hcju/isabelleps huggingface dataset:
splitFace_empty_ram2_ram1_in_f12: fixes g :: "graph"   and ram1 :: "nat"   and ram2 :: "nat"   and oldF :: "face"   and f12 :: "face"   and f21 :: "face"   and newGraph :: "graph" assumes "pre_splitFace g ram1 ram2 oldF []"   and "(f12, f21, newGraph) = splitFace g ram1 ram2 oldF []" shows "(ram2, ram1) \<in> \<E> f12"
lesub_step_typeD: fixes A :: "(nat \<times> 'b) set"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and B :: "(nat \<times> 'b) set"   and x :: "nat"   and y :: "'b" assumes "A {\<sqsubseteq>\<^bsub>r\<^esub>} B"   and "(x, y) \<in> A" shows "\<exists>y'. (x, y') \<in> B \<and> y \<sqsubseteq>\<^bsub>r\<^esub> y'"
Sup_lexord_rel: fixes I :: "'b set"   and k :: "'d \<Rightarrow> 'c"   and A :: "'b \<Rightarrow> 'd"   and B :: "'b \<Rightarrow> 'd"   and R :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and c :: "'d set \<Rightarrow> 'd"   and s :: "'d set \<Rightarrow> 'd" assumes "\<And>i. i \<in> I \<Longrightarrow> k (A i) = k (B i)"   and "R (c (A ` {a \<in> I. k (B a) = (SUP x\<in>I. k (B x))})) (c (B ` {a \<in> I. k (B a) = (SUP x\<in>I. k (B x))}))"   and "R (s (A ` I)) (s (B ` I))" shows "R (Sup_lexord k c s (A ` I)) (Sup_lexord k c s (B ` I))"
1: shows "rawpsubst (rawpsubst \<phi>1 (zip (map Var us) (map snd txs))) (zip (map fst txs) us) = rawpsubst (rawpsubst \<phi>1 (zip (map Var vs1) (map snd txs))) (zip (map fst txs) vs1)"
intro: fixes G :: "('f, 'd, 'g, 'e, 'h) galcon_scheme" assumes "\<And>y. y \<in> carrier \<Y>\<^bsub>G\<^esub> \<Longrightarrow> y \<sqsubseteq>\<^bsub>\<Y>\<^bsub>G\<^esub>\<^esub> \<pi>\<^sup>*\<^bsub>G\<^esub> (\<pi>\<^sub>*\<^bsub>G\<^esub> y)" shows "coretract_axioms G"
le_def: 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 mxs :: "nat"   and mxl :: "nat" shows "JVM_SemiType.le P mxs mxl \<equiv> fst (snd (JVM_SemiType.sl P mxs mxl))"
simps_1: fixes prec :: "nat"   and i :: "nat"   and k :: "nat"   and x :: "float" shows "lb_sin_cos_aux prec 0 i k x = 0"
TQ: fixes C :: "acom"   and e :: "(char list \<Rightarrow> int) \<Rightarrow> nat"   and s :: "char list \<Rightarrow> int" shows "preT C e s = e (postQ C s)"
Cons_parallelI1: fixes a :: "'c"   and b :: "'c"   and as :: "'c list"   and bs :: "'c list" assumes "a \<noteq> b" shows "a # as \<parallel> b # bs"
sinvar_mono_imp_negative_delete_edge_mono: fixes G :: "'v graph"   and X :: "('v \<times> 'v) set"   and Y :: "('v \<times> 'v) set"   and nP :: "'v \<Rightarrow> 'a" assumes "sinvar_mono"   and "wf_graph G"   and "X \<subseteq> Y"   and "\<not> sinvar (delete_edges G Y) nP" shows "\<not> sinvar (delete_edges G X) nP"
conductor_leI: fixes n :: "nat"   and \<chi> :: "nat \<Rightarrow> complex"   and d :: "nat" assumes "dcharacter n \<chi>"   and "dcharacter.induced_modulus n \<chi> d" shows "dcharacter.conductor n \<chi> \<le> d"
UN_mono: fixes A :: "'a set"   and B :: "'a set"   and f :: "'a \<Rightarrow> 'b set"   and g :: "'a \<Rightarrow> 'b set" assumes "A \<subseteq> B"   and "\<And>x. x \<in> A \<Longrightarrow> f x \<subseteq> g x" shows "\<Union> (f ` A) \<subseteq> \<Union> (g ` B)"
map_idx_nth: fixes i :: "nat"   and xs :: "'c list"   and f :: "'c \<Rightarrow> nat \<Rightarrow> 'd"   and n :: "nat" assumes "i < length xs" shows "map_idx f xs n ! i = f (xs ! i) (n + i)"
mod_to_zero_def: fixes x :: "integer"   and y :: "integer" shows "mod_to_zero x y = (let z = snd (Code_Numeral.divmod_abs x y) in if x < 0 then - z else z)"
simps_3: fixes x :: "name"   and a :: "name"   and P :: "pi" assumes "x \<sharp> a" shows "hnf (a<x>.P) = True"
discounted_sum: fixes I :: "'c set"   and r :: "real"   and f :: "'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> real" assumes "finite I" shows "\<forall>n w. (\<Sum>i\<in>I. discounted_value r (f i) n w) = discounted_value r (\<lambda>m x. \<Sum>i\<in>I. f i m x) n w"
vreal: shows "\<forall>\<^sub>F n in F. v n = ereal (real_of_ereal (v n))"
Cons_5: shows "(lookup_pair xs k = v) = ((k, v) \<in> set xs)"
image_def: fixes R :: "('d \<times> 'c) odlist"   and xs :: "'d odlist" shows "ODList.image R xs = ODList (List_local.image (toList R) (toList xs))"
partition_P: fixes P :: "'a \<Rightarrow> bool"   and xs :: "'a list"   and yes :: "'a list"   and no :: "'a list" assumes "partition P xs = (yes, no)" shows "(\<forall>p\<in>set yes. P p) \<and> (\<forall>p\<in>set no. \<not> P p)"
reduction_val: fixes v :: "db"   and v' :: "db" assumes "val v"   and "v \<rightarrow>\<^sub>v v'" shows "False"
trivial_hom: fixes H :: "('a, 'b) monoid_scheme"   and G :: "('c, 'd) monoid_scheme" assumes "Group.group H" shows "(\<lambda>x. \<one>\<^bsub>H\<^esub>) \<in> hom G H"
set_sel_1: fixes a :: "'b stream" shows "shd a \<in> sset a"
induct: fixes P :: "MIR.num \<Rightarrow> bool"   and a0 :: "MIR.num" assumes "\<And>i. P (C i)"   and "\<And>n c t. (\<And>x. P t) \<Longrightarrow> P (CN n c t)"   and "\<And>c s t. (\<And>x. P t) \<Longrightarrow> P (CF c s t)"   and "\<And>v. P (Bound v)"   and "\<And>v. P (Neg v)"   and "\<And>v va. P (Add v va)"   and "\<And>v va. P (Sub v va)"   and "\<And>v va. P (Mul v va)"   and "\<And>v. P (Floor v)" shows "P a0"
seteq_qle_def: fixes M :: "'a list list set"   and N :: "'a list list set" shows "M =\<^sub>\<simeq> N \<equiv> M \<subseteq>\<^sub>\<simeq> N \<and> N \<subseteq>\<^sub>\<simeq> M"
asInp_def: fixes qinp :: "'d \<Rightarrow> ('e, 'f, 'varSort, 'var, 'g) qTerm option" shows "asInp qinp \<equiv> lift asTerm qinp"
distinct_15: fixes x2 :: "'c"   and x5 :: "('b, 'c) fm" shows "Nom x2 \<noteq> (\<^bold>\<diamond> x5)"
aux: fixes x :: "int"   and y :: "int" assumes "0 \<le> x"   and "0 \<le> y" shows "\<exists>u v. u * x + v * y = gcd x y"
someI_ex: fixes P :: "'g \<Rightarrow> bool" assumes "\<exists>x. P x" shows "P (SOME x. P x)"
finrank_Span: fixes vs :: "'b list" assumes "set vs \<subseteq> V" shows "R_finrank (RSpan vs)"
mod_diff_right_eq: fixes divide :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and one :: "'e"   and modulo :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and euclidean_size :: "'e \<Rightarrow> nat"   and uminus :: "'e \<Rightarrow> 'e"   and a :: "'e"   and b :: "'e"   and c :: "'e" assumes "class.euclidean_ring_cancel divide plus minus zero times one modulo euclidean_size uminus" shows "modulo (minus a (modulo b c)) c = modulo (minus a b) c"
sup_left_top: fixes x :: "'a" shows "top \<squnion> x = top"
simps_1: fixes \<mu> :: "'c" shows "Trg \<^bold>\<langle>\<mu>\<^bold>\<rangle>\<^sub>0 = \<^bold>\<langle>\<mu>\<^bold>\<rangle>\<^sub>0"
2: shows "local.step' \<top> c' \<sqsubseteq> c'"
assms_1: shows "openin (top_of_set (affine hull T)) S"
power_diff': fixes n :: "nat"   and m :: "nat"   and x :: "'b" assumes "n \<le> m"   and "x \<noteq> (0::'b)" shows "x ^ (m - n) = x ^ m div x ^ n"
mon_env_def: fixes fg :: "('b, 'c, 'd, 'a, 'e) flowgraph_rec_scheme"   and ww :: "('c, 'f) label list el_step list" shows "mon_env fg ww \<equiv> mon_ww fg (map le_rem_s (env ww))"
deltaG_metric_completion_def: fixes uu :: "'a metric_completion itself" shows "deltaG uu = deltaG TYPE('a)"
iso_char: fixes \<mu> :: "'a arrow_of_spans_data" shows "V.iso \<mu> = (V.arr \<mu> \<and> C.iso (Chn \<mu>))"
disjoint_symmetry: fixes x :: "i"   and y :: "i" assumes "\<not> O x y" shows "\<not> O y x"
snd_conv: fixes x1 :: "'d"   and x2 :: "'c" shows "snd (x1, x2) = x2"
handshake_invL_def: shows "handshake_invL = (\<lambda>s. (atS (mutator m) hs_noop_locs s \<longrightarrow> sys_hs_type s\<down> = ht_NOOP) \<and> (atS (mutator m) hs_get_roots_locs s \<longrightarrow> sys_hs_type s\<down> = ht_GetRoots) \<and> (atS (mutator m) hs_get_work_locs s \<longrightarrow> sys_hs_type s\<down> = ht_GetWork) \<and> (atS (mutator m) ht_loaded_locs s \<longrightarrow> mut_hs_pending s\<down> \<longrightarrow> mut_hs_type s\<down> = sys_hs_type s\<down>) \<and> (atS (mutator m) hs_pending_loaded_locs s \<longrightarrow> mut_hs_pending s\<down> \<longrightarrow> sys_hs_pending m s\<down>) \<and> (atS (mutator m) hs_pending_locs s \<longrightarrow> mut_hs_pending s\<down>) \<and> (atS (mutator m) no_pending_mutations_locs s \<longrightarrow> filter (\<lambda>s. is_mw_Mutate s \<or> is_mw_Mutate_Payload s) (sys_mem_store_buffers (mutator m) s\<down>) = []))"
vrs_prm_tyvrs_def: fixes pi :: "(vrs \<times> vrs) list"   and a :: "tyvrs" shows "pi \<bullet> a \<equiv> a"
completion_ex_borel_measurable_real: fixes g :: "'c \<Rightarrow> real"   and M :: "'c measure" assumes "g \<in> borel_measurable (completion M)" shows "\<exists>g'\<in>borel_measurable M. AE x in M. g x = g' x"
2: fixes x :: "typ \<times> typ"   and xa :: "typ"   and y :: "typ"   and xb :: "(variable \<times> String.literal set \<Rightarrow> typ option) option"   and subs :: "((variable \<times> String.literal set) \<times> typ) list" assumes "a = b \<and> length Ts = length Us"   and "x \<in> set (zip Ts Us)"   and "(xa, y) = x"   and "xb = Some (map_of subs)" shows "raw_match' xa y (map_of subs) = map_option map_of (assoc_match xa y subs)"
nth_in_set_dropI: fixes j :: "nat"   and i :: "nat"   and xs :: "'c list" assumes "j \<le> i"   and "i < length xs" shows "xs ! i \<in> set (drop j xs)"
member_empty_empty: fixes dxs :: "'b set_dlist" assumes "ID CEQ('b) \<noteq> None" shows "(\<forall>x. \<not> DList_Set.member dxs x) = (dxs = DList_Set.empty)"
m_assoc: fixes x :: "'c"   and y :: "'c"   and z :: "'c" assumes "x \<in> carrier H"   and "y \<in> carrier H"   and "z \<in> carrier H" shows "x \<otimes>\<^bsub>H\<^esub> y \<otimes>\<^bsub>H\<^esub> z = x \<otimes>\<^bsub>H\<^esub> (y \<otimes>\<^bsub>H\<^esub> z)"
list_all2_Nil2: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'a list" shows "list_all2 P xs [] = (xs = [])"
typing_Ax_elim1: fixes \<Gamma> :: "(name \<times> ty) list"   and x :: "name"   and a :: "coname"   and B :: "ty"   and \<Delta> :: "(coname \<times> ty) list" assumes "\<Gamma> \<turnstile> Ax x a \<turnstile> ((a, B) # \<Delta>)" shows "(x, B) \<in> set \<Gamma>"
arrE: fixes f :: "('a \<Rightarrow> 'b, 'a \<Rightarrow> 'b) concrete_category.arr \<times> 'a" assumes "A_BxA.arr f" obtains "A_B.arr (fst f) \<and> A.arr (snd f)"
lookup_monom_mult: fixes c :: "'a"   and t :: "'b"   and p :: "'b \<Rightarrow>\<^sub>0 'a"   and v :: "'b" shows "lookup (punit.monom_mult c t p) v = (if punit.adds_pp t v then c * lookup p (punit.sminus v t) else (0::'a))"
recurrence_time_restricts_compact_flow': fixes t :: "real"   and t' :: "real"   and y :: "'a" assumes "t < t'"   and "t \<in> existence_ivl0 y"   and "t' \<in> existence_ivl0 y"   and "flow0 y t = flow0 y t'" shows "range (flow0 y) = flow0 y ` {t..t'}"
collapse_1: fixes thread :: "thread" assumes "is_WAIT thread" shows "WAIT (ticket thread) = thread"
sqrt_real_def: fixes x :: "real" shows "sqrt_real x = (if x \<in> \<rat> \<and> 0 \<le> x then if x = 0 then [0] else let sx = sqrt x in [sx, - sx] else [])"
oalist_inv_map_val_raw: fixes rep_key_order :: "'a \<Rightarrow> 'b key_order"   and xs :: "('b \<times> 'c) list \<times> 'a"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd" assumes "oalist_raw.oalist_inv rep_key_order xs" shows "oalist_raw.oalist_inv rep_key_order (map_val_raw f xs)"
or_eq_not_not_and: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bit :: "'c \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and drop_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and take_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and "and" :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and or :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and xor :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and mask :: "nat \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and not :: "'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "class.ring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask uminus not" shows "or a b = not (and (not a) (not b))"
seg_facts_1: shows "P = \<Union> S \<union> P1 \<union> P2 \<union> Q"
subresultant_mat_dim_2: fixes j :: "nat"   and p :: "'b poly"   and q :: "'b poly" shows "dim_col (subresultant_mat j p q) = degree p - j + (degree q - j)"
fBexE: fixes A :: "'h fset"   and P :: "'h \<Rightarrow> bool" assumes "fBex A P" obtains x :: "'h" where "x |\<in>| A" and "P x"
iso_inv_iso: fixes f :: "('a S.E.term, 'a) S.arr" assumes "S.iso f" shows "S.iso (S.inv f)"
ide_trg: fixes \<mu> :: "'a arrow_of_spans_data" assumes "V.arr \<mu>" shows "V.ide (trg \<mu>)"
coeff_0_reflect_poly_0_iff: fixes p :: "'b poly" shows "(constant_term (reflect_poly p) = (0::'b)) = (p = 0)"
Si_at_top_integrable: fixes t :: "real" assumes "0 \<le> t" shows "interval_lebesgue_integrable lborel 0 \<infinity> (\<lambda>x. exp (- (x * t)) * (x * sin t + cos t) / (1 + x\<^sup>2))"
Map_in_Hom: fixes Obj :: "'b set"   and Hom :: "'b \<Rightarrow> 'b \<Rightarrow> 'c set"   and Id :: "'b \<Rightarrow> 'c"   and Comp :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and f :: "('b, 'c) Maps.arr" assumes "concrete_category Obj Hom Id Comp"   and "partial_magma.arr (concrete_category.COMP Obj Hom Comp) f" shows "Maps.Map f \<in> Hom (concrete_category.Dom f) (Maps.Cod f)"
Card_order_empty: fixes r :: "('c \<times> 'c) set" assumes "Card_order r" shows "|{}| \<le>o r"
exch_alw_InvGlobNonposImpRecordsNonpos: fixes s :: "('b, 't, 'loc) Combined.configuration stream" assumes "FullSpec s" shows "alw (\<lambda>s. cri.InvGlobNonposImpRecordsNonpos (exchange_config (shd s))) s"
offset_not_none: fixes k :: "nat"   and x :: "'a"   and xs :: "'a list" shows "(offset k x xs \<noteq> None) = (\<exists>ys zs. xs = ys @ x # zs)"
pp_fusion_rightneutral: fixes x :: "'a ppath" shows "pp_fusion x (Node (pp_last x)) = x"
rtc_rtranclI: fixes a :: "'a"   and b :: "'a" assumes "a \<rightarrow>\<^sup>* b" shows "(a, b) \<in> {(x, y). x \<rightarrow> y}\<^sup>*"
lfinite_ldrop: fixes n :: "enat"   and xs :: "'b llist" shows "lfinite (ldrop n xs) = (lfinite xs \<or> n = \<infinity>)"
PP_def: shows "PP \<equiv> \<lambda>t. subst P t xx"
s'_eq: fixes i :: "nat" shows "s' i = (\<lambda>z. \<Sum>y\<in>s' i ` space M - {0::'b}. indicat_real {x \<in> space M. s' i x = y} z *\<^sub>R y)"
38: fixes x :: "'a"   and y :: "'a" shows "- x \<squnion> (- - x \<squnion> y) = top \<squnion> y"
dpref_def: fixes X :: "'a set"   and Y :: "'a set" shows "dpref X Y = (\<forall>x\<in>X. \<exists>y\<in>Y. (x, y) \<in> Pd (Xd x))"
f_2: shows "continuous_on ({c..d} - g ` s) f"
lex_ord_pair_EqI: fixes comp :: "'a \<Rightarrow> 'a \<Rightarrow> order"   and xs :: "('a \<times> 'b) list"   and ys :: "('a \<times> 'c) list"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> order option" assumes "comparator comp"   and "comparator.oalist_inv_raw comp xs"   and "comparator.oalist_inv_raw comp ys"   and "\<And>k. k \<in> fst ` set xs \<union> fst ` set ys \<Longrightarrow> f k (comparator.lookup_pair comp xs k) (comparator.lookup_pair comp ys k) = Some Eq" shows "comparator.lex_ord_pair comp f xs ys = Some Eq"
rw: shows "(\<Sum>i\<le>b. monom (g i) i) mod f = (\<Sum>i\<le>b. monom (g i) i)"
fresh_def: fixes permute :: "perm \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "atom"   and x :: "'a" assumes "class.pt permute" shows "pt.fresh permute a x \<equiv> a \<notin> pt.supp permute x"
prems_4: shows "Max (enumerate_aexp_inputs (Times a1 a2)) < A"
distinct_mset_mono: fixes D' :: "'a multiset"   and D :: "'a multiset" assumes "D' \<subseteq># D"   and "distinct_mset D" shows "distinct_mset D'"
HPhase1or2Write_HInv4d_q: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and d :: "Disk"   and q :: "Proc" assumes "HPhase1or2Write s s' p d"   and "HInv4d s q"   and "p \<noteq> q" shows "HInv4d s' q"
inf_absorb2: fixes inf :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and y :: "'d"   and x :: "'d" assumes "class.semilattice_inf inf less_eq less"   and "less_eq y x" shows "inf x y = y"
comp_arr_inv: fixes f :: "'b"   and g :: "'b" assumes "C'.inverse_arrows f g" shows "C' f g = C'.dom g"
fB_C_2: shows "C \<subseteq> f ` B"
valid_2: shows "valid_reads m\<^sub>s\<^sub>b' ts\<^sub>s\<^sub>b'"
simps_5: fixes l1 :: "value"   and v :: "'a aexp"   and va :: "'a aexp" shows "local.less_aexp_aux (L l1) (Times v va) = True"
find_first_some_index: fixes i :: "nat"   and xs :: "'a list" assumes "i < length xs"   and "distinct xs" shows "find_first (xs ! i) xs = Some i"
dfree_Var_simps: fixes x :: "'e" shows "dfree (Var x) = True"
uMult_assoc: fixes x :: "nat discr"   and y :: "nat discr"   and z :: "nat discr" shows "x *\<^sub># y *\<^sub># z = x *\<^sub># (y *\<^sub># z)"
cenvalid_def2: fixes A :: "triple set"   and P :: "assn"   and e :: "expr"   and Q :: "vassn" shows "A |\<Turnstile>\<^sub>e (P, e, Q) = (\<forall>n. |\<Turnstile>n: A \<longrightarrow> (\<forall>s v t. s -e\<succ>v-n\<rightarrow> t \<longrightarrow> P s \<longrightarrow> Q v t))"
simps_1: shows "linorder.sorted (\<le>lex) \<epsilon> = True"
suc_AboveS: fixes B :: "'a set" assumes "B \<subseteq> Field r"   and "local.AboveS B \<noteq> {}" shows "local.suc B \<in> local.AboveS B"
that: fixes L :: "real" assumes "L-lipschitz_on {a..b} (flow t0 x)" shows "thesis"
coincide_sym: fixes \<sigma> :: "('b \<times> 'b trm) list"   and \<eta> :: "('b \<times> 'b trm) list"   and V :: "'b set" assumes "coincide_on \<sigma> \<eta> V" shows "coincide_on \<eta> \<sigma> V"
simps_1: fixes a :: "(('fsym, 'tp) efsym, ('psym, 'tp) epsym) atm" shows "GE.wtL (Pos a) = GE.wtA a"
adopt_node_is_l_adopt_node: shows "l_adopt_node ShadowRootClass.type_wf ShadowRootClass.known_ptr ShadowRootClass.known_ptrs Shadow_DOM.get_parent Shadow_DOM.adopt_node Shadow_DOM.adopt_node_locs Shadow_DOM.get_child_nodes Shadow_DOM.get_owner_document"
p_points_def: fixes l :: "p_line" shows "p_points l = {p. p_incident l p}"
assms_2: fixes x :: "'a" assumes "x \<in> fmdom' m" shows "P x = Q x"
Elem_Fun_Lambda: fixes F :: "ZF"   and U :: "ZF"   and V :: "ZF" assumes "Elem F (Fun U V)" shows "\<exists>f. F = Lambda U f"
functor_axioms: shows "functor hom'.comp hom.comp (\<lambda>\<mu>'. d\<^sub>1 \<star> \<mu>' \<star> e\<^sub>0)"
AndL_inv': fixes F :: "'b formula"   and G :: "'b formula"   and \<Gamma> :: "'b formula multiset"   and \<Delta> :: "'b formula multiset"   and n :: "nat" assumes "F \<^bold>\<and> G, \<Gamma> \<Rightarrow> \<Delta> \<down> n" shows "F, G, \<Gamma> \<Rightarrow> \<Delta> \<down> n"
pt_open_on_imp_open_on_LEXEQ: fixes Q :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool" assumes "pt_open_on Q (SEQ A)" shows "open_on (LEXEQ P) Q (SEQ A)"
A_dest_2: fixes A :: "'a set"   and y :: "'a" assumes "A \<in> {{y \<in> X. (y, x) \<in> r \<and> (x, y) \<in> r} |x. x \<in> X}"   and "y \<in> A" shows "(y, SOME y. y \<in> A) \<in> r"
Collect_mono: fixes P :: "'b::type \<Rightarrow> bool"   and Q :: "'b::type \<Rightarrow> bool" assumes "\<And>x::'b::type. (P::'b::type \<Rightarrow> bool) x \<longrightarrow> (Q::'b::type \<Rightarrow> bool) x" shows "Collect (P::'b::type \<Rightarrow> bool) \<subseteq> Collect (Q::'b::type \<Rightarrow> bool)"
less_sb: shows "\<forall>i\<in>read_tmps sb. i < n"
roots_of_2_def: fixes p :: "int poly" shows "roots_of_2 p = concat (map roots_of_2_irr (factors_of_int_poly p))"
order_mult_of: assumes "finite (carrier R)" shows "order (mult_of R) = order R - 1"
iso_tuple_snd_update_def: fixes isom :: "('a, 'b, 'c) tuple_isomorphism"   and f :: "'c \<Rightarrow> 'c" shows "iso_tuple_snd_update isom f = abst isom \<circ> apsnd f \<circ> repr isom"
h5: shows "\<exists>!x. x \<in> set (sorted_list_of_set (fset (document_ptr_kinds h))) \<and> h \<turnstile> Heap_Error_Monad.bind (get_disconnected_nodes x) (\<lambda>children. return (node_ptr \<in> set children)) \<rightarrow>\<^sub>r True"
split_list_first: fixes x :: "'b"   and xs :: "'b list" assumes "x \<in> set xs" shows "\<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set ys"
assms: shows "\<not> (dep (prob_proj PROB (prob_dom PROB - vs)))\<^sup>+\<^sup>+ v v'"
thr_init_fin_list_state: fixes s :: "status"   and \<sigma> :: "('b, 'c) locks \<times> (('c \<Rightarrow> ('a \<times> 'b \<Rightarrow>f nat) option) \<times> 'd) \<times> ('c \<Rightarrow> 'e wait_set_status option) \<times> 'c set"   and t :: "'c" shows "thr (init_fin_lift_state s \<sigma>) t = map_option (\<lambda>(x, ln). ((s, x), ln)) (thr \<sigma> t)"
fall2_imp_alw_index: fixes \<phi> :: "(('St, 'U, 'C) state \<times> ('U, 'C, 'Out) Noninterference.aprop set) stream list \<Rightarrow> (('St, 'U, 'C) state \<times> ('U, 'C, 'Out) Noninterference.aprop set) stream list \<Rightarrow> bool"   and \<psi> :: "(('St, 'U, 'C) state \<times> ('U, 'C, 'Out) Noninterference.aprop set) stream list \<Rightarrow> (('St, 'U, 'C) state \<times> ('U, 'C, 'Out) Noninterference.aprop set) stream list \<Rightarrow> bool" assumes "\<And>\<pi> \<pi>'. wfp UNIV \<pi> \<and> wfp UNIV \<pi>' \<longrightarrow> \<phi> [] [\<pi>, \<pi>'] = f (stateOf \<pi>) (stateOf \<pi>') \<and> \<psi> [] [\<pi>, \<pi>'] = g (stateOf \<pi>) (stateOf \<pi>')" shows "fall2 (\<lambda>\<pi>' \<pi> \<pi>l. imp (alw (\<phi> \<pi>l)) (alw (\<psi> \<pi>l)) (\<pi>l @ [\<pi>, \<pi>'])) [] = (\<forall>\<pi> \<pi>'. wfp UNIV \<pi> \<and> wfp UNIV \<pi>' \<and> stateOf \<pi> = s0 \<and> stateOf \<pi>' = s0 \<longrightarrow> (\<forall>i. (\<forall>j\<le>i. f (fst (\<pi> !! j)) (fst (\<pi>' !! j))) \<longrightarrow> g (fst (\<pi> !! i)) (fst (\<pi>' !! i))))"
simps_3: fixes x :: "'sz"   and y :: "'sz"   and z :: "'sz"   and \<theta> :: "('sf, 'sz) trm" shows "PUrename x y (DiffAssign z \<theta>) = DiffAssign (swap x y z) (TUrename x y \<theta>)"
n_def: shows "n = nat \<lfloor>x\<rfloor>"
bind_def: fixes xs :: "'h list"   and f :: "'h \<Rightarrow> 'g list" shows "List.bind xs f = concat (map f xs)"
verit_comp_simplify_2: fixes a :: "'a" shows "a \<le> a"
ipresIGVar_def: fixes h :: "'l \<Rightarrow> 'm"   and MOD :: "('n, 'o, 'p, 'q, 'r, 's, 'l, 't, 'u) model_scheme"   and MOD' :: "('v, 'w, 'p, 'x, 'y, 's, 'm, 'z, 'aa) model_scheme" shows "ipresIGVar h MOD MOD' \<equiv> \<forall>xs x. h (igVar MOD xs x) = igVar MOD' xs x"
connected_iff_connected_component_eq: fixes S :: "'c set" shows "connected S = (\<forall>x\<in>S. \<forall>y\<in>S. connected_component_set S x = connected_component_set S y)"
smult_smult_mat: fixes A :: "complex mat"   and nr :: "nat"   and n :: "nat"   and k :: "complex"   and l :: "complex" assumes "A \<in> carrier_mat nr n" shows "k \<cdot>\<^sub>m (l \<cdot>\<^sub>m A) = k * l \<cdot>\<^sub>m A"
oalist_inv_raw_tl: fixes x :: "'o"   and xs :: "('a \<times> 'c) list" assumes "oalist_inv_raw x xs" shows "oalist_inv_raw x (tl xs)"
if_cong: fixes b :: "bool"   and c :: "bool"   and x :: "'h"   and u :: "'h"   and y :: "'h"   and v :: "'h" assumes "b = c"   and "c \<Longrightarrow> x = u"   and "\<not> c \<Longrightarrow> y = v" shows "(if b then x else y) = (if c then u else v)"
simps_1: fixes r :: "real" shows "eln (ennreal r) = (if r \<le> 0 then - \<infinity> else ereal (ln r))"
Beta: fixes r :: "dB"   and s :: "dB"   and ss :: "dB list" assumes "WN (r[s/0] \<degree>\<degree> ss)" shows "WN (Abs r \<degree> s \<degree>\<degree> ss)"
obsp_obs_eq: fixes n :: "'node"   and S :: "'node set" shows "obsp n (\<lambda>x. x \<in> S) = (\<lambda>x. x \<in> obs n S)"
med0n0i_def: shows "med0n0i \<equiv> id"
circ_isotone: fixes x :: "'a"   and y :: "'a" assumes "x \<le> y" shows "top \<le> top"
worklists_vacant_to_trans: fixes summary :: "'a \<Rightarrow> 'a \<Rightarrow> 'b antichain"   and results_in :: "'c \<Rightarrow> 'b \<Rightarrow> 'c"   and c :: "('a, 'c) configuration"   and t :: "'c"   and t' :: "'c" assumes "dataflow_topology summary results_in"   and "dataflow_topology.worklists_vacant_to summary results_in c t"   and "t' \<le> t" shows "dataflow_topology.worklists_vacant_to summary results_in c t'"