hcju/isabelleps huggingface dataset:
set_inorder: fixes t :: "('a \<times> 'b) tree" shows "set (Tree2.inorder t) = Tree2.set_tree t"
strongBisimWeakCong: fixes P :: "pi"   and Q :: "pi" assumes "P \<sim> Q" shows "P \<simeq> Q"
unit_simps_5: fixes a :: "'a" assumes "C.obj a" shows "D.cod (F.unit a) = F a"
interaction_bounded_by_mono: fixes "consider" :: "'a \<Rightarrow> bool"   and gpv :: "('b, 'a, 'c) gpv"   and n :: "enat"   and m :: "enat" assumes "interaction_bounded_by consider gpv n"   and "n \<le> m" shows "interaction_bounded_by consider gpv m"
contour_integral_split_linepath: fixes a :: "complex"   and b :: "complex"   and f :: "complex \<Rightarrow> complex"   and c :: "complex" assumes "continuous_on (closed_segment a b) f"   and "c \<in> closed_segment a b" shows "contour_integral (linepath a b) f = contour_integral (linepath a c) f + contour_integral (linepath c b) f"
integral_complex_of_real: fixes M :: "'a measure"   and f :: "'a \<Rightarrow> real" shows "CLINT x|M. complex_of_real (f x) = complex_of_real (integral\<^sup>L M f)"
weak_eq: fixes x :: "('a, 'b) pp \<times> nat"   and y :: "('a, 'b) pp \<times> nat" assumes "nat_term_compare cmp x y = Eq" shows "x = y"
fst_splitAt_upt: fixes j :: "nat"   and i :: "nat"   and k :: "nat" assumes "j \<le> i"   and "i < k" shows "fst (splitAt i [j..<k]) = [j..<i]"
arr_eval: fixes f :: "'c local.term" assumes "Arr f" shows "arr \<lbrace>f\<rbrace>"
seqlI: fixes P :: "'e \<times> 'f \<Rightarrow> bool"   and s :: "('g \<Rightarrow> 'e) \<times> 'f"   and i :: "'g" assumes "P (fst s i, snd s)" shows "seql i P s"
iIN_set_def: shows "iIN_set \<equiv> {[n\<dots>,d] |n d. True}"
carrier_FactGroup: fixes G :: "('e, 'f) monoid_scheme"   and N :: "'e set" shows "carrier (G Mod N) = (#>\<^bsub>G\<^esub>) N ` carrier G"
iso_dghm_Obj_HomDom_if_Obj_HomCod: fixes b :: "V" assumes "b \<in>\<^sub>\<circ> \<BB>\<lparr>Obj\<rparr>" obtains a :: "V" where "a \<in>\<^sub>\<circ> \<AA>\<lparr>Obj\<rparr>" and "b = \<FF>\<lparr>ObjMap\<rparr>\<lparr>a\<rparr>"
valid_sops_stmt: fixes i :: "nat"   and s :: "stmt"   and t :: "nat"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "(stmt \<times> nat) memref list"   and \<D> :: "bool"   and \<O> :: "nat set \<times> (nat \<Rightarrow> bool option)" assumes "i < length ts"   and "ts ! i = ((s, t), is, \<theta>, sb, \<D>, \<O>)" shows "valid_sops_stmt t s"
min_product_set_def: fixes X :: "'b fset set set" shows "\<Otimes>\<^sub>m X = Finite_Set.fold (\<otimes>\<^sub>m) {{||}} X"
predicate2D_vimage2p: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'd"   and S :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'a"   and y :: "'b" assumes "R \<le> BNF_Def.vimage2p f g S"   and "R x y" shows "S (f x) (g y)"
a_lcomm: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<in> carrier G"   and "y \<in> carrier G"   and "z \<in> carrier G" shows "x \<oplus> (y \<oplus> z) = y \<oplus> (x \<oplus> z)"
dgrad_p_set_insert: fixes F :: "('t \<Rightarrow>\<^sub>0 'c) set"   and d :: "'a \<Rightarrow> nat"   and m :: "nat"   and f :: "'t \<Rightarrow>\<^sub>0 'd" assumes "F \<subseteq> dgrad_p_set d m" obtains n :: "nat" where "m \<le> n" and "f \<in> dgrad_p_set d n" and "F \<subseteq> dgrad_p_set d n"
U_par_s_id: shows "1\<^sub>\<sigma> \<cdot> 1\<^sub>\<pi> \<parallel> U = U"
jAction_simAbs_cong: fixes jkbp :: "'b \<Rightarrow> ('b, 'c, 'd) GC list"   and envInit :: "'e list"   and envAction :: "'e \<Rightarrow> 'f list"   and envTrans :: "'f \<Rightarrow> ('b \<Rightarrow> 'd) \<Rightarrow> 'e \<Rightarrow> 'e"   and envVal :: "'e \<Rightarrow> 'c \<Rightarrow> bool"   and jview :: "'b \<Rightarrow> 'e Trace \<Rightarrow> 'g"   and envObs :: "'b \<Rightarrow> 'e \<Rightarrow> 'h"   and jviewInit :: "'b \<Rightarrow> 'h \<Rightarrow> 'g"   and jviewIncr :: "'b \<Rightarrow> 'h \<Rightarrow> 'g \<Rightarrow> 'g"   and simf :: "'e Trace \<Rightarrow> 'i"   and simRels :: "'b \<Rightarrow> ('i \<times> 'i) set"   and simVal :: "'i \<Rightarrow> 'c \<Rightarrow> bool"   and simAbs :: "'j \<Rightarrow> 'i set"   and simObs :: "'b \<Rightarrow> 'j \<Rightarrow> 'h"   and simInit :: "'b \<Rightarrow> 'h \<Rightarrow> 'j"   and simTrans :: "'b \<Rightarrow> 'j \<Rightarrow> 'j list"   and simAction :: "'b \<Rightarrow> 'j \<Rightarrow> 'd list"   and t :: "'e Trace"   and ec :: "'j"   and a :: "'b"   and ec' :: "'j" assumes "AlgSimIncrEnvironment jkbp envInit envAction envTrans envVal jview envObs jviewInit jviewIncr simf simRels simVal simAbs simObs simInit simTrans simAction"   and "t \<in> PreEnvironmentJView.jkbpC jkbp envInit envAction envTrans envVal jview"   and "simAbs ec = simf ` {t' \<in> PreEnvironmentJView.jkbpC jkbp envInit envAction envTrans envVal jview. jview a t' = jview a t}"   and "simAbs ec = simAbs ec'" shows "set (simAction a ec) = set (simAction a ec')"
ge_eq_refl: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a" assumes "(=) \<le> R" shows "R x x"
image_last_index: fixes A :: "'a set"   and xs :: "'a list"   and x :: "'a" assumes "A \<subseteq> set (xs @ [x])" shows "last_index (xs @ [x]) ` A = (if x \<in> A then insert (length xs) (last_index xs ` (A - {x})) else last_index xs ` A)"
coincide_fml_def: fixes \<phi> :: "('d, 'e, 'f) formula" shows "coincide_fml \<phi> = (\<forall>\<nu> \<nu>' I J. Iagree I J (SIGF \<phi>) \<longrightarrow> Vagree \<nu> \<nu>' (FVF \<phi>) \<longrightarrow> (\<nu> \<in> fml_sem I \<phi>) = (\<nu>' \<in> fml_sem J \<phi>))"
map_comp: fixes g1 :: "'j \<Rightarrow> 'h"   and g2 :: "'k \<Rightarrow> 'i"   and f1 :: "'l \<Rightarrow> 'j"   and f2 :: "'m \<Rightarrow> 'k"   and v :: "'l \<times> 'm" shows "map_prod g1 g2 (map_prod f1 f2 v) = map_prod (g1 \<circ> f1) (g2 \<circ> f2) v"
resultant_impl_main: fixes F :: "nat \<Rightarrow> 'b fract poly"   and n :: "nat \<Rightarrow> nat"   and \<delta> :: "nat \<Rightarrow> nat"   and f :: "nat \<Rightarrow> 'b fract"   and k :: "nat"   and \<beta> :: "nat \<Rightarrow> 'b fract"   and G1 :: "'b poly"   and G2 :: "'b poly"   and div_exp :: "'b \<Rightarrow> 'b \<Rightarrow> nat \<Rightarrow> 'b" assumes "subresultant_prs_locale2 F n \<delta> f k \<beta> G1 G2"   and "\<And>x y n. to_fract x ^ n / to_fract y ^ (n - 1) \<in> range to_fract \<Longrightarrow> to_fract (div_exp x y n) = to_fract x ^ n / to_fract y ^ (n - 1)" shows "resultant_impl_main div_exp G1 G2 = resultant G1 G2"
1: shows "[\<^bold>\<forall>x y. (\<^bold>\<lambda>z. \<lparr>F,x\<^sup>P,y\<^sup>P,z\<^sup>P\<rparr>) \<^bold>=\<^sub>1 (\<^bold>\<lambda>z. \<lparr>G,x\<^sup>P,y\<^sup>P,z\<^sup>P\<rparr>) in v]"
wqo_on_Plus: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a set"   and Q :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and B :: "'b set" assumes "wqo_on P A"   and "wqo_on Q B" shows "wqo_on (sum_le P Q) (A <+> B)"
prems_2: shows "v \<notin> ifex_var_set e"
Ints_abs: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and a :: "'b" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one"   and "a \<in> ring_1.Ints one plus zero minus" shows "abs a \<in> ring_1.Ints one plus zero minus"
simplex_top_face: fixes p :: "nat"   and s' :: "(nat \<Rightarrow> nat) set"   and n :: "nat" assumes "0 < p"   and "\<forall>x\<in>s'. x n = p" shows "ksimplex p n s' = (\<exists>s a. ksimplex p (Suc n) s \<and> a \<in> s \<and> s' = s - {a})"
Red_in_Sup: fixes Ns :: "'a set llist" assumes "chain (\<rhd>) Ns" shows "Sup_llist Ns - Liminf_llist Ns \<subseteq> Red_F (Sup_llist Ns)"
conj_assoc: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(x \<^bold>\<squnion> y) \<^bold>\<squnion> z = x \<^bold>\<squnion> y \<^bold>\<squnion> z"
SLN_right: fixes P :: "assn" shows "P * SLN = P"
fold_congs_7: fixes r :: "('a, 'b) state_scheme"   and r' :: "('a, 'b) state_scheme"   and v' :: "('a \<times> 'a) set"   and f :: "('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set"   and f' :: "('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set" assumes "r = r'"   and "back_edges r' = v'"   and "\<And>v. v' = v \<Longrightarrow> f v = f' v" shows "back_edges_update f r = back_edges_update f' r'"
eulerian_poly_Suc: fixes n :: "nat" shows "eulerian_poly (Suc n) = (let p = eulerian_poly n in [:0::'b, 1::'b, - (1::'b):] * pderiv p + p * [:1::'b, of_nat n:])"
strict_implies_order: fixes a :: "'d multiset"   and b :: "'d multiset" assumes "a \<subset># b" shows "a \<subseteq># b"
satisfies_atom_set'_stronger: fixes as :: "'a atom set"   and v :: "nat \<Rightarrow> 'a" assumes "v \<Turnstile>\<^sub>a\<^sub>e\<^sub>s as" shows "v \<Turnstile>\<^sub>a\<^sub>s as"
cases: fixes a :: "cld list" assumes "acyclic_clds a" obtains P :: "cld list" where "a = P" and "\<forall>ctx fqn. (\<exists>ctx' cld. find_cld P ctx fqn (Some (ctx', cld))) \<longrightarrow> (\<exists>nn. path_length P ctx (cl_fqn fqn) nn)"
measure_well_com: fixes n :: "nat"   and M :: "nat \<Rightarrow> complex mat"   and S :: "com list"   and k :: "nat" assumes "well_com (Measure n M S)"   and "k < n" shows "well_com (S ! k)"
gamma_num': fixes i :: "int" shows "i \<in> \<gamma> (num' i)"
minus_mult_left: fixes a :: "'a"   and b :: "'a" shows "- (a * b) = - a * b"
assms_3: fixes x :: "nat \<Rightarrow> 'a" shows "dim_row (matricize I (T x)) = nr"
bij_betw_iff_bijections: fixes f :: "'q \<Rightarrow> 'r"   and A :: "'q set"   and B :: "'r set" shows "bij_betw f A B = (\<exists>g. (\<forall>x\<in>A. f x \<in> B \<and> g (f x) = x) \<and> (\<forall>y\<in>B. g y \<in> A \<and> f (g y) = y))"
rotate_length01: fixes xs :: "'a list"   and n :: "nat" assumes "length xs \<le> 1" shows "rotate n xs = xs"
drefinesD: fixes \<phi> :: "'c \<Rightarrow> 'd"   and G :: "'c \<Rightarrow> bool"   and A :: "bool \<Rightarrow> ('d \<Rightarrow> real) \<Rightarrow> 'd \<Rightarrow> real"   and B :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real"   and P :: "'d \<Rightarrow> real"   and Q :: "'d \<Rightarrow> real" assumes "drefines \<phi> G A B"   and "unitary P"   and "unitary Q"   and "P \<tturnstile> wp A Q" shows "\<guillemotleft> G \<guillemotright> && (P \<circ> \<phi>) \<tturnstile> wp B (Q \<circ> \<phi>)"
assms_3: shows "\<forall>\<phi>1\<in>set \<phi>1s. \<exists>\<phi>2\<in>set \<phi>2s. prv (imp \<phi>1 \<phi>2)"
assoc: fixes a :: "'c"   and b :: "'c"   and c :: "'c" shows "gcd (gcd a b) c = gcd a (gcd b c)"
perspective_from_point_desargues_config: fixes A :: "Points"   and B :: "Points"   and C :: "Points"   and A' :: "Points"   and B' :: "Points"   and C' :: "Points"   and R :: "Points" assumes "are_perspective_from_point A B C A' B' C' R"   and "line A B \<noteq> line A' B'"   and "line A C \<noteq> line A' C'"   and "line B C \<noteq> line B' C'" shows "desargues_config A B C A' B' C' (Pascal_Property.inter (line B C) (line B' C')) (Pascal_Property.inter (line A C) (line A' C')) (Pascal_Property.inter (line A B) (line A' B')) R"
uminus_eq: fixes x :: "'a" shows "- x = - x \<squnion> bot"
bounded_linear_flip_blinfun: shows "bounded_linear flip_blinfun"
imethds_cases: fixes im :: "qtname \<times> mhead"   and G :: "prog"   and I :: "qtname"   and sig :: "sig"   and i :: "iface" assumes "im \<in> DeclConcepts.imethds G I sig"   and "iface G I = Some i"   and "ws_prog G" obtains "table_of (map (\<lambda>(s, mh). (s, I, mh)) (imethods i)) sig = Some im"   | J :: "qtname" where "J \<in> set (isuperIfs i)" and "im \<in> DeclConcepts.imethds G J sig"
Pi_mem: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and B :: "'b \<Rightarrow> 'c set"   and x :: "'b" assumes "f \<in> Pi A B"   and "x \<in> A" shows "f x \<in> B x"
Vfrom_in_bundle_4: fixes i :: "V"   and j :: "V"   and A :: "V"   and B :: "V" assumes "i \<in>\<^sub>\<circ> j"   and "A = A"   and "B = B" shows "Ord (rank j)"
Pow_singleton_iff: fixes X :: "'b set"   and Y :: "'b set" shows "(Pow X = {Y}) = (X = {} \<and> Y = {})"
left_impl_times: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and imp_l :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and one :: "'b"   and a :: "'b"   and b :: "'b" assumes "class.left_complemented_monoid_algebra times inf imp_l less_eq less one" shows "times (imp_l a b) a = times (imp_l b a) b"
IH: assumes "Z \<subseteq> V" shows "Z' \<subseteq> V"
leof_trans: fixes a :: "'b nres"   and X :: "'b set"   and c :: "'b nres" assumes "a \<le>\<^sub>n RES X"   and "RES X \<le>\<^sub>n c" shows "a \<le>\<^sub>n c"
path_join_eq: fixes g1 :: "real \<Rightarrow> 'b"   and g2 :: "real \<Rightarrow> 'b" assumes "path g1"   and "path g2" shows "path (g1 +++ g2) = (pathfinish g1 = pathstart g2)"
arctan_unique: fixes x :: "real"   and y :: "real" assumes "- (pi / 2) < x"   and "x < pi / 2"   and "tan x = y" shows "arctan y = x"
blinfun_inverse_left: fixes f :: "'b \<Rightarrow>\<^sub>L 'b"   and f' :: "'b \<Rightarrow>\<^sub>L 'b" shows "(f o\<^sub>L f' = 1\<^sub>L) = (f' o\<^sub>L f = 1\<^sub>L)"
NotR_inv: fixes \<Gamma> :: "'b formula multiset"   and F :: "'b formula"   and \<Delta> :: "'b formula multiset" assumes "\<Gamma> \<Rightarrow> \<^bold>\<not> F, \<Delta>" shows "F, \<Gamma> \<Rightarrow> \<Delta>"
SucState_StepConf: fixes TS :: "('d \<times> (('d, 'e) expr \<times> ('f data \<Rightarrow> bool) \<times> 'e set \<times> 'f pupdate) \<times> 'd) set"   and ST :: "('d, 'e, 'f) status"   and S :: "'d"   and A :: "('d, 'e, 'f) seqauto" assumes "TS \<in> HPT ST"   and "S \<in> StepConf (HA ST) (Conf ST) TS"   and "A \<in> the (CompFun (HA ST) S)" shows "\<exists>x. x \<in> States A \<and> x \<in> StepConf (HA ST) (Conf ST) TS"
mod_eq_nat1E: fixes m :: "nat"   and q :: "nat"   and n :: "nat" assumes "m mod q = n mod q"   and "n \<le> m" obtains s :: "nat" where "m = n + q * s"
ev_blocks_partD: fixes m :: "nat"   and A :: "'a mat"   and i :: "nat"   and j :: "nat"   and k :: "nat" assumes "local.ev_blocks_part m A"   and "i < j"   and "j < k"   and "k < m"   and "A $$ (k, k) = A $$ (i, i)" shows "A $$ (j, j) = A $$ (i, i)"
prems: shows "y \<in> UNIV\<^sup>\<star>"
OrdP_SUCC_I0: fixes k :: "tm" shows "{OrdP k} \<turnstile> OrdP (SUCC k)"
val_Zp_mult0: fixes x :: "nat \<Rightarrow> int"   and y :: "nat \<Rightarrow> int" assumes "x \<in> carrier Zp"   and "x \<noteq> \<zero>"   and "y \<in> carrier Zp"   and "y \<noteq> \<zero>" shows "val_Zp (x \<otimes> y) = val_Zp x + val_Zp y"
wnf_lemma_3: fixes x :: "'a"   and z :: "'a"   and y :: "'a" shows "(- x \<squnion> - z) \<sqinter> (- - x \<squnion> - y) \<squnion> - - x = - - x \<squnion> - y"
run_altc_sample_state: fixes altc_sample :: "'c \<Rightarrow> ('d \<Rightarrow> 'a) \<Rightarrow> 'a"   and p :: "'c"   and f :: "'d \<Rightarrow> ('b, 'a) stateT"   and s :: "'b" shows "run_state (altc_sample_state altc_sample p f) s = altc_sample p (\<lambda>x. run_state (f x) s)"
induced_automorphism_automorphism: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'b"   and C0 :: "'b set" assumes "OpposedThinChamberComplexFoldings X f g C0" shows "ChamberComplexAutomorphism X (OpposedThinChamberComplexFoldings.induced_automorphism X f g)"
bilinear_conv_bounded_bilinear: fixes h :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" shows "bilinear h = bounded_bilinear h"
length_derivation_shift: fixes \<NN> :: "symbol set"   and \<TT> :: "symbol set"   and \<RR> :: "(symbol \<times> symbol list) set"   and \<SS> :: "symbol"   and D :: "(nat \<times> symbol \<times> symbol list) list"   and left :: "nat"   and right :: "nat" assumes "CFG \<NN> \<TT> \<RR> \<SS>" shows "length (derivation_shift D left right) = length D"
inject_3: fixes x31 :: "hfm"   and x32 :: "hfm"   and y31 :: "hfm"   and y32 :: "hfm" shows "(x31 HOR x32 = y31 HOR y32) = (x31 = y31 \<and> x32 = y32)"
f: shows "f \<in> A \<rightarrow>\<^sub>E X \<and> bij_betw f A X"
wl: fixes loc :: "'loc" shows "c_work (prop_config c1 p) loc = {#}\<^sub>z"
that: fixes c :: "real" assumes "a \<le> c"   and "c \<le> b"   and "((\<lambda>x. g x *\<^sub>R f x) has_integral integral {c..b} f) {a..b}" shows "thesis"
gp_fin: shows "AE x in M. ennreal (f x) * nn_transfer_operator gp x \<noteq> \<infinity>"
lexext_irrefl: fixes xs :: "'b list"   and gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<forall>x\<in>set xs. \<not> gt x x" shows "\<not> lexext gt xs xs"
tObsC_initial_1: fixes t :: "('a, 'es, 'as) BEState Trace" shows "tFirst (tObsC t) = envObsC (es (tFirst t))"
intro: fixes G_invar :: "'a \<Rightarrow> bool"   and G_adj :: "'a \<Rightarrow> 'b \<Rightarrow> ('b \<times> nat) list"   and G_empty :: "'a"   and G_add_edge :: "'b \<times> 'b \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a"   and M_empty :: "'c"   and M_update :: "'b \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'c"   and M_delete :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and M_lookup :: "'c \<Rightarrow> 'b \<Rightarrow> 'b option"   and M_invar :: "'c \<Rightarrow> bool"   and Q_empty :: "'d"   and Q_update :: "'b \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> 'd"   and Q_delete :: "'b \<Rightarrow> 'd \<Rightarrow> 'd"   and Q_invar :: "'d \<Rightarrow> bool"   and Q_lookup :: "'d \<Rightarrow> 'b \<Rightarrow> nat option"   and Q_is_empty :: "'d \<Rightarrow> bool"   and Q_getmin :: "'d \<Rightarrow> 'b \<times> nat"   and G_\<alpha>w :: "'a \<Rightarrow> 'b set \<Rightarrow> nat"   and G_\<alpha>g :: "'a \<Rightarrow> 'b ugraph"   and g :: "'a" assumes "Prim_Impl_Defs G_invar G_adj G_empty G_add_edge M_empty M_update M_delete M_lookup M_invar Q_empty Q_update Q_delete Q_invar Q_lookup Q_is_empty Q_getmin G_\<alpha>w G_\<alpha>g"   and "Prim_Impl_axioms G_invar g" shows "Prim_Impl G_invar G_adj G_empty G_add_edge M_empty M_update M_delete M_lookup M_invar Q_empty Q_update Q_delete Q_invar Q_lookup Q_is_empty Q_getmin g G_\<alpha>w G_\<alpha>g"
one_ow_def: fixes U :: "'a set"   and one :: "'a" shows "one_ow U one \<equiv> neutral_ow U one"
dimacs_model_to_abs_atom': fixes x :: "nat"   and dimacs_M :: "int list"   and M :: "nat \<Rightarrow> bool" assumes "0 < x"   and "- int x \<in> set dimacs_M"   and "distinct (map dimacs_lit_to_var dimacs_M)" shows "\<not> dimacs_model_to_abs dimacs_M M x"
fp_cop_F_all: fixes d :: "'d"   and ds :: "'d set" assumes "d \<in> ds"   and "d \<notin> Xd ` cop ds" shows "Field (Pd d) \<subseteq> fp_cop_F ds"
submodule_sum_def: fixes N1 :: "'a vec set"   and N2 :: "'a vec set" shows "subspace_sum N1 N2 = (\<lambda>(x, y). x + y) ` {(x, y). x \<in> N1 \<and> y \<in> N2}"
body': shows "P \<turnstile> \<langle>body,(h\<^sub>2, l\<^sub>2(this \<mapsto> Addr a, pns [\<mapsto>] vs), sh\<^sub>2),b\<^sub>2\<rangle> \<rightarrow>* \<langle>ef,(h\<^sub>3, l\<^sub>2 ++ l\<^sub>3, sh\<^sub>3),b\<^sub>3\<rangle>"
vprojection_in_VUnionI: fixes A :: "V"   and I :: "V"   and F :: "V \<Rightarrow> V"   and f :: "V"   and i :: "V" assumes "A \<subseteq>\<^sub>\<circ> vproduct I F"   and "f \<in>\<^sub>\<circ> A"   and "i \<in>\<^sub>\<circ> I" shows "f\<lparr>i\<rparr> \<in>\<^sub>\<circ> \<Union>\<^sub>\<circ> (\<Union>\<^sub>\<circ> (\<Union>\<^sub>\<circ> A))"
abs_eq: fixes x :: "'a set" shows "ccspan x = Abs_clinear_space (closure (cspan x))"
start_prog_Start_sees_start_method: fixes P :: "jvm_prog"   and Mm :: "char list \<Rightarrow> ((staticb \<times> ty list \<times> ty \<times> jvm_method) \<times> char list) option"   and C :: "char list"   and M :: "char list" assumes "P \<turnstile> Object sees_methods Mm" shows "class_add P (start_class C M) \<turnstile> Start sees start_m, Static :  []\<rightarrow>Void = (1, 0, [Invokestatic C M 0, Return], []) in Start"
acyclic_2b_def: fixes x :: "'a" shows "acyclic_2b x \<equiv> \<forall>y. y \<squnion> y\<^sup>T \<le> x \<and> asymmetric y \<longrightarrow> y\<^sup>\<star> \<sqinter> y\<^sup>T\<^sup>\<star> = (1::'a)"
fps_deriv_radical': fixes r :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and k :: "nat"   and a :: "'a fps" assumes "r (Suc k) (fps_nth a 0) ^ Suc k = fps_nth a 0"   and "fps_nth a 0 \<noteq> (0::'a)" shows "fps_deriv (fps_radical r (Suc k) a) = fps_deriv a / (of_nat (Suc k) * fps_radical r (Suc k) a ^ k)"
star_assoc: fixes a :: "assn"   and b :: "assn"   and c :: "assn" shows "a * b * c = a * (b * c)"
measurable_add_dim: fixes i :: "'d"   and I :: "'d set"   and M :: "'d \<Rightarrow> 'e measure" shows "(\<lambda>(f, y). f(i := y)) \<in> Pi\<^sub>M I M \<Otimes>\<^sub>M M i \<rightarrow>\<^sub>M Pi\<^sub>M (insert i I) M"
subst_ident: fixes x :: "'a"   and t :: "('b, 'a) Term.term"   and u :: "('b, 'a) Term.term" assumes "x \<notin> vars_term t" shows "t \<cdot> subst x u = t"
Collect_empty_eq_bot: fixes P :: "'b \<Rightarrow> bool" shows "(Collect P = {}) = (P = \<bottom>)"
SUP_sup_distrib: fixes A :: "V set"   and f :: "V \<Rightarrow> V"   and g :: "V \<Rightarrow> V" assumes "small A" shows "(SUP x\<in>A. f x \<squnion> g x) = \<Squnion> (f ` A) \<squnion> \<Squnion> (g ` A)"
has_initial_arrows_to_functor_1: fixes C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and D :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and F :: "'d \<Rightarrow> 'c"   and G :: "'c \<Rightarrow> 'd"   and \<phi> :: "'d \<Rightarrow> 'c \<Rightarrow> 'd"   and \<psi> :: "'c \<Rightarrow> 'd \<Rightarrow> 'c"   and y :: "'d" assumes "meta_adjunction C D F G \<phi> \<psi>"   and "partial_magma.ide D y" shows "initial_arrow_to_functor C D G y (F y) (meta_adjunction.\<eta> D F \<phi> y)"
return_time_ex1: fixes P :: "'a set"   and x :: "'a" assumes "returns_to P x"   and "closed P" shows "\<exists>!t. 0 < t \<and> t \<in> existence_ivl0 x \<and> flow0 x t \<in> P \<and> (\<forall>s\<in>{0<..<t}. flow0 x s \<notin> P)"
qAbs_alphaAbs_qSwap_qAFresh: fixes X :: "('b, 'c, 'varSort, 'var, 'd) qTerm"   and ys :: "'varSort"   and x' :: "'var"   and x :: "'var" assumes "qGood X"   and "qAFresh ys x' X" shows "qAbs ys x X $= qAbs ys x' (X #[[x' \<and> x]]_ys)"
rhoare_trans: fixes R1 :: "('a \<times> 'b) set"   and T1 :: "('a \<times> 'a) set"   and T2 :: "('b \<times> 'b) set"   and R2 :: "('b \<times> 'c) set"   and T3 :: "('c \<times> 'c) set" assumes "{R1} T1, T2 {> R1}"   and "{R2} T2, T3 {> R2}" shows "{R1 O R2} T1, T3 {> R1 O R2}"
set_eq_code_2: fixes A :: "'b set"   and B :: "'b set" shows "set_eq (Complement A) (Complement B) = set_eq A B"
l\<^sub>2\<^sub>3: shows "instance_of\<^sub>l l\<^sub>2 l\<^sub>3"
inclusion_of_generators_def: shows "\<F>C.inclusion_of_generators \<equiv> \<lambda>f. if C.arr f then \<F>C.mkarr \<^bold>\<langle>f\<^bold>\<rangle> else \<F>C.null"
a1: shows "s' = snd (fst (branch_instr instr s)) \<and> get_S (cpu_reg_val PSR s) = 0"
Xsub: shows "X closure_of (topspace X \<inter> U) \<subseteq> X interior_of (topspace X \<inter> T)"
wf_inv_image: fixes r :: "('d \<times> 'd) set"   and f :: "'e \<Rightarrow> 'd" assumes "wf r" shows "wf (inv_image r f)"
ReflectLAt_def: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and M :: "'a"   and P' :: "'a"   and P :: "'a"   and A :: "'a"   and B :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong" shows "Tarski_neutral_dimensionless.ReflectLAt Bet Cong M P' P A B \<equiv> (Tarski_neutral_dimensionless.Midpoint Bet Cong M P P' \<and> Tarski_neutral_dimensionless.Col Bet A B M) \<and> (Tarski_neutral_dimensionless.Perp Bet Cong A B P P' \<or> P = P')"
add_less_abs_positive_iff: fixes b :: "real"   and a :: "real" assumes "\<bar>b\<bar> < \<bar>a\<bar>" shows "(0 < a + b) = (0 < a)"
upt_add_eq_append': fixes i :: "nat"   and j :: "nat"   and k :: "nat" assumes "i \<le> j"   and "j \<le> k" shows "[i..<k] = [i..<j] @ [j..<k]"
ereal_less_le: fixes y :: "real"   and a :: "ereal"   and x :: "real" assumes "ereal y < a"   and "x \<le> y" shows "ereal x < a"
intro: fixes G :: "('c, 'd) ParityGame_scheme"   and p :: "Player"   and P :: "'c llist"   and \<sigma> :: "'c \<Rightarrow> 'c" assumes "ParityGame.path_conforms_with_strategy G p P \<sigma>" shows "vmc_path_axioms G P p \<sigma>"