hcju/isabelleps huggingface dataset:
infinite_convex_domain_iff: fixes f :: "'b \<Rightarrow> ereal" assumes "\<forall>x. f x = \<infinity> \<or> f x = - \<infinity>" shows "Lower_Semicontinuous.convex_on UNIV f = convex (domain f)"
comm_ring_def: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c" shows "class.comm_ring plus zero minus uminus times \<equiv> class.ab_group_add plus zero minus uminus \<and> class.comm_semiring plus times"
permutes_inv: fixes p :: "'a \<Rightarrow> 'a"   and S :: "'a set" assumes "p permutes S" shows "inv' p permutes S"
good_context_2: fixes s :: "'b sparc_state"   and v1 :: "word32" assumes "good_context s \<and> fetch_instruction (delayed_pool_write s) = Inr v1 \<and> (\<nexists>v2. decode_instruction v1 = Inr v2)" shows "False"
nat_pow_Suc2: fixes G :: "('b, 'c) monoid_scheme"   and x :: "'b"   and n :: "nat" assumes "Group.monoid G"   and "x \<in> carrier G" shows "x [^]\<^bsub>G\<^esub> Suc n = x \<otimes>\<^bsub>G\<^esub> x [^]\<^bsub>G\<^esub> n"
dTransHd_le_1: fixes H :: "'b"   and d :: "'b" assumes "stone_algebra_class.comp_inf.equivalence H"   and "bounded_distrib_lattice_class.comp_inf.coreflexive (stone_algebra_class.comp_inf.rel_access (H \<sqinter> d) (H \<sqinter> d))" shows "bounded_distrib_lattice_class.comp_inf.coreflexive (stone_algebra_class.comp_inf.rel_access d H \<sqinter> d)"
certificate_valid_pubEK: fixes C :: "agent"   and EKi :: "nat"   and evs :: "event list" assumes "cert C EKi onlyEnc (priSK RCA) \<in> parts (knows Spy evs)"   and "evs \<in> set_pur" shows "EKi = pubEK C"
nullhomotopic_into_ANR_extension: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and T :: "'b set" assumes "closed S"   and "continuous_on S f"   and "ANR T"   and "f ` S \<subseteq> T"   and "S \<noteq> {}" shows "(\<exists>c. homotopic_with_canon (\<lambda>x. True) S T f (\<lambda>x. c)) = (\<exists>g. continuous_on UNIV g \<and> range g \<subseteq> T \<and> (\<forall>x\<in>S. g x = f x))"
card_eq: shows "card (H <#> J) = card H * card J"
bisimWeakBisimulation: fixes P :: "ccs"   and Q :: "ccs" assumes "P \<sim> Q" shows "P \<approx> Q"
rho_bound2_def: fixes C :: "nat \<Rightarrow> real \<Rightarrow> real" shows "rho_bound2 C = (\<forall>p s t. correct p t \<and> s \<le> t \<longrightarrow> (t - s) * (1 - \<rho>) \<le> C p t - C p s)"
mat_adjoint_def: fixes A :: "'b mat" shows "mat_adjoint A \<equiv> mat_of_rows (dim_row A) (map conjugate (cols A))"
strictly_maximal_wrt_def: fixes A :: "'a"   and CA :: "'a literal multiset" shows "strictly_maximal_wrt A CA = (\<forall>B\<in>atms_of CA. B < A)"
induct: fixes P :: "'b formula \<Rightarrow> bool"   and a0 :: "'b formula" assumes "\<And>k. P (Atom k)"   and "P \<bottom>"   and "\<And>F G. \<lbrakk>P (\<^bold>\<not> F); P (\<^bold>\<not> G)\<rbrakk> \<Longrightarrow> P (\<^bold>\<not> (F \<^bold>\<and> G))"   and "\<And>F G. \<lbrakk>P (\<^bold>\<not> F); P (\<^bold>\<not> G)\<rbrakk> \<Longrightarrow> P (\<^bold>\<not> (F \<^bold>\<or> G))"   and "\<And>F. P F \<Longrightarrow> P (\<^bold>\<not> (\<^bold>\<not> F))"   and "\<And>F G. \<lbrakk>P F; P (\<^bold>\<not> G)\<rbrakk> \<Longrightarrow> P (\<^bold>\<not> (F \<^bold>\<rightarrow> G))"   and "\<And>v. P (\<^bold>\<not> (Atom v))"   and "P (\<^bold>\<not> \<bottom>)"   and "\<And>F G. \<lbrakk>P F; P G\<rbrakk> \<Longrightarrow> P (F \<^bold>\<and> G)"   and "\<And>F G. \<lbrakk>P F; P G\<rbrakk> \<Longrightarrow> P (F \<^bold>\<or> G)"   and "\<And>F G. \<lbrakk>P (\<^bold>\<not> F); P G\<rbrakk> \<Longrightarrow> P (F \<^bold>\<rightarrow> G)" shows "P a0"
backward_simulation_def: fixes step1 :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and step2 :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and final1 :: "'b \<Rightarrow> bool"   and final2 :: "'c \<Rightarrow> bool"   and order :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and match :: "'d \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool" shows "backward_simulation step1 step2 final1 final2 order match \<equiv> (semantics step1 final1 \<and> semantics step2 final2) \<and> well_founded order \<and> backward_simulation_axioms step1 step2 final1 final2 order match"
step: fixes xs :: "'a llist" assumes "lfinite xs"   and "\<And>zs. llength zs < llength xs \<Longrightarrow> P zs" shows "P xs"
assms_2: shows "x \<sharp> \<alpha> \<prec> P"
assms_1: shows "fv\<^sub>s\<^sub>s\<^sub>t A \<inter> bvars\<^sub>s\<^sub>s\<^sub>t A = {}"
gen_closed: shows "b \<in> carrier (H\<lparr>carrier := generate H {b}\<rparr>)"
index_first: fixes i :: "nat"   and xs :: "'c list"   and x :: "'c" assumes "i < index xs x" shows "x \<noteq> xs ! i"
basis_in_supI: fixes \<E> :: "'a set"   and B :: "'a set"   and \<E>' :: "'a set" assumes "\<E> \<subseteq> insert x X"   and "B \<subseteq> \<E>'"   and "\<E>' \<subseteq> \<E>"   and "\<E> \<subseteq> insert x X"   and "X_insert.basis_in \<E> B" shows "X_insert.basis_in \<E>' B"
bi_unique_union_r: fixes T :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and a :: "'e set"   and a' :: "'f set"   and b :: "'e set"   and b' :: "'f set"   and xr :: "'f set" assumes "bi_unique T"   and "rel_set T a a'"   and "rel_set T b b'"   and "rel_set T (a \<union> b) xr" shows "a' \<union> b' = xr"
fp_cop_F_range_inv': fixes x :: "'a"   and ds :: "'b set" assumes "x \<in> fp_cop_F ds" shows "x \<in> Field (Pd (Xd x)) \<and> Xd x \<in> ds"
insertl_0_eq: fixes x :: "'a"   and xs :: "'a list" shows "insertl 0 x xs = x # xs"
hyps_4: shows "pi \<bullet> (\<parallel><B>\<parallel>) = \<parallel><B>\<parallel>"
N_vector_top_pp: fixes x :: "'a" shows "N (x * top) * top = - - (x * top)"
ss'_def: shows "ss' = filter_min (adds\<^sub>t) (map (\<lambda>b. term_of_pair (punit.lt (rep_list b), i)) bs)"
at_within_empty: fixes "open" :: "'b set \<Rightarrow> bool"   and a :: "'b" assumes "class.topological_space open" shows "topological_space.at_within open a {} = bot"
prod_rel_domain: fixes A :: "('d \<times> 'f) set"   and B :: "('e \<times> 'g) set" shows "Domain (A \<times>\<^sub>r B) = Domain A \<times> Domain B"
sel_2: fixes xs :: "'b llist" assumes "\<not> lnull xs" shows "ltl (lalternate xs) = (case xs of LCons x xs \<Rightarrow> lalternate (ltl xs))"
llist_all2_rsp: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and S :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and T :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'c llist"   and y :: "'d llist"   and a :: "'c llist"   and b :: "'d llist" assumes "\<forall>x y. R x y \<longrightarrow> (\<forall>a b. R a b \<longrightarrow> S x a = T y b)"   and "llist_all2 R x y"   and "llist_all2 R a b" shows "llist_all2 S x a = llist_all2 T y b"
rel_fun_def: fixes A :: "'d \<Rightarrow> 'f \<Rightarrow> bool"   and B :: "'e \<Rightarrow> 'g \<Rightarrow> bool" shows "rel_fun A B = (\<lambda>f g. \<forall>x y. A x y \<longrightarrow> B (f x) (g y))"
C: shows "0 < offs_num (length ns) xs index key mi ma i"
awalk_appendI: fixes u :: "'a"   and p :: "('a \<times> 'a) list"   and v :: "'a"   and q :: "('a \<times> 'a) list"   and w :: "'a" assumes "G.awalk u p v"   and "G.awalk v q w" shows "G.awalk u (p @ q) w"
"1_2": shows "P (e l1) s = (\<not> P (e l2) s)"
integrableI_nn_integral_finite: fixes f :: "'d \<Rightarrow> real"   and M :: "'d measure"   and x :: "real" assumes "f \<in> borel_measurable M"   and "AE x in M. 0 \<le> f x"   and "\<integral>\<^sup>+ x. ennreal (f x) \<partial>M = ennreal x" shows "integrable M f"
affected_inverse: fixes f :: "'a perm" shows "affected (inverse f) = affected f"
configuration_token: fixes x :: "nat"   and n :: "nat" assumes "x \<le> n" shows "x \<in> \<UU>.configuration (\<UU>.token_run x n) n"
min_wpl_is_optimal: fixes t :: "int tree"   and i :: "int"   and j :: "int" assumes "inorder t = [i..j]" shows "local.min_wpl i j \<le> local.wpl i j t"
distinct_183: fixes x101 :: "expr"   and x102 :: "char list option"   and x103 :: "char list"   and x104 :: "expr list"   and x121 :: "expr"   and x122 :: "expr" shows "Call x101 x102 x103 x104 \<noteq> x121;; x122"
ccttas: shows "thread_oks (redT_updT' TS TA) TAS"
all_final_except_mono': fixes ts :: "('e, 'f) locks \<times> (('f \<Rightarrow> ('x \<times> 'e \<Rightarrow>f nat) option) \<times> 'g) \<times> ('f \<Rightarrow> 'h wait_set_status option) \<times> 'f set"   and A :: "'f set"   and B :: "'f set" assumes "all_final_except ts A"   and "\<And>x. x \<in> A \<Longrightarrow> x \<in> B" shows "all_final_except ts B"
nnf_cnf_form: fixes F :: "'a formula" assumes "is_nnf F" shows "is_nnf (cnf_form_of F)"
exhaust: fixes y :: "'d stream" obtains x1 :: "'d" and x2 :: "'d stream" where "y = x1 ## x2"
assms_4: shows "initReachable c \<and> <\<bottom>, outM v> \<in># msgs c"
fresh_floatariths_fresh_floatarith: fixes fas :: "floatarith list"   and i :: "nat"   and fa :: "floatarith" assumes "fresh_floatariths fas i"   and "fa \<in> set fas" shows "fresh_floatarith fa i"
at_prm_rev_eq1: fixes pi1 :: "('a \<times> 'a) list"   and pi2 :: "('a \<times> 'a) list" assumes "at TYPE('a)"   and " pi1 \<triangleq> pi2 " shows " rev pi1 \<triangleq> rev pi2 "
concE: fixes w :: "'b list"   and A :: "'b list set"   and B :: "'b list set" assumes "w \<in> A @@ B" obtains u :: "'b list" and v :: "'b list" where "u \<in> A" and "v \<in> B" and "w = u @ v"
weakly_preserves_hcomp: fixes f :: "'a"   and g :: "'a" assumes "ide f"   and "ide g"   and "src f = trg g" shows "S.isomorphic (S.UP f \<star>\<^sub>S S.UP g) (S.UP (f \<star> g))"
degree_poly_lift: fixes p :: "'c poly" shows "degree (poly_lift p) = degree p"
snapshot_produces_marker: fixes t :: "('a, 'b, 'c) event list"   and i :: "nat"   and p :: "'a"   and cid :: "nat"   and q :: "'a" assumes "trace init t final"   and "\<not> ps (S t i) p \<noteq> None"   and "ps (S t (Suc i)) p \<noteq> None"   and "channel cid = Some (p, q)" shows "Marker \<in> set (msgs (S t (Suc i)) cid) \<or> ps (S t i) q \<noteq> None"
simps: fixes a1 :: "Value"   and a2 :: "CValue" shows "similar'_base a1 a2 = (a1 = \<bottom> \<and> a2 = \<bottom>)"
prems: shows "V \<notin> fv (RI (C,e) ; D # Cs \<leftarrow> e\<^sub>1)"
coeff_0_prod_list: fixes xs :: "'a poly list" shows "coeff (prod_list xs) 0 = (\<Prod>p\<leftarrow>xs. coeff p 0)"
weaken_pre_empty: fixes r :: "('d, 'e, 'f) ann" shows "weaken_pre r {} = r"
is_tiny_dghm_op: shows "op_dghm \<FF> : op_dg \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>t\<^sub>i\<^sub>n\<^sub>y\<^bsub>\<alpha>\<^esub> op_dg \<BB>"
not_Some_eq: fixes x :: "'a option" shows "(\<forall>y. x \<noteq> \<lfloor>y\<rfloor>) = (x = \<bottom>)"
strong_fusion_eq: fixes F :: "i \<Rightarrow> bool" assumes "\<exists>x. F x" shows "(\<sigma>x. F x) = (THE x. (\<forall>y. F y \<longrightarrow> P y x) \<and> (\<forall>y. P y x \<longrightarrow> (\<exists>z. F z \<and> O y z)))"
le_inf_iff: fixes inf :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "'e"   and y :: "'e"   and z :: "'e" assumes "class.semilattice_inf inf less_eq less" shows "less_eq x (inf y z) = (less_eq x y \<and> less_eq x z)"
polychainI: fixes xs :: "('a \<times> 'a) list" assumes "\<And>i. Suc i < length xs \<Longrightarrow> snd (xs ! i) = fst (xs ! Suc i)" shows "polychain xs"
D: shows "con_comp_map D E p q x = Some (the (con_comp_map D E p q x))"
firstPosAppend: fixes a :: "'b"   and l :: "'b list"   and l' :: "'b list" assumes "a \<in> set l" shows "firstPos a l = firstPos a (l @ l')"
map_pair_cong: fixes xs :: "('f \<times> 'g) list"   and f :: "'f \<times> 'g \<Rightarrow> 'f \<times> 'h"   and g :: "'f \<times> 'g \<Rightarrow> 'f \<times> 'h" assumes "\<And>kv. kv \<in> set xs \<Longrightarrow> f kv = g kv" shows "map_pair f xs = map_pair g xs"
balL_cl: fixes l :: "('a, 'b) rbt"   and r :: "('a, 'b) rbt"   and k :: "'a"   and v :: "'b" assumes "cl_inv' l"   and "cl_inv r"   and "cl r = B" shows "cl_inv (balL l k v r)"
map_update_def: fixes x :: "'e"   and y :: "'f option"   and f :: "'e \<Rightarrow> 'f option" shows "map_update x y f = f(x := y)"
I_1: shows "[u]\<^sub>\<R> = Regions.region \<X> I r"
distinct_delete_aux: fixes ps :: "('b \<times> 'c) list"   and k :: "'b" assumes "distinct (map fst ps)" shows "distinct (map fst (AList.delete_aux k ps))"
fsymsPB_def: fixes \<Phi> :: "('c, 'd) lit list set" shows "fsymsPB \<Phi> = \<Union> {fsymsC c |c. c \<in> \<Phi>}"
relhoare_conj_right_cartesian: fixes pre :: "('a \<times> 'b) set"   and Ra :: "('a \<times> 'a) set"   and I :: "'a set"   and Rc :: "('b \<times> 'b) set"   and J :: "'b set"   and post :: "('a \<times> 'b) set" assumes "{Domain pre} Ra {> I}"   and "{Range pre} Rc {> J}"   and "{pre} Ra, Rc {> post}" shows "{pre} Ra, Rc {> post \<inter> I \<times> J}"
commute: fixes U :: "'a set"   and f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "abel_semigroup_ow U f"   and "a \<in> U"   and "b \<in> U" shows "f a b = f b a"
any_moves_remain_winning_Odd: fixes p :: "'a list"   and \<sigma> :: "'a list \<Rightarrow> 'a"   and m :: "'a" assumes "even (length p)"   and "strategy_winning_by_Odd \<sigma> p" shows "strategy_winning_by_Odd \<sigma> (p @ [m])"
rphil_phil: fixes N :: "nat"   and e :: "dining_event"   and i :: "nat"   and s :: "nat" assumes "DiningPhilosophers N"   and "e \<in> Tr\<^sub>r\<^sub>p i s" shows "phil e = i"
aux: shows "submatrix A I J $$ (a, b) = A $$ (pick I a, pick J b)"
ring_distrib2: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<in> carrier R"   and "y \<in> carrier R"   and "z \<in> carrier R" shows "(y \<plusminus> z) \<cdot>\<^sub>r x = y \<cdot>\<^sub>r x \<plusminus> z \<cdot>\<^sub>r x"
simps: fixes a1 :: "char list"   and a2 :: "char list"   and P :: "'a prog" shows "P \<turnstile> a1 \<prec>\<^sup>1 a2 = (\<exists>C D rest. a1 = C \<and> a2 = D \<and> class P C = \<lfloor>(D, rest)\<rfloor> \<and> C \<noteq> Object)"
of_real_poly_map_pCons: fixes a :: "real"   and p :: "real poly" shows "map_poly of_real (pCons a p) = pCons (of_real a) (map_poly of_real p)"
linorder_cases': fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "class.linorder less_eq less" obtains "less_eq x y"   | "less y x"
reds_preserves_sconf: fixes E :: "char list \<Rightarrow> ty option"   and es :: "expr list"   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 es' :: "expr list"   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 Ts :: "ty list" assumes "P,E \<turnstile> \<langle>es,s\<rangle> [\<rightarrow>] \<langle>es',s'\<rangle>"   and "P,E,hp s \<turnstile> es [:] Ts"   and "P,E \<turnstile> s \<surd>"   and "wwf_prog P" shows "P,E \<turnstile> s' \<surd>"
limit_elem: fixes x :: "'b"   and f :: "'b set \<Rightarrow> 'b set"   and n :: "nat"   and X :: "'b set" assumes "x \<in> funpower f n X" shows "x \<in> limit f X"
Forall_def: fixes x :: "'c"   and P :: "('a, 'b, 'c) formula" shows "Forall x P = ! (Exists x (! P))"
cut_of_rat: fixes q :: "rat" assumes "0 < q" shows "Dedekind_Real.cut {r. 0 < r \<and> r < q}"
advantage_def: fixes \<A> :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool spmf" shows "local.ddh.advantage \<A> = \<bar>spmf (local.ddh.ddh_0 \<A>) True - spmf (local.ddh.ddh_1 \<A>) True\<bar>"
F_def: shows "[\<^bold>\<box>(\<^bold>\<forall>x. \<lparr>F,x\<^sup>P\<rparr> \<^bold>\<equiv> (\<^bold>\<exists>!y. \<^bold>\<A>\<phi> y (x\<^sup>P))) in v]"
hyps: fixes v :: "'v" shows "is_trail v xs v' = (is_path v xs v' \<and> distinct xs \<and> set xs \<inter> set (rev_path xs) = {})"
to_nat_eq: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and one :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and Rep :: "'a \<Rightarrow> int"   and Abs :: "int \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a" assumes "class.mod_type times minus plus zero uminus one less_eq less Rep Abs" shows "(mod_type.to_nat Rep x = mod_type.to_nat Rep y) = (x = y)"
word_and_mask_shiftl: fixes x :: "'a word"   and n :: "nat"   and m :: "nat" shows "x && (mask n << m) = (x >> m) && mask n << m"
l8_9: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Per Bet Cong A B C"   and "Tarski_neutral_dimensionless.Col Bet A B C" shows "A = B \<or> C = B"
desargues_config_2D_rkA'B'PQ_rkA'C'PQ_rkB'C'PQ_1: fixes A :: "Points"   and B :: "Points"   and C :: "Points"   and A' :: "Points"   and B' :: "Points"   and C' :: "Points"   and P :: "Points"   and \<alpha> :: "Points"   and \<beta> :: "Points"   and \<gamma> :: "Points"   and Q :: "Points" assumes "desargues_config_2D A B C A' B' C' P \<alpha> \<beta> \<gamma>"   and "rk {A', P} = 2"   and "rk {B', P} = 2"   and "rk {C', P} = 2"   and "4 \<le> rk {A, B, C, A', B', C', P, Q}" shows "rk {A', B', P, Q} = 4"
pivot_tableau_normalized: fixes s :: "('i, 'a) state"   and x\<^sub>i :: "nat"   and x\<^sub>j :: "nat" assumes "\<triangle> (\<T> s)"   and "x\<^sub>i \<in> lvars (\<T> s)"   and "x\<^sub>j \<in> rvars_eq (eq_for_lvar (\<T> s) x\<^sub>i)" shows "\<triangle> (\<T> (pivot x\<^sub>i x\<^sub>j s))"
jordan_block_pow_dim_1: fixes n :: "nat"   and a :: "'b"   and r :: "nat" shows "dim_row (jordan_block n a ^\<^sub>m r) = n"
aHom_inv_inv: fixes F :: "('e, 'f) aGroup_scheme"   and G :: "('g, 'h) aGroup_scheme"   and f :: "'e \<Rightarrow> 'g"   and a :: "'e" assumes "aGroup F"   and "aGroup G"   and "f \<in> aHom F G"   and "a \<in> carrier F" shows "f (-\<^sub>a\<^bsub>F\<^esub> a) = -\<^sub>a\<^bsub>G\<^esub> f a"
eq_matI: fixes B :: "'a mat"   and A :: "'a mat" assumes "\<And>i j. \<lbrakk>i < dim_row B; j < dim_col B\<rbrakk> \<Longrightarrow> A $$ (i, j) = B $$ (i, j)"   and "dim_row A = dim_row B"   and "dim_col A = dim_col B" shows "A = B"
nat_less_real_le: fixes n :: "nat"   and m :: "nat" shows "(n < m) = (real n + 1 \<le> real m)"
FunctorCompMapsTo3: fixes F :: "('a, 'g, 'b, 'h, 'c, 'i) Functor"   and G :: "('g, 'd, 'h, 'e, 'i, 'f) Functor"   and f :: "'b"   and X :: "'a"   and Y :: "'a" assumes "f maps\<^bsub>CatDom (F ;;: G)\<^esub> X to Y"   and "F \<approx>>;;; G"   and "PreFunctor (F ;;: G)" shows "F ;;: G ## f maps\<^bsub>CatCod (F ;;: G)\<^esub> F ;;: G @@ X to F ;;: G @@ Y"
reach_loc: shows "s\<turnstile> l reachable_from (s@@objLoc cf a)"
in_KeysE: fixes s :: "'e"   and F :: "('e \<Rightarrow>\<^sub>0 'f) set" assumes "s \<in> Keys F" obtains f :: "'e \<Rightarrow>\<^sub>0 'f" where "s \<in> keys f" and "f \<in> F"
distinct_rem_mid: fixes xs :: "'c list"   and x :: "'c"   and ys :: "'c list" assumes "distinct (xs @ [x] @ ys)" shows "distinct (xs @ ys)"
extension_onI: fixes A :: "'b set"   and s :: "('b \<times> 'b) set"   and r :: "('b \<times> 'b) set" assumes "\<And>x y. \<lbrakk>x \<in> A; y \<in> A; (x, y) \<in> s\<rbrakk> \<Longrightarrow> (x, y) \<in> r" shows "extension_on A r s"
permute_set_eq_image: fixes p :: "perm"   and X :: "'a set" shows "p \<bullet> X = permute p ` X"
one_assn_def: shows "emp \<equiv> Abs_assn (Assn (\<lambda>h. addrOf h = {}))"
dec_I1_def: fixes \<AA> :: "interp"   and fs :: "WS1S_Nameful.fo list"   and x :: "WS1S_Nameful.fo" shows "dec_I1 \<AA> fs x = fMin (index fs x\<^bsup>\<AA>\<^esup>FO)"
e: shows "e = (\<lambda>s. Discrete.log (nat (s ''x'')) + 1)"
ring_def: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c" shows "class.ring plus zero minus uminus times \<equiv> class.ab_group_add plus zero minus uminus \<and> class.semiring plus times"
span_UNIV: shows "vec.span (columns (Gram_Schmidt_matrix A)) = UNIV"
ide_char: fixes a :: "'d" shows "D.ide a = (D.arr a \<and> D.dom a = a \<and> D.cod a = a)"
constructible_radical_sqrt: fixes M :: "point" assumes "M \<in> constructible" shows "abscissa M \<in> radical_sqrt \<and> ordinate M \<in> radical_sqrt"
rel_envirD: fixes p :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and f :: "'g \<Rightarrow> 'e"   and g :: "'g \<Rightarrow> 'f"   and x :: "'g" assumes "rel_envir p f g" shows "p (f x) (g x)"
bst_mrbst_join: fixes t' :: "'a tree"   and t1 :: "'a tree"   and t2 :: "'a tree" assumes "t' \<in> set_pmf (mrbst_join t1 t2)"   and "bst t1"   and "bst t2"   and "\<And>x y. \<lbrakk>x \<in> set_tree t1; y \<in> set_tree t2\<rbrakk> \<Longrightarrow> x < y" shows "bst t'"
that: fixes n :: "nat" assumes "\<sigma> [m..] \<approx> \<tau> [n..]"   and "\<forall>j<n. \<exists>k<m. \<sigma> [k..] \<approx> \<tau> [j..]" shows "thesis"
labeled_entailment_lifting: fixes NL1 :: "('f \<times> 'l) set"   and NL2 :: "('f \<times> 'l) set" shows "(\<Union> (\<G>_F_L ` NL1) \<Turnstile>G \<Union> (\<G>_F_L ` NL2)) = (no_labels.\<G>_Fset (fst ` NL1) \<Turnstile>G no_labels.\<G>_Fset (fst ` NL2))"
zero_order_3: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and n :: "'b" assumes "class.canonically_ordered_monoid_add plus zero less_eq less" shows "\<not> less n zero"
det3_rotate: fixes a :: "real \<times> real"   and b :: "real \<times> real"   and c :: "real \<times> real" shows "det3 a b c = det3 c a b"
insert_invar: fixes q :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree list"   and e :: "'a"   and a :: "'b" assumes "SkewBinomialHeapStruc.invar q" shows "SkewBinomialHeapStruc.invar (SkewBinomialHeapStruc.insert e a q)"
subset_MST: shows "is_subset_MST w rg ({(v, u)} \<union> A)"
assms_5: shows "older_seniors x n = older_seniors x (Suc n)"
assms: shows "run_one_step d i (s, vs, ves, $Unop_f x251 x252) = (s', vs', res)"
Un_commute: fixes A :: "'a set"   and B :: "'a set" shows "A \<union> B = B \<union> A"
inject_2: fixes x21 :: "LambdaType.type"   and x22 :: "LambdaType.type"   and y21 :: "LambdaType.type"   and y22 :: "LambdaType.type" shows "(x21 \<Rightarrow> x22 = y21 \<Rightarrow> y22) = (x21 = y21 \<and> x22 = y22)"
exhaust: fixes y :: "('b, 'a) L_state" obtains x1 :: "'a \<times> 'a fs_set \<times> 'b" where "y = AC x1"   | x2 :: "'a fs_set \<times> 'b" where "y = EF x2"
collect_interrupts_def: fixes ias :: "'b interrupt_action list" shows "collect_interrupts ias = foldr collect_interrupt ias {}"
dom_fun_upd: fixes f :: "'b \<Rightarrow> 'c option"   and x :: "'b"   and y :: "'c option" shows "dom (f(x := y)) = (if y = None then dom f - {x} else insert x (dom f))"
map_vector_0: fixes v :: "('a, 'd) Finite_Cartesian_Product.vec" shows "(map_vector hom v = 0) = (v = 0)"
par_strict_not_col_2: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "A B ParStrict C D" shows "\<not> Col B C D"
hyps: fixes y :: "hf" assumes "(y, x) \<in> hmem_rel"   and "coding_hf y" shows "HR y \<lbrakk>Goedel_I.HF y\<rbrakk>e"
gen_wf_Entry_def: shows "gen_wf_Entry \<equiv> map_fun Rep_gen_cfg_wf id gen_Entry"
treap_of_subset: fixes A :: "('a \<times> 'b) set" shows "set_tree (treap_of A) \<subseteq> A"
discounted_value_def: fixes r :: "real"   and X :: "nat \<Rightarrow> 'b \<Rightarrow> real" shows "discounted_value r X = (\<lambda>n w. discount_factor r n w * X n w)"
p_res_ring_0: shows "carrier (residue_ring (p ^ 0)) = {0}"
assms_3: assumes "\<not> b"   and "\<not> b'" shows "ERETURN S2 \<le> \<Down>\<^sub>E E R S2'"
HEndPhase0_Inv2c: fixes s :: "state"   and s' :: "state"   and p :: "Proc" assumes "Inv2c s"   and "HEndPhase0 s s' p"   and "Inv2a s"   and "Inv1 s" shows "Inv2c s'"