hcju/isabelleps huggingface dataset:
bidirected_digraph: shows "bidirected_digraph (with_proj (subdivide H e w)) (swap_in (parcs (subdivide H e w)))"
simps_1: fixes c :: "'d \<Rightarrow> bool"   and f :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and \<sigma> :: "'d" shows "foldli [] c f \<sigma> = \<sigma>"
powr_zero_eq_one: fixes x :: "'b" shows "x powr (0::'b) = (if x = (0::'b) then 0::'b else (1::'b))"
a_closed: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier R"   and "y \<in> carrier R" shows "x \<oplus> y \<in> carrier R"
assms: fixes v :: "'a"   and P :: "'a llist" assumes "v \<in> A"   and "vmc_path G P v p \<sigma>" shows "visits_via P A W"
invertible_transpose: fixes A :: "(('a, 'b) vec, 'b) vec" assumes "invertible A" shows "invertible (Finite_Cartesian_Product.transpose A)"
term_ok'_incr_boundvars: fixes \<Theta> :: "((String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)) \<times> Core.term set"   and t :: "Core.term"   and lev :: "nat" assumes "term_ok' (sig \<Theta>) t" shows "term_ok' (sig \<Theta>) (incr_boundvars lev t)"
in_subset_eq_grounding_of_clss_is_ground_cls: fixes C :: "'a literal multiset"   and CC :: "'a literal multiset set"   and DD :: "'a literal multiset set" assumes "C \<in> CC"   and "CC \<subseteq> grounding_of_clss DD" shows "is_ground_cls C"
assms_1: shows "theta \<subseteq> Wretr (theta \<union> Wbis)"
continuous_attains_sup: fixes s :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "compact s"   and "s \<noteq> {}"   and "continuous_on s f" shows "\<exists>x\<in>s. \<forall>y\<in>s. f y \<le> f x"
exists_subterm_iff_exists_subterm': fixes P :: "Core.term \<Rightarrow> bool"   and t :: "Core.term" shows "exists_subterm P t = exists_subterm' P t"
valid_node_cases: fixes n :: "'node" assumes "valid_node n" obtains "n = (_Entry_)"   | "n = (_Exit_)"   | "inner_node n"
Image_empty2: fixes R :: "('c \<times> 'b) set" shows "R `` {} = {}"
sbintrunc_minus_simps_2: fixes n :: "nat" assumes "0 < n" shows "signed_take_bit n (- 1) = - 1 mod 2 + 2 * signed_take_bit (n - 1) (- 1)"
cases: fixes a1 :: "('a, 'b) Proc"   and a2 :: "('a, 'b) Proc"   and Enc :: "'a \<Rightarrow> 'b" assumes "(a1, a2) \<in> encoding.indRelR Enc" obtains S :: "'a" where "a1 = SourceTerm S" and "a2 = TargetTerm (Enc S)"
restrict_merge_1: fixes I :: "'d set"   and J :: "'d set"   and x :: "'d \<Rightarrow> 'e"   and y :: "'d \<Rightarrow> 'e" assumes "I \<inter> J = {}" shows "restrict (merge I J (x, y)) I = restrict x I"
max_add_distrib_left: fixes x :: "'b multiset"   and y :: "'b multiset"   and z :: "'b multiset" shows "subset_mset.max x y + z = subset_mset.max (x + z) (y + z)"
on_l: shows "{u, v} \<subseteq> circline_set l1 \<inter> circline_set l2"
a0n_agreement_init: shows "init a0n \<subseteq> a0n_agreement"
parts_fst: fixes X :: "msg"   and Y :: "msg"   and H :: "msg set" assumes "\<lbrace>X, Y\<rbrace> \<in> parts H" shows "X \<in> parts H"
pqs_def: shows "pqs \<equiv> {(p, q). 0 < q \<and> coprime p q \<and> \<bar>\<xi> - real_of_int p / real_of_int q\<bar> < 1 / real_of_int q powr (2 + \<delta> * \<epsilon> / (1 + \<epsilon>))}"
o_case_sum: fixes h :: "'d \<Rightarrow> 'c"   and f :: "'a \<Rightarrow> 'd"   and g :: "'b \<Rightarrow> 'd" shows "h \<circ> case_sum f g = case_sum (h \<circ> f) (h \<circ> g)"
xor_closed: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier L"   and "y \<in> carrier L" shows "x \<oplus> y \<in> carrier L"
choice_iff: fixes Q :: "'i \<Rightarrow> 'j \<Rightarrow> bool" shows "(\<forall>x. \<exists>y. Q x y) = (\<exists>f. \<forall>x. Q x (f x))"
assms_4: shows "f \<in> borel_measurable (nat_filtration n)"
sublist_take: fixes n :: "nat"   and xs :: "'c list" shows "sublist (take n xs) xs"
assms_2: shows "ipresIGSwapAbs hA MOD MOD'"
axioms_2: fixes V\<^sub>B :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and H\<^sub>B :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<a>\<^sub>B :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<i>\<^sub>B :: "'c \<Rightarrow> 'c"   and src\<^sub>B :: "'c \<Rightarrow> 'c"   and trg\<^sub>B :: "'c \<Rightarrow> 'c"   and V\<^sub>C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and H\<^sub>C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and \<a>\<^sub>C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and \<i>\<^sub>C :: "'d \<Rightarrow> 'd"   and src\<^sub>C :: "'d \<Rightarrow> 'd"   and trg\<^sub>C :: "'d \<Rightarrow> 'd"   and V\<^sub>D :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and H\<^sub>D :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and \<a>\<^sub>D :: "'e \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and \<i>\<^sub>D :: "'e \<Rightarrow> 'e"   and src\<^sub>D :: "'e \<Rightarrow> 'e"   and trg\<^sub>D :: "'e \<Rightarrow> 'e"   and F :: "'c \<Rightarrow> 'd"   and \<Phi>\<^sub>F :: "'c \<times> 'c \<Rightarrow> 'd"   and G :: "'d \<Rightarrow> 'e"   and \<Phi>\<^sub>G :: "'d \<times> 'd \<Rightarrow> 'e" assumes "composite_pseudofunctor V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>\<^sub>F G \<Phi>\<^sub>G" shows "bicategory V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C"
axioms_3: fixes ts :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'c \<times> 'd \<times> nat set \<times> 'e) list"   and \<S> :: "nat \<Rightarrow> bool option"   and valid :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'f list \<times> 'd \<times> nat set \<times> ('g \<Rightarrow> 'h option)) list \<Rightarrow> bool" assumes "initial\<^sub>v ts \<S> valid" shows "unowned_shared \<S> ts"
ifex_ordered_implied: fixes a :: "('a \<Rightarrow> bool) \<Rightarrow> bool"   and b :: "'a ifex" assumes "(a, b) \<in> bf_ifex_rel" shows "ifex_ordered b"
Map_in_Hom: fixes f :: "(bool, unit) arr" assumes "arr f" shows "Map f \<in> (if Dom f = Cod f then {()} else {})"
coefficients_head: fixes p :: "'b poly" shows "last (coefficients p) = head p"
permutation_lists_fc_succ: shows "permutation (lists_fc_succ xss)"
isFDERIV_I: fixes n :: "nat"   and xs :: "nat list"   and fas :: "floatarith list"   and vs :: "real list" assumes "\<And>i j. \<lbrakk>i < n; j < n\<rbrakk> \<Longrightarrow> isDERIV (xs ! i) (fas ! j) vs"   and "length fas = n"   and "length xs = n" shows "isFDERIV n xs fas vs"
combine_infmatrix_def: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd" shows "combine_infmatrix f \<equiv> \<lambda>A B j i. f (A j i) (B j i)"
gd: shows "\<forall>x. a < x \<and> x < b \<longrightarrow> g differentiable at x"
NSLIMSEQ_const: fixes k :: "'a" shows "(\<lambda>n. k) \<longlonglongrightarrow>\<^sub>N\<^sub>S k"
lm071: shows "finestpart {} = {} \<and> card \<circ> finestpart = card"
tree0: fixes gamma :: "'b"   and subs :: "'b \<Rightarrow> 'b set" shows "(0, gamma) \<in> tree subs gamma"
cod_char: fixes f :: "'a" shows "R.cod f = (if R.arr f then cod f else null)"
M1_cardeq_rcosetGM1g: fixes G :: "('a, 'b) monoid_scheme"   and p :: "nat"   and a :: "nat"   and m :: "nat"   and M1 :: "'a set"   and M :: "'a set set"   and g :: "'a" assumes "sylow_central G p a m M1 M"   and "g \<in> carrier G" shows "card (M1 #>\<^bsub>G\<^esub> g) = card M1"
finite_subset_induct: fixes F :: "'c set"   and A :: "'c set"   and P :: "'c set \<Rightarrow> bool" assumes "finite F"   and "F \<subseteq> A"   and "P {}"   and "\<And>a F. \<lbrakk>finite F; a \<in> A; a \<notin> F; P F\<rbrakk> \<Longrightarrow> P (insert a F)" shows "P F"
sel_1: fixes x1 :: "('a \<times> 'b \<times> 'b) set"   and x2 :: "'b set" shows "edges (LG x1 x2) = x1"
arrow: shows "D.ide (Go (C.dom f)) \<and> \<guillemotleft>\<epsilon>o (C.dom f) : F (Go (C.dom f)) \<rightarrow>\<^sub>C C.dom f\<guillemotright>"
rel_setI: fixes A :: "'g set"   and B :: "'h set"   and R :: "'g \<Rightarrow> 'h \<Rightarrow> bool" assumes "\<And>x. x \<in> A \<Longrightarrow> \<exists>y\<in>B. R x y"   and "\<And>y. y \<in> B \<Longrightarrow> \<exists>x\<in>A. R x y" shows "rel_set R A B"
le: shows "(n div x ^ pm + x * int pm) div int p \<le> x"
of_weak_ranking_indifference: fixes A :: "'b set"   and xs :: "'b set list"   and x :: "'b"   and y :: "'b" assumes "A \<in> set xs"   and "x \<in> A"   and "y \<in> A" shows "of_weak_ranking xs x y"
elims_2: fixes x :: "('c \<Rightarrow> 'd \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'd \<Rightarrow> bool)"   and xa :: "'c match_expr list"   and xb :: "action"   and xc :: "'d" assumes "match_list x xa xb xc" obtains \<gamma> :: "('c \<Rightarrow> 'd \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'd \<Rightarrow> bool)" and m :: "'c match_expr" and ms :: "'c match_expr list" and a :: "action" and p :: "'d" where "x = \<gamma>" and "xa = m # ms" and "xb = a" and "xc = p" and "if matches \<gamma> m a p then True else match_list \<gamma> ms a p"
mult_sub_right_one: fixes x :: "'c \<times> 'c \<Rightarrow> 'd" shows "x \<preceq> x \<odot> mone"
strong_reduction_bisimulations_impl_inverse_is_simulation: fixes Rel :: "('b \<times> 'b) set"   and Cal :: "'b processCalculus" assumes "strong_reduction_simulation Rel Cal \<and> (\<forall>P Q Q'. (P, Q) \<in> Rel \<and> Q \<longmapsto>Cal Q' \<longrightarrow> (\<exists>P'. P \<longmapsto>Cal P' \<and> (P', Q') \<in> Rel))" shows "strong_reduction_simulation (Rel\<inverse>) Cal"
lrev_body3_strict: fixes r :: "'b llist \<rightarrow> 'b llist \<rightarrow> 'b llist" shows "lrev_body3\<cdot>r\<cdot>\<bottom> = \<bottom>"
simps: 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 a1 :: "ty"   and a2 :: "ty" shows "P \<turnstile> a1 \<le> a2 = ((\<exists>T. a1 = T \<and> a2 = T) \<or> (\<exists>C D. a1 = Class C \<and> a2 = Class D \<and> P \<turnstile> C \<preceq>\<^sup>* D) \<or> (\<exists>C. a1 = NT \<and> a2 = Class C))"
lemDirectionFromTo: fixes p :: "'a Point"   and dir :: "'a Vector" shows "direction line joining p to p \<leadsto> dir = dir"
Cones_SET_eq_\<iota>_img_cones: fixes a' :: "'c" assumes "C.ide a'" shows "Cones.SET a' = \<iota> ` D.cones a'"
assms: fixes i :: "'c" assumes "i \<in> S" shows "f i \<in> M \<rightarrow>\<^sub>M count_space UNIV"
nonintSI_nonintS: fixes st0 :: "'a"   and do :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'a"   and out :: "'a \<Rightarrow> 'b \<Rightarrow> 'd"   and GH :: "'b set"   and GL :: "'b set" shows "GM_sec_model.nonintSI st0 do out GH GL = GM_sec_model.nonintS st0 do out GH GL"
mkdisjoint_mon_conv: fixes A :: "nat \<Rightarrow> 'b set"   and B :: "'b set" assumes "A\<up>B" shows "\<Union> (range (mkdisjoint A)) = B"
bind_get: fixes f :: "'b \<Rightarrow> 'm"   and g :: "'s \<Rightarrow> 'm" shows "bind (get f) g = get (\<lambda>s. bind (f s) g)"
swap_fm: fixes a :: "nat"   and L :: "real list"   and b :: "nat"   and F :: "atom fm" assumes "a < length L"   and "b < length L" shows "eval F L = eval (swap_fm a b F) (swap_list a b L)"
lemma_2_reachability_s_i: fixes t :: "'a set"   and s :: "'b set"   and f :: "'b \<Rightarrow> 'a" assumes "finite t"   and "card t < card s" shows "\<not> FactoredSystem.inj f s t"
shiftr_mask2: fixes n :: "nat"   and m :: "nat" assumes "n \<le> LENGTH('b)" shows "mask n >> m = mask (n - m)"
det_swaprows_mat: fixes k :: "nat"   and n :: "nat"   and l :: "nat" assumes "k < n"   and "l < n"   and "k \<noteq> l" shows "det (swaprows_mat n k l) = - (1::'b)"
subalgebra_sigma_finite: fixes N :: "bool stream measure" assumes "subalgebra M N" shows "sigma_finite_subalgebra M N"
r2f_f2r_galois_var: fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c set"   and R :: "'a \<Rightarrow> ('b \<times> 'c) set" shows "(\<R> \<circ> f = R) = (\<F> \<circ> R = f)"
assms: shows "Broadcast (i, Insert e n) \<in> set (history j)"
ts_fct_ow_def: fixes U\<^sub>1 :: "'a set"   and \<tau>\<^sub>1 :: "'a set \<Rightarrow> bool"   and U\<^sub>2 :: "'b set"   and \<tau>\<^sub>2 :: "'b set \<Rightarrow> bool" shows "ts_fct_ow U\<^sub>1 \<tau>\<^sub>1 U\<^sub>2 \<tau>\<^sub>2 \<equiv> topological_space_ow U\<^sub>1 \<tau>\<^sub>1 \<and> first_countable_topology_ow U\<^sub>2 \<tau>\<^sub>2"
case_prod_conv: fixes f :: "'e \<Rightarrow> 'f \<Rightarrow> 'd"   and a :: "'e"   and b :: "'f" shows "(case (a, b) of (c, d) \<Rightarrow> f c d) = f a b"
Gcd_fin_eq_Gcd: fixes one :: "'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and gcd :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and lcm :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and Gcd :: "'b set \<Rightarrow> 'b"   and Lcm :: "'b set \<Rightarrow> 'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and A :: "'b set" assumes "class.semiring_Gcd one zero times gcd lcm Gcd Lcm divide plus minus unit_factor normalize"   and "finite A" shows "semiring_gcd.Gcd_fin one zero gcd A = Gcd A"
mult_isor: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<le> y" shows "x ; z \<le> y ; z"
max_subgraph_def: fixes P :: "('a, 'b) pre_digraph \<Rightarrow> bool"   and H :: "('a, 'b) pre_digraph" shows "max_subgraph P H \<equiv> subgraph H G \<and> P H \<and> (\<forall>H'. H' \<noteq> H \<and> subgraph H H' \<longrightarrow> \<not> (subgraph H' G \<and> P H'))"
suffixI: fixes ys :: "'b list"   and zs :: "'b list"   and xs :: "'b list" assumes "ys = zs \<cdot> xs" shows "xs \<le>s ys"
nomulti_digraphI_app_iso: fixes hom :: "('c, 'd, 'e, 'f) digraph_isomorphism" assumes "iG.digraph_isomorphism hom" shows "nomulti_digraph (app_iso hom (app_iso hom G))"
trm_simp_2: fixes A :: "'b trm"   and B :: "'b trm"   and C :: "'b trm"   and D :: "'b trm" assumes "App A B = App C D" shows "A = C"
induct: fixes P :: "int list \<Rightarrow> bool"   and a0 :: "int list" assumes "P []"   and "\<And>x. P [x]"   and "\<And>x y xs. (x < y \<Longrightarrow> P (y # xs)) \<Longrightarrow> P (x # y # xs)" shows "P a0"
a4: shows "F' \<in> set_offending_flows \<lparr>nodes = V, edges = E'\<rparr> nP"
hom_imp_img_comm_group: fixes h :: "'a \<Rightarrow> 'e"   and H :: "('e, 'f) monoid_scheme" assumes "h \<in> hom G H" shows "comm_group (H\<lparr>carrier := h ` carrier G, one := h \<one>\<rparr>)"
card_times_pmf_of_set_bind: fixes X :: "'a set"   and f :: "'a \<Rightarrow> 'b pmf" assumes "finite X" shows "real (card X) * pmf (pmf_of_set X \<bind> f) x = (\<Sum>y\<in>X. pmf (f y) x)"
rel_conversep: fixes R1 :: "'g \<Rightarrow> 'd \<Rightarrow> bool"   and R2 :: "'h \<Rightarrow> 'e \<Rightarrow> bool" shows "rel_gpv R1\<inverse>\<inverse> R2\<inverse>\<inverse> = (rel_gpv R1 R2)\<inverse>\<inverse>"
fps_lr_one_over_one_minus_fps_X_squared_1: shows "fps_left_inverse ((1 - fps_X)\<^sup>2) (1::'b) = Abs_fps (\<lambda>n. of_nat (n + 1))"
FDOM_eff_subset_prob_dom_pair: fixes a :: "('a, 'b) fmap \<times> ('a, 'c) fmap"   and PROB :: "(('a, 'b) fmap \<times> ('a, 'c) fmap) set" assumes "a \<in> PROB" shows "fmdom' (snd a) \<subseteq> prob_dom PROB"
Max_ge_iff: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and A :: "'c set"   and x :: "'c" assumes "class.linorder less_eq less"   and "finite A"   and "A \<noteq> {}" shows "less_eq x (linorder.Max less_eq A) = (\<exists>a\<in>A. less_eq x a)"
nonneg: fixes x :: "real" assumes "0 \<le> x" shows "0 \<le> f x"
cod_object: fixes CC :: "('b, 'c, 'd) category_scheme"   and f :: "'c" assumes "category CC"   and "f \<in> Ar\<^bsub>CC\<^esub>" shows "Cod\<^bsub>CC\<^esub> f \<in> Ob\<^bsub>CC\<^esub>"
translation_subtract_diff: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a"   and s :: "'a set"   and t :: "'a set" assumes "class.ab_group_add plus zero minus uminus" shows "(\<lambda>x. minus x a) ` (s - t) = (\<lambda>x. minus x a) ` s - (\<lambda>x. minus x a) ` t"
nth_map_matrix: fixes f :: "'e \<Rightarrow> 'b"   and x :: "(('e, 'c) vec, 'd) vec"   and i :: "'d"   and j :: "'c" shows "map_matrix f x $ i $ j = f (x $ i $ j)"
zero_im_sheaf_def: shows "zero_im_sheaf \<equiv> \<lambda>V. \<zero>\<^bsub>f \<^sup>\<inverse> S V\<^esub>"
Class_def: shows "Class = (\<lambda>a\<in>M. {b \<in> M. (b, a) \<in> E(\<eta>)})"
NOTRIGHT_elim: fixes a :: "coname"   and M :: "trm"   and B :: "ty" assumes "<a>:M \<in> NOTRIGHT (NOT B) (\<parallel>(B)\<parallel>)" obtains x' :: "name" and M' :: "trm" where "M = NotR x'.M' a" and "fic (NotR x'.M' a) a" and "x':M' \<in> \<parallel>(B)\<parallel>"
assms_2: shows "\<forall>n. #\<^sub>\<le> sub a n \<le> #\<^sub>\<le> sub b n"
fls_shifted_times_simps_2: fixes n :: "int"   and f :: "'b fls"   and g :: "'b fls" shows "fls_shift n f * g = fls_shift n (f * g)"
cases: fixes a :: "'e"   and r :: "('f \<times> 'e) set" assumes "a \<in> Range r" obtains a :: "'f" and b :: "'e" where "aa__ = b" and "(a, b) \<in> r"
Id_on_iff: fixes x :: "'e"   and y :: "'e"   and A :: "'e set" shows "((x, y) \<in> Id_on A) = (x = y \<and> x \<in> A)"
not_dvd_pderiv: fixes p :: "'a poly" assumes "Polynomial.degree p \<noteq> 0" shows "\<not> p dvd pderiv p"
it_to_list_def: fixes it :: "'b \<Rightarrow> ('c \<Rightarrow> bool) \<Rightarrow> ('d \<Rightarrow> 'd list \<Rightarrow> 'd list) \<Rightarrow> 'e list \<Rightarrow> 'a"   and s :: "'b" shows "it_to_list it s \<equiv> it s (\<lambda>_. True) (\<lambda>x l. l @ [x]) []"
homeomorphic_space_trans: fixes X1 :: "'a topology"   and X2 :: "'b topology"   and X3 :: "'c topology" assumes "X1 homeomorphic_space X2"   and "X2 homeomorphic_space X3" shows "X1 homeomorphic_space X3"
spmf_eq_0_set_spmf: fixes p :: "'c spmf"   and x :: "'c" shows "(spmf p x = 0) = (x \<notin> set_spmf p)"
open_ball: fixes x :: "'a"   and d :: "real" shows "open {y. dist x y < d}"
that: fixes f :: "'a" assumes "\<guillemotleft>f : a \<rightarrow> a'\<guillemotright> \<and> local.iso f" shows "thesis"
thm_2_3_7_indets: fixes ord :: "('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow> ('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow> bool"   and ord_strict :: "('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow> ('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow> bool"   and fs :: "(('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) list"   and b :: "nat" assumes "pm_powerprod ord ord_strict"   and "pm_powerprod.is_GB_cofactor_bound ord (set fs) b" shows "(1 \<in> ideal (set fs)) = (1 \<in> set (ordered_term.Macaulay_list ord ord_strict (pm_powerprod.deg_shifts ord (\<Union> (indets ` set fs)) b fs)))"
paths_both: fixes xs :: "'a list"   and t :: "'a ttree"   and t' :: "'a ttree" shows "(xs \<in> paths (t \<otimes>\<otimes> t')) = (\<exists>ys\<in>paths t. \<exists>zs\<in>paths t'. xs \<in> ys \<otimes> zs)"
emeasure_uniform_measure: fixes A :: "'c set"   and M :: "'c measure"   and B :: "'c set" assumes "A \<in> sets M"   and "B \<in> sets M" shows "emeasure (uniform_measure M A) B = emeasure M (A \<inter> B) / emeasure M A"
DBM_reset_mono: fixes M :: "nat \<Rightarrow> nat \<Rightarrow> 'c DBMEntry"   and n :: "nat"   and k :: "nat"   and d :: "'c"   and M' :: "nat \<Rightarrow> nat \<Rightarrow> 'c DBMEntry"   and i :: "nat"   and j :: "nat" assumes "DBM_reset M n k d M'"   and "i \<le> n"   and "j \<le> n"   and "i \<noteq> k"   and "j \<noteq> k" shows "M' i j \<le> M i j"
less_imp_le: fixes x :: "'t \<Rightarrow>\<^sub>0 'c"   and y :: "'t \<Rightarrow>\<^sub>0 'c" assumes "ord_strict_p x y" shows "ord_p x y"
conjE2_PL: fixes A :: "'a set"   and p :: "'a"   and q :: "'a" assumes "A \<turnstile> p .\<and>. q" shows "A \<turnstile> q"
divides_imp_divides_mult: fixes a :: "'a list"   and b :: "'a list" assumes "a \<in> carrier (K [X])"   and "b \<in> carrier (K [X]) - {\<zero>\<^bsub>K [X]\<^esub>}"   and "a divides\<^bsub>K [X]\<^esub> b" shows "a divides\<^bsub>Ring_Divisibility.mult_of (K [X])\<^esub> b"
always_eventually_pigeonhole: fixes k :: "nat"   and P :: "nat \<Rightarrow> nat \<Rightarrow> bool" shows "(\<forall>i. \<exists>n\<ge>i. \<exists>m\<le>k. P m n) = (\<exists>m\<le>k. \<forall>i. \<exists>n\<ge>i. P m n)"
dflt_ops_impl: shows "StdMap hm_basic.dflt_ops"
bezout_prime: fixes p :: "nat"   and a :: "nat" assumes "prime p"   and "\<not> p dvd a" shows "\<exists>x y. a * x = Suc (p * y)"
total_linorder_matrix: fixes f :: "'c \<times> 'c \<Rightarrow> 'd" shows "matrix_bounded_idempotent_semiring.total f = (\<forall>i. \<exists>j. f (i, j) = top)"
map_cong0: fixes x :: "'b list"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "\<And>z. z \<in> set x \<Longrightarrow> f z = g z" shows "map f x = map g x"
sup_continuous_lfp: fixes F :: "'b \<Rightarrow> 'b" assumes "sup_continuous F" shows "lfp F = (SUP i. (F ^^ i) bot)"
l_set_tag_name_get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_set_tag_name_get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M DocumentClass.type_wf set_tag_name set_tag_name_locs DocumentClass.known_ptr get_child_nodes get_child_nodes_locs"
evalRR_simps_3: fixes p :: "nat \<Rightarrow> 'a"   and q :: "nat \<Rightarrow> 'a"   and x :: "'a" assumes "p \<in> carrier P"   and "q \<in> carrier P"   and "x \<in> carrier R" shows "eval R R id x (p \<otimes>\<^bsub>P\<^esub> q) = eval R R id x p \<otimes> eval R R id x q"
ms1: shows "matrix_seq d (matrix_sum d (\<lambda>n. denote_while_n (x1 0) (x1 1) (denote x2) n \<rho>\<^sub>1))"
Maclaurin_lemma: fixes h :: "real"   and f :: "real \<Rightarrow> real"   and j :: "nat \<Rightarrow> real"   and n :: "nat" assumes "0 < h" shows "\<exists>B. f h = (\<Sum>m<n. j m / fact m * h ^ m) + B * (h ^ n / fact n)"
strict_iff_order: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" assumes "ordering less_eq less" shows "less a b = (less_eq a b \<and> a \<noteq> b)"
WT_coreD_foldl_spmf_cpoke: fixes \<I>_adv :: "('a, 'b) \<I>"   and \<I>_usr :: "('c, 'd) \<I>"   and I :: "'e \<Rightarrow> bool"   and core :: "('e, 'f, 'a, 'c, 'b, 'd) core"   and s' :: "'e"   and p :: "'e spmf"   and es :: "'f list" assumes "WT_core \<I>_adv \<I>_usr I core"   and "s' \<in> set_spmf (foldl_spmf (cpoke core) p es)"   and "\<forall>s\<in>set_spmf p. I s" shows "I s'"
inext_nth_Suc_mono: fixes I :: "nat set"   and n :: "nat" shows "I \<rightarrow> n \<le> I \<rightarrow> Suc n"
LAssRedsThrow: fixes e :: "char list exp"   and s :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and a :: "char list exp"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and V :: "char list" assumes "P \<turnstile> \<langle>e,s\<rangle> \<rightarrow>* \<langle>throw a,s'\<rangle>" shows "P \<turnstile> \<langle>V:=e,s\<rangle> \<rightarrow>* \<langle>throw a,s'\<rangle>"
unsigned_word_eqI: fixes v :: "'c word"   and w :: "'c word" assumes "unsigned v = unsigned w" shows "v = w"
simps_5: fixes uu :: "variable \<times> typ"   and uv :: "nat"   and v :: "nat" shows "bind_fv2 uu uv (Bv v) = Bv v"
rel_simps_70: shows "\<not> (0::'b) < (0::'b)"
assigned_slot_def: shows "assigned_slot \<equiv> l_find_slot\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_assigned_slot Shadow_DOM.get_parent get_shadow_root get_mode get_attribute get_tag_name Shadow_DOM.first_in_tree_order"
add: fixes f :: "'f \<Rightarrow> 'g"   and x :: "'f"   and y :: "'f" assumes "additive f" shows "f (x + y) = f x + f y"