hcju/isabelleps huggingface dataset:
E: shows "\<not> (\<exists>v\<in>sinks_aux I D (D ` set ys) zs. (v, D z) \<in> I) \<longrightarrow> ipurge_tr_rev_aux I D (insert (D z) U) (xs @ ys @ zs) = ipurge_tr_rev_aux I D (insert (D z) U) (xs @ ipurge_tr_aux I D (D ` set ys) zs)"
length_pathVertices_eq: fixes u :: "nat"   and p :: "(nat \<times> nat) list" shows "length (pathVertices u p) = length p + 1"
inter_overappr_def: fixes X :: "'b set"   and Y :: "'b set" shows "inter_overappr X Y = SPEC (\<lambda>R. X \<inter> Y \<subseteq> R \<and> R \<subseteq> X)"
Clos_Suc: fixes x :: "nat"   and y :: "nat"   and rel :: "(nat \<times> nat) set"   and n :: "nat" assumes "(x, y) \<in> Clos rel n" shows "(x, y) \<in> Clos rel (Suc n)"
simps_7: fixes r :: "'a pfm"   and p :: "'a pfm" shows "reduce ([r]\<^sub>! p) = reduce' (reduce r) (reduce p)"
z_neutral: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and z :: "'b"   and x :: "'b"   and y :: "'b" assumes "abel_semigroup_set f z" shows "f (f x y) z = f x y"
writes_not_ok: fixes h :: "'l"   and f :: "('l, 'm, 'n) prog"   and S :: "('l, 'm, 'o) prog set"   and h' :: "'l" assumes "\<not> h \<turnstile> ok f" shows "writes S f h h'"
store_sub1_result: fixes instr :: "sparc_operation \<times> inst_operand list"   and rd :: "word5"   and s_val :: "word1"   and s :: "'b sparc_state" shows "snd (store_sub1 instr rd s_val s) = False"
Ide_Nmlize_Can: fixes t :: "'a local.term" assumes "Can t" shows "Ide \<^bold>\<lfloor>t\<^bold>\<rfloor>"
prod_list_prod_nth: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and xs :: "'c list" assumes "class.monoid_add plus zero" shows "prod_list xs = prod ((!) xs) {0..<length xs}"
monom_mult_list: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and \<alpha> :: "'c \<Rightarrow> 'd"   and m :: "('c \<times> nat) list"   and n :: "('c \<times> nat) list" assumes "class.linorder less_eq less" shows "linorder.eval_monom_list \<alpha> (linorder.monom_mult_list less m n) = linorder.eval_monom_list \<alpha> m * linorder.eval_monom_list \<alpha> n"
countable_PiE: fixes I :: "'a set"   and F :: "'a \<Rightarrow> 'b set" assumes "finite I"   and "\<And>i. i \<in> I \<Longrightarrow> countable (F i)" shows "countable (Pi\<^sub>E I F)"
SemWhileT: fixes b :: "BExpr"   and s :: "Var \<Rightarrow> Val"   and c :: "IMP"   and n :: "nat"   and r :: "Var \<Rightarrow> Val"   and m :: "nat"   and t :: "Var \<Rightarrow> Val"   and k :: "nat" assumes "evalB b s"   and " s , c \<rightarrow>\<^sub>n  r "   and " r , While b c \<rightarrow>\<^sub>m  t "   and "k = max n m + 1" shows " s , While b c \<rightarrow>\<^sub>k  t "
finite_total_preorder_on_iff: fixes A :: "'b set"   and R :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "finite_total_preorder_on A R = (total_preorder_on A R \<and> finite A)"
inf_ccSup: fixes B :: "'a set"   and a :: "'a" assumes "countable B" shows "inf a (Sup B) = Sup (inf a ` B)"
ZObis_matchT_ZO_rev: fixes s :: "('test, 'atom) com"   and t :: "('test, 'atom) com" assumes "s \<approx>01 t" shows "matchT_ZO t s"
max_Var_floatariths_ode_d_fa: fixes cxs :: "floatarith list"   and ys :: "floatarith list"   and z :: "nat"   and i :: "nat" assumes "0 < length ode_e"   and "max_Var_floatariths ode_e \<le> length ode_e"   and "length cxs = length ode_e"   and "length ys = length ode_e"   and "max (max_Var_floatariths cxs) (max_Var_floatariths ys) \<le> z" shows "max_Var_floatariths (ode_d_fa i cxs ys) \<le> z"
WEAK_EX_list_def: fixes Xi :: "'b list"   and P_impl :: "'b \<Rightarrow> bool nres" shows "WEAK_EX_list Xi P_impl \<equiv> WHILE (\<lambda>(xa, xb). \<not> xb \<and> xa \<noteq> []) (\<lambda>(xa, xb). RETURN (hd xa, tl xa) \<bind> (\<lambda>(xc, xd). P_impl xc \<bind> (\<lambda>xe. RETURN (xd, xe \<or> xb)))) (Xi, False) \<bind> (\<lambda>(x, y). RETURN y)"
empty_in_\<L>\<^sub>P: shows "[] \<in> \<L>\<^sub>P"
spectrum_imp_cxt_quotient_ring: fixes R :: "'c set"   and addition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and multiplication :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and unit :: "'c"   and \<pp> :: "'c set" assumes "comm_ring R addition multiplication zero unit"   and "\<pp> \<in> comm_ring.spectrum R addition multiplication zero unit" shows "Comm_Ring.quotient_ring (R\<setminus>\<pp>) R addition multiplication zero unit"
reachable_def: fixes G :: "nat graph_rec"   and v :: "nat" assumes "cfg_doms G" shows "cfg_doms.reachable G v \<equiv> v \<in> (g_E G)\<^sup>* `` {0}"
Units_inv_closed: fixes x :: "'a" assumes "x \<in> Units (add_monoid R)" shows "inv\<^bsub>add_monoid R\<^esub> x \<in> carrier R"
closed_segment_subset_domain: fixes t0 :: "real"   and t :: "real" assumes "t0 \<in> T"   and "t \<in> T" shows "{t0--t} \<subseteq> T"
fconverse_vsingleton: fixes a :: "V"   and b :: "V" shows "(set {[a, b]\<^sub>\<circ>})\<inverse>\<^sub>\<bullet> = set {[b, a]\<^sub>\<circ>}"
mult_aux: fixes a :: "nat"   and b :: "nat" assumes "a \<in> totatives n"   and "b \<in> totatives n" shows "\<chi> (a * b) = \<chi> a * \<chi> b"
nodes_target_elim: fixes q :: "'d"   and A :: "('e, 'f, 'd, 'g) FSM_scheme" assumes "q \<in> nodes A" obtains r :: "(('e \<times> 'f) \<times> 'd) list" and p :: "'d" where "p = initial A" and "path A r p" and "q = target r p"
vals_nonequiv_def: fixes K :: "('c, 'd) Ring_scheme"   and n :: "nat"   and vv :: "nat \<Rightarrow> 'c \<Rightarrow> ant" shows "vals_nonequiv K n vv = (valuations K n vv \<and> (\<forall>j\<le>n. \<forall>l\<le>n. j \<noteq> l \<longrightarrow> \<not> v_equiv K (vv j) (vv l)))"
inverse_scaleR_times: fixes v :: "num"   and w :: "num"   and a :: "'b" shows "(1 / numeral v) *\<^sub>R (numeral w * a) = (numeral w / numeral v) *\<^sub>R a"
limit_suffix: fixes n :: "nat"   and x :: "nat \<Rightarrow> 'b" shows "limit (suffix n x) = limit x"
alpha_fp: fixes x :: "'a" shows "(\<nu> x = x) = (x \<cdot> (0::'a) = (0::'a))"
run_get_writer: fixes get :: "('g \<Rightarrow> 'd) \<Rightarrow> 'd"   and f :: "'g \<Rightarrow> ('e, 'f, 'd) writerT" shows "run_writer (get_writer get f) = get (\<lambda>s. run_writer (f s))"
op_succ_def: fixes E :: "('c \<times> 'b) set"   and u :: "'c" shows "op_succ E u \<equiv> E `` {u}"
simps_1: shows "unbox Ubx1 = map_option OpUbx1 \<circ> unbox_ubx1"
Cn_conj_bis: fixes Cn :: "'b set \<Rightarrow> 'b set"   and true_PL :: "'b"   and false_PL :: "'b"   and imp_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and not_PL :: "'b \<Rightarrow> 'b"   and conj_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and disj_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and equiv_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and p :: "'b"   and q :: "'b" assumes "Supraclassical_logic Cn true_PL false_PL imp_PL not_PL conj_PL disj_PL equiv_PL" shows "Cn {conj_PL p q} = Cn {p, q}"
corec1_Inl_sum: fixes s1 :: "'e \<Rightarrow> ('d, 'd JF1 + 'e, 'd JF2 + 'f) F1"   and s2 :: "'f \<Rightarrow> ('d, 'd JF1 + 'e, 'd JF2 + 'f) F2" shows "unfold1 (case_sum (F1map id Inl Inl \<circ> dtor1) s1) (case_sum (F2map id Inl Inl \<circ> dtor2) s2) \<circ> Inl = id"
rel_pmf_measureD: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'c pmf"   and q :: "'d pmf"   and A :: "'c set" assumes "rel_pmf R p q" shows "measure_pmf.prob p A \<le> measure_pmf.prob q {y. \<exists>x\<in>A. R x y}"
periodN_D3: fixes w :: "'b list"   and n :: "nat" assumes "periodN w n" shows "w \<le>p take n w \<cdot> w"
simps: fixes a :: "'a list" shows "hb_consistent a = (a = [] \<or> (\<exists>xs y. a = xs @ [y] \<and> hb_consistent xs \<and> (\<forall>x\<in>set xs. \<not> y \<prec> x)))"
univ_poly_subfield_of_consts: fixes K :: "'a list set" assumes "subfield K (K [X])" shows "subfield (UP.poly_of_const ` K) (K [X]\<^bsub>K [X]\<^esub>)"
case_sum_o_inj_2: fixes f :: "'c \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" shows "f (+) g \<circ> Inr = g"
apartment_min_galleryD_gallery: fixes X :: "'a set set"   and xs :: "'a set list" assumes "X \<in> \<A>"   and "SimplicialComplex.min_maxsimpchain X xs" shows "SimplicialComplex.maxsimpchain X xs"
allowed_redex_def: fixes t :: "'b trm"   and C :: "'b eclause"   and \<sigma> :: "('b \<times> 'b trm) list" shows "allowed_redex t C \<sigma> = (\<not> (\<exists>s\<in>trms_ecl C. occurs_in (t \<lhd> \<sigma>) (s \<lhd> \<sigma>)))"
close_eq_def: fixes s :: "predicate \<times> 'b list \<Rightarrow> bool" shows "close_eq s = (\<lambda>x. case x of predAtm p xs \<Rightarrow> s (p, xs) | Eq a b \<Rightarrow> a = b)"
aext_not: fixes P :: "'b upred"   and x :: "'b \<Longrightarrow> 'a" shows "(\<not> P) \<oplus>\<^sub>p x = (\<not> P \<oplus>\<^sub>p x)"
induced_Fiber_simp: fixes a :: "'a" assumes "a \<in> S" shows "induced (Class a) = \<alpha> a"
in_pair_def: fixes p :: "'e \<times> 'e"   and A :: "('e \<times> 'e) set" shows "(p \<in>\<^sub>p A) = (p \<in> A \<union> prod.swap ` A)"
maxsimpchainD_maxsimp: fixes xs :: "'a set list"   and x :: "'a set" assumes "folding.gallery xs"   and "x \<in> set xs" shows "folding.chamber x"
delete_maybe_def: fixes i :: "'b"   and D :: "('b, 'a) message set"   and s :: "'a" shows "delete_maybe i D s = (if Delete (DeleteMessage i) \<in> D then None else Some s)"
const_defined: fixes X :: "'a state \<times> 'a state \<Rightarrow> 'b" assumes "const X" shows "const (\<delta> X)"
exhaust_sel: fixes tllist :: "('f, 'g) tllist" obtains "tllist = TNil (terminal tllist)"   | "tllist = TCons (thd tllist) (ttl tllist)"
normA: fixes x :: "'a" assumes "x \<in> A" shows "norm x = 1"
inj_on_map_eq_map: fixes f :: "'a \<Rightarrow> 'b"   and xs :: "'a list"   and ys :: "'a list" assumes "inj_on f (set xs \<union> set ys)" shows "(map f xs = map f ys) = (xs = ys)"
powr_le_one_le: fixes x :: "real"   and y :: "real" assumes "0 < x"   and "x \<le> 1"   and "1 \<le> y" shows "x powr y \<le> x"
add_nonpos_neg: fixes a :: "'c multiset"   and b :: "'c multiset" assumes "a \<subseteq># {#}"   and "b \<subset># {#}" shows "a + b \<subset># {#}"
shuffles_commutes: fixes xs :: "'c list"   and ys :: "'c list" shows "shuffles xs ys = shuffles ys xs"
bind_returns_heap_E3: fixes h :: "'l"   and f :: "('l, 'm, 'o) prog"   and g :: "'o \<Rightarrow> ('l, 'm, 'n) prog"   and h' :: "'l"   and x :: "'o" assumes "h \<turnstile> Heap_Error_Monad.bind f g \<rightarrow>\<^sub>h h'"   and "h \<turnstile> f \<rightarrow>\<^sub>r x"   and "pure f h" shows "h \<turnstile> g x \<rightarrow>\<^sub>h h'"
iT_Plus_mem_iff2: fixes x :: "nat"   and k :: "nat"   and I :: "nat set" shows "(x + k \<in> I \<oplus> k) = (x \<in> I)"
weaken: fixes G :: "prog"   and A :: "'b triple set"   and ts' :: "'b triple set"   and ts :: "'b triple set" assumes "G,A|\<turnstile>ts'"   and "ts \<subseteq> ts'" shows "G,A|\<turnstile>ts"
observableE: fixes ob :: "'e \<Rightarrow> 'f"   and P :: "'e set"   and s :: "'e"   and s' :: "'e" assumes "observable ob P"   and "ob s = ob s'"   and "s' \<in> P" shows "s \<in> P"
mem_close: fixes P :: "('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list" assumes "List.member SG P"   and "P \<noteq> \<phi>" shows "List.member (close SG \<phi>) P"
simps_1: fixes var :: "nat" shows "grab_atoms var TrueF = Some []"
assms: shows "inv_after_move (as, lm) (s, l, Bk # r) ires"
cdi_path_swap: fixes \<pi>' :: "nat \<Rightarrow> 'n"   and j :: "nat"   and \<pi> :: "nat \<Rightarrow> 'n"   and k :: "nat" assumes "is_path \<pi>'"   and "j cd\<^bsup>\<pi>\<^esup>\<rightarrow> k"   and "\<pi> =\<^bsub>j\<^esub> \<pi>'" shows "j cd\<^bsup>\<pi>'\<^esup>\<rightarrow> k"
relative_toE: fixes P :: "'b set \<Rightarrow> bool"   and U :: "'b set"   and S :: "'b set"   and Q :: "'b set \<Rightarrow> bool" assumes "(P relative_to U) S"   and "\<And>S. P S \<Longrightarrow> Q (U \<inter> S)" shows "Q S"
winding_number_lt_half_lemma: fixes \<gamma> :: "real \<Rightarrow> complex"   and z :: "complex"   and a :: "complex"   and b :: "real" assumes "valid_path \<gamma>"   and "z \<notin> path_image \<gamma>"   and "a \<bullet> z \<le> b"   and "path_image \<gamma> \<subseteq> {w. b < a \<bullet> w}" shows "Re (winding_number \<gamma> z) < 1 / 2"
T: shows "T \<in> P \<rightarrow>\<^sub>M Q"
Restrp_antisymmetric: shows "antisymmetric UNIV ((\<bowtie>) \<restriction> A)"
normalize_poly_def: fixes p :: "(char list multiset \<times> int) multiset" shows "PAC_Polynomials.normalize_poly p = fold_mset add_to_coefficient {#} p"
three_ins_2: fixes t :: "'a"   and i :: "'a"   and e :: "'a" shows "t \<in> set [i, t, e]"
exhaust: fixes y :: "order" obtains "y = Eq"   | "y = Lt"   | "y = Gt"
exp_add: fixes x :: "'b"   and y :: "'b" shows "exp (x + y) = exp x * exp y"
mult_div_mod_eq: fixes b :: "'a"   and a :: "'a" shows "b * (a div b) + a mod b = a"
step: shows "\<Gamma>\<turnstile> (p # ps @ cs, css, s) \<rightarrow> (cs', (pnorm @ cs, pabr @ cs) # css, t)"
lessE: fixes a :: "'k multiset"   and b :: "'k multiset" assumes "a \<subset># b" obtains c :: "'k multiset" where "b = a + c" and "c \<noteq> {#}"
enum_ex: shows "enum_ex = (\<lambda>P. ex_n_lists enum_ex_b (\<lambda>bs. P (the \<circ> map_of (zip enum_a bs))) (length enum_a))"
ide_def: fixes a :: "'a" shows "ide a \<equiv> a \<cdot>\<^sup>o\<^sup>p a \<noteq> null \<and> (\<forall>f. (f \<cdot>\<^sup>o\<^sup>p a \<noteq> null \<longrightarrow> f \<cdot>\<^sup>o\<^sup>p a = f) \<and> (a \<cdot>\<^sup>o\<^sup>p f \<noteq> null \<longrightarrow> a \<cdot>\<^sup>o\<^sup>p f = f))"
mcont_SUP: fixes Inf :: "'f set \<Rightarrow> 'f"   and Sup :: "'f set \<Rightarrow> 'f"   and inf :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and less_eq :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and less :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and sup :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and bot :: "'f"   and top :: "'f"   and lub :: "'g set \<Rightarrow> 'g"   and ord :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and f :: "'g \<Rightarrow> 'h set"   and g :: "'g \<Rightarrow> 'h \<Rightarrow> 'f" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "mcont lub ord \<Union> (\<subseteq>) f"   and "\<And>y. mcont lub ord Sup less_eq (\<lambda>x. g x y)" shows "mcont lub ord Sup less_eq (\<lambda>x. Sup (g x ` f x))"
satisfies_eq_iff: fixes v :: "nat \<Rightarrow> 'a"   and x :: "nat"   and p :: "linear_poly" shows "v \<Turnstile>\<^sub>e (x, p) \<equiv> v x = p \<lbrace> v \<rbrace>"
disj: shows "{Inl vid1, Inr vid1} \<inter> BVO ODE = {}"
simps: fixes x :: "int\<^sub>\<bottom>"   and y :: "int\<^sub>\<bottom>" shows "enumFromToS\<cdot>x\<cdot>y = Stream\<cdot>(enumFromToStep\<cdot>y)\<cdot>(up\<cdot>x)"
sim: fixes p :: "'s1 spmf"   and q :: "'s2 spmf"   and a :: "'a"   and res :: "'s2"   and b :: "'b"   and s' :: "'s2" assumes "S p q"   and "a \<in> A"   and "res \<in> set_spmf q"   and "(b, s') \<in> set_spmf (callee2 res a)" shows "trace_eq_simcl S (cond_spmf_fst (p \<bind> (\<lambda>s. callee1 s a)) b) (cond_spmf_fst (q \<bind> (\<lambda>s. callee2 s a)) b)"
has_codomain_char: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<mu> :: "'b arrow_of_spans_data" assumes "span_vertical_category C" shows "(partial_magma.codomains (span_vertical_category.vcomp C) \<mu> \<noteq> {}) = arrow_of_spans C \<mu>"
simps: fixes a :: "'node SDG_node"   and n :: "'node SDG_node" shows "(a \<in> sum_SDG_slice1 n) = (a = n \<and> valid_SDG_node n \<or> (\<exists>n'' n'. a = n'' \<and> n'' s\<longrightarrow>\<^bsub>cd\<^esub> n' \<and> n' \<in> sum_SDG_slice1 n) \<or> (\<exists>n'' V n'. a = n'' \<and> n'' s-V\<rightarrow>\<^sub>d\<^sub>d n' \<and> n' \<in> sum_SDG_slice1 n) \<or> (\<exists>n'' p n'. a = n'' \<and> n'' s-p\<rightarrow>\<^bsub>call\<^esub> n' \<and> n' \<in> sum_SDG_slice1 n) \<or> (\<exists>n'' p V n'. a = n'' \<and> n'' s-p:V\<rightarrow>\<^bsub>in\<^esub> n' \<and> n' \<in> sum_SDG_slice1 n) \<or> (\<exists>n'' p n'. a = n'' \<and> n'' s-p\<rightarrow>\<^bsub>sum\<^esub> n' \<and> n' \<in> sum_SDG_slice1 n))"
homeomorphic_map_closedness_eq: fixes X :: "'c topology"   and Y :: "'d topology"   and f :: "'c \<Rightarrow> 'd"   and U :: "'c set" assumes "homeomorphic_map X Y f" shows "closedin X U = (U \<subseteq> topspace X \<and> closedin Y (f ` U))"
simps_3: fixes vs :: "'b list"   and bs :: "'b list"   and a :: "tm" shows "Itm vs bs (tm.Neg a) = - Itm vs bs a"
between_mem_segment: fixes a :: "'a"   and b :: "'a"   and x :: "'a" shows "between (a, b) x = (x \<in> closed_segment a b)"
mult_isol_equiv_subdistl: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "class.near_dioid plus times less_eq less" shows "(\<forall>x y z. less_eq x y \<longrightarrow> less_eq (times z x) (times z y)) = (\<forall>x y z. less_eq (times z x) (times z (plus x y)))"
denote_while_n_sum_positive: fixes M0 :: "complex mat"   and M1 :: "complex mat"   and DS :: "complex mat \<Rightarrow> complex mat"   and \<rho> :: "complex mat"   and n :: "nat" assumes "M0 \<in> carrier_mat d d"   and "M1 \<in> carrier_mat d d"   and "adjoint M1 * M1 \<le>\<^sub>L 1\<^sub>m d"   and "\<And>\<rho>. \<lbrakk>\<rho> \<in> carrier_mat d d; partial_density_operator \<rho>\<rbrakk> \<Longrightarrow> positive (DS \<rho>) \<and> trace (DS \<rho>) \<le> trace \<rho> \<and> DS \<rho> \<in> carrier_mat d d"   and "partial_density_operator \<rho>"   and "\<rho> \<in> carrier_mat d d" shows "positive (matrix_sum d (\<lambda>n. denote_while_n M0 M1 DS n \<rho>) n)"
set_subset_i_append: fixes xs :: "'b list"   and f :: "nat \<Rightarrow> 'b" shows "set xs \<subseteq> range (xs \<frown> f)"
step_to_steps: fixes Cal :: "'b processCalculus"   and P :: "'b"   and P' :: "'b" assumes "P \<longmapsto>Cal P'" shows "P \<longmapsto>Cal* P'"
induct: fixes x1 :: "'a"   and x2 :: "'b list"   and x3 :: "'a"   and t :: "('a \<times> 'b \<times> 'a) set"   and P :: "'a \<Rightarrow> 'b list \<Rightarrow> 'a \<Rightarrow> bool" assumes "(x1, x2, x3) \<in> trcl t"   and "\<And>c. P c [] c"   and "\<And>c a c' w c''. \<lbrakk>(c, a, c') \<in> t; (c', w, c'') \<in> trcl t; P c' w c''\<rbrakk> \<Longrightarrow> P c (a # w) c''" shows "P x1 x2 x3"
sub_good: fixes w :: "bval"   and v :: "val"   and v' :: "val" assumes "w \<in> good v"   and "v' \<sqsubseteq> v" shows "w \<in> good v'"
splitAt_distinct_ab: fixes vs :: "'c list"   and a :: "'c list"   and b :: "'c list"   and ram :: "'c" assumes "distinct vs"   and "(a, b) = splitAt ram vs" shows "set a \<inter> set b = {}"
a_get_ancestors_si_locs_def: fixes get_parent_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_host_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_get_root_node_si\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_ancestors_si_locs get_parent_locs get_host_locs = get_parent_locs \<union> get_host_locs"
zariski_open_is_subset: fixes U :: "nat set set" assumes "cring0.is_zariski_open U" shows "U \<subseteq> cring0.spectrum"
l2_inv8_step3: fixes Ra :: "fid_t"   and A :: "agent"   and B :: "agent"   and gny :: "msg" shows "{l2_inv8 \<inter> l2_inv1 \<inter> l2_inv2 \<inter> l2_inv4'} l2_step3 Ra A B gny {> l2_inv8}"
simps_1: shows "PX4d D1 = linord_of_list [X4.Xd1', X4.Xd1]"
x_2: shows "infinite {n. birkhoff_sum f n x \<in> {real n * expectation f..real n * expectation f + 1}}"
linear_limit_linear: fixes f :: "nat \<Rightarrow> 'c \<Rightarrow> 'd"   and F :: "'c \<Rightarrow> 'd" assumes "\<And>n. linear (f n)"   and "f \<midarrow>pointwise\<rightarrow> F" shows "linear F"
eq: fixes b :: "'b" assumes "b \<in> B" shows "{x \<in> A. f (inv p\<^sub>A x) = b} = p\<^sub>A ` {x \<in> A. f x = b}"
mus: shows "mus = fst (mus_adjuster f n_gs [] (0\<^sub>v n)) # tl mus"
unit_def: fixes a :: "'a" shows "F.unit a \<equiv> THE \<psi>. \<guillemotleft>\<psi> : F.map\<^sub>0 a \<Rightarrow>\<^sub>D F a\<guillemotright> \<and> D.iso \<psi> \<and> \<psi> \<cdot>\<^sub>D \<i>\<^sub>D[F.map\<^sub>0 a] = (F \<i>\<^sub>C[a] \<cdot>\<^sub>D \<Phi>\<^sub>F (a, a)) \<cdot>\<^sub>D (\<psi> \<star>\<^sub>D \<psi>)"
simps_1: fixes A :: "nat \<Rightarrow> nat \<Rightarrow> 'b" shows "gauss_jordan A 0 = Some A"
rep_eq: fixes x :: "'b semilattice_set" shows "set_fold1 x = semilattice_set.F (semilattice_set_apply x)"
Entry_Exit_edge: shows "\<exists>a. valid_edge a \<and> sourcenode a = Entry \<and> targetnode a = Exit \<and> kind a = (\<lambda>s. False)\<^sub>\<surd>"
atLeastLessThan_upt: fixes i :: "nat"   and j :: "nat" shows "{i..<j} = set [i..<j]"
get_elements_by_tag_name_def: shows "get_elements_by_tag_name = (\<lambda>ptr tag. Heap_Error_Monad.bind (to_tree_order ptr) (map_filter_M (\<lambda>ptr. case cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr of None \<Rightarrow> return None | Some element_ptr \<Rightarrow> Heap_Error_Monad.bind (get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t element_ptr tag_name) (\<lambda>this_tag_name. if this_tag_name = tag then return (Some element_ptr) else return None))))"
nth_visible_eff: fixes s :: "('c, 'd option) insert_message list"   and n :: "nat"   and i :: "'c extended" assumes "nth_visible s n = Inr i" shows "extended_to_set i \<subseteq> I ` set s"
\<chi>\<sigma>': shows "instInpP \<chi>\<sigma> 0 \<tau> = exi z' (cnj (subst \<tau> (Var z') out) (subst \<chi>\<sigma> (Var z') inp))"
dc: fixes u :: "nat discr"   and v :: "nat discr" shows "undiscr u + undiscr v = undiscr v + undiscr u"
sc_symcl: fixes a :: "'a"   and b :: "'a" shows "(a \<leftrightarrow> b) = ((a, b) \<in> {(x, y). x \<rightarrow> y}\<^sup>\<leftrightarrow>)"
path_entry_hd_edge: fixes G :: "nat graph_rec"   and pa :: "nat list"   and p :: "nat" assumes "cfg_doms G"   and "path_entry (g_E G) pa p"   and "pa \<noteq> []" shows "(hd pa, p) \<in> g_E G"
Push_inject1: fixes i :: "'a + nat"   and f :: "nat \<Rightarrow> 'a + nat"   and j :: "'a + nat"   and g :: "nat \<Rightarrow> 'a + nat" assumes "Push i f = Push j g" shows "i = j"
simps_5: fixes F :: "'b formula"   and G :: "'b formula" shows "to_mini_formula (F \<^bold>\<and> G) = (to_mini_formula F \<^bold>\<rightarrow> to_mini_formula G \<^bold>\<rightarrow> \<bottom>) \<^bold>\<rightarrow> \<bottom>"
append_eq_append_conv: fixes xs :: "'a list"   and ys :: "'a list"   and us :: "'a list"   and vs :: "'a list" assumes "\<parallel>xs\<parallel> = \<parallel>ys\<parallel> \<or> \<parallel>us\<parallel> = \<parallel>vs\<parallel>" shows "(xs @ us = ys @ vs) = (xs = ys \<and> us = vs)"
ftf_syn21: fixes a :: "'a event"   and u :: "'a event list"   and t :: "'a event list"   and A :: "'a event set" shows "(a \<in> set u \<and> a \<notin> set t \<or> a \<in> set t \<and> a \<notin> set u) \<and> a \<in> A \<longrightarrow> setinterleaving (u, A, t) = {}"
has_pullbacks_def: shows "J'J.A.has_pullbacks = (\<forall>f0 f1. J'J.A.cospan f0 f1 \<longrightarrow> (\<exists>p0 p1. J'J.A.has_as_pullback f0 f1 p0 p1))"
one_less_Fract_iff: fixes b :: "'b"   and a :: "'b" assumes "(0::'b) < b" shows "(1 < Fract a b) = (b < a)"
gauss_jordan_carrier_1: fixes A :: "'c mat"   and nr :: "nat"   and nc :: "nat"   and B :: "'c mat"   and nc' :: "nat"   and A' :: "'c mat"   and B' :: "'c mat" assumes "A \<in> carrier_mat nr nc"   and "B \<in> carrier_mat nr nc'"   and "gauss_jordan A B = (A', B')" shows "A' \<in> carrier_mat nr nc"
finite_Collect_conjI: fixes P :: "'g \<Rightarrow> bool"   and Q :: "'g \<Rightarrow> bool" assumes "finite {x. P x} \<or> finite {x. Q x}" shows "finite {x. P x \<and> Q x}"
D2_def: fixes r :: "('a \<times> 'a) set"   and d :: "('b \<times> ('a \<times> 'b) list) \<times> ('b \<times> ('a \<times> 'b) list) \<times> ('b \<times> ('a \<times> 'b) list) \<times> 'b \<times> ('a \<times> 'b) list" shows "D2 r d = (let (\<tau>, \<sigma>, \<sigma>', \<tau>') = d in Decreasing_Diagrams.D r (labels \<tau>) (labels \<sigma>) (labels \<sigma>') (labels \<tau>'))"
intro: fixes X :: "'a set"   and is_open :: "'a set \<Rightarrow> bool"   and S :: "'a set" assumes "topological_space X is_open"   and "ind_topology_axioms X S" shows "ind_topology X is_open S"