hcju/isabelleps huggingface dataset:
until_unfold: fixes S :: "'a set"   and s0 :: "'a"   and \<delta> :: "'a \<Rightarrow> 'a set"   and AP :: "'b set"   and L :: "'a \<Rightarrow> 'b set"   and AP' :: "'b set"   and \<phi> :: "('a \<times> 'b set) stream list \<Rightarrow> bool"   and \<psi> :: "('a \<times> 'b set) stream list \<Rightarrow> bool"   and \<pi>l :: "('a \<times> 'b set) stream list" assumes "Shallow S s0 \<delta> AP L AP'" shows "Shallow.until \<phi> \<psi> \<pi>l = (\<psi> \<pi>l \<or> \<phi> \<pi>l \<and> Shallow.until \<phi> \<psi> (map stl \<pi>l))"
pgen_lossless_gpv_map_gpv_id: fixes fail :: "ennreal"   and \<I> :: "('a, 'b) \<I>"   and f :: "'d \<Rightarrow> 'c"   and gpv :: "('d, 'a, 'b) gpv" shows "pgen_lossless_gpv fail \<I> (map_gpv f id gpv) = pgen_lossless_gpv fail \<I> gpv"
normal_form_unique: fixes p :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'a"   and q :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'a" assumes "(punit.red (punit.reduced_GB F))\<^sup>*\<^sup>* p q"   and "\<not> punit.is_red (punit.reduced_GB F) q" shows "normal_form F p = q"
real_cond_exp_diff: fixes f :: "'a \<Rightarrow> real"   and g :: "'a \<Rightarrow> real" assumes "integrable M f"   and "integrable M g" shows "AE x in M. real_cond_exp M F (\<lambda>x. f x - g x) x = real_cond_exp M F f x - real_cond_exp M F g x"
norm_Pair: fixes a :: "'c"   and b :: "'d" shows "norm (a, b) = sqrt ((norm a)\<^sup>2 + (norm b)\<^sup>2)"
r_consist_upto_def: fixes r_psi :: "recf" shows "r_consist_upto r_psi \<equiv> Let (Cn 4 r_ifeq [Cn 4 r_psi [recf.Id 4 2, recf.Id 4 0], Cn 4 r_nth [recf.Id 4 3, recf.Id 4 0], recf.Id 4 1, r_constn 3 1]) (Pr 2 (r_constn 1 0))"
project_mono: fixes F :: "'c program"   and G :: "'c program"   and C :: "'c set" assumes "F \<le> G" shows "project h C F \<le> project h C G"
order_less_irrefl: fixes x :: "'e" shows "\<not> x < x"
empty_prefix_of_all: fixes l :: "'b list" shows "[] \<preceq> l"
same_genPrefix_genPrefix: fixes r :: "('a \<times> 'a) set"   and xs :: "'a list"   and ys :: "'a list"   and zs :: "'a list" assumes "refl r" shows "((xs @ ys, xs @ zs) \<in> genPrefix r) = ((ys, zs) \<in> genPrefix r)"
eq_iff: fixes a :: "nat"   and b :: "nat" shows "(a = b) = (a ddvd b)"
setops_subterm_trms: fixes t :: "('fun, 'b) Term.term"   and S :: "('fun, 'b) stateful_strand_step list"   and s :: "('fun, 'b) Term.term" assumes "t \<in> pair ` setops\<^sub>s\<^sub>s\<^sub>t S"   and "s \<sqsubset> t" shows "s \<sqsubseteq>\<^sub>s\<^sub>e\<^sub>t trms\<^sub>s\<^sub>s\<^sub>t S"
islimpt_subset: fixes x :: "'b"   and S :: "'b set"   and T :: "'b set" assumes "x islimpt S"   and "S \<subseteq> T" shows "x islimpt T"
assms: shows "low_equal s1 s2 \<and> t1 = snd (fst (div_write_new_val i result temp_V s1)) \<and> t2 = snd (fst (div_write_new_val i result temp_V s2))"
legs_form_cospan_1: shows "C.cospan \<nu>.leg0 \<pi>.leg1"
eq_onp_eqD: fixes P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b" assumes "eq_onp P = Q" shows "P x = Q x x"
simps_3: fixes C :: "char list"   and e :: "expr" shows "fv (\<lparr>C\<rparr>e) = fv e"
add_fract: fixes b :: "'c"   and d :: "'c"   and a :: "'c"   and c :: "'c" assumes "b \<noteq> (0::'c)"   and "d \<noteq> (0::'c)" shows "Fraction_Field.Fract a b + Fraction_Field.Fract c d = Fraction_Field.Fract (a * d + c * b) (b * d)"
U_def: shows "U \<equiv> \<lambda>P T. product_lists (map (term_variants P) T)"
cont_Done: fixes s :: "'state"   and i :: "nat" shows "local.cont Done s i = Done"
simps_3: fixes uu :: "('b \<Rightarrow> 'a) option" shows "Applicative.ap uu None = None"
c_snd_le_arg: fixes u :: "nat" shows "c_snd u \<le> u"
r_cancel: fixes y :: "'a"   and x :: "'a"   and z :: "'a" shows "(y ** x = z ** x) = (y = z)"
path_cons_elim: fixes A :: "('c, 'd) nba"   and a :: "'c \<times> 'd"   and r :: "('c \<times> 'd) list"   and p :: "'d" assumes "path A (a # r) p" obtains "fst a \<in> alphabet A \<and> snd a \<in> transition A (fst a) p" and "path A r (snd a)"
function_zero_def: fixes S :: "'d set"   and R :: "('e, 'f) ring_scheme" shows "function_zero S R = (\<lambda>x\<in>S. \<zero>\<^bsub>R\<^esub>)"
is_prefix_of_append: fixes p :: "'a list"   and a :: "'a list"   and b :: "'a list" assumes "is_prefix p (a @ b)" shows "is_prefix p a \<or> (\<exists>b'. b' \<noteq> [] \<and> is_prefix b' b \<and> p = a @ b')"
disjoint_UN: fixes I :: "'b set"   and F :: "'b \<Rightarrow> 'c set set" assumes "\<And>i. i \<in> I \<Longrightarrow> disjoint (F i)"   and "disjoint_family_on (\<lambda>i. \<Union> (F i)) I" shows "disjoint (\<Union> (F ` I))"
cases: fixes x :: "'a atom" obtains var :: "nat" and b :: "'a" where "x = Leq var__ b__"   | var :: "nat" and b :: "'a" where "x = Geq var b"
C2: fixes k :: "nat" shows "upper_asymptotic_density (C k) \<le> (\<Sum>n. upper_asymptotic_density (A n))"
rep_eq: fixes x :: "perm"   and xa :: "'c set['d]" shows "set_bset (x \<bullet> xa) = x \<bullet> set_bset xa"
pre_between_rotate: fixes vs :: "'a list"   and ram1 :: "'a"   and ram2 :: "'a"   and n :: "nat" assumes "pre_between vs ram1 ram2" shows "pre_between (rotate n vs) ram1 ram2"
assms_4: shows "\<forall>(k, \<gamma>)\<in>one_chain1. line_integral_exists F basis \<gamma>"
Shallow_Idle_axioms: shows "Shallow_Idle L AP' S s0 \<delta> AP f Sink"
lfp_lowerbound: fixes f :: "'b \<times> 'c \<Rightarrow> 'b \<times> 'c"   and A :: "'b \<times> 'c" assumes "less_eq_prod (f A) A" shows "less_eq_prod (prod.lfp f) A"
subst_mgt_comp: fixes \<theta> :: "'d \<Rightarrow> ('b, 'a) Term.term"   and \<delta> :: "'a \<Rightarrow> ('b, 'c) Term.term" shows "\<theta> \<preceq>\<^sub>\<circ> \<theta> \<circ>\<^sub>s \<delta>"
FunctorId: fixes X :: "'o1" assumes "X \<in> obj\<^bsub>CatDom F\<^esub>" shows "\<exists>Y\<in>obj\<^bsub>CatCod F\<^esub>. F ## id\<^bsub>CatDom F\<^esub> X = id\<^bsub>CatCod F\<^esub> Y"
hyp: fixes env :: "('a, 'b, 'c) env" assumes "lookup env xs = None" shows "update (xs @ y # ys) opt env = env"
Infinitesimal_mult_disj: fixes x :: "'b star"   and y :: "'b star" assumes "x * y \<in> Infinitesimal" shows "x \<in> Infinitesimal \<or> y \<in> Infinitesimal"
J_2: fixes n :: "nat" shows "J n \<subseteq> I"
right_minus: fixes a :: "'a"   and b :: "'a" assumes "a \<in> R"   and "b \<in> R" shows "a \<cdot> comm.additive.inverse b = comm.additive.inverse (a \<cdot> b)"
simps_1: fixes nat :: "nat"   and nat' :: "nat" shows "(enat nat = enat nat') = (nat = nat')"
ta: shows "T\<^sub>p_on_rand' BIT (type0 init x y) u = 15 / 10"
g_comp_def: fixes c_ls :: "nat"   and key :: "nat" shows "g_comp c_ls key = (let n = c_fst key; x = c_snd key; m = c_snd n; m1 = c_fst m; m2 = c_snd m in if c_assoc_have_key c_ls (c_pair m2 x) = 0 then let y = c_assoc_value c_ls (c_pair m2 x) in if c_assoc_have_key c_ls (c_pair m1 y) = 0 then let z = c_assoc_value c_ls (c_pair m1 y) in c_cons (c_pair key z) c_ls else c_ls else c_ls)"
f_inv_unique: fixes x :: "'a"   and x' :: "'a"   and x'' :: "'a" assumes "x \<in> carrier K - {\<zero>}"   and "x' \<in> carrier K"   and "x'' \<in> carrier K"   and "x' \<cdot>\<^sub>r x = 1\<^sub>r"   and "x'' \<cdot>\<^sub>r x = 1\<^sub>r" shows "x' = x''"
vector_smult_hom: fixes x :: "'a"   and v :: "('a, 'd) Finite_Cartesian_Product.vec" shows "hom x *s map_vector hom v = map_vector hom (x *s v)"
in_snapshot_def: fixes r :: "'g" shows "in_snapshot r = (\<lambda>s. black r s \<or> (\<exists>x. (x grey_protects_white r) s))"
f2: shows "pderiv p = q * pderiv (p div q) + p div q * pderiv q"
injectivity: fixes x :: "'a"   and y :: "'a" assumes "hom x = hom y" shows "x = y"
countable_lists: fixes A :: "'a set" assumes "countable A" shows "countable (lists A)"
star_if_stepsn: fixes cs :: "com \<times> (char list \<Rightarrow> int)"   and n :: "nat"   and cs' :: "com \<times> (char list \<Rightarrow> int)" assumes "cs \<rightarrow>(n) cs'" shows "cs \<rightarrow>* cs'"
divide_add_eq_iff: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and z :: "'a"   and x :: "'a"   and y :: "'a" assumes "class.division_ring divide inverse one times plus zero minus uminus"   and "z \<noteq> zero" shows "plus (divide x z) y = divide (plus x (times y z)) z"
FG_advice_\<mu>LTL_1: fixes \<phi> :: "'b ltln"   and X :: "'b ltln set" shows "\<phi>[X]\<^sub>\<mu> \<in> \<mu>LTL"
valid_regions_I_cong: fixes X :: "'d set"   and k :: "'d \<Rightarrow> nat"   and I :: "'d \<Rightarrow> intv"   and r :: "('d \<times> 'd) set"   and I' :: "'d \<Rightarrow> intv" assumes "valid_region X k I r"   and "\<forall>x\<in>X. I x = I' x" shows "region X I r = region X I' r \<and> valid_region X k I' r"
absolutely_integrable_scaleR_left: fixes S :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and c :: "real" assumes "f absolutely_integrable_on S" shows "(\<lambda>x. c *\<^sub>R f x) absolutely_integrable_on S"
set_map: fixes f :: "'b \<Rightarrow> 'a"   and v :: "'b tree" shows "set_tree (map_tree f v) = f ` set_tree v"
simps_15: shows "cospan_shape.comp cospan_shape.TT cospan_shape.AA = cospan_shape.Null"
star_rtc_least_eq: fixes x :: "'a list set"   and y :: "'a list set" assumes "p_one \<union> x \<union> p_prod y y = y" shows "p_star x \<subseteq> y"
induct: fixes P :: "('c, 'b) ipurge_rec \<Rightarrow> bool"   and a0 :: "('c, 'b) ipurge_rec" assumes "\<And>I D U xs X. P \<lparr>Pol = I, Map = D, Doms = U, List = xs, ListOp = Some [], Set = ipurge_ref_aux I D U [] X\<rparr> \<Longrightarrow> P \<lparr>Pol = I, Map = D, Doms = U, List = xs, ListOp = None, Set = X\<rparr>"   and "\<And>I D U x xs ys X. \<lbrakk>\<exists>u\<in>U. (u, D x) \<in> I \<Longrightarrow> P \<lparr>Pol = I, Map = D, Doms = U, List = ipurge_tr I D (D x) xs, ListOp = Some ys, Set = ipurge_ref I D (D x) xs X\<rparr>; \<not> (\<exists>u\<in>U. (u, D x) \<in> I) \<Longrightarrow> P \<lparr>Pol = I, Map = D, Doms = U, List = xs, ListOp = Some (ys @ [x]), Set = X\<rparr>\<rbrakk> \<Longrightarrow> P \<lparr>Pol = I, Map = D, Doms = U, List = x # xs, ListOp = Some ys, Set = X\<rparr>"   and "\<And>I D U ys X. P \<lparr>Pol = I, Map = D, Doms = U, List = [], ListOp = Some ys, Set = X\<rparr>" shows "P a0"
P: shows "{x \<in> space (M1 \<Otimes>\<^sub>M M2). P (fst x) (snd x)} \<in> sets (M1 \<Otimes>\<^sub>M M2)"
U1_def: shows "U1 = {q x |x. x \<in> - A}"
map_mat_def: fixes f :: "'d \<Rightarrow> 'c"   and A :: "'d mat" shows "map_mat f A \<equiv> mat (dim_row A) (dim_col A) (\<lambda>ij. f (A $$ ij))"
part_equivp_symp: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "part_equivp R"   and "R x y" shows "R y x"
p: shows "p = F * (1 - fps_X) ^ (k + 1)"
cases: fixes x :: "'addr val \<times> 'addr val" obtains i1 :: "32 word" and i2 :: "32 word" where "x = (Intg i1, Intg i2)"   | v2 :: "'addr val" where "x = (Unit, v2a__)"   | v2 :: "'addr val" where "x = (Null, v2b__)"   | v :: "bool" and v2 :: "'addr val" where "x = (Bool v__, v2c__)"   | v :: "'addr" and v2 :: "'addr val" where "x = (Addr vb__, v2)"   | v1 :: "'addr val" where "x = (v1a__, Unit)"   | v1 :: "'addr val" where "x = (v1b__, Null)"   | v1 :: "'addr val" and v :: "bool" where "x = (v1c__, Bool vc__)"   | v1 :: "'addr val" and v :: "'addr" where "x = (v1, Addr v)"
rbt_delete: fixes t :: "('b \<times> color) tree"   and x :: "'b" assumes "rbt t" shows "rbt (delete x t)"
upset_anti_iff: fixes dual :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and y :: "'b"   and x :: "'b" assumes "class.preorder_with_dual dual less_eq less" shows "(ord.upset less_eq y \<subseteq> ord.upset less_eq x) = less_eq x y"
takeWhile_dropWhile_id: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b list" shows "takeWhile P xs \<cdot> dropWhile P xs = xs"
same_cs_if_not_recv: 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 cid :: "nat" assumes "distributed_system channel"   and "distributed_system.next channel trans send recv c ev c'"   and "\<not> isRecv ev" shows "fst (cs c cid) = fst (cs c' cid)"
dfs_algo_def: fixes \<Sigma>i :: "'a set"   and R :: "('a \<times> 'a) set" shows "dfs_algo \<Sigma>i R \<equiv> \<lparr>wa_cond = dfs_cond, wa_step = dfs_step R, wa_initial = dfs_initial \<Sigma>i, wa_invar = dfs_invar \<Sigma>i R\<rparr>"
list_of_vec_index: fixes v :: "'c vec"   and j :: "nat" shows "list_of_vec v ! j = v $ j"
fiX_lbound: fixes f :: "nat \<Rightarrow> real"   and p :: "nat"   and l :: "nat" shows "f p - \<Delta> \<le> fiX f p l"
V_def: shows "V = (\<lambda>y. if y \<in> J then \<Inter> {U i |i. i \<in> I \<and> x i = y} else UNIV)"
lt: fixes i :: "'n" shows "0 < (A1n *v y) $h i"
assms_4: shows "\<forall>n\<in>V. even (degree n G)"
SUP_image: fixes g :: "'d \<Rightarrow> 'b \<times> 'c"   and f :: "'e \<Rightarrow> 'd"   and A :: "'e set" shows "Sup_prod (g ` f ` A) = Sup_prod ((g \<circ> f) ` A)"
Aexp_subst: fixes Aexp :: "exp \<Rightarrow> Arity \<rightarrow> Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and e :: "exp"   and y :: "Vars.var"   and x :: "Vars.var"   and a :: "Arity" assumes "SubstArityAnalysis Aexp" shows "Aexp e[y::=x]\<cdot>a \<sqsubseteq> env_delete y (Aexp e\<cdot>a) \<squnion> esing x\<cdot>(up\<cdot>0)"
intro: fixes hom :: "'b \<Rightarrow> 'c" assumes "idom_hom hom"   and "zero_hom_0 hom" shows "inj_idom_hom hom"
take_equalityI: fixes xs :: "'d list"   and ys :: "'d list" assumes "\<forall>i. take i xs = take i ys" shows "xs = ys"
simps_28: fixes f1 :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> k \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'a"   and f2 :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'a"   and f3 :: "(nat \<Rightarrow> nat option) \<Rightarrow> nat \<Rightarrow> 'a"   and f4 :: "nat \<Rightarrow> nat \<Rightarrow> 'a"   and f5 :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'a"   and x31 :: "nat \<Rightarrow> nat option"   and x32 :: "nat" shows "(case Rerr x31 x32 of Rreq x xa xb xc xd xe xf xg \<Rightarrow> f1 x xa xb xc xd xe xf xg | Rrep x xa xb xc xd \<Rightarrow> f2 x xa xb xc xd | Rerr x xa \<Rightarrow> f3 x xa | Newpkt x xa \<Rightarrow> f4 x xa | Pkt x xa xb \<Rightarrow> f5 x xa xb) = f3 x31 x32"
unat_of_nat_ctz_mw: fixes w :: "'b word" shows "unat (word_of_nat (word_ctz w)) = word_ctz w"
update_simps_23: fixes T :: "(nat \<times> linear_poly) list"   and s :: "('a, 'b) state" shows "\<U> (\<T>_update T s) = \<U> s"
h4: shows "[p = x] (mod 4) \<and> x < 4"
POT_def: fixes to :: "'b nat_term_order" shows "POT to = Abs_nat_term_order (pot_comp (nat_term_compare to))"
cong: fixes g :: "'d \<Rightarrow> 'a"   and h :: "'d \<Rightarrow> 'a" assumes "\<And>a. g a = h a" shows "G g = G h"
fbrelation: shows "fpairs (\<D>\<^sub>\<circ> f) = \<D>\<^sub>\<circ> f"
assms: shows "h \<turnstile> get_tag_name element_ptr \<rightarrow>\<^sub>r children"
set_intros_1: fixes x1 :: "'a" shows "x1 \<in> atoms (Atom x1)"
permute_minus_cancel_1: fixes permute :: "perm \<Rightarrow> 'c \<Rightarrow> 'c"   and p :: "perm"   and x :: "'c" assumes "class.pt permute" shows "permute p (permute (- p) x) = x"
fps_one_to_fls: shows "fps_to_fls 1 = 1"
concat_bit_take_bit_eq: fixes n :: "nat"   and b :: "int" shows "concat_bit n (take_bit n b) = concat_bit n b"
nfa_reach_rtrancl: fixes i :: "bool list"   and j :: "bool list" assumes "nfa_is_node A i" shows "(\<exists>bss. nfa_reach A i bss j \<and> list_all (is_alph n) bss) = ((i, j) \<in> (succsr (subset_succs A))\<^sup>*)"
modifies_def: fixes vars :: "char list set"   and s\<^sub>1 :: "char list \<Rightarrow> int \<Rightarrow> int"   and s\<^sub>2 :: "char list \<Rightarrow> int \<Rightarrow> int" shows "modifies vars s\<^sub>1 s\<^sub>2 = (\<forall>x. x \<notin> vars \<longrightarrow> s\<^sub>1 x = s\<^sub>2 x)"
lam_eval_prod: fixes a :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and b :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and c :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and h :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "ide a"   and "ide b"   and "ide c"   and "\<guillemotleft>h : a \<rightarrow> exp b c\<guillemotright>" shows "\<Lambda> a b c (local.comp (eval b c) (HF'.prod h b)) = h"
l_set_mode_get_shadow_root_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and set_mode :: "(_) shadow_root_ptr \<Rightarrow> shadow_root_mode \<Rightarrow> ((_) heap, exception, unit) prog"   and set_mode_locs :: "(_) shadow_root_ptr \<Rightarrow> ((_) heap, exception, unit) prog 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" shows "l_set_mode_get_shadow_root type_wf set_mode set_mode_locs get_shadow_root get_shadow_root_locs \<equiv> l_set_mode type_wf set_mode set_mode_locs \<and> l_get_shadow_root type_wf get_shadow_root get_shadow_root_locs \<and> l_set_mode_get_shadow_root_axioms set_mode_locs get_shadow_root_locs"
unstream_remove1_trans_False: fixes b :: "'e"   and g :: "('e, 'f) generator"   and s :: "'f" shows "unstream (remove1_trans b g) (False, s) = unstream g s"
leg1_simps_4: shows "trg (tab\<^sub>1 h) = trg h"
case_1: fixes f1 :: "'l"   and f2 :: "'m \<Rightarrow> 'l" shows "(case None of None \<Rightarrow> f1 | Some x \<Rightarrow> f2 x) = f1"
id2: shows "map_of (concat l) x = map_of (l ! bounded_hashcode_nat (length l) x) x"
sdom_cons_v: fixes v :: "exp"   and x :: "nat"   and \<rho> :: "(nat \<times> exp) list" assumes "FV v = {}" shows "sdom ((x, v) # \<rho>) = insert x (sdom \<rho>)"
simps_2: fixes callee :: "'e \<Rightarrow> 'f \<Rightarrow> ('d \<times> 'e) spmf"   and p :: "'e spmf"   and a :: "'f"   and b :: "'d"   and xs :: "('f \<times> 'd) list"   and x :: "'f" shows "trace_callee callee p ((a, b) # xs) x = trace_callee callee (cond_spmf_fst (p \<bind> (\<lambda>s. callee s a)) b) xs x"
Domain_Int_subset: fixes A :: "('e \<times> 'f) set"   and B :: "('e \<times> 'f) set" shows "Domain (A \<inter> B) \<subseteq> Domain A \<inter> Domain B"
assms: fixes x :: "'a" assumes "x \<in> keys t" shows "f x = monomial (1::'b) (monomial 1 x)"
fun_prod_commute: fixes f :: "'c \<Rightarrow> 'b"   and g :: "'d \<Rightarrow> 'c"   and A :: "'d set" assumes "f (1::'c) = (1::'b)"   and "\<And>x y. f (x * y) = f x * f y" shows "f (prod g A) = (\<Prod>a\<in>A. f (g a))"
ex_points_anga: fixes a :: "'p \<Rightarrow> 'p \<Rightarrow> 'p \<Rightarrow> bool" assumes "QCongAACute a" shows "\<exists>A B C. a A B C"
dim2: shows "\<forall>n. matrix_sum d (\<lambda>n. denote_while_n (x1 0) (x1 1) (denote x2) n \<rho>\<^sub>2) n \<in> carrier_mat d d"
eq: fixes i :: "real"   and j :: "real" assumes "i \<in> I"   and "j \<in> I" shows "ind_ring_morphisms (W i) (W i \<inter> W j) (s i) = ind_ring_morphisms (W j) (W i \<inter> W j) (s j)"
lub_unique: fixes cl :: "'b potype"   and S :: "'b set"   and x :: "'b"   and L :: "'b" assumes "CL cl"   and "S \<subseteq> pset cl"   and "isLub S cl x"   and "isLub S cl L" shows "x = L"
Scons_inject_lemma1: fixes M :: "('a, 'b) node set"   and N :: "('a, 'b) node set"   and M' :: "('a, 'b) node set"   and N' :: "('a, 'b) node set" assumes "Scons M N \<subseteq> Scons M' N'" shows "M \<subseteq> M'"
coprime_def': fixes p :: "'a"   and q :: "'a" shows "coprime p q \<equiv> \<forall>r. r dvd p \<longrightarrow> r dvd q \<longrightarrow> is_unit r"
DDH_bound1: fixes \<G> :: "'b cyclic_group"   and \<A> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool spmf" assumes "ddh_ext \<G>" shows "\<bar>spmf (ddh.DDH0 \<G> \<A>) True - spmf (ddh.DDH1 \<G> \<A>) True\<bar> \<le> \<bar>spmf (ddh.ddh_0 \<G> \<A>) True - spmf (ddh.ddh_1 \<G> \<A>) True\<bar> + \<bar>spmf (ddh.ddh_1 \<G> \<A>) True - spmf (ddh.DDH1 \<G> \<A>) True\<bar>"
conj_one_right: fixes x :: "'a" shows "x \<^bold>\<squnion> \<zero> = x"
Check_check: fixes rhs_eq_val :: "(nat, 'a) mapping \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> nat \<times> linear_poly \<Rightarrow> 'a" assumes "RhsEqVal rhs_eq_val" shows "Check check"
collinear_def: fixes z1 :: "complex"   and z2 :: "complex"   and z3 :: "complex" shows "collinear z1 z2 z3 = (z1 = z2 \<or> is_real ((z3 - z1) / (z2 - z1)))"
f: fixes i :: "'c" assumes "i \<in> I" shows "f i \<in> S i"
log_of_power_le: fixes m :: "nat"   and b :: "real"   and n :: "nat" assumes "real m \<le> b ^ n"   and "1 < b"   and "0 < m" shows "log b (real m) \<le> real n"
continuous_map_from_subtopology: fixes X :: "'c topology"   and X' :: "'d topology"   and f :: "'c \<Rightarrow> 'd"   and S :: "'c set" assumes "continuous_map X X' f" shows "continuous_map (subtopology X S) X' f"
ORLEFT_eqvt_coname: fixes pi :: "(coname \<times> coname) list"   and A :: "ty"   and B :: "ty"   and X :: "ntrm set"   and Y :: "ntrm set" shows "pi \<bullet> ORLEFT (A OR B) X Y = ORLEFT (A OR B) (pi \<bullet> X) (pi \<bullet> Y)"
inv_comm: fixes x :: "'a"   and y :: "'a" assumes "x ** y = one" shows "y ** x = one"
sbh_step_prog_configs_eq: fixes "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "'a memref list"   and m :: "nat \<Rightarrow> nat"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option"   and \<S> :: "nat \<Rightarrow> bool option"   and is' :: "instr list"   and \<theta>' :: "nat \<Rightarrow> nat option"   and sb' :: "'a memref list"   and m' :: "nat \<Rightarrow> nat"   and \<D>' :: "bool"   and \<O>' :: "nat set"   and \<R>' :: "nat \<Rightarrow> bool option"   and \<S>' :: "nat \<Rightarrow> bool option" assumes "(is, \<theta>, sb, m, \<D>, \<O>, \<R>, \<S>) \<rightarrow>\<^sub>s\<^sub>b\<^sub>h (is', \<theta>', sb', m', \<D>', \<O>', \<R>', \<S>')" shows "prog_configs sb' = prog_configs sb"
lhd_conv_lnth: fixes xs :: "'b llist" assumes "\<not> lnull xs" shows "lhd xs = lnth xs 0"
zminimal: shows "index (fst (TSdet init h (as @ [x] @ bs @ [x] @ cs) (length (as @ [x] @ bs @ [x]) + i))) z = Min (index (fst (TSdet init h (as @ [x] @ bs @ [x] @ cs) (length (as @ [x] @ bs @ [x]) + i))) ` {xa. xa < (as @ [x] @ bs @ [x] @ cs) ! (length (as @ [x] @ bs @ [x]) + i) in fst (TSdet init h (as @ [x] @ bs @ [x] @ cs) (length (as @ [x] @ bs @ [x]) + i)) \<and> count_list (take (index (snd (TSdet init h (as @ [x] @ bs @ [x] @ cs) (length (as @ [x] @ bs @ [x]) + i))) ((as @ [x] @ bs @ [x] @ cs) ! (length (as @ [x] @ bs @ [x]) + i))) (snd (TSdet init h (as @ [x] @ bs @ [x] @ cs) (length (as @ [x] @ bs @ [x]) + i)))) xa \<le> 1})"
equiv_range_permutation: fixes A :: "'c set"   and B :: "'d set" shows "equiv (A \<rightarrow>\<^sub>E B) (range_permutation A B)"
subprob_measure_le_1: fixes s :: "'s"   and X :: "'s stream set" shows "T.prob s X \<le> 1"
build_psimp_2: fixes k :: "'b"   and s :: "real"   and ks :: "'b list"   and x :: "(real, 'b) vec"   and y :: "(real, 'b) vec"   and zs :: "(real, 'b) vec list"   and l :: "(real, 'b) vec list"   and m :: "real"   and r :: "(real, 'b) vec list" assumes "(k, s) = widest_spread ks (x # y # zs)"   and "(l, m, r) = partition_by_median k (x # y # zs)"   and "build_dom (ks, l)"   and "build_dom (ks, r)" shows "build ks (x # y # zs) = Node k m (build ks l) (build ks r)"
natural_transformation_axioms: shows "natural_transformation \<F>\<^sub>SC.comp \<F>\<^sub>SC.comp DoS.map DoS.map DoS.map"
tfilter'_code_2: fixes b' :: "unit \<Rightarrow> 'b"   and P :: "'a \<Rightarrow> bool"   and a :: "'a"   and tr :: "('a, 'b) tllist" shows "TLList.tfilter' b' P (TCons a tr) = (if P a then TCons a (TLList.tfilter' b' P tr) else TLList.tfilter' b' P tr)"
pt_abs_fun_inst: assumes "pt TYPE('a) TYPE('b)"   and "at TYPE('b)" shows "pt TYPE('b \<Rightarrow> 'a noption) TYPE('b)"