hcju/isabelleps huggingface dataset:
simps_6: fixes \<A> :: "'a \<Rightarrow> bool"   and F :: "'a formula"   and G :: "'a formula" shows "\<A> \<Turnstile> F \<^bold>\<rightarrow> G = (\<A> \<Turnstile> F \<longrightarrow> \<A> \<Turnstile> G)"
surj_to_def: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and B :: "'d set" shows "surj_to f A B = (f ` A = B)"
is_least_fixpoint_def: fixes f :: "'a \<Rightarrow> 'a"   and x :: "'a" shows "is_least_fixpoint f x \<equiv> f x = x \<and> (\<forall>y. f y = y \<longrightarrow> x \<le> y)"
simps_3: fixes x2 :: "'b" shows "Ref x2 \<noteq> Null"
sum_squares_le_zero_iff: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and abs :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and x :: "'c"   and y :: "'c" assumes "class.linordered_ring_strict plus zero minus uminus less_eq less abs times" shows "less_eq (plus (times x x) (times y y)) zero = (x = zero \<and> y = zero)"
f'_def: shows "f' = fds (\<lambda>n. if n = Suc 0 then 0::'a else fds_nth f n)"
le_imp_less_or_eq: fixes x :: "'a word"   and y :: "'a word" assumes "x \<le>s y" shows "x <s y \<or> x = y"
simps_9: fixes f :: "'e \<Rightarrow> 'd"   and x2 :: "'e" shows "map_option f (Some x2) = Some (f x2)"
numeral_eq_one_iff: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and n :: "num" assumes "class.semiring_char_0 one times plus zero" shows "(numeral.numeral one plus n = one) = (n = num.One)"
\<alpha>: shows "\<alpha> = weight \<Gamma> b - d_IN (h 0) b"
rep_eq: fixes x :: "'c Bonk_Schramm_extension"   and xa :: "'c Bonk_Schramm_extension" shows "dist x xa = extended_distance (rep_Bonk_Schramm_extension x) (rep_Bonk_Schramm_extension xa)"
min_gallery_simps_2: fixes X :: "'b set set"   and x :: "'b set" assumes "ChamberComplex X" shows "SimplicialComplex.min_maxsimpchain X [x] = SimplicialComplex.maxsimp X x"
isis': shows "snd (TSdet init h (as @ [x] @ bs @ [x] @ cs) (length (as @ [x] @ bs @ [x]) + i)) = snd (config\<^sub>p (rTS h) init (as @ [x] @ bs @ [x] @ take i cs))"
bounds_of_interval_inject: fixes x :: "'a interval"   and y :: "'a interval" shows "(bounds_of_interval x = bounds_of_interval y) = (x = y)"
nodes_alt_def: fixes A :: "('d, 'c) nba" shows "nodes A = \<Union> (reachable A ` {p. p \<in> initial A})"
Image_iff: fixes b :: "'e"   and r :: "('f \<times> 'e) set"   and A :: "'f set" shows "(b \<in> r `` A) = (\<exists>x\<in>A. (x, b) \<in> r)"
dom: shows "f \<in> {i. i < 2 ^ n} \<rightarrow>\<^sub>E {0, 1}"
mult_DirProd: fixes G :: "('e, 'g) monoid_scheme"   and H :: "('f, 'h) monoid_scheme"   and g :: "'e"   and h :: "'f"   and g' :: "'e"   and h' :: "'f" shows "(g, h) \<otimes>\<^bsub>G \<times>\<times> H\<^esub> (g', h') = (g \<otimes>\<^bsub>G\<^esub> g', h \<otimes>\<^bsub>H\<^esub> h')"
assms: shows "T : A \<mapsto>\<^bsub>dg_Set \<alpha>\<^esub> B"
disj'_1: shows "bvars\<^sub>s\<^sub>t (S @ S') \<inter> (subst_domain \<delta> \<union> range_vars \<delta>) = {}"
finprod_UN_disjoint: fixes I :: "'d set"   and A :: "'d \<Rightarrow> 'e set"   and g :: "'e \<Rightarrow> 'a" assumes "finite I"   and "\<And>i. i \<in> I \<Longrightarrow> finite (A i)"   and "pairwise (\<lambda>i j. disjnt (A i) (A j)) I"   and "\<And>i x. \<lbrakk>i \<in> I; x \<in> A i\<rbrakk> \<Longrightarrow> g x \<in> carrier G" shows "finprod G g (\<Union> (A ` I)) = (\<Otimes>i\<in>I. finprod G g (A i))"
euclidean_representation: fixes x :: "'b" shows "(\<Sum>b\<in>Basis. (x \<bullet> b) *\<^sub>R b) = x"
enat_defs_5: fixes m :: "enat"   and n :: "enat" shows "(m \<le> n) = (case n of enat n1 \<Rightarrow> case m of enat m1 \<Rightarrow> m1 \<le> n1 | \<infinity> \<Rightarrow> False | \<infinity> \<Rightarrow> True)"
assms: shows "(0, v) \<in> E\<^sup>*"
fix_least: fixes F :: "'c \<rightarrow> 'c"   and x :: "'c" assumes "F\<cdot>x = x" shows "fix\<cdot>F \<sqsubseteq> x"
mat_adj_adj: fixes A :: "complex \<times> complex \<times> complex \<times> complex" shows "mat_adj (mat_adj A) = A"
induct: fixes P :: "'b poly \<Rightarrow> bool"   and a0 :: "'b poly" assumes "\<And>c p. P (CN c 0 p)"   and "\<And>v. P (v)\<^sub>p"   and "\<And>v. P (Bound v)"   and "\<And>v va. P (Add v va)"   and "\<And>v va. P (Sub v va)"   and "\<And>v va. P (Mul v va)"   and "\<And>v. P (Neg v)"   and "\<And>v va. P (Pw v va)"   and "\<And>v vc vb. P (CN v (Suc vc) vb)" shows "P a0"
ipurge_ref_all: fixes D :: "'b \<Rightarrow> 'a"   and X :: "'b set"   and xs :: "'b list"   and u :: "'a"   and I :: "('a \<times> 'a) set" assumes "\<not> (\<exists>v\<in>D ` (X \<union> set xs). (u, v) \<in> I)" shows "ipurge_ref I D u xs X = X"
trace_append_is_append_trace: fixes sig :: "'i signature"   and e' :: "'j \<times> ('i \<times> 'k) list"   and e :: "'l \<times> ('i \<times> 'k) list" shows "trace sig (append_exec e' e) = append (trace sig e) (trace sig e')"
simps: fixes ires :: "cell list"   and rs :: "nat"   and l :: "cell list"   and r :: "cell list" shows "wcode_goon_right_moving_2 ires rs (l, r) = (\<exists>ml mr ln rn. l = Oc \<up> ml @ Bk # Bk # Bk \<up> ln @ Oc # ires \<and> r = Oc \<up> mr @ Bk \<up> rn \<and> ml + mr = Suc rs)"
extf_singleton_nil_if_\<delta>_eq_\<epsilon>: fixes f :: "'s"   and gt :: "('s, 'v) tm \<Rightarrow> ('s, 'v) tm \<Rightarrow> bool"   and s :: "('s, 'v) tm" assumes "\<delta> = \<epsilon>" shows "extf f gt [s] []"
Abs_image: fixes Rep :: "'d \<Rightarrow> 'e"   and Abs :: "'e \<Rightarrow> 'd"   and A :: "'e set" assumes "type_definition Rep Abs A" shows "Abs ` A = UNIV"
convergent_real_imp_convergent_ereal_1: fixes a :: "nat \<Rightarrow> real" assumes "convergent a" shows "convergent (\<lambda>n. ereal (a n))"
quasi_isometry_onD_2: fixes lambda :: "real"   and C :: "real"   and X :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and x :: "'c"   and y :: "'c" assumes "lambda C -quasi_isometry_on X f"   and "x \<in> X"   and "y \<in> X" shows "1 / lambda * dist x y - C \<le> dist (f x) (f y)"
order_linear: fixes x :: "'b"   and a :: "'b" shows "order x [:- a, 1::'b:] = (if x = a then 1 else 0)"
ideD_1: fixes a :: "'a" assumes "L'.C.ide a" shows "L'.C.arr a"
round_mono: fixes x :: "'a"   and y :: "'a" assumes "x \<le> y" shows "round x \<le> round y"
SDG_edge_valid_SDG_node_1: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and Exit :: "'b"   and Def :: "'b \<Rightarrow> 'c set"   and Use :: "'b \<Rightarrow> 'c set"   and ParamDefs :: "'b \<Rightarrow> 'c list"   and ParamUses :: "'b \<Rightarrow> 'c set list"   and n :: "'b SDG_node"   and Vopt :: "'c option"   and popt :: "('f \<times> bool) option"   and n' :: "'b SDG_node" assumes "SDG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit Def Use ParamDefs ParamUses"   and "SDG.SDG_edge sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses n Vopt popt n'" shows "SDG.valid_SDG_node sourcenode targetnode kind valid_edge procs n"
qmpt_density: fixes h :: "'a \<Rightarrow> ennreal" assumes "h \<in> borel_measurable (density M h)"   and "AE x in density M h. h x \<noteq> 0"   and "AE x in density M h. h x \<noteq> \<infinity>" shows "qmpt (density (density M h) h) T"
has_derivative_at_left_at_right': fixes g :: "real \<Rightarrow> 'a"   and g' :: "real \<Rightarrow> 'a"   and x :: "real" assumes "(g has_derivative g') (at x within {..x})"   and "(g has_derivative g') (at x within {x..})" shows "(g has_derivative g') (at x)"
trans: fixes b :: "'e multiset"   and a :: "'e multiset"   and c :: "'e multiset" assumes "b \<subseteq># a"   and "c \<subseteq># b" shows "c \<subseteq># a"
vimage_ident: fixes Y :: "'b set" shows "(\<lambda>x. x) -` Y = Y"
nodup_step: fixes a :: "'a"   and x :: "'a"   and ls :: "'a list" shows "nodup a (x # ls) = (if x = a then a \<notin> set ls else nodup a ls)"
moebius_dilatation_inverse: fixes a :: "real" assumes "0 < a" shows "moebius_inv (moebius_dilatation a) = moebius_dilatation (1 / a)"
image_subset_of_target: fixes f :: "'c \<Rightarrow> 'd"   and S :: "'c set"   and T :: "'d set" assumes "Set_Theory.map f S T" shows "f ` S \<subseteq> T"
NotifyWatchesLoopWatchCharacterizationEffect: fixes state :: "State"   and M :: "(Literal \<times> bool) list"   and Wl :: "nat list"   and literal :: "Literal"   and decision :: "bool"   and newWl :: "nat list" assumes "InvariantWatchesEl (getF state) (getWatch1 state) (getWatch2 state)"   and "InvariantWatchesDiffer (getF state) (getWatch1 state) (getWatch2 state)"   and "InvariantConsistent (getM state)"   and "InvariantUniq (getM state)"   and "InvariantWatchCharacterization (getF state) (getWatch1 state) (getWatch2 state) M"   and "\<forall>c. c \<in> set Wl \<longrightarrow> 0 \<le> c \<and> c < length (getF state)"   and "getM state = M @ [(opposite literal, decision)]"   and "uniq Wl"   and "\<forall>c. c \<in> set Wl \<longrightarrow> Some literal = getWatch1 state c \<or> Some literal = getWatch2 state c" shows "let state' = notifyWatches_loop literal Wl newWl state in \<forall>c. c \<in> set Wl \<longrightarrow> (\<forall>w1 w2. Some w1 = getWatch1 state' c \<and> Some w2 = getWatch2 state' c \<longrightarrow> watchCharacterizationCondition w1 w2 (getM state') (getF state' ! c) \<and> watchCharacterizationCondition w2 w1 (getM state') (getF state' ! c))"
classD: shows "class P D = \<lfloor>(Bs', fs', ms')\<rfloor>"
case_prod_beta: fixes f :: "'e \<Rightarrow> 'f \<Rightarrow> 'd"   and p :: "'e \<times> 'f" shows "(case p of (x, xa) \<Rightarrow> f x xa) = f (fst p) (snd p)"
neighbors_adj: fixes u :: "nat" shows "n u \<subseteq> adjacent_nodes u"
C_def: shows "C = (\<lambda>c. - sqrt (2 / e) / 3 * (2 * e + 3 * c))"
fps_compose_Lambert_W: shows "fps_Lambert_W oo fps_X * fps_exp 1 = fps_X"
inf_idem: fixes x :: "'e multiset" shows "x \<inter># x = x"
Adec_eph: fixes X :: "msg"   and Y :: "nonce_t"   and H :: "msg set" assumes "Aenc X (epubK Y) \<in> analz H"   and "epriK Y \<in> synth (analz H)" shows "X \<in> analz H"
get_root_node_si_no_parent: fixes h :: "(_) heap"   and node_ptr :: "(_) node_ptr" assumes "h \<turnstile> get_parent node_ptr \<rightarrow>\<^sub>r None" shows "h \<turnstile> get_root_node_si (cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r node_ptr) \<rightarrow>\<^sub>r cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r node_ptr"
integral_cong_AE: fixes f :: "'c \<Rightarrow> 'd"   and M :: "'c measure"   and g :: "'c \<Rightarrow> 'd" assumes "f \<in> borel_measurable M"   and "g \<in> borel_measurable M"   and "AE x in M. f x = g x" shows "integral\<^sup>L M f = integral\<^sup>L M g"
mapl_G_cong: fixes x :: "('n, 'o, 'p, 'q, 'r, 's, 't) G"   and l1 :: "'n \<Rightarrow> 'u"   and l1' :: "'n \<Rightarrow> 'u"   and l2 :: "'o \<Rightarrow> 'v"   and l2' :: "'o \<Rightarrow> 'v" assumes "\<And>z. z \<in> set1_G x \<Longrightarrow> l1 z = l1' z"   and "\<And>z. z \<in> set2_G x \<Longrightarrow> l2 z = l2' z" shows "mapl_G l1 l2 x = mapl_G l1' l2' x"
in_components_maximal: fixes c :: "'c set"   and s :: "'c set" shows "(c \<in> components s) = (c \<noteq> {} \<and> c \<subseteq> s \<and> connected c \<and> (\<forall>d. d \<noteq> {} \<and> c \<subseteq> d \<and> d \<subseteq> s \<and> connected d \<longrightarrow> d = c))"
simps_2: fixes ops :: "'b arith_ops_record"   and xs :: "'b list" shows "minus_poly_i ops xs [] = xs"
ts: shows "ts = ts\<^sub>1 @ ts\<^sub>2"
elims: fixes x :: "('a, 'b) rbt"   and y :: "nat" assumes "black_depth x = y" obtains "x = Leaf" and "y = 0"   | l :: "('a, 'b) rbt" and k :: "'a" and v :: "'b" and r :: "('a, 'b) rbt" where "x = Node l__ R k__ v__ r__" and "y = black_depth l__"   | l :: "('a, 'b) rbt" and k :: "'a" and v :: "'b" and r :: "('a, 'b) rbt" where "x = Node l B k v r" and "y = black_depth l + 1"
simps_3: fixes x2 :: "'b set" shows "RES x2 \<noteq> FAILi"
class_Object: fixes wf_md :: "'a prog \<Rightarrow> char list \<Rightarrow> 'a mdecl \<Rightarrow> bool"   and P :: "'a prog" assumes "wf_prog wf_md P" shows "\<exists>C fs ms. class P Object = \<lfloor>(C, fs, ms)\<rfloor>"
set_Cons_def: fixes A :: "'b set"   and XS :: "'b list set" shows "set_Cons A XS = {x # xs |x xs. x \<in> A \<and> xs \<in> XS}"
abel_semigroup_axioms: shows "abel_semigroup (\<inter>#)"
sigma_eqI: fixes M :: "'b set set"   and \<Omega> :: "'b set"   and N :: "'b set set" assumes "M \<subseteq> Pow \<Omega>"   and "N \<subseteq> Pow \<Omega>"   and "sigma_sets \<Omega> M = sigma_sets \<Omega> N" shows "sigma \<Omega> M = sigma \<Omega> N"
assms_3: shows "are_all_operators_applicable (\<phi>\<^sub>s \<Psi> s) ops' \<and> STRIPS_Semantics.are_all_operator_effects_consistent ops'"
order_on_defs_5: fixes A :: "'c set"   and r :: "('c \<times> 'c) set" shows "well_order_on A r \<equiv> linear_order_on A r \<and> wf (r - Id)"
Pdrop_maximal: fixes G :: "('c, 'd) ParityGame_scheme"   and P :: "'c llist"   and v0 :: "'c"   and n :: "nat" assumes "vm_path G P v0" shows "Digraph.maximal_path G (ldropn n P)"
integrable_iff_bounded: fixes M :: "'d measure"   and f :: "'d \<Rightarrow> 'e" shows "integrable M f = (f \<in> borel_measurable M \<and> \<integral>\<^sup>+ x. ennreal (norm (f x)) \<partial>M < \<infinity>)"
bdd_above_image_antimono: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set" assumes "antimono f"   and "bdd_below A" shows "bdd_above (f ` A)"
wf_fun_pair_negchecks_map: fixes X :: "'a set"   and A :: "('b, 'a) strand_step list"   and Y :: "'a list"   and F :: "(('b, 'a) Term.term \<times> ('b, 'a) Term.term) list"   and M :: "(('b, 'a) Term.term \<times> ('b, 'a) Term.term) list list" assumes "wf\<^sub>s\<^sub>t X A" shows "wf\<^sub>s\<^sub>t X (map (\<lambda>G. \<forall>Y\<langle>\<or>\<noteq>: (F @ G)\<rangle>\<^sub>s\<^sub>t) M @ A)"
inj_on_image: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set set" assumes "inj_on f (\<Union> A)" shows "inj_on ((`) f) A"
card_Diff_pos_decomp: fixes qs :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list"   and P :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<Rightarrow> bool" shows "card {(h, U). (h, U) \<in> set qs - set (qs\<^sub>+) \<and> P h} = card {h. (h, {}) \<in> set qs \<and> P h}"
ef: shows "f = [A,ef,B]\<^bsub>T\<^esub>"
epigraph_subset_iff: fixes f :: "'b \<Rightarrow> ereal"   and g :: "'b \<Rightarrow> ereal" shows "(Epigraph UNIV f \<subseteq> Epigraph UNIV g) = (\<forall>x. g x \<le> f x)"
solution_set_def: fixes A :: "(('b, 'c) vec, 'd) vec"   and b :: "('b, 'd) vec" shows "solution_set A b = {x. is_solution x A b}"
qSwap_preserves_qGood1: fixes X :: "('c, 'd, 'varSort, 'var, 'e) qTerm"   and zs :: "'varSort"   and x :: "'var"   and y :: "'var" assumes "qGood X" shows "qGood (X #[[x \<and> y]]_zs)"
dres_Sup_chain_cases: fixes M :: "'a dres set" assumes "is_chain M" obtains "M \<subseteq> {dSUCCEED}" and "Sup M = dSUCCEED"   | r :: "'a" where "M \<subseteq> {dSUCCEED, dRETURN r}" and "dRETURN r \<in> M" and "Sup M = dRETURN r"   | "dFAIL \<in> M" and "Sup M = dFAIL"
bh: shows "berlekamp_hensel p n f = hensel_lifting p n f gs"
pCons_1: shows "a \<noteq> (0::'a) \<or> p \<noteq> 0"
sets_stat: fixes K :: "'a \<Rightarrow> 'a pmf"   and C :: "'a set" shows "sets (MC_syntax.stat K C) = sets (count_space UNIV)"
graph_lemma: fixes X :: "'b list set" shows "\<BB>\<^sub>F X = {hd (Dec \<BB>\<^sub>F X x) |x. x \<in> X \<and> x \<noteq> \<epsilon>}"
i_set_def: shows "i_set \<equiv> iFROM_set \<union> iTILL_set \<union> iIN_set \<union> iMOD_set \<union> iMODb_set"
runit_in_hom: fixes C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and T :: "'c \<times> 'c \<Rightarrow> 'c"   and \<alpha> :: "'c \<times> 'c \<times> 'c \<Rightarrow> 'c"   and \<iota> :: "'c"   and a :: "'c" assumes "monoidal_category C T \<alpha> \<iota>"   and "partial_magma.ide C a" shows "partial_magma.in_hom C (monoidal_category.runit C T \<alpha> \<iota> a) (T (a, monoidal_category.unity C \<iota>)) a"
simps_151: fixes x9 :: "Term.var"   and x61 :: "unop"   and x62 :: "expr" shows "Acc x9 \<noteq> UnOp x61 x62"
method_exit_cases: fixes n :: "'node" assumes "method_exit n" obtains "n = (_Exit_)"   | a :: "'edge" and Q :: "('var \<Rightarrow> 'val option) \<times> 'ret \<Rightarrow> bool" and f :: "('var \<Rightarrow> 'val option) \<Rightarrow> ('var \<Rightarrow> 'val option) \<Rightarrow> 'var \<Rightarrow> 'val option" and p :: "'pname" where "n = sourcenode a" and "valid_edge a" and "kind a = Q\<hookleftarrow>\<^bsub>p\<^esub>f"
varsOppositeLiteralList: fixes clause :: "Literal list" shows "vars (oppositeLiteralList clause) = vars clause"
fmlookup_restrict_set: fixes A :: "'e set"   and m :: "('e, 'd) fmap"   and x :: "'e" shows "fmlookup (fmrestrict_set A m) x = (if x \<in> A then fmlookup m x else None)"
C\<pi>\<rho>x': fixes x :: "'a" shows "(C\<pi> o\<^sub>C\<^sub>L C\<rho>) *\<^sub>V ket x = (case (\<pi> \<circ>\<^sub>m \<rho>) x of None \<Rightarrow> 0 | Some i \<Rightarrow> ket i)"
term_variants_pred_inv_Var_2: fixes P :: "'a \<Rightarrow> 'a list"   and t :: "('a, 'b) Term.term"   and x :: "'b" shows "term_variants_pred P t (Var x) = (t = Var x)"
sum_interval_cong: fixes m :: "'b"   and n :: "'b"   and P :: "'b \<Rightarrow> 'c"   and Q :: "'b \<Rightarrow> 'c" assumes "\<And>i. \<lbrakk>m \<le> i; i \<le> n\<rbrakk> \<Longrightarrow> P i = Q i" shows "sum P {m..n} = sum Q {m..n}"
Hadamard's_inequality: fixes A :: "real mat"   and n :: "nat" assumes "A \<in> carrier_mat n n" shows "\<bar>det A\<bar> \<le> sqrt (prod_list (map sq_norm (rows A)))"
ipresIGWlsAbs_termMOD: fixes hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'h"   and MOD :: "('i, 'j, 'varSort, 'sort, 'k, 'l, 'm, 'h, 'n) model_scheme" shows "ipresIGWlsAbs hA termMOD MOD = ipresWlsAbs hA MOD"
intra_path_slp: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and n :: "'b"   and as :: "'a list"   and n' :: "'b" assumes "CFG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main"   and "CFG.intra_path sourcenode targetnode kind valid_edge n as n'" shows "CFG.same_level_path' sourcenode targetnode kind valid_edge get_return_edges n as n'"
dom_closed: fixes f :: "'a" assumes "Arr f" shows "Arr (B.dom f)"
nat_pow_distrib: fixes x :: "'a"   and y :: "'a"   and n :: "nat" assumes "x \<in> carrier R"   and "y \<in> carrier R" shows "(x \<otimes> y) [^] n = x [^] n \<otimes> y [^] n"
empty_path_empty_set: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and C :: "char list" shows "{Cs. Subobjs P C Cs \<and> length Cs \<le> 0} = {}"
finsert_fminus_if: fixes x :: "'c"   and A :: "'c fset"   and B :: "'c fset" shows "finsert x A |-| B = (if x |\<in>| B then A |-| B else finsert x (A |-| B))"
left_uniqueI: fixes A :: "'b \<Rightarrow> 'c \<Rightarrow> bool" assumes "\<And>x y z. \<lbrakk>A x z; A y z\<rbrakk> \<Longrightarrow> x = y" shows "left_unique A"
exhaust: fixes y :: "'b mrexp2" obtains "y = Zero2"   | "y = One2"   | x31 :: "bool" and x32 :: "'b" where "y = Atom2 x31 x32"   | x41 :: "'b mrexp2" and x42 :: "'b mrexp2" and x43 :: "bool" and x44 :: "bool" where "y = Plus2 x41 x42 x43 x44"   | x51 :: "'b mrexp2" and x52 :: "'b mrexp2" and x53 :: "bool" and x54 :: "bool" where "y = Times2 x51 x52 x53 x54"   | x61 :: "'b mrexp2" and x62 :: "bool" where "y = Star2 x61 x62"
edg_simps_2: fixes \<Delta> :: "('a, 'b) network_scheme"   and x :: "'a"   and y :: "'a"   and v :: "'a antiparallel_edges.vertex" assumes "antiparallel_edges \<Delta>" shows "antiparallel_edges.edg \<Delta> (antiparallel_edges.Edge x y) v = (v = antiparallel_edges.Vertex y \<and> edge \<Delta> x y)"
nop_instr_def: fixes instr :: "sparc_operation \<times> inst_operand list" shows "nop_instr instr \<equiv> return ()"
P_k_pre_not_Nil: fixes G :: "('c, 'd) Graph_scheme"   and v0 :: "'c"   and v1 :: "'c"   and paths :: "'c list set"   and P_new :: "'c list"   and sep_size :: "nat"   and P_k_pre :: "'c list"   and y :: "'c"   and P_k_post :: "'c list" assumes "ProofStepInduct_NonTrivial_P_k_pre G v0 v1 paths P_new sep_size P_k_pre y P_k_post" shows "P_k_pre \<noteq> []"
X_XD_smallest_XH_largest: fixes ds :: "'b set"   and X :: "'a set" assumes "stable_on ds X" shows "X = XD_smallest ds X \<inter> XH_largest ds X"
inject: fixes x1 :: "'a interval"   and x2 :: "nat"   and y1 :: "'a interval"   and y2 :: "nat" shows "(IdxInterval x1 x2 = IdxInterval y1 y2) = (x1 = y1 \<and> x2 = y2)"
rep_eq: fixes x :: "'c"   and xa :: "'d" shows "alist.impl_of (single_alist_entry x xa) = [(x, xa)]"
commute: fixes a :: "'a"   and b :: "'a" shows "a \<sqinter> b = b \<sqinter> a"
length_nat_to_bv_length: fixes x :: "nat"   and y :: "nat" assumes "nat_to_bv_length x y \<noteq> []" shows "length (nat_to_bv_length x y) = y"
not_in_butlast: fixes x :: "'a"   and ys :: "'a list" assumes "x \<in> set ys"   and "x \<notin> set (butlast ys)" shows "x = last ys"
phis'_wf: fixes g :: "('b, 'a, 'c) gen_cfg_wf"   and n :: "'b"   and v :: "'a \<times> 'b \<times> Def"   and args :: "('a \<times> 'b \<times> Def) list" assumes "gen_wf.phis' g (n, v) = Some args" shows "length (gen_wf_predecessors g n) = length args"
emb_preserve_mono: fixes J :: "nat set"   and L :: "nat set"   and X :: "(nat \<Rightarrow> 'a) set"   and Y :: "(nat \<Rightarrow> 'a) set" assumes "J \<subseteq> L"   and "L \<subseteq> UNIV"   and "X \<in> sets (Pi\<^sub>M J M)"   and "Y \<in> sets (Pi\<^sub>M J M)"   and "PF.emb L J X \<subseteq> PF.emb L J Y" shows "X \<subseteq> Y"
point_surj: fixes M :: "point" shows "Point (abscissa M) (ordinate M) = M"
distrib: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "class.comm_semiring plus times" shows "times (plus a b) c = plus (times a c) (times b c)"
lhs_bound: shows "\<bar>spmf (R1_14 M (c0, c1) \<bind> D) True - spmf (S1_14 M out \<bind> D) True\<bar> \<le> \<bar>spmf (pair_spmf coin_spmf coin_spmf \<bind> (\<lambda>(m0, m1). R1_OT12 (m0, m1) c0 \<bind> (\<lambda>view. A1 view (m0, m1)))) True - spmf (pair_spmf coin_spmf coin_spmf \<bind> (\<lambda>(m0, m1). S1_OT12 (m0, m1) () \<bind> (\<lambda>view. A1 view (m0, m1)))) True\<bar> + \<bar>spmf (pair_spmf coin_spmf coin_spmf \<bind> (\<lambda>(m0, m1). R1_OT12 (m0, m1) c1 \<bind> (\<lambda>view. A2 view (m0, m1)))) True - spmf (pair_spmf coin_spmf coin_spmf \<bind> (\<lambda>(m0, m1). S1_OT12 (m0, m1) () \<bind> (\<lambda>view. A2 view (m0, m1)))) True\<bar> + \<bar>spmf (pair_spmf coin_spmf coin_spmf \<bind> (\<lambda>(m0, m1). R1_OT12 (m0, m1) c1 \<bind> (\<lambda>view. A3 view (m0, m1)))) True - spmf (pair_spmf coin_spmf coin_spmf \<bind> (\<lambda>(m0, m1). S1_OT12 (m0, m1) () \<bind> (\<lambda>view. A3 view (m0, m1)))) True\<bar>"
pickBinp_alphaBinp: fixes binp :: "'e \<Rightarrow> ('f, 'g, 'varSort, 'var, 'h) qAbs set option" assumes "goodBinp binp" shows "(pickBinp binp, pickBinp binp) \<in> alphaBinp"
approachable_lt_le2: fixes Q :: "'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> real"   and P :: "'a \<Rightarrow> bool" shows "(\<exists>d>0. \<forall>x. Q x \<longrightarrow> f x < d \<longrightarrow> P x) = (\<exists>d>0. \<forall>x. f x \<le> d \<longrightarrow> Q x \<longrightarrow> P x)"
a_zpm: fixes m :: "int" shows "ant m + - \<infinity> = - \<infinity>"