hcju/isabelleps huggingface dataset:
prove_nonneg_fuel_mono: fixes prnt :: "(nat \<times> nat \<times> char list) list"   and i :: "nat"   and p :: "nat"   and fa :: "floatarith"   and YSS :: "(real \<times> real pdevs) list list" assumes "prove_nonneg prnt i p (slp_of_fas [fa]) YSS" shows "prove_nonneg prnt (Suc i) p (slp_of_fas [fa]) YSS"
inf_assoc: fixes inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "class.semilattice_inf inf less_eq less" shows "inf (inf x y) z = inf x (inf y z)"
monad_catch_state_optionT: fixes return :: "'a option \<Rightarrow> 'b"   and bind :: "'b \<Rightarrow> ('a option \<Rightarrow> 'b) \<Rightarrow> 'b"   and get :: "('c \<Rightarrow> 'b) \<Rightarrow> 'b"   and put :: "'c \<Rightarrow> 'b \<Rightarrow> 'b" assumes "monad_state return bind get put" shows "monad_catch_state (return_option return) (bind_option return bind) (fail_option return) (catch_option return bind) (get_option get) (put_option put)"
transient_extend_set_imp_project_transient: fixes G :: "'c program"   and C :: "'c set"   and A :: "'a set" assumes "G \<in> transient (C \<inter> extend_set h A)"   and "G \<in> stable C" shows "project h C G \<in> transient (project_set h C \<inter> A)"
strict_implies_order: fixes a :: "'f"   and b :: "'f" assumes "a < b" shows "a \<le> b"
sorted_remdups: fixes xs :: "'a list" assumes "sorted xs" shows "sorted (remdups xs)"
boundedI: fixes A :: "'a set"   and x :: "'a" assumes "finite A"   and "A \<noteq> {}"   and "\<And>a. a \<in> A \<Longrightarrow> a \<le> x" shows "Max A \<le> x"
subst_list_singleton_fv_subset: fixes v :: "'a"   and t :: "('b, 'a) Term.term"   and E :: "(('b, 'a) Term.term \<times> ('b, 'a) Term.term) list" shows "(\<Union>x\<in>set (subst_list (subst v t) E). fv (fst x) \<union> fv (snd x)) \<subseteq> fv t \<union> (\<Union>x\<in>set E. fv (fst x) \<union> fv (snd x))"
UNIV_7: shows "UNIV = {1, 2, 3, 4, 5, 6, 7}"
frechet_derivative_componentwise: fixes f :: "'c \<Rightarrow> real"   and a :: "'c"   and v :: "'c" assumes "f differentiable at a" shows "frechet_derivative f (at a) v = (\<Sum>i\<in>Basis. v \<bullet> i * frechet_derivative f (at a) i)"
C_def: shows "C = max (max C0 C1) (max (D / (a * B 0)) (C0 / a))"
that: fixes g :: "real \<Rightarrow> 'a" assumes "arc g"   and "pathstart g \<in> U"   and "pathfinish g \<in> V"   and "g ` {0<..<1} \<subseteq> S" shows "thesis"
assms_3: fixes K :: "'a set" assumes "K \<in> s" shows "K \<noteq> {}"
case_option_conv_if: fixes f :: "'b"   and g :: "'c \<Rightarrow> 'b"   and v :: "'c option" shows "(case v of None \<Rightarrow> f | \<lfloor>x\<rfloor> \<Rightarrow> g x) = (if \<exists>a. v = \<lfloor>a\<rfloor> then g (the v) else f)"
partition_tailrec: fixes f :: "'c \<Rightarrow> bool"   and as :: "'c list" shows "partition_tailrec f as = (filter f as, filter (\<lambda>x. \<not> f x) as)"
g: shows "(a, b) \<in> set (shift (adjust (tm_of abc_twice @ shift (mopup (Suc 0)) (length (tm_of abc_twice) div 2)) (Suc ((length (tm_of abc_twice) + 16) div 2))) 12)"
isCont_Ln': fixes z :: "'a"   and f :: "'a \<Rightarrow> complex" assumes "isCont f z"   and "f z \<notin> \<real>\<^sub>\<le>\<^sub>0" shows "isCont (\<lambda>x. Ln (f x)) z"
finite_primepow_factors: fixes x :: "'a" assumes "x \<noteq> (0::'a)" shows "finite (primepow_factors x)"
not_redundant: fixes g :: "'g" shows "\<not> notriv.redundant g"
RP_sound: fixes Sts :: "('a literal multiset set \<times> 'a literal multiset set \<times> 'a literal multiset set) llist" assumes "chain (\<leadsto>RP) Sts"   and "{#} \<in> N_of_state (Liminf_state Sts) \<union> P_of_state (Liminf_state Sts) \<union> Q_of_state (Liminf_state Sts)" shows "\<not> satisfiable (\<G>_Fset (N_of_state (lhd Sts) \<union> P_of_state (lhd Sts) \<union> Q_of_state (lhd Sts)))"
assms_2: shows "\<forall>(v, a)\<in>set (effect_of op). \<forall>(v', a')\<in>set (effect_of op). v \<noteq> v' \<or> a = a'"
simps: fixes a1 :: "'f inference set \<times> ('f \<times> 'l) set"   and a2 :: "'f inference set \<times> ('f \<times> 'l) set" shows "(a1 \<leadsto>LGC a2) = ((\<exists>N1 N M N2 M' T. a1 = (T, N1) \<and> a2 = (T, N2) \<and> N1 = N \<union> M \<and> N2 = N \<union> M' \<and> M \<subseteq> Red_F_\<G> (N \<union> M') \<and> active_subset M' = {}) \<or> (\<exists>T2 T1 T' N1 N C L N2. a1 = (T1, N1) \<and> a2 = (T2, N2) \<and> T2 = T1 \<union> T' \<and> N1 = N \<union> {(C, L)} \<and> N2 = N \<union> {(C, active)} \<and> L \<noteq> active \<and> T' = no_labels.Inf_between (fst ` active_subset N) {C}) \<or> (\<exists>T1 T2 \<iota> N2 N1 M. a1 = (T1, N1) \<and> a2 = (T2, N2) \<and> T1 = T2 \<union> {\<iota>} \<and> N2 = N1 \<union> M \<and> active_subset M = {} \<and> \<iota> \<in> no_labels.Red_I_\<G> (fst ` (N1 \<union> M))) \<or> (\<exists>T1 T2 T' N. a1 = (T1, N) \<and> a2 = (T2, N) \<and> T1 = T2 \<union> T' \<and> T' \<inter> no_labels.Inf_from (fst ` active_subset N) = {}))"
qe_inv: shows "\<forall>bs p. qfree p \<longrightarrow> qfree (qe p) \<and> Ifm vs bs (qe p) = Ifm vs bs (E p)"
vector_restrict_comp_conv: fixes x :: "'a"   and y :: "'a" assumes "vector x" shows "x \<sqinter> y \<le> x[[y]]"
Least_le: fixes less_eq :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and less :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and P :: "'f \<Rightarrow> bool"   and k :: "'f" assumes "class.wellorder less_eq less"   and "P k" shows "less_eq (ord.Least less_eq P) k"
InvDomCod_1: fixes f :: "'b" assumes "ciso f" shows "dom Cinv f = cod f"
has_sgnx_def: fixes f :: "real \<Rightarrow> real"   and c :: "real"   and F :: "real filter" shows "(f has_sgnx c) F = (\<forall>\<^sub>F x in F. sgn (f x) = c)"
\<theta>_bigo: shows "\<theta> \<in> O(\<lambda>x. x)"
if_rule: fixes b :: "bool"   and S1 :: "'a nres"   and \<Phi> :: "'a \<Rightarrow> bool"   and S2 :: "'a nres" assumes "b \<Longrightarrow> S1 \<le> SPEC \<Phi>"   and "\<not> b \<Longrightarrow> S2 \<le> SPEC \<Phi>" shows "(if b then S1 else S2) \<le> SPEC \<Phi>"
scf_list_bef_i_aft: fixes scf :: "nat \<Rightarrow> nat"   and bef :: "'a list"   and i :: "'a"   and aft :: "'a list" shows "scf_list scf (bef @ i # aft) = scf_list scf bef @ replicate (scf (length bef)) i @ scf_list (\<lambda>i. scf (Suc (length bef + i))) aft"
Gt_lt_conv: fixes comp :: "'b \<Rightarrow> 'b \<Rightarrow> order"   and x :: "'b"   and y :: "'b" assumes "comparator comp" shows "(comp x y = Gt) = lt_of_comp comp y x"
h_plus_le_g: fixes i :: "nat"   and e :: "'v \<times> 'v" shows "h_plus i e \<le> g e"
lub_insert: fixes x :: "'a"   and S :: "'a set" shows "lub (insert x S) = x \<squnion> lub S"
nonzero_times_divide_eq: fixes a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" assumes "a \<in> carrier R"   and "b \<in> carrier R"   and "c \<in> carrier R"   and "d \<in> carrier R"   and "b \<noteq> \<zero>"   and "d \<noteq> \<zero>" shows "a \<oslash> b \<otimes> (c \<oslash> d) = a \<otimes> c \<oslash> (b \<otimes> d)"
residue_of_sum': fixes a :: "nat \<Rightarrow> int"   and b :: "nat \<Rightarrow> int"   and k :: "nat" shows "(a \<oplus> b) k = (a k + b k) mod p ^ k"
reads_consistent_mem_eq_on_unforwarded_non_volatile_reads_drop: fixes A :: "nat set"   and W :: "nat set"   and m' :: "nat \<Rightarrow> nat"   and m :: "nat \<Rightarrow> nat"   and sb :: "'b memref list"   and \<O> :: "nat set" assumes "\<forall>a\<in>A \<union> W. m' a = m a"   and "unforwarded_non_volatile_reads (dropWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb) W \<subseteq> A"   and "acquired_reads True (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb) \<O> \<subseteq> A"   and "reads_consistent False \<O> m sb" shows "reads_consistent False \<O> m' sb"
has_limits_def: fixes uu :: "'b" shows "D.J.has_limits uu \<equiv> \<forall>J. category J \<longrightarrow> D.J.has_limits_of_shape J"
problem_plan_bound_ge_min_pls: fixes PROB :: "(('c, bool) fmap \<times> ('c, bool) fmap) set"   and s :: "('c, bool) fmap"   and as :: "(('c, bool) fmap \<times> ('c, bool) fmap) list"   and k :: "nat" assumes "finite PROB"   and "s \<in> valid_states PROB"   and "as \<in> valid_plans PROB"   and "problem_plan_bound PROB \<le> k" shows "Inf (PLS s as) \<le> problem_plan_bound PROB"
Sup_eq_Inf: fixes A :: "'a set" shows "\<Squnion> A = \<Sqinter> {b. \<forall>a\<in>A. a \<le> b}"
u_eq: shows "Q $$ (0, 0) * Q $$ (1, 1) + Q $$ (1, 0) * - Q $$ (0, 1) = u"
r_def: shows "r = (\<lambda>t. (\<MM> t - ln t) / (t * (ln t)\<^sup>2))"
valuate_nonneg: fixes p :: "linear_poly"   and v :: "nat \<Rightarrow> 'a" assumes "\<forall>x\<in>vars p. (0 < coeff p x \<longrightarrow> (0::'a) \<le> v x) \<and> (coeff p x < 0 \<longrightarrow> v x \<le> (0::'a))" shows "(0::'a) \<le> p \<lbrace> v \<rbrace>"
v_ale_diff: fixes v :: "'a \<Rightarrow> ant"   and x :: "'a"   and y :: "'a" assumes "valuation K v"   and "x \<in> carrier K"   and "y \<in> carrier K"   and "x \<noteq> \<zero>"   and "v x \<le> v y" shows "0 \<le> v (y \<cdot>\<^sub>r x\<^bsup>\<hyphen> K\<^esup>)"
is_BaseDim: fixes dim_ty_sem :: "'d itself \<Rightarrow> (int, sdim) dimvec" assumes "class.basedim_type dim_ty_sem" shows "is_BaseDim (dim_ty_sem TYPE('d))"
accept: fixes \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and m :: "'c match_expr"   and p :: "'d"   and \<Gamma> :: "char list \<Rightarrow> 'c rule list option" assumes "matches \<gamma> m p" shows "\<Gamma>,\<gamma>,p\<turnstile> \<langle>[Rule m Accept], Undecided\<rangle> \<Rightarrow> Decision FinalAllow"
onode_sos_src_is_net_state: fixes \<sigma> :: "nat \<Rightarrow> 'd"   and p :: "'e net_state"   and a :: "'f node_action"   and s' :: "(nat \<Rightarrow> 'd) \<times> 'e net_state"   and S :: "(((nat \<Rightarrow> 'd) \<times> 'e) \<times> 'f seq_action \<times> (nat \<Rightarrow> 'd) \<times> 'e) set" assumes "((\<sigma>, p), a, s') \<in> onode_sos S" shows "\<exists>i \<zeta> R. p = NodeS i \<zeta> R"
FactGroup_universal_kernel: fixes N :: "'a set" assumes "N \<lhd> G"   and "N \<subseteq> kernel G H h" obtains g :: "'a set \<Rightarrow> 'c" where "g \<in> hom (G Mod N) H" and "\<And>x. x \<in> carrier G \<Longrightarrow> g (N #> x) = h x"
ih_3: shows "InvariantWatchListsContainOnlyClausesFromF (getWatchList state') (getF state')"
edka3_def: shows "edka3 \<equiv> let cf = c in while\<^sub>T (\<lambda>(cf, brk). \<not> brk) (\<lambda>(cf, uu_). assert (RGraph c s t cf) \<bind> (\<lambda>_. find_shortest_augmenting_spec_cf cf \<bind> (\<lambda>p. case p of None \<Rightarrow> return (cf, True) | Some p \<Rightarrow> assert (p \<noteq> []) \<bind> (\<lambda>_. assert (Graph.isShortestPath cf s p t) \<bind> (\<lambda>_. resCap_cf_impl cf p \<bind> (\<lambda>bn. augment_cf_impl cf p bn \<bind> (\<lambda>cf. assert (RGraph c s t cf) \<bind> (\<lambda>_. return (cf, False))))))))) (cf, False) \<bind> (\<lambda>(cf, uu_). assert (RGraph c s t cf) \<bind> (\<lambda>_. Let (flow_of_cf cf) return))"
dghm_up_ArrMap_app: fixes x :: "V"   and \<CC> :: "V"   and I :: "V"   and \<AA> :: "V \<Rightarrow> V"   and \<phi> :: "V \<Rightarrow> V" assumes "x \<in>\<^sub>\<circ> \<CC>\<lparr>Arr\<rparr>" shows "dghm_up I \<AA> \<CC> \<phi>\<lparr>ArrMap\<rparr>\<lparr>x\<rparr> = (\<lambda>i\<in>\<^sub>\<circ>I. \<phi> i\<lparr>ArrMap\<rparr>\<lparr>x\<rparr>)"
cbv_eval: fixes t1 :: "lam"   and t2 :: "lam"   and t3 :: "lam" assumes "t1 \<longrightarrow>cbv t2"   and "t2 \<Down> t3" shows "t1 \<Down> t3"
the_index_last_distinct: fixes xs :: "'b list" assumes "distinct xs \<and> xs \<noteq> []" shows "the_index xs (last xs) = length xs - 1"
netmask_initmissing_netgmap: fixes s :: "'s net_state" shows "netmask (net_ips s) (initmissing (netgmap sr s)) = netgmap sr s"
le_Inf_iff: fixes b :: "'b \<times> 'c"   and A :: "('b \<times> 'c) set" shows "less_eq_prod b (Inf_prod A) = (\<forall>a\<in>A. less_eq_prod b a)"
one_mat_def: fixes n :: "nat" shows "1\<^sub>m n \<equiv> mat n n (\<lambda>(i, j). if i = j then 1::'d else (0::'d))"
LIST_FOREACH_autoref: fixes \<Phi> :: "'e set \<Rightarrow> 'f \<Rightarrow> bool"   and Rv :: "('c \<times> 'e) set"   and R\<sigma> :: "('d \<times> 'f) set" shows "(LIST_FOREACH', LIST_FOREACH \<Phi>) \<in> \<langle>\<langle>Rv\<rangle>list_rel\<rangle>nres_rel \<rightarrow> (R\<sigma> \<rightarrow> bool_rel) \<rightarrow> (Rv \<rightarrow> R\<sigma> \<rightarrow> \<langle>R\<sigma>\<rangle>nres_rel) \<rightarrow> R\<sigma> \<rightarrow> \<langle>R\<sigma>\<rangle>nres_rel"
genby_add_closed: fixes g :: "'b"   and S :: "'b set"   and h :: "'b" assumes "g \<in> \<langle>S\<rangle>"   and "h \<in> \<langle>S\<rangle>" shows "g + h \<in> \<langle>S\<rangle>"
order_refl: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c" assumes "class.preorder less_eq less" shows "less_eq x x"
supp_dist_remove: fixes p :: "'a \<Rightarrow> real"   and x :: "'a" assumes "p x \<noteq> 0"   and "p x \<noteq> 1" shows "supp (dist_remove p x) = supp p - {x}"
HStartBallot_maxBalInp: fixes s :: "state"   and s' :: "state"   and q :: "Proc"   and b :: "nat"   and v :: "InputsOrNi" assumes "HStartBallot s s' q"   and "maxBalInp s b v" shows "maxBalInp s' b v"
node_ptr_kinds_eq_h2: shows "node_ptr_kinds h3 = node_ptr_kinds h2"
mrNoneMT: fixes r :: "'a"   and p :: "'a list"   and C :: "'a \<Rightarrow> 'b \<Rightarrow> 'c option"   and x :: "'b" assumes "r \<in> set p"   and "applied_rule_rev C x p = \<bottom>" shows "x \<notin> dom (C r)"
configuration_non_empty: fixes \<Sigma> :: "'c set"   and \<delta> :: "'d \<Rightarrow> 'c \<Rightarrow> 'd"   and q\<^sub>0 :: "'d"   and w :: "nat \<Rightarrow> 'c"   and x :: "nat"   and n :: "nat" assumes "semi_mojmir \<Sigma> \<delta> q\<^sub>0 w"   and "x \<le> n" shows "semi_mojmir_def.configuration \<delta> q\<^sub>0 w (semi_mojmir_def.token_run \<delta> q\<^sub>0 w x n) n \<noteq> {}"
HUN_I: fixes a :: "hf"   and A :: "hf"   and b :: "hf"   and B :: "hf \<Rightarrow> hf" assumes "a \<^bold>\<in> A"   and "b \<^bold>\<in> B a" shows "b \<^bold>\<in> \<Squnion>RepFun A B"
card_Mi_le_floor_div_2_NVLi: fixes L :: "'b \<Rightarrow> nat"   and E :: "('b \<times> 'b) set"   and V :: "'b set"   and M :: "('b \<times> 'b) set"   and i :: "nat" assumes "OSC L E \<and> matching V E M \<and> 1 < i" shows "card (matching_i i V E M L) \<le> N V L i div 2"
intros_3: fixes N :: "nonce_t" shows "cNonce N \<in> cpayload"
et_2: fixes v :: "'a"   and e :: "'a"   and t :: "'a" assumes "is_vector v"   and "e \<le> v ; - v\<^sup>T"   and "t \<le> v ; v\<^sup>T" shows "e ; t\<^sup>T = 0"
assms: fixes x :: "real" assumes "x \<in> {0<..<1}" shows "f x = g x"
tllist_all2_rel: fixes R1 :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and Abs1 :: "'g \<Rightarrow> 'h"   and Rep1 :: "'h \<Rightarrow> 'g"   and R2 :: "'i \<Rightarrow> 'i \<Rightarrow> bool"   and Abs2 :: "'i \<Rightarrow> 'j"   and Rep2 :: "'j \<Rightarrow> 'i"   and r :: "('g, 'i) tllist"   and s :: "('g, 'i) tllist" assumes "Quotient3 R1 Abs1 Rep1"   and "Quotient3 R2 Abs2 Rep2" shows "tllist_all2 R1 R2 r s = (tllist_all2 R1 R2 r r \<and> tllist_all2 R1 R2 s s \<and> tmap Abs1 Abs2 r = tmap Abs1 Abs2 s)"
red_singleton: fixes f :: "'a \<Rightarrow>\<^sub>0 'c"   and p :: "'a \<Rightarrow>\<^sub>0 'c"   and q :: "'a \<Rightarrow>\<^sub>0 'c" shows "punit.red {f} p q = (\<exists>t. punit.red_single p q f t)"
ring_simprules_16: fixes R :: "('d, 'e) ring_scheme"   and x :: "'d" assumes "ring R"   and "x \<in> carrier R" shows "x \<oplus>\<^bsub>R\<^esub> \<ominus>\<^bsub>R\<^esub> x = \<zero>\<^bsub>R\<^esub>"
assms_1: shows "vars p \<inter> {i..<i + j} = {}"
Rel_app: fixes A :: "'c \<Rightarrow> 'e \<Rightarrow> bool"   and B :: "'d \<Rightarrow> 'f \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd"   and g :: "'e \<Rightarrow> 'f"   and x :: "'c"   and y :: "'e" assumes "Transfer.Rel (A ===> B) f g"   and "Transfer.Rel A x y" shows "Transfer.Rel B (f x) (g y)"
PosetComplex_def: fixes P :: "'a set" shows "PosetComplex P \<equiv> \<Union>x\<in>P. {{y. pseudominimal_in (P .\<^bold>\<le> x) y}}"
simps_1: fixes f :: "'c \<Rightarrow> 'b \<Rightarrow> 'b" shows "fold f [] = id"
zero_less_iff_neq_zero: fixes plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and n :: "'e" assumes "class.canonically_ordered_monoid_add plus zero less_eq less" shows "less zero n = (n \<noteq> zero)"
memb_def: fixes h :: "'b \<Rightarrow> int"   and x :: "'b"   and t :: "'b Tree" shows "memb h x t \<equiv> case tlookup h (h x) t of None \<Rightarrow> False | Some z \<Rightarrow> x = z"
sum_in_spanI: fixes q :: "'b \<Rightarrow> 'b"   and B :: "'b set" shows "(\<Sum>b\<in>B. q b * b) \<in> ideal B"
M2_def: shows "M2 = M.summset {#(if ia = j then lc j - lc i * c else lc ia) \<cdot>\<^sub>v us[i := us ! i + c \<cdot>\<^sub>v us ! j] ! ia. ia \<in># mset_set ({0..<length us} - {i, j})#}"
inorder_ins: fixes t :: "'a bro"   and h :: "nat"   and a :: "'a" assumes "t \<in> T h"   and "sorted (inorder t)" shows "inorder (ins a t) = ins_list a (inorder t)"
renaming_is_injective: fixes A :: "'at set"   and L1 :: "'at Literal"   and L2 :: "'at Literal" assumes "rename_literal A L1 = rename_literal A L2" shows "L1 = L2"
satisfies_bounded_lderiv0: fixes \<AA> :: "'i"   and a :: "'a"   and x :: "'x" assumes "wf0 (#\<^sub>V \<AA>) a"   and "#\<^sub>V \<AA> = size x"   and "lformula0 a" shows "(\<AA> \<Turnstile>\<^sub>b lderiv0 x a) = (CONS x \<AA> \<Turnstile>\<^sub>0 a)"
bdd_hm_lookup_rule: fixes bdd :: "bdd"   and i :: "nat"   and t :: "nat"   and e :: "nat"   and p :: "nat option"   and bddi :: "bddi" assumes "dcl bdd (i, t, e) = p" shows "<is_bdd_impl bdd bddi> hm_lookup (i, t, e) (dcli bddi) <\<lambda>r. is_bdd_impl bdd bddi * \<up> (r = p)>\<^sub>t"
lookup_env_delete_None: fixes x :: "'b"   and m :: "'b \<Rightarrow> 'a" shows "env_delete x m x = \<bottom>"
equal_tllist_def: fixes x :: "('a, 'b) tllist"   and y :: "('a, 'b) tllist" shows "equal_class.equal x y = (x = y)"
mod_0_imp_sub_1_div_conv: fixes n :: "nat"   and m :: "nat" assumes "n mod m = 0" shows "(n - Suc 0) div m = n div m - Suc 0"
sup2_def: fixes a :: "'a"   and b :: "'a" shows "pseudo_hoop_algebra.sup2 (\<sqinter>) (r\<rightarrow>) (l\<rightarrow>) a b = ((a l\<rightarrow> b) r\<rightarrow> b) \<sqinter> ((b l\<rightarrow> a) r\<rightarrow> a)"
fun_upd_twist: fixes a :: "'c"   and c :: "'c"   and m :: "'c \<Rightarrow> 'd"   and b :: "'d"   and d :: "'d" assumes "a \<noteq> c" shows "m(a := b, c := d) = m(c := d, a := b)"
set_product: fixes xs :: "'d list"   and ys :: "'e list" shows "set (List.product xs ys) = set xs \<times> set ys"
currentD_IN': fixes f :: "'v \<times> 'v \<Rightarrow> ennreal"   and x :: "'v" assumes "current \<Gamma> f"   and "x \<notin> B \<Gamma>" shows "d_IN f x = 0"
Sup_enat_def: fixes A :: "enat set" shows "\<Squnion> A = (if A = {} then 0 else if finite A then Max A else \<infinity>)"
vsv_vlrestriction_vinsert: fixes r :: "V"   and a :: "V"   and A :: "V" assumes "vsv r"   and "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> r" shows "r \<restriction>\<^sup>l\<^sub>\<circ> vinsert a A = vinsert \<langle>a, r\<lparr>a\<rparr>\<rangle> (r \<restriction>\<^sup>l\<^sub>\<circ> A)"
subset: shows "contained.Class ` G \<subseteq> contained.Partition"
simps_1: fixes p :: "nat"   and x :: "real" shows "power_up p x 0 = 1"
freeword_funlift_lconjby: fixes f :: "'b \<Rightarrow> 'a"   and x :: "'b freeword"   and g :: "'b freeword" shows "freeword_funlift f (lconjby x g) = lconjby (freeword_funlift f x) (freeword_funlift f g)"
ray_cases: fixes \<E> :: "'b set"   and \<P> :: "'b set set"   and betw :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and a :: "'b"   and b :: "'b" assumes "MinkowskiBetweenness \<E> \<P> betw"   and "x \<in> MinkowskiBetweenness.ray \<P> betw a b" shows "betw a x b \<or> betw a b x \<or> x = b"
lv_relI: fixes x :: "real list" assumes "length x = DIM('c)" shows "(x, eucl_of_list x) \<in> lv_rel"
bij_betw_ide_Obj_4: fixes a :: "(bool, unit) arr" assumes "a \<in> Collect ide" shows "MkIde (Dom a) = a"
dghm_up: shows "dghm_up I \<AA> \<CC> \<phi> : \<CC> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> dg_prod I \<AA>"
meet_cont2: fixes x :: "'c" shows "cont ((\<sqinter>) x)"
Aw_def': fixes rs :: "nat list"   and y :: "nat"   and shared_weights :: "bool" assumes "deep_model_correct_params_y rs y" shows "deep_model_correct_params_y.Aw rs y = deep_model_correct_params_y.A shared_weights rs y (deep_model_correct_params_y.witness_weights shared_weights rs)"
simply_connected_imp_path_connected: fixes S :: "'c set" assumes "simply_connected S" shows "path_connected S"
map_leI: fixes f :: "'f \<Rightarrow> 'g option"   and g :: "'f \<Rightarrow> 'g option" assumes "\<And>k v. f k = \<lfloor>v\<rfloor> \<Longrightarrow> g k = \<lfloor>v\<rfloor>" shows "f \<subseteq>\<^sub>m g"
incl_incl_of_2: fixes S :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<AA> :: "('c \<times> 'c) set"   and a :: "'b"   and b :: "'b" assumes "set_category S \<AA>"   and "set_category.incl_in S \<AA> a b" shows "partial_magma.in_hom S (set_category_given_img.mkArr S (set_category.some_img S \<AA>) \<AA> (set_category_data.set S (set_category.some_img S \<AA>) a) (set_category_data.set S (set_category.some_img S \<AA>) b) (\<lambda>x. x)) a b"
simps: fixes m :: "nat"   and rs :: "nat"   and l :: "cell list"   and r :: "cell list" shows "wadjust_goon_left_moving m rs (l, r) = (wadjust_goon_left_moving_B m rs (l, r) \<or> wadjust_goon_left_moving_O m rs (l, r))"
b: shows "B \<in> Cn A .\<bottom>. \<phi>"
segment_inc_less: fixes W :: "'a set"   and a :: "'a"   and y :: "'a"   and x :: "'a" assumes "W \<subseteq> carrier D"   and "a \<in> carrier D"   and "y \<in> W"   and "x \<in> segment (Iod D W) a"   and "y \<prec> x" shows "y \<in> segment (Iod D W) a"
terminal_ttl: fixes xs :: "('b, 'a) tllist" shows "terminal (ttl xs) = terminal xs"
map_comp: fixes g :: "'c \<Rightarrow> 'b"   and f :: "'d \<Rightarrow> 'c"   and v :: "'a \<Rightarrow> 'd" shows "(g \<circ>\<circ>\<circ> (\<circ>)) f v = g \<circ> f \<circ> v"
ndepth_def: fixes n :: "('a, 'b) node" shows "ndepth n \<equiv> case Rep_Node n of (f, x) \<Rightarrow> LEAST k. f k = Inr 0"
eq_fin_remove1: fixes w\<^sub>1 :: "'action list"   and w\<^sub>2 :: "'action list"   and c :: "'action" assumes "w\<^sub>1 =\<^sub>F w\<^sub>2" shows "remove1 c w\<^sub>1 =\<^sub>F remove1 c w\<^sub>2"
norm_memb_eq_rep: fixes f :: "'c local.term set"   and t :: "'c local.term" assumes "\<F>C.arr f"   and "t \<in> f" shows "\<^bold>\<parallel>t\<^bold>\<parallel> = \<F>C.rep f"
map_keys: fixes f :: "'a \<Rightarrow> 'c \<Rightarrow> 'b"   and t :: "('a, 'c) rbt" shows "keys (RBT_Impl.map f t) = keys t"
is_span: shows "C.span trg_f.cod.leg0 trg_f.cod.leg1"
P'_def: shows "P' \<equiv> \<lambda>b. f b = FALSE"
sublist_ext: fixes lo :: "nat"   and hi :: "nat"   and xs' :: "'c list"   and xs :: "'c list" assumes "\<forall>i. lo \<le> i \<and> i \<le> hi \<longrightarrow> xs' ! i = xs ! i"   and "length xs' = length xs"   and "lo \<le> hi"   and "hi < length xs" shows "WB_Sort.sublist xs' lo hi = WB_Sort.sublist xs lo hi"
invariant3pI: fixes I :: "'g set"   and r :: "'g \<Rightarrow> 'h \<Rightarrow> 'g \<Rightarrow> bool" assumes "\<And>s tl s'. \<lbrakk>s \<in> I; r s tl s'\<rbrakk> \<Longrightarrow> s' \<in> I" shows "invariant3p r I"
distinct_143: fixes x5 :: "floatarith"   and x151 :: "floatarith"   and x152 :: "nat" shows "Cos x5 \<noteq> x151 ^\<^sub>e x152"
pdevs_val_pdevs_of_list_inl2E: fixes e :: "nat \<Rightarrow> real"   and X :: "(real \<times> real) pdevs" assumes "e \<in> UNIV \<rightarrow> {- 1..1}" obtains e' :: "nat \<Rightarrow> real" where "pdevs_val e X = pdevs_val e' (pdevs_of_list (inl X))" and "e' \<in> UNIV \<rightarrow> {- 1..1}"
lm10: fixes pair :: "'c \<times> 'd set"   and N :: "'c set"   and \<Omega> :: "'d set"   and a :: "('c \<times> 'd set) set" assumes "pair \<in> N \<times> (Pow \<Omega> - {{}})" shows "summedBid (pseudoAllocation a <| (N \<times> finestpart \<Omega>)) pair = summedBid (bidMaximizedBy a N \<Omega> Elsee 0) pair"
pair_pmf_def: fixes A :: "'b pmf"   and B :: "'c pmf" shows "pair_pmf A B = A \<bind> (\<lambda>x. B \<bind> (\<lambda>y. return_pmf (x, y)))"
intros_1: fixes \<sigma> :: "'c \<Rightarrow> 'd" shows "ce \<sigma> Skip \<sigma>"