hcju/isabelleps huggingface dataset:
dynmethd_Some_rec_cases: fixes G :: "prog"   and statC :: "qtname"   and dynC :: "qtname"   and sig :: "sig"   and dynM :: "qtname \<times> methd"   and c :: "class" assumes "dynmethd G statC dynC sig = Some dynM"   and "class G dynC = Some c"   and "ws_prog G" obtains "methd G statC sig = Some dynM"   | statM :: "qtname \<times> methd" where "methd G statC sig = Some statM" and "methd G dynC sig = Some dynM" and "statM \<noteq> dynM" and "G \<turnstile> qmdecl sig dynM overrides qmdecl sig statM"   | "dynC \<noteq> Object" and "dynmethd G statC (super c) sig = Some dynM"
fps_monom_poly_code: fixes c :: "'b"   and k :: "nat" shows "fps_monom_poly c k = (if k = 0 then 1 else pCons (0::'b) (fps_monom_poly_aux c k))"
nAct_eq_proj: fixes n :: "nat"   and active :: "'a \<Rightarrow> cnf \<Rightarrow> bool"   and c :: "'a"   and t :: "cnf llist"   and tCMP :: "'a \<Rightarrow> cnf \<Rightarrow> 'b" assumes "\<not> (\<exists>i\<ge>n. active c (lnth t i))" shows "dynamic_component.nAct tCMP active c (enat n) t = llength (dynamic_component.proj tCMP active c t)"
SN_order_pair_def: fixes S :: "('b \<times> 'b) set"   and NS :: "('b \<times> 'b) set" shows "SN_order_pair S NS \<equiv> order_pair S NS \<and> SN_ars S"
clauseFalseRemove: fixes clause :: "Literal list"   and valuation :: "Literal list"   and literal :: "Literal" assumes "clauseFalse clause valuation" shows "clauseFalse (removeAll literal clause) valuation"
cycle_Cons: fixes x :: "'d"   and xs :: "'d list" shows "cycle (x # xs) = x ## cycle (xs @ [x])"
f1: shows "\<not> 0 \<le> Sup (f ` X) + - 1 * Sup (g ` X)"
intros_3: fixes m1 :: "'a"   and m2 :: "'a" shows "lformula0 (Less m1 m2)"
steps_1_step: fixes \<pi> :: "Syntax.term list"   and e :: "Syntax.term"   and m :: "mode"   and \<pi>' :: "Syntax.term list"   and e' :: "Syntax.term" shows "\<lless>\<pi>, e\<ggreater> m\<rightarrow>1 \<lless>\<pi>', e'\<ggreater> = \<lless>\<pi>, e\<ggreater> m\<rightarrow> \<lless>\<pi>', e'\<ggreater>"
PseudoHA_SAs: fixes SA :: "('d, 'e, 'f) seqauto"   and D :: "'f data" shows "SAs (PseudoHA SA D) = {SA}"
simps_1: fixes \<sigma> :: "('e, 'f, 'd) subst"   and i :: "'e" shows "SFV \<sigma> (Inl i) = (case SFunctions \<sigma> i of None \<Rightarrow> {} | Some x \<Rightarrow> FVT x)"
wellFoundedLexLessState1: fixes Vbl :: "nat set" assumes "finite Vbl" shows "wf (lexLessState1 Vbl)"
lve: shows "lve \<equiv> lift_valid_edge valid_edge sourcenode targetnode kind Entry Exit"
axioms_4: fixes J :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and D :: "'b \<Rightarrow> 'c"   and a :: "'c"   and \<pi> :: "'b \<Rightarrow> 'c" assumes "product_cone J C D a \<pi>" shows "limit_cone J C D a \<pi>"
sum_SucD: fixes f :: "'a \<Rightarrow> nat"   and A :: "'a set"   and n :: "nat" assumes "sum f A = Suc n" shows "\<exists>a\<in>A. 0 < f a"
scale: fixes r :: "'a"   and b :: "('a, 'm) vec" shows "f (r *s b) = r *s f b"
intro: fixes S :: "'b set" assumes "CoxeterSystem S"   and "CoxeterComplex_axioms S" shows "CoxeterComplex S"
supdate_glob_s_def: fixes s :: "s"   and k :: "nat"   and v :: "v" shows "supdate_glob_s s k v = s\<lparr>s.globs := (s.globs s)[k := (s.globs s ! k)\<lparr>g_val := v\<rparr>]\<rparr>"
h1: shows "freeIn var (quadratic_sub var a b c d (Neq p)) = freeIn var (Or (fm.Atom (Less (- A * B))) (fm.Atom (Neq (A\<^sup>2 - B\<^sup>2 * c))))"
polish_pmpt_axioms_def: fixes M :: "'b measure" shows "polish_pmpt_axioms M \<equiv> sets M = sets borel"
C_def: shows "C = (\<lambda>x. x mod int (nat p)) ` B"
factorize_polys_map_coprime_pderiv: fixes ps :: "rat poly list"   and fs :: "rat poly list"   and data :: "(rat \<times> (nat \<times> nat) list) list"   and fss :: "real poly list"   and f :: "real poly" assumes "factorize_polys ps = (fs, data)"   and "fss = map real_of_rat_poly fs"   and "f \<in> set fss" shows "comm_monoid_mult_class.coprime f (pderiv (prod_list fss))"
f_join_aux_length: fixes xs :: "'b list"   and n :: "nat"   and I :: "nat set" shows "length (f_join_aux xs n I) = card (I \<inter> {n..<n + length xs})"
continuous_at_setdist: fixes x :: "'c"   and S :: "'c set" shows "isCont (\<lambda>y. setdist {y} S) x"
bin_snd_mismatch: shows "i.\<alpha> \<cdot> [i.c\<^sub>1] \<le>p y \<cdot> i.\<alpha>"
q_2: fixes x :: "'a" shows "poly q x = poly p (z + x)"
finsum_zero: fixes A :: "'c set" shows "(\<Oplus>i\<in>A. \<zero>) = \<zero>"
simps_1: fixes N :: "nat" shows "freevars (VAR N) = {N}"
groupF_empty_elem: fixes x :: "'a list"   and f :: "'a \<Rightarrow> 'b"   and xs :: "'a list" assumes "x \<in> set (groupF f xs)" shows "x \<noteq> []"
trg_vcomp: fixes \<nu> :: "'a"   and \<mu> :: "'a" assumes "seq \<nu> \<mu>" shows "I.map (\<nu> \<cdot> \<mu>) = I.map \<nu>"
const_of_f32: fixes cs :: "e list"   and \<S> :: "s_context"   and \<C> :: "t_context" assumes "const_list cs"   and "\<S>\<bullet>\<C> \<turnstile> cs : [] _> [T_f32]" shows "\<exists>c. cs = [$C ConstFloat32 c]"
bin_rest_NOT: fixes x :: "int" shows "NOT x div 2 = NOT (x div 2)"
dghm_cn_cov_comp_ArrMap_app: fixes \<alpha> :: "V"   and \<BB> :: "V"   and \<CC> :: "V"   and \<GG> :: "V"   and \<AA> :: "V"   and \<FF> :: "V"   and a :: "V" assumes "\<GG> : op_dg \<BB> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<CC>"   and "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>"   and "a \<in>\<^sub>\<circ> \<AA>\<lparr>Arr\<rparr>" shows "(\<GG> \<^sub>D\<^sub>G\<^sub>H\<^sub>M\<circ> \<FF>)\<lparr>ArrMap\<rparr>\<lparr>a\<rparr> = \<GG>\<lparr>ArrMap\<rparr>\<lparr>\<FF>\<lparr>ArrMap\<rparr>\<lparr>a\<rparr>\<rparr>"
compare_sym: fixes x :: "'a"   and y :: "'a" shows "compare\<cdot>x\<cdot>y = (case compare\<cdot>y\<cdot>x of LT \<Rightarrow> GT | EQ \<Rightarrow> EQ | GT \<Rightarrow> LT)"
interaction_bound_fixp_induct: fixes P :: "(('h, 'i, 'j) gpv \<Rightarrow> enat) \<Rightarrow> bool"   and "consider" :: "'i \<Rightarrow> bool" assumes "lfp.admissible P"   and "P (\<lambda>_. 0)"   and "\<And>interaction_bound'. \<lbrakk>P interaction_bound'; \<And>gpv. interaction_bound' gpv \<le> interaction_bound consider gpv; \<And>gpv. interaction_bound' gpv \<le> (\<Squnion>generat\<in>set_spmf (the_gpv gpv). case generat of Pure x \<Rightarrow> 0 | IO out c \<Rightarrow> if consider out then eSuc (\<Squnion>input. interaction_bound' (c input)) else \<Squnion>input. interaction_bound' (c input))\<rbrakk> \<Longrightarrow> P (\<lambda>gpv. \<Squnion>generat\<in>set_spmf (the_gpv gpv). case generat of Pure x \<Rightarrow> 0 | IO out c \<Rightarrow> if consider out then eSuc (\<Squnion>input. interaction_bound' (c input)) else \<Squnion>input. interaction_bound' (c input))" shows "P (interaction_bound consider)"
cap_object_def: fixes cap :: "cdl_cap" shows "cap_object cap \<equiv> if cap_has_object cap then THE obj_id. cap_objects cap = {obj_id} else undefined"
Next_geq_not_member: fixes "next" :: "'b \<Rightarrow> 'b"   and S :: "'b set"   and arb :: "'b"   and s :: "'b" assumes "fresh next"   and "finite S"   and "fresh.Next next arb S \<le> s" shows "s \<notin> S"
uv_adm: shows "(u, v) \<in> adm_edges f l"
ereal_add_cancel_right: fixes a :: "ereal"   and b :: "ereal"   and c :: "ereal" assumes "a \<noteq> - \<infinity>" shows "(b + a = c + a) = (a = \<infinity> \<or> b = c)"
inf_prefix_length_maximality1: fixes xs :: "'b list"   and ys :: "'b list" assumes "inf_prefix_length xs ys \<noteq> min (length xs) (length ys)" shows "xs ! inf_prefix_length xs ys \<noteq> ys ! inf_prefix_length xs ys"
SesKey_is_session_key: fixes Tgs_B :: "agent"   and A :: "agent"   and SesKey :: "nat"   and T :: "nat"   and evs :: "event list" assumes "Crypt (shrK Tgs_B) \<lbrace>Agent A, Agent Tgs_B, Key SesKey, Number T\<rbrace> \<in> parts (knows Spy evs)"   and "Tgs_B \<notin> bad"   and "evs \<in> kerbV" shows "SesKey \<notin> range shrK"
ba_limsup_bound_1: shows "ereal (1 / real_of_int (B * (B + 1))) \<le> limsup (\<lambda>x. ereal (real_of_int (b x) / real_of_int (a x)))"
that: fixes q :: "'b list" assumes "awalk u q v" shows "thesis"
disc_market_pos_stock_def: fixes M :: "'c measure"   and F :: "nat \<Rightarrow> 'c measure"   and Mkt :: "('c, 'd) discrete_market"   and pos_stock :: "'d" shows "disc_market_pos_stock M F Mkt pos_stock \<equiv> disc_equity_market M F \<and> disc_market_pos_stock_axioms F Mkt pos_stock"
r_e2rv_def: shows "r_e2rv \<equiv> r_pdec2"
INF_greatest: fixes Inf :: "'d set \<Rightarrow> 'd"   and Sup :: "'d set \<Rightarrow> 'd"   and inf :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sup :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and bot :: "'d"   and top :: "'d"   and A :: "'e set"   and u :: "'d"   and f :: "'e \<Rightarrow> 'd" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "\<And>i. i \<in> A \<Longrightarrow> less_eq u (f i)" shows "less_eq u (Inf (f ` A))"
disc_unequal: fixes v :: "'v"   and w :: "'v" assumes "v \<in> dom (discovered s)"   and "w \<in> dom (discovered s)"   and "v \<noteq> w" shows "\<delta> s v \<noteq> \<delta> s w"
WTCond1: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and E :: "char list \<Rightarrow> ty option"   and e :: "char list exp"   and e\<^sub>1 :: "char list exp"   and T\<^sub>1 :: "ty"   and e\<^sub>2 :: "char list exp"   and T\<^sub>2 :: "ty" assumes "P,E \<turnstile> e :: Boolean"   and "P,E \<turnstile> e\<^sub>1 :: T\<^sub>1"   and "P,E \<turnstile> e\<^sub>2 :: T\<^sub>2"   and "P \<turnstile> T\<^sub>1 \<le> T\<^sub>2"   and "P \<turnstile> T\<^sub>2 \<le> T\<^sub>1 \<longrightarrow> T\<^sub>2 = T\<^sub>1" shows "P,E \<turnstile> if (e) e\<^sub>1 else e\<^sub>2 :: T\<^sub>2"
collapse_refine: fixes s :: "'a list \<times> nat list \<times> ('a \<Rightarrow> node_state option) \<times> (nat \<times> 'a set) list"   and p :: "'a set list"   and D :: "'a set"   and pE :: "('a \<times> 'a) set"   and v :: "'a"   and v' :: "'a" assumes "(s, p, D, pE) \<in> GS_rel"   and "(v, v') \<in> Id"   and "v' \<in> \<Union> (set p)" shows "collapse_impl v s \<le> \<Down> GS_rel (RETURN (collapse v' (p, D, pE)))"
index_mat_of_cols_list: fixes i :: "nat"   and nr :: "nat"   and j :: "nat"   and cs :: "complex list list" assumes "i < nr"   and "j < length cs" shows "Tensor.mat_of_cols_list nr cs $$ (i, j) = cs ! j ! i"
transPI: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<And>x y z. \<lbrakk>r x y; r y z\<rbrakk> \<Longrightarrow> r x z" shows "transp r"
bind_spmf_pmf_assoc: fixes p :: "'e pmf"   and f :: "'e \<Rightarrow> 'd spmf"   and g :: "'d \<Rightarrow> 'c spmf" shows "p \<bind> f \<bind> g = p \<bind> (\<lambda>x. f x \<bind> g)"
c1: shows "(\<exists>l\<in>set eq. evalUni (elimVarUni_atom L' (EqUni l)) x) = (\<exists>(a', b', c')\<in>set eq. evalUni (elimVarUni_atom L' (EqUni (a', b', c'))) x)"
integrable_mult_right: fixes c :: "'b"   and M :: "'c measure"   and f :: "'c \<Rightarrow> 'b" assumes "c \<noteq> (0::'b) \<Longrightarrow> integrable M f" shows "integrable M (\<lambda>x. c * f x)"
le_HCollect_iff: fixes A :: "hf"   and P :: "hf \<Rightarrow> bool"   and B :: "hf" shows "(A \<le> HCollect P B) = (A \<le> B \<and> HBall A P)"
distinct_5: fixes ty1' :: "ty"   and ty2' :: "ty" shows "TUnit \<noteq> ty1'\<rightarrow>ty2'"
assms_1: shows "c \<equiv> ASSERT \<Phi> \<bind> (\<lambda>_. m)"
put_M\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a_ok: fixes h :: "(_) heap"   and ptr :: "(_) character_data_ptr"   and setter :: "('v \<Rightarrow> 'v) \<Rightarrow> 's option RCharacterData_scheme \<Rightarrow> 's option RCharacterData_scheme"   and v :: "'v" assumes "CharacterDataClass.type_wf h"   and "ptr |\<in>| character_data_ptr_kinds h" shows "h \<turnstile> ok put_M\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a ptr setter v"
inner_dirsum_doubleD: fixes M :: "'a set"   and N :: "'a set" assumes "add_independentS [M, N]" shows "M \<oplus> N = M + N"
D_2: shows "(ik\<^sub>e\<^sub>s\<^sub>t A \<cdot>\<^sub>s\<^sub>e\<^sub>t \<I>) \<union> ik\<^sub>e\<^sub>s\<^sub>t D \<turnstile>\<^sub>c t \<cdot> \<I>"
rel_cong: fixes x :: "'a list"   and ya :: "'a list"   and y :: "'b list"   and xa :: "'b list"   and R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and Ra :: "'a \<Rightarrow> 'b \<Rightarrow> bool" assumes "x = ya"   and "y = xa"   and "\<And>z yb. \<lbrakk>z \<in> set ya; yb \<in> set xa\<rbrakk> \<Longrightarrow> R z yb = Ra z yb" shows "list_all2 R x y = list_all2 Ra ya xa"
simps_1: fixes \<mu> :: "'a" shows "E.Src \<^bold>\<langle>\<mu>\<^bold>\<rangle>\<^sub>0 = \<^bold>\<langle>\<mu>\<^bold>\<rangle>\<^sub>0"
fps_mult_nth_conv_upto_subdegree_right: fixes f :: "'a fps"   and g :: "'a fps"   and n :: "nat" shows "fps_nth (f * g) n = (\<Sum>i = 0..n - subdegree g. fps_nth f i * fps_nth g (n - i))"
projI: fixes x :: "'a" assumes "x \<in> affine hull S" shows "proj x \<in> affine hull S"
meet_le1: fixes x :: "'a"   and y :: "'a" shows "x \<sqinter> y \<sqsubseteq> x"
ennreal_spmf_bind: fixes p :: "'c spmf"   and f :: "'c \<Rightarrow> 'b spmf"   and x :: "'b" shows "ennreal (spmf (p \<bind> f) x) = \<integral>\<^sup>+ xa. ennreal (spmf (f xa) x) \<partial>measure_spmf p"
right_coset_singular_relboundary: fixes p :: "nat"   and X :: "'b topology"   and S :: "'b set" shows "(#>\<^bsub>relcycle_group p X S\<^esub>) (singular_relboundary_set p X S) = homologous_rel_set p X S"
spmf_scale_spmf: fixes r :: "real"   and p :: "'a spmf"   and x :: "'a" shows "spmf (scale_spmf r p) x = max 0 (min (inverse (weight_spmf p)) r) * spmf p x"
can_simps_3: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and t :: "'b bicategorical_language.term"   and u :: "'b bicategorical_language.term" assumes "bicategory V H \<a> \<i> src trg"   and "bicategorical_language.Ide V src trg t"   and "bicategorical_language.Ide V src trg u"   and "bicategorical_language.Nmlize V t = bicategorical_language.Nmlize V u" shows "partial_magma.cod V (bicategory.can V H \<a> \<i> src trg u t) = Coherence.evaluation_map.eval V H \<a> \<i> src trg (\<lambda>\<mu>. if partial_magma.arr V \<mu> then \<mu> else partial_magma.null V) u"
N_3: shows "{x. \<not> (x \<in> \<Omega> \<longrightarrow> f x = g x)} \<subseteq> N"
ncols_def: fixes A :: "(('b, 'c) vec, 'd) vec" shows "ncols A = CARD('c)"
pi_dom: fixes it :: "('d \<Rightarrow> bool) \<Rightarrow> ('e \<times> 'f \<Rightarrow> 'd \<Rightarrow> 'd) \<Rightarrow> 'd \<Rightarrow> 'd"   and it' :: "('g \<Rightarrow> bool) \<Rightarrow> ('e \<times> 'f \<Rightarrow> 'g \<Rightarrow> 'g) \<Rightarrow> 'g \<Rightarrow> 'g" assumes "proper_it it it'" shows "proper_it (map_iterator_dom it) (map_iterator_dom it')"
pw_bind_nofail: fixes M :: "'b nres"   and f :: "'b \<Rightarrow> 'a nres" shows "nofail (M \<bind> f) = (nofail M \<and> (\<forall>x. inres M x \<longrightarrow> nofail (f x)))"
Suc_1: fixes u :: "'a list"   and v :: "'a list"   and w :: "'a list" assumes "length u = n"   and "length v = n"   and "length w = n" shows "foldr (\<lambda>a. case a of (x, y) \<Rightarrow> (+) (x * y)) (zip (map (\<lambda>xy. fst xy + snd xy) (zip u v)) w) (0::'a) = foldr (\<lambda>a. case a of (x, y) \<Rightarrow> (+) (x * y)) (zip u w) (0::'a) + foldr (\<lambda>a. case a of (x, y) \<Rightarrow> (+) (x * y)) (zip v w) (0::'a)"
chain_imp_not_SN_on: fixes f :: "nat \<Rightarrow> 'b"   and r :: "('b \<times> 'b) set"   and i :: "nat" assumes "\<forall>i. (f i, f (Suc i)) \<in> r" shows "\<not> SN_on r {f i}"
cring_lucas_lehmer_ring: shows "cring lucas_lehmer_ring"
prems_5: shows "m -as'\<rightarrow>* (_Low_)"
pt_bij: fixes pi :: "('e \<times> 'e) list"   and x :: "'d"   and y :: "'d" assumes "pt TYPE('d) TYPE('e)"   and "at TYPE('e)" shows "(pi \<bullet> x = pi \<bullet> y) = (x = y)"
map_prod_imageI: fixes a :: "'d"   and b :: "'e"   and R :: "('d \<times> 'e) set"   and f :: "'d \<Rightarrow> 'f"   and g :: "'e \<Rightarrow> 'g" assumes "(a, b) \<in> R" shows "(f a, g b) \<in> map_prod f g ` R"
list_member_filterD: fixes P :: "'a \<Rightarrow> bool"   and xs :: "'a list"   and x :: "'a" assumes "list_member (filter P xs) x" shows "list_member xs x"
step: fixes x :: "nat"   and z :: "nat"   and rel :: "(nat \<times> nat) set"   and n :: "nat"   and y :: "nat" assumes "(x, z) \<in> Clos rel n"   and "(z, y) \<in> Clos rel n"   and "z < n" shows "(x, y) \<in> Clos rel n"
scale_scale: fixes a :: "'a"   and b :: "'a"   and x :: "'c" shows "a *b b *b x = (a * b) *b x"
rt_fresher_single: fixes ip :: "nat"   and rt1 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option"   and rt2 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option" assumes "rt1 \<sqsubseteq>\<^bsub>ip\<^esub> rt2" shows "the (rt1 ip) \<sqsubseteq> the (rt2 ip)"
simps_1: fixes uu :: "'a rexp" shows "local.le_rexp Zero uu = True"
RangePI: fixes r :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'a"   and b :: "'b" assumes "r a b" shows "Rangep r b"
app_2: fixes \<Gamma>2 :: "name list"   and s1 :: "name"   and s2 :: "name" assumes "length \<Gamma>1 = length \<Gamma>2"   and "distinct \<Gamma>1"   and "distinct \<Gamma>2"   and "wellformed' (length \<Gamma>1) t\<^sub>2"   and "fBall (frees t\<^sub>2 |\<union>| fset_of_list \<Gamma>1) (\<lambda>y. y \<le> s1)"   and "fdisjnt (fset_of_list \<Gamma>1) (frees t\<^sub>2)"   and "fBall (frees t\<^sub>2 |\<union>| fset_of_list \<Gamma>2) (\<lambda>y. y \<le> s2)"   and "fdisjnt (fset_of_list \<Gamma>2) (frees t\<^sub>2)" shows "alpha_equiv (fmap_of_list (zip \<Gamma>1 \<Gamma>2)) (fst (run_state (term_to_nterm \<Gamma>1 t\<^sub>2) s1)) (fst (run_state (term_to_nterm \<Gamma>2 t\<^sub>2) s2))"
Rep_Node: fixes x :: "('b, 'a) node" shows "Rep_Node x \<in> Node"
prv_eqv_imp_transi: fixes \<phi>1 :: "'fmla"   and \<phi>2 :: "'fmla"   and \<chi> :: "'fmla" assumes "\<phi>1 \<in> fmla"   and "\<phi>2 \<in> fmla"   and "\<chi> \<in> fmla"   and "prv (eqv \<phi>1 \<phi>2)"   and "prv (imp \<phi>2 \<chi>)" shows "prv (imp \<phi>1 \<chi>)"
is_monom_vars_monom_var: fixes l :: "linear_poly" assumes "is_monom l" shows "vars l = {monom_var l}"
elims_3: fixes x :: "value"   and xa :: "value" assumes "\<not> x < xa" obtains s :: "String.literal" and n :: "int" where "x = Str s" and "xa = Num n"   | s1 :: "String.literal" and s2 :: "String.literal" where "x = Str s1" and "xa = Str s2" and "\<not> s1 < s2"   | n1 :: "int" and n2 :: "int" where "x = Num n1" and "xa = Num n2" and "\<not> n1 < n2"
fx: fixes x :: "'a" assumes "x \<in> S" shows "(\<Sum>i\<in>B. (f x \<bullet> i) *\<^sub>R i) = f x"
semiring_0_cancel_ow_def: fixes U :: "'b set"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" shows "semiring_0_cancel_ow U plus minus zero times \<equiv> semiring_ow U plus times \<and> cancel_comm_monoid_add_ow U plus minus zero"
intro: fixes \<U> :: "'b set"   and \<A> :: "'b set multiset"   and \<V> :: "'b set"   and \<B> :: "'b set multiset" assumes "proper_sub_incidence_system \<U> \<A> \<V> \<B>"   and "design \<V> \<B>" shows "proper_sub_design \<U> \<A> \<V> \<B>"
zero_and_eq: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bit :: "'b \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and drop_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and take_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and "and" :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and or :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and xor :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and mask :: "nat \<Rightarrow> 'b"   and a :: "'b" assumes "class.semiring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask" shows "and zero a = zero"
\<Phi>o_\<Psi>o: fixes a' :: "'t" assumes "S'.ide a'" shows "\<Phi>o (\<Psi>o a') = a'"
path_offset: fixes g :: "real \<Rightarrow> 'a"   and z :: "'a" shows "path (\<lambda>t. g t - z) = path g"
get_child_nodes\<^sub>c\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>_\<^sub>d\<^sub>a\<^sub>t\<^sub>a\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def: fixes uu :: "(_) character_data_ptr"   and uv :: "unit" shows "get_child_nodes\<^sub>c\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>_\<^sub>d\<^sub>a\<^sub>t\<^sub>a\<^sub>_\<^sub>p\<^sub>t\<^sub>r uu uv = return []"
path_component_disjoint: fixes S :: "'a set"   and a :: "'a"   and b :: "'a" shows "disjnt (path_component_set S a) (path_component_set S b) = (a \<notin> path_component_set S b)"
funcsetI_permutes: fixes f :: "'e \<Rightarrow> 'e"   and S :: "'e set" assumes "f permutes S" shows "f \<in> S \<rightarrow> S"
c'_insertion'': fixes x :: "real" shows "insertion (nth_default 0 (xs' @ x # xs)) (isolate_variable_sparse p var 0) = c'"
fold_simps_1: fixes f :: "'d \<Rightarrow> 'c \<Rightarrow> 'c"   and s :: "'c" shows "List.fold f [] s = s"
run_properties_1: fixes \<phi> :: "'a ltl"   and i :: "nat" shows "fst (run (\<A>.\<delta>\<^sub>\<A> \<Sigma>) (\<A>.\<iota>\<^sub>\<A> \<phi>) w i) = foldl \<up>af (Abs \<phi>) (prefix i w)"
scalar_mult_mat: fixes x :: "'b"   and y :: "'b" shows "x *k mat y = mat (x * y)"
simps_1: fixes c :: "'a"   and f :: "'a poly" shows "graeffe_poly_impl_main c f 0 = f"
ring_hom_add: fixes h :: "'a \<Rightarrow> 'b"   and R :: "('a, 'c) ring_scheme"   and S :: "('b, 'd) ring_scheme"   and x :: "'a"   and y :: "'a" assumes "h \<in> ring_hom R S"   and "x \<in> carrier R"   and "y \<in> carrier R" shows "h (x \<oplus>\<^bsub>R\<^esub> y) = h x \<oplus>\<^bsub>S\<^esub> h y"
inherits_member: fixes G :: "prog"   and C :: "qtname"   and m :: "qtname \<times> memberdecl" assumes "G \<turnstile> C inherits m" shows "G \<turnstile> m member_of C"
axioms_3: fixes \<S> :: "nat \<Rightarrow> bool option"   and ts :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'b memref list \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list" assumes "valid_ownership \<S> ts" shows "read_only_reads_unowned ts"
finfun_upd_triv: fixes f :: "('a, 'b) finfun"   and x :: "'a" shows "finfun_update f x (finfun_apply f x) = f"
reachable_def: shows "reachable \<equiv> \<lambda>A I. {x. reachablep A I x}"
count_greater_eq_one_iff: fixes M :: "'d multiset"   and x :: "'d" shows "(1 \<le> count M x) = (x \<in># M)"
image_eqI: fixes b :: "'a"   and f :: "'b \<Rightarrow> 'a"   and x :: "'b"   and A :: "'b set" assumes "b = f x"   and "x \<in> A" shows "b \<in> f ` A"
l_known_ptrs_axioms: shows "l_known_ptrs known_ptr known_ptrs"
step_abs: fixes s :: "'b"   and s' :: "'b" assumes "s \<in> wa_invar WB"   and "s \<in> wa_cond WB"   and "(s, s') \<in> wa_step WB" shows "(\<alpha>2 s, \<alpha>2 s') \<in> wa_step WA"
rcs_in_set_rcs: fixes G :: "('e, 'f) Group_scheme"   and H :: "'e set"   and a :: "'e" assumes "Group G"   and "G \<guillemotright> H "   and "a \<in> carrier G" shows "H \<bullet>\<^bsub>G\<^esub> a \<in> set_rcs G H"
subs: shows "{0<--<t - r / 2} \<subseteq> T"
open_iff: fixes m :: "'a llist set" shows "(m open) = (m \<in> topo (\<Union>s\<in>A\<^sup>\<star>. {infsuff A s}))"
echelon_form_Hermite_of: fixes ass :: "'a \<Rightarrow> 'a"   and res :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bezout :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a \<times> 'a \<times> 'a"   and A :: "(('a, 'b) vec, 'c) vec" assumes "ass_function ass"   and "res_function res"   and "is_bezout_ext bezout" shows "echelon_form (Hermite_of A ass res bezout)"