hcju/isabelleps huggingface dataset:
rconjby_lconjby: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a" assumes "class.group_add minus plus zero uminus" shows "plus (plus (uminus x) (minus (plus x y) x)) x = y"
prod_tOp_mem: fixes I :: "'d set"   and A :: "'d \<Rightarrow> ('e, 'f) Ring_scheme"   and X :: "'d \<Rightarrow> 'e"   and Y :: "'d \<Rightarrow> 'e" assumes "\<forall>k\<in>I. Ring (A k)"   and "X \<in> carr_prodag I A"   and "Y \<in> carr_prodag I A" shows "prod_tOp I A X Y \<in> carr_prodag I A"
m_def: fixes p :: "PrIds" shows "m p = (rtrner rmCh!p, ires!p)"
vimageD: fixes a :: "'e"   and f :: "'e \<Rightarrow> 'f"   and A :: "'f set" assumes "a \<in> f -` A" shows "f a \<in> A"
start: shows "items_le k (\<J> k 0) = Gen (paths_le k (\<P> k 0))"
assms_4: shows "\<exists>!i. n \<le> i \<and> latestAct_cond c t n' i"
kD_update_unchanged: fixes rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and dip :: "nat"   and dsn :: "nat"   and dsk :: "k"   and flag :: "f"   and hops :: "nat"   and nhip :: "nat"   and pre :: "nat set" assumes "rt = update rt dip (dsn, dsk, flag, hops, nhip, pre)" shows "dip \<in> kD rt"
exec_all_conf: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> nat \<times> nat \<times> instr list \<times> (nat \<times> nat \<times> char list \<times> nat \<times> nat) list) list) list"   and \<sigma> :: "nat option \<times> (nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (val list \<times> val list \<times> char list \<times> char list \<times> nat) list"   and \<sigma>' :: "nat option \<times> (nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (val list \<times> val list \<times> char list \<times> char list \<times> nat) list"   and \<sigma>'' :: "nat option \<times> (nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (val list \<times> val list \<times> char list \<times> char list \<times> nat) list" assumes "P \<turnstile> \<sigma> -jvm\<rightarrow> \<sigma>'"   and "P \<turnstile> \<sigma> -jvm\<rightarrow> \<sigma>''" shows "P \<turnstile> \<sigma>' -jvm\<rightarrow> \<sigma>'' \<or> P \<turnstile> \<sigma>'' -jvm\<rightarrow> \<sigma>'"
cut_ge_absorb: fixes I :: "'a set"   and t :: "'a" shows "I \<down>\<ge> t \<down>\<ge> t = I \<down>\<ge> t"
\<tau>lifting_inv_axioms: shows "\<tau>lifting_inv final r \<tau>move P"
more_arith_simps_4: fixes a :: "'a" shows "a + - a = (0::'a)"
frameResChainComm: fixes xvec :: "name list"   and yvec :: "name list"   and F :: "'b frame" shows "\<lparr>\<nu>*xvec\<rparr>\<lparr>\<nu>*yvec\<rparr>F \<simeq>\<^sub>F \<lparr>\<nu>*yvec\<rparr>\<lparr>\<nu>*xvec\<rparr>F"
assms_2: shows "typ_of1 (map (\<lambda>T. tsubstT T \<rho>) Ts) (tsubst t \<rho>) = Some ty'"
infinite_mono: fixes S :: "'e set"   and T :: "'e set" assumes "infinite S"   and "\<And>x. x \<in> S \<Longrightarrow> x \<in> T" shows "infinite T"
dghm_const_ObjMap_app: fixes x :: "V"   and \<CC> :: "V"   and \<DD> :: "V"   and a :: "V"   and f :: "V" assumes "x \<in>\<^sub>\<circ> \<CC>\<lparr>Obj\<rparr>" shows "dghm_const \<CC> \<DD> a f\<lparr>ObjMap\<rparr>\<lparr>x\<rparr> = a"
\<phi>_def: fixes v :: "'a" shows "\<phi> v = (\<psi> ^^ vdist v) v"
div_free_CHAOS: fixes A :: "'b set" shows "D (CHAOS A) = {}"
length_helper: fixes p :: "nat"   and q :: "nat"   and m :: "bit list"   and M :: "bit list" assumes "prime p"   and "prime q"   and "(length (nat_to_bv (p * q)) - Suc 0) mod 8 \<noteq> 0"   and "MGF (sha1 (generate_M' (sha1 m) salt)) (length (generate_DB (generate_PS (length (nat_to_bv (p * q)) - Suc 0) (length (sha1 (generate_M' (sha1 m) salt)))))) \<noteq> []"   and "length (sha1 M) + sLen + 16 \<le> length (nat_to_bv (p * q)) - Suc 0" shows "length (remzero (maskedDB_zero (bvxor (generate_DB (generate_PS (length (nat_to_bv (p * q)) - Suc 0) (length (sha1 (generate_M' (sha1 m) salt))))) (MGF (sha1 (generate_M' (sha1 m) salt)) (length (generate_DB (generate_PS (length (nat_to_bv (p * q)) - Suc 0) (length (sha1 (generate_M' (sha1 m) salt)))))))) (length (nat_to_bv (p * q)) - Suc 0) @ sha1 (generate_M' (sha1 m) salt) @ BC)) < length (nat_to_bv (p * q))"
path_component_of_discrete_topology: fixes U :: "'a set"   and x :: "'a" shows "path_component_of_set (discrete_topology U) x = (if x \<in> U then {x} else {})"
nfa_steps_quantify_nfa: fixes N :: "bool list bdd list \<times> bool list"   and n :: "nat"   and w :: "bool list list"   and q :: "bool list"   and v :: "nat" assumes "wf_nfa N (Suc n)"   and "list_all (is_alph n) w"   and "nfa_is_node N q"   and "v \<le> n" shows "set_of_bv (nfa_steps (quantify_nfa v N) q w) = (\<Union>xs\<in>{x. length x = length w}. set_of_bv (nfa_steps N q (insertll v xs w)))"
row_to_poly_zero: fixes ts :: "'a list" shows "row_to_poly ts (0\<^sub>v (length ts)) = 0"
intros_1: fixes order :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b"   and xs :: "'b list"   and ys :: "'b list" assumes "order x y"   and "length xs = length ys" shows "lexp order (x # xs) (y # ys)"
prepare_def: fixes cfg :: "'a \<times> (program \<Rightarrow> unit)"   and ast :: "AST.module list"   and \<phi> :: "expr ltlc" shows "prepare cfg ast \<phi> \<equiv> let (prog, g\<^sub>0) = setUp ast; (APs, \<phi>\<^sub>i) = ltl_convert \<phi> in ((prog, APs, g\<^sub>0), \<phi>\<^sub>i)"
unconst_const: fixes size :: "'b \<Rightarrow> nat"   and frees :: "'b \<Rightarrow> name fset"   and subst :: "'b \<Rightarrow> (name, 'b) fmap \<Rightarrow> 'b"   and "consts" :: "'b \<Rightarrow> name fset"   and app :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unapp :: "'b \<Rightarrow> ('b \<times> 'b) option"   and const :: "name \<Rightarrow> 'b"   and unconst :: "'b \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'b"   and unfree :: "'b \<Rightarrow> name option"   and name :: "name" assumes "class.pre_term size frees subst consts app unapp const unconst free unfree" shows "unconst (const name) = Some name"
f3: shows "\<forall>s sa sb w wa wb sc. \<not> low_equal s sa \<or> sb \<noteq> snd (fst (write_reg w wa wb s)) \<or> sc \<noteq> snd (fst (write_reg w wa wb sa)) \<or> low_equal sb sc"
floor_rat_def: fixes x :: "rat" shows "\<lfloor>x\<rfloor> = (THE z. rat_of_int z \<le> x \<and> x < rat_of_int (z + 1))"
leg1_composite: shows "f_src.leg1 = f.leg1 \<cdot> f_src.prj\<^sub>1"
map_iterator_abs_genord_rule_P: fixes \<alpha> :: "'e \<Rightarrow> 'f"   and invar :: "'e \<Rightarrow> bool"   and iti :: "('g \<Rightarrow> bool) \<Rightarrow> ('h \<times> 'e \<Rightarrow> 'g \<Rightarrow> 'g) \<Rightarrow> 'g \<Rightarrow> 'g"   and m :: "'h \<Rightarrow> 'f option"   and R :: "'h \<times> 'f \<Rightarrow> 'h \<times> 'f \<Rightarrow> bool"   and I :: "'h set \<Rightarrow> 'g \<Rightarrow> bool"   and \<sigma>0 :: "'g"   and c :: "'g \<Rightarrow> bool"   and f :: "'h \<times> 'e \<Rightarrow> 'g \<Rightarrow> 'g"   and P :: "'g \<Rightarrow> bool" assumes "map_iterator_abs_genord \<alpha> invar iti m R"   and "I (dom m) \<sigma>0"   and "\<And>k v it \<sigma>. \<lbrakk>c \<sigma>; k \<in> it; invar v; m k = Some (\<alpha> v); it \<subseteq> dom m; I it \<sigma>; \<forall>k' v'. k' \<in> it - {k} \<and> invar v' \<and> m k' = Some (\<alpha> v') \<longrightarrow> R (k, \<alpha> v) (k', \<alpha> v'); \<forall>k' v'. k' \<notin> it \<and> invar v' \<and> m k' = Some (\<alpha> v') \<longrightarrow> R (k', \<alpha> v') (k, \<alpha> v)\<rbrakk> \<Longrightarrow> I (it - {k}) (f (k, v) \<sigma>)"   and "\<And>\<sigma>. I {} \<sigma> \<Longrightarrow> P \<sigma>"   and "\<And>\<sigma> it. \<lbrakk>it \<subseteq> dom m; it \<noteq> {}; \<not> c \<sigma>; I it \<sigma>; \<forall>k v k' v'. k \<notin> it \<and> invar v \<and> m k = Some (\<alpha> v) \<and> k' \<in> it \<and> invar v' \<and> m k' = Some (\<alpha> v') \<longrightarrow> R (k, \<alpha> v) (k', \<alpha> v')\<rbrakk> \<Longrightarrow> P \<sigma>" shows "P (iti c f \<sigma>0)"
suminf: fixes f :: "'b \<Rightarrow> 'c"   and X :: "nat \<Rightarrow> 'b" assumes "bounded_linear f"   and "summable X" shows "f (\<Sum>n. X n) = (\<Sum>n. f (X n))"
c_cons_is_pr: shows "c_cons \<in> PrimRec2"
exp_of_real: fixes x :: "real" shows "exp (of_real x) = of_real (exp x)"
lincomb_list_elem: fixes Vs :: "'a vec list"   and i :: "nat" assumes "set Vs \<subseteq> carrier_vec n" shows "lincomb_list (\<lambda>j. if i = j then 1::'a else (0::'a)) Vs = (if i < length Vs then Vs ! i else 0\<^sub>v n)"
hcomp_simps\<^sub>W\<^sub>C_1: fixes \<nu> :: "'a"   and \<mu> :: "'a" assumes "\<nu> \<star> \<mu> \<noteq> null" shows "arr (\<nu> \<star> \<mu>)"
length_append: fixes xs :: "'e list"   and ys :: "'e list" shows "length (xs @ ys) = length xs + length ys"
wf_digraph_axioms: shows "wf_digraph (del_vert u)"
snd_normalize_quot_nonzero: fixes x :: "'a \<times> 'a" shows "snd (normalize_quot x) \<noteq> (0::'a)"
toplevel_summands_ACI_norm_alt: fixes r :: "'b rexp" shows "toplevel_summands (ACI_norm_alt r) = ACI_norm_alt ` toplevel_summands r"
ab: shows "int CARD('q) \<le> int CARD('pq)"
enum_0: fixes p :: "nat"   and n :: "nat"   and base :: "nat \<Rightarrow> nat"   and upd :: "nat \<Rightarrow> nat"   and s :: "(nat \<Rightarrow> nat) set" assumes "kuhn_simplex p n base upd s" shows "kuhn_simplex.enum base upd 0 = base"
rtranclp_idemp: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "r\<^sup>*\<^sup>*\<^sup>*\<^sup>* = r\<^sup>*\<^sup>*"
that: shows "x \<in>\<^sub>i range_tm e t + real_interval i"
"2_1": fixes x :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list \<times> ('Var \<Rightarrow> 'Val)" assumes "sound_mode_use x"   and "x \<rightarrow>\<^bsub>ns\<^esub> gc'" shows "sound_mode_use gc'"
that: fixes g :: "'a \<Rightarrow>\<^sub>C 'b" assumes "\<And>x. x \<in> cbox a b \<Longrightarrow> apply_bcontfun g x = f x"   and "\<And>x. apply_bcontfun g x = f (clamp a b x)" shows "thesis"
assms_5: shows "set qs \<subseteq> {a, b}"
unsatisfiableFormulaWithSingleLiteralClauses: fixes F :: "Literal list list"   and c :: "Literal list" assumes "\<not> satisfiable (F @ val2form (oppositeLiteralList c))"   and "\<not> clauseTautology c" shows "formulaEntailsClause F c"
isomorphic_unit_right: fixes f :: "'a" assumes "ide f" shows "f \<star> src f \<cong> f"
state_measure'_def: fixes V :: "nat set"   and V' :: "nat set"   and \<Gamma> :: "nat \<Rightarrow> pdf_type"   and \<rho> :: "nat \<Rightarrow> val" shows "state_measure' V V' \<Gamma> \<rho> = distr (state_measure V \<Gamma>) (state_measure (V \<union> V') \<Gamma>) (\<lambda>\<sigma>. merge V V' (\<sigma>, \<rho>))"
face_cycle_set_parts: fixes a :: "'b"   and b :: "'b" shows "G.face_cycle_set a = G.face_cycle_set b \<or> G.face_cycle_set a \<inter> G.face_cycle_set b = {}"
diff_numeral_simps_2: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and one :: "'b"   and m :: "num"   and n :: "num" assumes "class.neg_numeral minus plus zero uminus" shows "minus (numeral.numeral one plus m) (uminus (numeral.numeral one plus n)) = numeral.numeral one plus (m + n)"
wf_trm_Var: fixes x :: "'b" shows "wf\<^sub>t\<^sub>r\<^sub>m (Var x)"
singular_relboundary_diff: fixes p :: "nat"   and X :: "'b topology"   and S :: "'b set"   and a :: "((nat \<Rightarrow> real) \<Rightarrow> 'b) \<Rightarrow>\<^sub>0 int"   and b :: "((nat \<Rightarrow> real) \<Rightarrow> 'b) \<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)"
uncountable_bij_betw: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and B :: "'b set" assumes "bij_betw f A B"   and "uncountable B" shows "uncountable A"
keys_focus: fixes X :: "'c set"   and p :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd" shows "keys (focus X p) = (\<lambda>t. except t (- X)) ` keys p"
finsum_ldistr: fixes A :: "'c set"   and a :: "'a"   and f :: "'c \<Rightarrow> 'a" assumes "finite A"   and "a \<in> carrier R"   and "f \<in> A \<rightarrow> carrier R" shows "finsum R f A \<otimes> a = (\<Oplus>i\<in>A. f i \<otimes> a)"
span_Gram_Schmidt_matrix: fixes A :: "(('a, 'c) vec, 'b) vec" shows "span (columns A) = span (columns (Gram_Schmidt_matrix A))"
nprv_clear5_3: fixes \<phi>1 :: "'fmla"   and \<phi>2 :: "'fmla"   and \<phi>4 :: "'fmla"   and \<phi>5 :: "'fmla"   and \<psi> :: "'fmla"   and \<phi>3 :: "'fmla" assumes "nprv {\<phi>1, \<phi>2, \<phi>4, \<phi>5} \<psi>"   and "\<phi>1 \<in> fmla"   and "\<phi>2 \<in> fmla"   and "\<phi>3 \<in> fmla"   and "\<phi>4 \<in> fmla"   and "\<phi>5 \<in> fmla"   and "\<psi> \<in> fmla" shows "nprv {\<phi>1, \<phi>2, \<phi>3, \<phi>4, \<phi>5} \<psi>"
D_imp_front_tickFree: fixes s :: "'b event list"   and P :: "'b process" assumes "s \<in> D P" shows "front_tickFree s"
map_scale_eq_mult_scalar: fixes c :: "'c"   and p :: "'t \<Rightarrow>\<^sub>0 'c" shows "c \<cdot> p = monomial c (0::'a) \<odot> p"
wprepare_loop1: fixes b :: "cell list"   and mr :: "nat"   and m :: "nat"   and lm :: "nat list"   and list :: "cell list"   and rn :: "nat" assumes "rev b @ Oc \<up> mr = Oc \<up> Suc m @ Bk # Bk # <lm>"   and "b \<noteq> []"   and "0 < mr"   and "Oc # list = Oc \<up> mr @ Bk \<up> rn" shows "wprepare_loop_start_on_rightmost m lm (Oc # b, list)"
conjr1_iso: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<le> y" shows "z \<rhd> x \<le> z \<rhd> y"
closure_insert: fixes x :: "'d"   and s :: "'d set" shows "closure (insert x s) = insert x (closure s)"
Cons_rev: fixes a :: "'b"   and u :: "'b list" shows "a # rev u = rev (snocs u [a])"
step_0: fixes l :: "cell list"   and r :: "cell list"   and p :: "(action \<times> nat) list \<times> nat" shows "step (0, l, r) p = (0, l, r)"
trivial_integer_mod_group: fixes n :: "nat" shows "trivial_group (Z n) = (n = 1)"
yivp_2: shows "Y\<^sub>2 t\<^sub>0 = s"
in_parts_UnE: fixes c :: "msg"   and G :: "msg set"   and H :: "msg set" assumes "c \<in> parts (G \<union> H)" obtains "c \<in> parts G"   | "c \<in> parts H"
check_no_overlap_alt: fixes \<gamma> :: "'a set \<Rightarrow> 'b \<Rightarrow> bool"   and ft :: "('a, 'c) flow_entry_match list" shows "check_no_overlap \<gamma> ft = check_no_overlap2 \<gamma> ft"
Axiom_of_Choice: fixes A :: "V" obtains f :: "V" where "\<And>x. \<lbrakk>x \<in>\<^sub>\<circ> A; x \<noteq> 0\<rbrakk> \<Longrightarrow> f\<lparr>x\<rparr> \<in>\<^sub>\<circ> x" and "vsv f"
join_idem: fixes x :: "'b" shows "x \<squnion> x = x"
mon_pl_def: fixes w :: "('a set \<times> 'a set) list" shows "mon_pl w \<equiv> foldl (\<union>) {} (map (\<lambda>e. fst e \<union> snd e) w)"
prod_mset_prime_factorization_weak: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and x :: "'b" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "x \<noteq> zero" shows "normalize (comm_monoid_mult.prod_mset times one (factorial_semiring.prime_factorization zero times one normalize x)) = normalize x"
scaleRe_ivl_coll_spec_def: fixes l :: "ereal"   and u :: "ereal"   and X :: "('d \<times> 'e) set" shows "scaleRe_ivl_coll_spec l u X = sets_of_coll X \<bind> (\<lambda>XS. FORWEAK XS (RETURN op_empty_coll) (\<lambda>X. scaleRe_ivl_spec l u X \<bind> (\<lambda>I. RETURN (mk_coll I))) (\<lambda>X X'. RETURN (X' \<union> X)))"
unfinished_dnodes'_def: fixes w :: "'b \<times> 'b \<Rightarrow> enat"   and s :: "'b"   and V :: "'b \<Rightarrow> 'c option" shows "Dijkstra.unfinished_dnodes' w s V \<equiv> Dijkstra.unfinished_dnodes w s (dom V)"
absorb_iff1: fixes b :: "'d"   and a :: "'d" shows "(b \<le> a) = (a \<squnion> b = a)"
hom_mult: fixes G :: "('d, 'e) monoid_scheme"   and H :: "('f, 'g) monoid_scheme"   and h :: "'d \<Rightarrow> 'f"   and x :: "'d"   and y :: "'d" assumes "group_hom G H h"   and "x \<in> carrier G"   and "y \<in> carrier G" shows "h (x \<otimes>\<^bsub>G\<^esub> y) = h x \<otimes>\<^bsub>H\<^esub> h y"
cast_element_ptr_not_character_data_ptr_1: fixes element_ptr :: "(_) element_ptr"   and character_data_ptr :: "(_) character_data_ptr" shows "cast element_ptr \<noteq> cast character_data_ptr"
f: fixes x :: "'a" assumes "x \<in> A" shows "inj_on (f x) (B x) \<and> f x ` B x \<subseteq> C x"
no_ofail_def: fixes P :: "'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b option" shows "no_ofail P f \<equiv> \<forall>s. P s \<longrightarrow> f s \<noteq> None"
of_int_ceiling_le_add_one: fixes r :: "'c" shows "of_int \<lceil>r\<rceil> \<le> r + (1::'c)"
rev_permutations_of_set: fixes A :: "'b set" shows "rev ` permutations_of_set A = permutations_of_set A"
Fin: fixes G :: "prog"   and s0 :: "st"   and c1 :: "stmt"   and n :: "nat"   and x1 :: "abopt"   and s1 :: "st"   and c2 :: "stmt"   and s2 :: "state"   and s3 :: "state" assumes "G\<turnstile>Norm s0 \<midarrow>c1\<midarrow>n\<rightarrow> (x1, s1)"   and "G\<turnstile>Norm s1 \<midarrow>c2\<midarrow>n\<rightarrow> s2"   and "s3 = (if \<exists>err. x1 = Some (Error err) then (x1, s1) else abupd (abrupt_if (x1 \<noteq> None) x1) s2)" shows "G\<turnstile>Norm s0 \<midarrow>c1 Finally c2\<midarrow>n\<rightarrow> s3"
at_left_minus: fixes a :: "real" shows "at_left a = filtermap uminus (at_right (- a))"
one_dim_iso_of_zero': fixes x :: "'b" assumes "one_dim_iso x = (0::'a)" shows "x = (0::'b)"
Sup_Un_distrib: fixes A :: "V set"   and B :: "V set" assumes "small A"   and "small B" shows "\<Squnion> (A \<union> B) = \<Squnion> A \<union>\<^sub>\<circ> \<Squnion> B"
gt_sym_antisym: fixes f :: "'s"   and g :: "'s" assumes "f >\<^sub>s g" shows "\<not> g >\<^sub>s f"
pair_in_Id_conv: fixes a :: "'e"   and b :: "'e" shows "((a, b) \<in> Id) = (a = b)"
inner_vec_def: fixes x :: "('a, 'b) Finite_Cartesian_Product.vec"   and y :: "('a, 'b) Finite_Cartesian_Product.vec" shows "x \<bullet> y = (\<Sum>i\<in>UNIV. x $h i \<bullet> y $h i)"
us: shows "us = getFrN (map snd txs) (map fst txs) [\<phi>] (length txs)"
is_regular_zero_sheaf_spec: fixes U :: "'a set set" assumes "is_zariski_open U" shows "is_regular (zero_sheaf_spec U) U"
rs_def: shows "rs = the \<circ> map_of (map (\<lambda>(x, y). (y, x)) ts)"
fps_0th_higher_deriv: fixes n :: "nat"   and f :: "'a fps" shows "(fps_deriv ^^ n) f $ 0 = fact n * f $ n"
nz: fixes i :: "nat" assumes "i < m + n" shows "M $$ (i, \<pi> i) \<noteq> 0"
gen_model_worlds: fixes M :: "('f, 'g, 'e) KripkeStructure"   and w :: "'e" shows "worlds (gen_model M w) = worlds M \<inter> (\<Union> (range (relations M)))\<^sup>* `` {w}"
AddSA_CompFun_the: fixes SA :: "('a, 'b, 'c) seqauto"   and A :: "('a, 'b, 'c) hierauto"   and S :: "'a" assumes "States SA \<inter> HAStates A = {}"   and "S \<in> HAStates A" shows "the (CompFun (A [++] (S, SA)) S) = insert SA (the (CompFun A S))"
that: fixes k :: "'c \<Rightarrow> nat" assumes "valid_abstraction A (clk_set A) (\<lambda>x. real (k x))" shows "thesis"
n_in_sup: shows "N \<in> Sup_llist (lmap Red_F Ns)"
ceiling_le_one: fixes x :: "'a" shows "(\<lceil>x\<rceil> \<le> 1) = (x \<le> (1::'a))"
assms: shows "rt ip = Some (dsn, dsk, flag, hops, nhip)"
t_list_of_A_bound: fixes t :: "'b tree" shows "t_list_of_A t \<le> 2 * height t * size t"
ord_canc: fixes s :: "'a"   and u :: "'a"   and t :: "'a" assumes "s + u \<preceq> t + u" shows "s \<preceq> t"
hyps_2: fixes x :: "'a"   and Q :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "x \<in> A"   and "\<forall>xa y. xa \<in> A \<and> y \<in> A \<and> (P xa y \<or> P x xa) \<longrightarrow> Q xa y" shows "af A Q"
prim_invar2_init_def: fixes Q :: "'v \<Rightarrow> enat"   and \<pi> :: "'v \<Rightarrow> 'v option" shows "prim_invar2_init Q \<pi> \<equiv> Q = initQ \<and> \<pi> = init\<pi>"
asprod_0_x: fixes x :: "ant" shows "0 *\<^sub>a x = 0"
neg_prod_le: fixes c :: "'a"   and x :: "'a" assumes "c < (0::'a)" shows "c * x \<le> (0::'a) \<equiv> (0::'a) \<le> x"
a_append_child_def: fixes ptr :: "(_) object_ptr"   and child :: "(_) node_ptr" shows "local.a_append_child ptr child = insert_before ptr child None"
k_\<Lambda>_PBD_axioms_def: fixes u_block_size :: "int" shows "k_\<Lambda>_PBD_axioms u_block_size \<equiv> 2 \<le> u_block_size"
space_rT: fixes \<omega> :: "'c stream"   and K :: "'c \<Rightarrow> 'c pmf"   and x :: "'c" shows "(\<omega> \<in> space (MC_syntax.rT K x)) = MC_syntax.enabled K x \<omega>"
two_cong_0_iff_int: fixes p :: "int" assumes "0 < p" shows "[2 = 0] (mod p) = (p = 1 \<or> p = 2)"
prems_2: fixes xs :: "'a list" assumes "xs \<in> set (xs__ # xss)" shows "list_succ xs = perm_restrict f (set xs)"
borel_measurable_count_space: fixes f :: "'b \<Rightarrow> 'c"   and S :: "'b set" shows "f \<in> borel_measurable (count_space S)"
inj_mapI: fixes f :: "'g \<Rightarrow> 'h" assumes "inj f" shows "inj (map f)"
lQuot_union: fixes a :: "'b"   and A :: "'b list set"   and B :: "'b list set" shows "lQuot a (A \<union> B) = lQuot a A \<union> lQuot a B"
lookup_map_default: fixes x :: "'b"   and d :: "'a"   and f :: "'a \<Rightarrow> 'a"   and m :: "('b, 'a) mapping" shows "Mapping.lookup (Mapping.map_default x d f m) x = Some (f (Mapping.lookup_default d m x))"
input2state_not_None: fixes i :: "value list"   and x :: "nat" assumes "input2state i $ x \<noteq> None" shows "x < length i"
degree_fps_poly_swap2_eq: fixes p :: "'b poly fps"   and d :: "nat"   and n :: "nat" assumes "\<And>n. degree (p $ n) \<le> d"   and "0 < d \<or> p $ n \<noteq> 0"   and "degree (p $ n) = d" shows "degree (fps_poly_swap2 p) = d"
equality: fixes r :: "('m, 'n, 'o, 'p) RShadowRoot_scheme"   and r' :: "('m, 'n, 'o, 'p) RShadowRoot_scheme" assumes "RObject.nothing r = RObject.nothing r'"   and "RDocument.nothing r = RDocument.nothing r'"   and "doctype r = doctype r'"   and "document_element r = document_element r'"   and "disconnected_nodes r = disconnected_nodes r'"   and "RShadowRoot.nothing r = RShadowRoot.nothing r'"   and "mode r = mode r'"   and "RShadowRoot.child_nodes r = RShadowRoot.child_nodes r'"   and "RShadowRoot.more r = RShadowRoot.more r'" shows "r = r'"
l_repr_independence: fixes G :: "('c, 'd) monoid_scheme"   and y :: "'c"   and x :: "'c"   and H :: "'c set" assumes "Group.group G"   and "y \<in> x <#\<^bsub>G\<^esub> H"   and "x \<in> carrier G"   and "subgroup H G" shows "x <#\<^bsub>G\<^esub> H = y <#\<^bsub>G\<^esub> H"
C: fixes \<sigma>' :: "'c"   and \<sigma> :: "'d" assumes "(\<sigma>', \<sigma>) \<in> R" shows "(c' \<sigma>', c \<sigma>) \<in> bool_rel"
lex: shows "\<forall>n. Re (trace (f n)) \<le> Re x \<and> Im x = 0"
rel_ipurge_aux_def: fixes P :: "'a process"   and I :: "('b \<times> 'b) set"   and D :: "'a \<Rightarrow> 'b"   and U :: "'b set" shows "rel_ipurge_aux P I D U \<equiv> {(xs, ys). xs \<in> traces P \<and> ys \<in> traces P \<and> ipurge_tr_rev_aux I D U xs = ipurge_tr_rev_aux I D U ys}"
subst_eq: shows "Fsubst (seq2fml (\<Gamma>, \<Delta>)) \<sigma> = seq2fml (Ssubst (\<Gamma>, \<Delta>) \<sigma>)"
image_eqI: fixes b :: "'g"   and f :: "'h \<Rightarrow> 'g"   and x :: "'h"   and A :: "'h set" assumes "b = f x"   and "x \<in> A" shows "b \<in> f ` A"
vec_simps_15: fixes ai :: "'a"   and bi :: "'a" shows "ai * bi = bi * ai"
has_bochner_integral_sum: fixes I :: "'a set"   and M :: "'b measure"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and x :: "'a \<Rightarrow> 'c" assumes "\<And>i. i \<in> I \<Longrightarrow> has_bochner_integral M (f i) (x i)" shows "has_bochner_integral M (\<lambda>x. \<Sum>i\<in>I. f i x) (sum x I)"
obv: fixes a :: "'b \<times> 'c" shows "a = (fst a, snd a)"
numeral_sqr: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and k :: "num" assumes "class.semiring_numeral one times plus" shows "numeral.numeral one plus (Num.sqr k) = times (numeral.numeral one plus k) (numeral.numeral one plus k)"