hcju/isabelleps huggingface dataset:
supd_lupd: fixes vn :: "lname"   and v :: "val"   and s :: "state" shows "supd lupd(vn\<mapsto>v) s = (fst s, lupd(vn\<mapsto>v) (snd s))"
pref_iffI: fixes F :: "'b \<Rightarrow> bool"   and G :: "'b \<Rightarrow> bool" assumes "|~ F \<longrightarrow> G"   and "|~ G \<longrightarrow> F" shows "|~ F = G"
timpls_transformable_to'_trans: fixes TI :: "('e set \<times> 'e set) list"   and s :: "(('f, 'g, 'e) prot_fun, 'h) Term.term"   and t :: "(('f, 'g, 'e) prot_fun, 'h) Term.term"   and u :: "(('f, 'g, 'e) prot_fun, 'h) Term.term" assumes "timpls_transformable_to' TI s t"   and "timpls_transformable_to' TI t u" shows "timpls_transformable_to' TI s u"
mk_rtrancl_list_main: fixes r :: "'b \<Rightarrow> 'b list"   and subsumes :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and fin :: "'b set"   and finl :: "'b list"   and todo :: "'b list" assumes "relation_subsumption_list r subsumes"   and "fin = set finl" shows "set (mk_rtrancl_list_main subsumes r todo finl) = relation_subsumption_impl.mk_rtrancl_main r subsumes set (\<lambda>x. case x of h # t \<Rightarrow> (h, t)) (@) todo fin"
awalk_cyc_decomp_has_prop: fixes G :: "('c, 'd) pre_digraph"   and u :: "'c"   and p :: "'d list"   and v :: "'c" assumes "wf_digraph G"   and "pre_digraph.awalk G u p v"   and "\<not> distinct (pre_digraph.awalk_verts G u p)" shows "wf_digraph.is_awalk_cyc_decomp G p (wf_digraph.awalk_cyc_decomp G p)"
INT_subset_iff: fixes B :: "'b set"   and A :: "'c \<Rightarrow> 'b set"   and I :: "'c set" shows "(B \<subseteq> \<Inter> (A ` I)) = (\<forall>i\<in>I. B \<subseteq> A i)"
productiveI: fixes g :: "'c \<Rightarrow> ('d, 'c) step" assumes "\<And>s. s \<in> productive_on g" shows "productive g"
infnorm_Max: fixes x :: "'b" shows "infnorm x = (MAX i\<in>Basis. \<bar>x \<bullet> i\<bar>)"
broadcast_only_valid_msgs: fixes pre :: "('a \<times> ('a, 'b) operation) event list"   and m :: "'a \<times> ('a, 'b) operation"   and i :: "nat" assumes "pre @ [Broadcast m] prefix of i" shows "\<exists>state. apply_operations pre = Some state \<and> valid_behaviours state m"
\<nu>_def: shows "\<F>C.\<nu> \<equiv> \<lambda>f. if \<F>C.arr f then \<F>C.mkarr (\<F>C.COD f\<^bold>\<down>) \<cdot> f else \<F>C.null"
HSem_ignores_fresh_restr'': fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and S :: "Vars.var set"   and \<rho> :: "Vars.var \<Rightarrow> 'Value" assumes "fv \<Gamma> \<subseteq> S" shows "(\<lbrace>\<Gamma>\<rbrace>\<rho>) f|` S = \<lbrace>\<Gamma>\<rbrace>\<rho> f|` S"
emeasure_eq_measure: fixes A :: "'b set" shows "emeasure (M \<bind> f) A = ennreal (Sigma_Algebra.measure (M \<bind> f) A)"
S3Forward: fixes p :: "PrIds"   and rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState" shows "\<turnstile> RPCFwd crCh rmCh rst p \<and> HNext rmhist p \<and> $S3 rmhist p \<and> unchanged (e p, c p, m p) \<longrightarrow> S4 rmhist p$ \<and> unchanged (rmhist!p)"
set_disconnected_nodes_typess_preserved: fixes w :: "((_) heap, exception, unit) prog"   and object_ptr :: "(_) document_ptr"   and h :: "(_) heap"   and h' :: "(_) heap" assumes "w \<in> set_disconnected_nodes_locs object_ptr"   and "h \<turnstile> w \<rightarrow>\<^sub>h h'" shows "ShadowRootClass.type_wf h = ShadowRootClass.type_wf h'"
closed_err_lift2_conv: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b err" shows "closed (err A) (lift2 f) = (\<forall>x\<in>A. \<forall>y\<in>A. x \<squnion>\<^bsub>f\<^esub> y \<in> err A)"
id: shows "A ^\<^sub>m k = P * B ^\<^sub>m k * Q"
distinct_1: fixes x2 :: "'a" shows "\<turnstile> \<noteq> \<lbrakk>x2\<rbrakk>"
enum_Suc: fixes p :: "nat"   and n :: "nat"   and base :: "nat \<Rightarrow> nat"   and upd :: "nat \<Rightarrow> nat"   and s :: "(nat \<Rightarrow> nat) set"   and i :: "nat" assumes "kuhn_simplex p n base upd s"   and "i < n" shows "kuhn_simplex.enum base upd (Suc i) = (kuhn_simplex.enum base upd i)(upd i := Suc (kuhn_simplex.enum base upd i (upd i)))"
codomains_comp: fixes g :: "('c \<Rightarrow> 's, 'c \<Rightarrow> 's) concrete_category.arr"   and f :: "('c \<Rightarrow> 's, 'c \<Rightarrow> 's) concrete_category.arr" assumes "Cop_S.seq g f" shows "Cop_S.codomains (Cop_S.comp g f) = Cop_S.codomains g"
induct: fixes P1 :: "('c, 'd, 'a, 'b, 'e) qTerm \<Rightarrow> bool"   and P2 :: "('c, 'd, 'a, 'b, 'e) qAbs \<Rightarrow> bool"   and qTerm :: "('c, 'd, 'a, 'b, 'e) qTerm"   and qAbs :: "('c, 'd, 'a, 'b, 'e) qAbs" assumes "\<And>x1 x2. P1 (qVar x1 x2)"   and "\<And>x1 x2 x3. \<lbrakk>\<And>x2a x2aa. \<lbrakk>x2a \<in> range x2; x2aa \<in> set_option x2a\<rbrakk> \<Longrightarrow> P1 x2aa; \<And>x3a x3aa. \<lbrakk>x3a \<in> range x3; x3aa \<in> set_option x3a\<rbrakk> \<Longrightarrow> P2 x3aa\<rbrakk> \<Longrightarrow> P1 (qOp x1 x2 x3)"   and "\<And>x1 x2 x3. P1 x3 \<Longrightarrow> P2 (qAbs x1 x2 x3)" shows "P1 qTerm \<and> P2 qAbs"
e_def: fixes k :: "nat" shows "e k = (\<Sum>Y | Y \<subseteq> {..<n} \<and> card Y = k. prod x Y)"
B: shows "wr \<in> w2 \<otimes>\<^bsub>\<alpha>\<^esub> w3"
len'_def: fixes a :: "'b array" shows "Array.len' a = Array.len a \<bind> (\<lambda>n. return (of_nat n))"
4: shows "kronecker_product (M_mat signs1 subs1) (M_mat signs2 subs2) $$ (i, j) = z (subs1 ! (i div length subs2)) (signs1 ! (j div length signs2)) * z (subs2 ! (i mod length subs2)) (signs2 ! (j mod length signs2))"
subspace_sum: fixes scale :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and A :: "'c set"   and B :: "'d set"   and f :: "'d \<Rightarrow> 'c" assumes "module scale"   and "module.subspace scale A"   and "\<And>x. x \<in> B \<Longrightarrow> f x \<in> A" shows "sum f B \<in> A"
L0: fixes vs :: "e list"   and lholed :: "Lholed"   and es' :: "e list"   and es :: "e list" assumes "const_list vs"   and "lholed = LBase vs es'" shows "Lfilled 0 lholed es (vs @ es @ es')"
simps_1: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "R ^^ 0 = (=)"
prems_2: shows "\<delta>_states \<delta> \<inter> \<delta>_states \<delta>' = {}"
absolute_extensor_imp_AR: fixes S :: "'a set" assumes "\<And>f U T. \<lbrakk>continuous_on T f; f ` T \<subseteq> S; closedin (top_of_set U) T\<rbrakk> \<Longrightarrow> \<exists>g. continuous_on U g \<and> g ` U \<subseteq> S \<and> (\<forall>x\<in>T. g x = f x)" shows "AR S"
step: fixes y :: "'a" shows "f\<cdot>y\<cdot>(foldl\<cdot>f\<cdot>z\<cdot>xs) = foldl\<cdot>f\<cdot>(f\<cdot>z\<cdot>y)\<cdot>xs"
simps_111: fixes x101 :: "Term.var"   and x102 :: "expr"   and x41 :: "expr"   and x42 :: "ref_ty" shows "x101:=x102 \<noteq> x41 InstOf x42"
congruentD: fixes r :: "('a \<times> 'a) set"   and f :: "'a \<Rightarrow> 'b"   and y :: "'a"   and z :: "'a" assumes "f respects r"   and "(y, z) \<in> r" shows "f y = f z"
iex_subset_imp_iex: fixes A :: "nat set"   and P :: "nat \<Rightarrow> bool"   and B :: "nat set" assumes "\<diamond> t A. P t"   and "A \<subseteq> B" shows "\<diamond> t B. P t"
iFROM_iTILL_iIN_conv: fixes n :: "nat"   and n' :: "nat" assumes "n \<le> n'" shows "[n\<dots>] \<inter> [\<dots>n'] = [n\<dots>,n' - n]"
fls_integral_const': fixes x :: "'a" assumes "inverse (1::'a) = (1::'a)" shows "fls_integral (fls_const x) = fls_const x * fls_X"
Suc_n_not_n: fixes n :: "nat" shows "Suc n \<noteq> n"
cross_add_left: fixes x :: "(real, 3) vec"   and y :: "(real, 3) vec"   and z :: "(real, 3) vec" shows "(x + y) \<times> z = x \<times> z + y \<times> z"
nn_integral_RealVal: fixes f :: "val \<Rightarrow> ennreal" assumes "f \<in> borel_measurable (stock_measure REAL)" shows "integral\<^sup>N (stock_measure REAL) f = \<integral>\<^sup>+ x. f (RealVal x) \<partial>lborel"
BinopElim1: fixes M :: "Var list \<times> (int \<times> Instr) list \<times> int"   and l :: "int"   and os :: "Val list"   and S :: "(Var \<times> Val) list"   and h :: "(Addr \<times> Class \<times> (Field \<times> Val) list) list"   and n :: "nat"   and ll :: "int"   and t :: "Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list"   and f :: "Val \<Rightarrow> Val \<Rightarrow> Val" assumes "(M, l, (os, S, h), n, ll, t) \<in> Step"   and "get_ins M l = Some (binop f)" shows "\<exists>v w ops. os = v # w # ops \<and> n = Suc 0 \<and> t = (f v w # ops, S, h) \<and> ll = l + 1"
su1_4: shows "dist (flow0 x su1) u < duv"
normal_series_extend: fixes H :: "'a set"   and \<HH> :: "'a set list" assumes "normal_series (G\<lparr>carrier := H, carrier := H\<rparr>) \<HH>"   and "H \<lhd> G\<lparr>carrier := H\<rparr>" shows "normal_series (G\<lparr>carrier := H\<rparr>) (\<HH> @ [carrier (G\<lparr>carrier := H\<rparr>)])"
d1: shows "pi \<bullet> fresh_fun h = pi \<bullet> h a'"
eventually_division_filter: fixes P :: "('a \<times> 'a set) set \<Rightarrow> bool"   and s :: "'a set" shows "eventually P (division_filter s) = (\<exists>g. gauge g \<and> (\<forall>p. p tagged_division_of s \<and> g fine p \<longrightarrow> P p))"
pos_mod_sign: fixes b :: "'b"   and a :: "'b" assumes "(0::'b) < b" shows "(0::'b) \<le> a mod b"
awalk_vertex_props: fixes u :: "'a"   and p :: "('a \<times> 'a) list"   and v :: "'a"   and P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" assumes "S.awalk u p v"   and "p \<noteq> []"   and "\<And>w. w \<in> set (pawalk_verts u p) \<Longrightarrow> P w \<or> Q w"   and "P u"   and "Q v" shows "\<exists>e\<in>set p. P (fst e) \<and> Q (snd e)"
mset_unplusm_dist_cases: fixes s :: "'a"   and A :: "'a multiset"   and B :: "'a multiset"   and C :: "'a multiset" assumes "{#s#} + A = B + C" obtains "B = {#s#} + (B - {#s#})" and "A = B - {#s#} + C"   | "C = {#s#} + (C - {#s#})" and "A = B + (C - {#s#})"
lead_coeff_smult: fixes c :: "'b"   and p :: "'b poly" shows "lead_coeff (smult c p) = c * lead_coeff p"
weak_triangle: fixes x :: "'a"   and z :: "'a"   and y :: "'a" shows "f x z \<le> sqrt 2 * max (f x y) (f y z)"
not_in_keys_iff_lookup_eq_zero: fixes k :: "'d"   and f :: "'d \<Rightarrow>\<^sub>0 'e" shows "(k \<notin> keys f) = (lookup f k = (0::'e))"
extendLevel_Nsteps: fixes belowN :: "fmla stream"   and n :: "nat"   and N :: "fmla stream"   and m :: "nat" assumes "i.R belowN = {\<phi>. max_depth \<phi> < n}"   and "i.R N = {\<phi>. max_depth \<phi> = n}" shows "\<exists>belowNext Next. (extendLevel ^^ m) (belowN, N) = (belowNext, Next) \<and> i.R belowNext = {\<phi>. max_depth \<phi> < n + m} \<and> i.R Next = {\<phi>. max_depth \<phi> = n + m}"
list_subset_sexp: fixes A :: "('b, unit) Old_Datatype.node set set" assumes "A \<subseteq> sexp" shows "list A \<subseteq> sexp"
cod_simps_2: shows "\<nu>.cod.leg0 = \<nu>.leg0"
separation_t0: fixes "open" :: "'c set \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "class.t0_space open" shows "(x \<noteq> y) = (\<exists>U. open U \<and> (x \<in> U) \<noteq> (y \<in> U))"
a: shows "list_all2 (\<approx>) (map f xa) (map f x)"
Pf_def: fixes y :: "hf" shows "Pf y \<equiv> \<exists>s k. Prf s k y"
out2__conga: fixes B :: "'p"   and A' :: "'p"   and A :: "'p"   and C' :: "'p"   and C :: "'p" assumes "B Out A' A"   and "B Out C' C" shows "A B C CongA A' B C'"
less_imp_1: fixes x :: "real"   and y :: "real" assumes "x \<in> C"   and "y \<in> C"   and "x < y" shows "f' x * (y - x) \<le> f y - f x"
intros_4: fixes F :: "'b formula"   and H :: "'b formula"   and G :: "'b formula" shows "((F::'b formula) \<^bold>\<rightarrow> (H::'b formula)) \<^bold>\<rightarrow> ((G::'b formula) \<^bold>\<rightarrow> H) \<^bold>\<rightarrow> (F \<^bold>\<or> G) \<^bold>\<rightarrow> H \<in> AX10"
simps_1: fixes \<phi> :: "'b ltln"   and \<psi> :: "'b ltln"   and w :: "nat \<Rightarrow> 'b set" shows "\<F>\<G> (\<phi> and\<^sub>n \<psi>) w = \<F>\<G> \<phi> w \<union> \<F>\<G> \<psi> w"
subset_insert_iff: fixes A :: "'b set"   and x :: "'b"   and B :: "'b set" shows "(A \<subseteq> insert x B) = (if x \<in> A then A\<setminus>{x} \<subseteq> B else A \<subseteq> B)"
mod_mult_self1: fixes divide :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and one :: "'e"   and modulo :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and euclidean_size :: "'e \<Rightarrow> nat"   and a :: "'e"   and c :: "'e"   and b :: "'e" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size" shows "modulo (plus a (times c b)) b = modulo a b"
good_context_all: fixes s :: "'b sparc_state"   and s'' :: "'b sparc_state" assumes "good_context s \<and> s'' = delayed_pool_write s" shows "(get_trap_set s = {} \<or> reset_trap_val s \<noteq> False \<or> get_ET (cpu_reg_val PSR s) \<noteq> 0) \<and> ((\<exists>e. fetch_instruction s'' = Inl e) \<or> (\<exists>v1 v2. fetch_instruction s'' = Inr v1 \<and> decode_instruction v1 = Inr v2 \<and> (annul_val s'' = True \<or> annul_val s'' = False \<and> (\<forall>v1' v2'. fetch_instruction s'' = Inr v1' \<and> decode_instruction v1' = Inr v2' \<longrightarrow> supported_instruction (fst v2') = True) \<and> (fst v2 \<noteq> ctrl_type RETT \<or> fst v2 = ctrl_type RETT \<and> (get_ET (cpu_reg_val PSR s'') = 1 \<or> get_ET (cpu_reg_val PSR s'') \<noteq> 1 \<and> get_S (cpu_reg_val PSR s'') \<noteq> 0 \<and> get_WIM_bit (nat ((uint (get_CWP (cpu_reg_val PSR s'')) + 1) mod NWINDOWS)) (cpu_reg_val WIM s'') = 0 \<and> get_addr (snd v2) s'' AND 3 = 0)))))"
elims_2: fixes x :: "atom"   and xa :: "real list" assumes "aEval x xa" obtains p :: "real mpoly" and L :: "real list" where "x = Eq p__" and "xa = La__" and "insertion (nth_default 0 La__) p__ = 0"   | p :: "real mpoly" and L :: "real list" where "x = Less pa__" and "xa = Lb__" and "insertion (nth_default 0 Lb__) pa__ < 0"   | p :: "real mpoly" and L :: "real list" where "x = Leq pb__" and "xa = Lc__" and "insertion (nth_default 0 Lc__) pb__ \<le> 0"   | p :: "real mpoly" and L :: "real list" where "x = Neq p" and "xa = L" and "insertion (nth_default 0 L) p \<noteq> 0"
hyps_2: fixes m :: "nat"   and q :: "'a \<Rightarrow> 'a" assumes "swapidseq m q" shows "swapidseq (n + m) (p \<circ> q)"
Map_in_Hom: fixes f :: "(bool, 'arr list) concrete_category.arr" assumes "arr f" shows "Map f \<in> Collect (G.path (Dom f) (Cod f))"
map_lasso_run: fixes f :: "'f \<Rightarrow> 'e"   and L :: "'f lasso" shows "run_of_lasso (map_lasso f L) = (f \<circ>\<circ> run_of_lasso) L"
signed_ucast_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 push_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and drop_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and take_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and "and" :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and or :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and xor :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and mask :: "nat \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and not :: "'b \<Rightarrow> 'b"   and w :: "'d word" assumes "class.ring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask uminus not" shows "ring_1.signed one plus zero minus (UCAST('d \<rightarrow> 'c) w) = ring_bit_operations.signed_take_bit zero times one bit take_bit or mask not (LENGTH('c) - Suc 0) (semiring_1.unsigned one plus zero w)"
bijections_in_domain_permutation: fixes A :: "'a set"   and B :: "'b set" assumes "finite A"   and "finite B"   and "card A = card B" shows "{f \<in> A \<rightarrow>\<^sub>E B. bij_betw f A B} \<in> {f \<in> A \<rightarrow>\<^sub>E B. bij_betw f A B} // domain_permutation A B"
degree_pcompose: fixes p :: "'b poly"   and q :: "'b poly" shows "degree (p \<circ>\<^sub>p q) = degree p * degree q"
function_factors_left: fixes g :: "'d \<Rightarrow> 'e"   and f :: "'d \<Rightarrow> 'f" shows "(\<forall>x y. g x = g y \<longrightarrow> f x = f y) = (\<exists>h. f = h \<circ> g)"
monotone_parametric: fixes A :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and B :: "'h \<Rightarrow> 'i \<Rightarrow> bool" assumes "bi_total A" shows "rel_fun (rel_fun A (rel_fun A (=))) (rel_fun (rel_fun B (rel_fun B (=))) (rel_fun (rel_fun A B) (=))) monotone monotone"
asem_def: fixes x :: "char list"   and e :: "aexp"   and S :: "(char list \<Rightarrow> 'a) option" shows "asem x e S = (case S of None \<Rightarrow> None | Some S \<Rightarrow> Some (S(x := aval' e S)))"
visits_via_def: fixes P :: "'c llist"   and A :: "'c set"   and W :: "'c set" shows "G'.visits_via P A W \<equiv> \<exists>n. enat n < llength P \<and> P $ n \<in> W \<and> lset (ltake (enat n) P) \<subseteq> A"
undo_factorize_correct: fixes p :: "rat poly"   and c :: "rat"   and fs :: "(rat poly \<times> nat) list"   and ftrs :: "rat poly list"   and x :: "real" assumes "factorize_rat_poly_monic p = (c, fs)"   and "\<And>f p. (f, p) \<in> set fs \<Longrightarrow> f \<in> set ftrs" shows "undo_factorize (c, map (\<lambda>(f, pow). (index_of ftrs f, pow)) fs) (sign_vec ftrs x) = squash (rpoly p x)"
merkle_interfaceD: fixes h :: "'c \<Rightarrow> 'd"   and bo :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and m :: "'c \<Rightarrow> 'c \<Rightarrow> 'c option" assumes "merkle_interface h bo m" shows "merge_on UNIV h bo m"
Shallow_axioms: shows "Shallow UNIV s0 \<delta> UNIV L UNIV"
typed_no_annotated_stop: fixes aexp_vars :: "'a \<Rightarrow> 'b set"   and bexp_vars :: "'c \<Rightarrow> 'b set"   and some_val :: "'d"   and some_val' :: "'d"   and ev\<^sub>A :: "('b \<Rightarrow> 'd) \<Rightarrow> 'a \<Rightarrow> 'd"   and ev\<^sub>B :: "('b \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> bool"   and dma :: "'b \<Rightarrow> Sec"   and \<Gamma> :: "'b \<Rightarrow> Sec option"   and c :: "('b, 'a, 'c) Stmt"   and \<Gamma>' :: "'b \<Rightarrow> Sec option" assumes "sifum_types aexp_vars bexp_vars some_val some_val' ev\<^sub>A ev\<^sub>B"   and "sifum_types.has_type aexp_vars bexp_vars dma ev\<^sub>A ev\<^sub>B \<Gamma> c \<Gamma>'" shows "\<not> sifum_types.has_annotated_stop c"
extend_def: fixes ord :: "WS1S_Prelim.order"   and b :: "'a" shows "extend ord b \<equiv> \<lambda>(bs1, bs2). case ord of FO \<Rightarrow> (b # bs1, bs2) | SO \<Rightarrow> (bs1, b # bs2)"
assms: shows "retraction_map (nsphere p) (nsphere p) f"
Gchain_def: fixes n :: "nat"   and g :: "nat \<Rightarrow> ('e set, 'f) Group_scheme" shows "Gchain n g = (\<forall>l\<le>n. Group (g l))"
assms_2: fixes x :: "'a"   and y :: "'a" assumes "x \<in> X"   and "y \<in> X" shows "\<exists>G. geodesic_segment_between G x y \<and> (\<forall>z\<in>G. infdist z X \<le> C)"
rec_newrgt3_def: shows "rec_newrgt3 = Cn 2 rec_quo [recf.id 2 1, Cn 2 (constn 2) [recf.id 2 0]]"
circ_decompose_7: fixes x :: "'a"   and y :: "'a" shows "(x \<squnion> y)\<^sup>\<star> = x\<^sup>\<star> * y\<^sup>\<star> * (x \<squnion> y)\<^sup>\<star>"
supset_glbound_in_ofI: fixes b :: "'b set"   and P :: "'b set set"   and x :: "'b set"   and y :: "'b set" assumes "b \<in> P"   and "dual_order.lbound_of x y b"   and "\<And>a. \<lbrakk>a \<in> P; dual_order.lbound_of x y a\<rbrakk> \<Longrightarrow> order.greater_eq a b" shows "dual_order.glbound_in_of P x y b"
fmran'_alt_def: fixes m :: "('e, 'd) fmap" shows "fmran' m = fset (fmran m)"
False: shows "\<not> (\<forall>t\<in>{0..1} - {1 / 2}. Re ((poly p \<circ> circlepath z0 r) t) = 0)"
hnr_pre_pure_conv: fixes \<Gamma> :: "assn"   and P :: "bool"   and c :: "'a Heap"   and \<Gamma>' :: "assn"   and R :: "'b \<Rightarrow> 'a \<Rightarrow> assn"   and a :: "'b nres" shows "hn_refine (\<Gamma> * \<up> P) c \<Gamma>' R a = (P \<longrightarrow> hn_refine \<Gamma> c \<Gamma>' R a)"
bs: fixes b :: "'a poly" assumes "b \<in> set bs" shows "0 < degree b"
weakly_preserves_trg: fixes \<mu> :: "'c" assumes "C.arr \<mu>" shows "D.isomorphic (local.map (trg\<^sub>C \<mu>)) (trg\<^sub>D (local.map \<mu>))"
assms_3: shows "II \<in> local.shift cl n ` P n"
add_less_cancel_left: fixes c :: "complex"   and a :: "complex"   and b :: "complex" shows "(c + a <\<^sub>\<complex> c + b) = (a <\<^sub>\<complex> b)"
consequence_relation_family_axioms: shows "consequence_relation_family Bot_F Q (\<lambda>q N1 N2. entails_q q (no_labels.\<G>_Fset_q q N1) (no_labels.\<G>_Fset_q q N2))"
lists_drop_emp': fixes us :: "'b list list"   and C :: "'b list set" assumes "us \<in> lists C" shows "filter (\<lambda>x. x \<noteq> \<epsilon>) us \<in> lists C\<^sub>+"
compose_eq: fixes x :: "'a"   and A :: "'a set"   and g :: "'c \<Rightarrow> 'b"   and f :: "'a \<Rightarrow> 'c" assumes "x \<in> A" shows "compose A g f x = g (f x)"
add_mset_subseteq_single_iff: fixes a :: "'a"   and M :: "'a multiset"   and b :: "'a" shows "(add_mset a M \<subseteq># \<LM> b  \<RM>) = (M = \<Empt> \<and> a = b)"
cases: fixes r :: "'b Vector" obtains tdir :: "'b" and xdir :: "'b" and ydir :: "'b" and zdir :: "'b" where "r = \<lparr>tdir = tdir, xdir = xdir, ydir = ydir, zdir = zdir\<rparr>"
Ext_foo_no_hide: fixes sig :: "sig"   and old :: "qtname \<times> methd"   and P :: "bool" assumes "tprg,sig\<turnstile> (Ext, snd Ext_foo) hides old" shows "P"
minus_unique: fixes y :: "'c \<Rightarrow> 'a"   and x :: "'c \<Rightarrow> 'a"   and y' :: "'c \<Rightarrow> 'a" assumes "y \<oplus>\<^bsub>F\<^esub> x = \<zero>\<^bsub>F\<^esub>"   and "x \<oplus>\<^bsub>F\<^esub> y' = \<zero>\<^bsub>F\<^esub>"   and "x \<in> carrier F"   and "y \<in> carrier F"   and "y' \<in> carrier F" shows "y = y'"
readCArray_def: fixes array :: "nat \<Rightarrow> 'a"   and index :: "nat" shows "readCArray array index \<equiv> array index"
submonoid_is_monoid: fixes H :: "'e set"   and G :: "('e, 'f) monoid_scheme" assumes "submonoid H G"   and "Group.monoid G" shows "Group.monoid (G\<lparr>carrier := H\<rparr>)"
nres_transfer_2: shows "nres_of dFAIL = FAIL"
equiv_trans_le_trans: fixes t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and u :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and v :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" assumes "equiv_trans t u"   and "le_trans u v" shows "le_trans t v"
ball_UNIV: fixes P :: "'c \<Rightarrow> bool" shows "Ball UNIV P = All P"
iAll_def: fixes I :: "nat set"   and P :: "nat \<Rightarrow> bool" shows "iAll I P \<equiv> \<forall>t\<in>I. P t"
wf_rderiv0: fixes idx :: "'n"   and a :: "'a"   and x :: "'x" assumes "wf0 idx a" shows "local.wf idx (rderiv0 x a)"
C_def: shows "C = {x \<in> {(int p * int q - 1) div 2<..int p * int q - 1}. x mod int p \<in> {(int p - 1) div 2<..int p - 1} \<and> x mod int q \<in> {0<..(int q - 1) div 2}}"
sturm_tarski_R: fixes p :: "real poly"   and q :: "real poly" shows "taq {x. poly p x = 0} q = changes_R_smods p (pderiv p * q)"
vec_smult_distr_right: fixes \<alpha> :: "'b"   and \<beta> :: "'b"   and as :: "'b list" shows "vec_smult (\<alpha> + \<beta>) as = vec_plus (vec_smult \<alpha> as) (vec_smult \<beta> as)"
wcode_main_first_part_len: shows "length t_wcode_main_first_part = 24"
intros: fixes a :: "'a set"   and b :: "'a set"   and TI :: "('a set \<times> 'a set) set"   and t :: "(('b, 'c, 'a) prot_fun, 'd) Term.term"   and s :: "(('b, 'c, 'a) prot_fun, 'd) Term.term" assumes "(a, b) \<in> TI"   and "term_variants_pred ((\<lambda>_. [])(Abs a := [Abs b])) t s" shows "(t, s) \<in> timpl_closure'_step TI"
induct: fixes \<phi> :: "'a stream \<Rightarrow> bool"   and x :: "'a stream"   and P :: "'a stream \<Rightarrow> bool" assumes "ev \<phi> x"   and "\<And>xs. \<phi> xs \<Longrightarrow> P xs"   and "\<And>xs. \<lbrakk>ev \<phi> (stl xs); P (stl xs)\<rbrakk> \<Longrightarrow> P xs" shows "P x"
card_1_singletonE: fixes A :: "'b set" assumes "card A = 1" obtains x :: "'b" where "A = {x}"
less_complexI: fixes x :: "complex"   and y :: "complex" assumes "Re x < Re y"   and "Im x = Im y" shows "x < y"
kop_contrav: fixes f :: "'d \<Rightarrow> 'e set"   and g :: "'e \<Rightarrow> 'c set" shows "op\<^sub>K (f \<circ>\<^sub>K g) = op\<^sub>K g \<circ>\<^sub>K op\<^sub>K f"