hcju/isabelleps huggingface dataset:
dual_assertion_conjunctive: fixes x :: "'a" assumes "x \<in> assertion" shows "x ^ o \<in> conjunctive.conjunctive (\<sqinter>) (\<sqinter>) (*)"
assms_1: shows "rk {A, B, C, P} = 3"
fold_insert: fixes A :: "'d set"   and x :: "'d"   and z :: "'e" assumes "finite A"   and "x \<notin> A" shows "Finite_Set.fold f z (insert x A) = f x (Finite_Set.fold f z A)"
dvdm_imp_div_mod: fixes u :: "int poly"   and g :: "int poly" assumes "u dvdm g" shows "\<exists>q r. g = q * u + smult m r"
simps_4: fixes \<sigma> :: "(char \<Rightarrow> trm option) \<times> (char \<Rightarrow> trm option) \<times> (char \<Rightarrow> fml option) \<times> (char \<Rightarrow> game option)"   and U :: "variable set"   and f :: "char"   and \<theta> :: "trm" shows "usubstappt \<sigma> U (trm.Func f \<theta>) = (case usubstappt \<sigma> U \<theta> of None \<Rightarrow> undeft | Some \<sigma>\<theta> \<Rightarrow> case SFuncs \<sigma> f of None \<Rightarrow> Aterm (trm.Func f \<sigma>\<theta>) | Some r \<Rightarrow> if FVT r \<inter> U = {} then usubstappt (dotsubstt \<sigma>\<theta>) {} r else undeft)"
rhs_step_bind_SPEC: fixes \<Phi> :: "'c \<Rightarrow> bool"   and x' :: "'c"   and m :: "'d nres"   and R :: "('d \<times> 'e) set"   and f' :: "'c \<Rightarrow> 'e nres" assumes "\<Phi> x'"   and "m \<le> \<Down> R (f' x')" shows "m \<le> \<Down> R (SPEC \<Phi> \<bind> f')"
exactly_one_factorization: fixes f :: "'a poly" assumes "f \<noteq> 0" shows "\<exists>!cfs. factorization Irr_Mon f cfs"
set_mset_Sup: fixes A :: "'k multiset set" assumes "subset_mset.bdd_above A" shows "set_mset (Sup A) = \<Union> (set_mset ` A)"
WT_callee: fixes s :: "'s2" assumes "I2 s" shows "\<I> \<turnstile>c oracle2 s \<surd>"
singleton_conv: fixes a :: "'b" shows "{x. x = a} = {a}"
bi_uniqueDl: fixes A :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and x :: "'e"   and y :: "'f"   and z :: "'e" assumes "bi_unique A"   and "A x y"   and "A z y" shows "x = z"
csqrt_1_diff_eq: fixes x :: "real" shows "csqrt (1 - (complex_of_real x)\<^sup>2) = (if x\<^sup>2 \<le> 1 then complex_of_real (sqrt (1 - x\<^sup>2)) else \<i> * complex_of_real (sqrt (x\<^sup>2 - 1)))"
independent_vnmD3: fixes \<alpha> :: "real"   and p :: "'a pmf"   and C :: "'a pmf set"   and q :: "'a pmf"   and r :: "'a pmf"   and P :: "('a pmf \<times> 'a pmf) set" assumes "\<alpha> \<in> {0<..1}"   and "p \<in> C"   and "q \<in> C"   and "r \<in> C"   and "independent_vnm C P"   and "mix_pmf \<alpha> p r \<succeq>[P] mix_pmf \<alpha> q r" shows "p \<succeq>[P] q"
ascending_chain_condition: fixes I :: "nat \<Rightarrow> 'a set" assumes "\<forall>n. ideal (I n)"   and "\<forall>n. I n \<subseteq> I (n + 1)" shows "\<exists>n. I n = I (n + 1)"
case_2: fixes f1 :: "'a"   and f2 :: "'b \<Rightarrow> 'a"   and x2 :: "'b" shows "(case \<lfloor>x2\<rfloor> of None \<Rightarrow> f1 | \<lfloor>x\<rfloor> \<Rightarrow> f2 x) = f2 x2"
wellformed_transaction_cases_5: fixes T :: "('e, 'f, 'g, 'h) prot_transaction"   and l :: "'h strand_label"   and x :: "(('e, 'f, 'g) prot_fun, (('e, 'f, 'g) prot_fun, 'f prot_atom) Term.term \<times> nat) stateful_strand_step" assumes "wellformed_transaction T"   and "(l, x) \<in> set (transaction_send T)" shows "\<exists>t. x = send\<langle>t\<rangle>"
NomP: fixes A :: "'b set"   and H :: "(('c, 'b) fm list \<times> 'b) set"   and i :: "'b"   and a :: "'b" assumes "Hintikka A H"   and "Nom a at i in' H" shows "\<not> (\<^bold>\<not> Nom a) at i in' H"
gbm_invar_split_1: fixes g :: "'m1" assumes "gbm_invar g" shows "m1.invar g"
qcr_dlist_def: fixes x :: "'b list"   and y :: "'b dlist" shows "qcr_dlist x y = (y = abs_dlist x)"
echelon_form_imp_upper_triagular: fixes A :: "(('b, 'c) vec, 'c) vec" assumes "echelon_form A" shows "upper_triangular A"
D: shows "\<D>\<^sub>\<circ> (xs #\<^sub>\<circ> x) = n"
Fvars_ball: fixes x :: "'var"   and \<phi> :: "'fmla"   and t :: "'trm" assumes "x \<in> var"   and "\<phi> \<in> fmla"   and "t \<in> trm" shows "Fvars (ball x t \<phi>) = Fvars \<phi> \<union> FvarsT t - {x}"
fls_inverse_eq_0_iff': fixes f :: "'a fls" shows "(inverse f = 0) = (inverse (fls_nth f (fls_subdegree f)) = (0::'a))"
CConcStartA: fixes x :: "int \<times> (int \<times> int \<times> protocol) \<times> (int \<times> int \<times> protocol) \<times> DummyContent"   and a :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators"   and b :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list" assumes "x \<in> dom (Cp a)" shows "x \<in> dom (Cp (list2FWpolicy (a # b)))"
2: shows "val_Zp (x \<ominus> y) = val_Zp (y \<ominus> x)"
d: fixes f :: "'a fps" assumes "f \<in> A" shows "d dvd f"
antisymI: fixes r :: "('b \<times> 'b) set" assumes "\<And>x y. \<lbrakk>(x, y) \<in> r; (y, x) \<in> r\<rbrakk> \<Longrightarrow> x = y" shows "antisym r"
map_upds_Some_insert: fixes tab :: "'b \<Rightarrow> 'a option"   and xs :: "'b list"   and ys :: "'a list"   and vn :: "'b"   and z :: "'a"   and u :: "'b"   and v :: "'a" assumes "(tab(xs [\<mapsto>] ys)) vn = Some z" shows "\<exists>z. (tab(u \<mapsto> v, xs [\<mapsto>] ys)) vn = Some z"
cmp_components_are_iso: fixes f :: "'c"   and g :: "'c" assumes "C.ide f"   and "C.ide g"   and "src\<^sub>C f = trg\<^sub>C g" shows "D.iso (\<Phi>\<^sub>H (f, g))"
game3_def: fixes \<A> :: "(bool, bool list \<times> bool list + bool list \<times> bool list \<times> 'hash, (bool list \<times> bool list \<times> 'hash) option + bool list option) gpv" shows "game3 \<A> \<equiv> key_gen \<bind> (\<lambda>key. coin_spmf \<bind> (\<lambda>b. exec_gpv (oracle_encrypt3 key b \<oplus>\<^sub>O oracle_decrypt2 key) \<A> (False, Map_empty) \<bind> (\<lambda>(b', bad, D). return_spmf (b = b', bad))))"
rel_frontier_sing: fixes a :: "'a" shows "rel_frontier {a} = {}"
simps_1: fixes mul_p :: "'b"   and u :: "'b"   and curr_p :: "'b"   and i :: "nat" shows "power_polys mul_p u curr_p (Suc i) = curr_p # power_polys mul_p u (curr_p * mul_p mod u) i"
vconn_implies_hpar_3: fixes \<mu> :: "'b"   and f :: "'b"   and g :: "'b" assumes "\<guillemotleft>\<mu> : f \<Rightarrow>\<^sub>D g\<guillemotright>" shows "src\<^sub>D g = src\<^sub>D f"
vars_unE: fixes x :: "nat"   and a :: "'a mpoly"   and b :: "'a mpoly" assumes "x \<in> vars (a * b)" obtains "x \<in> vars a"   | "x \<in> vars b"
assms_1: shows "(\<S>, \<A>) \<Rightarrow>\<^sup>\<bullet>\<^sub>c\<^sup>* (\<S>', \<A>')"
vconn_implies_hpar_4: fixes \<mu> :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and f :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and g :: "('a, 'a set) Maps.arr arrow_of_spans_data" assumes "Span.in_hom \<mu> f g" shows "Span.trg g = Span.trg f"
igWls_termMOD: shows "igWls termMOD = wls"
fun_upds_apply: fixes m :: "'d \<Rightarrow> 'c option"   and xs :: "'d list"   and ys :: "'c list"   and x :: "'d" shows "(m(xs [\<mapsto>] ys)) x = (let xs' = take (length ys) xs in if x \<in> set xs' then Some (ys ! index xs' x) else m x)"
cc'_def: shows "cc' \<equiv> \<lambda>x. if x \<in> T' then c' x else 0"
chkchop_same_def: fixes ground_heads_var :: "'a \<Rightarrow> 'b set"   and gt_sym :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and arity_sym :: "'b \<Rightarrow> enat"   and arity_var :: "'a \<Rightarrow> enat"   and extf :: "'b \<Rightarrow> (('b, 'a) tm \<Rightarrow> ('b, 'a) tm \<Rightarrow> bool) \<Rightarrow> ('b, 'a) tm list \<Rightarrow> ('b, 'a) tm list \<Rightarrow> bool"   and gt :: "('b, 'a) tm \<Rightarrow> ('b, 'a) tm \<Rightarrow> bool"   and t :: "('b, 'a) tm"   and s :: "('b, 'a) tm" assumes "epo ground_heads_var gt_sym arity_sym arity_var extf" shows "epo.chkchop_same gt t s = (if is_Var (head t) then is_Hd t \<or> epo.chkchop gt (chop t) s else epo.chkchop gt t s)"
add_increasing: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'d"   and b :: "'d"   and c :: "'d" assumes "class.ordered_comm_monoid_add plus zero less_eq less"   and "less_eq zero a"   and "less_eq b c" shows "less_eq b (plus a c)"
transfer_correctness: fixes T1 :: "('d, 'f) tree_automaton_rec"   and T2 :: "('e, 'f) tree_automaton_rec"   and P :: "('d \<times> 'e) set \<times> ('d \<times> 'e) list \<Rightarrow> bool" assumes "\<forall>s. s \<in> wa_invar (dfs_algo (ta_initial T1 \<times> ta_initial T2) (f_succ (\<delta>_prod (ta_rules T1) (ta_rules T2)))) \<and> s \<notin> wa_cond (dfs_algo (ta_initial T1 \<times> ta_initial T2) (f_succ (\<delta>_prod (ta_rules T1) (ta_rules T2)))) \<longrightarrow> P s" shows "\<forall>sc. sc \<in> wa_invar (frp_algo T1 T2) \<and> sc \<notin> wa_cond (frp_algo T1 T2) \<longrightarrow> P (frp_\<alpha> sc)"
plusU_zeroU_right: fixes fmapU :: "(udom \<rightarrow> udom) \<rightarrow> udom\<cdot>'b \<rightarrow> udom\<cdot>'b"   and plusU :: "udom\<cdot>'b \<rightarrow> udom\<cdot>'b \<rightarrow> udom\<cdot>'b"   and zeroU :: "udom\<cdot>'b"   and m :: "udom\<cdot>'b" assumes "class.functor_zero_plus fmapU plusU zeroU" shows "plusU\<cdot>m\<cdot>zeroU = m"
nat_seqs_def: fixes R :: "('a, 'b) ring_scheme" shows "R\<^bsup>\<omega>\<^esup> \<equiv> function_ring UNIV R"
ord_resolve_max_side_prems: fixes CAs :: "'a literal multiset list"   and DA :: "'a literal multiset"   and AAs :: "'a multiset list"   and As :: "'a list"   and \<sigma> :: "'s"   and E :: "'a literal multiset" assumes "ord_resolve CAs DA AAs As \<sigma> E" shows "length CAs \<le> size DA"
P9: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x l\<rightarrow> y r\<rightarrow> z = y r\<rightarrow> x l\<rightarrow> z"
measure_preserving_is_quasi_measure_preserving: fixes f :: "'b \<Rightarrow> 'c"   and M :: "'b measure"   and N :: "'c measure" assumes "f \<in> measure_preserving M N" shows "f \<in> quasi_measure_preserving M N"
iIN_card: fixes n :: "nat"   and d :: "nat" shows "card [n\<dots>,d] = Suc d"
IntI: fixes c :: "'k"   and A :: "'k set"   and B :: "'k set" assumes "c \<in> A"   and "c \<in> B" shows "c \<in> A \<inter> B"
BoxDmdDmdBox: fixes F :: "'a \<Rightarrow> bool"   and G :: "'a \<Rightarrow> bool" shows "\<turnstile> \<box>\<diamond>F \<and> \<diamond>\<box>G \<longrightarrow> \<box>\<diamond>(F \<and> G)"
assms_3: shows "ipassmt (Iface (p_iiface p)) = Some p_ips \<and> p_src p \<in> ipcidr_union_set (set p_ips)"
null_def: shows "C1.null = (THE n. \<forall>f. n \<cdot>\<^sub>1 f = n \<and> f \<cdot>\<^sub>1 n = n)"
flip: shows "bounded_bilinear (\<lambda>x y. y ** g x)"
"3_3": assumes "x \<noteq> y"   and "\<not> rel x y"   and "\<And>xa. \<lbrakk>xa \<in> set (x # xs); P xa\<rbrakk> \<Longrightarrow> False" shows "filter P (merge_wrt rel (x # xs) ys) = filter P ys"
lActive_greater_active_all: fixes n' :: "nat"   and c :: "'id"   and t :: "nat \<Rightarrow> cnf" assumes "\<forall>n''>n'. \<not> \<parallel>c\<parallel>\<^bsub>t n''\<^esub>" shows "\<not> (\<exists>n>\<langle>c \<and> t\<rangle>. \<parallel>c\<parallel>\<^bsub>t n\<^esub>)"
defect_def: fixes N :: "'a quasinorm" shows "defect N = fst (Rep_quasinorm N)"
reciprocal_sum: fixes P :: "'a \<Rightarrow> 'b poly"   and p :: "nat"   and A :: "'a set" assumes "\<And>k. degree (P k) \<le> p" shows "reciprocal_poly p (sum P A) = (\<Sum>k\<in>A. reciprocal_poly p (P k))"
simple_proto_conjunct_None: fixes p1 :: "protocol"   and p2 :: "protocol"   and pkt :: "8 word" assumes "simple_proto_conjunct p1 p2 = None" shows "\<not> (match_proto p1 pkt \<and> match_proto p2 pkt)"
real_polynomial_function_diff: fixes f :: "'c \<Rightarrow> real"   and g :: "'c \<Rightarrow> real" assumes "real_polynomial_function f"   and "real_polynomial_function g" shows "real_polynomial_function (\<lambda>x. f x - g x)"
u_def: shows "u = (\<lambda>n. if n = 0 then x else y)"
vfsequence_vrange_vproduct: fixes A :: "V \<Rightarrow> V" assumes "\<And>i. i \<in>\<^sub>\<circ> vcard [a, a]\<^sub>\<circ> \<Longrightarrow> [a, a]\<^sub>\<circ>\<lparr>i\<rparr> \<in>\<^sub>\<circ> A i" shows "[a, a]\<^sub>\<circ> \<in>\<^sub>\<circ> vproduct (vcard [a, a]\<^sub>\<circ>) A"
test6: fixes \<tau> :: "'b state \<times> 'b state"   and A :: "'b state \<times> 'b state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>"   and B :: "'b state \<times> 'b state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>" assumes "\<tau> \<Turnstile> not A" shows "\<tau> |\<noteq> A and B"
converged_cd_same_suc_return: fixes \<pi> :: "nat \<Rightarrow> 'n"   and \<pi>' :: "nat \<Rightarrow> 'n"   and l :: "nat"   and n' :: "nat" assumes "is_path \<pi>"   and "is_path \<pi>'"   and "\<pi> 0 = \<pi>' 0"   and "\<forall>k k'. cs\<^bsup>\<pi>\<^esup> k = cs\<^bsup>\<pi>'\<^esup> k' \<and> l cd\<^bsup>\<pi>\<^esup>\<rightarrow> k \<longrightarrow> \<pi> (Suc k) = \<pi>' (Suc k')"   and "\<pi>' n' = local.return" shows "\<exists>l'. cs\<^bsup>\<pi>\<^esup> l = cs\<^bsup>\<pi>'\<^esup> l'"
Let_preservation: fixes \<Gamma> :: "binding list"   and p :: "pat"   and t\<^sub>1 :: "trm"   and t\<^sub>2 :: "trm"   and T :: "POPLmarkRecord.type"   and ts :: "trm list" assumes "\<Gamma> \<turnstile> (LET p = t\<^sub>1 IN t\<^sub>2) : T"   and "\<turnstile> p \<rhd> t\<^sub>1 \<Rightarrow> ts" shows "\<Gamma> \<turnstile> t\<^sub>2[0 \<mapsto>\<^sub>s ts] : T"
weight_spmf_eq_nn_integral_spmf: fixes p :: "'a spmf" shows "ennreal (weight_spmf p) = (\<Sum>\<^sup>+ x. ennreal (spmf p x))"
digraph_map_isoI: fixes G :: "('c, 'd) pre_digraph"   and M :: "'d pre_map"   and hom :: "('c, 'd, 'e, 'f) digraph_isomorphism" assumes "digraph_map G M"   and "pre_digraph.digraph_isomorphism G hom" shows "digraph_map (app_iso hom G) (pre_digraph_map.map_iso G M hom)"
sstart_sets: fixes S :: "'b set"   and xs :: "'b list" shows "sstart S xs \<in> sets (stream_space (count_space UNIV))"
inverse_equality: fixes u :: "'a"   and v :: "'a" assumes "u \<cdot> v = \<one>"   and "v \<cdot> u = \<one>"   and "u \<in> M"   and "v \<in> M" shows "inverse u = v"
UN_subset_iff: fixes A :: "'f \<Rightarrow> 'e set"   and I :: "'f set"   and B :: "'e set" shows "(\<Union> (A ` I) \<subseteq> B) = (\<forall>i\<in>I. A i \<subseteq> B)"
sum_map_def: fixes f :: "'d \<Rightarrow> 'b"   and g :: "'e \<Rightarrow> 'c"   and x :: "'d + 'e" shows "sum_map f g x \<equiv> case x of Inl v \<Rightarrow> Inl (f v) | Inr v' \<Rightarrow> Inr (g v')"
weakenR': fixes \<Gamma> :: "'b formula multiset"   and \<Delta> :: "'b formula multiset"   and n :: "nat"   and F :: "'b formula" assumes "\<Gamma> \<Rightarrow> \<Delta> \<down> n" shows "\<Gamma> \<Rightarrow> F, \<Delta> \<down> n"
that: fixes c :: "('a, 'b) chart"   and r :: "real" assumes "c \<in> atlas"   and "x \<in> domain c"   and "domain c \<subseteq> U"   and "codomain c = ball (apply_chart c x) r"   and "0 < r" shows "thesis"
LPHIL0_def: shows "LPHIL0 = (\<mu> x. picks 0 (N - 1) \<rightarrow> picks 0 0 \<rightarrow> putsdown 0 0 \<rightarrow> putsdown 0 (N - 1) \<rightarrow> x)"
CastFail\<^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 :: "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 :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and l :: "val list"   and D :: "char list"   and fs :: "char list \<times> char list \<Rightarrow> val option"   and C :: "char list" assumes "P \<turnstile>\<^sub>1 \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>addr a,(h, l)\<rangle>"   and "h a = \<lfloor>(D, fs)\<rfloor>"   and "(D, C) \<notin> (subcls1 P)\<^sup>*" shows "P \<turnstile>\<^sub>1 \<langle>Cast C e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>THROW ClassCast,(h, l)\<rangle>"
AbstrLevels_S8_A5: fixes i :: "AbstrLevelsID" assumes "sS8 \<in> AbstrLevel i" shows "sA5 \<notin> AbstrLevel i"
wf_sees_clinit: fixes wf_md :: "'a prog \<Rightarrow> char list \<Rightarrow> 'a mdecl \<Rightarrow> bool"   and P :: "'a prog"   and C :: "char list"   and a :: "'a class" assumes "wf_prog wf_md P"   and "class P C = \<lfloor>a\<rfloor>" shows "\<exists>m. P \<turnstile> C sees clinit, Static :  []\<rightarrow>Void = m in C"
part_def: fixes v :: "'d"   and \<R> :: "'d set set" shows "[v]\<^sub>\<R> \<equiv> THE R. R \<in> \<R> \<and> v \<in> R"
fin_make_untimed_def: fixes x :: "'d list list" shows "fin_make_untimed x \<equiv> concat x"
less_ack2: fixes j :: "nat"   and i :: "nat" shows "j < ack i j"
inj_inj_on: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set" assumes "inj f" shows "inj_on f A"
ThrowReds: fixes e :: "expr"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and e' :: "expr"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option" assumes "P,E \<turnstile> \<langle>e,s\<rangle> \<rightarrow>* \<langle>e',s'\<rangle>" shows "P,E \<turnstile> \<langle>throw e,s\<rangle> \<rightarrow>* \<langle>throw e',s'\<rangle>"
set_append: fixes xs :: "'a list"   and ys :: "'a list" shows "set (xs @ ys) = set xs \<union> set ys"
subtopology_eq_discrete_topology_finite: fixes X :: "'a topology"   and S :: "'a set" assumes "t1_space X"   and "S \<subseteq> topspace X"   and "finite S" shows "subtopology X S = discrete_topology S"
simple_proto_conjunct_Some: fixes p1 :: "protocol"   and p2 :: "protocol"   and proto :: "protocol"   and pkt :: "8 word" assumes "simple_proto_conjunct p1 p2 = Some proto" shows "match_proto proto pkt = (match_proto p1 pkt \<and> match_proto p2 pkt)"
ground_reflexion_is_sound: fixes C :: "'a eclause"   and D :: "'a eclause"   and \<sigma> :: "('a \<times> 'a trm) list"   and k :: "inferences"   and C' :: "'a literal set"   and \<theta> :: "('a \<times> 'a trm) list" assumes "finite (cl_ecl C)"   and "reflexion C D \<sigma> k C'"   and "ground_clause (subst_cl (cl_ecl D) \<theta>)" shows "clause_entails_clause (subst_cl (subst_cl (cl_ecl C) \<sigma>) \<theta>) (subst_cl (cl_ecl D) \<theta>)"
MkArr_expansion: fixes f :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr" assumes "arr f" shows "f = MkArr (Dom f) (Cod f) (restrict (Map f) (Dom f))"
mat_delete_def: fixes A :: "'a mat"   and i :: "nat"   and j :: "nat" shows "mat_delete A i j \<equiv> Matrix.mat (dim_row A - 1) (dim_col A - 1) (\<lambda>(i', j'). A $$ (if i' < i then i' else Suc i', if j' < j then j' else Suc j'))"
collapse_1: fixes tllist :: "('f, 'g) tllist" assumes "is_TNil tllist" shows "TNil (terminal tllist) = tllist"
t_id_t_proper_1: fixes v :: "i"   and \<tau> :: "\<kappa>"   and \<tau>' :: "\<kappa>" shows "[\<tau> \<^bold>= \<tau>' \<^bold>\<rightarrow> (\<^bold>\<exists>\<beta>. \<beta>\<^sup>P \<^bold>= \<tau>) in v]"
shift2_inv_preserved: fixes d :: "nat"   and qs :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list" assumes "shift2_inv d qs" shows "shift2_inv (Suc d) (fst (shift1 (qs, {q \<in> set qs. poly_deg (fst q) = d \<and> m < card (snd q)})))"
adjoint_equivalence_in_bicategory_def: 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"   and g :: "'b"   and \<eta> :: "'b"   and \<epsilon> :: "'b" shows "adjoint_equivalence_in_bicategory V H \<a> \<i> src trg f g \<eta> \<epsilon> \<equiv> equivalence_in_bicategory V H \<a> \<i> src trg f g \<eta> \<epsilon> \<and> adjunction_in_bicategory V H \<a> \<i> src trg f g \<eta> \<epsilon>"
diff_epsilon: fixes f :: "'a \<Rightarrow> real" assumes "\<And>x y. \<lbrakk>le x y; le y x\<rbrakk> \<Longrightarrow> f x = f y" shows "\<exists>\<epsilon>>0. is_vnm_utility (\<lambda>x. u x - \<epsilon> * f x)"
Max_by_dest: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "finite A"   and "A \<noteq> {}" shows "Max_by f A \<in> A \<and> f (Max_by f A) = Max (f ` A)"
poly_mult_left_cancel: fixes p :: "'a list"   and q :: "'a list"   and r :: "'a list" shows "(poly (p *** q) = poly (p *** r)) = (poly p = poly [] \<or> poly q = poly r)"
lgc_invar_lgc_init: fixes TNs :: "('f inference set \<times> ('f \<times> 'l) set) llist" assumes "\<not> lnull TNs"   and "active_subset (snd (lhd TNs)) = {}"   and "\<forall>\<iota>\<in>Inf_F. prems_of \<iota> = [] \<longrightarrow> \<iota> \<in> fst (lhd TNs)" shows "lgc_invar TNs 0"
cones_map_ide: fixes \<chi> :: "'j \<Rightarrow> 'b"   and a :: "'b" assumes "\<chi> \<in> \<chi>a.D.cones a" shows "\<chi>a.D.cones_map a \<chi> = \<chi>"
rep_eq: fixes f :: "nat \<Rightarrow> complex"   and n :: "nat" shows "poly.coeff (power_sum_puzzle_existence.Q' f n) = (\<lambda>k. if n < k then 0 else (- 1) ^ (n - k) * power_sum_puzzle_existence.e' f n (n - k))"
eventually_nhds: fixes "open" :: "'a set \<Rightarrow> bool"   and P :: "'a \<Rightarrow> bool"   and a :: "'a" assumes "class.topological_space open" shows "eventually P (topological_space.nhds open a) = (\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x))"
mtf2_q_after: fixes A :: "'a list"   and q :: "'a"   and n :: "nat" assumes "distinct A"   and "q \<in> set A" shows "index (mtf2 n q A) q = index A q - n"
generate_singleton_one: fixes a :: "'a" assumes "generate G {a} = {\<one>}" shows "a = \<one>"
ipd_2: shows "\<forall>n\<in>{k'..<m'}. \<pi>' n \<noteq> ipd (\<pi> k)"
intros_1: fixes \<iota> :: "nat"   and m :: "'id \<Rightarrow> 'val" shows "SRabbr (skip\<^bsub>\<iota>\<^esub>) m [] None m"
substC_def: fixes \<pi> :: "Preliminaries.var \<Rightarrow> 'a trm"   and c :: "('a, 'b) lit list" shows "substC \<pi> c = map (substL \<pi>) c"
x1_def: shows "x1 = (1 - u) *\<^sub>R x0 + u *\<^sub>R x3"
cp_2: shows "pathfinish cp = p + complex_of_real (h p)"
pred_set: fixes P :: "'b \<Rightarrow> bool" shows "list_all P = (\<lambda>x. Ball (set x) P)"
add_local_ring_at_def: shows "add_local_ring_at \<equiv> cq.add_rel"
matrix_to_iarray_mat: fixes k :: "'b" shows "matrix_to_iarray (mat k) = mat_iarray k CARD('c)"
ofilter_def: fixes A :: "'a set" shows "r.ofilter A \<equiv> A \<subseteq> Field r \<and> (\<forall>a\<in>A. r.under a \<subseteq> A)"
compact_valid_path_image: fixes g :: "real \<Rightarrow> 'a" assumes "valid_path g" shows "compact (path_image g)"
Icc_subset_Ici_iff: fixes l :: "'a multiset"   and h :: "'a multiset"   and l' :: "'a multiset" shows "(subset_mset.atLeastAtMost l h \<subseteq> subset_mset.atLeast l') = (\<not> l \<subseteq># h \<or> l' \<subseteq># l)"
Confidentiality_lemma: fixes A :: "agent"   and authK :: "nat"   and servK :: "nat"   and B :: "agent"   and Ts :: "nat"   and evs :: "event list" assumes "Says Tgs A (Crypt authK \<lbrace>Key servK, Agent B, Number Ts, Crypt (shrK B) \<lbrace>Agent A, Agent B, Key servK, Number Ts\<rbrace>\<rbrace>) \<in> set evs"   and "Key authK \<notin> analz (knows Spy evs)"   and "servK \<in> symKeys"   and "A \<notin> bad"   and "B \<notin> bad"   and "evs \<in> kerbIV"   and "Key servK \<in> analz (knows Spy evs)" shows "expiredSK Ts evs"
lift_SD_iff': fixes p :: "'b pmf"   and q :: "'b pmf"   and i :: "'a" assumes "p \<in> lotteries_on alts"   and "q \<in> lotteries_on alts"   and "i \<in> agents'" shows "SD (R' i) p q = (i \<notin> agents \<or> SD (R i) p q)"
single_addition_pos: fixes f :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and z :: "'e"   and i :: "'e \<Rightarrow> 'e"   and a :: "'e"   and n :: "int"   and m :: "int" assumes "commute_bpp f (aug_pm_set z i {a})"   and "assoc_bpp (aug_pm_set z i {a}) f"   and "ipp_cond1 {a} i"   and "ipp_cond2 z {a} i f"   and "ipp_cond3 z i"   and "inv_ipp z i f {a}"   and "commute_bpp f (aug_pm_set z i {a})"   and "zeroA z i f {a} z"   and "0 < n"   and "0 < m" shows "n\<Odot>a\<^bsub>f,i,z\<^esub> \<^sub>f+ m\<Odot>a\<^bsub>f,i,z\<^esub> = (n + m)\<Odot>a\<^bsub>f,i,z\<^esub>"