hcju/isabelleps huggingface dataset:
hm_is_it'_def: fixes l :: "('a \<times> 'b) list list"   and ht :: "('a, 'b) hashtable"   and l' :: "('a \<times> 'b) list"   and it :: "nat \<times> ('a \<times> 'b) list \<times> ('a, 'b) hashtable" shows "hm_is_it' l ht l' it \<equiv> is_hashtable l ht * \<up> (let (i, r, ht') = it in ht = ht' \<and> l' = concat (take i l) @ rev r \<and> distinct (map fst l') \<and> i \<le> length l \<and> (r = [] \<longrightarrow> i = 0))"
that: fixes q_in_bs :: "bool" assumes "(ic, q_in_bs, p, q) \<in> set ps" shows "thesis"
convex_scaled: fixes S :: "real set"   and c :: "'a" assumes "convex S" shows "convex ((\<lambda>x. x *\<^sub>R c) ` S)"
simps: fixes a1 :: "'a list"   and a2 :: "'a list" shows "lexordp_eq a1 a2 = ((\<exists>ys. a1 = [] \<and> a2 = ys) \<or> (\<exists>x y xs ys. a1 = x # xs \<and> a2 = y # ys \<and> x < y) \<or> (\<exists>x y xs ys. a1 = x # xs \<and> a2 = y # ys \<and> \<not> x < y \<and> \<not> y < x \<and> lexordp_eq xs ys))"
dom_in_domains: fixes f :: "'b" assumes "C'.domains f \<noteq> {}" shows "C'.dom f \<in> C'.domains f"
a_psi: shows "square_free_factorization_int f = (a, psi)"
incr_boundvars_def: fixes inc :: "nat"   and t :: "Core.term" shows "incr_boundvars inc t = incr_bv inc 0 t"
order_strict_order: fixes x :: "'a" assumes "order x" shows "strict_order (x \<sqinter> - (1::'a))"
ceq_set_def: shows "CEQ('b set) = (case ID CEQ('b) of None \<Rightarrow> None | Some x \<Rightarrow> Some set_eq)"
meldUniq_mset: fixes q :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree list"   and q' :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree list" assumes "SkewBinomialHeapStruc.queue_invar q"   and "SkewBinomialHeapStruc.queue_invar q'" shows "SkewBinomialHeapStruc.queue_to_multiset (SkewBinomialHeapStruc.meldUniq q q') = SkewBinomialHeapStruc.queue_to_multiset q + SkewBinomialHeapStruc.queue_to_multiset q'"
simps_33: fixes x101 :: "expr"   and x102 :: "char list option"   and x103 :: "char list"   and x104 :: "expr list"   and x1 :: "char list" shows "Call x101 x102 x103 x104 \<noteq> new x1"
not_squarefree_0: shows "\<not> squarefree (0::'b)"
while_hyp_change: fixes n :: "nat"   and z :: "nat"   and i :: "nat"   and j :: "nat"   and z' :: "nat" assumes "\<forall>x\<le>n. 0 < x \<longrightarrow> s (e_snoc (local.adverse z i j \<triangleright> x) 0) \<noteq> s (local.adverse z i j \<triangleright> x) \<or> s (e_snoc (local.adverse z i j \<triangleright> x) 1) \<noteq> s (local.adverse z i j \<triangleright> x)" shows "\<forall>x\<le>Suc n. local.adverse z i j x = local.adverse z' i j x"
mult2: fixes x :: "'a" assumes "x \<in> carrier R" shows "x \<oplus> x = \<guillemotleft>2\<guillemotright> \<otimes> x"
seq_fold: fixes cs :: "command list"   and \<Pi> :: "(interface, 'a) chain"   and p1 :: "assertion"   and p2 :: "assertion" assumes "length cs = chainlen \<Pi>"   and "p1 = asn (pre \<Pi>)"   and "p2 = asn (post \<Pi>)"   and "\<And>i. i < chainlen \<Pi> \<Longrightarrow> prov_triple (asn (fst3 (nthtriple \<Pi> i)), cs ! i, asn (thd3 (nthtriple \<Pi> i)))" shows "prov_triple (p1, foldr (;;) cs Skip, p2)"
assms_3: shows "InvariantQCharacterization (getConflictFlag state) (getQ state) (getF state) (getM state)"
less_option_None_is_Some: fixes x :: "'b option" assumes "None < x" shows "\<exists>z. x = Some z"
pre: shows "pre_adjust \<langle>l, (a, lv), r\<rangle>"
aaa_sub: shows "poss (add_mset A AA) \<subseteq># map_clause f C"
inf_sup_ord_3: fixes inf :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and sup :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and x :: "'e"   and y :: "'e" assumes "class.lattice inf less_eq less sup" shows "less_eq x (sup x y)"
filter_tab_None: fixes t :: "'d \<Rightarrow> 'c option"   and k :: "'d"   and p :: "'d \<Rightarrow> 'c \<Rightarrow> bool" assumes "t k = None" shows "filter_tab p t k = None"
U: shows "U \<equiv> T \<union> T' \<union> reduced_letter_set S w"
list_int_nat: fixes id :: "'b"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and k :: "nat"   and x :: "'c"   and xs :: "'c list" assumes "Matrix_Tensor.mult id f" shows "0 < k \<longrightarrow> (x # xs) ! k = xs ! nat (int k + - 1)"
int_of_nat_raw: fixes m :: "nat" shows "int_of_nat_raw m = (m, 0)"
frechet_derivative_works: fixes f :: "'b \<Rightarrow> 'c"   and net :: "'b filter" shows "(f differentiable net) = (f has_derivative frechet_derivative f net) net"
map_snd_apfst: fixes x :: "'d \<Rightarrow> 'c"   and l :: "('d \<times> 'b) list" shows "map snd (map (apfst x) l) = map snd l"
pmf_of_set_lottery: fixes A :: "'b set"   and B :: "'b set" assumes "A \<noteq> {}"   and "finite A"   and "A \<subseteq> B" shows "pmf_of_set A \<in> lotteries_on B"
simps_2: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ"   and f :: "('fsym, 'tp) efsym"   and Tl :: "('fsym, 'tp) efsym trm list" shows "GE.int \<xi> (Fn f Tl) = eintF f (map (GE.int \<xi>) Tl)"
lprefix_conv_lappend: fixes xs :: "'c llist"   and ys :: "'c llist" shows "xs \<sqsubseteq> ys = (\<exists>zs. ys = lappend xs zs)"
flip: fixes prod :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" assumes "bounded_cbilinear prod" shows "bounded_cbilinear (\<lambda>x y. prod y x)"
simps_2: fixes A :: "'b set"   and n :: "nat" shows "nth_least A (Suc n) = nth_least (A - {least A}) n"
fst_apply_icrit: fixes crit :: "nat \<times> 'e \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list \<Rightarrow> ('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g \<Rightarrow> ('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g \<Rightarrow> bool"   and d :: "'a \<Rightarrow> nat"   and gs :: "(('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list"   and bs :: "(('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list"   and hs :: "(('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list"   and m :: "nat"   and data :: "nat \<times> 'e"   and p :: "('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g"   and q :: "('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g"   and q_in_bs :: "bool"   and ps :: "(bool \<times> (('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) \<times> ('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list" assumes "icrit_spec crit"   and "dickson_grading d"   and "fst ` (set gs \<union> set bs \<union> set hs) \<subseteq> dgrad_p_set d m"   and "unique_idx (gs @ bs @ hs) data"   and "is_Groebner_basis (fst ` set gs)"   and "p \<in> set hs"   and "q \<in> set gs \<union> set bs \<union> set hs"   and "fst p \<noteq> 0"   and "fst q \<noteq> 0"   and "(True, q_in_bs, p, q) \<in> set (apply_icrit crit data gs bs hs ps)" shows "crit_pair_cbelow_on d m (fst ` (set gs \<union> set bs \<union> set hs)) (fst p) (fst q)"
frontier_subset_retraction: fixes S :: "'a set"   and T :: "'a set"   and f :: "'a \<Rightarrow> 'a" assumes "bounded S"   and "frontier S \<subseteq> T"   and "continuous_on (closure S) f"   and "f ` S \<subseteq> T"   and "\<And>x. x \<in> T \<Longrightarrow> f x = x" shows "S \<subseteq> T"
assms: fixes \<phi> :: "('state \<times> 'aprop set) stream list \<Rightarrow> bool" assumes "\<phi> \<in> \<phi>s" shows "\<phi> \<pi>l"
bl_to_bin_False: fixes bl :: "bool list" shows "bl_to_bin (False # bl) = bl_to_bin bl"
induced_automorphism_simplex_image_1: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'b"   and C0 :: "'b set"   and C :: "'b set"   and x :: "'b set" assumes "OpposedThinChamberComplexFoldings X f g C0"   and "C \<in> f \<turnstile> ChamberComplex.chamber_system X"   and "order.greater_eq C x" shows "OpposedThinChamberComplexFoldings.induced_automorphism X f g ` x = g ` x"
1: shows "rawpsubst (rawpsubst \<phi> (zip (map Var ws) (map snd txs))) (zip (map fst txs) ws) = rawpsubst (rawpsubst \<phi> (zip (map Var us) (map snd txs))) (zip (map fst txs) us)"
antisym: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c" assumes "ordering less_eq less"   and "less_eq a b"   and "less_eq b a" shows "a = b"
in_domain: fixes f :: "'b \<Rightarrow> 'b"   and T :: "real set"   and S :: "'b set"   and \<phi> :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and s :: "'b" assumes "local_flow f T S \<phi>"   and "s \<in> S" shows "(\<lambda>t. \<phi> t s) \<in> T \<rightarrow> S"
big_imp_small: fixes \<pi> :: "char list \<Rightarrow> com option"   and cs :: "com \<times> (char list \<Rightarrow> int \<Rightarrow> int)"   and t :: "char list \<Rightarrow> int \<Rightarrow> int" assumes "\<pi>: cs \<Rightarrow> t" shows "small_steps \<pi> cs (Some (SKIP, t))"
release_append: fixes sb :: "'b memref list"   and xs :: "'b memref list"   and \<S> :: "nat \<Rightarrow> bool option"   and \<R> :: "nat \<Rightarrow> bool option" shows "release (sb @ xs) (dom \<S>) \<R> = release xs (dom (share sb \<S>)) (release sb (dom \<S>) \<R>)"
local_lipschitz_temp_dyn: fixes a :: "real"   and L :: "real" assumes "0 < a" shows "local_lipschitz UNIV UNIV (\<lambda>t. temp_vec_field a L)"
osc_property: fixes ur :: "('s \<times> 's) set"   and s1 :: "'s"   and s1' :: "'s"   and \<alpha> :: "'e list" assumes "osc ur"   and "s1 \<in> S\<^bsub>SES\<^esub>"   and "s1' \<in> S\<^bsub>SES\<^esub>"   and "\<alpha> \<upharpoonleft> C\<^bsub>\<V>\<^esub> = []"   and "reachable SES s1"   and "reachable SES s1'"   and "enabled SES s1' \<alpha>"   and "(s1', s1) \<in> ur" shows "\<exists>\<alpha>'. \<alpha>' \<upharpoonleft> C\<^bsub>\<V>\<^esub> = [] \<and> \<alpha>' \<upharpoonleft> V\<^bsub>\<V>\<^esub> = \<alpha> \<upharpoonleft> V\<^bsub>\<V>\<^esub> \<and> enabled SES s1 \<alpha>'"
less: fixes t :: "'a" assumes "t \<in> T" shows "l t < b"
assms_2: shows "next \<in> Mapping.keys (phis g)"
free_vars_expr_code: fixes e :: "expr" shows "free_vars e = free_vars_expr_code e"
euler_maclaurin_strong_int_aux: fixes b :: "int" assumes "a \<le> b" shows "(\<Sum>k = a..b. f (real_of_int k)) - F (real_of_int b) - T (real_of_int b) = EM_remainder' N (fs N) (real_of_int a) (real_of_int b) + (C - C')"
vpa_valid_call_list_valid_return_list_vpra: fixes cs :: "'edge list"   and cs' :: "'edge list"   and n :: "'node"   and n' :: "'node" assumes "valid_path_aux cs cs'"   and "valid_call_list cs n"   and "valid_return_list cs' n'" shows "valid_path_rev_aux cs' (rev cs)"
less_INF_D: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and y :: "'b"   and f :: "'c \<Rightarrow> 'b"   and A :: "'c set"   and i :: "'c" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "less y (Inf (f ` A))"   and "i \<in> A" shows "less y (f i)"
dec_interp_Inl: fixes i :: "nat"   and FO :: "nat set"   and n :: "nat"   and x :: "('a \<times> bool list) list" assumes "i \<in> FO"   and "i < n" shows "\<exists>p. dec_interp n FO x ! i = Inl p"
S5MemUnch: fixes p :: "PrIds"   and rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState" shows "\<turnstile> [RNext rmCh mm ires p]_(m p) \<and> $S5 rmhist p \<longrightarrow> unchanged m p"
diverge_lfilter_LNil: fixes xs :: "'c llist"   and P :: "'c \<Rightarrow> bool" assumes "\<forall>x\<in>lset xs. \<not> P x" shows "lfilter P xs = LNil"
intros: fixes \<phi> :: "'a stream \<Rightarrow> bool"   and xs :: "'a stream" assumes "\<phi> xs"   and "alw \<phi> (stl xs)" shows "alw \<phi> xs"
bcomp_correct: fixes x :: "bexp \<times> bool \<times> int"   and cfs :: "(int \<times> (char list \<Rightarrow> int) \<times> int list) list" assumes "bcomp x \<Turnstile> cfs\<box>"   and "0 \<le> snd (snd x)" shows "bpred x (cfs ! 0) (cfs ! (length cfs - Suc 0))"
R23sD: fixes implem :: "chan \<Rightarrow> msg"   and s :: "l2_state"   and t :: "l3_state" assumes "pfslvl3 implem"   and "(s, t) \<in> pfslvl3.R23s implem" shows "ik s = ik t \<inter> payload \<and> secret s = secret t \<and> progress s = progress t \<and> l1_state.signals s = l1_state.signals t \<and> chan s = semivalid_implem.abs implem (ik t) \<and> l2_state.bad s = l3_state.bad t"
simps: fixes x :: "'a"   and t :: "'a tree"   and dl :: "nat" shows "local.delete x (t, dl) = (case del x t of None \<Rightarrow> (t, dl) | Some t' \<Rightarrow> if bal_d (size t') (dl + 1) then (t', dl + 1) else (balance_tree t', 0))"
well_order_induct: fixes P :: "'b \<Rightarrow> bool"   and a :: "'b" assumes "\<And>x. \<forall>y. y \<noteq> x \<and> (y, x) \<in> s \<longrightarrow> P y \<Longrightarrow> P x" shows "P a"
ipp_cond3_def: fixes z :: "'e"   and i :: "'e \<Rightarrow> 'e" shows "ipp_cond3 z i = (\<^sub>i- z = z)"
abs_eq: fixes x :: "int \<Rightarrow> 'a" assumes "eq_onp (\<lambda>f. \<forall>\<^sub>\<infinity>n. f (- int n) = (0::'a)) x x" shows "fls_regpart (Abs_fls x) = Abs_fps (\<lambda>n. x (int n))"
astI_def: 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" shows "ast_problem.astI problem \<equiv> case problem of (D, I, G, \<delta>) \<Rightarrow> I"
arrI: fixes f :: "'s"   and a :: "'s"   and b :: "'s" assumes "\<guillemotleft>f : a \<rightarrow>\<^sub>S b\<guillemotright>" shows "S.arr f"
Q_defined: fixes u :: "'v"   and v :: "'v" assumes "(u, v) \<in> edges rg"   and "u \<notin> S (A Q \<pi>)"   and "v \<in> S (A Q \<pi>)" shows "Q u \<noteq> \<infinity>"
face_cycle_succ_closed: fixes a :: "'b" shows "(H.face_cycle_succ a \<in> arcs H) = (a \<in> arcs H)"
pre: shows "(l = PAodv-:4 \<or> l = PAodv-:5 \<or> l = PRreq-:0 \<or> l = PRreq-:2) \<and> sip (\<sigma> i) \<noteq> oip (\<sigma> i) \<longrightarrow> oip (\<sigma> i) \<in> kD (rt (\<sigma> (sip (\<sigma> i)))) \<and> osn (\<sigma> i) \<le> nsqn (rt (\<sigma> (sip (\<sigma> i)))) (oip (\<sigma> i)) \<and> (nsqn (rt (\<sigma> (sip (\<sigma> i)))) (oip (\<sigma> i)) = osn (\<sigma> i) \<longrightarrow> the (dhops (rt (\<sigma> (sip (\<sigma> i)))) (oip (\<sigma> i))) \<le> hops (\<sigma> i) \<or> the (flag (rt (\<sigma> (sip (\<sigma> i)))) (oip (\<sigma> i))) = Aodv_Basic.inv)"
vars_clause_def: fixes C :: "('d, 'c) Term.term literal multiset" shows "vars_clause C = \<Union> (set_mset (image_mset vars_lit C))"
simps_7: fixes t :: "'b EV.term" shows "\<lbrace>\<^bold>\<r>\<^bold>[t\<^bold>]\<rbrace> = D.\<rr> \<lbrace>t\<rbrace>"
ran_def: fixes m :: "'h \<Rightarrow> 'g option" shows "ran m = {b. \<exists>a. m a = Some b}"
equalityCE: fixes A :: "'c set"   and B :: "'c set"   and c :: "'c" assumes "A = B" obtains "c \<in> A" and "c \<in> B"   | "c \<notin> A" and "c \<notin> B"
agree_sub: fixes A :: "('c + 'c) set"   and B :: "('c + 'c) set"   and \<nu> :: "(real, 'c) vec \<times> (real, 'c) vec"   and \<omega> :: "(real, 'c) vec \<times> (real, 'c) vec" assumes "A \<subseteq> B"   and "Vagree \<nu> \<omega> B" shows "Vagree \<nu> \<omega> A"
strips_equivalent_to_sas_plus_ii: fixes \<Psi> :: "('c, 'd) sas_plus_problem"   and \<psi> :: "('c, 'd) sas_plus_operator list list" assumes "is_valid_problem_sas_plus \<Psi>"   and "SAS_Plus_Semantics.is_parallel_solution_for_problem \<Psi> \<psi>" shows "list_all (list_all (\<lambda>op. ListMem op ((\<phi> \<Psi> )\<^sub>\<O>))) (\<phi>\<^sub>P \<Psi> \<psi>)"
continuous_on_closed_vimage: fixes s :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "closed s" shows "continuous_on s f = (\<forall>B. closed B \<longrightarrow> closed (f -` B \<inter> s))"
max_def: fixes a :: "'a"   and b :: "'a" shows "ordered_powerprod_lin.max a b = (if a \<preceq> b then b else a)"
not_dvd_imp_N_ge: fixes n :: "nat"   and a :: "int" assumes "\<not> int n dvd a"   and "0 < n" shows "1 / q ^ n \<le> N a"
d_def: shows "d = (if 0 < d' then d' / 2 else cmod (z - of_int (round d')) / 2)"
loc_Op2_def: shows "loc_Op2 = (\<lambda>f. PrimRecOp_last (\<lambda>x. 0) (loc_Op1 f))"
t: fixes nl :: "nat"   and ml :: "nat"   and nr :: "nat"   and mr :: "nat"   and rn :: "nat" shows "(c = Bk \<up> nl @ Oc # Oc \<up> ml @ Bk # Oc \<up> Suc m \<and> [] = Bk \<up> nr @ Oc \<up> mr @ Bk \<up> rn \<and> ml + mr = Suc (Suc rs) \<and> 0 < mr \<and> 0 < nl + nr) = False"
gauss_cnj_1: shows "gauss_cnj 1 = 1"
minus_def: fixes x :: "'a"   and y :: "'a" shows "x - y = - (- x \<squnion> y)"
l_to_tree_order_si\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def: fixes to_tree_order_si :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_shadow_root :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, (_) shadow_root_ptr option) prog"   and get_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool" shows "l_to_tree_order_si\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M to_tree_order_si get_child_nodes get_child_nodes_locs get_shadow_root get_shadow_root_locs type_wf known_ptr \<equiv> l_get_child_nodes type_wf known_ptr get_child_nodes get_child_nodes_locs \<and> l_get_shadow_root type_wf get_shadow_root get_shadow_root_locs \<and> l_to_tree_order_si\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms to_tree_order_si get_child_nodes get_shadow_root"
model_def: fixes C :: "'b set"   and G :: "('b Standard_Constant, 'b + 'c) labeled_graph"   and T :: "('b Standard_Constant allegorical_term \<times> 'b Standard_Constant allegorical_term) set" shows "model C G T \<equiv> standard' C G \<and> (\<forall>S\<in>T. G \<Turnstile> S)"
isomorphic_def: fixes a :: "'a"   and a' :: "'a" shows "(a \<cong>\<^sub>C a') = (\<exists>f. \<guillemotleft>f : a \<Rightarrow>\<^sub>C a'\<guillemotright> \<and> C.iso f)"
supp_removeAll: fixes x :: "atom"   and xs :: "atom list" shows "supp (removeAll x xs) = supp xs - {x}"
d: shows "(r|\<tau>| + r|\<sigma>'| -s r \<down>l \<tau>, r|\<tau>| + r|\<sigma>|) \<in> mul_eq r"
wf_bs_polyadd: fixes bs :: "'a list"   and p :: "poly"   and q :: "poly" shows "wf_bs bs p \<and> wf_bs bs q \<longrightarrow> wf_bs bs (p +\<^sub>p q)"
simps_1: fixes n :: "nat"   and rai :: "int poly \<times> rat \<times> rat" shows "invariant_2 (Irrational n rai) = (invariant_1_2 rai \<and> n = card {y. y \<le> real_of_1 rai \<and> poly (real_of_int_poly (poly_real_alg_1 rai)) y = 0})"
imp_step: fixes \<sigma> :: "nat \<Rightarrow> bool"   and cnf :: "lit uprod set"   and l\<^sub>1 :: "lit"   and l\<^sub>2 :: "lit" assumes "sat_cnf \<sigma> cnf"   and "sat_lit \<sigma> l\<^sub>1"   and "(l\<^sub>1, l\<^sub>2) \<in> imp_graph cnf" shows "sat_lit \<sigma> l\<^sub>2"
get_child_nodes\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def: fixes document_ptr :: "(_) document_ptr"   and uu :: "unit" shows "get_child_nodes\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r document_ptr uu = Heap_Error_Monad.bind (get_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t document_ptr document_element) (\<lambda>doc_elem. case doc_elem of None \<Rightarrow> return [] | Some element_ptr \<Rightarrow> return [cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r element_ptr])"
less_add_eq_less: fixes k :: "nat"   and l :: "nat"   and m :: "nat"   and n :: "nat" assumes "order.greater l k"   and "m + l = k + n" shows "order.greater n m"
ENF_offending_case2: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and G :: "'v graph"   and nP :: "'v \<Rightarrow> 'a" assumes "sinvar_all_edges_normal_form P"   and "sinvar G nP" shows "{} = set_offending_flows G nP"
Frct_def: fixes p :: "int \<times> int" shows "Frct p = Rat.Fract (fst p) (snd p)"
order_le_less: fixes x :: "'a"   and y :: "'a" shows "(x \<le> y) = (x < y \<or> x = y)"
Empty_is_rbt: fixes less :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "ord.is_rbt less rbt.Empty"
fps_one_mult_1: fixes f :: "'a fps" shows "1 * f = f"
minus_less_iff_ennreal: fixes b :: "ennreal"   and a :: "ennreal"   and c :: "ennreal" assumes "b < top"   and "b \<le> a" shows "(a - b < c) = (a < c + b)"
onll_def: fixes \<Gamma> :: "'d \<Rightarrow> ('b, 'c, 'd, 'e) seqp"   and P :: "('a \<times> 'e) \<times> 'f \<times> 'a \<times> 'e \<Rightarrow> bool" shows "onll \<Gamma> P \<equiv> \<lambda>((\<xi>, p), a, \<xi>', p'). \<forall>l\<in>labels \<Gamma> p. \<forall>l'\<in>labels \<Gamma> p'. P ((\<xi>, l), a, \<xi>', l')"
mahler_measure_poly_explicit: fixes c :: "complex"   and as :: "complex list" shows "mahler_measure_poly (Polynomial.smult c (\<Prod>a\<leftarrow>as. [:- a, 1:])) = cmod c * (\<Prod>a\<leftarrow>as. max 1 (cmod a))"
locally_sym_def: fixes R :: "('b \<times> 'b) set"   and A :: "'b multiset" shows "locally_sym R A \<equiv> \<forall>t u. t \<in># A \<longrightarrow> u \<in># A \<longrightarrow> (t, u) \<in> R \<longrightarrow> (u, t) \<in> R"
finite_Union: fixes X :: "'a set set" assumes "finite X"   and "X \<subseteq> sigma_sets \<Omega> M" shows "\<Union> X \<in> sigma_sets \<Omega> M"
f': shows "simple_function M (\<lambda>x. f x * indicator (space M) x)"
continuous_on_diff_ereal: fixes A :: "'a set"   and f :: "'a \<Rightarrow> ereal"   and g :: "'a \<Rightarrow> ereal" assumes "continuous_on A f"   and "continuous_on A g"   and "\<And>x. x \<in> A \<Longrightarrow> \<bar>f x\<bar> \<noteq> \<infinity>"   and "\<And>x. x \<in> A \<Longrightarrow> \<bar>g x\<bar> \<noteq> \<infinity>" shows "continuous_on A (\<lambda>z. f z - g z)"
SystemVCut2_def: shows "SystemVCut2 = (Geq (f0 fid1) (Const 0) \<rightarrow> [[EvolveODE (OProd (OSing vid1 (f0 fid1)) (OSing vid2 (Var vid1))) TT]]Geq (f0 fid1) (Const 0))"
meta_cong_bigtheta: fixes F :: "'a filter"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "\<Theta>[F](f) \<equiv> \<Theta>[F](g)" shows "o[F](f) \<equiv> o[F](g)"
is_processT5_S2a: fixes s :: "'a event list"   and X :: "'a event set"   and P :: "'a process"   and c :: "'a event" assumes "(s, X) \<in> F P"   and "(s, X \<union> {c}) \<notin> F P" shows "(s @ [c], {}) \<in> F P"
fis_singleton_fthe_elem: fixes A :: "'a fset" shows "fis_singleton A = (A = {|fthe_elem A|})"
map_poly_eval_poly: fixes p :: "int poly" shows "of_int_poly p = eval_poly (\<lambda>a. [:of_int a:]) p [:0::'a, 1::'a:]"
set_same: fixes r :: "'a ref"   and x :: "'a"   and y :: "'a"   and h :: "heap" shows "Ref.set r x (Ref.set r y h) = Ref.set r x h"
simps_2: fixes p :: "fm"   and q :: "fm" shows "isrlfm (Or p q) = (isrlfm p \<and> isrlfm q)"
iFROM_I: fixes n :: "nat"   and x :: "nat" assumes "n \<le> x" shows "x \<in> [n\<dots>]"
is_relevant_entry_def: fixes P :: "'a prog"   and i :: "instr"   and pc :: "nat"   and e :: "ex_entry" shows "is_relevant_entry P i pc e = (let (f, t, C, h, d) = e in is_relevant_class i P C \<and> pc \<in> {f..<t})"
fun_upd_same: fixes f :: "'c \<Rightarrow> 'b"   and x :: "'c"   and y :: "'b" shows "(f(x := y)) x = y"
T_on_on': fixes step :: "'b \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> 'b"   and t :: "'b \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> nat"   and s0 :: "'b"   and qs :: "'c list"   and A :: "('b \<Rightarrow> 'e) \<times> ('b \<times> 'e \<Rightarrow> 'c \<Rightarrow> 'd \<times> 'e)"   and x :: "'e" shows "On_Off.T step t s0 qs (On_Off.off2 step A (s0, x) qs) = On_Off.T_on' step t A (s0, x) qs"
nres_rel_def: fixes R :: "('i \<times> 'j) set" shows "\<langle>R\<rangle>nres_rel \<equiv> {(c, a). c \<le> \<Down> R a}"
xor_left_inverse: fixes x :: "'g"   and y :: "'g" shows "x \<oplus> x \<oplus> y = y"