hcju/isabelleps huggingface dataset:
diff_outputs_ctx_def: fixes e1 :: "((nat \<times> nat) \<times> transition) fset"   and e2 :: "((nat \<times> nat) \<times> transition) fset"   and s1 :: "nat"   and s2 :: "nat"   and t1 :: "'a transition_scheme"   and t2 :: "'b transition_scheme" shows "diff_outputs_ctx e1 e2 s1 s2 t1 t2 = (if Outputs t1 = Outputs t2 then False else \<exists>p c1 r. obtains s1 c1 e1 0 <> p \<and> obtains s2 r e2 0 <> p \<and> (\<exists>i. can_take_transition t1 i r \<and> can_take_transition t2 i r \<and> evaluate_outputs t1 i r \<noteq> evaluate_outputs t2 i r))"
sig_inv_set_closed_plus: fixes r :: "'t \<Rightarrow>\<^sub>0 'b"   and j :: "nat"   and s :: "'t \<Rightarrow>\<^sub>0 'b" assumes "r \<in> sig_inv_set' j"   and "s \<in> sig_inv_set' j" shows "r + s \<in> sig_inv_set' j"
assms_7: shows "fst ` X1 \<subseteq> K"
simps_2: fixes f :: "'d \<Rightarrow> 'b"   and t :: "'a"   and b :: "bool" shows "convert_new_thread_action f (ThreadExists t b) = ThreadExists t b"
refl: fixes carrier :: "'c set"   and le :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c" assumes "Order_Predicates.preorder_on carrier le"   and "x \<in> carrier" shows "le x x"
rel_inject_3: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and x3 :: "'b"   and y3 :: "'c" shows "rel_rexp R (Atom x3) (Atom y3) = R x3 y3"
fo_rg_inv: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and s :: "nat"   and t :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'a" assumes "NPreflow c s t f" shows "Network.flow_of_cf c (residualGraph c f) = f"
rep_eq: fixes x :: "'b"   and xa :: "'b fset" shows "(x |\<in>| xa) = (x \<in> \<langle>xa\<rangle>\<^sub>f)"
1: shows "w * (F \<sqinter> w\<^sup>T\<^sup>\<star>) * e\<^sup>T * e \<le> F"
is_rbt_impl_of: fixes t :: "'b set_rbt" assumes "ID ccompare \<noteq> None" shows "ord.is_rbt cless (mapping_rbt.impl_of t)"
that: fixes v\<^sub>1 :: "'item list" assumes "w\<^sub>1 @ v\<^sub>1 =\<^sub>F w\<^sub>2" shows "thesis"
split_Edge2_eq_0I: fixes y :: "'v vertex"   and x :: "'v"   and f :: "'v \<times> 'v \<Rightarrow> ennreal"   and z :: "'v" assumes "\<And>z. y \<noteq> Vertex x" shows "split f (y, Edge x z) = 0"
\<theta>_simps_3: shows "trg \<theta>' = src r"
projl_def: fixes sum :: "'g + 'h" shows "projl sum = (case sum of Inl x1 \<Rightarrow> x1)"
\<rho>'_ide_simp: fixes a :: "'d" assumes "D.ide a" shows "D.\<rho>' a = D.runit' a"
ordLeq_transitive: fixes r :: "('a \<times> 'a) set"   and r' :: "('b \<times> 'b) set"   and r'' :: "('c \<times> 'c) set" assumes "r \<le>o r'"   and "r' \<le>o r''" shows "r \<le>o r''"
maxR_domain: fixes x :: "'a"   and y :: "'a"   and A :: "'a set" assumes "{x, y} \<subseteq> A" shows "maxR x y \<in> A"
p1: shows "\<Theta>,\<Gamma> \<turnstile> mk_all x \<tau> B"
select_convs_2: fixes sprFst :: "('a, 'b, 'c, 'd, 'e) BEState"   and sprLst :: "('a, 'b, 'c, 'd, 'e) BEState"   and sprCRel :: "(('a, 'b, 'c, 'd, 'e) BEState \<times> ('a, 'b, 'c, 'd, 'e) BEState) set"   and more :: "'f" shows "sprLst \<lparr>sprFst = sprFst, sprLst = sprLst, sprCRel = sprCRel, \<dots> = more\<rparr> = sprLst"
max_enat_simps_3: fixes q :: "enat" shows "max 0 q = q"
projxx: fixes a :: "nat"   and x :: "nat"   and y :: "nat"   and e :: "nat"   and f :: "nat list"   and bs :: "nat list"   and init :: "nat list" assumes "a \<in> {x, y}"   and "a \<noteq> e"   and "e \<in> {x, y}"   and "f \<in> {[], [e]}"   and "set (bs @ [a] @ f) \<subseteq> {x, y}"   and "Lxy init {x, y} \<in> {[x, y], [y, x]}" shows "a < e in fst (config\<^sub>p (rTS []) (Lxy init {x, y}) ((bs @ [a] @ f) @ [a]))"
map_is_empty_def: fixes m :: "'c \<Rightarrow> 'd option" shows "map_is_empty m = (m = Map.empty)"
less_imp_neq: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.order less_eq less"   and "less x y" shows "x \<noteq> y"
assms_2: fixes P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi" assumes "\<Psi> \<rhd> P \<doteq> Q" shows "\<Psi> \<rhd> C P \<approx> C Q"
ergodic_conservative_def: fixes M :: "'b measure"   and T :: "'b \<Rightarrow> 'b" shows "ergodic_conservative M T \<equiv> ergodic_qmpt M T \<and> conservative M T"
simps_3: fixes c :: "poly"   and e :: "tm" shows "plusinf (Eq (CNP 0 c e)) = Parametric_Ferrante_Rackoff.conj (eq (CP c)) (eq e)"
nth_default_Cons_Suc: fixes dflt :: "'b"   and x :: "'b"   and xs :: "'b list"   and n :: "nat" shows "nth_default dflt (x # xs) (Suc n) = nth_default dflt xs n"
equals_cod_if_value_is_ide: fixes A :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and g :: "'b" assumes "constant_transformation A B g"   and "partial_magma.ide B g" shows "constant_transformation.map A B g = constant_functor.map A B (partial_magma.cod B g)"
Ide_Cod: fixes t :: "'c local.term" assumes "Arr t" shows "Ide (Cod t)"
opI: fixes a :: "'a" assumes "a \<in> S" shows "\<exists>T. openin (top_of_set S) T \<and> a \<in> T \<and> (\<forall>x\<in>T. \<forall>y\<in>T. P x \<and> P y \<and> Q x \<longrightarrow> Q y)"
decycle_def: fixes E :: "'b uprod set"   and u :: "'b"   and p :: "'b uprod list" shows "decycle E u p \<equiv> epath E u p u \<and> 2 < length p \<and> distinct p"
inj_on_arc_to_ends: shows "inj_on (arc_to_ends G) (pre_digraph.arcs G)"
slexer_better_simp: fixes r :: "'b rexp"   and c :: "'b"   and s :: "'b list" shows "slexer r (c # s) = (case slexer (fst (simp (deriv c r))) s of None \<Rightarrow> None | Some v \<Rightarrow> Some (injval r c (snd (simp (deriv c r)) v)))"
comp_target_ide: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and b :: "'b"   and f :: "'b" assumes "regular_weak_composition V H"   and "b \<in> weak_composition.targets V H f"   and "partial_magma.ide V f" shows "category.isomorphic V (H b f) f"
HEndPhase2_Inv2a: fixes s :: "state"   and s' :: "state"   and p :: "Proc" assumes "Inv2a s"   and "HEndPhase2 s s' p" shows "Inv2a s'"
groundc\<sigma>: shows "ground\<^sub>l\<^sub>s (C \<cdot>\<^sub>l\<^sub>s sub_of_denot E)"
and_num_4294967167_128: shows "4294967167 AND 128 = 0"
QF_ne: fixes j :: "nat"   and k :: "nat"   and i :: "nat" assumes "j < Suc k" shows "QF k (j, i) \<noteq> {}"
initially_undefined_context_check_def: fixes e :: "((nat \<times> nat) \<times> transition) fset"   and r :: "nat"   and s :: "nat" shows "initially_undefined_context_check e r s = (\<forall>t a. obtains s a e 0 <> t \<longrightarrow> a $ r = None)"
the_equality: fixes P :: "'g \<Rightarrow> bool"   and a :: "'g" assumes "P a"   and "\<And>x. P x \<Longrightarrow> x = a" shows "(THE x. P x) = a"
insert_partition: fixes x :: "'a set"   and F :: "'a set set" assumes "x \<notin> F"   and "\<forall>c1\<in>insert x F. \<forall>c2\<in>insert x F. c1 \<noteq> c2 \<longrightarrow> c1 \<inter> c2 = {}" shows "x \<inter> \<Union> F = {}"
resChainI: fixes Rel :: "(pi \<times> pi) set"   and P :: "pi"   and Q :: "pi"   and lst :: "name list" assumes "eqvt Rel"   and "\<And>R S x. (R, S) \<in> Rel \<Longrightarrow> (<\<nu>x>R, <\<nu>x>S) \<in> Rel"   and "P \<leadsto>[Rel] Q" shows "resChain lst P \<leadsto>[Rel] resChain lst Q"
cm_preserves_inv_implications_nonneg: fixes c0 :: "('loc, 't) configuration"   and c1 :: "('loc, 't) configuration"   and loc :: "'loc"   and t :: "'t"   and n :: "int" assumes "next_change_multiplicity' c0 c1 loc t n"   and "inv_implications_nonneg c0" shows "inv_implications_nonneg c1"
assms_3: shows "isRecv ev \<longrightarrow> getId ev \<noteq> i"
hom_signof: fixes p :: "nat \<Rightarrow> nat" shows "of_int (signof p) = signof p"
nth_map_matrix: fixes f :: "'e \<Rightarrow> 'b"   and x :: "(('e, 'c) Finite_Cartesian_Product.vec, 'd) Finite_Cartesian_Product.vec"   and i :: "'d"   and j :: "'c" shows "Finite_Cartesian_Product.map_matrix f x $h i $h j = f (x $h i $h j)"
leq_n: fixes p :: "nat \<Rightarrow> nat"   and n :: "nat"   and i :: "nat" assumes "\<forall>i. 0 < p i \<longrightarrow> 1 \<le> i \<and> i \<le> n"   and "(\<Sum>i\<le>n. p i * i) = n" shows "p i * i \<le> n"
from_Gromov_completion_continuous_3: fixes x :: "'c Gromov_completion" assumes "x \<in> range to_Gromov_completion" shows "isCont from_Gromov_completion x"
berlekamp_mat_def: fixes u :: "'b mod_ring poly" shows "berlekamp_mat u = (let n = degree u; mul_p = power_poly_f_mod u [:0, 1:] CARD('b); xks = power_polys mul_p u 1 n in mat_of_rows_list n (map (\<lambda>cs. let coeffs_cs = coeffs cs; k = n - length (coeffs cs) in coeffs cs @ replicate k 0) xks))"
gallery_append_reduce1: fixes xs :: "'a set list"   and ys :: "'a set list" assumes "folding.gallery (xs @ ys)" shows "folding.gallery xs"
poly_add_normalize_2: fixes p1 :: "'a list"   and p2 :: "'a list" assumes "set p1 \<subseteq> carrier R"   and "set p2 \<subseteq> carrier R" shows "R.poly_add p1 p2 = R.poly_add p1 (R.normalize p2)"
eventually_at_rightI: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and "open" :: "'a set \<Rightarrow> bool"   and a :: "'a"   and b :: "'a"   and P :: "'a \<Rightarrow> bool" assumes "class.order_topology less_eq less open"   and "\<And>x. x \<in> ord.greaterThanLessThan less a b \<Longrightarrow> P x"   and "less a b" shows "eventually P (topological_space.at_within open a (ord.greaterThan less a))"
get_return_edges_valid: fixes a :: "'a"   and a' :: "'a" assumes "valid_edge a"   and "a' \<in> get_return_edges a" shows "valid_edge a'"
trans_rtrancl_eq_reflcl: fixes A :: "('a \<times> 'a) set" assumes "trans A" shows "A\<^sup>* = A\<^sup>="
c_k_def: fixes i :: "int"   and j :: "int"   and k :: "int" assumes "i < j"   and "k \<in> {i + 1..j}" shows "c_k i j k = w i j + c i (k - 1) + c k j"
wf_instantiate_action_schema: fixes a :: "ast_action_schema"   and args :: "object list" assumes "action_params_match a args"   and "wf_action_schema a" shows "wf_ground_action (instantiate_action_schema a args)"
hausdorff_distance_nonneg: fixes A :: "'b set"   and B :: "'b set" shows "0 \<le> hausdorff_distance A B"
parallel_3: fixes j :: "nat"   and x :: "nat"   and i :: "nat" assumes "\<phi> j x \<down> \<and> \<phi> i x \<up>" shows "parallel i j x \<down>= prod_encode (1, the (\<phi> j x))"
push_down_state_rank_oldest_token: fixes \<Sigma> :: "'c set"   and \<delta> :: "'d \<Rightarrow> 'c \<Rightarrow> 'd"   and q\<^sub>0 :: "'d"   and w :: "nat \<Rightarrow> 'c"   and q :: "'d"   and n :: "nat"   and i :: "nat" assumes "semi_mojmir \<Sigma> \<delta> q\<^sub>0 w"   and "semi_mojmir_def.state_rank \<Sigma> \<delta> q\<^sub>0 w q n = Some i" shows "\<exists>x. semi_mojmir_def.oldest_token \<delta> q\<^sub>0 w q n = Some x"
assms: fixes y :: "nat"   and x :: "nat" shows "P (y + x)"
SUP_le_iff: fixes f :: "'c \<Rightarrow> 'b"   and A :: "'c set"   and u :: "'b" shows "(\<Squnion>(f ` A) \<le> u) = (\<forall>i\<in>A. f i \<le> u)"
ln_fact_bounds: fixes n :: "nat" assumes "0 < n" shows "\<bar>ln (fact n) - real n * ln (real n) + real n\<bar> \<le> 1 + ln (real n)"
homeomorphism_translation: fixes a :: "'b"   and S :: "'b set" shows "homeomorphism ((+) a ` S) S ((+) (- a)) ((+) a)"
ball_eq_empty: fixes x :: "'a"   and e :: "real" shows "(ball x e = {}) = (e \<le> 0)"
ls: shows "ELam x e = ELam x (subst y v e')"
homeomorphic_path_connected_space: fixes X :: "'b topology"   and Y :: "'c topology" assumes "X homeomorphic_space Y" shows "path_connected_space X = path_connected_space Y"
WEAK_ALL_def: fixes I :: "'b \<Rightarrow> bool"   and X :: "'b set"   and P :: "'b \<Rightarrow> bool nres" shows "WEAK_ALL\<^bsup>I\<^esup> X P = WHILE\<^bsup>\<lambda>(Y, b). b \<longrightarrow> (\<forall>x\<in>X - Y. I x)\<^esup> (\<lambda>(X, b). b \<and> X \<noteq> {}) (\<lambda>(X, b). ASSERT (X \<noteq> {}) \<bind> (\<lambda>_. op_set_npick_remove X \<bind> (\<lambda>(x, X'). P x \<bind> (\<lambda>b'. RETURN (X', b' \<and> b))))) (X, True) \<bind> (\<lambda>(uu_, b). RETURN b)"
subgame_Digraph: fixes V' :: "'a set" shows "Digraph (subgame V')"
Sem_eval_cases_4: fixes s :: "Var \<Rightarrow> Val"   and b :: "BExpr"   and c :: "IMP"   and n :: "nat"   and t :: "Var \<Rightarrow> Val" assumes " s , While b c \<rightarrow>\<^sub>n  t " obtains na :: "nat" and r :: "Var \<Rightarrow> Val" and m :: "nat" where "n = Suc (max na m)" and "evalB b s" and " s , c \<rightarrow>\<^sub>na  r " and " r , While b c \<rightarrow>\<^sub>m  t "   | "n = Suc 0" and "t = s" and "\<not> evalB b s"
elims_3: fixes proper_interval :: "'a option \<Rightarrow> 'a option \<Rightarrow> bool"   and x :: "'a"   and xa :: "'a list" assumes "\<not> proper_intrvl.exhaustive_above proper_interval x xa" obtains xb :: "'a" where "x = xb__" and "xa = []" and "\<not> \<not> proper_interval (Some xb__) None"   | xb :: "'a" and y :: "'a" and ys :: "'a list" where "x = xb" and "xa = y # ys" and "\<not> (\<not> proper_interval (Some xb) (Some y) \<and> proper_intrvl.exhaustive_above proper_interval y ys)"
sym_sum_inter_num: fixes b1 :: "'a set" assumes "b1 \<in># \<B>" shows "\<Sum>\<^sub># (image_mset ((|\<inter>|) b1) (remove1_mset b1 \<B>)) = \<k> * (\<Lambda> * (\<v> - 1) div (\<k> - 1) - 1)"
eq_mono: fixes Aexp :: "exp \<Rightarrow> Arity \<rightarrow> Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and Aheap :: "(Vars.var \<times> exp) list \<Rightarrow> exp \<Rightarrow> Arity \<rightarrow> Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and a :: "Arity"   and a' :: "Arity"   and v1 :: "Value"   and v2 :: "Value" assumes "ArityAnalysisLetSafe Aexp Aheap"   and "a \<sqsubseteq> a'"   and "eq a' v1 v2" shows "eq a v1 v2"
iso_def: fixes f :: "'a arrow_of_spans_data" shows "local.iso f \<equiv> \<exists>g. inverse_arrows f g"
a'A: shows "emb a A \<in> A \<rightarrow> carrier K"
Pi_I: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and B :: "'c \<Rightarrow> 'd set" assumes "\<And>x. x \<in> A \<Longrightarrow> f x \<in> B x" shows "f \<in> Pi A B"
elims: fixes x :: "('dyn, 'ubx1, 'ubx2) unboxed"   and y :: "'dyn" assumes "norm_unboxed x = y" obtains d :: "'dyn" where "xa__ = OpDyn d" and "y = d"   | x :: "'ubx1" where "xa__ = OpUbx1 xb__" and "y = box_ubx1 xb__"   | x :: "'ubx2" where "xa__ = OpUbx2 x" and "y = box_ubx2 x"
rbt_product_def: fixes rbt1 :: "('a, 'b) rbt"   and rbt2 :: "('c, 'd) rbt" shows "local.rbt_product rbt1 rbt2 = rbtreeify (local.alist_product (RBT_Impl.entries rbt1) (RBT_Impl.entries rbt2))"
intros_2: fixes l :: "'b llist"   and A :: "'b set"   and a :: "'b" assumes "l \<in> A\<^sup>\<star>"   and "a \<in> A" shows "a ## l \<in> A\<^sup>\<star>"
good_imp_wf_ruleset: fixes rs :: "'b rule list"   and \<gamma> :: "('b \<Rightarrow> 'c \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'c \<Rightarrow> bool)"   and p :: "'c" assumes "good_ruleset rs" shows "wf_ruleset \<gamma> p rs"
tseytin_assmt_def: fixes F :: "'a formula" shows "tseytin_assmt F \<equiv> let SF = remdups (subformulae F) in zip (nfresh (atoms F) (length SF)) SF"
simps_24: fixes f1 :: "'b"   and f2 :: "'b"   and f3 :: "'b"   and f4 :: "'b"   and f5 :: "char list \<Rightarrow> 'b" shows "(case Integer of Void \<Rightarrow> f1 | Boolean \<Rightarrow> f2 | Integer \<Rightarrow> f3 | NT \<Rightarrow> f4 | Class x \<Rightarrow> f5 x) = f3"
ode_ops_rel_def: shows "ode_ops_rel = {(init, ode_ops). init = init_ode_ops True True ode_ops \<or> init = init_ode_ops True False ode_ops \<or> init = init_ode_ops False False ode_ops}"
intros_2: fixes P :: "'a stream \<Rightarrow> bool"   and \<omega> :: "'a stream"   and n :: "nat" assumes "\<not> P \<omega>"   and "ev_at P n (stl \<omega>)" shows "ev_at P (Suc n) \<omega>"
ts_l_def: shows "\<S>\<bullet>\<C>\<lparr>label := [ts'] @ label \<C>\<rparr> \<turnstile> lfilledk : [] _> ts_l"
InterruptedException_def: shows "InterruptedException = STR ''java/lang/InterruptedException''"
sublinear_wp_Bind: fixes a :: "'b \<Rightarrow> bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and f :: "'a \<Rightarrow> 'b" assumes "\<And>s. sublinear (wp (a (f s)))" shows "sublinear (wp (Bind f a))"
t2: shows "term_ok \<Theta> (Abs_fv x \<tau>' s $ Fv x \<tau>')"
list_pair_induct: fixes P :: "('a \<times> 'b) list \<Rightarrow> bool"   and lista :: "('a \<times> 'b) list" assumes "P []"   and "\<And>a b list. P list \<Longrightarrow> P ((a, b) # list)" shows "P lista"
append_in_lists_conv: fixes xs :: "'b list"   and ys :: "'b list"   and A :: "'b set" shows "(xs @ ys \<in> lists A) = (xs \<in> lists A \<and> ys \<in> lists A)"
Step1_5_3a: fixes p :: "PrIds"   and rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState" shows "\<turnstile> \<box>(ImpNext p \<and> [HNext rmhist p]_(c p, r p, m p, rmhist!p)) \<and> \<box>(\<forall>l. $MemInv mm l) \<and> \<box>$ImpInv rmhist p \<and> ImpLive p \<longrightarrow> \<box>\<diamond>S1 rmhist p"
le_max_iff_disj: fixes z :: "'t"   and x :: "'t"   and y :: "'t" shows "(z \<preceq>\<^sub>t ord_term_lin.max x y) = (z \<preceq>\<^sub>t x \<or> z \<preceq>\<^sub>t y)"
single_rt_fresher: fixes rt1 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and ip :: "nat"   and rt2 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option" assumes "the (rt1 ip) \<sqsubseteq> the (rt2 ip)" shows "rt1 \<sqsubseteq>\<^bsub>ip\<^esub> rt2"
pfinpref_def: fixes A :: "'b set"   and s :: "'b llist" shows "pfinpref A s \<equiv> finpref A s - {LNil}"
r_prod_inj: fixes r1 :: "('e, 'd) ta_rule"   and r2 :: "('f, 'd) ta_rule"   and r1' :: "('e, 'd) ta_rule"   and r2' :: "('f, 'd) ta_rule" assumes "rhsl r1 = rhsl r2"   and "length (rhsq r1) = length (rhsq r2)"   and "rhsl r1' = rhsl r2'"   and "length (rhsq r1') = length (rhsq r2')"   and "r_prod r1 r2 = r_prod r1' r2'" shows "r1 = r1' \<and> r2 = r2'"
times_divide_times_eq: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and x :: "'c"   and y :: "'c"   and z :: "'c"   and w :: "'c" assumes "class.field divide inverse times one plus zero minus uminus" shows "times (divide x y) (divide z w) = divide (times x z) (times y w)"
SPRAY_def: fixes \<E> :: "'b set"   and \<P> :: "'b set set"   and x :: "'b" assumes "MinkowskiPrimitive \<E> \<P>" shows "MinkowskiPrimitive.SPRAY \<P> x \<equiv> {R \<in> \<P>. x \<in> R}"
l2_inv4_def: shows "l2_inv4 \<equiv> {s. \<forall>Ra A B gny Nr. hmac \<langle>Number 0, Exp Gen (NonceF (Ra $ nx)), gny, Messages.Agent B, Messages.Agent A\<rangle> (hmac (NonceF (Ra $ ni)) Nr) \<in> parts (extr (bad s) (ik s) (chan s)) \<longrightarrow> guessed_runs Ra = \<lparr>role = Init, owner = A, partner = B\<rparr> \<longrightarrow> A \<notin> bad s \<and> B \<notin> bad s \<longrightarrow> (\<exists>Rb. guessed_runs Rb = \<lparr>role = Resp, owner = B, partner = A\<rparr> \<and> in_progressS (progress s Rb) {xny, xni, xnr, xgnx, xgny, xsk} \<and> sklvl1.guessed_frame Rb xgny = Some gny \<and> sklvl1.guessed_frame Rb xnr = Some Nr \<and> sklvl1.guessed_frame Rb xni = Some (NonceF (Ra $ ni)) \<and> sklvl1.guessed_frame Rb xgnx = Some (Exp Gen (NonceF (Ra $ nx))))}"
linear_dependent_inj_imageD: fixes f :: "'b \<Rightarrow> 'b"   and s :: "'b set" assumes "linear (*s) (*s) f"   and "dependent (f ` s)"   and "inj_on f (span s)" shows "dependent s"
pre_get_pending_def: fixes gds :: "('a, 'b) gen_dfs_struct"   and V0 :: "'a set"   and s :: "'b" shows "gen_dfs_defs.pre_get_pending gds V0 s \<equiv> rwof (gds_init gds) (gen_dfs_defs.gen_cond gds V0) (gen_dfs_defs.gen_step gds V0) s \<and> \<not> gds_is_empty_stack gds s \<and> \<not> gds_is_break gds s"
CH_range': fixes c :: "'x"   and B :: "'x set" assumes "c \<in> CH B" shows "c \<in> B"
lowered_def: fixes agents :: "'a set"   and alts :: "'b set"   and scf :: "('a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'b set"   and agents' :: "'a set"   and alts' :: "'b set" assumes "scf_lowering agents alts scf agents' alts'" shows "scf_lowering.lowered agents alts scf agents' alts' = scf \<circ> lift_pref_profile agents' alts' agents alts"
intros_1: fixes g :: "'g"   and \<phi> :: "'val" assumes "unnecessaryPhi g \<phi>" shows "\<phi> \<in> reachable g \<phi>"
B_def: shows "B = {x \<in> space M. \<forall>n. f ((T ^^ n) x) \<le> f ((T ^^ (n + 1)) x)}"
bind_commute_spmf: fixes p :: "'b spmf"   and q :: "'c spmf"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'a spmf" shows "p \<bind> (\<lambda>x. q \<bind> f x) = q \<bind> (\<lambda>y. p \<bind> (\<lambda>x. f x y))"
assms_2: fixes y :: "'b"   and y' :: "'a" assumes "REMOVE_INTERNAL (x = y)"   and "(y', y) \<in> Ra" shows "(f' y', f $ y) \<in> Rr"
fin_append_com_Nil1: fixes y :: "'d list"   and z :: "nat \<Rightarrow> 'd" shows "fin_inf_append [] (fin_inf_append y z) = fin_inf_append ([] @ y) z"
that: fixes g :: "complex \<Rightarrow> complex" assumes "g holomorphic_on S"   and "\<And>z. z \<in> S \<Longrightarrow> f z = exp (g z)" shows "thesis"
h1: shows "invertible_mat (M_mat_R (signs_smash signs1 signs2) (subsets_smash_R (length qs1) subsets1 subsets2))"
valid_6: shows "valid_data_dependency ts\<^sub>s\<^sub>b'"
Digamma_real_strict_mono: fixes x :: "real"   and y :: "real" assumes "0 < x"   and "x < y" shows "Digamma x < Digamma y"
inject: fixes x1 :: "'k"   and x2 :: "'l"   and y1 :: "'k"   and y2 :: "'l" shows "((x1, x2) = (y1, y2)) = (x1 = y1 \<and> x2 = y2)"
assms: shows "S : A \<mapsto>\<^bsub>dg_Rel \<alpha>\<^esub> B"
3: shows "to_fun f (b \<oplus> a) = fa \<oplus> f'a \<otimes> b \<oplus> c \<otimes> to_fun (X_poly R [^]\<^bsub>P\<^esub> 2) b"
hyps: fixes ys :: "'a list" shows "(xs = ys) = (length xs = length ys \<and> (\<forall>i<length xs. xs ! i = ys ! i))"
simps_5: fixes \<phi>\<^sub>1 :: "'a fm"   and \<phi>\<^sub>2 :: "'a fm" shows "nqfree (Or \<phi>\<^sub>1 \<phi>\<^sub>2) = (nqfree \<phi>\<^sub>1 \<and> nqfree \<phi>\<^sub>2)"
leg0_in_hom_2: shows "\<guillemotleft>f : f \<Rightarrow> f\<guillemotright>"
wfI_min: fixes R :: "('d \<times> 'd) set" assumes "\<And>x Q. x \<in> Q \<Longrightarrow> \<exists>z\<in>Q. \<forall>y. (y, z) \<in> R \<longrightarrow> y \<notin> Q" shows "wf R"
field_le_mult_one_interval: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "\<And>z. \<lbrakk>less zero z; less z one\<rbrakk> \<Longrightarrow> less_eq (times z x) y" shows "less_eq x y"
g_def: shows "is_increasing g \<and> s'' = take_subseq s' g"
strict_implies_order: fixes b :: "'t"   and a :: "'t" assumes "lt_of_nat_term_order cmp_term b a" shows "le_of_nat_term_order cmp_term b a"
min_def: fixes a :: "'t"   and b :: "'t" shows "ord_term_lin.min a b = (if a \<preceq>\<^sub>t b then a else b)"
meet_commute: fixes x :: "'b"   and y :: "'b" shows "x \<sqinter> y = y \<sqinter> x"
right: fixes t1 :: "('a, 'b) tm"   and t2 :: "('a, 'b) tm" shows "App t1 t2 \<rightarrow>\<^sub>e\<^sub>m\<^sub>b t2"
RD_bounds_sublistD_i: fixes Pi :: "(('c, bool) fmap \<times> ('c, bool) fmap) set"   and x :: "nat" assumes "finite Pi"   and "\<forall>y. y \<in> MPLS Pi \<longrightarrow> y \<le> x"   and "x \<in> MPLS Pi" shows "x \<le> Max {length p - 1 |p. valid_path Pi p \<and> distinct p}"
b_e_type_comp_conc: fixes \<C> :: "t_context"   and es :: "b_e list"   and t1s :: "t list"   and t2s :: "t list"   and es' :: "b_e list"   and t3s :: "t list" assumes "\<C> \<turnstile> es : t1s _> t2s"   and "\<C> \<turnstile> es' : t2s _> t3s" shows "\<C> \<turnstile> es @ es' : t1s _> t3s"
nMTcharn: fixes C :: "'b \<Rightarrow> 'c \<Rightarrow> 'd option"   and p :: "'b list" shows "none_MT_rules C p = (\<forall>r\<in>set p. dom (C r) \<noteq> {})"