hcju/isabelleps huggingface dataset:
w: shows "w = geodesic_segment_param Gxy y (Gromov_product_at y x z)"
pair_no_loops: fixes G :: "'b pair_pre_digraph"   and e :: "'b \<times> 'b" assumes "pair_loopfree_digraph G"   and "e \<in> parcs G" shows "fst e \<noteq> snd e"
G_def: fixes f :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and z :: "'c"   and p :: "'d \<Rightarrow> 'c"   and I :: "'d set" assumes "comm_monoid_set f z" shows "comm_monoid_set.G f z p I \<equiv> if finite {x \<in> I. p x \<noteq> z} then comm_monoid_set.F f z p {x \<in> I. p x \<noteq> z} else z"
update_def: fixes rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option"   and ip :: "nat"   and r :: "nat \<times> k \<times> f \<times> nat \<times> nat" shows "update rt ip r \<equiv> case rt ip of None \<Rightarrow> rt(ip \<mapsto> r) | Some s \<Rightarrow> if \<pi>\<^sub>2 s < \<pi>\<^sub>2 r then rt(ip \<mapsto> r) else if \<pi>\<^sub>2 s = \<pi>\<^sub>2 r \<and> (\<pi>\<^sub>5 r < \<pi>\<^sub>5 s \<or> \<pi>\<^sub>4 s = Aodv_Basic.inv) then rt(ip \<mapsto> r) else if \<pi>\<^sub>3 r = unk then rt(ip \<mapsto> (\<pi>\<^sub>2 s, snd r)) else rt(ip \<mapsto> s)"
in_gamma_update: fixes num' :: "int \<Rightarrow> 'b"   and plus' :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<gamma> :: "'b \<Rightarrow> int set"   and s :: "char list \<Rightarrow> int"   and S :: "'b Abs_State.st"   and i :: "int"   and a :: "'b"   and x :: "char list" assumes "Abs_Int num' plus' \<gamma>"   and "s \<in> \<gamma>_st \<gamma> S"   and "i \<in> \<gamma> a" shows "s(x := i) \<in> \<gamma>_st \<gamma> (update S x a)"
ereal_minus_1: fixes r :: "real"   and p :: "real" shows "ereal r - ereal p = ereal (r - p)"
row_to_poly_zero: fixes ts :: "'t list" shows "row_to_poly ts (0\<^sub>v (length ts)) = 0"
emb_at_Cons: fixes p :: "dir list"   and d :: "dir"   and ps :: "(dir list \<times> dir) list"   and t :: "('c, 'd) tm" shows "emb_at ((p, d) # ps) t = emb_step_at p d (emb_at ps t)"
dec_i_def: fixes ba :: "'b list \<times> 'b list" shows "dec_i ba = (case ba of (b, a) \<Rightarrow> (tl b, hd b # a))"
entries_impl_of: fixes t :: "('a, 'b) RBT.rbt" shows "RBT_Impl.entries (impl_of t) = entries t"
rcos_mult_step3: fixes H :: "'c set"   and G :: "('c, 'd) monoid_scheme"   and x :: "'c"   and y :: "'c" assumes "H \<lhd> G"   and "x \<in> carrier G"   and "y \<in> carrier G" shows "H <#>\<^bsub>G\<^esub> (H #>\<^bsub>G\<^esub> x) #>\<^bsub>G\<^esub> y = H #>\<^bsub>G\<^esub> x \<otimes>\<^bsub>G\<^esub> y"
hd_snd: fixes V :: "nat set"   and V' :: "nat set"   and \<Gamma> :: "nat \<Rightarrow> pdf_type"   and \<delta> :: "(nat \<Rightarrow> val) \<Rightarrow> ennreal"   and e :: "expr"   and f :: "(nat \<Rightarrow> val) \<Rightarrow> val \<Rightarrow> ennreal" assumes "(V, V', \<Gamma>, \<delta>) \<turnstile>\<^sub>d e \<Rightarrow> f" shows "(V, V', \<Gamma>, \<delta>) \<turnstile>\<^sub>d Snd $$ e \<Rightarrow> (\<lambda>\<rho> y. \<integral>\<^sup>+ x. f \<rho> <|x, y|> \<partial>stock_measure (the (expr_type \<Gamma> (Fst $$ e))))"
collect_locks'_def: fixes las :: "'c lock_actions" shows "collect_locks' las \<equiv> {l. must_acquire_lock (las $ l)}"
last_coeffs_eq_coeff_degree: fixes p :: "'b poly" assumes "p \<noteq> 0" shows "last (coeffs p) = lead_coeff p"
that: fixes C' :: "(('a list \<times> nat) multiset \<Rightarrow> 'a) ring set" assumes "C' \<subseteq> extensions"   and "C = law_restrict ` C'" shows "thesis"
5: fixes y :: "'a"   and y' :: "'a" assumes "y \<in> V"   and "y' \<in> V" shows "norm (g y' - g y) \<le> 2 * norm (y' - y)"
flow0_def: fixes x0 :: "'a"   and t :: "real" shows "flow0 x0 t = ll_on_open.flow UNIV (\<lambda>_. f) X 0 x0 t"
lu_2: shows "lim (\<lambda>n. u n / real n) = lw"
b: shows "(supp x - as \<inter> supp x) \<sharp>* p"
approx_floatarith_Elem: fixes p :: "nat"   and ra :: "floatarith"   and VS :: "((real \<times> real pdevs) \<times> real) list"   and X :: "(real \<times> real pdevs) \<times> real"   and e :: "nat \<Rightarrow> real"   and vs :: "real list" assumes "approx_floatarith p ra VS = Some X"   and "e \<in> funcset UNIV {- 1..1}"   and "vs \<in> aforms_err e VS" shows "interpret_floatarith ra vs \<in> aform_err e X"
monomial_uminus: fixes s :: "'d"   and c :: "'e" shows "- monomial c s = monomial (- c) s"
MPair_def: fixes X :: "msg"   and Y :: "msg" shows "MPair X Y = Abs_Msg (\<Union>U\<in>Rep_Msg X. \<Union>V\<in>Rep_Msg Y. msgrel `` {MPAIR U V})"
disjSB: fixes r :: "real" shows "disjnt (S r) (B r)"
wcode_double_case: fixes m :: "nat"   and ires :: "cell list"   and rs :: "nat"   and n :: "nat" shows "\<exists>stp ln rn. steps0 (Suc 0, Bk # Bk \<up> m @ Oc # Oc # ires, Bk # Oc \<up> Suc rs @ Bk \<up> n) t_wcode_main stp = (Suc 0, Bk # Bk \<up> ln @ Oc # ires, Bk # Oc \<up> Suc (2 * rs + 2) @ Bk \<up> rn)"
sym: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "sympartp r x y" shows "sympartp r y x"
simps: fixes I :: "('b, 'c, 'a) interp"   and ODE :: "('b, 'a) ODE"   and sol :: "(real, 'a) vec" shows "mk_xode I ODE sol = (sol, ODE_sem I ODE sol)"
simps_1: fixes i :: "nat"   and k :: "nat" shows "loose_bvar (Bv i) k = (k \<le> i)"
naturality: fixes f :: "'c \<times> 'c" assumes "C.VV.arr f" shows "\<Phi>\<^sub>G (C.VV.cod f) \<cdot>\<^sub>D G.H\<^sub>DoFF.map f = G.FoH\<^sub>C.map f \<cdot>\<^sub>D \<Phi>\<^sub>G (C.VV.dom f)"
rev_vsubsetD: fixes c :: "V"   and a :: "V"   and b :: "V" assumes "c \<in>\<^sub>\<circ> a"   and "a \<subseteq>\<^sub>\<circ> b" shows "c \<in>\<^sub>\<circ> b"
awalk_cyc_decomp_has_prop: fixes u :: "'a"   and p :: "'b list"   and v :: "'a" assumes "awalk u p v"   and "\<not> distinct (awalk_verts u p)" shows "is_awalk_cyc_decomp p (awalk_cyc_decomp p)"
cases: fixes ci :: "('addr, 'heap) check_instr"   and P :: "(nat \<times> nat \<times> 'addr instr list \<times> (nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list) prog"   and ins :: "'addr instr list"   and xt :: "(nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list"   and t :: "'thread_id"   and a1 :: "'heap"   and a2 :: "'addr val list \<times> 'addr val list \<times> nat \<times> 'addr option"   and a3 :: "('addr, 'thread_id, 'heap) jvm_thread_action"   and a4 :: "'heap"   and a5 :: "'addr val list \<times> 'addr val list \<times> nat \<times> 'addr option" assumes "exec_meth ci P ins xt t a1 a2 a3 a4 a5" obtains ta :: "('addr, 'thread_id, 'heap) jvm_thread_action" and xcp :: "'addr option" and h' :: "'heap" and stk' :: "'addr val list" and loc' :: "'addr val list" and pc' :: "nat" and pc :: "nat" and h :: "'heap" and stk :: "'addr val list" and loc :: "'addr val list" where "a1 = ha__" and "a2 = (stkb__, locb__, pcb__, None)" and "a3 = ta" and "a4 = h'" and "a5 = (stk', loc', pc'b__, xcpb__)" and "(ta, xcpb__, h', [(stk', loc', undefined, undefined, pc'b__)]) \<in> exec_instr (ins ! pcb__) P t ha__ stkb__ locb__ undefined undefined pcb__ []" and "pcb__ < length ins" and "ci_app ci (ins ! pcb__) P ha__ stkb__ locb__ undefined undefined pcb__ []"   | h :: "'heap" and xcp :: "'addr" and pc :: "nat" and pc' :: "nat" and d :: "nat" and stk :: "'addr val list" and loc :: "'addr val list" where "a1 = h" and "a2 = (stk, loc, pc, \<lfloor>xcp\<rfloor>)" and "a3 = \<lbrace>\<rbrace>" and "a4 = h" and "a5 = (Addr xcp # drop (length stk - d) stk, loc, pc', None)" and "match_ex_table P (cname_of h xcp) pc xt = \<lfloor>(pc', d)\<rfloor>" and "d \<le> length stk"
rel_interior_convex_nonempty_aux: fixes S :: "'a set" assumes "convex S"   and "(0::'a) \<in> S" shows "rel_interior S \<noteq> {}"
le_SPEC_UNIV_rule: fixes m :: "'a nres" assumes "m \<le> SPEC (\<lambda>_. True)" shows "m \<le> RES UNIV"
stirling_weak_bigo: shows "(\<lambda>x. sum_upto (\<lambda>x. ln (real x)) x - x * ln x) \<in> O(\<lambda>x. x)"
exe_wf_subclass_def: fixes cs :: "('a \<times> 'a) list" shows "exe_wf_subclass cs = (trans (set cs) \<and> exe_antisym cs \<and> Refl (set cs))"
supported_instruction_def: fixes instr :: "sparc_operation" shows "supported_instruction instr \<equiv> if instr \<in> {load_store_type LDSB, load_store_type LDUB, load_store_type LDUBA, load_store_type LDUH, load_store_type LD, load_store_type LDA, load_store_type LDD, load_store_type STB, load_store_type STH, load_store_type ST, load_store_type STA, load_store_type STD, sethi_type SETHI, nop_type NOP, logic_type ANDs, logic_type ANDcc, logic_type ANDN, logic_type ANDNcc, logic_type ORs, logic_type ORcc, logic_type ORN, logic_type XORs, logic_type XNOR, shift_type SLL, shift_type SRL, shift_type SRA, arith_type ADD, arith_type ADDcc, arith_type ADDX, arith_type SUB, arith_type SUBcc, arith_type SUBX, arith_type UMUL, arith_type SMUL, arith_type SMULcc, arith_type UDIV, arith_type UDIVcc, arith_type SDIV, ctrl_type SAVE, ctrl_type RESTORE, call_type CALL, ctrl_type JMPL, ctrl_type RETT, sreg_type RDY, sreg_type RDPSR, sreg_type RDWIM, sreg_type RDTBR, sreg_type WRY, sreg_type WRPSR, sreg_type WRWIM, sreg_type WRTBR, load_store_type FLUSH, bicc_type BE, bicc_type BNE, bicc_type BGU, bicc_type BLE, bicc_type BL, bicc_type BGE, bicc_type BNEG, bicc_type BG, bicc_type BCS, bicc_type BLEU, bicc_type BCC, bicc_type BA, bicc_type BN} then True else False"
sel_1: fixes x21 :: "'c"   and x22 :: "'c llist" shows "un_LCons1 (LCons x21 x22) = x21"
natural_transformation_axioms: shows "natural_transformation J (\<cdot>\<^sub>B) Fa.map (F \<circ> D) (F \<circ> \<chi>)"
intpow_diff: shows "fls_X_intpow (df - 1) = X * fls_X_inv"
norm_sum: fixes f :: "'d \<Rightarrow> 'c"   and A :: "'d set" shows "norm (sum f A) \<le> (\<Sum>i\<in>A. norm (f i))"
evenperm_comp: fixes p :: "'a \<Rightarrow> 'a"   and q :: "'a \<Rightarrow> 'a" assumes "permutation p"   and "permutation q" shows "evenperm (p \<circ> q) = (evenperm p = evenperm q)"
curry_preserves_transformations: fixes F :: "'a1 \<times> 'a2 \<Rightarrow> 'b"   and G :: "'a1 \<times> 'a2 \<Rightarrow> 'b"   and \<tau> :: "'a1 \<times> 'a2 \<Rightarrow> 'b" assumes "natural_transformation (\<cdot>\<^sub>A\<^sub>1\<^sub>x\<^sub>A\<^sub>2) (\<cdot>\<^sub>B) F G \<tau>" shows "natural_transformation (\<cdot>\<^sub>A\<^sub>1) (\<cdot>\<^sub>[\<^sub>A\<^sub>2,\<^sub>B\<^sub>]) (local.curry F F F) (local.curry G G G) (local.curry F G \<tau>)"
simps: fixes n :: "nat"   and m :: "nat"   and f :: "int vec list \<times> int vec list"   and mu :: "int iarray iarray"   and ds :: "int iarray"   and i :: "nat"   and fs :: "int vec list" shows "LLL.LLL_impl_inv n m (f, mu, ds) i fs = (list_repr i f (map ((!) fs) [0..<m]) \<and> LLL.d_repr n m ds fs \<and> LLL.mu_repr n m mu fs)"
cod_dom: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and f :: "'a" assumes "category C"   and "partial_magma.arr C f" shows "partial_magma.cod C (partial_magma.dom C f) = partial_magma.dom C f"
simps_1: fixes uu :: "'d set" shows "as_proj [] uu = []"
Cons_2: shows "C \<in> f \<turnstile> \<C>"
lens_indep_def: fixes X :: "'a \<Longrightarrow> 'b"   and Y :: "'c \<Longrightarrow> 'b" shows "X \<bowtie> Y \<equiv> (\<forall>\<sigma> v u. put\<^bsub>X\<^esub> (put\<^bsub>Y\<^esub> \<sigma> v) u = put\<^bsub>Y\<^esub> (put\<^bsub>X\<^esub> \<sigma> u) v) \<and> (\<forall>\<sigma> v. get\<^bsub>X\<^esub> (put\<^bsub>Y\<^esub> \<sigma> v) = get\<^bsub>X\<^esub> \<sigma>) \<and> (\<forall>\<sigma> u. get\<^bsub>Y\<^esub> (put\<^bsub>X\<^esub> \<sigma> u) = get\<^bsub>Y\<^esub> \<sigma>)"
rel_option_flip: fixes R :: "'b \<Rightarrow> 'a \<Rightarrow> bool" shows "rel_option (flip R) = flip (rel_option R)"
antisym: fixes b :: "('a, 'b) Term.term"   and a :: "('a, 'b) Term.term" assumes "b \<sqsubseteq> a"   and "a \<sqsubseteq> b" shows "a = b"
map_comp0: fixes g1 :: "'j \<Rightarrow> 'g"   and f1 :: "'d \<Rightarrow> 'j"   and g2 :: "'k \<Rightarrow> 'h"   and f2 :: "'e \<Rightarrow> 'k"   and g3 :: "'l \<Rightarrow> 'i"   and f3 :: "'f \<Rightarrow> 'l" shows "F1map (g1 \<circ> f1) (g2 \<circ> f2) (g3 \<circ> f3) = F1map g1 g2 g3 \<circ> F1map f1 f2 f3"
simps_1: fixes A :: "bool"   and uu :: "bool"   and uv :: "unit" shows "Dom (MkArr A uu uv) = A"
one_in_subset: fixes H :: "'a set" assumes "H \<subseteq> carrier (G\<lparr>carrier := H__\<rparr>)"   and "H \<noteq> {}"   and "\<forall>a\<in>H. inv\<^bsub>G\<lparr>carrier := H__\<rparr>\<^esub> a \<in> H"   and "\<forall>a\<in>H. \<forall>b\<in>H. a \<otimes>\<^bsub>G\<lparr>carrier := H__\<rparr>\<^esub> b \<in> H" shows "\<one>\<^bsub>G\<lparr>carrier := H__\<rparr>\<^esub> \<in> H"
Mprefix_def: fixes A :: "'b set"   and P :: "'b \<Rightarrow> 'b process" shows "Mprefix A P \<equiv> Abs_process ({(tr, ref). tr = [] \<and> ref \<inter> ev ` A = {}} \<union> {(tr, ref). tr \<noteq> [] \<and> hd tr \<in> ev ` A \<and> (\<exists>a. ev a = hd tr \<and> (tl tr, ref) \<in> F (P a))}, {d. d \<noteq> [] \<and> hd d \<in> ev ` A \<and> (\<exists>a. ev a = hd d \<and> tl d \<in> D (P a))})"
ChiPlus_def: fixes A :: "('a, 'b, 'c) hierauto" shows "ChiPlus A \<equiv> (ChiRel A)\<^sup>+"
p_t_stop: shows "delayed_safe_distance.other.p delayed_safe_distance.other.t_stop = delayed_safe_distance.other.p_max"
pred_inject_2: fixes P1 :: "'d \<Rightarrow> bool"   and P2 :: "'e \<Rightarrow> bool"   and P3 :: "'f \<Rightarrow> bool"   and a :: "'e"   and aa :: "'f" shows "pred_generat P1 P2 P3 (IO a aa) = (P2 a \<and> P3 aa)"
with_def: fixes plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and zeroa :: "'d" shows "dim.with plus zero zeroa \<equiv> \<lambda>scale V. if \<exists>b. (with zeroa zero plus scale : \<guillemotleft>independent\<guillemotright> b) \<and> (with zero plus scale : \<guillemotleft>span\<guillemotright> b) = (with zero plus scale : \<guillemotleft>span\<guillemotright> V) then card (SOME b. (with zeroa zero plus scale : \<guillemotleft>independent\<guillemotright> b) \<and> (with zero plus scale : \<guillemotleft>span\<guillemotright> b) = (with zero plus scale : \<guillemotleft>span\<guillemotright> V)) else 0"
remdups_adj_append: fixes xs\<^sub>1 :: "'a list"   and x :: "'a"   and xs\<^sub>2 :: "'a list" shows "remdups_adj (xs\<^sub>1 @ x # xs\<^sub>2) = remdups_adj (xs\<^sub>1 @ [x]) @ tl (remdups_adj (x # xs\<^sub>2))"
widen_Iface: fixes G :: "prog"   and I :: "qtname"   and T :: "ty" assumes "G\<turnstile>Iface I\<preceq>T" shows "T = Class Object \<or> (\<exists>J. T = Iface J)"
flow_network_axioms: fixes i :: "nat" shows "flow_network \<Delta> (f_i i)"
intros_2: fixes c :: "'f"   and a :: "'g"   and c' :: "'f"   and t :: "('f \<times> 'g \<times> 'f) set"   and w :: "'g list"   and c'' :: "'f" assumes "(c, a, c') \<in> t"   and "(c', w, c'') \<in> trcl t" shows "(c, a # w, c'') \<in> trcl t"
le_foldseq: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and n :: "nat"   and s :: "nat \<Rightarrow> 'b"   and t :: "nat \<Rightarrow> 'b" assumes "\<forall>a b c d. a \<le> b \<and> c \<le> d \<longrightarrow> f a c \<le> f b d"   and "\<forall>i\<le>n. s i \<le> t i" shows "foldseq f s n \<le> foldseq f t n"
2: fixes t :: "real" assumes "1 < t * 2" shows "u + (2 * t - 1) * v \<le> v + (2 * t - 1) * w"
ext: shows "\<forall>Q. (\<forall>x\<in>A. \<forall>y\<in>A. P x y \<longrightarrow> Q x y) \<and> qo_on Q A \<longrightarrow> wfp_on (strict Q) A"
LeadsTo_cancel1: fixes F :: "'c program"   and A :: "'c set"   and B :: "'c set"   and A' :: "'c set"   and B' :: "'c set" assumes "F \<in> A \<longmapsto>w B \<union> A'"   and "F \<in> B \<longmapsto>w B'" shows "F \<in> A \<longmapsto>w B' \<union> A'"
that: fixes TA :: "(nat, 'L) tree_automaton_rec" assumes "L = ta_lang TA"   and "complete_tree_automaton TA A" shows "thesis"
sel_1: fixes x1 :: "'e" shows "projl (Inl x1) = x1"
exp_mod_exp: 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 modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and division_segment :: "'a \<Rightarrow> 'a"   and m :: "nat"   and n :: "nat" assumes "class.unique_euclidean_semiring_with_nat divide plus minus zero times one modulo euclidean_size division_segment" shows "modulo (power.power one times (numeral.numeral one plus (num.Bit0 num.One)) m) (power.power one times (numeral.numeral one plus (num.Bit0 num.One)) n) = times (zero_neq_one.of_bool one zero (m < n)) (power.power one times (numeral.numeral one plus (num.Bit0 num.One)) m)"
swapEnv_updEnv: fixes ys :: "'varSort"   and y1 :: "'var"   and y2 :: "'var"   and rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm set option"   and x :: "'var"   and X :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and xs :: "'varSort" shows "rho [x \<leftarrow> X]_xs &[y1 \<and> y2]_ys = rho &[y1 \<and> y2]_ys [x @xs[y1 \<and> y2]_ys \<leftarrow> X #[y1 \<and> y2]_ys]_xs"
cInf_eq_Min: fixes X :: "'a set" assumes "finite X"   and "X \<noteq> {}" shows "Inf X = Min X"
reaches_step_2: fixes y :: "'e"   and z :: "'e"   and s :: "'f process_name \<Rightarrow> ('g, 'f, 'h, 'e) Model.local_state"   and x :: "'e" assumes "(y reaches z) s"   and "(x points_to y) s" shows "(x reaches z) s"
wf_prog_staticD: fixes G :: "prog"   and C :: "qtname"   and c :: "class"   and sig :: "sig"   and old :: "qtname \<times> methd"   and new :: "methd" assumes "wf_prog G"   and "class G C = Some c"   and "C \<noteq> Object"   and "methd G (super c) sig = Some old"   and "Protected \<le> accmodi old"   and "table_of (methods c) sig = Some new" shows "is_static new = is_static old"
borel_measurable_induct: fixes u :: "'a \<Rightarrow> ennreal"   and M :: "'a measure"   and P :: "('a \<Rightarrow> ennreal) \<Rightarrow> bool" assumes "u \<in> borel_measurable M"   and "\<And>f g. \<lbrakk>f \<in> borel_measurable M; g \<in> borel_measurable M; \<And>x. x \<in> space M \<Longrightarrow> f x = g x; P g\<rbrakk> \<Longrightarrow> P f"   and "\<And>A. A \<in> sets M \<Longrightarrow> P (indicator A)"   and "\<And>u c. \<lbrakk>c < top; u \<in> borel_measurable M; \<And>x. x \<in> space M \<Longrightarrow> u x < top; P u\<rbrakk> \<Longrightarrow> P (\<lambda>x. c * u x)"   and "\<And>u v. \<lbrakk>u \<in> borel_measurable M; \<And>x. x \<in> space M \<Longrightarrow> u x < top; P u; v \<in> borel_measurable M; \<And>x. x \<in> space M \<Longrightarrow> v x < top; \<And>x. x \<in> space M \<Longrightarrow> u x = 0 \<or> v x = 0; P v\<rbrakk> \<Longrightarrow> P (\<lambda>x. v x + u x)"   and "\<And>U. \<lbrakk>\<And>i. U i \<in> borel_measurable M; \<And>i x. x \<in> space M \<Longrightarrow> U i x < top; \<And>i. P (U i); incseq U; u = Sup (range U)\<rbrakk> \<Longrightarrow> P (Sup (range U))" shows "P u"
dom_comp: fixes g :: "'a arrow_of_spans_data"   and f :: "'a arrow_of_spans_data" assumes "V.seq g f" shows "V.dom (g \<bullet> f) = V.dom f"
step: fixes x :: "'a" assumes "x \<le> ccpo_class.fixp f"   and "x \<le> f x"   and "P x" shows "P (f x)"
hyps_2: shows "liftAll (\<lambda>a. qAFreshAbs ys (y @ys[z1 \<and> z2]_zs) (a $[[z1 \<and> z2]]_zs) = qAFreshAbs ys y a) binp"
main: shows "real (logD fs) \<le> real (m * m) * log base (real N)"
leg0_composite_1: shows "HfHgh.dom.leg0 = fgh.\<pi>.leg0 \<cdot> fgh.Prj\<^sub>0\<^sub>0"
eval_fds_add: fixes f :: "'b fds"   and s :: "'b"   and g :: "'b fds" assumes "fds_converges f s"   and "fds_converges g s" shows "eval_fds (f + g) s = eval_fds f s + eval_fds g s"
cmp_components_are_iso: fixes f :: "'b"   and g :: "'b" assumes "D.ide f"   and "D.ide g"   and "src\<^sub>D f = trg\<^sub>D g" shows "C.iso (F.right_cmp (f, g))"
lconf_cong: fixes G :: "prog"   and x :: "lname \<Rightarrow> val option"   and s :: "st"   and l :: "'a \<Rightarrow> val option"   and L :: "'a \<Rightarrow> ty option" shows "G,set_locals x s\<turnstile>l[\<Colon>\<preceq>]L = G,s\<turnstile>l[\<Colon>\<preceq>]L"
insert_iff: fixes a :: "'a"   and b :: "'a"   and A :: "'a set" shows "(a \<in> b \<triangleright> A) = (a = b \<or> a \<in> A)"
assms_2: fixes i :: "V" assumes "i \<in>\<^sub>\<circ> I"   and "a \<in>\<^sub>\<circ> f i" shows "R"
carrierE: fixes x :: "'b"   and T :: "'b set set" assumes "x \<in> carrier\<^bsub>T\<^esub>" obtains t :: "'b set" where "t open\<^bsub>T\<^esub>" and "x \<in> t"
finprod_infinite: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'a" assumes "infinite A" shows "finprod G f A = \<one>"
in_\<R>\<^sub>1: shows "\<langle>c, mds, mem\<rangle> \<R>\<^sup>1\<^bsub>\<Gamma>',\<S>',P'\<^esub> \<langle>c', mds', mem'\<rangle>"
earlyCongStructCong: fixes P :: "pi"   and Q :: "pi" assumes "P \<equiv>\<^sub>s Q" shows "P \<sim>\<^sup>s\<^sub>e Q"
A41_DAcc_level1: shows "DAcc level1 sA41 = {sA22}"
tauLaw3SimRight: fixes Rel :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and \<Psi> :: "'b"   and \<alpha> :: "'a prefix" assumes "eqvt Rel"   and "\<And>\<Psi> xvec Tvec. length xvec = length Tvec \<Longrightarrow> (\<Psi>, P[xvec::=Tvec], \<tau>.Q[xvec::=Tvec]) \<in> Rel"   and "\<And>\<Psi> P Q R S. \<lbrakk>\<Psi> \<rhd> P \<sim> Q; (\<Psi>, Q, R) \<in> Rel; \<Psi> \<rhd> R \<sim> S\<rbrakk> \<Longrightarrow> (\<Psi>, P, S) \<in> Rel"   and "\<And>\<Psi>. (\<Psi>, P, \<tau>.Q) \<in> Rel" shows "\<Psi> \<rhd> \<alpha>\<cdot>P \<leadsto><Rel> \<alpha>\<cdot>(\<tau>.Q)"
assms: fixes x :: "'a" shows "\<alpha> (f x) \<le> F x"
refine: shows "(impl_red_dfs_hash, red_dfs) \<in> \<langle>Id\<rangle>slg_rel \<rightarrow> \<langle>Id\<rangle>hs.rel \<rightarrow> \<langle>Id\<rangle>hs.rel \<rightarrow> Id \<rightarrow> \<langle>\<langle>Id\<rangle>hs.rel \<times>\<^sub>r red_wit_rel Id\<rangle>nres_rel"
step_t_z: fixes A :: "('e \<times> ('f, 'g) cconstraint \<times> 'h \<times> 'f list \<times> 'e) set \<times> ('e \<Rightarrow> ('f, 'g) cconstraint)"   and l :: "'e"   and Z :: "('f \<Rightarrow> 'g) set" shows "A \<turnstile> \<langle>l, Z\<rangle> \<leadsto> \<langle>l, (Z \<inter> {u. u \<turnstile> inv_of A l})\<^sup>\<up> \<inter> {u. u \<turnstile> inv_of A l}\<rangle>"
convex_EpigraphI: fixes s :: "'b set"   and f :: "'b \<Rightarrow> ereal" assumes "Lower_Semicontinuous.convex_on s f"   and "convex s" shows "convex (Epigraph s f)"
llist_all2_LNil1: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b llist" shows "llist_all2 P LNil xs = (xs = LNil)"
map_id: fixes t :: "'a fset" shows "id |`| t = t"
InvariantWatchesElNotifyWatchesLoop: fixes state :: "State"   and Wl :: "nat list"   and literal :: "Literal"   and newWl :: "nat list" assumes "InvariantWatchesEl (getF state) (getWatch1 state) (getWatch2 state)"   and "\<forall>c. c \<in> set Wl \<longrightarrow> 0 \<le> c \<and> c < length (getF state)" shows "let state' = notifyWatches_loop literal Wl newWl state in InvariantWatchesEl (getF state') (getWatch1 state') (getWatch2 state')"
limit_mat_add: fixes X :: "nat \<Rightarrow> complex mat"   and A :: "complex mat"   and m :: "nat"   and Y :: "nat \<Rightarrow> complex mat"   and B :: "complex mat" assumes "limit_mat X A m"   and "limit_mat Y B m" shows "limit_mat (\<lambda>k. X k + Y k) (A + B) m"
gso'_def: fixes i :: "nat" shows "gso' i = d i \<cdot>\<^sub>v gso i"
Exists_def: fixes p :: "hf"   and q :: "hf" shows "Exists p q \<equiv> \<exists>x x' y v. Form x \<and> VarNonOccForm v y \<and> AbstForm v 0 x x' \<and> p = q_Imp x y \<and> q = q_Imp (q_Ex x') y"
SUP_emeasure_incseq: fixes A :: "nat \<Rightarrow> 'a set"   and M :: "'a measure" assumes "range A \<subseteq> sets M"   and "incseq A" shows "(\<Squnion>n. emeasure M (A n)) = emeasure M (\<Union> (range A))"
UP_subring_monom: fixes S :: "'a set"   and a :: "'a"   and n :: "nat" assumes "subring S R"   and "a \<in> S" shows "monom (UP R) a n = monom (UP (R\<lparr>carrier := S\<rparr>)) a n"
mHom_0: fixes M :: "('h, 'i, 'j) Module_scheme"   and R :: "('i, 'k) Ring_scheme"   and N :: "('l, 'i, 'm) Module_scheme"   and f :: "'h \<Rightarrow> 'l" assumes "R module M"   and "R module N"   and "f \<in> mHom R M N" shows "f \<zero>\<^bsub>M\<^esub> = \<zero>\<^bsub>N\<^esub>"
sub: shows "P \<turnstile> C \<preceq>\<^sup>* D"
ccw_sefl': fixes a :: "real \<times> real"   and b :: "real \<times> real" shows "ccw a b a"