hcju/isabelleps huggingface dataset:
BinOpThrow\<^sub>2\<^sub>1: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> nat exp) list) list"   and e\<^sub>1 :: "nat exp"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> val list"   and v\<^sub>1 :: "val"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> val list"   and e\<^sub>2 :: "nat exp"   and e :: "nat exp"   and s\<^sub>2 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> val list"   and bop :: "bop" assumes "P \<turnstile>\<^sub>1 \<langle>e\<^sub>1,s\<^sub>0\<rangle> \<Rightarrow> \<langle>Val v\<^sub>1,s\<^sub>1\<rangle>"   and "P \<turnstile>\<^sub>1 \<langle>e\<^sub>2,s\<^sub>1\<rangle> \<Rightarrow> \<langle>throw e,s\<^sub>2\<rangle>" shows "P \<turnstile>\<^sub>1 \<langle>e\<^sub>1 \<guillemotleft>bop\<guillemotright> e\<^sub>2,s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw e,s\<^sub>2\<rangle>"
rel_cases: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'a tree"   and b :: "'b tree" assumes "rel_tree R a b" obtains x1 :: "nat" and x2 :: "'a" and y1 :: "nat" and y2 :: "'b" where "a = Leaf x1 x2" and "b = Leaf y1 y2" and "x1 = y1" and "R x2 y2"   | x1a :: "nat" and x2a :: "'a tree" and x3 :: "'a tree" and y1a :: "nat" and y2a :: "'b tree" and y3 :: "'b tree" where "a = Node x1a x2a x3" and "b = Node y1a y2a y3" and "x1a = y1a" and "rel_tree R x2a y2a" and "rel_tree R x3 y3"
assms_9: shows "\<not> is_sig_red (\<prec>\<^sub>t) (=) G q"
val_ifex: fixes b :: "'d bool_expr"   and s :: "'d \<Rightarrow> bool" shows "val_ifex (ifex_of b) s = val_bool_expr b s"
tagged_division_ofD_2: fixes s :: "('c \<times> 'c set) set"   and i :: "'c set"   and x :: "'c"   and K :: "'c set" assumes "s tagged_division_of i"   and "(x, K) \<in> s" shows "x \<in> K"
l6_13_1: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and P :: "'a"   and A :: "'a"   and B :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Out Bet P A B"   and "Tarski_neutral_dimensionless.Le Bet Cong P A P B" shows "Bet P A B"
cUn_cinsert_right: fixes A :: "'d cset"   and a :: "'d"   and B :: "'d cset" shows "cUn A (cinsert a B) = cinsert a (cUn A B)"
while_k_SE: fixes k :: "nat" shows "while_C = while_k k"
trivial_RModule: assumes "\<And>r. r \<in> R \<Longrightarrow> r \<cdot> (0::'m) = (0::'m)" shows "RModule R (\<cdot>) 0"
f4_sel_aux_subset: fixes t :: "'a"   and ps :: "((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list" shows "set (f4_sel_aux t ps) \<subseteq> set ps"
opt_mru_round_refines: fixes r :: "nat"   and S :: "process set"   and v :: "val"   and d_f :: "process \<Rightarrow> val option"   and Q :: "process set" shows "{lv_ref_rel \<inter> (Vinv1 \<inter> SV_inv3 \<inter> SV_inv4) \<times> UNIV} sv_round r S v d_f, opt_mru_round r Q S v d_f {> lv_ref_rel}"
Diff_mono: fixes A :: "'c set"   and C :: "'c set"   and D :: "'c set"   and B :: "'c set" assumes "A \<subseteq> C"   and "D \<subseteq> B" shows "A - B \<subseteq> C - D"
r_search_enum_recfn: shows "recfn 2 r_search_enum"
io_path_states_prefix: fixes M :: "('d, 'e, 'f) FSM"   and io1 :: "('d \<times> 'e) list"   and tr1 :: "'f list"   and q :: "'f"   and io2 :: "('d \<times> 'e) list"   and tr2 :: "'f list" assumes "observable M"   and "path M (io1 || tr1) q"   and "length tr1 = length io1"   and "path M (io2 || tr2) q"   and "length tr2 = length io2"   and "prefix io1 io2" shows "tr1 = take (length tr1) tr2"
sel_3: fixes x :: "quat"   and y :: "quat" shows "Im2 (x + y) = Im2 x + Im2 y"
aadd_poss_pos: fixes x :: "ant"   and y :: "ant" assumes "0 < x"   and "0 \<le> y" shows "0 < x + y"
S': shows "S' \<in> carrier_mat (dim_row (Mod_Type_Connect.from_hma\<^sub>m A)) (dim_col (Mod_Type_Connect.from_hma\<^sub>m A))"
subs_NL: shows "set (zip (prems_of \<iota>) Ll) \<subseteq> NL"
butlast_rest_bl2bin_aux: fixes bl :: "bool list"   and w :: "int" assumes "bl \<noteq> []" shows "bl_to_bin_aux (butlast bl) w = bl_to_bin_aux bl w div 2"
even_xor_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 push_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and drop_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and take_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and "and" :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and or :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and xor :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and mask :: "nat \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.semiring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask" shows "dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) (xor a b) = (dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) a = dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) b)"
no_induced_modulus: shows "\<not> (\<exists>d<n. induced_modulus d)"
tedious_reasoning: fixes M0 :: "'a multiset"   and a :: "'a" assumes "M0 \<in> Wellfounded.acc (mult1 r)"   and "\<And>b. (b, a) \<in> r \<Longrightarrow> \<forall>M\<in>Wellfounded.acc (mult1 r). add_mset b M \<in> Wellfounded.acc (mult1 r)"   and "\<forall>M. (M, M0) \<in> mult1 r \<longrightarrow> add_mset a M \<in> Wellfounded.acc (mult1 r)" shows "add_mset a M0 \<in> Wellfounded.acc (mult1 r)"
abs_fun_pi: fixes pi :: "('b \<times> 'b) list"   and a :: "'b"   and x :: "'a" assumes "pt TYPE('a) TYPE('b)"   and "at TYPE('b)" shows "pi \<bullet> [a].x = [(pi \<bullet> a)].(pi \<bullet> x)"
openin: fixes x :: "'d topology" shows "openin x \<in> {L. istopology L}"
elims_1: fixes x :: "('b \<Rightarrow> bool) \<times> ('b \<Rightarrow> 'c)"   and xa :: "'c \<Rightarrow> 'b"   and y :: "bool" assumes "wf_disc_sel x xa = y" obtains disc :: "'b \<Rightarrow> bool" and sel :: "'b \<Rightarrow> 'c" and C :: "'c \<Rightarrow> 'b" where "x = (disc, sel)" and "xa = C" and "y = ((\<forall>a. disc a \<longrightarrow> C (sel a) = a) \<and> (\<forall>a. sel (C a) = a))"
norm_sum: fixes minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and dist :: "'d \<Rightarrow> 'd \<Rightarrow> real"   and norm :: "'d \<Rightarrow> real"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and uminus :: "'d \<Rightarrow> 'd"   and scaleR :: "real \<Rightarrow> 'd \<Rightarrow> 'd"   and sgn :: "'d \<Rightarrow> 'd"   and uniformity :: "('d \<times> 'd) filter"   and "open" :: "'d set \<Rightarrow> bool"   and f :: "'e \<Rightarrow> 'd"   and A :: "'e set" assumes "class.real_normed_vector minus dist norm plus zero uminus scaleR sgn uniformity open" shows "norm (comm_monoid_add.sum plus zero f A) \<le> (\<Sum>i\<in>A. norm (f i))"
ACI_map_respects: fixes r :: "'d rexp"   and s :: "'d rexp"   and f :: "'d \<Rightarrow> 'e" assumes "r ~ s" shows "map_rexp f r ~ map_rexp f s"
assms: shows "card (Pow X) = 1 \<or> card (Pow X) = 2"
punit_pp_of_term: shows "MPoly_Type_Class.punit.pp_of_term = (\<lambda>x. x)"
Seq_NoFaultStuckD2: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c1 :: "('e, 'd, 'f) com"   and c2 :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate"   and F :: "'f set" assumes "\<Gamma>\<turnstile> \<langle>Seq c1 c2,s\<rangle> \<Rightarrow>\<notin>{Stuck} \<union> Fault ` F" shows "\<forall>t. \<Gamma>\<turnstile> \<langle>c1,s\<rangle> \<Rightarrow> t \<longrightarrow> t \<notin> {Stuck} \<union> Fault ` F \<longrightarrow> \<Gamma>\<turnstile> \<langle>c2,t\<rangle> \<Rightarrow>\<notin>{Stuck} \<union> Fault ` F"
eq_iff: fixes a :: "'x \<Rightarrow>\<^sub>0 nat"   and b :: "'x \<Rightarrow>\<^sub>0 nat" shows "(a = b) = (a \<preceq> b \<and> b \<preceq> a)"
iso_fcomp: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'d \<Rightarrow> 'b" assumes "mono f"   and "mono g" shows "mono (f \<circ> g)"
canonical_fun_def: fixes U :: "'a set set"   and x :: "'a set \<Rightarrow> ('a \<times> 'a) set" shows "st.canonical_fun U x = st.class_of U x"
eq_onp_same_args: fixes P :: "'k \<Rightarrow> bool"   and x :: "'k" shows "eq_onp P x x = P x"
fstTS_t3': shows "snd (TSdet init h \<sigma> (Suc (Suc (length as + length bs + i)))) = rev (take i cs) @ [x] @ rev bs @ [x] @ rev as @ h"
mgu_gives_wellformed_subst: fixes s :: "('b, 'a) Term.term"   and t :: "('b, 'a) Term.term"   and \<theta> :: "'a \<Rightarrow> ('b, 'a) Term.term" assumes "mgu s t = Some \<theta>" shows "wf\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<theta>"
LI_in_measure_trans: fixes S\<^sub>1 :: "('fun, 'var) strand_step list"   and \<theta>\<^sub>1 :: "'var \<Rightarrow> ('fun, 'var) Term.term"   and S\<^sub>2 :: "('fun, 'var) strand_step list"   and \<theta>\<^sub>2 :: "'var \<Rightarrow> ('fun, 'var) Term.term" assumes "(S\<^sub>1, \<theta>\<^sub>1) \<leadsto>\<^sup>+ (S\<^sub>2, \<theta>\<^sub>2)" shows "((S\<^sub>2, \<theta>\<^sub>2), S\<^sub>1, \<theta>\<^sub>1) \<in> measure\<^sub>s\<^sub>t"
zero_less_floor: fixes x :: "'b" shows "(0 < \<lfloor>x\<rfloor>) = ((1::'b) \<le> x)"
isUbD2: fixes R :: "'a set"   and S :: "'a set"   and x :: "'a" assumes "isUb R S x" shows "S *<= x"
bind_duplicate: fixes m :: "'m"   and f :: "'a \<Rightarrow> 'a \<Rightarrow> 'm" shows "bind m (\<lambda>x. bind m (f x)) = bind m (\<lambda>x. f x x)"
cod_trg_eq_dom_trg: shows "HfHgh.cod.trg = HfHgh.dtrg"
fmadd_empty_2: fixes m :: "('c, 'd) fmap" shows "m ++\<^sub>f {$$} = m"
compact_attains_inf: 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" assumes "class.linorder_topology less_eq less open"   and "topological_space.compact open S"   and "S \<noteq> {}" shows "\<exists>s\<in>S. \<forall>t\<in>S. less_eq s t"
ideD_2: fixes a :: "'a" assumes "C.ide a" shows "C.dom a = a"
CompDiag_Cod_Diag: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and t :: "'a local.term" assumes "monoidal_language C"   and "monoidal_language.Diag C t" shows "monoidal_language.CompDiag C (monoidal_language.Cod C t) t = t"
path_image_subset_reversepath: fixes g :: "real \<Rightarrow> 'a" shows "path_image (reversepath g) \<subseteq> path_image g"
alw_ev_ks: shows "alw (ev (\<lambda>ks. shd (stl ks) \<noteq> shd ks)) ks"
set_finals_splitFace: fixes f :: "face"   and g :: "graph"   and u :: "nat"   and v :: "nat"   and vs :: "nat list" assumes "f \<in> \<F> g"   and "\<not> final f" shows "set (finals (snd (snd (splitFace g u v f vs)))) = set (finals g)"
inject_3: fixes x41 :: "'a formula"   and x42 :: "'a formula"   and y41 :: "'a formula"   and y42 :: "'a formula" shows "(x41 \<^bold>\<and> x42 = y41 \<^bold>\<and> y42) = (x41 = y41 \<and> x42 = y42)"
C: fixes pa :: "'b"   and xsa :: "'a list"   and ls :: "'a list"   and eq :: "'a"   and gs :: "'a list" shows "fst (snd (group_part_aux pa xsa (ls, eq, gs))) = fold merge (filter (\<lambda>x. f x = pa) xsa) eq"
inverse_arrows: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<mu> :: "'b arrow_of_spans_data" assumes "span_vertical_category C"   and "partial_magma.arr (span_vertical_category.vcomp C) \<mu>"   and "category.iso C (Chn \<mu>)" shows "category.inverse_arrows (span_vertical_category.vcomp C) \<mu> \<lparr>Chn = category.inv C (Chn \<mu>), Dom = Cod \<mu>, Cod = Dom \<mu>\<rparr>"
is_unit_monom_0: fixes a :: "'b" assumes "a \<noteq> (0::'b)" shows "is_unit (monom a 0)"
rep_eq: fixes x :: "('d, 'e) uexpr"   and xa :: "'c \<Longrightarrow> 'e" shows "\<lbrakk>x \<restriction>\<^sub>e xa\<rbrakk>\<^sub>e = (\<lambda>b. \<lbrakk>x\<rbrakk>\<^sub>e (create\<^bsub>xa\<^esub> b))"
awlast_if_cas: fixes u :: "'a"   and p :: "'b list"   and v :: "'a" assumes "cas u p v" shows "awlast u p = v"
simps_1: shows "\<A>s [] = \<lfloor>{}\<rfloor>"
bind_returns_result_E: fixes h :: "'m"   and f :: "('m, 'n, 'p) prog"   and g :: "'p \<Rightarrow> ('m, 'n, 'o) prog"   and y :: "'o" assumes "h \<turnstile> Heap_Error_Monad.bind f g \<rightarrow>\<^sub>r y" obtains x :: "'p" and h' :: "'m" where "h \<turnstile> f \<rightarrow>\<^sub>r x" and "h \<turnstile> f \<rightarrow>\<^sub>h h'" and "h' \<turnstile> g x \<rightarrow>\<^sub>r y"
word_not_not: fixes x :: "'a Word.word" shows "~~ ~~ x = x"
prime_factorization_unique'': fixes S :: "'a set"   and f :: "'a \<Rightarrow> nat"   and n :: "'a" assumes "S = {p. 0 < f p}"   and "finite S"   and "\<forall>p\<in>S. prime p"   and "normalize n = (\<Prod>p\<in>S. p ^ f p)" shows "S = prime_factors n \<and> (\<forall>p. prime p \<longrightarrow> f p = multiplicity p n)"
sorted_weighted_gauss_Ico_div2: fixes n :: "nat"   and f :: "nat \<Rightarrow> nat" assumes "\<And>i j. \<lbrakk>i \<le> j; j < n\<rbrakk> \<Longrightarrow> f j \<le> f i" shows "(\<Sum>i = 0..<n. (i + 1) * f i) \<le> (n + 1) * sum f {0..<n} div 2"
of_nat_nat: fixes 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 z :: "int" assumes "class.ring_1 one times plus zero minus uminus"   and "0 \<le> z" shows "semiring_1.of_nat one plus zero (nat z) = ring_1.of_int one plus zero minus z"
runit_eqI: fixes C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and T :: "'d \<times> 'd \<Rightarrow> 'd"   and \<alpha> :: "'d \<times> 'd \<times> 'd \<Rightarrow> 'd"   and \<iota> :: "'d"   and f :: "'d"   and a :: "'d" assumes "monoidal_category C T \<alpha> \<iota>"   and "partial_magma.in_hom C f (T (a, monoidal_category.unity C \<iota>)) a"   and "T (f, monoidal_category.unity C \<iota>) = C (T (a, \<iota>)) (\<alpha> (a, monoidal_category.unity C \<iota>, monoidal_category.unity C \<iota>))" shows "f = monoidal_category.runit C T \<alpha> \<iota> a"
rep_eq: fixes x :: "'e"   and xa :: "'e multiset" shows "count (add_mset x xa) = (\<lambda>b. if b = x then Suc (count xa b) else count xa b)"
Icc_subset_Ici_iff: fixes l :: "'b multiset"   and h :: "'b multiset"   and l' :: "'b multiset" shows "(subset_mset.atLeastAtMost l h \<subseteq> subset_mset.atLeast l') = (\<not> l \<subseteq># h \<or> l' \<subseteq># l)"
return_pmf_None_eq_mk_lossless: fixes p :: "'c spmf" shows "(return_pmf None = mk_lossless p) = (p = return_pmf None)"
slice1_sum_slice1: fixes nx :: "'node SDG_node"   and n :: "'node SDG_node"   and p :: "'pname"   and n' :: "'node SDG_node" assumes "nx \<in> sum_SDG_slice1 n"   and "n s-p\<rightarrow>\<^bsub>sum\<^esub> n'" shows "nx \<in> sum_SDG_slice1 n'"