hcju/isabelleps huggingface dataset:
l_adopt_node_axioms_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptrs :: "(_) heap \<Rightarrow> bool"   and get_parent :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr option) prog"   and adopt_node :: "(_) document_ptr \<Rightarrow> (_) node_ptr \<Rightarrow> ((_) heap, exception, unit) prog"   and adopt_node_locs :: "(_) object_ptr option \<Rightarrow> (_) document_ptr \<Rightarrow> (_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_owner_document :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) document_ptr) prog" shows "l_adopt_node_axioms type_wf known_ptrs get_parent adopt_node adopt_node_locs get_child_nodes get_owner_document \<equiv> ((\<forall>h node document_ptr h'. writes (adopt_node_locs |h \<turnstile> get_parent node|\<^sub>r |h \<turnstile> get_owner_document (cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r node)|\<^sub>r document_ptr) (adopt_node document_ptr node) h h') \<and> (\<forall>w parent owner_document document_ptr h h'. w \<in> adopt_node_locs parent owner_document document_ptr \<longrightarrow> h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> object_ptr_kinds h = object_ptr_kinds h')) \<and> (\<forall>w parent owner_document document_ptr h h'. w \<in> adopt_node_locs parent owner_document document_ptr \<longrightarrow> h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> type_wf h = type_wf h') \<and> (\<forall>h document_ptr child. h \<turnstile> ok adopt_node document_ptr child \<longrightarrow> child |\<in>| node_ptr_kinds h) \<and> (\<forall>h owner_document node h' ptr children children'. h \<turnstile> adopt_node owner_document node \<rightarrow>\<^sub>h h' \<longrightarrow> h \<turnstile> get_child_nodes ptr \<rightarrow>\<^sub>r children \<longrightarrow> h' \<turnstile> get_child_nodes ptr \<rightarrow>\<^sub>r children' \<longrightarrow> known_ptrs h \<longrightarrow> type_wf h \<longrightarrow> set children' \<subseteq> set children)"
prems: shows "s \<in> fst (simp (Times r1 r2)) \<rightarrow> v"
conj_commute: fixes a :: "'a::refinement_lattice"   and b :: "'a::refinement_lattice" shows "(a::'a::refinement_lattice) \<iinter> (b::'a::refinement_lattice) = b \<iinter> a"
assms: shows "fst (list_of_oalist xs) \<noteq> []"
Int_Diff_Un: fixes A :: "'a set"   and B :: "'a set" shows "A \<inter> B \<union> (A - B) = A"
A'_def: shows "A' \<equiv> \<langle>rb, a, ra\<rangle>"
ring_iso_memE_5: fixes h :: "'c \<Rightarrow> 'd"   and R :: "('c, 'e) ring_scheme"   and S :: "('d, 'f) ring_scheme" assumes "h \<in> ring_iso R S" shows "bij_betw h (carrier R) (carrier S)"
A1: fixes p :: "'b fm" assumes "tautology p" shows "\<turnstile> p"
erfc_real_less_cancel: fixes a :: "real"   and b :: "real" shows "(erfc a < erfc b) = (b < a)"
le_rem_id_simps_2: shows "le_rem_s \<circ> ENV = id"
bij_ex1_imp_the_shift: fixes f :: "'a \<Rightarrow> 'b"   and P :: "'b \<Rightarrow> bool" assumes "bij f"   and "\<exists>!y. P y" shows "(THE x. P (f x)) = inv f (THE y. P y)"
CConcStartaux: fixes a :: "((int \<times> int) set set, int) Combinators"   and x :: "int \<times> (int \<times> int) \<times> (int \<times> int) \<times> DummyContent"   and aa :: "((int \<times> int) set set, int) Combinators" assumes "C a x = \<bottom>" shows "(C a \<Oplus> C aa) x = C aa x"
Norm_def: fixes N :: "'a quasinorm"   and x :: "'a" shows "Norm N x = enn2real (eNorm N x)"
independent_vnm_def: fixes C :: "'a pmf set"   and P :: "('a pmf \<times> 'a pmf) set" shows "independent_vnm C P = (\<forall>p\<in>C. \<forall>q\<in>C. \<forall>r\<in>C. \<forall>\<alpha>\<in>{0<..1}. p \<succeq>[P] q = mix_pmf \<alpha> p r \<succeq>[P] mix_pmf \<alpha> q r)"
equivE: fixes A :: "'a set"   and r :: "('a \<times> 'a) set" assumes "equiv A r" obtains "refl_on A r" and "sym r" and "trans r"
iso_set_trans: fixes h :: "'d \<Rightarrow> 'e"   and G :: "('d, 'f) monoid_scheme"   and H :: "('e, 'g) monoid_scheme"   and i :: "'e \<Rightarrow> 'h"   and I :: "('h, 'i) monoid_scheme" assumes "h \<in> Group.iso G H"   and "i \<in> Group.iso H I" shows "i \<circ> h \<in> Group.iso G I"
add_idem': fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c" assumes "class.near_dioid plus times less_eq less" shows "plus x x = x"
strategy_attracts_viaI: fixes v0 :: "'a"   and p :: "Player"   and \<sigma> :: "'a \<Rightarrow> 'a"   and A :: "'a set"   and W :: "'a set" assumes "\<And>P. vmc_path G P v0 p \<sigma> \<Longrightarrow> visits_via P A W" shows "strategy_attracts_via p \<sigma> v0 A W"
elims: fixes x :: "'b"   and xa :: "'b list"   and y :: "nat option" assumes "find_first x xa = y" obtains uu_ :: "'b" where "xb__ = uu_" and "xa = []" and "yb__ = None"   | x :: "'b" and y :: "'b" and ys :: "'b list" where "xb__ = x" and "xa = y # ys" and "yb__ = (if x = y then Some 0 else map_option Suc (find_first x ys))"
RAT_F1: shows "map of_int_hom.vec_hom fs' = (map of_int_hom.vec_hom fs)[i := map of_int_hom.vec_hom fs ! i - rat_of_int c \<cdot>\<^sub>v map of_int_hom.vec_hom fs ! j]"
a: shows "\<Gamma> \<turnstile> M \<turnstile> \<Delta>"
Int_Diff: fixes A :: "'a set"   and B :: "'a set"   and C :: "'a set" shows "A \<inter> B - C = A \<inter> (B - C)"
A: shows "Deliver (mo, Append mo e2) \<in> set pre \<or> (\<exists>mo2. Deliver (mo, Store e2 mo2 mo) \<in> set pre)"
set_spmf_I_N: fixes prime_set :: "nat set"   and N :: "nat"   and e :: "nat"   and d :: "nat" assumes "rsa_base prime_set"   and "((N, e), d) \<in> set_spmf (rsa_base.I prime_set)" obtains P :: "nat" and Q :: "nat" where "N = P * Q" and "P \<noteq> Q" and "prime P" and "prime Q" and "coprime e ((P - 1) * (Q - 1))" and "d = nat (fst (bezw e ((P - 1) * (Q - 1))) mod int ((P - 1) * (Q - 1)))"
grounding_of_cls_ground: fixes C :: "'a literal multiset" assumes "is_ground_cls C" shows "grounding_of_cls C = {C}"
mem_val_alt_10_unchanged2: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state"   and pa :: "word36"   and addr :: "word36"   and val :: "word8" assumes "mem_equal s1 s2 pa" shows "mem_val_alt 10 ((pa AND 68719476732) + 2) (s1\<lparr>mem := (mem s1)(10 := mem s1 10(addr \<mapsto> val), 11 := (mem s1 11)(addr := None))\<rparr>) = mem_val_alt 10 ((pa AND 68719476732) + 2) (s2\<lparr>mem := (mem s2)(10 := mem s2 10(addr \<mapsto> val), 11 := (mem s2 11)(addr := None))\<rparr>)"
monoI: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'c" assumes "class.order less_eq less"   and "\<And>x y. less_eq x y \<Longrightarrow> f x \<le> f y" shows "order.mono less_eq f"
equals_flowI: fixes T' :: "real set"   and x0 :: "'a"   and z :: "real \<Rightarrow> 'a"   and t :: "real" assumes "t0 \<in> T'"   and "is_interval T'"   and "T' \<subseteq> existence_ivl t0 x0"   and "(z solves_ode f) T' X"   and "z t0 = flow t0 x0 t0"   and "t \<in> T'" shows "z t = flow t0 x0 t"
if_eq_eq: fixes v :: "'a" assumes "(if v = (1::'a) then 0 else 1) = 0" shows "v = (1::'a)"
node_deliver_messages_distinct: fixes xs :: "('a \<times> ('a, 'b) operation) event list"   and i :: "nat" assumes "xs prefix of i" shows "distinct (node_deliver_messages xs)"
sorted_hyp_var2: shows "\<forall>q1<length (sorted_list_of_set {x. \<exists>q\<in>set qs. rpoly q x = 0}). sorted_list_of_set {x. \<exists>q\<in>set qs. rpoly q x = 0} ! q1 \<le> sorted_list_of_set {x. \<exists>q\<in>set qs. rpoly q x = 0} ! (length (sorted_list_of_set {x. \<exists>q\<in>set qs. rpoly q x = 0}) - 1)"
prod_nonneg: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and A :: "'d set"   and f :: "'d \<Rightarrow> 'c" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "\<forall>a\<in>A. less_eq zero (f a)" shows "less_eq zero (comm_monoid_mult.prod times one f A)"
sum_power2_eq_zero_iff: fixes 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 times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and x :: "'a"   and y :: "'a" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "(plus (power.power one times x 2) (power.power one times y 2) = zero) = (x = zero \<and> y = zero)"
ub_set_def: fixes X :: "'a set" shows "ub_set X = {y. \<forall>x\<in>X. x \<le> y}"
pbij_lens_mwb: fixes x :: "'d \<Longrightarrow> 'e" assumes "pbij_lens x" shows "mwb_lens x"
assms_12: shows "(b2 * A2 - B2 * A1)\<^sup>2 + (b1 * A2 - B1 * A1)\<^sup>2 \<noteq> 0"
Suc: fixes P :: "'a hrel" shows "\<lbrakk>power.power II (;;) P i\<rbrakk>\<^sub>e = (\<lambda>b. b \<in> {a. \<lbrakk>P\<rbrakk>\<^sub>e a} ^^ i)"
equal_eq: fixes equal :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "equal equal" shows "equal = (=)"
add_left_cancel: fixes plus :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and a :: "'f"   and b :: "'f"   and c :: "'f" assumes "class.cancel_semigroup_add plus" shows "(plus a b = plus a c) = (b = c)"
StaticOperationCallTE: fixes \<Gamma> :: "String.literal \<rightharpoonup>\<^sub>f 'a OCL_Types.type"   and \<tau> :: "'a OCL_Types.type"   and op :: "String.literal"   and as :: "'a expr list"   and \<sigma> :: "'a OCL_Types.type" assumes "\<Gamma> \<turnstile>\<^sub>E StaticOperationCall \<tau> op as : \<sigma>" obtains \<pi> :: "'a OCL_Types.type list" and oper :: "String.literal \<times> 'a OCL_Types.type \<times> (String.literal \<times> 'a OCL_Types.type \<times> param_dir) list \<times> 'a OCL_Types.type \<times> bool \<times> 'a expr option" where "\<sigma> = oper_type oper" and "\<Gamma> \<turnstile>\<^sub>L as : \<pi>" and "static_operation \<tau> op \<pi> oper"
simps: fixes a :: "'b monoid list \<times> ('b \<Rightarrow> 'b) list" shows "exact_seq a = ((\<exists>G1 G2 f. a = ([G2, G1], [f]) \<and> group_hom G1 G2 f) \<or> (\<exists>G K l g q H h. a = (H # G # K # l, h # g # q) \<and> exact_seq (G # K # l, g # q) \<and> Group.group H \<and> h \<in> hom G H \<and> kernel G H h = g ` carrier K))"
Units_r_inv_ex: fixes x :: "'a" assumes "x \<in> Units G" shows "\<exists>y\<in>carrier G. x \<otimes> y = \<one>"
"2_3": shows "\<not> P1 a \<and> x = filter (\<lambda>x. \<not> P1 x) l2"
simps_2: fixes x :: "('b, 'c) Combinators"   and xs :: "('b, 'c) Combinators list"   and l :: "'b set list" shows "all_in_list (x # xs) l = (in_list x l \<and> all_in_list xs l)"
iso_tuple_UNIV_I: fixes x :: "'e" shows "x \<in> UNIV \<equiv> True"
valid_Pred: fixes P :: "'state"   and \<phi> :: "'pred" shows "P \<Turnstile> Pred \<phi> = P \<turnstile> \<phi>"
hyps: fixes qs :: "'b list"   and tab :: "'a \<Rightarrow> 'b option"   and x :: "'a"   and y :: "'b" assumes "length ps = length qs" shows "tab(ps [\<mapsto>] qs, x \<mapsto> y) = tab(ps @ [x] [\<mapsto>] qs @ [y])"
preserves_arr: fixes f :: "'j" assumes "J.arr f" shows "C.arr (D f)"
ctxt: shows "\<Theta>' = \<Theta> \<union> (\<Union>p\<in>Procs. \<Union>Z. {(P p Z, p, Q p Z, A p Z)})"
assms_1: shows "(f \<longlongrightarrow> l) (at x within S)"
length_cruel: fixes A :: "('b list \<Rightarrow> 'c) \<times> ('b list \<times> 'c \<Rightarrow> 'b \<Rightarrow> (nat \<times> nat list) \<times> 'c)"   and s :: "'b list \<times> 'c"   and n :: "nat" shows "length (cruel A s n) = n"
cmod_Lam: fixes i :: "nat" assumes "i \<in> I" shows "cmod (Lam i) = 1"
mapAssoc: fixes S :: "'b sequent"   and R :: "'b sequent"   and c :: "'b sequent list" shows "map (extend S) (map (extend R) c) = map (extend (extend S R)) c"
seqI': fixes f :: "('a2 \<Rightarrow> 'b, 'a2 \<Rightarrow> 'b) concrete_category.arr"   and a :: "('a2 \<Rightarrow> 'b, 'a2 \<Rightarrow> 'b) concrete_category.arr"   and b :: "('a2 \<Rightarrow> 'b, 'a2 \<Rightarrow> 'b) concrete_category.arr"   and g :: "('a2 \<Rightarrow> 'b, 'a2 \<Rightarrow> 'b) concrete_category.arr"   and c :: "('a2 \<Rightarrow> 'b, 'a2 \<Rightarrow> 'b) concrete_category.arr" assumes "\<guillemotleft>f : a \<rightarrow>\<^sub>[\<^sub>A\<^sub>2\<^sub>,\<^sub>B\<^sub>] b\<guillemotright>"   and "\<guillemotleft>g : b \<rightarrow>\<^sub>[\<^sub>A\<^sub>2\<^sub>,\<^sub>B\<^sub>] c\<guillemotright>" shows "A2_B.seq g f"
ALength_\<tau>execI: fixes ci :: "('addr, 'heap) check_instr"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and e :: "'addr expr1"   and h :: "'heap"   and s :: "'addr val list \<times> 'addr val list \<times> nat \<times> 'addr option"   and s' :: "'addr val list \<times> 'addr val list \<times> nat \<times> 'addr option" assumes "\<tau>exec_move ci P t e h s s'" shows "\<tau>exec_move ci P t (e\<bullet>length) h s s'"
exhaust: fixes y :: "'e\<cdot>'f resT" obtains "y = \<bottom>"   | a :: "'e" where "y = Done\<cdot>a"   | app :: "'e\<cdot>'f resT\<cdot>'f" where "y = More\<cdot>app"
term_ok_Types_typ_ok: fixes \<Theta> :: "((String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)) \<times> Core.term set"   and t :: "Core.term"   and \<tau> :: "typ" assumes "wf_theory \<Theta>"   and "term_ok \<Theta> t"   and "\<tau> \<in> Types t" shows "typ_ok \<Theta> \<tau>"
exec_step_ind_Push: fixes v :: "val"   and P :: "jvm_prog"   and h :: "heap"   and stk :: "val list"   and loc :: "val list"   and C\<^sub>0 :: "char list"   and M\<^sub>0 :: "char list"   and pc :: "nat"   and ics :: "init_call_status"   and frs :: "frame list"   and sh :: "sheap" shows "exec_step_ind (StepI (Push v)) P h stk loc C\<^sub>0 M\<^sub>0 pc ics frs sh (None, h, (v # stk, loc, C\<^sub>0, M\<^sub>0, Suc pc, ics) # frs, sh)"
p_Suc': fixes x :: "'s option"   and y :: "'s option"   and n :: "nat" shows "K2.p x y (Suc n) = measure_pmf.expectation (K' x) (\<lambda>x'. K2.p x' y n)"
map_S_comp: fixes l1 :: "'p \<Rightarrow> 'j"   and l2 :: "'q \<Rightarrow> 'k"   and co1 :: "'r \<Rightarrow> 'l"   and co2 :: "'s \<Rightarrow> 'm"   and contra1 :: "'n \<Rightarrow> 't"   and contra2 :: "'o \<Rightarrow> 'u"   and l1' :: "'b \<Rightarrow> 'p"   and l2' :: "'d \<Rightarrow> 'q"   and co1' :: "'e \<Rightarrow> 'r"   and co2' :: "'f \<Rightarrow> 's"   and contra1' :: "'t \<Rightarrow> 'g"   and contra2' :: "'u \<Rightarrow> 'h" shows "map_S l1 l2 co1 co2 contra1 contra2 \<circ> map_S l1' l2' co1' co2' contra1' contra2' = map_S (l1 \<circ> l1') (l2 \<circ> l2') (co1 \<circ> co1') (co2 \<circ> co2') (contra1' \<circ> contra1) (contra2' \<circ> contra2)"
spec_def: fixes s :: "('b, 'loc \<times> 't) Exchange.configuration stream" shows "cri.spec s \<equiv> holds cri.init_config s \<and> alw cri.next s"
chain_on_def: fixes A :: "'b set"   and P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and C :: "'b set" shows "chain_on P C A = (C \<subseteq> A \<and> (\<forall>x\<in>C. \<forall>y\<in>C. P\<^sup>=\<^sup>= x y \<or> P\<^sup>=\<^sup>= y x))"
seqlsimp: fixes i :: "'a"   and P :: "'b \<times> 'c \<Rightarrow> bool"   and \<sigma> :: "'a \<Rightarrow> 'b"   and p :: "'c" shows "seql i P (\<sigma>, p) = P (\<sigma> i, p)"
rep_eq: fixes x :: "'c \<Rightarrow>\<^sub>0 'd"   and xa :: "'c \<Rightarrow>\<^sub>0 'd" shows "poly_mapping.lookup (x + xa) = (\<lambda>k. poly_mapping.lookup x k + poly_mapping.lookup xa k)"
angle_construction_1: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and A' :: "'a"   and B' :: "'a"   and P :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "\<not> Tarski_neutral_dimensionless.Col Bet A B C"   and "\<not> Tarski_neutral_dimensionless.Col Bet A' B' P" shows "\<exists>C'. Tarski_neutral_dimensionless.CongA Bet Cong A B C A' B' C' \<and> Tarski_neutral_dimensionless.OS Bet A' B' C' P"
path_length_n_finite_set: fixes wf_md :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list \<Rightarrow> char list \<Rightarrow> char list \<times> ty list \<times> ty \<times> char list list \<times> expr \<Rightarrow> bool"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and C :: "char list"   and n :: "nat" assumes "wf_prog wf_md P" shows "finite {Cs. Subobjs P C Cs \<and> length Cs = n}"
step: fixes A :: "('e \<times> ('f, 'g) cconstraint \<times> 'h \<times> 'f list \<times> 'e) set \<times> ('e \<Rightarrow> ('f, 'g) cconstraint)"   and l :: "'e"   and Z :: "('f \<Rightarrow> 'g) set"   and l' :: "'e"   and Z' :: "('f \<Rightarrow> 'g) set"   and l'' :: "'e"   and Z'' :: "('f \<Rightarrow> 'g) set" assumes "A \<turnstile> \<langle>l, Z\<rangle> \<leadsto> \<langle>l', Z'\<rangle>"   and "A \<turnstile> \<langle>l', Z'\<rangle> \<leadsto>* \<langle>l'', Z''\<rangle>" shows "A \<turnstile> \<langle>l, Z\<rangle> \<leadsto>* \<langle>l'', Z''\<rangle>"
IH_3: assumes "\<langle>c: t \<doteq> s\<rangle> \<in> set S" shows "fv t \<union> fv s \<subseteq> fv\<^sub>s\<^sub>s\<^sub>t S"
poly_mod_prime_def: fixes p :: "int" shows "poly_mod_prime p \<equiv> prime p"
prefs_undefined': fixes agents :: "'a set"   and alts :: "'b set"   and R :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and i :: "'a" assumes "pref_profile_wf agents alts R"   and "i \<notin> agents" shows "R i = (\<lambda>_ _. False)"
ReachableInExecution: fixes trans :: "'a \<Rightarrow> 'b \<Rightarrow> 'c messageValue \<Rightarrow> 'b"   and sends :: "'a \<Rightarrow> 'b \<Rightarrow> 'c messageValue \<Rightarrow> ('a, 'c) message \<Rightarrow> nat"   and start :: "'a \<Rightarrow> 'b"   and exec :: "('a, 'c, 'b) configuration list"   and trace :: "('a, 'c) message list"   and i :: "nat"   and j :: "nat" assumes "execution trans sends start exec trace"   and "i < length exec"   and "j \<le> i" shows "asynchronousSystem.reachable trans sends (exec ! j) (exec ! i)"
rec_eq_def: shows "rec_eq = CN rec_minus [CN (constn 1) [recf.Id 2 0], CN rec_add [rec_minus, rec_swap rec_minus]]"
g_def: shows "g = (\<lambda>x. real_cond_exp M Invariants f x - f x)"
mult_unit_dvd_iff: fixes b :: "'a"   and a :: "'a"   and c :: "'a" assumes "is_unit b" shows "(a * b dvd c) = (a dvd c)"
fold_pderiv_set_flatten_PLUS: fixes w :: "'a list"   and X :: "'a rexp set" assumes "w \<noteq> []"   and "finite X" shows "fold pderiv_set w {flatten PLUS X} = fold pderiv_set w X"
bindET_bindET: fixes m :: "'i\<cdot>('f, 'g) errorT"   and h :: "'i \<rightarrow> 'h\<cdot>('f, 'g) errorT"   and k :: "'h \<rightarrow> 'e\<cdot>('f, 'g) errorT" shows "bindET\<cdot>(bindET\<cdot>m\<cdot>h)\<cdot>k = bindET\<cdot>m\<cdot>(\<Lambda> x. bindET\<cdot>(h\<cdot>x)\<cdot>k)"
avg_def: fixes ops :: "(nat, 'a, 'b) set_ops_scheme"   and s :: "'a" assumes "MyContext ops" shows "MyContext.avg ops s \<equiv> case MyContext.avg_aux ops s of (c, s) \<Rightarrow> s div c"
assms_9: shows "roots = (\<lambda>(a, b, c). if a = 0 \<and> b \<noteq> 0 then {- c / b} else if a \<noteq> 0 \<and> 0 \<le> b\<^sup>2 - 4 * a * c then {(- b + sqrt (b\<^sup>2 - 4 * a * c)) / (2 * a)} \<union> {(- b - sqrt (b\<^sup>2 - 4 * a * c)) / (2 * a)} else {})"
add_rquot: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<preceq>\<^sub>R y" shows "(D x z \<and> x \<oplus> z = y) = (z = rquot y x)"
alphaAbs_qAbs_imp_all_distinct_qAFresh: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm"   and xs :: "'varSort"   and x :: "'var"   and xs' :: "'varSort"   and x' :: "'var"   and X' :: "('d, 'e, 'varSort, 'var, 'f) qTerm" assumes "qGood X"   and "qAbs xs x X $= qAbs xs' x' X'" shows "xs = xs' \<and> (\<forall>y. y \<notin> {x, x'} \<and> qAFresh xs y X \<and> qAFresh xs y X' \<longrightarrow> X #[[y \<and> x]]_xs #= X' #[[y \<and> x']]_xs)"
add_strict_mono: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" assumes "class.strict_ordered_ab_semigroup_add plus less_eq less"   and "less a b"   and "less c d" shows "less (plus a c) (plus b d)"
in_homI: fixes f :: "(('c + ('j \<Rightarrow> 'c)) set, 'c + ('j \<Rightarrow> 'c) \<Rightarrow> 'c + ('j \<Rightarrow> 'c)) concrete_category.arr"   and a :: "(('c + ('j \<Rightarrow> 'c)) set, 'c + ('j \<Rightarrow> 'c) \<Rightarrow> 'c + ('j \<Rightarrow> 'c)) concrete_category.arr"   and b :: "(('c + ('j \<Rightarrow> 'c)) set, 'c + ('j \<Rightarrow> 'c) \<Rightarrow> 'c + ('j \<Rightarrow> 'c)) concrete_category.arr" assumes "S.arr f"   and "S.dom f = a"   and "S.cod f = b" shows "\<guillemotleft>f : a \<rightarrow>\<^sub>S b\<guillemotright>"
P2: shows "Bet B' C' C1 \<and> Cong C' C1 B C"
set_generate': fixes A :: "nat"   and B :: "nat"   and a :: "nat list"   and b :: "nat list" shows "set (generate' A B a b) = {(x, y). (x, y) \<noteq> (zeroes (length a), zeroes (length b)) \<and> (x, y) \<in> set (Algorithm.gen2 A B a b)}"
reds: shows "P,E \<turnstile> \<langle>es,s\<rangle> [\<rightarrow>]* \<langle>es'',s''\<rangle>"
update_convs_1: fixes ok' :: "bool \<Rightarrow> bool"   and ok :: "bool"   and more :: "'c" shows "ok_update ok' \<lparr>ok = ok, \<dots> = more\<rparr> = \<lparr>ok = ok' ok, \<dots> = more\<rparr>"
wp_eval_7: fixes a :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and b :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and P :: "'a \<Rightarrow> real" shows "wp (a ;; b) P = (wp a \<circ> wp b) P"
valid_def: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and F :: "'c set"   and P :: "'b set"   and c :: "('b, 'a, 'c) com"   and Q :: "'b set"   and A :: "'b set" shows "\<Gamma> \<Turnstile>\<^bsub>/F\<^esub> P c Q, A \<equiv> \<forall>s t c'. \<Gamma> \<turnstile> (c, s) \<rightarrow>\<^sup>* (c', t) \<longrightarrow> final (c', t) \<longrightarrow> s \<in> Normal ` P \<longrightarrow> t \<notin> Fault ` F \<longrightarrow> c' = Skip \<and> t \<in> Normal ` Q \<or> c' = Throw \<and> t \<in> Normal ` A"
homotopy_equivalent_space_imp_isomorphic_relative_homology_groups: fixes X :: "'b topology"   and Y :: "'c topology"   and f :: "'b \<Rightarrow> 'c"   and S :: "'b set"   and T :: "'c set"   and g :: "'c \<Rightarrow> 'b"   and p :: "int" assumes "continuous_map X Y f"   and "f ` S \<subseteq> T"   and "continuous_map Y X g"   and "g ` T \<subseteq> S"   and "homotopic_with (\<lambda>h. h ` S \<subseteq> S) X X (g \<circ> f) id"   and "homotopic_with (\<lambda>k. k ` T \<subseteq> T) Y Y (f \<circ> g) id" shows "relative_homology_group p X S \<cong> relative_homology_group p Y T"
strips_state_to_state_inverse_is_iii: fixes \<Psi> :: "('c, 'd) sas_plus_problem"   and v :: "'c"   and s :: "'c \<Rightarrow> 'd option"   and a :: "'d"   and a' :: "'d" assumes "is_valid_problem_sas_plus \<Psi>"   and "v \<in> set (\<Psi>\<^sub>\<V>\<^sub>+)"   and "s v = Some a"   and "a \<in> \<R>\<^sub>+ \<Psi> v"   and "a' \<in> \<R>\<^sub>+ \<Psi> v"   and "(\<phi>\<^sub>S \<Psi> s) (v, a) = Some True"   and "(\<phi>\<^sub>S \<Psi> s) (v, a') = Some True" shows "a = a'"
quot_of_ratfps_1: shows "quot_of_ratfps 1 = (1, 1)"
simps_2: fixes prec :: "nat"   and n :: "nat"   and I :: "float interval list"   and a :: "float interval list" shows "compute_bound_poly prec (poly.Bound n) I a = round_interval prec (I ! n - a ! n)"
pair_spmf_return_spmf1: fixes x :: "'a"   and q :: "'b spmf" shows "pair_spmf (return_spmf x) q = map_spmf (Pair x) q"