hcju/isabelleps huggingface dataset:
ipresSubstAll_def: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'b"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'c"   and MOD :: "('d, 'e, 'varSort, 'f, 'g, 'var, 'b, 'c, 'h) model_scheme" shows "ipresSubstAll h hA MOD \<equiv> ipresSubst h MOD \<and> ipresSubstAbs h hA MOD"
filter_def: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and F :: "'b set" shows "ord.filter less_eq F \<equiv> F \<noteq> {} \<and> (\<forall>x\<in>F. \<forall>y\<in>F. \<exists>z\<in>F. less_eq z x \<and> less_eq z y) \<and> (\<forall>x\<in>F. \<forall>y. less_eq x y \<longrightarrow> y \<in> F)"
queue_invar_simps_4: fixes q :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree list"   and t :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree" assumes "SkewBinomialHeapStruc.queue_invar q"   and "t \<in> set q" shows "SkewBinomialHeapStruc.tree_invar t"
Mul_Auxk_def: shows "Mul_Auxk \<equiv> \<lambda>s. l s < Queue s \<or> M s ! k s \<noteq> Black \<or> \<not> BtoW (E s ! ind s, M s) \<or> obc s \<subset> Blacks (M s)"
measurable_measure_pmf: fixes M :: "'a \<Rightarrow> 'b pmf" shows "(\<lambda>x. measure_pmf (M x)) \<in> count_space UNIV \<rightarrow>\<^sub>M subprob_algebra (count_space UNIV)"
chop_elem_not_Nil: fixes ys :: "'b list"   and n :: "nat"   and xs :: "'b list" assumes "ys \<in> set (chop n xs)" shows "ys \<noteq> []"
add2: fixes B :: "'d"   and k :: "'d"   and b :: "'d"   and a :: "'d" assumes "B \<equiv> k + b" shows "a + B \<equiv> k + (a + b)"
hyps_4: shows "finite_ReZ_segments (subpath 0 s g) z"
prems_2: shows "wf_formula (length I) (FEXISTS \<phi>)"
set_restrictionI: fixes f :: "'a set \<Rightarrow> 'a set"   and P :: "'a \<Rightarrow> bool" assumes "f = (\<lambda>A. {x \<in> A. P x})" shows "set_restriction f"
evaln_Skip_eq: fixes G :: "prog"   and s :: "state"   and n :: "nat"   and s' :: "state" shows "G\<turnstile>s \<midarrow>Skip\<midarrow>n\<rightarrow> s' = (s = s')"
funion_image_bind_eq: fixes f :: "'b \<Rightarrow> 'a fset"   and M :: "'b fset" shows "ffUnion (f |`| M) = fbind M f"
prj_simps_11: shows "C.seq \<pi>\<rho>.prj\<^sub>1 \<p>\<^sub>0[\<nu>.leg0, \<pi>.leg1 \<cdot> \<pi>\<rho>.prj\<^sub>1]"
nths_weave: fixes xs :: "'a list"   and A :: "nat set"   and ys :: "'a list" assumes "length xs = card {a \<in> A. a < length xs + length ys}"   and "length ys = card {a \<in> - A. a < length xs + length ys}" shows "nths (weave A xs ys) A = xs \<and> nths (weave A xs ys) (- A) = ys"
m_eq: shows "m = size Y\<^sub>\<prec> + size Y\<^sub>\<succeq>"
GREATEST: fixes x :: "'a" assumes "f x = x" shows "x \<le> a"
simple_merge_option: fixes getIJ :: "(nat set \<times> 'b option list set) set \<Rightarrow> (nat set \<times> 'b option list set) set \<Rightarrow> nat set \<Rightarrow> nat set \<times> nat set"   and a :: "'b option"   and b :: "'b option" assumes "getIJ getIJ" shows "(New_max.merge_option (a, b) = None) = (a = None \<and> b = None)"
inj_on_chamber_system: fixes X :: "'b set set"   and Y :: "'c set set"   and f :: "'b \<Rightarrow> 'c" assumes "ChamberComplexIsomorphism X Y f" shows "inj_on ((`) f) (ChamberComplex.chamber_system X)"
inner_zero_right: fixes x :: "'a" shows "inner x (0::'a) = 0"
less_3: fixes xs :: "'a list" assumes "ls = ls' @ xs"   and "xs \<noteq> []" shows "thesis"
is_category: fixes C :: "'ag \<Rightarrow> 'ag \<Rightarrow> 'ag" assumes "dual_category C" shows "category (dual_category.comp C)"
set_2: fixes x21 :: "'a"   and x22 :: "'a list" shows "list.set (x21 # x22) = insert x21 (list.set x22)"
simulation_steps: fixes a :: "'a"   and as :: "'a list"   and b :: "'b" assumes "A.steps (a # as)"   and "equiv' a b" shows "\<exists>bs. B.steps (b # bs) \<and> list_all2 equiv' as bs"
inf_le2: fixes x :: "'d upred"   and y :: "'d upred" shows "y \<sqsubseteq> (x \<and> y)"
hcomp_in_hhom': fixes \<mu> :: "'d"   and \<nu> :: "'d"   and a :: "'d"   and c :: "'d" assumes "D.arr \<mu>"   and "D.arr \<nu>"   and "src\<^sub>D \<mu> = a"   and "trg\<^sub>D \<nu> = c"   and "src\<^sub>D \<nu> = trg\<^sub>D \<mu>" shows "\<guillemotleft>\<nu> \<star>\<^sub>D \<mu> : a \<rightarrow>\<^sub>D c\<guillemotright>"
takeWhile_eq_take: fixes P :: "'c \<Rightarrow> bool"   and xs :: "'c list" shows "takeWhile P xs = take (length (takeWhile P xs)) xs"
leg0_uniquely_isomorphic_1: shows "tab\<^sub>0 (cod \<mu>) \<star> \<mu>.chine \<cong> tab\<^sub>0 (local.dom \<mu>)"
cr_linear_poly_def: shows "cr_linear_poly \<equiv> \<lambda>x y. x = Rep_linear_poly y"
assms_3: shows "Q * F \<Longrightarrow>\<^sub>t B"
measurable_lborel1: fixes M :: "'a measure" shows "M \<rightarrow>\<^sub>M lborel = borel_measurable M"
WT_elim_cases_10: 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 E :: "char list \<Rightarrow> ty option"   and e :: "expr"   and e\<^sub>1 :: "expr"   and e\<^sub>2 :: "expr"   and T :: "ty" assumes "P,E \<turnstile> if (e) e\<^sub>1 else e\<^sub>2 :: T" obtains "P,E \<turnstile> e :: Boolean" and "P,E \<turnstile> e\<^sub>1 :: T" and "P,E \<turnstile> e\<^sub>2 :: T"
transitivity_lemma: fixes \<Gamma>' :: "binding list"   and Sa :: "ty"   and Ta :: "ty" assumes "\<Gamma>'\<turnstile>Sa<:Q"   and "\<Gamma>'\<turnstile>Q<:Ta" shows "\<Gamma>'\<turnstile>Sa<:Ta"
execute_serial_plan_sas_plus_is_execute_parallel_plan_sas_plus: fixes \<Psi> :: "('a, 'b) sas_plus_problem"   and \<psi> :: "('a, 'b) sas_plus_operator list" assumes "is_valid_problem_sas_plus \<Psi>"   and "SAS_Plus_Semantics.is_serial_solution_for_problem \<Psi> \<psi>" shows "SAS_Plus_Semantics.is_parallel_solution_for_problem \<Psi> (List_Supplement.embed \<psi>)"
not_less: fixes x :: "'b"   and y :: "'b" shows "(\<not> order.greater y x) = order.greater_eq x y"
hcpow_mult: fixes r :: "complex star"   and s :: "complex star"   and n :: "nat star" shows "(r * s) pow n = r pow n * s pow n"
IH: shows "\<exists>ns. cs\<^bsup>\<pi>\<^esup> k = cs\<^bsup>\<pi>\<^esup> m @ ns @ [\<pi> k]"
refl: fixes carrier :: "'a set"   and le :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a" assumes "Order_Predicates.preorder_on carrier le"   and "x \<in> carrier" shows "le x x"
scaleC_diff_left: fixes a :: "complex"   and b :: "complex"   and x :: "'b" shows "(a - b) *\<^sub>C x = a *\<^sub>C x - b *\<^sub>C x"
unit_circle_fix_id_moebius: shows "unit_circle_fix id_moebius"
length_merge_adj: fixes xss :: "'c list list"   and m :: "nat" assumes "even (length xss)"   and "\<forall>xs\<in>set xss. length xs = m" shows "\<forall>xs\<in>set (merge_adj xss). length xs = 2 * m"
that_2: fixes r :: "'rule"   and h :: "'form"   and c :: "('form, 'var) antecedent" assumes "n = Rule r"   and "p = Hyp h c" shows "thesis"
simps_2: fixes pi :: "('a \<times> 'a) list"   and p :: "pat"   and q :: "pat" shows "pi \<bullet> \<langle>\<langle>p, q\<rangle>\<rangle> = \<langle>\<langle>pi \<bullet> p, pi \<bullet> q\<rangle>\<rangle>"
assms_7: fixes s :: "'a" assumes "s \<in> S"   and "\<not> compo s" shows "nF s = 0 \<or> nF s = 2"
RG_proj_mult_rightdelta': fixes g' :: "'g"   and r :: "'a"   and g :: "'g"   and s :: "'a" assumes "g' \<in> G" shows "RG_proj (r \<delta>\<delta> g * (s \<delta>\<delta> g')) = RG_proj (r \<delta>\<delta> g) * (s \<delta>\<delta> g')"
mult_less_cancel_left: 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 c :: "'c"   and a :: "'c"   and b :: "'c" assumes "class.linordered_ring_strict plus zero minus uminus less_eq less abs times" shows "less (times c a) (times c b) = ((less_eq zero c \<longrightarrow> less a b) \<and> (less_eq c zero \<longrightarrow> less b a))"
R1_not_in_NUM: shows "\<R> \<notin> NUM"
add_commute: fixes a :: "'a"   and b :: "'a" shows "a \<squnion> b = b \<squnion> a"
nretkn': shows "\<pi>' (k' + n') \<noteq> local.return"
Cod_Ide: fixes V :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and src :: "'a \<Rightarrow> 'a"   and trg :: "'a \<Rightarrow> 'a"   and t :: "'a local.term" assumes "bicategorical_language V src trg"   and "bicategorical_language.Ide V src trg t" shows "bicategorical_language.Cod V src trg t = t"
mcont_mono: fixes luba :: "'c set \<Rightarrow> 'c"   and orda :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and lubb :: "'d set \<Rightarrow> 'd"   and ordb :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd" assumes "mcont luba orda lubb ordb f" shows "monotone orda ordb f"
preserves_cod: fixes f :: "'a \<times> 'a \<times> 'a" assumes "B.VVV.arr f" shows "B.cod (B.HoVH f) = B.HoVH (B.VVV.cod f)"
vec_plusI_def: fixes pl :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and v :: "'c list"   and w :: "'c list" shows "vec_plusI pl v w = map (\<lambda>xy. pl (fst xy) (snd xy)) (zip v w)"
hyps_1: shows "n_leaves (reflect t1) = n_leaves t1"
abs_conv_abscissa_newman: shows "abs_conv_abscissa fds_newman \<le> 1"
assms_4: fixes y :: "'a" assumes "y \<in> set (local.filter_min xs)"   and "x \<notin> set (local.filter_min xs)"   and "rel y x" shows "thesis"
simps_1: fixes c :: "('Var, 'AExp, 'BExp) Stmt" shows "cxt_to_stmt [] c = c"
eval\<^sub>t\<^sub>s_bij: fixes b_of_a :: "'c \<Rightarrow> 'd"   and G :: "char list \<Rightarrow> 'c list \<Rightarrow> bool"   and p :: "char list"   and E :: "char list \<Rightarrow> 'c"   and F :: "char list \<Rightarrow> 'c list \<Rightarrow> 'c"   and ts :: "fterm list" assumes "bij b_of_a" shows "G_conv b_of_a G p (eval\<^sub>t\<^sub>s (E_conv b_of_a E) (F_conv b_of_a F) ts) = G p (eval\<^sub>t\<^sub>s E F ts)"
a: fixes x :: "nat" assumes "x \<le> 0" shows "x = 0"
plus_fun_apply: fixes f :: "'b \<Rightarrow> 'a"   and g :: "'b \<Rightarrow> 'a"   and x :: "'b" shows "(f + g) x = f x + g x"
allRdBlks_finite: fixes s :: "state"   and s' :: "state"   and bk :: "BlockProc" assumes "HInv1 s"   and "\<forall>p. allRdBlks s' p \<subseteq> insert bk (allRdBlks s p)" shows "\<forall>p. finite (allRdBlks s' p)"
image_iff: fixes z :: "'h"   and f :: "'i \<Rightarrow> 'h"   and A :: "'i set" shows "(z \<in> f ` A) = (\<exists>x\<in>A. z = f x)"
h0: fixes x :: "'n" assumes "x \<in> S" shows "0 \<le> h x"
ext_compat_list_def: fixes ext :: "('b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'b list \<Rightarrow> 'b list \<Rightarrow> bool" shows "ext_compat_list ext \<equiv> ext ext \<and> ext_compat_list_axioms ext"
abs_eq: fixes x :: "'a list" shows "fset_of_list x = Abs_fset (set x)"
simps_3: fixes var :: "nat"   and p :: "real mpoly" shows "freeIn var (fm.Atom (Leq p)) = (var \<notin> vars p)"
identify_block_cong: fixes n :: "nat"   and k :: "nat"   and A :: "'b mat"   and B :: "'b mat"   and i :: "nat" assumes "same_upto n k A B"   and "k < n"   and "i < k" shows "??.Jordan_Normal_Form_Existence.identify_block A i = ??.Jordan_Normal_Form_Existence.identify_block B i"
disc_1: fixes xs :: "'b llist" assumes "lnull xs" shows "lnull (lalternate xs)"
card_Diff_singleton_if: fixes A :: "'d set"   and x :: "'d" assumes "finite A" shows "card (A - {x}) = (if x \<in> A then card A - 1 else card A)"
split_mult_pos_le: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "class.ordered_ring plus minus zero less_eq less times uminus"   and "less_eq zero a \<and> less_eq zero b \<or> less_eq a zero \<and> less_eq b zero" shows "less_eq zero (times a b)"
all_waiting_implies_deadlock: fixes s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" assumes "lock_thread_ok (locks s) (thr s)"   and "\<And>t x. \<lbrakk>thr s t = \<lfloor>(x, no_wait_locks)\<rfloor>; \<not> final x; wset s t = None\<rbrakk> \<Longrightarrow> t \<turnstile> \<langle>x, shr s\<rangle> \<wrong> \<and> (\<forall>LT. t \<turnstile> \<langle>x, shr s\<rangle> LT \<wrong> \<longrightarrow> (\<exists>lt\<in>LT. must_wait s t lt (dom (thr s))))"   and "\<And>t x ln l. \<lbrakk>thr s t = \<lfloor>(x, ln)\<rfloor>; \<not> waiting (wset s t); 0 < ln $ l\<rbrakk> \<Longrightarrow> \<exists>l'. 0 < ln $ l' \<and> \<not> may_lock (locks s $ l') t"   and "\<And>t x w. thr s t = \<lfloor>(x, no_wait_locks)\<rfloor> \<Longrightarrow> wset s t \<noteq> \<lfloor>PostWS w\<rfloor>" shows "deadlock s"
terminates_Normal_elim_cases_6: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and b :: "'e set"   and c1 :: "('e, 'd, 'f) com"   and c2 :: "('e, 'd, 'f) com"   and s :: "'e" assumes "\<Gamma>\<turnstile>Cond b c1 c2 \<down> Normal s" obtains "s \<in> b" and "\<Gamma>\<turnstile>c1 \<down> Normal s"   | "s \<notin> b" and "\<Gamma>\<turnstile>c2 \<down> Normal s"
abs_not_less_zero: fixes abs :: "'c \<Rightarrow> 'c"   and 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 a :: "'c" assumes "class.ordered_ab_group_add_abs abs plus zero minus uminus less_eq less" shows "\<not> less (abs a) zero"
right_idem: fixes a :: "'e"   and b :: "'e" shows "max (max a b) b = max a b"
Abs_bset_inverse: fixes y :: "'c set" assumes "y \<in> {A. |A| <o natLeq +c |UNIV|}" shows "set_bset (Abs_bset y) = y"
fresh_star_permute_iff: fixes p :: "perm"   and a :: "atom set"   and x :: "'a" shows "(p \<bullet> a) \<sharp>* (p \<bullet> x) = a \<sharp>* x"
insert_Un: fixes x :: "'a"   and A :: "'a set"   and B :: "'a set" shows "insert x (A \<union> B) = insert x A \<union> insert x B"
NS_mono_one: fixes s :: "('f, 'b) Term.term"   and t :: "('f, 'b) Term.term"   and f :: "'f"   and ss1 :: "('f, 'b) Term.term list"   and ss2 :: "('f, 'b) Term.term list" assumes "NS s t" shows "NS (Fun f (ss1 @ s # ss2)) (Fun f (ss1 @ t # ss2))"
components_nonoverlap: fixes c :: "'c set"   and s :: "'c set"   and c' :: "'c set" assumes "c \<in> components s"   and "c' \<in> components s" shows "(c \<inter> c' = {}) = (c \<noteq> c')"
FactGroup_def: fixes G :: "('c, 'd) monoid_scheme"   and H :: "'c set" shows "G Mod H = \<lparr>carrier = rcosets\<^bsub>G\<^esub> H, monoid.mult = (<#>\<^bsub>G\<^esub>), one = H\<rparr>"
diff_fun_scaleR: fixes a :: "'a \<Rightarrow> real"   and b :: "'a \<Rightarrow> 'd" assumes "diff_fun k charts a"   and "diff_fun k charts b" shows "diff_fun k charts (\<lambda>x. a x *\<^sub>R b x)"
simps_2: fixes f :: "'d \<Rightarrow> ('a, 'b, 'c) prog"   and x :: "'d"   and xs :: "'d list" shows "map_M f (x # xs) = Heap_Error_Monad.bind (f x) (\<lambda>y. Heap_Error_Monad.bind (map_M f xs) (\<lambda>ys. return (y # ys)))"
1: fixes g :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" assumes "g \<in> G" shows "\<exists>F' q. finite F' \<and> F' \<subseteq> F \<and> g = (\<Sum>f\<in>F'. q f * f) \<and> (\<forall>f\<in>F'. poly_deg (q f * f) \<le> b)"
invh_join: fixes l :: "('c \<times> color) tree"   and r :: "('c \<times> color) tree" assumes "invh l"   and "invh r"   and "bheight l = bheight r" shows "invh (join l r) \<and> bheight (join l r) = bheight l"
left_commute: fixes b :: "bool"   and a :: "bool"   and c :: "bool" shows "(b \<and> a \<and> c) = (a \<and> b \<and> c)"
hypnat_infinite_has_nonstandard_iff: fixes A :: "nat set" shows "infinite A = (hypnat_of_nat ` A \<subset> *s* A)"
cl_subset: fixes X :: "'a set" assumes "X \<subseteq> \<E>" shows "X \<subseteq> \<E>.cl X"
simps_2: fixes y :: "'a"   and x :: "'a" shows "local.efficient_power y x (Suc 0) = f x y"
trivial_relative_homology_group_topspace: fixes p :: "int"   and X :: "'b topology" shows "trivial_group (relative_homology_group p X (topspace X))"
suminf_le_const: fixes f :: "nat \<Rightarrow> 'c"   and x :: "'c" assumes "summable f"   and "\<And>n. sum f {..<n} \<le> x" shows "suminf f \<le> x"
length_summarize_pdevs_list: fixes a :: "nat"   and b :: "nat \<Rightarrow> 'a list \<Rightarrow> bool"   and c :: "nat"   and d :: "'a pdevs list" shows "length (summarize_pdevs_list a b c d) = length d"
R_OK: fixes a :: "'a"   and b :: "'a" assumes "a \<in> S_a"   and "b \<in> S_b" shows "R a b"
WSE_7: shows "ms\<^sub>1 = msx @ mx # tl ms\<^sub>2"
connected_component_trans: fixes S :: "'a set"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "connected_component S x y"   and "connected_component S y z" shows "connected_component S x z"
Field_Un: fixes r :: "('a \<times> 'a) set"   and s :: "('a \<times> 'a) set" shows "Field (r \<union> s) = Field r \<union> Field s"
two_cube_boundary_is_boundary: fixes C :: "real \<times> real \<Rightarrow> real \<times> real" shows "boundary_chain (boundary C)"
1: shows "dims (dematricize rmodes (matricize rmodes T) (dims T)) = dims T"
p'_pos_iff: fixes a :: "real"   and v :: "real"   and x :: "real" shows "(0 < movement.p' a v x) = (if 0 < a then - v / a < x else if a < 0 then x < - v / a else 0 < v)"
PhiWhileOp_Monotone: fixes b :: "BExpr"   and \<Phi> :: "((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list) \<times> ((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list) \<times> (Location \<times> Location) set \<Rightarrow> bool" shows "Monotone (PhiWhileOp b \<Phi>)"
discovered_not_stack_imp_finished: fixes x :: "'v" assumes "x \<in> dom (discovered s)"   and "x \<notin> set (stack s)" shows "x \<in> dom (finished s)"
lsbis2_def: fixes B1 :: "'e set"   and B2 :: "'d set"   and s1 :: "'e \<Rightarrow> ('f, 'e, 'd) F1"   and s2 :: "'d \<Rightarrow> ('g, 'e, 'd) F2" shows "lsbis2 B1 B2 s1 s2 = \<Union> (snd ` {(R1, R2) |R1 R2. sbis B1 B2 s1 s2 R1 R2})"
assms_2: shows "\<lbrakk>M; map (\<lambda>d. \<langle>check: pair (t, s) \<doteq> pair d\<rangle>\<^sub>s\<^sub>t) Di @ map (\<lambda>d. \<forall>[]\<langle>\<or>\<noteq>: [(pair (t, s), pair d)]\<rangle>\<^sub>s\<^sub>t) (filter (\<lambda>d. d \<notin> set Di) D)\<rbrakk>\<^sub>d \<I>"
permutes_in_image: fixes p :: "'a \<Rightarrow> 'a"   and S :: "'a set"   and x :: "'a" assumes "p permutes S" shows "(p x \<in> S) = (x \<in> S)"
\<alpha>_inner: shows "inner_prod \<alpha> \<alpha> = 1"
path_components_of_maximal: fixes C :: "'c set"   and X :: "'c topology"   and S :: "'c set" assumes "C \<in> path_components_of X"   and "path_connectedin X S"   and "\<not> disjnt C S" shows "S \<subseteq> C"
coeff_one: fixes n :: "nat" shows "coeff P \<one>\<^bsub>P\<^esub> n = (if n = 0 then \<one> else \<zero>)"
consec_in_exclusive2: fixes i :: "nat_int"   and j :: "nat_int"   and n :: "nat" shows "consec i j \<and> n \<^bold>\<in> j \<longrightarrow> n \<^bold>\<notin> i"
single_inheritance: fixes A :: "qtname"   and B :: "qtname"   and G :: "prog"   and C :: "qtname" assumes "G\<turnstile>A\<prec>\<^sub>C1B"   and "G\<turnstile>A\<prec>\<^sub>C1C" shows "B = C"
inject_3: fixes x31 :: "('a, 'b, 'c, 'd) seqp"   and x32 :: "('a, 'b, 'c, 'd) seqp"   and y31 :: "('a, 'b, 'c, 'd) seqp"   and y32 :: "('a, 'b, 'c, 'd) seqp" shows "(x31 \<oplus> x32 = y31 \<oplus> y32) = (x31 = y31 \<and> x32 = y32)"
finite_sub_comp_iff_card_eq_IDirProd: fixes H :: "'a set"   and J :: "'a set" assumes "subgroup H G"   and "subgroup J G"   and "finite H"   and "finite J" shows "(card (IDirProd G H J) = card J * card H) = complementary H J"
vc_mono: fixes P' :: "(char list \<Rightarrow> int) \<Rightarrow> enat"   and P :: "(char list \<Rightarrow> int) \<Rightarrow> enat"   and C :: "acom" assumes "\<And>s. P' s \<le> P s"   and "vc C P" shows "vc C P'"
Deduct_with_False_def: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and eql :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and cnj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and imp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and all :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and exi :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and fls :: "'c"   and num :: "'b set"   and prv :: "'c \<Rightarrow> bool" shows "Deduct_with_False var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi fls num prv \<equiv> Syntax_with_Connectives_False var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi fls \<and> Deduct var trm fmla Var FvarsT substT Fvars subst num eql cnj imp all exi prv \<and> Deduct_with_False_axioms imp fls prv"
subspace_cotangent_space: fixes p :: "'a" shows "subspace (cotangent_space p)"
program_safe_reach_upto_def: fixes program_step :: "(nat \<Rightarrow> nat option) \<Rightarrow> 'b \<Rightarrow> 'b \<times> instr list \<Rightarrow> bool"   and n :: "nat"   and safe :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> unit \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list \<times> (nat \<Rightarrow> nat) \<times> (nat \<Rightarrow> bool option) \<Rightarrow> bool"   and c\<^sub>0 :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> unit \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list \<times> (nat \<Rightarrow> nat) \<times> (nat \<Rightarrow> bool option)" shows "program_safe_reach_upto program_step n safe c\<^sub>0 \<equiv> \<forall>k c l. k \<le> n \<longrightarrow> program_trace program_step c 0 k \<longrightarrow> c 0 = c\<^sub>0 \<longrightarrow> l \<le> k \<longrightarrow> safe (c l)"
swap_sym: fixes zs :: "'varSort"   and x :: "'var"   and y :: "'var"   and X :: "('d, 'e, 'varSort, 'var, 'f) qTerm set" shows "X #[x \<and> y]_zs = X #[y \<and> x]_zs"
fip_ind_to_f: fixes clds :: "(ctx \<times> cld) list" shows "\<forall>fs. fields_in_path clds fs \<longrightarrow> fields_in_path_f clds = fs"
l2x_def: shows "l2x = \<lceil>log 2 (real_of_int x)\<rceil>"
\<tau>red1r_Throw: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and h :: "'heap"   and a :: "'addr"   and xs :: "'addr val list"   and s' :: "'addr expr1 \<times> 'addr val list" shows "\<tau>red1gr uf P t h (Throw a, xs) s' = (s' = (Throw a, xs))"
neg_divide_le_eq: fixes c :: "'a"   and b :: "'a"   and a :: "'a" assumes "c < (0::'a)" shows "(b / c \<le> a) = (a * c \<le> b)"
ale: fixes n :: "int"   and m :: "int" shows "(n \<le> m) = (ant n \<le> ant m)"
surjective: fixes r :: "('b, 'c, 'd, 'e, 'f) functor_scheme" shows "r = \<lparr>om = r \<^bsub>\<o>\<^esub>, am = r \<^bsub>\<a>\<^esub>, \<dots> = functor.more r\<rparr>"
power_increasing_iff: assumes "1 < 2" shows "(2 ^ l \<le> 2 ^ m) = (l \<le> m)"
rb_aux_no_zero_red: fixes z :: "nat" assumes "is_pot_ord"   and "hom_grading dgrad"   and "is_regular_sequence fs" shows "snd (rb_aux (([], Koszul_syz_sigs fs, map Inr [0..<length fs]), z)) = z"
vec_index_of_poly_n: fixes i :: "nat"   and n :: "nat"   and p :: "'b poly" assumes "i < n" shows "vec_of_poly_n p n $ i = (if i < n - Suc (degree p) then 0::'b else coeff p (n - i - 1))"
woa_in_fs: shows "wrap_iso_arcs hom f \<in> iso_arcs hom ` A \<rightarrow> iso_arcs hom ` A"
simps_1: fixes A :: "'a mat" shows "swap_col_to_front A 0 = A"
ord_eq_le_eq_trans: fixes a :: "'g"   and b :: "'g"   and c :: "'g"   and d :: "'g" assumes "a = b"   and "b \<le> c"   and "c = d" shows "a \<le> d"
scaleR_vector_assoc: fixes c :: "real"   and A :: "(('d, 'f) vec, 'e) vec"   and x :: "('d, 'f) vec" shows "c *\<^sub>R (A *v x) = c *\<^sub>R A *v x"
D_inv: fixes r :: "('a \<times> 'a) set"   and \<tau>' :: "'a list"   and \<sigma> :: "'a list"   and \<tau> :: "'a list"   and \<sigma>' :: "'a list" assumes "trans r"   and "irrefl r"   and "(r|\<tau>'| -s r \<down>l \<sigma>, r|\<tau>|) \<in> mul_eq r"   and "(r|\<sigma>'| -s r \<down>l \<tau>, r|\<sigma>|) \<in> mul_eq r" shows "Decreasing_Diagrams.D r \<tau> \<sigma> \<sigma>' \<tau>'"