hcju/isabelleps huggingface dataset:
closed: fixes A :: "'b set" assumes "closed_csubspace A" shows "closed A"
intros_3: fixes g :: "'f \<Rightarrow> ('e, 'f) step"   and s :: "'f"   and a :: "'e"   and s' :: "'f" assumes "g s = Yield a s'"   and "s' \<in> terminates_on g" shows "s \<in> terminates_on g"
nsets_image_funcset: fixes g :: "'b \<Rightarrow> 'c"   and S :: "'b set"   and T :: "'c set"   and k :: "nat" assumes "g \<in> S \<rightarrow> T"   and "inj_on g S" shows "(`) g \<in> [S]\<^bsup>k\<^esup> \<rightarrow> [T]\<^bsup>k\<^esup>"
set_intros_1: fixes x31 :: "'c"   and x32 :: "'d" shows "x31 \<in> set1_step (Yield x31 x32)"
len_num1: fixes uu :: "1 itself" shows "len_of uu = 1"
vfsequence_vrange_vconsI: fixes xs :: "V"   and X :: "V"   and x :: "V" assumes "vfsequence xs"   and "\<R>\<^sub>\<circ> xs \<subseteq>\<^sub>\<circ> X"   and "x \<in>\<^sub>\<circ> X" shows "\<R>\<^sub>\<circ> (xs #\<^sub>\<circ> x) \<subseteq>\<^sub>\<circ> X"
assms_2: shows "wf'\<^sub>s\<^sub>s\<^sub>t {} (unlabel A) \<and> fv\<^sub>l\<^sub>s\<^sub>s\<^sub>t A \<inter> bvars\<^sub>l\<^sub>s\<^sub>s\<^sub>t A = {}"
assms: fixes a :: "nat"   and b :: "nat" shows "P ((a + 1) * (1 + b))"
deterministic: fixes a :: "'a"   and a' :: "'a" assumes "valid_edge a"   and "valid_edge a'"   and "sourcenode a = sourcenode a'"   and "targetnode a \<noteq> targetnode a'" shows "\<exists>Q Q'. kind a = (Q)\<^sub>\<surd> \<and> kind a' = (Q')\<^sub>\<surd> \<and> (\<forall>s. (Q s \<longrightarrow> \<not> Q' s) \<and> (Q' s \<longrightarrow> \<not> Q s))"
subst_apply_pair_pair: fixes \<I> :: "'e \<Rightarrow> ('c, 'd) Term.term"   and t :: "('c, 'e) Term.term"   and s :: "('c, 'e) Term.term" shows "(t, s) \<cdot>\<^sub>p \<I> = (t \<cdot> \<I>, s \<cdot> \<I>)"
nn_integral_disjoint_pair: fixes f :: "'b \<Rightarrow> ennreal"   and M :: "'b measure"   and B :: "'b set"   and C :: "'b set" assumes "f \<in> borel_measurable M"   and "B \<in> sets M"   and "C \<in> sets M"   and "B \<inter> C = {}" shows "set_nn_integral M (B \<union> C) f = set_nn_integral M B f + set_nn_integral M C f"
thelubE: fixes below :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and S :: "nat \<Rightarrow> 'b"   and l :: "'b" assumes "class.cpo below"   and "po.chain below S"   and "po.lub below (range S) = l" shows "po.is_lub below (range S) l"
inv_fn_o_fn_is_id: fixes f :: "'b \<Rightarrow> 'b"   and n :: "nat" assumes "bij f" shows "inv f ^^ n \<circ> f ^^ n = (\<lambda>x. x)"
cases: fixes f :: "'a \<Rightarrow> 'b"   and a1 :: "'a set"   and a2 :: "'a formula"   and a3 :: "'b" assumes "hom_graph f a1 a2 a3" obtains "a1 = {}" and "a2 = \<bottom>" and "a3 = \<bottom>"   | "a1 = {}" and "a2 = \<top>" and "a3 = \<top>"   | i :: "'a" and S :: "'a set" and x :: "'a formula" and a :: "'b" and y :: "'a formula" and b :: "'b" where "a1 = insert i S" and "a2 = ifte (var i) x y" and "a3 = ifte (f i) a b" and "i \<notin> S" and "hom_graph f S x a" and "hom_graph f S y b"
intro: fixes L :: "'b \<Rightarrow> 'a list set"   and init :: "'a rexp \<Rightarrow> 'b"   and delta :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and final :: "'b \<Rightarrow> bool" assumes "\<And>r. L (init r) = lang r"   and "\<And>a s. L (delta a s) = Deriv a (L s)"   and "\<And>s. final s = ([] \<in> L s)" shows "rexp_DA init delta final L"
induct: fixes P :: "'b tree23 \<Rightarrow> 'b \<Rightarrow> 'b tree23 \<Rightarrow> 'b \<Rightarrow> 'b upD \<Rightarrow> bool"   and a0 :: "'b tree23"   and a1 :: "'b"   and a2 :: "'b tree23"   and a3 :: "'b"   and a4 :: "'b upD" assumes "\<And>l a m b r. P l a m b (TD r)"   and "\<And>t1 a t2 b t3 c t4. P t1 a \<langle>t2, b, t3\<rangle> c (UF t4)"   and "\<And>t1 a t2 b t3 c t4 d t5. P t1 a \<langle>t2, b, t3, c, t4\<rangle> d (UF t5)"   and "\<And>a b d v. P a b \<langle>\<rangle> d (UF v)" shows "P a0 a1 a2 a3 a4"
simps_1: fixes f\<^sub>1 :: "'c val \<Rightarrow> 'c val"   and r\<^sub>2 :: "'b rexp"   and f\<^sub>2 :: "'c val \<Rightarrow> 'c val" shows "simp_Times (One, f\<^sub>1) (r\<^sub>2, f\<^sub>2) = (r\<^sub>2, F_Times1 f\<^sub>1 f\<^sub>2)"
compl_fam_def: fixes S :: "'d \<Rightarrow> 'a set"   and I :: "'d set" shows "compl_fam S I = (\<forall>i\<in>I. complementary (S i) (IDirProds G S (I - {i})))"
nonconsequence_remainder: fixes A :: "'a set"   and \<phi> :: "'a" shows "(A .\<bottom>. \<phi> = {A}) = (\<not> A \<turnstile> \<phi>)"
subseqs_set_subset: fixes ys :: "'d list"   and xs :: "'d list" assumes "ys \<in> set (subseqs xs)" shows "set ys \<subseteq> set xs"
convert_def: fixes fml :: "'b fml" shows "convert fml = (let ps = remdups (poly_list fml) in (map_fml (index_of ps) fml, ps))"
l2_inv7I: fixes x :: "l2_state" assumes "extr (bad x) (ik x) (chan x) \<subseteq> synth (analz generators)" shows "x \<in> l2_inv7"
dvd_smult_cancel: fixes p :: "'a poly"   and a :: "'a"   and q :: "'a poly" assumes "p dvd smult a q"   and "a \<noteq> (0::'a)" shows "p dvd q"
lookup_fresh: fixes X :: "W.tvar"   and \<theta> :: "(W.tvar \<times> ty) list" assumes "X \<sharp> \<theta>" shows "lookup \<theta> X = TVar X"
isDERIV_max_VarI: fixes fa :: "floatarith"   and i :: "nat"   and j :: "nat"   and xs :: "real list" assumes "max_Var_floatarith fa \<le> i"   and "isDERIV j fa xs" shows "isDERIV i fa xs"
divides_ff_coeff: fixes p :: "'b fract poly"   and n :: "'b"   and i :: "nat" assumes "set (coeffs p) \<subseteq> range to_fract"   and "divides_ff (to_fract n) (coeff p i)" shows "\<exists>m. coeff p i = to_fract n * to_fract m"
sinvar_monoI: fixes sinvar :: "'b graph \<Rightarrow> ('b \<Rightarrow> 'c) \<Rightarrow> bool" assumes "SecurityInvariant_preliminaries sinvar" shows "SecurityInvariant_withOffendingFlows.sinvar_mono sinvar"
invh_baliR: fixes l :: "(('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree"   and r :: "(('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree"   and a :: "'c \<times> 'd" assumes "invh l"   and "invh r"   and "bheight l = bheight r" shows "invh (baliR l a r)"
mk_linear_orders_total_on_decisive_on: fixes A :: "'a set"   and f :: "'a set \<Rightarrow> 'a set"   and r :: "'a list"   and C :: "'a set" assumes "f_range_on A f"   and "r \<in> mk_linear_orders f C A"   and "A \<subseteq> set r"   and "iia_on A f" shows "decisive_on A f"
simps_3: fixes \<Gamma> :: "char list set"   and e\<^sub>1 :: "expr"   and e\<^sub>2 :: "expr"   and s\<^sub>1 :: "stmt"   and s\<^sub>2 :: "stmt" shows "wfs \<Gamma> (IfEq e\<^sub>1 e\<^sub>2 s\<^sub>1 s\<^sub>2) = (wfe \<Gamma> e\<^sub>1 \<and> wfe \<Gamma> e\<^sub>2 \<and> wfs \<Gamma> s\<^sub>1 \<and> wfs \<Gamma> s\<^sub>2)"
tree\<^sub>i_bal: fixes u :: "'a up\<^sub>i" assumes "bal_up\<^sub>i u" shows "bal (tree\<^sub>i u)"
iterate_size_abort_correct: fixes it :: "(nat \<Rightarrow> bool) \<Rightarrow> ('c \<Rightarrow> nat \<Rightarrow> nat) \<Rightarrow> nat \<Rightarrow> nat"   and S0 :: "'c set"   and n :: "nat" assumes "set_iterator it S0" shows "iterate_size_abort it n = min n (card S0) \<and> finite S0"
valid_states_def: fixes prob :: "(('c, 'e) fmap \<times> ('c, 'f) fmap) set" shows "valid_states prob \<equiv> {s. fmdom' s = prob_dom prob}"
rotate14: fixes A14 :: "fmla"   and A1 :: "fmla"   and A2 :: "fmla"   and A3 :: "fmla"   and A4 :: "fmla"   and A5 :: "fmla"   and A6 :: "fmla"   and A7 :: "fmla"   and A8 :: "fmla"   and A9 :: "fmla"   and A10 :: "fmla"   and A11 :: "fmla"   and A12 :: "fmla"   and A13 :: "fmla"   and H :: "fmla set"   and B :: "fmla" assumes "insert A14 (insert A1 (insert A2 (insert A3 (insert A4 (insert A5 (insert A6 (insert A7 (insert A8 (insert A9 (insert A10 (insert A11 (insert A12 (insert A13 H))))))))))))) \<turnstile> B" shows "insert A1 (insert A2 (insert A3 (insert A4 (insert A5 (insert A6 (insert A7 (insert A8 (insert A9 (insert A10 (insert A11 (insert A12 (insert A13 (insert A14 H))))))))))))) \<turnstile> B"
sets: shows "sets (M1 \<Otimes>\<^sub>M M2) = sets M"
rev_bl_order_bl_to_bin: fixes xs :: "bool list"   and ys :: "bool list" assumes "length xs = length ys" shows "rev_bl_order True xs ys = (bl_to_bin (rev xs) \<le> bl_to_bin (rev ys)) \<and> rev_bl_order False xs ys = (bl_to_bin (rev xs) < bl_to_bin (rev ys))"
blind_source_tree_cases: fixes x1ba :: "'c \<Rightarrow> 'b"   and xa :: "'c rose_tree" shows "blind_source_tree x1ba xa = (case xa of Tree (x3a, x2ba) \<Rightarrow> Tree\<^sub>m (Blinded (Content (x1ba x3a, map (hash_source_tree x1ba) x2ba))))"
strict_order_iff: fixes b :: "'a"   and a :: "'a" shows "(b \<sqsubset> a) = (a = ord.max (\<sqsubseteq>) a b \<and> a \<noteq> b)"
mostly_zero: fixes k :: "nat"   and j :: "nat" assumes "k < length hs"   and "j < length hs" shows "(css ! j \<bullet>\<currency>\<currency> hfvss ! j) ((1::'f) \<delta>\<delta> hs ! k) = (if j = k then css ! k \<bullet>\<sharp>\<cdot> vs else (0::'v))"
clock_numbering_2: fixes X :: "'d set"   and v :: "'d \<Rightarrow> nat"   and n :: "nat"   and not_in_X :: "'d" assumes "Regions X v n not_in_X" shows "\<forall>k\<le>n. 0 < k \<longrightarrow> (\<exists>c\<in>X. v c = k)"
isomorphic_def: fixes a :: "('a S.E.term, 'a) S.arr \<times> ('a S.E.term, 'a) S.arr"   and a' :: "('a S.E.term, 'a) S.arr \<times> ('a S.E.term, 'a) S.arr" shows "S.VV.isomorphic a a' = (\<exists>f. S.VV.in_hom f a a' \<and> S.VV.iso f)"
preserves_assoc_1: fixes f :: "'c"   and g :: "'c"   and h :: "'c" assumes "C.ide f"   and "C.ide g"   and "C.ide h"   and "src\<^sub>C f = trg\<^sub>C g"   and "src\<^sub>C g = trg\<^sub>C h" shows "local.map \<a>\<^sub>C[f, g, h] = cmp (f, g \<star>\<^sub>C h) \<cdot>\<^sub>D (local.map f \<star>\<^sub>D cmp (g, h)) \<cdot>\<^sub>D \<a>\<^sub>D[local.map f, local.map g, local.map h] \<cdot>\<^sub>D (D.inv (cmp (f, g)) \<star>\<^sub>D local.map h) \<cdot>\<^sub>D D.inv (cmp (f \<star>\<^sub>C g, h))"
finter_finsert_right_ifffempty: fixes a :: "'c"   and A :: "'c fset"   and B :: "'c fset" assumes "a |\<notin>| A" shows "A |\<inter>| finsert a B = A |\<inter>| B"
sched_typing_axioms: shows "sched_typing cfg la_en la_ex cfg' la_en' la_ex' ty_local"
Exit_no_strong_postdominator: fixes n :: "'node"   and as :: "'edge list" assumes "(_Exit_) strongly-postdominates n"   and "n -as\<rightarrow>* (_Exit_)" shows "False"
make_set_def: fixes k :: "nat"   and p :: "nat \<Rightarrow> 'a \<Rightarrow> bool" shows "make_set k p = {x. \<exists>j. p j x \<and> j < k}"
bintrunc_rest: fixes n :: "nat"   and bin :: "int" shows "take_bit n (take_bit n bin div 2) = take_bit n bin div 2"
in_components_maximal: fixes c :: "'b set"   and s :: "'b set" shows "(c \<in> components s) = (c \<noteq> {} \<and> c \<subseteq> s \<and> connected c \<and> (\<forall>d. d \<noteq> {} \<and> c \<subseteq> d \<and> d \<subseteq> s \<and> connected d \<longrightarrow> d = c))"
f0: shows "prob0 3 (Matrix.mat 8 (Suc 0) (\<lambda>(i, j). [[\<phi> $$ (0, 0) / complex_of_real (sqrt 2), \<phi> $$ (Suc 0, 0) / complex_of_real (sqrt 2), \<phi> $$ (Suc 0, 0) / complex_of_real (sqrt 2), \<phi> $$ (0, 0) / complex_of_real (sqrt 2), 0, 0, 0, 0]] ! j ! i)) (Suc 0) = 1 / 2"
ctxt_2: shows "\<forall>A\<in>\<Gamma>. typ_of A = Some propT"
proj_not_same_active: fixes n :: "nat"   and n' :: "enat"   and t :: "cnf llist"   and c :: "'id" assumes "enat n \<le> n'"   and "\<not> lfinite t \<or> n' - 1 < llength t"   and "\<pi>\<^bsub>c\<^esub>ltake n' t \<noteq> \<pi>\<^bsub>c\<^esub>ltake (enat n) t" shows "\<exists>k\<ge>n. enat k < n' \<and> enat k < llength t \<and> \<parallel>c\<parallel>\<^bsub>lnth t k\<^esub>"
subst_img_comp_subset''': fixes \<theta>1 :: "'b \<Rightarrow> ('a, 'b) Term.term"   and \<theta>2 :: "'b \<Rightarrow> ('a, 'b) Term.term" shows "subterms\<^sub>s\<^sub>e\<^sub>t (subst_range (\<theta>1 \<circ>\<^sub>s \<theta>2)) - range Var \<subseteq> subterms\<^sub>s\<^sub>e\<^sub>t (subst_range \<theta>2) - range Var \<union> (subterms\<^sub>s\<^sub>e\<^sub>t (subst_range \<theta>1) - range Var \<cdot>\<^sub>s\<^sub>e\<^sub>t \<theta>2)"
as0_11: shows "As0 \<cdot>al \<eta>0 = As"
assms_2: shows "(i, Store e1 mo i) \<in> set (node_deliver_messages xs)"
s\<^sub>2: shows "s\<^sub>2 = s\<^sub>1(r \<mapsto> (\<sigma>, \<tau>, \<E> [subst (VE v) x e]))"
set_intros_2: fixes ya :: "'c \<times> 'c list_R1.list_R1"   and x :: "unit + 'c \<times> 'c list_R1.list_R1"   and yc :: "'c list_R1.list_R1"   and yd :: "'c" assumes "ya \<in> setr x"   and "yc \<in> snds ya"   and "yd \<in> list_R1.set_list_R1 yc" shows "yd \<in> list_R1.set_list_R1 (list_R1.list_R1 x)"
emeasure_subprob_space_less_top: fixes M :: "'b pmf"   and A :: "'b set" shows "emeasure (measure_pmf M) A \<noteq> top"
allnonzero: shows "\<forall>q. List.member (fst (factorize_polys qs)) q \<longrightarrow> q \<noteq> 0"
bit_of_bl_iff: fixes bs :: "bool list"   and n :: "nat" shows "bit (of_bl bs) n = (rev bs ! n \<and> n < LENGTH('b) \<and> n < length bs)"
normal: fixes t :: "'t"   and x :: "'x" assumes "thr s t = \<lfloor>(x, no_wait_locks)\<rfloor>"   and "\<not> final x"   and "wset s t = None" shows "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 (final_threads s)))"
DBM_val_bounded_len'3: fixes v :: "'c \<Rightarrow> nat"   and u :: "'c \<Rightarrow> 'd"   and m :: "nat \<Rightarrow> nat \<Rightarrow> 'd DBMEntry"   and n :: "nat"   and vs :: "nat list"   and c1 :: "'c"   and c2 :: "'c" assumes "u \<turnstile>\<^bsub>v,n\<^esub> m"   and "cnt 0 vs \<le> 1"   and "v c1 \<le> n"   and "v c2 \<le> n"   and "\<forall>k\<in>set vs. 0 < k \<longrightarrow> k \<le> n \<and> (\<exists>c. v c = k)" shows "dbm_entry_val u (Some c1) (Some c2) (len m (v c1) (v c2) vs)"
Inr_inject: fixes x :: "'c"   and y :: "'c" assumes "Inr x = Inr y" shows "x = y"
lfp_upperbound: fixes x :: "'c"   and f :: "'c \<Rightarrow> 'c" assumes "\<And>y. x \<le> f y" shows "x \<le> lfp f"
simps_2: fixes l :: "('c, 'd) tree"   and k :: "'c"   and v :: "'d"   and r :: "('c, 'd) tree" shows "tree_sorted (Node l k v r) = ((\<forall>x\<in>tree_set l. x < k) \<and> Ball (tree_set r) ((<) k) \<and> tree_sorted l \<and> tree_sorted r)"
block_closed: fixes S :: "'c set"   and P :: "'c set set"   and a :: "'c"   and A :: "'c set" assumes "Set_Theory.partition S P"   and "a \<in> A"   and "A \<in> P" shows "a \<in> S"
assms_4: shows "v11 (T\<lparr>ArrVal\<rparr>)"
conc_def: fixes w :: "'a list"   and x :: "nat \<Rightarrow> 'a" shows "w \<frown> x \<equiv> \<lambda>n. if n < length w then w ! n else x (n - length w)"
map_def: fixes f :: "'a" shows "\<epsilon>oF\<epsilon>'G.map f = (if A.arr f then FG.\<epsilon> (A.cod f) \<cdot>\<^sub>A (F \<circ> F'G'.\<epsilon> \<circ> G) f else A.null)"
image_Collect_mem: fixes f :: "'c \<Rightarrow> 'b"   and S :: "'c set" shows "{f x |x. x \<in> S} = f ` S"
lim_Si: shows "Liminf_upto_llist Ns (enat (Suc i)) = lnth Ns (Suc i)"
intros_2: fixes \<phi> :: "'d stream \<Rightarrow> bool"   and xs :: "'d stream" assumes "ev \<phi> (stl xs)" shows "ev \<phi> xs"
rc1: shows "root_cond (p1, l1, r1) x"
is_unique: fixes R :: "'c set"   and addition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and multiplication :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and unit :: "'c"   and I :: "'c set"   and J :: "'c set" assumes "local_ring R addition multiplication zero unit"   and "max_lideal I R addition multiplication zero unit"   and "max_lideal J R addition multiplication zero unit" shows "I = J"
of_safe_unsafe_match_eq: fixes m :: "of_match_field set"   and p :: "(32, 'a) simple_packet_ext_scheme" assumes "all_prerequisites m" shows "OF_match_fields m p = Some (OF_match_fields_unsafe m p)"
cos_i_times: fixes z :: "complex" shows "cos (\<i> * z) = (exp z + inverse (exp z)) / 2"
sigma_finite_measure_axioms: shows "sigma_finite_measure (distr N T g)"
power_up_nonneg_iff: fixes prec :: "nat"   and b :: "real"   and n :: "nat" shows "(0 \<le> power_up prec b n) = (even n \<or> 0 \<le> b)"
stage_nb: fixes p :: "fm"   and a :: "real"   and bs :: "real list"   and c :: "int"   and e :: "MIR.num"   and d :: "int" assumes "iszlfm p (a # bs)"   and "0 < c"   and "numbound0 e" shows "bound0 (stage p d (e, c))"
Lp_space: fixes p :: "real"   and M :: "'a measure" assumes "0 < p" shows "lspace M (ennreal p) = {f \<in> borel_measurable M. integrable M (\<lambda>x. \<bar>f x\<bar> powr p)}"
True: fixes F :: "('a \<Rightarrow> bool) \<Rightarrow> bool" assumes "is_filter F" shows "F (\<lambda>x. True)"
subst_cls_list_nth: fixes subst_atm :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and id_subst :: "'c"   and comp_subst :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and renamings_apart :: "'b literal multiset list \<Rightarrow> 'c list"   and atm_of_atms :: "'b list \<Rightarrow> 'b"   and i :: "nat"   and Cs :: "'b literal multiset list"   and \<tau> :: "'c" assumes "substitution subst_atm id_subst comp_subst renamings_apart atm_of_atms"   and "i < length Cs" shows "substitution_ops.subst_cls_list subst_atm Cs \<tau> ! i = substitution_ops.subst_cls subst_atm (Cs ! i) \<tau>"
Red_I_to_Inf: fixes N :: "'f set" shows "Red_I N \<subseteq> Inf"
ideD_1: fixes C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c" assumes "category C"   and "partial_magma.ide C a" shows "partial_magma.arr C a"
jnf: shows "jordan_nf A (triangular_to_jnf_vector A)"
eq_game_plus_inverse: fixes x :: "game" shows "eq_game (plus_game x (neg_game x)) zero_game"
i1: shows " P \<longrightarrow>\<^sub>1 P'"
ideal_points_different_1: fixes u :: "complex_homo"   and v :: "complex_homo"   and i1 :: "complex_homo"   and i2 :: "complex_homo" assumes "u \<in> unit_disc"   and "v \<in> unit_disc"   and "u \<noteq> v"   and "ideal_points (poincare_line u v) = {i1, i2}" shows "i1 \<noteq> i2"
classes_above_ifields: fixes C :: "char list"   and P :: "'b prog"   and P' :: "'b prog" assumes "classes_above P C \<inter> classes_changed P P' = {}" shows "ifields P C = ifields P' C"
assms: fixes x1 :: "'a1"   and x2 :: "'a2"   and x3 :: "'a3" assumes "\<And>y1 y2 y3. h1 y1 < h1 x1 \<Longrightarrow> phi y1 y2 y3"   and "\<And>y1 y2 y3. \<lbrakk>h1 y1 \<le> h1 x1; h2 y2 < h2 x2\<rbrakk> \<Longrightarrow> phi y1 y2 y3"   and "\<And>y1 y2 y3. \<lbrakk>h1 y1 \<le> h1 x1; h2 y2 \<le> h2 x2; h3 y3 < h3 x3\<rbrakk> \<Longrightarrow> phi y1 y2 y3" shows "phi x1 x2 x3"
get_scdom_component_contains_get_dom_component: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and sc :: "(_) object_ptr list"   and ptr' :: "(_) object_ptr" assumes "heap_is_wellformed h"   and "DocumentClass.type_wf h"   and "DocumentClass.known_ptrs h"   and "h \<turnstile> get_scdom_component ptr \<rightarrow>\<^sub>r sc"   and "ptr' \<in> set sc" obtains c :: "(_) object_ptr list" where "h \<turnstile> get_dom_component ptr' \<rightarrow>\<^sub>r c" and "set c \<subseteq> set sc"
ub_def: shows "ub = \<N>\<lbrace>\<Gamma>\<rbrace>"
cbig: shows "(\<sigma>', coll) \<in> cbig P \<sigma>1"
mojmir_accept_iff_rabin_accept: shows "accept\<^sub>R' = accept\<^sub>R (\<delta>\<^sub>R, q\<^sub>R, {(fail\<^sub>R \<union> merge\<^sub>R j, succeed\<^sub>R j) |j. j < max_rank\<^sub>R}) w"
vcomp_ide_cod: fixes A :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and B :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and F :: "'d \<Rightarrow> 'e"   and G :: "'d \<Rightarrow> 'e"   and \<tau> :: "'d \<Rightarrow> 'e" assumes "natural_transformation A B F G \<tau>" shows "vertical_composite.map A B \<tau> G = \<tau>"
WbisT_matchT_MT_rev: fixes s :: "('test, 'atom) com"   and t :: "('test, 'atom) com" assumes "s \<approx>wT t" shows "matchT_MT t s"
eligible0': shows "eligible S \<tau> As0' (D0' + negs (mset As0'))"
c_len_is_pr: shows "c_len \<in> PrimRec1"
insert_compr: fixes a :: "'a"   and B :: "'a set" shows "insert a B = {x. x = a \<or> s2p B x}"
pedersen_base_def: fixes \<G> :: "'a cyclic_group" shows "pedersen_base \<G> \<equiv> prime (order \<G>)"
commutative: fixes x :: "'b"   and y :: "'b" assumes "x \<in> R'"   and "y \<in> R'" shows "x +' y = y +' x"
nonzero_inverse_inverse_eq: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> '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" assumes "class.division_ring divide inverse one times plus zero minus uminus"   and "a \<noteq> zero" shows "inverse (inverse a) = a"
map_gals_1: fixes Cs :: "'b set list" assumes "SimplicialComplex.min_maxsimpchain W (C # Cs)" shows "pgallery (f \<Turnstile> (C # Cs))"
word_and_le': fixes b :: "'a word"   and c :: "'a word"   and a :: "'a word" assumes "b \<le> c" shows "a && b \<le> c"
eq_None_iff_not_dom: fixes f :: "'b \<Rightarrow> 'a option"   and x :: "'b" shows "(f x = None) = (x \<notin> dom f)"
some_third_chamber_def: fixes C :: "'a set"   and D :: "'a set"   and z :: "'a set" shows "some_third_chamber C D z \<equiv> SOME E. E \<in> X - {C, D} \<and> z \<lhd> E"
completely_multiplicative_function_axioms: shows "completely_multiplicative_function \<chi>\<^sub>0"
sig_trd_term_def: fixes d :: "'a \<Rightarrow> nat" shows "sig_trd_term d = {(x, y). punit.dgrad_p_set_le d {rep_list (snd x)} (insert (rep_list (snd y)) (rep_list ` set bs)) \<and> fst x \<in> keys (rep_list (snd x)) \<and> fst y \<in> keys (rep_list (snd y)) \<and> fst x \<prec> fst y}"
len: shows "real (length c) \<le> 6 * log 2 (real p) - 4"
finite_UNIV_bool_def: shows "finite_UNIV = Phantom(bool) True"
preserves_ide: fixes a :: "'a" assumes "ide a" shows "ide (\<I> \<otimes> a)"
emeasure_eq_measure: fixes M :: "'b measure"   and A :: "'b set" assumes "finite_measure M" shows "emeasure M A = ennreal (Sigma_Algebra.measure M A)"
wp_assign_locals_eq: fixes \<pi> :: "char list \<Rightarrow> com option"   and l :: "char list \<Rightarrow> int \<Rightarrow> int"   and Q :: "(char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool"   and s :: "char list \<Rightarrow> int \<Rightarrow> int" shows "wp \<pi> (Assign_Locals l) Q s = Q <l|s>"
scan_nth: fixes i :: "nat"   and f :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and xs :: "'e list"   and a :: "'d" assumes "i < length (scan f xs a)" shows "scan f xs a ! i = fold f (take (Suc i) xs) a"
tw_cmpser_is_w_cmpser: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a set" assumes "tw_cmpser G n f" shows "w_cmpser G n f"
preservation_of_barbs_and_closures_2: fixes Rel :: "('a \<times> 'a) set"   and CWB :: "('a, 'b) calculusWithBarbs" assumes "rel_preserves_binary_pred Rel (\<lambda>a. hasBarb a CWB)" shows "rel_preserves_binary_pred (Rel\<^sup>+) (\<lambda>a. hasBarb a CWB)"
INTERP_def: fixes S :: "'b literal multiset \<Rightarrow> 'b literal multiset"   and N :: "'b literal multiset set" assumes "ground_resolution_with_selection S" shows "ground_resolution_with_selection.INTERP S N = \<Union> (ground_resolution_with_selection.production S N ` N)"
bad_eq: shows "map_spmf bad p = map_spmf bad q"
simps: fixes a :: "'e"   and G :: "('e, 'f) monoid_scheme"   and gens :: "'e set" shows "(a \<in> \<langle>gens\<rangle>\<^bsub>G\<^esub>) = (a = \<one>\<^bsub>G\<^esub> \<or> (\<exists>x. a = x \<and> x \<in> gens) \<or> (\<exists>x. a = inv\<^bsub>G\<^esub> x \<and> x \<in> \<langle>gens\<rangle>\<^bsub>G\<^esub>) \<or> (\<exists>x y. a = x \<otimes>\<^bsub>G\<^esub> y \<and> x \<in> \<langle>gens\<rangle>\<^bsub>G\<^esub> \<and> y \<in> \<langle>gens\<rangle>\<^bsub>G\<^esub>))"
Gromov_converging_at_boundaryI: fixes a :: "'b"   and u :: "nat \<Rightarrow> 'b" assumes "\<And>M. \<exists>N. \<forall>n\<ge>N. \<forall>m\<ge>N. M \<le> Gromov_product_at a (u m) (u n)" shows "Gromov_converging_at_boundary u"
span_mono: fixes A :: "'b set"   and B :: "'b set" assumes "A \<subseteq> B" shows "cspan A \<subseteq> cspan B"
StatEq: fixes \<Psi>' :: "'b"   and S :: "('a, 'b, 'c) psi"   and T :: "('a, 'b, 'c) psi"   and \<Psi>'' :: "'b" assumes "(\<Psi>', S, T) \<in> Rel"   and "\<Psi>' \<simeq> \<Psi>''" shows "(\<Psi>'', S, T) \<in> Rel"
mds_yields_stable_types_def: fixes mds :: "Mode \<Rightarrow> 'Var set" shows "mds_yields_stable_types mds \<equiv> \<forall>x. x \<in> mds AsmNoWrite \<union> mds AsmNoReadOrWrite \<longrightarrow> (\<forall>v\<in>\<C>_vars x. v \<in> mds AsmNoWrite \<union> mds AsmNoReadOrWrite)"
assms_3: shows "f \<squnion> f\<^sup>T \<le> w \<squnion> w\<^sup>T"