hcju/isabelleps huggingface dataset:
bit_exp_iff: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bit :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and m :: "nat"   and n :: "nat" assumes "class.semiring_bits plus minus zero times one divide modulo bit" shows "bit (power.power one times (numeral.numeral one plus (num.Bit0 num.One)) m) n = (power.power one times (numeral.numeral one plus (num.Bit0 num.One)) m \<noteq> zero \<and> m = n)"
nontriv_solution_power_nat: fixes z :: "nat \<times> nat"   and n :: "nat" assumes "nontriv_solution z"   and "0 < n" shows "nontriv_solution (pell_power z n)"
bij_betw_singleton: fixes f :: "'a \<Rightarrow> 'b"   and x :: "'a"   and y :: "'b" shows "bij_betw f {x} {y} = (f x = y)"
nsum_jointfun: fixes A :: "('d, 'e) aGroup_scheme"   and n :: "nat"   and f :: "nat \<Rightarrow> 'd"   and m :: "nat"   and g :: "nat \<Rightarrow> 'd" assumes "aGroup A"   and "\<forall>j\<le>n. f j \<in> carrier A"   and "\<forall>j\<le>m. g j \<in> carrier A" shows "\<Sigma>\<^sub>e A jointfun n f m g Suc (n + m) = \<Sigma>\<^sub>e A f n \<plusminus>\<^bsub>A\<^esub> \<Sigma>\<^sub>e A g m"
focus_Polys: fixes p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and X :: "'b set" assumes "p \<in> P[X]" shows "focus X p = (\<Sum>t\<in>keys p. monomial (monomial (lookup p t) 0) t)"
rep_eq: fixes x :: "'a blinop" shows "($) x = blinfun_apply (Rep_blinop x)"
less_le_trans: fixes x :: "'a \<times> ('a, 'b) operation"   and y :: "'a \<times> ('a, 'b) operation"   and z :: "'a \<times> ('a, 'b) operation" assumes "hb x y"   and "weak_hb y z" shows "hb x z"
bind_returns_heap_E2: fixes h :: "'x"   and f :: "('x, 'y, 'aa) prog"   and g :: "'aa \<Rightarrow> ('x, 'y, 'z) prog"   and h' :: "'x" assumes "h \<turnstile> Heap_Error_Monad.bind f g \<rightarrow>\<^sub>h h'"   and "pure f h" obtains x :: "'aa" where "h \<turnstile> f \<rightarrow>\<^sub>r x" and "h \<turnstile> g x \<rightarrow>\<^sub>h h'"
divide_eq1: fixes h :: "real \<Rightarrow> real"   and F :: "real filter"   and f :: "real \<Rightarrow> real"   and g :: "real \<Rightarrow> real" assumes "\<forall>\<^sub>F x in F. h x \<noteq> 0" shows "(f \<in> L F (\<lambda>x. g x / h x)) = ((\<lambda>x. f x * h x) \<in> L F g)"
L: shows "\<forall>x y. P x \<and> P y \<and> g x = g y \<longrightarrow> f x = f y"
countable': fixes x :: "'a" shows "countable {y. (x, y) \<in> support_flow (\<Squnion> Y)}"
is_alignedE_pre: fixes w :: "'b word"   and n :: "nat" assumes "is_aligned w n" shows "\<exists>q. w = 2 ^ n * word_of_nat q \<and> q < 2 ^ (LENGTH('b) - n)"
has_field_derivative_imp_has_derivative: fixes f :: "'a \<Rightarrow> 'a"   and D :: "'a"   and F :: "'a filter" assumes "(f has_field_derivative D) F" shows "(f has_derivative (*) D) F"
inv: fixes a :: "'a" assumes "A.ide a" shows "B.inverse_arrows (\<tau> a) (\<tau>' a)"
inj_on_proj_verts_H: shows "inj_on ((`) proj_verts_H) H.sccs_verts"
get_component_separates_tree_order_general: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and c :: "(_) object_ptr list"   and ptr'' :: "(_) object_ptr"   and to'' :: "(_) object_ptr list"   and ptr' :: "(_) object_ptr"   and c' :: "(_) object_ptr list" assumes "heap_is_wellformed h"   and "type_wf h"   and "known_ptrs h"   and "h \<turnstile> get_component ptr \<rightarrow>\<^sub>r c"   and "h \<turnstile> to_tree_order ptr'' \<rightarrow>\<^sub>r to''"   and "ptr'' \<in> set c"   and "h \<turnstile> get_component ptr' \<rightarrow>\<^sub>r c'"   and "ptr' \<notin> set c" shows "set to'' \<inter> set c' = {}"
PhiWhile_def: fixes b :: "BExpr"   and \<Phi> :: "(Var \<Rightarrow> Val) \<times> (Var \<Rightarrow> Val) \<Rightarrow> bool" shows "PhiWhile b \<Phi> = FIX (PhiWhileOp b \<Phi>)"
ds_subid_aux: fixes x :: "'a"   and y :: "'a" shows "domain_semigroup.ds_ord ans_d (\<cdot>) (ans_d x \<cdot> y) y"
add_cancr: fixes times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and D :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and z :: "'c"   and y :: "'c" assumes "class.cancellative_partial_semigroup times D"   and "D x z"   and "D y z"   and "times x z = times y z" shows "x = y"
ac_Zp_res: fixes k :: "nat"   and m :: "nat"   and x :: "nat \<Rightarrow> int" assumes "k < m"   and "x \<in> carrier Zp"   and "x \<noteq> \<zero>" shows "p_residue k (ac_Zp x m) = ac_Zp x k"
fls_uminus_nth: fixes f :: "'a fls"   and n :: "int" shows "fls_nth (- f) n = - fls_nth f n"
exhaust: fixes x :: "'edge" assumes "x \<in> E" shows "\<exists>a b. joins a b x"
diff_closed: fixes x :: "'a"   and y :: "'a" assumes "vectorspace E"   and "x \<in> V"   and "y \<in> V" shows "x - y \<in> V"
Pd_shuffle_to_top_range: fixes Xd :: "'d \<Rightarrow> 'e"   and Xh :: "'d \<Rightarrow> 'f"   and Pd :: "'e \<Rightarrow> ('d \<times> 'd) set"   and Ch :: "'f \<Rightarrow> 'd set \<Rightarrow> 'd set"   and ds' :: "'e set"   and Y :: "'d set"   and d :: "'e" assumes "Contracts Xd Xh Pd Ch" shows "Field (Contracts.Pd_shuffle_to_top Xd Pd ds' Y d) \<subseteq> {x. Xd x = d}"
mayDiverge_If_L: fixes tst :: "'test"   and s :: "'state"   and c1 :: "('test, 'atom) com"   and c2 :: "('test, 'atom) com" assumes "tval tst s"   and "mayDiverge c1 s" shows "mayDiverge (if tst then c1 else c2) s"
F_strict_def: shows "local.F_strict = {f \<in> {0..<n} \<rightarrow> {0..<m}. (\<forall>i. i \<notin> {0..<n} \<longrightarrow> f i = i) \<and> strict_mono_on f {0..<n}}"
word_le_plus: fixes a :: "'b word"   and b :: "'b word"   and c :: "'b word" assumes "a < a + b"   and "c < b" shows "a \<le> a + c"
inner_sum_left: fixes f :: "'d \<Rightarrow> 'b"   and A :: "'d set"   and y :: "'b" shows "inner (sum f A) y = (\<Sum>x\<in>A. inner (f x) y)"
wfI_min: fixes R :: "('b \<times> 'b) set" assumes "\<And>x Q. x \<in> Q \<Longrightarrow> \<exists>z\<in>Q. \<forall>y. (y, z) \<in> R \<longrightarrow> y \<notin> Q" shows "wf R"
row_ith: fixes i :: "'a"   and A :: "'a sq_mtx" shows "\<r>\<o>\<w> i A = A $$ i"
rel_mono_strong: fixes R :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "'f \<rightharpoonup>\<^sub>f 'd"   and y :: "'f \<rightharpoonup>\<^sub>f 'e"   and Ra :: "'d \<Rightarrow> 'e \<Rightarrow> bool" assumes "fmrel R x y"   and "\<And>z yb. \<lbrakk>z \<in> fmran' x; yb \<in> fmran' y; R z yb\<rbrakk> \<Longrightarrow> Ra z yb" shows "fmrel Ra x y"
hj_2: fixes y :: "real" shows "h (j y) = y"
gcd_0_0: shows "some_gcd (0::'a) (0::'a) = (0::'a)"
simps: fixes f :: "Val \<Rightarrow> Val \<Rightarrow> bool"   and e1 :: "Expr"   and e2 :: "Expr" shows "BVars (compB f e1 e2) = EVars e1 \<union> EVars e2"
diamond_def: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "diamond R = commute R R"
rel_frext_def: fixes a :: "'c \<Longrightarrow> 'b"   and P :: "'c hrel" shows "a:[P]\<^sup>+ = {a}:[P \<oplus>\<^sub>r a]"
rel_cases: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'a option"   and b :: "'b option" assumes "rel_option R a b" obtains "a = None" and "b = None"   | x :: "'a" and y :: "'b" where "a = Some x" and "b = Some y" and "R x y"
homology_coefficients: fixes X :: "'a topology"   and a :: "'a" assumes "topspace X = {a}" shows "homology_group 0 X \<cong> integer_group"
widens_Cons: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list"   and x :: "ty"   and xs :: "ty list"   and ys :: "ty list" shows "P \<turnstile> (x # xs) [\<le>] ys = (\<exists>z zs. ys = z # zs \<and> subtype P x z \<and> P \<turnstile> xs [\<le>] zs)"
Pi_cong_sets: fixes I :: "'d set"   and J :: "'d set"   and M :: "'d \<Rightarrow> 'e set"   and N :: "'d \<Rightarrow> 'e set" assumes "I = J"   and "\<And>x. x \<in> I \<Longrightarrow> M x = N x" shows "Pi I M = Pi J N"
reads_def: fixes S :: "('x \<Rightarrow> 'x \<Rightarrow> bool) set"   and getter :: "('x, 'y, 'z) prog"   and h :: "'x"   and h' :: "'x" shows "reads S getter h h' = ((\<forall>P\<in>S. reflp P \<and> transp P) \<and> ((\<forall>P\<in>S. P h h') \<longrightarrow> preserved getter h h'))"
simps_1: fixes e :: "'b equation" shows "vars_of_lit (Pos e) = vars_of_eq e"
map_comp: fixes g1 :: "'e \<Rightarrow> 'c"   and g2 :: "'f \<Rightarrow> 'd"   and f1 :: "'g \<Rightarrow> 'e"   and f2 :: "'h \<Rightarrow> 'f"   and v :: "('g, 'h) blindable\<^sub>m" shows "map_blindable\<^sub>m g1 g2 (map_blindable\<^sub>m f1 f2 v) = map_blindable\<^sub>m (g1 \<circ> f1) (g2 \<circ> f2) v"
FAccRedsStatic: fixes e :: "char list exp"   and s :: "heap \<times> sfields \<times> sheap"   and b :: "bool"   and a :: "nat"   and s' :: "heap \<times> sfields \<times> sheap"   and b' :: "bool"   and P :: "(char list list \<times> char list exp) prog"   and C :: "char list"   and fs :: "Objects.fields"   and F :: "char list"   and t :: "ty"   and D :: "char list" assumes "P \<turnstile> \<langle>e,s,b\<rangle> \<rightarrow>* \<langle>addr a,s',b'\<rangle>"   and "hp s' a = \<lfloor>(C, fs)\<rfloor>"   and "P \<turnstile> C has F,Static:t in D" shows "P \<turnstile> \<langle>e\<bullet>F{D},s,b\<rangle> \<rightarrow>* \<langle>THROW IncompatibleClassChangeError,s',b'\<rangle>"
simps_10: fixes f1 :: "'d"   and f2 :: "num \<Rightarrow> 'd"   and f3 :: "num \<Rightarrow> 'd"   and x2 :: "num" shows "(case num.Bit0 x2 of num.One \<Rightarrow> f1 | num.Bit0 x \<Rightarrow> f2 x | num.Bit1 x \<Rightarrow> f3 x) = f2 x2"
writes_union_right_I: fixes S' :: "('a, 'b, 'c) prog set"   and f :: "('a, 'b, 'd) prog"   and h :: "'a"   and h' :: "'a"   and S :: "('a, 'b, 'c) prog set" assumes "writes S' f h h'" shows "writes (S \<union> S') f h h'"
c_rec: fixes n :: "nat" assumes "N \<le> n" shows "c (n + 1) = c n * a n - B * b n"
continuous_on_inversediff: fixes z :: "'a"   and S :: "'a set" assumes "z \<notin> S" shows "continuous_on S (\<lambda>w. (1::'a) / (w - z))"
intro: fixes G :: "('c, 'd) ring_scheme" assumes "comm_group (add_monoid G)" shows "abelian_group_axioms G"
free_vars_integrate_vars: fixes \<Gamma> :: "nat \<Rightarrow> pdf_type"   and vs :: "nat list"   and e :: "cexpr" shows "free_vars (integrate_vars \<Gamma> vs e) = free_vars e - set vs"
gso'_def: fixes i :: "nat"   and fsa :: "int vec iarray"   and gsa :: "int vec iarray"   and dmusa :: "int iarray iarray" shows "gso_array.gso' i fsa gsa dmusa = (if i = 0 then fsa !! 0 else Let (dmusa !! 0 !! 0 \<cdot>\<^sub>v fsa !! i - dmusa !! i !! 0 \<cdot>\<^sub>v fsa !! 0) (gso_array.while i 1 gsa dmusa))"
assms_3: shows "eoutKnowCorrect A (kKS m)"
oreach_equiv_obeh_states: fixes S :: "('f, 'e) spec" shows "oreach S = \<Union> (set ` obeh S)"
supp_distinguishing_formula: fixes P :: "'state"   and Q :: "'state" assumes "\<not> P =\<cdot> Q" shows "supp (distinguishing_formula P Q) \<subseteq> supp P"
z_def: shows "z \<equiv> getFr [xx, yy] [t] []"
big_mult: fixes f1 :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g1 :: "'a \<Rightarrow> 'b"   and f2 :: "'a \<Rightarrow> 'b"   and g2 :: "'a \<Rightarrow> 'b" assumes "f1 \<in> O[F](g1)"   and "f2 \<in> O[F](g2)" shows "(\<lambda>x. f1 x * f2 x) \<in> O[F](\<lambda>x. g1 x * g2 x)"
Sup_bot_conv_2: fixes A :: "'a set" shows "(\<bottom> = \<Squnion>A) = (\<forall>x\<in>A. x = \<bottom>)"
rep_eq: fixes x :: "uint16" shows "bit x = bit (Rep_uint16 x)"
match_2: fixes h :: "'a"   and g :: "'a"   and f :: "'a" assumes "seq h (g \<cdot> f)"   and "seq g f" shows "seq h g"
equiv_up_to_trans: fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> bool"   and c :: "com"   and c' :: "com"   and c'' :: "com" assumes "P \<Turnstile> c \<sim> c'"   and "P \<Turnstile> c' \<sim> c''" shows "P \<Turnstile> c \<sim> c''"
term_ok': shows "term_ok \<Theta> (subst_bvs (map (\<lambda>(x, y). Fv x y) vs) t)"
even_Suc_Suc_iff: fixes n :: "nat" shows "gcd_nat.greater_eq (Suc (Suc n)) 2 = gcd_nat.greater_eq n 2"
graph_from_list_foldl: fixes l :: "('a \<times> 'a) list" shows "graph_from_list l = fold ins_edge l graph_empty"
may_lock_upd_locks_conv: fixes l :: "('b \<times> nat) option"   and t :: "'b"   and Ls :: "lock_action list" assumes "lock_actions_ok l t Ls" shows "may_lock (upd_locks l t Ls) t = may_lock l t"
inInter_def: fixes r :: "'c rexp"   and s :: "'c rexp" shows "inInter r s = (case Inters (rexp.Inter r s) of None \<Rightarrow> Zero | Some x \<Rightarrow> INTERSECT x)"
chine_assoc_in_hom: shows "\<guillemotleft>\<mu>\<nu>_\<pi>_\<rho>.chine_assoc : HHH\<mu>\<nu>\<pi>\<rho>.chine \<rightarrow>\<^sub>C HH\<mu>\<nu>H\<pi>\<rho>.chine\<guillemotright>"
class_of_def: fixes U :: "'a set set"   and s :: "'a set \<Rightarrow> ('a \<times> 'a) set" shows "st.class_of U s \<equiv> eq.Class (U, s)"
sup_ge1: fixes x :: "'a"   and y :: "'a" shows "x \<le> sup x y"
simps_4: fixes V :: "'c set"   and t :: "('d, 'c) Term.term"   and t' :: "('d, 'c) Term.term"   and S :: "('d, 'c) stateful_strand_step list" shows "wf'\<^sub>s\<^sub>s\<^sub>t V (\<langle>t := t'\<rangle> # S) = (fv t' \<subseteq> V \<and> wf'\<^sub>s\<^sub>s\<^sub>t (V \<union> fv t) S)"
check: shows "check_instr' (ins ! pc) P h stk loc C M pc frs"
Grp_apply: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and x :: "'b"   and y :: "'c" shows "Grp A f x y = (y = f x \<and> x \<in> A)"
list_of_vec_of_poly: fixes p :: "'b poly" shows "list_of_vec (vec_of_poly p) = (if p = 0 then [0::'b] else rev (coeffs p))"
assms_3: fixes xi :: "'c"   and x :: "'a" assumes "(xi, x) \<in> Ra"   and "x \<in> s" shows "(fi xi, f $ x) \<in> Rb"
run_shift_elim: fixes r :: "'action list"   and s :: "'action stream"   and p :: "'state" assumes "R.run (r @- s) p" obtains "R.path r p" and "R.run s (target r p)"
vector_7_3: fixes x1 :: "'b"   and x2 :: "'b"   and x3 :: "'b"   and x4 :: "'b"   and x5 :: "'b"   and x6 :: "'b"   and x7 :: "'b" shows "vector [x1, x2, x3, x4, x5, x6, x7] $ 3 = x3"
happen_implies_can_occur: fixes channel :: "nat \<Rightarrow> ('d \<times> 'd) option"   and trans :: "'d \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and c :: "('d, 'e, 'f) configuration"   and ev :: "('d, 'e, 'f) event"   and c' :: "('d, 'e, 'f) configuration" assumes "distributed_system channel"   and "distributed_system.next channel trans send recv c ev c'" shows "distributed_system.can_occur channel trans send recv ev c"
strict_coboundedI2: fixes b :: "'c word"   and c :: "'c word"   and a :: "'c word" assumes "b <s c" shows "signed.min a b <s c"
PiM_comb_seq: fixes i :: "nat" shows "distr (Pi\<^sub>M UNIV (\<lambda>i. M) \<Otimes>\<^sub>M Pi\<^sub>M UNIV (\<lambda>i. M)) (Pi\<^sub>M UNIV (\<lambda>i. M)) (\<lambda>(\<omega>, \<omega>'). comb_seq i \<omega> \<omega>') = Pi\<^sub>M UNIV (\<lambda>i. M)"
rep_eq: fixes x :: "'g"   and xa :: "('e, 'f, 'g) formula" shows "Rep_formula (Act x xa) = Act\<^sub>\<alpha> x (Rep_formula xa)"
Collector: shows "\<turnstile> Collector  \<lbrace>False\<rbrace>"
meet_double_iso: fixes x :: "'a"   and y :: "'a"   and w :: "'a"   and z :: "'a" assumes "x \<le> y" shows "w \<cdot> x \<cdot> z \<le> w \<cdot> y \<cdot> z"
strict_arr_char': fixes f :: "'c local.term set" shows "\<F>\<^sub>SC.arr f = (arr f \<and> D f = f)"
beh2_def: fixes s1 :: "'f \<Rightarrow> ('d, 'f, 'g) F1"   and s2 :: "'g \<Rightarrow> ('e, 'f, 'g) F2"   and a :: "'g" shows "beh2 s1 s2 a = (\<Union>n. Lev2 s1 s2 n a, Lab2 s1 s2 a)"
PO_imp_refl_on: fixes cl :: "'b potype" assumes "PO cl" shows "refl_on (pset cl) (order cl)"
LL: fixes l :: "'b"   and r :: "'b" assumes "l \<in> S_Min_Lt"   and "r \<in> S_Max_Lt" shows "l < r"
no_edges_imp_all_nodes_initial: fixes V :: "node fset"   and \<Lambda> :: "node \<Rightarrow> assertion_gadget" shows "initials (Graph V \<Lambda> []) = V"
sort_leq_trans: fixes cs :: "(String.literal \<times> String.literal) set"   and x :: "String.literal set"   and y :: "String.literal set"   and z :: "String.literal set" assumes "wf_subclass_loc cs"   and "sort_leq cs x y"   and "sort_leq cs y z" shows "sort_leq cs x z"
Min_eqI: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a set"   and x :: "'a" assumes "class.linorder less_eq less"   and "finite A"   and "\<And>y. y \<in> A \<Longrightarrow> less_eq x y"   and "x \<in> A" shows "linorder.Min less_eq A = x"
list_sub_implies_member: fixes Z :: "'a set" shows "\<forall>a x. set (a # x) \<subseteq> Z \<longrightarrow> a \<in> Z"
elims_1: fixes x :: "('b \<times> nat) tree"   and y :: "bool" assumes "ltree x = y" obtains "x = \<langle>\<rangle>" and "y"   | l :: "('b \<times> nat) tree" and a :: "'b" and n :: "nat" and r :: "('b \<times> nat) tree" where "x = \<langle>l, (a, n), r\<rangle>" and "y = (min_height r \<le> min_height l \<and> n = min_height r + 1 \<and> ltree l \<and> ltree r)"
x_2: shows "(negatex \<circ> sqprojection \<circ> (\<lambda>w. (f \<circ> (\<lambda>x. x $ 1)) w - (g \<circ> (\<lambda>x. x $ 2)) w)) x = x"
closed_arc_image: fixes g :: "real \<Rightarrow> 'a" assumes "arc g" shows "closed (path_image g)"
to_gbg_alt_3: fixes m :: "'b" shows "g_V0 (to_gbg_ext m) = V0"
Gromov_converging_at_boundaryI: fixes a :: "'a"   and u :: "nat \<Rightarrow> 'a" assumes "\<And>M. \<exists>N. \<forall>n\<ge>N. \<forall>m\<ge>N. M \<le> Gromov_product_at a (u m) (u n)" shows "Gromov_converging_at_boundary u"
relAPP_def: fixes f :: "('d \<times> 'e) set \<Rightarrow> 'b"   and x :: "('d \<times> 'e) set" shows "\<langle>x\<rangle>f \<equiv> f x"
order_refl: fixes x :: "'c DBMEntry" shows "x \<preceq> x"
prefixToLevel_def: fixes l :: "nat"   and t :: "('a \<times> bool) list" shows "prefixToLevel l t \<equiv> prefixToLevel_aux t l 0"
shadow_root_eq_h3: fixes ptr' :: "(_) element_ptr"   and children :: "(_) shadow_root_ptr option" shows "h3 \<turnstile> get_shadow_root ptr' \<rightarrow>\<^sub>r children = h' \<turnstile> get_shadow_root ptr' \<rightarrow>\<^sub>r children"
assms_1: fixes x :: "'a"   and y :: "'a" shows "f x = f y \<longrightarrow> g x = g y"
ring_n_tOp_K_tOp: fixes n :: "nat"   and P :: "nat \<Rightarrow> ('a \<Rightarrow> ant) set"   and x :: "'a"   and y :: "'a" assumes "distinct_pds K n P"   and "x \<in> carrier (O\<^bsub>K P n\<^esub>)"   and "y \<in> carrier (O\<^bsub>K P n\<^esub>)" shows "x \<cdot>\<^sub>r\<^bsub>O\<^bsub>K P n\<^esub>\<^esub> y = x \<cdot>\<^sub>r y"
rel_envTE: fixes R :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and M :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and m :: "('e, 'g) envT"   and m' :: "('f, 'h) envT" assumes "local.rel_envT R M m m'" obtains f :: "'e \<Rightarrow> 'g" and g :: "'f \<Rightarrow> 'h" where "m = EnvT f" and "m' = EnvT g" and "(R ===> M) f g"
bisim: shows "\<forall>s1 s2. R s1 s2 \<longrightarrow> wf_state s1 \<and> wf_state s2 \<and> final s1 = final s2 \<and> (\<forall>x\<in>set (\<sigma> n). BNF_Greatest_Fixpoint.image2p post post R (post (delta x s1)) (post (delta x s2)))"
assoc_in_hom_1: fixes f :: "'b"   and g :: "'b"   and h :: "'b" assumes "D.ide f"   and "D.ide g"   and "D.ide h"   and "src\<^sub>D f = trg\<^sub>D g"   and "src\<^sub>D g = trg\<^sub>D h" shows "\<guillemotleft>\<a>\<^sub>D[f, g, h] : src\<^sub>D h \<rightarrow>\<^sub>D trg\<^sub>D f\<guillemotright>"
project_sets_def: fixes X :: "'c set"   and b :: "'c"   and y :: "real" shows "project_sets X b y = SPEC (\<lambda>ivl. X \<inter> plane b y \<subseteq> ivl \<and> ivl \<subseteq> plane b y)"
l4_18: 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"   and C' :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "A \<noteq> B"   and "Tarski_neutral_dimensionless.Col Bet A B C"   and "Cong A C A C'"   and "Cong B C B C'" shows "C = C'"
nonzero_neg_divide_eq_eq: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and b :: "'c"   and a :: "'c"   and c :: "'c" assumes "class.division_ring divide inverse one times plus zero minus uminus"   and "b \<noteq> zero" shows "(uminus (divide a b) = c) = (uminus a = times c b)"
transpose_def: fixes A :: "(('b, 'd) vec, 'c) vec" shows "Finite_Cartesian_Product.transpose A = (\<chi>i j. A $ j $ i)"
has_bochner_integral_nn_integral: fixes f :: "'a \<Rightarrow> real"   and M :: "'a measure"   and x :: "real" assumes "f \<in> borel_measurable M"   and "AE x in M. 0 \<le> f x"   and "0 \<le> x"   and "\<integral>\<^sup>+ x. ennreal (f x) \<partial>M = ennreal x" shows "has_bochner_integral M f x"
QR_decomposition_def: fixes A :: "(('d, 'e) vec, 'f) vec" shows "QR_decomposition A = (let Q = divide_by_norm (Gram_Schmidt_matrix A) in (Q, Finite_Cartesian_Product.transpose Q ** A))"
simps_2: fixes v1 :: "'d \<times> nat"   and e :: "('d \<times> nat) edge"   and es :: "('d \<times> nat) edge list"   and v2 :: "'d \<times> nat"   and subs :: "(('d \<times> nat) \<times> 'd \<times> nat) set" shows "SubRel.ces v1 (e # es) v2 subs = ((v1 = src e \<or> (v1, src e) \<in> subs\<^sup>+) \<and> SubRel.ces (tgt e) es v2 subs)"
cr_blinfun_def: shows "cr_blinfun \<equiv> \<lambda>x y. x = blinfun_apply y"
a: shows "a ` {0..<m} \<subseteq> carrier_vec n"
simps_3: fixes Pi :: "(('c, 'd) fmap \<times> ('c, 'd) fmap) set"   and s1 :: "('c, 'd) fmap"   and s2 :: "('c, 'd) fmap"   and rest :: "('c, 'd) fmap list" shows "valid_path Pi (s1 # s2 # rest) = (s1 \<in> valid_states Pi \<and> (\<exists>a. a \<in> Pi \<and> exec_plan s1 [a] = s2) \<and> valid_path Pi (s2 # rest))"
MakeNT_def: fixes \<eta> :: "('d, 'e, 'f, 'g, 'h, 'i) NatTrans" shows "MakeNT \<eta> \<equiv> \<lparr>NTDom = NTDom \<eta>, NTCod = NTCod \<eta>, NatTransMap = restrict (($$) \<eta>) (obj\<^bsub>NTCatDom \<eta>\<^esub>)\<rparr>"
combine_ncols: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'b"   and A :: "'c matrix"   and q :: "nat"   and B :: "'d matrix" assumes "f (0::'c) (0::'d) = (0::'b)"   and "ncols A \<le> q"   and "ncols B \<le> q" shows "ncols (combine_matrix f A B) \<le> q"
bisim1CallParams: fixes P :: "'a prog"   and h :: "'heap"   and ps :: "'addr expr1 list"   and ps' :: "'addr expr1 list"   and xs :: "'addr val list"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and xcp :: "'addr option"   and obj :: "'addr expr1"   and M :: "String.literal"   and v :: "'addr val" assumes "P,ps,h \<turnstile> (ps', xs) [\<leftrightarrow>] (stk, loc, pc, xcp)" shows "P,obj\<bullet>M(ps),h \<turnstile> (Val v\<bullet>M(ps'), xs) \<leftrightarrow> (stk @ [v], loc, length (compE2 obj) + pc, xcp)"
continuous_on_eval_fps: fixes f :: "'a fps" shows "continuous_on (eball (0::'a) (fps_conv_radius f)) (eval_fps f)"
w\<^sub>f'_def: shows "w\<^sub>f' = f\<^sub>0gh\<^sub>1.p\<^sub>1 \<star> TTfgh_TfTgh.chine"
rbt_delete_def: fixes less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and k :: "'b"   and t :: "('b, 'c) rbt" shows "ord.rbt_delete less k t = paint color.B (ord.rbt_del less k t)"
dom_cod: fixes f :: "('c \<Rightarrow> 's, 'c \<Rightarrow> 's) concrete_category.arr" assumes "Cop_S.arr f" shows "Cop_S.dom (Cop_S.cod f) = Cop_S.cod f"
map_simp_ide: fixes a :: "'c" assumes "Cop.ide a" shows "\<Phi>\<Psi>'.map a = \<Psi>'.map a \<cdot>\<^sub>S \<Phi> a"
pgen_lossless_gpv_parametric: shows "rel_fun (=) (rel_fun (rel_\<I> (BNF_Def.Grp UNIV id) (=)) (rel_fun (rel_gpv (BNF_Def.Grp UNIV f) (BNF_Def.Grp UNIV id)) (=))) pgen_lossless_gpv pgen_lossless_gpv"
nres_of_nress_impl_map: fixes f :: "'d \<Rightarrow> 'c nres"   and x :: "'d list" shows "nres_of_nress_impl (map f x) = rec_list (RETURN []) (\<lambda>x xs r. f x \<bind> (\<lambda>fx. r \<bind> (\<lambda>r. RETURN (fx # r)))) x"
bounded_iff: fixes b :: "'b"   and c :: "'b"   and a :: "'b" shows "(b + c \<le> a) = (b \<le> a \<and> c \<le> a)"