hcju/isabelleps huggingface dataset:
one_chain_line_integral_eq_line_integral_on_sudivision'_1: fixes one_chain1 :: "(int \<times> (real \<Rightarrow> real \<times> real)) set"   and subdiv :: "(int \<times> (real \<Rightarrow> real \<times> real)) set"   and F :: "real \<times> real \<Rightarrow> real \<times> real"   and basis :: "(real \<times> real) set" assumes "chain_subdiv_chain one_chain1 subdiv"   and "boundary_chain one_chain1"   and "boundary_chain subdiv"   and "\<forall>(k, \<gamma>)\<in>one_chain1. line_integral_exists F basis \<gamma>"   and "\<forall>(k, \<gamma>)\<in>subdiv. valid_path \<gamma>"   and "finite one_chain1"   and "finite basis" shows "one_chain_line_integral F basis one_chain1 = one_chain_line_integral F basis subdiv"
lang_lderivs: fixes n :: "nat"   and r :: "'b rexp"   and ws :: "'a list" assumes "local.wf n r"   and "wf_word n ws" shows "lang n (lderivs ws r) = lQuots ws (lang n r)"
rtrancl_trancl_trancl: fixes x :: "'b"   and y :: "'b"   and r :: "('b \<times> 'b) set"   and z :: "'b" assumes "(x, y) \<in> r\<^sup>*"   and "(y, z) \<in> r\<^sup>+" shows "(x, z) \<in> r\<^sup>+"
indep_card_eq_dim_span: fixes B :: "'a set" assumes "independent B" shows "finite B \<and> card B = dim (span B)"
fresh_1: fixes a :: "name"   and x2 :: "'d"   and x1 :: "('d, 'e, 'f) psi" shows "a \<sharp> x2 \<prec>' x1 = (a \<sharp> x2 \<and> a \<sharp> x1)"
2: shows "enat i < llength (ltake (enat k) (lproject A w))"
set_image_le_iff: fixes A :: "'a set"   and f :: "'a \<Rightarrow> V"   and B :: "V" assumes "small A" shows "(ZFC_in_HOL.set (f ` A) \<subseteq>\<^sub>\<circ> B) = (\<forall>x\<in>A. f x \<in>\<^sub>\<circ> B)"
assms_2: shows "v \<in> \<langle>G\<rangle>"
assms_2: fixes y :: "'a" assumes "y \<in> A" shows "cmp_f y \<le> cmp_f x"
hom_restrict: fixes h :: "'a \<Rightarrow> 'd"   and H :: "('d, 'e) monoid_scheme"   and t :: "'a \<Rightarrow> 'd" assumes "h \<in> hom G H"   and "\<And>g. g \<in> carrier G \<Longrightarrow> h g = t g" shows "t \<in> hom G H"
exhaust_sel: fixes list :: "'b list" obtains "list = \<epsilon>"   | "list = hd list # tl list"
null_def: fixes xs :: "'e list" shows "List.null xs = (xs = [])"
Mxne: shows "{y \<in> S. y \<le> x} \<noteq> {}"
resolution_class_size2: fixes S :: "'a set multiset" assumes "S \<in># \<P>" shows "int (size S) = \<v> div \<k>"
points_index_one_not_unique_block: fixes B :: "'b set multiset"   and ps :: "'b set"   and bl :: "'b set"   and bl' :: "'b set" assumes "B index ps = 1"   and "ps \<subseteq> bl"   and "bl \<in># B"   and "bl' \<in># remove1_mset bl B" shows "\<not> ps \<subseteq> bl'"
card_of_Card_order: fixes A :: "'d set" shows "Card_order |A|"
r: assumes "a divides b"   and "\<not> b divides a" shows "P"
intro: fixes hom :: "'b \<Rightarrow> 'c" assumes "monoid_add_hom hom"   and "inj_semigroup_add_hom hom" shows "inj_monoid_add_hom hom"
det_row_add': fixes i :: "'d"   and j :: "'d"   and A :: "(('e, 'd) vec, 'd) vec"   and q :: "'e" assumes "i \<noteq> j" shows "det (row_add A i j q) = det A"
handleE'_def: fixes f :: "'d \<Rightarrow> (('g + 'f) \<times> 'd) \<times> bool"   and handler :: "'g \<Rightarrow> 'd \<Rightarrow> (('e + 'f) \<times> 'd) \<times> bool" shows "f <handle2> handler \<equiv> do v \<leftarrow> f; case v of Inl e \<Rightarrow> handler e | Inr v' \<Rightarrow> return (Inr v') od"
ideal_R_R: shows "ideal R R (+) (\<cdot>) \<zero> \<one>"
commute: fixes a :: "'a multiset"   and b :: "'a multiset" shows "a \<union># b = b \<union># a"
smult_distribs_1: fixes a :: "'a"   and b :: "'a"   and p :: "'a poly" shows "smult (a + b) p = smult a p + smult b p"
listall_cons: fixes P :: "'a \<Rightarrow> bool"   and x :: "'a"   and xs :: "'a list" assumes "P x"   and "listall P xs" shows "listall P (x # xs)"
num_succtran_zero: fixes G :: "'a graph"   and v :: "'a" assumes "succ_tran G v = {}" shows "num_reachable G v = 0"
select_convs_1: fixes newOID :: "CTypeId \<Rightarrow> nat"   and newAID :: "Arraytype \<Rightarrow> nat"   and vals :: "Location \<Rightarrow> Value"   and more :: "'a" shows "newOID \<lparr>newOID = newOID, newAID = newAID, vals = vals, \<dots> = more\<rparr> = newOID"
DERIV_odd_real_root: fixes n :: "nat"   and x :: "real" assumes "odd n"   and "x \<noteq> 0" shows "(root n has_real_derivative inverse (real n * root n x ^ (n - Suc 0))) (at x)"
servTicket_authentic_Tgs: fixes B :: "agent"   and A :: "agent"   and servK :: "nat"   and Ts :: "msg"   and evs :: "event list" assumes "Crypt (shrK B) \<lbrace>Agent A, Agent B, Key servK, Ts\<rbrace> \<in> parts (knows Spy evs)"   and "B \<noteq> Tgs"   and "B \<notin> bad"   and "evs \<in> kerbV" shows "\<exists>authK. Says Tgs A \<lbrace>Crypt authK \<lbrace>Key servK, Agent B, Ts\<rbrace>, Crypt (shrK B) \<lbrace>Agent A, Agent B, Key servK, Ts\<rbrace>\<rbrace> \<in> set evs"
rb_aux_top_irred: fixes args :: "(('t \<Rightarrow>\<^sub>0 'b) list \<times> 't list \<times> (('t \<Rightarrow>\<^sub>0 'b) \<times> ('t \<Rightarrow>\<^sub>0 'b) + nat) list) \<times> nat"   and b :: "'t \<Rightarrow>\<^sub>0 'b" assumes "rword_strict = rw_rat_strict"   and "rb_aux_inv (fst args)"   and "b \<in> set (fst (fst (rb_aux args)))"   and "\<And>b0. b0 \<in> set (fst (fst args)) \<Longrightarrow> \<not> is_sig_red (\<preceq>\<^sub>t) (=) (set (fst (fst args)) - {b0}) b0" shows "\<not> is_sig_red (\<preceq>\<^sub>t) (=) (set (fst (fst (rb_aux args))) - {b}) b"
valid1: shows "valid_net' (insert_weights shared_weights (deep_model_l rs) wd)"
induct: fixes x1 :: "'a set"   and x2 :: "'a list"   and P :: "'a set \<Rightarrow> 'a list \<Rightarrow> bool" assumes "local.independent x1 x2"   and "\<And>K. P K []"   and "\<And>u K Us. \<lbrakk>u \<in> carrier R; u \<notin> Span K Us; local.independent K Us; P K Us\<rbrakk> \<Longrightarrow> P K (u # Us)" shows "P x1 x2"
r_into_trancl: fixes r :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'d"   and b :: "'d" assumes "r a b" shows "r\<^sup>+\<^sup>+ a b"
convex_alt: fixes s :: "'a set" shows "convex s = (\<forall>x\<in>s. \<forall>y\<in>s. \<forall>u. 0 \<le> u \<and> u \<le> 1 \<longrightarrow> (1 - u) *\<^sub>R x + u *\<^sub>R y \<in> s)"
simps_9: fixes \<phi> :: "'a ltln"   and \<psi> :: "'a ltln" shows "rewrite_X_enat (\<phi> W\<^sub>n \<psi>) = combine mk_weak_until (rewrite_X_enat \<phi>) (rewrite_X_enat \<psi>)"
bound_compare_defs_6: fixes lt :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and c :: "'b"   and b :: "'a option" shows "\<rhd>\<^sub>l\<^sub>b lt c b \<equiv> case b of None \<Rightarrow> True | Some b' \<Rightarrow> lt b' c"
assms: shows "(\<delta> \<circ>\<^sub>s \<tau>) x = Fun c []"
finite_measureI: fixes M :: "'b measure" assumes "emeasure M (space M) \<noteq> \<infinity>" shows "finite_measure M"
IfTrue: fixes b :: "bexp"   and s :: "char list \<Rightarrow> int \<Rightarrow> int"   and \<pi> :: "char list \<Rightarrow> com option"   and c\<^sub>1 :: "com"   and t :: "char list \<Rightarrow> int \<Rightarrow> int"   and c\<^sub>2 :: "com" assumes "bval b s"   and "\<pi>: (c\<^sub>1, s) \<Rightarrow> t" shows "\<pi>: (IF b THEN c\<^sub>1 ELSE c\<^sub>2, s) \<Rightarrow> t"
bij_betw_quotient_domain_permutation_eq: fixes A :: "'a set"   and B :: "'b set" assumes "finite A"   and "finite B"   and "card A = card B" shows "{f \<in> A \<rightarrow>\<^sub>E B. bij_betw f A B} // domain_permutation A B = {{f \<in> A \<rightarrow>\<^sub>E B. bij_betw f A B}}"
Int: shows "(\<lambda>x. if x \<in> {x \<in> S. f x \<noteq> (0::'b)} \<inter> {x \<in> T. f x \<noteq> (0::'b)} then f x else (0::'b)) absolutely_integrable_on UNIV"
all2': fixes v :: "(real, 'sz) vec \<times> (real, 'sz) vec" assumes "\<exists>sola. sol 0 = sola 0 \<and> (\<exists>t. v = mk_v I (OVar vid1) (sola 0, bb) (sola t) \<and> 0 \<le> t \<and> (sola solves_ode (\<lambda>a. ODEs I vid1)) {0..t} {x. mk_v I (OVar vid1) (sola 0, bb) x \<in> Contexts I pid1 UNIV \<and> mk_v I (OVar vid1) (sola 0, bb) x \<in> Contexts I pid3 UNIV})" shows "v \<in> Contexts I pid2 UNIV"
ide_dom: fixes f :: "'a \<times> 'a" assumes "VV.arr f" shows "VV.ide (VV.dom f)"
arrI: fixes f1 :: "'a"   and f2 :: "'a" assumes "arr f1"   and "arr f2" shows "CC.arr (f1, f2)"
NoCycles_def: fixes F :: "('d, 'e, 'f) seqauto set"   and G :: "'d \<Rightarrow> ('d, 'e, 'f) seqauto set option" shows "NoCycles F G = (\<forall>S\<in>Pow (\<Union> (States ` F)). S \<noteq> {} \<longrightarrow> (\<exists>s\<in>S. S \<inter> \<Union> (States ` the (G s)) = {}))"
prepend_def: fixes vs :: "nat list"   and f :: "nat \<Rightarrow> nat option" shows "vs \<odot> f \<equiv> \<lambda>x. if x < length vs then Some (vs ! x) else f (x - length vs)"
CFG_SSA_wf_axioms: shows "CFG_SSA_wf \<alpha>e \<alpha>n invar inEdges' Entry defs uses phis"
mv_box_halt_cond: fixes m :: "nat"   and n :: "nat"   and a :: "nat"   and b :: "nat list"   and lm :: "nat list" assumes "m \<noteq> n"   and "mv_box_inv (a, b) m n lm"   and "a = 3" shows "b = lm[n := lm ! m + lm ! n, m := 0]"
set_finsert: fixes x :: "'c"   and A :: "'c fset" assumes "x |\<in>| A" obtains B :: "'c fset" where "A = finsert x B" and "x |\<notin>| B"
assms_2: fixes m :: "nat"   and n :: "nat" shows "u (m + n) \<le> u m * u n"
FAssRedsVal: fixes e\<^sub>1 :: "char list exp"   and s\<^sub>0 :: "heap \<times> sfields \<times> sheap"   and b\<^sub>0 :: "bool"   and a :: "nat"   and s\<^sub>1 :: "heap \<times> sfields \<times> sheap"   and b\<^sub>1 :: "bool"   and P :: "(char list list \<times> char list exp) prog"   and e\<^sub>2 :: "char list exp"   and v :: "val"   and h\<^sub>2 :: "heap"   and l\<^sub>2 :: "sfields"   and sh\<^sub>2 :: "sheap"   and b\<^sub>2 :: "bool"   and C :: "char list"   and F :: "char list"   and t :: "ty"   and D :: "char list"   and fs :: "Objects.fields" assumes "P \<turnstile> \<langle>e\<^sub>1,s\<^sub>0,b\<^sub>0\<rangle> \<rightarrow>* \<langle>addr a,s\<^sub>1,b\<^sub>1\<rangle>"   and "P \<turnstile> \<langle>e\<^sub>2,s\<^sub>1,b\<^sub>1\<rangle> \<rightarrow>* \<langle>Val v,(h\<^sub>2, l\<^sub>2, sh\<^sub>2),b\<^sub>2\<rangle>"   and "P \<turnstile> C has F,NonStatic:t in D"   and "\<lfloor>(C, fs)\<rfloor> = h\<^sub>2 a" shows "P \<turnstile> \<langle>e\<^sub>1\<bullet>F{D} := e\<^sub>2,s\<^sub>0,b\<^sub>0\<rangle> \<rightarrow>* \<langle>unit,(h\<^sub>2(a \<mapsto> (C, fs((F, D) \<mapsto> v))), l\<^sub>2, sh\<^sub>2),b\<^sub>2\<rangle>"
init_q: shows "ego.q 0 = s\<^sub>e"
dickson_gradingD2': fixes d :: "'a \<Rightarrow> nat"   and n :: "nat" assumes "dickson_grading d" shows "wqo_on (adds) {x. d x \<le> n}"
assert_bound_code_def: shows "assert_bound_code = Update.assert_bound update_code"
InitBlockReds_aux: fixes e :: "char list exp"   and s :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and e' :: "char list exp"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and V :: "char list"   and T :: "ty" assumes "P \<turnstile> \<langle>e,s\<rangle> \<rightarrow>* \<langle>e',s'\<rangle>" shows "\<forall>h l h' l' v. s = (h, l(V \<mapsto> v)) \<longrightarrow> s' = (h', l') \<longrightarrow> P \<turnstile> \<langle>{V:T; V:=Val v;; e},(h, l)\<rangle> \<rightarrow>* \<langle>{V:T; V:=Val (the (l' V));; e'},(h', l'(V := l V))\<rangle>"
Rep_nonzero: fixes x :: "'b nonzero" shows "Rep_nonzero x \<in> UNIV - {0::'b}"
axioms_1: fixes S :: "'c set"   and is_open :: "'c set \<Rightarrow> bool"   and \<FF> :: "'c set \<Rightarrow> 'd set"   and \<rho> :: "'c set \<Rightarrow> 'c set \<Rightarrow> 'd \<Rightarrow> 'd"   and b :: "'d"   and add_str :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and mult_str :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and zero_str :: "'c set \<Rightarrow> 'd"   and one_str :: "'c set \<Rightarrow> 'd" assumes "sheaf_of_rings S is_open \<FF> \<rho> b add_str mult_str zero_str one_str" shows "presheaf_of_rings S is_open \<FF> \<rho> b add_str mult_str zero_str one_str"
inject_2: fixes x2 :: "(val \<times> val) list"   and y2 :: "(val \<times> val) list" shows "(VFun x2 = VFun y2) = (x2 = y2)"
induct: fixes x :: "('a, 'b, 'c) Tree\<^sub>\<alpha>"   and P :: "('a, 'b, 'c) Tree\<^sub>\<alpha> \<Rightarrow> bool" assumes "hereditarily_fs x"   and "\<And>tset\<^sub>\<alpha>. \<lbrakk>finite (supp tset\<^sub>\<alpha>); \<And>t\<^sub>\<alpha>. t\<^sub>\<alpha> \<in> set_bset tset\<^sub>\<alpha> \<Longrightarrow> hereditarily_fs t\<^sub>\<alpha>; \<And>t\<^sub>\<alpha>. t\<^sub>\<alpha> \<in> set_bset tset\<^sub>\<alpha> \<Longrightarrow> P t\<^sub>\<alpha>\<rbrakk> \<Longrightarrow> P (Conj\<^sub>\<alpha> tset\<^sub>\<alpha>)"   and "\<And>t\<^sub>\<alpha>. \<lbrakk>hereditarily_fs t\<^sub>\<alpha>; P t\<^sub>\<alpha>\<rbrakk> \<Longrightarrow> P (Not\<^sub>\<alpha> t\<^sub>\<alpha>)"   and "\<And>\<phi>. P (Pred\<^sub>\<alpha> \<phi>)"   and "\<And>t\<^sub>\<alpha> \<alpha>. \<lbrakk>hereditarily_fs t\<^sub>\<alpha>; P t\<^sub>\<alpha>\<rbrakk> \<Longrightarrow> P (Act\<^sub>\<alpha> \<alpha> t\<^sub>\<alpha>)" shows "P x"
finite_characters: shows "finite (characters (G\<lparr>carrier := H'\<rparr>))"
set_bd_2: fixes x :: "('c, 'a, 'b) var0" shows "|set2_var0 x| \<le>o bd_pre_var0"
finite_total_preorder_enumeration: fixes X :: "'d set"   and r :: "('d \<times> 'd) set" assumes "finite X"   and "total_on X r"   and "refl_on X r"   and "trans r" obtains f :: "'d \<Rightarrow> nat" where "\<forall>x\<in>X. \<forall>y\<in>X. (f x \<le> f y) = ((x, y) \<in> r)"
sset_orbits_list: fixes f :: "'b \<Rightarrow> 'b"   and xs :: "'b list" assumes "permutation f" shows "sset (orbits_list f xs) = orbit f ` set xs"
measurable_bounded_by_integrable_imp_lebesgue_integrable: fixes f :: "'a \<Rightarrow> 'b"   and S :: "'a set"   and g :: "'a \<Rightarrow> real" assumes "f \<in> borel_measurable (lebesgue_on S)"   and "integrable (lebesgue_on S) g"   and "\<And>x. x \<in> S \<Longrightarrow> norm (f x) \<le> g x"   and "S \<in> sets lebesgue" shows "integrable (lebesgue_on S) f"
flip_at_simps_1: fixes a :: "'a"   and b :: "'a" shows "(a \<leftrightarrow> b) \<bullet> a = b"
ideal_nsum_closed: fixes I :: "'a set"   and n :: "nat"   and f :: "nat \<Rightarrow> 'a" assumes "ideal R I"   and "\<forall>j\<le>n. f j \<in> I" shows "\<Sigma>\<^sub>e R f n \<in> I"
GatewayReq_L1: fixes req :: "nat \<Rightarrow> reqType list"   and stop :: "nat \<Rightarrow> stopType list"   and a :: "nat \<Rightarrow> aType list"   and lose :: "nat \<Rightarrow> bool list"   and t :: "nat"   and k :: "nat"   and d :: "nat"   and ack :: "nat \<Rightarrow> GatewayStatus list"   and dt :: "nat \<Rightarrow> ECall_Info list"   and i :: "nat \<Rightarrow> ECall_Info list"   and vc :: "nat \<Rightarrow> vcType list" assumes "msg (Suc 0) req"   and "msg (Suc 0) stop"   and "msg (Suc 0) a"   and "ts lose"   and "req (t + 3 + k) = [send]"   and "\<forall>j\<le>2 * d + (4 + k). lose (t + j) = [False]"   and "\<forall>m\<le>k. ack (t + 2 + m) = [connection_ok]"   and "GatewayReq req dt a stop lose d ack i vc" shows "ack (t + 3 + k) = [sending_data]"
that: fixes x :: "'a"   and y :: "'a" assumes "xy = x * y"   and "x \<in>\<^sub>i X"   and "y \<in>\<^sub>i Y" shows "thesis"
lad_on_def: fixes A :: "'a set"   and f :: "'a set \<Rightarrow> 'a set" shows "lad_on A f = (\<forall>B\<subseteq>A. \<forall>C\<subseteq>B. card (f C) \<le> card (f B))"
assms_1: shows "fds_deriv f = fds_deriv g"
parallelD2: fixes x :: "'c list"   and y :: "'c list" assumes "x \<parallel> y" shows "\<not> prefix y x"
fmdom'_filter: fixes P :: "'c \<Rightarrow> bool"   and m :: "('c, 'd) fmap" shows "fmdom' (fmfilter P m) = Set.filter P (fmdom' m)"
eucl_truncate_up_mono: fixes x :: "'a"   and y :: "'a"   and p :: "nat" assumes "x \<le> y" shows "eucl_truncate_up p x \<le> eucl_truncate_up p y"
ascP_f_singleton: fixes f :: "'b list \<Rightarrow> 'b list"   and x :: "'b" assumes "ascP f" shows "f [x] = f [] @ [x]"
perm_count_conv: fixes A :: "'b list"   and B :: "'b list" shows "(A <~~> B) = (\<forall>x. PermutationLemmas.count x A = PermutationLemmas.count x B)"
monoid_comm_monoidI: fixes G :: "('c, 'd) monoid_scheme" assumes "Group.monoid G"   and "\<And>x y. \<lbrakk>x \<in> carrier G; y \<in> carrier G\<rbrakk> \<Longrightarrow> x \<otimes>\<^bsub>G\<^esub> y = y \<otimes>\<^bsub>G\<^esub> x" shows "Group.comm_monoid G"
comp_in_homE: fixes g :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and f :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and a :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and c :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "\<guillemotleft>local.comp g f : a \<rightarrow> c\<guillemotright>" obtains b :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" where "\<guillemotleft>f : a \<rightarrow> b\<guillemotright>" and "\<guillemotleft>g : b \<rightarrow> c\<guillemotright>"
add_cong: fixes m :: "int"   and x :: "int"   and y :: "int" assumes "residues m" shows "x mod m \<oplus>\<^bsub>residue_ring m\<^esub> y mod m = (x + y) mod m"
simps_3: fixes a :: "poscheckvariant"   and t :: "('a, 'b) Term.term"   and s :: "('a, 'b) Term.term"   and \<theta> :: "'b \<Rightarrow> ('a, 'b) Term.term" shows "\<langle>a: t \<doteq> s\<rangle> \<cdot>\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<theta> = \<langle>a: (t \<cdot> \<theta>) \<doteq> s \<cdot> \<theta>\<rangle>"
cases: fixes x :: "tm" obtains m :: "name" and xa :: "tm" where "atom m \<sharp> xa" and "x = xa"
bitlen_normal_mantissa: fixes x :: "('a, 'b) IEEE.float" shows "bitlen \<bar>normal_mantissa x\<bar> = int (Suc LENGTH('b))"
sum_leq: fixes F :: "'a set" assumes "finite F"   and "F \<subseteq> A" shows "sum f F \<le> sum g F"
tt_monomial: fixes c :: "'c"   and v :: "'a" assumes "c \<noteq> (0::'c)" shows "punit.tt (monomial c v) = v"
porder_on_def: fixes A :: "'b set"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "porder_on A r = (refl_onP A r \<and> transp r \<and> antisymp r)"
l_set_disconnected_nodes_get_host_def: fixes set_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_host_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_set_disconnected_nodes_get_host set_disconnected_nodes_locs get_host_locs \<equiv> \<forall>ptr h h'. \<forall>w\<in>set_disconnected_nodes_locs ptr. h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> (\<forall>r\<in>get_host_locs. r h h')"
assms_2: shows "\<rho> x \<sqsubseteq> y"
g: shows "mcont lub ord lSup (\<sqsubseteq>) g"
T_vrestr_meas_1: fixes M :: "'b measure"   and T :: "'b \<Rightarrow> 'b"   and A :: "'b set" assumes "qmpt M T"   and "A \<in> sets M" shows "qmpt.vimage_restr M T A \<in> sets M"
ring_simprules_20: fixes x :: "'a" assumes "x \<in> carrier R" shows "\<ominus> (\<ominus> x) = x"
h': shows "h2 \<turnstile> forall_M (\<lambda>shadow_root_ptr. Heap_Error_Monad.bind (get_host shadow_root_ptr) (\<lambda>host. Heap_Error_Monad.bind (Heap_Error_Monad.bind (get_child_nodes (cast\<^sub>e\<^sub>l\<^sub>e\<^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 host)) (forall_M remove)) (\<lambda>_. Heap_Error_Monad.bind (Heap_Error_Monad.bind (get_child_nodes (cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^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 shadow_root_ptr)) (forall_M (append_child (cast\<^sub>e\<^sub>l\<^sub>e\<^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 host)))) (\<lambda>_. remove_shadow_root host)))) shadow_root_ptrs \<rightarrow>\<^sub>h h'"
subspace_def: fixes K :: "('g, 'h) ring_scheme"   and W :: "'i set"   and V :: "('g, 'i, 'j) module_scheme" shows "subspace K W V \<equiv> vectorspace K V \<and> LinearCombinations.submodule K W V"
proper_implies_not_part: fixes x :: "i"   and y :: "i" assumes "M"   and "PP x y" shows "\<not> P y x"
group_reduced_homology_group: fixes p :: "int"   and X :: "'b topology" shows "Group.group (reduced_homology_group p X)"
map_option_is_None: fixes f :: "'d \<Rightarrow> 'c"   and opt :: "'d option" shows "(map_option f opt = None) = (opt = None)"
\<sigma>\<^sub>T\<^sub>O\<^sub>Y_labels: fixes \<xi> :: "state"   and p :: "(state, msg, pseqp, pseqp label) seqp"   and i :: "nat" assumes "(\<xi>, p) \<in> \<sigma>\<^sub>T\<^sub>O\<^sub>Y i" shows "labels \<Gamma>\<^sub>T\<^sub>O\<^sub>Y p = {PToy-:0}"
A_def: shows "A \<equiv> sum_upto (\<lambda>n. moebius_mu n / real n)"
tuple_is_extensional: fixes f :: "'a"   and g :: "'a"   and h :: "'a"   and k :: "'a" assumes "\<not> C.commutative_square f g h k" shows "\<langle>h \<lbrakk>f, g\<rbrakk> k\<rangle> = C.null"
representsD_2: fixes p :: "int poly"   and x :: "'a" assumes "p represents x" shows "ipoly p x = (0::'a)"
finite_pderivs_lang_UNIV: fixes r :: "'b rexp" shows "finite (pderivs_lang UNIV r)"
e_notin: shows "(v, l, u) \<notin> arcs G"
Scons_not_Leaf: fixes M :: "('c, 'd) Old_Datatype.node set"   and N :: "('c, 'd) Old_Datatype.node set"   and a :: "'c" shows "Scons M N \<noteq> Leaf a"
finite_keys: fixes f :: "'a \<Rightarrow>\<^sub>0 'b" shows "finite (Poly_Mapping.keys f)"
indpt: shows "\<not> module.lin_dep class_ring (module_vec TYPE('a) (dim_row (submatrix A I J))) (set (cols (submatrix A I J)))"
return_pure: fixes x :: "'n"   and h :: "'l" shows "pure (return x) h"
simps_38: fixes f1 :: "'c \<Rightarrow> 'b"   and f2 :: "'d \<Rightarrow> 'b"   and f3 :: "('c, 'd) fm \<Rightarrow> 'b"   and f4 :: "('c, 'd) fm \<Rightarrow> ('c, 'd) fm \<Rightarrow> 'b"   and f5 :: "('c, 'd) fm \<Rightarrow> 'b"   and f6 :: "'d \<Rightarrow> ('c, 'd) fm \<Rightarrow> 'b"   and x2 :: "'d" shows "(case Nom x2 of Pro x \<Rightarrow> f1 x | Nom x \<Rightarrow> f2 x | \<^bold>\<not> x \<Rightarrow> f3 x | x \<^bold>\<or> xa \<Rightarrow> f4 x xa | \<^bold>\<diamond> x \<Rightarrow> f5 x | \<^bold>@ x xa \<Rightarrow> f6 x xa) = f2 x2"
prems_8: shows "set (p # bs) \<subseteq> set (identify_blocks A k)"
geodesic_segmentD: fixes G :: "'b set"   and x :: "'b"   and y :: "'b"   and t :: "real" assumes "geodesic_segment_between G x y" shows "\<exists>g. g t = x \<and> g (t + dist x y) = y \<and> isometry_on {t..t + dist x y} g \<and> G = g ` {t..t + dist x y}"
prime_field: fixes \<G> :: "'d cyclic_group"   and a :: "nat" assumes "ot_base \<G>"   and "a < order \<G>"   and "a \<noteq> 0" shows "coprime a (order \<G>)"
convex_hull_empty_2: fixes A :: "'a vec set" assumes "A \<subseteq> carrier_vec n" shows "(convex_hull A = {}) = (A = {})"
hconfD: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'a) list) list"   and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and a :: "nat"   and obj :: "char list \<times> (char list \<times> char list \<Rightarrow> val option)" assumes "P \<turnstile> h \<surd>"   and "h a = \<lfloor>obj\<rfloor>" shows "P,h \<turnstile> obj \<surd>"
evaluate_iff_1: fixes eval :: "v sem_env \<Rightarrow> exp0 \<Rightarrow> 'b state \<Rightarrow> 'b state \<times> (v, v) result"   and eval_list :: "v sem_env \<Rightarrow> exp0 list \<Rightarrow> 'b state \<Rightarrow> 'b state \<times> (v list, v) result"   and eval_match :: "v sem_env \<Rightarrow> v \<Rightarrow> (pat \<times> exp0) list \<Rightarrow> v \<Rightarrow> 'b state \<Rightarrow> 'b state \<times> (v, v) result"   and env :: "v sem_env"   and st :: "'b state"   and e :: "exp0"   and r :: "'b state \<times> (v, v) result" assumes "eval eval eval_list eval_match" shows "evaluate True env st e r = (r = eval env e st)"
3: shows "\<forall>i j. i < row_length A1 * row_length B1 \<and> j < length A2 * length B2 \<longrightarrow> ((A1 \<otimes> B1) \<circ> (A2 \<otimes> B2)) ! j ! i = scalar_product (vec_vec_Tensor (row A1 (i div row_length B1)) (row B1 (i mod row_length B1))) (vec_vec_Tensor (col A2 (j div length B2)) (col B2 (j mod length B2)))"