hcju/isabelleps huggingface dataset:
simps_2: fixes i :: "nat"   and "is" :: "nat list"   and lab_c :: "t list list"   and ts :: "t list" shows "same_lab_h (i # is) lab_c ts = (if length lab_c \<le> i then None else if lab_c ! i = ts then same_lab_h is lab_c (lab_c ! i) else None)"
permutes_imp_bij: fixes p :: "'c \<Rightarrow> 'c"   and S :: "'c set" assumes "p permutes S" shows "bij_betw p S S"
cV: shows "carrier KA.NC.V = carrier_vec (n + m)"
adj_wall_imp_ex1_wall: fixes C :: "'a set"   and D :: "'a set"   and H0 :: "'a set set set" assumes "C \<sim> D"   and "H0 \<in> (\<Union>(f, g)\<in>foldpairs. {{f \<turnstile> folding_g.\<C>, g \<turnstile> folding_g.\<C>}})"   and "separated_by H0 C D" shows "\<exists>!H. H \<in> (\<Union>(f, g)\<in>foldpairs. {{f \<turnstile> folding_g.\<C>, g \<turnstile> folding_g.\<C>}}) \<and> separated_by H C D"
snd_apfst: fixes f :: "'c \<Rightarrow> 'b"   and x :: "'c \<times> 'a" shows "snd (apfst f x) = snd x"
inv_comp_left_1: fixes g :: "'a \<times> 'a"   and f :: "'a \<times> 'a" assumes "VV.iso (VV.comp g f)"   and "VV.iso g" shows "VV.inv (VV.comp g f) = VV.comp (VV.inv f) (VV.inv g)"
sum_Suc_diff: fixes m :: "nat"   and n :: "nat"   and f :: "nat \<Rightarrow> 'a" assumes "m \<le> Suc n" shows "(\<Sum>i = m..n. f (Suc i) - f i) = f (Suc n) - f m"
Subseqs_insert: fixes xs :: "'a list"   and A :: "'a list set" shows "Subseqs (insert xs A) = set (subseqs xs) \<union> Subseqs A"
ground_term_def: fixes t :: "'b trm" shows "ground_term t = (vars_of t = {})"
ord_gpv_reflI: fixes f :: "('d, 'e, 'f) gpv" shows "ord_gpv f f"
rem_implicit_pres_ops_valid_states: fixes prob :: "(char list \<times> nat option \<times> char list list) list \<times> nat list \<times> (nat \<times> nat) list \<times> (char list \<times> (nat \<times> nat) list \<times> ((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list \<times> nat) list" shows "ast_problem.valid_states (rem_implicit_pres_ops prob) = ast_problem.valid_states prob"
greaterThanLessThan_iff: fixes i :: "'a"   and l :: "'a"   and u :: "'a" shows "(i \<in> {l<..<u}) = (l < i \<and> i < u)"
elims_3: fixes x :: "(('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree" assumes "\<not> invc2 x" obtains l :: "(('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree" and a :: "'c \<times> 'd" and uu_ :: "tcolor \<times> 'c \<times> 'd" and r :: "(('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree" where "x = \<langle>l, (a, uu_), r\<rangle>" and "\<not> (invc l \<and> invc r)"
applyUnitPropagate_def: fixes state :: "State" shows "applyUnitPropagate state = (let state' = assertLiteral (hd (getQ state)) False state in state'\<lparr>getQ := tl (getQ state')\<rparr>)"
Cons_1: fixes stateA :: "State"   and stateB :: "State"   and newWl :: "nat list" assumes "InvariantWatchesEl (getF stateA) (getWatch1 stateA) (getWatch2 stateA)"   and "\<forall>c. c \<in> set Wl' \<longrightarrow> 0 \<le> c \<and> c < length (getF stateA)"   and "getM stateA = getM stateB"   and "getF stateA = getF stateB"   and "getWatch1 stateA = getWatch1 stateB"   and "getWatch2 stateA = getWatch2 stateB"   and "getConflictFlag stateA = getConflictFlag stateB"   and "getSATFlag stateA = getSATFlag stateB" shows "let state' = notifyWatches_loop literal Wl' newWl stateA; state'' = notifyWatches_loop literal Wl' newWl stateB in getM state' = getM state'' \<and> getF state' = getF state'' \<and> getSATFlag state' = getSATFlag state'' \<and> getConflictFlag state' = getConflictFlag state''"
mk_monoid_simps_1: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and x :: "'a" assumes "class.comm_semiring_1 times one plus zero" shows "(x \<in> carrier \<lparr>carrier = UNIV - {zero}, monoid.mult = times, one = one\<rparr>) = (x \<noteq> zero)"
assms_3: fixes i :: "'a" shows "f (ket i) = g (ket i)"
is_red_singleton_not_0: fixes f :: "'t \<Rightarrow>\<^sub>0 'c"   and p :: "'t \<Rightarrow>\<^sub>0 'c" assumes "is_red {f} p" shows "f \<noteq> 0"
new_character_data_ptr_not_in_any_children: fixes p :: "(_) object_ptr" assumes "p |\<in>| object_ptr_kinds h'" shows "cast\<^sub>c\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>_\<^sub>d\<^sub>a\<^sub>t\<^sub>a\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r new_character_data_ptr \<notin> set |h' \<turnstile> get_child_nodes p|\<^sub>r"
len: shows "length cms\<^sub>1 = length cms\<^sub>2"
homeomorphic_maps_swap: fixes X :: "'a topology"   and Y :: "'b topology" shows "homeomorphic_maps (prod_topology X Y) (prod_topology Y X) (\<lambda>(x, y). (y, x)) (\<lambda>(y, x). (x, y))"
conj_commute: fixes a :: "'b::refinement_lattice"   and b :: "'b::refinement_lattice" shows "(a::'b::refinement_lattice) \<iinter> (b::'b::refinement_lattice) = b \<iinter> a"
path_component_refl_eq: fixes S :: "'a set"   and x :: "'a" shows "path_component S x x = (x \<in> S)"
simps: fixes prec :: "nat"   and x :: "float" shows "ub_exp prec x = (if 0 < x then float_divr prec 1 (lb_exp prec (- x)) else if x < - 1 then power_up_fl prec (ub_exp_horner prec (get_odd (prec + 2)) 1 1 (float_divr prec x (- floor_fl x))) (nat (- int_floor_fl x)) else ub_exp_horner prec (get_odd (prec + 2)) 1 1 x)"
finite_imp_finite_component_Keys: fixes F :: "('t \<Rightarrow>\<^sub>0 'd) set" assumes "finite F" shows "finite (component_of_term ` Keys F)"
simps_13: fixes f1 :: "'a"   and f2 :: "'a"   and f3 :: "'a"   and f4 :: "'a" shows "(case Private of Private \<Rightarrow> f1 | Package \<Rightarrow> f2 | Protected \<Rightarrow> f3 | Public \<Rightarrow> f4) = f1"
at_step_def: fixes f :: "nat \<Rightarrow> rat \<times> rat"   and n :: "nat"   and a :: "'a" shows "local.at_step f n a \<equiv> \<forall>i. bnd_get ((bnd_update ^^ i) a) = f (n + i)"
chain_def: fixes C :: "nat \<Rightarrow> 'b set" shows "chain C = (\<forall>i. C i \<subseteq> C (i + 1))"
symmetric: fixes P :: "pi"   and Q :: "pi" assumes "P \<sim>\<^sup>s Q" shows "Q \<sim>\<^sup>s P"
UP_pre_univ_propI: fixes R :: "('a, 'b) ring_scheme"   and S :: "('c, 'd) ring_scheme"   and h :: "'a \<Rightarrow> 'c" assumes "cring R"   and "cring S"   and "h \<in> ring_hom R S" shows "UP_pre_univ_prop R S h"
wf_prog_hidesD: fixes G :: "prog"   and new :: "qtname \<times> mdecl"   and old :: "qtname \<times> mdecl" assumes "G\<turnstile> new hides old"   and "wf_prog G" shows "accmodi old \<le> accmodi new \<and> is_static old"
fps_nonzero_nth_minimal: fixes f :: "'a fps" shows "(f \<noteq> 0) = (\<exists>n. fps_nth f n \<noteq> (0::'a) \<and> (\<forall>m<n. fps_nth f m = (0::'a)))"
map_upds_Nil2: fixes m :: "'a \<Rightarrow> 'b option"   and as :: "'a list" shows "m(as [\<mapsto>] []) = m"
timed_progress_axioms: shows "timed_progress nfv fv sat progress"
dlex_pp_lin: fixes s :: "('a, 'b) pp"   and t :: "('a, 'b) pp" shows "dlex_pp s t \<or> dlex_pp t s"
rsd_def: fixes x :: "'b \<times> 'c" shows "prod.rsd x = less_eq_prod (times_prod' x top_prod) x"
prv_imp_mp: fixes \<phi> :: "'fmla"   and \<chi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "\<chi> \<in> fmla"   and "prv (imp \<phi> \<chi>)"   and "prv \<phi>" shows "prv \<chi>"
intensional_simps_5: fixes P :: "'a \<Rightarrow> bool" shows "\<not> P = P \<equiv> #False"
diff_diff_add: fixes plus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and minus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and a :: "'k"   and b :: "'k"   and c :: "'k" assumes "class.cancel_ab_semigroup_add plus minus" shows "minus (minus a b) c = minus a (plus b c)"
f8: shows "ucast (get_CWP (cpu_reg_val PSR s2)) = fst (fst (get_curr_win () s2)) \<and> s2 = snd (fst (get_curr_win () s2))"
abs_eq: fixes x :: "'d set" shows "ccspan x = Abs_clinear_space (closure (cspan x))"
sets_completionI_sub: fixes N' :: "'b set"   and M :: "'b measure"   and N :: "'b set" assumes "N' \<in> null_sets M"   and "N \<subseteq> N'" shows "N \<in> sets (completion M)"
DBM_reset'_resets: fixes n :: "nat"   and v :: "'e \<Rightarrow> nat"   and cs :: "'e list"   and u :: "'e \<Rightarrow> 'f"   and M :: "nat \<Rightarrow> nat \<Rightarrow> 'f DBMEntry"   and d :: "'f" assumes "\<forall>k\<le>n. 0 < k \<longrightarrow> (\<exists>c. v c = k)"   and "\<forall>c. 0 < v c \<and> (\<forall>x y. v x \<le> n \<and> v y \<le> n \<and> v x = v y \<longrightarrow> x = y)"   and "\<forall>c\<in>set cs. v c \<le> n"   and "u \<turnstile>\<^bsub>v,n\<^esub> reset' M n cs v d" shows "\<forall>c\<in>set cs. u c = d"
bind_ask: fixes f :: "'r \<Rightarrow> 'm"   and g :: "'a option \<Rightarrow> 'm" shows "bind (ask f) g = ask (\<lambda>r. bind (f r) g)"
prems_1: shows "timpls_transformable_to' TI (Fun f T) t"
SUPSEQ_regular: fixes A :: "'b list set" assumes "regular A" shows "regular (SUPSEQ A)"
component_of_lt_ge: fixes n :: "nat"   and p :: "'t \<Rightarrow>\<^sub>0 'c" assumes "is_pot_ord"   and "proj_poly_syz n p \<noteq> 0" shows "n \<le> component_of_term (lt p)"
injectionsFromEmptyAreEmpty: fixes Z :: "'e set" shows "set (injections_alg [] Z) = {{}}"
minus_divide_divide: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.field divide inverse times one plus zero minus uminus" shows "divide (uminus a) (uminus b) = divide a b"
nat_mirror_nat_mirror_ident: fixes n :: "nat"   and l :: "nat"   and r :: "nat" assumes "n \<le> l + r" shows "nat_mirror (nat_mirror n l r) l r = n"
Class_widen_Class: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'a) list) list"   and C :: "char list"   and D :: "char list" shows "subtype P (Class C) (Class D) = P \<turnstile> C \<preceq>\<^sup>* D"
sup_commute: fixes x :: "'a"   and y :: "'a" shows "x \<sqinter> y = y \<sqinter> x"
sumlist_def: fixes xs :: "'a vec list" shows "M.sumlist xs \<equiv> foldr (+) xs (0\<^sub>v n)"
parZero: fixes P :: "pi" shows "P \<parallel> \<zero> \<sim>\<^sup>s\<^sub>l P"
hgpgm: fixes x :: "'a" shows "h x = gp x + gm x"
tst': shows "thr s2 t = \<lfloor>(x2, ln)\<rfloor>"
is_processT5: fixes P :: "'b process" shows "\<forall>s X Y. (s, X) \<in> F P \<and> (\<forall>c. c \<in> Y \<longrightarrow> (s @ [c], {}) \<notin> F P) \<longrightarrow> (s, X \<union> Y) \<in> F P"
interleavings_filter_eq1: fixes xs :: "'a list"   and ys :: "'a list"   and zs :: "'a list"   and P :: "'a \<Rightarrow> bool" assumes "interleavings xs ys zs"   and "(\<forall>x\<in>set xs. P x) \<and> (\<forall>y\<in>set ys. \<not> P y)" shows "filter P zs = xs"
Min_antimono: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and M :: "'a set"   and N :: "'a set" assumes "class.linorder less_eq less"   and "M \<subseteq> N"   and "M \<noteq> {}"   and "finite N" shows "less_eq (linorder.Min less_eq N) (linorder.Min less_eq M)"
emb_prj_emb: fixes x :: "'e" assumes "DEFL('e) \<sqsubseteq> DEFL('f)" shows "EMB('f)\<cdot>(PRJ('f)\<cdot>(EMB('e)\<cdot>x)) = EMB('e)\<cdot>x"
step_Normal_elim_cases_13: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and b :: "'e set"   and c :: "('e, 'd, 'f) com"   and cs :: "('e, 'd, 'f) com list"   and css :: "(('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list"   and s :: "'e"   and u :: "('e, 'd, 'f) com list \<times> (('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list \<times> ('e, 'f) xstate" assumes "\<Gamma>\<turnstile> (While b c # cs, css, Normal s) \<rightarrow> u" obtains "u = (c # While b c # cs, css, Normal s)" and "s \<in> b"   | "u = (cs, css, Normal s)" and "s \<notin> b"
incseqD: fixes f :: "nat \<Rightarrow> 'd"   and i :: "nat"   and j :: "nat" assumes "incseq f"   and "i \<le> j" shows "f i \<le> f j"
distinct_selsort: fixes x :: "real \<times> real"   and xs :: "(real \<times> real) list" shows "distinct (ccw.selsort x xs) = distinct xs"
nn_integral_sfirst_finite: fixes H :: "'s set"   and s :: "'s" assumes "finite (acc_on (- H) `` {s})"   and "almost_everywhere (T s) (ev (HLD H))" shows "\<integral>\<^sup>+ x. ennreal_of_enat (sfirst (HLD H) (s ## x)) \<partial>T s \<noteq> \<infinity>"
\<phi>2s': shows "\<phi>2s' = remdups \<phi>2s"
transp_def: fixes r :: "'x \<Rightarrow> 'x \<Rightarrow> bool" shows "transp r = (\<forall>x y z. r x y \<longrightarrow> r y z \<longrightarrow> r x z)"
abc_Hoare_plus_halt: fixes P :: "nat list \<Rightarrow> bool"   and A :: "abc_inst list"   and Q :: "nat list \<Rightarrow> bool"   and B :: "abc_inst list"   and S :: "nat list \<Rightarrow> bool" assumes "{P} A {Q}"   and "{Q} B {S}" shows "{P} A [+] B {S}"
IH: fixes ys :: "'a list" assumes "traverse cast_Dyn xs = Some ys" shows "list_all (\<lambda>x. typeof x = None) xs"
impossible_Cons: fixes xs :: "'a list"   and ys :: "'a list"   and x :: "'a" assumes "length xs \<le> length ys" shows "(xs = x # ys) = False"
con_rews_4: fixes nat :: "nat" shows "ValKN\<cdot>nat \<noteq> \<bottom>"
with_proj_def: fixes G :: "'a pair_pre_digraph" shows "with_proj G = \<lparr>verts = pverts G, arcs = parcs G, tail = fst, head = snd\<rparr>"
a_set_val_def: fixes ptr :: "(_) character_data_ptr"   and v :: "char list" shows "a_set_val ptr v = Heap_Error_Monad.bind (get_M\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a ptr val) (\<lambda>m. put_M\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a ptr val_update v)"
maddux_3_12: fixes x :: "'a"   and y :: "'a" shows "x \<sqinter> - y \<squnion> x \<sqinter> y = x"
inject_2: fixes x2 :: "'a"   and y2 :: "'a" shows "(Abrupt x2 = Abrupt y2) = (x2 = y2)"
scalar_prod_right_unit: fixes i :: "nat"   and n :: "nat"   and v :: "'c vec" assumes "i < n" shows "v \<bullet> unit_vec n i = v $ i"
asym_lex_prod: fixes R :: "('g \<times> 'g) set"   and S :: "('h \<times> 'h) set" assumes "asym R"   and "asym S" shows "asym (R <*lex*> S)"
euler_imp_connected: fixes a :: "'a \<times> nat \<times> 'a"   and u :: "'a"   and p :: "('a \<times> nat \<times> 'a) list"   and v :: "'a" assumes "H.euler_trail a u p v" shows "Digraph_Component.connected (add_arc a)"
sets_measure_of: fixes A :: "'c set set"   and \<Omega> :: "'c set"   and \<mu> :: "'c set \<Rightarrow> ennreal" assumes "A \<subseteq> Pow \<Omega>" shows "sets (measure_of \<Omega> A \<mu>) = sigma_sets \<Omega> A"
ennreal_ge_nat_imp_PInf: fixes x :: "ennreal" assumes "\<And>N. of_nat N \<le> x" shows "x = \<infinity>"
graph_rel_id: shows "\<langle>Id, Id\<rangle>graph_rel = Id"
assms_4: shows "\<not> Separation G v0 v1 S"
inf_top_right: fixes x :: "'b upred" shows "(x \<and> true) = x"
openin_open_subtopology: fixes X :: "'c topology"   and S :: "'c set"   and T :: "'c set" assumes "openin X S" shows "openin (subtopology X S) T = (openin X T \<and> T \<subseteq> S)"
delete_def: fixes k :: "'a"   and m :: "(uint32, ('a, 'b) assoc_list) RBT.rbt" shows "HashMap_Impl.delete k m \<equiv> rm_map_entry (hashcode k) (\<lambda>v. case v of None \<Rightarrow> None | Some lm \<Rightarrow> let lm' = lm.delete k lm in if lm.isEmpty lm' then None else Some lm') m"
vsv_vsubset_vimageE: fixes B :: "V"   and A :: "V" assumes "B \<subseteq>\<^sub>\<circ> r `\<^sub>\<circ> A" obtains C :: "V" where "C \<subseteq>\<^sub>\<circ> A" and "B = r `\<^sub>\<circ> C"
WAC_def: shows "WAC \<equiv> \<lparr>wa_cond = condc, wa_step = stepc, wa_initial = initialc, wa_invar = invar_addc \<inter> {s. \<alpha> s \<in> wa_invar WAA}\<rparr>"
map\<^sub>0_in_hom_1: fixes a :: "'a" assumes "C.obj a" shows "\<guillemotleft>\<sigma>.map\<^sub>0 a : G.map\<^sub>0 a \<rightarrow>\<^sub>D F.map\<^sub>0 a\<guillemotright>"
closure_convex_hull: fixes S :: "'a set" assumes "compact S" shows "closure (convex hull S) = convex hull S"
s_u: shows "finite s \<and> s \<noteq> {} \<and> s \<subseteq> S \<and> sum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = (0::'a)"
less_irrefl: fixes x :: "'c list" shows "\<not> strict_prefix x x"
clean_decreasing: fixes A :: "('b, 'a) nba"   and w :: "'b stream"   and V :: "(nat \<times> 'a) set" shows "clean A w V \<subseteq> V"
assms: shows "arrow_to_functor (\<cdot>\<^sub>C) (\<cdot>\<^sub>D) G x y' g"
summable_imp_Bseq: fixes f :: "nat \<Rightarrow> 'a" assumes "summable f" shows "Bseq f"
ex_k: shows "\<exists>k. f $$ (M + int k) \<noteq> (0::'a)"
mult2_impl_sound: fixes sns :: "'a \<Rightarrow> 'a \<Rightarrow> bool \<times> bool"   and xs :: "'a list"   and ys :: "'a list" shows "mult2_impl sns xs ys = ((mset xs, mset ys) \<in> mult2_alt_s {(x, y). snd (sns x y)} {(x, y). fst (sns x y)}, (mset xs, mset ys) \<in> mult2_alt_ns {(x, y). snd (sns x y)} {(x, y). fst (sns x y)})"
parPres: fixes R :: "('a, 'b, 'c) psi"   and \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and Rel :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set"   and Rel' :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set" assumes "\<And>A\<^sub>R \<Psi>\<^sub>R. \<lbrakk>extractFrame R = \<langle>A\<^sub>R, \<Psi>\<^sub>R\<rangle>; A\<^sub>R \<sharp>* \<Psi>; A\<^sub>R \<sharp>* P; A\<^sub>R \<sharp>* Q\<rbrakk> \<Longrightarrow> (\<Psi> \<otimes> \<Psi>\<^sub>R, P, Q) \<in> Rel"   and "eqvt Rel"   and "eqvt Rel'"   and "\<And>\<Psi>' S T. (\<Psi>', S, T) \<in> Rel \<Longrightarrow> insertAssertion (extractFrame T) \<Psi>' \<hookrightarrow>\<^sub>F insertAssertion (extractFrame S) \<Psi>'"   and "\<And>\<Psi>' S T. (\<Psi>', S, T) \<in> Rel \<Longrightarrow> \<Psi>' \<rhd> S \<leadsto>[Rel] T"   and "\<And>\<Psi>' S T \<Psi>''. (\<Psi>', S, T) \<in> Rel \<Longrightarrow> (\<Psi>' \<otimes> \<Psi>'', S, T) \<in> Rel"   and "\<And>\<Psi>' S T A\<^sub>U \<Psi>\<^sub>U U. \<lbrakk>(\<Psi>' \<otimes> \<Psi>\<^sub>U, S, T) \<in> Rel; extractFrame U = \<langle>A\<^sub>U, \<Psi>\<^sub>U\<rangle>; A\<^sub>U \<sharp>* \<Psi>'; A\<^sub>U \<sharp>* S; A\<^sub>U \<sharp>* T\<rbrakk> \<Longrightarrow> (\<Psi>', S \<parallel> U, T \<parallel> U) \<in> Rel'"   and "\<And>\<Psi>' S T xvec. \<lbrakk>(\<Psi>', S, T) \<in> Rel'; xvec \<sharp>* \<Psi>'\<rbrakk> \<Longrightarrow> (\<Psi>', \<lparr>\<nu>*xvec\<rparr>S, \<lparr>\<nu>*xvec\<rparr>T) \<in> Rel'"   and "\<And>\<Psi>' S T \<Psi>''. \<lbrakk>(\<Psi>', S, T) \<in> Rel; \<Psi>' \<simeq> \<Psi>''\<rbrakk> \<Longrightarrow> (\<Psi>'', S, T) \<in> Rel" shows "\<Psi> \<rhd> P \<parallel> R \<leadsto>[Rel'] Q \<parallel> R"
totient_less_not_prime: fixes n :: "nat" assumes "\<not> prime n"   and "1 < n" shows "totient n < n - 1"
span_eq: fixes S :: "('b, 'c) vec set"   and T :: "('b, 'c) vec set" shows "(vec.span S = vec.span T) = (S \<subseteq> vec.span T \<and> T \<subseteq> vec.span S)"
pfa_drop_def: shows "pfa_drop \<equiv> \<lambda>(src, sn) si (dst, dn). blit src si dst 0 (sn - si) \<bind> (\<lambda>_. return (dst, sn - si))"
cr_uint8_def: shows "cr_uint8 \<equiv> \<lambda>x y. x = Rep_uint8 y"
index_first: fixes i :: "nat"   and xs :: "'b list"   and x :: "'b" assumes "i < index xs x" shows "x \<noteq> xs ! i"
distinct_card: fixes xs :: "'c list" assumes "distinct xs" shows "card (set xs) = length xs"
INT_I: fixes A :: "'c set"   and b :: "'d"   and B :: "'c \<Rightarrow> 'd set" assumes "\<And>x. x \<in> A \<Longrightarrow> b \<in> B x" shows "b \<in> \<Inter> (B ` A)"
enum_all_num1_def: fixes P :: "num1 \<Rightarrow> bool" shows "enum_class.enum_all P = P 1"
prefix_takeI: fixes a :: "'c list"   and b :: "'c list"   and n :: "nat" assumes "prefix a b"   and "length a \<le> n" shows "prefix a (take n b)"
eSwap_simp1: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model"   and s :: "'sort"   and X :: "'c"   and zs :: "'varSort"   and z1 :: "'var"   and z2 :: "'var" assumes "igWls MOD s X" shows "eSwap MOD zs z1 z2 (OK X) = OK (igSwap MOD zs z1 z2 X)"
inf_glb: fixes L :: "('a, 'b) gorder_scheme"   and A :: "'a set" assumes "weak_complete_lattice L"   and "A \<subseteq> carrier L" shows "is_glb L (\<Sqinter>\<^bsub>L\<^esub>A) A"
equi_desc_descA_2: fixes v :: "i"   and \<phi> :: "\<nu> \<Rightarrow> \<o>" shows "[(\<^bold>\<exists>y. y\<^sup>P \<^bold>= (\<^bold>\<iota>x. \<phi> x)) \<^bold>\<rightarrow> (\<^bold>\<iota>x. \<phi> x) \<^bold>= (\<^bold>\<iota>x. \<^bold>\<A>\<phi> x) in v]"
assms: fixes i :: "nat" assumes "i \<le> n" shows "X i \<in> M \<rightarrow>\<^sub>M N"
base: fixes x :: "real" assumes "x\<^sub>0 \<le> x"   and "x \<le> x\<^sub>1" shows "P x"
comp_in_homE: fixes g :: "'a \<times> 'a"   and f :: "'a \<times> 'a"   and a :: "'a \<times> 'a"   and c :: "'a \<times> 'a" assumes "CC.in_hom (CC.comp g f) a c" obtains b :: "'a \<times> 'a" where "CC.in_hom f a b" and "CC.in_hom g b c"
open_iff: fixes m :: "'b set" shows "(m open\<^bsub>{{}}\<^esub>) = (m = {})"
card_Un_le: fixes A :: "'b set"   and B :: "'b set" shows "card (A \<union> B) \<le> card A + card B"
ExPre_def: fixes D :: "('c, 'd) Order_scheme"   and a :: "'c" shows "ExPre D a = (\<exists>x. x \<in> carrier D \<and> x \<prec>\<^bsub>D\<^esub> a \<and> \<not> (\<exists>y\<in>carrier D. x \<prec>\<^bsub>D\<^esub> y \<and> y \<prec>\<^bsub>D\<^esub> a))"
fold_A_min_gallery_betw_map: fixes F :: "'a set"   and G :: "'a set"   and Fs :: "'a set list" assumes "chamber F"   and "chamber G"   and "order.greater_eq F z"   and "face_distance z G = chamber_distance F G"   and "min_gallery (F # Fs @ [G])" shows "min_gallery (fold_A \<Turnstile> (F # Fs @ [G]))"
linear_imp_scale: fixes D :: "'a \<Rightarrow> ('a, 'b) vec" assumes "Vector_Spaces.linear (*) (*s) D" obtains d :: "('a, 'b) vec" where "D = (\<lambda>x. x *s d)"
rev_ordsetmap_strict: fixes a :: "'a"   and b :: "'a" assumes "a \<in> P"   and "b \<in> P"   and "codomain.greater (f b) (f a)" shows "b \<succ> a"
prems: shows "\<Gamma>|\<^bsub>M\<^esub>\<turnstile> \<langle>Catch c1 c2,Normal s\<rangle> \<Rightarrow>\<notin>{Stuck}"
assms: fixes i :: "'b" shows "(\<lambda>x. f x i) \<in> borel_measurable M"
elims: fixes x :: "op"   and y :: "dynamic list \<Rightarrow> dynamic" assumes "eval x = y" obtains "x = OpNeg" and "y = eval_Neg"   | "x = OpAdd" and "y = eval_Add"   | "x = OpMul" and "y = eval_Mul"
IH: assumes "insert_ops (xs @ ys)" shows "\<exists>A\<subseteq>set (map fst ys). set (interp_ins (xs @ ys)) = set (interp_ins xs) \<union> A"
zgcd_gt1: fixes i :: "int"   and j :: "int" assumes "1 < gcd i j" shows "1 < \<bar>i\<bar> \<and> 1 < \<bar>j\<bar> \<or> \<bar>i\<bar> = 0 \<and> 1 < \<bar>j\<bar> \<or> 1 < \<bar>i\<bar> \<and> \<bar>j\<bar> = 0"
integral_distr: fixes g :: "'d \<Rightarrow> 'e"   and M :: "'d measure"   and N :: "'e measure"   and f :: "'e \<Rightarrow> 'f" assumes "g \<in> M \<rightarrow>\<^sub>M N"   and "f \<in> borel_measurable N" shows "integral\<^sup>L (distr M N g) f = LINT x|M. f (g x)"
seqI: fixes f :: "'b"   and g :: "'b" assumes "B.arr f"   and "B.arr g"   and "B.dom g = B.cod f" shows "B.seq g f"
alt_group_mult: fixes n :: "nat" shows "(\<otimes>\<^bsub>alt_group n\<^esub>) = (\<circ>)"
the1_equality': fixes P :: "'e \<Rightarrow> bool"   and a :: "'e" assumes "\<exists>\<^sub>\<le>\<^sub>1 x. P x"   and "P a" shows "(THE x. P x) = a"
c2_c2': shows "c2 \<subseteq>\<^sub>g c2'"
cast\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>2\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_def: fixes document :: "('m, 'n, 'o, 't option) RDocument_scheme" shows "cast document = RObject.extend (RObject.truncate document) (Inr (Inl (RObject.more document)))"
preserves_hom: fixes f :: "'c"   and a :: "'c"   and b :: "'c" assumes "Cop.in_hom f a b" shows "\<guillemotleft>F f : F a \<rightarrow>\<^sub>S F b\<guillemotright>"
qe_interior\<^sub>1_def: fixes \<phi> :: "atom fm" shows "qe_interior\<^sub>1 \<phi> = (let as = DLO.atoms\<^sub>0 \<phi>; lbs = lbounds as; ubs = ubounds as; ebs = ebounds as; intrs = concat (map (\<lambda>l. map (\<lambda>u. And (Atom (Less l u)) (subst\<^sub>2 l u \<phi>)) ubs) lbs) in list_disj (inf\<^sub>- \<phi> # inf\<^sub>+ \<phi> # intrs @ map (subst \<phi>) ebs))"
pivotal_def: fixes i :: "indi"   and b :: "alt" shows "pivotal i b \<equiv> \<exists>P. Extreme P b \<and> b \<cdot>< P i \<and> b \<cdot>< F P \<and> F (P(i := mktop (P i) b)) <\<cdot> b"