hcju/isabelleps huggingface dataset:
cp\<^sub>P\<^sub>R\<^sub>E: fixes self :: "'\<AA> state \<times> '\<AA> state \<Rightarrow> '\<alpha>0"   and a1 :: "'\<AA> state \<times> '\<AA> state \<Rightarrow> '\<alpha>1"   and a2 :: "'\<AA> state \<times> '\<AA> state \<Rightarrow> '\<alpha>2"   and a3 :: "'\<AA> state \<times> '\<AA> state \<Rightarrow> '\<alpha>3"   and \<tau> :: "'\<AA> state \<times> '\<AA> state" shows "PRE self a1 a2 a3 \<tau> = PRE (\<lambda>_. self \<tau>) (\<lambda>_. a1 \<tau>) (\<lambda>_. a2 \<tau>) (\<lambda>_. a3 \<tau>) \<tau>"
q_inter: fixes Q :: "'a set"   and Q' :: "'a set" assumes "proj.quorum Q"   and "proj.quorum Q'"   and "Q \<inter> I \<noteq> {}"   and "Q' \<inter> I \<noteq> {}" shows "Q \<inter> Q' \<inter> I \<noteq> {}"
topology_eq: fixes T1 :: "'d topology"   and T2 :: "'d topology" shows "(T1 = T2) = (\<forall>S. openin T1 S = openin T2 S)"
minus_eq_enat_iff: fixes a :: "enat"   and k :: "nat"   and i :: "nat" shows "(a - enat k = enat i) = (\<exists>j. a = enat j \<and> j - k = i)"
sectionI: fixes g :: "'s"   and f :: "'s" assumes "ide (g \<cdot> f)" shows "section f"
updown_def: fixes dm :: "nat"   and lm :: "nat"   and \<alpha> :: "(nat \<times> int) list \<Rightarrow> real" shows "updown dm lm \<alpha> = updown' dm lm dm \<alpha>"
bheight_baliR: fixes l :: "('b \<times> color) tree"   and r :: "('b \<times> color) tree"   and a :: "'b" assumes "bheight l = bheight r" shows "bheight (baliR l a r) = Suc (bheight l)"
times_diff_plus_sym_hmset: fixes a :: "hmultiset"   and c :: "hmultiset"   and b :: "hmultiset" shows "a * (c - b) + a * b = a * (b - c) + a * c"
C0_prems_i: shows "(C0, active) \<in> set (prems_of \<iota>)"
minGraphProps11: fixes g :: "graph" assumes "minGraphProps g" shows "distinct (normFaces (faces g))"
bigo_mult3: fixes f :: "'c \<Rightarrow> 'd"   and h :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd"   and j :: "'c \<Rightarrow> 'd" assumes "f \<in> O(h)"   and "g \<in> O(j)" shows "f * g \<in> O(h * j)"
R'_2: shows "set R' \<subseteq> set (tl R'_walk)"
axioms_2: fixes I :: "'c set"   and R :: "'c set"   and addition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and multiplication :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and unit :: "'c" assumes "lideal I R addition multiplication zero unit" shows "lideal_axioms I R multiplication"
split_conv: fixes f :: "'d \<Rightarrow> 'e \<Rightarrow> 'b"   and a :: "'d"   and b :: "'e" shows "(case (a, b) of (c, d) \<Rightarrow> f c d) = f a b"
power_inverse: 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 a :: "'a"   and n :: "nat" assumes "class.division_ring divide inverse one times plus zero minus uminus" shows "power.power one times (inverse a) n = inverse (power.power one times a n)"
symmetric_not_admissable: assumes "\<Lambda> * (\<v> - 1) \<noteq> \<k> * (\<k> - 1)" shows "\<not> symmetric_bibd \<V> \<B> \<k> \<Lambda>"
select_convs_6: fixes igWls :: "'f \<Rightarrow> 'c \<Rightarrow> bool"   and igWlsAbs :: "'a \<times> 'f \<Rightarrow> 'h \<Rightarrow> bool"   and igVar :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and igAbs :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'h"   and igOp :: "'g \<Rightarrow> ('d \<Rightarrow> 'c option) \<Rightarrow> ('e \<Rightarrow> 'h option) \<Rightarrow> 'c"   and igFresh :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and igFreshAbs :: "'a \<Rightarrow> 'b \<Rightarrow> 'h \<Rightarrow> bool"   and igSwap :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'c"   and igSwapAbs :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'h \<Rightarrow> 'h"   and igSubst :: "'a \<Rightarrow> 'c \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'c"   and igSubstAbs :: "'a \<Rightarrow> 'c \<Rightarrow> 'b \<Rightarrow> 'h \<Rightarrow> 'h"   and more :: "'i" shows "igFresh \<lparr>igWls = igWls, igWlsAbs = igWlsAbs, igVar = igVar, igAbs = igAbs, igOp = igOp, igFresh = igFresh, igFreshAbs = igFreshAbs, igSwap = igSwap, igSwapAbs = igSwapAbs, igSubst = igSubst, igSubstAbs = igSubstAbs, \<dots> = more\<rparr> = igFresh"
distinct_1: fixes x1 :: "'a"   and x2 :: "'b blindable\<^sub>h" shows "Unblinded x1 \<noteq> Blinded x2"
fmdrop_idle: fixes x :: "'e"   and B :: "'e \<rightharpoonup>\<^sub>f 'f" assumes "x |\<notin>| fmdom B" shows "fmdrop x B = B"
lift_CFG: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> 'c edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and Def :: "'b \<Rightarrow> 'd set"   and Use :: "'b \<Rightarrow> 'd set"   and state_val :: "'c \<Rightarrow> 'd \<Rightarrow> 'e"   and Exit :: "'b" assumes "CFGExit_wf sourcenode targetnode kind valid_edge Entry Def Use state_val Exit" shows "CFG src trg (lift_valid_edge valid_edge sourcenode targetnode kind Entry Exit) NewEntry"
outputCases': fixes a :: "name"   and b :: "name"   and P :: "pi"   and Rs :: "residual" assumes "a{b}.P \<longmapsto> Rs" obtains aa :: "name" and ba :: "name" and P :: "pi" where "a = aa \<and> b = ba \<and> Pb__ = P" and "Rs = aa[ba] \<prec> P"
simps_4: fixes uu :: "'b \<Rightarrow> 'b list \<Rightarrow> bool"   and uv :: "'b"   and uw :: "'b list" shows "uv # uw \<simeq> {[], [], uu} = False"
11: shows "set_dom (F \<^bsub>\<a>\<^esub> g) = F \<^bsub>\<o>\<^esub> A"
rrep_prrep: shows "paodv i \<TTurnstile> onl \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V (\<lambda>(\<xi>, l). l \<in> {PRrep-:2..PRrep-:7} \<longrightarrow> dip \<xi> \<in> kD (rt \<xi>) \<and> sqn (rt \<xi>) (dip \<xi>) = dsn \<xi> \<and> the (dhops (rt \<xi>) (dip \<xi>)) = Suc (hops \<xi>) \<and> the (flag (rt \<xi>) (dip \<xi>)) = val \<and> the (nhop (rt \<xi>) (dip \<xi>)) \<in> kD (rt \<xi>))"
hta_lookup_f_def: fixes f :: "'b"   and H :: "('a, 'b, 'c) hashedTa_scheme" shows "hta_lookup_f f H \<equiv> hll_idx.lookup f (the (hta_idx_f H))"
rec_pi_def: shows "rec_pi = Cn 1 rec_dummy_pi [recf.id 1 0, recf.id 1 0]"
atLeastLessThan_rev_at_least_Suc_atMost: fixes g :: "nat \<Rightarrow> 'b"   and n :: "nat"   and m :: "nat" shows "prod g {n..<m} = (\<Prod>i = Suc n..m. g (m + n - i))"
hd_last_sorted: fixes x :: "real \<times> real"   and xs :: "(real \<times> real) list" assumes "ccw.sortedP x xs"   and "1 < length xs" shows "ccw x (hd xs) (last xs)"
inj_on_add': fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and A :: "'a set" assumes "class.cancel_semigroup_add plus" shows "inj_on (\<lambda>b. plus b a) A"
OUTfromVCorrect1_data21: shows "OUTfromVCorrect1 data21"
all_squarefree: shows "\<forall>q. List.member (fst (factorize_polys qs)) q \<longrightarrow> rsquarefree q"
runit_naturality: fixes f :: "'a" assumes "arr f" shows "\<r>\<^sub>1[cod f] \<cdot> (f \<otimes>\<^sub>1 \<I>\<^sub>1) = f \<cdot> \<r>\<^sub>1[local.dom f]"
sel_1: fixes xs :: "'c llist"   and P :: "'c \<Rightarrow> bool" assumes "\<not> lnull xs"   and "P (lhd xs)" shows "lhd (ltakeWhile P xs) = lhd xs"
Rep_monom_inverse: fixes x :: "'c monom" shows "Abs_monom (Rep_monom x) = x"
real_of_def: fixes x :: "complex" shows "real_of x = eucl_of_list [Re x, Im x]"
P1: shows "Bet B A A0 \<and> Cong A A0 B'' A'' \<and> Bet B C C0 \<and> Cong C C0 B'' C'' \<and> Bet B'' A'' A2 \<and> Cong A'' A2 B A \<and> Bet B'' C'' C2 \<and> Cong C'' C2 B C \<and> Cong A0 C0 A2 C2"
d: shows "singular_chain p (subtopology X (topspace X \<inter> S - topspace X \<inter> U)) d"
no_renegotiation_union: fixes x :: "'a"   and h :: "'c"   and X :: "'a set"   and Y :: "'a set" assumes "Xd x \<notin> Xd ` Ch h X"   and "x \<in> X" shows "x \<notin> Ch h (X \<union> Y)"
restrict0_def: fixes A :: "'d set"   and f :: "'d \<Rightarrow> 'c"   and x :: "'d" shows "restrict0 A f x = (if x \<in> A then f x else (0::'c))"
integrable_max: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> real"   and g :: "'c \<Rightarrow> real" assumes "integrable M f"   and "integrable M g" shows "integrable M (\<lambda>x. max (f x) (g x))"
inf_img_fin_dom': fixes f :: "'b \<Rightarrow> 'a"   and A :: "'b set" assumes "finite (f ` A)"   and "infinite A" shows "\<exists>y\<in>f ` A. infinite (f -` {y} \<inter> A)"
ccw'_not_coll_4: fixes a :: "real \<times> real"   and b :: "real \<times> real"   and c :: "real \<times> real" assumes "ccw' a b c" shows "det3 b c a \<noteq> 0"
word_add_increasing: fixes p :: "'b word"   and w :: "'b word"   and x :: "'b word" assumes "p + w \<le> x"   and "p \<le> p + w" shows "p \<le> x"
FL_bisimilarity_implies_equivalence_Act: fixes L :: "'b \<times> 'c fs_set \<times> 'c \<Rightarrow> 'c fs_set"   and satisfies :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and transition :: "'d \<Rightarrow> ('b, 'd) residual \<Rightarrow> bool"   and effect_apply :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and f :: "'c"   and F :: "'c fs_set"   and \<alpha> :: "'b"   and x :: "('f, 'e, 'b, 'c) formula"   and P :: "'d"   and Q :: "'d" assumes "indexed_effect_nominal_ts TYPE('a) L satisfies transition effect_apply"   and "f \<in>\<^sub>f\<^sub>s F"   and "bn \<alpha> \<sharp>* (F, f)"   and "effect_nominal_ts.is_FL_formula L (L (\<alpha>, F, f)) x"   and "\<And>P Q. effect_nominal_ts.FL_bisimilar satisfies transition effect_apply L (L (\<alpha>, F, f)) P Q \<Longrightarrow> effect_nominal_ts.FL_valid satisfies transition effect_apply P x = effect_nominal_ts.FL_valid satisfies transition effect_apply Q x"   and "effect_nominal_ts.FL_bisimilar satisfies transition effect_apply L F P Q"   and "effect_nominal_ts.FL_valid satisfies transition effect_apply P (Act f \<alpha> x)" shows "effect_nominal_ts.FL_valid satisfies transition effect_apply Q (Act f \<alpha> x)"
number_partition_of_def: fixes A :: "'c set"   and B :: "'d set"   and F :: "('c \<Rightarrow> 'd) set" shows "number_partition_of A B F = univ (\<lambda>f. image_mset card (mset_set ((\<lambda>b. {x \<in> A. f x = b}) ` B - {{}}))) F"
INF_cong: fixes A :: "'c set"   and B :: "'c set"   and C :: "'c \<Rightarrow> 'd"   and D :: "'c \<Rightarrow> 'd"   and Inf :: "'d set \<Rightarrow> 'd" assumes "A = B"   and "\<And>x. s2p B x \<Longrightarrow> C x = D x" shows "Inf (\<P> C A) = Inf (\<P> D B)"
hf_to_ide_ide_to_hf: fixes a :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "a \<in> Collect ide" shows "hf_to_ide (ide_to_hf a) = a"
l_def: shows "l \<equiv> liminf (\<lambda>n. ereal (p n * f n / f (Suc n) - p (Suc n)))"
ub_ln2: fixes prec :: "nat" shows "ln 2 \<le> real_of_float (ub_ln2 prec)"
union_single_eq_member: fixes x :: "'a"   and M :: "'a multiset"   and N :: "'a multiset" assumes "x, M = N" shows "x \<in># N"
phi_def: shows "phi \<equiv> init \<and> \<box>[act]_x"
consec_vn_v2: shows "consec (lan vn) (lan v2)"
asIMOD_igSubstIGOp: fixes SEM :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'b) semDom" shows "igSubstIGOp (asIMOD SEM)"
val_type_eq_BOOL: fixes x :: "val" shows "(val_type x = BOOL) = (x \<in> range BoolVal)"
map_map: fixes f :: "'e \<Rightarrow> 'd"   and g :: "'f \<Rightarrow> 'e"   and xs :: "'f list" shows "map f (map g xs) = map (f \<circ> g) xs"
prefix_of_take_prefix: fixes i :: "nat"   and \<sigma> :: "'a trace" shows "prefix_of (take_prefix i \<sigma>) \<sigma>"
abs_eq: fixes x :: "'a \<times> 'a scheduler" shows "action (abs_cfg x) = (case x of (s, x) \<Rightarrow> action_sch x)"
valid_pub_OR: shows "(x0, x1) \<in> valid_pub_OR"
hcomp_simps_3: fixes \<nu> :: "'a arrow_of_spans_data"   and \<mu> :: "'a arrow_of_spans_data" assumes "hseq \<nu> \<mu>" shows "local.dom (\<nu> \<star> \<mu>) = local.dom \<nu> \<star> local.dom \<mu>"
Cauchy_uniform_iff: fixes X :: "nat \<Rightarrow> 'a" shows "Cauchy X = (\<forall>P. eventually P uniformity \<longrightarrow> (\<exists>N. \<forall>n\<ge>N. \<forall>m\<ge>N. P (X n, X m)))"
eq_idx_for_lvar: fixes x :: "nat"   and t :: "(nat \<times> linear_poly) list" assumes "x \<in> lvars t" shows "eq_idx_for_lvar t x < length t \<and> lhs (t ! eq_idx_for_lvar t x) = x"
assms_2: shows "xvec \<sharp>* \<Psi>"
deq: shows "degree ([a, 1::'a] %^ Suc n *** p) = degree ([a, 1::'a] %^ n *** ([a, 1::'a] *** p))"
less_eq_pratt2_def: fixes x :: "pratt2"   and y :: "pratt2" shows "x \<le> y \<equiv> x + y = y"
h_4: shows "{x--z} = h ` {0..dist x z}"
regularAmicPair_sym: fixes n :: "nat"   and m :: "nat" assumes "regularAmicPair n m" shows "regularAmicPair m n"
segment_step_2D: fixes x :: "'b"   and f :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "x \<in> segment f a (f b)" shows "x \<in> segment f a b \<or> x = b"
y_def: shows "y = 1 / (2 * x + 1)"
not_sdistinct_decomp: fixes w :: "'b stream" assumes "\<not> sdistinct w" obtains u :: "'b list" and v :: "'b list" and a :: "'b" and w' :: "'b stream" where "w = u @- a ## v @- a ## w'"
zero_add2: fixes x :: "'b" shows "total_partial.add2 (Some (0::'b)) (Some x) = Some x"
parallel_converg_either: fixes i :: "nat"   and x :: "nat"   and j :: "nat" shows "(\<phi> i x \<down> \<or> \<phi> j x \<down>) = (parallel i j x \<down>)"
remdups_adj_append: fixes xs\<^sub>1 :: "'g list"   and x :: "'g"   and xs\<^sub>2 :: "'g list" shows "remdups_adj (xs\<^sub>1 @ x # xs\<^sub>2) = remdups_adj (xs\<^sub>1 @ [x]) @ tl (remdups_adj (x # xs\<^sub>2))"
fresh_FDERIV_floatarith: fixes ode_e :: "floatarith"   and i :: "nat"   and ds :: "floatarith list" assumes "fresh_floatarith ode_e i"   and "fresh_floatariths ds i"   and "length ds = DIM('a)" shows "fresh_floatarith (FDERIV_floatarith ode_e [0..<DIM('a)] ds) i"
set_iterator_genord_I: fixes l0 :: "'a list"   and S :: "'a set"   and R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and iti :: "('b \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'b" assumes "distinct l0"   and "S = set l0"   and "sorted_wrt R l0"   and "iti = foldli l0" shows "set_iterator_genord iti S R"
process_prop_5_def: fixes P :: "('e list \<times> 'e set) set \<times> 'e list set" shows "process_prop_5 P \<equiv> \<forall>xs x. xs \<in> snd P \<longrightarrow> xs @ [x] \<in> snd P"
Hadjicostas_Chapman_formula_aux: assumes "- 2 < Re (- 1)" shows "Hadjicostas_integral (- 1) = Gamma (- 1 + 2) * pre_zeta 1 (- 1 + 2)"
a_pos_i: fixes m :: "int" assumes "0 < m" shows "ant m * \<infinity> = \<infinity>"
le_infE: fixes inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and a :: "'b"   and b :: "'b" assumes "class.semilattice_inf inf less_eq less"   and "less_eq x (inf a b)" obtains "less_eq x a" and "less_eq x b"
carrier_submanifold: shows "sub.carrier = U x \<inter> src.carrier"
fv_delete_heap: fixes \<Gamma> :: "(Vars.var \<times> 'a) list"   and x :: "Vars.var"   and e :: "'a" assumes "map_of \<Gamma> x = Some e" shows "fv (delete x \<Gamma>, e) \<union> {x} \<subseteq> fv (\<Gamma>, Var x)"
cover_3: fixes i :: "nat" shows "emeasure (density M f) (A i) \<noteq> \<infinity>"
bfs_correct: fixes src :: "nat"   and dst :: "nat" assumes "src \<in> V"   and "finite V" shows "bfs src dst \<le> (spec x. case x of None \<Rightarrow> \<not> local.connected src dst | Some p \<Rightarrow> isShortestPath src p dst)"
distinct_25: fixes x4 :: "'b val"   and x5 :: "'b val" shows "val.Right x4 \<noteq> val.Left x5"
ipresIGSubstAbs_def: fixes h :: "'c \<Rightarrow> 'd"   and hA :: "'e \<Rightarrow> 'f"   and MOD :: "('g, 'h, 'varSort, 'sort, 'i, 'j, 'c, 'e, 'k) model_scheme"   and MOD' :: "('l, 'm, 'varSort, 'n, 'o, 'j, 'd, 'f, 'p) model_scheme" shows "ipresIGSubstAbs h hA MOD MOD' \<equiv> \<forall>ys Y y us s A. igWls MOD (asSort ys) Y \<and> igWlsAbs MOD (us, s) A \<longrightarrow> hA (igSubstAbs MOD ys Y y A) = igSubstAbs MOD' ys (h Y) y (hA A)"
has_integral_on_superset: fixes f :: "'b \<Rightarrow> 'c"   and i :: "'c"   and S :: "'b set"   and T :: "'b set" assumes "(f has_integral i) S"   and "\<And>x. x \<notin> S \<Longrightarrow> f x = (0::'c)"   and "S \<subseteq> T" shows "(f has_integral i) T"
elims_3: fixes x :: "char list"   and xa :: "char list" assumes "\<not> internal_iface_name_match x xa" obtains i :: "char" and "is" :: "char list" where "x = ia__ # is__" and "xa = []" and "\<not> (ia__ = CHR ''+'' \<and> is__ = [])"   | uu_ :: "char" and uv_ :: "char list" where "x = []" and "xa = uu_ # uv_"   | i :: "char" and "is" :: "char list" and p_i :: "char" and p_is :: "char list" where "x = i # is" and "xa = p_i # p_is" and "\<not> (if i = CHR ''+'' \<and> is = [] then True else p_i = i \<and> internal_iface_name_match is p_is)"
sup_st_def: shows "(\<squnion>) \<equiv> map_fun rep_st (map_fun rep_st St) (map2_st_rep (\<squnion>))"
coreflexive_mult_closed: fixes x :: "'a"   and y :: "'a" assumes "coreflexive x"   and "coreflexive y" shows "coreflexive (x * y)"
simps: fixes args :: "Monitor.args"   and nt :: "nat"   and tp :: "nat"   and tss :: "nat queue"   and len :: "nat"   and maskL :: "bool list"   and maskR :: "bool list"   and a1_map :: "('a option list, nat) mapping"   and a2_map :: "(nat, ('a option list, nat + nat) mapping) mapping"   and "done" :: "'a option list set list"   and done_length :: "nat" shows "shift_mmuaux args nt (tp, tss, len, maskL, maskR, a1_map, a2_map, done, done_length) = Let (linearize (takeWhile_queue (\<lambda>t. enat t + right (args_ivl args) < enat nt) tss)) (foldl (\<lambda>aux _. eval_step_mmuaux aux) (tp, tss, len, maskL, maskR, a1_map, a2_map, done, done_length))"
swap_def: fixes p :: "'h \<times> 'g" shows "prod.swap p = (snd p, fst p)"
pre_FaceDiv_between2: fixes g' :: "graph"   and ram1 :: "nat"   and ram2 :: "nat"   and f :: "face" assumes "pre_splitFace g' ram1 ram2 f []" shows "between (vertices f) ram2 ram1 \<noteq> []"
hd_fusion_def: fixes g :: "('c, 'd) generator"   and s :: "'d" shows "hd_fusion g s = hd (list.unfoldr g s)"
remove_nth_len: fixes i :: "nat"   and xs :: "'c list" assumes "i < length xs" shows "length xs = Suc (length (remove_nth i xs))"
mark_guards_bseq: fixes f :: "'f"   and c1 :: "('d, 'e, 'g) com"   and c2 :: "('d, 'e, 'g) com" shows "mark_guards f (bseq c1 c2) = bseq (mark_guards f c1) (mark_guards f c2)"
sandwich_commutes: fixes A :: "'o"   and B :: "'o"   and C :: "'o"   and a :: "'a"   and f :: "'a" assumes "A \<in> Ob"   and "B \<in> Ob"   and "C \<in> Ob"   and "a \<in> F \<^bsub>\<o>\<^esub> A"   and "f \<in> Hom B C" shows "F \<^bsub>\<a>\<^esub> f \<odot> \<sigma>(A,a) B = \<sigma>(A,a) C \<odot> Hom(A,_) \<^bsub>\<a>\<^esub> f"
roots2: fixes c :: "complex poly"   and l :: "nat" assumes "(c, l) \<in> set cs"   and "k < l" shows "\<forall>x\<in>sphere 0 (1 / R). poly c x \<noteq> 0"
ideal_sum_mult: fixes R :: "('d, 'e) Ring_scheme"   and A :: "'d set"   and B :: "'d set" assumes "Ring R"   and "A \<subseteq> carrier R"   and "A \<noteq> {}"   and "ideal R B" shows "ideal R (sum_mult R A B)"
singletonI: fixes a :: "'g" shows "a \<in> {a}"
single_gen: fixes a :: "'a" assumes "ag_single_gen A a" shows "carrier A = {g. \<exists>n. g = n\<triangleright>a\<^bsub>A\<^esub>}"
degree_charpoly: fixes A :: "('b, 'c) sq_matrix" shows "degree (charpoly A) = CARD('c)"
n: shows "UP.dimension n (poly_of_const ` K) (carrier (K [X]))"
free_tv_mk_scheme: fixes t :: "typ" shows "free_tv (mk_scheme t) = free_tv t"
consistencyD: fixes f :: "'d set \<Rightarrow> 'd set"   and C :: "'d set"   and B :: "'d set" assumes "consistency f"   and "C \<subseteq> B"   and "f B \<subseteq> C" shows "f B = f C"
card_prod: shows "CARD('c \<times> 'd) = CARD('c) * CARD('d)"
pmod_degree: fixes K :: "'a list set"   and p :: "'a list list"   and q :: "'a list list" assumes "subfield K (K [X])"   and "p \<in> carrier (K [X]\<^bsub>K [X]\<^esub>)"   and "q \<in> carrier (K [X]\<^bsub>K [X]\<^esub>)"   and "q \<noteq> []" shows "UP.pmod p q = [] \<or> degree (UP.pmod p q) < degree q"
seqI': fixes f :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and a :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and b :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and g :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and c :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "\<guillemotleft>f : a \<rightarrow> b\<guillemotright>"   and "\<guillemotleft>g : b \<rightarrow> c\<guillemotright>" shows "seq g f"
infsetsum_nat': fixes f :: "nat \<Rightarrow> 'b" assumes "f abs_summable_on UNIV" shows "infsetsum f UNIV = (\<Sum>n. f n)"
flowD_finite_OUT: fixes f :: "'v \<times> 'v \<Rightarrow> ennreal"   and x :: "'v" assumes "flow \<Delta> f"   and "x \<noteq> source \<Delta>"   and "x \<noteq> sink \<Delta>" shows "d_OUT f x \<noteq> \<top>"
abs_diff_triangle_ineq: fixes abs :: "'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c"   and c :: "'c"   and d :: "'c" assumes "class.ordered_ab_group_add_abs abs plus zero minus uminus less_eq less" shows "less_eq (abs (minus (plus a b) (plus c d))) (plus (abs (minus a c)) (abs (minus b d)))"
hd_one: fixes n :: "nat" assumes "0 < n" shows "hd (nat_to_bv_helper n []) = \<one>"
acyclic_on_def: fixes U :: "'b set" shows "acyclic_on U = (\<lambda>r. \<forall>x\<in>U. (x, x) \<notin> on U r\<^sup>+)"
abbd: fixes a :: "real"   and b :: "real pdevs" assumes "(a, b) \<in> set as" shows "degree b \<le> degree_aforms as"
nsub_RI_iconf: fixes e :: "'a exp"   and sh :: "sheap" assumes "\<not> sub_RI e" shows "iconf sh e"
poly_eval_prop: shows "\<forall>x. poly [:c, b, a:] x = a * x\<^sup>2 + b * x + c"
cross_ratio_equal_1: fixes p :: "proj2"   and q :: "proj2"   and r :: "proj2" assumes "proj2_Col p q r"   and "p \<noteq> q"   and "r \<noteq> p"   and "r \<noteq> q" shows "cross_ratio p q r r = 1"
noteq_def: fixes r :: "'a array"   and s :: "'b array" shows "r =!!= s = (TYPEREP('a) \<noteq> TYPEREP('b) \<or> addr_of_array r \<noteq> addr_of_array s)"
invar_def: shows "invar = (\<lambda>seed (\<sigma>1, b) \<sigma>2. \<not> b \<and> dom \<sigma>1 = h seed ` dom \<sigma>2 \<and> (\<forall>x\<in>dom \<sigma>2. \<sigma>1 (h seed x) = map_option (Pair x) (\<sigma>2 x)))"
is_auto_reducedD: fixes B :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set"   and b :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" assumes "punit.is_auto_reduced B"   and "b \<in> B" shows "\<not> punit.is_red (B - {b}) b"
sqr_pos: fixes x :: "real" assumes "x \<noteq> 0" shows "sgn x * sgn x = 1"
division_of_subset: fixes p :: "'a set set"   and q :: "'a set set" assumes "p division_of \<Union> p"   and "q \<subseteq> p" shows "q division_of \<Union> q"
trans: fixes le :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "dpo le"   and "le x y"   and "le y z" shows "le x z"
separatedin_Un_2: fixes X :: "'c topology"   and S :: "'c set"   and T :: "'c set"   and U :: "'c set" shows "separatedin X (S \<union> T) U = (separatedin X S U \<and> separatedin X T U)"
map_eq_zero_iff: fixes f :: "'c \<Rightarrow> 'b"   and p :: "'a \<Rightarrow>\<^sub>0 'c" shows "(Poly_Mapping.map f p = 0) = (\<forall>k\<in>keys p. f (lookup p k) = (0::'b))"
sum_divisors_code: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'b" assumes "0 < n" shows "(\<Sum>d | d dvd n. f d) = fold_atLeastAtMost_nat (\<lambda>d acc. if d dvd n then f d + acc else acc) 1 n (0::'b)"