hcju/isabelleps huggingface dataset:
split: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "char list \<Rightarrow> 'b MFOTL.trm list \<Rightarrow> 'a"   and f2 :: "'b MFOTL.trm \<Rightarrow> 'b MFOTL.trm \<Rightarrow> 'a"   and f3 :: "'b MFOTL.formula \<Rightarrow> 'a"   and f4 :: "'b MFOTL.formula \<Rightarrow> 'b MFOTL.formula \<Rightarrow> 'a"   and f5 :: "'b MFOTL.formula \<Rightarrow> 'a"   and f6 :: "\<I> \<Rightarrow> 'b MFOTL.formula \<Rightarrow> 'a"   and f7 :: "\<I> \<Rightarrow> 'b MFOTL.formula \<Rightarrow> 'a"   and f8 :: "'b MFOTL.formula \<Rightarrow> \<I> \<Rightarrow> 'b MFOTL.formula \<Rightarrow> 'a"   and f9 :: "'b MFOTL.formula \<Rightarrow> \<I> \<Rightarrow> 'b MFOTL.formula \<Rightarrow> 'a"   and formula :: "'b MFOTL.formula" shows "P (case formula of formula.Pred x xa \<Rightarrow> f1 x xa | formula.Eq x xa \<Rightarrow> f2 x xa | formula.Neg x \<Rightarrow> f3 x | formula.Or x xa \<Rightarrow> f4 x xa | formula.Exists x \<Rightarrow> f5 x | formula.Prev x xa \<Rightarrow> f6 x xa | formula.Next x xa \<Rightarrow> f7 x xa | formula.Since x xa xb \<Rightarrow> f8 x xa xb | formula.Until x xa xb \<Rightarrow> f9 x xa xb) = ((\<forall>x11 x12. formula = formula.Pred x11 x12 \<longrightarrow> P (f1 x11 x12)) \<and> (\<forall>x21 x22. formula = formula.Eq x21 x22 \<longrightarrow> P (f2 x21 x22)) \<and> (\<forall>x3. formula = formula.Neg x3 \<longrightarrow> P (f3 x3)) \<and> (\<forall>x41 x42. formula = formula.Or x41 x42 \<longrightarrow> P (f4 x41 x42)) \<and> (\<forall>x5. formula = formula.Exists x5 \<longrightarrow> P (f5 x5)) \<and> (\<forall>x61 x62. formula = formula.Prev x61 x62 \<longrightarrow> P (f6 x61 x62)) \<and> (\<forall>x71 x72. formula = formula.Next x71 x72 \<longrightarrow> P (f7 x71 x72)) \<and> (\<forall>x81 x82 x83. formula = formula.Since x81 x82 x83 \<longrightarrow> P (f8 x81 x82 x83)) \<and> (\<forall>x91 x92 x93. formula = formula.Until x91 x92 x93 \<longrightarrow> P (f9 x91 x92 x93)))"
modusponens_def: fixes s :: "thm"   and s' :: "thm" shows "modusponens s s' \<equiv> case concl s of Imp p q \<Rightarrow> let p' = concl s' in if fol_equal p p' then Thm q else Thm Truth | _ \<Rightarrow> Thm Truth"
is_partial_heap_def: fixes cmp :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and heap :: "int \<Rightarrow> 'a"   and l :: "int"   and m :: "int"   and r :: "int" shows "is_partial_heap cmp heap l m r = (\<forall>x. bounded m r x \<longrightarrow> bounded m r (parent l x) \<longrightarrow> cmp (heap (parent l x)) (heap x))"
distinct_5: shows "b\<^sub>c \<noteq> c\<^sub>c"
eucl_of_list_map_const: fixes xs :: "'e list"   and c :: "real" assumes "length xs = DIM('f)" shows "eucl_of_list (map (\<lambda>x. c) xs) = c *\<^sub>R One"
accessible_def: fixes M :: "'b dfa" assumes "dfa M" shows "dfa.accessible M \<equiv> {q. dfa.left_lang M q \<noteq> {}}"
Cons_2: fixes kv :: "'b \<times> 'c" assumes "kv \<in> set (x # xs)" shows "f kv = g kv"
symmetricI_app_iso: fixes hom :: "('c, 'd, 'e, 'f) digraph_isomorphism" assumes "iG.digraph_isomorphism hom"   and "symmetric (app_iso hom G)" shows "symmetric (app_iso hom (app_iso hom G))"
dim_basis: fixes A :: "'c set" assumes "finite A"   and "basis A" shows "dim = card A"
info_rel_internal: fixes I :: "('e \<times> 'h) set"   and S :: "('f \<times> 'g) set" shows "info_rel I S = (I \<times>\<^sub>r S) O br snd top"
field_lbound_gt_zero: fixes d1 :: "'b"   and d2 :: "'b" assumes "(0::'b) < d1"   and "(0::'b) < d2" shows "\<exists>e>0::'b. e < d1 \<and> e < d2"
maxBound_def: shows "maxBound \<equiv> last enum_class.enum"
eq_iff: fixes x :: "'a mod_ring"   and y :: "'a mod_ring" shows "(to_int_mod_ring x = to_int_mod_ring y) = (x = y)"
map_sum_sel_1: fixes s :: "'d + 'e"   and f :: "'d \<Rightarrow> 'f"   and g :: "'e \<Rightarrow> 'g" assumes "isl s" shows "projl (map_sum f g s) = f (projl s)"
le_fun_def: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" shows "(f \<le> g) = (\<forall>x. f x \<le> g x)"
le_lt_convs_4: fixes comp :: "'a \<Rightarrow> 'a \<Rightarrow> order"   and x :: "'a"   and y :: "'a" assumes "comparator comp" shows "(comp x y = Lt) = lt_of_comp comp x y"
ss_proj_def: fixes ss :: "('d, 'e) fmap set"   and vs :: "'d set" shows "ss_proj ss vs \<equiv> fmrestrict_set vs ` ss"
SUP_upper2: fixes i :: "'c"   and A :: "'c set"   and u :: "'d \<times> 'e"   and f :: "'c \<Rightarrow> 'd \<times> 'e" assumes "i \<in> A"   and "less_eq_prod u (f i)" shows "less_eq_prod u (Sup_prod (f ` A))"
assms: shows "\<Psi> \<rhd> P \<leadsto>\<^sub>w<Rel> Q"
assms_2: shows "\<forall>\<^sup>\<infinity>n. f n = g n"
RECT_rule: fixes body :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b"   and V :: "('a \<times> 'a) set"   and pre :: "'a \<Rightarrow> bool"   and x :: "'a"   and M :: "'a \<Rightarrow> 'b" assumes "trimono body"   and "wf V"   and "pre x"   and "\<And>f x. \<lbrakk>\<And>x'. \<lbrakk>pre x'; (x', x) \<in> V\<rbrakk> \<Longrightarrow> f x' \<le> M x'; pre x; REC\<^sub>T body = f\<rbrakk> \<Longrightarrow> body f x \<le> M x" shows "REC\<^sub>T body x \<le> M x"
disjCond: fixes A :: "'b set"   and B :: "'b set" assumes "\<And>x. \<lbrakk>x \<in> A; x \<in> B\<rbrakk> \<Longrightarrow> False" shows "A \<inter> B = {}"
update_simps_3: fixes ip :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and r :: "nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set" assumes "ip \<in> kD rt"   and "sqn rt ip = \<pi>\<^sub>2 r"   and "\<pi>\<^sub>5 r < the (dhops rt ip)" shows "update rt ip r = rt(ip \<mapsto> addpre r (\<pi>\<^sub>7 (the (rt ip))))"
index_add_mat_2: fixes A :: "'d mat"   and B :: "'d mat" shows "dim_row (A + B) = dim_row B"
normalized_prod_msetI: fixes A :: "'a multiset" assumes "\<And>a. a \<in># A \<Longrightarrow> normalize a = a" shows "normalize (\<Prod>\<^sub># A) = \<Prod>\<^sub># A"
Class_left_inverse: fixes G :: "'c set"   and composition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit :: "'c"   and E :: "('c \<times> 'c) set"   and a :: "'c" assumes "group_congruence G composition unit E"   and "a \<in> G" shows "monoid_congruence.quotient_composition G composition E (equivalence.Class G E (monoid.inverse G composition unit a)) (equivalence.Class G E a) = equivalence.Class G E unit"
row_rw: shows "Matrix.row (submatrix A I UNIV) i = Matrix.row A (pick I i)"
bind_ASSERT_eq_if: fixes \<Phi> :: "bool"   and m :: "'b nres" shows "ASSERT \<Phi> \<bind> (\<lambda>_. m) = (if \<Phi> then m else FAIL)"
wca_on_def: fixes A :: "'b set"   and f :: "'b set \<Rightarrow> 'b set" shows "wca_on A f = (\<forall>(x, y)\<in>rwp_on A f. \<forall>B\<subseteq>A. x \<in> f B \<and> y \<in> B \<longrightarrow> y \<in> f B)"
nth_index: fixes x :: "'b"   and xs :: "'b list" assumes "x \<in> set xs" shows "xs ! index xs x = x"
insert_3: assumes "\<And>a1 a2. \<lbrakk>a1 \<in> A; a2 \<in> A; a1 \<noteq> a2\<rbrakk> \<Longrightarrow> keys (f a1) \<inter> keys (f a2) = {}" shows "keys (sum f A) = (\<Union>a\<in>A. keys (f a))"
iFROM_union': fixes n :: "nat"   and n' :: "nat" shows "[n\<dots>] \<union> [n'\<dots>] = [min n n'\<dots>]"
priv_are_new_vars: fixes x :: "'Var\<^sub>C"   and i :: "nat" assumes "x \<in> priv_mem\<^sub>C ! i"   and "i < length cms" shows "x \<notin> range var\<^sub>C_of"
pair_fusion: fixes x :: "i"   and z :: "i"   and y :: "i" shows "(P x z \<and> P y z) \<and> (\<forall>w. P w z \<longrightarrow> O w x \<or> O w y) \<longrightarrow> (\<sigma>z. z = x \<or> z = y) = z"
simps: fixes a1 :: "('d \<times> ('e, 'f) cconstraint \<times> 'g \<times> 'e list \<times> 'd) set \<times> ('d \<Rightarrow> ('e, 'f) cconstraint)"   and a2 :: "'d"   and a3 :: "'e \<Rightarrow> 'f"   and a4 :: "'g"   and a5 :: "'d"   and a6 :: "'e \<Rightarrow> 'f" shows "a1 \<turnstile> \<langle>a2, a3\<rangle> \<rightarrow>\<^bsub>a4\<^esub> \<langle>a5, a6\<rangle> = (\<exists>A l g a r l' u u'. a1 = A \<and> a2 = l \<and> a3 = u \<and> a4 = a \<and> a5 = l' \<and> a6 = u' \<and> A \<turnstile> l \<longrightarrow>\<^bsup>g,a,r\<^esup> l' \<and> u \<turnstile> g \<and> u' \<turnstile> inv_of A l' \<and> u' = [r\<rightarrow>(0::'f)]u)"
pcr_cr_eq: shows "pcr_transaction\<^sub>h = cr_transaction\<^sub>h"
atMost_Suc_shift: fixes g :: "nat \<Rightarrow> 'a"   and n :: "nat" shows "sum g {..Suc n} = g 0 + (\<Sum>i\<le>n. g (Suc i))"
mHom_ex_zero: fixes N :: "('g, 'b, 'h) Module_scheme" assumes "R module N" shows "mzeromap M N \<in> mHom R M N"
CeqEnd: fixes a :: "((int \<times> int) set set, int) Combinators list"   and x :: "int \<times> (int \<times> int) \<times> (int \<times> int) \<times> DummyContent"   and b :: "((int \<times> int) set set, int) Combinators list" assumes "a \<noteq> []"   and "x \<in> dom (C (list2FWpolicy a))" shows "C (list2FWpolicy (b @ a)) x = C (list2FWpolicy a) x"
distinct_map_filter: fixes f :: "'g \<Rightarrow> 'f"   and xs :: "'g list"   and P :: "'g \<Rightarrow> bool" assumes "distinct (map f xs)" shows "distinct (map f (filter P xs))"
intro: fixes G :: "('e, 'f) pre_digraph"   and M :: "'f pre_map" assumes "fin_digraph G"   and "bidirected_digraph G (edge_rev M)"   and "digraph_map_axioms G M" shows "digraph_map G M"
naturality: fixes f :: "'c" assumes "C.arr f" shows "G (C.cod f) \<cdot>\<^sub>D G f = G f \<cdot>\<^sub>D G (C.dom f)"
bintr_bin_clr_le: fixes n :: "nat"   and m :: "nat"   and w :: "int" shows "take_bit n (bin_sc m False w) \<le> take_bit n w"
finfun_upd_apply_same: fixes f :: "('b, 'a) finfun"   and a :: "'b"   and b :: "'a" shows "finfun_apply (finfun_update f a b) a = b"
jumpF_linear_comp_2: fixes c :: "real"   and f :: "real \<Rightarrow> real"   and b :: "real"   and x :: "real" assumes "c \<noteq> 0" shows "jumpF (f \<circ> (\<lambda>x. c * x + b)) (at_right x) = (if 0 < c then jumpF f (at_right (c * x + b)) else jumpF f (at_left (c * x + b)))"
phl_cons1: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and w :: "'c"   and y :: "'c"   and z :: "'c" assumes "class.near_dioid plus times less_eq less"   and "less_eq x w"   and "less_eq (times w y) (times y z)" shows "less_eq (times x y) (times y z)"
supp_map_transform: fixes ae :: "Vars.var \<Rightarrow> 'c\<^sub>\<bottom>"   and \<Gamma> :: "(Vars.var \<times> exp) list" shows "supp (map_transform trans ae \<Gamma>) \<subseteq> supp \<Gamma>"
mds\<^sub>s_consistent: shows "mds_consistent mds\<^sub>s \<Gamma>\<^sub>0"
rr: shows "restrict_top i vr vl = restrict i vr vl"
nvs_hom_meet: fixes x :: "'a"   and y :: "'a" shows "d ((x \<sqinter> nc) \<cdot> nc \<sqinter> (y \<sqinter> nc) \<cdot> nc) = d ((x \<sqinter> nc) \<cdot> nc) \<sqinter> d ((y \<sqinter> nc) \<cdot> nc)"
Abs_SkewBinomialHeap_inverse: fixes y :: "unit + ('c, 'd) BsSkewBinomialHeapStruc.BsSkewElem" assumes "y \<in> {q. BsSkewBinomialHeapStruc.bs_invar q}" shows "Rep_SkewBinomialHeap (Abs_SkewBinomialHeap y) = y"
restr_ucon: fixes F :: "nat uprod set" assumes "F \<subseteq> E" shows "uconnected F = Restr (uconnected F) V \<union> nat_rel"
cohomotopically_trivial_retraction_null_gen: fixes U :: "'c set"   and Q :: "('b \<Rightarrow> 'c) \<Rightarrow> bool"   and P :: "('a \<Rightarrow> 'c) \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c" assumes "\<And>f. \<lbrakk>continuous_on T f; f ` T \<subseteq> U; Q f\<rbrakk> \<Longrightarrow> P (f \<circ> h)"   and "\<And>f. \<lbrakk>continuous_on S f; f ` S \<subseteq> U; P f\<rbrakk> \<Longrightarrow> Q (f \<circ> k)"   and "\<And>h k. (\<And>x. x \<in> T \<Longrightarrow> h x = k x) \<Longrightarrow> Q h = Q k"   and "\<And>f g. \<lbrakk>continuous_on S f; f ` S \<subseteq> U; P f\<rbrakk> \<Longrightarrow> \<exists>c. homotopic_with_canon P S U f (\<lambda>x. c)"   and "continuous_on T f"   and "f ` T \<subseteq> U"   and "Q f" obtains c :: "'c" where "homotopic_with_canon Q T U f (\<lambda>x. c)"
that: fixes nx :: "'node SDG_node"   and nsx :: "'node SDG_node list" assumes "matched nx nsx n'"   and "n \<in> set nsx"   and "nx s-p\<rightarrow>\<^bsub>sum\<^esub> CFG_node (parent_node n')" shows "thesis"
UP_subring_eval_closed: fixes S :: "'a set"   and g :: "nat \<Rightarrow> 'a"   and a :: "'a" assumes "subring S R"   and "g \<in> carrier (UP (R\<lparr>carrier := S\<rparr>))"   and "a \<in> S" shows "to_fun g a \<in> S"
eq_refl: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.preorder (less_eq::'b \<Rightarrow> 'b \<Rightarrow> bool) (less::'b \<Rightarrow> 'b \<Rightarrow> bool)"   and "(x::'b) = (y::'b)" shows "(less_eq::'b \<Rightarrow> 'b \<Rightarrow> bool) (x::'b) (y::'b)"
dvd_div_eq_cancel: fixes a :: "'a"   and c :: "'a"   and b :: "'a" assumes "a div c = b div c"   and "c dvd a"   and "c dvd b" shows "a = b"
rm_vars_ground_supports: fixes \<theta> :: "'a \<Rightarrow> ('b, 'a) Term.term"   and X :: "'a set" assumes "ground (subst_range \<theta>)" shows "rm_vars X \<theta> supports \<theta>"
cases: fixes x :: "'b list" obtains xa :: "'b" and xs :: "'b list" where "x = xa # xs"   | "x = []"
hom_decompose_prod_root_unity: fixes p :: "real poly" shows "decompose_prod_root_unity (map_poly of_real p) = map_prod id (map_poly of_real) (decompose_prod_root_unity p)"
not_prime_unit: 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 unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and x :: "'b" assumes "class.normalization_semidom divide plus minus zero times one unit_factor normalize"   and "dvd.dvd times x one" shows "\<not> normalization_semidom.prime zero times one normalize x"
assert_def: fixes return :: "'e \<Rightarrow> 'd"   and bind :: "'d \<Rightarrow> ('e \<Rightarrow> 'd) \<Rightarrow> 'd"   and fail :: "'d"   and P :: "'e \<Rightarrow> bool"   and m :: "'d" shows "monad_fail_base.assert return bind fail P m = bind m (\<lambda>x. if P x then return x else fail)"
comp_fun_idem_apply': fixes f :: "('b, 'c) comp_fun_idem" shows "comp_fun_idem (comp_fun_idem_apply f)"
Chain_sub: fixes X :: "'a set" assumes "Chain D X" shows "X \<subseteq> carrier D"
heap_merge: fixes t1 :: "'b tree"   and t2 :: "'b tree" assumes "heap t1"   and "heap t2" shows "heap (merge t1 t2)"
det: shows "\<forall>\<sigma>. small P \<sigma> = {} \<or> (\<exists>\<sigma>'. small P \<sigma> = {\<sigma>'})"
pp_of_term_def: fixes v :: "'t" shows "pp_of_term v = fst (pair_of_term v)"
that: fixes U :: "'a set"   and V :: "'a set" assumes "open U"   and "open V"   and "S \<subseteq> U"   and "T \<subseteq> V"   and "U \<inter> V = {}" shows "thesis"
word_div_eq_1_iff: fixes n :: "'b word"   and m :: "'b word" shows "(n div m = 1) = (m \<le> n \<and> unat n < 2 * unat m)"
align\<^sub>d_M: shows "local.aligned\<^sub>d e M"
less_eq_language_def: fixes r :: "'b language"   and s :: "'b language" shows "(r \<le> s) = (Plus r s = s)"
pnPlus_pnorm: fixes r :: "'a rexp"   and s :: "'a rexp" shows "pnPlus (pnorm r) (pnorm s) = pnorm (Plus r s)"
extremum_unique: fixes a :: "'a word" shows "(a \<le> 0) = (a = 0)"
not_all_zero: shows "\<not> vector_all_zero_from_index (mod_type_class.to_nat j, vec_to_iarray (row i A))"
fun_relE2: fixes f :: "'c \<Rightarrow> 'd"   and f' :: "'c \<Rightarrow> 'e"   and Rv :: "('d \<times> 'e) set"   and t :: "'d"   and x :: "'c" assumes "(f, f') \<in> Id \<rightarrow> Rv"   and "t = f x" shows "(t, f' x) \<in> Rv"
nn_integral_Markov_inequality: fixes u :: "'c \<Rightarrow> ennreal"   and M :: "'c measure"   and A :: "'c set"   and c :: "ennreal" assumes "u \<in> borel_measurable M"   and "A \<in> sets M" shows "emeasure M ({x \<in> space M. 1 \<le> c * u x} \<inter> A) \<le> c * \<integral>\<^sup>+ x. u x * indicator A x \<partial>M"
swap_cs_Send_RecvMarker: fixes channel :: "nat \<Rightarrow> ('a \<times> 'a) option"   and trans :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and c :: "('a, 'b, 'c) configuration"   and ev :: "('a, 'b, 'c) event"   and d :: "('a, 'b, 'c) configuration"   and ev' :: "('a, 'b, 'c) event"   and e :: "('a, 'b, 'c) configuration"   and d' :: "('a, 'b, 'c) configuration"   and e' :: "('a, 'b, 'c) configuration"   and i :: "nat" assumes "distributed_system channel"   and "distributed_system.next channel trans send recv c ev d"   and "distributed_system.next channel trans send recv d ev' e"   and "isSend ev"   and "isRecvMarker ev'"   and "distributed_system.next channel trans send recv c ev' d'"   and "distributed_system.next channel trans send recv d' ev e'" shows "cs e i = cs e' i"
ordertype_UNIV_\<omega>\<omega>: shows "ordertype UNIV (lenlex less_than) = \<omega> \<up> \<omega>"
mult_divide_mult_cancel_left_if: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and c :: "'a"   and a :: "'a"   and b :: "'a" assumes "class.field divide inverse times one plus zero minus uminus" shows "divide (times c a) (times c b) = (if c = zero then zero else divide a b)"
3: shows "\<forall>f\<in>set (facesAt g v). triangle f \<or> |vertices f| = 4 \<or> 5 \<le> |vertices f|"
is_ascii_upChar: fixes str :: "char list" assumes "is_ascii str" shows "is_ascii (upChar str)"
PiE: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and B :: "'b \<Rightarrow> 'c set"   and x :: "'b" assumes "f \<in> Pi A B" obtains "f x \<in> B x"   | "x \<notin> A"
least_prefixpoint_char: fixes f :: "'a \<Rightarrow> 'a"   and x :: "'a" shows "is_least_prefixpoint f x = (has_least_prefixpoint f \<and> x = p\<mu> f)"
PfP_subst: fixes t :: "tm"   and j :: "name"   and w :: "tm" shows "(PfP t)(j::=w) = PfP (subst j w t)"
strict_implies_order: fixes ko :: "'a key_order"   and b :: "'a"   and a :: "'a" assumes "ko.lt ko b a" shows "ko.le ko b a"
set_cases_2: fixes e :: "'c"   and a :: "('d, 'c) step" assumes "e \<in> set2_step a" obtains "a = Skip e"   | z1 :: "'d" where "a = Yield z1 e"
\<omega>_limit_set_in_compact_connected: fixes K :: "'a set"   and x :: "'a" assumes "compact K"   and "K \<subseteq> X"   and "x \<in> X"   and "trapped_forward x K" shows "connected (\<omega>_limit_set x)"
is_break_refine: fixes s :: "('a, 'd) simple_state_scheme"   and s' :: "('a, 'd) simple_state_scheme" assumes "(s, s') \<in> rel" shows "is_break parami s = is_break parami s'"
dom_simp: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b local.term set" assumes "free_monoidal_category C"   and "partial_magma.arr (free_monoidal_category.comp C) f" shows "partial_magma.dom (free_monoidal_category.comp C) f = free_monoidal_category.mkarr C (monoidal_language.Dom C (free_monoidal_category.rep C f))"
cong_bigtheta: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and g :: "'b \<Rightarrow> 'c" assumes "f \<in> \<Theta>[F](g)" shows "o[F](f) = o[F](g)"
hom_inv: fixes x :: "(((nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real) \<Rightarrow>\<^sub>0 int) set" assumes "x \<in> carrier (relative_homology_group (int n) (nsphere n) (equator n))" shows "hom_boundary (int n) (nsphere n) (equator n) (inv\<^bsub>relative_homology_group (int n) (nsphere n) (equator n)\<^esub> x) = inv\<^bsub>homology_group (int n - 1) (subtopology (nsphere n) (equator n))\<^esub> hom_boundary (int n) (nsphere n) (equator n) x"
coding_hf_hpair: fixes x :: "hf"   and y :: "hf" shows "coding_hf \<langle>x, y\<rangle> = ((\<exists>i. \<langle>x, y\<rangle> = ord_of i) \<or> coding_hf x \<and> coding_hf y)"
unit_SE_def: fixes e :: "'c" shows "unit\<^sub>S\<^sub>E e = (\<lambda>\<sigma>. Some (e, \<sigma>))"
eqvt_1: fixes p :: "perm"   and a :: "exp"   and aa :: "Vars.var"   and ab :: "Vars.var" shows "p \<bullet> a[aa::=ab] = (p \<bullet> a)[(p \<bullet> aa)::=(p \<bullet> ab)]"
4: fixes \<I>' :: "'var \<Rightarrow> ('fun, 'var) Term.term" assumes "subst_domain \<I>' = \<V> \<and> ground (subst_range \<I>')" shows "\<lbrakk>{}; set (unlabel []) \<cdot>\<^sub>p\<^sub>s\<^sub>e\<^sub>t \<I>'; unlabel \<A>\<rbrakk>\<^sub>s \<I>' = (\<exists>B\<in>set (tr\<^sub>p\<^sub>c \<A> []). \<lbrakk>{}; unlabel B\<rbrakk>\<^sub>d \<I>')"
eucl_le: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and sgn :: "'b \<Rightarrow> 'b"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and inner :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and Basis :: "'b set"   and abs :: "'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.ordered_euclidean_space Inf Sup minus dist norm plus zero uminus scaleR sgn uniformity open inner Basis abs inf sup less_eq less" shows "less_eq x y = (\<forall>i\<in>Basis. inner x i \<le> inner y i)"
p0: fixes p :: "nat \<Rightarrow> nat" assumes "p \<in> {p. p permutes {0..<n}} - {id}" shows "signof p * (\<Prod>i = 0..<n. A $$ (i, p i)) = (0::'a)"
simps_2: fixes x :: "'i"   and \<theta> :: "('h + 'j, 'i) trm"   and \<sigma> :: "'j \<Rightarrow> ('h, 'i) trm" shows "OsubstFO (OSing x \<theta>) \<sigma> = OSing x (TsubstFO \<theta> \<sigma>)"
a_lcomm: fixes G :: "('c, 'd) ring_scheme"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "abelian_monoid G"   and "x \<in> carrier G"   and "y \<in> carrier G"   and "z \<in> carrier G" shows "x \<oplus>\<^bsub>G\<^esub> (y \<oplus>\<^bsub>G\<^esub> z) = y \<oplus>\<^bsub>G\<^esub> (x \<oplus>\<^bsub>G\<^esub> z)"
dist_commute_lessI: fixes dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and y :: "'b"   and x :: "'b"   and e :: "real" assumes "class.metric_space dist uniformity open"   and "dist y x < e" shows "dist x y < e"
induct: fixes P :: "poly \<Rightarrow> bool"   and a0 :: "poly" assumes "\<And>c. P (C c)"   and "\<And>c n p. \<lbrakk>P c; P p\<rbrakk> \<Longrightarrow> P (CN c n p)"   and "\<And>v. P (Bound v)"   and "\<And>v va. P (Add v va)"   and "\<And>v va. P (Sub v va)"   and "\<And>v va. P (Mul v va)"   and "\<And>v. P (Neg v)"   and "\<And>v va. P (Pw v va)" shows "P a0"
SETComp: fixes f :: "ZF"   and g :: "ZF" assumes "f \<approx>>\<^bsub>SET\<^esub> g" shows "f ;;\<^bsub>SET\<^esub> g = f |o| g"
word_and_nth: fixes x :: "'b word"   and y :: "'b word"   and n :: "nat" shows "(x AND y) !! n = (x !! n \<and> y !! n)"
conth: fixes s :: "'b set"   and h :: "'b \<Rightarrow> 'c"   and t :: "'c set"   and k :: "'c \<Rightarrow> 'b" assumes "Retracts s h t k" shows "continuous_on s h"
bres_galois_imp: fixes x :: "'c \<times> 'd"   and y :: "'c \<times> 'd"   and z :: "'c \<times> 'd" assumes "less_eq_prod (times_prod' x y) z" shows "less_eq_prod y (prod.bres x z)"
1: shows "openin (top_of_set S) {b. \<exists>T. openin (top_of_set S) T \<and> b \<in> T \<and> (\<forall>x\<in>T. P x \<longrightarrow> Q x)}"
some_third_chamberD_adj: fixes C :: "'a set"   and z :: "'a set"   and D :: "'a set" assumes "chamber C"   and "z \<lhd> C" shows "C \<sim> some_third_chamber C D z"
tr_def_2: shows "length (w1 @ w2) = length tr"