hcju/isabelleps huggingface dataset:
gseq_iff: fixes f :: "nat \<Rightarrow> 'c"   and g :: "nat \<Rightarrow> 'c"   and A :: "'c set" shows "((f, g) \<in> mbs.gseq A) = (f \<in> SEQ A \<and> g \<in> SEQ A \<and> (\<exists>i. size (g i) < size (f i) \<and> (\<forall>j<i. f j = g j)))"
oalist_inv_raw_map2_val_pair: fixes xs :: "('a \<times> 'e) list"   and ys :: "('a \<times> 'f) list"   and g :: "('a \<times> 'e) list \<Rightarrow> ('a \<times> 'g) list"   and h :: "('a \<times> 'f) list \<Rightarrow> ('a \<times> 'g) list"   and f :: "'a \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> 'g" assumes "oalist_inv_raw xs"   and "oalist_inv_raw ys"   and "map2_val_compat g"   and "map2_val_compat h" shows "oalist_inv_raw (map2_val_pair f g h xs ys)"
simps: fixes ops :: "'c arith_ops_record"   and a :: "'c"   and b :: "'c" shows "arith_ops.gcd_eucl_i ops a b = (if b = arith_ops_record.zero ops then arith_ops_record.normalize ops a else arith_ops.gcd_eucl_i ops b (arith_ops_record.modulo ops a b))"
basis_gen_hull_free: fixes G :: "'b list set" shows "\<langle>\<BB>\<^sub>F G\<rangle> = \<langle>G\<rangle>\<^sub>F"
extgsimp: fixes \<sigma> :: "'d"   and l1 :: "'e"   and l2 :: "'f" shows "extg ((\<sigma>, l1), l2) = (\<sigma>, l1, l2)"
finvimage_fcomp: fixes g :: "V"   and f :: "V"   and x :: "V" shows "(g \<circ>\<^sub>\<bullet> f) -`\<^sub>\<bullet> x = f -`\<^sub>\<bullet> g -`\<^sub>\<bullet> x"
Conseq_sound: fixes pre :: "'b set"   and pre' :: "'b set"   and rely :: "('b \<times> 'b) set"   and rely' :: "('b \<times> 'b) set"   and guar' :: "('b \<times> 'b) set"   and guar :: "('b \<times> 'b) set"   and post' :: "'b set"   and post :: "'b set"   and P :: "'b com" assumes "pre \<subseteq> pre'"   and "rely \<subseteq> rely'"   and "guar' \<subseteq> guar"   and "post' \<subseteq> post"   and "\<Turnstile> P sat [pre', rely', guar', post']" shows "\<Turnstile> P sat [pre, rely, guar, post]"
OclAllInstances_at_post_defined: fixes \<tau> :: "'d state \<times> 'd state"   and H :: "'d \<Rightarrow> \<langle>'e\<rangle>\<^sub>\<bottom>" shows "\<tau> \<Turnstile> \<delta> H .allInstances()"
p: shows "\<forall>x\<in>s - {a}. x j = p"
enn2ereal_of_nat: fixes n :: "nat" shows "enn2ereal (of_nat n) = ereal (real n)"
card_Timess_pderivs_lang_le: fixes s :: "'a rexp"   and A :: "'a list set"   and r :: "'a rexp" shows "card (Timess (pderivs_lang A r) s) \<le> card (pderivs_lang A r)"
p: fixes p :: "('i, 'pkt_ext) tagged_packet_scheme" assumes "(\<Gamma>::char list \<Rightarrow> 'i common_primitive rule list option),(\<gamma>::'i common_primitive \<Rightarrow> ('i, 'pkt_ext) tagged_packet_scheme \<Rightarrow> bool),(p::('i, 'pkt_ext) tagged_packet_scheme)\<turnstile> \<langle>(rs::'i common_primitive rule list), Undecided\<rangle> \<Rightarrow> Decision FinalAllow \<and> (newpkt::('i, 'pkt_ext) tagged_packet_scheme \<Rightarrow> bool) p" shows "(common_matcher, in_doubt_allow),(p::('i, 'pkt_ext) tagged_packet_scheme)\<turnstile> \<langle>(preprocess::'i common_primitive rule list \<Rightarrow> 'i common_primitive rule list) (rs::'i common_primitive rule list), Undecided\<rangle> \<Rightarrow>\<^sub>\<alpha> Decision FinalAllow \<and> (newpkt::('i, 'pkt_ext) tagged_packet_scheme \<Rightarrow> bool) p"
MAX_def: fixes B :: "'a list set" shows "MAX B = (SOME b. max_cond B b)"
vec_of_basis_enum_add: fixes b1 :: "'b"   and b2 :: "'b" shows "vec_of_basis_enum (b1 + b2) = vec_of_basis_enum b1 + vec_of_basis_enum b2"
cond_L6: fixes b :: "'b upred"   and P :: "('a, 'b) uexpr"   and Q :: "('a, 'b) uexpr"   and R :: "('a, 'b) uexpr" shows "P \<triangleleft> b \<triangleright> (Q \<triangleleft> b \<triangleright> R) = P \<triangleleft> b \<triangleright> R"
card1_tr0: fixes A :: "'h set"   and a :: "'h" assumes "finite A"   and "card A = Suc 0"   and "a \<in> A" shows "{a} = A"
\<Phi>a_mapsto: fixes S :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and S' :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<AA> :: "('d \<times> 'd) set"   and \<phi> :: "'b \<Rightarrow> 'c"   and f :: "'b" assumes "two_set_categories_bij_betw_Univ S S' \<AA> \<phi>"   and "partial_magma.arr S f" shows "two_set_categories_bij_betw_Univ.\<Phi>a S \<AA> \<phi> f \<in> set_category_data.set S' (set_category.some_img S' \<AA>) (two_set_categories_bij_betw_Univ.\<Phi>o S S' \<AA> \<phi> (partial_magma.dom S f)) \<rightarrow> set_category_data.set S' (set_category.some_img S' \<AA>) (two_set_categories_bij_betw_Univ.\<Phi>o S S' \<AA> \<phi> (partial_magma.cod S f))"
return_nondetT_def: shows "Monad_Overloading.return = nondetM_base.return_nondet Monad_Overloading.return (\<lambda>x. {#x#})"
mono_Sync_D3: fixes P :: "'c process"   and Q :: "'c process"   and A :: "'c set"   and S :: "'c process" assumes "P \<sqsubseteq> Q" shows "min_elems (D (P \<lbrakk>A\<rbrakk> S)) \<subseteq> T (Q \<lbrakk>A\<rbrakk> S)"
llist_all2_lnthD: fixes P :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and xs :: "'e llist"   and ys :: "'f llist"   and n :: "nat" assumes "llist_all2 P xs ys"   and "enat n < llength xs" shows "P (lnth xs n) (lnth ys n)"
EF: shows "D = E @ derivation_shift F 0 (Suc j) \<and> LeftDerivation (take (length b1) (b1 @ [X] @ b2)) E (take j c) \<and> LeftDerivation (drop (Suc (length b1)) (b1 @ [X] @ b2)) F (drop (Suc j) c)"
arg_minus_ii: shows "arg (- \<i>) = - pi / 2"
simps_27: fixes vb :: "char list"   and vc :: "aexp"   and vd :: "'b"   and v :: "'b acom"   and va :: "'b acom" shows "(vb ::= vc {vd} \<sqsubseteq> v;; va) = False"
succsr_isNode: fixes succs :: "'d \<Rightarrow> 'd list"   and isNode :: "'d \<Rightarrow> bool"   and invariant :: "'e \<Rightarrow> bool"   and ins :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and memb :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and empt :: "'e"   and nodeAbs :: "'d \<Rightarrow> 'f"   and x :: "'d"   and y :: "'d" assumes "DFS succs isNode invariant ins memb empt nodeAbs"   and "(x, y) \<in> (DFS.succsr succs)\<^sup>*"   and "isNode x" shows "isNode y"
infsetsum_altdef: fixes f :: "'d \<Rightarrow> 'c"   and A :: "'d set" shows "infsetsum f A = set_lebesgue_integral (count_space UNIV) A f"
is_natural_2: fixes A :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and B :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and F :: "'d \<Rightarrow> 'e"   and G :: "'d \<Rightarrow> 'e"   and \<tau> :: "'d \<Rightarrow> 'e"   and f :: "'d" assumes "natural_transformation A B F G \<tau>"   and "partial_magma.arr A f" shows "B (\<tau> (partial_magma.cod A f)) (F f) = \<tau> f"
transfer_start': fixes P :: "bool"   and Q :: "bool" assumes "P"   and "Transfer.Rel (\<longrightarrow>) P Q" shows "Q"
lemLineContainsBasepoint: fixes p :: "'a Point"   and q :: "'a Point" shows "inLine p line joining p to q"
exP0_P': shows "adjoint exM0 * P' * exM0 = P'"
eqvt_apply: fixes p :: "perm"   and f :: "'b \<Rightarrow> 'a"   and x :: "'b" shows "p \<bullet> f x \<equiv> (p \<bullet> f) (p \<bullet> x)"
eval_polyexp_Cons: fixes x :: "'b \<times> nat \<times> 'b"   and xs :: "('b \<times> nat \<times> 'b) list" shows "eval_polyexp (x # xs) = (\<lambda>n. (case x of (a, k, b) \<Rightarrow> a * of_nat n ^ k * b ^ n) + eval_polyexp xs n)"
xs_def: shows "xs = xs' #\<^sub>\<circ> x"
trop_ueval: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'a"   and x :: "('c, 'b) uexpr"   and y :: "('d, 'b) uexpr"   and z :: "('e, 'b) uexpr"   and b :: "'b" shows "\<lbrakk>trop f x y z\<rbrakk>\<^sub>e b = f (\<lbrakk>x\<rbrakk>\<^sub>e b) (\<lbrakk>y\<rbrakk>\<^sub>e b) (\<lbrakk>z\<rbrakk>\<^sub>e b)"
ord_spmf_measureD: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and p :: "'a spmf"   and q :: "'a spmf"   and A :: "'a set" assumes "ord_spmf R p q" shows "Sigma_Algebra.measure (measure_spmf p) A \<le> Sigma_Algebra.measure (measure_spmf q) {y. \<exists>x\<in>A. R x y}"
path_take_from: fixes l1 :: "'vtx"   and l2 :: "'vtx"   and xs :: "('vtx \<times> 'lbl \<times> 'vtx) list"   and m :: "nat"   and l2' :: "'vtx" assumes "path l1 l2 xs"   and "m < length xs"   and "(case xs ! m of (s, l, t) \<Rightarrow> s) = l2'" shows "path l1 l2' (take m xs)"
ball_eq_sym: fixes S :: "('a \<times> 'b) set"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and g :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" assumes "\<forall>(x, y)\<in>S. f x y = g x y" shows "\<forall>(x, y)\<in>S. g x y = f x y"
f'_def: fixes x :: "'a" shows "f' x = (if x \<in> A then f x else (0::'b))"
mod_in_carrier: fixes a :: "'a mod_ring poly" shows "a mod f \<in> carrier R"
simps_2: fixes V :: "'c set"   and t :: "('d, 'c) Term.term"   and S :: "('d, 'c) strand_step list" shows "wf\<^sub>s\<^sub>t V (receive\<langle>t\<rangle>\<^sub>s\<^sub>t # S) = (fv t \<subseteq> V \<and> wf\<^sub>s\<^sub>t V S)"
exec_move_FAcc: fixes pc :: "nat"   and e :: "'addr expr1"   and ci :: "('addr, 'heap) check_instr"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and F :: "String.literal"   and D :: "String.literal"   and h :: "'heap"   and stk :: "'addr val list"   and loc :: "'addr val list"   and xcp :: "'addr option" assumes "pc < length (compE2 e)" shows "exec_move ci P t (e\<bullet>F{D}) h (stk, loc, pc, xcp) = exec_move ci P t e h (stk, loc, pc, xcp)"
InvariantQCharacterizationAfterAssertLiteral: fixes state :: "State"   and literal :: "Literal"   and decision :: "bool" assumes "InvariantConsistent (getM state @ [(literal, decision)])"   and "InvariantWatchListsContainOnlyClausesFromF (getWatchList state) (getF state)"   and "InvariantWatchListsUniq (getWatchList state)"   and "InvariantWatchListsCharacterization (getWatchList state) (getWatch1 state) (getWatch2 state)"   and "InvariantWatchesEl (getF state) (getWatch1 state) (getWatch2 state)"   and "InvariantWatchesDiffer (getF state) (getWatch1 state) (getWatch2 state)"   and "InvariantWatchCharacterization (getF state) (getWatch1 state) (getWatch2 state) (getM state)"   and "InvariantConflictFlagCharacterization (getConflictFlag state) (getF state) (getM state)"   and "InvariantQCharacterization (getConflictFlag state) (getQ state) (getF state) (getM state)" shows "let state' = assertLiteral literal decision state in InvariantQCharacterization (getConflictFlag state') (removeAll literal (getQ state')) (getF state') (getM state')"
ts_okD: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> bool"   and ts :: "'c \<Rightarrow> ('d \<times> 'f \<Rightarrow>f nat) option"   and m :: "'e"   and t :: "'c"   and x :: "'d"   and ln :: "'f \<Rightarrow>f nat" assumes "ts_ok P ts m"   and "ts t = \<lfloor>(x, ln)\<rfloor>" shows "P t x m"
of_int_eq_0_iff: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and z :: "int" assumes "class.ring_char_0 one times plus zero minus uminus" shows "(ring_1.of_int one plus zero minus z = zero) = (z = 0)"
trimmed_real_def: fixes x :: "real" shows "trimmed x = (x \<noteq> 0)"
differential_zero_maxmin: fixes x :: "'a"   and S :: "'a set"   and f :: "'a \<Rightarrow> real"   and f' :: "'a \<Rightarrow> real" assumes "x \<in> S"   and "open S"   and "(f has_derivative f') (at x)"   and "(\<forall>y\<in>S. f y \<le> f x) \<or> (\<forall>y\<in>S. f x \<le> f y)" shows "f' = (\<lambda>v. 0)"
cone_decomp_exact: fixes X :: "'x set"   and qs :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list"   and k :: "nat"   and T :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set" assumes "finite X"   and "valid_decomp X qs"   and "standard_decomp k qs"   and "cone_decomp T qs" shows "cone_decomp T (exact X k qs)"
\<Gamma>_reductive: fixes \<gamma> :: "'a inference" assumes "\<gamma> \<in> \<Gamma>" shows "concl_of \<gamma> < main_prem_of \<gamma>"
sets_PiM_subset_borel: shows "sets (Pi\<^sub>M UNIV (\<lambda>_. borel)) \<subseteq> sets borel"
factorization_unique_aux: fixes M :: "nat multiset"   and N :: "nat multiset" assumes "\<forall>p\<in>#M. prime p"   and "\<forall>p\<in>#N. prime p"   and "(\<Prod>i\<in>#M. i) dvd (\<Prod>i\<in>#N. i)" shows "M \<subseteq># N"
lderivs_Not: fixes xs :: "'a list"   and r :: "'b rexp" shows "lderivs xs (rexp.Not r) = rexp.Not (lderivs xs r)"
fourthh: assumes "a < 0" shows "\<exists>x. \<forall>y<x. a * y\<^sup>2 + b * y + c < 0"
dfs_\<alpha>_def: fixes S :: "'d set \<times> 'd list" shows "dfs_\<alpha> S \<equiv> let (\<Sigma>, W) = S in (\<Sigma>, set W)"
mono_body_fixp: fixes F :: "('a \<Rightarrow> 'b Heap) \<Rightarrow> 'a \<Rightarrow> 'b Heap" assumes "\<And>x. mono_Heap (\<lambda>f. F f x)" shows "heap.fixp_fun F = F (heap.fixp_fun F)"
signed_minus_1: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b" assumes "class.ring_1 one times plus zero minus uminus" shows "ring_1.signed one plus zero minus (- 1) = uminus one"
A: shows "\<forall>n\<in>{..length ys}. \<exists>Y. (ipurge_tr I (c_dom D) u (c_tr step out (foldl step s\<^sub>0 xs) (take n ys)), Y) \<in> futures (c_process step out s\<^sub>0) (c_tr step out s\<^sub>0 xs)"
preserves_hom: fixes A :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and B :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and F :: "'d \<Rightarrow> 'e"   and G :: "'d \<Rightarrow> 'e"   and \<tau> :: "'d \<Rightarrow> 'e"   and f :: "'d"   and a :: "'d"   and b :: "'d" assumes "natural_transformation A B F G \<tau>"   and "partial_magma.in_hom A f a b" shows "partial_magma.in_hom B (\<tau> f) (F a) (G b)"
append_eq_conv_conj: fixes xs :: "'c list"   and ys :: "'c list"   and zs :: "'c list" shows "(xs @ ys = zs) = (xs = take (length xs) zs \<and> ys = drop (length xs) zs)"
coprime_Suc_0_right: fixes n :: "nat" shows "coprime n (Suc 0)"
eq_iff: fixes a :: "'d multiset"   and b :: "'d multiset" shows "(a = b) = (b \<subseteq># a \<and> a \<subseteq># b)"
IH: assumes "\<forall>x\<in>set xs. \<forall>y\<in>set xs. g x = g y \<longrightarrow> f x = f y" shows "f ` set (remdups' g xs) = f ` set xs"
other_def: fixes P :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and I :: "'d set"   and \<sigma> :: "'d \<Rightarrow> 'c"   and \<sigma>' :: "'d \<Rightarrow> 'c" shows "other P I \<sigma> \<sigma>' \<equiv> \<forall>i. if i \<in> I then \<sigma>' i = \<sigma> i else P (\<sigma> i) (\<sigma>' i)"
Sup_inf: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and B :: "'a set"   and a :: "'a" assumes "class.complete_distrib_lattice Inf Sup inf less_eq less sup bot top" shows "inf (Sup B) a = Sup ((\<lambda>b. inf b a) ` B)"
mult_oner: fixes x :: "'c \<times> 'd" shows "times_prod' x one_prod = x"
that: fixes p :: "int" assumes "prime p"   and "p dvd k" shows "thesis"
tiny_digraph_dg_SemiCAT: fixes \<beta> :: "V" assumes "\<Z> \<beta>"   and "\<alpha> \<in>\<^sub>\<circ> \<beta>" shows "tiny_digraph \<beta> (dg_SemiCAT \<alpha>)"
sup_absorb2: fixes x :: "'c multiset"   and y :: "'c multiset" assumes "x \<subseteq># y" shows "x \<union># y = y"
all_info: shows "set (characterize_consistent_signs_at_roots_copr p qs) \<subseteq> set [[1], [- 1]]"
simps: fixes a1 :: "'a"   and a2 :: "'a" shows "rel_12_23 a1 a2 = (a1 = \<^bold>1 \<and> a2 = \<^bold>2 \<or> a1 = \<^bold>2 \<and> a2 = \<^bold>3)"
composable: shows "src (local.dom \<nu>) = trg (local.dom \<pi>)"
Int_mono: fixes A :: "'a set"   and C :: "'a set"   and B :: "'a set"   and D :: "'a set" assumes "A \<subseteq> C"   and "B \<subseteq> D" shows "A \<inter> B \<subseteq> C \<inter> D"
vimage_insert: fixes f :: "'b \<Rightarrow> 'c"   and a :: "'c"   and B :: "'c set" shows "f -` insert a B = f -` {a} \<union> f -` B"
xx_2: shows "xa \<in> set (if fst (sports r) = 0 \<and> snd (sports r) = - 1 then [None] else if fst (sports r) \<le> snd (sports r) then map (\<lambda>x. Some (pfxm_prefix x, ~~ pfxm_mask x)) (wordinterval_CIDR_split_prefixmatch (WordInterval (fst (sports r)) (snd (sports r)))) else [])"
assms_2: assumes "\<phi> \<pi>l" shows "\<chi>"
step: shows "(if is_final_dstate St then let (uu_, uu_, Q, uu_) = St in Some (map fst Q) else self_call (deterministic_RP_step St)) = Some St'"
distinct_1: fixes x2 :: "'c" shows "\<bottom> \<noteq> \<lfloor>x2\<rfloor>"
bdd_below_empty: fixes less_eq :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and less :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool" assumes "class.preorder (less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (less::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool)" shows "preorder.bdd_below (less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) {}"
comm_mult_left_mono: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<le> b"   and "(0::'a) \<le> c" shows "c * a \<le> c * b"
eventually_elim2: fixes P :: "'e \<Rightarrow> bool"   and F :: "'e filter"   and Q :: "'e \<Rightarrow> bool"   and R :: "'e \<Rightarrow> bool" assumes "eventually P F"   and "eventually Q F"   and "\<And>i. \<lbrakk>P i; Q i\<rbrakk> \<Longrightarrow> R i" shows "eventually R F"
association_ends_unique: fixes \<C> :: "classes1"   and "from" :: "String.literal"   and role :: "String.literal"   and end\<^sub>1 :: "classes1 \<times> nat \<times> enat \<times> bool \<times> bool"   and end\<^sub>2 :: "classes1 \<times> nat \<times> enat \<times> bool \<times> bool" assumes "association_ends' classes assocs \<C> from role end\<^sub>1"   and "association_ends' classes assocs \<C> from role end\<^sub>2" shows "end\<^sub>1 = end\<^sub>2"
antisym_conv2: fixes x :: "'k multiset"   and y :: "'k multiset" assumes "x \<subseteq># y" shows "(\<not> x \<subset># y) = (x = y)"
ring_simprules_12: fixes R :: "('c, 'd) ring_scheme"   and x :: "'c" assumes "ring R"   and "x \<in> carrier R" shows "\<one>\<^bsub>R\<^esub> \<otimes>\<^bsub>R\<^esub> x = x"
continuous_on_mult_right: fixes s :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and c :: "'b" assumes "continuous_on s f" shows "continuous_on s (\<lambda>x. f x * c)"
set_integral_norm_bound: fixes M :: "'a measure"   and k :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "set_integrable M k f" shows "norm (set_lebesgue_integral M k f) \<le> \<integral>x\<in>k. norm (f x)\<partial>M"
some_gcd_0_2: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and b :: "'a" assumes "class.comm_semiring_1 times one plus zero" shows "dvd.is_gcd times (dvd.some_gcd times zero b) zero b"
iso_img_block_orig_exists: fixes \<V> :: "'c set"   and \<B> :: "'c set multiset"   and \<V>' :: "'d set"   and \<B>' :: "'d set multiset"   and bij_map :: "'c \<Rightarrow> 'd"   and x :: "'d set" assumes "incidence_system_isomorphism \<V> \<B> \<V>' \<B>' bij_map"   and "x \<in># \<B>'" shows "\<exists>bl. bl \<in># \<B> \<and> x = bij_map ` bl"
simps: fixes ls :: "lit set" shows "extend ls = (if vars_of_cnf cnf \<subseteq> var ` ls then ls else let x = SOME x. x \<in> vars_of_cnf cnf - var ` ls in if consistent (ls \<union> G\<^sup>* `` {Pos x}) then extend (ls \<union> G\<^sup>* `` {Pos x}) else extend (ls \<union> G\<^sup>* `` {Neg x}))"
hyperUn_assoc: fixes A :: "'b set option"   and B :: "'b set option"   and C :: "'b set option" shows "A \<squnion> B \<squnion> C = A \<squnion> (B \<squnion> C)"
update_convs_1: fixes verts' :: "'a set \<Rightarrow> 'a set"   and verts :: "'a set"   and arcs :: "('a \<times> 'a) set"   and more :: "'b" shows "verts_update verts' \<lparr>verts = verts, arcs = arcs, \<dots> = more\<rparr> = \<lparr>verts = verts' verts, arcs = arcs, \<dots> = more\<rparr>"
substNext_def: fixes g :: "'g" shows "uninst.substNext (usesOf \<circ> uses) (\<lambda>g. Mapping.lookup (phis g)) g \<equiv> \<lambda>v. if v = uninst_code.chooseNext uses phis g then uninst.substitution (usesOf \<circ> uses) (\<lambda>g. Mapping.lookup (phis g)) g else v"
abel_semigroup_def: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" shows "abel_semigroup f \<equiv> semigroup f \<and> abel_semigroup_axioms f"
setdist_eq_0_closedin: fixes U :: "'c set"   and S :: "'c set"   and x :: "'c" assumes "closedin (top_of_set U) S"   and "x \<in> U" shows "(setdist {x} S = 0) = (S = {} \<or> x \<in> S)"
simps_2: fixes G :: "'a graph"   and v :: "'a"   and v' :: "'a"   and es :: "('a \<times> 'a) list" shows "delete_edges_list G ((v, v') # es) = delete_edges_list (delete_edge v v' G) es"
generate_valid_stateful_policy_IFSACS_select_simps_2: fixes G :: "'b graph"   and M :: "'b SecurityInvariant_configured list"   and edgesList :: "('b \<times> 'b) list" shows "flows_fix (generate_valid_stateful_policy_IFSACS G M edgesList) = edges G"
geometric_pmf_prob_atMost: fixes p :: "real"   and n :: "nat" assumes "p \<in> {0<..1}" shows "measure_pmf.prob (geometric_pmf p) {..n} = 1 - (1 - p) ^ (n + 1)"
assms_1: shows "\<I>' \<turnstile>res res \<surd>"
cases: fixes x :: "('b \<times> color) tree \<times> 'b \<times> ('b \<times> color) tree" obtains t1 :: "('b \<times> color) tree" and a :: "'b" and t2 :: "('b \<times> color) tree" and b :: "'b" and t3 :: "('b \<times> color) tree" and c :: "'b" and t4 :: "('b \<times> color) tree" where "x = (R (R t1 a__ t2__) b__ t3__, c__, t4__)"   | a :: "'b" and t2 :: "('b \<times> color) tree" and b :: "'b" and t3 :: "('b \<times> color) tree" and c :: "'b" and t4 :: "('b \<times> color) tree" where "x = (R \<langle>\<rangle> aa__ (R t2a__ ba__ t3a__), ca__, t4a__)"   | v :: "('b \<times> color) tree" and vc :: "'b" and vb :: "('b \<times> color) tree" and a :: "'b" and t2 :: "('b \<times> color) tree" and b :: "'b" and t3 :: "('b \<times> color) tree" and c :: "'b" and t4 :: "('b \<times> color) tree" where "x = (R (B v__ vc__ vb__) ab__ (R t2b__ b t3), c, t4)"   | a :: "'b" and t2 :: "('b \<times> color) tree" where "x = (\<langle>\<rangle>, ac__, t2c__)"   | v :: "'b" and vb :: "('b \<times> color) tree" and a :: "'b" and t2 :: "('b \<times> color) tree" where "x = (B \<langle>\<rangle> va__ vba__, ad__, t2d__)"   | va :: "'b \<times> color" and a :: "'b" and t2 :: "('b \<times> color) tree" where "x = (\<langle>\<langle>\<rangle>, vaa__, \<langle>\<rangle>\<rangle>, ae__, t2e__)"   | va :: "'b \<times> color" and v :: "('b \<times> color) tree" and ve :: "'b" and vd :: "('b \<times> color) tree" and a :: "'b" and t2 :: "('b \<times> color) tree" where "x = (\<langle>\<langle>\<rangle>, vab__, B vd__ ve__ vd\<rangle>, af__, t2f__)"   | vc :: "('b \<times> color) tree" and vf :: "'b" and ve :: "('b \<times> color) tree" and v :: "'b" and vb :: "('b \<times> color) tree" and a :: "'b" and t2 :: "('b \<times> color) tree" where "x = (B (B vca__ vf__ vea__) vg__ vbb__, ag__, t2g__)"   | vc :: "('b \<times> color) tree" and vf :: "'b" and ve :: "('b \<times> color) tree" and va :: "'b \<times> color" and a :: "'b" and t2 :: "('b \<times> color) tree" where "x = (\<langle>B vcb__ vfa__ veb__, vac__, \<langle>\<rangle>\<rangle>, ah__, t2h__)"   | vc :: "('b \<times> color) tree" and vf :: "'b" and ve :: "('b \<times> color) tree" and va :: "'b \<times> color" and v :: "('b \<times> color) tree" and vh :: "'b" and vg :: "('b \<times> color) tree" and a :: "'b" and t2 :: "('b \<times> color) tree" where "x = (\<langle>B vcc__ vf ve, va, B vh__ vh vg\<rangle>, ai__, t2i__)"   | v :: "('b \<times> color) tree" and vc :: "'b" and vb :: "('b \<times> color) tree" and a :: "'b" and t2 :: "('b \<times> color) tree" where "x = (B v vc vb, a, t2)"
face_of_polyhedron_subset_facet: fixes S :: "'b set"   and C :: "'b set" assumes "polyhedron S"   and "C face_of S"   and "C \<noteq> {}"   and "C \<noteq> S" obtains F :: "'b set" where "F facet_of S" and "C \<subseteq> F"
intro: fixes U\<^sub>1 :: "'a set"   and le\<^sub>1 :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and ls\<^sub>1 :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and U\<^sub>2 :: "'b set"   and le\<^sub>2 :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and ls\<^sub>2 :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "order_ow U\<^sub>1 le\<^sub>1 ls\<^sub>1"   and "linorder_ow U\<^sub>2 le\<^sub>2 ls\<^sub>2" shows "order_linorder_ow U\<^sub>1 le\<^sub>1 ls\<^sub>1 U\<^sub>2 le\<^sub>2 ls\<^sub>2"