hcju/isabelleps huggingface dataset:
heap_is_wellformed_def: shows "heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M = (\<lambda>h. local.CD.a_acyclic_heap h \<and> local.CD.a_all_ptrs_in_heap h \<and> local.CD.a_distinct_lists h \<and> local.CD.a_owner_document_valid h)"
rupture_is_field_iff_pirreducible: fixes K :: "'a set"   and p :: "'a list" assumes "subfield K R"   and "p \<in> carrier (K [X])" shows "field (Rupt K p) = pirreducible K p"
rel_cases: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'c list"   and b :: "'d list" assumes "list_all2 R a b" obtains "a = []" and "b = []"   | x1 :: "'c" and x2 :: "'c list" and y1 :: "'d" and y2 :: "'d list" where "a = x1 # x2" and "b = y1 # y2" and "R x1 y1" and "list_all2 R x2 y2"
intro: fixes A :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and F :: "'a \<Rightarrow> 'b" assumes "faithful_functor A B F"   and "full_functor A B F" shows "fully_faithful_functor A B F"
is_strict_rewrite_ord_irrefl: fixes rel :: "'t \<times> ('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> 't \<times> ('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> bool"   and x :: "'t \<times> ('a \<Rightarrow>\<^sub>0 'b)" assumes "is_strict_rewrite_ord rel" shows "\<not> rel x x"
harm_expansion: fixes n :: "nat"   and N :: "nat" assumes "0 < n"   and "0 < N" shows "harm n = ln (real n) + euler_mascheroni + 1 / (2 * real n) - (\<Sum>i = 1..N. bernoulli (2 * i) / (2 * real i * real n ^ (2 * i))) - harm_remainder N n"
lm102: fixes x :: "'b"   and X :: "'b set"   and Y :: "'b set" assumes "x \<in> X \<union> Y" shows "(X <| Y) x = chi X x"
lunit_char_4: fixes f :: "'a" assumes "ide f" shows "\<exists>!\<mu>. \<guillemotleft>\<mu> : trg f \<star> f \<Rightarrow> f\<guillemotright> \<and> trg f \<star> \<mu> = (\<i>[trg f] \<star> f) \<cdot> local.inv \<a>[trg f, trg f, f]"
stationary_distribution_def: fixes K :: "'a \<Rightarrow> 'a pmf"   and N :: "'a pmf" shows "MC_syntax.stationary_distribution K N = (N = N \<bind> K)"
wf_subset: fixes r :: "('g \<times> 'g) set"   and p :: "('g \<times> 'g) set" assumes "wf r"   and "p \<subseteq> r" shows "wf p"
strict_trans: fixes b :: "'b Word.word"   and a :: "'b Word.word"   and c :: "'b Word.word" assumes "b <s a"   and "c <s b" shows "c <s a"
hmset_of_enat_1: shows "hmset_of_enat 1 = 1"
nth_append_take: fixes i :: "nat"   and xs :: "'c list"   and y :: "'c"   and ys :: "'c list" assumes "i \<le> length xs" shows "(take i xs @ y # ys) ! i = y"
pair_commute_pmf: fixes x :: "'d pmf"   and y :: "'e pmf" shows "pair_pmf x y = map_pmf (\<lambda>(x, y). (y, x)) (pair_pmf y x)"
natural_transformation_axioms: shows "natural_transformation (\<cdot>\<^sub>A) (\<cdot>\<^sub>A) A.map A.map A.map"
lhs: shows "sqrt (Sup {(f i)\<^sup>2 |i. i \<in> A}) = \<bar>f i\<bar>"
H_is_subgroup: fixes G :: "('a, 'b) monoid_scheme"   and p :: "nat"   and a :: "nat"   and m :: "nat"   and M1 :: "'a set"   and M :: "'a set set" assumes "sylow_central G p a m M1 M" shows "subgroup {g \<in> carrier G. M1 #>\<^bsub>G\<^esub> g = M1} G"
vc_sound: fixes C :: "acom"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> bool" assumes "vc C Q" shows "\<turnstile> {pre C Q} strip C {Q}"
subst_cls_mset_comp_subst: fixes CC :: "'a literal multiset multiset"   and \<tau> :: "'s"   and \<sigma> :: "'s" shows "CC \<cdot>cm (\<tau> \<odot> \<sigma>) = CC \<cdot>cm \<tau> \<cdot>cm \<sigma>"
nprv_exiE0: fixes x :: "'var"   and \<phi> :: "'fmla"   and F :: "'fmla set"   and \<psi> :: "'fmla" assumes "exi x \<phi> \<in> F"   and "nprv (insert \<phi> F) \<psi>"   and "F \<subseteq> fmla"   and "finite F"   and "\<phi> \<in> fmla"   and "x \<in> var"   and "\<psi> \<in> fmla"   and "\<forall>x\<in>F. x \<notin> Fvars x"   and "x \<notin> Fvars \<psi>" shows "nprv F \<psi>"
group_ringD: shows "ActingGroup.group_ring = {x. supp (aezfun x) \<subseteq> G}"
word_upto_set_eq: fixes a :: "'a word"   and b :: "'a word"   and x :: "'a word" assumes "a \<le> b" shows "(x \<in> set (word_upto a b)) = (a \<le> x \<and> x \<le> b)"
validn_valid: fixes C :: "Class"   and m :: "Method"   and l :: "int"   and A :: "(Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool"   and B :: "(Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<times> Val \<Rightarrow> bool"   and I :: "(Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool" assumes "\<forall>K. \<Turnstile>\<^sub>K \<lbrace> A \<rbrace> C , m , l \<lbrace> B \<rbrace> I " shows " \<Turnstile> \<lbrace> A \<rbrace> C , m , l \<lbrace> B \<rbrace> I"
butlast_rev: fixes xs :: "'g list" shows "butlast (rev xs) = rev (tl xs)"
distinct_count_atmost_1: fixes x :: "'b list" shows "distinct x = (\<forall>a. count (mset x) a = (if a \<in> set x then 1 else 0))"
induct: fixes P :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a list \<Rightarrow> bool"   and a0 :: "'a \<Rightarrow> 'b"   and a1 :: "'a list" assumes "\<And>f. P f []"   and "\<And>f x xs. P f (filter (\<lambda>y. f x \<noteq> f y) xs) \<Longrightarrow> P f (x # xs)" shows "P a0 a1"
distinct_11: fixes x3 :: "int"   and x4 :: "int" shows "Greater' x3 \<noteq> Smaller' x4"
assms_7: shows "trace (\<rho>\<^sub>1 + \<rho>\<^sub>2) \<le> 1"
transitionCases: fixes P :: "pi"   and \<alpha> :: "freeRes"   and P' :: "pi"   and F :: "freeRes \<Rightarrow> pi \<Rightarrow> bool" assumes "P \<Longrightarrow>\<^sup>^\<alpha> \<prec> P'"   and "P \<Longrightarrow>\<alpha> \<prec> P' \<Longrightarrow> F \<alpha> P'"   and "F (\<tau>) P" shows "F \<alpha> P'"
prefix_length_less: fixes xs :: "'c list"   and ys :: "'c list" assumes "strict_prefix xs ys" shows "length xs < length ys"
reachable_imp_dfs: fixes y :: "'rep"   and xs :: "'rep list" assumes "k_isNode y"   and "list_all k_isNode xs"   and "y \<in> KBPAlg.reachable (set xs)" shows "\<exists>y'. simAbs y' = simAbs y \<and> k_memb y' (KBPAlg.dfs k_empt xs)"
DIAndCurry12_def: shows "DIAndCurry12 = \<lparr>SFunctions = Map.empty, SPredicates = Map.empty, SContexts = \<lambda>C. if C = pid1 then Some (Pc (Inl pid1)) else if C = pid2 then Some (Pc (Inl pid2) \<rightarrow> (Pc (Inl pid1) && Pc (Inl pid2))) else None, SPrograms = Map.empty, SODEs = Map.empty\<rparr>"
dependent_explicit: fixes s :: "'b set" shows "m1.dependent s = (\<exists>t u. finite t \<and> t \<subseteq> s \<and> (\<Sum>v\<in>t. u v *a v) = (0::'b) \<and> (\<exists>v\<in>t. u v \<noteq> (0::'a)))"
cResB: fixes P :: "pi"   and a :: "name"   and x :: "name"   and P' :: "pi"   and y :: "name"   and C :: "'a" assumes "P \<longmapsto> a<x> \<prec> P'"   and "y \<noteq> a"   and "y \<noteq> x"   and "y \<sharp> C"   and "\<And>C. F C P a x P'" shows "F C (<\<nu>y>P) a x (<\<nu>y>P')"
independent_inj_image: fixes S :: "'b set" assumes "vs1.independent S"   and "inj f" shows "vs2.independent (f ` S)"
intD: fixes A :: "'b \<Rightarrow> bool"   and w :: "'b" assumes "\<turnstile> A" shows "A w"
cases: fixes a1 :: "'action stream"   and a2 :: "'state" assumes "R.run a1 a2" obtains a :: "'action" and p :: "'state" and r :: "'action stream" where "a1 = a ## r" and "a2 = p" and "ren a p" and "R.run r (ex a p)"
None_eq_map_option_iff: fixes f :: "'f \<Rightarrow> 'e"   and x :: "'f option" shows "(None = map_option f x) = (x = None)"
mode_update_add_anno: fixes aexp_vars :: "'a \<Rightarrow> 'b set"   and bexp_vars :: "'c \<Rightarrow> 'b set"   and some_val :: "'d"   and some_val' :: "'d"   and ev\<^sub>A :: "('b \<Rightarrow> 'd) \<Rightarrow> 'a \<Rightarrow> 'd"   and ev\<^sub>B :: "('b \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> bool"   and dma :: "'b \<Rightarrow> Sec"   and mds :: "Mode \<Rightarrow> 'b set"   and \<Gamma> :: "'b \<Rightarrow> Sec option"   and upd :: "'b ModeUpd" assumes "sifum_types aexp_vars bexp_vars some_val some_val' ev\<^sub>A ev\<^sub>B"   and "sifum_types.mds_consistent dma mds \<Gamma>" shows "sifum_types.mds_consistent dma (sifum_lang.update_modes upd mds) (sifum_types.add_anno dma \<Gamma> upd)"
cless_eq_set_def: shows "cless_eq_set = cless_eq"
REP_simps_3: fixes A :: "('a, 'a set) Maps.arr" assumes "Maps.arr A" shows "trg (Maps.REP A) = Maps.Cod A"
to_nat_def: fixes a :: "'b" shows "Bij_Nat.to_nat a = (SOME i. univ_list ! i = a \<and> i < length univ_list)"
adds_ppE: fixes pair_of_term :: "'a \<Rightarrow> 'b \<times> 'c"   and term_of_pair :: "'b \<times> 'c \<Rightarrow> 'a"   and t :: "'b"   and v :: "'a" assumes "term_powerprod pair_of_term term_of_pair"   and "term_powerprod.adds_pp pair_of_term t v" obtains u :: "'a" where "v = term_powerprod.splus pair_of_term term_of_pair t u"
cases: fixes a :: "'c"   and r :: "('c \<times> 'c) set" assumes "a \<in> Wellfounded.acc r" obtains x :: "'c" where "a = x" and "\<And>y. (y, x) \<in> r \<Longrightarrow> y \<in> Wellfounded.acc r"
not_empty_irreducible: fixes C :: "'a set"   and K :: "'a \<Rightarrow> 'a pmf" assumes "C \<in> UNIV // MC_syntax.communicating K" shows "C \<noteq> {}"
atLeast_upt: fixes n :: "nat" shows "{..<n} = set [0..<n]"
finite_init_pred: fixes prb :: "('d, 'e, 'f) pre_RedBlack" assumes "finite_RedBlack prb" shows "finite (pred (init_conf prb))"
elims_3: fixes x :: "nat \<times> cell list \<times> cell list" assumes "\<not> is_final x" obtains s :: "nat" and l :: "cell list" and r :: "cell list" where "x = (s, l, r)" and "s \<noteq> 0"
rel_converter_neg_distr: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and A' :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and R :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and R' :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and B :: "'g \<Rightarrow> 'k \<Rightarrow> bool"   and B' :: "'k \<Rightarrow> 'i \<Rightarrow> bool"   and C :: "'h \<Rightarrow> 'l \<Rightarrow> bool"   and C' :: "'l \<Rightarrow> 'j \<Rightarrow> bool" assumes "left_unique A"   and "right_total A"   and "right_unique A'"   and "left_total A'"   and "left_unique R"   and "right_total R"   and "right_unique R'"   and "left_total R'" shows "rel_converter (A OO A') (B OO B') (C OO C') (R OO R') \<le> rel_converter A B C R OO rel_converter A' B' C' R'"
that: fixes u :: "'a list"   and v :: "'a list" assumes "u \<in> A"   and "v \<in> B"   and "w = u @ v" shows "thesis"
strategy_attracts_lset: fixes A :: "'a set"   and W :: "'a set" assumes "strategy_attracts p \<sigma> A W"   and "P $ n \<in> A" shows "lset P' \<inter> W \<noteq> {}"
pure_conj_right': fixes Q :: "'a \<Rightarrow> bool"   and P' :: "'a \<Rightarrow> bool"   and Q' :: "bool" shows "(Q \<and>* (\<lambda>s. P' s \<and> Q')) = (\<lambda>s. Q' \<and> (Q \<and>* P') s)"
comp_in_homI: fixes f :: "('a, 'a set) Maps.arr"   and a :: "('a, 'a set) Maps.arr"   and b :: "('a, 'a set) Maps.arr"   and g :: "('a, 'a set) Maps.arr"   and c :: "('a, 'a set) Maps.arr" assumes "Maps.in_hom f a b"   and "Maps.in_hom g b c" shows "Maps.in_hom (g \<odot> f) a c"
hom_mult: fixes x :: "int poly"   and y :: "int poly" shows "of_int_poly (x * y) = of_int_poly x * of_int_poly y"
prems: fixes a :: "'a"   and a' :: "'a" assumes "a \<in> insert x F"   and "a' \<in> insert x F"   and "a \<noteq> a'" shows "\<langle>f a, f a'\<rangle> = 0"
listspE: fixes A :: "'a \<Rightarrow> bool"   and x :: "'a"   and l :: "'a list" assumes "listsp A (x # l)" obtains "A x" and "listsp A l"
distributed_cong_density: fixes f :: "'d \<Rightarrow> ennreal"   and g :: "'d \<Rightarrow> ennreal"   and N :: "'d measure"   and M :: "'e measure"   and X :: "'e \<Rightarrow> 'd" assumes "AE x in N. f x = g x"   and "g \<in> borel_measurable N"   and "f \<in> borel_measurable N" shows "distributed M N X f = distributed M N X g"
emeasure_le_1: fixes M :: "'c measure"   and S :: "'c set" assumes "prob_space M" shows "emeasure M S \<le> 1"
simps_1: fixes step :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'a"   and t :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> nat"   and s :: "'a" shows "On_Off.T step t s [] [] = 0"
B_def: shows "B = [b, b', b'']\<^sub>\<circ>"
Tan_def: fixes x :: "floatarith" shows "Tan x = Sin x / Cos x"
field_tOp_assoc: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<in> carrier K"   and "y \<in> carrier K"   and "z \<in> carrier K" shows "x \<cdot>\<^sub>r y \<cdot>\<^sub>r z = x \<cdot>\<^sub>r (y \<cdot>\<^sub>r z)"
B_A_step: fixes a' :: "'c"   and b' :: "'c"   and a :: "'b" assumes "C a' b'"   and "sim2 a a'"   and "PB a"   and "PC a'" shows "\<exists>b. B a b \<and> sim2 b b'"
obj_ty_def: fixes obj :: "char list \<times> (char list \<times> char list \<Rightarrow> val option)" shows "obj_ty obj \<equiv> Class (fst obj)"
abs_eq: fixes xa :: "int"   and x :: "int extended \<times> int extended" shows "in_ivl xa (abs_ivl x) = (case x of (l, h) \<Rightarrow> l \<le> Fin xa \<and> Fin xa \<le> h)"
rel_witness_spmf2: fixes A :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and p :: "'d spmf"   and q :: "'e spmf" assumes "rel_spmf A p q" shows "rel_spmf (\<lambda>(a, b') b. b = b' \<and> A a b') (rel_witness_spmf A (p, q)) q"
carrier_subgroup_generated_subset: fixes A :: "'a set" shows "carrier (subgroup_generated B A) \<subseteq> carrier B"
at_le: fixes "open" :: "'b set \<Rightarrow> bool"   and s :: "'b set"   and t :: "'b set"   and x :: "'b" assumes "class.topological_space open"   and "s \<subseteq> t" shows "topological_space.at_within open x s \<le> topological_space.at_within open x t"
inject: fixes x1 :: "'a"   and x2 :: "'b"   and y1 :: "'a"   and y2 :: "'b" shows "(Monoid_ap x1 x2 = Monoid_ap y1 y2) = (x1 = y1 \<and> x2 = y2)"
replicate_eq_replicate: fixes m :: "nat"   and x :: "'a"   and n :: "nat"   and y :: "'a" shows "(x \<up> m = y \<up> n) = (m = n \<and> (m \<noteq> 0 \<longrightarrow> x = y))"
unfold1_def: fixes s1 :: "'e \<Rightarrow> ('d, 'e, 'f) F1"   and s2 :: "'f \<Rightarrow> ('d, 'e, 'f) F2"   and x :: "'e" shows "unfold1 s1 s2 x = Abs_JF1 ((Equiv_Relations.proj (lsbis1 carT1 carT2 strT1 strT2) \<circ> beh1 s1 s2) x)"
NEG_def: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and B :: "'a \<Rightarrow> 'b \<Rightarrow> bool" shows "Lifting.NEG A B \<equiv> B \<le> A"
ideal_def: fixes R :: "('d, 'e) Ring_scheme"   and I :: "'d set" shows "ideal R I = (R +> I \<and> (\<forall>r\<in>carrier R. \<forall>x\<in>I. r \<cdot>\<^sub>r\<^bsub>R\<^esub> x \<in> I))"
eq: shows "a \<cdot>| L = b \<cdot>| L"
closed_consistent: fixes e :: "exp" assumes "fv e = {}" shows "consistent (\<bottom>, \<bottom>, 0, [], []) ([], e, [])"
less_le_trans: fixes x :: "'b"   and y :: "'b"   and z :: "'b" assumes "x < y"   and "y \<le> z" shows "x < z"
P_def: shows "P = (\<lambda>t. Re ((g1 +++ g2) t - z) = 0 \<and> 0 < t \<and> t < 1 \<or> t = 0 \<or> t = 1)"
lsbis1_incl: fixes B1 :: "'d set"   and B2 :: "'e set"   and s1 :: "'d \<Rightarrow> ('f, 'd, 'e) F1"   and s2 :: "'e \<Rightarrow> ('g, 'd, 'e) F2" shows "lsbis1 B1 B2 s1 s2 \<subseteq> B1 \<times> B1"
lt_eq: fixes x :: "'a"   and y :: "'a" assumes "x \<in> D"   and "y \<in> D" shows "(cmp x y = LESS) = (cmp y x = GREATER)"
neg_numeral_le_numeral: fixes 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 sgn :: "'c \<Rightarrow> 'c"   and abs :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and m :: "num"   and n :: "num" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less_eq (uminus (numeral.numeral one plus m)) (numeral.numeral one plus n)"
gap_relabel_measure: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and s :: "nat"   and t :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'a"   and l :: "nat \<Rightarrow> nat"   and u :: "nat" assumes "Height_Bounded_Labeling c s t f l"   and "Network.relabel_precond c t f l u" shows "Network.sum_heights_measure c (Network.gap_relabel_effect c f l u) < Network.sum_heights_measure c l"
circline_type_cmat_def: fixes H :: "complex \<times> complex \<times> complex \<times> complex" shows "circline_type_cmat H = sgn (Re (mat_det H))"
map_index'_is_NilD: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'c \<Rightarrow> 'b"   and xs :: "'c list" assumes "map_index' n f xs = []" shows "xs = []"
HPhase0Read_valueChosen: fixes s :: "state"   and s' :: "state"   and q :: "Proc"   and d :: "Disk"   and v :: "InputsOrNi" assumes "HPhase0Read s s' q d"   and "valueChosen s v"   and "v \<in> Inputs" shows "valueChosen s' v"
rotate_minus1': fixes n :: "nat"   and ls :: "'c list"   and ms :: "'c list" assumes "0 < n"   and "rotate n ls = ms" shows "rotate (n - 1) ls = rotate (|ms| - 1) ms"
atms_of_def: fixes C :: "'b literal multiset" shows "atms_of C = atm_of ` set_mset C"
tau_int: fixes x :: "'a" shows "\<tau> x \<le> x"
size_single: fixes b :: "'b" shows "size {#b#} = 1"
l_set_disconnected_nodes_get_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   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" shows "l_set_disconnected_nodes_get_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf get_disconnected_nodes get_disconnected_nodes_locs set_disconnected_nodes set_disconnected_nodes_locs \<equiv> l_get_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf get_disconnected_nodes get_disconnected_nodes_locs \<and> l_set_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf set_disconnected_nodes set_disconnected_nodes_locs"
IH: assumes "\<not> is_all t" shows "strip_all_vars (mk_all_list vs t) = rev (map snd vs)"
inversion_def: fixes dc :: "nat option \<times> bool list" shows "inversion dc = map (\<lambda>c. (payer dc = Some c) \<noteq> (coin dc c \<noteq> coin dc (c + 1))) [0..<n]"
execute_parallel_operators_strips_none_if_contraposition: fixes s :: "'b \<Rightarrow> bool option"   and ops :: "'b strips_operator list"   and v :: "'b" assumes "execute_parallel_operator s ops v \<noteq> None" shows "(\<exists>op\<in>set ops. v \<in> set (add_effects_of op) \<or> v \<in> set (delete_effects_of op)) \<or> s v \<noteq> None"
inorder_split: fixes t :: "('b \<times> nat) tree" shows "Tree2.inorder (split t) = Tree2.inorder t"
\<Phi>_def: fixes f :: "'t" shows "\<Psi> f = (if S'.arr f then S.mkArr (S.set (\<Psi>o (S'.dom f))) (S.set (\<Psi>o (S'.cod f))) (\<Psi>a f) else S.null)"
function_smult_closed: fixes a :: "nat \<Rightarrow> int"   and f :: "(nat \<Rightarrow> int) \<Rightarrow> nat \<Rightarrow> int" assumes "a \<in> carrier Zp"   and "f \<in> carrier (function_ring (carrier Zp) Zp)" shows "a \<odot>\<^bsub>function_ring (carrier Zp) Zp\<^esub> f \<in> carrier (function_ring (carrier Zp) Zp)"
normalizability_E: fixes a :: "'c"   and b :: "'c"   and A :: "('c \<times> 'c) set" assumes "(a, b) \<in> A\<^sup>!" obtains "(a, b) \<in> A\<^sup>*" and "b \<in> NF A"
semiconfluentp_def: fixes r :: "'d \<Rightarrow> 'd \<Rightarrow> bool" shows "semiconfluentp r = (r\<inverse>\<inverse> OO r\<^sup>*\<^sup>* \<le> r\<^sup>*\<^sup>* OO r\<inverse>\<inverse>\<^sup>*\<^sup>*)"
cnj_mult_self: fixes k :: "nat" assumes "coprime k n" shows "cnj (\<chi> k) * \<chi> k = 1"
le_less_trans: fixes x :: "'a word"   and y :: "'a word"   and z :: "'a word" assumes "x \<le>s y"   and "y <s z" shows "x <s z"
hyps: fixes xs :: "'a llist" assumes "enat (n + m) < llength xs" shows "lnth (ldropn n xs) m = lnth xs (m + n)"
assms_4: shows "finite (\<Union> (block_nominals ` extend A S f n))"
sup_set_def: fixes A :: "'a set"   and B :: "'a set" shows "A \<union> B = Collect (sup (\<lambda>x. x \<in> A) (\<lambda>x. x \<in> B))"
not_strict_par2: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p"   and X :: "'p" assumes "A B Par C D"   and "Col A B X"   and "Col C D X" shows "Col A B D"
to_nat_from_nat_id: fixes x :: "nat" assumes "x < CARD('d)" shows "mod_type_class.to_nat (mod_type_class.from_nat x) = x"
k_and_op_are: shows "(k, op) \<in> {0..<length \<pi>} \<times> set (\<Pi>\<^sub>\<O>)"
le_fin_eq_fin_transp: fixes w\<^sub>1 :: "'item list"   and w\<^sub>2 :: "'item list"   and w\<^sub>3 :: "'item list" assumes "w\<^sub>1 \<preceq>\<^sub>F w\<^sub>2"   and "w\<^sub>2 =\<^sub>F w\<^sub>3" shows "w\<^sub>1 \<preceq>\<^sub>F w\<^sub>3"
cong_mult_self_poly: fixes a :: "'c poly"   and m :: "'c poly" shows "[a * m = 0] (mod m)"
multi_winker: fixes k :: "nat" shows "\<beta> \<squnion> k \<times> \<alpha> = \<beta>"
progress: fixes \<Gamma> :: "char list \<Rightarrow> ty"   and c :: "com"   and s :: "char list \<Rightarrow> val" assumes "\<Gamma> \<turnstile> c"   and "\<Gamma> \<turnstile> s"   and "c \<noteq> SKIP" shows "\<exists>cs'. (c, s) \<rightarrow> cs'"
cong_mult_self_right: fixes b :: "'a"   and a :: "'a" shows "[b * a = 0::'a] (mod a)"
locally_included_def: fixes R1 :: "('b \<times> 'b) set"   and R2 :: "('b \<times> 'b) set"   and A :: "'b multiset"   and B :: "'b multiset" shows "locally_included R1 R2 A B \<equiv> \<forall>t u. t \<in># A \<longrightarrow> u \<in># B \<longrightarrow> (t, u) \<in> R1 \<longrightarrow> (t, u) \<in> R2"
dlo_axioms_def: fixes le :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "dlo_axioms le \<equiv> \<forall>x y. le x y \<or> le y x"
discr_discr0: fixes c :: "('test, 'atom) com" assumes "discr c" shows "discr0 c"
someI2_ex: fixes P :: "'c \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> bool" assumes "\<exists>a. P a"   and "\<And>x. P x \<Longrightarrow> Q x" shows "Q (SOME x. P x)"
f'': fixes x :: "real" assumes "x \<in> C" shows "(f' has_real_derivative f'' x) (at x)"
induced_modulus_altdef2: fixes d :: "nat" assumes "d dvd d__"   and "d \<noteq> 1" shows "\<Phi>.induced_modulus d = (\<exists>\<Phi>'. dcharacter d \<Phi>' \<and> (\<forall>k. \<Phi> k = \<Phi>' k * principal_dchar d__ k))"
map\<^sub>0_simps_3: fixes a :: "'d" assumes "D.obj a" shows "trg\<^sub>D (\<eta>'.map\<^sub>0 a) = I\<^sub>D.map\<^sub>0 a"
le_supI1: fixes sup :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and less_eq :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and less :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and x :: "'b::type"   and a :: "'b::type"   and b :: "'b::type" assumes "class.semilattice_sup (sup::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (less::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool)"   and "(less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (x::'b::type) (a::'b::type)" shows "(less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (x::'b::type) ((sup::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (a::'b::type) (b::'b::type))"
qball: shows "\<forall>a\<in>proots q. P a"
Output: fixes a :: "name"   and b :: "name"   and P :: "pi" shows "a{b}.P \<Longrightarrow>\<^sub>l\<^sup>^a[b] \<prec> P"
subg_asubg: fixes G :: "('c, 'd) aGroup_scheme"   and H :: "'c set" assumes "b_ag G \<guillemotright> H " shows "G +> H"
assms_4: assumes "m < n \<and> x < y"   and "x = m"   and "y \<noteq> m"   and "y \<noteq> n" shows "P (arr y) (arr n)"