hcju/isabelleps huggingface dataset:
all_shared_append: fixes xs :: "'b memref list"   and ys :: "'b memref list" shows "all_shared (xs @ ys) = all_shared xs \<union> all_shared ys"
ex_un_null: shows "\<exists>!n. \<forall>f. B n f = n \<and> B f n = n"
connected_component_UNIV: fixes x :: "'a" shows "connected_component_set UNIV x = UNIV"
3: shows "\<not> minarc h \<le> - minarc h"
inject: fixes x :: "real"   and ya :: "real" shows "(Rmin x = Rmin ya) = (x = ya)"
sum_pos: fixes I :: "'a set"   and f :: "'a \<Rightarrow> 'b multiset" assumes "finite I"   and "I \<noteq> {}"   and "\<And>i. i \<in> I \<Longrightarrow> {#} \<subset># f i" shows "{#} \<subset># comm_monoid_add.sum (+) {#} f I"
identity_functor_axioms: shows "identity_functor (\<cdot>\<^sub>C)"
purge_Cons_in: fixes u :: "'a"   and G :: "'a set"   and c :: "'b"   and ucl :: "('a \<times> 'b) list" assumes "u \<notin> G" shows "GM_sec_model.purge G ((u, c) # ucl) = (u, c) # GM_sec_model.purge G ucl"
cont_eff_def: fixes cf :: "('test, 'atom, 'choice) cmd \<times> 'state"   and b :: "nat" shows "cont_eff cf b = snd (wt_cont_eff (fst cf) (snd cf) b)"
red_setI: fixes f :: "'a \<Rightarrow>\<^sub>0 'd"   and F :: "('a \<Rightarrow>\<^sub>0 'd) set"   and p :: "'a \<Rightarrow>\<^sub>0 'd"   and q :: "'a \<Rightarrow>\<^sub>0 'd"   and t :: "'a" assumes "f \<in> F"   and "punit.red_single p q f t" shows "punit.red F p q"
cm_neg_lu: shows "continuous_map (lsphere n) (usphere n) neg"
mod_pure_conv: fixes R :: "('a \<times> 'b) set"   and a :: "'b"   and b :: "'a"   and h :: "heap"   and as :: "nat set" shows "((h, as) \<Turnstile> pure R a b) = (as = {} \<and> (b, a) \<in> R)"
False: shows "\<nexists>k. f \<bullet> ns k = \<zero>"
SC3Q: fixes p :: "('b, 'c) Combinators list"   and l :: "'b set list" assumes "singleCombinators p" shows "singleCombinators (qsort p l)"
intros: fixes X :: "'k set"   and Y :: "'k set"   and R :: "('k \<times> 'k) set" assumes "finite X"   and "finite Y"   and "Y \<noteq> {}"   and "\<And>x. x \<in> X \<Longrightarrow> \<exists>xa\<in>Y. (x, xa) \<in> R" shows "(X, Y) \<in> max_ext R"
fds_mangoldt_times_zeta: shows "fds mangoldt * fds_zeta = fds (\<lambda>x. of_real (ln (real x)))"
real_of_vreal_def: shows "real_of_vreal = inv vreal_of_real"
F_Bot: shows "F Bot = {(s, X). front_tickFree s}"
HFail_HInv4b: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and q :: "Proc" assumes "HFail s s' p"   and "HInv4b s q" shows "HInv4b s' q"
INF_less_iff: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and f :: "'b \<Rightarrow> 'a"   and A :: "'b set"   and a :: "'a" assumes "class.complete_linorder Inf Sup inf less_eq less sup bot top" shows "less (Inf (f ` A)) a = (\<exists>x\<in>A. less (f x) a)"
cases: fixes a :: "'b"   and r :: "('b \<times> 'b) set" assumes "a \<in> Wellfounded.acc r" obtains x :: "'b" where "a = x" and "\<And>y. (y, x) \<in> r \<Longrightarrow> y \<in> Wellfounded.acc r"
primepow_power_iff_nat: fixes p :: "nat"   and n :: "nat" assumes "0 < p" shows "primepow (p ^ n) = (primepow p \<and> 0 < n)"
par_Int_bot: fixes P :: "'a process"   and A :: "'a set" shows "(P \<lbrakk>A\<rbrakk> \<bottom>) = \<bottom>"
assms_7: shows "det3 a b d \<noteq> 0"
eq\<^sub>g: shows "(g \<star> \<theta>\<^sub>g) \<cdot> \<a>[g, tab\<^sub>0 g, w\<^sub>g] \<cdot> (g.tab \<star> w\<^sub>g) = ((g \<star> \<theta>\<^sub>g') \<cdot> \<a>[g, tab\<^sub>0 g, w\<^sub>g'] \<cdot> (g.tab \<star> w\<^sub>g')) \<cdot> \<beta>\<^sub>g"
simps_2: fixes x2 :: "'c"   and y2 :: "'c" shows "(Inr x2 = Inr y2) = (x2 = y2)"
intro: fixes \<phi> :: "Formula.formula" assumes "Formula.future_bounded \<phi>" shows "future_bounded_mfodl \<phi>"
STEP: shows "(\<Sigma>, \<Sigma>') \<in> brw_step \<delta>"
sort_les_def: fixes cs :: "('a \<times> 'a) set"   and s1 :: "'a set"   and s2 :: "'a set" shows "sort_les cs s1 s2 = (sort_leq cs s1 s2 \<and> \<not> sort_leq cs s2 s1)"
IH: fixes s :: "'variable \<Rightarrow> bool option" assumes "are_all_operators_applicable s ops"   and "are_all_operator_effects_consistent ops"   and "are_all_operators_non_interfering ops" shows "execute_parallel_operator s ops = execute_serial_plan s ops"
sort_oalist_def: fixes xs :: "('a \<times> 'c) list" shows "cmp2.sort_oalist xs = foldr cmp2.update_by_pair xs []"
zone_delay_def: fixes Z :: "('c \<Rightarrow> 'd) set" shows "Z\<^sup>\<up> = {u \<oplus> d |u d. u \<in> Z \<and> (0::'d) \<le> d}"
x: shows "x \<in> results_gpv \<Gamma> gpv"
nxt: fixes q :: "hf"   and x :: "'a" assumes "q \<in> states M" shows "nxt M q x \<in> states M"
strict_coboundedI2: fixes b :: "'a"   and c :: "'a"   and a :: "'a" assumes "b < c" shows "min a b < c"
composition_series_simple_group: shows "simple_group G = (\<HH> = [{\<one>\<^bsub>G\<^esub>}, carrier G])"
lift_map_def: fixes i :: "nat" shows "lift_map i \<equiv> \<lambda>(s, f, uu). (insert_map i s f, uu)"
verit_minus_simplify_2: fixes a :: "'c" shows "a - (0::'c) = a"
Ipoly_CInt: fixes bs :: "'a list"   and i :: "int" shows "\<lparr>(i)\<^sub>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> = of_int i"
mem_convex_alt: fixes S :: "'c set"   and x :: "'c"   and y :: "'c"   and u :: "real"   and v :: "real" assumes "convex S"   and "x \<in> S"   and "y \<in> S"   and "0 \<le> u"   and "0 \<le> v"   and "0 < u + v" shows "(u / (u + v)) *\<^sub>R x + (v / (u + v)) *\<^sub>R y \<in> S"
det_def: fixes A :: "'d mat" shows "Determinant.det A = (if dim_row A = dim_col A then \<Sum>p | p permutes {0..<dim_row A}. signof p * (\<Prod>i = 0..<dim_row A. A $$ (i, p i)) else (0::'d))"
vec_hom_smult: fixes ev :: "int"   and v :: "int vec" shows "of_int_hom.vec_hom (ev \<cdot>\<^sub>v v) = of_int ev \<cdot>\<^sub>v of_int_hom.vec_hom v"
opeV: shows "openin (top_of_set T) (f ` (S \<inter> V))"
assms: fixes x :: "'a" assumes "x \<in> set xs" shows "h \<circ> g x = f x \<circ> h"
bit_unset_bit_iff: 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 m :: "nat"   and a :: "'b"   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)"
nu_par_aux3: fixes x :: "'a"   and y :: "'a" shows "\<nu> (\<nu> x \<parallel> \<tau> y) = \<nu> x \<parallel> \<tau> y"
less_two_pow_divD: fixes x :: "nat"   and n :: "nat"   and m :: "nat" assumes "x < 2 ^ n div 2 ^ m" shows "m \<le> n \<and> x < 2 ^ (n - m)"
distinct_names_f_def: fixes P :: "cld list" shows "distinct_names_f P = distinct (map class_name_f P)"
closed_map_imp_quotient_map: fixes S :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and T :: "'c set" assumes "continuous_on S f"   and "T \<subseteq> f ` S"   and "\<And>T. closedin (top_of_set S) T \<Longrightarrow> closedin (top_of_set (f ` S)) (f ` T)" shows "openin (top_of_set S) (S \<inter> f -` T) = openin (top_of_set (f ` S)) T"
P_t_sum_P_O: fixes x :: "'key \<times> 'message list" shows "prob ((t \<circ> OB) -` {t (OB x)} \<inter> msgs) = (\<Sum>obs\<in>t -` {t (OB x)} \<inter> OB ` msgs. prob (OB -` {obs} \<inter> msgs))"
trie_odlist_lookup_def: shows "trie_odlist_lookup = (\<lambda>M. lookup_trie M \<circ> toList)"
gfp_eq_fixp: fixes f :: "'a \<Rightarrow> 'a" assumes "monotone (\<lambda>x y. y \<le> x) (\<lambda>x y. y \<le> x) f" shows "gfp f = ccpo.fixp Inf (\<lambda>x y. y \<le> x) f"
"1_3": shows "set_of todo \<union> fin \<subseteq> R\<^sup>* `` init"
substitutes_onD: fixes A :: "'d set"   and f :: "'d set \<Rightarrow> 'd set"   and B :: "'d set"   and a :: "'d"   and b :: "'d" assumes "substitutes_on A f"   and "B \<subseteq> A"   and "a \<in> A"   and "b \<in> A"   and "b \<notin> f (insert b B)" shows "b \<notin> f (insert a (insert b B))"
bell10_index_1: shows "|\<beta>\<^sub>1\<^sub>0\<rangle> $$ (0, 0) = complex_of_real (1 / sqrt 2)"
assms_3: shows "descartes_roots_test a b p = 0"
OKI_inj: fixes inp :: "'index \<Rightarrow> 'c option"   and inp' :: "'index \<Rightarrow> 'c option" shows "(OKI inp = OKI inp') = (inp = inp')"
normal_Int_subgroup: fixes H :: "'a set"   and N :: "'a set" assumes "subgroup H (G\<lparr>carrier := S\<rparr>)"   and "N \<lhd> G\<lparr>carrier := S\<rparr>" shows "N \<inter> H \<lhd> G\<lparr>carrier := S, carrier := H\<rparr>"
gp: shows "\<Gamma>' \<approx> (x, A) # \<Gamma>"
list_emb_append_mono: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b list"   and xs' :: "'b list"   and ys :: "'b list"   and ys' :: "'b list" assumes "list_emb P xs xs'"   and "list_emb P ys ys'" shows "list_emb P (xs @ ys) (xs' @ ys')"
keys_update_with: fixes v :: "'b"   and k :: "'a"   and f :: "'b \<Rightarrow> 'b"   and al :: "('a, 'b) alist" shows "keys (update_with v k f al) = insert k (keys al)"
"case": fixes f :: "real \<Rightarrow> real \<Rightarrow> 'b"   and x1 :: "real"   and x2 :: "real" shows "(case Complex x1 x2 of Complex x xa \<Rightarrow> f x xa) = f x1 x2"
prems_4: shows "set (e # Es) \<subseteq> E"
pfp_pfp: fixes f :: "'b \<Rightarrow> 'b"   and x0 :: "'b"   and x :: "'b" assumes "pfp f x0 = Some x" shows "f x \<le> x"
strict_iff_order: fixes b :: "'c list"   and a :: "'c list" shows "strict_suffix b a = (suffix b a \<and> a \<noteq> b)"
cross7_minus_right: fixes x :: "(real, 7) vec"   and y :: "(real, 7) vec" shows "x \<times>\<^sub>7 - y = - (x \<times>\<^sub>7 y)"
tdghm_dghm_comp_components_4: fixes \<NN> :: "V"   and \<HH> :: "V" shows "(\<NN> \<circ>\<^sub>T\<^sub>D\<^sub>G\<^sub>H\<^sub>M\<^sub>-\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<HH>)\<lparr>NTDGDom\<rparr> = \<HH>\<lparr>HomDom\<rparr>"
ge_one_powr_ge_zero: fixes x :: "real"   and a :: "real" assumes "1 \<le> x"   and "0 \<le> a" shows "1 \<le> x powr a"
segment_eq_empty_1: fixes a :: "'a"   and b :: "'a" shows "closed_segment a b \<noteq> {}"
list_eq_sum: fixes m :: "real"   and xs :: "real list" shows "sum_list (list_eq xs m) = m * real (length (list_eq xs m))"
mult_matrix_zero_left: fixes fadd :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and fmul :: "'d \<Rightarrow> 'c \<Rightarrow> 'b"   and A :: "'c matrix" assumes "fadd (0::'b) (0::'b) = (0::'b)"   and "\<forall>a. fmul (0::'d) a = (0::'b)" shows "mult_matrix fmul fadd 0 A = 0"
HPhase1or2ReadThen_valueChosen2: fixes s :: "state"   and s' :: "state"   and q :: "Proc"   and d :: "Disk"   and pp :: "Proc"   and D :: "Disk set"   and b :: "nat"   and p :: "Proc" assumes "HPhase1or2ReadThen s s' q d pp"   and "\<forall>d\<in>D. b \<le> bal (disk s d p) \<and> (\<forall>q. phase s q = 1 \<and> b \<le> mbal (dblock s q) \<and> hasRead s q d p \<longrightarrow> (\<exists>br\<in>blocksRead s q d. b \<le> bal (block br)))" shows "\<forall>d\<in>D. b \<le> bal (disk s' d p) \<and> (\<forall>q. phase s' q = 1 \<and> b \<le> mbal (dblock s' q) \<and> hasRead s' q d p \<longrightarrow> (\<exists>br\<in>blocksRead s' q d. b \<le> bal (block br)))"
case_optionE: fixes P :: "bool"   and Q :: "'x \<Rightarrow> bool"   and x :: "'x option" assumes "case x of None \<Rightarrow> P | Some y \<Rightarrow> Q y" obtains "x = None" and "P"   | y :: "'x" where "x = Some y" and "Q y"
assms_2: assumes "0 < degree_m f"   and "\<And>g h. \<lbrakk>degree_m g < degree_m f; degree_m h < degree_m f\<rbrakk> \<Longrightarrow> Mp f \<noteq> Mp (g * h)" shows "thesis"
induct: fixes P :: "v list \<Rightarrow> nat \<Rightarrow> bool"   and a0 :: "v list"   and a1 :: "nat" assumes "\<And>n. P [] n"   and "\<And>v va. P (v # va) 0"   and "\<And>e es n. P es n \<Longrightarrow> P (e # es) (Suc n)" shows "P a0 a1"
simps_3: fixes v :: "'b"   and va :: "'b"   and vb :: "'c"   and l :: "'b set list" shows "smaller (AllowPortFromTo v va vb) DenyAll l = False"
bind_lunit: fixes x :: "'d"   and f :: "'d \<Rightarrow> 'c option" shows "Option.bind (Some x) f = f x"
sup_mono: fixes a :: "'b multiset"   and c :: "'b multiset"   and b :: "'b multiset"   and d :: "'b multiset" assumes "a \<subseteq># c"   and "b \<subseteq># d" shows "a \<union># b \<subseteq># c \<union># d"
SI_def: fixes \<V> :: "'b V_rec"   and Tr :: "'b list set" shows "SI \<V> Tr \<equiv> \<forall>\<alpha> \<beta>. \<forall>c\<in>C\<^bsub>\<V>\<^esub>. \<beta> @ \<alpha> \<in> Tr \<and> \<alpha> \<upharpoonleft> C\<^bsub>\<V>\<^esub> = [] \<longrightarrow> \<beta> @ [c] @ \<alpha> \<in> Tr"
cnorm_ge_square: fixes a :: "real"   and x :: "'a" shows "(a \<le> norm x) = (a \<le> 0 \<or> complex_of_real (a\<^sup>2) \<le> cinner x x)"
remove_child_preserves_known_ptrs: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and child :: "(_) node_ptr"   and h' :: "(_) heap" assumes "Shadow_DOM.heap_is_wellformed h"   and "ShadowRootClass.type_wf h"   and "ShadowRootClass.known_ptrs h"   and "h \<turnstile> Shadow_DOM.remove_child ptr child \<rightarrow>\<^sub>h h'" shows "ShadowRootClass.known_ptrs h'"
zero_one_infty_not_equal_5: shows "\<infinity>\<^sub>h \<noteq> 0\<^sub>h"
ring_hom_closed: fixes h :: "'e \<Rightarrow> 'f"   and R :: "('e, 'g) ring_scheme"   and S :: "('f, 'h) ring_scheme"   and x :: "'e" assumes "h \<in> ring_hom R S"   and "x \<in> carrier R" shows "h x \<in> carrier S"
closedin_compact_space: fixes X :: "'d topology"   and S :: "'d set" assumes "compact_space X"   and "closedin X S" shows "compactin X S"
simps: fixes a1 :: "('procS, 'procT) Proc"   and a2 :: "('procS, 'procT) Proc"   and SRel :: "('procS \<times> 'procS) set"   and TRel :: "('procT \<times> 'procT) set" shows "a1 \<R>\<lbrakk>\<cdot>\<rbrakk>R<SRel,TRel> a2 = ((\<exists>S. S \<in>S a1 \<and> \<lbrakk>S\<rbrakk> \<in>T a2) \<or> (\<exists>S1 S2. S1 \<in>S a1 \<and> S2 \<in>S a2 \<and> (S1, S2) \<in> SRel) \<or> (\<exists>T1 T2. T1 \<in>T a1 \<and> T2 \<in>T a2 \<and> (T1, T2) \<in> TRel))"
P_def: shows "P = (\<lambda>a. if a \<in> (\<Oplus>G\<leftarrow>[W', U]. G) then W'\<oplus>U\<leftarrow> a ! 1 else (0::'v))"
assms: shows "{x \<in> \<Omega>. P x} \<in> M"
bot_eq_true: shows "R (false \<turnstile> true) = R true"
hcomp_assoc: fixes \<mu> :: "('a E.term, 'a) arr"   and \<nu> :: "('a E.term, 'a) arr"   and \<tau> :: "('a E.term, 'a) arr" assumes "arr \<mu>"   and "arr \<nu>"   and "arr \<tau>"   and "src \<mu> = trg \<nu>"   and "src \<nu> = trg \<tau>" shows "(\<mu> \<star> \<nu>) \<star> \<tau> = \<mu> \<star> \<nu> \<star> \<tau>"
inf_sup_absorb: fixes x :: "'a"   and y :: "'a" shows "x \<sqinter> (x + y) = x"
simps_2: fixes k :: "'a"   and p :: "'b"   and l :: "('a \<times> 'b) tree"   and k1 :: "'a"   and p1 :: "'b"   and r :: "('a \<times> 'b) tree" shows "ins k p \<langle>l, (k1, p1), r\<rangle> = (if k < k1 then case ins k p l of \<langle>l2, (k2, p2), r2\<rangle> \<Rightarrow> if p1 \<le> p2 then \<langle>\<langle>l2, (k2, p2), r2\<rangle>, (k1, p1), r\<rangle> else \<langle>l2, (k2, p2), \<langle>r2, (k1, p1), r\<rangle>\<rangle> else if k1 < k then case ins k p r of \<langle>l2, (k2, p2), r2\<rangle> \<Rightarrow> if p1 \<le> p2 then \<langle>l, (k1, p1), \<langle>l2, (k2, p2), r2\<rangle>\<rangle> else \<langle>\<langle>l, (k1, p1), l2\<rangle>, (k2, p2), r2\<rangle> else \<langle>l, (k1, p1), r\<rangle>)"
residue_mult: fixes k :: "nat"   and x :: "int"   and y :: "int" shows "x \<otimes>\<^bsub>residue_ring (p ^ k)\<^esub> y = x * y mod p ^ k"
Liminf_le_Limsup: fixes F :: "'b filter"   and f :: "'b \<Rightarrow> 'c" assumes "F \<noteq> bot" shows "Liminf F f \<le> Limsup F f"
usc_limsup: fixes x0 :: "'b"   and f :: "'b \<Rightarrow> ereal" shows "usc_at x0 f = (\<forall>x. x \<longlonglongrightarrow> x0 \<longrightarrow> limsup (f \<circ> x) \<le> f x0)"
integral_component_lbound: fixes f :: "'a \<Rightarrow> 'b"   and a :: "'a"   and b :: "'a"   and B :: "real"   and k :: "'b" assumes "f integrable_on cbox a b"   and "\<forall>x\<in>cbox a b. B \<le> f x \<bullet> k"   and "k \<in> Basis" shows "B * content (cbox a b) \<le> integral (cbox a b) f \<bullet> k"
intro: fixes A :: "'b llist set"   and N :: "enat" assumes "\<And>e. e \<in> A \<Longrightarrow> \<exists>p. lprefix (llist_of p) e \<and> (\<forall>e'. lprefix (llist_of p) e' \<longrightarrow> llength e' = 2 * N \<longrightarrow> e' \<in> A)" shows "GaleStewartDefensiveStrategies.closed_GSgame_axioms A N"
distinct_7: shows "UMinusOp \<noteq> numeric_unop.ToIntegerOp"
bind_return_pmf: fixes x :: "'b"   and f :: "'b \<Rightarrow> 'a pmf" shows "return_pmf x \<bind> f = f x"
inv_Characters: fixes G :: "('c, 'd) monoid_scheme"   and \<chi> :: "'c \<Rightarrow> complex" assumes "character G \<chi>" shows "inv\<^bsub>Characters G\<^esub> \<chi> = inv_character \<chi>"
strict_subset_subsumption_redundant_clause: fixes D :: "'a literal multiset"   and \<sigma> :: "'s"   and C :: "'a literal multiset" assumes "D \<cdot> \<sigma> \<subset># C"   and "is_ground_subst \<sigma>" shows "C \<in> sr.Rf (grounding_of_cls D)"
assms_2: fixes s :: "'b set" assumes "s \<in> set ss" shows "local.subspace s"
intros: fixes ta' :: "('g, 'h, 'x, 'i, 'j, 'k) thread_action"   and ta :: "('g, 'h, 'l, 'i, 'j, 'k) thread_action" assumes "collect_locks' \<lbrace>ta'\<rbrace>\<^bsub>l\<^esub> \<subseteq> collect_locks \<lbrace>ta\<rbrace>\<^bsub>l\<^esub>"   and "collect_cond_actions \<lbrace>ta'\<rbrace>\<^bsub>c\<^esub> \<subseteq> collect_cond_actions \<lbrace>ta\<rbrace>\<^bsub>c\<^esub>"   and "collect_interrupts \<lbrace>ta'\<rbrace>\<^bsub>i\<^esub> \<subseteq> collect_interrupts \<lbrace>ta\<rbrace>\<^bsub>i\<^esub>" shows "actions_subset ta' ta"
rel_mono: fixes R1 :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and R1a :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and R2 :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and R2a :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and R3 :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and R3a :: "'g \<Rightarrow> 'h \<Rightarrow> bool" assumes "R1 \<le> R1a"   and "R2 \<le> R2a"   and "R3 \<le> R3a" shows "rel_pre_GF\<^sub>M R1 R2 R3 \<le> rel_pre_GF\<^sub>M R1a R2a R3a"
not_less_iff_gr_or_eq: fixes y :: "'b"   and x :: "'b" shows "(\<not> y < x) = (x < y \<or> x = y)"
nact_exists: fixes n :: "nat"   and c :: "'id"   and t :: "nat \<Rightarrow> cnf" assumes "\<exists>i\<ge>n. \<parallel>c\<parallel>\<^bsub>t i\<^esub>" shows "\<exists>i\<ge>n. \<parallel>c\<parallel>\<^bsub>t i\<^esub> \<and> \<not> (\<exists>k\<ge>n. k < i \<and> \<parallel>c\<parallel>\<^bsub>t k\<^esub>)"
liftL_type_1: fixes \<Gamma> :: "nat \<Rightarrow> Syntax.type"   and \<Delta> :: "nat \<Rightarrow> Syntax.type"   and t :: "trm"   and T :: "Syntax.type"   and U :: "Syntax.type" assumes "\<Gamma> , \<Delta> \<turnstile>\<^sub>T t : T" shows "\<forall>k. \<Gamma>\<langle>k:U\<rangle> , \<Delta> \<turnstile>\<^sub>T liftL_trm t k : T"
bound_on_all_plans_bounds_problem_plan_bound_charles: fixes P :: "(('c, bool) fmap \<times> ('c, bool) fmap) set \<Rightarrow> bool"   and f :: "(('c, bool) fmap \<times> ('c, bool) fmap) set \<Rightarrow> nat" assumes "\<forall>PROB as s. P PROB \<and> finite PROB \<and> s \<in> valid_states PROB \<and> as \<in> valid_plans PROB \<longrightarrow> (\<exists>as'. exec_plan s as = exec_plan s as' \<and> subseq as' as \<and> length as' < f PROB)" shows "\<forall>PROB. P PROB \<and> finite PROB \<longrightarrow> problem_plan_bound_charles PROB < f PROB"
wf_plan_act_is_sound': fixes \<pi> :: "plan_action" assumes "wf_plan_action \<pi>" shows "sound_opr (resolve_instantiate \<pi>) (pddl_opr_to_act (resolve_instantiate \<pi>))"
of_int_0: 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" assumes "class.ring_1 one times plus zero minus uminus" shows "ring_1.of_int one plus zero minus 0 = zero"
to_nat_less_nrows: fixes a :: "'b"   and A :: "(('c, 'd) Finite_Cartesian_Product.vec, 'b) Finite_Cartesian_Product.vec" shows "mod_type_class.to_nat a < nrows A"
fpxs_val_def: fixes f :: "'b fpxs" shows "fpxs_val f = rat_of_int (fls_subdegree (fls_of_fpxs f)) / rat_of_nat (fpxs_root_order f)"
UN_least: fixes A :: "'c set"   and B :: "'c \<Rightarrow> 'd set"   and C :: "'d set" assumes "\<And>x. s2p A x \<Longrightarrow> B x \<subseteq> C" shows "\<mu> (\<P> B A) \<subseteq> C"
support_postList: fixes xs :: "(char list \<times> ((char list \<Rightarrow> int) \<Rightarrow> nat)) list" shows "support (postList xs) \<subseteq> lesvars xs"
simps_2: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a" shows "msort f [x] = [x]"
arr_iff_in_hom: fixes f :: "(bool, 'arr list) concrete_category.arr" shows "arr f = \<guillemotleft>f : local.dom f \<rightarrow> cod f\<guillemotright>"
s: fixes i :: "'c" assumes "i \<in> I" shows "subgroup (S i) G"
n_absorb2: fixes x :: "'a"   and y :: "'a" shows "!x \<cdot> (!x \<oplus> !y) = !x"
ifex_upd_other: fixes v :: "'a"   and i :: "'a ifex"   and a :: "'a \<Rightarrow> bool"   and any :: "bool" assumes "v \<notin> ifex_var_set i" shows "val_ifex i (a(v := any)) = val_ifex i a"
abs_minus_commute: fixes abs :: "'e \<Rightarrow> 'e"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and uminus :: "'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and a :: "'e"   and b :: "'e" assumes "class.ordered_ab_group_add_abs abs plus zero minus uminus less_eq less" shows "abs (minus a b) = abs (minus b a)"
bi_unique_mod_ring_rel32_1: shows "bi_unique mod_ring_rel32"
indicator_simps_1: fixes x :: "'d"   and S :: "'d set" assumes "x \<in> S" shows "indicator S x = (1::'e)"
word_le_def: fixes a :: "'a word"   and b :: "'a word" shows "(a \<le> b) = (uint a \<le> uint b)"
state_rank_eq_rank_SOME: fixes q :: "'b"   and n :: "nat" shows "state_rank q n = (if configuration q n \<noteq> {} then rank (SOME x. x \<in> configuration q n) n else None)"
Rangep_Range_eq: fixes r :: "('b \<times> 'a) set" shows "Rangep (\<lambda>x xa. (x, xa) \<in> r) = (\<lambda>x. x \<in> Range r)"
eventually_prod_filter: fixes P :: "'c \<times> 'd \<Rightarrow> bool"   and F :: "'c filter"   and G :: "'d filter" shows "eventually P (F \<times>\<^sub>F G) = (\<exists>Pf Pg. eventually Pf F \<and> eventually Pg G \<and> (\<forall>x y. Pf x \<longrightarrow> Pg y \<longrightarrow> P (x, y)))"
u_3: fixes i :: "nat" shows "ereal (u i) < b"
TUren_frechet: fixes I :: "('sf, 'b, 'sz) interp"   and \<theta> :: "('sf, 'sz) trm"   and x :: "'sz"   and y :: "'sz"   and \<nu> :: "(real, 'sz) vec"   and \<nu>' :: "(real, 'sz) vec" assumes "is_interp I"   and "dfree \<theta>" shows "frechet I (TUrename x y \<theta>) \<nu> \<nu>' = frechet I \<theta> (RSadj x y \<nu>) (RSadj x y \<nu>')"
dim_span_eq_card_independent: fixes B :: "'c set" assumes "vs2.independent B" shows "vs2.dim (vs2.span B) = card B"
P_invariant_steps: fixes a :: "'a"   and as :: "'a list" assumes "steps (a # as)" shows "list_all P as"
simps_1: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'c"   and a :: "'c" shows "foldl f a [] = a"
finite_pderivs_lang_UNIV1: fixes r :: "'a rexp" shows "finite (pderivs_lang UNIV1 r)"
grr: fixes S :: "'b set"   and c :: "nat"   and a :: "'b" assumes "Suc (card S) \<le> c" shows "card (a \<triangleright> S) \<le> c"
homotopic_with_symD: fixes P :: "('b \<Rightarrow> 'c) \<Rightarrow> bool"   and X :: "'b topology"   and Y :: "'c topology"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "homotopic_with P X Y f g" shows "homotopic_with P X Y g f"