hcju/isabelleps huggingface dataset:
cases: fixes a :: "'b formula" assumes "a \<in> AXH" obtains F :: "'b formula" and G :: "'b formula" where "a = Fa__ \<^bold>\<rightarrow> G__ \<^bold>\<rightarrow> Fa__"   | F :: "'b formula" and G :: "'b formula" where "a = (Fb__ \<^bold>\<rightarrow> Fb__ \<^bold>\<rightarrow> Ga__) \<^bold>\<rightarrow> Fb__ \<^bold>\<rightarrow> Ga__"   | F :: "'b formula" and G :: "'b formula" and H :: "'b formula" where "a = (Fc__ \<^bold>\<rightarrow> Gb__ \<^bold>\<rightarrow> H__) \<^bold>\<rightarrow> Gb__ \<^bold>\<rightarrow> Fc__ \<^bold>\<rightarrow> H__"   | G :: "'b formula" and H :: "'b formula" and F :: "'b formula" where "a = (G \<^bold>\<rightarrow> H) \<^bold>\<rightarrow> (F \<^bold>\<rightarrow> G) \<^bold>\<rightarrow> F \<^bold>\<rightarrow> H"
trpl_code_simp: fixes l :: "nat"   and lm :: "'b"   and tp :: "(action \<times> nat) list" shows "trpl_code (steps0 (Suc 0, Bk \<up> l, <lm>) tp 0) = rec_exec rec_conf [code tp, bl2wc (<lm>), 0]"
proper: fixes ct' :: "'s \<Rightarrow> 's pmf"   and ct :: "'s \<Rightarrow> 's pmf" assumes "(ct', ct) \<in> R" shows "proper ct'"
lift_prod_list: fixes xs :: "'a list" shows "lift (prod_list xs) = prod_list (map lift xs)"
distinguish_attach: fixes d :: "('h, 'i) distinguisher"   and conv :: "('h, 'i, 'j, 'k) converter"   and res :: "('j, 'k) resource" shows "connect d (conv \<rhd> res) = connect (absorb d conv) res"
subst_cong: fixes \<sigma> :: "'c \<Rightarrow> ('d, 'e) Term.term"   and \<sigma>' :: "'c \<Rightarrow> ('d, 'e) Term.term"   and \<theta> :: "'e \<Rightarrow> ('d, 'f) Term.term"   and \<theta>' :: "'e \<Rightarrow> ('d, 'f) Term.term" assumes "\<sigma> = \<sigma>'"   and "\<theta> = \<theta>'" shows "\<sigma> \<circ>\<^sub>s \<theta> = \<sigma>' \<circ>\<^sub>s \<theta>'"
of_nat_mono_maybe: fixes x :: "nat"   and y :: "nat" assumes "x < 2 ^ LENGTH('b)"   and "y < x" shows "word_of_nat y < word_of_nat x"
le_less: fixes x :: "'x \<Rightarrow>\<^sub>0 nat"   and y :: "'x \<Rightarrow>\<^sub>0 nat" shows "(x \<preceq> y) = (x \<prec> y \<or> x = y)"
DIAndSG1_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.DIAndSG1 vid1 pid1 pid3 = (Pc pid1 \<rightarrow> ([[$\<alpha> vid1]]Pc pid3) \<rightarrow> [[$\<alpha> vid1]]Pc pid1)"
is_sig_red_top_addsI: fixes f :: "'t \<Rightarrow>\<^sub>0 'b"   and F :: "('t \<Rightarrow>\<^sub>0 'b) set"   and p :: "'t \<Rightarrow>\<^sub>0 'b"   and sing_reg :: "'t \<Rightarrow> 't \<Rightarrow> bool" assumes "f \<in> F"   and "rep_list f \<noteq> 0"   and "rep_list p \<noteq> 0"   and "punit.lt (rep_list f) adds punit.lt (rep_list p)"   and "ord_term_lin.is_le_rel sing_reg"   and "sing_reg (punit.lt (rep_list p) \<oplus> lt f) (punit.lt (rep_list f) \<oplus> lt p)" shows "is_sig_red sing_reg (=) F p"
intros: fixes j :: "name"   and i :: "name"   and A :: "fm" assumes "atom j \<sharp> (i, A)" shows "A(i::=Zero) IMP SyntaxN.All i (SyntaxN.All j (A IMP A(i::=Var j) IMP A(i::=Eats (Var i) (Var j)))) IMP SyntaxN.All i A \<in> induction_axioms"
simps_3: fixes bs :: "real list"   and n :: "nat"   and c :: "int"   and a :: "MIR.num" shows "Inum bs (CN n c a) = real_of_int c * bs ! n + Inum bs a"
right_gpv_def: shows "right_gpv \<equiv> corec_gpv (\<lambda>gpv. map_spmf (map_generat id Inr (\<lambda>uu input. case input of Inl a \<Rightarrow> Inl Fail | Inr input' \<Rightarrow> Inr (uu input'))) (the_gpv gpv))"
\<omega>_sum_Cons: fixes \<alpha> :: "V"   and \<alpha>s :: "V list" shows "\<omega>_sum (\<alpha> # \<alpha>s) = \<omega> \<up> \<alpha> + \<omega>_sum \<alpha>s"
self_append_conv2: fixes ys :: "'a list"   and xs :: "'a list" shows "(ys = append xs ys) = (xs = [])"
compact_open: fixes S :: "'b set" shows "(compact S \<and> open S) = (S = {})"
dg_in_ArrE: fixes f :: "V" assumes "f \<in>\<^sub>\<circ> \<AA>\<lparr>Arr\<rparr>" obtains a :: "V" and b :: "V" where "f : a \<mapsto>\<^bsub>\<AA>\<^esub> b" and "a \<in>\<^sub>\<circ> \<AA>\<lparr>Obj\<rparr>" and "b \<in>\<^sub>\<circ> \<AA>\<lparr>Obj\<rparr>"
lm144: fixes A :: "'b set" shows "infinite A = (card (Pow A) = 0)"
valid_msg_def: fixes msg :: "nat" shows "valid_msg msg \<equiv> msg \<in> {..<q}"
f_range: fixes r :: "('d \<times> 'd) set"   and linear_cf :: "'d set \<Rightarrow> 'd set"   and X :: "'d set" assumes "linear_cf r linear_cf" shows "f_range_on X linear_cf"
less_self: fixes t :: "'c list"   and a :: "'c" shows "t < t @ [a]"
strong_formula_induct: fixes P :: "formula \<Rightarrow> bool"   and A :: "formula" assumes "\<forall>A. (\<forall>B. size B < size A \<longrightarrow> P B) \<longrightarrow> P A" shows "P A"
nhop_update: fixes rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and dip :: "nat"   and dsn :: "nat"   and dsk :: "k"   and flg :: "f"   and hops :: "nat"   and sip :: "nat" assumes "rt \<noteq> update rt dip (dsn, dsk, flg, hops, sip, {})" shows "the (nhop (update rt dip (dsn, dsk, flg, hops, sip, {})) dip) = sip"
MC_action_defs_2: fixes send :: "state \<Rightarrow> PrIds \<Rightarrow> (memOp, Vals) chan"   and rcv :: "state \<Rightarrow> PrIds \<Rightarrow> (rpcOp, Vals) chan"   and cst :: "state \<Rightarrow> PrIds \<Rightarrow> mClkState"   and p :: "PrIds" shows "MClkFwd send rcv cst p = ($Calling send p \<and> $(cst!p) = #clkA \<and> Call rcv p MClkRelayArg<arg<send!p>> \<and> (cst!p)$ = #clkB \<and> unchanged (rtrner send!p))"
mult_vec_zero_left: fixes p :: "'t \<Rightarrow>\<^sub>0 'c" shows "0 ** p = 0"
poly_lift_hom_X_minus_nat_pow: fixes S :: "('e, 'f) ring_scheme"   and \<phi> :: "'a \<Rightarrow> 'e"   and a :: "'a"   and n :: "nat" assumes "cring S"   and "\<phi> \<in> ring_hom R S"   and "a \<in> carrier R" shows "poly_lift_hom R S \<phi> (X_poly_minus R a [^]\<^bsub>UP R\<^esub> n) = X_poly_minus S (\<phi> a) [^]\<^bsub>UP S\<^esub> n"
case_prodE: fixes c :: "'b \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'b \<times> 'd" assumes "case p of (a, b) \<Rightarrow> c a b" obtains x :: "'b" and y :: "'d" where "p = (x, y)" and "c x y"
zero_in_unit_circle_set: shows "0\<^sub>h \<notin> circline_set unit_circle"
expand: fixes x :: "'a" shows "f\<cdot>x = g\<cdot>x \<union> f\<cdot>(r\<cdot>x)"
distinct_1: fixes x21 :: "'a"   and x22 :: "'a list" shows "NO_CYC \<noteq> REACH x21 x22"
\<Phi>o_in_hom: fixes a' :: "'c" assumes "C.ide a'" shows "\<guillemotleft>\<Phi>o a' : S.mkIde (Hom.set (a', a)) \<rightarrow>\<^sub>S S.mkIde (\<iota> ` D.cones a')\<guillemotright>"
enum_ex_fin_string: shows "enum_class.enum_ex \<equiv> fin_string_enum_ex"
complete_monitor: fixes \<pi> :: "'a prefix"   and \<sigma> :: "'a trace"   and v :: "'b option list"   and i :: "nat" assumes "prefix_of \<pi> \<sigma>"   and "wf_tuple nfv fv v"   and "\<And>\<sigma>. prefix_of \<pi> \<sigma> \<Longrightarrow> sat \<sigma> (map the v) i" shows "\<exists>\<pi>'. prefix_of \<pi>' \<sigma> \<and> (i, v) \<in> M \<pi>'"
push_forward_in_tangent_space: fixes p :: "'c" assumes "p \<in> dest.carrier" shows "inv.push_forward ` dest.tangent_space p \<subseteq> src.tangent_space (f' p)"
varMonotoneAux: fixes b :: "BExpr"   and \<Phi> :: "(Var \<Rightarrow> Val) \<times> (Var \<Rightarrow> Val) \<Rightarrow> bool"   and s :: "Var \<Rightarrow> Val"   and t :: "Var \<Rightarrow> Val"   and \<Psi> :: "(Var \<Rightarrow> Val) \<times> (Var \<Rightarrow> Val) \<Rightarrow> bool" assumes "(b, \<Phi>, s, t) \<in> var"   and "\<And>s t. \<Phi> (s, t) \<Longrightarrow> \<Psi> (s, t)" shows "(b, \<Psi>, s, t) \<in> var"
\<tau>_def: fixes f :: "'a" shows "\<tau> f \<equiv> if arr f then \<t>[local.dom f] else null"
prob_space_axioms: shows "prob_space (Pi\<^sub>M UNIV (\<lambda>_. M))"
distinct_update_with_aux: fixes v :: "'b"   and k :: "'a"   and f :: "'b \<Rightarrow> 'b"   and ps :: "('a \<times> 'b) list" shows "distinct (map fst (update_with_aux v k f ps)) = distinct (map fst ps)"
left_mult_one_mat': fixes A :: "'c mat"   and n :: "nat" assumes "dim_row A = n" shows "1\<^sub>m n * A = A"
spr_knowledge: fixes s :: "('a, 'es, 'as) BEState \<times> ('a, 'es, 'as) BEState"   and cec :: "(('a, 'es, 'as) BEState \<times> ('a, 'es, 'as) BEState) odlist"   and a :: "'a" assumes "s \<in> toSet cec" shows "toSet (spr_knowledge envObs cec a s) = relations (spr_repMC (toSet cec)) a `` {s}"
Union_Strong_dlHPP_Bisim: fixes d :: "'d"   and PP :: "nat set"   and R :: "(('exp, 'id) MWLsCom list \<times> ('exp, 'id) MWLsCom list) set"   and R' :: "(('exp, 'id) MWLsCom list \<times> ('exp, 'id) MWLsCom list) set" assumes "SdlHPPB d PP R"   and "SdlHPPB d PP R'"   and "Arefl R \<inter> Arefl R' \<subseteq> {[]}" shows "SdlHPPB d PP (R \<union> R')"
bilateral_substitutes_on_doctor_separable_on_unilateral_substitutes_on: fixes A :: "'x set"   and f :: "'x set \<Rightarrow> 'x set" assumes "bilateral_substitutes_on A f"   and "doctor_separable_on A f"   and "f_range_on A f" shows "unilateral_substitutes_on A f"
adjoin_id: fixes H :: "'a set"   and a :: "'a" assumes "subgroup H G"   and "a \<in> H" shows "adjoin G H a = H"
in_hom_char: fixes f :: "'c local.term set"   and a :: "'c local.term set"   and b :: "'c local.term set" shows "\<guillemotleft>f : a \<rightarrow>\<^sub>S b\<guillemotright> = (\<F>\<^sub>SC.arr a \<and> \<F>\<^sub>SC.arr b \<and> \<F>\<^sub>SC.arr f \<and> \<guillemotleft>f : a \<rightarrow> b\<guillemotright>)"
is_singleton_the_elem: fixes A :: "'g set" shows "is_singleton A = (A = {the_elem A})"
unlessE_def: fixes P :: "bool"   and f :: "'d \<Rightarrow> (('e + unit) \<times> 'd) \<times> bool" shows "unlessE P f \<equiv> if P then returnOk () else f"
carrier_qring: fixes I :: "'a set" assumes "ideal R I" shows "carrier (R /\<^sub>r I) = set_rcs (b_ag R) I"
sets_bind: fixes M :: "'a measure"   and f :: "'a \<Rightarrow> 'b measure"   and N :: "'b measure" assumes "\<And>x. x \<in> space M \<Longrightarrow> sets (f x) = sets N"   and "space M \<noteq> {}" shows "sets (M \<bind> f) = sets N"
count_roots_between: fixes ps :: "real poly list"   and p :: "real poly"   and a :: "real"   and b :: "real" assumes "sturm_seq ps p"   and "p \<noteq> 0"   and "a \<le> b" shows "sign_changes ps a - sign_changes ps b = card {x. a < x \<and> x \<le> b \<and> poly p x = 0}"
Aij: fixes i :: "'n"   and j :: "'n" shows "A $h i $h j = from_hma\<^sub>m A $$ (Bij_Nat.to_nat i, Bij_Nat.to_nat j)"
steps: shows "P,C \<turnstile> [C] \<sqsubseteq> Cs @ [C']"
fmrel_to_subtuple: fixes r :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and xm :: "'f \<rightharpoonup>\<^sub>f 'd"   and ym :: "'f \<rightharpoonup>\<^sub>f 'e" assumes "fmrel r xm ym" shows "subtuple r xm ym"
load_sub2_low_equal: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state"   and t1 :: "'b sparc_state"   and address :: "word32"   and rd :: "word5"   and curr_win :: "'b word"   and w :: "word32"   and t2 :: "'b sparc_state" assumes "low_equal s1 s2 \<and> t1 = snd (fst (load_sub2 address 10 rd curr_win w s1)) \<and> t2 = snd (fst (load_sub2 address 10 rd curr_win w s2))" shows "low_equal t1 t2"
obvzer: shows "(\<Sum>s\<in>set (map (consistent_sign_vec qs) (characterize_root_list_p p)) \<inter> {s. z_R I s = 0}. z_R I s * rat_of_int (int (card {x. poly p x = 0 \<and> consistent_sign_vec qs x = s}))) = 0"
simps_2: fixes v :: "val"   and r :: "proc \<times> val" shows "\<gamma> (Some v) r = v"
assms_2: shows "lexext_rev gt (y # ys) (x # xs)"
encoding: fixes x :: "\<kappa>"   and F :: "\<Pi>\<^sub>1" shows "[[\<lbrace>x,F\<rbrace> \<^bold>\<rightarrow> \<^bold>\<box>\<lbrace>x,F\<rbrace>]]"
one_unique: fixes u :: "'c \<Rightarrow> 'a" assumes "u \<in> carrier F"   and "\<And>x. x \<in> carrier F \<Longrightarrow> u \<otimes>\<^bsub>F\<^esub> x = x" shows "u = \<one>\<^bsub>F\<^esub>"
PO_rhoare_defs_2: fixes r :: "('e \<times> 'g) set"   and s :: "('g \<times> 'f) set" shows "r O s = {(x, z). \<exists>y. (x, y) \<in> r \<and> (y, z) \<in> s}"
cong_0_1_nat: fixes n :: "nat" shows "[0 = 1] (mod n) = (n = 1)"
eq_prefix_TT: fixes xs :: "[:'a:]"   and ys :: "[:'a:]" assumes "eq\<cdot>xs\<cdot>ys = TT" shows "prefix\<cdot>xs\<cdot>ys = TT"
TensorDiag_preserves_Diag_3: fixes t :: "'a local.term"   and u :: "'a local.term" assumes "Diag t"   and "Diag u" shows "Cod (t \<^bold>\<lfloor>\<^bold>\<otimes>\<^bold>\<rfloor> u) = Cod t \<^bold>\<lfloor>\<^bold>\<otimes>\<^bold>\<rfloor> Cod u"
weak_sharing_consis: fixes i :: "nat"   and p :: "'a"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "'a memref list"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option" assumes "i < length (t # ts)"   and "(t # ts) ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>)" shows "weak_sharing_consistent \<O> sb"
diagonal_admits_SNF_imp_bezout_ring_JNF: assumes "\<forall>A n. A \<in> carrier_mat n n \<and> isDiagonal_mat A \<longrightarrow> (\<exists>P Q. P \<in> carrier_mat n n \<and> Q \<in> carrier_mat n n \<and> invertible_mat P \<and> invertible_mat Q \<and> Smith_normal_form_mat (P * A * Q))" shows "OFCLASS('b, bezout_ring_class)"
length_tp: fixes ly :: "nat list"   and ap :: "abc_inst list"   and tp :: "(action \<times> nat) list" assumes "ly = layout_of ap"   and "tp = tm_of ap" shows "start_of ly (length ap) = Suc (length tp div 2)"
subgroup_of_subgroup_generated: fixes G :: "('b, 'c) monoid_scheme"   and H :: "'b set"   and B :: "'b set" assumes "Group.group G"   and "H \<subseteq> B"   and "subgroup H G" shows "subgroup H (subgroup_generated G B)"
ls_trans_l: fixes R :: "('b \<times> 'b) set"   and A :: "'b multiset"   and B :: "'b multiset" assumes "locally_sym R (A + B)" shows "locally_sym R A"
that: fixes F :: "('a \<times> 'b \<times> 'a) set"   and I :: "('a \<times> 'b \<times> 'a) set set" assumes "(F, I) \<in> snd (snd R)"   and "accepting_pair\<^sub>G\<^sub>R_LTS (fst R) (fst (snd R)) (F, I) w" shows "thesis"
assms: shows "c \<notin>\<^sub>\<circ> B"
Float_mantissa_exponent: fixes f :: "Float.float" shows "Float (mantissa f) (Float.exponent f) = f"
Par2Chain: fixes Q :: "pi"   and Q' :: "pi"   and P :: "pi" assumes "Q \<Longrightarrow>\<^sub>\<tau> Q'" shows "P \<parallel> Q \<Longrightarrow>\<^sub>\<tau> P \<parallel> Q'"
refl: fixes a :: "'b multiset" shows "a \<subseteq># a"
simps_1: shows "Partial_Recursive.list_decode 0 = []"
exhaust: fixes y :: "SN_rel_ext_type" obtains "y = top_s"   | "y = top_ns"   | "y = normal_s"   | "y = normal_ns"
aff_dim_empty: fixes S :: "'b set" shows "(S = {}) = (aff_dim S = - 1)"
is_balanced_def: shows "is_balanced \<equiv> (\<forall>x\<in>{0, 1}. f x = x) \<or> is_swap"
containsI: fixes x :: "'b"   and A :: "'b set" assumes "x \<in> A" shows "Predicate_Compile.contains A x"
FP_def: fixes F :: "'b program" shows "FP F \<equiv> {s. F \<in> stable {s}}"
connected_componentI: fixes T :: "'b set"   and S :: "'b set"   and x :: "'b"   and y :: "'b" assumes "connected T"   and "T \<subseteq> S"   and "x \<in> T"   and "y \<in> T" shows "connected_component S x y"
get_scdom_component_ok: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr" assumes "heap_is_wellformed h"   and "DocumentClass.type_wf h"   and "DocumentClass.known_ptrs h"   and "ptr |\<in>| object_ptr_kinds h" shows "h \<turnstile> ok get_scdom_component ptr"
atLeast_eq_iff: fixes x :: "'a"   and y :: "'a" shows "({x..} = {y..}) = (x = y)"
\<delta>: fixes p :: "fm"   and bs :: "real list" assumes "iszlfm p bs" shows "d_\<delta> p (\<delta> p) \<and> 0 < \<delta> p"
d: shows "distributed M (count_space (X ` space M)) X (\<lambda>x. ennreal (if x \<in> X ` space M then prob {xa \<in> space M. X xa = x} else 0))"
subseq_Cons': fixes x :: "'a"   and xs :: "'a list"   and ys :: "'a list" assumes "subseq (x # xs) ys" shows "subseq xs ys"
complex_roots_int_def: fixes p :: "int poly" shows "complex_roots_int p = complex_roots_complex (of_int_poly p)"
SEM_def: fixes c :: "'b com"   and S :: "'b set" shows "SEM c S \<equiv> \<Union> (sem c ` S)"
ASSERT_leI: fixes \<Phi> :: "bool"   and M :: "'ma"   and M' :: "'ma" assumes "\<Phi>"   and "\<Phi> \<Longrightarrow> M \<le> M'" shows "bind (ASSERT \<Phi>) (\<lambda>_. M) \<le> M'"
cod_char: fixes f :: "('a \<Rightarrow> 'b, 'a \<Rightarrow> 'b) concrete_category.arr" shows "cod f = (if arr f then MkIde (Cod f) else null)"
agents_eq: shows "agents = {ia. (R(i := Ri')) ia \<noteq> (\<lambda>_ _. False)}"
r'_def: shows "r' = (\<Sum>i\<in>Basis. (real (r (b' i)) / real p) *\<^sub>R i)"
simple_match_ip_Accept: fixes d :: "'a word" assumes "simple_match_ip (src m) s" shows "(runFw s d c (SimpleRule m Accept # rs) = Decision FinalAllow) = (simple_match_ip (dst m) d \<or> runFw s d c rs = Decision FinalAllow)"
not_KeyCryptKey_cardSK: fixes cardSK :: "nat"   and evs :: "event list"   and K :: "nat" assumes "cardSK \<notin> symKeys"   and "\<forall>C. cardSK \<noteq> priEK C"   and "evs \<in> set_cr"   and "Key cardSK \<notin> analz (knows Spy evs)" shows "\<not> KeyCryptKey cardSK K evs"
1: shows "(set_mset A, set_mset B) \<in> max_strict \<or> B = {#} \<and> A = {#}"
Z99: shows "Col A B D \<longrightarrow> P Q TS B C"
Sup_distl: fixes x :: "'c \<times> 'd"   and Y :: "('c \<times> 'd) set" shows "times_prod' x (Sup_prod Y) = Sup_prod (times_prod' x ` Y)"
act2: fixes p :: "'b" shows "\<alpha> (1::'a) p = p"
design_iso_points_indices_imp: fixes \<V> :: "'c set"   and \<B> :: "'c set multiset"   and \<V>' :: "'d set"   and \<B>' :: "'d set multiset"   and bij_map :: "'c \<Rightarrow> 'd"   and x :: "int"   and t :: "int" assumes "design_isomorphism \<V> \<B> \<V>' \<B>' bij_map"   and "x \<in> incidence_system.point_indices \<V> \<B> t" shows "x \<in> incidence_system.point_indices \<V>' \<B>' t"
nn_integral_cmult: fixes f :: "'k \<Rightarrow> ennreal"   and M :: "'k measure"   and c :: "ennreal" assumes "f \<in> borel_measurable M" shows "\<integral>\<^sup>+ x. c * f x \<partial>M = c * integral\<^sup>N M f"
genPrefixp_genPrefix_eq: fixes r :: "('a \<times> 'a) set" shows "genPrefixp (\<lambda>x xa. (x, xa) \<in> r) = (\<lambda>x xa. (x, xa) \<in> genPrefix r)"
lookup_extend_indets_subst_aux: fixes t :: "'b \<Rightarrow>\<^sub>0 'c" shows "lookup (\<Sum>y\<in>keys t. monomial (lookup t y) (Some y)) = (\<lambda>x. case x of None \<Rightarrow> 0::'c | Some y \<Rightarrow> lookup t y)"
valid_own_sharing: shows "valid_ownership_and_sharing \<S>\<^sub>s\<^sub>b ts\<^sub>s\<^sub>b"
fun_app_def: fixes f :: "'b" shows "SMT.fun_app f = f"
bisim1TryCatchThrow: fixes P :: "'b prog"   and h :: "'heap"   and e2 :: "'addr expr1"   and a :: "'addr"   and xs :: "'addr val list"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and e :: "'addr expr1"   and C :: "String.literal"   and V :: "nat" assumes "P,e2,h \<turnstile> (Throw a, xs) \<leftrightarrow> (stk, loc, pc, \<lfloor>a\<rfloor>)" shows "P,try e catch(C V) e2,h \<turnstile> (Throw a, xs) \<leftrightarrow> (stk, loc, Suc (Suc (length (compE2 e) + pc)), \<lfloor>a\<rfloor>)"
int2: shows "integrable M (\<lambda>x. f x * max (- g x) 0)"
11: shows "sum_class.sum ((w \<sqinter> - (top * selected_edge h j g * w\<^sup>T\<^sup>\<star>) \<squnion> (w \<sqinter> top * selected_edge h j g * w\<^sup>T\<^sup>\<star>)\<^sup>T) \<sqinter> g) = sum_class.sum (w \<sqinter> g)"
code: fixes x :: "octo" shows "- x = Octo (- Ree x) (- Im1 x) (- Im2 x) (- Im3 x) (- Im4 x) (- Im5 x) (- Im6 x) (- Im7 x)"
SeqConseq: fixes P :: "'d set"   and P' :: "'d set"   and \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and \<Theta> :: "'e \<Rightarrow> ('d, 'e, 'f) ann list option"   and F :: "'f set"   and a :: "('d, 'e, 'f) ann"   and c :: "('d, 'e, 'f) com"   and Q' :: "'d set"   and A' :: "'d set"   and Q :: "'d set"   and A :: "'d set" assumes "P \<subseteq> P'"   and "\<Gamma>, \<Theta> \<tturnstile>\<^bsub>/F\<^esub> P' a c Q', A'"   and "Q' \<subseteq> Q"   and "A' \<subseteq> A" shows "\<Gamma>, \<Theta> \<tturnstile>\<^bsub>/F\<^esub> P a c Q, A"
wf'_3: shows "nonneg_cexpr (shift_var_set (set vs')) (case_nat REAL \<Gamma>) f"
hyps_3: assumes "y \<in> loc_reach lc" shows "z \<in> loc_reach lc"
SF_alt: fixes A :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and v :: "'a \<Rightarrow> 'b" shows "\<turnstile> SF(A)_v = (\<diamond>\<box>\<not> Enabled \<langle>A\<rangle>_v \<or> \<box>\<diamond>\<langle>A\<rangle>_v)"
ih4: fixes x :: "name"   and M :: "trm" assumes "x:M \<in> \<parallel>(B)\<parallel>" shows "SNa M"
arr_cod_iff_arr: fixes f :: "'a \<times> 'a" shows "H.VV.arr (H.VV.cod f) = H.VV.arr f"
B_eq_17_main: fixes i :: "nat" assumes "2 \<le> i"   and "i \<le> k" shows "h i = (- 1) ^ (n 1 + n i + i + 1) / f i * (\<Prod>l\<leftarrow>[3..<Suc (Suc i)]. \<alpha> l / \<beta> l) \<and> h i \<noteq> 0"
SXcptC_def: fixes xn :: "xname" shows "SXcptC xn = (SXcpt xn, \<lparr>access = Public, cfields = [], methods = SXcpt_mdecls, init = Skip, super = if xn = Throwable then Object else SXcpt Throwable, superIfs = []\<rparr>)"
borel_measurable_continuous_onI: fixes f :: "'b \<Rightarrow> 'c" assumes "continuous_on UNIV f" shows "f \<in> borel_measurable borel"
has_left_def: fixes xs :: "int list"   and j :: "nat" shows "has_left xs j = (\<exists>i<j. xs ! i < xs ! j)"
flow_has_derivative: fixes t :: "real"   and x0 :: "'d" assumes "t \<in> existence_ivl0 x0" shows "(flow0 x0 has_derivative (\<lambda>i. i *\<^sub>R ode (flow0 x0 t))) (at t)"
inorder_n2: fixes l :: "'c bro"   and a :: "'c"   and r :: "'c bro" shows "inorder (n2 l a r) = inorder l @ a # inorder r"
carrier_stalk_def: shows "stfx.carrier_stalk \<equiv> dlim \<O>\<^sub>Y \<rho>\<^sub>Y (target.neighborhoods (f x))"
open_unresolvable_not_ed_ou_card: fixes X :: "'b set" assumes "open_unresolvable X"   and "\<not> ed_ou X" shows "card {f. CK_nf_pos f} = 5"
pre_ranks_pre_oldest_token_Min_state: fixes q :: "'b"   and q' :: "'b"   and n :: "nat" assumes "\<not> sink q"   and "q = \<delta> q' (w n)"   and "configuration q (Suc n) \<noteq> {}" shows "(state_rank q' n = Some (Min (pre_ranks (\<lambda>q. state_rank q n) (w n) q))) = (oldest_token q' n = Some (Min (pre_oldest_tokens q n)))"
bigthetaI: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b" assumes "f \<in> O[F](g)"   and "f \<in> \<Omega>[F](g)" shows "f \<in> \<Theta>[F](g)"
sup_idem: fixes x :: "'c \<times> 'c \<Rightarrow> 'd" shows "x \<oplus> x = x"
fin_vc_elems: fixes c :: "nat set" assumes "c \<in> vc" shows "finite c"
E21: fixes F :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool" shows "\<turnstile> \<box>\<diamond>\<box>F = \<diamond>\<box>F"
SKIPt: fixes s :: "char list \<Rightarrow> int" shows "\<down>\<^sub>s (SKIP, s) = s"
invs_imp_InvGlobVacantUptoImpliesNrec: fixes s :: "('c, 'd) configuration stream" assumes "holds InvGlobalIncomingInfoUpright s"   and "holds InvGlobalRecordCount s"   and "holds (\<lambda>c. \<forall>t. 0 \<le> zcount (c_records c) t) s" shows "holds InvGlobVacantUptoImpliesNrec s"
SA: fixes S :: "'a set" assumes "S \<in> A" shows "open S \<and> connected S \<and> S \<noteq> {}"
h3: shows "lose t = [False] \<longrightarrow> ack t = [y] \<and> i1 t = b t \<and> vc t = [] \<and> st_out t = y"
sappend_def: fixes Al :: "'e list set"   and Bl :: "'e list set" shows "Al @@ Bl = {al @ bl |al bl. al \<in> Al \<and> bl \<in> Bl}"
remove_max_inv: fixes l :: "'b"   and m :: "'a"   and l' :: "'b" assumes "\<not> is_empty l"   and "inv l"   and "(m, l') = remove_max l" shows "inv l'"
gdpr_three: fixes h :: "char list"   and l :: "location"   and Igraph :: "infrastructure \<Rightarrow> igraph"   and d :: "(actor \<times> actor set) \<times> nat" assumes "h \<in> gdpr_actors"   and "l \<in> gdpr_locations"   and "owns (Igraph gdpr_scenario) l (Actor h) d" shows "gdpr_Kripke \<turnstile> AG {x. \<forall>l\<in>gdpr_locations. owns (Igraph x) l (Actor h) d}"
guessed_frame_dom_spec: fixes R :: "fid_t" shows "dom (guessed_frame R) = domain (role (guessed_runs R))"
measurable_on_UNIV: fixes S :: "'c set"   and f :: "'c \<Rightarrow> 'd" shows "((\<lambda>x. if x \<in> S then f x else (0::'d)) measurable_on UNIV) = (f measurable_on S)"
\<Z>_srng_\<Z>_csgrp_vplus: fixes \<alpha> :: "V"   and \<SS> :: "V" assumes "\<Z>_srng \<alpha> \<SS>" shows "\<Z>_csgrp \<alpha> [\<SS>\<lparr>\<A>\<rparr>, \<SS>\<lparr>vplus\<rparr>]\<^sub>\<circ>"