hcju/isabelleps huggingface dataset:
Gcd_eqI: fixes one :: "'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and gcd :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and lcm :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and Gcd :: "'b set \<Rightarrow> 'b"   and Lcm :: "'b set \<Rightarrow> 'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and a :: "'b"   and A :: "'b set" assumes "class.semiring_Gcd one zero times gcd lcm Gcd Lcm divide plus minus unit_factor normalize"   and "normalize a = a"   and "\<And>b. b \<in> A \<Longrightarrow> dvd.dvd times a b"   and "\<And>c. (\<And>b. b \<in> A \<Longrightarrow> dvd.dvd times c b) \<Longrightarrow> dvd.dvd times c a" shows "Gcd A = a"
ms: shows "measure_space (space (Pi\<^sub>M I M)) (sets (Pi\<^sub>M I M)) \<mu>"
w_defs_3: shows "w3 = fst (bv_chop (snd (bv_chop (snd (bv_chop w (Suc l))) (Suc j))) i)"
new_element_no_child_nodes: fixes h :: "(_) heap"   and new_element_ptr :: "(_) element_ptr"   and h' :: "(_) heap" assumes "h \<turnstile> new_element \<rightarrow>\<^sub>r new_element_ptr"   and "h \<turnstile> new_element \<rightarrow>\<^sub>h h'" shows "h' \<turnstile> get_child_nodes (cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r new_element_ptr) \<rightarrow>\<^sub>r []"
rep_eq: fixes x :: "'a \<Rightarrow>\<^sub>L 'b" shows "norm x = onorm (blinfun_apply x)"
Next'_preserves_invs_3: fixes c0 :: "('b, 't, 'loc) Combined.configuration"   and c1 :: "('b, 't, 'loc) Combined.configuration" assumes "cri.InvMsgInGlob (exchange_config c0)"   and "inv_init_imp_prop_safe c0"   and "InvGlobPointstampsEq c0"   and "Next' c0 c1"   and "\<forall>p. prop_invs (prop_config c0 p)" shows "InvGlobPointstampsEq c1"
assert_demonic: fixes p :: "'b \<Rightarrow> bool"   and r :: "'b \<Rightarrow> 'a \<Rightarrow> bool" shows "{. p .} \<circ> [: r :] = {. p .} \<circ> [: \<lambda>x y. p x \<and> r x y :]"
ssubst_Upd: fixes z :: "char list"   and u :: "sterm"   and t :: "sterm"   and l :: "Label"   and s :: "sterm" shows "[z \<rightarrow> u] Upd t l s = Upd ([z \<rightarrow> u] t) l ([z \<rightarrow> u] s)"
Diff: fixes a :: "'a set"   and b :: "'a set" assumes "a \<in> generated_ring"   and "b \<in> generated_ring" shows "a - b \<in> generated_ring"
sum_list_nonposI: fixes xs :: "'a list" assumes "list_all (\<lambda>x. x \<le> (0::'a)) xs" shows "sum_list xs \<le> (0::'a)"
prefix_code_2: fixes x :: "'c"   and xs :: "'c list" shows "prefix (x # xs) [] = False"
euclidean_all_zero_iff: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and inner :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and Basis :: "'a set"   and x :: "'a" assumes "class.euclidean_space minus dist norm plus zero uminus scaleR sgn uniformity open inner Basis" shows "(\<forall>u\<in>Basis. inner x u = 0) = (x = zero)"
sum_to_tup_def: shows "sum_to_tup = (\<Lambda> f. (\<Lambda> x. f\<cdot>(Discr (Inl (undiscr x))), \<Lambda> x. f\<cdot>(Discr (Inr (undiscr x)))))"
that: fixes z :: "'a" assumes "r y1 z"   and "r y2 z" shows "thesis"
execn_Normal_elim_cases_13: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and s :: "'e"   and n :: "nat"   and t :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile> \<langle>Throw,Normal s\<rangle> =n\<Rightarrow> t" obtains "t = Abrupt s"
gcd_eq_nitpick_gcd: fixes x :: "nat"   and y :: "nat" shows "gcd x y = nat_gcd x y"
bisim1CastFail: fixes P :: "'b prog"   and h :: "'heap"   and T :: "ty"   and e :: "'addr expr1"   and xs :: "'addr val list"   and v :: "'addr val" shows "P,Cast T e,h \<turnstile> (Throw (addr_of_sys_xcpt ClassCast), xs) \<leftrightarrow> ([v], xs, length (compE2 e), \<lfloor>addr_of_sys_xcpt ClassCast\<rfloor>)"
chain_imageI: fixes le_a :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and Y :: "'a set"   and le_b :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b" assumes "Complete_Partial_Order.chain le_a Y"   and "\<And>x y. \<lbrakk>x \<in> Y; y \<in> Y; le_a x y\<rbrakk> \<Longrightarrow> le_b (f x) (f y)" shows "Complete_Partial_Order.chain le_b (f ` Y)"
lprod_2_2: fixes a :: "'b"   and b :: "'b"   and R :: "('b \<times> 'b) set"   and c :: "'b" assumes "(a, b) \<in> R" shows "([a, c], [c, b]) \<in> lprod R"
simps_2: fixes n :: "nat"   and c :: "int"   and t :: "Ferrack.num" shows "numgcdh (CN n c t) = (\<lambda>g. gcd c (numgcdh t g))"
mult2_alt_map: fixes X :: "'a multiset"   and Y :: "'a multiset"   and ns :: "('a \<times> 'a) set"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b"   and ns' :: "('b \<times> 'b) set"   and s :: "('a \<times> 'a) set"   and s' :: "('b \<times> 'b) set"   and b :: "bool" assumes "\<And>x y. \<lbrakk>x \<in># X; y \<in># Y; (x, y) \<in> ns\<rbrakk> \<Longrightarrow> (f x, g y) \<in> ns'"   and "\<And>x y. \<lbrakk>x \<in># X; y \<in># Y; (x, y) \<in> s\<rbrakk> \<Longrightarrow> (f x, g y) \<in> s'"   and "(X, Y) \<in> mult2_alt b ns s" shows "(image_mset f X, image_mset g Y) \<in> mult2_alt b ns' s'"
conditionally_complete_dual: fixes A :: "'a set" assumes "conditionally_complete A (\<sqsubseteq>)" shows "conditionally_complete A (\<lambda>x y. y \<sqsubseteq> x)"
wf_Int1: fixes r :: "('c \<times> 'c) set"   and r' :: "('c \<times> 'c) set" assumes "wf r" shows "wf (r \<inter> r')"
inf_left_commute: fixes x :: "'b multiset"   and y :: "'b multiset"   and z :: "'b multiset" shows "x \<inter># (y \<inter># z) = y \<inter># (x \<inter># z)"
S4S5S6LeadstoS6: fixes rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState"   and p :: "PrIds"   and sigma :: "behavior" assumes "sigma \<Turnstile> S4 rmhist p \<and> ires!p = #NotAResult \<leadsto> S4 rmhist p \<and> ires!p \<noteq> #NotAResult \<or> S5 rmhist p"   and "sigma \<Turnstile> S4 rmhist p \<and> ires!p \<noteq> #NotAResult \<leadsto> S5 rmhist p"   and "sigma \<Turnstile> S5 rmhist p \<leadsto> S6 rmhist p" shows "sigma \<Turnstile> S4 rmhist p \<or> S5 rmhist p \<or> S6 rmhist p \<leadsto> S6 rmhist p"
simps_49: fixes x71 :: "nat"   and x72 :: "msg"   and x61 :: "msg"   and x62 :: "msg" shows "Crypt x71 x72 \<noteq> \<lbrace>x61, x62\<rbrace>"
weakenR_set: fixes \<Gamma> :: "'b formula multiset"   and \<Delta> :: "'b formula multiset"   and F :: "'b formula multiset" assumes "\<Gamma> \<Rightarrow> \<Delta>" shows "\<Gamma> \<Rightarrow> F + \<Delta>"
left_idem: fixes a :: "nat"   and b :: "nat" shows "gcd a (gcd a b) = gcd a b"
dotprod_pointwise_le_right: fixes z :: "nat list"   and u :: "nat list"   and v :: "nat list" assumes "length z = length u"   and "length u = length v"   and "\<forall>i<length v. u ! i \<le> v ! i" shows "z \<bullet> u \<le> z \<bullet> v"
abs_minus_commute: fixes abs :: "'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 less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" assumes "ordered_ab_group_add_abs abs plus zero minus uminus less_eq less" shows "abs (minus a b) = abs (minus b a)"
prod_pOp_def: fixes I :: "'c set"   and A :: "'c \<Rightarrow> ('d, 'e) aGroup_scheme" shows "prod_pOp I A = (\<lambda>f\<in>carr_prodag I A. \<lambda>g\<in>carr_prodag I A. \<lambda>x\<in>I. f x \<plusminus>\<^bsub>A x\<^esub> g x)"
Times_Int_Times: fixes A :: "'d set"   and B :: "'e set"   and C :: "'d set"   and D :: "'e set" shows "A \<times> B \<inter> C \<times> D = (A \<inter> C) \<times> (B \<inter> D)"
right_neutral: fixes a :: "nat" shows "gcd a 0 = a"
assms_4: fixes x :: "real" assumes "x \<in> closed_segment a b" shows "0 \<le> f x"
Class_self: fixes a :: "'a set \<times> 'b" assumes "a \<in> Sigma I \<FF>" shows "a \<in> rel.Class a"
negfn: fixes n :: "nat" shows "f ` {x \<in> S. norm x \<le> real n \<and> (\<exists>T. open T \<and> x \<in> T \<and> (\<forall>y\<in>S \<inter> T. norm (f y - f x) \<le> (real n + 1) * norm (y - x)))} \<in> null_sets lebesgue"
jkbpSemDet: shows "\<forall>a gc gc'. gc \<in> set (jkbp a) \<and> gc' \<in> set (jkbp a) \<and> t \<in> jkbpC \<longrightarrow> guard gc = guard gc' \<or> \<not> (MC, t \<Turnstile> guard gc \<and> MC, t \<Turnstile> guard gc')"
shunt1: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(x \<sqinter> y \<le> z) = (x \<le> - y \<squnion> z)"
abs_dist_diff_le: fixes dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "class.metric_space dist uniformity open" shows "\<bar>dist a b - dist b c\<bar> \<le> dist a c"
assms_1: shows "y < f a \<and> f b < y \<or> f a < y \<and> y < f b"
elims_2: fixes r :: "'d \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "'d list"   and xa :: "'e"   and xb :: "'e" assumes "rel_comp_pow r x xa xb" obtains i :: "'d" and xc :: "'e" and y :: "'e" where "x = [ia__]" and "xa = xc__" and "xb = y" and "r ia__ xc__ y"   | i :: "'d" and v :: "'d" and va :: "'d list" and xc :: "'e" and z :: "'e" where "x = i # v # va" and "xa = xc" and "xb = z" and "\<exists>y. r i xc y \<and> rel_comp_pow r (v # va) y z"
i_aggregate_i_drop_mult: fixes n :: "nat"   and k :: "nat"   and f :: "nat \<Rightarrow> 'd"   and ag :: "'d list \<Rightarrow> 'd" shows "i_aggregate (f \<Up> (n * k)) k ag = i_aggregate f k ag \<Up> n"
assms_1: shows "timpl_closure_set M TI \<turnstile>\<^sub>c t"
simps: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a1 :: "'a list"   and a2 :: "'a list" shows "list_emb P a1 a2 = ((\<exists>ys. a1 = [] \<and> a2 = ys) \<or> (\<exists>xs ys y. a1 = xs \<and> a2 = y # ys \<and> list_emb P xs ys) \<or> (\<exists>x y xs ys. a1 = x # xs \<and> a2 = y # ys \<and> P x y \<and> list_emb P xs ys))"
assms: shows "\<Gamma> \<turnstile> e, Roll eP', eV : Mu \<alpha> \<tau>"
set_cod: fixes U :: "'b set"   and f :: "'b set_arrow" shows "Cod\<^bsub>set_cat U\<^esub> f = set_cod f"
refine: fixes prog :: "program"   and g :: "gState" shows "nres_of (nexts_code_aux prog g) \<le> nexts prog g"
abs_eq: fixes x :: "'a \<Rightarrow> 'b option"   and xa :: "'a" assumes "eq_onp (\<lambda>m. finite (dom m)) x x" shows "fmdrop xa (map_of_fmap x) = map_of_fmap (map_drop xa x)"
u1: shows "subst_term (zip (map fst insts) (map2 Fv fresh_idns (map snd (map fst insts)))) t = fold (\<lambda>single. subst_term [single]) (zip (map fst insts) (map2 Fv fresh_idns (map snd (map fst insts)))) t"
assms_5: assumes "m < n \<and> x < y"   and "x = n"   and "y \<noteq> m"   and "y \<noteq> n" shows "P (arr m) (arr y)"
hyps: fixes xs :: "'a list" assumes "length xs < length xsa__" shows "sset (orbits_list f xs) = orbit f ` set xs"
mem_Collect_eq: fixes a :: "'b"   and P :: "'b \<Rightarrow> bool" shows "(a \<in> p2s P) = P a"
le_cInf_iff: fixes S :: "'b set"   and a :: "'b" assumes "S \<noteq> {}"   and "bdd_below S" shows "(a \<le> Inf S) = (\<forall>x\<in>S. a \<le> x)"
intro: fixes hom :: "'a \<Rightarrow> 'b" assumes "field_hom hom" shows "field_hom_0' hom"
cod_cod: fixes f :: "'a" assumes "arr f" shows "cod (cod f) = cod f"
set_eq_iff: fixes A :: "'e set"   and B :: "'e set" shows "(A = B) = (\<forall>x. (x \<in> A) = (x \<in> B))"
distinct_2: fixes x21 :: "'a tree23"   and x22 :: "'a"   and x23 :: "'a tree23s"   and x1 :: "'a tree23" shows "TTs x21 x22 x23 \<noteq> T x1"
edka_partial_refine: shows "edka_partial \<le> \<Down> Id fofu"
bernoulli_stream_pseudo_prob_rec': fixes n :: "nat"   and x :: "bool stream" assumes "pseudo_proj_True n x = x" shows "prob (pseudo_proj_True (Suc n) -` {x}) = prob (pseudo_proj_True n -` {x}) * prob_component p x n"
pred_inject_1: fixes P :: "'c \<Rightarrow> bool" shows "llist_all P LNil"
word_rep_drop: fixes bl :: "bool list" shows "to_bl (of_bl bl) = replicate (LENGTH('b) - length bl) False @ drop (length bl - LENGTH('b)) bl"
simps: fixes t1 :: "taylor_model"   and t2 :: "taylor_model" shows "tm_sub t1 t2 = tm_add t1 (tm_neg t2)"
field_cont_on_path_cd: shows "continuous_on {c..d} (\<lambda>f_var. F (f_var *\<^sub>R i + g f_var) \<bullet> i)"
real_imag_0: fixes a :: "complex" assumes "is_real a"   and "is_imag a" shows "a = 0"
elims_2: fixes x :: "char list \<Rightarrow> 'a"   and xa :: "char list \<Rightarrow> 'a list \<Rightarrow> 'a"   and xb :: "char list \<Rightarrow> 'a list \<Rightarrow> bool"   and xc :: "fterm literal" assumes "eval\<^sub>l x xa xb xc" obtains E :: "char list \<Rightarrow> 'a" and F :: "char list \<Rightarrow> 'a list \<Rightarrow> 'a" and G :: "char list \<Rightarrow> 'a list \<Rightarrow> bool" and p :: "char list" and ts :: "fterm list" where "x = E__" and "xa = F__" and "xb = G__" and "xc = Pos p__ ts__" and "G__ p__ (eval\<^sub>t\<^sub>s E__ F__ ts__)"   | E :: "char list \<Rightarrow> 'a" and F :: "char list \<Rightarrow> 'a list \<Rightarrow> 'a" and G :: "char list \<Rightarrow> 'a list \<Rightarrow> bool" and p :: "char list" and ts :: "fterm list" where "x = E" and "xa = F" and "xb = G" and "xc = Neg p ts" and "\<not> G p (eval\<^sub>t\<^sub>s E F ts)"
thms_excluded_middle_rule: fixes p :: "'b pl"   and H :: "'b pl set"   and q :: "'b pl" assumes "insert p H |- q"   and "insert (p -> false) H |- q" shows "H |- q"
rev: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and z :: "'b"   and xs :: "'b list" assumes "comm_monoid_list f z" shows "monoid_list.F f z (rev xs) = monoid_list.F f z xs"
l_get_dom_component_get_parent\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_get_dom_component_get_parent\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M heap_is_wellformed parent_child_rel DocumentClass.type_wf DocumentClass.known_ptr DocumentClass.known_ptrs to_tree_order get_parent get_parent_locs get_child_nodes get_child_nodes_locs get_dom_component is_strongly_dom_component_safe is_weakly_dom_component_safe get_root_node get_root_node_locs get_ancestors get_ancestors_locs get_disconnected_nodes get_element_by_id get_elements_by_class_name get_elements_by_tag_name first_in_tree_order get_attribute get_attribute_locs"
int_transE_snv: fixes T :: "'fsym trm"   and \<xi> :: "Preliminaries.var \<Rightarrow> univ" assumes "wt T"   and "I.wtE \<xi>"   and "\<And>\<sigma>. infTp \<sigma> \<or> (\<forall>x\<in>nvT T. tpOfV x \<noteq> \<sigma>)" shows "Ik.int (transE \<xi>) T = I.int \<xi> T"
wls_subst_Var_simp1: fixes ys :: "'varSort"   and Y :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set"   and xs :: "'varSort"   and y :: "'var"   and x :: "'var" assumes "wls (asSort ys) Y"   and "ys \<noteq> xs \<or> y \<noteq> x" shows "Var xs x #[Y / y]_ys = Var xs x"
simps_2: fixes v :: "'c"   and tail :: "'c list" shows "of_list_tree (v # tail) = T v (of_list_tree tail) E"
split: fixes P :: "'d \<Rightarrow> bool"   and f1 :: "'e \<Rightarrow> 'd"   and f2 :: "'e \<Rightarrow> 'd"   and decision :: "'e decision" shows "P (case decision of allow x \<Rightarrow> f1 x | deny x \<Rightarrow> f2 x) = ((\<forall>x1. decision = allow x1 \<longrightarrow> P (f1 x1)) \<and> (\<forall>x2. decision = deny x2 \<longrightarrow> P (f2 x2)))"
frees_all_frees: fixes t :: "'b" shows "frees t |\<subseteq>| all_frees t"
mod_base: fixes base :: "nat"   and n :: "nat"   and d :: "nat \<Rightarrow> nat" assumes "digits base"   and "\<And>i. i < n \<Longrightarrow> d i < base"   and "0 < n" shows "digits.from_digits base n d mod base = d 0"
rank_uniqify: fixes q :: "('c, 'd) SkewBinomialTree list" assumes "rank_skew_invar q" shows "rank_invar (uniqify q)"
p_irred: shows "\<forall>(E, k)\<in>set (P @ P'). j < k \<longrightarrow> is_irreducible [C] E"
weaken_hnr_post_triv: fixes \<Gamma> :: "assn"   and P :: "assn" shows "WEAKEN_HNR_POST \<Gamma> P P"
knowledge_equiv: fixes X :: "msg"   and A :: "agent"   and evs :: "event list"   and B :: "agent" assumes "X \<in> knows A evs"   and "Notes A X \<notin> set evs"   and "X \<notin> {Key (priEK A), Key (priSK A), Key (shrK A)}" shows "X \<in> knows B evs"
ASum_eq: fixes n :: "nat"   and f :: "nat \<Rightarrow> ant"   and g :: "nat \<Rightarrow> ant" assumes "\<forall>j\<le>n. f j \<in> Z\<^sub>\<infinity>"   and "\<forall>j\<le>n. g j \<in> Z\<^sub>\<infinity>"   and "\<forall>j\<le>n. f j = g j" shows "ASum f n = ASum g n"
assms_3: shows "ys \<noteq> zs \<or> y \<noteq> z"
liftI_def: fixes f :: "nat \<Rightarrow> nat \<Rightarrow> (nat \<times> int) list \<Rightarrow> rat array \<Rightarrow> unit Heap"   and d :: "nat"   and a :: "rat array" shows "liftI f d a = foldr (\<lambda>p n. n \<bind> (\<lambda>_. f d (lm - level p) p a)) (gridgen (start dm) ({0..<dm} - {d}) lm) (return ())"
assms_2: shows "i \<notin> net_tree_ips n"
snoc_2: shows "filter P xs = (ys @ [y]) @ zs"
is_processT5_S3: fixes s :: "'a event list"   and P :: "'a process"   and c :: "'a event" assumes "(s, {}) \<in> F P"   and "(s @ [c], {}) \<notin> F P" shows "(s, {c}) \<in> F P"
exp_uminus_Im: fixes z :: "complex" shows "exp (- Im z) \<le> exp (cmod z)"
assignsE_const_simp: fixes e :: "expr"   and v :: "val" assumes "constVal e = Some v" shows "assignsE e = {}"
C1: fixes S :: "('a, 'b, 'c) psi"   and T :: "('a, 'b, 'c) psi"   and U :: "('a, 'b, 'c) psi" shows "(\<Psi>, S \<parallel> T \<parallel> U, S \<parallel> (T \<parallel> U)) \<in> Rel"
abs_eq: fixes x :: "'a \<Rightarrow> 'b option" assumes "eq_onp (\<lambda>m. finite (dom m)) x x" shows "fmdom (map_of_fmap x) = Abs_fset (dom x)"
diff_invariant_eq: fixes I :: "'b \<Rightarrow> bool"   and f :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and U :: "'b \<Rightarrow> real set"   and S :: "'b set"   and t\<^sub>0 :: "real"   and G :: "'b \<Rightarrow> bool" shows "diff_invariant I f U S t\<^sub>0 G = (\<forall>s. I s \<longrightarrow> (\<forall>X\<in>Sols f U S t\<^sub>0 s. \<forall>t\<in>U s. (\<forall>\<tau>\<in>down (U s) t. G (X \<tau>)) \<longrightarrow> I (X t)))"
simps_2: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'c spmf"   and p :: "'c spmf"   and x :: "'d"   and xs :: "'d list" shows "foldl_spmf f p (x # xs) = foldl_spmf f (p \<bind> (\<lambda>a. f a x)) xs"
monic_mult: fixes p :: "'a poly"   and q :: "'a poly" assumes "monic p"   and "monic q" shows "monic (p * q)"
simps_4: fixes as :: "(Vars.var \<times> exp) list"   and e :: "exp" shows "isVal (Terms.Let as e) = False"
axioms_2: 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_isomorphism A B F G \<tau>" shows "natural_isomorphism_axioms A B \<tau>"
neighbor_invar_axioms: shows "neighbor_invar c s t f l n"
zer_def: shows "zer \<equiv> abs_trm zer_raw"
continuity_lem_lt0_expanded: fixes a :: "real"   and r :: "real"   and b :: "real"   and c :: "real" assumes "a * r\<^sup>2 + b * r + c < 0" shows "\<exists>y'>r. \<forall>x\<in>{r<..y'}. a * x\<^sup>2 + b * x + c < 0"
AbstrLevels_A8_A81: fixes i :: "AbstrLevelsID" assumes "sA8 \<in> AbstrLevel i" shows "sA81 \<notin> AbstrLevel i"
positions_in_row_length: fixes w :: "('b \<times> bool list) list"   and i :: "nat" assumes "\<exists>!p. p < length w \<and> snd (w ! p) ! i" shows "the_elem (positions_in_row w i) < length w"
1130: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(- (x \<squnion> y) \<squnion> z) \<sqinter> x = z \<sqinter> x"
assms: shows "(sig_red sing_reg top_tail F)\<^sup>*\<^sup>* p q"
hom_gauss_jordan_main: fixes A :: "'a mat"   and nr :: "nat"   and nc :: "nat"   and B :: "'a mat"   and nc2 :: "nat"   and i :: "nat"   and j :: "nat" assumes "A \<in> carrier_mat nr nc"   and "B \<in> carrier_mat nr nc2" shows "gauss_jordan_main (mat\<^sub>h A) (mat\<^sub>h B) i j = map_prod mat\<^sub>h mat\<^sub>h (gauss_jordan_main A B i j)"
hyp: shows "(l = PAodv-:6 \<or> l = PAodv-:7 \<or> l = PRrep-:0 \<or> l = PRrep-:1) \<and> sip (\<sigma>' i) \<noteq> dip (\<sigma>' i)"
suff_cond_safe_dist2_def: fixes s\<^sub>e :: "real"   and v\<^sub>e :: "real"   and a\<^sub>e :: "real"   and s\<^sub>o :: "real"   and v\<^sub>o :: "real"   and a\<^sub>o :: "real" shows "suff_cond_safe_dist2 s\<^sub>e v\<^sub>e a\<^sub>e s\<^sub>o v\<^sub>o a\<^sub>o \<equiv> let D2 = discriminant s\<^sub>e v\<^sub>e a\<^sub>e s\<^sub>o v\<^sub>o a\<^sub>o in \<not> (a\<^sub>e < a\<^sub>o \<and> v\<^sub>o < v\<^sub>e \<and> 0 \<le> D2 \<and> v\<^sub>e - a\<^sub>e / a\<^sub>o * v\<^sub>o < sqrt D2)"
PO_m1_refines_trans_m1a: shows "{R1a1} TS.trans m1a, TS.trans m1 {> R1a1}"
rt_fresh_asI: fixes dip :: "nat"   and rt1 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and rt2 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option" assumes "rt1 \<sqsubseteq>\<^bsub>dip\<^esub> rt2"   and "rt2 \<sqsubseteq>\<^bsub>dip\<^esub> rt1" shows "rt1 \<approx>\<^bsub>dip\<^esub> rt2"
lass_val_of_iconf: fixes e :: "'a exp"   and a :: "'a \<times> val"   and sh :: "sheap" assumes "lass_val_of e = \<lfloor>a\<rfloor>" shows "iconf sh e"
bounded_clinear_cinner_right: fixes a :: "'a" shows "bounded_clinear (cinner a)"
absorb_iff2: fixes b :: "'e"   and a :: "'e" shows "(b \<le> a) = (inf_class.inf a b = b)"
InlInr_InlInl_Inr_DataSpace: shows "DataSpace [Part UNIV (Inl \<circ> Inr), Part UNIV (Inl \<circ> Inl), Part UNIV Inr]"
simps_2: fixes n :: "'b"   and l :: "'b tree"   and v :: "'b"   and va :: "'b tree"   and vb :: "'b tree"   and vc :: "nat"   and h :: "nat" shows "delete_max (MKT n l (MKT v va vb vc) h) = (let (n', r') = delete_max (MKT v va vb vc) in (n', mkt_bal_l n l r'))"
has_typ1_imp_no_loose_bvar: fixes Ts :: "typ list"   and t :: "Core.term"   and ty :: "typ" assumes "Ts \<turnstile>\<^sub>\<tau> t : ty" shows "\<not> loose_bvar t (length Ts)"
reaches_refl: fixes r :: "'e"   and s :: "'f process_name \<Rightarrow> ('g, 'f, 'h, 'e) Model.local_state" shows "(r reaches r) s"
runit_tensor: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and T :: "'a \<times> 'a \<Rightarrow> 'a"   and \<alpha> :: "'a \<times> 'a \<times> 'a \<Rightarrow> 'a"   and \<iota> :: "'a"   and a :: "'a"   and b :: "'a" assumes "monoidal_category C T \<alpha> \<iota>"   and "partial_magma.ide C a"   and "partial_magma.ide C b" shows "monoidal_category.runit C T \<alpha> \<iota> (T (a, b)) = C (T (a, monoidal_category.runit C T \<alpha> \<iota> b)) (\<alpha> (a, b, monoidal_category.unity C \<iota>))"
mset_set_Union: fixes A :: "'c set"   and B :: "'c set" assumes "finite A"   and "finite B"   and "A \<inter> B = {}" shows "mset_set (A \<union> B) = mset_set A + mset_set B"
io_targets_observable_singleton_ex: fixes M :: "('d, 'e, 'f) FSM"   and io :: "('d \<times> 'e) list"   and q1 :: "'f" assumes "observable M"   and "io \<in> LS M q1" shows "\<exists>q2. io_targets M q1 io = {q2}"
mult_exp_mod_exp_eq: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bit :: "'b \<Rightarrow> nat \<Rightarrow> bool"   and m :: "nat"   and n :: "nat"   and a :: "'b" assumes "class.semiring_bits plus minus zero times one divide modulo bit"   and "m \<le> n" shows "modulo (times a (power.power one times (numeral.numeral one plus (num.Bit0 num.One)) m)) (power.power one times (numeral.numeral one plus (num.Bit0 num.One)) n) = times (modulo a (power.power one times (numeral.numeral one plus (num.Bit0 num.One)) (n - m))) (power.power one times (numeral.numeral one plus (num.Bit0 num.One)) m)"