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The lemma named "poly_eval_indexed_pmult'" assumes that "P" is in the Pring set of "R" and "I", "g" is a closed function in "R", and "i" is in "I". It shows that the polynomial evaluation of "P" multiplied by the indexed polynomial of "i" in "R" and "S" using function "g" is equal to the polynomial evaluation of "P" in...
Mapping a function 'f' over the range of a restricted association list 'V' in 'Γ' is equivalent to restricting 'V' in the association list that results from mapping the function 'f' over the range of 'Γ'.
If a state 's' is in the reachable set of a transition system 'T', then there exists a behavior 'b' in the behavior set of 'T' such that the state 's' is in the set of 'b'.
For the starting program with parameters P, C, and M, in the context of heap h and Start, an empty map is soundly type-checked.
The lemma named "init_model" assumes that there does not exist any 'n' prime such that the condition 'latestAct_cond' holds for 'nid', 't', 'n', and 'n' prime. It also assumes that 'nid' is parallel at 't' 'n'. Under these assumptions, it shows that the blockchain at 't' 'n' for 'nid' is empty.
The unsigned integer representation of a numeral in binary form as a word of type 'a is equal to the numeral in binary form truncated to the length of type 'a.
, under the assumption that for any integer 'i' less than 'm', 'c i' is less than or equal to 'c' prime 'i', it can be shown that the multi_arithmetic progression of 'm', 'start', 'steps', and 'c' is less than or equal to the multi_arithmetic progression of 'm', 'start', 'steps', and 'c' prime.
The set of variables in the strands of the received map is equal to the set of free variables in the function applied to the set of strands.
The lemma named "bind_return" assumes that the function 'f' is measurable with respect to the measure space 'M' and the subprobability algebra 'N', and that 'x' is an element of the space 'M'. It then shows that binding the return function on 'M' with 'x' and 'f' is equal to applying 'f' to 'x'.
For any state sigma and any assignment rho, the transition relation from sigma to the identity relation under the assignment rho results in the composition of rho and sigma.
The shortest path distance from source 's' to vertex 'v' is less than or equal to the shortest path distance from source 's' to vertex 'u' plus the weight of the edge from 'u' to 'v'.
If a function G preserves another function f, then the projection of function G on a set C with respect to function h also preserves the identity function.
If a transition 't' is in the transitions of the canonical separator of a machine 'M' for states 'q1' and 'q2', and 'q1' and 'q2' are states in 'M', and the source of 't' is the pair of states '(s1, s2)', then there exists a transition 't1' in 'M' such that the source of 't1' is 's1' and the input and output of 't1' ar...
For a given interference I, domain D, user u, transition function step, output out, state s, and list xs, the sinks of the transition function (c_tr) with these parameters is equal to the sinks of the domain D with user u and list xs.
: Given the following conditions: 1. p is not equal to 0, 2. for any q in the set qs, p and q are coprime, 3. the signs are distinct, 4. the consistent signs at roots of p and qs are a subset of the set of signs, 5. for any l in the set of subsets and any i in the set l, i is less than the length of qs, it can be shown...
The lemma "vwalk_join_hd" assumes that paths p and q are joinable. It shows that the head of the joined path (p joined with q) is equal to the head of path p.
The lemma named "phi_cases" assumes the following: 1. The function phi is applied to the transition from state 's' to state 's'' with action 'a' and output 'ou'. 2. The step from state 's' with action 'a' results in output 'ou' and state 's''. 3. The state 's' is reachable. Under these assumptions, one of the followi...
The lemma Amicable_Example_Descartes shows that 9363584 and 9437056 are amicable numbers.
If the graph G is right total, and if the functions alpha_e, alpha_n, invar, inEdges', Entry, defs, uses, and phis are all equivalent under G to their respective counterparts alpha_e2, alpha_n2, invar2, inEdges2, Entry2, defs2, uses2, and phis2, then the function allDefs is also equivalent under G to the function allDe...
The lemma named "drest_smap_drest_smap_drest" establishes that for any fixed variables x, s, and vs, the restriction of the finite map x to the set vs is a submap of s if and only if it is also a submap of the restriction of the finite map s to the set vs.
The lemma named "backward_finite_path_connected_aux" assumes that there is a backward finite path rooted at 'x' in relation 'r'. It shows that the transposed relation of 'x' composed with 'r' and then composed with the transposed relation of 'x' is less than or equal to the reflexive transitive closure of 'x' plus the ...
If a Van Emde Boas tree "t" of size "n" satisfies the invariant "invar_vebt", then the set of elements in the Van Emde Boas tree after deleting an element "x" is equal to the set of elements in the original Van Emde Boas tree minus the set containing the element "x".
If A satisfies the constraint of being pure according to predicate P, then applying P to the pure form of A is true.
The relation "mn_LE" is well-founded.
If integer 'i' and integer 'j' are defined, then the sum of 'i' and 'j' is equal to the sum of the values of 'i' and 'j'.
If x is greater than or equal to zero and y is greater than zero, then the result of x divided by y is also greater than or equal to zero.
The sequence "seqll" with parameters "i" and "P" applied to a triple consisting of a pair (\<sigma>, p), an action "a", and another pair (\<sigma>', p') is equal to "P" applied to a triple consisting of a pair (\<sigma> i, p), the same action "a", and another pair (\<sigma>' i, p').
The prime omega function of 1 equals 0.
"for all x, if x is an essential property of God, then there exists an x that is an essential property of God."
If the binary split of a number 'c' at position 'n' results in a pair '(a, b)', then 'b' is equal to the operation 'take_bit' applied to 'n' and 'c', where 'take_bit' is a function that takes a natural number and an integer and returns an integer.
If "x" is not equal to "a", "y" is not equal to "b", and either "x" is not equal to "y" and "x" is not equal to "b", or "a" is not equal to "b" and "a" is not equal to "y", then the set containing "x" and "a" is not equal to the set containing "y" and "b".
The lemma "SA_Units_closed" assumes that a function 'f' belongs to the unit group of the semialgebraic function ring 'SA' of degree 'n'. It shows that this function 'f' also belongs to the carrier of the semialgebraic function ring 'SA' of degree 'n'.
The null set is defined as true.
If x is less than or equal to y, y is less than or equal to z, and z is less than or equal to x, then x equals y and y equals z.
If a set A is small and for every element x in A, the set B of x is also small, then the Cartesian product of A and B (denoted as Sigma A B) is also small.
The lemma "lowlink_set_not_empty" assumes that "v" is in the domain of the "discovered" set "s". It shows that the "lowlink_set" of "s" and "v" is not empty.
If 'a' belongs to the set of hyperreal infinite numbers and 'a' is greater than zero, then the hyperreal logarithm of 'a' with respect to 'x' is equal to the standard part function of the natural logarithm of 'x' divided by the standard part function of the natural logarithm of 'a'.
The lemma named "sub_eq_zero" assumes that "z" is not equal to infinity. It shows that the homogeneous subtraction of "z" from itself equals zero.
If the value of a database entry for a given user "u" and two conditions "c" and "c'" is less than or equal to "b", and "b" is less than or equal to "b'", then the value of the database entry for the same user "u" and conditions "c" and "c'" is less than or equal to "b'".
The length of the postorder traversal of a tree is equal to the size of the tree.
The set of variables in context C is finite.
The lemma "cross_once_notin" assumes that Q and R are both in a set P, a is in Q, b is in both Q and R, a is not equal to b, and Q is not equal to R. It then shows that a is not in R.
A while loop with condition 'b' and process 'P' is equivalent to the reflexive-transitive closure of the process 'P' under the condition 'b', followed by the assertion that 'b' is not true.
The lemma named "dimension_is_inj" assumes that the dimensions of the embedded algebra E over the field K are n and m. It shows that n must be equal to m.
A tree 't' is heap ordered if and only if for every element 'x' in the multiset converted from the tree 't', the priority of the tree 't' is less than or equal to the second element of 'x'.
If the pair (a, r t) is adjacent to the list (x # xs), then the pair (a, r x) is also adjacent to the list (x # xs).
The lemma named "lim_sequence_aux" is measurable and it assumes the following: 1. "p" is a policy. 2. For any "x" in the space "M", "f(x)" is a policy. 3. For any natural number "n", the function "f" applied to the first element of the first pair "x", then applied to "n", and finally applied to the pair formed by the ...
The lemma "subst_b_check_s" is fixing the variables s, b, cs, css, v, and τ. It assumes that b is well-formed under the empty context and that B is the set containing bv. It then shows three things: 1. If s is type-checked under the contexts Θ, Φ, B, G, and D and is inferred to have type τ, then the substitution of b ...
The conjunction (denoted by *) of any proposition "P" with "false" results in "false".
The lemma "is_head_reduction_imp_contains_head_reduction" shows that if "t" is a head reduction, then "t" contains a head reduction.
The lemma "Proc_CFG_edge_no_lhs_equal" assumes that there is a program control flow graph edge from label 'l' to node 'n' with edge type 'et' in the program 'prog', and that the variable 'V' is not in the left-hand side of the label of the program at 'l'. Under these assumptions, it shows that the state value of the va...
The theorem named "wls_psubst_vsubst_compose" assumes that "X" is a well-sorted term of sort "s" and "rho" is a well-sorted environment. It shows that the parallel substitution of "y1" for "y" in "X" of sort "ys", followed by the substitution of the environment "rho", is equal to the substitution in "X" of the environm...
The lemma named "identity_rules" assumes the following: 1. G is a graph. 2. The identity rules L are maintained in graph G. 3. The first element of each edge in graph G is a subset of L. The lemma shows that there exists a function f such that: 1. The composition of f with itself is equal to f. 2. The identity relatio...
The set of natural numbers from 0 to less than 4 is equal to the set containing 0, 1, 2, and 3.
The lemma "similar_mat_witI" assumes that the product of matrices P and Q equals the identity matrix of order n, the product of matrices Q and P also equals the identity matrix of order n, and matrix A equals the product of matrix P, matrix B, and matrix Q. It also assumes that matrices A, B, P, and Q are all n by n ma...
The lemma named "wfG_cons_append" is about well-formedness of context in MiniSail. It fixes a variable 'b' of type 'b'. It assumes that a context consisting of a list with an element (x', b', c') followed by a context Gamma', appended with another list with an element (x, b, c) followed by a context Gamma, is well-form...
The lemma named "homeomorphic_affinity" is about a set S of real normed vectors. It assumes that a constant "c" is not equal to zero. It shows that the set S is homeomorphic to the image of the set S under the function that maps each element "x" to "a + c * x".
If you substitute term 't' for variable 'i' in formula 'A', and then substitute term 'u' for variable 'i' in the resulting formula, it is equivalent to substituting term 'subst i u t' for variable 'i' in formula 'A'.
The lemma named "fps_inverse_compose" assumes that the zeroth element of b (in a field 'a) is zero and the zeroth element of a is not zero. Under these assumptions, it shows that the composition of the inverse of a with b is equal to the inverse of the composition of a with b.
For every element 'c' in the set of cuts of 'xs', 'c' is either increasing or decreasing.
The lemma named "alpha'_ide" assumes that "f", "g", and "h" are identities and that the source of "f" is the target of "g" and the source of "g" is the target of "h". It shows that the alpha' of the tuple (f, g, h) is equal to the source-target association of f, g, and h.
"nsp" is a one-step process.
The lemma named "ls_insls_map_of'" assumes the following conditions: 1. "ls" is a hash table. 2. "ls" has distinct elements. 3. "ld" is a hash table. 4. "ld" has distinct elements. 5. The length of "ld" is greater than 1. 6. "n" is less than the length of "ls". Under these assumptions, it shows that the map of the co...
If a node is left-aligned with a list consisting of a left subtree and separator, a subtree and separator, a right subtree and separator, and a right list, and if the left separator is aligned with a new subtree and a value, and if this value is aligned with a new right subtree and right separator, then the node is lef...
Two formal Laurent series 'f' and 'g' are equal if and only if their coefficients are equal for all indices 'n'.
If set S2 is a subset of set S1, then the restriction of function m to set S2 is less than or equal to the restriction of function m to set S1.
If the depth-first search (dfs) of the composition of two functions f and g on a list xs is distinct, then the depth-first search of the function g on the same list xs is also distinct.
A thread must wait if and only if the following conditions are met: - In the case where "lt" is of the form "Inl l", there exists a thread "t'" in the set "Ts" such that "t" is not equal to "t'" and "t'" has a lock on the lock "l" in the state "s". - In the case where "lt" is of the form "Inr (Inl t')", the thread "t'...
"if 'a' and 'not a' are true, then 'not b' must also be true."
: Assuming that the substitution of 'n' in 't' with 'Fvar s' and 'Fvar p' equals 'Fvar x', and 'x' is not equal to 's' or 'p', it can be shown that 't' is equal to 'Fvar x'.
If proposition p is equal to proposition q, and proposition q is equal to proposition r, then proposition p is equal to proposition r, in the context of a given variable v.
If a document pointer "document_ptr" is updated with a doctype "doctype_update" and a value "v" in a heap "h" to produce a new heap "h'", then the well-formedness of types in the original heap "h" is preserved in the new heap "h'".
The lemma named "small_step_fragmentsLS" assumes that there is a small step transition from state 's' to state 's'' under action 'α'. It shows that the set of fragments in the left-to-right sequential order of state 's'' is a subset of the set of fragments in the left-to-right sequential order of state 's'.
The lemma named "Melem_subset_eq_sum_list" assumes that "x" is an element of the multiset "T". It shows that "x" is a subset of the sum of the list "T".
If the index 'i' is less than the number of rows 'nr' and the dimension of the vector function 'f' at 'i' equals the number of columns 'nc', then the row 'i' of the matrix created by the row function 'f' with 'nr' rows and 'nc' columns is equal to the function 'f' at 'i'.
In the context of a Zermelo universe, for any given beta that is a Zermelo universe and any alpha that is an element of beta, the set of all co-final sequences of alpha is an element of the set determined by beta.
If the set of elements in list 'xs' is a subset of set 'A', and the set of elements in list 'ys' is also a subset of set 'A', and list 'xs' is less than or equal to list 'ys' according to relation 'r', and 'p' is less than the size of list 'xs', and element 'x' is less than or equal to the 'p'th element of list 'ys' ac...
The order topology is indeed a topology for any set X.
The lemma PNT1_imp_PNT2: assumes that the prime counting function is asymptotic to the function x / ln x at infinity, shows that the function of the product of the prime counting function and the natural logarithm of the prime counting function is asymptotic to the function x at infinity.
The lemma named "transpose_of_prod" is about complex matrices M and N. It assumes that the number of columns in matrix M is equal to the number of rows in matrix N. It then shows that the transpose of the product of M and N is equal to the product of the transpose of N and the transpose of M.
If "h" and "h'" are disjoint and "S" is the domain of "h", then the restriction of the addition of "h" and "h'" to "S" is equal to "h".
For all 'y', if the double modal diamond of 'x' applied to 'y' is less than or equal to the modal diamond of 'x' applied to 'y', then 'x' is Noetherian if and only if 'x' is Loebian.
If the check on a state 's' in a context 'G' is successful, then executing 'exec_d' in the same context 'G' with a normal state 's' will not result in a TypeError.
In the context of group homomorphisms, the kernel of a homomorphism from group G to group H is a normal subgroup of G.
If the range of a function 'f' is finite, then the image of any set 'A' under the function 'f' is also finite.
The lemma named "those_someD" assumes that the function "those" applied to the list "xs" equals "Some ys". Under this assumption, it shows that the list "xs" is equal to the list "ys" mapped with the function "Some".
The lemma named "extensional_funcset_def" defines that an extensional function set from set S to set T is the intersection of the set of all functions from S to T and the set of all extensional functions on S.
For all real numbers 'x', if 'x' is greater than or equal to 'a' and less than or equal to 'b', then the polynomial 'p' evaluated at 'x' is greater than 0. This is equivalent to either 'a' being greater than 'b' or the polynomial 'p' evaluated at 'a' being greater than 0, and for all 'x', if 'x' is greater than or equa...
: for all elements 'x' in the set 'xs', if 'x' is equal to 'y', then 'xs' is equal to a list that repeats 'y' for the length of 'xs'.
The union of the set obtained from the weak ranking according to the less than or equal to order is equal to the carrier.
The butterfly function, when applied to zero and any other value 'a', will always return zero.
If for all 'z' and 's', 'P' prime of 'z' and 's' implies 'P' of 'z' and 's', and if 'C' totally corrects from 'P' to 'Q' through 'c', then 'C' also totally corrects from 'P' prime to 'Q' through 'c'.
The lemma named "card_inversion" assumes that "xs" is in the image of the function "inversion" applied to "dc_crypto". It shows that the cardinality (or size) of the set of all "dc" in "dc_crypto" such that the "inversion" of "dc" equals "xs", is equal to twice "n".
If 'f' is a current in a network 'Γ', then the incoming flow of 'f' at a node 'x' is less than or equal to the weight of 'x' in the network 'Γ'.
If a function 'f' is simple Bochner integrable over a measure 'M', then the simple Bochner integral of 'f' over 'M' is equal to the Lebesgue integral of 'f' over 'M'.
Lemma subtr_rootR_in: Assuming that "tr1" is a subtree of "tr2" with respect to the set "ns", it can be shown that the root of "tr2" is an element of the set "ns".
The lemma named "expectation_X" assumes that "i" is an element of "I". It shows that the expectation of "X i" is equal to the expectation of "Y".
The lemma named "eqButUIDf_not_UID'" assumes two conditions. The first condition, "eqButUIDf frds frds1", means that the two friend request sets, "frds" and "frds1", are equal except for the user ID function. The second condition, "(uid,uid') \<notin> {(UID1,UID2), (UID2,UID1)}", means that the pair of user IDs "(uid, ...
The lemma named "sum_length_partition_by_median" assumes that a tuple (l, m, r) is equal to the result of partitioning the list "ps" by its median at position "k". It shows that the length of the list "ps" is equal to the sum of the lengths of the lists "l" and "r".
If a list "l" is a fragment of set "St", then there exist lists "pre" and "suff" such that "pre" is a prefix of "l", "suff" is a suffix of "pre", and "suff" is an element of set "St".