hcju/isabelleps huggingface dataset:
intro: fixes set_disconnected_nodes :: "(_) document_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and set_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" assumes "\<And>h document_ptr disc_nodes h'. h \<turnstile> set_disconnected_nodes document_ptr disc_nodes \<rightarrow>\<^sub>h h' \<Longrightarrow> h' \<turnstile> get_disconnected_nodes document_ptr \<rightarrow>\<^sub>r disc_nodes"   and "\<And>ptr ptr' w h h' r. \<lbrakk>ptr \<noteq> ptr'; w \<in> set_disconnected_nodes_locs ptr; h \<turnstile> w \<rightarrow>\<^sub>h h'; r \<in> get_disconnected_nodes_locs ptr'\<rbrakk> \<Longrightarrow> r h h'" shows "l_set_disconnected_nodes_get_disconnected_nodes_axioms get_disconnected_nodes get_disconnected_nodes_locs set_disconnected_nodes set_disconnected_nodes_locs"
fun_graph_def: fixes f :: "'a \<Rightarrow> 'b" shows "fun_graph f = {(x, f x) |x. True}"
assms_4: shows "h \<turnstile> P x \<rightarrow>\<^sub>r True"
ladder_stepdown_n: fixes L :: "(nat \<times> nat \<times> nat) list"   and L' :: "(nat \<times> nat \<times> nat) list"   and index :: "nat" assumes "1 < length L"   and "L' = ladder_stepdown L"   and "index < length L'" shows "ladder_n L' index = ladder_n L (Suc index) - ladder_stepdown_diff L"
t_stop_pos: assumes "v\<^sub>e \<noteq> 0" shows "0 < ego2.t_stop"
numgcdh0: fixes t :: "MIR.num"   and m :: "int" assumes "numgcdh t m = 0" shows "m = 0"
eq_upto_Suc: fixes g :: "nat \<Rightarrow> 'b"   and B :: "(nat \<Rightarrow> 'b) set"   and f :: "nat \<Rightarrow> 'b"   and i :: "nat" assumes "g \<in> eq_upto B f i"   and "g i = f i" shows "g \<in> eq_upto B f (Suc i)"
mpoly_coeff_Const: fixes c :: "'b"   and m :: "nat \<Rightarrow>\<^sub>0 nat" shows "MPoly_Type.coeff (Const c) m = (if m = 0 then c else (0::'b))"
degree_divisor1: fixes f :: "'b mod_ring poly"   and d :: "nat" assumes "irreducible f"   and "degree f = d" shows "f dvd monom 1 1 ^ CARD('b) ^ d - monom 1 1"
a_l_repr_independence: fixes y :: "'a"   and x :: "'a"   and H :: "'a set" assumes "y \<in> x <+ H"   and "x \<in> carrier R"   and "subgroup H (add_monoid R)" shows "x <+ H = y <+ H"
fixed: shows "(x ## x' \<and> x + x' = y) \<and> y ## y' \<and> y + y' = z"
state_iso_def: shows "state_iso \<equiv> \<lambda>(f, g). type_definition f g UNIV"
pullback_ide_self: fixes a :: "'a" assumes "C.ide a" shows "\<p>\<^sub>0[a, a] = \<p>\<^sub>1[a, a]"
ES_valid_def: fixes ES :: "'a ES_rec" shows "ES_valid ES \<equiv> es_inputs_are_events ES \<and> es_outputs_are_events ES \<and> es_inputs_outputs_disjoint ES \<and> traces_contain_events ES \<and> traces_prefixclosed ES"
map2_val_compat'_id: shows "map2_val_compat' id"
comp_char: fixes g :: "'a \<times> 'a \<times> 'a"   and f :: "'a \<times> 'a \<times> 'a" shows "B.VxVxV.comp g f = (if B.seq (fst g) (fst f) \<and> B.VxV.seq (snd g) (snd f) then (fst g \<cdot>\<^sub>B fst f, B.VxV.comp (snd g) (snd f)) else (B.null, B.VxV.null))"
rep_eq: fixes x :: "'c" shows "blinfun_apply (ode_d2 x) = (if x \<in> Csafe then ode_d 1 x else (\<lambda>_. 0))"
set2fun_def: fixes X :: "'a set" shows "set2fun X = (\<lambda>x. if s2p X x then \<eta> x else {})"
cs_done_implies_both_snapshotted_2: fixes t :: "('a, 'b, 'c) event list"   and i :: "nat"   and cid :: "nat"   and p :: "'a"   and q :: "'a" assumes "trace init t final"   and "snd (cs (S t i) cid) = Done"   and "i < length t"   and "channel cid = Some (p, q)" shows "ps (S t i) q \<noteq> None"
commutative_def: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'c" shows "commutative f \<equiv> \<forall>x y. f x y = f y x"
dg_Arr_in_Vset: fixes \<beta> :: "V" assumes "\<Z> \<beta>"   and "\<alpha> \<in>\<^sub>\<circ> \<beta>" shows "\<CC>\<lparr>Arr\<rparr> \<in>\<^sub>\<circ> Vset \<beta>"
congruence_rule_def: fixes t :: "'c"   and l :: "'c" shows "congruence_rule t l = transl_rule (A_Cmp (A_Cmp (A_Lbl t) (A_Lbl l)) (A_Lbl t) \<sqsubseteq> A_Lbl l)"
atMost_def: fixes u :: "'c list" shows "sublist_order.atMost u \<equiv> {x. sublist x u}"
simple_valid_cfg_iff: fixes K :: "'a \<Rightarrow> 'a pmf set"   and S :: "'a set"   and ct :: "'a \<Rightarrow> 'a pmf"   and s :: "'a" assumes "Finite_Markov_Decision_Process K S"   and "ct \<in> Pi S K" shows "(Finite_Markov_Decision_Process.simple K S ct s \<in> Finite_Markov_Decision_Process.valid_cfg K S) = (s \<in> S)"
case_pu_inf: fixes w1 :: "32 Word.word"   and P :: "32 Word.word \<Rightarrow> 32 Word.word \<Rightarrow> bool"   and w2 :: "32 Word.word" assumes "w1 = POS_INF \<Longrightarrow> P w1 w2"   and "w2 = POS_INF \<Longrightarrow> P w1 w2"   and "\<lbrakk>w1 = NEG_INF; w2 = NEG_INF\<rbrakk> \<Longrightarrow> P w1 w2"   and "\<lbrakk>w1 = NEG_INF; w2 \<noteq> POS_INF; w2 \<noteq> NEG_INF\<rbrakk> \<Longrightarrow> P w1 w2"   and "\<lbrakk>w1 \<noteq> POS_INF; w1 \<noteq> NEG_INF; w2 = NEG_INF\<rbrakk> \<Longrightarrow> P w1 w2"   and "\<lbrakk>w1 \<noteq> POS_INF; w1 \<noteq> NEG_INF; w2 \<noteq> POS_INF; w2 \<noteq> NEG_INF\<rbrakk> \<Longrightarrow> P w1 w2" shows "P w1 w2"
subseteq_guards_While: fixes c :: "('d, 'e, 'f) com"   and b :: "'d set"   and c' :: "('d, 'e, 'f) com" assumes "c \<subseteq>\<^sub>g While b c'" shows "\<exists>c''. c = While b c'' \<and> (c'' \<subseteq>\<^sub>g c')"
allQ_clearQuantifiers: fixes \<phi> :: "atom fm"   and xs :: "real list" assumes "\<And>xs. eval (clearQuantifiers \<phi>) xs = eval \<phi> xs" shows "eval (clearQuantifiers (AllQ \<phi>)) xs = eval (AllQ \<phi>) xs"
associated_trans: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<sim> b"   and "b \<sim> c"   and "a \<in> carrier G"   and "c \<in> carrier G" shows "a \<sim> c"
code: shows "compP\<^sub>2 P\<^sub>1,C,M,0 \<rhd> compE\<^sub>2 body"
existence_ivl_initial_time: fixes t0 :: "real"   and x0a :: "'a"   and x0 :: "'a \<Rightarrow>\<^sub>L 'a" assumes "t0 \<in> Y.existence_ivl0 x0a"   and "x0 \<in> UNIV" shows "t0 \<in> Y.mvar.existence_ivl x0a t0 x0"
valid_eval_list: fixes env :: "v sem_env"   and s :: "'a state"   and es :: "exp0 list" shows "evaluate_list True env s es (eval_list' env es s)"
reachable_init: fixes s :: "'c"   and A :: "('c, 'd) automaton"   and I :: "'d \<Rightarrow> bool" assumes "s \<in> init A" shows "s \<in> reachable A I"
uniformly_continuous_on_const: fixes s :: "'a set"   and c :: "'b" shows "uniformly_continuous_on s (\<lambda>x. c)"
eucl_truncate_down_prod_def: fixes p :: "nat"   and a :: "'a \<times> 'b" shows "eucl_truncate_down p a = (eucl_truncate_down p (fst a), eucl_truncate_down p (snd a))"
igConsIPresIGWlsSTR_imp_igConsIPresIGWls: fixes Delta :: "('b, 'c, 'd, 'e, 'f) signature"   and MOD :: "('b, 'c, 'd, 'e, 'f, 'g, 'h, 'i, 'j) model_scheme" assumes "FixSyn TYPE('a) Delta"   and "FixSyn.igConsIPresIGWlsSTR Delta MOD" shows "FixSyn.igConsIPresIGWls Delta MOD"
empty_not_add_mset: fixes a :: "'a"   and A :: "'a multiset" shows "{#} \<noteq> add_mset a A"
rsemimodule5: fixes s :: "'b" shows "\<lambda> (s, 0::'a) = \<dd>"
"case": fixes f :: "'f \<Rightarrow> 'e"   and x :: "'f" shows "(case object_ptr.Ext x of object_ptr.Ext x \<Rightarrow> f x) = f x"
simps_3: fixes a1 :: "aexp"   and a2 :: "aexp" shows "acomp (Plus a1 a2) = acomp a1 @ acomp a2 @ [ADD]"
winding_number_triangle: fixes z :: "complex"   and a :: "complex"   and b :: "complex"   and c :: "complex" assumes "z \<in> interior (convex hull {a, b, c})" shows "winding_number (linepath a b +++ linepath b c +++ linepath c a) z = (if 0 < Im ((b - a) * cnj (b - z)) then 1 else - 1)"
simps: fixes Y :: "char list" shows "upChar Y = (if Y \<noteq> [] \<and> 97 \<le> of_char (last Y) \<and> of_char (last Y) < 122 then butlast Y @ [char_of (of_char (last Y) + 1)] else Y @ ''a'')"
interpret_floatarith_subst_floatarith_idem: fixes fa :: "floatarith"   and vs :: "real list"   and s :: "nat \<Rightarrow> nat" assumes "max_Var_floatarith fa \<le> length vs"   and "\<And>j. j < max_Var_floatarith fa \<Longrightarrow> vs ! s j = vs ! j" shows "interpret_floatarith (subst_floatarith (\<lambda>i. Var (s i)) fa) vs = interpret_floatarith fa vs"
ccApprox_below_iff: fixes t :: "Vars.var ttree"   and G :: "CoCalls" shows "(ccApprox t \<sqsubseteq> G) = (\<forall>xs\<in>paths t. ccFromList xs \<sqsubseteq> G)"
ceq_pred_def: shows "CEQ('a Predicate.pred) = map_option predicate_eq (ID CEQ('a))"
not_neutral_contains_not_neutral: fixes g :: "'c \<Rightarrow> bool"   and A :: "'c set" assumes "F g A \<noteq> True" obtains a :: "'c" where "a \<in> A" and "g a \<noteq> True"
gallery_betw_map: fixes X :: "'b set set"   and Y :: "'c set set"   and f :: "'b \<Rightarrow> 'c"   and C :: "'b set"   and Cs :: "'b set list"   and D :: "'b set" assumes "ChamberComplexMorphism X Y f"   and "SimplicialComplex.maxsimpchain X (C # Cs @ [D])" shows "SimplicialComplex.maxsimpchain Y (f ` C # f \<Turnstile> Cs @ [f ` D])"
finite_params_extend: fixes S :: "('a, 'b) fm set"   and f :: "nat \<Rightarrow> ('a, 'b) fm"   and n :: "nat" assumes "finite (params S)" shows "finite (params (extend S f n))"
minus_def: fixes x :: "'a" shows "- x = Abs (- Rep x mod n)"
simps_7: fixes \<phi> :: "'a ltln"   and \<psi> :: "'a ltln" shows "rewrite_X_enat (\<phi> U\<^sub>n \<psi>) = combine mk_until (rewrite_X_enat \<phi>) (rewrite_X_enat \<psi>)"
l: assumes "degree p \<le> k - 1 \<and> (\<forall>x y. (x, y) \<in> set (zip (map (unity_root k) (map int [0..<k])) ws) \<longrightarrow> poly p x = y)"   and "degree (finite_fourier_poly ws) \<le> k - 1 \<and> (\<forall>x y. (x, y) \<in> set (zip (map (unity_root k) (map int [0..<k])) ws) \<longrightarrow> poly (finite_fourier_poly ws) x = y)" shows "p = finite_fourier_poly ws"
before_verticesFrom: fixes f :: "face"   and v :: "nat"   and w :: "nat" assumes "distinct (vertices f)"   and "v \<in> \<V> f"   and "w \<in> \<V> f"   and "v \<noteq> w" shows "before (verticesFrom f v) v w"
modifies_sym: fixes vs :: "char list set"   and a :: "char list \<Rightarrow> int \<Rightarrow> int"   and b :: "char list \<Rightarrow> int \<Rightarrow> int" assumes "modifies vs a b" shows "modifies vs b a"
elims: fixes x :: "nat"   and xa :: "nat list"   and y :: "(nat \<times> nat) convnet" assumes "deep_model' x xa = y" obtains Y :: "nat" where "x = Y__" and "xa = []" and "y = Input Y__"   | Y :: "nat" and r :: "nat" and rs :: "nat list" where "x = Y" and "xa = r # rs" and "y = Pool (deep_model Y r rs) (deep_model Y r rs)"
vdiffE: fixes x :: "V"   and A :: "V"   and B :: "V" assumes "x \<in>\<^sub>\<circ> A -\<^sub>\<circ> B" obtains "x \<in>\<^sub>\<circ> A" and "x \<notin>\<^sub>\<circ> B"
step_Fault_end: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and c\<^sub>1 :: "('b, 'a, 'c) com"   and s :: "('b, 'c) xstate"   and c\<^sub>1' :: "('b, 'a, 'c) com"   and s' :: "('b, 'c) xstate"   and f :: "'c" assumes "\<Gamma>\<turnstile> (c\<^sub>1, s) \<rightarrow> (c\<^sub>1', s')"   and "s' = Fault f" shows "s = Fault f \<or> (\<exists>g c x. redex c\<^sub>1 = Guard f g c \<and> s = Normal x \<and> x \<notin> g)"
simps_6: fixes b :: "nat"   and \<phi> :: "Formula.formula" shows "Formula.fvi b (formula.Neg \<phi>) = Formula.fvi b \<phi>"
echelon_form_upt_k_def: fixes A :: "(('d, 'e) vec, 'f) vec"   and k :: "nat" shows "echelon_form_upt_k A k = ((\<forall>i. is_zero_row_upt_k i k A \<longrightarrow> \<not> (\<exists>j>i. \<not> is_zero_row_upt_k j k A)) \<and> (\<forall>i j. i < j \<and> \<not> is_zero_row_upt_k i k A \<and> \<not> is_zero_row_upt_k j k A \<longrightarrow> (LEAST n. A $ i $ n \<noteq> (0::'d)) < (LEAST n. A $ j $ n \<noteq> (0::'d))))"
assms_1: fixes x :: "'a"   and y :: "'a" shows "boa x y \<longrightarrow> boa' x y"
simps_2: fixes r :: "('a \<times> 'a) set"   and n :: "nat" shows "lexn r (Suc n) = map_prod (\<lambda>(x, xs). x # xs) (\<lambda>(x, xs). x # xs) ` (r <*lex*> lexn r n) \<inter> {(xs, ys). length xs = Suc n \<and> length ys = Suc n}"
add_cf_val_hi: fixes c :: "nat \<times> (nat \<Rightarrow> 'a)"   and d :: "nat \<times> (nat \<Rightarrow> 'a)" assumes "fst c < fst d" shows "snd (add_cf S c d) (fst d) = snd d (fst d)"
ereal_uminus_reorder_2: fixes a :: "ereal"   and b :: "ereal" shows "(- a < b) = (- b < a)"
one_tag_const_def: shows "1 \<equiv> \<tau>\<^sub>c\<^sub>s\<^sub>t (1::'b)"
triangle_def: fixes A :: "Points"   and B :: "Points"   and C :: "Points" shows "triangle A B C \<equiv> distinct3 A B C \<and> line A B \<noteq> line A C"
assms_2: shows "S \<subseteq> r \<down>s T"
absorb_iff2: fixes b :: "'a"   and a :: "'a" shows "(b \<le> a) = (inf a b = b)"
comp_in_homI: fixes f :: "'a"   and a :: "'a"   and b :: "'a"   and g :: "'a"   and c :: "'a" assumes "\<guillemotleft>f : a \<Rightarrow>\<^sub>C b\<guillemotright>"   and "\<guillemotleft>g : b \<Rightarrow>\<^sub>C c\<guillemotright>" shows "\<guillemotleft>g \<cdot>\<^sub>C f : a \<Rightarrow>\<^sub>C c\<guillemotright>"
agg_delta: fixes j :: "'d"   and f :: "'d \<Rightarrow> 'c" shows "aggregation.sum_0 (\<lambda>l. if l = j then f l else zero) {l. True} = f j + bot"
dvd_dvd_smult: fixes a :: "'b"   and b :: "'b"   and f :: "'b poly"   and g :: "'b poly" assumes "a dvd b"   and "f dvd g" shows "smult a f dvd smult b g"
tendsto_def: fixes f :: "'b \<Rightarrow> 'a"   and l :: "'a"   and F :: "'b filter" shows "(f \<longlongrightarrow> l) F = (\<forall>S. open S \<longrightarrow> l \<in> S \<longrightarrow> (\<forall>\<^sub>F x in F. f x \<in> S))"
entropy_data_processing: fixes X :: "'a \<Rightarrow> 'd"   and f :: "'d \<Rightarrow> 'e" assumes "simple_function M X" shows "\<H>(f \<circ> X) \<le> \<H>(X)"
weak_action_modality_not_tau: fixes \<alpha> :: "'act"   and x :: "('idx, 'pred, 'act) formula" assumes "\<alpha> \<noteq> \<tau>" shows "\<langle>\<langle>\<alpha>\<rangle>\<rangle>x = weak_tau_modality (Act \<alpha> (weak_tau_modality x))"
Func_empty: fixes B :: "'d set" shows "Func {} B = {\<lambda>x. undefined}"
C_App1: fixes E :: "trm \<Rightarrow> trm"   and u :: "trm" assumes "E \<in> ctxt" shows "(\<lambda>t. E t \<bullet> u) \<in> ctxt"
a_get_owner_document_tups_def: shows "local.a_get_owner_document_tups = [(is_element_ptr\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r, local.a_get_owner_document\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r \<circ> the \<circ> cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r), (is_character_data_ptr\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r, local.a_get_owner_document\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r \<circ> the \<circ> cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r), (is_document_ptr\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r, local.a_get_owner_document\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r \<circ> the \<circ> cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r)]"
prems_1: shows "no_write_to_read_only_memory \<S> (x # sb)"
yoneda_lemma_def: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and S :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<phi> :: "'a \<times> 'a \<Rightarrow> 'a \<Rightarrow> 'b"   and F :: "'a \<Rightarrow> 'b"   and a :: "'a" shows "yoneda_lemma C S \<phi> F a \<equiv> (category C \<and> dual_category C) \<and> replete_set_category S \<and> set_valued_functor (dual_category.comp C) S F \<and> yoneda_functor_fixed_object C S \<phi> a"
mem_eq: shows "\<forall>x. type_max (to_total \<Gamma> x) mem\<^sub>1 = Low \<longrightarrow> mem\<^sub>1 x = mem\<^sub>2 x"
cccFix_nonrec_cong: fixes cccexp1 :: "exp \<Rightarrow> Arity \<rightarrow> (Vars.var \<Rightarrow> Arity\<^sub>\<bottom>) \<times> CoCalls"   and e :: "exp"   and cccexp2 :: "exp \<Rightarrow> Arity \<rightarrow> (Vars.var \<Rightarrow> Arity\<^sub>\<bottom>) \<times> CoCalls"   and x :: "Vars.var" assumes "cccexp1 e = cccexp2 e" shows "CoCallArityAnalysis.cccFix_nonrec cccexp1 x e = CoCallArityAnalysis.cccFix_nonrec cccexp2 x e"
accessible_from_commonD: fixes G :: "prog"   and S :: "qtname"   and m :: "qtname \<times> memberdecl"   and C :: "qtname" assumes "G \<turnstile> m of C accessible_from S" shows "G \<turnstile> m member_of C \<and> G \<turnstile> Class C accessible_in pid S"
case_eq_if: fixes f1 :: "'b"   and f2 :: "'c \<Rightarrow> 'c llist \<Rightarrow> 'b"   and llist :: "'c llist" shows "(case llist of LNil \<Rightarrow> f1 | LCons x xa \<Rightarrow> f2 x xa) = (if lnull llist then f1 else f2 (lhd llist) (ltl llist))"
independent_from_def: fixes d :: "nat"   and Y :: "'a \<times> 'a pdevs" shows "independent_from d Y = msum_aform d (0::'a, zero_pdevs) Y"
assms_1: shows "\<forall>a. lossless_spmf (D a)"
fls_divide_X_inv': fixes f :: "'a fls" assumes "inverse (1::'a) = (1::'a)" shows "f / fls_X_inv = fls_shift (- 1) f"
sat_primal_def: fixes A :: "'a mat"   and b :: "'a vec" shows "sat_primal A b = {x. [A*\<^sub>vx]\<le>b}"
split_face_edges_f21_bet_vs: fixes f :: "face"   and ram1 :: "nat"   and ram2 :: "nat"   and f12 :: "face"   and f21 :: "face" assumes "pre_split_face f ram1 ram2 []"   and "(f12, f21) = split_face f ram1 ram2 []"   and "between (vertices f) ram2 ram1 = []" shows "\<E> f21 = {(ram1, ram2), (ram2, ram1)}"
im_set_mono: fixes f :: "'e \<Rightarrow> 'f"   and A :: "'e set"   and B :: "'f set"   and A1 :: "'e set"   and A2 :: "'e set" assumes "f \<in> A \<rightarrow> B"   and "A1 \<subseteq> A2"   and "A2 \<subseteq> A" shows "f ` A1 \<subseteq> f ` A2"
dg_prod_3_is_arrI: fixes a :: "V"   and b :: "V"   and f :: "V"   and a' :: "V"   and b' :: "V"   and f' :: "V"   and a'' :: "V"   and b'' :: "V"   and f'' :: "V" assumes "f : a \<mapsto>\<^bsub>\<AA>\<^esub> b"   and "f' : a' \<mapsto>\<^bsub>\<BB>\<^esub> b'"   and "f'' : a'' \<mapsto>\<^bsub>\<CC>\<^esub> b''" shows "[f, f', f'']\<^sub>\<circ> : [a, a', a'']\<^sub>\<circ> \<mapsto>\<^bsub>\<AA> \<times>\<^sub>D\<^sub>G\<^sub>3 \<BB> \<times>\<^sub>D\<^sub>G\<^sub>3 \<CC>\<^esub> [b, b', b'']\<^sub>\<circ>"
connected_finite_iff_sing: fixes S :: "'b set" assumes "connected S" shows "finite S = (S = {} \<or> (\<exists>a. S = {a}))"
sum_upto_inverse_bigo: shows "sum_upto (\<lambda>i. 1 / real i) \<in> O(ln)"
ln_one_plus_pos_lower_bound: fixes x :: "real" assumes "0 \<le> x"   and "x \<le> 1" shows "x - x\<^sup>2 \<le> ln (1 + x)"
all_envir_def: fixes p :: "'c \<Rightarrow> bool" shows "all_envir p = (\<forall>x. p x)"
h1b_helper_les: fixes les :: "(real \<times> real \<times> real) list" assumes "\<forall>(a, b, c)\<in>set les. \<exists>x. \<forall>y<x. a * y\<^sup>2 + b * y + c < 0" shows "\<exists>y. \<forall>x<y. \<forall>(a, b, c)\<in>set les. a * x\<^sup>2 + b * x + c < 0"
dvd_div_gt0: fixes d :: "nat"   and n :: "nat" assumes "d dvd n"   and "0 < n" shows "0 < n div d"
list_max_def: fixes xs :: "'t list" shows "list_max xs \<equiv> foldl ord_term_lin.max min_term xs"
Zfun_le: fixes g :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and f :: "'b \<Rightarrow> 'd" assumes "Zfun g F"   and "\<forall>x. norm (f x) \<le> norm (g x)" shows "Zfun f F"
cont_def: fixes luba :: "'a set \<Rightarrow> 'a"   and orda :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and lubb :: "'b set \<Rightarrow> 'b"   and ordb :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b" shows "cont luba orda lubb ordb f = (\<forall>Y. Complete_Partial_Order.chain orda Y \<longrightarrow> Y \<noteq> {} \<longrightarrow> f (luba Y) = lubb (f ` Y))"
zip_replicate2: fixes xs :: "'b list"   and n :: "nat"   and y :: "'c" shows "zip xs (replicate n y) = map (\<lambda>x. (x, y)) (take n xs)"
divides_trans: fixes a :: "'a list"   and b :: "'a list"   and c :: "'a list" assumes "a divides\<^bsub>poly_ring R\<^esub> b"   and "b divides\<^bsub>poly_ring R\<^esub> c"   and "a \<in> carrier (poly_ring R)" shows "a divides\<^bsub>poly_ring R\<^esub> c"
bigo_const_mult3: fixes c :: "'c"   and f :: "'d \<Rightarrow> 'c" assumes "c \<noteq> (0::'c)" shows "f \<in> O(\<lambda>x. c * f x)"
ref: shows "refl_on verts (Restr (uconnected F) verts)"
ld_ld_1_less: fixes x :: "real"   and y :: "real" assumes "0 < x"   and "0 < y" shows "log 2 x + log 2 y + 1 < 2 * log 2 (x + y)"
wlsAbs_nchotomy: fixes xs :: "'varSort"   and s :: "'sort"   and A :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set" assumes "wlsAbs (xs, s) A" shows "isInBar (xs, s) \<and> (\<exists>x X. wls s X \<and> A = Abs xs x X)"
intros_1: fixes C :: "'b literal set"   and S :: "'b literal set set" assumes "C \<in> S" shows "S \<turnstile> C"
\<alpha>_def: fixes g :: "'m" shows "\<alpha> g \<equiv> \<lambda>(u, v). case M_lookup g u of None \<Rightarrow> \<infinity> | Some l \<Rightarrow> if \<exists>d. (d, v) \<in> set l then enat (SOME d. (d, v) \<in> set l) else \<infinity>"
invariant_initE: fixes A :: "('c, 'd) automaton"   and I :: "'d \<Rightarrow> bool"   and P :: "'c \<Rightarrow> bool"   and s :: "'c" assumes "A \<TTurnstile> (I \<rightarrow>) P"   and "s \<in> init A" shows "P s"
measurable_CI: shows "I \<in> borel_measurable (restrict_space borel {0<..<1})"
A: fixes k :: "nat" shows "offset k x xs \<noteq> None \<longrightarrow> (\<exists>ys zs. xs = ys @ x # zs)"
simps_2: fixes ground_heads_var :: "'a \<Rightarrow> 'b set"   and gt_sym :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and \<epsilon> :: "nat"   and \<delta> :: "nat"   and extf :: "'b \<Rightarrow> (('b, 'a) tm \<Rightarrow> ('b, 'a) tm \<Rightarrow> bool) \<Rightarrow> ('b, 'a) tm list \<Rightarrow> ('b, 'a) tm list \<Rightarrow> bool"   and arity_sym :: "'b \<Rightarrow> enat"   and arity_var :: "'a \<Rightarrow> enat"   and wt_sym :: "'b \<Rightarrow> nat"   and s :: "('b, 'a) tm"   and t :: "('b, 'a) tm" assumes "kbo_std ground_heads_var gt_sym \<epsilon> \<delta> extf arity_sym arity_var wt_sym" shows "kbo_std.wt ground_heads_var \<delta> arity_sym wt_sym (App s t) = kbo_std.wt ground_heads_var \<delta> arity_sym wt_sym s - \<delta> + kbo_std.wt ground_heads_var \<delta> arity_sym wt_sym t"
integral_zero: fixes M :: "'b measure" shows "LINT x|M. (0::'a) = (0::'a)"
poly_smult_zero_iff: fixes a :: "'a"   and p :: "'a poly"   and x :: "'a" shows "(poly (smult a p) x = (0::'a)) = (a = (0::'a) \<or> poly p x = (0::'a))"
simps: fixes V :: "node fset"   and \<Lambda> :: "node \<Rightarrow> assertion_gadget"   and E :: "(node fset \<times> command_gadget \<times> node fset) list" shows "(Graph V \<Lambda> E)^V = V"
C: shows "(Asset n, \<lbrace>Num 4, PubKey D\<rbrace>) \<in> s"
GF_advice_sync: fixes n :: "nat"   and index :: "nat \<Rightarrow> nat"   and w :: "nat \<Rightarrow> 'b set"   and formula :: "nat \<Rightarrow> 'b ltln"   and X :: "'b ltln set" assumes "\<And>i. i < n \<Longrightarrow> \<exists>j. suffix (index i + j) w \<Turnstile>\<^sub>n af (formula i) (w [index i \<rightarrow> index i + j])[X]\<^sub>\<nu>" shows "\<exists>k. \<forall>i<n. index i \<le> k \<and> suffix k w \<Turnstile>\<^sub>n af (formula i) (w [index i \<rightarrow> k])[X]\<^sub>\<nu>"
fps_nth_fps_const: fixes c :: "'b"   and n :: "nat" shows "fps_const c $ n = (if n = 0 then c else (0::'b))"