hcju/isabelleps huggingface dataset:
sub_less_def: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and top :: "'b"   and x :: "'b"   and y :: "'b" assumes "class.subset_extended minus uminus inf sup bot less_eq less top" shows "less (uminus x) (uminus y) = (sup (uminus x) (uminus y) = uminus y \<and> sup (uminus y) (uminus x) \<noteq> uminus x)"
eq_eqvt: fixes pi :: "(atom \<times> atom) list"   and x :: "'a"   and y :: "'a" shows "pi \<bullet> (x = y) = (pi \<bullet> x = pi \<bullet> y)"
assms_3: shows "z \<cdot> a' \<le> a \<cdot> z"
length_append: fixes xs :: "'b list"   and ys :: "'b list" shows "\<^bold>|xs \<cdot> ys\<^bold>| = \<^bold>|xs\<^bold>| + \<^bold>|ys\<^bold>|"
lsb0: fixes x :: "'b word"   and y :: "'b word" shows "(x OR y) !! 0 = (x !! 0 \<or> y !! 0) \<and> (x AND y) !! 0 = (x !! 0 \<and> y !! 0) \<and> (x XOR y) !! 0 = (x !! 0 \<noteq> y !! 0) \<and> NOT x !! 0 = (\<not> x !! 0)"
simps: fixes I :: "\<I>"   and dt :: "nat"   and t :: "nat"   and a :: "'a"   and b :: "'b" shows "check_before I dt (t, a, b) = (enat t + right I < enat dt)"
sets_measure_pmf: fixes p :: "'d pmf" shows "measure_pmf.events p = UNIV"
multiplicity_diff_pow: fixes p :: "nat"   and x :: "int"   and y :: "int"   and n :: "nat" assumes "prime p"   and "int p dvd x - y"   and "\<not> int p dvd x"   and "\<not> int p dvd y"   and "2 < p"   and "x \<noteq> y"   and "0 < n" shows "multiplicity (int p) (x ^ n - y ^ n) = multiplicity (int p) (x - y) + multiplicity p n"
dirichlet_prod_altdef1: fixes f :: "nat \<Rightarrow> 'b"   and g :: "nat \<Rightarrow> 'b" shows "dirichlet_prod f g = (\<lambda>n. \<Sum>d | d dvd n. f (n div d) * g d)"
fst: fixes m :: "nat" assumes "\<And>m. m \<le> Suc n \<Longrightarrow> geom_proc m x = geom_proc m y"   and "m \<le> n" shows "x !! m = y !! m"
diagonal_to_Smith_aux_PQ: fixes P :: "(('a, 'c) vec, 'c) vec"   and A :: "(('a, 'b) vec, 'c) vec"   and Q :: "(('a, 'b) vec, 'b) vec"   and B :: "(('a, 'b) vec, 'c) vec"   and bezout :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a \<times> 'a \<times> 'a"   and P' :: "(('a, 'c) vec, 'c) vec"   and B' :: "(('a, 'b) vec, 'c) vec"   and Q' :: "(('a, 'b) vec, 'b) vec"   and k :: "nat" assumes "P ** A ** Q = B"   and "invertible P"   and "invertible Q"   and "is_bezout_ext bezout"   and "(P', B', Q') = diagonal_to_Smith_aux_PQ [0..<k] bezout (P, B, Q)"   and "isDiagonal B"   and "k < min (nrows A) (ncols A)" shows "B' = P' ** A ** Q' \<and> invertible P' \<and> invertible Q' \<and> B' = diagonal_to_Smith_aux B [0..<k] bezout"
top_left_normalize: fixes a :: "'b" shows "gcd (0::'b) a = normalize a"
p_natpow_prod_Suc_1: fixes m :: "nat" shows "[p] \<cdot> \<one> \<otimes> [p] \<cdot> \<one> [^] m = [p] \<cdot> \<one> [^] Suc m"
CX3h_substitutes: fixes h :: "H1" shows "substitutes (CX3h h)"
list_all2_eq: fixes xs :: "'b list"   and ys :: "'b list" shows "(xs = ys) = list_all2 (=) xs ys"
inv_comp_left_1: fixes g :: "'a"   and f :: "'a" assumes "local.iso (g \<cdot> f)"   and "local.iso g" shows "local.inv (g \<cdot> f) = local.inv f \<cdot> local.inv g"
mC_ZOC_def: fixes cval :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and indis :: "('b \<times> 'b) set"   and aval :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and tval :: "'d \<Rightarrow> 'b \<Rightarrow> bool"   and theta :: "(('d, 'c, 'a) cmd \<times> ('d, 'c, 'a) cmd) set"   and c :: "('d, 'c, 'a) cmd"   and d :: "('d, 'c, 'a) cmd"   and s :: "'b"   and t :: "'b"   and I0 :: "nat set"   and P :: "nat set set"   and F :: "nat set \<Rightarrow> nat set" assumes "PL_Indis cval indis" shows "PL_Indis.mC_ZOC aval tval cval indis theta c d s t I0 P F \<equiv> PL_Indis.mC_ZOC_part c d s t I0 P F \<and> inj_on F P \<and> PL_Indis.mC_ZOC_wt aval tval cval c d s t I0 P F \<and> PL_Indis.mC_ZOC_eff_cont0 aval tval cval indis theta c d s t I0 F \<and> PL_Indis.mC_ZOC_eff_cont aval tval cval indis theta c d s t I0 P F"
fresh_ssubst: fixes V :: "name set"   and a :: "atom"   and t :: "tm" assumes "finite V"   and "a \<sharp> p \<bullet> V"   and "a \<sharp> t" shows "a \<sharp> ssubst t V F"
z_s: shows "z < x in fst (TSdet init h (as @ [x] @ bs @ [x] @ cs) (length (as @ [x] @ bs @ [x]) + i))"
rtranclp_eq_rtrancl_tab_nil: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" shows "r\<^sup>*\<^sup>* x y = rtrancl_tab r [] x y"
abc_event: shows "\<forall>\<^sub>F n in sequentially. B * b n = c n * a n - c (n + 1) \<and> \<bar>real_of_int (c (n + 1))\<bar> < real_of_int (a n) / 2"
continuous_map_eq_image_closure_subset_gen: fixes X :: "'a topology"   and Y :: "'b topology"   and f :: "'a \<Rightarrow> 'b" shows "continuous_map X Y f = (f ` topspace X \<subseteq> topspace Y \<and> (\<forall>S. f ` (X closure_of S) \<subseteq> Y closure_of f ` S))"
simps: fixes a :: "'a set \<Rightarrow> 'a set" shows "IK_nf a = (a = id \<or> a = I \<or> a = K \<or> a = I \<circ> K \<or> a = K \<circ> I \<or> a = I \<circ> K \<circ> I \<or> a = K \<circ> I \<circ> K)"
is_standard_operator_rem_implicit: fixes n :: "char list"   and p :: "(nat \<times> nat) list"   and vp :: "((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list"   and v :: "nat" assumes "is_standard_operator (n, p, vp, v)" shows "is_standard_operator (rem_implicit_pres (n, p, vp, v))"
mset_union_diff_comm: fixes t :: "'a"   and S :: "'a multiset"   and T :: "'a multiset" assumes "t \<in># S" shows "T + (S - {#t#}) = T + S - {#t#}"
validc_elim: fixes a :: "coname"   and B :: "ty"   and \<Delta> :: "(coname \<times> ty) list" assumes "validc ((a, B) # \<Delta>)" shows "validc \<Delta> \<and> a \<sharp> \<Delta>"
cond_abs: fixes s :: "'b" assumes "s \<in> wa_invar WB"   and "s \<in> wa_cond WB" shows "\<alpha>2 s \<in> wa_cond WA"
wls_subst_vsubst_trans: fixes s :: "'sort"   and X :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set"   and ys :: "'varSort"   and Y :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set"   and y1 :: "'var"   and y :: "'var" assumes "wls s X"   and "wls (asSort ys) Y"   and "fresh ys y1 X" shows "X #[y1 // y]_ys #[Y / y1]_ys = X #[Y / y]_ys"
semigroup_add_ow_axioms: shows "semigroup_add_ow U\<^sub>V\<^sub>S (+\<^sub>V\<^sub>S)"
case_distrib: fixes h :: "'b \<Rightarrow> 'a"   and f1 :: "'b"   and f2 :: "'c \<Rightarrow> 'b"   and withERR :: "'c withERR" shows "h (case withERR of ERR \<Rightarrow> f1 | OK x \<Rightarrow> f2 x) = (case withERR of ERR \<Rightarrow> h f1 | OK x \<Rightarrow> h (f2 x))"
get_owner_document_split: fixes P :: "((_) heap, exception, (_) document_ptr) prog \<Rightarrow> bool"   and xs :: "(((_) object_ptr \<Rightarrow> bool) \<times> ((_) object_ptr \<Rightarrow> unit \<Rightarrow> ((_) heap, exception, (_) document_ptr) prog)) list"   and ptr :: "(_) object_ptr" shows "P (invoke (a_get_owner_document_tups @ xs) ptr ()) = ((known_ptr ptr \<longrightarrow> P (get_owner_document ptr)) \<and> (\<not> known_ptr ptr \<longrightarrow> P (invoke xs ptr ())))"
Rep_boolean_subset_cases: fixes y :: "'b" assumes "y \<in> {x. \<exists>y. x = - y}" obtains x :: "'b boolean_subset" where "y = Rep_boolean_subset x"
Top_st_def: shows "\<top> = FunDom (\<lambda>x. \<top>) []"
assms: shows "prefix B (A \<cdot>\<^sub>l\<^sub>s\<^sub>s\<^sub>t \<theta>)"
A: shows "Auth_Data (priAK n) b \<in> parts (A \<union> spies evsFR4) \<longrightarrow> (\<exists>m run. IntMapK (S (Card m, m, run)) = Some b)"
inf_sup_aci_4: fixes inf :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sup :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and x :: "'d"   and y :: "'d" assumes "class.lattice inf less_eq less sup" shows "inf x (inf x y) = inf x y"
is_le_rel_def: fixes rel :: "'t \<Rightarrow> 't \<Rightarrow> bool" shows "ord_term_lin.is_le_rel rel = (rel = (=) \<or> rel = (\<preceq>\<^sub>t) \<or> rel = (\<prec>\<^sub>t))"
float_distinct_7: fixes a :: "('a, 'b) IEEE.float" shows "\<not> (is_infinity a \<and> IEEE.is_zero a)"
hyps_2: fixes y :: "'a" assumes "y \<in> set zs" shows "compare cmp z y \<noteq> Greater"
hyps_3: assumes "\<forall>x\<in>A. \<forall>y\<in>A. f x y = (if y = x then g x else (0::'b))" shows "\<forall>x\<in>A. sum (f x) A = g x"
least_unique: fixes x :: "'a"   and A :: "'a set"   and y :: "'a" assumes "least L x A"   and "least L y A" shows "x = y"
lfilter_K_True: fixes xs :: "'c llist" shows "lfilter (\<lambda>_. True) xs = xs"
index_less: fixes x :: "'c"   and xs :: "'c list"   and n :: "nat" assumes "x \<in> set xs"   and "length xs \<le> n" shows "index xs x < n"
plus_dim1: fixes A :: "'b tensor"   and B :: "'b tensor" assumes "dims A = dims B" shows "dims (A + B) = dims A"
assms_4: fixes v :: "'t"   and w :: "'t" assumes "snd (pair_of_term v) \<le> snd (pair_of_term w)" shows "snd (rep_nat_term v) \<le> snd (rep_nat_term w)"
path_image_part_circlepath: fixes s :: "real"   and t :: "real"   and z :: "complex"   and r :: "real" assumes "s \<le> t" shows "path_image (part_circlepath z r s t) = {z + complex_of_real r * exp (\<i> * complex_of_real x) |x. s \<le> x \<and> x \<le> t}"
nonsquare_D: fixes D :: "nat" assumes "pell D" shows "\<not> is_square D"
E_bernoulli3: fixes p :: "real"   and f :: "bool \<Rightarrow> real pmf" assumes "0 < p"   and "p < 1"   and "finite (set_pmf (bernoulli_pmf p \<bind> f))" shows "E (bernoulli_pmf p \<bind> f) = E (f True) * p + E (f False) * (1 - p)"
jkbpCSt_jkbpCS_subset: fixes t :: "'b Trace" shows "clock_jkbpCSt t \<subseteq> Clock.jkbpCS"
fpxs_val_power: fixes f :: "'b fpxs"   and n :: "nat" assumes "f \<noteq> 0 \<or> 0 < n" shows "fpxs_val (f ^ n) = rat_of_nat n * fpxs_val f"
lc_f: shows "lookup (monom_mult (lc p / lc f) (lp p - lp f) f) (lt p) = lookup p (lt p)"
Lcm_insert: fixes one :: "'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and gcd :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and lcm :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and Gcd :: "'b set \<Rightarrow> 'b"   and Lcm :: "'b set \<Rightarrow> 'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and a :: "'b"   and A :: "'b set" assumes "class.semiring_Gcd one zero times gcd lcm Gcd Lcm divide plus minus unit_factor normalize" shows "Lcm (insert a A) = lcm a (Lcm A)"
match_pres_distinct: fixes pres :: "('a \<times> 'b) list"   and s :: "'a \<Rightarrow> 'b option" assumes "distinct (map fst pres)" shows "Solve_SASP.match_pres pres s = (Map.map_of pres \<subseteq>\<^sub>m s)"
rel_ANNOT_eq: fixes t :: "'c"   and R :: "('d \<times> 'c) set" shows "t \<equiv> t ::: R"
capacity_outside: fixes e :: "'v \<times> 'v" assumes "e \<notin> \<^bold>E" shows "capacity \<Delta> e = 0"
seq_extract_split: fixes i :: "nat"   and xs :: "'b list" assumes "i \<le> length xs" shows "{0..<i} \<upharpoonleft>\<^sub>l xs @ {i..<length xs} \<upharpoonleft>\<^sub>l xs = xs"
vector_export_comp_unit: fixes x :: "'a"   and y :: "'a" shows "x \<sqinter> top \<sqinter> top \<sqinter> y = x \<sqinter> top \<sqinter> y"
I_intT_def: fixes wtFsym :: "'a \<Rightarrow> bool"   and wtPsym :: "'b \<Rightarrow> bool"   and arOf :: "'a \<Rightarrow> 'c list"   and resOf :: "'a \<Rightarrow> 'c"   and parOf :: "'b \<Rightarrow> 'c list"   and \<Phi> :: "('a, 'b) lit list set"   and infTp :: "'c \<Rightarrow> bool"   and intT :: "'c \<Rightarrow> univ \<Rightarrow> bool"   and intF :: "'a \<Rightarrow> univ list \<Rightarrow> univ"   and intP :: "'b \<Rightarrow> univ list \<Rightarrow> bool"   and \<sigma> :: "'c"   and a :: "univ" assumes "ModelIkMcalc wtFsym wtPsym arOf resOf parOf \<Phi> infTp intT intF intP" shows "ModelIkMcalc.I_intT infTp intT \<sigma> a \<equiv> infTp \<sigma> \<longrightarrow> intT \<sigma> a"
inject_1: fixes x11 :: "'f"   and x12 :: "'g"   and x13 :: "'h"   and y11 :: "'f"   and y12 :: "'g"   and y13 :: "'h" shows "(NewThread x11 x12 x13 = NewThread y11 y12 y13) = (x11 = y11 \<and> x12 = y12 \<and> x13 = y13)"
dvds_eq_abseq: fixes l :: "int"   and k :: "int" shows "(l dvd k \<and> k dvd l) = (\<bar>l\<bar> = \<bar>k\<bar>)"
pfun_sum_pdom_res: fixes f :: "('a, 'b) pfun"   and A :: "'a set" assumes "finite (pdom f)" shows "pfun_sum (A \<lhd>\<^sub>p f) = pfun_sum f - pfun_sum ((- A) \<lhd>\<^sub>p f)"
prev_in_basics_2: fixes u :: "'a" assumes "u \<in> {a<..b}" shows "prev_in A u < u"
add_minus_id: fixes a :: "'b"   and b :: "'b"   and p\<^sub>1 :: "'b point"   and p\<^sub>2 :: "'b point" assumes "nonsingular a b"   and "on_curve a b p\<^sub>1"   and "on_curve a b p\<^sub>2" shows "ell_field_class.add a (ell_field_class.add a p\<^sub>1 p\<^sub>2) (opp p\<^sub>2) = p\<^sub>1"
LeftDerives1_X_is_part_of_rule: fixes \<delta> :: "symbol list"   and i :: "nat"   and r :: "symbol \<times> symbol list"   and a :: "symbol list"   and X :: "symbol"   and b :: "symbol list"   and \<alpha> :: "symbol list"   and N :: "symbol"   and \<beta> :: "symbol list" assumes "LeftDerives1 \<delta> i r (a @ [X] @ b)"   and "splits_at \<delta> i \<alpha> N \<beta>"   and "\<And>\<beta>. \<lbrakk>\<delta> = a @ [X] @ \<beta>; length a < i; is_word (a @ [X]); LeftDerives1 \<beta> (i - length a - 1) r b\<rbrakk> \<Longrightarrow> False"   and "\<And>\<alpha>. \<lbrakk>\<delta> = \<alpha> @ [X] @ b; LeftDerives1 \<alpha> i r a\<rbrakk> \<Longrightarrow> False" shows "\<exists>u v. a = \<alpha> @ u \<and> b = v @ \<beta> \<and> snd r = u @ [X] @ v"
Cons: fixes ys :: "('t \<Rightarrow>\<^sub>0 'b) list" shows "set ys \<subseteq> set (comp_red_basis_aux xs ys)"
card_nyinitcls_gext: fixes s :: "state"   and s' :: "state"   and G :: "prog"   and n :: "nat" assumes "snd s\<le>|snd s'"   and "card (nyinitcls G s) \<le> n" shows "card (nyinitcls G s') \<le> n"
img_fst: fixes a :: "'a"   and b :: "'b"   and S :: "('a \<times> 'b) set" assumes "(a, b) \<in> S" shows "a \<in> fst ` S"
intros_7: fixes uu :: "'b \<Rightarrow> 'c"   and uv :: "'b option"   and uw :: "'b option" shows "dbm_entry_val uu uv uw \<infinity>"
simps_3: fixes \<phi> :: "'b ltln"   and \<psi> :: "'b ltln" shows "list_dnf (\<phi> and\<^sub>n \<psi>) = list_dnf \<phi> \<otimes>\<^sub>l list_dnf \<psi>"
distinct_287: fixes x14 :: "floatarith"   and x151 :: "floatarith"   and x152 :: "nat" shows "Ln x14 \<noteq> Power x151 x152"
fps_compose_sub_distrib: fixes a :: "'b fps"   and b :: "'b fps"   and c :: "'b fps" shows "a - b oo c = (a oo c) - (b oo c)"
d4_const_def: shows "d4_const \<equiv> \<d> 4"
zj: shows "[0..<j] = [0..<jj] @ [jj]"
relhoare_refl: fixes R :: "('a \<times> 'a) set" shows "{Id} R, R {> Id}"
prems_5: shows "(ad_, a'd_) \<in> \<langle>R\<rangle>ltlr_rel \<rightarrow> \<langle>R\<rangle>ltlr_rel \<rightarrow> Rv \<rightarrow> Rv \<rightarrow> Rv"
piecewise_continuous_on_congI: fixes a :: "'a"   and b :: "'a"   and I :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "piecewise_continuous_on a b I f"   and "\<And>x. x \<in> {a..b} - I \<Longrightarrow> g x = f x" shows "piecewise_continuous_on a b I g"
norm_minus_commute: fixes a :: "'c"   and b :: "'c" shows "norm (a - b) = norm (b - a)"
diff_matrix_def: fixes A :: "'b matrix"   and B :: "'b matrix" shows "A - B = combine_matrix (-) A B"
assms: fixes i :: "nat"   and j :: "nat" assumes "i < j"   and "i < length xs"   and "j < length xs" shows "xs ! i \<noteq> xs ! j"
instanceF_def: fixes w :: "vbl"   and body :: "formula" shows "instanceF w body = subF (\<lambda>v. case v of zeroX \<Rightarrow> w | nextX v \<Rightarrow> v) body"
stutter_selectionD_infinite: fixes s :: "nat set"   and w :: "'a llist"   and i :: "nat" assumes "stutter_selection s w"   and "enat i < llength w"   and "finite s"   and "Max s < i" shows "w ?! i = w ?! Max s"
ltln_weak_to_strong_1: fixes w :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltln"   and \<psi> :: "'b ltln" assumes "w \<Turnstile>\<^sub>n \<phi> W\<^sub>n \<psi>"   and "\<not> w \<Turnstile>\<^sub>n G\<^sub>n \<phi>" shows "w \<Turnstile>\<^sub>n \<phi> U\<^sub>n \<psi>"
not_gr_zero: fixes plus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and zero :: "'g"   and less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and n :: "'g" assumes "class.canonically_ordered_monoid_add plus zero less_eq less" shows "(\<not> less zero n) = (n = zero)"
subspace_def: fixes U :: "'c set"   and V :: "'c set" shows "Subspace.subspace U V \<equiv> (U \<noteq> {} \<and> U \<subseteq> V) \<and> (\<forall>x y. x \<in> U \<longrightarrow> y \<in> U \<longrightarrow> x + y \<in> U) \<and> (\<forall>x a. x \<in> U \<longrightarrow> a \<cdot> x \<in> U)"
lens_override_plus: fixes X :: "'a \<Longrightarrow> 'b"   and Y :: "'c \<Longrightarrow> 'b"   and S\<^sub>1 :: "'b"   and S\<^sub>2 :: "'b" assumes "X \<bowtie> Y" shows "S\<^sub>1 \<triangleleft>\<^bsub>X +\<^sub>L Y\<^esub> S\<^sub>2 = S\<^sub>1 \<triangleleft>\<^bsub>X\<^esub> S\<^sub>2 \<triangleleft>\<^bsub>Y\<^esub> S\<^sub>2"
tl_stack_hd_tree_path: fixes G :: "('c, 'd) graph_rec_scheme"   and param :: "('c, ('c, 'e) state_scheme, 'e) gen_parameterization"   and s :: "('c, 'e) state_scheme"   and v :: "'c" assumes "DFS_invar G param s"   and "stack s \<noteq> []"   and "v \<in> set (tl (stack s))" shows "(v, hd (stack s)) \<in> (tree_edges s)\<^sup>+"
Infinitesimal_subset_HFinite: shows "Infinitesimal \<subseteq> HFinite"
sum_eq: fixes x :: "'a"   and t :: "'a" shows "x + t = (0::'a) \<equiv> x = - t"
neg_def: fixes \<phi> :: "'fmla" shows "neg \<phi> = imp \<phi> fls"
poly_eval_zero: fixes a :: "'c \<Rightarrow> 'b" shows "poly_eval a 0 = (0::'b)"
const_is_pr: fixes n :: "nat" shows "(\<lambda>x. n) \<in> PrimRec1"
eq_matI: fixes B :: "'m mat"   and A :: "'m 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"
c'_def: shows "c' \<equiv> complex_of_real (2 * pi) * \<i>"
from_vec_to_vec: fixes M :: "('a, 'b) sq_matrix" shows "from_vec (to_vec M) = M"
1: fixes is1 :: "nat list"   and is2 :: "nat list" assumes "is1 \<lhd> input_sizes m1"   and "is2 \<lhd> input_sizes m2" shows "(\<Prod>k<length inputs. inputs ! k $ ((is1 @ is2) ! k)) = (\<Prod>k<length inputs1. inputs1 ! k $ (is1 ! k)) * (\<Prod>k<length inputs2. inputs2 ! k $ (is2 ! k))"
samet_def: fixes fgto :: "(nat, nat list list list) trie option"   and ags :: "nat list list list" shows "samet fgto ags = (case fgto of None \<Rightarrow> False | Some tfgs \<Rightarrow> let tags = tries_of_list hash ags in all_tries (\<lambda>fg. list_ex (iso_test fg) (lookup_tries tags (hash fg))) tfgs \<and> all_tries (\<lambda>ag. list_ex (iso_test ag) (lookup_tries tfgs (hash ag))) tags)"
assms_5: shows "(a1, r1) = euclidean_circle (of_ocircline H1)"
DataSpace_def: fixes L :: "'a set list" shows "DataSpace.DataSpace L = (distinct L \<and> (\<forall>D1\<in>set L. \<forall>D2\<in>set L. D1 \<noteq> D2 \<longrightarrow> D1 \<inter> D2 = {}) \<and> \<Union> (set L) = UNIV)"
Ah_2: shows "(f w \<in> A) = (f (pseudo_proj_True n w) \<notin> A)"
bad: fixes s' :: "'s"   and x :: "'a" assumes "I s'"   and "\<not> bad s'"   and "enat (count s') < n + enat (count s)"   and "consider x"   and "x \<in> outs_\<I> \<I>" shows "spmf (map_spmf (bad \<circ> snd) (callee s' x)) True \<le> k"
fresh_atm_1: fixes a :: "name"   and b :: "name" shows "a \<sharp> b = (a \<noteq> b)"
AR_def: fixes S :: "'c set" shows "AR S \<equiv> \<forall>U S'. S homeomorphic S' \<and> closedin (top_of_set U) S' \<longrightarrow> S' retract_of U"
fbrelation_def: fixes r :: "V" shows "fbrelation r \<equiv> fpairs r = r"
bex_simps_8: fixes A :: "'a set"   and P :: "'a \<Rightarrow> bool" shows "(\<not> (\<exists>x\<in>A. P x)) = (\<forall>x\<in>A. \<not> P x)"
pbdom_in_hom: fixes f :: "('a, 'a set) Maps.arr"   and g :: "('a, 'a set) Maps.arr" assumes "Maps.cospan f g" shows "Maps.in_hom (Maps.pbdom f g) (Maps.pbdom f g) (Maps.pbdom f g)"
continuous_on_finite: fixes S :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "finite S" shows "continuous_on S f"
drop_sconstE: fixes n :: "nat"   and w :: "'c list"   and y :: "'c"   and p :: "nat" assumes "drop n w @- sconst y = sconst y"   and "p < length w"   and "\<not> p < n" shows "w ! p = y"
inf_llist_lnth: fixes xs :: "'b llist" assumes "\<not> lfinite xs" shows "inf_llist (lnth xs) = xs"
Gauss_Jordan_in_ij_iarrays_PA_def: fixes A' :: "'b iarray iarray \<times> 'b iarray iarray"   and i :: "nat"   and j :: "nat" shows "Gauss_Jordan_in_ij_iarrays_PA A' i j = (let P = fst A'; A = snd A'; n = least_non_zero_position_of_vector_from_index (column_iarray j A) i; interchange_A = interchange_rows_iarray A i n; interchange_P = interchange_rows_iarray P i n; P' = mult_row_iarray interchange_P i ((1::'b) / interchange_A !! i !! j) in (IArray.of_fun (\<lambda>s. if s = i then P' !! s else row_add_iarray P' s i (- interchange_A !! s !! j) !! s) (nrows_iarray A), Gauss_Jordan_in_ij_iarrays A i j))"
squareCI: fixes v :: "state \<Rightarrow> 'a"   and t :: "state"   and s :: "state"   and A :: "state \<times> state \<Rightarrow> bool" assumes "v t \<noteq> v s \<Longrightarrow> A (s, t)" shows "(s, t) \<Turnstile> [A]_v"
trancl_unfold_right: fixes r :: "('c \<times> 'c) set" shows "r\<^sup>+ = r\<^sup>* O r"
argmin2_argmin: fixes xs :: "'a list"   and f :: "'a \<Rightarrow> 'b" assumes "xs \<noteq> []" shows "argmin2 f xs = (argmin f xs, f (argmin f xs))"
gcd_gauss_int_def: shows "gcd \<equiv> normalization_euclidean_semiring_class.gcd"
absorb1: fixes a :: "'a"   and b :: "'a" assumes "a \<le> b" shows "a \<sqinter> b = a"
Interval'_split: fixes P :: "'b interval option \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" shows "P (Interval' a b) = ((\<forall>ivl. a \<le> b \<longrightarrow> lower ivl = a \<longrightarrow> upper ivl = b \<longrightarrow> P (Some ivl)) \<and> (\<not> a \<le> b \<longrightarrow> P None))"
simps_2: shows "zhmset_of_enat \<infinity> = \<omega>\<^sub>z"
integer_of_natural_of_nat: fixes n :: "nat" shows "integer_of_natural (of_nat n) = of_nat n"
prv_imp_mp: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and num :: "'b set"   and eql :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and cnj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and imp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and all :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and exi :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and prv :: "'c \<Rightarrow> bool"   and \<phi> :: "'c"   and \<chi> :: "'c" assumes "Deduct var trm fmla Var FvarsT substT Fvars subst num eql cnj imp all exi prv"   and "\<phi> \<in> fmla"   and "\<chi> \<in> fmla"   and "prv (imp \<phi> \<chi>)"   and "prv \<phi>" shows "prv \<chi>"
walk_array_foldl: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and a :: "'c array"   and x :: "'b" shows "walk_array f a x = foldl f x (list_of_array a)"
Strong_dlHPP_Bisimulation_def: fixes d :: "'d"   and PP :: "nat set"   and R :: "('com list \<times> 'com list) set" shows "Strong_dlHPP_Bisimulation d PP R \<equiv> sym R \<and> trans R \<and> (\<forall>(V, V')\<in>R. length V = length V') \<and> (\<forall>(V, V')\<in>R. \<forall>i<length V. NDC d (V ! i) \<or> IDC d (V ! i) (htchLoc (pp (V ! i)))) \<and> (\<forall>(V, V')\<in>R. \<forall>i<length V. \<forall>m1 m1' m2 \<alpha> p. \<langle>V ! i,m1\<rangle> \<rightarrow>\<lhd>\<alpha>\<rhd> \<langle>p,m2\<rangle> \<and> m1 \<sim>\<^bsub>d,htchLocSet PP\<^esub> m1' \<longrightarrow> (\<exists>p' \<alpha>' m2'. \<langle>V' ! i,m1'\<rangle> \<rightarrow>\<lhd>\<alpha>'\<rhd> \<langle>p',m2'\<rangle> \<and> stepResultsinR p p' R \<and> (\<alpha>, \<alpha>') \<in> R \<and> dhequality_alternative d PP (pp (V ! i)) m2 m2'))"