hcju/isabelleps huggingface dataset:
hyps: fixes y :: "('a, 'b) tm" assumes "hsize y < hsize s"   and "\<And>w. \<lbrakk>y \<unrhd>\<^sub>e\<^sub>m\<^sub>b w; subst \<rho> w \<unrhd>\<^sub>e\<^sub>m\<^sub>b u; \<forall>w'. w \<rightarrow>\<^sub>e\<^sub>m\<^sub>b w' \<longrightarrow> \<not> subst \<rho> w' \<unrhd>\<^sub>e\<^sub>m\<^sub>b u\<rbrakk> \<Longrightarrow> thesis"   and "subst \<rho> y \<unrhd>\<^sub>e\<^sub>m\<^sub>b u" shows "thesis"
minus_fold_remove: fixes A :: "'b set"   and B :: "'b set" assumes "finite A" shows "B - A = Finite_Set.fold Set.remove B A"
remdups_eq_nil_iff: fixes x :: "'d list" shows "(remdups x = []) = (x = [])"
isGlbD2: fixes R :: "'a set"   and S :: "'a set"   and x :: "'a"   and y :: "'a" assumes "isGlb R S x"   and "y \<in> S" shows "x \<le> y"
rtrancl_induct: fixes a :: "'b"   and b :: "'b"   and r :: "('b \<times> 'b) set"   and P :: "'b \<Rightarrow> bool" assumes "(a, b) \<in> r\<^sup>*"   and "P a"   and "\<And>y z. \<lbrakk>(a, y) \<in> r\<^sup>*; (y, z) \<in> r; P y\<rbrakk> \<Longrightarrow> P z" shows "P b"
redT_has_locks_inv: fixes final :: "'a \<Rightarrow> bool"   and r :: "('d, 'b, 'a, 'c, 'e, 'f) semantics"   and convert_RA :: "'d \<Rightarrow>f nat \<Rightarrow> 'f list"   and ls :: "('d, 'b) locks"   and ts :: "'b \<Rightarrow> ('a \<times> 'd \<Rightarrow>f nat) option"   and m :: "'c"   and ws :: "'b \<Rightarrow> 'e wait_set_status option"   and "is" :: "'b set"   and t :: "'b"   and ta :: "('d, 'b, 'a, 'c, 'e, 'f) thread_action"   and ls' :: "('d, 'b) locks"   and ts' :: "'b \<Rightarrow> ('a \<times> 'd \<Rightarrow>f nat) option"   and m' :: "'c"   and ws' :: "'b \<Rightarrow> 'e wait_set_status option"   and is' :: "'b set"   and t' :: "'b"   and l :: "'d" assumes "multithreaded_base.redT final r convert_RA (ls, (ts, m), ws, is) (t, ta) (ls', (ts', m'), ws', is')"   and "t \<noteq> t'" shows "has_locks (ls $ l) t' = has_locks (ls' $ l) t'"
arr_iff_in_hom: fixes f :: "'s" shows "arr f = \<guillemotleft>f : local.dom f \<rightarrow> cod f\<guillemotright>"
NotInDom_cong: fixes H :: "fm set"   and t :: "tm"   and t' :: "tm"   and r :: "tm"   and r' :: "tm" assumes "H \<turnstile> t EQ t'"   and "H \<turnstile> r EQ r'" shows "H \<turnstile> NotInDom t r IFF NotInDom t' r'"
parPresAux: fixes P :: "ccs"   and Rel :: "(ccs \<times> ccs) set"   and Q :: "ccs"   and R :: "ccs"   and Rel' :: "(ccs \<times> ccs) set"   and T :: "ccs"   and Rel'' :: "(ccs \<times> ccs) set" assumes "P \<leadsto>\<^sup>^<Rel> Q"   and "(P, Q) \<in> Rel"   and "R \<leadsto>\<^sup>^<Rel'> T"   and "(R, T) \<in> Rel'"   and "\<And>P' Q' R' T'. \<lbrakk>(P', Q') \<in> Rel; (R', T') \<in> Rel'\<rbrakk> \<Longrightarrow> (P' \<parallel> R', Q' \<parallel> T') \<in> Rel''" shows "P \<parallel> R \<leadsto>\<^sup>^<Rel''> Q \<parallel> T"
fMin_def: shows "fMin \<equiv> semilattice_fset.F min"
Domain_empty_iff: fixes r :: "('c \<times> 'd) set" shows "(Domain r = {}) = (r = {})"
le_option_def: fixes pos :: "bool"   and x :: "int option"   and y :: "int option" shows "le_option pos x y = (case x of None \<Rightarrow> case y of None \<Rightarrow> True | Some j \<Rightarrow> \<not> pos | Some i \<Rightarrow> case y of None \<Rightarrow> pos | Some j \<Rightarrow> i \<le> j)"
put_nondetT_def: shows "put = nondetM_base.put_nondet put"
seq_char: fixes g :: "('obj, 'obj) concrete_category.arr"   and f :: "('obj, 'obj) concrete_category.arr" shows "seq g f = (arr f \<and> f = g)"
assign_eval\<^sub>w: fixes x :: "'Var"   and e :: "'AExp"   and mds :: "Mode \<Rightarrow> 'Var set"   and mem :: "'Var \<Rightarrow> 'Val" shows "\<langle>x \<leftarrow> e, mds, mem\<rangle> \<leadsto> \<langle>Stop, mds, mem(x := ev\<^sub>A mem e)\<rangle>"
simplex_like_def: fixes P :: "'b set" shows "codomain.simplex_like P \<equiv> finite P \<and> (\<exists>f A. OrderingSetIso (\<^bold>\<le>*) (\<^bold><*) dual_order.greater_eq dual_order.greater P f \<and> f ` P = Pow A)"
part_witness_UNIV: shows "UNIV = set [part_witness.a, part_witness.b, part_witness.c]"
dil: fixes k :: "real" shows "P (moebius_dilatation k)"
distrib_right: fixes a :: "'c"   and b :: "'c"   and c :: "'c" shows "(a + b) * c = a * c + b * c"
multiplicity_eqI: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and p :: "'a"   and n :: "nat"   and x :: "'a" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "dvd.dvd times (power.power one times p n) x"   and "\<not> dvd.dvd times (power.power one times p (Suc n)) x" shows "factorial_semiring.multiplicity times one p x = n"
is_path_split': fixes v :: "'v"   and p1 :: "('v \<times> 'w \<times> 'v) list"   and u :: "'v"   and w :: "'w"   and u' :: "'v"   and p2 :: "('v \<times> 'w \<times> 'v) list"   and v' :: "'v" shows "is_path v (p1 @ (u, w, u') # p2) v' = (is_path v p1 u \<and> (u, w, u') \<in> E \<and> is_path u' p2 v')"
eventually_nat_real: fixes P :: "real \<Rightarrow> bool" assumes "eventually P at_top" shows "\<forall>\<^sub>F x in sequentially. P (real x)"
iMODb_div_ge: fixes m :: "nat"   and k :: "nat"   and r :: "nat"   and c :: "nat" assumes "0 < m"   and "m \<le> k" shows "[ r, mod m, c ] \<oslash> k = [r div k\<dots>,(r + m * c) div k - r div k]"
msetext_dersh_mono_strong: fixes ys :: "'b list"   and xs :: "'b list"   and gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and gt' :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<forall>y\<in>set ys. \<forall>x\<in>set xs. gt y x \<longrightarrow> gt' y x"   and "msetext_dersh gt ys xs" shows "msetext_dersh gt' ys xs"
real_interval_power: fixes a :: "float interval"   and b :: "nat" shows "real_interval (a ^ b) = real_interval a ^ b"
1: shows "u \<le>\<^sub>F\<^sub>I w\<^sub>1"
zoom_def: fixes l :: "'g \<Longrightarrow> 'e"   and m :: "('g, 'f) state" shows "l \<rhd> m = State (\<lambda>b. case run_state m (get\<^bsub>l\<^esub> b) of (c, a) \<Rightarrow> (c, put\<^bsub>l\<^esub> b a))"
m: shows "punit.is_monic_set (punit.reduced_GB F)"
p_boundsI': fixes x :: "real"   and y :: "real" assumes "(\<Sum>i<k. as ! i * bs ! i powr x) < 1 \<and> 1 < (\<Sum>i<k. as ! i * bs ! i powr y)" shows "p \<in> {y<..<x}"
zero_le: fixes plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "'e" assumes "class.canonically_ordered_monoid_add plus zero less_eq less" shows "less_eq zero x"
ag_r_zero: fixes A :: "('e, 'f) aGroup_scheme"   and x :: "'e" assumes "aGroup A"   and "x \<in> carrier A" shows "x \<plusminus>\<^bsub>A\<^esub> \<zero>\<^bsub>A\<^esub> = x"
adv_P1_def: fixes funct_14_OT :: "bool \<times> bool \<times> bool \<times> bool \<Rightarrow> bool \<times> bool \<Rightarrow> (unit \<times> bool) spmf"   and m1 :: "bool \<times> bool \<times> bool \<times> bool"   and m2 :: "bool \<times> bool"   and D :: "'v_14_OT1 \<Rightarrow> bool spmf" shows "OT_14_sim.adv_P1 funct_14_OT m1 m2 D \<equiv> \<bar>spmf (R1_14_OT m1 m2 \<bind> D) True - spmf (funct_14_OT m1 m2 \<bind> (\<lambda>(o1, o2). S1_14_OT m1 o1 \<bind> D)) True\<bar>"
program_equalityI: fixes F :: "'c program"   and G :: "'c program" assumes "Init F = Init G"   and "Acts F = Acts G"   and "AllowedActs F = AllowedActs G" shows "F = G"
spr_simAbs_tObsC_abs: fixes t :: "('a, 'es, 'as) BEState Trace"   and ec :: "(('a, 'es, 'as) BEState \<times> ('a, 'es, 'as) BEState) odlist \<times> (('a, 'es, 'as) BEState \<times> ('a, 'es, 'as) BEState) odlist"   and a :: "'a" assumes "t \<in> SPR.jkbpC"   and "spr_simAbs ec = SPRdet.sim_equiv_class a t" shows "toSet (fst ec) = tObsC_abs t"
b: shows "\<forall>a. N \<noteq> Ax x a"
not_0: fixes a :: "ordinal" assumes "additive_principal a" shows "0 < a"
ml_def: shows "ml = max (return_time {x \<in> S. s x = 0} x / 2) (return_time {x \<in> S. s x = 0} x - e)"
simps_3: fixes v :: "nat" shows "degreen (Bound v) = (\<lambda>m. 0)"
in_rel_def: fixes R :: "('d \<times> 'e) set"   and x :: "'d"   and y :: "'e" shows "in_rel R x y \<equiv> (x, y) \<in> R"
subspace_Inter: fixes f :: "'b set set" assumes "Ball f subspace" shows "subspace (\<Inter> f)"
sorted_map_same: fixes f :: "'e \<Rightarrow> 'd"   and g :: "'e list \<Rightarrow> 'd"   and xs :: "'e list" shows "sorted (map f (filter (\<lambda>x. f x = g xs) xs))"
n\<phi>_3: shows "Fvars \<phi> = {}"
assms_5: assumes "nsqn rt1 dip = nsqn rt2 dip"   and "the (dhops rt2 dip) \<le> the (dhops rt1 dip)" shows "P rt1 rt2 dip"
hsubsetD: fixes A :: "hf"   and B :: "hf"   and c :: "hf" assumes "A \<le> B"   and "c \<^bold>\<in> A" shows "c \<^bold>\<in> B"
fbox_demodalisation3: fixes y :: "'a"   and x :: "'a"   and z :: "'a" shows "(d y \<le>  |x] d z) = \<lparr>d y\<rparr>x\<lparr>x \<cdot> d z\<rparr>"
lt_p_gr: fixes b :: "'t \<Rightarrow>\<^sub>0 'b" assumes "b \<in> set bs" shows "lt b \<prec>\<^sub>t lt (poly_of_pair p)"
that: fixes g :: "'a" assumes "ide (g \<cdot> f)" shows "thesis"
pCons_as_add: shows "pCons a p = [:a:] + pCons 0 p"
finite_set_has_max2: fixes A :: "'a list set" assumes "finite A"   and "A \<noteq> {}" shows "\<exists>max\<in>A. \<forall>a\<in>A. length a \<le> length max"
type_universe_type: fixes w :: "val"   and t :: "pdf_type" shows "(w \<in> type_universe t) = (val_type w = t)"
gen_lossless_gpv_Done: fixes co :: "bool"   and \<I> :: "('a, 'b) \<I>"   and x :: "'c" shows "gen_lossless_gpv co \<I> (Generative_Probabilistic_Value.Done x)"
wlsSEM_def: fixes SEM :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'b, 'c) semDom_scheme" shows "wlsSEM SEM \<equiv> sWlsNE SEM \<and> sWlsDisj SEM \<and> sOpPrSWls SEM"
list_set_autoref_filter: fixes R :: "('e \<times> 'f) set" shows "(filter, op_set_filter) \<in> (R \<rightarrow> bool_rel) \<rightarrow> \<langle>R\<rangle>list_set_rel \<rightarrow> \<langle>R\<rangle>list_set_rel"
drop_f_join_eq2: fixes I :: "nat set"   and xs :: "'b list"   and n :: "nat" assumes "card (I \<down>< length xs) \<le> n" shows "xs \<Join>\<^sub> I \<up> n = []"
return_def: fixes x :: "'a" shows "return x = heap (Pair x)"
asm: shows "falsifies\<^sub>c G (C\<^sub>1 \<cdot>\<^sub>l\<^sub>s (\<lambda>x. \<epsilon> (''1'' @ x)))"
smember_code: fixes x :: "'d"   and y :: "'d"   and s :: "'d stream" shows "smember x (y ## s) = (if x = y then True else smember x s)"
proj_setI: fixes y :: "'b"   and A :: "'b set"   and x :: "'b" assumes "y \<in> A"   and "dist x y \<le> infdist x A" shows "y \<in> proj_set x A"
ImpN: fixes H :: "('a, 'b) fm set"   and p :: "('a, 'b) fm"   and q :: "('a, 'b) fm" assumes "Hintikka H"   and "(\<^bold>\<not> (p \<^bold>\<longrightarrow> q)) \<in> H" shows "p \<in> H \<and> (\<^bold>\<not> q) \<in> H"
bisimilar_is_bisimulation: shows "is_bisimulation (\<sim>\<cdot>)"
trading_strategy_def: fixes p :: "'c \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> real" shows "trading_strategy p = (portfolio p \<and> (\<forall>asset\<in>support_set p. borel_predict_stoch_proc F (p asset)))"
ft_dclI: fixes P :: "cld list"   and ctx :: "ctx"   and dcl :: "char list"   and ctx' :: "ctx"   and cld :: "cld" assumes "find_cld P ctx (fqn_def dcl) (Some (ctx', cld))" shows "find_type P ctx (cl_fqn (fqn_def dcl)) (Some (ty_def ctx' dcl))"
V: fixes z :: "complex" assumes "z \<in> v" shows "((\<lambda>w. f w / (w - z)) has_contour_integral h z) \<gamma>"
gcs_comm: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and lcs :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and s :: "'a"   and t :: "'a" assumes "class.lcs_powerprod plus minus zero lcs" shows "lcs_powerprod.gcs plus minus lcs s t = lcs_powerprod.gcs plus minus lcs t s"
induct: fixes P :: "('a, 'k) aformula \<Rightarrow> bool"   and a0 :: "('a, 'k) aformula" assumes "\<And>\<phi>. P (FNot \<phi>)"   and "\<And>\<phi> \<psi>. \<lbrakk>P \<phi>; P \<psi>\<rbrakk> \<Longrightarrow> P (FOr \<phi> \<psi>)"   and "\<And>\<phi> \<psi>. \<lbrakk>P \<phi>; P \<psi>\<rbrakk> \<Longrightarrow> P (FAnd \<phi> \<psi>)"   and "\<And>b \<phi>. P \<phi> \<Longrightarrow> P (FEx b \<phi>)"   and "\<And>b \<phi>. P \<phi> \<Longrightarrow> P (FAll b \<phi>)"   and "\<And>b. P (FBool b)"   and "\<And>v. P (FBase v)" shows "P a0"
acceptable_rate: fixes Mkt :: "('b, 'c) discrete_market"   and r :: "real"   and risk_free_asset :: "'c" assumes "rsk_free_asset Mkt r risk_free_asset" shows "- 1 < r"
beta_reduction_is_parallel_reduction: fixes A :: "'a trm"   and B :: "'a trm" assumes "A \<rightarrow>\<beta> B" shows "A >> B"
ldropn_lnull: fixes xs :: "'b llist"   and n :: "nat" assumes "lnull xs" shows "ldropn n xs = LNil"
rtrancl_impl_def: fixes gen_succ :: "('b \<times> 'b) list \<Rightarrow> 'b list \<Rightarrow> 'b list"   and un :: "'b list \<Rightarrow> 'c \<Rightarrow> 'c"   and memb :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and emp :: "'c"   and rel :: "('b \<times> 'b) list" shows "rtrancl_impl gen_succ un memb emp rel = (let succ = gen_succ rel; n = length rel in (\<lambda>as. relpow_impl succ un memb as emp n))"
simps_2: fixes x :: "('a, 'b) aformula" shows "nFAND [x] = x"
valid_sops_stmt_sb: fixes i :: "nat"   and s :: "stmt"   and t :: "nat"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> 'a option"   and sb :: "(stmt \<times> nat) memref list"   and \<D> :: "'b"   and \<O> :: "'c"   and s' :: "stmt"   and t' :: "nat" assumes "i < length ts"   and "ts ! i = ((s, t), is, \<theta>, sb, \<D>, \<O>)"   and "(s', t') \<in> prog_configs sb" shows "valid_sops_stmt t' s'"
tm_comp_fetch_second_inst: fixes B :: "(action \<times> nat) list"   and sa :: "nat"   and x :: "cell"   and a :: "action"   and s :: "nat"   and A :: "(action \<times> nat) list" assumes "fetch B sa x = (a, s)"   and "tm_wf0 A"   and "sa \<noteq> 0"   and "s \<noteq> 0" shows "fetch (A |+| B) (sa + length A div 2) x = (a, s + length A div 2)"
RedThrowNull: fixes s :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and 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" shows "P \<turnstile> \<langle>throw null,s\<rangle> \<rightarrow> \<langle>THROW NullPointer,s\<rangle>"
disj_commute: fixes x :: "'b"   and y :: "'b" shows "x OR y = y OR x"
reduction_word: fixes q :: "'state"   and v :: "'action stream" assumes "q \<in> nodes"   and "run v q" obtains u :: "'action stream" and w :: "'action stream" where "R.run w q" and "v =\<^sub>I u" and "u \<preceq>\<^sub>I w" and "lproject visible (llist_of_stream u) = lproject visible (llist_of_stream w)"
H_R1: fixes p :: "'a"   and q :: "'a" shows "H p (R p q) q"
assms_4: shows "g_V0 G' \<subseteq> reachable"
scopeExtRight: fixes x :: "name"   and P :: "ccs"   and Rel :: "(ccs \<times> ccs) set"   and Q :: "ccs" assumes "x \<sharp> P"   and "\<And>y R T. y \<sharp> R \<Longrightarrow> (R \<parallel> \<lparr>\<nu>y\<rparr>T, \<lparr>\<nu>y\<rparr>(R \<parallel> T)) \<in> Rel" shows "P \<parallel> \<lparr>\<nu>x\<rparr>Q \<leadsto>[Rel] \<lparr>\<nu>x\<rparr>(P \<parallel> Q)"
gt_right_comm: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Gt Bet Cong A B C D" shows "Tarski_neutral_dimensionless.Gt Bet Cong A B D C"
add_increasing2: fixes plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and c :: "'e"   and b :: "'e"   and a :: "'e" assumes "class.ordered_comm_monoid_add plus zero less_eq less"   and "less_eq zero c"   and "less_eq b a" shows "less_eq b (plus a c)"
assms_3: shows "mat (row_length M2) (length M2) M2"
sort_defs_1: fixes sub :: "('a \<times> 'b) set"   and s1 :: "'a set"   and s2 :: "'b set" shows "sort_leq sub s1 s2 = (\<forall>c\<^sub>2\<in>s2. \<exists>c\<^sub>1\<in>s1. class_leq sub c\<^sub>1 c\<^sub>2)"
inner_diff_right: fixes x :: "'b"   and y :: "'b"   and z :: "'b" shows "inner x (y - z) = inner x y - inner x z"
UnD1: fixes c :: "'b"   and A :: "'b set"   and B :: "'b set" assumes "c \<in> A \<union> B"   and "c \<notin> A" shows "c \<in> B"
return_returns_result: fixes h :: "'l"   and x :: "'n"   and y :: "'n" shows "h \<turnstile> return x \<rightarrow>\<^sub>r y = (x = y)"
id_aux2: fixes p :: "('b set set, 'c) Combinators list"   and a :: "('b set set, 'd) Combinators"   and s :: "('b set set, 'c) Combinators" assumes "noDenyAll p"   and "\<forall>s. s \<in> set p \<longrightarrow> disjSD_2 a s"   and "\<not> member DenyAll a"   and "DenyAllFromTo (first_srcNet s) (first_destNet s) \<oplus> DenyAllFromTo (first_destNet s) (first_srcNet s) \<oplus> s \<in> set (insertDenies p)" shows "disjSD_2 a (DenyAllFromTo (first_srcNet s) (first_destNet s) \<oplus> DenyAllFromTo (first_destNet s) (first_srcNet s) \<oplus> s)"
abelian_subgroup_generated: fixes G :: "('b, 'c) monoid_scheme"   and S :: "'b set" assumes "Group.group G"   and "comm_group G" shows "comm_group (subgroup_generated G S)"
DIAndCut34Elim2_def: fixes vid1 :: "'a"   and vid2 :: "'a"   and vid3 :: "'a"   and fid1 :: "'b"   and fid2 :: "'b"   and fid3 :: "'b"   and pid1 :: "'c"   and pid2 :: "'c"   and pid3 :: "'c"   and pid4 :: "'c" assumes "ids vid1 vid2 vid3 fid1 fid2 fid3 pid1 pid2 pid3 pid4" shows "ids.DIAndCut34Elim2 vid1 pid3 pid4 = (([[$\<alpha> vid1]](Pc pid3 && Pc pid4) \<rightarrow> Pc pid4) \<rightarrow> ([[$\<alpha> vid1]]Pc pid3 && Pc pid4) \<rightarrow> [[$\<alpha> vid1]]Pc pid4)"
homeomorphic_maps_prod: fixes X :: "'e topology"   and Y :: "'f topology"   and X' :: "'g topology"   and Y' :: "'h topology"   and f :: "'e \<Rightarrow> 'g"   and g :: "'f \<Rightarrow> 'h"   and f' :: "'g \<Rightarrow> 'e"   and g' :: "'h \<Rightarrow> 'f" shows "homeomorphic_maps (prod_topology X Y) (prod_topology X' Y') (\<lambda>(x, y). (f x, g y)) (\<lambda>(x, y). (f' x, g' y)) = (topspace (prod_topology X Y) = {} \<and> topspace (prod_topology X' Y') = {} \<or> homeomorphic_maps X X' f f' \<and> homeomorphic_maps Y Y' g g')"
prj_chine_1: shows "THfgh.p\<^sub>0 \<star> THfgh.chine \<cong> h.chine \<star> THfgh.p\<^sub>0"
simps_2: fixes f :: "'v \<times> 'v"   and fs :: "('v \<times> 'v) list"   and keep :: "('v \<times> 'v) list"   and G :: "'v graph"   and nP :: "'v \<Rightarrow> 'a" shows "minimalize_offending_overapprox (f # fs) keep G nP = (if sinvar (delete_edges_list G (fs @ keep)) nP then minimalize_offending_overapprox fs keep G nP else minimalize_offending_overapprox fs (f # keep) G nP)"
\<beta>\<^sub>h: shows "\<guillemotleft>\<beta>\<^sub>h : tab\<^sub>1 h \<star> w\<^sub>h \<rightarrow> tab\<^sub>1 h \<star> w\<^sub>h'\<guillemotright>"
edom_iff2: fixes m :: "'e \<Rightarrow> 'd"   and a :: "'e" shows "(m a = \<bottom>) = (a \<notin> edom m)"
sum_upto_asymptotics_lift_nat_real: fixes f :: "nat \<Rightarrow> real"   and g :: "real \<Rightarrow> real"   and h :: "real \<Rightarrow> real" assumes "(\<lambda>n. sum f {1..n} - g (real n)) \<in> O(\<lambda>n. h (real n))"   and "(\<lambda>n. g (real n) - g (real (Suc n))) \<in> O(\<lambda>n. h (real n))"   and "(\<lambda>n. h (real n)) \<in> O(\<lambda>n. h (real (Suc n)))"   and "\<And>x. 1 \<le> x \<Longrightarrow> 0 < h x"   and "mono_on g {1..} \<or> mono_on (\<lambda>x. - g x) {1..}"   and "mono_on h {1..} \<or> mono_on (\<lambda>x. - h x) {1..}" shows "\<exists>c>0. \<forall>x\<ge>1. \<bar>sum_upto f x - g x\<bar> \<le> c * h x"
reduce_root_inorder: fixes t :: "'b btree" shows "inorder (reduce_root t) = inorder t"
IH: fixes x :: "pat"   and y :: "value"   and env :: "(name, value) fmap" assumes "name = name' \<and> length ps = length vs"   and "x \<in> set ps"   and "y \<in> set vs"   and "vmatch x y = Some env" shows "fmdom env = patvars x"
rel_set_def: fixes R :: "'d \<Rightarrow> 'e \<Rightarrow> bool" shows "rel_set R = (\<lambda>A B. (\<forall>x\<in>A. \<exists>y\<in>B. R x y) \<and> (\<forall>y\<in>B. \<exists>x\<in>A. R x y))"
simps_2: fixes x :: "complex"   and y :: "complex" shows "Im (x + y) = Im x + Im y"
jumpF_pathfinish_def: fixes g :: "real \<Rightarrow> complex"   and z :: "complex" shows "jumpF_pathfinish g z = jumpF (\<lambda>t. Im (g t - z) / Re (g t - z)) (at_left 1)"
wt_is_type: fixes E :: "env"   and dt :: "loc \<Rightarrow> ty option"   and v :: "Term.term"   and T :: "tys" assumes "E,dt\<Turnstile>v\<Colon>T" shows "wf_prog (prg E) \<longrightarrow> dt = empty_dt \<longrightarrow> (case T of Inl T \<Rightarrow> is_type (prg E) T | Inr Ts \<Rightarrow> Ball (set Ts) (is_type (prg E)))"
insert_remove: fixes A :: "'b set"   and g :: "'b \<Rightarrow> 'c"   and x :: "'b" assumes "finite A" shows "prod g (insert x A) = g x * prod g (A - {x})"
points_index_count_min: fixes n :: "nat"   and bl :: "'a set"   and ps :: "'a set" assumes "n \<le> multiplicity bl"   and "ps \<subseteq> bl" shows "n \<le> \<B> index ps"
cases: fixes x :: "H1 \<times> X3 set" obtains uu_ :: "H1" where "x = (uu_, {X3.Xd1})"   | uv_ :: "H1" where "x = (uv_, {X3.Xd1'})"   | uw_ :: "H1" where "x = (uw_, {X3.Xd2})"   | ux_ :: "H1" where "x = (ux_, {X3.Xd1, X3.Xd1'})"   | uy_ :: "H1" where "x = (uy_, {X3.Xd1, X3.Xd2})"   | uz_ :: "H1" where "x = (uz_, {X3.Xd1', X3.Xd2})"   | va_ :: "H1" where "x = (va_, {X3.Xd1, X3.Xd1', X3.Xd2})"   | vb_ :: "H1" where "x = (vb_, {})"
subgroup_refl_on_LCoset_rel: fixes H :: "'g set" assumes "Group H \<and> order.greater_eq G H" shows "refl_on G (LCoset_rel H)"
lins2D_1: shows "distinct (Inl v # \<pi>)"
mult_eq_0_iff: fixes a :: "'b"   and b :: "'b" shows "(a * b = (0::'b)) = (a = (0::'b) \<or> b = (0::'b))"
hyps_3: shows "(\<Otimes>x\<in>A. a) = a [^] card A"
intros: fixes t :: "('s, 'v) tm"   and s :: "('s, 'v) tm" assumes "head t = head s"   and "chksubs (>\<^sub>t) t s"   and "\<forall>f\<in>local.ground_heads (head t). extf f (>\<^sub>t) (args t) (args s)" shows "gt_same t s"
index_add_mat_2: fixes A :: "'m mat"   and B :: "'m mat" shows "dim_row (A + B) = dim_row B"
3: shows "arrow_from_functor (\<cdot>) (\<cdot>\<^sub>[\<^sub>J\<^sub>,\<^sub>C\<^sub>]) \<Delta>.map (G (J_C.MkIde D)) (J_C.MkIde D) (Adj.\<epsilon> (J_C.MkIde D))"
drop_shift_preserves_T: fixes k :: "nat"   and \<rho> :: "val set list"   and \<tau> :: "ty" assumes "k \<le> length \<rho>" shows "T \<tau> (drop k \<rho>) = T (shift k 0 \<tau>) \<rho>"
F1set1_bd: fixes x :: "('d, 'e, 'f) F1" shows "|F1set1 x| \<le>o IFbd"
fvs_simp_6: fixes P :: "'b trm" shows "fvs (Fst P) = fvs P"
Solutions_snd_not_0: fixes a :: "nat list"   and b :: "nat list"   and x :: "nat list"   and y :: "nat list" assumes "hlde a b"   and "(x, y) \<in> Linear_Diophantine_Equations.Solutions a b"   and "nonzero x" shows "nonzero y"
out_cong_cong: fixes B :: "'p"   and A :: "'p"   and A0 :: "'p"   and E :: "'p"   and D :: "'p"   and D0 :: "'p" assumes "B Out A A0"   and "E Out D D0"   and "Cong B A E D"   and "Cong B A0 E D0" shows "Cong A A0 D D0"
Abs_eint_inject: fixes x :: "int option"   and y :: "int option" assumes "x \<in> UNIV"   and "y \<in> UNIV" shows "(Abs_eint x = Abs_eint y) = (x = y)"
op2_rep2_left2_right2_strict_4: fixes pat :: "[:'r:]" shows "rep2\<cdot>pat\<cdot>\<bottom> = \<bottom>"
remove_key_single: fixes v :: "'b"   and n :: "'c" shows "remove_key v (Poly_Mapping.single v n) = 0"
remdups_eq_nil_right_iff: fixes x :: "'a list" shows "([] = remdups x) = (x = [])"
minus_int_def: shows "(-) \<equiv> (Rep_Integ ---> Rep_Integ ---> Abs_Integ) (\<lambda>(x, y) (u, v). (x + v, y + u))"
digit_to_char_simps_11: fixes n :: "nat" assumes "9 < n" shows "digit_to_char n = CHR ''X''"
single_zero: fixes k :: "'b" shows "Poly_Mapping.single k (0::'c) = 0"