hcju/isabelleps huggingface dataset:
while_bot_def: fixes b :: "'a upred"   and P :: "'a hrel" shows "while\<^sub>\<bottom> b do P od = (\<mu> X \<bullet> P ;; X \<triangleleft> b \<triangleright>\<^sub>r II)"
gbm_add_node_def: fixes v :: "'V"   and g :: "'m1" shows "gbm_add_node v g \<equiv> case m1.lookup v g of None \<Rightarrow> m1.update v (m2.empty ()) g | Some x \<Rightarrow> g"
wp_while: fixes p :: "'a"   and r :: "'a"   and x :: "'a" assumes "d p \<cdot> d r \<le> wp x p" shows "d p \<le> wp (while r do x od) (d p \<cdot> ad r)"
Collect_restrict: fixes X :: "'a set"   and P :: "'a \<Rightarrow> bool" shows "{x \<in> X. P x} \<subseteq> X"
g_def: fixes x :: "real" shows "g x = ln_Gamma x - (x - 1 / 2) * ln x + x"
simps_6: fixes f1 :: "qtname \<Rightarrow> 'a"   and f2 :: "ty \<Rightarrow> int \<Rightarrow> 'a"   and x21 :: "ty"   and x22 :: "int" shows "(case Arr x21 x22 of CInst x \<Rightarrow> f1 x | Arr x xa \<Rightarrow> f2 x xa) = f2 x21 x22"
enum_inj: fixes i :: "nat"   and j :: "nat" assumes "i \<le> n"   and "j \<le> n" shows "(t.enum i = t.enum j) = (i = j)"
intro: fixes d :: "'a"   and gt :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "SN_one_mono_ordered_semiring_1 d gt" shows "matrix_carrier gt d"
foundation16': fixes \<tau> :: "'a state \<times> 'a state"   and X :: "'a state \<times> 'a state \<Rightarrow> 'b" shows "(\<tau> \<Turnstile> \<delta> X) = (X \<tau> \<noteq> invalid \<tau> \<and> X \<tau> \<noteq> null \<tau>)"
cor_mult: fixes a :: "real"   and b :: "real" shows "cor (a * b) = cor a * cor b"
inverse_arrows_def: fixes f :: "'s"   and g :: "'s" shows "inverse_arrows f g \<equiv> ide (g \<cdot> f) \<and> ide (f \<cdot> g)"
dg_length: shows "vcard \<AA> = 4\<^sub>\<nat>"
widen_NT2: fixes G :: "prog"   and S :: "ty" assumes "G\<turnstile>S\<preceq>NT" shows "S = NT"
abs_eq: fixes x :: "complex_homo_coords" shows "~\<^sub>h (abs_complex_homo x) = abs_complex_homo (~\<^sub>h\<^sub>c x)"
exec_meth_ci_appD: fixes addr2thread_id :: "'a \<Rightarrow> 'b"   and thread_id2addr :: "'b \<Rightarrow> 'a"   and spurious_wakeups :: "bool"   and empty_heap :: "'c"   and allocate :: "'c \<Rightarrow> htype \<Rightarrow> ('c \<times> 'a) set"   and typeof_addr :: "'c \<Rightarrow> 'a \<Rightarrow> htype option"   and heap_read :: "'c \<Rightarrow> 'a \<Rightarrow> addr_loc \<Rightarrow> 'a val \<Rightarrow> bool"   and heap_write :: "'c \<Rightarrow> 'a \<Rightarrow> addr_loc \<Rightarrow> 'a val \<Rightarrow> 'c \<Rightarrow> bool"   and ci :: "('a, 'c) check_instr"   and P :: "(nat \<times> nat \<times> 'a instr list \<times> (nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list) prog"   and ins :: "'a instr list"   and xt :: "(nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list"   and t :: "'b"   and h :: "'c"   and stk :: "'a val list"   and loc :: "'a val list"   and pc :: "nat"   and ta :: "('a, 'b, 'c) jvm_thread_action"   and h' :: "'c"   and fr' :: "'a val list \<times> 'a val list \<times> nat \<times> 'a option" assumes "JVM_heap_base.exec_meth addr2thread_id thread_id2addr spurious_wakeups empty_heap allocate typeof_addr heap_read heap_write ci P ins xt t h (stk, loc, pc, None) ta h' fr'" shows "ci_app ci (ins ! pc) P h stk loc undefined undefined pc []"
mk_regular2_def: fixes P :: "'b \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and F :: "'b \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a"   and I :: "'a set" shows "mk_regular2 P F I = I \<union> {F q x y |q x y. x \<in> I \<and> y \<in> I \<and> P q x y}"
le_of_nat_term_order_plus_monotone: fixes to :: "'a nat_term_order"   and s :: "'a"   and t :: "'a"   and u :: "'a" assumes "le_of_nat_term_order to s t" shows "le_of_nat_term_order to (u + s) (u + t)"
IH: assumes "insert_ops ops"   and "insert_seq None xs"   and "insert_ops xs"   and "set xs \<subseteq> set ops" shows "\<forall>i\<in>set (map fst xs). i \<in> set (interp_ins ops)"
divide_minus_left: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and 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 a :: "'c"   and b :: "'c" assumes "class.division_ring divide inverse one times plus zero minus uminus" shows "divide (uminus a) b = uminus (divide a b)"
hull_UNIV: fixes S :: "'a set \<Rightarrow> bool" shows "S hull UNIV = UNIV"
lengthLxy: shows "length (Lxy init {x, y}) = 2"
one_smult_vec: fixes v :: "'c vec" shows "(1::'c) \<cdot>\<^sub>v v = v"
exists_induced_map: shows "\<exists>w. f.is_induced_map w"
hyps: fixes z :: "'a" shows "sum_list xs + z \<le> fold (\<lambda>x y. eucl_truncate_up p (x + y)) xs z"
SetDC_singleton: fixes a :: "'b \<Rightarrow> bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and x :: "'b" shows "SetDC a (\<lambda>_. {x}) = a x"
coset_assoc: fixes x :: "'a"   and y :: "'a"   and H :: "'a set" assumes "x \<in> carrier G"   and "y \<in> carrier G"   and "H \<subseteq> carrier G" shows "x <# (H #> y) = x <# H #> y"
repeat_mset_eq_empty_iff: fixes n :: "nat"   and A :: "'a multiset" shows "(repeat_mset n A = {#}) = (n = 0 \<or> A = {#})"
abs_norm_cancel: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and a :: "'a" assumes "class.real_normed_vector minus dist norm plus zero uminus scaleR sgn uniformity open" shows "\<bar>norm a\<bar> = norm a"
new\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M_def: shows "new\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M = Heap_Error_Monad.bind get_heap (\<lambda>h. Heap_Error_Monad.bind (return (new\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t h)) (\<lambda>(new_ptr, h'). Heap_Error_Monad.bind (return_heap h') (\<lambda>_. return new_ptr)))"
map_upd_cong_ext: fixes tab :: "'b \<Rightarrow> 'a option"   and z :: "'b"   and tab' :: "'b \<Rightarrow> 'a option"   and x :: "'b"   and y :: "'a" assumes "tab z = tab' z" shows "(tab(x \<mapsto> y)) z = (tab'(x \<mapsto> y)) z"
predecessor_point': fixes x :: "'a"   and s :: "'a"   and e :: "'a" assumes "path x"   and "point s"   and "point e"   and "e ; s\<^sup>T \<le> x" shows "x ; s = e"
cap_has_object_def: fixes cap :: "cdl_cap" shows "cap_has_object cap \<equiv> case cap of NullCap \<Rightarrow> False | _ \<Rightarrow> True"
WT_parallel_oracle: fixes \<I>l :: "('a, 'b) \<I>"   and left :: "'c \<Rightarrow> 'a \<Rightarrow> ('b \<times> 'c) spmf"   and sl :: "'c"   and \<I>r :: "('d, 'e) \<I>"   and right :: "'f \<Rightarrow> 'd \<Rightarrow> ('e \<times> 'f) spmf"   and sr :: "'f" assumes "\<I>l \<turnstile>c left sl \<surd>"   and "\<I>r \<turnstile>c right sr \<surd>" shows "\<I>l \<oplus>\<^sub>\<I> \<I>r \<turnstile>c (left \<ddagger>\<^sub>O right) (sl, sr) \<surd>"
diff: fixes x :: "'a"   and y :: "'a" shows "f (x - y) = f x - f y"
disjoint_iff_not_equal: fixes A :: "'e set"   and B :: "'e set" shows "(A \<inter> B = {}) = (\<forall>x\<in>A. \<forall>y\<in>B. x \<noteq> y)"
dom_def: fixes f :: "'a" shows "C.dom f = (if C.domains f \<noteq> {} then SOME a. a \<in> C.domains f else C.null)"
vsvD_1: fixes r :: "V" assumes "vsv r" shows "vbrelation r"
add_divide_eq_if_simps_1: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and 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 a :: "'b"   and b :: "'b"   and z :: "'b" assumes "class.division_ring divide inverse one times plus zero minus uminus" shows "plus a (divide b z) = (if z = zero then a else divide (plus (times a z) b) z)"
extendRule_def: fixes forms :: "sequent"   and R :: "sequent list \<times> sequent" shows "extendRule forms R \<equiv> (map (extend forms) (fst R), extend forms (snd R))"
Abs1_eq_iff_all_3: fixes a :: "'b"   and x :: "'a"   and b :: "'b"   and y :: "'a"   and z :: "'c" shows "([[atom a]]lst. x = [[atom b]]lst. y) = (\<forall>c. atom c \<sharp> z \<longrightarrow> atom c \<sharp> (a, b, x, y) \<longrightarrow> (a \<leftrightarrow> c) \<bullet> x = (b \<leftrightarrow> c) \<bullet> y)"
fps_mod_def: fixes f :: "'b fps"   and g :: "'b fps" shows "f mod g = (if g = 0 then f else let h = unit_factor g in fps_cutoff (subdegree g) (f * inverse h) * h)"
combine_conditions_on_sets_and_pairs_of_relations: fixes RelA :: "(('procS, 'procT) Proc \<times> ('procS, 'procT) Proc) set"   and CondA :: "('procS, 'procT) Proc \<times> ('procS, 'procT) Proc \<Rightarrow> bool"   and RelB :: "(('procS, 'procT) Proc \<times> ('procS, 'procT) Proc) set"   and CondB :: "('procS, 'procT) Proc \<times> ('procS, 'procT) Proc \<Rightarrow> bool"   and Cond :: "(('procS, 'procT) Proc \<times> ('procS, 'procT) Proc) set \<Rightarrow> bool"   and Rel :: "(('procS, 'procT) Proc \<times> ('procS, 'procT) Proc) set" assumes "\<forall>(P, Q)\<in>RelA. CondA (P, Q)"   and "\<forall>(P, Q)\<in>RelB. CondB (P, Q)"   and "Cond Rel \<and> Rel \<subseteq> RelA \<and> Rel \<subseteq> RelB" shows "Cond Rel \<and> (\<forall>(P, Q)\<in>Rel. CondA (P, Q)) \<and> (\<forall>(P, Q)\<in>Rel. CondB (P, Q))"
preserves_arr: fixes f :: "'c" assumes "C.arr f" shows "D.arr (G f)"
inline_def: fixes callee :: "'c \<Rightarrow> 'f \<Rightarrow> ('g \<times> 'c, 'd, 'e) gpv"   and c :: "('b, 'f, 'g) gpv"   and s :: "'c" shows "inline callee c s = inline_aux callee (Inl (c, s))"
goedel_first_strong: assumes "\<omega>consistent" shows "\<not> prv \<phi>G \<and> \<not> prv (neg \<phi>G) \<and> isTrue \<phi>G"
sumPres: fixes P :: "pi"   and Q :: "pi"   and R :: "pi" assumes "P \<sim> Q" shows "P \<oplus> R \<sim> Q \<oplus> R"
sorted_lr_of_tran_s3: fixes f :: "('b \<times> 32 simple_match \<times> char list \<times> simple_action) list"   and s :: "char list list" assumes "sorted_descending (map fst f)" shows "sorted_descending (map fst (lr_of_tran_s3 s f))"
fls_X_times_conv_shift_2: fixes f :: "'a fls" shows "f * fls_X = fls_shift (- 1) f"
gcd_greatest_iff: 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"   and b :: "'b"   and c :: "'b" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize" shows "dvd.dvd times a (gcd b c) = (dvd.dvd times a b \<and> dvd.dvd times a c)"
elementOnCurrentLevel: fixes a :: "'a"   and M :: "('a \<times> bool) list"   and d :: "bool" assumes "a \<notin> set (elements M)" shows "elementLevel a (M @ [(a, d)]) = currentLevel (M @ [(a, d)])"
boxNotInit: fixes F :: "'a \<Rightarrow> bool" shows "\<box>\<not> F \<equiv> \<box>\<not> Init F"
ZF2m_def: fixes T :: "(ZF, ZF) Axioms" shows "ZF2m T \<equiv> inv_into {TermEquivCl' T A e B |A e B. True} m2ZF"
f: shows "integrable (M1 \<Otimes>\<^sub>M M2) (\<lambda>(x, y). f x y)"
Nset_injTr0: fixes f :: "nat \<Rightarrow> nat"   and n :: "nat" assumes "f \<in> {i. i \<le> Suc n} \<rightarrow> {i. i \<le> Suc n}"   and "inj_on f {i. i \<le> Suc n}"   and "f (Suc n) = Suc n" shows "f \<in> {i. i \<le> n} \<rightarrow> {i. i \<le> n} \<and> inj_on f {i. i \<le> n}"
R2_idem: shows "R2 \<circ> R2 = R2"
LLL_inv_wD_4: fixes fs :: "int vec list"   and i :: "nat" assumes "LLL_invariant_weak fs"   and "i < m" shows "fs ! i \<in> carrier_vec n"
surjI: fixes g :: "'b \<Rightarrow> 'a"   and f :: "'a \<Rightarrow> 'b" assumes "\<And>x. g (f x) = x" shows "surj g"
repb_node: shows "repb node = hd (filter (\<lambda>sn. repNodes_eq sn node low high repa) (prx @ node # sfx))"
conj_commute: fixes P :: "bool"   and Q :: "bool" shows "(P \<and> Q) = (Q \<and> P)"
sds_wf: fixes R :: "'agent \<Rightarrow> 'alt \<Rightarrow> 'alt \<Rightarrow> bool" assumes "is_pref_profile R" shows "RD R \<in> lotteries"
assms_2: shows "prob1 n v i \<noteq> 0"
face_cycle_eq: fixes b :: "'b"   and a :: "'b" assumes "b \<in> G.face_cycle_set a" shows "G.face_cycle_set b = G.face_cycle_set a"
size_M': fixes \<psi> :: "'a ltln" assumes "\<psi> \<in> M" shows "size (\<psi>[N]\<^sub>\<Sigma>\<^sub>1) \<le> size \<phi>"
contk: shows "continuous_on ({0..1} \<times> S) k"
CanonicalCatCat': fixes T :: "(ZF, ZF) Axioms" assumes "ZFAxioms T" shows "Category_axioms (CanonicalCat' T)"
f_Exec_State_Idle_replicate_NoMsg_gr_output: fixes localState :: "'a \<Rightarrow> 'b"   and output_fun :: "'a \<Rightarrow> 'c message_af"   and trans_fun :: "'d message_af \<Rightarrow> 'a \<Rightarrow> 'a"   and m :: "nat"   and c :: "'a"   and n :: "nat" assumes "State_Idle localState output_fun trans_fun (localState (f_Exec_Comp trans_fun NoMsg\<^bsup>m\<^esup> c))"   and "m < n" shows "output_fun (f_Exec_Comp trans_fun NoMsg\<^bsup>n\<^esup> c) = NoMsg"
treewidth_tree: fixes G :: "nat Graph" assumes "Tree G" shows "Graph.treewidth G \<le> 1"
diag_block_pow_mat: fixes As :: "'b mat list"   and n :: "nat" assumes "Ball (set As) square_mat" shows "diag_block_mat As ^\<^sub>m n = diag_block_mat (map (\<lambda>A. A ^\<^sub>m n) As)"
nso_nostep: fixes l :: "'b"   and i :: "nat" assumes "\<not> rstep l s i"   and "can_step l (s i)" shows "next_step_of l (Suc i) = next_step_of l i"
mapM_return: fixes f :: "'c \<Rightarrow> 'a + 'b"   and xs :: "'c list"   and ys :: "'b list" assumes "mapM f xs = Inr ys" shows "ys = map (projr \<circ> f) xs \<and> (\<forall>x\<in>set xs. \<forall>e. f x \<noteq> Inl e)"
AddSA_HARoot: fixes SA :: "('d, 'e, 'f) seqauto"   and HA :: "('d, 'e, 'f) hierauto"   and S :: "'d" assumes "States SA \<inter> HAStates HA = {}"   and "S \<in> HAStates HA" shows "HARoot (HA [++] (S, SA)) = HARoot HA"
simps_4: fixes n :: "nat"   and f :: "recf"   and gs :: "recf list" shows "wellf (Cn n f gs) = (0 < n \<and> (\<forall>g\<in>set gs. arity g = n \<and> wellf g) \<and> arity f = length gs \<and> wellf f)"
C_ne_max_dist: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and src :: "nat"   and dst :: "nat"   and PRED :: "nat \<Rightarrow> nat option"   and C :: "nat set"   and N :: "nat set"   and d :: "nat" assumes "nf_invar c src dst PRED C N d"   and "C \<noteq> {}" shows "d \<le> Graph.max_dist c src"
collect_disj: fixes P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" shows "{x. P x \<or> Q x} = {x. P x} \<union> {x. Q x}"
not_lt_iff_ge_zmset: fixes x :: "'a zmultiset"   and y :: "'a zmultiset" shows "(\<not> x < y) = (y \<le> x)"
Vagree_repv: fixes \<omega> :: "variable \<Rightarrow> real"   and x :: "variable"   and d :: "real" shows "Vagree \<omega> (repv \<omega> x d) (- {x})"
H: shows "continuous_on UNIV (\<lambda>x. H (fst x) (snd x))"
pq: shows "[:f, e, 0, 1:] = [:- x1, 1:] * q"
simps_4: fixes x21 :: "char list"   and x22 :: "'a list"   and x11 :: "char list"   and x12 :: "'a list" shows "Neg x21 x22 \<noteq> Pos x11 x12"
frontier_real_atLeast: fixes a :: "real" shows "frontier {a..} = {a}"
insert: fixes A :: "'c set"   and x :: "'c"   and g :: "'c \<Rightarrow> 'a" assumes "finite A"   and "x \<notin> A" shows "sum_0 g (insert x A) = g x + sum_0 g A"
HT'_def: fixes \<pi> :: "char list \<Rightarrow> com option"   and P :: "(char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool"   and c :: "(char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> com"   and Q :: "(char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool" shows "HT' \<pi> P c Q \<equiv> \<forall>s\<^sub>0. P s\<^sub>0 \<longrightarrow> wp \<pi> (c s\<^sub>0) (Q s\<^sub>0) s\<^sub>0"
read_instr_cache_def: fixes state :: "'b sparc_state"   and va :: "word32" shows "read_instr_cache state va \<equiv> let tag = ucast (va >> 12); offset0 = ucast va AND 4092; offset1 = offset0 OR 1; offset2 = offset0 OR 2; offset3 = offset0 OR 3; r0 = icache_val (tag, offset0) state; r1 = icache_val (tag, offset1) state; r2 = icache_val (tag, offset2) state; r3 = icache_val (tag, offset3) state in if r0 = None \<or> r1 = None \<or> r2 = None \<or> r3 = None then None else let byte0 = case r0 of Some v \<Rightarrow> v; byte1 = case r1 of Some v \<Rightarrow> v; byte2 = case r2 of Some v \<Rightarrow> v; byte3 = case r3 of Some v \<Rightarrow> v in Some ((((ucast byte0 << 24) OR (ucast byte1 << 16)) OR (ucast byte2 << 8)) OR ucast byte3)"
ltln_expand_WeakUntil: fixes \<xi> :: "nat \<Rightarrow> 'a set"   and \<phi> :: "'a ltln"   and \<psi> :: "'a ltln" shows "\<xi> \<Turnstile>\<^sub>n \<phi> W\<^sub>n \<psi> = \<xi> \<Turnstile>\<^sub>n \<psi> or\<^sub>n (\<phi> and\<^sub>n X\<^sub>n (\<phi> W\<^sub>n \<psi>))"
elims_2: fixes x :: "'a set \<Rightarrow> 'b \<Rightarrow> bool"   and xa :: "('a, 'c) flow_entry_match list" assumes "no_overlaps x xa" obtains uu_ :: "'a set \<Rightarrow> 'b \<Rightarrow> bool" where "x = uu_" and "xa = []"   | \<gamma> :: "'a set \<Rightarrow> 'b \<Rightarrow> bool" and a :: "('a, 'c) flow_entry_match" and as :: "('a, 'c) flow_entry_match list" where "x = \<gamma>" and "xa = a # as" and "no_overlaps \<gamma> as \<and> (\<forall>b\<in>set as. ofe_prio a = ofe_prio b \<longrightarrow> \<not> (\<exists>p\<in>UNIV. \<gamma> (ofe_fields a) p \<and> \<gamma> (ofe_fields b) p))"
complete_imp_closed: fixes S :: "'d set" assumes "complete S" shows "closed S"
TIP_PP: fixes \<phi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "Fvars \<phi> = {}"   and "isTrue (PP \<langle>\<phi>\<rangle>)" shows "prv \<phi>"
Jacobi_1_eq_1: fixes p :: "int" assumes "p \<noteq> 0" shows "Jacobi 1 p = 1"
iTILL_iFROM_subset_conv: fixes n' :: "nat"   and n :: "nat" shows "([\<dots>n'] \<subseteq> [n\<dots>]) = (n = 0)"
gState_progress_rel_gState_invI2: fixes g :: "'b gState_scheme"   and g' :: "'b gState_scheme"   and prog :: "program" assumes "(g, g') \<in> gState_progress_rel prog" shows "gState_inv prog g'"
term_ok_translate_code_step: fixes cto :: "(String.literal \<times> typ) list"   and tao :: "(String.literal \<times> nat) list"   and sa :: "(String.literal \<times> String.literal) list \<times> (String.literal \<times> (String.literal \<times> String.literal set list) list) list"   and ax :: "Core.term list"   and p :: "Core.term" assumes "exe_sig_conds (ExeSignature cto tao sa)" shows "(term_ok (translate_theory (ExeTheory (ExeSignature cto tao sa) ax)) p \<and> typ_of p = Some propT) = (term_ok' (translate_signature (ExeSignature cto tao sa)) p \<and> typ_of p = Some propT)"
right_unit_mult_rng_of_frac: fixes r :: "'a"   and s :: "'a" assumes "(r, s) \<in> carrier rel" shows "(r |\<^bsub>rel\<^esub> s) \<otimes>\<^bsub>rec_monoid_rng_of_frac\<^esub> \<one>\<^bsub>rec_monoid_rng_of_frac\<^esub> = (r |\<^bsub>rel\<^esub> s)"
scaleR_nonpos_nonneg: fixes a :: "real"   and x :: "'b" assumes "a \<le> 0"   and "(0::'b) \<le> x" shows "a *\<^sub>R x \<le> (0::'b)"
convergentI: fixes X :: "nat \<Rightarrow> 'b"   and L :: "'b" assumes "X \<longlonglongrightarrow> L" shows "convergent X"
graph_nodes_it_axioms_def: fixes \<alpha> :: "'d \<Rightarrow> ('e, 'f) graph"   and invar :: "'d \<Rightarrow> bool"   and nodes_list_it :: "'d \<Rightarrow> ('e list \<Rightarrow> bool) \<Rightarrow> ('e \<Rightarrow> 'e list \<Rightarrow> 'e list) \<Rightarrow> 'e list \<Rightarrow> 'e list" shows "graph_nodes_it_axioms \<alpha> invar nodes_list_it \<equiv> \<forall>g. invar g \<longrightarrow> set_iterator (nodes_list_it g) (nodes (\<alpha> g))"
natural_transformation_axioms: shows "natural_transformation (\<cdot>\<^sup>o\<^sup>p) (\<cdot>\<^sub>S) (Y a) F \<T>e.map"
elims: fixes x :: "(('a \<times> 'b \<times> 'a) list \<times> 'b) option"   and y :: "('a \<times> 'b \<times> 'a) list option" assumes "mpath' x = y" obtains "x = None" and "y = None"   | p :: "('a \<times> 'b \<times> 'a) list" and w :: "'b" where "x = Some (p, w)" and "y = Some p"
strict_trans2: fixes b :: "'e"   and a :: "'e"   and c :: "'e" assumes "b < a"   and "c \<le> b" shows "c < a"
enat_ord_number_1: fixes m :: "num"   and n :: "num" shows "(numeral m \<le> numeral n) = (numeral m \<le> numeral n)"
gfp_def: fixes f :: "'b \<times> 'c \<Rightarrow> 'b \<times> 'c" shows "prod.gfp f = Sup_prod {u. less_eq_prod u (f u)}"
character_extension_exists: fixes H :: "'a set"   and \<chi> :: "'a \<Rightarrow> complex" assumes "subgroup H G"   and "character (G\<lparr>carrier := H\<rparr>) \<chi>" obtains \<chi>' :: "'a \<Rightarrow> complex" where "character G \<chi>'" and "\<And>x. x \<in> H \<Longrightarrow> \<chi>' x = \<chi> x"
assms_1: shows "(a, b) \<in> r\<^sup>*"
abs_eq: fixes satisfies :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and transition :: "'b \<Rightarrow> ('d, 'b) residual \<Rightarrow> bool"   and effect_apply :: "'e \<Rightarrow> 'b \<Rightarrow> 'b"   and L :: "'d \<times> 'e fs_set \<times> 'e \<Rightarrow> 'e fs_set"   and xa :: "'b"   and x :: "('f, 'c, 'd, 'e) Tree" assumes "effect_nominal_ts satisfies transition effect_apply L" shows "effect_nominal_ts.FL_valid_Tree\<^sub>\<alpha> satisfies transition effect_apply xa (abs_Tree\<^sub>\<alpha> x) = effect_nominal_ts.FL_valid_Tree satisfies transition effect_apply xa x"
strip_acom: fixes f :: "'c \<Rightarrow> 'b"   and c :: "'c acom" shows "strip (ACom.map_acom f c) = strip c"
star_idemp: fixes A :: "'a list set" shows "star (star A) = star A"
notsdom_notsdominate: fixes i :: "nat"   and j :: "nat" assumes "\<not> strict_dom i j"   and "j < length start" shows "non_strict_dominate i j"
hom_char: fixes a :: "'a"   and b :: "'a" shows "Left_a.hom a b = hom a b \<inter> Collect (left a)"
1: shows "\<ominus>\<^bsub>residue_ring (p ^ k)\<^esub> (y mod p ^ k) \<oplus>\<^bsub>residue_ring (p ^ k)\<^esub> y = \<zero>\<^bsub>residue_ring (p ^ k)\<^esub>"
subset_rel_interior: fixes S :: "'a set"   and T :: "'a set" assumes "S \<subseteq> T"   and "affine hull S = affine hull T" shows "rel_interior S \<subseteq> rel_interior T"
irreducible\<^sub>d_smult_not_zero_divisor_left: fixes c :: "'b"   and p :: "'b poly" assumes "\<not> zero_divisor c"   and "irreducible\<^sub>d (smult c p)" shows "irreducible\<^sub>d p"
phiArg_disjoint_paths_extend: fixes g :: "'g"   and r :: "'val"   and V :: "'var"   and s :: "'val"   and n :: "'node"   and m :: "'node"   and ns :: "'node list"   and ms :: "'node list"   and \<phi>\<^sub>r :: "'val" assumes "var g r = V"   and "var g s = V"   and "r \<in> allVars g"   and "s \<in> allVars g"   and "V \<in> oldDefs g n"   and "V \<in> oldDefs g m"   and "g \<turnstile> n-ns\<rightarrow>defNode g r"   and "g \<turnstile> m-ms\<rightarrow>defNode g s"   and "set ns \<inter> set ms = {}"   and "phiArg g \<phi>\<^sub>r r" obtains ns' :: "'node list" where "g \<turnstile> n-ns @ ns'\<rightarrow>defNode g \<phi>\<^sub>r" and "set (butlast (ns @ ns')) \<inter> set ms = {}"
even_mult_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 a :: "'a"   and b :: "'a" assumes "class.semiring_parity plus minus zero times one divide modulo" shows "dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) (times a b) = (dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) a \<or> dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) b)"
inv_CONJ: fixes f :: "int \<Rightarrow> int" assumes "bij f" shows "inv (CONJ f) = CONJ (inv f)"