hcju/isabelleps huggingface dataset:
rtrancl_into_trancl2: fixes a :: "'d"   and b :: "'d"   and r :: "('d \<times> 'd) set"   and c :: "'d" assumes "(a, b) \<in> r"   and "(b, c) \<in> r\<^sup>*" shows "(a, c) \<in> r\<^sup>+"
put_inef: fixes x :: "'b \<Longrightarrow> 'c"   and \<sigma> :: "'c"   and v :: "'b" assumes "ief_lens x" shows "put\<^bsub>x\<^esub> \<sigma> v = \<sigma>"
mult_not_zero: fixes a :: "'a"   and b :: "'a" assumes "a * b \<noteq> bot" shows "a \<noteq> bot \<and> b \<noteq> bot"
epi_iff_subset: fixes f :: "'b \<Rightarrow> 'c"   and G :: "('b, 'd) monoid_scheme"   and G' :: "('c, 'e) monoid_scheme" shows "(f \<in> epi G G') = (f \<in> hom G G' \<and> carrier G' \<subseteq> f ` carrier G)"
assms_3: shows "\<Gamma> \<turnstile> e : Low"
l_i: fixes G :: "('e, 'f) Group_scheme"   and a :: "'e" assumes "Group G"   and "a \<in> carrier G" shows "\<rho>\<^bsub>G\<^esub> a \<cdot>\<^bsub>G\<^esub> a = \<one>\<^bsub>G\<^esub>"
range_composition: fixes f :: "'f \<Rightarrow> 'd"   and g :: "'e \<Rightarrow> 'f" shows "range (\<lambda>x. f (g x)) = f ` range g"
dy_fake_msg_mono_ik: fixes G :: "msg set"   and H :: "msg set"   and bad :: "agent set"   and C :: "chan set" assumes "G \<subseteq> H" shows "dy_fake_msg bad G C \<subseteq> dy_fake_msg bad H C"
openin_relative_to: fixes X :: "'d topology"   and S :: "'d set" shows "openin X relative_to S = openin (subtopology X S)"
ptrm_nprv_all_imp_exi: fixes \<sigma> :: "'fmla"   and n :: "nat"   and \<phi> :: "'fmla" assumes "\<sigma> \<in> ptrm n"   and "\<phi> \<in> fmla" shows "nprv {all out (imp \<sigma> \<phi>)} (exi out (cnj \<sigma> \<phi>))"
poly_deg_hom_components_le: fixes q :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" assumes "q \<in> hom_components p" shows "poly_deg q \<le> poly_deg p"
domA_map_of_Some_the: fixes x :: "'a"   and \<Gamma> :: "('a \<times> 'b) list" assumes "x \<in> domA \<Gamma>" shows "map_of \<Gamma> x = Some (the (map_of \<Gamma> x))"
simps_8: fixes b :: "nat"   and \<phi> :: "'c MFOTL.formula"   and I :: "\<I>"   and \<psi> :: "'c MFOTL.formula" shows "MFOTL.fvi b (formula.Since \<phi> I \<psi>) = MFOTL.fvi b \<phi> \<union> MFOTL.fvi b \<psi>"
some_gcd_def: fixes a :: "'a"   and b :: "'a" shows "some_gcd a b \<equiv> SOME x. is_gcd x a b"
assms: fixes x :: "nat" assumes "x \<in> lc_subst s" shows "f1 x = f2 x"
mult_scalar_monom_mult: fixes c :: "'c"   and t :: "'a"   and p :: "'a \<Rightarrow>\<^sub>0 'c"   and q :: "'a \<Rightarrow>\<^sub>0 'c" shows "local.punit.mult_scalar (local.punit.monom_mult c t p) q = local.punit.monom_mult c t (local.punit.mult_scalar p q)"
in_hom_def: fixes f :: "'b"   and a :: "'b"   and b :: "'b" shows "\<guillemotleft>f : a \<rightarrow>\<^sub>B b\<guillemotright> \<equiv> a \<in> B.domains f \<and> b \<in> B.codomains f"
bij_betw_cong: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b"   and A' :: "'b set" assumes "\<And>a. a \<in> A \<Longrightarrow> f a = g a" shows "bij_betw f A A' = bij_betw g A A'"
rtc_implies_cs: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "r\<^sup>*\<^sup>* x y" shows "relation.cs r x y"
l8_7: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Per Bet Cong A B C"   and "Tarski_neutral_dimensionless.Per Bet Cong A C B" shows "B = C"
HFinite_FreeUltrafilterNat: fixes X :: "nat \<Rightarrow> 'a" assumes "star_n X \<in> HFinite" shows "\<exists>u. \<forall>\<^sub>F n in \<U>. norm (X n) < u"
inv_solve_right: fixes G :: "('c, 'd) monoid_scheme"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "Group.group G"   and "a \<in> carrier G"   and "b \<in> carrier G"   and "c \<in> carrier G" shows "(a = b \<otimes>\<^bsub>G\<^esub> inv\<^bsub>G\<^esub> c) = (b = a \<otimes>\<^bsub>G\<^esub> c)"
s1: fixes d :: "'i word" shows "runFw ip1 d c rs = runFw ip2 d c rs"
m2: fixes x :: "'av"   and y :: "'av" assumes "x < y" shows "m y < m x"
is_lub_def: fixes S :: "'b set"   and x :: "'b" shows "S <<| x = (S <| x \<and> (\<forall>u. S <| u \<longrightarrow> x \<sqsubseteq> u))"
simWeakSim: fixes P :: "ccs"   and Rel :: "(ccs \<times> ccs) set"   and Q :: "ccs" assumes "P \<leadsto>[Rel] Q" shows "P \<leadsto>\<^sup>^<Rel> Q"
assms_3: fixes xs :: "nat list"   and rest :: "(recf \<times> nat list \<times> nat list) list" assumes "valid ((g, xs, []) # rest)" shows "correct ((g, xs, []) # rest, None)"
scaleR: fixes fa :: "'a \<Rightarrow>\<^sub>L 'b"   and r :: "real"   and b :: "'a" shows "blinfun_apply fa (r *\<^sub>R b) = r *\<^sub>R blinfun_apply fa b"
cd_diff_l: fixes n :: "nat"   and m :: "nat"   and x :: "nat" assumes "n \<le> m" shows "cd x m n = cd x (m - n) n"
h: shows "k \<le> card B + card C - card (B \<union> C)"
set_enforce_spmf: fixes P :: "'c \<Rightarrow> bool"   and p :: "'c spmf" shows "set_spmf (enforce_spmf P p) = {a \<in> set_spmf p. P a}"
infsuff_appE: fixes r :: "'b llist"   and A :: "'b set"   and t :: "'b llist" assumes "r \<in> A\<^sup>\<star>"   and "t \<in> infsuff A r" obtains s :: "'b llist" where "s \<in> A\<^sup>\<omega>" and "t = r @@ s"
invert_opposite_sides_of_square: fixes f :: "'a \<times> 'a"   and g :: "'a \<times> 'a"   and h :: "'a \<times> 'a"   and k :: "'a \<times> 'a" assumes "CC.seq f g"   and "CC.comp f g = CC.comp h k"   and "CC.iso f"   and "CC.iso k" shows "CC.seq g (CC.inv k) \<and> CC.seq (CC.inv f) h \<and> CC.comp g (CC.inv k) = CC.comp (CC.inv f) h"
j: shows "j + 1 < length (remdups_adj (map ((\<inter>) K) \<GG>))"
assms_2: fixes x :: "'a" assumes "\<gamma> x t n" shows "\<gamma>' t n"
pochhammer_prod: fixes a :: "'a"   and n :: "nat" shows "pochhammer a n = (\<Prod>i = 0..<n. a + of_nat i)"
chain_boundary_sum': fixes I :: "'b set"   and p :: "nat"   and g :: "'b \<Rightarrow> ((nat \<Rightarrow> real) \<Rightarrow> 'c) \<Rightarrow>\<^sub>0 int" assumes "finite I" shows "chain_boundary p (sum' g I) = sum' (chain_boundary p \<circ> g) I"
split_zip: fixes P :: "('d \<times> 'e) list \<Rightarrow> prop" shows "(\<And>x. PROP P x) \<equiv> (\<And>y z. length y = length z \<Longrightarrow> PROP P (y || z))"
estep_S: fixes c :: "(recf \<times> nat list \<times> nat list) list \<times> nat option"   and xs :: "nat list"   and ls :: "nat list"   and fs :: "(recf \<times> nat list \<times> nat list) list"   and rv :: "nat option" assumes "c = ((S, xs, ls) # fs, rv)"   and "recfn (length xs) (fst (hd (fst c)))" shows "estep (encode_config c) = encode_config (step c)"
simps_2: fixes f :: "('d, 'e) bintree \<Rightarrow> 'c"   and xs :: "('d, 'e) bintree option list" shows "dfs f (None # xs) = dfs f xs"
nxtActI: fixes n :: "nat"   and c :: "'id"   and t :: "nat \<Rightarrow> cnf" assumes "\<exists>i\<ge>n. \<parallel>c\<parallel>\<^bsub>t i\<^esub>" shows "n \<le> \<langle>c \<rightarrow> t\<rangle>\<^bsub>n\<^esub> \<and> \<parallel>c\<parallel>\<^bsub>t \<langle>c \<rightarrow> t\<rangle>\<^bsub>n\<^esub>\<^esub> \<and> \<not> (\<exists>k\<ge>n. latestAct_cond c t \<langle>c \<rightarrow> t\<rangle>\<^bsub>n\<^esub> k)"
disc_fct_def: fixes f :: "'c \<Rightarrow> 'd" shows "disc_fct f \<equiv> countable (range f)"
B_def: shows "B \<equiv> E - \<Union> (H.edges_of_walk ` paths_with_new)"
inj_on_fun_updI: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and y :: "'c"   and x :: "'b" assumes "inj_on f A"   and "y \<notin> f ` A" shows "inj_on (f(x := y)) A"
fail_writer_def: fixes fail :: "'f" shows "fail_writer fail = WriterT fail"
Spy_see_shrK: fixes evs :: "event list"   and A :: "agent" assumes "evs \<in> woolam" shows "(Key (shrK A) \<in> parts (knows Spy evs)) = (A \<in> bad)"
fixed_point_congruence: fixes p :: "nat"   and a :: "nat" assumes "order (G\<lparr>carrier := P\<rparr>) = p ^ a"   and "prime p"   and "finite (subgroups_of_size (p ^ a))" shows "card (subgroups_of_size (p ^ a)) mod p = card conjP.fixed_points mod p"
monotone_if_bot: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and bound :: "'b"   and ord :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c"   and bot :: "'c" assumes "class.preorder less_eq less"   and "\<And>x y. \<lbrakk>less_eq x y; \<not> less_eq x bound\<rbrakk> \<Longrightarrow> ord (f x) (f y)"   and "\<And>x. \<not> less_eq x bound \<Longrightarrow> ord bot (f x)"   and "ord bot bot" shows "monotone less_eq ord (\<lambda>x. if less_eq x bound then bot else f x)"
assms_2: shows "\<forall>a\<in>A. f a = g a"
S1: shows "PrimRecOp g h \<in> PrimRec2"
R_def: shows "R = (\<lambda>S. {x. jumpF h (at_right x) \<noteq> 0 \<and> x \<in> S})"
ag_r_zero: fixes x :: "'a" assumes "x \<in> carrier A" shows "x \<plusminus> \<zero> = x"
spmf_map: fixes f :: "'b \<Rightarrow> 'a"   and p :: "'b spmf"   and x :: "'a" shows "spmf (map_spmf f p) x = Sigma_Algebra.measure (measure_spmf p) (f -` {x})"
assms_3: shows "\<forall>v\<in>dom s. the (s v) \<in> \<R>\<^sub>+ \<Psi> v"
prefix: shows "lprefix (lSup (llist_of_tllist ` A)) (llist_of_tllist xs)"
SUP_UN_eq2: fixes r :: "'f \<Rightarrow> ('d \<times> 'e) set"   and S :: "'f set" shows "(SUP i\<in>S. (\<lambda>x y. (x, y) \<in> r i)) = (\<lambda>x y. (x, y) \<in> \<Union> (r ` S))"
opt_no_group: fixes F :: "atom fm"   and L :: "real list" shows "eval ((push_forall \<circ> nnf \<circ> unpower 0 \<circ> clearQuantifiers) F) L = eval F L"
uminus_eq_eq: fixes x :: "'a" shows "- x = x \<leadsto> bot"
iMODb_iMOD_subset_same: fixes r :: "nat"   and m :: "nat"   and c :: "nat" shows "[ r, mod m, c ] \<subseteq> [ r, mod m ]"
ideD_2: fixes a :: "'a" assumes "S.ide a" shows "S.dom a = a"
fm_restrict: fixes n :: "nat"   and y :: "'i \<Rightarrow> 'e" shows "fm n (restrict y (J n)) = fm n y"
that: fixes C :: "'a set" assumes "x \<in> C"   and "open C"   and "compact (closure C)"   and "closure C \<subseteq> carrier" shows "thesis"
M_def: shows "M = (MAX y\<in>B. card (f -` {y} \<inter> A))"
lipschitz_incr: fixes s :: "real"   and h :: "real" assumes "s \<in> {t 0..t1}"   and "h \<in> {0..t1 - s}" shows "L-lipschitz_on (cball (x s) \<bar>r\<bar>) (incr h s)"
sel_2: fixes x1 :: "real"   and x2 :: "real"   and x3 :: "real"   and x4 :: "real"   and x5 :: "real"   and x6 :: "real"   and x7 :: "real"   and x8 :: "real" shows "Im1 (Octo x1 x2 x3 x4 x5 x6 x7 x8) = x2"
lengths_partition_by_median_2_2: fixes ps :: "(real, 'a) vec list"   and l :: "(real, 'a) vec list"   and m :: "real"   and r :: "(real, 'a) vec list"   and k :: "'a" assumes "1 < length ps"   and "\<forall>k. distinct (map (\<lambda>p. p $ k) ps)"   and "(l, m, r) = partition_by_median k ps" shows "length l < length ps"
mutual_information_nonneg_simple: fixes X :: "'a \<Rightarrow> 'd"   and Y :: "'a \<Rightarrow> 'e" assumes "simple_function M X"   and "simple_function M Y" shows "0 \<le> \<I>(X ; Y)"
hyp: shows "dim_row A = dim_row B \<and> dim_col A = dim_col B \<and> mat_to_list A = mat_to_list B"
apex_composite: shows "fg.apex = f.leg0 \<down>\<down> g.leg1"
equality: fixes r :: "('c, 'd, 'e) pre_digraph_scheme"   and r' :: "('c, 'd, 'e) pre_digraph_scheme" assumes "verts r = verts r'"   and "arcs r = arcs r'"   and "tail r = tail r'"   and "head r = head r'"   and "more r = more r'" shows "r = r'"
extends_imp_wf_sub_rel_of: fixes sub :: "('a \<times> nat) \<times> 'a \<times> nat"   and subs' :: "(('a \<times> nat) \<times> 'a \<times> nat) set" assumes "extends g subs sub subs'" shows "wf_sub_rel_of g subs'"
vend_ge_100: fixes r :: "nat \<Rightarrow>f value option"   and l :: "String.literal"   and i :: "value list" assumes "possible_steps drinks 1 r l i = {|(2, vend)|}" shows "\<not>? value_gt (Some (Num 100)) (r $ 2) = trilean.true"
step_Normal_elim_cases_6: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and nrms :: "('e, 'd, 'f) com list"   and abrs :: "('e, 'd, 'f) com list"   and css :: "(('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list"   and s :: "'e"   and u :: "('e, 'd, 'f) com list \<times> (('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list \<times> ('e, 'f) xstate" assumes "\<Gamma>\<turnstile> ([], (nrms, abrs) # css, Abrupt s) \<rightarrow> u" obtains "u = (abrs, css, Normal s)"
wprepare_loop_goon_in_middle_Bk_False3_1: fixes lm :: "nat list"   and m :: "nat"   and b :: "cell list"   and a :: "cell"   and lista :: "cell list" assumes "lm \<noteq> []"   and "wprepare_loop_start_in_middle m lm (b, Bk # a # lista)" shows "wprepare_loop_goon_in_middle m lm (Bk # b, a # lista)"
continuous_within_ball: fixes x :: "'b"   and s :: "'b set"   and f :: "'b \<Rightarrow> 'c" shows "continuous (at x within s) f = (\<forall>e>0. \<exists>d>0. f ` (ball x d \<inter> s) \<subseteq> ball (f x) e)"
join_closed: fixes L :: "('c, 'd) gorder_scheme"   and x :: "'c"   and y :: "'c" assumes "weak_upper_semilattice L"   and "x \<in> carrier L"   and "y \<in> carrier L" shows "x \<squnion>\<^bsub>L\<^esub> y \<in> carrier L"
i_shrink_nth: fixes f :: "nat \<Rightarrow> 'd message_af"   and k :: "nat"   and n :: "nat" shows "(f \<div> k) n = last_message (f \<Up> (n * k) \<Down> k)"
ln: fixes x :: "'b list" shows "length x = length [n..<n + length x]"
T1: fixes u :: "'a"   and \<omega> :: "'a" assumes "C.ide u"   and "\<guillemotleft>\<omega> : C.dom \<omega> \<Rightarrow>\<^sub>C r \<star>\<^sub>C u\<guillemotright>" shows "\<exists>w \<theta> \<nu>. C.ide w \<and> \<guillemotleft>\<theta> : f \<star>\<^sub>C w \<Rightarrow>\<^sub>C u\<guillemotright> \<and> \<guillemotleft>\<nu> : C.dom \<omega> \<Rightarrow>\<^sub>C g \<star>\<^sub>C w\<guillemotright> \<and> C.iso \<nu> \<and> ((r \<star>\<^sub>C \<theta>) \<cdot>\<^sub>C \<a>\<^sub>C[r, f, w] \<cdot>\<^sub>C (\<rho> \<star>\<^sub>C w)) \<cdot>\<^sub>C \<nu> = \<omega>"
Rep_formula_inverse: fixes x :: "('a, 'b, 'c) formula" shows "Abs_formula (Rep_formula x) = x"
semiring_normalization_rules_25: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and a :: "'a"   and c :: "'a"   and d :: "'a" assumes "class.comm_semiring_1 times one plus zero" shows "plus a (plus c d) = plus (plus a c) d"
hn_refine_cons: fixes P :: "assn"   and P' :: "assn"   and c :: "'c Heap"   and Q :: "assn"   and R :: "'d \<Rightarrow> 'c \<Rightarrow> assn"   and m :: "'d nres"   and Q' :: "assn"   and R' :: "'d \<Rightarrow> 'c \<Rightarrow> assn" assumes "P \<Longrightarrow>\<^sub>t P'"   and "hn_refine P' c Q R m"   and "Q \<Longrightarrow>\<^sub>t Q'"   and "\<And>x y. R x y \<Longrightarrow>\<^sub>t R' x y" shows "hn_refine P c Q' R' m"
mult_left_mono_neg: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and b :: "'c"   and a :: "'c"   and c :: "'c" assumes "class.ordered_ring plus minus zero less_eq less times uminus"   and "less_eq b a"   and "less_eq c zero" shows "less_eq (times c a) (times c b)"
infinite_Ioo_iff: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a" assumes "class.dense_linorder less_eq less" shows "infinite (ord.greaterThanLessThan less a b) = less a b"
bf_biimp_def: fixes a :: "'b \<Rightarrow> bool"   and b :: "'b \<Rightarrow> bool" shows "bf_biimp a b = bf_ite a b (bf_not b)"
complement_blocks_def: shows "\<B>\<^sup>C \<equiv> image_mset block_complement \<B>"
expunge_delete_ids_imply_messages_same: fixes i :: "'a"   and "is" :: "'a set"   and xs :: "('a \<times> ('a, 'b) operation) event list"   and j :: "nat"   and e1 :: "'b"   and mo :: "'a"   and ir :: "'a"   and e2 :: "'b" assumes "i \<in> is"   and "xs prefix of j"   and "(i, Expunge e1 mo i) \<in> set (node_deliver_messages xs)"   and "(ir, Delete is e2) \<in> set (node_deliver_messages xs)" shows "e1 = e2"
g_def: shows "g = (\<lambda>p. prod (f p) {0..<Suc l})"
preserves_ide: fixes a :: "'a" assumes "ide a" shows "Span.ide (SPN a)"
ergodic_qmpt_axioms: shows "ergodic_qmpt M T"
r2s_union_hom: fixes R :: "('b \<times> 'c) set"   and S :: "('b \<times> 'c) set" shows "r2s (R \<union> S) = r2s R \<union> r2s S"
coin_coin: fixes S3 :: "bool"   and m1 :: "bool" shows "map_spmf (\<lambda>S0. S0 \<oplus> S3 \<oplus> m1) coin_spmf = coin_spmf"
ereal_less_eq_4: fixes r :: "real" shows "(ereal r \<le> 0) = (r \<le> 0)"
partition_on_split2: fixes A :: "'a set"   and P :: "'a set set"   and Q :: "'a set set" assumes "partition_on A (P \<union> Q)" shows "partition_on (\<Union> Q) Q"
B: shows "n \<notin> bad_pwd \<union> bad_shakey \<inter> bad_prikey"
HFinite_bounded_hcmod: fixes x :: "complex star"   and y :: "real star" assumes "x \<in> HFinite"   and "y \<le> hcmod x"   and "0 \<le> y" shows "y \<in> HFinite"
prv_all_imp_gen: fixes x :: "'var"   and \<phi> :: "'fmla"   and \<chi> :: "'fmla" assumes "x \<notin> Fvars \<phi>"   and "prv (imp \<phi> \<chi>)" shows "prv (imp \<phi> (all x \<chi>))"
sets_eq_imp_space_eq: fixes M :: "'b measure"   and M' :: "'b measure" assumes "sets M = sets M'" shows "space M = space M'"
distinct_3: fixes x1 :: "nat"   and x31 :: "expr"   and x32 :: "expr" shows "Var x1 \<noteq> LET x31 IN x32"
assms: fixes S :: "'a set" assumes "Separation G v0 v1 S" shows "n \<le> card S"
Apply_def: fixes f :: "'b \<Rightarrow> 'b" shows "Apply f \<equiv> \<lambda>ab P s. P (f s)"
fequalityI: fixes A :: "'b fset"   and B :: "'b fset" assumes "A |\<subseteq>| B"   and "B |\<subseteq>| A" shows "A = B"
comp_ide_arr: fixes b :: "'a \<times> 'a"   and f :: "'a \<times> 'a" assumes "B.VV.ide b"   and "B.VV.seq b f" shows "B.VV.comp b f = f"
bequiv_up_to_sym: fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> bool"   and b :: "bexp"   and b' :: "bexp" shows "(P \<Turnstile> b <\<sim>> b') = (P \<Turnstile> b' <\<sim>> b)"
path_imp_reversepath: fixes g :: "real \<Rightarrow> 'b" assumes "path g" shows "path (reversepath g)"
PORI_1: shows "(\<lambda>i. ValTT) \<in> unlr POR.delta"
minorM_iarrays_def: fixes A :: "'c iarray iarray"   and i :: "nat"   and j :: "nat" shows "minorM_iarrays A i j = tabulate2 (nrows_iarray A) (ncols_iarray A) (\<lambda>k l. if k = i \<and> l = j then 1::'c else if k = i \<or> l = j then 0::'c else A !! k !! l)"
rStatImp: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi" assumes "(\<Psi>, P, Q) \<in> X" shows "\<Psi> \<rhd> P \<lessapprox>\<^sub>w<X> Q"
max_fun_diff_def: fixes f :: "'b \<Rightarrow> 'd"   and g :: "'b \<Rightarrow> 'd" shows "s.max_fun_diff f g \<equiv> s.maxim {a \<in> Field s. f a \<noteq> g a}"
sat_list_distinct: fixes t :: "'a ifex"   and u :: "('a \<times> bool) list" assumes "ifex_no_twice t"   and "ifex_sat_list t = Some u" shows "distinct (map fst u)"
llist_all2_lsetD2: fixes P :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and xs :: "'e llist"   and ys :: "'f llist"   and y :: "'f" assumes "llist_all2 P xs ys"   and "y \<in> lset ys" shows "\<exists>x\<in>lset xs. P x y"
lins2I: fixes \<pi> :: "(node + node fset \<times> command_gadget \<times> node fset) list"   and G :: "diagram"   and S :: "node fset" assumes "distinct \<pi>"   and "set \<pi> = (fset G^V - fset S) <+> set G^E"   and "\<And>i j v e. \<lbrakk>i < length \<pi>; j < length \<pi>; \<pi> ! i = Inl v; \<pi> ! j = Inr e; v |\<in>| fst3 e\<rbrakk> \<Longrightarrow> i < j"   and "\<And>j k w e. \<lbrakk>j < length \<pi>; k < length \<pi>; \<pi> ! j = Inr e; \<pi> ! k = Inl w; w |\<in>| thd3 e\<rbrakk> \<Longrightarrow> j < k" shows "\<pi> \<in> lins2 S G"
coprime_mult_self_right_iff: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and 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 a :: "'b"   and c :: "'b"   and b :: "'b" assumes "class.algebraic_semidom divide plus minus zero times one" shows "algebraic_semidom.coprime times one (times a c) (times b c) = (dvd.dvd times c one \<and> algebraic_semidom.coprime times one a b)"
Union_rel_def: fixes L :: "('a \<times> 'c set) set"   and S :: "('c \<times> 'b set) set" shows "\<langle>L, S\<rangle>Union_rel = L O \<langle>S\<rangle>set_rel O br \<Union> top"
ainvf_aHom: fixes F :: "('c, 'd) aGroup_scheme"   and G :: "('e, 'f) aGroup_scheme"   and f :: "'c \<Rightarrow> 'e" assumes "aGroup F"   and "aGroup G"   and "bijec\<^bsub>F,G\<^esub> f" shows "ainvf\<^bsub>F,G\<^esub> f \<in> aHom G F"
Utn_def: shows "Utn = to_nat_on (\<Union> (range J))"
weakCongE: fixes P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and \<sigma> :: "(name list \<times> 'a list) list"   and \<Psi> :: "'b" assumes "P \<doteq>\<^sub>c Q"   and "wellFormedSubst \<sigma>" shows "\<Psi> \<rhd> P[<\<sigma>>] \<doteq> Q[<\<sigma>>]"
closed_segment_iv_subset_domain: fixes t0 :: "real"   and x0 :: "'a"   and t :: "real"   and s :: "real" assumes "t0 \<in> T"   and "x0 \<in> X"   and "t \<in> T"   and "s \<in> {t0--t}" shows "s \<in> T"
bij_betw_inv_into_right: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and A' :: "'b set"   and a' :: "'b" assumes "bij_betw f A A'"   and "a' \<in> A'" shows "f (inv_into A f a') = a'"
coll_commute: fixes a :: "real \<times> real"   and b :: "real \<times> real" shows "coll 0 a b = coll 0 b a"
upper_abs_interval: fixes x :: "'a interval" shows "upper (abs_interval x) = max \<bar>lower x\<bar> \<bar>upper x\<bar>"
bnds_power: fixes l1 :: "float"   and u1 :: "float"   and prec :: "nat"   and n :: "nat" shows "\<forall>x l u. (l1, u1) = float_power_bnds prec n l u \<and> x \<in> {real_of_float l..real_of_float u} \<longrightarrow> real_of_float l1 \<le> x ^ n \<and> x ^ n \<le> real_of_float u1"
l_inv: fixes x :: "'a" shows "local.inv x ** x = one"
2: fixes c' :: "('test, 'atom) com"   and s' :: "'state" assumes "(c, s) \<rightarrow>*c (c', s')"   and "s' \<approx> t'"   and "discr c'" shows "thesis"
f'_ge_f: fixes x :: "nat" assumes "x\<^sub>0 \<le> x" shows "f x \<le> gc1 * f' x"
conf_hext: fixes h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and h' :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and v :: "val"   and T :: "ty" assumes "h \<unlhd> h'"   and "P,h \<turnstile> v :\<le> T" shows "P,h' \<turnstile> v :\<le> T"
assms_1: fixes p :: "nat" assumes "p \<in># prime_factorization n" shows "x [^] (n div p) \<noteq> \<one>"
det_lower_triangular: fixes n :: "nat"   and A :: "'b mat" assumes "\<And>i j. \<lbrakk>i < j; j < n\<rbrakk> \<Longrightarrow> A $$ (i, j) = (0::'b)"   and "A \<in> carrier_mat n n" shows "det A = prod_list (diag_mat A)"
intro: fixes scale :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and Basis :: "'d set" assumes "vector_space scale"   and "finite_dimensional_vector_space_axioms scale Basis" shows "finite_dimensional_vector_space scale Basis"
are_endpoints_in_S': fixes p :: "proj2"   and q :: "proj2"   and a :: "proj2"   and b :: "proj2" assumes "p \<noteq> q"   and "a \<noteq> b"   and "p \<in> S"   and "q \<in> S"   and "a \<in> hyp2 \<union> S"   and "b \<in> hyp2 \<union> S"   and "proj2_set_Col {p, q, a, b}" shows "p = endpoint_in_S a b \<and> q = endpoint_in_S b a \<or> q = endpoint_in_S a b \<and> p = endpoint_in_S b a"
summable_iff_shift: fixes f :: "nat \<Rightarrow> 'b"   and k :: "nat" shows "summable (\<lambda>n. f (n + k)) = summable f"
ce_sets_containing_n_ce: fixes n :: "nat" shows "{p. n \<in> nat_to_ce_set p} \<in> ce_sets"
ltl_llist_of_stream: fixes xs :: "'c stream" shows "ltl (llist_of_stream xs) = llist_of_stream (stl xs)"
simps_2: fixes oid :: "'b"   and v :: "'b" shows "insert_spec [] (oid, Some v) = []"
simps_2: fixes uu :: "'a" shows "decr [uu] = True"
h: shows "h = \<^bold>g [^] x1 \<otimes> g' [^] x2"