hcju/isabelleps huggingface dataset:
not_grd_dgr: fixes a :: "'a"   and D :: "'d \<times> 'b \<Rightarrow> 'c \<Rightarrow> 'a set"   and i :: "'d" shows "(a \<in> - grd (step D) i) = (\<forall>j. a \<notin> grd (D (i, j)))"
finite_Collect_disjI: fixes P :: "'d \<Rightarrow> bool"   and Q :: "'d \<Rightarrow> bool" shows "finite {x. P x \<or> Q x} = (finite {x. P x} \<and> finite {x. Q x})"
map_simp_ide: fixes a :: "'a" assumes "ide a" shows "\<rr> a = \<r>[a]"
invertible_implies_rref_id: fixes A :: "(('b, 'c) vec, 'c) vec" assumes "invertible A" shows "Gauss_Jordan A = mat (1::'b)"
inject: fixes x1 :: "'a"   and x2 :: "'b"   and x3 :: "('a, 'b) BsSkewBinomialHeapStruc.BsSkewBinomialTree list"   and y1 :: "'a"   and y2 :: "'b"   and y3 :: "('a, 'b) BsSkewBinomialHeapStruc.BsSkewBinomialTree list" shows "(BsSkewBinomialHeapStruc.Element x1 x2 x3 = BsSkewBinomialHeapStruc.Element y1 y2 y3) = (x1 = y1 \<and> x2 = y2 \<and> x3 = y3)"
boundedI: fixes b :: "'c"   and a :: "'c"   and c :: "'c" assumes "b \<le> a"   and "c \<le> a" shows "sup b c \<le> a"
vec_nth: fixes x :: "('e, 'd) vec" shows "($) x \<in> UNIV"
Nil: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "genPrefixp r [] []"
prems: fixes n :: "nat" assumes "\<And>l. n \<le> l \<Longrightarrow> card (level A w (2 * Suc k) l) \<le> card (nodes A) - Suc k" shows "thesis"
wf_update: fixes agents :: "'a set"   and alts :: "'b set"   and R :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and i :: "'a"   and Ri' :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "pref_profile_wf agents alts R"   and "i \<in> agents"   and "total_preorder_on alts Ri'" shows "pref_profile_wf agents alts (R(i := Ri'))"
I_def: shows "I \<equiv> integral {real_of_int a..real_of_int b} f"
cases: fixes x :: "sat_plan_variable" obtains t :: "nat" and k :: "nat" where "x = Operator t__ k__"   | t :: "nat" and k :: "nat" where "x = State t k"
subst_pp_zero: fixes f :: "'e \<Rightarrow> ('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd" shows "subst_pp f 0 = 1"
reflp_llist_all2: fixes R :: "'e \<Rightarrow> 'e \<Rightarrow> bool" assumes "reflp R" shows "reflp (llist_all2 R)"
TryThrow\<^sub>1: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> nat exp) list) list"   and e\<^sub>1 :: "nat exp"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> val list"   and a :: "nat"   and h\<^sub>1 :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and ls\<^sub>1 :: "val list"   and D :: "char list"   and fs :: "char list \<times> char list \<Rightarrow> val option"   and C :: "char list"   and i :: "nat"   and e\<^sub>2 :: "nat exp" assumes "P \<turnstile>\<^sub>1 \<langle>e\<^sub>1,s\<^sub>0\<rangle> \<Rightarrow> \<langle>Throw a,(h\<^sub>1, ls\<^sub>1)\<rangle>"   and "h\<^sub>1 a = \<lfloor>(D, fs)\<rfloor>"   and "(D, C) \<notin> (subcls1 P)\<^sup>*" shows "P \<turnstile>\<^sub>1 \<langle>try e\<^sub>1 catch(C i) e\<^sub>2,s\<^sub>0\<rangle> \<Rightarrow> \<langle>Throw a,(h\<^sub>1, ls\<^sub>1)\<rangle>"
fmrestrict_fset_drop: fixes S :: "'c fset"   and b :: "'c"   and m :: "('c, 'd) fmap" shows "fmrestrict_fset S (fmdrop b m) = fmrestrict_fset (S |-| {|b|}) m"
pt_fun_inst: assumes "pt TYPE('a) TYPE('b)"   and "pt TYPE('c) TYPE('b)"   and "at TYPE('b)" shows "pt TYPE('a \<Rightarrow> 'c) TYPE('b)"
separating_left: fixes A :: "'a set"   and A' :: "'a set"   and \<Gamma> :: "('a, 'b) graph_scheme"   and B :: "'a set" assumes "A \<subseteq> A'" shows "separating_gen \<Gamma> A B A'"
comp_null_1: fixes f :: "'s" shows "S.null \<cdot> f = S.null"
hyps: fixes x :: "'a" shows "ldropn n (iterates f x) = iterates f ((f ^^ n) x)"
mkM_kripke: fixes T :: "'s Trace set" shows "kripke (mkM T)"
elims_2: fixes x :: "'a \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> bool"   and xa :: "('a \<times> 'c) list \<times> 'o"   and xb :: "('a \<times> 'd) list \<times> 'o" assumes "prod_ord_raw x xa xb" obtains f :: "'a \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> bool" and xs :: "('a \<times> 'c) list" and ox :: "'o" and ys :: "('a \<times> 'd) list \<times> 'o" where "x = f" and "xa = (xs, ox)" and "xb = ys" and "prod_ord_pair ox f xs (sort_oalist_aux ox ys)"
open_segment_eq: fixes a :: "'b"   and b :: "'b"   and c :: "'b"   and d :: "'b" shows "(open_segment a b = open_segment c d) = (a = b \<and> c = d \<or> {a, b} = {c, d})"
fsafe: fixes f :: "'sz"   and f' :: "('sf + 'sz, 'sc, 'sz) formula" assumes "SPredicates \<sigma> f = Some f'" shows "fsafe f'"
leg_simps_1: shows "C.arr \<eta>.dom.leg0"
csum_absorb2: fixes r2 :: "('c \<times> 'c) set"   and r1 :: "('d \<times> 'd) set" assumes "Cinfinite r2"   and "r1 \<le>o r2" shows "r1 +c r2 =o r2"
interpret_floatarith_FDERIV_floatariths_append: fixes fas :: "floatarith list"   and x :: "'e"   and ramsch :: "real list" assumes "isFDERIV DIM('e) [0..<DIM('e)] fas (list_of_eucl x @ ramsch)"   and "max_Var_floatariths fas \<le> DIM('e) + length ramsch"   and "length fas = DIM('e)" shows "((\<lambda>x. einterpret fas (list_of_eucl x @ ramsch)) has_derivative (\<lambda>d. einterpret (FDERIV_floatariths fas [0..<DIM('e)] (map floatarith.Var [DIM('e) + length ramsch..<2 * DIM('e) + length ramsch])) (list_of_eucl x @ ramsch @ list_of_eucl d))) (at x)"
vec_elements_h_def: fixes v :: "('c, 'd) Finite_Cartesian_Product.vec" shows "vec_elements_h v = range (($h) v)"
indspace_el_eq_on_1ddh_imp_eq_on_rddh: fixes HmodG :: "'g set"   and f :: "('f, 'g) aezfun \<Rightarrow> 'v"   and f' :: "('f, 'g) aezfun \<Rightarrow> 'v"   and h :: "'g"   and r :: "'f" assumes "HmodG \<subseteq> H"   and "H = (\<Union>h\<in>HmodG. G + {h})"   and "f \<in> indV"   and "f' \<in> indV"   and "\<forall>h\<in>HmodG. f ((1::'f) \<delta>\<delta> h) = f' ((1::'f) \<delta>\<delta> h)"   and "h \<in> H" shows "f (r \<delta>\<delta> h) = f' (r \<delta>\<delta> h)"
simps_2: fixes v :: "variable"   and T :: "typ" shows "fv (Fv v T) = {(v, T)}"
pseudo_ordered_set_axioms: shows "pseudo_ordered_set P (\<sqsubseteq>)"
sets_embed_eq_vimage_algebra: fixes f :: "'c \<Rightarrow> 'd"   and M :: "'c measure" assumes "inj_on f (space M)" shows "sets (embed_measure M f) = sets (vimage_algebra (f ` space M) (the_inv_into (space M) f) M)"
piecewise_differentiable_add: fixes f :: "'a \<Rightarrow> 'b"   and i :: "'a set"   and g :: "'a \<Rightarrow> 'b" assumes "f piecewise_differentiable_on i"   and "g piecewise_differentiable_on i" shows "(\<lambda>x. f x + g x) piecewise_differentiable_on i"
locally_path_connected_components: fixes S :: "'b set"   and c :: "'b set" assumes "locally path_connected S"   and "c \<in> components S" shows "locally path_connected c"
0: fixes k :: "nat" shows "0 \<le> (1 - \<alpha>) ^ k"
wt_While: fixes tst :: "'test"   and c :: "('test, 'atom, 'choice) cmd"   and s :: "'state"   and i :: "nat" shows "wt (While tst c) s i = 1"
Deriv_union: fixes a :: "'b"   and A :: "'b list set"   and B :: "'b list set" shows "Deriv a (A \<union> B) = Deriv a A \<union> Deriv a B"
wf_lex_prod: fixes ra :: "('g \<times> 'g) set"   and rb :: "('h \<times> 'h) set" assumes "wf ra"   and "wf rb" shows "wf (ra <*lex*> rb)"
mk_eqcl'_Some2: fixes i :: "nat"   and xs :: "nat option list"   and k :: "nat"   and l :: "nat"   and ii :: "nat"   and j :: "nat"   and T :: "bool list list" assumes "i < length xs"   and "k < l" shows "(mk_eqcl' xs ii j l T ! i = Some k) = (xs ! i = Some k)"
collapse: fixes list :: "'d list" assumes "list \<noteq> []" shows "hd list # tl list = list"
insert_Id_AllowedActs: fixes F :: "'c program" shows "insert Id (AllowedActs F) = AllowedActs F"
prefixes_diverg_add: fixes t :: "nat"   and j :: "nat"   and d :: "nat" assumes "local.prefixes t j \<up>" shows "local.prefixes (t + d) j \<up>"
SCConcEnd: fixes xs :: "('b, 'c) Combinators list"   and xa :: "('b, 'c) Combinators" assumes "singleCombinators (xs @ [xa])" shows "singleCombinators xs"
in_Keys: fixes s :: "'b"   and F :: "('b \<Rightarrow>\<^sub>0 'c) set" shows "(s \<in> Keys F) = (\<exists>f\<in>F. s \<in> keys f)"
map_transfer: fixes R1b :: "'b \<Rightarrow> 'd \<Rightarrow> bool"   and S1d :: "'c \<Rightarrow> 'e \<Rightarrow> bool"   and R2b :: "'f \<Rightarrow> 'h \<Rightarrow> bool"   and S2d :: "'g \<Rightarrow> 'i \<Rightarrow> bool" shows "((R1b ===> S1d) ===> (R2b ===> S2d) ===> rel_prod R1b R2b ===> rel_prod S1d S2d) map_prod map_prod"
independent_rotate1: fixes Us :: "'a list"   and Vs :: "'a list" assumes "local.independent K (Us @ Vs)" shows "local.independent K (rotate1 Us @ Vs)"
Cons_5: shows "\<forall>c. c \<in> set (clause # Wl') \<longrightarrow> Some literal = getWatch1 state c \<or> Some literal = getWatch2 state c"
greatest_Lower_below: fixes L :: "('c, 'd) gorder_scheme"   and i :: "'c"   and A :: "'c set"   and x :: "'c" assumes "is_glb L i A"   and "x \<in> A"   and "A \<subseteq> carrier L" shows "i \<sqsubseteq>\<^bsub>L\<^esub> x"
transpose_empty: fixes xs :: "'b list list" shows "(transpose xs = []) = (\<forall>x\<in>set xs. x = [])"
slens_compl_def: fixes a :: "<'e, 'd> \<Longleftrightarrow> 'f" shows "-\<^sub>L a = \<lparr>view = \<C>\<^bsub>a\<^esub>, coview = \<V>\<^bsub>a\<^esub>\<rparr>"
weights_unit_def: shows "weights = \<lparr>w = Suc \<circ> snd, w0 = 1, pr_strict = \<lambda>(uu_, n) (uu_, m). m < n, least = \<lambda>_. True, scf = \<lambda>_ _. 1\<rparr>"
distinct_mset_add: fixes L :: "'b multiset"   and L' :: "'b multiset" shows "distinct_mset (L + L') = (distinct_mset L \<and> distinct_mset L' \<and> L \<inter># L' = {#})"
integer_shiftl_def: fixes x :: "integer"   and n :: "integer" shows "integer_shiftl x n = (if n < 0 then undefined x n else push_bit (nat_of_integer n) x)"
psubst_fls: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and eql :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and cnj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and imp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and all :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and exi :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and fls :: "'c"   and txs :: "('b \<times> 'a) list" assumes "Syntax_with_Connectives_False var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi fls"   and "snd ` set txs \<subseteq> var"   and "fst ` set txs \<subseteq> trm" shows "Generic_Syntax.psubst var Var FvarsT Fvars subst fls txs = fls"
path_to_def: fixes M :: "'b dfa"   and q :: "hf" assumes "dfa M" shows "dfa.path_to M q \<equiv> SOME u. u \<in> dfa.left_lang M q"
simps: fixes a :: "'s Trace"   and jp :: "'a \<Rightarrow> ('obs, 'aAct, 'c) Protocol" shows "(a \<in> jpTraces jp) = ((\<exists>s. a = tInit s \<and> s \<in> set envInit) \<or> (\<exists>t eact aact s. a = t \<leadsto> s \<and> t \<in> jpTraces jp \<and> eact \<in> set (envAction (tLast t)) \<and> (\<forall>x. aact x \<in> set (actJP jp t x)) \<and> s = envTrans eact aact (tLast t)))"
dgrad_p_set_le_red: fixes d :: "'a \<Rightarrow> nat"   and F :: "('t \<Rightarrow>\<^sub>0 'c) set"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" assumes "dickson_grading d"   and "red F p q" shows "dgrad_p_set_le d {q} (insert p F)"
supportE_if: fixes b :: "(char list \<Rightarrow> int) \<Rightarrow> bool"   and A :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> nat"   and B :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> nat" shows "supportE (\<lambda>l s. if b s then A l s else B l s) \<subseteq> supportE A \<union> supportE B"
inf_le2: fixes inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "class.semilattice_inf inf less_eq less" shows "less_eq (inf x y) y"
ide_char: fixes a :: "'d" shows "Dop.ide a = D.ide a"
SD_singleton_right: fixes x :: "'a"   and q :: "'a pmf" assumes "x \<in> carrier"   and "q \<in> lotteries_on carrier" shows "SD le q (return_pmf x) = (\<forall>y\<in>set_pmf q. le y x)"
SQ: shows "(S, Q) = Smith_1x2_eucl_JNF A"
finite_enumerate_step: fixes S :: "'a set"   and n :: "nat" assumes "finite S"   and "Suc n < card S" shows "enum S n < enum S (Suc n)"
subset_eq_diff_conv: fixes A :: "'c multiset"   and C :: "'c multiset"   and B :: "'c multiset" shows "(A - C \<subseteq># B) = (A \<subseteq># B + C)"
simps_4: fixes M :: "'b trm"   and N :: "'b trm"   and v :: "'b" shows "unify (M \<cdot> N) (Var v) = (if Var v \<prec> M \<cdot> N then None else Some [(v, M \<cdot> N)])"
simps_2: fixes et :: "('var, 'val, 'ret, 'pname) edge_kind"   and ets :: "('var, 'val, 'ret, 'pname) edge_kind list"   and s :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" shows "preds (et # ets) s = (pred et s \<and> preds ets (transfer et s))"
ccINF_greatest: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and A :: "'c set"   and u :: "'b"   and f :: "'c \<Rightarrow> 'b" assumes "class.countable_complete_lattice Inf Sup inf less_eq less sup bot top"   and "countable A"   and "\<And>i. i \<in> A \<Longrightarrow> less_eq u (f i)" shows "less_eq u (Inf (f ` A))"
diff_right: fixes a :: "'c \<Rightarrow>\<^sub>L 'b"   and b :: "'c"   and b' :: "'c" shows "blinfun_apply a (b - b') = blinfun_apply a b - blinfun_apply a b'"
carrier_Gp: fixes G :: "('e, 'f) Group_scheme"   and H :: "'e set" assumes "Group G"   and "G \<guillemotright> H " shows "carrier (\<natural>\<^bsub>G\<^esub>H) = H"
ord_option_map2: fixes ord :: "'h \<Rightarrow> 'i \<Rightarrow> bool"   and x :: "'h option"   and f :: "'j \<Rightarrow> 'i"   and y :: "'j option" shows "ord_option ord x (map_option f y) = ord_option (\<lambda>x y. ord x (f y)) x y"
cdf_diff_eq: fixes x :: "real"   and y :: "real" assumes "x < y" shows "cdf M1 y - cdf M1 x = M1.prob {x<..y}"
q63_infinite_def: shows "q63 = (\<lambda>k. k\<^sup>2 mod 63) ` {0..}"
upd_nested_step_def: fixes d :: "'c"   and f :: "'c \<Rightarrow> 'c"   and x :: "'a \<times> 'b"   and m :: "('a, ('b, 'c) mapping) mapping" shows "upd_nested_step d f x m = (case x of (k, k') \<Rightarrow> case Mapping.lookup m k of None \<Rightarrow> Mapping.update k (Mapping.update k' d Mapping.empty) m | Some m' \<Rightarrow> case Mapping.lookup m' k' of None \<Rightarrow> Mapping.update k (Mapping.update k' d m') m | Some v \<Rightarrow> Mapping.update k (Mapping.update k' (f v) m') m)"
comp: fixes x :: "'a"   and b :: "'b" assumes "x \<in> A"   and "b \<in> Basis" shows "0 \<le> f x \<bullet> b"
cart_basis_def: shows "cart_basis = {axis i (1::'b) |i. i \<in> UNIV}"
sc: shows "h \<turnstile> get_scdom_component (cast\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r document_ptr) \<rightarrow>\<^sub>r sc"
monoid_homomorphism_axioms: shows "monoid_homomorphism \<eta> M (\<cdot>) \<one> M' (\<cdot>') \<one>'"
ChiPlus_ChiRel_int: fixes T :: "'a"   and S :: "'a"   and A :: "('a, 'b, 'c) hierauto" assumes "(T, S) \<in> ChiPlus A" shows "ChiPlus A `` {T} \<inter> ChiRel A `` {S} = ChiRel A `` {S}"
assms_2: fixes a :: "'a"   and w :: "'a vec" assumes "v = vCons a w" shows "thesis"
semiring_normalization_rules_18: fixes times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and lx :: "'c"   and rx :: "'c"   and ry :: "'c" assumes "class.comm_semiring_1 times one plus zero" shows "times lx (times rx ry) = times (times lx rx) ry"
bags_continuous: fixes s :: "'c"   and u :: "'c"   and t :: "'c" assumes "s \<in> V\<^bsub>T\<^esub>"   and "u \<in> V\<^bsub>T\<^esub>"   and "t \<in> set (s \<leadsto>\<^bsub>T\<^esub> u)" shows "bag s \<inter> bag u \<subseteq> bag t"
emptyI: fixes b :: "real"   and a :: "real"   and P :: "real \<Rightarrow> bool" assumes "b \<le> a" shows "finite_Psegments P a b"
I_def: shows "I \<equiv> {f \<in> Basis \<rightarrow>\<^sub>E \<rat> \<times> \<rat>. box (a' f) (b' f) \<subseteq> M}"
weight_alt: fixes w :: "'a set \<Rightarrow> nat"   and g :: "'a ugraph" shows "weight w g = sum w (uedge ` edges g)"
cinner_diff_left: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "\<langle>x - y, z\<rangle> = \<langle>x, z\<rangle> - \<langle>y, z\<rangle>"
select_convs_1: fixes Obj :: "'a set"   and Mor :: "'b set"   and Dom :: "'b \<Rightarrow> 'a"   and Cod :: "'b \<Rightarrow> 'a"   and Id :: "'a \<Rightarrow> 'b"   and Comp :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and more :: "'c" shows "obj\<^bsub>\<lparr>Obj = Obj, Mor = Mor, Dom = Dom, Cod = Cod, Id = Id, Comp = Comp, \<dots> = more\<rparr>\<^esub> = Obj"
cindex_poly_taq: fixes p :: "real poly"   and a :: "real"   and b :: "real"   and q :: "real poly" shows "taq {x. poly p x = 0 \<and> a < x \<and> x < b} q = cindex_poly a b (pderiv p * q) p"
comp_assoc: fixes h :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr"   and g :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr"   and f :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr" shows "(h \<cdot> g) \<cdot> f = h \<cdot> g \<cdot> f"
case_distrib: fixes h :: "'c \<Rightarrow> 'b"   and f :: "'d \<Rightarrow> 'e \<Rightarrow> 'c"   and prod :: "'d \<times> 'e" shows "h (case prod of (x, xa) \<Rightarrow> f x xa) = (case prod of (x1, x2) \<Rightarrow> h (f x1 x2))"
create_name_gba_impl_def: shows "create_name_gba_impl \<equiv> \<lambda>x. create_graph_impl x \<bind> (\<lambda>xa. cr_rename_gba_impl xa x)"
Inf_insert: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and a :: "'a"   and A :: "'a set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "Inf (insert a A) = inf a (Inf A)"
thetaSeqZOT_sym: fixes indis :: "('a \<times> 'a) set"   and tval :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'a \<Rightarrow> 'a" assumes "PL_Indis indis" shows "sym (PL_Indis.thetaSeqZOT tval aval indis)"
map_conv_bind_option: fixes f :: "'f \<Rightarrow> 'e"   and x :: "'f option" shows "map_option f x = x \<bind> Some \<circ> f"
rquot_dfs_invariant: fixes A :: "nat bdd list \<times> bool list"   and n :: "nat"   and x :: "nat" assumes "rquot_DFS A n"   and "dfa_is_node A x" shows "rquot_invariant A (rquot_dfs A n x)"
UNpart1_def: fixes cval :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and indis :: "('b \<times> 'b) set"   and cl :: "('c, 'd, 'a) cmd list"   and dl :: "('e, 'f, 'g) cmd list"   and P :: "nat \<Rightarrow> nat set set" assumes "PL_Indis cval indis" shows "PL_Indis.UNpart1 cl dl P \<equiv> \<Union>n\<in>theNFTBoth cl dl. PL_Indis.shift cl n ` P n"
list_of_vec_mult: fixes c :: "complex"   and v :: "complex vec" shows "list_of_vec (c \<cdot>\<^sub>v v) = map ((*) c) (list_of_vec v)"
frag_cmul_one: fixes x :: "'b \<Rightarrow>\<^sub>0 int" shows "frag_cmul 1 x = x"
OrdP_Mem_lemma: fixes z :: "name"   and k :: "tm"   and l :: "tm"   and H :: "fm set" assumes "atom z \<sharp> (k, l)"   and "insert (OrdP k) H \<turnstile> l IN k" shows "insert (OrdP k) H \<turnstile> l SUBS k AND All2 z l (Var z SUBS l)"
assms: shows "\<D>\<^sub>\<circ> r \<noteq> 0"
Ord_is_Transset: fixes i :: "V" assumes "Ord i" shows "Transset i"
atLeastLessThan_eq_atLeastAtMost_diff: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" assumes "class.order less_eq less" shows "ord.atLeastLessThan less_eq less a b = ord.atLeastAtMost less_eq a b - {b}"
splitFace_split: fixes f :: "face"   and g :: "graph"   and v :: "nat"   and a :: "nat"   and f' :: "face"   and nvl :: "nat list" assumes "f \<in> \<F> (snd (snd (splitFace g v a f' nvl)))" shows "f \<in> \<F> g \<or> f = fst (splitFace g v a f' nvl) \<or> f = fst (snd (splitFace g v a f' nvl))"
map_def: fixes f :: "'c" shows "G'\<eta>F'o\<eta>'.map f = (if C.arr f then (G' \<circ> FG.\<eta> \<circ> F') (C.cod f) \<cdot>\<^sub>C F'G'.\<eta> f else C.null)"
infnorm_eq_0: fixes x :: "'a" shows "(infnorm x = 0) = (x = (0::'a))"
T2: fixes w :: "i"   and x :: "\<kappa>"   and F :: "\<Pi>\<^sub>1" shows "[\<lbrace>x,F\<rbrace> in w] = (\<exists>r o\<^sub>1. Some r = Semantics.d\<^sub>1 F \<and> Some o\<^sub>1 = Semantics.d\<^sub>\<kappa> x \<and> o\<^sub>1 \<in> Semantics.en r)"
LIM_I: fixes a :: "'c"   and f :: "'c \<Rightarrow> 'd"   and L :: "'d" assumes "\<And>r. 0 < r \<Longrightarrow> \<exists>s>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < s \<longrightarrow> norm (f x - L) < r" shows "f \<midarrow>a\<rightarrow> L"
Func_def: fixes A :: "'a set"   and B :: "'b set" shows "Func A B = {f. (\<forall>a\<in>A. f a \<in> B) \<and> (\<forall>a. a \<notin> A \<longrightarrow> f a = undefined)}"
R01s_def: shows "R01s \<equiv> {(s, s'). s = \<lparr>ik = ik s', secret = secret s'\<rparr>}"
fundamental_theorem: fixes \<Gamma> :: "(name \<times> ty) list"   and t :: "trm"   and \<tau> :: "ty"   and \<theta> :: "(name \<times> trm) list" assumes "\<Gamma> \<turnstile> t : \<tau>"   and "\<theta> closes \<Gamma>" shows "\<theta><t> \<in> RED \<tau>"
rel_frontier_of_polyhedron_alt: fixes S :: "'b set" assumes "polyhedron S" shows "rel_frontier S = \<Union> {F. F face_of S \<and> F \<noteq> S}"
inter_add_left2: fixes x :: "'a"   and N :: "'a multiset"   and M :: "'a multiset" assumes "x \<in># N" shows "add_mset x M \<inter># N = add_mset x (M \<inter># remove1_mset x N)"
exists_root: fixes M :: "(('a list \<times> nat) multiset \<Rightarrow> 'a) ring"   and P :: "'a list" assumes "M \<in> extensions"   and "\<And>L. \<lbrakk>L \<in> extensions; M \<lesssim> L\<rbrakk> \<Longrightarrow> law_restrict L = law_restrict M"   and "P \<in> carrier (poly_ring R)" shows "ring.splitted M (\<sigma> P)"
valid_failure: fixes ioprog :: "'d \<Rightarrow> 'c \<Rightarrow> ('b \<times> 'c) option"   and a :: "'d"   and \<sigma> :: "'c"   and S :: "'d list"   and M :: "'b list \<Rightarrow> 'c \<Rightarrow> (bool \<times> 'c) option" assumes "ioprog a \<sigma> = None" shows "(\<sigma> \<Turnstile> bind\<^sub>S\<^sub>E (mbind (a # S) ioprog) M) = (\<sigma> \<Turnstile> M [])"
CollectI: fixes P :: "'a \<Rightarrow> bool"   and a :: "'a" assumes "P a" shows "a \<in> _Collect x (P x)"
intro: fixes G :: "('a, 'b) graph_rec_scheme"   and param :: "('a, ('a, 'c) state_scheme, 'c) gen_parameterization"   and s :: "('a, 'c) state_scheme" assumes "param_DFS G"   and "DFS_invar_axioms G param s" shows "DFS_invar G param s"
infsetsum_mono_neutral_left: fixes f :: "'a \<Rightarrow> real"   and A :: "'a set"   and g :: "'a \<Rightarrow> real"   and B :: "'a set" assumes "f abs_summable_on A"   and "g abs_summable_on B"   and "\<And>x. x \<in> A \<Longrightarrow> f x \<le> g x"   and "A \<subseteq> B"   and "\<And>x. x \<in> B - A \<Longrightarrow> 0 \<le> g x" shows "infsetsum f A \<le> infsetsum g B"
renaming: fixes c :: "name"   and P :: "pi"   and a :: "name"   and b :: "name" assumes "c \<sharp> P" shows "P[a::=b] = ([(c, a)] \<bullet> P)[c::=b]"
degree_rebase_poly_le: fixes p :: "'b mod_ring poly" shows "degree (#p) \<le> degree p"
add_diff_self_ennreal: fixes a :: "ennreal"   and b :: "ennreal" shows "a + (b - a) = (if a \<le> b then b else a)"
extended_Gromov_distance_def: fixes x :: "'b Gromov_completion"   and y :: "'b Gromov_completion" shows "extended_Gromov_distance x y = (if x \<in> Gromov_boundary \<or> y \<in> Gromov_boundary then \<infinity> else ereal (dist (inv to_Gromov_completion x) (inv to_Gromov_completion y)))"
iv2: shows "height_tree (Node e2 a2 r ts2) = r"