hcju/isabelleps huggingface dataset:
lossless_sample_coprime: fixes prime_set :: "nat set"   and N :: "nat" assumes "rsa_base prime_set"   and "\<not> prime N"   and "2 < N" shows "lossless_spmf (sample_coprime N)"
ring_hom_memE_4: fixes h :: "'e \<Rightarrow> 'f"   and R :: "('e, 'g) ring_scheme"   and S :: "('f, 'h) ring_scheme" assumes "h \<in> ring_hom R S" shows "h \<one>\<^bsub>R\<^esub> = \<one>\<^bsub>S\<^esub>"
sum: fixes g :: "'b \<Rightarrow> ('a, 'm) vec"   and S :: "'b set" shows "f (sum g S) = (\<Sum>a\<in>S. f (g a))"
cobounded2: fixes b :: "'g multiset"   and a :: "'g multiset" shows "b \<subseteq># a \<union># b"
integral_emptyI_2: fixes b :: "real"   and a :: "real"   and f :: "real \<Rightarrow> 'a" assumes "b < a" shows "integral {a..b} f = (0::'a)"
cts_step_uniformly_continuous: fixes a :: "real"   and b :: "real" assumes "a < b" shows "isUCont (cts_step a b)"
abs_ge_minus_self: fixes abs :: "'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"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a" assumes "class.ordered_ab_group_add_abs abs plus zero minus uminus less_eq less" shows "less_eq (uminus a) (abs a)"
scalar_nop: shows "nop A []\<^sub>\<circ> f"
path_connected_imp_connected: fixes S :: "'b set" assumes "path_connected S" shows "connected S"
less_extended_def: fixes x :: "'b extended"   and y :: "'b extended" shows "(x < y) = (x \<le> y \<and> \<not> y \<le> x)"
rl: fixes ts1 :: "'a set list"   and ts2 :: "'a set list" assumes "direct_decomp A (ts1 @ ts2)"   and "{} \<notin> set ts2" shows "direct_decomp (sum_list ` listset ts1) ts1"
simps: fixes a1 :: "pi"   and a2 :: "pi"   and Rel :: "(pi \<times> pi) set" shows "((a1, a2) \<in> bangRel Rel) = ((\<exists>P Q. a1 = !P \<and> a2 = !Q \<and> (P, Q) \<in> Rel) \<or> (\<exists>R T P Q. a1 = R \<parallel> P \<and> a2 = T \<parallel> Q \<and> (R, T) \<in> Rel \<and> (P, Q) \<in> bangRel Rel) \<or> (\<exists>P Q a. a1 = <\<nu>a>P \<and> a2 = <\<nu>a>Q \<and> (P, Q) \<in> bangRel Rel))"
bit_unset_bit_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 uminus :: "'a \<Rightarrow> 'a"   and not :: "'a \<Rightarrow> 'a"   and m :: "nat"   and a :: "'a"   and n :: "nat" 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 "bit (ring_bit_operations.unset_bit one push_bit and not m a) n = (bit a n \<and> m \<noteq> n)"
cong: fixes X :: "'b set set"   and Y :: "'c set set"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "ChamberComplexMorphism X Y f"   and "fun_eq_on g f (\<Union> X)" shows "ChamberComplexMorphism X Y g"
upd_cols_single: fixes M :: "('c, 'd) sq_matrix"   and i :: "'d"   and f :: "'d \<Rightarrow> ('c, 'd) vec" shows "upd_cols M {i} f = upd_col M i (f i)"
domA_from_set: fixes x :: "'b"   and e :: "'c"   and h :: "('b \<times> 'c) list" assumes "(x, e) \<in> set h" shows "x \<in> domA h"
blit'_def: fixes src :: "'b array"   and si :: "integer"   and dst :: "'b array"   and di :: "integer"   and len :: "integer" shows "blit' src si dst di len = blit src (nat_of_integer si) dst (nat_of_integer di) (nat_of_integer len)"
inf_commute: fixes x :: "'a"   and y :: "'a" shows "x \<sqinter> y = y \<sqinter> x"
orderI: fixes a :: "'a"   and b :: "'a" assumes "a = a \<cdot> b" shows "a \<le> b"
iMODb_iMOD_subset_conv: fixes c' :: "nat"   and r' :: "nat"   and m' :: "nat"   and r :: "nat"   and m :: "nat" assumes "0 < c'" shows "([ r', mod m', c' ] \<subseteq> [ r, mod m ]) = (r' mod m = r mod m \<and> r \<le> r' \<and> m' mod m = 0)"
sorted_filter_less_subset_take: fixes xs :: "'a list"   and i :: "nat" assumes "sorted xs"   and "i < length xs" shows "{#x \<in># mset xs. x < xs ! i#} \<subseteq># mset (take i xs)"
cring_simprules_24: fixes R :: "('a, 'b) ring_scheme"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "cring R"   and "x \<in> carrier R"   and "y \<in> carrier R"   and "z \<in> carrier R" shows "x \<otimes>\<^bsub>R\<^esub> (y \<otimes>\<^bsub>R\<^esub> z) = y \<otimes>\<^bsub>R\<^esub> (x \<otimes>\<^bsub>R\<^esub> z)"
f_shrink_defs_2: fixes s :: "'b list"   and k :: "nat"   and ag :: "'b list \<Rightarrow> 'b" shows "f_aggregate s k ag \<equiv> map ag (list_slice s k)"
of_int_power_eq_of_int_cancel_iff: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and x :: "int"   and b :: "int"   and w :: "nat" assumes "class.ring_char_0 one times plus zero minus uminus" shows "(ring_1.of_int one plus zero minus x = power.power one times (ring_1.of_int one plus zero minus b) w) = (x = b ^ w)"
minimal_element_axioms: shows "minimal_element (\<lambda>x y. size x < size y) A"
gcd: shows "gcd (e - e') (order \<G>) = 1"
Gcd_fin_mult: 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 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 set"   and b :: "'b" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize"   and "finite A" shows "semiring_gcd.Gcd_fin one zero gcd (times b ` A) = normalize (times b (semiring_gcd.Gcd_fin one zero gcd A))"
assms_4: shows "unwind_cont (last \<Delta>s) {last \<Delta>s, \<Delta>e}"
hyps: fixes k :: "'a"   and t :: "('a, 'b) trie" assumes "(k, t) \<in> set kvs"   and "invar_trie t" shows "(dom (lookup_trie t) = {}) = is_empty_trie t"
path_join: fixes g1 :: "real \<Rightarrow> 'c"   and g2 :: "real \<Rightarrow> 'c" assumes "pathfinish g1 = pathstart g2" shows "path (g1 +++ g2) = (path g1 \<and> path g2)"
simps: fixes i :: "nat" shows "pred (Suc i) = i"
pivotandupdate_valuation_xi: fixes s :: "('i, 'a) state"   and x\<^sub>i :: "nat"   and x\<^sub>j :: "nat"   and c :: "'a" assumes "\<triangle> (\<T> s)"   and "\<nabla> s"   and "x\<^sub>i \<in> lvars (\<T> s)"   and "x\<^sub>j \<in> rvars_eq (eq_for_lvar (\<T> s) x\<^sub>i)" shows "Mapping.lookup (\<V> (pivot_and_update x\<^sub>i x\<^sub>j c s)) x\<^sub>i = Some c"
double_sum_eq_sum: shows "(\<Sum>twoCube\<in>two_chain. \<Sum>(k, g)\<in>boundary twoCube. real_of_int k * line_integral F {i} g) = (\<Sum>twoCube\<in>two_chain. integral (cubeImage twoCube) (\<lambda>a. - partial_vector_derivative (\<lambda>x. F x \<bullet> i) j a))"
vconverse_vdoubleton: fixes a :: "V"   and b :: "V"   and c :: "V"   and d :: "V" shows "(set {\<langle>a, b\<rangle>, \<langle>c, d\<rangle>})\<inverse>\<^sub>\<circ> = set {\<langle>b, a\<rangle>, \<langle>d, c\<rangle>}"
length_rotate1: fixes xs :: "'c list" shows "|rotate1 xs| = |xs|"
comp: fixes f :: "'i \<Rightarrow> 'g"   and g :: "'j \<Rightarrow> 'h"   and h :: "'e \<Rightarrow> 'i"   and i :: "'f \<Rightarrow> 'j" shows "map_sum f g \<circ> map_sum h i = map_sum (f \<circ> h) (g \<circ> i)"
Tn_integral_preserving_2: fixes f :: "'a \<Rightarrow> 'b"   and n :: "nat" assumes "integrable M f" shows "LINT x|M. f ((T ^^ n) x) = integral\<^sup>L M f"
comp_assoc: fixes f :: "'d \<Rightarrow> 'b"   and g :: "'c \<Rightarrow> 'd"   and h :: "'a \<Rightarrow> 'c" shows "f \<circ> g \<circ> h = f \<circ> (g \<circ> h)"
infinite: fixes A :: "'a set" assumes "infinite A" shows "sorted_list_of_set A = []"
in_segment_inner_rot2: fixes x :: "'a"   and a :: "'a"   and b :: "'a"   and y :: "'a" assumes "x \<in> {a--b}"   and "y \<in> {a--b}" shows "(x - y) \<bullet> rot (a - b) = 0"
ennreal_minus_pos_iff: fixes a :: "ennreal"   and b :: "ennreal" assumes "a < top \<or> b < top"   and "0 < a - b" shows "b < a"
mat_addrow_def: fixes ad :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and mul :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c"   and k :: "nat"   and l :: "nat"   and A :: "'c mat" shows "mat_addrow_gen ad mul a k l A = mat (dim_row A) (dim_col A) (\<lambda>(i, j). if k = i then ad (mul a (A $$ (l, j))) (A $$ (i, j)) else A $$ (i, j))"
lsc_liminf: fixes x0 :: "'b"   and f :: "'b \<Rightarrow> ereal" shows "lsc_at x0 f = (\<forall>x. x \<longlonglongrightarrow> x0 \<longrightarrow> f x0 \<le> liminf (f \<circ> x))"
bex_UNIV: fixes P :: "'b \<Rightarrow> bool" shows "Bex UNIV P = Ex P"
get_ancestors_never_empty: fixes h :: "(_) heap"   and child :: "(_) object_ptr"   and ancestors :: "(_) object_ptr list" assumes "Shadow_DOM.heap_is_wellformed h"   and "h \<turnstile> Shadow_DOM.get_ancestors child \<rightarrow>\<^sub>r ancestors" shows "ancestors \<noteq> []"
in_homI: fixes f :: "'d"   and a :: "'d"   and b :: "'d" assumes "D'.arr f"   and "D'.dom f = a"   and "D'.cod f = b" shows "D'.in_hom f a b"
comp_converter_map_converter2: fixes conv :: "('h, 'i, 'l, 'm) converter"   and f :: "'l \<Rightarrow> 'n"   and g :: "'o \<Rightarrow> 'm"   and h :: "'p \<Rightarrow> 'j"   and k :: "'k \<Rightarrow> 'q"   and conv' :: "('n, 'o, 'p, 'q) converter" shows "conv \<odot> map_converter f g h k conv' = map_converter id id h k (map_converter id id f g conv \<odot> conv')"
elims_3: fixes x :: "'b set"   and xa :: "'b interrupt_action list" assumes "\<not> interrupt_actions_ok x xa" obtains "is" :: "'b set" and ia :: "'b interrupt_action" and ias :: "'b interrupt_action list" where "x = is" and "xa = ia # ias" and "\<not> (interrupt_action_ok is ia \<and> interrupt_actions_ok (redT_updI is ia) ias)"
lookup_tabulate: fixes xs :: "'d list"   and f :: "'d \<Rightarrow> 'c"   and x :: "'d" shows "Mapping.lookup (Mapping.tabulate xs f) x = (if x \<in> set xs then Some (f x) else None)"
divides_refl: fixes G :: "('c, 'd) monoid_scheme"   and a :: "'c" assumes "Group.monoid G"   and "a \<in> carrier G" shows "a divides\<^bsub>G\<^esub> a"
continuous_on_powr': fixes s :: "'a set"   and f :: "'a \<Rightarrow> real"   and g :: "'a \<Rightarrow> real" assumes "continuous_on s f"   and "continuous_on s g"   and "\<forall>x\<in>s. 0 \<le> f x \<and> (f x = 0 \<longrightarrow> 0 < g x)" shows "continuous_on s (\<lambda>x. f x powr g x)"
sup_top_right: fixes x :: "'a upred" shows "(x \<or> true) = true"
pochhammer_Suc_prod_rev: fixes a :: "'a"   and n :: "nat" shows "pochhammer a (Suc n) = (\<Prod>i = 0..n. a + of_nat (n - i))"
mod_0_abs_less_imp_0: fixes a :: "int"   and m :: "int" assumes "[a = 0] (mod m)"   and "\<bar>a\<bar> < m" shows "a = 0"
comm_monoid_set_axioms: shows "comm_monoid_set (*) (1::'a)"
partner_unique: fixes R :: "fid_t"   and R'' :: "fid_t"   and R' :: "fid_t" assumes "partner_runs R R''"   and "partner_runs R R'" shows "R' = R''"
simps: fixes a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" shows "mat_det (a, b, c, d) = a * d - b * c"
a_subset: fixes H :: "'a set"   and G :: "('a, 'b) ring_scheme" assumes "additive_subgroup H G" shows "H \<subseteq> carrier G"
ir_If_False_False: fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and b :: "bexp"   and b' :: "bexp"   and c\<^sub>2 :: "com"   and c\<^sub>2' :: "com"   and Q\<^sub>2 :: "(char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and c\<^sub>1 :: "com"   and c\<^sub>1' :: "com" assumes "ir_hoare (\<lambda>s s'. P s s' \<and> \<not> bval b s \<and> \<not> bval b' s') c\<^sub>2 c\<^sub>2' Q\<^sub>2" shows "ir_hoare P (IF b THEN c\<^sub>1 ELSE c\<^sub>2) (IF b' THEN c\<^sub>1' ELSE c\<^sub>2') Q\<^sub>2"
disj_conj_abs: fixes P :: "'b upred"   and Q :: "'b upred" shows "(P \<or> P \<and> Q) = P"
fold_random_permutation_foldl: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'b" assumes "finite A" shows "fold_random_permutation f x A = map_pmf (foldl (\<lambda>x y. f y x) x) (pmf_of_set (permutations_of_set A))"
assms_4: shows "eval_until (args_ivl args) nt auxlist = (res', auxlist')"
S2: shows "c_fold (x # ls) = c_pair x (c_fold ls)"
subst_exi: fixes \<phi> :: "'fmla"   and t :: "'trm"   and x :: "'var"   and y :: "'var" assumes "\<phi> \<in> fmla"   and "t \<in> trm"   and "x \<in> var"   and "y \<in> var"   and "x \<noteq> y"   and "x \<notin> FvarsT t" shows "subst (exi x \<phi>) t y = exi x (subst \<phi> t y)"
boundedI: fixes a :: "'c"   and b :: "'c"   and c :: "'c" assumes "a \<le> b"   and "a \<le> c" shows "a \<le> min b c"
assoc: fixes a :: "'c list"   and b :: "'c list"   and c :: "'c list" shows "(a \<cdot> b) \<cdot> c = a \<cdot> b \<cdot> c"
c1c2: shows "path_image c1 \<inter> path_image c2 = {a, b}"
is_unit_triv: fixes a :: "'b" assumes "a \<noteq> (0::'b)" shows "is_unit [:a:]"
dag_setofD: fixes p :: "ref"   and low :: "ref \<Rightarrow> ref"   and high :: "ref \<Rightarrow> ref"   and t :: "dag"   and n :: "ref" assumes "Dag p low high t"   and "n \<in> set_of t" shows "(\<exists>nt. Dag n low high nt) \<and> (\<forall>nt. Dag n low high nt \<longrightarrow> set_of nt \<subseteq> set_of t)"
projection_intersection_neutral: fixes l :: "'a list"   and X :: "'a set"   and Y :: "'a set" assumes "set l \<subseteq> X" shows "l \<upharpoonleft> (X \<inter> Y) = l \<upharpoonleft> Y"
assms_6: shows "(a2, r2) = euclidean_circle (of_ocircline H2)"
loop_stepB: fixes P\<^sub>1 :: "'a set set"   and P\<^sub>2 :: "'a set set"   and B\<^sub>1 :: "'a set"   and B\<^sub>2 :: "'a set"   and V :: "'a set"   and u :: "'a" assumes "inv\<^sub>3 P\<^sub>1 P\<^sub>2 B\<^sub>1 B\<^sub>2 V"   and "B\<^sub>1 = {}"   and "u \<in> V \<inter> L" shows "inv\<^sub>3 P\<^sub>1 P\<^sub>2 {u} B\<^sub>2 (V - {u})"
10: shows "row_length B1 = row_length (B1 \<circ> B2)"
Max_Int_le2: fixes A :: "'a set"   and B :: "'a set" assumes "A \<inter> B \<noteq> {}"   and "finite B" shows "Max (A \<inter> B) \<le> Max B"
sat_\<Phi>: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ" assumes "wtE \<xi>" shows "satPB \<xi> tPB"
assms_6: shows "[e * d = 1] (mod P - 1)"
hb_cross_node_delivery: fixes m1 :: "'a"   and m2 :: "'a"   and i :: "nat"   and j :: "nat" assumes "hb m1 m2"   and "Broadcast m1 \<in> set (history i)"   and "Broadcast m2 \<in> set (history j)"   and "i \<noteq> j" shows "Deliver m1 \<in> set (history j)"
foldl_spmf_pair_right: fixes f :: "'s \<Rightarrow> 't \<Rightarrow> 's spmf"   and l :: "'r"   and r :: "'s"   and es :: "'t list" shows "foldl_spmf (\<lambda>(l, r) e. map_spmf (Pair l) (f r e)) (return_spmf (l, r)) es = map_spmf (Pair l) (foldl_spmf f (return_spmf r) es)"
value_aux: fixes i :: "nat" shows "value_flow \<Delta> (f_aux i) = ennreal ((1 - (1 / 2) ^ i) * \<alpha>')"
not_eta_reducible_eta_norm: fixes t :: "Core.term" shows "\<not> eta_reducible (eta_norm t)"
continuous_at_right_imp_inf_continuous: fixes f :: "'a \<Rightarrow> 'b" assumes "mono f"   and "\<And>x. continuous (at_right x) f" shows "inf_continuous f"
v11_vconverse: shows "v11 ((\<FF>\<lparr>ObjMap\<rparr>)\<inverse>\<^sub>\<circ>)"
len_nat_to_bv_pos: fixes a :: "nat" assumes "1 < a" shows "0 < length (nat_to_bv a)"
real_divide_square_eq: fixes r :: "real"   and a :: "real" shows "r * a / (r * r) = a / r"
DisjI: fixes A :: "fm"   and B :: "fm" assumes "se_fm A"   and "se_fm B" shows "se_fm (A OR B)"
doesnt_read_or_modify_doesnt_modify: fixes c :: "('Var, 'AExp, 'BExp) Stmt"   and x :: "'Var" assumes "doesnt_read_or_modify c x" shows "doesnt_modify c x"
minimal_antichain_path_weightp_distinct: fixes weights :: "'a \<Rightarrow> 'a \<Rightarrow> 'b antichain"   and l1 :: "'a"   and l2 :: "'a" assumes "graph weights" shows "minimal_antichain {xs. graph.path_weightp weights l1 l2 xs} = minimal_antichain {xs. graph.path_weightp_distinct weights l1 l2 xs}"
simps_4: fixes \<psi>\<^sub>1 :: "'b ltln"   and \<psi>\<^sub>2 :: "'b ltln"   and Y :: "'b ltln set" shows "(\<psi>\<^sub>1 U\<^sub>n \<psi>\<^sub>2)[Y]\<^sub>\<Sigma>\<^sub>1 = \<psi>\<^sub>1[Y]\<^sub>\<Sigma>\<^sub>1 U\<^sub>n \<psi>\<^sub>2[Y]\<^sub>\<Sigma>\<^sub>1"
WT_real: shows "\<I>_real \<oplus>\<^sub>\<I> \<I>_common \<turnstile>res real \<surd>"
sets_vimage_algebra2: fixes f :: "'b \<Rightarrow> 'c"   and X :: "'b set"   and M :: "'c measure" assumes "f \<in> X \<rightarrow> space M" shows "sets (vimage_algebra X f M) = {f -` A \<inter> X |A. A \<in> sets M}"
chain_boundary_singular_subdivision: fixes p :: "nat"   and X :: "'c topology"   and c :: "((nat \<Rightarrow> real) \<Rightarrow> 'c) \<Rightarrow>\<^sub>0 int" assumes "singular_chain p X c" shows "chain_boundary p (singular_subdivision p c) = singular_subdivision (p - Suc 0) (chain_boundary p c)"
intros_1: fixes A :: "agent" shows "cAgent A \<in> cpayload"
measurable_trace_in': fixes ss :: "'a set" shows "Measurable.pred (borel \<Otimes>\<^sub>M count_space UNIV \<Otimes>\<^sub>M stream_space S) (\<lambda>(t, s, \<omega>). trace_in ss t s \<omega>)"
divide_le_eq_1_pos: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sgn :: "'c \<Rightarrow> 'c"   and abs :: "'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less zero a" shows "less_eq (divide b a) one = less_eq b a"
visits_via_monotone: fixes P :: "'c llist"   and A :: "'c set"   and W :: "'c set"   and A' :: "'c set" assumes "visits_via P A W"   and "A \<subseteq> A'" shows "visits_via P A' W"
that: fixes b :: "'a"   and g :: "'b \<Rightarrow> 'a option" assumes "f = Abs_finfun (map_default b g)"   and "finite (dom g)"   and "b \<notin> ran g" shows "thesis"
Units_one_closed: shows "\<one>\<^bsub>residue_ring (int p)\<^esub> \<in> Units (residue_ring (int p))"
dickson_lessD3: fixes ord :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and ord_strict :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and d :: "'a \<Rightarrow> nat"   and m :: "nat"   and s :: "'a"   and t :: "'a" assumes "gd_powerprod ord ord_strict"   and "gd_powerprod.dickson_less ord_strict d m s t" shows "ord_strict s t"
set_ww: shows "set w = {q. \<exists>t. (q, t) \<in> set (step (hd b) (a ! hd b)) \<and> t \<squnion>\<^bsub>f\<^esub> a ! q \<noteq> a ! q} \<union> set (tl b)"
list_Previous_def: fixes xs :: "'d list"   and init :: "'d"   and n :: "nat" shows "xs\<^bsup>\<leftarrow>' init\<^esup> n \<equiv> case n of 0 \<Rightarrow> init | Suc n' \<Rightarrow> xs ! n'"
vp_slp_Append: fixes n :: "'node"   and as :: "'edge list"   and n'' :: "'node"   and as' :: "'edge list"   and n' :: "'node" assumes "n -as\<rightarrow>\<^sub>\<surd>* n''"   and "n'' -as'\<rightarrow>\<^bsub>sl\<^esub>* n'" shows "n -as @ as'\<rightarrow>\<^sub>\<surd>* n'"
is_extensional: fixes f :: "'c local.term set" assumes "\<not> arr f" shows "D f = \<F>\<^sub>SC.null"
nxtActI: fixes n :: "nat"   and c :: "'id"   and t :: "nat \<Rightarrow> cnf" assumes "\<exists>i\<ge>n. \<parallel>c\<parallel>\<^bsub>t i\<^esub>" shows "n \<le> \<langle>c \<rightarrow> t\<rangle>\<^bsub>n\<^esub> \<and> \<parallel>c\<parallel>\<^bsub>t \<langle>c \<rightarrow> t\<rangle>\<^bsub>n\<^esub>\<^esub> \<and> \<not> (\<exists>k\<ge>n. k < \<langle>c \<rightarrow> t\<rangle>\<^bsub>n\<^esub> \<and> \<parallel>c\<parallel>\<^bsub>t k\<^esub>)"
distinct_1: fixes x1 :: "'b btree"   and x21 :: "'b btree"   and x22 :: "'b"   and x23 :: "'b btree" shows "T\<^sub>i x1 \<noteq> Up\<^sub>i x21 x22 x23"
adjunction_unit_determines_counit: fixes f :: "('a S.E.term, 'a) S.arr"   and g :: "('a S.E.term, 'a) S.arr"   and \<eta> :: "('a S.E.term, 'a) S.arr"   and \<epsilon> :: "('a S.E.term, 'a) S.arr"   and \<epsilon>' :: "('a S.E.term, 'a) S.arr" assumes "adjunction_in_bicategory (\<cdot>\<^sub>S) (\<star>\<^sub>S) S.\<a> S.\<i> S.src S.trg f g \<eta> \<epsilon>"   and "adjunction_in_bicategory (\<cdot>\<^sub>S) (\<star>\<^sub>S) S.\<a> S.\<i> S.src S.trg f g \<eta> \<epsilon>'" shows "\<epsilon> = \<epsilon>'"
x_def: shows "x \<equiv> getFr [xx, yy, zz, z] [t] []"
redT_updTs_new_thread: fixes ts :: "'i \<Rightarrow> ('g \<times> 'h \<Rightarrow>f nat) option"   and tas :: "('i, 'g, 'j) new_thread_action list"   and t :: "'i"   and x :: "'g"   and w :: "'h \<Rightarrow>f nat" assumes "redT_updTs ts tas t = \<lfloor>(x, w)\<rfloor>"   and "thread_oks ts tas"   and "ts t = None" shows "\<exists>m. NewThread t x m \<in> set tas \<and> w = no_wait_locks"
hyps: fixes A :: "'a mat" assumes "A \<in> carrier_mat n m"   and "I < n" shows "swap_row_to_front A I = mat n m (\<lambda>a. case a of (i, j) \<Rightarrow> if i = 0 then A $$ (I, j) else if i \<le> I then A $$ (i - 1, j) else A $$ (i, j))"
vintersection_function_mono: fixes f :: "V \<Rightarrow> V"   and A :: "V"   and B :: "V" assumes "mono f" shows "f (A \<inter>\<^sub>\<circ> B) \<subseteq>\<^sub>\<circ> f A \<inter>\<^sub>\<circ> f B"
iteratei_rule_P: fixes S :: "('c, unit) RBT.rbt"   and I :: "'c set \<Rightarrow> 'd \<Rightarrow> bool"   and \<sigma>0 :: "'d"   and c :: "'d \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and P :: "'d \<Rightarrow> bool" assumes "rs.invar S"   and "I (rs.\<alpha> S) \<sigma>0"   and "\<And>x it \<sigma>. \<lbrakk>c \<sigma>; x \<in> it; it \<subseteq> rs.\<alpha> S; I it \<sigma>\<rbrakk> \<Longrightarrow> I (it - {x}) (f x \<sigma>)"   and "\<And>\<sigma>. I {} \<sigma> \<Longrightarrow> P \<sigma>"   and "\<And>\<sigma> it. \<lbrakk>it \<subseteq> rs.\<alpha> S; it \<noteq> {}; \<not> c \<sigma>; I it \<sigma>\<rbrakk> \<Longrightarrow> P \<sigma>" shows "P (rs.iteratei S c f \<sigma>0)"
Fall_def: fixes p :: "pvar"   and \<phi> :: "'a dfmla" shows "Fall p \<phi> \<equiv> Neg (Exi p (Neg \<phi>))"
fmrestrict_fset_null: fixes m :: "('c, 'd) fmap" shows "fmrestrict_fset {||} m = fmempty"
refl: fixes A :: "'a set"   and h :: "'a \<Rightarrow> 'b"   and bo :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and m :: "'a \<Rightarrow> 'a \<Rightarrow> 'a option"   and x :: "'a" assumes "merge_on A h bo m"   and "x \<in> UNIV" shows "bo x x"
sum_nonneg: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "\<And>x. x \<in> A \<Longrightarrow> bot \<le> f x" shows "bot \<le> sup_monoid.sum f A"
valid_abs_init_1: fixes problem :: "(char list \<times> nat option \<times> char list list) list \<times> nat list \<times> (nat \<times> nat) list \<times> (char list \<times> (nat \<times> nat) list \<times> ((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list \<times> nat) list"   and v :: "nat" assumes "abs_ast_prob problem" shows "(ast_problem.abs_ast_initial_state problem v \<noteq> None) = (v \<in> set (ast_problem.abs_ast_variable_section problem))"
arrays_on_Ref_set: fixes p :: "'b ref"   and x :: "'b"   and h :: "heap"   and t :: "typerep"   and i :: "nat" shows "arrays (Ref.set p x h) t i = arrays h t i"