G i)"
bij_betw_partitions_of: fixes A :: "'c set"   and B :: "'d set" assumes "finite A"   and "finite B" shows "bij_betw (partitions_of A B) ({f \<in> A \<rightarrow>\<^sub>E B. f ` A = B} // range_permutation A B) {P. partition_on A P \<and> card P = card B}"
poly_subst_times: fixes f :: "'e \<Rightarrow> ('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd"   and p :: "('e \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd"   and q :: "('e \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd" shows "poly_subst f (p * q) = poly_subst f p * poly_subst f q"
if_split_mem2: fixes a :: "'a"   and Q :: "bool"   and x :: "'a set"   and y :: "'a set" shows "(a \<in> (if Q then x else y)) = ((Q \<longrightarrow> a \<in> x) \<and> (\<not> Q \<longrightarrow> a \<in> y))"
natural_isomorphism_axioms: shows "natural_isomorphism C.CC.comp (\<cdot>\<^sub>D) FoT\<^sub>C.map FoT\<^sub>C.map FoT\<^sub>C.map"
Complete_item_end: fixes x :: "item"   and k :: "nat"   and Y :: "item set" assumes "x \<in> Complete k Y" shows "item_end x = k \<or> x \<in> Y"
lens_indepI: fixes x :: "'a \<Longrightarrow> 'c"   and y :: "'b \<Longrightarrow> 'c" assumes "\<And>u v \<sigma>. put\<^bsub>x\<^esub> (put\<^bsub>y\<^esub> \<sigma> v) u = put\<^bsub>y\<^esub> (put\<^bsub>x\<^esub> \<sigma> u) v"   and "\<And>v \<sigma>. get\<^bsub>x\<^esub> (put\<^bsub>y\<^esub> \<sigma> v) = get\<^bsub>x\<^esub> \<sigma>"   and "\<And>u \<sigma>. get\<^bsub>y\<^esub> (put\<^bsub>x\<^esub> \<sigma> u) = get\<^bsub>y\<^esub> \<sigma>" shows "x \<bowtie> y"
gS_in_sets_leb: shows "g ` S \<in> sets lebesgue"
a: fixes f :: "'a \<Rightarrow> real \<times> real" shows "(\<Sum>i\<in>Basis. fst (f i) *\<^sub>R i) = a f"
\<E>\<tau>_in_Fa: fixes \<tau> :: "'c \<Rightarrow> 's" assumes "natural_transformation (\<cdot>\<^sup>o\<^sup>p) (\<cdot>\<^sub>S) (Y a) F \<tau>" shows "\<E> \<tau> \<in> F.SET a"
validated_part_def: fixes atom_ordering :: "('a \<times> 'a) set"   and I :: "'a set"   and C :: "'a Literal set" assumes "propositional_atoms atom_ordering" shows "propositional_atoms.validated_part I C = {L \<in> C. propositional_atoms.validate_literal I L}"
tdev_def: fixes x :: "'a pdevs" shows "tdev x = (\<Sum>i<degree x. \<bar>pdevs_apply x i\<bar>)"
strategyproof': fixes R1 :: "'agent \<Rightarrow> 'alt \<Rightarrow> 'alt \<Rightarrow> bool"   and R2 :: "'agent \<Rightarrow> 'alt \<Rightarrow> 'alt \<Rightarrow> bool"   and i :: "'agent"   and j :: "'agent" assumes "is_pref_profile R1"   and "is_pref_profile R2"   and "i \<in> agents"   and "j \<in> agents"   and "anonymous_profile R2 = anonymous_profile R1 - {#weak_ranking (R1 i)#} + {#weak_ranking (R2 j)#}" shows "\<not> scf R1 \<prec>[Fishb (R1 i)] scf R2"
finite_dimensional_vector_space_ow_axioms_def: fixes U\<^sub>V\<^sub>S :: "'c set"   and plus\<^sub>V\<^sub>S :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero\<^sub>V\<^sub>S :: "'c"   and scale :: "'d \<Rightarrow> 'c \<Rightarrow> 'c"   and basis :: "'c set" shows "finite_dimensional_vector_space_ow_axioms U\<^sub>V\<^sub>S plus\<^sub>V\<^sub>S zero\<^sub>V\<^sub>S scale basis \<equiv> (basis \<subseteq> U\<^sub>V\<^sub>S \<and> finite basis) \<and> (with (0::'d) zero\<^sub>V\<^sub>S plus\<^sub>V\<^sub>S scale : \<guillemotleft>independent\<guillemotright> basis) \<and> (with zero\<^sub>V\<^sub>S plus\<^sub>V\<^sub>S scale : \<guillemotleft>span\<guillemotright> basis) = U\<^sub>V\<^sub>S"
lebesgue_measurable_diff_null: fixes N :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "N \<in> null_sets lebesgue" shows "(f \<in> borel_measurable (lebesgue_on (- N))) = (f \<in> borel_measurable lebesgue)"
B: shows "B \<in> carrier_mat (n - 1) (n - 1)"
deg_pm_superset: fixes s :: "'b \<Rightarrow>\<^sub>0 'c"   and A :: "'b set" assumes "keys s \<subseteq> A"   and "finite A" shows "deg_pm s = sum (lookup s) A"
MakeFormP_Disj: fixes x :: "tm"   and y :: "tm" shows "{} \<turnstile> MakeFormP (Q_Disj x y) x y"
invalidate_clone': fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> assn"   and x :: "'a"   and y :: "'b" shows "R x y \<Longrightarrow>\<^sub>A invalid_assn R x y * R x y * true"
HIm_def: fixes q :: "quat" shows "HIm q \<equiv> vector [Im1 q, Im2 q, Im3 q]"
multiset_induct: fixes P :: "'k multiset \<Rightarrow> bool"   and M :: "'k multiset" assumes "P {#}"   and "\<And>x M. P M \<Longrightarrow> P (add_mset x M)" shows "P M"
no_state_change_if_no_event: fixes channel :: "nat \<Rightarrow> ('a \<times> 'a) option"   and trans :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and c :: "('a, 'b, 'c) configuration"   and ev :: "('a, 'b, 'c) event"   and c' :: "('a, 'b, 'c) configuration"   and p :: "'a" assumes "distributed_system channel"   and "distributed_system.next channel trans send recv c ev c'"   and "occurs_on ev \<noteq> p" shows "states c p = states c' p \<and> ps c p = ps c' p"
invar_children: fixes e :: "'c"   and a :: "'d"   and r :: "nat"   and ts :: "('c, 'd) BinomialTree list" assumes "tree_invar (Node e a r ts)" shows "queue_invar ts"
prob_Collect_eq_0: fixes s :: "'a cfg"   and P :: "'a cfg stream \<Rightarrow> bool" assumes "{x \<in> space (MC.T s). P x} \<in> MC.T.events s" shows "(MC.T.prob s {x \<in> space (MC.T s). P x} = 0) = (AE x in MC.T s. \<not> P x)"
assms_2: fixes xs :: "nat list"   and rest :: "(recf \<times> nat list \<times> nat list) list" assumes "valid ((f, xs, []) # rest)" shows "correct ((f, xs, []) # rest, None)"
T_inorder2_Leaf: fixes xs :: "'b list" shows "T_inorder2 \<langle>\<rangle> xs = 1"
mat_mult_row_def: fixes m :: "(('a, 'c) vec, 'd) vec"   and m' :: "(('a, 'b) vec, 'c) vec"   and f :: "'d" shows "mat_mult_row m m' f = (\<chi>c. \<Sum>k\<in>UNIV. m $ f $ k * m' $ k $ c)"
v_2: shows "[v]\<^sub>\<R> = region X I J r"
xcpt_eff_def: fixes i :: "instr"   and P :: "'a prog"   and pc :: "nat"   and \<tau> :: "ty list \<times> ty err list"   and et :: "ex_table" shows "xcpt_eff i P pc \<tau> et = (let (ST, LT) = \<tau> in map (\<lambda>(f, t, C, h, d). (h, \<lfloor>(Class C # drop (length ST - d) ST, LT)\<rfloor>)) (relevant_entries P i pc et))"
USdBsym: fixes d :: "'d" shows "sym \<approx>\<^bsub>d\<^esub>"
pr_in_hom_2: fixes a :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and b :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "ide a"   and "ide b" shows "\<guillemotleft>pr1 a b : HF'.prod a b \<rightarrow> a\<guillemotright>"
lhs: shows "Hom(A,_) \<^bsub>\<a>\<^esub> (g \<bullet> f) = \<lparr>set_dom = Hom A (Dom f), set_func = restrict ((\<bullet>) (g \<bullet> f)) (Hom A (Dom f)), set_cod = Hom A (Cod g)\<rparr>"
cadd_cmult_distrib: fixes i :: "V"   and j :: "V"   and k :: "V" shows "(i \<oplus> j) \<otimes> k = i \<otimes> k \<oplus> j \<otimes> k"
tyenv_eq: shows "mem\<^sub>1 =\<^bsub>\<Gamma>\<^esub> mem\<^sub>2"
axioms_3: fixes get_disconnected_document :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) document_ptr) prog"   and get_disconnected_document_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and type_wf :: "(_) heap \<Rightarrow> bool"   and get_host :: "(_) shadow_root_ptr \<Rightarrow> ((_) heap, exception, (_) element_ptr) prog"   and get_host_locs :: "((_) 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 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 get_tag_name :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, char list) prog"   and heap_is_wellformed :: "(_) heap \<Rightarrow> bool"   and parent_child_rel :: "(_) heap \<Rightarrow> ((_) object_ptr \<times> (_) object_ptr) set"   and heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) heap \<Rightarrow> bool" assumes "l_get_host_wf\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_disconnected_document get_disconnected_document_locs known_ptr type_wf get_host get_host_locs get_shadow_root get_shadow_root_locs get_child_nodes get_child_nodes_locs get_disconnected_nodes get_disconnected_nodes_locs get_tag_name heap_is_wellformed parent_child_rel heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M" shows "l_get_shadow_root type_wf get_shadow_root get_shadow_root_locs"
assms_4: fixes y :: "'a" assumes "y \<in> Y" shows "AE x\<in>X' in lborel. norm (f y x) \<le> g x"
prevVertex_in_edges: fixes f :: "face"   and v :: "nat" assumes "distinct (vertices f)"   and "v \<in> \<V> f" shows "(f\<^bsup>-1\<^esup> \<bullet> v, v) \<in> \<E> f"
assms_4: shows "roots = (\<lambda>(a, b, c). if a = 0 \<and> b \<noteq> 0 then {- c / b} else if a \<noteq> 0 \<and> 0 \<le> b\<^sup>2 - 4 * a * c then {(- b + sqrt (b\<^sup>2 - 4 * a * c)) / (2 * a)} \<union> {(- b - sqrt (b\<^sup>2 - 4 * a * c)) / (2 * a)} else {})"
itrm_ext_id: fixes extra_rule :: "'a itrm \<Rightarrow> 'a itrm \<Rightarrow> bool"   and y :: "'a itrm" shows "itrm_cong (special_idiom.idiom_ext_rule extra_rule) (Pure \<I> \<diamondop> y) y"
assms_1: shows "cfg1 \<turnstile> m \<mapsto> cfg2"
sep_add_assoc: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and sep_disj :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "class.pre_sep_algebra plus zero sep_disj"   and "sep_disj x y"   and "sep_disj y z"   and "sep_disj x z" shows "plus (plus x y) z = plus x (plus y z)"
of_rat_real_alg: fixes x :: "rat" shows "real_of_rat x = real_of (of_rat_real_alg x)"
row_length_constant: fixes nr :: "nat"   and nc :: "nat"   and A :: "'a list list"   and j :: "nat" assumes "mat nr nc A"   and "j < length A" shows "length (A ! j) = row_length A"
cr_real_def: shows "cr_real \<equiv> \<lambda>x y. realrel x x \<and> Real.Real x = y"
star_inductl_var: fixes x :: "'a"   and y :: "'a" assumes "x \<cdot> y \<le> y" shows "x\<^sup>\<star> \<cdot> y \<le> y"
lookupc_nmaps: fixes \<theta>_n :: "(name \<times> coname \<times> trm) list"   and x :: "name"   and c :: "coname"   and P :: "trm"   and a :: "coname" assumes "\<theta>_n nmaps x to Some (c, P)" shows "lookupc x a \<theta>_n = P[c\<turnstile>c>a]"
refl: shows "refl_on (vertices G) (getRel S_Idt G)"
tendsto_within_subset: fixes f :: "'b \<Rightarrow> 'd"   and l :: "'d filter"   and x :: "'b"   and S :: "'b set"   and T :: "'b set" assumes "filterlim f l (at x within S)"   and "T \<subseteq> S" shows "filterlim f l (at x within T)"
true_Guard1: fixes g :: "'c \<Rightarrow> bool"   and a :: "('d, 'c) alpha_rp_scheme"   and P :: "('d, 'c) action"   and b :: "('d, 'c) alpha_rp_scheme" assumes "g (alpha_rp.more a) = True" shows "action.relation_of g `&` P (a, b) = action.relation_of P (a, b)"
card_prod_def: fixes a :: "nat"   and b :: "nat" shows "card_prod a b = a * b"
atLeastLessThan_rev_at_least_Suc_atMost: fixes g :: "nat \<Rightarrow> 'c"   and n :: "nat"   and m :: "nat" shows "sum g {n..<m} = (\<Sum>i = Suc n..m. g (m + n - i))"
wf_prog_idecl: fixes G :: "prog"   and I :: "qtname"   and i :: "iface" assumes "iface G I = Some i"   and "wf_prog G" shows "wf_idecl G (I, i)"
alphas_1: fixes bs :: "atom set"   and x :: "'a"   and R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> atom set"   and p :: "perm"   and cs :: "atom set"   and y :: "'a" shows "(bs, x) \<approx>set R f p (cs, y) = (f x - bs = f y - cs \<and> (f x - bs) \<sharp>* p \<and> R (p \<bullet> x) y \<and> p \<bullet> bs = cs)"
ball_eball_mono: fixes r :: "real"   and r' :: "ereal"   and z :: "'a" assumes "ereal r \<le> r'" shows "ball z r \<subseteq> eball z r'"
Inf_lower: fixes x :: "'b \<times> 'c"   and A :: "('b \<times> 'c) set" assumes "x \<in> A" shows "less_eq_prod (Inf_prod A) x"
path_f: shows "path_image \<gamma> \<subseteq> s - zeros_f"
comb_sys_h: shows "snd (snd (reduce_system_R p (combine_systems_R p (qs1, calculate_data_R p qs1) (qs2, calculate_data_R p qs2)))) = snd (snd (reduce_system_R p (qs1 @ qs2, M_mat_R (get_signs_R (snd (combine_systems_R p (qs1, calculate_data_R p qs1) (qs2, calculate_data_R p qs2)))) (get_subsets_R (snd (combine_systems_R p (qs1, calculate_data_R p qs1) (qs2, calculate_data_R p qs2)))), get_subsets_R (snd (combine_systems_R p (qs1, calculate_data_R p qs1) (qs2, calculate_data_R p qs2))), get_signs_R (snd (combine_systems_R p (qs1, calculate_data_R p qs1) (qs2, calculate_data_R p qs2))))))"
step_in_A: shows "snd ` set (step pc (\<tau>s ! pc)) \<subseteq> A"
M2: shows "mat m m (\<lambda>(i, j). A $$ (if i < n then i + k + l else if i < k + n then i - n + l else i - (k + n), j)) \<in> carrier_mat m m"
two_chain_valid_valid_cubes: shows "\<forall>two_cube\<in>two_chain. valid_two_cube two_cube"
base_le: fixes a :: "nat \<Rightarrow> nat"   and j :: "nat" assumes "a \<in> s" shows "base j \<le> a j"
finite_list': fixes A :: "'b set" assumes "finite A" obtains xs :: "'b list" where "A = set xs" and "distinct xs" and "length xs = card A"
2: shows "\<turnstile> concl (Thm (Imp p (Imp (Imp p p) p)))"
NTCatDom: fixes \<eta>1 :: "('g, 'h, 'i, 'j, 'k, 'l) NatTrans"   and \<eta>2 :: "('g, 'h, 'i, 'j, 'k, 'l) NatTrans" assumes "\<eta>1 \<approx>>\<bullet> \<eta>2" shows "NTCatDom \<eta>1 = NTCatDom \<eta>2"
head_is_vert: fixes u :: "nat"   and v :: "nat" assumes "(u, v) \<in> g_E G" shows "v \<in> set (g_V G)"
a_closure: fixes antidomain_op :: "'b \<Rightarrow> 'b"   and one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'b" assumes "class.antidomain_left_monoid antidomain_op one times" shows "antidomain_left_monoid.am_d antidomain_op (antidomain_op x) = antidomain_op x"
append_prefix: fixes xs :: "'a list"   and ys :: "'a list" shows "(xs @ ys \<le> xs) = (ys \<le> [])"
Extra_ax_def: shows "Extra_ax \<equiv> {\<lbrakk>\<guillemotleft>extra_axiom\<guillemotright>\<rbrakk>e0}"
Min_Di'_le: fixes i :: "nat"   and e :: "nat"   and y :: "nat list" assumes "i < m"   and "e \<in> Di' i y"   and "length y \<le> n" shows "Min (Di' i y) \<le> Max (set b)"
arr_mkArr: fixes A :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr set"   and B :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr set"   and F :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr \<Rightarrow> (hf set, hf \<Rightarrow> hf) concrete_category.arr" shows "arr (mkArr A B F) = (A \<subseteq> Univ \<and> |A| <o natLeq \<and> B \<subseteq> Univ \<and> |B| <o natLeq \<and> F \<in> A \<rightarrow> B)"
transp_on_converse_simp: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set" shows "transp_on P\<inverse>\<inverse> A = transp_on P A"
measure_preservingE_1: fixes f :: "'b \<Rightarrow> 'c"   and M :: "'b measure"   and N :: "'c measure" assumes "f \<in> measure_preserving M N" shows "f \<in> M \<rightarrow>\<^sub>M N"
writer_map_def: shows "writer_map \<equiv> \<Lambda> w a. \<mu> uu. writer_abs oo sprod_map\<cdot>(u_map\<cdot>w)\<cdot>(u_map\<cdot>a) oo writer_rep"
im_zero: fixes smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and V :: "'b set"   and smult' :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and T :: "'b \<Rightarrow> 'c" assumes "VectorSpaceHom smult V smult' T" shows "T (0::'b) = (0::'c)"
sort_eqv_trans: fixes x :: "String.literal set"   and y :: "String.literal set"   and z :: "String.literal set" assumes "sort_eqv cs x y"   and "sort_eqv cs y z" shows "sort_eqv cs x z"
signed_take_bit_eq_concat_bit: fixes n :: "nat"   and k :: "int" shows "signed_take_bit n k = concat_bit n k (- of_bool (bit k n))"
rank_invar_not_empty_hd: fixes t :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree"   and bq :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree list" assumes "SkewBinomialHeapStruc.rank_invar (t # bq)"   and "bq \<noteq> []" shows "SkewBinomialHeapStruc.rank t < SkewBinomialHeapStruc.rank (hd bq)"
pivot_len: shows "length (pivot_positions A) \<le> dim_row A"
worst_chain_bad: fixes gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and p :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b" assumes "wf {(x, y). gt y x}"   and "bad p x" shows "inf_chain p (worst_chain p gt)"
F2set2_Lev1: fixes xf :: "(bd_type_F + bd_type_F) list"   and s1 :: "'d \<Rightarrow> ('f, 'd, 'e) F1"   and s2 :: "'e \<Rightarrow> ('g, 'd, 'e) F2"   and n :: "nat"   and xe :: "'d"   and xb :: "'e"   and xa :: "'d" assumes "xf \<in> Lev1 s1 s2 n xe"   and "rv1 s1 s2 xf xe = Inr xb"   and "xa \<in> F2set2 (s2 xb)" shows "xf @ [Inl (tobd_F22 s2 xb xa)] \<in> Lev1 s1 s2 (Suc n) xe"
m_le_length: shows "mod_type_class.to_nat m < IArray.length (vec_to_iarray A)"
1: shows "prv (imp \<phi> (instInp Pinp \<langle>\<phi>\<rangle>))"
iso_1: fixes x :: "'a" shows "d x \<cdot> 1\<^sub>\<pi> \<parallel> 1\<^sub>\<sigma> = d x"
res_def: fixes \<eta> :: "nat" shows "CIPHER.KEY.res \<eta> = RES (CIPHER.KEY.key_oracle \<eta> \<oplus>\<^sub>O CIPHER.KEY.key_oracle \<eta>) None"
assms_1: shows "\<forall>\<^sub>F x in at_top. 0 \<le> l x"
quorum_def: fixes Q :: "'a set" shows "i1.orig.quorum Q \<equiv> \<forall>p\<in>Q \<inter> W. \<exists>Sl\<in>slices p. Sl \<subseteq> Q"
set_awalk_verts: fixes u :: "'a"   and p :: "'b list"   and v :: "'a" assumes "H.awalk u p v" shows "set (H.awalk_verts u p) = {u} \<union> set (map (tail (G \<restriction> S)) p) \<union> set (map (head (G \<restriction> S)) p)"