hcju/isabelleps huggingface dataset:
start_prog_Start_sees_methods: fixes P :: "jvm_prog"   and Mm :: "char list \<Rightarrow> ((staticb \<times> ty list \<times> ty \<times> jvm_method) \<times> char list) option"   and C :: "char list"   and M :: "char list" assumes "P \<turnstile> Object sees_methods Mm" shows "class_add P (start_class C M) \<turnstile> Start sees_methods Mm ++ (map_option (\<lambda>m. (m, Start)) \<circ> map_of [start_method C M, start_clinit])"
add_def: fixes a :: "'a"   and p\<^sub>1 :: "'a point"   and p\<^sub>2 :: "'a point" shows "local.add a p\<^sub>1 p\<^sub>2 = (case p\<^sub>1 of Infinity \<Rightarrow> p\<^sub>2 | Point x\<^sub>1 y\<^sub>1 \<Rightarrow> case p\<^sub>2 of Infinity \<Rightarrow> p\<^sub>1 | Point x\<^sub>2 y\<^sub>2 \<Rightarrow> if x\<^sub>1 = x\<^sub>2 then if y\<^sub>1 = \<ominus> y\<^sub>2 then Infinity else let l = (\<guillemotleft>3\<guillemotright> \<otimes> x\<^sub>1 [^] 2 \<oplus> a) \<oslash> (\<guillemotleft>2\<guillemotright> \<otimes> y\<^sub>1); x\<^sub>3 = l [^] 2 \<ominus> \<guillemotleft>2\<guillemotright> \<otimes> x\<^sub>1 in Point x\<^sub>3 (\<ominus> y\<^sub>1 \<ominus> l \<otimes> (x\<^sub>3 \<ominus> x\<^sub>1)) else let l = (y\<^sub>2 \<ominus> y\<^sub>1) \<oslash> (x\<^sub>2 \<ominus> x\<^sub>1); x\<^sub>3 = l [^] 2 \<ominus> x\<^sub>1 \<ominus> x\<^sub>2 in Point x\<^sub>3 (\<ominus> y\<^sub>1 \<ominus> l \<otimes> (x\<^sub>3 \<ominus> x\<^sub>1)))"
BoundR_def: fixes x3 :: "subject"   and x1 :: "name"   and x2 :: "pi" shows "x3\<guillemotleft>x1\<guillemotright> \<prec> x2 \<equiv> Abs_residual (BoundR_Rep x3 ([x1].x2))"
axioms_1: fixes x :: "'c \<Longrightarrow> 'd" assumes "mwb_lens x" shows "weak_lens x"
wajsberg2: fixes a :: "'a"   and b :: "'a" shows "(a r\<rightarrow> b) l\<rightarrow> b = (b r\<rightarrow> a) l\<rightarrow> a"
edges_undef_def: fixes l :: "('c \<times> 'd \<times> 'e) list"   and w :: "'c \<times> 'd \<Rightarrow> 'f" shows "edges_undef l w \<equiv> \<forall>u v d. (u, v, d) \<in> set l \<longrightarrow> w (u, v) = \<infinity>"
imp_ex: fixes P :: "'c \<Rightarrow> bool"   and Q :: "bool" shows "((\<exists>x. P x) \<longrightarrow> Q) = (\<forall>x. P x \<longrightarrow> Q)"
mat_O_mult_beta: shows "mat_O *\<^sub>v \<beta> = - \<beta>"
spanning_subset_independent: fixes B :: "'c set"   and A :: "'c set" assumes "B \<subseteq> A"   and "vs2.independent A"   and "A \<subseteq> vs2.span B" shows "A = B"
partition_aux: fixes X :: "'a set"   and r :: "'a \<times> 'a \<Rightarrow> bool"   and xs :: "'a list" assumes "equiv X (rel_ext r)"   and "set xs \<subseteq> X" shows "fst (partition_aux r xs) = [] \<and> set ` set (snd (partition_aux r xs)) = set xs // Restr (rel_ext r) (set xs)"
remove_key_sum: fixes k :: "'a"   and f :: "'a \<Rightarrow>\<^sub>0 'b" shows "remove_key k f + monomial (lookup f k) k = f"
map_cong': fixes xs :: "'a list"   and h :: "'b \<Rightarrow> 'a"   and ys :: "'b list"   and f :: "'a \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "xs = map h ys"   and "\<And>y. y \<in> set ys \<Longrightarrow> f (h y) = g y" shows "map f xs = map g ys"
sum_power2_gt_zero_iff: fixes 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 sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and x :: "'b"   and y :: "'b" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less zero (plus (power.power one times x 2) (power.power one times y 2)) = (x \<noteq> zero \<or> y \<noteq> zero)"
gval_fold_rev_false: fixes G :: "'a gexp list"   and s :: "'a \<Rightarrow> value option" assumes "gval (fold gAnd (rev G) (Bc True)) s = false" shows "gval (fold gAnd G (Bc True)) s = false"
sconf': shows "P,E \<turnstile> s'' \<surd>"
cm_conds: shows "\<forall>(loc, t)\<in>#\<^sub>zhd (c_msg (exchange_config c0) p q). \<exists>t'. t' \<in>\<^sub>A frontier (c_imp (prop_config c0 q) loc) \<and> t' \<le> t"
neutral: fixes A :: "'c set"   and g :: "'c \<Rightarrow> 'a" assumes "\<forall>x\<in>A. g x = \<^bold>1" shows "F g A = \<^bold>1"
height_bal_subtrees_merge: fixes as :: "('c btree \<times> 'c) list"   and a :: "'c btree"   and bs :: "('d btree \<times> 'd) list"   and b :: "'d btree" assumes "height (Node as a) = height (Node bs b)"   and "bal (Node as a)"   and "bal (Node bs b)" shows "\<forall>x\<in>set (subtrees as) \<union> {a}. height x = height b"
taylor_deg_1_eval''': fixes R :: "('a, 'b) ring_scheme"   and f :: "nat \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "UP_cring R"   and "f \<in> carrier (UP R)"   and "a \<in> carrier R"   and "b \<in> carrier R"   and "c = UP_cring.to_fun R (shift 2 (UP_cring.taylor R a f)) (\<ominus>\<^bsub>R\<^esub> b)"   and "b \<otimes>\<^bsub>R\<^esub> derivative R f a = UP_cring.to_fun R f a" shows "UP_cring.to_fun R f (a \<ominus>\<^bsub>R\<^esub> b) = c \<otimes>\<^bsub>R\<^esub> b [^]\<^bsub>R\<^esub> 2"
Arg2pi_diff: fixes w :: "complex"   and z :: "complex" assumes "w \<noteq> 0"   and "z \<noteq> 0" shows "Arg2pi w - Arg2pi z = (if Arg2pi z \<le> Arg2pi w then Arg2pi (w / z) else Arg2pi (w / z) - 2 * pi)"
exhaust: fixes y :: "finite_5" obtains "y = a\<^sub>1"   | "y = a\<^sub>2"   | "y = a\<^sub>3"   | "y = a\<^sub>4"   | "y = a\<^sub>5"
space_density: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> ennreal" shows "space (density M f) = space M"
apply_affected: fixes f :: "'b perm"   and a :: "'b" shows "(apply f a \<in> affected f) = (a \<in> affected f)"
holds_smap: fixes P :: "'a \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'a"   and w :: "'b stream" shows "holds P (smap f w) = holds (P \<circ> f) w"
lookup_tensor0: fixes "is" :: "nat list"   and ds :: "nat list" assumes "is \<lhd> ds" shows "lookup (tensor0 ds) is = (0::'b)"
max_norm_witness: fixes f :: "'b poly" shows "\<exists>i. \<parallel>f\<parallel>\<^sub>\<infinity> = \<bar>coeff f i\<bar>"
OrdP_IN_SUCC_Iff: fixes H :: "fm set"   and y :: "tm"   and x :: "tm" assumes "H \<turnstile> OrdP y" shows "H \<turnstile> SUCC x IN SUCC y IFF x IN y"
Abs_inverse: fixes y :: "int" assumes "y \<in> uints LENGTH('a)" shows "uint (word_of_int y) = y"
ccSUP_commute: fixes A :: "'d set"   and B :: "'e set"   and f :: "'d \<Rightarrow> 'e \<Rightarrow> 'a" assumes "countable A"   and "countable B" shows "(SUP i\<in>A. Sup (f i ` B)) = (SUP j\<in>B. SUP i\<in>A. f i j)"
taut_split_subst: fixes x :: "'a \<Longrightarrow> 'b"   and P :: "'b upred" assumes "vwb_lens x" shows "`P` = (\<forall>v. `P\<lbrakk>\<guillemotleft>v\<guillemotright>/x\<rbrakk>`)"
assms: shows "small_steps \<pi> cs\<^sub>1 (Some (SKIP, s\<^sub>2))"
smc_in_ArrE: fixes f :: "V" assumes "f \<in>\<^sub>\<circ> \<BB>\<lparr>Arr\<rparr>" obtains a :: "V" and b :: "V" where "f : a \<mapsto>\<^bsub>\<BB>\<^esub> b" and "a \<in>\<^sub>\<circ> \<BB>\<lparr>Obj\<rparr>" and "b \<in>\<^sub>\<circ> \<BB>\<lparr>Obj\<rparr>"
vector_meet_comp_x: fixes x :: "'a" shows "x ; 1 ; x\<^sup>T = x ; 1 \<cdot> 1 ; x\<^sup>T"
indep: shows "indep_vars (\<lambda>_. borel) X (insert i I)"
intro: fixes U :: "'a set"   and one :: "'a"   and zero :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" assumes "zero_neq_one_ow U one zero"   and "comm_semiring_0_ow U plus zero times"   and "comm_monoid_mult_ow U times one" shows "comm_semiring_1_ow U times one plus zero"
simps_3: fixes iface :: "iface"   and ipassmt :: "iface \<Rightarrow> ('b word \<times> nat) list option"   and m :: "'b common_primitive match_expr"   and rs :: "'b common_primitive rule list"   and allowed :: "'b wordinterval"   and denied1 :: "'b wordinterval" shows "no_spoofing_algorithm_executable iface ipassmt (Rule m Drop # rs) allowed denied1 = no_spoofing_algorithm_executable iface ipassmt rs allowed (wordinterval_union denied1 (wordinterval_setminus (get_all_matching_src_ips_executable iface m) allowed))"
h3: assumes "Av = 0"   and "Bv = 0" shows "(Av + Bv * sqrt Cv \<le> 0) = (Av \<le> 0 \<and> Bv\<^sup>2 * Cv \<le> Av\<^sup>2 \<or> Bv \<le> 0 \<and> Av\<^sup>2 \<le> Bv\<^sup>2 * Cv)"
mem_box_2: fixes x :: "'b"   and a :: "'b"   and b :: "'b" shows "(x \<in> cbox a b) = (\<forall>i\<in>Basis. a \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> b \<bullet> i)"
is_span: shows "C.span \<eta>.cod.leg0 \<eta>.cod.leg1"
is_sup_join: fixes x :: "'a"   and y :: "'a" shows "is_sup x y (x \<squnion> y)"
rep_eq: fixes x :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and xa :: "'d set['f]"   and xb :: "'e set['f]" shows "rel_bset x xa xb = rel_set x (set_bset xa) (set_bset xb)"
word_l_diffs_4: fixes w :: "'b word"   and x :: "'b word"   and z :: "'b word" assumes "w + x \<le> z"   and "x \<le> w + x" shows "w \<le> z - x"
that: fixes g :: "'a" assumes "quasi_inverses f g" shows "thesis"
star_of_minus: fixes x :: "'a" shows "star_of (- x) = - star_of x"
intro: fixes L :: "('c, 'd) gorder_scheme" assumes "weak_partial_order L"   and "weak_lower_semilattice_axioms L" shows "weak_lower_semilattice L"
class_of_0_eq: fixes U :: "'a set"   and U' :: "'a set" assumes "U \<in> I"   and "U' \<in> I" shows "\<lfloor>U, \<zero>\<^bsub>U\<^esub>\<rfloor> = \<lfloor>U', \<zero>\<^bsub>U'\<^esub>\<rfloor>"
lc_q'': shows "punit.lc (rep_list q') = punit.lc (rep_list q)"
fds_nth_zeta: shows "fds_nth fds_zeta = (\<lambda>n. if n = 0 then 0::'a else (1::'a))"
map_simp_ide: fixes a :: "'d" assumes "D.ide a" shows "\<epsilon>FoF\<eta>.map a = (\<epsilon>.map \<circ> F) a \<cdot>\<^sub>C (F \<circ> \<eta>.map) a"
get_host_pure: fixes element_ptr :: "(_) shadow_root_ptr"   and h :: "(_) heap" shows "pure (get_host element_ptr) h"
drop_suffix_css_step': fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and cs :: "('e, 'd, 'f) com list"   and cssxs :: "(('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list"   and s :: "('e, 'f) xstate"   and cs' :: "('e, 'd, 'f) com list"   and css'xs :: "(('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list"   and t :: "('e, 'f) xstate"   and css :: "(('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list"   and xs :: "(('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list"   and css' :: "(('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list" assumes "\<Gamma>\<turnstile> (cs, cssxs, s) \<rightarrow> (cs', css'xs, t)"   and "cssxs = css @ xs"   and "css'xs = css' @ xs" shows "\<Gamma>\<turnstile> (cs, css, s) \<rightarrow> (cs', css', t)"
index_mult_mat_1: fixes i :: "nat"   and A :: "'a Matrix.mat"   and j :: "nat"   and B :: "'a Matrix.mat" assumes "i < dim_row A"   and "j < dim_col B" shows "(A * B) $$ (i, j) = Matrix.row A i \<bullet> Matrix.col B j"
assms_3: fixes P :: "'prog"   and \<sigma> :: "'state"   and \<sigma>' :: "'state" assumes "Q P \<sigma>"   and "\<sigma>' \<in> small P \<sigma>" shows "Q P \<sigma>'"
preserves_hhom: fixes \<mu> :: "'a" assumes "arr \<mu>" shows "\<guillemotleft>S.UP \<mu> : UP.map\<^sub>0 (src \<mu>) \<rightarrow>\<^sub>S UP.map\<^sub>0 (trg \<mu>)\<guillemotright>"
dvd_smult_iff: fixes a :: "'a"   and p :: "'a poly"   and q :: "'a poly" assumes "a \<noteq> (0::'a)" shows "(p dvd smult a q) = (p dvd q)"
correct_def: fixes m1 :: "'a \<times> 'a"   and m2 :: "bool" shows "mal_def.correct m1 m2 = (protocol_ot m1 m2 = mal_def.trusted_party m1 m2)"
le_fininf_concat_start: fixes w :: "'action list"   and w\<^sub>1 :: "'action list"   and w\<^sub>2 :: "'action stream" shows "(w @ w\<^sub>1 \<preceq>\<^sub>F\<^sub>I w @- w\<^sub>2) = (w\<^sub>1 \<preceq>\<^sub>F\<^sub>I w\<^sub>2)"
homeomorphismI: fixes S :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and T :: "'c set"   and g :: "'c \<Rightarrow> 'b" assumes "continuous_on S f"   and "continuous_on T g"   and "f ` S \<subseteq> T"   and "g ` T \<subseteq> S"   and "\<And>x. x \<in> S \<Longrightarrow> g (f x) = x"   and "\<And>y. y \<in> T \<Longrightarrow> f (g y) = y" shows "homeomorphism S T f g"
height_slim_node: fixes k :: "nat"   and a :: "'b"   and h :: "nat" shows "height (slim_node k a h) = h"
qfree_lift_nnf_qe: fixes qe :: "atom fm \<Rightarrow> atom fm"   and \<phi> :: "atom fm" assumes "\<And>\<phi>. nqfree \<phi> \<Longrightarrow> qfree (qe \<phi>)" shows "qfree (DLO.lift_nnf_qe qe \<phi>)"
wf_since_aux_Nil: fixes \<phi>' :: "Formula.formula"   and \<psi>' :: "Formula.formula"   and \<sigma> :: "(String.literal \<times> event_data list) trace"   and V :: "String.literal \<Rightarrow> (nat \<Rightarrow> event_data list set) option"   and R :: "event_data list set"   and I :: "\<I>"   and n :: "nat"   and b :: "bool" assumes "fv \<phi>' \<subseteq> fv \<psi>'" shows "wf_since_aux \<sigma> V R (init_args I n (fv \<phi>') (fv \<psi>') b) \<phi>' \<psi>' (init_msaux (init_args I n (fv \<phi>') (fv \<psi>') b)) 0"
PInitReq: fixes evsPIReq :: "event list"   and Transaction :: "msg"   and M :: "agent"   and C :: "agent"   and OrderDesc :: "nat"   and PurchAmt :: "nat"   and Chall_C :: "nat"   and LID_M :: "nat" assumes "evsPIReq \<in> set_pur"   and "Transaction = \<lbrace>Agent M, Agent C, Number OrderDesc, Number PurchAmt\<rbrace>"   and "Nonce Chall_C \<notin> used evsPIReq"   and "Chall_C \<notin> range CardSecret"   and "Chall_C \<notin> range PANSecret"   and "Notes C \<lbrace>Number LID_M, Transaction\<rbrace> \<in> set evsPIReq" shows "Says C M \<lbrace>Number LID_M, Nonce Chall_C\<rbrace> # evsPIReq \<in> set_pur"
diff_right_commute: fixes a :: "'c"   and c :: "'c"   and b :: "'c" shows "a - c - b = a - b - c"
the_Abs_def: fixes prot_fun :: "('b, 'c, 'a) prot_fun" shows "the_Abs prot_fun = (case prot_fun of Abs x4 \<Rightarrow> x4)"
prime_elemD2: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and p :: "'a"   and a :: "'a" assumes "class.algebraic_semidom divide plus minus zero times one"   and "comm_semiring_1.prime_elem times one zero p"   and "dvd.dvd times a p"   and "\<not> dvd.dvd times a one" shows "dvd.dvd times p a"
exec_eq_exec_n: fixes P :: "instr list"   and c :: "int \<times> (char list \<Rightarrow> int) \<times> int list"   and c' :: "int \<times> (char list \<Rightarrow> int) \<times> int list" shows "(P \<turnstile> c \<rightarrow>* c') = (\<exists>n. P \<turnstile> c \<rightarrow>^n c')"
igSwapBinp_def: fixes MOD :: "('f, 'g, 'h, 'i, 'j, 'k, 'l, 'e, 'm) model_scheme"   and zs :: "'h"   and z1 :: "'k"   and z2 :: "'k"   and binp :: "'d \<Rightarrow> 'e option" shows "igSwapBinp MOD zs z1 z2 binp \<equiv> lift (igSwapAbs MOD zs z1 z2) binp"
repeatable_both_both: fixes t :: "'a ttree"   and t' :: "'a ttree" assumes "repeatable t" shows "t \<otimes>\<otimes> t' \<otimes>\<otimes> t = t \<otimes>\<otimes> t'"
terminates_Normal_elim_cases_9: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and c :: "'b \<Rightarrow> ('b, 'a, 'c) com"   and s :: "'b" assumes "\<Gamma>\<turnstile>DynCom c \<down> Normal s" obtains "\<Gamma>\<turnstile>c s \<down> Normal s"
SN_both_mono_ordered_semiring_1_axioms: shows "SN_both_mono_ordered_semiring_1 1 (\<lambda>x y. y < x) pos_arctic"
transpose_mat_def: fixes A :: "'b mat" shows "transpose_mat A \<equiv> mat (dim_col A) (dim_row A) (\<lambda>(i, j). A $$ (j, i))"
p'_def: shows "p' = drop \<^bold>|s\<^bold>| w"
map_add_empty: fixes m :: "'g \<Rightarrow> 'h option" shows "m ++ Map.empty = m"
omega_continuousD: fixes A :: "'b set"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'b"   and b :: "'b"   and c :: "nat \<Rightarrow> 'b" assumes "omega_continuous A less_eq f"   and "b \<in> A"   and "range c \<subseteq> A"   and "monotone (\<le>) less_eq c"   and "extreme_bound A less_eq (range c) b" shows "extreme_bound A less_eq (f ` range c) (f b)"
set_zip_rightI: fixes x :: "'a"   and ys :: "'a list"   and xs :: "'b list" assumes "x \<in> set ys"   and "length ys \<le> length xs" shows "\<exists>z. (z, x) \<in> set (zip xs ys)"
ipath_restrict_eq: fixes E :: "('b \<times> 'b) set"   and r :: "nat \<Rightarrow> 'b" shows "ipath (Restr E (E\<^sup>* `` {r 0})) r = ipath E r"
that: fixes C :: "'a set" assumes "countable C"   and "C \<subseteq> K"   and "pairwise (\<lambda>i j. disjnt (cball (a i) (r i)) (cball (a j) (r j))) C"   and "\<And>i. i \<in> K \<Longrightarrow> \<exists>j. j \<in> C \<and> \<not> disjnt (cball (a i) (r i)) (cball (a j) (r j)) \<and> cball (a i) (r i) \<subseteq> ball (a j) (5 * r j)" shows "thesis"
all: shows "\<forall>xs ys. as @ [a] = xs @ ys \<longrightarrow> \<not> same_level_path_rev_aux cs ys \<or> upd_rev_cs cs ys \<noteq> []"
strong_induct: fixes x1 :: "lam"   and x2 :: "lam"   and P :: "'a \<Rightarrow> lam \<Rightarrow> lam \<Rightarrow> bool"   and z :: "'a" assumes "x1 \<rightarrow>\<^sub>s x2"   and "\<And>rs rs' x y. listrelp (\<lambda>x1 x2. x1 \<rightarrow>\<^sub>s x2 \<and> (\<forall>x. P x x1 x2)) rs rs' \<Longrightarrow> P y (Var x \<degree>\<degree> rs) (Var x \<degree>\<degree> rs')"   and "\<And>x ss ss' r r' y. \<lbrakk>x \<sharp> y; x \<sharp> (ss, ss'); r \<rightarrow>\<^sub>s r'; \<And>z. P z r r'; listrelp (\<lambda>x1 x2. x1 \<rightarrow>\<^sub>s x2 \<and> (\<forall>x. P x x1 x2)) ss ss'\<rbrakk> \<Longrightarrow> P y ((Lam [x].r) \<degree>\<degree> ss) ((Lam [x].r') \<degree>\<degree> ss')"   and "\<And>x s ss t r y. \<lbrakk>x \<sharp> y; x \<sharp> (s, ss, t); r[x::=s] \<degree>\<degree> ss \<rightarrow>\<^sub>s t; \<And>z. P z (r[x::=s] \<degree>\<degree> ss) t\<rbrakk> \<Longrightarrow> P y ((Lam [x].r) \<degree> s \<degree>\<degree> ss) t" shows "P z x1 x2"
p_eq_p1_p2: fixes x1 :: "'a"   and x2 :: "'b"   and y1 :: "'a"   and y2 :: "'b"   and n :: "nat" shows "p (x1, x2) (y1, y2) n = K1.p x1 y1 n * K2.p x2 y2 n"
nths_rep_2_n: shows "nths (replicate n (2::'a)) {n} = []"
subset_subseq: fixes X :: "'a set"   and ys :: "'a list" shows "(X \<subseteq> set ys) = (\<exists>xs. X = set xs \<and> subseq xs ys)"
Hausdorff_space_retraction_map_image: fixes X :: "'c topology"   and Y :: "'d topology"   and r :: "'c \<Rightarrow> 'd" assumes "retraction_map X Y r"   and "Hausdorff_space X" shows "Hausdorff_space Y"
insert_subsetI: fixes x :: "'c"   and A :: "'c set"   and X :: "'c set" assumes "x \<in> A"   and "X \<subseteq> A" shows "insert x X \<subseteq> A"
asym_greater: fixes cmp :: "'b \<Rightarrow> 'b \<Rightarrow> comp"   and a :: "'b"   and b :: "'b" assumes "comparator cmp"   and "cmp a b = Greater" shows "cmp b a \<noteq> Greater"
HPhase0Read_allBlocks: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and d :: "Disk" assumes "HPhase0Read s s' p d" shows "allBlocks s' \<subseteq> allBlocks s"
explicationFWB3: shows "\<forall>c x. \<I>pv (World c) \<sqinter> (\<^bold>\<not>FWB x) c"
Vfbasis: shows "set Vfbasis \<subseteq> V \<and> V = R_scalar_mult.RSpan UNIV (\<sharp>\<cdot>) Vfbasis \<and> R_scalar_mult.R_lin_independent UNIV (\<sharp>\<cdot>) Vfbasis"
If_def: fixes P :: "bool"   and x :: "'a"   and y :: "'a" shows "if P then x else y \<equiv> THE z. (P = True \<longrightarrow> z = x) \<and> (P = False \<longrightarrow> z = y)"
card_image_le: fixes A :: "'e set"   and f :: "'e \<Rightarrow> 'f" assumes "finite A" shows "card (f ` A) \<le> card A"
c1: shows "homotopic_with_canon (\<lambda>h. True) S U (g1 \<circ> f1) (\<lambda>x. c1)"
Int_ac_4: fixes A :: "'a set"   and B :: "'a set"   and C :: "'a set" shows "A \<inter> (B \<inter> C) = B \<inter> (A \<inter> C)"
Rep_nonempty_list: fixes x :: "'d nonempty_list" shows "Rep_nonempty_list x \<in> {xs. xs \<noteq> []}"
simps_2: fixes insts :: "((variable \<times> typ) \<times> Core.term) list"   and idn :: "variable"   and T :: "typ" shows "subst_term insts (Fv idn T) = the_default (Fv idn T) (lookup (\<lambda>x. x = (idn, T)) insts)"
elims_3: fixes x :: "'a Matrix.mat" assumes "\<not> square_mat x" obtains A :: "'a Matrix.mat" where "x = A" and "dim_col A \<noteq> dim_row A"
wls_fresh_swap_id: fixes s :: "'sort"   and X :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set"   and xs :: "'varSort"   and x1 :: "'var"   and x2 :: "'var" assumes "wls s X"   and "fresh xs x1 X"   and "fresh xs x2 X" shows "X #[x1 \<and> x2]_xs = X"
frac_eq_eq: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and y :: "'b"   and z :: "'b"   and x :: "'b"   and w :: "'b" assumes "class.field divide inverse times one plus zero minus uminus"   and "y \<noteq> zero"   and "z \<noteq> zero" shows "(divide x y = divide w z) = (times x z = times w y)"
dvd_prod: fixes A :: "'c set"   and a :: "'d"   and f :: "'c \<Rightarrow> 'd" assumes "\<exists>b\<in>A. a dvd f b"   and "finite A" shows "a dvd prod f A"
circ_circ_mult: fixes x :: "'a" shows "(1::'a)\<^sup>\<circ> * x\<^sup>\<circ> = x\<^sup>\<circ>\<^sup>\<circ>"
cf_h_def: fixes f :: "'e \<Rightarrow> 'f" shows "cf_h f = (\<lambda>c. (fst c, cmp f (snd c)))"
that: fixes g :: "'a \<Rightarrow> 'b" assumes "continuous_on U g"   and "g ` U \<subseteq> C"   and "\<And>x. x \<in> S \<Longrightarrow> g x = f x" shows "thesis"
tendsto_If_within_closures: fixes x :: "'a"   and s :: "'a set"   and t :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and l :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "x \<in> s \<union> closure s \<inter> closure t \<Longrightarrow> (f \<longlongrightarrow> l x) (at x within s \<union> closure s \<inter> closure t)"   and "x \<in> t \<union> closure s \<inter> closure t \<Longrightarrow> (g \<longlongrightarrow> l x) (at x within t \<union> closure s \<inter> closure t)"   and "x \<in> s \<union> t" shows "((\<lambda>x. if x \<in> s then f x else g x) \<longlongrightarrow> l x) (at x within s \<union> t)"
simps_2: fixes p :: "perm"   and t :: "('e, 'f, 'g) Tree" shows "p \<bullet> tNot t = tNot (p \<bullet> t)"
A_splay_max_ub: fixes t :: "'b tree" assumes "t \<noteq> \<langle>\<rangle>" shows "A_splay_max t \<le> 3 * (\<phi> t - 1) + 1"
less_imp_not_less: fixes x :: "'a"   and y :: "'a" assumes "x < y" shows "(\<not> y < x) = True"
other_bothE: fixes P :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and i :: "'e"   and \<sigma> :: "'e \<Rightarrow> 'd"   and \<sigma>' :: "'e \<Rightarrow> 'd" assumes "other P {i} \<sigma> \<sigma>'" obtains "\<sigma>' i = \<sigma> i" and "\<forall>j. j \<noteq> i \<longrightarrow> P (\<sigma> j) (\<sigma>' j)"
blinfun_of_matrix_apply: fixes a :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and x :: "'b" shows "blinfun_apply (blinfun_of_matrix a) x = (\<Sum>i\<in>Basis. \<Sum>j\<in>Basis. (x \<bullet> j * a i j) *\<^sub>R i)"
subgroup_self: fixes G :: "('f, 'g) monoid_scheme" assumes "Group.group G" shows "subgroup (carrier G) G"