hcju/isabelleps huggingface dataset:
refine: fixes n :: "nat"   and Xi :: "real list \<times> real list"   and X :: "'b set"   and Mi :: "nat"   and M :: "nat" assumes "DIM_precond TYPE('b) n"   and "(Xi, X) \<in> lvivl_rel"   and "(Mi, M) \<in> nat_rel" shows "(nres_of (split_spec_exact_ivl_rel n Mi Xi), split_spec_ivl M X) \<in> \<langle>lvivl_rel \<times>\<^sub>r lvivl_rel\<rangle>nres_rel"
select_trap_def: fixes uu :: "unit" shows "select_trap uu \<equiv> do traps \<leftarrow> gets get_trap_set; rt_val \<leftarrow> gets reset_trap_val; psr_val \<leftarrow> gets (cpu_reg_val PSR); et_val \<leftarrow> gets (\<lambda>s. get_ET psr_val); modify emp_trap_set; if rt_val = True then return () else if et_val = 0 then do set_err_mode True; set_exe_mode False; fail () od else if data_store_error \<in> traps then do write_cpu_tt 43; return () od else if instruction_access_error \<in> traps then do write_cpu_tt 33; return () od else if r_register_access_error \<in> traps then do write_cpu_tt 32; return () od else if instruction_access_exception \<in> traps then do write_cpu_tt 1; return () od else if privileged_instruction \<in> traps then do write_cpu_tt 3; return () od else if illegal_instruction \<in> traps then do write_cpu_tt 2; return () od else if fp_disabled \<in> traps then do write_cpu_tt 4; return () od else if cp_disabled \<in> traps then do write_cpu_tt 36; return () od else if unimplemented_FLUSH \<in> traps then do write_cpu_tt 37; return () od else if window_overflow \<in> traps then do write_cpu_tt 5; return () od else if window_underflow \<in> traps then do write_cpu_tt 6; return () od else if mem_address_not_aligned \<in> traps then do write_cpu_tt 7; return () od else if fp_exception \<in> traps then do write_cpu_tt 8; return () od else if cp_exception \<in> traps then do write_cpu_tt 40; return () od else if data_access_error \<in> traps then do write_cpu_tt 41; return () od else if data_access_exception \<in> traps then do write_cpu_tt 9; return () od else if tag_overflow \<in> traps then do write_cpu_tt 10; return () od else if division_by_zero \<in> traps then do write_cpu_tt 42; return () od else if trap_instruction \<in> traps then do ticc_trap_type \<leftarrow> gets ticc_trap_type_val; write_cpu_tt (word_cat 1 ticc_trap_type); return () od else return () od"
var_def2: shows "B = Fraction_Field.Fract 1 x"
eq_iff: fixes a :: "'d"   and b :: "'d" shows "(a = b) = (b \<le> a \<and> a \<le> b)"
pp_of_term_def: fixes v :: "'a \<times> 'b" shows "pprod.pp_of_term v = fst v"
valid_atomics: fixes t :: "'a tree"   and i :: "nat"   and j :: "nat" assumes "t \<in> set (local.atomics i j)"   and "0 < i"   and "j \<le> length as" shows "local.valid t"
A93_NotSourceSet_level1: shows "\<forall>C\<in>AbstrLevel level1. sA93 \<notin> Sources level1 C"
parent_child_rel_impl: shows "parent_child_rel = local.CD.a_parent_child_rel"
assms_2: shows "gauss_int_norm z = p\<^sup>2"
add_root: fixes z :: "'b list"   and w :: "'b list" shows "z \<cdot> w \<in> z* = w \<in> z*"
cinner_ket_left: fixes i :: "'b"   and \<psi> :: "'b ell2" shows "ket i \<bullet>\<^sub>C \<psi> = Rep_ell2 \<psi> i"
open_left: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and "open" :: "'a set \<Rightarrow> bool"   and S :: "'a set"   and x :: "'a"   and y :: "'a" assumes "class.linorder_topology less_eq less open"   and "open S"   and "x \<in> S"   and "less y x" shows "\<exists>b. less b x \<and> ord.greaterThanAtMost less_eq less b x \<subseteq> S"
AssumeH_7: fixes A1 :: "fm"   and A2 :: "fm"   and A3 :: "fm"   and A4 :: "fm"   and A5 :: "fm"   and A6 :: "fm"   and A7 :: "fm"   and H :: "fm set" shows "insert A1 (insert A2 (insert A3 (insert A4 (insert A5 (insert A6 (insert A7 H)))))) \<turnstile> A7"
extend_Join_eq_extend_D: fixes F :: "'a program"   and G :: "'c program"   and H :: "'a program" assumes "extend h F \<squnion> G = extend h H" shows "H = F \<squnion> project h UNIV G"
nearly_healthy_def: fixes t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" shows "nearly_healthy t = ((\<forall>P. unitary P \<longrightarrow> unitary (t P)) \<and> (\<forall>P Q. unitary P \<longrightarrow> unitary Q \<longrightarrow> P \<tturnstile> Q \<longrightarrow> t P \<tturnstile> t Q))"
linord_of_list_linord_of_listP: fixes xy :: "'d \<times> 'd"   and xs :: "'d list" shows "(xy \<in> linord_of_list xs) = linord_of_listP (fst xy) (snd xy) xs"
homotopic_loops_refl: fixes s :: "'b set"   and p :: "real \<Rightarrow> 'b" shows "homotopic_loops s p p = (path p \<and> path_image p \<subseteq> s \<and> pathfinish p = pathstart p)"
wf_syscls_def: fixes P :: "'a prog" shows "wf_syscls P \<equiv> {Object} \<union> sys_xcpts \<subseteq> set (map fst P)"
simple_split_2: fixes S :: "('c, 'd) strand_step list"   and S' :: "('c, 'd) strand_step list" assumes "simple (S @ S')" shows "simple S'"
bind_eq_return_pmf_None: fixes p :: "'e spmf"   and f :: "'e \<Rightarrow> 'd spmf" shows "(p \<bind> f = return_pmf None) = (\<forall>x\<in>set_spmf p. f x = return_pmf None)"
less_hf_def: fixes A :: "hf"   and B :: "hf" shows "(A < B) = (A \<le> B \<and> A \<noteq> B)"
funpow_in_orbit: fixes s :: "'b"   and f :: "'b \<Rightarrow> 'b"   and t :: "'b"   and n :: "nat" assumes "s \<in> orbit f t" shows "(f ^^ n) s \<in> orbit f t"
dsumM_def: fixes R :: "('j, 'k) Ring_scheme"   and I :: "'h set"   and A :: "'h \<Rightarrow> ('i, 'j, 'l) Module_scheme" shows "\<^bsub>R\<^esub>\<Sigma>\<^sub>d\<^bsub>I\<^esub> A = \<lparr>carrier = carr_dsumag I A, pop = prod_pOp I A, mop = prod_mOp I A, zero = prod_zero I A, sprod = prodM_sprod R I A\<rparr>"
Cons_1: fixes s :: "'a" assumes "(s, ts) \<in> seq ars" shows "(s, map (Pair True) ts) \<in> conv ars \<and> lst_conv (s, map (Pair True) ts) = lst (s, ts) \<and> labels (s, ts) = labels_conv (s, map (Pair True) ts)"
stable_constrains_stable: fixes F :: "'b program"   and C :: "'b set"   and A :: "'b set" assumes "F \<in> stable C"   and "F \<in> C \<inter> A co A" shows "F \<in> stable (C \<inter> A)"
io_diff: shows "io \<in> {io \<in> L M1. io \<notin> L M2}"
find_lambda: shows "\<lbrace>\<lambda>s. True\<rbrace> find_lambda \<lbrace>\<lambda>s. l s = lambda\<rbrace>"
finite_nonzero_count: fixes M :: "'a multiset" shows "finite {t. 0 < count M t}"
ih_2: assumes "n \<noteq> 0"   and "M \<in># hmsetmset (local.encode (e + 1) (n div base))" shows "\<exists>e'\<ge>e + 1. M = local.encode 0 e'"
ide_mkIde_finite: fixes A :: "'s set" assumes "A \<subseteq> Univ"   and "finite A" shows "ide (mkIde A)"
wens_Union_inductive_step: fixes T :: "'b set"   and B :: "'b set"   and F :: "'b program"   and G :: "'b program"   and X :: "'b set"   and act :: "('b \<times> 'b) set"   and Y :: "'b set" assumes "T - B \<subseteq> awp F T"   and "\<And>X. X \<in> wens_set F B \<Longrightarrow> T \<inter> X - B \<subseteq> awp G (T \<inter> X)"   and "X \<in> wens_set F B"   and "act \<in> Acts F"   and "Y \<subseteq> X"   and "T \<inter> X = T \<inter> Y" shows "wens (F \<squnion> G) act Y \<subseteq> wens F act X \<and> T \<inter> wens F act X = T \<inter> wens (F \<squnion> G) act Y"
split_i_j_x_def: fixes i :: "nat"   and j :: "nat"   and x :: "(nat, 'a) mapping" shows "split_i_j_x i j x = (vec i \<langle>x\<rangle>, vec (j - i) (\<lambda>y. \<langle>x\<rangle> (y + i)))"
map_def: fixes gf :: "'c \<times> 'c" shows "local.map gf = (if CopxC.arr gf then S.mkArr (local.set (CopxC.dom gf)) (local.set (CopxC.cod gf)) (\<phi> (CopxC.cod gf) \<circ> (\<lambda>h. snd gf \<cdot> h \<cdot> fst gf) \<circ> \<psi> (CopxC.dom gf)) else S.null)"
sym_Un: fixes r :: "('a \<times> 'a) set"   and s :: "('a \<times> 'a) set" assumes "sym r"   and "sym s" shows "sym (r \<union> s)"
split: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "linear_poly \<Rightarrow> 'b \<Rightarrow> 'a"   and f2 :: "linear_poly \<Rightarrow> 'b \<Rightarrow> 'a"   and ns_constraint :: "'b ns_constraint" shows "P (case ns_constraint of LEQ_ns x xa \<Rightarrow> f1 x xa | GEQ_ns x xa \<Rightarrow> f2 x xa) = ((\<forall>x11 x12. ns_constraint = LEQ_ns x11 x12 \<longrightarrow> P (f1 x11 x12)) \<and> (\<forall>x21 x22. ns_constraint = GEQ_ns x21 x22 \<longrightarrow> P (f2 x21 x22)))"
snapshot_state_unchanged: 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"   and p :: "'d" assumes "distributed_system channel"   and "distributed_system.next channel trans send recv c ev c'"   and "ps c p \<noteq> None" shows "ps c p = ps c' p"
ucast_up_ucast_id: fixes w :: "'b word" assumes "is_up UCAST('b \<rightarrow> 'c)" shows "UCAST('c \<rightarrow> 'b) (UCAST('b \<rightarrow> 'c) w) = w"
lborel_f_def: fixes n :: "nat" shows "lborel_f n = Pi\<^sub>M {..<n} (\<lambda>b. lborel)"
icdf0: shows "\<nexists>i'. f 0 cd\<^bsup>\<pi>'\<^esup>\<rightarrow> i'"
assms: shows "dbm_entry_val u None (Some c) (min a b)"
simps_4: fixes w :: "nat"   and t\<^sub>1 :: "'a tree"   and v :: "nat"   and va :: "'a tree"   and vb :: "'a tree"   and a :: "'a" shows "sibling (Node w t\<^sub>1 (Node v va vb)) a = (if a \<in> alphabet t\<^sub>1 then sibling t\<^sub>1 a else if a \<in> alphabet (Node v va vb) then sibling (Node v va vb) a else a)"
simps_1: fixes z :: "nat"   and i :: "nat"   and j :: "nat" shows "local.prefixes z i j 0 = [i]"
list_add_lo: fixes t :: "'b tree"   and a :: "'b" assumes "braun t" shows "list (add_lo a t) = a # list t"
sel_stable: fixes \<rho> :: "'s"   and C :: "'a literal multiset" assumes "is_renaming \<rho>" shows "S (C \<cdot> \<rho>) = S C \<cdot> \<rho>"
modif: shows "\<forall>Z. \<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>(P' Z) c (Q' Z),(A' Z)"
iD: fixes i :: "nat" assumes "i < DIM('a)" shows "isDERIV (xs ! i) fa vs"
relcompI: fixes r :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and a :: "'g"   and b :: "'h"   and s :: "'h \<Rightarrow> 'i \<Rightarrow> bool"   and c :: "'i" assumes "r a b"   and "s b c" shows "(r OO s) a c"
simps_1: fixes \<sigma> :: "nat \<Rightarrow> val"   and x :: "nat" shows "expr_sem \<sigma> (Var x) = return_val (\<sigma> x)"
vconn_implies_hpar_3: fixes \<mu> :: "'c"   and f :: "'c"   and g :: "'c" assumes "\<guillemotleft>\<mu> : f \<rightarrow>\<^sub>C g\<guillemotright>" shows "src\<^sub>C g = src\<^sub>C f"
elem: fixes x :: "'c"   and xs :: "'c list" shows "ListMem x (x # xs)"
poly_mult_inv: fixes p :: "('c monom \<times> 'd) list"   and q :: "('c monom \<times> 'd) list" assumes "poly_inv p"   and "poly_inv q" shows "poly_inv (poly_mult p q)"
1: fixes i :: "nat" assumes "enat i < llength w" shows "(w \<bar>\<bar> iterates Suc 0) ?! i = (w ?! i, i)"
simple_match_to_of_match_single_def: fixes m :: "(32, 'b) simple_match_scheme"   and iif :: "char list option"   and prot :: "protocol"   and sport :: "(16 word \<times> 16 word) option"   and dport :: "(16 word \<times> 16 word) option" shows "simple_match_to_of_match_single m iif prot sport dport \<equiv> uncurry L4Src ` option2set sport \<union> uncurry L4Dst ` option2set dport \<union> IPv4Proto ` (case prot of ProtoAny \<Rightarrow> {} | Proto p \<Rightarrow> {p}) \<union> IngressPort ` option2set iif \<union> IPv4Src ` option2set (toprefixmatch (src m)) \<union> IPv4Dst ` option2set (toprefixmatch (dst m)) \<union> {EtherType 2048}"
indets_times_subset: fixes p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and q :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" shows "indets (p * q) \<subseteq> indets p \<union> indets q"
cases: fixes a1 :: "value"   and a2 :: "value" assumes "veq_structure a1 a2" obtains uu_ :: "(Term_Class.term \<times> sterm) list" and uv_ :: "(name, value) fmap" and uw_ :: "(Term_Class.term \<times> sterm) list" and ux_ :: "(name, value) fmap" where "a1 = Vabs uu_ uv_" and "a2 = Vabs uw_ ux_"   | uy_ :: "(name, (Term_Class.term \<times> sterm) list) fmap" and uz_ :: "name" and va_ :: "(name, value) fmap" and vb_ :: "(name, (Term_Class.term \<times> sterm) list) fmap" and vc_ :: "name" and vd_ :: "(name, value) fmap" where "a1 = Vrecabs uy_ uz_ va_" and "a2 = Vrecabs vb_ vc_ vd_"   | ts :: "value list" and us :: "value list" and name :: "name" where "a1 = Vconstr name ts" and "a2 = Vconstr name us" and "list_all2 veq_structure ts us"
\<Delta>'''_sel_4: shows "sink \<Delta>'''.\<Delta>''' = sink \<Delta>'''"
a: shows "x:M \<in> BINDINGn B (\<parallel><B>\<parallel>)"
trm_simps_2: fixes f :: "'a" assumes "arr f" shows "local.dom \<t>[f] = local.dom f"
reachable_awalk: fixes u :: "'a"   and v :: "'a" shows "(u \<rightarrow>\<^sup>*\<^bsub>G\<^esub> v) = (\<exists>p. G.awalk u p v)"
lt_ord_p: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" assumes "lt p \<prec>\<^sub>t lt q" shows "p \<prec>\<^sub>p q"
direct_decomp_singleton: fixes A :: "'a set" shows "direct_decomp A [A]"
clock_numbering'_1: shows "\<forall>c\<in>clk_set A. 0 < v c"
add_mset_add_single: fixes a :: "'b"   and A :: "'b multiset" shows "a, A = A + (a, {#})"
group_l_invI: assumes "\<And>x. x \<in> carrier tetrahedral_group \<Longrightarrow> \<exists>y\<in>carrier tetrahedral_group. y \<otimes>\<^bsub>tetrahedral_group\<^esub> x = \<one>\<^bsub>tetrahedral_group\<^esub>" shows "Group.group tetrahedral_group"
shared_weight_net_Pool: fixes m1 :: "real mat convnet"   and m2 :: "real mat convnet" assumes "shared_weight_net m1"   and "shared_weight_net m2"   and "count_weights True (remove_weights m1) = count_weights True (remove_weights m2)"   and "\<And>x. x < count_weights True (remove_weights m1) \<Longrightarrow> extract_weights True m1 x = extract_weights True m2 x" shows "shared_weight_net (Pool m1 m2)"
zero_neq_neg_one: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a" assumes "class.ring_char_0 one times plus zero minus uminus" shows "zero \<noteq> uminus one"
swapBinp_def2: fixes xs :: "'varSort"   and x1 :: "'var"   and x2 :: "'var"   and binp :: "'e \<Rightarrow> ('f, 'g, 'varSort, 'var, 'h) qAbs set option" shows "binp %%[x1 \<and> x2]_xs = asBinp (qSwapBinp xs x1 x2 (pickBinp binp))"
leq_min_inf: fixes m :: "nat"   and n :: "nat" shows "m \<le> n \<longrightarrow> Min {m..n} = m"
restrict_top_bf: fixes i :: "'a ifex"   and "is" :: "'a ifex list"   and vr :: "'a"   and fi :: "('a \<Rightarrow> bool) \<Rightarrow> bool"   and vl :: "bool"   and ass :: "'a \<Rightarrow> bool" assumes "i \<in> set is"   and "lowest_tops is = Some vr"   and "ifex_ordered i"   and "\<And>ass. fi ass = val_ifex i ass" shows "val_ifex (restrict_top i vr vl) ass = bf_restrict vr vl fi ass"
extended_fin_sat: fixes S :: "'b formula set" assumes "fin_sat S" shows "fin_sat (extended S)"
IH: assumes "\<forall>a\<in>set l. case a of (v, a) \<Rightarrow> \<forall>b\<in>set l. case b of (v', a') \<Rightarrow> v \<noteq> v' \<or> a = a'"   and "(v, a) \<in> set l" shows "map_of l v = Some a"
assms: shows "H \<turnstile> x EQ HPair (HPair Zero (HPair Zero Zero)) y"
finite_imp_compact_convex_hull: fixes S :: "'a set" assumes "finite S" shows "compact (convex hull S)"
take_bit_not_eq_mask_diff: fixes 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 divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bit :: "'b \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and drop_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and take_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and "and" :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and or :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and xor :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and mask :: "nat \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and not :: "'b \<Rightarrow> 'b"   and n :: "nat"   and a :: "'b" assumes "class.ring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask uminus not" shows "take_bit n (not a) = minus (mask n) (take_bit n a)"
lnull_prepend: fixes xs :: "'b list"   and ys :: "'b llist" shows "lnull (prepend xs ys) = (xs = [] \<and> lnull ys)"
brI: fixes a :: "'c"   and \<alpha> :: "'d \<Rightarrow> 'c"   and c :: "'d"   and I :: "'d \<Rightarrow> bool" assumes "a = \<alpha> c"   and "I c" shows "(c, a) \<in> br \<alpha> I"
fps_conv_radius_deriv: fixes f :: "'a fps" shows "fps_conv_radius f \<le> fps_conv_radius (fps_deriv f)"
empty_iff: fixes c :: "'a" shows "(c \<in> {}) = False"
\<A>1_def: shows "\<A>1 = {A \<in> Pow X. card A < k} \<times> X"
sum_inverse_squares_expansion_bigo_simple: shows "(\<lambda>n. (\<Sum>i = 1..n. 1 / (real i)\<^sup>2) - (pi\<^sup>2 / 6 - 1 / real n + 1 / (2 * (real n)\<^sup>2))) \<in> O(\<lambda>x. 1 / real x ^ 3)"
IGRep_def: shows "IGRep = IrrFinGroupRepresentation G fgsmult"
quotient_of_unique: fixes r :: "rat" shows "\<exists>!p. r = Fract (fst p) (snd p) \<and> 0 < snd p \<and> coprime (fst p) (snd p)"
snd_spp_of: fixes r :: "'t \<Rightarrow>\<^sub>0 'b" shows "snd (spp_of r) = rep_list r"
float_defs_3: fixes a :: "('a, 'b) IEEE.float" shows "is_zero a = (IEEE.exponent a = 0 \<and> fraction a = 0)"
map_of_remove_filtered_SomeD: fixes fs :: "(('d \<times> 'e) \<times> 'b \<times> 'c) list"   and F :: "'d"   and C :: "'e"   and a :: "'b"   and T :: "'c"   and Q :: "('d \<times> 'e) \<times> 'b \<times> 'c \<Rightarrow> bool"   and P :: "'c \<Rightarrow> 'f" assumes "map_of fs (F, C) = \<lfloor>(a, T)\<rfloor>"   and "Q ((F, C), a, T)" shows "map_of (map (\<lambda>((F, D), b, T). (F, P T)) (filter (\<lambda>((F, D), b, T). Q ((F, D), b, T) \<and> D = C) fs)) F = \<lfloor>P T\<rfloor>"
assms_1: fixes N :: "nat" shows "f N \<le> g N"
hyps: fixes v\<^sub>1 :: "'item list"   and v\<^sub>2 :: "'item list" assumes "u @ v\<^sub>1 =\<^sub>F u @ v\<^sub>2" shows "v\<^sub>1 =\<^sub>F v\<^sub>2"
asym_less: fixes cmp :: "'b \<Rightarrow> 'b \<Rightarrow> comp"   and a :: "'b"   and b :: "'b" assumes "comparator cmp"   and "cmp a b = Less" shows "cmp b a \<noteq> Less"
quotient_map_snd: fixes X :: "'d topology"   and Y :: "'e topology" shows "quotient_map (prod_topology X Y) Y snd = (topspace X = {} \<longrightarrow> topspace Y = {})"
mu_square: fixes f :: "'a \<Rightarrow> 'a" assumes "isotone f"   and "has_least_fixpoint f"   and "has_least_fixpoint (f \<circ> f)" shows "\<mu> f = \<mu> (f \<circ> f)"
Gromov_extension_inside_space: fixes f :: "'d \<Rightarrow> 'c"   and x :: "'d" shows "Gromov_extension f (to_Gromov_completion x) = to_Gromov_completion (f x)"
trace_eq'_resource_of_oracle: fixes A :: "'a set"   and callee1 :: "'c \<Rightarrow> 'a \<Rightarrow> ('b \<times> 'c) spmf"   and s1 :: "'c"   and callee2 :: "'d \<Rightarrow> 'a \<Rightarrow> ('b \<times> 'd) spmf"   and s2 :: "'d" shows "A \<turnstile>\<^sub>R RES callee1 s1 \<approx> RES callee2 s2 = A \<turnstile>\<^sub>C callee1(s1) \<approx> callee2(s2)"
mark_guards_simps_6: fixes f :: "'f"   and b :: "'d set"   and c :: "('d, 'e, 'g) com" shows "mark_guards f (While b c) = While b (mark_guards f c)"
f2r_uncurry: fixes f :: "'a \<Rightarrow> 'b set" shows "\<R> f = (Collect \<circ> case_prod) (s2p \<circ> f)"
1: fixes \<Gamma>' :: "'a formula list" assumes "set \<Gamma>' \<subseteq> \<Gamma> \<and> ({} \<turnstile> \<^bold>\<And>\<Gamma>' \<^bold>\<rightarrow> F)" shows "thesis"
disj_simps_1: fixes Q :: "fm" shows "Ferrack.disj T Q = T"
prems_3: shows "edom ae \<subseteq> domA \<Gamma> \<union> upds S"
simps_2: fixes xs :: "char list list"   and n :: "int" shows "merge_coeffs0 [(xs, n)] = (if n = 0 then [] else [(xs, n)])"
cis_pi_half: shows "cis (pi / 2) = \<i>"
l_get_shadow_root_safe\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and get_shadow_root_safe :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, (_) shadow_root_ptr option) prog"   and get_shadow_root_safe_locs :: "(_) element_ptr \<Rightarrow> (_) shadow_root_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_shadow_root :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, (_) shadow_root_ptr option) prog"   and get_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_mode :: "(_) shadow_root_ptr \<Rightarrow> ((_) heap, exception, shadow_root_mode) prog"   and get_mode_locs :: "(_) shadow_root_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_get_shadow_root_safe\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms type_wf get_shadow_root_safe get_shadow_root_safe_locs get_shadow_root get_shadow_root_locs get_mode get_mode_locs \<equiv> type_wf = ShadowRootClass.type_wf \<and> get_shadow_root_safe = l_get_shadow_root_safe\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_shadow_root_safe get_shadow_root get_mode \<and> get_shadow_root_safe_locs = l_get_shadow_root_safe\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_shadow_root_safe_locs get_shadow_root_locs get_mode_locs"
good_freeCons_3: fixes delta :: "'f"   and inp :: "'d \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm set option"   and binp :: "'e \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qAbs set option"   and xs :: "'varSort"   and x :: "'var" shows "Op delta inp binp \<noteq> Var xs x"
foldl_spmf_helper2: fixes p :: "'e \<Rightarrow> 'c"   and f :: "'c \<Rightarrow> 'd \<Rightarrow> 'e"   and q :: "'e \<Rightarrow> 'd"   and g :: "'d \<Rightarrow> 'f \<Rightarrow> 'd spmf"   and acc :: "'e spmf"   and es :: "'f list" assumes "\<And>x y. p (f x y) = x"   and "\<And>x y. q (f x y) = y"   and "\<And>x. f (p x) (q x) = x" shows "foldl_spmf (\<lambda>a e. map_spmf (f (p a)) (g (q a) e)) acc es = acc \<bind> (\<lambda>acc'. map_spmf (f (p acc')) (foldl_spmf g (return_spmf (q acc')) es))"
getRel_hom_map: fixes G :: "('b, 'c) labeled_graph"   and l :: "'b"   and f :: "'c \<Rightarrow> 'd" assumes "graph G" shows "getRel l (map_graph_fn G f) = map_prod f f ` getRel l G"
VDM_lowerm: fixes n :: "nat"   and c :: "IMP"   and A :: "(Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool"   and m :: "nat" assumes " \<Turnstile>\<^sub>n c : A "   and "m \<le> n" shows " \<Turnstile>\<^sub>m c : A "
InvCompIso: fixes C :: "('a, 'b, 'c) Category_scheme"   and f :: "'b" assumes "Category C"   and "ciso\<^bsub>C\<^esub> f" shows "Cinv\<^bsub>C\<^esub> f \<approx>>\<^bsub>C\<^esub> f"
verit_comp_simplify1_1: fixes a :: "'c" shows "(a < a) = False"
set_in_convex_hull_list: fixes Vs :: "'a vec list" assumes "set Vs \<subseteq> carrier_vec n" shows "set Vs \<subseteq> convex_hull_list Vs"
heap_is_wellformed_def: shows "Shadow_DOM.heap_is_wellformed \<equiv> l_heap_is_wellformed\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_heap_is_wellformed Shadow_DOM.get_child_nodes get_disconnected_nodes get_shadow_root get_tag_name"
underS_empty: fixes a :: "'a"   and r :: "('a \<times> 'a) set" assumes "a \<notin> Field r" shows "underS r a = {}"
p_strict_antimono: fixes a :: "real"   and v :: "real"   and x :: "real"   and y :: "real"   and s0 :: "real" assumes "braking_movement a v"   and "x < y"   and "movement.t_stop a v \<le> x" shows "movement.p a v s0 y < movement.p a v s0 x"
extremum_uniqueI: fixes a :: "'b list" assumes "suffix a []" shows "a = []"
is_associated_string_def: fixes M :: "('a, 'b) message set"   and s :: "('a, 'b option) insert_message list" shows "is_associated_string M s \<equiv> consistent M \<and> set s = to_woot_character M ` insert_messages M \<and> (\<forall>a. a_conditions (insert_messages M) a \<longrightarrow> sorted_wrt (<) (map a (ext_ids s)))"