hcju/isabelleps huggingface dataset:
lconcat_lappend: fixes xss :: "'c llist llist"   and yss :: "'c llist llist" assumes "lfinite xss" shows "lconcat (lappend xss yss) = lappend (lconcat xss) (lconcat yss)"
take_bit_rec: fixes 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 divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bit :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and drop_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and take_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and n :: "nat"   and a :: "'a" assumes "class.semiring_bit_shifts plus minus zero times one divide modulo bit push_bit drop_bit take_bit" shows "take_bit n a = (if n = 0 then zero else plus (times (take_bit (n - 1) (divide a (numeral.numeral one plus (num.Bit0 num.One)))) (numeral.numeral one plus (num.Bit0 num.One))) (modulo a (numeral.numeral one plus (num.Bit0 num.One))))"
completely_multiplicative_nonzero: fixes f :: "nat \<Rightarrow> complex"   and d :: "nat" assumes "completely_multiplicative_function f"   and "d \<noteq> 0"   and "\<And>p. prime p \<Longrightarrow> f p \<noteq> 0" shows "f d \<noteq> 0"
integ_iexp: shows "complex_integrable M (\<lambda>x. iexp (t * x))"
inductive_on_imp_wfp_on: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set" assumes "inductive_on P A" shows "wfp_on P A"
openin_cases: fixes y :: "'c set \<Rightarrow> bool" assumes "y \<in> {L. istopology L}" obtains x :: "'c topology" where "y = openin x"
V_simps_1: fixes \<phi> :: "('b, 'c, 'd) trans \<Rightarrow> bool"   and f :: "('b, 'c, 'd) trans \<Rightarrow> 'a" shows "BD_Security.V \<phi> f [] = []"
A1: shows "c_unfold (length ls) (c_fold ls) = ls"
HInv4d_def: fixes s :: "state"   and p :: "Proc" shows "HInv4d s p = (\<forall>bk\<in>blocksOf s p. \<exists>D\<in>MajoritySet. \<forall>d\<in>D. bal bk \<le> mbal (disk s d p))"
prems_2: fixes S' :: "POPLmark.type" assumes "\<Gamma> \<turnstile> U <: S"   and "TVarB U \<Colon> \<Gamma> \<turnstile> s : S'"   and "TVarB U \<Colon> \<Gamma> \<turnstile> S' <: U'" shows "P"
unique_connecting_path_set_1: fixes G :: "('c, 'd) Graph_scheme"   and v :: "'c"   and w :: "'c" assumes "Tree G"   and "v \<in> V\<^bsub>G\<^esub>"   and "w \<in> V\<^bsub>G\<^esub>" shows "v \<in> set (v \<leadsto>\<^bsub>G\<^esub> w)"
sum_rw: shows "sum f (insert a (UNIV - {a} - {b})) = f a + sum f (UNIV - {a} - {b})"
da_elim_cases_49: fixes Env :: "env"   and B :: "lname set"   and D :: "qtname"   and c :: "stmt"   and A :: "assigned" assumes "Env\<turnstile> B \<guillemotright>\<langle>Body D c\<rangle>\<guillemotright> A" obtains C :: "assigned" where "Env\<turnstile> B \<guillemotright>\<langle>c\<rangle>\<guillemotright> C" and "jumpNestingOkS {Ret} c" and "Result \<in> nrm C" and "nrm A = B" and "brk A = (\<lambda>l. UNIV)"
assms_1: shows "set (coeffs p) \<subseteq> range to_fract"
Step: fixes s :: "'state"   and a :: "'act" assumes "reach s" shows "reach (snd (step s a))"
product_coneI: fixes a :: "'a"   and \<pi> :: "'j \<Rightarrow> 'a" assumes "D.limit_cone a \<pi>" shows "product_cone J (\<cdot>) D a \<pi>"
foreach_reflexive_transitive: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> bool"   and S :: "'d set"   and a\<^sub>1 :: "'c"   and a\<^sub>2 :: "'c" assumes "reflp r"   and "transp r"   and "foreach f S a\<^sub>1 a\<^sub>2"   and "\<And>x a\<^sub>1 a\<^sub>2. \<lbrakk>x \<in> S; f x a\<^sub>1 a\<^sub>2\<rbrakk> \<Longrightarrow> r a\<^sub>1 a\<^sub>2" shows "r a\<^sub>1 a\<^sub>2"
inf_sup_distrib2: fixes y :: "'d upred"   and z :: "'d upred"   and x :: "'d upred" shows "((y \<or> z) \<and> x) = (y \<and> x \<or> z \<and> x)"
eventually_nhds_in_nhd: fixes x :: "'a"   and s :: "'a set" assumes "x \<in> interior s" shows "\<forall>\<^sub>F y in nhds x. y \<in> s"
select_convs_2: fixes vars :: "char list list"   and body :: "stmt"   and more :: "'a" shows "body \<lparr>vars = vars, body = body, \<dots> = more\<rparr> = body"
vfinite_vunionD_2: fixes A :: "V"   and B :: "V" assumes "vfinite (A \<union>\<^sub>\<circ> B)" shows "vfinite B"
I_qe_FM\<^sub>1: fixes as :: "atom list"   and xs :: "real list" assumes "\<forall>a\<in>set as. is_Less a"   and "\<forall>a\<in>set as. depends\<^sub>R a" shows "R.I (qe_FM\<^sub>1 as) xs = (\<exists>x. \<forall>a\<in>set as. I\<^sub>R a (x # xs))"
sel_3: fixes x21 :: "'b"   and x22 :: "'b list" shows "tl (x21 # x22) = x22"
cring_simprules_17: fixes R :: "('a, 'b) ring_scheme"   and x :: "'a" assumes "cring R"   and "x \<in> carrier R" shows "x \<oplus>\<^bsub>R\<^esub> \<ominus>\<^bsub>R\<^esub> x = \<zero>\<^bsub>R\<^esub>"
assms_3: fixes a :: "'a"   and b :: "'a" assumes "reachable a"   and "P a"   and "a \<rightarrow> b" shows "P b"
fps_of_poly_fps_X: shows "fps_of_poly [:0::'b, 1::'b:] = fps_X"
tr_rev_step_2: assumes "ps c p \<noteq> None" shows "ps c' p \<noteq> None"
meet_3_col_1: fixes A :: "Points"   and B :: "Points"   and m :: "Lines"   and n :: "Lines"   and P :: "Points" assumes "meet_3_in (line A B) m n P" shows "col A B P"
eq_iff: fixes a :: "'q multiset"   and b :: "'q multiset" shows "(a = b) = (b \<subseteq># a \<and> a \<subseteq># b)"
pd_sub_def: shows "pd_sub = (\<lambda>G t. G \<subseteq> F \<and> card G = t \<and> pairwise disjnt G \<and> {} \<notin> G)"
lookup_eq_zero_in_keys_contradict: fixes f :: "'d \<Rightarrow>\<^sub>0 'c"   and k :: "'d" assumes "lookup f k = (0::'c)" shows "k \<notin> keys f"
map_simp_ide: fixes a :: "'a" assumes "Adj.C.ide a" shows "\<tau>.map a = G' (Adj.\<epsilon> a) \<cdot>\<^sub>D Adj'.\<eta> (G a)"
changes_gt_smods_def: fixes a :: "real"   and p :: "real poly"   and q :: "real poly" shows "changes_gt_smods a p q = (let ps = smods p q in changes_poly_at ps a - changes_poly_pos_inf ps)"
\<R>_int: fixes R :: "('a \<Rightarrow> real) set"   and R' :: "('a \<Rightarrow> real) set" assumes "R \<in> \<R>"   and "R' \<in> \<R>"   and "R \<noteq> R'" shows "R \<inter> R' = {}"
c1_2: shows "B1 \<in> carrier_mat nr1 n2"
deconstruct_assm_set: fixes k :: "'b"   and Z :: "'b set"   and F :: "'b formula"   and \<A> :: "'b \<Rightarrow> bool" assumes "\<And>\<A>. line_assm \<A> (k \<triangleright> Z) \<turnstile> F" shows "line_assm \<A> Z \<turnstile> F"
get_parent_ptr_in_heap: fixes h :: "(_) heap"   and node_ptr :: "(_) node_ptr" assumes "h \<turnstile> ok Shadow_DOM.get_parent node_ptr" shows "node_ptr |\<in>| node_ptr_kinds h"
assms_4: shows "F \<in> I \<rightarrow> carrier R"
\<B>_def: shows "\<B> \<equiv> {B'. B \<subseteq> B' \<and> B' \<subseteq> A \<and> \<not> B' \<turnstile> \<phi>}"
monad_alt_writerT: assumes "monad_alt return bind alt" shows "monad_alt local.return_writer local.bind_writer local.alt_writer"
inj_on_implies_distinct: fixes enum :: "nat \<Rightarrow> 'c"   and A :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and B :: "'d set" assumes "bij_betw enum {0..<card A} A"   and "f \<in> A \<rightarrow>\<^sub>E B"   and "inj_on f A" shows "distinct (sequence_of A enum f)"
integral_norm_bound_ennreal: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> 'd" assumes "integrable M f" shows "ennreal (norm (integral\<^sup>L M f)) \<le> \<integral>\<^sup>+ x. ennreal (norm (f x)) \<partial>M"
elims_2: fixes x :: "'b comparator"   and xa :: "'b list" assumes "Sorting_Algorithms.sorted x xa" obtains cmp :: "'b comparator" where "xa__ = cmpa__" and "xa = []"   | cmp :: "'b comparator" and x :: "'b" where "xa__ = cmpb__" and "xa = [xb__]"   | cmp :: "'b comparator" and y :: "'b" and x :: "'b" and xs :: "'b list" where "xa__ = cmp" and "xa = y # x # xs" and "compare cmp y x \<noteq> Greater \<and> Sorting_Algorithms.sorted cmp (x # xs)"
sum4: fixes p :: "nat" assumes "prime p" shows "summable (\<lambda>n. norm (fds_nth f p / nat_power p s) ^ Suc n)"
S_nonneg: fixes i :: "nat" shows "0 \<le> S i"
ipurge_tr_ref_aux_futures: fixes P :: "'e process"   and I :: "('f \<times> 'f) set"   and D :: "'e \<Rightarrow> 'f"   and ys :: "'e list"   and Y :: "'e set"   and xs :: "'e list"   and U :: "'f set" assumes "secure P I D"   and "(ys, Y) \<in> futures P xs" shows "(ipurge_tr_aux I D U ys, ipurge_ref_aux I D U ys Y) \<in> futures P xs"
conjugate_vec_def: fixes v :: "'c vec" shows "conjugate v = vec (dim_vec v) (\<lambda>i. conjugate (v $ i))"
div_numeral_Bit0_float: fixes x :: "real"   and n :: "num" assumes "x / numeral n \<in> float" shows "x / numeral (num.Bit0 n) \<in> float"
terminationRule: fixes P :: "nat \<Rightarrow> bool"   and msrFn :: "nat \<Rightarrow> nat"   and m :: "nat" assumes "\<And>n. P n \<Longrightarrow> \<not> P (Suc n) \<or> P (Suc n) \<and> msrFn (Suc n) < msrFn n"   and "P m" shows "\<exists>n. P n \<and> \<not> P (Suc n)"
carrier_is_subring: shows "subring (carrier (poly_ring R)) (poly_ring R)"
eq_OO: fixes R :: "'d \<Rightarrow> 'e \<Rightarrow> bool" shows "(=) OO R = R"
env_restr_evalHeap_noop: fixes h :: "('d \<times> 'e) list"   and S :: "'d set"   and eval :: "'e \<Rightarrow> 'f" assumes "domA h \<subseteq> S" shows "evalHeap h eval f|` S = evalHeap h eval"
soundness: fixes fs :: "formula list" assumes "fs \<in> deductions PC" shows "validS fs"
intro: fixes f :: "'d \<Rightarrow> 'e" assumes "additive f"   and "antilinear_axioms f" shows "antilinear f"
coboundedI: fixes A :: "'b set"   and a :: "'b" assumes "finite A"   and "a \<in> A" shows "Min A \<le> a"
offsets_is_def: fixes ins :: "instr list" shows "offsets_is ins = \<Union> (offsets ` set ins)"
hyp_distinct: shows "\<forall>ps\<in>set (all_partitions_list xs). distinct ps"
sibling_reciprocal: fixes t :: "'a tree"   and a :: "'a"   and b :: "'a" assumes "consistent t"   and "sibling t a = b" shows "sibling t b = a"
inverse2: fixes g :: "'e \<Rightarrow> 'b" shows "U2 (C2 g) = g"
SFAssRedsNone: fixes e\<^sub>2 :: "char list exp"   and s\<^sub>0 :: "heap \<times> sfields \<times> sheap"   and b\<^sub>0 :: "bool"   and v :: "val"   and h\<^sub>2 :: "heap"   and l\<^sub>2 :: "sfields"   and sh\<^sub>2 :: "sheap"   and b\<^sub>2 :: "bool"   and P :: "(char list list \<times> char list exp) prog"   and C :: "char list"   and F :: "char list"   and D :: "char list" assumes "P \<turnstile> \<langle>e\<^sub>2,s\<^sub>0,b\<^sub>0\<rangle> \<rightarrow>* \<langle>Val v,(h\<^sub>2, l\<^sub>2, sh\<^sub>2),b\<^sub>2\<rangle>"   and "\<nexists>b t. P \<turnstile> C has F,b:t in D" shows "P \<turnstile> \<langle>C\<bullet>\<^sub>sF{D} := e\<^sub>2,s\<^sub>0,b\<^sub>0\<rangle> \<rightarrow>* \<langle>THROW NoSuchFieldError,(h\<^sub>2, l\<^sub>2, sh\<^sub>2),False\<rangle>"
carr_prodag_mem_eq: fixes I :: "'h set"   and A :: "'h \<Rightarrow> ('i, 'j) aGroup_scheme"   and X :: "'h \<Rightarrow> 'i"   and Y :: "'h \<Rightarrow> 'i" assumes "\<forall>k\<in>I. aGroup (A k)"   and "X \<in> carr_prodag I A"   and "Y \<in> carr_prodag I A"   and "\<forall>l\<in>I. X l = Y l" shows "X = Y"
Zm_def: fixes R :: "('f, 'g) Ring_scheme"   and e :: "'e" shows "Zm R e = \<lparr>carrier = {e}, pop = \<lambda>x\<in>{e}. \<lambda>y\<in>{e}. e, mop = \<lambda>x\<in>{e}. e, zero = e, sprod = \<lambda>r\<in>carrier R. \<lambda>x\<in>{e}. e\<rparr>"
index_map_mat_3: fixes f :: "'c \<Rightarrow> 'b"   and A :: "'c mat" shows "dim_col (map_mat f A) = dim_col A"
ccpo_Sup_upper: fixes Sup :: "'a set \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a set"   and x :: "'a" assumes "class.ccpo Sup less_eq less"   and "Complete_Partial_Order.chain less_eq A"   and "x \<in> A" shows "less_eq x (Sup A)"
vfinite_vlepoll_trans: fixes A :: "V"   and B :: "V" assumes "vfinite A"   and "B \<lesssim>\<^sub>\<circ> A" shows "vfinite B"
bindWT_bindWT: fixes m :: "'i\<cdot>('f, 'g) writerT"   and h :: "'i \<rightarrow> 'h\<cdot>('f, 'g) writerT"   and k :: "'h \<rightarrow> 'e\<cdot>('f, 'g) writerT" shows "bindWT\<cdot>(bindWT\<cdot>m\<cdot>h)\<cdot>k = bindWT\<cdot>m\<cdot>(\<Lambda> x. bindWT\<cdot>(h\<cdot>x)\<cdot>k)"
map\<^sub>0_in_hom_1: fixes a :: "'a" assumes "C.obj a" shows "\<guillemotleft>\<rho>\<^sub>0 a : G.map\<^sub>0 a \<rightarrow>\<^sub>D H.map\<^sub>0 a\<guillemotright>"
simps_1: fixes ss :: "'t list"   and bs :: "('t \<times> ('a \<Rightarrow>\<^sub>0 'b)) list"   and a :: "'t \<times> ('a \<Rightarrow>\<^sub>0 'b)"   and b :: "'t \<times> ('a \<Rightarrow>\<^sub>0 'b)" shows "new_syz_sigs_spp ss bs (Inl (a, b)) = ss"
Rep_one_or_two_cases: fixes y :: "'a" assumes "y \<in> one_or_two" obtains x :: "'a one_or_two" where "y = Rep_one_or_two x"
simps_1: fixes x1 :: "'a"   and y1 :: "'a" shows "(SourceTerm x1 = SourceTerm y1) = (x1 = y1)"
identity_arrow_of_spans_axioms: shows "identity_arrow_of_spans (\<cdot>) f"
rStep: fixes xvec :: "name list"   and yvec :: "name list"   and x :: "name"   and y :: "name" assumes "length xvec = length yvec"   and "P xvec yvec" shows "P (x # xvec) (y # yvec)"
remove_Union_def: fixes A :: "'a set"   and X :: "'b set"   and B :: "'b \<Rightarrow> 'a set" shows "remove_Union A X B = A - \<Union> (B ` X)"
fst_prod_decode: fixes z :: "nat" shows "pdec1 z = z - triangle (Max_triangle z)"
negligible_diff: fixes s :: "'c set"   and t :: "'c set" assumes "negligible s" shows "negligible (s - t)"
noteq: shows "aEvalUni At x \<noteq> aEvalUni At y"
i_even_power: fixes n :: "nat" shows "\<i> ^ (n * 2) = (- 1) ^ n"
ThrowThrow\<^sub>1: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> nat exp) list) list"   and e :: "nat exp"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> val list"   and e' :: "nat exp"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> val list" assumes "P \<turnstile>\<^sub>1 \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw e',s\<^sub>1\<rangle>" shows "P \<turnstile>\<^sub>1 \<langle>throw e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw e',s\<^sub>1\<rangle>"
ideal_eq_UNIV_iff_contains_one: fixes B :: "'a set" shows "(ideal B = UNIV) = ((1::'a) \<in> ideal B)"
transitive: fixes P :: "pi"   and Q :: "pi"   and R :: "pi" assumes "P \<sim>\<^sub>e Q"   and "Q \<sim>\<^sub>e R" shows "P \<sim>\<^sub>e R"
one_plus_numeral: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and n :: "num" assumes "class.numeral plus" shows "plus one (numeral.numeral one plus n) = numeral.numeral one plus (num.One + n)"
odiff_0_right: fixes x :: "V" shows "odiff x 0 = x"
ltl_llist_inf: fixes xs :: "'a llist"   and ys :: "'a llist" assumes "xs \<noteq> LNil"   and "ys \<noteq> LNil"   and "lhd xs = lhd ys" shows "ltl (xs \<sqinter> ys) = ltl xs \<sqinter> ltl ys"
lookup_power_zero: fixes p :: "'c \<Rightarrow>\<^sub>0 'b"   and k :: "nat" shows "lookup (p ^ k) (0::'c) = lookup p (0::'c) ^ k"
rel_gpv''_refl_strong: fixes R :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and gpv :: "('d, 'e, 'f) gpv"   and A :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and C :: "'e \<Rightarrow> 'e \<Rightarrow> bool" assumes "\<And>x. x \<in> results_gpv (\<I>_uniform UNIV {x. Domainp R x}) gpv \<Longrightarrow> A x x"   and "\<And>x. x \<in> outs_gpv (\<I>_uniform UNIV {x. Domainp R x}) gpv \<Longrightarrow> C x x"   and "R \<le> (=)" shows "rel_gpv'' A C R gpv gpv"
Ord_vdoubleton_in_Vset_succI: fixes \<alpha> :: "V"   and a :: "V"   and b :: "V" assumes "Ord \<alpha>"   and "a \<in>\<^sub>\<circ> Vset \<alpha>"   and "b \<in>\<^sub>\<circ> Vset \<alpha>" shows "set {a, b} \<in>\<^sub>\<circ> Vset (ZFC_in_HOL.succ \<alpha>)"
RBT_HM_2: shows "m' \<in> {hm. HashMap_Impl.invar hm}"
denormalize_shift: fixes f :: "float"   and m :: "int"   and e :: "int" assumes "f = Float m e"   and "f \<noteq> 0" obtains i :: "nat" where "m = mantissa f * 2 ^ i" and "e = exponent f - int i"
simps_4: fixes a :: "nat rexp"   and b :: "nat rexp"   and \<sigma> :: "nat \<Rightarrow> nat rexp option" shows "subst (Plus a b) \<sigma> = Plus (subst a \<sigma>) (subst b \<sigma>)"
order_le_less_subst1: fixes a :: "'c"   and f :: "'d \<Rightarrow> 'c"   and b :: "'d"   and c :: "'d" assumes "a \<le> f b"   and "b < c"   and "\<And>x y. x < y \<Longrightarrow> f x < f y" shows "a < f c"
observations_wellformed: shows "\<O>\<^sub>s \<noteq> {}"
lad_on_def: fixes A :: "'b set"   and f :: "'b set \<Rightarrow> 'b set" shows "lad_on A f = (\<forall>B\<subseteq>A. \<forall>C\<subseteq>B. card (f C) \<le> card (f B))"
measurable_restrict: fixes I :: "'a set"   and X :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and N :: "'b measure"   and M :: "'a \<Rightarrow> 'c measure" assumes "\<And>i. i \<in> I \<Longrightarrow> X i \<in> N \<rightarrow>\<^sub>M M i" shows "(\<lambda>x. \<lambda>i\<in>I. X i x) \<in> N \<rightarrow>\<^sub>M Pi\<^sub>M I M"
val_Zp_dist_res_eq: fixes a :: "nat \<Rightarrow> int"   and b :: "nat \<Rightarrow> int"   and k :: "nat" assumes "a \<in> carrier Zp"   and "b \<in> carrier Zp"   and "eint (int k) < val_Zp_dist a b" shows "a k = b k"
isMaxFlow_def: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and s :: "nat"   and t :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'a" assumes "Network c s t" shows "Network.isMaxFlow c s t f \<equiv> Flow c s t f \<and> (\<forall>f'. Flow c s t f' \<longrightarrow> Flow.val c s f' \<le> Flow.val c s f)"
fst_snd_Gauss_Jordan_column_k_PA_eq: fixes P :: "(('a, 'c) vec, 'c) vec"   and i :: "nat"   and A :: "(('a, 'b) vec, 'c) vec"   and k :: "nat" shows "fst (snd (Gauss_Jordan_column_k_PA (P, i, A) k)) = fst (Gauss_Jordan_column_k (i, A) k)"
square_less: fixes a :: "real"   and n :: "real" assumes "a < n"   and "- n < a" shows "a\<^sup>2 < n\<^sup>2"
singleton: fixes x :: "'a" shows "\<Sqinter> {x} = x"
completeness: fixes p :: "(nat, nat) fm" assumes "valid p" shows "\<turnstile> p"
msog_append: fixes xs :: "(accent \<times> 'a) list"   and ys :: "(accent \<times> 'a) list" shows "ms_of_greek (xs @ ys) = image_mset (adj_msog [] ys) (ms_of_greek xs) + image_mset (adj_msog xs []) (ms_of_greek ys)"
connectedI: fixes "open" :: "'b set \<Rightarrow> bool"   and U :: "'b set" assumes "class.topological_space open"   and "\<And>A B. \<lbrakk>open A; open B; A \<inter> U \<noteq> {}; B \<inter> U \<noteq> {}; A \<inter> B \<inter> U = {}; U \<subseteq> A \<union> B\<rbrakk> \<Longrightarrow> False" shows "topological_space.connected open U"
l_assigned_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms_def: fixes assigned_nodes :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and flatten_dom :: "((_) heap, exception, unit) prog"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_tag_name :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, char list) prog"   and get_root_node :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr) prog"   and get_host :: "(_) shadow_root_ptr \<Rightarrow> ((_) heap, exception, (_) element_ptr) prog"   and find_slot :: "bool \<Rightarrow> (_) node_ptr \<Rightarrow> ((_) heap, exception, (_) element_ptr option) prog"   and remove :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, unit) prog"   and append_child :: "(_) object_ptr \<Rightarrow> (_) node_ptr \<Rightarrow> ((_) heap, exception, unit) prog"   and remove_shadow_root :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, unit) prog" shows "l_assigned_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms assigned_nodes flatten_dom get_child_nodes get_tag_name get_root_node get_host find_slot remove append_child remove_shadow_root \<equiv> assigned_nodes = l_assigned_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_assigned_nodes get_child_nodes get_tag_name get_root_node get_host find_slot \<and> flatten_dom = l_assigned_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_flatten_dom get_child_nodes get_tag_name get_root_node get_host find_slot remove append_child remove_shadow_root"
depth_swapSyms_neither: fixes t :: "'a tree"   and c :: "'a"   and a :: "'a"   and b :: "'a" assumes "consistent t"   and "c \<noteq> a"   and "c \<noteq> b" shows "depth (swapSyms t a b) c = depth t c"
validTransition_3: fixes P :: "pi"   and P' :: "pi" assumes "P \<longmapsto> \<tau> \<prec> P'"   and "valid P" shows "valid P'"
assms_2: fixes x :: "'b"   and \<sigma> :: "'a" shows "RETURN (fi x \<sigma>) \<le> f x \<sigma>"
join_related: fixes x :: "'a"   and y :: "'a" assumes "x \<sqsubseteq> y" shows "x \<squnion> y = y"
inj_on_def: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set" shows "inj_on f A = (\<forall>x\<in>A. \<forall>y\<in>A. f x = f y \<longrightarrow> x = y)"
is_nth_power_mult_coprime_natD_2: fixes a :: "nat"   and b :: "nat"   and n :: "nat" assumes "coprime a b"   and "is_nth_power n (a * b)"   and "0 < a"   and "0 < b" shows "is_nth_power n b"
countably_compact_def: fixes U :: "'b set" shows "countably_compact U = (\<forall>A. countable A \<longrightarrow> (\<forall>a\<in>A. open a) \<longrightarrow> U \<subseteq> \<Union> A \<longrightarrow> (\<exists>T\<subseteq>A. finite T \<and> U \<subseteq> \<Union> T))"
pd_node2: fixes y :: "'n"   and x :: "'n" assumes "y pd\<rightarrow> x" shows "x \<in> local.nodes"
intros_3: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and Exit :: "'b"   and Def :: "'b \<Rightarrow> 'c set"   and Use :: "'b \<Rightarrow> 'c set"   and ParamDefs :: "'b \<Rightarrow> 'c list"   and ParamUses :: "'b \<Rightarrow> 'c set list"   and n\<^sub>0 :: "'b SDG_node"   and ns :: "'b SDG_node list"   and n\<^sub>1 :: "'b SDG_node"   and p :: "'f"   and n\<^sub>2 :: "'b SDG_node"   and ns' :: "'b SDG_node list"   and n\<^sub>3 :: "'b SDG_node"   and n\<^sub>4 :: "'b SDG_node"   and V :: "'c"   and a :: "'a"   and a' :: "'a" assumes "SDG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit Def Use ParamDefs ParamUses"   and "SDG.matched sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses n\<^sub>0 ns n\<^sub>1"   and "SDG.SDG_edge sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses n\<^sub>1 None (Some (p, True)) n\<^sub>2"   and "SDG.matched sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses n\<^sub>2 ns' n\<^sub>3"   and "SDG.SDG_edge sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses n\<^sub>3 None (Some (p, False)) n\<^sub>4 \<or> SDG.SDG_edge sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses n\<^sub>3 (Some V) (Some (p, False)) n\<^sub>4"   and "valid_edge a"   and "a' \<in> get_return_edges a"   and "sourcenode a = parent_node n\<^sub>1"   and "targetnode a = parent_node n\<^sub>2"   and "sourcenode a' = parent_node n\<^sub>3"   and "targetnode a' = parent_node n\<^sub>4" shows "SDG.matched sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses n\<^sub>0 (ns @ n\<^sub>1 # ns' @ [n\<^sub>3]) n\<^sub>4"
singular_relboundary_imp_relcycle: fixes p :: "nat"   and X :: "'b topology"   and S :: "'b set"   and c :: "((nat \<Rightarrow> real) \<Rightarrow> 'b) \<Rightarrow>\<^sub>0 int" assumes "singular_relboundary p X S c" shows "singular_relcycle p X S c"
simps_6: fixes ux :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and S :: "(char list \<Rightarrow> int) \<Rightarrow> char list \<Rightarrow> int"   and uy :: "(char list \<Rightarrow> int) \<Rightarrow> nat"   and b :: "bexp"   and C :: "acom"   and s :: "char list \<Rightarrow> int" shows "postQ ({(ux, S, uy)} WHILE b DO C) s = S s"