hcju/isabelleps huggingface dataset:
induct: fixes A :: "'g \<Rightarrow> bool"   and x :: "'g list"   and P :: "'g list \<Rightarrow> bool" assumes "listsp A x"   and "P []"   and "\<And>a l. \<lbrakk>A a; listsp A l; P l\<rbrakk> \<Longrightarrow> P (a # l)" shows "P x"
V_def: shows "V \<equiv> {v. \<forall>x\<in>\<X>. (0::'b) \<le> v x}"
removeAll_subseteq_remove1_mset: fixes M :: "'b multiset"   and x :: "'b" shows "removeAll_mset x M \<subseteq># remove1_mset x M"
chainD: fixes ord :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and S :: "'a set"   and x :: "'a"   and y :: "'a" assumes "Complete_Partial_Order.chain ord S"   and "x \<in> S"   and "y \<in> S" shows "ord x y \<or> ord y x"
bulkload_tabulate: fixes xs :: "'c list" shows "Mapping.bulkload xs = Mapping.tabulate [0..<length xs] ((!) xs)"
cSup_cInf: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and S :: "'c set" assumes "class.conditionally_complete_lattice Inf Sup inf less_eq less sup"   and "S \<noteq> {}"   and "preorder.bdd_above less_eq S" shows "Sup S = Inf {x. \<forall>s\<in>S. less_eq s x}"
normalize_positive_dst_ports_nnf: fixes m :: "'b common_primitive match_expr"   and m' :: "'b common_primitive match_expr" assumes "normalized_nnf_match m"   and "\<not> has_disc_negated is_Dst_Ports False m"   and "m' \<in> set (normalize_positive_dst_ports m)" shows "normalized_nnf_match m'"
quasi_inverses_some_quasi_inverse_1: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and f :: "'b" assumes "bicategory V H \<a> \<i> src trg"   and "bicategory.equivalence_map V H \<a> \<i> src trg f" shows "bicategory.quasi_inverses V H \<a> \<i> src trg f (bicategory.some_quasi_inverse V H \<a> \<i> src trg f)"
update_simps_5: fixes ip :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option"   and r :: "nat \<times> k \<times> f \<times> nat \<times> nat" assumes "ip \<in> kD rt"   and "\<pi>\<^sub>3 r = unk"   and "(\<pi>\<^sub>2 r = 0) = (\<pi>\<^sub>3 r = unk)" shows "update rt ip r = rt(ip \<mapsto> (\<pi>\<^sub>2 (the (rt ip)), \<pi>\<^sub>3 r, \<pi>\<^sub>4 r, \<pi>\<^sub>5 r, \<pi>\<^sub>6 r))"
d_delta_arctan_upper_11_pos: fixes x :: "real" assumes "x \<noteq> 0" shows "0 < diff_delta_arctan_upper_11 x"
lookup_default_bind_pmf_aux': fixes p :: "'d pmf"   and f :: "'d \<Rightarrow> 'c pmf"   and x :: "'c" shows "Mapping.lookup_default 0 (bind_pmf_aux p f (set_pmf p)) x = pmf (p \<bind> f) x"
quant_equiv_iff: fixes x :: "'a['b, 'c]"   and y :: "'a['d, 'c]" shows "(x \<cong>\<^sub>Q y) = (\<lbrakk>x\<rbrakk>\<^sub>Q = \<lbrakk>y\<rbrakk>\<^sub>Q \<and> QD('b) = QD('d))"
fv_regex_from_mregex: fixes \<phi>s :: "Formula.formula list"   and mr :: "mregex" assumes "ok (length \<phi>s) mr" shows "fv_regex (from_mregex mr \<phi>s) \<subseteq> \<Union> (fv ` set \<phi>s)"
hom_mult: fixes x :: "'a"   and y :: "'a" shows "hom (x ** y) = hom x +++ hom y"
replicate_le_diff2: fixes k :: "nat"   and m :: "nat"   and n :: "nat"   and x :: "'d" assumes "k \<le> m"   and "m \<le> n" shows "x\<^bsup>m - k\<^esup> @ x\<^bsup>n - m\<^esup> = x\<^bsup>n - k\<^esup>"
inverse_pow_pow: fixes G :: "('a, 'b) cyclic_group_scheme"   and a :: "'a"   and r :: "nat" assumes "Cyclic_Group.cyclic_group G"   and "a \<in> carrier G" shows "inv\<^bsub>G\<^esub> (a [^]\<^bsub>G\<^esub> r) = inv\<^bsub>G\<^esub> a [^]\<^bsub>G\<^esub> r"
flip_def: fixes f :: "'b \<Rightarrow> 'a \<Rightarrow> 'c" shows "flip f = (\<lambda>b a. f a b)"
singular_relboundary_add: fixes p :: "nat"   and X :: "'c topology"   and S :: "'c set"   and a :: "((nat \<Rightarrow> real) \<Rightarrow> 'c) \<Rightarrow>\<^sub>0 int"   and b :: "((nat \<Rightarrow> real) \<Rightarrow> 'c) \<Rightarrow>\<^sub>0 int" assumes "singular_relboundary p X S a"   and "singular_relboundary p X S b" shows "singular_relboundary p X S (a + b)"
stable_onI: fixes ds :: "D2 set"   and X :: "X3 set" assumes "Theorem_9.individually_rational_on ds X"   and "Theorem_9.stable_no_blocking_on ds X" shows "Theorem_9.stable_on ds X"
iT_Plus_neg_card: fixes I :: "nat set"   and k :: "nat" shows "card (I \<oplus>- k) = card (I \<down>\<ge> k)"
partition_on_insert_elements: fixes A :: "'a set"   and B :: "'a set"   and P :: "'a set set"   and f :: "'a \<Rightarrow> 'a set" assumes "A \<inter> B = {}"   and "partition_on B P"   and "f \<in> A \<rightarrow>\<^sub>E P" shows "partition_on (A \<union> B) ((\<lambda>X. X \<union> {x \<in> A. f x = X}) ` P)"
wcount_union: fixes f :: "'k \<Rightarrow> nat"   and M :: "'k multiset"   and N :: "'k multiset"   and a :: "'k" shows "wcount f (M + N) a = wcount f M a + wcount f N a"
assms_3: fixes i :: "nat" assumes "i < dim_vec c" shows "(A\<^sup>T *\<^sub>v x) $ i = c $ i"
trans_MapOps_update_def: shows "trans_MapOps_update \<equiv> \<lambda>((cec, aec), obs) v m. trie_update_with' (map fst (toList cec)) m empty_trie (\<lambda>m. trie_update_with' (map snd (toList cec)) m empty_trie (\<lambda>m. trie_update_with' (map fst (toList aec)) m empty_trie (\<lambda>m. trie_update_with' (map snd (toList aec)) m Mapping.empty (Mapping.update obs v))))"
CFG_call_determ: fixes sourcenode :: "'g \<Rightarrow> 'h"   and targetnode :: "'g \<Rightarrow> 'h"   and kind :: "'g \<Rightarrow> ('i, 'j, 'k, 'l) edge_kind"   and valid_edge :: "'g \<Rightarrow> bool"   and Entry :: "'h"   and get_proc :: "'h \<Rightarrow> 'l"   and get_return_edges :: "'g \<Rightarrow> 'g set"   and procs :: "('l \<times> 'i list \<times> 'i list) list"   and Main :: "'l"   and Def :: "'h \<Rightarrow> 'i set"   and Use :: "'h \<Rightarrow> 'i set"   and ParamDefs :: "'h \<Rightarrow> 'i list"   and ParamUses :: "'h \<Rightarrow> 'i set list"   and a :: "'g"   and Q :: "('i \<Rightarrow> 'j option) \<times> 'k \<Rightarrow> bool"   and r :: "'k"   and p :: "'l"   and fs :: "(('i \<Rightarrow> 'j option) \<Rightarrow> 'j option) list"   and a' :: "'g"   and Q' :: "('i \<Rightarrow> 'j option) \<times> 'k \<Rightarrow> bool"   and r' :: "'k"   and p' :: "'l"   and fs' :: "(('i \<Rightarrow> 'j option) \<Rightarrow> 'j option) list"   and s :: "(('i \<Rightarrow> 'j option) \<times> 'k) list" assumes "CFG_wf sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Def Use ParamDefs ParamUses"   and "valid_edge a"   and "kind a = Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs"   and "valid_edge a'"   and "kind a' = Q':r'\<hookrightarrow>\<^bsub>p'\<^esub>fs'"   and "sourcenode a = sourcenode a'"   and "CFG.pred (kind a) s"   and "CFG.pred (kind a') s" shows "a = a'"
keys_def: shows "keys \<equiv> map_fun lookup id (\<lambda>f. {k. f k \<noteq> (0::'b)})"
notKSP: shows "sKS m \<notin> specKeysSecrets P"
kth_res_equals_def: fixes k :: "nat"   and n :: "int"   and a :: "nat \<Rightarrow> int" shows "kth_res_equals k n a = (a k = n)"
set_disconnected_nodes_get_ancestors_si: fixes ptr :: "(_) document_ptr"   and h :: "(_) heap"   and h' :: "(_) heap" shows "\<forall>w\<in>set_disconnected_nodes_locs ptr. h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> (\<forall>r\<in>get_ancestors_si_locs. r h h')"
cons_pre_rulet: fixes P :: "assn"   and P' :: "assn"   and c :: "'a Heap"   and Q :: "'a \<Rightarrow> assn" assumes "P \<Longrightarrow>\<^sub>t P'"   and "<P'> c <Q>\<^sub>t" shows "<P> c <Q>\<^sub>t"
lm049: fixes XX :: "('c \<times> 'd set) set" assumes "XX \<in> partitionValuedUniverse" shows "{} \<notin> Range XX"
simps: fixes a1 :: "'a"   and a2 :: "'a" shows "hull_relp a1 a2 = (\<exists>p x. a1 = p \<bullet> x \<and> a2 = x)"
ComplI: fixes c :: "'a"   and A :: "'a set" assumes "c \<in> A \<Longrightarrow> False" shows "c \<in> - A"
Confidentiality_B: fixes B :: "agent"   and Tk :: "nat"   and A :: "agent"   and K :: "nat"   and evs :: "event list" assumes "Crypt (shrK B) \<lbrace>Number Tk, Agent A, Key K\<rbrace> \<in> parts (knows Spy evs)"   and "Notes Spy \<lbrace>Number Tk, Key K\<rbrace> \<notin> set evs"   and "A \<notin> bad"   and "B \<notin> bad"   and "evs \<in> bankerberos" shows "Key K \<notin> analz (knows Spy evs)"
obj_char: fixes a :: "('a, 'a set) Maps.arr arrow_of_spans_data" shows "Span.obj a = (Span.ide a \<and> a = Span.mkObj (Chn a))"
keys_eq_Nil_iff: fixes rbt :: "'b set_rbt" shows "(keys rbt = []) = (rbt = RBT_Set2.empty)"
trans: fixes r :: "'b"   and s :: "'b"   and t :: "'b" assumes "r = s"   and "s = t" shows "r = t"
induct: fixes P :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'a list \<Rightarrow> bool"   and a0 :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a1 :: "'a list"   and a2 :: "'a list" assumes "\<And>R x xs y ys. \<lbrakk>R x y \<Longrightarrow> P R xs (y # ys); \<not> R x y \<Longrightarrow> P R (x # xs) ys\<rbrakk> \<Longrightarrow> P R (x # xs) (y # ys)"   and "\<And>R xs. P R xs []"   and "\<And>R v va. P R [] (v # va)" shows "P a0 a1 a2"
pOp_assocTr42: fixes a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" assumes "a \<in> carrier M"   and "b \<in> carrier M"   and "c \<in> carrier M"   and "d \<in> carrier M" shows "a \<plusminus> b \<plusminus> c \<plusminus> d = a \<plusminus> (b \<plusminus> c) \<plusminus> d"
assms_1: fixes i :: "nat" assumes "i < n" shows "degree (f i) \<le> 1"
image_projection_PiE: fixes i :: "'d"   and I :: "'d set"   and S :: "'d \<Rightarrow> 'c set" shows "(\<lambda>f. f i) ` Pi\<^sub>E I S = (if Pi\<^sub>E I S = {} then {} else if i \<in> I then S i else {undefined})"
size_12: fixes x61 :: "'a formula"   and x62 :: "'a formula" shows "size (x61 \<^bold>\<rightarrow> x62) = size x61 + size x62 + Suc 0"
sum_wcount_Int: fixes A :: "'k set"   and f :: "'k \<Rightarrow> nat"   and N :: "'k multiset" assumes "finite A" shows "sum (wcount f N) (A \<inter> set_mset N) = sum (wcount f N) A"
Deterministic_f_Exec: fixes trans_fun :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and localState :: "'e \<Rightarrow> 'f"   and c1 :: "'e"   and c2 :: "'e"   and xs :: "'d list" assumes "Deterministic_Trans_Fun trans_fun localState"   and "localState c1 = localState c2"   and "xs \<noteq> []" shows "f_Exec_Comp trans_fun xs c1 = f_Exec_Comp trans_fun xs c2"
domains_char: fixes f :: "('o, 'a) arr" shows "domains f = (if arr f then {local.dom f} else {})"
vimage_vunion_right: fixes r :: "V"   and A :: "V"   and B :: "V" shows "r `\<^sub>\<circ> (A \<union>\<^sub>\<circ> B) = r `\<^sub>\<circ> A \<union>\<^sub>\<circ> r `\<^sub>\<circ> B"
Lm_read: fixes a :: "'b"   and r :: "(bool \<times> 'b) rexp" shows "Lm (read a r) = {w \<in> Lm r. w \<noteq> [] \<and> hd w = a}"
rev_take_Suc: fixes j :: "nat"   and xs :: "'a list" assumes "j < length xs" shows "rev (take (Suc j) xs) = xs ! j # rev (take j xs)"
inj_on_of_nat: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and N :: "nat set" assumes "class.semiring_char_0 one times plus zero" shows "inj_on (semiring_1.of_nat one plus zero) N"
satA_cong: fixes at :: "('fsym, 'psym) atm"   and \<xi>1 :: "Preliminaries.var \<Rightarrow> 'univ"   and \<xi>2 :: "Preliminaries.var \<Rightarrow> 'univ" assumes "\<And>x. x \<in> varsA at \<Longrightarrow> \<xi>1 x = \<xi>2 x" shows "satA \<xi>1 at = satA \<xi>2 at"
chain_lappend: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b llist"   and ys :: "'b llist" assumes "chain R xs"   and "chain R ys"   and "R (llast xs) (lhd ys)" shows "chain R (lappend xs ys)"
deriv: fixes P :: "'a eclause set"   and S :: "'a eclause set"   and C :: "'a eclause"   and S' :: "'a eclause set"   and \<sigma> :: "('a \<times> 'a trm) list"   and C' :: "'a literal set" assumes "\<forall>x. x \<in> P \<longrightarrow> derivable_ecl x S"   and "derivable C P S' \<sigma> FirstOrder C'" shows "derivable_ecl C S"
Ioc_inj: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c"   and c :: "'c"   and d :: "'c" assumes "class.linorder less_eq less" shows "(ord.greaterThanAtMost less_eq less a b = ord.greaterThanAtMost less_eq less c d) = (less_eq b a \<and> less_eq d c \<or> a = c \<and> b = d)"
closedin_compact_eq: fixes S :: "'a set"   and T :: "'a set" assumes "compact S" shows "closedin (top_of_set S) T = (compact T \<and> T \<subseteq> S)"
polymul_degreen: fixes p :: "poly"   and n0 :: "nat"   and q :: "poly"   and n1 :: "nat"   and m :: "nat" assumes "isnpolyh p n0"   and "isnpolyh q n1"   and "m \<le> min n0 n1" shows "degreen (p *\<^sub>p q) m = (if p = 0\<^sub>p \<or> q = 0\<^sub>p then 0 else degreen p m + degreen q m)"
summable: shows "summable (\<lambda>p. ln (real (p + 1)) * (1 / real (p * (p - 1))))"
distinct_3: fixes x1 :: "'b"   and x31 :: "'b match_expr"   and x32 :: "'b match_expr" shows "Match x1 \<noteq> MatchAnd x31 x32"
division_contains: fixes s :: "'c set set"   and i :: "'c set" assumes "s division_of i" shows "\<forall>x\<in>i. \<exists>k\<in>s. x \<in> k"
dual_upset: shows "\<up> = ord.downset (\<lambda>x y. y \<le> x)"
ordinal_plus_monoR: fixes y :: "ordinal"   and y' :: "ordinal"   and x :: "ordinal" assumes "y \<le> y'" shows "x + y \<le> x + y'"
has_integral_integral_lebesgue: fixes f :: "'c \<Rightarrow> 'd" assumes "integrable lebesgue f" shows "(f has_integral integral\<^sup>L lebesgue f) UNIV"
simps_2: fixes f :: "'aq \<Rightarrow> 'ar \<Rightarrow> 'ap"   and x1 :: "'aq"   and x2 :: "'ar" shows "(case (x1, x2) of (x, xa) \<Rightarrow> f x xa) = f x1 x2"
dist_nonneg_bounded: fixes l :: "real"   and x :: "real"   and u :: "real"   and y :: "real" assumes "l \<le> x"   and "x \<le> u"   and "l \<le> y"   and "y \<le> u" shows "dist x y \<le> u - l"
leq_antisym: fixes x :: "'d spmf"   and y :: "'d spmf" assumes "ord_spmf (=) x y"   and "ord_spmf (=) y x" shows "x = y"
ins_size_2: fixes bal_i :: "nat \<Rightarrow> nat \<Rightarrow> bool"   and n :: "nat"   and d :: "nat"   and x :: "'b"   and t :: "'b tree"   and t' :: "'b tree" assumes "RBTi1 TYPE('b) bal_i"   and "RBTi1.ins bal_i n d x t = Unbal t'" shows "size t' = size t + 1"
assms_3: shows "\<bar>int (KD_Tree.size_kdt l) - int (KD_Tree.size_kdt r)\<bar> \<le> 1"
pdivides_imp_splitted: fixes R :: "('c, 'd) ring_scheme"   and p :: "'c list"   and q :: "'c list" assumes "field R"   and "p \<in> carrier (poly_ring R)"   and "q \<in> carrier (poly_ring R)"   and "q \<noteq> []"   and "ring.splitted R q"   and "p pdivides\<^bsub>R\<^esub> q" shows "ring.splitted R p"
taken_def: fixes p :: "'g"   and l :: "'h"   and s :: "('i, 'h, 'g, 'j, 'k) system_state" shows "taken p l s = (cTKN (GST s p) = Some l)"
bprv_\<phi>R_eqv: shows "bprv (eqv \<phi>R (neg (PP' \<langle>\<phi>R\<rangle>)))"
ipset_from_cidr_0: fixes ip :: "'b word" shows "ipset_from_cidr ip 0 = UNIV"
pairwise_def: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and S :: "'b set" shows "pairwise R S = (\<forall>x\<in>S. \<forall>y\<in>S. x \<noteq> y \<longrightarrow> R x y)"
bs_empty_correct: fixes h :: "unit + ('a, 'b) BsSkewBinomialHeapStruc.BsSkewElem" shows "(h = BsSkewBinomialHeapStruc.bs_empty) = (BsSkewBinomialHeapStruc.bs_to_mset h = {#})"
dyn_array_new_sz_def: fixes dflt :: "'b"   and n :: "nat" shows "dyn_array_new_sz dflt n \<equiv> Array.new n dflt"
simps_7: fixes V :: "char list"   and e :: "expr" shows "fv (V:=e) = {V} \<union> fv e"
strand_subst_img_bound: fixes \<delta> :: "'c \<Rightarrow> ('d, 'c) Term.term"   and S :: "('d, 'c) strand_step list" assumes "subst_domain \<delta> \<union> range_vars \<delta> \<subseteq> fv\<^sub>s\<^sub>t S"   and "(subst_domain \<delta> \<union> range_vars \<delta>) \<inter> bvars\<^sub>s\<^sub>t S = {}" shows "range_vars \<delta> \<subseteq> fv\<^sub>s\<^sub>t (S \<cdot>\<^sub>s\<^sub>t \<delta>)"
\<phi>_5: shows "csupport_on carrier \<phi> \<inter> carrier \<subseteq> W"
Inf_has_prem: fixes \<iota> :: "'f inference" assumes "\<iota> \<in> Inf" shows "prems_of \<iota> \<noteq> []"
lunit_char_2: fixes a :: "'d" assumes "D.ide a" shows "\<I>\<^sub>1 \<otimes>\<^sub>D \<l>\<^sub>1[a] = (\<iota>\<^sub>1 \<otimes>\<^sub>D a) \<cdot>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[\<I>\<^sub>1, \<I>\<^sub>1, a]"
leq_trans: fixes y :: "'a"   and x :: "'a"   and z :: "'a" assumes "y \<le> x"   and "z \<le> y" shows "z \<le> x"
ccEmpty_below: fixes G :: "CoCalls" shows "ccEmpty \<sqsubseteq> G"
equivp_reflp: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b" assumes "equivp R" shows "R x x"
name_fresh_left: fixes a :: "name"   and pi :: "(name \<times> name) list"   and x :: "'a" shows "a \<sharp> pi \<bullet> x = rev pi \<bullet> a \<sharp> x"
prv_cnjER: fixes \<phi> :: "'fmla"   and \<chi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "\<chi> \<in> fmla"   and "prv (cnj \<phi> \<chi>)" shows "prv \<chi>"
split_of_bool_asm: fixes one :: "'a"   and zero :: "'a"   and P :: "'a \<Rightarrow> bool"   and p :: "bool" assumes "class.zero_neq_one one zero" shows "P (zero_neq_one.of_bool one zero p) = (\<not> (p \<and> \<not> P one \<or> \<not> p \<and> \<not> P zero))"
intro: fixes U\<^sub>1 :: "'a set"   and inf\<^sub>1 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and le\<^sub>1 :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and ls\<^sub>1 :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and U\<^sub>2 :: "'b set"   and inf\<^sub>2 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and le\<^sub>2 :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and ls\<^sub>2 :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "semilattice_inf_ow U\<^sub>1 inf\<^sub>1 le\<^sub>1 ls\<^sub>1"   and "semilattice_inf_ow U\<^sub>2 inf\<^sub>2 le\<^sub>2 ls\<^sub>2" shows "semilattice_inf_pair_ow U\<^sub>1 inf\<^sub>1 le\<^sub>1 ls\<^sub>1 U\<^sub>2 inf\<^sub>2 le\<^sub>2 ls\<^sub>2"
factors_equal: fixes a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" assumes "a = b"   and "c = d" shows "a + c = b + d"
antisym: fixes b :: "'a"   and a :: "'a" assumes "b \<preceq> a"   and "a \<preceq> b" shows "a = b"
covering_space_lift_homotopy: fixes C :: "'c set"   and p :: "'c \<Rightarrow> 'd"   and S :: "'d set"   and U :: "'e set"   and h :: "real \<times> 'e \<Rightarrow> 'd"   and f :: "'e \<Rightarrow> 'c" assumes "covering_space C p S"   and "continuous_on ({0..1} \<times> U) h"   and "h ` ({0..1} \<times> U) \<subseteq> S"   and "\<And>y. y \<in> U \<Longrightarrow> h (0, y) = p (f y)"   and "continuous_on U f"   and "f ` U \<subseteq> C" obtains k :: "real \<times> 'e \<Rightarrow> 'c" where "continuous_on ({0..1} \<times> U) k" and "k ` ({0..1} \<times> U) \<subseteq> C" and "\<And>y. y \<in> U \<Longrightarrow> k (0, y) = f y" and "\<And>z. z \<in> {0..1} \<times> U \<Longrightarrow> h z = p (k z)"
simps: fixes B :: "ty"   and C :: "ty"   and X :: "ntrm set"   and Y :: "ctrm set"   and Z :: "ntrm set"   and U :: "ctrm set" shows "IMPRIGHT (B IMP C) X Y Z U = {<b>:ImpR x.<a>.M b |x a b M. fic (ImpR x.<a>.M b) b \<and> (\<forall>z P. x \<sharp> (z, P) \<and> z:P \<in> Z \<longrightarrow> x:M{a:=z.P} \<in> X) \<and> (\<forall>c Q. a \<sharp> (c, Q) \<and> <c>:Q \<in> U \<longrightarrow> <a>:M{x:=<c>.Q} \<in> Y)}"
step_Cons': fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and ccs :: "('e, 'd, 'f) com list"   and css :: "(('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' :: "(('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list"   and t :: "('e, 'f) xstate"   and c :: "('e, 'd, 'f) com"   and cs :: "('e, 'd, 'f) com list" assumes "\<Gamma>\<turnstile> (ccs, css, s) \<rightarrow> (cs', css', t)"   and "ccs = c # cs" shows "\<exists>css''. css' = css'' @ css \<and> (if css'' = [] then \<exists>p. cs' = p @ cs else \<exists>pnorm pabr. css'' = [(pnorm @ cs, pabr @ cs)])"
module_is_ag: fixes M :: "('h, 'i, 'j) Module_scheme"   and R :: "('i, 'k) Ring_scheme" assumes "R module M" shows "aGroup M"
tauActionE_1: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and P' :: "('a, 'b, 'c) psi" assumes "\<Psi> \<rhd> \<tau>.P \<longmapsto> \<tau> \<prec> P'" shows "\<Psi> \<rhd> P \<sim> P'"
strips_equivalent_to_sas_plus: fixes \<Psi> :: "('c, 'd) sas_plus_problem"   and \<psi> :: "('c, 'd) sas_plus_operator list list" assumes "is_valid_problem_sas_plus \<Psi>"   and "SAS_Plus_Semantics.is_parallel_solution_for_problem \<Psi> \<psi>" shows "STRIPS_Semantics.is_parallel_solution_for_problem (\<phi> \<Psi> ) (\<phi>\<^sub>P \<Psi> \<psi>)"
AboveS_def: fixes r :: "('d \<times> 'd) set"   and A :: "'d set" shows "AboveS r A \<equiv> {b \<in> Field r. \<forall>a\<in>A. b \<noteq> a \<and> (a, b) \<in> r}"
vproduct_eq_vemptyI: fixes i :: "V"   and I :: "V"   and A :: "V \<Rightarrow> V" assumes "i \<in>\<^sub>\<circ> I"   and "A i = 0" shows "vproduct I A = 0"
pairwise_rel_prime1: shows "\<forall>q. List.member qs1 q \<longrightarrow> coprime p q"
minus_less_iff: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less" shows "less (uminus a) b = less (uminus b) a"
\<gamma>1: shows "\<gamma>1 \<in> conv ars"
emeasure_cball: fixes r :: "real"   and c :: "'c" assumes "0 \<le> r" shows "emeasure lborel (cball c r) = ennreal (unit_ball_vol (real DIM('c)) * r ^ DIM('c))"
transfer_WHILEI: fixes f :: "'a \<Rightarrow> 'mc"   and F :: "'a \<Rightarrow> 'ma"   and I :: "'a \<Rightarrow> bool"   and b :: "'a \<Rightarrow> bool"   and x :: "'a" assumes "\<And>x. \<alpha> (f x) \<le> F x" shows "\<alpha> (cWHILEI I b f x) \<le> aWHILEI I b F x"
types_preserved_load: fixes \<S> :: "s_context"   and \<C> :: "t_context"   and k :: "i32"   and t :: "t"   and tp :: "(tp \<times> sx) option"   and a :: "nat"   and off :: "nat"   and ts :: "t list"   and ts' :: "t list"   and v :: "v" assumes "\<S>\<bullet>\<C> \<turnstile> [$C ConstInt32 k, $Load t tp a off] : ts _> ts'"   and "typeof v = t" shows "\<S>\<bullet>\<C> \<turnstile> [$C v] : ts _> ts'"
bdd_above_Int2: fixes B :: "'b multiset set"   and A :: "'b multiset set" assumes "subset_mset.bdd_above B" shows "subset_mset.bdd_above (A \<inter> B)"
intro: fixes A :: "'c set"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "compatible_ordering A less_eq less"   and "pseudo_ordered_set A less_eq" shows "pseudo_ordering A less_eq less"
linear_independent_injective_image: fixes f :: "'b \<Rightarrow> 'c"   and s :: "'b set" assumes "linear (*a) (*b) f"   and "vs1.independent s"   and "inj_on f (vs1.span s)" shows "vs2.independent (f ` s)"
eq_tpoly_def: fixes q :: "('v pvar, hmultiset) tpoly"   and p :: "('v pvar, hmultiset) tpoly" shows "(q =\<^sub>p p) = (\<forall>A. legal_zpassign A \<longrightarrow> eval_tpoly A (map_tpoly (\<lambda>x. x) zhmset_of q) = eval_tpoly A (map_tpoly (\<lambda>x. x) zhmset_of p))"
xy_2: shows "y = rat_of_int c / rat_of_int d"
pair_complete_imp_directed: fixes A :: "'b set"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "pair_complete A less_eq" shows "directed A less_eq"
elem_in_singleton: fixes a :: "'a" shows "[a]\<^sub>v $ 0 = a"
iam_lookup_abs2: fixes k :: "nat"   and l :: "'a option list" assumes "\<not> k < length l" shows "iam_of_list l k = None"
simple_not_triv: fixes G :: "('c, 'd) monoid_scheme" assumes "simple_group G" shows "carrier G \<noteq> {\<one>\<^bsub>G\<^esub>}"
uexpr_eq_iff: fixes e :: "('b, 'c) uexpr"   and f :: "('b, 'c) uexpr" shows "(e = f) = (\<forall>b. \<lbrakk>e\<rbrakk>\<^sub>e b = \<lbrakk>f\<rbrakk>\<^sub>e b)"
simps_10: fixes t :: "'a local.term"   and u :: "'a local.term"   and v :: "'a local.term" shows "Can \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[t, u, v\<^bold>] = (Can t \<and> Can u \<and> Can v \<and> Src t = Trg u \<and> Src u = Trg v)"