hcju/isabelleps huggingface dataset:
AssociationClassEndCallTE: fixes \<Gamma> :: "String.literal \<rightharpoonup>\<^sub>f 'a OCL_Types.type"   and src :: "'a expr"   and k :: "call_kind"   and role :: "String.literal"   and \<tau> :: "'a OCL_Types.type" assumes "\<Gamma> \<turnstile>\<^sub>E AssociationClassEndCall src k role : \<tau>" obtains \<A> :: "'a" and "end" :: "'a \<times> nat \<times> enat \<times> bool \<times> bool" where "k = DotCall" and "\<tau> = class_assoc_end_type end" and "\<Gamma> \<turnstile>\<^sub>E src : \<langle>\<A>\<rangle>\<^sub>\<T>[1]" and "association_class_end \<A> role end"
node_silent_change_only: fixes i :: "nat"   and R\<^sub>i :: "nat set" shows "\<langle>i : opaodv i \<langle>\<langle>\<^bsub>i\<^esub> qmsg : R\<^sub>i\<rangle>\<^sub>o \<Turnstile>\<^sub>A (\<lambda>\<sigma> _. oarrivemsg (\<lambda>_ _. True) \<sigma>, other (\<lambda>_ _. True) {i} \<rightarrow>) globala (\<lambda>(\<sigma>, a, \<sigma>'). a \<noteq> \<tau> \<longrightarrow> \<sigma>' i = \<sigma> i)"
proots_within_pCons_1_iff_2: fixes a :: "'a"   and s :: "'a set" shows "proots_within [:a, - (1::'a):] s = (if a \<in> s then {a} else {})"
prems_1: shows "as_1 @ as_2 = a # as"
in_CompE: fixes h :: "'a"   and G :: "'a set"   and F :: "'a set" assumes "h \<in> Maps.Comp G F" obtains f :: "'a" and g :: "'a" where "is_iso_class F" and "is_iso_class G" and "f \<in> F" and "g \<in> G" and "g \<star> f \<cong> h"
graph_finite: fixes ks :: "int list"   and l :: "int" shows "finite {x. dioph_is_node ks l x}"
eq_iff_4: fixes var :: "Syntax.var"   and "term" :: "Syntax.term"   and vara :: "Syntax.var"   and terma :: "Syntax.term" shows "(Rec var term = Rec vara terma) = ([[atom var]]lst. term = [[atom vara]]lst. terma)"
is_run_def: fixes r :: "nat \<Rightarrow> 'a node" shows "gba.is_run r \<equiv> r 0 \<in> gba.V0 \<and> ipath gba.E r"
plussub_def: fixes x :: "'b"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'a"   and y :: "'c" shows "x \<squnion>\<^bsub>f\<^esub> y = f x y"
inj_on_generator: shows "inj_on (([^]) \<^bold>g) {..<order G}"
less_eq_Suc_le: fixes n :: "nat"   and m :: "nat" shows "(n < m) = (Suc n \<le> m)"
sturm_squarefree'_def: fixes p :: "real poly" shows "sturm_squarefree' p = (let d = gcd p (pderiv p) in map (\<lambda>p'. p' div d) (sturm p))"
singular_face_def: fixes p :: "nat"   and k :: "nat"   and f :: "(nat \<Rightarrow> real) \<Rightarrow> 'c" shows "singular_face p k f \<equiv> restrict (f \<circ> simplical_face k) (standard_simplex (p - Suc 0))"
intro: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and u_block_size :: "int"   and grouping :: "int"   and min_index :: "int" assumes "block_design point_set block_collection u_block_size"   and "t_packing_design_axioms point_set block_collection u_block_size grouping min_index" shows "t_packing_design point_set block_collection u_block_size grouping min_index"
linear_orderable_6_implies_8: fixes x :: "'a" assumes "linear_orderable_6 x" shows "asymmetric (x\<^sup>+) \<and> strict_linear (x\<^sup>+)"
measurable_emeasure_Pair: fixes Q :: "('c \<times> 'a) set"   and N :: "'c measure" assumes "Q \<in> sets (N \<Otimes>\<^sub>M M)" shows "(\<lambda>x. emeasure M (Pair x -` Q)) \<in> borel_measurable N"
from_list_correct'_1: fixes l :: "('b \<times> 'b \<times> nat) list" assumes "valid_graph_rep l" shows "invar (from_list l)"
sum_distrib_right: fixes f :: "'c \<Rightarrow> 'a"   and A :: "'c set"   and r :: "'a" shows "sum f A * r = (\<Sum>n\<in>A. f n * r)"
set_eq_fold: fixes x :: "'b"   and xs :: "'b list" shows "Max (set (x # xs)) = fold max xs x"
h_minus_finite: fixes i :: "nat"   and e :: "'v \<times> 'v" shows "h_minus i e \<noteq> \<top>"
Norm_triangular_ineq_diff: fixes x :: "'b"   and N :: "'b quasinorm"   and y :: "'b" assumes "x \<in> space\<^sub>N N" shows "Norm N (x - y) \<le> defect N * Norm N x + defect N * Norm N y"
assms_10: shows "phiArg g \<phi>\<^sub>r r"
boundary_def: fixes X :: "'b set" shows "boundary X = closure X - interior X"
flow_initial_time_if: fixes x0 :: "'a" shows "flow t0 x0 t0 = (if t0 \<in> T \<and> x0 \<in> X then x0 else (0::'a))"
m_1: shows "\<forall>t\<in>S_Min_Le. t \<le> m"
assms_2: assumes "seq g f"   and "seq f g"   and "g \<cdot> f = local.dom f"   and "g \<cdot> f = cod g" shows "T"
assms: fixes x :: "'a"   and y :: "'a" assumes "(x, y) \<in> (rsp_on A f)\<^sup>+" shows "(y, x) \<notin> rwp_on A f"
get_root_node_same_no_parent_parent_child_rel: fixes h :: "(_) heap"   and ptr' :: "(_) object_ptr" assumes "heap_is_wellformed h"   and "type_wf h"   and "known_ptrs h"   and "h \<turnstile> get_root_node ptr' \<rightarrow>\<^sub>r ptr'" shows "\<nexists>p. (p, ptr') \<in> parent_child_rel h"
crdt_ops_ref_exists: fixes pre :: "('a \<times> 'b) list"   and oid :: "'a"   and oper :: "'b"   and suf :: "('a \<times> 'b) list"   and deps :: "'b \<Rightarrow> 'a set"   and ref :: "'a" assumes "crdt_ops (pre @ (oid, oper) # suf) deps"   and "ref \<in> deps oper" shows "ref \<in> fst ` set pre"
leq_max_sup: fixes m :: "nat"   and n :: "nat" shows "m \<le> n \<longrightarrow> Max {m..n} = n"
invertible_image_lemma_2: fixes a :: "'a" assumes "invertible a"   and "a \<in> M" shows "\<eta> (inverse a) \<cdot>' \<eta> a = \<one>'"
closed_Compl: fixes S :: "'a set" assumes "open S" shows "closed (- S)"
ev_2: shows "\<forall>\<^sub>F n in F. (f n + c) / (g n + d) \<le> u"
ground_subterm: fixes t :: "('b, 'a) Term.term"   and t' :: "('b, 'a) Term.term" assumes "fv t = {}"   and "t' \<sqsubseteq> t" shows "fv t' = {}"
poly_of_vars_simps_1: fixes x :: "char list"   and xs :: "char list multiset" shows "poly_of_vars (add_mset x xs) = Var (\<phi> x) * poly_of_vars xs"
lcp_first_mismatch': fixes a :: "'b"   and b :: "'b"   and u :: "'b list" assumes "a \<noteq> b" shows "u \<cdot> [a] \<and>\<^sub>p u \<cdot> [b] = u"
dim_empty: fixes scale :: "'b \<Rightarrow> 'c \<Rightarrow> 'c" assumes "vector_space scale" shows "vector_space.dim scale {} = 0"
map_default_def: fixes b :: "'a"   and f :: "'b \<Rightarrow> 'a option"   and a :: "'b" shows "map_default b f a \<equiv> case f a of None \<Rightarrow> b | Some b' \<Rightarrow> b'"
dcommutes: shows "\<forall>act\<in>Acts F. \<forall>s\<in>T. \<forall>t. (s, t) \<in> relcl L \<longrightarrow> s \<in> B \<or> t \<in> B \<or> (funof act s, funof act t) \<in> relcl L"
seq_lexord_irrefl: fixes R :: "('b \<times> 'b) set" assumes "irrefl R" shows "irrefl (seq_lexord R)"
sum_set_cond_linear: fixes P :: "'a set \<Rightarrow> bool"   and f :: "'a set \<Rightarrow> 'b set"   and I :: "'c set"   and S :: "'c \<Rightarrow> 'a set" assumes "\<And>A B. \<lbrakk>P A; P B\<rbrakk> \<Longrightarrow> P (A + B)"   and "P {0::'a}"   and "\<And>A B. \<lbrakk>P A; P B\<rbrakk> \<Longrightarrow> f (A + B) = f A + f B"   and "f {0::'a} = {0::'b}"   and "\<And>i. i \<in> I \<Longrightarrow> P (S i)" shows "f (sum S I) = sum (f \<circ> S) I"
simps_2: fixes a :: "real" shows "Im (cis a) = sin a"
that: fixes p :: "'a set set" assumes "p division_of cbox a b"   and "cbox c d \<in> p" shows "thesis"
dim_eq_card: fixes B :: "('b, 'c) vec set"   and V :: "('b, 'c) vec set" assumes "vec.span B = vec.span V"   and "vec.independent B" shows "vec.dim V = card B"
simps: fixes q :: "'b"   and n :: "nat" shows "configuration q n = {x. x \<le> n \<and> token_run x n = q}"
min_union_idem: fixes A :: "'b fset set" shows "A \<union>\<^sub>m A = min_set A"
minus_float_def: fixes a :: "('c, 'd) IEEE.float"   and b :: "('c, 'd) IEEE.float" shows "a - b = fsub To_nearest a b"
isFun_def: fixes f :: "ZF" shows "isFun f \<equiv> \<forall>x y1 y2. Elem (Opair x y1) f \<and> Elem (Opair x y2) f \<longrightarrow> y1 = y2"
ide_dom: fixes f :: "'a1" assumes "C1.arr f" shows "C1.ide (C1.dom f)"
swap_list: fixes a :: "nat"   and L :: "'a list"   and b :: "nat"   and xa :: "nat" assumes "a < length L"   and "b < length L" shows "nth_default (0::'a) (L[b := L ! a, a := L ! b]) (swap a b xa) = nth_default (0::'a) L xa"
intros_2: fixes n :: "nat"   and l :: "'b tree23"   and r :: "'b tree23"   and p :: "'b" assumes "full n l"   and "full n r" shows "full (Suc n) \<langle>l, p, r\<rangle>"
Yield: fixes g :: "'d \<Rightarrow> ('c, 'd) step"   and s :: "'d"   and x :: "'c"   and s' :: "'d" assumes "g s = Yield x s'" shows "productive_onp g s"
lookupb_unicity: fixes x :: "name"   and a :: "coname"   and \<theta>_c :: "(coname \<times> name \<times> trm) list"   and c :: "coname"   and P :: "trm"   and y :: "name"   and b :: "coname" assumes "lookupb x a \<theta>_c c P = Ax y b"   and "b \<sharp> (\<theta>_c, c, P)"   and "y \<sharp> \<theta>_c" shows "x = y \<and> a = b"
connected_def: fixes G :: "('a, 'b) graph" assumes "valid_unMultigraph G" shows "valid_unMultigraph.connected G \<equiv> \<forall>v\<in>nodes G. \<forall>v'\<in>nodes G. v \<noteq> v' \<longrightarrow> (\<exists>ps. valid_graph.is_path G v ps v')"
path_sourcenode: fixes n :: "'node"   and as :: "'edge list"   and n' :: "'node" assumes "n -as\<rightarrow>* n'"   and "as \<noteq> []" shows "hd (sourcenodes as) = n"
shrK_neq_priK: fixes b :: "keymode"   and C :: "agent"   and A :: "agent" shows "privateKey b C \<noteq> shrK A"
axioms: fixes intT :: "'a \<Rightarrow> univ \<Rightarrow> bool" assumes "CM.Tstruct intT" shows "M.Tstruct intT"
post_def: fixes C :: "'a acom" shows "post C = last (annos C)"
surjective: fixes r :: "('b, 'c) list_graph_scheme" shows "r = \<lparr>nodesL = nodesL r, edgesL = edgesL r, \<dots> = list_graph.more r\<rparr>"
mt: fixes P :: "'a \<Rightarrow> real"   and Q :: "'a \<Rightarrow> real" assumes "unitary P"   and "unitary Q"   and "P \<tturnstile> Q" shows "t P \<tturnstile> t Q"
Union_included_in_supp: fixes S :: "'a set" assumes "finite S" shows "\<Union> (supp ` S) \<subseteq> supp S"
add_assoc: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<in> E"   and "y \<in> E"   and "z \<in> E" shows "x + y + z = x + (y + z)"
filtermap_filtermap: fixes f :: "'d \<Rightarrow> 'c"   and g :: "'e \<Rightarrow> 'd"   and F :: "'e filter" shows "filtermap f (filtermap g F) = filtermap (\<lambda>x. f (g x)) F"
T_on__as_sum: fixes step :: "'b \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> 'b"   and t :: "'b \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> nat"   and A :: "('b \<Rightarrow> 'e) \<times> ('b \<times> 'e \<Rightarrow> 'c \<Rightarrow> 'd \<times> 'e)"   and s0 :: "'b"   and rs :: "'c list" shows "On_Off.T_on' step t A (s0, fst A s0) rs = (\<Sum>n<length rs. On_Off.T_on' step t A (On_Off.config' step A (s0, fst A s0) (take n rs)) [rs ! n])"
blinding_respects_hashes_def: fixes h :: "'e \<Rightarrow> 'f"   and bo :: "'e \<Rightarrow> 'e \<Rightarrow> bool" shows "blinding_respects_hashes h bo \<equiv> bo \<le> vimage2p h h (=)"
has_integral_open_interval: fixes f :: "'c \<Rightarrow> 'd"   and y :: "'d"   and a :: "'c"   and b :: "'c" shows "(f has_integral y) (box a b) = (f has_integral y) (cbox a b)"
translation_Int: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b"   and s :: "'b set"   and t :: "'b set" assumes "class.ab_group_add plus zero minus uminus" shows "plus a ` (s \<inter> t) = plus a ` s \<inter> plus a ` t"
chain_subset_def: fixes C :: "'b set set" shows "chain\<^sub>\<subseteq> C = (\<forall>A\<in>C. \<forall>B\<in>C. A \<subseteq> B \<or> B \<subseteq> A)"
is_inf_lower: fixes x :: "'c"   and y :: "'c"   and inf :: "'c" assumes "is_inf x y inf" obtains "inf \<sqsubseteq> x" and "inf \<sqsubseteq> y"
completionI: fixes A :: "'b set"   and S :: "'b set"   and N :: "'b set"   and N' :: "'b set"   and M :: "'b measure" assumes "A = S \<union> N"   and "N \<subseteq> N'"   and "N' \<in> null_sets M"   and "S \<in> sets M" shows "A \<in> {uu_. \<exists>S N N'. uu_ = S \<union> N \<and> S \<in> sets M \<and> N' \<in> null_sets M \<and> N \<subseteq> N'}"
assms_2: assumes "\<And>x. x \<in> A \<Longrightarrow> r x x" shows "thesis"
remove_fset_def: shows "remove_fset \<equiv> (id ---> rep_fset ---> abs_fset) removeAll"
gen_cong_eval: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "rel' (step.gen_cong R) x y" shows "step.gen_cong R (eval' x) (eval' y)"
intro: fixes ccw :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and S :: "'a set" assumes "ccw_system123 ccw S"   and "ccw_system4_axioms ccw S" shows "ccw_system4 ccw S"
valid_path_aux_split_1: fixes cs :: "'edge list"   and as :: "'edge list"   and as' :: "'edge list" assumes "valid_path_aux cs (as @ as')" shows "valid_path_aux cs as"
Int_insert_right_if0: fixes a :: "'e"   and A :: "'e set"   and B :: "'e set" assumes "a \<notin> A" shows "A \<inter> insert a B = A \<inter> B"
normFace_def: shows "normFace \<equiv> \<lambda>f. verticesFrom f (minVertex f)"
Rcomb: fixes P :: "'prog"   and coll1 :: "'coll"   and coll2 :: "'coll" shows "R P (combine coll1 coll2) = (R P coll1 \<and> R P coll2)"
index_zero_mat_1: fixes i :: "nat"   and nr :: "nat"   and j :: "nat"   and nc :: "nat" assumes "i < nr"   and "j < nc" shows "0\<^sub>m nr nc $$ (i, j) = (0::'d)"
random_permutation_of_set: fixes A :: "'a set" assumes "finite A"   and "A \<noteq> {}" shows "pmf_of_set (permutations_of_set A) = pmf_of_set A \<bind> (\<lambda>x. pmf_of_set (permutations_of_set (A - {x})) \<bind> (\<lambda>xs. return_pmf (x # xs)))"
walk_first_edge': fixes v :: "'a"   and xs :: "'a list" assumes "walk (v # xs)"   and "xs \<noteq> []" shows "v \<rightarrow> hd xs"
collect_locksE: fixes l :: "'a"   and las :: "'a lock_actions" assumes "l \<in> collect_locks las" obtains "Lock \<in> set (las $ l)"
assms: fixes x :: "'n" assumes "x \<in> S" shows "f x = (0::'a)"
eqvt_fresh_cong2: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd"   and a :: "atom"   and x :: "'b"   and y :: "'c" assumes "\<And>p x y. p \<bullet> f x y = f (p \<bullet> x) (p \<bullet> y)"   and "a \<sharp> x"   and "a \<sharp> y" shows "a \<sharp> f x y"
simps_2: fixes x :: "'c"   and a :: "'c"   and b :: "'d"   and ps :: "('c \<times> 'd) list" shows "del_list x ((a, b) # ps) = (if x = a then ps else (a, b) # del_list x ps)"
igSubstClsSTR_def: fixes Delta :: "('b, 'c, 'd, 'e, 'f) signature"   and MOD :: "('g, 'h, 'd, 'e, 'i, 'j, 'k, 'l, 'm) model_scheme" assumes "FixSyn TYPE('a) Delta" shows "FixSyn.igSubstClsSTR Delta MOD \<equiv> FixSyn.igSubstIGVar1STR Delta MOD \<and> FixSyn.igSubstIGVar2STR Delta MOD \<and> igSubstIGAbsSTR MOD \<and> FixSyn.igSubstIGOpSTR Delta MOD"
direct_lim_axioms: shows "direct_lim Y is_open\<^sub>Y \<O>\<^sub>Y \<rho>\<^sub>Y d add_str\<^sub>Y mult_str\<^sub>Y zero_str\<^sub>Y one_str\<^sub>Y {U. is_open\<^sub>Y U \<and> f x \<in> U}"
restrict_spmf_mono: fixes p :: "'a spmf"   and q :: "'a spmf"   and A :: "'a set" assumes "ord_spmf (=) p q" shows "ord_spmf (=) (p \<upharpoonleft> A) (q \<upharpoonleft> A)"
nat_of_omega: fixes n :: "V" assumes "n \<in>\<^sub>\<circ> \<omega>" obtains m :: "nat" where "n = m\<^sub>\<nat>"
ex_nonempty: fixes A :: "'c set" assumes "\<exists>x. x \<in> A" shows "A \<noteq> {}"
sys_initial_state_def: fixes s :: "('e, 'f, 'g, 'h) Model.local_state" shows "sys_initial_state s = ((\<forall>m. \<not> hs_pending s m \<and> ghost_hs_in_sync s m) \<and> ghost_hs_phase s = hp_IdleMarkSweep \<and> hs_type s = ht_GetRoots \<and> obj_mark ` ran (heap s) \<subseteq> {initial_mark} \<and> fA s = initial_mark \<and> fM s = initial_mark \<and> phase s = ph_Idle \<and> ghost_honorary_grey s = {} \<and> W s = {} \<and> (\<forall>p. mem_store_buffers s p = []) \<and> mem_lock s = None)"
step3_a: shows "DataRefinement ({. Loop'' .} \<circ> Q4''_a) RR_a RR_a QQ5_a"
nths_upt_eq_take: fixes l :: "'c list"   and n :: "nat" shows "nths l {..<n} = take n l"
distinct_9: fixes x2 :: "qtname"   and x4 :: "ty" shows "IfaceT x2 \<noteq> ArrayT x4"
update_arg_wf_tuples': fixes n :: "nat"   and hops :: "nat"   and nhip :: "nat"   and pre :: "nat set" assumes "Suc 0 \<le> n" shows "update_arg_wf (n, kno, val, hops, nhip, pre)"
assms: fixes x :: "'a" assumes "x \<in> A" shows "sum_of_2_squares_nat (f x)"
op_mult_is_pr: shows "(*) \<in> PrimRec2"
cases: fixes a1 :: "'a"   and a2 :: "'a"   and F :: "'a set"   and r :: "('a \<times> 'a) set" assumes "(a1, a2) \<in> rtrancl_on F r" obtains a :: "'a" where "a1 = a__" and "a2 = a__" and "a__ \<in> F"   | a :: "'a" and b :: "'a" and c :: "'a" where "a1 = a" and "a2 = c" and "(a, b) \<in> rtrancl_on F r" and "(b, c) \<in> r" and "c \<in> F"
atMost_upto: fixes n :: "nat" shows "{..n} = set [0..<Suc n]"
decode_minsky_def: fixes n :: "nat" shows "decode_minsky n = map (\<lambda>n. (c_fst n, decode_Op (c_snd n))) (nat_to_list n)"
simps: fixes as :: "nat"   and lm :: "nat list"   and s :: "nat"   and l :: "cell list"   and r :: "cell list"   and ires :: "cell list" shows "inv_after_write (as, lm) (s, l, r) ires = (\<exists>rn m lm1 lm2. lm = lm1 @ m # lm2 \<and> (if lm1 = [] then l = Oc \<up> m @ Bk # Bk # ires else Oc # l = Oc \<up> Suc m @ Bk # <rev lm1> @ Bk # Bk # ires) \<and> r = [Oc] @ <lm2> @ Bk \<up> rn)"
rel_intros_2: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and x2 :: "'b"   and y2 :: "'c" assumes "R x2 y2" shows "rel_err R (OK x2) (OK y2)"
induct: fixes \<Delta> :: "('a, 'b) network_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal" assumes "flow \<Delta> f" obtains "\<And>e. f e \<le> capacity \<Delta> e" and "\<And>x. \<lbrakk>x \<noteq> source \<Delta>; x \<noteq> sink \<Delta>\<rbrakk> \<Longrightarrow> KIR f x"
not_eq_last_diff_paths: fixes s :: "('a, 'b) fmap"   and PROB :: "(('a, 'b) fmap \<times> ('a, 'b) fmap) set"   and as :: "(('a, 'b) fmap \<times> ('a, 'b) fmap) list" assumes "s \<in> valid_states PROB"   and "as \<in> valid_plans PROB"   and "\<not> FactoredSystem.inj last (state_set (state_list s as)) (valid_states PROB)" shows "\<exists>slist_1 slist_2. slist_1 \<in> state_set (state_list s as) \<and> slist_2 \<in> state_set (state_list s as) \<and> last slist_1 = last slist_2 \<and> length slist_1 \<noteq> length slist_2"
dd_ent: shows "DD \<union> CC \<Turnstile> {E}"
simps: fixes a :: "'g set" shows "finite a = (a = {} \<or> (\<exists>A aa. a = insert aa A \<and> finite A))"
Sim: fixes \<alpha> :: "'a action"   and Q' :: "('a, 'b, 'c) psi" assumes "\<Psi> \<rhd> Q \<longmapsto> \<alpha> \<prec> Q'"   and "bn \<alpha> \<sharp>* P"   and "bn \<alpha> \<sharp>* Q"   and "bn \<alpha> \<sharp>* \<Psi>"   and "bn \<alpha> \<sharp>* subject \<alpha>"   and "distinct (bn \<alpha>)"   and "bn \<alpha> \<sharp>* C"   and "xvec \<sharp>* \<alpha>"   and "xvec \<sharp>* Q'" shows "\<exists>P'. \<Psi> \<rhd> P \<longmapsto> \<alpha> \<prec> P' \<and> (\<Psi>, P', Q') \<in> Rel"
prems: shows "Subobjs\<^sub>R P (hd ((D # Ds) @ C' # Cs')) ((D # Ds) @ C' # Cs')"
simps_1: fixes ts :: "'b set set" shows "partitioning [] ts = ts"
Gromov_completion_converge_to_boundary: fixes x :: "'c Gromov_completion"   and u :: "nat \<Rightarrow> 'c" assumes "x \<in> Gromov_boundary" shows "(\<lambda>n. to_Gromov_completion (u n)) \<longlonglongrightarrow> x = (Gromov_completion_rel u u \<and> abs_Gromov_completion u = x)"
while_option_None_invD: fixes b :: "'c \<Rightarrow> bool"   and c :: "'c \<Rightarrow> 'c"   and s :: "'c"   and r :: "('c \<times> 'c) set"   and I :: "'c \<Rightarrow> bool" assumes "while_option b c s = None"   and "wf r"   and "I s"   and "\<And>s. \<lbrakk>I s; b s\<rbrakk> \<Longrightarrow> I (c s)"   and "\<And>s. \<lbrakk>I s; b s\<rbrakk> \<Longrightarrow> (c s, s) \<in> r" shows "False"
lubx: shows "(\<forall>y. F y \<longrightarrow> P y x) \<and> (\<forall>z. (\<forall>y. F y \<longrightarrow> P y z) \<longrightarrow> P x z)"
itrm_xchng: fixes f :: "'a itrm"   and x :: "dB" shows "f \<diamondop> Pure x \<simeq> Pure (\<T> \<degree> x) \<diamondop> f"
sequence_of_function_of: fixes enum :: "nat \<Rightarrow> 'c"   and A :: "'c set"   and xs :: "'d list"   and B :: "'d set" assumes "bij_betw enum {0..<card A} A"   and "set xs \<subseteq> B"   and "length xs = card A" shows "sequence_of A enum (function_of A enum xs) = xs"
freeword_funlift_im: fixes x :: "'a freeword"   and S :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "x \<in> FreeGroup S" shows "freeword_funlift f x \<in> \<langle>f ` S\<rangle>"
c123_3: shows "lst_conv (fst (fst (conv_mirror (fst \<delta>1''', map (Pair True) (snd \<delta>1'''))), snd (conv_mirror (fst \<delta>1''', map (Pair True) (snd \<delta>1'''))) @ snd \<delta>3), snd (fst (conv_mirror (fst \<delta>1''', map (Pair True) (snd \<delta>1'''))), snd (conv_mirror (fst \<delta>1''', map (Pair True) (snd \<delta>1'''))) @ snd \<delta>3) @ snd (conv_mirror \<gamma>3)) = lst_conv (conv_mirror \<gamma>3)"
T_composition_borel_1: fixes M :: "'b measure"   and T :: "'b \<Rightarrow> 'b"   and f :: "'b \<Rightarrow> 'c" assumes "qmpt M T"   and "f \<in> borel_measurable M" shows "(\<lambda>x. f (T x)) \<in> borel_measurable M"
mat_leqb_eqc_for_EQ: fixes b :: "rat vec"   and i :: "nat"   and c :: "rat vec"   and A :: "rat mat" assumes "dim_vec b \<le> i"   and "i < dim_vec (b @\<^sub>v c)"   and "dim_row A = dim_vec b"   and "dim_vec c \<le> dim_col A" shows "mat_leqb_eqc A b c ! i = EQ (vec_to_lpoly (0\<^sub>v (dim_col A) @\<^sub>v row A\<^sup>T (i - dim_vec b))) (c $ (i - dim_vec b))"
psubstl_correct: fixes l :: "'a list"   and P1 :: "pol"   and mps :: "(mon \<times> pol) list"   and n :: "nat"   and P2 :: "pol" assumes "in_carrier l"   and "psubstl P1 mps n = Some P2"   and "Imon_pol_list l mps" shows "Ipol l P1 = Ipol l P2"
matrix_mul_rid: fixes A :: "(('a, 'b) vec, 'c) vec" shows "A ** mat (1::'a) = A"
XD_smallest_def: shows "XD_smallest = - (XH_largest - X)"
numer_cf7_deriv_def: shows "numer_cf7_deriv \<equiv> \<lambda>x. 7 * x ^ 6 + 336 * x ^ 5 + 7560 * x ^ 4 + 100800 * x ^ 3 + 831600 * x\<^sup>2 + 3991680 * x + 8648640"
S: shows "nfa_startnode (nfa_of_dfa A) = (replicate (length (fst A)) False)[0 := True]"
simps_35: fixes vd :: "char list"   and ve :: "aexp"   and vf :: "'a"   and v :: "'a"   and va :: "bexp"   and vb :: "'a acom"   and vc :: "'a" shows "(vd ::= ve {vf} \<sqsubseteq> {v} WHILE va DO vb {vc}) = False"
p_inf: fixes uminus :: "'c \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and x :: "'c" assumes "class.p_algebra uminus inf less_eq less sup bot top" shows "inf (uminus x) x = bot"
crb_lem_pos: fixes p :: "real poly"   and x :: "real" assumes "poly p x = 0"   and "p \<noteq> 0" shows "x < real_of_int (crb p)"
SigmaE2: fixes a :: "'f"   and b :: "'g"   and A :: "'f set"   and B :: "'f \<Rightarrow> 'g set" assumes "(a, b) \<in> Sigma A B" obtains "a \<in> A" and "b \<in> B a"
power_down_mono: fixes a :: "real"   and b :: "real"   and n :: "nat"   and prec :: "nat" assumes "0 \<le> a \<and> a \<le> b \<or> odd n \<and> a \<le> b \<or> even n \<and> a \<le> 0 \<and> b \<le> a" shows "power_down prec a n \<le> power_down prec b n"
verit_comp_simplify_1: fixes a :: "'c" shows "(a < a) = False"
vors_4: shows "set zs \<inter> set xs \<subseteq> {u} \<union> set as"
finite_measure_Union': fixes A :: "'a set"   and B :: "'a set" assumes "A \<in> sets N"   and "B \<in> sets N" shows "Sigma_Algebra.measure N (A \<union> B) = Sigma_Algebra.measure N A + Sigma_Algebra.measure N (B - A)"
l_div_eqn: fixes a :: "'a"   and x :: "'a"   and y :: "'a" assumes "a \<in> carrier G"   and "x \<in> carrier G"   and "y \<in> carrier G"   and "a \<cdot> x = a \<cdot> y" shows "x = y"
b_disjoint_less: fixes K :: "nat set"   and j0 :: "nat"   and i' :: "nat"   and j :: "nat"   and i :: "nat" assumes "K \<subseteq> {j0<..}"   and "finite K"   and "j0 \<le> card K"   and "i' < j"   and "i < i'"   and "j \<le> j0" shows "b (enum K i) (j0, i) \<inter> b (enum K i') (j0, i') = {}"
z1: shows "case z1 of (x, y) \<Rightarrow> x \<le> y \<and> x\<^sup>2 + y\<^sup>2 = p"
computational_adequacy_refines: fixes e1 :: "db"   and e2 :: "db" assumes "closed e1"   and "closed e2"   and "evalDdb e1\<cdot>env_empty_db = evalDdb e2\<cdot>env_empty_db" shows "e1 \<unlhd> e2"
compact_frontier: fixes S :: "'a set" assumes "compact S" shows "compact (frontier S)"