hcju/isabelleps huggingface dataset:
wp_under_wlp_Seq: fixes a :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and b :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" assumes "nearly_healthy (wlp a)"   and "healthy (wp b)"   and "nearly_healthy (wlp b)"   and "wp_under_wlp a"   and "wp_under_wlp b" shows "wp_under_wlp (a ;; b)"
image_gen_preserves_ncol: fixes A :: "'p"   and A' :: "'p"   and X :: "'p"   and Y :: "'p"   and B :: "'p"   and B' :: "'p"   and C :: "'p"   and C' :: "'p" assumes "A A' Reflect X Y"   and "B B' Reflect X Y"   and "C C' Reflect X Y"   and "\<not> Col A B C" shows "\<not> Col A' B' C'"
arcs_predecessor'': fixes a :: "nat"   and b :: "nat"   and i :: "nat"   and j :: "nat"   and xs :: "nat list" assumes "(a, b) \<in> set (arcs i j xs)"   and "a \<noteq> i" shows "\<exists>c. xs = [a] \<or> (\<exists>ys. xs = a # b # ys) \<or> (\<exists>ys. xs = ys @ [c, a] \<and> b = j) \<or> (\<exists>ys zs. xs = ys @ c # a # b # zs)"
reflect_poly_simps_4: fixes c :: "'a"   and p :: "'a poly" shows "reflect_poly (smult c p) = smult c (reflect_poly p)"
ordertype_eq_iff: fixes \<alpha> :: "V"   and r :: "('a \<times> 'a) set"   and A :: "'a set" assumes "Ord \<alpha>"   and "wf r"   and "small A"   and "total_on A r"   and "trans r" shows "(ordertype A r = \<alpha>) = (\<exists>f. bij_betw f A (elts \<alpha>) \<and> (\<forall>x\<in>A. \<forall>y\<in>A. (f x < f y) = ((x, y) \<in> r)))"
le_iff_sup: fixes x :: "'e multiset"   and y :: "'e multiset" shows "(x \<subseteq># y) = (x \<union># y = y)"
assms_5: fixes h :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'a"   and U :: "'x set"   and a :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'a" assumes "(h, U) \<in> set qs"   and "a \<in> cone (h, U)"   and "a \<in> ideal F" shows "a = 0"
MakeFtor: fixes F :: "('g, 'h, 'i, 'j, 'k, 'l, 'm) Functor_scheme" assumes "FunctorM F" shows "Functor (MakeFtor F)"
cs_induct: fixes P :: "(nat \<Rightarrow> 'n) \<Rightarrow> nat \<Rightarrow> bool"   and a0 :: "nat \<Rightarrow> 'n"   and a1 :: "nat" assumes "\<And>\<pi> n. (Ex (is_icdi n \<pi>) \<Longrightarrow> P \<pi> (The (is_icdi n \<pi>))) \<Longrightarrow> P \<pi> n" shows "P a0 a1"
not_nofail_inres: fixes S :: "'a nres"   and x :: "'a" assumes "\<not> nofail S" shows "inres S x"
ls_nds_rel_def_internal: fixes R :: "('a \<times> ('b, 'c) node_scheme) set" shows "ls_nds_rel R \<equiv> \<langle>R\<rangle>list_set_rel \<inter> {(uu_, s). inj_on name s}"
lucas_lehmer_sufficient: fixes p :: "nat" assumes "prime p"   and "odd p"   and "2 ^ p - 1 dvd gen_lucas_lehmer_sequence 4 (p - 2)" shows "prime (2 ^ p - 1)"
folding': fixes C :: "'a set" assumes "C \<in> f \<turnstile> folding.\<C>" shows "\<exists>!D. D \<in> folding.\<C> - f \<turnstile> folding.\<C> \<and> f ` D = C"
table_of_map_SomeI: fixes t :: "('d \<times> 'c) list"   and k :: "'d"   and x :: "'c"   and f :: "'c \<Rightarrow> 'e" assumes "table_of t k = Some x" shows "table_of (map (\<lambda>(k, x). (k, f x)) t) k = Some (f x)"
le_add_diff_inverse: fixes plus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and minus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and less_eq :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and less :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and zero :: "'k"   and times :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and one :: "'k"   and b :: "'k"   and a :: "'k" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "less_eq b a" shows "plus b (minus a b) = a"
simps: fixes S :: "'a literal multiset \<Rightarrow> 'a literal multiset"   and a1 :: "'a literal multiset list"   and a2 :: "'a literal multiset"   and a3 :: "'a multiset list"   and a4 :: "'a list"   and a5 :: "'s"   and a6 :: "'a literal multiset" shows "ord_resolve_rename S a1 a2 a3 a4 a5 a6 = (\<exists>CAs n AAs As DA \<rho> \<rho>s \<sigma> E. a1 = CAs \<and> a2 = DA \<and> a3 = AAs \<and> a4 = As \<and> a5 = \<sigma> \<and> a6 = E \<and> length CAs = n \<and> length AAs = n \<and> length As = n \<and> (\<forall>i<n. poss (AAs ! i) \<subseteq># CAs ! i) \<and> negs (mset As) \<subseteq># DA \<and> \<rho> = hd (renamings_apart (DA # CAs)) \<and> \<rho>s = tl (renamings_apart (DA # CAs)) \<and> ord_resolve S (CAs \<cdot>\<cdot>cl \<rho>s) (DA \<cdot> \<rho>) (AAs \<cdot>\<cdot>aml \<rho>s) (As \<cdot>al \<rho>) \<sigma> E)"
YFtorNTApp1: fixes X :: "'d"   and C :: "('d, 'e, 'f) LSCategory_scheme"   and f :: "'e" assumes "X \<in> obj\<^bsub>NTCatDom (YFtorNT C f)\<^esub>" shows "YFtorNT C f $$ X = Hom\<^bsub>C\<^esub>[X,f]"
1: fixes X :: "nat \<Rightarrow> rat"   and Y :: "nat \<Rightarrow> rat" assumes "realrel X Y"   and "\<exists>r>0. \<exists>k. \<forall>n\<ge>k. r < X n" shows "\<exists>r>0. \<exists>k. \<forall>n\<ge>k. r < Y n"
fds_abs_converges_shift: fixes c :: "'b"   and f :: "'b fds"   and z :: "'b" shows "fds_abs_converges (fds_shift c f) z = fds_abs_converges f (z - c)"
InvariantWatchCharacterizationAfterApplyUnitPropagate: fixes state :: "State" assumes "InvariantConsistent (getM state)"   and "InvariantUniq (getM state)"   and "InvariantWatchListsContainOnlyClausesFromF (getWatchList state) (getF state)"   and "InvariantWatchListsUniq (getWatchList state)"   and "InvariantWatchListsCharacterization (getWatchList state) (getWatch1 state) (getWatch2 state)"   and "InvariantWatchesEl (getF state) (getWatch1 state) (getWatch2 state)"   and "InvariantWatchesDiffer (getF state) (getWatch1 state) (getWatch2 state)"   and "InvariantWatchCharacterization (getF state) (getWatch1 state) (getWatch2 state) (getM state)"   and "InvariantQCharacterization (getConflictFlag state) (getQ state) (getF state) (getM state)"   and "getQ state \<noteq> []"   and "\<not> getConflictFlag state" shows "let state' = applyUnitPropagate state in InvariantWatchCharacterization (getF state') (getWatch1 state') (getWatch2 state') (getM state')"
pred_nat_def: shows "pred_nat = {(m, n). n = Suc m}"
otherE: fixes P :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and I :: "'f set"   and \<sigma> :: "'f \<Rightarrow> 'e"   and \<sigma>' :: "'f \<Rightarrow> 'e"   and R :: "('f \<Rightarrow> 'e) \<Rightarrow> ('f \<Rightarrow> 'e) \<Rightarrow> bool" assumes "other P I \<sigma> \<sigma>'"   and "\<lbrakk>\<forall>i\<in>I. \<sigma>' i = \<sigma> i; \<forall>j. j \<notin> I \<longrightarrow> P (\<sigma> j) (\<sigma>' j)\<rbrakk> \<Longrightarrow> R \<sigma> \<sigma>'" shows "R \<sigma> \<sigma>'"
boundedI: fixes b :: "'c"   and a :: "'c"   and c :: "'c" assumes "b \<le> a"   and "c \<le> a" shows "max b c \<le> a"
word_less_rbl: fixes x :: "'a word"   and y :: "'a word" shows "(x < y) = rev_bl_order False (rev (to_bl x)) (rev (to_bl y))"
remove_1_mset_id_iff_notin: fixes M :: "'a multiset"   and a :: "'a" shows "(remove1_mset a M = M) = (a \<notin># M)"
r_lazyifzero_recfn: fixes n :: "nat"   and j\<^sub>1 :: "nat"   and j\<^sub>2 :: "nat" shows "recfn (Suc (Suc n)) (r_lazyifzero n j\<^sub>1 j\<^sub>2)"
numeral_power_less_nat_cancel_iff: fixes x :: "num"   and n :: "nat"   and a :: "int" shows "(numeral x ^ n < nat a) = (numeral x ^ n < a)"
w1: shows "M.sumlist (map (\<lambda>i. h i \<cdot>\<^sub>v gs ! i) [0..<k]) \<in> carrier_vec n"
inv_translationI4: fixes r :: "nat set" assumes "finite r"   and "\<And>x. x < card r \<Longrightarrow> x \<in> r" shows "r = {0..<card r}"
coeff_nonzero_dim_vec_non_zero_2: fixes v :: "rat vec"   and i :: "nat" assumes "Abstract_Linear_Poly.coeff (vec_to_lpoly v) i \<noteq> 0" shows "i < dim_vec v"
statement69: fixes a :: "hyp2"   and b :: "hyp2"   and a' :: "hyp2"   and b' :: "hyp2"   and c :: "hyp2"   and c' :: "hyp2" assumes "a b \<congruent>\<^sub>K a' b'"   and "b c \<congruent>\<^sub>K b' c'"   and "a c \<congruent>\<^sub>K a' c'" shows "\<exists>J. is_K2_isometry J \<and> hyp2_cltn2 a J = a' \<and> hyp2_cltn2 b J = b' \<and> hyp2_cltn2 c J = c'"
wp_eval_20: fixes f :: "'b \<Rightarrow> bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and p :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and P :: "'a \<Rightarrow> real" shows "wlp (SetPC f p) P = (\<lambda>s. \<Sum>a\<in>supp (p s). p s a * wlp (f a) P s)"
analz_mpair: fixes X :: "msg"   and Y :: "msg"   and H :: "msg set" shows "analz (insert \<lbrace>X, Y\<rbrace> H) = insert \<lbrace>X, Y\<rbrace> (analz ({X, Y} \<union> H))"
V_valid_def: fixes v :: "'a V_rec" shows "V_valid v \<equiv> VN_disjoint v \<and> VC_disjoint v \<and> NC_disjoint v"
cod_comp: fixes g :: "'d"   and f :: "'d" assumes "D.seq g f" shows "D.cod (g \<cdot>\<^sub>D f) = D.cod g"
ineq_model_ground_subst: fixes F :: "(('b, 'a) Term.term \<times> ('b, 'a) Term.term) list"   and X :: "'a list"   and \<delta> :: "'a \<Rightarrow> ('b, 'a) Term.term"   and \<theta> :: "'a \<Rightarrow> ('b, 'a) Term.term" assumes "fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s F - set X \<subseteq> subst_domain \<delta>"   and "ground (subst_range \<delta>)"   and "ineq_model \<delta> X F" shows "ineq_model (\<delta> \<circ>\<^sub>s \<theta>) X F"
add_less_cancel_left: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and c :: "'c"   and a :: "'c"   and b :: "'c" assumes "class.ordered_ab_semigroup_add_imp_le plus minus less_eq less" shows "less (plus c a) (plus c b) = less a b"
permutation_delete_expand: fixes p :: "nat \<Rightarrow> nat"   and i :: "nat" shows "permutation_delete p i = (\<lambda>ia. if p (if ia < i then ia else Suc ia) < p i then p (if ia < i then ia else Suc ia) else p (if ia < i then ia else Suc ia) - Suc 0)"
nat_pow_Suc2: fixes x :: "'a"   and n :: "nat" assumes "x \<in> carrier R" shows "x [^] Suc n = x \<otimes> x [^] n"
proj_active_nth: fixes i :: "nat"   and t :: "cnf llist"   and c :: "'id" assumes "enat (Suc i) < llength t"   and "\<parallel>c\<parallel>\<^bsub>lnth t i\<^esub>" shows "lnth (\<pi>\<^bsub>c\<^esub>t) (the_enat \<langle>c #\<^bsub>enat i\<^esub>t\<rangle>) = \<sigma>\<^bsub>c\<^esub>lnth t i"
Nset_un: fixes n :: "nat" shows "{j. j \<le> Suc n} = {j. j \<le> n} \<union> {Suc n}"
initiali_def: fixes i :: "'d"   and OI :: "(('d \<Rightarrow> 'e) \<times> 'f) set"   and CI :: "('e \<times> 'f) set" shows "initiali i OI CI \<equiv> {(\<sigma> i, p) |\<sigma> p. (\<sigma>, p) \<in> OI} = CI"
choose_subspace_of_subspace: fixes n :: "nat"   and S :: "'c set" assumes "n \<le> dim S" obtains T :: "'c set" where "subspace T" and "T \<subseteq> span S" and "dim T = n"
ascP_Cons: fixes x :: "'c" shows "ascP ((#) x)"
not_empty: shows "space (P i (restrict x {0..<i})) \<noteq> {}"
preserves_dom: fixes f :: "'a" assumes "B.arr f" shows "B.dom (B.map f) = B.map (B.dom f)"
mul_ci: fixes l :: "'a list"   and P :: "pol"   and Q :: "pol" assumes "in_carrier l" shows "Ipol l (P \<langle>*\<rangle> Q) = Ipol l P \<otimes> Ipol l Q"
is_UP_monomE_2: fixes f :: "nat \<Rightarrow> nat \<Rightarrow> int" assumes "is_UP_monom f" shows "f = monom (UP Zp) (f (deg Zp f)) (deg Zp f)"
P: shows "pre @ [Broadcast (r, Store e2 mo r)] prefix of k"
prob_space_K: fixes x :: "'a" assumes "x \<in> space M" shows "prob_space (K x)"
intros_1: fixes i :: "nat" shows "([],  \<LM> At i  \<RM> \<Rightarrow>* \<LM> At i  \<RM>) \<in> Ax"
bags_union: fixes G :: "('e, 'f) Graph_scheme"   and T :: "('g, 'h) Graph_scheme"   and bag :: "'g \<Rightarrow> 'e set" assumes "TreeDecomposition G T bag" shows "\<Union> {bag t |t. t \<in> V\<^bsub>T\<^esub>} = V\<^bsub>G\<^esub>"
nMTeqSet: fixes p :: "'a list"   and s :: "'a list"   and C :: "'a \<Rightarrow> 'b \<Rightarrow> 'c option" assumes "set p = set s" shows "none_MT_rules C p = none_MT_rules C s"
dimacs_to_var_def: fixes v :: "nat" shows "dimacs_to_var v \<equiv> if v < 1 + n_ops * h then Operator ((v - 1) mod h) ((v - 1) div h) else let k = v - 1 - n_ops * h in State (k mod h) (k div h)"
Tree\<^sub>\<alpha>_free_5: fixes t1\<^sub>\<alpha> :: "('a, 'b, 'c, 'd) Tree\<^sub>\<alpha>"   and f :: "'d"   and \<alpha> :: "'c"   and t2\<^sub>\<alpha> :: "('a, 'b, 'c, 'd) Tree\<^sub>\<alpha>" shows "Not\<^sub>\<alpha> t1\<^sub>\<alpha> \<noteq> Act\<^sub>\<alpha> f \<alpha> t2\<^sub>\<alpha>"
fin: fixes x :: "'a" assumes "x \<in> topspace X" shows "\<exists>V. openin X V \<and> x \<in> V \<and> finite {i \<in> I. T i \<inter> V \<noteq> {}}"
Above_Field: fixes r :: "('a \<times> 'a) set"   and A :: "'a set" shows "Above r A \<subseteq> Field r"
wf_select_method_fun: fixes wf_md :: "(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 \<Rightarrow> char list \<Rightarrow> char list \<times> ty list \<times> ty \<times> char list list \<times> expr \<Rightarrow> bool"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and C :: "char list"   and Cs :: "char list list"   and M :: "char list"   and mthd :: "ty list \<times> ty \<times> char list list \<times> expr"   and Cs' :: "char list list"   and mthd' :: "ty list \<times> ty \<times> char list list \<times> expr"   and Cs'' :: "char list list" assumes "wf_prog wf_md P"   and "P \<turnstile> (C,Cs) selects M = mthd via Cs'"   and "P \<turnstile> (C,Cs) selects M = mthd' via Cs''" shows "mthd = mthd' \<and> Cs' = Cs''"
subring_props_7: fixes h1 :: "'a"   and h2 :: "'a" assumes "h1 \<in> K"   and "h2 \<in> K" shows "h1 \<oplus> h2 \<in> K"
subset: fixes R :: "('h, 'i) ring_scheme"   and N :: "'j set"   and M :: "('h, 'j, 'k) module_scheme" assumes "LinearCombinations.submodule R N M" shows "N \<subseteq> carrier M"
succ_inv_5: fixes s :: "('i, 'a) state"   and s' :: "('i, 'a) state"   and v :: "nat \<Rightarrow> 'a" assumes "\<triangle> (\<T> s) \<and> \<diamond> s \<and> \<Turnstile>\<^sub>n\<^sub>o\<^sub>l\<^sub>h\<^sub>s s \<and> \<nabla> s \<and> s \<succ>\<^sub>x s' \<and> \<B>\<^sub>i s' = \<B>\<^sub>i s \<and> \<U>\<^sub>c s' = \<U>\<^sub>c s" shows "v \<Turnstile>\<^sub>t \<T> s = v \<Turnstile>\<^sub>t \<T> s'"
fpxs_of_fls_mult: fixes f :: "'a fls"   and g :: "'a fls" shows "fpxs_of_fls (f * g) = fpxs_of_fls f * fpxs_of_fls g"
assms: shows "\<forall>\<^sub>F n in sequentially. real (card (A \<inter> {..<n})) \<le> l * real n"
iprev_le_mono_rev: fixes a :: "nat"   and I :: "nat set"   and b :: "nat" assumes "iprev a I \<le> iprev b I"   and "a \<in> I"   and "b \<in> I"   and "iMin I \<noteq> iprev b I" shows "a \<le> b"
map_converter_id_move_right: fixes f :: "'h \<Rightarrow> 'l"   and g :: "'m \<Rightarrow> 'i"   and f' :: "'l \<Rightarrow> 'j"   and g' :: "'k \<Rightarrow> 'm" shows "map_converter f g f' g' 1\<^sub>C = map_converter id id (f' \<circ> f) (g \<circ> g') 1\<^sub>C"
diff_divide_distrib: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and one :: "'c"   and times :: "'c \<Rightarrow> '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 a :: "'c"   and b :: "'c"   and c :: "'c" assumes "class.division_ring divide inverse one times plus zero minus uminus" shows "divide (minus a b) c = minus (divide a c) (divide b c)"
simple_function_iff_borel_measurable: fixes M :: "'b measure"   and f :: "'b \<Rightarrow> 'c" shows "simple_function M f = (finite (f ` space M) \<and> f \<in> borel_measurable M)"
cases: fixes r :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and a :: "'d" assumes "Domainp r a" obtains aa :: "'d" and b :: "'e" where "a = aa" and "r aa b"
ground_SCF: fixes t :: "('f, 'b) Term.term" shows "ground (SCF t) = ground t"
secrecy_lemma: fixes A :: "agent"   and B :: "agent"   and evs :: "event list"   and K :: "nat"   and na :: "msg"   and nb :: "msg" assumes "A \<notin> bad"   and "B \<notin> bad"   and "evs \<in> yahalom" shows "Says Server A \<lbrace>Crypt (shrK A) \<lbrace>Agent B, Key K, na, nb\<rbrace>, Crypt (shrK B) \<lbrace>Agent A, Key K\<rbrace>\<rbrace> \<in> set evs \<longrightarrow> Notes Spy \<lbrace>na, nb, Key K\<rbrace> \<notin> set evs \<longrightarrow> Key K \<notin> analz (knows Spy evs)"
1: shows "rawpsubst (subst \<phi> (Var u) x) (zip (map Var us) (map snd txs)) = rawpsubst \<phi> (zip (map Var us) (map snd txs) @ [(Var u, x)])"
bounded_by_mono: fixes a :: "real"   and P :: "'c \<Rightarrow> real"   and b :: "real" assumes "bounded_by a P"   and "a \<le> b" shows "bounded_by b P"
length_takefill: fixes fill :: "'b"   and n :: "nat"   and l :: "'b list" shows "length (takefill fill n l) = n"
le_eclose: fixes X :: "hf" shows "X \<le> eclose X"
linear_orderable_6_implies_3: fixes x :: "'a" assumes "x \<sqinter> top \<le> - top \<and> top \<squnion> id top = top" shows "x \<sqinter> top \<sqinter> (x \<sqinter> top) \<le> x \<sqinter> top \<and> x \<sqinter> top \<sqinter> id (x \<sqinter> top) = bot \<and> x \<sqinter> top \<squnion> id (x \<sqinter> top) = - top"
CFG_SSA_wf_def: fixes \<alpha>e :: "'d \<Rightarrow> ('e \<times> 'f \<times> 'e) set"   and \<alpha>n :: "'d \<Rightarrow> 'e list"   and invar :: "'d \<Rightarrow> bool"   and inEdges' :: "'d \<Rightarrow> 'e \<Rightarrow> ('e \<times> 'f) list"   and Entry :: "'d \<Rightarrow> 'e"   and defs :: "'d \<Rightarrow> 'e \<Rightarrow> 'g set"   and "uses" :: "'d \<Rightarrow> 'e \<Rightarrow> 'g set"   and phis :: "'d \<Rightarrow> 'e \<times> 'g \<Rightarrow> 'g list option" shows "CFG_SSA_wf \<alpha>e \<alpha>n invar inEdges' Entry defs uses phis \<equiv> CFG_SSA \<alpha>e \<alpha>n invar inEdges' Entry defs uses phis \<and> CFG_SSA_wf_axioms \<alpha>n invar inEdges' Entry defs uses phis"
aincc: shows "a \<in> connected_component_set (- S) a"
3: fixes xs :: "nat list" assumes "xs \<noteq> []" shows "hd xs + (sum_list xs - hd xs) = sum_list xs"
runit_simps_2: fixes f :: "'b" assumes "D.ide f" shows "src\<^sub>D \<r>\<^sub>D[f] = src\<^sub>D f"
non_empty_cycle_root_msc_cycle: fixes r :: "'a"   and x :: "'a" assumes "non_empty_cycle_root r x" shows "cycle x"
basic_monos_7: fixes A :: "'b set"   and B :: "'b set"   and x :: "'b" assumes "A \<subseteq> B" shows "x \<in> A \<longrightarrow> x \<in> B"
simps_1: fixes x21 :: "'g"   and x22 :: "'g list"   and y21 :: "'g"   and y22 :: "'g list" shows "(x21 # x22 = y21 # y22) = (x21 = y21 \<and> x22 = y22)"
extract_splitres_a_def: fixes r :: "('a, 'b) splitres" shows "extract_splitres_a r \<equiv> case Rep_splitres r of (l, a, s) \<Rightarrow> a"
weak_map_of_SomeI: fixes k :: "'a"   and x :: "'b"   and l :: "('a \<times> 'b) list" assumes "(k, x) \<in> set l" shows "\<exists>x. map_of l k = Some x"
DBInit_def: shows "DBInit \<equiv> BInit inp q1 mid \<and> BInit mid q2 out"
SDG_to_CFG_set_def: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and Exit :: "'b"   and Def :: "'b \<Rightarrow> 'c set"   and Use :: "'b \<Rightarrow> 'c set"   and ParamDefs :: "'b \<Rightarrow> 'c list"   and ParamUses :: "'b \<Rightarrow> 'c set list"   and S :: "'b SDG_node set" assumes "SDG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit Def Use ParamDefs ParamUses" shows "SDG.SDG_to_CFG_set S \<equiv> {m. CFG_node m \<in> S}"
Un_absorb: fixes A :: "'l set" shows "A \<union> A = A"
E: shows "Crypt (sesK (c * f)) (Number 0) \<in> synth (analz (A \<union> spies evsFR5))"
if_: fixes mds :: "Mode \<Rightarrow> 'Var set"   and annos :: "'Var ModeUpd list"   and c\<^sub>1 :: "('Var, 'AExp, 'BExp) Stmt"   and mds'' :: "Mode \<Rightarrow> 'Var set"   and c\<^sub>2 :: "('Var, 'AExp, 'BExp) Stmt"   and e :: "'BExp" assumes "\<turnstile> mds \<oplus> annos { c\<^sub>1 } mds''"   and "\<turnstile> mds \<oplus> annos { c\<^sub>2 } mds''"   and "bexp_vars e \<inter> mds GuarNoReadOrWrite = {}"   and "\<forall>v. \<C>_vars v \<inter> bexp_vars e \<noteq> {} \<longrightarrow> v \<notin> mds GuarNoReadOrWrite" shows "\<turnstile> mds { Stmt.If e c\<^sub>1 c\<^sub>2 \<otimes> annos } mds''"
assms_3: fixes c :: "'d"   and pred :: "'b" shows "P c (Pred pred)"
marriage_HV: fixes I :: "'a set"   and A :: "'a \<Rightarrow> 'b set" assumes "finite I"   and "\<forall>i\<in>I. finite (A i)"   and "\<forall>J\<subseteq>I. card J \<le> card (\<Union> (A ` J))" shows "\<exists>R. (\<forall>i\<in>I. R i \<in> A i) \<and> inj_on R I"
of_nat_add: fixes one :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and m :: "nat"   and n :: "nat" assumes "class.semiring_1 one times plus zero" shows "semiring_1.of_nat one plus zero (m + n) = plus (semiring_1.of_nat one plus zero m) (semiring_1.of_nat one plus zero n)"
vardicts_finite: fixes ss :: "(integer \<times> edge list) iarray"   and p :: "nat \<times> edgeIndex \<times> procArg list \<times> varDecl list" shows "finite (Collect (vardict_inv ss p))"
prems_2: shows "n -a # as\<rightarrow>* n'"
measurable_count_space: fixes f :: "'d \<Rightarrow> 'e"   and A :: "'d set" shows "f \<in> count_space A \<rightarrow>\<^sub>M count_space UNIV"
Pred\<^sub>\<alpha>_eqvt: fixes p :: "perm"   and f :: "'d"   and \<phi> :: "'b" shows "p \<bullet> Pred\<^sub>\<alpha> f \<phi> = Pred\<^sub>\<alpha> (p \<bullet> f) (p \<bullet> \<phi>)"
it_paths_prefix: fixes "is" :: "nat list"   and t :: "'a rose_tree"   and is' :: "nat list" assumes "is \<in> it_paths t"   and "prefix is' is" shows "is' \<in> it_paths t"
sum_incr2: fixes f :: "int \<times> nat \<Rightarrow> 'a"   and A :: "(int \<times> nat) set" shows "sum f (incr2 ` A) = (\<Sum>xy\<in>A. f (fst xy, snd xy + 1))"
basis_wf_Cons: fixes f :: "real \<Rightarrow> real"   and basis :: "(real \<Rightarrow> real) list" shows "basis_wf (f # basis) = (filterlim f at_top at_top \<and> (\<forall>g\<in>set basis. (\<lambda>x. ln (g x)) \<in> o(\<lambda>x. ln (f x))) \<and> basis_wf basis)"
NewHigh: fixes x :: "Var"   and G :: "((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool) set"   and C :: "Class" assumes "CONTEXT x = high" shows "G \<rhd> New x C : HighSec"
map_insort_is_insort_key: fixes m :: "('d, 'c) fmap"   and k :: "'d"   and v :: "'c"   and xs :: "'d list" assumes "m $$ k = None" shows "map (\<lambda>k'. (k', m(k $$:= v) $$! k')) (insort k xs) = insort_key fst (k, v) (map (\<lambda>k'. (k', m(k $$:= v) $$! k')) xs)"
cSup_add: fixes S :: "real set"   and b :: "real"   and c :: "real" assumes "S \<noteq> {}"   and "\<And>x. x \<in> S \<Longrightarrow> x \<le> b" shows "Sup S + c = Sup {x + c |x. x \<in> S}"
idcarr: shows "1\<^sub>m nr \<in> carrier_mat nr nr"
map: fixes f1 :: "'e \<Rightarrow> 'c"   and f2 :: "'f \<Rightarrow> 'd"   and x :: "'f" shows "map_phantom f1 f2 (Phantom('e) x) = Phantom('c) (f2 x)"
ncg_seqr: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "\<nu> x = \<nu> y" shows "\<nu> (x \<cdot> z) = \<nu> (y \<cdot> z)"
Abs_unat_inverse: fixes y :: "nat option" assumes "y \<in> UNIV" shows "Rep_unat (Abs_unat y) = y"
compact_imp_bounded: fixes U :: "'e set" assumes "compact U" shows "bounded U"
c_refusals: fixes X :: "('a \<times> 'b) set"   and step :: "'c \<Rightarrow> 'a \<Rightarrow> 'c"   and out :: "'c \<Rightarrow> 'a \<Rightarrow> 'b"   and s\<^sub>0 :: "'c"   and xps :: "('a \<times> 'b) list" shows "(X \<in> refusals (c_process step out s\<^sub>0) xps) = ((xps, X) \<in> c_failures step out s\<^sub>0)"
right_neutral: fixes a :: "'b" shows "a \<squnion> \<bottom> = a"
matrix_continuous_on_def: fixes T :: "real set"   and A :: "real \<Rightarrow> (('b, 'c) vec, 'd) vec" shows "matrix_continuous_on T A = (\<forall>t\<in>T. \<forall>\<epsilon>>0. \<exists>\<delta>>0. \<forall>\<tau>\<in>T. \<bar>\<tau> - t\<bar> < \<delta> \<longrightarrow> \<parallel>A \<tau> - A t\<parallel>\<^sub>o\<^sub>p \<le> \<epsilon>)"
order_eq: shows "\<forall>x\<in>s. order x p = order x q"