hcju/isabelleps huggingface dataset:
symP_def: fixes \<phi> :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "symP \<phi> \<equiv> \<forall>x y. \<phi> x y \<longrightarrow> \<phi> y x"
set_intros_1: fixes x1 :: "'a" shows "x1 \<in> set_Literal (Pos x1)"
lookup_some_append: fixes env :: "('f, 'g, 'h) env"   and xs :: "'h list"   and ys :: "'h list"   and e :: "('f, 'g, 'h) env" assumes "lookup env (xs @ ys) = Some e" shows "\<exists>e. lookup env xs = Some e"
assms: shows "Suc (mod_type_class.to_nat (GREATEST n. \<not> is_zero_row_upt_k n k A)) = nrows A"
has_field_derivative_at_within: fixes f :: "'a \<Rightarrow> 'a"   and f' :: "'a"   and x :: "'a"   and s :: "'a set" assumes "(f has_field_derivative f') (at x)" shows "(f has_field_derivative f') (at x within s)"
euler_char_def: shows "G.euler_char \<equiv> int (card (verts G)) - int (card (arcs G) div 2) + int (card G.face_cycle_sets)"
alw_unfold: fixes \<phi> :: "('state \<times> 'aprop set) stream list \<Rightarrow> bool"   and \<pi>l :: "('state \<times> 'aprop set) stream list" shows "alw \<phi> \<pi>l = (\<phi> \<pi>l \<and> alw \<phi> (map stl \<pi>l))"
bilinear_l_lin: fixes M1 :: "('g, 'a, 'h) Module_scheme"   and M2 :: "('i, 'a, 'h) Module_scheme"   and N :: "('j, 'a, 'h) Module_scheme"   and m1 :: "'g"   and m2 :: "'i"   and r :: "'a"   and f :: "'g \<times> 'i \<Rightarrow> 'j" assumes "R module M1"   and "R module M2"   and "R module N"   and "m1 \<in> carrier M1"   and "m2 \<in> carrier M2"   and "r \<in> carrier R"   and "bilinear_map f R M1 M2 N" shows "f (r \<cdot>\<^sub>s\<^bsub>M1\<^esub> m1, m2) = r \<cdot>\<^sub>s\<^bsub>N\<^esub> f (m1, m2)"
Abs_fnat_hom_0: shows "0 = word_of_nat 0"
init_fin_bisimulation: fixes t :: "'t" shows "bisimulation (r1.init_fin t) (r2.init_fin t) (init_fin_bisim t) (ta_bisim init_fin_bisim)"
strategy_attracts_not_outside: fixes v0 :: "'a"   and A :: "'a set"   and W :: "'a set"   and p :: "Player"   and \<sigma> :: "'a \<Rightarrow> 'a" assumes "v0 \<in> V - A - W"   and "strategy p \<sigma>" shows "\<not> strategy_attracts_via p \<sigma> v0 A W"
linorder_linear: fixes x :: "'h"   and y :: "'h" shows "x \<le> y \<or> y \<le> x"
sorted_wrt_conv_nth: fixes R :: "('b \<times> 'b) set"   and xs :: "'b list" shows "Linorder_Relations.sorted_wrt R xs = (\<forall>i j. i < j \<and> j < length xs \<longrightarrow> (xs ! i, xs ! j) \<in> R)"
simps: fixes outPorts :: "'a \<Rightarrow> 'b fset"   and vertices :: "'c fset"   and nodeOf :: "'c \<Rightarrow> 'a"   and v :: "'c"   and p :: "'b" shows "Vertex_Graph.valid_out_port outPorts vertices nodeOf (v, p) = (v |\<in>| vertices \<and> p |\<in>| outPorts (nodeOf v))"
chop_add2: fixes i :: "nat_int"   and x :: "nat"   and y :: "nat" shows "|i| = x + y \<longrightarrow> (\<exists>j k. N_Chop(i,j,k) \<and> |j| = x \<and> |k| = y)"
par_beta_body_refl: fixes u :: "sterm" assumes "body u" shows "\<exists>L. finite L \<and> (\<forall>s p. s \<notin> L \<and> p \<notin> L \<and> s \<noteq> p \<longrightarrow> (\<exists>t'. u\<^bsup>[Fvar s,Fvar p]\<^esup> => t' \<and> u = \<sigma>[s,p] t'))"
Gamma_real_nonneg: fixes x :: "real" assumes "0 < x" shows "0 \<le> Gamma x"
FDTs: shows "map_of (map (\<lambda>((F, D), b, T). (F, D, b, T)) FDTs) F = \<lfloor>(D, b, T)\<rfloor>"
simps_2: fixes lc :: "'b"   and q :: "'b poly"   and r :: "'b poly"   and d :: "'b poly"   and dr :: "nat" shows "pseudo_divmod_main lc q r d dr 0 = (q, r)"
simps: fixes s1 :: "('e, 'f, 'g, 'h) SLin_state_scheme"   and s2 :: "'i" shows "P1 (s1, s2) = (\<forall>p. status s1 p \<in> {Pending, Aborted} \<longrightarrow> fst (pending s1 p) = p)"
is_processT8: fixes P :: "'c process" shows "\<forall>s X. s \<in> D P \<longrightarrow> (s, X) \<in> F P"
stable_cardSuc: fixes r :: "('a \<times> 'a) set" assumes "Card_order r"   and "infinite (Field r)" shows "stable (cardSuc r)"
Shleg_altdef: fixes n :: "nat"   and x :: "real" shows "Shleg n x = Gen_Shleg n n x / fact n"
strict_trans1: fixes b :: "'b list"   and a :: "'b list"   and c :: "'b list" assumes "Sublist.prefix b a"   and "strict_prefix c b" shows "strict_prefix c a"
Empty_correct_1: shows "FingerTreeStruc.toList Empty = []"
bigtheta_sym: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and g :: "'b \<Rightarrow> 'c" shows "(f \<in> \<Theta>[F](g)) = (g \<in> \<Theta>[F](f))"
llist_of_tllist_inverse: fixes b :: "('f, 'g) tllist" shows "tllist_of_llist (terminal b) (llist_of_tllist b) = b"
hlam_arr_to_hfun_in_hexp: fixes a :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and b :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and c :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and f :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "ide a"   and "ide b"   and "ide c"   and "\<guillemotleft>f : local.prod a b \<rightarrow> c\<guillemotright>" shows "hlam (ide_to_hf a) (ide_to_hf b) (ide_to_hf c) (arr_to_hfun f) \<^bold>\<in> hexp (ide_to_hf a) (ide_to_hf (exp b c))"
length_subseqs: fixes xs :: "'g list" shows "length (subseqs xs) = 2 ^ length xs"
interaction_bounded_by_inline: fixes consider' :: "'d \<Rightarrow> bool"   and gpv :: "('e, 'd, 'f) gpv"   and p :: "enat"   and "consider" :: "'h \<Rightarrow> bool"   and callee :: "'g \<Rightarrow> 'd \<Rightarrow> ('f \<times> 'g, 'h, 'i) gpv"   and q :: "enat"   and s :: "'g" assumes "interaction_bounded_by consider' gpv p"   and "\<And>s x. consider' x \<Longrightarrow> interaction_bounded_by consider (callee s x) q"   and "\<And>s x. \<not> consider' x \<Longrightarrow> interaction_bounded_by consider (callee s x) 0" shows "interaction_bounded_by consider (inline callee gpv s) (p * q)"
cint_eq: fixes f :: "complex \<Rightarrow> complex" assumes "f holomorphic_on S - {z}" shows "contour_integral p f = contour_integral \<gamma> f"
hd_prog_last_prog_append_Prog\<^sub>s\<^sub>b: fixes p' :: "'b"   and xs :: "'b memref list"   and p\<^sub>1 :: "'b"   and p\<^sub>2 :: "'b"   and mis :: "instr list" assumes "hd_prog p' xs = p'"   and "last_prog p' xs = p\<^sub>1" shows "hd_prog p' (xs @ [Prog\<^sub>s\<^sub>b p\<^sub>1 p\<^sub>2 mis]) = p'"
SMP_subterms_eq: fixes M :: "('fun, 'var) Term.term set" shows "SMP (subterms\<^sub>s\<^sub>e\<^sub>t M) = SMP M"
select_convs_1: fixes node_properties :: "'a \<Rightarrow> 'b option"   and more :: "'c" shows "node_properties \<lparr>node_properties = node_properties, \<dots> = more\<rparr> = node_properties"
wf_mdecl_wwf_mdecl: fixes P :: "'a J_prog"   and C :: "String.literal"   and Md :: "(String.literal list \<times> 'a expr) mdecl" assumes "wf_J_mdecl P C Md" shows "wwf_J_mdecl P C Md"
set_sel_1: fixes a :: "'a stream" shows "shd a \<in> sset a"
AE_Invariant_func_is_AE_constant: fixes f :: "'a \<Rightarrow> 'c" assumes "f \<in> borel_measurable M"   and "AE x in M. f (T x) = f x" shows "\<exists>y. AE x in M. f x = y"
conjunctiveD: fixes x :: "'b"   and inf_b :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inf_c :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and times_abc :: "'b \<Rightarrow> 'c \<Rightarrow> 'd"   and y :: "'c"   and z :: "'c" assumes "x \<in> conjunctive.conjunctive inf_b inf_c times_abc" shows "times_abc x (inf_b y z) = inf_c (times_abc x y) (times_abc x z)"
set_remove1_subset: fixes x :: "'g"   and xs :: "'g list" shows "set (remove1 x xs) \<subseteq> set xs"
simps_2: fixes x :: "preal" shows "preal_min x preal.PInfty = x"
fls_compose_power_1_left: fixes d :: "nat" assumes "0 < d" shows "fls_compose_power 1 d = 1"
uminus_Assertion_def: fixes p :: "'b Assertion" shows "- p = abs_wpt (neg_assert {\<cdot> p })"
sets_lebesgue_inner_closed: fixes S :: "'a set"   and e :: "real" assumes "S \<in> sets lebesgue"   and "0 < e" obtains T :: "'a set" where "closed T" and "T \<subseteq> S" and "S - T \<in> lmeasurable" and "emeasure lebesgue (S - T) < ennreal e"
pseudoAllocation_def: fixes allocation :: "('a \<times> 'b set) set" shows "pseudoAllocation allocation \<equiv> \<Union> (omega ` allocation)"
Union_Strong_dlHPP_Bisim: fixes SR :: "(('a \<times> ('b \<Rightarrow> 'c)) \<times> 'a list \<times> 'a option \<times> ('b \<Rightarrow> 'c)) set"   and E :: "'e \<Rightarrow> ('b \<Rightarrow> 'c) \<Rightarrow> 'c"   and pp :: "'a \<Rightarrow> nat"   and DA :: "'b \<Rightarrow> 'f"   and lH :: "('f \<times> 'e \<times> nat) set"   and d :: "'f"   and PP :: "nat set"   and R :: "('a list \<times> 'a list) set"   and R' :: "('a list \<times> 'a list) set" assumes "WHATWHERE.Strong_dlHPP_Bisimulation SR E pp DA lH d PP R"   and "WHATWHERE.Strong_dlHPP_Bisimulation SR E pp DA lH d PP R'"   and "Arefl R \<inter> Arefl R' \<subseteq> {[]}" shows "WHATWHERE.Strong_dlHPP_Bisimulation SR E pp DA lH d PP (R \<union> R')"
directed_nonempty: fixes X :: "'a set" assumes "directed X" shows "X \<noteq> {}"
semidecisive_def: fixes scf :: "('c \<Rightarrow> ('d \<times> 'd) set) \<Rightarrow> ('d \<times> 'd) set"   and A :: "'d set"   and Is :: "'c set"   and C :: "'c set"   and x :: "'d"   and y :: "'d" shows "semidecisive scf A Is C x y \<equiv> C \<subseteq> Is \<and> (\<forall>P. profile A Is P \<and> (\<forall>i\<in>C. x \<^bsub>(P i)\<^esub>\<prec> y) \<and> (\<forall>i\<in>Is - C. y \<^bsub>(P i)\<^esub>\<prec> x) \<longrightarrow> x \<^bsub>(scf P)\<^esub>\<prec> y)"
calc_x_axis_intersection_in_unit_disc: fixes H :: "circline" assumes "is_poincare_line H"   and "intersects_x_axis H" shows "calc_x_axis_intersection H \<in> unit_disc"
nonbossy_def: fixes Xd :: "'d \<Rightarrow> 'e"   and Xh :: "'d \<Rightarrow> 'f"   and Pd :: "'e \<Rightarrow> ('d \<times> 'd) set"   and Ch :: "'f \<Rightarrow> 'd set \<Rightarrow> 'd set"   and ds :: "'e set"   and \<phi> :: "('e \<Rightarrow> ('d \<times> 'd) set) \<Rightarrow> ('f \<Rightarrow> 'd set \<Rightarrow> 'd set) \<Rightarrow> 'e set \<Rightarrow> 'd set" assumes "Contracts Xd Xh Pd Ch" shows "Contracts.nonbossy Xd Xh ds \<phi> = ((\<forall>Pd Pd' Ch. \<forall>d\<in>ds. ContractsWithBilateralSubstitutesAndIRC Xd Xh Pd Ch \<and> ContractsWithBilateralSubstitutesAndIRC Xd Xh (Pd(d := Pd')) Ch \<longrightarrow> Contracts.dX Xd (\<phi> Pd Ch ds) d = Contracts.dX Xd (\<phi> (Pd(d := Pd')) Ch ds) d \<longrightarrow> \<phi> Pd Ch ds = \<phi> (Pd(d := Pd')) Ch ds) \<and> (\<forall>Pd Ch Ch' h. ContractsWithBilateralSubstitutesAndIRC Xd Xh Pd Ch \<and> ContractsWithBilateralSubstitutesAndIRC Xd Xh Pd (Ch(h := Ch')) \<longrightarrow> Contracts.hX Xh (\<phi> Pd Ch ds) h = Contracts.hX Xh (\<phi> Pd (Ch(h := Ch')) ds) h \<longrightarrow> \<phi> Pd Ch ds = \<phi> Pd (Ch(h := Ch')) ds))"
i: shows "i \<in> set (\<sigma> (f, length ts))"
fls_divide_nth_below: fixes n :: "int"   and f :: "'b fls"   and g :: "'b fls" assumes "n < fls_subdegree f - fls_subdegree g" shows "(f / g) $$ n = (0::'b)"
simps_1: fixes s :: "'c stream" shows "s !! 0 = shd s"
intros_1: fixes s :: "'d \<times> 'e"   and A :: "('d \<times> 'e, 'f) automaton"   and S :: "'d \<Rightarrow> 'd \<Rightarrow> 'f \<Rightarrow> bool"   and U :: "'d \<Rightarrow> 'd \<Rightarrow> bool" assumes "s \<in> init A" shows "s \<in> oreachable A S U"
it': shows "set_iterator_genord iti (set lc) (\<lambda>x y. R (\<alpha> x) (\<alpha> y))"
assms: shows "length u\<^sub>1 = length v\<^sub>1"
rel_mono_strong: fixes R :: "'b \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'b pmf"   and y :: "'d pmf"   and Ra :: "'b \<Rightarrow> 'd \<Rightarrow> bool" assumes "rel_pmf R x y"   and "\<And>z yb. \<lbrakk>z \<in> set_pmf x; yb \<in> set_pmf y; R z yb\<rbrakk> \<Longrightarrow> Ra z yb" shows "rel_pmf Ra x y"
eval_runiq_in_Range: fixes R :: "('a \<times> 'b) set"   and a :: "'a" assumes "runiq R"   and "a \<in> Domain R" shows "R ,, a \<in> Range R"
gext_trans: fixes s :: "st"   and s' :: "st"   and s'' :: "st" assumes "s\<le>|s'"   and "s'\<le>|s''" shows "s\<le>|s''"
isTrue_neg: fixes \<phi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "Fvars \<phi> = {}" shows "isTrue \<phi> \<or> isTrue (neg \<phi>)"
assms_2: shows "list_all (\<lambda>P. index_free P i) Ps"
Node_K0_I: fixes a :: "'b + nat" shows "(\<lambda>k. Inr 0, a) \<in> Node"
NatTransFtor: fixes \<eta> :: "('g, 'h, 'i, 'j, 'k, 'l) NatTrans" assumes "NatTransP \<eta>" shows "Functor (NTDom \<eta>)"
error_free_abrupt_if1: fixes a :: "abopt"   and s :: "st"   and x :: "abrupt"   and p :: "bool" assumes "error_free (a, s)"   and "\<nexists>err. x = Error err" shows "error_free (abrupt_if p (Some x) a, s)"
frontier_of_openin: fixes X :: "'c topology"   and S :: "'c set" assumes "openin X S" shows "X frontier_of S = X closure_of S - S"
distinct_69: fixes x51 :: "'d"   and x52 :: "'a \<Rightarrow> 'b"   and x53 :: "('a, 'b, 'c, 'd) seqp"   and x10 :: "'c" shows "{x51}broadcast(x52) . x53 \<noteq> call(x10)"
scalar_prod_left_zero: fixes v :: "'b vec"   and n :: "nat" assumes "v \<in> carrier_vec n" shows "0\<^sub>v n \<bullet> v = (0::'b)"
is_wt_instance_of_condD': fixes arity :: "'a \<Rightarrow> nat"   and public :: "'a \<Rightarrow> bool"   and Ana :: "('a, 'b) Term.term \<Rightarrow> ('a, 'b) Term.term list \<times> ('a, 'b) Term.term list"   and \<Gamma> :: "('a, 'b) Term.term \<Rightarrow> ('a, 'c) Term.term"   and t :: "('a, 'b) Term.term"   and s :: "('a, 'b) Term.term" assumes "typed_model arity public Ana \<Gamma>"   and "intruder_model.wf\<^sub>t\<^sub>r\<^sub>m arity t"   and "intruder_model.wf\<^sub>t\<^sub>r\<^sub>m arity s"   and "is_wt_instance_of_cond \<Gamma> t s" shows "\<exists>\<delta>. typed_model.wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<Gamma> \<delta> \<and> (\<forall>t\<in>subst_range \<delta>. intruder_model.wf\<^sub>t\<^sub>r\<^sub>m arity t) \<and> t = s \<cdot> \<delta>"
top_coreflexive_def: shows "top \<equiv> Abs_coreflexive (1::'b)"
poly_of_pm_minus: fixes x :: "'b"   and p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'a"   and q :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'a" shows "poly_of_pm x (p - q) = poly_of_pm x p - poly_of_pm x q"
that: fixes y :: "'b" assumes "y \<in> f ` A"   and "infinite (A \<inter> f -` {y})" shows "thesis"
IJ_iff: fixes u :: "nat \<times> nat"   and k :: "nat" shows "(u \<in> IJ k) = (\<exists>j i. u = (j, i) \<and> i < j \<and> j \<le> k)"
parallel_fixp_induct_2_1: fixes orda :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and luba :: "'b set \<Rightarrow> 'b"   and ordb :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and lubb :: "'c set \<Rightarrow> 'c"   and F :: "('d \<Rightarrow> 'e \<Rightarrow> 'b) \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'b"   and G :: "('f \<Rightarrow> 'c) \<Rightarrow> 'f \<Rightarrow> 'c"   and f :: "'d \<Rightarrow> 'e \<Rightarrow> 'b"   and g :: "'f \<Rightarrow> 'c"   and P :: "('d \<Rightarrow> 'e \<Rightarrow> 'b) \<Rightarrow> ('f \<Rightarrow> 'c) \<Rightarrow> bool" assumes "partial_function_definitions orda luba"   and "partial_function_definitions ordb lubb"   and "\<And>x. monotone (fun_ord orda) orda (\<lambda>f. case x of (x, xa) \<Rightarrow> F (curry f) x xa)"   and "\<And>y. monotone (fun_ord ordb) ordb (\<lambda>g. G g y)"   and "f \<equiv> curry (ccpo.fixp (fun_lub luba) (fun_ord orda) (\<lambda>f (x, y). F (curry f) x y))"   and "g \<equiv> ccpo.fixp (fun_lub lubb) (fun_ord ordb) G"   and "ccpo.admissible (prod_lub (fun_lub luba) (fun_lub lubb)) (rel_prod (fun_ord orda) (fun_ord ordb)) (\<lambda>x. P (curry (fst x)) (snd x))"   and "P (\<lambda>_ y. luba {}) (\<lambda>_. lubb {})"   and "\<And>f g. P f g \<Longrightarrow> P (F f) (G g)" shows "P f g"
floor_log_eq_if: fixes x :: "nat"   and b :: "nat"   and y :: "nat" assumes "x div b = y div b"   and "1 < b"   and "0 < x"   and "1 \<le> x div b" shows "\<lfloor>log (real b) (real x)\<rfloor> = \<lfloor>log (real b) (real y)\<rfloor>"
bisim1Throw2: fixes P :: "'b prog"   and h :: "'heap"   and e :: "'addr expr1"   and a :: "'addr"   and xs :: "'addr val list" shows "P,throw e,h \<turnstile> (Throw a, xs) \<leftrightarrow> ([Addr a], xs, length (compE2 e), \<lfloor>a\<rfloor>)"
cols_mat_mul_map: fixes A :: "'b mat"   and B :: "'b mat" shows "cols (A * B) = map ((*\<^sub>v) A) (cols B)"
rel_funD2: fixes A :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and B :: "'h \<Rightarrow> 'i \<Rightarrow> bool"   and f :: "'g \<Rightarrow> 'h"   and g :: "'g \<Rightarrow> 'i"   and x :: "'g" assumes "(A ===> B) f g"   and "A x x" shows "B (f x) (g x)"
rangeB_def: shows "rangeB = space_as_set (B *\<^sub>S \<top>)"
dvd_refl: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and a :: "'b" assumes "class.comm_monoid_mult times one" shows "dvd.dvd times a a"
red_plus: fixes F :: "('t \<Rightarrow>\<^sub>0 'c) set"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c"   and r :: "'t \<Rightarrow>\<^sub>0 'c" assumes "red F p q" obtains s :: "'t \<Rightarrow>\<^sub>0 'c" where "(red F)\<^sup>*\<^sup>* (p + r) s" and "(red F)\<^sup>*\<^sup>* (q + r) s"
separate_compact_closed: fixes s :: "'a set"   and t :: "'a set" assumes "compact s"   and "closed t"   and "s \<inter> t = {}" shows "\<exists>d>0. \<forall>x\<in>s. \<forall>y\<in>t. d \<le> dist x y"
all_positions_sublist: fixes i :: "nat"   and xs :: "'a list"   and ys :: "'a list" assumes "i + length xs \<le> length ys"   and "\<forall>jj<length xs. ys ! (i + jj) = xs ! jj" shows "sublist_at xs ys i"
removeAll_id: fixes x :: "'b"   and xs :: "'b list" assumes "x \<notin> set xs" shows "removeAll x xs = xs"
id': shows "real_of_1 y = real_of_rat (Rat.Fract (- coeff p 0) (coeff p 1))"
unity_root_sum_0_right: fixes k :: "nat" assumes "0 < k" shows "unity_root_sum k 0 = of_nat k"
b: shows "[d2 = q + q + e2 * b + q * b - x * b + (q - r)] (mod q)"
that: shows "\<exists>Rep Abs. type_definition Rep Abs {0..<d}"
alphaAbs_all_distinct_qAFresh_def: fixes xs :: "'d"   and x :: "'e"   and X :: "('f, 'g, 'd, 'e, 'h) qTerm"   and xs' :: "'d"   and x' :: "'e"   and X' :: "('f, 'g, 'd, 'e, 'h) qTerm" shows "alphaAbs_all_distinct_qAFresh xs x X xs' x' X' \<equiv> xs = xs' \<and> (\<forall>y. y \<notin> {x, x'} \<and> qAFresh xs y X \<and> qAFresh xs y X' \<longrightarrow> X #[[y \<and> x]]_xs #= X' #[[y \<and> x']]_xs)"
hyp: shows "fst (snd (diagonal_to_Smith_aux_PQ [0..<k] bezout (P, P ** A ** Q, Q))) = fst (diagonal_to_Smith_aux_PQ [0..<k] bezout (P, P ** A ** Q, Q)) ** A ** snd (snd (diagonal_to_Smith_aux_PQ [0..<k] bezout (P, P ** A ** Q, Q))) \<and> invertible (fst (diagonal_to_Smith_aux_PQ [0..<k] bezout (P, P ** A ** Q, Q))) \<and> invertible (snd (snd (diagonal_to_Smith_aux_PQ [0..<k] bezout (P, P ** A ** Q, Q)))) \<and> fst (snd (diagonal_to_Smith_aux_PQ [0..<k] bezout (P, P ** A ** Q, Q))) = diagonal_to_Smith_aux B [0..<k] bezout"
assms: fixes j2 :: "'a"   and j1 :: "'a" assumes "j2 \<in> local.underS i"   and "(j1, j2) \<in> r" shows "R j1 initial_segment_of R j2"
insertCI: fixes a :: "'a"   and B :: "'a set"   and b :: "'a" assumes "a \<notin> B \<Longrightarrow> a = b" shows "a \<in> {b} \<union> B"
RF: shows "x \<notin> RF\<^sup>\<circ> (TER f)"
eta_reducible_Abs2: fixes t :: "Core.term"   and T :: "typ" assumes "\<nexists>f. t = f $ Bv 0"   and "\<not> eta_reducible (Abs T t)" shows "\<not> eta_reducible t"
oghoare_steps: fixes \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and c :: "('d, 'e, 'f) com"   and s :: "'d"   and c' :: "('d, 'e, 'f) com"   and t :: "'d"   and \<Theta> :: "'e \<Rightarrow> ('d, 'e, 'f) ann list option"   and F :: "'f set"   and a :: "('d, 'e, 'f) ann"   and Q :: "'d set"   and A :: "'d set" assumes "\<Gamma> \<turnstile> (c, Normal s) \<rightarrow>\<^sup>* (c', Normal t)"   and "\<Gamma>, \<Theta> \<turnstile>\<^bsub>/F\<^esub> a c Q, A"   and "s \<in> pre a" shows "\<exists>a'. \<Gamma>, \<Theta> \<turnstile>\<^bsub>/F\<^esub> a' c' Q, A \<and> t \<in> pre a' \<and> (\<forall>as. assertionsR \<Gamma> \<Theta> Q A a' c' as \<longrightarrow> assertionsR \<Gamma> \<Theta> Q A a c as) \<and> (\<forall>pm cm. atomicsR \<Gamma> \<Theta> a' c' (pm, cm) \<longrightarrow> atomicsR \<Gamma> \<Theta> a c (pm, cm))"
read_tmps_le: 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 "\<forall>i\<in>read_tmps sb. i < t"
anonymous_card: fixes P :: "'i \<Rightarrow> ('a \<times> 'a) set"   and P' :: "'i \<Rightarrow> ('a \<times> 'a) set"   and x :: "'a"   and y :: "'a" assumes "profile A Is P"   and "profile A Is P'"   and "hasw [x, y] A"   and "card {i \<in> Is. x \<^bsub>(P i)\<^esub>\<prec> y} = card {i \<in> Is. x \<^bsub>(P' i)\<^esub>\<prec> y}"   and "card {i \<in> Is. y \<^bsub>(P i)\<^esub>\<prec> x} = card {i \<in> Is. y \<^bsub>(P' i)\<^esub>\<prec> x}" shows "(x \<^bsub>(scf P)\<^esub>\<preceq> y) = (x \<^bsub>(scf P')\<^esub>\<preceq> y)"
irreducibleD: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and p :: "'b"   and a :: "'b"   and b :: "'b" assumes "class.comm_semiring_1 times one plus zero"   and "comm_semiring_1.irreducible times one zero p"   and "p = times a b" shows "dvd.dvd times a one \<or> dvd.dvd times b one"
le_mask_imp_and_mask: fixes x :: "'b word"   and n :: "nat" assumes "x \<le> mask n" shows "x AND mask n = x"
leD: fixes y :: "'t \<Rightarrow>\<^sub>0 'c"   and x :: "'t \<Rightarrow>\<^sub>0 'c" assumes "y \<preceq>\<^sub>p x" shows "\<not> x \<prec>\<^sub>p y"
\<pi>l': shows "\<pi>' (LEAST l'. \<pi>' l' = \<pi> l) = \<pi> l"
primitive_extractor_reassemble_preserves: fixes disc :: "'b \<Rightarrow> bool"   and sel :: "'b \<Rightarrow> 'c"   and C :: "'c \<Rightarrow> 'b"   and m :: "'b match_expr"   and P :: "'b match_expr \<Rightarrow> bool"   and as :: "'c negation_type list"   and ms :: "'b match_expr" assumes "wf_disc_sel (disc, sel) C"   and "normalized_nnf_match m"   and "P m"   and "P MatchAny"   and "primitive_extractor (disc, sel) m = (as, ms)"   and "\<And>m1 m2. P (MatchAnd m1 m2) = (P m1 \<and> P m2)"   and "\<And>ls1 ls2. P (alist_and' (ls1 @ ls2)) = (P (alist_and' ls1) \<and> P (alist_and' ls2))" shows "P (alist_and' (NegPos_map C as))"
brnL_Suc: fixes n :: "nat"   and cl :: "('test, 'atom, 'choice) cmd list" assumes "n < length cl" shows "brnL cl (Suc n) = brnL cl n + brn (cl ! n)"
Q_map: fixes x :: "'b H"   and l :: "'b \<Rightarrow> 'd" assumes "Q x" shows "Q (map_H l x)"
inverse_eq_infinity_iff_eq_zero: fixes x :: "ereal" shows "(1 / x = \<infinity>) = (x = 0)"
S_def: shows "S = norm (a /\<^sub>R (1 + b))"
spick_eq_pseudo_proj_False: fixes w :: "bool stream"   and n :: "nat" shows "spick w n False = pseudo_proj_False n w"
icache_mod_def: fixes c :: "word20 \<times> word12"   and val :: "word8"   and state :: "'b sparc_state" shows "icache_mod c val state \<equiv> state\<lparr>cache := cache state\<lparr>icache := icache (cache state)(c \<mapsto> val)\<rparr>\<rparr>"
1: shows "[\<^bold>\<exists>F. \<^bold>\<box>(\<^bold>\<forall>x. \<lparr>F,x\<^sup>P\<rparr> \<^bold>\<equiv> (\<^bold>\<exists>!y. \<^bold>\<A>\<phi> y (x\<^sup>P))) in v]"
mmonitorable_def: fixes \<phi> :: "'a MFOTL.formula" shows "mmonitorable \<phi> = (safe_formula \<phi> \<and> MFOTL.future_reach \<phi> \<noteq> \<infinity>)"
new_document_types_preserved: fixes h :: "(_) heap"   and h' :: "(_) heap" assumes "h \<turnstile> new_document \<rightarrow>\<^sub>h h'" shows "type_wf h = type_wf h'"
Catch: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "'d \<Rightarrow> ('e, 'd, 'f) ann list option"   and F :: "'f set"   and P\<^sub>1 :: "('e, 'd, 'f) ann"   and c\<^sub>1 :: "('e, 'd, 'f) com"   and Q :: "'e set"   and P\<^sub>2 :: "('e, 'd, 'f) ann"   and c\<^sub>2 :: "('e, 'd, 'f) com"   and A :: "'e set" assumes "\<Gamma>, \<Theta> \<turnstile>\<^bsub>/F\<^esub> P\<^sub>1 c\<^sub>1 Q, (pre P\<^sub>2)"   and "\<Gamma>, \<Theta> \<turnstile>\<^bsub>/F\<^esub> P\<^sub>2 c\<^sub>2 Q, A" shows "\<Gamma>, \<Theta> \<turnstile>\<^bsub>/F\<^esub> (AnnComp P\<^sub>1 P\<^sub>2) (Catch c\<^sub>1 c\<^sub>2) Q, A"