hcju/isabelleps huggingface dataset:
calculus_axioms: shows "calculus Bot Inf (\<Turnstile>Q) Red_I Red_F"
p_nonzero: shows "[p] \<cdot> \<one> \<in> nonzero Zp"
gen_new_keys_not_used: fixes evs :: "event list"   and K :: "nat"   and KK :: "nat set" assumes "evs \<in> set_mr"   and "Key K \<notin> used evs"   and "K \<in> symKeys" shows "K \<notin> keysFor (parts (Key ` KK \<union> knows Spy evs))"
reach_reach\<^sub>t_fst: fixes \<Sigma> :: "'b set"   and \<delta> :: "'a \<Rightarrow> 'b \<Rightarrow> 'a"   and q\<^sub>0 :: "'a" shows "reach \<Sigma> \<delta> q\<^sub>0 = fst ` reach\<^sub>t \<Sigma> \<delta> q\<^sub>0"
mimic_trans: fixes subst :: "('a, 'b, 'c) expr \<Rightarrow> 'c \<Rightarrow> ('a, 'b, 'c) expr \<Rightarrow> ('a, 'b, 'c) expr"   and s\<^sub>1 :: "'a \<Rightarrow> (('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('a, 'b, 'c) expr) option"   and s\<^sub>1' :: "'a \<Rightarrow> (('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('a, 'b, 'c) expr) option"   and s\<^sub>2 :: "'a \<Rightarrow> (('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('a, 'b, 'c) expr) option" assumes "substitution subst"   and "s\<^sub>1 \<approx> s\<^sub>1'"   and "(s\<^sub>1, s\<^sub>2) \<in> (substitution.steps subst)\<^sup>*" shows "\<exists>s\<^sub>2'. s\<^sub>2 \<approx> s\<^sub>2' \<and> (s\<^sub>1', s\<^sub>2') \<in> (substitution.steps subst)\<^sup>*"
prepend_cong1_base: fixes X :: "'b llist \<Rightarrow> bool"   and xs :: "'b llist" assumes "X xs" shows "prepend_cong1 X xs"
Cinfinite_csum: fixes r1 :: "('c \<times> 'c) set"   and r2 :: "('d \<times> 'd) set" assumes "Cinfinite r1 \<or> Cinfinite r2" shows "Cinfinite (r1 +c r2)"
update_trie_induct: fixes P :: "'e list \<Rightarrow> ('d, 'c) trie \<Rightarrow> bool"   and xs :: "'e list"   and t :: "('d, 'c) trie" assumes "\<And>v ps. P [] (Trie v ps)"   and "\<And>k ks v ps. (\<And>x. P ks x) \<Longrightarrow> P (k # ks) (Trie v ps)" shows "P xs t"
orthonormal_system_trigonometric_set: shows "orthonormal_system {- pi..pi} trigonometric_set"
evaln_abrupt: fixes G :: "prog"   and xc :: "abrupt"   and s :: "st"   and e :: "Term.term"   and n :: "nat"   and w :: "vals"   and s' :: "state" shows "G\<turnstile>(Some xc, s) \<midarrow>e\<succ>\<midarrow>n\<rightarrow> (w, s') = (s' = (Some xc, s) \<and> w = undefined3 e \<and> G\<turnstile>(Some xc, s) \<midarrow>e\<succ>\<midarrow>n\<rightarrow> (undefined3 e, (Some xc, s)))"
M': shows "M \<inter> iterates_above a \<in> Chains leq"
NormalizeD_sound: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and F :: "'f set"   and P :: "'e set"   and c :: "('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set" assumes "\<Gamma>,\<Theta>\<Turnstile>\<^sub>t\<^bsub>/F\<^esub> P Language.normalize c Q,A" shows "\<Gamma>,\<Theta>\<Turnstile>\<^sub>t\<^bsub>/F\<^esub> P c Q,A"
Int_insert_right_if1: fixes a :: "'c"   and A :: "'c set"   and B :: "'c set" assumes "a \<in> A" shows "A \<inter> insert a B = insert a (A \<inter> B)"
domIff: fixes a :: "'f"   and m :: "'f \<Rightarrow> 'g option" shows "(a \<in> dom m) = (m a \<noteq> None)"
Rats_of_int: fixes z :: "int" shows "of_int z \<in> \<rat>"
converse_unfold: fixes r :: "('b \<times> 'a) set" shows "r\<inverse> = {(y, x). (x, y) \<in> r}"
can_sync_def: fixes r :: "('d, 'a, 'b, 'c, 'e, 'f) semantics"   and t :: "'a"   and x :: "'b"   and m :: "'c"   and LT :: "('d + 'a + 'a) set" shows "multithreaded_base.can_sync r t x m LT \<equiv> \<exists>ta x' m'. r t (x, m) ta (x', m') \<and> LT = collect_waits ta"
wf_inv_image: fixes r :: "('e \<times> 'e) set"   and f :: "'f \<Rightarrow> 'e" assumes "wf r" shows "wf (inv_image r f)"
open_cblinfun_def: fixes S :: "('a \<Rightarrow>\<^sub>C\<^sub>L 'b) set" shows "open S = (\<forall>x\<in>S. \<forall>\<^sub>F (x', y) in uniformity. x' = x \<longrightarrow> y \<in> S)"
inject: fixes x1 :: "'a form multiset"   and x2 :: "'a form multiset"   and y1 :: "'a form multiset"   and y2 :: "'a form multiset" shows "(( x1 \<Rightarrow>* x2) = ( y1 \<Rightarrow>* y2)) = (x1 = y1 \<and> x2 = y2)"
Eq: fixes t1 :: "Formula.trm"   and t2 :: "Formula.trm"   and n :: "nat"   and \<sigma> :: "(String.literal \<times> event_data list) trace"   and j :: "nat"   and P :: "String.literal \<Rightarrow> nat option"   and V :: "String.literal \<Rightarrow> (nat \<Rightarrow> event_data list set) option"   and R :: "event_data list set" assumes "is_simple_eq t1 t2"   and "\<forall>x\<in>fv_trm t1. x < n"   and "\<forall>x\<in>fv_trm t2. x < n" shows "wf_mformula \<sigma> j P V n R (MRel (eq_rel n t1 t2)) (formula.Eq t1 t2)"
preserves_cones: fixes J :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and D :: "'a \<Rightarrow> 'c"   and a :: "'c"   and \<chi> :: "'a \<Rightarrow> 'c" assumes "Limit.cone J (\<cdot>\<^sub>C) D a \<chi>" shows "Limit.cone J (\<cdot>\<^sub>C) (C.map \<circ> D) (C.map a) (C.map \<circ> \<chi>)"
compl_le_compl_iff: fixes x :: "'b"   and y :: "'b" shows "(- x \<le> - y) = (y \<le> x)"
\<eta>_def: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and D :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and F :: "'c \<Rightarrow> 'b"   and G :: "'b \<Rightarrow> 'c"   and \<phi> :: "'c \<Rightarrow> 'b \<Rightarrow> 'c"   and \<psi> :: "'b \<Rightarrow> 'c \<Rightarrow> 'b" assumes "meta_adjunction C D F G \<phi> \<psi>" shows "meta_adjunction.\<eta> D F \<phi> \<equiv> transformation_by_components.map D D (identity_functor.map D) (meta_adjunction.\<eta>o F \<phi>)"
prems: shows "Subobjs P C (Cs @ C' # rev (D # Ds))"
assms: shows "i = None \<or> (\<exists>i'. i = Some i' \<and> i' \<in> fst ` set xs)"
simps_3: fixes u :: "'b trm"   and M :: "'b trm"   and N :: "'b trm" shows "u \<prec> M \<cdot> N = (u = M \<or> u = N \<or> u \<prec> M \<or> u \<prec> N)"
intro\<^sub>2: fixes \<Gamma> :: "'Var \<Rightarrow> Sec option"   and lc :: "(('Var, 'AExp, 'BExp) Stmt \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)"   and lc' :: "(('Var, 'AExp, 'BExp) Stmt \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)" assumes "(case lc of (x, xa) \<Rightarrow> (case x of (x, xa) \<Rightarrow> \<lambda>xaa ((xaaa, xaaaa), xaaaaa). \<langle>x, xa, xaa\<rangle> \<R>\<^sup>2\<^bsub>\<Gamma>\<^esub> \<langle>xaaa, xaaaa, xaaaaa\<rangle>) xa) lc'" shows "lc \<R>\<^sup>u\<^bsub>\<Gamma>\<^esub> lc'"
igSwapAllIPresIGWlsAll_def: fixes MOD :: "('c, 'd, 'varSort, 'sort, 'e, 'f, 'g, 'h, 'i) Iteration.model_scheme" shows "igSwapAllIPresIGWlsAll MOD \<equiv> igSwapIPresIGWls MOD \<and> igSwapAbsIPresIGWlsAbs MOD"
ord_FO_\<Gamma>_def: shows "ord_FO_\<Gamma> = {uu_. \<exists>CAs DA AAs As \<sigma> E. uu_ = Infer (mset CAs) DA E \<and> ord_resolve_rename CAs DA AAs As \<sigma> E}"
logical_monotonicity: fixes \<Gamma> :: "(name \<times> ty) list"   and s :: "trm"   and t :: "trm"   and T :: "ty"   and \<Gamma>' :: "(name \<times> ty) list" assumes "\<Gamma> \<turnstile> s is t : T"   and " \<Gamma> \<subseteq> \<Gamma>' "   and "valid \<Gamma>'" shows "\<Gamma>' \<turnstile> s is t : T"
iTL_defs_9: fixes t0 :: "nat"   and I :: "nat set"   and P :: "nat \<Rightarrow> bool" shows "iNext t0 I P \<equiv> P (inext t0 I)"
ord_cases: fixes i :: "'a" obtains j :: "'a" where "i = succ j" and "local.aboveS j \<noteq> {}"   | "isLim i"
f10: shows "s \<otimes> s'' \<otimes> (s' \<otimes> s'') \<otimes> (s' \<otimes> r \<otimes> r'') = s \<otimes> s' \<otimes> s'' \<otimes> (s' \<otimes> s'' \<otimes> (r \<otimes> r''))"
wp_loop_step_sound: fixes body :: "bool \<Rightarrow> ('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real"   and t :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real"   and P :: "'b \<Rightarrow> real"   and G :: "'b \<Rightarrow> bool" assumes "healthy (wp body)"   and "\<And>P. sound P \<Longrightarrow> sound (t P)"   and "sound P" shows "sound (wp (body ;; Embed t \<^bsub>\<guillemotleft> G \<guillemotright>\<^esub>\<oplus> Skip) P)"
encL: fixes S :: "'procS"   and SRel :: "('procS \<times> 'procS) set"   and TRel :: "('procT \<times> 'procT) set" shows "TargetTerm (\<lbrakk>S\<rbrakk>) \<R>\<lbrakk>\<cdot>\<rbrakk><SRel,TRel> SourceTerm S"
map_rel_obtain1: fixes m :: "'c \<Rightarrow> 'd option"   and n :: "'e \<Rightarrow> 'f option"   and K :: "('c \<times> 'e) set"   and V :: "('d \<times> 'f) set"   and l :: "'e"   and w :: "'f" assumes "(m, n) \<in> \<langle>K, V\<rangle>map_rel"   and "n l = Some w" obtains k :: "'c" and v :: "'d" where "m k = Some v" and "(k, l) \<in> K" and "(v, w) \<in> V"
a': shows "a' \<in> span ((+) (- z) ` S)"
hb: shows "ta_hb_consistent P E (lappend E' E'')"
atLeast0_atMost_Suc_shift: fixes g :: "nat \<Rightarrow> 'a"   and n :: "nat" shows "prod g {0..Suc n} = g 0 * prod (g \<circ> Suc) {0..n}"
r_totalizer: fixes d :: "nat"   and i :: "nat"   and x :: "nat" shows "eval (r_totalizer d) [i, x] = (if the (delay i x) = 0 then Some d else \<phi> i (e_take (the (delay i x)) x))"
mem_restrE: fixes A :: "'c list set"   and x :: "'c option list"   and n :: "nat"   and V :: "nat set" assumes "mem_restr A x"   and "wf_tuple n V x"   and "\<forall>i\<in>V. i < n" obtains y :: "'c list" where "y \<in> A" and "\<forall>i\<in>V. x ! i = Some (y ! i)"
fmeasurable_INT: fixes I :: "'b set"   and i :: "'b"   and F :: "'b \<Rightarrow> 'c set"   and M :: "'c measure" assumes "countable I"   and "i \<in> I"   and "\<And>i. i \<in> I \<Longrightarrow> F i \<in> sets M"   and "F i \<in> fmeasurable M" shows "\<Inter> (F ` I) \<in> fmeasurable M"
induct: fixes P :: "ctstate \<Rightarrow> 'b common_primitive match_expr \<Rightarrow> bool"   and a0 :: "ctstate"   and a1 :: "'b common_primitive match_expr" assumes "\<And>s x. P s (Match (CT_State x))"   and "\<And>s v. P s (Match (Src v))"   and "\<And>s v. P s (Match (Dst v))"   and "\<And>s v. P s (Match (IIface v))"   and "\<And>s v. P s (Match (OIface v))"   and "\<And>s v. P s (Match (Prot v))"   and "\<And>s v. P s (Match (Src_Ports v))"   and "\<And>s v. P s (Match (Dst_Ports v))"   and "\<And>s v. P s (Match (MultiportPorts v))"   and "\<And>s v. P s (Match (L4_Flags v))"   and "\<And>s v. P s (Match (Extra v))"   and "\<And>s m. P s m \<Longrightarrow> P s (MatchNot m)"   and "\<And>uu_. P uu_ MatchAny"   and "\<And>s m1 m2. \<lbrakk>P s m1; P s m2\<rbrakk> \<Longrightarrow> P s (MatchAnd m1 m2)" shows "P a0 a1"
ennreal_continuous_on_cmult: fixes c :: "ennreal"   and A :: "'a set"   and f :: "'a \<Rightarrow> ennreal" assumes "c < top"   and "continuous_on A f" shows "continuous_on A (\<lambda>x. c * f x)"
regular_cover_2: shows "\<Union> (U ` N) = (\<Union>i\<in>N. domain (\<psi> i))"
Diff_eq: fixes A :: "'g set"   and B :: "'g set" shows "A - B = A \<inter> - B"
row_rank_def: fixes A :: "(('b, 'c) vec, 'd) vec" shows "row_rank A = vec.dim (row_space A)"
fls_compose_power_diff: fixes f :: "'b fls"   and g :: "'b fls"   and d :: "nat" shows "fls_compose_power (f - g) d = fls_compose_power f d - fls_compose_power g d"
sccs_verts_conv: shows "sccs_verts = verts ` sccs"
assms_3: shows "u \<preceq>\<^sub>F\<^sub>I w"
rk_AA'Ca: fixes A :: "Points"   and A' :: "Points"   and C :: "Points"   and B :: "Points"   and P :: "Points"   and a :: "Points"   and B' :: "Points"   and C' :: "Points" assumes "3 \<le> rk {A, A', C}"   and "4 \<le> rk {A, B, P, a}"   and "rk {A, B, C, A', B', C', P} = 3" shows "4 \<le> rk {A, A', C, a}"
per_init_per: fixes D :: "'a set" shows "part_equiv (per_init D)"
simps_4: fixes n :: "nat"   and m :: "nat" shows "geod (Suc n) (Suc (2 * m)) = middle (geod n m) (geod n (Suc m))"
iT_Plus_Plus_neg_assoc: fixes b :: "nat"   and a :: "nat"   and I :: "nat set" assumes "b \<le> a" shows "I \<oplus> a \<oplus>- b = I \<oplus> a - b"
hyps: fixes f :: "nat \<Rightarrow> 'a" shows "convergent (\<lambda>n. f (n + m)) = convergent f"
card_lists_length_eq: fixes A :: "'c set"   and n :: "nat" assumes "finite A" shows "card {xs. set xs \<subseteq> A \<and> length xs = n} = card A ^ n"
RID\<^sub>GI_1: fixes s :: "'b \<Rightarrow> (('a \<Rightarrow> ('b, 'a, 'c) val option) \<times> ('a \<Rightarrow> ('b, 'a, 'c) val option) \<times> ('b, 'a, 'c) expr) option"   and r :: "'b"   and v :: "('a \<Rightarrow> ('b, 'a, 'c) val option) \<times> ('a \<Rightarrow> ('b, 'a, 'c) val option) \<times> ('b, 'a, 'c) expr" assumes "s r = Some v" shows "r \<in> RID\<^sub>G s"
hyps: assumes "\<And>k. k < n \<Longrightarrow> f k \<in> carrier_mat d d"   and "\<And>k. k < n \<Longrightarrow> g k \<in> carrier_mat d d"   and "\<And>k. k < n \<Longrightarrow> f k \<le>\<^sub>L g k" shows "matrix_sum d f n \<le>\<^sub>L matrix_sum d g n"
A_FactGroup_onto: assumes "h ` carrier G = carrier H" shows "(\<lambda>X. the_elem (h ` X)) ` carrier (G A_Mod a_kernel G H h) = carrier H"
member_conv_keys: fixes t :: "'a set_rbt" shows "member t = (\<lambda>x. x \<in> set (keys t))"
get_parent_locs_def: shows "get_parent_locs \<equiv> l_get_parent\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_parent_locs get_child_nodes_locs"
Acts_eq: fixes init :: "'b set"   and acts :: "('b \<times> 'b) set set"   and allowed :: "('b \<times> 'b) set set" shows "Acts (mk_program (init, acts, allowed)) = insert Id acts"
generalized_sfw_def: fixes l :: "('d simple_match \<times> 'e) list"   and p :: "('d, 'f) simple_packet_scheme" shows "generalized_sfw l p \<equiv> find (\<lambda>(m, a). simple_matches m p) l"
is_serial_solution_then_abstracted: fixes ops :: "(nat, nat) sas_plus_operator list" assumes "is_serial_solution_for_problem abs_prob ops" shows "\<forall>op\<in>set ops. \<exists>ast_op\<in>set ast\<delta>. op = abs_ast_operator ast_op"
openI: fixes S :: "'b set" assumes "\<And>x. x \<in> S \<Longrightarrow> \<exists>e>0. ball x e \<subseteq> S" shows "open S"
left_neutral: fixes a :: "'a multiset" shows "\<Empt> \<union># a = a"
hyp: shows "\<exists>\<^sub>\<infinity>n. w n \<in> S"
dist_midpoint_2: fixes b :: "'a"   and a :: "'a" shows "dist b (midpoint a b) = dist a b / 2"
finite_Diff: fixes A :: "'c set"   and B :: "'c set" assumes "finite A" shows "finite (A - B)"
intros: fixes M :: "'b multiset" assumes "\<forall>n. count M n < base" shows "local.well_base M"
isInAlternativeList: fixes aa :: "'b"   and a :: "('b, 'c) Combinators"   and p :: "('b, 'c) Combinators list" assumes "aa \<in> set (net_list_aux [a]) \<or> aa \<in> set (net_list_aux p)" shows "aa \<in> set (net_list_aux (a # p))"
hom_add_eq_zero: fixes x :: "'b poly"   and y :: "'b poly" assumes "x + y = 0" shows "pCons (0::'b) x + pCons (0::'b) y = 0"
rtrancl_trans: fixes x :: "'b"   and y :: "'b"   and r :: "('b \<times> 'b) set"   and z :: "'b" assumes "(x, y) \<in> r\<^sup>*"   and "(y, z) \<in> r\<^sup>*" shows "(x, z) \<in> r\<^sup>*"
private_fun_deduct_in_ik: fixes M :: "('fun, 'var) Term.term set"   and t :: "('fun, 'var) Term.term"   and f :: "'fun"   and T :: "('fun, 'var) Term.term list" assumes "M \<turnstile> t"   and "Fun f T \<sqsubseteq> t"   and "\<not> public f" shows "Fun f T \<sqsubseteq>\<^sub>s\<^sub>e\<^sub>t M"
homotopic_join_subpaths2: fixes s :: "'b set"   and u :: "real"   and v :: "real"   and g :: "real \<Rightarrow> 'b"   and w :: "real" assumes "homotopic_paths s (subpath u v g +++ subpath v w g) (subpath u w g)" shows "homotopic_paths s (subpath w v g +++ subpath v u g) (subpath w u g)"
fg_symmetric: shows "OpposedThinChamberComplexFoldings X g f D0"
comp_arr_point: fixes f :: "'s"   and x :: "'s" assumes "arr f"   and "\<guillemotleft>x : unity \<rightarrow> local.dom f\<guillemotright>" shows "f \<cdot> x = inv_into (hom unity (cod f)) img (Fun f (img x))"
poly_of_int_poly_of_real: fixes p :: "int poly"   and x :: "real" shows "poly (of_int_poly p) (of_real x) = of_real (poly (of_int_poly p) x)"
assms: fixes x :: "real" assumes "x \<in> A" shows "(f has_real_derivative f' x) (at x)"
sgn_sdiv_eq_sgn_mult: fixes a :: "int"   and b :: "int" assumes "a sdiv b \<noteq> 0" shows "sgn (a sdiv b) = sgn (a * b)"
equivE: fixes A :: "'b set"   and r :: "('b \<times> 'b) set" assumes "equiv A r" obtains "refl_on A r" and "sym r" and "trans r"
assms_5: shows "q * f\<^sup>T \<le> - e"
prWls_def: fixes g :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> ('varSort \<Rightarrow> 'a \<Rightarrow> 'b) \<Rightarrow> 'b"   and SEM :: "('c, 'd, 'e, 'sort, 'f, 'b, 'g) semDom_scheme" shows "prWls g SEM \<equiv> \<forall>s X val. wls s X \<and> sWlsVal SEM val \<longrightarrow> sWls SEM s (g X val)"
IH: shows "map_of (local.alist_product xs ys) (a, c) = (case map_of xs a of None \<Rightarrow> None | Some b \<Rightarrow> map_option (f a b c) (map_of ys c))"
matchers_map_def: fixes m :: "'d \<Rightarrow> ('e, 'f) Term.term option" shows "matchers_map m = matchers ((\<lambda>x. (Var x, the (m x))) ` dom m)"
fmsubset_restrict_set: fixes S :: "'d set"   and m :: "('d, 'e) fmap" shows "fmrestrict_set S m \<subseteq>\<^sub>f m"
hom_closed: fixes x :: "(((nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real) \<Rightarrow>\<^sub>0 int) set" assumes "x \<in> carrier (reduced_homology_group (int n - 1) (nsphere (n - Suc 0)))" shows "gp x \<in> carrier (relative_homology_group (int n) (subtopology (nsphere n) {x. x n \<le> 0}) {x. x n = 0})"
sympartp_dual: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "sympartp (\<lambda>x y. r y x) = sympartp r"
unlift'_equiv: fixes x :: "'a itrm"   and y :: "'a itrm"   and n :: "nat"   and i :: "nat" assumes "x \<simeq> y" shows "unlift' n x i \<leftrightarrow> unlift' n y i"
pos_le_pos_iff: fixes A :: "'a"   and B :: "'a" shows "(Pos A \<le> Pos B) = (A \<le> B)"
cases: fixes a :: "Predicate_Compile_Tests.char" assumes "is_D_or_E a" obtains x :: "Predicate_Compile_Tests.char" where "a = x" and "x = D \<or> x = E"
intros_4: fixes \<sigma> :: "'a \<Rightarrow> 'b"   and k :: "nat"   and \<sigma>' :: "'a \<Rightarrow> 'b"   and k' :: "nat"   and prob :: "('c, 'a, 'b, 'd) ifc_problem" assumes "((\<sigma>, k), \<sigma>', k') \<in> IFC_def.cp prob" shows "((\<sigma>', k'), \<sigma>, k) \<in> IFC_def.cp prob"
cases: fixes rh :: "'e \<Rightarrow> 'f"   and bo :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and a1 :: "('e, 'f) T\<^sub>m"   and a2 :: "('e, 'f) T\<^sub>m" assumes "blinding_of_T rh bo a1 a2" obtains x :: "('e, 'f, ('e, 'f) T\<^sub>m, 'f T\<^sub>h) F\<^sub>m" and y :: "('e, 'f, ('e, 'f) T\<^sub>m, 'f T\<^sub>h) F\<^sub>m" where "a1 = T\<^sub>m x" and "a2 = T\<^sub>m y" and "blinding_of_F rh bo (root_hash_T rh) (blinding_of_T rh bo) x y"
is_const_def: shows "is_const \<equiv> const 0 \<or> const 1"
iIN_imirror_ident: fixes n :: "nat"   and d :: "nat" shows "imirror [n\<dots>,d] = [n\<dots>,d]"
runit_simps_1: fixes f :: "'a" assumes "ide f" shows "arr \<r>[f]"
ok'_1: shows "term_ok \<Theta> (Abs \<tau> (f $ Bv 0))"
one_fds_def: shows "1 = fds (\<lambda>n. if n = 1 then 1::'b else (0::'b))"
x_c_lb1: fixes x :: "'a"   and y :: "'a" shows "x \<cdot> 1\<^sub>\<pi> \<parallel> (y \<cdot> 1\<^sub>\<pi>) \<le> x \<cdot> 1\<^sub>\<pi>"
sel_3: fixes x1 :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x2 :: "('a, 'b) state \<Rightarrow> (nat, 'a \<times> 'b) mapping"   and x3 :: "('a, 'b) state \<Rightarrow> (nat, 'a \<times> 'b) mapping"   and x4 :: "('a, 'b) state \<Rightarrow> nat \<Rightarrow> 'b option"   and x5 :: "('a, 'b) state \<Rightarrow> nat \<Rightarrow> 'b option"   and x6 :: "('a, 'b) state \<Rightarrow> nat \<Rightarrow> 'a"   and x7 :: "('a, 'b) state \<Rightarrow> nat \<Rightarrow> 'a"   and x8 :: "((nat, 'a \<times> 'b) mapping \<Rightarrow> (nat, 'a \<times> 'b) mapping) \<Rightarrow> ('a, 'b) state \<Rightarrow> ('a, 'b) state"   and x9 :: "nat \<Rightarrow> 'b \<Rightarrow> 'b atom"   and x10 :: "nat \<Rightarrow> 'b \<Rightarrow> 'b atom"   and x11 :: "rat \<Rightarrow> rat \<Rightarrow> bool" shows "Direction.UBI (Direction x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11) = x3"
trans: fixes a :: "real"   and b :: "real"   and c :: "real" assumes "P a b"   and "P b c"   and "a \<le> b"   and "b \<le> c" shows "P a c"
r1a3: fixes y :: "'a" assumes "y \<in> {flow0 x t1<--<b}" shows "{0<..<d} \<subseteq> existence_ivl0 y"
simps_1: fixes A :: "nat \<Rightarrow> 'at"   and S :: "'at Literal set set" shows "resolvents_sequence A S 0 = simplify S"
bounded_arc_image: fixes g :: "real \<Rightarrow> 'b" assumes "arc g" shows "bounded (path_image g)"
glbound_in_ofD_in: fixes P :: "'a set"   and x :: "'a"   and y :: "'a"   and b :: "'a" assumes "glbound_in_of P x y b" shows "b \<in> P"
blinding_of_blindable_simps2_1: fixes h :: "'c \<Rightarrow> 'd"   and bo :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" shows "blinding_of_blindable h bo (Unblinded x) (Unblinded y) = bo x y"
add_Suc: fixes m :: "nat"   and n :: "nat" shows "Suc m + n = Suc (m + n)"
monic_degree_map_poly_hom: fixes p :: "int poly" assumes "monic p" shows "degree (of_int_poly p) = degree p"
False: shows "\<not> (\<forall>n. x [^] n = \<one> \<longrightarrow> n = 0)"
max_eq_sum: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a" assumes "class.add_linorder plus minus zero less_eq less" obtains y :: "'a" where "ord.max less_eq a b = plus a y"
INF_inf_const2: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and I :: "'b set"   and f :: "'b \<Rightarrow> 'a"   and x :: "'a" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "I \<noteq> {}" shows "Inf ((\<lambda>i. inf (f i) x) ` I) = inf (Inf (f ` I)) x"
lsimulation_this: shows "lsimulation ((R1 O R2)\<inverse>) C A"
elims_1: fixes x :: "'a option"   and xa :: "'a option"   and y :: "bool" assumes "(x \<sqsubseteq> xa) = y" obtains xb :: "'a" and ya :: "'a" where "x = Some xb" and "xa = Some ya__" and "y = (xb \<sqsubseteq> ya__)"   | ya :: "'a option" where "x = None" and "xa = ya" and "y"   | uu_ :: "'a" where "x = Some uu_" and "xa = None" and "\<not> y"
solves_ode_vderivD: fixes y :: "real \<Rightarrow> 'b"   and f :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and T :: "real set"   and X :: "'b set" assumes "(y solves_ode f) T X" shows "(y has_vderiv_on (\<lambda>t. f t (y t))) T"
fg_2: shows "g piecewise_C1_differentiable_on f ` S"
FSbImorph_def: fixes h :: "'d \<Rightarrow> 'e"   and hA :: "'f \<Rightarrow> 'g"   and MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'h, 'd, 'f, 'i) model_scheme"   and MOD' :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'h, 'e, 'g, 'j) model_scheme" shows "FSbImorph h hA MOD MOD' \<equiv> ipresIGWlsAll h hA MOD MOD' \<and> ipresIGCons h hA MOD MOD' \<and> ipresIGFreshAll h hA MOD MOD' \<and> ipresIGSubstAll h hA MOD MOD'"
synth_crypt: fixes K :: "nat"   and X :: "msg"   and H :: "msg set" assumes "Crypt K X \<in> synth H" shows "Crypt K X \<in> H \<or> X \<in> synth H \<and> Key K \<in> H"
true_cls_empty: fixes I :: "'a set" shows "\<not> I \<Turnstile> {#}"
j0: shows "vec_norm (vs ! j) \<noteq> 0"
differentiable_bound: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and f' :: "'a \<Rightarrow> 'a \<Rightarrow> 'b"   and B :: "real"   and x :: "'a"   and y :: "'a" assumes "convex S"   and "\<And>x. x \<in> S \<Longrightarrow> (f has_derivative f' x) (at x within S)"   and "\<And>x. x \<in> S \<Longrightarrow> onorm (f' x) \<le> B"   and "x \<in> S"   and "y \<in> S" shows "norm (f x - f y) \<le> B * norm (x - y)"
induct: fixes P :: "k \<Rightarrow> bool"   and k :: "k" assumes "P kno"   and "P unk" shows "P k"
fermat_witness_imp_strong_fermat_witness: assumes "1 < n"   and "local.fermat_witness" shows "local.strong_fermat_witness"
strict_trans: fixes b :: "'a Word.word"   and a :: "'a Word.word"   and c :: "'a Word.word" assumes "b <s a"   and "c <s b" shows "c <s a"
pmf_pred_mono_strong: fixes P :: "'c \<Rightarrow> bool"   and p :: "'c pmf"   and P' :: "'c \<Rightarrow> bool" assumes "pred_pmf P p"   and "\<And>a. \<lbrakk>a \<in> set_pmf p; P a\<rbrakk> \<Longrightarrow> P' a" shows "pred_pmf P' p"