hcju/isabelleps huggingface dataset:
split_list: fixes x :: "'a"   and xs :: "'a list" assumes "x \<in> set xs" shows "\<exists>ys zs. xs = ys @ x # zs"
eheight_exp_bound: fixes n :: "nat" shows "eheight_exp n \<le> real (n + 3 choose 3) / 4"
filterlim_tendsto_neg_mult_at_bot: fixes f :: "'a \<Rightarrow> real"   and c :: "real"   and F :: "'a filter"   and g :: "'a \<Rightarrow> real" assumes "(f \<longlongrightarrow> c) F"   and "c < 0"   and "filterlim g at_top F" shows "LIM x F. f x * g x :> at_bot"
len_not_in_bv: fixes xs :: "bool list"   and n :: "nat" assumes "length xs \<noteq> n" shows "xs \<notin> set_pmf (bv n)"
term_poly_list_rel_remdups_mset: fixes p :: "char list list"   and p' :: "char list multiset" assumes "(p, p') \<in> term_poly_list_rel" shows "(p, remdups_mset p') \<in> term_poly_list_rel"
assms_2: shows "\<guillemotleft>f : F y \<rightarrow>\<^sub>C x\<guillemotright>"
natpermute_def: fixes n :: "nat"   and k :: "nat" shows "natpermute n k = {l. length l = k \<and> sum_list l = n}"
qtable_union: fixes n :: "nat"   and A :: "nat set"   and P :: "'a option list \<Rightarrow> bool"   and Q1 :: "'a option list \<Rightarrow> bool"   and X :: "'a option list set"   and Q2 :: "'a option list \<Rightarrow> bool"   and Y :: "'a option list set"   and Q :: "'a option list \<Rightarrow> bool" assumes "qtable n A P Q1 X"   and "qtable n A P Q2 Y"   and "\<And>x. \<lbrakk>wf_tuple n A x; P x\<rbrakk> \<Longrightarrow> Q x = (Q1 x \<or> Q2 x)" shows "qtable n A P Q (X \<union> Y)"
boundsu_def: fixes s :: "('b, 'a) state" shows "\<B>\<^sub>u s = map_option snd \<circ> Mapping.lookup (\<B>\<^sub>i\<^sub>u s)"
e: shows "E = \<Sum>\<^sub># (mset Cs) + D"
trg_inv: fixes \<mu> :: "'a" assumes "B.iso \<mu>" shows "trg\<^sub>B (B.inv \<mu>) = trg\<^sub>B \<mu>"
induct: fixes P :: "(nat \<times> 'a) list \<Rightarrow> (nat \<times> 'a) list \<Rightarrow> (nat \<times> (nat \<times> 'a) list) list \<Rightarrow> bool"   and a0 :: "(nat \<times> 'a) list"   and a1 :: "(nat \<times> 'a) list"   and a2 :: "(nat \<times> (nat \<times> 'a) list) list" assumes "\<And>c brr. P c [] brr"   and "\<And>c v va. P c (v # va) []"   and "\<And>c i a arr j b brr. \<lbrakk>i < j \<Longrightarrow> P c arr ((j, b) # brr); \<lbrakk>\<not> i < j; j < i\<rbrakk> \<Longrightarrow> P c ((i, a) # arr) brr; \<lbrakk>\<not> i < j; \<not> j < i\<rbrakk> \<Longrightarrow> P (addmult_spvec a c b) arr brr\<rbrakk> \<Longrightarrow> P c ((i, a) # arr) ((j, b) # brr)" shows "P a0 a1 a2"
functor_def: fixes A :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and F :: "'a \<Rightarrow> 'b" shows "functor A B F \<equiv> category A \<and> category B \<and> functor_axioms A B F"
assms_4: shows "h \<turnstile> get_parent child \<rightarrow>\<^sub>r Some parent"
q_def: shows "q = Min_mset (prime_factorization (2 ^ p - 1))"
path_via_def: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and C :: "char list"   and D :: "char list"   and Cs :: "char list list" shows "P \<turnstile> Path C to D via Cs  \<equiv> Subobjs P C Cs \<and> last Cs = D"
igFreshCls_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'd, 'e, 'f, 'g) model_scheme" shows "igFreshCls MOD \<equiv> igFreshIGVar MOD \<and> igFreshIGAbs1 MOD \<and> igFreshIGAbs2 MOD \<and> igFreshIGOp MOD"
set_child_nodes_impl: shows "set_child_nodes = Shadow_DOM.set_child_nodes"
simps_1: fixes s :: "'a FOL_Fitting.term"   and k :: "nat" shows "[][s/k] = []"
simps_2: fixes mem :: "var\<^sub>C \<Rightarrow> nat"   and x :: "var\<^sub>C" shows "ev\<^sub>A\<^sub>C mem (aexp\<^sub>C.Load x) = mem x"
sup_fun_def: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" shows "f \<squnion> g = (\<lambda>x. f x \<squnion> g x)"
subseq_strict_length: fixes x :: "'b list"   and y :: "'b list" assumes "subseq x y"   and "x \<noteq> y" shows "length x < length y"
is_ground_cls_list_is_ground_cls_sum_list: fixes Cs :: "'a literal multiset list" assumes "is_ground_cls_list Cs" shows "is_ground_cls (sum_list Cs)"
sum_constant_scale: fixes scale :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and y :: "'c"   and A :: "'d set" assumes "Modules.module scale" shows "(\<Sum>x\<in>A. y) = scale (of_nat (card A)) y"
app_def: fixes s :: "'a"   and \<sigma> :: "nat \<Rightarrow> 'a" shows "s ## \<sigma> \<equiv> %n. if n = 0 then s else \<sigma> (n - 1)"
hyps_2: shows "[] \<turnstile> t\<^sub>1 : (\<forall>X<:T\<^sub>11. T\<^sub>12)"
schedule_def: shows "schedule \<equiv> filter_act \<circ> snd"
safe_reac_direct_def': fixes safe :: "('a \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> unit \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list \<times> (nat \<Rightarrow> nat) \<times> (nat \<Rightarrow> bool option) \<Rightarrow> bool"   and cfg :: "('a \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> unit \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list \<times> (nat \<Rightarrow> nat) \<times> (nat \<Rightarrow> bool option)" shows "safe_reach_direct safe cfg \<equiv> \<forall>cfg'. cfg \<Rightarrow>\<^sub>d\<^sup>* cfg' \<longrightarrow> safe cfg'"
Gromov_converging_at_infinity_isometry: fixes f :: "'c \<Rightarrow> 'd"   and u :: "nat \<Rightarrow> 'c" assumes "isometry_on UNIV f" shows "Gromov_converging_at_boundary (\<lambda>n. f (u n)) = Gromov_converging_at_boundary u"
prop_nested_in_set: fixes D :: "'a"   and P :: "'b \<Rightarrow> 'a set"   and C :: "'b"   and A :: "'a \<Rightarrow> bool"   and B :: "'b \<Rightarrow> 'a \<Rightarrow> bool" assumes "D \<in> P C"   and "C \<in> {C. \<forall>D\<in>P C. A D \<or> B C D}" shows "A D \<or> B C D"
sublist_append: fixes l1 :: "'a list"   and l1' :: "'a list"   and l2 :: "'a list"   and l2' :: "'a list" assumes "subseq l1 l1'"   and "subseq l2 l2'" shows "subseq (l1 @ l2) (l1' @ l2')"
assms: shows "perm \<bullet> P \<Longrightarrow>\<^sub>\<tau> perm \<bullet> P'"
norm_le_zero_iff: fixes minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and dist :: "'e \<Rightarrow> 'e \<Rightarrow> real"   and norm :: "'e \<Rightarrow> real"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and uminus :: "'e \<Rightarrow> 'e"   and scaleR :: "real \<Rightarrow> 'e \<Rightarrow> 'e"   and sgn :: "'e \<Rightarrow> 'e"   and uniformity :: "('e \<times> 'e) filter"   and "open" :: "'e set \<Rightarrow> bool"   and x :: "'e" assumes "class.real_normed_vector minus dist norm plus zero uminus scaleR sgn uniformity open" shows "(norm x \<le> 0) = (x = zero)"
preserves_dom: fixes f :: "'a" assumes "arr f" shows "local.dom (trg f) = trg (local.dom f)"
Compl: fixes a :: "'c set"   and \<Omega> :: "'c set"   and M :: "'c set set" assumes "a \<in> smallest_ccdi_sets \<Omega> M" shows "\<Omega> - a \<in> smallest_ccdi_sets \<Omega> M"
image_comp: fixes f :: "'i \<Rightarrow> 'h"   and g :: "'j \<Rightarrow> 'i"   and r :: "'j set" shows "f ` g ` r = (f \<circ> g) ` r"
closure_subset: fixes S :: "'c set" shows "S \<subseteq> closure S"
countable: fixes A :: "'c set"   and M :: "'c measure" assumes "\<And>a. a \<in> A \<Longrightarrow> {a} \<in> sets M"   and "countable A" shows "A \<in> sets M"
step_7: shows "InvariantWatchListsCharacterization (getWatchList state') (getWatch1 state') (getWatch2 state')"
simps: 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 a1 :: "('a, 'b, 'c) configuration"   and a2 :: "('a, 'b, 'c) event list"   and a3 :: "('a, 'b, 'c) configuration" assumes "distributed_system channel" shows "distributed_system.trace channel trans send recv a1 a2 a3 = ((\<exists>c. a1 = c \<and> a2 = [] \<and> a3 = c) \<or> (\<exists>c ev c' t c''. a1 = c \<and> a2 = ev # t \<and> a3 = c'' \<and> distributed_system.next channel trans send recv c ev c' \<and> distributed_system.trace channel trans send recv c' t c''))"
L_sim2: fixes r :: "'b \<Rightarrow> 'v"   and i :: "'b"   and l :: "'L" assumes "range r \<subseteq> f ` V"   and "G'.L (r i) l" shows "L (fi (r i)) l"
Px_def: shows "Px \<equiv> \<lambda>x. \<integral>\<^sup>+ z. Pxy (x, z) \<partial>T"
widen_subcls: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and C :: "char list"   and D :: "char list" assumes "P \<turnstile> Path C to D unique" shows "P \<turnstile> Class C \<le> Class D"
mult_commute_abs: fixes c :: "'a" shows "(\<lambda>x. x * c) = (*) c"
cone_axioms: shows "YoD.cone (local.map a) (local.map \<circ> \<chi>)"
atLeastLessThan_empty_iff2: fixes a :: "'a word"   and b :: "'a word" shows "({} = signed.atLeastLessThan a b) = (\<not> a <s b)"
induct: fixes P :: "'b tree \<Rightarrow> bool"   and a0 :: "'b tree" assumes "\<And>l a r. (l \<noteq> \<langle>\<rangle> \<Longrightarrow> P l) \<Longrightarrow> P \<langle>l, a, r\<rangle>"   and "P \<langle>\<rangle>" shows "P a0"
strictly_preserves_\<alpha>: fixes f :: "'a"   and g :: "'a"   and h :: "'a" assumes "F.C.arr f"   and "F.C.arr g"   and "F.C.arr h" shows "F (F.C.assoc f g h) = F.D.assoc (F f) (F g) (F h)"
Tstruct_axioms: shows "CM.Tstruct Ik_intT"
stateful_matcher_tagged_def: shows "stateful_matcher_tagged \<equiv> \<lambda>m p. m = Established \<and> get_packet_tag p = TagEstablished \<or> m = New \<and> get_packet_tag p = TagNew \<or> m = IsSSH \<and> (\<exists>dst_port tag. p = SomePacket_tagged (22, dst_port, tag))"
prime_imp_prime_elem: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and p :: "'a" assumes "class.normalization_semidom divide plus minus zero times one unit_factor normalize"   and "normalization_semidom.prime zero times one normalize p" shows "comm_semiring_1.prime_elem times one zero p"
invertible_qmpt_def: fixes M :: "'b measure"   and T :: "'b \<Rightarrow> 'b" assumes "qmpt M T" shows "qmpt.invertible_qmpt M T \<equiv> bij T \<and> inv T \<in> M \<rightarrow>\<^sub>M M"
sscan_eq_scons: fixes f :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and xs :: "'c stream"   and a :: "'b"   and b :: "'b"   and w :: "'b stream" shows "(sscan f xs a = b ## w) = (f (shd xs) a = b \<and> sscan f (stl xs) (f (shd xs) a) = w)"
simps_2: fixes v :: "'fsym"   and va :: "'fsym trm list" shows "tT (Fn v va) = tNN (Fn v va)"
assms: fixes x :: "'a" shows "f *\<^sub>V x = t \<bullet>\<^sub>C x"
gt_l: fixes a :: "MIR.num" shows "isrlfm (rsplit gt a)"
assms: shows "s' = snd (fst (mul_instr instr s)) \<and> get_S (cpu_reg_val PSR s) = 0"
reduced_GB_Polys: fixes X :: "'x set"   and F :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set" assumes "finite X"   and "F \<subseteq> P[X]" shows "punit.reduced_GB F \<subseteq> P[X]"
Deterministic_Trans_Fun_def: fixes trans_fun :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and localState :: "'e \<Rightarrow> 'f" shows "Deterministic_Trans_Fun trans_fun localState \<equiv> \<forall>c1 c2 x. localState c1 = localState c2 \<longrightarrow> trans_fun x c1 = trans_fun x c2"
intro: fixes f :: "nat \<Rightarrow> 'a" assumes "f 0 = (0::'a)"   and "f 1 = (1::'a)"   and "\<And>a b. \<lbrakk>1 < a; 1 < b; coprime a b\<rbrakk> \<Longrightarrow> f (a * b) = f a * f b" shows "multiplicative_function f"
bs'carr: shows "set (bs[0 := bs ! 0 \<otimes> u]) \<subseteq> carrier G"
butlast_take: fixes n :: "nat"   and xs :: "'a list" assumes "n \<le> length xs" shows "butlast (take n xs) = take (n - 1) xs"
proots_count_power_n_n: fixes a :: "'b"   and n :: "nat"   and s :: "'b set" shows "proots_count ([:- a, 1::'b:] ^ n) s = (if a \<in> s \<and> 0 < n then n else 0)"
arcs_slim_in_G: shows "parcs slim \<subseteq> parcs G"
split_of_bool: fixes one :: "'b"   and zero :: "'b"   and P :: "'b \<Rightarrow> bool"   and p :: "bool" assumes "class.zero_neq_one one zero" shows "P (zero_neq_one.of_bool one zero p) = ((p \<longrightarrow> P one) \<and> (\<not> p \<longrightarrow> P zero))"
append_fun_le: fixes t1' :: "(val \<times> val) list"   and t1 :: "(val \<times> val) list"   and t2' :: "(val \<times> val) list"   and t2 :: "(val \<times> val) list" assumes "t1' \<lesssim> t1"   and "t2' \<lesssim> t2" shows "t1' @ t2' \<lesssim> t1 @ t2"
fs_def: shows "fs \<equiv> mset (berlekamp_hensel p n f)"
assms_2: assumes "P \<sigma> a"   and "\<forall>j. j \<notin> I \<longrightarrow> Q (\<sigma> j) (\<sigma>' j)" shows "R \<sigma> \<sigma>' a"
sig_red_single_lt_rep_list: fixes pair_of_term :: "'c \<Rightarrow> 'd \<times> nat"   and term_of_pair :: "'d \<times> nat \<Rightarrow> 'c"   and ord :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_strict :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_term :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and ord_term_strict :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and fs :: "('d \<Rightarrow>\<^sub>0 'e) list"   and sing_reg :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and top_tail :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'c \<Rightarrow>\<^sub>0 'e"   and q :: "'c \<Rightarrow>\<^sub>0 'e"   and f :: "'c \<Rightarrow>\<^sub>0 'e"   and t :: "'d" assumes "qpm_inf_term pair_of_term term_of_pair ord ord_strict ord_term ord_term_strict"   and "qpm_inf_term.sig_red_single pair_of_term term_of_pair ord ord_strict ord_term ord_term_strict fs sing_reg top_tail p q f t" shows "ord (ordered_term.lt fst ord (qpm_inf_term.rep_list term_of_pair fs q)) (ordered_term.lt fst ord (qpm_inf_term.rep_list term_of_pair fs p))"
\<phi>_right_support': fixes x :: "real"   and i :: "int"   and l :: "nat" assumes "x \<in> {real_of_int i / 2 ^ (l + 1)..real_of_int (i + 1) / 2 ^ (l + 1)}" shows "\<phi> (l, i) x = 1 - x * 2 ^ (l + 1) + real_of_int i"
alpha2: shows "(qbinp', qbinp'') \<in> alphaBinp"
3: shows "((d \<sqinter> - a)[[(H * (d \<sqinter> - a))\<^sup>\<star>]]) * (H * a * top) \<le> (H * (d \<sqinter> - a))\<^sup>\<star> * H * a * top"
fresh_1: fixes a :: "NominalSequents.var"   and x2 :: "nat"   and x1 :: "NominalSequents.var list" shows "a \<sharp> At x2 x1 = (a \<sharp> x2 \<and> a \<sharp> x1)"
1: shows "(\<forall>xs. x \<notin> varsOfS P xs) \<and> (\<forall>s'. \<forall>Y\<in>termsOfS P s'. fresh xs x Y) \<and> (\<forall>us s'. \<forall>B\<in>(if us = xs \<and> s' = s then {A} else {}) \<union> absOfS P (us, s'). freshAbs xs x B) \<and> (\<forall>rho\<in>envsOfS P. freshEnv xs x rho)"
ran_m_def: fixes N :: "('b, 'a) fmap" shows "ran_m N = the `# fmlookup N `# dom_m N"
red_plus: fixes F :: "('a \<Rightarrow>\<^sub>0 'd) set"   and p :: "'a \<Rightarrow>\<^sub>0 'd"   and q :: "'a \<Rightarrow>\<^sub>0 'd"   and r :: "'a \<Rightarrow>\<^sub>0 'd" assumes "punit.red F p q" obtains s :: "'a \<Rightarrow>\<^sub>0 'd" where "(punit.red F)\<^sup>*\<^sup>* (p + r) s" and "(punit.red F)\<^sup>*\<^sup>* (q + r) s"
module_vec_def: fixes ty :: "'a itself"   and n :: "nat" shows "module_vec ty n \<equiv> \<lparr>carrier = carrier_vec n, monoid.mult = undefined, one = undefined, zero = 0\<^sub>v n, add = (+), module.smult = (\<cdot>\<^sub>v)\<rparr>"
left_commute: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "a * (b * c) = b * (a * c)"
simps_9: fixes p :: "fm"   and q :: "fm" shows "decr0 (Iff p q) = Parametric_Ferrante_Rackoff.iff (decr0 p) (decr0 q)"
InfCard_le_cmult_eq: fixes \<kappa> :: "V"   and \<mu> :: "V" assumes "InfCard \<kappa>"   and "\<mu> \<le> \<kappa>"   and "\<mu> \<noteq> 0" shows "\<kappa> \<otimes> \<mu> = \<kappa>"
simps: fixes a1 :: "(char list \<times> pat) list"   and a2 :: "(char list \<times> trm) list"   and a3 :: "trm list" shows "(\<turnstile> a1 [\<rhd>] a2 \<Rightarrow> a3) = ((\<exists>fs. a1 = [] \<and> a2 = fs \<and> a3 = []) \<or> (\<exists>fs l t p ts fps us. a1 = (l, p) \<Colon> fps \<and> a2 = fs \<and> a3 = ts @ us \<and> fs\<langle>l\<rangle>\<^sub>? = \<lfloor>t\<rfloor> \<and> \<turnstile> p \<rhd> t \<Rightarrow> ts \<and> \<turnstile> fps [\<rhd>] fs \<Rightarrow> us))"
assms_4: shows "q2 \<in> reachable AB q1"
bigMaxU: shows "wmin (wmin (wtimes l\<^sub>1 l\<^sub>2) (wtimes u\<^sub>1 l\<^sub>2)) (wmin (wtimes l\<^sub>1 u\<^sub>2) (wtimes u\<^sub>1 u\<^sub>2)) \<equiv>\<^sub>L min (min (rl1 * rl2) (ru1 * rl2)) (min (rl1 * ru2) (ru1 * ru2))"
section_retraction_of_iso_1: fixes f :: "'d"   and g :: "'d" assumes "D.iso f"   and "D.ide (g \<cdot>\<^sub>D f)" shows "D.inverse_arrows f g"
comp_ipresIGSubstAll: fixes MOD :: "('k, 'l, 'varSort, 'sort, 'm, 'n, 'o, 'p, 'q) model_scheme"   and h :: "'o \<Rightarrow> 'r"   and hA :: "'p \<Rightarrow> 's"   and MOD' :: "('t, 'u, 'varSort, 'sort, 'v, 'n, 'r, 's, 'w) model_scheme"   and h' :: "'r \<Rightarrow> 'x"   and hA' :: "'s \<Rightarrow> 'y"   and MOD'' :: "('z, 'aa, 'varSort, 'ab, 'ac, 'n, 'x, 'y, 'ad) model_scheme" assumes "igWlsAbsIsInBar MOD"   and "ipresIGWlsAll h hA MOD MOD'"   and "ipresIGSubstAll h hA MOD MOD'"   and "ipresIGSubstAll h' hA' MOD' MOD''" shows "ipresIGSubstAll (h' \<circ> h) (hA' \<circ> hA) MOD MOD''"
div_normalize: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and a :: "'a" assumes "class.normalization_semidom divide plus minus zero times one unit_factor normalize" shows "divide a (normalize a) = unit_factor a"
vD_flag_val: fixes dip :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option" assumes "dip \<in> vD rt" shows "the (flag rt dip) = val"
pnPlus_flatten_PLUS: fixes X1 :: "'a rexp set"   and X2 :: "'a rexp set" assumes "X1 \<noteq> {}"   and "X2 \<noteq> {}"   and "finite X1"   and "finite X2"   and "\<forall>x\<in>X1 \<union> X2. \<not> is_Zero x \<and> \<not> is_Plus x" shows "pnPlus (flatten PLUS X1) (flatten PLUS X2) = flatten PLUS (X1 \<union> X2)"
depth_le_height: fixes t :: "'a tree"   and a :: "'a" shows "depth t a \<le> height t"
intD: fixes A :: "'a \<Rightarrow> bool"   and w :: "'a" assumes "\<turnstile> A" shows "A w"
elims: fixes x :: "('b \<times> nat) tree"   and y :: "nat" assumes "lvl x = y" obtains "x = \<langle>\<rangle>" and "y = 0"   | uu_ :: "('b \<times> nat) tree" and uv_ :: "'b" and lv :: "nat" and uw_ :: "('b \<times> nat) tree" where "x = \<langle>uu_, (uv_, lv), uw_\<rangle>" and "y = lv"
stmt_inj_term_def: fixes c :: "stmt" shows "\<langle>c\<rangle> = In1r c"
mono_sup: fixes sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b"   and A :: "'a"   and B :: "'a" assumes "class.semilattice_sup sup less_eq less"   and "order.mono less_eq f" shows "sup (f A) (f B) \<le> f (sup A B)"
singular_simplex_empty: fixes X :: "'c topology"   and p :: "nat"   and f :: "(nat \<Rightarrow> real) \<Rightarrow> 'c" assumes "topspace X = {}" shows "\<not> singular_simplex p X f"
result_conform: shows "\<forall>s t. t \<in> Modif (init s) \<longrightarrow> return' s t = return s t"
antisym_conv2: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.order (less_eq::'b \<Rightarrow> 'b \<Rightarrow> bool) (less::'b \<Rightarrow> 'b \<Rightarrow> bool)"   and "(less_eq::'b \<Rightarrow> 'b \<Rightarrow> bool) (x::'b) (y::'b)" shows "(\<not> (less::'b \<Rightarrow> 'b \<Rightarrow> bool) (x::'b) (y::'b)) = (x = y)"
ws_subint1_induct: fixes G :: "prog"   and I :: "qtname"   and P :: "qtname \<Rightarrow> bool" assumes "is_iface G I"   and "ws_prog G"   and "\<And>I i. iface G I = Some i \<and> (\<forall>J\<in>set (isuperIfs i). G\<turnstile>I\<prec>I1J \<and> P J \<and> is_iface G J) \<Longrightarrow> P I" shows "P I"
distributed_distr_eq_density: fixes M :: "'c measure"   and N :: "'d measure"   and X :: "'c \<Rightarrow> 'd"   and f :: "'d \<Rightarrow> ennreal" assumes "distributed M N X f" shows "distr M N X = density N f"
that: fixes j :: "'a" assumes "j \<in> I"   and "j \<noteq> i" shows "thesis"
intro: fixes 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 set_child_nodes :: "(_) object_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_parent :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr option) prog"   and get_parent_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_owner_document :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) document_ptr) prog"   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 set_disconnected_nodes :: "(_) document_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and remove_child :: "(_) object_ptr \<Rightarrow> (_) node_ptr \<Rightarrow> ((_) heap, exception, unit) prog"   and remove_child_locs :: "(_) object_ptr \<Rightarrow> (_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and remove :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, unit) prog"   and type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and known_ptrs :: "(_) heap \<Rightarrow> bool"   and heap_is_wellformed :: "(_) heap \<Rightarrow> bool"   and parent_child_rel :: "(_) heap \<Rightarrow> ((_) object_ptr \<times> (_) object_ptr) set" assumes "l_remove_child_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_child_nodes get_child_nodes_locs set_child_nodes set_child_nodes_locs get_parent get_parent_locs get_owner_document get_disconnected_nodes get_disconnected_nodes_locs set_disconnected_nodes set_disconnected_nodes_locs remove_child remove_child_locs remove type_wf known_ptr known_ptrs heap_is_wellformed parent_child_rel"   and "l_heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr type_wf get_child_nodes get_child_nodes_locs get_disconnected_nodes get_disconnected_nodes_locs heap_is_wellformed parent_child_rel" shows "l_remove_child_wf2\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_child_nodes get_child_nodes_locs set_child_nodes set_child_nodes_locs get_parent get_parent_locs get_owner_document get_disconnected_nodes get_disconnected_nodes_locs set_disconnected_nodes set_disconnected_nodes_locs remove_child remove_child_locs remove type_wf known_ptr known_ptrs heap_is_wellformed parent_child_rel"
del_point_def: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and p :: "'b" assumes "incidence_system point_set block_collection" shows "incidence_system.del_point point_set p \<equiv> point_set - {p}"
that: fixes c :: "'fun" assumes "Fun c [] \<sqsubseteq> fresh_pgwt S (\<Gamma> t)"   and "c \<notin> S" shows "thesis"
fresh_prodD_2: fixes a :: "'b"   and x :: "'c"   and y :: "'d" assumes "a \<sharp> (x, y)" shows "a \<sharp> y"
Err_any_conv: 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 T :: "ty err" shows "P \<turnstile> Err \<le>\<^sub>\<top> T = (T = Err)"